## 6 TTL Estimation
### Results under S-Type Offset
#### 6.1.2 Impact of Lateral Offset  
Keeping the longitudinal offsets unchanged as in the previous subsection, the impact of different lateral offsets is shown in Figure 9 and Table 4 in the case that the pointing angle  
Figure 8: Input (red) and estimated (dashed blue) TTL error as well as their difference (black) under different pointing angle noise (the same noise level was applied to all pointing angles of the two satellites in each case). The TTL error of GRACE-FO on 1 April 2021 (green) was estimated by AEI [26]. (a) \(0.01\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\); (b) \(0.1\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\); (c) \(1\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\); (d) \(10\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\).  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Pointing Angle Noise**} & \multicolumn{4}{c}{**Coupling Factor**} \\ \cline{2-5}  (\(\theta_{y,1}\), \(\theta_{y,2}\), \(\theta_{z,1}\), \(\theta_{z,2}\)) & \(p_{y,1}\) (\(\SIUnitSymbolMicro rad\)) & \(p_{z,1}\) (\(\SIUnitSymbolMicro rad\)) & \(p_{y,2}\) (\(\SIUnitSymbolMicro rad\)) & \(p_{z,2}\) (\(\SIUnitSymbolMicro rad\)) \\ \hline (a) \(0.01\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\) & \(-487.7\pm 1.8\) & \(502.9\pm 2.7\) & \(-508.8\pm 1.9\) & \(489.1\pm 3.0\) \\ (b) \(0.1\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\) & \(-487.6\pm 1.9\) & \(500.9\pm 2.7\) & \(-509.2\pm 1.9\) & \(487.6\pm 3.0\) \\ (c) \(1\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\) & \(-420.1\pm 2.1\) & \(366.9\pm 2.8\) & \(-451.2\pm 2.1\) & \(340.9\pm 3.0\) \\ (d) \(10\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\) & \(-30.5\pm 0.8\) & \(18.1\pm 0.8\) & \(-34.0\pm 0.8\) & \(15.4\pm 0.8\) \\ \hline \hline
**Input Value** & \(-500.0\) & \(500.0\) & \(-500.0\) & \(500.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Input and estimated TTL coupling factors under different pointing angle noise.  
noise level was set to \(0.1\,\mathrm{\SIUnitSymbolMicro rad/Hz^{1/2}}\). As the lateral offsets increase, the true TTL error (red) grows and so does the estimation error (black) at high frequencies, which are limited by the white angle noise, while the estimation error at low frequencies does not significantly change. Considering that the high frequency noise can be filtered out, the lateral offset is not a strict condition in the case of \(0.1\,\mathrm{\SIUnitSymbolMicro rad/Hz^{1/2}}\) angle noise from the perspective of TTL estimation in this study.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Lateral Offset** & \multicolumn{4}{c}{**Coupling Factor**} \\ \cline{2-5} (\(\Delta y_{1},\Delta z_{1},\Delta y_{2},\Delta z_{2}\)) & \(p_{y,1}\) (\(\SIUnitSymbolMicro\)/rad) & \(p_{z,1}\) (\(\SIUnitSymbolMicro\)/rad) & \(p_{y,2}\) (\(\SIUnitSymbolMicro\)/rad) & \(p_{z,2}\) (\(\SIUnitSymbolMicro\)/rad) \\ \hline (a) \(100\,\mathrm{\SIUnitSymbolMicro m}\) & \(-87.7\pm 1.8\) & \(102.1\pm 2.7\) & \(-108.5\pm 1.9\) & \(89.6\pm 3.0\) \\ (b) \(500\,\mathrm{\SIUnitSymbolMicro m}\) & \(-487.6\pm 1.9\) & \(500.9\pm 2.7\) & \(-509.2\pm 1.9\) & \(487.6\pm 3.0\) \\ (c) \(2000\,\mathrm{\SIUnitSymbolMicro m}\) & \(-1986.9\pm 1.9\) & \(1994.9\pm 2.9\) & \(-2007.8\pm 2.0\) & \(1978.2\pm 3.2\) \\ (d) \(5000\,\mathrm{\SIUnitSymbolMicro m}\) & \(-4988.4\pm 2.3\) & \(4976.3\pm 3.4\) & \(-5008.7\pm 2.3\) & \(4962.4\pm 3.8\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Estimated TTL coupling factors under different lateral offsets (S-Type offsets).  
Figure 9: Input and estimated TTL error as well as their difference under different lateral offsets (the same value was applied to both \(\Delta y\) and \(\Delta z\) of the two satellites in each case). (a) \(0.1\,\mathrm{mm}\); (b) \(0.5\,\mathrm{mm}\); (c) \(2.0\,\mathrm{mm}\); (d) \(5.0\,\mathrm{mm}\).