Traveling wave packets are key coherent features contributing to the dynamics of several advective flows. This work introduces the Hilbert proper orthogonal decomposi- tion (HPOD) to distill these features from flow field data, leveraging their mathematical representation as modulated traveling waves. The HPOD is a complex-valued extension of the proper orthogonal decomposition, where the Hilbert transform of the dataset is used to compute its analytic signal. Two versions of the technique are explored and compared: the conventional HPOD, computing the analytic signal in time, and a novel space-only HPOD, computing it along the advection direction. The HPOD is shown to extract wave packets with amplitude and frequency modulation in time and space. Its broadband nature offers an alternative to spectrally pure decompositions when instantaneous, local wave characteristics are important. The space-only version, leveraging space/time equivalence in traveling waves to swap temporal operations with spatial ones, is proven mathematically equivalent to its conventional counterpart. The two HPOD versions are characterized and validated on three datasets ordered by complexity: a two-dimensional (2D) direct numerical simulation of a laminar bluff-body wake with periodic vortex shedding; a large eddy simulation of a turbulent jet with intermittent, highly modulated wave packets; and a 2D particle image velocimetry experiment of a turbulent jet with measurement errors and no temporal resolution. In advecting flows, both HPOD versions deliver practically identical complex-valued advecting wave-packet structures, characterized by spatiotempo- ral amplification and decay, wave modulation, and intermittency phenomena in turbulent flow cases, such as in turbulent jets. The space-only variant allows the extraction of these structures from temporally under-resolved datasets, typical of snapshot particle image velocimetry
Raiola et al. (2026) studied this question.