‟ M2 constituent of ocean tide loading displacements from VLBI CONT14 hourly sessions

: Several studies prove that ocean tide loading (OTL) displacements can be observed with 5 space geodetic techniques. In this study, the amplitudes and Greenwich phase lags for each 6 coordinate component, i.e., radial, west, and south of the principal lunar semidiurnal tide, M 2 of OTL 7 displacements were estimated at the very long baseline interferometry (VLBI) sites of the 15 days 8 long continuous VLBI campaign, CONT14, carried out by the International VLBI Service for Geodesy 9 and Astrometry (IVS). In the estimation of the amplitudes and Greenwich phase lags of the M 2 tidal 10 constituent, hourly VLBI station coordinate time series were used as observations derived through 11 analyzing 1 hour VLBI sessions of the CONT14 campaign. In the analysis of hourly sessions of the 12 CONT14 campaign, to derive accurate hourly station coordinates, troposphere delays estimated from 13 daily sessions were reduced from the observations a priori to the analysis. The estimated amplitudes 14 and Greenwich phase lags of the M 2 constituent of OTL displacements were compared with the 15 predictions the state-of-the-art ocean tide models, among others, FES2012 (Lyard et al. 2006, Carrère 16 et al. 2012), FES2014 (Carrère et al. 2016) and TPXO8 (Egbert and Erofeeva 2002, Egbert et al. 2010). 17 Both the amplitudes and the phases between CONT14 estimates and ocean tide models agree well 18 for the M 2 tide at all the sites and in most of the coordinate components. The RMS misfits of the M 2 19 tide of OTL displacements in all coordinate components between CONT14 and ocean tide models 20 over coastal sites are found about two times larger than those of inland sites. This result confirms the 21 modeling insufficiencies in shallow waters of ocean tide models which cause an accuracy restriction 22 of OTL displacement predictions around coastal regions.


Introduction 26
The seafloor pressure variations due to the ocean tide loading (OTL) cause position and tidal 27 frequency-dependent harmonic displacements on the Earth crust, the so-called OTL displacements. 28 OTL displacements can be predicted by convolution software, e.g., OLFG/OLMPP (Scherneck 1991), 29 SPOTL (Agnew 1996), NLOADF (Agnew 1997 The geodetic VLBI observatories (stations) contributing to the CONT14 campaign are listed in Table 1. 164 The 17 VLBI stations are separated into inland (7) stations and coastal (10) stations, depending on 165 whether the station is located within 150 km from the coastline or not (Table 1). Although the VLBI 166 stations at Zelenchukskaya (Russia) and Matera (Italy) are located within 150 km from the coastline, 167 they are considered as inland stations in this paper due to the very low tidal amplitudes at the Black 168 Sea and the Mediterranean Sea. 169

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In contrast to 24-hour session analysis, the main disadvantage of analyzing the observations of 174 1 hour VLBI sessions is that reliable station positions cannot be estimated due to the high correlation 175 between troposphere delays and station positions (e.g. Rothacher and Beutler 1998, 176 Teke et al. 2013). To show the amount of shared variances (degree of linear relationship) between 177 the ZWD and radial coordinates estimated simultaneously from 1 hour sessions of the CONT14 178 campaign, ZWD from 24 hour sessions were subtracted from those of 1 hour sessions, 179 ΔZWD1H-24H=ZWD1H-ZWD24H. Then, Pearson correlation coefficients were calculated between 180 ΔZWD1H-24H and the radial coordinates of the stations for each 1 hour session. All negative 181 correlations between ZWD1H-ZWD24H and radial estimates from 1 hour session analysis at the VLBI 182 stations of CONT14 are found between -0.74 and -0.93. The p-values of these correlations are below 183 0.05 indicating that they are all statistically significant. It is inferred that, as a rule of thumb, 1 cm 184 ZWD1H variation propagates into the radial positions approximately 1.5 to 2.5 cm when 1 hour 185 session is analyzed as well as these two parameters are simultaneously estimated (see e.g. correlation between ZWD1H -ZWD24H and radial positions is found as -0.80 at this station.

