The combinatorics of scattering in layered media

TL;DR

This paper develops a combinatorial tree-based approach to derive explicit formulas for wave reflection and transmission in layered media, addressing a longstanding gap in time domain Green's function analysis.

Contribution

It introduces a novel tree representation of scattering sequences, enabling exact formulas for Green's functions in layered media.

Findings

Derived explicit formulas for Green's functions

Solved the combinatorial obstacle in scattering sequence analysis

Enhanced understanding of wave behavior in layered structures

Abstract

Reflection and transmission of waves in piecewise constant layered media are important in various imaging modalities and have been studied extensively. Despite this, no exact time domain formulas for the Green's functions have been established. Indeed, there is an underlying combinatorial obstacle: the analysis of scattering sequences. In the present paper we exploit a representation of scattering sequences in terms of trees to solve completely the inherent combinatorial problem, and thereby derive new, explicit formulas for the reflection and transmission Green's functions.