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.     

Wave data assimilation with the Kalman filter

A new data assimilation method for ocean waves is presented, based on an efficient low-rank approximation to the Kalman filter. Both the extended Kalman filter and a truncated second-order filter are implemented. In order to explicitly estimate past wind corrections based on current wave measurements, the filter is extended to a fixed-lag Kalman smoother for the wind fields. The filter is tested in a number of synthetic experiments with simple geometries. Propagation experiments with errors in the boundary condition showed that the KF was able to accurately propagate forecast errors, resulting in spatially varying error correlations, which would be impossible to model with time-independent assimilation methods like OI. An explicit comparison with an OI assimilation scheme showed that the KF also is superior in estimating the sea state at some distance from the observations. In experiments with errors in the driving wind, the modeled error estimates were also in agreement with the actual forecast errors. The bias in the state estimate, which is introduced through the nonlinear dependence of the waves on the driving wind field, was largely removed by the second-order filter, even without actually assimilating data. Assimilation of wave observations resulted in an improved wave analysis and in correction of past wind fields. The accuracy of this wind correction depends strongly on the actual place and time of wave generation, which is correctly modeled by the error estimate supplied by the Kalman filter. In summary, the KF approach is shown to be a reliable assimilation scheme in these simple experiments, and has the advantage over other assimilation methods that it supplies explicit dynamical error estimates.     

Data assimilation into a Princeton Ocean Model of the Mediterranean Sea using advanced Kalman filters

This study investigates the effectiveness of the Singular Evolutive Extended Kalman filter (SEEK) and its variants (SEIK and SFEK filters) for data assimilation into a Princeton Ocean Model (POM) of the Mediterranean Sea. The SEEK filters are sub-optimal Kalman filters based on the approximation of the filter's error covariance matrices by singular low-rank matrices, reducing in this way extensive computational burden. At the initialization, the filters error covariance matrix is parameterized by a set of multivariate empirical orthogonal functions (EOFs) which describe the dominant modes of the system's variability. The Mediterranean model is implemented on a 1/4° × 1/4° horizontal grid with 25 sigma levels and is forced with 6-hour ECMWF re-analysis atmospheric data. Several twin experiments, in which pseudo-observations of altimetric data and/or data profiles were assimilated, were first performed to evaluate the filters performances and to study their sensitivities to different parameters and setups. The results of these experiments were very encouraging and helped in setting up an effective configuration for the assimilation of real data in near-real time situation. In the hindcast experiments, Topex/Poseidon and ERS weekly sea level anomaly data were first assimilated during 1993 and the filters solution was evaluated against independent Reynolds sea surface temperature (SST) analysis. The assimilation system was able to significantly enhance the consistency between the model and the assimilated data, although the improvement with respect to independent SST data was significantly less pronounced. The model SST was only improved after including SST data in the assimilation system.     

Sequential data assimilation in an upwelling influenced estuary

We have evaluated the impact of assimilating chlorophyll, nitrate, phosphate, silicate and ammonium into a coupled 1D hydrodynamic ecosystem model (GOTM-ERSEM) in an upwelling influenced estuary. The assimilation method chosen is the Ensemble Kalman Filter (EnKF), which has been demonstrated to improve field estimates of key variables (chlorophyll, nutrients) for bulk algal bloom prediction. The 1D model has been set up for a central station inside the Ría de Vigo (Spain). Data from bi-weekly surveys are used to constrain the model. Temperature and salinity profiles are used to ensure the correct representation of the water structure through a relaxation scheme. Chlorophyll extracts and nutrients at three depths are assimilated sequentially during 1 year simulation (1991). The assimilation period includes episodes of active upwelling and downwelling. All five assimilated variables are successfully constrained and represent a large improvement on the reference simulation (without assimilation). Small divergences can be related to poorly resolved physical processes in the model. The assimilation was further evaluated by comparing observed biomass partitioning with model results. Diatoms accounted for the largest biomass update and the largest improvement in terms of percentage of variance explained (R²). This is particularly significant as they represent the 46% of the yearly integrated observed biomass of the planktonic autotrophs. Nonetheless the R² value was low for all phytoplankton groups. Bacteria and nanoflagellates showed an improvement with respect to their yearly Root Mean Square (RMS), while the other functional groups worsen or remained unaffected. Chlorophyll assimilation was responsible for most of the impact on the phytoplankton biomass with small contributions from the silicate. It had minor impact on the updates of nutrients which in turn corrected the state variables related to the detrital pool. In this current setting, combined assimilation of chlorophyll and nutrients is not sufficient to produce a skillful simulation of the phytoplankton succession.     

