^{1}

^{2}

^{*}

^{3}

^{1}

^{2}

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

The site displacement due to ocean tidal loading is regarded as one of the largest uncertainties in precise geodetic positioning measurements, among which the effect of minor ocean tides (MOT), except for the 11 main tidal constituents, are sometimes neglected in routine precise global positioning system (GPS) data processing. We find that MOT can cause large vertical loading displacements with peak-to-peak variations reaching more than 8 mm at coastal/island stations. The impact of MOT on the 24-hour GPS solution is slightly larger than the magnitude of MOT loading itself, with peak-to-peak displacement variation at about 10 mm for the horizontal and 30 mm for the vertical components. We also find that the vertical velocity of all the selected stations in the Southwest Pacific was reduced by more than 10% after considering the MOT effect, while stations with weighted root mean square reduced data account for 62%, 59%, and 36% for the up, east, and north components respectively, in particular for most coastal/island stations. Furthermore, MOT correction could significantly reduce the annual signal of the global stacked east component, the near fortnightly and the long-term periodic signals in the up component. The power of some anomalous harmonics of 1.04 cycle per year is also decreased to some extent. These results further proved the benefits of MOT correction in precise GPS data processing.

Time variable deformation of the Earth caused by ocean tides could reach up to 100 mm at some special coast regions [

The OTL at a particular location on the Earth caused by a given tidal harmonic is computed by integrating the tide height with Green's function over the whole ocean area [

Since early February 2006, Agnew has posted a fortran routine hardisp.f by considering a total of 141 constituent tides from the 11 main tides, which then became a conventional implementation in November to calculate local site displacement due to OTL, and Hugentobler found that completely neglecting the other ocean tides and nodal modulations with only the 11 main tides may lead to errors of up to 5 mm weighted root mean square (WRMS) at high latitudes using the GOT00.2 ocean model [

Thanks to the efforts of many OTL researchers, hardisp.f is being continuously updated online. Until now, it includes a total of 342 constituent tides whose amplitudes and phases are found by spline interpolation of the tidal admittances based on the 11 main tides, and has been implemented in most of the ACs' analysis strategy [

In the spectral domain, it is well known that there is strong correlation between stations' seasonal variation, especially the vertical annual displacement and the surface displacement induced by redistribution of environmental loads [

Finally, Ray

In this paper, we first determine the magnitude and spatial distribution of global IGS station's displacement caused by MOT. The OTL modeling method including the MOT correction is then implemented in GAMIT by expanding the 11 main ocean tides into 342 constituents. Based on both the original and the modified GAMIT software, the GPS data of 109 globally distributed IGS stations spanning from June, 1998 to December, 2010 has been reprocessed with state of the art models according to IERS Conventions 2010. Finally, quantitative analyses have been done on two sets of GPS coordinate time series in both time and frequency domains to evaluate the contributions of MOT to global GPS coordinate time series. Results of this paper may provide numerical support to the recommended data processing strategy in the IERS Conventions for crustal movement and interpretation of geophysical signal, as well as the target accuracy of ITRF to achieve 1 mm in position and 0.1 mm/a in velocity [

In practice, the 3-D site displacement due to OTL is calculated by:
_{cj}_{cj}_{j}

The amplitudes and phases for other tidal component can be calculated from the above 11 main tides by a variety of approximation methods. For instance, if one wishes to correct for the modulating effect of the 18.6-year lunar node, then:
_{k}_{k}

Here, we use the conventional routine hardisp.f recommend by IERS Conventions 2010 to model the MOT effects [

To investigate the impact of MOT on the position of global IGS stations, we implement the above recommended conventional OTL correction method in GAMIT 10.4 by substituting the embedded OTL module with hardisp.f. Then a contrast experiment has been designed by reprocessing the GPS data of 109 evenly distributed global IGS stations using the original (Experiment A) and improved GAMIT (Experiment B) [

During the data reprocessing, satellite orbits, earth orientation parameters (EOPs), site coordinates together with the tropospheric delay and the horizontal gradient parameters are resolved simultaneously. Loose constraints are implemented on the stations, among which the constraints of IGS core stations are set as 5 cm, and non-core stations are 1 dm. Where possible, ambiguities are fixed to integers [

Using the method outlined in Section 2.1, we obtained the loading time series caused by MOT for the selected 109 stations.

