Regional gravity modeling in terms of spherical base functions

Journal of Geodesy - Tập 81 - Trang 17-38 - 2006
Michael Schmidt1, Martin Fengler2, Torsten Mayer-Gürr3, Annette Eicker3, Jürgen Kusche4, Laura Sánchez1, Shin-Chan Han5
1Deutsches Geodätisches Forschungsinstitut (DGFI), Munich, Germany
2Geomathematics Group, Technical University of Kaiserslautern, Kaiserslautern, Germany
3Institute of Theoretical Geodesy, University of Bonn, Bonn, Germany
4Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Delft, The Netherlands
5Geodetic Science, Department of Geological Sciences, Ohio State University, Columbus, USA

Tóm tắt

This article provides a survey on modern methods of regional gravity field modeling on the sphere. Starting with the classical theory of spherical harmonics, we outline the transition towards space-localizing methods such as spherical splines and wavelets. Special emphasis is given to the relations among these methods, which all involve radial base functions. Moreover, we provide extensive applications of these methods and numerical results from real space-borne data of recent satellite gravity missions, namely the Challenging Minisatellite Payload (CHAMP) and the Gravity Recovery and Climate Experiment (GRACE). We also derive high-resolution gravity field models by effectively combining space-borne and surface measurements using a new weighted level-combination concept. In addition, we outline and apply a strategy for constructing spatio-temporal fields from regional data sets spanning different observation periods.

