Bayesian probabilistic arguments are used to derive idealized equations for finding the best analysis for numerical weather prediction. These equations are compared with those from other published methods in the light of the physical characteristics of the NWP analysis problem; namely the predetermined nature of the basis for the analysis, the need for approximation because of large-order systems, the underdeterminacy of the problem when using observations alone, and the availability of prior relationships to resolve the underdeterminacy.

Prior relationships result from (1) knowledge of the time evolution of the model (which together with the use of a time distribution of observations constitutes four-dimensional data assimilation); (2) knowledge that the atmosphere varies slowly (leading to balance relationships); (3) other nonlinear relationships coupling parameters and scales in the atmosphere.

Methods discussed include variational techniques, smoothing splines, Kriging, optimal interpolation, successive corrections, constrained initialization, the Kalman-Bucy filter, and adjoint model data assimilation. They are all shown to relate to the idealized analysis, and hence to each other. Opinions are given on when particular methods might be more appropriate. By comparison with the idealized method some insight is gained into appropriate choices of parameters in the practical methods.