# Hind Oubanas, Irstea

## On the model error treatment in variational DA using the ’nuisance’ parameter approach

##### Coauthors

Igor Gejadze, Victor Shutyaev#### Abstract:

Variational data assimilation problem is formulated as an optimal control problem for a model governed by nonlinear PDEs. The input data contains the observation and background errors. In addition, there is a ’model’ error, which may include errors of different nature. In hydraulic applications this error is usually associated with the uncertainties in model parameters.

The concept of ’nuisance parameters’ is well known both in the classical and Bayesian estimation theories. By definition, these are the parameters which affect the estimates of the control variables under interest (valuable controls), but have little practical value by themselves. A few methods for the treatment of such parameters currently exist.

Following the above concept we suggest a ’generalized’ observation covariance V to be used in the classical data assimilation cost function. It has been proved that, from the point of view of the estimation error in the valuable controls, this approach is equivalent to the control vector extension approach (if the tangent linear hypothesis is valid). The covariance V is represented by a set of its largest eigenvalue/eigenvector pairs. Since V depends on the current estimate of valuable controls, it may be needed to recompute these eigenpairs a few times (2-3) during the minimization procedure, which is particularly convenient to implement in the framework of the Gauss-Newton method.

Furthermore, considering the model error as a composition of valuable and nuisance components we introduce a novel mixed DA formulation, where the valuable component is included into the extended control vector, and the nuisance component is treated by introducing the corresponding generalized observation covariance. This treatment allows the model error time evolution to be accounted for, while keeping the computationally feasible control vector size. The developed approach has been verified by numerical tests involving the nonlinear convection-diffusion-reaction equation.