Structure functions correspond to the background error covariance functions. They play a key role in the current data assimilation schemes used in operational centers. In particular these functions contribute to spread the information from the observations. The spatio-temporal variability of these functions is highly flow dependent. The structure functions can be estimated from an ensemble method e.g. from an ensemble of perturbed assimilations.

However the huge memory size that would be required to store these functions is beyond the capacity of modern super-computer. This issue can be solved by modeling structure functions. A possible model is based on a diffusion operator. Actually it is known that solution of the diffusion equation with initial state concentrated at a point, on the real line, is a Gaussian. This kind of function can be used as a preliminary model of structure functions. However the diffusion coefficient is not known ; more generally the local diffusion tensor has to be estimated.

This issue has been recently resolved by estimating the local diffusion tensor directly from an ensemble. This method has been successfully tested in order to represent horizontal component of the background error correlation functions in the frame of the MOCAGE-PALM chemical global assimilation system. This work has been done with researchers from the CERFACS. The method has also been tested to represent the transport uncertainty in CO2 fluxes retrieval at mesoscale. The local diffusion tensor have been estimated from a PEARP   ensemble.

Left panel: horizontal component of ozone forecast error correlation functions estimated form an ensemble over one month (during Austral winter). Right panel: the correlation obtained from the model based on diffusion operator whose local diffusion tensor have been estimated from the same previous ensemble.