Invariant imbedding for the wave equation in three dimensions and the applications to the direct and inverse problems

Wave splitting and invariant imbedding is used to obtain a one-parameter family of scattering problems and the associated reflection operator for the wave equation in three dimensions. The velocity is assumed to be at least twice differentiable. Existence and smoothness properties of the reflection operator are proven and a smoothed version of the imbedding equation for the kernel of the reflection operator is developed. The practical use of these imbedding equations in a layer-stripping approach (that can be implemented numerically) to the direct and inverse problem is shown.