Section: New Results

Image assimilation

Sequences of images, such as satellite acquisitions, display structures evolving in time. This information is recognized of major interest by forecasters (meteorologists, oceanographers, etc.) in order to improve the information provided by numerical models. However, the satellite images are mostly assimilated in geophysical models on a point-wise basis, discarding the space-time coherence visualized by the evolution of structures such as clouds. Assimilating in an optimal way image data is of major interest and this issue should be considered in two ways:

• from the model's viewpoint, the location of structures on the observations is used to control the state vector.

• from the image's viewpoint, a model of the dynamics and structures is built from the observations.

Model error and motion estimation

Participants : Isabelle Herlin, Dominique Béréziat [UPMC] .

Data assimilation technics are used to retrieve motion from image sequences. These methods require a model of the underlying dynamics, displayed by the evolution of image data. In order to quantify the approximation linked to the chosen dynamic model, an error term is included in the evolution equation of motion and a weak formulation of 4D-Var data assimilation is designed. The cost function to be minimized depends simultaneously on the initial motion field, at the beginning of the studied temporal window, and on the error value at each time step. The result allows to assess the model error and analyze its impact on motion estimation. The approach is used to estimate geophysical forces (gravity, Coriolis, diffusion) from images in order to better assess the surface dynamics and forecast the displacement of structures like oilspill.

Tracking of structures from an image sequence

Participants : Isabelle Herlin, Yann Lepoittevin, Dominique Béréziat [UPMC] .

The research concerns an approach to estimate velocity on an image sequence and simultaneously segment and track a given structure. It relies on the underlying dynamics' equations of the studied physical system. A data assimilation method is designed to solve evolution equations of image brightness, those of motion's dynamics. The method is for instance applied on meteorological satellite data, in order to track tropical clouds on image sequences and estimate their motion, as seen on Fig. 2 .

Data assimilation is performed either with a 4D-Var variational approach or with a Kalman ensemble method. In the last case, the initial ensemble is obtained from a set of optical flow methods of the literature with various parameters values.

Various ways for representing the structures are studied and compared.

• For the variational approach, we consider: 1- a distance map modeling the tracked structures, which is added to the state vector, 2- anisotropic regularization terms, which are added to the cost function minimized during the assimilation process, 3- covariances between pixels, which are included in the background error covariance matrix.

• For the filtering approach, we focus either on domain decomposition or on explicit localization, which are both related to the displayed structures.

Applying POD on a model output dabase for defining a reduced motion model

Participants : Isabelle Herlin, Etienne Huot.

Dimension reduction may be obtained by determining a small size reduced basis computed by Proper Orthogonal Decomposition (POD) of a motion fields database and applying the Galerkin projection. This database is constructed for characterizing accurately the surface circulation of the studied area, so that linear combinations of the basis elements obtained by POD accurately describe the motion function observed on satellite image sequences. The database includes the geostrophic motion fields obtained from Sea Level Anomaly reanalysis maps that are available from the MyOcean European project website ( http://marine.copernicus.eu/ ). Fig. 3 displays such SLA maps and the associated motion fields.

Image assimilation with the POD reduced model allows estimating motion as displayed on Fig. 4 .

Rain nowcasting from radar image acquisitions

Participants : Isabelle Herlin, Yann Lepoittevin.

This research concerns the design of an operational method for rainfall nowcasting that aims at mitigating flash floods. The nowcasting method is composed of two main components:

• a data assimilation method, based on radar images, estimates the state of the atmosphere: this is the estimation phase.

• a forecast method uses this estimation to extrapolate the state of the atmosphere in the future: this is the forecast phase.

The method is transfered to the industrial company Weather Measures.

Current research concerns the use of object components in the state vector in order to get an improved motion estimation and a better localization of endangered regions. Assimilation of pluviometers measures in the nowcasting system is also investigated.