Factorization of the wave equation in a nonplanar stratified medium

The wave equation is considered for a stratified medium where the stratifications are in the form of a family of nested C2 surfaces along which the velocity c is constant (c varying only in a direction normal to the surfaces). On each surface c is constant, the solution u of the wave equation is decomposed into an outgoing wave component u+ and an incoming wave component u−. The associated outgoing and incoming wave conditions are expressed in terms of integral operators (kernels being time-dependent single and double layer potential type terms) operating on u and the normal derivative ∂u/∂n on each surface. Using the decomposition the scalar wave equation is split into a vector system involving the components u+ and u−, the vector system decoupling in a region where c is constant. Such a splitting is useful for the inverse problem where a reflection operator relating the outgoing wave to an incoming wave can be defined, and this in turn can be used to determine c.