A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation

Abstract

An ensemble Kalman filter may be considered for the 4D assimilation of atmospheric data. In this paper, an efficient implementation of the analysis step of the filter is proposed. It employs a Schur (elementwise) product of the covariances of the background error calculated from the ensemble and a correlation function having local support to filter the small (and noisy) background-error covariances associated with remote observations. To solve the Kalman filter equations, the observations are organized into batches that are assimilated sequentially. For each batch, a Cholesky decomposition method is used to solve the system of linear equations. The ensemble of background fields is updated at each step of the sequential algorithm and, as more and more batches of observations are assimilated, evolves to eventually become the ensemble of analysis fields. A prototype sequential filter has been developed. Experiments are performed with a simulated observational network consisting of 542 radiosonde and 615 satellite-thickness profiles. Experimental results indicate that the quality of the analysis is almost independent of the number of batches (except when the ensemble is very small). This supports the use of a sequential algorithm. A parallel version of the algorithm is described and used to assimilate over 100 000 observations into a pair of 50-member ensembles. Its operation count is proportional to the number of observations, the number of analysis grid points, and the number of ensemble members. In view of the flexibility of the sequential filter and its encouraging performance on a NEC SX-4 computer, an application with a primitive equations model can now be envisioned.

Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

“…Localization is generally done either explicitly, considering only the observations from a region surrounding the location of the analysis [29,20,30,1,36,37], or implicitly, by multiplying the entries in P b by a distance-dependent function that decays to zero beyond a certain distance, so that observations do not affect the model state beyond that distance [21,17,45]. We will follow the explicit approach here, doing a separate analysis for each spatial grid point of the model.…”

“…Furthermore, many or all of the blocks that make up R may be unchanged from one analysis time to the next, so that their inverses need not be recomputed each time. Based on these considerations, the number of operations required (at each grid point) for this step in a typical application should be proportional to kℓ, multiplied by a factor related to the typical block size of R. (21). Here ρ > 1 is a multiplicative covariance inflation factor, as described at the end of the previous section.…”

“…However, as explained below, for efficiency one should make the batches as large as possible while still meeting the first condition.) Then at Step 3, instead of selecting (overlapping) submatrices ofȳ b This approach has similarities with the "sequential" approach of [21] and [45], in which observations are divided into uncorrelated batches and a separate analysis is done for each batch; the analysis is done in the observation space whose dimension is the number of observations in a batch. However, in the sequential approach, the analysis ensemble for one batch of observations is used as the background ensemble for the next batch of observations.…”

Total Energy Singular Vectors for Atmospheric Chemical Transport Models

“…As explained by Fisher [6], the Kalman filter is too expensive to be a practical assimilation method for large-scale systems. The ensemble Kalman filter [7] is a feasible approach which approximates the Kalman filter covariance matrix by a MonteCarlo-type technique.…”

Operational Implementation of Variational Data Assimilation

“…Section 5 discusses some avenues being explored in current research. This includes the ensemble Kalman filter (Evensen, 1997;Houtekamer and Mitchell, 2001) or representer algorithms (Bennett and Thorburn, 1982;Xu and Daley, 2002).…”