Helmholtz–Hodge decomposition and the analysis of 2D vector field ensembles

The Helmholtz–Hodge decomposition is a very useful instrument for vector field interpretation because it traditionally distinguishes conservative (curl-free) components from mass-preserving (divergence-free) components. In this work, we propose to use the Helmholtz–Hodge decomposition as a basic tool for the analysis of an ensemble of 2D vector fields. Given a vector field ensemble E, our approach firstly generates the corresponding curl-free, divergence-free, and harmonic component ensembles by using the Natural Helmholtz–Hodge decomposition to each vector field in E. With these ensembles in hand, our proposal not only quantifies, via a statistical analysis, how much each component ensemble is point-wisely correlated to the original vector field ensemble, but also allows to investigate the uncertainty of curl-free, divergence-free, and harmonic components separately. We test the proposed method with a synthetic vector field ensemble and with a simulated vector field ensemble.