Sequential Bayesian Gaussian Mixture Linear Inversion of Seismic Data for Elastic and Reservoir Properties Estimation

We propose here a methodology to sequentially simulate elastic and reservoir properties, such as impedances, porosity, and facies, in order to obtain a set of reservoir models conditioned by seismic data. The sequential inversion approach provides multiple simulations obtained as solution of a Bayesian linear inverse problem where we assume that rock properties are distributed according to a Gaussian mixture model, i.e. a linear combination of Gaussian distributions. The weights of the Gaussian components are the probabilities of the facies. The main advantage of this approach is that the assumption of Gaussian mixture distribution of reservoir properties overcomes the common Gaussian assumption allowing the description of the multimodal behavior of the data. Furthermore the solution of the Bayesian Gaussian mixture linear inverse problem preserves analytical tractability. The linear inverse theory results valid in the Gaussian case have been extended to the multimodal case, by deriving the analytical expression of means, covariance matrices, and weights of the Gaussian mixture conditional distribution of the model. We describe here how to compute the analytical solution and implement a sequential approach in the simulation algorithm. We then apply the sequential Gaussian mixture linearized inversion to layer maps extracted from a 3D geophysical model of a clastic reservoir located in the North Sea. Impedances are first inverted from seismic data, then porosity and facies are simulated from impedance maps. This application provides a complete set of reservoir models with multimodal rock properties.