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

Abstract

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).

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

“…In the spirit of the linear slip model [50] used to describe the seismic response of fractures in rock, the contact condition over Γ is given by a 3 × 3 symmetric matrix, K(ξ), of specific stiffness coefficients -synthesizing the spatially-varying nature of its rough and possibly multi-phase interface. Assuming time-harmonic seismic illumination, K is taken to be complex-valued with (K) 0 in order to allow for energy dissipation at the interface and to ensure the well-posedness of the forward scattering problem [47].…”

“…Over the past two decades, however, a number of sampling methods have emerged that consider the nonlinear nature of the inverse problem in an iteration-free way. In the context of extended scatterers, examples of such paradigm include the linear sampling method (LSM) [20,14,28], the factorization method (FM) [34,12], the generalized linear sampling method (GLSM) [2,47,17], the concept of topological sensitivity (TS) [27,18,7] and the subspace migration technique [44]. Among these, the TS and GLSM approaches have been recently adapted to permit elastic-wave sensing of heterogeneous fractures [46,47].…”

“…As indicated earlier, there is a mounting interest in medical diagnosis, target recognition, seismology, and energy production [15,40,56,45] to develop hybrid sensing schemes that also reveal the boundary condition of hidden anomalies. In this vein, it is noteworthy that some anomaly-indicator functionals -such as those featured by the FM [34], (G)LSM [16,47] and TS [46] are largely insensitive to the (unknown) boundary condition of a hidden anomaly. For instance, [15,9] demonstrate that the LSM is successful in reconstructing electromagnetic obstacles and cracks regardless of their boundary condition.…”

“…In light of the above developments, the twin aim of (non-iterative) seismic imaging and interfacial characterization of heterogeneous fractures is pursued sequentially via a 3-step approach where: (1) the shape reconstruction of penetrable fractures is effected, without prior knowledge of their contact condition, via recently established GLSM [47] and TS [46] approaches to elastic waveform tomography of discontinuity surfaces; (2) given the reconstructed fracture geometryΓ, the fracture opening displacement (FOD) profile is recovered via a double-layer potential representation mapping the FOD to the scattered field observations, and (3) the specific stiffness profile (as given by its normal, shear, and mixed-mode components) -is resolved from the knowledge of FOD andΓ by making use of the traction boundary integral equation written forΓ. To help construct a robust inverse solution, a three-step regularization scheme is also devised to minimize the error due to: (i) compactness of the double-layer potential map used to recover the FOD; (ii) inexact geometric reconstruction of the fracture surface, and (iii) presence (if any) of areas onΓ with near-zero FOD values.…”

On the elastic anatomy of heterogeneous fractures in rock

“…This is accomplished in a laboratory setting by monitoring the full-field interaction of ultrasonic shear waves (propagating through granite specimens) with stationary and advancing fractures via a recently acquired 3D Scanning Laser Doppler Vibrometer (SLDV) that is capable of monitoring triaxial particle velocity, with frequencies up to 1MHz, over the surface of rock specimens with 0.1mm spatial resolution and O(nm) displacement accuracy. Looking forward, the full-field seismic observations such as those presented herein may not only help decipher the true physics of a fracture interface and shine light on fidelity of classical interface models, but may also provide the ground truth toward validating the next generation of seismic imaging tools for simultaneous reconstruction and interfacial characterization of fractures in rock from remote sensory data [14,15,16].…”