2001A&A...366..636B

We present a detailed study of the Δ-variance as a method to quantify molecular cloud structure. The Δ-variance was introduced by Stutzki et al. (1998A&A...336..697S) to analyze the drift behaviour of scalar functions and is used to characterize the spatial structure of observed molecular cloud images. For fractional Brownian motion structures (fBm-fractals), characterized by a power law power spectrum and random phases, the Δ-variance allows to determine the power spectral index β. We present algorithms to determine the Δ-variance for discretely sampled maps and study the influence of white noise, beam smoothing and the finite spatial extent of the maps. We find that for images with β>3, edge effects can bias the structure parameters when determined by means of a Fourier transform analysis. In contrast, the Δ-variance provides a reliable estimate for the spectral index β, if determined in the spatial domain. The effects of noise and beam smoothing are analytically represented in a leading order approximation. This allows to use the Δ-variance of observed maps even at scales where the influence of both effects becomes significant, allowing to derive the spectral index β over a wider range and thus more reliably than possible otherwise. The Δ-variance is applied to velocity integrated spectral line maps of several clouds observed in rotational transitions of ¹²CO and ¹³CO. We find that the spatial structure of the emission is well characterized by a power law power spectrum in all cases. For linear scales larger than ∼0.5pc the spectral index is remarkably uniform for the different clouds and transitions observed (2.5≤β≤2.8). Significantly larger values (βγ3) are found for observations made with higher linear resolution toward the molecular cloud Mcld 123.5+24.9 in the Polaris Flare, indicating a smoother spatial structure of the emission at small scales (<0.5pc).