Fast spherical harmonic synthesis and co-synthesis with applications in gravity field modeling
Fast algorithms for spherical harmonic synthesis and co-synthesis have been developed. Synthesis and co-synthesis are defined as the application to a vector of the design matrix (matrix of partial derivatives, Jacobian matrix) and of the transposed design matrix, respectively. The algorithms are applicable to a variety of functional models including the cases of irregularly distributed data points, arbitrary sensor orientations, and data contaminated by colored noise. The performance of the proposed algorithms is of the order of O(N) for large number N of data points. The fast synthesis and co-synthesis algorithms can be efficiently used for data simulation and, in combination with the pre-conditioned conjugate gradient method (PCCG), for the least-squares inversion of data into model parameters. Importantly, the PCCG method allows one to avoid an assembling of the normal matrix. If, however, the normal matrix has to be computed explicitly, the application of the proposed algorithms can also be beneficial. With the fast synthesis and co-synthesis, the number of operations required for the computation of the normal matrix is of the order of O (NL max 2 ) for large N, where L max is the maximum degree of the spherical harmonic expansion. The explicitly computed normal matrix may be used, in particular, for the combination of the estimated model parameters with other data sets (possibly, to be acquired later).
KeywordsSpherical harmonics gravity field co-synthesis normal matrix data combination.
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