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A linear algorithm for computing the spherical harmonic coefficients of the potential from a constant-density polyhedron

Dr Olivier Jamet(1)and Ms Emilie Thomas(2)

(1) Ecole Nationale des Sciences Géographiques, 6 et 8 av. Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée CEDEX, France
(2) Université de Franche-Comté, 16, Route de Gray, 25030 Besançon CEDEX, France


The availability of gradiometric measurements along the tracks of the forthcoming low orbiting satellite GOCE will bring several new challenges to geodesists and geophysicists -- such as the integration of this new kind of measurements in the gravity field determination techniques, the inversion of medium scale geophysical phenonmena (e.g. seamounts, rifts, fracture zones) from this new gravity data, etc. In this paper, we present a new computational method for the determination of the spherical harmonic coefficients of the contribution to the potential of a constant density polyhedron of arbitrary shape. Up to now, only one solution is known to this problem. It was proposed by R.A. Werner (1997). This solution has a complexity function of the square of the number of computed spherical harmonic coefficients. This rather high complexity prohibits the processing of realistic geologic models (polyhedra with tens of thousands of faces) at a degree and order compatible with GOCE accuracy (over 400). The complexity of the algorithm we developped is linear as a function of the number of spherical harmonic coefficients to be computed, and as a function of the complexity of the polyhedron (its number of edges). This low complexity will allow to produce simulated spherical harmonic developpements of the potential at high degree and order, from realistic geological models. Such simulated models will be of great value for assessing the quality of geophysical inversion processes. We aim for instance at using a complete geological model of the crust (from surface down to Moho) over a 100km wide area for assessing the signature of several density discontinuities at GOCE altitude. We will present the principle of the algorithm and discuss the possible strategies for assessing the numerical accuracy of the computation.

Reference : R.A. Werner (1997) Spherical harmonic coefficients for the potential of a constant-density polyhedron, Computer & Geosciences, vol. 23, n. 10, pp. 1071-1077


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Keywords: ESA European Space Agency - Agence spatiale europeenne, observation de la terre, earth observation, satellite remote sensing, teledetection, geophysique, altimetrie, radar, chimique atmospherique, geophysics, altimetry, radar, atmospheric chemistry