Comparison of five different ways to assimilate data for a simplistic weakly nonlinear parameter estimation problem

Abstract

Four non-iterative parameter estimation methods, applying four different ways to assimilate data, are compared to the (iterative) randomized maximum likelihood (RML) method on problems where the forward model is weakly nonlinear and simplistic. Both original and slightly modified versions of established data assimilation methods used for reservoir history matching are considered. Within the Gauss-linear regime, these methods are equivalent in the sense that they all sample correctly from the posterior pdf. The modification consists of replacing the global Kalman gain by a local gain similar to the one applied with RML. This modification, and the simplicity of the forward model considered, allows for asymptotic calculations, correct to lowest order in the quantity controlling the forward-model nonlinearity, to be performed for the non-iterative methods as well as for RML. Comparison of the resulting asymptotic parameter estimates explicitly reveals interesting relations between different ways to assimilate data, that goes beyond the Gauss-linear regime. Results from extensive numerical experiments with the modified methods and with the original methods support the asymptotic results when considering the mean results over a large number of randomized experimental settings where also the forward models vary. For randomized experiments with fixed forward model settings, trends from the asymptotic results were not visible in the numerical results. Although the nonlinear forward models considered are not closely related to reservoir models, the knowledge obtained should be useful also when the latter type of models are considered.