## 3 ABP Determination Models and BPE Computing Method
### Ideal ABP Determination Model
On the basis of the above definitions, now, we can deduce the ideal ABP determination model. Referring to Figure 3, let us denote the ECF coordinate representation of the GPV of the ECF-emitted beam \(L1\) as \(L1_{\rm ECF,out}\). According to Section 2.2, the ECF-emitted beam emerges along the positive \(z_{a}\)-axis from the origin \(A\) of the ECF coordinate system. Thus, \(L1_{\rm ECF,out}\) can be represented in the following form:  
\[L1_{\rm ECF,out}=\left[ \begin{array}{cccccccc}0&0&1&0&0&0&1\end{array} \right]^ {\rm T} \tag{4}\]  
In the process of 1-D LFS, we assume the initial scan azimuth of the ECF is \(\phi_{\rm f0}\) (\(\phi_{\rm f0}\) is measured counterclockwise from the positive \(x_{a}\)-axis), the angular velocity of the ECF is \(\omega\), and the initial scanning time of the current scan swath is \(t_{0}\). Therefore, for any moment \(t\), the FRA coordinate representation of the GPV \(L1\) can be obtained as follows:  
\[\left\{ \begin{array}{l}L1_{\rm FRA,out}=M_{\rm ECF}^{\rm FRA}(t)\mathbf{L}1_{\rm ECF,out}\\ \\ M_{\rm ECF}^{\rm FRA}(t)=\begin{bmatrix}\mathbf{Rot}_{z}(\mathbf{t})\mathbf{Rot}_{\rm ECF }^{\rm FRA}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 1}\\ \mathbf{O}_{3\times 3}&\mathbf{Rot}_{z}(\mathbf{t})\mathbf{Rot}_{\rm ECF0}^{\rm FRA}&A_{\rm FRA}(t) \\ \mathbf{O}_{1\times 3}&\mathbf{O}_{1\times 3}&1\end{bmatrix}\end{array} \right. \tag{5}\]  
where \(\mathbf{Rot}_{\rm ECF0}^{\rm FRA}\) represents the 3-by-3 ATM from the \"zero-position\" ECF coordinate system to the FRA coordinate system, \(\mathbf{Rot}_{z}(t)\) represents the vector rotation matrix rotating the \"zero-position\" ECF through an angle of \(\left(\phi_{\rm f0}+\omega(t-t_{0})\right)\) around the \(z_{b}\)-axis of the FRA coordinate system, the angle of \(\left(\phi_{\rm f0}+\omega(t-t_{0})\right)\) is the scan azimuth of the ECF at moment \(t\), and \(A_{\rm FRA}(t)\) is the FRA coordinate representation of the origin \(A\) of the ECF system. Here, \(\mathbf{Rot}_{z}(t)\) and \(A_{\rm FRA}(t)\) can be explicitly computed as follows:  
\[\left\{ \begin{array}{l}\mathbf{Rot}_{z}(t)=\left[\begin{array}{ccc}c(t)&-s(t)& 0\\ s(t)&c(t)&0\\ 0&0&1\end{array}\right]\\ \\ \mathbf{A}_{\rm FRA}(t)=\mathbf{Rot}_{z}(t)\mathbf{Rot}_{\rm ICS}^{\rm FRA}\left[\begin{array} []{c}x_{A0}-x_{B}\\ y_{A0}-y_{B}\\ z_{A0}-z_{B}\end{array}\right]_{\rm ICS}\end{array} \right. \tag{6}\]  
where \(c(t)=\cos\bigl{(}\phi_{\rm f0}+\omega(t-t_{0})\bigr{)}\) and \(s(t)=\sin\bigl{(}\phi_{\rm f0}+\omega(t-t_{0})\bigr{)}\). To obtain the GPV \(L2_{\rm SSR,out}\) representing the beam reflected off the SSR, we can first transform the ECF-emitted beam \(L1_{\rm FRA,out}\) defined in the FRA coordinate system into the incident beam \(L1_{\rm SSR,in}\) defined in the SSR coordinate system. Using a similar calculation process to that shown above, \(L1_{\rm SSR,in}\) can be obtained as follows:  
\[\left\{ \begin{array}{l}L1_{\rm SSR,in}=M_{\rm FRA}^{\rm SSR}L1_{\rm FRA,out }\\ \\ M_{\rm FRA}^{\rm SSR}=\left[\begin{array}{ccc}\mathbf{Rot}_{\rm FRA}^{\rm SSR}& \mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 1}\\ \mathbf{O}_{3\times 3}&\mathbf{Rot}_{\rm FRA}^{\rm SSR}&\mathbf{B}_{\rm SSR}\\ \mathbf{O}_{1\times 3}&\mathbf{O}_{1\times 3}&1\end{array}\right]\end{array} \right. \tag{7}\]  
where \(\mathbf{Rot}_{\rm FRA}^{\rm SSR}\) is the 3-by-3 ATM from the FRA coordinate system to the SSR coordinate system and \(\mathbf{B}_{\rm SSR}\) is the SSR coordinate representation of the origin \(B\) of the FRA coordinate system. Assuming that the point of incidence (POI) where the incident beam \(L1_{\text{SSR,in}}\) falls on the SSR is \(p_{1}\), and the coordinates of \(p_{1}\) in the SSR coordinate system are \(\boldsymbol{p}_{1,\text{SSR}}\), according to the incidence geometry, \(\boldsymbol{p}_{1,\text{SSR}}\) takes the following form:  
\[\left[ \begin{array}{c}\boldsymbol{p}_{1,x}\\ \boldsymbol{p}_{1,y}\\ \boldsymbol{p}_{1,z}\end{array} \right]_{\text{SSR}}=\left[ \begin{array}{c} \boldsymbol{r}1_{x}\\ \boldsymbol{r}1_{y}\\ \boldsymbol{r}1_{z}\end{array}