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This paper presents an alternative numerical scheme for a class of forward and inverse scattering problems based on invariant imbedding method [Corones et al., J. Acoust. Soc. Am. 74, 1535–1541 (1983)]. The algorithm presented in this paper is decoupled more than the previous implementations and hence, readily amenable to parallel processing. It is also shown that the effect of this decoupling on error can be expressed in terms of the estimated reflection coefficients, facilitating the decision to accept or reject any computed parameter.

We derive a layer‐recursive algorithm for reconstructing an equal‐travel‐time layered acoustic medium from its transmission response to an impulsive plane wave incident from below. The medium is bounded below by an infinite half‐space from which it is probed, and it is bounded above by a free surface at which its transmission response is measured. This problem arises in the nondestructive evaluation of layered composite materials, in which the transmission response is measured. The algorithm is similar in form to the Levinson algorithm, which reconstructs a layered medium from its reflection response; it is closely related to the lattice algorithm for computing the QR factorization of a matrix. We relate the new algorithm to the Levinson, fast Cholesky, and lattice algorithms, all of which are well‐known fast algorithms for signal processing.

A discrete layered absorbing acoustic medium is probed with an impulsive plane wave at normal incidence, and the reflection response of the medium is measured. The absorbing nature of the medium is specified by a complex wave speed, corresponding to a constant Q. A fast layer stripping algorithm for recovering the profile of the complex wave speed as a function of depth is obtained. The algorithm works by recursively computing the reflection response of that portion of the medium below the current depth, thus stripping away the effects of that portion of the medium at and above the current depth. The algorithm is, in principle, exact, includes the effects of multiple reflections, and admits a clear physical interpretation of its operation. Several computer simulations illustrate the performance of the algorithm on synthetic data.

The Schur algorithm is a signal processing algorithm which works on a layer‐stripping principle. Its time‐domain version, the fast Cholesky recursion, is a fast and efficient algorithm well‐suited for high‐speed data processing. In this paper, these algorithms are applied to the inverse problem for a continuous layered acoustic medium. Three different excitations of the medium are considered: impulsive plane waves at normal incidence, impulsive plane waves at oblique incidence, and spherical waves emanating from an impulsive point source. The fast algorithms obtained for each of these problems seem to be computationally superior to past work done on these problems that employed Gelfand–Levitan theory to reconstruct the potential of a Schrodinger equation.

Two different layer recursive algorithms for reconstructing an inhomogeneous viscoelastic medium are presented. A viscoelastic medium is one whose stress–strain relation is not instantaneous, in that the present stress depends on past strains. Hence, the inverse scattering problem for such media involves the reconstruction of functions varying in time as well as in space; thus it is much more difficult than the inverse problem for simple acoustic media. Both algorithms reconstruct an inhomogeneous viscoelastic medium whose wave speed c(z) and elastic modulus G(z,t) vary with position; variations in density ρ(z) can also be reconstructed using the first algorithm. The first algorithm requires as data the steady‐state reflection response of the medium to a point harmonic source at a wide range of frequencies. The second algorithm requires as data only the reflection response to a normally incident plane wave, but applies only to layered media whose relaxation times are less than the two‐way travel time thro...

A continuous layered medium is probed by sinusoidal spherical waves from a point harmonic source, and the reflection response of the medium is measured. By performing this experiment at two different source frequencies, the separate density and sound‐speed profiles ρ(z) and c(z) are recovered. Two experimental configurations are considered: the case in which the inhomogeneous medium is confined between two infinite homogeneous half‐spaces and is probed from the upper half‐space, and the case in which the inhomogeneous medium is bounded above by a free‐surface, at which the source is located. Layer‐stripping fast algorithm solutions to both of these inverse problems are obtained and illustrated with examples. The algorithms are interpreted physically as constructing distributions of image sources that simulate the medium response at each depth.