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<62657474657220> Tj 1 0 0 1 378.48 582.24 Tm <756E6465727374616E64696E6720> Tj 1 0 0 1 428.64 582.24 Tm <6F6620> Tj 1 0 0 1 437.28 582.00 Tm <74686520> Tj 1 0 0 1 450.24 582.00 Tm <53696D706C6520> Tj 1 0 0 1 474.96 582.00 Tm <616E6420> Tj 1 0 0 1 489.60 582.00 Tm <4F7264696E61727920> Tj 1 0 0 1 520.32 582.00 Tm <4B726967696E67> Tj 0.48 Tc 1 0 0 1 189.36 572.88 Tm <616C676F726974686D7320> Tj 1 0 0 1 227.28 573.12 Tm <666F7420> Tj 1 0 0 1 239.76 573.60 Tm <67656F6C6F676963616C20> Tj 1 0 0 1 277.20 573.12 Tm <656E67696E656572732C20> Tj 1 0 0 1 314.88 573.12 Tm <617320> Tj 1 0 0 1 325.20 573.12 Tm <74686520> Tj 1 0 0 1 338.64 573.12 Tm <74776F20> Tj 1 0 0 1 353.76 573.12 Tm <6D6F73742D7573656420> Tj 1 0 0 1 391.20 573.60 Tm <67656F737461746973746963616C20> Tj 1 0 0 1 439.44 572.64 Tm <746563686E697175657320> Tj 1 0 0 1 479.52 573.36 Tm <28696E636C7564656420> Tj 1 0 0 1 513.60 572.64 Tm <496E64696361746F72> Tj 0.48 Tc 1 0 0 1 189.60 563.52 Tm <6C28696C67696E6720> Tj 1 0 0 1 214.56 563.52 Tm <617320> Tj 1 0 0 1 223.44 563.52 Tm <74686520> Tj 1 0 0 1 235.68 563.52 Tm <7468697264662E20> Tj 1 0 0 1 257.76 563.52 Tm <54686520> Tj 1 0 0 1 271.68 563.52 Tm <636F6E636C7573696F6E20> Tj 1 0 0 1 309.12 563.52 Tm <696E636C7564657320> Tj 1 0 0 1 338.64 564.24 Tm <70726F706F73616C732C20> Tj 1 0 0 1 374.64 564.24 Tm <70726573656E74656420> Tj 1 0 0 1 409.44 563.52 Tm <696E20> Tj 1 0 0 1 416.88 563.52 Tm <666F757220> Tj 1 0 0 1 432.00 563.28 Tm <73746570732C20> Tj 1 0 0 1 453.36 563.28 Tm <666F7220> Tj 1 0 0 1 464.16 563.04 Tm <74686520> Tj 1 0 0 1 476.64 563.04 Tm <646574656D696E6174696F6E20> Tj 1 0 0 1 524.88 563.04 Tm <6F6620> Tj 1 0 0 1 532.80 563.04 Tm <746865> Tj 0.48 Tc 1 0 0 1 189.60 553.92 Tm <6C616772616E676520> Tj 1 0 0 1 222.72 554.16 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 267.36 554.16 Tm <76616C756520> Tj 1 0 0 1 287.52 554.16 Tm <696E20> Tj 1 0 0 1 295.92 553.92 Tm <616E7920> Tj 1 0 0 1 309.84 553.92 Tm <307264696E61747920> Tj 1 0 0 1 340.56 554.16 Tm <4B746967696E6720> Tj 1 0 0 1 366.00 553.92 Tm <6571756174696F6E2E> Tj 0.48 Tc 1 0 0 1 189.12 536.16 Tm <4B657920> Tj 1 0 0 1 204.48 536.16 Tm <776F726473272E20> Tj 1 0 0 1 230.88 536.64 Tm <67656F737461746973746963732C20> Tj 1 0 0 1 280.08 536.16 Tm <6B726967696E672C20> Tj 1 0 0 1 308.88 536.16 Tm <6C696E65617220> Tj 1 0 0 1 330.48 536.16 Tm <657374696D61746F722C20> Tj 1 0 0 1 368.16 536.16 Tm <4C616772616E676520> Tj 1 0 0 1 404.40 535.92 Tm <6D756C7469706C696572> Tj 0.96 Tc /F3 26.0 Tf 1 0 0 1 51.12 745.20 Tm <4C696E65617269747920> Tj 1 0 0 1 160.80 744.96 Tm <616E6420> Tj 1 0 0 1 209.52 745.68 Tm <4C616772616E676520> Tj 1 0 0 1 321.84 745.68 Tm <4C696E656172> Tj 0.48 Tc /F3 12.0 Tf 1 0 0 1 51.84 491.28 Tm <312E20> Tj 1 0 0 1 66.00 491.28 Tm <494E54524F44554354494F4E> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 50.40 476.88 Tm <54686520> Tj 1 0 0 1 69.12 476.88 Tm <6B726967696E6720> Tj 1 0 0 1 101.52 476.88 Tm <6D6574686F6420> Tj 1 0 0 1 136.56 477.12 Tm <697320> Tj 1 0 0 1 146.64 477.12 Tm <636F6E7369646572656420> Tj 1 0 0 1 196.08 477.12 Tm <6120> Tj 1 0 0 1 203.28 477.60 Tm <70726F677065737369766520> Tj 1 0 0 1 254.64 477.36 Tm <696E746572706F2D> Tj 0.24 Tc 1 0 0 1 50.40 466.80 Tm <6C6174696F6E20> Tj 1 0 0 1 78.00 466.80 Tm <6D6574686F6420> Tj 1 0 0 1 113.76 466.80 Tm <666F7220> Tj 1 0 0 1 129.36 466.80 Tm <74686520> Tj 1 0 0 1 146.40 466.80 Tm <657374696D61666C6F6E20> Tj 1 0 0 1 194.64 466.80 Tm <6F6620> Tj 1 0 0 1 206.16 466.80 Tm <6120> Tj 1 0 0 1 214.32 466.80 Tm <726567696F6E616C697A656420> Tj 1 0 0 1 269.04 466.80 Tm <766172692D> Tj 0.24 Tc 1 0 0 1 50.64 456.48 Tm <61626C6520> Tj 1 0 0 1 73.92 456.48 Tm <617420> Tj 1 0 0 1 87.84 456.48 Tm <73656C656374656420> Tj 1 0 0 1 127.44 456.96 Tm <6772696420> Tj 1 0 0 1 150.24 456.96 Tm <706F696E74732E20> Tj 1 0 0 1 185.52 456.72 Tm <54686520> Tj 1 0 0 1 207.60 456.72 Tm <7465726D20> Tj 1 0 0 1 234.00 456.72 Tm <726567696F6E616C697A6564> Tj 0.24 Tc 1 0 0 1 50.64 446.16 Tm <7561726961626C6520> Tj 1 0 0 1 88.80 446.16 Tm <72656665727320> Tj 1 0 0 1 115.92 446.16 Tm <746F20> Tj 1 0 0 1 126.96 446.16 Tm <7375636820> Tj 1 0 0 1 149.04 446.16 Tm <7661726961626C65732C20> Tj 1 0 0 1 192.48 446.40 Tm <776869636820> Tj 1 0 0 1 220.80 446.40 Tm <63616E20> Tj 1 0 0 1 237.84 446.40 Tm <626520> Tj 1 0 0 1 250.32 446.64 Tm <7265676172646564> Tj 0.24 Tc 1 0 0 1 50.40 435.84 Tm <746F20> Tj 1 0 0 1 60.48 435.84 Tm <626520> Tj 1 0 0 1 72.72 436.08 Tm <72616E646F6D20> Tj 1 0 0 1 108.48 435.84 Tm <7661726961626C6520> Tj 1 0 0 1 144.72 436.08 Tm <696E20> Tj 1 0 0 1 155.28 436.08 Tm <74686520> Tj 1 0 0 1 170.64 436.08 Tm <696E66696E6974656C7920> Tj 1 0 0 1 210.00 436.08 Tm <736D616C6C20> Tj 1 0 0 1 235.20 436.08 Tm <737572726F756E64696E67> Tj 0.24 Tc 1 0 0 1 50.64 425.52 Tm <6F6620> Tj 1 0 0 1 61.44 425.52 Tm <6120> Tj 1 0 0 1 68.88 426.00 Tm <706172746963756C617220> Tj 1 0 0 1 114.24 425.76 Tm <6C6F636174696F6E2C20> Tj 1 0 0 1 153.36 425.76 Tm <62757420> Tj 1 0 0 1 170.64 425.76 Tm <746865726520> Tj 1 0 0 1 195.60 425.76 Tm <657869737420> Tj 1 0 0 1 218.16 425.76 Tm <6120> Tj 1 0 0 1 225.60 426.00 Tm <66756E6374696F6E20> Tj 1 0 0 1 264.00 425.76 Tm <61626F7665> Tj 0.24 Tc 1 0 0 1 50.40 415.20 Tm <74686520> Tj 1 0 0 1 65.76 415.20 Tm <726567696F6E20> Tj 1 0 0 1 94.56 415.20 Tm <776869636820> Tj 1 0 0 1 122.64 415.20 Tm <63616E20> Tj 1 0 0 1 139.92 415.44 Tm <72656C61746520> Tj 1 0 0 1 166.08 415.20 Tm <746865736520> Tj 1 0 0 1 190.32 415.44 Tm <7661726961626C657320> Tj 1 0 0 1 231.36 415.44 Tm <746F20> Tj 1 0 0 1 242.16 415.44 Tm <6561636820> Tj 1 0 0 1 264.24 415.44 Tm <6F746865722E> Tj 0.24 Tc 1 0 0 1 50.40 404.88 Tm <54686520> Tj 1 0 0 1 69.60 404.88 Tm <7465726D20> Tj 1 0 0 1 93.60 406.32 Tm <2272616E646F6D2220> Tj 1 0 0 1 137.28 405.12 Tm <617373756D657320> Tj 1 0 0 1 177.36 405.12 Tm <7468617420> Tj 1 0 0 1 197.52 405.12 Tm <74686520> Tj 1 0 0 1 213.84 405.60 Tm <70726F626162696C69747920> Tj 1 0 0 1 263.76 405.12 Tm <6F6620> Tj 1 0 0 1 275.04 405.12 Tm <746865> Tj 0.24 Tc 1 0 0 1 49.92 394.80 Tm <7661726961626C6520> Tj 1 0 0 1 86.40 394.80 Tm <6265696E2420> Tj 1 0 0 1 112.32 394.80 Tm <736D616C6C657220> Tj 1 0 0 1 146.88 394.80 Tm <7468616E20> Tj 1 0 0 1 168.72 394.80 Tm <6120> Tj 1 0 0 1 176.16 395.04 Tm <7265616C20> Tj 1 0 0 1 195.12 395.04 Tm <6E756D62657220> Tj 1 0 0 1 231.84 395.04 Tm <65786973747320> Tj 1 0 0 1 258.48 395.04 Tm <666F7220> Tj 1 0 0 1 273.36 395.04 Tm <616E79> Tj 0.24 Tc 1 0 0 1 50.40 384.48 Tm <7265616C20> Tj 1 0 0 1 69.12 384.48 Tm <6E756D6265722E> Tj 0.24 Tc 1 0 0 1 57.36 370.32 Tm <4B64676E672C20> Tj 1 0 0 1 92.64 370.08 Tm <617320> Tj 1 0 0 1 103.92 370.32 Tm <6120> Tj 1 0 0 1 110.88 370.32 Tm <737461746973746963616C20> Tj 1 0 0 1 153.60 370.32 Tm <657374696D6174696F6E20> Tj 1 0 0 1 200.16 370.56 Tm <6D6574686F642C20> Tj 1 0 0 1 237.12 370.56 Tm <697320> Tj 1 0 0 1 246.48 370.56 Tm <6E616D656420> Tj 1 0 0 1 277.68 370.56 Tm <61662D> Tj 0.24 Tc 1 0 0 1 50.16 360.00 Tm <74657220> Tj 1 0 0 1 66.24 360.00 Tm <536F75746820> Tj 1 0 0 1 95.04 360.00 Tm <416672696B61616E7320> Tj 1 0 0 1 140.88 360.24 Tm <656E67696E65657220> Tj 1 0 0 1 180.96 360.24 Tm <4B726C6765752C20> Tj 1 0 0 1 213.12 360.24 Tm <77686F20> Tj 1 0 0 1 234.96 360.24 Tm <666972737420> Tj 1 0 0 1 256.80 360.24 Tm <6170706C696564> Tj 0.24 Tc 1 0 0 1 50.40 349.68 Tm <616E6420> Tj 1 0 0 1 68.64 349.68 Tm <64657363726962656420> Tj 1 0 0 1 112.08 349.68 Tm <7468697320> Tj 1 0 0 1 130.32 349.92 Tm <6D6574686F646F6C6F677220> Tj 1 0 0 1 186.72 349.92 Tm <746F20> Tj 1 0 0 1 197.52 349.92 Tm <657374696D61746520> Tj 1 0 0 1 235.20 350.40 Tm <676F6C6420> Tj 1 0 0 1 255.60 349.92 Tm <636F6E63656E2D> Tj 0.24 Tc 1 0 0 1 50.16 339.36 Tm <74726174696F6E7320> Tj 1 0 0 1 85.20 339.60 Tm <696E20> Tj 1 0 0 1 95.76 339.36 Tm <6F726520> Tj 1 0 0 1 111.36 339.60 Tm <6465706F736974732E20> Tj 1 0 0 1 151.68 339.60 Tm <5468697320> Tj 1 0 0 1 172.08 339.60 Tm <77617320> Tj 1 0 0 1 190.56 339.60 Tm <666F6C6C6F77656420> Tj 1 0 0 1 228.00 339.60 Tm <627920> Tj 1 0 0 1 240.00 339.84 Tm <6675727468657220> Tj 1 0 0 1 271.92 339.60 Tm <7468652D> Tj 0.24 Tc 1 0 0 1 50.40 329.04 Tm <6F7265746963616C20> Tj 1 0 0 1 85.92 329.04 Tm <646576656C6F706D656E7420> Tj 1 0 0 1 144.00 329.28 Tm <6F6620> Tj 1 0 0 1 155.52 329.28 Tm <74686520> Tj 1 0 0 1 172.56 329.28 Tm <6D6574686F642C20> Tj 1 0 0 1 211.68 329.76 Tm <707265646F6D696E616E666C7920> Tj 1 0 0 1 277.92 329.28 Tm <6279> Tj 0.24 Tc 1 0 0 1 50.16 318.96 Tm <4672656E636820> Tj 1 0 0 1 82.56 318.72 Tm <736369656E74697374732C20> Tj 1 0 0 1 127.92 318.96 Tm <657370656369616C6C7920> Tj 1 0 0 1 171.84 318.96 Tm <4D61746865726F6E6C20> Tj 1 0 0 1 216.72 321.60 Tm <312E20> Tj 1 0 0 1 225.12 319.20 Tm <46696E616C6C792C20> Tj 1 0 0 1 258.48 318.96 Tm <7365766572616C> Tj 0.00 Tc 1 0 0 1 50.16 308.64 Tm <6B726967696E6720> Tj 1 0 0 1 81.84 308.64 Tm <746563686E697175657320> Tj 1 0 0 1 129.60 308.88 Tm <6861766520> Tj 1 0 0 1 150.96 308.64 Tm <6265656E20> Tj 1 0 0 1 173.28 308.64 Tm <646566696E656420> Tj 1 0 0 1 206.16 308.64 Tm <7769746820> Tj 1 0 0 1 227.04 308.88 Tm <646966666572656E7420> Tj 1 0 0 1 264.72 308.88 Tm <657175612D> Tj 0.24 Tc 1 0 0 1 50.16 298.32 Tm <74696F6E7320> Tj 1 0 0 1 74.88 299.28 Tm <287375636820> Tj 1 0 0 1 101.28 298.32 Tm <617320> Tj 1 0 0 1 114.00 298.32 Tm <53696D706C652C20> Tj 1 0 0 1 150.00 298.32 Tm <4F7264696E6172792C20> Tj 1 0 0 1 193.44 298.32 Tm <616E6420> Tj 1 0 0 1 212.88 298.56 Tm <496E64696361746F7220> Tj 1 0 0 1 254.88 298.56 Tm <4974756967696E672C> Tj 0.48 Tc 1 0 0 1 50.40 288.00 Tm <616E6420> Tj 1 0 0 1 69.12 288.00 Tm <6F7468657273292E> Tj 0.00 Tc 1 0 0 1 57.12 273.84 Tm <4B7269676C6E6720> Tj 1 0 0 1 90.72 273.60 Tm <657374696D6174696F6E20> Tj 1 0 0 1 137.76 273.84 Tm <697320> Tj 1 0 0 1 147.60 273.84 Tm <626173656420> Tj 1 0 0 1 175.44 273.84 Tm <6F6E20> Tj 1 0 0 1 188.88 273.84 Tm <74686520> Tj 1 0 0 1 204.96 273.84 Tm <6170706C69636174696F6E20> Tj 1 0 0 1 254.88 273.84 Tm <6F6620> Tj 1 0 0 1 265.68 273.84 Tm <65786973742D> Tj 0.24 Tc 1 0 0 1 50.16 263.52 Tm <696E6720> Tj 1 0 0 1 65.04 263.52 Tm <6D6561737572656D656E747320> Tj 1 0 0 1 130.32 264.48 Tm <28736F2D63616C6C656420> Tj 1 0 0 1 173.52 264.96 Tm <22636F6E74726F6C20> Tj 1 0 0 1 209.52 264.00 Tm <706F696E747322292C20> Tj 1 0 0 1 247.68 263.52 Tm <77686F736520> Tj 1 0 0 1 276.72 263.76 Tm <696E2D> Tj 0.00 Tc 1 0 0 1 49.92 253.20 Tm <666C75656E636520> Tj 1 0 0 1 84.72 253.20 Tm <6F6E20> Tj 1 0 0 1 99.84 253.20 Tm <74686520> Tj 1 0 0 1 117.60 253.20 Tm <657374696D6174696F6E20> Tj 1 0 0 1 166.32 253.44 Tm <697320> Tj 1 0 0 1 178.32 253.20 Tm <65787072657373656420> Tj 1 0 0 1 224.88 253.44 Tm <7769746820> Tj 1 0 0 1 247.68 253.44 Tm <776569676874696E67> Tj 0.24 Tc 1 0 0 1 50.16 242.64 Tm <636F656666696369656E74732E20> Tj 1 0 0 1 103.68 242.88 Tm <5468697320> Tj 1 0 0 1 126.48 242.88 Tm <657374696D6174696F6E20> Tj 1 0 0 1 174.72 242.88 Tm <616C736F20> Tj 1 0 0 1 195.84 242.88 Tm <617373756D657320> Tj 1 0 0 1 236.16 242.88 Tm <74686520> Tj 1 0 0 1 253.20 242.88 Tm <73617469736661632D> Tj 0.24 Tc 1 0 0 1 50.16 232.56 Tm <74696F6E20> Tj 1 0 0 1 69.36 232.56 Tm <6F6620> Tj 1 0 0 1 79.92 232.56 Tm <74686520> Tj 1 0 0 1 96.00 232.56 Tm <72656C6576616E7420> Tj 1 0 0 1 132.72 232.56 Tm <63726974657269612E20> Tj 1 0 0 1 168.72 232.80 Tm <5468697320> Tj 1 0 0 1 190.56 232.80 Tm <6D65616E7320> Tj 1 0 0 1 221.28 232.80 Tm <7468617420> Tj 1 0 0 1 240.72 232.80 Tm <74686520> Tj 1 0 0 1 256.80 232.80 Tm <657374696D612D> Tj 0.24 Tc 1 0 0 1 50.16 222.24 Tm <74696F6E20> Tj 1 0 0 1 72.72 222.24 Tm <6D75737420> Tj 1 0 0 1 100.56 222.24 Tm <626520> Tj 1 0 0 1 116.88 222.48 Tm <756E62696173656420> Tj 1 0 0 1 162.24 222.24 Tm <616E6420> Tj 1 0 0 1 184.32 222.96 Tm <706572666F726D656420> Tj 1 0 0 1 235.92 222.48 Tm <736F20> Tj 1 0 0 1 251.52 222.48 Tm <7468617420> Tj 1 0 0 1 274.56 222.48 Tm <746865> Tj 0.24 Tc 1 0 0 1 49.68 211.92 Tm <76617269616E636520> Tj 1 0 0 1 88.32 211.92 Tm <6265747765656E20> Tj 1 0 0 1 125.76 211.92 Tm <74686520> Tj 1 0 0 1 141.84 212.16 Tm <6D6561737572656420> Tj 1 0 0 1 187.20 212.16 Tm <616E6420> Tj 1 0 0 1 206.16 212.16 Tm <657374696D6174656420> Tj 1 0 0 1 250.08 212.16 Tm <76616C75657320> Tj 1 0 0 1 279.84 212.16 Tm <6174> Tj 0.24 Tc 1 0 0 1 49.92 201.60 Tm <74686520> Tj 1 0 0 1 66.00 201.60 Tm <73656C656374656420> Tj 1 0 0 1 102.48 202.08 Tm <706F696E747320> Tj 1 0 0 1 131.76 201.60 Tm <697320> Tj 1 0 0 1 141.84 201.60 Tm <617320> Tj 1 0 0 1 153.84 201.84 Tm <6C6F7720> Tj 1 0 0 1 171.36 201.60 Tm <617320> Tj 1 0 0 1 183.12 202.32 Tm <706F737369626C652E20> Tj 1 0 0 1 223.92 201.84 Tm <5375636820> Tj 1 0 0 1 248.40 201.84 Tm <6120> Tj 1 0 0 1 255.36 201.84 Tm <76616C756520> Tj 1 0 0 1 280.56 201.84 Tm <6973> Tj 0.24 Tc 1 0 0 1 50.16 191.28 Tm <616C736F20> Tj 1 0 0 1 70.56 191.28 Tm <63616C6C656420> Tj 1 0 0 1 98.88 191.28 Tm <74686520> Tj 1 0 0 1 115.68 192.72 Tm <226B726967696E6720> Tj 1 0 0 1 151.92 191.28 Tm <76617269616E6365222C20> Tj 1 0 0 1 197.28 191.52 Tm <776869636820> Tj 1 0 0 1 226.08 191.52 Tm <726570726573656E747320> Tj 1 0 0 1 274.56 191.52 Tm <746865> Tj 0.24 Tc 1 0 0 1 49.92 180.96 Tm <696E746572706F6C6174696F6E20> Tj 1 0 0 1 111.60 181.44 Tm <7175616C69747920> Tj 1 0 0 1 147.36 180.96 Tm <616E6420> Tj 1 0 0 1 170.16 180.96 Tm <63616E20> Tj 1 0 0 1 191.52 181.20 Tm <626520> Tj 1 0 0 1 208.56 181.20 Tm <636F6D706172656420> Tj 1 0 0 1 257.52 180.96 Tm <7769746820> Tj 1 0 0 1 282.96 181.20 Tm <61> Tj 0.24 Tc 1 0 0 1 50.16 170.64 Tm <63726F73732D76616C69646174696F6E20> Tj 1 0 0 1 120.24 170.64 Tm <746563686E697175652E> Tj 0.24 Tc 1 0 0 1 56.88 156.24 Tm <496E20> Tj 1 0 0 1 67.68 156.24 Tm <657665727920> Tj 1 0 0 1 91.68 156.24 Tm <6B726967696E6720> Tj 1 0 0 1 123.60 156.24 Tm <746563686E69717565732C20> Tj 1 0 0 1 174.24 156.24 Tm <74686520> Tj 1 0 0 1 189.60 156.72 Tm <70726F63656475726520> Tj 1 0 0 1 235.44 156.48 Tm <666F7220> Tj 1 0 0 1 249.60 156.48 Tm <6D696E696D697A2D> Tj 0.24 Tc 1 0 0 1 49.92 145.92 Tm <696E6720> Tj 1 0 0 1 66.72 145.92 Tm <74686520> Tj 1 0 0 1 84.00 145.92 Tm <657374696D6174696F6E20> Tj 1 0 0 1 132.96 145.92 Tm <6572726F7220> Tj 1 0 0 1 158.88 146.16 Tm <696E636C7564657320> Tj 1 0 0 1 198.96 146.16 Tm <6120> Tj 1 0 0 1 207.36 145.92 Tm <7661726961626C6520> Tj 1 0 0 1 245.76 146.16 Tm <63616C6C656420> Tj 1 0 0 1 274.56 146.16 Tm <746865> Tj 0.24 Tc 1 0 0 1 50.16 136.32 Tm <224C616772616E676520> Tj 1 0 0 1 96.48 135.60 Tm <6D756C7469706C696361746F72222E20> Tj 1 0 0 1 162.72 135.84 Tm <54686520> Tj 1 0 0 1 183.60 136.32 Tm <707572706F736520> Tj 1 0 0 1 223.20 135.84 Tm <6F6620> Tj 1 0 0 1 235.68 135.84 Tm <74686520> Tj 1 0 0 1 253.68 135.84 Tm <616E616C79736973> Tj 0.24 Tc 1 0 0 1 49.68 125.76 Tm <70726573656E74656420> Tj 1 0 0 1 94.32 125.28 Tm <6865726520> Tj 1 0 0 1 115.68 125.52 Tm <697320> Tj 1 0 0 1 125.76 125.28 Tm <746F20> Tj 1 0 0 1 136.80 125.28 Tm <646573637269626520> Tj 1 0 0 1 175.20 125.52 Tm <74686520> Tj 1 0 0 1 191.04 125.52 Tm <6D617468656D61746963616C20> Tj 1 0 0 1 251.52 125.52 Tm <6D65616E696E67> Tj 0.00 Tc 1 0 0 1 49.92 114.96 Tm <6F6620> Tj 1 0 0 1 59.76 114.96 Tm <74686520> Tj 1 0 0 1 75.12 115.20 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 132.96 115.20 Tm <616E6420> Tj 1 0 0 1 150.96 115.20 Tm <69747320> Tj 1 0 0 1 163.44 115.20 Tm <726F6C6520> Tj 1 0 0 1 182.16 115.20 Tm <696E20> Tj 1 0 0 1 192.48 115.20 Tm <6D696E696D697A696E6774686520> Tj 1 0 0 1 258.00 115.20 Tm <6B726967696E67> Tj 0.24 Tc 1 0 0 1 49.44 104.64 Tm <7661666C616E63652E> Tj 0.24 Tc 1 0 0 1 50.16 81.84 Tm <4E4146544120> Tj 1 0 0 1 70.08 81.84 Tm <353920> Tj 1 0 0 1 79.44 82.32 Tm <283120> Tj 1 0 0 1 84.96 80.40 Tm <2920> Tj 1 0 0 1 88.80 81.84 Tm <333120> Tj 1 0 0 1 96.24 83.76 Tm <2D333720> Tj 1 0 0 1 108.00 82.32 Tm <283230303929> Tj 0.48 Tc /F3 12.0 Tf 1 0 0 1 305.52 491.52 Tm <322E20> Tj 1 0 0 1 320.88 491.52 Tm <4B524947494E4720> Tj 1 0 0 1 380.40 491.28 Tm <50524F50455254494553> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 305.28 477.84 Tm <497B6E67696E6720> Tj 1 0 0 1 339.60 477.84 Tm <746563686E697175657320> Tj 1 0 0 1 389.28 477.84 Tm <62656C6F6E6720> Tj 1 0 0 1 421.20 477.60 Tm <746F20> Tj 1 0 0 1 433.20 477.60 Tm <6120> Tj 1 0 0 1 442.08 477.60 Tm <73657420> Tj 1 0 0 1 457.68 477.36 Tm <6F6620> Tj 1 0 0 1 469.20 477.60 Tm <6C696E65617220> Tj 1 0 0 1 498.00 477.36 Tm <616C676F726974686D73> Tj 0.24 Tc 1 0 0 1 305.28 467.28 Tm <7468617420> Tj 1 0 0 1 325.68 467.28 Tm <61726520> Tj 1 0 0 1 342.00 467.28 Tm <626173656420> Tj 1 0 0 1 370.80 467.28 Tm <6F6E20> Tj 1 0 0 1 384.96 467.28 Tm <74686520> Tj 1 0 0 1 401.28 467.28 Tm <6C6561737420> Tj 1 0 0 1 424.80 467.04 Tm <7371756172657320> Tj 1 0 0 1 461.52 467.04 Tm <6D6574686F642E20> Tj 1 0 0 1 500.16 467.04 Tm <54686520> Tj 1 0 0 1 519.84 466.80 Tm <73656C65632D> Tj 0.24 Tc 1 0 0 1 305.28 456.72 Tm <74696F6E20> Tj 1 0 0 1 324.24 456.72 Tm <6F6674686520> Tj 1 0 0 1 349.92 456.72 Tm <617070726F70726961746520> Tj 1 0 0 1 402.24 456.72 Tm <746563686E6971756520> Tj 1 0 0 1 446.16 456.48 Tm <697320> Tj 1 0 0 1 455.28 456.48 Tm <626173656420> Tj 1 0 0 1 483.12 456.24 Tm <6F6E20> Tj 1 0 0 1 495.84 456.24 Tm <74686520> Tj 1 0 0 1 511.20 456.72 Tm <70726F7065722D> Tj 0.24 Tc 1 0 0 1 305.28 446.16 Tm <7469657320> Tj 1 0 0 1 323.28 446.16 Tm <6F6620> Tj 1 0 0 1 334.32 446.40 Tm <74686520> Tj 1 0 0 1 350.40 446.40 Tm <72616E646F6D20> Tj 1 0 0 1 387.12 446.16 Tm <7661726961626C6520> Tj 1 0 0 1 424.08 445.92 Tm <7468617420> Tj 1 0 0 1 444.00 445.92 Tm <697320> Tj 1 0 0 1 454.32 445.92 Tm <746F20> Tj 1 0 0 1 465.36 445.92 Tm <626520> Tj 1 0 0 1 478.56 445.92 Tm <696E746572706F6C6174656420> Tj 1 0 0 1 533.28 445.68 Tm <6279> Tj 0.24 Tc 1 0 0 1 305.28 435.84 Tm <686967726E672E20> Tj 1 0 0 1 341.04 435.84 Tm <54686520> Tj 1 0 0 1 360.72 435.60 Tm <746563686E697175657320> Tj 1 0 0 1 410.64 435.60 Tm <61726520> Tj 1 0 0 1 427.20 435.36 Tm <626173656420> Tj 1 0 0 1 456.24 435.36 Tm <6F6E20> Tj 1 0 0 1 470.16 435.12 Tm <74686520> Tj 1 0 0 1 487.20 435.12 Tm <657374696D6174696F6E20> Tj 1 0 0 1 535.44 435.12 Tm <6F66> Tj 0.00 Tc 1 0 0 1 304.80 425.04 Tm <776569676874696E6720> Tj 1 0 0 1 349.20 425.04 Tm <636F656669696369656E747320> Tj 1 0 0 1 399.60 424.80 Tm <7769746820> Tj 1 0 0 1 422.16 424.80 Tm <616E20> Tj 1 0 0 1 436.32 424.80 Tm <617373756D7074696F6E20> Tj 1 0 0 1 490.08 424.56 Tm <6F6620> Tj 1 0 0 1 501.36 424.80 Tm <756E6269617365642D> Tj 0.24 Tc 1 0 0 1 305.04 414.48 Tm <6E6573732E20> Tj 1 0 0 1 330.72 414.48 Tm <4561636820> Tj 1 0 0 1 355.92 414.24 Tm <6861726420> Tj 1 0 0 1 380.40 414.24 Tm <6461746120> Tj 1 0 0 1 402.24 414.24 Tm <68617320> Tj 1 0 0 1 420.96 414.24 Tm <69747320> Tj 1 0 0 1 435.36 414.00 Tm <6F76726E20> Tj 1 0 0 1 456.24 414.00 Tm <776569676874696E6720> Tj 1 0 0 1 500.64 414.00 Tm <636F656666696369656E74> Tj 0.24 Tc 1 0 0 1 305.52 404.16 Tm <283122292C20> Tj 1 0 0 1 320.88 403.20 Tm <776869636820> Tj 1 0 0 1 348.24 403.20 Tm <726570726573656E747320> Tj 1 0 0 1 395.28 402.96 Tm <74686520> Tj 1 0 0 1 410.40 403.20 Tm <696E666C75656E636520> Tj 1 0 0 1 450.96 402.72 Tm <6F6620> Tj 1 0 0 1 461.04 402.72 Tm <6120> Tj 1 0 0 1 467.52 403.20 Tm <706172746963756C617220> Tj 1 0 0 1 512.16 402.72 Tm <6461746120> Tj 1 0 0 1 532.56 402.72 Tm <6F6E> Tj 0.24 Tc 1 0 0 1 305.04 392.64 Tm <74686520> Tj 1 0 0 1 320.16 392.64 Tm <76616C756520> Tj 1 0 0 1 345.12 392.64 Tm <6F6620> Tj 1 0 0 1 355.20 392.64 Tm <74686520> Tj 1 0 0 1 370.56 392.64 Tm <66696E616C20> Tj 1 0 0 1 391.68 392.64 Tm <657374696D6174696F6E20> Tj 1 0 0 1 438.72 392.40 Tm <617420> Tj 1 0 0 1 448.80 392.40 Tm <74686520> Tj 1 0 0 1 464.40 392.16 Tm <73656C656374656420> Tj 1 0 0 1 500.40 392.64 Tm <6772696420> Tj 1 0 0 1 519.60 392.16 Tm <6E6F64652E> Tj 0.24 Tc 1 0 0 1 305.04 382.32 Tm <54686520> Tj 1 0 0 1 323.28 382.32 Tm <72656C6174696F6E7368697020> Tj 1 0 0 1 376.32 382.08 Tm <6265747765656E20> Tj 1 0 0 1 413.04 382.08 Tm <74686520> Tj 1 0 0 1 428.40 381.84 Tm <6578697374696E6720> Tj 1 0 0 1 463.20 382.56 Tm <28686172642920> Tj 1 0 0 1 491.52 381.60 Tm <6461746120> Tj 1 0 0 1 511.92 381.60 Tm <616E6420> Tj 1 0 0 1 529.92 381.60 Tm <746865> Tj 0.24 Tc 1 0 0 1 305.28 371.76 Tm <657374696D6174696F6E20> Tj 1 0 0 1 351.60 372.00 Tm <706F696E7420> Tj 1 0 0 1 375.84 371.76 Tm <68617320> Tj 1 0 0 1 392.64 371.52 Tm <6265656E20> Tj 1 0 0 1 414.96 371.52 Tm <65787072657373656420> Tj 1 0 0 1 459.12 371.28 Tm <6279766172696F6772616D20> Tj 1 0 0 1 516.24 371.28 Tm <76616C756573> Tj 0.24 Tc 1 0 0 1 305.52 361.92 Tm <28696620> Tj 1 0 0 1 315.60 361.20 Tm <776520> Tj 1 0 0 1 329.52 361.20 Tm <737570706F736520> Tj 1 0 0 1 366.96 360.96 Tm <74686520> Tj 1 0 0 1 382.32 360.96 Tm <6578697374656E636520> Tj 1 0 0 1 423.12 360.96 Tm <6F6620> Tj 1 0 0 1 433.20 360.96 Tm <696E7472696E73696320> Tj 1 0 0 1 470.88 360.72 Tm <6879706F7468657369732920> Tj 1 0 0 1 522.00 360.72 Tm <6F7220> Tj 1 0 0 1 533.04 360.72 Tm <6279> Tj 0.24 Tc 1 0 0 1 305.28 350.40 Tm <636F76617269616E636520> Tj 1 0 0 1 352.56 350.64 Tm <696E20> Tj 1 0 0 1 363.36 350.40 Tm <6361736520> Tj 1 0 0 1 384.00 350.40 Tm <6F667365636F6E6420> Tj 1 0 0 1 426.24 350.16 Tm <6F7264657220> Tj 1 0 0 1 451.92 350.16 Tm <73746174696F6E61726974792E20> Tj 1 0 0 1 504.48 350.16 Tm <5375636876616C2D> Tj 0.24 Tc 1 0 0 1 305.04 340.08 Tm <75657320> Tj 1 0 0 1 322.32 339.84 Tm <646573637269626520> Tj 1 0 0 1 360.48 339.84 Tm <74686520> Tj 1 0 0 1 376.32 339.84 Tm <7370617469616C20> Tj 1 0 0 1 407.28 339.84 Tm <646570656E64656E636520> Tj 1 0 0 1 460.08 339.60 Tm <616E6420> Tj 1 0 0 1 478.56 339.60 Tm <74686520> Tj 1 0 0 1 494.16 339.60 Tm <696E666C75656E636520> Tj 1 0 0 1 535.20 339.36 Tm <6F66> Tj 0.24 Tc 1 0 0 1 305.04 329.28 Tm <74686520> Tj 1 0 0 1 320.64 329.76 Tm <706172746963756C617220> Tj 1 0 0 1 365.28 329.28 Tm <6C6F636174696F6E20> Tj 1 0 0 1 401.28 329.28 Tm <696E20> Tj 1 0 0 1 412.08 329.04 Tm <7465726D7320> Tj 1 0 0 1 439.44 329.04 Tm <6F6620> Tj 1 0 0 1 449.52 329.04 Tm <69747320> Tj 1 0 0 1 462.48 328.80 Tm <64697374616E636520> Tj 1 0 0 1 500.40 328.80 Tm <616E6420> Tj 1 0 0 1 518.88 328.80 Tm <64697265632D> Tj 0.24 Tc 1 0 0 1 305.04 318.72 Tm <74696F6E20> Tj 1 0 0 1 324.24 318.72 Tm <66726F6D20> Tj 1 0 0 1 347.52 318.72 Tm <74686520> Tj 1 0 0 1 363.60 318.72 Tm <657374696D6174656420> Tj 1 0 0 1 407.52 318.48 Tm <6C6F636174696F6E20> Tj 1 0 0 1 444.48 319.20 Tm <28706F696E74292E> Tj 0.24 Tc 1 0 0 1 311.52 306.24 Tm <4120> Tj 1 0 0 1 321.84 306.00 Tm <7370617469616C20> Tj 1 0 0 1 353.76 306.00 Tm <6D6F64656C20> Tj 1 0 0 1 383.52 306.00 Tm <626173656420> Tj 1 0 0 1 412.56 305.76 Tm <6F6E20> Tj 1 0 0 1 426.96 305.76 Tm <6120> Tj 1 0 0 1 435.12 305.76 Tm <6C617267657220> Tj 1 0 0 1 463.68 305.76 Tm <6E756D62657220> Tj 1 0 0 1 501.36 305.52 Tm <6F6620> Tj 1 0 0 1 512.88 305.52 Tm <636F6E74726F6C> Tj 0.24 Tc 1 0 0 1 304.80 295.92 Tm <706F696E74732C20> Tj 1 0 0 1 337.68 295.44 Tm <6120> Tj 1 0 0 1 345.60 295.44 Tm <6869676C657220> Tj 1 0 0 1 375.84 295.44 Tm <766172696F6772616D20> Tj 1 0 0 1 422.64 295.44 Tm <72616E67652C20> Tj 1 0 0 1 452.40 294.96 Tm <6120> Tj 1 0 0 1 460.32 295.20 Tm <6C6F77657220> Tj 1 0 0 1 486.96 294.96 Tm <6E756767657420> Tj 1 0 0 1 518.16 294.96 Tm <6566666563742C> Tj 0.24 Tc 1 0 0 1 305.04 285.12 Tm <616E6420> Tj 1 0 0 1 324.48 284.88 Tm <776974686F757420> Tj 1 0 0 1 360.72 284.88 Tm <616E69736F74726F707920> Tj 1 0 0 1 409.68 284.88 Tm <697320> Tj 1 0 0 1 420.48 284.88 Tm <757375616C6C7920> Tj 1 0 0 1 454.80 284.64 Tm <6D75636820> Tj 1 0 0 1 482.64 284.64 Tm <6D6F726520> Tj 1 0 0 1 508.56 284.64 Tm <72656C6961626C652E> Tj 0.24 Tc 1 0 0 1 305.04 274.56 Tm <5265676172646C65737320> Tj 1 0 0 1 355.20 274.32 Tm <6F6620> Tj 1 0 0 1 366.96 274.56 Tm <74686520> Tj 1 0 0 1 384.24 274.32 Tm <7370617469616C20> Tj 1 0 0 1 416.40 274.32 Tm <646570656E64656E636520> Tj 1 0 0 1 470.64 274.08 Tm <64657363726962656420> Tj 1 0 0 1 515.76 273.84 Tm <627920> Tj 1 0 0 1 529.68 273.84 Tm <746865> Tj 0.24 Tc 1 0 0 1 304.56 263.76 Tm <766172696F6772616D20> Tj 1 0 0 1 354.24 263.76 Tm <6F7220> Tj 1 0 0 1 369.36 263.76 Tm <74686520> Tj 1 0 0 1 388.80 263.76 Tm <636F76617269616E636520> Tj 1 0 0 1 439.92 263.76 Tm <66756E6374696F6E2C20> Tj 1 0 0 1 483.84 263.28 Tm <74686520> Tj 1 0 0 1 502.56 263.28 Tm <776569676C74696E67> Tj 0.00 Tc 1 0 0 1 305.04 253.20 Tm <636F656666696369656E7420> Tj 1 0 0 1 354.96 253.20 Tm <617420> Tj 1 0 0 1 369.84 253.20 Tm <6120> Tj 1 0 0 1 381.60 253.68 Tm <706172746963756C617220> Tj 1 0 0 1 430.80 253.20 Tm <6C6F636174696F6E20> Tj 1 0 0 1 471.60 252.96 Tm <697320> Tj 1 0 0 1 486.00 252.96 Tm <6578636C75736976656C7920> Tj 1 0 0 1 538.32 252.72 Tm <61> Tj 0.24 Tc 1 0 0 1 304.80 242.64 Tm <6D65617375726520> Tj 1 0 0 1 345.60 242.64 Tm <6F6620> Tj 1 0 0 1 357.60 242.64 Tm <7466726520> Tj 1 0 0 1 375.36 242.64 Tm <64697374616E636520> Tj 1 0 0 1 414.72 242.40 Tm <616E6420> Tj 1 0 0 1 435.12 242.40 Tm <6F7269656E7461666C6F6E20> Tj 1 0 0 1 485.52 242.16 Tm <6265747765656E20> Tj 1 0 0 1 524.64 242.16 Tm <64617461> Tj 0.24 Tc 1 0 0 1 304.80 232.56 Tm <706F696E74732E20> Tj 1 0 0 1 337.44 232.08 Tm <5468697320> Tj 1 0 0 1 359.76 232.08 Tm <6D65616E7320> Tj 1 0 0 1 390.72 232.08 Tm <7468617420> Tj 1 0 0 1 410.64 231.84 Tm <74686520> Tj 1 0 0 1 427.20 231.84 Tm <6D6561737572656D656E7420> Tj 1 0 0 1 488.40 231.60 Tm <76616C75657320> Tj 1 0 0 1 518.64 231.60 Tm <617420> Tj 1 0 0 1 529.68 231.60 Tm <746865> Tj 0.24 Tc 1 0 0 1 305.04 221.52 Tm <6F6273657276656420> Tj 1 0 0 1 346.80 222.00 Tm <706F696E747320> Tj 1 0 0 1 377.76 221.52 Tm <646F20> Tj 1 0 0 1 392.16 221.52 Tm <6E6F7420> Tj 1 0 0 1 409.92 221.52 Tm <696E666C75656E636520> Tj 1 0 0 1 452.16 221.04 Tm <74686520> Tj 1 0 0 1 468.96 221.04 Tm <766172696F6772616D20> Tj 1 0 0 1 516.72 221.04 Tm <6F7220> Tj 1 0 0 1 529.68 221.04 Tm <746865> Tj 0.24 Tc 1 0 0 1 305.04 210.96 Tm <636F76617269616E636520> Tj 1 0 0 1 353.28 210.96 Tm <76616C7565732E20> Tj 1 0 0 1 386.88 210.96 Tm <4869676C657220> Tj 1 0 0 1 418.56 210.72 Tm <766172696F6772616D20> Tj 1 0 0 1 465.60 210.48 Tm <76616C75657320> Tj 1 0 0 1 496.56 211.44 Tm <2877697468696E20> Tj 1 0 0 1 529.68 210.48 Tm <746865> Tj 0.24 Tc 1 0 0 1 304.80 200.40 Tm <72616E676520> Tj 1 0 0 1 333.84 200.40 Tm <6F6620> Tj 1 0 0 1 346.80 200.40 Tm <696E666C75656E63652920> Tj 1 0 0 1 393.84 200.40 Tm <696E64696361746520> Tj 1 0 0 1 432.96 199.92 Tm <616E20> Tj 1 0 0 1 448.56 199.92 Tm <696E6372656173696E6720> Tj 1 0 0 1 497.76 201.12 Tm <226F726465722220> Tj 1 0 0 1 533.76 199.92 Tm <6F72> Tj 0.24 Tc 1 0 0 1 304.80 189.60 Tm <657374696D6174696F6E20> Tj 1 0 0 1 353.28 189.84 Tm <72656C696162696C6974792C20> Tj 1 0 0 1 398.88 189.60 Tm <692E652E2C20> Tj 1 0 0 1 418.08 189.60 Tm <6869676C72657220> Tj 1 0 0 1 448.56 189.36 Tm <76616C75657320> Tj 1 0 0 1 479.04 189.36 Tm <666F7220> Tj 1 0 0 1 495.36 189.12 Tm <616E7920> Tj 1 0 0 1 513.60 189.60 Tm <7061697220> Tj 1 0 0 1 534.96 189.12 Tm <6F66> Tj 0.24 Tc 1 0 0 1 304.56 179.52 Tm <706F696E747320> Tj 1 0 0 1 337.92 179.04 Tm <696E64696361746520> Tj 1 0 0 1 378.48 179.04 Tm <6120> Tj 1 0 0 1 389.76 179.28 Tm <6772656174657220> Tj 1 0 0 1 426.24 178.80 Tm <696E746572616374696F6E20> Tj 1 0 0 1 478.56 178.56 Tm <6265747765656E20> Tj 1 0 0 1 519.84 178.56 Tm <74686F7365> Tj 0.24 Tc 1 0 0 1 304.80 168.48 Tm <6C6F636174696F6E7320> Tj 1 0 0 1 350.88 168.48 Tm <737570706F73696E6720> Tj 1 0 0 1 402.00 168.24 Tm <7468617420> Tj 1 0 0 1 426.72 168.24 Tm <746865697220> Tj 1 0 0 1 454.56 168.00 Tm <73657061726174696F6E20> Tj 1 0 0 1 507.60 168.00 Tm <64697374616E6365> Tj 0.00 Tc 1 0 0 1 304.80 157.68 Tm <736D616C6C657220> Tj 1 0 0 1 343.20 157.92 Tm <7468616E20> Tj 1 0 0 1 369.12 157.68 Tm <74686520> Tj 1 0 0 1 388.80 157.92 Tm <72616E676520> Tj 1 0 0 1 419.28 157.68 Tm <6F6620> Tj 1 0 0 1 433.68 157.68 Tm <696E666C75656E63652E20> Tj 1 0 0 1 481.20 157.44 Tm <416E20> Tj 1 0 0 1 499.92 157.44 Tm <6164646974696F6E616C> Tj 0.24 Tc 1 0 0 1 304.80 147.12 Tm <657374696D6174696F6E20> Tj 1 0 0 1 357.12 147.60 Tm <7175616C69747920> Tj 1 0 0 1 393.36 147.12 Tm <63616E20> Tj 1 0 0 1 415.44 147.12 Tm <626520> Tj 1 0 0 1 432.96 147.12 Tm <7265616368656420> Tj 1 0 0 1 474.00 146.88 Tm <627920> Tj 1 0 0 1 491.52 146.64 Tm <74686520> Tj 1 0 0 1 512.16 146.88 Tm 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Malutt and D. Bali6 ORIGINAL SCIENTIFIC PAPERThe equations of $imple and Otdinaty Ktiging are compared to outline their dilferences in the estimationprocedure. Emphasis is given to the lagtange multiplicator as a variable that allows the minimization ofvaliance in Otdinaty Ktiging. The matrices and lineat systems ol kriging, which are nost olten performed inthe background of the conputer mapping algorithm, are analyzed and presented in detail. The intention is toshow the impoftance ol geomathematics in one of the basic geological tasks, mapping. Fufihermore, adetailed presentation of equation sets provides a better understanding of the Simple and Ordinary Krigingalgorithms fot geological engineers, as the two most-used geostatistical techniques (included Indicatorl(ilging as the thirdf. The conclusion includes proposals, presented in four steps, for the detemination of thelagrange multiplicator value in any 0rdinaty Ktiging equation.Key words'. geostatistics, kriging, linear estimator, Lagrange multiplier Linearity and Lagrange Linear 1. INTRODUCTIONThe kriging method is considered a progpessive interpo-lation method for the estimaflon of a regionalized vari-able at selected grid points. The term regionalizeduariable refers to such variables, which can be regardedto be random variable in the infinitely small surroundingof a particular location, but there exist a function abovethe region which can relate these variables to each other.The term "random" assumes that the probability of thevariable bein$ smaller than a real number exists for anyreal number.Kdgng, as a statistical estimation method, is named af-ter South Afrikaans engineer Krlgeu, who first appliedand described this methodologr to estimate gold concen-trations in ore deposits. This was followed by further the-oretical development of the method, predominanfly byFrench scientists, especially Matheronl 1. Finally, severalkriging techniques have been defined with different equa-tions (such as Simple, Ordinary, and Indicator Ituiging,and others).Kriglng estimation is based on the application of exist-ing measurements (so-called "control points"), whose in-fluence on the estimation is expressed with weightingcoefficients. This estimation also assumes the satisfac-tion of the relevant criteria. This means that the estima-tion must be unbiased and performed so that thevariance between the measured and estimated values atthe selected points is as low as possible. Such a value isalso called the "kriging variance", which represents theinterpolation quality and can be compared with across-validation technique.In every kriging techniques, the procedure for minimiz-ing the estimation error includes a variable called the"Lagrange multiplicator". The purpose of the analysispresented here is to describe the mathematical meaningof the multiplicator and its role in minimizingthe krigingvaflance.NAFTA 59 (1 ) 31 -37 (2009) 2. KRIGING PROPERTIESI{nging techniques belong to a set of linear algorithmsthat are based on the least squares method. The selec-tion ofthe appropriate technique is based on the proper-ties of the random variable that is to be interpolated byhigrng. The techniques are based on the estimation ofweighting coefiicients with an assumption of unbiased-ness. Each hard data has its ovrn weighting coefficient(1"), which represents the influence of a particular data onthe value of the final estimation at the selected grid node.The relationship between the existing (hard) data and theestimation point has been expressed byvariogram values(if we suppose the existence of intrinsic hypothesis) or bycovariance in case ofsecond order stationarity. Suchval-ues describe the spatial dependence and the influence ofthe particular location in terms of its distance and direc-tion from the estimated location (point).A spatial model based on a larger number of controlpoints, a higler variogram range, a lower nugget effect,and without anisotropy is usually much more reliable.Regardless of the spatial dependence described by thevariogram or the covariance function, the weigltingcoefficient at a particular location is exclusively ameasure of tfre distance and orientaflon between datapoints. This means that the measurement values at theobserved points do not influence the variogram or thecovariance values. Higler variogram values (within therange of influence) indicate an increasing "order" orestimation reliability, i.e., higlrer values for any pair ofpoints indicate a greater interaction between thoselocations supposing that their separation distancesmaller than the range of influence. An additionalestimation quality can be reached by the regulardistribution of the control points. 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<6B726967696E6720> Tj 1 0 0 1 92.88 539.28 Tm <626173656420> Tj 1 0 0 1 120.72 539.28 Tm <6F6E20> Tj 1 0 0 1 134.16 539.28 Tm <6E20> Tj 1 0 0 1 142.32 539.76 Tm <706F696E747320> Tj 1 0 0 1 171.60 539.52 Tm <69733A> Tj 0.96 Tc /F3 12.0 Tf 1 0 0 1 47.52 516.24 Tm <7A7220> Tj 1 0 0 1 58.32 517.44 Tm <3D6974272C27322C> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 47.52 488.64 Tm <7768657265> Tj 0.48 Tc 1 0 0 1 47.76 475.20 Tm <29226920> Tj 1 0 0 1 79.20 475.20 Tm <697320> Tj 1 0 0 1 86.64 475.20 Tm <74686520> Tj 1 0 0 1 100.08 475.20 Tm <776569676874696E6720> Tj 1 0 0 1 137.04 475.20 Tm <636F656666696369656E7420> Tj 1 0 0 1 175.20 475.20 Tm <666F7220> Tj 1 0 0 1 186.96 475.20 Tm <6120> Tj 1 0 0 1 193.92 475.68 Tm <706172746963756C617220> Tj 1 0 0 1 229.44 475.20 Tm <6C6F636174696F6E20> Tj 1 0 0 1 259.92 476.16 Tm <2269223B> Tj 0.48 Tc 1 0 0 1 47.52 458.16 Tm <5A6A20> Tj 1 0 0 1 78.96 458.16 Tm <61726520> Tj 1 0 0 1 92.64 458.16 Tm <6B6E6F776E20> Tj 1 0 0 1 118.08 458.16 Tm <76616C7565732C20> Tj 1 0 0 1 145.20 458.40 Tm <74686520> Tj 1 0 0 1 158.88 458.16 Tm <736F2D63616C6C656420> Tj 1 0 0 1 193.68 459.12 Tm <22636F6E74726F6C20> Tj 1 0 0 1 223.92 458.64 Tm <706F696E74732220> Tj 1 0 0 1 251.76 458.64 Tm <2868617264> Tj 0.48 Tc 1 0 0 1 78.96 448.80 Tm <64617461293B> Tj 0.48 Tc 1 0 0 1 47.28 431.76 Tm <5A6B20> Tj 1 0 0 1 78.96 431.76 Tm <697320> Tj 1 0 0 1 86.40 431.76 Tm <74686520> Tj 1 0 0 1 99.84 431.76 Tm <76616C756520> Tj 1 0 0 1 121.20 431.76 Tm <657374696D6174656420> Tj 1 0 0 1 158.64 431.76 Tm <627920> Tj 1 0 0 1 169.44 431.76 Tm <6B726967696E672E> Tj 0.24 Tc 1 0 0 1 47.52 414.48 Tm <546865736520> Tj 1 0 0 1 76.56 414.24 Tm <6571756174696F6E7320> Tj 1 0 0 1 121.44 414.48 Tm <726570726573656E7420> Tj 1 0 0 1 165.84 414.48 Tm <74686520> Tj 1 0 0 1 182.64 414.48 Tm <73797374656D20> Tj 1 0 0 1 216.00 414.48 Tm <6F6620> Tj 1 0 0 1 227.52 414.48 Tm <6C696E65617220> Tj 1 0 0 1 255.84 414.48 Tm <6B7267696E67> Tj 0.24 Tc 1 0 0 1 47.76 403.92 Tm <6571756174696F6E7320> Tj 1 0 0 1 91.20 403.68 Tm <7468617420> Tj 1 0 0 1 110.40 403.92 Tm <697320> Tj 1 0 0 1 120.24 403.92 Tm <64657363726962656420> Tj 1 0 0 1 164.16 403.92 Tm <696E20> Tj 1 0 0 1 174.72 403.92 Tm <6E756D65726F757320> Tj 1 0 0 1 220.56 403.92 Tm <626F6F6B732C20> Tj 1 0 0 1 251.76 403.92 Tm <652E672E2C20> Tj 1 0 0 1 270.96 403.92 Tm <7265662D> Tj 0.48 Tc 1 0 0 1 47.76 393.36 Tm <6572656E636573332720> Tj 1 0 0 1 86.64 395.76 Tm <342720> Tj 1 0 0 1 93.36 395.76 Tm <352E20> Tj 1 0 0 1 100.56 395.76 Tm <3720> Tj 1 0 0 1 107.28 393.36 Tm <616C3A6420> Tj 1 0 0 1 125.28 393.36 Tm <6F74686572732E> Tj 0.24 Tc 1 0 0 1 54.48 379.20 Tm <45717561666C6F6E20> Tj 1 0 0 1 95.52 378.96 Tm <332E3220> Tj 1 0 0 1 111.12 379.20 Tm <63616E20> Tj 1 0 0 1 127.68 379.20 Tm <626520> Tj 1 0 0 1 139.44 379.20 Tm <7772697474656E20> Tj 1 0 0 1 171.84 379.20 Tm <617320> Tj 1 0 0 1 182.88 379.20 Tm <74686520> Tj 1 0 0 1 198.00 379.20 Tm <6D617472697820> Tj 1 0 0 1 228.00 379.20 Tm <4571756174696F6E20> Tj 1 0 0 1 269.04 379.20 Tm <332E332E> Tj 0.24 Tc 1 0 0 1 47.52 368.64 Tm <496E20> Tj 1 0 0 1 59.76 368.64 Tm <626F746820> Tj 1 0 0 1 83.28 368.64 Tm <6F6620> Tj 1 0 0 1 95.04 368.64 Tm <746865736520> Tj 1 0 0 1 120.96 368.64 Tm <6D617472696365732C20> Tj 1 0 0 1 164.64 368.64 Tm <74686520> Tj 1 0 0 1 181.20 368.64 Tm <76616C75657320> Tj 1 0 0 1 212.16 368.64 Tm <61726520> Tj 1 0 0 1 229.68 368.64 Tm <65787072657373656420> Tj 1 0 0 1 276.24 368.64 Tm <6173> Tj 0.24 Tc 1 0 0 1 47.28 358.08 Tm <766172696F6772616D20> Tj 1 0 0 1 93.84 357.84 Tm <76616C7565732C20> Tj 1 0 0 1 126.96 358.08 Tm <692E652E2C20> Tj 1 0 0 1 145.92 358.08 Tm <746865736520> Tj 1 0 0 1 171.12 358.08 Tm <76616C75657320> Tj 1 0 0 1 201.84 358.08 Tm <646570656E6420> Tj 1 0 0 1 236.64 358.08 Tm <6F6E6C7920> Tj 1 0 0 1 258.00 358.08 Tm <6F6E20> Tj 1 0 0 1 272.16 358.08 Tm <746865> Tj 0.24 Tc 1 0 0 1 47.52 347.52 Tm <64697374616C63657320> Tj 1 0 0 1 91.68 347.52 Tm <616E6420> Tj 1 0 0 1 112.08 347.52 Tm <6F7269656E746174696F6E7320> Tj 1 0 0 1 167.28 347.52 Tm <6265747765656E20> Tj 1 0 0 1 206.40 347.52 Tm <74686520> Tj 1 0 0 1 224.16 347.52 Tm <636F6E74726F6C20> Tj 1 0 0 1 258.48 348.00 Tm <706F696E7473> Tj 0.24 Tc 1 0 0 1 47.52 336.96 Tm <616E6420> Tj 1 0 0 1 69.12 336.96 Tm <6E6F7420> Tj 1 0 0 1 89.04 336.96 Tm <6F6E20> Tj 1 0 0 1 105.36 336.96 Tm <746865697220> Tj 1 0 0 1 131.04 336.96 Tm <76616C7565732E20> Tj 1 0 0 1 166.56 336.96 Tm <54686520> Tj 1 0 0 1 188.64 336.96 Tm <746869726420> Tj 1 0 0 1 215.76 337.20 Tm <6D617472697820> Tj 1 0 0 1 249.60 337.20 Tm <696E636C75646573> Tj 0.24 Tc 1 0 0 1 47.28 326.40 Tm <776569676874696E6720> Tj 1 0 0 1 89.28 326.16 Tm <636F65667B696369656E74732C20> Tj 1 0 0 1 141.12 326.40 Tm <776869636820> Tj 1 0 0 1 168.72 326.40 Tm <61726520> Tj 1 0 0 1 184.08 326.40 Tm <73696D706C7920> Tj 1 0 0 1 214.56 326.40 Tm <657374696D6174656420> Tj 1 0 0 1 257.76 326.40 Tm <66726F6D20> Tj 1 0 0 1 280.32 326.40 Tm <61> Tj 0.24 Tc 1 0 0 1 47.52 315.84 Tm <73797374656D20> Tj 1 0 0 1 79.20 315.84 Tm <7769746820> Tj 1 0 0 1 100.80 317.04 Tm <226E2220> Tj 1 0 0 1 115.68 315.84 Tm <6571756174696F6E7320> Tj 1 0 0 1 159.36 315.84 Tm <7769746820> Tj 1 0 0 1 180.96 317.04 Tm <226E2720> Tj 1 0 0 1 195.84 315.84 Tm <756E6B6E6F776E20> Tj 1 0 0 1 238.08 315.84 Tm <7661726961626C65732E> Tj 0.00 Tc /F3 12.0 Tf 1 0 0 1 47.04 295.68 Tm <74416C2E746C6C3D20> Tj 1 0 0 1 80.16 295.68 Tm <74426C> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 54.00 278.88 Tm <54686520> Tj 1 0 0 1 75.60 278.88 Tm <6D657466726F6420> Tj 1 0 0 1 113.04 278.64 Tm <6F6620> Tj 1 0 0 1 126.00 278.88 Tm <6B726967696E6720> Tj 1 0 0 1 160.80 278.88 Tm <696E636C7564657320> Tj 1 0 0 1 201.84 278.64 Tm <7365766572616C20> Tj 1 0 0 1 236.16 278.64 Tm <746563686E69717565732E> Tj 0.24 Tc 1 0 0 1 47.04 268.32 Tm <546865736520> Tj 1 0 0 1 76.80 268.08 Tm <61726520> Tj 1 0 0 1 94.80 268.08 Tm <53696D706C6520> Tj 1 0 0 1 128.64 268.32 Tm <4B726967726E672C20> Tj 1 0 0 1 166.08 268.08 Tm <4F7264696E61727920> Tj 1 0 0 1 208.80 268.32 Tm <497B7280696E672C20> Tj 1 0 0 1 246.48 268.32 Tm <496E64696361746F72> Tj 0.00 Tc 1 0 0 1 47.04 257.76 Tm <4B726967726E672C20> Tj 1 0 0 1 85.20 257.76 Tm <4D756C7469706C6520> Tj 1 0 0 1 125.04 257.76 Tm <496E64696361746F7220> Tj 1 0 0 1 168.72 257.76 Tm <4B726967696E672C20> Tj 1 0 0 1 206.64 257.52 Tm <556E6976657273616C20> Tj 1 0 0 1 251.76 257.76 Tm <4966746967696E672C> Tj 0.00 Tc 1 0 0 1 47.04 247.20 Tm <4952466B20> Tj 1 0 0 1 76.08 247.20 Tm <4B726967696E672C20> Tj 1 0 0 1 117.60 247.20 Tm <4C6F676E6F726D616C20> Tj 1 0 0 1 171.84 247.20 Tm <49667480696E672C20> Tj 1 0 0 1 213.60 246.96 Tm <616E6420> Tj 1 0 0 1 237.60 247.20 Tm <4469736A756E6374697665> Tj 0.00 Tc 1 0 0 1 47.04 236.40 Tm <4B7269676C6E672E20> Tj 1 0 0 1 86.16 236.40 Tm <496E20> Tj 1 0 0 1 100.80 236.40 Tm <74686520> Tj 1 0 0 1 119.76 236.40 Tm <666F6C6C6F77696E6720> Tj 1 0 0 1 164.16 236.40 Tm <636F6D70617269736F6E20> Tj 1 0 0 1 220.80 236.40 Tm <6F6620> Tj 1 0 0 1 234.48 236.40 Tm <73696D706C6520> Tj 1 0 0 1 269.04 236.40 Tm <616E64> Tj 0.24 Tc 1 0 0 1 47.28 225.84 Tm <6F7264696E61727920> Tj 1 0 0 1 87.60 225.84 Tm <6B726967696E672C20> Tj 1 0 0 1 123.60 225.84 Tm <776520> Tj 1 0 0 1 138.96 225.84 Tm <77696C6C20> Tj 1 0 0 1 158.40 225.84 Tm <646573637269626520> Tj 1 0 0 1 198.00 225.84 Tm <74686520> Tj 1 0 0 1 215.52 225.84 Tm <65666665637420> Tj 1 0 0 1 241.68 225.84 Tm <63617573656420> Tj 1 0 0 1 274.80 225.84 Tm <6279> Tj 0.24 Tc 1 0 0 1 47.04 215.28 Tm <696E636C7564696E6720> Tj 1 0 0 1 88.80 215.04 Tm <6F6620> Tj 1 0 0 1 98.64 215.04 Tm <74686520> Tj 1 0 0 1 114.00 215.28 Tm <6C696E65617220> Tj 1 0 0 1 140.88 215.28 Tm <636F656666696369656E7420> Tj 1 0 0 1 185.76 215.28 Tm <696E20> Tj 1 0 0 1 196.08 215.28 Tm <74686520> Tj 1 0 0 1 211.68 215.28 Tm <6B726967696E6720> Tj 1 0 0 1 243.60 215.28 Tm <6571756174696F6E73> Tj 0.24 Tc 1 0 0 1 47.04 204.72 Tm <746F20> Tj 1 0 0 1 58.08 204.72 Tm <6163686965766520> Tj 1 0 0 1 91.68 204.72 Tm <756E6269617365646E6573732E> Tj 0.48 Tc /F3 11.0 Tf 1 0 0 1 46.80 186.00 Tm <332E312E20> Tj 1 0 0 1 69.60 186.00 Tm <53696D706C6520> Tj 1 0 0 1 107.76 186.24 Tm <4B726967696E6720> Tj 1 0 0 1 147.36 186.00 Tm <7468656F7279> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 47.04 171.60 Tm <53696D706C6520> Tj 1 0 0 1 82.08 171.84 Tm <6B726967696E672C20> Tj 1 0 0 1 119.76 171.60 Tm <617320> Tj 1 0 0 1 134.16 171.84 Tm <69747320> Tj 1 0 0 1 149.76 171.84 Tm <6E616D6520> Tj 1 0 0 1 178.56 171.84 Tm <696D706C6965732C20> Tj 1 0 0 1 217.68 171.84 Tm <697320> Tj 1 0 0 1 230.16 171.60 Tm <74686520> Tj 1 0 0 1 248.88 171.60 Tm <7369726E706C657374> Tj 0.24 Tc 1 0 0 1 46.80 161.04 Tm <6B726967696E6720> Tj 1 0 0 1 79.20 161.04 Tm <746563686E697175652E20> Tj 1 0 0 1 126.24 161.28 Tm <54686520> Tj 1 0 0 1 145.20 161.04 Tm <66756C6C20> Tj 1 0 0 1 161.76 161.04 Tm <6D617472697820> Tj 1 0 0 1 192.72 161.04 Tm <6571756174696F6E20> Tj 1 0 0 1 232.08 161.04 Tm <69733A> Tj 0.24 Tc 1 0 0 1 303.12 761.76 Tm <7768657265> Tj 0.48 Tc 1 0 0 1 303.12 750.96 Tm <7920> Tj 1 0 0 1 337.44 750.48 Tm <61726520> Tj 1 0 0 1 350.64 750.72 Tm <74686520> Tj 1 0 0 1 364.08 750.72 Tm <766172696F6772616D20> Tj 1 0 0 1 402.24 750.48 Tm <76616C7565733B> Tj 0.48 Tc 1 0 0 1 302.88 733.44 Tm <5A722E2E2E5A6E20> Tj 1 0 0 1 337.44 733.44 Tm <61726520> Tj 1 0 0 1 351.12 733.44 Tm <6B6E6F776E20> Tj 1 0 0 1 377.04 733.44 Tm <6D6561737572656420> Tj 1 0 0 1 414.72 733.44 Tm <76616C75657320> Tj 1 0 0 1 439.92 732.96 Tm <617420> Tj 1 0 0 1 449.04 733.92 Tm <706F696E74733B> Tj 0.48 Tc 1 0 0 1 302.88 716.40 Tm <5A20> Tj 1 0 0 1 337.68 716.40 Tm <697320> Tj 1 0 0 1 344.88 716.40 Tm <74686520> Tj 1 0 0 1 358.80 717.60 Tm <706F696E7420> Tj 1 0 0 1 378.96 716.16 Tm <617420> Tj 1 0 0 1 387.36 716.16 Tm <776869636820> Tj 1 0 0 1 410.88 716.16 Tm <6E657720> Tj 1 0 0 1 427.44 716.16 Tm <76616C75657320> Tj 1 0 0 1 452.64 715.92 Tm <61726520> Tj 1 0 0 1 466.32 715.92 Tm <657374696D6174656420> Tj 1 0 0 1 503.04 715.92 Tm <66726F6D> Tj 0.48 Tc 1 0 0 1 337.68 707.04 Tm <6B6E6F776E20> Tj 1 0 0 1 363.84 707.28 Tm <28686172642920> Tj 1 0 0 1 387.84 706.80 Tm <6461746120> Tj 1 0 0 1 405.60 707.28 Tm <28322C2E20> Tj 1 0 0 1 418.08 706.80 Tm <2E20> Tj 1 0 0 1 420.96 706.80 Tm <2E322C292E> Tj 0.24 Tc 1 0 0 1 303.12 689.76 Tm <4B6467696E6720> Tj 1 0 0 1 336.00 689.76 Tm <7573657320> Tj 1 0 0 1 357.84 689.52 Tm <64696D656E73696F6E6C65737320> Tj 1 0 0 1 420.72 689.76 Tm <706F696E7420> Tj 1 0 0 1 445.44 689.28 Tm <6461746120> Tj 1 0 0 1 466.08 689.04 Tm <7468617420> Tj 1 0 0 1 485.04 689.28 Tm <726570726573656E7420> Tj 1 0 0 1 528.00 689.04 Tm <746865> Tj 0.24 Tc 1 0 0 1 302.88 678.96 Tm <76616C75657320> Tj 1 0 0 1 331.92 678.96 Tm <6F6620> Tj 