# The Use of SMAP-Reflectometry in Science Applications: Calibration and Capabilities
## 3 Calibration of SMAP-R Signals  
The mathematical equations that describe the collected power over a scattering surface of the GPS signal (i.e., DDM) were defined in [35; 36] and can easily be found in the literature. Those equations are \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multicolumn{2}{c}{**N = 80\%**} & \multicolumn{2}{c}{**N = 75\%**} & \multicolumn{2}{c}{**N = 70\%**} & \multicolumn{2}{c}{**N = 65\%**} & \multicolumn{2}{c}{**N = 60\%**} \\ \cline{2-11}  & \(\mu\) (km) & \(\sigma\) (km) & \(\mu\) (km) & \(\sigma\) (km) & \(\mu\) (km) & \(\sigma\) (km) & \(\mu\) (km) & \(\sigma\) (km) & \(\mu\) (km) & \(\sigma\) (km) \\ \hline
**Sea Ice** & 2.8 & 0.9 & 2.4 & 0.8 & 2.1 & 0.6 & 1.8 & 0.3 & 1.8 & 0.25 \\
**Lake** & 2.7 & 0.8 & 2.6 & 0.9 & 2.3 & 0.7 & 1.5 & 0.25 & 1.5 & 0.26 \\
**Wetland** & 9.6 & 2.8 & 3.3 & 1.4 & 2.8 & 0.5 & 2.4 & 0.34 & 2.1 & 0.28 \\
**Arid Land** & 13.3 & 5.8 & 6.4 & 4.2 & 5.3 & 2.6 & 4.9 & 2.3 & 4.3 & 2.4 \\
**Low Vegetation** & 20.2 & 6.6 & 12.5 & 5.6 & 12.1 & 4.2 & 11.9 & 3.9 & 8.6 & 3.3 \\
**High Vegetation** & 43.1 & 7.3 & 38.9 & 6.5 & 26.6 & 4.5 & 26.2 & 4.1 & 23.5 & 8.5 \\
**Ocean** & 45.3 & 8.4 & 44.8 & 7.9 & 44.6 & 7.6 & 42.3 & 6.8 & 41.1 & 6.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean size of the scattering area (diameter) of GPS reflections over a variety of scenes and N values as measured by SMAP-R.  
presented here to illustrate the SMAP-R differences with respect to other missions. Equation (1) describes the reflected GPS signal that would be received from a surface dominated by incoherent scattering. \[P_{incoth}(\tau,f_{d})=T_{i}^{2}\int\frac{P_{tx}G_{tx}G_{rx}\Big{(}\overrightarrow{ \rho}\Big{)}\lambda^{2}\Lambda^{2}\Big{(}\tau-\tau_{s}\Big{(}\overrightarrow{ \rho}\Big{)}\Big{)}}{(4\pi)^{3}R_{rx}^{2}\Big{(}\overrightarrow{\rho}\Big{)}R _{tx}^{2}\Big{(}\overrightarrow{\rho}\Big{)}}\times\big{|}S\big{(}f_{d}\Big{(} \overrightarrow{\rho}\Big{)}-f\big{)}\big{|}^{2}\sigma_{0}\Big{(}\overrightarrow {\rho}\Big{)}d^{2}\rho \tag{1}\]  
where  
* \(\overrightarrow{\rho}\) defines the different surface pixels within the scattering area;
* T\({}_{\rm i}\) is the coherent integration time;
* P\({}_{\rm tx}\) is the GPS transmitted power;
* G\({}_{\rm rx}\) is the GPS transmitter antenna gain;
* G\({}_{\rm rx}\) is the receiver antenna gain;
* \(\lambda\) is the GPS signal wavelength (at GPS-L2C is 24.42 cm);
* R\({}_{\rm rx}\) corresponds to the distance from the transmitter to a particular surface pixel (\(\overrightarrow{\rho}\));
* R\({}_{\rm rx}\) corresponds to the distance from the receiver to a particular surface pixel (\(\overrightarrow{\rho}\));
* The sinc (S) function defines the spread of the signal in Doppler;
* The delta \(\Lambda\) function defines the spread of the signal in time;
* \(\sigma_{0}\) (\(\overrightarrow{\rho}\)) is the bistatic radar cross-section of a rough surface and is defined as:
* \(\sigma_{0}\) (\(\overrightarrow{\rho}\)) = \(\pi\)!\(\mathcal{R}(\overrightarrow{\rho})\)!\(\frac{2^{4}\mathcal{(} \overrightarrow{\rho})}{\mathrm{q}_{\mathrm{z}}^{4}(\overrightarrow{\rho})} \)!\(\varphi\)(-\frac{\mathrm{q}_{\mathrm{z}}(\overrightarrow{\rho})}{ \mathrm{q}_{\mathrm{z}}(\overrightarrow{\rho})}),  
where \(\mathcal{R}\) is the Fresnel reflection coefficient, \(\varphi\) is the probability density function of the surface slopes and \(\mathrm{q}_{\mathrm{z}}\) and \(\mathrm{q}_{\mathrm{\perp}}\) are the tangential and perpendicular components of the bisector vector or scattering vector \(\overrightarrow{\mathrm{q}}\). Equation (2) describes the reflected GPS signal that would be received from a surface dominated by coherent scattering [15; 36].