The iterative ensemble Kalman filter (IEnKF) was recently proposed to improve the performance of ensemble Kalman filtering with strongly nonlinear geophysical models. IEnKF can be used as a lag-one smoother and extended to a fixed-lag smoother: the iterative ensemble Kalman smoother (IEnKS ). IEnKS is an ensemble variational method. It does not require the use of the tangent linear of the evolution and observation models, nor the adjoint of these models: the required sensitivities (gradient and Hessian) are obtained from the ensemble. Looking for the optimal performance, we consider a quasi-static algorithm, out of the many possible extensions. IEnKS is explored on the Lorenz'95 model and on a 2D turbulence model. As a logical extension of IEnKF, IEnKS significantly outperforms standard Kalman filters and smoothers in strongly nonlinear regimes. In mildly nonlinear regimes (typically synoptic scale meteorology), its filtering performance is marginally but clearly better than the standard ensemble Kalman filter, and it keeps improving as the length of the temporal data assimilation window is increased. For long windows, its smoothing performance very significantly outranks the standard smoothers, which is believed to stem from the variational but flow-dependent nature of the algorithm. For very long windows, the use of a multiple data assimilation variant of the scheme, where observations are assimilated several times, is advocated. This paves the way for finer re-analysis freed from the static prior assumption of 4D-Var, but also partially freed from the Gaussian assumptions that usually impede standard ensemble Kalman filtering and smoothing.

Both ensemble filtering and variational data assimilation methods have proven being useful in the joint estimation of state variables and parameters of geophysical models. Yet, their respective benefits and drawbacks in this task are distinct. An ensemble variational method, known as the iterative ensemble Kalman smoother (IEnKS), has recently been introduced. It is based on an adjoint-free variational but flow-dependent scheme. As such, IEnKS is a candidate tool for joint state and parameter estimation that may inherit the benefits from both the ensemble filtering and variational approaches.

In this study , an augmented state IEnKS is tested on the estimation of the forcing parameter of the Lorenz'95 model. Since joint state and parameter estimation is especially useful in applications where the forcings are uncertain but nevertheless determining, typically in atmospheric chemistry, the augmented state IEnKS is tested on a new low-order model that combines the Lorenz'95 model, representing its meteorological part, and the advection diffusion of a tracer for its chemical part. In these experiments, IEnKS is compared to the ensemble Kalman filter, the ensemble Kalman smoother and a 4D-Var method, that are considered choices to solve these joint estimation problems. In this low-order model context, IEnKS is shown to significantly outperform those methods, for any length of the data assimilation window, and for present time analysis as well as retrospective analysis. Besides, the performance of IEnKS is even more striking on parameter estimation, whereas getting close to the same performance with 4D-Var is likely to require both a long data assimilation window and a complex modeling of the background statistics.

Based on Verdandi , Polyphemus and the “Urban Air Quality Analysis” software, data assimilation was further developed at urban scale. The Best Linear Unbiased Estimator (BLUE) is computed to merge the outputs of the ADMS Urban model and the observations of a sparse monitoring network . We improved the modeling of the covariance of the model state error. The assimilation was applied for part of Paris (see Fig. ) and for Paris region, in the context of the PRIMEQUAL project PREQUALIF (“Multidisciplinary Program on Air Quality Research in Île-de-France”).

It was applied to nitrogen dioxide, particulate matter and black carbon. Specific investigations were carried out to estimate the variance of the a posteriori error and to determine the impact of each monitoring station on the final results.

To estimate the amount of radionuclides and the temporal profile of the source term released in the atmosphere during the accident of the Fukushima Daiichi nuclear power plant in March 2011, inverse modeling techniques have been used and have proven their ability in this context. In a previous study, the lower bounds of the caesium-137 and iodine-131 source terms were estimated with such techniques, using activity concentration observations. The importance of an objective assessment of prior errors (the observation errors and the background errors) was emphasised for a reliable inversion. In such critical context where the meteorological conditions can make the source term partly unobservable and where only a few observations are available, such prior estimation techniques are mandatory, the retrieved source term being very sensitive to this estimation.

