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Frequency-time analysis, low-rank reconstruction and denoising of turbulent flows using SPOD
Akhil Nekkanti, Oliver T. Schmidt
TL;DR
The paper examines how SPOD can support low-rank reconstruction, denoising, frequency-time analysis, and prewhitening of turbulent-jet data. It compares frequency-domain inversion with time-domain projection, introduces eigenvalue-threshold denoising and convolution-based frequency-time analysis, and finds that reconstruction and processing preferences depend on the application.
Problem
SPOD-based reconstruction is ambiguous because SPOD relies on spectral estimation, while frequency-time analysis becomes computationally intractable when time-varying coefficients are computed in the frequency domain.
Method
The paper demonstrates frequency-domain SPOD inversion, time-domain projection, hard-threshold denoising of SPOD eigenvalues, and convolution-based frequency-time analysis on turbulent-jet LES data.
Results
The time-domain approach better captures individual-snapshot dynamics, whereas the frequency-domain approach is effective for denoising and frequency-time analysis while retaining mode-frequency correspondence.
Takeaways & Limitations
SPOD-based frequency-time analysis links high-energy events in the turbulent jet to intermittent spatially coherent structures resembling leading SPOD modes.
Takeaways & Limitations
The analysis uses pressure data from a turbulent jet and exploits a rotationally symmetric jet’s individual azimuthal Fourier components.
Abstract
from arXiv · showhide
Four different applications of spectral proper orthogonal decomposition (SPOD): low-rank reconstruction, denoising, frequency-time analysis, and prewhitening are demonstrated on large-eddy simulation data of a turbulent jet. SPOD-based low-rank reconstruction can be performed by direct inversion of a truncated SPOD. This spectral inversion problem, however, is ambiguous since SPOD relies on spectral estimation. We demonstrate SPOD-based flow field reconstruction using direct inversion of the SPOD algorithm (frequency-domain approach) and propose an alternative approach based on projection of the time series data onto the modes (time-domain approach). While the SPOD optimally represents the flow in a statistical sense, the time-domain approach seeks an optimal reconstruction of each instantaneous flow field. We further propose a SPOD-based denoising strategy that is based on hard-thresholding of the SPOD eigenvalues. The proposed strategy achieves significant noise reduction while facilitating drastic data compression. In contrast to standard methods of frequency-time analysis such as wavelet transform, a proposed SPOD-based approach yields a spectrogram that characterizes the temporal evolution of spatially coherent flow structures. In the frequency-domain, time-varying expansion coefficients can be obtained by basing the SPOD on a sliding window. This approach, however, is computationally intractable, and an alternative strategy based on convolution in the time-domain is presented. When applied to the turbulent jet data, SPOD-based frequency-time analysis reveals that the intermittent occurrence of large-scale coherent structures is directly associated with high-energy events. This work suggests that the time-domain approach is preferable for low-rank reconstruction of individual snapshots, and the frequency-domain approach for denoising and frequency-time analysis.
1. Introduction
The paper motivates SPOD as a frequency-domain modal decomposition for extracting essential features from turbulent-flow data and applies it to reconstruction, denoising, frequency-time analysis, and prewhitening.
- Motivation: Modal decomposition techniques address the curse of dimensionality by extracting essential flow features and providing low-dimensional data representations.POD modes optimally represent variance or energy and are coherent in space at zero time lag.
- SPOD background: SPOD computes energy-ranked orthogonal modes at each frequency from estimates of the cross-spectral density matrix.For statistically stationary flows, it combines POD’s optimality and orthogonality with DMD’s temporal monochromaticity.
- Denoising: SPOD-based denoising targets measurement noise that can impair physical analysis, especially when computing spatial derivatives or resolving small-scale structures.The study applies SPOD denoising to simulation data with high levels of additive Gaussian noise and combines advantages of temporal filtering and POD-based denoising.
- Frequency-time analysis: Frequency-time analysis represents a signal’s frequency content as a function of time, helping identify intermittent high- or low-energy events and their wave characteristics.The paper contrasts established tools including wavelet transforms and short-time Fourier transforms with an SPOD-based approach.
- Prewhitening: Prewhitening removes serial correlations through filtering that produces a flat power spectrum and is used for trend detection.The paper situates prewhitening in atmospheric and geophysical applications.
- Study scope: The four applications are demonstrated on large-eddy simulation data of a turbulent jet.The applications are low-dimensional reconstruction, denoising, frequency-time analysis, and prewhitening.
