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Academic Editor: Fabrizio Lamberti

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

The Earth's rotation undergoes changes with the influence of geophysical factors, such as Earth's surface fluid mass redistribution of the atmosphere, ocean and hydrology. However, variations of Earth Rotation Parameters (ERP) are still not well understood, particularly the short-period variations (e.g., diurnal and semi-diurnal variations) and their causes. In this paper, the hourly time series of Earth Rotation Parameters are estimated using Global Positioning System (GPS), Global Navigation Satellite System (GLONASS), and combining GPS and GLONASS data collected from nearly 80 sites from 1 November 2012 to 10 April 2014. These new observations with combining different satellite systems can help to decorrelate orbit biases and ERP, which improve estimation of ERP. The high frequency variations of ERP are analyzed using a de-trending method. The maximum of total diurnal and semidiurnal variations are within one milli-arcseconds (mas) in Polar Motion (PM) and 0.5 milli-seconds (ms) in UT1-UTC. The semidiurnal and diurnal variations are mainly related to the ocean tides. Furthermore, the impacts of satellite orbit and time interval used to determinate ERP on the amplitudes of tidal terms are analyzed. We obtain some small terms that are not described in the ocean tide model of the IERS Conventions 2010, which may be caused by the strategies and models we used or the signal noises as well as artifacts. In addition, there are also small differences on the amplitudes between our results and IERS convention. This might be a result of other geophysical excitations, such as the high-frequency variations in atmospheric angular momentum (AAM) and hydrological angular momentum (HAM), which needs more detailed analysis with more geophysical data in the future.

The Earth Rotation Parameters (ERPs) are changing due to geophysical excitation (such as redistribution of geophysical fluid mass) as well as lunisolar gravitational torque, including the motion of the rotation axis (Polar motion, PM) and its change rate (Length of Day, LOD). In addition, some larger crustal activities may also affect Earth's rotation. Hopkin [

The International Earth Rotation and Reference System Service (IERS) estimates ERPs at a sub-millimeter precision. Using VLBI, SLR, Lunar Laser Ranging (LLR), GPS, and Doppler Orbitography by Radiopositioning Integrated on Satellite (DORIS). However, Most of the IERS ERP series, such as IERS C04 [

Up to now, about one hundred stations are available to track both GPS and GLONASS satellites simultaneously. The precision of ERP is related more to the distribution of the satellites than the number of stations [

First of all, these stations are core sites of International Terrestrial Reference Frame (ITRF) [

The uninterrupted and continuous tracking stations established by IGS allow us to accumulate a large number of observations from GPS and GLONASS to estimate ERP from a few hours of data. This is a great advantage when compared to other technologies, such as SLR or VLBI, which does not have continuous observations. In this paper, ERPs are estimated for every hour with GPS and GLONASS observations, individually and by a combined GPS and GLONASS method. The theory and methods to determinate ERP were introduced in details in the references [

All GNSS data analysis was executed using Bernese 5.0 software [

In the estimation of ERP, we divide a long interval (e.g., one day or 3 day arc) into several sub-interval of equal length (2 or 4 h). Then in any sub-interval [_{i}_{i}_{+1}], the ERP can be represented as following:
_{i}_{i}_{i}

According to the strategy described in Section 2.2, firstly, we estimate the daily ERP for one month and then compare them to IGS published values in order to assess the accuracy of our process strategy. The Root Mean Square (RMS) difference of PM and UT1-UTC is about 0.23 mas (0.31 mas) and 0.017 ms (0.027 ms) when compared our daily ERP from GPS (GLONASS) to the IGS values, respectively. These figures show a result with a good enough precision when considering that only about 80 sites are involved. Then ERPs are estimated with a frequency of 1 h for 526 days since 11 January 2012, but only the independent sets of ERPs are used here, that is to say the ERP are used every 2 h or 4 h. To give us a first indication of how larger is the high frequency variations, the ERP series are compared to IGS published values, too. This seems to be a strange comparison, because the ERP estimated with a sub-interval of 2 h from GPS, GLONASS and combined GPS + GLONASS contains many subdaily signals, however, the IGS published values do not contain the subdaily signal. Thus, to operate an appropriate and informative comparison, we firstly compute the daily average of the hourly ERPs for each day. Then the daily average series of ERPs are compared with the IGS values (at UTC 12:00:00). As we note, this comparison was operated only to give us a first indication of how larger is the high frequency variations. The statistic information of the differences between daily average results and IGS published values (at UTC 12:00:00) are shown in

