  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.
