An earlier review paper of analysis methods for numerical weather prediction showed how many methods could be derived from the same basic analysis equation, expressing the Bayesian probability that any state is the true state. Prior probabilities depend on the error covariance of the background prior estimate, and these covariances figure in most solutions. This paper extends the analysis of iterative solution methods (one of which is known as the 'Successive Correction' method). In a simple example the methods are compared with a direct solution: the 'Optimal Interpolation' method. It is shown that the iterative methods can be equally 'optimal'. If the number of iterations is limited, or the background weighting is omitted, then the methods normalizing the increments in observation space are shown to be more reliable than those normalizing in grid-point space. The traditional successive-correction method has a grid-point space normalization. The covariances in these methods perform a filtering function. It is possible to replace them by a filter acting on the analysis increments. Iterative methods using such a filter are derived, and are shown to correspond exactly to the iterative methods using covariance functions. A simple and efficient recursive filter is described and applied to the same example. The analysis using a two-pass filter is almost identical to that using a 'second-order auto-regressive' covariance function. A filter with many passes corresponds to a Gaussian-shaped covariance function. Approximate filters can be devised to model the effect of observational-error correlations, with an accuracy adequate in view of the lack of knowledge of the real correlations. With this filter, the iterative methods can be extended to deal effectively with data from remote-sensing instruments. Published iterative methods (many originally empirically derived) are reviewed. and fitted into the optimal theory. Practical applications are discussed.
