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 backward and forward nudging (or **back and forth nudging, BFN**) consists in initially solving the forward equations with a nudging term, and then using the final state thus obtained as initial condition to solve the same equations in a backward direction with a feedback term (with the opposite sign compared to the feedback term of forward nudging). After resolution of this backward equation, one obtains an estimate of the initial state of the system. This process is then repeated in an iterative way until convergence of the initial state is achieved.

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 to the real initial state. The algorithm has been tested on non-linear models, i.e. the Lorenz equations and the quasi-geostrophic model and a comparison with the 4D-VAR algorithm was achieved. Twice less iterations 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 to obtain a correct initial state, with a smaller number of iterations than in a variational method and an easier implementation. This can be of great value in the present context of more and more complex models and increasingly diversified observations of the ocean or the atmosphere.

We have recently compared this algorithm to the 4D-VAR method on toy models
such as the Lorenz' and 1D viscous Burgers' equations, but also on a layered
quasi-geostrophic ocean model. The numerical convergence of the BFN
algorithm is always achieved in a few iterations, and in the case of twin
experiments, it provides a good estimation of the initial condition and very
good forecasts (see Figure 1). This work is partly presented in [2].

We already proved the theoretical convergence for a linear compact operator,
provided that some hypothesis on the spatial distribution of the
observations and the observation operator were satisfied. We are currently
working on the theoretical point of view for nonlinear models, and already
obtained some a priori estimations on the Lorenz system. We are also doing
many numerical experiments on a shallow water model (see Figure
2), and exhaustive comparisons with other data assimilation
schemes on a Burgers model (see [3]).

Figure 1: BFN on a Burgers model: evolution
in time of the RMS difference between the reference trajectory and the
perturbed trajectory derived from the noised observations of the system at
time t=0 (plain line), and between the reference trajectory and the
identified trajectories for the BFN (dotted line), the 4D-VAR (dash-dotted
line) and the BFN-preprocessed 4D-VAR (dashed-line) algorithms.

Figure 2: BFN on a shallow-water model: data assimilation period = 1 month,
forecast period = 3 months; model states at the end of the forecast period:
true state (left), no assimilation (center), BFN assimilation
(right).

You can download here the following movies, corresponding to figure 2 (BFN
scheme on the left, no assimilation in the middle, true state on the right):