# Error Correction of Water Vapor Radiometers for VLBI Observations in Deep-Space Networks
## 3 Error Source Analysis and Calibration
### Temperature Noise Error of Antenna Feeder System  
The following figure shows the noise error contribution of each component of the radiometer receiving system. The noise error after the antenna feeder can be compensated by the receiver gain error calibration method, as described in Section 3.2. However, since the noise source power enters the signal path after the feeding horn, it is impossible to correct the luminance temperature error introduced by the non-thermally stable feeding horn using the noise standard calibration method. An important system noise contribution is related to the antenna feed system. Figure 8 shows the noise contribution of each component of the WVR system. The system noise temperature converted to the input end is:  
\[T_{REC}=T_{1}+\frac{T_{2}}{G_{1}}+\frac{T_{3}}{G_{1}G_{2}}+\frac{T_{4}}{G_{1}G_ {2}G_{3}}+\frac{T_{i}}{G_{1}LG_{i-1}} \tag{8}\]  
As shown in Figure 9, the noise temperature of the antenna \(T_{1}\) has a great impact on the noise temperature of the entire system, and the subsequent gain \(G\) is large enough. A typical loss of a waveform feedforward horn running at 20-30 GHz is less than \(L=1\) dB. When the physical temperature of the feedforward horn changes by 10 \({}^{\circ}\)C, the system noise  
Figure 8: Contribution of WVR components of system noise temperature \(T_{REC}\).  
Figure 7: The error of measuring blackbody after receiver gain error calibration was reduced by nearly 1 K.  
increases by approximately 2 K, resulting in the absolute brightness temperature error of 2 K.  
Therefore, the best way is to keep the antenna system at a constant temperature like the receiver. However, because the WVR antenna used in this project is large and exposed, it is difficult to keep the antenna at a constant temperature. Therefore, the temperature measurement probe can only be installed near the feeder in order to test the temperature variation of the antenna, thus eliminating this error.  
Figure 9 shows the relationship between the measured feed temperature and the difference between the measured WVR and the blackbody temperature. The measured values show that there is a clear linear relationship between them. By fitting a large number of measurements, the following relationship can be obtained:  
\[\Delta T_{f}=T_{w}-T_{B}=\Delta T_{(APT)}\times 0.21 \tag{9}\]  
This relationship is very consistent with the previous analysis, where the measurement brightness temperature error of 2 K is caused by a 10 \({}^{\circ}\)C change in the feed temperature. Therefore, we need to add the error correction of the antenna and feeder to the final output brightness temperature. Combining Equations (5) and (6), the following equations can be obtained:  
\[\begin{array}{l}T_{Ai}^{\prime}=T_{Ai}-\Delta T_{f}\\ =\frac{V_{Ai}}{G_{50}\cdot\frac{V_{Ni}}{V_{Nb0}}}-(T_{REC0}+\Delta T_{REC})- \Delta T_{APT1}*0.21\end{array} \tag{10}\]  
where \(T_{Ai}\)\({}^{\prime}\) is the output bright temperature value data of the \(i\)-th group after correction of the antenna error, and \(\Delta T_{APT1}\) is the measured physical temperature data at the feed source of the \(i\)-th group.  
In summary, the measured data on 4 March 2017 were calibrated by using the three calibration methods described above. The results are shown in Figure 10, where Figure 10a shows the measurement error without calibration, Figure 10b shows the measurement error after system absolute calibration, Figure 10c shows the measurement error after receiver gain error calibration, and Figure 10d shows the measurement error after calibration of the temperature noise error of the antenna feeder system.  
Figure 9: The relationship between the measured feed temperature and the difference between the measured WVR and the blackbody temperature.  
After the above three calibrations, the brightness temperature measurement accuracy of the vapor radiometer in the zenith direction can reach 0.2 K in a one-month scale (root mean square error) and 0.05 K in a 1000 s scale.