Bayesian methods are commonly used for geophysical inverse problems, such as seismic and rock-physics inversion, for the prediction of petroelastic properties. Bayesian inversion is based on Bayes’ theorem and combines the information from a prior distribution and a likelihood function; in geophysical applications, the prior model generally includes the available geologic information about the model variables, whereas the likelihood includes the geophysical models that link the model to the data. The goal of Bayesian inversion is to estimate the posterior distribution of the model variables conditioned by the measured data. The focus is on the prior model and its parameters. Typically, the parameters of the prior distributions are assumed to be fixed, for example, the mean and standard deviation of the prior distribution of petroelastic properties in seismic inversion or the facies proportions and transition probabilities in facies classification. I have studied the posterior distribution of the model given the data in a Bayesian setting using multiple prior models. The posterior distribution is assessed by summing the contributions of all of the likelihood functions of the model given the data, using different sets of parameters, weighted by the probabilities of the parameters. I apply the mathematical formulation in different problems, including log-facies classification, seismic-facies classification, and petrophysical property prediction and using different methods for the prior model generation such as transition matrices, training images, and Gaussian mixture models with multiple modes. The results show that multiple prior models can match the data and that the uncertainty in the prior parameters should be accounted for in the posterior distribution of the reservoir properties.
An In-Class Demonstration of Bayesian Inference
