Design of control set in the framework of variational data assimilation

Solving data assimilation problems under uncertainty in driving conditions, model parameters and domain geometry may require a careful design of the control set. One task is to prevent oversizing which results into a dramatic rise of computational expenses. Another task is to avoid certain compositions of the control vector, for which the control problem becomes particularly ill-posed, or the robustness of the solution process is compromised.
In the first part, a method for quantifying the performance of a chosen (active) control set, being a subset of the full set of the model inputs, is presented. Based on this quantification, all possible compositions of the control vector can be ranked, and a safe and sufficient control set can be revealed. This method is generalized to the case of 'integral' controls, i.e. auxiliary inputs which may not coincide with the original model inputs. For example, as such one may consider the averaged or restricted values of the original spatially or temporally distributed model variables or their invariants. To illustrate the method, numerical experiments involving the fully nonlinear Saint-Venant hydraulic network model SIC2 are presented. In these experi- ments the inflow discharge estimation problem under uncertainty in the river bed level and the distributed bed roughness is considered.
In the second part, a method for implicit treatment of uncertainty in some model inputs by inflating the observation covariance matrix is presented. The equivalency theorem substantiating the method has been proved. This method embodies the idle control technique, which is similar to the 'nuisance parameter' treatment technique known in statistics. We also con- sider a case with a biased model error. In the corresponding mixed formulation, the spatially distributed mean error is included into the active control vector, whereas the time dependent fluctuations around the mean are subjected to the implicit treatment. Numerical experiments for the 1D generalized Burgers' equation illustrate the presented theory. In these experiments the model error related to uncertainty in the advection coefficient has been considered. Some results showing the efficiency of this method involving the hydraulic model SIC2 are also pre- sented.
Finally, the prospects of combining these two methods and the future work are discussed.

References:

Gejadze I. and Malaterre P.-O. Design of the control set in the framework of variational data assimilation. Journal of Computational Physics 325 (2016), pp. 358-379.

Gejadze I, Oubanas H., and Shutyaev V. Implicit treatment of model error using inflated obser- vation covariance. Quarterly Journal of the Royal Meteorological Society, (05/2017), accepted.