Journal of Geodesy

, Volume 70, Issue 1–2, pp 2–12 | Cite as

Fourier geoid determination with irregular data

  • Michael G. Sideris
Article

Abstract

The fast Fourier transform (FFT) and, recently, the fast Hartley transform (FHT) have been extensively used by geodesists for efficient geoid determination. For this kind of efficiency, data must be given on a regular grid and, consequently, a pre-processing step of interpolation is required when only point measurements are available. This paper presents a way of computing a grid of geoid undulations N without explicitly gridding the data. The method is applicable to all FFT or FHT techniques of geoid or terrain effects determination, and it works with planar as well as spherical formulas. This method can be used not only for, e.g., computing a grid of undulations from irregular gravity anomalies Δg but it also lends itself to other applications, such as the gridding of gravity anomalies and, since the contribution of each data point is computed individually, the update of N- or Δg-grids as soon as new point measurements become available. In the case that there are grid cells which contain no measurements, the results of gravity interpolation or geoid estimation can be drastically improved by incorporating into the procedure a frequency-domain interpolating function. In addition to numerical results obtained using a few simple interpolating functions, the paper presents briefly the mathematical formulas for recovering missing grid values and for transforming values from one grid to another which might be rotated and/or scaled with respect to the first one. The geodetic problems where these techniques may find applications are pointed out throughout the paper.

Keywords

Fourier Transform Fast Fourier Transform Grid Cell Point Measurement Gravity Anomaly 

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References

  1. Blais, J.A.R., 1982, A Synthesis of Kriging Estimation Methods,Manuscripta Geodaetica Vol. 7, pp. 325–352.Google Scholar
  2. Bracewell, R.N., 1986a,The Fast Hartley Transform, Oxford Engineering Science Series No. 19, Oxford University Press.Google Scholar
  3. Bracewell, R.N., 1986b,The Fourier Transform and its Applications, Second edition, revised. McGraw-Hill, New-York.Google Scholar
  4. Cressie, N., 1992,Spatial Data Analysis, John Wiley & Sons, Inc., New York.Google Scholar
  5. Dudgeon, D.E. and Mersereau, R.M., 1984,Multidimensional Digital Signal Processing, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.Google Scholar
  6. Eren, K., 1980, Spectral Analysis of GEOS-3 Altimeter Data and Frequency Domain Collocation, OSU Report No. 297, The Ohio State University, Department of Geodetic Science, Columbus, Ohio.Google Scholar
  7. Haagmans, R., de Min, E. and 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 Vol. 18, pp. 227–241.Google Scholar
  8. Lancaster, P. and Salkauskas, K., 1986,Curve and Surface Fitting — An Introduction, Academic Press, Inc., London.Google Scholar
  9. Li, Y.C. and Sideris, M.G. 1992, The Fast Hartley Transform and Its Application in Physical Geodesy,Manuscripta Geodaetica Vol. 17, pp. 281–387.Google Scholar
  10. Li. Y.C., 1993, Optimized Spectral Geoid Determination, UCGE Report No. 20050, The University of Calgary, Department of Geomatics Engineering, Calgary, Alberta.Google Scholar
  11. Marks II, R.J., 1991,Introduction to Shannon Sampling and Interpolation Theory, Springer-Verlag, New York.Google Scholar
  12. Meskó, A., 1984,Digital Filtering: Applications in Geophysical Exploration for Oil. Akadémiai Kiadó, Budapest.Google Scholar
  13. Oppenheim, A.V. and Schafer, R.W., 1989,Discrete-Time Signal Processing, Prentice Hall, Englewood Cliffs, N.J. 07632.Google Scholar
  14. Schafer, R.W. and Rabiner, L.R., 1973, A Digital Signal Processing Approach to Interpolation,Proceedings of the IEEE Vol. 61, No. 6, pp. 692–702.Google Scholar
  15. Schwarz, K.P., Sideris, M.G. and Forsberg, R., 1990, The Use of FFT techniques in physical geodesy,Geophysical Journal International Vol. 100, pp. 485–514.Google Scholar
  16. Sideris, M.G. and Li, Y.C., 1993, Gravity Field Convolutions Without Windowing and Edge Effects,Bulletin Géodésique Vol. 67, pp. 107–118.Google Scholar
  17. Sideris, M.G. and Tziavos, I.N., 1988, FFT-Evaluation and Applications of Gravity-Field Convolution Integrals With Mean and Point Data,Bulletin Géodésique Vol. 62, pp. 521–540.Google Scholar
  18. Soumekh, M., 1988, Band-Limited Interpolation From Unevenly Spaced Sampled Data,IEEE Transactions on Acoustics, Speech, and Signal Processing Vol. 36, No. 1, pp. 110–121.Google Scholar
  19. Sünkel, H., 1981, Cardinal Interpolation, OSU Report No. 312, The Ohio State University, Department of Geodetic Science, Columbus, Ohio.Google Scholar
  20. Tziavos, I.N., 1993, Numerical Considerations of FFT Methods in Gravity Field Modelling, Wissenschaftliche Arbeiten der Fachrichtung Vermessungswesen der Universität Hannover Nr. 188, Hannover.Google Scholar
  21. Vermeer, M., 1992, A Frequency Domain Approach to Optimal Geophysical Data Gridding,Manuscripta Geodaetica, Vol. 17, pp. 141–154.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Michael G. Sideris
    • 1
  1. 1.Department of Geomatics EngineeringThe University of CalgaryCalgaryCanada

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