Exact single-sided inverse scattering in three-dimensions

TL;DR

This paper proves an exact integral equation for single-sided inverse scattering in three dimensions, applicable to various physical systems, and demonstrates its equivalence to the Wapenaar iteration algorithm, including all internal reflections.

Contribution

It provides a simple, physical proof of an exact integral equation for 3D single-sided inverse scattering, extending applicability beyond acoustic waves to other time-reversal invariant systems.

Findings

The integral equation is exact and includes all internal multiple reflections.

It is equivalent to the Wapenaar iteration algorithm.

Applicable to diverse fields like seismology, electronics, and quantum physics.

Abstract

During the past three years, Wapenaar, Snieder, Broggini and others have developed an algorithm to compute the Green's function for any point inside a medium to points on the surface from measurements on that surface only. Their algorithm is based on focusing an incoming wavefield to a single point in order to create a virtual source at the focus. The procedure has been justified only by heuristic arguments. In this paper I am using simple physical arguments to prove an integral equation for single-sided, higher-dimensional inverse scattering. This integral equation is equivalent to the Wapenaar iteration algorithm. The equation will be exact, including all internal multiple reflections. The derivation makes use of time-invariance but does not use the explicit form of the wave equation. It is therefore not only applicable to the acoustic wave equation, but also to other time-reversal invariant systems. Potential areas include seismology, electronics, microwave and ultrasonic inverse scattering as well as quantum physics. The simplicity and generality of the argument will make this paper accessible to researchers from all the mentioned fields.