## 1. Introduction

The Kalman filter algorithm is currently one of the most popular approaches to solving the problem of data assimilation (Ghil and Malanotte-Rizzolli, 1991). An equation for the conditional mean is solved in the method to obtain an optimal estimate of the current atmospheric state with observational data and a forecast model, which is in general nonlinear (Jazwinsky, 1970). The equation is difficult to solve, but in simplified versions it can be reduced to equations for the 1st and 2nd moments. The simplifications are based on linearization about some basic state (the extended Kalman filter) or expansion into a power series in terms of the estimation error (second-order truncated filters). The random fields considered are assumed to be Gaussian (Jazwinsky, 1970).

The ensemble approach today is a leading tool in applying the Kalman filter to data assimilation. It was first proposed by Evensen (1994) and later developed by Burgers *et al.* (1998), Evensen (2003, 2007) and Houtekamer and Mitchell (1998, 2001, 2005). This approach allows calculating covariance matrices of estimation errors for nonlinear forecast models. In this case a version of the extended Kalman filter (EKF) is used, where forecast error covariances are estimated with an ensemble of forecasts.

Implementation of the ensemble algorithm causes many problems due to the limited number of ensemble members and a necessity to obtain an ensemble with such a covariance matrix that corresponds to the analysis error covariances. To overcome these problems, EKF algorithms that generate random observation errors (perturbed observations) can be used (Burgers *et al.*, 1998). Such versions of the EKF were considered in the publications by Houtekamer and Mitchell (1998, 2005) and Burgers *et al.* (1998), but Whitaker and Hamill (2002) showed that the sampling error can be great. To avoid this effect the authors proposed an ensemble square root filter that allows implementation of the EKF without perturbed observations. Such an approach to determination of the analysis error ensemble is commonly referred to as ‘deterministic’. Tippett *et al.* (2003) give an overview of ensemble filters using the deterministic approach. It has been shown that all the approaches using the square root Kalman filter obtain ensembles that differ one from another, but all of them are used to calculate the same covariance matrix. One more version of the deterministic approach was described by Sakov *et* *al.* (2008). They proposed a simplified version of the ensemble square root filter.

To avoid the problems that occur due to the limited number of ensemble members the EKF may be applied locally. A method of localization, to limit the correlation radius, was proposed by Houtekamer and Mitchell (1998). These ideas were developed further by Houtekamer and Mitchell (2005) and Hunt *et al.* (2007), where the authors proposed using the algorithm in the subdomains.

Another significant problem is to avoid a false decrease in the forecast error variance being estimated using the ensemble, which results in the so-called time divergence of the algorithm (Lorenc, 2003). To solve this problem non-zero model noises have to be specified, but their exact values are unknown.

The EKF is a technically sophisticated algorithm operating with high-order matrices. In this article, an efficient algorithm of data assimilation for nonlinear models based on the EKF is proposed. The basic idea is taken from automatic control theory (Krasovskii *et al.*, 1979). A simple algorithm, called the *π*-algorithm, proposed by Krasovskii *et al.* (1979) was considered in (Klimova, 2008a) for its possible application to data assimilation. Klimova (2008b) generalized the *π*-algorithm so that it could be used with an ensemble of forecasts (the ensemble *π*-algorithm). In the present article the formulae of the ensemble *π*-algorithm are derived from assumptions that are more general than those in Klimova (2008b). The operation count of the algorithm is close to that of the local ensemble transform Kalman filter (LETKF; Hunt *et al.*, 2007; Szunyogh *et al.*, 2008), but its formulae differ from those of the LETKF. In particular, the ensemble *π*-algorithm does not require calculating an ensemble that corresponds to the analysis error covariances because it is done automatically. As in LETKF, all operations in the ensemble *π*-algorithm are performed with matrices of the same order as that of the ensemble.

The ensemble *π*–algorithm formulae for nonlinear model and data operators are derived in section 2. A modification of the ensemble *π*-algorithm to be used in ensemble forecasting is given in section 3. Section 4 provides a comparative analysis of the *π*-algorithm and LETKF. Section 5 presents the results of numerical experiments on model data assimilation with the one-dimensional Burgers' equation, and section 6 presents the main conclusions of the article.