1 0 0 1 341.52 678.96 Tm <74686520> Tj 1 0 0 1 356.88 678.96 Tm <726567696F6E616C697A65647661726961626C652E20> Tj 1 0 0 1 448.56 678.72 Tm <496E20> Tj 1 0 0 1 459.36 678.48 Tm <53696D706C6520> Tj 1 0 0 1 491.04 678.48 Tm <4B6467696E672C20> Tj 1 0 0 1 526.08 678.48 Tm <69746973> Tj 0.24 Tc 1 0 0 1 303.36 668.40 Tm <617373756D656420> Tj 1 0 0 1 343.44 668.40 Tm <7468617420> Tj 1 0 0 1 362.64 668.40 Tm <74686520> Tj 1 0 0 1 378.24 668.40 Tm <726567696F6E616C697A656420> Tj 1 0 0 1 431.76 668.16 Tm <7661726961626C6520> Tj 1 0 0 1 468.00 667.92 Tm <68617320> Tj 1 0 0 1 485.52 667.92 Tm <7365636F6E6420> Tj 1 0 0 1 517.68 667.92 Tm <6F72646572> Tj 0.24 Tc 1 0 0 1 303.36 657.84 Tm <73746174696F6E6172792C20> Tj 1 0 0 1 349.44 657.84 Tm <74686520> Tj 1 0 0 1 364.80 657.84 Tm <657863657074656420> Tj 1 0 0 1 403.20 657.60 Tm <76616C756520> Tj 1 0 0 1 427.68 657.60 Tm <697320> Tj 1 0 0 1 437.28 657.60 Tm <657665723572776865726520> Tj 1 0 0 1 486.72 657.36 Tm <636F6E7374616E7420> Tj 1 0 0 1 525.12 657.36 Tm <616E64> Tj 0.24 Tc 1 0 0 1 303.12 647.04 Tm <6B6E6F776E20> Tj 1 0 0 1 334.80 644.16 Tm <6C6974226C20> Tj 1 0 0 1 357.36 648.48 Tm <3D20> Tj 1 0 0 1 365.52 646.80 Tm <4F5D20> Tj 1 0 0 1 376.32 646.32 Tm <2E20> Tj 1 0 0 1 381.36 646.80 Tm <61726120> Tj 1 0 0 1 400.08 646.80 Tm <74686520> Tj 1 0 0 1 416.40 646.56 Tm <636F76617269616E636520> Tj 1 0 0 1 464.16 646.32 Tm <66756E6374696F6E20> Tj 1 0 0 1 502.32 646.32 Tm <697320> Tj 1 0 0 1 512.40 646.32 Tm <6B6E6F776E> Tj 0.72 Tc 1 0 0 1 302.88 630.72 Tm <5B412A2C616C3D436F745C5A2878292C32285529295D20> Tj 1 0 0 1 408.00 632.88 Tm <2E20> Tj 1 0 0 1 414.48 632.88 Tm <7A757274747265726D6F72652C20> Tj 1 0 0 1 475.68 632.64 Tm <7768656E20> Tj 1 0 0 1 502.80 632.64 Tm <7468697320> Tj 1 0 0 1 523.44 632.64 Tm <657374692D> Tj 0.24 Tc 1 0 0 1 302.88 622.80 Tm <6D6174696F6E20> Tj 1 0 0 1 336.00 622.80 Tm <697320> Tj 1 0 0 1 346.56 623.04 Tm <706572666F726D656420> Tj 1 0 0 1 395.04 622.56 Tm <617420> Tj 1 0 0 1 406.08 622.32 Tm <74686520> Tj 1 0 0 1 422.64 622.32 Tm <636F6E74726F6C20> Tj 1 0 0 1 456.00 622.56 Tm <706F696E742C20> Tj 1 0 0 1 484.32 622.08 Tm <74686520> Tj 1 0 0 1 500.88 622.08 Tm <6572726F7220> Tj 1 0 0 1 526.32 622.08 Tm <63616E> Tj 0.24 Tc 1 0 0 1 303.12 612.00 Tm <616C736F20> Tj 1 0 0 1 322.80 612.00 Tm <626520> Tj 1 0 0 1 335.76 612.00 Tm <63616C63756C6174656420> Tj 1 0 0 1 381.60 611.76 Tm <617420> Tj 1 0 0 1 391.92 611.76 Tm <74686520> Tj 1 0 0 1 407.76 612.24 Tm <706F696E7420> Tj 1 0 0 1 432.96 611.52 Tm <61733A> Tj 0.24 Tc /F3 7.0 Tf 1 0 0 1 302.88 594.00 Tm <6520> Tj 1 0 0 1 309.12 596.88 Tm <2D2D20> Tj 1 0 0 1 316.32 594.48 Tm <285A2A6E20> Tj 1 0 0 1 334.80 596.16 Tm <2D20> Tj 1 0 0 1 342.00 594.00 Tm <5A2E2274692C2E7422612920> Tj 1 0 0 1 524.88 593.76 Tm <28332E3529> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 302.88 574.32 Tm <496620> Tj 1 0 0 1 311.04 574.32 Tm <746865726520> Tj 1 0 0 1 335.28 574.32 Tm <697320> Tj 1 0 0 1 344.88 574.32 Tm <6E6F20> Tj 1 0 0 1 358.32 574.32 Tm <65787465726E616C20> Tj 1 0 0 1 394.80 574.08 Tm <647269667420> Tj 1 0 0 1 415.44 574.08 Tm <696E20> Tj 1 0 0 1 426.24 573.84 Tm <74686520> Tj 1 0 0 1 441.60 573.84 Tm <7661726961626C6520> Tj 1 0 0 1 478.08 573.60 Tm <616E6420> Tj 1 0 0 1 496.56 573.60 Tm <74686520> Tj 1 0 0 1 512.16 573.60 Tm <73756D20> Tj 1 0 0 1 533.04 573.60 Tm <6F66> Tj 0.24 Tc 1 0 0 1 303.12 563.52 Tm <616C6C20> Tj 1 0 0 1 315.84 563.52 Tm <776569676874696E6720> Tj 1 0 0 1 359.04 563.52 Tm <636F656666696369656E747320> Tj 1 0 0 1 409.20 563.52 Tm <697320> Tj 1 0 0 1 420.48 563.28 Tm <312C20> Tj 1 0 0 1 430.56 563.28 Tm <756E6269617365646E65737320> Tj 1 0 0 1 491.28 563.04 Tm <697320> Tj 1 0 0 1 501.60 563.04 Tm <61636869657665642E> Tj 0.24 Tc 1 0 0 1 302.64 552.96 Tm <54686520> Tj 1 0 0 1 322.56 552.96 Tm <646966666572656E636520> Tj 1 0 0 1 366.96 552.96 Tm <6265747765656E20> Tj 1 0 0 1 405.12 552.72 Tm <616C6C20> Tj 1 0 0 1 418.32 552.72 Tm <74686520> Tj 1 0 0 1 434.88 552.72 Tm <6D6561737572656420> Tj 1 0 0 1 480.24 552.24 Tm <616E6420> Tj 1 0 0 1 499.68 552.24 Tm <657374696D61746564> Tj 0.24 Tc 1 0 0 1 302.64 542.40 Tm <76616C75657320> Tj 1 0 0 1 331.44 542.40 Tm <697320> Tj 1 0 0 1 341.04 542.16 Tm <63616C6C656420> Tj 1 0 0 1 367.92 542.16 Tm <74686520> Tj 1 0 0 1 383.28 542.16 Tm <657374696D6174696F6E20> Tj 1 0 0 1 433.20 541.92 Tm <6572726F7220> Tj 1 0 0 1 457.68 541.68 Tm <6F7220> Tj 1 0 0 1 468.96 541.92 Tm 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Tc 1 0 0 1 301.68 306.72 Tm <7928322C20> Tj 1 0 0 1 318.48 308.64 Tm <2D20> Tj 1 0 0 1 324.96 306.48 Tm <5A2C292D20> Tj 1 0 0 1 339.36 306.48 Tm <292C2C2A20> Tj 1 0 0 1 354.48 306.48 Tm <79285A5E20> Tj 1 0 0 1 371.28 308.40 Tm <2D20> Tj 1 0 0 1 377.76 306.24 Tm <5A2C292E20> Tj 1 0 0 1 393.60 306.24 Tm <372C20> Tj 1 0 0 1 403.44 306.24 Tm <2B20> Tj 1 0 0 1 410.16 306.00 Tm <2E2E2E2B20> Tj 1 0 0 1 425.52 306.24 Tm <7928322C20> Tj 1 0 0 1 442.08 307.92 Tm <2D20> Tj 1 0 0 1 448.56 305.76 Tm <5A2C292E20> Tj 1 0 0 1 464.40 305.76 Tm <292E2C20> Tj 1 0 0 1 474.96 306.96 Tm <3D20> Tj 1 0 0 1 482.16 306.00 Tm <79285A5E20> Tj 1 0 0 1 498.72 307.92 Tm <2D20> Tj 1 0 0 1 505.20 305.76 Tm <5A29> Tj 0.24 Tc 1 0 0 1 301.92 285.12 Tm <54686520> Tj 1 0 0 1 322.08 284.88 Tm <65737469726E6174696F6E20> Tj 1 0 0 1 370.32 284.88 Tm <697320> Tj 1 0 0 1 381.36 284.64 Tm <74686520> Tj 1 0 0 1 398.40 284.64 Tm <73656C656374656420> Tj 1 0 0 1 435.60 284.88 Tm <706F696E7420> Tj 1 0 0 1 461.52 284.16 Tm <7468617420> 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0 0 1 202.08 125.04 Tm <6C20> Tj 1 0 0 1 206.16 127.20 Tm <3D20> Tj 1 0 0 1 213.36 125.04 Tm <6C72747A2220> Tj 1 0 0 1 233.52 134.40 Tm <2D7A746C> Tj 0.96 Tc 1 0 0 1 182.16 116.64 Tm <4920> Tj 1 0 0 1 202.08 116.64 Tm <7420> Tj 1 0 0 1 213.36 116.64 Tm <7420> Tj 1 0 0 1 250.08 116.64 Tm <6920> Tj 1 0 0 1 268.56 126.00 Tm <28333429> Tj 0.48 Tc 1 0 0 1 145.92 106.32 Tm <7628322E20> Tj 1 0 0 1 162.48 110.16 Tm <2D322E292920> Tj 1 0 0 1 187.92 108.96 Tm <70272E3120> Tj 1 0 0 1 213.36 104.88 Tm <6C5E72372E20> Tj 1 0 0 1 233.52 110.16 Tm <2D7A2929> Tj 0.48 Tc /F3 11.0 Tf 1 0 0 1 46.56 82.08 Tm <3332> Tj 0.00 Tc /F3 9.0 Tf 1 0 0 1 462.24 81.36 Tm <4E4146544120> Tj 1 0 0 1 482.40 81.36 Tm <353920> Tj 1 0 0 1 491.76 82.08 Tm <283120> Tj 1 0 0 1 497.52 80.16 Tm <2920> Tj 1 0 0 1 501.12 81.36 Tm <333120> Tj 1 0 0 1 508.56 83.28 Tm <2D733720> Tj 1 0 0 1 520.08 82.08 Tm <283230303929> Tj ET endstream endobj 19 0 obj 31332 endobj 20 0 obj << /Length 21 0 R >> stream In the text below, the detailed meaning of linearity inSimple and Ordinary Krigrng, and how such linearity isachieved, will be described.3. I\,IATHEITIATICAL FUNDANIENTALSOF KRIGINGThe principle of kriging is shown most simply with setsof equations that define the method. Kriging is applied tothe estimation of the values of a regionalized variable at aselected location (Z),based on surrounding existingval-ues (Zr). Each such location is assiggred a relevant weig;ht-ing coefficient (1",), and tJre calculation of this is the mostdemanding part of the kriging algorithm. The value of areqionalized variable cal be defined as: Z, = Z(x,) (3.1) wherexi is the value at the known location.Moreover, the value of a regionalized variable estimatedby kriging based on n points is: zr =it','2,where )"i is the weighting coefficient for a particular location "i";Zj are known values, the so-called "control points" (harddata);Zk is the value estimated by kriging.These equations represent the system of linear krgingequations that is described in numerous books, e.g., ref-erences3' 4' 5. 7 al:d others.Equaflon 3.2 can be written as the matrix Equation 3.3.In both of these matrices, the values are expressed asvariogram values, i.e., these values depend only on thedistalces and orientations between the control pointsand not on their values. The third matrix includesweighting coef{icients, which are simply estimated from asystem with "n" equations with "n' unknown variables. tAl.tll= tBl The metfrod of kriging includes several techniques.These are Simple Krigrng, Ordinary I{ring, IndicatorKrigrng, Multiple Indicator Kriging, Universal Iftiging,IRFk Kriging, Lognormal Ifting, and DisjunctiveKriglng. In the following comparison of simple andordinary kriging, we will describe the effect caused byincluding of the linear coefficient in the kriging equationsto achieve unbiasedness.3.1. Simple Kriging theorySimple kriging, as its name implies, is the sirnplestkriging technique. The full matrix equation is: wherey are the variogram values;Zr...Zn are known measured values at points;Z is the point at which new values are estimated fromknown (hard) data (2,. . .2,).Kdging uses dimensionless point data that represent thevalues of the regionalizedvariable. In Simple Kdging, itisassumed that the regionalized variable has second orderstationary, the excepted value is ever5rwhere constant andknown lit"l = O] . ara the covariance function is known[A*,al=Cot\Z(x),2(U))] . zurttrermore, when this esti-mation is performed at the control point, the error canalso be calculated at the point as:e -- (Z*n - Z."ti,.t"a) (3.5)If there is no external drift in the variable and the sum ofall weighting coefficients is 1, unbiasedness is achieved.The difference between all the measured and estimatedvalues is called the estimation error or kriging varianceand it is expressed as: d2_ \'(7 _ 7 \2ZJ\'rcal 'estima@dt i (3.6) It can also be shown as the second square ofthe varianceor the standard error of the estimation:o ="17 (3.7)In an ideal case, kriging tries to calculate tJle optimalweiglting coefficients that will lead to the minimal esti-mation error. Such coefficients, which lead to an estima-tion of unbiasedness with minimal variance. arecalculated by solving of the matrix equations system. Ifthe matrices in Equation 3.4 is represented by linearequations, it can be written as:./ (2, - Z,) . 1 j + y (Z 1 - Z). )"2 + .. + t(2, - Z,)' ),, = y(2, - Z)T(2, - Z,). \ + y(22 - Zr) - 1" + ... + y(2, - Z,). 1, = y(2, - Z) (3.8)y(2, - Z,)- ),,* y(Z^ - Z,). 7, + ...+ y(2, - Z,). )., = y(Z^ - Z)The estirnation is the selected point that can be repre-sented by the following equation: (3.s) Moreover, for Equation 3.8 to be considered unbiased,an additional condition must b^e fulfilled, i.e., the sum ofall weighting coefficients is f (),1, = 1). This condition isachieved by adding new conditions to the kriging matri-ces like Lagrange multiplicator in OrdinaryKrigingequa-Uons. (3.2) (3.3) z: y(2,-2,)l | ),,1 lvQ,-Z)ly(2, -2.) I l r, l = lrtz" -ztlI t t i (34)v(2. -2.)) p'.1 l^r7. -z)) 32 NAFTA 59 (1 ) 31 -s7 (2009) endstream endobj 21 0 obj 4126 endobj 22 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 23 0 obj << /Filter /FlateDecode /Length 112 >> stream xMA 0 ^/ʲKRK@M1'0J\[nUy!JT`AWxPi. 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0.48 Tc 1 0 0 1 57.84 534.24 Tm <63657320> Tj 1 0 0 1 74.40 534.48 Tm <697420> Tj 1 0 0 1 82.80 534.24 Tm <63616E20> Tj 1 0 0 1 100.56 534.24 Tm <626520> Tj 1 0 0 1 112.56 534.24 Tm <7772697474656E20> Tj 1 0 0 1 145.92 534.48 Tm <61733A> Tj 0.48 Tc 1 0 0 1 103.92 517.44 Tm <7928322C20> Tj 1 0 0 1 120.48 519.36 Tm <2D20> Tj 1 0 0 1 126.96 517.20 Tm <5A222920> Tj 1 0 0 1 162.24 517.44 Tm <7928322C20> Tj 1 0 0 1 178.80 519.36 Tm <2D20> Tj 1 0 0 1 185.28 517.20 Tm <7A2C29> Tj 0.96 Tc 1 0 0 1 102.96 505.20 Tm <7928322C2D322C2920> Tj 1 0 0 1 161.52 505.44 Tm <7928322C2D322C29> Tj 0.48 Tc 1 0 0 1 102.96 481.20 Tm <7928322C20> Tj 1 0 0 1 121.20 483.36 Tm <2D20> Tj 1 0 0 1 127.68 481.20 Tm <7A222920> Tj 1 0 0 1 161.28 481.44 Tm <7928322C20> Tj 1 0 0 1 179.52 483.36 Tm <2D20> Tj 1 0 0 1 186.00 481.20 Tm <7A2C29> Tj 1.20 Tc 1 0 0 1 273.60 458.16 Tm <28332E3120> Tj 1 0 0 1 287.04 456.96 Tm <3129> Tj 0.24 Tc 1 0 0 1 57.60 438.24 Tm <54686520> Tj 1 0 0 1 76.56 438.24 Tm <6E756D62657220> Tj 1 0 0 1 113.28 438.24 Tm <6F6620> Tj 1 0 0 1 123.60 438.24 Tm <776569676874696E6720> Tj 1 0 0 1 166.80 438.48 Tm <636F656666696369656E747320> Tj 1 0 0 1 217.20 438.48 Tm <616E6420> Tj 1 0 0 1 235.92 438.48 Tm <636F6E74726F6C20> Tj 1 0 0 1 268.80 438.96 Tm <706F696E7473> Tj 0.24 Tc 1 0 0 1 57.84 427.68 Tm <63616E20> Tj 1 0 0 1 74.88 427.68 Tm <626520> Tj 1 0 0 1 87.36 427.68 Tm <7665727920> Tj 1 0 0 1 107.76 427.92 Tm <6C617267652C20> Tj 1 0 0 1 133.44 427.68 Tm <62757420> Tj 1 0 0 1 150.72 427.68 Tm <636F6E74656D706F72612D727920> Tj 1 0 0 1 212.64 427.92 Tm <636F6D70757465727320> Tj 1 0 0 1 260.64 427.92 Tm <63616E20> Tj 1 0 0 1 278.16 427.92 Tm <7375632D> Tj 0.24 Tc 1 0 0 1 57.84 417.12 Tm <6365737366756C6C7920> Tj 1 0 0 1 97.92 417.12 Tm <736F6C766520> Tj 1 0 0 1 122.88 417.12 Tm <6E756D65726963616C6C7920> Tj 1 0 0 1 177.12 417.12 Tm <64656D616E64696E6720> Tj 1 0 0 1 228.24 417.36 Tm <7461736B732E20> Tj 1 0 0 1 257.52 417.36 Tm <54686520> Tj 1 0 0 1 277.92 417.36 Tm <657374692D> Tj 0.24 Tc 1 0 0 1 57.60 406.56 Tm <6D6174696F6E20> Tj 1 0 0 1 93.60 406.56 Tm <63616E20> Tj 1 0 0 1 114.00 406.56 Tm <626520> Tj 1 0 0 1 129.84 407.04 Tm <706572666F726D656420> Tj 1 0 0 1 180.96 406.56 Tm <73696D706C7920> Tj 1 0 0 1 214.80 406.80 Tm <627920> Tj 1 0 0 1 230.88 406.80 Tm <63616C63756C6174696E6720> Tj 1 0 0 1 282.24 406.80 Tm <746865> Tj 0.24 Tc 1 0 0 1 57.60 396.00 Tm <696E666C75656E636520> Tj 1 0 0 1 98.64 396.00 Tm <6F6620> Tj 1 0 0 1 108.96 396.00 Tm <616C6C20> Tj 1 0 0 1 121.44 396.00 Tm <74686520> Tj 1 0 0 1 137.28 396.00 Tm <636F6E74726F6C20> Tj 1 0 0 1 169.68 396.48 Tm <706F696E747320> Tj 1 0 0 1 198.72 396.24 Tm <776569676874656420> Tj 1 0 0 1 238.56 396.24 Tm <627920> Tj 1 0 0 1 251.04 396.24 Tm <746865697220> Tj 1 0 0 1 273.84 396.24 Tm <6173736F2D> Tj 0.24 Tc 1 0 0 1 57.84 385.44 Tm <63696174656420> Tj 1 0 0 1 84.96 385.44 Tm <636F656666696369656E747320> Tj 1 0 0 1 134.16 385.44 Tm <6163636F7264696E6720> Tj 1 0 0 1 177.36 385.68 Tm <746F20> Tj 1 0 0 1 187.68 385.68 Tm <45717561746920> Tj 1 0 0 1 216.00 385.68 Tm <6F6E20> Tj 1 0 0 1 228.72 385.68 Tm <332E322C20> Tj 1 0 0 1 246.96 385.68 Tm <617320> Tj 1 0 0 1 258.00 385.68 Tm <666F6C6C6F777320> Tj 1 0 0 1 288.72 385.68 Tm <3A> Tj 1.20 Tc /F3 7.0 Tf 1 0 0 1 57.36 368.40 Tm <5A20> Tj 1 0 0 1 65.04 369.84 Tm <3D20> Tj 1 0 0 1 72.48 368.40 Tm <4C74275A722B20> Tj 1 0 0 1 99.60 368.40 Tm <292C322732322B2E2E2E2B4C6E277A6E> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 273.36 369.12 Tm <28332E313229> Tj 0.24 Tc 1 0 0 1 57.36 350.16 Tm <54686520> Tj 1 0 0 1 76.56 350.16 Tm <63616C63756C6174696F6E20> Tj 1 0 0 1 125.52 350.16 Tm <6F6620> Tj 1 0 0 1 136.32 350.16 Tm <74686520> Tj 1 0 0 1 152.40 350.16 Tm <657374696D6174696F6E20> Tj 1 0 0 1 199.20 350.16 Tm <76617269616E636520> Tj 1 0 0 1 237.84 350.40 Tm <696E636C7564657320> Tj 1 0 0 1 276.48 350.16 Tm <6164642D> Tj 0.24 Tc 1 0 0 1 57.36 340.32 Tm <696E6720> Tj 1 0 0 1 72.96 339.60 Tm <74686520> Tj 1 0 0 1 88.80 339.60 Tm <4C616772616E676520> Tj 1 0 0 1 130.32 339.60 Tm <636F656666696369656E743A> Tj 1.20 Tc 1 0 0 1 57.36 322.08 Tm <6F2720> Tj 1 0 0 1 68.64 323.52 Tm <3A20> Tj 1 0 0 1 76.32 322.32 Tm <292C2E2779285A2E2D7A292B20> Tj 1 0 0 1 128.40 322.32 Tm <292E2C277928322C2D7A292B20> Tj 1 0 0 1 187.68 322.32 Tm <2B29222C277928322C2D7A292B6D20> Tj 1 0 0 1 273.36 323.04 Tm <2833313329> Tj 0.24 Tc 1 0 0 1 57.60 304.08 Tm <4865726520> Tj 1 0 0 1 79.68 303.84 Tm <61726520> Tj 1 0 0 1 95.28 303.84 Tm <73686F776E20> Tj 1 0 0 1 124.80 304.08 Tm <74686520> Tj 1 0 0 1 139.92 304.08 Tm <74776F20> Tj 1 0 0 1 156.96 304.56 Tm <70726F6261626C7920> Tj 1 0 0 1 197.04 304.32 Tm <6D6F73742D7573656420> Tj 1 0 0 1 243.12 304.32 Tm <6B726967696E6720> Tj 1 0 0 1 274.80 304.08 Tm <746563682D> Tj 0.24 Tc 1 0 0 1 57.36 293.28 Tm <6E69717565732E20> Tj 1 0 0 1 91.20 293.28 Tm <496E20> Tj 1 0 0 1 102.24 293.76 Tm <67656E6572616C2C20> Tj 1 0 0 1 138.48 293.52 Tm <696E20> Tj 1 0 0 1 149.52 293.52 Tm <616C6C20> Tj 1 0 0 1 162.24 293.52 Tm <6C696E65617220> Tj 1 0 0 1 189.60 293.52 Tm <6B726967696E6720> Tj 1 0 0 1 222.00 293.52 Tm <746563686E697175657320> Tj 1 0 0 1 270.96 294.48 Tm <2873696E6365> Tj 0.24 Tc 1 0 0 1 57.36 282.72 Tm <7468657920> Tj 1 0 0 1 77.52 282.72 Tm <61726520> Tj 1 0 0 1 93.60 282.72 Tm <636F6E73747261696E656420> Tj 1 0 0 1 146.16 282.96 Tm <6F7074696D697A6174696F6E20> Tj 1 0 0 1 201.84 283.44 Tm <70726F626C656D73292C20> Tj 1 0 0 1 250.56 282.96 Tm <74686520> Tj 1 0 0 1 266.40 282.96 Tm <6173736F63692D> Tj 0.24 Tc 1 0 0 1 57.60 272.16 Tm <6174656420> Tj 1 0 0 1 78.48 272.16 Tm <6571756174696F6E7320> Tj 1 0 0 1 122.64 272.40 Tm <63616E20> Tj 1 0 0 1 139.92 272.40 Tm <626520> Tj 1 0 0 1 153.12 272.40 Tm <6469766964656420> Tj 1 0 0 1 186.72 272.40 Tm <696E746F20> Tj 1 0 0 1 205.92 272.40 Tm <74776F20> Tj 1 0 0 1 223.68 272.88 Tm <70617274733A> Tj 0.24 Tc 1 0 0 1 65.52 259.44 Tm <612920> Tj 1 0 0 1 75.84 259.68 Tm <696E20> Tj 1 0 0 1 86.88 259.44 Tm <6F6E6520> Tj 1 0 0 1 104.64 260.16 Tm <7061727420> Tj 1 0 0 1 125.04 259.68 Tm <6F6620> Tj 1 0 0 1 135.60 259.68 Tm <74686520> Tj 1 0 0 1 151.44 259.68 Tm <6571756174696F6E732C20> Tj 1 0 0 1 198.24 259.68 Tm <74686520> Tj 1 0 0 1 214.08 259.92 Tm <7370617469616C20> Tj 1 0 0 1 245.04 259.68 Tm <646570656E64656E6365> Tj 0.24 Tc 1 0 0 1 77.76 249.84 Tm <287370617469616C20> Tj 1 0 0 1 116.88 249.12 Tm <636F7272656C6174696F6E2920> Tj 1 0 0 1 174.48 249.12 Tm <6F6620> Tj 1 0 0 1 190.32 249.12 Tm <74686520> Tj 1 0 0 1 211.44 249.36 Tm <6D6561737572656420> Tj 1 0 0 1 261.84 249.12 Tm <6461746120> Tj 1 0 0 1 288.00 249.36 Tm <6973> Tj 0.24 Tc 1 0 0 1 77.52 238.32 Tm <63616C63756C617465642C20> Tj 1 0 0 1 125.52 238.56 Tm <757375616C6C7920> Tj 1 0 0 1 158.64 238.56 Tm <7573696E6720> Tj 1 0 0 1 184.56 238.56 Tm <6120> Tj 1 0 0 1 191.52 238.56 Tm <766172696F6772616D3B> Tj 0.24 Tc 1 0 0 1 65.04 225.60 Tm <622920> Tj 1 0 0 1 78.24 225.84 Tm <74686520> Tj 1 0 0 1 95.76 225.60 Tm <6F7468657220> Tj 1 0 0 1 122.16 226.32 Tm <7061727420> Tj 1 0 0 1 144.48 225.84 Tm <6F6620> Tj 1 0 0 1 156.48 225.84 Tm <74686520> Tj 1 0 0 1 174.00 225.84 Tm <6571756174696F6E7320> Tj 1 0 0 1 219.60 226.08 Tm <696E636C7564657320> Tj 1 0 0 1 259.68 225.84 Tm <646966666572656E74> Tj 0.24 Tc 1 0 0 1 77.28 215.04 Tm <636F6E73747261696E74732C20> Tj 1 0 0 1 132.24 215.28 Tm <726573756C74696E6720> Tj 1 0 0 1 174.00 215.52 Tm <696E20> Tj 1 0 0 1 186.96 215.28 Tm <74686520> Tj 1 0 0 1 204.96 215.28 Tm <73756D20> Tj 1 0 0 1 228.00 215.28 Tm <6F6620> Tj 1 0 0 1 240.48 215.28 Tm <616C6C20> Tj 1 0 0 1 254.88 215.28 Tm <776569676874696E67> Tj 0.24 Tc 1 0 0 1 77.28 204.72 Tm <636F656666696369656E7420> Tj 1 0 0 1 122.64 204.72 Tm <6265696E6720> Tj 1 0 0 1 148.32 204.72 Tm <657175616C20> Tj 1 0 0 1 174.00 204.72 Tm <746F20> Tj 1 0 0 1 186.24 204.96 Tm <6C2E> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 57.60 183.60 Tm <342E20> Tj 1 0 0 1 73.20 183.60 Tm <4558414D504C455320> Tj 1 0 0 1 147.36 183.60 Tm <4F4620> Tj 1 0 0 1 168.24 183.84 Tm <54484520> Tj 1 0 0 1 199.20 183.84 Tm <43414C43554C4154494F4E> Tj 0.24 Tc 1 0 0 1 72.48 171.36 Tm <4F4620> Tj 1 0 0 1 93.60 171.36 Tm <4B524947494E4720> Tj 1 0 0 1 153.12 171.36 Tm <495C2C4941545249434553> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 57.12 157.92 Tm <54686520> Tj 1 0 0 1 75.60 157.92 Tm <6D6F64656C20> Tj 1 0 0 1 104.40 158.16 Tm <70726573656E74656420> Tj 1 0 0 1 148.56 157.92 Tm <696E20> Tj 1 0 0 1 159.12 157.92 Tm <7265662E3220> Tj 1 0 0 1 178.80 157.92 Tm <697320> Tj 1 0 0 1 188.16 157.92 Tm <7573656420> Tj 1 0 0 1 210.96 157.92 Tm <617320> Tj 1 0 0 1 222.24 157.92 Tm <6120> Tj 1 0 0 1 229.44 157.92 Tm <7369726E706C6520> Tj 1 0 0 1 260.16 157.92 Tm <6578616D706C65> Tj 0.24 Tc 1 0 0 1 57.36 147.12 Tm <696E20> Tj 1 0 0 1 67.92 147.12 Tm <74686520> Tj 1 0 0 1 83.28 147.12 Tm <666F6C6C6F77696E6720> Tj 1 0 0 1 124.08 147.12 Tm <73756273656374696F6E732E20> Tj 1 0 0 1 179.04 147.36 Tm <546865726520> Tj 1 0 0 1 206.40 147.36 Tm <697320> Tj 1 0 0 1 216.00 147.36 Tm <6E6F20> Tj 1 0 0 1 228.96 147.36 Tm <726561736F6E20> Tj 1 0 0 1 259.68 147.36 Tm <77687920> Tj 1 0 0 1 279.12 147.36 Tm <74686973> Tj 0.24 Tc 1 0 0 1 57.36 136.56 Tm <6D6F64656C20> Tj 1 0 0 1 87.36 136.56 Tm <697320> Tj 1 0 0 1 98.16 136.56 Tm <62657474657220> Tj 1 0 0 1 126.96 136.56 Tm <7468616E20> Tj 1 0 0 1 150.00 136.56 Tm <6F7468657220> Tj 1 0 0 1 176.16 136.80 Tm <73696D696C617220> Tj 1 0 0 1 210.24 136.80 Tm <73696D706C6520> Tj 1 0 0 1 242.16 136.80 Tm <6D6F64656C7320> Tj 1 0 0 1 276.72 136.80 Tm <77697468> Tj 0.24 Tc 1 0 0 1 57.36 126.00 Tm <7365766572616C20> Tj 1 0 0 1 92.64 126.00 Tm <636F6E74726F6C20> Tj 1 0 0 1 128.40 126.48 Tm <706F696E74732C20> Tj 1 0 0 1 163.68 126.00 Tm <65786365707420> Tj 1 0 0 1 196.08 126.24 Tm <7468617420> Tj 1 0 0 1 218.40 126.24 Tm <74686520> Tj 1 0 0 1 237.36 126.24 Tm <617574686F727320> Tj 1 0 0 1 275.76 126.24 Tm <68617665> Tj 0.24 Tc 1 0 0 1 57.36 115.44 Tm <636C6561726C7920> Tj 1 0 0 1 90.24 115.92 Tm <70726573656E74656420> Tj 1 0 0 1 137.76 115.44 Tm <616C6C20> Tj 1 0 0 1 153.12 115.44 Tm <74686520> Tj 1 0 0 1 171.60 115.44 Tm <7370617469616C20> Tj 1 0 0 1 205.44 115.44 Tm <6461746120> Tj 1 0 0 1 228.48 115.44 Tm <76616C75657320> Tj 1 0 0 1 260.88 115.44 Tm <616E6420> Tj 1 0 0 1 282.00 115.68 Tm <746865> Tj 0.24 Tc 1 0 0 1 56.88 104.88 Tm <766172696F6772616D20> Tj 1 0 0 1 102.96 104.88 Tm <76616C75657320> Tj 1 0 0 1 132.48 105.12 Tm <666F7220> Tj 1 0 0 1 147.36 104.88 Tm <7468656D2E20> Tj 1 0 0 1 174.48 105.12 Tm <5468697320> Tj 1 0 0 1 196.32 105.12 Tm <6D65616E7320> Tj 1 0 0 1 226.80 105.12 Tm <7468617420> Tj 1 0 0 1 246.00 105.12 Tm <7769746820> Tj 1 0 0 1 267.60 105.12 Tm <7375636820> Tj 1 0 0 1 290.40 105.12 Tm <61> Tj 0.48 Tc 1 0 0 1 57.60 81.84 Tm <4E4146544120> Tj 1 0 0 1 77.76 81.84 Tm <353920> Tj 1 0 0 1 86.88 80.40 Tm <3020> Tj 1 0 0 1 92.64 80.40 Tm <2920> Tj 1 0 0 1 96.24 81.84 Tm <333120> Tj 1 0 0 1 103.68 83.76 Tm <2D333720> Tj 1 0 0 1 115.44 82.32 Tm <283230303929> Tj 0.24 Tc 1 0 0 1 312.72 518.64 Tm <636C65617220> Tj 1 0 0 1 336.24 518.40 Tm <616E6420> Tj 1 0 0 1 354.96 518.40 Tm <73696D706C6520> Tj 1 0 0 1 385.68 518.64 Tm <6D6F64656C2C20> Tj 1 0 0 1 417.36 518.40 Tm <776520> Tj 1 0 0 1 431.76 518.40 Tm <6361726C20> Tj 1 0 0 1 449.04 518.40 Tm <666F63757320> Tj 1 0 0 1 474.24 518.16 Tm <696E20> Tj 1 0 0 1 485.04 518.16 Tm <74686520> Tj 1 0 0 1 501.12 518.16 Tm <616E616C7973697320> Tj 1 0 0 1 537.60 518.16 Tm <62652D> Tj 0.24 Tc 1 0 0 1 312.72 508.08 Tm <6C6F7720> Tj 1 0 0 1 330.24 508.08 Tm <6F6E20> Tj 1 0 0 1 343.44 508.08 Tm <74686520> Tj 1 0 0 1 359.52 508.08 Tm <6368616C67657320> Tj 1 0 0 1 396.00 507.84 Tm <7468617420> Tj 1 0 0 1 415.68 507.84 Tm <7375636820> Tj 1 0 0 1 438.48 507.84 Tm <6461746120> Tj 1 0 0 1 459.60 507.60 Tm <63616E20> Tj 1 0 0 1 477.36 507.60 Tm <636175736520> Tj 1 0 0 1 503.76 507.84 Tm <696E20> Tj 1 0 0 1 515.04 507.60 Tm <646966666572656E74> Tj 0.24 Tc 1 0 0 1 312.72 497.52 Tm <6B726967696E6720> Tj 1 0 0 1 346.32 497.52 Tm <746563686E6971756520> Tj 1 0 0 1 391.92 497.28 Tm <6571756174696F6E732E20> Tj 1 0 0 1 439.92 497.28 Tm <54686520> Tj 1 0 0 1 460.32 497.04 Tm <63616C63756C6174696F6E20> Tj 1 0 0 1 510.48 497.04 Tm <7461736B7320> Tj 1 0 0 1 537.36 497.04 Tm <617265> Tj 0.24 Tc 1 0 0 1 312.72 486.96 Tm <636F6D706172656420> Tj 1 0 0 1 359.28 486.96 Tm <7573696E6720> Tj 1 0 0 1 386.16 486.96 Tm <53696D706C6520> Tj 1 0 0 1 419.76 486.96 Tm <4B726967696E6720> Tj 1 0 0 1 454.80 487.44 Tm <286265737420> Tj 1 0 0 1 478.80 486.72 Tm <6C696E65617220> Tj 1 0 0 1 507.60 486.72 Tm <657374696D61746F722C> Tj 0.24 Tc 1 0 0 1 312.72 476.40 Tm <616262726576696174656420> Tj 1 0 0 1 365.28 476.40 Tm <424C452920> Tj 1 0 0 1 390.24 476.40 Tm <6F7220> Tj 1 0 0 1 402.24 476.40 Tm <4F7264696E61727920> Tj 1 0 0 1 443.76 476.16 Tm <4966746967696E6720> Tj 1 0 0 1 477.60 476.88 Tm <286265737420> Tj 1 0 0 1 500.88 476.16 Tm <6C696E65617220> Tj 1 0 0 1 528.24 476.16 Tm <756E62692D> Tj 0.24 Tc 1 0 0 1 312.72 465.84 Tm <6173656420> Tj 1 0 0 1 335.04 465.84 Tm <657374696D61746F722C20> Tj 1 0 0 1 381.12 465.84 Tm <616262726576696174656420> Tj 1 0 0 1 433.44 465.60 Tm <424C5545292E> Tj 0.24 Tc 1 0 0 1 312.96 447.12 Tm <342E312E20> Tj 1 0 0 1 335.52 447.12 Tm <446966666572656E636520> Tj 1 0 0 1 390.48 446.88 Tm <696E20> Tj 1 0 0 1 403.44 446.88 Tm <744A726520> Tj 1 0 0 1 422.88 446.88 Tm <657374696D6174696F6E20> Tj 1 0 0 1 478.80 446.64 Tm <76617269616E636520> Tj 1 0 0 1 524.16 446.64 Tm <77697468> Tj 0.24 Tc 1 0 0 1 335.52 435.12 Tm <53696D706C6520> Tj 1 0 0 1 373.44 435.12 Tm <616E6420> Tj 1 0 0 1 395.52 435.12 Tm <4F7264696E61727920> Tj 1 0 0 1 443.04 434.88 Tm <4B726967696E6720> Tj 1 0 0 1 482.64 434.88 Tm <746563686E6971756573> Tj 0.24 Tc 1 0 0 1 312.72 420.24 Tm <426173656420> Tj 1 0 0 1 341.76 420.24 Tm <6F6E20> Tj 1 0 0 1 355.44 420.00 Tm <74686520> Tj 1 0 0 1 372.00 420.24 Tm <6461746120> Tj 1 0 0 1 393.12 420.24 Tm <666F7220> Tj 1 0 0 1 408.48 420.00 Tm <74686520> Tj 1 0 0 1 424.56 420.00 Tm <766172696F6772616D20> Tj 1 0 0 1 470.88 419.76 Tm <76616C75657320> Tj 1 0 0 1 500.88 420.00 Tm <666F7220> Tj 1 0 0 1 516.00 419.76 Tm <74686520> Tj 1 0 0 1 532.56 419.76 Tm <636F6E2D> Tj 0.24 Tc 1 0 0 1 312.48 409.44 Tm <74726F6C20> Tj 1 0 0 1 330.00 409.92 Tm <706F696E747320> Tj 1 0 0 1 359.04 409.44 Tm <74616B656E20> Tj 1 0 0 1 385.20 409.44 Tm <66726F6D20> Tj 1 0 0 1 408.24 409.44 Tm <46696775726520> Tj 1 0 0 1 438.48 409.44 Tm <3120> Tj 1 0 0 1 445.44 409.20 Tm <616E6420> Tj 1 0 0 1 463.68 409.20 Tm <7265662E3220> Tj 1 0 0 1 483.60 409.20 Tm <74686520> Tj 1 0 0 1 499.20 409.20 Tm <6D6174726963657320> Tj 1 0 0 1 538.32 409.20 Tm <666F72> Tj 0.24 Tc 1 0 0 1 312.72 399.12 Tm <4F7264696E61727920> Tj 1 0 0 1 353.76 399.12 Tm <4B7280696E6720> Tj 1 0 0 1 387.60 399.84 Tm <284571756174696F6E20> Tj 1 0 0 1 432.24 398.88 Tm <332E6C20> Tj 1 0 0 1 446.64 398.88 Tm <6C20> Tj 1 0 0 1 451.68 397.68 Tm <2920> Tj 1 0 0 1 456.96 398.88 Tm <697320> Tj 1 0 0 1 467.04 398.64 Tm <73686F776E20> Tj 1 0 0 1 497.52 398.64 Tm <61733A> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 312.24 379.44 Tm <7C20> Tj 1 0 0 1 323.52 381.84 Tm <3020> Tj 1 0 0 1 342.48 381.84 Tm <31322E3635> Tj 0.96 Tc /F3 13.0 Tf 1 0 0 1 312.24 363.60 Tm <4920> Tj 1 0 0 1 315.84 369.84 Tm <727A6F7320> Tj 1 0 0 1 349.68 369.60 Tm <6F> Tj 0.24 Tc /F3 8.0 Tf 1 0 0 1 312.24 354.48 Tm <7C20> Tj 1 0 0 1 316.80 357.84 Tm <32312E353420> Tj 1 0 0 1 342.24 357.60 Tm <31342E3435> Tj 1.50 Tc /F3 12.0 Tf 1 0 0 1 312.24 342.48 Tm <497220> Tj 1 0 0 1 350.16 345.60 Tm <31> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 312.48 322.32 Tm <4D617472697820> Tj 1 0 0 1 343.44 320.64 Tm <5B425D20> Tj 1 0 0 1 358.08 322.32 Tm <63616E20> Tj 1 0 0 1 375.36 322.32 Tm <626520> Tj 1 0 0 1 388.56 322.32 Tm <63616C63756C6174656420> Tj 1 0 0 1 434.16 322.08 Tm <61733A> Tj 0.72 Tc 1 0 0 1 312.48 312.96 Tm <2D20> Tj 1 0 0 1 317.52 311.04 Tm <6C6E20> Tj 1 0 0 1 325.68 311.04 Tm <74686520> Tj 1 0 0 1 339.84 311.04 Tm <312420> Tj 1 0 0 1 349.92 311.04 Tm <726F773A> Tj 0.48 Tc 1 0 0 1 312.48 301.44 Tm <3078302E3338303520> Tj 1 0 0 1 347.76 301.68 Tm <2B20> Tj 1 0 0 1 356.40 301.68 Tm <31322E363578302E3439363420> Tj 1 0 0 1 406.56 301.68 Tm <2B20> Tj 1 0 0 1 414.48 301.68 Tm <323120> Tj 1 0 0 1 423.84 301.44 Tm <2E353478302E3132333220> Tj 1 0 0 1 465.36 301.20 Tm <2B20> Tj 1 0 0 1 474.00 301.20 Tm <3178282D20> Tj 1 0 0 1 490.56 301.20 Tm <302E3933313929> Tj 0.72 Tc 1 0 0 1 313.20 293.52 Tm <3D20> Tj 1 0 0 1 321.36 292.32 Tm <382E303031> Tj 0.48 Tc 1 0 0 1 312.48 276.96 Tm <2D20> Tj 1 0 0 1 317.52 275.04 Tm <496E20> Tj 1 0 0 1 325.92 275.04 Tm <6F7468657220> Tj 1 0 0 1 346.80 275.28 Tm <726F777320> Tj 1 0 0 1 366.48 275.52 Tm <28326E302C20> Tj 1 0 0 1 382.80 275.04 Tm <3372642C20> Tj 1 0 0 1 396.00 275.04 Tm <347468292C20> Tj 1 0 0 1 411.36 275.04 Tm <74686520> Tj 1 0 0 1 425.04 274.80 Tm <73616D6520> Tj 1 0 0 1 447.12 275.28 Tm <70726F63656475726520> Tj 1 0 0 1 486.24 274.80 Tm <697320> Tj 1 0 0 1 493.92 274.80 Tm <6170706C6965642E> Tj 0.24 Tc 1 0 0 1 312.48 257.28 Tm <467572746865726D6F72652C20> Tj 1 0 0 1 371.04 257.28 Tm <74686520> Tj 1 0 0 1 385.92 257.28 Tm <76617269616C636520> Tj 1 0 0 1 423.84 257.04 Tm <6F6620> Tj 1 0 0 1 433.68 257.04 Tm <4F7264696E61727920> Tj 1 0 0 1 474.00 257.04 Tm <4B726967696E6720> Tj 1 0 0 1 506.64 256.80 Tm <63616E20> Tj 1 0 0 1 523.20 256.80 Tm <626520> Tj 1 0 0 1 535.44 256.80 Tm <63616C2D> Tj 0.24 Tc 1 0 0 1 312.48 246.72 Tm <63756C6174656420> Tj 1 0 0 1 346.08 246.72 Tm <6163636F7264696E6720> Tj 1 0 0 1 390.24 246.72 Tm <746F3A> Tj 0.96 Tc 1 0 0 1 312.24 228.24 Tm <6F7A6F6E20> Tj 1 0 0 1 328.80 229.68 Tm <3A20> Tj 1 0 0 1 336.24 228.24 Tm <31722779287A722D7A2920> Tj 1 0 0 1 380.88 228.24 Tm <2B20> Tj 1 0 0 1 388.56 228.24 Tm <292E322779287A722D5A292B2E20> Tj 1 0 0 1 447.60 228.00 Tm <2B312C277928322C20> Tj 1 0 0 1 482.64 229.92 Tm <2D5A2920> Tj 1 0 0 1 498.96 227.76 Tm <2B20> Tj 1 0 0 1 506.16 227.76 Tm <6D20> Tj 1 0 0 1 534.24 228.24 Tm <283420> Tj 1 0 0 1 542.88 227.76 Tm <3229> Tj 0.24 Tc 1 0 0 1 312.24 208.56 Tm <5468697320> Tj 1 0 0 1 333.36 208.32 Tm <76617269616E636520> Tj 1 0 0 1 371.52 208.56 Tm <697320> Tj 1 0 0 1 380.88 208.32 Tm <6F323D362E373020> Tj 1 0 0 1 418.08 208.32 Tm <6D322C20> Tj 1 0 0 1 434.64 208.08 Tm <7768657265617320> Tj 1 0 0 1 472.08 208.08 Tm <74686520> Tj 1 0 0 1 487.68 207.84 Tm <7374616E6461726420> Tj 1 0 0 1 528.48 208.08 Tm <6572726F72> Tj 0.48 Tc 1 0 0 1 312.24 196.80 Tm <697320> Tj 1 0 0 1 321.60 196.56 Tm <6F3D322E353920> Tj 1 0 0 1 355.20 196.56 Tm <6D2E20> Tj 1 0 0 1 368.64 196.56 Tm <4E6F746520> Tj 1 0 0 1 390.00 196.56 Tm <747472617420> Tj 1 0 0 1 408.96 192.48 Tm <292C3120> Tj 1 0 0 1 429.36 198.00 Tm <3A31286265636175736520> Tj 1 0 0 1 481.20 196.08 Tm <6F6620> Tj 1 0 0 1 491.04 196.08 Tm <74686520> Tj 1 0 0 1 506.40 196.32 Tm <6E756D62657220> Tj 1 0 0 1 542.40 196.08 Tm <6F66> Tj 0.24 Tc 1 0 0 1 312.48 182.16 Tm <646563696D616C20> Tj 1 0 0 1 348.24 182.64 Tm <706C6163657320> Tj 1 0 0 1 378.00 182.16 Tm <757365642C20> Tj 1 0 0 1 403.68 181.92 Tm <74686520> Tj 1 0 0 1 420.00 181.92 Tm <6572726F7220> Tj 1 0 0 1 444.72 181.92 Tm <697320> Tj 1 0 0 1 455.76 181.92 Tm <2A6C6F2F6F6F292E20> Tj 1 0 0 1 487.92 181.92 Tm <4E6F772C20> Tj 1 0 0 1 511.20 181.92 Tm <4571756174696F6E> Tj 0.24 Tc 1 0 0 1 312.48 171.60 Tm <342E4920> Tj 1 0 0 1 330.48 171.60 Tm <63616E20> Tj 1 0 0 1 348.00 171.60 Tm <626520> Tj 1 0 0 1 361.92 171.60 Tm <72657772697474656E20> Tj 1 0 0 1 404.16 171.36 Tm <696E20> Tj 1 0 0 1 416.40 171.36 Tm <74686520> Tj 1 0 0 1 433.20 171.36 Tm <666F726D20> Tj 1 0 0 1 457.68 171.12 Tm <6F6620> Tj 1 0 0 1 469.20 171.12 Tm <74686520> Tj 1 0 0 1 486.24 171.12 Tm <53696D706C6520> Tj 1 0 0 1 519.60 171.12 Tm <4B726967696E67> Tj 0.24 Tc 1 0 0 1 312.24 161.04 Tm <6D617472697820> Tj 1 0 0 1 344.16 161.52 Tm <70726573656E74656420> Tj 1 0 0 1 390.00 160.80 Tm <696E20> Tj 1 0 0 1 402.24 160.80 Tm <4571756174696F6E20> Tj 1 0 0 1 445.44 160.80 Tm <342E3320> Tj 1 0 0 1 463.20 161.52 Tm <2874686520> Tj 1 0 0 1 483.12 160.56 Tm <6C6F636174696F6E7320> Tj 1 0 0 1 525.36 160.56 Tm <6F6620> Tj 1 0 0 1 536.88 160.56 Tm <746865> Tj 0.24 Tc 1 0 0 1 312.48 150.24 Tm <636F6E74726F6C20> Tj 1 0 0 1 345.12 150.72 Tm <706F696E747320> Tj 1 0 0 1 374.88 150.24 Tm <61726520> Tj 1 0 0 1 390.72 150.24 Tm <6B65707420> Tj 1 0 0 1 412.32 150.00 Tm <74686520> Tj 1 0 0 1 428.40 150.00 Tm <73616D6520> Tj 1 0 0 1 453.36 150.00 Tm <617320> Tj 1 0 0 1 465.36 150.00 Tm <696E20> Tj 1 0 0 1 476.40 150.00 Tm <46696775726520> Tj 1 0 0 1 506.40 150.00 Tm <32293A> Tj 0.96 Tc /F3 8.0 Tf 1 0 0 1 534.48 363.36 Tm <28342E3129> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 323.28 132.00 Tm <3020> Tj 1 0 0 1 341.76 132.24 Tm <31322E363520> Tj 1 0 0 1 368.40 132.24 Tm <323135343120> Tj 1 0 0 1 394.32 130.32 Tm <5B302E333830353120> Tj 1 0 0 1 435.36 130.32 Tm <7C20> Tj 1 0 0 1 444.00 131.76 Tm <422E65336520> Tj 1 0 0 1 466.80 130.32 Tm <49> Tj 0.72 Tc /F3 13.0 Tf 1 0 0 1 317.28 120.24 Tm <31322E367320> Tj 1 0 0 1 349.44 120.00 Tm <6F20> Tj 1 0 0 1 369.36 120.00 Tm <31342E3432312E31302E2B6F6F2B20> Tj 1 0 0 1 424.08 119.52 Tm <6A3D6C20> Tj 1 0 0 1 441.60 119.76 Tm <6F2E7373727320> Tj 1 0 0 1 468.96 114.00 Tm <4920> Tj 1 0 0 1 534.00 118.08 Tm <722B2E7374> Tj 0.48 Tc /F3 9.0 Tf 1 0 0 1 316.56 108.00 Tm <32312E353420> Tj 1 0 0 1 342.00 108.00 Tm <31342E343520> Tj 1 0 0 1 375.36 107.76 Tm <3020> Tj 1 0 0 1 386.16 104.64 Tm <2E6C20> Tj 1 0 0 1 394.32 104.64 Tm <4C20> Tj 1 0 0 1 399.12 107.76 Tm <302E313233323120> Tj 1 0 0 1 435.36 104.64 Tm <4C31732E333531656C> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 427.44 627.12 Tm <646570746820> Tj 1 0 0 1 450.96 625.92 Tm <696C313020> Tj 1 0 0 1 466.32 626.88 Tm <66656521> Tj 0.48 Tc /F3 9.0 Tf 1 0 0 1 427.44 618.24 Tm <647562696E616933332E353320> Tj 1 0 0 1 476.16 618.00 Tm <726865747261> Tj 0.00 Tc /F3 13.0 Tf 1 0 0 1 447.12 589.68 Tm <34> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 318.96 571.44 Tm <4669672E20> Tj 1 0 0 1 335.28 571.68 Tm <312E20> Tj 1 0 0 1 344.40 571.68 Tm <4469736C7269627574696F6E20> Tj 1 0 0 1 390.48 571.44 Tm <6F6620> Tj 1 0 0 1 400.32 571.44 Tm <636F6E74726F6C20> Tj 1 0 0 1 429.36 571.20 Tm <616E6420> Tj 1 0 0 1 445.92 571.20 Tm <657374696D6174696F6E20> Tj 1 0 0 1 487.68 571.68 Tm <706F696E747320> Tj 1 0 0 1 513.60 571.20 Tm <696E20> Tj 1 0 0 1 522.72 571.20 Tm <61> Tj 0.48 Tc 1 0 0 1 318.72 562.08 Tm <636F6F7264696E61746520> Tj 1 0 0 1 361.68 562.08 Tm <73797374656D> Tj 0.48 Tc 1 0 0 1 318.72 552.72 Tm <536C2E20> Tj 1 0 0 1 330.48 552.72 Tm <312E20> Tj 1 0 0 1 339.12 552.72 Tm <526173706F72656420> Tj 1 0 0 1 376.08 552.72 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 413.76 552.72 Tm <6920> Tj 1 0 0 1 417.36 552.72 Tm <746F646B6520> Tj 1 0 0 1 439.68 552.96 Tm <70726F636A656E6520> Tj 1 0 0 1 473.04 552.24 Tm <7520> Tj 1 0 0 1 480.00 552.24 Tm <6B6F6F7264696E61746E6F6D> Tj 0.24 Tc 1 0 0 1 318.72 543.36 Tm <73757374617675> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 538.80 81.60 Tm <3333> Tj ET endstream endobj 30 0 obj 30369 endobj 31 0 obj << /Length 32 0 R >> stream l, r(2.- z,)IrG"-2,)lx'' -',tLi 3.2. Ordinary kriging theoryAll others kriging techniques add some constraints to thematrices, to minimize the error oi&), and these tech-niques are unbiasedness estimations. Generally, thesefactors would describe some external limit (restriction)on the input data, which cannot simply be observed inthe measured values. The most-used kriging technique isprobably ordinary kriging, and we therefore analyse theconstraint factor in Ordinary lklging equations, calledthe Lagr ang e multiplicator.As discussed above, if the sum of all weighting coefficientis 1, Expression 3.8 can be written as:y(2, - Zr) . \ + y (Z 1 - Zr) . 1, + ... + y (2. - Z,). )., + m = y (z 1 - z)y(2, - z,) . \+ y(22 - Z"). 7' + ... + y(Z' - Z,). )" + n = y(Z' - Z)y(2, - z,). ).1+ y(2, - Z"). l' +,.. + y(2, - z,). ),, + m = y(2, - z)\+12+...+7,+0=1(3.10)If such a system of linear equations is shown as a matri-ces it can be written as: y(2, - Z") y(2, - z,)y(2,-2,) y(2,-2,)y(2, - z") y(2, - z,) (3.1 1)The number of weighting coefficients and control pointscan be very large, but contempora-ry computers can suc-cessfully solve numerically demanding tasks. The esti-mation can be performed simply by calculating theinfluence of all the control points weighted by their asso-ciated coefficients according to Equati on 3.2, as follows : Z = Lt'Zr+ ),2'22+...+Ln'zn (3.12) The calculation of the estimation variance includes add-ing the Lagrange coefficient:o' : ),.'y(Z.-z)+ ).,'y(2,-z)+ +)",'y(2,-z)+m (313)Here are shown the two probably most-used kriging tech-niques. In general, in all linear kriging techniques (sincethey are constrained optimization problems), the associ-ated equations can be divided into two parts:a) in one part of the equations, the spatial dependence(spatial correlation) of the measured data iscalculated, usually using a variogram;b) the other part of the equations includes differentconstraints, resulting in the sum of all weightingcoefficient being equal to l.4. EXAMPLES OF THE CALCULATIONOF KRIGING I\,IATRICESThe model presented in ref.2 is used as a sirnple examplein the following subsections. There is no reason why thismodel is better than other similar simple models withseveral control points, except that the authors haveclearly presented all the spatial data values and thevariogram values for them. This means that with such aNAFTA 59 0 ) 31 -37 (2009) clear and simple model, we carl focus in the analysis be-low on the chalges that such data can cause in differentkriging technique equations. The calculation tasks arecompared using Simple Kriging (best linear estimator,abbreviated BLE) or Ordinary Iftiging (best linear unbi-ased estimator, abbreviated BLUE).4.1. Difference in tJre estimation variance withSimple and Ordinary Kriging techniquesBased on the data for the variogram values for the con-trol points taken from Figure 1 and ref.2 the matrices forOrdinary Kring (Equation 3.l l ) is shown as: | 0 12.65I rzos o| 21.54 14.45Ir 1 Matrix [B] can be calculated as:- ln the 1$ row:0x0.3805 + 12.65x0.4964 + 21 .54x0.1232 + 1x(- 0.9319)= 8.001- In other rows (2n0, 3rd, 4th), the same procedure is applied.Furthermore, the varialce of Ordinary Kriging can be cal-culated according to:ozon : 1r'y(zr-z) + ).2'y(zr-Z)+. +1,'y(2, -Z) + m (4 2)This variance is o2=6.70 m2, whereas the standard erroris o=2.59 m. Note ttrat ),1 :1(because of the number ofdecimal places used, the error is *lo/oo). Now, Equation4.I can be rewritten in the form of the Simple Krigingmatrix presented in Equation 4.3 (the locations of thecontrol points are kept the same as in Figure 2): (4.1) 0 12.65 21541 [0.38051 | B.e3e I12.6s o 14.421.10.+oo+ j=l o.ssrs I r+.st21.54 14.45 0 .l L 0.12321 L1s.351el depth il10 fee! dubinai33.53 rhetra 4 Fig. 1. Dislribution of control and estimation points in acoordinate systemSl. 1. 