2. Methodology
The methodology estimates SPOD from segmented, windowed Fourier realizations, then reconstructs data either by frequency-domain inversion or time-domain projection with configurable truncation. Overlapping-block reconstruction uses window-weighted averaging, while the time-domain approach is computationally efficient and applicable to new data and individual snapshots.
- SPOD estimation: SPOD estimates the cross-spectral density by segmenting statistically stationary data into overlapping blocks, applying a Hamming window when blocks overlap, and Fourier transforming each block.The blocks provide ensemble members under the ergodic hypothesis, while windowing reduces spectral leakage.
- SPOD estimation: At each frequency, SPOD modes and energies are obtained as eigenvectors and eigenvalues of the cross-spectral density matrix.The modes are orthogonal in the space-time inner product and, at a given frequency, in the spatial inner product.
- Frequency-domain reconstruction: The frequency-domain method reconstructs Fourier realizations from modal expansion coefficients, applies the inverse weighted Fourier transform, and then reconstructs the time series.Partial reconstructions are obtained by zeroing selected modal coefficients before inverse transformation.
- Frequency-domain reconstruction: Overlapping blocks create reconstruction ambiguity, so snapshots are averaged from two block reconstructions weighted by their relative window values.The two candidate reconstructions correspond to the same snapshot’s positions in adjacent overlapping blocks.
- Time-domain reconstruction: The time-domain method projects data onto the SPOD modal basis using an oblique projection rather than exploiting space-time orthogonality.It is computationally efficient and can be applied to new data not used to compute SPOD and to individual snapshots.
- Truncation: Truncation is implemented by modifying expansion coefficients, including aggressive rank-k truncation below the number of numerically nonzero eigenvalues.This framework supports low-rank approximation and other partial reconstruction strategies.
3. Applications of SPOD: Low-rank reconstruction, denoising, frequency-time analysis, and prewhitening
Using turbulent-jet LES data, the paper compares SPOD-based frequency- and time-domain reconstructions and applies truncation to reconstruction and denoising. The time-domain approach better reconstructs individual flow fields, while frequency-domain processing supports energy correction and denoising analysis.
- Data and SPOD representation: The study analyzes pressure fluctuations in an isothermal subsonic turbulent jet using the helical m=1 azimuthal component and Strouhal-number frequencies.The LES has Re=0.45×10^6, M_j=0.4, and T_j/T_∞=1.0.
- Data and SPOD representation: The first 10 SPOD modes contain 80% of total energy, while the first mode contains 30%; energy is more concentrated at low frequencies.At St=0.5, the leading mode represents a Kelvin–Helmholtz wavepacket, whereas suboptimal modes are multi-lobed.
- Low-rank reconstruction: For a fixed number of modes, time-domain reconstruction captures more energy and approximates the data better than frequency-domain reconstruction.The time-domain method’s oblique projection provides the best least-squares approximation, whereas frequency-domain reconstruction can overpredict boundary-block norms because of windowing.
- Low-rank reconstruction: Frequency-domain reconstruction recovers SPOD mode energies after adding the residual energy of truncated eigenvalues, with remaining differences largely caused by windowing.Both methods recover the original instantaneous field when all modes are retained, while time-domain reconstructions preserve more detail at low rank.
- Denoising: SPOD truncation is used for denoising because additive noise occupies identifiable spectral regions, while time-domain local optimality tends to reconstruct the noise.In the reported comparison, low-pass filtering performs better for x≳10, whereas SPOD filtering performs better farther downstream; near x≳2.5 its SNR is marginally lower than the unfiltered data.
- Frequency-time analysis: Frequency-time diagrams identify intervals when coherent SPOD structures resemble the instantaneous flow, linking maxima to high-energy events and intermittency of large-scale structures.The approach uses spatially coherent, energetic modes to characterize temporal evolution rather than only scalar frequency content.
4. Summary and Conclusions
The paper demonstrates SPOD applications for reconstruction, denoising, and frequency-time analysis, comparing frequency- and time-domain approaches on turbulent-jet LES data. It recommends the time-domain approach for individual-snapshot reconstruction and the frequency-domain approach for denoising and frequency-time analysis.
- Low-rank reconstruction: SPOD reconstructions accurately capture integral energy, while time-domain reconstructions better capture instantaneous dynamics and retain more energy for a fixed modal count.Frequency-domain reconstructions preserve SPOD orthogonality and frequency-mode correspondence, whereas time-domain reconstructions optimize individual snapshots.
