This study aims at the theoretical development of a method of “four-dimensional analysis,” namely the numerical variational analysis. The three basic types of variational formalism in the numerical variational analysis method are discussed. The basic formalisms are categorized into three areas: (1) “timewise localized” formalism, (2) formalism with strong constraint, and (3) formalism with weak constraint. Exact satisfaction of selected prognostic equations were formulated as constraints in the functionals for the first two formalisms. However, only the second formalism contains explicitly the time variation terms in the Euler equations. The third formalism is characterized by the subsidiary condition which requires that the prognostic or diagnostic equations must be approximately satisfied. The variational formalisms and the associated Euler-Lagrange equations are obtained in the form of finite-difference analogs. In this article, the filtering of cach formalism and the uniqueness of solutions o... Abstract This study aims at the theoretical development of a method of “four-dimensional analysis,” namely the numerical variational analysis. The three basic types of variational formalism in the numerical variational analysis method are discussed. The basic formalisms are categorized into three areas: (1) “timewise localized” formalism, (2) formalism with strong constraint, and (3) formalism with weak constraint. Exact satisfaction of selected prognostic equations were formulated as constraints in the functionals for the first two formalisms. However, only the second formalism contains explicitly the time variation terms in the Euler equations. The third formalism is characterized by the subsidiary condition which requires that the prognostic or diagnostic equations must be approximately satisfied. The variational formalisms and the associated Euler-Lagrange equations are obtained in the form of finite-difference analogs. In this article, the filtering of cach formalism and the uniqueness of solutions o...