Bayesian model error method for the passive inverse scattering problem

Abstract

This paper focuses on the passive inverse scattering problem, which uses passive measurements corresponding to randomly distributed incident sources to recover the shape of the sound-soft obstacle from a Bayesian perspective. Due to the unpredictability and randomness of incident sources, the classical Bayesian inversion framework may be unable to capture the likelihood involving the passive forward model for this inverse problem. We present the Bayesian model error method (BMEM), a novel passive imaging technique, to overcome this difficulty. The cross-correlations and the Helmholtz–Kirchhoff identity are specifically used to build an approximate active scattering model. This approximate model and the model error that it produces can be combined effectively by the suggested BMEM. The well-posedness of the posterior measure in the BMEM is proved. To further estimate the model error, an online scheme is utilized in conjunction with a preconditioned Crank–Nicolson Markov Chain Monte Carlo method to numerically approximate the posterior. Numerical experiments illustrate the effectiveness of the proposed method and also show that the online evaluation of model error can significantly improve reconstruction accuracy.