## 1 Introduction  
The receiver antenna, as a key device in the Global Navigation Satellite System (GNSS) applications, can introduce biases into the carrier-phase observations. Such phase biases, also known as phase center corrections (PCC), cannot be neglected for high-precision positioning, and therefore, antenna calibration is an essential research field in GNSS. Most of the existing antenna calibration methods are widely used to calibrate high-grade antennas such as geodetic antennas but are not well suited for low-cost antennas such as patch antennas. The large-scale application of the low-cost antennas requires that the antenna calibration method shall be easy to conduct, and in addition, centimeter-level PCCs can degrade the success rate of integer ambiguity resolution during a classical antenna calibration process. To address these problems, this paper proposes a relative field antenna calibration method specifically designed for the low-cost GNSS antennas. Anechoic chamber measurement is a method to accurately measure various characteristics of antennas. It measures an antenna in a room with few reflected signals, which can be considered an ideal environment. Mechanical instruments are used to control the rotation and tilt of the antenna under test and to measure the phase delay experienced by the signal as it is incident from different directions to the antenna [1]. The anechoic chamber measurement method enables the antenna calibrations to be performed under near-ideal conditions, but the cost of the chamber construction and other equipment is high. Another disadvantage of this method is that the use of non-real satellite signals during the calibration process may introduce some deviations into the calibration results [2; 3]. Absolute field calibration methods collect real GNSS observations in a field environment using a reference antenna and an antenna under test simultaneously [4; 5; 6; 7; 8]. By using a specialized calibration robot that can keep an antenna in a constant position while changing its attitude, operators are able to orient the zenith of the antenna to a particular satellite in the sky at any moment following a designed calibration strategy. The benefits of the absolute field calibrations are as follows: (1) by continuously changing the attitude of the antenna under test, observations that fully cover the antenna hemisphere can be rapidly (about 4-6 h) collected, which is much faster than the relative field calibration method as will be mentioned later; (2) the entire antenna hemisphere, including the low-elevation regions, can be calibrated; and (3) the calibration results of the antenna under test are less dependent on the PCCs of the reference antenna since in a short time span, the direction of a selected satellite is almost constant with respect to the reference antenna, while the satellite signal is projected into the antenna under test from diverse angles [9]. [PERSON] and [PERSON] [9] pointed out that the results of absolute field calibration are primarily affected by near-field multipath. [PERSON] et al. [5] and [PERSON] and [PERSON] [6] carried out a differencing operation on single- or double-differenced observations between two consecutive mean sidereal days, which can eliminate multipath errors and errors related to geometric distances; this method requires two data collections of hours on two consecutive days, during which the attitude of the antenna under test changes according to the same strategy but using a different initial attitude. [PERSON] et al. [10] and [PERSON] et al. [11] employed an epoch-wise approach, i.e., performing time difference with double-differenced observations at two epochs spaced several seconds apart; this epoch-wise approach can avoid the changing meteorological conditions over the span of two days. It should be noted that since the rotation of the antenna under test also introduces changes in the carrier-phase observations, the carrier-phase ambiguities cannot be eliminated by time-differencing operations.