A stochastic filter for fluid motion tracking

In this paper we present a method for the tracking of fluid flows velocity fields. The technique we propose is formalized within sequential Bayesian filter framework. The filter we propose here combines an Itô diffusion process coming from a stochastic formulation of the vorticity-velocity form of Navier-Stokes equation and discrete measurements extracted from an image sequence. The resulting tracker provides robust and consistent estimations of instantaneous motion fields along the whole image sequence. In order to handle a state space of reasonable dimension for the s-tochastic filtering problem, we represent the motion field as a combination of adapted basis functions. The used basis functions ensue from a mollification of Biot-Savart integral and a discretization of the vorticity and divergence maps of the fluid vector field. The efficiency of the method is demonstrated on a long real world sequence showing a vortex launch at tip of airplane wing. 1

Combining analog method and ensemble data assimilation: application to the Lorenz-63 chaotic system

International audienceNowadays, ocean and atmosphere sciences face a deluge of data from space, in situ monitoring as well as numerical simulations. The availability of these different data sources offer new opportunities, still largely underexploited, to improve the understanding,modeling and reconstruction of geophysical dynamics. The classical way to reconstruct the space-time variations of a geophysical system from observations relies on data assimilation methods using multiple runs of the known dynamical model. This classical framework may have severe limitations including its computational cost, the lack of adequacy of the model with observed data, modeling uncertainties. In this paper, we explore an alternative approach and develop a fully data-driven framework, which combines machine learning and statistical sampling to simulate the dynamics of complex system. As a proof concept, we address the assimilation of the chaotic Lorenz-63 model. We demonstrate that a nonparametric sampler from a catalog of historical datasets, namely a nearest neighbor or analog sampler, combined with a classical stochastic data assimilation scheme, the ensemble Kalman filter and smoother, reach state-of-the-art performances, without online evaluations of the physical model

Vortex and source particles for fluid motion estimation

acuzol,memin¡ Abstract. In this paper we propose a new motion estimator for image sequences depicting fluid flows. The proposed estimator is based on the Helmholtz decomposition of vector fields. This decomposition consists in representing the velocity field as a sum of a divergence free component and a curl free component. In order to provide a low dimensional solution, both components are approximated using a discretization of the vorticity and divergence maps through regularized Dirac measure. The resulting so called irrotational and solenoidal fields consist then in linear combinations of basis functions obtained through a convolution product of the Green kernel gradient and the vorticity map or the divergence map respectively. The coefficient values and the basis function parameters are obtained as the minimizer of a functional relying on an integrated version of mass conservation principle of fluid mechanics. Results are provided on real world sequences.

Suivi de mouvement fluide par filtrage stochastique.

National audienceNous présentons dans cet article une méthode pour le suivi temporel de champs de déplacements décrivant des mouvements fluides. La technique proposée est formalisée dans un cadre de filtrage bayésien séquentiel. Le filtre que nous proposons combine un processus de diffusion de Itô issu d'une formulation stochastique de l'équation de Navier- Stokes sous sa forme vorticité-vitesse, et des observations discrètes extraites de la séquence d'images. L'algorithme de suivi qui en découle fournit des estimations précises et robustes des champs de déplacements instantanés tout au long de la séquence. En vue de définir un espace d'état de dimension raisonnable adaptée au problème de filtrage stochastique, nous représentons le champ de déplacements par une combinaison de fonctions de base adaptées. Ces fonctions de base sont issues de l'intégration de Biot-Savart à partir d'une discrétisation régularisée de la vorticité et de la divergence du champ de vecteurs. L'efficacité de la méthode est démontrée sur une longue séquence réelle montrant l'évolution de vortex générés à l'extrémité d'une aile d'avion

A stochastic filtering technique for fluid flows velocity fields tracking

International audienceIn this paper we present a method for the temporal tracking of fluid flows velocity fields. The technique we propose is formalized within a sequential Bayesian filtering framework. The filtering model combines an Itˆo diffusion process coming from a stochastic formulation of the vorticity-velocity form of the Navier-Stokes equation and discrete measurements extracted from the image sequence. In order to handle a state space of reasonable dimension, the motion field is represented as a combination of adapted basis functions, derived from a discretization of the vorticity map of the fluid flow velocity field. The resulting non linear filtering problem is solved with the particle filter algorithm in continuous time. An adaptive dimensional reduction method is applied to the filtering technique, relying on dynamical systems theory. The efficiency of the tracking method is demonstrated on synthetic and real world sequences

Image assimilation with the weighted ensemble Kalman filter

International audienceWe present a sequential data assimilation method based on the combination of two assimilation techniques: the ensemble Kalman filter (EnKF) and the particle filter. Both techniques are based on Monte Carlo sampling allowing approximate solving of non linear stochastic filtering problems. However, while the EnKF is still based on a Gaussian assumption, the particle filter does not rely on such an assumption but is known to be less efficient when the number of available samples is small. In practice, both techniques are combined in the sense that the sampling step of the particle filter is based on the EnKF technique, followed by a weighting of samples using observations. The association of these two approaches is a step toward an efficient application of ensemble techniques to high-dimensional and non linear / non Gaussian problems, such as those encountered in meteorology or oceanography. We show the performances of this new approach on high-dimensional problems where the goal is to filter turbulent velocity fields from image observations. The assimilation technique associates a non linear stochastic dynamical model to linear observations extracted from the image sequences, or directly to the image data through a non linear observation operator. The method has been validated on a synthetic sequence, and applied to real oceanographic satellite image sequences of SST (sea surface temperature). This work corresponds to an extension of the preliminary study published in Tellus (N. Papadakis, E. Memin, A. Cuzol, N. Gengembre. Data assimilation with the Weighted Ensemble Kalman Filter. Tellus A, vol.62(5), p. 673-697, 2010). In a second part, we present a way to improve the assimilation when the time step between observations (images) is very long. We make use of a conditional simulation technique in order to reduce dynamical discontinuities produced in that case by the sequential techniques