An inverse moving point source problem in inhomogeneous acoustic media: uniqueness, stability and reconstruction

1 Introduction

• Earlier moving-source methods largely assume homogeneous media, while an inhomogeneous-medium method cited here recovers only partial trajectory geometry at low frequencies.The paper studies joint trajectory-and-intensity recovery in the inhomogeneous setting.Source: §1 · pp. 1–2
• Scattered tails in inhomogeneous media may couple receiver measurements to source history, complicating joint recovery of trajectory and intensity.Source: §1 · p. 2
• Whole-boundary measurements over a finite time interval uniquely determine the moving source through temporal regularization and continuation outside trajectory traces.Source: §1 · p. 2
• Under suitable propagation and rank assumptions, waveforms from four fixed receivers outside the source region yield conditional joint Lipschitz stability.The assumptions can be realized by small perturbations of a homogeneous background medium.Source: §1 · p. 2
• The paper develops a Volterra–Euler scheme with an error estimate and uses a spectral reconstruction method for general inhomogeneous media.Numerical experiments with noisy four-receiver data demonstrate the effectiveness of the proposed methods.Source: §1 · p. 2

• For pairs (a_j,λ_j) satisfying (1.5) and activity condition (2.1), with causal solutions of (1.1) sourced by F_j=λ_j(t)δ_{a_j(t)}, the theorem concludes λ1(t)=λ2(t) and a1(t)=a2(t) almost everywhere on (0,T0).Source: §2 · p. 3
• Removing finitely many compact Lipschitz trajectory curves leaves a connected complement and preserves the infimum of travel times between points outside them.This geometric result supports applying continuation away from trajectory traces.Source: §2 · p. 4
• Temporal mollification, exterior wave-energy estimates, and one-sided unique continuation show that the source difference vanishes in the visible region.The visible region is defined by t+r(x)<T; this yields equality of the source distributions there.Source: §2 · p. 5
• For finite sums of moving point sources, equal boundary traces determine the complete source distributions when T≥T0+R0, but collisions, cancellation, or label ambiguity may obstruct individual recovery.The extension concerns equality of total source distributions, not necessarily unique identification of each source.Source: §2 · p. 5

3 Inverse moving source problem: four-point measurements

• The section studies conditional joint Lipschitz stability of trajectory and intensity recovery from pressure waveforms at four fixed receivers outside the source region.Theorem 2 considers four distinct receivers and admissible source pairs with common prescribed endpoints.Source: §3 · p. 6
• The Green function is assumed to have a direct arrival plus a C1 scattered tail, with no additional singular arrivals before the observation horizon.The travel-time geometry is also required to be uniformly nondegenerate.Source: §3 · p. 6
• For sufficiently small perturbations of a homogeneous background, the propagation and rank conditions hold; a regular tetrahedron of receivers works when the source region is a sufficiently small central ball.The rank condition must hold in the homogeneous reference medium.Source: §3 · p. 6
• The analysis derives a Volterra identity for receiver waveforms, whose uniform Lipschitz regularity on arrival intervals supports waveform comparison.The proof also uses a tail-subtracted quantity and a matrix factorization with uniformly bounded inverses.Source: §3 · pp. 7–8

4.1 Volterra–Euler scheme

• The Volterra–Euler scheme evaluates the history tail from reconstructed source history, solves for current velocity and intensity, then advances the trajectory by Euler steps.An error estimate applies when the scheme completes all steps without a failed check; the reconstructed endpoint is not imposed, and its error is bounded by Cqerr.Source: §4.1 · pp. 8–10

4.2 Spectral reconstruction

• The spectral method fits complete receiver-waveform spectra using the full outgoing Green function, accounting for direct propagation and medium scattering.It represents trajectory and intensity with clamped cubic B-splines and uses bounded variable projection for intensity with trajectory-coefficient updates.Source: §4.2 · p. 10

4.3 Numerical experiments

• The spectral experiments demonstrate reconstruction in a known inhomogeneous medium without imposing the Volterra–Euler scheme’s time-domain propagation assumptions.The Volterra–Euler experiments use a finite-time Green function known explicitly.Source: §4 · p. 8, §4.3 · p. 11
• At 10% noise, Volterra–Euler trajectory errors range from 5.06% to 7.63%, and intensity errors range from 4.44% to 4.88%; all 16 marches complete without a failed safeguard.Reported errors use the raw piecewise-linear trajectory and piecewise-constant intensity.Source: §4.3 · p. 13, Table 1 · p. 13
• At 10% noise, spectral trajectory errors are 11.78%–19.83% and intensity errors are 1.78%–3.09%; independent parameter selection reduces trajectory errors in ten of twelve noisy cases and intensity errors in all twelve.The ellipse has slightly larger trajectory errors at 3% and 5% noise than the coupled-parameter baseline.Source: §4.3 · pp. 14–15, Table 2 · p. 15

A.1 Causal solutions with point sources

• The moving point source belongs to L1(R;H−2) because each finite Radon measure has a uniformly bounded H−2 norm.The source acts on test functions through its trajectory and intensity.Source: §A.1 · p. 15
• A source supported in t ≥ 0 has a unique causal distributional solution, obtained through the wave propagator and uniqueness by duality.The solution satisfies the stated wave equation and zero initial values.Source: §A.1 · pp. 15–16
• Compactly supported sources propagate at finite speed with respect to the acoustic metric g.A source at (y,s) can influence (x,t) only when t ≥ s + dg(x,y).Source: §A.1 · pp. 15–16
• The operator A = −n−1∆ is nonnegative and self-adjoint on L2(R3,ndx), with domain H2.The associated quadratic form is the integral of |∇v|2, and multiplication by n±1 is bounded on the Sobolev spaces used.Source: §A.1 · p. 15
• By the spectral theorem [27, Appendix A], the map H−2 → H−1 is uniformly bounded whenever 0 ≤ t − s ≤ L.At spectral value zero, the quotient is defined by continuity as t − s.Source: §A.1 · p. 16

