Reconstruction of the shape of a convex defect from a scattered wave field in the ray approximation

Abstract

The problem of reconstructing the shape of a convex defect in a solid body from the results of measurements of the amplitude of back-scattering of an ultrasound wave is considered. It is assumed that the length-scale of the defect is much larger than the wavelength, which allows the problem to be considered in the ray approximation. It has been shown [1] that, using such an approach, the problem being investigated reduces to the well-known Minkowski problem: for a given Gaussian surface curvature reconstruct the shape of a closed convex surface. It has been shown [2] that under specified conditions a unique convex surface exists which has the Gaussian curvature of a given continuous function. However, an algorithm allowing one to construct such a convex surface does not exist. In this paper a numerical method is developed which enables one to implement the reconstruction of the required convex surface. As examples we consider the reconstruction of ellipsoids of revolution with various eccentricities, and also a nearly-cylindrical surface.