Introduction

Obtaining accurate Sea Surface Height (SSH) is significant for human living, especially for those live along ocean coasts. Furthermore, it is also important to coastal ecosystems and coastal morphology1,2. SSH has been obtained primarily with tide gauges during the last centuries. While for the last three decades, satellite altimetry has been the dominating technique3,4,5. The satellite altimetry has unique advantages in monitoring sea surface topography in open ocean research such as ocean circulation6. However, due to tidal aliasing effect, it’s spatial and temporal resolution are insufficient to observe complex and rapidly changing dynamics which makes it difficult to use near the coast7. Meanwhile, tide gauges are affected by both sea level and land surface changes since the measurements are related to a benchmark on the land, where they were established. For these reasons, it is difficult to use traditional tide gauges for sea-level studies in tectonically active areas or applications related to changes in the global ocean volume, e.g., the global sea level budget8. These applications need absolute observations of sea level, in other words, measurements with respect to a terrestrial reference frame9.

In recent years, the reflected signals of Global Navigation Satellite System (GNSS) are used to retrieve a range of geophysical parameters such as SSH, soil moisture, ocean wind speed, etc.10,11,12,13,14,15,16,17,18. This technique is also termed GNSS-Reflectometry (GNSS-R), which exploits conventional receivers or low-cost, low-mass, and low-power characteristics of GNSS-R receivers. Different GNSS-R based methods have been proposed in the literature to measure a range of geophysical parameters. Basically, either a single antenna or two separate antennas are used for the direct and reflected signal reception. A number of researchers have investigated GNSS-based ocean surface altimetry by using the measurement of signal arrival time obtained by either mounting the receiver on an aircraft or fixing it on the ground such as on a bridge19,20. Anderson et al. first proposed to use the interference pattern in the recorded Signal-to-Noise Ratio (SNR) to estimate SSH21. Later, Larson et al. applied the SNR method to SSH of the nearby ocean during three months22. Santamaría-Gómez et al. developed an approach to extract SNR data dominated by sea-surface reflections and to remove SNR frequency changes caused by the dynamic sea surface23. They successfully demonstrated its ability to estimate local SSH and improved accuracy. Nevertheless, many of their studies used single frequency or single system. With constantly developing and perfecting of other navigation satellite systems such as Galileo and BDS, can we improve the accuracy of SSH estimation using multi frequencies or multi-systems? There are still many challenging problems to be resolved24.

In this paper, we propose an improvement of the method based on the SNR analysis of a single antenna. And the focus is on GNSS-R ocean surface altimetry based on the SNR data of multiple satellite constellations and multiple frequencies, namely BDS (L1, L5, L7), GPS (L1, L2, L5) and GLONASS (L1, L2). The purpose is to exploit measurements diversity to achieve a performance gain. The structure of the next part of this paper is organized as follows. Firstly, the fundamental theory of SSH GNSS-R and contains a wavelet analysis in detail are given. Thereafter the data introduction is presented. Then SSH estimation results by using individual GNSS-R and multi-GNSS-R are firstly shown. The SNR data collected by an IGS station are used for evaluation and analysis. Tide gauge observations are used as ground-truth data for performance comparison. Furthermore, the initial discrete SSH estimation results are used to reconstruct the tidal waveform by using wavelet de-noising technique. Finally, the concluding remarks are declared.

Theory and Methods

Methods of GNSS-R

Multipath propagation is one of the main GNSS error sources, which constrains high-precision positioning. The multipath caused by the difference of the phase between the direct and reflected signals will affect GNSS observations and give rise to oscillations in the observations. As a result, the SNR data recorded by a GNSS receiver with a single antenna will contain information about the interference, antenna height and hence SSH variation. Figure 1 shows the diagram of SSH estimation based on the SNR method. Here h(i) is the antenna height relative to reflection surface, and θ(i) is the angle between the direct signal and the instantaneous sea surface of a specific scattering point. In Fig. 1, the reflected signal has one excess phase delay when compared to the direct signal, which is related to the antenna height h(i).

Figure 1
figure 1

Diagram of GNSS-R for SSH variation.

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Figure 2 shows an example of the SNR variations recorded by a BDS receiver. The thin blue fast-varying curve is the observed data, while the red thick smooth curve is the fitted SNR of the direct signal. As we can see, the multipath SNR is proportional to the difference between the observed curve and the fitted curve. It can be seen that the multipath SNR during the rising and setting of the satellite is significantly higher than that elsewhere, as shown in the two black ellipses.

