A stochastic algorithm for fault inverse problems in elastic half space with proof of convergence

Abstract: A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in [11]. We show in this paper how it can be used to solve the fault inverse problem, where a planar fault in elastic half-space and a slip on that fault have to be reconstructed from noisy surface displacement measurements. With the parameter giving the plane containing the fault denoted by m and the regularization parameter for the linear part of the inverse problem denoted by C, both modeled as random variables, we derive a formula for the posterior marginal of m. Modeling C as a random variable allows to sweep through a wide range of possible values which was shown to be superior to selecting a fixed value [11]. We prove that this posterior marginal of m is convergent as the number of measurement points and the dimension of the space for discretizing slips increase. Simply put, our proof only assumes that the regularized discrete error functional for processing measurements relates to an order 1 quadrature rule and that the union of the finite-dimensional spaces for discretizing slips is dense. Our proof relies on trace class operator theory to show that an adequate sequence of determinants is uniformly bounded. We also explain how our proof can be extended to a whole class of inverse problems, as long as some basic requirements are met. Finally, we show numerical simulations that illustrate the numerical convergence of our algorithm.