## 5 Data Simulation
### SCA1B Simulation  
The official GRACE-FO SCA1B data provide rotation quaternions from the inertial frame to the satellite frame. The matrix rotating from IF to SF can be derived from the quaternions:  
\[\mathbf{R}_{\text{IF}}^{\text{SF}}= \begin{bmatrix}q_{0}^{2}+q_{1}^{2}-q_{2}^{2}-q_ {3}^{2}&2(q_{1}q_{2}+q_{0}q_{3})&2(q_{1}q_{3}-q_{0}q_{2})\\ 2(q_{1}q_{2}-q_{0}q_{3})&q_{0}^{2}-q_{1}^{2}+q_{2}^{2}-q_{3}^{2}&2(q_{2}q_{3}+ q_{0}q_{1})\\ 2(q_{1}q_{3}+q_{0}q_{2})&2(q_{2}q_{3}-q_{0}q_{1})&q_{0}^{2}-q_{1}^{2}-q_{2}^{2 }+q_{3}^{2}\end{bmatrix} , \tag{33}\]  
where \(q=(q_{0}\), \(q_{1}\), \(q_{2}\), \(q_{3})\) represents the quaternions. In order to calculate the pointing angles, the rotation matrix from IF to LOSF is needed, which can be written as  
\[\mathbf{R}_{\text{IF}}^{\text{LOSF}}= \begin{bmatrix}\mathbf{e}_{\text{LOSF},x}&\mathbf{e}_ {\text{LOSF},y}&\mathbf{e}_{\text{LOSF},z}\end{bmatrix} ^{\text{T}}. \tag{34}\]  
Taking SAT1 as an example, the axes of LOSF in Equation (34) are defined as  
\[\mathbf{e}_{\text{LOSF}_{1},x} =\frac{\mathbf{r}_{2}-\mathbf{r}_{1}}{|\mathbf{r}_{2}-\mathbf{r}_{1}|}, \tag{35}\] \[\mathbf{e}_{\text{LOSF}_{1},y} =\frac{\mathbf{e}_{\text{LOSF}_{1},x}\times\mathbf{r}_{1}}{|\mathbf{e}_{\text {LOSF}_{1},x}\times\mathbf{r}_{1}|},\] \[\mathbf{e}_{\text{LOSF}_{1},z} =\mathbf{e}_{\text{LOSF}_{1},x}\times\mathbf{e}_{\text{LOSF}_{1},y},\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Orbital Element** & **SAT1** & **SAT2** \\ \hline Semi-major Axis & 6862.266 km & 6862.709 km \\ Eccentricity & 0.000961 & 0.000907 \\ Inclination & 89.088\({}^{\circ}\) & 89.088\({}^{\circ}\) \\ Argument of Periapsis & 79.795\({}^{\circ}\) & 77.920\({}^{\circ}\) \\ Longitude of the Ascending Node & 98.494\({}^{\circ}\) & 98.497\({}^{\circ}\) \\ True Anomaly & 163.952\({}^{\circ}\) & 164.201\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial orbital elements of the two satellites. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Model in Forward Procedure** & **Model in POD Procedure** \\ \hline Static Gravity Field & EIGEN-6C4 (\(180\times 180\)) & GGM05C (\(180\times 180\)) \\ Solid Earth Tides & IERS Conventions (2010) & IERS Conventions (2010) \\ Ocean Tides & FES2014b (\(180\times 180\)) & EOT11a (\(120\times 120\)) \\ Solid Earth Pole Tides & IERS Conventions (2010) & IERS Conventions (2010) \\ Ocean Pole Tides & Desai (2002) (\(180\times 180\)) & Desai (2002) (\(60\times 60\)) \\ Atmosphere \& Oceanic Variability & AOD1B RLD6 (\(180\times 180\)) & Not Considered \\ N-Body Perturbations & JPL DE440 & JPL DE440 \\ Non-Gravitational Forces & ACT1A \& SCA1B RL04 & Simulated ACC1B \& SCA1B \\ General Relativistic Perturbations & IERS Conventions (2010) & IERS Conventions (2010) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Force models for orbit integration in the forward procedure and the POD procedure. where \(r_{1}\) and \(r_{2}\) represent the position vectors of SAT1 and SAT2 in the inertial frame, respectively. The indices 1 and 2 in Equation (35) should be exchanged for SAT2. For the GRACE-FO mission, the inertial position vectors can be found in the GNIIB data. Then the rotation matrix from SF to LOSF reads  
\[\mathbf{R}_{\rm SF}^{\rm LOSF}=\mathbf{R}_{\rm IF}^{\rm LOSF}\Big{(}\mathbf{R}_{\rm IF}^{ \rm SF}\Big{)}^{\rm T}. \tag{36}\]  
The pointing angles can then be computed as  
\[\theta_{x} =\arctan\!\left(\frac{R_{23}}{R_{33}}\right), \tag{37}\] \[\theta_{y} =-\arcsin\!\left(R_{13}\right),\] \[\theta_{z} =\arctan\!\left(\frac{R_{12}}{R_{11}}\right),\]  
where \(R_{ij}\) represents the entry in row \(i\) and column \(j\) of \(\mathbf{R}_{\rm SF}^{\rm LOSF}\). The pointing angles calculated from the GRACE-FO GNIIB and SCA1B data are considered to be error-free and were converted into the rotation matrix transforming the non-gravitational accelerations from SF to IF with the calculated trajectory states. These pointing angles were also used to compute the true TTL error, which was then added to the instantaneous range.