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Lars Nerger

Abstract

Ensemble square root filters can either assimilate all observations that are available at a given time at once, or assimilate the observations in batches or one at a time. For large-scale models, the filters are typically applied with a localized analysis step. This study demonstrates that the interaction of serial observation processing and localization can destabilize the analysis process, and it examines under which conditions the instability becomes significant. The instability results from a repeated inconsistent update of the state error covariance matrix that is caused by the localization. The inconsistency is present in all ensemble Kalman filters, except for the classical ensemble Kalman filter with perturbed observations. With serial observation processing, its effect is small in cases when the assimilation changes the ensemble of model states only slightly. However, when the assimilation has a strong effect on the state estimates, the interaction of localization and serial observation processing can significantly deteriorate the filter performance. In realistic large-scale applications, when the assimilation changes the states only slightly and when the distribution of the observations is irregular and changing over time, the instability is likely not significant.

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Paul Kirchgessner, Lars Nerger, and Angelika Bunse-Gerstner

Abstract

In data assimilation applications using ensemble Kalman filter methods, localization is necessary to make the method work with high-dimensional geophysical models. For ensemble square root Kalman filters, domain localization (DL) and observation localization (OL) are commonly used. Depending on the localization method, appropriate values have to be chosen for the localization parameters, such as the localization length and the weight function. Although frequently used, the properties of the localization techniques are not fully investigated. Thus, up to now an optimal choice for these parameters is a priori unknown and they are generally found by expensive numerical experiments. In this study, the relationship between the localization length and the ensemble size in DL and OL is studied using twin experiments with the Lorenz-96 model and a two-dimensional shallow-water model. For both models, it is found that the optimal localization length for DL and OL depends linearly on an effective local observation dimension that is given by the sum of the observation weights. In the experiments no influence of the model dynamics on the optimal localization length was observed. The effective observation dimension defines the degrees of freedom that are required for assimilating observations, while the ensemble size defines the available degrees of freedom. Setting the localization radius such that the effective local observation dimension equals the ensemble size yields an adaptive localization radius. Its performance is tested using a global ocean model. The experiments show that the analysis quality using the adaptive localization is similar to the analysis quality of an optimally tuned constant localization radius.

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Lars Nerger, Tijana Janjić, Jens Schröter, and Wolfgang Hiller

Abstract

In recent years, several ensemble-based Kalman filter algorithms have been developed that have been classified as ensemble square root Kalman filters. Parallel to this development, the singular “evolutive” interpolated Kalman (SEIK) filter has been introduced and applied in several studies. Some publications note that the SEIK filter is an ensemble Kalman filter or even an ensemble square root Kalman filter. This study examines the relation of the SEIK filter to ensemble square root filters in detail. It shows that the SEIK filter is indeed an ensemble square root Kalman filter. Furthermore, a variant of the SEIK filter, the error subspace transform Kalman filter (ESTKF), is presented that results in identical ensemble transformations to those of the ensemble transform Kalman filter (ETKF), while having a slightly lower computational cost. Numerical experiments are conducted to compare the performance of three filters (SEIK, ETKF, and ESTKF) using deterministic and random ensemble transformations. The results show better performance for the ETKF and ESTKF methods over the SEIK filter as long as this filter is not applied with a symmetric square root. The findings unify the separate developments that have been performed for the SEIK filter and the other ensemble square root Kalman filters.

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Julian Tödter, Paul Kirchgessner, Lars Nerger, and Bodo Ahrens

Abstract

This work assesses the large-scale applicability of the recently proposed nonlinear ensemble transform filter (NETF) in data assimilation experiments with the NEMO ocean general circulation model. The new filter constitutes a second-order exact approximation to fully nonlinear particle filtering. Thus, it relaxes the Gaussian assumption contained in ensemble Kalman filters. The NETF applies an update step similar to the local ensemble transform Kalman filter (LETKF), which allows for efficient and simple implementation. Here, simulated observations are assimilated into a simplified ocean configuration that exhibits globally high-dimensional dynamics with a chaotic mesoscale flow. The model climatology is used to initialize an ensemble of 120 members. The number of observations in each local filter update is of the same order resulting from the use of a realistic oceanic observation scenario. Here, an importance sampling particle filter (PF) would require at least 106 members. Despite the relatively small ensemble size, the NETF remains stable and converges to the truth. In this setup, the NETF achieves at least the performance of the LETKF. However, it requires a longer spinup period because the algorithm only relies on the particle weights at the analysis time. These findings show that the NETF can successfully deal with a large-scale assimilation problem in which the local observation dimension is of the same order as the ensemble size. Thus, the second-order exact NETF does not suffer from the PF’s curse of dimensionality, even in a deterministic system.

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Tijana Janjić, Lars Nerger, Alberta Albertella, Jens Schröter, and Sergey Skachko

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.

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Qinghua Yang, Martin Losch, Svetlana N. Losa, Thomas Jung, and Lars Nerger

Abstract

The sensitivity of assimilating sea ice thickness data to uncertainty in atmospheric forcing fields is examined using ensemble-based data assimilation experiments with the Massachusetts Institute of Technology General Circulation Model (MITgcm) in the Arctic Ocean during November 2011–January 2012 and the Met Office (UKMO) ensemble atmospheric forecasts. The assimilation system is based on a local singular evolutive interpolated Kalman (LSEIK) filter. It combines sea ice thickness data derived from the European Space Agency’s (ESA) Soil Moisture Ocean Salinity (SMOS) satellite and Special Sensor Microwave Imager/Sounder (SSMIS) sea ice concentration data with the numerical model. The effect of representing atmospheric uncertainty implicit in the ensemble forcing is assessed by three different assimilation experiments. The first two experiments use a single deterministic forcing dataset and a different forgetting factor to inflate the ensemble spread. The third experiment uses 23 members of the UKMO atmospheric ensemble prediction system. It avoids additional ensemble inflation and is hence easier to implement. As expected, the model-data misfits are substantially reduced in all three experiments, but with the ensemble forcing the errors in the forecasts of sea ice concentration and thickness are smaller compared to the experiments with deterministic forcing. This is most likely because the ensemble forcing results in a more plausible spread of the model state ensemble, which represents model uncertainty and produces a better forecast.

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Xi Liang, Qinghua Yang, Lars Nerger, Svetlana N. Losa, Biao Zhao, Fei Zheng, Lin Zhang, and Lixin Wu

Abstract

Sea surface temperature (SST) data from the Copernicus Marine Environment Monitoring Service are assimilated into a pan-Arctic ice–ocean coupled model using the ensemble-based local singular evolutive interpolated Kalman (LSEIK) filter. This study found that the SST deviation between model hindcasts and independent SST observations is reduced by the assimilation. Compared with model results without data assimilation, the deviation between the model hindcasts and independent SST observations has decreased by up to 0.2°C at the end of summer. The strongest SST improvements are located in the Greenland Sea, the Beaufort Sea, and the Canadian Arctic Archipelago. The SST assimilation also changes the sea ice concentration (SIC). Improvements of the ice concentrations are found in the Canadian Arctic Archipelago, the Beaufort Sea, and the central Arctic basin, while negative effects occur in the west area of the eastern Siberian Sea and the Laptev Sea. Also, sea ice thickness (SIT) benefits from ensemble SST assimilation. A comparison with upward-looking sonar observations reveals that hindcasts of SIT are improved in the Beaufort Sea by assimilating reliable SST observations into light ice areas. This study illustrates the advantages of assimilating SST observations into an ice–ocean coupled model system and suggests that SST assimilation can improve SIT hindcasts regionally during the melting season.

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