Linearization Stability for an Inverse Problem in Several-Dimensional Wave Propagation

Abstract

Inverse problems in wave propagation arise from the physical notion that the response of a mechanical continuum to a specified excitation should reflect the properties of the medium through which the excited waves travel, even though the waves are eventually measured at a remote location. We consider the formal linearization of a simple model inverse problem in several-dimensional wave propagation, in which the density distribution of a linear fluid is to be recovered from its remotely measured response to an incident plane-wave excitation, assuming the sound velocity distribution to be known and constant. We make reasonable choices for norms (error measures), and give simple examples establishing the ill-posed nature of the problem and indicating the mechanism of instability. We then show how the problem may be regularized (rendered well-posed) by the introduction of minimally stringent a priori constraints, motivated by a (crude) approximation to the singular value decomposition of the linearized problem.