The tidal displacement field at Earth's surface determined using global GPS observations


Corresponding author: B. F. Chao, Institute of Earth Sciences, Academia Sinica, Taipei 11529, Taiwan. (


[1] We investigate the 3-D tidal displacement field on Earth's surface recorded globally by 456 continuous global positioning system (GPS) stations of IGS spanning 1996–2011, for eight principal diurnal and semidiurnal tidal constituents. In-phase and quadrature amplitudes of the residual tidal displacements, after removal of an a priori body tide model, are estimated using the precise point positioning (PPP) technique on the daily GPS data; the resultant daily estimates are combined to derive final estimates for each tide at each station. The results are compared with the predictions of eight recent global ocean tide models, separately for coastal (307) and inland (149) stations. We show that GPS can provide tidal displacement estimates accurate to the level of 0.12 mm (horizontal) and 0.24 mm (vertical) for the lunar-only constituents (M2, N2, O1, and Q1) and less favorably for solar-related tidal constituents (S2, K2, K1, and P1), although improved by ambiguity resolution. Most recent ocean tide models fit the GPS estimates equally well on the global scale but do not agree well between them in certain coastal areas, especially for the vertical displacements, suggesting the existence of model uncertainties near shallow seas. The tidal residuals for the inland stations after removing both body tides and ocean tidal loading (OTL) furthermore show clear continental-scale spatial coherence, implying deficiencies of the a priori body tide modeling in catching lateral heterogeneity in elastic as well as inelastic properties in the Earth's deep interior. We assert that the GPS tidal displacement estimates now achieve sufficient accuracy to potentially provide constraints on the Earth's structure.

1 Introduction

[2] Our present knowledge of the physical structure of the Earth's interior has mainly come from seismological studies. Modern geodetic observations with ever-increasing accuracy and resolution are poised to augment this knowledge by providing independent constraints on existing models [e.g., Latychev et al., 2009]. In particular, tidal measurements have previously reached the millimeter-level accuracy by very long baseline interferometry (VLBI) and global positioning system (GPS) [Petrov and Ma, 2003; Thomas et al., 2007], and better than 0.1 µgal precision by superconducting gravimeters [Baker and Bos, 2003; Boy et al., 2003].

[3] The tidal deformation occurring at any given location on Earth's surface consists mainly of two superposing parts. The primary part is the body tides (or the solid Earth tides) caused by the direct lunar-solar gravitation. The body tides follow the lunar-solar tidal potential in spatial pattern, typically amounting to tens of centimeters in amplitude as modulated by the respective Love numbers. Secondary to the body tides is the loading effect of the water mass redistribution associated with the ocean tides, referred to as the ocean tidal loading (OTL) here. OTL deformations have irregular spatial patterns that depend strongly on the ocean tide behavior around the observing location and may reach up to over ten cm in coastal regions.

[4] The body tides and OTL are of the same tidal periods: semidiurnal, diurnal, and long period. For a given tidal constituent, the amplitude and phase can be conveniently represented by phasors on the complex plane (Figure 1). Let Zobs be the observed tidal displacement phasor at a given location (whether coastal or inland) for a given component (east, north, or up) and tidal constituent (diurnal or semidiurnal), Zbody and ZOTL be the corresponding quantity predicted by the a priori body tide model and the selected OTL model, respectively. We shall call Zbody + ZOTL the “theoretical” value Zth. The residual Zres = ZobsZth is the key quantity that would then signify the errors in the combined body tide and OTL model predictions (plus the observational errors of course).

Figure 1.

A schematic depiction of the relationship among the various phasor vectors Zobs, Zbody, ZOTL, Zth, and Zres. The horizontal axis represents the in-phase component, and the vertical axis the out-of-phase component relative to the tidal potential.

[5] The increasing number of globally distributed, continuous stations along with the advances in technology as well as in the method of data analysis and modeling make the GPS technique particularly effective in studying tidal displacements in the sense of the high precision and spatial resolution not readily achievable by other geodetic techniques. With the main purpose of evaluating and discriminating ocean tide models, previous studies on GPS tidal deformation focused on specific coastal areas where ocean tides are insufficiently modeled [Ito et al., 2009; Khan and Tscherning, 2001; King et al., 2005; Vergnolle et al., 2008; Yeh et al., 2011]. The “observation versus model” differences are attributed mainly to the errors in the ocean tide models and imperfect accuracy in the GPS measurement.

[6] Yuan et al. [2009] used GPS data from a dense, continuous (albeit local) network in Hong Kong to evaluate the internal precision of GPS tidal displacement estimates. The results then showed that the misfits for the major semidiurnal and diurnal constituents (except K1 and K2) are less than 0.5 and 1.0 mm, respectively, for the horizontal and vertical components, implying that the GPS measurement error may not be a limiting factor to scrutinize body tide and OTL models.

[7] More recently, Ito and Simons [2011] made inferences on the asthenospheric structure beneath the western United States based on OTL displacements derived from regional GPS observations, under the assumption that body tides are sufficiently well modeled. Using GPS observations over the western half of the United States but spanning a much longer time span with finely tuned methodology, Yuan and Chao [2012] succeeded in demonstrating the precision of GPS tidal displacement estimates down to the level of ~0.1 mm (horizontal) and ~0.3 mm (vertical). Their results of Zres revealed clear, coherent continental-scale spatial patterns (in the western United States) and nondiminishing amplitudes even well inland, signifying errors not only in OTL but also in the body tide model which does not account for the lateral heterogeneities in the Earth's deep interior.

[8] On the global scale, GPS network observations have been conducted by Schenewerk et al. [2001], who estimated the vertical tidal displacements of 353 globally distributed GPS stations for eight major semidiurnal and diurnal constituents using 3 years of observations and found large-scale systematic observation-versus-model differences. The estimation accuracy of their tidal displacement (vertical only) was at the time inadequate for scrutinizing ocean tide models (or body tide models for that matter) [King et al., 2005; Petrov and Ma, 2003]. Based on 25 global (but sparse) VLBI stations colocated with GPS, Thomas et al. [2007] found no apparent latitudinal or longitudinal dependence in their estimates for Zres.

