bayesian inversion

The goal of this study is to invert the seismic data directly to porosity as well as to quantify the associated uncertainty in one of the carbonate reservoirs located in southwestern Iran. Probabilistic inverse methods are able to present the model parameters as a posterior probability distribution function by combining the probability density function of the model prior information and the likelihood model. The likelihood density function is defined based on the noise model. The answer to the probabilistic inverse problem is a posterior distribution function that is not only consistent with the prior model but also is constrained to the seismic data. In this study, one of the sampling methods based on Markov chain Monte Carlo is used, which is able to generate realizations of the desired model parameter by sampling from the posterior distribution function. Unlike deterministic inverse methods, which provide only one answer for the model parameters, in probabilistic inverse methods the realizations generated from the posterior distribution function allow statistical analysis and model uncertainty quantification. The results of the implementation of the proposed method on synthetic seismic data showed that the estimation of noise variance has a significant effect on the results of probabilistic inversion and the uncertainty of the model realizations. The underestimation of the noise variance leads to fitting the noise on the data and subsequently generates artifacts on the output realizations. The overestimation of the noise variance provides smooth realizations with higher uncertainty. In the latter case, the reference porosity model is in the 95% confidence interval in contrast to the former case. Therefore, care should be taken in estimating the noise variance in probabilistic inverse methods. Considering a calibrated rock physics model for the carbonate reservoir under study, which is the main core in a direct inversion approach, the proposed algorithm was applied to the seismic traces adjacent to the four well logs. The uncertainty of the porosity was quantified in each well location. The correlation coefficient of the mean of the porosity realizations and the true porosity in four well locations were approximated about 79%, 63%, 51% and 67%. The consistency of the results obtained from the inversion with the observed porosity at the well locations indicates the good performance of the algorithm in estimating the porosity and its associated uncertainty. Due to the ability of the probabilistic inverse methods in a direct inverse of seismic data to the petrophysical properties, and their applicability in being performed in a parallel structure in processing clusters, these algorithms can be used in reservoir characterization of 2D and 3D data.

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.