# Analysis and Discussion on the Optimal Noise Model of Global GNSS Long-Term Coordinate Series Considering Hydrological Loading

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. GNSS Data

#### 2.2. Hydrological Loading Data

#### 2.3. Theoretical Method

## 3. Results

#### 3.1. Before Hydrological Loading Correction

#### 3.2. Hydrological Loading Displacement

#### 3.3. After Hydrological Loading Correction

## 4. Discussion

#### 4.1. Velocity and Velocity Uncertainty Analysis

#### 4.2. Amplitude Analysis

## 5. Conclusions

- Pure white noise cannot represent the optimal noise model for the long-term series of global IGS reference station coordinates. The noise models of global IGS reference station coordinate time series are diverse, and the N, E, and U directions show different optimal noise model characteristics. Before the hydrological loading correction, the optimal noise model for 46.4% of the station components is WN+FN, the optimal noise model for 28.4% of the station components is WN+PL, and the 22.6% components is the GGM noise model. There is also a small number of components represented as WN+GGM and WN+RW model combinations.
- Experiments show that the hydrological loading does cause changes in the noise characteristics of IGS stations. After calculating the hydrological loading correction, the ratio of WN+FN in the optimal noise model increased significantly (50.1%). Among them, the largest proportion of the optimal noise model in the U direction is still the GGM noise model (47.6%), followed by WN+PL.
- When studying the influence of hydrological loading on the velocity of the stations and its uncertainty, it is found that the hydrological loading has little effect on the velocity uncertainty of the IGS long-term coordinate series, but it will affect the velocity of the stations, especially in its vertical direction, its velocity influence value can reach up to 1.8 mm. Therefore, when estimating the vertical velocity, the influence of hydrological loading must be considered.
- Different complex noise models will affect the station velocity and velocity uncertainty. For WN+FN, 85% of the stations velocity influence value are within 1 mm. For velocity uncertainty, its influence in the vertical direction is more obvious, and the maximum value can be close to 2 mm. When analyzing the WN + RW noise model combination, its impact on the GNSS time series velocity uncertainty is quite different from other combined noise models. Therefore, when studying the influence of the world’s optimal noise model on time series, this will be the next step worth pondering.
- The hydrological loading will have a certain impact on the annual and semi-annual amplitudes of global IGS stations, which is mainly reflected in the vertical direction. The magnitude of the impact varies from station to station, mainly related to the environment around the station. The annual amplitude motion caused is greater than the half-annual amplitude motion. After adding hydrological loading correction, it cannot completely reduce the annual and half-annual amplitude movement of the station. The amplitude of some stations not only does not decrease, but shows an increasing trend. When considering the influence of different noise models on the annual and half-annual amplitude difference before and after hydrological loading correction, it is found that the influence value is very small and can be ignored.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AR1 | First-order autogressive noise |

