The relation of the Allan- and Delta-variance to the continuous wavelet transform

TL;DR

This paper establishes a mathematical connection between the Allan- and Delta-variance methods and the wavelet transform, demonstrating that these variances are essentially the variances of wavelet coefficients, thus linking structure analysis techniques.

Contribution

It reveals that Allan- and Delta-variances are equivalent to the variances of wavelet transform coefficients, providing a new perspective on their application in molecular cloud structure analysis.

Findings

Allan- and Delta-variances are variances of wavelet coefficients

Provides a link between variance-based and wavelet-based structure analysis

Enhances understanding of molecular cloud structure characterization

Abstract

This paper is understood as a supplement to the paper by [Stutzki et al, 1998], where we have shown the usefulness of the Allan-variance and its higher dimensional generalization, the Delta-variance, for the characterization of molecular cloud structures. In this study we present the connection between the Allan- and Delta-variance and a more popular structure analysis tool: the wavelet transform. We show that the Allan- and Delta-variances are the variances of wavelet transform coefficients.