[Paper Review] Optimal frequency resolution for spectral proper orthogonal decomposition

Liam Heidt, Tim Colonius|arXiv (Cornell University)|Feb 24, 2024

Image and Signal Denoising Methods4 citations

TL;DR

This paper demonstrates that the frequency resolution in spectral proper orthogonal decomposition (SPOD) critically affects mode accuracy, with previously used fixed resolutions introducing significant bias at physically important frequencies. The authors propose a physics-informed adaptive frequency-resolution algorithm that dynamically adjusts resolution based on spatial mode variation, yielding substantially less biased SPOD modes than standard constant-resolution methods, particularly in turbulent jet flows.

ABSTRACT

We demonstrate that accurate computation of the spectral proper orthogonal decomposition (SPOD) critically depends on the choice of frequency resolution. Using both artificially generated data and large-eddy simulation data of a turbulent subsonic jet, we show that the optimal choice depends on how rapidly the SPOD modes change in space at adjacent frequencies. Previously employed values are found to be too high, resulting in unnecessarily biased results at physically important frequencies. A physics-informed adaptive frequency-resolution SPOD algorithm is developed that provides substantially less biased SPOD modes than the standard constant resolution method.

Motivation & Objective

• To identify the limitations of fixed-frequency-resolution SPOD in accurately capturing coherent structures in statistically stationary flows.
• To investigate how the spatial variation of SPOD modes across adjacent frequencies influences the optimal frequency resolution.
• To develop a physics-informed adaptive frequency-resolution algorithm that reduces bias in SPOD mode estimation.
• To evaluate the statistical convergence and sensitivity of the widely used alignment metric between SPOD and resolvent modes.
• To demonstrate the superiority of the adaptive method over constant-resolution SPOD using artificial data and large-eddy simulation of a turbulent subsonic jet.

Proposed method

• The authors use artificial data with known SPOD spectra and modes to systematically test the impact of frequency resolution Δf on mode bias.
• They apply Welch’s periodogram method with variable block size and windowing to estimate the cross-spectral density (CSD) at each frequency bin.
• A physics-informed adaptive frequency-resolution algorithm is developed that adjusts Δf based on the rate of change of SPOD modes across adjacent frequencies, using a convergence criterion to guide resolution changes.
• The method employs a modified adaptive procedure that reduces resolution at low Strouhal numbers and increases it at higher frequencies, based on mode similarity and variance reduction.
• The standard SPOD computation is performed using the method-of-snapshots to solve the eigenvalue problem for the CSD matrix, with energy-ranked modes extracted from the eigenvectors.
• The alignment metric between SPOD and resolvent modes is used to quantify mode similarity, with statistical convergence assessed across varying data lengths.

Experimental results

Research questions

• RQ1 How does the choice of frequency resolution Δf affect the bias in SPOD mode estimation, particularly when modes vary rapidly across adjacent frequencies?
• RQ2 What is the optimal frequency resolution for SPOD when the underlying modes exhibit significant spatial variation over the frequency band?
• RQ3 Can a physics-informed adaptive frequency-resolution algorithm reduce bias in SPOD modes compared to standard constant-resolution methods?
• RQ4 How sensitive is the alignment metric between SPOD and resolvent modes to statistical convergence and bias in the SPOD estimation?
• RQ5 To what extent does increasing data length improve the convergence and reliability of the alignment metric between SPOD and resolvent modes?

Key findings

• Previously used fixed frequency resolutions are too high, introducing significant bias in SPOD modes at physically important frequencies, especially where mode shapes change rapidly.
• The adaptive frequency-resolution algorithm reduces bias substantially, with the modified adaptive method achieving excellent alignment between SPOD and resolvent modes across all frequencies and data lengths.
• The alignment metric between SPOD and resolvent modes is highly sensitive to bias and statistical convergence, with alignment increasing by up to ≈0.2 as data length increases from 5,000 to 20,000 snapshots.
• The optimal Δf decreases with increasing data length, consistent with reduced variance, and practitioners should increase resolution while mode shapes remain stable and noise decreases.
• The adaptive method outperforms fixed-resolution SPOD, particularly in capturing mode structure at low and intermediate Strouhal numbers, where mode variation is most pronounced.
• The study demonstrates that the alignment metric requires substantial data to converge, and conclusions based on alignment alone should be treated with caution due to high variance and slow convergence.