# Reciprocity and representation theorems for flux- and field-normalised decomposed wave fields

## TL;DR

This paper develops reciprocity and representation theorems for flux- and field-normalised decomposed wave fields, providing new integral formulas useful for reflection imaging and multiple scattering analysis in wave propagation.

## Contribution

It systematically analyzes decomposition operators and derives reciprocity and representation theorems for decomposed wave fields, including Kirchhoff-Helmholtz integrals.

## Key findings

- Derived reciprocity theorems for decomposed wave fields.
- Established representation theorems including Kirchhoff-Helmholtz integrals.
- Applied results to reflection imaging and multiple scattering scenarios.

## Abstract

We consider wave propagation problems in which there is a preferred direction of propagation. To account for propagation in preferred directions, the wave equation is decomposed into a set of coupled equations for waves that propagate in opposite directions along the preferred axis. This decomposition is not unique. We discuss flux-normalised and field-normalised decomposition in a systematic way, analyse the symmetry properties of the decomposition operators and use these symmetry properties to derive reciprocity theorems for the decomposed wave fields, for both types of normalisation. Based on the field-normalised reciprocity theorems, we derive representation theorems for decomposed wave fields. In particular we derive double- and single-sided Kirchhoff-Helmholtz integrals for forward and backward propagation of decomposed wave fields. The single-sided Kirchhoff-Helmholtz integrals for backward propagation of field-normalised decomposed wave fields find applications in reflection imaging, accounting for multiple scattering.