Singular Evolutive Interpolated Kalman Research Articles

Ensemble Kalman filter inference of spatially-varying Manning’s n coefficients in the coastal ocean

Ensemble Kalman (EnKF) filtering is an established framework for large scale state estimation problems. EnKFs can also be used for state-parameter estimation, using the so-called “Joint-EnKF” approach. The idea is simply to augment the state vector with the parameters to be estimated and assign invariant dynamics for the time evolution of the parameters. In this contribution, we investigate the efficiency of the Joint-EnKF for estimating spatially-varying Manning’s n coefficients used to define the bottom roughness in the Shallow Water Equations (SWEs) of a coastal ocean model.Observation System Simulation Experiments (OSSEs) are conducted using the ADvanced CIRCulation (ADCIRC) model, which solves a modified form of the Shallow Water Equations. A deterministic EnKF, the Singular Evolutive Interpolated Kalman (SEIK) filter, is used to estimate a vector of Manning’s n coefficients defined at the model nodal points by assimilating synthetic water elevation data. It is found that with reasonable ensemble size (O(10)), the filter’s estimate converges to the reference Manning’s field. To enhance performance, we have further reduced the dimension of the parameter search space through a Karhunen-Loéve (KL) expansion. We have also iterated on the filter update step to better account for the nonlinearity of the parameter estimation problem. We study the sensitivity of the system to the ensemble size, localization scale, dimension of retained KL modes, and number of iterations. The performance of the proposed framework in term of estimation accuracy suggests that a well-tuned Joint-EnKF provides a promising robust approach to infer spatially varying seabed roughness parameters in the context of coastal ocean modeling.

Efficient ensemble forecasting of marine ecology with clustered 1D models and statistical lateral exchange: application to the Red Sea

Forecasting the state of large marine ecosystems is important for many economic and public health applications. However, advanced three-dimensional (3D) ecosystem models, such as the European Regional Seas Ecosystem Model (ERSEM), are computationally expensive, especially when implemented within an ensemble data assimilation system requiring several parallel integrations. As an alternative to 3D ecological forecasting systems, we propose to implement a set of regional one-dimensional (1D) water-column ecological models that run at a fraction of the computational cost. The 1D model domains are determined using a Gaussian mixture model (GMM)-based clustering method and satellite chlorophyll-a (Chl-a) data. Regionally averaged Chl-a data is assimilated into the 1D models using the singular evolutive interpolated Kalman (SEIK) filter. To laterally exchange information between subregions and improve the forecasting skills, we introduce a new correction step to the assimilation scheme, in which we assimilate a statistical forecast of future Chl-a observations based on information from neighbouring regions. We apply this approach to the Red Sea and show that the assimilative 1D ecological models can forecast surface Chl-a concentration with high accuracy. The statistical assimilation step further improves the forecasting skill by as much as 50%. This general approach of clustering large marine areas and running several interacting 1D ecological models is very flexible. It allows many combinations of clustering, filtering and regression technics to be used and can be applied to build efficient forecasting systems in other large marine ecosystems.

A hybrid ensemble-OI Kalman filter for efficient data assimilation into a 3-D biogeochemical model of the Mediterranean

A hybrid ensemble data assimilation scheme (HYBRID), combining a flow-dependent with a static background covariance, was developed and implemented for assimilating satellite (SeaWiFS) Chl-a data into a marine ecosystem model of the Mediterranean. The performance of HYBRID was assessed against a model free-run, the ensemble-based singular evolutive interpolated Kalman (SEIK) and its variant with static covariance (SFEK), with regard to the assimilated variable (Chl-a) and non-assimilated variables (dissolved inorganic nutrients). HYBRID was found more efficient than both SEIK and SFEK, reducing the Chl-a error by more than 40% in most areas, as compared to the free-run. Data assimilation had a positive overall impact on nutrients, except for a deterioration of nitrates simulation by SEIK in the most productive area (Adriatic). This was related to SEIK pronounced update in this area and the phytoplankton limitation on phosphate that lead to a built up of excess nitrates. SEIK was found more efficient in productive and variable areas, where its ensemble exhibited important spread. SFEK had an effect mostly on Chl-a, performing better than SEIK in less dynamic areas, adequately described by the dominant modes of its static covariance. HYBRID performed well in all areas, due to its “blended” covariance. Its flow-dependent component appears to track changes in the system dynamics, while its static covariance helps maintaining sufficient spread in the forecast. HYBRID sensitivity experiments showed that an increased contribution from the flow-dependent covariance results in a deterioration of nitrates, similar to SEIK, while the improvement of HYBRID with increasing flow-dependent ensemble size quickly levels off.

