衛(wèi)星重力測量

衛(wèi)星重力測量-基礎(chǔ)、模型化方法與數(shù)據(jù)處理算法作者簡介：張傳定，男，1966年04月出生，1996年09月師從于解放軍信息工程大學陸仲連教授，于2000年12月獲博士學位。摘要論文的中心內(nèi)容是衛(wèi)星重力測量中如何由星載傳感器獲得的觀測數(shù)據(jù)恢復地球重力場這一過程的模型化問題。旨在吸取前人的研究成果，提出更加合理的數(shù)據(jù)處理模型。論文最突出的貢獻是，改造并完善了大地重力學、空間大地測量、衛(wèi)星軌道力學等學科模型化的理論與方法以適應衛(wèi)星重力測量這一新型觀測技術(shù)。作者的主要工作和創(chuàng)新點有：.在綜合衛(wèi)星重力測量有關(guān)最新研究成果的基礎(chǔ)上，系統(tǒng)地論述了動態(tài)加速度測量、衛(wèi)星重力梯度測量的基本原理；論證了它們的測量精度與姿態(tài)角加速度的關(guān)系以及衛(wèi)星重力測量系統(tǒng)最終恢復地球重力場能力的判定準則；深入理解并掌握了現(xiàn)行SSTSGG衛(wèi)星CHAMP、GRACEGOCE各項指標及恢復地球重力場各頻段的精度指標。.簡要介紹了衛(wèi)星重力測量中所涉及到的曲線坐標系下矢量、張量與曲線坐標之間的微分關(guān)系、坐標系之間的變換關(guān)系以及它們的矩陣表示。詳細研究了在地球重力場確定中常用的關(guān)于研究點P和流動點Q相互關(guān)聯(lián)的球極坐標系，給出了球極坐標系下地球引力位V關(guān)于P點和關(guān)于Q點的微分公式以及它們與球坐標系下局部微分算子的關(guān)系。深入研究了關(guān)于P和Q兩點局部導數(shù)算子的相互作用問題，得到了擾動場元之間核函數(shù)和協(xié)方差函數(shù)的解析與級數(shù)展開式，首次給出了較為實用的明晰表達式。此結(jié)果是對物理大地測量學關(guān)于這一論題的補充和完善。這項工作是本文的一個創(chuàng)新點。.詳細推導了地球、衛(wèi)星、加速度傳感器檢驗荷載這一特殊限定性三體問題的運動方程；指出星載加速度傳感器的輸出就是衛(wèi)星所受非引力加速度和檢驗荷載相對于衛(wèi)星中心地球引力的潮汐力之差；進而得到了由星載加速度傳感器的比力測量和GPS跟蹤測量數(shù)據(jù)直接恢復地球引力矢量的理論公式。.通過對扭秤、旋轉(zhuǎn)梯度儀工作原理的考察和Molodensky關(guān)于垂線偏差推求高程異常的論述以及目前業(yè)已發(fā)現(xiàn)水平梯度分量的某種組合是球面正交函數(shù)系的事實，作者明確指出，在地球重力場的研究中，水平方向觀測量的組合應作為復數(shù)使用。擾動場元觀測量的復數(shù)表達是本文立論和各種模型化（建模）工作的思想基礎(chǔ)，也是本文最為突出的創(chuàng)新點。.在§2.7中，直接由體球諧函數(shù)水平梯度的復表示定義并證明了描述地球引力位直到二階水平梯度所需的球面正交函數(shù)系。它們關(guān)于緯度的函數(shù)是Legender函數(shù)及其導數(shù)的擬線性組合，可由目前熟知有關(guān)Legender函數(shù)及其導數(shù)的遞推公式給予賦值。連同球諧函數(shù)構(gòu)成了描述引力矢量、引力梯度張量所需的正交函數(shù)系。因而，利用它們可將引力矢量、引力梯度張量的復分量表達成一致的形式。.利用衛(wèi)星重力測量數(shù)據(jù)恢復地球重力場，若從邊值問題理論上可將其歸結(jié)為平均軌道面上衛(wèi)星重力測量超定邊值問題。通常又將利用單個邊值條件確定擾動位問題稱為單定問題。在§3中，先以重力異常為例，類比依次給出直到二階梯度球域單定連續(xù)邊值問題恢復地球引力位系數(shù)的理論公式及其外部解析解和向下延拓截斷核函數(shù)解；接著導出離散網(wǎng)格平均重力異常對應的簡單調(diào)和分析公式和最小二乘調(diào)和分析公式；然后推廣得到廣義梯度調(diào)和分析公式和超定邊值問題的最小方差解、最小二乘解。并證明了最小方差解等價于單定邊值問題調(diào)和分析解的頻域加權(quán)平均；最小二乘解等價于單定離散邊值問題最小二乘調(diào)和分析解法方程相加所得的解。廣義調(diào)和分析方法所需的有關(guān)勒讓德函數(shù)及其導數(shù)的積分遞推公式在§3.6中給出。.首次定義并推導出了水平一階和二階梯度平滑因子、。在概念上澄清了它們與熟知的面球諧函數(shù)平滑因子是不同的。盡管、與相差不大，但在實踐上應嚴格區(qū)分它們，這樣邏輯上才是嚴格的。與觀測量的對應關(guān)系是使用時應按格網(wǎng)均值數(shù)據(jù)類型，采用相應的平滑因子。.由于水平擾動場元之間的協(xié)方差并非各向同性，導致協(xié)方差矩陣結(jié)構(gòu)復雜（子矩陣不是Toeplitz循環(huán)陣），不能利用變換矩陣將其降階，無法付諸實踐，迄今尚無最小二乘配置理論應用于水平擾動場元觀測量的模型化公式和數(shù)據(jù)處理方法。作者研究發(fā)現(xiàn)，利用水平梯度的復組合，即復數(shù)表示后，擾動重力場元復組合之間的協(xié)方差函數(shù)盡管還是各向異性，但它們對應的協(xié)方差矩陣卻具有分塊Toeplitz循環(huán)陣的結(jié)構(gòu)，因而水平分量復組合的配置問題與重力異常的配置問題相似，可以利用傅立葉變換矩陣進行降階處理。這表明，必須將最小二乘配置理論拓展，以適應復數(shù)信號的配置問題，本文將其稱為最小二乘復配置。作者將最小二乘配置理論拓展為既能處理復信號又能處理實信號的配置模型，得到了最小二乘復配置解所需的公式。結(jié)合衛(wèi)星重力測量觀測量，詳細研究了重力場元復分量之間協(xié)方差函數(shù)的級數(shù)展開式、擾動引力位系數(shù)與復分量間的協(xié)方差關(guān)系。然后，利用最小二乘復配置理論和重力場復分量之間協(xié)方差函數(shù)表示，得到了與重力異常配置解接近于一致的各類單定離散邊值問題的最小二乘復配置廣義調(diào)和分析解和超定離散邊值問題的最小二乘復配置廣義調(diào)和分析解。