# Sea Level Estimation Based on GNSS Dual-Frequency Carrier Phase Linear Combinations and SNR

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Fundamentals of the GNSS-R Altimetry Using the Multipath Effect

_{d}and A

_{m}are the amplitudes of the direct and reflected signals, respectively, and δφ(t) is the phase caused by the excess range of the reflected signal compared with the direct one. The factors A and $\tilde{\psi}(t)$ represent the amplitude and phase of the composite signals received by the antenna, and ψ(t) represents the phase of the direct signal. The factor β(t) is the composite excess phase with respect to the direct phase, h is the vertical height between the antenna and the sea surface, θ is the satellite elevation angle and λ is the electromagnetic wave length of the GNSS signal.

#### 2.2. Sea Level Based on SNR Data

_{noise}is the noise power. In order to solve for reflector height (h), it is necessary to isolate the last item from Equation (6) by removing the first two items using a fifth-order polynomial detrending. The detrending of the SNR becomes

#### 2.3. Sea Level Based on Carrier Phase-Combination Measurements

_{i}represents ionospheric delays. T represents tropospheric delays, and Δ accounts for all other carrier-independent effects. To isolate the multipath, the L1–L2 combination of carrier phase measurements, shown in Equation (12), is used. For simplicity, this combination is referred to as L4 in this paper.

_{m}$\ll $A

_{d}, after substituting Equations (4) and (5) into Equation (14), Equation (14) can be further simplified as follows:

_{i}= 2h/λ

_{i}. Similar to the SNR method, a spectral analysis, such as LSP, is used to deal with the dL4 data because of the irregular sampling intervals of sinθ. As dL4 is the linear combination of L1 and L2, shown as a linear combination of two quasi-sinusoidal functions in Equation (15), there will be two peaks in its spectrogram (Figure 3b,d). The peak at the lower frequency is caused by L2, and the peak at the higher frequency is caused by L1. Their different frequencies depend on the carrier wavelengths, and L2 peaks are usually used because they show clearer spectral peaks than L1.

#### 2.4. Theoretical Model for Calculating Reflector Height

## 3. Experiment and Analysis of Results

#### 3.1. Brief Introduction to Station SC02

^{®}NETR9 receiver and TRM29659.00 choke-ring antenna (Trimble Inc., Sunnyvale, CA, USA) with a hemispherical radome. However, it only provides GPS measurements which contain a pseudo-random code with a single frequency (C/A1), a precise code (P2), a carrier phase with two frequencies (L1 and L2) and the SNR data at two frequencies (S1 and S2).

^{®}air acoustic sensor (Aquatrak Corp., Sanford, FL, USA) in a protective well. Sea level data at this TG are recorded at 6-min sampling intervals and were obtained from National Oceanic and Atmospheric Administration (NOAA) for comparison.

#### 3.2. Data Processing Strategies

#### 3.3. Results Analysis

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Simplified schematic diagram of the ground-based global navigation satellite system reflectometry (GNSS-R) altimetry system with one antenna and the multi-path effect of direct and reflected signals. Definitions: ARP, antenna reference point; GPS, global positioning system. TGZ: tide gauge zero.

**Figure 2.**(

**a**) GPS signal-to-noise ratio (SNR)1 observation at the SC02 GPS station; (

**b**) multipath pattern after detrending; (

**c**) Lomb–Scargle Periodogram for the PRN27 GPS satellite on 6 and 8 January 2015.

**Figure 3.**(

**a**) Multipath error of double-frequency phase combination; (

**b**) Lomb–Scargle Periodogram from satellite PRN30 recorded on 2 and 3 January 2015; (

**c**) Simulated double-frequency phase multipath using the software developed by [39]; (

**d**) Lomb–Scargle Periodogram of the simulated L4 multipath.

**Figure 4.**Graphical illustration of the linear relationship between peak frequencies and antenna height from the model for the elevation range 5–12°.

**Figure 5.**Photograph showing GPS station SC02 and the surrounding environment at Friday Harbor Port in Washington State, USA (image from http://www.sonel.org/spip.php?page=gps&idStation=2689).

**Figure 6.**Sea level changes over a 31-day period, as determined from GPS L1 and L2 SNR via double-frequency combination calculated from Equation (8) and the linear model for elevation range 5–15°. Tide gauge (TG) data are also shown (black curves) for the same 31-day period, for comparison.

**Figure 7.**Correlation of GPS dual-frequency phase combination and SNR estimations for the elevation range 5–15° using a linear regression model with tide gauge (TG) observations.

**Figure 8.**Correlation of GPS dual-frequency phase combination and SNR estimations for the elevation range 5–15°, using Equation (8) with tide gauge (TG) observations.

**Figure 9.**Residuals of sea surface height differences compared between SNR and geometry-free phase estimations for the elevation range 5–15° and tide gauge (TG) observations.

**Table 1.**Statistical analysis of residual data between sea level changes derived from the GPS multipath effect and tide gauge (TG) observations over the one month period of 1–31 January 2015 with different elevations ranges.

SNR1 | SNR2 | L4d | L4-Model | S2-Model | ||
---|---|---|---|---|---|---|

5°–12° | RMSE(cm) | 19.32 | 14.01 | 14.68 | 18.30 | 17.63 |

Corr-coeff^{@} | 0.9753 | 0.9822 | 0.9822 | 0.9701 | 0.9696 | |

5°–15° | RMSE(cm) | 14.69 | 12.36 | 13.81 | 13.67 | 13.17 |

Corr-coeff^{@} | 0.9778 | 0.9831 | 0.9810 | 0.9810 | 0.9831 |

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**MDPI and ACS Style**

Wang, N.; Xu, T.; Gao, F.; Xu, G. Sea Level Estimation Based on GNSS Dual-Frequency Carrier Phase Linear Combinations and SNR. *Remote Sens.* **2018**, *10*, 470.
https://doi.org/10.3390/rs10030470

**AMA Style**

Wang N, Xu T, Gao F, Xu G. Sea Level Estimation Based on GNSS Dual-Frequency Carrier Phase Linear Combinations and SNR. *Remote Sensing*. 2018; 10(3):470.
https://doi.org/10.3390/rs10030470

**Chicago/Turabian Style**

Wang, Nazi, Tianhe Xu, Fan Gao, and Guochang Xu. 2018. "Sea Level Estimation Based on GNSS Dual-Frequency Carrier Phase Linear Combinations and SNR" *Remote Sensing* 10, no. 3: 470.
https://doi.org/10.3390/rs10030470