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\section{Introduction}  Turbulence in the interstellar medium is ubiquitous and self-similar across many orders of magnitude \citep[]{brandenberg13}. Within molecular clouds, turbulence appears to play an essential role in the star formation process, regulating the efficiency at which stars form, seeding filaments and over-densities, and even potentially setting the stellar initial mass function \citep{padoan13, Offner14}. \citep{padoan13,Offner14}.  While the presence of supersonic motions is readily verified and has been studied using molecular spectral lines for several decades \citep{larson81}, the origin, energy injection scale, means of sustenance, and rate of dissipation remain debated. Moreover, molecular clouds display significant variation in bulk properties, ongoing star formation, and morphology. Consequently, it seems highly likely that these differences impact the turbulent properties of the gas and leave signatures — but if so they are difficult to identify observationally. Detailed study of the turbulence within molecular clouds is confounded by a variety of factors including observational resolution, projection effects, complex gas chemistry, and variable local conditions \citep[e.g.,][and references therein]{beaumont13}. Any one molecular tracer only samples a limited set of gas densities and scales, so that reconstructing cloud kinematics reliably involves assembling a variety of tracers across different densities and scales \citep[e.g.]{gaches15}.  %The gas also changes phase when it is dissociated or ionized,  Many studies of cloud structure instead rely on a single gas tracer like CO, which is bright and exhibits widespread emission that reflects the underlying H$_2$ distribution \citep{bolatto13,heyer15}. Connecting such emission data to underlying turbulence and bulk cloud properties, however, is non-trivial. A variety of statistics have been proposed throughout the literature to characterize spectral data cubes and distill the complex emission information into more manageable 1D or 2D forms \citep[e.g.,]{heyer97,rosolowsky99,rosolowsky08, burkurt09}. \citep[e.g.,]{heyer97,rosolowsky99,rosolowsky08,burkurt09}.  However, in most cases, the utility of the statistic and its interpretation is not well constrained. Numerical simulations, which supply full 6-D information $(x,y,z, v_x, v_y, v_z)$, provide a means to study turbulence and constrain cloud properties. Prior studies have investigated how the turbulent power spectrum, inertial driving range, and fraction of compressive motions have influenced star formation \citep{klessen01,bate09b,federrath08}. Other studies have connected simulated turbulent properties to observables such as CO emission by performing radiative post-processing \citep[e.g.,]{padoan01,bertram14}. In some cases, this procedure is able to identify theoretical models that have good agreement with a given observation. Consequently, the most effective way to study turbulence in molecular clouds is by comparing observations with ``synthetic observations", in which the emission from the simulated gas is calculated via radiative transfer post-processing \citep{Offner2008b,Offner12}. Recently, \citep{yeremi14} and \citp{koch15} performed parameter studies of magnetohydrodynamic simulations in order to assess the sensitivity of common astrostatistics to changes in cloud velocity dispersion, virial parameter, driving scale, and magnetic field strength. They found that some statistics were responsive to changes in the temperature, virial parameter, mach number and inertial driving range. These results suggest that certain statistics may also be sensitive to energy input and environmental variation due to ongoing star formation.