Iterative inverse scattering algorithms: Methods of computing Fréchet derivatives

Abstract

Iterative approaches to the nonlinear inverse scattering problem generally attempt to find the scattering distribution that best predicts the data by minimizing a global error norm (e.g., the mean-square error) which quantifies the misfit between a set of measured data and data predicted on the basis of a forward calculation. A crucial quantity in this minimization is the Fréchet derivative of the error norm which tells us how to update the current estimate of the scattering distribution to reduce the global error at each iteration. This paper demonstrates how to compute the Fréchet derivative using three different, but fundamentally equivalent, methods: the conventional adjoint method, the Lagrange multiplier method, and the integral equation method. The first two begin with the wave equation, while the latter method is based on a Lippmann–Schwinger integral equation. These techniques are not only far more efficient, but also numerically less error prone, than “brute force” methods for computing derivatives based, for example, on finite differences. For simplicity, a variational approach is employed in which the fields and scattering distribution are represented by continuous functions, but the finite-dimensional (discretized) problem is shown to follow directly from the continuous-space results.