Application of a SEEK filter to a 1D biogeochemical model of the Ligurian Sea: Twin experiments and real in-situ data assimilation

The Singular Evolutive Extended Kalman (SEEK) filter has been implemented to assimilate in-situ data in a 1D coupled physical-ecosystem model of the Ligurian Sea. The biogeochemical model describes the partly decoupled nitrogen and carbon cycles of the pelagic food web. The GHER hydrodynamic model (1D version) is used to represent the physical forcings. The data assimilation scheme (SEEK filter) parameterizes the error statistics by means of a set of empirical orthogonal functions (EOFs). Twin experiments are first performed with the aim to choose the suitable experimental protocol (observation and estimation vectors, number of EOFs, frequency of the assimilation,…) and to assess the SEEK filter performances. This protocol is then applied to perform real data assimilation experiments using the DYFAMED data base. By assimilating phytoplankton observations, the method has allowed to improve not only the representation of the phytoplankton community, but also of other variables such as zooplankton and bacteria that evolve with model dynamics and that are not corrected by the data assimilation scheme. The validation of the assimilation method and the improvement of model results are studied by means of suitable error measurements.     

Multigrid state vector for data assimilation in a two-way nested model of the Ligurian Sea

A system of two nested models composed by a coarse resolution model of the Mediterranean Sea, an intermediate resolution model of the Provençal Basin and a high resolution model of the Ligurian Sea is coupled with a Kalman-filter based assimilation method. The state vector for the data assimilation is composed by the temperature, salinity and elevation of the three models. The forecast error is estimated by an ensemble run of 200 members by perturbing initial condition and atmospheric forcings. The 50 dominant empirical orthogonal functions (EOF) are taken as the error covariance of the model forecast. This error covariance is assumed to be constant in time. Sea surface temperature (SST) and sea surface height (SSH) are assimilated in this system.     

Gaussian anamorphosis extension of the DEnKF for combined state parameter estimation: Application to a 1D ocean ecosystem model

We consider the problem of combined state-parameter estimations in biased nonlinear models with non-Gaussian extensions of the Deterministic Ensemble Kalman Filter (DEnKF). We focus on the particular framework of ocean ecosystem models. Such models present important obstacles to the use of data assimilation methods based on Kalman filtering due to the non-linearity of the models, the constraints of positiveness that apply to the variables and parameters, and the non-Gaussian distribution of the variables in which they result.We present extensions of the DEnKF dealing with these difficulties by introducing a nonlinear change of variables (anamorphosis function) in order to execute the analysis step with Gaussian transformed variables and parameters. Several strategies to build the anamorphosis functions are investigated and compared within the framework of twin experiments realized in a simple 1D ocean ecosystem model. A solution to the problem of the specification of the observation error for transformed observations is suggested. The study highlights the inability of the plain DEnKF with a simple post-processing of the negative values to properly estimate parameters when constraints of positiveness apply to the variables. It goes on to show that the introduction of the Gaussian anamorphosis can remedy these assimilation biases.     

High-resolution synthetic monitoring by a 4-dimensional variational data assimilation system in the northwestern North Pacific

A data assimilation system is applied to the integrated monitoring of oceanic states in the northwestern North Pacific by combining a high resolution ocean general circulation model with an adjoint method. A comparison of assimilation results with observations shows that the system is better able to represent synoptic features of ocean circulation than do models or data alone. Furthermore, meso-scale features associated with frontal structures and eddies, which are often seen in the Kuroshio and Oyashio extension regions and the Sea of Japan, are better defined in the assimilation results. These features suggest that our 4D-VAR high-resolution data assimilation system is capable of providing time series data which satisfy the model physics and fit the observations, and hence the ocean state derived from our system has greater information content than that obtained from earlier methods.     

