# Geometric Processing and Accuracy Verification of Zhuhai-1 Hyperspectral Satellites
## 2 Methods  
### Geometric Calibration  
Although the OHS payload is a planar CMOS payload, the imaging principle of each spectral segment of the OHS conforms to that of linear array push-broom imaging. The imaging geometric positioning model can be expressed as follows [3; 4; 5; 6; 7; 8; 9; 10; 11]:  
\[\left[ \begin{array}{c}X\\ Y\\ Z\end{array} \right]_{\text{WGS84}}=\left[ \begin{array}{c}X_{s}\\ Y_{s}\\ Z_{s}\end{array} \right]_{\text{WGS84}}+m\mathbf{R}_{\text{J2000}}^{\text{WGS84}} \mathbf{R}_{body}^{\text{J2000}}\mathbf{R}_{u}\mathbf{R}_{cam}^{body}\left[ \begin{array}{c}x-x_{0}-\Delta x\\ y-y_{0}-\Delta y\\ -f\end{array} \right] \tag{1}\]  
In the above collinearity equation, \(\left( \begin{array}{ccc}X&Y&Z\end{array} \right)_{\text{WGS84}}^{T}\) represents the ground coordinates of the point in the World Geodetic System 1984 (WGS84) geocentric coordinate system and \(\left( \begin{array}{ccc}X_{s}&Y_{s}&Z_{s}\end{array} \right)_{\text{WGS84}}^{T}\) indicates the position of the OHS with respect to the WGS84 geocentric coordinate system. Furthermore, m denotes the scaling factor, \(\mathbf{R}_{A}^{B}\) denotes the rotation matrix for converting the coordinate system A to coordinate system B, \(\left(x_{0},y_{0}\right)\) is the principal point position, \(f\) is the focal length, \(\left(\Delta x,\Delta y\right)\) denotes the interior distortion effects, and \(\mathbf{R}_{U}\) is the offset matrix. The OHS geometric calibration model mainly considers the compensation of the load installation error, attitude and orbit system error, and camera distortion. The orbit system error and attitude system error are equivalent and can be compensated for via unified modeling [6; 7; 8; 9]. The load installation error has the same influence on geometric positioning as the attitude system error; hence, it can also be equivalent to the attitude system error. Therefore, the offset matrix shown in Equation (1) can be used to compensate for the attitude system error, and the influence of the load installation error and attitude \begin{table}
\begin{tabular}{c c c} \hline \hline  & Total satellite mass & 67 kg \\  & Orbit height & 500 km \\  & Orbit inclination angle & 98\({}^{\circ}\) \\ Satellite Platform & Regression cycle & 5 days \\  & Global positioning system (GPS) positioning precision & 15 m \\  & Attitude accuracy & 15\({}^{\circ}\) (3\(\sigma\)) \\  & Attitude maneuver & \(\pm\)45\({}^{\circ}\)/80 s \\  & Attitude stability & 0.002\({}^{\circ}\)/s (1\(\sigma\)) \\ \hline  & Detector size & 4.4 \(\mu\)m \\  & Field of view (FOV) & 20.5\({}^{\circ}\) \\  & Spectral range & 400–1000 nm \\ Satellite Payload & Quantitative level & \(\geq\)12 bits \\  & Band number & 32 \\  & Signal Noise Ratio (SNR) & \(\geq\)300 dB \\  & Ground sample distance & 10 m \\  & Ground swath & 150 km \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Orbita hyperspectral satellite (OHS) parameters. and orbit system error on geometric positioning can be eliminated simultaneously. \(\mathbf{R}_{u}\) is defined as follows:  
\[\mathbf{R}_{U}=\left[ \begin{array}{ccc}\cos\varphi&0&\sin\varphi\\ 0&1&0\\ -\sin\varphi&0&\cos\varphi\end{array} \right]\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\omega&-\sin\omega\\ 0&\sin\omega&\cos\omega\end{array} \right]\left[ \begin{array}{ccc}\cos \kappa&-\sin\kappa&0\\ \sin\kappa&\cos\kappa&0\\ 0&0&1\end{array} \right] \tag{2}\]  
For camera distortion, the OHS single-band uses several lines in the CMOS array to simulate time delay integration (TDI) Charge-coupled Device (CCD) push-broom; hence, the distortion model of the single-band can incorporate the directional angle model shown in Equation (3) [12; 13; 14; 15]:  
\[\left\{ \begin{array}{l}\tan(\psi_{x})=a_{0}+a_{1}s+a_{2}s^{2}+\cdots+a_{i}s^ {i}\\ \tan(\psi_{y})=b_{0}+b_{1}s+b_{2}s^{2}+\cdots+b_{j}s^{j}\end{array} \right.,j\leq 5 \tag{3}\]  
where \(s\) is the image column. Therefore, the geometric calibration model of the OHS single spectral band is shown in Equation (4):  
\[\left[ \begin{array}{c}X\\ Y\\ Z\end{array} \right]_{\text{WGS84}}=\left[