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Backward nudging

We now assume that we have a final condition in equation (3.1) instead of an initial condition. This leads to the following backward equation:

$\displaystyle \frac{d\tilde{X}}{dt} = F(\tilde{X}), \quad T>t>0,$ (3.3)

with a final condition $ \tilde{X}(T) = \tilde{x}_T$ . 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 [15]. If we apply nudging to this backward model with a feedback term of the opposite sign (in order to have a well posed problem), we obtain

$\displaystyle \frac{d\tilde{X}}{dt} = F(\tilde{X})-K'(X_{obs}-C(\tilde{X})), \quad T>t>0,$ (3.4)

where $ K'$ is the backward nudging matrix.

The backward integration of this equation provides a state vector at time $ t=0$ , which can be seen as an identified initial condition for our data assimilation period.


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