An inverse problem for a layered medium with a point source

Suppose q(z) is a smooth function on [0,∞) whose odd order derivatives are zero at z=0. Take r = (x, y, z) and let U(r, t) be the solution of the the IBVP Utt − Uxx − Uyy − Uzz + q(z)U = 0 for r ∈ R, z ≥ 0, t ∈ R Uz(x, y, z=0, t) = δ(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.