Data assimilation refers to any methodology that uses partial
observational data and the dynamics of a system for estimating the
model state or its parameters. We consider here a non classical
approach to data assimilation based in null controllability
introduced in [Puel, C. R. Math. Acad. Sci. Paris335 (2002) 161–166] and [Puel, SIAM J. Control Optim.48 (2009) 1089–1111] and we apply it to oceanography.
More precisely, we are interested in developing this methodology
to recover the unknown final state value (state value at the end of the measurement period) in a quasi-geostrophic
ocean model from satellite altimeter data, which allows in fact to
make better predictions of the ocean circulation. The main idea of
the method is to solve several null controllability problems for the adjoint system in
order to obtain projections of the final state on a reduced basis.
Theoretically, we have to prove the well posedness of the
involved systems associated to the method and we also need an
observability property to show the existence of null controls for the adjoint system. To
this aim, we use a global Carleman inequality for the associated
velocity-pressure formulation of the problem which was previously
proved in [Fernández-Cara et al., J. Math. Pures Appl.83
(2004) 1501–1542]. We present numerical simulations using a regularized
version of this data assimilation methodology based on null
controllability for elements of a reduced spectral basis.
After proving the convergence of the regularized solutions, we
analyze the incidence of the observatory size and noisy data in
the recovery of the initial value for a quality prediction.