A&A 585, A98 (2016) Astronomy DOI: 10.1051/0004-6361/201525899 & c ESO 2015 Astrophysics Wavelet-based cross-correlation analysis of structure scaling in turbulent clouds Tigran G. Arshakian1,2 and Volker Ossenkopf1 1 I. Physikalisches Institut, Universität zu Köln, Zülpicher Strasse 77, 50937 Köln, Germany e-mail: [arshakian;ossk]@ph1.uni-koeln.de 2 Byurakan Astrophysical Observatory, 378433 Aragatsotn prov., and Isaac Newton Institute of Chile, Armenian Branch, Armenia Received 13 February 2015 / Accepted 12 September 2015 ABSTRACT Aims. We propose a statistical tool to compare the scaling behaviour of turbulence in pairs of molecular cloud maps. Using artificial maps with well-defined spatial properties, we calibrate the method and test its limitations to apply it ultimately to a set of observed maps. Methods. We develop the wavelet-based weighted cross-correlation (WWCC) method to study the relative contribution of structures of different sizes and their degree of correlation in two maps as a function of spatial scale, and the mutual displacement of structures in the molecular cloud maps. Results. We test the WWCC for circular structures having a single prominent scale and fractal structures showing a self-similar be- haviour without prominent scales. Observational noise and a finite map size limit the scales on which the cross-correlation coefficients and displacement vectors can be reliably measured. For fractal maps containing many structures on all scales, the limitation from ob- servational noise is negligible for signal-to-noise ratios ∼>5. We propose an approach for the identification of correlated structures in the maps, which allows us to localize individual correlated structures and recognize their shapes and suggest a recipe for recovering enhanced scales in self-similar structures. The application of the WWCC to the observed line maps of the giant molecular cloud G 333 allows us to add specific scale information to the results obtained earlier using the principal component analysis. The WWCC confirms the chemical and excitation similarity of 13CO and C18O on all scales, but shows a deviation of HCN at scales of up to 7 pc. This can be interpreted as a chemical transition scale. The largest structures also show a systematic offset along the filament, probably due to a large-scale density gradient. Conclusions. The WWCC can compare correlated structures in different maps of molecular clouds identifying scales that represent structural changes, such as chemical and phase transitions and prominent or enhanced dimensions. Key words. methods: data analysis – methods: statistical – ISM: structure – ISM: clouds 1. Introduction tracers, to identify those scales it is essential to compare the scal- ing behaviour of various tracers in a same region. The charac- The interstellar medium has a complex dynamic structure on all teristic scales could be: e.g. chemical transition scales (see e.g. scales (from sub-parsecs to at least tens of parsecs) as a result Glover et al. 2010), when comparing molecular line maps of dif- of various physical processes occurring in the multi-scale turbu- ferent species; dynamical scales, for the formation of coherent lent cascade of molecular and atomic gas. High-resolution and structures (Goodman et al. 1998); IR penetration scales which high dynamic ranges of observations of emission line transitions show up in dust continuum maps at different wavelengths (e.g. and continuum emission of interstellar clouds provide evidence Abergel et al. 1996); ambipolar diffusion scales for dynamical for clumpy structures on all scales (see e.g. Stutzki & Guesten coupling between ionized and neutral particles, seen when com- 1990; Roman-Duval et al. 2011) and anisotropic clouds such as paring maps of ionized and neutral species (McKee et al. 2010; shells and filaments (e.g. Men’shchikov et al. 2010; Deharveng Li et al. 2012); and dissipation scales, when comparing channel et al. 2010). Observations of line transitions, total, and polarized maps of individual atomic or molecular lines (Falgarone et al. continuum emission provide valuable information about physi- 1998). cal conditions (density, temperature, magnetic field) and kine- To address these issues, we use two different starting points. matics of a multi-phase gas distributed on different scales. The wavelet analysis has been proven to be a powerful tool The correlation between different structures measured in in- for detecting structures on different spatial scales in time se- terstellar clouds, e.g. contours of different chemical tracers or the ries and in two-dimensional maps. This analysis was used e.g. density, temperature, and velocity peaks, as a function of their in the Δ-variance analysis measuring the amount of structure in size can be used to quantify commonalities and differences in the a molecular cloud map as a function of scale and to determine formation of these structures. Understanding the commonalities the slope of the power spectrum of the cloud scaling (Stutzki and differences seen in different tracers, different velocity com- et al. 1998). A combination of a wavelet filtering with the cross- ponents, or different excitation conditions as a function of scale correlation function was first proposed by Nesme-Ribes et al. length helps to infer the underlying physical processes in the (1995) to study solar activity. Frick et al. (2001) improved the turbulent cascade in interstellar clouds. As many processes have method introducing wavelet cross-correlation function to study characteristic scales, but only show their signatures in particular the correlation between galactic images as a function of scale. Article published by EDP Sciences A98, page 1 of 24 A&A 585, A98 (2016) This method and its modifications were successfully applied to by computing its variance study solar physics, ionosphere fluctuations, images of astro- nomical objects, and all-sky surveys (e.g. Vielva et al. 2006; 2 2 σΔ(l) = [F(x, l) − F(l)] dx, (2) Liu & Zhang 2006; Tabatabaei & Berkhuijsen 2010; Roux et al. 2012; Tabatabaei et al. 2013). x,y These approaches, however, are not designed for recovering ∗ the displacement of structures in the data sets on scale-by-scale where the symbol represents the convolution and F(l)istheav- basis and accounting for a variable noise distribution across the erage intensity of the map. For uniform, zero average data with- maps and irregular boundaries. Patrikeev et al. (2006)usedan out boundaries, this is equivalent to the wavelet power spectrum. anisotropic wavelet transform to isolate spiral features in images The wavelet is composed of a positive core, ψc,andanega- of M 51 for different tracers and analysed their location and pitch tive annulus, ψa i.e. angle as a function of radius and azimuth, but they did not cross- ψ(x, l) = ψ (x, l) + ψ (x, l). (3) correlate the features seen in different tracers. The wavelet-based c a cross-correlation method can be improved by inheriting the cor- Core and annulus are normalized to an integrated weight of responding formalism from the wavelet-based Δ-variance anal- unity, so that the integral over the whole wavelet cancels to zero. ysis and combining it with the cross-correlation analysis. To ac- For a fast computation of the convolution in Eq. (1), Stutzki count for an uneven distribution of noise in the maps, Ossenkopf et al. (1998) used the multiplication in Fourier space, but Bensch et al. (2008a, hereafter O08) implemented a weighting function et al. (2001) showed that this can lead to considerable errors in the improved Δ-variance method to analyse arbitrary data sets from edge effects due to the incompatibility of the implicit as- of molecular clouds. The weighting function corrects for the sumption of periodic maps in the Fourier transform and the contribution of data points with a varying signal-to-noise ratio boundaries of real observed maps. Bensch et al. (2001) sug- and allows us to perform a proper treatment of map edges, inde- gested a filter that changes its shape closer to the map boundaries pendent of their shape, and calculations in Fourier space, which by truncating it beyond the map edges and changing the ampli- speeds up the computation by making use of the fast Fourier tude of core and annulus to retain the wavelet normalization con- transform. Moreover, O08 suggested an optimal shape of the dition. While this improves the edge treatment, its weakness is wavelet filter for analysing the turbulent structures. that the convolution of the map can no longer be performed in We make use of the advantages of weighting functions and Fourier space. develop a weighted wavelet cross-correlation (WWCC) method Ossenkopf et al. (2008a) suggested a modification that over- to recover the correlation and displacement between structures comes the edge treatment problems, and simultaneously deals of molecular clouds as a function of scale, in which no assump- with the effect of observational uncertainties and irregular map tion about the noise or boundaries are made. boundaries and allows for a computation in the Fourier domain. The paper is organized as follows. Section 2 introduces the To fix the filter function, they increase the map size when the Δ-variance and cross-correlation methods. The WWCC method filter extends beyond the map edges and apply a zero-padding to is presented in Sect. 3. The application of the WWCC to sim- the extended map area. The re-normalization of the filter is ac- ulated circular structures and fractal structures is described in complished by introducing the complementary weighting func- Sect. 4. The application of the WWCC to observed emission line tion w(x) > 0 inside the valid map and w(x) = 0 in the zero- maps of the giant molecular cloud G 333 is presented in Sect. 5. padded region. When using a generalized weighting function The discussion and conclusions are presented in Sect. 6. 0 ≤ w(x) ≤ 1, this can characterize the significance of every individual data point, including the effects of noise and obser- 2.