Inverse scattering for time-varying one-dimensional layered media: algorithms and applications

We formulate and present algorithms and applications for the inverse scattering problem for a one-dimensional layered medium whose reflection coefficients vary with time. We show that given the reflection responses to impulsive plane waves incident on the medium at different times, this problem can be solved either by solving a series of nested non-Toeplitz systems of equations, or by propagating a set of coupled layer-stripping algorithms with reflection coefficients varying in time. All multiple scattering effects are included here. We show that our results reduce to previous results in the special case of time-invariant media. Applications include the two-dimensional (2D) inverse resistivity problem of reconstructing the 2D resistivity ρ(x, y) of a 2D medium from surface measurements, and the reconstruction of rapidly curing layered media. We also present a time-varying discrete Miura transform linking the time-varying discrete wavesystem inverse scattering problem to a time-varying discrete Schrödinger equation, and a new important feasibility condition on the reflection responses, which seems to be new in the context of the 2D inverse resistivity problem. A numerical example is provided.