Covariance kernels investigation from diffusive wave equations for data assimilation in hydrology

In data assimilation (DA), the estimation of the background error covariance operator is a classical and still open topic. However, this operator is often modeled using empirical information. In order to exploit at best the potential of the knowledge of the physics, the present study proposes a method to derive covariance operators from the underlying equations. In addition, Green's kernels can be used to model covariance operators and are naturally linked to them. Therefore, Green's kernels of equations representing physics can provide physically-derived estimates of the background error covariance operator, and also physically-consistent parameters. In this context, the present covariance operators are used in a variational DA (VDA) process of altimetric data to infer bathymetry in the Saint-Venant equations. In order to investigate these new physically-derived covariance operators, the associated VDA results are compared to the VDA results using classical operators with physically-consistent and arbitrary parameters. The physically-derived operators and physically-consistent exponential operator provide better accuracy and faster convergence than empirical operators, especially during the first iterations of the VDA optimization process.