Bayesian Update of Wave Equation Based Seismic Inversion Using Geological Prior Information and Scenario Testing

Reservoir characterisation is a data driven process which involves the integration of different datasets to describe the subsurface. One of the difficulties of integrating geological data with the wave-equation based seismic inversion
is that geological information is invariably interpreted as a layer-based model, whereas the wave-equation is defined and solved on a grid. Mapping a layer-based model space onto a grid-based space leads to highly non-Gaussian, multi-modal distribution functions, even when the layer-based properties have simple Gaussian distributions. In this paper an analytic method is presented that translates the prior layer-model based distributions to grid-based prior distributions. From the unconstrained seismic inversion result a Gaussian likelihood function is constructed and the method to find the maximum a posterior estimate (MAP) and its uncertainty is described. As geological prior information we use well data, a geological concept of the environment of deposition and structural seismic interpretation in the form of some horizons to guide the prior model in between wells. Given the prior model, a measure for the probability of the data is formulated. When this process is repeated for various prior scenarios, the probability of the scenario, given the data, can be calculated for every location.