Development of NAVDAS-AR: non-linear formulation and outer loop tests

NAVDAS-AR is the 4DVAR extension of the United States Navy's operational 3DVAR data assimilation system, NAVDAS (Naval Research Laboratory Atmospheric Variational Data Assimilation System), where AR stands for accelerated representer. Like NAVDAS, NAVDAS-AR is cast in observation space, and is an observation space formulation of the 4DVAR algorithm, in contrast to the European Centre for Medium-Range Weather Forecasts 4DVAR system, which is an analysis space algorithm. In this paper we show how an inner loop linear solution can be augmented by an outer loop iteration procedure that introduces non-linear effects in the form of a first-order Taylor series expansion. The non-linear problem is then solved iteratively as a sequence of linear problems. We show results of the outer loop strategy, where the first loop, the `linear' solution, is modified in the second outer loop by both a Taylor series term in the calculation of the observation innovations, and a linearized form of the background model that is linearized about an analysis `trajectory' from the solution of the first loop. We also show results of how the outer loop strategy allows better assimilation of observations of surface wind speed and precipitable water, compared to the operational NAVDAS.