UC San Diego

In this dissertation the problem of data assimilation in stochastic nonlinear systems is formulated using path integrals. Each path represents a time evolution of the model states, and the time independent model parameters. In the path integral, every possible path is integrated over with each path weighted by P(XIY) ̃exp[-A₀(X,Y)], where A₀(X,Y) is the action, which quantifies how likely it is that the given path X was actually realized in the experiment which produced the observed time series Y. The goal of data assimilation is to combine information from a measurement time series with a dynamical model to make statistical estimates or predictions of model states and parameters. Both the measurements and the dynamical model may be noisy, and this fact is incorporated by using a probabilistic formulation for P(XIY), the posterior path distribution conditioned on the observed time series. With an expression for P(XIY) it is possible to express expectation values, conditioned upon the observations, of any function of the path as a path integral over all possible paths. The path integrals can then be numerically approximated using a Markov chain Monte Carlo method such as the Metropolis method. This method is discussed and applied to two example systems: the Colpitts oscillator circuit, and the Lorenz 96 toy atmosphere model. By studying the characteristics of the action as a function of the path, properties of the data assimilation problem can be deduced. For instance, if the surface in path space defined by the action is rough with many local minima with similar values of action, then the data assimilation problem is not well-defined. If more observations are made which rule out regions of path space that were previously likely, then the surface may become smoother with a single minimum. By examining the shape of the action, the question of how many measurements are needed to fully reconstruct the model state can be answered. It is also important to examine the shape of the action in the vicinity of the global minimum to find the level of uncertainty in state and parameter estimates. These ideas are illustrated with the Lorenz 96 system as an example