As previously described, the concept of the GNSS tide gauge can be realized in different ways. In this study, two techniques were investigated: 1) the SNR analysis using one zenith-looking antenna connected to a geodetic GNSS receiver and 2) the geodetic phase delay analysis using both a zenith-looking and a nadir-looking antenna, each connected to a geodetic GNSS receiver (see Figure 2). In both analysis techniques, the satellite signals reflected off the sea surface are used to estimate the sea level. However, the techniques take advantage of two different satellite observations, i.e., the SNR and the phase delay data, recorded by one or both of the receivers.

Both the SNR and the phase delay analysis have been described before, e.g., the SNR analysis in [14, 16] and the phase delay analysis in [22, 24]. However, for the sake of completeness, both analysis methods are summarized below.

### 3.1 The SNR analysis method

GNSS antennas are by design sensitive to the direct satellite signals, suppressing the unwanted signals that are reflected in the surrounding environment before reaching the antenna. Nevertheless, some of the reflected signals interfere with the direct signals which affect the GNSS observables recorded by the receiver (see Figure 2 (left) for an illustration of a zenith-pointing antenna). This effect, known as multipath, is one of the major error sources in high-accuracy positioning, and there are several studies on how to mitigate the effect, e.g., [9, 10, 25]. In addition, there are studies on how to use the reflected signals to measure properties of the reflecting surface, e.g., sea level [13, 14, 16], soil moisture [26], and snow depth [27].

When a GNSS satellite moves across the sky, the phase difference in the receiver between the direct and the reflected satellite signals changes, creating an interference pattern. This pattern is especially visible in the SNR data recorded by the receiver, and as an example, the SNR data from two satellite observations affected by multipath are presented in Figure 3 (left). Note that the SNR data in Figure 3 (left) are smoothed with a 30-s moving average filter (the original SNR data have a resolution of 0.25 dB-Hz and a sampling rate of 1 s) to improve the visibility. This is, however, not necessary for the SNR analysis.

There are two main characteristics of the SNR data in Figure 3 (left): the SNR multipath oscillations and the overall trend. The multipath effect is decreasing with increasing satellite elevation angle, i.e., the amplitude of the interference pattern decreases with increasing satellite elevation angle. This decrease depends on the amplitude (signal strength) of the reflected signal and the antenna gain pattern. For low satellite elevation angles, the reflected signal is mostly RHCP due to the higher amplitude of the RHCP Fresnel reflection coefficient compared to the LHCP coefficient (see e.g., [22, 25]). However, the RHCP reflection coefficient decreases with increasing elevation angle.

The overall trend of the SNR arc (see Figure 3, left) depends on the receiver-satellite distance, the atmospheric attenuation, and the receiving antenna gain pattern. In order to isolate the multipath contribution to the SNR observation, the overall trend can be removed by either fitting and removing a low-order polynomial (see e.g., [28]) or by filtering the SNR signal (e.g., [29]). The remaining detrended SNR (*δ* SNR), which consists of the multipath oscillations, can be described by

\delta \text{SNR}=Acos(\frac{4\pi {h}_{r}}{{\lambda}_{i}}sin\epsilon +\phi )

(1)

where *A* is the amplitude, *h*_{
r
} is the distance between the reflecting surface and the antenna phase center (also called reflector height; see Figure 2, left), *λ*_{
i
} is the carrier wavelength of the GNSS signal, *ε* is the satellite elevation angle, and *φ* is a phase offset.

Assuming that the reflector height (e.g., the sea level) is not changing during the satellite arc and that the reflector (e.g., the sea surface) is horizontal, the frequency of the multipath oscillations (2*h*_{
r
}/*λ*_{
i
}) is constant with respect to the sine of the satellite elevation angle. It is also possible to model a reflector height that is changing during a satellite arc [15]. However, this is not necessary for stations with a sub-diurnal tidal range of less than about 2.7 m [16].

By spectral analysis of the *δ* SNR data as a function of sine of elevation angle, it is possible to derive the dominant multipath frequency. This is usually performed using the Lomb-Scargle periodogram (LSP), since it can handle unevenly spaced samples (the SNR data are evenly sampled in time, but not as a function of sine of elevation). As an example, Figure 3 (right) depicts the LSP results after analysis of the SNR data in Figure 3 (left) (note that the SNR data were first detrended). The peaks of the LSP in Figure 3 (right) correspond to the dominant multipath frequencies and thus the reflector heights. The reflector height is negatively correlated with sea level, e.g., a large reflector height, or high multipath frequency, corresponds to a large distance between the antenna and the sea surface, which means a low sea level.

### 3.2 The phase delay analysis method

To achieve high-accuracy positioning with GNSS, analysis of phase delay data is necessary. For the GNSS tide gauge, the setup consists of one zenith-looking antenna, recording the direct satellite signals which are RHCP, and one nadir-looking antenna, recording the satellite signals that are reflected off the sea surface which are mostly LHCP (see Figure 2 (right) for an illustration). As previously described, the RHCP satellite signal is still mostly RHCP for low satellite elevations after reflection. However, the amplitude of the LHCP reflection coefficient is increasing with increasing satellite elevation angle, and for elevation angles over about 8°, the amplitude of the LHCP reflection coefficient is larger than the amplitude of the RHCP reflection coefficient (see e.g., [22, 25]).

