# Some Approaches for Light and Color on the Surface of Mars
## 3 Results
### Modeling Light on Mars' Surface from CCT  
#### 3.2.1 Proposal for a Model Analogous to the One Adopted by CIE for Daylight [3]  
As indicated before, to model light on Mars' surface, we followed the method proposed by [PERSON] et al. [1] and adopted by CIE (clause 4.1.2 in [3]), which was developed from 622 experimental measurements of spectral power distributions on Earth's surface and used CCT as the only input parameter. The main difference in our current model is that it starts from 3139 theoretical spectral irradiances obtained from the COMIMART model (see Section 3.1) rather than from experimental measurements. First, we normalized the 3139 total spectral irradiances to the irradiance at 560 nm. Second, using the method introduced in [17] we computed their CCTs and \(x\),\(y\) chromaticity coordinates. Next, the values of the \(x\) coordinate were fitted to a third-order polynomial of the inverse of CCT with a high accuracy (R\({}^{2}\) = 0.9994), obtaining Equation (2). Depending on the CCT value, the method proposed by CIE [1,3] used two different equations with the same format as Equation (2). However, for lights on Mars' surface, the range in CCT is shorter than that for lights on Earth, and Equation (2) is sufficient to obtain the \(x\) coordinate values from CCT. Equation (3) shows a parabolic function fitting values of the \(y\) coordinate from values of the \(x\) coordinate for the 3139 total spectral irradiances (R\({}^{2}\) = 0.9907). Next, Equations (4) and (5) provide values of coefficients \(M_{1}\) and \(M_{2}\) from \(x\),\(y\) chromaticity coordinates. Finally, Equation (6) provides the spectral power distribution of light on Mars' surface \(S_{M}(\lambda)\) from the mean \(S_{0}(\lambda)\) and two first eigenvectors, \(V_{1}(\lambda)\) and \(V_{2}(\lambda)\), computed by principal component analysis. From these two first eigenvectors, we can explain 99.96% of the information in the original 3139 total spectral irradiance data. Table A1 provides values of functions \(S_{0}(\lambda)\), \(V_{1}(\lambda)\), and \(V_{2}(\lambda)\) from 300 nm to 830 nm in steps of 5 nm. Figure 11 shows the functions \(S_{0}(\lambda)\), \(V_{1}(\lambda)\), and \(V_{2}(\lambda)\), with each one normalized to its corresponding norm. Previously mentioned Equations (2)-(6) constitute our proposal for modeling lights on Mars' surface (sol-light illuminants) for a selected CCT in the range of 2333 K-5868 K (see the Excel file in the Supplementary Materials), and are as follows:  
\[x=\frac{7.1528\times 10^{8}}{T_{cp}^{3}}-\frac{8.3336\times 10^{5}}{T_{cp}^{2}}+ \frac{0.9416\times 10^{3}}{T_{cp}}+0.1845 \tag{2}\]  
\[y=-1.6343\;x^{2}+1.7752\;x-0.0669 \tag{3}\]  
\[M_{1}=\frac{-9.0797+59.0649\;x-36.9329\;y}{-0.3418-1.3063\;x+4.7227\;y} \tag{4}\]  
\[M_{2}=\frac{13.5231+35.7428\;x-73.4435\;y}{-0.3418-1.3063\;x+4.7227\;y} \tag{5}\]  
\[S_{M}(\lambda)=S_{0}(\lambda)+M_{1}V_{1}(\lambda)+M_{2}V_{2}(\lambda) \tag{6}\]  
Table 1 shows basic statistics on the accuracy of the sol-light proposed model (Equations (2)-(6)) for the 3139 total spectral irradiances generated from COMIMART. Specifically, we compared each total spectral irradiance generated by COMIMART with the total spectral irradiance computed from our model (CCT computed by the method in [17]), using GFC (Equation (1)) [24] and \(\Delta E_{\text{u}^{\prime}\text{v}^{\prime}}\) units [25]. Figure 12 shows the total spectral irradiances for the median (0.9999839) and worst (0.9996321) GFC values. As we can see, the accuracy of our proposed model to compute sol-light illuminants is very high. The average distance between original and reconstructed total spectral irradiances is 0.00090 \(\Delta E_{\text{u}^{\prime}\text{v}^{\prime}}\) units, which is below the value of a just-noticeable chromaticity difference at 50% probability (0.0013 \(\Delta E_{\text{u}^{\prime}\text{v}^{\prime}}\) units [25]).