  We present a $^{13}\mathrm{CO} (J = 1 \rightarrow 0)$ mapping survey of 12
nearby galaxies from the CARMA STING sample. The line intensity ratio
$\mathcal{R} \equiv I[^{12}\mathrm{CO} (J = 1 \rightarrow
0)]/I[^{13}\mathrm{CO} (J = 1 \rightarrow 0)]$ is derived to study the
variations in molecular gas properties. For 11 galaxies where it can be
measured with high significance, the spatially resolved $\mathcal{R}$ on
(sub-)kiloparsec scales varies by up to a factor of 3--5 within a galaxy. Lower
$\mathcal{R}$ values are usually found in regions with weaker $^{12}\rm CO$. We
attribute this apparent trend to a bias against measuring large $\mathcal{R}$
values when $^{12}\rm CO$ is weak. Limiting our analysis to the $^{12}\rm CO$
bright regions that are less biased, we do not find $\mathcal{R}$ on (sub)kpc
scales correlate with galactocentric distance, velocity dispersion or the star
formation rate. The lack of correlation between SFR and $\mathcal{R}$ indicates
that the CO optical depth is not sensitive to stellar energy input, or that any
such sensitivity is easily masked by other factors. Extending the analysis to
all regions with $\rm ^{12}CO$ emission by spectral stacking, we find that 5
out of 11 galaxies show higher stacked $\mathcal{R}$ for galactocentric radii
of $\gtrsim 1$ kpc and $\Sigma_{\mathrm{SFR}} \lesssim 0.1 \rm \ M_{sun}
yr^{-1} kpc^{-2}$, which could result from a greater contribution from diffuse
gas. Moreover, significant galaxy-to-galaxy variations are found in
$\mathcal{R}$, but the global $\mathcal{R}$ does not strongly depend on dust
temperature, inclination, or metallicity of the galaxy.
