## 2 Multipath Theory and Signal-to-Noise Ratio  
Most GNSS receivers currently use code division multiple access (CDMA) methods to split signal processing into multiple channels, allowing multiple satellites to be tracked at the same time. Most GNSS receivers use a code tracking loop or the delay-lock-loop (DLL) and the phase-lock loop (PLL) to track each satellite. These devices ensure that incoming carrier phases and codes are matched (or locked) to receiver-generated phases and codes and that the match is maintained throughout the tracking of continually received signals. The received signal is then aligned with a locally generated replica utilising a delayed lock loop. In the presence of multipaths, the signal entering the PLL becomes a composite signal between the direct signal and the reflected signal. In the GNSS receiver, the multipath manifests in many ways, such as through the additional propagation path, changes in the amplitude of the signal, changes in the phase rate akin to the additional path, the electrical-magnetic properties of the surrounding surfaces and the changes in the polarization of the signal. To infer multipaths, geodesists use the so-called signal-to-noise ratio (SNR) records available in most GNSS receivers and reported in RINEX files. The SNR records are directly related to the carrier phase error [38]. However, different GNSS receiver manufacturers implement proprietary methodologies and definitions in recording the SNR, making it problematic to compare SNR values acquired with different receivers. In this study, we use the SNR as a proxy for the existence of a carrier phase multipath signature [49]. The dominant frequency and amplitude content of the SNR time series can be analysed to derive the qualitative measure of the multipath error that appears in the carrier phase observations. The inherent frequency content can be examined via wavelet analysis [34; 35; 50]. Figure 1 depicts the multipath phenomenon as received by a GNSS antenna. The radio signal arrives via two possible routes at the GNSS antenna, one through a direct route and the other (specularly) reflected from the horizontal reflector surface. Reflected signals arrive at GNSS receivers coherently or incoherently, depending on the roughness of the surface. Smooth surfaces create coherently reflected signals, which are referred to as specularly reflected signals [51]. The majority of the reflected signal occurs within the first Fresnel zone (see Section 3) about the specular point. Rough surfaces create incoherent reflected signals, which are referred to as diffusely scattered signals. Typically, multipaths come in many forms through signal diffraction, where the incoming direct signal is diffracted at the edges of an obstruction, and most multipaths involve specular multipaths where the GNSS antenna is placed above an idealized infinite horizontal reflector surface [9]. Let us assume that there's only one reflected signal to deal with. In this case, the direct signal \(A_{d}\), the one reflected signal, \(A_{r}\), and the composite signal, \(A_{c}\) can be described after [49] by  
\[S_{c}=A_{d}\cos\phi_{d}+\alpha\ A_{d}\ \cos\left(\phi_{d}+\Delta\phi_{r}\right) \tag{1}\]  
where \(A_{d}\) and \(\Delta\phi_{r}\) are the amplitude and the shifted phase of the direct signal, respectively, and \(\alpha\) is the damping factor that relates the reflected signal \(A_{r}\) with the direct signal's amplitude, i.e., \(A_{r}=\alpha A_{d}\). The amplitude \(A_{c}\) of the composite signal, \(S_{c}\), is estimated from:  
\[A_{c}=A_{d}\ \sqrt{1+2\alpha\cos\left(\Delta\phi_{r}\right)+\alpha^{2}} \tag{2}\]  
Figure 1: Multipath geometry. The multipath is the interference between the direct and the reflected signal shown in the idealized horizontal surface. H is the height of the GNSS antenna.