Information-based seismic amplitude inversion

Abstract

Amplitude variation with offset (AVO) inversion, particularly for more than two model parameters, is a highly ill-posed problem and, hence, regularization is indispensable. For common regularization such as norm-regularization, learning can be forged ahead via gradient descent. Alternatively, learning can be achieved by performing gradient descent on the objective and projects the result back into the space of the regularizer; this allows the use of atypical regularization functions. Here, we propose a regularized inverse problem to mitigate the Ill-posedness of the amplitude inversion. The regularization is added to measure the difference in information between the a priori probability density function and the predicted probability density of the inverted parameters, and therefore enhances the consistency between the inversion results and the a priori geological information. Information theory provides a collection of contrast functions which quantify the divergence from one probability distribution to another, such as the relative entropy. The a priori density is approximated by a Gaussian mixture model, obtained from well logs and rock physics data. The mixture model is a density estimator, providing the statistical properties of the model parameters of interest. The likelihood of the data and the divergence are combined in an augmented Lagrangian scheme, to obtain a stable solution that best generate the recorded seismic data and satisfy the geological constraints conveyed by the a priori probability density function. The proposed inversion scheme is applied to the anisotropy AVO inversion, for estimating the elastic and seismic anisotropy parameters of shale formations. Compared to the unconstrained minimization, the P- and S-wave velocity, and ε are better recovered, moreover, density and Thomsen's δ are well-constrained.