# COMS Visible Channel Calibration Using Moon Observation Data
## 2 Data and Methods
### Methodology
#### 2.2.1 Computation of the ROLO Irradiance  
ROLO measurements deliver very good relative performances from one measurement condition to another. However, its ability to provide absolute accuracy is not reliable; therefore, the ROLO model is best employed using a multi-temporal approach for relative calibration change. With a long-term set of observations, relative sensor response trending retrieved by ROLO will have residuals approaching 0.1%. To monitor the long-term behavior of the instrument, we defined a coefficient, P(t), in Equation (1) that was solely dependent on the instrument response. At a given time, the instrument state could be used as a reference point, and the Moon irradiance value was then used to monitor instrument evolution with respect to this reference. \[\mathrm{P(t)}=\frac{\mathrm{I_{Instrument}}}{\mathrm{I_{ROLO}}}, \tag{1}\]  
where \(\mathrm{I_{Instrument}}\) is Moon irradiance measured by the sensor, and \(\mathrm{I_{ROLO}}\) is Moon irradiance computed from the ROLO model under the same conditions. Data inputs for the irradiance model (\(\mathrm{I_{ROLO}}\)) were derived from processed ROLO lunar images that were calibrated to exo-atmospheric radiance and totaled using all pixels on the lunar disk (including the unilluminated portions); this provided disk-equivalent irradiance. For a given wavelength, ROLO irradiance is a function of the effective disk albedo,  
\[\mathrm{I(\lambda)}=\mathrm{A_{\lambda}}\;\times\;\mathrm{E_{Sun}(\lambda)} \;\times\;\mathrm{\Omega_{M}}\;/\;\pi, \tag{2}\]  
where \(\mathrm{A_{\lambda}}\) is the disk-equivalent reflectance (albedo), \(\mathrm{\Omega_{M}}\) is the solid angle of the Moon at a standard distance (=6.4236 \(\times\) 10\({}^{-5}\) steradian), and \(\mathrm{E_{Sun}(\lambda)}\) is solar spectral irradiance (W/m\({}^{2}\cdot\upmu\)m) [21]. The standard Moon-Sun and Moon-Earth distances are 1 AU and 384,400 km, respectively. For a given  
Figure 1: Level 1A (Local Area (LA) mode) Moon observation data (**a**) Normal Moon observation data (**b**) Moon image with Earth on the edge of LA Image (1–2 times annually). detector, \(\mathrm{E_{SUN}}\) is defined as the average spectral solar irradiance over the spectral response function for the detector,  
\[\mathrm{E_{SUN}}=\ \frac{\int_{\lambda_{1}}^{\lambda_{2}}\mathrm{E_{Sun}(\lambda)} \ \times\ \mathrm{R(\lambda)}\ \mathrm{d\lambda}}{\int_{\lambda_{1}}^{\lambda_{2}}\mathrm{R( \lambda)}\ \mathrm{d\lambda}}\Big{(}\mathrm{W\!\cdot\!m^{-2}\!\cdot\!\mathrm{\upmu m^{-1}} \Big{)}}, \tag{3}\]  
where \(\lambda\) is a wavelength (\(\upmu\)m) in the range [\(\lambda_{1}\), \(\lambda_{2}\)] over which the detector has a significant quantum efficiency, \(\mathrm{R(\lambda)}\) is the system relative spectral response for the band, and the limits for the integrals are the wavelengths over which the spectral response is measured. To calculate the ROLO equivalent irradiance (\(\mathrm{I_{ROLO\_E}}\)) over the spectral band of the instrument, the unitary irradiances were averaged through the wavelength band of the instrument,  
\[\mathrm{I_{ROLO\_E}}=\ \frac{\int_{\lambda_{1}}^{\lambda_{2}}\mathrm{I( \lambda)}\times\Phi(\lambda)\ \mathrm{d\lambda}}{\int_{\lambda_{1}}^{\lambda_{2}}\Phi(\lambda)\ \mathrm{d \lambda}}\Big{(}\mathrm{W\!\cdot\!m^{-2}\!\cdot\!sr^{-1}\cdot\upmu m^{-1}} \Big{)}, \tag{4}\]  
where \(\mathrm{I(\lambda)}\) is Irradiance (\(\mathrm{W/m^{2}\!\cdot\!sr\!\cdot\!\upmu m}\)) and \(\Phi(\lambda)\)is the detector spectral response (unitless). The empirical analytical form of the equivalent reflectance of the entire lunar disk (regardless of illuminated fraction) is a function of the primary geometric variables of illumination and observation of the Moon for a given wavelength [21],  
\[ \begin{split}\mathrm{lnA_{\lambda}}=\ \sum_{i=0}^{3}\mathrm{a_{ik}g^{i} }\ +\ \sum_{j=0}^{3}\mathrm{b_{jk}}\Phi^{2j-1}\ +\ C_{1}\theta+\ C_{2}\phi+\ C_{3}\phi\theta+\ C_{4}\Phi\phi+\\ \mathrm{d_{1k}e^{-g/P_{1}}}+\ \mathrm{d_{2k}e^{-g/P_{2}}}+\ \mathrm{d_{3k} \cos}\big{[}(\mathrm{g-P_{3}})/\ \mathrm{P_{4}}\big{]}\,\end{split}