## 4 Soil Moisture Retrieval from Reflection Measurements: A Background on the Discipline
### LHCP + RHCP-Based Soil Moisture Retrieval  
The retrieval algorithm described above is based on the assumptions of having a smooth reflection surface. Nonetheless, in order to better take into account the effects of the soil roughness, which makes the reflection different from specular, another approach is needed, which exploits the availability of both the LHCP and the RHCP SNR measurements. The roughness of a surface impacts its capability of reflecting an incident electromagnetic field along a principal direction (reflection angle); this capability is typically quantified in terms of the so-called radar cross-section RCS [35]. The RCS of an object is in turn a function of: (i) the object dimensions and shape; (ii) the electromagnetic wave incident angle; and (iii) the reflecting material (through the so-called normalized radar cross-section (NRCS), \(\sigma^{o}\)). The NRCS is a function of the dielectric properties of the material and separates into a horizontal and a vertical polarization component. For these reasons, we expect it to be possible to extract the dielectric constant of the soil by estimating the NRCS for the two circular polarizations of the reflected GNSS signals (it is well known that the two LH and RH circular polarizations of an electromagnetic wave can be written as combinations of the two linear polarizations) [33]. The SNR of the reflected signals, which was expressed in Equation (2) for the LHCP reflection, can be expressed for both the LHCP and the RHCP reflections as:  
\[SNR_{refl,l} = \frac{G^{t}G^{r}_{refl,l}\lambda^{2}\left|\sigma_{lr}\right|}{ \left(4\pi\right)^{3}{R_{A}}^{2}{R_{B}}^{2}}\frac{P^{t}}{P_{N,l}^{r}} \tag{6}\] \[SNR_{refl,r} = \frac{G^{t}G^{r}_{refl,r}\lambda^{2}\left|\sigma_{rr}\right|}{ \left(4\pi\right)^{3}{R_{A}}^{2}{R_{B}}^{2}}\frac{P^{t}}{P_{N,r}^{r}} \tag{7}\]  
where the notation is the one adopted in Equation (2), while the parameters \(\sigma_{ij}\) represent the RCS for the circular polarized components of the incident and the reflected waves, for which the subscripts are such that \(i\) indicates the polarization of the reflected signal and \(j\) indicates the polarization of the incident wave. Furthermore:  
\[\sqrt{\sigma_{lr}} = \frac{\sqrt{A}}{2}\left(\sqrt{\sigma_{hh}^{o}}+\sqrt{\sigma_{vv} ^{o}}\right) \tag{8}\] \[\sqrt{\sigma_{rr}} = \frac{\sqrt{A}}{2}\left(\sqrt{\sigma_{hh}^{o}}-\sqrt{\sigma_{vv} ^{o}}\right) \tag{9}\]  
where \(\sigma_{hh}^{o}\), \(\sigma_{vv}^{o}\) are the horizontal and vertical polarization components of the NRCS and \(A\) is the total illuminated area, or glistening zone, which depends on the reflection geometry [22, 35]. The NRCS is a key parameter in reflection theory: through its estimation, the characteristics of the reflecting surface can be retrieved, but to do that, a good model of the reflecting system is required. A detailed analysis of the NRCS and of the effects of the geometry (incidence angle) is available in [36], where also the soil inhomogeneity is taken into account. The more accurate the model applied is, the more accurate the soil estimate will be. However, at this stage of the work, a simple approximated model has been considered, without a thorough study of the soil characteristics, as for instance the soil roughness. Combining Equations (6) and (7) with Equations (8) and (9), the ratio between Equations (6) and (7) can be written as:  
\[\frac{SNR_{refl,l}}{SNR_{refl,r}}=\frac{\left|\sqrt{\sigma_{hh}^{o}}+\sqrt{ \sigma_{vv}^{o}}\right|}{\left|\sqrt{\sigma_{hh}^{o}}-\sqrt{\sigma_{vv}^{o}} \right|}\cdot C^{{}^{\prime}} \tag{10}\]  
where \(C^{{}^{\prime}}=\frac{G^{r}_{refl,l}}{P_{N,l}^{r}}\cdot\frac{P_{N,r}^{r}}{G^{ r}_{refl,r}}\) is a calibration constant similar to \(C\) in Equation (5). The parameters \(\sigma_{hh}^{o}\) and \(\sigma_{vv}^{o}\) are functions, in particular, of the soil dielectric constant and the incidence angle (the satellite elevation); it can be stated that:  
\[ \begin{cases}\sigma_{hh}^{o}=f_{1}(\epsilon_{r},\theta)\\ \sigma_{vv}^{o}=f_{2}(\epsilon_{r},\theta)\end{cases}