... density

The HVC component was computed by integrating over the channels between -168.5 and -96.4 km s^-1.

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... broadening

Here we consider that the spectral response follows a Gaussian function.

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... constant

Here we make the assumption that the depth of the cloud is small compared to the distance to the cloud, which allows us to do the integral in Cartesian coordinates and not take into account the fact that a beam sees structures of increasing physical size with distance.

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... maps

The standard deviation of  $\sigma(x,y)$ is not given here and not used in this analysis as it is more affected by noise.

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... fields

As shown by Brunt & Low (2004) and Esquivel & Lazarian (2005) this is only true for low contrast density fields ( $\sigma_n/\langle n\rangle \leq 1$).

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... them

Miville-Deschênes et al. (2003b); Brunt & Low (2004) showed that the level of correlation between the density and velocity does not modify significantly the statistics of projected quantities.

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... spectrum

From our tests on cubes with 0.1<H/L<10 and a spectral index $\gamma=-3.4$ we did not observe significant changes of the power spectrum after the modification of the fBm using Eq. (11).

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...$\sigma_n/\langle n\rangle$)

Both the average density $\langle n\rangle$ and the standard deviation of thedensity $\sigma_n$ depend on D but not their ratio.

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... values

These relatively low contrasts give an indication that the velocity power spectrum obtained from the centroid velocity is not affected significantly by density fluctuations.

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