Problematic:
Current methods in data assimilation allow the computation of the first two moments (mean and variance) of some variables' probability density functions, once these methods are interpreted in a statistical framework. If the physical model is non-linear, it is necessary to search for a finer description of probability density functions.
Objectives:
Theoretically, the probability density describing the state of the system is given by the solution of the Fokker-Planck equation associated to the physical model. Obviously, it is not possible to solve this advection-diffusion equation if the model possesses a high degree of freedom. In that case, other methods must be devised.
The objective is, like in the context of environmental forecasting, to develop and apply these other methods. Ensemble forecasting, as it begins to be used in meteorology, is a possible way. In such a context, the use of methods like particular filtering (as they are developed in the Sigma2 project, future ASPI project) can be investigated, with an obstacle to solve: the systems' dimensionality.
Monte-Carlo methods, set on reduced models coming from multivariate representations of the output in function of the input variables, are possible and interesting approaches.