Reconstruction of relaxation modulus for viscoelastic medium with complete data

The inverse scattering problem for viscoelastic medium with wave impedance mismatch at the rear boundary is studied in this paper. The propagation operators of the viscoelastic medium are defined first and the integro-differential equations of these operators are derived via the invariant imbedding technique. For the inverse scattering problem, a new inversion procedure to reconstruct the relaxation modulus of the viscoelastic medium is developed, which utilizes a complete set of data, namely the one-side reflection data for one round trip through the viscoelastic slab. These data are complete in the sense that they can be extended to arbitrary time t in the inversion procedure. This inverse method is implemented numerically on several problems at the end of the paper. The more general case of a medium consisting of a stack of homogeneous viscoelastic medium layers is also considered.