Linear and Nonlinear Inverse Scattering

In this paper we discuss one dimensional scattering and inverse scattering for the Helmholtz equation on the half line from the point of view of the layer stripping. By full or nonlinear scattering, we mean the transformation between the index of refraction (actually half of its logarithmic derivative) and the reflection coefficient. We refer to this mapping as nonlinear scattering, because the mapping itself is nonlinear. Another appropriate name is multiple scattering, as this model includes the affects of multiple reflections. By linear scattering we mean the Born, or single scattering, approximation. This is the Frechet derivative of the full scattering transform at the constant index of refraction, which can be calculated to be exactly the Fourier transform. In [6], we introduced a variant of layer stripping based on causality and the Riesz transform, rather than on trace formulas – see [2], [3],[4],or [7],for other approaches to layer stripping. A byproduct of our layer stripping formalism was the discovery of a nonlinear Plancherel equality, which plays a role in the analysis of the inverse scattering problem, analogous to that played by the linear Plancherel equality in developing the theory of the Fourier transform. That linear-nonlinear analogy sets the theme for this work. In the next section, we review the main results of [6], including a brief derivation of the nonlinear Plancherel equality. We show how this equality suggests a natural metric for measuring the distance between reflection coefficients and show that the scattering and inverse scattering maps become homeomorphisms when we use this metric. In section 2, we exhibit a nonlinear Riesz transform , which plays for the nonlinear inverse scattering problem the same role in enforcing causality as the linear Riesz transform 1 plays in signal processing. We then construct a numerical inverse scattering algorithm 2 based on the nonlinear Riesz transform. A rough statement of the results there (compare theorems 2.1 and 2.2 with theorems 2.3 and 2.4) is that the inverse scattering transform is a Fourier decomposition, with the addition of Fourier modes replaced by the composition of these modes as conformal maps of the unit disk. Notice that such a composition rule preserves the nonlinear constraint that the modulus of the reflection coefficient remain less than one. In the final section we prove two linear and nonlinear Paley-Wiener theorems and a Shannon sampling theorem. One of our PaleyWeiner theorems (theorem 3.2) appears to be new even in the linear case.