# 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
\(I_{S}\) is the backscattering energy, \(I_{T}\) is the incident energy, \(I_{T(1)}\) is the initial incident energy of the EM wave, and \(I_{T(k)}\) is the residual energy value at the \(k\)-th scattering. \(\delta\) is the backscatter coefficient; \(\overset{\rightarrow}{N}\) is the normal vector of the surface facet; \(R_{k}\) is the instantaneous slant range; \(K^{d}\) is the diffuse coefficient; \(\frac{k^{d}}{\pi}\) indicates the diffuse scattering mechanism to the surrounding area; \(K^{f}\) is the specular reflection coefficient; and \(K^{s}\) is the specular index and is used to control scattering energy along the receiving direction within the projected range. \(\rho\) indicates the polarimetric scatter coefficient, \(\varepsilon\) is the relative permittivity, \(\overset{\rightarrow}{v}\) is the direction of EM wave incidence, and \(\overset{\rightarrow}{r}\) is the direction of EM wave specular reflection. \(\overset{\rightarrow}{v^{\prime\prime}}\) is the direction of energy reception and \(K_{los}\) is the energy decay coefficient. \(I_{end}\) is the is the energy decay threshold, and it can be used to terminate the tracking of the complex multiple scattering path of each discrete EM wave. \(\left[\frac{1}{R^{d}}\right]\) is the propagation distance attenuation factor, which is specifically set according to the relationship that the fourth power of the propagation distance of EM waves is inversely proportional to the backscattered energy. We mainly use the ray tracing algorithm to realize the path tracing of multiple scattering of EM waves, and the key points are as follows. When the EM wave is at \(k\)-th scattering, the reflection direction and the intersection point of the main energy axis are taken as the new direction of incidence and the launching starting position of \(k+1\)-th scattering, respectively, so as to complete the scattering simulation of the EM wave in a new round. In describing the polarization state of EM waves, it is necessary to set up the global and local coordinate systems. Vertical polarization means that the electric field vector of the EM wave is parallel to the incident plane, and horizontal polarization means that the electric field vector of the EM wave is perpendicular to the incident plane. Combined with the small perturbation model theory, the target polarimetric scatter process is regarded as the coordinate conversion process from the incident wave to the scatter wave. Usually, the depolarization effect is generated throughout the multiple scattering processes of the EM wave and the target backscatter field calculation is chosen to be carried out in the local coordinate system, while the target backscatter field in the global coordinate system is recorded by the sensor, and the depolarization effect can be represented in the process of the conversion between the local scattering coordinate system and the global transmitting coordinates. Because the above polarization coordinate system conversion is not the focus of this paper, we only briefly describe the functional relationship between the polarimetric scatter coefficient, the local incident angle, and the relative permittivity; the detailed derivation process can be referred to our previously published article [16]. Parallel or vertically polarized incident waves in the local coordinate system can only produce horizontally or vertically polarized reflective waves, and according to the boundary condition of continuity of the tangential component of the EM field at the interface of the incident and reflective wave components, we can represent the polarization effect in the global coordinate system and express the vertically and horizontally polarized scattering coefficients at the intersection of the first scattering in the local coordinate system by Equation (8). \[\left\{