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<28342E3529> Tj 0.48 Tc 1 0 0 1 55.92 456.24 Tm <32312E35342720> Tj 1 0 0 1 79.68 456.00 Tm <31312B20> Tj 1 0 0 1 94.08 456.00 Tm <31342E34322720> Tj 1 0 0 1 116.88 456.00 Tm <4C7A20> Tj 1 0 0 1 127.20 457.44 Tm <3D20> Tj 1 0 0 1 134.88 456.00 Tm <31342E3432> Tj 0.24 Tc 1 0 0 1 55.44 435.36 Tm <54686520> Tj 1 0 0 1 74.16 435.12 Tm <776569676874696E6720> Tj 1 0 0 1 117.36 435.12 Tm <636F656666696369656E7420> Tj 1 0 0 1 162.96 435.12 Tm <63616E20> Tj 1 0 0 1 180.24 435.12 Tm <626520> Tj 1 0 0 1 193.20 435.12 Tm <65787072657373656420> Tj 1 0 0 1 238.32 435.12 Tm <61733A> Tj 0.72 Tc 1 0 0 1 57.12 417.36 Tm <2C20> Tj 1 0 0 1 74.16 419.52 Tm <382D32312E35342720> Tj 1 0 0 1 109.68 419.52 Tm <295C20> Tj 1 0 0 1 127.92 417.36 Tm <2C20> Tj 1 0 0 1 143.76 419.28 Tm <352E363620> Tj 1 0 0 1 160.80 421.68 Tm <2D31342E343520> Tj 1 0 0 1 187.68 421.68 Tm <272E3133> Tj 1.50 Tc /F3 5.0 Tf 1 0 0 1 55.68 414.24 Tm <415E3D20> Tj 1 0 0 1 114.24 417.36 Tm <2D20> Tj 1 0 0 1 126.72 414.24 Tm <412D3D> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 59.76 412.08 Tm <2720> Tj 1 0 0 1 86.40 407.76 Tm <31322E363520> Tj 1 0 0 1 132.00 411.84 Tm <2720> Tj 1 0 0 1 162.00 407.76 Tm <3132363520> Tj 1 0 0 1 277.20 407.52 Tm <28342E3629> Tj 0.72 Tc 1 0 0 1 56.64 394.80 Tm <2220> Tj 1 0 0 1 72.24 398.16 Tm <31342E34322D31342E34322E292E22> Tj 0.24 Tc 1 0 0 1 60.72 390.72 Tm <2720> Tj 1 0 0 1 92.40 386.40 Tm <32312E3534> Tj 0.24 Tc 1 0 0 1 55.20 368.16 Tm <54686520> Tj 1 0 0 1 82.32 368.16 Tm <666F6C6C6F77696E6720> Tj 1 0 0 1 131.52 367.92 Tm <636F656666696369656E7420> Tj 1 0 0 1 184.80 367.92 Tm <76616C75657320> Tj 1 0 0 1 222.96 367.92 Tm <61726520> Tj 1 0 0 1 247.44 367.92 Tm <63616C63756C617465643A> Tj 0.96 Tc 1 0 0 1 54.24 356.88 Tm <49743D20> Tj 1 0 0 1 72.48 356.88 Tm <302E3334373B20> Tj 1 0 0 1 101.52 356.88 Tm <6C7A3A20> Tj 1 0 0 1 122.16 356.64 Tm <4F2E3438333B317320> Tj 1 0 0 1 162.48 358.32 Tm <3D20> Tj 1 0 0 1 171.60 356.64 Tm <302E30383820> Tj 1 0 0 1 198.96 357.60 Tm <286E6F746520> Tj 1 0 0 1 222.24 356.64 Tm <7468617420> Tj 1 0 0 1 241.44 356.64 Tm <4C313D20> Tj 1 0 0 1 262.80 356.64 Tm <302E393138292E> Tj 0.00 Tc 1 0 0 1 55.20 346.32 Tm <5468697320> Tj 1 0 0 1 77.52 346.32 Tm <6D65616E7320> Tj 1 0 0 1 108.72 346.08 Tm <7468617420> Tj 1 0 0 1 128.88 346.08 Tm <74686520> Tj 1 0 0 1 145.44 346.32 Tm <6D617472697820> Tj 1 0 0 1 177.12 346.08 Tm <6571756174696F6E20> Tj 1 0 0 1 217.44 346.32 Tm <696E20> Tj 1 0 0 1 229.20 346.08 Tm <53696D706C6520> Tj 1 0 0 1 262.56 346.08 Tm <6B7267696E67> Tj 0.24 Tc 1 0 0 1 54.96 335.76 Tm <666F7220> Tj 1 0 0 1 69.84 335.76 Tm <74686520> Tj 1 0 0 1 85.68 336.00 Tm <706F696E747320> Tj 1 0 0 1 115.44 335.52 Tm <646973747269627574656420> Tj 1 0 0 1 165.36 335.52 Tm <617320> Tj 1 0 0 1 177.12 335.76 Tm <696E20> Tj 1 0 0 1 188.16 335.76 Tm <46696775726520> Tj 1 0 0 1 219.12 335.52 Tm <3120> Tj 1 0 0 1 226.08 335.52 Tm <69733A> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 317.76 559.20 Tm <576520> Tj 1 0 0 1 334.80 559.20 Tm <73656C656374656420> Tj 1 0 0 1 373.68 559.20 Tm <746865736520> Tj 1 0 0 1 400.56 558.96 Tm <746872656520> Tj 1 0 0 1 426.96 558.72 Tm <76616C75657320> Tj 1 0 0 1 458.64 558.72 Tm <666F7220> Tj 1 0 0 1 475.68 558.48 Tm <74686520> Tj 1 0 0 1 493.92 558.48 Tm <73616D6520> Tj 1 0 0 1 520.80 558.48 Tm <6D6174726978> Tj 0.24 Tc 1 0 0 1 311.04 548.64 Tm <6571756174696F6E7320> Tj 1 0 0 1 355.92 548.64 Tm <746F20> Tj 1 0 0 1 367.92 548.64 Tm <696C6C7573747261746520> Tj 1 0 0 1 409.68 548.40 Tm <74686520> Tj 1 0 0 1 426.48 548.40 Tm <6661637420> Tj 1 0 0 1 445.68 548.16 Tm <7468617420> Tj 1 0 0 1 466.08 547.92 Tm <74686520> Tj 1 0 0 1 483.12 547.92 Tm <73656C656374696F6E20> Tj 1 0 0 1 524.16 547.92 Tm <6F6620> Tj 1 0 0 1 535.68 547.92 Tm <746865> Tj 0.24 Tc 1 0 0 1 310.80 538.32 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 368.40 538.08 Tm <766166756520> Tj 1 0 0 1 393.36 538.08 Tm <697320> Tj 1 0 0 1 403.20 537.84 Tm <74686520> Tj 1 0 0 1 419.04 537.84 Tm <6D6F737420> Tj 1 0 0 1 443.04 537.84 Tm <696D706F7274616E7420> Tj 1 0 0 1 488.16 537.60 Tm <666163746F7220> Tj 1 0 0 1 515.52 537.36 Tm <696E20> Tj 1 0 0 1 526.56 537.36 Tm <4F7264692D> Tj 0.24 Tc 1 0 0 1 310.80 527.52 Tm <6E61727920> Tj 1 0 0 1 332.88 527.52 Tm <4B726967696E6720> Tj 1 0 0 1 366.72 527.52 Tm <63616C63756C6174696F6E732C20> Tj 1 0 0 1 423.12 527.28 Tm <692E652E2C20> Tj 1 0 0 1 441.60 527.04 Tm <696E20> Tj 1 0 0 1 452.88 527.04 Tm <6D696E696D697A696E6720> Tj 1 0 0 1 503.76 526.80 Tm <6F6620> Tj 1 0 0 1 514.80 526.80 Tm <74686520> Tj 1 0 0 1 531.12 526.80 Tm <657374692D> Tj 0.24 Tc 1 0 0 1 310.80 516.96 Tm <6D6174696F6E20> Tj 1 0 0 1 343.20 516.96 Tm <6572726F7220> Tj 1 0 0 1 368.16 517.68 Tm <286B726967696E6720> Tj 1 0 0 1 402.72 516.72 Tm <76617269616E6365292E20> Tj 1 0 0 1 446.88 516.48 Tm <54686520> Tj 1 0 0 1 465.84 516.24 Tm <656E7469726520> Tj 1 0 0 1 492.96 516.24 Tm <63616C63756C6174696F6E20> Tj 1 0 0 1 541.44 516.24 Tm <6973> Tj 0.24 Tc 1 0 0 1 310.56 506.88 Tm <706572666F726D656420> Tj 1 0 0 1 357.84 506.40 Tm <6D616E75616C6C792C20> Tj 1 0 0 1 401.04 506.16 Tm <776869636820> Tj 1 0 0 1 429.12 505.92 Tm <6D616B657320> Tj 1 0 0 1 459.36 505.92 Tm <697420> Tj 1 0 0 1 467.76 506.16 Tm <706F737369626C6520> Tj 1 0 0 1 505.20 505.68 Tm <746F20> Tj 1 0 0 1 516.24 505.68 Tm <6F627365727665> Tj 0.24 Tc 1 0 0 1 310.56 495.84 Tm <74686520> Tj 1 0 0 1 328.08 495.84 Tm <6368616E67657320> Tj 1 0 0 1 366.00 495.84 Tm <696E20> Tj 1 0 0 1 378.48 495.60 Tm <74686520> Tj 1 0 0 1 395.52 495.60 Tm <6D6174726963657320> Tj 1 0 0 1 436.80 495.36 Tm <616E6420> Tj 1 0 0 1 456.72 495.36 Tm <6C696E65617220> Tj 1 0 0 1 485.52 495.12 Tm <73797374656D7320> Tj 1 0 0 1 523.44 495.12 Tm <6F6620> Tj 1 0 0 1 535.44 495.12 Tm <746865> Tj 0.48 Tc 1 0 0 1 310.56 485.52 Tm <6B726967696E6720> Tj 1 0 0 1 343.20 486.24 Tm <284571756174696F6E7320> Tj 1 0 0 1 391.92 485.04 Tm <332E313020> Tj 1 0 0 1 413.28 485.04 Tm <616E6420> Tj 1 0 0 1 431.76 484.80 Tm <332E3133292E20> Tj 1 0 0 1 458.88 484.80 Tm <54686520> Tj 1 0 0 1 477.84 484.56 Tm <646973747269627574696F6E20> Tj 1 0 0 1 530.40 484.56 Tm <616E612D> Tj 0.24 Tc 1 0 0 1 310.56 474.96 Tm <6C7973656420> Tj 1 0 0 1 334.56 474.96 Tm <696E636C7564657320> Tj 1 0 0 1 372.48 474.72 Tm <666F757220> Tj 1 0 0 1 392.40 474.48 Tm <636F6E74726F6C20> Tj 1 0 0 1 424.80 474.72 Tm <706F696E74732E20> Tj 1 0 0 1 456.48 474.24 Tm <54686520> Tj 1 0 0 1 474.96 474.24 Tm <756E6B6E6F776E20> Tj 1 0 0 1 516.72 474.00 Tm <76616C756520> Tj 1 0 0 1 541.44 474.00 Tm <6973> Tj 0.24 Tc 1 0 0 1 310.56 464.40 Tm <6C6F636174656420> Tj 1 0 0 1 344.16 464.40 Tm <696E20> Tj 1 0 0 1 355.92 464.16 Tm <74686520> Tj 1 0 0 1 372.48 464.16 Tm <63656E74726520> Tj 1 0 0 1 402.00 463.92 Tm <6F6620> Tj 1 0 0 1 413.28 463.92 Tm <6120> Tj 1 0 0 1 421.20 464.40 Tm <706F6C79676F6E20> Tj 1 0 0 1 457.68 463.68 Tm <616E6420> Tj 1 0 0 1 476.88 463.68 Tm <74686520> Tj 1 0 0 1 493.44 463.44 Tm <657374696D6174696F6E20> Tj 1 0 0 1 541.20 463.44 Tm <6973> Tj 0.24 Tc 1 0 0 1 310.32 454.32 Tm <706572666F726D656420> Tj 1 0 0 1 357.84 453.60 Tm <7769746820> Tj 1 0 0 1 379.20 453.60 Tm <4F7264696E61727920> Tj 1 0 0 1 420.24 453.60 Tm <4B726967696E6720> Tj 1 0 0 1 454.08 454.08 Tm <2846696775726520> Tj 1 0 0 1 486.72 453.12 Tm <32292E> Tj 0.24 Tc 1 0 0 1 310.56 439.92 Tm <496E20> Tj 1 0 0 1 323.04 439.92 Tm <46696775726520> Tj 1 0 0 1 354.24 439.92 Tm <322C20> Tj 1 0 0 1 366.24 439.68 Tm <74686520> Tj 1 0 0 1 383.52 439.68 Tm <72656C617469766520> Tj 1 0 0 1 418.80 439.44 Tm <64697374616E63657320> Tj 1 0 0 1 462.24 439.20 Tm <6265747765656E20> Tj 1 0 0 1 500.88 439.20 Tm <74686520> Tj 1 0 0 1 518.16 438.96 Tm <636F6E74726F6C> Tj 0.24 Tc 1 0 0 1 310.32 429.84 Tm <706F696E747320> Tj 1 0 0 1 339.84 430.08 Tm <28353020> Tj 1 0 0 1 355.92 429.36 Tm <6D2920> Tj 1 0 0 1 370.08 429.12 Tm <63616E20> Tj 1 0 0 1 387.12 429.12 Tm <626520> Tj 1 0 0 1 399.60 429.12 Tm <726561642E20> Tj 1 0 0 1 424.08 428.88 Tm <54686520> Tj 1 0 0 1 442.56 428.64 Tm <766172696F677261726E20> Tj 1 0 0 1 488.16 428.64 Tm <76616C75657320> Tj 1 0 0 1 517.44 428.40 Tm <61726520> Tj 1 0 0 1 533.52 428.40 Tm <63616C2D> Tj 0.24 Tc 1 0 0 1 310.56 418.56 Tm <63756C6174656420> Tj 1 0 0 1 343.68 418.80 Tm <66726F6D20> Tj 1 0 0 1 366.96 418.56 Tm <746865736520> Tj 1 0 0 1 391.92 418.32 Tm <64697374616E6365732E20> Tj 1 0 0 1 437.04 418.32 Tm <5468697320> Tj 1 0 0 1 458.16 418.08 Tm <766172696F6772616D20> Tj 1 0 0 1 504.24 418.08 Tm <6D6F64656C20> Tj 1 0 0 1 533.52 417.84 Tm <686173> Tj 0.24 Tc 1 0 0 1 310.32 408.00 Tm <74686520> Tj 1 0 0 1 326.16 408.00 Tm <666F6C6C6F77696E6720> Tj 1 0 0 1 367.20 408.00 Tm <6368617261637465726973746963733A> Tj 0.00 Tc 1 0 0 1 311.52 396.24 Tm <2E20> Tj 1 0 0 1 317.76 394.08 Tm <54686520> Tj 1 0 0 1 339.84 393.84 Tm <6578706572696D656E74616C20> Tj 1 0 0 1 400.56 393.60 Tm <766172696F6772616D20> Tj 1 0 0 1 449.76 393.60 Tm <697320> Tj 1 0 0 1 462.96 393.36 Tm <617070726F7869726E6174656420> Tj 1 0 0 1 527.52 393.12 Tm <627920> Tj 1 0 0 1 543.60 393.12 Tm <61> Tj 0.24 Tc 1 0 0 1 318.00 383.52 Tm <73706865726963616C20> Tj 1 0 0 1 359.52 383.52 Tm <6D6F64656C3B> Tj 0.48 Tc 1 0 0 1 311.28 372.24 Tm <2E20> Tj 1 0 0 1 317.76 370.08 Tm <54686520> Tj 1 0 0 1 336.72 369.84 Tm <73696C6C20> Tj 1 0 0 1 352.08 369.84 Tm <697320> Tj 1 0 0 1 363.36 369.84 Tm <493A> Tj 0.24 Tc 1 0 0 1 311.28 358.56 Tm <2E20> Tj 1 0 0 1 317.76 356.40 Tm <54686520> Tj 1 0 0 1 336.24 356.40 Tm <76612D72696F6772616D20> Tj 1 0 0 1 382.56 356.40 Tm <72616E676520> Tj 1 0 0 1 408.96 356.16 Tm <697320> Tj 1 0 0 1 419.04 356.16 Tm <324F4F20> Tj 1 0 0 1 438.00 355.92 Tm <6D3B> Tj 0.24 Tc 1 0 0 1 311.28 345.12 Tm <2E20> Tj 1 0 0 1 317.76 342.96 Tm <54686520> Tj 1 0 0 1 336.48 342.96 Tm <6E75808069657420> Tj 1 0 0 1 367.20 342.72 Tm <65666665637420> Tj 1 0 0 1 392.16 342.72 Tm <646F657320> Tj 1 0 0 1 414.48 342.72 Tm <6E6F7420> Tj 1 0 0 1 431.28 342.24 Tm <6170706C7920> Tj 1 0 0 1 457.44 342.96 Tm <284F292E> Tj 0.48 Tc 1 0 0 1 310.08 327.12 Tm <342E332E4C2C20> Tj 1 0 0 1 339.36 326.88 Tm <4C616772616E676520> Tj 1 0 0 1 383.28 326.64 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 444.96 326.40 Tm <76616C756520> Tj 1 0 0 1 471.84 326.16 Tm <302E4F36> Tj 0.24 Tc 1 0 0 1 309.84 313.20 Tm <496E20> Tj 1 0 0 1 328.56 312.96 Tm <7468697320> Tj 1 0 0 1 354.72 312.96 Tm <73756273656374696F6E2C20> Tj 1 0 0 1 412.80 312.72 Tm <74686520> Tj 1 0 0 1 435.84 312.48 Tm <76616C756520> Tj 1 0 0 1 468.24 312.24 Tm <6F6620> Tj 1 0 0 1 486.24 312.24 Tm <74686520> Tj 1 0 0 1 509.28 312.24 Tm <4C616772616E6765> Tj 0.24 Tc 1 0 0 1 309.84 302.64 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 367.92 302.64 Tm <697320> Tj 1 0 0 1 377.76 302.40 Tm <74616B656E20> Tj 1 0 0 1 403.92 302.40 Tm <66726F6D20> Tj 1 0 0 1 426.96 302.16 Tm <7265666572656E6365332E20> Tj 1 0 0 1 474.72 301.92 Tm <5468697320> Tj 1 0 0 1 496.08 301.68 Tm <76616C756520> Tj 1 0 0 1 521.28 301.68 Tm <616C6C6F7773> Tj 0.24 Tc 1 0 0 1 309.84 292.08 Tm <757320> Tj 1 0 0 1 323.28 291.84 Tm <746F20> Tj 1 0 0 1 335.04 291.84 Tm <6163686965766520> Tj 1 0 0 1 369.60 291.84 Tm <74686520> Tj 1 0 0 1 386.16 291.84 Tm <6D696E696D756D20> Tj 1 0 0 1 431.52 291.36 Tm <6B726967696E6720> Tj 1 0 0 1 464.40 291.12 Tm <76617269616E636520> Tj 1 0 0 1 503.28 291.12 Tm <7769746820> Tj 1 0 0 1 525.60 291.12 Tm <4F7264692D> Tj 0.24 Tc 1 0 0 1 309.60 281.52 Tm <6E6172794B726967696E672C20> Tj 1 0 0 1 366.00 281.28 Tm <776869636820> Tj 1 0 0 1 393.60 281.28 Tm <697320> Tj 1 0 0 1 402.72 281.04 Tm <776879776520> Tj 1 0 0 1 435.36 281.04 Tm <7573656420> Tj 1 0 0 1 457.92 280.80 Tm <7468697320> Tj 1 0 0 1 475.68 280.56 Tm <76616C756520> Tj 1 0 0 1 500.16 280.80 Tm <696E20> Tj 1 0 0 1 510.48 280.56 Tm <74686520> Tj 1 0 0 1 525.84 280.56 Tm <6D616E2D> Tj 0.00 Tc 1 0 0 1 309.60 270.96 Tm <75616C20> Tj 1 0 0 1 328.56 270.72 Tm <63616C63756C6174696F6E20> Tj 1 0 0 1 380.64 270.48 Tm <6F6620> Tj 1 0 0 1 393.84 270.48 Tm <74686520> Tj 1 0 0 1 412.80 270.48 Tm <6B726967696E6720> Tj 1 0 0 1 448.32 270.24 Tm <6571756174696F6E7320> Tj 1 0 0 1 495.36 270.00 Tm <616E6420> Tj 1 0 0 1 516.72 270.00 Tm <666F7220> Tj 1 0 0 1 534.48 270.00 Tm <746865> Tj 0.24 Tc 1 0 0 1 309.84 260.16 Tm <63616C6962726174696F6E20> Tj 1 0 0 1 357.84 260.16 Tm <63616C63756C6174696F6E20> Tj 1 0 0 1 405.84 260.40 Tm <706572666F726D656420> Tj 1 0 0 1 452.40 259.44 Tm <7769746820> Tj 1 0 0 1 473.28 259.44 Tm <646966666572656E7420> Tj 1 0 0 1 510.72 259.44 Tm <76616C75657320> Tj 1 0 0 1 539.76 259.20 Tm <6F66> Tj 0.00 Tc 1 0 0 1 309.60 249.60 Tm <74686520> Tj 1 0 0 1 325.68 249.60 Tm <4C6167726172676520> Tj 1 0 0 1 366.72 249.60 Tm <6D756C7469706C696361746F722E> Tj 0.24 Tc 1 0 0 1 309.36 235.68 Tm <54686520> Tj 1 0 0 1 328.08 235.68 Tm <6D6174726963657320> Tj 1 0 0 1 366.96 235.68 Tm <666F7220> Tj 1 0 0 1 381.60 235.44 Tm <4F7264696E61727920> Tj 1 0 0 1 422.16 235.20 Tm <4B726967696E6720> Tj 1 0 0 1 454.80 235.20 Tm <696E20> Tj 1 0 0 1 465.60 234.96 Tm <7468697320> Tj 1 0 0 1 483.84 234.96 Tm <6361736520> Tj 1 0 0 1 504.24 234.96 Tm <61726520> Tj 1 0 0 1 520.08 234.72 Tm <73686F776E> Tj 0.24 Tc 1 0 0 1 309.60 224.88 Tm <617320> Tj 1 0 0 1 321.84 224.88 Tm <6120> Tj 1 0 0 1 329.52 224.88 Tm <636F766172696172636520> Tj 1 0 0 1 377.04 224.88 Tm <6D6174726963657320> Tj 1 0 0 1 417.12 225.60 Tm <28696E76657274656420> Tj 1 0 0 1 457.20 224.40 Tm <746F20> Tj 1 0 0 1 468.24 224.40 Tm <6120> Tj 1 0 0 1 475.44 224.16 Tm <766172696F6772616D293A> Tj 0.24 Tc 1 0 0 1 309.12 137.76 Tm <496E20> Tj 1 0 0 1 321.36 137.76 Tm <7468697320> Tj 1 0 0 1 341.28 137.76 Tm <6578616D706C6520> Tj 1 0 0 1 380.16 137.52 Tm <6F6620> Tj 1 0 0 1 391.68 137.52 Tm <6120> Tj 1 0 0 1 400.08 137.52 Tm <726567756C617220> Tj 1 0 0 1 434.40 137.28 Tm <646973747269627574696F6E20> Tj 1 0 0 1 488.64 137.04 Tm <6F6620> Tj 1 0 0 1 500.16 137.04 Tm <74686520> Tj 1 0 0 1 516.96 137.04 Tm <636F6E74726F6C> Tj 0.24 Tc 1 0 0 1 309.12 127.20 Tm <706F696E74732C20> Tj 1 0 0 1 342.24 126.48 Tm <74686520> Tj 1 0 0 1 359.28 126.48 Tm <6D6174726963657320> Tj 1 0 0 1 400.32 124.56 Tm <5B2C315D20> Tj 1 0 0 1 413.52 126.24 Tm <616E6420> Tj 1 0 0 1 433.92 124.32 Tm <5B425D20> Tj 1 0 0 1 450.00 126.96 Tm <284571756174696F6E20> Tj 1 0 0 1 495.60 125.76 Tm <332E332920> Tj 1 0 0 1 516.00 126.00 Tm <696E636C756465> Tj 0.24 Tc 1 0 0 1 309.36 115.92 Tm <636F6E7374616E7420> Tj 1 0 0 1 348.48 115.92 Tm <76616C7565732E20> Tj 1 0 0 1 381.36 115.92 Tm <496620> Tj 1 0 0 1 390.48 115.68 Tm <74686520> Tj 1 0 0 1 407.04 115.68 Tm <6D617472697820> Tj 1 0 0 1 438.48 115.44 Tm <6571756174696F6E20> Tj 1 0 0 1 478.56 115.44 Tm <697320> Tj 1 0 0 1 488.88 115.20 Tm <7772697474656E20> Tj 1 0 0 1 522.24 115.20 Tm <696E20> Tj 1 0 0 1 534.00 115.20 Tm <746865> Tj 0.24 Tc 1 0 0 1 309.12 105.60 Tm <6C696E65617220> Tj 1 0 0 1 336.48 105.36 Tm <73767374656D2E20> Tj 1 0 0 1 371.28 105.36 Tm <697420> Tj 1 0 0 1 379.92 105.36 Tm <69733A> Tj 0.48 Tc /F3 10.0 Tf 1 0 0 1 66.00 318.48 Tm <6F20> Tj 1 0 0 1 84.72 318.48 Tm <31322E363520> Tj 1 0 0 1 111.84 318.48 Tm <32312E35343120> Tj 1 0 0 1 137.28 316.80 Tm <6C6F2E3334376C20> Tj 1 0 0 1 173.76 316.80 Tm <4920> Tj 1 0 0 1 182.16 318.48 Tm <532E65736520> Tj 1 0 0 1 205.20 324.72 Tm <276C> Tj 1.20 Tc /F3 13.0 Tf 1 0 0 1 58.32 306.48 Tm <31322E363520> Tj 1 0 0 1 92.16 306.48 Tm <6F20> Tj 1 0 0 1 112.08 306.48 Tm <31343432312E31302E2B65656C3D6C20> Tj 1 0 0 1 180.00 306.24 Tm <6F737372736C20> Tj 1 0 0 1 276.72 304.80 Tm <652E7A74> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 59.28 294.48 Tm <32312E353420> Tj 1 0 0 1 84.48 294.48 Tm <31342E343520> Tj 1 0 0 1 117.84 294.24 Tm <3020> Tj 1 0 0 1 128.88 291.36 Tm <4931302E3038383120> Tj 1 0 0 1 173.76 291.36 Tm <4C31352E337331656C> Tj 0.24 Tc 1 0 0 1 54.96 274.08 Tm <342E332E20> Tj 1 0 0 1 77.76 273.84 Tm <43616C63756C6174696F6E20> Tj 1 0 0 1 138.00 273.84 Tm <6F6620> Tj 1 0 0 1 150.72 273.84 Tm <4F7264696E61727920> Tj 1 0 0 1 198.24 273.84 Tm <4B726967696E6720> Tj 1 0 0 1 238.08 273.84 Tm <6D61747269636573> Tj 0.24 Tc 1 0 0 1 77.28 262.32 Tm <7769746820> Tj 1 0 0 1 102.48 262.08 Tm <6120> Tj 1 0 0 1 111.12 262.08 Tm <726567756C617220> Tj 1 0 0 1 150.00 262.08 Tm <646973747269627574696F6E20> Tj 1 0 0 1 212.40 262.08 Tm <6F6620> Tj 1 0 0 1 225.12 261.84 Tm <636F6E74726F6C> Tj 0.24 Tc 1 0 0 1 77.28 250.80 Tm <706F696E747320> Tj 1 0 0 1 111.84 250.32 Tm <616E6420> Tj 1 0 0 1 133.44 250.56 Tm <766172696174696F6E20> Tj 1 0 0 1 181.44 250.32 Tm <6F6620> Tj 1 0 0 1 193.92 250.32 Tm <74686520> Tj 1 0 0 1 213.36 250.32 Tm <4C616772616E6765> Tj 0.24 Tc 1 0 0 1 77.28 238.80 Tm <6D756C7469706C696361746F72> Tj 0.24 Tc 1 0 0 1 54.72 225.12 Tm <496E20> Tj 1 0 0 1 66.24 224.88 Tm <7468697320> Tj 1 0 0 1 85.68 224.88 Tm <73756273656374696F6E2C20> Tj 1 0 0 1 136.80 224.88 Tm <776520> Tj 1 0 0 1 151.44 224.88 Tm <75736520> Tj 1 0 0 1 169.44 224.88 Tm <6120> Tj 1 0 0 1 177.36 224.88 Tm <6E657720> Tj 1 0 0 1 197.52 224.88 Tm <646973747269627574696F6E20> Tj 1 0 0 1 251.28 224.64 Tm <6F6620> Tj 1 0 0 1 262.32 224.64 Tm <636F6E74726F6C> Tj 0.24 Tc 1 0 0 1 54.48 214.80 Tm <706F696E747320> Tj 1 0 0 1 87.12 215.28 Tm <2846696775726520> Tj 1 0 0 1 122.64 214.32 Tm <322920> Tj 1 0 0 1 136.80 214.32 Tm <616E6420> Tj 1 0 0 1 158.16 214.32 Tm <6E657720> Tj 1 0 0 1 180.24 214.08 Tm <76616C75657320> Tj 1 0 0 1 212.64 214.08 Tm <666F7220> Tj 1 0 0 1 230.40 214.08 Tm <74686520> Tj 1 0 0 1 248.88 214.08 Tm <76612D72696F677261726E> Tj 0.24 Tc 1 0 0 1 54.48 203.76 Tm <6D6F64656C2E20> Tj 1 0 0 1 90.48 203.76 Tm <54686520> Tj 1 0 0 1 113.52 204.00 Tm <707572706F736520> Tj 1 0 0 1 155.28 203.76 Tm <697320> Tj 1 0 0 1 169.44 203.52 Tm <746F20> Tj 1 0 0 1 184.56 204.00 Tm <70726573656E7420> Tj 1 0 0 1 223.44 203.52 Tm <6368616E67657320> Tj 1 0 0 1 264.00 203.52 Tm <696E20> Tj 1 0 0 1 279.12 203.52 Tm <744A6C65> Tj 0.24 Tc 1 0 0 1 54.48 193.20 Tm <4C616772616E676520> Tj 1 0 0 1 96.96 193.20 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 156.72 192.96 Tm <616E6420> Tj 1 0 0 1 176.64 192.96 Tm <6576616C7561746520> Tj 1 0 0 1 215.04 192.96 Tm <746865697220> Tj 1 0 0 1 239.04 192.96 Tm <696E666C75656E636520> Tj 1 0 0 1 281.76 192.96 Tm <6F6E> Tj 0.24 Tc 1 0 0 1 54.48 182.40 Tm <74686520> Tj 1 0 0 1 70.56 182.40 Tm <657374696D6174656420> Tj 1 0 0 1 114.00 182.40 Tm <76616C75652E> Tj 0.24 Tc 1 0 0 1 61.44 169.20 Tm <54686520> Tj 1 0 0 1 86.40 169.20 Tm <746872656520> Tj 1 0 0 1 117.12 169.20 Tm <6578616D706C657320> Tj 1 0 0 1 165.60 169.44 Tm <70726573656E74656420> Tj 1 0 0 1 216.48 169.20 Tm <696E636C75646520> Tj 1 0 0 1 256.80 168.96 Tm <646966666572656E74> Tj 0.24 Tc 1 0 0 1 54.24 158.64 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 113.04 158.64 Tm <76616C7565732E20> Tj 1 0 0 1 146.64 158.64 Tm <496E20> Tj 1 0 0 1 159.12 158.64 Tm <74686520> Tj 1 0 0 1 175.68 158.64 Tm <66697273742C20> Tj 1 0 0 1 199.68 158.64 Tm <74686520> Tj 1 0 0 1 216.96 158.64 Tm <4C616772616E676520> Tj 1 0 0 1 258.96 158.40 Tm <76616C756520> Tj 1 0 0 1 285.12 158.40 Tm <6973> Tj 0.24 Tc 1 0 0 1 54.48 148.08 Tm <302E30362C20> Tj 1 0 0 1 79.20 148.08 Tm <74616B656E20> Tj 1 0 0 1 105.60 148.08 Tm <66726F6D20> Tj 1 0 0 1 129.12 148.32 Tm <7075626C697368656420> Tj 1 0 0 1 174.00 147.84 Tm <776F726B332E20> Tj 1 0 0 1 205.20 148.08 Tm <486F77657665722C20> Tj 1 0 0 1 246.24 147.84 Tm <696E20> Tj 1 0 0 1 257.52 147.84 Tm <7466726520> Tj 1 0 0 1 273.84 147.84 Tm <7375622D> Tj 0.24 Tc 1 0 0 1 54.24 137.52 Tm <73657175656E7420> Tj 1 0 0 1 90.24 137.28 Tm <74776F20> Tj 1 0 0 1 108.24 137.28 Tm <6578616D706C657320> Tj 1 0 0 1 150.48 137.28 Tm <6F6620> Tj 1 0 0 1 161.04 137.52 Tm <6B726967696E6720> Tj 1 0 0 1 193.68 137.28 Tm <63616C63756C6174696F6E732C20> Tj 1 0 0 1 249.60 137.28 Tm <74686520> Tj 1 0 0 1 265.20 137.28 Tm <76616C756573> Tj 0.24 Tc 1 0 0 1 54.00 126.96 Tm <666F7220> Tj 1 0 0 1 69.84 126.72 Tm <74686520> Tj 1 0 0 1 86.88 126.96 Tm <4C616772616E676520> Tj 1 0 0 1 129.12 126.96 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 188.16 126.72 Tm <61726520> Tj 1 0 0 1 205.44 126.72 Tm <302E3920> Tj 1 0 0 1 222.72 126.72 Tm <616E6420> Tj 1 0 0 1 242.16 128.64 Tm <2D302E392E20> Tj 1 0 0 1 266.88 126.72 Tm <5468657365> Tj 0.24 Tc 1 0 0 1 54.24 116.40 Tm <74776F20> Tj 1 0 0 1 72.96 116.16 Tm <76616C75657320> Tj 1 0 0 1 104.16 116.16 Tm <61726520> Tj 1 0 0 1 121.68 116.16 Tm <636F6E7369646572656420> Tj 1 0 0 1 172.32 116.16 Tm <74686520> Tj 1 0 0 1 189.60 116.16 Tm <65787472656D6520> Tj 1 0 0 1 227.28 116.16 Tm <76616C75657320> Tj 1 0 0 1 258.24 116.16 Tm <7468617420> Tj 1 0 0 1 278.88 115.92 Tm <746865> Tj 0.24 Tc 1 0 0 1 54.00 105.84 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 112.32 105.60 Tm <63616E20> Tj 1 0 0 1 129.84 105.60 Tm <74616B6520> Tj 1 0 0 1 150.72 105.60 Tm <696E20> Tj 1 0 0 1 161.52 106.08 Tm <70726163666C63652E> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 54.00 82.80 Tm <3334> Tj 0.00 Tc /F3 14.0 Tf 1 0 0 1 426.96 758.64 Tm <61> Tj 0.00 Tc 1 0 0 1 438.72 746.64 Tm <7A32> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 317.28 594.48 Tm <4669672E20> Tj 1 0 0 1 333.36 594.48 Tm <322E20> Tj 1 0 0 1 342.72 594.48 Tm <526567756C617220> Tj 1 0 0 1 374.40 594.24 Tm <646973747269627574696F6E20> Tj 1 0 0 1 420.00 594.00 Tm <6F6620> Tj 1 0 0 1 429.36 594.00 Tm <74686520> Tj 1 0 0 1 444.00 593.76 Tm <636F6E74726F6C20> Tj 1 0 0 1 473.04 594.24 Tm <706F696E7473> Tj 0.48 Tc 1 0 0 1 317.04 585.12 Tm <536C2E20> Tj 1 0 0 1 328.32 585.12 Tm <322E20> Tj 1 0 0 1 337.44 585.12 Tm <50726176696C616E20> Tj 1 0 0 1 367.92 585.12 Tm <726173706F72656420> Tj 1 0 0 1 402.24 584.88 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 439.44 584.64 Tm <746F64616B61> Tj 0.00 Tc /F3 14.0 Tf 1 0 0 1 500.64 684.00 Tm <61> Tj 0.48 Tc 1 0 0 1 487.68 672.24 Tm <7A33> Tj 0.00 Tc /F3 9.0 Tf 1 0 0 1 469.92 81.84 Tm <4E4146544120> Tj 1 0 0 1 490.08 81.60 Tm <353920> Tj 1 0 0 1 499.20 82.08 Tm <283120> Tj 1 0 0 1 504.72 80.16 Tm <2920> Tj 1 0 0 1 508.56 81.60 Tm <333120> Tj 1 0 0 1 516.00 83.52 Tm <2D333720> Tj 1 0 0 1 527.76 82.08 Tm <283230306529> Tj ET endstream endobj 40 0 obj 29849 endobj 41 0 obj << /Length 42 0 R >> stream | 0 12.65 21.541| L1f [ 8.e3e 'llrz.os o 14421 | ,,1 =l ossrs IL21.54 14.45 0 I 1,1.1 115.351e1 In this matrix, the values of the variograms between theestimation and control points are changed (i.e., valuesfor matrix [B] from Equation 3.3 are changed). Thismeans that the locations of the estimation points couldbe changed from those in Figure 1. Moreover, it impliesthat the estimation variance is changed (to c2=7.631 m2),as well as the standard error (o=2.76 m). Both are in-creased.4.2. Clnange in weighting coefficient values inSimple KrigingLet us now analyse the case in which Simple Iftingmaintains the locations of the control and estirnationpoints, as in the previous example. In this way, the samevariogram values for all points are held equal in both thekriging techniques examined here. It is clear that the val-ues for the weighting coefficient in Simple Kriging havebeen changed and can be calculated according the equa-tion: (4.4) When this equation is presented as linear formulae, thelinear system is:12.65.),2 + 21.54. 12 = B12.65.11+14.45.\=5.66 (4.5)21.54' 11+ 14.42' Lz = 14.42The weighting coefficient can be expressed as:, 8-21.54' )\ , 5.66 -14.45 '.13A^= - A-=' 12.65 ' 1265 (4.6)" 14.42-14.42.)."' 21.54The following coefficient values are calculated:It= 0.347; lz: O.483;1s = 0.088 (note that L1= 0.918).This means that the matrix equation in Simple krgingfor the points distributed as in Figure 1 is: We selected these three values for the same matrixequations to illustrate the fact that the selection of themultiplicator vafue is the most important factor in Ordi-nary Kriging calculations, i.e., in minimizing of the esti-mation error (kriging variance). The entire calculation isperformed manually, which makes it possible to observethe changes in the matrices and linear systems of thekriging (Equations 3.10 and 3.13). The distribution ana-lysed includes four control points. The unknown value islocated in the centre of a polygon and the estimation isperformed with Ordinary Kriging (Figure 2).In Figure 2, the relative distances between the controlpoints (50 m) can be read. The variograrn values are cal-culated from these distances. This variogram model hasthe following characteristics:. The experimental variogram is approxirnated by aspherical model;. The sill is I:. The va-riogram range is 2OO m;. The nuiet effect does not apply (O).4.3.L, Lagrange multiplicator value 0.O6In this subsection, the value of the Lagrangemultiplicator is taken from reference3. This value allowsus to achieve the minimum kriging variance with Ordi-naryKriging, which is whywe used this value in the man-ual calculation of the kriging equations and for thecalibration calculation performed with different values ofthe Lagrarge multiplicator.The matrices for Ordinary Kriging in this case are shownas a covariarce matrices (inverted to a variogram): In this example of a regular distribution of the controlpoints, the matrices [,1] and [B] (Equation 3.3) includeconstant values. If the matrix equation is written in thelinear svstem. it is: o 12.65 21.541 lo.347l I S.ese 'l12.65 o 14421.10.+eel=l ossrsl e.zt21.54 14.45 0 I10.0881 L15.3s1el 4.3. Calculation of Ordinary Kriging matriceswith a regular distribution of controlpoints and variation of the LagrangemultiplicatorIn this subsection, we use a new distribution of controlpoints (Figure 2) and new values for the va-riograrnmodel. The purpose is to present changes in tJleLagrange multiplicator and evaluate their influence onthe estimated value.The three examples presented include differentmultiplicator values. In the first, the Lagrange value is0.06, taken from published work3. However, in tfre sub-sequent two examples of kriging calculations, the valuesfor the Lagrange multiplicator are 0.9 and -0.9. Thesetwo values are considered the extreme values that themultiplicator can take in pracflce.34 az2 Fig. 2. Regular distribution of the control pointsSl. 2. Pravilan raspored kontrolnih todaka a z3 NAFTA 59 (1 ) 31 -37 (200e) endstream endobj 42 0 obj 4020 endobj 43 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 44 0 obj << /Filter /FlateDecode /Length 109 >> stream xMA @н?> "(%izMu $닆M:H6F Rȣ! 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<776569676C74696E6720> Tj 1 0 0 1 246.00 696.96 Tm <636F656666696369656E74> Tj 0.24 Tc 1 0 0 1 51.12 685.20 Tm <616E6420> Tj 1 0 0 1 69.60 685.20 Tm <74686520> Tj 1 0 0 1 85.92 685.20 Tm <6572726F7220> Tj 1 0 0 1 110.16 685.44 Tm <76617269616E63653A> Tj 0.72 Tc 1 0 0 1 50.88 667.20 Tm <6F5A7820> Tj 1 0 0 1 66.48 668.64 Tm <3A302E323520> Tj 1 0 0 1 90.72 669.84 Tm <27302E363320> Tj 1 0 0 1 110.16 667.68 Tm <2B20> Tj 1 0 0 1 117.60 667.68 Tm <302E323527302E363320> Tj 1 0 0 1 153.60 667.92 Tm <2B20> Tj 1 0 0 1 161.04 667.92 Tm <302E323527302E363320> Tj 1 0 0 1 197.04 668.16 Tm <2B20> Tj 1 0 0 1 204.48 668.16 Tm <302E323520> Tj 1 0 0 1 224.40 668.40 Tm <302E363320> Tj 1 0 0 1 240.48 668.40 Tm <2B20> Tj 1 0 0 1 247.92 668.40 Tm <302E303620> Tj 1 0 0 1 264.72 670.08 Tm <3D20> Tj 1 0 0 1 272.40 668.64 Tm <302E3639> Tj 0.96 Tc 1 0 0 1 266.88 660.00 Tm <28342E313029> Tj 0.24 Tc 1 0 0 1 51.36 641.52 Tm <342E332E322E20> Tj 1 0 0 1 80.40 641.76 Tm <4C616772616E676520> Tj 1 0 0 1 124.32 642.00 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 186.00 642.48 Tm <76616C756520> Tj 1 0 0 1 212.88 642.72 Tm <4F2E39> Tj 0.00 Tc 1 0 0 1 51.36 627.36 Tm <496E20> Tj 1 0 0 1 64.08 627.36 Tm <7468697320> Tj 1 0 0 1 84.48 627.60 Tm <6578616D706C652C20> Tj 1 0 0 1 126.72 627.84 Tm <74686520> Tj 1 0 0 1 144.24 628.08 Tm <4C616772616E676520> Tj 1 0 0 1 186.96 628.32 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 246.72 628.56 Tm <697320> Tj 1 0 0 1 258.00 629.04 Tm <676976656E20> Tj 1 0 0 1 284.40 628.80 Tm <61> Tj 0.24 Tc 1 0 0 1 51.12 617.28 Tm <706F73696669766520> Tj 1 0 0 1 86.16 616.80 Tm <76616C6520> Tj 1 0 0 1 105.84 617.04 Tm <6F6620> Tj 1 0 0 1 116.40 617.28 Tm <302E392E20> Tj 1 0 0 1 135.36 617.28 Tm <496E20> Tj 1 0 0 1 146.64 617.28 Tm <7375636820> Tj 1 0 0 1 169.44 617.52 Tm <63617365732C20> Tj 1 0 0 1 197.52 617.76 Tm <74686520> Tj 1 0 0 1 213.36 617.76 Tm <636F76617269616E636520> Tj 1 0 0 1 261.12 618.24 Tm <6D6174726978> Tj 0.48 Tc /F3 7.0 Tf 1 0 0 1 51.36 606.24 Tm <6C733A> Tj 0.48 Tc /F3 9.0 Tf 1 0 0 1 305.28 766.32 Tm <6F5A7820> Tj 1 0 0 1 321.12 768.00 Tm <3A302E32732E282D302E333237352920> Tj 1 0 0 1 384.96 766.80 Tm <2B20> Tj 1 0 0 1 392.40 766.80 Tm <302E323520> Tj 1 0 0 1 409.20 769.20 Tm <27282D302E333237352920> Tj 1 0 0 1 448.80 767.04 Tm <2B20> Tj 1 0 0 1 456.24 767.04 Tm <302E323520> Tj 1 0 0 1 473.04 769.20 Tm <2E282D302E333237355C20> Tj 1 0 0 1 512.88 767.28 Tm <2B> Tj 0.48 Tc 1 0 0 1 328.08 754.32 Tm <302E32352E20> Tj 1 0 0 1 347.76 755.04 Tm <282D302E333237352920> Tj 1 0 0 1 384.48 754.80 Tm <2B20> Tj 1 0 0 1 391.68 755.04 Tm <282D302E392920> Tj 1 0 0 1 415.44 756.24 Tm <3D20> Tj 1 0 0 1 422.88 756.96 Tm <2D302E393832> Tj 0.96 Tc 1 0 0 1 521.52 744.24 Tm <28342E313629> Tj 0.24 Tc 1 0 0 1 305.28 723.36 Tm <416C6C20> Tj 1 0 0 1 320.88 723.36 Tm <746872656520> Tj 1 0 0 1 346.56 723.36 Tm <6578616D706C657320> Tj 1 0 0 1 389.76 724.08 Tm <70726573656E74656420> Tj 1 0 0 1 435.60 723.84 Tm <696E20> Tj 1 0 0 1 447.84 723.84 Tm <53756273656374696F6E20> Tj 1 0 0 1 498.48 724.08 Tm <342E3320> Tj 1 0 0 1 516.00 724.08 Tm <636C6561726C79> Tj 0.24 Tc 1 0 0 1 306.00 712.56 Tm <73686F7720> Tj 1 0 0 1 329.76 712.80 Tm <7468617420> Tj 1 0 0 1 348.48 712.80 Tm <74686520> Tj 1 0 0 1 363.60 713.04 Tm <4C616772616E676520> Tj 1 0 0 1 403.92 713.04 Tm <76616C756520> Tj 1 0 0 1 428.64 713.04 Tm <6F6620> Tj 1 0 0 1 438.48 713.04 Tm <302E303620> Tj 1 0 0 1 459.84 714.00 Tm <28617320> Tj 1 0 0 1 473.76 713.76 Tm <676976656E20> Tj 1 0 0 1 497.76 713.52 Tm <696E20> Tj 1 0 0 1 507.84 713.52 Tm <7265662E5320> Tj 1 0 0 1 527.76 713.52 Tm <616E64> Tj 0.24 Tc 1 0 0 1 306.00 702.00 Tm <73686F776E20> Tj 1 0 0 1 336.72 702.24 Tm <696E20> Tj 1 0 0 1 348.24 702.24 Tm <46696775726520> Tj 1 0 0 1 378.72 702.48 Tm <322920> Tj 1 0 0 1 390.48 702.48 Tm <697320> Tj 1 0 0 1 400.80 702.48 Tm <74686520> Tj 1 0 0 1 417.12 702.48 Tm <76616C756520> Tj 1 0 0 1 443.04 702.48 Tm <617420> Tj 1 0 0 1 453.60 702.48 Tm <776869636820> Tj 1 0 0 1 482.64 702.72 Tm <74686520> Tj 1 0 0 1 499.20 702.72 Tm <657374696D6174696F6E> Tj 0.24 Tc 1 0 0 1 305.52 691.44 Tm <76617269616E636520> Tj 1 0 0 1 344.40 691.68 Tm <6F6620> Tj 1 0 0 1 355.20 691.68 Tm <4F7264696E61727920> Tj 1 0 0 1 396.24 691.92 Tm <4966746967696E6720> Tj 1 0 0 1 429.84 691.92 Tm <697320> Tj 1 0 0 1 439.92 691.92 Tm <6D696E696D616C2E20> Tj 1 0 0 1 480.96 692.16 Tm <54686520> Tj 1 0 0 1 500.16 692.40 Tm <726561736F6E20> Tj 1 0 0 1 531.60 692.40 Tm <666F72> Tj 0.24 Tc 1 0 0 1 305.76 680.88 Tm <7468697320> Tj 1 0 0 1 324.00 680.88 Tm <616E6420> Tj 1 0 0 1 341.76 681.12 Tm <686F7720> Tj 1 0 0 1 360.96 681.12 Tm <776520> Tj 1 0 0 1 374.64 681.12 Tm <63616E20> Tj 1 0 0 1 391.44 681.36 Tm <73656C65637420> Tj 1 0 0 1 417.12 681.36 Tm <74686520> Tj 1 0 0 1 432.24 681.36 Tm <6D6F737420> Tj 1 0 0 1 455.52 681.36 Tm <617070726F70726961746520> Tj 1 0 0 1 507.12 681.60 Tm <76616C756520> Tj 1 0 0 1 531.60 681.84 Tm <666F72> Tj 0.00 Tc 1 0 0 1 306.00 670.32 Tm <74686520> Tj 1 0 0 1 321.36 670.56 Tm <4C616772616E676520> Tj 1 0 0 1 361.92 670.80 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 419.76 670.80 Tm <61726520> Tj 1 0 0 1 435.36 670.80 Tm <616E616C7973656420> Tj 1 0 0 1 474.48 671.04 Tm <696E20> Tj 1 0 0 1 485.04 671.04 Tm <74686520> Tj 1 0 0 1 500.40 671.04 Tm <6E65787420> Tj 1 0 0 1 520.56 671.28 Tm <636861702D> Tj 0.24 Tc 1 0 0 1 306.00 659.76 Tm <746572732C20> Tj 1 0 0 1 327.60 660.00 Tm <74686520> Tj 1 0 0 1 343.68 660.00 Tm <44697363757373696F6E20> Tj 1 0 0 1 393.12 660.24 Tm <616E6420> Tj 1 0 0 1 411.84 660.24 Tm <436F6E636C7573696F6E20> Tj 1 0 0 1 462.48 660.24 Tm <6F6620> Tj 1 0 0 1 472.80 660.48 Tm <74686520> Tj 1 0 0 1 489.12 660.48 Tm <616E616C797369732E> Tj 0.48 Tc /F3 12.0 Tf 1 0 0 1 306.48 637.92 Tm <352E20> Tj 1 0 0 1 321.84 638.16 Tm <44495343555353494F4E20> Tj 1 0 0 1 404.64 638.40 Tm <4F4620> Tj 1 0 0 1 425.76 638.40 Tm <544845> Tj 0.24 Tc 1 0 0 1 320.88 625.68 Tm <495C7B41544845495C2C4941544943414C20> Tj 1 0 0 1 429.36 626.16 Tm <4D45414E494E475320> Tj 1 0 0 1 501.84 626.40 Tm <4F46> Tj 0.24 Tc 1 0 0 1 320.88 613.44 Tm <54484520> Tj 1 0 0 1 351.60 613.68 Tm <4C414752414E474520> Tj 1 0 0 1 426.72 613.68 Tm <4D554C5449504C494341544F52> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 306.24 599.04 Tm <54686520> Tj 1 0 0 1 324.72 599.04 Tm <726573756C747320> Tj 1 0 0 1 356.16 599.04 Tm <6F6620> Tj 1 0 0 1 366.48 599.28 Tm <4368617074657220> Tj 1 0 0 1 402.96 599.52 Tm <3420> Tj 1 0 0 1 410.88 599.28 Tm <636C6561726C7920> Tj 1 0 0 1 440.88 599.28 Tm <73686F7720> Tj 1 0 0 1 465.12 599.52 Tm <7468617420> Tj 1 0 0 1 484.32 599.52 Tm <696E20> Tj 1 0 0 1 494.88 599.52 Tm <74686520> Tj 1 0 0 1 510.48 599.76 Tm <6170706C6963612D> Tj 0.24 Tc 1 0 0 1 306.24 588.48 Tm <74696F6E20> Tj 1 0 0 1 325.20 588.48 Tm <6F6620> Tj 1 0 0 1 335.52 588.48 Tm <6F7264696E61727920> Tj 1 0 0 1 374.40 588.72 Tm <6B726967696E672C20> Tj 1 0 0 1 408.96 588.72 Tm <617320> Tj 1 0 0 1 420.72 588.72 Tm <6120> Tj 1 0 0 1 427.68 588.72 Tm <737461746973746963616C20> Tj 1 0 0 1 470.64 589.44 Tm <70726F63656475726520> Tj 1 0 0 1 516.48 589.20 Tm <666F7220> Tj 1 0 0 1 530.88 589.20 Tm <746865> Tj 0.24 Tc 1 0 0 1 306.48 577.92 Tm <65737469726E6174696F6E20> Tj 1 0 0 1 354.72 578.16 Tm <6F6620> Tj 1 0 0 1 365.76 578.64 Tm <67656F6C6F676963616C20> Tj 1 0 0 1 410.88 578.40 Tm <6F7220> Tj 1 0 0 1 423.60 578.40 Tm <6F7468657220> Tj 1 0 0 1 449.04 578.16 Tm <7661726961626C65732C20> Tj 1 0 0 1 493.92 578.40 Tm <74686520> Tj 1 0 0 1 510.24 578.64 Tm <76616C756520> Tj 1 0 0 1 536.40 578.64 Tm <6F66> Tj 0.00 Tc 1 0 0 1 306.24 567.36 Tm <74686520> Tj 1 0 0 1 321.60 567.60 Tm <4C616772616E676520> Tj 1 0 0 1 361.92 567.60 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 419.28 568.08 Tm <706C61797320> Tj 1 0 0 1 443.76 567.84 Tm <6120> Tj 1 0 0 1 450.48 567.60 Tm <637269746963616C20> Tj 1 0 0 1 481.44 567.84 Tm <726F6C6520> Tj 1 0 0 1 500.16 568.08 Tm <696E20> Tj 1 0 0 1 510.24 568.08 Tm <74686520> Tj 1 0 0 1 525.36 568.08 Tm <66696E616C> Tj 0.24 Tc 1 0 0 1 306.24 556.80 Tm <726573756C742E> Tj 0.24 Tc 1 0 0 1 313.44 542.40 Tm <54686520> Tj 1 0 0 1 332.64 542.40 Tm <74776F20> Tj 1 0 0 1 350.88 542.40 Tm <616363657074656420> Tj 1 0 0 1 390.48 542.64 Tm <76612D6C75657320> Tj 1 0 0 1 420.48 542.64 Tm <6F6620> Tj 1 0 0 1 431.28 542.64 Tm <74686520> Tj 1 0 0 1 447.36 542.88 Tm <4C616772616E676520> Tj 1 0 0 1 488.88 542.88 Tm <6D756C7469706C696361746F72> Tj 0.24 Tc 1 0 0 1 306.24 531.84 Tm <7573656420> Tj 1 0 0 1 329.52 531.84 Tm <746F20> Tj 1 0 0 1 340.56 531.84 Tm <6D696E696D697A6520> Tj 1 0 0 1 383.28 532.08 Tm <74686520> Tj 1 0 0 1 398.88 532.08 Tm <76617269616E636520> Tj 1 0 0 1 437.76 532.32 Tm <696E20> Tj 1 0 0 1 449.04 532.08 Tm <74686520> Tj 1 0 0 1 465.12 532.80 Tm <70726576696F757320> Tj 1 0 0 1 504.72 532.32 Tm <6578616D706C6573> Tj 0.24 Tc 1 0 0 1 306.72 521.04 Tm <64696666657265642C20> Tj 1 0 0 1 344.40 521.28 Tm <7769746820> Tj 1 0 0 1 365.76 521.28 Tm <6F6E6520> Tj 1 0 0 1 383.52 521.52 Tm <6E6567617469766520> Tj 1 0 0 1 420.72 522.48 Tm <282D302E393331392920> Tj 1 0 0 1 463.92 521.52 Tm <616E6420> Tj 1 0 0 1 482.64 521.52 Tm <74686520> Tj 1 0 0 1 498.72 521.76 Tm <6F7468657220> Tj 1 0 0 1 523.44 522.48 Tm <706F73692D> Tj 0.24 Tc 1 0 0 1 306.48 510.48 Tm <7469766520> Tj 1 0 0 1 326.40 511.44 Tm <28302E3036292E20> Tj 1 0 0 1 358.80 510.72 Tm <54686520> Tj 1 0 0 1 379.92 511.20 Tm <70726F6F6620> Tj 1 0 0 1 407.28 510.96 Tm <7468617420> Tj 1 0 0 1 428.64 510.96 Tm <626F746820> Tj 1 0 0 1 452.64 510.96 Tm <76616C75657320> Tj 1 0 0 1 484.56 510.96 Tm <61726520> Tj 1 0 0 1 502.80 511.20 Tm <636F727265637420> Tj 1 0 0 1 537.36 511.44 Tm <6973> Tj 0.24 Tc 1 0 0 1 306.72 499.92 Tm <73686F776E20> Tj 1 0 0 1 337.44 500.16 Tm <627920> Tj 1 0 0 1 350.88 500.16 Tm <76617279696E6720> Tj 1 0 0 1 385.92 500.40 Tm <6F6620> Tj 1 0 0 1 396.72 500.40 Tm <76616C756520> Tj 1 0 0 1 423.12 500.40 Tm <302E303620> Tj 1 0 0 1 445.68 500.40 Tm <696E20> Tj 1 0 0 1 457.68 500.40 Tm <74686520> Tj 1 0 0 1 474.48 500.64 Tm <6B726967696E6720> Tj 1 0 0 1 507.60 500.88 Tm <6D61747269636573> Tj 0.24 Tc 1 0 0 1 306.96 490.32 Tm <2853756273656374696F6E20> Tj 1 0 0 1 360.24 489.84 Tm <342E33292E20> Tj 1 0 0 1 382.80 489.84 Tm <4120> Tj 1 0 0 1 392.64 489.84 Tm <6368616E676520> Tj 1 0 0 1 425.28 490.08 Tm <696E20> Tj 1 0 0 1 437.04 489.84 Tm <7468697320> Tj 1 0 0 1 456.00 489.84 Tm <76616C756520> Tj 1 0 0 1 481.92 490.08 Tm <696E647563656420> Tj 1 0 0 1 519.60 490.32 Tm <616E20> Tj 1 0 0 1 533.28 490.32 Tm <696E2D> Tj 0.24 Tc 1 0 0 1 306.72 479.04 Tm <63726561736520> Tj 1 0 0 1 336.00 479.28 Tm <696E20> Tj 1 0 0 1 346.80 479.04 Tm <74686520> Tj 1 0 0 1 362.64 479.28 Tm <657374696D6174696F6E20> Tj 1 0 0 1 409.44 479.28 Tm <76617269616E636520> Tj 1 0 0 1 448.32 480.24 Tm <2853756273656374696F6E20> Tj 1 0 0 1 500.64 479.76 Tm <342E332E322920> Tj 1 0 0 1 528.00 479.76 Tm <6F7220> Tj 1 0 0 1 540.00 479.76 Tm <61> Tj 0.24 Tc 1 0 0 1 306.72 468.48 Tm <6E6567617469766520> Tj 1 0 0 1 344.88 469.44 Tm <286D617468656D61746963616C6C7920> Tj 1 0 0 1 415.92 468.96 Tm <696D706F737369626C6520> Tj 1 0 0 1 462.96 467.76 Tm <2920> Tj 1 0 0 1 468.72 468.96 Tm <76617269616E636520> Tj 1 0 0 1 508.56 469.92 Tm <2820> Tj 1 0 0 1 511.44 469.20 Tm <5375627365632D> Tj 0.72 Tc 1 0 0 1 306.72 457.92 Tm <74696F6E20> Tj 1 0 0 1 326.16 458.16 Tm <342E332E33292E> Tj 0.24 Tc 1 0 0 1 313.92 443.76 Tm <4C657420> Tj 1 0 0 1 330.48 443.76 Tm <757320> Tj 1 0 0 1 343.92 443.76 Tm <6578616D696E6520> Tj 1 0 0 1 382.56 443.76 Tm <6F6E636520> Tj 1 0 0 1 405.60 444.00 Tm <616761696E20> Tj 1 0 0 1 431.04 444.00 Tm <74686520> Tj 1 0 0 1 447.60 444.00 Tm <6D617468656D61746963616C20> Tj 1 0 0 1 508.56 444.24 Tm <6D65616E696E67> Tj 0.24 Tc 1 0 0 1 306.96 432.96 Tm <6F6620> Tj 1 0 0 1 318.24 432.96 Tm <74686520> Tj 1 0 0 1 334.80 433.20 Tm <4C616772616E676520> Tj 1 0 0 1 377.04 433.44 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 435.84 433.44 Tm <696E20> Tj 1 0 0 1 447.60 433.44 Tm <6B726967696E6720> Tj 1 0 0 1 480.96 433.44 Tm <6571756174696F6E732E20> Tj 1 0 0 1 528.48 433.92 Tm <546865> Tj 0.24 Tc 1 0 0 1 306.48 422.40 Tm <76616C756520> Tj 1 0 0 1 331.92 422.64 Tm <617420> Tj 1 0 0 1 342.24 422.64 Tm <74686520> Tj 1 0 0 1 358.56 422.64 Tm <657374696D6174696F6E20> Tj 1 0 0 1 405.60 423.36 Tm <706F696E7420> Tj 1 0 0 1 430.80 422.88 Tm <63616E20> Tj 1 0 0 1 447.84 422.88 Tm <626520> Tj 1 0 0 1 461.04 422.88 Tm <65787072657373656420> Tj 1 0 0 1 506.16 423.12 Tm <61733A> Tj 1.44 Tc /F3 8.0 Tf 1 0 0 1 306.72 404.40 Tm <5A2872293D702B65287229> Tj 0.96 Tc 1 0 0 1 528.72 405.60 Tm <28352E3129> Tj 0.24 Tc 1 0 0 1 306.72 385.20 Tm <7768657265> Tj 0.48 Tc 1 0 0 1 307.44 374.40 Tm <7020> Tj 1 0 0 1 338.40 374.40 Tm <697320> Tj 1 0 0 1 345.84 374.40 Tm <74686520> Tj 1 0 0 1 359.52 374.16 Tm <636F727265637420> Tj 1 0 0 1 387.36 374.88 Tm <2862757420> Tj 1 0 0 1 403.44 374.40 Tm <7265616C6C7920> Tj 1 0 0 1 424.56 374.40 Tm <756E6B6E6F776E2920> Tj 1 0 0 1 462.24 374.64 Tm <6D65616E20> Tj 1 0 0 1 484.32 374.64 Tm <6F6620> Tj 1 0 0 1 492.96 374.64 Tm <74686520> Tj 1 0 0 1 506.64 374.64 Tm <656E74697265> Tj 0.48 Tc 1 0 0 1 338.40 365.52 Tm <706F70756C6174696F6E3B> Tj 0.48 Tc 1 0 0 1 307.20 347.28 Tm <6528722920> Tj 1 0 0 1 338.64 347.52 Tm <697320> Tj 1 0 0 1 346.08 347.52 Tm <74686520> Tj 1 0 0 1 359.76 347.52 Tm <637572766520> Tj 1 0 0 1 381.60 347.52 Tm <616E6420> Tj 1 0 0 1 397.44 347.76 Tm <6D65616E20> Tj 1 0 0 1 419.28 347.76 Tm <76616C756520> Tj 1 0 0 1 440.88 347.76 Tm <6E6F726D616C697A656420> Tj 1 0 0 1 482.64 347.76 Tm <61626F757420> Tj 1 0 0 1 504.96 348.00 Tm <74686520> Tj 1 0 0 1 518.16 348.00 Tm <76616C7565> Tj 0.48 Tc 1 0 0 1 338.40 338.16 Tm <3020> Tj 1 0 0 1 344.88 338.16 Tm <6F6E20> Tj 1 0 0 1 356.16 338.16 Tm <74686520> Tj 1 0 0 1 369.84 338.16 Tm <637572766520> Tj 1 0 0 1 391.44 338.16 Tm <6F6674686520> Tj 1 0 0 1 413.76 338.40 Tm <7370617469616C20> Tj 1 0 0 1 439.44 338.40 Tm <72616E646F6D20> Tj 1 0 0 1 468.96 338.40 Tm <66756E6374696F6E2E> Tj 0.24 Tc 1 0 0 1 307.20 320.64 Tm <557375616C6C792C20> Tj 1 0 0 1 342.48 320.64 Tm <776520> Tj 1 0 0 1 356.64 320.88 Tm <646F20> Tj 1 0 0 1 369.36 320.88 Tm <6E6F7420> Tj 1 0 0 1 385.68 320.88 Tm <6B6E6F7720> Tj 1 0 0 1 410.88 320.88 Tm <74686520> Tj 1 0 0 1 426.48 321.12 Tm <7265616C20> Tj 1 0 0 1 444.96 321.12 Tm <6D65616E20> Tj 1 0 0 1 470.16 320.88 Tm <76616C756520> Tj 1 0 0 1 495.12 321.12 Tm <6F6620> Tj 1 0 0 1 504.96 321.12 Tm <74686520> Tj 1 0 0 1 520.80 321.36 Tm <656E74697265> Tj 0.24 Tc 1 0 0 1 307.20 309.84 Tm <706F70756C6174696F6E20> Tj 1 0 0 1 357.36 310.56 Tm <2870292E20> Tj 1 0 0 1 374.40 309.84 Tm <5468657265666F72652C20> Tj 1 0 0 1 422.40 309.84 Tm <74686520> Tj 1 0 0 1 439.44 309.84 Tm <657374696D6174696F6E20> Tj 1 0 0 1 487.44 310.08 Tm <697320> Tj 1 0 0 1 498.24 310.08 Tm <6D6F726520> Tj 1 0 0 1 524.16 310.08 Tm <6F6674656E> Tj 0.24 Tc 1 0 0 1 307.20 299.28 Tm <706572666F726D656420> Tj 1 0 0 1 353.76 299.04 Tm <7769746820> Tj 1 0 0 1 374.40 299.04 Tm <74686520> Tj 1 0 0 1 389.52 299.04 Tm <4F7264696E61727920> Tj 1 0 0 1 429.84 299.28 Tm <4B726967696E6720> Tj 1 0 0 1 462.48 299.28 Tm <746563686E6971756520> Tj 1 0 0 1 505.68 299.52 Tm <7468616E20> Tj 1 0 0 1 526.56 299.52 Tm <77697468> Tj 0.24 Tc 1 0 0 1 307.44 288.24 Tm <53696D706C6520> Tj 1 0 0 1 339.84 288.48 Tm <4B726967696E672C20> Tj 1 0 0 1 375.60 288.72 Tm <7573696E6720> Tj 1 0 0 1 401.28 288.48 Tm <6F6E6C7920> Tj 1 0 0 1 421.68 288.72 Tm <74686520> Tj 1 0 0 1 437.52 288.72 Tm <6C6F63616C20> Tj 1 0 0 1 459.84 288.72 Tm <6D65616E20> Tj 1 0 0 1 485.76 288.72 Tm <626173656420> Tj 1 0 0 1 513.60 288.96 Tm <6F6E20> Tj 1 0 0 1 527.04 288.96 Tm <64617461> Tj 0.24 Tc 1 0 0 1 307.44 277.68 Tm <656E636F6D70617373656420> Tj 1 0 0 1 368.64 277.92 Tm <627920> Tj 1 0 0 1 382.56 277.92 Tm <74686520> Tj 1 0 0 1 399.60 278.16 Tm <73656172636820> Tj 1 0 0 1 431.28 278.16 Tm <72616469757320> Tj 1 0 0 1 462.72 278.16 Tm <61726F756E6420> Tj 1 0 0 1 497.28 278.40 Tm <74686520> Tj 1 0 0 1 514.32 278.40 Tm <65737469726E612D> Tj 0.24 Tc 1 0 0 1 307.44 267.12 Tm <74696F6E20> Tj 1 0 0 1 328.08 267.60 Tm <706F696E742E20> Tj 1 0 0 1 357.36 267.36 Tm <4D6F72656F7665722C20> Tj 1 0 0 1 403.44 267.60 Tm <7365636F6E642D6F7264657220> Tj 1 0 0 1 464.16 267.60 Tm <73746174696F6E617269747920> Tj 1 0 0 1 516.72 267.84 Tm <697320> Tj 1 0 0 1 528.24 267.84 Tm <616C736F> Tj 0.24 Tc 1 0 0 1 307.68 256.56 Tm <617373756D656420> Tj 1 0 0 1 347.52 256.80 Tm <666F7220> Tj 1 0 0 1 361.92 256.80 Tm <6B726967696E677769746820> Tj 1 0 0 1 414.48 257.04 Tm <74686520> Tj 1 0 0 1 430.08 257.04 Tm <636F76617269616E636520> Tj 1 0 0 1 477.36 257.04 Tm <43226F6620> Tj 1 0 0 1 499.20 257.04 Tm <747D726520> Tj 1 0 0 1 514.56 257.28 Tm <657374696D612D> Tj 0.24 Tc 1 0 0 1 307.44 246.00 Tm <74696F6E20> Tj 1 0 0 1 326.64 246.00 Tm <697320> Tj 1 0 0 1 336.24 246.24 Tm <5A2878292E20> Tj 1 0 0 1 359.04 250.56 Tm <5E7468697320> Tj 1 0 0 1 381.12 246.24 Tm <73746174696F6E617269747920> Tj 1 0 0 1 432.24 246.24 Tm <63616E20> Tj 1 0 0 1 449.28 246.24 Tm <626520> Tj 1 0 0 1 462.48 246.48 Tm <6578707265737365643A> Tj 0.48 Tc 1 0 0 1 307.68 227.76 Tm <432C28722E2C2B2920> Tj 1 0 0 1 336.24 229.44 Tm <3D20> Tj 1 0 0 1 343.44 228.00 Tm <455B6528722C2920> Tj 1 0 0 1 366.00 230.40 Tm <276528722C295D20> Tj 1 0 0 1 389.04 229.68 Tm <3D20> Tj 1 0 0 1 396.48 228.24 Tm <432C28722C2D20> Tj 1 0 0 1 420.00 226.32 Tm <6420> Tj 1 0 0 1 529.44 229.20 Tm <28352E3229> Tj 0.24 Tc 1 0 0 1 307.44 208.80 Tm <5468697320> Tj 1 0 0 1 330.72 208.80 Tm <636F6E646974696F6E20> Tj 1 0 0 1 374.88 208.80 Tm <63616E20> Tj 1 0 0 1 393.12 209.04 Tm <626520> Tj 1 0 0 1 407.52 209.04 Tm <64657363726962656420> Tj 1 0 0 1 452.64 209.04 Tm <7769746820> Tj 1 0 0 1 475.44 209.04 Tm <6120> Tj 1 0 0 1 483.60 209.04 Tm <766172696F6772616D20> Tj 1 0 0 1 531.60 210.24 Tm <28696E2D> Tj 0.24 Tc 1 0 0 1 307.92 198.00 Tm <737465616420> Tj 1 0 0 1 334.08 198.24 Tm <6F6620> Tj 1 0 0 1 345.84 198.24 Tm <636F76617269616E63652920> Tj 1 0 0 1 397.20 198.48 Tm <7768656E20> Tj 1 0 0 1 423.84 198.48 Tm <74686520> Tj 1 0 0 1 440.88 198.48 Tm <73746174696F6E617269747920> Tj 1 0 0 1 492.72 198.72 Tm <697320> Tj 1 0 0 1 503.52 199.20 Tm <70726573656E746564> Tj 0.72 Tc 1 0 0 1 307.92 187.44 Tm <61733A> Tj 0.72 Tc 1 0 0 1 307.44 165.84 Tm <792C285C2C742C2920> Tj 1 0 0 1 336.48 167.04 Tm <3D20> Tj 1 0 0 1 350.88 168.00 Tm <27454C6528722C2920> Tj 1 0 0 1 376.80 168.00 Tm <2D8028722C296C2D20> Tj 1 0 0 1 406.80 167.28 Tm <3D767A285C2D722920> Tj 1 0 0 1 529.68 166.80 Tm <28352E3329> Tj 0.00 Tc /F3 6.0 Tf 1 0 0 1 344.64 159.60 Tm <7A> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 307.92 140.16 Tm <54686520> Tj 1 0 0 1 328.08 140.16 Tm <617272617920> Tj 1 0 0 1 354.24 140.40 Tm <6F6620> Tj 1 0 0 1 365.76 140.40 Tm <4E20> Tj 1 0 0 1 376.08 140.40 Tm <6D6561737572656D656E747320> Tj 1 0 0 1 442.32 140.64 Tm <5A287874292E2E2E322878222920> Tj 1 0 0 1 496.08 140.64 Tm <617420> Tj 1 0 0 1 507.60 140.88 Tm <6C6F636174696F6E73> Tj 0.24 Tc 1 0 0 1 307.44 129.60 Tm <78722E20> Tj 1 0 0 1 319.92 129.60 Tm <2E20> Tj 1 0 0 1 322.80 129.60 Tm <2E782E20> Tj 1 0 0 1 335.28 129.84 Tm <697320> Tj 1 0 0 1 344.88 129.60 Tm <617373756D65642E20> Tj 1 0 0 1 387.36 129.84 Tm <546872656520> Tj 1 0 0 1 414.24 130.08 Tm <636F6E646974696F6E7320> Tj 1 0 0 1 460.80 130.08 Tm <6D75737420> Tj 1 0 0 1 484.08 130.08 Tm <626520> Tj 1 0 0 1 496.56 130.08 Tm <73617469736669656420> Tj 1 0 0 1 533.76 130.56 Tm <666F72> Tj 0.24 Tc 1 0 0 1 308.16 119.04 Tm <616E20> Tj 1 0 0 1 320.64 119.04 Tm <4F7264696E61727920> Tj 1 0 0 1 360.96 119.28 Tm <4B726967696E6720> Tj 1 0 0 1 393.84 119.28 Tm <65737469726E6174696F6E2C20> Tj 1 0 0 1 443.04 119.28 Tm <626173656420> Tj 1 0 0 1 470.40 119.52 Tm <6F6E20> Tj 1 0 0 1 483.12 119.52 Tm <74686520> Tj 1 0 0 1 498.00 119.76 Tm <76616C756520> Tj 1 0 0 1 522.72 119.76 Tm <6F6620> Tj 1 0 0 1 532.56 119.76 Tm <746865> Tj 0.24 Tc 1 0 0 1 307.68 107.52 Tm <7661726961626C6520> Tj 1 0 0 1 344.16 107.76 Tm <5A20> Tj 1 0 0 1 353.52 108.72 Tm <285A2920> Tj 1 0 0 1 368.16 107.76 Tm <617420> Tj 1 0 0 1 378.96 107.76 Tm <616E7920> Tj 1 0 0 1 396.24 108.00 Tm <756E73616D706C656420> Tj 1 0 0 1 446.40 108.00 Tm <6C6F636174696F6E20> Tj 1 0 0 1 482.40 108.24 Tm <786F3A> Tj 0.48 Tc 1 0 0 1 77.04 589.44 Tm <302E343920> Tj 1 0 0 1 98.88 589.44 Tm <302E343920> Tj 1 0 0 1 120.96 589.68 Tm <302E3331> Tj 0.48 Tc 1 0 0 1 79.44 577.44 Tm <312E3020> Tj 1 0 0 1 99.12 577.44 Tm <302E333120> Tj 1 0 0 1 120.72 577.68 Tm <302E3439> Tj 0.48 Tc 1 0 0 1 77.52 565.20 Tm <302E333120> Tj 1 0 0 1 101.52 565.44 Tm <312E3020> Tj 1 0 0 1 120.72 565.44 Tm <302E3439> Tj 0.48 Tc 1 0 0 1 77.04 552.96 Tm <302E343920> Tj 1 0 0 1 98.88 552.96 Tm <302E343920> Tj 1 0 0 1 123.36 553.44 Tm <312E30> Tj 0.72 Tc 1 0 0 1 79.68 540.96 Tm <312E3020> Tj 1 0 0 1 101.52 541.20 Tm <312E3020> Tj 1 0 0 1 123.36 541.20 Tm <312E30> Tj 0.48 Tc 1 0 0 1 58.56 304.80 Tm <312E3020> Tj 1 0 0 1 78.00 304.80 Tm <302E343920> Tj 1 0 0 1 99.84 305.04 Tm <302E343920> Tj 1 0 0 1 121.92 305.28 Tm <302E3331> Tj 0.48 Tc 1 0 0 1 56.16 292.56 Tm <302E343920> Tj 1 0 0 1 80.64 293.04 Tm <312E3020> Tj 1 0 0 1 100.08 292.80 Tm <302E333120> Tj 1 0 0 1 121.68 293.04 Tm <302E3439> Tj 0.48 Tc 1 0 0 1 56.16 280.56 Tm <30343920> Tj 1 0 0 1 78.48 280.80 Tm <302E333120> Tj 1 0 0 1 102.48 281.04 Tm <312E3020> Tj 1 0 0 1 121.68 281.04 Tm <302E3439> Tj 0.48 Tc 1 0 0 1 56.40 268.56 Tm <302E333120> Tj 1 0 0 1 78.00 268.56 Tm <302E343920> Tj 1 0 0 1 99.84 268.80 Tm <302E343920> Tj 1 0 0 1 124.32 269.04 Tm <312E30> Tj 0.48 Tc 1 0 0 1 58.80 260.40 Tm <27742E3020> Tj 1 0 0 1 80.64 256.56 Tm <312E3020> Tj 1 0 0 1 102.48 256.80 Tm <312E3020> Tj 1 0 0 1 124.32 257.04 Tm <312E30> Tj 0.96 Tc 1 0 0 1 267.36 566.40 Tm <28342E3120> Tj 1 0 0 1 280.80 566.16 Tm <3129> Tj 0.72 Tc 1 0 0 1 268.32 281.76 Tm <28342E313429> Tj 0.24 Tc 1 0 0 1 51.12 518.16 Tm <416761696E2C20> Tj 1 0 0 1 81.12 518.16 Tm <74686520> Tj 1 0 0 1 97.44 518.40 Tm <6D6174726963657320> Tj 1 0 0 1 138.00 516.96 Tm <5B5D2C3120> Tj 1 0 0 1 151.44 518.64 Tm <616E6420> Tj 1 0 0 1 170.88 516.96 Tm <5B425D20> Tj 1 0 0 1 186.24 518.88 Tm <636F6E7461696E20> Tj 1 0 0 1 220.80 519.12 Tm <636F6E7374616E7420> Tj 1 0 0 1 259.92 519.36 Tm <76616C7565732C> Tj 0.48 Tc 1 0 0 1 51.84 507.60 Tm <617320> Tj 1 0 0 1 64.08 507.60 Tm <696E20> Tj 1 0 0 1 75.84 507.60 Tm <53756273656374696F6E20> Tj 1 0 0 1 126.00 508.08 Tm <342E332E312E20> Tj 1 0 0 1 153.84 508.32 Tm <546865736520> Tj 1 0 0 1 182.16 508.56 Tm <6D6174726963657320> Tj 1 0 0 1 222.24 508.56 Tm <61726520> Tj 1 0 0 1 238.80 508.80 Tm <726570726573656E746564> Tj 0.24 Tc 1 0 0 1 51.60 497.04 Tm <627920> Tj 1 0 0 1 64.32 497.28 Tm <6C696E65617220> Tj 1 0 0 1 91.68 497.28 Tm <6571756174696F6E7320> Tj 1 0 0 1 135.84 497.52 Tm <696E20> Tj 1 0 0 1 146.64 497.52 Tm <74686520> Tj 1 0 0 1 162.72 497.76 Tm <666F6C6C6F77696E6720> Tj 1 0 0 1 203.76 498.00 Tm <73797374656D3A> Tj 0.48 Tc 1 0 0 1 51.60 480.96 Tm <312720> Tj 1 0 0 1 59.28 480.96 Tm <302E323520> Tj 1 0 0 1 76.08 481.20 Tm <2B20> Tj 1 0 0 1 83.76 480.96 Tm <302E34392720> Tj 1 0 0 1 102.96 481.20 Tm <302E323520> Tj 1 0 0 1 119.76 481.44 Tm <2B20> Tj 