## Appendix A Variable Retrieval
### Soil Moisture Retrieval
#### a.1.2 Zenith Total Delay Retrieval from InSAR  
Synthetic Aperture Radar data can be profitably used to monitor the tropospheric water vapor with unprecedented spatial resolution by means of interferometric techniques (e.g., References [9; 10] ). The basic idea behind SAR interferometry is simple. Satellite Radars transmit Electro Magnetic (EM) signals toward the earth, which are scattered by different targets, such as man made structures, bare soil, vegetation, bouncing back toward the satellite. These \"back scattered\" signals are captured by satellite sensors, at different times, according to their distance from the satellite and are used to generate radar images of the earth's surface. The key information for ZTD retrieval is the phase of the returned signal, measured for each single target in the scene. This phase is the number of cycles of the EM wave from the satellite to the target and then back to the satellite, plus a contribution that depend on the specific electric and geometric properties of the target. The phase is proportional to the Optical Path Length (OPL) \(R\) from the sensor, located in \(S(t)\) (t being the sensing time), to the target, located in \(P\):  
\[R(P;t)=\int_{S}^{P}cdt=\int_{S}^{P}\frac{c}{v(l;t)}dl=\int_{S}^{P}n(l;t)dl=R_ {0}(P;t)+R_{N}(P;t) \tag{1}\]  
where \(v(l;t\) is the actual velocity of the EM signal through the atmosphere, \(c\) is the velocity of light in vacuum, \(n(l;t)\) is the refraction index along the path \(R_{0}(P)=|S(P;t)-P|\) is the sensor-target geometric distance and \(R_{N}=\int_{S}^{P}(n(l;t)-1)dl=10^{-6}\int_{S}^{P}N(l;t)dl\) is the extra path due to the variation of the EM signal velocity with in the atmosphere with respect to vacuum, expressed as the integral of the refractivity index \(N=10^{6}(n-1)\). The phase is then:  
\[\phi(P;t)=\frac{2\pi f_{0}}{c}\cdot 2R(P;t)+\phi_{0}(P), \tag{10}\]  
where \(f_{0}\) the Radar frequency, \(\phi_{0}(P)\) is the target own phase. The phase \(\phi\) is not usable to estimate the optical path length, \(R(P;t)\), since \(\phi_{0}(P)\) is unknown. However, this contribution is removed in the difference between the phases of the same target observed in two different epochs:  
\[\Delta\phi(P)=\frac{4\pi f_{0}\cdot\Delta R(P)}{c}\simeq\frac{4\pi f_{0}}{c} \cdot\left(\Delta R_{0}(P)+\Delta R_{N}(P;t)\right), \tag{11}\]  
This is the interferometric phase, that is modelled as the sum of the contributions due difference in the geometrical distances between the target and the satellite (at the two epochs) and the difference in the integrals of the refractivity indexes along the path [62], henceforth leading to the defined Atmospheric Phase Screen (APS). In deriving (11) from (10) and (10) we have assumed that the sensor-target path does not change appreciably, to give a variation of the refractive index. The retrieval of the APS from (11) implies:  
1--the precise knowledge of \(\Delta R_{0}(P)\) the differential position of the satellite, the orbit and of each target in the scene, with an error much lower than the wavelength (5.7 cm for Sentinel-1),  
2--the stability of the target dielectric and geometric properties with time, that is assuming \(\phi_{0}(P)\) constant,  
3--the separation of the ionospheric contribution, that is also affecting the propagation delay,  
4--the reconstruction of the absolute phase field, from that measured by the data, that is known in modulus \(2\pi\). All these issues has been addressed in these past decades and good solutions have been found, though leading to some limitations. The most challenging is indeed the second, the requirement of the target dielectric and geometrical stability in a time interval of many days. It has been shown (e.g., References [38; 63]) the existence of natural targets, named Permanent Scatterers, that are stable over very long periods, encompassing years, so stable that can be used to track sub-millimetric crustal deformations.