## 2 Theoretical Model
### Ground Surface Modeling  
The geometrical optics limit of the Kirchhoff approximation is a high frequency limit. Thus, it assumes the roughness scale is significantly larger than the electromagnetic wavelength. In other words, the surface is locally smooth. However, this is not always the case on land surfaces. For example, the wavelength of the Global Positioning System (GPS) L1 signal is 19 cm; however, many land surfaces, including our validation sites, are not smooth at this scale. To address this, we propose using two scales for the random roughness. This proposal was discussed by [PERSON] et al. [22] for GPS scattering for ocean surfaces. Splitting the random roughness into two scales makes the model contain three scales of surface height. These surface height scales are of (1) large deterministic scale \(\zeta_{\text{dem}}\), (2) intermediate random scale \(\zeta_{\text{L}}\), which accounts for the long-wave DEM error, and (3) small random scale \(\zeta_{\text{S}}\), which accounts for the short-wave DEM error. The deterministic scale \(\zeta_{\text{dem}}\) is the same as in [11]. The normalized BRCS is expressed as [11] (Equation (43))  
\[ \begin{split}\sigma_{0}(\vec{\rho})=\frac{|\mathcal{R}(\vec{\rho} )|^{2}q^{4}(\vec{\rho})}{4\pi q_{z}^{2}(\vec{\rho})}&\int\exp \big{[}\text{i}k\big{(}R_{\text{rs}}^{\prime}+R_{\text{st}}^{\prime}-R_{\text{ rs}}^{\prime\prime}-R_{\text{st}}^{\prime\prime}\big{)}|_{\zeta=0}\big{]}\\ &\times\left\langle\exp\big{[}-\text{i}q_{z}^{\prime}\zeta^{\prime }+\text{i}q_{z}^{\prime\prime}\zeta^{\prime\prime}\big{)}\right\rangle\text{d} \vec{\zeta}.\end{split}  \tag{3}\]The notation of prime and double prime indicates the evaluation at \(\vec{r}^{\prime}\) and \(\vec{r}^{\prime\prime}\), respectively, which are two arbitrary vectors on the surface \(\Sigma_{0}\). Details of the surface \(\Sigma_{0}\) are discussed in [11]. The quantities \(q_{z}\) and \(\vec{q}_{\perp}\) are components of \(\vec{q}\) defined in Equation (2). The components are expressed mathematically as  
\[\vec{q}=\vec{q}_{\perp}+\hat{z}q_{z}. \tag{4}\]  
The quantity \(q\) is the amplitude of the vector \(\vec{q}\). The vectors \(\vec{p}\) and \(\vec{\xi}\) are defined as  
\[\vec{\xi}=\vec{r}^{\prime}-\vec{r}^{\prime\prime} \tag{5}\]  
\[\vec{\rho}=(\vec{r}^{\prime}+\vec{r}^{\prime\prime})/2 \tag{6}\]  
The height \(\zeta\) is divided into three variables and is expressed as  
\[\zeta=\zeta_{\text{dem}}+\zeta_{\text{L}}+\zeta_{\text{S}}. \tag{7}\]  
where \(\zeta_{\text{dem}}\) is the height given by the DEM. The variables \(\zeta_{\text{L}}\) and \(\zeta_{\text{S}}\) are the long-wave and short-wave components, respectively, of the height below the DEM resolution. The quantity \(\zeta_{\text{dem}}\) is deterministic, whereas \(\zeta_{\text{L}}\) and \(\zeta_{\text{S}}\) are stochastic. Assuming \(\zeta_{\text{L}}\) and \(\zeta_{\text{S}}\) are mutually independent spatial random processes, and using Equation (7), we can write Equation (3) as  
\[ \begin{split}\sigma_{0}(\vec{\rho}\,)=\frac{|\mathcal{R}(\vec{ \rho}\,)|^{2}q^{4}(\vec{\rho}\,)}{4\pi q_{z}^{2}(\vec{\rho}\,)}& \int\exp\bigl{[}\mathrm{i}k\bigl{(}R_{\text{rs}}^{\prime}+R_{ \text{st}}^{\prime}-R_{\text{rs}}^{\prime\prime}-R_{\text{st}}^{\prime\prime} \bigr{)}\bigr{|}_{\zeta=0}\bigr{]}\\ &\times\Phi_{\text{DEM}}(\vec{\rho},\vec{\xi})\Phi_{\text{L}}( \vec{\rho},\vec{\xi})\Phi_{\text{S}}(\vec{\rho},\vec{\xi})\,\text{d}\vec{\xi} \end{split}  \tag{8}\]  
where \(\Phi_{X}(\vec{\rho},\vec{\xi})\) is the double point characteristic function of of \(\zeta_{X}\). The subscript \(X\) is defined as \(X\in\{\text{DEM},\text{L},\text{S}\}\).