# Bounding Volume Hierarchy-Assisted Fast SAR Image Simulation Based on Spatial Segmentation
## 2 BVH-Assisted Computation of the Target Multiple Backscatter Field
### Calculation of the Multiple Backscatter Field
\begin{array}{l}\rho_{hh}=\frac{\cos\left(\overrightarrow{v}\cdot \overrightarrow{N}\right)-\sqrt{\epsilon-\sin^{2}\left(\overrightarrow{v} \cdot\overrightarrow{N}\right)}}{\cos\left(\overrightarrow{v}\cdot \overrightarrow{N}\right)+\sqrt{\epsilon-\sin^{2}\left(\overrightarrow{v} \cdot\overrightarrow{N}\right)}}\\ \rho_{vv}=\frac{\left(\epsilon-1\right)\left(\sin^{2}\left(\overrightarrow {v}\cdot\overrightarrow{N}\right)-\epsilon\left(1+\sin^{2}\left( \overrightarrow{v}\cdot\overrightarrow{N}\right)\right)\right)}{\left( \epsilon\cos\left(\overrightarrow{v}\cdot\overrightarrow{N}\right)+\sqrt{ \epsilon-\sin^{2}\left(\overrightarrow{v}\cdot\overrightarrow{N}\right) }\right)^{2}}\\ \rho_{hv}=0\end{array} \right. \tag{8}\]  
where \(\rho_{vv}\) is the horizontal polarimetric scatter coefficient, \(\rho_{vv}\) is the vertical polarimetric scatter coefficient, and \(\rho_{hv}\) is the cross-polarimetric scatter coefficient. In the local coordinate system, \(\overrightarrow{v}\) is the direction of EM wave incidence, \(\overrightarrow{N}\) is the normal vector of the surface facet, and \(\epsilon\) is the relative permittivity of the surface facet. From the coordinates of the intersection point \(P_{k}\) calculated during the multiple scattering processing, the slant range of the corresponding multiple scattering can be found. Then, the slant range of the multiple scattering of the EM wave can be brought into the SAR impulse response signal to obtain the phase, which contains two parts, the Doppler phase and the range phase, which are used in the compression imaging process. The echo signal of the EM wave is limited by two-dimensional azimuthal and range envelopes in real time during the recording process. In the stop-and-go model, the real-time phase of the \(k\)-th scattering of the EM wave can be obtained in Equation (9). \[\left\{ \begin{array}{l}R_{k}=\left(\begin{array}{cc}\sqrt{\left(S^{\prime}- P_{0}^{\prime}\right)^{2}}&+\sqrt{\left(P_{0}-P_{1}^{\prime}\right)^{2}}+ \\ \cdot\cdot\cdot\cdot\cdot\cdot&+\sqrt{\left(P_{k-1}^{\prime}-P_{k}^{\prime} \right)^{2}}+\end{array}\right.\sqrt{\left(P_{k}^{\prime}-S^{\prime}\right)^{2 }}\end{array} \right)/2 \tag{9}\]  
where \(R_{k}\) is half of the total distance of multiple scattering of EM wave transmission; \(S^{\prime}\) is the instantaneous sensor position; \(P_{0}^{\prime}\)-\(P_{k}^{\prime}\) is the intersection point of multiple scattering; \(Pha(k)\) is the multiple scattering phase; \(f_{0}\) and \(K_{r}\) are the signal carrier frequency and chirp rate, respectively; \(\tau\) and \(\eta\) are the fast time and slow time, respectively. Combining Equations (7)-(9) can simultaneously obtain the amplitude and phase of the EM wave when scattering occurs, specifically referring to Equation (10) to vectorially superpose the amplitude and phase of the multiple scattering field to obtain the backscatter field of the EM wave, including multiple scattering. \[\left\{ \begin{array}{l}\left\langle Pha(k)\right\rangle=\left[w_{r}(\tau-2R_ {k}(\eta)/c)\times w_{a}(\eta-\eta_{c})\right]\times Pha(k)\\ E_{s}=\sum\limits_{k=1}^{M}\delta_{k}\cdot e^{\left(Pha(k)\right)}=\sum\limits_ {k=1}^{M}\left[\delta_{k}\cdot\cos(\left\langle Pha(k)\right\rangle)+i\cdot \delta_{k}\cdot\sin(\left\langle Pha(k)\right\rangle)\right]\end{array} \right. \tag{10}\]where \(E_{s}\) is the backscatter field of the EM wave; \(\delta_{k}\) and \(\langle Pha(k)\rangle\) are the amplitude and phase of the EM wave after two-dimensional envelope constraints when the \(k\)-th scattering occurs, respectively; and \(M\) is the total number of scattering.