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