Data assimilation of depth-distributed satellite chlorophyll-α in two Mediterranean contrasting sites

A new approach for processing the remote sensing chlorophyll-α (Chl-α) before assimilating into an ecosystem model is applied in two contrasting, regarding productivity and nutrients availability, Mediterranean sites: the DYFAMED and POSEIDON E1-M3A fixed point open ocean observatories. The new approach derives optically weighted depth-distributed Chl-α profiles from satellite data based on the model simulated Chl-α vertical distribution and light attenuation coefficient. We use the 1D version of the operational ecological 3D POSEIDON model, based on the European Regional Seas Ecosystem Model (ERSEM). The required hydrodynamic properties are obtained (off-line) from the POSEIDON operational 3D hydrodynamic Mediterranean basin scale model. The data assimilation scheme is the singular evolutive interpolated Kalman (SEIK) filter, the ensemble variant of the singular evolutive extended Kalman (SEEK) filter. The performance of the proposed assimilation approach was evaluated against the Chl-α satellite data and the seasonal averages of available in situ data for nitrate, phosphate and Chl-α. An improvement of the model simulated near-surface and subsurface maximum Chl-α concentrations is obtained, especially at the DYFAMED site. Model nitrate is improved with assimilation, particularly with the new approach assimilating depth-distributed Chl-α, while model phosphate is slightly worse after assimilation. Additional sensitivity experiments were performed, showing a better performance of the new approach under different scenarios of model Chl-α deviation from pseudo-observations of surface Chl-α.

A Comparison of Ensemble Kalman Filters for Storm Surge Assimilation

Abstract This study evaluates and compares the performances of several variants of the popular ensemble Kalman filter for the assimilation of storm surge data with the advanced circulation (ADCIRC) model. Using meteorological data from Hurricane Ike to force the ADCIRC model on a domain including the Gulf of Mexico coastline, the authors implement and compare the standard stochastic ensemble Kalman filter (EnKF) and three deterministic square root EnKFs: the singular evolutive interpolated Kalman (SEIK) filter, the ensemble transform Kalman filter (ETKF), and the ensemble adjustment Kalman filter (EAKF). Covariance inflation and localization are implemented in all of these filters. The results from twin experiments suggest that the square root ensemble filters could lead to very comparable performances with appropriate tuning of inflation and localization, suggesting that practical implementation details are at least as important as the choice of the square root ensemble filter itself. These filters also perform reasonably well with a relatively small ensemble size, whereas the stochastic EnKF requires larger ensemble sizes to provide similar accuracy for forecasts of storm surge.

Improving Short-Range Ensemble Kalman Storm Surge Forecasting Using Robust Adaptive Inflation

Abstract This paper presents a robust ensemble filtering methodology for storm surge forecasting based on the singular evolutive interpolated Kalman (SEIK) filter, which has been implemented in the framework of the H∞ filter. By design, an H∞ filter is more robust than the common Kalman filter in the sense that the estimation error in the H∞ filter has, in general, a finite growth rate with respect to the uncertainties in assimilation. The computational hydrodynamical model used in this study is the Advanced Circulation (ADCIRC) model. The authors assimilate data obtained from Hurricanes Katrina and Ike as test cases. The results clearly show that the H∞-based SEIK filter provides more accurate short-range forecasts of storm surge compared to recently reported data assimilation results resulting from the standard SEIK filter.

