  Measuring the mass distribution of infrared dark clouds (IRDCs) over the wide
dynamic range of their column densities is a fundamental obstacle in
determining the initial conditions of high-mass star formation and star cluster
formation. We present a new technique to derive high-dynamic-range,
arcsecond-scale resolution column density data for IRDCs and demonstrate the
potential of such data in measuring the density variance - sonic Mach number
relation in molecular clouds. We combine near-infrared data from the
UKIDSS/Galactic Plane Survey with mid-infrared data from the Spitzer/GLIMPSE
survey to derive dust extinction maps for a sample of ten IRDCs. We then
examine the linewidths of the IRDCs using 13CO line emission data from the
FCRAO/Galactic Ring Survey and derive a column density - sonic Mach number
relation for them. For comparison, we also examine the relation in a sample of
nearby molecular clouds. The presented column density mapping technique
provides a very capable, temperature independent tool for mapping IRDCs over
the 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 direct
measurement of the relationship between the column density dispersion,
\sigma_{N/<N>}, and sonic Mach number, M_s, in molecular clouds. We detect
correlation between the variables with about 3-sigma confidence. We derive the
relation \sigma_{N/<N>} = (0.047 \pm 0.016) Ms, which is suggestive of the
correlation coefficient between the volume density and sonic Mach number,
\sigma_{\rho/<\rho>} = (0.20^{+0.37}_{-0.22}) Ms, in which the quoted
uncertainties indicate the 3-sigma range. When coupled with the results of
recent numerical works, the existence of the correlation supports the picture
of weak correlation between the magnetic field strength and density in
molecular clouds (i.e., B ~ \rho^{0.5}).
