We refer the reader to several sources and briefly summarize the results here. Example analyses computed using the open-source implementation spod are introduced both to illustrate the choice of estimation parameters and to provide guidance regarding the interpretation of results.Īn aim of this paper is to make SPOD accessible to practitioners, but a thorough understanding and disambiguation between variants of the approach requires some theoretical background. III, we detail the SPOD algorithm and discuss tradeoffs related to the choice of spectral estimation parameters.
II by briefly reviewing the theory behind POD and contrasting the standard and spectral versions of it. We hope that, by doing so, we facilitate its understanding and use by the broader fluids community. Thus the purpose of this paper is to expose the SPOD–Welch algorithm in detail and provide examples of applying it to both computational and experimental data. The key algorithm behind SPOD is Welch’s method, an averaging technique that consistently and accurately estimates the CSD from a time series. Our terminology here stems from earlier usage and the fact that the kernel is the CSD.Īlthough the “POD part” of SPOD is conceptually and algorithmically straightforward (eigenvectors of a matrix), the “spectral” part of SPOD is less so. We caution that the term “SPOD” has been applied to a different technique proposed by Sieber et al. In addition, there are useful connections between SPOD modes and a resolvent analysis of a forced, linear operator that provide for additional ways to interpret (and model) turbulent flows. For the case of stationary data from stochastic processes, SPOD combines the advantages of DMD in terms of expressing temporal correlation among the resulting structures, with the optimality associated with POD itself. In a recent paper, we reviewed the properties of SPOD and determined the relationships between SPOD and DMD. On the other hand, since at least the late 1980s, the space-only version has become dominant to the point that the space–time and frequency–space versions are sometimes neglected in favor of the dynamic mode decomposition (DMD), Cronos–Koopman analysis, and other techniques. Spectral POD is hardly new: much of the original literature on POD stemming from Lumley is agnostic as to the choice of domain and inner product, and the frequency-based version we discuss in this paper has been widely applied in the intervening years. Mathematically, SPOD modes are the eigenvectors of a cross-spectral density (CSD) tensor at each frequency. SPOD is a special case of a more general space–time decomposition under the assumption of statistically stationary data (meaning that the mean and variance do not change in time), and the resulting modes are harmonic in time, and computed one frequency at a time from the data. As a consequence, they are optimal at expressing spatiotemporal coherence in the data.
What distinguishes SPOD from “standard” POD is that the modes vary in both space and time and are orthogonal under a space–time inner product, rather than only space. Like other variants of POD, the SPOD finds an optimal orthogonal basis for the data in the sense that a subset of the modes captures a larger fraction of the total energy (variance) in the data than any other orthogonal basis. Alongside other operator- and data-driven decompositions employed in fluid mechanics, the resulting modes can be used for a variety of purposes, from classification to reduced-order modeling to control. S pectral proper orthogonal decomposition (spectral POD, or SPOD) is an empirical method to extract coherent structures, or modes, from flow data. Vector eigenfunction, discrete eigenvector Spatial coordinates, first Cartesian coordinate
Total number of degrees of freedom, Mach numberĬross-spectral density tensor, sample cross-spectral density matrix Covariance tensor, discrete sample covariance matrix
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