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

Ashkan Javaherian, Seyed Kamaledin Setarehdan

TL;DR

The paper develops a rigorous time-domain framework for forward and inverse acoustics with finite-sized surfaces, establishing an equivalence between analytic monopole/dipole surface formulations and their full-waveform counterparts. It introduces trace and extension/restriction operators that map between near-surface sources and boundary measurements, and derives the adjoint and time-reversal operators consistent with Dirichlet data, enabling accurate amplitude modeling for applications like therapeutic ultrasound and photoacoustic tomography. The work provides a full discretization strategy and numerical validation against analytical solutions, demonstrating that force-based dipole representations preserve near-field and angular sensitivity better than monopole approximations. These results have practical impact for ultrasound tomography and PAT, offering improved forward/adjoint operators for inversion and optimization with finite-size receivers. Overall, the framework advances amplitude-accurate forward modeling for complex transducer geometries and supports enhanced inverse problem solutions in biomedical acoustics.

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-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 a constant factor-with the interior-field time-reversed form of the dipole integral formula, evaluated on the receiver surfaces. This study has 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.

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

TL;DR

The paper develops a rigorous time-domain framework for forward and inverse acoustics with finite-sized surfaces, establishing an equivalence between analytic monopole/dipole surface formulations and their full-waveform counterparts. It introduces trace and extension/restriction operators that map between near-surface sources and boundary measurements, and derives the adjoint and time-reversal operators consistent with Dirichlet data, enabling accurate amplitude modeling for applications like therapeutic ultrasound and photoacoustic tomography. The work provides a full discretization strategy and numerical validation against analytical solutions, demonstrating that force-based dipole representations preserve near-field and angular sensitivity better than monopole approximations. These results have practical impact for ultrasound tomography and PAT, offering improved forward/adjoint operators for inversion and optimization with finite-size receivers. Overall, the framework advances amplitude-accurate forward modeling for complex transducer geometries and supports enhanced inverse problem solutions in biomedical acoustics.

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-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 a constant factor-with the interior-field time-reversed form of the dipole integral formula, evaluated on the receiver surfaces. This study has 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.
Paper Structure (46 sections, 8 theorems, 101 equations, 9 figures, 1 algorithm)

This paper contains 46 sections, 8 theorems, 101 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Wave Equation in the Time Domain
  3. Cauchy Conditions for Unique Solution
  4. Green's Function Solution to the Wave Equation
  5. Primary Solution
  6. Kirchhoff-Helmholtz Solution
  7. Rayleigh-Sommerfeld solution
  8. Monopole and Dipole Formulas in Terms of Actions of the Causal Green's Function on the Surface Source
  9. Monopole Formula
  10. Dipole formula
  11. Full-Waveform Approximation of the Wave Equation in the Time Domain
  12. System of Wave Equations in Terms of Source $s$
  13. System of Wave Equations in Terms of Regularized Source $\mathcal{S}$
  14. Definition of the Regularized Source $\mathcal{S}$ in Terms of the Surface Integral Formulas
  15. Wavefield Representation in Terms of $\rho_0 \frac{\partial \boldsymbol{u}}{\partial t} \cdot \boldsymbol{n}$ Compactly Supported on an Infinite Plane
  16. ...and 31 more sections

Key Result

Lemma 1

Let $f \in C_0^{\infty}(\mathbb{R}^d \times [0,T])$ be a field consisting of a smooth component plus a smeared jump across the receiver surface $\partial \nu_r \cap \partial \Omega$, where the jump decays as one moves away from the receiver surface and whose smoothness depends on the bandwidth param where $\boldsymbol{n}_{r,+}$ is the outward-pointing unit normal vector on $\partial \nu_r \cap \pa

Figures (9)

  • Figure 1: (a) A single emitter point and 39 receiver points arranged on a hemisphere with a radius of 5.6 cm, (b) the source pulse, $s$, represented in the time domain, (c) the source pulse, $s$, shown in the frequency domain, decomposed into amplitude and phase components.
  • Figure 2: (a) Amplitude and (b) phase of the action of the causal Green's function on a source defined at a single point, evaluated at receiver 10 across all chosen frequencies. (c) Amplitude and (d) phase of the action approximated at a single frequency of 1 MHz, evaluated across all receiver points.
  • Figure 3: (a) A disk-shaped emitter and 64 receiver points, ordered by varying $\theta$, $\varphi$, and $r$. (b) Source pulse $u^{\boldsymbol{n}}$ under rigid-baffle conditions. (c) Source pulse $p$ under soft-baffle conditions.
  • Figure 4: Wavefields approximated on the plane $\boldsymbol{x}^1 = 2.94$ cm at a single time $t = 45\mu\text{s}$, following the excitation of the disk-shaped emitter by the source pulse $u^{\boldsymbol{n}}$ (shown in Figure \ref{['fig:3b']}). The emitter disk's center is positioned at the origin of the Cartesian coordinates, as indicated in yellow in Figure \ref{['fig:3a']} (not shown here). (a) Analytic solution of Eq. \ref{['eq:gr-monopole']} using Field II; (b) Full-waveform approximation using Algorithm \ref{['alg:1']} and a mass source discretized via Eq. \ref{['eq:mass-u-dis']}.
  • Figure 5: Wavefields approximated and recorded in time at receiver points following the excitation of the disk-shaped emitter by the source pulse $u^{\boldsymbol{n}}$ (shown in Figure \ref{['fig:3b']}). Receiver points: (a) 1, (b) 5, (c) 9, (d) 13. The monopole integral formula \ref{['eq:gr-monopole']} was approximated using both the full-waveform approach and the Field II toolbox.
  • ...and 4 more figures

Theorems & Definitions (32)

  • Remark 1
  • Remark 2
  • Definition 1
  • Remark 3
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 22 more