Figure 2
figure 2

SNR variations observed by a BDS receiver.

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SNR is one of the main observables of the GNSS receiver, which is mainly related to the satellite signal transmission power, antenna gain, and multipath effect. As the elevation angle increases, the antenna gain raises, subsequently, the direct signal SNR increases. In this paper, we use the observed SNR data from the GNSS receiver to estimate SSH. As described in22, the SNR is given by

$$\begin{array}{rcl}SNR & = & \frac{{A}^{2}}{2{P}_{n}}\\ & = & \frac{1}{2{P}_{n}}({A}_{d}^{2}+{A}_{r}^{2}+2{A}_{d}{A}_{r}\,\cos \,\delta \varphi (t))\end{array}$$
(1)

where Pn is the noise power, Ad indicates amplitude of direct signal, Ar refers amplitude of the reflected signal, δϕ(t) is the reflection excess phase with respect to direct phase (also known as the interferometric phase), given by

$$\delta \varphi (t)=\frac{4\pi h}{\lambda }\,\sin \,\theta (t)$$
(2)

where h stands for antenna height from the specular scattering point on the reflected surface, λ is the carrier wavelength, θ(t) represents the elevation angle. Since (i) GNSS antennas are designed to filter reflected signals, (ii) the reflected signal is attenuated upon reflection, and (iii) the azimuth angles we choose to make reflected signals come from the sea surface, we can assume that

$${A}_{d}\gg {A}_{r}$$
(3)

Therefore, the direct signal determines the overall trend of the SNR observations (see Fig. 2). So, we can use a low-order polynomial to fit the observed SNR data and then remove the trend. As a result, a detrended SNR time series is produced, which can be used to estimate the surface parameters such as SSH. As described in22, after removing the SNR trend, the remaining SNR can be approximated as

$$dSNR=A\,\cos (\frac{4\pi h}{\lambda }\,\sin \,\theta (t)+\phi )$$
(4)

where φ indicates a phase offset, A denotes the amplitude which is given by

$$A=\frac{{A}_{d}\cdot {A}_{r}}{{P}_{n}}$$
(5)

By defining ω = sin θ and f = 2h/λ, Eq. (4) can be rewritten as

$$dSNR=A\,\cos (2\pi f\omega +\phi )$$
(6)

Due to the tidal effect, the antenna height relative to the surface may change smoothly by around 0.72 m during half an hour at the selected observation station. Thus, the frequency of the detrended SNR signal varies gently with time. However, the mean frequency of the signal would be dominant, which basically corresponds to the antenna height at the middle of the observation time period.

From Eq. (6) and the definition of f, the antenna height is associated with the frequency of the detrended SNR signal by

$$h=f\lambda /2$$
(7)

Spectral analysis is applied to obtain the spectral peak frequency of the time series. In this paper, the Lomb-Scargle periodogram (LSP) method is used since it can handle unevenly spaced samples22. The recorded SNR data are typically evenly sampled in time, but the sine of elevation angle is not evenly distributed. The LSP is a well-known algorithm for detecting and characterizing periodicity in unevenly-sampled time-series, even more, has seen particularly wide use within the astronomy community25. As described in25, the LSP method calculates the power spectral density by

$${P}_{x}(f)=\frac{1}{2{\delta }^{2}}\{\frac{{[\sum ({X}_{i}-\overline{X})\cos \omega ({t}_{i}-\tau )]}^{2}}{\sum {\cos }^{2}\omega ({t}_{i}-\tau )}+\frac{{[\sum ({X}_{i}-\overline{X})\cos \omega ({t}_{i}-\tau )]}^{2}}{\sum {\sin }^{2}\omega ({t}_{i}-\tau )}\}$$
(8)

where \(\overline{X}\) and δ2 are the mean and the variance of the observed sequence respectively, ω denotes the angular frequency, and τ symbolizes the phase which can be inferred by

$$\tan (2\omega \tau )=\sum \sin (2\omega {t}_{i})/\sum \cos (2\omega {t}_{i})$$
(9)

As an example of obtaining the peak frequency using LSP method, Fig. 3 shows the results of LSP analysis of the detrended SNR time series from the SNR data observed by BDS satellite PRN09 on two different days. The elevation angles change from 5° to 20° and from 20° to 5° for day of year (DOY) 191 and DOY 193, respectively. The peak spectral frequency can be clearly observed. The peak frequency difference is caused by the variation in the SSH over the two various observation periods. The estimated peak frequency, besides, Eq. (7) are applied to obtain the antenna height.