利用復數(shù)表示，解決了最小二乘配置理論難以用于重力場水平觀測量這一瓶頸問題。.基于攝動力的ST、W分解，首次給出了衛(wèi)星受攝運動方程的反解公式。它只需對軌道根數(shù)求一次時間微分，便可求得攝動力的S、T、W分解（不含中心引力）。與由衛(wèi)星的地心位置矢量求二次導數(shù)得衛(wèi)星所受力（含中心引力）的直接方法相比，該方法因自然扣除中心引力且只需求時間一次導數(shù)，理論上精度要高一些，它具有Hill方程類似的作用，可用ST、W的時間序列與非心引力矢量的ST、W分解建立聯(lián)系，得到時域觀測方程。.成功地將高斯型求積公式用于常微分方程的數(shù)值解中，得到了高斯型隱式龍格庫塔方法（GRK）4級7階GRK法的精度已與目前較為常用的10級8階RK法的精度相當。任意級GRK法的權(quán)系數(shù)恒為正值，這就是GRK法精度高的本質(zhì)所在。利用Legender多項式的零點可以得到任意階GRK法的權(quán)系數(shù)，而傳統(tǒng)高階RK法的權(quán)系數(shù)只能手工推算，而且同牛頓柯斯特積分一樣，高階公式是不穩(wěn)定的。.通過地面跟蹤動力法觀測方程的建立，得到了SST觀測的動力法觀測方程。指出盡管SST測量是地面跟蹤的空間拓展，但SST測量因增加了非引力攝動比力加速度觀測量，其觀測方程與地面跟蹤觀測方程相比，具有質(zhì)的差異，這正是SST能以較高精度恢復地球重力場的優(yōu)勢所在。.利用最小二乘調(diào)和分析公式推導了確定恢復重力場最高階次的理論準則，其水平敏感度準則系作者給出，利用累積代表誤差和截斷誤差曲線的交點，即可求定各類觀測量敏感地球重力場的最高階次。.簡要論述了衛(wèi)星重力梯度測量數(shù)據(jù)向下延拓的截斷核譜組合解及其截斷階次和頻域加權(quán)準則。假定梯度測量數(shù)據(jù)各分量間以及相鄰采樣點間獨立不相關(guān)，則頻域加權(quán)與階次無關(guān)，只與獨立觀測量的個數(shù)有關(guān)。.配置方法的缺陷是，在數(shù)據(jù)范圍和分辨率一定時，增加觀測量類型，意味著協(xié)方差矩陣的維數(shù)升高。若是全張量梯度測量數(shù)據(jù)，則協(xié)方差矩陣的維數(shù)將是單個分量協(xié)方差矩陣的5倍。為解決矩陣降階問題，在§7.3中，闡述了最小二乘矢量、張量配置的思想及其算法公式，為利用衛(wèi)星重力測量數(shù)據(jù)逼近局部重力場提供了可用于實踐的數(shù)據(jù)處理方法。關(guān)鍵詞：地球重力場，衛(wèi)星重力測量，Toeplitz矩陣，正交函數(shù)系，復數(shù)表示，邊值問題，調(diào)和分析，平滑因子，最小二乘配置，最小二乘復配置，最小二乘矢量配置，最小二乘張量配置，攝動運動方程ABSTRACTThisdissertationismainlyfocusedonthemodelingprocessoftherecoveryoftheearth'sgravityfieldfromobservationsacquiredbysensorsonboardsatellites,withtheaimofsummarizingtheworksdonebeforetheauthorandestablishingstillmoreappropriatedataprocessingmodels.Themostimportantcontributionthisdissertationpresentsisthatthemodelingtheoriesandmethodsofgeodeticgravimetry,spacegeodesyandsatelliteorbitdynamicshavebeenreconstructedandperfectedsoastoadapttothenewsurveyingtechniquesofsatellite-bornegravimetry.Themainworkandinnovationsdonebytheauthorarepresentedasfollows:Onthebasisofthegeneralizationofthelateststudiesofsatellite-bornegravimetry,theauthordiscussestheprinciplesofdynamicaccelerationmeasurementandsatellite-bornegradiometrysystematically,anddemonstratesboththerelationshipbetweenaccuracyandangleaccelerationandthedeterminantrulesoftheabilityoftheearth'sgravityfieldrecoveryfromsatellite-bornegravimetry.ThentheauthorhasadeepinquiryintothetargetsofthecurrentSSTandSGGsatellitessuchasCHAMPGRACEandGOCEaswellastheaccuracytargetsofdifferentfrequenciesoftheearth'sgravityfieldrecovery.Theauthorthenintroducesthedifferentialrelationshipbetweenvectorsaswellastensorsinacurvecoordinatesystemandcurvecoordinates,andthetransformationrelationshipbetweencoordinatesystemsandtheirmatrixexpressions.AfterstudyingthefrequentlyusedsphericalpolarcoordinatesystemconcerningthecalculatingpointPandthemobilepointQindetail,theauthorpresentsthedifferentialformulaeoftheearth'sgravitationalpotentialVwithrespecttothepointsPandQandtheirrelationshipwithpartialdifferentialoperatorsinasphericalcoordinatesystem.ThenaftertheauthorresearchestheinteractionproblemofthepartialderivativeoperatorswithrespecttothetwopointsPandQ,analyticalexpressionsaswellasseriesexpansionsofthekernelfunctionwithindisturbinggravityfieldelementsaredetermined,andtheirexplicitexpressionswhichcanbeappliedpracticallyaregivenforthefirsttime.Thisachievement,beinganinnovationofthedissertation,isasupplementandperfectionofthetopicinphysicalgeodesy.Theauthordeducesthemotionequationofthespeciallyrestrictedthree-bodyproblemthatconsistsoftheearth,satelliteandaccelerationsensortestpayloadindetail,andpointsoutthattheoutputoftheaccelerationsensoronboardthesatelliteistheverydifferenceofthenon-gravitationaccelerationactingonthesatelliteandthetidalforcesofthetestpayloadrelativetothegravitationatthecenterofthesatellite,andthusobtainsthetheoreticalformulaoftheearth'sgravityvectorrecoveryfrommeasurementoftheaccelerationsensoronboardsatelliteandtheGPStrackingobservations.TheauthorpointsoutthatthecombinationofhorizontalobservableshouldbeusedintheformofthecomplexnumberinthestudyofgravityfieldthroughaninspectionoftheprinciplesoftorsionbalanceandrotationalgradiometerandthediscussionoftheacquisitionofheightanomalyfromverticaldeflectionsaccordingtoMolodensky's,aswellasthefactthatsomecombinationofhorizontalgradientcomponentsareorthogonalfunctionsonasphericalsurface.Thecomplexexpressionoftheobservationsofdisturbinggravityelementsisthebasisoftheargumentofthisdissertationandallkindsofmodelingprocessesanditisalsothemostprominentinnovation.Insection2.7,accordingtothecomplexexpressionofthehorizontalgradientofsphericalharmonic,thesphericalorthogonalfunctionfamilyneededforthedescriptionoftheearth'sgravitypotentialupto2degreehorizontalgradientsisdefinedandproveddirectly.Thefunctions,latitudebeingparameter,arethequasilinearcombinationoftheLegenderfunctionsandtheirderivatives,whichcanbeevaluatedbytherecursiveformulaeoftheLegenderfunctionanditsderivatives.Therefore,withthesefunctions,thecomplexcomponentsofthegravityvectorandthegravitygradienttensorcanbeexpressedinaconsistentform.Whenitcomestothegravityfieldrecoveryfromsatellite-bornegravimetryobservationsthroughthetheoryoftheboundaryvalueproblem,thisproblemcanbereducedtotheover-determinedboundaryvalueproblemofsatellite-bornegravimetryonanaverageorbitalplane.Usuallythedeterminationofthedisturbinggravitypotentialwithasingleboundaryvalueconditioniscalledasingle-determinedproblem.Insection3,takingthegravityanomalyforexamplefirstly,theauthorpresentstheoreticalformulaeoftherecoveryoftheearth'sgravitypotentialcoefficients,anditsexternalanalyticalsolutionaswellasthetruncatedkernelsolutionofdownwardcontinuationfromuptotwodegreegradientbyanalogy,astothesingle