ABSTRACT

Using the scattered elastic wavefield, a method to derive the power spectral density (PSD) of the heterogeneous compliance distribution, along the plane of a single fracture, is formulated. The method involves estimation of the stress field at the fracture depth from the scattered wavefield followed by least-squares optimization of the PSD of the stress field. We consider a 2D geometry and incident plane waves. The derivations are made in the frequency-wavenumber domain. To derive the relationship between the scattered wavefield and the PSD of the heterogeneous compliance, perturbation theory is used. In the forward modeling, the scattering response of the heterogeneous compliance is estimated, assuming a stationary random process. Numerical tests of the proposed method of PSD estimation offer important insights. The estimated PSD of the compliance (given by the standard deviation δ and the correlation length lc, assuming a Gaussian distribution) as a statistical measure of the heterogeneous fracture compliance is robust against fracture depth uncertainties. Furthermore, the sparse spatial sampling of the wavefield is not problematic as long as the data contain those wavenumber components which represent the slope of the PSD of the stress field. However, when the sampling interval is too large compared to the correlation length of the heterogeneous compliance distribution, it is difficult to estimate accurately the PSD of the compliance due to spatial aliasing and small amplitude variation within the available wavenumber. For a given spatial sampling, the use of higher frequencies results in a stronger amplitude variation in the scattered wavefield. This leads to better estimates of the PSD of the compliance via a least-squares optimization. In the presence of white noise, a change in the PSD of the stress field at the fracture depth significantly affects the PSD estimates. In this case, a data-driven amplitude correction improves the estimates of δ and lc.

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