2016 
Garnier, F., Brankart, J. M., Brasseur, P., & Cosme, E. (2016). Stochastic parameterizations of biogeochemical uncertainties in a 1/4 degrees NEMO/PISCES model for probabilistic comparisons with ocean color data. Journal Of Marine Systems, 155, 59–72.
Abstract: In spite of recent advances, biogeochemical models are still unable to represent the full complexity of natural ecosystems. Their formulations are mainly based on empirical laws involving many parameters. Improving biogeochemical models therefore requires to properly characterize model uncertainties and their consequences. Subsequently, this paper investigates the potential of using random processes to simulate some uncertainties of the 1/4 degrees coupled PhysicalBiogeochemical NEMO/PISCES model of the North Atlantic ocean. Starting from a deterministic simulation performed with the original PISCES formulation, we propose a generic method based on AR(1) random processes to generate perturbations with temporal and spatial correlations. These perturbations are introduced into the model formulations to simulate 2 classes of uncertainties: the uncertainties on biogeochemical parameters and the uncertainties induced by unresolved scales in the presence of nonlinear processes. Using these stochastic parameterizations, a probabilistic version of PISCES is designed and a 60member ensemble simulation is performed. With respect to the simulation of chlorophyll, the relevance of the probabilistic configuration and the impacts of these stochastic parameterizations are assessed. In particular, it is shown that the ensemble simulation is in good agreement with the SeaWIFS ocean color data. Using these observations, the statistical consistency (reliability) of the ensemble is evaluated with rank histograms. Finally, the benefits expected from the probabilistic description of uncertainties (model error) are discussed in the context of future ocean color data assimilation. (C) 2015 Elsevier B.V. All rights reserved.


2015 
Brankart, J. M., Candille, G., Garnier, F., Calone, C., Melet, A., Bouttier, P. A., et al. (2015). A generic approach to explicit simulation of uncertainty in the NEMO ocean model. Geoscientific Model Development, 8(5), 1285–1297.
Abstract: In this paper, a generic implementation approach is presented, with the aim of transforming a deterministic ocean model (like NEMO) into a probabilistic model. With this approach, several kinds of stochastic parameterizations are implemented to simulate the nondeterministic effect of unresolved processes, unresolved scales and unresolved diversity. The method is illustrated with three applications, showing that uncertainties can produce a major effect in the circulation model, in the ecosystem model, and in the sea ice model. These examples show that uncertainties can produce an important effect in the simulations, strongly modifying the dynamical behaviour of these three components of ocean systems.


Candille, G., Brankart, J. M., & Brasseur, P. (2015). Assessment of an ensemble system that assimilates Jason1/Envisat altimeter data in a probabilistic model of the North Atlantic ocean circulation. Ocean Science, 11(3), 425–438.
Abstract: A realistic circulation model of the North Atlantic ocean at 0.25 degrees resolution (NATL025 NEMO configuration) has been adapted to explicitly simulate model uncertainties. This is achieved by introducing stochastic perturbations in the equation of state to represent the effect of unresolved scales on the model dynamics. The main motivation for this work is to develop ensemble data assimilation methods, assimilating altimetric data from past missions Jason1 and Envisat. The assimilation experiment is designed to provide a description of the uncertainty associated with the Gulf Stream circulation for years 2005/2006, focusing on frontal regions which are predominantly affected by unresolved dynamical scales. An ensemble based on such stochastic perturbations is first produced and evaluated using alongtrack altimetry observations. Then each ensemble member is updated by a square root algorithm based on the SEEK (singular evolutive extended Kalman) filter (Brasseur and Verron, 2006). These three elements – stochastic parameterization, ensemble simulation and 4D observation operator – are then used together to perform a 4D analysis of alongtrack altimetry over 10day windows. Finally, the results of this experiment are objectively evaluated using the standard probabilistic approach developed for meteorological applications (Toth et al., 2003; Candille et al., 2007). The results show that the free ensemble – before starting the assimilation process – correctly reproduces the statistical variability over the Gulf Stream area: the system is then pretty reliable but not informative (null probabilistic resolution). Updating the free ensemble with altimetric data leads to a better reliability with an information gain of around 30% (for 10day forecasts of the SSH variable). Diagnoses on fully independent data (i.e. data that are not assimilated, like temperature and salinity profiles) provide more contrasted results when the free and updated ensembles are compared.


Yan, Y., Barth, A., Beckers, J. M., Candille, G., Brankart, J. M., & Brasseur, P. (2015). Ensemble assimilation of ARGO temperature profile, sea surface temperature, and altimetric satellite data into an eddy permitting primitive equation model of the North Atlantic Ocean. Journal Of Geophysical ResearchOceans, 120(7), 5134–5157.
Abstract: Sea surface height, sea surface temperature, and temperature profiles at depth collected between January and December 2005 are assimilated into a realistic eddy permitting primitive equation model of the North Atlantic Ocean using the Ensemble Kalman Filter. Sixty ensemble members are generated by adding realistic noise to the forcing parameters related to the temperature. The ensemble is diagnosed and validated by comparison between the ensemble spread and the model/ observation difference, as well as by rank histogram before the assimilation experiments. An incremental analysis update scheme is applied in order to reduce spurious oscillations due to the model state correction. The results of the assimilation are assessed according to both deterministic and probabilistic metrics with independent/semiindependent observations. For deterministic validation, the ensemble means, together with the ensemble spreads are compared to the observations, in order to diagnose the ensemble distribution properties in a deterministic way. For probabilistic validation, the continuous ranked probability score (CRPS) is used to evaluate the ensemble forecast system according to reliability and resolution. The reliability is further decomposed into bias and dispersion by the reduced centered random variable (RCRV) score in order to investigate the reliability properties of the ensemble forecast system. The improvement of the assimilation is demonstrated using these validation metrics. Finally, the deterministic validation and the probabilistic validation are analyzed jointly. The consistency and complementarity between both validations are highlighted.


