# Validating Precise Orbit Determination from Satellite-Borne GPS Data of Haiyang-2D
## 2 Methods
### Carrier Differential Validation Method  
The carrier differential validation is a comparison between the distance obtained by using carrier-phase data of adjacent epochs and the distance calculated by coordinates of the GPS satellite and the coordinates of LEO; the orbits of LEO are determined by the RD method. The influence of errors is then eliminated through making difference to obtain the residuals of validation.  
HY2 is a dual-frequency receiver, and satellite-borne GPS observations of HY-2D include two frequency carriers (\(L_{1}\) and \(L_{2}\)) [24; 25; 26], so the linear combination method is used to obtain the Ionosphere-Free combination [27] observation as:  
\[L_{IF}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}L_{1}-\frac{f_{2}^{2}}{f_{1}^{2}- f_{2}^{2}}L_{2} \tag{1}\]  
where \(f_{1}\) and \(f_{2}\) are the frequencies of \(L_{1}\) and \(L_{2}\), respectively; \(L_{1}\) and \(L_{2}\) are the carrier-phase observations of the two frequencies; \(L_{IF}\) is the Ionosphere-Free combination observation.  
The carrier-phase observations corresponding to the two frequencies at epoch \(t_{i}\) are \(\varphi_{1}(t_{i})\) and \(\varphi_{2}(t_{i})\), respectively, and \(\mathbf{c}\) is the speed of light. The linear combination (\(LC\)) of phase data is [27]:  
\[LC_{i}=2.54573\times\varphi_{1}(t_{i})\times\frac{\mathbf{c}}{1575.42\times 1000000} -1.54573\times\varphi_{2}(t_{i})\times\frac{\mathbf{c}}{1227.6\times 1000000} \tag{2}\]  
where \(LC_{i}\) is the \(LC\) observation at epoch \(t_{i}\), \(\varphi_{1}(t_{i})\) and \(\varphi_{2}(t_{i})\) are the carrier-phase observations of the two frequencies at epoch \(t_{i}\).  
The difference between the \(LC\) observations in the two adjacent epochs is found to obtain the carrier distance difference \(\Delta LC_{i}\):  
\[\Delta LC_{i}=LC_{i+1}-LC_{i} \tag{3}\]  
where \(LC_{i}\) is the \(LC\) observation at epoch \(t_{i}\), and \(LC_{i+1}\) is the \(LC\) observation at epoch \(t_{i+1}\).  
The geometrical distance \(\rho_{i}\) between the HY-2D orbit and the GPS satellite orbit can be calculated as:  
\[\rho_{i}=\sqrt{\left(x_{h}-x^{g}\right)^{2}+\left(y_{h}-y^{g}\right)^{2}+ \left(z_{h}-z^{g}\right)^{2}} \tag{4}\]  
where \(x_{h}\), \(y_{h}\), \(z_{h}\) represent the position of HY-2D obtained by POD, \(x^{g}\), \(y^{g}\), \(z^{g}\) represent the position of the GPS satellite, and \(\rho_{i}\) is the geometrical distance.  
The difference of geometrical distances between two adjacent epochs:  
\[\Delta\rho_{i}=\rho_{i+1}-\rho_{i} \tag{5}\]  
where \(\rho_{i}\) is the geometrical distance at epoch \(t_{i}\), and \(\rho_{i+1}\) is the geometrical distance at epoch \(t_{i+1}\).  
The difference between two adjacent epochs is found by using the carrier-phase distance difference and geometrical distance difference, that is, Equation (3) minus Equation (5):  
\[\Delta_{i}=\Delta LC_{i}-\Delta\rho_{i} \tag{6}\]  
Equation (6) is used to calculate \(\Delta_{i}\) and \(\Delta_{i+1}\) of two adjacent epochs and find the difference again:  
\[e_{i}=\Delta_{i+1}-\Delta_{i} \tag{7}\]  
The residuals can be obtained, and the effect of errors such as satellite clock bias and ambiguity are eliminated.  
The core idea of carrier differential validation is that the carrier distance variation and geometrical distance variation between one GPS satellite that is not participating in LEO POD and the LEO satellite at adjacent epochs should be the same in theory. The influence of other errors is eliminated by finding the difference, and the residuals are obtained as a result of the orbital accuracy validation.