## 3 Basic Principle of Calculating Tropospheric Zenith Delay
### The Principle of ERA5 Data Set Calculation to the Path Layer Delay  
The Saatamoinen method and the integration method are commonly used to calculate ZTD from ECMWF reanalysis data. In [11], [PERSON] points out that the accuracy of the integration method to calculate ZTD is better than the Saatamoinen method. Therefore, this paper uses the integration method to calculate the ZTD of ERA5 data. The applied formula for calculation of the ZTD reads:  
\[ZTD_{i}=10^{-6}\times\int Nds=10^{-6}\times\sum_{1}^{i}N_{i}\times\Delta s_{i} +ZTD_{top} \tag{8}\]  
where \(ZTD_{i}\) is the zenith tropospheric delay on the isobaric surface of the \(i\)th layer calculated from the top layer; \(ZTD_{top}\) is the ZTD above the top layer; \(N\) is the total atmospheric refractive index; \(N_{i}\) is the atmospheric refractive index of the ith layer; \(\Delta s_{i}\) is the height difference between level \(i\) and level \(i\)-1, with the topmost level \(\Delta s=0\). Since there is almost no wet delay above the top level, the \(ZTD_{top}\) can be replaced by the tropospheric zenith dry delay (ZHD) above the top layer, calculated by the Saatamoinen model. The calculation method of atmospheric refractive index is as follows:  
\[N=\frac{k_{1}(P-e)}{T}+\frac{k_{2}e}{T}+\frac{k_{3}e}{T^{2}} \tag{9}\]  
\[e=h\times P/0.622 \tag{10}\]  
Among them:  
\[k_{1}=77.604\text{ K}/\text{P}_{a}\]  
\[k_{2}=64.79\text{ K}/\text{P}_{a}\]  
\[k_{3}=337600\text{ K}^{2}/\text{P}_{a}\]  
where \(P\) is atmospheric pressure (unit:nPa); \(e\) is water vapor pressure (unit:nPa); \(T\) is temperature, and \(h\) is specific humidity. Because the height system of ECMWF is geopotential height, while the height system of GPS station is geodetic height, in order to facilitate comparative analysis, it is necessary to convert the geopotential height into geodetic height. In [18], [PERSON] and [PERSON] et al. introduced specific elevation system conversion meth  
Figure 1: Interpolation of 4 points around.  
ods. After unifying the elevation system, the atmospheric refractive index of each layer can be calculated by using the meteorological parameters of each layer. In order to improve the accuracy of calculating ZTD by the integration method, the meteorological parameters of each layer should take the mean value of the meteorological parameters of the two adjacent isobaric surfaces [19]. Only ZTD on the ECMWF barometric stratified data grid can be obtained by the integration method. To obtain ZTD of the GNSS station location, ZTD of the four nearest grid points around the station should be obtained by bilinear interpolation. Because the height of the GNSS station and its surrounding four grid points from ECMWF reanalysis data are different, it is necessary to reduce ZTD calculated from ECMWF reanalysis data to the same height as the GNSS station. Next, bilinear interpolation is used to perform horizontal interpolation. In [20], when ZTD is scaled in the vertical section, the second order polynomial is used to fit the change of ZTD with height. This method will cause the precision loss of ZTD in the process of fitting and interpolation.  
In this paper, ZTD of the grid point height at the station is directly calculated by the integration method, and then ZTD of the GNSS station height is interpolated by the bilinear interpolation level [21]. When calculating ZTD of the station height by integration, this method needs to obtain the meteorological elements of the grid point at the station height. When the height of the station is higher than the lowest level of the meteorological reanalysis data, the meteorological elements of the station height are obtained by linear interpolation of the meteorological elements of the two adjacent layers of the station height. When the height of the station is lower than the lowest level of the meteorological reanalysis data, the meteorological elements of the lowest level need to be extrapolated [22]. In [23], specific methods for the derivation of meteorological elements are provided.