  We use the COMPLETE Survey's observations of the Perseus star-forming region
to assess and intercompare three methods for measuring column density in
molecular clouds: extinction mapping (NIR); thermal emission mapping (FIR); and
mapping the intensity of CO isotopologues. The structures shown by all three
tracers are morphologically similar, but important differences exist.
Dust-based measures give similar, log-normal, distributions for the full
Perseus region, once careful calibration corrections are made. We also compare
dust- and gas-based column density distributions for physically-meaningful
sub-regions of Perseus, and we find significant variations in the distributions
for those regions. Even though we have used 12CO data to estimate excitation
temperatures, and we have corrected for opacity, the 13CO maps seem unable to
give column distributions that consistently resemble those from dust measures.
We have edited out the effects of the shell around the B-star HD 278942. In
that shell's interior and in the parts where it overlaps the molecular cloud,
there appears to be a dearth of 13CO, likely due either to 13CO not yet having
had time to form in this young structure, and/or destruction of 13CO in the
molecular cloud. We conclude that the use of either dust or gas measures of
column density without extreme attention to calibration and artifacts is more
perilous than even experts might normally admit. And, the use of 13CO to trace
total column density in detail, even after proper calibration, is unavoidably
limited in utility due to threshold, depletion, and opacity effects. If one's
main aim is to map column density, then dust extinction seems the best probe.
Linear fits amongst column density tracers are given, quantifying the inherent
uncertainties in using one tracer (when compared with others). [abridged]
