Divergence-Free Motion Estimation

International audienceThis paper describes an innovative approach to estimate motion from image observations of divergence-free flows. Unlike most state-of-the-art methods, which only minimize the divergence of the motion field, our approach utilizes the vorticity-velocity formalism in order to construct a motion field in the subspace of divergence free functions. A 4DVAR-like image assimilation method is used to generate an estimate of the vorticity field given image observations. Given that vorticity estimate, the motion is obtained solving the Poisson equation. Results are illustrated on synthetic image observations and compared to those obtained with state-of-the-art methods, in order to quantify the improvements brought by the presented approach. The method is then applied to ocean satellite data to demonstrate its performance on the real images

Assimilation de données et méthodes adjointes pour la géophysique

Mathematical models are essential for understanding the dynamics of the atmosphere and ocean. But if they were our only source of information no forecast would be possible, mainly due to the lack of a consistent initial condition. On the other hand observation of these systems are available in a larger and larger quantity, thanks to the numerous observation satellites that now cruises around our planet. These observations are often indirect and incomplete, and therefore do not provide a thorough knowledge of the state of the observed systems. Finally statistics are available on the fields of atmospheric variables, their variability and consistency in time and space. This information is also part of the data that will be used. As we shall see later the use of these statistics is very important to improve the forecast. I therefore present, in this paper, methods for combining all or part of this information in order to improve forecasts and knowledge of the behaviour of such systems. These methods are based most often on a solid mathematical theory, but applying them in a realistic setting is not always easy. That is why I think important to accompany the developments we make until operational applications or quasi-operational, in order to demonstrate their feasibility.Les modèles mathématiques sont importants pour la compréhension de la dynamique de l'atmosphère et de l'océan. Mais si ils étaient notre seule source d'information aucune prévision ne serait possible faute, notamment au manque de la connaissance d'une condition initiale cohérente. On dispose également d'observations de ces systèmes en nombre de plus en plus important, notamment grâce aux nombreux satellites d'observation qui croisent maintenant au large de notre planète. Ces observations sont souvent indirectes et incomplètes, et de ce fait ne fournissent pas non plus, à elles seules, une connaissance approfondie de l'état du milieu considéré. Et pour finir, on dispose de statistiques sur les champs des variables atmosphériques, leur variabilité, leur cohérence en temps et en espace. Je présente donc, dans ce document, des méthodes permettant de combiner tout ou partie de ces informations afin d'améliorer la prévision et la connaissance du fonctionnement de ces systèmes. Ces méthodes se basent le plus souvent sur une théorie mathématique solide, mais les appliquer dans un contexte réaliste n'est pas toujours chose aisée. C'est pourquoi on gardera le souci d'accompagner les développements que nous effectuons jusqu'à des applications opérationnelles ou quasi opérationnelles afin de démontrer la faisabilité de ceux ci

Variational assimilation of fluid motion from image sequence

International audienceIn this paper, a variational technique derived from optimal control theory is used in order to realize a dynamically consistent motion estimation of a whole fluid image sequence. The estimation is conducted through an iterative process involving a forward integration of a given dynamical model followed by a backward integration of an adjoint evolution law. By combining physical conservation laws and image observations, a physically grounded temporal consistency is imposed, and the quality of the motion estimation is significantly improved. The method is validated on two synthetic image sequences provided by numerical simulation of fluid flows and on real world meteorological examples