# Radiometric Cross-Calibration of GF-1 PMS Sensor with a New BRDF Model
## 3 Methodology  
### BRDF Model  
#### 3.1.1 Construction Method of BRDF Model  
In this paper, a new BRDF correction method based on TOA reflectance of time-series MODIS images is proposed for cross calibration which is mainly used for large-view zenith angle image pairs. Compared with the existing BRDF models, there are several advantages of this new BRDF construction method. Firstly, this method uses TOA reflectance as an input parameter, and the influence of atmospheric orientation as well as the change of surface direction is taken into account. Secondly, the BRDF model in this paper is constructed by using MODIS TOA reflectance data of calibrated test site on sunny days all year round, while MODIS BRDF products are calculated from multi-date, atmospherically corrected data measured over 16-day periods [31]. Thirdly, the new BRDF model includes the various solar angles and satellite observation angles from time-series MODIS images of the calibration test sites, rather than based on the given solar angle. The method is based on the following assumptions: For a calibration test site with clear weather and uniform and stable surface, the TOA reflectance at vertical observation angle is assumed to have a high stability throughout the year, and the TOA reflectance differences of different dates are mainly caused by the differences of observation angles and solar angles. Based on these assumptions, a BRDF model of calibration test site based on TOA reflectance is constructed by acquiring MODIS TOA reflectance from different solar and observation angles. One of the difficulties to be solved in this method is how to screen TOA reflectance on sunny days and cloudy days. A flow chart of the BRDF model construction method is shown in Figure 4, including seven steps. 1. The time-series MODIS satellite images of the calibration site were collected, and the corresponding DN value, calibration coefficients, and imaging geometry angles were calculated. 2. The time series of MODIS TOA reflectance of bands 1-4 and TOA bright temperature (BT) of band 31 are calculated based on calibration coefficients and BT calculation equation. The TOA reflectance of bands 1-4 are calculated by Equation (3): \[\rho_{\text{i}}=(DN*gain_{i}^{ref}+offset_{i}^{ref})/\cos\theta_{\text{s}}\] (3) where \(\rho_{\text{i}}\) is the TOA reflectance of \(i\)-th MODIS band. The \(gain_{i}^{ref}\) and \(offset_{i}^{ref}\) are the reflectance gain coefficient and reflectance offset coefficient of MODIS, which can be obtained from the MODIS file. The parameter \(\theta_{\text{s}}\) is the solar zenith angle of MODIS image over the test site. The TOA BT of band 31 is calculated by Equation (4): \[BT=\frac{\text{hc}}{k\lambda}\frac{1}{\ln(\frac{2 hc^{2}}{\lambda^{2}}\frac{1} {L}+1)}\] (4) where \(h\) is Planck constant, \(k\) is [PERSON] constant, \(c\) is light velocity, \(\lambda\) is wavelength, and \(L\) is radiance calculated by Equation (5): \[L_{\text{i}}=DN*gain_{i}^{rad}+offset_{i}^{rad}\] (5) where, \(L_{\text{i}}\) is the radiance of \(i\)-th band of MODIS. The \(gain_{i}^{rad}\) and \(offset_{i}^{rad}\) are the radiance gain coefficient and radiance offset coefficient of \(i\)-th band of MODIS. 3. The envelope brightness temperature (EBT) is established by TOA BT. In geometry, the envelope of a family of curves is a curve with at least one tangent to each line of the family (the family of curves is the infinite set of some curves, and they have some specific relations.) Then, the image whose difference between EBT and TOA BT is less than 10 K is selected (details can be found in Appendix A). The variation coefficient (VC) (the ratio of the standard deviation and mean reflectance) of gray value of the calibration site image in the MODIS band 1 was calculated, and the image with VC less than 4% was selected. 5. Eliminate the images with solar zenith angle greater than 55\({}^{\circ}\).