基于窗口分析法的地形表达误差提取方法

基于窗口分析法的地形表达误差提取方法

1ewell3环境保护a.差分模型（dem）是a.差分模型的特征（a.特征，2007）。dem可以继承区域函数的异常行为。错误是由记录方法和异常行为的一部分组成的（脚趾，1998；加罗，1998）。第二面反思的错误（etr）是第二面反思的错误，即第二面反思的错误（g，1997）。Highaccuracysurfacemodelling(HASM)wasconstructedintermsofthefundamentaltheoremofsurfaces(Yueetal.,2007).FormerresearchesindicatedthatHASMismoreaccuratethantheclassicalmethodsincludingTIN,IDW,SplineandKriging(Yueetal.,2008),butwhenevaluatingtheDEMerror,weonlyconsideredinterpolationRMSEignoringEtr.InordertogiveDEMacomprehensiveevaluation,thetotalDEMsimulationerrorwaspresentedinthispaper,whichiscalculatedintermsoferrorpropagationtheory.CanonicalsurfaceandDong-Zhi-YuaninGansuwereselectedtocomparativelyanalyzethetotalDEMerrorsofHASMandtheclassicalmethods.2通过外在热价反应remreponusinvi必要sremm药3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3日aswellagra内部循环/沙加尔地区saratorpoheningsix.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.4.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.Ifasurfaceisagraphofafunctionz=f(x,y),theGaussequationsetscanbeformulatedas,where,hrepresentsDEMresolutionTheiterativesimulationstepsaresummarizedasfollows:①ComputingthefirstandsecondfundamentalcoefficientsE,F,G,L,N,aswellascoefficientsofHASMequations.②Forn≥0,wecangettheDEMbysolvingtheHASMequations.③Theiterativeprocessisrepeateduntilsimulationaccuracyissatisfied.3同文本相关接口之一—ETRCOMPUTATIONFig.1accountsfortheexistenceofEtrona2Dprofile.Thethicklinerepresentsthetruegroundsurface,whileAandCaretwoadjacentelevationpointsfreefromerror.TheelevationdiscrepancybetweentheelevationmodelsurfaceandthetruesurfaceistheEtr.αrepresentstheslopeatA;xisavariable,representingthedistancebetweenAandD.Therefore,Etrcanbeformulatedas,Letfirst-orderderivativeofequalzero,wecanobtainthelocationofD,wherehasthebiggestvalue.Eq.(3)revealedthatatthemiddlepointon,itwouldbemostpossibletogetthemaximumEtr.Intermsofthisanalysis,themaximumEtrbetweenAandCcanbecomputedwithEq.(4).whereHA,HB,HCrespectivelyrepresenttheelevationvaluesatA,B,C.Inthe3DDEM,themaximumEtrwithinagridcanbeexpressedas,where,Hcistheelevationvalueatthecenterofthegrid;HLU,HLB,HRUandHRB,respectively,representelevationvaluesatthefourcomersofthegrid.AssumingtheDEMresolutiontobeh,a3×3kernelwindowisemployedtocalculatetheEtrwithEq.(6)(Fig.2).IfthekernelwindowmovesthroughthewholeDEMmatrixcellbycellandcomputestheEtrvalueofeachcell,wecanderiveanexpectedEtrmatrixwiththeresolutionof2h.Ifthekernelwindowisextendedto5×5,7×7,...,asetofDEMEtrmatriceswiththeresolutionsof4h,6h,...,canbederivedrespectively(Fig.2).Therootmeansquareerror(RMSE)oftheEtrmatricescanbecalculatedwithEq.(7).whereRow,ColumrespectivelyrepresenttherowandcolumnofEtrmatrix.Therefore,intermsoferrorpropagationtheorem,thetotalDEMerror(mD)canbeformulatedas,wheremI,mT,mSrepresentinterpolationRMSE,EtrandsamplingRMSErespectively.ThetotalDEMerrorcanberegardedasthefirstcriterionforassessingDEMaccuracy.4哈马罗伦斯4.1regrsionmo对于which3.3.3.3.3.3.3.3.3方法,德国自然资本市场,自然资本市场,自然资本市场,自然正义公民,自然正义公民,可能是re实践经济公约,并非选择能源,以uyratchratchratchratchratchratchratchratchratchratchratchratchratchrace.3.4.3.3.3.3.3.3.3.4.3.4.3.3.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.4.