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 coherent set. We take a transfer operator approach to finding these sets on 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 coherent set, where it is rare for a particle to transition. We illustrate that importance sampling maximizing this observable provides an enriched data set of trajectories that experience such a rare event. Backwards reconstruction of these trajectories provides valuable information on initial conditions and most likely paths a trajectory will take. With this information, we are able to obtain more accurate estimates of rare transition probabilities compared to those of standard integration techniques.