Microlocal regularity of an inverse problem for the multidimensional wave equation

Abstract

Many physical processes such as reflection seismology, oil exploration, and ground-penetrating radar may be modeled as inverse problems for the multidimensional acoustic wave equation with point energy sources. The inverse problem is to identify the coefficients from the knowledge of boundary measurements of the solution. In this research we formulate an inverse problem for the wave equation with constant wave speed as a functional equation involving a forward map, which maps the coefficient (density) to the boundary value of the solution (excess pressure). We begin by examining some fundamental results in nonsmooth microlocal analysis. Rauch's lemma on the algebraic property of microlocal Sobolev spaces and a Beals-Reed linear propagation of singularities theorem are extended. We then present a trace regularity theorem which indicates that with microlocal restrictions against tangential oscillations in the coefficient, the boundary value is just as regular as the solution itself. The trace theorem also gives the first hint of the appropriate domain and range for the forward map. However, compared to the one dimensional case, much more overall smoothness has to be imposed to assure the optimal regularity of timelike traces. The Hadamard theory on progressing wave expansions is employed to study the fundamental solution to the linear acoustic wave equation. To establish the regularity of the solution, the solutions of transport equations are investigated by applying the Rauch-type results. The central result for the regularity of the inverse problem is an upper bound for the linearized forward map with nonsmooth reference density. In order to establish this regularity result, a dual technique is developed which dramatically reduces the difficulties of the inverse problem. Our method has the potential to obtain some regularity results even for the important nonsmooth reference velocity case. Similar analyses could result in a continuity result and a differentiability result for the forward map. These regularity properties are obviously crucial in the design and analysis of the algorithms for solving the inverse problem