A null controllability data assimilation methodology applied to a large scale ocean circulation model^*

¹ Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago, Casilla 307, Correo 2, Santiago, Chile. galina.garcia@usach.cl
² Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3 Correo 3, Santiago and Centro de Modelamiento Matemático (UMI 2807 CNRS), FCFM Universidad de Chile, Santiago, Chile. axosses@dim.uchile.cl
³ Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin-en-Yvelines, 78035 Versailles, France. jppuel@math.uvsq.fr

Received: 7 August 2009
Revised: 19 April 2010

Abstract

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. Paris 335 (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.