An inverse problem for a layered medium with a point source

Suppose q (z) is a smooth function on [0, infinity) whose odd order derivatives are zero at z = 0. Take r = (x, y, z) and let U(r, t) be the solution of the IBVP U-tt - U-xx - U-yy - U-zz + q(z)U = 0, for r is an element of R-3, Z greater than or equal to 0, t is an element of R U-z(x, y, z = 0, t) = delta(x, y, t), U(r, t) = 0, for t < 0. We show that q(z) may be recovered from a knowledge of U (a, b, 0, t) for t varying over an interval and fixed a, b, by reducing the problem to a one-dimensional inverse problem. This reduction is not trivial since U (r, t) depends on r and t and not just on z and t.