This paper presents a computationally fast algorithm for estimating, both,the system and observation noise covariances of nonlinear dynamics, that can beused in an ensemble Kalman filtering framework. The new method is amodification of Belanger's recursive method, to avoid an expensivecomputational cost in inverting error covariance matrices of product ofinnovation processes of different lags when the number of observations becomeslarge. When we use only product of innovation processes up to one-lag, thecomputational cost is indeed comparable to a recently proposed method byBerry-Sauer's. However, our method is more flexible since it allows for usinginformation from product of innovation processes of more than one-lag. Extensive numerical comparisons between the proposed method and both theoriginal Belanger's and Berry-Sauer's schemes are shown in various examples,ranging from low-dimensional linear and nonlinear systems of SDE's and40-dimensional stochastically forced Lorenz-96 model. Our numerical resultssuggest that the proposed scheme is as accurate as the original Belanger'sscheme on low-dimensional problems and has a wider range of more accurateestimates compared to Berry-Sauer's method on L-96 example.
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