## 4 Soil Moisture Retrieval from Reflection Measurements: A Background on the Discipline
### LHCP-Based Soil Moisture Retrieval  
In the simplified hypothesis of a specular reflection, the soil reflection index can be estimated from the ratio between the reflected and the directly incident signal power. This quantity can be evaluated as the ratio of the estimated signal-to-noise ratios (SNR) on the reflected and on the direct channel, respectively. In this way, the differences in the hardware receiving chains of the reflected and the direct signals can be compensated. Once the soil reflection index is known, the dielectric constant can be retrieved. Thus, in order to estimate the soil dielectric constant from the GNSS reflection measurement, it is needed to _a priori_ know the satellite elevation (incidence angle) and to have at least a rough knowledge of the soil composition. Then, the reflection index must be estimated, which is a function of the reflected power percentage after the incidence. In what follows, the concepts explained so far will be provided in formulas. If the reflecting surface can be well approximated as a perfectly smooth surface, then a specular reflection can be assumed. In such a case, neglecting any surface roughness and, therefore, any non-coherent components in the reflection, the reflected GNSS signals results in being mainly LHCP, in particular considering the satellites with close to the zenith elevation. For the direct signal propagating in free space, the SNR is directly proportional to the transmitted power \(P^{t}\) and the transmitter gain \(G^{t}\), the receiver gain for the direct signal chain \(G^{r}_{dir}\), the signal wave-length \(\lambda\) and the processing gain \(G_{D}\). Then, it is inversely proportional to the transmitter-receiver distance \(R\) and to the noise power, which, for the direct signal receiving chain, is indicated as \(P^{r}_{N,dir}\). Thus:  
\[SNR_{dir}=\frac{P^{t}G^{t}}{4\pi R^{2}}\cdot\frac{G^{r}_{dir}\lambda^{2}G_{D}} {4\pi P^{r}_{N,dir}} \tag{1}\]  
Similarly, for the LHCP reflected channel, the SNR can be expressed as the power ratio between the reflected signal and the noise related to that channel. It can be written replacing in Equation (1) \(G^{r}_{dir}\) with the receiver gain through the appropriate chain \(G^{r}_{refl,l}\) and the traveled distance \(R\) with the sum of the distances from the satellite to the reflection point (\(R_{A}\)) and back to the receiver (\(R_{B}\)). Furthermore, it is necessary to account for the additional path loss due to the reflection, which can be written as \(\frac{1}{4}\left(\left|\Gamma_{vv}\right|+\left|\Gamma_{hh}\right|\right)^{2}\)[33], where \(\left|\Gamma_{vv}\right|\) and \(\left|\Gamma_{hh}\right|\) are the reflection indexes for the vertical and the horizontal polarizations, respectively, which combine together in the case of circular polarization. The noise power in the reflected signal chain is \(P^{r}_{N,l}\). Thus:  
\[SNR_{refl,l}=\frac{1}{4}\frac{P^{t}G^{t}}{4\pi(R_{A}+R_{B})^{2}}\cdot\frac{G^ {r}_{refl,l}\lambda^{2}G_{D}}{4\pi P^{r}_{N,l}}\cdot(\left|\Gamma_{vv}\right|+ \left|\Gamma_{hh}\right|)^{2} \tag{2}\]  
where the subscript \(l\) in \(SNR_{refl,l}\) and \(P^{r}_{N,l}\) refers to the LHCP reflected polarization. As said above, the reflection index is a function of both the reflecting surface characteristics and the incidence angle; therefore, it can be expressed as a function of the soil dielectric constant \(\epsilon_{r}\) and the satellite elevation angle \(\theta\):  
\[\Gamma_{vv}(\epsilon_{r},\theta)=\frac{\sin\theta-\sqrt{\epsilon_{r}-\cos^{2} \theta}}{\sin\theta+\sqrt{\epsilon_{r}-\cos^{2}\theta}} \tag{3}\]  
\[\Gamma_{hh}(\epsilon_{r},\theta)=\frac{\epsilon_{r}\sin\theta-\sqrt{\epsilon_ {r}-\cos^{2}\theta}}{\epsilon_{r}\sin\theta+\sqrt{\epsilon_{r}-\cos^{2} \theta}} \tag{4}\]  
In order to evaluate the soil dielectric constant, from which the soil moisture can be retrieved if there is some knowledge of the soil composition, the ratio between the reflected SNR in Equation (2) and the direct SNR in Equation (1) is computed.