2014 
Brankart, J. M. (2014). Traitement des incertitudes en océanographie. Habilitation thesis, Université Joseph Fourier, Grenoble.
Abstract: Ce mémoire aborde le problème général du traitement des incertitudes en océanographie, en le considérant de façon transverse, tant dans le cadre du problème direct (modélisation des océans) que du problème inverse (méthodes d'assimilation de données). Les questions abordées sont en particulier les méthodes de simulations d'ensemble, avec paramétrisation stochastiques des incertitudes, et les méthodes de réduction des incertitudes par assimilation données (sous hypothèse gaussienne et nongaussienne).
Keywords: océanographie, incertitudes, assimilation de données


Gaultier, L., Djath, B., Verron, J., Brankart, J. M., Brasseur, P., & Melet, A. (2014). Inversion of submesoscale patterns from a highresolution Solomon Sea model: Feasibility assessment. Journal Of Geophysical ResearchOceans, 119(7), 4520–4541.
Abstract: A highresolution realistic numerical model of the Solomon Sea, which exhibits a high level of variability at mesoscales and submesoscales, is used to explore new avenues for data assimilation. Image data assimilation represents a powerful methodology to integrate information from highresolution observations such as satellite sea surface temperature or chlorophyll, or highresolution altimetric sea surface height that will be observed in the forthcoming SWOT mission. The present study investigates the feasibility and accuracy of the inversion of the dynamical submesoscale information contained in highresolution images of sea surface temperature (SST) or salinity (SSS) to improve the estimation of oceanic surface currents. The inversion method is tested in the context of twin experiments, with SST and SSS data provided by a model of the Solomon Sea. For that purpose, synthetic tracer images are built by binarizing the norm of the gradient of SST, SSS or spiciness. The binarized tracer images are compared to the dynamical image which is derived from the FiniteSize Lyapunov Exponents. The adjustment of the dynamical image to the tracer image provides the optimal correction to be applied on the surface velocity field. The method is evaluated by comparing the result of the inversion to the reference model solution. The feasibility of the inversion of various images (SST, SSS, both SST and SSS or spiciness) is explored on two small areas of the Solomon Sea. We show that errors in the surface velocity field can be substantially reduced through the inversion of tracer images.


2013 
Brankart, J. M. (2013). Impact of uncertainties in the horizontal density gradient upon low resolution global ocean modelling. Ocean Modelling, 66, 64–76.
Abstract: In this study, it is shown (i) that, as a result of the nonlinearity of the seawater equation of state, unresolved scales represent a major source of uncertainties in the computation of the largescale horizontal density gradient from the largescale temperature and salinity fields, and (ii) that the effect of these uncertain ties can be simulated using random processes to represent unresolved temperature and salinity fluctuations. The results of experiments performed with a low resolution global ocean model show that this parameterization has a considerable effect on the average largescale circulation of the ocean, especially in the regions of intense mesoscale activity. The largescale flow is less geostrophic, with more intense associated vertical velocities, and the average geographical position of the main temperature and salinity fronts is more consistent with observations. In particular, the simulations suggest that the stochastic effect of the unresolved temperature and salinity fluctuations on the largescale density field may be sufficient to explain why the Gulf Stream pathway systematically overshoots in nonstochastic low resolution ocean models. (c) 2013 Elsevier Ltd. All rights reserved.


Doron, M., Brasseur, P., Brankart, J. M., Losa, S. N., & Melet, A. (2013). Stochastic estimation of biogeochemical parameters from Globcolour ocean colour satellite data in a North Atlantic 3D ocean coupled physicalbiogeochemical model. Journal Of Marine Systems, 117, 81–95.
Abstract: Biogeochemical parameters remain a major source of uncertainty in coupled physicalbiogeochemical models of the ocean. In a previous study (Doron et al., 2011), a stochastic estimation method was developed to estimate a subset of biogeochemical model parameters from surface phytoplankton observations. The concept was tested in the context of idealised twin experiments performed with a 1/4 resolution model of the North Atlantic ocean. The method was based on ensemble simulations describing the model response to parameter uncertainty. The statistical estimation process relies on nonlinear transformations of the estimated space to cope with the nonGaussian behaviour of the resulting joint probability distribution of the model state variables and parameters. In the present study, the same method is applied to real ocean colour observations, as delivered by the sensors SeaWiFS, MERIS and MODIS embarked on the satellites OrbView2, Envisat and Aqua respectively. The main outcome of the present experiments is a set of regionalised biogeochemical parameters. The benefit is quantitatively assessed with an objective norm of the misfits, which automatically adapts to the different ecological regions. The chlorophyll concentration simulated by the model with this set of optimally derived parameters is closer to the observations than the reference simulation using uniform values of the parameters. In addition, the interannual and seasonal robustness of the estimated parameters is tested by repeating the same analysis using ocean colour observations from several months and several years. The results show the overall consistency of the ensemble of estimated parameters, which are also compared to the results of an independent study. (C) 2013 Elsevier B.V. All rights reserved.


