We first try our algorithm on simulated data. We consider a basic model, the shallow water model (or Saint-Venant's equations), representing quite well the temporal evolution of geophysical flows. This model is detailed in section 3.3.3 (with different parameters), or in [23].
This model is then coupled with an advection-diffusion equation, modeling the fact that the concentration of a passive tracer is transported by the fluid velocity:
 |
(4.18) |
is the concentration of the passive tracer (e.g. chlorophyll in oceans). We also add to this equation an initial condition 
. We consider then a trajectory of this shallow water model coupled with a concentration equation, from which a concentration image is extracted every 100 time steps (in order to reproduce the time sampling of the satellite images).
Two consecutive images are extracted from these simulated data, and we apply our algorithm to these two images, with the aim of identifying the entire velocity field. As shown in [23], our algorithm quickly extracts very accurate velocity fields. This is mainly due to the combination of a multi-grid approach and an efficient optimization scheme (no a priori information and no linearization). The registration between two images is almost perfect after a few iterations, and the identified velocity field reproduces very well the global structure of the true velocity (a rotating vortex in a translation field in our experiments).
Concerning the regularization, we can note that the best results correspond to the 
norm (see equation (4.5)). The most physical regularization is probably 
(see equation (4.6)), as we expect a null divergence velocity field in geophysical flows. But the decrease of the cost function is not as good as for some other regularizations. Considering that the images are acquired every 100 time steps only, the velocity we want to identify between these two images is a time Lagrangian integration of many instantaneous velocities, and it cannot have a divergence equal to zero.
We also present an interesting application of the identification process. Assume that we have a particular object in the first image, e.g. a characteristic structure, that has been manually selected. In our case, we can identify one specific vortex. We can then limit the identification process to a region around this object. This region is propagated from one pair of images to the next one by the mean of the identified velocity. This allows us to track this object in time, in a fully automatic way.