We propose to extend the use of these techniques to the estimation of prior errors when assimilating observations from several data sets . The aim is to compute an estimate of the caesium-137 source term jointly using all available data about this radionuclide, such as activity concentrations in the air, but also daily fallout measurements and total cumulated fallout measurements. It is crucial to properly and simultaneously estimate the background errors and the prior errors relative to each data set. A proper estimation of prior errors is also a necessary condition to reliably estimate the a posteriori uncertainty of the estimated source term. Using such techniques, we retrieve a total released quantity of caesium-137 in the interval $11.6-19.3$ PBq with an estimated standard deviation range of $15-20\%$ depending on the method and the data sets. The “blind” time intervals of the source term have also been strongly mitigated compared to the first estimations with only activity concentration data.

The Chernobyl nuclear accident, and more recently the Fukushima accident, highlighted that the largest source of error on consequences assessment is the source term, including the time evolution of the release rate and its distribution between radioisotopes. Inverse modeling methods, which combine environmental measurements and atmospheric dispersion models, have proven being efficient in assessing source term due to an accidental situation. Most existing approaches are designed to use air sampling measurements and some of them also use deposition measurements . Some studies have been conceived to use dose rate measurements, but none of the developed methods were carried out to assess the complex source term of a real accident situation like the Fukushima accident. However, dose rate measurements are generated by the most widespread measurement system and, in the event of a nuclear accident, these data constitute the main source of measurements of the plume and radioactive fallout during releases. This study , proposes a method to use dose rate measurements as part of an inverse modeling approach to assess source terms. The method is proven efficient and reliable when applied to the accident at the Fukushima Daiichi Nuclear Power Plant (FD-NPP). The emissions for the eight main isotopes have been assessed. Accordingly, 105.9 PBq of ${}^{131}$I, 35.8 PBq of ${}^{132}$I, 15.5 PBq of ${}^{137}$Cs and 12,134 PBq of noble gases were released. The events at FD-NPP (such as venting, explosions, etc.) known to have caused atmospheric releases are well identified in the retrieved source term. The estimated source term is validated by comparing simulations of atmospheric dispersion and deposition with environmental observations. In total, it was found that for 80 % of the measurements, simulated and observed dose rates agreed within a factor of 2. Changes in dose rates over time have been overall properly reconstructed, especially in the most contaminated areas to the northwest and south of the FD-NPP. A comparison with observed atmospheric activity concentration and surface deposition shows that the emissions of caesiums and ${}^{131}$I are realistic but that ${}^{132}$I and ${}^{132}$Te are probably underestimated and noble gases are likely overestimated. Finally, an important outcome of this study is that the method proved to be perfectly suited to emergency management and could contribute to improve emergency response in the event of a nuclear accident.

The emissions of non-methane volatile organic compounds (VOCs) over western Europe for the year 2005 are estimated via inverse modeling by assimilation of in situ observations of concentration and they are subsequently compared to a standard emission inventory. The study  focuses on fifteen VOC species: five aromatics, six alkanes, two alkenes, one alkyne and one biogenic diene. The inversion relies on a validated fast adjoint of the chemical transport model used to simulate the fate and transport of these VOCs. The assimilated ground-based measurements over Europe are provided by the European Monitoring and Evaluation Programme (EMEP) network. The background emissions errors and the prior observational errors are estimated by maximum likelihood approaches. The positivity assumption on the VOC emission fluxes is pivotal for a successful inversion and this maximum likelihood approach consistently accounts for the positivity of the fluxes. For most species, the retrieved emissions lead to a significant reduction of the bias, which underlines the misfit between the standard inventories and the observed concentrations. The results are validated through a forecast test and a cross-validation test. An estimation of the posterior uncertainty is also provided. It is shown that the statistically consistent non-Gaussian approach, based on a reliable estimation of the errors, offers the best performance. The efficiency in correcting the inventory depends on the lifetime of the VOCs and the accuracy of the boundary conditions. In particular, it is shown that the use of in situ observations using a sparse monitoring network to estimate emissions of isoprene is inadequate because its short chemical lifetime significantly limits the spatial radius of influence of the monitoring data. For species with longer lifetime (a few days), successful, albeit partial, emission corrections can reach regions hundreds of kilometres away from the stations. Domainwide corrections of the emissions inventories of some VOCs are significant, with underestimations on the order of a factor of two for propane, ethane, ethylene and acetylene.