Fontana, C., Brasseur, P., & Brankart, J. M. (2013). Toward a multivariate reanalysis of the North Atlantic Ocean biogeochemistry during 19982006 based on the assimilation of SeaWiFS chlorophyll data. Ocean Science, 9(1), 37–56.
Abstract: Today, the routine assimilation of satellite data into operational models of ocean circulation is mature enough to enable the production of global reanalyses describing the ocean circulation variability during the past decades. The expansion of the “reanalysis” concept from ocean physics to biogeochemistry is a timely challenge that motivates the present study. The objective of this paper is to investigate the potential benefits of assimilating satelliteestimated chlorophyll data into a basinscale threedimensional coupled physicalbiogeochemical model of the North Atlantic. The aim is on the one hand to improve forecasts of ocean biogeochemical properties and on the other hand to define a methodology for producing datadriven climatologies based on coupled physicalbiogeochemical modeling. A simplified variant of the Kalman filter is used to assimilate ocean color data during a 9year period. In this frame, two experiments are carried out, with and without anamorphic transformations of the state vector variables. Data assimilation efficiency is assessed with respect to the assimilated data set, nitrate of the World Ocean Atlas database and a derived climatology. Along the simulation period, the nonlinear assimilation scheme clearly improves the surface analysis and forecast chlorophyll concentrations, especially in the North Atlantic bloom region. Nitrate concentration forecasts are also improved thanks to the assimilation of ocean color data while this improvement is limited to the upper layer of the water column, in agreement with recent related literature. This feature is explained by the weak correlation taken into account by the assimilation between surface phytoplankton and nitrate concentrations deeper than 50 meters. The assessment of the nonlinear assimilation experiments indicates that the proposed methodology provides the skeleton of an assimilative system suitable for reanalyzing the ocean biogeochemistry based on ocean color data.


Gaultier, L., Verron, J., Brankart, J. M., Titaud, O., & Brasseur, P. (2013). On the inversion of submesoscale tracer fields to estimate the surface ocean circulation. Journal Of Marine Systems, 126, 33–42.
Abstract: In this paper, we demonstrate the feasibility of inverting the information contained in oceanic submesoscales, such as the ones evidenced in tracer observations of sea surface temperature (SST), to improve the description of mesoscale dynamics provided by altimetric observations. A small region of the Western Mediterranean Sea is chosen as a test case. From a SST snapshot of the region in July 2004, information is extracted to improve the velocity field as computed by geostrophy from the AVISO altimetric data at the same location and time. Image information is extracted from SST using a binarization of the SST gradients. Similarly, image information is extracted from the dynamic topography using finite size Lyapunov exponents (FSLE). The inverse problem is formulated in a Bayesian framework and expressed in terms of a cost function measuring the misfits between the two images. The large amount of information which is already available from ocean color satellites or which will be available from highresolution altimetric satellites such as SWOT, is a strong motivation for this work. Moreover, the image data assimilation approach which is explored here, is a possible strategy for handling the huge amount of satellite data imprinted by small scale information. (c) 2012 Elsevier B.V. All rights reserved.


Meinvielle, M., Brankart, J. M., Brasseur, P., Barnier, B., Dussin, R., & Verron, J. (2013). Optimal adjustment of the atmospheric forcing parameters of ocean models using sea surface temperature data assimilation. Ocean Science, 9(5), 867–883.
Abstract: In ocean general circulation models, nearsurface atmospheric variables used to specify the atmospheric boundary condition remain one of the main sources of error. The objective of this research is to constrain the surface forcing function of an ocean model by sea surface temperature (SST) data assimilation. For that purpose, a set of corrections for ERAinterim (hereafter ERAi) reanalysis data is estimated for the period of 19892007, using a sequential assimilation method, with ensemble experiments to evaluate the impact of uncertain atmospheric forcing on the ocean state. The control vector of the assimilation method is extended to atmospheric variables to obtain monthly mean parameter corrections by assimilating monthly SST and sea surface salinity (SSS) climatological data in a low resolution global configuration of the NEMO model. In this context, the careful determination of the prior probability distribution of the parameters is an important matter. This paper demonstrates the importance of isolating the impact of forcing errors in the model to perform relevant ensemble experiments. The results obtained for every month of the period between 1989 and 2007 show that the estimated parameters produce the same kind of impact on the SST as the analysis itself. The objective is then to evaluate the longterm time series of the forcing parameters focusing on trends and mean error corrections of airsea fluxes. Our corrections tend to equilibrate the net heatflux balance at the global scale (highly positive in ERAi database), and to remove the potentially unrealistic negative trend (leading to ocean cooling) in the ERAi net heat flux over the whole time period. More specifically in the intertropical band, we reduce the warm bias of ERAi data by mostly modifying the latent heat flux by wind speed intensification. Consistently, when used to force the model, the corrected parameters lead to a better agreement between the mean SST produced by the model and mean SST observations over the period of 19892007 in the intertropical band.


2012 
Brankart, J. M., Testut, C. E., Beal, D., Doron, M., Fontana, C., Meinvielle, M., et al. (2012). Towards an improved description of ocean uncertainties: effect of local anamorphic transformations on spatial correlations. Ocean Science, 8(2), 121–142.
Abstract: The objective of this paper is to investigate if the description of ocean uncertainties can be significantly improved by applying a local anamorphic transformation to each model variable, and by making the assumption of joint Gaussianity for the transformed variables, rather than for the original variables. For that purpose, it is first argued that a significant improvement can already be obtained by deriving the local transformations from a simple histogram description of the marginal distributions. Two distinctive advantages of this solution for large size applications are the conciseness and the numerical efficiency of the description. Second, various oceanographic examples are used to evaluate the effect of the resulting piecewise linear local anamorphic transformations on the spatial correlation structure. These examples include (i) stochastic ensemble descriptions of the effect of atmospheric uncertainties on the ocean mixed layer, and of wind uncertainties or parameter uncertainties on the ecosystem, and (ii) nonstochastic ensemble descriptions of forecast uncertainties in current sea ice and ecosystem preoperational developments. The results indicate that (i) the transformation is accurate enough to faithfully preserve the correlation structure if the joint distribution is already close to Gaussian, and (ii) the transformation has the general tendency of increasing the correlation radius as soon as the spatial dependence between random variables becomes nonlinear, with the important consequence of reducing the number of degrees of freedom in the uncertainties, and thus increasing the benefit that can be expected from a given observation network.


