Data assimilation technique based on the ensemble π-algorithm

A practical realization of the data assimilation algorithm based on the Kalman filter in its non-simplified form is impossible for modern forecast models because of the high dimension of the associated sets of equations and nonlinearity of predicted processes. The main direction in the Kalman filter realization is an ensemble approach. Under the assumption of ergodicity of random forecast errors, a so-called π-algorithm can be considered, which is alternative to the ensemble Kalman filter and where probabilistic averaging is replaced by averaging over time. In the present paper, we suggest a generalization of the π-algorithm based on the ensemble approach. The algorithm is easy to implement; however, its applicability to the data assimilation problems, convergence, and relation to the Kalman filter are still to be studied. The applicability of the ensemble π-algorithm to the data assimilation problem is considered by an example of a simple one-dimensional advection equation. The use of such a simple equation allows us to compare the classical Kalman filter algorithm with various practical approaches to its realization.