We tackle the problem of coupling a geophysical simulation model with data coming from image processing. It needs to define the image observation space and to design an operator to transform results from the image space to the model space. In this study, we use a shallow-water oceanographic circulation model developed at MHI. We propose a processing chain first based on an image processing step relying on a dedicated motion estimation operator, and then a data assimilation step of the estimated velocity. We illustrate the method on different results without and with assimilation. OBJECTIVESIn the framework of numerical forecasting for the evolution of geophysical fluid, we are interested in the assimilation of data coming from images. In order to forecast the behavior of geophysical fluids we need: a forecast model to describe the evolution of a state variable (generally it is a non-linear PDE system) and observations spatially and temporally distributed. Data assimilation provides a mathematical solution to combine data and models. During last decades model quality increased significantly. But forecast quality is not directly linked to model quality. To increase forecast quality it is also necessary to increase the amount and the quality of observations. Therefore, images -particularly images coming from spatial remote sensing -provide a huge amount of information. Using images in a data assimilation framework raises several difficulties:1. First it is necessary to define which image space is relevant according to model specialists.2. Then, it is necessary to construct an image operator dedicated to the problematic and coherent to the physical behavior. Moreover it is necessary to define a set of norm in order to quantify the influence of image information along the assimilation process.3. And finally, it is necessary to construct an operator to compare image space and model space.In the study presented in this paper, we are interested in oceanographic circulation forecasting. Oceanographic circulation is ruled by fluid mechanic. Most of oceanographic circulation models are heavy 3D models based on primitive equations [1]. They correspond to an approximation of Navier-Stokes equation associated to a nonlinear state equation coupling salinity, temperature and 3D velocity. Nevertheless, it exists simplified models based on shallow-water approximation [2,3]. They rely on a so called 1.5 layer representation of the ocean: the sea surface is represented by a mixed layer interfaced to the atmosphere and a deeper layer. Equation ruling the circulation are then:

Apparent motion estimated on satellite data is used for example to compute the wind field in meteorology, and surface currents in oceanography. The satellite images display turbulent fluids with strong rotational patterns at different spatial and temporal scales. This specificity necessitates devising adapted methods, allowing to control the divergence and curl of the retrieved motion field. Vector spline methods are very adapted to that purpose. The vector spline problem is defined as finding a motion field that satisfies a temporal conservation equation at selected control points and that minimizes a regularity constraint in all the image domain. An exact solution of this problem can be found for the 2nd order div-curl regularity constraint. The retrieval of the solution does not require an iterative minimization procedure: a dense matrix must be inverted to compute the spline's coefficients. This matrix unfortunately becomes large and ill-conditioned as the number of control points increases, making the vector spline approach unsuitable for processing large satellite images. This paper presents a method called "Partition of Unity and Optical Flow" (PUOF), based on a decomposition of the spatial domain: local vector splines are computed in subdomains of the image, then merged using a partition of unity algorithm. The resulting motion field is a good approximation of the exact vector spline solution, and its retrieval is numerically stable and computationally affordable even when processing large data sets, as demonstrated by results obtained on sequences of synthetic and meteorological images.

International audienceSatellite image sequences visualize important patterns of the atmospheric and oceanographic circulation. Assessing motion from these data thus has a strong potential for improving the performances of the forecast models. Representing a vector field by a vector spline has been proven efficient for fluid motion assessment: the vector spline formulation makes it possible to initially select the locations where the conservation equation has to be taken into account; it efficiently implements the 2nd order div-curl regularity, advocated for turbulent fluids. The scientific contribution of this article is to formulate vector splines in a multiscale scheme, with the double objective of assessing motion even in the case of large displacements and capturing the spectrum of spatial scales associated to turbulent flows. The proposed method only requires the inversion of a band matrix, which is performed by an efficient numerical scheme making the method tractable for large satellite image sequences