The ERP series directly estimated from GPS and GLONASS contains both low and high frequency variations. Apart from the subdaily variations, a long period term is also included in the ERP series we acquired. To obtain the subdaily variations in ERPs, the long period term is removed in the ERPs generated from GPS and GLONASS. This can be accomplished by subtracting a prior trend which is also an ERP series but without high frequency variations [_{high_freq}_{trend}_{d}

It is important to note that the series of high frequency variations of ERP contains both sub-daily variations and estimation error or other noises. The error from the empirical model we used and the detrending method may also alias into the high frequency variations. The high frequency variations in PM and UT1-UTC are shown in

As the results in

To analyze the spectrum hidden in the series, the high frequency variations are converted to the frequency domain using a Fast Fourier Transform. Each high frequency variations series including X-pole, Y-pole and UT1-UTC can be expressed by a Fourier series independently as follows [_{k}_{i,xp}, _{i,xp}, _{i,yp}, _{i,yp}, _{i,ut} and _{i,ut} of X-pole, Y-pole and UT1-UTC of Fourier analysis are converted to amplitudes as follows:
_{i,amp} is the amplitude of signal at the frequency _{i,pro} and _{i,ret} is the amplitudes of prograde and retrograde PM, respectively. First of all, we estimated the amplitudes of prograde and retrograde PM and UT1-UTC from different observations with 2 h sub-interval. As an example,

To operate an appropriate comparison for these tidal terms, we firstly sum all the terms in the IERS tidal model and evaluate these whole tidal variations at the same epochs as our ERP series. The whole tidal variation series is regarded as a reference tidal variation series which then will be processed with the same method applied in our ERP series including the de-trending and FFT.

The differences of other terms whose period are not close to 12 h or 24 h are relatively small, but still exist. The most possible explanation is the different orbital period of GPS satellite (about 12 h) and the period of GLONASS (about 11 h 15 min). Although there is no algebraic relation between the orbital elements and tidal terms, the prograde tidal terms correspond to the translation of the orbital plane of each satellite. That is to say, the systematic changes of satellite in the orbit elements will lead to changes in the prograde tidal terms. For the GPS satellite, the period is about 12 h and the revolution is about 24 h in the inertial system, which is a systematic signal that propagates into those tidal terms with the same period. Furthermore, we can see that the terms of GNSS (combined GPS and GLONASS) at 24.058 h and 24.023 h is improved when compared to GPS only, because the difference between GPS and IERS is larger than that of GNSS. However, for those terms which are not so close to 12 h or 24 h (e.g., 26.859 h, 25.802 h), the combined result is dominated by GPS. In total, the standard deviation of the difference between GPS only and IERS 2010 is 18.33 μas. While the standard deviation of the difference between GNSS and IERS 2010 is 17.93 μas, which means the combined result is a little bit better than GPS only, overall. As a conclusion, the GPS satellite system has little impact on those tidal terms whose period is close to 24 h and 12 h. Thus, involving different satellite systems can reduce this impact in the theory, but our result is far to certify this adequately which need more investigation in the future.

In this manuscript, we also estimated the ERP with 4 h sub-interval, namely ERP at a temporal resolution of 2 h.

