Eulerian and Lagrangian observability of point vortex flows

We study the observability of one and two-point vortex flow from one or two Eulerian or Lagrangian observations. By observability, we mean the ability to determine the locations and strengths of the vortices from the time history of the observations. An Eulerian observation is a measurement of the velocity of the flow at a fixed point in the domain of the flow. A Lagrangian observation is the measurement of the position of a particle moving with the fluid. To determine observability, we introduce the observability and the strong observability rank conditions and compute them for the various vortex configurations and observations in this idealized setting. We find that vortex flows with Lagrangian observations tend to be more observable then the same flows with Eulerian observations. We also simulate extended Kalman filters for the various vortex configurations and observations and find that they perform poorly when the ORC or the strong observability rank condition fails to hold.