Summary
Regional geoid undulations are determined from CHAMP data using various locally supported basis functions to assess their respective efficiency, accuracy and multi-resolution representation properties. These functions include (biharmonic) B-spline tensor wavelets (with or without compression), multiquadrics (with or without flexible centering and predetermined smoothing) and radially symmetric truncated polynomials.
It is concluded that the B-spline wavelet model is the computationally most efficient approach. The non-periodic variation of the B-spline wavelets allows one to handle data on a bounded domain with small edge effects, and the piecewise linear version allows one to model the geoid using a patch-wise approach. The use of multiquadrics without centering in the data points and predetermined smoothing constant allows handling of heterogeneously distributed data using global optimization. The linear multiquadrics model fits the data best when comparing the residuals of different models with a fixed number of unknowns. For an efficient data synthesis the nonlinear models are best suited due to their far smaller number of basis functions. The smoothest surface was obtained using the nonlinear polynomial approach, whereas the multiquadrics show peaks and the wavelet models show horizontal and vertical edges in their representations. The linear B-spline wavelets are biharmonic, and the approach is capable of an efficient multi-resolution representation of regional gravity field models combining satellite (CHAMP, GRACE, GOCE) and in-situ data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Barthelmes F (1986) On the approximation of the outer earth gravity field by point masses with optimised positions (in German). Veröffentlichungen des Zentralinstitut für Physik der Erde, 92, Potsdam, Germany.
Chui CK, Quak E (1992) Wavelets on a bounded interval. in: Braess D, Schumaker L, eds, Numerical Methods in Approximation Theory, Vol 9: 53–77.
Franke R (1982) Scattered data interpolation: Tests of some methods. Math Comps 38(157): 181–199.
Grafarend E, Schaffrin B (1993) Adjustment Computations in Linear Models (in German), Bibliograph. Inst., Mannheim, Germany.
Hales S, Levesley J (2000) Multi-level approximation to scattered data using inverse multiquadrics. in: Cohen A, Rabut C, Schumaker L, eds, Curve and Surface Fitting: SaintMalo 1999, Vanderbilt University Press, Nashville/TN: 247–254.
Han S-C, Jekeli C, Shum CK (2002) Efficient gravity field recovery using in-situ disturbing potential observables from CHAMP. Geophys Res Letters 29(16): 36–41.
Han S, Shum CK, Jekeli C, Braun A, Chen Y, Kuo C (2003) CHAMP mean and temporal gravity field solutions and geophysical constraint studies. 2nd CHAMP Sci Meeting, GeoForschungsZentrum (GFZ), Potsdam, Sept 2003.
Hardy R (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76: 1905–1915.
Hardy R, Göpfert W (1975) Least-squares prediction of gravity anomalies, geoidal undulations, and deflections of the vertical by multiquadric harmonic functions. Geophys Res Letters 2(10): 423–426.
Kaschenz J (2002) Multidimensional spectral analysis with problem-oriented frequencies (in German). Diploma Thesis; Inst of Astronomical, Physical and Math Geodesy, Technical University of Berlin, Germany.
Kaschenz J (2003) Data modelling using various frequency series (in German). Z für Vermessungswesen (ZfV) 128(4): 260–265.
Mautz R (2001) On the determination of frequencies in time series solving nonlinear adjustment problems (in German). German Geodetic Comm, C-532, Munich, Germany.
Mautz R (2002) Solving nonlinear adjustment problems by global optimization. Bollettino di Geodesia e Scienze Affini 61(2): 123–134.
Mautz R, Schaffin B, Schmidt M, Shum CK (2002) Representation of spatial functions in geodesy using B-spline wavelets with compact support. Proc of the Heiskanen Symposium in Geodesy (Jekeli C, Shum CK, eds), Columbus/Ohio, Oct 2002.
Mautz R, Schaffrin B, Kaschenz J (2003) Biharmonic spline wavelets versus generalized multi-quadrics for continuous surface representations. IUGG General Assembly, IAG-Symp G4, Sapporo, Japan, July 2003.
Schaffrin B, Mautz R, Shum CK, Tseng HZ (2003) Towards a spherical pseudo-wavelet basis for geodetic applications. Computer-Aided Civil and Infrastructure Engineering 18(5): 369–378.
Schmidt M, Fabert O, Shum CK (2002) Multi-resolution representation of the gravity field using spherical wavelets. Proc of the Heiskanen Symposium in Geodesy (Jekeli C, Shum CK, eds.), Columbus/Ohio, Oct 2002.
Stollnitz E, DeRose T, Salesin D (1996) Wavelets for Computer Graphics. M aufmann, San Francisco/CA.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mautzl, R., Schaffrin, B., Shum, C.K., Han, SC. (2005). 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. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26800-6_37
Download citation
DOI: https://doi.org/10.1007/3-540-26800-6_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22804-2
Online ISBN: 978-3-540-26800-0
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)