Incorporating prior knowledge in observability-based path planning for ocean sampling

Observability-based path planning of autonomous sampling platforms for flow estimation is a technique by which candidate trajectories are evaluated based on their ability to enhance the observability of underlying flow-field parameters. Until now, observability-based path planning has focused primarily on forward-in-time integration. We present a novel approach that makes use of the background error covariance at the current time to account properly for uncertainty of the underlying flow. The reduced Hessian of an optimal, linear data-assimilation strategy properly accounts for prior knowledge in the linear case and must be full rank to infer the initial state. The reduced Hessian represents an observability Gramian augmented with an inverse prior covariance. We extend this concept to the nonlinear case to yield a new criterion for scoring candidate trajectories: the empirical augmented unobservability index. Solving the differential covariance Riccati equation of the Kalman Filter for deterministic dynamics also properly accounts for prior knowledge in the linear case, but at a later time. The solution to this equation reveals the important distinctions between observability-based, augmented observability-based, and anticipated covariance-based path planning. Path planning based on this unobservability index in the presence of prior information yields the desired behavior in numerical experiments of a guided Lagrangian sensor in a two-vortex flow pertinent to ocean sampling.