## Section: New Results

### Development of New Methods for Data Assimilation

#### A Nudging-Based Data Assimilation Method: the Back and Forth Nudging

Participants : Didier Auroux, Jacques Blum, Maëlle Nodet.

The back and forth nudging algorithm (see [39] , [38] ), has been recently introduced for simplicity reasons, as it does not require any linearization, or adjoint equation, or minimization process in comparison with variational schemes, but nevertheless it provides a new estimation of the initial condition at each iteration.

We have studied its convergence properties as well as its efficiency in the case of numerical experiment with a 2D shallow water model. Comparisons with the 4D-VAR have been performed. Finally, we also studied a hybrid method, by considering a few iterations of the BFN algorithm as a preprocessing tool for the 4D-VAR algorithm. We have shown that the BFN algorithm is extremely powerful in the very first iterations, and also that the hybrid method can both improve notably the quality of the identified initial condition by the 4D-VAR scheme and reduce the number of iterations needed to achieve convergence [1] , [24] .

We considered from a theoretical point of view the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Bürgers' equation). We showed that for non viscous equations (both linear transport and Bürgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Bürgers' equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered [41] . Several comparisons with the 4D-VAR and quasi-inverse algorithms have also been performed on this equation. The application of the BFN algorithm to OPA-NEMO ocean model is currently under investigation. The first experiments are very encouraging.

Finally, within the standard nudging framework, we considered the definition of an innovation term that takes into account the measurements and respects the symmetries of the physical model. We proved the convergence of the estimation error to zero on a linear approximation of the system (a 2D shallow-water model). It boils down to estimating the fluid velocity in a water-tank system using only SSH measurements. The observer is very robust to noise and easy to tune. The general nonlinear case has been illustrated by numerical experiments, and the results have been compared with the standard nudging techniques.

#### Variational Data Assimilation with Control of Model Error

Participant : Arthur Vidard.

One of the main limitation of the current operational variational data assimilation techniques is that they assume the model to be perfect mainly because of computing cost issues. Numerous researches have been carried out to reduce the cost of controlling model errors by controlling the correction term only in certain privileged directions or by controlling only the systematic and time correlated part of the error.

Both the above methods consider the model errors as a forcing term in the model equations. Trémolet (2006) describes another approach where the full state vector (4D field : 3D spatial + time) is controlled. Because of computing cost one cannot obviously control the model state at each time step. Therefore, the assimilation window is split into sub-windows, and only the initial conditions of each sub-window are controlled, the junctions between each sub-window being penalized. One interesting property is that, in this case, the computation of the gradients, for the different sub-windows, are independent and therefore can be done in parallel.

We are implementing this method in a realistic Oceanic framework using OPAVAR/ NEMOVAR as part of the VODA ANR project.

#### Variational data assimilation for locally nested models.

Participants : Éric Blayo, Laurent Debreu, Emilie Neveu.

The objectives are to study the mathematical formulation of variational data assimilation for locally nested models and to conduct numerical experiments for validation.

The state equations of the optimality system have been written for the general case of two embedded grids, for which several kinds of control (initial conditions, boundary conditions) have been proposed. Both one way and two way interactions have been studied. This last year, we worked on integration of non linear grid interactions in the algorithm. Additionally, the problem of specification of background error covariances matrices has been studied (see [18] ).

In the ANR MSDAG project and Emilie Neveu's PhD, we continue to work on the subject. Our main interest is on the use of multiscale optimization methods for data assimilation. The idea is to apply a multigrid algorithm to the solution of the optimization problem. One of the focus is on the design of resolution dependent observation operators. The other key point is in the adaptation of the original FAS (Full Approximation Scheme) multigrid algorithm to local mesh refinement. The applications will be done in the context of the assimilation of images (see 6.4 ).

#### Variational Data Assimilation and Control of Model's Parameters and Numerical Schemes.

Participant : Eugène Kazantsev.

In this section, we focused our attention on the data assimilation techniques devoted to identification of external model's parameters and parametrizations.

First, the attention was paid to control the bottom topography by variational data assimilation in frames of time-dependent motion governed by non-linear barotropic ocean model [8] . Assimilation of artificially generated data allows to measure the influence of various error sources and to classify the impact of noise that is present in observational data and model parameters. The choice of assimilation window is discussed. Assimilating noisy data with longer windows provides higher accuracy of identified topography. The topography identified once by data assimilation can be successfully used for other model runs that start from other initial conditions and are situated in other parts of the model's attractor.

Second, the numerical scheme of the shallow-water model at the boundary was controlled by assimilating data of a high resolution model. It was shown in [9] that control of approximation of boundary derivatives and interpolations can increase the model's accuracy in boundary regions and improve the solution in general. On the other hand, optimal boundary schemes obtained in this way, may not approximate derivatives in a common sense. Particular study was performed in [10] on the example of one-dimensional wave equation to understand this phenomenon. In this simple case, we see that the control changes the interval's length (and, consequently, all approximations of derivatives) in order to compensate numerical error in the wave velocity.

To illustrate advantages of the optimal parametrization of model's operators in the boundary region, we perform several classical experiments with a linearized shallow-water model in a square box. We run the reference model on a fine resolution grid and assimilate the data of this run into a coarse resolution model controlling it's boundary.

The first one concerns spurious oscillations that appear in numerical solution when the Munk boundary layer is not resolved by the model's grid. The Munk layer's width in this experiment was taken as 60 km. One can see in figure 1 (left) that strong oscillations are present in the solution on the model on 133 km grid while they are absent both in the finer grid solutions (45 and 15 km) and in the solution with the optimal boundary.

The second experiment is devoted to the study of inertia-gravity and Rossby waves simulated by the same model. We measure the difference between the reference solution, obtained on the high-resolution grid (h = 15km ) and solutions on a coarser grids. One can see in the figure 1 (right) that at the coarse (133 km) resolution the difference reaches as high values as 1000. At the medium resolution (45 km), the difference is smaller, but with increasing tendency. When the parametrization of the model's boundary is optimal, the solution is close to the reference one with no increasing tendency even at a low resolution.