Spatially Restricted Integrals in Gradiometric Boundary Value Problems

Open access

Spatially Restricted Integrals in Gradiometric Boundary Value Problems

The spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • Albertella A. Sansò F. and Sneeuw N. (1999) Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere. Journal of Geodesy Vol. 73 436-447.

  • Albertella A. Migliaccio F. and Sansò F. (2002) GOCE: The Earth Field by Space Gradiometry. Celestial Mechanics and Dynamical Astronomy Vol. 83 1-15.

  • Balmino G. Perosanz F. Rummel R. Sneeuw N. Sünkel H. and Woodworth P. (1998) European Views on Dedicated Gravity Field Missions: GRACE and GOCE. An Earth Sciences Division Consultation Document ESA ESD-MAG-REP-CON-001.

  • Balmino G. Perosanz F. Rummel R. Sneeuw N. and Sünkel H. (2001) CHAMP GRACE and GOCE: Mission Concepts and Simulations. Bollettino di Geofisica Teoricae Applicata Vol. 40 3-4 309-320.

  • Eshagh M. (2008) Non-singular expressions for the vector and the gradient tensor of gravitation in a geocentric spherical frame Computers & Geosciences Vol. 34 1762-1768.

  • Eshagh M. (2009a) On satellite gravity gradiometry Doctoral dissertation in Geodesy TRITA-TEC-PHD-09004 ISSN 1653-4468Royal Institute of Technology (KTH) Stockholm Sweden.

  • Eshagh M. (2009b) Complementary studies in satellite gravity gradiometry Postdoctoral report in Geodesy TRITA-TEC-RR 09-006 ISSN 1653-4484 ISBN 13:978-91-85539-47-5 Royal Institute of Technology (KTH) Stockholm Sweden.

  • Eshagh M. (2009c) The effect of polar gaps on the solutions of gradiometric boundary value problems Artificial Satellites Vol. 43 No. 3 97-108.

  • Eshagh M. (2010) Alternative expression for gravity gradients in local north-oriented frame and tensor spherical harmonics Acta Geophysica Vol. 58 215-243.

  • Grünbaum F.A. Longhi L. and Perlstadt M. (1982) Differential operators commuting with finite convolution integral operators: some non-abelian examples. SIAM Journal of Applied Mathematics Vol. 42 941-955.

  • Heiskanen W. and Moritz H. (1967) Physical Geodesy W. H. Freeman and company San Francisco and London.

  • Hwang C. (1991) Orthogonal Functions Over the Oceans and Applications to the Determination of Orbit Error Geoid and Sea Surface Topography from Satellite Altimetry PhD dissertation JPL 958121 OSURF 720426 229 pp Dec 1991.

  • Ilk K.H. (1983) Ein Beitrag zur Dynamik ausgedehnter Körper-Gravitationswechselwirkung. Deutsche Geodätische Kommission Reihe C Heft Nr. 288 München.

  • Kim M.C. and Tapley B. (2000) Formation of surface spherical harmonic normal matrices and application to high-degree geopotential modeling Journal of Geodesy Vol. 74 359-375.

  • Koop R. (1993) Global gravity field modeling using satellite gravity gradiometry. Publ Geodesy New series No. 38. Netherland Geodetic Commission Delft.

  • Mainville A. (1986) The altimetry-gravimetry problem using orthonormal base functions Report No. 373 Dept. of Geod Sci The Ohio state University Columbus Ohio.

  • Martinec Z. (2003) Green's function solution to spherical gradiometric boundaryvalue problems Journal of Geodesy Vol. 77 41-49.

  • Miranian L. (2004) Slepian functions on the sphere generalized Gaussian quadrature rule. Inverse Problems Vol. 20 877-892.

  • Pail R. Plank G. and Schuh W. D. (2001) Spatially restricted data distribution on the sphere: the method of orthonormalized functions and applications Journal of Geodesy Vol. 75 44-56.

  • Paul M.K. (1978) Recurrence relations for integrals of associated Legendre functions Bulletin Geod esique Vol. 52 177-190.

  • Rummel R. (1997) Spherical spectral properties of the Earth gravitational potential and its first and second derivatives Geodetic boundary value problems in view of the one centimeter geoid Lecture notes in Earth sciences Edited by Sanso F. and Rummel R. p.359-401.

  • Rummel R. Sanso F. Gelderen M. Koop R. Schrama E. Brovelli M. Migiliaccio F. and Sacerdote F. (1993) Spherical harmonic analysis of satellite gradiometry. Publ Geodesy New Series No. 39 Netherlands Geodetic Commission Delft.

  • Sebilleau D. (1998) On the computation of the integrated products of three spherical harmonics Journal of Physics A: Mathematical and General Vol. 31 7157-7168.

  • Slepian D. (1983) Some comments on Fourier-analysis uncertainty and modeling SIAM Vol. 25 379-393.

  • Simons M. Solomon S.C. and Hager B. H. (1997) Localization of gravity and topography: constraints on the tectonic and mantle dynamics of Venus Geophysical Journal International Vol. 131 24-44.

  • Simons F.J. Dahlen F.A. and Wieczorek M.A. (2006) Spatiospectral concentration on a sphere SIAM Review Vol. 48 3 504-536.

  • van Gelderen M. and Rummel R. (2001) The solution of the general boundary value problem by least-squares Journal of Geodesy Vol. 75 1-11.

  • van Gelderen M. and Rummel R. (2002) Corrections to "The solution of the general geodetic boundary value problem by least squares". Journal of Geodesy Vol. 76 121-122.

  • Varshalovich D.A. Moskalev A.N. and Khersonskii V.K. (1989) Quantum theory of angular momentum. World Scientific Publ Singapore.

  • Wieczorek M.A. and Simons F.J. (2005) Localized spectral analysis on the sphere Geophysical Journal International Vol. 162 655-675.

  • Xu Y.L. (1996) Fast evaluation of the Gaunt coefficients Mathematical Computations Vol. 65 1601-1612.

  • Zerilli F.J. (1970) Tensor harmonics in canonical form for gravitational radiation and other application. Journal of mathematical Physics Vol. 11 2203-2208.

Journal information
Impact Factor

CiteScore 2018: 0.61

SCImago Journal Rank (SJR) 2018: 0.211
Source Normalized Impact per Paper (SNIP) 2018: 0.728

Cited By
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 193 130 6
PDF Downloads 87 62 4