Turbulence plays a major role in the formation and evolution of molecular clouds .
The problem is that turbulent velocities are convolved with the density of an observed region .
To correct for this convolution , we investigate the relation between the turbulence spectrum of model clouds , and the statistics of their synthetic observations obtained from Principal Component Analysis ( PCA ) .
We apply PCA to spectral maps generated from simulated density and velocity fields , obtained from hydrodynamic simulations of supersonic turbulence , and from fractional Brownian motion fields with varying velocity , density spectra , and density dispersion .
We examine the dependence of the slope of the PCA pseudo structure function , \alpha _ { \mathrm { PCA } } , on intermittency , on the turbulence velocity ( \beta _ { v } ) and density ( \beta _ { n } ) spectral indexes , and on density dispersion .
We find that PCA is insensitive to \beta _ { n } and to the log-density dispersion \sigma _ { s } , provided \sigma _ { s } \leq 2 .
For \sigma _ { s } > 2 , \alpha _ { PCA } increases with \sigma _ { s } due to the intermittent sampling of the velocity field by the density field .
The PCA calibration also depends on intermittency .
We derive a PCA calibration based on fBms with \sigma _ { s } \leq 2 and apply it to 367 ^ { 13 } CO spectral maps of molecular clouds in the Galactic Ring Survey .
The average slope of the PCA structure function , \langle \alpha _ { \mathrm { PCA } } \rangle = 0.62 \pm 0.2 , is consistent with the hydrodynamic simulations and leads to a turbulence velocity exponent of \langle \beta _ { v } \rangle = 2.06 \pm 0.6 for a non-intermittent , low density dispersion flow .
Accounting for intermittency and density dispersion , the coincidence between the PCA slope of the GRS clouds and the hydrodynamic simulations suggests \beta _ { v } \simeq 1.9 , consistent with both Burgers and compressible intermittent turbulence .