An efficient algorithm for stochastic ensemble smoothing

The state of the environment using a mathematical model and observational data using a data assimilation procedure is assessed. The Kalman ensemble filter is one of the widespread data assimilation algorithms at present. An important component of the data assimilation procedure is the assessment not only of the predicted values, but also of the parameters that are not described by the model. A single improvement procedure from observational data in the Kalman ensemble filter may not provide a required accuracy. In this regard, the ensemble smoothing algorithm, in which data from a certain time interval are used to estimate values at a given time, is becoming increasingly popular. This paper considers a generalization of the previously proposed algorithm, which is a version of the Kalman stochastic ensemble filter. The generalized algorithm is an ensemble smoothing algorithm, in which smoothing is performed for the average value of a sample and then the ensemble of perturbations is transformed. The transformation matrix proposed in the paper is used to estimate both the predicted value and the parameter. An important advantage of the algorithm is its locality, which makes it possible to estimate a parameter in a given domain. The paper provides a rationale for the applicability of this algorithm to the implementation of ensemble smoothing. Test calculations were performed with the proposed numerical algorithm with a 1-dimensional model of transport and diffusion of passive impurity. The algorithm proposed is effective and can be used to assess the state of the environment.