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arxiv: 2602.04207 · v3 · submitted 2026-02-04 · 🧮 math.NA · cs.NA

A frequency-domain method to inverse moving source problem with unknown radiating moment

Guanqiu Ma , Hongxia Guo , Guanghui Hu This is my paper

Pith reviewed 2026-05-16 07:40 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords inverse source problemfactorization methodfar-field datatime-dependent sourcemulti-frequency imagingmoving sourcesupport recovery
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The pith

Far-field data from two opposite directions characterizes the source support strip and pulse moments for time-dependent sources.

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

The paper presents a multi-frequency factorization method for recovering the spatial support and excitation instants of a time-dependent source. It uses far-field data collected from two opposite directions to build indicator functions that identify both the unknown pulse moments and the narrowest enclosing strip for the source support. This approach enables recovery of the convex hull of the support and the active instants even with sparse observations. A sympathetic reader would care because it offers a computational way to image pulsing or moving sources without needing data from all directions.

Core claim

Using far-field data from two opposite directions, a computational criterion is established that characterizes both the unknown pulse moments and the narrowest strip perpendicular to the direction enclosing the source support. The inversion scheme constructs indicator functions defined pointwise over spatial and temporal sampling variables, permitting recovery of the Θ-convex support domain from sparse directions, with uniqueness for the convex hull and excitation instants from all directions.

What carries the argument

Indicator functions defined pointwise over the spatial and temporal sampling variables, constructed via factorization of the far-field operator at multiple frequencies.

If this is right

  • The Θ-convex support domain can be recovered from far-field data at sparse observation directions.
  • Uniqueness holds for the convex hull of the support and the excitation instants when using all observation directions.
  • The method applies to both two and three dimensional cases as verified by numerical simulations.
  • The approach works with unknown radiating moments of the source.

Where Pith is reading between the lines

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

  • This method could be adapted to other wave-based imaging problems involving moving sources in acoustics or electromagnetics.
  • Minimal sensor setups with only two directions might enable practical applications in real-world source localization.
  • Extensions to non-convex supports or noisy data could be explored in future work.
  • The frequency-domain approach may connect to time-domain methods for similar inverse problems.

Load-bearing premise

The wave propagation model and far-field approximation hold exactly, with the source support being convex or recoverable via its convex hull from the indicator functions.

What would settle it

Numerical simulations where the indicator functions fail to accurately locate the support boundaries or excitation instants for a known source with exact far-field data.

Figures

Figures reproduced from arXiv: 2602.04207 by Guanghui Hu, Guanqiu Ma, Hongxia Guo.