Furthermore, in order to analyze the impact of the length of sub-interval on the tidal amplitudes generated from the high frequency variations, the amplitudes of high frequency variations are estimated from both 2 h sub-interval and 4 h sub-interval. The amplitudes of prograde, retrograde PM and UT1-UTC generated from 2 h and 4 h sub-interval are compared. As we all know, the period terms (including diurnal and semidiurnal variations) generated from the high frequency variations are mainly the consequence of redistribution of geophysical fluid mass, especially the ocean tide. The tidal potential generated by moon and sun changes the ocean currents that cause the redistribution of earth fluid mass, which in return caused the variation in earth rotation. For instance, the period of 1.0024 days we obtain from the high frequency variations in ERP is one of the components of the ocean tides (which may correspond to _{1} in IERS tide model). As a comparison, the prograde PM from IERS convention 2010 is shown in

Some of the period of these terms may be slightly different from the IERS Conventions. The most reasonable explanation is these period terms are not separated correctly due to the length of data we use. For example, the semidiurnal term of λ2 (0.5092406d) and L2 (0.5079842d) from the IERS might merge into one term at the period of 0.5079 d in our result. This should be improved if more than years of GNSS data were processed. For instance, to separate the largest 30 tidal terms completely given in IERS Convention, at least three or more years of GNSS data are needed. In the

Apart from the time interval, there are various other factors that may impact on the estimation of ERP as well as the amplitudes of tidal terms. In this section, the influence of the prior information of ERP, ocean loading model, mapping function, and weighting function are discussed.

The changes in ERP will leads to changes in the high frequency variations. Thus, the impact on amplitudes is also analyzed.

In this paper, the hourly ERPs time series are determined using GPS, GLONASS, and combining GPS and GLONASS data collected from 1 November 2012 to 10 April 2014. The observations come from nearly 80 IGS sites with a globally uniform distribution. We assess the precision of our ERP results by comparing them to the IGS solution. Our findings indicate that the accuracy of ERP can be improved by combining GPS and GLONASS. In addition, the correlation between the orbit model and ERP will also be reduced by combing these two different navigation systems. The high frequency variations in ERP are further studied using a de-trending method based on smoothness priors operating, like a time-varying FIR high pass filter. Our results show that most of the high frequency variations are within 1 mas for PM and within 0.5 ms for UT1-UTC. Furthermore, the spectrum of high frequency ERP variations is analyzed using a Fast Fourier Transform. As a result, we obtained several period terms for both PM and UT1-UTC around about 8 h, 12 h and 24 h bands. Most of these terms are components of ocean tides. By comparing the amplitudes generated from different time intervals, the result shows that the amplitude will be compressed or reduced when estimated ERP with a bit longer time interval. The amplitudes generated from different satellite systems have also been discussed. The results show that different orbits do have some effects on the tidal terms as we show with some figures as evidence, but our results are far from being able to certify this adequately and how this impact exactly happens need more investigation in the future. In addition, there are also some small terms that occurred in the high frequency variations of ERP. This may be caused by the models and strategies that we used in the data processing as well as artifacts. Some differences in the amplitudes from IERS may be caused by other geophysical excitations, such as the thermally driven atmospheric and high-frequency hydrological variations, which will be analyzed in the future with more geophysical data.

This research is supported by the National Keystone Basic Research Program (MOST 973) (No. 2012CB72000), National “863 Project” of China (No. 2008AA12Z308), National Natural Science Foundation of China (No. 41374012, 11173050 and 11373059), Main Direction Project of Chinese Academy of Sciences (Grant no. KJCX2-EW-T03) and Shanghai Science and Technology Commission Project (No. 12DZ2273300). We appreciate for the data from IGS and IERS as well as the advice from the reviewers.

All authors contributed to the work presented in this paper. Erhu Wei and Shuanggen Jin were in charge of research and analysis. Lihua Wan designed the experiments and processed data, as well as wrote the draft. Erhu Wei and Shuanggen Jin revised the manuscript. Lihua Wan, Wenjie Liu, Yali Yang and Zhenghong Hu performed the experiments.

The authors declare no conflict of interest.

Distribution of GPS and GLONASS stations in this study. Red are the stations that track both GPS and GLONASS system. Blue are the station only track GPS system.

The principle of estimating hourly ERPs before (_{i+3} in the left are not equal to the ERP_{i+3} in the right.

Long term variation of PM in X coordinate form GPS with 2 h sub-interval. Upper left is the enlargement section of 23–27 January 2013. Red line represents the original ERP series and green line represents the trend which is considered as long term variation.