Duchez, A., Verron, J., Brankart, J. M., Ourmieres, Y., & Fraunie, P. (2012). Monitoring the Northern Current in the Gulf of Lions with an observing system simulation experiment. Scientia Marina, 76(3), 441–453.
Abstract: The coastal circulation in the Gulf of Lions (GoL) is influenced by the Northern Current (NC), forced by a complex wind system and also affected by important river discharges from the Rhone River. Correct modelling of this current is therefore important for obtaining a good representation of the gulf circulation. An observing system simulation experiment using the SEEK filter data assimilation method was used in a regional 1/16 degrees configuration of the GoL in the NEMO model. The synthetic observation database used for the experiment comprised altimetric data in addition to insitu temperature and salinity profiles. Statistical diagnostics and other physical criteria based on the improvement of NC representation were set up in order to assess the quality of this experiment. Comparisons between the free 1/16 degrees simulation and the experience with assimilation show that data assimilation significantly improved the description of the characteristics of the NC as well as its seasonal and mesoscale variability, which in turn improved the description of the water exchanges between the coastal region and the open sea.


Freychet, N., Cosme, E., Brasseur, P., Brankart, J. M., & Kpemlie, E. (2012). Obstacles and benefits of the implementation of a reducedrank smoother with a high resolution model of the tropical Atlantic Ocean. Ocean Science, 8(5), 797–811.
Abstract: Most of oceanographic operational centers use threedimensional data assimilation schemes to produce reanalyses. We investigate here the benefits of a smoother, i.e. a fourdimensional formulation of statistical assimilation. A squareroot sequential smoother is implemented with a tropical Atlantic Ocean circulation model. A simple twin experiment is performed to investigate its benefits, compared to its corresponding filter. Despite model's nonlinearities and the various approximations used for its implementation, the smoother leads to a better estimation of the ocean state, both on statistical (i.e. mean error level) and dynamical points of view, as expected from linear theory. Smoothed states are more in phase with the dynamics of the reference state, an aspect that is nicely illustrated with the chaotic dynamics of the North Brazil Current rings. We also show that the smoother efficiency is strongly related to the filter configuration. One of the main obstacles to implement the smoother is then to accurately estimate the error covariances of the filter. Considering this, benefits of the smoother are also investigated with a configuration close to situations that can be managed by operational center systems, where covariances matrices are fixed (optimal interpolation). We define here a simplified smoother scheme, called halffixed basis smoother, that could be implemented with current reanalysis schemes. Its main assumption is to neglect the propagation of the error covariances matrix, what leads to strongly reduce the cost of assimilation. Results illustrate the ability of this smoother to provide a solution more consistent with the dynamics, compared to the filter. The smoother is also able to produce analyses independently of the observation frequency, so the smoothed solution appears more continuous in time, especially in case of a low frenquency observation network.


Juza, M., Penduff, T., Brankart, J. M., & Barnier, B. (2012). Estimating the distortion of mixed layer property distributions induced by the Argo sampling. Journal Of Operational Oceanography, 5(1), 45–58.
Abstract: This global study evaluates how the varying geometry of the Argo array of profiling floats has affected the actual distributions of mixed layer depth (MLD), temperature (MLT) and heat content (MLHC) annual cycles between 2004 and 2009. These quantities' monthly distributions are computed regionally from a global 1/4 degrees simulation with and without Argolike subsampling, and the subsequent medians are compared. Argolike subsampling is shown to bias the medians of MLD, MLT and MLHC distributions by about +/ 10m, +/ 1 degrees C, 1GJ/m(2), respectively, with maximum values reaching +/ 100m, +/ 5 degrees C, 5GJ/m(2) in certain regions and months. MLD distributions are most distorted where and when the array geometry is irregular, and where MLD distributions are far from Gaussian. The differences between medians of subsampled and fullysampled distributions are also compared to the actual width of fullysampled MLHC distributions in every monthly regional bin to evaluate the intrinsic accuracy of the array. Comparing results from several periods (20042005,20062007 and 20082009), it is shown that Argobased estimates of mixed layer statistics have improved when the array reached its target density at the end of 2007.


Melet, A., Verron, J., & Brankart, J. M. (2012). Potential outcomes of glider data assimilation in the Solomon Sea: Control of the water mass properties and parameter estimation. Journal Of Marine Systems, 94, 232–246.
Abstract: Steerable underwater gliders are a recent addition to ocean observing systems. Gliders were deployed in the Solomon Sea to improve our knowledge of this potentially important region for Pacific climate. In this study, we explore the potential use of glider data assimilation to control some properties of the ocean state estimation, chosen here to be Solomon Sea thermohaline misfits due to an erroneous tidalmixing parameterization. Ocean observing system simulation experiments involving several scenarios of glider deployment show that the fleet design can strongly impact the control efficiency. A fairly good control of the Solomon Sea mass field can be achieved with a somewhat unrealistic fleet of 50 gliders. With a more realistic configuration of 10 gliders, the performance depends on the space and time distribution of the vehicles. Substantial control is achieved when glider trajectories are coordinated to collect informationrich data. As a complement, glider data assimilation was used to directly correct the model: the uncertain tidal mixing parameter is estimated through assimilation of data provided by the 10 coordinated gliders using an ensemble simulation method. This promising strategy allows an accurate estimation of the parameter and therefore yields an efficient correction of the errors in Solomon Sea thermohaline properties. (C) 2011 Elsevier B.V. All rights reserved.


