利用动力学方法解算GRACE时变重力场研究
|
摘要
本文利用动力学方法建立GRACE(Gravity Recovery And Climate Experiment)K波段距离变率(KBRR)观测、轨道观测与重力场系数的观测方程,通过GRACE Level 1B观测数据,成功解算出全球月时变重力场模型——IGG时变重力场模型,并将2008—2009年的解算结果与GRACE三大数据处理机构美国德克萨斯大学空间中心CSR(Center for Space Research)、美国宇航局喷气推进实验室JPL(Jet Propulsion Laboratory)和德国地学研究中心GFZ(GeoForschungs Zentrum)发布的最新全球时变重力场模型进行详细对比分析.结果表明:IGG结果在全球质量异常、中国及周边地区质量异常的趋势变化、全球质量异常均方差、2~60每阶位系数差值以及亚马逊流域和撒哈拉沙漠等典型区域平均质量异常等方面与CSR、JPL和GFZ解算的RL05结果较为一致.其中,IGG解算结果在2~20阶与CSR、GFZ和JPL最新解算结果基本一致,20~40阶IGG解算结果与GFZ、JPL单位最新解算结果较为接近,大于40阶IGG结果介于CSR与GFZ、JPL之间;亚马逊流域平均质量异常周年振幅IGG、CSR、GFZ和JPL获取到的结果分别为17.6±1.1cm、18.9±1.2cm、17.8±0.9cm和18.9±1.0cm等效水柱高.利用撒哈拉沙漠地区的平均质量异常做反演精度评定,IGG、CSR、GFZ和JPL的时变重力场获取到的平均质量异常均方差分别为1.1cm、0.9cm、0.8cm和1.2cm,表明IGG解算结果与CSR、GFZ和JPL最新发布的RL05结果在同一精度水平.
The Gravity Recovery and Climate Experiment(GRACE)mission can significantly improve our knowledge of the temporal variability of the Earth′s gravity field.We intend to obtain monthly gravity field solutions based on dynamic approach(variational equations approach)from GPS-derived positions of GRACE satellites and K-band range-rate measurements.Moreover,these solutions will validate through GRACE RL05 products published monthlygravity field solutions by the GRACE project,which including CSR(Center for Space Research),JPL(Jet Propulsion Laboratory)and GFZ(GeoForschungsZentrum).Based on the theory of the dynamic approach,the processing began with the computation of purely dynamic orbits by fitting GNV1 Borbits computed by the GRACE team using a reduced dynamic strategy.This step aimed to calibrate the GRACE A and B accelerometer data and the initial state vectors of these two satellites.Note that gravity field model parameters were not adjusted in this part.In the next step,the purely dynamic orbits were used as the reference orbit for the calculation of residual orbits,and were also applied to calculate the nominal value of Kband range-rate.In addition,the partial derivatives with respect to the initial state vectors,accelerometer parameters and geopentional coefficients were computed simultaneously with the integration of the purely dynamic orbits.Finally,normal equations of orbit positions and K-band range-rate measurements were synthesized using a fixed data weighting ratio to combine the both information matrices for the best combined solution.The monthly gravity field solution obtained though above procedures was named as the IGG temporal gravity field model.IGG temporal gravity field models were compared with GRACE RL05 products in following aspects:(i)monthly gravity field solutions in February,2008 and September,2009;(ii)the root mean squares of the global mass anomaly during 2008 to 2009;(iii)the trend of the mass anomaly in China and its nearby regions within 2008 to 2009;(iv)the square root degree difference variance with respect to the mean gravity field GIF48 model from2008 to 2009;(v)time-series changes in the mean water storage in the region of the Amazon Basin and the Sahara Desert between 2008 and 2009.The results showed that IGG solutions were almost consistent with GRACE RL05 products in above aspects(i)—(iii).For aspect(iv),IGG solutions were in good agreement with latest published GRACE RL05 products at the degree from 2to 20,and were at the same accuracy level with GFZ and JPL at the degree from 20 to 40,but were between CSR and GFZ(or JPL)at the degree exceeding 40.According to aspect(v),changes in the annual amplitude of mean water storage in the Amazon Basin were 17.6±1.1cm for IGG,18.9±1.2cm for CSR,17.8±0.9cm for GFZ and 18.9±1.0cm for JPL in terms of equivalent water height,respectively.The root mean squares of the mean mass anomaly in Sahara were 1.1cm,0.9cm,0.8cm and 1.2cm for temporal gravity field models of IGG,CSR,GFZ and JPL,respectively.Overall,it was noticeable that IGG temporal gravity field solutions were at the same accuracy level with the latest temporal gravity field solutions published by CSR,GFZ and JPL.