High frequency variations of ERP series generated from GPS with two hour sub-interval observations. (

Amplitudes of prograde and retrograde semidiurnal PM from 2 h sub-interval. Prograde (

Spectrum analysis results of high frequency variations in ERP from combined GPS and GLONASS. Spectrum analysis of X-pole (

Amplitudes of prograde diurnal and semidiurnal PM. (

Difference between our daily averages of ERPs and IGS values.

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GPS only | 1.5465 | −1.7256 | 0.4079 | 1.6519 | −1.2389 | 0.4609 | 0.1071 | −0.4033 | 0.0842 |

GLONASS only | 2.2049 | −1.8712 | 0.6763 | 2.4367 | −2.3954 | 0.6834 | 0.0953 | −0.5087 | 0.1021 |

GPS/GLONASS | 1.5632 | −1.7111 | 0.3832 | 1.4584 | −1.1252 | 0.4301 | 0.1001 | −0.4140 | 0.0833 |

Statistics result of high frequency variations of ERP from different observations.

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2 h | GPS only | 1.5293 | −1.5849 | 0.4597 | 1.7599 | −1.7781 | 0.4112 | 0.6151 | −0.6398 | 0.2379 |

GLONASS only | 2.2588 | −2.8308 | 0.5609 | 2.0416 | −2.1159 | 0.5404 | 0.6120 | −0.6541 | 0.2409 | |

GPS/GLONASS | 1.5637 | −1.4488 | 0.4423 | 1.4535 | −1.4054 | 0.3891 | 0.5933 | −0.6253 | 0.2348 | |

3 days arc | 1.4471 | −1.7828 | 0.4477 | 1.5848 | −1.1956 | 0.3630 | 0.5893 | −0.5846 | 0.2329 | |

4 h | GPS only | 1.7325 | −1.763 | 0.4673 | 1.5932 | −1.5914 | 0.3857 | 0.6031 | −0.6091 | 0.2385 |

GLONASS only | 2.2294 | −2.5976 | 0.5535 | 2.2496 | −1.7075 | 0.5026 | 0.6031 | −0.6685 | 0.2401 | |

GPS/GLONASS | 1.4408 | −1.3897 | 0.4504 | 1.3427 | −1.2922 | 0.3680 | 0.5890 | −0.5966 | 0.2340 | |

3 days arc | 1.4012 | −1.4215 | 0.4374 | 1.3648 | −1.1626 | 0.3545 | 0.5788 | −0.5921 | 0.2331 |

Coefficients of sine and cosine of PM and UT1-UTC form different systems.

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Coefficient generated from GPS observations | |||||||

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0.03723 | 26.859 | 1.467 | −15.679 | 14.017 | 1.073 | −3.50 | 1.69 |

0.03876 | 25.802 | 72.239 | 40.790 | −11.676 | 79.050 | 4.86 | −10.96 |

0.04156 | 24.058 | −15.409 | −100.467 | −137.990 | −40.446 | 73.33 | −18.50 |

0.04163 | 24.023 | −10.194 | −32.503 | 14.023 | 22.829 | 110.24 | 34.56 |

0.04181 | 23.918 | −21.341 | −118.263 | 34.166 | −60.994 | −38.18 | 30.43 |

0.04187 | 23.883 | 12.936 | 2.107 | −10.607 | −0.289 | 4.38 | 4.84 |

0.07898 | 12.662 | 26.751 | −28.465 | −23.818 | 1.166 | 2.75 | −1.08 |

0.08051 | 12.421 | −21.771 | 247.721 | 160.688 | −52.034 | −5.65 | 14.07 |

0.08331 | 12.003 | −115.643 | 98.433 | 52.203 | −39.150 | −69.59 | 17.48 |

0.08356 | 11.967 | −17.791 | −7.095 | 17.330 | 22.052 | −8.64 | −1.32 |

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Coefficient generated from GLONASS observations | |||||||

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0.03723 | 26.859 | 0.376 | −10.878 | 8.841 | −15.765 | −3.66 | 1.58 |

