## Section: New Results

### Tracking and data assimilation

#### Stochastic filtering for fluid motion tracking

Participants : Sébastien Béyou, Anne Cuzol, Sai Gorthi, Etienne Mémin.

We investigated the study of a recursive Bayesian filter for tracking velocity fields of fluid flows. The filter combines an Ito diffusion process associated to 2D vorticity-velocity formulation of Navier-Stokes equation and discrete image error reconstruction measurements. In contrast to usual filters designed for visual tracking problems, our filter combines a continuous law for the description of the vorticity evolution with discrete image measurements. We resort to a Monte-Carlo approximation based on particle filtering. The designed tracker provides a robust and consistent estimation of instantaneous motion fields along the whole image sequence.

When the likelihood of the measurement can be modeled as Gaussian law, we have also investigated the use of the so-called ensemble Kalman filtering for fluid tracking problems. This kind of filters introduced for the analysis of geophysical fluids is based on the Kalman filter update equation. Nevertheless, unlike traditional Kalman filtering setting, the covariances of the estimation errors, required to compute the so-called Kalman gain, relies on an ensemble of forecasts. Such a process gives rise to a Monte Carlo approximation for a family of non-linear stochastic filters enabling to handle state spaces of large dimension. We have recently proposed an extension of this technique that combines sequential importance sampling and the propagation law of a n ensemble Kalman filter. This technique leads to an ensemble Kalman filter with an improved efficiency. This year we have investigated the introduction of a nonlinear direct image measurements operator within this ensemble Kalman scheme. This modification of the filter provides very good results on 2D numerical and experimental flows even in the presence of strong noises. We are currently assessing its application to oceanic satellite images for the recovering of ocean streams. We are studying also the impact on the stochastic dynamics of turbulent noise defined as auto-similar Gaussian random fields and the introduction within an incremental ensemble analysis scheme of multiscale motion measurements. This work has been published in conference issues [28] , [27] , [35] .

#### Reduced-order models for flows representation from image data

Participants : Patrick Héas, Cédric Herzet, Etienne Mémin, Véronique Souchaud.

One of the possibilities to neglect the influence of some degrees of freedom over the main characteristics of a flow consists in representing it as a sum of $K$-orthonormal spatial basis functions weighted with temporal coefficients. To determine the basis function of this expansion one of the usual approaches relies on the Karhunen-Loeve decomposition (refered as proper orthogonal decomposition – POD – in the fluid mechanics domain). In practice, the spatial basis functions, also called modes, are the eigen vectors of an empirical auto-correlation matrix which is built from “snapshots" of the considered physical process.

In this axis of work we focus on the case where one does not have a direct access to snapshots of the considered physical process. Instead, the POD has to be built from the partial and noisy observation of the physical process. Instances of such scenarios include situations where real instantaneous vector-field snapshots are estimated from a sequence of *images*. We have been working on several approaches dealing with such a new paradigm. A first approach consists in extending standard penalized motion-estimation algorithms to the case where the sought velocity field is constrained to span a low-dimensional subspace [38] . Giving a probabilistic interpretation to this problem, we have designed novel optimization procedures in the framework of maximum a posteriori estimation problem. This work has led to the publication of a paper in the Gretsi conference. We are currently working on an EM-algorithm implementation of this approach.

In a second approach, we are considering the design of the POD as the solution of a minimum least squares estimation problem based on the distribution of the (unknown) velocity field given a sequence of images. This alternative formulation allowed us to take explicitly the uncertainty on the velocity field into account into our optimization process. We are currently working on several practical implementations of this problem, relying on Monte-Carlo integration and Krylov subspaces.

In a third axis we have studied two variational data assimilation techniques for the estimation of low order dynamical models for fluid flows. Both methods are built from optimal control recipes and rely on POD representation associated to Galerkin projection of the Navier Stokes equations. The proposed techniques differ in the control variables they involve. The first one introduces a weak dynamical model defined only up to an additional uncertainty time dependent function whereas the second one, handles a strong dynamical constraint in which the coefficients of the dynamical system constitute the control variables. Both choices correspond to different approximations of the relation between the reduced basis on which is expressed the motion feld and the basis components that have been neglected in the reduced order model construction. The techniques have been assessed on numerical data and for real experimental conditions with noisy Image Velocimetry data. This work has been presented in several conferences. A journal paper has been recently accepted with minor changes to the journal of computational Physics

#### Optimal control techniques for the coupling of large eddy dynamical systems and image data

Participants : Dominique Heitz, Etienne Mémin, Cordelia Robinson, Yin Yang.

This work aims at investigating the use of optimal control techniques for the coupling of Large Eddies Simulation (LES) techniques and 2D image data. The objective is to reconstruct a 3D flow from a set of simultaneous time resolved 2D image sequences visualizing the flow on a set of 2D plans enlightened with laser sheets. This approach will be experimented on shear layer flows and on wake flows generated on the wind tunnel of Irstea Rennes. Within this study we whish also to explore techniques to enrich large-scale dynamical models by the introduction of uncertainty terms or through the definition of subgrid models from the image data. This research theme is related to the issue of turbulence characterization from image sequences. Instead of predefined turbulence models, we aim here at tuning from the data the value of coefficients involved in traditional LES subgrid models or in longer-term goal to learn empirical subgrid models directly from image data. An accurate modeling of this term is essential for Large Eddies Simulation as it models all the non resolved motion scales and their interactions with the large scales.

