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Measuring the mass distribution of infrared dark clouds (IRDCs) over the widedynamic range of their column densities is a fundamental obstacle indetermining the initial conditions of high-mass star formation and star clusterformation. We present a new technique to derive high-dynamic-range,arcsecond-scale resolution column density data for IRDCs and demonstrate thepotential of such data in measuring the density variance - sonic Mach numberrelation in molecular clouds. We combine near-infrared data from theUKIDSS/Galactic Plane Survey with mid-infrared data from the Spitzer/GLIMPSEsurvey to derive dust extinction maps for a sample of ten IRDCs. We thenexamine the linewidths of the IRDCs using 13CO line emission data from theFCRAO/Galactic Ring Survey and derive a column density - sonic Mach numberrelation for them. For comparison, we also examine the relation in a sample ofnearby molecular clouds. The presented column density mapping techniqueprovides a very capable, temperature independent tool for mapping IRDCs overthe column density range equivalent to A_V=1-100 mag at a resolution of 2".Using the data provided by the technique, we present the first directmeasurement of the relationship between the column density dispersion,\sigma_{N/<N>}, and sonic Mach number, M_s, in molecular clouds. We detectcorrelation between the variables with about 3-sigma confidence. We derive therelation \sigma_{N/<N>} = (0.047 \pm 0.016) Ms, which is suggestive of thecorrelation coefficient between the volume density and sonic Mach number,\sigma_{\rho/<\rho>} = (0.20^{+0.37}_{-0.22}) Ms, in which the quoteduncertainties indicate the 3-sigma range. When coupled with the results ofrecent numerical works, the existence of the correlation supports the pictureof weak correlation between the magnetic field strength and density inmolecular clouds (i.e., B ~ \rho^{0.5}).

High-dynamic-range extinction mapping of infrared dark clouds: Dependence of density variance with sonic Mach number in molecular clouds