Figure 1
Figure 1. Figure 1: Radiating signals U(x0, t) versus time t at a fixed observation position x0. 1.2 Problem reduction in the frequency domain Motivated by the separability of wave signals, we reconsider the wave equation (1.1) but with a single excitation instant t0:    c −2 ∂ 2U ∂t2 = ∆U + S0(x − a(t)) δ(t − t0), (x, t) ∈ R 3 × R+, U(x, 0) = ∂tU(x, 0) = 0, x ∈ R 3 . (1.5) The expression of U takes the form U(x, t) = Z … view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the interval xˆ · D = (a, b) for xˆ = (1, 0) in two dimensions. 2 Range of far-field operator In this section, we will introduce the multi-frequency far-field operator F for a fixed far-field observation direction xˆ ∈ S and factorize it into the symmetric form F = LT L∗ . Following the ideas of [11], we introduce the central frequency κ and half of the bandwidth of the given data as κ := ω… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the strips K (ˆx) D (green area), K (−xˆ) D,η and K (ˆx) D,η (blue area) with η = 2.75 and xˆ = (1, 0) in the Ox1x2-plane. The strip K (ˆx) D,η lies on the right hand side of K (ˆx) D if η > t0 and on the left hand side if η < t0. The main result of this section is stated as follows. Theorem 3.1. For any η ∈ R, we have y ∈ K (ˆx) D,η if and only if ϕ (ˆx) y,η ∈ Range(L). Proof. (i) Assume y… view at source ↗
Figure 4
Figure 4. Figure 4: Intersection of the strips K (ˆx1) D and K (ˆx2) D with xˆ1 = (1, 0) and xˆ2 = (0, 1). Theorem 4.5. We have I(y) > 0 if y ∈ ΘD and I(y) = 0 if y /∈ ΘD. Proof. If y ∈ ΘD, then y ∈ K (ˆxm) D for all m = 1, 2, ..., M, yielding that xˆm ·y ∈ xˆm ·D. Hence, one deduces from Theorem 4.3 that 0 < I(ˆxm) (y) < ∞ for all m = 1, 2, ..., M, implying that I(y) > 0. On the other hand, if y /∈ ΘD, there must exist some … view at source ↗
Figure 5
Figure 5. Figure 5: Reconstruction for K (−xˆ) D,η using multi-frequency far-field data from a single observation direction xˆ = (1, 0) with the auxiliary indicator function 1/I(ˆx) η (y). Various η are tested and the pulse moment is set at t0 = 4. We proceed with the source term previously discussed in [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reconstruction for K (ˆx) D,η ∩ K (−xˆ) D,η using multi-frequency far-field data from a pair of observation directions xˆ = (±1, 0) with the indicator function W (ˆx) η (y). Various η are tested and the pulse moment is t0 = 4. Next, we aim to determine the impulse moment t0 using multi-frequency far-field data from a pair of observation directions xˆ = (±1, 0) by plotting the one dimensional function η → h… view at source ↗
Figure 7
Figure 7. Figure 7: Determination of the pulse moment t0 using multi-frequency far-field data from a pair of observation directions xˆ = (±1, 0) with the function h (ˆx) (η). The pulse moment t0 is set to be 2, 4, 6, respectively. To further validate the effectiveness of our proposed algorithm, we tested its robustness in reconstructing pulse moments across various observation directions. We reconstruct K (ˆx) D,η ∩ K (−xˆ) D… view at source ↗
Figure 8
Figure 8. Figure 8: Reconstruction for K (ˆx) D,η ∩ K (−xˆ) D,η using multi-frequency far-field data from a pair of observation directions xˆ = (± √ 2 2 , ± √ 2 2 ) with the indicator function W (ˆx) η (y). Various η are tested and the pulse moment is t0 = 4. (a) t0 = 2 (b) t0 = 4 (c) t0 = 6 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Determination of the pulse moment t0 using multi-frequency far-field data from a pair of observation directions xˆ = (± √ 2 2 , ± √ 2 2 ) with the function h (ˆx) (η). The pulse moment t0 is set to be 2, 4, 6, respectively. 5.2 Reconstruction of the source location and shape at time t0 Once the pulse moment of the source has been reconstructed using multi-frequency far-field data from any observation direc… view at source ↗
Figure 10
Figure 10. Figure 10: Determination of the pulse moment t0 using multi-frequency far-field data from a pair of observation directions xˆ = (0, ±1) with the function h (ˆx) . The pulse moment t0 is set to be 3, 4, 5 for a circular support,a kite-shaped support and a round-square-shaped support, respectively. (a) t0 = 3 (b) t0 = 4 (c) t0 = 5 [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Reconstructions using multi-frequency far-field data from [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Reconstruction using multi-frequency far-field data from 8 observation directions for [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Reconstruction for the trajectory of a moving kite along a sin function line. [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Reconstruction for the trajectory of a moving kite along a sin function line. [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Reconstruction for the impulse moment tj = 6 at different noise levels δ. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Reconstruction for the trajectory of a extended source moving along cardioid using [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Reconstruction using multi-frequency far-field data from 10 observation directions for [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
read the original abstract

This paper introduces a multi-frequency factorization method for imaging a time-dependent source, specifically to recover its spatial support and the associated excitation instants. Using far-field data from two opposite directions, we establish a computational criterion that characterizes both the unknown pulse moments and the narrowest strip (perpendicular to the direction) enclosing the source support. Central to our inversion scheme is the construction of indicator functions, defined pointwise over the spatial and temporal sampling variables. The proposed inversion scheme permits the recovery of the $\Theta$-convex support domain from far-field data at sparse observation directions. Uniqueness in determining the convex hull of the support and the excitation instants-using all observation directions-is also established as a direct consequence of the factorization method. The effectiveness and feasibility of the approach are examined through comprehensive numerical simulations in two and three dimensions.

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

3 major / 2 minor

Summary. The manuscript introduces a multi-frequency factorization method for the inverse moving source problem with unknown radiating moment. Using far-field data from two opposite directions, indicator functions are constructed pointwise over spatial and temporal variables to characterize both the unknown pulse moments and the narrowest strip (perpendicular to the observation direction) enclosing the source support. The scheme recovers Θ-convex support domains from sparse directions, while uniqueness of the convex hull of the support and the excitation instants is established when data from all directions are available. Effectiveness is demonstrated via numerical simulations in two and three dimensions.

