Filtering with state space localized Kalman gain

Abstract

It is well known that particle filters and Markov chain Monte Carlo methods are too expensive for high dimensional models; more robust techniques such as low rank versions of the Kalman filter are usually applied in large scale models. Common feature of all the ensemble Kalman type filters is that the model state and parameters are updated using the covariance matrix between the model states and the measurements, often estimated by a finite number of samples. In many nonlinear problems a global linearization of the model may be inadequate to describe the relationship between the observations and the model state and parameters. Hence the Kalman type updates may cause severe bias in the posterior estimates. In this paper we investigate the effect of using a local linearization. Motivated by local Gaussian density estimation we describe ways to estimate the covariance locally in the state space resulting in local ensemble filters. This is not to be confused with covariance localization which often refers to localization in the physical space. The filters are tested on the three dimensional Lorenz model and a three layer petroleum reservoir type model.The results clearly show that using a Kalman update based on local covariances allows resolution of small scale structures in the posterior distribution that cannot be resolved using a global covariance matrix. In particular we show that the new local filters are able to approximate posterior distributions with multiple modes contrary to the ensemble Kalman filter.