Troupin, C., Barth, A., Sirjacobs, D., Ouberdous, M., Brankart, J. M., Brasseur, P., et al. (2012). Generation of analysis and consistent error fields using the Data Interpolating Variational Analysis (DIVA). Ocean Modelling, 5253, 90–101.
Abstract: The Data Interpolating Variational Analysis (DIVA) is a method designed to interpolate irregularlyspaced, noisy data onto any desired location, in most cases on regular grids. It is the combination of a particular methodology, based on the minimisation of a cost function, and a numerically efficient method, based on a finiteelement solver. The cost function penalises the misfit between the observations and the reconstructed field, as well as the regularity or smoothness of the field. The method bears similarities to the smoothing splines, where the second derivatives of the field are also penalised. The intrinsic advantages of the method are its natural way to take into account topographic and dynamic constraints (coasts, advection, etc.) and its capacity to handle large data sets, frequently encountered in oceanography. The method provides gridded fields in two dimensions, usually in horizontal layers. Threedimension fields are obtained by stacking horizontal layers. In the present work, we summarize the background of the method and describe the possible methods to compute the error field associated to the analysis. In particular, we present new developments leading to a more consistent error estimation, by determining numerically the real covariance function in DIVA, which is never formulated explicitly, contrarily to Optimal Interpolation. The real covariance function is obtained by two concurrent executions of DIVA, the first providing the covariance for the second. With this improvement, the error field is now perfectly consistent with the inherent background covariance in all cases. A twodimension application using salinity measurements in the Mediterranean Sea is presented. Applied on these measurements, Optimal Interpolation and DIVA provided very similar gridded fields (correlation: 98.6%, RMS of the difference: 0.02). The method using the real covariance produces an error field similar to the one of OI, except in the coastal areas. (C) 2012 Elsevier Ltd. All rights reserved.


Ubelmann, C., Verron, J., Brankart, J. M., Brasseur, P., & Cosme, E. (2012). Assimilating altimetric data to control the tropical instability waves: an observing system simulation experiment study. Ocean Dynamics, 62(6), 867–880.
Abstract: Tropical instability waves (TIWs) are not easily simulated by ocean circulation models primarily because such waves are very sensitive to wind forcing. In this study, we investigate the impact of assimilating sea surface height (SSH) observations on the control of TIWs in an observing system simulation experiment (OSSE) context based on a regional model configuration of the tropical Atlantic. A Kalman filtering method with suitable adaptations is found to be successful when altimetric data are assimilated in conjunction with sea surface temperature and some in situ temperature/salinity profiles. In this rather realistic system, the TIW phase is roughly controlled with a single nadir observing satellite. However, a right correction of the TIW structure and amplitude requires at least two nadir observing satellites or a wide swath observing satellite. The significant impact of orbital parameters is also demonstrated: in particular, the Jason or GFO satellite orbits are found to be more suitable than the ENVISAT orbit. More generally, it is found that as soon as adequate subsampling exists (with periods of 510 days), the length of the repetitivity cycle of orbits does not have a significant impact.


2011 
Brankart, J. M., Cosme, E., Testut, C. E., Brasseur, P., & Verron, J. (2011). Efficient Local Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to a BasinScale Ocean Turbulent Flow. Monthly Weather Review, 139(2), 474–493.
Abstract: In largesized atmospheric or oceanic applications of square root or ensemble Kalman filters, it is often necessary to introduce the prior assumption that longrange correlations are negligible and force them to zero using a local parameterization, supplementing the ensemble or reducedrank representation of the covariance. One classic algorithm to perform this operation consists of taking the Schur product of the ensemble covariance matrix by a local support correlation matrix. However, with this parameterization, the square root of the forecast error covariance matrix is no more directly available, so that any observational update algorithm requiring this square root must include an additional step to compute local square roots from the Schur product. This computation generates an additional numerical cost or produces highrank square roots, which may deprive the observational update from its original efficiency. In this paper, it is shown how efficient local square root parameterizations can be obtained, for use with a specific square root formulation (i.e., eigenbasis algorithm) of the observational update. Comparisons with the classic algorithm are provided, mainly in terms of consistency, accuracy, and computational complexity. As an application, the resulting parameterization is used to estimate maps of dynamic topography characterizing a basinscale ocean turbulent flow. Even with this moderatesized system (a 2200kmwide square basin with 100kmwide mesoscale eddies), it is observed that more than 1000 ensemble members are necessary to faithfully represent the global correlation patterns, and that a local parameterization is needed to produce correct covariances with moderatesized ensembles. Comparisons with the exact solution show that the use of local square roots is able to improve the accuracy of the updated ensemble mean and the consistency of the updated ensemble variance. With the eigenbasis algorithm. optimal adaptive estimates of scaling factors for the forecast and observation error covariance matrix can also be obtained locally at negligible additional numerical cost. Finally, a comparison of the overall computational cost illustrates the decisive advantage that efficient local square root parameterizations may have to deal simultaneously with a larger number of observations and avoid data thinning as much as possible.


Doron, M., Brasseur, P., & Brankart, J. M. (2011). Stochastic estimation of biogeochemical parameters of a 3D ocean coupled physicalbiogeochemical model: Twin experiments. Journal Of Marine Systems, 87(34), 194–207.
Abstract: In a 3D ocean coupled physicalbiogeochemical model, implemented on the North Atlantic at 1/4 and including six biogeochemical variables, three parameters (phytoplankton maximal growth rate, phytoplankton mortality rate and zooplankton maximal grazing rate) are assumed to be stochastic and have regional variations. Ensemble simulations (200 members, lasting 30 days during the spring bloom) show that the phytoplankton concentration is sensitive to the parameterization, with strong spatial heterogeneity, combined to a nonlinear and nonGaussian behavior. Within the Kalman filter theory, parameter estimation can be done, in the framework of optimal estimate with Gaussian assumptions and reduced rank approximation, when the state vector is augmented with the uncertain parameters. Twin data assimilation experiments, using surface phytoplankton as observations, were performed either in the linear framework or introducing a nonlinear local transformation (anamorphosis). The anamorphosis is performed using a piecewise linear change of variables (applied to all biogeochemical quantities) remapping the percentiles of the empirical marginal distribution provided by the ensemble on the percentiles of the Gaussian distribution. Nonlinear parameter estimation performed better than linear estimation: on the 39 estimated parameters. there is a reduction in the variance obtained with the nonlinear analysis, compared to the variance obtained with the linear analysis, except for 2 parameters. The reduction is better than 60% in 80% of these cases. The anamorphosis is also useful to define an objective error norm for the biogeochemical variables. (C) 2011 Elsevier BM. All rights reserved.


