Representer-based variational data assimilation in a spectral element shallow water model on the cubed-sphere grid

A representer-based variational data assimilation system is newly developed for the spectral element shallow water model in the High Order Method Modeling Environment. This study includes the development of tangent linear and adjoint codes and a background error covariance model. The correctness of the developed codes were checked by various ways such as linearity tests for tangent linear codes, adjoint tests for adjoint codes and symmetric tests for representer functions, which are four-dimensional covariance functions in observation-space. Then, direct and indirect representer-based data assimilation systems were constructed and evaluated by performing a series of identical twin experiments, where synthetic data were obtained from a reference run (nature run) and assimilated to correct initial conditions. The characteristics of the covariance model according to the different horizontal scales were evaluated by a suite of single-observation experiments. The results show satisfactory behaviours for both direct and indirect representer-based variational data assimilation methods, which indicates that they are ready to be further developed as a full-fledged four-dimensional variational data assimilation system as next step.