AIC | Akaike Information Criterion |

BIC | Bayesian Information Criterion |

BPPL | band pass+powerlaw noise |

CMONOC | Crustal Movement Observation Network of China |

CME | common mode error |

CORS | Continuously Operating Reference Stations |

CGCS2000 | China Geodetic Coordinate System 2000 |

CF | center of figure |

EOST | School and Observatory of Earth Sciences |

FN | flicker noise |

FOGMRW | first order Gauss–Markov+random walk noise |

GNSS | Global Navigation Satellite System |

GGM | generalized Gauss–Markov noise |

GFZ | German Research Centre for Geosciences |

HYDL | hydrological loading |

IGS | International GNSS Service |

IMLS | International Mass Loading Service |

LSDM | Land Surface Discharge Model |

NASA | National Aeronautics and Space Administration |

PL | power-law noise |

RW | random walk noise |

SOPAC | Scripps Orbit and Permanent Array Center |

VW | variable white noise |

WGS84 | World Geodetic System 1984 |

WN | white noise |

## References

- Blewitt, G.; Lavallée, D.; Clarke, P.; Nurutdinov, K. A new global mode of Earth deformation: Seasonal cycle detected. Science
**2001**, 294, 2342–2345. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Williams, S.D.P.; Bock, Y.; Fang, P.; Jamason, P.; Nikolaidis, R.M.; Prawirodirdjo, L.; Miller, M.; Johnson, D.J. Error analysis of continuous GPS position time series. J. Geophys. Res. Solid Earth
**2004**, 109, B03412. [Google Scholar] [CrossRef] [Green Version] - Langbein, J. Noise in GPS Displacement Measurements from Southern California and Southern Nevada. J. Geophys. Res. Solid Earth
**2008**, 113, 1–12. [Google Scholar] [CrossRef] - Van Dam, T.; Altamimi, Z.; Collilieux, X.; Ray, J. Topographically induced height errors in predicted atmospheric loading effects. J. Geophys. Res. Solid Earth
**2010**, 115, B07415. [Google Scholar] [CrossRef] - Tian, Y.; Shen, Z. Extracting the regional common-mode component of GPS station position time series from dense continuous network. J. Geophys. Res. Solid Earth
**2016**, 121, 1080–1096. [Google Scholar] [CrossRef] - Gu, Y.; Yuan, L.; Fan, D.; You, W.; Su, Y. Seasonal crustal vertical deformation induced by environmental mass loading in mainland China derived from GPS, GRACE and surface loading models. Adv. Space Res.
**2017**, 59, 88–102. [Google Scholar] [CrossRef] - Chanard, K.; Fleitout, L.; Calais, E.; Rebischung, P.; Avouac, J. Toward a global horizontal and vertical elastic load deformation model derived from GRACE and GNSS station position time series. J. Geophys. Res. Solid Earth
**2018**, 123, 3225–3237. [Google Scholar] [CrossRef] - Klos, A.; Bos, M.S.; Bogusz, J. Detecting time-varying seasonal signal in gps position time series with different noise levels. GPS Solut.
**2018**, 22, 21. [Google Scholar] [CrossRef] [Green Version] - Klos, A.; Olivares, G.; Teferle, F.N.; Hunegnaw, A.; Bogusz, J. On the combined effect of periodic signals and colored noise on velocity uncertainties. GPS Solut.
**2018**, 22, 1–13. [Google Scholar] [CrossRef] [Green Version] - Klos, A.; Gruszczynska, M.; Bos, M.S.; Boy, J.-P.; Bogusz, J. Estimates of vertical errors for IGS ITRF2014 stations by applying the improved singular spectrum analysis method and environmental loading models. Pure Appl. Geophys.
**2018**, 175, 1823–1840. [Google Scholar] [CrossRef] - Yuan, P.; Li, Z.; Jiang, W.; Ma, Y.; Chen, W.; Sneeuw, N. Influences of environmental loading corrections on the nonlinear variations and velocity uncertainties for the reprocessed global positioning system height time series of the crustal movement observation network of China. Remote Sens.
**2018**, 10, 958. [Google Scholar] [CrossRef] [Green Version] - Ming, F.; Yang, Y.; Zeng, A.; Zhao, B. Spatiotemporal filtering for regional GPS network in China using independent component analysis. J. Geod.
**2017**, 91, 419–440. [Google Scholar] [CrossRef] - Ma, C.; Li, F.; Zhang, S.-K.; Lei, J.-T.; E, D.-C.; Hao, W.-F.; Zhang, Q.-C. The coordinate time series analysis of continuous GPS stations in the Antarctic Peninsula with consideration of common mode error. Chin. J. Geophys.
**2016**, 59, 2783–2795. [Google Scholar] - Wang, M.; Shen, Z.; Dong, D. Effects of non-tectonic crustal deformation on continuous GPS position time series and correction to them. Chin. J. Geophys.
**2005**, 48, 1045–1052. [Google Scholar] [CrossRef] - Ray, J.; Altamimi, Z.; Collilieux, X.; Van Dam, T. Anomalous harmonics in the spectra of GPS position estimates. GPS Solut.
**2008**, 12, 55–64. [Google Scholar] [CrossRef] - Beavan, R.J. Noise properties of continuous GPS data from concrete pillar geodetic monuments in New Zealand, and comparison with data from deep drilled braced monuments. J. Geophys. Res. Solid Earth
**2005**, 110, B8. [Google Scholar] [CrossRef] - Johnson, H.O.; Agnew, D.C. Monument motion and measurements of crustal velocities. Geophys. Res. Lett.
**1995**, 22, 2905–2908. [Google Scholar] [CrossRef] - Mao, A.; Harrison, C.G.A.; Dlxon, T.H. Noise in GPS Coordinate Time Series. J. Geophys. Res. Atmos.
**1999**, 104, 2797–2816. [Google Scholar] [CrossRef] [Green Version] - Williams, S.D.P. The Effect of Coloured Noise on the Uncertainties of Rates Estimated from Geodetic Time Series. J. Geod.
**2003**, 76, 483–494. [Google Scholar] [CrossRef] - He, X.; Bos, M.S.; MontilletiD, J.