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arxiv: 2212.04466 · v24 · submitted 2022-12-08 · 🧮 math.NA · cs.NA

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

Ashkan Javaherian , Seyed Kamaledin Setarehdan This is my paper

Pith reviewed 2026-05-24 10:05 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords acoustic wave equationfull-waveform approximationadjoint operatorKirchhoff-Helmholtz integralreception operatorsDirichlet boundary dataphotoacoustic tomographytherapeutic ultrasound
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The pith

Under Dirichlet boundary data the adjoint of the acoustic forward operator coincides with the time-reversed dipole integral on receiver surfaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first shows that the analytic monopole and dipole surface integrals are exactly equivalent to their full-waveform approximations for finite apertures. It then defines reception operators that extract boundary measurements from free-space pressure fields obtained by solving the wave equation. Using this trace mapping the adjoint of the forward operator is derived, and the central result follows: under the standard assumption of Dirichlet-type boundary data the adjoint equals a constant multiple of the interior time-reversed dipole integral evaluated on the receivers. This equivalence supplies a practical route to accurate forward modeling and adjoint-state inversion when amplitude fidelity matters.

Core claim

The paper establishes equivalence between the analytic expressions of the monopole and dipole integral formulations and their full-waveform approximations. It introduces reception operators that map free-space pressure wavefields onto measured fields restricted to the boundary. The adjoint of the forward operator is derived from this mapping. 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.

What carries the argument

Reception operators that map free-space pressure wavefields to boundary measurements, from which the adjoint is obtained and shown to match the time-reversed dipole integral.

If this is right

  • Forward modeling of finite acoustic apertures can use the full-waveform approximations in place of the classical integrals.
  • Adjoint-state methods for inverse problems become implementable by simple time reversal on the receiver surfaces.
  • Amplitude-preserving reconstructions are available for therapeutic ultrasound optimization and attenuation imaging.
  • The same adjoint construction applies directly to photoacoustic tomography under the stated boundary assumption.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result may allow existing time-reversal codes to serve as adjoint operators without additional surface-integral machinery.
  • If the Dirichlet assumption is relaxed, a different scaling or correction term would be needed to recover the adjoint.
  • The same trace-mapping approach could be examined for other linear wave equations where surface sources appear.

Load-bearing premise

Boundary data are of Dirichlet type so that pressure is prescribed on the surface.

What would settle it

A direct numerical comparison, for a known Dirichlet boundary setup, of the adjoint operator computed by the derived formula versus the interior time-reversed dipole integral; any systematic mismatch falsifies the claimed coincidence.

Figures

Figures reproduced from arXiv: 2212.04466 by Ashkan Javaherian, Seyed Kamaledin Setarehdan.

