## 4 TTL Model
For each satellite, the \(x\)-axis of LOSF coincides with the LOS, while the \(x\)-axis of the SF is slightly misaligned with the LOS due to the variability of satellite attitudes and the \(z\)-axes of both frames are roughly in the nadir direction. where \(c_{i}=\cos\theta_{i}\), \(s_{i}=\sin\theta_{i}\), \(i=x,y,z\); the matrices \(\mathbf{R}_{x}(\theta_{x})\), \(\mathbf{R}_{y}(\theta_{y})\) and \(\mathbf{R}_{z}(\theta_{z})\) denote the rotations around the \(x\)-, \(y\)- and \(z\)-axes by angles \(\theta_{x}\), \(\theta_{y}\), and \(\theta_{z}\), respectively. Substitution of Equation (24) into Equation (23) yields  
\[\delta\rho_{\text{TTL},i}\approx-c_{y}c_{z}\Delta x-c_{y}s_{z}\Delta y+s_{y} \Delta z, \tag{25}\]  
where the index \(i\) on the right-hand side is dropped. Since the angles are small, Taylor expansion can be applied. In general, terms with order higher than two are omitted; thus, we have  
\[\delta\rho_{\text{TTL},i}\approx\left(\frac{1}{2}\theta_{y}^{2}+\frac{1}{2} \theta_{z}^{2}\right)\Delta x-\theta_{z}\Delta y+\theta_{y}\Delta z-\Delta x, \tag{26}\]  
where we find \(\Delta x/2\) is the quadratic coupling factor of the pitch and yaw angles, \(-\Delta y\) is the linear coupling factor of the yaw angle, and \(\Delta z\) is the linear coupling factor of the pitch angle. The constant term is not important to the biased range. With the subscript \(1\) and \(2\) indicating the SAT1 and SAT2, respectively, the total TTL error of the two satellites reads  
\[ \begin{split}\delta\rho_{\text{TTL}}&\approx \left(\frac{1}{2}\theta_{y,1}^{2}+\frac{1}{2}\theta_{z,1}^{2}\right)\Delta x_{1 }-\theta_{z,1}\Delta y_{1}+\theta_{y,1}\Delta z_{1}-\Delta x_{1}\\ &+\left(\frac{1}{2}\theta_{y,2}^{2}+\frac{1}{2}\theta_{z,2}^{2} \right)\Delta x_{2}-\theta_{z,2}\Delta y_{2}+\theta_{y,2}\Delta z_{2}-\Delta x _{2}.\end{split}  \tag{27}\]  
In the case that \(\Delta x_{i}\) (\(i=1,2\)) is small, like the GRACE-FO LRI, the quadratic coupling terms can be omitted [14], i.e.,  
\[\delta\rho_{\text{TTL}}\approx-\theta_{z,1}\Delta y_{1}+\theta_{y,1}\Delta z_{ 1}-\Delta x_{1}-\theta_{z,2}\Delta y_{2}+\theta_{y,2}\Delta z_{2}-\Delta x_{2}. \tag{28}\]  
In practice, the pointing angles we use for TTL coupling estimation are measured ones, which unavoidably contain biases and noise. A static angle offset is irrelevant for the linear coupling TTL error since it only introduces an additional constant term, which is not important as mentioned above. However, taking the quadratic coupling terms into account, additional linear terms appear in the expression of the TTL error because of the biased angles. In this situation, the TTL error for each satellite reads  
\[ \begin{split}\delta\rho_{\text{TTL},i}&\approx \left(\frac{1}{2}\theta_{y,i}^{2}+\frac{1}{2}\theta_{z,i}^{2}\right)\Delta x_{i }-\theta_{z,i}\Delta y_{i}+\theta_{y,i}\Delta z_{i}-\Delta x_{i}\\ &\approx\left(\frac{1}{2}\left(\theta_{y,i}-\Delta\theta_{y,i} \right)^{2}+\frac{1}{2}\left(\theta_{z,i}-\Delta\theta_{z,i}\right)^{2}\right) \Delta x_{i}-\left(\theta_{z,i}-\Delta\theta_{z,i}\right)\Delta y_{i}+\left( \theta_{y,i}-\Delta\theta_{y,i}\right)\Delta z_{i}-\Delta x_{i}\\ &=\left(\frac{1}{2}\theta_{y,i}^{2}+\frac{1}{2}\theta_{z,i}^{2} \right)\Delta x_{i}+\theta_{z,i}(-\Delta\theta_{z,i}\Delta x_{i}-\Delta y_{i} )+\theta_{y,i}(-\Delta\theta_{y,i}\Delta x_{i}+\Delta z_{i})+\Delta\theta_{z, i}\Delta y_{i}-\Delta\theta_{y,i}\Delta z_{i}-\Delta x_{i},\end{split}  \tag{29}\]  
where \(\theta_{y,i}\) and \(\theta_{z,i}\) denote the measured pitch and yaw angles and \(\Delta\theta_{y,i}\) and \(\Delta\theta_{z,i}\) denote the pitch and yaw angle biases between the measured and the true angles, respectively, and they satisfy the equations \(\hat{\theta}_{y,i}\approx\theta_{y,i}+\Delta\theta_{y,i}\) and \(\hat{\theta}_{z,i}\approx\theta_{z,i}+\Delta\theta_{z,i}\). The additional linear terms are in direct proportion to \(\Delta x_{i}\) and the angle biases.