Impact of Formulation of Cost Function and Constraints on Three-Dimensional Variational Data Assimilation

Abstract

Three-dimensional variational data assimilation (3DVAR) analysis is an important method used at many operational and research institutes in meteorology, for example, the National Centers for Environmental Prediction (NCEP) and the European Centre for Medium-Range Weather Forecasts (ECMWF). In 3DVAR analysis, different forms of cost functions and constraints (e.g., geostrophic balance) have been used. However, the impacts of these different forms of cost functions, covariances, and constraints on the 3DVAR solutions have not been completely analyzed due to their complexity. Using the Fourier analysis where the Fourier transformation is applicable, the impacts of different forms of cost functions and some commonly used physical constraints are demonstrated. In the particular case of geostrophic balance as the constraint, the large-scale motion of a 3DVAR analysis could be in geostrophic balance, but the mesoscale solution may be nearly unchanged if the penalty terms and the forms of Jb (terms related to the background field in 3DVAR cost functions) and Jo (related to the observation field) are chosen properly. This conclusion shows that the penalization of geostrophic imbalance can be used for mesoscale data assimilation without serious damage to the mesoscale features. More important for constructing a 3DVAR system, this paper also demonstrates that some formulations of Jb can produce physically unexpected solutions. The theory is illustrated using numerical experiments.