0.03876 | 25.802 | 80.968 | 38.645 | −6.116 | 75.489 | 5.05 | −11.05 |

0.04156 | 24.058 | −48.380 | −89.783 | −168.672 | −20.558 | 70.73 | −13.14 |

0.04163 | 24.023 | 6.600 | 4.006 | −6.790 | 7.666 | 102.72 | 33.18 |

0.04181 | 23.918 | −61.392 | −105.817 | 28.908 | −26.770 | −37.08 | 26.60 |

0.04187 | 23.883 | −15.947 | 23.678 | −12.532 | 26.309 | 4.39 | 4.73 |

0.07898 | 12.662 | 26.652 | −27.803 | −29.961 | −4.131 | 2.66 | −1.28 |

0.08051 | 12.421 | −33.054 | 252.585 | 171.502 | −69.196 | −5.21 | 14.34 |

0.08331 | 12.003 | −121.566 | 57.062 | 67.587 | −30.709 | −46.14 | 11.70 |

0.08356 | 11.967 | −19.862 | −47.684 | 1.254 | 12.439 | −8.23 | −0.25 |

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Coefficient generated from combined GPS/GLONASS observations | |||||||

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0.03723 | 26.859 | 1.300 | −14.456 | 11.975 | −0.178 | −3.46 | 1.61 |

0.03876 | 25.802 | 75.395 | 36.629 | −13.648 | 80.423 | 4.95 | −11.00 |

0.04156 | 24.058 | −24.198 | −100.922 | −150.297 | −40.290 | 72.34 | −16.68 |

0.04163 | 24.023 | −4.638 | −19.448 | 13.318 | 17.320 | 107.23 | 33.86 |

0.04181 | 23.918 | −31.986 | −113.014 | 28.017 | −50.420 | −38.00 | 28.82 |

0.04187 | 23.883 | 5.503 | 10.046 | −13.297 | 9.722 | 4.12 | 4.82 |

0.07898 | 12.662 | 27.813 | −26.343 | −27.294 | 1.034 | 2.65 | −1.20 |

0.08051 | 12.421 | −24.588 | 248.340 | 162.571 | −56.315 | −5.57 | 14.13 |

0.08331 | 12.003 | −126.292 | 84.260 | 50.162 | −32.255 | −59.07 | 14.64 |

0.08356 | 11.967 | −22.706 | −19.336 | 14.093 | 15.941 | −7.85 | −1.33 |

Amplitudes of prograde and retrograde PM from different models. The unit is μas.

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0.03723 | 26.859 | 14.902 | 12.507 | 13.228 | 17.975 |

0.03876 | 25.802 | 80.064 | 81.367 | 81.865 | 79.081 |

0.04156 | 24.058 | 33.644 | 52.383 | 40.610 | 48.038 |

0.04163 | 24.023 | 24.105 | 8.945 | 17.568 | 3.416 |

0.04181 | 23.918 | 86.622 | 80.504 | 81.671 | 94.152 |

0.04187 | 23.883 | 8.967 | 18.832 | 13.935 | 15.268 |

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0.07898 | 12.662 | 14.150 | 11.312 | 14.5431 | 11.388 |

29.104 | 32.727 | 29.975 | 29.180 | ||

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0.08051 | 12.421 | 57.057 | 65.249 | 58.952 | 58.288 |

204.765 | 212.812 | 206.067 | 206.179 | ||

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0.08331 | 12.003 | 80.775 | 76.319 | 81.068 | 19.649 |

84.473 | 77.124 | 82.025 | 89.557 | ||

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0.08356 | 11.967 | 12.397 | 24.749 | 17.053 | 4.382 |

20.568 | 28.280 | 19.501 | 20.549 |

Tidal Amplitudes of PM. ↑ represents increased, ↓ represents decreased.

