# Error Correction of Water Vapor Radiometers for VLBI Observations in Deep-Space Networks
## 3 Error Source Analysis and Calibration
### Oblique Path Calibration  
As the measurement elevation angle decreases, the measurement accuracy of the microwave radiometer also decreases due to the effect of background noise on the main and side flaps of the antenna. There has been a challenge in evaluating the measurement accuracy of the microwave radiometer at low elevation angles and correcting the errors at low elevation angles. In this paper, based on the characteristics of atmospheric optical thickness, we investigate the optical thickness law calculated by using the brightness temperature measured by the microwave radiometer, and give an accurate evaluation of the measurement capability of the microwave radiometer at a low elevation angle indirectly. According to the atmospheric transmission theory, the optical thickness of different zenith angles under clear weather is linearly related to the tangent of the zenith angle. The optical thickness of each zenith angle can be computed as:  
\[t(q,j)=-ln\left(\frac{T_{m}(q)-2.73}{T_{m}(q)-T_{sky}(q,j)}\right) \tag{11}\]  
where \(T_{m}(\theta)\) is the average radiant light temperature of the atmosphere at elevation \(\theta\), and \(T_{sky}(\theta,\varphi)\) is the radiant light temperature of the sky at elevation \(\theta\) and azimuth \(\varphi\). The average radiant temperature of the atmosphere can be given by an empirical formula, and the radiant brightness temperature of the sky can be measured by a radiometer. This is illustrated below by an observation experiment in April 2017 that was carried out in an open area with clear weather. In the experiment, an inclination calibration was first successfully performed, which ensured the accuracy of the brightness temperature data in the zenith direction. Figure 11 compares the measured optical thickness and the calculated theoretical optical thickness from an elevation angle of 5\({}^{\circ}\) to 90\({}^{\circ}\). The measured optical thickness is measured by scanning the brightness temperature at different elevation angles and is calculated by Equation (8). The theoretical optical thickness is calculated from radiosonde data at the same address. From Figure 11, we found that:  
Figure 10: Calibration results of the measured data on 4 March 2017. (**a**) shows the measurement error without calibration, (**b**) shows the measurement error after system absolute calibration, (**c**) shows the measurement error after receiver gain error calibration, and (**d**) shows the measurement error after calibration of the temperature noise error of the antenna feeder system. (1) The measured optical thickness of the WVR is very close to the theoretical optical thickness when the elevation angle is greater than 60 degrees. Therefore, the brightness temperature measurements at elevation angles of 60-90 degrees are less influenced by external factors, such as sidelobes and the curvature of the Earth, without considering other factors. It can be considered that the measurement error is within an acceptable range;  
(2) At elevation angles of 5 to 60 degrees, the smaller the variation of the optical thickness with elevation angle, the larger the deviation from the theoretical value. At an elevation angle of 10 to 60 degrees, the deviation has a certain relationship with the elevation angle, and the relationship can be fitted with a quadratic curve. At elevation 5 to 10 degrees, there seems to be little regularity in the deviation, which may be due to the influence of the ground feature. Generally, when using WVR, VLBI uses data above 10 degrees. Therefore, in a practical measurement, we fit the law of optical thickness deviation by a large number of measured data, which can be used to correct the influence of the inclined path on the brightness temperature at an elevation angle of 10-90 degrees. By measuring the brightness temperature of different azimuths and elevations, we calculate and establish the measured optical thickness matrix, and then calculate the optical thickness deviation matrix.