Validation of a hybrid optimal interpolation and Kalman filter scheme for sea surface temperature assimilation

A hybrid data assimilation scheme designed for operational assimilation of satellite sea surface temperatures (SST) into an ocean model has been developed and validated against in-situ observations. The scheme consists of an optimal interpolation (OI) part and a greatly simplified Kalman filter (KF) part.The OI is performed only in the longitudinal and latitudinal directions. A climatological field is used as a background field for the interpolation. It is constructed by fitting daily averages of satellite SST to the annual mean, annual, and semiannual harmonics in a 20 km by 20 km grid. The background error covariance is approximated by a spatially varying two-dimensional exponential covariance model. The parameters of the covariance model are fitted to the deviations of the satellite data from the background field using data from a full year.The simplified KF uses ocean model forecasts as a background field. It is based on the assumption that it is possible to neglect horizontal SST covariances in the filter and that the typical time scale for vertical mixing in the mixed layer is much shorter than the average time between observations. We therefore assume that the error variance in a column of water is evenly spread out throughout the mixed layer. The result of these simplifications is a computationally very efficient KF.A one year validation of the scheme is performed for year 2001 using an operational eddy resolving ocean model covering the North Sea and the Baltic Sea. It is found that assimilation of sea surface temperature data reduces the model root mean square error from 1.13 °C to 0.70 °C. The hybrid scheme is found to reduce the root mean square error slightly more than the simplified KF without OI to 0.66 °C. The inclusion of spatially varying satellite error variances does not improve the performance of the scheme significantly.     

Improving the physics of a coupled physical–biogeochemical model of the North Atlantic through data assimilation: Impact on the ecosystem

Several studies on coupled physical–biogeochemical models have shown that major deficiencies in the biogeochemical fields arise from the deficiencies in the physical flow fields. This paper examines the improvement of the physics through data assimilation, and the subsequent impact on the ecosystem response in a coupled model of the North Atlantic. Sea surface temperature and sea surface height data are assimilated with a sequential method based on the SEEK filter adapted to the coupling needs. The model domain covers the Atlantic from 20°S to 70°N at eddy-permitting resolution. The biogeochemical model is a NPZD-DOM model based on the P3ZD formulation. The results of an annual assimilated simulation are compared with an annual free simulation.With assimilation, the representation of the mixed layer depth is significantly improved in mid latitudes, even though the mixed layer depth is generally overestimated compared to the observations. The representation of the mean and variance of the currents is also significantly improved.The nutrient input in the euphotic zone is used to assess the data assimilation impact on the ecosystem. Data assimilation results in a 50% reduction of the input due to vertical mixing in mid-latitudes, and in a four- to six-fold increase of the advective fluxes in mid-latitudes and subtropics. Averaged zonally, the net impact is a threefold increase for the subtropical gyre, and a moderate (20–30%) decrease at mid and high latitudes.Surface chlorophyll concentration increases along the subtropical gyre borders, but little changes are detected at mid and high latitudes. An increase of the primary production appears along the Gulf Stream path, but it represents only 12% on average for mid and high latitudes. In the subtropical gyre centre, primary production is augmented but stays underestimated (20% of observations). These experiments show the benefits of physical data assimilation in coupled physical–biogeochemical applications.     

