## 2 Materials and Methods
### TDS-1 Data Set
#### 2.1.2 Methodology: TDS-1 Data Pre-Processing  
It is well known that TDS-1 GNSS-R data is notably affected by attitude control issues when the satellite enters into eclipse, as Sun sensors are not able to provide attitude information. TDS-1 DDMs may also be affected by the presence of the direct signal within the delay and Doppler window where the reflected signal is being tracked, thus affecting the observables. TDS-1 GNSS-R data can also be affected by anomalously high noise, as in Figure 1a, or by radio frequency interference (RFI), as in Figure 1e in battlefronts, or in regions with high density of jammers (e.g., Asia). All these data have to be first screened and discarded. Then, the remaining data have to be re-compensated for antenna pattern effects which, as illustrated in Figure 4, strongly modulates the received signal power. Figure 4 scatter is due to antenna pattern azimuthal asymmetry (all \(\varphi\) angles overlaid for a given \(\theta\)) and platform attitude uncertainty. The variation of the distance transmitter-specular reflection point plus specular reflection point to the receiver (\"distance factor\", or DF) is also included at this stage. This uncertainty may be the largest contributor to the scattering encountered in the GNSS-R data, although --as it will be discussed later-- it is not the only source. Figure 4: Antenna pattern as a function of the incidence angle. Figure 3: **(a)** Pixels corresponding to different ranges of DDM delay spread (\(\sigma_{\tau}\)) and DDM Doppler frequency spread (\(\sigma_{f_{d}}\)) parameters. **(b)** Surface topography flag derived from a Digital Elevation Model. White pixels correspond to pixels that may be affected by topography. Note that it is nearly the negative of Figure 3a. The next step includes the elimination of pixels that are likely affected by surface topography effects (Figure 3b), and the compensation of the vegetation effects using the SMOS-derived Vegetation Optical Depth (VOD). To do that, a simple tau-omega model is assumed for the reflectivity [2]:  
\[\Gamma(\theta_{i},\text{SM},\text{VOD})=\Gamma(\theta_{i},\text{SM})\cdot e^{-HR }.e^{-2\cdot\text{VOD}/\cos(\theta_{i})} \tag{2}\]  
In Equation (2), \(\Gamma(\theta_{i},\text{SM})\) is the Fresnel reflectivity, that depends on the soil moisture (SM) and on the incidence angle (\(\theta_{i}\)). As a difference from the tau-omega model used in microwave radiometry at L-band, the \"2\" term in the exponential accounts for the two-way propagation through the vegetation layer. However, since the polarization-dependence of the VOD information is not available, it is neglected. It is worth to mention that the above model holds for most vegetation types, except for very dense vegetation and at large incidence angles, when multiple scattering cannot be neglected. In that case, the \(\tau\)-\(\omega\) model fails, and refined models are needed [8]. Small scale surface roughness (as opposed to the large scale surface topography effects) is accounted for using the \(HR\) parameter used in SMAP [9]. Last, it is important to emphasize that mixed pixel effects are not accounted for, and are believed to be the second largest contributor to the scatter of the data. This effect is studied in more detail in Sections 3.2 and 3.3, where SMOS L3 SM, and L4 SM data downscaled at 1 km [10], and point measurements are used for inter-comparison purposes. However, while mixed pixel effects are already accounted for in the SMOS SM processor, and SMOS SM values may not be available in case of strong heterogeneity, it is also important to emphasize that the resolution of SMOS products (native resolution from 35 to 50 km, re-gridded at 25 km) is significantly different from that of GNSS-R products, as it can be as small as a few hundred meters from a Low Earth Orbit (LEO), in case of coherent scattering, as it is determined by the size of the first Fresnel zone [5].