Figure 3
figure 3

Detrended SNR spectrum produced by Lomb-Scargle periodogram analysis.

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Wavelet De-noising

The concept of wavelet was first proposed by Morlet in 198426. Later, under the help of Grossmann, Morlet formalized the continuous wavelet transform (CWT)27 as follow

$${W}_{\psi }\,f(a,b)=\langle \,f,{\psi }_{a,b}\rangle ={a}^{-1/2}{\int }_{-\infty }^{+\infty }f(t)\overline{\psi }(\frac{t-b}{a})dt$$
(10)

where a indicates the scale parameter, b refers to the time parameter, ψ(t) is an analyzing wavelet, and \(\overline{\psi }(\bullet )\) symbolizes the complex conjugate of ψ(•). The inverse transform of CWT is given by

$$f(t)=\frac{1}{{C}_{\psi }}{\int }_{-\infty }^{+\infty }\{{W}_{\psi }\,f(a,b)[{a}^{-1/2}\psi (\frac{t-b}{a})db]\}\frac{da}{{a}^{2}}$$
(11)

where a, b and t are continuous. However, they must be discretized in analyzing real data. The discrete method is called discrete wavelet transform (DWT). In practice, the scale a and the time b are discretized as following

$$a={a}_{0}^{j},\,b=k{a}_{0}^{j}{b}_{0}$$
(12)

where j and k are integers. And the continuous wavelet function Wψf(a, b) become the DWT, given by

$$\begin{array}{ccc}{C}_{j,k}={\int }_{-\infty }^{+\infty }f(t){\overline{\psi }}_{j,k}(t)dt & with & {\psi }_{j,k}(t)={2}^{-j/2}({2}^{-j}t-k)\end{array}$$
(13)

The inverse transform of DWT is computed by

$$f(t)=\sum _{j\in Z}\sum _{K\in Z}{C}_{j,k}{\psi }_{j,k}(t)$$
(14)

Mallat proposed a fast DWT algorithm in 198928, which decompose and reconstruct the signal using the wavelet filters. The decomposition algorithm is given by

$$\begin{array}{rcl}{A}_{j}[\,f(t)] & = & \mathop{\sum _{k}H(2t-k){A}_{j-1}[\,f(t)]}\limits^{{A}_{0}[f(t)]=f(t)}\\ {D}_{j}[\,f(t)] & = & \sum _{k}G(2t-k){A}_{j-1}[\,f(t)]\end{array}\}$$
(15)

where t = 1, 2 …, N is the discrete time serial number, N denotes the signal length, f(t) symbolizes the original signal, j = 1, 2 …, J is the decomposition level, J is the maximum decomposition level; H and G are the wavelet low-frequency and high-frequency pass decomposition filters, respectively; Aj and Dj are the wavelet coefficients of f(t) in the low-frequency and the high-frequency part of the Jth layer, respectively. Figure 4 shows the wavelet decomposition filter model.

Figure 4
figure 4

Wavelet decomposition filter model.

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Each signal has the following relationship: S = A1 + D1; A1 = A2 + D2; A2 = A3 + D3; A3 = A4 + D4. As an example of decomposition, Fig. 5 shows the results of wavelet decomposition under Db6 wavelet with decomposition level 3, and the original signal S is the in-situ SSH estimation of this paper. In addition, the relationship of each signal is: S = a3 + d3 + d2 + d1, a3 = d3 + d2 + d1.

Figure 5
figure 5

Results of wavelet decomposition of Db6 wavelet under 3rd level.

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The reconstruction is given by

$${A}_{j}[f(t)=\sum _{k}h(2t-k){A}_{j+1}[f(t)]+\sum _{k}g(2t-k){D}_{j+1}[f(t)]]$$
(16)

where h and g represents the wavelet low pass and high pass reconstruction filter, respectively. Other symbols are the same as Eq. (15). Figure 6 shows the wavelet reconstruction filter model.

