The aim of data assimilation is to combine the observations and models, in order to retrieve a coherent and precise state of the system from a set of discrete spacetime data, and then to provide reliable forecasts of its evolution. Data assimilation covers all the mathematical and numerical techniques in which the observed information is accumulated into the model state by taking advantage of consistency constraints with laws of time evolution and physical properties, and which allow us to blend as optimally as possible all the sources of information coming from theory, models and other types of data [74,35,106].
Nudging is a data assimilation method that uses dynamical relaxation to adjust a model towards observations. The standard nudging algorithm consists of adding to the state equations of a dynamical system a feedback term proportional to the difference between the observation and the equivalent quantity computed by integration of the state equations. The model then appears as a weak constraint, and the nudging term forces the state variables to fit as well as possible to the observations. This forcing term in the model dynamics has a tunable coefficient that represents the relaxation time scale. This coefficient is chosen by numerical experimentation so as to keep the nudging terms small in comparison to the state equations, and large enough to force the model to the observations. The nudging term can also be seen as a penalty term, which penalizes the system if the model is too far from the observations. Note that in the linear case, the standard nudging method is nothing else than the Luenberger observer, also called asymptotic observer [81].
The nudging method is a flexible assimilation technique, and computationally much more economical than variational data assimilation methods [77]. First used in meteorology [69], the nudging method has been successfully introduced in oceanography in a quasi-geostrophic model [109,111,39] and has been applied to a mesoscale model of the atmosphere with synoptic-scale data [103]. The nudging coefficients can be optimized by a variational method [102,119], where a parameter estimation approach is proposed to obtain optimal nudging coefficients, in the sense that the difference between the model solution and the observations is as small as possible. A comparison between optimal nudging and Kalman filtering can be found in [112]. A drawback of this optimal nudging technique is that it requires the computation of the adjoint state of the model equations, which is not necessary in the standard nudging method.
The backward nudging algorithm consists of solving backwards in time the state equations of the model, starting from the observation of the system state at the final time of the assimilation period. A nudging term, with the opposite sign compared to the standard nudging algorithm, is added to the state equations, and the final state computed in the backward integration is in fact an approximation of the initial state of the system [15].
The Back and Forth Nudging (BFN) algorithm, introduced in [20], consists of solving first the forward nudging equation, and then the model equation backwards in time with a relaxation term (with the opposite sign in comparison with the relaxation term introduced in the forward equation). The initial condition of this backward integration is the final state obtained by the standard nudging method. After integration of this backward equation, one obtains an estimate of the initial state of the system. We then repeat these forward and backward integrations (with the relaxation terms) until convergence of the algorithm.
Such a forward-backward assimilation technique had already been introduced in [105,104]. In that algorithm, at each observation time, the values predicted by the model for the observed parameters were just replaced by the observed values. This corresponds to the particular case of our BFN algorithm where the nudging coefficients go to infinity.
The BFN algorithm can be compared to the four-dimensional variational algorithm (4D-VAR, see e.g. [77]), which also consists of a sequence of forward and backward integrations. In our algorithm it is useless to linearize the system, even for nonlinear problems, and the backward system is not the adjoint equation but the model equations, with an extra feedback term that stabilizes the numerical integration of this ill-posed backward problem.
Let us finally mention another back and forth data assimilation method, called the quasi-inverse method [74]. In that method, there are no nudging terms, and in the backward integration, the sign of the dissipation terms is changed for stability reasons. The idea of introducing relaxation (or nudging) terms in our algorithm enables us to keep the dissipation terms with the correct sign in the backward integration, as the nudging terms have a stabilizing role.
In this chapter, we first present the standard nudging algorithm in a general case (nonlinear model), then the nudging algorithm applied to the corresponding backward model, and finally we introduce the Back and Forth Nudging algorithm. We then present some theoretical convergence results in simplified cases (full observations) on various types of models: linear models, transport equations (both linear and nonlinear, with or without viscosity). Then, we present the numerical application of this algorithm to various physical models. Finally, nudging can be seen as a particular type of observer, and we define a specific nudging-based observer for a shallow-water model, allowing us to preserve the natural symmetries of the model, to reduce the sensitivity to the observation noise, and also to correct the non-observed variables with the observed ones. Several conclusions and perspectives are given at the end of this chapter.