The Gravity Recovery and Climate Experiment(GRACE)mission can significantly improve our knowledge of the temporal variability of the Earth′s gravity field.We intend to obtain monthly gravity field solutions based on dynamic approach(variational equations approach)from GPS-derived positions of GRACE satellites and K-band range-rate measurements.Moreover,these solutions will validate through GRACE RL05 products published monthlygravity field solutions by the GRACE project,which including CSR(Center for Space Research),JPL(Jet Propulsion Laboratory)and GFZ(GeoForschungsZentrum).Based on the theory of the dynamic approach,the processing began with the computation of purely dynamic orbits by fitting GNV1 Borbits computed by the GRACE team using a reduced dynamic strategy.This step aimed to calibrate the GRACE A and B accelerometer data and the initial state vectors of these two satellites.Note that gravity field model parameters were not adjusted in this part.In the next step,the purely dynamic orbits were used as the reference orbit for the calculation of residual orbits,and were also applied to calculate the nominal value of Kband range-rate.In addition,the partial derivatives with respect to the initial state vectors,accelerometer parameters and geopentional coefficients were computed simultaneously with the integration of the purely dynamic orbits.Finally,normal equations of orbit positions and K-band range-rate measurements were synthesized using a fixed data weighting ratio to combine the both information matrices for the best combined solution.The monthly gravity field solution obtained though above procedures was named as the IGG temporal gravity field model.IGG temporal gravity field models were compared with GRACE RL05 products in following aspects:(i)monthly gravity field solutions in February,2008 and September,2009;(ii)the root mean squares of the global mass anomaly during 2008 to 2009;(iii)the trend of the mass anomaly in China and its nearby regions within 2008 to 2009;(iv)the square root degree difference variance with respect to the mean gravity field GIF48 model from2008 to 2009;(v)time-series changes in the mean water storage in the region of the Amazon Basin and the Sahara Desert between 2008 and 2009.The results showed that IGG solutions were almost consistent with GRACE RL05 products in above aspects(i)—(iii).For aspect(iv),IGG solutions were in good agreement with latest published GRACE RL05 products at the degree from 2to 20,and were at the same accuracy level with GFZ and JPL at the degree from 20 to 40,but were between CSR and GFZ(or JPL)at the degree exceeding 40.According to aspect(v),changes in the annual amplitude of mean water storage in the Amazon Basin were 17.6±1.1cm for IGG,18.9±1.2cm for CSR,17.8±0.9cm for GFZ and 18.9±1.0cm for JPL in terms of equivalent water height,respectively.The root mean squares of the mean mass anomaly in Sahara were 1.1cm,0.9cm,0.8cm and 1.2cm for temporal gravity field models of IGG,CSR,GFZ and JPL,respectively.Overall,it was noticeable that IGG temporal gravity field solutions were at the same accuracy level with the latest temporal gravity field solutions published by CSR,GFZ and JPL.
引文
Berry M M,Healy L M.2004.Implementation of gauss-jacksonintegration for orbit propagation.The Journal of the AstronauticalSciences,52(3):331-357.
Bettadpur S.2009.Recommendation for a-priori bias&scaleparameters for Level-1BACC data(version 2).GRACE TN-02Version2.
Bettadpur S.2012.UTCSR Level-2processing stands document forLevel-2 product release 005.Center for Space Research,University of Texes at Austin.
Biancale R,Bode A.2006.Mean annual and seasonal atmospherictide models based on 3-hourly and 6-hourly ECMWF surfacepressure data.Potsdam:Deutsches GeoForschungsZentrumGFZ,Scientific Technical Report STR 06/01,doi:10.2312/GFZ.b103-06011.
Bruinsma S,Lemoine J,Biancale R,et al.2010.CNES/GRGS 10-day gravity field models(release 2)and their evaluation.Adv.Space Res.,45(4):587-601,doi:10.1016/j.asr.2009.10.012.
Case K,Kruizinga G,Wu S.2010.GRACE Level 1Bdata productuser handbook.Jet Propulsion Laboratory,California Instituteof Technology.
Cheng M K,Tapley B D.2004.Variations in the earth's oblatenessduring the past 28years.J.Geophys.Res.,109(B9):B09402,doi:10.1029/2004JB003028.