Figure 1
Figure 1. Figure 1: (a) A single emitter point and 39 receiver points positioned on a hemi￾sphere with a radius of 5.6cm, (b) source pulse, s, in the time domain, (c) source pulse, s, decomposed into amplitude and phase in the frequency domain. 6.1.2. Result [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Analytic action of the causal Green’s function on a source defined on a single point. The analytic action approximated on receiver 10 and at all chosen frequencies, a) amplitude, b) phase. The analytic action approximated at the single frequency 1MHz and on all receiver points, c) amplitude d) phase. approximated using the full-field algorithm 1 and using a mass source discretised in space and time by Eq. … view at source ↗
Figure 3
Figure 3. Figure 3: a) A disc-shaped emitter and 64 receiver points. The positions are ordered by changing θ, φ and r. b) source pulse u n (rigid-baffle condition), c) source pulse p (soft-baffle condition). In addition, the azimuthal angles are set θ ∈ {π/4, 3π/4, 5π/4, 7π/4}. The receiver positions are ordered based on changing θ, φ and r in order. Accordingly, for receiver points 1-4, r = 6.5cm and φ = 0 are set, and the p… view at source ↗
Figure 4
Figure 4. Figure 4: The wavefield approximated on the plane x 1 = 3.1cm and at a single time 45µs after excitation of the disc-shaped emitter by a source pulse u n, which is shown in figure 3(b). The emitter disc’s centre is placed on the origin of the Cartesian coordinates, as shown in figure 3(a) by the yellow colour. (Not shown here.) a) Field II, b) Full-field (RE= 7.10%). by a factor 4. Consider that for both analytic an… view at source ↗
Figure 5
Figure 5. Figure 5: The wavefield approximated and recorded in time on the receivers after an excitation of the disc-shaped emitter by a source pulse u n, which is shown in figure 3(b). Receiver points: a) 1, b) 5, c) 9, d) 13. The full-field approach and the Field II toolbox were used to approximate the monopole integral formula (32). the dipole integral formula (34), which is equivalent to a summation of the actions of the … view at source ↗
Figure 6
Figure 6. Figure 6: The relative error (RE) of the wavefield approximated by the full-field approach in time and on the chosen receiver points after an excitation of the emitter disc by a source pulse u n, which is shown in figure 3(b). The wavefield analytically calculated using the Field II toolbox is used as the benchmark. The receiver positions are ordered by changing θ, φ and r. On-grid sampling. As explained in section … view at source ↗
Figure 7
Figure 7. Figure 7: The wavefield approximated on the plane x 1 = 3.1cm and at a time 45µs after excitation of the disc-shaped emitter by a source pulse p, which is shown in figure 3(c). The centre of the emitter disc is placed on the origin of the Cartesian coordinates, as shown in figure 3(a). (Not shown here.) a) Field II (far-field term) + Analytic (near-field term) , b) Full-field using a mass source discretised using Eq… view at source ↗
Figure 8
Figure 8. Figure 8: The wavefield approximated on the receivers in time after an excitation of the disc-shaped emitter by a source pulse p, which is shown in figure 3(c). Receiver points: a) 1, b) 5, c) 9, d) 13. The full-field approach that uses algorithm 1 and a momentum source defined by Eq. (52) approximates directly the original dipole formula (34). The Field II toolbox is used to approximate the far-field term of the di… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper establishes an equivalence between the analytic expressions of the monopole and dipole integral formulations (via Kirchhoff-Helmholtz and Rayleigh-Sommerfeld identities) and their full-waveform approximations for the acoustic wave equation with surface sources. It defines reception operators as trace maps from free-space pressure solutions of the wave equation onto boundary-restricted measured fields, then derives the adjoint of this forward operator. Under the explicitly stated assumption of Dirichlet-type boundary data, the adjoint is shown to coincide, up to a constant factor, with the interior-field time-reversed form of the dipole integral formula evaluated on the receiver surfaces. The work targets improved forward approximations and adjoint-based inverse problems in acoustics.

Significance. If the derivations hold, the conditional equivalence supplies a parameter-free link between boundary-integral representations and full-waveform modeling for finite apertures, preserving amplitude information needed for therapeutic ultrasound optimization, attenuation reconstruction, and photoacoustic tomography. The explicit adjoint construction under a standard practical assumption enables direct use of time-reversal techniques without hidden regularity requirements or unstated limit interchanges. The paper's strength is the upfront declaration of scope, rendering the central claim falsifiable by direct numerical comparison of the two operator forms.

minor comments (2)
  1. [Abstract] The abstract would be strengthened by a single sentence stating the precise scaling factor in the adjoint identity or the section where the equivalence is proved.
  2. Figure captions should explicitly label the Dirichlet boundary data case when comparing forward and adjoint fields.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the provided report, so we offer no point-by-point rebuttals. We will address any editorial or minor suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the acoustic wave equation and standard Kirchhoff-Helmholtz / Rayleigh-Sommerfeld integral identities, constructs explicit reception (trace) operators, and obtains the adjoint via the usual duality pairing. The central claim is an explicit conditional statement: under the declared Dirichlet boundary assumption the adjoint equals (up to scaling) the time-reversed interior dipole integral. No parameter is fitted to data and then relabeled a prediction, no self-citation supplies a uniqueness theorem or ansatz, and the equivalence is not obtained by redefinition. The chain is therefore self-contained against external functional-analysis benchmarks for adjoints of boundary-trace operators.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or additional axioms are stated beyond the Dirichlet boundary assumption and the claimed equivalence.

axioms (2)
  • domain assumption Equivalence holds between analytic monopole/dipole integral expressions and their full-waveform approximations
    Stated as the starting point of the study in the abstract.
  • domain assumption Dirichlet-type boundary data is the relevant practical case
    Invoked to obtain the adjoint coincidence result.

pith-pipeline@v0.9.0 · 5724 in / 1287 out tokens · 23560 ms · 2026-05-24T10:05:28.754605+00:00 · methodology

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