Direct power spectral density estimation from structure functions without Fourier transforms

TL;DR

This paper presents a novel method to estimate power spectral densities directly from structure functions in real space, bypassing Fourier transforms, and validates it across various turbulence data sources.

Contribution

It introduces a new framework for power spectrum estimation from structure functions without Fourier transforms, applicable to turbulence and astrophysical data.

Findings

Accurately recovers expected power-law behaviors in turbulence data.

Validates the method against analytical and observational data.

Demonstrates robustness across different turbulence regimes.

Abstract

Second-order structure functions and power spectral densities are popular tools in the study of statistical properties across scales, particularly for the analysis of turbulent flows. Although intimately related, analyses primarily use one or the other. We introduce a framework for estimating the power spectrum using the second-order structure function without applying Fourier transforms -- enabling one to take advantage of the real-space structure function calculations. We validate and showcase this method, comparing it to classical Fourier power spectrum estimates determined from analytical calculations, fractional Brownian motion, turbulence simulations, and space-physics and astrophysical observations of turbulence. We show that this method is able to robustly obtain the expected power law behaviour where we use turbulence ranges as test-cases.