Direct power spectral density estimation from structure functions without Fourier transforms

TL;DR

This work develops and validates a framework to estimate the angle-averaged power spectral density $\mathcal{E}_{D}(k)$ directly from the second-order structure function $\bar{S}_{D}(\ell)$ without Fourier transforms, by introducing the equivalent spectrum $\mathcal{E}_{D}^{S}(k_{e}) = \frac{1}{2b} \ell^{2} \frac{\mathrm{d}\bar{S}_{D}(\ell)}{\mathrm{d}\ell}|_{\ell=b/k_{e}}$ and relating it to the true spectrum via a bias factor $\mathcal{B}$ and a wavenumber mapping $b$. The authors provide analytic results for pure power laws and model spectra with energy-containing scales, plus a general interpretation of $\mathcal{E}_{D}^{S}$ as a non-local filter of $\mathcal{E}_{D}(k)$, explaining why a universal $b$ cannot exist. A practical debiasing strategy using $\mathcal{B}^{\mathrm{pow}}(\beta_{est}, b)$ with a local slope estimate $\beta_{est}$ is proposed, and the method is validated through fractional Brownian motion, solar wind data, interstellar-medium imaging, and 3D turbulence simulations. The approach remains robust in the presence of missing data and can be implemented with smoothing and binning to mitigate numerical noise. Overall, the paper offers a versatile, real-space alternative to FFT-based PSD estimation that preserves spectral information across inertial and energy-containing ranges and demonstrates practical utility across disciplines.

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.

Figures (16)

• Figure 1: The amplitude bias \ref{['eqn:powerlaw_bias']} for a pure power law angle-integrated spectrum \ref{['eqn:powerlaw_spectrum']}, for Euclidean dimensions $D=1$, $D=2$, and $D=3$. The line contours correspond to linear spacings of 0.25 starting from $\Bpow=0.25$ to $1.75$ with $\Bpow=1$ represented by the solid gray line. The solid vertical gray line represents the (Kolmogorov) power law slope of $5/3$.
• Figure 2: The amplitude bias \ref{['eqn:powerlaw_bias']} with fixed $b=1$ for a pure power law angle-integrated spectrum \ref{['eqn:powerlaw_spectrum']}, for Euclidean dimensions $D=1$ (dashed), $D=2$ (dotted), and $D=3$ (dash-dotted). The solid vertical gray line represents the (Kolmogorov) power law slope of $-5/3$.
• Figure 3: Filter function $\mathcal{G}_{D}(k\ell)$ normalized by its maximum (in the generated grid) to [-1, 1] for $D=1,2,3$. The black dash-dotted, dotted, and dashed contours correspond to $k \ell=1$, $k \ell=\sqrt{2D}$, and $k \ell=\sqrt{2D+2}$ respectively.
• Figure 4: The location of the first peaks ($\tilde{b} = x_{\text{peak}} = \text{argmax} \{ \widetilde{\mathcal{G}}_{D}\bracket{x} \}$) of the filter $\widetilde{\mathcal{G}}_{D}(x) \sim x^{2-D/2} \mathcal{J}_{D/2}(x)$ as a function of dimension $D$. The first peak locations (red crosses) are estimated using numerical methods and the black solid line is the linear fit. The dashed and dotted black lines are some reasonable approximations. The red, green, and blue dash-dotted lines correspond to the estimates of $b$ from \ref{['tab:b_factors']}. Note, the estimates from \ref{['tab:b_factors']} correspond to the integrated influence of $\mathcal{G}_{D}(k\ell)$ with $\mathcal{E}_{D}(k)$ rather than just one part of the integrand.
• Figure 5: $b$ factor ($b_{\mathrm{p}}$) corresponding to aligning the position of the peaks of $\widetilde{\mathcal{E}}_{D}^{S}(k_{\mathrm{e}})$ with $\mathcal{E}_{D}(k)$ using the model spectrum \ref{['eqn:exp_spectrum']} for different $\beta$ and $k_0$ values (circles). Note, the circle markers are coloured representing different $k_0$ (shown by the colourbar), but $\bp$ appears almost independent of $k_0$. The black horizontal lines correspond to $1$ (dash-dot), $2D-2$ (solid), $2D$ (dotted), and $2D+2$ (dashed) as discussed in \ref{['sec:filter_function']} with $\bpest$ denoted as "est." where appropriate. The $\beta$-dependent thick black dashed line is the empirical formula \ref{['eqn:empirical_b_estimate']} ($\bpemp$, denoted as "emp."). The solid vertical gray line represents the (Kolmogorov) power law slope of $-5/3$.