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The discrete inverse scattering problem for viscoelastic medium is studied in this paper. It is assumed that the relaxation modulus varies only with time t. The object of this paper is to develop a method to reconstruct the relaxation modulus with less measurement data than before. The propagation operators of the viscoelastic medium are defined first and the imbedding equations governing the behavior of the propagation operators are derived with the invariant imbedding techniques. Using the finite difference method, these equations can be discretized to obtain a system of linear algebraic equations about the propagation operators and the material modulus. For the inverse scattering problem, it is assumed that the reflection data obtained from the scattering experiments are only available on one side of the medium and for one round trip through the viscoelastic slab. To reconstruct the unknown relaxation modulus, an inversion procedure is developed using this set of data that are complete in the sense that they can be extended to arbitrary time t and the other scattering and propagation operators can also be determined by the inversion procedure described in this paper. The inversion algorithm is implemented numerically on several examples at the end of the paper. It can be seen that the obtained curves of the material modulus coincide with the original relaxation modulus very well.

In this paper, the inverse problem for the viscoelastic medium is investigated in the time domain, in which the wave impedance of the medium is discontinuous at the rear interface. The differentio-integral equations governing the behavior of the scattering and propagation operators are utilized to reconstruct the relaxation modulus of the viscoelastic medium. A new approach, in which only the one-side measurement reflection data for one round trip through the viscoelastic layer, is developed. The numerical examples are given at the end of the paper. It is shown that the curves of the reconstructed moduli coincide very well with the original relaxation moduli.