Uncertainty quantification is critical in the inversion of CO${}_2$ surface fluxes from atmospheric concentration measurements. We estimate the main hyperparameters of the error covariance matrices for a priori fluxes and CO${}_2$ concentrations, that is, the variances and the correlation lengths, using real, continuous hourly CO${}_2$ concentration data in the context of the Ring 2 experiment of the North American Carbon Program Mid Continent Intensive. Several criteria, namely maximum likelihood (ML), general cross-validation (GCV) and ${\chi }^2$ test are compared for the first time under a realistic setting in a mesoscale CO${}_2$ inversion. It is shown  that the optimal hyperparameters under the ML criterion assure perfect ${\chi }^2$ consistency of the inverted fluxes. Inversions using the ML error variances estimates rather than the prescribed default values are less weighted by the observations, because the default values underestimate the model-data mismatch error, which is assumed to be dominated by the atmospheric transport error. As for the spatial correlation length in prior flux errors, the Ring 2 network is sparse for GCV and this method fails to reach an optimum. In contrast, the ML estimate (e.g. an optimum of 20 km for the first week of June 2007) does not support long spatial correlations that are usually assumed in the default values.

This study  investigates the potential impact of future ground-based lidar networks on analysis and short-term forecasts of PM${}_{10}$. To do so, an Observing System Simulation Experiment (OSSE) is built for PM${}_{10}$ data assimilation using optimal interpolation over Europe for one month in 2001. First, we estimate the efficiency of the assimilation of lidar network measurements in improving PM${}_{10}$ concentration analysis and forecast. It is compared to the efficiency of assimilating concentration measurements from the AirBase ground network, which includes about 500 stations in western Europe. It is found that the assimilation of lidar observations is more efficient at improving PM${}_{10}$ concentrations in terms of root mean square error and correlation after 12 hours of assimilation than the assimilation of AirBase measurements. Moreover, the spatial and temporal influence of the assimilation of lidar observations is larger and longer. In our experiments, the assimilation of lidar products improves PM${}_{10}$ forecast for 108 hours against 60 hours for AirBase assimilation. The results show a potentially powerful impact of the future lidar networks. Secondly, since a lidar is a very costly instrument, a sensitivity study on the number of required lidars is performed to help defining an optimal lidar network for PM${}_{10}$ forecast. The results suggest 12 lidar stations over western Europe, because a network with 26 lidar stations is more expensive and offers a limited improvement (less than $1\mu g{m}^{-3}$ of root mean square error on average) over the lidar network. A comparison of two networks with 12 lidar stations at different locations does not lead to substantial differences.

Both reduction and emulation were applied to the dynamic air quality model Polair3D from Polyphemus. The reduction relied on proper orthogonal decomposition (POD) on the input data and on the state vector. The dimension of the reduced subspace for the input data is about 80, while the dimension of the reduced state vector is less than 10. The projection of the state vector on its reduced subspace can be carried out before every integration time step, so that one can reproduce a full state trajectory (in time) using the reduced model.

Significant advances were made to emulate the reduced model, which requires about 90 inputs (reduced input data and reduced state vector) and computes about 10 outputs (reduced state vector). 90 inputs is however a large number to build an emulator using a classical approaches. Promising results were however obtained with radial basis functions and an adapted kriging-based method.