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Consequently, the classical Gauss-Markov least squares adjustment cannot decorrelate these two 197 parameters when they are estimated simultaneously in sub-daily e.g. 1 hour intervals. Thus,198 troposphere delays and antenna coordinates propagate into each other and resulting in unreliable 199 estimates. For example, the cyan and grey lines in Figure  To overcome this restriction, external troposphere slant delays L estimated from daily (24 hour) 210 sessions were reduced from the observations of hourly sessions a priori to the adjustment and 211 residual troposphere delays were not estimated. The troposphere delay model from Davis  used. For all solutions of hourly sessions, the ZHD were calculated at observation epochs with total 221 surface pressure values measured at the VLBI sites (Saastamoinen 1972, Saastamoinen 1973, 222 Davis et al. 1985. ZWD were estimated at 20-minute intervals with relative loose constraints as 223 1.5 cm after 20 minutes from 24-hour sessions. Note that in the analysis of these 24-hour sessions 224 FES2004 ocean tide model (Lyard et al. 2006) displacements are introduced to the a priori station 225 coordinates (solution: Case 1, see Table 3 in Section 4). Then, these ZWD estimated from 24 hour 226 VLBI sessions were linearly interpolated to observation epochs of hourly sessions. ZHD and ZWD 227 were mapped with the Vienna Mapping Functions 1 (VMF1, Böhm et al. 2006) to get the slant 228 hydrostatic and slant wet delays. Similar to ZWD, troposphere total north and east gradients at 229 1 hour intervals with relative loose constraints as 1 mm after 1 hour estimated from 24 hour VLBI 230 sessions were linearly interpolated to the observation epochs of hourly sessions. Then, azimuthally 231 asymmetric troposphere delays were calculated through mapping these horizontal total north and 232 east gradients to slant direction using the third term of Equation (1) where the gradient mapping 233 function by Chen and Herring (1997) was used. Finally, troposphere slant delays L from each VLBI 234 observation (delay) of the hourly sessions were subtracted a priori to the parameter estimation.   Table 2. 348  Hartebeesthoek is that the amplitudes of these components are small. 360 The best agreement of the estimated radial amplitudes of M2 tide with respect to those of ocean tide 361 models were found at the sites Onsala, Badary, Wettzell, Zelenchukskaya, Matera, Tsukuba, and 362 Yarragade varying between 0.01 and 0.40 mm whereas the worst agreement is seen at Ny-Ålesund, 363 Yebes, and Fortaleza sites with the radial amplitude differences of 1.48 mm with respect to FES2014, 364 0.93 mm and 1.50 mm with respect to TPXO8, respectively. At most of the stations, the estimated 365 tangential amplitudes of M2 tide with respect to those of ocean tide models vary from 0.03 mm (at 366 Matera) to 0.75 mm (at Fortaleza). It is worth to note that there is a large M2 tide radial amplitude 367 difference of about 0.7 mm between the TPXO8 model and both FES2014 and FES2012 models at    Yarragadee, the radial amplitudes of vector differences are larger relative to tangential components 406 and range from 0.7 to 1.6 mm. The worst agreement between CONT14 estimates and the ocean tide 407 models in terms of M2 tide phasor vector differences is found at the site Fortaleza varying in all 408 coordinate components between 0.7 mm to 1.6 mm (see Figure 6).  When the a priori ocean tide model is switched between FES2014 and GOT00.2, the level of change 420 of the estimated OTL displacements of the corresponding site is tested. A relatively older and less 421 accurate model, GOT00.2 is selected to get more significant differences to FES2014. It is worth to 422 note that for both of the hourly solutions (Case 1 and Case 2), the reduced external troposphere The differences of the OTL displacement estimates reveal a tide-like harmonic behavior (see 436 Figure 7). The peak-to-peak amplitudes of the differences of OTL displacements range from  Table 3 and explained in the below items. 445 ▪ To quantify the effect of using different a priori ocean tide models i.e. FES2014 or GOT00.2 in 446 the analysis of 1 hour sessions the amplitudes of phasor vector differences of M2 tide between 447 Case 1 and Case 2 are calculated (see Table 3 and Figure 9). 