I am employed by the Atmospheric Chemistry Division of the National Center for Atmospheric Research. For the last several years I have been working on applications of mathematical principals of estimation and inverse problem theory to large-scale, non-linear problems arising in numerical modeling of the Earth's atmosphere. This subject is often referred to as data assimilation. The overall goal of atmospheric data assimilation is to integrate observations, which are usually sparse and irregular, and theoretical models to derive a coherent global picture of the atmosphere and  to predict its evolution.The mathematical basis of data assimilation is estimation theory or inverse problem theory:

"Humans were naked worms; yet they had an internal model of the world. In the course of time up to the present, this model has been updated many times, following the development of new experimental possibilities or the development of their intellect. Sometimes the updating has been only quantitative, sometimes it has been qualitative. Inverse problem theory tries to describe the rules human beings should use for quantitative updatings."

Or, if you prefer, "Inverse theory is an organized set of mathematical techniques for reducing data to obtain useful information about physical world on the basis of inferences drawn from observations.

Main ideas of estimation theory were developed in 1940s by Norbert Weiner in the US and Andrey Kolmogorov in Russia. Weiner's interest in the subject originated from his assignment to develop an optimal mathematical strategy for shooting down German war planes during WWII.

  

Since then, ideas and methods of estimation theory have been extensively used in mechanical and electrical engineering, navigation, robotics, financial modeling. systems design, numerical weather prediction and other areas. Fairly recently, people started applying the same principles to modeling atmospheric photochemistry, and that has been the main area of my current research.