Data Assimilation is the process of melding physical models with observations of the environment. As with many things in science, there are numerous ways to attack this problem. One of the preferred ways in meteorology is now a dynamic assimilation methods referred to as Kalman filtering. I will try to present a very VERY simplistic overview of the technique, so please do not email me with corrections if you happen to have a PhD and want to tell me how over simplified this page is. There are a few basic steps in a Kalman filter that will help you understand how they work:

Kalman Filtering Steps:

1) Take a best estimate of the environment, and use a numerical model to predict forward one timestep. 2) At the next timestep, you can now take the observations you have, convert them into environmental data, and combine them with the prediction in such a way that the errors in the observation system are weighed against the errors in the prediction model, so that a "optimal" estimate can be made. 3) now you can run the model forward one step using the current best estimate. 4) repeat steps 2-3 until the assimilation period is complete.

The following images is a graphical representation of an Ensemble Filter, where multiple model predictions are used in the filtering process, decreasing random errors and allowing computational speedup by reducing the rank of the Kalman Gain Matrix.

This image taken from "Dynamic Data Assimilation" by John M. Lewis, S. Lakshmivarahan, and Sudarshan Dhall.

Additional resources about Kalman filter research in meteorology can be found here.