Bayesian linearized AVO inversion for prediction of Rock Parameters

Summary: In this study, a novel approach for linearized amplitude versus offset (AVO) inversion in a Bayesian framework is presented. Objective is to estimate the posterior distribution of three elastic parameters, P wave velocity, S wave velocity and rock density. The methodology is based on the convolutional model and a weak contrast linearized approximation of Zoeppritz equation for PP waves. In this study, assuming a priori Gaussian distribution for input parameters, and also, a Gaussian distribution for seismic misfit function, Bayesian equation also yields a Gaussian distribution for posterior parameters, which can be analytically computed. This analytic solution is a rather fast approach for inversion of elastic parameters along with their uncertainty distribution. This methodology is tested on both synthetic and field data sets and in both cases yields reasonable solutions. In the current study, assuming a weak contrast model for rock properties, a linearized AVO approximation of Zoeppritz equation is used in a Bayesian framework to invert prestack seismic data for the above-mentioned three elastic parameters. The methodology is tested on both synthetic and field data sets. The results show preferably good matches with the true data.

Introduction: Inversion of seismic AVO is a way to estimate elastic parameters from prestack seismic data. This technique can be solved in both nonlinear (Dahl and Ursin, 1991) and linear (Smith and Gildow, 1987) approaches. Lortzer and Berkhout (1993) also used the same methodology as presented here, but they used the relative contrast of elastic parameters instead of their absolute values.

Methodology and Approaches: Assuming a Gaussian distribution for elastic parameters and seismic noise, and also, a linearized formulation for forward modeling, distribution for posterior parameters will also be Gaussian. Based on this methodology, the mean and covariance matrices of prior distribution are estimated from well data. Then using the linearized formulation of AVO, the mean and covariance matrices of observed seismic data are estimated. Having the statistical parameters of prior and likelihood functions, the statistical parameters of posterior distribution is analytically yielded based on Bayesian formulation. The covariance matrix of posterior distribution gives an estimate of the uncertainty in the elastic parameters.

Results and Conclusions: A Bayesian AVO inversion method was proposed and tested on both synthetic and field data sets. In case of synthetic data, the estimated parameters fitted the true values almost exactly. The result for the field dataset was also reasonable and matched the well log data relatively well except in some locations where prestack seismic data were not preconditioned very well. The initial models used for this methodology does not need to be detailed at all and very simple initial models such as constant or linear values lead to good estimation of the posterior distribution. Therefore, this approach can be a good choice for generation of pseudowells where not a rich dataset is available.