Abstract
Thin-plate splines — well known for their flexibility and fidelity in representing experimental data — are especially suited for the numerical evaluation of geodetic integrals in the area where these are most sensitive to the data, i.e. in the immediate vicinity of the computation point. Quadrature rules that are exact for thin-plate splines interpolating randomly spaced data are derived for the inner zone contribution (to a planar approximation) to Stokes's formula, to the formulae of Vening Meinesz and to theL 1 gradient operator in the analytical continuation solution of Molodensky's problem.
The quadrature method is demonstrated by calculating the inner zone contribution to height anomalies in a mountainous area of Lesotho and carrying out a comparison with GPS-derived heights. Height anomalies are recovered with an accuracy of 6 cm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burley, AJ, GS Kimbell, DJ Patrick, G Turnbull and R Kashambuzi (1982) A Gravity Survey of Lesotho, Overseas Geology and Mineral Resources Report No 60, HMSO, London
Dent MC, SD Lynch and RE Schulze (1989) Mapping Mean Rainfall and Other Rainfall Statistics over Southern Africa. WRC Report No 109/1/89, Water Research Commission, Pretoria
Freeden W (1981) On approximation by harmonic splines. Manuscripta Geodaetica 6(2):193–244
Greggor KN (1991) Report on a GPS Check Survey of the Tunnel Route within Lesotho on the LHWP Project — April/May 1991. Unpublished report, Department of Surveying and Mapping, University of Pretoria
Heiskanen WA and H Moritz (1967) Physical Geodesy, WH Freeman, San Francisco
Kearsley AHW (1976) The computation of deflections of the vertical from gravity anomalies. Reports from the School of Surveying Unisurv S-15, University of New South Wales, Sydney
Kearsley AHW (1986) The determination of precise geoid height differences using ring integration. Bollettino di Geodesia e Scienze Affini 45(2):151–174
Meinguet J (1979) Multivariate interpolation at arbitrary points made simple, Journal of Applied Mathematics and Physics (ZAMP) 30:292–304
Merry CL (1977) Gravity and the South African height system, South African Survey Journal, 16(1):44–53
Moritz H (1980) Advanced Physical Geodesy, Herbert Wichmann, Karlsruhe
Rapp RH (1981) The Earth's Gravity Field to Degree and Order 180 Using Seasat Altimeter Data, Terrestrial Gravity Data and Other Data. Report No 322, Dept of Geodetic Science and Surveying, Ohio State University
Schwarz KP, MG Sideris and R Forsberg (1990) The use of FFT techniques in physical geodesy, Geophysical Journal International 100:485–514
Spiegel MR (1968) Mathematical Handbook of Formulas and Tables, Schaum's Outline Series, McGraw-Hill, New York
Van Gysen H (1988a) Splines and local approximation of the earth's gravity field, PhD thesis, University of Cape Town.
Van Gysen H (1988b) Thin-plate spline quadrature of geodetic integrals, presented at the Chapman Conference on Progress in the Determination of the Earth's Gravity Field, Fort Lauderdale, Florida, September 1988
Van Gysen H and CL Merry (1987) The Gravity Field in Southern Africa. Technical Report TR-3, Dept of Surveying, University of Cape Town
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van Gysen, H. Thin-plate spline quadrature of geodetic integrals. Bulletin Géodésique 68, 173–179 (1994). https://doi.org/10.1007/BF00808291
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00808291