The standard nudging algorithm consists in adding to the state equations of a dynamical system a feedback term, which is proportional to the difference between the observation and its equivalent quantity computed by the resolution of the state equations. The model appears then as a weak constraint, and the nudging term forces the state variables to fit as well as possible to the observations.
The back and forth nudging algorithm consists in solving first the forward nudging equation and then a backward equation, where the feedback term which is added to the state equations has an opposite sign to the one introduced in the forward equation. The initial state of this backward resolution is the final state obtained by the standard nudging method. After resolution of this backward equation, one obtains an estimate of the initial state of the system. We repeat these forward and backward resolutions (with the feedback terms) until convergence of the algorithm.
This algorithm has been compared to the 4D-VAR algorithm, which consists also in a sequence of forward and backward resolutions. In our algorithm, it is useless to linearize the system and the backward system is not the adjoint equation but the model equation, with an extra feedback term that stabilizes the resolution of this ill-posed backward resolution.
We have proved on a linear model that, provided that the feedback term is large enough as well as the assimilation period, we have convergence of the algorithm to the real initial state. Numerical results have been performed both on the Lorenz equations, on viscous Burgers equation, on a layered quasi-geostrophic model and on a shallow water model. Twice less iterations than the 4D-VAR are necessary to obtain the same level of convergence. The effect of errors on the observations and on the model has also been studied.
This algorithm is hence very promising in order to obtain a correct trajectory, with a smaller number of iterations than in a variational method and a very easy implementation.
Références :
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3) D. Auroux, J. Blum : Back and forth nudging algorithm for data assimilation problems, CRAS Paris Ser.1 340 (2005) 873-878
4) Zou, X.; Navon, I.-M.; Le Dimet, F.-X.: An Optimal Nudging Data Assimilation Scheme Using Parameter Estimation, Quart. J. Roy. Met. Soc. 118, 1193-1186 (1992)
5) Kalnay, E. et al: Application of the quasi-inverse method to data assimilation, Month. Weather Review, 128, 864-875 (2000)
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