A smooth global model for scattering in layered media

TL;DR

This paper introduces a smooth, unified model for calculating the Green's function in layered media with discontinuous coefficients, simplifying the analysis of wave scattering and data dependence in acoustic imaging.

Contribution

It presents a novel system of equations with smooth coefficients that captures all layered media configurations and reveals a new family of orthogonal polynomials related to the scattering process.

Findings

A single system governs all n-layered media simultaneously.

The system can be solved explicitly using separation of variables.

A new family of orthogonal polynomials on the disk is introduced.

Abstract

Layered media have been studied extensively both for their importance in imaging technologies and as an example of a hyperbolic PDE with discontinuous coefficients. From the perspective of acoustic imaging, the time limited impulse response at the boundary, or boundary Green's function, represents measured data, and the objective is to determine coefficients, which encode physical parameters, from the data. The present paper resolves two fundamental open problems for layered media: (1) how to compute the time limited Green's function in the presence of discontinuous coefficients; and (2) to determine precisely how data depends on coefficients. We show that there exists a single system of equations in $3n$-dimensional space that governs the parameterized family of all $n$-layered media simultaneously. The alternate system has smooth coefficients, can be solved directly by separation of variables, and recovers the impulse response at the boundary for the original equations. The analysis brings to light an exotic laplacian---hybrid between the euclidean and hyperbolic laplacians---that plays a central role in the scattering process. Its eigenfunctions comprise a new family of orthogonal polynomials on the disk. These serve as building blocks for a universal wavefield in terms of which the dependence of data on coefficients has a simple description: reflection data is obtained by sampling a translate of the wavefield on the integer lattice and then pushing forward by a linear functional, where the translate and pushforward correspond to reflectivity and layer depth vectors, respectively.