Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

TL;DR

This paper presents a method for uniquely determining piecewise constant wave speeds in a 1D Helmholtz equation from limited frequency data, with an exact reconstruction algorithm under certain conditions and effective approximations otherwise.

Contribution

It introduces a novel direct reconstruction algorithm for inverse scattering with piecewise constant wave speeds, linking scattering data to automorphisms of the unit disk and orthogonal polynomials.

Findings

Algorithm achieves exact reconstruction with sufficient frequency range and spacing.

Numerical examples demonstrate effective approximations for general wave speeds.

Theoretical insight connects scattering data to automorphisms and orthogonal polynomials.

Abstract

This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies and the jump points of the wave speed are equally spaced with respect to travel time. Numerical examples show that the algorithm works also in the general case of arbitrary wave speed (either with jumps or continuously varying etc.) giving progressively more accurate approximations as the range of recorded frequencies increases. A key underlying theoretical insight is to associate scattering data to compositions of automorphisms of the unit disk, which are in turn related to orthogonal polynomials on the unit circle. The algorithm exploits the three-term recurrence of orthogonal polynomials to reduce the required computation.