# Analysis and Demonstration of First Cross-Support Interferometry Tracking in China Mars Mission
## 2 Theorem and Methods
### Basic Theorem of DeltaDOR  
VLBI basics are discussed in many references and tutorials [34], wherein a brief theorem of DeltaDOR is described. The observing geometry of DeltaDOR is shown in Figure 2[35]. VLBI system measures the geometric delay of the signal send from the target, extragalactic quasar or spacecraft, between the two long-distance separated stations. The line connecting the receiving stations is called baseline. Observable time delay is a function of the known baseline vector and the direction to the target [36]. \[\tau=\frac{\mathbf{B}\cdot\mathbf{S}}{c}=\frac{\mathrm{Bcos}\theta}{c} \tag{1}\]  
where \(\mathbf{B}\) is baseline vector, and \(\mathbf{S}\) is the unit vector the target. Differential observable is related to determination of the relative angular position of a spacecraft with respect to a compact calibration radio source on the plane of the sky. \[\Delta\tau=\frac{\mathbf{B}\cdot\left(\mathbf{S}_{\mathrm{SC}}-\mathbf{S}_{ egrs}\right)}{c} \tag{2}\]  
The accuracy of VLBI delay observable is affected by two main kinds of factors, the random error and systematic error. Random error is dependent on signal-to-noise ratio (SNR) and synthetic bandwidth; other main reasons for errors in accuracy are variation in propagation path length for the signals arriving at the VLBI antennas, mainly including instrument changes, troposphere and ionosphere media delay, station clock drift, and other  
Figure 2: DeltaDOR Observation Geometry. constant errors, such as locations of the VLBI antennas and their axes of rotations, and Earth orientation. In order to minimize many sources of common systematic error, including media variations, instrumental changes, and errors associated with lack of knowledge of the exact observing geometry, observation of calibration quasar is induced, with signal recording channel and observing schedule specially designed, e.g., spacecraft tones signal locates the center of the quasar signal channels. Obviously, temporal and spatial angular separation between the measurements of the reference source and the target spacecraft play a key important role in the accuracy of DeltaDOR. Since most components contributing to observable errors are linearly proportional to the temporal and angular separation, it is expected that we will minimize the effects of error by reducing the temporal and angular separation between the spacecraft and the calibration quasar [37]. Usually, in a conventional DeltaDOR measurement, a typical spacecraft-quasar angular separation of 6 degrees and per scan alter period of 10 min is advised; in addition, compact radio sources having a flux density of typically at least 0.5 Jy at X band are advised for calibration, where a jansky (Jy) = \(10^{-26}\) W\(\cdot\)m\({}^{-2}\cdot\)Hz\({}^{-1}\). Take observation sequence quasar-spacecraft-quasar as example for discussion. Time tag \(t_{1}\),\(t_{3}\) correspond to calibration quasar, and \(t_{2}\) is the spacecraft observation. After correlation, we obtain observable delay of each scan:  
\[\tau_{grs}(t_{1}) = \tau_{grs-geo}(t_{1})+\tau_{egrs-inst}(t_{1})+\tau_{egrs-clock}(t _{1})+\tau_{egrs-EOP}(t_{1}) \tag{3}\] \[+\tau_{egrs-pos}(t_{1})+\tau_{egrs-atmo}(t_{1})+\tau_{egrs-iomo}(t _{1})\]  
\[\tau_{SC}(t_{2}) = \tau_{SC-geo}(t_{2})+\tau_{SC-inst}(t_{2})+\tau_{SC-clock}(t_{2} )+\tau_{SC-EOP}(t_{2}) \tag{4}\] \[+\tau_{SC-pos}(t_{2})+\tau_{SC-atmo}(t_{2})+\tau_{SC-iomo}(t_{2})\]  
\[\tau_{egrs}(t_{3}) = \tau_{egrs-geo}(t_{3})+\tau_{egrs-inst}(t_{3})+\tau_{egrs-clock}( t_{3})+\tau_{egrs-EOP}(t_{3}) \tag{5}\] \[+\tau_{egrs-pos}(t_{3})+\tau_{egrs-atmo}(t_{3})+\tau_{egrs-iomo}(t _{3})\]  
Item \(\tau_{egrs-geo}\) is the geometry delay of the calibration quasar; the quasar position is very accurately known, so the accuracy of \(\tau_{egrs-geo}\) is in the order of pico seconds.