Srinivasan, A., Chassignet, E. P., Bertino, L., Brankart, J. M., Brasseur, P., Chin, T. M., et al. (2011). A comparison of sequential assimilation schemes for ocean prediction with the HYbrid Coordinate Ocean Model (HYCOM): Twin experiments with static forecast error covariances. Ocean Modelling, 37(34), 85–111.
Abstract: We assess and compare four sequential data assimilation methods developed for HYCOM in an identical twin experiment framework. The methods considered are Multivariate Optimal Interpolation (MVOI), Ensemble Optimal Interpolation (EnOI), the fixed basis version of the Singular Evolutive Extended Kalman Filter (SEEK) and the Ensemble Reduced Order Information Filter (EnROIF). All methods can be classified as statistical interpolation but differ mainly in how the forecast error covariances are modeled. Surface elevation and temperature data sampled from an 1/12 degrees Gulf of Mexico HYCOM simulation designated as the truth are assimilated into an identical model starting from an erroneous initial state, and convergence of assimilative runs towards the truth is tracked. Sensitivity experiments are first performed to evaluate the impact of practical implementation choices such as the state vector structure, initialization procedures, correlation scales, covariance rank and details of handling multivariate datasets, and to identify an effective configuration for each assimilation method. The performance of the methods are then compared by examining the relative convergence of the assimilative runs towards the truth. All four methods show good skill and are able to enhance consistency between the assimilative and truth runs in both observed and unobserved model variables. Prediction errors in observed variables are typically less than the errors specified for the observations, and the differences between the assimilated products are small compared to the observation errors. For unobserved variables, RMS errors are reduced by 50% relative to a nonassimilative run and differ between schemes on average by about 5%. Dynamical consistency between the updated state space variables in the data assimilation algorithm, and the data adequately sampling significant dynamical features are the two crucial components for reliable predict:ions. The experiments presented here suggest that practical implementation details can have at least as much an impact on the accuracy of the assimilated product as the choice of assimilation technique itself. We also present a discussion of the numerical implementation and the computational requirements for the use of these methods in large scale applications. (C) 2011 Elsevier Ltd. All rights reserved.


Titaud, O., Brankart, J. M., & Verron, J. (2011). On the use of FiniteTime Lyapunov Exponents and Vectors for direct assimilation of tracer images into ocean models. Tellus Series ADynamic Meteorology And Oceanography, 63(5), 1038–1051.
Abstract: Satellite ocean tracer images, of sea surface temperature (SST) and ocean colour images, for example, show patterns like fronts and filaments that characterize the flow dynamics. These patterns can be described using Lagrangian tools such as FiniteTime Lyapunov Exponents (FTLE) or FiniteTime Lyapunov Vectors (FTLV). In recent years, several studies have investigated the possibility of directly assimilating structured data from satellite images into numerical models. In this paper, we exploit specific properties of FTLE and FTLV to define observation operators that can be used in a direct ocean tracer image assimilation scheme. In an idealized context, we show that highresolution SST and ocean colour images can be exploited to correct velocity fields using FTLE or FTLV.


2010 
Beal, D., Brasseur, P., Brankart, J. M., Ourmieres, Y., & Verron, J. (2010). Characterization of mixing errors in a coupled physical biogeochemical model of the North Atlantic: implications for nonlinear estimation using Gaussian anamorphosis. Ocean Science, 6(1), 247–262.
Abstract: In biogeochemical models coupled to ocean circulation models, vertical mixing is an important physical process which governs the nutrient supply and the plankton residence in the euphotic layer. However, vertical mixing is often poorly represented in numerical simulations because of approximate parameterizations of subgrid scale turbulence, wind forcing errors and other misrepresented processes such as restratification by mesoscale eddies. Getting a sufficient knowledge of the nature and structure of these errors is necessary to implement appropriate data assimilation methods and to evaluate if they can be controlled by a given observation system. In this paper, Monte Carlo simulations are conducted to study mixing errors induced by approximate wind forcings in a threedimensional coupled physicalbiogeochemical model of the North Atlantic with a 1/4 degrees horizontal resolution. An ensemble forecast involving 200 members is performed during the 1998 spring bloom, by prescribing perturbations of the wind forcing to generate mixing errors. The biogeochemical response is shown to be rather complex because of nonlinearities and threshold effects in the coupled model. The response of the surface phytoplankton depends on the region of interest and is particularly sensitive to the local stratification. In addition, the statistical relationships computed between the various physical and biogeochemical variables reflect the signature of the nonGaussian behaviour of the system. It is shown that significant information on the ecosystem can be retrieved from observations of chlorophyll concentration or sea surface temperature if a simple nonlinear change of variables (anamorphosis) is performed by mapping separately and locally the ensemble percentiles of the distributions of each state variable on the Gaussian percentiles. The results of idealized observational updates (performed with perfect observations and neglecting horizontal correlations) indicate that the implementation of this anamorphosis method into sequential assimilation schemes can substantially improve the accuracy of the estimation with respect to classical computations based on the Gaussian assumption.


