Varying dimensional Bayesian acoustic waveform inversion for 1D semi-infinite heterogeneous media

Abstract

This paper introduces a methodology to infer the spatial variation of the acoustic characteristics of a 1D vertical elastic heterogeneous earth model via a Bayesian calibration approach, given a prescribed sequence of loading and the corresponding time history response registered at the ground level. This involves solving an inverse problem that maps the ground seismic response onto a random profile of the ground stratigraphy (i.e. a 1D continuous spatial random field). From a Bayesian point of view, the solution to an inverse problem is fully characterized by a posterior density function of the forward model random parameters, which explicitly overcomes the solution's non-uniqueness. This subsurface earth model is parameterized using a Bayesian partition model, where the number of soil layers, the location of the layers' interfaces, and their corresponding mechanical characteristics are defined as random variables. The partition model approach to an inverse problem is closely related to a Bayesian model selection problem, where the likely dimensionality of the inverse problem (number of unknowns) is inferred conditioned on the experimental observations. The main benefit of the proposed approach is that the explicit regularization of the inverted profile by global damping procedures is not required. A Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm is used to sample the target posterior of varying dimension, dependent on the number of layers. A synthetic case study is provided to indicate the applicability of the proposed methodology.