Dahle C,Flechtner F,Gruber C,et al.2013.GFZ GRACE Level-2processing standards document for Level-2product release 0005:revised edition.Potsdam:Deutsches GeoForschungsZentrumGFZ,doi:10.2312/GFZ.b103-1202-25.
Desai S D.2002.Observing the pole tide with satellite altimetry.J.Geophys.Res.,107(C11):7-1-7-13,doi:10.1029/2001JC001224.
Feng W,Zhong M,Lemoine J M,et al.2013.Evaluation ofgroundwater depletion in North China using the GravityRecovery and Climate Experiment(GRACE)data and groundbased measurements.Water Resour.Res.,49(4):2110-2118,doi:10.1002/wrcr.20192.
Flechtner F,Dobslaw H.2013.GRACE AOD1B productdescription document for product release 05.GFZ GermanResearch Centre for Geosciences.
Geruo A,Wahr J,Zhong S J.2013.Computations of theviscoelastic response of a 3-D compressible Earth to surfaceloading:an application to Glacial Isostatic Adjustment inAntarctica and Canada.Geophys.J.Int.,192(2):557-572,doi:10.1093/gji/ggs030.
Jggi A,Beutler G,Meyer U,et al.2012.AIUB-GRACE02S:Status of GRACE gravity field recovery using the celestialmechanics approach.//Kenyon S,Pacino M C,Marti U eds.Geodesy for Planet Earth,International Association of GeodesySymposia 136.Berlin Heidelberg:Springer,doi:10.1007/978-3-642-20338-1_20.
Jekeli C.1981.Alternative methods to smooth the earth's gravityfield.Report No.327,Reports of the Department of GeodeticScience and Surveying.Columbus:The Ohio State University.
Kim J.2000.Simulation study of a low-low satellite-to-satellitetracking mission[Ph.D.thesis].Texas:The University ofTexas at Austin.
Liu X.2008.Global gravity field recovery from satellite-to-satellitetracking data with the acceleration approach[Ph.D.thesis].Delft:Delft University of Technology.
Liu X,Ditmar P,Siemes C,et al.2010.DEOS Mass Transportmodel(DTM-1)based on GRACE satellite data:methodologyand validation.Geophys.J.Int.,181(2):769-788,doi:10.1111/j.1365-246X.2010.04533.x.
Luo J.2003.Theory and methodology of earth gravity fielddetermination using satellite-to-satellite tracking[Ph.D.thesis](in Chinese).Wuhan:Wuhan University.
Mayer-Gürr T.2006.Gravitationsfeldbestimmung aus der Analysekurzer Bahnboegen am Beispiel der Satellitenmissioncn CHAMPund GRACE[Ph.D.thesis].Bonn:Institute für TheoretischeGeodsie der Universitt Bonn.
Ran J.2014.Theory,methodology and application of recovery usinglow-low tracking gravity satellite data[Ph.D.thesis](inChinese).Wuhan:Institute of Geodesy and Geophysics.Chinese Academy Sciences.
Ran J J,Xu H Z,Shen Y Z,et al.2012.Expected accuracy of theglobal gravity field for next GRACE satellite gravity mission.Chinese J.Geophsys.(in Chinese),55(9):2898-2908,doi:10.6038/j.issn.0001-5733.2012.09.009.
Ran J J,Xu H Z,Zhong M,et al.2014.Global temporal gravityfiled recovery using GRACE data.Chinese J.Geophysics.(inChinese),57(4):1032-1040,doi:10.6038/cjg20140402.
Rieser D,Mayer-Gürr T,Savcenko R,et al.2012.The ocean tidemodel EOT11ain spherical harmonics representation.TechnicalNote.
Rodell M,Velicogna I,Famiglietti J S.2009.Satellite-basedestimates of groundwater depletion in India.Nature,460(7258):999-102,doi:10.1038/nature08238.
Shen Y Z,Chen Q J,Hsu H,et al.2013.A modified short arcapproach for recovering gravity field model.Presented at theGRACE Science Team Meeting,University of Texas,Austin,Oct.22-25.
Swenson S,Wahr J.2006.Post-processing removal of correlatederrors in GRACE data.Geophys.Res.Lett.,33(8):L08402,doi:10.1029/2005GL025285.
Swenson S,Chambers D,Wahr J.2008.Estimating geocentervariations from a combination of GRACE and ocean modeloutput.J.Geophys.Res.,113(B8):B08410,doi:10.1029/2007JB005338.
