Solutions to Linear Inverse Problems on the Sphere by Tikhonov Regularization, Wiener filtering and Spectral Smoothing and Combination — A Comparison

Abstract

Solutions to Linear Inverse Problems on the Sphere by Tikhonov Regularization, Wiener filtering and Spectral Smoothing and Combination — A ComparisonSolutions to linear inverse problems on the sphere, common in geodesy and geophysics, are compared for Tikhonov's method of regularization, Wiener filtering and spectral smoothing and combination as well as harmonic analysis. It is concluded that Wiener and spectral smoothing, although based on different assumptions and target functions, yield the same estimator. Also, provided that the extra information on the signal and error degree variances is available, the standard Tikhonov method is inferior to the other methods, which, in contrast to Tikhonov's approach, match the spectral errors and signals in an optimum way. We show that the corresponding Tikhonov matrix for optimum regularization can only be determined approximately. Moreover, as Tikhonov's method solves an integral equation, it is less computationally efficient than the other methods, which use forward integration. Also harmonic analysis uses direct integration and is not hampered, as previous methods, with spectral leakage. Spectral combination, in addition to filtering, has the advantage of combining different data sets by least squares spectral weighting.