1 0 0 1 127.20 481.20 Tm <302E34392720> Tj 1 0 0 1 146.16 481.44 Tm <302E323520> Tj 1 0 0 1 163.20 481.68 Tm <2B20> Tj 1 0 0 1 170.64 481.44 Tm <302E33312720> Tj 1 0 0 1 188.88 481.68 Tm <302E323520> Tj 1 0 0 1 205.92 481.92 Tm <2B20> Tj 1 0 0 1 213.60 481.92 Tm <4F273920> Tj 1 0 0 1 225.84 483.36 Tm <3D20> Tj 1 0 0 1 233.04 481.92 Tm <312E34373235> Tj 0.72 Tc 1 0 0 1 52.08 468.72 Tm <302E34392E302E323520> Tj 1 0 0 1 87.84 468.96 Tm <2B20> Tj 1 0 0 1 95.04 468.96 Tm <312E302E323520> Tj 1 0 0 1 119.76 469.44 Tm <2B20> Tj 1 0 0 1 127.20 469.20 Tm <30333120> Tj 1 0 0 1 142.32 471.60 Tm <27302E323520> Tj 1 0 0 1 162.48 469.44 Tm <2B20> Tj 1 0 0 1 169.92 469.44 Tm <302E343927302E323520> Tj 1 0 0 1 205.92 469.92 Tm <2B20> Tj 1 0 0 1 213.36 469.68 Tm <4F2E3920> Tj 1 0 0 1 225.84 471.12 Tm <3D312E34373235> Tj 0.48 Tc 1 0 0 1 52.08 456.72 Tm <302E34392E20> Tj 1 0 0 1 71.04 456.72 Tm <302E327320> Tj 1 0 0 1 87.84 456.96 Tm <2B20> Tj 1 0 0 1 95.52 456.96 Tm <302E333120> Tj 1 0 0 1 110.40 459.36 Tm <2720> Tj 1 0 0 1 113.76 456.96 Tm <302E323520> Tj 1 0 0 1 130.56 457.20 Tm <2B20> Tj 1 0 0 1 137.76 457.44 Tm <3127302E323520> Tj 1 0 0 1 162.48 457.44 Tm <2B20> Tj 1 0 0 1 169.92 457.44 Tm <302E34392720> Tj 1 0 0 1 189.12 457.68 Tm <302E323520> Tj 1 0 0 1 205.92 457.68 Tm <2B20> Tj 1 0 0 1 213.36 457.68 Tm <4F2E3920> Tj 1 0 0 1 225.84 459.12 Tm <3D20> Tj 1 0 0 1 233.04 457.92 Tm <312E3437323520> Tj 1 0 0 1 267.84 458.64 Tm <283427313229> Tj 0.48 Tc 1 0 0 1 52.08 444.72 Tm <302E333120> Tj 1 0 0 1 66.96 447.12 Tm <2E20> Tj 1 0 0 1 70.32 444.72 Tm <302E323520> Tj 1 0 0 1 87.12 444.96 Tm <2B20> Tj 1 0 0 1 94.80 444.96 Tm <302E34392720> Tj 1 0 0 1 113.76 444.96 Tm <302E323520> Tj 1 0 0 1 130.56 445.20 Tm <2B20> Tj 1 0 0 1 138.24 445.20 Tm <302E34392720> Tj 1 0 0 1 157.20 445.20 Tm <302E323520> Tj 1 0 0 1 174.00 445.44 Tm <2B20> Tj 1 0 0 1 181.20 445.44 Tm <312720> Tj 1 0 0 1 189.12 445.44 Tm <4F2E323520> Tj 1 0 0 1 205.92 445.68 Tm <2B20> Tj 1 0 0 1 213.36 445.68 Tm <302E3920> Tj 1 0 0 1 225.84 447.12 Tm <3D20> Tj 1 0 0 1 233.04 445.92 Tm <312E34373235> Tj 0.96 Tc 1 0 0 1 51.84 432.72 Tm <312E302E323520> Tj 1 0 0 1 76.32 432.72 Tm <2B20> Tj 1 0 0 1 83.52 432.96 Tm <312720> Tj 1 0 0 1 91.44 432.72 Tm <302E323520> Tj 1 0 0 1 108.24 432.96 Tm <2B20> Tj 1 0 0 1 115.44 432.96 Tm <312E302E323520> Tj 1 0 0 1 140.16 433.20 Tm <2B20> Tj 1 0 0 1 147.36 433.20 Tm <312E302E323520> Tj 1 0 0 1 171.84 433.44 Tm <2B20> Tj 1 0 0 1 179.28 433.44 Tm <302E302E3920> Tj 1 0 0 1 200.88 434.88 Tm <3D20> Tj 1 0 0 1 208.08 433.68 Tm <31> Tj 0.24 Tc 1 0 0 1 52.08 411.12 Tm <54686520> Tj 1 0 0 1 70.80 411.36 Tm <6B6467696E6720> Tj 1 0 0 1 103.20 411.36 Tm <76617269616E636520> Tj 1 0 0 1 142.08 411.60 Tm <666F7220> Tj 1 0 0 1 157.68 411.60 Tm <614C616772616E676520> Tj 1 0 0 1 205.20 411.84 Tm <76616C756520> Tj 1 0 0 1 230.40 412.08 Tm <6F6620> Tj 1 0 0 1 240.96 412.08 Tm <302D3920> Tj 1 0 0 1 257.28 412.56 Tm <69733A> Tj 0.72 Tc 1 0 0 1 51.84 393.12 Tm <6F5A7820> Tj 1 0 0 1 67.44 394.56 Tm <3A20> Tj 1 0 0 1 75.12 393.36 Tm <4F2E323527312E343720> Tj 1 0 0 1 110.16 393.60 Tm <2B20> Tj 1 0 0 1 117.84 393.60 Tm <4F2E323527312E343720> Tj 1 0 0 1 153.12 393.84 Tm <2B20> Tj 1 0 0 1 160.56 393.84 Tm <4F2E323520> Tj 1 0 0 1 177.84 396.24 Tm <27312E343720> Tj 1 0 0 1 195.84 394.08 Tm <2B20> Tj 1 0 0 1 203.52 394.08 Tm <4F323527312E343720> Tj 1 0 0 1 238.56 394.32 Tm <2B20> Tj 1 0 0 1 246.00 394.56 Tm <4F2E3920> Tj 1 0 0 1 258.48 395.76 Tm <3D312E32363735> Tj 0.96 Tc 1 0 0 1 268.08 386.40 Tm <28342E313329> Tj 0.24 Tc 1 0 0 1 52.32 367.68 Tm <342E332E332E20> Tj 1 0 0 1 81.60 367.68 Tm <4C616772616E676520> Tj 1 0 0 1 125.28 368.16 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 186.96 368.40 Tm <76616C756520> Tj 1 0 0 1 213.60 370.32 Tm <2D302E39> Tj 0.24 Tc 1 0 0 1 52.32 353.52 Tm <496E20> Tj 1 0 0 1 63.12 353.52 Tm <74686520> Tj 1 0 0 1 78.72 353.52 Tm <6C61737420> Tj 1 0 0 1 96.72 353.52 Tm <6578616D706C652C20> Tj 1 0 0 1 137.04 354.00 Tm <74686520> Tj 1 0 0 1 152.64 354.00 Tm <4C616772616E676520> Tj 1 0 0 1 193.68 354.24 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 251.28 354.72 Tm <697320> Tj 1 0 0 1 260.88 354.96 Tm <676976656E20> Tj 1 0 0 1 285.36 354.72 Tm <61> Tj 0.24 Tc 1 0 0 1 52.32 342.96 Tm <6E6567617469766520> Tj 1 0 0 1 88.56 342.96 Tm <76616C756520> Tj 1 0 0 1 113.52 343.20 Tm <6F6620> Tj 1 0 0 1 123.60 345.12 Tm <2D302E3920> Tj 1 0 0 1 144.96 344.40 Tm <286F757220> Tj 1 0 0 1 165.12 343.44 Tm <696E74656E74696F6E20> Tj 1 0 0 1 205.44 343.68 Tm <77617320> Tj 1 0 0 1 224.40 343.92 Tm <746F20> Tj 1 0 0 1 235.20 343.92 Tm <73656C65637420> Tj 1 0 0 1 261.60 343.92 Tm <6120> Tj 1 0 0 1 269.04 344.16 Tm <73696D692D> Tj 0.24 Tc 1 0 0 1 52.32 332.40 Tm <6C617220> Tj 1 0 0 1 67.44 332.40 Tm <76616C756520> Tj 1 0 0 1 93.84 332.40 Tm <617320> Tj 1 0 0 1 106.80 332.64 Tm <7468617420> Tj 1 0 0 1 127.44 332.64 Tm <6170706C69656420> Tj 1 0 0 1 162.48 332.88 Tm <696E20> Tj 1 0 0 1 174.72 332.88 Tm <53756273656374696F6E20> Tj 1 0 0 1 225.60 333.36 Tm <342E31292E20> Tj 1 0 0 1 248.64 333.60 Tm <5468656E2C20> Tj 1 0 0 1 276.96 333.60 Tm <746865> Tj 0.24 Tc 1 0 0 1 52.32 321.60 Tm <6B726967726E6720> Tj 1 0 0 1 84.72 321.84 Tm <6D6174726963657320> Tj 1 0 0 1 124.32 322.08 Tm <6172653A> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 52.32 233.76 Tm <417320> Tj 1 0 0 1 66.96 233.76 Tm <696E20> Tj 1 0 0 1 79.44 233.76 Tm <74686520> Tj 1 0 0 1 96.72 234.24 Tm <70726576696F757320> Tj 1 0 0 1 137.28 234.24 Tm <74776F20> Tj 1 0 0 1 156.72 234.24 Tm <6578616D706C65732C20> Tj 1 0 0 1 203.04 234.72 Tm <6D6174726963657320> Tj 1 0 0 1 244.32 233.04 Tm <5B2C315D20> Tj 1 0 0 1 258.24 234.96 Tm <616E6420> Tj 1 0 0 1 278.64 233.28 Tm <5B425D> Tj 0.24 Tc 1 0 0 1 52.56 223.20 Tm <6861766520> Tj 1 0 0 1 74.88 223.20 Tm <636F6E7374616E7420> Tj 1 0 0 1 114.00 223.44 Tm <636F76617269616E636520> Tj 1 0 0 1 161.28 223.68 Tm <76616C7565732E20> Tj 1 0 0 1 193.68 224.16 Tm <4D6F72656F7665722C20> Tj 1 0 0 1 238.32 224.16 Tm <696620> Tj 1 0 0 1 246.48 224.16 Tm <74686520> Tj 1 0 0 1 262.56 224.40 Tm <6D6174726978> Tj 0.24 Tc 1 0 0 1 52.80 212.40 Tm <697320> Tj 1 0 0 1 62.88 212.40 Tm <737562737469747574656420> Tj 1 0 0 1 113.28 212.64 Tm <7769746820> Tj 1 0 0 1 134.64 213.12 Tm <6C696E65617220> Tj 1 0 0 1 162.48 213.12 Tm <6571756174696F6E732C20> Tj 1 0 0 1 209.28 213.36 Tm <697420> Tj 1 0 0 1 217.92 213.36 Tm <6265636F6D657320> Tj 1 0 0 1 258.00 213.60 Tm <74686520> Tj 1 0 0 1 274.32 213.60 Tm <7379732D> Tj 0.48 Tc 1 0 0 1 52.80 201.84 Tm <74656D3A> Tj 0.48 Tc 1 0 0 1 52.56 185.52 Tm <3120> Tj 1 0 0 1 57.12 187.92 Tm <2E20> Tj 1 0 0 1 60.24 185.52 Tm <302E323520> Tj 1 0 0 1 77.04 185.76 Tm <2B20> Tj 1 0 0 1 84.72 185.52 Tm <302E34392E20> Tj 1 0 0 1 103.92 185.76 Tm <302E323520> Tj 1 0 0 1 120.72 186.00 Tm <2B20> Tj 1 0 0 1 128.16 186.00 Tm <302E34392E20> Tj 1 0 0 1 147.36 186.00 Tm <302E323520> Tj 1 0 0 1 164.16 186.24 Tm <2B20> Tj 1 0 0 1 171.60 186.24 Tm <302E33312720> Tj 1 0 0 1 189.84 186.24 Tm <302E323520> Tj 1 0 0 1 206.88 186.48 Tm <2B20> Tj 1 0 0 1 214.08 186.96 Tm <282D302E392920> Tj 1 0 0 1 237.60 188.16 Tm <3D20> Tj 1 0 0 1 246.00 188.88 Tm <2D302E33323735> Tj 0.48 Tc 1 0 0 1 53.04 173.28 Tm <6F20> Tj 1 0 0 1 58.56 173.52 Tm <34392E20> Tj 1 0 0 1 72.00 173.52 Tm <302E323520> Tj 1 0 0 1 88.80 173.52 Tm <2B20> Tj 1 0 0 1 96.00 173.52 Tm <312E20> Tj 1 0 0 1 103.92 173.52 Tm <302E323520> Tj 1 0 0 1 120.72 173.76 Tm <2B20> Tj 1 0 0 1 128.16 173.76 Tm <302E333120> Tj 1 0 0 1 143.28 176.16 Tm <2E20> Tj 1 0 0 1 146.64 173.76 Tm <302E323520> Tj 1 0 0 1 163.44 174.00 Tm <2B20> Tj 1 0 0 1 170.88 174.00 Tm <3020> Tj 1 0 0 1 176.64 174.24 Tm <34392E20> Tj 1 0 0 1 190.08 174.24 Tm <302E323520> Tj 1 0 0 1 206.88 174.24 Tm <2B20> Tj 1 0 0 1 214.08 174.72 Tm <282D302E392920> Tj 1 0 0 1 237.84 177.36 Tm <2D2D20> Tj 1 0 0 1 246.00 176.64 Tm <2D302E33323735> Tj 0.48 Tc 1 0 0 1 53.04 161.28 Tm <302E34392E20> Tj 1 0 0 1 72.00 161.28 Tm <302E323520> Tj 1 0 0 1 88.80 161.52 Tm <2B20> Tj 1 0 0 1 96.48 161.28 Tm <302E333120> Tj 1 0 0 1 111.36 163.92 Tm <2E20> Tj 1 0 0 1 114.72 161.52 Tm <302E323520> Tj 1 0 0 1 131.52 161.76 Tm <2B20> Tj 1 0 0 1 138.72 161.76 Tm <3120> Tj 1 0 0 1 143.28 164.16 Tm <2E20> Tj 1 0 0 1 146.64 161.76 Tm <302E323520> Tj 1 0 0 1 163.44 162.00 Tm <2B20> Tj 1 0 0 1 170.88 161.76 Tm <302E34392720> Tj 1 0 0 1 190.08 162.00 Tm <302E323520> Tj 1 0 0 1 206.88 162.24 Tm <2B20> Tj 1 0 0 1 214.32 162.72 Tm <282D302E392920> Tj 1 0 0 1 237.84 163.68 Tm <3D20> Tj 1 0 0 1 246.00 164.64 Tm <2D302E33323735> Tj 0.48 Tc 1 0 0 1 53.04 149.04 Tm <302E33312720> Tj 1 0 0 1 71.28 149.28 Tm <302E323520> Tj 1 0 0 1 88.08 149.28 Tm <2B20> Tj 1 0 0 1 95.76 149.28 Tm <302E34392720> Tj 1 0 0 1 114.72 149.52 Tm <302E323520> Tj 1 0 0 1 131.52 149.52 Tm <2B20> Tj 1 0 0 1 139.20 149.52 Tm <3020> Tj 1 0 0 1 144.72 149.76 Tm <34392720> Tj 1 0 0 1 158.16 149.76 Tm <302E323520> Tj 1 0 0 1 174.96 149.76 Tm <2B20> Tj 1 0 0 1 182.16 150.00 Tm <312720> Tj 1 0 0 1 190.08 149.76 Tm <302E323520> Tj 1 0 0 1 206.88 150.00 Tm <2B20> Tj 1 0 0 1 214.32 150.96 Tm <282D302E392920> Tj 1 0 0 1 237.84 151.68 Tm <3D2D20> Tj 1 0 0 1 250.80 150.24 Tm <4F2733323735> Tj 1.20 Tc 1 0 0 1 52.80 137.04 Tm <312E302E323520> Tj 1 0 0 1 77.28 137.28 Tm <2B312E302E323520> Tj 1 0 0 1 109.20 137.52 Tm <2B20> Tj 1 0 0 1 116.40 137.52 Tm <312E302E323520> Tj 1 0 0 1 141.12 137.76 Tm <2B20> Tj 1 0 0 1 148.32 137.76 Tm <312E302E323520> Tj 1 0 0 1 172.80 137.76 Tm <2B20> Tj 1 0 0 1 180.24 137.76 Tm <3027282D302E392920> Tj 1 0 0 1 212.88 139.44 Tm <3D31> Tj 0.72 Tc 1 0 0 1 268.80 126.96 Tm <28342E313529> Tj 0.24 Tc 1 0 0 1 53.04 106.32 Tm <54686520> Tj 1 0 0 1 72.00 106.32 Tm <6B726967696E6720> Tj 1 0 0 1 104.16 106.32 Tm <76617269616E636520> Tj 1 0 0 1 142.80 106.80 Tm <697320> Tj 1 0 0 1 152.64 106.80 Tm <7468656E3A> Tj 0.48 Tc 1 0 0 1 53.52 83.28 Tm <4E4146544120> Tj 1 0 0 1 73.44 83.28 Tm <353920> Tj 1 0 0 1 82.80 84.00 Tm <283120> Tj 1 0 0 1 88.32 82.08 Tm <2920> Tj 1 0 0 1 92.16 83.52 Tm <333120> Tj 1 0 0 1 99.60 85.20 Tm <2D333720> Tj 1 0 0 1 111.12 84.00 Tm <283230303929> Tj 0.48 Tc /F3 11.0 Tf 1 0 0 1 534.72 85.44 Tm <3335> Tj ET endstream endobj 50 0 obj 36437 endobj 51 0 obj << /Length 52 0 R >> stream | 1.0I 0.4ei 0.4elnari*"'L 1.0 1.0.25 + 0.49.0.25 + 0.49.0.25 + 0.31 '0.25 + 0.06 = 0.63250.49'0.25 + 1'0.25 + 0.31 .0.25 + 0.49'0.25 + 0.06 = 0.63250.49. 0.25 + 0.31.0.25 + 1'0.25 + O.49. 0.25 + 0.06 = 0.6325 (4.9)0.31' 0.25 + 0.49'0.25 + 0.49' 0.25 + 1'0.25 + 0.06 = 063251. O.25 + 1. 025 + 1. 025 + 1. 0.25 + 0' 0.06 = 1Now, it is possible to calculate the weiglting coefficientand the error variance:oZx :0.25 '0.63 + 0.25'0.63 + 0.25'0.63 + 0.25 0.63 + 0.06 = 0.69(4.10)4.3.2. Lagrange multiplicator value O.9In this example, the Lagrange multiplicator is given aposifive vale of 0.9. In such cases, the covariance matrixls: oZx :0.2s.(-0.3275) + 0.25 '(-0.3275) + 0.25 .(-0.3275\ +0.25. (-0.3275) + (-0.9) = -0.982(4.16)All three examples presented in Subsection 4.3 clearlyshow that the Lagrange value of 0.06 (as given in ref.S andshown in Figure 2) is the value at which the estimationvariance of Ordinary Iftiging is minimal. The reason forthis and how we can select the most appropriate value forthe Lagrange multiplicator are analysed in the next chap-ters, the Discussion and Conclusion of the analysis.5. DISCUSSION OF THEI\{ATHEI\,IATICAL MEANINGS OFTHE LAGRANGE MULTIPLICATORThe results of Chapter 4 clearly show that in the applica-tion of ordinary kriging, as a statistical procedure for theestirnation of geological or other variables, the value ofthe Lagrange multiplicator plays a critical role in the finalresult.The two accepted va-lues of the Lagrange multiplicatorused to minimize the variance in the previous examplesdiffered, with one negative (-0.9319) and the other posi-tive (0.06). The proof that both values are correct isshown by varying of value 0.06 in the kriging matrices(Subsection 4.3). A change in this value induced an in-crease in the estimation variance (Subsection 4.3.2) or anegative (mathematically impossible ) variance ( Subsec-tion 4.3.3).Let us examine once again the mathematical meaningof the Lagrange multiplicator in kriging equations. Thevalue at the estimation point can be expressed as: Z(r)=p+e(r) (5.1) wherep is the correct (but really unknown) mean of the entirepopulation;e(r) is the curve and mean value normalized about the value0 on the curve ofthe spatial random function.Usually, we do not know the real mean value of the entirepopulation (p). Therefore, the estimation is more oftenperformed with the Ordinary Kriging technique than withSimple Kriging, using only the local mean based on dataencompassed by the search radius around the estirna-tion point. Moreover, second-order stationarity is alsoassumed for krigingwith the covariance C"of t}re estima-tion is Z(x). ^this stationarity can be expressed:C,(r.,+) = E[e(r,) 'e(r,)] = C,(r,- d (5.2)This condition can be described with a variogram (in-stead of covariance) when the stationarity is presentedas:y,(\,t,) = 'ELe(r,) -(r,)l- =vz(\-r) (5.3)zThe array of N measurements Z(xt)...2(x") at locationsxr. . .x. is assumed. Three conditions must be satisfied foran Ordinary Kriging estirnation, based on the value of thevariable Z (Z) at any unsampled location xo: 0.49 0.49 0.311.0 0.31 0.490.31 1.0 0.490.49 0.49 1.01.0 1.0 1.0 1.0 0.49 0.49 0.310.49 1.0 0.31 0.49049 0.31 1.0 0.490.31 0.49 0.49 1.0't.0 1.0 1.0 1.0 (4.1 1) (4.14) Again, the matrices [],1 and [B] contain constant values,as in Subsection 4.3.1. These matrices are representedby linear equations in the following system:1' 0.25 + 0.49' 0.25 + 0.49' 0.25 + 0.31' 0.25 + O'9 = 1.47250.49.0.25 + 1.0.25 + 031 '0.25 + 0.49'0.25 + O.9 =1.47250.49. 0.2s + 0.31 ' 0.25 + 1'0.25 + 0.49' 0.25 + O.9 = 1.4725 (4'12)0.31 . 0.25 + 0.49' 0.25 + 0.49' 0.25 + 1' O.25 + 0.9 = 1.47251.0.25 + 1' 0.25 + 1.0.25 + 1.0.25 + 0.0.9 = 1The kdging variance for aLagrange value of 0-9 is:oZx : O.25'1.47 + O.25'1.47 + O.25 '1.47 + O25'1.47 + O.9 =1.2675(4.13)4.3.3. Lagrange multiplicator value -0.9In the last example, the Lagrange multiplicator is given anegative value of -0.9 (our intention was to select a simi-lar value as that applied in Subsection 4.1). Then, thekrigrng matrices are: As in the previous two examples, matrices [,1] and [B]have constant covariance values. Moreover, if the matrixis substituted with linear equations, it becomes the sys-tem:1 . 0.25 + 0.49. 0.25 + 0.49. 0.25 + 0.31' 0.25 + (-0.9) = -0.3275o 49. 0.25 + 1. 0.25 + 0.31 . 0.25 + 0 49. 0.25 + (-0.9) -- -0.32750.49. 0.25 + 0.31 . 0.25 + 1 . 0.25 + 0.49' 0.25 + (-0.9) = -0.32750.31' 0.25 + 0.49' 0.25 + 0 49' 0.25 + 1' 0.25 + (-0.9) =- O'32751.0.25 +1.0.25 + 1.0.25 + 1.0.25 + 0'(-0.9) =1(4.15)The kriging variance is then:NAFTA 59 (1 ) 31 -37 (2009) 35 endstream endobj 52 0 obj 4607 endobj 53 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 54 0 obj << /Filter /FlateDecode /Length 108 >> stream xM 0Oئs+!""Amk&R$5 |m`Z3)7 ɑ@Z*UFcƏ3v#eCY>d/z] endstream endobj 55 0 obj << /Type /Page /MediaBox [ 0 0 595.20 841.68 ] /Parent 13 0 R /Resources << /Font << /F3 7 0 R >> /XObject << /Obj46 46 0 R /Obj47 47 0 R /Obj48 48 0 R >> /Subtype 51 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 53 0 R 54 0 R 49 0 R ] >> endobj 56 0 obj << /Type /XObject /Subtype /Image /Width 1240 /Height 1753 /BitsPerComponent 8 /ColorSpace /DeviceGray /Filter /DCTDecode /Length 46024 >> stream JFIF (B+(%%(Q:=0B`Ued_U][jxjqs[]gɺǙ(#(N++Nn]n  }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyzw!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?΃NlORTt_?G_Ώ6?G͏}:<j}:<j}:<?x~tbx~tyѽ?:7QKzѸzѸQFh4fњ3Fh4fњ3Fh444f4fњ3Fi3K3Fh4fњ3Fh4fњ3Ef3Fh3Fh%.i3@4fњ3Fhњ3E44fњ3%ffE4QEQKKE%:#)#*sx/(sJ(/4Qڗ>gџz\Tf4RFiE4RK>QE-QGKEQ\њ3KFO.O>oSF?t{xt{xty?K?̓I|?ߝdoΏ6Oi|?~f:Oߝtѿ:O:Oߝtѿ:<~tyFQ|hD4}o>7SOQ(T} >?k}~ >?GglzʏM꿕n*>7?o*?&O/O'O/O/SFOI'ғGq:?yΏ7cMo??yP}?Q?֣Wj??֣[j??֤?Zq<xK?Z]_?xG&Oߝ<׏mQҏ真? r~Ch؃CiX俐_obOʏ{oWi;Z4jZ?iSۑQAo<RKE-s<4a'}yQ+1GخcGcG}yQ;ߘ?wɿ1G.?瑣SQI瑣#GgmG٧G&<Ϳ*_<6<*dѓ4>ttnoiw~?.?.|}?GoΗGoΏ5tyG|?F6Oѣ͓z?i|?F6Oѣ͓z?hd}_:_ѣΓz7i|~f:_ߝi?~ttѿ:_/G%D?iLi3}_Q O >/hGR}Eh?GQ}ȤCr?Q<Eh?/QGL)<cGL`(?x0x0yxE|Ə=rgD'hb }O6_>4yϰ'CGCG?"<|ǿD4eh-Go>gJ4QޓRQIE9~IjrՁNo<4RޖҖ(>c?>xL,NAD6'ҖhZ J-E\nXg>Ne+'@7` z42Hqigrӭ"i9l Jw4@ 88GB){SJ1V)S{:'zGBFCREgR:vC%ʌ(bI  (J֢?-RE-QKEQE-RE-QKEQKEQE%QE%RQEF24X Nz{IҬr0ʣ!V ѵ=1ކe n`=@'(?&┫~OfQIE'RQE%QE%RQIEPi?'GJJ(CA-ެ Z+?[%|#X4R){=iE5mQ+ddf dn JLRH>P$>g̠}{?%9rJ|F㼪/ҀXly8*D 9'TmXG%s)E<})쫱hzwjs2!T*O9PhC,2#qRށaC1t ʂ5)oqQ(1$j8J83 g^B3˹?J#mVRdr Ev aF3i"a4&DQn֜~rvgƢUY =qP(Pi('䜞`Ss<`Eh~4PF|W{/Fs/?j@"N ItWQfqp$ѷ 5q2%qJdQۆ9e\,(҇u$O8|ǧMqSʃ 6bKIridcS̓;4xᅳW2ɍL=rУa_R>} 7BoU?<_[}[}Ѱ}aOchѰ?ƍ?4BFУaU.꿘aaKɣcz~loCF4F?.~TlF~Tl7KߕQߕ]6?cq*6?cq*M?&?tT>T`ӷFB8i#qЌh5mIF|PA94$MWۜcE C>rRSn? 4y1z{R͸Ǔ!Du(x垹ԂF[9ޅBdӥ( L$ѶgTWK*]Fdy AH!) 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<70726F76656E20> Tj 1 0 0 1 338.16 673.68 Tm <7468617420> Tj 1 0 0 1 358.56 673.92 Tm <67656F737461746973746963616C20> Tj 1 0 0 1 417.60 673.20 Tm <696E746572706F6C6174696F6E20> Tj 1 0 0 1 476.88 673.44 Tm <28617320> Tj 1 0 0 1 492.72 672.48 Tm <77656C6C20> Tj 1 0 0 1 513.84 672.24 Tm <617320> Tj 1 0 0 1 527.04 672.24 Tm <73746F2D> Tj 0.24 Tc 1 0 0 1 49.92 662.16 Tm <556E6269617365646E65737320> Tj 1 0 0 1 113.52 662.64 Tm <28636F6E646974696F6E20> Tj 1 0 0 1 161.04 661.68 Tm <322920> Tj 1 0 0 1 173.76 661.68 Tm <697320> Tj 1 0 0 1 185.28 661.68 Tm <66756C66696C6C656420> Tj 1 0 0 1 222.00 661.68 Tm <696620> Tj 1 0 0 1 231.84 661.44 Tm <74686520> Tj 1 0 0 1 249.60 661.44 Tm <666F6C6C6F77696E6720> Tj 1 0 0 1 304.80 663.12 Tm <6368617374696320> Tj 1 0 0 1 336.96 663.12 Tm <65737469726E6174696F6E2920> Tj 1 0 0 1 387.12 662.88 Tm <697320> Tj 1 0 0 1 396.72 662.64 Tm <74686520> Tj 1 0 0 1 412.32 662.64 Tm <6265737420> Tj 1 0 0 1 432.72 662.40 Tm <617070726F61636820> Tj 1 0 0 1 475.20 661.92 Tm <746F20> Tj 1 0 0 1 486.00 661.92 Tm <6D617070696E6720> Tj 1 0 0 1 525.36 662.16 Tm <67656F2D> Tj 0.48 Tc 1 0 0 1 49.92 651.60 Tm <6571756174696F6E20> Tj 1 0 0 1 89.28 651.36 Tm <697320> Tj 1 0 0 1 99.12 651.36 Tm <76616C69643A20> Tj 1 0 0 1 304.56 652.80 Tm <6C6F676963616C20> Tj 1 0 0 1 334.08 652.56 Tm <7661726961626C657320> Tj 1 0 0 1 375.84 653.04 Tm <28652E672E2C20> Tj 1 0 0 1 397.92 652.08 Tm <4965666572656E63657320> Tj 1 0 0 1 444.48 654.24 Tm <312C382C652720> Tj 1 0 0 1 464.88 654.00 Tm <31302C20> Tj 1 0 0 1 475.20 654.00 Tm <3132292E> Tj 0.48 Tc /F3 12.0 Tf 1 0 0 1 49.44 628.32 Tm <2C6C327420> Tj 1 0 0 1 67.68 620.88 Tm <226C6C20> Tj 1 0 0 1 80.40 629.52 Tm <3D20> Tj 1 0 0 1 87.60 628.32 Tm <7020> Tj 1 0 0 1 94.80 628.80 Tm <2B20> Tj 1 0 0 1 104.16 626.16 Tm <743420> Tj 1 0 0 1 111.84 629.28 Tm <3D20> Tj 1 0 0 1 119.28 624.48 Tm <695E2C20> Tj 1 0 0 1 138.48 630.24 Tm <2E20> Tj 1 0 0 1 141.60 627.84 Tm <723772312C2C272C3120> Tj 1 0 0 1 169.68 628.32 Tm <2D> Tj 1.20 Tc /F3 5.0 Tf 1 0 0 1 119.28 619.68 Tm <693D37> Tj 0.24 Tc /F3 8.0 Tf 1 0 0 1 49.44 600.24 Tm <4D696E696D697A696E6720> Tj 1 0 0 1 98.88 600.00 Tm <74686520> Tj 1 0 0 1 114.00 600.00 Tm <6572726F7220> Tj 1 0 0 1 138.48 600.72 Tm <28636F6E646974696F6E20> Tj 1 0 0 1 183.36 599.76 Tm <332920> Tj 1 0 0 1 193.92 599.76 Tm <64656D616E647320> Tj 1 0 0 1 234.72 599.52 Tm <74686520> Tj 1 0 0 1 250.08 599.52 Tm <73656C656374696F6E> Tj 0.24 Tc 1 0 0 1 49.44 589.20 Tm <6F6620> Tj 1 0 0 1 60.24 588.96 Tm <74686520> Tj 1 0 0 1 76.32 588.96 Tm <6D6F737420> Tj 1 0 0 1 100.56 588.72 Tm <617070726F70726961746520> Tj 1 0 0 1 153.36 588.72 Tm <76616C75657320> Tj 1 0 0 1 182.88 588.72 Tm <666F7220> Tj 1 0 0 1 198.00 588.48 Tm <74686520> Tj 1 0 0 1 214.32 588.48 Tm <636F656666696369656E747320> Tj 1 0 0 1 264.48 588.24 Tm <6C2E722C20> Tj 1 0 0 1 276.24 588.24 Tm <2E20> Tj 1 0 0 1 279.12 588.24 Tm <2E20> Tj 1 0 0 1 282.24 588.24 Tm <2E20> Tj 1 0 0 1 285.12 587.28 Tm <2C> Tj 0.24 Tc 1 0 0 1 48.48 578.64 Tm <292E6E616E6420> Tj 1 0 0 1 76.80 578.40 Tm <74686520> Tj 1 0 0 1 91.92 578.40 Tm <4C616772616E676520> Tj 1 0 0 1 132.48 578.16 Tm <6D756C666C706C696361746F7220> Tj 1 0 0 1 189.60 578.16 Tm <326D2E546865697220> Tj 1 0 0 1 233.52 577.68 Tm <6F7074696D697A6174696F6E> Tj 0.24 Tc 1 0 0 1 49.44 567.84 Tm <63616E20> Tj 1 0 0 1 71.28 567.84 Tm <626520> Tj 1 0 0 1 89.04 567.84 Tm <65787072657373656420> Tj 1 0 0 1 138.72 567.60 Tm <666F7220> Tj 1 0 0 1 158.40 567.60 Tm <6561636820> Tj 1 0 0 1 185.04 567.84 Tm <706F696E7420> Tj 1 0 0 1 214.80 567.36 Tm <617320> Tj 1 0 0 1 231.36 567.36 Tm <74686520> Tj 1 0 0 1 252.00 567.12 Tm <66756E6374696F6E> Tj 1.20 Tc 1 0 0 1 48.72 557.28 Tm <4C28662C2C2C2E2E2E2C292E5E2C6D293A> Tj 0.48 Tc 1 0 0 1 270.72 531.60 Tm <28352E3529> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 48.72 503.52 Tm <54686520> Tj 1 0 0 1 69.36 503.28 Tm <6F7074696D697A6174696F6E20> Tj 1 0 0 1 127.20 503.04 Tm <6F6620> Tj 1 0 0 1 139.44 503.04 Tm <74686520> Tj 1 0 0 1 157.20 503.04 Tm <636F656666696369656E747320> Tj 1 0 0 1 209.28 503.76 Tm <282C31722C2E2E2E2C20> Tj 1 0 0 1 237.12 502.80 Tm <2C7472222920> Tj 1 0 0 1 253.20 502.56 Tm <616E6420> Tj 1 0 0 1 273.36 502.56 Tm <746865> Tj 0.24 Tc 1 0 0 1 48.72 492.96 Tm <4C616772616E676520> Tj 1 0 0 1 89.28 492.72 Tm <76616C756520> Tj 1 0 0 1 114.72 493.44 Tm <286D2920> Tj 1 0 0 1 131.28 492.72 Tm <697320> Tj 1 0 0 1 141.36 492.48 Tm <616368696576656420> Tj 1 0 0 1 180.00 492.48 Tm <627920> Tj 1 0 0 1 192.72 492.48 Tm <736F6C76696E6720> Tj 1 0 0 1 224.64 492.24 Tm <74686520> Tj 1 0 0 1 240.48 492.24 Tm <646966666572656E7469616C> Tj 0.24 Tc 1 0 0 1 90.24 484.08 Tm <414C> Tj 0.72 Tc 1 0 0 1 48.72 477.60 Tm <6571756174696F6E20> Tj 1 0 0 1 90.96 471.12 Tm <41553D4F2E> Tj 0.24 Tc 1 0 0 1 48.48 454.56 Tm <496620> Tj 1 0 0 1 57.12 454.32 Tm <776520> Tj 1 0 0 1 71.76 454.32 Tm <6C6F6F6B20> Tj 1 0 0 1 93.60 454.08 Tm <616761696E20> Tj 1 0 0 1 119.04 454.08 Tm <617420> Tj 1 0 0 1 130.08 454.08 Tm <74686520> Tj 1 0 0 1 146.16 453.84 Tm <626173696320> Tj 1 0 0 1 171.36 453.84 Tm <6B726967696E6720> Tj 1 0 0 1 204.24 453.84 Tm <6D6174726963657320> Tj 1 0 0 1 244.56 454.56 Tm <284571756174696F6E> Tj 0.24 Tc 1 0 0 1 48.48 443.76 Tm <332E332920> Tj 1 0 0 1 68.40 443.76 Tm <616E6420> Tj 1 0 0 1 87.60 443.52 Tm <73686F7720> Tj 1 0 0 1 112.56 443.52 Tm <7468656D20> Tj 1 0 0 1 137.52 443.52 Tm <7573696E6720> Tj 1 0 0 1 163.68 443.28 Tm <636F76616669616E6365732C20> Tj 1 0 0 1 219.12 443.04 Tm <74686520> Tj 1 0 0 1 235.44 443.04 Tm <6B726967696E6720> Tj 1 0 0 1 268.56 442.80 Tm <657374692D> Tj 0.24 Tc 1 0 0 1 48.24 433.20 Tm <6D6174696F6E20> Tj 1 0 0 1 80.64 433.20 Tm <69733A> Tj 0.48 Tc /F3 7.0 Tf 1 0 0 1 48.24 414.96 Tm <432E20> Tj 1 0 0 1 58.56 414.96 Tm <292C6F20> Tj 1 0 0 1 69.12 416.16 Tm <3D20> Tj 1 0 0 1 76.80 414.72 Tm <366F> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 270.00 414.24 Tm <28352E3629> Tj 0.24 Tc 1 0 0 1 48.00 395.76 Tm <54686520> Tj 1 0 0 1 66.48 395.52 Tm <7661726961626C657320> Tj 1 0 0 1 108.00 395.28 Tm <4320> Tj 1 0 0 1 117.12 395.28 Tm <616E6420> Tj 1 0 0 1 136.08 395.28 Tm <436F20> Tj 1 0 0 1 148.56 395.28 Tm <63616E20> Tj 1 0 0 1 165.84 395.04 Tm <626520> Tj 1 0 0 1 178.56 395.52 Tm <70726573656E74656420> Tj 1 0 0 1 223.44 395.04 Tm <617320> Tj 1 0 0 1 235.44 395.04 Tm <6D617472696365733A> Tj 1.50 Tc /F3 11.0 Tf 1 0 0 1 47.76 354.24 Tm <633D> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 269.52 353.76 Tm <28352E3729> Tj 0.48 Tc 1 0 0 1 269.04 286.32 Tm <28352E3829> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 46.80 240.96 Tm <54686520> Tj 1 0 0 1 65.28 240.72 Tm <657870656374656420> Tj 1 0 0 1 104.16 240.48 Tm <776569676874696E6720> Tj 1 0 0 1 146.64 240.24 Tm <636F656666696369656E7420> Tj 1 0 0 1 191.76 240.24 Tm <63616E20> Tj 1 0 0 1 208.56 240.00 Tm <7468656E20> Tj 1 0 0 1 229.44 240.00 Tm <626520> Tj 1 0 0 1 241.92 240.00 Tm <63616C63756C61746564> Tj 0.48 Tc 1 0 0 1 47.28 231.12 Tm <287573696E6720> Tj 1 0 0 1 75.36 230.16 Tm <4571756174696F6E20> Tj 1 0 0 1 117.12 229.92 Tm <352E362920> Tj 1 0 0 1 136.80 229.68 Tm <61733A> Tj 0.72 Tc /F3 10.0 Tf 1 0 0 1 76.80 378.48 Tm <6328302920> Tj 1 0 0 1 109.44 378.48 Tm <6328342D7A2920> Tj 1 0 0 1 167.04 378.24 Tm <6328342D322C2920> Tj 1 0 0 1 209.52 378.24 Tm <31> Tj 0.72 Tc 1 0 0 1 67.20 366.48 Tm <632832322D32312920> Tj 1 0 0 1 118.56 366.24 Tm <6328302920> Tj 1 0 0 1 166.08 366.00 Tm <632832322D322C2920> Tj 1 0 0 1 209.28 366.00 Tm <31> Tj 0.72 Tc 1 0 0 1 66.96 342.48 Tm <6328322C2D32312920> Tj 1 0 0 1 108.48 342.24 Tm <63287A222D7A2920> Tj 1 0 0 1 176.16 342.00 Tm <6328302920> Tj 1 0 0 1 209.28 341.76 Tm <31> Tj 1.50 Tc 1 0 0 1 82.08 330.24 Tm <31313130> Tj 0.96 Tc /F3 9.0 Tf 1 0 0 1 46.80 211.20 Tm <696F20> Tj 1 0 0 1 57.12 212.64 Tm <3D20> Tj 1 0 0 1 64.80 211.20 Tm <432D2727436F> Tj 0.24 Tc 1 0 0 1 46.32 189.84 Tm <54686520> Tj 1 0 0 1 65.52 189.60 Tm <657374696D6174696F6E20> Tj 1 0 0 1 112.80 189.36 Tm <617420> Tj 1 0 0 1 123.36 189.36 Tm <74686520> Tj 1 0 0 1 139.68 189.36 Tm <73656C656374656420> Tj 1 0 0 1 175.92 189.60 Tm <706F696E7420> Tj 1 0 0 1 200.88 189.12 Tm <69733A> Tj 0.96 Tc 1 0 0 1 47.04 175.68 Tm <3A2C> Tj 0.72 Tc 1 0 0 1 46.32 170.88 Tm <5A28786F3120> Tj 1 0 0 1 66.48 172.08 Tm <3D20> Tj 1 0 0 1 73.92 170.64 Tm <312C2E20> Tj 1 0 0 1 84.96 170.64 Tm <3731722C292B2E2E2E2B20> Tj 1 0 0 1 120.48 170.40 Tm <292C2C20> Tj 1 0 0 1 130.08 172.56 Tm <2E20> Tj 1 0 0 1 132.96 170.40 Tm <5A28782C6C> Tj 0.24 Tc 1 0 0 1 311.52 639.12 Tm <426173656420> Tj 1 0 0 1 339.60 638.88 Tm <6F6E20> Tj 1 0 0 1 352.08 638.64 Tm <7375636820> Tj 1 0 0 1 374.16 638.64 Tm <616E616C797365732C20> Tj 1 0 0 1 414.72 638.40 Tm <697420> Tj 1 0 0 1 422.40 638.40 Tm <697320> Tj 1 0 0 1 431.76 638.16 Tm <636F6E636C7564656420> Tj 1 0 0 1 477.36 637.68 Tm <7468617420> Tj 1 0 0 1 495.84 637.68 Tm <6B726967696E6720> Tj 1 0 0 1 527.76 637.44 Tm <63616E> Tj 0.24 Tc 1 0 0 1 304.32 628.32 Tm <626520> Tj 1 0 0 1 318.24 628.32 Tm <75736566756C6C7920> Tj 1 0 0 1 355.20 628.08 Tm <6170706C69656420> Tj 1 0 0 1 390.48 627.84 Tm <746F20> Tj 1 0 0 1 402.96 627.84 Tm <636173657320> Tj 1 0 0 1 429.12 627.60 Tm <7769746820> Tj 1 0 0 1 451.92 627.36 Tm <6120> Tj 1 0 0 1 460.56 627.36 Tm <6D696E696D756D20> Tj 1 0 0 1 506.40 627.12 Tm <6F6620> Tj 1 0 0 1 519.36 627.12 Tm <6C4F20> Tj 1 0 0 1 533.28 626.88 Tm <6F72> Tj 0.24 Tc 1 0 0 1 304.56 617.76 Tm <6576656E20> Tj 1 0 0 1 326.64 617.76 Tm <313520> Tj 1 0 0 1 338.88 617.76 Tm <6461746120> Tj 1 0 0 1 359.28 618.00 Tm <706F696E74732E20> Tj 1 0 0 1 390.72 617.28 Tm <486F77657665722C20> Tj 1 0 0 1 431.52 616.80 Tm <6172727920> Tj 1 0 0 1 448.32 616.80 Tm <726570726573656E74617469766520> Tj 1 0 0 1 510.72 616.32 Tm <73746174697374692D> Tj 0.24 Tc 1 0 0 1 304.56 607.20 Tm <63616C20> Tj 1 0 0 1 319.92 607.20 Tm <6461746173657420> Tj 1 0 0 1 354.00 606.96 Tm <6F6620> Tj 1 0 0 1 365.04 607.20 Tm <67656F6C6F676963616C20> Tj 1 0 0 1 409.44 606.48 Tm <7661726961626C657320> Tj 1 0 0 1 451.44 606.24 Tm <6D75737420> Tj 1 0 0 1 476.64 606.00 Tm <696E636C75646520> Tj 1 0 0 1 511.20 605.76 Tm <617420> Tj 1 0 0 1 522.48 605.76 Tm <6C65617374> Tj 0.24 Tc 1 0 0 1 304.56 596.64 Tm <333020> Tj 1 0 0 1 318.24 596.40 Tm <6461746120> Tj 1 0 0 1 338.88 596.88 Tm <706F696E74732E20> Tj 1 0 0 1 371.04 596.40 Tm <5468697320> Tj 1 0 0 1 392.64 596.16 Tm <6D65616E7320> Tj 1 0 0 1 423.36 595.92 Tm <7468617420> Tj 1 0 0 1 442.56 595.44 Tm <776520> Tj 1 0 0 1 456.96 595.44 Tm <63616E6E6F7420> Tj 1 0 0 1 488.40 595.20 Tm <636F6E636C75646520> Tj 1 0 0 1 528.96 594.96 Tm <746865> Tj 0.24 Tc 1 0 0 1 303.84 585.84 Tm <76616C756520> Tj 1 0 0 1 330.72 585.84 Tm <6F6620> Tj 1 0 0 1 342.72 585.84 Tm <74686520> Tj 1 0 0 1 360.00 585.84 Tm <7265616C20> Tj 1 0 0 1 380.64 586.08 Tm <706F70756C6174696F6E20> Tj 1 0 0 1 430.80 585.12 Tm <6D65616E20> Tj 1 0 0 1 458.88 585.84 Tm <286578706563746174696F6E2920> Tj 1 0 0 1 516.96 584.64 Tm <6261736564> Tj 0.24 Tc 1 0 0 1 304.32 575.52 Tm <6F6E6C7920> Tj 1 0 0 1 325.68 575.28 Tm <6F6E20> Tj 1 0 0 1 339.84 575.28 Tm <74686520> Tj 1 0 0 1 356.88 575.28 Tm <757375616C6C7920> Tj 1 0 0 1 390.96 575.04 Tm <617661696C61626C6520> Tj 1 0 0 1 432.00 574.56 Tm <64617461736574732E20> Tj 1 0 0 1 473.28 574.32 Tm <466F7220> Tj 1 0 0 1 491.04 574.08 Tm <7468697320> Tj 1 0 0 1 510.48 574.08 Tm <726561736F6E2C> Tj 0.24 Tc 1 0 0 1 304.08 564.96 Tm <74686520> Tj 1 0 0 1 319.68 564.72 Tm <736F2D63616C6C656420> Tj 1 0 0 1 360.00 565.92 Tm <226C6F63616C20> Tj 1 0 0 1 385.68 564.48 Tm <6D65616E2220> Tj 1 0 0 1 414.24 564.24 Tm <697320> Tj 1 0 0 1 423.84 564.00 Tm <6D6F737420> Tj 1 0 0 1 447.36 563.76 Tm <6F6674656E20> Tj 1 0 0 1 470.88 563.52 Tm <6170706C6965642C20> Tj 1 0 0 1 506.88 563.28 Tm <776869636820> Tj 1 0 0 1 534.72 563.28 Tm <6973> Tj 0.24 Tc 1 0 0 1 304.08 554.16 Tm <63616C63756C6174656420> Tj 1 0 0 1 349.92 553.92 Tm <6F6E6C7920> Tj 1 0 0 1 370.56 553.92 Tm <66726F6D20> Tj 1 0 0 1 394.32 553.68 Tm <74686520> Tj 1 0 0 1 410.40 553.68 Tm <6861726420> Tj 1 0 0 1 433.92 553.20 Tm <6461746120> Tj 1 0 0 1 455.28 553.20 Tm <656E636F6D70617373656420> Tj 1 0 0 1 515.52 552.72 Tm <627920> Tj 1 0 0 1 528.72 552.72 Tm <746865> Tj 0.24 Tc 1 0 0 1 304.08 543.60 Tm <736561726368696E6720> Tj 1 0 0 1 348.72 543.60 Tm <72616469692C20> Tj 1 0 0 1 375.84 543.36 Tm <616E6420> Tj 1 0 0 1 395.76 543.12 Tm <7468697320> Tj 1 0 0 1 415.68 543.12 Tm <6661766F75727320> Tj 1 0 0 1 451.44 542.64 Tm <74686520> Tj 1 0 0 1 468.72 542.40 Tm <4F7264696E61727920> Tj 1 0 0 1 511.20 542.40 Tm <49667569676C6E67> Tj 0.24 Tc 1 0 0 1 303.84 533.04 Tm <746563686E697175652E> Tj 0.24 Tc 1 0 0 1 310.80 519.60 Tm <5468657265666F72652C20> Tj 1 0 0 1 357.36 519.12 Tm <776520> Tj 1 0 0 1 371.76 519.12 Tm <6D75737420> Tj 1 0 0 1 396.00 518.88 Tm <626520> Tj 1 0 0 1 408.72 518.88 Tm <7665727920> Tj 1 0 0 1 429.60 518.64 Tm <6361726566756C20> Tj 1 0 0 1 461.28 518.16 Tm <746F20> Tj 1 0 0 1 472.56 518.16 Tm <616E616C79736520> Tj 1 0 0 1 506.64 517.92 Tm <74686520> Tj 1 0 0 1 522.24 517.92 Tm <766172692D> Tj 0.24 Tc 1 0 0 1 303.84 508.80 Tm <61626C657320> Tj 1 0 0 1 328.32 508.80 Tm <7769746820> Tj 1 0 0 1 349.92 508.56 Tm <4F7264696E61727920> Tj 1 0 0 1 391.20 508.56 Tm <4B7269676E6720> Tj 1 0 0 1 424.80 508.08 Tm <6571756174696F6E732E20> Tj 1 0 0 1 471.84 507.84 Tm <4D6F737420> Tj 1 0 0 1 495.60 507.60 Tm <6F6620> Tj 1 0 0 1 506.40 507.36 Tm <74686520> Tj 1 0 0 1 522.00 507.36 Tm <766172692D> Tj 0.24 Tc 1 0 0 1 303.60 498.24 Tm <61626C657320> Tj 1 0 0 1 327.84 498.24 Tm <61726520> Tj 1 0 0 1 343.20 498.24 Tm <74686520> Tj 1 0 0 1 358.56 498.00 Tm <7374616E6461726420> Tj 1 0 0 1 398.88 497.76 Tm <656C656D656E747320> Tj 1 0 0 1 438.72 497.52 Tm <6F6620> Tj 1 0 0 1 448.56 497.28 Tm <616C6C20> Tj 1 0 0 1 460.80 497.28 Tm <6B726967696E6720> Tj 1 0 0 1 492.72 497.04 Tm <746563686E69717565732C> Tj 0.24 Tc 1 0 0 1 303.36 487.92 Tm <62757420> Tj 1 0 0 1 322.80 487.68 Tm <746865726520> Tj 1 0 0 1 349.92 487.68 Tm <697320> Tj 1 0 0 1 362.40 487.44 Tm <6120> Tj 1 0 0 1 372.24 487.44 Tm <756E6971756520> Tj 1 0 0 1 406.56 487.20 Tm <7661726961626C652C20> Tj 1 0 0 1 448.56 486.96 Tm <74686520> Tj 1 0 0 1 467.04 486.72 Tm <4C616772616E676520> Tj 1 0 0 1 511.20 487.20 Tm <286C696E65617229> Tj 0.24 Tc 1 0 0 1 303.36 477.36 Tm <6D756C7469706C696361746F722C20> Tj 1 0 0 1 362.88 477.12 Tm <74686520> Tj 1 0 0 1 378.24 476.88 Tm <726F6C6520> Tj 1 0 0 1 397.20 476.64 Tm <616E6420> Tj 1 0 0 1 415.44 476.64 Tm <696D706F7274616C636520> Tj 1 0 0 1 466.56 476.16 Tm <6F6620> Tj 1 0 0 1 476.40 476.16 Tm <776869636820> Tj 1 0 0 1 504.00 476.16 Tm <68617320> Tj 1 0 0 1 521.28 475.92 Tm <6265656E> Tj 0.24 Tc 1 0 0 1 303.60 466.80 Tm <64697363757373656420> Tj 1 0 0 1 347.28 466.56 Tm <696E20> Tj 1 0 0 1 357.60 466.56 Tm <74686520> Tj 1 0 0 1 372.72 466.80 Tm <70726576696F757320> Tj 1 0 0 1 411.36 466.08 Tm <63686170746572732E20> Tj 1 0 0 1 452.64 465.84 Tm <486F7720> Tj 1 0 0 1 474.00 465.60 Tm <63616C20> Tj 1 0 0 1 490.32 465.36 Tm <776520> Tj 1 0 0 1 504.00 465.36 Tm <63616C63756C617465> Tj 0.24 Tc 1 0 0 1 303.12 456.24 Tm <6E756D65726963616C6C7920> Tj 1 0 0 1 356.16 456.00 Tm <74686520> Tj 1 0 0 1 372.24 456.00 Tm <6D6F737420> Tj 1 0 0 1 396.24 455.76 Tm <617070726F70726961746520> Tj 1 0 0 1 448.80 455.28 Tm <76616C756520> Tj 1 0 0 1 473.76 455.04 Tm <666F7220> Tj 1 0 0 1 488.88 455.04 Tm <7468697320> Tj 1 0 0 1 507.12 454.80 Tm <7661726961626C65> Tj 0.24 Tc 1 0 0 1 303.60 446.64 Tm <286D292C20> Tj 1 0 0 1 322.80 445.68 Tm <692E652E2C20> Tj 1 0 0 1 340.32 445.44 Tm <74686520> Tj 1 0 0 1 355.20 445.44 Tm <76616C756520> Tj 1 0 0 1 379.92 445.20 Tm <7468617420> Tj 1 0 0 1 398.40 445.20 Tm <77696C6C20> Tj 1 0 0 1 415.68 445.20 Tm <726573756C7420> Tj 1 0 0 1 442.56 444.96 Tm <696E20> Tj 1 0 0 1 452.88 444.72 Tm <74686520> Tj 1 0 0 1 468.24 444.72 Tm <6C6561737420> Tj 1 0 0 1 490.08 444.48 Tm <6B726967696E6720> Tj 1 0 0 1 521.76 444.24 Tm <766172692D> Tj 0.24 Tc 1 0 0 1 303.12 435.12 Tm <616E63653F> Tj 0.24 Tc 1 0 0 1 310.08 421.44 Tm <54686520> Tj 1 0 0 1 329.76 421.20 Tm <616E7377657220> Tj 1 0 0 1 363.36 421.20 Tm <68617320> Tj 1 0 0 1 381.60 420.96 Tm <6265656E20> Tj 1 0 0 1 405.12 420.72 Tm <736F7567687420> Tj 1 0 0 1 436.80 420.48 Tm <696E20> Tj 1 0 0 1 448.56 420.48 Tm <74686520> Tj 1 0 0 1 465.12 420.72 Tm <70726F63656475726520> Tj 1 0 0 1 512.16 420.00 Tm <6F6620> Tj 1 0 0 1 523.44 420.00 Tm <72616E2D> Tj 0.24 Tc 1 0 0 1 303.12 410.88 Tm <646F6D20> Tj 1 0 0 1 324.96 410.64 Tm <73616D706C696E672C20> Tj 1 0 0 1 368.64 410.64 Tm <692E652E2C20> Tj 1 0 0 1 386.88 410.40 Tm <74686520> Tj 1 0 0 1 402.48 410.16 Tm <73656C656374696F6E20> Tj 1 0 0 1 442.56 409.68 Tm <6F6620> Tj 1 0 0 1 452.88 409.68 Tm <6120> Tj 1 0 0 1 460.32 409.68 Tm <72616E646F6D20> Tj 1 0 0 1 496.80 409.44 Tm <7365656420> Tj 1 0 0 1 518.40 409.44 Tm <6E756D2D> Tj 0.24 Tc 1 0 0 1 302.64 400.32 Tm <62657220> Tj 1 0 0 1 321.36 400.32 Tm <66726F6D20> Tj 1 0 0 1 346.32 400.08 Tm <74686520> Tj 1 0 0 1 364.08 400.08 Tm <73657420> Tj 1 0 0 1 380.64 399.84 Tm <6F6620> Tj 1 0 0 1 392.88 400.08 Tm <706F737369626C6520> Tj 1 0 0 1 432.96 400.32 Tm <2865787065637465642920> Tj 1 0 0 1 479.76 399.12 Tm <76616C75657320> Tj 1 0 0 1 510.96 398.88 Tm <666F7220> Tj 1 0 0 1 527.76 398.88 Tm <746865> Tj 0.24 Tc 1 0 0 1 302.64 390.00 Tm <4C616772616E676520> Tj 1 0 0 1 343.68 389.76 Tm <6D756C7427706C696361746F722E20> Tj 1 0 0 1 403.20 389.28 Tm <497420> Tj 1 0 0 1 411.60 389.28 Tm <68617320> Tj 1 0 0 1 428.88 389.04 Tm <6265656E20> Tj 1 0 0 1 451.44 388.80 Tm <73686F776E20> Tj 1 0 0 1 481.44 388.56 Tm <7468617420> Tj 1 0 0 1 500.40 388.56 Tm <7468697320> Tj 1 0 0 1 518.40 388.32 Tm <76616C7565> Tj 0.00 Tc 1 0 0 1 302.88 379.44 Tm <63616E20> Tj 1 0 0 1 320.88 379.20 Tm <626520> Tj 1 0 0 1 334.56 379.20 Tm <6E656761746976652C20> Tj 1 0 0 1 374.88 378.96 Tm <62757420> Tj 1 0 0 1 392.64 378.72 Tm <74686520> Tj 1 0 0 1 408.96 378.72 Tm <76616C75657320> Tj 1 0 0 1 439.44 378.48 Tm <666F7220> Tj 1 0 0 1 455.04 378.24 Tm <74686520> Tj 1 0 0 1 471.84 378.24 Tm <6B726C67696E6720> Tj 1 0 0 1 504.72 378.00 Tm <76617269616E6365> Tj 0.24 Tc 1 0 0 1 302.64 368.88 Tm <63616E6E6F7420> Tj 1 0 0 1 335.52 368.64 Tm <626520> Tj 1 0 0 1 349.92 368.64 Tm <6E6567617469766520> Tj 1 0 0 1 389.04 369.36 Tm <286D617468656D61746963616C6C7920> Tj 1 0 0 1 460.80 367.68 Tm <696D706F737369626C65292E20> Tj 1 0 0 1 517.20 367.44 Tm <496620> Tj 1 0 0 1 527.28 367.20 Tm <746865> Tj 0.24 Tc 1 0 0 1 302.16 358.32 Tm <76616C756520> Tj 1 0 0 1 326.88 358.32 Tm <697320> Tj 1 0 0 1 336.48 358.32 Tm <72616E646F6D6C7920> Tj 1 0 0 1 380.16 357.84 Tm <73616D706C656420> Tj 1 0 0 1 418.32 357.60 Tm <6D616E7920> Tj 1 0 0 1 444.24 357.36 Tm <74696D657320> Tj 1 0 0 1 469.68 357.12 Tm <66726F6D20> Tj 1 0 0 1 492.96 356.88 Tm <736F6D6520> Tj 1 0 0 1 517.44 356.88 Tm <696E7465722D> Tj 0.48 Tc 1 0 0 1 302.16 347.04 Tm <76616C20> Tj 1 0 0 1 317.52 348.00 Tm <28652E672E20> Tj 1 0 0 1 337.44 345.12 Tm <5B2D312C31292C20> Tj 1 0 0 1 369.60 346.80 Tm <6F6E6520> Tj 1 0 0 1 386.88 346.56 Tm <76616C756520> Tj 1 0 0 1 411.60 346.32 Tm <77696C6C20> Tj 1 0 0 1 429.36 346.56 Tm <70726F6475636520> Tj 1 0 0 1 467.28 345.84 Tm <74686520> Tj 1 0 0 1 483.12 345.84 Tm <6C6561737420> Tj 1 0 0 1 505.20 346.08 Tm <706F737369626C65> Tj 0.24 Tc 1 0 0 1 302.40 336.72 Tm <687267696E6720> Tj 1 0 0 1 336.00 336.48 Tm <76612D7269616E63652E20> Tj 1 0 0 1 378.72 336.24 Tm <497420> Tj 1 0 0 1 389.04 336.00 Tm <7468656E20> Tj 1 0 0 1 411.84 336.00 Tm <72656D61696E7320> Tj 1 0 0 1 450.72 335.52 Tm <746F20> Tj 1 0 0 1 463.20 335.28 Tm <73656C65637420> Tj 1 0 0 1 491.04 335.04 Tm <74686520> Tj 1 0 0 1 508.32 335.28 Tm <696E74657276616C> Tj 0.00 Tc 1 0 0 1 301.92 325.92 Tm <776964746820> Tj 1 0 0 1 328.32 325.68 Tm <616E6420> Tj 1 0 0 1 346.08 325.68 Tm <74686520> Tj 1 0 0 1 361.20 325.68 Tm <6E756D62657220> Tj 1 0 0 1 396.96 325.44 Tm <6F6620> Tj 1 0 0 1 406.80 325.20 Tm <73616D706C696E67732E20> Tj 1 0 0 1 454.80 324.72 Tm <486F7720> Tj 1 0 0 1 475.68 324.72 Tm <7468697320> Tj 1 0 0 1 493.44 324.48 Tm <697320> Tj 1 0 0 1 502.80 324.48 Tm <646F6E6520> Tj 1 0 0 1 525.60 324.24 Tm <63616E> Tj 0.24 Tc 1 0 0 1 301.92 315.36 Tm <626520> Tj 1 0 0 1 315.12 315.36 Tm <64657363726962656420> Tj 1 0 0 1 359.28 315.12 Tm <696E20> Tj 1 0 0 1 370.08 315.12 Tm <666F757220> Tj 1 0 0 1 390.72 314.88 Tm <73746570733A> Tj 0.24 Tc 1 0 0 1 303.36 302.40 Tm <312E20> Tj 1 0 0 1 313.20 302.40 Tm <426173656420> Tj 1 0 0 1 342.24 301.92 Tm <6F6E20> Tj 1 0 0 1 356.16 301.92 Tm <657870657269656E63652C20> Tj 1 0 0 1 406.80 301.68 Tm <697420> Tj 1 0 0 1 415.68 301.44 Tm <63616E20> Tj 1 0 0 1 433.44 301.20 Tm <626520> Tj 1 0 0 1 446.64 301.20 Tm <617373756D656420> Tj 1 0 0 1 487.44 300.72 Tm <7468617420> Tj 1 0 0 1 506.88 300.72 Tm <76616C756520> Tj 1 0 0 1 532.32 300.48 Tm <6F66> Tj 0.24 Tc 1 0 0 1 313.92 291.60 Tm <74686520> Tj 1 0 0 1 334.56 291.60 Tm <4C616772616E676520> Tj 1 0 0 1 380.64 291.36 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 443.52 290.64 Tm <73686F756C6420> Tj 1 0 0 1 480.00 290.40 Tm <6F6363757220> Tj 1 0 0 1 510.96 290.16 Tm <696E20> Tj 1 0 0 1 526.80 290.16 Tm <746865> Tj 0.24 Tc 1 0 0 1 313.92 280.56 Tm <696E74657276616C20> Tj 1 0 0 1 349.20 278.64 Tm <5B2D6620> Tj 1 0 0 1 362.16 279.36 Tm <2C6C5D2E20> Tj 1 0 0 1 378.48 280.08 Tm <536F6D6520> Tj 1 0 0 1 404.64 279.84 Tm <76616C75657320> Tj 1 0 0 1 434.40 279.60 Tm <63616E20> Tj 1 0 0 1 451.44 279.36 Tm <626520> Tj 1 0 0 1 463.92 279.36 Tm <7665727920> Tj 1 0 0 1 484.56 279.12 Tm <636C6F736520> Tj 1 0 0 1 508.32 279.12 Tm <746F20> Tj 1 0 0 1 519.60 278.88 Tm <4F20> Tj 1 0 0 1 528.00 279.84 Tm <286173> Tj 0.24 Tc 1 0 0 1 313.92 269.76 Tm <73686F776E20> Tj 1 0 0 1 348.48 269.76 Tm <696E20> Tj 1 0 0 1 364.32 269.52 Tm <6F757220> Tj 1 0 0 1 386.40 269.52 Tm <6578616D706C65292E20> Tj 1 0 0 1 434.64 269.04 Tm <466F7220> Tj 1 0 0 1 456.00 268.80 Tm <746865736520> Tj 1 0 0 1 485.52 268.56 Tm <726561736F6E732C20> Tj 1 0 0 1 528.24 268.32 Tm <7765> Tj 0.24 Tc 1 0 0 1 313.92 259.20 Tm <636F6E736964657220> Tj 1 0 0 1 357.36 258.96 Tm <7468617420> Tj 1 0 0 1 381.36 258.96 Tm <74686520> Tj 1 0 0 1 401.76 258.72 Tm <7374617274696E6720> Tj 1 0 0 1 441.12 258.72 Tm <706F696E7420> Tj 1 0 0 1 470.40 258.24 Tm <696E20> Tj 1 0 0 1 485.76 258.00 Tm <74686520> Tj 1 0 0 1 505.92 258.00 Tm <72616E646F6D> Tj 0.24 Tc 1 0 0 1 313.68 248.64 Tm <73616D706C696E6720> Tj 1 0 0 1 354.24 248.64 Tm <666F7220> Tj 1 0 0 1 368.40 248.40 Tm <74686520> Tj 1 0 0 1 383.52 248.40 Tm <4C616772616E676520> Tj 1 0 0 1 423.84 247.92 Tm <76616C756520> Tj 1 0 0 1 448.08 247.68 Tm <73686F756C6420> Tj 1 0 0 1 478.80 247.44 Tm <626520> Tj 1 0 0 1 491.04 247.44 Tm <73657420> Tj 1 0 0 1 505.20 247.20 Tm <617420> Tj 1 0 0 1 515.04 247.20 Tm <4F2E4F20> Tj 1 0 0 1 529.68 247.20 Tm <3120> Tj 1 0 0 1 534.72 247.20 Tm <2E> Tj 0.24 Tc 1 0 0 1 301.92 236.16 Tm <322E20> Tj 1 0 0 1 312.72 236.16 Tm <496E20> Tj 1 0 0 1 324.00 235.92 Tm <74686520> Tj 1 0 0 1 339.84 235.92 Tm <6E65787420> Tj 1 0 0 1 360.72 235.68 Tm <737465702C20> Tj 1 0 0 1 383.76 235.44 Tm <74686520> Tj 1 0 0 1 399.84 235.44 Tm <7374617274696E6720> Tj 1 0 0 1 434.64 234.96 Tm <76616C756520> Tj 1 0 0 1 459.84 234.96 Tm <6D75737420> Tj 1 0 0 1 484.08 234.72 Tm <626520> Tj 1 0 0 1 497.28 234.72 Tm <646563726561736564> Tj 0.48 Tc 1 0 0 1 313.92 226.32 Tm <28696E20> Tj 1 0 0 1 327.36 225.36 Tm <737465707320> Tj 1 0 0 1 351.84 225.12 Tm <6F6620> Tj 1 0 0 1 361.68 227.04 Tm <2D302E3035292C20> Tj 1 0 0 1 393.60 224.88 Tm <692E652E2C20> Tj 1 0 0 1 411.12 224.88 Tm <697420> Tj 1 0 0 1 419.28 224.64 Tm <6265636F6D657320> Tj 1 0 0 1 458.64 224.40 Tm <6E656761746976652E20> Tj 1 0 0 1 497.52 224.16 Tm <416761696E2C20> Tj 1 0 0 1 526.32 223.92 Tm <746865> Tj 0.24 Tc 1 0 0 1 313.68 214.80 Tm <617070726F70726961746520> Tj 1 0 0 1 368.16 214.56 Tm <6B726967696E6720> Tj 1 0 0 1 402.00 214.08 Tm <76617269616E636520> Tj 1 0 0 1 442.56 213.84 Tm <63616E20> Tj 1 0 0 1 461.52 213.60 Tm <626520> Tj 1 0 0 1 476.16 213.60 Tm <63616C63756C6174656420> Tj 1 0 0 1 523.68 213.36 Tm <616E64> Tj 0.24 Tc 1 0 0 1 313.20 204.24 Tm <726567697374657265642E20> Tj 1 0 0 1 360.72 204.00 Tm <5375636820> Tj 1 0 0 1 384.48 203.76 Tm <6120> Tj 1 0 0 1 391.20 204.00 Tm <70726F63656475726520> Tj 1 0 0 1 437.28 203.28 Tm <73686F756C6420> Tj 1 0 0 1 468.24 203.04 Tm <626520> Tj 1 0 0 1 480.72 203.04 Tm <726570656174656420> Tj 1 0 0 1 519.60 202.80 Tm <756E74696C> Tj 0.24 Tc 1 0 0 1 313.20 193.68 Tm <74686520> Tj 1 0 0 1 329.04 193.68 Tm <6B726967696E6720> Tj 1 0 0 1 361.20 193.20 Tm <76617269616E636520> Tj 1 0 0 1 399.60 193.20 Tm <697320> Tj 1 0 0 1 409.44 193.44 Tm <706F7369746976652E20> Tj 1 0 0 1 447.84 192.72 Tm <5573696E6720> Tj 1 0 0 1 474.48 192.48 Tm <7468697320> Tj 1 0 0 1 492.96 192.72 Tm <70726F6365647572652C> Tj 0.24 Tc 1 0 0 1 313.20 182.88 Tm <74686520> Tj 1 0 0 1 329.52 182.88 Tm <76616C756520> Tj 1 0 0 1 355.20 182.64 Tm <6F6620> Tj 1 0 0 1 366.48 182.64 Tm <6D20> Tj 1 0 0 1 377.76 182.40 Tm <7769746820> Tj 1 0 0 1 399.60 182.40 Tm <74686520> Tj 1 0 0 1 416.16 182.40 Tm <6C6561737420> Tj 1 0 0 1 439.20 182.16 Tm <6B7280696E6720> Tj 1 0 0 1 471.84 181.68 Tm <76617269616E636520> Tj 1 0 0 1 511.20 181.44 Tm <63616E20> Tj 1 0 0 1 529.20 181.44 Tm <6265> Tj 0.24 Tc 1 0 0 1 313.20 172.32 Tm <73656C656374656420> Tj 1 0 0 1 350.16 173.04 Tm <2846696775726520> Tj 1 0 0 1 382.80 171.84 Tm <3320> Tj 1 0 0 1 390.72 171.84 Tm <6C656674292E> Tj 0.24 Tc 1 0 0 1 301.20 159.60 Tm <332E20> Tj 1 0 0 1 311.76 159.84 Tm <54686520> Tj 1 0 0 1 330.24 160.08 Tm <706F73697469766520> Tj 1 0 0 1 365.52 159.36 Tm <7369646520> Tj 1 0 0 1 385.44 159.12 Tm <6F6620> Tj 1 0 0 1 395.52 159.12 Tm <74686520> Tj 1 0 0 1 411.12 159.12 Tm <4C616772616E676520> Tj 1 0 0 1 451.68 158.64 Tm <76616C756520> Tj 1 0 0 1 476.64 158.40 Tm <73686F756C6420> Tj 1 0 0 1 508.08 158.16 Tm <7468656E20> Tj 1 0 0 1 528.96 158.16 Tm <6265> Tj 0.24 Tc 1 0 0 1 313.20 149.04 Tm <636865636B65642C20> Tj 1 0 0 1 353.04 148.80 Tm <616761696E20> Tj 1 0 0 1 377.76 148.80 Tm <696E6372656173696E6720> Tj 1 0 0 1 423.84 148.32 Tm <74686520> Tj 1 0 0 1 439.20 148.08 Tm <76616C756520> Tj 1 0 0 1 463.92 148.08 Tm <696E20> Tj 1 0 0 1 474.96 147.84 Tm <737465707320> Tj 1 0 0 1 499.44 147.60 Tm <6F6620> Tj 1 0 0 1 510.48 147.84 Tm <2B4F2E4F532E> Tj 0.24 Tc 1 0 0 1 312.48 138.48 Tm <416761696E2C20> Tj 1 0 0 1 343.20 138.24 Tm <6120> Tj 1 0 0 1 351.84 138.24 Tm <726564756374696F6E20> Tj 1 0 0 1 396.48 138.00 Tm <696E20> Tj 1 0 0 1 408.72 137.76 Tm <74686520> Tj 1 0 0 1 425.52 137.76 Tm <76617269616E636520> Tj 1 0 0 1 464.88 137.28 Tm <77696C6C20> Tj 1 0 0 1 484.08 137.28 Tm <626520> Tj 1 0 0 1 498.24 137.28 Tm <6F627365727665642C> Tj 0.24 Tc 1 0 0 1 312.96 127.92 Tm <616E6420> Tj 1 0 0 1 334.08 127.68 Tm <74686520> Tj 1 0 0 1 352.32 128.16 Tm <70726F63656475726520> Tj 1 0 0 1 401.28 127.20 Tm <73686F756C6420> Tj 1 0 0 1 435.36 126.96 Tm <626520> Tj 1 0 0 1 450.72 126.96 Tm <636F6E74696E75656420> Tj 1 0 0 1 498.00 126.72 Tm <756E74696C20> Tj 1 0 0 1 522.96 126.48 Tm <74686973> Tj 0.24 Tc 1 0 0 1 312.72 117.36 Tm <726564756374696F6E20> Tj 1 0 0 1 357.36 117.12 Tm <76616C69736865732E20> Tj 1 0 0 1 400.80 116.88 Tm <54686520> Tj 1 0 0 1 421.20 116.64 Tm <666972737420> Tj 1 0 0 1 442.56 116.40 Tm <74696D6520> Tj 1 0 0 1 465.36 116.16 Tm <7468617420> Tj 1 0 0 1 486.00 115.92 Tm <74686520> Tj 1 0 0 1 502.80 115.92 Tm <76617269616E6365> Tj 0.24 Tc 1 0 0 1 312.72 106.80 Tm <696E637265617365732C20> Tj 1 0 0 1 357.84 106.56 Tm <74686520> Tj 1 0 0 1 373.92 106.32 Tm <63616C63756C6174696F6E20> Tj 1 0 0 1 422.88 106.08 Tm <697320> Tj 1 0 0 1 432.96 105.84 Tm <73746F7070656420> Tj 1 0 0 1 469.92 106.56 Tm <2846696775726520> Tj 1 0 0 1 502.56 105.36 Tm <3320> Tj 1 0 0 1 510.48 105.36 Tm <7269676C7274292E> Tj 0.48 Tc /F3 10.0 Tf 1 0 0 1 268.32 210.72 Tm <28352E6529> Tj 0.96 Tc /F3 8.0 Tf 1 0 0 1 262.32 170.16 Tm <28352E313029> Tj 0.00 Tc /F3 9.0 Tf 1 0 0 1 46.08 149.28 Tm <54686520> Tj 1 0 0 1 65.28 149.28 Tm <6F7074696D616C20> Tj 1 0 0 1 100.08 149.04 Tm <776569676C7274696E6720> Tj 1 0 0 1 143.28 148.80 Tm <636F656666696369656E7420> Tj 1 0 0 1 188.88 148.56 Tm <616E6420> Tj 1 0 0 1 207.36 148.56 Tm <4C616772616E676520> Tj 1 0 0 1 248.40 148.32 Tm <76616C756520> Tj 1 0 0 1 273.60 148.32 Tm <616C2D> Tj 0.24 Tc 1 0 0 1 46.08 138.72 Tm <6C6F7720> Tj 1 0 0 1 63.12 138.72 Tm <74686520> Tj 1 0 0 1 78.96 138.48 Tm <63616C63756C6174696F6E20> Tj 1 0 0 1 127.68 138.24 Tm <6F6620> Tj 1 0 0 1 138.00 138.24 Tm <74686520> Tj 1 0 0 1 153.60 138.24 Tm <6C6561737420> Tj 1 0 0 1 175.68 138.48 Tm <706F737369626C6520> Tj 1 0 0 1 213.60 138.00 Tm <7374616E6461726420> Tj 1 0 0 1 254.16 137.76 Tm <6B726967696E67> Tj 0.24 Tc 1 0 0 1 45.84 128.16 Tm <76617269616E636520> Tj 1 0 0 1 84.24 127.92 Tm <7573696E6720> Tj 1 0 0 1 109.92 127.68 Tm <74686520> Tj 1 0 0 1 126.00 127.68 Tm <6571756174696F6E2E> Tj 0.24 Tc /F3 8.0 Tf 1 0 0 1 45.60 109.92 Tm <6F28786F2920> Tj 1 0 0 1 66.00 111.36 Tm <3D20> Tj 1 0 0 1 73.44 109.68 Tm <4328302920> Tj 1 0 0 1 90.24 111.84 Tm <2D20> Tj 1 0 0 1 96.96 109.68 Tm <312220> Tj 1 0 0 1 107.52 109.44 Tm <636F20> Tj 1 0 0 1 116.64 109.68 Tm <2B20> Tj 1 0 0 1 123.84 109.44 Tm <6D> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 45.84 84.96 Tm <3336> Tj 1.20 Tc /F3 8.0 Tf 1 0 0 1 261.84 109.20 Tm <28352E313129> Tj 0.00 Tc 1 0 0 1 461.52 82.80 Tm <4E4146544120> Tj 1 0 0 1 481.68 82.56 Tm <353920> Tj 1 0 0 1 490.80 82.80 Tm <283120> Tj 1 0 0 1 496.56 80.88 Tm <2920> Tj 1 0 0 1 500.40 82.56 Tm <333120> Tj 1 0 0 1 507.60 84.24 Tm <2D333720> Tj 1 0 0 1 519.36 82.80 Tm <283230303929> Tj ET endstream endobj 61 0 obj 33698 endobj 62 0 obj << /Length 63 0 R >> stream 1 . 