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Spherical harmonic analysis, aliasing, and filtering

Abstract

The currently practiced methods of harmonic analysis on the sphere are studied with respect to aliasing and filtering. It is assumed that a function is sampled on a regular grid of latitudes and longitudes. Then, transformations to and from the Cartesian plane yield formulations of the aliasing error in terms of spherical harmonic coefficients. The following results are obtained: 1) The simple quadratures method and related methods are biased even with band-limited functions. 2) A new method that eliminates this bias is superior to Colombo's method of least squares in terms of reducing aliasing. 3) But, a simple modification of the least-squares model makes it identical to the new method as one is the dual of the other. 4) The essential elimination of aliasing can only be effected with spherical cap averages, not with the often used constant angular block averages.

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References

  1. Abramowitz M, Stegun IA (eds.) (1972) Handbook of Mathematical Functions. Dover Publications.

  2. Albertella A, Sacerdote F (1995) Spectral analysis of block averaged data in geopotential global model determination. To appear in Manuscripta Geodaetica.

  3. Albertella A, Sacerdote F, Sansò F (1993) Geodetic calculus with block-averages observations on the sphere. Surveys in Geophysics 14: 395–402.

    Google Scholar 

  4. Colombo OL (1981) Numerical methods for harmonic analysis on the sphere. Report no.310, Department of Geodetic Science, The Ohio State University.

  5. Gaposchkin EM (1980) Averaging on the surface of a sphere. J Geophys Res 85(B6):3187–3193.

    Google Scholar 

  6. Gleason DM (1987) Developing an optimally estimated earth gravity model to degree and order 360 from a global set of 30′ by 30′ mean surface gravity anomalies. Manuscripta Geodaetica 12:253–267.

    Google Scholar 

  7. Goldstein JD (1978) Application of Fourier techniques to the computation of spherical harmonic coefficients. The Analytic Sciences Corp., Tech. Info. Mem. TIM-868-3.

  8. Heiskanen WA, Moritz H (1967) Physical Geodesy. W.H. Freeman, San Francisco.

    Google Scholar 

  9. Hobson EW (1965) The Theory of Spherical and Ellipsoidal Harmonics. Chelsea Publishing.

  10. Hwang C (1993) Fast algorithm for the formation of normal equations in a least-squares spherical harmonic analysis by FFT. Manuscripta Geodaetica 18(1):46–52.

    Google Scholar 

  11. Jekeli C (1981) Alternative methods to smooth the earth's gravity field. Report No.327, Department of Geodetic Science and Surveying, Ohio State University.

  12. Kaula WM (1993) Higher Order Statistics of Planetary Gravities and Topographies. Geophys Res Lett 20.

  13. Pavlis NK (1988) Modeling and estimation of a low degree geopotential model from terrestrial gravity data. Report No.386, Department of Geodetic Science and Surveying, The Ohio State University.

  14. Priestley MB (1981) Spectral Analysis and Time Series. Academic Press.

  15. Rapp RH (1977) Determination of Potential Coefficients to Degree 52 from 5° mean gravity anomalies. Bulletin Géodésique 51(4):301–323.

    Google Scholar 

  16. Rapp RH, Pavlis NK (1990) The development and analysis of geopotential coefficient models to spherical harmonic degree 360. J Geophys Res 95(B13):21885–21911.

    Google Scholar 

  17. Ricardi LJ, Burrows ML (1972) A recurrence technique for expanding a function in spherical harmonics. IEEE Trans Comp June:583–585.

  18. Schmitz DR, Cain JC (1983) Geomagnetic Spherical Harmonic Analysis, 1. Techniques. J Geophys Res 88(B2):1222–1228.

    Google Scholar 

  19. Sansò F (1990) On the aliasing problem in the spherical harmonic analysis. Bulletin Géodésique 64:313–330.

    Google Scholar 

  20. Sjöberg L (1980) A recurrence relation for the βn-function. Bulletin Géodésique 54(1):69–72.

    Google Scholar 

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Jekeli, C. Spherical harmonic analysis, aliasing, and filtering. Journal of Geodesy 70, 214–223 (1996). https://doi.org/10.1007/BF00873702

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Keywords

  • Harmonic Analysis
  • Regular Grid
  • Aliasing
  • Related Method
  • Quadrature Method