Adaptive parameterization in Bayesian inversions using transdimensional methods

Geothermal energy is a crucial component of the global transition to sustainable and green energy systems due to its renewable and long-term availability. In order to study potential resources, we need to describe the subsurface by solving inverse problems. The complexity and uncertainty of these problems require the use of probabilistic inversion approaches that repeatedly solve partial differential equations over a grid of parameters describing the subsurface domain. Frequently, the high dimensionality of the parameter space to be inferred implies prohibitive computational times and reduces the sensitivity of each parameter as the grid is refined. In this work, we implement and discuss adaptive parametrization strategies in Bayesian inversions. We model the thermal conductivity structure of 2D sections of the Earth's upper mantle and perform Markov chain Monte Carlo (MCMC) inversions to recover the thermal conductivity as a probability distribution based on the likelihood of the temperature measurements. To verify the solution, we first parametrize the physical properties of the subsurface domain equal to the high-dimensional finite element grid. In order to determine the optimal metaparameters on the run we rely on adaptive MCMC techniques that accelerate the convergence and reduce the risk of getting trapped in local minima. We then use a new parametrization based on the physical structure of the geological faults of the mantle that reduces the dimensionality of the problem. By relying on transdimensional sampling through reversible-jump MCMC, we consider the number of parameters as an unknown of the inversion. In these methods, the algorithm is allowed to increase the number of parameters to invert when the solutions found are not accurate enough and to decrease it when the accuracy of the solution is not significantly affected. Our results show that we recover the thermal conductivity structure both with and without adaptive parametrization, and the performance is improved when using transdimensionality. Moreover, the proposed transdimensional inversion decreases or increases the number of parameters locally, thereby providing an efficient and robust method for addressing the often challenging lack of information on subsurface heterogeneity.

Keywords: geothermal energy; Markov chain Monte Carlo; reversible jump MCMC; transdimensional inversion; adaptive parametrization; finite elements; Poisson equation.