Hybrid 4DVAR with a local ensemble tangent linear model: Application to the Shallow-Water Model

An ensemble-based tangent linear model (TLM) is described and tested in data assimilation experiments using a global shallow-water model (SWM). A hybrid variational data assimilation system was developed with a 4D variational (4DVAR) solver that could be run either with a conventional TLM or a local ensemble TLM (LETLM) that propagates analysis corrections using only ensemble statistics. An offline ensemble Kalman filter (EnKF) is used to generate and maintain the ensemble. The LETLM uses data within a local influence volume, similar to the local ensemble transform Kalman filter, to linearly propagate the state variables at the central grid point. After tuning the LETLM with offline 6-h forecasts of analysis corrections, cycling experiments were performed that assimilated randomly located SWM height observations, based on a truth run with forced bottom topography. The performance using the LETLM is similar to that of the conventional TLM, suggesting that a well-constructed LETLM could free 4D variational methods from dependence on conventional TLMs. This is a first demonstration of the LETLM application within a context of a hybrid-4DVAR system applied to a complex two-dimensional fluid dynamics problem. Sensitivity tests are included that examine LETLM dependence on several factors including length of cycling window, size of analysis correction, spread of initial ensemble perturbations, ensemble size, and model error. LETLM errors are shown to increase linearly with correction size in the linear regime, while TLM errors increase quadratically. As nonlinearity (or forecast model error) increases, the two schemes asymptote to the same solution.