On the inverse problem of the 3D telegraph equation

Abstract

In this paper, an approach to solving the inverse problem for the telegraph equation is examined in detail for the special case where the telegraph equation is given by with one source generating an instant spherical wave. By splitting the wave into up-going and down-going components and observing the reflective wave (down-going component) on the surface , one can obtain q(x). The first part of this paper is to prove the existence of wave splitting (the direct problem). For the second part, a layer-stripping method and asymptotic analysis will be applied to approximate the solution of the inverse problem. The fact that the leading coefficients of the asymptotic expansion of yield q(x) is a necessary condition for the solution of the ill-posed problem which involves the inverse Dirichlet operator. The ill-posedness of recovering the reflective wave is also mentioned.