## 2 Height Factor Model for GNSS Tomography
### Construction of the Height Factor Model
#### 2.3.1 Isotropic Component of SWD for Inside Signals  
ZWD, considered as the isotropic component, is related to the water vapor above the GNSS station. \[ZWD=\int\ N_{w}dh \tag{5}\]  
Generally, Equation (5) is discretized into the following form in GNSS tomography [26]:  
\[ZWD=\sum_{i=1}^{n}N_{w}(h_{i})\cdot h_{i}, \tag{6}\]  
Figure 2: Two-dimensional schematic diagram of the height factor model (HFM) to illustrate the estimation method of sectional slant wet delay (SWD) for top and side signals. where \(N_{w}(h_{i})\) represents the wet refractivity of the \(ith\) layer and \(h_{i}\) denotes the thickness of the \(ith\) layer. The ZWD from the station to any height \(h\) above the site can be correctly estimated using Equation (6). Therefore, we can establish the HFM to analyze the relationship between the height \(h\) and the sectional ZWD, which may contribute to estimating the isotropic component of the SWD of inside signals. Radiosonde data, preserving a high accuracy and optimal vertical resolution of meteorological parameters along the tropospheric column, are used to construct the HFM in this paper. The detailed steps are given below. 1. The initial radiosonde data is encrypted to produce uniform data with a vertical resolution of 100 m and the water vapor density of the interpolation points is calculated using the cubic spline interpolation algorithm [37]. In the processing of radiosonde observations, the vertical resolution of radiosonde data at different remote sensing time is variable, i.e., the resolution of data in the lower layer is higher than that in the upper layer. It should be noted that observations with an even distribution can generate a more accurate fitting model. Therefore, the initial radiosonde data should first be encrypted to produce uniform data. 2. The partial ZWD is computed from the surface to a certain height using Equation (6). 3. The isotropic height factor \(\lambda_{iso}\) is defined, which represents the ratio between the partial content of ZWD and the total ZWD, as shown in Equation (7). \[\lambda_{iso}(h)=\frac{ZWD_{h}}{ZWD_{total}},\] (7) where \(ZWD_{h}\) denotes the partial ZWD from the surface to the height \(h\), and \(ZWD_{total}\) derived from the GNSS data is the total ZWD of the stations. Figure 3 shows the relationship between the isotropic height factor and the height based on the 30-year radiosonde data from different months in Hong Kong. 4. The function relationship between the \(\lambda_{iso}(h)\) and \(h\) is analyzed. In view of the variety of atmospheric water vapor, owing to the fact that the radiosonde balloon in Hong Kong is launched twice a day, at 00:00 UTC and 12:00 UTC, the radiosonde data cannot provide sufficient observations for daily-scale analysis. In addition, according to the periodic characteristics of ZTD and precipitable water vapor (PWV) [1,38], the relationship between the isotropic height factor and the height using 30 years of radiosonde data from 12 months is presented in Figure 3. Besides, the long-term trends of the isotropic height factor will not have an important impact on the tomography results. On the other hand, the wet refractivity water vapor decreases exponentially with height in the troposphere, and it is found from Equation (7) that the isotropic height factor \(\lambda_{iso}\) is related to the wet refractivity in the zenith direction. In the work of [PERSON] and [PERSON] [14], [PERSON] et al. [39], and [PERSON] et al. [27], a similar scale factor or truncation factor \(\lambda\) is introduced, the trend of which changes with the height, as analyzed based on the exponential relationship in all three studies. 5. Accordingly, the HFM is established as follows: \[\lambda_{iso}(h)=a_{1}e^{b_{1}h}+a_{2}e^{b_{2}h},\] (8) where \(a_{1}\), \(b_{1}\), \(a_{2}\), and \(b_{2}\) represent the coefficients of the HFM, which are determined by least-squares fitting [27].