Data Assimilation - Data Assimilation As Statistical Estimation

Data Assimilation As Statistical Estimation

In data assimilation applications, the analysis and forecasts are best thought of as probability distributions. The analysis step is an application of the 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 unrealistically expensive, so various approximations operating on simplified representations of the probability distributions are used instead. If the probability distributions are normal, they can be represented by their mean and covariance, which gives rise to the Kalman filter. However it is not feasible to maintain the covariance because of the large number of degrees of freedom in the state, so various approximations are used instead.

Many methods represent the probability distributions only by the mean and impute some covariance instead. In the basic form, such analysis step is known as optimal statistical interpolation. Adjusting the initial value of the mathematical model instead of changing the state directly at the analysis time is the essence of the variational methods, 3DVAR and 4DVAR. Nudging, also known as Newtonian relaxation or 4DDA, is essentially the same as proceeding in continuous time rather than in discrete analysis cycles (the Kalman-Bucy filter), again with imputing simplified covariance.

Ensemble Kalman filters represent the probability distribution by an ensemble of simulations, and the covariance is approximated by sample covariance.

In addition to weather forecasting, other uses of DA include trajectory estimation for the Apollo program, GPS, and atmospheric chemistry.