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

A data assimilation (DA) system has been developed for the operational circulation model of the German Federal Maritime and Hydrographic Agency (BSH) in order to improve the forecast of hydrographic characteristics in the North and Baltic Seas. It is based on the local Singular Evolutive Interpolated Kalman (SEIK) filter algorithm and assimilation of the NOAA AVHRR-derived sea surface temperature (SST). The DA system allows one to improve the agreement of the SST forecast with the satellite observations by 27% on average over the period of October 2007–September 2008. However, a sensitivity analysis of the forecasting system performance shows a significant impact of initial model error statistics on ice fields and bottom temperature. A reinitialisation of model error covariances in accordance with seasonality of the model error statistics was required in order to maintain the predictive skill with respect to these variables. The success of the DA system is quantified by the comparison with independent data from MARNET stations as well as sea ice concentration measurements. In addition, the Maximum Entropy approach is used to assess the system performance and the prior and posterior model error statistics.

Assessing a robust ensemble-based Kalman filter for efficient ecosystem data assimilation of the Cretan Sea

An application of an ensemble-based robust filter for data assimilation into an ecosystem model of the Cretan Sea is presented and discussed. The ecosystem model comprises two on-line coupled sub-models: the Princeton Ocean Model (POM) and the European Regional Seas Ecosystem Model (ERSEM). The filtering scheme is based on the Singular Evolutive Interpolated Kalman (SEIK) filter which is implemented with a time-local H∞ filtering strategy to enhance robustness and performances during periods of strong ecosystem variability. Assimilation experiments in the Cretan Sea indicate that robustness can be achieved in the SEIK filter by introducing an adaptive inflation scheme of the modes of the filter error covariance matrix.Twin-experiments are performed to evaluate the performance of the assimilation system and to study the benefits of using robust filtering in an ensemble filtering framework. Pseudo-observations of surface chlorophyll, extracted from a model reference run, were assimilated every two days. Simulation results suggest that the adaptive inflation scheme significantly improves the behavior of the SEIK filter during periods of strong ecosystem variability.

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

The operational ocean prediction model for the North and Baltic Seas of the German Maritime and Hydrographic Agency (BSH) is augmented with a multivariate data assimilation (DA) system. We report on the implementation and performance of the scheme which is based on ensemble forecasting. Here we apply the localised Singular Evolutive Interpolated Kalman (SEIK) filter for assimilating the NOAA AVHRR-derived sea surface temperature (SST) data.Results are presented for two periods: October 2007 is used for calibration and March 2011 for the analysis of the performance in a pre-operational phase. The major forecast improvement is found to be a reduction in the local temperature bias. As compared with the regular BSH forecast without assimilation, the root mean square difference between the predicted SST and satellite observations is reduced on average from 0.87°C to 0.53°C for March 2011. The quality of the predicted fields that were not assimilated (velocities, sea level and salinity) is preserved as is confirmed by independent in situ data. The results have required adjustment of the conditional data error statistics.The experiments conducted with different timing and frequency of data assimilation and variable forecasting periods show that the DA system corrects systematic model uncertainties and, due to memory to the corrections, improves prediction over periods of up to 5days. The results also explicitly illustrate a lower quality of the AVHRR daytime product and reveal low informative influence of the data on the forecasting system when daytime SSTs are assimilated additionally to midnight observations.

A Unification of Ensemble Square Root Kalman Filters

AbstractIn 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.

Data Assimilation within the Advanced Circulation (ADCIRC) Modeling Framework for Hurricane Storm Surge Forecasting

AbstractAccurate, real-time forecasting of coastal inundation due to hurricanes and tropical storms is a challenging computational problem requiring high-fidelity forward models of currents and water levels driven by hurricane-force winds. Despite best efforts in computational modeling there will always be uncertainty in storm surge forecasts. In recent years, there has been significant instrumentation located along the coastal United States for the purpose of collecting data—specifically wind, water levels, and wave heights—during these extreme events. This type of data, if available in real time, could be used in a data assimilation framework to improve hurricane storm surge forecasts. In this paper a data assimilation methodology for storm surge forecasting based on the use of ensemble Kalman filters and the advanced circulation (ADCIRC) storm surge model is described. The singular evolutive interpolated Kalman (SEIK) filter has been shown to be effective at producing accurate results for ocean models using small ensemble sizes initialized by an empirical orthogonal function analysis. The SEIK filter is applied to the ADCIRC model to improve storm surge forecasting, particularly in capturing maximum water levels (high water marks) and the timing of the surge. Two test cases of data obtained from hindcast studies of Hurricanes Ike and Katrina are presented. It is shown that a modified SEIK filter with an inflation factor improves the accuracy of coarse-resolution forecasts of storm surge resulting from hurricanes. Furthermore, the SEIK filter requires only modest computational resources to obtain more accurate forecasts of storm surge in a constrained time window where forecasters must interact with emergency responders.

