## 1. Introduction

Data assimilation is a procedure for producing an optimal combination of model outputs and observations. The combined result should be closer to the true state than either the model forecast or the observation are. However, the quality of data assimilation depends crucially on the estimation accuracy of the forecast and observational error covariance matrices. If these matrices are estimated appropriately, then the analysis states can be generated by minimizing an objective function, which is technically straightforward and can be accomplished using existing engineering solutions (Reichle, 2008).

The ensemble Kalman filter (EnKF) is a popular sequential data assimilation approach, which has been widely studied and applied since its introduction by Evensen (1994a, 1994b). In EnKF, the forecast error covariance matrix is estimated as the sampling covariance matrix of the ensemble forecast states, which is usually underestimated due to the limited ensemble size and model error. This may eventually lead to the divergence of the EnKF assimilation scheme (e.g. Anderson and Anderson, 1999; Constantinescu *et al.*, 2007).

One of the forecast error covariance matrix inflation techniques is additive inflation, in which a noise is added to the ensemble forecast states that samples the probability distribution of model error (Hamill and Whitaker, 2005). Another widely used forecast error covariance matrix inflation technique is multiplicative inflation, that is, to multiply the matrix by an appropriate factor.

In early studies of multiplicative inflation, researchers determined the inflation factor by repeated experimentation and chose a value according to their prior knowledge. Hence such experimental tuning is rather empirical and subjective. Wang and Bishop (2003) proposed an on-line estimation method for the inflation factor of the forecast error covariance matrix in a model with a linear observational operator. Building on that work, Li *et al.*(2009) further developed the algorithm. All these methods are based on the first moment estimation of the squared observation-minus-forecast residual, which was first introduced by Dee (1995). Anderson (2007, 2009) used a Bayesian approach to covariance matrix inflation for the spatially independent observational errors, and Miyoshi (2011) further simplified Anderson's inflation approach by making a number of additional simplifying assumptions.

In practice, the observational error covariance matrix may also need to be adjusted (Liang *et al.*, 2011). Zheng (2009) and Liang *et al.*(2011) proposed an approach to simultaneously optimize the inflation factor of the forecast error covariance matrix and the adjustment factor of the observational error covariance matrix. Their approach is based on the optimization of the likelihood function of the observation-minus-forecast residual, an idea proposed by Dee and colleagues (Dee and Da Silva, 1999; Dee *et al.*, 1999). However, the likelihood function of the observation-minus-forecast residual is nonlinear and involves the computationally expensive determinant and inverse of the residual's covariance matrix. In this article, the second-order least squares (SLS; Wang and Leblanc, 2008) statistic of the squared observation-minus-forecast residual is introduced as the objective function instead. The main advantage of the SLS objective function is that it is a quadratic function of the factors, and therefore the closed forms of the estimators of the inflation factors can be obtained. Compared with the method proposed by Liang *et al.*(2011), the computational cost is greatly reduced.

Another innovation of this article is to propose a new structure for the forecast error covariance matrix that is different from the sampling covariance matrix of the ensemble forecast states used in the conventional EnKF. In the ideal situation, an ensemble forecast state is assumed to be a random vector with the true state as its population mean. Hence it is more appropriate to define the ensemble forecast error by the ensemble forecast states minus true state rather than by the perturbed forecast states minus their ensemble mean (Evensen, 2003). This is because in a model with large error and limited ensemble size, the ensemble mean of the forecast states can be very different from the true state. Therefore, the sampling covariance matrix of the ensemble forecast states can be very different from the true forecast error covariance matrix. As a result, the estimated analysis state can be substantially inaccurate. However, in reality the true state is unknown, and the analysis state is a better estimate of the true state than the forecast state. Therefore, in this article we propose to use the information feedback from the analysis state to update the forecast error covariance matrix. In fact, our proposed forecast error covariance matrix is a combination of multiplicative and additive inflation. Bai and Li (2011) also used the feedback from the analysis state to improve assimilation but in a different way.

This article consists of four sections. Section 2 proposes an EnKF scheme with a new structure for the forecast error covariance matrix and its adaptive estimation procedure based on the second-order least squares method. Section 3 presents the assimilation results on the Lorenz model with a correlated observational system. Conclusions and discussion are provided in section 4.