Sources and Sinks of Rare Trajectories in 2-Dimensional Velocity Fields Identified by Importance Sampling

We use importance sampling in a redefined way to highlight and investigate rare events in the form of trajectories trapped inside a target almost invariant set. We take a transfer operator approach to finding almost invariant sets of a reconstructed 2-dimensional flow of the atmosphere from wind velocity fields provided by the Portable University Model of the Atmosphere. Motivated by extreme value theory, we consider an observable $\phi(x) = -\log(d(x,\gamma))$ maximized at the center $\gamma$ of a chosen target almost invariant set. We illustrate that importance sampling using this observable provides an enriched data set of trajectories that experience a rare event, in this case defined as ending near $\gamma$. Backwards reconstruction of these trajectories provides valuable information on initial conditions and most likely paths a trajectory will take toward a rare event. In this specific setting, one can think of an almost invariant set as a region in the atmosphere where only a small number of particles move in and out of the region. For short time intervals, these regions are represented physically by individual eddies in the atmosphere; while longer time intervals suggest formation by eddy interaction such as particles trapped between the spin of two eddies. Particle movement in and interaction with these sets can provide useful information on the most probable path of a storm (in the former case), or origin of pollution (in the latter), for example.