Spherical Harmonic Analysis via Bayesian Inference

Abstract:

The real spherical harmonics form a compact, simple and commonly used set of basis functions for describing fields in tomographic inverse problems. It is therefore often useful to perform spherical harmonic analysis on data to represent it in the spherical harmonic parametrisation. Most existing algorithms, based on Fourier transforms, require that data be interpolated to a regular grid; this is not appropriate for the sparse, irregularly distributed data found in many geophysical applications. Instead, this work casts the problem of spherical harmonic analysis as an inverse problem, and applies the methods of Bayesian inference to overcome regularization problems in the inversion. This allows irregular data to be easily handled, and directly provides error estimates for the inverted spherical harmonic parameters. Synthetic tests have shown that this method easily handles relatively large amounts of added Gaussian noise. So far, this method has been applied to estimate the power in each harmonic degree for tomographic maps of the deep mantle based on PKP-PKIKP and PcP-P differential travel times, showing that they agree at global length scales despite local heterogeneity results being heavily influenced by data coverage. This potentially allows for simple heuristic arguments to constrain the global variation in core-mantle boundary topography based on the similarity between PKP and PcP derived tomographic maps.