No abstract

Wave splitting and invariant imbedding is used to obtain a one-parameter family of scattering problems and the associated reflection operator for the wave equ& tion 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.

The technique for splitting waves into up and down components is applied to solutions of the generalized wave equation in three dimensions. In particular, the splitting associated with the principal part of the equation is applied to the fundamental solution (Green function). The equations for the up and down wave components are obtained q d the singular parts removed. The use of these reduced wave components together with their equations. boundary data and initial data, in the inverse problem (where the reflected field 6 produced by a point-impulse source, exterior to the scattering medium) is discussed.Furthermore, we will assume that the scattering medium lies below the plane x3 = 0, and that the region x3 0 is free space, i.e.where co is a constant. For convenience we will set p = 1 for x3 > 0.is free space with c = CO. b = q = 0 and p = 1 here.Note for a compact scatterer lying in the slab -L e x3 < 0, then the region x3 < -LThe following notation will be employed x = (5, 1 3 ) 5 = (Xl, x2)

The factorization of the wave equation into a coupled system involving up-and down-going wave components is obtained for the case where the field quantities are multivariate functions of spatial variables, but the velocity c is a function of the z variable only. The form of the reflection operator is derived and the quadratic differential-integral equation satisfied by its kernel is obtained.

For pt.I see ibid., vol.6, p.1075 (1990). The results of a previous work on invariant imbedding and wave-splitting applied to the wave equation, are extended to the wave-splitting of the Green function. The system of equations for the up- and down-going wave components G+ and G- of green function (associated with a fixed point impulsive source, exterior to the scattering medium) are obtained. The (short-time) asymptotic behaviour of the wave components G+, G- are derived. The application of the system of equations for G+, G- to the layer stripping process is examined, and the consequent utilization of this process to the inverse problem is treated. Some of the problems in the numerical implementation of such a procedure are examined, and remaining additional analysis along these lines, that remains to be investigated, is given.