### Zero baseline data collection

#### Field experiments

Two ComNav K508 receivers were connected to one antenna using a GEMS signal splitter (PN:GS18). These two receivers have the capacity of tracking GPS L1, L2 and L5 signals, BDS B1, B2 and B3 signals, as well as the two GLONASS signals. As these two receivers tracked the satellite signals at the same antenna phase center, observations from the two receivers could be assumed as a baseline having a distance of “zero” units. The time series results of the final baseline processing reveal the precision of the two satellite navigation systems, as compared to the known distance of zero units. 24 h (one whole BDS orbital period, UTC 00:00:00–23:59:59) of data were collected on April 7th 2014. A Leica AR 25 choke ring antenna was located on a pillar on the roof of the Faculty of Science and Engineering building, University of Nottingham Ningbo China (UNNC) (Fig. 2). The cable connecting the Leica AR25 choke ring antenna and the signal splitter is around 50 m long, and two 3 m long antenna cables were used to connect the splitter with the two ComNav receivers. Both receivers’ sampling rate and the cut-off angle were set as 1 Hz and 15°, respectively. The zero-baseline data were processed using the software developed at UNNC.

### Zero baseline error analysis

Considering the two receivers’ site names as *i* and *j*, and the two tracked satellites’ names as *p* and *q*. Considering the tropospheric delay, ionospheric delay, multipath error, satellite and receiver clock errors, as well as the random receiver noise errors, these two receiver observation equations to satellite *p* can be written as:

$$ \kern0.5em {\phi}_i^p={\rho}_i^p+c\ \left(\delta {t}^p-\delta {t}_i\right)+{N}_i^p+{T}_i^p+{I}_i^p+{M}_i^p+{\xi}_i^p\kern0.5em $$

(1)

$$ {\phi}_j^p={\rho}_j^p+c\ \left(\delta {t}^p-\delta {t}_j\right)+{N}_j^p+{T}_j^p+{I}_j^p+{M}_j^p+{\xi}_j^p $$

(2)

where *ϕ* is the carrier phase measurement; *ρ* is the geometrical range between satellite *p* and receivers *i* and *j*; *c* is the speed of light in a vacuum. *δt*^{p} is the satellite *p* clock error; *δt*_{
i
} and *δt*_{
j
} are the receiver *i* and receiver *j* clock errors, respectively; \( {N}_i^p \)and \( {N}_j^p \) are the integer ambiguities of receivers *i* and *j* from satellite *p* carrier phase measurements, respectively. \( {T}_i^p \) and \( {T}_j^p \) are the tropospheric delays of receivers *i* and *j* from satellite *p*, respectively; \( {I}_i^p \) and \( {I}_j^p \) are the ionospheric delays from satellite *p* to receivers *i* and *j*, respectively; \( {M}_i^p \) and \( {M}_j^p \) are the multipath noise of receivers *i* and *j* from satellite *p*. Receivers *i* and *j* have the same tropospheric and ionospheric delay when they track the same satellite over a zero baseline. Choke ring antennas have the capacity of resisting most of the multipath. The remaining multipath impacts equally onto receivers *i* and *j* in this zero-baseline scenario. The single difference equation between the two receivers can be written as:

$$ {\phi}_{i,j}^p={\rho}_{i,j}^p- c\delta {t}_{i,j}+{N}_{i,j}^p+{\xi}_{i,j}^p\kern0.5em $$

(3)

Considering another satellite *q*, a second single difference equation can be written as:

$$ {\phi}_{i,j}^q={\rho}_{i,j}^q- c\delta {t}_{i,j}+{N}_{i,j}^q+{\xi}_{i,j}^q\kern0.50em $$

(4)

where the general relationship *g*_{i, j} = *g*_{
j
} − *g*_{
i
}. The tropospheric, ionospheric and multipath errors could be totally eliminated by differencing between the receivers’ carrier phase data. The random errors which reflect the quality of the signals, single difference integer ambiguity and the single difference receivers’ clock error remain. Through calculating the difference between eq. 4 minus eq. 3, the double difference observation is derived as:

$$ {\phi}_{i,j}^{p,q}={\rho}_{i,j}^{p,q}+{N}_{i,j}^{p,q}+{\xi}_{i,j}^{p,q}\kern0.75em $$

(5)

where the general relationship for all the individual components is \( {g}_{i,j}^{p,q} \). \( {g}_{i,j}^{p,q} \)represents the double difference operator, \( {g}_{i,j}^{p,q}={g}_j^q-{g}_j^p-{g}_i^q+{g}_i^p \).

Baseline resolution is also based on the satellite position. The satellites’ position derived by the ephemeris include errors, which can reduce the precision of the baseline resolution. The relationship of baseline precision and satellite orbital precision is:

$$ \delta b=\frac{\delta s}{\rho}\cdotp b $$

(6)

where *δb* is the error of the baseline, *δs* is the satellite orbital error, *ρ* is the distance between the tracked satellite to the receiver *b* is the length of the baseline. In the zero-baseline scenario, the error of the satellite orbit can be eliminated through differencing (Eq. 6).

Linearizing \( {\rho}_{i,j}^{p,q} \) and rewriting eq. 5 results in:

$$ {\phi}_{i,j}^{p,q}=\left[{l}_{i,j}^{p,q}\kern0.5em {m}_{i,j}^{p,q}\kern0.5em {n}_{i,j}^{p,q}\right]\cdotp {\left[\delta x\kern0.5em \delta y\kern0.5em \delta z\right]}^T+{N}_{i,j}^{p,q}+{\xi}_{i,j}^{p,q} $$

(7)

\( \left[{l}_{i,j}^{p,q}\kern0.5em {m}_{i,j}^{p,q}\kern0.5em {n}_{i,j}^{p,q}\right] \) is the double difference distance from the satellites to the receivers, which could be calculated by the receivers’ approximate coordinates and the tracked satellites’ coordinates. [*δx δy δz*] is the correction of the baseline’s 3D distance, \( {N}_{i,j}^{p,q} \) is the double difference integer ambiguity, \( {\xi}_{i,j}^{p,q} \) is the double difference random noise. The precision of the baseline correction is not only determined by the observation noise, but also the coefficient \( \left[{l}_{i,j}^{p,q}\kern0.5em {m}_{i,j}^{p,q}\kern0.5em {n}_{i,j}^{p,q}\right] \), after the double difference integer ambiguity \( {N}_{i,j}^{p,q} \) is correctly fixed. We use variance of unit weight during the least squares adjustment. The coefficients reflect the efficiency of different satellite geometry. In this paper, the final positional error is a combination of the observation random noise and the satellites’ geometrical impact. Distinguishing between these error sources is the topic of ongoing work. This paper pays more attention to the difference of GPS-only and BDS-only position precision and the improvement through the integration of GPS and BDS data.