Alex Miltenberger , Lijing Wang, Tapan Mukerji, Jef Caers

Today, most probabilistic geophysical inverse problems are formulated using one of two methods: (1) conditional probability and Bayes’ Theorem, or (2) Tarantola’s theory of intersecting probability densities. More recently, a third inverse problem formulation based on pushforward probability measures was proposed, termed “data-consistent inversion”. Many practical problems can be cast into any of these three formulations, but the choice of formulation may change the posterior. To help geophysicists understand how the formulation affects the posterior, we present a mathematical comparison of the three formulations accompanied by simple physical example to illustrate the difference between posteriors from each method. The distinguishing feature of the data-consistent formulation is that it yields a posterior that maps onto the measurement distribution, whereas the traditional formulations do not. Then we propose a flexible Sequential Monte Carlo algorithm for solving practical subsurface modeling problems conditioned to geophysical data. This algorithm uses density estimation techniques to fit a generative probabilistic model to the posterior. The algorithm is demonstrated using a synthetic gravity inversion example. The results are then compared to results from the classical Bayesian formulation solved using Markov Chain Monte Carlo. The comparison shows that even though the difference between the data-consistent posterior and Bayesian posterior can be small in practice, the lack of conditional probabilities in the data-consistent formulation can make it simpler to fit generative probabilistic models to the solution of the inverse problem.