## 4 Results and Discussion
### Impact of Attitude Model
#### 4.1.2 Validation and Discussion  
Due to the stable attitude control for GRACE-FO, the difference in the phase center vectors between two solutions is small (seen Figure 4). The average RMS of phase center vectors in the x, y, z direction is 4.72 mm, 3.88 mm and 5.08 mm, respectively. To analyze the influence of the attitude model, two sets of orbit and clock offset using nominal and measured attitude data for GRCC are generated. The impact of different attitude models on POCD is given in Figure 5, and Table 2 summarizes  
The difference in clock offset using nominal and measured attitude is shown in Figure 6. The average RMS can reach 0.023 ns, which is an important difference for users who wish to use the products of the LEO-based enhancement system. We introduce a signal-in-space  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Spacecraft**} & \multirow{2}{*}{**Solutions**} & \multirow{2}{*}{**Residual**} & \multicolumn{4}{c}{**Overlap Comparison (cm)**} & \multicolumn{4}{c}{**JPL Comparison (cm)**} \\ \cline{3-10}  & & & **R** & **T** & **N** & **3D** & **R** & **T** & **N** & **3D** \\ \hline \multirow{2}{*}{GRCC} & Nominal ATT & 7.95 & 0.75 & 1.56 & 1.25 & 2.15 & 2.07 & 2.08 & 1.15 & 3.18 \\  & Measured ATT & 7.93 & 0.75 & 1.55 & 1.24 & 2.14 & 2.06 & 2.07 & 1.15 & 3.17 \\ \multirow{2}{*}{CS01} & Nominal ATT & 9.37 & 0.72 & 1.44 & 1.01 & 1.93 & / & / & / & / \\  & Measured ATT & / & / & / & / & / & / & / & / & / \\ \hline \hline \end{tabular}
\end{table}
Table 2: The RMS of residual and orbital comparison using nominal and measured attitude for GRCC and CS01.  
Figure 4: The difference in phase center vectors in CTS frame using nominal and measured attitude. DOY 273, 2018.  
Figure 5: RMS of orbital differences for GRCC and CS01 using nominal and measured attitude model.  
range error (SISRE) to quantify this impact. Orbital errors \(\Delta\mathbf{r}=(\Delta r_{R},\Delta r_{T},\Delta r_{N})\), which are described in the \(R\), \(T\) and \(N\) directions, can be expressed as a weighted average of RMS errors. These orbital errors vary with the orientation of the line-of-sight vector \(\mathbf{e}\) and the user location. The combined orbit and clock SISRE can be expressed by [7; 43]  
\[\text{SISRE}=\sqrt{[RMS(w_{R}\Delta r_{R}-\Delta cdt)]^{2}+w_{A,C}^{2}\Big{(} \big{(}RMS\,\Delta r_{T}\big{)}^{2}+\big{(}RMS\,\Delta r_{N}\big{)}^{2}\Big{)}}\]  
where \(\Delta cdt\) is the error in the clock offset; the weight factors \(w_{R}^{2}\) and \(w_{A,C}^{2}\) are altitude-dependent and we compute the weight factors using the scheme from [43]. For the GRCC satellite, \(w_{R}^{2}=0.4557\) and \(w_{A,C}^{2}=0.6294\). For the CS01 satellite, \(w_{R}^{2}=0.5153\) and \(w_{A,C}^{2}=0.6060\). The combined orbit and clock SISRE for GRCC caused by the attitude model is 0.69 cm. On one hand, the attitude control of GRACE and GRACE-FO is very stable [13; 44], and the measured attitude model is close to the nominal attitude model. On the other hand, the results also indicate that the potential to implement measured and nominal attitude in SHAOOD software is high. However, the errors of the attitude model can be partly absorbed in the orbit and clock error. Therefore, it is still necessary to use the precise measured attitude model, if available.