Fast and accurate computation of spherical harmonic coefficients from satellite gravity gradiometry data

A very efficient approach for computing spherical harmonic coefficients from satellite gravity gradiometry (SGG) data has been developed. The core of the proposed approach is a combination of the method of conjugate gradients with preconditioning on the one hand and an extremely fast algorithm for synthesis (application of the design matrix to a vector) and co-synthesis (application of the transposed design matrix to a vector) on the other. The high performance of the synthesis and co-synthesis is achieved by introducing an intermediate step, where computations are made on a regular three-dimensional (3-D) spherical grid. As a result, the Legendre functions can be computed for all the points at a given latitude only once and 1-D fast-Fourier techniques can be fully exploited. Transition to the true observation points is carried out by means of a 3-D spline interpolation. It is expected that the technique will be able to invert the full set of SGG data from the GOCE satellite mission (12-month data, four tensor components, 1-s sampling) in only a few hours on an SGI Origin 3800 computer with 16 processing elements. This corresponds approximately to 1 or 2 days of computation on a Pentium-IV PC. The choice of a relatively coarse 3-D spherical grid improves the efficiency even further, at the cost of minor errors in the solution. In this mode, the proposed approach can be used for quick-look data analysis.