## 2 Height Factor Model for GNSS Tomography
### Construction of the Height Factor Model
#### 2.3.2 Anisotropic Component of SWD for Inside Signals  
Gradient delays, characterizing the anisotropic water vapor distribution, have been intensively studied by a large number of scholars in VLBI analysis [28, 32, 33]. In the deduction of [PERSON] and [PERSON] [32], the two gradients mentioned above are obtained from the vertical integral of the water vapor refractivity gradient with height:  
\[G_{\alpha}=10^{-6}.\int_{0}^{H_{\textit{top}}}N_{\alpha}(h)hdh, \tag{10}\]  
where \(\alpha\) denotes the azimuth direction (e.g., \(NS\) or \(EW\)), \(N_{\alpha}(h)\) is the water vapor refractivity gradient at height \(h\), and \(H_{\textit{top}}\) represents the top of the troposphere in the integration. Besides, the prevailing assumption that the refractivity gradient decreases exponentially with height was proposed in the work of [PERSON] and [PERSON] [33]. \[\frac{N_{\alpha}(h)}{N_{S}}=e^{-\frac{h}{H_{\textit{scale}}}}, \tag{11}\]  
where \(N_{S}\) is the refractivity gradient at the surface and \(H_{\textit{scale}}\) denotes the scale height of the neutral atmosphere, with a value of approximately 2 km, according to the local ground meteorological data [40]. The integral of Equation (10) can be written with the following expression:  
\[G_{\alpha}(h)=10^{-6}N_{S}\bigg{[}H_{\textit{scale}}^{2}+e^{-\frac{H_{\textit{ top}}}{H_{\textit{scale}}}}\cdot\big{(}-H_{\textit{scale}}^{2}-H_{\textit{top}} \cdot H_{\textit{scale}}\big{)}\bigg{]}. \tag{12}\]  
In this section, an analysis of the gradient delay between the integral signal (e.g., \(OP\) in Figure 2) and inside signal (e.g., \(OQ\) in Figure 2) is performed. The anisotropic gradient delay of signal \(OP\) and \(OQ\) can be calculated with Equation (2). \[\Delta L_{G}^{OP}=mf_{g}(\varepsilon)\cdot\big{(}G_{NS}^{OP}\cdot\cos(\alpha) +G_{EW}^{OP}\cdot\sin(\alpha)\big{)}, \tag{13}\]  
\[\Delta L_{G}^{OQ}=mf_{g}(\varepsilon)\cdot\big{(}G_{NS}^{OQ}\cdot\cos(\alpha) +G_{EW}^{OQ}\cdot\sin(\alpha)\big{)}, \tag{14}\]  
where \(\Delta L_{G}^{OP}\) and \(\Delta L_{G}^{OQ}\) represent the anisotropic component of SWD of integral signal \(OP\) and inside signal \(OQ\), respectively, and \(mf_{g}(\varepsilon)=\frac{1}{\sin(\varepsilon)\cdot\tan(\varepsilon)+0.003}\) denotes the horizontal gradient mapping function [33]. \(G_{\alpha}^{OP}\) and \(G_{\alpha}^{OQ}\), the wet gradients of two rays, can be computed by using Equation (12). Figure 4: The (**a**) root mean square error (RMSE) and (**b**) R-square of the least-squares fitting results of different months. Therefore, we can define the anisotropic height factor \(\lambda_{aniso}\) to estimate the relationship between the \(\Delta L_{G}^{OP}\) and \(\Delta L_{G}^{QQ}\). \[\lambda_{aniso}\Big{(}h_{QQ^{\prime}}\Big{)}=\frac{\Delta L_{G}^{QQ}}{\Delta L_ {G}^{OP}}=\frac{H_{scale}^{2}+e^{\frac{h_{QQ^{\prime}}}{H_{scale}}}\left(-H_{ scale}^{2}-h_{QQ^{\prime}}\cdot H_{scale}\right)}{H_{scale}^{2}+e^{\frac{H_{ top}}{H_{scale}}}\left(-H_{scale}^{2}-H_{top}\cdot H_{scale}\right)}, \tag{15}\]  
where \(h_{QQ^{\prime}}\) and \(H_{top}\) represent the height of the zenith projection of the slant path \(OQ\) and \(OP\) along the zenith, respectively. During the tomography periods in Hong Kong in August 2017, the tomography top boundary \(H_{top}\) was determined to be 11 km from Figure 1, and \(H_{scale}\) was set to 2 km, according to the ground meteorological data in Hong Kong. Therefore, the anisotropic height factor \(\lambda_{aniso}\Big{(}h_{QQ^{\prime}}\Big{)}\) only has a functional relationship with height \(h_{QQ^{\prime}}\). For instance, \(h_{QQ^{\prime}}\) increases equidistantly from 0 to 11 km with the interval of 0.1 km, and the corresponding \(\lambda_{iso}\Big{(}h_{QQ^{\prime}}\Big{)}\) could be calculated by Equation (15). With the sample data of \(h_{QQ}\) and \(\lambda_{iso}\Big{(}h_{QQ^{\prime}}\Big{)}\), the red curve in Figure 5 could be drawn.