The focus in this project is on environmental data assimilation -- broadly defined as the process of characterizing the state of an environmental system, using all relevant information. A class of sequential data assimilation techniques known as ensemble methods provides a promising way to deal with the issues of nonlinearity, high dimensionality, and uncertainty, which are all important in environmental applications. Ensemble methods effectively construct low dimensional approximations of complex nonlinear systems from a relatively small number of random replicates (model simulations driven by random initial conditions and model inputs). The approximations used by ensemble data assimilation methods can be viewed as projections of the original state space on continually changing random subspaces. These subspaces have structure that reflects the influence of physical constraints. But there is also a strong element of chance in the approximation process. This randomness seems to provide robustness and flexibility that cannot be so readily achieved with deterministic approximation methods, especially for large nonlinear problems. Although ensemble data assimilation is being applied to environmental problems with increasing frequency there is no body of theory that explains when or why it works. In particular, researchers do not understand why projections onto random linear subspaces can yield improved approximations for highly nonlinear problems. This project will advance understanding of ensemble estimation and provide a sound theoretical basis for the application and improvement of ensemble methods. It will combine theoretical analysis and computational experiments in a staged approach that gradually moves from simple to more complex problems. The project will rely on recent mathematical advances in random matrix theory and on the availability of parallel computing resources that are especially well-suited to ensemble applications.
The earth sciences are in the midst of an important transformation, due in large part to a dramatic increase in the quantity and quality of available information. Remote sensing has already had a significant impact on meteorology, hydrology and oceanography. Assimilation of all this new information is a challenging task. Nevertheless, the potential benefits are substantial, particularly at a time when climate change, population pressures on natural resources, and major modifications in global element cycles are attracting increasing attention. This project will investigate very efficient but poorly understood methods for assimilating and merging large amounts of diverse satellite data. The mathematical issues to be investigated are relevant to applications ranging from cell phone networks to oceanography. This project will support two major education/outreach activities: 1) involvement of undergraduate researchers in development of mathematical teaching laboratories and in earth science research projects, 2) a public outreach partnership with the Current Science & Technology Center (CS&T) at the Boston Museum of Science (including exhibits, multimedia presentations, and hands-on workshops). All of these programs will bring research results to a broad and diverse audience.
