dependence. The side length of each mesh is $\Delta L$ and the ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$ number density of the $(i, j)$-th mesh in the supermesh is $n_{i, j}$. The mean intensity of the emerging radiation is given by $$\langle I_{\rm out} \rangle = \frac{I_0}{N_y} \sum_j^{N_y} \exp[- \sigma_{\rm HI} {\cal N}_j], 
  \label{eq:exact}$$ where $\sigma_{\rm HI}$ is the ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$ cross-section and ${\cal N}_j$ is the ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$ column density of the $j$-th line, i.e. ${\cal N}_j = \sum_i^{N_x} n_{i,j} \Delta L$. [ In our supermesh approximation, we use the mean ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$ number density, $\langle n \rangle = \sum_{i,j} n_{i, j} / (N_x N_y)$, to estimate the mean intensity of the emerging radiation $\langle I_{\rm out} \rangle$. Doing this introduces some error as we will show below. In order to understand the accuracy and the nature of the supermesh approximation, we compare the mean intensity of the emerging radiation by the supermesh approximation to that calculated by using the meshes. We first consider the Taylor series expansion of the mean intensity of the emerging radiation when we solve the RT on the supermesh: ]{} $$\begin{aligned}
\langle I_{\rm out} \rangle_{\rm mean} &=& I_0 
  \exp(-\sigma_{\rm HI}\langle {\cal N} \rangle ) \nonumber \\
  & = 