## 2 Materials and Methods
### Skin Model  
Both Monte Carlo datasets were created using a two-layer skin model similar to the one proposed by [15], with an epidermis thickness of 60 \(\upmu\)m [30] and a dermis thickness of 2 mm. This model matches the one proposed by [PERSON] et al. and is consistent with the approximate maximum penetration depth of visible light in skin [37]. The upper layer is assumed to contain only melanin with volume fraction \(f_{mel}\) and the lower layer to contain only blood with volume fraction \(f_{blood}\). Other skin contents (e.g., water and fat) are ignored, since their absorption coefficients are orders of magnitude lower than those of melanin and hemoglobin in the 520-600 nm range of interest [38].  
The absorption coefficients for 100% melanin in the epidermis layer are given by [PERSON] and [PERSON] [39] as:  
\[\mu_{a,\ epi}(\lambda)=5\times 10^{10}\lambda^{-3.33}\;(\text{cm}^{-1}) \tag{3}\]  
Figure 1: Effects of convolution with a 15 nm FWHM Gaussian filter. (**a**) Example spectrum before and after convolution. The other plots show molar extinction coefficients for oxy- and deoxyhemoglobin (**b**) before and (**c**) after convolution. Convolution flattens and broadens the distinctive peaks in the oxyhemoglobin curve, most significantly in the higher wavelength lobe, bringing it closer to the deoxyhemoglobin curve.  
where the wavelength, \(\lambda\), is in nanometers. These values are multiplied by \(f_{mel}\) to arrive at the actual absorption of light in the epidermis. The value of \(f_{mel}\) can range from 1.3% to 6.3% for light-skinned adults, 11% to 16% for moderately pigmented adults, and 18% to 43% for darkly pigmented adults [30]. Absorption in the dermis is modeled as a combination of absorption by deoxyhemoglobin (Hb) and oxyhemoglobin (HbO2):  
\[\mu_{a,\ dermis}(\lambda)=\varepsilon_{HbO2}(\lambda)C_{HbO2}+\varepsilon_{Hb} (\lambda)C_{Hb} \tag{4}\]  
where \(\varepsilon_{HbO2}\) and \(\varepsilon_{Hb}\) are the absorption coefficients for Hb and HbO2, respectively, given by [PERSON] [40], and \(C_{HbO2}\) and \(C_{Hb}\) are the concentrations of these two substances in the blood. This equation can be expressed in a more convenient form that clearly illustrates the role of SO2 in the absorbance:  
\[\mu_{a,\ dermis}(\lambda)=((\varepsilon_{HbO2}(\lambda)-\varepsilon_{Hb}( \lambda))SO_{2}+\varepsilon_{Hb})T \tag{5}\]  
where \(T\) is the total concentration of hemoglobin in the blood and \(SO_{2}=C_{HbO2}/T\).  
Scattering coefficients, the same used for both the epidermis and the dermis, are provided by [PERSON] et al. [41] based on experimentation with intralipid 10%. These values were found to closely match the scattering behavior of biological tissues found by [PERSON] et al. [42]. Figure 2 shows the absorption and scattering coefficients plotted as functions of wavelength over the visible range. Anisotropy factors, also provided by [PERSON] et al. [41], are computed as:  
\[g(\lambda)=1.1-0.58\times 10^{-3}\lambda\ (\text{cm}^{-1}) \tag{6}\]  
Given the small variation in the wavelength over the range of interest, the index of refraction was set to a constant 1.4 for both layers [22; 43].  
Figure 2: Scattering and absorption coefficients for the skin model used in this study.