Stable Estimation of Observation Error Covariance Matrix by Bayesian Method

We develop a Bayesian technique for estimating parameters in the observation noise covariance matrix for ensemble data assimilation. We design posterior distribution by ensemble-approximated likelihood and a Wishart prior distribution and present an iterative algorithm for parameter estimation. As the algorithm by Ueno and Nakamura (2012) for maximum likelihood estimation, the present algorithm is identified as the EM algorithm for a Gaussian mixture model and can estimate a number of parameters in the observation noise covariance matrix. The advantage of the proposed method is that Rt can be estimated online, and more importantly, temporal smoothness of Rt can be controlled by adequately choosing the prior covariance S and its weight λ. In addition, the weight λ is objectively estimated by maximizing the marginal likelihood. We present an application to a coupled atmosphere-ocean model on the conditions that Rt is a diagonal matrix (Rt = diag rt), scalar multiplication of fixed matrices (Rt = αt Σ), and Rt has no specific structure. We verify that the proposed algorithm works well and that several iterations are necessary. By assuming the prior covariance matrix to be the previous estimate, S = ˆRt-1, we obtain Rt that varies smoothly in time. When Rt has no specific structure, we need to be careful to assume a positive definite matrix as the prior covariance matrix S. The proposed algorithm enables us to estimate parameters in Rt, the number of which may become exceedingly large for data assimilation.