## 1 Introduction
### Motivation and Contributions of This Paper
\begin{split}\text{c}\tau_{\text{max}}&=d_{\text{ DIF}}\left(t\right)-d_{\text{LOS}}\left(t\right)\\ &=\sqrt{\left(x_{\text{D}}\left(t\right)-x_{\text{R}}\left(t \right)\right)^{2}+\left(y_{\text{D}}\left(t\right)-y_{\text{R}}\left(t\right) \right)^{2}+z_{\text{R}}^{2}\left(t\right)}\\ &+\sqrt{\left(x_{\text{D}}\left(t\right)-x_{\text{T}}\left(t \right)\right)^{2}+\left(y_{\text{D}}\left(t\right)-y_{\text{T}}\left(t\right) \right)^{2}+z_{\text{T}}^{2}\left(t\right)}-d_{\text{LOS}}\left(t\right)\\ &=\sqrt{x_{\text{D}}^{2}\left(t\right)+\left(y_{\text{D}}\left(t \right)-y_{\text{R}}\left(t\right)\right)^{2}+z_{\text{R}}^{2}\left(t\right)} \\ &+\sqrt{x_{\text{D}}^{2}\left(t\right)+\left(y_{\text{D}}\left(t \right)-y_{\text{T}}\left(t\right)\right)^{2}+z_{\text{T}}^{2}\left(t\right)} -d_{\text{LOS}}\left(t\right)\end{split}  \tag{4}\]  
For simplification, we take this scattering region constrained by Equation (4) as an ellipse with half major axis length \(l_{\text{l}}\) and half minor axis length \(l_{\text{s}}\), where \(l_{\text{l}}\) and \(l_{\text{s}}\) are calculated through resolving Equation (4) with conditions \(x_{\text{D}}\left(t\right)=0\) and \(y_{\text{D}}\left(t\right)=0\), respectively, and as shown in Equations (6) and (7), where  
\[\xi\left(t\right)=d_{\text{LOS}}\left(t\right)+\text{c}\tau_{\text{max}} \tag{5}\]\[ \begin{split} l_{1}\left(t\right)&=\xi\left(t\right) \left[\,\xi^{2}\left(t\right)-2\,{y_{\text{R}}}^{2}\left(t\right)+4\,{y_{\text{R} }}\left(t\right)\,{y_{\text{T}}}\left(t\right)-2\,{y_{\text{T}}}^{2}\left(t \right)\right]^{-1}\\ &\cdot\left[-\xi^{2}\left(t\right)+{y_{\text{R}}}^{2}\left(t \right)-2\,{y_{\text{R}}}\left(t\right)\,{y_{\text{T}}}\left(t\right)+{y_{ \text{T}}}^{2}\left(t\right)+{z_{\text{R}}}^{2}\left(t\right)-2\,{z_{\text{R}} }\left(t\right)\,{z_{\text{T}}}\left(t\right)+{z_{\text{T}}}^{2}\left(t \right)\right]^{\frac{1}{2}}\\ &\cdot\left[-\xi^{2}\left(t\right)+{y_{\text{R}}}^{2}\left(t \right)-2\,{y_{\text{R}}}\left(t\right)\,{y_{\text{T}}}\left(t\right)+{y_{ \text{T}}}^{2}\left(t\right)+{z_{\text{R}}}^{2}\left(t\right)+2\,{z_{\text{R} }}\left(t\right)\,{z_{\text{T}}}\left(t\right)+{z_{\text{T}}}^{2}\left(t \right)\right]^{\frac{1}{2}}\\ &\qquad\qquad l_{\text{s}}\left(t\right)=\frac{1}{\xi\left(t \right)}\left[-\left(\xi\left(t\right)+{z_{\text{R}}}\left(t\right)-{z_{\text {T}}}\left(t\right)\right)\left(\xi\left(t\right)-{z_{\text{R}}}\left(t\right)+ {z_{\text{T}}}\left(t\right)\right)\right.\\ &\qquad\qquad\qquad\left.\left({z_{\text{R}}}\left(t\right)-\xi \left(t\right)+{z_{\text{T}}}\left(t\right)\right)\left(\xi\left(t\right)+{z_{ \text{R}}}\left(t\right)+{z_{\text{T}}}\left(t\right)\right)\right]^{\frac{1 }{2}}\end{split}  \tag{6}\]  
Thus the area of the scattering region is calculated as in Equation (8). \[S\left(t\right)=\pi l_{\text{s}}\left(t\right)l_{\text{t}}\left(t\right) \tag{8}\]  
It's to be mentioned that, the error of calculating the region area through this approximation, i.e., taking the scattering region as an ellipse, is under 0.1% in the presented scenarios. Calculation examples can be seen in Appendix A. We assume that the scatterers are obey regional UD, thus we have the joint PDF of the time variant X coordinate \(x_{\text{D}}\left(t\right)\), and the time variant Y coordinate \(y_{\text{D}}\left(t\right)\) of the scatterer at the moment \(t\), as in Equation (9). \[f_{X_{\text{D}}Y_{\text{D}}}\left(x_{\text{D}},y_{\text{D}},t\right)|_{z_{ \text{D}}=0}=\frac{1}{S\left(t\right)} \tag{9}\]