[9] Following the same methodology as Yuan and Chao [2012], the present study uses data from 456 globally distributed continuous GPS stations spanning up to 16 years to determine the 3-D crustal displacements for eight major semidiurnal and diurnal tides to the extent feasible. As in Yuan and Chao [2012], precisions as high as submillimeter level are achieved. The results are compared against the latest OTL model predictions in detail, while the spatial coherence found in Zres is examined in consideration of possible error sources, separately for coastal and inland stations. This allows further evaluation of the OTL models and constraints on the body tide models in terms of the lateral heterogeneities of the Earth's interior structure.

2 Estimation of GPS Tidal Displacements

2.1 Data Sets and a Priori Tide Models

[10] The GPS observation data are available at the global data centers of the International GNSS (Global Navigation Satellite System) Service (IGS). We collect those over the 16 year period from 1 January 1996 to 31 December 2011. Overall there are 456 GPS stations, of which 303 have more than 3000 daily solutions after removing outliers, whereas 82 stations have 1000–2000 daily solutions. We shall divide these stations into two categories: 307 coastal stations located within 150 km from the nearest coastline which would be significantly influenced by OTL (with modeling errors) and 149 inland stations otherwise, where the OTL modeling error (at short wavelengths) can be safely neglected at our required precision of ~0.1 mm [Penna et al., 2008].

[11] The a priori body tide model values Zbody that are removed are according to the standard model of the IERS Conventions 2010 [Petit and Luzum, 2010], as implemented in the GIPSY software used for our solution. This model is the basic 1-D (radial-stratified) Preliminary Reference Earth Model (PREM) [Dziewonski and Anderson, 1981] but further developed based on Mathews et al. [1997] and Dehant et al. [1999]. The contributions from the Earth's ellipticity (hence a latitudinal dependence) and the Coriolis force due to Earth rotation [Mathews et al., 1995; Wahr, 1981], the free core nutation resonance [Mathews, 2001], the resonance in the deformation due to OTL [Wahr and Sasao, 1981], and the mantle inelasticity [Dehant et al., 1999; Mathews et al., 1997] have been taken into account. The mantle inelasticity at the tidal periods is only moderately constrained but compatible with the nutation observations [Dehant and Defraigne, 1997]: That included in the IERS Conventions 2010 uses the mantle Q model developed by Widmer et al. [1991] assuming an ω0.15 frequency dependence between the tidal periods and the reference period of 200s.

[12] The OTL model displacements ZOTL are calculated using the SPOTL software [Agnew, 1997] that convolves the input ocean tide model with the Green's function [Farrell, 1972] calculated from three different Earth models (see section 3.2). Eight recent global ocean tide models are used, all based on the same ocean radar altimetry observations from the TOPEX/Poseidon and the follow-on Jason-1 satellite missions: CSR4.0 [Eanes and Shuler, 1999], NAO99b [Matsumoto et al., 2000], FES2004 [Lyard et al., 2006], TPXO7.2 (update of Egbert and Erofeeva [2002]), HAMTIDE11a [Taguchi et al., 2010], DTU10 [Cheng and Andersen, 2011], EOT11a (update of Savcenko and Bosch [2008]), GOT4.7 (update of Ray [1999]).

2.2 Estimation Methodology

[13] There have been two classes of approach for estimating GPS tidal displacements: kinematic and static [King, 2006]. The kinematic approach uses the kinematic GPS techniques to produce subdaily (typically 1–4 h) position time series that can then be analyzed by means of conventional harmonic analyses [Ito et al., 2009; Khan and Tscherning, 2001; Vergnolle et al., 2008]. The static approach, on the other hand, includes the harmonic displacement coefficients of the targeted tidal constituents into the processing of the daily GPS data as additional parameters after removal of Zbody values. The deduced daily harmonic parameters and their full extracted variance-covariance matrices are subsequently stacked to solve for the final (residual) tidal displacements [King et al., 2005; Schenewerk et al., 2001; Yuan et al., 2009].

[14] In the present paper we employ the static approach which in general produces more robust results as demonstrated previously by Yuan and Chao [2012], with an improvement in the scheme specifically aimed at extracting the tidal signals as follows.

[15] Neglecting three long-period tidal terms, the locally referenced 3-D displacement Δck (k = 1, 2, and 3 denoting the local east, north, and up components, respectively) due to OTL can be modeled as a sum of displacements from the eight principal constituents (the semidiurnal M2, S2, K2, and N2 and the diurnal K1, O1, P1, and Q1) [Petit and Luzum, 2010]

display math(1)

where Ak,j and Φk,j denote the amplitude and phase (relative to Greenwich and positive lags) of the jth tidal constituent in the kth direction; ωj is the constituent angular frequency; fj and μj are called the nodal modulation corrections in amplitude and phase respectively; and χj is the astronomical argument at reference time t0 here chosen to be J2000.

[16] To allow linear parameter estimation using, e.g., least squares, equation (1) is expanded into sine and cosine terms

display math(2)


display math(3)

Thus, in contrast to the conventional GPS data analysis, an additional set of 48 tidal displacement parameters are simultaneously estimated for each station.

[17] In order to make the nodal modulation correction to the amplitude and phase of a main constituent, one needs to assume the same amplitude ratios rjl and phase differences, as predicted by the tidal potential, between a major constituent j and its satellite constituents jl, with the relation [Foreman, 1977]

display math(4)

where Δjl = χjl(t0) − χj(t0). Expanding equation (4), the following explicit formulae can be derived for fj and μj

display math(5)

[18] The nodal modulation corrections are significant for K1, K2, O1, and Q1 (up to 30% for K2), but relatively small for the other four constituents.

[19] The practical estimation strategy is realized in two steps. First, the harmonic coefficients of tidal displacements are estimated in the daily GPS data processing. The harmonic parameters and the extracted full variance-covariance matrices from daily solutions are then combined using a Kalman filter to obtain the final estimates of OTL displacements [Dong et al., 1997].