The dimension reduction was applied to the outputs of the urban air quality model ADMS Urban, which is a static model with low-dimensional inputs and high-dimensional outputs. A proper orthogonal decomposition (POD) on the outputs allowed us to drastically reduce their dimension, from ${10}^4$ to just a few scalars. The emulation of the reduced model itself was successfully carried out with radial basis functions or an adapted kriging-based method. The resulting reduced/emulated model exhibited meaningful response to all variables. Its performance compared to observations was the same as the original model. The computational cost of the full model is about 8 minutes on 16 cores (for a single time step), while the reduced/emulated model requires only 50 ms on one core .

Dimension reduction is applied to an image model, composed of Lagrangian constancy of velocity and transport of image brightness. Waveforms basis are obtained on the image domain for subspaces of images, motion fields and divergence-free motion fields, as eigenvectors of quadratic functions. Image assimilation with th reduced model allows to estimate velocity fields satisfying space-time properties defined by user and traduced as a quadratic function. This approach also solves the issue of complex geographical domains and the difficulty of applying boundary conditions on these domains. Results are obtained with a reduced dimension of motion to a few scalars, to be compared with the original problem that has the size of image domain , , .

Nowadays, it is standard procedure to generate an ensemble of simulations for a meteorological forecast. Usually, meteorological centers produce a single forecast, out of the ensemble forecasts, computing the ensemble mean (where every model receives an equal weight). It is however possible to apply aggregation methods. When new observations are available, the meteorological centers also compute analyses. Therefore, we can apply the ensemble forecast of analyses. Ensembles of forecasts for wind velocity and mean sea level pressure, from Météo France, were aggregated. Preliminary results show significant improvements for mean sea level pressure.

An important issue is the estimation of the uncertainties associated with the aggregated forecasts. One investigated direction relies on the framework of machine learning, with the aggregation of an ensemble of probability density functions instead of the point forecasts of the ensemble.

Another direction is to reformulate the aggregation problem in a filtering problem for the weights. The weights are supposed to satisfy some dynamics with unknown model error, which defines the state equation of a filter. An observation equation compares the aggregated forecast with the observations (or analyses) with known observational error variance. The filter finally computes estimates for the weights and quantifies their uncertainties. We applied a Kalman filter and a minimax filter for air quality forecasting. We also introduced a criterion that the filter results should satisfy if they are representative of the uncertainties .

We carried out a sensitivity analysis of the dispersion of radionuclides during Fukushima disaster. We considered the dispersion at regional scale, with the Eulerian transport model Polair3D from Polyphemus. The sensitivities to most input parameters were computed using the Morris method (with 8 levels and 100 trajectories). The influences of 19 scalar parameters were quantified. The scalar parameters were additive terms or multiplicative factors applied to 1D, 2D or 3D fields such as emission rates, precipitations, cloud height, wind velocity. It was shown that, depending on the output quantities of interest (various aggregated atmospheric and ground dose rates), the sensitivity to the inputs may greatly vary. Very few parameters show low sensitivity in any case. The vertical diffusion coefficient, the scavenging factors, the winds and precipitation intensity were found to be the most influential inputs. Most input variables related to the source term (emission rates, emission dates) also had a strong influence.

This research addresses the issue of divergence-free motion estimation on an image sequence, acquired over a given temporal window. Unlike most state-of-the-art technics, which constrain the divergence to be small thanks to Tikhonov regularization terms, a method that imposes a null value of divergence of the estimated motion is defined.

Motion is either characterized by its vorticity value or by its coefficients on a divergence-free basis and assumed to satisfy the Lagragian constancy hypothesis. An image assimilation method, based on the 4D-Var technic, is defined that estimates motion as a compromise between the evolution equations of vorticity or projection coefficients and the observed sequence of images.

The method is applied on Sea Surface Temperature (SST) images acquired over Black Sea by NOAA-AVHRR sensors. The divergence-free assumption is roughly valid on these acquisitions, due to the small values of vertical velocity at the surface.