Tài liệu tham khảo

Bjerhammer A (1967) On the energy integral for satellites. Rep. of the R. Inst. of Techn. Sweden, Stockholm Cui J (1995) Finite pointset methods on the sphere and their application in physical geodesy. PhD thesis, University of Kaiserslautern, Mathematics Department, Geomathematics Group Driscoll JR, Healy RM (1994) Computing Fourier transforms and convolutions on the 2-sphere. Adv Appl Math 15:202–250 Eicker A, Mayer-Gürr T, Ilk K-H (2006) A global CHAMP gravity field by merging of regional refinement patches. Adv Geosci (submitted) (Proceedings of the Joint CHAMP/GRACE Science Meeting) Fengler MJ (2005) Vector spherical harmonic and vector wavelet based non-linear Galerkin schemes for solving the incompressible Navier-Stokes equation on the sphere. PhD thesis (submitted), University of Kaiserslautern, Mathematics Department, Geomathematics Group Fengler MJ, Freeden W, Gutting M (2004a) The Kaiserslautern geopotential model SWITCH-03 from Orbit Pertubations of the Satellite CHAMP and its comparison to the models EGM96, UCPH2002_02_0.5, EIGEN-1S, and EIGEN-2. Geophys J Int 157:499–514 Fengler MJ, Freeden W, Kusche J (2004b) Multiscale geopotential solutions from CHAMP orbits and accelerometry. In: Reigber C, Lühr H, Schwintzer P, Wickert J (eds) Earth observation with CHAMP, results from three years in orbit. Springer, Berlin Heidelberg New York, pp 139–144 Fengler MJ, Freeden W, Gutting M (2005) The spherical Bernstein wavelet. Schriften zur Funktionalanalysis und Geomathematik, 20, University of Kaiserslautern, Mathematics Department, Geomathematics Group Fengler MJ, Michel D, Michel V (2006a) Harmonic spline-wavelets on the 3-dimensional ball and their application to the reconstruction of the Earth’s density distribution from gravitational data at arbitrarily shaped orbits. ZAMM (accepted) Fengler MJ, Freeden W, Kohlhaas A, Michel V, Peters T (2006b) Wavelet modelling of regional and temporal variations of the Earth’s gravitational potential observed by GRACE. J Geod (accepted) Förste C, Flechtner F, Schmidt R, Meyer U, Stubenvoll R, Barthelmes F, König R, Neumayer K-H, Rothacher M, Reigber Ch, Biancale R, Bruinsma S, Lemoine J-M, Raimondo JC (2005) A new high resolution global gravity field model derived from combination of GRACE and CHAMP mission and altimetry/gravimetry surface gravity data. Presented at EGU General Assembly 2005, Vienna, Austria Freeden W (1981) On approximation by harmonic splines. manuscr geod 6:193–244 Freeden W (1999) Multiscale modelling of spaceborne geodata. Teubner, Stuttgart Freeden W, Schreiner M (1995) New wavelet methods for approximating harmonic functions. In: Sansò F (ed) Geodetic theory today. Springer, Berlin Heidelberg New York, pp 112–121 Freeden W, Windheuser U (1996) Spherical wavelet transform and its discretization. Adv Comput Math 5:51–94 Freeden W, Michel V (2004) Multiscale potential theory (with applications to Earth’s sciences). Birkhäuser Verlag, Boston Freeden W, Gervens T, Schreiner M (1998a) Constructive approximation on the sphere (with applications to geomathematics). Clarendon Press, Oxford Freeden W, Glockner O, Schreiner M (1998b) Spherical panel clustering and its numerical aspects. J Geod 72:586–599 Freeden W, Michel D, Michel V (2005) Local multiscale approximations of geostrophic flow: theoretical background and aspects of scientific computing. Mar Geod 28:313–329 Han SC (2003) Efficient global gravity determination from satellite-to-satellite tracking (SST). PhD thesis, Geodetic and Geoinformation Science, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, Columbus Han SC, Shum CK, Jekeli C, Alsdorf D (2005) Improved estimation of terrestrial water storage changes from GRACE. Geophys Res Lett 32:L07302. DOI 10.1029/2005GL022382 Han SC, Shum CK, Jekeli C (2006) Precise estimation of in situ geopotential differences from GRACE low-low satellite-to-satellite tracking and accelerometry data. J Geophys Res 111:B4411. DOI 10.1029/2005JB003719 Heiskanen W, Moritz H (1967) Physical geodesy. Freeman, San Francisco Holschneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys Earth Planet Int 135:107–124 Ilk KH, Löcher A (2005) The use of the energy balance relations for validation of gravity field models and orbit determination. In: Sansò F (ed) A window on the future of geodesy. Springer, Berlin Heidelberg New York, pp 494–499 Jekeli C (1981) Alternative methods to smooth the Earth’s gravity field. Report 327, Department of Geodetic Science, The Ohio State University, Columbus Jekeli C (1999) The determination of gravitational potential differences from satellite-to-satellite tracking. Cel Mech Dyn Astr 75:85–100 Kaula WM (1966) Theory of satellite geodesy. Blaisdell, Waltham Kern M, Schwarz KP, Sneeuw N (2003) A study on the combination of satellite, airborne, and terrestrial gravity data. J Geod 77:217–225 Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, Berlin Heidelberg New York Koch KR, Kusche J (2001) Regularization of geopotential determination from satellite data by variance components. J Geod 76:259–268 Kusche J (2002) Inverse Probleme bei der Gravitationsfeldbestimmung mittels SST- und SGG-Satellitenmissionen. German Geodetic Commission, Series C, 548, Munich Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, NASA/TP-1998-206861, National Aeronautics and Space Administration, Maryland Li TH (1999) Multiscale representation and analysis of spherical data by spherical wavelets. SIAM J Sci Comput 21:924–953 van Loon J, Kusche J (2005) Stochastic model validation of satellite gravity data: A test with CHAMP pseudo-observations. