On the nature of spectral proper orthogonal decomposition and related modal decompositions

TL;DR

This paper analytically explores spectral proper orthogonal decomposition (SPOD), clarifying its relation to other POD methods, and presents an alternative formulation based on classical ideas to enhance understanding.

Contribution

It provides an analytical comparison of SPOD with related POD approaches and introduces an alternative formulation rooted in classical concepts.

Findings

SPOD is closely related to POD with segmentation and Fourier transform.

SPOD is equivalent to snapshot POD with time delay embedding.

The paper clarifies controversial issues surrounding SPOD.

Abstract

The spectral proper orthogonal decomposition (SPOD) is a newly introduced extension of snapshot POD that recently gained attention but also brought up controversial issues. Within the first proposition, the approach was mainly presented in a methodological and phenomenological way. The present paper will detail the relations between SPOD and related POD approaches from an analytical point of view. To allow for a better grasp of the approach, an alternative formulation is given that is based on the classic idea from Lumley that was carried on by George. As will be shown, SPOD is closely related to POD with a prior segmentation and Fourier transformation in time. Moreover, the SPOD is shown to be equivalent to snapshot POD combined with time delay embedding.