A Full-waveform Approximation of Finite-Sized Acoustic Apertures: Forward and Adjoint Wavefields

TL;DR

This paper develops a full-waveform approximation framework for finite-sized acoustic apertures, establishing equivalence with integral formulations and deriving adjoint operators for improved inverse problem solutions.

Contribution

It introduces reception operators and derives the adjoint of the forward operator, enhancing amplitude modeling in acoustic inverse problems.

Findings

Established equivalence between integral formulations and full-waveform approximations.

Derived the adjoint operator as a time-reversed dipole integral formula.

Facilitates improved solutions for inverse problems like ultrasound and photoacoustic tomography.

Abstract

The acoustic wave equation governs wave propagation induced by either volumetric radiation sources, or by surface sources of monopole or dipole type. For surface sources, boundary value problems yield wavefield representations via the Kirchhoff-Helmholtz or Rayleigh-Sommerfeld integrals. This study begins by establishing an equivalence between the analytic expressions of the associated monopole and dipole integral formulations and their full-waveform approximations. Leveraging this equivalence, we introduce reception operators that map free space pressure wavefields-obtained by solving the wave equation-onto measured fields restricted to the boundary. Building on this trace mapping, we derive the adjoint of the forward operator. We show that, under the common practical assumption of Dirichlet-type boundary data, the adjoint operator coincides-up to a constant factor-with the interior-field time-reversed form of the dipole integral formula, evaluated on the receiver surfaces. This study aims to advance the approximation of forward problems and the solution of inverse problems in acoustics, with a particular focus on applications that require accurate amplitude modeling, including therapeutic ultrasound optimization, attenuation reconstruction, and photoacoustic tomography.