one state of a deterministic nonlinear system can result in large differences in a later state. At any update time, data assimilation usually takes a forecast (also known as the first guess, or background information) and applies a correction to the forecast based on a set of observed data and estimated errors that are present in both the observations and the forecast itself. The difference between the forecast and the observations at that time is called the departure or the innovation (as it provides new information to the data assimilation process). A weighting factor is applied to the innovation to determine how much of a correction should be made to the forecast based on the new information from the observations. The best estimate of the state of the system based on the correction to the forecast determined by a weighting factor times the innovation is called the analysis. In one dimension, computing the analysis could be as simple as forming a weighted average of a forecasted and observed value. In multiple dimensions the problem becomes more difficult. Much of the work in data assimilation is focused on adequately estimating the appropriate weighting factor based on intricate knowledge of the errors in the system. The measurements are usually made of a real-world system, rather than of the model's incomplete representation of that system, and so a special function called the observation operator (usually depicted by h() for a nonlinear operator or H for its linearization) is needed to map the modeled variable to a form that can be directly compared with the observation. Data assimilation as statistical estimation One of the common mathematical philosophical perspectives is to view data assimilation as a Bayesian estimation problem. From this perspective, the analysis step is an application of Bayes' theorem and the overall assimilation procedure is an example of recursive Bayesian estimation. However, the probabilistic analysis is usually simplified to a computationally feasible form. Advancing the probability distribution in time would be done exactly in the general case by the Fokker–Planck equation, but that is not feasible for high-dimensional systems; so, various approximations operating on simplified representations of the probability distributions are used instead. Often the probability distributions are assumed Gaussian so that they can be represented by their mean and covariance, which gives rise to the Kalman filter. Many methods represent the probability distributions only by the mean and input some pre-calculated covariance.