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.     

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.     

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.     

On the key role of nutrient data to constrain a coupled physical–biogeochemical assimilative model of the North Atlantic Ocean

A sequential assimilative system has been implemented into a coupled physical–biogeochemical model (CPBM) of the North Atlantic basin at eddy-permitting resolution (1/4°), with the long-term goal of estimating the basin scale patterns of the oceanic primary production and their seasonal variability. The assimilation system, which is based on the SEEK filter [Brasseur, P., Verron, J., 2006. The SEEK filter method for data assimilation in oceanography: a synthesis. Ocean Dynamics. doi: 10.1007/s10236-006-0080-3], has been adapted to this CPBM in order to control the physical and biogeochemical components of the coupled model separately or in combination. The assimilated data are the satellite Topex/Poseidon and ERS altimetric data, the AVHRR Sea Surface Temperature observations, and the Levitus climatology for salinity, temperature and nitrate.In the present study, different assimilation experiments are conducted to assess the relative usefulness of the assimilated data to improve the representation of the primary production by the CPBM. Consistently with the results obtained by Berline et al. [Berline, L., Brankart, J-M., Brasseur, P., Ourmières, Y., Verron, J., 2007. Improving the physics of a coupled physical–biogeochemical model of the North Atlantic through data assimilation: impact on the ecosystem. J. Mar. Syst. 64 (1–4), 153–172] with a comparable assimilative model, it is shown that the assimilation of physical data alone can improve the representation of the mixed layer depth, but the impact on the ecosystem is rather weak. In some situations, the physical data assimilation can even worsen the ecosystem response for areas where the prior nutrient distribution is significantly incorrect. However, these experiments also show that the combined assimilation of physical and nutrient data has a positive impact on the phytoplankton patterns by comparison with SeaWiFS ocean colour data, demonstrating the good complementarity between SST, altimetry and in situ nutrient data. These results suggest that more intensive in situ measurements of biogeochemical nutrients are urgently needed at basin scale to initiate a permanent monitoring of oceanic ecosystems.     

Application of an Ensemble Kalman filter to a 1-D coupled hydrodynamic-ecosystem model of the Ligurian Sea

The Ensemble Kalman filter (EnKF) has been applied to a 1-D complex ecosystem model coupled with a hydrodynamic model of the Ligurian Sea. In order to improve the performance of the EnKF, an ensemble subsampling strategy has been used to better represent the covariance matrices and a pre-analysis step for correcting the non-normality of the members distribution has been implemented. Twin experiments have been realized to assess the performance of the developed tool and a real data assimilation experiment has been conducted to hindcast the ecosystem at the Dyfamed site during the year 2000. Finally the performance of the EnKF has been compared with a Singular Evolutive Extended Kalman (SEEK) filter with a fixed basis. We conclude that, on one hand, there is a benefit in using the subsampling strategy and the lognormal transformation with the EnKF, and on the other hand, this filter presents better performance than the fixed basis version of the SEEK filter. However, it also incurs a large computational cost.     

Using data assimilation in laboratory experiments of geophysical flows

Data assimilation is used in numerical simulations of laboratory experiments in a stratified, rotating fluid. The experiments are performed on the large Coriolis turntable (Grenoble, France), which achieves a high degree of similarity with the ocean, and the simulations are performed with a two-layer shallow water model, using the SEEK method for data assimilation. Since the flow is measured with a high level of precision and resolution, a detailed analysis of a forecasting system is feasible. The procedure is tested for the baroclinic instability of a two-layer vortex. It is shown that the method of data assimilation properly captures the details of the initial flow perturbation. Then it is shown by sensitivity studies that an error of the model, attributed to small non-hydrostatic effects, can be distinguished from the error growth associated with the intrinsic instability of the system. Finally, it is experimentally demonstrated that the velocity field in the lower layer can be well controlled by data assimilation limited to the top layer.     

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.     

