Kriging, splines, conditional simulation, Bayesian inversion and Ensemble Kalman filtering

Abstract

This chapter discusses, from a theoretical point of view, how the geostatistical approach relates to other commonly-used models for inversion or data assimilation in the petroleum industry. The formal relationship between point Kriging and splines or radial basis functions is first presented. The generalizations of Kriging to the estimation of average values or values affected by measurement errors are also addressed. Two algorithms are often used for conditional simulation: The "rough plus smooth" approach consists of adding a smooth correction to a non-conditional simulation, whilst sequential Gaussian simulation allows the point-by-point construction of the realizations. As with Kriging, conditional simulation can be applied to average values or to data affected by measurement errors. Geostatistical inversion generates high-resolution realizations of vertical impedance traces constrained by seismic amplitudes. If the relationship between impedance and amplitude data is linearized, geostatistical inversion is a particular case of Bayesian inversion. Because of the non-linearity of production data vis-à-vis the variables of the earth model, their assimilation is harder than that of seismic data. Ensemble Kalman filtering, if considered from a geostatistical viewpoint, consists of using a large number-or ensemble-of realizations to calculate empirical covariances between the dynamic data and the parameters of the geostatistical model. These covariances are then used in the equations for interpolating the mismatch between simulated and new production data using a coKriging-like formalism. Interestingly, most of these techniques can be expressed using the same generic equation by which an initial model not honouring some newly arrived data is made conditional to these data by adding a (co-)Kriged interpolation of the data mismatches to the initial model. In spite of their similar equations, Bayesian inversion, geostatistics and ensemble Kalman filtering have a different approach to the inference of the covariance models used by these equations.