## 6 Effects of Multipath on GNSS Carrier Post-Fit Residuals
### Measurement of SNR Frequency Content Using the Wavelet Transform  
GNSS signals have the properties of nonstationarity characteristics, which vary over time. As constituents of the GNSS signal recorded in the GNSS receiver, multipaths also shows a nonstationarity process. To understand and localize the footprint of multipaths, the customarily applied Fourier analysis is not ideal in detecting and localizing the dominant frequency in a signal. This is because the Fourier transforms typically capture the global features of the signal due to the Fourier basis functions, i.e., sines and cosines. Hence, no local features of the signal are captured. We have applied wavelet analysis strategies to measure the multipath frequency and power content of the residual \(SNR_{n}\) time series with a sampling interval at \(\delta t\) and \(n=0\),..., \(N-1\). Wavelet analysis may be more suited than classic Fourier analysis by revealing a time series in terms of frequency and time information, which is particularly advantageous when the signal is nonstationary. Figure 12: Single-point positioning for site KBG2, which has no Eccosorb materials mounted to its antenna again using different GNSS constellations. Results are shown in blue for GPS (**a**), in red for Galileo (**b**), in dark green for GLONASS (**c**) and cyan for BeiDou (**d**), and in the magenta for COMBINED (**e**). The positioning accuracy using GPS in the north-south direction is 2.0 m and in the east-west direction, it is 1.7 m; for Galileo, it is 0.9 m and 0.9 m respectively; for GLONASS, it is 7.6 m and 4.0 m; and the lowest accuracy comes again from BeiDou, with 7.1 m and 8.0 m. We have also derived a combined GPS + GLONASS + Galileo + BeiDou solution with a north-south direction accuracy of 2.0 m and an east-west direction accuracy of 1.6 m.  
GNSS multipath is one of the signal types that has nonstationary properties [34; 35; 60], since its frequency changes with the rate of change of the satellite elevation angle. The purpose of wavelet spectra estimation is to elucidate the periodic multipath error and, in so doing, estimate how this multipath error affects GNSS positioning. We use the continuous wavelet function \(\psi(S)_{o}\) which satisfies with zero mean the localization at both times and frequencies [61], and integrating the wavelet basis over a certain time becomes close to zero. One of the wavelet functions that satisfies the condition of locality is a Morlet wavelet shown in Equation (8), a wavelet produced by modulating a complex sine wave by a Gaussian function. In this analysis, we heavily adapted the wavelet analysis strategies used in [50] and recently expanded by [34; 35]. \[\psi(x)=\pi^{-1/4}e^{i\omega_{o}x}e^{(-x^{2})/2} \tag{8}\]  
The continuous wavelet transform \(W_{n}\), also a complex function, of a discrete sequence the residual SNR, i.e., \(\delta SNR_{n^{\prime}}\), is defined as the convolution of \(\delta SNR_{n^{\prime}}\) with a varying scale and translating along the time index of the Morlet wavelet basis \(\psi(x)\):  
\[W_{n}(S)=\sum_{n^{\prime}=0}^{N-1}\delta SNR_{n^{\prime}}\psi*\left[\frac{(n^{ \prime}-n)\delta t}{s}\right] \tag{9}\]  
Once the wavelet function is selected, we can compute the wavelet transform for a set of scales to form a localized frequency and amplitude of the \(\delta SNR\) time series, where \((*)\) is a complex conjugate operator. It is convenient to compute the wavelet transform at different scales expressed as a fractional power of two,  
\[S_{j}=s_{o}2^{|ij},\quad j=0,1,...,J,\qquad\qquad J=\delta j^{-1}log_{2}(N \delta t/s_{o}) \tag{10}\]  
where \(s_{o}\) is the smallest resolvable scale, approximately twice the sampling interval, and \(J\) is the largest scale. For the Morlet wavelet, the choice, \(\delta j\), depends on the width of the spectral space of the wavelet, and we adopted the same value as [50] at 0.125.The Morlet continuous wavelet transform, which is implemented in the Python package, was utilized _pyCWT_[50; 62] with the power spectral density correction of [60].