Abstract
Kaula’s rule of thumb has been used in producing geopotential models from space geodetic measurements, including the most recent models from satellite gravity missions CHAMP. Although Xu and Rummel (Manuscr Geod 20 8–20, 1994b) suggested an alternative regularization method by introducing a number of regularization parameters, no numerical tests have ever been conducted. We have compared four methods of regularization for the determination of geopotential from precise orbits of COSMIC satellites through simulations, which include Kaula’s rule of thumb, one parameter regularization and its iterative version, and multiple parameter regularization. The simulation results show that the four methods can indeed produce good gravitational models from the precise orbits of centimetre level. The three regularization methods perform much better than Kaula’s rule of thumb by a factor of 6.4 on average beyond spherical harmonic degree 5 and by a factor of 10.2 for the spherical harmonic degrees from 8 to 14 in terms of degree variations of root mean squared errors. The maximum componentwise improvement in the root mean squared error can be up to a factor of 60. The simplest version of regularization by multiplying a positive scalar with a unit matrix is sufficient to better determine the geopotential model. Although multiple parameter regularization is theoretically attractive and can indeed eliminate unnecessary regularization for some of the harmonic coefficients, we found that it only improved its one parameter version marginally in this COSMIC example in terms of the mean squared error.
Similar content being viewed by others
References
Barry D (1986) Nonparametric Bayesian regression. Ann Statist 14:934–953
Bouman J (1998) Quality of regularization methods. DEOS Report No.98.2, Delft University Press, Delft
Bouman J (2000) Quality assessment of satellite-based global gravity field models. PhD Dissertation, Delft University of Technology, Delft
Chao BF, Palvis EC, Hwang C, Liu C-C, Shum CK, Tseng C-L, Yang M (2000) COSMIC: geodetic applications in improving Earth’s gravity model. Terrest Atmos Ocean Sci 11:365–378
Cook AH (1963) The contribution of observations of satellites to the determination of the Earth’s gravitational potential. Space Sci Rev 2:355–437
Ditmar P, Kusche J, Klees R (2003) Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues. J Geod 77:465–477
Douglas BC, Goad CC, Morrison FF (1980) Determination of the geopotential from satellite-to-satellite tracking data. J Geophys Res B85:5471–5480
Fletcher R (2000) Practical methods of optimization. 2nd edn. Wiley, Chichester
Garcia RV, Jekeli C (2002). On the inversion of GRACE observables for local geoid determination. In: Ádám J, Schwarz K-P (eds). Vistas for geodesy in the new millenium. Springer, Berlin Heidelberg New York, pp 156–161
Gruber M (1998) Improving efficiency by shrinkage. Marcel Dekker, New York
Guier WH (1963) Determination of the non-zonal harmonics of the geopotential from satellite Doppler data. Nature 200:124–125
Hansen PC (1990) Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank. SIAM J Sci Comput 11:503–518
Hemmerle W (1975) An explicit solution for generalized ridge regression. Technometrics 17:309–314
Hemmerle W, Brantle TF (1978) Explicit and constrained generalized ridge estimation. Technometrics 20:109–120
Hoerl AE, Kennard RW (1970a) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12:55–67
Hoerl AE, Kennard RW (1970b) Ridge regression: applications to nonorthogonal problems. Technometrics 12:69–82
Hwang C (2001) Gravity recovery using COSMIC GPS data: application of orbital perturbation theory. J Geod 75:117–136
Ilk K (1985) On the regional mapping of gravitation with two satellites. In: Proceedings of the 1st Hotine-Marussi symposium on mathematical geodesy, pp 807–831
Izsak IG (1963) Tesseral harmonics in the geopotential. Nature 199:137–139
Izsak IG (1964) Tesseral harmonics of the geopotential and corrections to station coordinates. J Geophys Res B69:2621–2630
Jekeli C (1999) The determination of gravitational potential differences from satellite-to-satellite tracking. Celes Mech Dynam Astron 75:85–101
Jekeli C, Upadhyay TN (1990) Gravity estimation from STAGE, a satellite-to-satellite tracking mission. J Geophys Res B95:10973–10985
Kaula WM (1961) A geoid and world geodetic system based on a combination of gravimetric, astrogeodetic and satellite data. J Geophys Res B66:1799–1811
Kaula WM (1966) Theory of satellite geodesy. Blaisdell, London
King-Hele DG (1958) The effect of the earth’s oblateness on the orbit of a near satellite. Proc R Soc Lond A247:49–72
Koch K-R, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76:259–268
Koop R (1993) Global gravity field modelling using satellite gravity gradiometry. Netherlands Geodetic Commission, Publication Geodesy (New Series), No. 38, Delft
