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

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 monopole and dipole integral formulations and their full-waveform approximations. Leveraging this equivalence, we introduce reception operators that map free space-time 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 an inverse constant factor--with the time-reversed form of the interior-field dipole integral formula, evaluated on the boundary. These findings have significant implications for both forward and inverse problems in acoustics, particularly in applications requiring accurate amplitude modeling, such as therapeutic ultrasound optimization, attenuation reconstruction, and photoacoustic tomography.