Ray-Born inversion for fracture parameters

SUMMARY The purpose of this study is to investigate the possibility of inverting for the fracture parameters using the ray-Born approximation. The fracture parameters normal and tangential weaknesses ΔN and ΔT, derived using a low-frequency effective medium theory, are indicators of the crack density and fracture infill (liquid-filled or dry). In our implementation of ray-Born inversion, ΔN and ΔT are unknown model parameters. We treat the background medium as isotropic, and its elastic properties are assumed to be already determined using other conventional methods prior to inversion. In our current implementation, we assume that the fractures are vertical and the orientation of the symmetry axis of the effective anisotropic medium or the fracture normal is known. We formulate the inverse problem in a least-squares sense and apply the conjugate-gradient method to minimize the error function. To overcome the problem of non-uniqueness in the inversion and to ensure that the inverted model parameters are physically meaningful and unique, we added an analytical function of model parameters to the error function. We show, using a numerical example, that the fracture parameters cannot be reliably obtained from single-component seismic data (vertical or horizontal). Simultaneous inversion of both components is needed for more robust estimates of ΔN and ΔT. The inversion scheme was implemented in the frequency domain, which allowed us to experiment with the number of frequencies used in the inversion. We found that a very small number of frequency samples covering the spectrum of the source wavelet could produce reasonable estimates of ΔN and ΔT. We tested our method on a synthetic data set generated using ray-Born forward modelling. For the experiment shown in the paper, the least-squares inversion scheme produced encouraging results. We also applied our method to a synthetic data set generated using a 3-D anisotropic finite-difference code. The results indicated that our method could predict the parameter trends, but with some inaccuracies in the values of ΔN and ΔT.