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In the ensemble Kalman filter, the forecast error covariance matrix is estimated as the sampling covariance matrix of the forecast ensemble. However, it is well known that such estimations may be far from the true forecast error covariance matrix. In this paper, an inflation approach on forecast error covariance matrix based on the maximum likelihood estimation theory is developed and compared to an existing time-dependent inflation method and the best-tuned constant inflation. Our method was first tested on a 40-variable Lorenz model using spatially correlated observation errors. Specifically, when the observation error variance is incorrectly specified, our proposed method can simultaneously inflate on both forecast and observation error covariance matrices. We then assessed our approach on the two-dimensional Shallow Water Equation model with higher state dimensions and a larger correlated observation system. The results confirmed that our method is effective in retrieving the true states and correcting observation error variances.

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Grey wolf optimizer (GWO) algorithm is a swarm intelligence optimization technique that is recently developed to mimic the hunting behavior and leadership hierarchy of grey wolves in nature. It has been successfully applied to many real world applications. In the GWO algorithm, ''C'' is an important parameter which favoring exploration. At present, the researchers are few study the parameter ''C'' in GWO algorithm. In addition, during the evolution process, the other individuals in the population move towards to the α, β, and δ wolves which are to accelerate convergence. However, GWO is easy to trap in the local optima. This paper presents a modified parameter ''C'' strategy to balance between exploration and exploitation of GWO. Simultaneously, a new random opposition-based learning strategy is proposed to help the population jump out of the local optima. The experiments on 23 widely used benchmark test functions with various features, 30 benchmark problems from IEEE CEC 2014 Special Session, and three engineering design optimization problems. The results reveal that the proposed algorithm shows better or at least competitive performance against other compared algorithms on not only global optimization but also engineering design optimization problems.

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