# Radiometric Cross-Calibration of GF-1 PMS Sensor with a New BRDF Model
## 3 Methodology
### Cross-Calibration Based on BRDF Model  
The GF-1 PMS sensor is cross-calibrated in this paper, taking MODIS and OLI as the reference sensor, respectively.  
Based on the proposed BRDF model above, the conventional \(\rho_{R}^{m}\) TOA reflectance of reference sensor has been replaced by the \(\rho_{R}^{m,0}\) term.  
\[\rho_{R}^{m,0}=\rho_{R}^{m}BRDF \tag{7}\]  
The SBAF (spectral band adjustment factors) was calculated to compensate the spectral response difference, and the SBAF coefficient was defined as Equation (8):  
\[SBAF=\frac{\rho_{PMS}^{m}}{\rho_{R}^{m}}=\frac{\rho_{PMS}^{s}}{\rho_{R}^{s}} \tag{8}\]  
where \(\rho_{PMS}^{m}\) and \(\rho_{R}^{m}\) are the measured TOA reflectance for PMS and reference sensor, respectively. These values represent the reflectance directly acquired from multispectral sensor imagery by converting calibrated DNs to TOA reflectance. The \(\rho_{PMS}^{s}\) and \(\rho_{R}^{s}\) are the simulated TOA reflectance for PMS and reference sensor, respectively. They refer to the reflectance obtained from integrating the RSR of a multispectral sensor with a hyper-spectral profile target. The PMS and reference sensor term can be expressed as Equation (9):  
\[\left\{\begin{array}{l}\rho_{PMS}^{s}=\frac{\int\rho(\lambda)RSR_{PMS}( \lambda)d\lambda}{\int RSR_{PMS}(\lambda)d\lambda}\\ \rho_{R}^{s}=\frac{\int\rho(\lambda)RSR_{R}(\lambda)d\lambda}{\int RSR_{R}( \lambda)d\lambda}\end{array}\right. \tag{9}\]  
The hyper-spectral profile \(\rho(\lambda)\) was simulated using a MODTRAN radiative transfer model, according to PMS observation geometry, as well as ground and atmospheric parameters measured over the test site. Then, the SBAF can then be defined as Equation (10):  
\[SBAF=\frac{\rho_{PMS}^{s}}{\rho_{R}^{s}}=\frac{[\int\rho(\lambda)RSR_{PMS}( \lambda)d\lambda]/[\int RSR_{PMS}(\lambda)d\lambda]}{[\int\rho(\lambda)RSR_{R} (\lambda)d\lambda]/[\int RSR_{R}(\lambda)d\lambda]} \tag{10}\]  
Then, the PMS TOA reflectance over the test site can be calculated as Equation (11):  
\[\rho_{PMS}^{m}=SBAF\times\rho_{R}^{m,0} \tag{11}\]  
The PMS TOA radiance can be calculated based on solar zenith angle and TOA reflectance according to the Equation (12):  
\[L_{PMS}^{m}=\frac{\rho_{PMS}^{m}ESUN_{PMS}\cos\theta_{s}}{\pi d^{2}} \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Site** & **Band** & \(\mathbf{f_{iso}}\) & \(\mathbf{f_{geo}}\) & \(\mathbf{f_{vol}}\) \\ \hline \multirow{4}{*}{Dunhuang} & Blue & 0.2864 & 0.0525 & 0.0509 \\  & Green & 0.3041 & 0.0487 & 0.0609 \\ \cline{1-1}  & Red & 0.2032 & \(-\)0.0206 & 0.2616 \\ \cline{1-1}  & NIR & 0.2548 & 0.037 & 0.0967 \\ \hline \multirow{4}{*}{Golmud} & Blue & 0.2839 & 0.0171 & 0.1043 \\ \cline{1-1}  & Green & 0.3213 & 0.0066 & 0.1306 \\ \cline{1-1}  & Red & 0.1684 & \(-\)0.0257 & 0.2344 \\ \cline{1-1}  & NIR & 0.2226 & 0.0079 & 0.1291 \\ \hline \hline \end{tabular}
\end{table}
Table 3: BRDF model parameters based on the MODIS TOA reflectance over the Dunhuang and Golmud test site.  
where \(L_{PMS}^{m}\) is the spectral radiance at the sensor's aperture (W/(m\({}^{2}\) sr \(\upmu\)m)), \(d\) is the Earth-Sun distance measured in astronomical units (AUs), \(ESUN_{PMS}\) represents the mean exo-atmospheric solar irradiance (W/(m\({}^{2}\) \(\upmu\)m)), and \(\theta_{s}\) is the solar zenith angle measured in \({}^{\circ}\).  
The PMS calibration coefficient gain is defined as Equation (13):  
\[gain_{PMS}=\frac{L_{PMS}^{m}}{DN_{PMS}} \tag{13}\]