Abstract. 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 fracture's 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 wellposedness of the forward problem leads to an admissibility condition on the fracture's (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).
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 Green's 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 nearboundary 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.
Differential evolution indicators are introduced for 3D spatiotemporal imaging of micromechanical processes in complex materials where progressive variations due to manufacturing and/or aging are housed in a highly scattering background of a-priori unknown or uncertain structure. In this vein, a three-tier imaging platform is established where: (1) the domain is periodically (or continuously) subject to illumination and sensing in an arbitrary configuration; (2) sequential sets of measured data are deployed to distill segment-wise scattering signatures of the domain's internal structure through carefully constructed, non-iterative solutions to the scattering equation; and (3) the resulting solution sequence is then used to rigorously construct an imaging functional carrying appropriate invariance with respect to the unknown stationary components of the background e.g., pre-existing interstitial boundaries and bubbles. This gives birth to differential indicators that specifically recover the 3D support of micromechanical evolution within a network of unknown scatterers. The direct scattering problem is formulated in the frequency domain where the background is comprised of a random distribution of monolithic fragments. The constituents are connected via highly heterogeneous interfaces of unknown elasticity and dissipation which are subject to spatiotemporal evolution. The support of internal boundaries are sequentially illuminated by a set of incident waves and thusinduced scattered fields are captured over a generic observation surface. The performance of the proposed imaging indicator is illustrated through a set of numerical experiments for spatiotemporal reconstruction of progressive damage zones featuring randomly distributed cracks and bubbles.
The concept of topological sensitivity (TS) is extended to enable simultaneous 3D reconstruction of fractures with unknown boundary condition and characterization of their interface by way of elastic waves.Interactions between the two surfaces of a fracture, due to e.g. presence of asperities, fluid, or proppant, are described via the Schoenberg's linear slip model. The proposed TS sensing platform is formulated in the frequency domain, and entails point-wise interrogation of the subsurface volume by infinitesimal fissures endowed with interfacial stiffness. For completeness, the featured elastic polarization tensor -central to the TS formula -is mathematically described in terms of the shear and normal specific stiffness (κ s , κ n ) of a vanishing fracture. Simulations demonstrate that, irrespective of the contact condition between the faces of a hidden fracture, the TS (used as a waveform imaging tool) is capable of reconstructing its geometry and identifying the normal vector to the fracture surface without iterations. On the basis of such geometrical information, it is further shown via asymptotic analysis -assuming "low frequency" elasticwave illumination, that by certain choices of (κ s , κ n ) characterizing the trial (infinitesimal) fracture, the ratio between the shear and normal specific stiffness along the surface of a nearly-panar (finite) fracture can be qualitatively identified. This, in turn, provides a valuable insight into the interfacial condition of a fracture at virtually no surcharge -beyond the computational effort required for its imaging. The proposed developments are integrated into a computational platform based on a regularized boundary integral equation (BIE) method for 3D elastodynamics, and illustrated via a set of numerical experiments.
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