[20] We process the GPS data in daily segments using the technique of precise point positioning (PPP) in the GIPSY/OASIS-II software (Version 6.1) [Zumberge et al., 1997]. The advantage of PPP is that the absolute tidal displacements can be directly estimated from a large amount of GPS data with a modest computational cost. The a priori body tides and the (equilibrium) pole tide are removed at the observation level following the IERS Conventions 2010 [Petit and Luzum, 2010]. Along with station-specific unknown parameters (station position, receiver clock, tropospheric delay parameters, and phase biases) in the standard GPS data analysis, we additionally estimate the in-phase and quadrature amplitudes of the (residual) tidal displacements of eight major constituents—the semidiurnal M2, S2, K2, and N2 and the diurnal K1, O1, P1 and Q1, for all three components east, north, and up as stated above.

[21] We shall adopt the OTL predictions based on the FES2004 ocean tide model [Lyard et al., 2006] as the a priori OTL values in our daily GPS data processing. The amplitudes and phase lags of the ZOTL in the instantaneous center of mass (CM) reference frame are computed for each station using SPOTL as stated above. To avoid numerical instabilities, the a priori tidal parameter constraints of 5 and 10 mm are respectively applied to the horizontal and vertical components, except for the constituents K1 and K2 where looser constraints of 10 and 20 mm respectively are applied to reconcile the GPS orbit-related errors specifically at these two frequencies [King et al., 2005; Yuan et al., 2009].

[22] The nodal modulation corrections are also included in the daily processing, again according to the IERS Conventions 2010 [Petit and Luzum, 2010], which considers a total of 331 minor tides whose amplitudes and phases are determined by spline interpolation of tidal admittances based on the a priori FES2004 OTL values of 11 main tidal constituents (the above-mentioned eight plus three long-period constituents of Mf, Mm, and Ssa).

[23] We use reprocessed fiducial-free satellite orbits, clocks, and Earth orientation products provided by the Jet Propulsion Laboratory (JPL). The impact of the IGS final satellite orbits and clocks products is described in section 2.3. We assign a 5-min sampling interval, an elevation cutoff angle of 7°, and no elevation-dependent weighting. The a priori hydrostatic and wet zenith delays are calculated from the ECMWF field, using the VMF1 mapping functions [Boehm et al., 2006]; the wet zenith delays and their gradients are estimated as random walk parameters with the process noise values of math formula and math formula (where h is hour), respectively. The receiver clock is modeled as a white noise process updated at each epoch. Carrier-phase ambiguities can also be resolved at this stage, to be discussed in section 2.3.

[24] Daily tidal displacement estimates of each tidal constituent vary significantly from day to day with large uncertainties (typically about 5 and 10 mm in the horizontal and vertical components, respectively). Moreover, they are highly correlated with companion constituents due to the nearness of their frequencies. As the tidal constituents can hardly be resolved from a single daily solution, a scheme of stacking or combining daily estimates is implemented to obtain the final estimates with sufficient accuracy. Thus, we extract tidal displacement estimates and their variance-covariance matrices from daily solutions and combine them using a Kalman filter.

[25] We exclude daily solutions whose station coordinate uncertainties are greater than 10 mm in any of the three components. In the combination, we also use the daily estimates of unit variance to reject the outliers and rescale the daily covariance matrices to produce a final unit variance in two iterations, a process resulting in only ~5% rejections of the daily solutions. During the first iteration, the daily variance-covariance matrices are initially scaled by a factor of 30, and the unit variances are calculated and saved. In the second iteration, the variance-covariance matrices are rescaled by the respective unit variances calculated from the first iteration to obtain the final estimates and their formal errors.

[26] The nodal modulations are already accounted for using the a priori FES2004 OTL values in the daily GPS data processing as mentioned above, so only those for the residual tidal displacements are necessary in the combination. Theoretically the nodal modulation corrections should be applied to each of the daily residual tidal displacement estimates for all constituents. In practice, it is inappropriate to do so due to the high correlations between the daily estimates of the constituents. On the other hand, neither is it appropriate to apply the corrections to the final estimates obtained from the long-term (over 16 years) observations, as they would undulate with an 18.6 year period. As a compromise, the corrections are assumed to be constant and equal to their value at the midpoint of each year. The daily solutions are first combined into yearly batches, to which the nodal modulation corrections are applied. The yearly corrected solutions are then combined further to derive the final parameter estimates. From our test results, having the nodal corrections for the residual tidal displacements indeed improves our estimates for K1, K2, and O1.

2.3 Impact of Ambiguity Resolution and Orbit Products

[27] Since the PPP technique requires the satellite-dependent parameters to remain fixed, any errors in satellite orbits and clocks would propagate directly into the station-specific parameter estimates and hence contaminate the tidal displacement estimates. To consider these possible errors, we compare the processed GPS solutions using IGS final satellite orbit and clock products with those of JPL as described in section 2.2. Figure 2 shows the differences in root mean square (RMS, see section 3.1) misfits of ZobsZbody against the GOT4.7's ZOTL under the two schemes, but after removing the geocenter motion effect (note that the JPL orbit estimates are given w.r.t. the CM frame, whereas the IGS orbit estimates are w.r.t CF; see section 2.5 later). Overall, the IGS PPP estimates have only slightly larger RMS misfits than the JPL PPP estimates, suggesting a negligible impact of different orbit and clock products. We use JPL ambiguity-fixed estimates in the following unless otherwise specified.

Figure 2.

RMS misfits of the GPS tidal residuals ZobsZbody against the GOT4.7 ZOTL predictions. IGS_PPP and JPL_PPP represent the PPP estimates without ambiguity resolution using IGS and JPL satellite orbit and clock products, respectively, and JPL_AMB indicates the ambiguity-fixed estimates using JPL satellite orbit and clock products. The effect of the tidal geocenter motion has been removed beforehand.

[28] Initially, the integer ambiguities are left unresolved in the daily PPP solutions, since ambiguity resolution traditionally requires GPS observations from at least two stations. After obtaining the daily PPP solutions for all the stations in a GPS network, double difference of simultaneous GPS observations from multiple stations can be formed for ambiguity resolution in order to improve solution accuracy [Blewitt, 1989]. However, such data processing demands excessive computation, especially for global network analysis, as the processing time of the full-network ambiguity resolution is generally scaled by O(n4). Fortunately, Bertiger et al. [2010] developed a single receiver phase ambiguity resolution method with computational load of O(n), which allows rapid processing of large networks.