Improving assimilation of SeaWiFS data by the application of bias correction with a local SEIK filter

Ocean-biogeochemical models show typically significant errors in the representation of chlorophyll concentrations. The model state can be improved by the assimilation of satellite chlorophyll data with algorithms based on the Kalman filter. However, these algorithms do usually not account for the possibility that the model prediction contains systematic errors in the form of model bias. Accounting explicitly for model biases can improve the assimilation performance. To study the effect of bias estimation on the estimation of surface chlorophyll concentrations, chlorophyll data from the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) are assimilated on a daily basis into the NASA Ocean Biogeochemical Model (NOBM). The assimilation is performed by the ensemble-based SEIK filter combined with an online bias correction scheme. The SEIK filter is simplified here by the use of a static error covariance matrix. The performance of the filter algorithm is assessed by comparison with independent in situ data over the 7-year period 1998–2004. The bias correction results in significant improvements of the surface chlorophyll concentrations compared to the assimilation without bias estimation. With bias estimation, the daily surface chlorophyll estimates from the assimilation show about 3.3% lower error than SeaWiFS data. In contrast, the error in the global surface chlorophyll estimate without bias estimation is 10.9% larger than the error of SeaWiFS data.     

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.     

Altimeter assimilation in the OCCAM global model: Part I: A twin experiment

An altimeter data assimilation scheme has been tested in the OCCAM (Ocean Circulation and Climate Advanced Modelling) global 1/4°, 36-level model using a twin experiment format. The Cooper and Haines displacement scheme is used. The method works well in most regions and depths. Currents and densities in the top 1000 m generally improve by over 50–70% after 5 months of sea level assimilation every 15 days. Below 1000 m, an error reduction of up to 50% is achieved. The errors remain low during a further 60-day run without assimilation. Diagnostics for the North Atlantic, the Tropical Pacific and the Antarctic Circumpolar Current are shown alongside the global averages.The main problems encountered were in weakly stratified regions of the Antarctic and Arctic seas. A scale selective filter is developed to avoid assimilating scales much larger than the local deformation radius, and this avoids the adverse assimilation effects in the southern oceans. A companion paper uses this scheme to assimilate TOPEX and ERS-1 altimeter maps.     

Forecast verification of a 3D model of the Mediterranean Sea. The use of discrete wavelet transforms and EOFs in the skill assessment of spatial forecasts

The quality assessment of a nested model system of the Mediterranean Sea is realised. The model has two zooms in the Provençal Basin and in the Ligurian Sea, realised with a two-way nesting approach. The experiment lasts for nine weeks, and at each week sea surface temperature (SST) and sea level anomaly are assimilated. The quality assessment of the surface temperature is done in a spatio-temporal approach, to take into account the high complexity of the SST distribution. We focus on the multi-scale nature of oceanic processes using two powerful tools for spatio-temporal analysis, wavelets and Empirical Orthogonal Functions (EOFs). We apply two-dimensional wavelets to decompose the high-resolution model and observed SST into different spatial scales. The Ligurian Sea model results are compared to observations at each of those spatial scales, with special attention on how the assimilation affects the model behaviour. We also use EOFs to assess the similarities between the Mediterranean Sea model and the observed SST. The results show that the assimilation mainly affects the model large-scale features, whereas the small scales show little or no improvement and sometimes, even a decrease in their skill. The multiresolution analysis reveals the connection between large- and small-scale errors, and how the choice of the maximum correlation length of the assimilation scheme affects the distribution of the model error among the different spatial scales.     

Multivariate assimilation in MERCATOR project: New statistical parameters from forecast error estimation

The new operational prototype of Mercator (french Global Ocean Data Assimilation Experiment contribution) is composed of a North Atlantic primitive equation ocean model OPA (Ocean Parallel Algorithm between 20°S and 70°N, [Madec, G., P. Delecluse, M. Imbard and C. Lévy (1998). OPA8.1 ocean general circulation model reference manuel. Notes du pôle de modélisation IPSL. n°11: 91p]) and of a multivariate and multidata assimilation scheme [De Mey, P. and M. Benkiran (2002). “A multivariate reduced-order optimal interpolation method and its application in Mediterranean basin-scale circulation.” Ocean Forecasting : Conceptual basis and application, Pinardi, N., Springer Verlag.] This system has already given some significant improvements from previous Mercator configurations (M. Benkiran, personal communication). However some biases on ocean state still remain in the tropics where the reduced-order optimal interpolation scheme is suspected to be ill-parameted in the model forecast error. Indeed the guess error covariance matrix is decomposed into an error variance value and a spatio-temporal correlation function which are assumed to have some “good” properties (spatial homogeneity of the correlation function, constant ratio between signal and error variance). This study shows how we can use ensemble methods to validate these assumptions. We can see that the correlation function can reach negative values locally, mostly in regions of high variability contradictory with the homogeneous hypothesis. The reduced space used in the operational configuration is based on the signal seasonal Empirical Orthogonal Functions (EOFs). An empirical relationship between signal and error variance has been set and the correlation function is the same on every dimension of the reduced space. By projection of the estimated guess error variance onto the reduced space, we find a repartition of this quantity quite different to what was set in the system. The error statistics is found to be inhomogeneous compared to hypothesis made in the assimilation scheme. These two new parameters tested separately in the assimilation scheme gives significant improvements of the forecast and analysis results. This is particularly obvious in the tropics. But relationship between signal and error statistics (as assumed in the optimal interpolation) is found to be complex.     

