## 6 Effects of Multipath on GNSS Carrier Post-Fit Residuals
### Measurement of SNR Frequency Content Using the Wavelet Transform
The result of the convolution between the Morlet wavelet and the residual SNR will help in the construction of the \(\delta SNR\) showing the amplitude of any features with respect to scale and how this amplitude also changes with time; i.e., \(|W_{n}(S)|\) is the amplitude, and the phase can be constructed from both the real and imaginary part of the \(W_{n}(S)\) and the wavelet power spectrum provided as \(|W_{n}(S)|^{2}\). Transforming the residual SNR time series using wavelet transform reveals the temporal features or dynamics and extracts spectral specific (frequency) content of the signal over time. The raw SNR was filtered for zero-value occurrences, with sample 1 Hz data including significant multipath peaks, as demonstrated in this example in Figure 13. Before we applied the wavelet representations to the SNR times series, we separated the SNR contributions to the direct signal and the residual SNR components to enhance the spectral analyses of the SNR more robustly by demeaning the time series. The steadily changing trend in the SNR represents the direct signal. Typically, the residual SNR is estimated by fitting a polynomial to the recorded SNR values at the two close-by stations on 1 March 2021 for GPS satellite PRN27, selected according to its Fresnel footprint; see Figure 2. Our choice of the polynomial fit is based on the minimum RMS between the model polynomial to the raw SNR. Figure 13 depicts how to isolate the quasi-oscillation--a proxy to the existence of multipath by removing a polynomial fit to the raw SNR time series. This quasi-oscillation is more prominent for station KBG2, which has no Eccosorb material, compared to station KBG1, which has Eccosorb material; see Figure 13.  
We show the wavelet power spectra of the \(\delta SNR\) to study the multipath contributions closely due to the specific obstructions to the two GNSS stations shown in Figures 3 and 4. The SNR data are typically reported on a logarithmic scale (dB-Hz); here we converted to a linear scale using \(10^{(SNR[db-Hz]/20)}\). Following Equations (9) and (10), we examined the properties of \(\delta SNR\) over a range of scales (frequencies). One can define the scale-averaged wavelet power between two wavelet scale bands, \(s_{1}\) and \(s_{2}\), as:  
\[W_{n}^{2}=\frac{\delta j\delta t}{C_{\delta}}\sum_{j=1}^{j=2}\frac{|W_{n}(s_{ j})|^{2}}{s_{j}} \tag{11}\]  
The constant factor \(C_{\delta}=0.776\) is chosen for the Morlet mother wavelet, as reported in [50]. The amplitude of the \(\delta SNR\), i.e., \(A_{r}\) (compared with Equation (5)) for selected scale bands within a certain time period range can be estimated taking the square root of Equation (11) over all scales:  
\[A_{r}(t)=\sqrt{W_{n}^{2}}=\sqrt{\frac{\delta j\delta t}{C_{\delta}}\sum_{j=0} ^{j=J}\frac{|(W_{n}s_{j})|^{2}}{s_{j}}} \tag{12}\]  
Figure 13: Raw SNR time series where the direct and indirect signals are superimposed as observed by the Alloy Trimble reference receiver for the L1, the GPS signal on 1 March, for GPS satellite PRN27. (**a**) Raw SNR as recorded by station KBC1. (**b**) Raw SNR as recorded by station KBG2, where the direct signal for both stations dominates the main signal. We have applied the ninth-order polynomial as depicted in (**c**) for KBG1 and (**d**) for KBG2. After subtracting the polynomial fit of the model from the raw SNR values, \(\delta SNR\) is shown in (**e**) for KBG1 and (**f**) for KBG2. Horizontal axes are the time of day in units of hours. The vertical axes are SNRs in units of dB-Hz. Figure 14 shows the average power of the residual SNR over all scales, applying Equation (12) between different period bands, i.e., 2-20, 20-40, 40-80, and 80-120 s, for two consecutive DoYs 060 and 061 (1 and 2 March 2021). Figure 14, which shows the scale-average of the wavelet power as a function of time, shows distinct significant wavelet power. In all cases, station KBG2 manifests higher power than KBG1.