-P.; Fernandes, R. Investigation of the noise properties at low frequencies in long GNSS time series. J. Geod.
**2019**, 93, 1271–1282. [Google Scholar] [CrossRef] - Wang, W.; Qiao, X.; Wang, D.; Chen, Z.; Yu, P.; Lin, M.; Chen, W. Spatiotemporal noise in GPS position time-series from Crustal Movement Observation Network of China. Geophys. J. Int.
**2019**, 216, 1560–1577. [Google Scholar] [CrossRef] - Yuan, L.-G.; Ding, X.-L.; Chen, W.; Kwok, S.; Chan, S.-B.; Hung, P.-S.; Chau, K.-T. Characteristics of Daily Position Time Series from Hong Kong GPS Fiducial Network. Chin. J. Geophys.
**2008**, 51, 1372–1384. [Google Scholar] [CrossRef] - Jiang, Z.; Zhang, P.; Bi, J.; Liu, L. Velocity Estimation on the Colored Noise Properties of CORS Network in China Based on the CGCS2000 Frame. Acta Geod. Cartogr. Sin.
**2010**, 39, 355–363. [Google Scholar] - Farrell, W.E. Deformation of the earth by surface loads. Rev. Geophys.
**1972**, 10, 761–797. [Google Scholar] [CrossRef] - Li, C.; Huang, S.; Chen, X.; Van Dam, T.; Fok, H.S.; Zhao, Q.; Wu, W.; Wang, X. Quantitative Evaluation of Environmental Loading Induced Displacement Products for Correcting GNSS Time Series in CMONOC. Remote Sens.
**2020**, 12, 594. [Google Scholar] [CrossRef] [Green Version] - Gong, G.; Hua, X.; He, X.; Shu, Y.; Ma, M. Analysis of Regional Characteristics of Environment Load Effect in GPS Coordinate Time Series. J. Geod. Geodyn.
**2017**, 37, 961–967. [Google Scholar] - Collilieux, X.; Van Dam, T.; Ray, J.; Coulot, D.; Métivier, L.; Altamimi, Z. Strategies to mitigate aliasing of loading signals while estimating GPS frame parameters. J. Geod.
**2012**, 86, 1–14. [Google Scholar] [CrossRef] - Amiri-Simkooei, A.R.; Tiberius, C.C.J.M.; Teunissen, P.J.G. Assessment of noise in GPS coordinate time series: Methodology and results. J. Geophys. Res. Atmos.
**2007**, 112. [Google Scholar] [CrossRef] [Green Version] - Bogusz, J.; Klos, A. On the significance of periodic signals in noise analysis of GPS station coordinates time series. GPS Solut.
**2016**, 20, 655–664. [Google Scholar] [CrossRef] [Green Version] - Van Dam, T.M.; Wanr, J. Modeling environment loading effects: A review. Phys. Chem. Earth
**1998**, 23, 1077–1087. [Google Scholar] [CrossRef] - Liao, H.H.; Zhong, M.; Zhou, X.H. Climate driven annual vertical deformation of the solid earth calculated from GRACE. Chin. J. Geophys.
**2010**, 53, 321–328. [Google Scholar] [CrossRef] - Jiang, W.; Xia, C.; Li, Z.; Guo, Q.; Zhang, S. Analysis of environmental loading effects on regional GPS coordinate time series. Acta Geod. Cartogr. Sin.
**2014**, 43, 1217–1223. [Google Scholar] - Wu, S.; Nie, G.; Meng, X.; Liu, J.; He, Y.; Xue, C.; Li, H. Comparative Analysis of the Effect of the Loading Series from GFZ and EOST on Long-Term GPS Height Time Series. Remote Sens.
**2020**, 12, 2822. [Google Scholar] [CrossRef] - Dill, R.; Dobslaw, H. Numerical simulations of global scale high-resolution hydrological crustal deformations. J. Geophys. Res. Solid Earth
**2013**, 118, 5008–5017. [Google Scholar] [CrossRef] - Dill, R. Hydrological model LSDM for operational Earth rotation and gravity field variations. GFZ
**2008**. [Google Scholar] [CrossRef] - Hagemann, S.; Dümenil, L. A parametrization of the lateral waterflow for the global scale. Clim. Dyn.
**1998**, 14, 17–31. [Google Scholar] [CrossRef] - Bevis, M.; Brown, A. Trajectory models and reference frames for crustal motion geodesy. J. Geod.
**2014**, 88, 283–311. [Google Scholar] [CrossRef] [Green Version] - Bos, M.S.; Fernandes, R.M.S.; Williams, S.D.P.; Bastos, L. Fast error analysis of continuous GNSS observations with missing data. J. Geod.
**2013**, 87, 351–360. [Google Scholar] [CrossRef] [Green Version] - Akaike, H.T. A new look at the statistical model identification. IEEE Trans. Autom. Control
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Schwarz, G. Estimating the Dimension of a Model. Ann. Stat.
**1978**, 6, 31–38. [Google Scholar] [CrossRef] - Bos, M.S.; Fernandes, R.M.S. Hector User Manual Version 1.6. 2016. Available online: http://segal.ubi.pt/hector/manual_1.7.2.pdf (accessed on 22 January 2021).
- Sun, H.P.; Ducarme, B.; Dehant, V. Effect of the Atmospheric Pressure on Surface Displacement. J. Geod.
**1995**, 70, 131–139. [Google Scholar] [CrossRef] - Van Dam, T.; Collilieux, X.; Wuite, J.; Altamimi, Z.; Ray, J. Nontidal ocean loading: Amplitudes and potential effects in GPS height time series. J. Geod.
**2012**, 86, 1043–1057. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; Yue, J.; Li, W.; Lu, D.; Li, X. A comparison of hydrological deformation using GPS and global hydrological model for the Eurasian plate. Adv. Space Res.
**2017**, 60, 587–596. [Google Scholar] [CrossRef] - Wu, S.; Nie, G.; Liu, J.; Xue, C.; Wang, J.; Li, H.; Peng, F. Analysis of deterministic and stochastic models of GPS stations in the crustal movement observation network of China. Adv. Space Res.
**2019**, 64, 335–351. [Google Scholar] [CrossRef] - Zhang, J.; Bock, Y.; Johnson, H.; Fang, P.; Williams, S.; Genrich, J.; Wdowinski, S.; Behr, J. Southern Californie Permanent GPS Geodetic Array: Error Analysis of Daily Position Estimates and Site Velocities. J. Geophys. Res. Solid Earth
**1997**, 102, 18035–18055. [Google Scholar] [CrossRef] - Huang, L. Noise properties in time series of coordinate component at GPS fiducial stations. J. Geod. Geodyn.
**2006**, 26, 31–33. [Google Scholar]