Brankart, J. M., Cosme, E., Testut, C. E., Brasseur, P., & Verron, J. (2010). Efficient Adaptive Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to the Control of Ocean Mesoscale Signals. Monthly Weather Review, 138(3), 932–950.
Abstract: In Kalman filter applications, an adaptive parameterization of the error statistics is often necessary to avoid filter divergence, and prevent error estimates from becoming grossly inconsistent with the real error. With the classic formulation of the Kalman filter observational update, optimal estimates of general adaptive parameters can only be obtained at a numerical cost that is several times larger than the cost of the state observational update. In this paper, it is shown that there exists a few types of important parameters for which optimal estimates can be computed at a negligible numerical cost, as soon as the computation is performed using a transformed algorithm that works in the reduced control space defined by the square root or ensemble representation of the forecast error covariance matrix. The set of parameters that can be efficiently controlled includes scaling factors for the forecast error covariance matrix, scaling factors for the observation error covariance matrix, or even a scaling factor for the observation error correlation length scale. As an application, the resulting adaptive filter is used to estimate the time evolution of ocean mesoscale signals using observations of the ocean dynamic topography. To check the behavior of the adaptive mechanism, this is done in the context of idealized experiments, in which model error and observation error statistics are known. This ideal framework is particularly appropriate to explore the illconditioned situations (inadequate prior assumptions or uncontrollability of the parameters) in which adaptivity can be misleading. Overall, the experiments show that, if used correctly, the efficient optimal adaptive algorithm proposed in this paper introduces useful supplementary degrees of freedom in the estimation problem, and that the direct control of these statistical parameters by the observations increases the robustness of the error estimates and thus the optimality of the resulting Kalman filter.


Cosme, E., Brankart, J. M., Verron, J., Brasseur, P., & Krysta, M. (2010). Implementation of a reduced rank squareroot smoother for high resolution ocean data assimilation. Ocean Modelling, 33(12), 87–100.
Abstract: Optimal smoothers enable the use of future observations to estimate the state of a dynamical system. In this paper, a squareroot smoother algorithm is presented, extended from the Singular Evolutive Extended Kalman (SEEK) filter, a squareroot Kalman filter routinely used for ocean data assimilation. With this filter algorithm, the smoother extension appears almost costfree. A modified algorithm implementing a particular parameterization of model error is also described. The smoother is applied with an ocean circulation model in a doublegyre, 1/4 degrees configuration, able to represent midlatitude mesoscale dynamics. Twin experiments are performed: the true fields are drawn from a simulation at a 1/6 degrees resolution, and noised. Then, altimetric satellite tracks and sparse vertical profiles of temperature are extracted to form the observations. The smoother is efficient in reducing errors, particularly in the regions poorly covered by the observations at the filter analysis time. It results in a significant reduction of the global error: the Root Mean Square Error in Sea Surface Height from the filter is further reduced by 20% by the smoother. The actual smoothing of the global error through time is also verified. Three essential issues are then investigated: (i) the time distance within which observations may be favourably used to correct the state estimates is found to be 8 days with our system. (ii) The impact of the model error parameterization is stressed. When this parameterization is spuriously neglected, the smoother can deteriorate the state estimates. (iii) Iterations of the smoother over a fixed time interval are tested. Although this procedure improves the state estimates over the assimilation window, it also makes the subsequent forecast worse than the filter in our experiment. (C) 2009 Elsevier Ltd. All rights reserved.


2009 
Brankart, J. M., Ubelmann, C., Testut, C. E., Cosme, E., Brasseur, P., & Verron, J. (2009). Efficient Parameterization of the Observation Error Covariance Matrix for Square Root or Ensemble Kalman Filters: Application to Ocean Altimetry. Monthly Weather Review, 137(6), 1908–1927.
Abstract: In the Kalman filter standard algorithm, the computational complexity of the observational update is proportional to the cube of the number y of observations (leading behavior for large y). In realistic atmospheric or oceanic applications, involving an increasing quantity of available observations, this often leads to a prohibitive cost and to the necessity of simplifying the problem by aggregating or dropping observations. If the filter error covariance matrices are in square root form, as in square root or ensemble Kalman filters, the standard algorithm can be transformed to be linear in y, providing that the observation error covariance matrix is diagonal. This is a significant drawback of this transformed algorithm and often leads to an assumption of uncorrelated observation errors for the sake of numerical efficiency. In this paper, it is shown that the linearity of the transformed algorithm in y can be preserved for other forms of the observation error covariance matrix. In particular, quite general correlation structures (with analytic asymptotic expressions) can be simulated simply by augmenting the observation vector with differences of the original observations, such as their discrete gradients. Errors in ocean altimetric observations are spatially correlated, as for instance orbit or atmospheric errors along the satellite track. Adequately parameterizing these correlations can directly improve the quality of observational updates and the accuracy of the associated error estimates. In this paper, the example of the North Brazil Current circulation is used to demonstrate the importance of this effect, which is especially significant in that region of moderate ratio between signal amplitude and observation noise, and to show that the efficient parameterization that is proposed for the observation error correlations is appropriate to take it into account. Adding explicit gradient observations also receives a physical justification. This parameterization is thus proved to be useful to ocean data assimilation systems that are based on square root or ensemble Kalman filters, as soon as the number of observations becomes penalizing, and if a sophisticated parameterization of the observation error correlations is required.


Lauvernet, C., Brankart, J. M., Castruccio, F., Broquet, G., Brasseur, P., & Verron, J. (2009). A truncated Gaussian filter for data assimilation with inequality constraints: Application to the hydrostatic stability condition in ocean models. Ocean Modelling, 27(12), 1–17.
Abstract: In many data assimilation problems, the state variables are subjected to inequality constraints. These constraints often contain valuable information that must be taken into account in the estimation process. However, with linear estimation methods (like the Kalman filter), there is no way to incorporate optimally that kind of additional information. In this study, it is shown that an optimal filter dealing with inequality constraints can be formulated under the assumption that the probability distributions are truncated Gaussian distributions. The statistical tools needed to implement this truncated Gaussian filter are described. It is also shown how the filter can be adapted to work in a reduced dimension space, and flow it can be simplified following several additional hypotheses. As an application, the truncated Gaussian assumption is shown to be adequate to deal with the condition of hydrostatic stability in ocean analyses. First, a detailed evaluation of the method is made using a onedimensional zcoordinate model of the mixed layer: particular attention is paid to the parameterization of the probability distribution, the accuracy of the estimation and the sensitivity to the observation system. In a second step, the method is applied to a threedimensional hybrid coordinate ocean model (HYCOM) of the Bay of Biscay (at a 1/15 degrees resolution), to show that it is efficient enough to be applied to real size problems. These examples also demonstrate that the algorithm can deal with the hydrostatic stability condition in isopycnic coordinates as well as in zcoordinates. (C) 2008 Elsevier Ltd. All rights reserved.


