## 1 Introduction
### [PERSON]'s Model  
[PERSON] et al. (2013) employs a uniform, random volume model of the forest medium to assess the sensitivity of power and interferometric synthetic aperture radar to tree height and vegetation density, considering extinction along with speckle and thermal noise. [PERSON]'s InSAR experiments support a simple model of forest vegetation as a homogeneous, randomly oriented volume ([PERSON], 2013; [PERSON], 2013). The model radar power, phase, coherence, and interferometric complex cross correlation as a function of vegetation height for extinction 0.2 dBm\({}^{-1}\), resulting from performing the integrals of [PERSON] et al. in 1996 [PERSON] et al. (2013), omitting the ground's contribution for simplicity is given as:  
\[\begin{split} Power\ \alpha\ \rho_{N}\ \frac{1-e^{-\gamma h_{\rm p}}}{\gamma}\\ Phase=\left[\frac{\gamma\text{\small{s}}\text{\small{s}}\text{ \small{s}}\text{\small{  
\(\eta\) is the azimuthal angle measured in the horizontal plane. The \(W_{\eta}\) function results from the process of synthesizing the aperture.  
\[W_{\eta}=e^{ik_{0}(2k_{z}-\tau_{A})}\frac{sinin\ [(n+1)]k_{0}vT/2}{sinin\ \left(\frac{k_{0}vT}{2}\right)} \tag{5}\]  
The range resolution function \(W_{\tau}\) is given by  
\[Ref(t)*\int_{-\infty}^{\infty}G(\omega)e^{-i\omega t}d\omega\Big{|}_{-}\tau= \int\left[\int_{-\infty}^{\infty}G(\omega)e^{-i\omega x}d\omega\ \right]\ Ref(x-\tau)\ dx=W_{\tau}(\tau)e^{-i\omega_{0}\tau} \tag{6}\]  
where the transmitted pulse, with Fourier transform G(\(\omega\)), is centered at time t = 0, and \(\omega_{0}\) is the center frequency of the transmitted signal. \(W_{\tau}\) is symmetric about its maximum, which occurs when its argument is zero. ### [PERSON]'s Model  
[PERSON]'s model introduces a discrete interferometric model of a random medium layer intended for radar interferometry applications. The forest is represented by defining tree trunks, branches, and leaves using randomly oriented, lossy dielectric particles with specified area and orientation. [PERSON] and [PERSON] (2009) examine the issue of scattering of a time-harmonic electromagnetic wave by N discrete identical lossy dielectric scatterers with random positions and orientations within a layer [17]. Scatterers are regarded as autonomous entities; hence, adjacent particles may not be aligned. Initially, an approximation equation for the coherent field and Green's function is delineated as [17]  
\[\underline{E}(\underline{x},0)=\underline{f}_{T}(\underline{i})\frac{e^{ik_{ 0}\tau}}{R} \tag{7}\]  
where \(k_{0}=\omega\sqrt{\mu_{0}\epsilon_{0}}\) is the wave number, \(\underline{f}_{T}(\underline{i})\) is the antenna pattern of the transmitter, and \(R\) is the distance. The far field from transmitter A on the interface of the layer of the random medium electric field is shown in Figure 3.  
Refs. [21,22] find the correlation of the fluctuating component of the scattering field as  
\[<E(x_{1})E^{*}(x_{2})>=\int_{\underline{g}}d\underline{s}\rho(s)\underline{e} _{se}(x_{1},s)\underline{e}_{se}^{*}(x_{2},s) \tag{8}\]  
Figure 3: Layer of a random medium [17]. where \(e_{se}(x,s)\) is the scattered field at \(x\) due to a scatterer at s and \(\rho(\text{s})\) is the density of particles. The correlation of the field is given as [17]  
\[C(x_{1},x_{2})=<E(x_{1})E^{*}(x_{2})>=\rho_{0}\int se_{se}(x_{1},s)e_{se}(x_{2},s) \tag{9}\]  
where \(\rho(s)=\rho_{0}\) is assumed. Figure 4 illustrates the four trajectories of the bi-static field. These four options manifest as reflection coefficients during the correlation of fields. The aforementioned equations and Figures 3 and 4 yield the four routes of the bi-static field for a narrow-band signal. \[C_{\alpha}(x_{1},\ x_{2})=\sum_{\alpha=1}^{N}\left[C_{\alpha}^{1}+C_{\alpha}^{ 2}+C_{\alpha}^{3}+C_{\alpha}^{4}\right] \tag{10}\]  
where \(\alpha=1\) for the trunk, \(\alpha=2\) for the branch, and \(\alpha=3\) for leaves.