# Analysis of GNSS Displacements in Europe and Their Comparison with Hydrological Loading Models

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. CNES-CLS Multi-GNSS Orbits and Clock Products

#### 2.2. Time Series Analysis

#### 2.2.1. Selection of Stations

#### 2.2.2. Parameter Estimation

#### 2.2.3. Interannual Polynomial Model

#### 2.3. Hydrological Loading Computations

## 3. Results

#### 3.1. Tectonic Velocity

#### 3.2. Annual Signal

#### 3.3. Comparison with Hydrological Models and GRACE

#### 3.4. Principal Component Analysis of the Interannual Signal

#### 3.5. Frequency Content and Interannual Variations

## 4. Discussion

#### 4.1. Interannual Signal in GNSS Time Series

#### 4.2. Importance of Draconitic Adjustment

#### 4.3. Model Phase Advance over GNSS Seasonal Signal

#### 4.4. Common Mode Estimation in GNSS

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Number of satellites and (

**b**) number of stations used in MG3 product computation as a function of time.

**Figure 3.**(

**a**) GNSS station network and some characteristics in terms of (

**b**) station time span, (

**c**) station availability over time, and (

**d**) daily solution in each station.

**Figure 4.**Velocity field of cats_d in (

**a**) horizontal and (

**b**) vertical direction, along with ICE-6G_D GIA model of [68]. The yellow dots in (

**a**) are stations for which the horizontal velocity is lower than $1\mathrm{mm}/\mathrm{y}$.

**Figure 5.**(

**a**,

**b**) Difference in velocity field between cats_d and cats. The panel (

**c**) represents the number of stations where the vertical velocity was, respectively, overestimated (red) or underestimated (blue) by cats_d compared to cats. (

**d**–

**f**) are the same for the difference between cats_d and tiasd, and (

**g**–

**i**), between cats_d and the EPOS solution.

**Figure 6.**Uncertainty of (

**a**–

**c**) horizontal and (

**d**–

**f**) vertical velocity field for cats_d, cats, and tiasd.

**Figure 7.**(

**a**) Phase and (

**b**) amplitude of cats_d along with their respective uncertainties (

**c**,

**d**) estimated using CATS software. The colour bar of the phase map indicates the month of maximum displacement towards the up direction.

**Figure 8.**(

**a**–

**d**) Difference in annual solar cycle between cats_d and cats. Panels (

**b**,

**d**) represent, respectively, the number of stations where cats is in phase advance (blue) or delay (red) with respect to cats_d and the number of stations with larger (blue) or smaller (red) amplitude than cats_d. (

**e**–

**h**) are the same for the difference between cats_d and tiasd.

**Figure 9.**Differences in phase and amplitude of annual solar cycle and the associated histograms as described in Figure 8 between cats_d and GLDAS2.1 (

**a**–

**d**), MERRA2 (

**e**–

**h**), and GRACE (

**i**–

**l**).

**Figure 10.**PCA (EOFs and associated PCs) of GNSS and the three loading models’ (including GRACE) residual time series between 2010 and 2020, where the trend and the seasonal signal have been removed: (

**a**–

**d**) first component, (

**e**–

**h**) second component, and (

**i**–

**l**) third component. The percentage of the total variance corresponding to each mode is given in each plot. The PCs are scaled to unit variance and plotted for the period 2010 to 2020, while the corresponding EOFs are given in terms of the correlation between the initial time series and the PCs.

**Figure 11.**Lomb–Scargle periodograms for the three loading models and the detrended MG3 GNSS solution. The periodograms of all stations were stacked and then divided by the number of stations in order to obtain mean periodograms for each solution. The vertical black lines indicate the solar cycle harmonics, and the vertical green lines indicate the draconitic harmonics.

**Figure 12.**Coefficient of correlation between annual solar wave and draconitic first harmonic for cosine and sine terms of tiasd as a function of the length of the time series.

MG3 Products | |
---|---|

Gravity field | EIGEN-GRGS.RL04.MEAN-FIELD [47] |

Ocean tides (gravity) | FES2014b (Finite Element Solution) [48] |

Planet ephemerides | de421bdlf.ad [49] |

Relativistic acceleration | Schwarzschild and geodetic precession and Lense–Thirring |

Antex | IGSR3.atx [50] |

Mean pole (C21/S21) | IERS conventions (from geopotential model) [51] |

Subdaily EOP model | [52] |

Atmospheric tides (S1/S2) | [53] |

Ocean tide loading | FES2014b [48] |

Centre of mass correction | FES2014b [48] |

Solid tides (station) | IERS conventions [51] |

Reference frame | IGS_R3 [IGSMAIL-8026] |

Galileo ponderation | $3.5$ cm/ 1 m |

GPS ponderation | $3.5$ mm/ 60 cm |

GLONASS ponderation | $3.5$ cm/ 2 m |

cats_d | cats | tiasd | |
---|---|---|---|

Draconitic frequencies adjustment | Yes | No | Yes |

Interannual variation | – | – | polynomials |

Stochastic model | WH + PL noise | WH + PL noise | WH noise |

**Table 3.**PL noise mean spectral index for the each of the two CATS-estimated solutions with or without WH noise estimation along PL noise.

East | North | Up | |
---|---|---|---|

cats_d (WH + PL) | $-0.89$ | $-1.03$ | $-0.77$ |

cats_d (PL) | $-0.69$ | $-0.84$ | $-0.63$ |

cats (PL) | $-0.86$ | $-0.99$ | $-0.75$ |

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**MDPI and ACS Style**

Michel, A.; Santamaría-Gómez, A.; Boy, J.-P.; Perosanz, F.; Loyer, S. Analysis of GNSS Displacements in Europe and Their Comparison with Hydrological Loading Models. *Remote Sens.* **2021**, *13*, 4523.
https://doi.org/10.3390/rs13224523

**AMA Style**

Michel A, Santamaría-Gómez A, Boy J-P, Perosanz F, Loyer S. Analysis of GNSS Displacements in Europe and Their Comparison with Hydrological Loading Models. *Remote Sensing*. 2021; 13(22):4523.
https://doi.org/10.3390/rs13224523

**Chicago/Turabian Style**

Michel, Alexandre, Alvaro Santamaría-Gómez, Jean-Paul Boy, Félix Perosanz, and Sylvain Loyer. 2021. "Analysis of GNSS Displacements in Europe and Their Comparison with Hydrological Loading Models" *Remote Sensing* 13, no. 22: 4523.
https://doi.org/10.3390/rs13224523