## 2 Description of the GNSS-R Experiment
Finally, after averaging, power DDM's \(\langle\left|Y^{\text{ref}}\right|^{2}\rangle\) were obtained as a result [18; 30]:  
\[\langle\left|Y^{\text{ref}}\right|^{2}\rangle=\frac{\text{P}_{\text{t}}\lambda^ {2}}{(4\pi)^{3}}\iint\frac{\text{D}_{\text{T}}(\overset{\rightarrow}{\rho}) \text{D}_{\text{R}}(\overset{\rightarrow}{\rho})\left|X(\tau-(\text{R}_{0}+ \text{R})/\text{c},\text{f}_{\text{D}}(\overset{\rightarrow}{\rho})-\text{f}_{ \text{c}})\right|^{2}\sigma^{0}(\overset{\rightarrow}{\rho})}{\text{R}_{\text{ T}}^{2}(\overset{\rightarrow}{\rho})\text{R}_{\text{R}}^{2}(\overset{ \rightarrow}{\rho})}\text{d}^{2}\rho, \tag{1}\]  
where \(\text{P}_{\text{t}}\) is the transmitted power, \(\text{D}_{\text{T}}\) and \(\text{D}_{\text{R}}\) are the transmitter and receiver antenna directivities at the point \(\overset{\rightarrow}{\rho}\) over the Earth's surface, \(\text{R}_{\text{T}}\) and \(\text{R}_{\text{R}}\) are the distance from the transmitter and the receiver to point \(\overset{\rightarrow}{\rho}\), \(\left|X(\tau,\text{f})\right|^{2}\) is the Woodward Ambiguity Function (WAF), and \(\sigma^{0}\) is scattering coefficient. The coherent integration time was set to \(\text{T}_{\text{coh}}\) = 5 ms and the number of incoherent averaging \(\text{N}_{\text{inc}}\) = 5 to increase the Signal-to-Noise Ratio (SNR). Lower levels of \(\text{T}_{\text{coh}}\) were tested, and it was found that \(\text{T}_{\text{coh}}\) = 5 ms is the optimum for this experiment configuration. The spatial resolution associated to the waveforms' peak is linked to the first Fresnel zone (\(\sim\)1 km\({}^{2}\)) in the case of specular scattering, and limited by the glistening zone in the case of the non-specular scattering, related to the area of the first iso-delay ellipse, that is \(\sim\)1800 km\({}^{2}\); plus the degradation due to: (a) the platform's movement (in the SMAP's case this degradation is \(\left|\overset{\rightarrow}{\text{V}_{\text{SMAP}}}\right|\cdot\text{T}_{ \text{c}}\cdot\text{N}_{\text{inc}}=7.48\) km/s \(\cdot\) 0.005 s \(\cdot\) 5 = 0.187 km); (b) the antenna rotation; this degradation is unique to this scenario, and as a consequence the point at antenna boresight moves \(\sim\)18 km during the integration time. Figure 1: The Soil Moisture Active Passive (SMAP) mission was successfully launched on 31 January 2015 into a Sun-Synchronous Orbit (SSO) with a local time of ascending node (LTAN) of 06:00 h, and an orbit reference height of \(\sim\)685 km. The experiment set-up of the Global Navigation Satellite Systems Reflectometry (GNSS-R) experiment on-board the SMAP mission is represented in this figure. Scattered GPS signals were collected by the SMAP’s radar antenna at an elevation angle \(\theta_{\text{e}}\)\(\sim\)55\({}^{\circ}\). The gain of the H-Pol and V-Pol antenna was \(\sim\)36 dB. Data processing was performed on-ground at NASA’s Jet Propulsion Laboratory (JPL). Waveforms corresponding to the Doppler central frequency were obtained for further assessment over land and cryosphere. Averaged waveforms with the associated RMS at each lag are represented in Figure 3 for both polarizations and different terrain types. To describe the shape of the waveforms quantitatively and to account for the power differences and spreading of the signal, the following parameters are used: Signal-to-Noise Ratio (SNR), Polarimetric Ratio (PR), and waveforms' trailing and leading edges width. The SNR is estimated as in [31], and defined as [3]:  
\[\mathrm{SNR}=\frac{\langle\left|\mathrm{Y^{ref}}\right|^{2}\rangle}{\langle \left|\mathrm{Y^{noise}}\right|^{2}\rangle}=\frac{\langle\left|\mathrm{Y^{ ref}}\right|^{2}\rangle}{\mathrm{k_{B}TB}}, \tag{2}\]  
where \(\mathrm{k_{B}}\) is the Boltzmann constant, \(\mathrm{T}\) is the equivalent noise temperature of the down-looking chain, and B is the signal bandwidth.