448 ▪ To get only the M2 tide OTL displacements, in addition to the FES2014, the GOT00.2 449 displacements, except M2, are reduced from the estimated hourly position time series. If the 450 FES2014 or the GOT00.2 models would contain errors, these tides would not be completely 451 reduced at the end and would affect the estimated M2 tide amplitudes and phases especially 452 due to the short time period of 15 days. The effect of reducing FES2014 or GOT00.2 tidal 453 displacements (except M2) from the hourly position series, on the estimated M2 tide, is 454 quantified through producing the amplitudes of the phasor vector differences between Case 1 455 and Case 3 (see Table 3 and Figure 9). 456 ▪ To quantify the effect of using different a priori ocean tide models i.e. GOT00.2 or FES2014 both 457 in the analysis of 1 hour sessions and in the reduction of all the other tides, except M2, from the 458 estimated OTL hourly displacements, the amplitudes of phasor vector differences between 459 Case 2 and Case 3 are calculated (see Table 3 and Figure 9). 460 ▪ When FES2014 or FES2004 ocean tide model displacements are introduced to the a priori 461 station coordinates in the analysis of 24 hour sessions, two different sets of troposphere delays 462 are estimated (e.g. see Figure 8 and readers are referred to the supplementary material for the 463 plots of the ZWD differences at the other VLBI sites when different a priori ocean tide models 464 are used in the analysis of 24 hour sessions). The level of influence of these two types of 465 troposphere delays estimated from 24 hour sessions and reduced from the observations of 466 1 hour sessions, on the estimated M2 tide of 1 hour sessions is evaluated. Thus, the estimated 467 M2 tides are compared in terms of the amplitudes phasor vector differences between Case 1 468 and Case 4 (see Table 3 and Figure 9). 469 campaign between four different cases of parameterization (see Table 3). The effect of using 476 different a priori ocean tide models i.e. FES2014 or GOT00. (ZWD) to vary in -2 and 2 mm (see e.g. Figure 8 for the station Ny Alesund and the supplementary 506 material for the other stations). Root-mean-square of the estimated ZWD differences (purple 507 horizontal lines in Figure 9) and the M2 tide phasor vector differences in radial components between 508 Case 1 and Case 4 reveal a statistically significant strong positive correlation of 0.89. These findings 509 suggest that the larger errors of ocean tide models (e.g. FES2014 and FES2004) at coastal sites 510 relative to the inland sites, propagate into the troposphere delays estimated from 24 hour sessions, 511 which in turn these errors in the troposphere delays cause larger differences in the M2 tide phasor 512 vectors of OTL displacements estimated from the 1 hour sessions. 513 To assess the overall agreement between CONT14 estimates and ocean tide models the 514 root-mean-square (RMS) of the amplitudes of the vector differences between observed and modeled 515 phasor vectors across the stations for the j 'th tidal constituent (M2) and the k 'th coordinate  Table 4. Hereupon, the term "RMS misfits" refers to the RMS of the amplitudes of 519 phasor vector differences between the CONT14 estimates and those of ocean tide models i.e. 520 FES2014, FES2012 and TPXO8 across all the stations. 521 Table 4. RMS misfits of M2 tide calculated from Equation (5) in mm between CONT14 phasor vectors 522 from the solutions of Case 1, Case 4 and those of ocean tide models (i.e. FES2014, FES2012, and 523 TPXO8) over coastal (10 stations, see Table 1), inland (7 stations otherwise, see Table 1 The RMS misfits of the M2 tide in the radial component between CONT14 and ocean tide models 526 over coastal sites are found about 0.70-0.96 mm both from the solutions of Case 1 and Case 4 which 527 is significantly larger than those of inland sites i.e. varying in 0.34-0.51 mm. The RMS misfits in both 528 radial and tangential components over coastal sites are found about two times larger than those of 529 inland sites which is valid for both solutions of Case 1 and Case 4 (see Table 4). When all sites are 530 considered, the RMS misfits of the M2 tide between CONT14 and ocean tide models are within 531 0.66-0.81 mm in radial and 0.21-0.40 mm in tangential components. The best agreement between 532 CONT14 and the selected ocean tide models for M2 tide is seen in all the components of inland sites 533 estimated from the Case 4 solution (see Table 3) with the RMS misfits ranging in 0.10-0.41 mm. 534

Conclusions 535