- Denoising: SPOD denoising uses a hard threshold above the noise floor, reducing noise while retaining substantial original flow-field energy.Noise is mainly captured by higher modes at low frequencies and by all modes at high frequencies.
- Frequency-time analysis: SPOD-based frequency-time analysis characterizes the temporal evolution of spatially coherent structures and avoids the intractable frequency-domain computation through convolution.The convolution strategy is mathematically equivalent in the limit of the continuously-discrete SPOD problem.
- Frequency-time analysis: Jet-data analysis confirms highly intermittent flow behavior, with most energy concentrated at low frequencies, St > 0.2.Comparisons between total flow energy and spectrograms connect high-energy events with the frequency-time representation.
- Practical recommendation: The recommended division of labor is time-domain reconstruction for individual snapshots and frequency-domain denoising and frequency-time analysis.For frequency-time analysis, convolution enables efficient computation of time-continuous expansion coefficients.
Appendix A. Effect of different parameters on the flow field reconstruction
Appendix A examines reconstruction choices caused by overlapping spectral-estimation blocks and windowing. Window-weighted averaging gives the smallest frequency-domain reconstruction error, while time-domain truncations better approximate the data for a fixed number of modes.
- Overlap and reconstruction ambiguity: 50% block overlap reduces spectral-estimate variance but creates ambiguity when reconstructing snapshots from overlapping segments.Several left-right block-combination rules are considered for resolving this ambiguity.
- Overlap and reconstruction ambiguity: The window-weighted average of left and right reconstructions produces the smallest error among the six tested reconstruction options.This rule is used for frequency-domain reconstruction throughout the paper.
- Reconstruction dimensionality: Full-dimensional reconstructions are generally accurate, so differences between reconstruction choices primarily affect truncated series.The appendix attributes the distinctions among reconstruction rules to low-dimensional truncation.
- Windowing effects: Windowing causes sudden jumps in local reconstruction energy, while modal-basis dimension directly controls pressure-norm accuracy.For a fixed number of modes, the time-domain approach provides a better approximation than the frequency-domain approach.
- Eigenvalue spectra: For a rectangular window without overlap, leading eigenvalues of one-mode frequency- and time-domain reconstructions are indistinguishable from those of the full data.The comparison uses the leading three eigenvalues for clarity.
- Reconstruction comparisons: Rectangular-window reconstructions compare frequency- and time-domain results using space-time and spatial norms across full, 10^129-, 3^129-, and 1^129-mode bases.The figure also shows summed SPOD mode energies and marks non-overlapping blocks.
Appendix B. Projection-based frequency-time analysis: effect of choice of basis and correspondence to convolution-based approach
Appendix B shows that projection-based frequency-time analysis depends on the modal basis because SPOD modes are non-orthogonal in the spatial norm. Using all SPOD modes yields a more accurate description of coherent-structure intermittency than using only the analyzed modes.
- Choice of modal basis: Projection outcomes differ between a basis containing only analyzed modes and a basis containing all available SPOD modes because spatial non-orthogonality prevents equivalent projections.The two choices are one mode at each frequency versus all available modes.
- Choice of modal basis: A one-mode-per-frequency basis produces a banded frequency-time diagram unlike the reference based on all SPOD modes.Its maxima are not predominantly located in the low-frequency regime, St > 0.2.
- Diagnostic comparisons: The appendix compares frequency-time diagrams from a one-mode basis and leading-mode representations, including PSD and cross-correlation diagnostics.The moving-mean analysis uses the same Hamming windowing function as SPOD.
- Choice of modal basis: The one-mode-per-frequency basis broadens the expansion-coefficient peak and lowers its correlation with the convolution-based reference.These effects indicate loss of mode-frequency correspondence.
- Correspondence to convolution: The time-domain approach should use all SPOD modes because this produces a more accurate description of coherent-structure intermittency.Moving averages of time-domain spectrograms resemble those obtained with the convolution approach.
Appendix C. Frequency-domain approach based frequency-time analysis: effect of overlap and correspondence to convolution-based approach
Appendix C compares convolution-based and frequency-domain frequency-time analysis under differing overlap requirements. The convolution method uses precomputed overlapping SPOD modes, whereas the frequency-domain comparison requires nearly complete overlap and computationally reduced data.
- Overlap and computational cost: The convolution method uses precomputed SPOD modes with 50% overlap, while time-continuous frequency-domain coefficients require n_ovlp = n_fft − 1.The latter requirement is computationally intractable, so the comparison reduces the spatial data resolution.