## 2 Expressions Correction and Monte Carlo Verification  
We first comment on several algebraic errors in the equations published in [12], where Equations (A1) and (A2) express the transmittance and the photon-path length of the transmitted photons, respectively, in terms of the optical properties of an isotropic scattering medium. These equations should have appeared as follows, where the corrections are underserved:  
\[\Gamma_{T}(r)=  
There also are two errors in the previously-published Equations (12b) and (23) [12]. The amended equations are shown below, again with the changes underscored:  
\[\Gamma_{T}(n,\rho)\approx\frac{e^{-\mu n}}{4\pi L}\sum_{m=1}^{L}(-1)^{m+1}[1- \cos(2\pi m\,/\,L)]\times e^{-(n-2)\underline{n}^{2}/6\,\underline{l}^{2}}( \frac{1}{n-2})e^{-3\,\underline{p}^{2}/2(n-2)} \tag{3}\]  
\[\Gamma_{T}(l)\approx\frac{e^{-\Sigma_{J}}}{6(L+1)}\sum_{m=1}^{L+1}(-1)^{m+1} \{[1-\cos(2\pi m\,/\,L+1)]\}\times(\frac{1}{3}\{2+\cos[\pi m\,/\,(L+1)]\})^{ \underline{n}_{m}-1} \tag{4}\]  
Correspondingly, there is a typo in the captions of Figures 2 and 3 in [12], where the parameter \(\mathcal{L}\) is in error by a factor of\(\sqrt{2}\). For example, \(\mathcal{L}\) should be 10 instead of \(10\sqrt{2}\) for the results shown in Figure 1.  
Second, many applications involve non-absorbing media, for which it is desirable to have a simple expression for the transmittance (and reflectance), including for the general case of anisotropic scattering. Equation (A4) in [12] provides an expression for the transmittance for an absorbing, isotropic scattering medium, _viz._,  
\[T_{0}(\Sigma_{a},\Sigma_{s})=\frac{e^{-2\Sigma_{a}/\Sigma_{s}}}{\sqrt{24 \Sigma_{a}\,/\,\Sigma_{s}}}[\frac{\cosh(\sqrt{24\Sigma_{a}\,/\,\Sigma_{s}})-1 }{\sinh(3\Sigma_{a}\Sigma_{s}d^{*})}] \tag{5}\]  
where \(\Sigma_{s}\) and \(\Sigma_{a}\) are the scattering and absorption coefficients of the slab, respectively, and \(d^{\prime}\) is the corrected sample thickness defined as \(d^{\prime}=d+\sqrt{2}\,\Sigma_{s}^{-1}\), where \(d\) is the physical thickness of the slab. It can be shown that the limit of Equation (A4) when \(\Sigma_{a}\)\(>\) 0 is:  
\[\Gamma_{o}=\frac{\sqrt{2}}{d\Sigma_{s}+\sqrt{2}}=\frac{\sqrt{2}}{OD+\sqrt{2}} \tag{6}\]  
where \(OD\) is the optical depth of the media defined as the product of the slab thickness (\(d\)) and the scattering coefficient (\(\Sigma_{s}\)). Figure 2 shows the transmittance predicted by Equation (6) compared with Monte Carlo simulations for both isotropic and anisotropic media. Mie theory was used to calculate the  
Figure 1: Comparison of the radial distribution of transmitted photons at large g.  
phase functions in the Monte Carlo simulations. For the cases of anisotropic media, the scattering coefficients were scaled using a similarity relationship developed for RWT [10, 11]. Following this similarity relationship, an effective scattering coefficient ( \(\Sigma_{s}\) ) was calculated according to \(\Sigma_{s}^{{}^{\prime}}=(1-g)\Sigma_{s}\Big{/}\sqrt{1+g}\) and used in Equation (6), where g is the anisotropy factor of the medium. The data shown in Figure 2 were obtained by simulating the multiple scattering process and recording the transmitted photons using a \(\pi\)/2 acceptance angle. The simulations were conducted for various particle concentrations assuming the following scatterers with various g: micron-sized water droplets in air (g = 0.77), polystyrene particles in water (g = 0.92), and hydrocarbon fuel droplets in air (g = 0.55). Note that the different phase functions can have the same g. We have performed simulations with different phase functions with the same g, and the results indicated that the transmission is less sensitive to the exact profile of the phase function than to g. As can be seen, Equation (6) predicts the transmittance with good accuracy across a wide range of conditions (in terms of \(OD\) and type of scatterer). We expect it to be a useful expression in the study of multiple scattering problems, especially in the design of experiments.