... structure

When dealing with projected quantities, one has to carefully distinguish correlation functions and power spectra. When a power-law approximation is good for both of them, the spectral index of correlation functions gets steeper by one due to projection, while the 2D projected power spectrum retains the spectral index of the underlying 3D spectrum.

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... cloud

From here on we drop the index z in the notation for the line-of-sight component of the velocity because we consider only this component.

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...Lazarian & Esquivel (2003)

In contrast to the original definition, we have not included the constant factor X in the centroid definition so that the weighted centroids have the dimension of a velocity-times-column-density here instead of velocity-times-intensity. This keeps the equations in the following sections somewhat shorter.

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...)

As discussed by Stutzki et al. (1998) it is easy to show that this fBm property violates the often used hypothesis that the fractal dimension decreases by one in projection (Peitgen & Saupe 1988).

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... a power law

For spatial separations corresponding to wavenumbers smaller than the cut-off wavenumber given by the finite sampling of any system.

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...$\Delta$-variance

The same applies to the structure function, but in a limited spectral range.

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...%

The exact value of the convergence criterion is not important, because it only changes the number of required iterations. We found that the results obtained for smaller error limits cannot be distinguished by eye from the 1% limit results.

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