Interpretation of saturation and porosity from inverted 4D seismic parameters

In this note we propose two alternative approaches for stochastic rock physics inversion of 4D seismic data. Both methods use the approach of approximate local marginal likelihood coming from a Bayesian inversion scheme. We describe the concept of approximate local marginal likelihood, and put forward arguments for the approach, using arguments from the linear estimation methods, bounding of error, and also a more heuristic argument through information theory. Within the framework we propose two alternative approximations, which differ with respect to which level we use the approximation. The fastest and simplest approach applies the concept of local approximate marginal likelihood directly to porosity and saturation, the slightly more time consuming approach applies the concept of approximate local marginal likelihood to the seismic parameters and will give more accurate results for porosity and saturation.