UC San Diego

Data assimilation has wide ranging applications, including neuroscience, oceanography and climate science. In this dissertation we will examine data assimilation as a tool for systems of partial differential equations on a discretized spacial grid, using simple geophysical models as a twin for our study. We will use the 1 layer shallow water equations (SWE), and describe how to extend the method to a 2 layer SWE. Although we only used the SWE for this dissertation, we examine how we would use the barotropic vorticity equations (BVE) as the twin in the same study. We will examine two different methods for performing data assimilation on chaotic systems. The first method relies on the measurements to smooth the synchronization manifold, allowing a nonlinear optimizer to correctly determine the most likely path, or the path which minimizes the cost function. The second method we call Metropolis-Hastings Monte Carlo (MHMC) integration scheme. MHMC also allows retention of a group of path samples whose statistics reflect the probability of each path, allowing histograms of state vector values for analysis or inputs to particle filter methods for prediction. The study uses MHMC with the SWE as twin. in this chapter we will examine a data set used for the study. We then describe he various numbers of state vectors needed as data, and the increase in the quality of the fit. We determine the number of state vectors needed as measurements to accurately predict the unmeasured ones