A theoretical foundation is developed for the active seismic reconstruction of fractures endowed with spatially varying interfacial conditions (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, or illumination frequency. This is accomplished through the paradigm of the Fω-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) farfield 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 to the establishment of the applicability of the Fω-factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such a 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).
Bibliographical noteFunding Information:
The second author kindly acknowledges the support provided by the National Science Foundation (grant #1536110) and the University of Minnesota Supercomputing Institute.
- elastic waves
- hydraulic fractures
- inverse scattering
- linear sampling method
- seismic imaging
- specific stiffness