**Figure 1.**Distribution of 671 giobal GNSS reference stations and their coordinate time series length in 21 years (2000–2021).

**Figure 2.**The optimal noise model distribution in the North (N), East (E), Up (U) directions and total components of global International Global Navigation Satellite System (GNSS) Service (IGS) stations before hydrological loading correction. (

**a**), (

**b**) and (

**c**) respectively show the optimal noise model percentage in the N, E and U directions. The optimal noise model percentage of total components shown in (

**d**).

**Figure 3.**Distribution diagram of hydrological loading displacement series at the three IGS stations ACP1, ALGO, and ALRT from 2000 to 2021.

**Figure 4.**A statistical diagram of the RMS value of hydrological loading displacement in the N, E, and U directions and their maximum RMS values. (

**a**), (

**b**) and (

**c**) respectively show the RMS value of hydrological loading displacement in the N, E, and U directions and their maximum RMS values are shown in (

**d**).

**Figure 5.**The optimal noise model distribution in the N, E, U directions and total components of global IGS stations after hydrological loading correction. (

**a**), (

**b**) and (

**c**) respectively show the optimal noise model percentage in the N, E and U directions and the optimal noise model percentage of total components are shown in (

**d**).

**Figure 6.**The difference between the velocity and its uncertainty estimation under the optimal noise model of global IGS stations. (

**a**,

**c**,

**e**) show the difference between the velocity under the optimal noise model of global IGS stations. (

**b**,

**d**,

**f**) show the difference between the velocity uncertainty estimation under the optimal noise model of global IGS stations.

