deletions | additions       diff --git a/subsection_Fourier_Statistics_VCA_VCS__.tex b/subsection_Fourier_Statistics_VCA_VCS__.tex  index a8c78de..4bf03f5 100644  --- a/subsection_Fourier_Statistics_VCA_VCS__.tex  +++ b/subsection_Fourier_Statistics_VCA_VCS__.tex 
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Velocity Channel Analysis (VCA) and Velocity Coordinate Spectrum (VCS) are techniques that isolate how fluctuations in velocity contribute to differences between spectral cubes (cite Authorea text?) by fitting a one-dimensional Power Spectrum to power laws. For runs W1T2t0.2 and W2T2t0, we compute the three-dimensional Power Spectrum of their the Fourier transforms. To obtain the VCA, we calculate a one-dimensional Power spectrum by integrating the 3D Power Spectrum over the velocity channels, and then radially averaging the obtained 2D Power Spectrum over the two-dimensional spatial frequencies. A portion (trying to emphasize how only a portion is fit to the power law) of the now 1-D Power Spectrum is then fit to a single-power law. We derive the VCS from different integration techniques. Here, we reduce the 3-D Power Spectrum to one dimension by averaging over the spatial frequencies. Since the VCS yields two distinct power law relations, we fit the 1D power spectrum to a segmented linear model similar to Koch et al. (2015), (using the method established by ???). Thus, VCA yields a 1D Power Spectrum as a function of spatial frequency, while VCS yields a 1D Power Spectrum as a function of velocity-channel frequency (frequency equivalent of velocity).  We show the VCA and VCS  power law fits for VCA in Figure 5, runs W1T2t0.2  and VCS W2T2t0  in Figure Figures 5 and  6. VCA generates similarly sloped power laws for our Power Spectra, but there is also a constant horizontal offset between them. VCS produces similar shaped  Feedback appears to be injecting energy into the turbulent molecular cloud.