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

TL;DR

This paper develops a robust, frequency-independent seismic imaging method for detecting heterogeneous fractures with varying contact conditions using the generalized linear sampling method (GLSM) in elastodynamics.

Contribution

It introduces a new fracture indicator functional based on GLSM that localizes fractures accurately regardless of contact condition and measurement errors.

Findings

The proposed method is insensitive to fracture contact conditions.

The GLSM-based indicator outperforms classical linear sampling in numerical tests.

The approach is computationally efficient with a non-iterative minimization.

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 $F_\sharp$-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 fracture's (linearized) contact parameters. This in turn contributes toward establishing the applicability of the $F_\sharp$-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).