A Multiscale MCMC Approach to Joint Geophysical Inversion: Tackling Dimensionality, Solver Integration, and Multiscale Data Fusion

Modern geophysical inversion increasingly relies on combining datasets that differ in resolution, sensitivity, and physical nature, but integrating such multiscale information remains challenging for Bayesian frameworks based on Markov chain Monte Carlo (MCMC) algorithms. Here we introduce and investigate a general MCMC framework that addresses these challenges by extending the concept of coupled parallel chains. The method enables parallel sampling at multiple discretization levels, with reversible swap proposals ensuring consistent and physically meaningful information exchange between coarse and fine scales. A key advantage is the ability to assign independent, scale-specific MCMC algorithms, priors, and forward solvers at each resolution, allowing parameterizations and proposals to be fully tailored to the data and physics represented at that scale. This structural flexibility, together with the use of regular tesselation schemes, improves convergence and sampling efficiency, mitigates the curse of dimensionality, and facilitates straightforward parallelization and integration of numerical solvers. The framework is applicable to a wide range of data-fusion scenarios, including multiple datasets that probe the same model parameters at different resolutions, datasets informing distinct but physically related parameters, and truly multiscale processes operating across multiple spatial or temporal scales. Numerical experiments confirm that the algorithm yields models that remain physically consistent across scales, while enhancements such as adaptive proposals, blockwise updates, and covariance-informed moves further improve performance. The results demonstrate that the method provides a rigorous, flexible, and computationally efficient framework to tackle multiscale and multi-observable Bayesian inversions for geophysical imaging.