## 2 Relevant Propagation Modeling Theory
### The Theory of Three-Dimensional \(RT\)  
When the wavelength of electromagnetic waves is very short and the medium changes slowly in space, the propagation and scattering characteristics of electromagnetic waves are localized. That is, the field in the vicinity of the observation point need not be solved according to the distribution over the entire initial surface but only by the field over a certain portion of that surface. This allows us to assume that the wavefront of an electromagnetic wave is viewed locally as the same as the wavefront of a plane wave. Therefore, in the high-frequency condition, it can be assumed that  
\[\overset{\rightarrow}{E}=\overset{\rightarrow}{E}_{0}e^{-jk_{d}\varphi}\quad \overset{\rightarrow}{H}=\overset{\rightarrow}{H}_{0}e^{-jk_{d}\varphi} \tag{14}\]  
where \(\varphi=\varphi(x,y,z)\) is a function of spatial position, and it is also an eigenfunction that determines the equiphase plane. \(\overset{\rightarrow}{E}_{0}\) and \(\overset{\rightarrow}{H}_{0}\) are also functions of spatial position.  
Substituting Equation (14) into the Maxwell equation for the passive region and using the geometrical-optical approximation of the high-frequency field, \(\lambda\to 0\), \(k_{d}\rightarrow\infty\), \(|\
abla\varepsilon|\lambda/\varepsilon\ll 1\), and \(|\
abla\mu|\lambda/\mu\ll 1\).  
We obtained the field equations based on the geometrical optics approximation.  
\[\
abla\varphi\times\overset{\rightarrow}{H}_{0}+c\varepsilon\overset{ \rightarrow}{E}_{0}=0 \tag{15}\]  
\[\
abla\varphi\times\overset{\rightarrow}{E}_{0}-c\mu\overset{\rightarrow} {H}_{0}=0 \tag{16}\]  
\[\overset{\rightarrow}{E}_{0}\cdot\
abla\varphi=0 \tag{17}\]  
\[\overset{\rightarrow}{H}_{0}\cdot\
abla\varphi=0 \tag{18}\]  
In urban microcellular environments, \(\mu\) is mostly constant and \(\mu=\mu_{0}\). Therefore, Equations (15) and (16) can be written as  
\[\overset{\rightarrow}{E}_{0}=Z_{0}\overset{\rightarrow}{H}_{0}\times\
abla \varphi/n \tag{19}\]  
\[Z_{0}\overset{\rightarrow}{H}_{0}=\
abla\varphi/n\times\overset{\rightarrow} {E}_{0} \tag{20}\]  
where \(n=\sqrt{\varepsilon/\varepsilon_{0}}\) is the refractive index of the medium and \(Z_{0}=\sqrt{\mu/\varepsilon}\) is the characteristic impedance of the medium. Equations (19) and (20) show that \(\overset{\rightarrow}{E}_{0}\), \(\overset{\rightarrow}{H}_{0}\), and \(\
abla\varphi\) are mutually orthogonal vectors; thus, the geometric optical field is locally a plane wave.  
Bringing Equation (16) into Equation (15) yields  
\[\
abla\varphi\times(\
abla\varphi\times\overset{\rightarrow}{E}_{0})+n^{2} \overset{\rightarrow}{E}_{0}=0 \tag{21}\]  
and  
\[\
abla\varphi\times(\
abla\varphi\times\overset{\rightarrow}{E}_{0})=( \overset{\rightarrow}{E}_{0}\cdot\
abla\varphi)-(\
abla\varphi)^{2}\overset{ \rightarrow}{E}_{0} \tag{22}\]  
Therefore,  
\[[n^{2}-(\
abla\varphi)^{2}]\overset{\rightarrow}{E}_{0}=0 \tag{23}\]  
As \(E_{0}\) has a non-zero solution, there must exist  
\[(\
abla\varphi)^{2}=\
abla\varphi\cdot\
abla\varphi=n^{2} \tag{24}\]  
where \(\varphi\) is referred to as the eikonal, which includes the variation in the electrical parameters of the medium. Equation (24) is referred to as the eikonal equation. This equation leads to the eikonal or phase function \(\varphi(x,y,z)\), which is a fundamental equation in geometric optics. Based on [PERSON]'s principle, the ray between two points is the curve that the optical path \(\int_{c}nds\) takes as the extreme value. In a homogeneous medium, the optical path is proportional to the geometric distance. The straight line between two points is the shortest; thus, the optical path consists of a straight line.