## 2 Materials and Methods
### Model of Antenna-Induced Phase Bias  
It is determined by the nature of the antenna that the signals incident from different directions suffer from different phase biases. Specifically, the antenna-induced phase bias refers to the distance between the antenna phase center (APC) and the antenna reference point (ARP) [19]. The APC is not fixed, and its position depends on the incident direction and carrier frequency of signal. The antenna-induced phase bias can be considered to consist of two components, namely, the phase center offset (\(PCO\)) and the phase center variation (\(PCV\)). The \(PCO\), which is the distance between the ARP and an arbitrarily selected mean phase center, is introduced to be able to be used alone and eliminate most phase biases [12]. However, some research has also used a unified term, namely, the phase center correction (PCC), to denote the antenna-induced phase bias [2; 11]. The relationship between the PCC, \(PCO\), and \(PCV\) can be expressed as  
\[\xi(azi,ele)=PCO\cdot\overrightarrow{e}+PCV(azi,ele) \tag{1}\]  
where \(\xi\) denotes the PCC, i.e., the antenna-induced phase bias; \(\overrightarrow{e}\) denotes the unit line-of-sight (LOS) vector from receiver to satellite; and \(azi\) and \(ele\) denote the azimuth and elevation of satellite, respectively. \(\xi\) depends on the incident direction and carrier frequency of signal. The incident direction can be represented by \(azi\) and \(ele\), while the frequency is dropped in this equation to improve conciseness since we concentrate on the single-frequency antennas in this contribution. The antenna-induced phase bias is a major obstacle to the use of low-cost antennas for high-precision applications. In applications that utilize the precise carrier-phase observations, antenna-induced phase biases need to be eliminated by applying a predetermined antenna phase pattern, which can describe the distribution of antenna-induced phase biases in an antenna frame [21]. The process of determine the antenna phase pattern is called antenna calibration. In field calibration approaches, there are always two antennas collecting GNSS observations: one reference antenna and one antenna under test, denoted as antennas 1 and 2, respectively. If the antenna-induced phase biases are not negligible, the carrier-phase observations of satellite \(s\) measured on the two antennas can be expressed as  
\[ \begin{array}{l}\lambda\cdot\phi_{1}^{s}=r_{1}^{s}+c(\delta t_{1}-\delta t^ {s})-I_{1}^{s}+T_{1}^{s}+\lambda N_{1}^{s}+\xi_{1}\big{(}azI_{1}^{s},ele_{1}^ {s}\big{)}+\varepsilon_{1}^{s}\\ \lambda\cdot\phi_{2}^{s}=r_{2}^{s}+c(\delta t_{2}-\delta t^{s})-I_{2}^{s}+T_{2 }^{s}+\lambda N_{2}^{s}+\xi_{2}\big{(}azI_{2}^{s},ele_{2}^{s}\big{)}+\varepsilon _{2}^{s}\end{array}  \tag{2}\]where \(\phi\) denotes the carrier-phase observation; subscripts 1 and 2 denote the index of the receiver antennas; superscript \(s\) denotes the index of the satellite; \(r\) denotes the distance from the phase center of the satellite transmitting antenna to that of the receiver antenna; \(\lambda\) denotes the carrier-phase wavelength; \(\delta t\) denotes the clock delay; \(c\) denotes the speed of light; \(I\) and \(T\) denote the ionosphere delay and troposphere delay, respectively; \(N\) denotes the carrier-phase ambiguity, which should be an integer; and \(\varepsilon\) denotes the measurement noise. For the relative field calibration method, the baseline length is usually only a few meters, so the signals from a same satellite to the two antennas can be considered parallel. Furthermore, if the north reference points (NRPs) of both antennas keep pointing to the same direction, then the two incident directions would have the same azimuth and elevation in the frames of the two antennas, i.e., \(azi_{1}^{s}=azi_{2}^{s}\), \(ele_{1}^{s}=ele_{2}^{s}\). So, we can construct a relative phase pattern, i.e., differenced phase pattern between the antenna under test and the reference antenna, describing the relative PCCs between the two antennas.