# Impact of Satellite Attitude on Altimetry Calibration with Microwave Transponders
## 3 Satellite Attitude and Transponder Calibration
### Calibration Accounting for Satellite Attitude
The rest of the corrections (i.e., atmospheric delays, loading effects, etc.) are applied as usual. Following Equation (10), the geometric range \(R_{0}(t)\) is defined as:  
\[R_{0}(t)=\sqrt{\left(X^{\text{TRP}}-X^{\text{APC}}(t)\right)^{2}+\left(Y^{ \text{TRP}}-Y^{\text{APC}}(t)\right)^{2}+\left(Z^{\text{TRP}}-Z^{\text{APC}}( t)\right)^{2}}. \tag{13}\]  
It is evident from Equation (13) that the evaluation of the geometric range \(R_{0}(t)\) requires the calculation of APC coordinates in EFRS. The APC coordinate vector in EFRS, denoted as \(\mathbf{V}_{\text{EFRS}}^{\text{APC}}=[X^{\text{APC}},Y^{\text{APC}},Z^{ \text{APC}}]^{\text{T}}\), can be determined using the relation:  
\[\mathbf{V}_{\text{EFRS}}^{\text{APC}}=\mathbf{V}_{\text{EFRS}}^{\text{CoG}}+ \Delta\mathbf{V}_{\text{EFRS}}, \tag{14}\]  
where \(\mathbf{V}_{\text{EFRS}}^{\text{CoG}}\) is the CoG coordinate vector and \(\Delta\mathbf{V}_{\text{EFRS}}\) the CoG-APC baseline vector in EFRS. The vector \(\mathbf{V}_{\text{EFRS}}^{\text{CoG}}\) corresponds to the satellite position as defined by orbit determination. The baseline vector \(\Delta\mathbf{V}_{\text{EFRS}}\) depends on the satellite attitude and is transformed from SAT system [Q: \(x_{B}y_{B}z_{B}\)] to EFRS system [O: \(XYZ\)] using the equation:  
\[\Delta\mathbf{V}_{\text{EFRS}}=\mathbf{R}_{\text{SAT}\rightarrow\text{EFRS}} \Delta\mathbf{V}_{\text{SAT}}=\mathbf{R}_{\text{IRS}\rightarrow\text{EFRS}} \mathbf{R}_{\text{SAT}\rightarrow\text{IRS}}\Delta\mathbf{V}_{\text{SAT}}. \tag{15}\]  
The calculation of rotation matrix \(\mathbf{R}_{\text{IRS}\rightarrow\text{EFRS}}\) for the transformation between IRS and EFRS is described in the IERS conventions 2010 [43; 44]. The calculation of \(\mathbf{R}_{\text{SAT}\rightarrow\text{IRS}}\) can be performed as already discussed in Section 3.2, and the evaluation of vector \(\Delta\mathbf{V}_{\text{SAT}}\) is based on the spacecraft geometry and mass properties that are provided in the corresponding mission documentation. The APC and CoG coordinates in the SAT system for Jason-3 and Sentinel-6A MF are given in Table 1. During the spacecraft operation, the CoG is subject to a minor displacement over time due to events such as fuel consumption (Figure 6). The precise evaluation of \(\Delta\mathbf{V}_{\text{SAT}}(t)\) at a specific time epoch \(t\) is performed as follows:  
\[\Delta\mathbf{V}_{\text{SAT}}(t)=\Delta\mathbf{V}_{\text{SAT}}(t_{0})+ \delta\mathbf{V}_{\text{SAT}}(t). \tag{16}\]  
The first term on the right-hand side of Equation (16) denotes the APC-CoG baseline vector as measured prior to the satellite deployment, and the second term denotes its temporal variations due to the CoG displacement [37]. The Jason-3 \(x_{B}\)-axis component of the CoG displacement in the SAT system is given in Figure 6, along with the spacecraft mass change. In line with Equation (11), the new datation bias, \(\widetilde{dt}\), after referring all parameters to APC is given by:  
\[\widetilde{dt}=\underset{t\in\mathbb{R}_{+}}{\text{arg min}}\ R(t)-\underset{ t\in\mathbb{R}_{+}}{\text{arg min}}\ R_{0}(t). \tag{17}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Reference Points** & \(\boldsymbol{x_{B}}\) **(m)** & \(\boldsymbol{y_{B}}\) **(m)** & \(\boldsymbol{z_{B}}\) **(m)** \\ \hline \multirow{2}{*}{Jason-3} & Spacecraft center of gravity & 1.0023 & 0.0000 & \(-\)0.0021 \\  & Altimeter phase center & 1.6390 & 0.0000 & 0.6644 \\ \hline \multirow{2}{*}{Sentinel-6A MF} & Spacecraft center of gravity & 1.5274 & \(-\)0.0073 & 0.0373 \\  & Altimeter phase center & 2.5240 & 0.0001 & 0.5650 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coordinates of reference points (CoG and APC) in the SAT system for Jason-3 and Sentinel-6A MF [37; 45]. We finally define the attitude effect on range and datation bias per cycle (denoted \(\delta B\) and \(\delta T\), respectively) as the bias difference before and after accounting for non-ideal attitude, i.e.,:  
\[\delta B=\widetilde{B}-B, \tag{18}\]  
and  
\[\delta T=\widetilde{d}t-dt. \tag{19}\]