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3z=2sin(πx)sin(πy)+l,thecanonicalsurfacewasemployedtocomparativelyanalyzetheDEMerror,thusthe“true”outputvaluecanbepre-determinedtoavoiduncertaintycausedbyuncontrollabledataerrors.Itscomputationaldomainis×.TheRMSEofEtrmatriceswithdifferentresolutionswerecalculated.TheresultswereshowninTable1.TheregressionmodelthatrelatedtheresolutionandEtrwasgeneratedwithSPSS,whichappearstohaveanon-linear(quadratic)correlation(Fig.3andEq.(9)).Varianceanalysisshowedthatthecurvethroughthesamplingpointscompletelyindicatesthattheregressionmodelhasagoodfitness(F=8541954,P=0).InadditiontothenewlydevelopedHASM,therearemanyothermethodsofsurfacemodelling,whicharewidelyusedinvariousGISapplications,includingIDW,SplineandKriging.ThesefourmodelswereemployedtocomparativelyanalyzeDEMerrors.Inthecomputationaldomain,25pointswereusedasthesamplingdata.Alltheclassicalinterpolationmethodswereperformedusingthemoduleof3DanalystinArcGIS9.2withthedefaultparameters.TheinterpolationRMSEwascalculatedwithEq.(10).TheresultswereshowninTable2.where,Row,Columrespectivelyrepresenttherowandcolumnofthelatticesmatrix;sfi,j,fi,jrepresentthesimulationvalueandtruevalueatthelattice(i,j)respectively.BasedonEq.(8),thetotalDEMerrorswerecalculated(Table3).ResultsindicatedthatHASMismoreobviouslyaffectedbyEtrthantheclassicalmethods(Table2and3).AlthoughEtrhaslittleinfluenceontheclassicalmethods,HASMisstillmoreaccuratethantheclassicalmethods.Whentheresolutionis1/64,theHASMinterpolationerrorisbiggerthanthatofresolutionof1/32(Table2).ButwhentakingtheEtrintoaccount(Table3),theHASMtotalerrorbecomessmallerwiththeresolutionincreasing(gridcellsizebecomingsmaller),whichindicatesthatifignoringtheimpactofresolutionontheterrainrepresentation,wemightobtainwrongconclusions.ConsideringallDEMsderivedfromtheclassicalmethods,thetotalDEMerrorincreasesasthegridcellsizedecreases.Therefore,underthesamesamplingdataandanyresolution,HASMismoreaccuratethantheclassicalmethods.4.2acculace.3.3indexh-pcrwhichmillamesims.完善的制度环境见表5Dong-Zhi-Yuanexperimentalstationwasselectedforthisstudy.ItwaslocatedinGansuprovinceofChinawithareaof2260km2.Thegeographiccoordinatesofitscentralpointare107.88°Eand36.03°N.OneofthetopographicmapswasutilizedinthisstudytoderiveDEMs.Thetopographicmapwithscale1:5000hasa25m-vertical-intervalbetweeneachcontourline.Itselevationsvaryfrom1195mto1530m.Thescannedtopographicmapwasenlargedthreetimesinordertoenhancethedigitizationaccuracy.ContoursweretracedusingthemoduleofEditorinArcGIS9.2.Afterthat,theelevationsweresampledfromthecontoursusingdatamanagementtools.Theelevationswereinterpolatedwithresolutionof5m.RMSEofEtrmatriceswithdifferentanalysisresolutionswerecalculated.TheresultswereshowninTable4.TheregressionmodelthatrelatedtheresolutionandEtrwasgeneratedwithSPSS(Fig.4andEq.(11)),whichappearstohavelinearcorrelation.Varianceanalysisshowsthatthecurvethroughthesamplingpointscompletelyindicatesthattheregressionmodelhasagoodfitness(F=44713.618,P=0).TheinterpolationaccuracyofterrainrepresentationwasevaluatedagainstRMSEat30%randomlyselectedelevationsascheckpoints(Table5).AssumingtheDEMsamplingerrortobezero,thecorrespondingtotalDEMerrorwascalculatedwithEq.