A Non-Decoupled Time-Domain Direct Sampling Method for Inverse Elastic Medium Scattering

Abstract: This work is concerned with an inverse medium problem for elastic waves, in which unknown inhomogeneities are reconstructed from time-resolved boundary measurements. We propose a novel time-domain direct sampling method for locating scatterers from a single incident source, without imposing specific assumptions on the temporal profile of the excitation. In particular, the imaging functional introduces a time-shifted correlation strategy that replaces the traditional $P$-$S$ wave decomposition with a travel-time alignment mechanism, thereby enabling direct imaging from the coupled elastic wave field. To analyze the proposed time-domain imaging functional, we employ Parseval's identity for the Fourier--Laplace transform and reformulate the functional in the frequency domain. By exploiting properties of modified Bessel functions, we characterize the asymptotic behavior of the imaging functional and show that it attains its maximum at the target location, which enables reliable identification of the scatterer. Rigorous theoretical justifications are provided to substantiate the effectiveness of the proposed method. Numerical experiments are also presented to demonstrate its performance and applicability.

• New technique achieves accurate 2D/3D inverse elastic medium scattering with a single incident source using a non-decoupled time-domain method.
• Process hinges on a unique travel-time alignment mechanism instead of traditional Helmholtz decomposition.
• Results demonstrate high-resolution scatterer localization in a variety of scenarios, including full aperture and noise contamination.

Overview and problem setting

The paper addresses the inverse medium problem in time-domain elastodynamics: reconstructing the compact support of an inhomogeneous elastic scatterer $D \subset \mathbb{R}^d$ ($d=2,3$) from time-resolved measurements of the scattered displacement field on a closed or open observation surface $\Gamma$. The background medium has constant Lamé parameters $(\lambda_2,\mu_2)$ and density $\rho_2$, while the inclusion carries $(\lambda_1,\mu_1)$ and a variable density $\rho_1$, with $\lambda_2/\lambda_1=\mu_2/\mu_1$ and strong convexity conditions imposed. The scattered field solves an initial-value problem for the Navier system driven by a causal incident wave.

Two features distinguish this contribution from prior work. First, it is, to the authors' knowledge, the first direct sampling method formulated directly on the coupled elastic wave field in the time domain; existing time-domain sampling techniques (linear sampling, factorization) had been developed for acoustic or electromagnetic models, with their extension to elasticity remaining open due to transmission eigenvalue difficulties. Second, the method requires no Helmholtz decomposition: rather than separating compressional (P) and shear (S) modes before imaging—a step that introduces substantial modeling complexity—the proposed functional uses a travel-time alignment mechanism that replaces P–S decoupling altogether. The reconstruction uses data from a single incident source, without any assumption on the temporal profile of the excitation, and involves only space–time integrals over $\Gamma$, so no iterative forward solves are needed.

The imaging functional

For sampling points $\bm{z}\in\widetilde{D}$ (a domain containing $d=2,3$0), the indicator is

$d=2,3$1

where $d=2,3$2 and $d=2,3$3 are the background P- and S-wave speeds, $d=2,3$4 and $d=2,3$5 are the far-field polarization projectors, and $d=2,3$6 provides exponential damping of the time signal. The functional correlates delayed boundary measurements with mode-dependent test functions, summing both wave types coherently—this is the non-decoupled design. It extends to elastodynamics the time-domain direct sampling methodology previously developed for acoustic [(2607.08067) references therein] and electromagnetic scattering.

Well-posedness via Fourier–Laplace analysis

The forward problem is analyzed in the half-plane $d=2,3$7. Using the Fourier–Laplace transform and Parseval's identity on the transformed line, the scattered field satisfies a frequency-domain Lippmann–Schwinger equation whose kernel is the elastic fundamental solution expressed through Hankel functions ($d=2,3$8) or exponentials ($d=2,3$9).

Well-posedness follows by a coercivity argument: testing the weak form with $\Gamma$0 yields a sesquilinear form coercive on $\Gamma$1 for $\Gamma$2, so Lax–Milgram applies uniformly on the transform line. Combined with Lubich-type convolution operator bounds, this yields the main regularity estimate:

$\Gamma$3

for any $\Gamma$4, $\Gamma$5, assuming $\Gamma$6 bounded below by a positive constant. This estimate is essential later: it controls the high-frequency tail of the imaging functional through the Sobolev regularity of the incident field.