Ourmieres, Y., Brasseur, P., Levy, M., Brankart, J. M., & Verron, J. (2009). On the key role of nutrient data to constrain a coupled physicalbiogeochemical assimilative model of the North Atlantic Ocean. Journal Of Marine Systems, 75(12), 100–115.
Abstract: A sequential assimilative system has been implemented into a coupled physicalbiogeochemical model (CPBM)of the North Atlantic basin at eddypermitting resolution (1/4 degrees), with the longterm 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/s1023600600803], 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, JM., Brasseur, P., Ourmieres, Y., Verron, J., 2007. Improving the physics of a coupled physicalbiogeochemical model of the North Atlantic through data assimilation: impact on the ecosystem. J. Mar. Syst. 64 (14),153172] 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. (c) 2008 Elsevier B.V. All rights reserved.


Skachko, S., Brankart, J. M., Castruccio, B. F., Brasseur, P., & Verron, J. (2009). Improved Turbulent AirSea Flux Bulk Parameters for Controlling the Response of the Ocean Mixed Layer: A Sequential Data Assimilation Approach. Journal Of Atmospheric And Oceanic Technology, 26(3), 538–555.
Abstract: Bulk formulations parameterizing turbulent airsea fluxes remain among the main sources of error in presentday ocean models. The objective of this study is to investigate the possibility of estimating the turbulent bulk exchange coefficients using sequential data assimilation. It is expected that existing ocean assimilation systems can use this method to improve the airsea fluxes and produce more realistic forecasts of the thermohaline characteristics of the mixed layer. The method involves augmenting the control vector of the assimilation scheme using the model parameters that are to be controlled. The focus of this research is on estimating two bulk coefficients that drive the sensible heat flux, the latent heat flux, and the evaporation flux of a global ocean model, by assimilating temperature and salinity profiles using horizontal and temporal samplings similar to those to be provided by the Argo float system. The results of twin experiments show that the method is able to correctly estimate the largescale variations in the bulk parameters, leading to a significant improvement in the atmospheric forcing applied to the ocean model.


Skandrani, C., Brankart, J. M., Ferry, N., Verron, J., Brasseur, P., & Barnier, B. (2009). Controlling atmospheric forcing parameters of global ocean models: sequential assimilation of sea surface MercatorOcean reanalysis data. Ocean Science, 5(4), 403–419.
Abstract: In the context of stand alone ocean models, the atmospheric forcing is generally computed using atmospheric parameters that are derived from atmospheric reanalysis data and/or satellite products. With such a forcing, the sea surface temperature that is simulated by the ocean model is usually significantly less accurate than the synoptic maps that can be obtained from the satellite observations. This not only penalizes the realism of the ocean longterm simulations, but also the accuracy of the reanalyses or the usefulness of the shortterm operational forecasts (which are key GODAE and MERSEA objectives). In order to improve the situation, partly resulting from inaccuracies in the atmospheric forcing parameters, the purpose of this paper is to investigate a way of further adjusting the state of the atmosphere (within appropriate error bars), so that an explicit ocean model can produce a sea surface temperature that better fits the available observations. This is done by performing idealized assimilation experiments in which MercatorOcean reanalysis data are considered as a reference simulation describing the true state of the ocean. Synthetic observation datasets for sea surface temperature and salinity are extracted from the reanalysis to be assimilated in a low resolution global ocean model. The results of these experiments show that it is possible to compute piecewise constant parameter corrections, with predefined amplitude limitations, so that longterm free model simulations become much closer to the reanalysis data, with misfit variance typically divided by a factor 3. These results are obtained by applying a Monte Carlo method to simulate the joint parameter/state prior probability distribution. A truncated Gaussian assumption is used to avoid the most extreme and nonphysical parameter corrections. The general lesson of our experiments is indeed that a careful specification of the prior information on the parameters and on their associated uncertainties is a key element in the computation of realistic parameter estimates, especially if the system is affected by other potential sources of model errors.


Ubelmann, C., Verron, J., Brankart, J. M., Cosme, E., & Brasseur, P. (2009). Impact of data from upcoming altimetric missions on the prediction of the threedimensional circulation in the tropical Atlantic Ocean. Journal Of Operational Oceanography, 2(1), 3–14.
Abstract: The use of Sea Surface Height (SSH) satellite measurements in ocean models is a key element the efficient control of the threedimensional circulation through data assimilation and therefore in the quality of operational oceanography products. This paper attempts to evaluate the impact of future satellite data, particularly from the upcoming JASON2, SARAL and SWOT missions, introduced into a model through a sophisticated data assimilation procedure. For this purpose, Observing System Simulation Experiments (OSSEs) are performed in the tropical Atlantic Ocean. The NEMO model is used (at a 1/4 degrees resolution) in a configuration covering the tropical Atlantic from 15 degrees S to 17 degrees N, and the assimilation scheme is a reducedorder Kalman (SEEK) filter. The study focuses principally on controlling of the circulation of the North Brazil Current and the propagation of Tropical Instability Waves (TIW). Among the orbits tested for altimetric satellites, the JASON2 orbit (10day repeat period) is found to give the best single satellite sampling for data assimilation. The addition of a second or third satellite to JASON2 is particularly useful in the TIW region and is even required to properly control the Brazil rings. A SWOT satellite would provide benefits that are equivalent to the contribution of two or three satellites, depending on the case.

