Inverse scattering solution for the spatially heterogeneous compliance of a single fracture

Characterizing the spatially heterogeneous fracture compliance through use of elastic waves has the potential to illuminate the hydraulic and mechanical properties along a fracture. We formulate the inverse scattering problem to estimate the heterogeneous compliance distribution along a single fracture embedded in a homogeneous background. For this purpose, we follow two steps: (1) estimating the stress field at the fracture depth from the wavefield at the receiver depth using forward and inverse wavefield extrapolation operators and (2) solving the relation between the scattered wavefield and the heterogeneous compliance distribution. The method assumes one-way wavefield and the absence of upgoing wavefield in the medium below the fracture. In the numerical tests, we consider a 2-D geometry and incident Gaussian beams. We model the spatially heterogeneous fracture compliance assuming a stationary random process. Our results show that the fracture compliance can be accurately estimated when the sampling interval is smaller than the correlation length of the spatially heterogeneous fracture compliance and when the input parameters viz. distance/depth and the medium velocity up to the fracture are known. In case these are not known, we suggest a data-driven approach to correct for the error. This leads to an accurate estimation of the spatially heterogeneous fracture compliance without exact knowledge of the input parameters. In case of noisy data, stable estimates can be obtained through proper regularization of the compliance function. The use of higher frequencies is beneficial against noise contamination, as higher frequencies produce stronger scattering. The use of a low-pass wavenumber filter suppresses the noise due to the evanescent waves. In this case, however, a lack of higher wavenumbers restricts the spatial resolution of the estimated compliance