Influences of the inflation factors generation in the main parameters of the ensemble smoother with multiple data assimilation

Abstract

The generation of the ensemble smoother with multiple data assimilation (ES-MDA) inflation factors has been the focus of several recent studies. These factors must be selected such that the sum of their inverses is equal to one. It guarantees the ES-MDA to correctly sample the posterior probability density function in the linear Gaussian case when the ensemble size goes to infinity. Concurrently, recent researches have shown a relationship between inflation factors and the final ensemble estimates quality. They have also suggested techniques to generate these factors based on methods derived from the discrepancy principle. However, a procedure to efficiently generate ES-MDA inflation factors remains an open problem. Additionally, the studies diverge on what regularization method suffices to produce ES-MDA inflation factors that provide optimal final results. Therefore, this study presents an investigation of the generation of ES-MDA inflation factors. Two main paths will be investigated: selecting them constant, equal to the number of assimilations, and in a geometrically decreasing order. When selecting them geometrically, two techniques will be used to generate the first inflation factor: the regular discrepancy principle and a Levenberg-Marquardt regularization scheme. The main objective of this study is to examine the error propagation during the multiple data assimilation of the ES-MDA and the estimates’ quality, considering only the generation of the inflation factors. Moreover, we numerically analyze their influence on the ES-MDA ensemble size and number of assimilations and how their choices affect the ES-MDA performance. The method will be evaluated in a synthetic two-dimensional history matching problem of waterflooding.