On Domain Localization in Ensemble-Based Kalman Filter Algorithms

Abstract

Ensemble Kalman filter methods are typically used in combination with one of two localization techniques. One technique is covariance localization, or direct forecast error localization, in which the ensemble-derived forecast error covariance matrix is Schur multiplied with a chosen correlation matrix. The second way of localization is by domain decomposition. Here, the assimilation is split into local domains in which the assimilation update is performed independently. Domain localization is frequently used in combination with filter algorithms that use the analysis error covariance matrix for the calculation of the gain like the ensemble transform Kalman filter (ETKF) and the singular evolutive interpolated Kalman filter (SEIK). However, since the local assimilations are performed independently, smoothness of the analysis fields across the subdomain boundaries becomes an issue of concern.To address the problem of smoothness, an algorithm is introduced that uses domain localization in combination with a Schur product localization of the forecast error covariance matrix for each local subdomain. On a simple example, using the Lorenz-40 system, it is demonstrated that this modification can produce results comparable to those obtained with direct forecast error localization. In addition, these results are compared to the method that uses domain localization in combination with weighting of observations. In the simple example, the method using weighting of observations is less accurate than the new method, particularly if the observation errors are small. Domain localization with weighting of observations is further examined in the case of assimilation of satellite data into the global finite-element ocean circulation model (FEOM) using the local SEIK filter. In this example, the use of observational weighting improves the accuracy of the analysis. In addition, depending on the correlation function used for weighting, the spectral properties of the solution can be improved.

Assimilating NOAA SST data into the BSH operational circulation model for the North and Baltic Seas: Inference about the data

“…6. One more approximation applied in order to reduce the ensemble size is localization (for overview see Janjić et al, 2011), in other words, sampling which is optimal to each particular subspace (fragment of space X t ), rather than to the global space. In practice, localisation algorithms applied in ensemble-based Kalman filters introduce a radius of data influence and a weighting of the observation influence based on an assumed correlation function (Nerger et al, 2006;Hunt et al, 2007;Miyoshi and Yamane, 2007).…”

Assimilating NOAA SST data into BSH operational circulation model for the North and Baltic Seas: Part 2. Sensitivity of the forecast's skill to the prior model error statistics

“…This makes the task of data assimilation (DA) into ocean models rather challenging (Brusdal et al, 2003;Penduff et al, 2002;Testut et al, 2003;Bertino and Lisaeter, 2008;Brasseur et al, 2005;Cummings et al, 2009;Storkey et al, 2010;Kurapov et al, 2011). It pertains not only the implementation of DA algorithms but also the approximation of the error statistics (Counillon et al, 2009;Janjić et al, 2011;Fu et al, 2011;Simon and Bertino, 2012;Lermusiaux, 2007), which in each case demands a study on its own. This is in full measure related to the development of a DA system for the operational forecasting model of the North and Baltic Seas run by the German Federal Maritime and Hydrographic Agency (BSH), which was described in Losa et al (2012).…”

Ensemble Kalman filter for the reconstruction of the Earth's mantle circulation

“…We also tested the domain localization strategy as described in Janjic et al (2011), since it is computationally more efficient and already implemented in PDAF. However, it led to a systematic failure of the assimilation.…”