# Error Correction of Water Vapor Radiometers for VLBI Observations in Deep-Space Networks
## 3 Error Source Analysis and Calibration
### Oblique Path Calibration
In this way, we can use the optical thickness deviation matrix to correct the brightness temperature values of different azimuths and elevations in the oblique path measurement, which greatly improves the brightness temperature measurement accuracy at lower elevations. The measurement airspace is divided into grids of 10 degree azimuth intervals and 1 degree pitch intervals, and the optical thickness deviation matrix \(\Delta\tau\) is established as Equation (9), where \(\Delta\tau_{(\theta,\varphi)}\) represents the optical thickness deviation of the antenna pointing at azimuth angle \(\varphi\) and elevation angle \(\theta\). In general, VLBI uses data above 10 degrees when using WVR. Therefore, in the actual measurement, we fit the law of optical thickness deviation by a large amount of measured data, which can be used to correct the effect of the tilted path on the brightness temperature at elevation angles of 10-90 degrees. By measuring the brightness temperature at different azimuth and elevation angles, we calculate and build the measured optical thickness matrix, and then calculate the optical thickness deviation matrix. In this way, we can use the optical thickness deviation matrix to correct the brightness temperature values of different azimuth angles and elevations in the oblique path measurement, thus greatly improving the brightness temperature measurement accuracy at low elevations. The measurement airspace is divided into grids of 10 degree azimuth intervals and 1 degree pitch intervals, and the optical thickness  
Figure 11: The optical thickness error at different elevation angles. deviation matrix \(\mathbf{\Delta\tau}\) is established as Equation (9), where \(\Delta\tau_{(\theta,\varphi)}\) represents the optical thickness deviation of the antenna pointing at azimuth angle \(\varphi\) and elevation angle \(\theta\). After the optical thickness at different elevation angles is measured, \(\Delta\tau_{(\theta,\varphi)}\) (the fitting value) can be obtained through binary primary regression. \[\mathbf{\Delta\tau}=\left[ \begin{array}{cccc}\Delta\tau_{10,0}&\Delta\tau_{10,10 }&\Delta\tau_{10,20}&\ldots&\Delta\tau_{10,350}\\ \Delta\tau_{11,0}&\Delta\tau_{11,10}&\Delta\tau_{11,20}&\ldots&\Delta\tau_{11, 350}\\ \Delta\tau_{12,0}&\Delta\tau_{12,10}&\Delta\tau_{12,20}&\ldots&\Delta\tau_{12, 350}\\ \ldots&\ldots&\ldots&\Delta\tau_{\theta,\varphi}&\ldots\\ \Delta\tau_{90,0}&\Delta\tau_{90,10}&\Delta\tau_{90,20}&\ldots&\Delta\tau_{90, 350}\end{array} \right] \tag{12}\]  
According to Equation (12),  
\[T_{\text{sky}}(\theta,\varphi)=T_{m}(\theta)-e^{\tau(\theta,\varphi)}\cdot(T_ {m}(\theta)-2.73) \tag{13}\]  
\[ \begin{array}{l}T_{\text{sky}}{}^{\prime}(\theta,\varphi)-T_{\text{sky}}( \theta,\varphi)\\ =e^{\tau^{\prime}(\theta,\varphi)}*(T_{m}(\theta)-2.73)-e^{\tau(\theta, \varphi)}*(T_{m}(\theta)-2.73)\\ =(T_{m}(\theta)-2.73)*e^{\Delta\tau(\theta,\varphi)}\end{array}  \tag{14}\]  
Then, we obtain,  
\[T_{\text{sky}}{}^{\prime}(\theta,\varphi)=T_{\text{sky}}(\theta,\varphi)+(T_ {m}(\theta)-2.73)*e^{\Delta\tau(\theta,\varphi)} \tag{15}\]  
where \(T_{\text{sky}}{}^{\prime}(\theta,\varphi)\) is the bright temperature after the correction of the oblique path, \(T_{\text{sky}}(\theta,\varphi)\) is the original bright temperature measured, and \(T_{m}\) can be calculated through the empirical model. After the step-by-step correction of the methods in Sections 3.1-3.4, we have achieved an accuracy of 20 mm (10 degrees above elevation) of the oblique path wet delay measurement. The error comparison method uses a two-station VLBI system to measure two stars with known precise positions in order to obtain more accurate atmospheric delays by the difference method. The related experimental procedure and results will be discussed in the follow-up papers.