The back and forth nudging algorithm applied to a shallow water model, comparison and hybridization with the 4D-VAR

We study in this paper a new data assimilation algorithm, called the Back and Forth Nudging (BFN). This scheme has been very 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 study its convergence properties as well as efficiency on a 2D shallow water model. All along the numerical experiments, comparisons with the standard variational algorithm (called 4D-VAR) are performed. Finally, a hybrid method is introduced, by considering a few iterations of the BFN algorithm as a preprocessing tool for the 4D-VAR algorithm. We show 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

Observers for compressible Navier-Stokes equation

We consider a multi-dimensional model of a compressible fluid in a bounded domain. We want to estimate the density and velocity of the fluid, based on the observations for only velocity. We build an observer exploiting the symmetries of the fluid dynamics laws. Our main result is that for the linearised system with full observations of the velocity field, we can find an observer which converges to the true state of the system at any desired convergence rate for finitely many but arbitrarily large number of Fourier modes. Our one-dimensional numerical results corroborate the results for the linearised, fully observed system, and also show similar convergence for the full nonlinear system and also for the case when the velocity field is observed only over a subdomain

Identification of velocity fields for geophysical fluids from a sequence of images

International audienceWe propose an algorithm to estimate the motion between two images. This algorithm is based on the nonlinear brightness constancy assumption. The number of unknowns is reduced by considering displacement fields that are piecewise linear with respect to each space variable, and the Jacobian matrix of the cost function to be minimized is assembled rapidly using a finite element method. Different regularization terms are considered, and a multiscale approach provides fast and efficient convergence properties. Several numerical results of this algorithm on simulated and real geophysical flows are presented and discussed

Symmetry-preserving nudging: theory and application to a shallow water model

One of the important topics in oceanography is the prediction of ocean circulation. The goal of data assimilation is to combine the mathematical information provided by the modeling of ocean dynamics with observations of the ocean circulation, e.g. measurements of the sea surface height (SSH). In this paper, we focus on a particular class of extended Kalman filters as a data assimilation method: nudging techniques, in which a corrective feedback term is added to the model equations. We consider here a standard shallow water model, and we define an innovation term that takes into account the measurements and respects the symmetries of the physical model. We prove the convergence of the estimation error to zero on a linear approximation of the system. 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 is illustrated by numerical experiments, and the results are compared with the standard nudging techniques

The Back and Forth Nudging algorithm for data assimilation problems: theoretical results on transport equations

International audienceIn this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Bürgers' equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), 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 Burgers' 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

An easy-to-implement and efficient data assimilation method for the identification of the initial condition: the Back and Forth Nudging (BFN) algorithm

International audienceThis paper deals with a new data assimilation algorithm called the Back and Forth Nudging. The standard nudging technique consists in adding to the model equations a relaxation term, which is supposed to force the model to the observations. The BFN algorithm consists of repeating forward and backward resolutions of the model with relaxation (or nudging) terms, that have opposite signs in the direct and inverse resolutions, so as to make the backward evolution numerically stable. We then applied the Back and Forth Nudging algorithm to a simple non-linear model: the 1D viscous Burgers' equations. The tests were carried out through several cases relative to the precision and density of the observations. These simulations were then compared with both the variational assimilation (VAR) and quasi-inverse (QIL) algorithms. The comparisons deal with the programming, the convergence, and time computing for each of these three algorithms

Several data assimilation methods for geophysical problems

International audienceIn this paper, we present an overview of various data assimilation methods, in order to identify the initial condition of a geophysical system and reconstruct its evolution in time and space. We first present the well known four dimensional variational adjoint method, the 4D-VAR algorithm, and then the four dimensional variational dual method, the 4D-PSAS algorithm, extended to nonlinear models. We present then an improved sequential data assimilation algorithm, the SEEK filter. We finally introduce a new simple algorithm, the Back and Forth Nudging. Some theoretical and numerical results about the BFN algorithm are finally given