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Forward nudging

In order to simplify the notations, we assume that the model equations have been discretized in space by a finite difference, finite element, or spectral discretization method. The time continuous model satisfies dynamical equations of the form:

$\displaystyle \frac{dX}{dt} = F(X), \quad 0<t<T,$ (3.1)

with an initial condition $ X(0) = x_0$ . In this equation, $ F$ represents all the linear or nonlinear operators of the model equation, including the spatial differential operators.

We will denote by $ C$ the observation operator, allowing us to compare the observations $ X_{obs}(t)$ with the corresponding $ C(X(t))$ , deduced from the state vector $ X(t)$ . The observation operator usually involves interpolation/extrapolation, and some change of variables. The various measurements are not extracted at the same location as the model gridpoints, leading to some necessary interpolation and extrapolation operators. Also, satellites do not observe the physical variables of the model (e.g. temperature, velocity, ...) but some other physical parameters, that can be related to the model state: for instance, many satellites measure radiances, that can be related to the sea surface height or temperature. We do not particularly assume that $ C$ is a linear operator.

If we apply nudging to the model (3.1), we obtain

$\displaystyle \frac{dX}{dt} = F(X)+K(X_{obs}-C(X)), \quad 0<t<T,$ (3.2)

with the same initial condition, and where $ K$ is the nudging (or gain) matrix. Note that it may also be a nudging scalar coefficient in some simple cases. 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. In the linear case (where F is a matrix, and C is a linear operator), the forward nudging method is nothing else than the Luenberger observer [81], also called asymptotic observer, where the matrix K can be chosen so that the error goes to zero when time goes to infinity. Unfortunately, in most geophysical applications, the assimilation period is not long enough to have the nudging method give good results.

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Next: Backward nudging Up: ``Back and Forth Nudging'' Previous: ``Back and Forth Nudging''   Contents
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