\subsubsection{FCRAO Molecular Line Emission}
We follow the analysis in \citet{Pineda_2008} to derive the equivalent V-band extinction from the $^{12}$CO (1-0) and $^{13}$CO (1-0) line observations. We start by assuming that the $^{12}CO$ is optically thick. We then calculate the excitation temperature using the main beam temperature at the peak of $^{12}CO$:

\begin{equation}
T_{\text{ex}} = \frac{5.5\,\text{[K]}}{\ln{\left(1+5.5\,\text{[K]}/\left(T_{\text{max}}(^{12}\text{CO})+0.82\,\text{[K]}\right)\right)}}\;\;\;\text{,}
\end{equation}

where $T_{\text{max}}(^{12}\text{CO})$ is the brightness temperature at the peak. If the excitation temperature of the $^{13}CO$ (1-0) line is the same as the $^{12}CO$ (1-0), we can calculate the optical depth of the $^{13}CO$ (1-0) line:

\begin{equation}
\tau(^{13}\text{CO}) = - \ln\left[1 - \frac{T_{\text{max}}(^{13}\text{CO})/5.3\,\text{[K]}}{1/\left(e^{5.3\,\text{[K]}/T_{\text{ex}}}-1\right) - 0.16}\right]
\end{equation}

Similarly, here $T_{\text{max}}(^{12}\text{CO})$ is the main beam brightness temperature at the peak of $^{13}CO$. Using the definition of column density in Rohlfs \& Wilson (1996), we can derive the $^{13}CO$ column density from the two equations above:

\begin{equation}
N(^{13}\text{CO}) = 3.0\times10^{14}\;\left[\frac{\tau(^{13}\text{CO})}{1-e^{-\tau(^{13}\text{CO})}}\right]\,\frac{W(^{13}\text{CO})}{1-e^{-5.3\,\text{[K]}/T_{\text{ex}}}}\;\text{[cm}^{-2}\text{]}\;\;\;\text{,}
\end{equation}

where $W(^{13}\text{CO})$ is the integrated intensity along the line of sight in units of [K km/s]. Similarly, we then fit the conversion from both $N(^{13}\text{CO})$ and $W(^{13}\text{CO})$ to the 2MASS extinction.

\begin{equation}
\begin{aligned}
A_V &= X_{N(^{13}\text{CO})}\,N(^{13}\text{CO}) \\
A_V &= X_{W(^{13}\text{CO})}\,W(^{13}\text{CO})
\end{aligned}
\end{equation}

Fig. 4 shows the scatter of the derived extinction based on $N(^{13}\text{CO})$. The dashed line shows the cutoff (lowest) value of column density measurable with $^{13}CO$. This cutoff value is 4.3 mag for the Ophiuchus cloud, compared to 1.67 mag for the Perseus. It is clear that no ``sub-population'' is seen here in the molecular line-based extinction. This suggests that the variation of the conversion factors inherent to the ``sub-populations'' seen in the thermal emission-based extinctions is the result of different dust-to-gas ratios in different parts of the cloud. This is consistent with the limit of using dust tracers for the column density, as presented in \citet{Goodman_2009}. For a region as complicated as the Ophiuchus (where multiple components with very different physical properties present), the variation in the dust-to-gas ratio affects the conversion from dust to the total column density.

Fig. 5 shows the distribution of the $CO$-based extinction. Although the profile does not seem as log-normal as other dust tracers, the mean and the variance agree with the 2MASS extinction measurements.