To have a better understanding of the global MOT effect,

In Section 3.1, we give the magnitude of stations' displacement caused by MOT. Through reprocessing the GPS data by using the method outlined in Section 2.2, we obtain the MOT effect on the 24-hour daily GPS solutions.

We observe that the magnitude of MOT on the 24-hour GPS solution is slightly larger than that of the MOT loading time series. The peak-to-peak daily displacement caused by MOT for inland station ALIC reaches 5 mm for the N, E component, and 15 mm for the U component (

It influences a lot on many coastal/island stations, for example, station ASC1 (Ascension Island), BAHR (Manama, Bahrain),

Here, we only estimate the velocity of IGS stations with continuous observation span of more than three years, that is 156 GPS weeks, so as to ensure the reliability of the statistics. The basic equation can be written as:
_{11}_{342}

From

Arnadottir

To investigate the impact of MOT on the non-linear characteristics of global IGS stations, we calculate the weighted root mean square (WRMS) of the coordinate time series for the 109 stations before and after considering the MOT effect. Here, we also only estimate the WRMS of IGS stations with continuous observation span of more than three years.

From

Until now, there is still big discrepancy between global IGS stations' non-linear position time series and the environmental loading induced displacement [

From

From

From

Another interesting result from

The effects of MOT are sometimes neglected in routine precise long-baseline GPS data processing. In this paper, we first determine the magnitude and spatial distribution of global IGS station's loading displacement caused by MOT. We find that MOT could only cause small variations in the horizontal loading displacement globally, with most of the STD being smaller than 1 mm. The vertical loading displacement caused by MOT is quite large, and becomes smaller with the increasing distance between station and the sea. This is obviously due to the characteristics of ocean tides. The peak-to-peak MOT induced loading displacement variations reaches more than 8 mm at the coastal/island stations, while most of the height variations for inland station are smaller than 1.5 mm, in particular for those in the Eurasia plate, which have almost the same magnitude as that of the E component.

Secondly, the OTL modeling method including the MOT correction is implemented in GAMIT. Through reprocessing the GPS data of 109 globally distributed IGS stations based on both the original and the modified GAMIT software, we find that the impact of MOT on the 24-hour GPS solution is slightly larger than the magnitude of MOT loading time series itself, and it also exhibits quite a different spatial pattern. There is no strong correlation between station's MOT induced 3-D displacement and its distance away from the sea. It influences a lot on many coastal/island stations (e.g., REYK) and also on some inland stations (e.g., WTZR), among which the STD and the absolute maximum of the GPS height caused by MOT reaching more than 5 mm and 20 mm respectively, with peak-to-peak displacement change at about 10 mm for the horizontal component and 30 mm for the vertical components. For stations located in North America and Eurasia plates, however, MOT only has a small impact on the GPS time series, with most of the 3-D STD smaller than 1 mm. We think that this may due to the interaction between MOT and the other GPS error sources at each epoch and in different places.

We then study the impact of MOT on the long-term velocity of global IGS stations. Our results show that the vertical velocity change caused by MOT is quite big, and also decrease with stations' distance away from the sea. 42% of the selected stations, including most coastal/island stations' vertical velocity change are larger than 10%, and the maximum velocity variation rate reaches more than 50% (e.g., station ASC1, BAHR). Besides, considering the MOT effect could reduce the vertical velocity of all the selected stations in the Southwest Pacific by more than 10%. Thus, we conclude that MOT may also be a potential candidate for the vertical motion of global IGS stations, in particular for those along the coast or in ocean areas, where MOT should be better corrected during high-precision GPS data processing so as to obtain more reliable vertical motion for geodynamic studies, such as PGR, sea level change,

