Prediction and error propagation for geophysical fluids

International audiencePredicting the evolution of geophysical fluids (ocean or atmosphere) has a great societal impact and therefore it's important to improve at best the prediction. But it makes sense only if we are able to product an information on the quality and fiability of the prediction. There are many types of errors: error on the model because it is an approximation of the true fluids, error on the observations and also error on the past prediction. In the algorithm of data asimilation these errors are mixed and produce an error on the retrieved initial condition that will be propagated to the prediction. Based on the theory of Optimal Control for partial differential equations, the variational approach for the assimilation of data gather all the available information in the so-called Optimality System (O.S.) which is the Euler-Lagrange equation of the optimization problem. Errors can be injected in the O.S. and we will be able to follow the propagation of the errors thanks to the Second Order Adjoint. In the talk we will establish a relation between the Hessiian of the cost function applied to the error on the optimal initial condition and the error of model and the error of observation. We will see how to estimate the covariance of the initial condition and to have an efficient choice of perturbations for "ensemble" prediction. A numerical example will be shown with an advection-diffusion type equation in the cases when advection (i.e. non linear processes) dominates and when diffusion (i.e. linear processes) dominates