Abstract
A general formula is developed and presented for transformations among geoidal undulation, gravity anomaly, gravity disturbance and other gravimetric quantities. Using a spectral form of the general formula, a criterion has been built in order to classify these transformations into forward and inverse transformations in this paper. Then, the two-dimensional convolution techniques are applied to the general formula to deal with the forward transformation while the two-dimensional deconvolution techniques are employed to treat the inverse transformation and evaluate the inverse general formula. Concepts of convolution and deconvolution are also reviewed in this paper. The stability and edge effect problems related to the deconvolution techniques are investigated using simulated data and numerical tests are done to quantify the stability of the deconvolution techniques for estimated gravity information. Finally, the marine gravity information for the Norwegian-Greenland Sea area has been derived from ERS-1 altimetry data using the deconvolution techniques.
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Balmino, G., Moynot, B., Sarrailh, M., Vales, N. [1987] Free Air Gravity Anomalies over the Oceans from SEASAT and GEOS-3 Altimetry Data. EOS Trans. AGU, 68, 17–19.
Blais, JAR [1993] Generalized Inverse Problems and Information Theory. Presentation at the International Association of Geodesy General Meeting, Beijing, China. Submitted for publication in Manuscripta Geodaetica.
Brigham, EO [1988] The Fast Fourier Transform and its Applications. Prentice-Hall, Inc.
Cordell, L., Grauch, VJS [1982] Reconciliation of the Discrete and Integral Fourier Transforms. Geophysics, Vol. 47, No., 2, 237–243.
Forsberg, R. [1984] A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modeling Report No. 355, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio.
Haagmans, R., de Min E, van Gelderen M. [1993] Fast Evaluation of Convolution Integrals on the Sphere using 1D-FFT and a Comparison with Existing Methods for Stokes' Integral. Manuscripta Geodaetica, 18: 227–241.
Haxby, WF [1987] Gravity Field of the World's Oceans, National Geophysical Data Center, NOAA, Boulder, CO.
Heck, B. [1979] Zur loken Geoidbestimmung aus terrestrischen Messungen vertikaler Schweregradienten. DGK, Reihe C, Heft Nr.259, Munchen.
Heiskanen, WA, Moritz, H [1967] Physical Geodesy. W.H. Freeman, San Francisco.
Hotine, M [1969] Mathematical Geodesy. ESSA Monograph No. 2, U. S. Department of Commerce, Washington, D. C.
Jordan, SK [1972] Self-consistent Statistical Models for the Gravity Anomaly, Vertical Deflections, and Undulation of the Geoid. Journal of Geophysical Research, Vol. 77, 3660–3670.
Jekeli, C. [1979] Global Accuracy Estimates of Point and Mean Undulation Differences Obtained from Gravity Disturbances, Gravity Anomalies and Potential Coefficients. Report No. 288, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio
Kanasewich, ER [1981] Time Sequence Analysis in Geophysics, 3rd edn. The University of Alberta Press, Edmonton, Alberta.
Knudsen, P., Baltazar, A., Tscherning, C. [1992] Altimetric Gravity Anomalies in the Norwegian-Greenland Sea — Preliminary Results from the ERS-1 35 Days Repeat Mission. Geophysical Research Letter, Vol. 19, No. 17, 1795–1798.
Mainville, A., Veronneau, M. [1990] Orthometric Heights Using GPS in Canada. 106 International Association of Geodesy Symposia, Determination of the Geoid, Present and Future.
Meissl, P. [1971] A Study of Covariance Functions Related to the Earth's Disturbing Potential. Report No. 151, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio.
Milbert, D.G. [1991] GEOID90: A Height-Resolution Geoid for the United States. EOS Trans. AGU, 72, 49–60
Molodenskij, MS, Eremeev, VF, Yurkina, MI [1962] Methods for Study of the External Gravitational Field and Figure of the Earth. Translated from Russian by the Israel Program for Scientific Translations for the Office of Technical Service, U. S. Department of Commerce, Washington, D. C., U. S. A, 1962.
Schwarz, K.P., Sideris, M.G., Forsberg, R. [1990] The Use of FFT Techniques in Physical Geodesy. Geophys. J. Int, 100, 485–514.
Sideris, MG [1984] Computation of Gravimetric Terrain Corrections Using Fast Fourier Transform Techniques. UCSE Report # 20007, Department of Geomatics Engineering, University of Calgary, Calgary, Alberta.
Sideris, MG., Li, Y.C [1993] Gravity Field Convolution without Windowing and Edge Effects. Bulletin Geodesique, 67: 107–118.
Sjoberg, LE, Fan, H. [1986] A Comparison of the Modified Stokes' Formula and Hotine's Formula in Physical Geodesy. The Department of Geodesy. Report No. 4, The Royal Institute of Technology, Stockholm.
Strang van Hees, G. [1990] Stokes Formula Using Fast Fourier Techniques. Manuscripta Geodaetica, 15: 235 - 239.
Vanicek, P., Krakiwsky, EJ [1982] Geodesy: the Concepts. North-Holland, Amsterdam.
Vanicek, P., Zhang, C., Sjoberg, LE [1992] A Comparison of Stokes's and Hotine's Approaches to Geoid Computation. Manuscripta Geodaetica, 17: 29–35.
Webster, GM [1978] Deconvolution. the Society of Exploration Geophysicists, Tulsa, Oklahoma.
Zhang, C., Blais, JAR [1993] Recovery of Gravity Disturbances from Satellite Altimetry by FFT Techniques: a Synthetic Study. Manuscripta Geodaetica, 18: 158–170.
Zhang, C. [1993] Recovery of Gravity Information from Satellite Altimetry and Associated Forward Geopotential Models. Ph.D. Dissertation, Department of Geomatics Engineering, University of Calgary, Calgary, Alberta.
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Zhang, C. A general formula and its inverse formula for gravimetric transformations by use of convolution and deconvolution techniques. Journal of Geodesy 70, 51–64 (1995). https://doi.org/10.1007/BF00863418
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DOI: https://doi.org/10.1007/BF00863418