## 3 ABP Determination Models and BPE Computing Method
### Actual ABP Determination Model  
Since neither the ECF coordinate system nor the FRA coordinate system undergoes thermal deformations, the GPV \(\widetilde{L}_{\text{FRA,out}}\) (i.e., the ECF-emitted beam defined in the FRA coordinate system in the presence or reflector thermal distortion effects) should have the same form as Equations (4) and (5):  
\[\left\{\begin{array}{l}\widetilde{L}_{\text{FRA,out}}=\mathbf{M}_{\text{ECF}}^{ \text{FRA}}(t)\widetilde{L}_{\text{ECF,out}}\\ \\ \widetilde{L}_{\text{ECF,out}}=\left[\begin{array}{cccccccc}0&0&1&0&0&0&1\end{array} \right]^{\text{T}}\end{array}\right. \tag{15}\]  
For the explicit expression of \(\mathbf{M}_{\text{ECF}}^{\text{FRA}}(t)\), please refer to Equations (5) and (6).  
To obtain the GPV \(\widetilde{L}_{\text{SSR}^{\prime},\text{out}}\) representing the beam reflected off the deformed SSR, we can first use the GTM \(\mathbf{M}_{\text{FRA}}^{\text{SRR}^{\prime}}\) to transform the ECF-emitted \(\widetilde{L}_{\text{FRA,out}}\) defined in the FRA coordinate system into the incident beam \(\widetilde{L}_{\text{SSR}^{\prime},\text{in}}\) defined in the actual SSR coordinate system. Here, \(\widetilde{\text{\emph{L1}}}_{\text{SSR}^{\prime},\text{in}}\) can be computed as follows (the calculation of \(M_{\text{FRA}}^{\text{SSR}^{\prime}}\) can be performed by referring to Equations (13) and (14)):  
\[\left\{\begin{array}{l}\widetilde{\text{\emph{L1}}}_{\text{SSR}^{\prime}, \text{in}}=M_{\text{FRA}}^{\text{SSR}^{\prime}}\widetilde{\text{\emph{L1}}}_{ \text{FRA,out}}\\ \\ M_{\text{FRA}}^{\text{SSR}^{\prime}}=\left[\begin{array}{ccc}\text{\emph{Rot }}_{\text{FRA}}^{\text{SSR}^{\prime}}&\text{\emph{O}}_{3\times 3}&\text{\emph{O}}_{3 \times 1}\\ \text{\emph{O}}_{3\times 3}&\text{\emph{Rot}}_{\text{FRA}}^{\text{SSR}^{\prime}}& \text{\emph{B}}_{\text{SSR}^{\prime}}\\ \text{\emph{O}}_{1\times 3}&\text{\emph{O}}_{1\times 3}&1\end{array}\right]\end{array}\right. \tag{16}\]  
where \(\text{\emph{Rot}}_{\text{FRA}}^{\text{SSR}^{\prime}}\) is the 3-by-3 ATM from the FRA coordinate system to the actual SSR coordinate system and \(\text{\emph{B}}_{\text{SSR}^{\prime}}\) is the SSR\({}^{\prime}\) coordinate representation of the origin \(B\) of the FRA coordinate system. Explicitly, \(\text{\emph{Rot}}_{\text{FRA}}^{\text{SSR}^{\prime}}\) and \(\text{\emph{B}}_{\text{SSR}^{\prime}}\) take the following form:  
\[\left\{\begin{array}{l}\text{\emph{Rot}}_{\text{FRA}}^{\text{SSR}^{\prime}}= \text{\emph{Rot}}_{\text{SSR}}^{\text{SSR}  
the impact of the reflector TDEs on the BPE, we define the angular distance between \(I_{\text{ICS,out}}\) and \(\tilde{I}_{\text{ICS,out}}\) as the thermally induced BPE \(\zeta\) (unit: rad):  
\[\zeta=\cos^{-1}\bigl{(}\tilde{I}_{\text{ICS,out}}\cdot I_{\text{ICS,out}}\bigr{)} =\cos^{-1}\biggl{(}\frac{\tilde{I}_{\text{ICS,out}}\cdot I_{\text{ICS,out}}}{ \lVert\tilde{I}_{\text{ICS,out}}\rVert\times\lVert I_{\text{ICS,out}}\rVert} \biggr{)}\in[0,\pi] \tag{20}\]  
where \(\tilde{I}_{\text{ICS,out}}\cdot I_{\text{ICS,out}}\) represents the inner product between \(\tilde{I}_{\text{ICS,out}}\) and \(I_{\text{ICS,out}}\) and \(\lVert\cdot\rVert\) represents the modulus of the vector.  
Considering that the reflector TDEs may lead to different BPEs within one complete 360\({}^{\circ}\) scan cycle of the ECF, we take the root-mean-square (RMS) of the BPEs obtained at different ECF scan azimuths as the overall thermally induced BPE:  
\[\xi_{RMSE}=\sqrt{\sum_{i=1}^{N}(\xi_{i})^{2}/N} \tag{21}\]  
where \(\xi_{RMSE}\) is the so-called root-mean-square error (RMSE) of the ABP and \(N\) is the number of BPEs obtained at different ECF scan azimuths.