Fatemeh Pourahmadian

Why the high-frequency inverse scattering by topological sensitivity may work.

This study deciphers the topological sensitivity (TS) as a tool for the reconstruction and characterization of impenetrable anomalies in the high-frequency regime. It is assumed that the anomaly is simply connected and convex, and that the measurements of the scattered field are of the far-field type. In this setting, the formula for TS—which quantifies the perturbation of a cost functional due to a point-like impenetrable scatterer—is expressed as a pair of nested surface integrals: one taken over the boundary of a hidden obstacle, and the other over the measurement surface. Using multipole expansion, the latter integral is reduced to a set of antilinear forms featuring Greens function and its gradient. The remaining expression is distilled by evaluating the scattered field on the surface of an obstacle via Kirchhoff approximation, and pursuing an asymptotic expansion of the resulting Fourier integral. In this way, the TS is found to survive upon three asymptotic lynchpins, namely (i) the near-boundary approximation for sampling points close to the ‘exposed’ surface of an obstacle; (ii) uniform expansions synthesizing the diffraction catastrophes for sampling points near caustic surfaces, lines and points; and (iii) stationary phase approximation. Within the framework of catastrophe theory, it is shown that, in the case of the full source aperture, the TS is asymptotically dominated by the (explicit) near-boundary term—which explains the previously reported reconstruction capabilities of this class of indicator functionals. The analysis further shows that, when the (Dirichlet or Neumann) character of an anomaly is unknown beforehand, the latter can be effectively exposed by assuming point-like Dirichlet perturbation and considering the sign of the leading term inside the reconstruction.

Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures

A theoretical foundation is developed for active seismic reconstruction of fractures endowed with spatially-varying interfacial condition (e.g. partially-closed fractures, hydraulic fractures). The proposed indicator functional carries a superior localization property with no significant sensitivity to the fractures contact condition, measurement errors, and illumination frequency. This is accomplished through the paradigm of the F-factorization technique and the recently developed Generalized Linear Sampling Method (GLSM) applied to elastodynamics. The direct scattering problem is formulated in the frequency domain where the fracture surface is illuminated by a set of incident plane waves, while monitoring the induced scattered field in the form of (elastic) far-field patterns. The analysis of the well-posedness of the forward problem leads to an admissibility condition on the fractures (linearized) contact parameters. This in turn contributes toward establishing the applicability of the F-factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such minimizer is then used to construct a robust fracture indicator function, whose performance is illustrated through a set of numerical experiments. For completeness, the results of the GLSM reconstruction are compared to those obtained by the classical linear sampling method (LSM).

A synoptic approach to the seismic sensing of heterogeneous fractures: from geometric reconstruction to interfacial characterization

A non-iterative waveform sensing approach is proposed toward (i) geometric reconstruction of penetrable fractures, and (ii) quantitative identification of their heterogeneous contact condition by seismic i.e. elastic waves. To this end, the fracture support Γ (which may be non-planar and unconnected) is first recovered without prior knowledge of the interfacial condition by way of the recently established approaches to non-iterative waveform tomography of heterogeneous fractures, e.g. the methods of generalized linear sampling and topological sensitivity. Given suitable approximation ˘ Γ of the fracture geometry, the jump in the displacement field across ˘ Γ i.e. the fracture opening displacement (FOD) profile is computed from remote sensory data via a regularized inversion of the boundary integral representation mapping the FOD to remote observations of the scattered field. Thus obtained FOD is then used as input for solving the traction boundary integral equation on ˘ Γ for the unknown (linearized) contact parameters. In this study, linear and possibly dissipative interactions between the two faces of a fracture are parameterized in terms of a symmetric, complex-valued matrix K(ξ) collecting the normal, shear, and mixed-mode coefficients of specific stiffness. To facilitate the high-fidelity inversion for K(ξ), a 3-step regularization algorithm is devised to minimize the errors stemming from the inexact geometric reconstruction and FOD recovery. The performance of the inverse solution is illustrated by a set of numerical experiments where a cylindrical fracture, endowed with two example patterns of specific stiffness coefficients, is illuminated by plane waves and reconstructed in terms of its geometry and heterogeneous (dissipative) contact condition.