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.     

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.     

Use of SeaWiFS data for light availability and parameter estimation of a phytoplankton production model of the Bay of Biscay

Processing SeaWiFS (Sea-viewing Wide Field-of-view Sensor) data provides useful information for the observation and modelling of the phytoplankton production of the Bay of Biscay. Empirical algorithms allow the retrieval of chlorophyll a and non-living Suspended Particulate Matter (SPM) concentrations. These data are used to constrain a coupled 3D physical–biogeochemical model of the Bay of Biscay continental shelf. Two issues are investigated, depending on the variable used, to constrain the winter to spring phytoplankton production for the year 2001. First, SPM data is used as forcing data to correct the corresponding state variable of our model. This allows the realistic simulation of the light limited bloom at the end of February 2001, as observed with SeaWiFS chlorophyll a images and from the NUTRIGAS field cruise. Second, chlorophyll a data is used for parameter estimation of the biogeochemical model. The ability of assimilating these data is tested to improve the simulation of strong blooms observed in late May 2001 in the Loire and Gironde plumes. A global optimization method (Evolutive Strategies) is adapted to the complete 3-D coupled model, in order to find the best set of parameters. The hydrological conditions during the bloom can be validated with data from the PEL01 field cruise. After selection of the most sensitive parameters, the method is tested with twin experiments. Then, the use of real SeaWiFS data reduces the model/data misfit by a factor of two, improving the simulation of bloom intensities and extensions. The sets of parameters retrieved in each plume are discussed.     

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.     

Assimilation of sea surface temperature in the POL Coastal Ocean Modelling System

A one-dimensional scheme is used to assimilate satellite Sea Surface Temperature data into the Proudman Oceanographic Laboratory Coastal Ocean Modelling System, set up in the Irish Sea with a fine resolution ( 1.8 km). The capabilities of the assimilation scheme are investigated using two different sets of satellite data, of lower and similar resolution to that of the model respectively. Comparison of results with independent data show that assimilation improves the modelled Sea Surface Temperature, but does not address model representation of the temperature vertical structure. It is concluded that for the Irish Sea and at the scales resolved by the model, the assimilation problem cannot be approached in a one-dimensional framework. It is also pointed out that forecast error needs to account explicitly for errors in the representation of the vertical structure of the thermal field.Three-dimensional methods that are suited for coastal systems are then suggested.     

非理想状态下无陀螺惯导系统的误差分析

非理想状态下，无陀螺惯导系统存在模型误差和系统误差。通过研究系统的数学模型，分析了系统的模型误差；在一种九加速度计配置方案的研究基础上，采用线性化Kalman滤波，抑制系统误差的积累。仿真结果表明，线性化Kalman滤波对系统误差有较好的抑制作用。     

Bayesian statistical data assimilation for ecosystem models using Markov Chain Monte Carlo

This study considers advanced statistical approaches for sequential data assimilation. These are explored in the context of nowcasting and forecasting using nonlinear differential equation based marine ecosystem models assimilating sparse and noisy non-Gaussian multivariate observations. The statistical framework uses a state space model with the goal of estimating the time evolving probability distribution of the ecosystem state. Assimilation of observations relies on stochastic dynamic prediction and Bayesian principles. In this study, a new sequential data assimilation approach is introduced based on Markov Chain Monte Carlo (MCMC). The ecosystem state is represented by an ensemble, or sample, from which distributional properties, or summary statistical measures, can be derived. The Metropolis-Hastings based MCMC approach is compared and contrasted with two other sequential data assimilation approaches: sequential importance resampling, and the (approximate) ensemble Kalman filter (including computational comparisons). A simple illustrative application is provided based on a 0-D nonlinear plankton ecosystem model with multivariate non-Gaussian observations of the ecosystem state from a coastal ocean observatory. The MCMC approach is shown to be straightforward to implement and to effectively characterize the non-Gaussian ecosystem state in both nowcast and forecast experiments. Results are reported which illustrate how non-Gaussian information originates, and how it can be used to characterize ecosystem properties.