Data assimilation technics are used to retrieve motion from image sequences. These methods require a model of the underlying dynamics, displayed by the evolution of image data. In order to quantify the approximation linked to the chosen dynamic model, an error term is included in the evolution equation of motion and a weak formulation of 4D-Var data assimilation is designed. The cost function to be minimized simultaneously depends on the initial motion field, at the begining of the studied temporal window, and on the error value at each time step. The result allows to assess the model error and analyze its impact on motion estimation.

The approach has been used to estimate the impact of geophysical forces (gravity, Coriolis, diffusion) and better assess the surface dynamics .

The research concerns an approach to estimate velocity on an image sequence and simultaneously segment and track a given structure. It relies on the underlying dynamics' equations of the studied physical system. A data assimilation method is designed to solve evolution equations of image brightness, those of motion's dynamics, and those of the distance map modeling the tracked structures. The method is applied on meteorological satellite data, in order to track tropical clouds on image sequences and estimate their motion, as seen on Fig. .

Quantification is obtained on synthetic experiments by comparing trajectories of characteristic points. The respective position of these points on the last image of the sequence for different methods may be compared to that obtained with ground truth as seen on Fig. .

Data assimilation is performed either with a 4D-Var variational approach , ,  or with an ensemble approach. In the last case, computation of the ensemble from optical flow methods of the literature is currently studied.

An iterative minimax method is developed for the problem of motion estimation from an image sequence. The main idea of the algorithm is to use the "bi-linear" structure of the Navier-Stokes equations and optical flow constraint in order to iteratively estimate the velocity. The algorithm consists of the following parts:

1) we construct a continuous image function $\widehat{I}$, solving the optical flow constraint, such that $\widehat{I}$ fits (in the sense of least-squares) the observed sequence of images. To do so, we set the velocity field in the optical flow constraint to be the current minimax estimate of the velocity field $𝐰$, obtained at the previous iteration of the algorithm, and construct the minimax estimate $\widehat{I}$ of the resulting linear advection equation using the observed image sequence as discrete measurements of the brightness function;

2) we plug the estimate of the image gradient, obtained out of pseudo-observations $\widehat{I}$ in 1), into the optical flow constraint and the current minimax estimate $𝐰$ of the velocity field into the non linear part of Navier-Stokes equations so that we end up with a system of linear PDEs, which represents an extended state equation: it contains a linear parabolic equation for the velocity field and linear advection equation for the image brightness function. We construct the minimax estimate of the velocity field from the extended state equation using again the observed image sequence as discrete measurements of the brightness function;

3) we use the minimax estimate of the velocity field obtained in 2) in order to start 1) again.

Alternatively, point 1) may be used to retrieve a continuous image function from sparse and noisy image snapshots, based on previous motion estimation with a 4D-Var technic as seen on Fig. , that displays ground truth, noisy image observation, image estimation at the end of the studied intervall.

In the field of forest fires risk management, important challenges exist in terms of people and goods preservation. Answering to strong needs from different actors (firefighters, foresters), researchers focus their efforts to develop operational decision support system tools that may forecast wildfire behavior. This requires the evaluation of model performance, but currently, simulation errors are not sufficiently qualified and quantified.

We consider that the proper evaluation of a model requires to apply it to a large number of fires – instead of carrying out a fine tuning on just one fire. We implemented a software to simulate a large number of fires (from the Prométhée database, http://www.promethee.com/) with the simulation model ForeFire (CNRS/University of Corsica) and evaluate the results with error measures . One simulation requires mainly the following data: the ignition point, the ground elevation, the vegetation cover and the wind field. See illustration in Fig. . We simulated 80 fires with four physical models, which proved that the most advanced models performed better overall, even though the input data is often inaccurate. We also carried out Monte Carlo simulations to evaluate the impact of the uncertainty in input data. We showed that the Monte Carlo approach led to a reliable forecasting system, which suggests that the probability densities derived from the simulations (see Fig. ) may be useful information for preventive actions in an operational context.