## 4 TTL Model
These terms can be omitted when \(\Delta x_{i}\) is not a large number since the largest angle biases are roughly of the order of \(10^{-3}\) rad. In contrast, the estimated linear TTL coupling factors will be apparently affected when \(\Delta x_{i}\) is up to \(1\) m. The least squares method is used for TTL coupling estimation in this paper. Six coupling factors \(\{p_{x,1},p_{y,1},p_{z,1},p_{x,2},p_{y,2},p_{z,2}\}\) can be estimated according to Equation (29) and the desired values are \(\{\Delta x_{1},-\Delta y_{1},\Delta z_{1},\Delta x_{2},-\Delta y_{2},\Delta z _{2}\}\) in the case with zero angle biases. As mentioned in Section 1, a band-pass filter is applied to the LRI range and the pointing angles in the estimation process, which yields  
\[ \begin{split}\text{BPF}(\rho_{\text{LRI}})&=\text{ BPF}(\rho_{\text{inst}})+\text{BPF}(\delta\rho_{\text{TTL}})+\text{BPF}(\delta\rho_{ \text{others}})\\ &\approx\text{BPF}(\delta\rho_{\text{NCS}})+\text{BPF}(\delta \rho_{\text{TTL}})+\text{BPF}(\delta\rho_{\text{LF}}),\end{split}  \tag{30}\]where the operator BPF refers to the band-pass filter. As an example, the cutoff frequencies are 40 mHz and 175 mHz in [10]. The band-pass-filtered instantaneous range is dominated by the term \(\delta\rho_{\text{NG}}\) caused by the non-gravitational forces. \(\delta\rho_{\text{LF}}\), the contribution of the laser frequency noise, expressed as \(\rho_{\text{inst}}\cdot\delta f_{1}/f_{1}\) according to Equation (16), is the main error term of the band-pass-filtered \(\delta\rho_{\text{others}}\). If the band-pass-filtered TTL error \(\delta\rho_{\text{TTL}}\) is significantly greater than the other two terms (e.g., a specific calibration maneuver has been performed), then we can directly use the following equation for TTL estimation:  
\[\text{BPF}(\rho_{\text{LRI}})=\text{BPF}(\delta\rho_{\text{TTL}})+\epsilon_{ \text{err}} \tag{31}\]  
where \(\epsilon_{\text{err}}\) is the small error term. The expression of TTL error is given in Equation (27). In general, the TTL error at the passband is probably lower than the terms of the non-gravitational forces or the laser frequency noise. As shown in Figure 4, the TTL error of GRACE-FO (green) is relatively small compared to the ranging signal and even lower than the laser frequency noise (red) at high frequencies. In the situation that the CM-VP offset in each satellite is set to 1.5 m, the TTL error may reach the level of the ranging signal at frequencies above 30 mHz, which is conducive to parameter estimation. Considering that the ranging signal induced by non-gravitational forces can be calculated from accelerometer measurements of both satellites by integrating the differential linear accelerations on the LOS twice, one can rewrite the observation equation as  
\[\text{BPF}(\rho_{\text{LRI}}-\delta\rho_{\text{NG}})\approx\text{BPF}\bigg{(} \rho_{\text{LRI}}-\iint(-a_{\text{LOS},1}-a_{\text{LOS},2})\text{d}t\bigg{)}= \text{BPF}(\delta\rho_{\text{TTL}})+\epsilon_{\text{err}} \tag{32}\]  
Figure 4: LRI ranging signal (blue), typical TTL error (magenta and green), and error contributed by laser frequency variations (red) in terms of ASD. Official GRACE-FO LRI1B v04 (version 04, or called RL04) data product on 1 April 2021 is used to plot the blue trace. TTL error can be computed from Equation (27) as long as the CM-VP offset and pointing angles of each satellite are provided. The magenta trace is plotted using the official SCA1B v04 data on 1 April 2021, assuming a large CM-VP offset for each satellite, which is similar to the CM-APC offset in the GRACE mission. The GRACE-FO’s TTL error (green) of the same day is provided in the LRI1B v54 generated by AEI [26]. The red trace shows relativistic effects at several integer multiples of orbital frequency (\(f_{\text{orb}}\approx 0.176\,\text{mHz}\)) and the error induced by random laser frequency noise which is generated according to Equation (1). in which \(a_{\text{LOS},i}\) (\(i=1,2\)) denotes the projection of the measured linear accelerations on the LOS for SAT\(i\).