2 is hnear and calculated from the val ues Z(x ,). . . Z(x ^); 6. CONCLUSION t=e(zu"t-Z^,-r0,,)' -^ (>^,-r) L- o Z. 2 is an unbiased estimation; In the previous chapters, we have shown why the Ordi-B. 2 minio,:zes the value of the mean square error naryKrigingtechniqueismoreappropriatefortheinter-expressed as nfz@) - 2r*ot)". fr:?ff"H'"ff:lhT::J:XT'"Hlfff'T5"t fl'flH;Lineanty (condition 1) is realized if the equation application of the kriging method to petroleum geologldata (porosity) collected from hydrocarbon reservoirs in2 :> ),, ' Z(r,) is satisfied. the Croatian part of the Pannonian Basin. It was thereproven that geostatistical interpolation (as well as sto-Unbiasedness (condition 2) is fulfilled if the following chastic estirnation) is the best approach to mapping geo-equation is valid: logical variables (e.g., Ieferences 1,8,e' 10, 12). ,l2t "ll = p + t4 = i^, . r7r1,,',1 -i=7 Minimizing the error (condition 3) demands the selectionof the most appropriate values for the coefficients l.r, . . . ,).nand the Lagrange mulflplicator 2m.Their optimizationcan be expressed for each point as the functionL(f,,,...,).^,m): (5.5) The optimization of the coefficients (,1r,..., ,tr") and theLagrange value (m) is achieved by solving the differentialALequation AU=O.If we look again at the basic kriging matrices (Equation3.3) and show them using covafiances, the kriging esti-mation is: C. ),o = 6o (5.6) The variables C and Co can be presented as matrices: c= (5.7) (5.8) The expected weighting coefficient can then be calculated(using Equation 5.6) as: c(0) c(4-z) c(4-2,) 1c(22-21) c(0) c(22-2,) 1c(2,-21) c(z"-z) c(0) 11110 io = C-''CoThe estimation at the selected point is::,Z(xo1 = 1,. 71r,)+...+ ),, . Z(x,l Based on such analyses, it is concluded that kriging canbe usefully applied to cases with a minimum of lO oreven 15 data points. However, arry representative statisti-cal dataset of geological variables must include at least30 data points. This means that we cannot conclude thevalue of the real population mean (expectation) basedonly on the usually available datasets. For this reason,the so-called "local mean" is most often applied, which iscalculated only from the hard data encompassed by thesearching radii, and this favours the Ordinary Ifuiglngtechnique.Therefore, we must be very careful to analyse the vari-ables with Ordinary Krigng equations. Most of the vari-ables are the standard elements of all kriging techniques,but there is a unique variable, the Lagrange (linear)multiplicator, the role and importalce of which has beendiscussed in the previous chapters. How cal we calculatenumerically the most appropriate value for this variable(m), i.e., the value that will result in the least kriging vari-ance?The answer has been sought in the procedure of ran-dom sampling, i.e., the selection of a random seed num-ber from the set of possible (expected) values for theLagrange mult'plicator. It has been shown that this valuecan be negative, but the values for the krlging variancecannot be negative (mathematically impossible). If thevalue is randomly sampled many times from some inter-val (e.g. [-1,1), one value will produce the least possiblehrging va-riance. It then remains to select the intervalwidth and the number of samplings. How this is done canbe described in four steps:1. Based on experience, it can be assumed that value ofthe Lagrange multiplicator should occur in theinterval [-f ,l]. Some values can be very close to O (asshown in our example). For these reasons, weconsider that the starting point in the randomsampling for the Lagrange value should be set at O.O 1 .2. In the next step, the starting value must be decreased(in steps of -0.05), i.e., it becomes negative. Again, theappropriate kriging variance can be calculated andregistered. Such a procedure should be repeated untilthe kriging variance is positive. Using this procedure,the value of m with the least kring variance can beselected (Figure 3 left).3. The positive side of the Lagrange value should then bechecked, again increasing the value in steps of +O.OS.Again, a reduction in the variance will be observed,and the procedure should be continued until thisreduction valishes. The first time that the varianceincreases, the calculation is stopped (Figure 3 riglrt). (5.e)(5.10) The optimal weiglrting coefficient and Lagrange value al-low the calculation of the least possible standard krigingvariance using the equation. o(xo) = C(0) - 1" co + m36 (5.11) NAFTA 59 (1 ) 31 -37 (2009) endstream endobj 63 0 obj 4523 endobj 64 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 65 0 obj << /Filter /FlateDecode /Length 128 >> stream xEA @ὧqwzvQ(oj&A\|.4r)h*Z׺op~F M$d4PΦub[*ŕV:x&Cũ [iIǚ v/C+E endstream endobj 66 0 obj << /Type /Page /MediaBox [ 0 0 595.20 841.68 ] /Parent 13 0 R /Resources << /Font << /F3 7 0 R >> /XObject << /Obj56 56 0 R /Obj57 57 0 R /Obj58 58 0 R /Obj59 59 0 R >> /Subtype 62 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 64 0 R 65 0 R 60 0 R ] >> endobj 67 0 obj << /Type /XObject /Subtype /Image /Width 1240 /Height 1753 /BitsPerComponent 8 /ColorSpace /DeviceGray /Filter /DCTDecode /Length 56888 >> stream JFIF (B+(%%(Q:=0B`Ued_U][jxjqs[]gɺǙ(#(N++Nn]n  }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyzw!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?ΏNB*UO?Lె9Wi"SHȤJibQLQ1F(bLRbRQ1F(.)1K1F)1F)qF(&)qF(bbQ1K1GF(bQ1F(bQ1F(bQ(bQ1K1F(b(bR`Q\RbڌQ1bQbP&)qI0(ҌQ1F(.⸈G*]VDZ4cX(U4斒JB)SEH%QIGj)(((()hJZ(((QEQEQEQEQEQEQEQE(((((QEQEQE褥QEQGz()?Z(sNii( 6 () QERQERREQKIKEQEQEQEQEQEQEQEQEQEQEQEQEQEQKEQEP((((((;A(QEQEQE%-5@>ԴwKIEiN4 (E%QEQEQEQEQE-RREQEQEt(QEQEQEQEQEQEQEQA)h4QE (((((((EP(((JZk֔QKKE-!ReR(KEQEQE%QEQEQERIKIKEQEQE(((((((((Z(QEQE())hZ((u((EQEQERR[֜(m-F)h ))i)i()((QEQEQEQE-QE%-QEQEQEQEQEQEQEQEQEQE(JZJZ())i(((4QKIGz)h((JZkvӨ=hw1ZBi(f(QGj3ERfњ3E4Q3E&hZ(JZ(QFh(4QFh4f3EEQE4f(fJZ)3E)(.hfњ(4f 4Q3Fh3K(&h<њ3Fh(4f4fL))sMcӞJ(Tෙurî*r_4VܗGͿ%o!4l9!4Xy?O/?O7?Ə툿lE<؋yIoҗ^?ߘ^?ߝ/~F蟑>/~Ght'htQ}/K}lR_*>/?mʏG%g~T}_QQ}GGk~T}_ʏQ}GGkr{s{kߠs{}l}Q >7AI >?AG(\'(\k\kz~=?AG} >?'C >?lz~O=?AI)~?''OQ:(e?ȣ}z㢏OtQۏ Oc\OQ\}}En:(}Gۮ:)>s=?G:(}E':(}E':?_t'Q!}zoIu=O? ? 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668.64 Tm <2821> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 323.76 655.92 Tm <2265> Tj 0.00 Tc /F3 6.0 Tf 1 0 0 1 325.44 651.60 Tm <286F> Tj 0.00 Tc 1 0 0 1 325.20 646.32 Tm <6F29> Tj 0.00 Tc 1 0 0 1 325.44 642.00 Tm <6329> Tj 0.00 Tc /F3 9.0 Tf 1 0 0 1 323.76 636.48 Tm <7A> Tj 0.24 Tc 1 0 0 1 60.24 503.52 Tm <342E20> Tj 1 0 0 1 70.56 503.76 Tm <497420> Tj 1 0 0 1 78.72 503.76 Tm <697320> Tj 1 0 0 1 88.08 503.76 Tm <69726E706F7274616E7420> Tj 1 0 0 1 132.96 503.76 Tm <746F20> Tj 1 0 0 1 143.52 504.00 Tm <6B6E6F7720> Tj 1 0 0 1 168.48 504.00 Tm <7468617420> Tj 1 0 0 1 187.44 504.00 Tm <69662C20> Tj 1 0 0 1 197.76 504.24 Tm <666F7220> Tj 1 0 0 1 212.16 504.00 Tm <74686520> Tj 1 0 0 1 227.28 504.24 Tm <666972737420> Tj 1 0 0 1 246.96 504.72 Tm <706F73697469766520> Tj 1 0 0 1 281.76 504.48 Tm <6D> Tj 0.24 Tc 1 0 0 1 72.48 494.16 Tm <286D3E30292C20> Tj 1 0 0 1 106.08 493.20 Tm <74686520> Tj 1 0 0 1 122.16 493.20 Tm <76617269616E636520> Tj 1 0 0 1 161.76 493.44 Tm <696D6D6564696174656C7920> Tj 1 0 0 1 217.44 493.68 Tm <7374612D727420> Tj 1 0 0 1 240.48 493.68 Tm <746F20> Tj 1 0 0 1 252.24 493.92 Tm <696E6372656173652C> Tj 0.24 Tc 1 0 0 1 72.00 482.64 Tm <7468656E20> Tj 1 0 0 1 93.36 482.64 Tm <697420> Tj 1 0 0 1 102.00 482.64 Tm <63616E20> Tj 1 0 0 1 119.04 482.88 Tm <626520> Tj 1 0 0 1 131.76 482.88 Tm <636F6E7369646572656420> Tj 1 0 0 1 180.96 483.12 Tm <7468617420> Tj 1 0 0 1 200.16 483.12 Tm <74686520> Tj 1 0 0 1 216.00 483.36 Tm <6D696E696D756D20> Tj 1 0 0 1 260.64 483.36 Tm <6B726967696E67> Tj 0.24 Tc 1 0 0 1 71.76 472.08 Tm <76617269616E636520> Tj 1 0 0 1 110.64 472.32 Tm <697320> Tj 1 0 0 1 120.72 472.32 Tm <666F756E6420> Tj 1 0 0 1 148.56 472.32 Tm <696E20> Tj 1 0 0 1 159.84 472.32 Tm <74686520> Tj 1 0 0 1 176.16 472.56 Tm <6E6567617469766520> Tj 1 0 0 1 213.60 472.56 Tm <7363616C6520> Tj 1 0 0 1 237.60 472.56 Tm <616E6420> Tj 1 0 0 1 256.32 472.80 Tm <72616E646F6D> Tj 0.24 Tc 1 0 0 1 72.24 461.52 Tm <73616D706C696E6720> Tj 1 0 0 1 113.76 461.52 Tm <637A726E20> Tj 1 0 0 1 131.52 461.76 Tm <636561736520> Tj 1 0 0 1 157.44 462.72 Tm <2846696775726520> Tj 1 0 0 1 189.84 462.00 Tm <3320> Tj 1 0 0 1 197.76 462.00 Tm <6C656674292E> Tj 0.24 Tc 1 0 0 1 52.08 448.80 Tm <576520> Tj 1 0 0 1 66.00 448.80 Tm <62656C6965766520> Tj 1 0 0 1 96.96 448.80 Tm <7468617420> Tj 1 0 0 1 115.68 449.04 Tm <74686520> Tj 1 0 0 1 130.80 449.28 Tm <666F757220> Tj 1 0 0 1 150.72 449.28 Tm <72756C657320> Tj 1 0 0 1 174.24 449.28 Tm <64697363757373656420> Tj 1 0 0 1 218.40 449.52 Tm <61626F766520> Tj 1 0 0 1 244.80 449.52 Tm <636F6D706C6574656C79> Tj 0.24 Tc 1 0 0 1 52.56 438.24 Tm <646573637269626520> Tj 1 0 0 1 90.24 438.24 Tm <74686520> Tj 1 0 0 1 105.84 438.48 Tm <636F727265637420> Tj 1 0 0 1 137.52 438.96 Tm <70726F63656475726520> Tj 1 0 0 1 183.36 438.72 Tm <666F7220> Tj 1 0 0 1 197.76 438.72 Tm <73656C656374696E6720> Tj 1 0 0 1 236.16 438.96 Tm <74686520> Tj 1 0 0 1 251.52 439.20 Tm <4C616772616E6765> Tj 0.24 Tc 1 0 0 1 52.32 427.68 Tm <6D756C7469706C696361746F7220> Tj 1 0 0 1 109.92 427.92 Tm <696E20> Tj 1 0 0 1 120.72 427.92 Tm <4F7264696E61727920> Tj 1 0 0 1 161.28 428.16 Tm <4B726967696E6720> Tj 1 0 0 1 194.16 428.40 Tm <65717561756F6E732E20> Tj 1 0 0 1 240.48 428.64 Tm <546875732C20> Tj 1 0 0 1 267.36 428.64 Tm <7573696E67> Tj 0.24 Tc 1 0 0 1 52.32 417.12 Tm <7468697320> Tj 1 0 0 1 70.56 417.12 Tm <746563686E6971756520> Tj 1 0 0 1 114.48 417.36 Tm <697320> Tj 1 0 0 1 124.08 417.36 Tm <6F6E6520> Tj 1 0 0 1 141.60 417.36 Tm <6F6620> Tj 1 0 0 1 151.92 417.36 Tm <74686520> Tj 1 0 0 1 167.04 417.36 Tm <6265737420> Tj 1 0 0 1 187.20 417.60 Tm <696E746572706F6C6174696F6E20> Tj 1 0 0 1 244.80 417.84 Tm <616C676F726974686D73> Tj 0.24 Tc 1 0 0 1 52.32 406.56 Tm <666F7220> Tj 1 0 0 1 67.20 406.56 Tm <6D617070696E6720> Tj 1 0 0 1 107.04 407.04 Tm <67656F6C6F676963616C20> Tj 1 0 0 1 150.72 406.80 Tm <7661726961626C65732E> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 52.32 385.68 Tm <41434B4E4F574C4544474D454E5453> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 52.32 372.48 Tm <5468697320> Tj 1 0 0 1 79.44 372.72 Tm <706170657220> Tj 1 0 0 1 112.32 372.48 Tm <726570726573656E747320> Tj 1 0 0 1 165.60 372.72 Tm <74686520> Tj 1 0 0 1 186.96 372.72 Tm <7468656F7265746963616C20> Tj 1 0 0 1 240.00 372.96 Tm <616E616C7973697320> Tj 1 0 0 1 282.48 373.20 Tm <6F66> Tj 0.24 Tc 1 0 0 1 52.32 362.16 Tm <67656F6D617468656D61746963616C20> Tj 1 0 0 1 129.12 362.16 Tm <6D6574686F647320> Tj 1 0 0 1 171.12 362.64 Tm <706572666F726D656420> Tj 1 0 0 1 221.28 362.40 Tm <696E20> Tj 1 0 0 1 234.96 362.40 Tm <324F4F3820> Tj 1 0 0 1 261.84 362.64 Tm <696E20> Tj 1 0 0 1 275.28 362.40 Tm <74776F> Tj 0.48 Tc 1 0 0 1 52.32 351.60 Tm <70726F6A656374733A> Tj 0.24 Tc 1 0 0 1 60.96 340.32 Tm <2D20> Tj 1 0 0 1 70.08 340.32 Tm <2273747261746967726170686963616C20> Tj 1 0 0 1 142.08 338.88 Tm <616E6420> Tj 1 0 0 1 163.20 339.36 Tm <67656F6D617468656D61746963616C20> Tj 1 0 0 1 240.24 339.36 Tm <726573656172636820> Tj 1 0 0 1 282.48 339.36 Tm <6F66> Tj 0.24 Tc 1 0 0 1 69.60 328.32 Tm <706574726F6C65756D20> Tj 1 0 0 1 115.92 328.56 Tm <67656F6C6F676963616C20> Tj 1 0 0 1 160.08 328.32 Tm <73797374656D7320> Tj 1 0 0 1 196.56 328.56 Tm <696E20> Tj 1 0 0 1 207.60 328.56 Tm <43726F6174696120> Tj 1 0 0 1 241.92 329.52 Tm <2870726F6A65637420> Tj 1 0 0 1 276.96 328.80 Tm <6E6F2E> Tj 0.48 Tc 1 0 0 1 70.80 317.52 Tm <3139352D313935313239332D3032333729222C20> Tj 1 0 0 1 166.80 317.76 Tm <66696E616E63656420> Tj 1 0 0 1 208.08 318.00 Tm <627920> Tj 1 0 0 1 223.44 318.00 Tm <74686520> Tj 1 0 0 1 241.92 318.00 Tm <4D696E697374727920> Tj 1 0 0 1 282.48 318.24 Tm <6F66> Tj 0.24 Tc 1 0 0 1 69.84 306.96 Tm <536369656E63652C20> Tj 1 0 0 1 108.72 306.96 Tm <456475636174696F6E2C20> Tj 1 0 0 1 159.36 307.20 Tm <616E6420> Tj 1 0 0 1 179.28 307.20 Tm <53706F72747320> Tj 1 0 0 1 211.68 307.44 Tm <6F6620> Tj 1 0 0 1 223.44 307.44 Tm <74686520> Tj 1 0 0 1 240.72 307.68 Tm <52657075626C696320> Tj 1 0 0 1 282.48 307.68 Tm <6F66> Tj 0.24 Tc 1 0 0 1 69.84 296.40 Tm <43726F617469613B> Tj 0.24 Tc 1 0 0 1 60.96 285.60 Tm <2D20> Tj 1 0 0 1 70.08 285.36 Tm <22496D70726F76656D656E747320> Tj 1 0 0 1 137.28 284.16 Tm <696E20> Tj 1 0 0 1 148.08 284.40 Tm <67656F6C6F676963616C20> Tj 1 0 0 1 192.00 284.40 Tm <696E746572707265746174696F6E20> Tj 1 0 0 1 253.44 284.64 Tm <6D6574686F6473> Tj 0.24 Tc 1 0 0 1 69.36 273.12 Tm <7769746820> Tj 1 0 0 1 90.24 273.12 Tm <74686520> Tj 1 0 0 1 105.60 273.60 Tm <707572706F736520> Tj 1 0 0 1 142.80 273.36 Tm <6F6620> Tj 1 0 0 1 152.88 273.60 Tm <696E6372656173696E6720> Tj 1 0 0 1 198.48 273.84 Tm <7265636F7665727920> Tj 1 0 0 1 236.40 273.84 Tm <696E20> Tj 1 0 0 1 247.20 273.84 Tm <73616E6473746F6E65> Tj 0.24 Tc 1 0 0 1 69.84 262.56 Tm <7265736572766F697273222C20> Tj 1 0 0 1 120.72 262.80 Tm <66696E616E63656420> Tj 1 0 0 1 159.12 262.80 Tm <616E6420> Tj 1 0 0 1 176.88 263.04 Tm <737570706F7274656420> Tj 1 0 0 1 222.24 263.28 Tm <627920> Tj 1 0 0 1 234.48 263.28 Tm <494E412D4F696C20> Tj 1 0 0 1 267.60 263.28 Tm <436F6D2D> Tj 0.24 Tc 1 0 0 1 69.60 252.48 Tm <70616E7920> Tj 1 0 0 1 94.08 252.24 Tm <506C632E20> Tj 1 0 0 1 113.28 252.24 Tm <5468697320> Tj 1 0 0 1 135.84 252.24 Tm <77617320> Tj 1 0 0 1 156.48 252.24 Tm <6120> Tj 1 0 0 1 165.12 252.48 Tm <636F6F706572617469766520> Tj 1 0 0 1 217.44 252.72 Tm <656E646561766F757220> Tj 1 0 0 1 265.44 252.72 Tm <6F6620> Tj 1 0 0 1 277.20 252.96 Tm <746865> Tj 0.24 Tc 1 0 0 1 69.84 241.44 Tm <4669656C6420> Tj 1 0 0 1 93.84 241.44 Tm <456E67696E656572696E6720> Tj 1 0 0 1 147.36 241.68 Tm <616E6420> Tj 1 0 0 1 166.08 241.92 Tm <5265736572766F697220> Tj 1 0 0 1 209.28 241.92 Tm <446576656C6F706D656E7420> Tj 1 0 0 1 267.60 242.16 Tm <446570742E> Tj 0.24 Tc 1 0 0 1 70.32 231.60 Tm <28494E4120> Tj 1 0 0 1 91.92 230.88 Tm <506C632E2C20> Tj 1 0 0 1 113.76 230.88 Tm <43726F617469612920> Tj 1 0 0 1 151.44 231.12 Tm <616E6420> Tj 1 0 0 1 170.64 231.12 Tm <74686520> Tj 1 0 0 1 187.44 231.36 Tm <496E7374697475746520> Tj 1 0 0 1 226.08 231.36 Tm <6F6620> Tj 1 0 0 1 237.60 231.36 Tm <47656F6C6F677220> Tj 1 0 0 1 274.56 231.60 Tm <616E64> Tj 0.24 Tc 1 0 0 1 70.08 220.08 Tm <47656F6C6F676963616C20> Tj 1 0 0 1 117.60 220.56 Tm <456E67696E656572696E6720> Tj 1 0 0 1 172.56 221.52 Tm <28466163756C666620> Tj 1 0 0 1 209.52 220.80 Tm <6F6620> Tj 1 0 0 1 221.28 220.80 Tm <4D696E696E672C20> Tj 1 0 0 1 257.04 221.04 Tm <47656F6C6F6772> Tj 0.24 Tc 1 0 0 1 70.08 209.52 Tm <6172726420> Tj 1 0 0 1 90.96 209.76 Tm <506574726F6C65756D20> Tj 1 0 0 1 138.96 210.00 Tm <456E67696E656572696E672C20> Tj 1 0 0 1 197.28 210.24 Tm <556E6976657273696C7920> Tj 1 0 0 1 245.28 210.24 Tm <6F6620> Tj 1 0 0 1 258.24 210.48 Tm <5A61677265622C> Tj 0.48 Tc 1 0 0 1 70.08 198.96 Tm <43726F61746961292E> Tj 0.48 Tc /F3 12.0 Tf 1 0 0 1 53.28 178.08 Tm <372E20> Tj 1 0 0 1 68.40 178.08 Tm <5245464552454E434553> Tj 0.00 Tc /F3 7.0 Tf 1 0 0 1 57.12 167.04 Tm <3120> Tj 1 0 0 1 64.08 167.04 Tm <42616C69362C20> Tj 1 0 0 1 82.80 167.04 Tm <4420> Tj 1 0 0 1 90.48 165.84 Tm <2C20> Tj 1 0 0 1 94.08 167.04 Tm <56656C69362C20> Tj 1 0 0 1 112.56 167.04 Tm <4A2E20> Tj 1 0 0 1 119.52 167.28 Tm <2620> Tj 1 0 0 1 127.20 167.28 Tm <4D616C7669362C20> Tj 1 0 0 1 150.96 167.28 Tm <543A20> Tj 1 0 0 1 160.08 167.28 Tm <53656C656374696F6E20> Tj 1 0 0 1 186.96 167.28 Tm <6F6620> Tj 1 0 0 1 194.40 167.52 Tm <74686520> Tj 1 0 0 1 204.72 167.52 Tm <6D6F737420> Tj 1 0 0 1 220.08 167.52 Tm <617070726F707269642E746520> Tj 1 0 0 1 254.64 167.76 Tm <696E746572706F6C6174696F6E> Tj 0.00 Tc 1 0 0 1 63.60 158.64 Tm <6D6574686F6420> Tj 1 0 0 1 86.88 157.20 Tm <666F7220> Tj 1 0 0 1 99.36 158.88 Tm <73616E6473746F6E6520> Tj 1 0 0 1 131.04 159.12 Tm <72657365666C6F69727320> Tj 1 0 0 1 162.24 159.12 Tm <696E20> Tj 1 0 0 1 171.12 159.12 Tm <74686520> Tj 1 0 0 1 182.88 159.12 Tm <4B6C6F6974617220> Tj 1 0 0 1 207.36 159.36 Tm <6F696C20> Tj 1 0 0 1 218.16 159.36 Tm <616E6420> Tj 1 0 0 1 231.84 159.60 Tm <67617320> Tj 1 0 0 1 244.32 157.92 Tm <4A69656C642E20> Tj 1 0 0 1 263.52 159.36 Tm <476564276F676961> Tj 0.48 Tc 1 0 0 1 63.84 150.24 Tm <43726F61746963612C20> Tj 1 0 0 1 92.88 150.48 Tm <36312C20> Tj 1 0 0 1 104.40 150.48 Tm <6C2E20> Tj 1 0 0 1 110.16 150.48 Tm <32372D333520> Tj 1 0 0 1 129.60 150.96 Tm <2832303038292E> Tj 0.24 Tc 1 0 0 1 56.88 140.40 Tm <3220> Tj 1 0 0 1 65.04 140.40 Tm <446F7273656C2C20> Tj 1 0 0 1 89.76 140.40 Tm <442E20> Tj 1 0 0 1 100.32 140.40 Tm <264C6120> Tj 1 0 0 1 119.76 140.64 Tm <4272656368652C20> Tj 1 0 0 1 145.68 140.64 Tm <543A20> Tj 1 0 0 1 155.52 140.64 Tm <456E7669726F6E6D656E74616C20> Tj 1 0 0 1 199.68 140.88 Tm <53616D706C696E6720> Tj 1 0 0 1 228.96 141.12 Tm <2620> Tj 1 0 0 1 236.40 141.12 Tm <4D6F6E69746F72696E6720> Tj 1 0 0 1 270.48 141.12 Tm <50696D6572> Tj 0.00 Tc 1 0 0 1 64.32 133.20 Tm <284B726967726E7020> Tj 1 0 0 1 89.52 131.04 Tm <2C20> Tj 1 0 0 1 94.32 132.24 Tm <617420> Tj 1 0 0 1 102.72 132.24 Tm <77656220> Tj 1 0 0 1 117.84 132.24 Tm <7369746520> Tj 1 0 0 1 130.80 132.48 Tm <6874746F3A2F2F7777772E6365652E76742E6564752F6577722F656E7669726F6E6D656E74616C2F7465287463682F736D704C6C6C6C6C6C6C3A> Tj 0.00 Tc 1 0 0 1 63.36 123.84 Tm <6B726977676C6C6C6C74656C676C6C6C746D6C2C> Tj 0.00 Tc 1 0 0 1 56.88 113.76 Tm <3320> Tj 1 0 0 1 65.04 113.76 Tm <486F686E2C20> Tj 1 0 0 1 86.88 113.76 Tm <4D2E452E3A20> Tj 1 0 0 1 106.80 114.00 Tm <47656F7374617469737469637320> Tj 1 0 0 1 144.48 114.00 Tm <616E6420> Tj 1 0 0 1 157.68 114.24 Tm <506574726F6C65756D20> Tj 1 0 0 1 189.12 114.24 Tm <47656F6C6F677420> Tj 1 0 0 1 215.52 114.48 Tm <597A6E20> Tj 1 0 0 1 229.92 114.48 Tm <4E6F737472616E6420> Tj 1 0 0 1 261.12 114.48 Tm <5265696E686F6C642C> Tj 0.24 Tc 1 0 0 1 63.60 105.60 Tm <4E657720> Tj 1 0 0 1 78.96 105.60 Tm <596F726B2C20> Tj 1 0 0 1 97.20 105.84 Tm <61303020> Tj 1 0 0 1 109.44 106.08 Tm <707020> Tj 1 0 0 1 120.72 106.08 Tm <283139383829> Tj 0.48 Tc 1 0 0 1 52.80 82.80 Tm <4E4146544120> Tj 1 0 0 1 72.96 82.56 Tm <353920> Tj 1 0 0 1 82.08 83.04 Tm <283120> Tj 1 0 0 1 87.84 81.12 Tm <2920> Tj 1 0 0 1 91.68 82.56 Tm <333120> Tj 1 0 0 1 99.12 84.48 Tm <2D333720> Tj 1 0 0 1 110.64 83.04 Tm <283230303929> Tj 0.48 Tc 1 0 0 1 311.52 506.88 Tm <3420> Tj 1 0 0 1 318.48 506.88 Tm <4A656E73656E2C20> Tj 1 0 0 1 342.48 506.88 Tm <4A4C2C4C616B652C20> Tj 1 0 0 1 376.08 506.88 Tm <4C572C20> Tj 1 0 0 1 393.60 506.88 Tm <436F72626574742C20> Tj 1 0 0 1 420.24 506.88 Tm <5056494D20> Tj 1 0 0 1 443.04 506.88 Tm <2620> Tj 1 0 0 1 450.72 506.88 Tm <476F6767696E2C20> Tj 1 0 0 1 476.16 506.88 Tm <444A3A537461746973746963736C6272506574726F2D> Tj 0.24 Tc 1 0 0 1 318.24 498.72 Tm <6C65756D20> Tj 1 0 0 1 334.56 498.72 Tm <456E67696E6565727320> Tj 1 0 0 1 365.28 498.72 Tm <616E6420> Tj 1 0 0 1 379.20 498.72 Tm <47656F736369656E746973747320> Tj 1 0 0 1 420.24 498.72 Tm <5072656E7469636520> Tj 1 0 0 1 447.84 498.72 Tm <48616C6C20> Tj 1 0 0 1 463.92 498.72 Tm <5054522C20> Tj 1 0 0 1 482.64 498.72 Tm <4E657720> Tj 1 0 0 1 499.20 498.72 Tm <4A65727365792C20> Tj 1 0 0 1 522.48 498.72 Tm <33393020> Tj 1 0 0 1 535.68 498.96 Tm <7070> Tj 0.48 Tc 1 0 0 1 318.72 490.56 Tm <286939653729> Tj 0.24 Tc 1 0 0 1 311.76 480.48 Tm <3520> Tj 1 0 0 1 318.72 480.48 Tm <4A6F75726E656C2C20> Tj 1 0 0 1 344.64 480.48 Tm <4120> Tj 1 0 0 1 352.08 480.48 Tm <4720> Tj 1 0 0 1 361.44 480.48 Tm <2620> Tj 1 0 0 1 368.88 480.48 Tm <4875696A6272656774732C20> Tj 1 0 0 1 404.16 480.48 Tm <4320> Tj 1 0 0 1 410.64 480.48 Tm <4A2E3A20> Tj 1 0 0 1 418.80 480.48 Tm <4D696E696E6720> Tj 1 0 0 1 441.12 480.48 Tm <47656F7374616C6C736C6C637320> Tj 1 0 0 1 479.52 480.48 Tm <41636164656D696320> Tj 1 0 0 1 511.44 480.48 Tm <50726573732C20> Tj 1 0 0 1 530.64 480.48 Tm <4C6F6E2D> Tj 0.48 Tc 1 0 0 1 318.48 472.08 Tm <646F6E2C36303020> Tj 1 0 0 1 345.60 472.56 Tm <707020> Tj 1 0 0 1 357.12 472.32 Tm <2831393738292E> Tj 0.48 Tc 1 0 0 1 311.76 462.00 Tm <3620> Tj 1 0 0 1 318.48 462.00 Tm <4B726967652C442E472E3A2C34737461746973746963616C617070726F616368746F736F6D6562617369636D696E6576616C756174696F6E70726F626C656D736F6E746865> Tj 0.24 Tc 1 0 0 1 318.48 453.84 Tm <57697477617465727372616E642C20> Tj 1 0 0 1 363.12 453.84 Tm <4A6F75726E616C20> Tj 1 0 0 1 386.88 453.84 Tm <6F6620> Tj 1 0 0 1 394.80 454.08 Tm <74686520> Tj 1 0 0 1 406.32 454.08 Tm <4368656E6963616C2C20> Tj 1 0 0 1 438.00 453.84 Tm <4D6574616C6C7572676963616C20> Tj 1 0 0 1 479.76 453.84 Tm <616E6420> Tj 1 0 0 1 492.72 453.84 Tm <4D696E696E6720> Tj 1 0 0 1 516.00 453.84 Tm <536F636965747920> Tj 1 0 0 1 539.28 454.08 Tm <6F66> Tj 0.48 Tc 1 0 0 1 318.48 445.68 Tm <536F75746820> Tj 1 0 0 1 337.44 445.68 Tm <4166726963612C20> Tj 1 0 0 1 360.00 445.68 Tm <35322C20> Tj 1 0 0 1 370.80 445.68 Tm <6931392D31333920> Tj 1 0 0 1 396.96 445.92 Tm <2831393531292E> Tj 0.24 Tc 1 0 0 1 311.52 435.36 Tm <3720> Tj 1 0 0 1 318.72 435.60 Tm <4C696562686F6C642C20> Tj 1 0 0 1 348.72 435.60 Tm <4120> Tj 1 0 0 1 356.16 435.60 Tm <4D20> Tj 1 0 0 1 364.56 434.40 Tm <2C20> Tj 1 0 0 1 368.16 435.60 Tm <526F7373692C20> Tj 1 0 0 1 387.84 435.60 Tm <5220> Tj 1 0 0 1 394.80 435.60 Tm <4520> Tj 1 0 0 1 403.20 435.60 Tm <2620> Tj 1 0 0 1 410.64 435.60 Tm <4B656D702C20> Tj 1 0 0 1 432.24 435.60 Tm <57502E3A20> Tj 1 0 0 1 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endstream endobj 72 0 obj 29159 endobj 73 0 obj << /Length 74 0 R >> stream m -0.3 -0.2 -0.1 0.01 0.1 -0.2 -0.1 0.01 0.1 v.z direciion of oo(E Lg(E-1U i te$tstoppinsP f, ,r- prekid tesia - i -'6girEv. the case when the Lagrange multiplicatoris decreased in negative valuessludaj promjene vrijednosti Lagrangeovogmultiplikatora prerna negalivnim vrijednostima {a ii minimu$ of vaiiancerninimurd varijanbe the case when the Lagrange multiplicatoris decreased in positive valuessludaj promjene vrijednostr Lagrangeovogmultiplikaiora prema pozitivnim vrijednostirna Fig. 3. Graphical representation of how to select the most appropriate value for the Lagrange multiplicator using randomsamplingSl. 3. Slikovni prikaz nadina odabira najprimjerenije vrijednosti Lagrangeovog multiplikatora metodom sludajnog uzorkovanja (!"e(oo)c)z 4. It is irnportant to know that if, for the first positive m(m>0), the variance immediately sta-rt to increase,then it can be considered that the minimum krigingvariance is found in the negative scale and randomsampling czrn cease (Figure 3 left).We believe that the four rules discussed above completelydescribe the correct procedure for selecting the Lagrangemultiplicator in Ordinary Kriging equauons. Thus, usingthis technique is one of the best interpolation algorithmsfor mapping geological variables.ACKNOWLEDGMENTSThis paper represents the theoretical analysis ofgeomathematical methods performed in 2OO8 in twoprojects:- "stratigraphical and geomathematical research ofpetroleum geological systems in Croatia (project no.195-1951293-0237)", financed by the Ministry ofScience, Education, and Sports of the Republic ofCroatia;- "Improvements in geological interpretation methodswith the purpose of increasing recovery in sandstonereservoirs", financed and supported by INA-Oil Com-pany Plc. This was a cooperative endeavour of theField Engineering and Reservoir Development Dept.(INA Plc., Croatia) and the Institute of Geologr andGeological Engineering (Faculff of Mining, Geolograrrd Petroleum Engineering, Universily of Zagreb,Croatia).7. REFERENCES1 Bali6, D , Veli6, J. & Malvi6, T: Selection of the most approprid.te interpolationmethod for sandstone resefloirs in the Kloitar oil and gas Jield. Ged'ogiaCroatica, 61, l. 27-35 (2008).2 Dorsel, D. &La Breche, T: Environmental Sampling & Monitoring Pimer(Krigrnp , at web site htto://www.cee.vt.edu/ewr/environmental/te(tch/smpLllllll:kriwglllltelgllltml,3 Hohn, M.E.: Geostatistics and Petroleum Geologt Yzn Nostrand Reinhold,New York, a00 pp (1988)NAFTA 59 (1 ) 31 -37 (2009) 4 Jensen, JL,Lake, LW, Corbett, PVIM & Goggin, DJ:StatisticslbrPetro-leum Engineers and Geoscientists Prentice Hall PTR, New Jersey, 390 pp(i9e7)5 Journel, A G & Huijbregts, C J.: Mining Geostallsllcs Academic Press, Lon-don,600 pp (1978).6 Krige,D.G.:,4statisticalapproachtosomebasicminevaluationproblemsontheWitwatersrand, Journal of the Chenical, Metallurgical and Mining Society ofSouth Africa, 52, i19-139 (1951).7 Liebhold, A M , Rossi, R E & Kemp, WP.: Geostatistics and geographic infor-mation $tstem in applied insecl ecolog\ Annual Review of Entomology, 38,303-i27 (1993\8 Malvil'[.: Kriging, cokriging or stochastical simulations, and the choice betweendeterministic or sequential approaches, Geologia Croalica (Zagreb), 61, 1,3747 (2008a\.9.Malvr6,'L: Izrorla karte poroznosti kri$ranjem u pjel1enjaikim leiiitima, primjeriz Savske deprestje (Production of aporosity map by kriging in sandstone reservoirs,case study Jromthe &na Depression),Kartografija i geoinfornactje (Zagreb),7,9,12-19 (2008b)10 Malvii, T & Durekovi6, M: Application of methods: mverse distance weighting,ordinary krigng and collocated cokriging in porosity oaluution, and comparison olresults on the Benicanci and Slari Grodac Fields in Croatia, Nafta (Zagreb), 54, 9,33L-340 (2003\.11 Matheron, G: Les Variables Rgionalises et leur Estimation Masson & Cie,Paris,306pp (196s).12 Smoljanovi6, S . &Malvi6,T.: Improvements in reservoir characteization appb)inggeostalistical modelling (estimatlon & stochastic simulations vs standard intetpola-tion methods), case stu$t from Croatia,Nafta (Zagreb),56,2,57-63 (2005) *Authors:Tomislav Malvi6, DSc, craduate in Geology, INA-Naftaplin, Reservoir Engi-neering and Field Development Department, Advisor,tomislavmalvic@ina. hrDavorin Bali6, Graduate in Geology, INA-Naftaplin, Reservoir Engineeringand Field Development Depaftment, Field Development Engineer,davorin.balic@ina. hr 37 endstream endobj 74 0 obj 4404 endobj 75 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 76 0 obj << /Filter /FlateDecode /Length 131 >> stream xE= A > M gG\ m#{7d@洷N3;t>of!{+G&Zr57sWE6|7} AGC9Wq]+ endstream endobj 77 0 obj << /Type /Page /MediaBox [ 0 0 595.20 841.68 ] /Parent 13 0 R /Resources << /Font << /F3 7 0 R >> /XObject << /Obj67 67 0 R /Obj68 68 0 R /Obj69 69 0 R /Obj70 70 0 R >> /Subtype 73 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 75 0 R 76 0 R 71 0 R ] >> endobj 78 0 obj << /Type /XObject /Subtype /Image /Width 1240 /Height 1753 /BitsPerComponent 8 /ColorSpace /DeviceGray /Filter /DCTDecode /Length 39112 >> stream JFIF (B+(%%(Q:=0B`Ued_U][jxjqs[]gɺǙ(#(N++Nn]n  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<6965646E616469627520> Tj 1 0 0 1 225.60 555.84 Tm <6F6269456E6F676120> Tj 1 0 0 1 258.24 555.84 Tm <6B726967696E67612E> Tj 0.48 Tc 1 0 0 1 189.12 537.84 Tm <4B6A75696E6520> Tj 1 0 0 1 217.68 537.84 Tm <72546569693A20> Tj 1 0 0 1 239.52 538.32 Tm <67656F73746174697374696B612C20> Tj 1 0 0 1 288.48 538.80 Tm <6A65646E6F737461766E6920> Tj 1 0 0 1 332.16 537.84 Tm <6B726967696E672C20> Tj 1 0 0 1 360.96 537.84 Tm <4C616772616E67656F7620> Tj 1 0 0 1 405.60 537.60 Tm <6D756C7469706C696B61746F722C20> Tj 1 0 0 1 453.36 537.36 Tm <6F6269646E6920> Tj 1 0 0 1 477.60 537.12 Tm <6B726967696E67> Tj 0.24 Tc /F3 8.0 Tf 1 0 0 1 239.28 473.76 Tm <6E61707265646E6F6D> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 305.76 487.20 Tm <322E20> Tj 1 0 0 1 321.12 486.96 Tm <53564F4A5354564120> Tj 1 0 0 1 392.40 486.96 Tm <4B524947494E4741> Tj 0.00 Tc /F3 9.0 Tf 1 0 0 1 305.04 468.72 Tm <5465686E696B6520> Tj 1 0 0 1 342.48 468.48 Tm <6B726967696E676120> Tj 1 0 0 1 380.64 468.72 Tm <707269706164616A7520> Tj 1 0 0 1 426.00 468.24 Tm <736B75707520> Tj 1 0 0 1 456.00 468.00 Tm <6C696E6561726E696820> Tj 1 0 0 1 498.24 467.76 Tm <616C676F726974616D61> Tj 0.24 Tc 1 0 0 1 305.04 458.16 Tm <6D65746F646520> Tj 1 0 0 1 338.40 458.16 Tm <6E616A6D616E6A696820> Tj 1 0 0 1 383.52 457.92 Tm <6B766164726174612E20> Tj 1 0 0 1 426.00 457.68 Tm <4F646162697220> Tj 1 0 0 1 457.68 457.44 Tm <7465686E696B6520> Tj 1 0 0 1 491.28 457.20 Tm <74656D656C6A6920> Tj 1 0 0 1 522.00 457.20 Tm <736520> Tj 1 0 0 1 532.80 457.20 Tm <6E61> Tj 0.24 Tc 1 0 0 1 305.28 447.36 Tm <73746F6861737469646B696D20> Tj 1 0 0 1 364.32 447.36 Tm <73766F6A737476696D6120> Tj 1 0 0 1 413.28 447.12 Tm <736C7564616A6E6520> Tj 1 0 0 1 451.68 446.88 Tm <766172696A61626C6520> Tj 1 0 0 1 492.72 446.88 Tm <6B6F6A7520> Tj 1 0 0 1 516.00 446.88 Tm <736520> Tj 1 0 0 1 528.96 446.88 Tm <5A656C69> Tj 0.24 Tc 1 0 0 1 305.04 437.04 Tm <696E746572706F6C697261746920> Tj 1 0 0 1 367.68 436.80 Tm <6B726967696E676F6D2E20> Tj 1 0 0 1 425.28 436.56 Tm <5465686E696B6520> Tj 1 0 0 1 470.64 436.32 Tm <736520> Tj 1 0 0 1 490.80 436.08 Tm <74656D656C6A6520> Tj 1 0 0 1 532.80 436.32 Tm <6E61> Tj 0.24 Tc 1 0 0 1 305.28 426.24 Tm <6F647265646976616E6A7520> Tj 1 0 0 1 358.32 426.24 Tm <74655A696E736B696820> Tj 1 0 0 1 399.12 426.00 Tm <6B6F65666963696A656E61746120> Tj 1 0 0 1 454.80 425.76 Tm <757A20> Tj 1 0 0 1 467.52 426.24 Tm <706F6472617A74786E696A6576616E6A65> Tj 0.24 Tc 1 0 0 1 305.04 415.68 Tm <6E65707269737472616E6F7374692E20> Tj 1 0 0 1 371.28 415.44 Tm <5072696C696B6F6D20> Tj 1 0 0 1 411.36 415.68 Tm <70726F636A656E6520> Tj 1 0 0 1 450.96 414.96 Tm <77696A65646E6F73746920> Tj 1 0 0 1 499.92 414.96 Tm <6B726967696E676F6D> Tj 0.24 Tc 1 0 0 1 305.04 405.12 Tm <7376616B6F6D6520> Tj 1 0 0 1 345.84 405.36 Tm <706F6461746B7520> Tj 1 0 0 1 384.72 404.88 Tm <756B6C6A7564656E6F6D20> Tj 1 0 0 1 438.48 404.64 Tm <7520> Tj 1 0 0 1 447.36 404.88 Tm <706F73747570616B20> Tj 1 0 0 1 491.04 404.40 Tm <646F64696A656C6A656E20> Tj 1 0 0 1 534.96 405.12 Tm <6A65> Tj 0.24 Tc 1 0 0 1 305.28 394.56 Tm <6F64726564656E6920> Tj 1 0 0 1 348.24 394.56 Tm <74655A696E736B6920> Tj 1 0 0 1 387.36 394.32 Tm <6B6F65666963696A656E7420> Tj 1 0 0 1 437.04 395.04 Tm <282C312920> Tj 1 0 0 1 452.64 394.08 Tm <6B6F6A696D20> Tj 1 0 0 1 481.92 394.80 Tm <6A6520> Tj 1 0 0 1 495.36 394.32 Tm <70726F63696A656E6A656E> Tj 0.24 Tc 1 0 0 1 305.04 384.00 Tm <6E6A65676F7620> Tj 1 0 0 1 340.08 384.00 Tm <75746A6563616A20> Tj 1 0 0 1 376.32 384.00 Tm <6E6120> Tj 1 0 0 1 394.80 383.76 Tm <756B75706E6920> Tj 1 0 0 1 433.92 383.52 Tm <726164756E20> Tj 1 0 0 1 466.56 383.28 Tm <77696A65646E6F73746920> Tj 1 0 0 1 520.32 383.28 Tm <746F646B65> Tj 0.24 Tc 1 0 0 1 304.80 373.92 Tm <70726F636A656E652E20> Tj 1 0 0 1 346.80 373.44 Tm <4F646E6F736920> Tj 1 0 0 1 379.68 373.44 Tm <697A6D65647520> Tj 1 0 0 1 413.28 373.68 Tm <706F73746F6A6569696820> Tj 1 0 0 1 459.12 372.96 Tm <77696A65646E6F73746920> Tj 1 0 0 1 506.88 372.96 Tm <6920> Tj 1 0 0 1 512.16 372.72 Tm <746F646B6520> Tj 1 0 0 1 537.36 372.72 Tm <75> Tj 0.24 Tc 1 0 0 1 305.04 362.88 Tm <6B6F6A6F6A20> Tj 1 0 0 1 332.16 362.88 Tm <736520> Tj 1 0 0 1 346.80 362.88 Tm <6F6E6120> Tj 1 0 0 1 367.68 363.36 Tm <70726F636A656E6A756A6520> Tj 1 0 0 1 420.96 362.64 Tm <697A7261692C6176616A7520> Tj 1 0 0 1 469.92 362.16 Tm <736520> Tj 1 0 0 1 484.08 362.16 Tm <77696A65646E6F7374696D61> Tj 0.24 Tc 1 0 0 1 304.56 352.32 Tm <766172696F6772616D6120> Tj 1 0 0 1 357.12 353.04 Tm <28616B6F20> Tj 1 0 0 1 378.96 352.56 Tm <70726574706F73746179696D6F20> Tj 1 0 0 1 444.00 352.32 Tm <706F73746F6A616E6A6520> Tj 1 0 0 1 491.52 351.60 Tm <696E7472696E69737469646B65> Tj 0.24 Tc 1 0 0 1 304.80 341.76 Tm <6869706F74657A65292C20> Tj 1 0 0 1 349.20 341.76 Tm <746520> Tj 1 0 0 1 359.52 341.76 Tm <726A65646520> Tj 1 0 0 1 384.00 341.76 Tm <6B6F766172696A616E636520> Tj 1 0 0 1 435.84 341.28 Tm <7520> Tj 1 0 0 1 444.48 341.28 Tm <736C7564616A7520> Tj 1 0 0 1 477.84 341.04 Tm <73746163696F6E6172616E6F737469> Tj 0.24 Tc 1 0 0 1 305.04 331.20 Tm <647275676F676120> Tj 1 0 0 1 344.40 331.20 Tm <726564612E20> Tj 1 0 0 1 371.52 331.20 Tm <4E6120> Tj 1 0 0 1 388.32 330.96 Tm <74616A20> Tj 1 0 0 1 404.16 330.96 Tm <6E6164696E20> Tj 1 0 0 1 432.96 330.72 Tm <737520> Tj 1 0 0 1 448.32 330.72 Tm <6F64726564656E6520> Tj 1 0 0 1 492.96 330.48 Tm <7A617669736E6F73746920> Tj 1 0 0 1 540.24 330.48 Tm <69> Tj 0.24 Tc 1 0 0 1 304.80 320.64 Tm <75746A6563616A6920> Tj 1 0 0 1 344.88 320.88 Tm <706F6A6564696E6520> Tj 1 0 0 1 390.72 320.40 Tm <6C6F6B6163696A6520> Tj 1 0 0 1 433.20 320.16 Tm <7320> Tj 1 0 0 1 447.12 319.92 Tm <6F627A69726F6D20> Tj 1 0 0 1 492.00 319.92 Tm <6E6120> Tj 1 0 0 1 511.68 319.92 Tm <6E6A657A696E75> Tj 0.24 Tc 1 0 0 1 304.80 310.08 Tm <7564616C6A656E6F737420> Tj 1 0 0 1 351.12 310.08 Tm <6920> Tj 1 0 0 1 355.92 309.84 Tm <736D6A657220> Tj 1 0 0 1 382.08 309.84 Tm <6F6420> Tj 1 0 0 1 394.56 309.84 Tm <746F646B6520> Tj 1 0 0 1 419.52 309.60 Tm <64696A6120> Tj 1 0 0 1 436.32 309.60 Tm <736520> Tj 1 0 0 1 446.64 309.36 Tm <77696A65646E6F737420> Tj 1 0 0 1 491.28 309.84 Tm <70726F636A656E6A756A652E> Tj 0.24 Tc 1 0 0 1 311.76 286.56 Tm <5520> Tj 1 0 0 1 323.04 287.04 Tm <70726176696C7520> Tj 1 0 0 1 357.84 286.56 Tm <6D6F64656C20> Tj 1 0 0 1 389.04 286.32 Tm <7320> Tj 1 0 0 1 397.44 286.32 Tm <766536696D20> Tj 1 0 0 1 426.96 286.32 Tm <62726F6A656D20> Tj 1 0 0 1 462.00 286.08 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 511.92 285.84 Tm <746F64616B612C> Tj 0.24 Tc 1 0 0 1 304.32 276.00 Tm <766536696D20> Tj 1 0 0 1 332.64 276.00 Tm <766172696F6772616D736B696D20> Tj 1 0 0 1 400.80 275.76 Tm <646F7365676F6D2C20> Tj 1 0 0 1 444.72 275.52 Tm <6D616E6A696D20> Tj 1 0 0 1 481.20 275.52 Tm <6F6473747570616E6A656D20> Tj 1 0 0 1 540.00 275.52 Tm <69> Tj 0.24 Tc 1 0 0 1 304.56 265.44 Tm <62657A20> Tj 1 0 0 1 323.28 265.44 Tm <75747764656E6520> Tj 1 0 0 1 365.52 265.44 Tm <616E697A6F74726F70696A6520> Tj 1 0 0 1 420.00 265.20 Tm <62697420> Tj 1 0 0 1 436.08 264.96 Tm <366520> Tj 1 0 0 1 449.04 265.44 Tm <70756E6F20> Tj 1 0 0 1 475.20 265.44 Tm <706F757A64616E696A692E20> Tj 1 0 0 1 527.28 264.96 Tm <42657A> Tj 0.24 Tc 1 0 0 1 305.52 254.88 Tm <6F627A3B726120> Tj 1 0 0 1 335.76 255.84 Tm <6A6520> Tj 1 0 0 1 349.20 254.88 Tm <6C6920> Tj 1 0 0 1 360.00 255.36 Tm <70726F73746F726E6120> Tj 1 0 0 1 407.28 254.64 Tm <76657A6120> Tj 1 0 0 1 431.52 254.40 Tm <6F706973616E6120> Tj 1 0 0 1 469.68 254.16 Tm <766172696F6772616D69726E6120> Tj 1 0 0 1 534.48 254.16 Tm <696C69> Tj 0.24 Tc 1 0 0 1 304.56 244.56 Tm <6B6F766172696A616E63616D6120> Tj 1 0 0 1 372.72 244.32 Tm <74655A696E736B6920> Tj 1 0 0 1 411.36 244.08 Tm <6B6F65666963696A656E7420> Tj 1 0 0 1 460.80 243.84 Tm <76657A617220> Tj 1 0 0 1 489.84 243.84 Tm <747A20> Tj 1 0 0 1 505.20 244.08 Tm <706F6A6564696E75> Tj 0.24 Tc 1 0 0 1 304.56 234.00 Tm <496F6B6163696A7520> Tj 1 0 0 1 347.04 233.76 Tm <69736B6C6A756469766F20> Tj 1 0 0 1 395.52 234.48 Tm <6A6520> Tj 1 0 0 1 412.08 233.52 Tm <6D6A65726120> Tj 1 0 0 1 446.16 233.28 Tm <6E6A657A6C6E6520> Tj 1 0 0 1 484.08 233.28 Tm <7564616C6A656E6F73746920> Tj 1 0 0 1 539.76 233.04 Tm <69> Tj 0.24 Tc 1 0 0 1 304.56 223.20 Tm <6F72696A656E746163696A6520> Tj 1 0 0 1 355.20 223.20 Tm <7520> Tj 1 0 0 1 364.56 222.96 Tm <6F646E6F737520> Tj 1 0 0 1 399.12 222.96 Tm <6E6120> Tj 1 0 0 1 413.04 222.96 Tm <746F646B7520> Tj 1 0 0 1 440.40 223.20 Tm <70726F636A656E652C20> Tj 1 0 0 1 483.60 222.48 Tm <6120> Tj 1 0 0 1 491.76 222.48 Tm <6E6520> Tj 1 0 0 1 505.20 222.48 Tm <6920> Tj 1 0 0 1 511.68 222.48 Tm <73747661726E65> Tj 0.24 Tc 1 0 0 1 304.08 212.64 Tm <77696A65646E6F73746920> Tj 1 0 0 1 357.12 212.88 Tm <706F6461746B6120> Tj 1 0 0 1 399.60 212.40 Tm <7520> Tj 1 0 0 1 412.80 212.40 Tm <746F6A20> Tj 1 0 0 1 431.52 212.16 Tm <746F646B692E20> Tj 1 0 0 1 462.96 212.16 Tm <506F7261737420> Tj 1 0 0 1 497.04 211.92 Tm <77696A65646E6F737469> Tj 0.24 Tc 1 0 0 1 304.08 202.08 Tm <766172696F6772616D6120> Tj 1 0 0 1 363.60 202.80 Tm <28756E7574617220> Tj 1 0 0 1 405.84 201.60 Tm <646F7365676120> Tj 1 0 0 1 444.96 202.08 Tm <70726F73746F726E6520> Tj 1 0 0 1 497.52 201.36 Tm <7A617669736E6F73746929> Tj 0.24 Tc 1 0 0 1 304.32 191.52 Tm <756B617A756A6520> Tj 1 0 0 1 344.64 191.52 Tm <6E6120> Tj 1 0 0 1 361.92 191.76 Tm <706F7261737420> Tj 1 0 0 1 396.24 192.48 Tm <22726564612720> Tj 1 0 0 1 429.12 191.04 Tm <696C6920> Tj 1 0 0 1 444.00 191.28 Tm <706F757A64616E6F73746920> Tj 1 0 0 1 503.04 191.28 Tm <70726F636A656E652C> Tj 0.24 Tc 1 0 0 1 304.56 180.72 Tm <6F646E6F736E6F20> Tj 1 0 0 1 343.92 180.72 Tm <7665646120> Tj 1 0 0 1 365.28 180.72 Tm <77696A65646E6F737420> Tj 1 0 0 1 411.36 180.48 Tm <646F626976656E6120> Tj 1 0 0 1 453.36 180.24 Tm <7A616E656B6920> Tj 1 0 0 1 486.24 180.48 Tm <70617220> Tj 1 0 0 1 504.00 180.48 Tm <706F6B617A756A65> Tj 0.24 Tc 1 0 0 1 304.32 170.16 Tm <6E6120> Tj 1 0 0 1 324.72 170.16 Tm <7665366520> Tj 1 0 0 1 352.80 170.16 Tm <6D656475736F626E6F20> Tj 1 0 0 1 413.04 169.92 Tm <6D656475646A656C6F76616E6A6520> Tj 1 0 0 1 490.08 169.44 Tm <746A6820> Tj 1 0 0 1 511.44 169.44 Tm <746F64616B612C> Tj 0.24 Tc 1 0 0 1 304.32 160.08 Tm <70726574706F737461766C6A616A75646920> Tj 1 0 0 1 381.60 159.36 Tm <646120> Tj 1 0 0 1 396.96 160.32 Tm <6A6520> Tj 1 0 0 1 411.36 159.36 Tm <6E6A69686F766120> Tj 1 0 0 1 448.32 159.12 Tm <7564616C6A656E6F737420> Tj 1 0 0 1 498.96 158.88 Tm <6D616E6A6120> Tj 1 0 0 1 531.84 158.88 Tm <6F64> Tj 0.24 Tc 1 0 0 1 304.56 149.04 Tm <646F7365676120> Tj 1 0 0 1 336.00 149.28 Tm <70726F73746F726E6520> Tj 1 0 0 1 380.88 148.80 Tm <7A617669736E6F73696C2E20> Tj 1 0 0 1 428.40 148.56 Tm <446F6461746E7520> Tj 1 0 0 1 468.00 148.32 Tm <6B76616C697465747520> Tj 1 0 0 1 507.36 148.80 Tm <70726F636A656E69> Tj 0.24 Tc 1 0 0 1 304.56 138.24 Tm <64617420> Tj 1 0 0 1 320.40 138.24 Tm <366520> Tj 1 0 0 1 331.44 138.72 Tm <70726176696C6E696A6920> Tj 1 0 0 1 372.00 138.24 Tm <726173706F72656420> Tj 1 0 0 1 412.80 138.24 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 460.56 137.76 Tm <746F64616B612E20> Tj 1 0 0 1 493.92 137.76 Tm <5A626F6720> Tj 1 0 0 1 517.44 137.76 Tm <73766F6A6968> Tj 0.24 Tc 1 0 0 1 304.32 127.92 Tm <6B76616C697465746120> Tj 1 0 0 1 348.24 127.92 Tm <6B726967696E6720> Tj 1 0 0 1 384.96 128.64 Tm <6A652C20> Tj 1 0 0 1 403.92 127.68 Tm <6B616F20> Tj 1 0 0 1 427.68 127.44 Tm <7374617469737469646B6120> Tj 1 0 0 1 479.04 127.20 Tm <696E746572706F6C6163696A736B61> Tj 0.24 Tc 1 0 0 1 304.08 117.12 Tm <7465686E696B612C20> Tj 1 0 0 1 342.00 117.12 Tm <6F706973616E6120> Tj 1 0 0 1 378.24 117.12 Tm <616B726F6E69726E6F6D20> Tj 1 0 0 1 431.28 115.92 Tm <3B2C424C5545222C20> Tj 1 0 0 1 469.20 116.64 Tm <6B6F6A6920> Tj 1 0 0 1 488.16 116.64 Tm <7520> Tj 1 0 0 1 497.04 116.64 Tm <656E676C65736B6F6D> Tj 0.24 Tc 1 0 0 1 304.08 106.56 Tm <7A6E61646920> Tj 1 0 0 1 329.28 105.84 Tm <2C2C4265737420> Tj 1 0 0 1 354.00 106.56 Tm <4C696E65617220> Tj 1 0 0 1 384.72 106.32 Tm <556E62696173656420> Tj 1 0 0 1 427.44 106.08 Tm <457374696D61746F72222E> Tj 0.72 Tc /F3 12.0 Tf 1 0 0 1 51.84 487.20 Tm <312E20> Tj 1 0 0 1 66.00 486.96 Tm <494D44> Tj 0.00 Tc /F3 9.0 Tf 1 0 0 1 50.16 473.76 Tm <506F73747570616B20> Tj 1 0 0 1 108.72 473.76 Tm <6B726967696E676120> Tj 1 0 0 1 162.24 473.52 Tm <736D6174726120> Tj 1 0 0 1 211.20 473.76 Tm <7365> Tj 0.24 Tc 1 0 0 1 50.16 463.20 Tm <696E746572706F6C6163696A736B6F6D20> Tj 1 0 0 1 130.32 463.20 Tm <6D65746F646F6D20> Tj 1 0 0 1 179.28 463.20 Tm <7A6120> Tj 1 0 0 1 196.80 463.68 Tm <70726F636A656E7520> Tj 1 0 0 1 242.64 463.20 Tm <77696A65646E6F737469> Tj 0.24 Tc 1 0 0 1 50.16 452.64 Tm <726567696F6E616C697A6972616E6520> Tj 1 0 0 1 117.12 452.64 Tm <766172696A61626C6520> Tj 1 0 0 1 157.92 452.64 Tm <7520> Tj 1 0 0 1 167.76 452.64 Tm <6F64616272616E696D20> Tj 1 0 0 1 218.88 452.64 Tm <746F646B616D6120> Tj 1 0 0 1 259.92 452.64 Tm <6D72654C652E> Tj 0.24 Tc 1 0 0 1 50.16 442.08 Tm <506F6420> Tj 1 0 0 1 70.32 442.56 Tm <706F6A6D6F6D20> Tj 1 0 0 1 110.40 442.08 Tm <72656720> Tj 1 0 0 1 124.80 442.08 Tm <696F6E616C697A697220> Tj 1 0 0 1 160.80 441.84 Tm <616E6520> Tj 1 0 0 1 181.92 442.08 Tm <7520> Tj 1 0 0 1 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<4E4146544120> Tj 1 0 0 1 485.28 83.04 Tm <353920> Tj 1 0 0 1 494.40 83.52 Tm <283120> Tj 1 0 0 1 500.16 81.36 Tm <2920> Tj 1 0 0 1 503.76 83.04 Tm <33382D343320> Tj 1 0 0 1 522.96 83.28 Tm <283230306529> Tj ET endstream endobj 84 0 obj 29320 endobj 85 0 obj << /Length 86 0 R >> stream t- Linearnost i Lagrangeov linearni multiptikator u jedna dilbama obiinoga kriginga T. Malui( i D. BaIi6 IZVORNI ZNASTVTNI CLANAKAnalizitane su jednadibe iednostavnoga i obiEnoga kfiginga kako bi se uoiila razlika u izraiunu procjene utoiki, te zorno pdkazalo na koji naEin se Lagrangeov multiplikator iskazuje kao veliiina u tehnici obiEnogakiginga koja minimizira variiancu ptocjene. lletalinom analizom matrica i linearnih jednadZbi, koje se ina6eautomatski izra6unavaju u pozadini ladunalnih programa za kartiranie, prikazana je vainost geomatematikeu jednoj od temelinih geolo5kih dielatnosti - kartitanju. l{adalie, izvoileniem niza jednadibi kroz Elanakptuia se na inienjetski na6in razumiievanie algoritama jednostavnoga i obidnoga kriginga, koje su danasdvije naj6e56e koli5tene geostatistidke tehnike {uz indikatotski ktiging kao tre6e). U zakljuEku su predloienipostupci koiima se moie odtediti najptimjetenija wijednost Lagrangeovog multiplikatora za bilo kojuiednadibu obiEnoga kriginga.Kjuine rTeii: geostatistika, jednostavni kriging, Lagrangeov multiplikator, obidni kriging naprednom 2. SVOJSTVA KRIGINGATehnike kriginga pripadaju skupu linearnih algoritamametode najmanjih kvadrata. Odabir tehnike temelji se nastohastidkim svojstvima sludajne varijable koju se Zeliinterpolirati krigingom. Tehnike se temelje naodredivanju teZinskih koeficijenata uz podraztxnijevanjenepristranosti. Prilikom procjene wijednosti krigingomsvakome podatku ukljudenom u postupak dodijeljen jeodredeni teZinski koeficijent (,1) kojim je procijenjennjegov utjecaj na ukupni radun wijednosti todkeprocjene. Odnosi izmedu postojeiih wijednosti i todke ukojoj se ona procjenjuje izrai,avaju se wijednostimavariograma (ako pretpostayimo postojanje intrinistidkehipoteze), te rjede kovarijance u sludaju stacionaranostidrugoga reda. Na taj nadin su odredene zavisnosti iutjecaji pojedine lokacije s obzirom na njezinuudaljenost i smjer od todke dija se wijednost procjenjuje.U pravilu model s ve6im brojem kontrolnih todaka,ve6im variogramskim dosegom, manjim odstupanjem ibez utwdene anizotropije bit 6e puno pouzdaniji. Bezobz;ra je li prostorna veza opisana variogramirna ilikovarijancama teZinski koeficijent vezar tz pojedinuIokaciju iskljudivo je mjera njezlne udaljenosti iorijentacije u odnosu na todku procjene, a ne i stvarnewijednosti podatka u toj todki. Porast wijednostivariograma (unutar dosega prostorne zavisnosti)ukazuje na porast "reda' ili pouzdanosti procjene,odnosno veda wijednost dobivena zaneki par pokazujena ve6e medusobno medudjelovanje tjh todaka,pretpostavljajudi da je njihova udaljenost manja oddosega prostorne zavisnosil. Dodatnu kvalitetu procjenidat 6e pravilniji raspored kontrolnih todaka. Zbog svojihkvaliteta kriging je, kao statistidka interpolacijskatehnika, opisana akronirnom ;,BLUE", koji u engleskomznadi ,,Best Linear Unbiased Estimator". 