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arxiv: 2507.02487 · v1 · submitted 2025-07-03 · ⚛️ physics.flu-dyn

Hilbert Proper Orthogonal Decomposition: a tool for educing advective wavepackets from flow field data

Marco Raiola , Jochen Kriegseis This is my paper

Pith reviewed 2026-05-19 06:46 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Hilbert proper orthogonal decompositiontravelling wavepacketsadvective flowsproper orthogonal decompositionanalytic signalturbulent jetsvortex sheddingflow field decomposition
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The pith

Hilbert proper orthogonal decomposition extracts modulated travelling wavepackets from advective flow data via the Hilbert transform.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces the Hilbert proper orthogonal decomposition (HPOD) to identify travelling wavepackets in flow field data by representing them as modulated travelling waves and applying the Hilbert transform to obtain their analytic signal. It develops and compares two versions: the conventional approach that computes the analytic signal in time, and a novel space-only version that applies the transform along the advection direction. The space-only variant is proved mathematically equivalent to the time-based one by exploiting space-time equivalence for travelling waves. Validation across a laminar bluff-body wake, a turbulent jet simulation, and particle image velocimetry measurements shows that both versions recover complex-valued structures exhibiting amplitude and frequency modulation, amplification, decay, and intermittency. The broadband character of HPOD provides an alternative to narrowband methods when instantaneous local wave features matter, and the space-only form works on temporally under-resolved data.

Core claim

The central claim is that the Hilbert proper orthogonal decomposition (HPOD), formed as a complex-valued extension of proper orthogonal decomposition using the Hilbert transform, distils advective wavepackets from flow field data. These wavepackets appear as structures with amplitude and frequency modulation in both time and space. The space-only HPOD, which replaces temporal operations with spatial ones along the advection direction, is mathematically equivalent to the conventional time-based version. When applied to a laminar bluff-body wake, a turbulent jet large-eddy simulation, and experimental particle image velocimetry data, both versions produce essentially identical complex-valued,,

What carries the argument

The Hilbert transform applied to the flow dataset to form its analytic signal, then inserted into the proper orthogonal decomposition to isolate modulated travelling waves.

Load-bearing premise

Advective flow features can be represented as modulated travelling waves whose analytic signal is recovered by the Hilbert transform applied either in time or along the advection direction.

What would settle it

Apply both the conventional and space-only HPOD to a single temporally well-resolved dataset of a known advective wavepacket flow and check whether the extracted structures differ substantially in their modulation or propagation properties.

Figures

Figures reproduced from arXiv: 2507.02487 by Jochen Kriegseis, Marco Raiola.