Tapley B,Ries J,Bettadpur S,et al.2005.GGM02—An improvedearth gravity field model from GRACE.J.Geod.,79(8):467-478,doi:10.1007/s00190-005-0480-z.
Tiwari V M,Wahr J,Swenson S.2009.Dwindling groundwaterresources in northern India,from satellite gravity observations.Geophys.Res.Lett.,36(18):L18401,doi:10.1029/2009GL039401.
Wahr J,Molenaar M,Bryan F.1998.Time variability of the Earth′sgravity field:Hydrological and oceanic effects and their possibledetection using GRACE.J.Geophys.Res.,103(B12):30205-30229.
Wang Z T.2005.Theory and methodology of earth gravity fieldrecovery by satellite-to-satellite tracking data[Ph.D.thesis](inChinese).Wuhan:Wuhan University.
Watkins M.2012.JPL Level-2 processing stands document forLevel-2product release 005,Jet Propulsion Laboratory.
Yi S,Sun W K.2014.Evaluation of glacier changes in highmountain Asia based on 10 year GRACE RL05 models.J.Geophys.Res.Solid Earth,119(3):2504-2517,doi:10.1002/2013JB010860.
You W.2011.Theory and methodology of earth′s gravitational field
model recovery by LEO data[Ph.D.thesis](in Chinese).Chengdu:Southwest Jiaotong University.
Zhang X F.2007.The earth's field model recovery on the basis ofsatellite-to-satellite tracking missions[Ph.D.thesis](in Chinese).Shanghai:Tongji University.
Zhao Q,Guo J,Hu Z,et al.2011.GRACE gravity field modelingwith an investigation on correlation between nuisance parametersand gravity field coefficients.Adv.Space Res.,47(10):1833-1850.doi:10.1016/j.asr.2010.11.041.
Zheng W.2007.Theory and methodology of earth′s gravitationalfield recovery based on satellite gravity measurement[Ph.D.thesis](in Chinese).Wuhan:Huazhong University of Scienceand Technology.
Zhou X H.2005.Study on satellite gravity and its application[Ph.D.thesis](in Chinese).Wuhan:Institute of Geodesy andGeophysics,Chinese Academy of Sciences.
Zou X C.2007.Theory of satellite orbit and earth gravity fielddetermination[Ph.D.thesis](in Chinese).Wuhan:WuhanUniversity.
罗佳.2003.利用卫星跟踪卫星确定地球重力场的理论和方法[博士论文].武汉:武汉大学.
冉将军.2014.低低跟踪模式重力卫星反演理论方法及应用[博士论文].武汉:中国科学院测量与地球物理研究所.
冉将军,许厚泽,沈云中等.2012.新一代GRACE重力卫星反演地球重力场的预期精度.地球物理学报,55(9):2898-2908,doi:10.6038/j.issn.0001-5733.2012.09.009.
冉将军,许厚泽,钟敏等.2014.利用GRACE重力卫星观测数据反演全球时变地球重力场模型.地球物理学报,57(4):1032-1040,doi:10.6038/cjg20140402.
王正涛.2005.卫星跟踪卫星测量确定地球重力场的理论与方法[博士论文].武汉:武汉大学.
游为.2011.应用低轨卫星数据反演地球重力场模型的理论和方法[博士论文].成都:西南交通大学.
张兴福.2007.应用低轨卫星跟踪数据反演地球重力场模型[博士论文].上海:同济大学.
郑伟.2007.基于卫星重力测量恢复地球重力场的理论和方法[博士论文].武汉:华中科技大学.
周旭华.2005.卫星重力及其应用研究[博士论文].武汉:中国科学院测量与地球物理研究所.
邹贤才.2007.卫星轨道理论与地球重力场模型的确定[博士论文].武汉:武汉大学.
Bettadpur S.2009.Recommendation for a-priori bias&scaleparameters for Level-1BACC data(version 2).GRACE TN-02Version2.
Bettadpur S.2012.UTCSR Level-2processing stands document forLevel-2 product release 005.Center for Space Research,University of Texes at Austin.
Biancale R,Bode A.2006.Mean annual and seasonal atmospherictide models based on 3-hourly and 6-hourly ECMWF surfacepressure data.Potsdam:Deutsches GeoForschungsZentrumGFZ,Scientific Technical Report STR 06/01,doi:10.2312/GFZ.b103-06011.