1. IMDPostupak kriginga smatra se interpolacijskom metodom za procjenu wijednostiregionalizirane varijable u odabranim todkama mreLe.Pod pojmom reg ionalizir ane u arij able podrazumijevase takva varijabla koja moZe biti smatrana sludajnomvarijablom u beskonadno malom podrudju okopojedinadne lokacije, ali gdje postoji funkcija kojom se ucijelom takvom podrudju moZe opisati odnos izmeduviSe tal{vih varijabli. Pod pojmom "sludajna'podrazumijeva se kako {erojatnost da varijabla budemanja od nekoga realnoga broja postoji za svaki takavbroj.I{rigrng kao statistiika metoda procjene je imenovanaprema juZnoafridkom inZenjeru Krigeuo koji ju je prviupotrijebio i opisao prilikom procjene koncentracijerudade u leZiStirna zlata. Zatrrn je wemenom uslijediodaljnji teoretski razvoj metode, najve6im dijelom ufrancuskih znanstvenika, poglavito Matheronall, te jerazvijen niz tehnika kriginga (poput jednostavnoga,obidnoga, indikatorskoga i drugih).Procjena krigingom se temelji na upotrebi postojedihmjerenja (tzv. kontrolnih todaka) diji je utjecaj izraienodgovaraju6im teZinskim koeficijentima. Takva procjenatakoder podrazumijeva da su zadovoljeni odredenikriteriji. Prema njima procjena mora biti nepristrana tenadinjena tako da je varijanca razlTke izmedu stvarnih iprocijenjenih wijednosti u odabranim todkamanajmanja moguia. Ta wijednost naziva se i varijancomkriginga, koja predstavlja proqeru kvalitete interpolacijete je usko povezana s postupkom kros-validacije.Upravo u postupku minimiziranja pogrjeske procjeneuvodi se u sve tehnike kriginga nova varijabla nazvana,,Lagraageovim multiplikatorom". 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<77696A65646E6F737420> Tj 1 0 0 1 109.68 580.32 Tm <726567696F6E616C697A6972616E6520> Tj 1 0 0 1 186.96 580.80 Tm <766172696A61626C6520> Tj 1 0 0 1 238.32 581.52 Tm <70726F63696A656E6A656E65> Tj 0.24 Tc 1 0 0 1 52.80 569.52 Tm <6B726967696E676F6D20> Tj 1 0 0 1 98.64 569.76 Tm <6E6120> Tj 1 0 0 1 111.60 569.76 Tm <74656D656C6A7520> Tj 1 0 0 1 146.16 570.00 Tm <6E20> Tj 1 0 0 1 154.56 570.00 Tm <6F6B6F6C6E696820> Tj 1 0 0 1 189.60 570.24 Tm <6D6A6572656E696820> Tj 1 0 0 1 230.40 570.48 Tm <746F64616B6120> Tj 1 0 0 1 260.40 571.44 Tm <6A653A> Tj 0.00 Tc /F3 14.0 Tf 1 0 0 1 52.32 545.76 Tm <7A7220> Tj 1 0 0 1 63.12 547.20 Tm <3D20> Tj 1 0 0 1 71.52 537.60 Tm <2C32312C2C20> Tj 1 0 0 1 90.00 548.16 Tm <2E322C> Tj 0.48 Tc /F3 9.0 Tf 1 0 0 1 53.04 517.44 Tm <47646A6520> Tj 1 0 0 1 75.60 517.68 Tm <73753A> Tj 0.72 Tc 1 0 0 1 271.44 547.20 Tm <28332E3229> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 52.08 503.76 Tm <292C6920> Tj 1 0 0 1 83.76 505.68 Tm <2D20> Tj 1 0 0 1 88.32 504.00 Tm <74655A696E736B6920> Tj 1 0 0 1 117.60 504.24 Tm <6B6F65666963696A656E746920> Tj 1 0 0 1 157.44 504.24 Tm <7A6120> Tj 1 0 0 1 167.76 504.24 Tm <7376616B7520> Tj 1 0 0 1 191.04 504.48 Tm <6C6F6B6163696A7520> Tj 1 0 0 1 220.56 503.52 Tm <2C2C6922> Tj 0.48 Tc 1 0 0 1 52.56 492.00 Tm <5A6920> Tj 1 0 0 1 83.76 493.92 Tm <2D20> Tj 1 0 0 1 88.80 492.00 Tm <6F6B6F6C6E6520> Tj 1 0 0 1 115.20 492.72 Tm <706F7A6E61746520> Tj 1 0 0 1 145.68 492.48 Tm <7672696A65646E6F7374692C20> Tj 1 0 0 1 186.48 492.72 Tm <747A762E20> Tj 1 0 0 1 200.64 492.72 Tm <6B6F6E74726F6C6E6520> Tj 1 0 0 1 235.92 492.72 Tm <746F646B65> Tj 0.48 Tc 1 0 0 1 52.56 480.00 Tm <5A6B20> Tj 1 0 0 1 84.00 481.92 Tm <2D20> Tj 1 0 0 1 88.32 480.24 Tm <7672696A65646E6F737420> Tj 1 0 0 1 125.76 480.72 Tm <70726F636A656E6A656E6120> Tj 1 0 0 1 169.92 480.72 Tm <6B726967696E676F6D> Tj 0.24 Tc 1 0 0 1 52.80 462.24 Tm <446F20> Tj 1 0 0 1 73.44 462.24 Tm <66696820> Tj 1 0 0 1 93.12 462.24 Tm <77696A65646E6F73746920> Tj 1 0 0 1 147.36 462.48 Tm <736520> Tj 1 0 0 1 165.84 462.72 Tm <646F6C617A6920> Tj 1 0 0 1 198.96 462.96 Tm <726A65536176616E6A656D20> Tj 1 0 0 1 258.96 463.20 Tm <73757374617661> Tj 0.24 Tc 1 0 0 1 52.80 451.68 Tm <6C696E6561726E696820> Tj 1 0 0 1 97.92 452.64 Tm <6A65646E61645A626920> Tj 1 0 0 1 147.84 452.16 Tm <6B726967696E67612C20> Tj 1 0 0 1 192.72 452.40 Tm <6B6F6A6520> Tj 1 0 0 1 218.64 452.40 Tm <737520> Tj 1 0 0 1 236.16 452.64 Tm <6F626A61766C6A656E6520> Tj 1 0 0 1 286.32 452.64 Tm <73> Tj 0.24 Tc 1 0 0 1 52.80 441.36 Tm <7072696D6A6572696D6120> Tj 1 0 0 1 107.52 441.36 Tm <6E6A69686F766520> Tj 1 0 0 1 144.72 441.60 Tm <75706F747265626520> Tj 1 0 0 1 189.12 441.84 Tm <7620> Tj 1 0 0 1 201.84 441.84 Tm <6E697A727420> Tj 1 0 0 1 226.32 441.84 Tm <6B6E6A6967612C20> Tj 1 0 0 1 261.60 442.08 Tm <6E70722E20> Tj 1 0 0 1 285.12 442.32 Tm <75> Tj 0.48 Tc 1 0 0 1 52.80 430.56 Tm <7261646F76696D6120> Tj 1 0 0 1 95.76 433.20 Tm <73273427352E3720> Tj 1 0 0 1 122.40 430.80 Tm <6920> Tj 1 0 0 1 127.92 430.80 Tm <6D6E6F67696D20> Tj 1 0 0 1 165.60 431.04 Tm <64727567696D612E> Tj 0.24 Tc 1 0 0 1 60.00 414.96 Tm <4A65646E61645A627520> Tj 1 0 0 1 110.40 415.20 Tm <332E3220> Tj 1 0 0 1 127.20 415.20 Tm <6D6F6775366520> Tj 1 0 0 1 162.00 416.40 Tm <6A6520> Tj 1 0 0 1 173.04 415.44 Tm <6E6170697361746920> Tj 1 0 0 1 210.00 415.68 Tm <7520> Tj 1 0 0 1 218.88 415.68 Tm <6F626C696B7520> Tj 1 0 0 1 249.12 415.92 Tm <6D61747269646E6968> Tj 0.24 Tc 1 0 0 1 52.08 404.64 Tm <6A65646E61645A626920> Tj 1 0 0 1 99.12 403.92 Tm <332E332E20> Tj 1 0 0 1 119.76 404.16 Tm <556E7574617220> Tj 1 0 0 1 154.56 404.16 Tm <6476696A7520> Tj 1 0 0 1 180.24 404.40 Tm <6F6420> Tj 1 0 0 1 195.12 404.40 Tm <74696820> Tj 1 0 0 1 211.20 404.64 Tm <6D61747269636120> Tj 1 0 0 1 248.88 405.60 Tm <285B415D20> Tj 1 0 0 1 268.08 404.88 Tm <6920> Tj 1 0 0 1 275.76 403.20 Tm <74426C29> Tj 0.24 Tc 1 0 0 1 52.56 393.36 Tm <77696A65646E6F73746920> Tj 1 0 0 1 100.32 393.36 Tm <737520> Tj 1 0 0 1 112.32 393.60 Tm <697A72615A656E6520> Tj 1 0 0 1 149.28 393.84 Tm <77696A65646E6F35647520> Tj 1 0 0 1 200.64 394.08 Tm <766172696F6772616D612C20> Tj 1 0 0 1 254.16 394.32 Tm <6F646E6F736E6F> Tj 0.24 Tc 1 0 0 1 53.04 382.80 Tm <6F7669736520> Tj 1 0 0 1 78.00 383.04 Tm <69736B6C6A756469766F20> Tj 1 0 0 1 122.64 383.04 Tm <6F20> Tj 1 0 0 1 131.28 383.28 Tm <6D656475736F626E6F6A20> Tj 1 0 0 1 187.68 383.52 Tm <7564616C6A656E6F73696C20> Tj 1 0 0 1 238.80 383.76 Tm <6920> Tj 1 0 0 1 245.52 383.76 Tm <6F72696A656E746163696A69> Tj 0.24 Tc 1 0 0 1 53.04 372.24 Tm <746F64616B6120> Tj 1 0 0 1 85.20 372.48 Tm <6E6120> Tj 1 0 0 1 99.84 372.48 Tm <6B6F6A69726E6120> Tj 1 0 0 1 133.44 373.20 Tm <706F73746F6A6520> Tj 1 0 0 1 168.24 372.72 Tm <77696A65646E6F7374692C20> Tj 1 0 0 1 221.04 373.20 Tm <6120> Tj 1 0 0 1 229.68 373.20 Tm <6E6520> Tj 1 0 0 1 244.32 373.20 Tm <6F20> Tj 1 0 0 1 253.44 373.20 Tm <6E6A69686F76696D> Tj 0.24 Tc 1 0 0 1 52.80 361.20 Tm <77696A65646E6F737469726E612E20> Tj 1 0 0 1 122.40 361.44 Tm <547265646120> Tj 1 0 0 1 155.04 361.68 Tm <6D61747269636120> Tj 1 0 0 1 196.80 362.64 Tm <28746C6C2920> Tj 1 0 0 1 221.28 361.92 Tm <736164725A6920> Tj 1 0 0 1 255.84 362.16 Tm <74655A696E736B65> Tj 0.24 Tc 1 0 0 1 53.04 350.64 Tm <6B6F65666963696A656E746520> Tj 1 0 0 1 103.44 350.88 Tm <6B6F6A6920> Tj 1 0 0 1 121.68 350.88 Tm <736520> Tj 1 0 0 1 132.72 350.88 Tm <6E6120> Tj 1 0 0 1 145.20 351.12 Tm <6B72616A7520> Tj 1 0 0 1 170.16 351.60 Tm <70726F636A656E6A756A7520> Tj 1 0 0 1 221.04 351.36 Tm <697A20> Tj 1 0 0 1 229.20 352.32 Tm <6A65646E6F737461726D6F6761> Tj 0.48 Tc 1 0 0 1 53.28 340.08 Tm <7375737461766120> Tj 1 0 0 1 87.84 339.36 Tm <2C2C6E2720> Tj 1 0 0 1 101.76 341.04 Tm <6A65646E61645A626920> Tj 1 0 0 1 147.12 340.32 Tm <7320> Tj 1 0 0 1 154.32 339.60 Tm <2C2C6E2720> Tj 1 0 0 1 169.20 340.56 Tm <6E65706F7A6E616E6963612E> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 52.80 319.20 Tm <74416C27742E726C20> Tj 1 0 0 1 78.72 322.80 Tm <3D20> Tj 1 0 0 1 85.92 319.44 Tm <74426C20> Tj 1 0 0 1 271.68 322.80 Tm <28332E3329> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 53.04 301.68 Tm <4B726967696E6720> Tj 1 0 0 1 92.64 301.92 Tm <6B616F20> Tj 1 0 0 1 116.88 302.16 Tm <6D65746F646120> Tj 1 0 0 1 157.68 302.16 Tm <736164725A6920> Tj 1 0 0 1 192.48 302.40 Tm <7669356520> Tj 1 0 0 1 217.92 302.64 Tm <7465686E696B612E20> Tj 1 0 0 1 261.36 302.88 Tm <546F20> Tj 1 0 0 1 281.04 302.88 Tm <7375> Tj 0.24 Tc 1 0 0 1 52.08 291.84 Tm <6A65646E6F73746172726E6920> Tj 1 0 0 1 104.64 291.36 Tm <6B726967696E672C20> Tj 1 0 0 1 139.68 291.60 Tm <6F6269646E6920> Tj 1 0 0 1 168.48 291.84 Tm <6B726967696E672C20> Tj 1 0 0 1 203.52 291.84 Tm <696E64696B61746F72736B6920> Tj 1 0 0 1 258.72 292.08 Tm <6B726967696E672C> Tj 0.24 Tc 1 0 0 1 52.80 280.56 Tm <76697365737472756B6920> Tj 1 0 0 1 96.96 280.80 Tm <696E64696B61746F72736B6920> Tj 1 0 0 1 151.68 281.04 Tm <6B726967696E672C20> Tj 1 0 0 1 186.00 281.28 Tm <756E697665727A616C6E6920> Tj 1 0 0 1 235.44 281.52 Tm <6B726967696E672C20> Tj 1 0 0 1 270.00 281.76 Tm <4952466B> Tj 0.24 Tc 1 0 0 1 53.52 270.72 Tm <28656E242E20> Tj 1 0 0 1 79.44 269.28 Tm <2C2C496E7472696E73696320> Tj 1 0 0 1 122.16 270.24 Tm <52616E646F6D20> Tj 1 0 0 1 161.76 270.48 Tm <46756E6374696F6E20> Tj 1 0 0 1 202.56 270.72 Tm <6F6620> Tj 1 0 0 1 213.60 270.72 Tm <4B20> Tj 1 0 0 1 223.68 270.72 Tm <4F72646572222920> Tj 1 0 0 1 258.72 270.96 Tm <6B726967696E672C> Tj 0.00 Tc 1 0 0 1 53.04 259.44 Tm <6C6F676E6F726D616C6E6920> Tj 1 0 0 1 110.16 259.68 Tm <6B726967696E6720> Tj 1 0 0 1 145.44 259.68 Tm <6920> Tj 1 0 0 1 153.84 259.92 Tm <6E656C696E6561726E6920> Tj 1 0 0 1 202.56 259.92 Tm <6469736A756E6B746972276D6920> Tj 1 0 0 1 258.96 260.40 Tm <6B726967696E672E> Tj 0.24 Tc 1 0 0 1 53.28 248.64 Tm <52657A756C74617420> Tj 1 0 0 1 92.64 248.88 Tm <646F646176616E6A6120> Tj 1 0 0 1 140.40 249.12 Tm <6C696E6561726E6F6720> Tj 1 0 0 1 184.56 249.36 Tm <6B6F65666963696A656E746120> Tj 1 0 0 1 237.60 249.60 Tm <7520> Tj 1 0 0 1 246.72 250.56 Tm <6A65646E61645A6265> Tj 0.24 Tc 1 0 0 1 53.28 238.08 Tm <6B726967696E67612C20> Tj 1 0 0 1 96.24 238.32 Tm <6120> Tj 1 0 0 1 106.56 238.32 Tm <7320> Tj 1 0 0 1 116.64 238.32 Tm <63696C6A656D20> Tj 1 0 0 1 147.60 239.04 Tm <706F7374697A616E6A6120> Tj 1 0 0 1 196.56 239.04 Tm <6E65707269737472616E6F7374692C20> Tj 1 0 0 1 264.96 239.04 Tm <62697420> Tj 1 0 0 1 282.24 239.28 Tm <3665> Tj 0.24 Tc 1 0 0 1 53.28 227.52 Tm <6F706973616E6120> Tj 1 0 0 1 96.00 227.76 Tm <6E6120> Tj 1 0 0 1 115.92 228.24 Tm <707269726E6A65727520> Tj 1 0 0 1 163.44 228.00 Tm <7573706F726564626520> Tj 1 0 0 1 217.44 229.20 Tm <6A65646E6F737461726D6F676120> Tj 1 0 0 1 288.24 228.72 Tm <69> Tj 0.24 Tc 1 0 0 1 53.28 216.96 Tm <6F6269646E6F676120> Tj 1 0 0 1 93.60 216.96 Tm <6B726967696E67612E> Tj 0.24 Tc 1 0 0 1 53.52 197.52 Tm <332E20> Tj 1 0 0 1 64.08 197.76 Tm <312E20> Tj 1 0 0 1 76.08 197.76 Tm <54656F72696A6120> Tj 1 0 0 1 112.56 199.20 Tm <6A65646E6F737461766E6F6720> Tj 1 0 0 1 182.64 198.48 Tm <6B726967696E6761> Tj 0.24 Tc 1 0 0 1 53.28 182.64 Tm <4A65646E6F737461766E6920> Tj 1 0 0 1 113.52 182.88 Tm <6B726967696E672C20> Tj 1 0 0 1 154.32 183.12 Tm <6B616B6F20> Tj 1 0 0 1 184.08 183.36 Tm <6D7520> Tj 1 0 0 1 206.88 183.36 Tm <7361726E6F20> Tj 1 0 0 1 238.08 183.60 Tm <696D6520> Tj 1 0 0 1 262.32 184.08 Tm <676F766F72692C> Tj 0.24 Tc 1 0 0 1 53.28 172.08 Tm <6E616A6A65646E6F73746172726E696A20> Tj 1 0 0 1 118.08 172.32 Tm <6120> Tj 1 0 0 1 123.84 173.28 Tm <6A20> Tj 1 0 0 1 127.68 172.32 Tm <6520> Tj 1 0 0 1 133.92 172.32 Tm <7465686E696B6120> Tj 1 0 0 1 168.00 172.56 Tm <6B726967696E67612E20> Tj 1 0 0 1 207.36 172.80 Tm <4D61747269646E6120> Tj 1 0 0 1 246.48 173.76 Tm <6A65646E61646C276261> Tj 0.48 Tc 1 0 0 1 53.28 161.52 Tm <7520> Tj 1 0 0 1 61.44 162.00 Tm <70756E6F6D6520> Tj 1 0 0 1 99.12 161.52 Tm <6F626C696B7520> Tj 1 0 0 1 128.88 159.60 Tm <246173693A> Tj 0.48 Tc 1 0 0 1 307.44 762.72 Tm <47646A6520> Tj 1 0 0 1 330.24 762.96 Tm <73753A> Tj 0.48 Tc 1 0 0 1 306.96 752.64 Tm <7920> Tj 1 0 0 1 358.08 754.32 Tm <2D20> Tj 1 0 0 1 362.64 752.40 Tm <7672696A65646E6F73746920> Tj 1 0 0 1 401.04 752.64 Tm <766172696F6772616D61> Tj 0.48 Tc 1 0 0 1 306.96 740.64 Tm <5A2C2E2E2E5A6E20> Tj 1 0 0 1 358.32 742.56 Tm <2D20> Tj 1 0 0 1 363.36 741.12 Tm <706F7A6E6174652C20> Tj 1 0 0 1 396.24 740.64 Tm <6D6A6572656E6A6520> Tj 1 0 0 1 428.88 740.88 Tm <7672696A65646E6F73746920> Tj 1 0 0 1 468.00 740.64 Tm <7520> Tj 1 0 0 1 474.24 740.88 Tm <746F646B616D61> Tj 0.48 Tc 1 0 0 1 306.96 728.64 Tm <5A20> Tj 1 0 0 1 358.08 730.56 Tm <2D20> Tj 1 0 0 1 362.64 728.88 Tm <746F646B6120> Tj 1 0 0 1 384.96 728.88 Tm <7520> Tj 1 0 0 1 391.68 728.88 Tm <6B6F6A6F6A20> Tj 1 0 0 1 410.88 728.88 Tm <736520> Tj 1 0 0 1 421.68 729.36 Tm <70726F636A656E6A756A6520> Tj 1 0 0 1 463.44 728.88 Tm <6E6F766120> Tj 1 0 0 1 482.64 728.88 Tm <7672696A65646E6F73746920> Tj 1 0 0 1 521.52 729.12 Tm <697A> Tj 0.72 Tc 1 0 0 1 358.32 719.28 Tm <6F6B6F6C6E69682C20> Tj 1 0 0 1 389.04 720.00 Tm <706F7A6E6174696820> Tj 1 0 0 1 421.92 720.00 Tm <28322C2E2E2E3229> Tj 0.24 Tc 1 0 0 1 307.20 705.60 Tm <4B726967696E6720> Tj 1 0 0 1 339.84 705.60 Tm <75706F747265626C6A61766120> Tj 1 0 0 1 394.80 705.60 Tm <62657A64696D656E7A696F6E616C6E6520> Tj 1 0 0 1 473.52 705.84 Tm <746F646B6173746520> Tj 1 0 0 1 510.72 706.32 Tm <706F6461746B65> Tj 0.24 Tc 1 0 0 1 307.44 694.80 Tm <6B6F6A6920> Tj 1 0 0 1 328.32 695.28 Tm <70726564737461766C6A616A7520> Tj 1 0 0 1 388.08 695.04 Tm <77696A65646E6F73746920> Tj 1 0 0 1 438.24 695.04 Tm <726567696F6E616C697A697220> Tj 1 0 0 1 487.44 695.04 Tm <616E6520> Tj 1 0 0 1 505.92 695.28 Tm <766172696A61626C652E> Tj 0.24 Tc 1 0 0 1 307.44 684.24 Tm <4B6F6420> Tj 1 0 0 1 329.52 685.20 Tm <6A65646E6F737461726D6F676120> Tj 1 0 0 1 396.96 684.48 Tm <6B726967696E676120> Tj 1 0 0 1 437.52 684.96 Tm <706F6472617A756D696A65766120> Tj 1 0 0 1 506.64 684.72 Tm <736520> Tj 1 0 0 1 521.76 684.72 Tm <646120> Tj 1 0 0 1 537.36 685.68 Tm <6A65> Tj 0.24 Tc 1 0 0 1 307.20 673.68 Tm <726567696F6E616C697A6972616E6120> Tj 1 0 0 1 380.40 673.92 Tm <766172696A61626C6120> Tj 1 0 0 1 427.20 673.92 Tm <6B6172616B746572697A6972616E6120> Tj 1 0 0 1 503.28 674.16 Tm <73746163696F6E61722D> Tj 0.24 Tc 1 0 0 1 307.44 663.36 Tm <6E6F53697520> Tj 1 0 0 1 336.00 663.36 Tm <647275676F676120> Tj 1 0 0 1 373.20 663.36 Tm <726564612C20> Tj 1 0 0 1 398.40 663.36 Tm <6F64656B6976616E6A6520> Tj 1 0 0 1 445.20 664.08 Tm <6A6520> Tj 1 0 0 1 456.48 663.36 Tm <73767567646A6520> Tj 1 0 0 1 491.28 663.60 Tm <6B6F6E7374616E746E6F20> Tj 1 0 0 1 542.64 663.60 Tm <69> Tj 0.24 Tc 1 0 0 1 307.68 652.56 Tm <706F7A6E61746F20> Tj 1 0 0 1 347.04 649.68 Tm <6C7028786C3D20> Tj 1 0 0 1 377.76 652.32 Tm <4F6C20> Tj 1 0 0 1 390.72 652.32 Tm <746520> Tj 1 0 0 1 403.20 653.28 Tm <6A6520> Tj 1 0 0 1 417.36 652.80 Tm <706F7A6E61746120> Tj 1 0 0 1 456.24 652.32 Tm <66756E6B63696A6120> Tj 1 0 0 1 496.56 652.56 Tm <6B6F766172696A616E6365> Tj 0.72 Tc 1 0 0 1 307.44 636.00 Tm <6C6374782C673D63287A2878292C322830295D20> Tj 1 0 0 1 404.64 638.64 Tm <2E20> Tj 1 0 0 1 410.64 638.88 Tm <49766161616C652C20> Tj 1 0 0 1 449.52 638.64 Tm 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364.80 581.52 Tm <6E6520> Tj 1 0 0 1 378.48 582.00 Tm <706F73746F6A6920> Tj 1 0 0 1 410.40 581.52 Tm <76616E6A736B6920> Tj 1 0 0 1 444.72 581.52 Tm <75746A6563616A20> Tj 1 0 0 1 476.88 582.48 Tm <28656E676C2E20> Tj 1 0 0 1 503.28 580.80 Tm <2C2C6472696674272920> Tj 1 0 0 1 535.20 581.76 Tm <6E61> Tj 0.24 Tc 1 0 0 1 307.20 570.72 Tm <766172696A61626C752C20> Tj 1 0 0 1 352.08 570.72 Tm <6120> Tj 1 0 0 1 361.44 570.96 Tm <7A62726F6A20> Tj 1 0 0 1 386.88 570.96 Tm <7376696820> Tj 1 0 0 1 407.76 570.96 Tm <74655A696E736B696820> Tj 1 0 0 1 450.72 570.96 Tm <6B6F65666963696A656E61746120> Tj 1 0 0 1 508.08 571.20 Tm <697A6E6F736920> Tj 1 0 0 1 538.56 571.20 Tm <312C> Tj 0.24 Tc 1 0 0 1 307.68 560.16 Tm <6F7374766172656E20> Tj 1 0 0 1 347.52 561.12 Tm <6A6520> Tj 1 0 0 1 360.24 560.40 Tm <75722C6A657420> Tj 1 0 0 1 385.20 560.40 Tm <6E65707269737472616E6F7374692E20> Tj 1 0 0 1 452.64 560.40 Tm <52617A6C696B6120> Tj 1 0 0 1 489.36 560.40 Tm <7376696820> Tj 1 0 0 1 510.72 560.88 Tm <70726176696820> Tj 1 0 0 1 542.88 560.64 Tm <69> Tj 0.24 Tc 1 0 0 1 307.44 549.84 Tm <70726F63696A656E6A656E696820> Tj 1 0 0 1 373.92 549.60 Tm <77696A65646E6F73746920> Tj 1 0 0 1 430.80 549.60 Tm <6E617A76613A656120> Tj 1 0 0 1 474.72 550.56 Tm <6A6520> Tj 1 0 0 1 493.44 550.08 Tm <706F67726A65356B6F6D> Tj 0.24 Tc 1 0 0 1 307.68 539.28 Tm <70726F636A656E6520> Tj 1 0 0 1 349.44 539.04 Tm <696C6920> Tj 1 0 0 1 360.24 539.04 Tm <766172696A616E636F6D20> Tj 1 0 0 1 413.28 539.04 Tm <6B726967696E676120> Tj 1 0 0 1 452.64 538.80 Tm <746520> Tj 1 0 0 1 463.20 538.80 Tm <736520> Tj 1 0 0 1 474.72 539.04 Tm <697A72615A61766120> Tj 1 0 0 1 512.40 539.04 Tm <6B616F3A> Tj 1.50 Tc /F3 5.0 Tf 1 0 0 1 307.68 506.64 Tm <5E325F> Tj 1.50 Tc /F3 7.0 Tf 1 0 0 1 327.60 518.16 Tm <5C727A20> Tj 1 0 0 1 359.04 518.40 Tm <2D3720> Tj 1 0 0 1 396.24 518.16 Tm <5C32> Tj 0.24 Tc /F3 4.0 Tf 1 0 0 1 327.60 512.88 Tm <5A695C4C706E766120> Tj 1 0 0 1 366.24 516.24 Tm <2D7072636369656E69656E6120> Tj 1 0 0 1 396.24 514.80 Tm <4920> Tj 1 0 0 1 399.84 514.32 Tm <69> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 526.56 504.96 Tm <28332E3629> Tj 0.24 Tc 1 0 0 1 307.68 479.76 Tm <496C6920> Tj 1 0 0 1 319.92 479.76 Tm <6B616F20> Tj 1 0 0 1 339.36 479.76 Tm <647275676920> Tj 1 0 0 1 365.52 479.76 Tm <6B6F72696A656E20> Tj 1 0 0 1 399.36 479.76 Tm <766172696A616E636520> Tj 1 0 0 1 442.32 479.76 Tm <6E617A76616E696D20> Tj 1 0 0 1 486.72 479.76 Tm <7374616E646172646E6F6D> Tj 0.24 Tc 1 0 0 1 307.68 469.44 Tm <706F67726A65536B6F6D20> Tj 1 0 0 1 362.40 469.44 Tm <70726F636A656E653A> Tj 0.24 Tc 1 0 0 1 307.44 450.72 Tm <6F20> Tj 1 0 0 1 315.36 452.16 Tm <3D20> Tj 1 0 0 1 322.56 444.48 Tm <2262272D20> Tj 1 0 0 1 526.56 450.00 Tm <405D29> Tj 0.24 Tc 1 0 0 1 307.68 431.28 Tm <5520> Tj 1 0 0 1 324.24 431.52 Tm <696465616C6E6F6D20> Tj 1 0 0 1 373.68 431.52 Tm <736C7564616A7520> Tj 1 0 0 1 413.52 431.76 Tm <6B726967696E6720> Tj 1 0 0 1 452.88 432.00 Tm <706F6B755361766120> Tj 1 0 0 1 503.28 431.76 Tm <697A726164756E617469> Tj 0.24 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1 379.68 107.04 Tm <6A20> Tj 1 0 0 1 383.52 106.08 Tm <65646E61645A62616D6120> Tj 1 0 0 1 440.64 106.08 Tm <6F6269646E6F676120> Tj 1 0 0 1 480.96 106.32 Tm <6B726967696E67612E> Tj 0.00 Tc /F3 8.0 Tf 1 0 0 1 53.52 81.84 Tm <4E4146544120> Tj 1 0 0 1 73.92 81.84 Tm <353920> Tj 1 0 0 1 83.04 82.32 Tm <283120> Tj 1 0 0 1 88.56 80.40 Tm <2920> Tj 1 0 0 1 92.40 81.84 Tm <33382D343320> Tj 1 0 0 1 111.36 82.56 Tm <283230303929> Tj 0.72 Tc /F3 11.0 Tf 1 0 0 1 534.96 83.28 Tm <3339> Tj ET endstream endobj 94 0 obj 28815 endobj 95 0 obj << /Length 96 0 R >> stream U daljnjem tekstu bit 6e detaljno opisano znadenjeIinearnosti kod tehnika jednostarrnoga i obidnogakriginga, te kako se takva linearnost postiZe.3. IT4ATEIT,IATIEKE OSNOVE KRIGINGANadelo rada kriginga najjednostavnije je prikazati nizornjednadZbi kojima je definiran. Krigingom se procjenjujewijednosti regionalizirane varijable na odabranojlokaciji (Zi, a na temelju postoje6ih okolnih wijednosti[.Z,). Svakoj od tih postoje6ih wijednosti pridruZen jeodgovaraju6i teZinski koeficijent (1'), a nadin njihovaproraduna je najzahtje'rrniji dio algoritma kriginga.Vrijednost regionalizirane varijable odredena je kao:Z, = Z(x,) (3.1)Gdje je x, lokacija u kojoj je oditana wijednost. Nadalje,wijednost regionalizirane varijable procijenjenekrigingom na temelju n okolnih mjerenih todaka je: zr = ,21,, .2,Gdje su: (3.2) ),i - teZinski koeficijenti za svaku lokaciju ,,i"Zi - okolne poznate vrijednosti, tzv. kontrolne todkeZk - vrijednost procjenjena krigingomDo fih wijednosti se dolazi rjeSavanjem sustavalinearnih jednadZbi kriginga, koje su objavljene sprimjerima njihove upotrebe v nizrt knjiga, npr. uradovima s'4'5.7 i mnogim drugima.JednadZbu 3.2 mogu6e je napisati u obliku matridnihjednadZbi 3.3. Unutar dviju od tih matrica ([A] i tBl)wijednosti su izraZene wijedno5du variograma, odnosnoovise iskljudivo o medusobnoj udaljenosil i orijentacijitodaka na kojirna postoje wijednosti, a ne o njihovimwijednostirna. Treda matrica (tll) sadrZi teZinskekoeficijente koji se na kraju procjenjuju iz jednostarmogasustava ,,n' jednadZbi s ,,n' nepoznanica.tAl't.rl = tBl (3.3)Kriging kao metoda sadrZi vi5e tehnika. To sujednostarrni kriging, obidni kriging, indikatorski kriging,visestruki indikatorski kriging, univerzalni kriging, IRFk(en$. ,,Intrinsic Random Function of K Order") kriging,lognormalni kriging i nelinearni disjunktir'mi kriging.Rezultat dodavanja linearnog koeficijenta u jednadZbekriginga, a s ciljem postizanja nepristranosti, bit 6eopisana na prirnjeru usporedbe jednostarmoga iobidnoga kriginga.3. 1. Teorija jednostavnog krigingaJednostavni kriging, kako mu sarno ime govori,najjednostarrnij a j e tehnika kriginga. Matridna jednadl'bau punome obliku $asi: Gdje su:y - vrijednosti variogramaZ,...Zn - poznate, mjerenje vrijednosti u todkamaZ - todka u kojoj se procjenjuje nova vrijednosti izokolnih, poznatih (2,...2)Kriging upotrebljava bezdimenzionalne todkaste podatkekoji predstavljaju wijednosti regionalizir ane varijable.Kod jednostarmoga kriginga podrazumijeva se da jeregionalizirana varijabla karakterizirana stacionar-noSiu drugoga reda, odekivanje je svugdje konstantno ipoznato lp(xl= Ol te je poznata funkcija kovarijancelctx,g=c(z(x),2(0)] . Ivaaale, ako se takva procjenanadini na kontrolnoj todki moZe se na njoj izradunati ipogfieSka:e = (20,"," - Zr**n1.*) (3.5)Nadalje, ako ne postoji vanjski utjecaj (engl. ,,drift') navarijablu, a zbroj svih teZinskih koeficijenata iznosi 1,ostvaren je ur,jet nepristranosti. Razlika svih pravih iprocijenjenih wijednosti nazva:ea je pogrje5komprocjene ili varijancom kriginga te se izraZava kao: ^2_ \rz -7 \2Zi\Lpnva -prccieniena I i (3.6) Ili kao drugi korijen varijance nazvanim standardnompogrjeSkom procjene:o = "b'- @])U idealnom sludaju kriging pokuSava izradunatioptimalne teZinske koeficijente koji 6e dovesti doprocjene minirnalne pogrje5ke procjene. Talryikoeficijenti koji dovode do nepristrane procjene uzminimalnu varijancu izradunati su rje5avanjem sustavamatridnih jednadZbi, poput onih prikazanih u jednadZbi3.4. Ako se ta matrica predstavi linearnirn jednadLbama,$asit 6e:t(2,- Zr) . Lt+ y(21- Zr) . 1" + .. + y(2.- Z) . Ln = yQ1- Z)TG, - Zr) . L j+ y(22 - Zr) . 1, + ... + y(2, - Z,) . l, = y(2, - Z) (3.8)y (2, - z.) . L + y (2, - Z"). l, + ... + y(2, - z,). ),, = y (2, - Z)Procjenu u Zeljenoj todki mogu6e je prikazati sljede6omiednadZbom: (3.e) I Y(2,- Z,)lilz,-z)lve.-z,t Da bi linearne jednadZbe 3.8 bile smatranenepristranirna mora biti zadovoljen uvjet da je zbrojteZinskih koeficijenata jednak 1, tj > 1,=!. Taj uvjetispunjen je dodavanjem novih uvjeta u matridnejednadZbe kriginga, poput dodavanja Lagrangeovogmultiplikatora u j ednadZbama obidnoga kriginga. 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0 1 85.68 739.20 Tm <2266616B746F7220> Tj 1 0 0 1 117.12 737.28 Tm <6F676620> Tj 1 0 0 1 130.80 737.28 Tm <616E6964656E6A612720> Tj 1 0 0 1 171.36 737.28 Tm <756E7574617220> Tj 1 0 0 1 202.80 737.28 Tm <7365626520> Tj 1 0 0 1 224.40 738.00 Tm <2820> Tj 1 0 0 1 227.28 737.28 Tm <656E242E2C2C636F6E73747261696E6620> Tj 1 0 0 1 297.60 741.36 Tm <2720> Tj 1 0 0 1 300.00 735.84 Tm <29> Tj 0.24 Tc 1 0 0 1 64.56 726.96 Tm <6B616B6F20> Tj 1 0 0 1 88.08 726.72 Tm <626920> Tj 1 0 0 1 99.12 726.72 Tm <736520> Tj 1 0 0 1 110.40 726.72 Tm <6D696E696D697A6972616C6120> Tj 1 0 0 1 167.28 727.20 Tm <706F67726A65536B616F276E7478292C20> Tj 1 0 0 1 236.64 726.96 Tm <706120> Tj 1 0 0 1 249.84 726.48 Tm <74696D6520> Tj 1 0 0 1 270.96 726.96 Tm <706F7374616A75> Tj 0.24 Tc 1 0 0 1 64.56 716.40 Tm <6E65707269737472616E6520> Tj 1 0 0 1 117.60 716.64 Tm <70726F636A656E652E20> Tj 1 0 0 1 160.80 716.16 Tm <54616B617620> Tj 1 0 0 1 190.08 716.16 Tm <66616B746F7220> Tj 1 0 0 1 219.60 715.92 Tm <7520> Tj 1 0 0 1 228.96 716.40 Tm <70726176696C7520> Tj 1 0 0 1 263.04 715.68 Tm <626920> Tj 1 0 0 1 275.28 715.68 Tm <74726562616F> Tj 0.24 Tc 1 0 0 1 64.56 705.60 Tm <6F70697361746920> Tj 1 0 0 1 97.68 705.60 Tm <6E656B6F20> Tj 1 0 0 1 122.64 705.60 Tm <76616E6A736B6F2C20> Tj 1 0 0 1 162.96 705.84 Tm <7072697669646E6F20> Tj 1 0 0 1 203.52 705.60 Tm <6E65756F646C6A69766F2C20> Tj 1 0 0 1 253.92 705.36 Tm <6F6772616E6964656E6A65> Tj 0.24 Tc 1 0 0 1 64.32 695.04 Tm <756C617A6E696820> Tj 1 0 0 1 98.64 695.52 Tm <706F646174616B6120> Tj 1 0 0 1 141.12 695.04 Tm <6B6F6A6520> Tj 1 0 0 1 161.76 695.04 Tm <6E696A6520> Tj 1 0 0 1 179.52 694.80 Tm <7669646C6A69766F20> Tj 1 0 0 1 212.88 694.80 Tm <76653620> Tj 1 0 0 1 229.20 694.80 Tm <697A20> Tj 1 0 0 1 239.28 694.56 Tm <73616D696820> Tj 1 0 0 1 267.84 694.56 Tm <6E6A69686F766968> Tj 0.24 Tc 1 0 0 1 64.08 684.48 Tm <77696A65646E6F7374692E20> Tj 1 0 0 1 119.28 684.48 Tm <4D6F69646120> Tj 1 0 0 1 154.56 684.48 Tm <6E616A646553366520> Tj 1 0 0 1 197.04 684.24 Tm <75706F747265626C6A6176616E6A6120> Tj 1 0 0 1 270.48 684.00 Tm <7465686E696B61> Tj 0.24 Tc 1 0 0 1 64.32 673.92 Tm <6B726967696E676120> Tj 1 0 0 1 105.36 674.64 Tm <6A6520> Tj 1 0 0 1 120.96 673.92 Tm <6F6269696E6920> Tj 1 0 0 1 154.80 673.68 Tm <6B726967696E672E20> Tj 1 0 0 1 195.84 673.68 Tm <53746F676120> Tj 1 0 0 1 226.80 673.68 Tm <366520> Tj 1 0 0 1 243.12 673.44 Tm <736520> Tj 1 0 0 1 259.44 673.68 Tm <72617A6D6F7472697469> Tj 0.24 Tc 1 0 0 1 64.32 663.36 Tm <6F6772616E69646176616A75366920> Tj 1 0 0 1 132.72 663.36 Tm <66616B746F7220> Tj 1 0 0 1 165.84 663.36 Tm <6B6F6A6920> Tj 1 0 0 1 189.84 663.12 Tm <736520> Tj 1 0 0 1 205.92 663.60 Tm <706F6A61766C6A756A6520> Tj 1 0 0 1 254.40 663.12 Tm <756E7574617220> Tj 1 0 0 1 290.64 662.88 Tm <746968> Tj 0.24 Tc 1 0 0 1 63.36 653.76 Tm <6A65646E61645A62692C20> Tj 1 0 0 1 111.60 652.80 Tm <6E617A76616E69726E20> Tj 1 0 0 1 154.56 652.80 Tm <4C616772616E67656F7520> Tj 1 0 0 1 209.76 652.56 Tm <6D756C7469706C696B61746F722E> Tj 0.24 Tc 1 0 0 1 70.80 640.08 Tm <56726174696D6F20> Tj 1 0 0 1 111.12 640.08 Tm <736520> Tj 1 0 0 1 125.28 640.08 Tm <7520> Tj 1 0 0 1 135.36 641.04 Tm <6A65646E61645A626520> Tj 1 0 0 1 185.28 639.84 Tm <6F6269646E6F676120> Tj 1 0 0 1 228.24 639.84 Tm <6B726967696E67612E20> Tj 1 0 0 1 270.48 639.84 Tm <416B6F20> Tj 1 0 0 1 293.28 639.60 Tm <7365> Tj 0.24 Tc 1 0 0 1 64.08 629.76 Tm <7A61646F766F6C6A6920> Tj 1 0 0 1 106.32 629.52 Tm <757B657420> Tj 1 0 0 1 131.28 629.52 Tm <646120> Tj 1 0 0 1 144.96 630.24 Tm <6A6520> Tj 1 0 0 1 158.16 629.52 Tm <7A62726F6A20> Tj 1 0 0 1 183.84 629.28 Tm <7376696820> Tj 1 0 0 1 205.20 629.28 Tm <74655A696E736B696820> Tj 1 0 0 1 248.40 629.28 Tm <6B6F65666963696A656E617461> Tj 0.48 Tc 1 0 0 1 63.12 619.92 Tm <6A65646E616B20> Tj 1 0 0 1 96.72 618.96 Tm <3120> Tj 1 0 0 1 103.92 618.96 Tm <7461646120> Tj 1 0 0 1 125.04 619.20 Tm <6C7A72617A20> Tj 1 0 0 1 148.80 618.96 Tm <332E3820> Tj 1 0 0 1 164.88 619.44 Tm <706F7374616A653A> Tj 0.72 Tc 1 0 0 1 63.60 601.92 Tm <5428322C2D20> Tj 1 0 0 1 85.20 603.60 Tm <5A2C292E20> Tj 1 0 0 1 100.80 603.36 Tm <31312B20> Tj 1 0 0 1 114.96 603.84 Tm <792832312D20> Tj 1 0 0 1 136.56 603.36 Tm <5A22292E20> Tj 1 0 0 1 152.40 603.36 Tm <312C20> Tj 1 0 0 1 162.24 603.36 Tm <2B20> Tj 1 0 0 1 177.12 603.36 Tm <2B20> Tj 1 0 0 1 184.08 603.60 Tm <7928322C2D20> Tj 1 0 0 1 205.92 603.36 Tm <5A2C292E20> Tj 1 0 0 1 221.76 603.12 Tm <292C2C20> Tj 1 0 0 1 231.60 603.12 Tm <6120> Tj 1 0 0 1 238.80 604.80 Tm <7020> Tj 1 0 0 1 248.64 604.32 Tm <3D20> Tj 1 0 0 1 255.84 603.36 Tm <7928322E2D20> Tj 1 0 0 1 277.44 603.12 Tm <5A29> Tj 0.48 Tc 1 0 0 1 63.36 589.92 Tm <7C20> Tj 1 0 0 1 68.16 591.84 Tm <28322C20> Tj 1 0 0 1 80.16 593.52 Tm <2D20> Tj 1 0 0 1 86.64 591.36 Tm <5A2C2920> Tj 1 0 0 1 97.92 593.52 Tm <2E20> Tj 1 0 0 1 101.04 591.36 Tm <4920> Tj 1 0 0 1 105.84 590.64 Tm <6A20> Tj 1 0 0 1 109.44 591.36 Tm <2B20> Tj 1 0 0 1 116.40 591.60 Tm <7920> Tj 1 0 0 1 120.96 591.84 Tm <28323220> Tj 1 0 0 1 132.96 593.52 Tm <2D20> Tj 1 0 0 1 139.44 591.36 Tm <5A2C292E20> Tj 1 0 0 1 155.52 591.36 Tm <4C2C20> Tj 1 0 0 1 165.12 591.36 Tm <2B20> Tj 1 0 0 1 171.84 591.12 Tm <2E2E2E20> Tj 1 0 0 1 180.00 591.36 Tm <2B20> Tj 1 0 0 1 187.20 591.60 Tm <7928322C20> Tj 1 0 0 1 204.00 593.28 Tm <2D20> Tj 1 0 0 1 210.24 591.12 Tm <5A2C2920> Tj 1 0 0 1 222.96 593.28 Tm <2E20> Tj 1 0 0 1 226.08 591.12 Tm <292E2C20> Tj 1 0 0 1 235.92 591.12 Tm <6120> Tj 1 0 0 1 243.12 591.12 Tm <3620> Tj 1 0 0 1 253.20 592.32 Tm <3D20> Tj 1 0 0 1 260.40 591.36 Tm <7928322220> Tj 1 0 0 1 276.96 593.04 Tm <2D20> Tj 1 0 0 1 283.20 590.88 Tm <5A29> Tj 0.48 Tc 1 0 0 1 63.36 565.68 Tm <5428322C20> Tj 1 0 0 1 80.16 569.52 Tm <2D20> Tj 1 0 0 1 86.40 567.36 Tm <5A72292E20> Tj 1 0 0 1 101.04 567.36 Tm <293120> Tj 1 0 0 1 109.20 567.36 Tm <2B20> Tj 1 0 0 1 116.16 567.60 Tm <7928322C20> Tj 1 0 0 1 132.96 569.28 Tm <2D20> Tj 1 0 0 1 139.20 567.36 Tm <5A722920> Tj 1 0 0 1 152.16 569.52 Tm <2E20> Tj 1 0 0 1 155.28 567.12 Tm <312C20> Tj 1 0 0 1 165.12 567.12 Tm <2B20> Tj 1 0 0 1 171.60 567.12 Tm <2E2E2E20> Tj 1 0 0 1 180.00 567.12 Tm <2B20> Tj 1 0 0 1 186.96 567.36 Tm <7928322C20> Tj 1 0 0 1 203.76 569.28 Tm <2D20> Tj 1 0 0 1 210.00 567.12 Tm <5A2C292E20> Tj 1 0 0 1 226.08 567.12 Tm <292C2C20> Tj 1 0 0 1 235.92 566.88 Tm <2B20> Tj 1 0 0 1 243.12 566.88 Tm <6D20> Tj 1 0 0 1 252.96 568.08 Tm <3D20> Tj 1 0 0 1 260.16 567.12 Tm <7928322C20> Tj 1 0 0 1 276.96 568.80 Tm <2D20> Tj 1 0 0 1 283.20 566.88 Tm <5A29> Tj 1.50 Tc 1 0 0 1 63.84 555.12 Tm <4C742B20> Tj 1 0 0 1 79.44 555.12 Tm <31322B20> Tj 1 0 0 1 101.28 555.12 Tm <2E2B20> Tj 1 0 0 1 111.60 555.12 Tm <6C2C2B6F3A31> Tj 0.96 Tc 1 0 0 1 279.36 543.36 Tm <28332E313029> Tj 0.24 Tc 1 0 0 1 63.12 525.12 Tm <416B6F20> Tj 1 0 0 1 90.24 524.88 Tm <7361646120> Tj 1 0 0 1 119.76 524.88 Tm <73757374617620> Tj 1 0 0 1 156.48 524.88 Tm <6C696E6561726E696820> Tj 1 0 0 1 203.76 525.60 Tm <6A65646E61645A626920> Tj 1 0 0 1 255.84 525.12 Tm <7072696B615A656D6F> Tj 0.24 Tc 1 0 0 1 63.36 514.56 Tm <6D61747269646E6F20> Tj 1 0 0 1 104.88 514.32 Tm <646F6269766120> Tj 1 0 0 1 135.84 514.32 Tm <73653A> Tj 0.24 Tc /F3 8.0 Tf 1 0 0 1 62.88 418.80 Tm <42726F6A20> Tj 1 0 0 1 86.88 418.56 Tm <74655A696E736B696820> Tj 1 0 0 1 131.52 418.56 Tm <6B6F65666963696A656E61746120> Tj 1 0 0 1 191.04 418.32 Tm <6F7669736920> Tj 1 0 0 1 216.24 418.32 Tm <6F20> Tj 1 0 0 1 226.56 418.08 Tm <62726F6A7520> Tj 1 0 0 1 255.60 418.08 Tm <6B6F6E74726F6C6E6968> Tj 0.24 Tc 1 0 0 1 62.88 408.00 Tm <746F64616B6120> Tj 1 0 0 1 92.88 408.00 Tm <6920> Tj 1 0 0 1 97.44 408.00 Tm <6D6F5A6520> Tj 1 0 0 1 122.16 407.76 Tm <736520> Tj 1 0 0 1 133.20 407.76 Tm <53697269746920> Tj 1 0 0 1 154.80 407.76 Tm <6E6120> Tj 1 0 0 1 167.04 407.76 Tm <776C6F20> Tj 1 0 0 1 185.28 407.76 Tm <76656C696B6520> Tj 1 0 0 1 212.64 407.76 Tm <6C756C617A6E6520> Tj 1 0 0 1 241.92 407.52 Tm <736B75706F76652C20> Tj 1 0 0 1 280.80 408.24 Tm <6A657220> Tj 1 0 0 1 295.44 407.28 Tm <75> Tj 0.24 Tc 1 0 0 1 62.88 397.44 Tm <64616E61536E6A6520> Tj 1 0 0 1 103.20 397.20 Tm <77696A656D6520> Tj 1 0 0 1 137.28 397.44 Tm <726164756E616C6E6920> Tj 1 0 0 1 180.00 397.68 Tm <70726F6772616D6920> Tj 1 0 0 1 222.00 397.20 Tm <7573706A65536E6F20> Tj 1 0 0 1 262.56 396.96 Tm <726A65536176616A75> Tj 0.24 Tc 1 0 0 1 62.40 386.88 Tm <76656C696B6520> Tj 1 0 0 1 92.88 386.88 Tm <6C696E6561726E6520> Tj 1 0 0 1 132.96 386.64 Tm <737573746176652E20> Tj 1 0 0 1 172.80 386.64 Tm <4E616B6F6E20> Tj 1 0 0 1 206.16 386.64 Tm <6F647265646976616E6A6120> Tj 1 0 0 1 261.84 386.40 Tm <74655A696E736B6968> Tj 0.24 Tc 1 0 0 1 62.64 376.56 Tm <6B6F65666963696A656E61746120> Tj 1 0 0 1 120.96 376.32 Tm <6D6F6775366520> Tj 1 0 0 1 157.68 377.04 Tm <6A6520> Tj 1 0 0 1 170.16 376.56 Tm <70726F636A656E7520> Tj 1 0 0 1 212.88 376.08 Tm <6E6164696E69746920> Tj 1 0 0 1 248.64 376.80 Tm <6A65646E6F737461726D6F> Tj 0.24 Tc 1 0 0 1 63.12 366.00 Tm <7A6272616A616A72316920> Tj 1 0 0 1 112.80 365.76 Tm <75746A6563616A20> Tj 1 0 0 1 149.28 365.76 Tm <7376696820> Tj 1 0 0 1 174.72 365.76 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 228.24 365.52 Tm <746F64616B6120> Tj 1 0 0 1 264.72 365.28 Tm <6F74655A616E6968> Tj 0.24 Tc 1 0 0 1 62.40 355.92 Tm <707269706164616A7564696D20> Tj 1 0 0 1 127.44 355.20 Tm <6B6F65666963696A656E74696D6120> Tj 1 0 0 1 195.36 355.92 Tm <28736B726136656E6F20> Tj 1 0 0 1 244.32 355.20 Tm <7072696B617A616E6F20> Tj 1 0 0 1 294.96 354.72 Tm <75> Tj 0.48 Tc 1 0 0 1 61.68 345.60 Tm <6A65646E61645A626920> Tj 1 0 0 1 107.04 344.64 Tm <332E3229746520> Tj 1 0 0 1 136.80 344.64 Tm <736520> Tj 1 0 0 1 148.56 344.64 Tm <646F626976613A> Tj 0.96 Tc /F3 9.0 Tf 1 0 0 1 278.16 327.12 Tm <28332E313229> Tj 0.24 Tc 1 0 0 1 62.40 309.36 Tm <6C7A726128756E20> Tj 1 0 0 1 103.68 309.12 Tm <766172696A616E636520> Tj 1 0 0 1 151.68 309.60 Tm <70726F636A656E6520> Tj 1 0 0 1 198.48 309.84 Tm <286B726967696E67612920> Tj 1 0 0 1 248.40 308.88 Tm <756B6C6A7564756A6520> Tj 1 0 0 1 297.36 308.64 Tm <69> Tj 0.24 Tc 1 0 0 1 62.40 298.56 Tm <646F646176616E6A6520> Tj 1 0 0 1 108.24 298.56 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 168.72 298.32 Tm <6D756C7469706C696B61746F726120> Tj 1 0 0 1 232.80 298.08 Tm <746520> Tj 1 0 0 1 242.88 295.92 Tm <246173693A> Tj 1.20 Tc 1 0 0 1 61.92 281.52 Tm <6F2720> Tj 1 0 0 1 73.44 282.96 Tm <3D20> Tj 1 0 0 1 80.88 281.52 Tm <29272C277928322C2D7A2920> Tj 1 0 0 1 125.52 281.52 Tm <2B20> Tj 1 0 0 1 132.96 281.52 Tm <313227792832322D5A292B2E2E2E2B312C277928322C2D5A2920> Tj 1 0 0 1 243.36 281.04 Tm <2B20> Tj 1 0 0 1 250.56 281.04 Tm <6D20> Tj 1 0 0 1 278.16 281.28 Tm <28332E313329> Tj 0.24 Tc 1 0 0 1 61.92 263.28 Tm <5072696B617A616E6520> Tj 1 0 0 1 108.96 263.04 Tm <737520> Tj 1 0 0 1 123.36 263.04 Tm <6476696A652C20> Tj 1 0 0 1 150.96 263.04 Tm <726E6F69646120> Tj 1 0 0 1 183.60 263.04 Tm <6E616A64653536652C20> Tj 1 0 0 1 226.32 262.80 Tm <7465686E696B6520> Tj 1 0 0 1 262.56 262.56 Tm <6B726967696E67612E> Tj 0.24 Tc 1 0 0 1 61.92 252.72 Tm <4E6F2C20> Tj 1 0 0 1 78.48 252.72 Tm <6F7036656E69746F20> Tj 1 0 0 1 116.40 252.72 Tm <6B6F6420> Tj 1 0 0 1 134.88 252.48 Tm <7376696820> Tj 1 0 0 1 153.84 252.48 Tm <6C696E6561726E696820> Tj 1 0 0 1 193.92 252.24 Tm <766172696A616E746920> Tj 1 0 0 1 231.36 252.24 Tm <7465686E696B6120> Tj 1 0 0 1 265.20 252.00 Tm <6B726967696E6761> Tj 0.24 Tc 1 0 0 1 62.40 242.88 Tm <287520> Tj 1 0 0 1 76.08 241.92 Tm <6B6F6A696D6120> Tj 1 0 0 1 110.88 241.92 Tm <736520> Tj 1 0 0 1 125.04 241.92 Tm <726A65356176616A7520> Tj 1 0 0 1 168.72 242.16 Tm <70726F626C656D6920> Tj 1 0 0 1 213.12 241.68 Tm <6F7074696D697A6163696A652920> Tj 1 0 0 1 272.88 241.44 Tm <737573746176> Tj 0.24 Tc 1 0 0 1 61.68 231.84 Tm <707269706164616A7536696820> Tj 1 0 0 1 118.08 232.08 Tm <6A65646E61645A626920> Tj 1 0 0 1 163.20 231.12 Tm <6D6F5A6520> Tj 1 0 0 1 188.64 231.12 Tm <736520> Tj 1 0 0 1 200.16 231.60 Tm <706F64696A656C69746920> Tj 1 0 0 1 240.24 230.88 Tm <7520> Tj 1 0 0 1 248.88 230.88 Tm <64766120> Tj 1 0 0 1 266.64 230.88 Tm <64696A656C613A> Tj 0.24 Tc 1 0 0 1 68.88 218.64 Tm <612920> Tj 1 0 0 1 83.04 218.64 Tm <4B726F7A20> Tj 1 0 0 1 108.72 219.36 Tm <6A6564616E20> Tj 1 0 0 1 139.20 218.40 Tm <64696F20> Tj 1 0 0 1 157.92 219.36 Tm <6A65646E61645A626920> Tj 1 0 0 1 206.40 218.40 Tm <726164756E6120> Tj 1 0 0 1 242.40 218.16 Tm <736520> Tj 1 0 0 1 257.52 218.64 Tm <70726F73746F726E61> Tj 0.24 Tc 1 0 0 1 61.68 208.08 Tm <7A617669736E6F737420> Tj 1 0 0 1 108.48 208.80 Tm <2870726F73746F726E6120> Tj 1 0 0 1 160.32 207.84 Tm <6B6F72656C6163696A612920> Tj 1 0 0 1 212.16 207.60 Tm <6D6A6572656E696820> Tj 1 0 0 1 257.04 208.08 Tm <706F646174616B612C> Tj 0.24 Tc 1 0 0 1 61.68 197.52 Tm <6E616A20> Tj 1 0 0 1 74.88 197.52 Tm <646535366520> Tj 1 0 0 1 100.08 197.28 Tm <6B6F7269737465366920> Tj 1 0 0 1 139.20 197.28 Tm <76612D72696F6772616D3B> Tj 0.24 Tc 1 0 0 1 68.40 184.80 Tm <622920> Tj 1 0 0 1 85.20 184.80 Tm <447275676920> Tj 1 0 0 1 117.12 184.56 Tm <64696F20> Tj 1 0 0 1 137.52 185.52 Tm <6A65646E61645A626920> Tj 1 0 0 1 188.88 184.32 Tm <736164724C61766120> Tj 1 0 0 1 233.52 184.32 Tm <72617A6E6520> Tj 1 0 0 1 264.96 184.08 Tm <6C696E6561726E65> Tj 0.24 Tc 1 0 0 1 61.44 174.24 Tm <6D756C7469706C696B61746F726520> Tj 1 0 0 1 129.84 174.00 Tm <6B6F6A696D6120> Tj 1 0 0 1 166.80 173.76 Tm <736520> Tj 1 0 0 1 182.88 174.24 Tm <706F7374695A6520> Tj 1 0 0 1 220.56 173.76 Tm <646120> Tj 1 0 0 1 237.36 174.48 Tm <6A6520> Tj 1 0 0 1 252.96 173.52 Tm <7A62726F6A20> Tj 1 0 0 1 282.24 173.52 Tm <73766968> Tj 0.24 Tc 1 0 0 1 61.44 163.44 Tm <74655A696E736B696820> Tj 1 0 0 1 102.96 163.44 Tm <6B6F65666963696A656E61746120> Tj 1 0 0 1 158.40 164.16 Tm <6A65646E616B20> Tj 1 0 0 1 192.00 163.20 Tm <312E> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 61.68 142.56 Tm <342E20> Tj 1 0 0 1 77.28 142.56 Tm <5052494D4A45524920> Tj 1 0 0 1 144.24 142.32 Tm <495A524143554E4520> Tj 1 0 0 1 213.84 142.08 Tm <495C34415452494341> Tj 0.24 Tc 1 0 0 1 76.32 130.32 Tm <4B524947494E4741> Tj 0.00 Tc /F3 9.0 Tf 1 0 0 1 61.20 116.88 Tm <4B616F20> Tj 1 0 0 1 81.60 116.64 Tm <6F676C65646E6920> Tj 1 0 0 1 116.64 117.12 Tm <706F6C617A6E6920> Tj 1 0 0 1 150.96 116.88 Tm <7072696D6A657220> Tj 1 0 0 1 187.68 116.64 Tm <697A726174756E6120> Tj 1 0 0 1 227.76 116.40 Tm <6D61747269636120> Tj 1 0 0 1 264.48 116.16 Tm <6B726967696E6761> Tj 0.24 Tc 1 0 0 1 60.96 106.56 Tm <707265757A657420> Tj 1 0 0 1 96.24 107.04 Tm <6A6520> Tj 1 0 0 1 108.24 106.56 Tm <70726F73746F726E6920> Tj 1 0 0 1 152.40 106.08 Tm <6D6F64656C20> Tj 1 0 0 1 182.88 106.32 Tm <706F646174616B6120> Tj 1 0 0 1 226.80 105.84 Tm <6B6F6A6920> Tj 1 0 0 1 247.20 105.60 Tm <737520> Tj 1 0 0 1 261.36 106.08 Tm <7072696B617A616C69> Tj 0.48 Tc /F3 11.0 Tf 1 0 0 1 60.96 83.52 Tm <3430> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 319.68 762.24 Tm <446F7273656C20> Tj 1 0 0 1 349.92 762.00 Tm <2620> Tj 1 0 0 1 359.28 762.00 Tm <4C6120> Tj 1 0 0 1 372.00 762.00 Tm <42726563686520> Tj 1 0 0 1 403.68 764.40 Tm <322E20> Tj 1 0 0 1 411.84 761.76 Tm <4E656D6120> Tj 1 0 0 1 438.24 762.00 Tm <706F7365626E696820> Tj 1 0 0 1 479.52 761.28 Tm <72617A6C6F676120> Tj 1 0 0 1 512.40 761.04 Tm <7A6153746F20> Tj 1 0 0 1 536.40 760.80 Tm <626920> Tj 1 0 0 1 546.96 760.80 Tm <74616A> Tj 0.24 Tc 1 0 0 1 319.68 751.68 Tm <6D6F64656C20> Tj 1 0 0 1 351.36 751.44 Tm <62696F20> Tj 1 0 0 1 370.32 751.44 Tm <626F6C6A6920> Tj 1 0 0 1 394.80 751.20 Tm <6F6420> Tj 1 0 0 1 411.12 751.20 Tm <64727567696820> Tj 1 0 0 1 443.76 751.68 Tm <6A65646E6F737461726D696820> Tj 1 0 0 1 504.24 750.72 Tm <726173706F7265646120> Tj 1 0 0 1 553.44 750.24 Tm <73> Tj 0.24 Tc 1 0 0 1 319.68 741.12 Tm <6E656B6F6C696B6F20> Tj 1 0 0 1 359.28 740.88 Tm <746F64616B612C20> Tj 1 0 0 1 392.88 740.64 Tm <6F73696D20> Tj 1 0 0 1 416.16 740.64 Tm <53746F20> Tj 1 0 0 1 431.76 740.40 Tm <737520> Tj 1 0 0 1 444.24 740.16 Tm <6175746F726920> Tj 1 0 0 1 472.08 740.16 Tm <776C6F20> Tj 1 0 0 1 490.56 740.88 Tm <6A61736E6F20> Tj 1 0 0 1 516.72 739.92 Tm <6E6176656C6920> Tj 1 0 0 1 544.32 739.68 Tm <737665> Tj 0.24 Tc 1 0 0 1 319.44 730.80 Tm <70726F73746F726E6520> Tj 1 0 0 1 368.88 730.56 Tm <706F6461746B6520> Tj 1 0 0 1 410.88 729.84 Tm <6D6A6572656E6520> Tj 1 0 0 1 452.64 729.36 Tm <766172696A61626C6520> Tj 1 0 0 1 497.04 729.36 Tm <746520> Tj 1 0 0 1 511.92 729.12 Tm <6C72696A65646E6F737469> Tj 0.24 Tc 1 0 0 1 319.20 719.76 Tm <766172696F6772616D6120> Tj 1 0 0 1 374.40 719.52 Tm <697A6D65647520> Tj 1 0 0 1 412.80 719.28 Tm <6E6A69682E20> Tj 1 0 0 1 438.96 719.04 Tm <5A6E61645C20> Tj 1 0 0 1 472.08 718.80 Tm <7320> Tj 1 0 0 1 483.36 718.56 Tm <74616B79696D20> Tj 1 0 0 1 518.16 719.52 Tm <6A61736E696D20> Tj 1 0 0 1 554.88 718.32 Tm <69> Tj 0.24 Tc 1 0 0 1 318.48 709.92 Tm <6A65646E6F737461726D696D20> Tj 1 0 0 1 379.92 708.96 Tm <6D6F64656C6F6D20> Tj 1 0 0 1 423.12 708.48 Tm <7520> Tj 1 0 0 1 432.48 708.48 Tm <64616C6A6E6A656D20> Tj 1 0 0 1 472.80 708.24 Tm <64696A656C7520> Tj 1 0 0 1 499.92 708.00 Tm <7261646120> Tj 1 0 0 1 522.72 708.00 Tm <6D6F5A6520> Tj 1 0 0 1 548.64 707.76 Tm <7365> Tj 0.24 Tc 1 0 0 1 319.44 698.64 Tm <75737265646F746F6469746920> Tj 1 0 0 1 375.12 698.16 Tm <73616D6F20> Tj 1 0 0 1 402.72 698.16 Tm <6E6120> Tj 1 0 0 1 418.08 698.40 Tm <70726F6D6A656E6520> Tj 1 0 0 1 463.68 697.68 Tm <6B6F6A6520> Tj 1 0 0 1 486.48 697.44 Tm <366520> Tj 1 0 0 1 500.40 697.44 Tm <74616B766920> Tj 1 0 0 1 525.84 697.68 Tm <706F6461746369> Tj 0.24 Tc 1 0 0 1 319.20 688.08 Tm <757A726F6B6F7661746920> Tj 1 0 0 1 369.12 687.84 Tm <7520> Tj 1 0 0 1 378.72 688.56 Tm <6A65646E61645A62616D6120> Tj 1 0 0 1 441.60 687.12 Tm <72617A626469746820> Tj 1 0 0 1 483.36 686.88 Tm <7465686E696B6120> Tj 1 0 0 1 520.08 686.64 Tm <6B726967696E67612E> Tj 0.24 Tc 1 0 0 1 319.20 677.28 Tm <5573706F726564656E6920> Tj 1 0 0 1 373.20 677.04 Tm <737520> Tj 1 0 0 1 388.08 677.04 Tm <726164756E736B6920> Tj 1 0 0 1 430.32 677.04 Tm <706F73747570636920> Tj 1 0 0 1 472.08 676.56 Tm <6B6F6420> Tj 1 0 0 1 491.04 677.28 Tm <6A65646E6F737471756E6F67612C> Tj 0.24 Tc 1 0 0 1 319.44 666.72 Tm <6B726967696E6720> Tj 1 0 0 1 350.16 666.72 Tm <6120> Tj 1 0 0 1 360.24 667.44 Tm <287465686E696B6520> Tj 1 0 0 1 398.64 666.48 Tm <6E616A626F6C6A20> Tj 1 0 0 1 428.64 666.24 Tm <65676120> Tj 1 0 0 1 448.80 666.00 Tm <6C696E6561726E6F676120> Tj 1 0 0 1 498.72 666.24 Tm <70726F636A20> Tj 1 0 0 1 521.28 665.52 Tm <656E6974656C6A20> Tj 1 0 0 1 548.40 665.28 Tm <6129> Tj 0.24 Tc 1 0 0 1 319.20 656.40 Tm <746520> Tj 1 0 0 1 333.12 656.16 Tm <6F6269746E6F676120> Tj 1 0 0 1 379.20 655.92 Tm <6B726967696E676120> Tj 1 0 0 1 422.64 656.64 Tm <287465686E696B6520> Tj 1 0 0 1 463.44 655.44 Tm <6E716A626F6C6A65676120> Tj 1 0 0 1 516.00 655.20 Tm <6C696E6561726E6F67> Tj 0.24 Tc 1 0 0 1 319.20 645.60 Tm <6E6520> Tj 1 0 0 1 328.80 646.08 Tm <7020> Tj 1 0 0 1 335.28 645.60 Tm <7220> Tj 1 0 0 1 339.36 645.60 Tm <697320> Tj 1 0 0 1 346.56 645.60 Tm <747220> Tj 1 0 0 1 353.52 647.04 Tm <716E6F20> Tj 1 0 0 1 369.36 