## 1. Introduction

The Ensemble Kalman Filter (EnKF) is a popular sequential data assimilation approach, which has been examined and applied in a number of studies since it was first introduced by Evensen (1994a, 1994b). Conceptually, by assimilating observations into the model, data assimilation can provide an optimal combination of model outputs and observations. However, the ‘optimality’ of the combination depends on the accuracy of the estimated forecast and observation error covariances; if the estimates are erroneous, the model updates will be suboptimal. Therefore, a correct description of forecast error and observation error covariances is crucial to the high analysis quality of EnKF (e.g. Evensen, 2003). Unfortunately, it is very difficult to have an exact idea of these statistical properties in practice, since the true states are never known and perfect statistics cannot be computed (Sénégas *et al.*, 2001).

In EnKF, the forecast error covariance matrix is estimated as the sampling covariance matrix of the forecast ensemble. However, past work on EnKF found that the sampling error in such estimations, resulting from finite-size ensembles, can generally lead to underestimation of the forecast ensemble covariance and eventually result in filter divergence (e.g. Anderson and Anderson, 1999; Constantinescu *et al.*, 2007). To compensate for this, covariance inflation for increasing ensemble variance is becoming popular. A simple and common way is to inflate the deviation from the forecast ensemble mean by a small constant factor for each ensemble member (Anderson and Anderson, 1999), in which the factor is chosen by repeated experimentation. In section 3.2, we compare our approach with this method.

Dee (1995) and his later work with colleagues (Dee and da Silva, 1999; Dee *et al.*, 1999) proposed a maximum likelihood estimation method for forecast error and observation error covariances. This method first parametrizes the forecast error and observation error covariance matrices, and then estimates the parameters by minimizing the −2log(likelihood) of the observation-minus-forecast residuals. However, their work did not suggest a reliable parametrization for the forecast error covariance matrix, and what is more important, the computation of the determinant in the −2log-likelihood is very difficult in the general case. Consequently, the method has not become very popular for estimating forecast and observation error covariances.

Using statistics of the observation-minus-forecast residuals described in Dee (1995), Wang and Bishop (2003, hereafter denoted W-B) proposed a method of estimating inflation factors on-line in an Ensemble Transform Kalman Filter (ETKF). Building on their work, Li *et al.* (2009) came up with another algorithm to estimate the inflation factor at each analysis step. Anderson (2007, 2009) also proposed a temporally varying inflation method and later a spatially and temporally varying inflation method using a hierarchical Bayesian approach. However, the work of Li *et al.* (2009) and Anderson (2007, 2009) are limited to the case of uncorrelated observation errors. In this article, we investigate how to estimate inflation factors when observation errors are spatially correlated and observation error statistics may not be correctly known.

As an extended application of the maximum likelihood theory developed in Dee (1995) and Dee and da Silva (1999), Zheng (2009) proposed a ‘multivariate covariance inflation’ extending the inflation factor to a time-dependent diagonal matrix. However, only a simple model and independent observation errors were tested for that work. We made this study to further develop the work of Zheng, in that the inflation method was tested on more realistic models with much higher dimensions and using spatially correlated observation errors. We also detected the capability of our approach of simultaneously inflating both forecast error and observation error covariance matrices when the observation error variance is wrongly specified. However, the inflation factors were constrained as scalar ones in this article. In appendix B, we describe an efficient way of computing the determinant in the −2log-likelihood. However, it is worth noting that this effective method is practical only when the estimated inflation factors are scalar parameters and the forecast error covariance matrix is of small rank (fortunately, this is the case in our study for the scalar inflation of the forecast error covariance matrix in EnKF). For more complicated parametrizations of the error covariances, different techniques would have to be used to make the computations practical.

This paper comprises five sections. Section 2 summarizes the W-B method and our approach. Section 3 presents the assimilation results on a low-order Lorenz model (Lorenz, 1996). Especially, our approach is compared with the W-B method for cases in which the observation error variance is exactly known. Section 4 provides the validation on a more realistic two-dimensional Shallow Water Equation model with a larger correlated observation system. Conclusions and discussions are given in section 5. Hereafter, we denote our method building on the maximum likelihood estimation theory as MLE. It is also worth noting that in this article the expression ‘inflation factor’ does not specifically represent a factor larger than 1; it can also refer to a factor smaller than 1.