## 2 Materials and Methods
### InSAR Dataset
The image will have less noise but less spatial resolution, changing from about 5 \(\times\) 20 m\({}^{2}\) to 300 \(\times\) 300 m\({}^{2}\) (in our case). Both filters are available in the SNAP toolbox. The unwrapping interferometry phase was computed using the algorithm proposed by [46]. On the hypothesis that all the phase contributions, except for the wet contribution, can be neglected or have been mitigated (see Equation (4)), we can estimate differential PWV maps from the unwrapping phase as  
Figure 1: GNSS tomographic 2D grid over Houston city (TX, USA). Blue triangles display the location of GPS stations used in the tomographic process, and red triangles are the stations used to obtain the absolute and calibrated InSAR PWV maps used to evaluate the tomographic solution spatial variability. The rectangles display the two footprints of Sentinel-1 used (ascending and descending orbits). The background colormap represents the elevation. The red dashed line indicates the vertical cross-section location used further on. The black line is the best track (determined by the National Hurricane Center) of Hurricane Harvey from 26 August to 29 August 2017 (hours in UTC). \[\Delta PWV=\Pi\cdot\frac{\lambda}{4\pi}\cdot\Delta\phi_{wet}\cdot\cos(\theta_{ look}) \tag{5}\]  
where \(\lambda\) is the radar wavelength corresponding to the central frequency of the Sentinel-1 signal, \(\theta_{look}\) is the look angle along the swath, and \(\Pi\) is a proportionality constant that transforms signal delay in a corresponding PWV amount [47], given by  
\[\Pi=\frac{10^{6}}{\rho\cdot R_{w}\cdot\left[\frac{k_{3}}{T_{m}}+k_{2}^{\prime} \right]} \tag{6}\]  
where \(\rho\) is the density of liquid water, \(R_{w}\) is the specific gas constant for water vapor, \(k_{2}^{\prime}\) and \(k_{3}\) are refractivity constants [48], and \(T_{m}\) is the weighted mean temperature of the atmosphere, given by  
\[T_{m}=\int\left(\frac{e}{T}\right)\mathrm{d}z/\int\left(\frac{e}{T^{2}}\right) \mathrm{d}z \tag{7}\]  
with \(e\) in hPa and \(T\) in K. We used the ERA5 data to calculate \(T_{m}\); details about this calculation can be seen in [49]. Note that while \"mm\" is commonly used for PWV, \"kg/m2\" is the correct physical unit. It aligns with the water budget equation and avoids confusion with ZWD, which is a delay computed as a distance. Our choice is consistent with [50]. Two essential aspects of InSAR data processing should be noted. Firstly, any geophysical quantity derived from the interferometric phase is inherently differential. Secondly, the unwrapping algorithm introduces an arbitrary constant to the phase. In order to obtain an absolute and calibrated quantity, the methodology proposed by [51] and improved by [52] was adopted here. This method is based on the least-squares approach to estimate single epochs and uses data from an NWP (in our case, from the ERA5 model) and GNSS observations (highlighted in red in Figure 1). It is important to note that there is no perfect method for deriving an absolute map from the InSAR time series; each technique has its own set of limitations. A comprehensive comparison of different methods, including their drawbacks, is detailed in [53]. It is now well established from a variety of studies that the InSAR technique can provide accurate PWV maps, achieving an accuracy of 1-2 mm at a spatial resolution of about 300 \(\times\) 300 m\({}^{2}\)[42; 43; 49; 54; 55; 56; 57]. Furthermore, this technique uses microwave signals (most common are L-, C-, and X-band), showing advantages over other techniques (like visible light sensors) since it can operate day and night and is not affected by weather conditions such as clouds, making it a reliable data source under various atmospheric conditions [42]. It is sensitive enough to detect small-scale atmospheric phenomena, such as localized water vapor variations, which other observation methods often miss. It is used here to validate and complement other atmospheric observations, enhancing this study's overall reliability and robustness.