Bruinsma S,Lemoine J,Biancale R,et al.2010.CNES/GRGS 10-day gravity field models(release 2)and their evaluation.Adv.Space Res.,45(4):587-601,doi:10.1016/j.asr.2009.10.012.
Case K,Kruizinga G,Wu S.2010.GRACE Level 1Bdata productuser handbook.Jet Propulsion Laboratory,California Instituteof Technology.
Cheng M K,Tapley B D.2004.Variations in the earth's oblatenessduring the past 28years.J.Geophys.Res.,109(B9):B09402,doi:10.1029/2004JB003028.
Dahle C,Flechtner F,Gruber C,et al.2013.GFZ GRACE Level-2processing standards document for Level-2product release 0005:revised edition.Potsdam:Deutsches GeoForschungsZentrumGFZ,doi:10.2312/GFZ.b103-1202-25.
Desai S D.2002.Observing the pole tide with satellite altimetry.J.Geophys.Res.,107(C11):7-1-7-13,doi:10.1029/2001JC001224.
Feng W,Zhong M,Lemoine J M,et al.2013.Evaluation ofgroundwater depletion in North China using the GravityRecovery and Climate Experiment(GRACE)data and groundbased measurements.Water Resour.Res.,49(4):2110-2118,doi:10.1002/wrcr.20192.
Flechtner F,Dobslaw H.2013.GRACE AOD1B productdescription document for product release 05.GFZ GermanResearch Centre for Geosciences.
Geruo A,Wahr J,Zhong S J.2013.Computations of theviscoelastic response of a 3-D compressible Earth to surfaceloading:an application to Glacial Isostatic Adjustment inAntarctica and Canada.Geophys.J.Int.,192(2):557-572,doi:10.1093/gji/ggs030.
Jggi A,Beutler G,Meyer U,et al.2012.AIUB-GRACE02S:Status of GRACE gravity field recovery using the celestialmechanics approach.//Kenyon S,Pacino M C,Marti U eds.Geodesy for Planet Earth,International Association of GeodesySymposia 136.Berlin Heidelberg:Springer,doi:10.1007/978-3-642-20338-1_20.
Jekeli C.1981.Alternative methods to smooth the earth's gravityfield.Report No.327,Reports of the Department of GeodeticScience and Surveying.Columbus:The Ohio State University.
Kim J.2000.Simulation study of a low-low satellite-to-satellitetracking mission[Ph.D.thesis].Texas:The University ofTexas at Austin.
Liu X.2008.Global gravity field recovery from satellite-to-satellitetracking data with the acceleration approach[Ph.D.thesis].Delft:Delft University of Technology.
Liu X,Ditmar P,Siemes C,et al.2010.DEOS Mass Transportmodel(DTM-1)based on GRACE satellite data:methodologyand validation.Geophys.J.Int.,181(2):769-788,doi:10.1111/j.1365-246X.2010.04533.x.
Luo J.2003.Theory and methodology of earth gravity fielddetermination using satellite-to-satellite tracking[Ph.D.thesis](in Chinese).Wuhan:Wuhan University.
Mayer-Gürr T.2006.Gravitationsfeldbestimmung aus der Analysekurzer Bahnboegen am Beispiel der Satellitenmissioncn CHAMPund GRACE[Ph.D.thesis].Bonn:Institute für TheoretischeGeodsie der Universitt Bonn.
Ran J.2014.Theory,methodology and application of recovery usinglow-low tracking gravity satellite data[Ph.D.thesis](inChinese).Wuhan:Institute of Geodesy and Geophysics.Chinese Academy Sciences.
Ran J J,Xu H Z,Shen Y Z,et al.2012.Expected accuracy of theglobal gravity field for next GRACE satellite gravity mission.Chinese J.Geophsys.(in Chinese),55(9):2898-2908,doi:10.6038/j.issn.0001-5733.2012.09.009.
Ran J J,Xu H Z,Zhong M,et al.2014.Global temporal gravityfiled recovery using GRACE data.Chinese J.Geophysics.(inChinese),57(4):1032-1040,doi:10.6038/cjg20140402.
Rieser D,Mayer-Gürr T,Savcenko R,et al.2012.The ocean tidemodel EOT11ain spherical harmonics representation.TechnicalNote.
Rodell M,Velicogna I,Famiglietti J S.2009.Satellite-basedestimates of groundwater depletion in India.Nature,460(7258):999-102,doi:10.1038/nature08238.
