A reduced order approach to four-dimensional variational data assimilation using proper orthogonal decomposition

Yanhua Cao, Jiang Zhu, Ionel Michael Navon, Zhendong Luo

Four dimensional variational data assimilation (4DVAR) is a powerful tool for data assimilation in meteorology and oceanography. However, a major hurdle in use of 4DVAR for realistic general circulation models is the dimension of the control space (generally equal to the size of the model state variable and typically of order 107 108) and the high computational cost in computing the cost function and its gradient that require integration model and its adjoint model). Current ways to obtain feasible implementations of 4D-Var consist mainly of the incremental method that consists in generating a succession of quadratic problems which can be solved in inner loop using a coarse resolution corrected by full model runs in few outer loops. However the method is characterized by the fact that the dimension of the control space remains very large in realistic applications.

In this paper, we propose a 4DVAR approach based on proper orthogonal decomposition (POD). POD is an efficient way to carry out reduced order modeling by identifying the few most energetic modes in a sequence of snapshots from a time-dependent system, and providing a means of obtaining a low-dimensional description of the system's dynamics. The POD based 4DVAR not only reduces the dimension of control space, but also reduces the size of dynamical model, both in dramatic ways. The novelty of our approach also consists in the inclusion of adaptability, applied when in the process of iterative control the new control variables depart significantly from the ones on which the POD model was based upon. In addition, these approaches also allow to conveniently constructing the adjoint model.

The proposed POD based 4DVAR methods are tested and demonstrated using a reduced gravity wave ocean model in Pacific domain in the context of identical twin data assimilation experiments. The results show that POD 4DVAR methods converge faster with a much smaller computational cost (less than 1/100 computer time of the full order 4DVAR). This study also shows that further research efforts in this direction are worth pursuing and may lead ultimately to a practical implementation in both operational NWP and ocean forecasts.