In this study, hourly VLBI station coordinate time series derived from the analysis of 1 hour VLBI 536 sessions of 15 days long continuous VLBI campaign, CONT14 were used to estimate the principal 537 lunar semidiurnal tide, M2 of OTL displacements. Only M2 tide is estimated in this study and the 538 station displacements caused by the remaining; long period, semi-diurnal and diurnal tides were 539 calculated from FES2014 model and reduced from the hourly station coordinates a priori to the 540 estimation. Intrinsic to this study, the troposphere delays estimated from 24-hour sessions were 541 reduced from the observations of hourly sessions a priori to the analysis. CONT14 estimates and those of ocean tide models are found below 0.5 mm at the inland sites i.e. 547 Badary, Wettzell, Zelenchukskaya, Matera, and Katherine as well as at the coastal sites Onsala and 548 Westford. The worst agreement between CONT14 estimates and ocean tide models for M2 tide in 549 terms of phasor vector differences is found at the site Fortaleza where the values in all coordinate 550 components are between 0.7 mm to 1.6 mm. This is due to the extremely humid weather at 551 Fortaleza which results in the unmodelled troposphere delays propagate into coordinate 552 components in the least-squares adjustment. Besides, the Fortaleza VLBI site is located at about 5 km 553 far away from the Atlantic ocean and its M2 amplitude of OTL displacements is about 36 mm in the 554 radial component, the largest radial amplitude relative to the other CONT14 VLBI sites. 555 As an overall assessment, the RMS of the amplitudes of M2 tide phasor vector differences between 556 the CONT14 estimates and those of ocean tide models i.e. FES2014, FES2012 and TPXO8 across all 557 the stations, namely RMS misfits are considered. The best agreement between CONT14 and the 558 selected ocean tide models for M2 tide is seen in all the components of inland sites with the RMS 559 misfits ranging in 0.1-0.4 mm. The RMS misfits of the M2 tide of OTL displacements in all coordinate 560 components between CONT14 and ocean tide models over coastal sites are found about two times 561 larger than those of inland sites. This result confirms the modeling insufficiencies in shallow waters of 562 ocean tide models which results in an accuracy restriction of OTL displacements around coastal 563 regions. The larger errors of ocean tide models (e.g. FES2014 and FES2004) at coastal sites relative to 564 the inland sites, propagate into the troposphere delays estimated from 24 hour sessions, which in 565 turn these errors in the troposphere delays cause larger differences in the M2 tide phasor vectors of 566 OTL displacements estimated from the 1 hour sessions. The larger discrepancies between the VLBI 567 observed and model-predicted M2 tides at coastal sites may not only be resulted from the errors of 568 ocean tide models at coastal regions but also additional errors can become from VLBI data and 569 analysis. Besides, hydrological loading and non-tidal ocean loading, which are not introduced a priori 570 to the station coordinates in the VLBI analysis of 1 hour sessions in this study, would contribute to 571 the increase in the misfits of phasor vectors between ocean tide models and VLBI at both coastal and 572 inland sites. Because, even these geophysical effects have periods much longer than one day, given 573 the short time span of CONT14 their contributions may not be negligible. 574 To conclude, the level of agreement between the VLBI observed and model-derived OTL 575 displacements can be significantly increased through reducing the 24-hour session troposphere Page23 delays from the observations of hourly sessions. As an outlook to improve the estimated accuracies 577 of semidiurnal and diurnal tidal constituents of OTL displacements from the analysis of the 578 observations of VLBI, further developments should be introduced to the present-day troposphere 579 delay models as well as to the geophysical models for the station position displacements. Besides, 580 the number of VLBI observations should be increased. Their temporal distribution over a session and 581 sky coverage should be homogenized. 582

Acknowledgments 583
This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK), 584 project number: 115Y244. The author acknowledges the International VLBI Service for Geodesy and 585 Astrometry (IVS) for providing the observations of the CONT14 campaign. 586