Lagrangian gradient regression for the detection of coherent structures from sparse trajectory data

Complex flows are often characterized using the theory of Lagrangian coherent structures (LCS), which leverages the motion of flow-embedded tracers to highlight features of interest. LCS are commonly employed to study fluid mechanical systems where flow tracers are readily observed, but they are broadly applicable to dynamical systems in general. A prevailing class of LCS analyses depends on reliable computation of flow gradients. The finite-time Lyapunov exponent (FTLE), for example, is derived from the Jacobian of the flow map, and the Lagrangian-averaged vorticity deviation (LAVD) relies on velocity gradients. Observational tracer data, however, are typically sparse (e.g. drifters in the ocean), making accurate computation of gradients difficult. While a variety of methods have been developed to address tracer sparsity, they do not provide the same information about the flow as gradient-based approaches. This work proposes a purely Lagrangian method, based on the data-driven machinery of regression, for computing instantaneous and finite-time flow gradients from sparse trajectories. The tool is demonstrated on a common analytical benchmark to provide intuition and demonstrate performance. The method is seen to effectively estimate gradients using data with sparsity representative of observable systems.