# The Implementation of Precise Point Positioning (PPP): A Comprehensive Review
## 2 Error Sources in PPP
### Antenna Phase Center Effects
#### 2.6.2 Receiver APC Errors  
The antenna phase center is inherently nonphysical and exhibits frequency dependence. As a result, the location of a receiver antenna typically references the ARP. As the IGS delineates, the ARP is where the antenna's base intersects its vertical axis of symmetry [40]. The receiver PCO represents the distance between the average APC and the ARP. Antenna manufacturers typically provide these PCO values, or at least the vertical PCO component, for each carrier frequency. Furthermore, the IGS ANTEX file includes calibrated PCO values for numerous recognized antenna models. The effective receiver PCO value (\(PCO^{\prime}\)) signifies the projection of the PCO vector within the antenna-fixed frame, \(\mathbf{PCO^{\prime}_{a}}\), on the negative LOS vector connecting the receiver and satellite antennas, as depicted in Figure 3[60].  
\[PCO^{\prime}=-\mathbf{u^{s}_{a}}\cdot(R^{e}_{a}\ \mathbf{PCO^{\prime}_{a}}) \tag{13}\]  
where the matrix \(R^{e}_{a}\) is the transformation matrix from the antenna-fixed frame to the ECEF frame.  
The IGS ANTEX format specifies the receiver PCO values in the Local-Level Frame (LLF), where the offsets are expressed in the north, east, and up directions. However, when applying the rotation, care must be taken in ordering the axes. Assuming that the \(\mathbf{PCO^{\prime}_{a}}\) vector is arranged such that its components are east (E), north (N), and up (U), respectively. Then, the rotation matrix \(R^{e}_{a}\) is equal to \(R^{e}_{l}\), which is the rotation from the LLF to the ECEF frame [61]  
\[R^{e}_{a}=R^{e}_{l}=\begin{bmatrix}-\sin(\lambda)&-\sin(\varphi)\cos(\lambda)& \cos(\varphi)\cos(\lambda)\\ \cos(\lambda)&-\sin(\varphi)\sin(\lambda)&\cos(\varphi)\sin(\lambda)\\ 0&\cos(\varphi)&\sin(\varphi)\end{bmatrix} \tag{14}\]  
where \(\varphi\) and \(\lambda\) are the latitude and longitude of the receiver, respectively.  
Figure 3: Illustration of the receiver antenna phase center errors: phase center offset (\(PCO^{\prime}\)) and phase center variation (\(PCV^{\prime}\)) (after [60]).  
The receiver PCV is depicted as a function of the satellite elevation and azimuth angles, as represented by the solid red curve in Figure 3. Calibrated PCV values for various antenna models can be extracted from the IGS ANTEX file. These values are computed at zenith angles (\(zenith\ angle=90^{\circ}-elevation\ angle\)) from 0 to 90\({}^{\circ}\) in 5\({}^{\circ}\) increments and are available in both azimuth-dependent and azimuth-independent formats. The azimuthal-dependent values cover the 360-degree azimuth range in 5\({}^{\circ}\) steps. Interpolation methods are employed to obtain PCV values at specific elevation and azimuth angles.  
The receiver APC errors are systematic errors in the GNSS observations with magnitudes ranging between 5 to 15 cm for the PCO and up to 3 cm for the PCV [7]. The receiver PCO error primarily influences the estimated height component. It is important to note that the receiver's APC corrections are often referenced to the ARP. If the target location is at a monument marker or geodetic point, an additional offset to the marker must be incorporated into \(\mathbf{PCO_{s}^{r}}\) in (13). Moreover, consistency must be preserved when comparing the PPP solution with a reference solution, ensuring that receiver APC corrections are uniformly applied or omitted across both solutions.  
The total receiver APC error can be written as  
\[APC_{r}=PCO_{r}+PCV_{r} \tag{15}\]  
This error is subtracted from both code and phase measurements to compensate for the receiver APC effects.