645.84 Tm <6720> Tj 1 0 0 1 376.80 645.84 Tm <7020> Tj 1 0 0 1 383.52 645.36 Tm <7220> Tj 1 0 0 1 387.12 645.36 Tm <6F20> Tj 1 0 0 1 392.16 645.36 Tm <636A20> Tj 1 0 0 1 399.36 645.36 Tm <656E697420> Tj 1 0 0 1 415.92 645.12 Tm <6520> Tj 1 0 0 1 421.44 645.12 Tm <6C6A20> Tj 1 0 0 1 426.48 645.12 Tm <612920> Tj 1 0 0 1 436.08 644.88 Tm <2E> Tj 0.24 Tc 1 0 0 1 319.44 627.60 Tm <342E312E20> Tj 1 0 0 1 342.00 627.36 Tm <52617A6C696B6120> Tj 1 0 0 1 382.56 627.36 Tm <766172696A616E636120> Tj 1 0 0 1 431.52 627.36 Tm <70726F636A656E6520> Tj 1 0 0 1 478.08 626.40 Tm <6B6F6420> Tj 1 0 0 1 499.92 626.40 Tm <7465686E696B61> Tj 0.24 Tc 1 0 0 1 340.56 616.80 Tm <6A65646E6F737461766E6F676120> Tj 1 0 0 1 416.40 615.12 Tm <6920> Tj 1 0 0 1 423.12 615.12 Tm <6F6269696E6F676120> Tj 1 0 0 1 471.12 614.64 Tm <6B726967696E6761> Tj 0.24 Tc 1 0 0 1 318.48 601.68 Tm <416B6F20> Tj 1 0 0 1 340.32 601.44 Tm <736520> Tj 1 0 0 1 353.52 601.44 Tm <6E6120> Tj 1 0 0 1 368.64 601.44 Tm <74656D656C6A7520> Tj 1 0 0 1 404.40 601.68 Tm <706F646174616B6120> Tj 1 0 0 1 448.80 600.72 Tm <697A20> Tj 1 0 0 1 460.56 600.72 Tm <736C696B6520> Tj 1 0 0 1 486.24 600.48 Tm <4920> Tj 1 0 0 1 494.88 600.48 Tm <746520> Tj 1 0 0 1 506.88 600.48 Tm <6C69742E3220> Tj 1 0 0 1 526.08 600.48 Tm <72617370695365> Tj 0.48 Tc 1 0 0 1 318.72 591.12 Tm <6D61747269646E6120> Tj 1 0 0 1 361.92 591.60 Tm <6A65646E61645A626120> Tj 1 0 0 1 412.56 590.64 Tm <6F6269646E6F6720> Tj 1 0 0 1 451.44 590.16 Tm <6B726967696E676120> Tj 1 0 0 1 492.24 590.88 Tm <28666F726D756C6120> Tj 1 0 0 1 534.72 589.68 Tm <332E313129> Tj 0.24 Tc 1 0 0 1 318.96 580.56 Tm <646F6269766120> Tj 1 0 0 1 349.92 580.32 Tm <73653A> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 329.76 563.76 Tm <3020> Tj 1 0 0 1 348.24 563.76 Tm <3134363520> Tj 1 0 0 1 375.12 563.52 Tm <32312C3534> Tj 0.48 Tc 1 0 0 1 322.32 551.76 Tm <31322C363520> Tj 1 0 0 1 355.92 551.52 Tm <3020> Tj 1 0 0 1 374.88 551.52 Tm <31342C3432> Tj 0.48 Tc 1 0 0 1 322.80 539.76 Tm <32312C353420> Tj 1 0 0 1 348.24 539.52 Tm <312B343520> Tj 1 0 0 1 381.60 539.28 Tm <30> Tj 1.50 Tc 1 0 0 1 330.00 527.52 Tm <313131> Tj 0.48 Tc /F3 10.0 Tf 1 0 0 1 317.04 331.92 Tm <4920> Tj 1 0 0 1 328.32 333.84 Tm <6F20> Tj 1 0 0 1 347.52 333.84 Tm <31322C367320> Tj 1 0 0 1 373.68 333.60 Tm <32312C3534276C20> Tj 1 0 0 1 399.60 332.16 Tm <6C6F2C73676F73276C20> Tj 1 0 0 1 440.64 331.92 Tm <4920> Tj 1 0 0 1 449.04 332.88 Tm <422C73737320> Tj 1 0 0 1 472.08 339.60 Tm <275D> Tj 1.20 Tc /F3 12.0 Tf 1 0 0 1 317.04 318.96 Tm <6C727A6F7320> Tj 1 0 0 1 354.48 321.36 Tm <6F20> Tj 1 0 0 1 373.20 321.36 Tm <312B34323120> Tj 1 0 0 1 399.36 318.96 Tm <7C20> Tj 1 0 0 1 403.92 321.12 Tm <6F2C2B736F2B6C3D6C20> Tj 1 0 0 1 446.88 319.68 Tm <6F756E2E726E20> Tj 1 0 0 1 474.00 318.96 Tm <49> Tj 0.96 Tc 1 0 0 1 317.04 306.72 Tm <6C7A742C732B20> Tj 1 0 0 1 346.80 309.60 Tm <742B2C2B7320> Tj 1 0 0 1 380.40 309.36 Tm <6F20> Tj 1 0 0 1 391.20 306.24 Tm <4A6C6F2C727A737A5D20> Tj 1 0 0 1 440.40 305.76 Tm <6C72732C7373726F6C> Tj 0.96 Tc /F3 8.0 Tf 1 0 0 1 537.36 544.32 Tm <28342E3129> Tj 1.20 Tc 1 0 0 1 278.88 436.56 Tm <283320> Tj 1 0 0 1 288.00 436.56 Tm <313129> Tj 0.48 Tc 1 0 0 1 318.48 505.20 Tm <4F646E6F736E6F20> Tj 1 0 0 1 359.52 504.96 Tm <7A61697A726174756E20> Tj 1 0 0 1 406.08 504.72 Tm <6D61747269636520> Tj 1 0 0 1 441.60 502.56 Tm <5B425D20> Tj 1 0 0 1 455.52 504.24 Tm <77696A6564693A> Tj 0.72 Tc 1 0 0 1 318.24 496.56 Tm <2D20> Tj 1 0 0 1 323.04 494.64 Tm <7A6120> Tj 1 0 0 1 333.84 494.64 Tm <312E20> Tj 1 0 0 1 342.72 494.64 Tm <726564> Tj 0.48 Tc 1 0 0 1 318.24 485.28 Tm <302720> Tj 1 0 0 1 327.36 485.28 Tm <302C3338303520> Tj 1 0 0 1 353.04 485.28 Tm <2B20> Tj 1 0 0 1 360.24 485.28 Tm <3120> Tj 1 0 0 1 364.08 485.04 Tm <322C363520> Tj 1 0 0 1 380.88 487.20 Tm <2E20> Tj 1 0 0 1 384.00 484.80 Tm <302C3439363420> Tj 1 0 0 1 408.72 484.80 Tm <2B20> Tj 1 0 0 1 416.64 484.80 Tm <32312C35342E20> Tj 1 0 0 1 438.96 484.56 Tm <302C313233322D20> Tj 1 0 0 1 469.92 484.32 Tm <3620> Tj 1 0 0 1 475.68 484.08 Tm <39333120> Tj 1 0 0 1 488.88 484.08 Tm <4920> Tj 1 0 0 1 495.60 485.52 Tm <3D20> Tj 1 0 0 1 503.04 483.12 Tm <51303031> Tj 0.72 Tc 1 0 0 1 318.24 475.44 Tm <2D20> Tj 1 0 0 1 322.80 473.76 Tm <746520> Tj 1 0 0 1 331.44 474.48 Tm <6A65646E616B6F20> Tj 1 0 0 1 362.88 473.52 Tm <6920> Tj 1 0 0 1 366.48 473.52 Tm <3261322E2C332E20> Tj 1 0 0 1 396.96 473.28 Tm <6920> Tj 1 0 0 1 400.56 473.28 Tm <342E20> Tj 1 0 0 1 409.68 473.04 Tm <7265642E> Tj 0.24 Tc 1 0 0 1 318.00 456.72 Tm <4E6164616C6A652C20> Tj 1 0 0 1 355.20 456.48 Tm <76612D72696A616E636120> Tj 1 0 0 1 396.96 456.24 Tm <6F6269646E6F6720> Tj 1 0 0 1 432.48 456.00 Tm <6B726967696E676120> Tj 1 0 0 1 469.92 455.52 Tm <736520> Tj 1 0 0 1 481.68 455.52 Tm <726164756E6120> Tj 1 0 0 1 513.84 455.76 Tm <7072656D613A> Tj 1.20 Tc 1 0 0 1 317.76 438.96 Tm <6F4C2C20> Tj 1 0 0 1 334.56 440.40 Tm <3D20> Tj 1 0 0 1 342.00 438.96 Tm <312C2E3728322C2D5A292B20> Tj 1 0 0 1 394.08 438.72 Tm <313227792832322D5A292B2E2E2E2B312C2E7928322C2D5A292B20> Tj 1 0 0 1 511.68 437.76 Tm <6D20> Tj 1 0 0 1 536.88 438.00 Tm <28342E3229> Tj 0.48 Tc 1 0 0 1 317.52 419.52 Tm <566172696A616E636120> Tj 1 0 0 1 360.96 419.28 Tm <697A6E6F736920> Tj 1 0 0 1 390.24 419.04 Tm <6F323D362C373020> Tj 1 0 0 1 429.12 418.80 Tm <6D322E20> Tj 1 0 0 1 447.36 419.04 Tm <706120> Tj 1 0 0 1 462.00 418.56 Tm <7374616E646172646E6120> Tj 1 0 0 1 514.32 418.56 Tm <706F67726A65736B61> Tj 0.24 Tc 1 0 0 1 317.52 407.76 Tm <6F3D322C353920> Tj 1 0 0 1 352.80 407.52 Tm <726E2E20> Tj 1 0 0 1 367.20 407.52 Tm <5A6170617A696D6F20> Tj 1 0 0 1 412.56 407.04 Tm <646120> Tj 1 0 0 1 425.04 407.76 Tm <6A6520> Tj 1 0 0 1 436.80 403.20 Tm <3E20> Tj 1 0 0 1 449.28 406.80 Tm <3120> Tj 1 0 0 1 456.96 408.24 Tm <3D6C20> Tj 1 0 0 1 470.88 407.52 Tm <287A626F6720> Tj 1 0 0 1 497.52 406.56 Tm <7A616F6B72754C6976616E6A61> Tj 0.24 Tc 1 0 0 1 318.24 393.36 Tm <72617A6C6C5E6B6120> Tj 1 0 0 1 353.52 393.36 Tm <697A6E6F736920> Tj 1 0 0 1 385.92 393.12 Tm <2B6C20> Tj 1 0 0 1 404.64 396.00 Tm <6F2F6F6F292E20> Tj 1 0 0 1 428.16 392.64 Tm <50726F6D6F7472696D6F20> Tj 1 0 0 1 486.24 392.16 Tm <7361646120> Tj 1 0 0 1 512.40 392.64 Tm <706F6E6135616E6A65> Tj 0.24 Tc 1 0 0 1 317.52 383.04 Tm <676F726E6A6520> Tj 1 0 0 1 350.88 382.80 Tm <6D61747269646E6520> Tj 1 0 0 1 394.80 383.28 Tm <6A65646E61645A626520> Tj 1 0 0 1 445.92 381.84 Tm <616B6F20> Tj 1 0 0 1 468.48 381.84 Tm <736520> Tj 1 0 0 1 484.08 381.84 Tm <7A6164724C6920> Tj 1 0 0 1 517.20 381.60 Tm <726173706F726564> Tj 0.24 Tc 1 0 0 1 317.76 372.48 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 366.72 372.00 Tm <746F64616B6120> Tj 1 0 0 1 398.64 371.76 Tm <746520> Tj 1 0 0 1 409.92 371.76 Tm <77696A65646E6F73746920> Tj 1 0 0 1 459.36 371.28 Tm <74655A696E736B696820> Tj 1 0 0 1 502.08 371.28 Tm <6B6F65666963696A656E617461> Tj 0.24 Tc 1 0 0 1 317.52 361.92 Tm <6B616B766920> Tj 1 0 0 1 343.92 361.68 Tm <737520> Tj 1 0 0 1 357.36 361.44 Tm <64616E6920> Tj 1 0 0 1 378.96 361.44 Tm <6E6120> Tj 1 0 0 1 392.64 361.44 Tm <736C69636920> Tj 1 0 0 1 413.76 361.20 Tm <312C20> Tj 1 0 0 1 424.08 361.20 Tm <616C6920> Tj 1 0 0 1 437.52 360.96 Tm <616B6F20> Tj 1 0 0 1 456.00 360.96 Tm <696820> Tj 1 0 0 1 467.52 360.72 Tm <726173706935656D6F20> Tj 1 0 0 1 514.80 360.48 Tm <7465686E696B6F6D> Tj 0.24 Tc 1 0 0 1 317.04 352.08 Tm <6A65646E6F737461766E6F6720> Tj 1 0 0 1 375.84 350.88 Tm <6B726967696E67612E20> Tj 1 0 0 1 415.92 350.64 Tm <5461646120> Tj 1 0 0 1 439.68 350.40 Tm <736520> Tj 1 0 0 1 451.44 350.16 Tm <646F626976613A> Tj 0.00 Tc /F3 9.0 Tf 1 0 0 1 317.28 288.00 Tm <5A6170617A69726E6F20> Tj 1 0 0 1 367.68 287.52 Tm <6B616B6F20> Tj 1 0 0 1 397.44 287.28 Tm <737520> Tj 1 0 0 1 415.68 287.28 Tm <7520> Tj 1 0 0 1 429.60 287.04 Tm <74616B766F6D20> Tj 1 0 0 1 469.44 286.80 Tm <6D61747269646E6F6D20> Tj 1 0 0 1 524.88 286.56 Tm <726164756E75> Tj 0.24 Tc 1 0 0 1 317.04 277.68 Tm <70726F6D696A656E6A656E6520> Tj 1 0 0 1 380.40 276.72 Tm <77696A65646E6F73746920> Tj 1 0 0 1 433.44 276.48 Tm <766172696F6772616D6120> Tj 1 0 0 1 489.60 276.00 Tm <7564616C6A656E6F73746920> Tj 1 0 0 1 544.56 275.76 Tm <6F64> Tj 0.24 Tc 1 0 0 1 317.04 266.88 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 370.56 266.40 Tm <746F64616B6120> Tj 1 0 0 1 407.04 266.16 Tm <646F20> Tj 1 0 0 1 425.76 265.92 Tm <746F646B6520> Tj 1 0 0 1 456.96 266.16 Tm <70726F636A656E652C20> Tj 1 0 0 1 504.72 265.44 Tm <6F646E6F736E6F20> Tj 1 0 0 1 549.36 265.20 Tm <75> Tj 0.24 Tc 1 0 0 1 316.08 256.32 Tm <6A65646E61645A626920> Tj 1 0 0 1 361.92 255.12 Tm <332E3320> Tj 1 0 0 1 378.48 255.12 Tm <77696A65646E6F73746920> Tj 1 0 0 1 427.20 254.64 Tm <6D61747269636520> Tj 1 0 0 1 463.20 252.72 Tm <5B425D2E20> Tj 1 0 0 1 481.20 254.40 Tm <546F20> Tj 1 0 0 1 495.84 254.40 Tm <7A6E61746920> Tj 1 0 0 1 520.32 254.16 Tm <646120> Tj 1 0 0 1 534.00 253.92 Tm <626920> Tj 1 0 0 1 545.04 253.92 Tm <7365> Tj 0.24 Tc 1 0 0 1 315.84 245.28 Tm <70726F6D696A656E696F20> Tj 1 0 0 1 365.28 245.04 Tm <706F6C6F5A616A20> Tj 1 0 0 1 400.80 244.32 Tm <746F746B6520> Tj 1 0 0 1 426.72 244.56 Tm <70726F636A656E6520> Tj 1 0 0 1 465.60 243.84 Tm <7520> Tj 1 0 0 1 473.76 243.84 Tm <726172726E696E6920> Tj 1 0 0 1 506.40 243.60 Tm <6E6120> Tj 1 0 0 1 519.60 243.60 Tm <736C69636920> Tj 1 0 0 1 540.24 243.36 Tm 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Teorija obiinoga krigingaSve tehnike kriginga (osirn jednostarmoga) imaju dodaneneki "faktor ogf anidenja' unutar sebe ( en$.,,constrainf ' )kako bi se minimizirala pogrjeSkao'ntx), pa time postajunepristrane procjene. Takav faktor u pravilu bi trebaoopisati neko vanjsko, prividno neuodljivo, ogranidenjeulaznih podataka koje nije vidljivo ve6 iz samih njihovihwijednosti. Moida najdeS6e upotrebljavanja tehnikakriginga je obiini kriging. Stoga 6e se razmotritiogranidavaju6i faktor koji se pojavljuje unutar tihjednadZbi, nazvanirn Lagrangeou multiplikator.Vratimo se u jednadZbe obidnoga kriginga. Ako sezadovolji u{et da je zbroj svih teZinskih koeficijenatajednak 1 tada lzraz 3.8 postaje:T(2,- Z,). 11+ y(21- Z"). 1, + + y(2,- Z,). ),, a p = y(2.- Z)| (2, - Z,) . I j + y (22 - Z,). L, + ... + y(2, - Z,) . )., a 6 = y(2" - Z)T(2, - Zr). )1 + y(2, - Zr) . 1, + ... + y(2, - Z,). ),, + m = y(2, - Z)Lt+ 12+ .+ l,+o:1(3.10)Ako sada sustav linearnih jednadZbi prikaZemomatridno dobiva se: Broj teZinskih koeficijenata ovisi o broju kontrolnihtodaka i moZe se Siriti na wlo velike lulazne skupove, jer udanaSnje wijeme radunalni programi uspjeSno rjeSavajuvelike linearne sustave. Nakon odredivanja teZinskihkoeficijenata mogu6e je procjenu nadiniti jednostarmozbrajajr1i utjecaj svih kontrolnih todaka oteZanihpripadajudim koeficijentima (skra6eno prikazano ujednadZbi 3.2)te se dobiva: (3.12) lzra(un varijance procjene (kriginga) ukljuduje idodavanje Lagrangeovog multiplikatora te $asi:o' = )','y(2,-z) + 12'y(22-Z)+...+1,'y(2,-Z) + m (3.13)Prikazane su dvije, rnoida najde56e, tehnike kriginga.No, op6enito kod svih linearnih varijanti tehnika kriginga(u kojima se rje5avaju problemi optimizacije) sustavpripadaju6ih jednadZbi moZe se podijeliti u dva dijela:a) Kroz jedan dio jednadZbi raduna se prostornazavisnost (prostorna korelacija) mjerenih podataka,naj de56e koriste6i va-riogram;b) Drugi dio jednadZbi sadrLava razne linearnemultiplikatore kojima se postiZe da je zbroj svihteZinskih koeficijenata jednak 1.4. PRIMJERI IZRACUNE I\4ATRICAKRIGINGAKao ogledni polazni primjer izratuna matrica krigingapreuzet je prostorni model podataka koji su prikazali40 Dorsel & La Breche 2. Nema posebnih razloga zaSto bi tajmodel bio bolji od drugih jednostarmih rasporeda snekoliko todaka, osim Sto su autori wlo jasno naveli sveprostorne podatke mjerene varijable te lrijednostivariograma izmedu njih. Znad\ s takyim jasnim ijednostarmim modelom u daljnjem dijelu rada moZe seusredotoditi samo na promjene koje 6e takvi podatciuzrokovati u jednadZbama razbdith tehnika kriginga.Usporedeni su radunski postupci kod jednostqunoga,kriging a (tehnike najbolj ega linearnoga procj enitelj a)te obitnoga kriginga (tehnike nqjboljega linearnogne p r is tr qno g p r o cj enit e lj a) .4.1. Razlika varijanca procjene kod tehnikajednostavnoga i obiinoga krigingaAko se na temelju podataka iz slike I te lit.2 raspiSematridna jednadZba obidnog kriginga (formula 3.11)dobiva se: 0 1465 21,5412,65 0 14,4221,54 1+45 0111 I o 12,6s 21,54'l lo,sgos'l I B,sss ']lrzos o 1+421 | o,+so+l=l oun.rn Ilzt,s+ t+,+s o Jlo,rzsz] lrs,ssrol (4.1) (3 11) Odnosno zaizratun matrice [B] wijedi:- za 1. red0' 0,3805 + 1 2,65 . 0,4964 + 21,54. 0,1232- 6 931 I = Q001- te jednako i 2a2.,3. i 4. red.Nadalje, va-rijanca obidnog kriginga se raduna prema:oL, = 1,.7(2,-Z)+ 12'y(22-Z)+...+1,.y(2,-Z)+ m (4.2)Varijanca iznosi o2=6,70 m2. pa standardna pogrjeskao=2,59 rn. Zapazimo da je > 1 =l (zbog zaokruLivanjarazll^ka iznosi +l o/oo). Promotrimo sada pona5anjegornje matridne jednadZbe ako se zadrLi rasporedkontrolnih todaka te wijednosti teZinskih koeficijenatakakvi su dani na slici 1, ali ako ih raspi5emo tehnikomjednostavnog kriginga. Tada se dobiva: Zapazirno kako su u takvom matridnom radunupromijenjene wijednosti variograma udaljenosti odkontrolnih todaka do todke procjene, odnosno ujednadZbi 3.3 wijednosti matrice [B]. To znati da bi sepromijenio poloZaj totke procjene u rarrnini na slici l, atirne bi porasla i varijanca procjene kojaizrrosio2:7,63Im2 (odnosno standardna pogrje5ka o=2,2G rn).4.2. Prcmjena vrijednosti teZinskihkoeficijenata kod jednostavnoga krigingaKrenimo sada promotriti sljede6i sludaj, a to je da su kodj ednostamo gkriginga zadr Z,ari poloL,aji svih kontrolnih itodke procjene kao i kod obidnog kriginga. Time suzadri,ane jednake variogramske wijednosti izmedu svihtodaka u obje promatrane tehnike kriginga. 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<736D6F20> Tj 1 0 0 1 426.96 688.08 Tm <6A7520> Tj 1 0 0 1 439.68 687.12 Tm <6B616F20> Tj 1 0 0 1 458.64 687.12 Tm <74616B767520> Tj 1 0 0 1 486.00 687.12 Tm <726173706973616C6920> Tj 1 0 0 1 526.08 687.36 Tm <6B726F7A> Tj 0.24 Tc 1 0 0 1 306.48 677.52 Tm <6A65646E61645A626520> Tj 1 0 0 1 353.52 676.80 Tm <746520> Tj 1 0 0 1 363.84 676.80 Tm <75706F7472696A6562696C6920> Tj 1 0 0 1 413.76 676.80 Tm <7A6120> Tj 1 0 0 1 426.00 676.80 Tm <6B616C6962726163696A7520> Tj 1 0 0 1 474.72 676.56 Tm <726164756E6120> Tj 1 0 0 1 507.12 676.56 Tm <7320> Tj 1 0 0 1 514.56 676.56 Tm <64727567696D> Tj 0.24 Tc 1 0 0 1 306.96 666.00 Tm <77696A20> Tj 1 0 0 1 321.60 666.00 Tm <65646E6F7374696D6120> Tj 1 0 0 1 368.64 666.24 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 429.12 666.00 Tm <6D756C7469706C696B61746F72612E> Tj 0.24 Tc 1 0 0 1 314.40 649.20 Tm <4D61747269636120> Tj 1 0 0 1 353.52 648.96 Tm <6F6269646E6F676120> Tj 1 0 0 1 397.68 649.20 Tm <6B726967696E676120> Tj 1 0 0 1 438.96 648.96 Tm <7A6120> Tj 1 0 0 1 455.52 648.96 Tm <6F76616920> Tj 1 0 0 1 478.56 648.96 Tm <736C7564616A20> Tj 1 0 0 1 509.76 648.96 Tm <697A72615A656E61> Tj 0.24 Tc 1 0 0 1 307.44 638.40 Tm <6B6F766172696A20> Tj 1 0 0 1 337.20 638.40 Tm <616E63616D6120> Tj 1 0 0 1 373.20 639.36 Tm <28696E7665727A6E6520> Tj 1 0 0 1 414.72 638.40 Tm <737520> Tj 1 0 0 1 426.96 638.40 Tm <76612D72696F6772616D752920> Tj 1 0 0 1 481.92 636.00 Tm <246173693A> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 313.68 620.16 Tm <312C3020> Tj 1 0 0 1 332.88 620.16 Tm <302C343920> Tj 1 0 0 1 354.72 620.16 Tm <302C343920> Tj 1 0 0 1 377.04 620.16 Tm <302C3331> Tj 0.48 Tc 1 0 0 1 311.04 607.92 Tm <302C343920> Tj 1 0 0 1 335.52 607.92 Tm <312C3020> Tj 1 0 0 1 355.20 607.92 Tm <302C333120> Tj 1 0 0 1 376.56 607.92 Tm <302C3435> Tj 0.48 Tc 1 0 0 1 311.04 595.92 Tm <302C343920> Tj 1 0 0 1 333.36 595.92 Tm <302C333120> Tj 1 0 0 1 357.36 595.92 Tm <312C3020> Tj 1 0 0 1 376.56 597.12 Tm <713439> Tj 0.48 Tc 1 0 0 1 311.52 583.92 Tm <302C333120> Tj 1 0 0 1 332.88 583.92 Tm <302C343920> Tj 1 0 0 1 354.72 583.68 Tm <302C343920> Tj 1 0 0 1 379.44 583.92 Tm <312C30> Tj 0.72 Tc 1 0 0 1 313.68 571.68 Tm <312C3020> Tj 1 0 0 1 335.52 571.92 Tm <312C3020> Tj 1 0 0 1 357.36 571.68 Tm <312C3020> Tj 1 0 0 1 379.44 571.68 Tm <312C30> Tj 0.48 Tc 1 0 0 1 526.08 595.68 Tm <28342E3829> Tj 0.24 Tc 1 0 0 1 307.44 548.16 Tm <5320> Tj 1 0 0 1 318.00 548.16 Tm <6F627A69726F6D20> Tj 1 0 0 1 357.84 548.16 Tm <646120> Tj 1 0 0 1 372.72 548.16 Tm <736520> Tj 1 0 0 1 385.92 548.16 Tm <7261646920> Tj 1 0 0 1 407.76 548.16 Tm <6F20> Tj 1 0 0 1 417.12 548.64 Tm <70726176696C6E6F6D20> Tj 1 0 0 1 465.36 548.16 Tm <726173706F726564752C20> Tj 1 0 0 1 516.24 548.16 Tm <756E75746172> Tj 0.24 Tc 1 0 0 1 307.20 537.12 Tm <6D61747269636520> Tj 1 0 0 1 342.00 535.44 Tm <5B5D223120> Tj 1 0 0 1 354.48 537.12 Tm <746520> Tj 1 0 0 1 364.56 535.44 Tm <5B425D20> Tj 1 0 0 1 378.72 537.84 Tm <286A65646E61645A626120> Tj 1 0 0 1 427.20 536.88 Tm <332E332920> Tj 1 0 0 1 446.16 536.88 Tm <646F626976656E6520> Tj 1 0 0 1 485.76 536.88 Tm <737520> Tj 1 0 0 1 497.76 536.88 Tm <6B6F6E7374616E746E65> Tj 0.24 Tc 1 0 0 1 306.96 526.32 Tm <77696A65646E6F7374692E20> Tj 1 0 0 1 365.04 526.56 Tm <416B6F20> Tj 1 0 0 1 392.88 526.32 Tm <736520> Tj 1 0 0 1 411.84 526.56 Tm <6D61747269646E6120> Tj 1 0 0 1 459.60 527.04 Tm <6A65646E61645A626120> Tj 1 0 0 1 514.56 526.32 Tm <72617370693565> Tj 0.24 Tc 1 0 0 1 307.20 515.76 Tm <6C696E6561726E696D20> Tj 1 0 0 1 351.36 515.76 Tm <6977617A696D6120> Tj 1 0 0 1 390.96 515.76 Tm <646F6269766120> Tj 1 0 0 1 421.92 515.76 Tm <73653A> Tj 0.72 Tc 1 0 0 1 306.96 498.24 Tm <312E20> Tj 1 0 0 1 314.88 498.24 Tm <4F2C323520> Tj 1 0 0 1 331.68 498.24 Tm <2B20> Tj 1 0 0 1 339.12 498.24 Tm <302C34392E20> Tj 1 0 0 1 358.32 498.24 Tm <302C323520> Tj 1 0 0 1 375.36 498.24 Tm <2B20> Tj 1 0 0 1 382.80 498.24 Tm <302C34392E20> Tj 1 0 0 1 401.76 498.24 Tm <302C323520> Tj 1 0 0 1 418.80 498.24 Tm <2B20> Tj 1 0 0 1 426.24 498.00 Tm <302C333120> Tj 1 0 0 1 441.36 500.40 Tm <2E20> Tj 1 0 0 1 444.72 498.00 Tm <302C323520> Tj 1 0 0 1 461.52 498.24 Tm <6120> Tj 1 0 0 1 468.96 498.00 Tm <362C363620> Tj 1 0 0 1 485.76 499.44 Tm <3D20> Tj 1 0 0 1 493.44 497.04 Tm <5136333235> Tj 0.96 Tc 1 0 0 1 307.44 486.24 Tm <302C34392E71323520> Tj 1 0 0 1 343.44 486.24 Tm <2B312E302C323520> Tj 1 0 0 1 375.36 486.24 Tm <2B302C333120> Tj 1 0 0 1 397.92 488.40 Tm <27302C323520> Tj 1 0 0 1 418.08 486.24 Tm <2B302C34392E302C323520> Tj 1 0 0 1 461.52 488.16 Tm <2D2C27362C36363D5136333235> Tj 0.48 Tc 1 0 0 1 307.44 474.00 Tm <302C34392720> Tj 1 0 0 1 326.40 474.00 Tm <302C323520> Tj 1 0 0 1 343.44 474.24 Tm <2B20> Tj 1 0 0 1 350.88 474.00 Tm <302C33312720> Tj 1 0 0 1 369.36 474.24 Tm <302C323520> Tj 1 0 0 1 386.16 474.24 Tm <2B20> Tj 1 0 0 1 393.36 474.24 Tm <312720> Tj 1 0 0 1 401.04 474.00 Tm <302C323520> Tj 1 0 0 1 418.08 474.24 Tm <2B20> Tj 1 0 0 1 425.52 474.00 Tm <4F2C34392E20> Tj 1 0 0 1 444.72 474.00 Tm <302C323520> Tj 1 0 0 1 461.52 474.00 Tm <2B20> Tj 1 0 0 1 468.96 474.00 Tm <302C303620> Tj 1 0 0 1 485.76 475.20 Tm <3D20> Tj 1 0 0 1 493.44 474.00 Tm <302C3633323520> Tj 1 0 0 1 526.08 474.72 Tm <28342E3929> Tj 0.48 Tc 1 0 0 1 307.44 462.00 Tm <302C33312720> Tj 1 0 0 1 325.68 462.00 Tm <302C323520> Tj 1 0 0 1 342.72 462.24 Tm <2B20> Tj 1 0 0 1 350.16 462.00 Tm <302C34392720> Tj 1 0 0 1 369.36 462.00 Tm <302C323520> Tj 1 0 0 1 386.16 462.24 Tm <2B20> Tj 1 0 0 1 393.60 462.00 Tm <302C34392720> Tj 1 0 0 1 412.80 462.00 Tm <302C323520> Tj 1 0 0 1 429.60 462.00 Tm <2B20> Tj 1 0 0 1 436.80 462.00 Tm <312E20> Tj 1 0 0 1 444.72 462.00 Tm <4F2C323520> Tj 1 0 0 1 461.52 462.00 Tm <2B20> Tj 1 0 0 1 468.96 461.76 Tm <302C303620> Tj 1 0 0 1 485.76 463.20 Tm <3D20> Tj 1 0 0 1 493.44 461.76 Tm <302C36333235> Tj 0.96 Tc 1 0 0 1 306.96 450.00 Tm <3127302C323520> Tj 1 0 0 1 331.92 450.00 Tm <2B20> Tj 1 0 0 1 338.88 450.24 Tm <3178302C323520> Tj 1 0 0 1 363.36 450.24 Tm <2B312E20> Tj 1 0 0 1 378.48 451.20 Tm <71323520> Tj 1 0 0 1 395.28 450.00 Tm <2B20> Tj 1 0 0 1 402.48 450.00 Tm <312E302C323520> Tj 1 0 0 1 427.20 450.00 Tm <2B20> Tj 1 0 0 1 434.64 450.00 Tm <302E302C303620> Tj 1 0 0 1 460.32 451.20 Tm <3D20> Tj 1 0 0 1 467.76 450.00 Tm <31> Tj 0.24 Tc 1 0 0 1 307.44 427.20 Tm <5361646120> Tj 1 0 0 1 330.48 428.16 Tm <6A6520> Tj 1 0 0 1 341.28 427.20 Tm <6D6F6775366520> Tj 1 0 0 1 377.04 427.20 Tm <697A726164746E61746920> Tj 1 0 0 1 421.44 426.96 Tm <766172696A616E637520> Tj 1 0 0 1 463.92 426.96 Tm <6B726967696E676120> Tj 1 0 0 1 501.36 427.44 Tm <7072656D613A> Tj 0.48 Tc 1 0 0 1 306.96 407.76 Tm <6F5A7820> Tj 1 0 0 1 322.80 408.96 Tm <3D20> Tj 1 0 0 1 330.48 407.76 Tm <6F2E32352E20> Tj 1 0 0 1 350.16 407.76 Tm <302E363320> Tj 1 0 0 1 366.48 407.76 Tm <2B20> Tj 1 0 0 1 373.92 407.76 Tm <302E323520> Tj 1 0 0 1 390.48 409.92 Tm <2E20> Tj 1 0 0 1 393.60 407.52 Tm <302E363320> Tj 1 0 0 1 409.92 407.76 Tm <2B20> Tj 1 0 0 1 417.36 407.52 Tm <3020> Tj 1 0 0 1 423.12 407.76 Tm <323520> Tj 1 0 0 1 433.92 409.92 Tm <2E20> Tj 1 0 0 1 437.28 407.52 Tm <302E363320> Tj 1 0 0 1 453.36 407.52 Tm <2B20> Tj 1 0 0 1 460.80 407.52 Tm <302E32352720> Tj 1 0 0 1 480.72 407.52 Tm <302E363320> Tj 1 0 0 1 496.80 407.52 Tm <2B20> Tj 1 0 0 1 504.24 407.52 Tm <302E303620> Tj 1 0 0 1 521.04 408.96 Tm <3D20> Tj 1 0 0 1 528.72 407.52 Tm <302E3639> Tj 0.96 Tc 1 0 0 1 373.20 399.36 Tm <28342E313029> Tj 0.24 Tc 1 0 0 1 307.44 382.08 Tm <342E332E322E20> Tj 1 0 0 1 336.24 381.84 Tm <5672696A65646E6F737420> Tj 1 0 0 1 385.92 382.08 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 450.48 381.84 Tm <6D756C7469706C696B61746F726120> Tj 1 0 0 1 517.92 381.84 Tm <4F2C39> Tj 0.24 Tc 1 0 0 1 307.20 366.24 Tm <5520> Tj 1 0 0 1 324.48 366.24 Tm <6F766F6D6520> Tj 1 0 0 1 362.16 366.96 Tm <7072696D6A65727520> Tj 1 0 0 1 410.64 366.48 Tm <7A6120> Tj 1 0 0 1 430.32 366.48 Tm <4C616772616E67656F7620> Tj 1 0 0 1 488.64 366.24 Tm <6D756C7469706C696B61746F72> Tj 0.24 Tc 1 0 0 1 307.44 355.68 Tm <6F64616272616E6120> Tj 1 0 0 1 353.52 356.64 Tm <6A6520> Tj 1 0 0 1 367.92 356.40 Tm <706F7A697469776120> Tj 1 0 0 1 412.80 355.68 Tm <77696A65646E6F737420> Tj 1 0 0 1 462.48 355.68 Tm <302C392E20> Tj 1 0 0 1 485.28 355.68 Tm <5461646120> Tj 1 0 0 1 512.64 355.68 Tm <6D617472696361> Tj 0.24 Tc 1 0 0 1 307.20 345.36 Tm <6B6F766172696A616E636920> Tj 1 0 0 1 357.12 342.96 Tm <246173693A> Tj 1.20 Tc 1 0 0 1 523.20 302.88 Tm <283431315C> Tj 0.24 Tc 1 0 0 1 307.20 254.64 Tm <506F6E6F722C6D6F20> Tj 1 0 0 1 351.60 254.64 Tm <737520> Tj 1 0 0 1 369.84 254.64 Tm <756E7574617220> Tj 1 0 0 1 406.80 254.64 Tm <6D61747269636520> Tj 1 0 0 1 447.84 252.72 Tm <746C2E6C20> Tj 1 0 0 1 466.32 254.40 Tm <746520> Tj 1 0 0 1 482.88 252.48 Tm <74426C20> Tj 1 0 0 1 502.80 254.88 Tm <7072696B617A616E65> Tj 0.48 Tc 1 0 0 1 307.20 244.08 Tm <6B6F6E73746172746E6520> Tj 1 0 0 1 356.40 244.08 Tm <77696A65646E6F7374692C20> Tj 1 0 0 1 406.56 244.08 Tm <6B616F20> Tj 1 0 0 1 424.08 243.84 Tm <7520> Tj 1 0 0 1 431.76 244.32 Tm <706F74706F2461766C6A7520> Tj 1 0 0 1 485.76 243.84 Tm <342E332E312E20> Tj 1 0 0 1 512.40 243.84 Tm <4B61646120> Tj 1 0 0 1 536.16 243.84 Tm <7365> Tj 0.24 Tc 1 0 0 1 306.96 233.76 Tm <676F726E6A6120> Tj 1 0 0 1 338.16 233.52 Tm <6D61747269646E6120> Tj 1 0 0 1 379.68 234.24 Tm <6A65646E61645A626120> Tj 1 0 0 1 428.16 233.28 Tm <7261737069536520> Tj 1 0 0 1 463.20 233.28 Tm <6C696E6561726E696D20> Tj 1 0 0 1 508.32 233.28 Tm <697A72617A69726E61> Tj 0.24 Tc 1 0 0 1 307.44 222.72 Tm <646F6269766120> Tj 1 0 0 1 338.40 222.72 Tm <736520> Tj 1 0 0 1 350.16 222.72 Tm <7375737461763A> Tj 0.48 Tc 1 0 0 1 306.96 204.96 Tm <3120> Tj 1 0 0 1 312.00 207.36 Tm <27302C323520> Tj 1 0 0 1 331.68 205.20 Tm <2B20> Tj 1 0 0 1 339.12 204.96 Tm <302C343927302C323520> Tj 1 0 0 1 375.12 205.20 Tm <2B20> Tj 1 0 0 1 382.56 204.96 Tm <302C34392720> Tj 1 0 0 1 401.76 204.96 Tm <302C323520> Tj 1 0 0 1 418.56 204.96 Tm <2B20> Tj 1 0 0 1 426.00 204.96 Tm <302C333120> Tj 1 0 0 1 441.12 207.12 Tm <2720> Tj 1 0 0 1 444.48 204.72 Tm <302C323520> Tj 1 0 0 1 461.28 204.96 Tm <2B20> Tj 1 0 0 1 468.72 204.72 Tm <302C3920> Tj 1 0 0 1 481.20 206.16 Tm <3D20> Tj 1 0 0 1 488.40 204.96 Tm <312C34373235> Tj 0.72 Tc 1 0 0 1 307.20 192.96 Tm <6F2C34392E302C323520> Tj 1 0 0 1 343.20 192.96 Tm <2B312E302C323520> Tj 1 0 0 1 375.12 192.96 Tm <2B20> Tj 1 0 0 1 382.56 192.96 Tm <302C333120> Tj 1 0 0 1 397.68 195.12 Tm <2E302C323520> Tj 1 0 0 1 417.84 192.96 Tm <2B20> Tj 1 0 0 1 425.28 192.72 Tm <302C34392E302C323520> Tj 1 0 0 1 461.28 192.72 Tm <2B20> Tj 1 0 0 1 468.72 192.72 Tm <302C3920> Tj 1 0 0 1 481.20 194.16 Tm <3D312C34373235> Tj 0.72 Tc 1 0 0 1 307.20 180.72 Tm <4F2C34392720> Tj 1 0 0 1 326.40 180.72 Tm <302C323520> Tj 1 0 0 1 343.20 180.96 Tm <2B20> Tj 1 0 0 1 350.64 180.72 Tm <302C33312720> Tj 1 0 0 1 369.12 180.72 Tm <302C323520> Tj 1 0 0 1 385.92 180.72 Tm <2B20> Tj 1 0 0 1 393.12 180.72 Tm <312E20> Tj 1 0 0 1 400.80 181.92 Tm <71323520> Tj 1 0 0 1 417.84 180.72 Tm <2B20> Tj 1 0 0 1 425.28 180.72 Tm <4F2C34392E20> Tj 1 0 0 1 444.48 180.48 Tm <302C323520> Tj 1 0 0 1 461.28 180.72 Tm <2B20> Tj 1 0 0 1 468.72 180.48 Tm <4F2C3920> Tj 1 0 0 1 481.20 181.92 Tm <3D20> Tj 1 0 0 1 488.40 180.48 Tm <312C3437323520> Tj 1 0 0 1 522.96 181.20 Tm <28342E313229> Tj 0.72 Tc 1 0 0 1 307.20 168.48 Tm <302C333120> Tj 1 0 0 1 322.32 170.88 Tm <2E302C323520> Tj 1 0 0 1 342.48 168.72 Tm <2B20> Tj 1 0 0 1 349.92 168.72 Tm <302C34392E302C323520> Tj 1 0 0 1 385.92 168.72 Tm <2B20> Tj 1 0 0 1 393.36 168.48 Tm <302C34392E302C323520> Tj 1 0 0 1 429.36 168.48 Tm <2B20> Tj 1 0 0 1 436.56 168.48 Tm <312E302C323520> Tj 1 0 0 1 461.28 168.48 Tm <2B20> Tj 1 0 0 1 468.72 168.48 Tm <302C3920> Tj 1 0 0 1 481.20 169.68 Tm <3D312C34373235> Tj 0.96 Tc 1 0 0 1 306.96 156.48 Tm <312E20> Tj 1 0 0 1 314.64 156.48 Tm <302C323520> Tj 1 0 0 1 331.68 156.48 Tm <2B20> Tj 1 0 0 1 338.64 156.48 Tm <312720> Tj 1 0 0 1 346.56 156.48 Tm <302C323520> Tj 1 0 0 1 363.36 156.48 Tm <2B20> Tj 1 0 0 1 370.56 156.48 Tm <312E20> Tj 1 0 0 1 378.48 156.48 Tm <302C323520> Tj 1 0 0 1 395.28 156.48 Tm <2B20> Tj 1 0 0 1 402.48 156.48 Tm <312E20> Tj 1 0 0 1 410.16 156.48 Tm <302C323520> Tj 1 0 0 1 427.20 156.48 Tm <2B20> Tj 1 0 0 1 434.64 156.24 Tm <302E20> Tj 1 0 0 1 443.76 156.24 Tm <302C3920> Tj 1 0 0 1 456.00 157.68 Tm <3D20> Tj 1 0 0 1 463.20 156.24 Tm <31> Tj 0.24 Tc 1 0 0 1 306.72 133.44 Tm <566172696A616E636120> Tj 1 0 0 1 349.20 133.68 Tm <6B726967696E676120> Tj 1 0 0 1 386.40 133.68 Tm <757A20> Tj 1 0 0 1 399.12 133.68 Tm <4C616772616E67656F7620> Tj 1 0 0 1 450.00 133.44 Tm <6B6F65666963696A656E7420> Tj 1 0 0 1 496.80 133.44 Tm <302C3920> Tj 1 0 0 1 513.12 133.44 Tm <697A6E6F73693A> Tj 0.72 Tc 1 0 0 1 306.96 114.24 Tm <6F5A7220> Tj 1 0 0 1 322.80 115.44 Tm <3D20> Tj 1 0 0 1 330.24 114.24 Tm <302C323527312C343720> Tj 1 0 0 1 365.52 114.24 Tm <2B20> Tj 1 0 0 1 372.96 114.24 Tm <302C323527312C343720> Tj 1 0 0 1 408.24 114.24 Tm <2B20> Tj 1 0 0 1 415.68 114.00 Tm <4F2C323527312C343720> Tj 1 0 0 1 450.96 114.00 Tm <2B20> Tj 1 0 0 1 458.40 114.00 Tm <302C323527312C343720> Tj 1 0 0 1 493.68 114.00 Tm <2B20> Tj 1 0 0 1 501.12 114.00 Tm <4F2C6720> Tj 1 0 0 1 513.60 115.44 Tm <3D312C32363735> Tj 0.72 Tc 1 0 0 1 522.96 105.12 Tm <28342E313329> Tj 0.96 Tc /F3 10.0 Tf 1 0 0 1 533.76 81.84 Tm <3431> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 109.44 524.40 Tm <32312C35343120> Tj 1 0 0 1 134.16 523.20 Tm <7C20> Tj 1 0 0 1 139.20 524.40 Tm <303334373120> Tj 1 0 0 1 170.88 523.20 Tm <5B20> Tj 1 0 0 1 179.28 524.40 Tm <612C73535320> Tj 1 0 0 1 202.32 523.44 Tm <49> Tj 0.96 Tc /F3 13.0 Tf 1 0 0 1 108.24 512.16 Tm <412C34326C786C6F2C346D20> Tj 1 0 0 1 159.84 506.40 Tm <7C20> Tj 1 0 0 1 163.92 513.84 Tm <3D6C20> Tj 1 0 0 1 177.12 512.40 Tm <6F2C7373727320> Tj 1 0 0 1 204.48 506.40 Tm <49> Tj 0.96 Tc /F3 10.0 Tf 1 0 0 1 115.44 500.16 Tm <3020> Tj 1 0 0 1 126.48 497.52 Tm <4920> Tj 1 0 0 1 134.16 497.52 Tm <4C302C3042425D20> Tj 1 0 0 1 170.88 497.52 Tm <3131352C3335316531> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 271.20 751.20 Tm <28342E3429> Tj 0.24 Tc 1 0 0 1 249.12 716.88 Tm <6C696E6561726E696D> Tj 0.24 Tc 1 0 0 1 271.20 678.48 Tm <283420> Tj 1 0 0 1 279.60 678.00 Tm <3529> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 52.56 574.08 Tm <4E616B6F6E20> Tj 1 0 0 1 85.68 574.08 Tm <697A726128756E6120> Tj 1 0 0 1 127.20 574.32 Tm <6B6F65666963696A656E746920> Tj 1 0 0 1 178.80 574.32 Tm <697A6E6F736520> Tj 1 0 0 1 210.96 574.56 Tm <292C7420> Tj 1 0 0 1 225.12 576.24 Tm <3A20> Tj 1 0 0 1 236.88 574.56 Tm <512C4A513720> Tj 1 0 0 1 262.08 573.60 Tm <3B20> Tj 1 0 0 1 269.52 574.56 Tm <5D612220> Tj 1 0 0 1 283.68 576.24 Tm <3D> Tj 0.72 Tc 1 0 0 1 52.56 562.80 Tm <4F2C3438333B20> Tj 1 0 0 1 83.04 562.80 Tm <6C7320> Tj 1 0 0 1 95.28 564.48 Tm <3D20> Tj 1 0 0 1 105.12 562.80 Tm <302C30383820> Tj 1 0 0 1 133.20 563.76 Tm <287A6170617A696D6F20> Tj 1 0 0 1 179.52 563.04 Tm <646120> Tj 1 0 0 1 192.24 564.00 Tm <6A6520> Tj 1 0 0 1 203.28 563.28 Tm <584C3D302C393138292E20> Tj 1 0 0 1 254.64 563.28 Tm <546F20> Tj 1 0 0 1 268.32 563.28 Tm <7A6E616469> Tj 0.24 Tc 1 0 0 1 52.80 552.24 Tm <646120> Tj 1 0 0 1 65.28 552.24 Tm <6D61747269646E6120> Tj 1 0 0 1 105.36 552.96 Tm <6A65646E61645A626120> Tj 1 0 0 1 151.68 553.20 Tm <6A65646E6F737461776F6720> Tj 1 0 0 1 210.72 552.72 Tm <6B726967696E676120> Tj 1 0 0 1 248.64 552.72 Tm <7A61706F6C6F7A2C616A> Tj 0.24 Tc 1 0 0 1 52.32 541.68 Tm <746F64616B6120> Tj 1 0 0 1 83.28 541.68 Tm <6B616B617620> Tj 1 0 0 1 110.40 542.64 Tm <6A6520> Tj 1 0 0 1 121.68 542.16 Tm <70726C6B617A616E20> Tj 1 0 0 1 161.76 541.92 Tm <6E6120> Tj 1 0 0 1 175.20 541.92 Tm <736C69636920> Tj 1 0 0 1 196.08 542.16 Tm <4920> Tj 1 0 0 1 203.04 542.40 Tm <676C6173693A> Tj 0.96 Tc 1 0 0 1 63.36 524.16 Tm <3020> Tj 1 0 0 1 82.08 524.16 Tm <31323635> Tj 0.24 Tc 1 0 0 1 55.92 512.16 Tm <3132363520> Tj 1 0 0 1 89.52 512.16 Tm <30> Tj 0.24 Tc 1 0 0 1 56.64 500.16 Tm <32312C353420> Tj 1 0 0 1 82.08 500.16 Tm <31342C3435> Tj 0.48 Tc /F3 10.0 Tf 1 0 0 1 52.32 479.76 Tm <342E332E6C7A72616475726E20> Tj 1 0 0 1 116.16 480.00 Tm <6D61747269636120> Tj 1 0 0 1 157.68 480.00 Tm <6F6269696E6F676120> Tj 1 0 0 1 205.44 480.24 Tm <6B726967696E676120> Tj 1 0 0 1 249.36 480.24 Tm <6B6F64> Tj 0.24 Tc 1 0 0 1 74.64 468.48 Tm <70726176696C6E6F676120> Tj 1 0 0 1 130.32 468.48 Tm <726173706F7265646120> Tj 1 0 0 1 182.88 468.48 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 239.04 468.48 Tm <746F69616B6120> Tj 1 0 0 1 275.04 468.48 Tm <757A> Tj 0.24 Tc 1 0 0 1 74.64 456.96 Tm <70726F6D6A656E7520> Tj 1 0 0 1 126.00 456.72 Tm <7672696A65646E6F7374696D6120> Tj 1 0 0 1 198.00 456.72 Tm <4C616772616E67656F766F67> Tj 0.24 Tc 1 0 0 1 74.64 444.72 Tm <6D756C7469706C696B61746F7261> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 52.32 430.32 Tm <5520> Tj 1 0 0 1 64.80 430.32 Tm <6F766F6D6520> Tj 1 0 0 1 97.68 430.80 Tm <706F74706F676C61766C6A7520> Tj 1 0 0 1 155.28 430.80 Tm <75706F7472696A65626C6A656E20> Tj 1 0 0 1 214.56 431.76 Tm <6A6520> Tj 1 0 0 1 228.24 431.04 Tm <6E6F766920> Tj 1 0 0 1 251.76 431.04 Tm <726173706F726564> Tj 0.24 Tc 1 0 0 1 52.56 419.76 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 106.80 419.76 Tm <746F64616B6120> Tj 1 0 0 1 144.48 420.96 Tm <28736C696B6120> Tj 1 0 0 1 177.12 420.24 Tm <322920> Tj 1 0 0 1 194.64 420.24 Tm <746520> Tj 1 0 0 1 211.68 420.24 Tm <647275676520> Tj 1 0 0 1 244.80 420.24 Tm <77696A65646E6F737469> Tj 0.24 Tc 1 0 0 1 52.08 409.20 Tm <766172696F6772616D736B6F6720> Tj 1 0 0 1 123.36 409.44 Tm <6D6F64656C612E20> Tj 1 0 0 1 165.84 409.44 Tm <43696C6A20> Tj 1 0 0 1 189.12 409.44 Tm <616E6166697A6520> Tj 1 0 0 1 225.84 409.68 Tm <62696F20> Tj 1 0 0 1 246.72 410.64 Tm <6A6520> Tj 1 0 0 1 263.28 410.16 Tm <7072696B617A> Tj 0.24 Tc 1 0 0 1 52.32 399.12 Tm <70726F6D6A656E6520> Tj 1 0 0 1 98.88 398.64 Tm <77696A65646E6F73746920> Tj 1 0 0 1 150.96 398.88 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 215.04 399.12 Tm <6D756C7469706C696B61746F726120> Tj 1 0 0 1 282.48 399.12 Tm <7465> Tj 0.24 Tc 1 0 0 1 52.32 388.32 Tm <75746A6563616A20> Tj 1 0 0 1 83.04 388.32 Tm <74616B766F676120> Tj 1 0 0 1 118.08 388.80 Tm <706F737475706B6120> Tj 1 0 0 1 161.04 388.56 Tm <6E6120> Tj 1 0 0 1 174.00 388.80 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Tj 1 0 0 1 256.56 353.76 Tm <736C6A6564656461> Tj 0.24 Tc 1 0 0 1 52.56 342.72 Tm <64766120> Tj 1 0 0 1 72.72 342.72 Tm <697A726164756E6120> Tj 1 0 0 1 114.48 342.72 Tm <77696A65646E6F73746920> Tj 1 0 0 1 165.84 342.96 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 228.72 343.20 Tm <6D756C7469706C696B61746F7261> Tj 0.48 Tc 1 0 0 1 52.32 332.16 Tm <697A6E6F73696C6520> Tj 1 0 0 1 91.68 332.16 Tm <737520> Tj 1 0 0 1 108.72 332.16 Tm <302C392C20> Tj 1 0 0 1 131.76 332.40 Tm <6F646E6F736E6F20> Tj 1 0 0 1 175.20 334.32 Tm <2D302C392E20> Tj 1 0 0 1 201.36 332.64 Tm <546520> Tj 1 0 0 1 218.40 332.64 Tm <6476696A6520> Tj 1 0 0 1 244.80 332.64 Tm <77696A65646E6F737469> Tj 0.24 Tc 1 0 0 1 52.56 321.60 Tm <736D617472616E6520> Tj 1 0 0 1 97.44 321.60 Tm <737520> Tj 1 0 0 1 112.08 322.08 Tm <707269626C695A6E696D20> Tj 1 0 0 1 162.00 321.60 Tm <656B737472656D696D6120> Tj 1 0 0 1 217.68 321.84 Tm <6B6F6A6520> Tj 1 0 0 1 240.48 321.84 Tm <746120> Tj 1 0 0 1 252.96 321.84 Tm <766172696A61626C61> Tj 0.24 Tc 1 0 0 1 52.32 311.04 Tm <6E616A646553366520> Tj 1 0 0 1 90.48 311.04 Tm <7520> Tj 1 0 0 1 98.88 311.52 Tm <7072616B736920> Tj 1 0 0 1 128.88 311.28 Tm <6D6F5A6520> Tj 1 0 0 1 154.08 311.52 Tm <706F7072696D6974692E> Tj 0.24 Tc 1 0 0 1 59.52 297.12 Tm <4F64616269726F6D20> Tj 1 0 0 1 111.36 297.12 Tm <7472696A7520> Tj 1 0 0 1 138.00 297.36 Tm <77696A65646E6F73746920> Tj 1 0 0 1 192.24 297.60 Tm <7A6120> Tj 1 0 0 1 209.52 298.32 Tm <6A65646E616B6520> Tj 1 0 0 1 252.24 297.60 Tm <6D61747269646E65> Tj 0.24 Tc 1 0 0 1 51.60 287.28 Tm <6A65646E61645A626520> Tj 1 0 0 1 100.32 286.80 Tm <4C2C656C6A656C6F20> Tj 1 0 0 1 131.76 286.80 Tm <736520> Tj 1 0 0 1 145.20 287.28 Tm <706F6B617A61746920> Tj 1 0 0 1 186.48 286.80 Tm <6B616B6F20> Tj 1 0 0 1 212.40 286.80 Tm <6F646162697220> Tj 1 0 0 1 244.56 287.04 Tm <77696A65646E6F737469> Tj 0.24 Tc 1 0 0 1 52.32 276.00 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 114.72 276.24 Tm <6D756C7469706C696B61746F726120> Tj 1 0 0 1 180.96 276.24 Tm <696D6120> Tj 1 0 0 1 201.60 276.72 Tm <707265737564616E20> Tj 1 0 0 1 246.96 276.48 Tm <75746A6563616A20> Tj 1 0 0 1 279.60 276.48 Tm <6E61> Tj 0.24 Tc 1 0 0 1 52.32 265.44 Tm <6D696E696D697A6972616E6A6520> Tj 1 0 0 1 118.32 266.16 Tm <706F67726A65736B6520> Tj 1 0 0 1 166.32 265.68 Tm <6B6F6420> Tj 1 0 0 1 189.36 265.92 Tm <697A726164756E6120> Tj 1 0 0 1 232.32 265.92 Tm <77696A65646E6F73746920> Tj 1 0 0 1 284.64 266.16 Tm <75> Tj 0.24 Tc 1 0 0 1 52.32 254.88 Tm <746F646B692E20> Tj 1 0 0 1 79.44 255.12 Tm <526164756E20> Tj 1 0 0 1 108.96 255.84 Tm <6A6520> Tj 1 0 0 1 120.48 255.12 Tm <697A766564656E20> Tj 1 0 0 1 155.52 255.12 Tm <7275646E6F2C20> Tj 1 0 0 1 186.72 255.36 Tm <6B616B6F20> Tj 1 0 0 1 211.20 255.12 Tm <626920> Tj 1 0 0 1 222.96 255.36 Tm <736520> Tj 1 0 0 1 235.20 255.36 Tm <6D6F676C6520> Tj 1 0 0 1 263.76 255.84 Tm <70726174697469> Tj 0.24 Tc 1 0 0 1 52.32 244.80 Tm <70726F6D6A656E6520> Tj 1 0 0 1 96.96 244.56 Tm <6B726F7A20> Tj 1 0 0 1 120.48 244.56 Tm <6D61747269646E6520> Tj 1 0 0 1 162.48 244.56 Tm <6920> Tj 1 0 0 1 169.20 244.56 Tm <6C696E6561726E6520> Tj 1 0 0 1 207.12 245.52 Tm <6A65646E61645A626520> Tj 1 0 0 1 255.60 244.80 Tm <6B726967696E6761> Tj 0.48 Tc 1 0 0 1 52.80 234.72 Tm <286A65646E61645A626520> Tj 1 0 0 1 102.96 233.76 Tm <332E313020> Tj 1 0 0 1 125.76 234.00 Tm <6920> Tj 1 0 0 1 132.72 234.00 Tm <332E3133292E20> Tj 1 0 0 1 161.28 234.00 Tm <416E616C697A6972616E6920> Tj 1 0 0 1 211.92 234.24 Tm <726173706F72656420> Tj 1 0 0 1 254.64 234.24 Tm <736164725A6920> Tj 1 0 0 1 284.88 234.24 Tm <34> Tj 0.24 Tc 1 0 0 1 52.32 223.20 Tm <6B6F6E74726F6C6E6520> Tj 1 0 0 1 100.08 223.20 Tm <746F646B6520> Tj 1 0 0 1 129.12 223.20 Tm <746520> Tj 1 0 0 1 143.04 223.20 Tm <736520> Tj 1 0 0 1 158.16 223.44 Tm <75706F747265626F6D20> Tj 1 0 0 1 211.68 223.44 Tm <6F6269646E6F676120> Tj 1 0 0 1 255.60 223.68 Tm <6B726967696E6761> Tj 0.24 Tc 1 0 0 1 52.32 212.88 Tm <70726F63696A656E696C6120> Tj 1 0 0 1 103.44 212.64 Tm <77696A65646E6F737420> Tj 1 0 0 1 152.40 212.64 Tm <7520> Tj 1 0 0 1 164.40 212.64 Tm <746F646B6920> Tj 1 0 0 1 191.76 212.88 Tm <736D6A655374656E6F6A20> Tj 1 0 0 1 243.60 213.12 Tm <7520> Tj 1 0 0 1 255.60 212.88 Tm <7372656469357475> Tj 0.48 Tc 1 0 0 1 52.32 202.32 Tm <706F6C69676F6E6120> Tj 1 0 0 1 91.68 202.80 Tm <28736C696B6120> Tj 1 0 0 1 117.60 202.08 Tm <32292E> Tj 0.24 Tc 1 0 0 1 59.28 188.16 Tm <4E6120> Tj 1 0 0 1 73.68 187.92 Tm <736C69636920> Tj 1 0 0 1 93.84 188.16 Tm <3220> Tj 1 0 0 1 102.72 187.92 Tm <6F7A6E61282E656E6120> Tj 1 0 0 1 142.80 189.12 Tm <6A6520> Tj 1 0 0 1 153.84 188.16 Tm <6D656475736F626E6120> Tj 1 0 0 1 206.16 188.40 Tm <72656C6174696D6120> Tj 1 0 0 1 245.76 188.64 Tm <7564616C6A656E6F7374> Tj 0.24 Tc 1 0 0 1 52.32 177.36 Tm <6B6F6E74726F6C6E696820> Tj 1 0 0 1 100.32 177.36 Tm <746F64616B6120> Tj 1 0 0 1 131.52 178.32 Tm <28353020> Tj 1 0 0 1 147.84 177.60 Tm <6D292C20> Tj 1 0 0 1 164.88 177.60 Tm <6E6120> Tj 1 0 0 1 178.08 177.60 Tm <74656D656C6A7520> Tj 1 0 0 1 212.40 177.84 Tm <6B6F6A6520> Tj 1 0 0 1 233.04 177.84 Tm <737520> Tj 1 0 0 1 246.00 177.84 Tm <697A726128743A696E617465> Tj 0.24 Tc 1 0 0 1 52.08 166.80 Tm <766172696F6772616D736B6520> Tj 1 0 0 1 115.68 166.80 Tm <77696A65646E6F7374692E20> Tj 1 0 0 1 170.16 167.04 Tm <54616B617620> Tj 1 0 0 1 201.12 167.04 Tm <766172696F6772616D736B6920> Tj 1 0 0 1 263.76 167.28 Tm <6D6F64656C> Tj 0.48 Tc 1 0 0 1 52.32 156.24 Tm <696D6120> Tj 1 0 0 1 71.28 156.24 Tm <736C6A656465636520> Tj 1 0 0 1 107.04 156.24 Tm <7672696A65646E6F7374693A> Tj 0.24 Tc 1 0 0 1 53.52 144.24 Tm <2E20> Tj 1 0 0 1 60.00 141.84 Tm <656B73706572696D656E74616C6E6120> Tj 1 0 0 1 136.80 142.32 Tm <6B72696C27756C6A6120> Tj 1 0 0 1 175.68 142.32 Tm <6170726F6B73696D6972616E6120> Tj 1 0 0 1 243.60 143.28 Tm <6A6520> Tj 1 0 0 1 257.52 142.32 Tm <736665726E696D> Tj 0.48 Tc 1 0 0 1 59.76 131.52 Tm <6D6F64656C6F6D3B> Tj 0.48 Tc 1 0 0 1 53.52 120.00 Tm <2E20> Tj 1 0 0 1 59.76 118.32 Tm <7072616720> Tj 1 0 0 1 81.60 117.84 Tm <6D6F64656C6120> Tj 1 0 0 1 114.72 118.80 Tm <6A6520> Tj 1 0 0 1 126.72 118.08 Tm <6C3B> Tj 0.48 Tc 1 0 0 1 53.52 106.56 Tm <2E20> Tj 1 0 0 1 60.00 104.16 Tm <646F73656720> Tj 1 0 0 1 86.88 104.40 Tm <6D6F64656C6120> Tj 1 0 0 1 119.76 105.36 Tm <6A6520> Tj 1 0 0 1 131.04 104.64 Tm <32303020> Tj 1 0 0 1 149.76 104.64 Tm <6D3B> Tj 0.48 Tc 1 0 0 1 52.56 81.36 Tm <4E4146544120> Tj 1 0 0 1 72.72 81.36 Tm <357320> Tj 1 0 0 1 81.84 81.84 Tm <283120> Tj 1 0 0 1 87.60 79.92 Tm <2920> Tj 1 0 0 1 91.20 81.36 Tm <33382D343320> Tj 1 0 0 1 110.40 81.84 Tm <283230303929> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 271.20 513.12 Tm <28342E3729> Tj 0.72 Tc 1 0 0 1 271.20 614.88 Tm <28342E3629> Tj ET endstream endobj 112 0 obj 29734 endobj 113 0 obj << /Length 114 0 R >> stream 0 12,6512,65 021,54 1+45 Kada se gornja matridna jednadZba raspiSeizrazirna dobiva se sustav:12,65. 12 + 21,54-72 = I12,65'11+ 14,45')," = $,$$21,54' )" t + 1 +42' 12 = 1 4,42TeZinski koeficijenti mogu se izrazitikao:. B-21,54.L , 5,66-14,45.,t3/\^ =' 12,65 12,65^ 14,42-14,42.L,"1 - 21.s4 . odstupanje modela ne postoji (0).4.3. 1. Vrijednost Lagrangeovog multiplikatora O,O6(kao u preuzetom primjeru)U ovom potpo$avlju wijednost Lagrangeovogmultiplikatora preuzeta je iz lit.3. Takva wijednostosigurava najmanju wijednost va-rijance procjeneobidnirn krigingom pa smo ju kao takvu raspisali krozjednadZbe te upotrijebili za kalibraciju raduna s drugimwij ednostima Lagrangeovog multiplikatora.Matrica obidnoga kriginga za ovai sludaj izraZenakovarij ancama (inverzne su va-riogramu) $asi: 1,0 0,49 0,49 0,310,49 1,0 0,31 0,450,49 0,31 1,0 q490,31 0,49 0,49 1,01,0 1,0 1,0 1,0 (4.8) S obzirom da se radi o pravilnom rasporedu, unutarmatrice []"1 te [B] (jednadZba 3.3) dobivene su konstantnewijednosti. Ako se matridna jednadZba raspi5elinearnim iwazima dobiva se:1. O,25 + 0,49. 0,25 + 0,49. 0,25 + 0,31 . 0,25 a 6,66 = Q63250,49.q25 +1.0,25 +0,31 '0,25 +0,49.0,25 -,'6,66=Q63250,49' 0,25 + 0,31' 0,25 + 1' 0,25 + O,49. 0,25 + 0,06 = 0,6325 (4.9)0,31' 0,25 + 0,49' 0,25 + 0,49' 0,25 + 1. O,25 + 0,06 = 0,63251'0,25 + 1x0,25 +1. q25 + 1.0,25 + 0.0,06 = 1Sada je mogu6e izradtnati varijancu kriginga prema:oZx = o.25. 0.63 + 0.25 . 0.63 + 0 25 . 0.63 + 0.25' 0.63 + 0.06 = 0.69(4.10)4.3.2. Vrijednost Lagrangeovog multiplikatora O,9U ovome primjeru za Lagrangeov multiplikatorodabrana je pozitiwa wijednost 0,9. Tada matricakovarijanci $asi: (411\ Ponor,mo su unutar matrice tl.l te tBl prikazanekonstartne wijednosti, kao u potpo$avlju 4.3.1. Kada segornja matridna jednadZba raspiSe linearnim izrazirnadobiva se sustav:1 '0,25 + 0,49'0,25 + 0,49' 0,25 + 0,31 ' 0,25 + 0,9 = 1,4725o,49.0,25 +1.0,25 + 0,31 .0,25 + 0,49.0,25 + 0,9 =1,4725O,49' 0,25 + 0,31' 0,25 + 1. q25 + O,49. 0,25 + O,9 = 1,4725 (4.12)0,31 .0,25 + 0,49.0,25 + 0,49.0,25 + 1.0,25 + 0,9 =1,47251. 0,25 + 1' 0,25 + 1. 0,25 + 1. 0,25 + 0. 0,9 = 1Varijanca kriginga uz Lagrangeov koeficijent 0,9 iznosi:oZr = 0,25'1,47 + 0,25'1,47 + O,25'1,47 + 0,25'1,47 + O,g =1,2675(4.13)41 21,541 | 03471 [ a,sSS IA,42lxlo,4m | =l o,ssrs I0 I L0,0BB] 115,351e1 (4.4)linearnim(4 5) Nakon izra(una koeficijenti iznose ),t : Q,JQ7 ; ]a" =O,483; ls = 0,088 (zapazimo da je XL=0,918). To znadida matridna jednadZba jednostawog kriginga zapoloz,ajtodaka kakav je prlkazan na slici I glasi: 0 12651265 021,54 14,45 4.3.lzradurn matrica obiinoga kriginga kodpravilnoga rasporeda kontrolnih toiaka uzpromjenu vrijednostima LagrangeovogmultiplikatoraU ovome potpoglavlju upotrijebljen je novi rasporedkontrolnih todaka (slika 2) te druge wijednostivariogramskog modela. Cilj anafize bio je prikazpromjene wijednosti Lagrangeovog multiplikatora teutjecaj takvoga postupka na procjenu.Analizirani su sludajevi s tri razlidite wijednostimultiplikatora. U prvomu Lagrangeova wijednost iznosi0,06, a preuzeta je iz objavljenoga primjera3. U sljededadva izraduna wijednosti Lagrangeovog multiplikatoraiznosile su 0,9, odnosno -0,9. Te dvije wijednostismatrane su pribliZnim ekstremima koje ta varijablanajdeS6e u praksi moZe poprimiti.Odabirom triju wijednosti za jednake matridnejednadZbe L,eljelo se pokazati kako odabir wijednostiLagrangeovog multiplikatora ima presudan utjecaj naminimiziranje pogrjeske kod izraduna wijednosti utodki. Radun je izveden rudno, kako bi se mogle pratitipromjene kroz matridne i linearne jednadZbe kriginga(jednadZbe 3.10 i 3.13). Analizirani raspored sadrZi 4kontrolne todke te se upotrebom obidnoga krigingaprocijenila wijednost u todki smjeStenoj u sredi5tupoligona (slika 2).Na slici 2 ozna(.ena je medusobna relatima udaljenostkontrolnih todaka (50 m), na temelju koje su izra(t:inatevariogramske wijednosti. 