## An inverse scattering problem for hyperbolic systems of first-order in semi-infinite media

##### Abstract

The quiescent inverse scattering problem for multi-component hyperbolic systems of first order in spatial one dimension involves the reconstruction, from boundary data for ƒÕ, the real inhomogeneity matrix B{z), under the assumption that ƒÕ satisfies the system of equations. Here ƒÕ = ƒÕ(z,t) is an n vector, and J4 is a known constant n x n diagonal matrix. The approach to be adopted in this work towards the solution of this problem deals with the scattering operator for L, rather than the above differential system itself. Thus the first significant result of this dissertation is the development of a set of integro/partial-differential equations (the time-dependent invariant imbedding equations) satisfied by the scattering operator. The initial value of this set of equations is obtained by computing the discontinuities in the Riemann-Green function for the operator L. These discontinuities are directly related to the off-diagonal elements of B{z), and thus the relation between the discontinuities in the Riemann-Green function and the elements of B{z) defines a local one-to-one correspondence between the elements of B{z) and the reflection and transmission functions, the latter being defined in terms of the Riemann-Green function (matrix). This one-to-one correspondence, which supplies an important link between the inverse problem data and the elements of B{z), is also used to induce a natural decomposition of B{z) in terms of the data to be used in the inversion scheme (e.g., right reflection data only, right reflection and left transmission data only, etc.). This one-to-one correspondence between the kernels of the scattering operator and the elements of B{z) constitutes the second significant result obtained here.
The major result of this dissertation is the solution of the inverse scattering problem for semi-infinite media, for which only the invariant imbedding equation for the left reflection function (matrix) is utilized. The algorithm described for this solution requires B{z) to be of the special form, where the Gij are known matrices, and the above partition of B{z) is defined in accord with the positive and negative eigenvalues of A. Thus recovering the elements of B21{z) at each z yields B{z). Several examples of the reconstruction scheme are also presented.
In addition, we fully analyze and describe the solution of the inverse scattering problem for two-component systems in both the semi-infinite and bounded geometries. The majority of the results described in this dissertation for this problem may also be found in the literature. Our emphasis is on the methods used to solve the inverse problem for the various geometries and a priori assumptions. This discussion includes, for instance, a brief description of the classic layer-peeling algorithm for semi-infinite media, as presented by Bube and Burridge, and the solution of the second-order hyperbolic equations by invariant imbedding for bounded media, as presented by Corones, Kristensson and Krueger. The method we employ later in this dissertation for the solution of the inverse scattering problem for n-component systems in semi-infinite media is an extension of the ideas used by Corones, Kristensson and Krueger, and includes the layer-peeling algorithm for two-component systems as a special case. Our major contribution for the solution of the inverse scattering problem for two-component systems is in collecting the various results and presenting them under one framework. Finally, we briefly discuss how it might be possible to extend this set of ideas to solve the general inverse scattering problem on bounded domains.
It should be emphasized that the results of this dissertation are obtained by working entirely in the time domain, as opposed to the frequency-domain approach that often is used for such inverse problems. Further the reconstruction algorithm to be presented for the elements of B(z) in the semi-infinite setting is non-iterative. We also assume throughout this dissertation that the given inverse problem data is noise-free, and hence no smoothing of the data will be considered.