## 4 Results  
### Calibration of Thermal Inertia Based on TLHS under Controlled Conditions--The Indoor Experiment  
Estimation of soil water content, SWC\({}_{\text{Kp}}\), requires investigating the temporal variability of soil thermal properties, as well as to parameterize the model shown in Equation (2). Soil bulk density, \(\rho\), measured at the dry condition was about 1330 kg m\({}^{-3}\), whereas the porosity, \(e\), was estimated in about 0.5 m\({}^{3}\) m\({}^{-3}\). These values were comparable to those obtained in the field (Table 1). Measurements of actual SWC through the ML3-ThetaProbe, SWC\({}_{\text{TP}}\), allowed calibrating Equation (2). The least square fitting method, by fixing \(e\) to the reported value, allowed estimating \(\epsilon=1.78\) and \(\mu=0.47\). These parameters were fitted through a generalized reduced gradient (GRG) non-linear solving method by maximizing \(r^{2}\) and minimizing the mean absolute error (MAE). Figure 2 shows the relationship between the thermal-inertia-based soil water content, SWC\({}_{\text{Kp}}\), and the corresponding soil water content measured by the ML3-ThetaProbe, SWC\({}_{\text{TP}}\). Metrics of SWC\({}_{\text{Kp}}\) vs.  
SWC\({}_{\rm TP}\) highlight good agreement (R\({}^{2}\) > 0.99, MAE = 0.01 m\({}^{3}\) m\({}^{-3}\), \(RMSE\) = 0.02 m\({}^{3}\) m\({}^{-3}\)); however, a moderate intercept value of \(\sim\)\(-\)0.03 m\({}^{3}\) m\({}^{-3}\) evidences a systematic underestimation of SWC\({}_{\rm KP}\). Simulations of SWC temporal dynamics were also performed by the Hydrus-2 Dmodel according to the schematization described in paragraph 3.5. Soil water contents simulated by Hydrus-2D (SWC\({}_{\rm H2D}\)) are plotted in Figure 3 together with those measured by both the balance (SWC\({}_{\rm m}\)) and ML3-ThetaProbe (SWC\({}_{\rm TP}\)). Additionally, the soil water contents estimated by the thermal inertia approach (SWC\({}_{\rm KP}\)) are also shown. A qualitative analysis of Figure 3 shows good performance of the Hydrus-2 Dmodel if compared to the measured soil water content, SWC\({}_{\rm m}\) and SWC\({}_{\rm TP}\). On the contrary, the thermal inertia model simulates values of SWCs lower than the corresponding measured ones, even if with a similar trend. Statistically, the \(RMSE\) associated to both the models (Hydrus-2D, TLHS) were appreciable and equal to \(RMSE\) = 0.02 m\({}^{3}\) m\({}^{-3}\) for the Hydrus model and \(RMSE\) = 0.05 m\({}^{3}\) m\({}^{-3}\) for the TLHS approaches. With reference to the model efficiency, E, more than satisfactory behavior has been observed for both the Hydrus-2D (\(E\) = 0.98) and TLHS (\(E\) = 0.82) models, which can therefore be considered suitable to predict the temporal dynamic of SWC in the surface soil layer. Figure 3: Temporal dynamics of simulated and measured soil water contents during the experimental laboratory trial. Figure 2: Thermal inertia (SWC\({}_{\rm KP}\)) vs. measured (SWC\({}_{\rm TP}\)) soil water content (grey circles). The 1:1 line is also shown (dashed line). Application of Thermal Inertia Based on Proximity Sensing Images under Natural Conditions--The Field Experiment  
With reference to a typical acquisition date (30 July), Figure 4 shows a true color composite image (left panel) and a color-infrared composite image (right panel) acquired at 13:21 (solar time) with the optical and the multispectral cameras, respectively. Vegetation cover is distinguishable by the reddish color (near infrared) in the color-infrared composite image. The very high spatial resolution (\(\sim\)0.03 m) of the proximity-sensing images pointed out some operational limits due to the shadows of trees and soil vegetation cover. The deployment of black and white reference targets (Figure 4) allowed the accurate in-reflectance calibration of the VIS/NIR images and thus of the albedo, \(\alpha_{SW}\). Bare soil is, on average, characterized by a lower \(\alpha_{SW}\) than soil covered by vegetation, while wet soil was slightly darker than dry soil (higher \(\alpha_{SW}\)).