## 2 Photometric and Melanopic Relationships among Different Standards
### Theoretical Considerations
If the photometric quantity is defined by E\({}_{\rm photopic,D65}=\rm K_{m}\times E_{e,photopic,D65}\), the same melanopic, \(\rm E_{ melanopic,D65}=\rm K_{ melanopic,D65}\times E_{e, melanopic,D65}\) for light source D\({}_{65}\) can be described in terms of equality as follows:  
\[ \begin{split}\rm E_{photopic,D65}=\rm E_{ melanopic,D65}\to 683\times\int_{\lambda=380}^{780}\rm SPD_{D65}( \lambda)\times V(\lambda)d\lambda=\ K_{ melanopic,D65}\times\\ \frac{\int_{\lambda=380}^{780}\rm SPD_{D65}(\lambda)\times S_{mel} (\lambda)d\lambda}{\int_{\lambda=380}^{780}\rm SPD_{D65}(\lambda)\times V( \lambda)d\lambda}\times\int_{\lambda=380}^{780}\rm SPD_{D65}(\lambda)\times V( \lambda)d\lambda\to K_{ melanopic,D65}=\frac{683}{0.906}=\\ 753.8\ \rm Im/W_{ melanopic,D65}.\end{split}  \tag{8}\]  
For any source:  
\[\rm EDI=753.8\times\int_{\lambda=380}^{780}\rm SPD(\lambda)\times S_{mel}( \lambda)d\lambda. \tag{9}\]  
In addition, EML's relation with photometric illuminance can be deduced as follows:  
\[\rm EDI=1.104\times MAF\times E_{photopic}. \tag{10}\]  
Using these conversion factors and photopic values of illuminance, transformations from one melanopic metric to another are easily calculated:  
\[\rm EML=1.104\times EDI. \tag{11}\]  
Similarly, [PERSON] et al. [13; 28] mathematically define the modelled spectral sensitivity of the human circadian system by the circadian light (\(\rm CL_{A}\)) expression in terms of the spectrally weighted lux per unit area. In this model, all known photoreceptors contribute to the spectral sensitivity of the circadian system [29]. There is a sudden transition in the modelled spectral efficiency at 497 nm. The \(\rm CL_{A}\) efficiency at shorter wavelengths reflects both ipRGC-melanopsin and S-cone sensitivities, whereas efficiency at longer wavelengths is modelled by the ipRGC-melanopsin spectral sensitivity alone. By definition, 1000 lux of CIE illuminant A equals 1000 lux on the \(\rm CL_{A}\) scale and is identified by the subscript \"A\", as in Equation (12):  
\[\rm CL_{A}=\left\{ \begin{array}{l}1548\left[\int M_{\rm CA}SPD_{ \lambda}d\lambda+\left(\begin{array}{c}\rm a_{b-y}\left(\int\frac{S_{h}}{ \rm m_{2}}\rm SPD_{\lambda}d\lambda-k\int\frac{V_{h}}{\rm m_{2}}\rm SPD_{ \lambda}d\lambda\right)\\ -\rm a_{rot}\left(1-e\frac{\int V_{h}\rm SPD_{\lambda}d\lambda}{\rm d\lambda} \right)\end{array}\right)\right]\text{if }f\int\frac{S_{h}}{\rm m_{2}}\rm SPD_{\lambda}d\lambda-k\int\frac{V_{h}}{\rm m _{2}}\rm SPD_{\lambda}d\lambda>0\\ 1548\int M_{\rm CA}SPD_{\lambda}d\lambda\qquad\text{if }\int\frac{S_{h}}{\rm m _{2}}\rm SPD_{\lambda}d\lambda-k\int\frac{V_{h}}{\rm m_{2}}\rm SPD_{\lambda}d \lambda\leq 0\\ \text{CS}=0.7-\frac{0.7}{1+\left(\frac{\rm CL_{A}}{\rm 0.85}\right)^{1.102 }}.\end{array} \right. \tag{12}\]  
\(\rm CL_{A}\) is the circadian light and CS is the circadian stimuli. The constant, 1548 lm/W, sets the normalization of \(\rm CL_{A}\) so that 2856 K blackbody radiation at 1000 lux has a \(\rm CL_{A}\) value of 1000 lux. \(\rm SPD_{\lambda}\) is the light source spectral irradiance distribution; \(\rm MC_{\lambda}\) is the melanopsin (corrected for crystalline lens transmittance) [30; 31]; \(\rm S_{\lambda}\) is the S-cone fundamental; \(\rm mp_{\lambda}\) is the macular pigment transmittance; \(\rm V_{\lambda}\) is the photonic luminous efficiency function; \(\rm V^{*}_{\lambda}\) is the scotopic luminous efficiency function; \(\rm RodSat\) is the half-saturation constant for bleaching (\(\rm rods=6.5\ W/m^{2}\), k = 0.2616, \(\rm a_{b-y}=0.700\), and \(\rm a_{rot}=3.300\)). The WELL standard, based on the model developed by [PERSON] et al. [15], and the CIE model assume equal-energetic and D\({}_{65}\) theoretical illuminants, respectively, as references for their calculations. This approach represents one of the main conceptual differences of these proposals from the [PERSON] et al. model, which adopts the A illuminant as its standard.