[29] Previous studies have investigated the impact of traditional double-difference ambiguity resolution on tidal displacement solutions. Thomas et al. [2007] demonstrated that ambiguity resolution with relatively low success rates (~ 30%–50%) has a marginal improvement on the tidal displacement estimates, whereas Yuan et al. [2009], based on results from a low-latitude local GPS network, showed that ambiguity resolution with ~ 97% success rates gives slight improvement.

[30] To examine the impact of ambiguity resolution on the tidal displacement estimates on a global scale, we perform single-receiver ambiguity resolution using the wide lane and phase bias information from JPL global GPS solutions [Bertiger et al., 2010]. The daily ambiguity-fixed tidal parameter solutions are also combined as described above for the PPP solutions. When compared with the model predictions, shown in Figure 2, ambiguity resolution in the GPS estimates improves the agreement, particularly in the east component for the four solar-related constituents S2, K2, K1, and P1 (for example, the RMS misfits are reduced by more than 50% in the K2 horizontal and in the K1 east.)

2.4 Convergence and Stability of Solutions

[31] To confirm the constituent convergence and stability, we compare the vector differences of the accumulated estimates (after the addition of each daily solution) with the final estimates for ZobsZbody of each of the eight tidal constituents. The nodal modulation corrections for the residual tidal displacements are not taken into account at this point. We use 303 stations with more than 3000 daily solutions; Figure 3 shows the medians of their vector differences as a function of the increasing number of daily solutions, showing rapid dropoff of the said differences.

Figure 3.

Tidal estimation convergence for the eight tidal constituents, showing the median of their vector differences, for the 303 stations that have more than 3000 daily solutions, between the accumulated and final estimates of Zobs as the number of daily solutions increases.

[32] Four lunar-only constituents, M2, N2, O1, and Q1, have the fastest convergence; their medians converge to below 0.1 mm (horizontal) and 0.2 mm (vertical) by upward of 1000 daily solutions. For S2 and P1, the corresponding convergence level is ~0.2 and 0.5 mm, while K1 and K2 have the slowest convergence, reaching ~0.1 and 0.4 mm after as many as 3000 daily solutions. This is the reason we only use stations with more than 1000 daily solutions in this work.

[33] On closer examination of individual station's convergence, we find notable seasonal fluctuations in the amplitudes of the slow-converging solar-related constituents, S2, K2, K1, and P1. Moreover, the convergence of K1 and K2 are generally site dependent and sometimes hardly reached, presumably due to the multipath effects at these precise periods [King et al., 2005; Yuan et al., 2009; Zhong et al., 2011]. The estimates of K1 and K2 should therefore be regarded with caution.

2.5 Geocenter Motion Correction

[34] The geocenter is the center of mass (CM) of the entire Earth system that includes the solid Earth and its fluid envelope [Blewitt, 2003]. Fixed in space in the absence of external force, Earth's CM is the point to which the dynamical motion of GPS satellites respond. Redistributed masses in, say, the ocean cause CM to undergo apparent periodic motion (in the opposite sense) relative to the center of figure (CF) of the solid Earth defined geometrically by the surface GPS stations. This geocenter motion needs to be corrected in high-precision geodetic applications such as ours.

[35] The predicted tidal geocenter motions, referred to CF, reach up to several millimeters; their component-wise differences among different global ocean tide models can reach up to 0.5 mm for major constituents, e.g., M2 and K1. We test the estimation of the geocenter correction using Zres under the following scenarios: (i) ZOTL predicted by different ocean tide models in CF; (ii) horizontal-only versus 3-D residuals; (iii) inland-only versus all GPS stations; (iv) IGS PPP, JPL PPP, or JPL ambiguity-fixed solutions. The first three scenarios found to have negligible impacts on our geocenter motion estimates (differences less than 0.1 mm), we choose the 3-D Zres with GOT4.7's ZOTL for all stations to estimate the geocenter correction. Figure 4 shows the results for the three different sets of solutions in (iv). Since both IGS and JPL global analysis centers adopt the a priori FES2004 geocenter correction (so we needed to add it back before comparing with the other solutions), it is not surprising that overall the geocenter correction estimates agree best with the FES2004 model. However, there are large biases between IGS and JPL solutions in the Z component for K1 (up to 0.4 mm) and O1 (0.2 mm), mostly due to the analysis center difference in the correction scheme for the minor tides and nodal modulations. The JPL ambiguity-fixed solutions are chosen as the final estimated geocenter correction to be removed (see Table 1).

Figure 4.

Phasor diagrams of geocenter motion due to ocean tides according to seven models (CSR4.0, triangles; NAO99b, diamonds; FES2004, circles; TPXO7.2, pentagons; HAMTIDE11a, hexagons; DTU10, squares; EOT11a, inverted triangles; GOT4.7, crosses; courtesy of M.S. Bos and H.G. Scherneck). The horizontal is the in-phase amplitude, the vertical the quadrature amplitude. The green, blue, and red error bars represent, respectively, the three-sigma formal errors for IGS PPP estimates, JPL PPP estimates, and JPL ambiguity-fixed estimates. The FES2004 model values adopted in JPL global analysis center are also given in gray solid stars for comparison. The scale is the same for all panels.

