# Bounding Volume Hierarchy-Assisted Fast SAR Image Simulation Based on Spatial Segmentation
## 2 BVH-Assisted Computation of the Target Multiple Backscatter Field
### BVH-Assisted Simulation of Electromagnetic Wave Transmission
\tag{1}\]  
where \(M\) and \(N\) are the number of azimuthal sampling points and slant range gates in corresponding SAR slant range image, respectively; \(x_{\max}\) and \(x_{\min}\) are the azimuthal coordinates of the scene; \(R_{\min}\) and \(R_{\max}\) are the near range and far range, respectively; \(y_{\max}\) and \(y_{\min}\) are the ground range coordinate of the scene; \(H\) is the platform height; \(\rho_{r}\) is the slant range sampling space; \(R_{I}\) is the instantaneous slant range within the zero Doppler plane; \(\theta_{j}\) is the instantaneous incidence angle within the zero Doppler plane; and \(\vv_{0}\) is the instantaneous incidence direction within the zero Doppler plane. (b) According to the azimuthal range of the scene, the flight speed of the platform, and pulse repetition frequency, the instantaneous position of the sensor within the zero Doppler plane is determined. Then, the instantaneous incidence direction can be determined by combining it with the incidence angle of EM waves. The number of slant range gates corresponds to the number of discrete EM waves within the zero Doppler plane. The ray casting theory is used to detect the collision between the EM waves and the target surface, and the intersection facets and intersection point coordinates are recorded. The process of  
Figure 1: The workflow of BVH-assisted EM wave transmission. determining the scanning reference is essentially the process of simulating the discrete EM wave transmission and recording the intersection point within the zero Doppler plane. \[\left\{ \begin{array}{l}S=\left[\begin{array}{cc}x_{i},&-H\cdot\tan\theta,&H \end{array}\right]\\ x_{i}=i*\rho_{a}+x_{\text{min}}\\ P=S+t\stackrel{{\rightarrow}}{{v}}_{0}\\ \left(P-\hat{\rho}\right)\cdot\stackrel{{\rightarrow}}{{N}}=0 \end{array} \right.\qquad i\leq M \tag{2}\]  
where \(S\) is the sensor position within the zero Doppler plane; \(x_{i}\) is the sensor azimuthal coordinate; \(\rho_{a}\) is the azimuthal sampling space; \(\theta\) is the incidence angle; \(P\) is the intersection point on the target surface; \(t\) is the propagation distance of the EM wave; \(\hat{P}\) is the coordinate of an arbitrary point on the target surface; and \(\stackrel{{\rightarrow}}{{N}}\) is the normal vector of the facet on the target surface. (2) Base scanning plane rotation. As shown in Figure 2, the zero Doppler plane scanning does not take into account the dynamic occlusion relationship between the target and the sensor within the corresponding synthetic aperture time and has not yet been able to reproduce the time-varying characteristic of the radar beam radiation area and the shadow area, so only the zero Doppler plane is used as the base scanning plane here. According to the beam pointing rotation characteristics in different SAR modes, the SAR equivalent beam rotation angle is calculated, the zero Doppler plane is used as the base scanning plane, and then the base scanning plane is rotated using the line between the sensor and the nadir point as the center axis to determine the real-time scanning plane. The specific process is as follows:  
(a) Considering the rotational characteristics of beam pointing in different SAR modes and the time-varying characteristics of the EM wave transmission direction within the beamwidth, we uniformly take half of the beamwidth corresponding to the synthetic aperture as the equivalent rotation angle. According to the calculated equivalent angle, the determined base scanning surface will be rotated forward or backward, which is essentially the overall rotation of the discrete EM wave within the base scanning surface. This can be expressed by Equation (3). \[\left\{ \begin{array}{l}\theta_{s}=\arctan(L_{s}/R_{0}/2)\\ \theta_{s-strip}=0.886\lambda/D_{a}/2\end{array} \right. \tag{3}\]  
Figure 2: Beam scanning model.