Hardy space on the polydisk and scattering in layered media

TL;DR

This paper uses Hardy spaces on the polydisk to provide an exact, direct framework for understanding scattering in layered media, revealing the nonlinear dependence of Green's functions and enabling precise calculation of reflection coefficients.

Contribution

It introduces an exact, explicit approach using Hardy space theory and Jacobi polynomial connections for analyzing layered media scattering, avoiding approximations or iterative methods.

Findings

Power spectrum of Green's function is approximately constant.

Derived formulas for reflection coefficients from amplitude data.

Provided a qualitative interpretation of scattering dependence on media parameters.

Abstract

Hardy space on the polydisk provides the setting for a global description of scattering in piecewise-constant layered media, giving a simple qualitative interpretation for the nonlinear dependence of the Green's function on reflection coefficients and layer depths. Using explicit formulas for amplitudes, we prove that the power spectrum of the Green's function is approximately constant. In addition we exploit a connection to Jacobi polynomials to derive formulas for computing reflection coefficients from partial amplitude data. Unlike most approaches to layered media, which variously involve scaling limits, approximations or iterative methods, the formulas and methods in the present paper are exact and direct.