  We study how the estimation of the sonic Mach number ($M_s$) from $^{13}$CO
linewidths relates to the actual 3D sonic Mach number. For this purpose we
analyze MHD simulations which include post-processing to take radiative
transfer effects into account. As expected, we find very good agreement between
the linewidth estimated sonic Mach number and the actual sonic Mach number of
the simulations for optically thin tracers. However, we find that opacity
broadening causes $M_s$ to be overestimated by a factor of ~ 1.16 - 1.3 when
calculated from optically thick $^{13}$CO lines. We also find that there is a
dependency on the magnetic field: super-Alfv\'enic turbulence shows increased
line broadening as compared with sub-Alfv\'enic turbulence for all values of
optical depth for supersonic turbulence. Our results have implications for the
observationally derived sonic Mach number--density standard deviation
($\sigma_{\rho/<\rho>}$) relationship, $\sigma^2_{\rho/<\rho>}=b^2M_s^2$, and
the related column density standard deviation ($\sigma_{N/<N>}$) sonic Mach
number relationship. In particular, we find that the parameter b, as an
indicator of solenoidal vs. compressive driving, will be underestimated as a
result of opacity broadening. We compare the $\sigma_{N/<N>}-M_s$ relation
derived from synthetic dust extinction maps and $^{13}$CO linewidths with
recent observational studies and find that solenoidally driven MHD turbulence
simulations have values of $\sigma_{N/<N>}$ which are lower than real molecular
clouds. This may be due to the influence of self-gravity which should be
included in simulations of molecular cloud dynamics.