Assimilation of altimetric data in the mid-latitude oceans using the Singular Evolutive Extended Kalman filter with an eddy-resolving, primitive equation model

A new data assimilation scheme has been elaborated for ocean circulation models based on the concept of an evolutive, reduced-order Kalman filter. The dimension of the assimilation problem is reduced by expressing the initial error covariance matrix as a truncated series of orthogonal perturbations. This error sub-space evolves during the assimilation so as to capture the growing modes of the estimation error. The algorithm has been formulated in quite a general fashion to make it tractable with a large variety of ocean models and measurement types. In the present paper, we have examined three possible strategies to compute the evolution of the error subspace in the so-called Singular Evolutive Extended Kalman (SEEK) filter: the steady filter considers a time-independent error sub-space, the apprentice filter progressively enriches the error sub-space with the information learned from the innovation vector after each analysis step, and the dynamical filter updates the error sub-space according to the model dynamics. The SEEK filter has been implemented to assimilate synthetic observations of the surface topography in a non-linear, primitive equation model that uses density as vertical coordinate. A simplified box configuration has been adopted to simulate a Gulf Stream-like current and its associated eddies and gyres with a resolution of 20 km in the horizontal, and 4 levels in the vertical. The concept of twin experiments is used to demonstrate that the conventional SEEK filter must be complemented by a learning mechanism in order to model the misrepresented tail of the error covariance matrix. An approach based on the vertical physics of the isopycnal model, is shown particularly robust to control the velocity field in deep layers with surface observations only. The cost of the method makes it a suitable candidate for large-size assimilation problems and operational applications.     

Estimates of vertical mixing during a Lagrangian experiment off the Galician coast

An adjoint 1-D model was used to determine vertical diffusivity coefficients from temperature profiles collected within a filament escaping from the Galician coast following an upwelling event. The optimisation scheme ended with relatively high diffusivity values within the thermocline (9×10⁻⁵ m² s⁻¹). Such high values are relevant for biogeochemical exchanges between surface and deep waters in stratified areas.The optimised values were several orders of magnitude higher than the bulk of diffusivity measurements recorded with a free-falling device; however, the optimisation solution was consistent with the arithmetic mean of the measurements in the thermocline (7.7×10⁻⁵ m² s⁻¹), giving more weight to the few largest values. Below the thermocline, the data assimilation method failed because of the three-dimensional nature of the advective field of the upwelling system. Ignoring this advective forcing in the model led to estimates that were two orders of magnitude too high.The results suggest that turbulent mixing is a random process where a few intense events determine the average mixing that drives the long-term evolution of the water column structure. This statistical property is very important when one wants to use instantaneous diffusivity measurements for modelling purposes.     

Investigation of the condition and the behavior of a modular floating structure by harnessing monitoring data

In this paper, the condition and the behavior of an installed and operating Modular Floating Structure (MFS) is investigated and assessed by harnessing field monitoring data and using collectively multiple Correlation Coefficients (CCs) between measured quantities. The examined MFS consists of five pairs of interconnected floating concrete modules and it functions as a floating breakwater. The field monitoring data are acquired through a sensor network deployed on one pair of modules (connected through two groups of connectors) of the MFS. A methodological data processing framework for data organization, manipulation and post-processing is developed and presented. This framework enables the quantification of the structure's condition at different time periods through the calculation of CCs: (a) between the incident wave height and the tensions of the mooring lines and (b) between the tensions of the mooring lines, considering various wave directions. Recorded data at three characteristic time periods during the structure's lifetime are used, namely: (i) before any failure (structure's initial condition), (ii) after the failure of the first connectors' group and (iii) after the failure of the second connectors' group. The data processing framework developed in the present paper is applied to the above recorded data in order to calculate CCs and, therefore, quantify the structure's condition, at the three aforementioned time periods. The quantification of the structure's initial condition resulted to conclusions that were consistent from a physical point of view with the most recently documented, available in-situ mapping of the mooring lines' configuration in the horizontal plane. By considering the structure's initial condition as a reference base for comparison, the effect of the connectors' failure on the CCs, used to quantify this condition, was also investigated and efficiently assessed. Specifically, the significant changes observed in the variation patterns of all examined CCs, when compared with their respective patterns corresponding to the structure's initial condition, demonstrated and confirmed the existence of significant reformation of the examined structural system resulting from the connectors' failure. In this way, the effectiveness of the joint utilization of CCs to assess the structure's condition was proved.