**Nonlinear/Non-Gaussian Time Series Estimation
**

Juan M. Restrepo

Uncertainty Quantification Group, Mathematics, Physics, and Atmospheric Sciences, University of Arizona

State estimation techniques are used in weather and climate prediction,
hydrogeology, seismology, as a way to blend model output and real
data in order to improve on predictions from the exclusive use of the
model or the data alone. This "forward strategy" is especially well suited
when the data is sparse in space and time (e.g. in climate) and when
uncertainties are present in model states or parameters. Better model initialization using these assimilation techniques has resulted in significant improvements in weather forecasting. Better parameter estimation via data assimilation in hydrogeology has improved transport modeling as well.

Techniques that are based upon least-squares ideas, such as the
family of Kalman Filter/Smoothers, or Variational Data Assimilation, are
optimal in linear/Gaussian problems. However, they fail in problems
in which nonlinearities are important and/or when Gaussianity
in the statistics cannot be assumed. Even linearization may fail, and so
do ensemble techniques that make nonlinear predictions but rely
on linear analyses. These comprise the practical state of the art, at least
in weather forecasting and in hydrogeology. I will describe these as
well as how failures arise in these methods.

We have created a number of nonlinear/non-Gaussian data assimilation techniques. Our present efforts are to make them computationally
practical as well as to use of these to do problems that are otherwise intractable using conventional means. One such application is in Lagrangian data assimilation: here we tackle the problem of blending data that has been sampled along paths, which when blended in traditional ways on Eulerian grids will lead to loss of critical features even though the estimates may be variance-minimizing.