Figure 6
figure 6

Wavelet reconstruction filter model.

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One important step of wavelet de-nosing is choosing the appropriate wavelet function Wψf(a, b). Wavelet functions such as Daubechies (dbN), Haar, and Symlets (SymN) are commonly used for wavelet analysis. To choose a suitable wavelet function, the performance of db6, Haar and Sym2 at different levels are analyzed in section “Improved SSH estimates with Wavelet De-noising”.

Data Introduction

GPS observations

To verify the feasibility and effectiveness of the proposed GNSS-R based SSH estimation method, the observation data of MAYG station (latitude: −12.78°, longitude: 45.26°, and height: −16.35 m), which is located in Mayotte, France near the Indian Ocean, are used. The TRIMBLE NETR9 receiver and TRM59800.00 antenna is installed in MAYG station, and the data sampling rate is 1 Hz. Because MAYG is one of the Multi-GNSS Experiment (MGEX) stations, so the data of GPS, GLONASS and BDS signals are all recorded. The navigation and observation data can be downloaded from the IGS website (http://www.igs.org/). The SNR data of BDS (L1, L5, L7; also called B1, B2, and B3), GPS (L1, L2, L5), and GLONASS (L1, L2) over a period of seven days from DOY 190 to DOY 196 in 2017 were used to evaluate the proposed method. The corresponding elevation angles range between 5°~20°, whereas the corresponding azimuth angles are between the values 20°~80° and 110°~170°, respectively. The elevation and azimuth angle selection are based on the fact that the multipath signal is strong at low elevation angle. In contrast, the reflected signals from the sea surface are of interest.

Tide gauge observations

The observation data produced by the Dzaoudzi tide gauge, which is about ten meters from the MAYG station, were used as the ground-truth data for evaluating the performance of the proposed SSH estimation method. The sampling rate of the Dzaoudzi tide gauge data is 1 min, which can be downloaded from the Intergovernmental Oceanographic Commission (IOC) website (http://www.ioc-unesco.org/).

Results and Discussion

SSH estimation results with Multi-GNSS

Figure 7 shows the in-situ SSH estimation results using the L1 signals of GPS, GLONASS, and BDS. The horizontal axis represents the DOY, and the tide gauge observations (black solid line) are also shown for comparison. The GNSS-R based SSH estimates are represented by solid circle, solid square, and solid inverted triangle, respectively for signals of three different satellites.

Figure 7
figure 7

Initial in-situ SSH estimation results respectively produced by (a) BDS-R, (b) GPS-R, and (c) GLONASS-R with different frequencies The black solid line indicates the time series derived from Dzaoudzi tide gauge.

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As can be seen from Fig. 7, there is a significant daily periodicity of in-situ SSH changes observed by tide gauge, which is mainly caused by tide and wind variability, as well as sea level pressure variability. There is a good agreement between tide gauge SSH observations and SSH estimates obtained by the GNSS-R methods over a period of seven days. Only a limited number of GNSS satellites can be seen over a certain period of time. Thus, SSH estimations are not uniformly distributed. In some cases, no in-situ SSH estimates are available over a quite long time interval. In general, the number of GPS satellites is the largest among the numbers of GPS, BDS, and GLONASS satellites. So, GPS-R has the highest temporal resolution for SSH estimation on average. Figure 8 shows the combined results of SSH estimates with all available GNSS satellites and frequencies. It is interesting to note that the time resolution of SSH is greatly improved.

Figure 8
figure 8

Initial in-situ SSH estimation results respectively produced by BDS-R, GPS-R, and GLONASS-R with different frequencies.

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Table 1 presents the in-situ SSH estimation performance in terms of RMSE and correlation coefficients with GPS, BDS, and GLONASS satellite signals of eight different frequency bands. It can be seen from Table 1 that GNSS-R with GPS L1 signal achieves the best performance, i.e., maximum correlation coefficient (0.91) and highest accuracy (RMSE = 0.23 m). Figure 9 shows the histogram of the in-situ SSH estimation errors of the three different navigation systems. It can be seen that the errors approximately have a normal distribution. The mean errors are all within 0.02 m, while the standard deviations are 0.31 m, 0.35 m, and 0.32 m for GPS-R, BDS-R, and GLONASS-R, respectively. The means are very close to zero, so the estimation can be considered as unbiased estimation. Figure 10 shows the scatter plot between the tide gauge observations and the GNSS-R SSH estimates. Clearly, the GNSS-R estimates are closely around the tide gauge observations. The RMSEs and correlation coefficients of the combined GNSS-R based estimates are 0.32 m and 0.83, respectively. Although the precision and correlation coefficient are slightly degraded compared with the best results using single frequency band GPS L1, the temporal resolution is increased significantly.

