## 4 Modeling and Inversion of GNSS-R Reflectivity
### Application to the GLORI Data  
The proposed model was applied to the reflectivity \(\Gamma_{RL}\), derived from data recorded at all elevations between 50\({}^{\circ}\) and 90\({}^{\circ}\). To evaluate the model linking \(\Gamma_{RL}\) to soil moisture and LAI, we implemented a 5-fold cross-validation approach. Our database set (90 samples, each of which corresponds to a GNSS-R mean measurement on a reference field and a given date) was divided into five disjoint folds of the same size. In these sequential folds, the first was used for validation, and the four remaining folds were used for model parameterization. As in the case of the \"repeated holdout\" method, the overall accuracy is given by the average of the values obtained from all runs.  
The calibrated parameters \(\alpha\), \(\beta\), and \(\delta\) can then be introduced into Equation (9), together with the known values of soil moisture, vegetation _LAI_ and elevation angle, to determine the expected GNSS-R reflectivity.  
The retrieved equation is written as:  
\[\Gamma_{p}(\theta)_{dB}=40.73\;Mv-\frac{0.421\;LAI}{\sin(\theta)}-19.04 \tag{10}\]where \(Mv\) is the soil moisture in (m\({}^{3}\)/m\({}^{3}\)).  
Figure 7 illustrates the comparison between the observed and modeled reflectivity over the training dataset. The determinant coefficient R\({}^{2}\) is 0.69, with an RMSE (root mean square error) equal to 2.9 dB. The points showing the highest discrepancy with the proposed model correspond to the corn reference fields having the highest vegetation height. The use of LAI as a proxy for optical thickness instead of vegetation water content induces some limitations related to the saturation of LAI for dense or high covers.  
If we consider the known \(\Gamma_{RL}\) reflectivity and the LAI, we can estimate the soil moisture values of the validation set by inverting Equation (10), and we obtain:  
\[Mv=0.025\ \Gamma_{p}(\theta)_{dB}+\frac{0.01\ LAI}{\sin(\theta)}+0.47 \tag{11}\]  
Figure 8 illustrates the comparison between the estimated and measured in situ soil moisture. The results illustrate an RMSE equal to 0.06 m\({}^{3}\)/m\({}^{3}\). This accuracy is in agreement with those retrieved by other studies using GNSS-R airborne measurements [20; 21; 22; 23].  
The results show that the tau-omega model has a high potential for the retrieval of the GNSS-R \(\Gamma_{RL}\) reflectivity. The model is based on the combined contributions from only two parameters: soil moisture and vegetation LAI. The highest simulated values of \(\Gamma_{RL}\) are retrieved for fields with a slight vegetation coverage (low LAI), in agreement with the observed signals. An increase in vegetation LAI leads to an increase in vegetation attenuation, thus decreasing the global simulated signal. When all types of vegetation are considered in the validation process, no large discrepancies are revealed between the different types of cover (Figure 7). It is interesting to note that a single set of parameters (\(\alpha\), \(\beta\), \(\delta\)) can be used for all types of vegetation cover (wheat, corn, etc.), from which it is possible to derive satisfactory results, with an RMSE equal to 2.9 dB and R\({}^{2}\) equal to 0.69 for simulation of GNSS-R reflectivity and only 0.06 m\({}^{3}\)/m\({}^{3}\) as RMSE for the soil moisture inversion from the GNSS-R data. In the case of the present study, the dataset was too small to separately consider the vegetation covers with different parameterizations of the tau-omega model. However, in the context of its potential application to low-resolution satellite observations (with CYGNSS for example), these results could justify the use of  
Figure 7: Modeled vs observed \(\Gamma_{RL}\) using the training dataset. RMSE = root mean square error.  
just one variable (e.g., the NDVI) to describe the vegetation for all agricultural areas, with only one parameterization for the inversion model.