Estimation of the Boundary of an Inclusion of Known Material from Scattering Data

Abstract

An interesting type of probabilistic inverse problem associated with the scattering of elastic waves from inhomogeneities in solids is that in which the prior statistics of the inhomogeneity is non-Gaussian. Here we consider a relatively simple example of such a problem in which the inhomogeneity is an inclusion with known material but unknown boundary. The appropriate measurement model for longitudinal-to-longitudinal backscatter has the typical signal-plus-noise form in which the noise is assumed to be Gaussian and the signal is given by the application of the Born approximation to an inclusion described by a characteristic function for a random domain. The so-called signal also incorporates the transfer function characterizing the response of the measurement system to a fictitious point scatterer. Assuming a discretized physical space represented by a lattice of points, the prior random properties of the characteristic function are described by a set of independent random variables, each of which has the values 0 and 1 at each point with prescribed probabilities. Thus, in a prior sense, all possible domains (including sets of separate domains) are represented (at least within the resolution implied by the lattice). Our problem is to find the most probable (in the posterior sense) characteristic function given the results of the scattering measurements. We have found that the direct maximization of the posterior probability with respect to the characteristic function is cornputationally intractable. We have devised a computationally convenient method obtained by maximizing the appropriate probability function of the characteristic function and the additive noise while regarding the measurement model as a continuous set of constraints to be handled by the conjugate vector method. With appropriate analytical manipulations we finally obtained a convex function of the Lagrange multiplier vector to be minimized by computational means. Using synthetic input test data, we have estimated the inclusion boundaries for a number of cases. Compared with conventional imaging techniques, the estimations of the boundary geometries are surprisingly good even for rather sparse sets of input data.