Moreover, the impact of MOT on the non-linear characteristics of global IGS stations is discussed. We find that in general MOT would only introduce small WRMS variations in the global IGS stations, even for the coastal/island stations, where the maximum variation rate is smaller than 5%. The U component exhibits the smallest impact, the N component ranks the second, while the E component ranks the third, with stations whose WRMS variation rates smaller than 1% account for 84%, 70%, and 62% of the selected stations. 62%, 59%, and 36% of the stations' WRMS reduced for the U, E and N components respectively after considering the MOT effect, among which the maximum reduction with magnitude of 1%∼5% gathered mostly along coasts or on the island. Nevertheless, implementing MOT correction could also increase the WRMS for some global IGS stations, in particular for the N component (e.g., station NOT1). Further work still needs to be done on whether the MOT modeling at the observation level of GPS data processing would narrow the gap between global IGS stations' non-linear position and the environmental loading induced displacement time series.

Finally, the spectral characteristics of global IGS stations that caused by MOT is analyzed. Our results show that MOT could only cause small periodic difference, although the impact on the coastal stations is slightly bigger than that on the inland stations, and the magnitude of the periodic difference in the vertical component is about 10-times larger than the horizontal components. We find that MOT correction could significantly reduce the annual signal of the global stacked E component, but that of the U and N components exhibit a significant increase, for example, the annual signal of the N component for station ARTU increased by 23%. Furthermore, the near fortnightly periodic signal and the long-term periodic signal with frequency equal to or smaller than 0.5 cpy in the U component show a significant decrease after correcting the MOT effect. This may indicate a more precise data processing strategy for reducing some of the high frequency periodic signals in the GPS height time series, and also support the recent finding that the unmodeled diurnal and sub-diurnal periodic signal would indeed propagate into spurious long-term signals. Finally, we find that MOT could reduce the power of most anomalous harmonics of 1.04

We thank the SOPAC for making the global GPS data freely available. We thank R. W. King and Elizabeth J. Petrie for providing advice and assistance in GPS data processing. We thank the MIT group in the US for providing the GAMIT/GLOBK software available. We are also grateful for the code of ocean tidal loading provided by D. Agnew and the CATS software provided by S. D. P. Williams. Figures in this paper are plotted with the GMT and MATLAB software. This research is supported by the Changjiang Scholars program and the National Natural Science Foundation of China (41374033).

Zhao Li and Weiping Jiang performed the analyses and prepared the paper. Wenwu Ding provided the technical guidance to the implementation of hardisp.f in GAMIT software. Liansheng Deng contributed to the discussion on the anomalous harmonics. Lifeng Peng helped the plotting of the figures.

The authors declare no conflict of interest.

_{2}and S

_{2}ocean tide models for the North Atlantic Ocean and adjacent seas from ERS-1 altimetry

_{2}ocean tide loading wave in Alaska: Vertical and horizontal displacements, modeled and observed

Selected Global IGS reference station distribution. Black texts indicate the name of stations that mentioned in this paper.

Averaged weekly MOT loading time series for station POTS (

Spatial distribution of the STD (

GPS coordinate time series for station ALIC (

Spatial distribution of the STD (

Spatial distribution of the MOT induced long-term velocity variation rate. From top to bottom are the U, E, and N components. Unit of the variation rate is in percentage (%). The black and white dots in the figure indicate that the velocity variation rate for the station is smaller than the minimum and larger than the maximum value on the scale.

Percentage of stations with MOT induced vertical velocity variation rate inside different ranges. Different color indicates different velocity variation rate range. Unit of variation rate is in percent (%).

Spatial distribution of the MOT induced WRMS variation rate. Black dot in the figure indicates that the WRMS variation rate for the station is smaller than the minimum value on the scale. From top to bottom are the U, E, and N components. Unit of the variation rate is in percentage (%).

Percentage of stations with the WRMS variation rate inside certain range. Different color indicates different WRMS variation rate in unit of %. (

PSD results for the U, N, and E components of station BAHR (top panels) and ARTU (bottom panels) before and after MOT correction. Vertical black and green dash lines represent harmonics of 1

Top: Global stacking PSD results for the filtered U, N, and E component before and after MOT correction. Bottom: PSD difference caused by MOT for each corresponding component. The vertical black and green dash lines in the figure have the same meaning as that in

Expanded view of