Adaptive Error Covariances Estimation Methods for Ensemble Kalman   Filters
  This paper presents a computationally fast algorithm for estimating, both, the system and observation noise covariances of nonlinear dynamics, that can be used in an ensemble Kalman filtering framework. The new method is a modification of Belanger's recursive method, to avoid an expensive computational cost in inverting error covariance matrices of product of innovation processes of different lags when the number of observations becomes large. When we use only product of innovation processes up to one-lag, the computational cost is indeed comparable to a recently proposed method by Berry-Sauer's. However, our method is more flexible since it allows for using information from product of innovation processes of more than one-lag.   Extensive numerical comparisons between the proposed method and both the original Belanger's and Berry-Sauer's schemes are shown in various examples, ranging from low-dimensional linear and nonlinear systems of SDE's and 40-dimensional stochastically forced Lorenz-96 model. Our numerical results suggest that the proposed scheme is as accurate as the original Belanger's scheme on low-dimensional problems and has a wider range of more accurate estimates compared to Berry-Sauer's method on L-96 example. 
Yicun Zhen and John Harlim
math.OC