Data assimilation is the process of combining observations from a wide variety of sources and forecast output from a weather prediction model. The resulting analysis is considered to be the 'best' estimate of the state of the atmosphere at a particular instant in time. The process of combining the observational and model information is accomplished within a Bayesian statistical framework where probability distributions associated with observations and forecasts are combined with dynamical constraints.
Most commonly, analyzed products are created using three-dimensional or four-dimensional (3DVAR, 4DVAR) variational data assimilation schemes. 3DVAR treats observations within some time interval about the target analysis time as occurring at the time of the analysis. 4DVAR uses the observations distributed about the target analysis time to estimate the value. (Very simplistically, think of a least squares fit of the observations.) An alternative 4DVAR approach is to use "nudging" where damped differences between the observations and the initial forecast guess, which is assumed to be 'perfect', are used to constrain the analysis toward the observations.
National Center for Atmospheric Research Staff (Eds). Last modified 20 Aug 2013. "The Climate Data Guide: Simplistic Overview of Reanalysis Data Assimilation Methods." Retrieved from https://climatedataguide.ucar.edu/climate-data/simplistic-overview-reanalysis-data-assimilation-methods.