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0 0 1 55.68 502.80 Tm <507269706164616A75366120> Tj 1 0 0 1 115.92 502.56 Tm <766172696A616E636120> Tj 1 0 0 1 165.12 502.56 Tm <7461646120> Tj 1 0 0 1 192.48 503.52 Tm <6A6520> Tj 1 0 0 1 210.48 502.56 Tm <6E6567617469766E6120> Tj 1 0 0 1 261.12 503.28 Tm <2853746F20> Tj 1 0 0 1 285.36 503.28 Tm <6A65> Tj 0.24 Tc 1 0 0 1 55.44 492.24 Tm <6D6174656D617469646B6920> Tj 1 0 0 1 110.88 492.24 Tm <6E656D6F677536652C20> Tj 1 0 0 1 158.64 492.48 Tm <706120> Tj 1 0 0 1 170.64 492.96 Tm <6A6520> Tj 1 0 0 1 181.20 492.24 Tm <6920> Tj 1 0 0 1 186.48 492.24 Tm <4C616772616E67656F7620> Tj 1 0 0 1 236.88 492.00 Tm <6D756C7469706C696B61746F72> Tj 0.24 Tc 1 0 0 1 55.44 481.68 Tm <6E65697370726176616E2920> Tj 1 0 0 1 108.00 481.68 Tm <697A6E6F73693A> Tj 1.50 Tc 1 0 0 1 60.72 469.20 Tm <2C20> Tj 1 0 0 1 78.72 469.20 Tm <5E5E2D> Tj 0.48 Tc 1 0 0 1 55.20 464.88 Tm <6F276F7820> Tj 1 0 0 1 71.04 466.08 Tm <3D20> Tj 1 0 0 1 78.48 464.88 Tm <6F2C323520> Tj 1 0 0 1 95.28 467.04 Tm <27282D302C333237352920> Tj 1 0 0 1 134.88 464.88 Tm <2B20> Tj 1 0 0 1 142.32 464.64 Tm <302C323520> Tj 1 0 0 1 159.12 467.04 Tm <27282D302C333237352920> Tj 1 0 0 1 198.72 464.88 Tm <2B20> Tj 1 0 0 1 206.16 464.64 Tm <302C323520> Tj 1 0 0 1 222.96 467.04 Tm <27282D302C333237352920> Tj 1 0 0 1 262.80 464.64 Tm <2B> Tj 0.48 Tc 1 0 0 1 78.00 452.88 Tm <6F2C32352E20> Tj 1 0 0 1 97.68 453.12 Tm <282D302C333237352920> Tj 1 0 0 1 134.40 452.88 Tm <2B20> Tj 1 0 0 1 141.60 453.12 Tm <282D302C392920> Tj 1 0 0 1 165.12 454.08 Tm <3D20> Tj 1 0 0 1 172.56 454.80 Tm <2D302C393832> Tj 0.72 Tc 1 0 0 1 271.20 441.12 Tm <28342E313629> Tj 0.24 Tc 1 0 0 1 55.20 422.40 Tm <497A20> Tj 1 0 0 1 65.52 422.40 Tm <73766120> Tj 1 0 0 1 82.08 422.40 Tm <74726920> Tj 1 0 0 1 95.04 422.64 Tm <7072696D6A65726120> Tj 1 0 0 1 135.36 422.64 Tm <7072696B617A616E6120> Tj 1 0 0 1 180.48 422.40 Tm <7520> Tj 1 0 0 1 188.88 422.64 Tm <706F74706F676C61766C6A7520> Tj 1 0 0 1 243.84 422.16 Tm <342E3320> Tj 1 0 0 1 259.44 422.88 Tm <6A61736E6F20> Tj 1 0 0 1 285.12 422.88 Tm <6A65> Tj 0.24 Tc 1 0 0 1 55.20 411.84 Tm <6B616B6F20> Tj 1 0 0 1 78.24 412.56 Tm <6A6520> Tj 1 0 0 1 89.04 411.60 Tm <77696A65646E6F737420> Tj 1 0 0 1 134.64 411.84 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 195.12 411.60 Tm <6B6F65666963696A656E746120> Tj 1 0 0 1 246.96 411.60 Tm <302C303620> Tj 1 0 0 1 269.28 412.32 Tm <286B616B6F> Tj 0.24 Tc 1 0 0 1 54.24 402.00 Tm <6A6520> Tj 1 0 0 1 64.56 401.28 Tm <6E61766564656E6F20> Tj 1 0 0 1 107.04 401.28 Tm <7520> Tj 1 0 0 1 114.96 401.28 Tm <6C69742E3320> Tj 1 0 0 1 132.00 401.28 Tm <6920> Tj 1 0 0 1 137.04 401.52 Tm <7072696B617A616E6F20> Tj 1 0 0 1 182.40 401.04 Tm <6E6120> Tj 1 0 0 1 195.36 401.04 Tm <736C69636920> Tj 1 0 0 1 215.04 401.04 Tm <322920> Tj 1 0 0 1 225.84 401.04 Tm <6120> Tj 1 0 0 1 232.80 401.04 Tm <746F20> Tj 1 0 0 1 242.40 401.76 Tm <6A6520> Tj 1 0 0 1 252.96 401.04 Tm <75707261766F20> Tj 1 0 0 1 285.12 400.80 Tm <7461> Tj 0.24 Tc 1 0 0 1 54.72 390.72 Tm <77696A65646E6F737420> Tj 1 0 0 1 101.52 390.96 Tm <67646A6520> Tj 1 0 0 1 122.16 391.44 Tm <6A6520> Tj 1 0 0 1 134.16 390.48 Tm <766172696A616E636120> Tj 1 0 0 1 177.36 390.72 Tm <6B726967696E676120> Tj 1 0 0 1 216.00 390.72 Tm <6D696E696D616C6E612E20> Tj 1 0 0 1 268.80 390.48 Tm <4E6164696E> Tj 0.24 Tc 1 0 0 1 55.20 380.16 Tm <697A626F726120> Tj 1 0 0 1 87.60 380.16 Tm <6F7064656E69746F20> Tj 1 0 0 1 129.12 380.16 Tm <6E616A707269726E6A6572656E696A6520> Tj 1 0 0 1 199.92 379.92 Tm <77696A65646E6F73746920> Tj 1 0 0 1 251.04 379.92 Tm <4C616772616E67652D> Tj 0.24 Tc 1 0 0 1 55.20 369.60 Tm <6F766F6720> Tj 1 0 0 1 77.28 369.60 Tm <6B6F65666963696A656E74612C20> Tj 1 0 0 1 132.72 369.60 Tm <616E61666C7A7272616E6920> Tj 1 0 0 1 180.24 369.36 Tm <737520> Tj 1 0 0 1 193.20 369.60 Tm <6B726F7A20> Tj 1 0 0 1 216.00 369.36 Tm <736C6A656465646120> Tj 1 0 0 1 252.72 369.84 Tm <706F2461766C6A612C> Tj 0.24 Tc 1 0 0 1 54.96 359.04 Tm <6F646E6F736E6F20> Tj 1 0 0 1 94.56 359.04 Tm <6469736B7573696A7520> Tj 1 0 0 1 136.56 359.04 Tm <6920> Tj 1 0 0 1 142.08 359.04 Tm <7A616B6C6A75646B6520> Tj 1 0 0 1 185.28 359.04 Tm <697A766564656E6520> Tj 1 0 0 1 224.16 359.04 Tm <697A20> Tj 1 0 0 1 234.96 358.80 Tm <616E6166697A652E> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 55.20 337.92 Tm <352E20> Tj 1 0 0 1 70.56 337.92 Tm <4449534B5553494A4120> Tj 1 0 0 1 144.48 337.68 Tm <4F20> Tj 1 0 0 1 157.68 337.92 Tm <495C7B415445495541544943584F4957> Tj 0.24 Tc 1 0 0 1 69.60 325.68 Tm <5A4E4143454E4A5520> Tj 1 0 0 1 141.84 325.44 Tm <4C414752414E47454F564F47> Tj 0.24 Tc 1 0 0 1 69.36 313.44 Tm <4D554C5449504C49497B41544F5241> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 54.72 299.52 Tm <6C7A20> Tj 1 0 0 1 68.40 299.52 Tm <72657A756C7461746120> Tj 1 0 0 1 111.36 299.28 Tm <7520> Tj 1 0 0 1 122.88 299.76 Tm <706F2461766C6A7520> Tj 1 0 0 1 167.28 299.28 Tm <3420> Tj 1 0 0 1 178.32 299.04 Tm <7669646C6A69766F20> Tj 1 0 0 1 214.32 300.00 Tm <6A6520> Tj 1 0 0 1 228.24 299.28 Tm <6B616B6F20> Tj 1 0 0 1 255.36 299.52 Tm <7072696C696B6F6D> Tj 0.24 Tc 1 0 0 1 54.48 288.96 Tm <75706F747265626520> Tj 1 0 0 1 96.96 288.72 Tm <6F6269646E6F676120> Tj 1 0 0 1 138.72 288.72 Tm <6B726967696E67612C20> Tj 1 0 0 1 180.24 288.72 Tm <6B616F20> Tj 1 0 0 1 200.16 288.48 Tm <7374617469737469646B6F6720> Tj 1 0 0 1 252.48 288.96 Tm <706F737475706B61> Tj 0.24 Tc 1 0 0 1 54.48 278.64 Tm <70726F636A20> Tj 1 0 0 1 76.80 278.16 Tm <656E652C20> Tj 1 0 0 1 97.44 278.16 Tm <6F647265646976616E6A20> Tj 1 0 0 1 142.80 278.16 Tm <6520> Tj 1 0 0 1 150.48 278.16 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 211.68 278.16 Tm <6D756C7469706C696B61746F726120> Tj 1 0 0 1 276.48 277.92 Tm <696D61> Tj 0.24 Tc 1 0 0 1 54.48 268.08 Tm <7072657375646E7520> Tj 1 0 0 1 97.92 267.60 Tm <76615A6E6F737420> Tj 1 0 0 1 132.96 267.60 Tm <6E6120> Tj 1 0 0 1 146.16 267.60 Tm <6B6F6E61646E6920> Tj 1 0 0 1 182.88 267.60 Tm <72657A756C7461742E> Tj 0.24 Tc 1 0 0 1 61.44 254.16 Tm <4973707261726D6520> Tj 1 0 0 1 100.32 254.16 Tm <77696A65646E6F73746920> Tj 1 0 0 1 148.56 254.16 Tm <7520> Tj 1 0 0 1 156.96 253.92 Tm <6F626120> Tj 1 0 0 1 174.72 254.40 Tm <7072696D6A65726120> Tj 1 0 0 1 214.80 253.92 Tm <6F6269646E6F676120> Tj 1 0 0 1 255.12 253.92 Tm <6B726967696E67612C> Tj 0.24 Tc 1 0 0 1 54.24 244.08 Tm <67646A6520> Tj 1 0 0 1 74.64 244.56 Tm <6A6520> Tj 1 0 0 1 86.40 243.60 Tm <74616A20> Tj 1 0 0 1 101.04 243.60 Tm <6D756C7469706C696B61746F7220> Tj 1 0 0 1 161.28 243.60 Tm <75706F7472696A65626C6A656E20> Tj 1 0 0 1 219.60 243.36 Tm <7A6120> Tj 1 0 0 1 232.56 243.36 Tm <6D696E696D697A6972616E6A65> Tj 0.24 Tc 1 0 0 1 54.00 233.04 Tm <766172696A616E636520> Tj 1 0 0 1 96.72 233.28 Tm <70726F636A656E652C20> Tj 1 0 0 1 140.16 232.80 Tm <62696C6520> Tj 1 0 0 1 160.08 232.80 Tm <737520> Tj 1 0 0 1 174.24 232.80 Tm <647275676164696A6520> Tj 1 0 0 1 217.68 234.72 Tm <2D20> Tj 1 0 0 1 223.68 233.76 Tm <6A65646E6120> Tj 1 0 0 1 252.00 232.80 Tm <6E65676174766E61> Tj 0.48 Tc 1 0 0 1 54.72 223.20 Tm <282D302C39333139292C20> Tj 1 0 0 1 101.28 222.24 Tm <6120> Tj 1 0 0 1 111.12 222.24 Tm <647275676120> Tj 1 0 0 1 140.40 222.72 Tm <706F7A697469726D6120> Tj 1 0 0 1 184.32 223.20 Tm <28302C3036292E20> Tj 1 0 0 1 216.96 222.24 Tm <446F6B617A20> Tj 1 0 0 1 248.88 222.24 Tm <6F20> Tj 1 0 0 1 258.72 222.00 Tm <6F646162697275> Tj 0.24 Tc 1 0 0 1 54.24 211.92 Tm <697370726177726520> Tj 1 0 0 1 94.08 211.68 Tm <77696A65646E6F73746920> Tj 1 0 0 1 143.52 212.16 Tm <7072696B617A616E20> Tj 1 0 0 1 184.32 212.64 Tm <6A6520> Tj 1 0 0 1 196.32 211.68 Tm <7661726972616E6A656D20> Tj 1 0 0 1 246.72 211.44 Tm <77696A65646E6F737469> Tj 0.24 Tc 1 0 0 1 54.00 201.36 Tm <6D756C7469706C696B61746F726120> Tj 1 0 0 1 125.76 201.12 Tm <6B726F7A20> Tj 1 0 0 1 155.28 201.12 Tm <6D61747269636520> Tj 1 0 0 1 197.52 201.12 Tm <7520> Tj 1 0 0 1 213.36 201.60 Tm <706F74706F676C61766C6A7520> Tj 1 0 0 1 275.76 200.88 Tm <342E332E> Tj 0.24 Tc 1 0 0 1 54.24 190.56 Tm <4F6473747570616E6A656D20> Tj 1 0 0 1 116.88 190.56 Tm <6F6420> Tj 1 0 0 1 132.48 190.56 Tm <6E616A7072696D6A6572656E696A6520> Tj 1 0 0 1 202.80 190.32 Tm <77696A65646E6F73746920> Tj 1 0 0 1 252.96 190.32 Tm <766172696A616E6361> Tj 0.48 Tc 1 0 0 1 54.00 180.48 Tm <70726F636A656E6520> Tj 1 0 0 1 100.32 179.76 Tm <366520> Tj 1 0 0 1 118.56 180.00 Tm <726173746920> Tj 1 0 0 1 147.84 180.72 Tm <28706F74706F676C61766C6A6520> Tj 1 0 0 1 210.96 179.76 Tm <342E332E322920> Tj 1 0 0 1 245.28 179.76 Tm <696C6920> Tj 1 0 0 1 262.80 180.00 Tm <706F7374617469> Tj 0.24 Tc 1 0 0 1 54.48 170.40 Tm <28706F74706F676C61766C6A20> Tj 1 0 0 1 103.68 169.20 Tm <6520> Tj 1 0 0 1 110.16 169.44 Tm <342E20> Tj 1 0 0 1 118.32 169.44 Tm <332E20> Tj 1 0 0 1 126.48 169.20 Tm <3320> Tj 1 0 0 1 132.48 168.24 Tm <2920> Tj 1 0 0 1 136.80 169.44 Tm <6E6567617469726D6120> Tj 1 0 0 1 179.28 170.16 Tm <2820> Tj 1 0 0 1 181.92 169.20 Tm <6D6174656D617469646B6920> Tj 1 0 0 1 236.64 169.20 Tm <6E656973707261766E6F20> Tj 1 0 0 1 284.40 168.00 Tm <292E> Tj 0.24 Tc 1 0 0 1 60.96 156.00 Tm <50726F6D6F7472696D6F20> Tj 1 0 0 1 121.20 156.72 Tm <6A6F5320> Tj 1 0 0 1 142.80 156.72 Tm <6A6564616C70757420> Tj 1 0 0 1 190.56 155.76 Tm <6D6174656D617469646B6F20> Tj 1 0 0 1 255.36 155.52 Tm <7A6E6164656E6A65> Tj 0.24 Tc 1 0 0 1 54.00 145.44 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 116.16 145.20 Tm <6D756C7469706C696B61746F726120> Tj 1 0 0 1 182.16 145.20 Tm <7520> Tj 1 0 0 1 191.52 145.92 Tm <6A65646E61644C6261726E6120> Tj 1 0 0 1 254.16 144.96 Tm <6F6269646E6F6761> Tj 0.24 Tc 1 0 0 1 54.00 134.88 Tm <6B726967696E67612E20> Tj 1 0 0 1 93.84 134.64 Tm <5672696A65646E6F73746920> Tj 1 0 0 1 144.00 134.64 Tm <7520> Tj 1 0 0 1 152.64 134.64 Tm <746F646B6920> Tj 1 0 0 1 176.40 134.88 Tm <70726F636A656E6520> Tj 1 0 0 1 216.24 134.64 Tm <6D6F5A6520> Tj 1 0 0 1 241.92 134.40 Tm <736520> Tj 1 0 0 1 254.16 134.88 Tm <70726C6B617A617474> Tj 0.48 Tc 1 0 0 1 54.00 124.32 Tm <6B616F3A> Tj 0.72 Tc 1 0 0 1 53.52 106.80 Tm <7A28722920> Tj 1 0 0 1 69.84 108.00 Tm <3D20> Tj 1 0 0 1 77.28 106.80 Tm <702B20> Tj 1 0 0 1 91.44 104.88 Tm <7B7229> Tj 0.48 Tc 1 0 0 1 53.76 82.32 Tm <3432> Tj 0.48 Tc /F3 8.0 Tf 1 0 0 1 312.00 761.28 Tm <47646A6520> Tj 1 0 0 1 334.56 761.28 Tm <73753A> Tj 0.48 Tc 1 0 0 1 312.00 751.44 Tm <7020> Tj 1 0 0 1 342.72 752.88 Tm <2D20> Tj 1 0 0 1 347.28 751.20 Tm <746F646E6120> Tj 1 0 0 1 370.32 751.44 Tm <28616C6920> Tj 1 0 0 1 382.56 750.96 Tm <73747561726E6F20> Tj 1 0 0 1 411.12 750.72 Tm <6E65706F7A6E6174612920> Tj 1 0 0 1 453.60 750.48 Tm <73726564696E6120> Tj 1 0 0 1 482.16 750.24 Tm <63696A656C6520> Tj 1 0 0 1 503.04 750.72 Tm <706F70756C6163696A653B> Tj 0.48 Tc 1 0 0 1 311.76 738.96 Tm <8028722920> Tj 1 0 0 1 342.72 740.88 Tm <2D20> Tj 1 0 0 1 347.76 739.44 Tm <706F67726A65356B6120> Tj 1 0 0 1 385.44 738.72 Tm <6E6F726D616C697A6972616E6120> Tj 1 0 0 1 435.60 738.48 Tm <6F6B6F20> Tj 1 0 0 1 450.96 738.48 Tm <7672696A65646E6F73746920> Tj 1 0 0 1 489.84 738.24 Tm <3020> Tj 1 0 0 1 496.56 738.24 Tm <6E6120> Tj 1 0 0 1 507.60 738.24 Tm <6B726976756C6A69> Tj 0.48 Tc 1 0 0 1 342.72 729.36 Tm <736C7564616A6E6520> Tj 1 0 0 1 374.88 729.84 Tm <70726F73746F726E6520> Tj 1 0 0 1 411.12 729.36 Tm <66756E6B63696A65> Tj 0.24 Tc 1 0 0 1 311.52 712.56 Tm <53747661726E6120> Tj 1 0 0 1 346.80 712.32 Tm <77696A65646E6F737420> Tj 1 0 0 1 392.40 712.32 Tm <6F64656B6976616E6A6120> Tj 1 0 0 1 440.16 712.80 Tm <28702920> Tj 1 0 0 1 453.84 712.08 Tm <7A6120> Tj 1 0 0 1 465.60 711.84 Tm <6D6A6572656E7520> Tj 1 0 0 1 503.52 711.60 Tm <766172696A61626C7520> Tj 1 0 0 1 543.84 711.60 Tm <75> Tj 0.24 Tc 1 0 0 1 311.52 702.00 Tm <73747661726E6F73746920> Tj 1 0 0 1 360.72 701.76 Tm <6E61726E20> Tj 1 0 0 1 385.68 702.48 Tm <6A6520> Tj 1 0 0 1 400.80 701.76 Tm <6E65706F7A6E6174612C20> Tj 1 0 0 1 454.32 701.76 Tm <706120> Tj 1 0 0 1 471.84 701.04 Tm <736520> Tj 1 0 0 1 487.92 701.28 Tm <7A61746F20> Tj 1 0 0 1 512.40 701.52 Tm <70726F636A656E61> Tj 0.24 Tc 1 0 0 1 311.28 691.44 Tm <6E616A646535366520> Tj 1 0 0 1 354.00 691.20 Tm <7261646920> Tj 1 0 0 1 378.24 690.96 Tm <7465686E696B6F6D20> Tj 1 0 0 1 426.24 690.72 Tm <6F6269646E6F676120> Tj 1 0 0 1 470.64 690.72 Tm <6B726967696E676120> Tj 1 0 0 1 512.64 691.44 Tm <28756D6A6573746F> Tj 0.24 Tc 1 0 0 1 310.56 681.60 Tm <6A65646E6F737461726D696D20> Tj 1 0 0 1 371.28 680.64 Tm <6B726967696E676F6D292E20> Tj 1 0 0 1 422.88 680.16 Tm <5520> Tj 1 0 0 1 432.24 680.16 Tm <6F6269646E6F6D20> Tj 1 0 0 1 471.60 680.16 Tm <6B726967696E677520> Tj 1 0 0 1 509.28 680.40 Tm <70726F73746F726E69> Tj 0.24 Tc 1 0 0 1 311.28 670.32 Tm <6D6F64656C20> Tj 1 0 0 1 342.96 670.32 Tm <726164756E6120> Tj 1 0 0 1 378.00 670.08 Tm <73726564696E7520> Tj 1 0 0 1 415.68 669.84 Tm <69736B6C6A756469766F20> Tj 1 0 0 1 461.52 669.60 Tm <697A20> Tj 1 0 0 1 474.00 669.84 Tm <706F646174616B6120> Tj 1 0 0 1 518.88 669.36 Tm <6B6F6A6920> Tj 1 0 0 1 540.48 669.12 Tm <7365> Tj 0.24 Tc 1 0 0 1 311.76 659.76 Tm <6E616C617A6520> Tj 1 0 0 1 346.08 659.76 Tm <7520> Tj 1 0 0 1 360.00 659.52 Tm <6F6B6F6C696E6920> Tj 1 0 0 1 397.44 659.28 Tm <746F646B6520> Tj 1 0 0 1 428.16 659.52 Tm <70726F636A656E652E20> Tj 1 0 0 1 475.68 659.04 Tm <546F20> Tj 1 0 0 1 493.20 659.52 Tm <6A6520> Tj 1 0 0 1 509.28 659.28 Tm <706F73747570616B> Tj 0.24 Tc 1 0 0 1 311.28 649.20 Tm <6F647265646976616E6A6520> Tj 1 0 0 1 364.80 648.96 Tm <747A762E20> Tj 1 0 0 1 382.56 648.96 Tm <496F6B616C6E6520> Tj 1 0 0 1 417.84 648.72 Tm <73726564696E6520> Tj 1 0 0 1 454.08 648.72 Tm <67646A6520> Tj 1 0 0 1 475.44 648.24 Tm <736520> Tj 1 0 0 1 487.92 648.24 Tm <7520> Tj 1 0 0 1 497.52 648.24 Tm <7376616B6F6A20> Tj 1 0 0 1 528.24 648.24 Tm <746F646B69> Tj 0.24 Tc 1 0 0 1 311.04 638.64 Tm <726E72654C2C6520> Tj 1 0 0 1 348.24 638.40 Tm <75706F747265626C6A61766120> Tj 1 0 0 1 412.80 638.16 Tm <737265646E6A6120> Tj 1 0 0 1 455.52 637.68 Tm <77696A65646E6F737420> Tj 1 0 0 1 509.28 638.16 Tm <706F646174616B61> Tj 0.24 Tc 1 0 0 1 311.28 627.84 Tm <6F627568766136656E696820> Tj 1 0 0 1 371.76 627.84 Tm <726164696A75736F726E20> Tj 1 0 0 1 421.44 627.84 Tm <707265747261696975616E6A612E20> Tj 1 0 0 1 491.52 627.12 Tm <4E6164616C6A652C20> Tj 1 0 0 1 532.80 626.88 Tm <6B6F64> Tj 0.24 Tc 1 0 0 1 311.04 617.76 Tm <7072696D6A656E6520> Tj 1 0 0 1 365.76 617.28 Tm <6B726967696E676120> Tj 1 0 0 1 417.12 616.80 Tm <646F6461746E6F20> Tj 1 0 0 1 468.48 616.56 Tm <736520> Tj 1 0 0 1 493.68 616.80 Tm <70726574706F737461766C6A61> Tj 0.24 Tc 1 0 0 1 311.28 606.48 Tm <73746163696F6E61726E6F737420> Tj 1 0 0 1 376.56 606.48 Tm <322E20> Tj 1 0 0 1 391.92 606.48 Tm <7265646120> Tj 1 0 0 1 418.80 606.24 Tm <746520> Tj 1 0 0 1 434.16 606.00 Tm <736520> Tj 1 0 0 1 450.48 606.00 Tm <7461646120> Tj 1 0 0 1 476.40 605.76 Tm <726E6F692C6520> Tj 1 0 0 1 506.88 605.76 Tm <697A72612874746E617474> Tj 0.24 Tc 1 0 0 1 311.04 596.16 Tm <6B6F766172696A616E636120> Tj 1 0 0 1 365.04 595.92 Tm <432C20> Tj 1 0 0 1 379.44 595.92 Tm <73766F6A737476656E6120> Tj 1 0 0 1 427.68 595.92 Tm <70726F636A656E6920> Tj 1 0 0 1 466.56 595.44 Tm <5A2878292E20> Tj 1 0 0 1 492.24 595.20 Tm <4F6E6120> Tj 1 0 0 1 513.12 595.20 Tm <736520> Tj 1 0 0 1 526.32 595.20 Tm <6D6F5A65> Tj 0.24 Tc 1 0 0 1 311.28 585.60 Tm <697A72617A6974696B616F3A> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 311.04 568.32 Tm <432228722C2C722C2920> Tj 1 0 0 1 339.60 569.52 Tm <3D20> Tj 1 0 0 1 346.80 568.32 Tm <455B6528722C292E6528722C296C3A20> Tj 1 0 0 1 399.84 567.84 Tm <434A722C2D20> Tj 1 0 0 1 423.12 565.92 Tm <2B29> Tj 0.72 Tc 1 0 0 1 529.68 567.84 Tm <28352E3229> Tj 0.24 Tc /F3 8.0 Tf 1 0 0 1 310.32 550.08 Tm <416B6F20> Tj 1 0 0 1 330.24 549.84 Tm <736520> Tj 1 0 0 1 341.52 549.84 Tm <6B6F766172696A616E636120> Tj 1 0 0 1 393.36 549.60 Tm <7A616D696A656E6920> Tj 1 0 0 1 431.52 549.36 Tm <766172696F6772616D6F6D20> Tj 1 0 0 1 491.28 548.88 Tm <73746163696F6E61726E6F7374> Tj 0.24 Tc 1 0 0 1 310.80 539.28 Tm <736520> Tj 1 0 0 1 322.32 539.28 Tm <6D6F5A6520> Tj 1 0 0 1 347.76 539.28 Tm <697A72617A6974696B616F3A> Tj 0.72 Tc 1 0 0 1 310.32 518.88 Tm <7920> Tj 1 0 0 1 314.88 516.48 Tm <7A7C312C72322920> Tj 1 0 0 1 339.36 519.84 Tm <3D20> Tj 1 0 0 1 346.56 512.16 Tm <292E20> Tj 1 0 0 1 356.88 518.40 Tm <2C6C4F722C2C20> Tj 1 0 0 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Vrijednost Lagrangeovog multiplikatora -O,9U zadnjem primjeru za Lagrangeov multiplikatorodabrana je negatir,'na wijednost od -0,09 (po uzoru naslidnu wijednost primijenjenu u potpoglavlju 4.1). Tadamatrica g;lasi: o,4e 0.31 1,ol fo,25l [-0,33'l (4.14) Kao i u prethodna dva primjera unutar matrica [tr] i tBlizradunate su konstantne kovarijance. Ako se gornjamatridna jednadZba raspi5e linearnim iztazima dobivase sustav:1. 0,25 + 0,49. 0,25 + 0,49' 0,25 + 0,31' 0,25 + (-0,9) = -0,3275o,49'o,25+1'0,25 +0,31 'q25 +0,49'0,25 +(-0,9) =-0,32750,49. 925 + 0,31 . 0,25 + 1.0,25 + 0,49.0,25 + (-0,9) = -0,32750,31 .0,25 + 0,49.0,25 + 0,49.0,25 + 1.0,25 + (-0,9) =-o,32751 - 0,25 + 1. q25 + 1. 0,25 + 1. 0,25 + 0'(-0,9) = 1(4.15)Pripadaju6a varijanca tada je negativna (Sto jematematidki nemogu6e, pa je i Lagrangeov multiplikatorneispravan) iznosi:, ^^-o'ox = o,25 '(-0,3275) + 0,25 '(-0,3275) + 0,25 '(-0,3275) +o,25. (-0,3275) + (-0,9) = -0,982(4.16)Iz sva tri primjera prikazana u potpoglavlju 4.3 jasno jekako je wijednost Lagrangeovog koeficijenta 0,06 (kakoje navedeno u lit.3 i prikazano na slici 2) a to je upravo tawijednost gdje je varijanca kriginga minimalna. Nadinizbora opdenito najprirnjerenije wijednosti Lagrange-ovog koeficijenta, anaflzrrani su kroz sljededa po$avlja,odnosno diskusiju i zakljudke izvedene iz anafize.5. DISKUSIJA O I\{ATEIUATICXOIWZNACENJU LAGRANGEOVOGMULTIPLII{ATORAlz rezultata u po$avlju 4 vidljivo je kako prilikomupotrebe obidnoga kriginga, kao statistidkog postupkaprocj ene, odredivanj e Lagrangeovog multiplikatora imapresudnu vaZnost na konadni rezultat.Isprarme wijednosti u oba primjera obidnoga kriginga,gdje je taj multiplikator upotrijebljen za minimiziranjevarijance procjene, bile su drugadije - jedna negatvna(-0,9319), a druga pozitirma (0,06). Dokaz o odabiruisprawre wijednosti prikazan je variranjem wijednostimultiplikatora kroz matrice u potpoglavlju 4.3.Odstupanjem od najprimjerenije wijednosti varijancaprocjene 6e rasti (potpoglavlje 4.3.2) ili postati(potpoglavlj e 4. 3. 3 ) negatirma ( matematidki neispravno ).Promotrimo joS jedalput matematidko znadenjeLagrangeovog multiplikatora u jednadLbarna obidnogakriginga. Vrijednosti u todki procjene moZe se prlkazattkao: z(r) = p+ {r)42 Gdje su:p - todna (ali stuarno nepoznata) sredina cijele populacije;(r) - pogrje5ka normalizirana oko vrijednosti 0 na krivuljisludajne prostorne funkcijeStvarna wijednost odekivanja (p) za mjerenu varijablu ustvarnosti narn je nepoznata, pa se zato procjenanajde56e radi tehnikom obidnoga kriginga (umjestojednostarmim krigingom). U obidnom krigingu prostornimodel raduna sredinu iskljudivo iz podataka koji senalaze u okolini todke procjene. To je postupakodredivanje tzv. Iokalne sredine gdje se u svakoj todkirnreL,e upotrebljava srednja wijednost podatakaobuhva6enih radijusorn pretraiiuanja. Nadalje, kodprimjene kriginga dodatno se pretpostavljastacionarnost 2. reda te se tada rnoi,e izra(ttnattkovarijanca C, svojstvena procjeni Z(x). Ona se moZeizrazitikao: C"(r,,r,) = E[e(r,).e(r,)l: CJr,- +) (5.2) Ako se kovarijanca zamijeni variogramom stacionarnostse moZe izrazitikao: y z|1,r2) = ). ,lOr,, - r(r,))" = y zV1- rz) (5.3) Pretpostavimo niz od N mjerenja Z(xr)...2(x^) napoznatim lokacijama xt...xn. Ako se iz tih podataka Zeliobidnim krigingom procijeniti wijednost varijable Z (2)na nekoj neuzrokovanoj lokaciji xo trebaju se zadovoljititri uvjeta:t. 2 treba biti linearan i izradunat iz niza Z(y ,). . . Z(y .\Z. 2^ ie nepristrana procjena;3. Z minirnizira wijednost srednje kvadratne pogrjeskeizraiene kao elzlx o1 - a* "l' .Linearnost (wjet 1) se postiZe ako je zadovoljen izraz2 =>7,'Z(r,).Nepristranost (uljet 2) je zadovoljena ako wijedi:t3.nnrlZV"ll= / + tt = )s,,. e1zg,1l* p =2^,. u* )t", =1 (5.4)Minimiziranj " po=grj"sL" (uvj et 3 )'rurru; *.' loredivanjenajboljih mogu6ih wijednosti koeficijenata 1,r, ...,1.. teLagrangeovog parametra 2m. Njihova optimizacija moZese izrazlti za svaku todku procjene u obliku funkcijeL(Lr,...,7,,m): Optimizacij a koeficij enata ( lr,. . .,1") i Lagrangeovog (m)parametra postiZe se rjeSavanjem diferencijalnejednadZbe .Ako se watimo na osnovnu matridnu jednadZbukriginga (jednadZba 3.3) te ju prikaZemo prekokovarijanci tada ona $asi:c. 1o -- co (5.6)Vrijednosti C i Co mogu se matridno prikazati kao: t=r(zv"t-2^,.ror)" -* (>^ -r) L- o (5s) /A l\ NAFTA 59 (1 ) 38-43 (2009) endstream endobj 123 0 obj 4432 endobj 124 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 125 0 obj << /Filter /FlateDecode /Length 88 >> stream x=A 0 yE^&ݻ=H1Eev:(GY IuT6$Ш|EÜr^| ) endstream endobj 126 0 obj << /Type /Page /MediaBox [ 0 0 595.20 841.68 ] /Parent 13 0 R /Resources << /Font << /F3 7 0 R >> /XObject << /Obj118 118 0 R /Obj119 119 0 R >> /Subtype 122 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 124 0 R 125 0 R 120 0 R ] >> endobj 127 0 obj << /Type /XObject 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417.84 Tm <6E6567617469276E6D6F6D> Tj 0.24 Tc 1 0 0 1 308.88 408.00 Tm <706F647275646A7520> Tj 1 0 0 1 350.64 407.52 Tm <6920> Tj 1 0 0 1 355.92 408.00 Tm <706F73747570616B20> Tj 1 0 0 1 398.88 407.52 Tm <736520> Tj 1 0 0 1 410.40 407.52 Tm <6D6F5A6520> Tj 1 0 0 1 435.60 407.76 Tm <7072656B696E75746920> Tj 1 0 0 1 478.56 408.00 Tm <28736C696B6120> Tj 1 0 0 1 504.48 407.28 Tm <3320> Tj 1 0 0 1 512.40 407.28 Tm <6C696A65766F292E> Tj 0.24 Tc 1 0 0 1 315.84 390.72 Tm <566A6572756A656D6F20> Tj 1 0 0 1 368.64 390.96 Tm <6B616B6F20> Tj 1 0 0 1 398.40 391.20 Tm <676F726E6A6120> Tj 1 0 0 1 434.40 390.72 Tm <3420> Tj 1 0 0 1 448.56 390.96 Tm <70726176696C6120> Tj 1 0 0 1 486.48 390.72 Tm <7520> Tj 1 0 0 1 500.88 391.20 Tm <706F7470756E6F737469> Tj 0.24 Tc 1 0 0 1 309.12 380.40 Tm <7A61646F766F6C6A6176616A7520> Tj 1 0 0 1 368.64 380.40 Tm <697370726176616E20> Tj 1 0 0 1 407.28 380.64 Tm <706F73747570616B20> Tj 1 0 0 1 449.52 380.16 Tm <6F647265646976616E6A6120> Tj 1 0 0 1 501.36 380.16 Tm <77696A65646E6F737469> Tj 0.24 Tc 1 0 0 1 309.12 369.84 Tm <4C616772616E67656F766F6720> Tj 1 0 0 1 371.28 369.84 Tm <6D756C7469706C696B61746F726120> Tj 1 0 0 1 437.28 369.84 Tm <7520> Tj 1 0 0 1 446.64 370.56 Tm <6A65646E61644C6261726E6120> Tj 1 0 0 1 509.28 369.60 Tm <6F6269646E6F6761> Tj 0.24 Tc 1 0 0 1 309.12 359.28 Tm <6B726967696E67612C20> Tj 1 0 0 1 351.60 359.28 Tm <6F646E6F736E6F20> Tj 1 0 0 1 393.36 359.76 Tm <7072696D6A656E7520> Tj 1 0 0 1 437.76 359.04 Tm <746520> Tj 1 0 0 1 450.48 359.04 Tm <7465686E696B6520> Tj 1 0 0 1 487.44 359.04 Tm <6B616F20> Tj 1 0 0 1 507.12 360.00 Tm <6A65646E6520> Tj 1 0 0 1 536.40 359.04 Tm <6F64> Tj 0.24 Tc 1 0 0 1 308.88 348.72 Tm <6E616A626F6C6A696820> Tj 1 0 0 1 358.08 348.72 Tm <696E746572706F6C6163696A736B696820> Tj 1 0 0 1 436.32 348.96 Tm <707269737475706120> Tj 1 0 0 1 483.84 348.48 Tm <7A6120> Tj 1 0 0 1 504.72 348.48 Tm <6B6172746972616E6A65> Tj 0.24 Tc 1 0 0 1 308.88 338.64 Tm <67656F6C6F356B696820> Tj 1 0 0 1 351.84 338.16 Tm <766172696A61626C692E> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 309.12 315.36 Tm <5A41493A544949414C41> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 308.88 299.76 Tm <52616420> Tj 1 0 0 1 334.08 300.24 Tm <70726564737461766C6A6120> Tj 1 0 0 1 389.28 299.52 Tm <74656F72696A736B7520> Tj 1 0 0 1 435.60 299.52 Tm <616E616C697A7520> Tj 1 0 0 1 474.24 299.76 Tm <67656F6D6174656D617469646B6968> Tj 0.24 Tc 1 0 0 1 308.88 289.20 Tm <6D65746F646120> Tj 1 0 0 1 342.48 289.20 Tm <6E6164696E6A20> Tj 1 0 0 1 368.64 289.20 Tm <656E7520> Tj 1 0 0 1 385.92 289.20 Tm <7520> Tj 1 0 0 1 393.84 289.20 Tm <323030382E20> Tj 1 0 0 1 420.00 289.44 Tm <676F64696E6920> Tj 1 0 0 1 448.08 288.96 Tm <7520> Tj 1 0 0 1 456.00 288.96 Tm <6F6B7669727520> Tj 1 0 0 1 485.52 288.72 Tm <64766120> Tj 1 0 0 1 502.08 289.20 Tm <70726F6A20> Tj 1 0 0 1 519.84 288.72 Tm <656B74613A> Tj 0.24 Tc 1 0 0 1 317.52 276.00 Tm <2D20> Tj 1 0 0 1 326.64 273.36 Tm <2C2C53747261746967726166736B6120> Tj 1 0 0 1 398.88 274.08 Tm <6920> Tj 1 0 0 1 414.96 274.32 Tm <67656F6D6174656D617469646B6120> Tj 1 0 0 1 498.00 273.84 Tm <69737472615A6976616E6A61> Tj 0.24 Tc 1 0 0 1 326.40 263.52 Tm <6E6166746E6F67656F6C6F536B696820> Tj 1 0 0 1 406.32 263.52 Tm <7375737461766120> Tj 1 0 0 1 450.48 263.28 Tm <7520> Tj 1 0 0 1 468.72 263.28 Tm <4872766174736B6F6A2220> Tj 1 0 0 1 526.80 264.00 Tm <2862726F6A> Tj 0.48 Tc 1 0 0 1 327.36 252.96 Tm <3139352D313935313239332D303233372920> Tj 1 0 0 1 426.72 252.96 Tm <66696E616E636972616E6F6720> Tj 1 0 0 1 494.88 252.72 Tm <6F6420> Tj 1 0 0 1 520.32 252.72 Tm <737472616E65> Tj 0.24 Tc 1 0 0 1 326.40 242.40 Tm <4D696E69737461727374766120> Tj 1 0 0 1 380.40 242.40 Tm <7A6E616E6F7374692C20> Tj 1 0 0 1 420.72 242.16 Tm <6F6272617A6F76616E6A20> Tj 1 0 0 1 467.52 242.16 Tm <6120> Tj 1 0 0 1 473.52 242.16 Tm <6920> Tj 1 0 0 1 477.84 242.16 Tm <53706F72746120> Tj 1 0 0 1 506.64 242.16 Tm <52483B> Tj 0.48 Tc 1 0 0 1 317.28 230.88 Tm <2D2C556E617072696A65646976616E6A6520> Tj 1 0 0 1 398.88 229.20 Tm <67656F6C6F356B696820> Tj 1 0 0 1 441.84 228.72 Tm <696E746572707265746163696A736B696820> Tj 1 0 0 1 515.28 228.72 Tm <6D65746F6461> Tj 0.24 Tc 1 0 0 1 326.16 218.40 Tm <7520> Tj 1 0 0 1 339.12 218.16 Tm <63696C6A7520> Tj 1 0 0 1 364.56 218.88 Tm <706F766536616E6A6120> Tj 1 0 0 1 413.76 218.40 Tm <69736372706B6120> Tj 1 0 0 1 452.88 218.16 Tm <756E7574617220> Tj 1 0 0 1 489.12 218.40 Tm <706A653564656E6A61646B6968> Tj 0.24 Tc 1 0 0 1 326.16 207.84 Tm <6C655A69537461272C20> Tj 1 0 0 1 363.36 207.84 Tm <66696E616E636972616E6F6720> Tj 1 0 0 1 420.96 207.60 Tm <6920> Tj 1 0 0 1 428.16 208.08 Tm <706F64757072746F6720> Tj 1 0 0 1 477.36 207.36 Tm <6F6420> Tj 1 0 0 1 492.72 207.36 Tm <737472616E6520> Tj 1 0 0 1 523.68 207.60 Tm <494E412D65> Tj 0.48 Tc 1 0 0 1 326.40 197.04 Tm <642E642E20> Tj 1 0 0 1 346.32 197.04 Tm <50726F6A656B7420> Tj 1 0 0 1 379.68 198.00 Tm <6A6520> Tj 1 0 0 1 391.20 197.04 Tm <73757261646E6A6120> Tj 1 0 0 1 432.48 196.80 Tm <53656B746F726120> Tj 1 0 0 1 468.96 196.80 Tm <7A6120> Tj 1 0 0 1 482.16 196.80 Tm <72617A7261647220> Tj 1 0 0 1 519.36 197.76 Tm <28494E412920> Tj 1 0 0 1 544.08 196.80 Tm <69> Tj 0.24 Tc 1 0 0 1 326.64 186.48 Tm <5A617620> Tj 1 0 0 1 342.72 186.48 Tm <6F646120> Tj 1 0 0 1 358.80 186.48 Tm <7A6120> Tj 1 0 0 1 370.56 186.72 Tm <67656F6C6F67696A7520> Tj 1 0 0 1 408.48 186.48 Tm <6920> Tj 1 0 0 1 412.56 186.72 Tm <67656F6C6F356B6F20> Tj 1 0 0 1 451.20 186.24 Tm <696E5A656E6A65727374766F20> Tj 1 0 0 1 504.48 187.20 Tm <2852474E46292E> Tj 0.00 Tc 1 0 0 1 308.88 156.24 Tm <2A> Tj 0.48 Tc 1 0 0 1 308.64 143.52 Tm <4175746F72693A> Tj 0.48 Tc 1 0 0 1 308.64 132.48 Tm <546F6D697366617620> Tj 1 0 0 1 346.56 132.48 Tm <4D616C7669362C20> Tj 1 0 0 1 376.80 132.48 Tm <4F722E20> Tj 1 0 0 1 389.04 132.24 Tm <73632E2C20> Tj 1 0 0 1 403.92 132.24 Tm <6469706C2E20> Tj 1 0 0 1 421.44 132.24 Tm <696E672E2C20> Tj 1 0 0 1 438.96 132.24 Tm <494E412D4E61667461706C696E2C20> Tj 1 0 0 1 485.76 132.00 Tm <53656B746F7220> Tj 1 0 0 1 510.00 132.24 Tm <7A6120> Tj 1 0 0 1 521.76 132.24 Tm <72617A72616F752C> Tj 0.48 Tc /F3 7.0 Tf 1 0 0 1 308.64 123.84 Tm <7361766A65746E696B2C20> Tj 1 0 0 1 339.60 124.08 Tm <746F6D69736C61766D616C76696340696E612E20> Tj 1 0 0 1 405.60 123.84 Tm <6872> Tj 0.48 Tc 1 0 0 1 309.12 112.80 Tm <4461766F72696E20> Tj 1 0 0 1 342.72 113.04 Tm <42616C69362C20> Tj 1 0 0 1 367.44 112.80 Tm <6469706C2E20> Tj 1 0 0 1 384.96 112.80 Tm <696E672E2C20> Tj 1 0 0 1 402.48 112.80 Tm <494E412D4E61667461706C696E2C20> Tj 1 0 0 1 449.28 112.56 Tm <53656B746F7220> Tj 1 0 0 1 473.52 112.56 Tm <7A6120> Tj 1 0 0 1 485.28 112.56 Tm <72617A726164752C20> Tj 1 0 0 1 514.32 112.56 Tm <536C755A626120> Tj 1 0 0 1 539.52 112.56 Tm <7A61> Tj 0.48 Tc 1 0 0 1 308.88 104.88 Tm <67656F6C6F536B7520> Tj 1 0 0 1 339.60 104.40 Tm <72617A7261647520> Tj 1 0 0 1 365.52 104.40 Tm <6C655A693574612C20> Tj 1 0 0 1 388.80 104.40 Tm <696E5A656E6A657220> Tj 1 0 0 1 414.96 104.40 Tm <7A6120> Tj 1 0 0 1 424.32 104.40 Tm <72617A7261647520> Tj 1 0 0 1 449.76 104.16 Tm <6C655A695374612C20> Tj 1 0 0 1 473.04 104.16 Tm <6461766F72696E2E62616C696340696E612E6872> Tj 0.00 Tc /F3 8.0 Tf 1 0 0 1 216.00 763.44 Tm <31> Tj 0.00 Tc 1 0 0 1 216.00 751.20 Tm <31> Tj 0.00 Tc 1 0 0 1 216.00 726.96 Tm <31> Tj 0.00 Tc 1 0 0 1 215.52 714.72 Tm <30> Tj 0.48 Tc 1 0 0 1 273.12 738.96 Tm <28352E3729> Tj 0.48 Tc 1 0 0 1 273.12 671.76 Tm <28352E3829> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 230.88 630.72 Tm <27777261692C656E6920> Tj 1 0 0 1 267.60 626.40 Tm <7072656B6F> Tj 0.72 Tc /F3 8.0 Tf 1 0 0 1 270.24 557.76 Tm <28352E313029> Tj 0.00 Tc /F3 7.0 Tf 1 0 0 1 10.32 548.64 Tm <31> Tj 0.00 Tc /F3 13.0 Tf 1 0 0 1 9.60 531.36 Tm <2B> Tj 0.00 Tc /F3 5.0 Tf 1 0 0 1 10.80 527.04 Tm <49> Tj 0.00 Tc /F3 14.0 Tf 1 0 0 1 10.32 515.04 Tm <49> Tj 0.00 Tc /F3 7.0 Tf 1 0 0 1 10.80 509.76 Tm <31> Tj 0.00 Tc 1 0 0 1 10.80 503.76 Tm <49> Tj 0.00 Tc /F3 14.0 Tf 1 0 0 1 9.60 491.28 Tm <49> Tj 0.00 Tc /F3 11.0 Tf 1 0 0 1 10.56 461.76 Tm <49> Tj 0.00 Tc /F3 8.0 Tf 1 0 0 1 10.80 454.32 Tm <74> Tj 0.00 Tc /F3 6.0 Tf 1 0 0 1 10.80 445.92 Tm <49> Tj 0.00 Tc /F3 10.0 Tf 1 0 0 1 9.36 439.20 Tm <6A> Tj 0.00 Tc /F3 5.0 Tf 1 0 0 1 10.80 428.40 Tm <49> Tj 0.00 Tc 1 0 0 1 10.80 421.92 Tm <74> Tj 0.00 Tc /F3 12.0 Tf 1 0 0 1 10.32 409.68 Tm <49> Tj 0.00 Tc /F3 8.0 Tf 1 0 0 1 10.32 403.20 Tm <49> Tj 0.00 Tc /F3 5.0 Tf 1 0 0 1 10.80 399.84 Tm <74> Tj 0.00 Tc /F3 10.0 Tf 1 0 0 1 10.32 392.40 Tm <28> Tj 0.00 Tc /F3 3.0 Tf 1 0 0 1 11.76 387.60 Tm <49> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 54.24 535.92 Tm <4F7074696D656C20> Tj 1 0 0 1 88.80 536.88 Tm <6E6920> Tj 1 0 0 1 99.36 535.92 Tm <74657A696E736B6920> Tj 1 0 0 1 135.12 536.16 Tm <6B6F65666963696A656E746120> Tj 1 0 0 1 186.24 536.16 Tm <746520> Tj 1 0 0 1 196.32 536.64 Tm <7072696D6A20> Tj 1 0 0 1 220.32 536.40 Tm <6572656E6920> Tj 1 0 0 1 244.08 536.40 Tm <4C616772616E67656F76> Tj 0.24 Tc 1 0 0 1 54.24 525.36 Tm <6D756C7469706C696B61746F7220> Tj 1 0 0 1 113.52 525.60 Tm <72657A756C746972617420> Tj 1 0 0 1 156.24 525.60 Tm <366520> Tj 1 0 0 1 168.00 525.60 Tm <7520> Tj 1 0 0 1 176.64 525.60 Tm <697A726164756E7520> Tj 1 0 0 1 217.20 525.84 Tm <6E616A6D616E6A6520> Tj 1 0 0 1 259.44 525.84 Tm <6D6F67756965> Tj 0.24 Tc 1 0 0 1 54.24 514.80 Tm <7374616E64696C2736687320> Tj 1 0 0 1 105.12 515.28 Tm <706F67726A65736B6520> Tj 1 0 0 1 149.28 515.04 Tm <6B726967696E676120> Tj 1 0 0 1 186.48 515.52 Tm <7072696B617A616E6520> Tj 1 0 0 1 231.36 515.28 Tm <6B616F3A> Tj 0.96 Tc /F3 8.0 Tf 1 0 0 1 54.00 497.04 Tm <6F28786F2920> Tj 1 0 0 1 74.40 498.48 Tm <3D3631393B20> Tj 1 0 0 1 98.64 499.44 Tm <2D692727636F2B6D> Tj 0.24 Tc /F3 12.0 Tf 1 0 0 1 54.48 476.40 Tm <362E20> Tj 1 0 0 1 71.28 476.64 Tm <5A414B4C4A5545414B> Tj 0.96 Tc /F3 8.0 Tf 1 0 0 1 270.00 497.76 Tm <28352E3120> Tj 1 0 0 1 283.68 497.76 Tm <3129> Tj 0.24 Tc /F3 9.0 Tf 1 0 0 1 54.00 462.48 Tm <5520> Tj 1 0 0 1 63.12 462.72 Tm <70726574686F647272696D20> Tj 1 0 0 1 115.92 462.96 Tm <706F676C61766C6A696D6120> Tj 1 0 0 1 167.52 462.96 Tm <706F6B617A616C6920> Tj 1 0 0 1 205.68 462.72 Tm <736D6F20> Tj 1 0 0 1 225.84 462.96 Tm <7A6135746F20> Tj 1 0 0 1 249.60 463.68 Tm <6A6520> Tj 1 0 0 1 260.16 462.72 Tm <7465686E696B61> Tj 0.24 Tc 1 0 0 1 54.24 451.92 Tm <6F6269646E6F676120> Tj 1 0 0 1 94.08 451.92 Tm <6B726967696E676120> Tj 1 0 0 1 130.56 452.40 Tm <7072696D6A6572656E696A6120> Tj 1 0 0 1 185.52 452.16 Tm <7A6120> Tj 1 0 0 1 196.80 452.40 Tm <696E746572706F6C6163696A7520> Tj 1 0 0 1 252.72 452.40 Tm <746F646B6173746968> Tj 0.24 Tc 1 0 0 1 54.00 441.84 Tm <706F646174616B6120> Tj 1 0 0 1 102.96 441.36 Tm <6E65676F20> Tj 1 0 0 1 131.52 441.60 Tm <6C6920> Tj 1 0 0 1 146.40 441.60 Tm <6F736E6F766E6120> Tj 1 0 0 1 190.32 441.60 Tm <7465686E696B6120> Tj 1 0 0 1 230.64 442.56 Tm <6A65646E6F737461726D6F6761> Tj 0.24 Tc 1 0 0 1 54.24 430.80 Tm <74616967696E67612E20> Tj 1 0 0 1 96.72 430.80 Tm <4E6164616C6A652C20> Tj 1 0 0 1 136.56 431.04 Tm <7520> Tj 1 0 0 1 147.12 431.04 Tm <6E656B696D20> Tj 1 0 0 1 179.04 431.04 Tm <72616E696A696D20> Tj 1 0 0 1 215.28 431.28 Tm <7261646F76696D6120> Tj 1 0 0 1 260.64 431.28 Tm <6B6F6A6920> Tj 1 0 0 1 282.00 431.28 Tm <7375> Tj 0.24 Tc 1 0 0 1 54.00 420.72 Tm <7072696B617A616C6920> Tj 1 0 0 1 94.80 420.24 Tm <75706F747265627520> Tj 1 0 0 1 136.56 420.24 Tm <6D65746F646520> Tj 1 0 0 1 170.88 420.24 Tm <6B726967696E676120> Tj 1 0 0 1 208.32 420.48 Tm <6E6120> Tj 1 0 0 1 221.76 420.48 Tm <6E6166746E6F67656F6C6F356B696D> Tj 0.24 Tc 1 0 0 1 54.00 409.92 Tm <706F64617463696D6120> Tj 1 0 0 1 112.32 410.64 Tm <28706F726F7A6E6F73742920> Tj 1 0 0 1 174.00 409.68 Tm <697A6D6A6572656E696D20> Tj 1 0 0 1 235.20 409.92 Tm <7520> Tj 1 0 0 1 253.92 409.92 Tm <6C655A695374696D61> Tj 0.24 Tc 1 0 0 1 54.00 399.12 Tm <7567696C6A696B6F766F726C6C6B6120> Tj 1 0 0 1 116.40 399.36 Tm <7520> Tj 1 0 0 1 126.72 399.12 Tm <6872766174736B6F6D20> Tj 1 0 0 1 177.84 399.12 Tm <64696A656C7520> Tj 1 0 0 1 206.64 399.36 Tm <50616E6F6E736B6F676120> Tj 1 0 0 1 262.08 399.36 Tm <62617A656E61> Tj 0.24 Tc 1 0 0 1 54.24 388.56 Tm <646F6B617A616E6F20> Tj 1 0 0 1 98.88 389.28 Tm <6A6520> Tj 1 0 0 1 111.36 388.56 Tm <6B616B6F20> Tj 1 0 0 1 137.28 388.56 Tm <737520> Tj 1 0 0 1 151.44 389.04 Tm <67656F7374617469737469646B6120> Tj 1 0 0 1 213.84 388.80 Tm <696E746572706F6C6163696A6120> Tj 1 0 0 1 271.68 389.76 Tm <28706120> Tj 1 0 0 1 289.20 389.04 Tm <69> Tj 0.24 Tc 1 0 0 1 54.24 378.00 Tm <73746F6861737469646B6120> Tj 1 0 0 1 113.76 378.48 Tm <70726F636A656E612920> Tj 1 0 0 1 165.36 378.24 Tm <6E616A7072696D6A6572656E696A6920> Tj 1 0 0 1 240.00 378.72 Tm <7072697374757020> Tj 1 0 0 1 282.72 378.48 Tm <7A61> Tj 0.24 Tc 1 0 0 1 54.24 367.44 Tm <6B6172746972616E6A6520> Tj 1 0 0 1 98.88 367.92 Tm <67656F6C6F536B696820> Tj 1 0 0 1 141.84 367.68 Tm <766172696A61626C6920> Tj 1 0 0 1 180.00 368.64 Tm <286E70722E20> Tj 1 0 0 1 202.56 367.92 Tm <6C69746572617475726120> Tj 1 0 0 1 245.52 370.32 Tm <312C382C20> Tj 1 0 0 1 258.96 370.32 Tm <652720> Tj 1 0 0 1 266.40 370.32 Tm <31302720> Tj 1 0 0 1 276.48 370.32 Tm <3132292E> Tj 0.24 Tc 1 0 0 1 61.20 353.76 Tm <54616B6F6465722C20> Tj 1 0 0 1 101.52 353.76 Tm <6E6120> Tj 1 0 0 1 115.20 353.76 Tm <74656D656C6A7520> Tj 1 0 0 1 150.24 353.76 Tm <74696820> Tj 1 0 0 1 164.88 354.00 Tm <6920> Tj 1 0 0 1 171.12 353.76 Tm <64727567696820> Tj 1 0 0 1 203.28 354.00 Tm <616E614A697A6120> Tj 1 0 0 1 235.92 354.00 Tm <7A616B6C6A7564656E6F20> Tj 1 0 0 1 283.92 354.96 Tm <6A65> Tj 0.24 Tc 1 0 0 1 54.24 343.20 Tm <6B616B6F20> Tj 1 0 0 1 81.36 343.20 Tm <7A6120> Tj 1 0 0 1 96.72 343.68 Tm <7072696D6A6572656E7520> Tj 1 0 0 1 150.72 343.44 Tm <75706F747265627520> Tj 1 0 0 1 196.08 343.44 Tm <6D65746F646120> Tj 1 0 0 1 233.76 343.68 Tm <6B726967696E676120> Tj 1 0 0 1 274.32 343.44 Tm <62726F6A> Tj 0.24 Tc 1 0 0 1 54.48 332.64 Tm <754A617A6E696820> Tj 1 0 0 1 88.08 333.12 Tm <706F646174616B6120> Tj 1 0 0 1 130.32 332.88 Tm <747265626120> Tj 1 0 0 1 155.04 332.64 Tm <6269746920> Tj 1 0 0 1 171.60 332.88 Tm <7665366920> Tj 1 0 0 1 191.04 332.88 Tm <6F6420> Tj 1 0 0 1 205.44 332.88 Tm <31302C20> Tj 1 0 0 1 220.56 333.36 Tm <706120> Tj 1 0 0 1 233.76 332.88 Tm <6920> Tj 1 0 0 1 240.24 332.88 Tm <313520> Tj 1 0 0 1 252.72 332.88 Tm <746F646B6173746968> Tj 0.24 Tc 1 0 0 1 53.76 321.84 Tm <7672696A65646E6F7374692E20> Tj 1 0 0 1 108.96 322.08 Tm <4E6F2C20> Tj 1 0 0 1 130.32 322.08 Tm <72657072657A656E74617469276E6D6920> Tj 1 0 0 1 201.36 322.08 Tm <7374617469737469646B6920> Tj 1 0 0 1 249.60 322.32 Tm <757A6F72616B20> Tj 1 0 0 1 286.32 322.32 Tm <75> Tj 0.00 Tc 1 0 0 1 54.96 311.28 Tm <616E616C697A6920> Tj 1 0 0 1 85.20 311.76 Tm <67656F6C6F536B696820> Tj 1 0 0 1 128.88 311.52 Tm <766172696A61626C6920> Tj 1 0 0 1 167.52 311.52 Tm <7376616B616B6F20> Tj 1 0 0 1 205.68 311.76 Tm <626920> Tj 1 0 0 1 217.68 311.76 Tm <74726562616F20> Tj 1 0 0 1 248.40 311.76 Tm <736164725A6176617469> Tj 0.24 Tc 1 0 0 1 54.24 300.72 Tm <333020> Tj 1 0 0 1 66.96 300.96 Tm <6920> Tj 1 0 0 1 71.52 300.72 Tm <7669356520> Tj 1 0 0 1 90.00 300.72 Tm <746F646B617374696820> Tj 1 0 0 1 131.04 300.96 Tm <77696A65646E6F7374692E20> Tj 1 0 0 1 181.44 301.20 Tm 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odabranoj todkije:2u ) = )-,' z(r,)+ - - -+ ).,' z(x,) Pokazali smo da ta wijednost moi,e biti i negatfirna nowijednosti va.rijance kriginga to ne mogu biti (inade jeradun besmislen). Nadalje, ako sludajno uzorkujemo tuwijednost iz nekoga intervala (npr. [-1,1]) ve6im brojempoku5aja jedna od njih 6e rezulttrati u najmanjojvarijanci kriginga. Nadin kako odrediti Sirinu intervalauzorkovanja i njihov broj moZe se prikazati kroz detirikoraka (slika 3):1. Iskustveno smo odredili da bi numeridki interval ukojem rnoLerno odekivati wijednost Larangeovogmultiplikatora pribliZno trebao biti [-1,1]. Takoder,ponekad ta velidina moZe biti wlo bliska wijednosti 0(kako je to prikazano). Zato je podetna wijednost za gen-erator sludajnih brojeva Lagrangeovog multiplikatorapostavljena na 0,01.2. U sljede6em koraku (velidina koraka bila bi 0,05) tawijednost smanjena je, te prelazi u negatirmo podrudje.lzradunata je ponovno pripadaju6a varijanca kdginga izablljeLena. Takav postupak bio bi ponavljan sve dokvarijanca kriginga ne bi postala negatiura. Na tomnegativnom intervalu wijednosti m bila bi izdvojena onas najmanjom varijancom (slika 3 lijevo).3. Tada bi wijednost m=O,Ol bila uve6ana zakorak0,05 u pozitirmom smjeru, ponovno prate6i smanjuje lise wijednost varijance. Ako da, nastavilo bi se s njezinimizradunom dok se ne zablJjeLi njezin porast i na taj nadinbio bi odreden njezin minirnum (nakon prvoga porasta,slika 3 desno).4. Yaino je upamtiti, ako izradunom za prvi po zltrvni mvarijanca odmah podinje rasti, tada se moZe smatrati daje minimum varijance kriginga odreden u negati'nmompodrudju i postupak se moZe prekinuti (slika 3 lijevo).Vjerujemo kako gornja 4 pravila u potpunostizadovoljavaju ispravan postupak odredivanja wijednostiLagrangeovog multiplikatora u jednadLbarna obidnogakriginga, odnosno primjenu te tehnike kao jedne odnajboljih interpolacijskih pristupa za kartiranjegeolo5kih varijabli.ZAI:TIIALARad predstavlja teorijsku analizu geomatematidkihmetoda nadinj enu u 2008. godini u okviru dva proj ekta:- ,,Stratigrafska i geomatematidka istraZivanjanaftnogeoloSkih sustava u Hrvatskoj" (broj195-1951293-0237) financiranog od straneMinistarstva znanosti, obrazovanj a i Sporta RH;-,Unaprijedivanje geolo5kih interpretacijskih metodau cilju pove6anja iscrpka unutar pje5denjadkihleZiSta', financiranog i poduprtog od strane INA-ed.d. Projekt je suradnja Sektora za razradr (INA) iZav oda za geologiju i geolo5ko inZenjerstvo (RGNF).*Autori:Tomisfav Malvi6, Or. sc., dipl. ing., INA-Naftaplin, Sektor za razraou,savjetnik, tomislavmalvic@ina. hrDavorin Bali6, dipl. ing., INA-Naftaplin, Sektor za razradu, SluZba zageoloSku razradu leZi5ta, inZenjer za razradu leZiSta, davorin.balic@ina.hr 1110 (5.7) (5.8) 'wrai,eni preko (5.10) 1+II1IIItIjItIIt(I Optimel ni tezinski koeficijenta te primj ereni Lagrangeovmultiplikator rezultirat 6e u izradunu najmanje moguiestandil'6hs pogrjeske kriginga prikazane kao: o(xo) =619; -i''co+m6. ZAKLJUEAK (5.1 1) U prethodrrim poglavljima pokazali smo za5to je tehnikaobidnoga kriginga primjerenija za interpolaciju todkastihpodataka nego li osnovna tehnika jednostarmogataiginga. Nadalje, u nekim ranijim radovima koji suprikazali upotrebu metode kriginga na naftnogeolo5kimpodatcima (poroznost) izmjerenim u leZiStimaugiljikovorllka u hrvatskom dijelu Panonskoga bazenadokazano je kako su geostatistidka interpolacija (pa istohastidka procjena) najprimjereniji pristup zakartiranje geoloSkih varijabli (npr. literatura 1,8, e' 10' 12).Takoder, na temelju tih i drugih anaJiza zakljudeno jekako za primjerenu upotrebu metoda kriginga brojuJaznih podataka treba biti ve6i od 10, pa i 15 todkastihvrijednosti. No, reprezentati'nmi statistidki uzorak uanalizi geoloSkih varijabli svakako bi trebao sadrZavati30 i vi5e todkastih wijednosti. To znadi da teSko moZemozakljuditi o stvarnome odekivanju populacijepredstavljene mjerenirn wijednostima, te je moguiejedino upotrebljavati tzv. lokalnu srednju wijednostpodataka ukljudenih kroz radijus pretraLivanja dwstihpodataka oko todke procjene. To istide kao primjerenufshniku obidnoga kriginga.Zbog toga treba s najve6om palnjorn prouditipalametre ukljudene u jednadZbu te tehnike. Ve6ina njihstandardna je za matridniprlkaz metode kriginga, ali sejavlja i jedan dodani dlan nazvarr linearnimmultiplikatorom Lagrangea. Njegovu ulogu i vaZnostprikazali smo u prethodnim po$avljirna. JoS jedinonismo odgovorili na pitanje kako se izradunavanajprimjerenija wijednosti toga parametra (m), odnosnowijednost koja 6e u konadnici rezultirati minimalnomvarijancom kriginga.Odgovor na to traZili smo u sludajnom uzorkovanju izskupa mogu6ih (odekivanih) wijednosti te velidine.NAFTA 5S (1 ) s8-43 (2009) 43 endstream endobj 135 0 obj 4792 endobj 136 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 137 0 obj << /Filter /FlateDecode /Length 149 >> stream xMA 1V{BڅMb|:)c0>U %xv`QNwuad'GBq'cIIfzx:8N)mdS5^r Q6QSU 7 endstream endobj 138 0 obj << /Type /Page /MediaBox [ 0 0 595.20 841.68 ] /Parent 13 0 R /Resources << /Font << /F3 7 0 R >> /XObject << /Obj127 127 0 R /Obj128 128 0 R /Obj129 129 0 R /Obj130 130 0 R /Obj131 131 0 R >> /Subtype 134 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 136 0 R 137 0 R 132 0 R ] >> endobj 13 0 obj << /Kids [ 12 0 R 24 0 R 35 0 R 45 0 R 55 0 R 66 0 R 77 0 R 89 0 R 99 0 R 108 0 R 117 0 R 126 0 R 138 0 R ] /Count 13 /Type /Pages >> endobj 14 0 obj << /Pages 13 0 R /Type /Catalog >> endobj xref 0 139 0000000000 65535 f 0000000015 00000 n 0000000110 00000 n 0000043104 00000 n 0000128839 00000 n 0000132908 00000 n 0000166935 00000 n 0000166957 00000 n 0000167060 00000 n 0000172029 00000 n 0000172050 00000 n 0000172129 00000 n 0000172307 00000 n 0002159557 00000 n 0002159712 00000 n 0000172582 00000 n 0000219053 00000 n 0000256513 00000 n 0000292681 00000 n 0000324069 00000 n 0000324092 00000 n 0000328273 00000 n 0000328295 00000 n 0000328374 00000 n 0000328559 00000 n 0000328842 00000 n 0000388369 00000 n 0000455489 00000 n 0000456604 00000 n 0000456875 00000 n 0000487300 00000 n 0000487323 00000 n 0000491302 00000 n 0000491324 00000 n 0000491403 00000 n 0000491606 00000 n 0000491903 00000 n 0000544686 00000 n 0000574454 00000 n 0000617122 00000 n 0000647027 00000 n 0000647050 00000 n 0000651125 00000 n 0000651147 00000 n 0000651226 00000 n 0000651408 00000 n 0000651691 00000 n 0000700858 00000 n 0000776846 00000 n 0000777109 00000 n 0000813602 00000 n 0000813625 00000 n 0000818287 00000 n 0000818309 00000 n 0000818388 00000 n 0000818569 00000 n 0000818852 00000 n 0000865051 00000 n 0000939395 00000 n 0000940017 00000 n 0000948696 00000 n 0000982450 00000 n 0000982473 00000 n 0000987051 00000 n 0000987073 00000 n 0000987152 00000 n 0000987353 00000 n 0000987650 00000 n 0001044713 00000 n 0001056944 00000 n 0001057549 00000 n 0001123273 00000 n 0001152488 00000 n 0001152511 00000 n 0001156970 00000 n 0001156992 00000 n 0001157071 00000 n 0001157275 00000 n 0001157572 00000 n 0001196859 00000 n 0001198037 00000 n 0001202211 00000 n 0001285971 00000 n 0001286231 00000 n 0001315607 00000 n 0001315630 00000 n 0001320321 00000 n 0001320343 00000 n 0001320422 00000 n 0001320648 00000 n 0001320959 00000 n 0001369194 00000 n 0001446338 00000 n 0001447208 00000 n 0001476079 00000 n 0001476102 00000 n 0001480355 00000 n 0001480377 00000 n 0001480456 00000 n 0001480632 00000 n 0001480915 00000 n 0001532987 00000 n 0001619092 00000 n 0001651783 00000 n 0001651807 00000 n 0001656464 00000 n 0001656487 00000 n 0001656567 00000 n 0001656726 00000 n 0001657004 00000 n 0001711272 00000 n 0001783881 00000 n 0001813673 00000 n 0001813697 00000 n 0001817828 00000 n 0001817851 00000 n 0001817931 00000 n 0001818092 00000 n 0001818370 00000 n 0001867042 00000 n 0001948075 00000 n 0001979100 00000 n 0001979124 00000 n 0001983613 00000 n 0001983636 00000 n 0001983716 00000 n 0001983877 00000 n 0001984155 00000 n 0002028791 00000 n 0002034354 00000 n 0002119919 00000 n 0002120541 00000 n 0002121385 00000 n 0002154032 00000 n 0002154056 00000 n 0002158905 00000 n 0002158928 00000 n 0002159008 00000 n 0002159231 00000 n trailer << /Size 139 /Info 1 0 R /Root 14 0 R /ID [] >> startxref 2159767 %%EOF