Figure 1
Figure 1. Figure 1: Energy content of each mode versus the total energy content of the sequence for the HPOD implementations and POD: (left) original time-resolved sequence; (right) sequence shuffled in time. resolved data. As expected, while the space-only HPOD retain the same energy distribution as in the unshuffled case, the conventional implementation in time cannot correctly complexify the sequence, converging, therefore… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between POD and HPOD modes – computed on the original time-resolved sequence, 100% of the sequence: first 3 pairs of spatial POD modes (1st column); first 3 spatial modes of the conventional HPOD (2nd column); first 3 spatial modes of the space-only HPOD (3rd column); phase plot of the temporal modes for HPOD implementations and for POD pairs (4th column) [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between POD and HPOD modes – computed on the original time-resolved sequence, 70% of the sequence: first 3 pairs of spatial POD modes (1st column); first 3 spatial modes of the conventional HPOD (2nd column); first 3 spatial modes of the space-only HPOD (3rd column); phase plot of the temporal modes for HPOD implementations and for POD pairs (4th column) [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between POD and HPOD modes – computed on the shuffled sequence, 100% of the sequence: first 3 pairs of spatial POD modes (1st column); first 3 spatial modes of the conventional HPOD (2nd column); first 3 spatial modes of the space-only HPOD (3rd column); phase plot of the temporal modes for HPOD implementations and for POD pairs (4th column) [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between POD and HPOD modes – computed on the shuffled sequence, 70% of the sequence: first 3 pairs of spatial POD modes (1st column); first 3 spatial modes of the conventional HPOD (2nd column); first 3 spatial modes of the space-only HPOD (3rd column); phase plot of the temporal modes for HPOD implementations and for POD pairs (4th column) [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Radial velocity component of the complex-valued spatial modes of the HPOD performed in the temporal direction: (left) 1st mode, (centre) 2nd mode, (right) 3rd mode. From top to bottom: real part of the spatial mode; imaginary part of the spatial mode; absolute value of the spatial mode; phase plot of the real versus imaginary part of the temporal mode [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Radial velocity component of the complex-valued spatial modes of the HPOD performed in the advective direction: (left) 1st mode, (centre) 2nd mode, (right) 3rd mode. From top to bottom: real part of the spatial mode; imaginary part of the spatial mode; absolute value of the spatial mode; phase plot of the real versus imaginary part of the temporal mode. confirms that the spatial patterns as well are analyt… view at source ↗
Figure 8
Figure 8. Figure 8: Spectrogram of the complex-valued temporal modes of the HPOD performed in the temporal direction (top) and histogram of the peak frequency through time (bottom). From left to right: (left) 1st mode, (centre) 2nd mode, (right) 3rd mode [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Spectrogram of the complex-valued temporal modes of the HPOD performed in the advective direction (top) and histogram of the peak frequency through time (bottom). From left to right: (left) 1st mode, (centre) 2nd mode, (right) 3rd mode. distribution around a specific spatial wavenumber around which the mode shows a more or less strong modulation in wavenumber. As the mode number is increased, this peak wav… view at source ↗
Figure 10
Figure 10. Figure 10: Spectrogram along 𝑥 of the complex-valued spatial modes of the conventional HPOD (top) and histogram of the peak spatial wavenumber through the advective direction (bottom). From left to right: (left) 1 st mode, (centre) 2nd mode, (right) 3rd mode [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Spectrogram along 𝑥 of the complex-valued temporal modes of the space-only HPOD (top) and histogram of the peak spatial wavenumber through the advective direction (bottom). From left to right: (left) 1 st mode, (centre) 2nd mode, (right) 3rd mode. implementations of the HPOD deliver temporal and spatial modes, which are approximately the same apart from being one the complex conjugate of the other. This b… view at source ↗
Figure 12
Figure 12. Figure 12: A sketch of the turbulent-jet experimental setup. wavenumber. Similarly, the conventional HPOD needs to have negative wavenumbers, which manifest as a phase opposition in the spatial patterns with respect to the space-only HPOD, characterized instead by positive wavenumbers. Notice that, both spatial and temporal modes delivered by the conventional and space-only HPOD are one the complex conjugate of the … view at source ↗
Figure 13
Figure 13. Figure 13: Radial component of the complex-value HPOD modes: (left) 1st mode, (centre) 2nd mode, (right) 3 rd mode. From top to bottom: real part of the spatial mode; imaginary part of the spatial mode; absolute value of the spatial mode; phase plot of the real versus imaginary part of the temporal mode. The first 3 complex-valued structures obtained from the decomposition are depicted in [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 14
Figure 14. Figure 14: Spectrogram along 𝑥 of the complex-valued temporal modes of the space-only HPOD (top) and histogram of the peak spatial frequency through the advective direction (bottom). From left to right: (left) 1 st mode, (centre) 2nd mode, (right) 3rd mode. is produced by multiplying these complex-valued spatial modes by the complex exponential exp(𝑖𝜑), where 𝜑 is the temporal phase of the wave motion. Frames of the… view at source ↗
Figure 15
Figure 15. Figure 15: Radial component of the oscillator model of the space-only HPOD modes: (left) 1st mode, (centre) 2nd mode, (right) 3rd mode. From top to bottom: different phases according to the oscillator model. amplitude, thus representing the temporal or spatial function of a complex-valued wavepacket. It is worth remarking that, differently from other complex-valued decomposition such as the spectral POD, the spectra… view at source ↗
read the original abstract

Travelling wavepackets are key coherent features contributing to the dynamics of several advective flows. This work introduces the Hilbert proper orthogonal decomposition (HPOD) to distil these features from flow field data, leveraging their mathematical representation as modulated travelling 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; a novel space-only HPOD, computing it along the advection direction. The HPOD is shown to extract wavepackets 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 travelling waves to swap temporal operations by spatial ones, is proved mathematically equivalent to its conventional counterpart. The two HPOD versions are characterized and validated on three datasets ordered by complexity: a 2D-DNS of a laminar bluff-body wake with periodic vortex shedding; an LES of a turbulent jet with intermittent, highly modulated wavepackets; and a 2D-PIV of a turbulent jet with measurement errors and no temporal resolution. In advecting flows, both HPOD versions deliver practically identical complex-valued advecting wavepacket structures, characterized by spatiotemporal amplification and decay, wave modulation and intermittency phenomena in turbulent flow cases, such as in turbulent jets. The space-only variant allows to extract these structures from temporally under-resolved datasets, typical of snapshot particle image velocimetry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces Hilbert Proper Orthogonal Decomposition (HPOD) as a complex-valued extension of POD for extracting advective wavepackets from flow data, representing them as modulated travelling waves via the analytic signal obtained from the Hilbert transform. It develops a conventional time-based version and a novel space-only version that applies the Hilbert transform along the advection direction. A mathematical equivalence proof is provided for the two versions by interchanging temporal and spatial operations under the travelling-wave ansatz. The method is validated on three datasets of increasing complexity: 2D DNS of a laminar bluff-body wake with periodic shedding, LES of a turbulent jet with intermittent modulated wavepackets, and temporally under-resolved 2D PIV of a turbulent jet. Both versions are shown to recover wavepackets exhibiting spatiotemporal amplification/decay, frequency modulation, and intermittency, with the space-only variant enabling extraction from snapshot PIV data.