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity, geoid and space missions. Springer, Berlin Heiodelbertg New York, pp 24–29 Maier T (2005) Wavelet-Mie-representations for solenoidal vector fields with applications to ionospheric geomagnetic data. J Appl Math 65:1888–1912 Marchenko AN (1998) Parameterization of the Earth’s gravity field – Point and line singularities. Lviv Astronomical and Geodetic Society, Lviv Mautz R, Schaffrin B, Shum CK, Han SC (2004) Regional geoid undulations from CHAMP represented by locally supported basis functions. In: Reigber C, Lühr H, Schwintzer P, Wickert J (eds) Earth observation with CHAMP, results from three years in orbit. Springer, Berlin Heidelberg New York, pp 230–236 Mayer C (2004) Wavelet modelling of the spherical inverse source problem with application to geomagnetism. Inverse problems 20:1713–1728 Mayer-Gürr T, Ilk KH, Eicker A, Feuchtinger M (2005) ITG-CHAMP01: a CHAMP gravity field model from short kinematical arcs of a one-year observation period. J Geod 78:462–480 Mayer-Gürr T, Eicker A, Ilk KH (2006) Gravity field recovery from GRACE-SST data of short arcs. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the Earth system from Space. Springer, Berlin Heidelberg New York, pp 131–148 Mertins A (1999) Signal analysis: wavelets, filter banks, time-frequency transforms and applications. Wiley, Chichester Moritz H (1980) Advanced physical geodesy. Wichmann, Karlsruhe Narcowich FJ, Ward JD (1996) Nonstationary wavelets on them-sphere for scattered data. Appl Comput Harmon Anal 3:324–336 Panet I, Jamet O, Diament M, Chambodut A (2005) Modelling the Earth’s gravity field using wavelet frames. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity, geoid and space missions. Springer, Berlin Heidelberg New York, pp 48–53 Pavlis NK, Holmes SA, Kenyon SC, Schmidt D, Trimmer R (2005) A preliminary gravitational model to degree 2160. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity, geoid and space missions. Springer, Berlin Heidelberg New York, pp 18–23 Reigber C, Schmidt R, Flechtner F, König R, Meyer U, Neumayer KH, Schwintzer P, Zhu SY (2005) An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. J Geodyn 39(1):1–10 Sánchez L (2003) Bestimmung der Höhenreferenzfläche für Kolumbien. Diploma thesis. Institute of Planetary Geodesy, Technical University of Dresden Sandwell DT, Smith WHF (1997) Marine gravity anomaly from Geosat and ERS-1 satellite altimetry. J Geophys Res 102(B5):10039–10054 (http://topex.ucsd.edu/www_html/ mar_grav.html) Sansò F, Tscherning CC (2003) Fast spherical collocation: theory and examples. J Geod 77:101–112. DOI 10.1007/s00190-002-0310-5 Schmidt M, Fabert O, Shum CK (2005a) Towards the estimation of a multi-resolution representattion of the gravity field based on spherical wavelets. In: Sansò F (ed) A window on the future of geodesy. Springer, Berlin Heidelberg New York, pp 362–367 Schmidt M, Kusche J, van Loon J, Shum CK, Han SC, Fabert O (2005b) Multi-resolution representation of regional gravity data. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity, Geoid and space missions. Springer, Berlin Heidelberg New York, pp 167–172 Schmidt M, Fabert O, Shum CK (2005c) On the estimation of a multi-resolution representation of the gravity field based on spherical harmonics and wavelets. J Geodyn 39:512–526 Schmidt M, Han SC, Kusche J, Sánchez L, Shum CK (2006) Regional high-resolution spatiotemporal gravity modeling from GRACE data using spherical wavelets. Geophys Res Lett 33:L08403. DOI 10.1029/2005GL025509 Schneider F (1996) The solution of linear inverse problems in satellite geodesy by means of spherical spline approximation. J Geod 71(1):2–15 Swenson S, Wahr J (2002) Methods for inferring regional surface-mass anomalies from satellite measurements of time variable gravity. J Geophys Res 107(B9):2193. DOI 10.1029/2001JB000576 Tapley BD, Bettadpur S, Watkins M, Reigber C (2004a) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31:L06619. DOI 10.1029/2003GL019285 Tapley BD, Bettadpur S, Ries J, Thompson P, Watkins M (2004b) GRACE measurements of mass variability in the Earth system. Science 305:503–505 Torge W (2001) Geodesy. de Gruyter, Berlin Wahr J, Molenaar M, Bryan F (1998) Time variability of the earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103(B12):30205–30229 Wang Y-M (1993) On the optimal combination of potential coefficient with terrestrial gravity data for FFT computations. manuscr geod 18:406–416