On the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses View Full Text


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Article Info

DATE

2013-11

AUTHORS

Yan Ming Wang, Xu Yang

ABSTRACT

This paper is devoted to the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses in the most rigorous way. Such an expansion can be used to compute gravimetric topographic effects for geodetic and geophysical applications. It can also be used to augment a global gravity model to a much higher resolution of the gravitational potential of the topography. A formulation for a spherical harmonic expansion is developed without the spherical approximation. Then, formulas for the spheroidal harmonic expansion are derived. For the latter, Legendre’s functions of the first and second kinds with imaginary variable are expanded in Laurent series. They are then scaled into two real power series of the second eccentricity of the reference ellipsoid. Using these series, formulas for computing the spheroidal harmonic coefficients are reduced to surface harmonic analysis. Two numerical examples are presented. The first is a spherical harmonic expansion to degree and order 2700 by taking advantage of existing software. It demonstrates that rigorous spherical harmonic expansion is possible, but the computed potential on the geoid shows noticeable error pattern at Polar Regions due to the downward continuation from the bounding sphere to the geoid. The second numerical example is the spheroidal expansion to degree and order 180 for the exterior space. The power series of the second eccentricity of the reference ellipsoid is truncated at the eighth order leading to omission errors of 25 nm (RMS) for land areas, with extreme values around 0.5 mm to geoid height. The results show that the ellipsoidal correction is 1.65 m (RMS) over land areas, with maximum value of 13.19 m in the Andes. It shows also that the correction resembles the topography closely, implying that the ellipsoidal correction is rich in all frequencies of the gravity field and not only long wavelength as it is commonly assumed. More... »

PAGES

909-921

References to SciGraph publications

  • 2007-02. A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling in JOURNAL OF GEODESY
  • 2013-04. Recursive computation of oblate spheroidal harmonics of the second kind and their first-, second-, and third-order derivatives in JOURNAL OF GEODESY
  • 2012-07. Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies in JOURNAL OF GEODESY
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  • 1997-01. On the error of analytical downward continuation of the earth's external gravitational potential on and inside the earth's surface in JOURNAL OF GEODESY
  • 2007-09. On the computation and approximation of ultra-high-degree spherical harmonic series in JOURNAL OF GEODESY
  • 1991. Review of Geoid Prediction Methods in Mountainous Regions in DETERMINATION OF THE GEOID
  • 2004-08. Proof of expansion of the reciprocal distance in spheroidal coordinates in TECHNICAL PHYSICS
  • 1995-11. Numerical problems in the computation of ellipsoidal harmonics in JOURNAL OF GEODESY
  • 2008-10. A comparison of different mass elements for use in gravity gradiometry in JOURNAL OF GEODESY
  • 2012-04. Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers in JOURNAL OF GEODESY
  • 2002-05. A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions in JOURNAL OF GEODESY
  • 2012-03. The US Gravimetric Geoid of 2009 (USGG2009): model development and evaluation in JOURNAL OF GEODESY
  • 2009-09. On the geoid–quasigeoid separation in mountain areas in JOURNAL OF GEODESY
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