Table 1. Estimates and Their Formal Errors of the Tidal Geocenter Motions for Eight Principal Tidal Constituents
TideX (mm)Y (mm)Z (mm)
M2−1.472 ± 0.0110.801 ± 0.0111.110 ± 0.0110.291 ± 0.011−1.169 ± 0.014−1.461 ± 0.014
S2−0.587 ± 0.011−0.327 ± 0.011−0.029 ± 0.0110.289 ± 0.011−0.193 ± 0.014−0.619 ± 0.014
N2−0.263 ± 0.0020.199 ± 0.0020.259 ± 0.002−0.117 ± 0.003−0.283 ± 0.003−0.291 ± 0.003
K2−0.135 ± 0.008−0.030 ± 0.008−0.014 ± 0.0080.156 ± 0.008−0.106 ± 0.010−0.205 ± 0.010
K1−1.797 ± 0.013−0.921 ± 0.013−0.895 ± 0.013−1.768 ± 0.013−1.074 ± 0.0174.378 ± 0.017
O1−1.350 ± 0.004−0.242 ± 0.004−0.877 ± 0.004−0.637 ± 0.004−0.183 ± 0.0062.871 ± 0.006
P1−0.606 ± 0.006−0.316 ± 0.006−0.292 ± 0.006−0.576 ± 0.006−0.366 ± 0.0081.437 ± 0.008
Q1−0.250 ± 0.001−0.021 ± 0.001−0.208 ± 0.001−0.042 ± 0.0010.042 ± 0.0020.452 ± 0.002

3 Results and Discussions

3.1 GPS Tidal Residuals and Accuracy Assessment

[36] The final values of the GPS tidal displacement estimates ZobsZbody in both CM and CF reference frames are provided in supplementary Tables S1–S2, respectively. We shall now assess their agreement against the various OTL model predictions ZOTL. The RMS misfits between them [Yuan et al., 2009], for the jth of the eight tidal constituents in the kth coordinate component is

display math(6)

where Zj,k,n = [AGPS(cos ΦGPS + i sin ΦGPS)j,k,n − Amodel(cos Φmodel + i sin Φmodel)j,k,n], n is the station index, A the amplitude, and Φ the phase lag. The RMS can be readily extended to the weighted RMS (WRMS) by

display math(7)

where σj,k,n is the formal error of the corresponding tidal displacement estimate. The two indicators lead to essentially consistent conclusions (see later). The comparison statistics are presented in Figure 5 and Table 2.

Figure 5.

RMS misfits of the GPS tidal displacement estimates ZobsZbody against the ocean-tide model predictions ZOTL, for the indicated 8 different models. The effect of the tidal geocenter motions have been removed beforehand. Note the scale for the vertical component is 3 times that of the horizontals.

Table 2. WRMS Misfits of the GPS Tidal Residuals ZobsZbody Against the GOT4.7 ZOTL Predictions, Separately for the Following: All (the Entire Set of 456 IGS Stations), Coast (the 307 Stations Within 150 km of Distance From Coastline), and Inland (the 149 Stations Otherwise) (unit: mm)

[37] At this point, we should point out the recognized GPS noise sources of concern to the determination accuracy for specific tide constituents. The GPS-related errors affecting subdaily tidal displacement estimates include inadequate orbit modeling, tropospheric mapping function and a priori hydrostatic delay errors, higher-order ionospheric delays, and multipath effects [King et al., 2008; Ray et al., 2008]. Fortunately, they appear not to map appreciably into lunar-related tidal frequencies, namely M2, N2, O1 and Q1 (Figure 3).

[38] Previous studies suggested that the large uncertainties for K1 and K2 estimates stem from the GPS satellite orbit errors and multipath effects, because the GPS constellation repeat period (one sidereal day) corresponds to the K1 period and the satellite orbital period (one half sidereal day) corresponds to the K2 period [King et al., 2005, 2008; Schenewerk et al., 2001; Zhong et al., 2010]. By examining long-running, short-baseline (<<1 km) time series, King and Williams [2009] further suggested that the majority of the GPS observation-versus-model tidal residuals of K1 and K2 may be due to satellite orbit mismodeling and GPS signal propagation effects (including tropospheric and higher-order ionospheric effects) rather than local site effects such as multipath. Presently the mechanisms responsible for the (extra) residuals in K1 and K2 are still an open question.

[39] Residual tropospheric delay errors (due to unmodeled hydrostatic delay and/or mapping function error) may bias the estimates of P1 (due to the closeness of P1 period to the tropical day or S1) and S2 (at half tropical day), and higher-order ionospheric effects (not taken into consideration in our analysis) may be partially responsible for the tidal estimates in the diurnal band. However, their subdaily characteristics at a wide range of stations are not yet well understood [Fritsche et al., 2005; Kedar et al., 2003]. Other possible error sources for P1 and S2 estimates may include local thermal perturbation of station monuments and antennas [e.g., Yan et al., 2009].

[40] Seen from Figure 5, the older ocean tide models of CSR4.0 and NAO99b make predictions that are in relatively poorer agreement with the (global) GPS estimates especially for M2, whereas the others essentially show equally good agreement. HAMIDE11a has relatively large RMS misfits in the north and vertical components for O1. Somewhat surprisingly, the most recent model TPXO7.2, which IERS Conventions 2010 recommend [Petit and Luzum, 2010], has relatively larger RMS misfits in the east and vertical for M2 and in the vertical for O1. We take GOT4.7 as our reference model in the following in light of its overall good agreement with our global GPS estimates, presumably owing to its improvement in shallow water tides.

[41] The corresponding WRMS statistics given in Table 2 show further that, with a few exceptions whose differences are insignificant, the WRMS misfits for the inland stations are invariably smaller than those for the coastal stations for all tidal constituents in all three components. For example, the M2 and O1 WRMS misfits in the vertical component are respectively 0.31 and 0.19 mm for inland stations, compared to coastal stations’ 0.85 and 0.47 mm. It is remarkable that the WRMS misfits of N2 and Q1 for inland stations are respectively as small as 0.07 and 0.04 mm for the horizontal and 0.11 and 0.08 mm for the vertical component, even less than the formal errors of our GPS estimates.

[42] The formal errors of our GPS tidal displacement estimates are similar for all tidal constituents, ranging from 0.04 to 0.13 mm in the horizontal and 0.15 to 0.49 mm in the vertical component, depending on the time span and quality of the observations. The constituent convergence as illustrated in Figure 3 can be used to assess the precision (or internal consistency) of our estimates; they give consistent precision levels for the two groups: (i) For the four lunar-only constituents M2, N2, O1, and Q1, whose estimates are insensitive to GPS systematic errors, the formal errors should be representative of the corresponding precisions. (ii) For the other four constituents S2, P1, K1, and K2, we can tell from Figure 3 that the formal errors are apparently too optimistic w.r.t. the true precision due to failure to account for temporal noises.