**Figure 7.**The percentage diagram of velocity influence value before and after hydrological loading correction.

**Figure 8.**The distribution of velocity and velocity uncertainty differences under different noise models. (

**a**) shows the distribution of velocity differences under different noise models and (

**b**) shows the distribution of velocity uncertainty differences under different noise models.

**Figure 9.**Distribution of the annual amplitudes before hydrological loading correction are shown (

**a**,

**c**,

**e**). (

**b**,

**d**,

**f**) show the half-annual amplitude before hydrological loading correction. (unit: mm).

**Figure 10.**Statistics of annual and half-annual amplitude differences before and after hydrological loading correction. (

**a**,

**c**,

**e**) show the statistics of annual amplitude differences before and after hydrological loading correction. The half-annual amplitude differences before and after hydrological loading correction are shown in (

**b**,

**d**,

**f**). (unit: mm).

**Figure 11.**Statistics of annual and half-annual amplitude difference under different noise models before and after hydrological loading correction. (

**a**) shows the statistics of annual amplitude difference under different noise models before and after hydrological loading correction and (

**b**) shows the half-annual amplitude difference under different noise models before and after hydrological loading correction. (unit: mm).

Velocity Difference (mm/Year) | Velocity Uncertainty Difference (mm/Year) | ||||||||
---|---|---|---|---|---|---|---|---|---|

(−2, −1) | (−1, 0) | (0, 1) | (1, 2) | (0, 0.2) | (0.2, 0.4) | (0.4, 0.8) | (0.6, 0.8) | (0.8, 1.0) | |

N | 2.2% | 51.1% | 38.5% | 2.2% | 65.7% | 20.1% | 9.2% | 4.0% | 0.75% |

E | 1.3% | 45.5% | 42.5% | 2.4% | 63.0% | 24.0% | 8.6% | 2.4% | 1.0% |

(0, 0.4) | (0.4, 0.8) | (0.8, 1.2) | (1.2, 1.6) | (1.6, 2.0) | |||||

U | 3.0% | 32.6% | 52.8% | 2.2% | 44.7% | 28.8% | 19.8% | 4.9% | 1.1% |

**Table 2.**Annual and half-annual amplitude difference percentage table before and after hydrological loading correction.

Annual Amplitude Difference (mm) | Half-Annual Amplitude Difference (mm) | |||||||
---|---|---|---|---|---|---|---|---|

(−2, −1) | (−1, 0) | (0, 1) | (1, 2) | (−0.4, −0.2) | (−0.2, 0) | (0, 0.2) | (0.2, 0.4) | |

N | 0.3% | 51% | 48.7% | 0.1% | 1.3% | 50% | 48.3% | 0.4% |

E | 0% | 49.6% | 50.3% | 0.1% | 0.6% | 52.1% | 47.3% | 0% |

(−3, 0) | (0, 3) | (3, 6) | (6, 9) | (−2, −1) | (−1, 0) | (0, 1) | (1, 2) | |

U | 30% | 67.5% | 2.2% | 0.4% | 1.2% | 47.2% | 51.4% | 0.2% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

He, Y.; Nie, G.; Wu, S.; Li, H.
Analysis and Discussion on the Optimal Noise Model of Global GNSS Long-Term Coordinate Series Considering Hydrological Loading. *Remote Sens.* **2021**, *13*, 431.
https://doi.org/10.3390/rs13030431

**AMA Style**

He Y, Nie G, Wu S, Li H.
Analysis and Discussion on the Optimal Noise Model of Global GNSS Long-Term Coordinate Series Considering Hydrological Loading. *Remote Sensing*. 2021; 13(3):431.
https://doi.org/10.3390/rs13030431

**Chicago/Turabian Style**

He, Yuefan, Guigen Nie, Shuguang Wu, and Haiyang Li.
2021. "Analysis and Discussion on the Optimal Noise Model of Global GNSS Long-Term Coordinate Series Considering Hydrological Loading" *Remote Sensing* 13, no. 3: 431.
https://doi.org/10.3390/rs13030431