## 7 Discussion  
In order to better compare the TTL estimation results under different conditions, the RMS values of the estimation error were computed after subtracting the mean since the constant term is meaningless for the biased range. The error of the TTL estimation is mainly contributed by the linear terms and can be written as  
\[\delta\rho_{\text{err},\text{TTL}}\approx\sum_{i=1}^{2}\big{(}\delta\theta_{y, i}(\theta_{y,i}p_{x,i}+p_{z,i})+\delta\theta_{z,i}(\theta_{z,i}p_{x,i}+p_{y,i})+ \theta_{z,i}\Delta p_{y,i}+\theta_{y,i}\Delta p_{z,i}\big{)}. \tag{54}\]  
Here, \(\Delta p_{k,i}\) (\(k=y,z\)) denotes the difference between the estimated linear coupling factor \(\hat{p}_{k,i}\) and the true value \(p_{k,i}\). The first two terms in Equation (54) are induced by the angle noise \(\delta\theta_{k,i}\), while the last two terms are due to the estimation error of the linear coupling factor \(\Delta p_{k,i}\). Figure 14 shows the RMS values of the estimation error under different pointing angle noise in the cases of S-type (blue) and L-type (magenta) longitudinal offsets. Figure 15  
Figure 13: Input and estimated TTL error as well as their difference under different laser frequency noise. (**a**) \(1\times\text{ASD}[\delta f_{\text{cav}}](f)\); (**b**) \(10\times\text{ASD}[\delta f_{\text{cav}}](f)\). \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Laser Frequency Noise** **(ASD Model)**} & \multicolumn{5}{c}{**Coupling Factor**} \\ \cline{2-7}  & \(p_{x,1}\) (mm/rad2) & \(p_{y,1}\) (mm/rad) & \(p_{z,1}\) (mm/rad) & \(p_{y,2}\) (mm/rad2) & \(p_{y,2}\) (\(\mu\)m/rad) & \(p_{z,2}\) (\(\mu\)m/rad) \\ \hline (a) \(1\times\text{ASD}[\delta f_{\text{cav}}](f)\) & \(1474.7\pm 14.8\) & \(-33.7\pm 2.7\) & \(-232.4\pm 14.7\) & \(1597.3\pm 12.8\) & \(-1654.1\pm 12.9\) & \(1162.8\pm 9.7\) \\ (b) \(10\times\text{ASD}[\delta f_{\text{cav}}](f)\) & \(1270.8\pm 118.6\) & \(-134.2\pm 21.5\) & \(22.9\pm 117.7\) & \(1902.4\pm 102.3\) & \(-1949.5\pm 103.2\) & \(1455.0\pm 77.5\) \\ \hline
**Input Value** & 1500 & -500 & 500 & 1500 & –500 & 500 \\
**Desired Value** & 1500 & -50 & –250 & 1500 & –1550 & 1100 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Input, desired, and estimated TTL coupling factors under different laser frequency noise. shows the RMS values of the estimation error under different lateral offsets and laser frequency noise in various cases. The dashed black lines in Figures 14 and 15 represent the RMS level of GRACE-FO's TTL error. The two traces in Figure 14 are flat at angle noise below \(0.3\upmu\mathrm{rad}/\mathrm{Hz}^{1/2}\) with the values lower than \(4\) nm probably because the ranging noise (mainly the laser frequency noise and the added noise introduced by the accelerometers) limits the TTL estimation, which yields similar estimation error for the coupling factors in the last two terms in Equation (54) under different angle noise. As the angle noise \(\delta\theta_{k,i}\) increases, the first two terms grow proportionally and the coupling factor error \(\Delta p_{k,i}\) in the last two terms also increases, leading to the rising parts of the traces. As the RMS values of the estimation error reach the levels of the input TTL error, the two traces gradually show a trend to become flat. It can be seen in Figure 14 that to eliminate the TTL error to the level lower than the TTL error of GRACE-FO, the angle noise should be lower than \(1\upmu\mathrm{rad}/\mathrm{Hz}^{1/2}\), as mentioned above, or lower than \(0.3\upmu\mathrm{rad}/\mathrm{Hz}^{1/2}\) for a better result. One can also see the obvious difference between the results with \(0.1\upmu\mathrm{rad}/\mathrm{Hz}^{1/2}\) (blue and magenta) and \(1\upmu\mathrm{rad}/\mathrm{Hz}^{1/2}\) angle noise (red and green) in Figure 15, so it is important to achieve low noise angle measurement for the estimation approach in this study.