## 2 Methods
### Relative Calibration Parameter Solving  
The statistical calibration principle can be applied to the virtual linear array sensor constructed from the original Bayer pattern. The probability distribution of scene DN values acquired by each detector is the same when there are enough imaging samples. The variability in the probability distribution of DNs across the detectors is caused by the inconsistent radiometric response of the detectors. For the discrete quantization of the optical remote sensing satellite sensor, the specific process is as follows:  
1. Calculate the DN number of the imaging data acquired by the \(i\)th detector, as in the following equations: \[PixelNums_{i}(k)=PixelNums_{i}(k)+1,\] (5) \[TotalPixelNums_{i}=TotalPixelNums_{i}+1,\] (6) where \(k\in[0,2^{n}-1]\), \(n\) is sensor quantization bits, \(PixelNums_{i}(k)\) is the number of DNs whose value of the \(i\)th detector of the virtual linear sensor is equal to \(k\). \(TotalPixelNums_{i}\) is the total number of DNs acquired by the \(i\)th detector of the virtual linear array.
2. Establish the cumulative probability distribution function, \(P_{i}(k)\), of each sensor detector according to Equation (7).
3. \[P_{i}(k)=\sum_{j=0}^{k}\frac{PixelNums_{i}(k)}{TotalPixelNums_{i}}.\] (7)
4. Set the cumulative probability distribution function of the imaging gray data of all the sensor detectors as the reference cumulative probability distribution function for each band, \(P_{G}(k)\), \(P_{B}(k)\), and \(P_{R}(k)\): \[P_{G}(k)=\sum_{k=0}^{2^{n}-1}\sum_{j}^{N_{total}}\frac{PixelNums_{j}(k)}{Total PixelNums_{j}},(j=1,4,5,8,9,12,13\ldots),\] (8)\[P_{B}(k)=\sum_{k=0}^{2^{n}-1}\sum_{j}^{N_{total}}\frac{PixelNum_{j}(k)}{Total PixelNum_{j}},(j=2,6,10,14\ldots), \tag{9}\]  
\[P_{R}(k)=\sum_{k=0}^{2^{n}-1}\sum_{j}^{N_{total}}\frac{PixelNum_{j}(k)}{Total PixelNum_{j}},(j=3,7,11,15\ldots), \tag{10}\]  
where \(N_{total}\) is the total number of the virtual linear array sensor detectors.
9. Obtain the imaging response model of each sensor detector by mapping the cumulative probability distribution function of each sensor detector to the reference cumulative probability distribution function based on Equations (11) and (12): \[P_{R}(k-x)\leq P_{i}(k)\leq P_{R}(k+y),\text{ and }\] (11)  
\[k=\begin{cases}k-x,&|P_{R}(k-x)-P_{i}(k)|-|P_{i}(k)-P_{R}(k+y)|\leq 0\\ k+y,&|P_{R}(k-x)-P_{i}(k)|-|P_{i}(k)-P_{R}(k+y)|>0^{\prime},\end{cases} \tag{12}\]  
where \(k\) is the current calibration gray-level, the range of values of \(x\) and \(y\) is \([0,2^{n}-1]\), and \(n\) is the satellite sensor image quantization bit.  
The relative calibration parameters obtained from steps 1-4 are the maps of the imaging DN values from each detector to the relative corrected DN values. The relative radiometric correction is achieved by locating each detector according to the Bayer pattern conversion and position determination rules from Sections 2.1 and 2.2, and the imaging DNs of each detector are then mapped to the new DNs based on the relative calibration parameters. This is an effective method to solve the non-linear calibration model of remote sensing sensors. A one-to-one correspondence between sensor input and output can be established within the sensor quantization range.