Gaussian process models—II. Lessons for discrete inversion

Abstract

Summary By starting from a general framework for probabilistic continuous inversion (developed in Part I) and introducing discrete basis functions, we obtain the well-known algorithms for probabilistic least-squares inversion set out by Tarantola & Valette (1982). In doing so, we establish a direct equivalence between the spatial covariance function that must be specified in continuous inversion, and the combination of basis functions and prior covariance matrix that must be chosen for discretised inversion. We show that the common choice of Tikhonov regularisation ($\mathbf {C_m^{-1}} = \sigma ^2\mathbf {I}$) arises from a delta-function spatial covariance, and that this lies behind many of the artefacts commonly associated with discretised inversion. We show that other choices of spatial covariance function can be used to generate regularisation matrices yielding substantially better results, and permitting localisation of features even if global basis functions are employed. We are also able to offer a straightforward explanation for the spectral leakage problem identified by Trampert & Snieder (1996).