## 2 Theoretical Model
### Ground Surface Modeling
Using Equations (5) and (6) and applying the approximation \(q_{z}^{\prime}=q_{z}^{\prime\prime}=q_{z}\), \(\Phi_{X}(\vec{\rho},\vec{\xi})\) is given as  
\[\Phi_{X}(\vec{\rho},\vec{\xi})=\left\langle\exp\bigl{[}-\mathrm{i}q_{z}\bigl{[} \zeta_{X}\Bigl{(}\vec{p}+\vec{\xi}/2\Bigr{)}-\zeta_{X}\Bigl{(}\vec{\rho}-\vec {\xi}/2\Bigr{)}\bigr{]}\right\}\right\rangle\,. \tag{9}\]  
For any infinitely differentiable function \(f\), such as a physical surface, the Taylor series approximation is given by  
\[f(\vec{r}^{\prime})-f(\vec{r}^{\prime\prime})=f\Bigl{(}\vec{\rho}+\vec{\xi}/ 2\Bigr{)}-f\Bigl{(}\vec{\rho}-\vec{\xi}/2\Bigr{)}=\
abla_{\perp}f(\vec{\rho} \,)\cdot\vec{\xi}+O(\xi^{3}). \tag{10}\]  
Assuming both \(\Phi_{\text{DEM}}\) and \(\Phi_{\text{L}}\) have limited regions of support and \(\zeta_{\text{DEM}}\) and \(\zeta_{\text{L}}\) are locally smooth, the Taylor series approximation in Equation (10) can be applied to both \(\Phi_{\text{DEM}}\) and \(\Phi_{\text{L}}\). However, the approximation cannot be applied to \(\Phi_{\text{S}}\) as \(\zeta_{\text{S}}\) can be non-smooth within the support region of \(\Phi_{\text{S}}\). Applying the approximation gives  
\[\Phi_{\text{DEM}}(\vec{\rho},\vec{\xi})=\exp\Bigl{[}-\mathrm{i}q_{z}\
abla_{ \perp}\zeta_{\text{DEM}}(\vec{\rho}\,)\cdot\vec{\xi}\Bigr{]} \tag{11}\]  
\[\Phi_{\text{L}}(\vec{\rho},\vec{\xi})=\left\langle\exp\Bigl{[}-\mathrm{i}q_{z} \
abla_{\perp}\zeta_{\text{L}}(\vec{\rho}\,)\cdot\vec{\xi}\Bigr{]}\right\rangle\,. \tag{12}\]  
To evaluate the characteristic function of the random process \(\zeta_{\text{S}}\), we assume it is Gaussian. Using the previous assumptions, the expectation of the short-wave component in Equation (8) is the characteristic function of two jointly Gaussian random variables, which can be written as  
\[\Phi_{\text{S}}(\vec{\rho},\vec{\xi})=\exp\left\{-q_{z}^{2}\sigma_{\text{S}}^{ 2}\Bigl{[}1-C\Bigl{(}\vec{\xi}\Bigr{)}\Bigr{]}\right\}=\exp\left[-q_{z}^{2} \sigma_{\text{S}}^{2}\right]\exp\left[q_{z}^{2}\sigma_{\text{S}}^{2}C\Bigl{(} \vec{\xi}\Bigr{)}\right] \tag{13}\]where \(\sigma_{\rm S}\) is the standard deviation of the short-wave process and \(C(\vec{\xi})\) is the normalized covariance function. Assuming \((q_{z}\sigma_{\rm S})^{2}<<1\), Equation (13) can be approximated using the Taylor series as  
\[\Phi_{\rm S}(\vec{\rho},\vec{\xi})\approx\exp\left[-q_{z}^{2}\sigma_{\rm S}^{2} \right]\left[1+q_{z}^{2}\sigma_{\rm S}^{2}C\left(\vec{\xi}\right)\right]. \tag{14}\]  
Substituting Equations (11), (12), and (14) into Equation (8) and using the definition of \(\vec{q}\) in Equation (2) gives  
\[ \begin{split}\sigma_{0}(\vec{\rho}\,)=\frac{|\mathcal{R}|^{2}q^{ 4}}{4\pi q_{z}^{2}}\exp\left[-q_{z}^{2}\sigma_{\rm S}^{2}\right]& \int\exp\left[-\mathrm{i}\vec{q}_{\perp}\cdot\vec{\xi}\right] \exp\left[-\mathrm{i}q_{z}\
abla_{\perp}\zeta_{\rm DEM}\cdot\vec{\xi}\right] \\ &\times\left\langle\exp\left[-\mathrm{i}q_{z}\
abla_{\perp}\zeta_ {\rm L}\cdot\vec{\xi}\right]\right\rangle\left[1+q_{z}^{2}\sigma_{\rm S}^{2} C\left(\vec{\xi}\right)\right]\mathrm{d}\vec{\xi}\,.\end{split}  \tag{15}\]  
Then, using [PERSON]'s theorem to take the expectation outside the integral and Fourier transform properties to evaluate the integral, Equation (15) can be simplified to  
\[ \begin{split}\sigma_{0}(\vec{\rho}\,)=\frac{\pi|\mathcal{R}|^{2} q^{4}}{q_{z}^{4}}\exp\left[-q_{z}^{2}\sigma_{\rm S}^{2}\right]\left\langle \delta\left(\
abla_{\perp}\zeta_{\rm L}+\frac{\vec{q}_{\perp}}{q_{z}}+\
abla_ {\perp}\zeta_{\rm DEM}\right)+\right.\\ \left.q_{z}^{2}\sigma_{\rm S}^{2}\psi^{5}\left(\
abla_{\perp}\zeta _{\rm L}+\frac{\vec{q}_{\perp}}{q_{z}}+\
abla_{\perp}\zeta_{\rm DEM}\right) \right\rangle\end{split}  \tag{16}\]  
where \(\delta\) is the Dirac delta function and \(\psi^{5}\) is the Fourier transform of the normalized covariance function scaled by \(1/q_{z}\) and it is expressed as  
\[\psi^{5}(\vec{\omega})=\int C(\vec{\xi}/q_{z})\exp\left(-\mathrm{i}\vec{\omega }\vec{\xi}\right)\mathrm{d}\vec{\xi}\,. \tag{17}\]  
Evaluating the expectation in Equation (16) gives  
\[