[43] On the other hand, the accuracy of our tidal estimates can only be quantified by comparison with the geophysical models. However, their comparison is limited by the uncertainties in the body tide and OTL models. For inland stations where the uncertainties in OTL predictions can be safely neglected at the ~0.1 mm (horizontal) and 0.2 mm (vertical) level (see below), their residuals show strong systematic, spatially coherent variations presumably due to the deficiency in the a priori body tide model. Thus, the WRMS misfit statistics of the observation-versus-model residuals of the 149 inland stations (Table 2) can be adopted to represent the limit of accuracy of our GPS estimates: we take such WRMS value and multiply it by a factor of 3 to be the (conservative) accuracy of our GPS estimates for Q1, P1, and K2. The accuracies of the other three lunar-only constituents (M2, N2, and O1) should be the same as Q1, whereas the accuracies of S2 and K1 are the same as P1 and K2, respectively. Thus, overall, the GPS tidal estimation accuracy can be considered to be better than 0.12 mm (horizontal) and 0.24 mm (vertical) for the lunar-only constituents (M2, N2, O1, and Q1), better than 0.69 mm (horizontal) and 1.29 mm (vertical) for S2 and P1, and better than 0.84 mm (horizontal) and 2.31 mm (vertical) for K2 and K1.

3.2 Spatial Pattern of Inland Residuals: Errors in Body Tides?

[44] Figures 6 and 7, respectively, for the M2 and O1 constituents, show in map view the phasor vectors Zres = ZobsZth in terms of amplitude and phase (ZOTL according to the reference model GOT4.7), at all 456 GPS stations (blue vectors indicating the coastal stations and red vectors the inland stations). The figures for the other six tidal constituents are provided as supplemental information. As depicted in Figure 1, the phase angles, increasing counterclockwise with time, is referenced w.r.t. that of the (real-valued) tidal potential. The horizontal axis represents the in-phase component and the vertical axis represents the out-of-phase component of Zres relative to the tidal potential. Note that the removed a priori Zbody itself includes nonzero out-of-phase components arising from the modeled mantle anelasticity as described above. Figure 8 zooms in on the continental North America and Europe for the M2 residuals (duplicated from Figure 6), where relatively high concentrations of stations are available.

Figure 6.

Spatial distribution of phasor vectors of residuals Zres = ZobsZth for M2 (where ZOTL is according to GOT4.7, shown as the background color scales for reference). The red vectors indicate the 149 inland GPS stations and blue the 307 coastal GPS stations. The horizontal axis represents the in-phase component, and the vertical axis the out-of-phase component relative to the tidal potential. The effect of the tidal geocenter motions have been removed beforehand.

Figure 7.

Same as Figure 6, but for the O1 tide.

Figure 8.

Same as Figure 6 but zooming-in on North America (left) and Europe (right).

[45] The Zbody that has been removed beforehand according to the IERS Conventions 2010 [Petit and Luzum, 2010] has uncertainties considered to be on the order of 1%. So the body tide displacement errors are on the 1 mm level. Figures 6-8 (and S1–S6) reveal long-wavelength spatial coherence (as opposed to random noises) in the inland Zres, which we argue to be a consequence of the deficiencies of the a priori body tide modeling, only to be revealed under the present high-precision processing of the GPS data. We note here that the tidal displacement variation due to the a priori model of the mantle anelasticity is only latitude-dependent (at least presently), the spatial coherence seen in our results depends upon both latitude and longitude.

[46] Recent studies have begun to consider the impacts on body tides due to the mantle's laterally heterogeneous structure that is not yet included in the conventional models. Metivier and Conrad [2008] showed that the vertical surface displacements due to convection-induced mantle heterogeneity may be no more than 0.3 mm. Latychev et al. [2009] found that the vertical perturbations due to the lateral heterogeneity of density and elastic structure reach about 1 mm at the diurnal and semidiurnal periods and that the effect of the elastic moduli, which have been ignored by Metivier and Conrad [2008], accounts for greater than 50% of the perturbation.

[47] Take in particular the example of M2. The spatially coherent signals in the inland Zres can reach up to 1 mm in magnitude for all three components, indeed consistent with the perturbations due to lateral heterogeneities in density and elastic moduli as quantified by Latychev et al. [2009]. The overall WRMS misfits for the horizontal components (0.32 mm east and 0.37 mm north) are slightly larger than that of the up (0.31 mm) (Table 2), implying comparable errors of Zbody in all components, given that so far there is no report (that we know of) about the horizontal body tide effects of the mantle heterogeneity. Interestingly, the M2 north component has slightly larger tidal residuals for inland stations than coastal stations (0.37 mm versus 0.35 mm, Table 2), suggesting the significant residuals attributable to body tide model errors.

[48] Furthermore, comparing Figures 6 and 7 with S1–S6 points to a general similarity in the spatial patterns of inland Zres among own tidal species in the following sense. In the semidiurnal band, M2 and N2 show almost the same phase lag patterns for all three components of Zres, despite their very distinct tidal strengths. For example, the largest spatially coherent residuals across East Africa reach 1.28 mm (M2) and 0.27 mm (N2) in the east component. For the north component, the large inland coherent spatial patterns appearing in Eurasia and the western United States see amplitudes of up to 0.70 mm (M2) and 0.16 mm (N2). In the up component, large continental-scale spatial patterns are also apparent in Eurasia and the western United States with amplitudes up to about 0.90 mm (M2) and 0.30 mm (N2).

[49] The east residuals of S2 show a spatial pattern consistent with those of M2 and N2, so are the north residuals except across South America. The S2 up residuals in North America have phase lags almost exactly opposite to those of M2 and N2, and the pattern in Asia shows differences. The spatial pattern of the K2 residuals is rather erratic in comparison.

