Designing Solutions to Geophysical Inverse Problems by Changing Variables

TL;DR

This paper investigates how changing variable parametrizations in geophysical inverse problems can lead to inconsistent Bayesian and deterministic solutions, impacting the reliability of these methods.

Contribution

It demonstrates that different parametrizations can cause inconsistent probability densities and solutions, challenging current practices in Bayesian and deterministic inversion.

Findings

Parametrization changes lead to inconsistent conditional probabilities.

Bayesian posterior solutions can vary dramatically with parametrization.

Deterministic solutions are also affected by parametrization choices.

Abstract

Geoscientists often solve inverse problems to estimate values of parameters of interest given relevant data sets. Bayesian inference solves these problems by combining probability distributions that describe uncertainties in both observations and unknown parameters, and we require that the solution provides unbiased uncertainty estimates in order to inform risk-based decisions. It has been known for over a century that employing different, but equivalent parametrisations of the same information can yield conditional probabilities that are mathematically inconsistent, a property referred to as the BK-inconsistency. Recently this inconsistency was shown to invalidate the solutions to physical problems found using several well-established methods of Bayesian inference. In this study, we explore the extent to which this inconsistency affects solutions to common geophysical problems. We demonstrate that changes in parametrisations result in inconsistent conditional probability densities, even though they represent exactly the same information. We show that this can affect Bayesian posterior solutions dramatically across various geoscientific problems using real and synthetic data. Given that deterministic inversion is often equivalent to finding the maximum a posteriori solution to specific Bayesian problems (the mathematical equations to be solved are identical), the BK-inconsistency also results in inconsistent solutions to deterministic inverse problems. Indeed, we show that solutions can potentially be designed, simply by changing the parametrisation. This study highlights that a careful rethinking of Bayesian inference and deterministic inversion may be required in physical problems: the effects that we demonstrate are likely to affect past and present inverse problem solutions in a variety of different fields of application.