## 3 Algorithm and Strategies of Data Processing
### Absolute PCO and PCV Separated from Relative Observation  
Forming a short baseline with the test antenna and an arbitrary antenna is required for the sake of removing most common errors, e.g., satellite orbit and clock errors, ionospheric and tropospheric propagation path delay errors, _etc._ The relative observables also verify that multipath is greatly reduced, however, this may lead the antenna phase center depending on the antenna placed on the base station (see Figure 2). To access the test antenna PCO and PCV in an absolute way, one can subtract the same baseline scenario's observables of two successive mean sidereal days with a rotation of the test antenna in the horizontal orientation on the second day [4]. Without a change of rotation, the phase center variations will also be eliminated when double differences are formed. Furthermore, tilting w.r.t a horizontal axis is also needed to avoid possible station-dependent \"Northern Hole\" effects, see Figure 4 Observation coverage on the antenna hemisphere when the antenna was fixed (24 h static, DoY2014207 in Wuhan). To meet the requirement of a large range of motions (rapidly rotating and tilting), a precise 6\({}^{\circ}\) of freedom robot (nominal repeatability precision better than 0.1 mm) is introduced in our calibration procedure as to improve calibration accuracy and sharply shorten the calibration running time. The complete data processing for GPS antenna calibration used in this study consists of three steps. The collected GPS data are firstly used to form well known double difference (DD) observation equations. After the DD ambiguities are fixed, then the triple difference (TD) between two successive epochs are performed for step of absolute PCOs derivation, followed by PCVs estimation derived from TD observation residuals. Our data processing procedure is different from that used in [5] for absolute antenna calibrations. While they derive an absolute test antenna's PCO and PCVs by forming the time difference of a single difference (TDSD) observable (one satellite only), this kind of calibration requires a common clock at the reference and test antennas, though it is well handled in the whole procedure. In addition, the TDSD model removes the ambiguities without taking ambiguity integer characteristics into account. In our procedure, the TDSD observations are further formed DD observations between two  
Figure 3: Multipath migration using sidereal day differences and epoch-wise TD. satellites which do not only remove receivers' clock bias, but also easily fix the DD ambiguity using so-called LAMBDA [20] based on integer least squares (ILS). As fixing correct ambiguities is a priority to achieving the highest accuracies and reliability of PCCs, we validate the estimated ambiguities using Ps-LAMBDA [21] based on ambiguity success rates. Before the start of the calibration procedure, several known factors must be mentioned: (1) The baseline between the reference and test XYZ monument positions were well-determined using GPS single frequency solutions on a short baseline; (2) Coordinate transformation. Although the repeatability precision of the introduced robot is announced to be better than 0.1 mm, the robot output Cartesian coordinate values is w.r.t robot local coordinate system, which is different from the GNSS one (e.g., WGS-84 in this paper) Therefore, a seven-parameter transformation program has been implemented to resolve the coordinate transformation parameters between the robot local coordinate system and WGS-84; (3) Time synchronization. Every visible satellite's elevation and azimuth w.r.t the antenna-fixed coordinate system (was set to parallel with robot local coordinate system) has to be computed in the course of the robot's rotating and tilting. Therefore, it is quite clear that those motions must be directly connected to GPS time for synchronization with the phase observations. Unfortunately, the time synchronization function in the control robot system is not available at the moment.