A Unification of Ensemble Square Root Kalman Filters

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 SEIK (Singular ``Evolutive'' Interpolated Kalman) 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

Conservation of Mass and Preservation of Positivity with Ensemble-Type Kalman Filter Algorithms

This paper considers the incorporation of constraints to enforce physically based conservation laws in the ensemble Kalman filter. In particular, constraints are used to ensure that the ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. In certain situations filtering algorithms such as the ensemble Kalman filter (EnKF) and ensemble transform Kalman filter (ETKF) yield updated ensembles that conserve mass but are negative, even though the actual states must be nonnegative. In such situations if negative values are set to zero, or a log transform is introduced, the total mass will not be conserved. In this study, mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate non-negativity constraints. Simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically plausible both for individual ensemble members and for the ensemble mean. In two examples, an update that includes a non-negativity constraint is able to properly describe the transport of a sharp feature (e.g., a triangle or cone). A number of implementation questions still need to be addressed, particularly the need to develop a computationally efficient quadratic programming update for large ensemble

Assimilation of dynamic topography in a global model

Absolute dynamic topography, i.e. the difference between time dependent multi-mission altimetric sea surface height and one of the most recent GOCE and GRACE based geoids, is assimilated in a global ocean general circulation model. To this end we apply an ensemble based Kalman technique, the "Error Subspace Transform Kalman Filter" (ESTKF). Here we present an update of our work. First of all the geoid is improved over previous versions. The ocean model now includes better dynamics and full sea-ice ocean interactions and more realistic surface forcing. Finally the assimilation method is augmented by a fixed lag smoother technique. This smoother allows to significantly improve the model performance, most strikingly in the first adjustment phase

Mass Conservation and Positivity Preservation with Ensemble-type Kalman Filter Algorithms

Maintaining conservative physical laws numerically has long been recognized as being important in the development of numerical weather prediction (NWP) models. In the broader context of data assimilation, concerted efforts to maintain conservation laws numerically and to understand the significance of doing so have begun only recently. In order to enforce physically based conservation laws of total mass and positivity in the ensemble Kalman filter, we incorporate constraints to ensure that the filter ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. We show that the analysis steps of ensemble transform Kalman filter (ETKF) algorithm and ensemble Kalman filter algorithm (EnKF) can conserve the mass integral, but do not preserve positivity. Further, if localization is applied or if negative values are simply set to zero, then the total mass is not conserved either. In order to ensure mass conservation, a projection matrix that corrects for localization effects is constructed. In order to maintain both mass conservation and positivity preservation through the analysis step, we construct a data assimilation algorithms based on quadratic programming and ensemble Kalman filtering. Mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate constraints. Some simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically plausible both for individual ensemble members and for the ensemble mean. The results show clear improvements in both analyses and forecasts, particularly in the presence of localized features. Behavior of the algorithm is also tested in presence of model error

A virtual centre at the interface of basic and applied weather and climate research

The Hans-Ertel Centre for Weather Research is a network of German universities, research institutes and the German Weather Service (Deutscher Wetterdienst, DWD). It has been established to trigger and intensify basic research and education on weather forecasting and climate monitoring. The performed research ranges from nowcasting and short-term weather forecasting to convective-scale data assimilation, the development of parameterizations for numerical weather prediction models, climate monitoring and the communication and use of forecast information. Scientific findings from the network contribute to better understanding of the life-cycle of shallow and deep convection, representation of uncertainty in ensemble systems, effects of unresolved variability, regional climate variability, perception of forecasts and vulnerability of society. Concrete developments within the research network include dual observation-microphysics composites, satellite forward operators, tools to estimate observation impact, cloud and precipitation system tracking algorithms, large-eddy-simulations, a regional reanalysis and a probabilistic forecast test product. Within three years, the network has triggered a number of activities that include the training and education of young scientists besides the centre's core objective of complementing DWD's internal research with relevant basic research at universities and research institutes. The long term goal is to develop a self-sustaining research network that continues the close collaboration with DWD and the national and international research community

SEIK - the unknown ensemble Kalman filter

The SEIK filter (Singular "Evolutive" Interpolated Kalman filter) hasbeen introduced in 1998 by D.T. Pham as a variant of the SEEK filter,which is a reduced-rank approximation of the Extended KalmanFilter. In recent years, it has been shown that the SEIK filter isan ensemble-based Kalman filter that uses a factorization rather thansquare-root of the state error covariance matrix. Unfortunately, theexistence of the SEIK filter as an ensemble-based Kalman filter withsimilar efficiency as the later introduced ensemble square-root Kalmanfilters, appears to be widely unknown and the SEIK filter is omittedin reviews about ensemble-based Kalman filters. To raise the attentionabout the SEIK filter as a very efficient ensemble-based Kalmanfilter, we review the filter algorithm and compare it with ensemblesquare-root Kalman filter algorithms. For a practical comparison theSEIK filter and the Ensemble Transformation Kalman filter (ETKF) areapplied in twin experiments assimilating sea level anomaly data intothe finite-element ocean model FEOM

A regulated localization method for ensemble-based Kalman filters

Data assimilation applications with large-scale numerical models exhibit extreme requirements on computational resources. Good scalability of the assimilation system is necessary to make these applications feasible. Sequential data assimilation methods based on ensemble forecasts, like ensemble-based Kalman filters, provide such good scalability, because the forecast of each ensemble member can be performed independently. However, this parallelism has to be combined with the parallelization of both the numerical model and the data assimilation algorithm. In order to simplify the implementation of scalable data assimilation systems based on existing numerical models, the Parallel Data Assimilation Framework PDAF (http://pdaf.awi.de) has been developed. PDAF provides support for implementing a data assimilation system with parallel ensemble forecasts and parallel numerical models. Further, it includes several optimized parallel filter algorithms, like the Ensemble Transform Kalman Filter. We will discuss the philosophy behind PDAF as well as features and scalability of data assimilation systems based on PDAF on the example of data assimilation with the finite element ocean model FEOM

The impact of the new gravity field models on the Mean Dynamic Ocean Topography and the derived geostrophic velocities

The absolute Mean Dynamic ocean Topography (MDT) can be determined from an accurate geoid model and a Mean Sea Surface (MSS). The MSS is derived using long-term time series of sea surface heights from multi-mission satellite altimetry. Recently, data from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite has become available. Now, GOCE and GRACE satellite data can be combined to obtain a geoid with higher accuracy and spatial resolution than before. The improvement in the geoid accuracy and resolution implies improvements in the resolution of MDT. From only 6 months of GOCE data, oceanographic fields like mean dynamic topography and geostrophic velocities are given in a fine spatial scale that has been poorly resolved previously. This is especially true in the areas of strong currents like Agulhas, Gulf, Kuroshio and Antarctic Circumpolar Current. Geostrophic velocities derived from only satellite data show very good agreement with geostrophic velocities measured by drifters. In addition the assimilation of this data set allows us to obtain all surface and subsurface ocean variables consistent with new MDT, giving promising results in comparison to the free model