variance of the column density, $\langle {\cal N}^2 \rangle - \langle {\cal N} \rangle^2$, could be very large in the inhomogeneous medium, we substantially overestimate the optical depth if we use Eqn. (\[eq:mean\]).

We can therefore in principle improve the approximation by estimating the variance of the column density. According to the central limit theorem, the variance of the column density for large $N_x$ can be expressed by using the variance of the density, if the density, $n_{i, j}$, is a sequence of independent and identically distributed random variables: $$\langle {\cal N}^2 \rangle - \langle {\cal N} \rangle^2 = N_x \left[\langle n^2 \rangle - \langle n \rangle^2\right]. 
\label{eq:centrallimit}$$ Using this relation, the mean intensity of the emerging radiation can be approximated as: $$\begin{aligned}
\langle I_{\rm out} \rangle_{\rm variance} &=& I_0 \biggl[\exp\left(-\sigma_{\rm HI} \langle {\cal N} \rangle \right) \nonumber \\
    && \hspace{1cm} + \frac{\sigma_{\rm HI}^2 N_x}{2}\left(\langle n^2 \rangle - \langle n \rangle^2\right)\biggl]. 
    \label{eq:smapprox}\end{aligned}$$ The effective column density for a ray segment that intersects the supermesh is hence $${\cal N}_{\rm eff} = - \frac{\ln\left[\exp\left(-\sigma_{\rm HI} \langle n \rangle h\right) 
+ \frac{\sigma_{\rm HI}^2 h}{\Delta L} \left(\langle n^2 \rangle - \langle n \rangle^2\right)\right]}{\sigma_{\rm HI}}, 
\label{eq:effectivecolumn}$$ where $h$ is the length of a ray segment. 