Li–Yang–Zhang (2021) justified a modified linear sampling method for recovering a locally rough penetrable interface — an unbounded interface that coincides with a plane outside a compact set — from point-source near-field data measured on a line segment above it, and posed two open questions: extending the method with full mathematical justification to non-locally rough (globally rough) interfaces, and simultaneously recovering the interface together with obstacles buried in the lower half-space. The first of these questions is the target of this task; the companion question is not part of it. For a globally rough interface the two pillars of the 2021 analysis fail — the reduction to a Lippmann–Schwinger equation on a bounded domain via auxiliary flat surfaces, and the equivalence with the Sommerfeld radiation condition — and no justified direct method exists for this case, although uniqueness of the interface from the same data has been proved (Wu–Yang, 2024). The problem is to construct and rigorously justify a direct sampling-type method, with an explicit admissible interface class, radiation condition, and spectral assumptions, that recovers a globally rough penetrable interface from such near-field data, together with a working numerical realization.

Background

Time-harmonic acoustic scattering in a two-layered medium is governed by the two-dimensional Helmholtz equation with a piecewise-constant wavenumber: an unbounded interface Γ={(x1,f(x1))}\Gamma=\{(x_1,f(x_1))\} separates an upper half-space (wavenumber k1k_1) from a lower half-space (wavenumber k2k_2), and across Γ\Gamma the total field and its normal derivative are continuous (transmission conditions), so the interface is penetrable rather than sound-soft. A locally rough interface coincides with a flat plane outside a compact set; a globally (non-locally) rough interface need not coincide with a plane outside any compact set. The inverse problem is to recover the shape and location of Γ\Gamma from measured scattered fields; here the data are near-field values, generated by point sources at a single frequency, taken on a bounded straight line segment a finite distance above the interface. The linear sampling method (LSM) is a qualitative, non-iterative imaging technique that probes a candidate point by testing whether a specially chosen field singularity at that point lies, approximately, in the range of the data operator; the indicator blows up off the scatterer and stays bounded on it, so no initial guess or prior shape information is needed. Lippmann–Schwinger equations are volume integral equations equivalent to the scattering problem; the Sommerfeld radiation condition prescribes the asymptotic decay of outgoing waves and is replaced, for unbounded rough structures, by an upward radiation condition, which imposes the prescribed decay only on propagation into the upper half-space. A transmission eigenvalue is a value of k12k_1^2 (the squared wavenumber in the upper region) at which the associated homogeneous interior transmission problem admits a nontrivial solution; at such frequencies the solvability arguments underlying sampling methods can break down, so the locally rough theory explicitly excludes them.

Problem Statement

Extend the modified linear sampling method — or construct an equally justified direct sampling-type method — with full mathematical justification to the recovery of a globally (non-locally) rough penetrable interface: an unbounded interface given as the graph of a bounded C2C^2 function that need not coincide with a plane outside any compact set, separating two half-spaces with constant wavenumbers, in the two-dimensional setting of the source work. The data are the scattered near-field values, generated by point sources at a single fixed frequency, measured on a bounded straight line segment at a finite distance above the interface. A complete answer must: (i) define the near-field operator equation (or an equivalent sampling indicator) for this setting, repairing or replacing the two devices on which the locally rough theory rests — the auxiliary-surface reduction to a Lippmann–Schwinger equation on a bounded domain, and the equivalence with the classical Sommerfeld radiation condition, both of which fail or require substantial modification when the interface need not coincide with a plane outside any compact set; (ii) prove a characterization theorem stating precisely for which admissible interface class, which radiation condition, and which spectral assumptions (in the locally rough theory, the exclusion of transmission eigenvalues) the indicator distinguishes points on the interface from points off it; and (iii) demonstrate the method numerically, including its stability under noisy data.

References

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