Because of the additional travel path of the reflected signals, as compared to the directly received signals, the nadir-looking antenna will in the analysis appear to be a virtual antenna located below the sea surface. The distance between the virtual antenna and the sea surface, *h*_{
a
}, is the same as the distance from the actual LHCP antenna to the sea surface (see Figure 2, right). This means that when there is a change in sea surface, the additional travel path of the reflected signal changes, and the LHCP antenna appears to change its vertical position. The height of the nadir-looking antenna over the sea surface (*h*_{
a
}) can from the geometry presented in Figure 2 (right) be related to the vertical baseline between the two antennas as 2*h*_{
a
}+*d*, where *d* is the vertical separation between the phase centers of the two antennas. Furthermore, the height of the nadir-looking antenna over the sea surface is directly proportional to the sea surface height.

There are several ways to analyze GNSS phase delay data in order to estimate the vertical baseline between the antennas (see e.g., [24, 30]). First consider the GNSS phase observation equation expressed in meters for a single receiver and satellite, denoted with subscripts *A* and *j*, respectively,

{\lambda}_{i}{\Phi}_{A}^{j}={\varrho}_{A}^{j}+c({\tau}_{A}-{\tau}^{j})+{Z}_{A}^{j}-{I}_{A}^{j}+{\lambda}_{i}{N}_{A}^{j}

(2)

where *λ*_{
i
} is the carrier wavelength of the GNSS signal, {\Phi}_{A}^{j} is the observed carrier phase in units of cycles, {\varrho}_{A}^{j} is the geometric range to the satellite, *c* is the speed of light in vacuum, *τ*_{
A
} is the receiver clock bias, *τ*^{j} is the satellite clock bias, {Z}_{A}^{j} is the delay caused by the neutral atmosphere (tropospheric delay), {I}_{A}^{j} is the ionospheric delay, and {N}_{A}^{j} is the phase ambiguity (including an integer number of wavelengths and unknown instrumental phase offsets from the satellite and the receiver).

Using Equation 2 for two receivers, denoted *A* and *B*, and forming the difference result in the single difference equation for each epoch

\phantom{\rule{-14.0pt}{0ex}}{\lambda}_{i}\Delta {\Phi}_{\mathit{\text{AB}}}^{j}=\Delta {\varrho}_{\mathit{\text{AB}}}^{j}+\mathrm{c\Delta}{\tau}_{\mathit{\text{AB}}}+\Delta {Z}_{\mathit{\text{AB}}}^{j}-\Delta {I}_{\mathit{\text{AB}}}^{j}+{\lambda}_{i}\Delta {N}_{\mathit{\text{AB}}}^{j}

(3)

where \Delta {\Phi}_{\mathit{\text{AB}}}^{j} is the difference between the measured phases expressed in cycles, \Delta {\varrho}_{\mathit{\text{AB}}}^{j} is the difference in geometry, *Δ* *τ*_{
A
B
} is the difference in receiver clock bias, \Delta {Z}_{\mathit{\text{AB}}}^{j} is the difference in tropospheric delay, \Delta {I}_{\mathit{\text{AB}}}^{j} is the difference in ionospheric delay, and \Delta {N}_{\mathit{\text{AB}}}^{j} is the phase ambiguity difference in cycles. Note that since the difference is taken with respect to the same satellite, the differential satellite clock bias term is not present in Equation 3. Additionally, for short baselines, the tropospheric and ionospheric effects can be assumed to cancel.

For short baselines, the term for the difference in geometry (see Equation 3) can be expressed in a local coordinate system using the azimuth, *α*, and elevation, *ε*, angle for each satellite as

\phantom{\rule{-14.0pt}{0ex}}\begin{array}{c}\Delta {\varrho}_{\mathit{\text{AB}}}^{j}=\mathrm{\Delta e}\phantom{\rule{0.3em}{0ex}}sin\left({\alpha}^{j}\right)cos\left({\epsilon}^{j}\right)+\mathrm{\Delta n}\phantom{\rule{0.3em}{0ex}}cos\left({\alpha}^{j}\right)cos\left({\epsilon}^{j}\right)\hfill \\ \phantom{\rule{4em}{0ex}}+\mathrm{\Delta v}\phantom{\rule{0.3em}{0ex}}sin\left({\epsilon}^{j}\right)\hfill \end{array}

(4)

where *Δ* *e*, *Δ* *n*, and *Δ* *v* are the east, north, and vertical components of the baseline between the two receivers, respectively. For a known horizontal baseline (the horizontal baseline for our GNSS tide gauge is zero, see Figure 1), the east and north components can be used to adjust the left side of Equation 3.

It is also possible to form additional differences to Equation 3, e.g., double differences (between two receivers and two satellites) and triple differences (between double differences at different epochs). The double difference equation is especially advantageous for GPS observations for which the double differenced phase ambiguity parameters become integers (the receiver clock bias terms also cancel out). However, this is not the case for GLONASS observations, since the visible satellites have different carrier frequencies. Additionally, one extra observation is needed to form double differences as compared to single differences, and for our configuration, the number of (reflected) observations is limited. For these two reasons, we thus focused on single difference analysis.