= (𝑟̇/𝑐)𝑓t,
(1) where 𝑟̇ is the spacecraft’s instantaneous range rate. This directly yields the spacecraft’s radial velocity from the line of sight of the ground station.
There are three different ways in which the signal can be transmitted and received; one-way, two-way and three-way mode, the latter of which is used for PRIDE observations. Three-way mode operates by having two ground stations used, one for transmission and another for reception of the spacecraft signal. Therefore, independent frequency standards are used at the two ground stations, typically hydrogen masers.

The stability of the spacecraft Ultra Stable Oscillator (USO) is of the order 10−13 [10], while the hydrogen masers at the ground station are less than 10−14 [11], both over τ = 10s. Since hydrogen masers are an order of magnitude more stable, three-way mode is preferred over one-way for ranging and Doppler observations with VLBI receivers.
The Doppler and range analysis of spacecraft does not give a direct measurement of the spacecraft’s angular position. Therefore, errors in the estimated angular position can arise if mistakes are made in the modelling of the forces on the spacecraft affecting the amplitude or phase of the received signal [12]. VLBI however provides direct measurements of the spacecraft’s angular position.
The PRIDE technique is a VLBI phase-referencing technique. Connecting radio telescopes in an interferometric array improves the resolution to overcome the limitations of diffraction at the long radio wavelengths. For example, considering Rayleigh diffraction, at an X-band wavelength of 3.6cm, the Hobart 26m radio telescope would have a limiting resolution of approximately 5.8 arcminutes. Meanwhile VLBI baselines of up to 10,000km can enable angular resolutions of up to 1 milliarcsecond.
VLBI observations are conducted through recording at each station separately, then correlating the data post-observation to form an interference fringe pattern. The signals first are shifted to compensate for the signal delay between the stations, which includes the geometric delay τg, as well as Doppler, propagation and instrumentation effects. The interferometer setup and fringe pattern formed for a generic two antenna setup is shown in Fig. 1 (left panel).
The output of the correlator can be mathematically described by Eqn 2.
𝑟= 𝐴0|𝑉|Δ𝜈cos (2𝜋𝒃𝝀⋅𝒌𝟎−ΦV), (2) where 𝐴0 is the effective antenna area, 𝑉is the complex visibility (the Fourier transform of the source brightness), Δ𝜈is the bandwidth, 𝒃𝝀is the baseline vector measured in units of wavelength, 𝒌𝟎is the direction to the image centre and ΦV is the error of the visibility phase.
For spacecraft tracking using the VLBI technique, the error in the predicted position of the spacecraft is calculated using the error of the visibility phase ΦV [6]. The model used in the correlator needs to account for corrections in the geometric delay τg, atmospheric and ionospheric effects, as well as errors in the instrumentation and clocks.
Solving for the phase delay in the interferometer response enables the error in the visibility phase to be determined. In VLBI phase-referencing techniques, such as PRIDE, the phase delay is calculated by observing a phase reference source. Conventionally, fringe-fitting and selfcalibration is performed on the target data in VLBI observations, however due to spacecraft having a relatively low SNR and the lack of accuracy around their position and structure, this is not reliable. Instead, the antennas alternate observations between the spacecraft and a quasar, as shown in Fig. 1 (right panel).
Quasars are distant galaxies that appear as bright point-like sources with their positions determined down to an accuracy of several tens of microarcseconds. The quasars are used as reference sources, and the fringe-fitting and self-calibration is performed on them. Thus, if the reference source is within a few degrees of the spacecraft, the delay, delay rate and phase errors of the spacecraft signal can be compensated for. Therefore, any residual error is due to the deviation of the spacecraft’s position from its previously calculated position.

In this way, the spacecraft’s position can be determined.