First tests have been conducted with two-dimensional Direct Numerical Simulations (DNS) of mixing layer coupled with noisy observations. By modifying the initial condition of the system, the proposed method recovers the state of an unknown function with good accuracy. This work has been published in the International Symposium on Turbulence and Shear Flow Phenomena (TSFP) 2011 [29] .

#### Free surface flows reconstruction and tracking

Participants : Benoît Combes, Dominique Heitz, Etienne Mémin.

Characterising a free-surface flow (space and time-dependent velocity and geometry) given observations/measures at successive times is an ubiquitous problem in fluid mechanic and in hydrology. Observations can consist of e.g. measurements of velocity, or like in this work of measurements of the geometry of the free-surface. Indeed, recently developed depth/range sensors allow to capture directly a rough 3D geometry of surfaces with high space and time resolution. The main purpose of this study is to evaluate the ability of the Kinect sensor to estimate time-dependent 3D free-surface geometries. Then, based on these observations and on a stochastic data assimilation method, we want to estimate both time dependent geometry and displacement field associated to a free-surface flow from a simple temporal sequence of Kinect data. This year we have demonstrated on real data the possibility to measure free surface flow geometry with kinect sensor and on synthetic data to estimate both time dependent geometry and displacement field associated to a free-surface flow from a simple temporal sequence of Kinect-like data. This work has been published in the new conference Flow Volume Reconstruction [22] . We intend to extend such a study to hydrological applications.

#### Stochastic filtering technique for the tracking of closed curves

Participants : Christophe Avenel, Etienne Mémin.

We have proposed a filtering methodology for the visual tracking of closed curves. Opposite to works of the literature related to this issue, we consider here a curve dynamical model based on a continuous time evolution law with different noise models. This led us to define three different stochastic differential equations that capture the uncertainty relative to curve motions. This new approach provides a natural understanding of classical level-set dynamics in terms of such uncertainties. These evolution laws have been combined with various color and motion measurements to define probabilistic state space models whose associated Bayesian filters can be handled with particle filters. This on going work will be continued within extensive curve tracking experiments and extended to the tracking of other very high dimensional entities such as vector fields and surfaces. This work has been published in conference proceedings and a journal article is conditionally accepted to minor changes for publication in a meteorological journal.

#### Sequential smoothing for fluid motion

Participants : Anne Cuzol, Etienne Mémin.

In parallel to the construction of stochastic filtering techniques for fluid motions, we have proposed a new sequential smoothing method within a Monte-Carlo framework. This smoothing aims at reducing the temporal discontinuities induced by the sequential assimilation of discrete time data into continuous time dynamical models. The time step between observations can indeed be long in environmental applications for instance, and much longer than the time step used to discretize the model equations. While the filtering aims at estimating the state of the system at observations times in an optimal way, the objective of the smoothing is to improve the estimation of the hidden state between observation times. The method is based on a Monte-Carlo approximation of the filtering and smoothing distributions, and relies on a simulation technique of conditioned diffusions. The proposed smoother can be applied to general non linear and multidimensional models. It has been applied to a turbulent flow in a high-dimensional context, in order to smooth the filtering results obtained from a particle filter with a proposal density built from an Ensemble Kalman procedure.

#### Stochastic fluid flows dynamics under Gaussian uncertainty

Participant : Etienne Mémin.

In this research axis we aim at devising stochastic Eulerian expression for the description of fluid flow evolution laws incorporating uncertainty on the particles location. Such an uncertainty modeled through the introduction of a random term allows taking into account approximations or truncation effects performed within the dynamics analytical constitution steps. This includes for instance the modeling of unresolved scales interaction in large eddies simulation (LES) or in Reynolds average numerical simulation (RANS), but also uncertainties attached to non uniform grid discretization. This model is mainly based on a stochastic version of the Reynolds transport theorem. Within this framework various simple expressions of the mean drift component can be exhibited for different models of the random field carrying the uncertainties we have on the flow. We aim at using such a formalization within image based data assimilation framework and to derive appropriate stochastic versions of geophysical flow dynamical modeling.

#### Variational assimilation of images for large scale fluid flow dynamics with uncertainty

Participants : Souleymane Kadri Harouna, Etienne Mémin.

In this work we explore the assimilation of a large scale representation of the flow dynamics with image data provided at a finer resolution. The velocity fields at large scales is described as a regular smooth components whereas the complement component is a highly oscillating random velocity field defined on the image grid but living at all the scales. Following this route we have started to assess the performances of a variational assimilation technique with direct image data observation. Preliminary results obtained for a wavelet based 2D Navier Stokes implementation and images of a passive scalar transported by the flow are very encouraging.