Kozai Y (1966) The Earth gravitational potential derived from satellite motion. Space Sci Rev 5:818–879
Kusche J (2002) Inverse Probleme bei der Gravitationsfeldbestimmung mittels SST- und SGG-Satellitenmissiones, Deutsche Geodätische Kommission, Nr.548, München
Kusche J, Klees R (2002) Regularization of gravity field estimation from satellite gravity gradients. J Geod 76:359–368
Marsh JG, Lerch FJ, Putney BH, Christodoulidis DC, Smith DE, Felsentreger TL, Sanches BV, Klosko SM, Pavlis EC, Martin TV, Robbins JW, Williamson RG, Colombo OL, Rowlands DD, Eddy W, Chandler NL, Rachlin KE, Patel GB, Bhati S, Chinn D (1988) A new gravitational model for the Earth from satellite tracking data: GEM-T1. J Geophys Res B93:6169–6215
Marsh JG, Lerch FJ, Putney BH, Felsentreger TL, Sanches B, Klosko SM, Patel GB, Robbins JW, Williamson RG, Engelis TL, Eddy W, Chandler NL, Chinn DS, Kapoor S, Rachlin KE, Braatz L, Pavlis EC (1990) The GEM-T2 gravitational model. J Geophys Res B95:22043–22071
Merson RH, King-Hele DG (1958) Use of artificial satellites to explore the Earth’s gravitational field: results from SPUTNIK 2 (1957β). Nature 182:640–641
Montenbruck O, Gill E (2001) Satellite orbits – models, methods, applications. Springer, Berlin Heidelberg New York
Moritz H (1980) Advanced physical geodesy. Herbert Wichmann Verlag, Karlsruhe
Morozov V (1984) Methods for solving incorrectly posed problems. Springer, Berlin Heidelberg, New York
Phillips DL (1962) A technique for the numerical solution of certain integral equations of the first kind. J Assoc Comput Mach 9:84–97
Reigber C, Balmino G, Schwintzer P, Biancale R, Bode A, Lemoine J-M, König R, Loyer S, Neumayer K-H, Marty J-C, Barthelmes F, Perosanz F, Zhu SY (2003) Global gravity field recovery using solely GPS tracking and accelerometer data from CHAMP. Space Sci Rev 108:55–66
Rowlands DD, Ray RD, Chinn DS, Lemoine FG (2002) Short-arc analysis of intersatellite tracking data in a gravity mapping mission. J Geod 76:307–316
Rummel R (1985). From the observational model to gravity parameter estimation. In: Schwarz KP (eds). Local gravity field approximation. The University of Calgary, Calgary, pp. 67–106
Rummel R, van Gelderen M (1995) Meissl scheme – spectral characteristics of physical geodesy. Manus Geod 20:379–385
Rummel R, Schwarz KP, Gerstl M (1979) Least squares collocation and regularization. Bull Geod 53:343–361
Rust BW, Burrus WR (1972) Mathematical programming and the numerical solution of linear equations. Elsevier, New York
Seeber G (2003) Satellite geodesy, 2nd edn. Walter de Gruyter, Berlin
Tarantola A (1987) Inverse problem theory. Elsevier, Amsterdam
Tikhonov AN (1963) Regularization of incorrectly posed problems. Sov Math 4:1624–1627
Tikhonov AN, Arsenin VY (1977) Solutions of Ill-posed Problems. Wiley, New York
Vinod H, Ullah A (1981) Recent advances in regression methods. Marcel Dekker Inc., New York
Visser PNAM (2005) Low-low satellite-to-satellite tracking: a comparison between analytical linear orbit perturbation theory and numerical integration. J Geod 79:160–166
Visser PNAM, van den Ijssel J, Koop R, Klees R (2001) Exploring gravity field determination from orbit perturbations of the European Gravity Mission GOCE. J Geod 75:89–98
Visser PNAM, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates. J Geod 77:207–216
Wagner CA (1983) Direct determination of gravitational harmonics from low-low GRAVSAT data. J Geophys Res B88:10309–10321
Wolff M (1969) Direct measurements of the Earth’s gravitational potential using a satellite pair. J Geophys Res 74:5295–5300
Wood SN (2000) Modelling and smoothing parameter estimation with multiple quadratic penalties. J R Statist Soc B62:413–428
Xu PL (1992a) Determination of surface gravity anomalies using gradiometric observables. Geophys J Int 110:321–332
Xu PL (1992b) The value of minimum norm estimation of geopotential fields. Geophys J Int 111:170–178
Xu PL (1998) Truncated SVD methods for linear discrete ill-posed problems. Geophys J Int 135:505–514
Xu PL, Rummel R (1992) A generalized regularization method with applications in determination of potential fields. In: Holota P, Vermeer M (eds) Proc 1st continental workshop on the geoid in Europe, Prague, pp 444–457
Xu PL, Rummel R (1994a) A simulation study of smoothness methods in recovery of regional gravity fields. Geophys J Int 117:472–486
Xu PL, Rummel R (1994b) A generalized ridge regression method with applications in determination of potential fields. Manus Geod 20:8–20
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, P., Fukuda, Y. & Liu, Y. Multiple Parameter Regularization: Numerical Solutions and Applications to the Determination of Geopotential from Precise Satellite Orbits. J Geodesy 80, 17–27 (2006). https://doi.org/10.1007/s00190-006-0025-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-006-0025-0