[50] In the diurnal band, the four constituents have spatial patterns that are similar to varied extent for the inland Zres. Those of O1 show the smoothest pattern across the globe. With large amplitudes across North and South Americas, the phase of the O1 east residuals has a consistent trend, whereas the north residuals are relatively smaller. Similarly but smaller than 0.1 mm (horizontal) and 0.3 mm (vertical), the Q1 residual spatial pattern can still be seen to be coherent. In contrast, P1 shows no apparent coherent residuals in the east component across North and South Americas. K1 has relatively large residuals with weak spatial coherence, but similar to the other three diurnal tides. We reiterate that the above behavior for various tidal constituents ought to be considered in conjunction with their possible systematic errors discussed above in section 3.1.

3.3 Spatial Pattern of Coastal Residuals: Deficiencies in OTL (Mostly)?

[51] In the presence of errors in Zbody, the deficiencies in ZOTL predominantly contribute to Zres of the coastal stations. Unlike body tides, the accuracy of ZOTL depends on several factors: the adopted ocean tide model, geophysical Green's function, coastline geography, and the scheme of numerical loading computation. The last two error sources amount only to 2%–5% and will not be addressed in detail here [Agnew, 1997; Bos and Baker, 2005]. Penna et al. [2008] have shown that the vertical ZOTL of M2 calculated from different software differ by 1–2 mm at coastal sites near complicated coastlines and shallow seas, but by less than 0.2 mm for stations more than ~150 km inland, which is the criteria we adopted for our designation of inland stations.

[52] The differences in ZOTL predicted by different Green's functions are also found to be less than 2% [Francis and Mazzega, 1990]. Here we experiment with the sensitivity w.r.t. the Green's function as follows. We calculate ZOTL predictions from three Green's functions provided by Endo and Okubo [1984], Farrell [1972], and Guo et al. [2004], that are respectively based on three benchmark Earth models: 1066A [Gilbert and Dziewonski, 1975], Gutenberg-Bullen A (G-B) [Alterman et al., 1961], and PREM [Dziewonski and Anderson,1981]. Normalized Green's functions for horizontal and vertical displacements are displayed in Figure 9. The three Green's function differ significant only in the near field where the angular distance θ is less than 1°, whereas the horizontal component is somewhat more sensitive in the far field than the vertical component.

Figure 9.

The Green's functions (for horizontal and vertical displacements) showing the response to a unit mass load (1 kg) at the surface at the angular distance θ (in logarithmic scale) from the observations point, for three different benchmark Earth models, G-B, 1066A, and PREM.

[53] In Figure 10, we compare the overall RMS misfits of ZOTL predicted by the three different Green's functions w.r.t. ZobsZbody for the 307 coastal stations. In general, the differences are insignificant in all three components for all tidal constituents except M2, which sees definite differences in the east and up components. The mean difference of PREM is the largest in the up component, but the smallest in the east component. We argue that this inconsistency is caused by the deficiencies in Zbody [Yuan and Chao, 2012]. Moreover, the WRMS misfit of coastal stations is about twice larger in the up (0.85 mm) than in the east (0.40 mm), while those of inland stations are almost the same for the east (0.32 mm) and up (0.31 mm) components. Thus, we can infer that the PREM model has the poorest overall performance in the up component, while the east inconsistency is attributed to the effects of the body tide errors. Nevertheless, it does not appear possible to discriminate readily among the Earth's shallow structure Green's functions based on GPS tidal displacement observations on the global scale.

Figure 10.

RMS misfits of the GPS tidal displacement estimates ZobsZbody against the GOT4.7 ZOTL using three different Green's fuctions (in Figure 9) for the 307 coastal stations.

[54] We now examine the largest contributor to the uncertainty of ZOTL, particularly for areas close to shallow seas, namely the deficiency in ocean tide models [Bos and Baker, 2005; Penna et al., 2008]. Here we compare the ZOTL predictions from six recent ocean tide models: FES2004, TPXO7.2, HAMTIDE11a, DTU10, EOT11a, and GOT4.7 (excluding the two older models, see section 3.1). The 3-D RMS misfits for almost all the 149 inland sites (see Figure 11) are less than 0.1 mm (horizontal) and 0.2 mm (vertical), for all tidal constituents except M2, which has anomalies at station WHIT (0.12 mm north and 0.25 mm up), SCH2 (0.17 mm up) and three nearly colocated sites in Kiruna (Sweden): KIR0 (0.21 mm), KIRU (0.21 mm), and KR0G (0.21mm) . The anomaly at these stations except SCH2 is traced to the large biases of TPXO7.2 predictions (although recommended by the IERS Conventions 2010) [Petit and Luzum, 2010] in comparison to the other five model predictions (Figure 5) which fit the GPS estimates equally well globally. We can conclude that the OTL prediction error can be safely neglected at the inland stations at the current (albeit high) accuracy of our GPS tidal displacement estimates.

Figure 11.

Distribution of RMS differences of ZOTL against ZobsZbody among six recent ocean tide models, for each tidal constituent as a function of the logarithmic distance from each station to the nearest coastline. The vertical line depicts the 150 km distance that separates inland from coastal.

[55] On the other hand, the ZOTL errors dominate at most coastal stations without apparent long-wavelength spatial coherence. The RMS misfits of the six ocean tide models for some coastal stations are quite large, up to 0.87 mm (east), 0.31 mm (north), and 2.37 mm (up) for M2. Moreover, the misfits w.r.t. GOT4.7 at coastal stations are much larger than the RMS disagreement among different ocean tide predictions and can reach up to 1.93 mm (east), 1.19 mm (north), and 4.11 mm (up) for M2. It clearly indicates that the accuracy of ocean tide models is still the limiting factor for coastal ZOTL predictions.

3.4 Atmospheric Tidal Loading (ATL)

[56] Mass loading effects due to the atmospheric S1 and S2 thermal tides (as opposed to gravitational tides) have also received attention in geodetic analyses [Tregoning and Watson, 2009; van Dam and Ray, 2008]. An S1 and S2 ATL model proposed by van Dam and Ray [2010] based on the model of Ray and Ponte [2003] is currently being considered to be included in the station displacement models of the IERS Conventions 2010 [Petit and Luzum, 2010]. Such ATL vertical deformation would reach 1–2 mm near Equator and diminish to negligible toward the poles, hence potentially exceeding the uncertainty of our GPS tidal observations. Horizontal deformations are smaller by an order of magnitude. It should be noted that the solar heating occurring at S1 and S2 periodicities can also affect the physical GPS monuments.

