Reconstruction of a compactly supported sound profile in the presence of a random background medium

In this paper, we present algorithms for reconstructing an unknown compact scatterer embedded in a random noisy background medium, given measurements of the scattered field and information about the background medium and the sound profile. We present six different methods for the solution of this inverse problem using different amounts of scattered data and prior information about the random background medium and the scatterer. The different inversion algorithms are defined by a combination of stochastic programming methods and Bayesian formulation. Our basic results show that if we have data for just one instance of the random background medium the best strategy is to invert for both random medium and unknown scatterer with appropriate regularization. However, if we have data for multiple instances of the medium it may be worth solving a coupled set of multiple inverse problems. We present several numerical results for inverting for various scatterer geometries under different inversion scenarios. The main take-away of our study is that one should invert for both unknown scatterer and random medium, with appropriate, prior-information based regularization. Furthermore, if data from multiple realizations of the background medium is available, then it may be beneficial to combine results from multiple inversions.
Comment: 40 pages