Convergence Tests for Transdimensional Markov Chains in Geoscience Imaging

TL;DR

This paper evaluates convergence tests for transdimensional Markov Chain Monte Carlo methods in geoscience imaging, proposing model conversions to enable classic convergence assessments despite variable parameter counts.

Contribution

It introduces three model conversion techniques to standardize transdimensional MCMC outputs, allowing effective convergence assessment in geoscience imaging applications.

Findings

Scalar model conversions preserve statistical properties.

Model conversions enable classic convergence tests.

Transdimensional MCMC can be effectively assessed with proposed methods.

Abstract

Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov Chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution, and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and have to be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles which are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization.