## 2 Problem Description
### Error Budget Analysis  
The potential sources of error in a \(\Delta\)DOR measurement encompass the received tone power-to-noise ratio and the effective bandwidth of the DOR tones, both of which contribute to random errors. Other error sources include the precision of the quasar delay measurement, the quasar position accuracy, the clock stability error, the instrumental phase ripple, the uncertainties in the Earth Rotation Parameters and atmospheric media delay error, each of which can potentially induce systematic errors. References [27; 28] have conducted an analysis on the typical influence of the above errors of Mars exploration in the Deep Space Network (DSN). At present, the fields of radio measurement technology and geodetic measurement technology are experiencing rapid advancement. As orbit prediction becomes increasingly precise, the integration time for correlation processing can extend to a magnitude of up to 10 s in CDSN interferometry systems. The equipment used for data recording and acquisition has evolved, allowing for a higher interferometry data sampling frequency. The accuracy of station coordinates has been improved to the mm level, and the prediction accuracy of UT1-UTC has also been improved to reduce the influence of Earth orientation error. Quasars selected as \(\Delta\)DOR reference sources possess positions that are recognized in the International Celestial Reference Frame (ICRF), where most catalog sources exhibit a positional accuracy better than 0.3 nrad [28]. For \(\Delta\)DOR, it is beneficial to select compact sources with an accurate position. Research results have shown that the troposphere zenith delay measurement accuracy has been improved to about 5 mm [28]. Thus, an analysis of the \(\Delta\)DOR measurement error effects in the CDSN system is given. The distance from the spacecraft to the receiver is about 150 million km, and the radio frequency of the downlink signal is 8.4 GHz. The effective transmitted tone power along the line-of-sight to the receiver is 40 dBW, and the modulation loss is about 21 dB. The diameters of the two antennas on the baseline are 65 m and 35 m, and their efficiencies and system noise temperature are both 0.55 and 50 K [29]. According to the CCSDS standard, the total spanned bandwidth is set to 38.25 MHz. The integration time of spacecraft and quasar signal correlation processing is 30 s and 300 s, respectively. The system loss factor stands at 0.8, which can vary depending on the implementation. The number of time-multiplexed frequency channels is four. The quasar position uncertainty in ICRF3 is about 0.2 nrad, and the number of quasar data samples per second is 8 \(\times\) 10\({}^{6}\) (corresponding to 8 MHz sample frequency). The projected baseline length is about 4000 km. The uncertainties in the  
Figure 1: The basic principle of \(\Delta\)DOR. baseline coordinates and orientation are 1 cm and 3 cm, respectively. The clock instability is \(10^{-14}\) and the instrumental phase ripple is 0.2 degrees. The elevations of the spacecraft and quasar are 20 degrees and 25 degrees, respectively, and their angular separation is 10 degrees. The zenith troposphere delay uncertainty is 0.5 cm. The sun-earth-source angle is 50 degrees, and the solar wind velocity is 400 km/s. The error budget for the X-band \(\Delta\)DOR in the CDSN interferometry system with the above conditions is shown in Figure 2, where RSS stands for the Root Sum Square of all the above ten errors and is always used as a system's total error criterion. As is evident from Figure 2, spacecraft thermal noise emerges as the predominant error source, accounting for approximately 93% of the overall error effect. The thermal noise error, which is believed to be a random error source, can be calculated using Formula (1) [27; 28], where \(\Delta f\) represents the effective bandwidth, and hardly changes under the CCSDS specification, and \(T\) is the integration time, \(P/N_{0}\) being the carrier-to-noise power density ratio, which is related to the downlink signal power.