# Error Correction of Water Vapor Radiometers for VLBI Observations in Deep-Space Networks
## 3 Error Source Analysis and Calibration
### Receiver Gain Error Calibration  
The effect of gain fluctuations on the stability of water vapor radiometers is direct and decisive. The water vapor radiation brightness temperature is usually tens of Kelvin. The total gain of the amplification link of the radiometer receiver is often more than 80 dB. In long-term operation, the gain fluctuation can easily exceed 1 dB. Therefore, the gain drift on the radiometer stability is very obvious, and must be suppressed in some way. The receiver gain drift is mainly caused by active devices (amplifiers and mixers), and active devices are mainly caused by changes in the ambient temperature. Gain drift with temperature (referred to as temperature drift) is an inherent characteristic of active devices and a key issue in RF circuit design. The receiver module of modern radiometers is usually placed in a constant temperature structure to ensure the stability of the amplifier's ambient  
Figure 4: The relationship between measurement error and ambient temperature. Figure 3: The brightness temperature of the water vapor radiometer at 23.8 GHz frequency and the brightness temperature of the black body were measured when the ambient temperature changed. temperature. Despite this, the receiver gain can still drift, resulting in 1 K level errors. Periodic correction methods are needed in order to reduce the drift error. Noise injection correction is currently a commonly used method, where noise power provided by a highly stable noise source is coupled to the antenna channel, and noise injection is controlled by the RF switch. This correction can be performed automatically at any time, so the gain fluctuation of the receiver can be corrected in real time. The noise injection radiometer is shown in Figure 5.  
During measurement, the RF switch is periodically switched on and off to control the injection of noise. (a) During the period when the noise source is turned off, the measurement equation is as follows:  
\[V_{A}=G_{S}(T_{A}+T_{REC}) \tag{1}\]  
where \(T_{A}\) is the input radiation brightness temperature of the antenna port, \(T_{REC}\) is the noise temperature of the system, \(G_{S}\) is the total gain of the system, and \(V_{A}\) is the output voltage value of the system after square-rate detection;  
(b) During the period when the noise source is turned on, the measurement equation is as follows:  
\[V_{AN}=G_{S}(T_{A}+T_{N}+T_{REC}) \tag{2}\]  
where \(T_{N}\) is the temperature of the standard noise (reference noise), and \(V_{AN}\) is the output voltage value of the system when adding noise. Combining Equations (1) and (2), we obtain  
\[G_{S}=\frac{V_{AN}-V_{A}}{T_{N}}=\frac{V_{N}}{T_{N}} \tag{3}\]  
\(V_{N}\) is the output voltage of standard noise (reference noise), which can be obtained by subtracting the input voltage \(V_{A}\) from the output voltage \(V_{AN}\). The gain obtained by Equation (3) can be compared with the gain at the time of calibration to obtain the change in gain for correction in the final measurement results. This is a real-time gain correction method. From Equation (3), it can be seen that the accuracy of gain measurement depends on the stability of the noise source \(T_{N}\). Therefore, a high stability noise source is generally selected and placed in a constant temperature structure to maintain the stability of the noise output. Figure 5: Block diagram of noise injection radiometer system. In practice, the gain of the receiving system and the input noise temperature will change. Assume that the gain of the system is \(G_{S0}\) and the noise temperature is \(T_{REC0}\) when the standard is just set. In the actual measurement, the gain of the system is \(G_{Si}\) and the noise temperature is \(T_{RECi}\). According to Equation (3), the following equation can be obtained. \[G_{S0}=\frac{V_{N0}}{T_{N}},\;G_{Si}=\frac{V_{Ni}}{T_{N}} \tag{4}\]  
where \(V_{N0}\) is the input voltage of the reference noise source at the calibration time, and \(V_{Ni}\) is the input voltage of the reference noise source at the actual measurement time.