Significance. If the central claims hold, HPOD supplies a broadband alternative to spectrally narrow decompositions when local instantaneous wave characteristics matter in advective flows. The explicit mathematical equivalence proof between time-HPOD and space-only HPOD is a clear strength, as is the systematic validation across laminar-to-turbulent and resolved-to-under-resolved datasets. The space-only formulation could meaningfully extend wavepacket analysis to common experimental PIV configurations lacking temporal resolution.

major comments (1)
  1. [Equivalence proof and §5.2–5.3] The equivalence proof (detailed after the method introduction, prior to the validation sections) starts from the ideal travelling-wave representation f(x,t) = Re[A(x-Ut) exp(i k (x-Ut))] with constant convection speed U. In the turbulent jet cases (LES §5.2 and PIV §5.3), local convection velocity varies spatially while wavepackets are intermittent and modulated in both space and time; under these conditions the time and space Hilbert operators do not commute exactly. The manuscript reports that the extracted structures are “practically identical,” but this observation alone does not establish rigorous interchangeability. Please either extend the proof to spatially varying U or supply quantitative metrics (e.g., normalized L2 difference or modal inner product between corresponding time-HPOD and space-HPOD modes) for the turbulent datasets.
minor comments (2)
  1. [Abstract] The abstract states that the method extracts wavepackets “with amplitude and frequency modulation in time and space,” yet no explicit quantitative measures (e.g., modulation indices or correlation with reference signals) are summarized; adding one or two such metrics would improve the abstract’s informativeness.
  2. [Validation sections] In the validation sections, the comparison between HPOD variants would benefit from a short table of similarity metrics (inner products, phase differences) rather than relying solely on visual inspection of mode shapes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Equivalence proof and §5.2–5.3] The equivalence proof (detailed after the method introduction, prior to the validation sections) starts from the ideal travelling-wave representation f(x,t) = Re[A(x-Ut) exp(i k (x-Ut))] with constant convection speed U. In the turbulent jet cases (LES §5.2 and PIV §5.3), local convection velocity varies spatially while wavepackets are intermittent and modulated in both space and time; under these conditions the time and space Hilbert operators do not commute exactly. The manuscript reports that the extracted structures are “practically identical,” but this observation alone does not establish rigorous interchangeability. Please either extend the proof to spatially varying U or supply quantitative metrics (e.g., normalized L2 difference or modal inner product between corresponding time-HPOD and space-HPOD modes) for the turbulent datasets.

    Authors: We agree that the equivalence is derived under the constant-convection-speed travelling-wave ansatz. In the turbulent jet cases the local velocity varies and the operators therefore do not commute exactly. The manuscript currently relies on visual agreement of the extracted structures. To strengthen the claim we will add, in the revised manuscript, quantitative comparisons for both the LES and PIV datasets: the normalized L2 difference between corresponding time-HPOD and space-HPOD modes together with their modal inner products. These metrics will be reported in §5.2 and §5.3. Extending the analytic proof to a spatially varying U would require a substantially more general framework that relaxes the constant-speed assumption; we therefore elect to supply the requested quantitative evidence instead. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence proof is self-contained mathematical derivation

full rationale

The paper's central derivation establishes mathematical equivalence between conventional (time-Hilbert) HPOD and space-only (streamwise-Hilbert) HPOD by invoking the space/time equivalence property of travelling waves, which is a standard analytic property of the Hilbert transform applied to advective signals of the form f(x,t) = Re[A(x-Ut)exp(i k (x-Ut))]. This is presented as an explicit proof rather than a reduction to fitted parameters, self-citations, or ansatz smuggling. Validation proceeds on independent datasets (DNS wake, LES jet, PIV jet) without the target result being presupposed in the inputs. No load-bearing self-citation chains or renaming of known results appear; the method is introduced as a direct complex-valued extension of POD. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests primarily on the domain assumption that flow features behave as modulated travelling waves amenable to Hilbert-transform analysis; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Advective flow features can be represented as modulated travelling waves
    Explicitly invoked in the abstract as the mathematical representation leveraged by HPOD.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

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    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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