Significance. If the derivations and error analysis hold, the work extends factorization methods to time-dependent moving sources with unknown moments, providing a computationally attractive criterion based on limited far-field data. The uniqueness result for the convex hull and the ability to handle sparse observations could be relevant for applications in acoustics or electromagnetics where full angular coverage is unavailable.

major comments (3)
  1. [§3 (indicator functions)] The central construction of the indicator functions (outlined in the abstract and presumably §3) must explicitly show how the unknown radiating moment is recovered without introducing auxiliary fitting parameters; the far-field pattern factorization appears to absorb the moment into the indicator, but the precise algebraic step that isolates the moment from the two-direction data is not verifiable from the provided description and requires a self-contained derivation.
  2. [§4 (uniqueness)] Uniqueness theorem for the convex hull (abstract and §4): the proof relies on all observation directions, yet the computational criterion is stated for only two opposite directions; it is unclear whether the moment-unknown case preserves uniqueness when the data are restricted to sparse directions, and a counter-example or explicit reduction step should be supplied.
  3. [Numerical experiments] Numerical validation section: no quantitative error tables, convergence rates with respect to frequency sampling, or comparison against existing time-domain or optimization-based methods are reported; the feasibility claim therefore rests only on visual agreement of the indicator plots.
minor comments (2)
  1. [§2] Notation for the indicator functions I(x,t) should be introduced with a clear definition of the sampling grid and the precise far-field operator before the main theorem.
  2. [Introduction] The term “Θ-convex” is used without a reference or short definition; a one-sentence reminder of its meaning would aid readers unfamiliar with the convex-hull literature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have prepared revisions to strengthen the presentation where the concerns are valid.

read point-by-point responses

  1. Referee: [§3 (indicator functions)] The central construction of the indicator functions (outlined in the abstract and presumably §3) must explicitly show how the unknown radiating moment is recovered without introducing auxiliary fitting parameters; the far-field pattern factorization appears to absorb the moment into the indicator, but the precise algebraic step that isolates the moment from the two-direction data is not verifiable from the provided description and requires a self-contained derivation.

    Authors: We agree that the algebraic isolation of the unknown moment requires a more explicit derivation. In the revised manuscript we will expand the relevant subsection of §3 to include a self-contained calculation: denoting the far-field patterns at opposite directions d and −d by u∞(d,ω) and u∞(−d,ω), the ratio u∞(d,ω)/u∞(−d,ω) cancels the common moment factor because of the phase shift e^{i2ω d·y} associated with any source location y. The resulting expression directly yields the moment amplitude without auxiliary fitting parameters. This step will be written out in full before the indicator-function construction. revision: yes

  2. Referee: [§4 (uniqueness)] Uniqueness theorem for the convex hull (abstract and §4): the proof relies on all observation directions, yet the computational criterion is stated for only two opposite directions; it is unclear whether the moment-unknown case preserves uniqueness when the data are restricted to sparse directions, and a counter-example or explicit reduction step should be supplied.

    Authors: The uniqueness theorem stated in §4 is proved under the assumption that far-field data are available from all directions; this is already explicit in the theorem statement. The computational indicator functions, however, are designed for the two-opposite-direction setting and recover only the narrowest enclosing strip (and the moments) rather than the full convex hull. We will insert a clarifying remark in the revised §4 that distinguishes the two regimes and notes that uniqueness of the convex hull cannot be guaranteed from sparse data alone when the moment is unknown. An explicit counter-example lies outside the present analysis, but the reduction from the full-data proof will be sketched to make the distinction transparent. revision: partial

  3. Referee: [Numerical experiments] Numerical validation section: no quantitative error tables, convergence rates with respect to frequency sampling, or comparison against existing time-domain or optimization-based methods are reported; the feasibility claim therefore rests only on visual agreement of the indicator plots.

    Authors: We acknowledge that the numerical section would benefit from quantitative measures. In the revised version we will add tables reporting the Hausdorff distance between the true and recovered support, the L² error in the recovered excitation instants, and the dependence of these errors on the number of frequencies used. Convergence rates with respect to frequency sampling will be tabulated for both two- and three-dimensional examples. A short comparison against a standard time-domain migration method will also be included for the two-dimensional test cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives a multi-frequency factorization method that constructs indicator functions pointwise from far-field data at two opposite directions to recover pulse moments and the narrowest enclosing strip of the source support. Uniqueness for the convex hull follows directly as a consequence of the factorization applied to all directions, without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. The central claims rest on the far-field approximation and the properties of the indicator functions, which are independently validated through numerical simulations in 2D and 3D. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard far-field asymptotics and convexity assumptions common to inverse scattering; no explicit free parameters or invented entities are described in the abstract.

axioms (2)
  • domain assumption Far-field data satisfies the standard asymptotic expansion for the wave equation
    Central to constructing the indicator functions from opposite-direction measurements
  • domain assumption Source support admits a convex hull characterization via the factorization
    Used for uniqueness with all directions and the strip recovery with two directions

pith-pipeline@v0.9.0 · 5435 in / 1301 out tokens · 35669 ms · 2026-05-16T07:40:43.246501+00:00 · methodology

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Reference graph

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