[57] In order to assess the sensitivity of our GPS tidal estimates to the ATL effect, we compare the S2 tidal displacement estimates with the S2 ATL predictions calculated by van Dam and Ray [2010]. It is found that the latter are too small in the horizontal component (less than 0.2 mm) to explain our residuals Zres as above. For the vertical residuals, barring some slight changes in the statistics of Zres given in Figure 12, applying the ATL model actually increases the RMS misfits from 1.17 to 1.34 mm, implying the possible existence of significant errors in the ATL predictions or other related sources. In addition, the P1 and K1 residuals may also include the effects of S1 ATL considering the nearness of their periods. From our analysis on the global scale, we assert that ATL is indiscernible from our tidal displacement estimates at the present level of accuracy. This awaits further investigations.

Figure 12.

Spatial distribution of phasor vectors of Zres for the S2 Up tidal displacement (same convention as before), before (black arrows), and after (brown arrows) removing the effects of modeled ATL displacements. The effect of tidal geocenter motions has been removed beforehand.

4 Conclusions

[58] We obtain and analyze, to the extent feasible with the present GPS data, the 3-D tidal displacements at 456 global, continuous IGS stations for eight principal semidiurnal and diurnal tidal constituents over the 16 year period from 1996 to 2011. We account for all the recognized GPS error sources and include the minor tides in the solutions and correct for the effects of the tidal geocenter motions.

[59] We separate the 456 global stations into inland (307) stations and coastal (149) stations, depending on whether the station is beyond or within 150 km from the coastline. We then make detailed comparisons w.r.t. the calculated tidal displacement field according to models of body tides as well as OTL, the two major sources of tidal displacement. The additional ATL effects for S1 and S2, although relatively small, remain undetermined at the present time in lack of proper modeling or relevant observations.

[60] Based on our error assessments, we assert that the accuracy in our GPS displacement estimates is better than 0.12 mm (horizontal) and 0.24 mm (vertical) for the lunar-only constituents (M2, N2, O1 and Q1), at 0.69 mm (horizontal) and 1.29 mm (vertical) for S2 and P1, and somewhat poorer for K2 and K1 due to inherent errors in satellite orbit modeling and multipath effects. We note that ambiguity resolution in the GPS data processing improves the precision, especially for the four solar-related constituents, S2, K2, K1, and P1.

[61] At such precision, we have demonstrated that the accuracy of our careful solution of the GPS tidal displacements exceeds that of the present a priori tidal models. This is true not only for OTL, as already treated by previous investigations [Ito and Simons, 2011; King et al., 2005; Yuan et al., 2009], but also for body tide models. In other words, the present OTL and body tide models are inadequate in the sense that they make predictions that have errors generally larger than those of our GPS solutions. Conversely, then, the GPS solutions provide quantitative source information and insights that can be employed to potentially improve the tidal models, for both OTL and body tides, along with Earth's internal structure models.

[62] In the case of the coastal GPS stations, we find that recent ocean tide models fit our GPS estimates of ZOTL equally well on the global scale, albeit with quite varied degree of agreement in certain coastal areas. This indicates the existence of considerable uncertainties in all the ocean tide models especially in shallow water seas. Although beyond our present scope, our results can thus potentially discriminate among the ocean tide models and provide the models with useful constraints, global or regional. The GPS tidal estimates under improved ocean tide models can presumably shed further light on the shallower lithospheric structure of the region as proposed by Ito and Simons [2011], although the degree of sensitivity for such applications needs further investigation [Yuan and Chao, 2012].

[63] In the case of body tides at the inland GPS stations where the OTL influences are minimal, our high-precision Zres estimates are evidently nonrandom and see continental-scale coherent patterns, particularly for the better-determined constituents M2 (WRMS < 0.37 mm), N2 (< 0.11 mm), and O1 (< 0.19 mm). We take this to be a strong evidence for the presence of errors, to the above-indicated level, of the a priori body tide model, in our case the IERS Conventions 2010. The inland tidal residuals Zres thus provide significant information on the solid Earth's deeper interior not properly modeled by the body tide models at present. Further improvements in the GPS observations are expected to serve as source data for inference for lateral heterogeneities of density and elastic moduli, as well as inelastic structures for both deep Earth interior and regional lithosphere at diurnal and semidiurnal time scales, for example, in the form of tidal tomography inversion [Metivier and Conrad, 2008; Latychev et al., 2009].

[64] Similar analysis scheme can be applied to GPS networks of different settings. For example, Yuan and Chao [2012] used data from the dense GPS array of PBO in the western United States and obtained results that would afford regional solutions for the (complex) Love numbers h and l with detailed spatial resolution. Finally, we note that accounting for the fine structures of improved body tide modeling can in turn lead to even better estimation of ZOTL in coastal regions [Yuan and Chao, 2012], further helping the cause of discriminating ocean tide models and inferring Earth's shallow structures.


[65] We are grateful to IGS and JPL for providing global GPS observations and products, to JPL for providing the GIPSY software, to Duncan Agnew who kindly made the SPOTL software freely available to the public, to Machiel Bos and Hans-Georg Scherneck for providing the ocean tidal geocenter motion coefficients, and to Matt King for useful discussions and kind help with tidal displacement estimation methodology. The providers of the various ocean tide models are acknowledged. We also thank Duncan Agnew and Jeff Freymueller for their constructive and thoughtful reviews. This work was completed during the tenure of author Linguo Yuan in the Institute of Earth Sciences of Academia Sinica and is supported by the National Science Council of Taiwan (No. 98-2116-M-001-015), National Natural Science Foundation of China (No. 41004001, No. 41104007 and No. 41174065), and State Key Laboratory of Geodesy and Earth's Dynamics (Institute of Geodesy and Geophysics, CAS) (No. SKLGED2013-2-2-E). Some of the figures were generated using the GMT software [Wessel and Smith, 1998].