Table 1 The precisions and correlation coefficients of SSH estimation based on BDS-R, GPS-R, and GLONASS-R.
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Figure 9
figure 9

Histogram of the in-situ SSH estimation errors of (a) BDS-R, (b) GPS-R, and (c) GLONASS-R.

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Figure 10
figure 10

Scatter plot of sea surface height estimates by BDS-R, GPS-R, and GLONASS-R. Tide gauge data are treated as ground-truth. The black line is X = Y line.

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Improved SSH estimates with Wavelet De-noising

As observed from Fig. 8, compared with the tide gauge observations, the GNSS-R based estimates are rather noisy. In order to improve the GNSS-R based in-situ SSH estimation precision, we exploit the wavelet de-noising method to reconstruct the tidal waveforms. To produce the best possible wavelet de-noising results, a number of parameters should be selected appropriately29,30. Three of the wavelets i.e. Db6, Haar, and Sym2 are used to reconstruct the tidal waveform. The number of levels for decomposition and reconstruction is set to be 1–8. Table 2 presents the RMSEs and correlation coefficients by using the three different wavelets. It can be seen that when the decomposition level is less than 5, the three different wavelets produce very similar performance. Db6 slightly outperforms the other two wavelets and Db6 with decomposition level of 3 achieves the minimum RMSE. Meanwhile, the decomposition level is greater than 4, the performance of RMSEs and correlation coefficients are degraded, as well as Sym2 with decomposition level of 8 achieves the maximal RMSE. Therefore, only results produced by Db6 wavelet with decomposition level 3 are presented in the following. Figure 11 shows the reconstructed tidal waveforms after wavelet de-noising. For comparison, the tide gauge observations, as well as the initial GNSS-R in-situ SSH estimates are also presented. Basically, there is a nice match between the observed and the reconstructed tidal waveforms. The histogram of SSH estimation errors is shown in Fig. 12. By comparing with the results have shown in Fig. 9, the in-situ SSH estimation error is significantly reduced and their distribution is more reasonable. The correlation coefficient and RMSE of the wavelet de-noising method are respectively 0.95 and 0.19 m, which are improved by 14.5% and 40.6% over the initial multi-GNSS-R in-situ SSH estimation results.

Table 2 The precisions and correlation coefficients after wavelet de-noising with different levels.
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Figure 11
figure 11

Time series of in-situ SSH estimates from initial results and de-noising results.

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Figure 12
figure 12

Histogram of SSH estimation errors with the de-noising method.

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Conclusions

In this paper, the SNR data of the reflected GNSS satellite signals of different frequency bands, namely BDS (L1, L5, L7), GPS (L1, L2, L5), and GLONASS (L1, L2), were employed to estimate the SSH. The initial in-situ SSH estimates were obtained through LSP spectral analysis on the detrended SNR time series. There is a good agreement between tide gauge SSH observations and SSH estimates obtained by the GPS, BDS and GLONASS data. Although the multi-GNSS combined precision and correlation coefficient are slightly degraded compared with GPS L1, the temporal resolution is increased significantly. Wavelet de-noising was applied to the initial in-situ SSH estimates to reconstruct the tidal waveform. Compared with tide gauge observations, The research findings demonstrated that the refined in-situ SSH estimation based on wavelet de-noising achieved a significant accuracy gain with the RMSE reduced from 0.23 m (GPS L1) and 0.32 m (multiple constellations and multiple frequencies) to 0.19 m, improving the accuracy by 17.4%, 40.6%, respectively. It is worthy of indicating that although these results are inferior to the other research (a few centimeters)31, the tidal change at the MAYG station in this paper is up to 3 m, while that of the previous research on other station is within 1 m. Therefore, the relative precision of our method is comparable to the previous research. Furthermore, since water levels and wave dynamics near the coast in storm conditions are very worth researching and discussing32,33, future study will focus on verifying the performance of this proposed method on estimating storm surge water levels.