Assimilation of geodetic dynamic ocean topography using ensemble based Kalman filter

Estimation of ocean circulation is investigated via assimilation of satellite measurements of the dynamic ocean topography (DOT) into the global finite-element ocean model (FEOM). The DOT was obtained by means of a geodetic approach from carefully cross-calibrated multi-mission altimeter data and GRACE gravity fields. The spectral consistency was achieved by consistently filtering both, the sea surface and the geoid. The filter length is determined by the spatial resolution of the gravity field and corresponds to approximately 241km half width for the GRACE-based gravity field model ITG-Grace03s.The assimilation of the geodetic DOT was performed by employing a local singular evolutive interpolated Kalman (SEIK) filter in combination with the method of weighting of observations. It is shown that this approach leads to a successful assimilation technique that reduced the RMS difference between the model and the data from 16cm to 5cm during one year of assimilation. The ocean model returns an optimized mean dynamic ocean topography. The effects of assimilation on transport estimates across several hydrographic World Ocean Circulation Experiment (WOCE) sections show improvements compared to the FEOM run without data assimilation. As a result of the assimilation, DOT estimates are available in the polar or coastal regions where the geodetic estimates from satellite data alone are not adequate. Furthermore, more realistic features of the ocean can be seen in these areas compared to those obtained using the filtered data fields.

Assimilation of sea ice motion in a finite‐element sea ice model

A finite‐element sea ice model (FESIM) is applied in a data assimilation study with the singular evolutive interpolated Kalman (SEIK) filter. The model has been configured for a regional Arctic domain and is forced with a combination of daily NCEP reanalysis data for 2‐m air temperature and 10‐m winds with monthly mean humidities from the ECMWF reanalysis and climatological fields for precipitation and cloudiness. We assimilate 3‐day mean ice drift fields derived from passive microwave satellite data. Based on multivariate covariances (which describe the statistical relationship between anomalies in different model fields), the sea ice drift data assimilation produces not only direct modifications of the ice drift but also updates for sea ice concentration and thickness, which in turn yield sustainable corrections of ice drift. We use observed buoy trajectories as an independent data set to validate the analyzed sea ice drift field. A good agreement between modeled and observed tracks is achieved already in the reference simulation. Application of the SEIK filter with satellite‐derived drift fields further improves the agreement. Spatial and temporal variability of ice thickness increases due to the assimilation procedure; a comparison to thickness data from a submarine‐based upward looking sonar indicates that the thickness distribution becomes more realistic. Validation with regard to satellite data shows that the velocity data assimilation has only a small effect on ice concentration, but a general improvement of the ice concentration within the pack is still evident.

Data assimilation with the Ensemble Kalman Filter and the SEIK filter applied to a finite element model of the North Atlantic

Currently there are different approaches to filter algorithms based on the Kalman filter. One of the most used filter algorithms is the Ensemble Kalman Filter (EnKF). It uses a Monte Carlo approach to the filtering problem. Another approach is given by the Singular Evolutive Extended Kalman (SEEK) and Singular Evolutive Interpolated Kalman (SEIK) filters. These filters operate explicitly on a low-dimensional error space which is represented by an ensemble of model states. The EnKF and the SEIK filter have been implemented within a parallel data assimilation framework in the Finite Element Ocean Model FEOM. In order to compare the filter performances of the algorithms, several data assimilation experiments are performed. The filter algorithms have been applied with a model configuration of FEOM for the North Atlantic to assimilate the sea surface height in twin experiments. The dependence of the filter estimates on the represented error subspace is discussed. In the experiments the SEIK algorithm provides better estimates than the EnKF. Furthermore, the SEIK filter is much cheaper in terms of computing time.