Shen Y Z,Chen Q J,Hsu H,et al.2013.A modified short arcapproach for recovering gravity field model.Presented at theGRACE Science Team Meeting,University of Texas,Austin,Oct.22-25.
Swenson S,Wahr J.2006.Post-processing removal of correlatederrors in GRACE data.Geophys.Res.Lett.,33(8):L08402,doi:10.1029/2005GL025285.
Swenson S,Chambers D,Wahr J.2008.Estimating geocentervariations from a combination of GRACE and ocean modeloutput.J.Geophys.Res.,113(B8):B08410,doi:10.1029/2007JB005338.
Tapley B,Ries J,Bettadpur S,et al.2005.GGM02—An improvedearth gravity field model from GRACE.J.Geod.,79(8):467-478,doi:10.1007/s00190-005-0480-z.
Tiwari V M,Wahr J,Swenson S.2009.Dwindling groundwaterresources in northern India,from satellite gravity observations.Geophys.Res.Lett.,36(18):L18401,doi:10.1029/2009GL039401.
Wahr J,Molenaar M,Bryan F.1998.Time variability of the Earth′sgravity field:Hydrological and oceanic effects and their possibledetection using GRACE.J.Geophys.Res.,103(B12):30205-30229.
Wang Z T.2005.Theory and methodology of earth gravity fieldrecovery by satellite-to-satellite tracking data[Ph.D.thesis](inChinese).Wuhan:Wuhan University.
Watkins M.2012.JPL Level-2 processing stands document forLevel-2product release 005,Jet Propulsion Laboratory.
Yi S,Sun W K.2014.Evaluation of glacier changes in highmountain Asia based on 10 year GRACE RL05 models.J.Geophys.Res.Solid Earth,119(3):2504-2517,doi:10.1002/2013JB010860.
You W.2011.Theory and methodology of earth′s gravitational field
model recovery by LEO data[Ph.D.thesis](in Chinese).Chengdu:Southwest Jiaotong University.
Zhang X F.2007.The earth's field model recovery on the basis ofsatellite-to-satellite tracking missions[Ph.D.thesis](in Chinese).Shanghai:Tongji University.
Zhao Q,Guo J,Hu Z,et al.2011.GRACE gravity field modelingwith an investigation on correlation between nuisance parametersand gravity field coefficients.Adv.Space Res.,47(10):1833-1850.doi:10.1016/j.asr.2010.11.041.
Zheng W.2007.Theory and methodology of earth′s gravitationalfield recovery based on satellite gravity measurement[Ph.D.thesis](in Chinese).Wuhan:Huazhong University of Scienceand Technology.
Zhou X H.2005.Study on satellite gravity and its application[Ph.D.thesis](in Chinese).Wuhan:Institute of Geodesy andGeophysics,Chinese Academy of Sciences.
Zou X C.2007.Theory of satellite orbit and earth gravity fielddetermination[Ph.D.thesis](in Chinese).Wuhan:WuhanUniversity.
罗佳.2003.利用卫星跟踪卫星确定地球重力场的理论和方法[博士论文].武汉:武汉大学.
冉将军.2014.低低跟踪模式重力卫星反演理论方法及应用[博士论文].武汉:中国科学院测量与地球物理研究所.
冉将军,许厚泽,沈云中等.2012.新一代GRACE重力卫星反演地球重力场的预期精度.地球物理学报,55(9):2898-2908,doi:10.6038/j.issn.0001-5733.2012.09.009.
冉将军,许厚泽,钟敏等.2014.利用GRACE重力卫星观测数据反演全球时变地球重力场模型.地球物理学报,57(4):1032-1040,doi:10.6038/cjg20140402.
王正涛.2005.卫星跟踪卫星测量确定地球重力场的理论与方法[博士论文].武汉:武汉大学.
游为.2011.应用低轨卫星数据反演地球重力场模型的理论和方法[博士论文].成都:西南交通大学.
张兴福.2007.应用低轨卫星跟踪数据反演地球重力场模型[博士论文].上海:同济大学.
郑伟.2007.基于卫星重力测量恢复地球重力场的理论和方法[博士论文].武汉:华中科技大学.
周旭华.2005.卫星重力及其应用研究[博士论文].武汉:中国科学院测量与地球物理研究所.
邹贤才.2007.卫星轨道理论与地球重力场模型的确定[博士论文].武汉:武汉大学.