(8)(Table6).Table5indicatedthatwhentheresolutionis20m,HASMhasabiggerinterpolationaccuracyloss,whichmightbecausedbythelocationdifferencesbetweenthesamplingpointsandthecorrespondingcentralpointsoflatticesofthesimulatedsurfaces.WhentakingEtrintoaccount,thetotalDEMerrorofHASMdecreaseswiththeresolutionincreasing(Table6).SoonlytakinginterpolationerrorasthecriterionforassessingDEMaccuracy,wemightnotgiveHASMacomprehensiveevaluation.Underanyresolution,HASMismoreaccuratethantheclassicalmethods.HASMmaybeanalternativeforDEMconstruction.5u3000eso文件InordertogiveDEMafullevaluation,EtrwaspresentedandregressionmodelthatrelatesEtrandDEMresolutionwasgeneratedwithSPSS.TotalDEMsimulationerrorwascalculatedintermsoferrorpropagationtheory.WeselectedthecanonicalsurfaceandDong-Zhi-YuantoanalyzethetotalDEMsimulationerrorofHASMandtheclassicalmethodsincludingIDW,SplineandKriging.Numericaltestandreal-worldexampleshowedthatwhentakingEtrintoaccount,wecangiveHASMafullevaluation.HASMismoreaccuratethantheclassicalmethods,whichcanbeconsideredasanalternativeforDEMconstruction.HASMhasahugecomputationcostbecauseitmustuseanequationsetforsimulatingeachlatticeofasurface.AnadaptivesimulationapproachwithgridselectionstrategiesishighlysignificantforHASM,whichisaverydesirablefeatureforanaccurateanalysisandanefficientsimulation(Berger,1989;Jesseeetal.,1998).AdaptivemethodforHASMconcentratescomputationaleffortwhereitismostneeded,whichmayreducemuchsimulationcostandvolumestoragethanglobalsimulation(Martin,1998;Ekevidetal.,2004).1地形复杂的误差数字高程模型(digitalelevationmodel,DEM)是对地球表面地形地貌的一种离散的数字表达(Yang,2007)。构建DEM常用方法之一是利用插值技术对已知数据内插完成。DEM精度表述为利用已知数据建立的DEM对真实地面描述的准确程度,DEM误差是衡量DEM精度的重要指标(Li,1991)。DEM误差可分为高程采样误差、插值误差和地形表达误差(terrainrepresentationerror,Etr)(Tang,2000;Damell等,2008)。在地形复杂度一定的情况下,Etr受DEM网格分辨率的影响(汤国安等,2001)。如何利用有限的采样数据获取高精度和高分辨率的DEM是降低DEM误差的关键技术。有曲面论定律可知,曲面有第一基本量和第二基本量确定(梅向明等,2003),据此,建立了高精度曲面模型(highaccuracysurfacemodelling,HASM)(岳天祥等,2004,2005)。HASM以高斯方程作为构建曲面的基本方程,将已知采样数据作为高斯方程的约束条件,利用最小二乘求算构建的方程就可以得到高精度曲面。以往研究表明,HASM插值精度较传统方法提高了多个数量级,但对DEM误差评价时仅考虑了插值误差(岳天祥等,2006;陈传法等,2009)。为了合理验证HASM构建的DEM误差,本文选择标准曲面和甘肃省董志塬作为研究对象,对DEM误差评价中引入Etr,并比较HASM与GIS常用的插值方法(IDW,Spline以及Kriging)获取的DEM误差。2dem方程的构建设曲面表达式为z=f(x,y),{(xi,yj)}是对计算区域Ω进行均匀正交剖分产生的网格,则HASM的最佳迭代模拟方程为(岳天祥等,2004)式中,h为网格分辨率;n为迭代次数;E,F,G为曲面第一基本量;L,N为曲面第二基本量;,(i=11,22,k=1,2)为第2类克里斯托费尔符号,它们表达式为:利用HASM构建DEM主要步骤为:①利用已知采样点计算第一基本量E,F,G,第二基本量L,N和HASM系数;②将公式(1)右端计算值作为已知量,左端变量作为未知量,求算HASM方程;③将算出的新值作为已知值,重复①—②计算过程直到满足精度要求。3窗口分析法etr计算Etr定义为当采样误差和插值误差为零时,模拟地面与实际地面的差异。如图1,表示实际地面的剖面,表示线性结构的剖面,则为因线性插值而带来的Etr。令α为点A处的坡度值,则:令BE的一阶导数为0,则BE取最大值时B点的位置可由(3)式计算:得出,,表明在的中点位置线性插值带来的Etr最大。同理,双线性插值在4个点的中点位置Etr最大,因此可以利用窗口分析法(Tang,2000)计算Etr。窗口分析法基本思想为:设Ha,Hb,Hc,Hd,Ho为对应地面点A,B,C,D以及该4个点中点O的高程真值,则Etr可以表达为:Etr=Ho-(Ha+Hb+Hc+Hd)/4。对于分辨率为h的栅格DEM,利用3×3窗口计算第(i,j)点的Etr公式为(图2):。显然窗口计算的Etr分辨率为2h。顺序移动该窗口计算整个DEM的Etr,可以得出对应分辨率的误差矩阵。同理,分析窗口扩大到5×5,7×7,…,可以计算分辨率为4h,6h,…,的误差矩阵。Etr使用中误差(RMSE)衡量,计算公式为:式中,Row,Colum分别为Etr矩阵的行数和列数;Etri,j为Etr矩阵中第i行j列的值。4dem误差的传播规律选择标准曲面z=2sin(πx)sin(πy)+1为研究对象,模拟区域为×。利用窗口分析法计算Etr,计算结果如表1。图3显示Etr(RMSE)随分辨率的降低而增大且呈明显的二次线性相关关系,回归方程为:RMSE=1.52×10-6-0.001h+2.516h2;R2=1(5)作拟合优度检验,方差分析表明:F=8541954,P=0,表明拟合曲线通过已知样点,拟合结果准确。均匀选择25个点作为已知采样点,用HASM以及GIS插值方法(IDW,Spline和Kriging)分别生成不同分辨率的DEM。由于各个网格点的真值可以通过曲面方程计算,因此可以准确计算不同分辨率DEM的插值中误差,计算结果如表2。由于采样误差为零,根据误差传播定律,DEM中误差mD的计算公式为:式中,mI为插值中误差,mT为Etr。借助公式(6)求算的DEM中误差结果如表3。从表2、表3可见,HASM受Etr影响较大;由于IDW,Spline和Kriging的插