A comparison of error subspace Kalman filters

Three advanced filter algorithms based on the Kalman filter are reviewed and presented in a unified notation. They are the well-known ensemble Kalman filter (EnKF), the singular evolutive extended Kalman (SEEK) filter, and the less common singular evolutive interpolated Kalman (SEIK) filter. For comparison, the mathematical formulations of the filters are reviewed in relation to the extended Kalman filter as error subspace Kalman filters. The algorithms are presented in their original form and possible variations are discussed. A comparison of the algorithms shows their theoretical capabilities for efficient data assimilation with large-scale non-linear systems. In particular, problems of the analysis equations are apparent in the original EnKF algorithm due to the Monte Carlo sampling of ensembles. Theoretically, the SEIK filter appears to be a numerically very efficient algorithm with high potential for use with non-linear models. The superiority of the SEIK filter is demonstrated on the basis of identical twin experiments using a shallow-water model with non-linear evolution. Identical initial conditions for all three filters allow for a consistent comparison of the data assimilation results. These show how choices of particular state ensembles and assimilation schemes lead to significant variations of the filter performance. This is related to different qualities of the predicted error subspaces, as is demonstrated in an examination of the predicted state covariance matrices.

A comparison of error subspace Kalman filters

Three advanced filter algorithms based on the Kalman filter are reviewed and presented in a unified notation. They are the well-known ensemble Kalman filter (EnKF), the singular evolutive extended Kalman (SEEK) filter, and the less common singular evolutive interpolated Kalman (SEIK) filter. For comparison, the mathematical formulations of the filters are reviewed in relation to the extended Kalman filter as error subspace Kalman filters. The algorithms are presented in their original form and possible variations are discussed. A comparison of the algorithms shows their theoretical capabilities for efficient data assimilation with large-scale non-linear systems. In particular, problems of the analysis equations are apparent in the original EnKF algorithm due to the Monte Carlo sampling of ensembles. Theoretically, the SEIK filter appears to be a numerically very efficient algorithm with high potential for use with non-linear models. The superiority of the SEIK filter is demonstrated on the basis of identical twin experiments using a shallow-water model with non-linear evolution. Identical initial conditions for all three filters allow for a consistent comparison of the data assimilation results. These show how choices of particular state ensembles and assimilation schemes lead to significant variations of the filter performance. This is related to different qualities of the predicted error subspaces, as is demonstrated in an examination of the predicted state covariance matrices.

A singular evolutive interpolated Kalman filter for efficient data assimilation in a 3-D complex physical–biogeochemical model of the Cretan Sea

A singular evolutive interpolated Kalman (SEIK) filter is used to assimilate pseudo-observations via twin simulation experiments in a complex three-dimensional coupled physical–biogeochemical model of the Cretan Sea. The simulation system comprises two on-line coupled sub-models: the three-dimensional Princeton Model and the European Regional Seas Ecosystem Model (ERSEM). In the SEIK filter, the estimation error is represented by an ensemble of state vectors, which are drawn randomly at every filtering step. In the twin experiments performed the predictions of the coupled model were corrected every 2 days using synthetic measurements extracted from a model reference run according to a network of 23 stations in the Cretan Sea. The filter is shown to be very efficient, with the assimilation results exhibiting a continuous decrease of the estimation error during the experimental period.

A simplified reduced order Kalman filtering and application to altimetric data assimilation in Tropical Pacific

Several studies have demonstrated the effectiveness of the singular evolutive extended Kalman (SEEK) filter and its interpolated variant called singular evolutive interpolated Kalman (SEIK) filter in their capacity to assimilate altimetric data into ocean models. However, these filters remain expensive for real operational assimilation. The purpose of this paper is to develop degraded forms of the SEIK filter which are less costly and yet perform reasonably well. Our approach essentially consists in simplifying the evolution of the correction basis of the SEIK filter, which is the most expensive part of this filter. To deal with model instabilities, we also introduce two adaptive tuning schemes to control the correction basis evolution and adjust the variable forgetting factor. Our filters have been implemented in a realistic setting of the OPA model over the tropical pacific zone and their performance studied through twin experiments in which the observations are taken to be synthetic altimeter data sampled on the sea surface. The SEIK filter is used as a reference for comparison. Our new filters perform nearly as well as the SEIK, but can be 2–30 times faster.