Asymptotic characterization of the indicator

The theoretical core reformulates $\Gamma$7 in the frequency domain via Parseval's identity, giving an integral over the line $\Gamma$8, $\Gamma$9, of squared moduli of boundary correlations involving phase factors $(\lambda_2,\mu_2)$0. Two auxiliary results drive the analysis:

Sphere-integral identities: integrals over $(\lambda_2,\mu_2)$1 of $(\lambda_2,\mu_2)$2 and $(\lambda_2,\mu_2)$3 are computed explicitly in terms of modified Bessel functions $(\lambda_2,\mu_2)$4 (or spherical modified Bessel functions $(\lambda_2,\mu_2)$5), using spherical harmonic expansions. These produce matrix kernels $(\lambda_2,\mu_2)$6 and $(\lambda_2,\mu_2)$7 evaluated at $(\lambda_2,\mu_2)$8.

Kernel asymptotics: for a spherical measurement surface of radius $(\lambda_2,\mu_2)$9, the interaction kernel $\rho_2$0 admits the expansion

$\rho_2$1

and decays as $\rho_2$2 when the sampling point moves away from the source point—an oscillatory-integral decay that underpins the localization property.

Under a small-scatterer regime $\rho_2$3, $\rho_2$4, with well-separated components at minimum distance $\rho_2$5, the Born approximation is validated: $\rho_2$6 in 2D and $\rho_2$7 in 3D as $\rho_2$8.

The main theorem then establishes the two-regime behavior of $\rho_2$9. If $(\lambda_1,\mu_1)$0 lies near scatterer $(\lambda_1,\mu_1)$1,

$(\lambda_1,\mu_1)$2

with positive quantities $(\lambda_1,\mu_1)$3 determined by the contrast and incident field over $(\lambda_1,\mu_1)$4, whereas if $(\lambda_1,\mu_1)$5 the functional decays like $(\lambda_1,\mu_1)$6. Hence $(\lambda_1,\mu_1)$7 attains local maxima at the scatterer locations and decays away from them, providing a rigorous identification mechanism. The expansion also exposes explicit resolution conditions: the observation radius must satisfy $(\lambda_1,\mu_1)$8, the scatterers must be small relative to wavelength ($(\lambda_1,\mu_1)$9), and well separated ($\rho_1$0 large). The analysis assumes $\rho_1$1 with $\rho_1$2 and $\rho_1$3; these regularity and smallness assumptions are load-bearing, and the theory is restricted to small, well-separated scatterers under Born approximation.

An appendix develops the companion non-decoupled frequency-domain direct sampling method, proving analogous near/far asymptotics: near a scatterer, $\rho_1$4 with constants $\rho_1$5 and $\rho_1$6, and far away $\rho_1$7.

Numerical experiments

Forward data are generated by finite elements with absorbing boundary truncation, excited by a point source modulated by a Ricker wavelet. Three studies are reported:

• Single source, 2D: three small square scatterers reconstructed from 48 receivers on a circle of radius 1.8, $\rho_1$8, 450 time steps. Against the total focusing method (TFM) baseline, the proposed direct sampling method (DSM) achieves visibly higher resolution across peak frequencies $\rho_1$9, with both methods improving at higher frequency.
• Stability: additive Gaussian noise at SNR levels from $\lambda_2/\lambda_1=\mu_2/\mu_1$0 down to $\lambda_2/\lambda_1=\mu_2/\mu_1$1 dB still yields high-resolution reconstructions, indicating robustness to severe noise contamination.
• Limited aperture: reconstructions degrade gracefully but progressively as the aperture narrows toward a single angular sector.

For an extended kite-shaped scatterer, a single source recovers only the illuminated portion; a multi-source variant averaging correlations over an array of 8–32 sources on a circle progressively completes the shape. In 3D, two small cubes are localized from slice plots and isosurfaces of $\lambda_2/\lambda_1=\mu_2/\mu_1$2 using one incident source, confirming dimensional scalability.

Limitations and open questions

Several restrictions qualify the results. The rigorous theory covers only the small-scatterer (Born) regime with finitely many, well-separated inclusions of diameter $\lambda_2/\lambda_1=\mu_2/\mu_1$3; no reconstruction guarantee is given for extended or strongly contrasting media, where multiple scattering invalidates the approximation. The asymptotic estimates require large observation radius and full (or sufficiently wide) aperture; limited-aperture performance is demonstrated numerically but not analyzed. The regularization parameter $\lambda_2/\lambda_1=\mu_2/\mu_1$4 enters all estimates, yet its optimal selection is not addressed. Finally, the paper notes that extending time-domain linear sampling or factorization methods to elastic waves remains open, and the question of quantitative recovery (contrast values, not just support) lies outside the scope of the qualitative method presented here.

Conclusion

This work supplies the first rigorously justified time-domain direct sampling method for inverse elastic medium scattering that operates directly on the coupled P–S wave field. By replacing Helmholtz decomposition with travel-time alignment and analyzing the resulting functional through Fourier–Laplace transform, Parseval's identity, and modified Bessel function asymptotics, the authors establish local-maxima identification of small scatterers together with explicit resolution conditions, supported by noise-robust 2D and 3D reconstructions requiring only a single incident source and no iterative forward solves.