A.2 Distributional pressure traces

• When the source support K is compact and disjoint from a boundary collar N, the convolved field Uη is smooth near Γ.Interior elliptic regularity and Sobolev embedding yield this smoothness, with seminorm bounds on each fixed compact time support.Source: §A.2 · p. 16
• The boundary pairing extends uniquely to a distribution on Γ×R, and temporal convolution commutes with taking its trace.The extension follows from the preceding bounds and the Schwartz kernel theorem.Source: §A.2 · p. 16
• Finite propagation makes the boundary trace vanish before dg(D0,Γ), so equality on Γ×(0,T) implies equality on Γ×(−∞,T).The positive travel-time gap follows from source support in D0×[0,T0].Source: §A.2 · p. 16

Appendix B Green function representation

• The source-parametrized Hadamard construction uses a normal neighborhood in which the exponential map is a diffeomorphism.The neighborhood is chosen uniformly over source points near the compact set D.Source: Appendix B · p. 16
• The construction uses geodesic normal coordinates and the Hadamard transport formulas to obtain smooth, time-independent coefficients with uniform bounds.The normal-coordinate density is defined by pulling back the metric volume.Source: Appendix B · p. 17
• For N ≥ m+2, the regular residual is jointly Cm; for m ≥ 4, its causal correction is jointly C1 with bounds uniform in the source point.The residual’s cone factor has derivatives extending continuously by zero at the vertex under the stated order condition.Source: Appendix B · pp. 16–17
• For four distinct receivers outside D, sufficiently small perturbations nε = 1+εq0 yield smooth travel times and positive amplitudes with C1-extendible tails.The result applies over the stated interval 0 < t < H.Source: Appendix B · p. 17

B.1 Proof of Proposition B.2

• The receivers are chosen outside a neighborhood of D, with Euclidean travel distances bounded by H.Metric comparison preserves the strict travel-time bound for sufficiently small |ε|.Source: §B.1 · p. 17
• For small |ε|, source-centered exponential maps are diffeomorphisms on uniform metric balls around the compact source region.The proof bounds the geodesic equation and its linearization, then uses injectivity and nonsingularity of the exponential map.Source: §B.1 · pp. 17–18
• The finite Hadamard–Riesz construction gives a local causal kernel on the resulting normal domain.Its phase uses the metric distance ρ = dgε(x,y).Source: §B.1 · p. 18
• Spatial and temporal cutoffs produce a jointly Cm forcing, and a causal solution cancels its residual before time bH.The correction is jointly C1 and uniformly controlled in the source point.Source: §B.1 · p. 18
• At each separated receiver, the travel time is less than H, the leading amplitude is positive, and the correction can be included in a C1 tail.Compactness supplies a positive lower bound on the amplitude and the required norm bounds.Source: §B.1 · p. 18
• Small |ε| preserves the uniform positive rank bound because the travel-time maps converge in C2(D) to their Euclidean counterparts.The smallest singular value is continuous under this convergence.Source: §B.1 · p. 18

C.1 Interior sampling and interpolation

• Retain all samples strictly inside each receiver’s arrival interval; when its endpoints are approximate, account for cropping at both ends.The 2εK cropping term is omitted when the endpoints are known exactly.Source: §C.1 · pp. 18–19
• The condition 4εK + 2∆ < θT0 guarantees at least two retained samples per receiver.The interval-length bound is |bI safe_i| ≥ θT0 − 4εK.Source: §C.1 · p. 19
• Piecewise-linear interpolation with constant extensions bounds waveform error using the sampling error, grid spacing, and endpoint gaps.The interpolation error between retained samples is bounded by δ + Lw∆; outside their range, the constant extensions use the corresponding endpoint gap.Source: §C.1 · p. 19

C.2 Proof of Theorem 3

• The proof uses lower bounds on pi(s) and σmin(M∗(s)), and an upper bound on ∥M∗(s)−1∥.The stated bounds are pi(s) ≥ p∗, σmin(M∗(s)) ≥ m∗, and ∥M∗(s)−1∥ ≤ KM.Source: §C.2 · p. 19
• Left-endpoint quadrature contributes an error of at most Ch when the exact tail integrand is uniformly Lipschitz in emission time.The proof attributes this Lipschitz property to the source bounds and C1 bounds on Vi.Source: §C.2 · p. 19
• The induction keeps current evaluations in K × [0,H] and verifies the checks before and after each solve.The proof chooses ε0 small enough relative to p∗ and m∗, then uses the error bounds to preserve the checks and keep the next node in K.Source: §C.2 · p. 20
• max 0≤n≤N en + max 0≤n<N dn ≤ Cqerr across all completed steps.The discrete Gronwall inequality bounds dn and position errors on accepted steps; the induction verifies that all steps are completed.Source: §C.2 · p. 20
• Arrival times are monotone for every receiver.The proof uses the piecewise-defined approximations and the chain rule almost everywhere.Source: §C.2 · pp. 20–21

Data Availability Statement

• Data supporting the numerical results are available from the corresponding author upon reasonable request.The experiments use synthetic data generated using the models and parameters described in the article.Source: Data Availability Statement · p. 21