A Full-waveform Approximation of Finite-Sized Acoustic Apertures: Forward and Adjoint Wavefields
This work advances forward and inverse problem solutions in acoustics, with applications in therapeutic ultrasound optimization, attenuation reconstruction, and photoacoustic tomography, but it appears incremental as it builds on existing integral formulations.
The study established an equivalence between analytic monopole/dipole integral formulations and their full-waveform approximations, introducing reception operators to map free space pressure wavefields to boundary-measured fields and deriving the adjoint operator, which coincides with the time-reversed dipole integral formula under Dirichlet-type boundary data.
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.