Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem

Abstract

Summary A key element in the solution of a geophysical inverse problem is the quantification of non‐uniqueness, that is, how much parameters of an inferred earth model can vary while fitting a set of measurements. A widely used approach is that of Bayesian inference, where Bayes' rule is used to determine the uncertainty of the earth model parameters a posteriori given the data. I describe here, a natural extension of Bayesian parameter estimation that accounts for the posterior probability of how complex an earth model is (specifically, how many layers it contains). This approach has a built‐in parsimony criterion: among all earth models that fit the data, those with fewer parameters (fewer layers) have higher posterior probabilities. To implement this approach in practice, I use a Markov chain Monte Carlo (MCMC) algorithm applied to the nonlinear problem of inverting DC resistivity sounding data to infer characteristics of a 1‐D earth model. The earth model is parametrized as a layered medium, where the number of layers and their resistivities and thicknesses are poorly known a priori. The algorithm obtains a sample of layered media from the posterior distribution; this sample measures non‐uniqueness in terms of how many layers are effectively resolved by the data and of the range of layer thicknesses and resistivities consistent with the data. Because the complexity of the model is effectively determined by the data, the solution does not need to be regularized. This is a desirable feature, because requiring the solution to be smooth beyond what is implied by prior information can lead to underestimating posterior uncertainty. Letting the number of layers be a free parameter, as done here, broadens the space of earth models possible a priori and makes the determination of posterior uncertainty less dependent on the parametrization.

Monte Carlo Inversion of SkyTEMTM AEM data from Lake Thetis, Western Australia

“…The methodology, previously described by Brodie and Sambridge (2012), was adapted from the 2D seismic tomography inversion work of Bodin and Sambridge (2009). Similar techniques in the geophysical literature include Malinverno (2002) and Minsley (2012).…”

Reconstructing plate-motion changes in the presence of finite-rotations noise

“…At first glance, models with a greater number of change points may seem prone to fit the data better, hence yielding higher posterior probability with respect to simpler (that is, less change points) models. However, it is well established that trans-dimensional Bayesian inference follows the principle of 'natural parsimony' 18 , where preference always falls on the least complex explanation of observations.…”

“…Here, we tackle noise in finite rotations by using a trans-dimensional hierarchical Bayesian framework [17][18][19][20][21] (Methods). We find that changes in the temporal trends of plate motions occur on timescales no shorter than a few million years, yielding simpler kinematic patterns and more plausible dynamics.…”

A method of spherical harmonic analysis in the geosciences via hierarchical Bayesian inference

“…Therefore, a criterion for an optimal model dimension selection, which here depends on the maximum degree of spherical harmonic expansion, has to be established. A transdimensional type of sampling has recently been applied in many inverse problems to determine optimal model selection, despite the high computational burden (Malinverno 2002;Sambridge et al 2006;Dettmer et al 2010;Tkalčić et al 2013). Alternatively, an information criterion (IC) can be used at lower computational cost (Schwarz 1978;Pachhai et al 2014).…”