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Diurnal terms in PM | ||||||||||

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0.03723 | 1.1191 | 15.60 | 4.53 | 15.88 ↑ | 4.00 ↓ | |||||

0.03741 | 1.1136 | 16.47 | 2.25 | 18.79 ↑ | 2.30 ↑ | |||||

0.03876 | 1.0751 | 81.33 | 10.44 | 80.08 ↓ | 10.55 ↑ | |||||

0.04028 | 1.0343 | 14.61 | 7.85 | 13.49 ↓ | 7.52 ↓ | |||||

0.04053 | 1.0281 | 7.76 | 10.46 | 7.83 ↑ | 10.64 ↑ | |||||

0.04071 | 1.0235 | 5.52 | 2.78 | 3.44 ↓ | 3.30 ↑ | |||||

0.04144 | 1.0054 | 29.28 | 17.09 | 30.55 ↑ | 17.70 ↑ | |||||

0.04150 | 1.0039 | 18.27 | 24.23 | 17.68 ↓ | 23.56 ↓ | |||||

0.04156 | 1.0024 | 45.89 | 75.46 | 50.18 ↑ | 75.57 ↑ | |||||

0.04163 | 1.0010 | 24.98 | 115.94 | 26.41 ↑ | 115.59 ↓ | |||||

0.04169 | 0.9995 | 13.50 | 181.02 | 18.81 ↑ | 181.14 ↑ | |||||

0.04175 | 0.9981 | 73.76 | 35.97 | 73.85 ↑ | 36.20 ↑ | |||||

0.04181 | 0.9966 | 81.77 | 53.97 | 80.60 ↓ | 53.40 ↓ | |||||

0.04187 | 0.9951 | 18.55 | 6.53 | 20.36 ↑ | 6.87 ↑ | |||||

0.04205 | 0.9908 | 13.70 | 8.56 | 13.83 ↑ | 8.09 ↓ | |||||

0.04303 | 0.9683 | 4.69 | 6.61 | 5.69 ↑ | 6.18 ↓ | |||||

0.04474 | 0.9313 | 7.04 | 18.02 | 7.54 ↑ | 18.47 ↑ | |||||

0.04633 | 0.8994 | 2.73 | 1.70 | 3.22 ↑ | 1.15 ↓ | |||||

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Semidiurnal terms in PM | ||||||||||

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0.07898 | 0.5276 | 13.93 | 28.41 | 3.17 | 13.73 ↓ | 29.61 ↑ | 3.09 ↓ | |||

0.08051 | 0.5176 | 60.15 | 204.04 | 15.87 | 60.67 ↑ | 203.24 ↓ | 15.95 ↑ | |||

0.08203 | 0.5079 | 6.79 | 10.06 | 3.89 | 7.42 ↑ | 9.70 ↓ | 3.64 ↓ | |||

0.08319 | 0.5009 | 18.88 | 72.00 | 15.33 | 21.27 ↑ | 71.42 ↓ | 15.27 ↓ | |||

0.08325 | 0.5005 | 10.63 | 52.81 | 10.59 | 10.77 ↑ | 51.56 ↓ | 10.88 ↑ | |||

0.08331 | 0.5001 | 77.81 | 89.58 | 61.38 | 83.89 ↑ | 88.47 ↓ | 61.77 ↑ | |||

0.08344 | 0.4994 | 43.52 | 24.42 | 26.38 | 46.13 ↑ | 21.57 ↓ | 26.50 ↑ | |||

0.08350 | 0.4990 | 41.44 | 3.21 | 8.51 | 41.12 ↓ | 3.54 ↑ | 8.18 ↓ | |||

0.08356 | 0.4987 | 16.48 | 21.79 | 8.13 | 16.41 ↓ | 24.04 ↑ | 8.43 ↑ |

Impact of different GNSS models on ERP. (

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(a) | 0.240 | 0.062 | 0.030 | 0.096 | 0.052 | 0.018 |

(b) | 0.470 | 0.173 | 0.830 | 0.280 | 0.040 | 0.016 |

(c) | 1.360 | 0.655 | 2.120 | 0.842 | 0.367 | 0.114 |

(d) | 0.230 | 0.070 | 1.090 | 0.201 | 0.020 | 0.007 |

Impact of different GNSS models on tidal amplitudes. (

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(a) | 3.17 | 1.73 | 7.41 | 4.02 | 0.45 | 0.27 |

(b) | 9.14 | 4.56 | 7.81 | 5.94 | 0.92 | 0.55 |

(c) | 19.93 | 10.21 | 59.63 | 33.28 | 13.83 | 8.77 |

(d) | 4.33 | 2.07 | 13.22 | 6.02 | 0.76 | 0.43 |