-determinedcontinuousboundaryvalueprobleminsphereregion.Theauthoralsopresentssimpleharmonicanalysisformulaeandleastsquaresharmonicanalysisformulaecorrespondingtodiscretegridmeangravityanomaly.Thentheseformulaearegeneralizedtoobtainwhatiscalledthegeneralizedgradientharmonicanalysismethodinthedissertation.Thisstepisfollowedbytheminimumvariancesolutionandleastsquaressolutiontotheover-determinedboundaryvalueproblem.Itisalsoprovedthattheminimumvariancesolutionisequivalenttotheweightedaverageinfrequencydomainoftheharmonicanalysissolutiontothesingledeterminedboundaryvalueproblem,andthattheleastsquaressolutionisequivalenttothesummationofthenormalequationsoftheleastsquaresharmonicanalysissolutiontothesingledetermineddiscreteboundaryvalueproblem.TherecursiveformulaeoftheLegenderanditsderivativesrequisiteforthegeneralizedharmonicanalysismethodaregiveninsection3.6.Itisforthefirsttimethatthehorizontalone-andtwo-degreegradientsmoothingfactors,aredefinedanddeduced.Andtheconceptionaldifferencebetween,andthewell-knownsurfacesphericalharmonicisclarified.Althoughthereissmalldifferencebetween,and,yettheyshouldbestrictlydistinguishedinpracticesoastokeepstrictinlogic.Thecorrespondingrelationofandobservableisshownasfollows:Inapplication,theselectionofthesmoothingfactorsshouldbebasedonthetypeofthegridmeanvalues.Onthegroundthatthecovariancebetweenthedisturbinggravityfieldelementsisnotisotropic,thecovariancematrixhasacomplicatedstructure(itssubmatrixbeingnotrepetitiveToeplitzmatrix)andthusitsdegreecannotbedegradedwithtransformationmatrixmethod,soitcannotbeputintopractice.Sofar,theapplicationofleastsquarescollocationtothemodelinganddataprocessingofthehorizontaldisturbinggravityfieldobservationshasnotbeenfoundinliterature.Afterstudyingthisphenomenon,theauthorfindsthat,thecovariancematrixofthecomplexexpressionsofhorizontalgradientsofthedisturbinggravityfieldelementshasastructureofblockwiserepetitiveToeplitzmatrix,thoughthecovariancewithinthedisturbinggravityfieldelementsisstillanisotropic.Therefore,thecollocationproblemofthecomplexcombinationofhorizontalcomponentsissimilartothatofgravityanomaly,whichmakesthedegradationbeabletobecompletedbyFFTtransformationmatrix.Thisindicatesthattheleastsquarescollocationtheorymustbedevelopedsoastoadapttothecollocationofcomplexsignals.Andthisiswhatiscalledthecomplexleastsquarescollocationinthedissertation.Theauthordevelopstheleastsquarescollocationintothecollocationmodelwhichcandealwithbothcomplexandrealsignals,thusobtainingformulaeforcomplexleastsquarescollocation.Inreferencetothesatellite-bornegravimetryobservations,theauthorstudiestheseriesexpansionofcovariancebetweenthegravityfieldelements'complexcomponentsandthecovariancerelationbetweenthedisturbinggravitypotentialcoefficients,andthecomplexcomponentsindetail.Then,throughapplyingthecomplexleastsquarescollocationtheoryandthecovarianceexpressionsbetweenthecomplexcomponentsofthegravityfield,theauthoracquiresthegeneralizedharmonicanalysissolutionofthecomplexleastsquarescollocationofallkindsofsingledetermineddiscreteboundaryvalueproblemsandsuchsolutiontoover-determineddiscreteboundaryvalueproblems.Thesolutionisnearlyconsistentwiththeresultofgravityanomalycollocation.Bymeansofthecomplexexpression,theauthorovercomesthebottleneckproblemthatitishardtoapplytheleastsquarescollocationtheorytothehorizontalobservationsofthegravityfield.BasedontheS,TandWdecompositionofperturbingforces,theformulaeofthereversesolutiontothesatellite'sperturbedmotionequationisgivenforthefirsttime.Aslongasthefirstdifferentialoftheorbitalelementswithrespecttotimeismade,theS,T,andWdecompositionofperturbingforces(excludingtheearth'scentralgravitation)canbeobtained.Comparedwiththedirectmethodofacquiringtheforcesexertedonsatellite(containingtheearth'scentralgravitation)fromthesecondderivativeofthesatellite'sgeocentricpositionvector,thismethodisstillmoreaccuratetheoreticallyinthatitexcludestheearth'scentralgravitationandjustneedsthefirstderivativewithrespecttotime.AndthismethodworksliketheHill'sequation,sotheobservationequationintimedomaincanbeobtainedbyrelatingthetimesequenceofS,TandWwiththeS,TandWdecompositionofnon-centralgravitation.TheauthorsuccessfullyappliestheGauss-typeintegralformulatothenumericalsolutiontotheconstantdifferentialequationandcreatestheGauss-typeimplicitRunge-KuttaMethod(GRK)ofwhichtheaccuracyoforder4anddegree7isalreadyequivalenttothatoftheRKmethodwithorder10anddegree8thatisincommonuse.TheGRKmethodofanyorderhasapositiveweightcoefficient,whichleadstoitshigheraccuracy.TheweightcoefficientsofGRKofanyordercanbeacquiredthroughthezeropointsoftheLegenderpolynomial,whereastheweightcoefficientsofthetraditionalRKofhigherorderscanonlybesolvedmanuallyand,liketheNewton-CoastIntegral,theformulaofhigherorderisunstable.Throughtheestablishmentofobservationequationofgroundtrackingdynamicmethod,thedynamicobservationequationofSSTisobtained.Theauthorpointsoutthatthereisonemoreobservable,thatistosay,theaccelerationofnon-gravitationalperturbingforcesintheSSTmeasurement,thoughtheSSTmeasurementistheextensionofgroundtracking.Therefore,theobservationequationoftheSSTmeasurementisessentiallydifferentfromthatofthegroundtracking,whichistheadvantageoftheSST'srecoveryoftheearth'sgravityfieldwithhigheraccuracy.Bymeansoftheleastsquaresharmonicanalysisformulae,thetheoreticalruleinthedeterminationofthehighestorderanddegreeoftheearth'sgravityfieldrecoveryisdeduced,inwhichthehorizontalsensitivityruleistheauthor's.Thehighestorderanddegreeofthegravityfieldsensedbyallkindsofobservation