JOURNAL OF GEOPHYSICAL RESEARCH: SOLID EARTH, VOL. 118, 2618–2632, doi:10.1002/jgrb.50159, 2013
The tidal displacement field at Earth’s surface determined using global GPS observations Linguo Yuan,1,2 Benjamin Fong Chao,2 Xiaoli Ding,3 and Ping Zhong1 Received 30 October 2012; revised 9 March 2013; accepted 11 March 2013; published 30 May 2013.
[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. Citation: Yuan, L., B. F. Chao, X. Ding, and P. Zhong (2013), The tidal displacement field at Earth’s surface determined using global GPS observations, J. Geophys. Res. Solid Earth, 118, 2618–2632, doi:10.1002/jgrb.50159.
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 mgal precision by superconducting gravimeters [Baker and Bos, 2003; Boy et al., 2003].
Additional supporting information may be found in the online version of this article. 1 Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, Chengdu, China. 2 Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan. 3 Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China. Corresponding author: B. F. Chao, Institute of Earth Sciences, Academia Sinica, Taipei 11529, Taiwan. (
[email protected]) ©2013. American Geophysical Union. All Rights Reserved. 2169-9313/13/10.1002/jgrb.50159
[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 = Zobs – Zth 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).
2618
YUAN ET AL.: TIDAL DISPLACEMENTS DETERMINED BY GPS Out-of-phase component
Phase delay In-phase component (Tidal potential reference phase zero)
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 o0.15 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
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YUAN ET AL.: TIDAL DISPLACEMENTS DETERMINED BY GPS
[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]).
[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]
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 variancecovariance 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]
(4)
Δck ¼
8 X
fj Ak; j cos oj t þ wj ðt0 Þ þ mj Φk; j
(1)
j¼1
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; oj is the constituent angular frequency; fj and mj are called the nodal modulation corrections in amplitude and phase respectively; and wj 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 Δck ¼
8 X
Ack; j cos oj t þ wj ðt0 Þ þ Ask; j sin oj t þ wj ðt0 Þ
(2)
j¼1
where Ack; j ¼ fj Ak; j cos Φk; j mj Ask; j ¼ fj Ak; j sin Φk; j mj
(3)
Thus, in contrast to the conventional GPS data analysis, an additional set of 48 tidal displacement parameters are simultaneously estimated for each station.
fj Ak ; j cos oj t þ wj ðt0 Þ þ mj Φk; j ¼ "
Ak; j cos oj t þ wj ðt0 Þ Φk; j þ
X
rjl cos oj t þ wj ðt0 Þ Φk; j þ Δjl
#
l
where Δjl = wjl(t0) wj(t0). Expanding equation (4), the following explicit formulae can be derived for fj and mj 2 fj ¼ 4 1 þ
X
!2 rjl cosΔjl
l
2 X
þ
3 rjl sinΔjl 6 7 l 7 mj ¼ arctan6 41 þ X r cosΔ 5 jl jl
X
!2 31=2 rjl sinΔjl 5
l
(5)
l
[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 variancecovariance 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].
2620
IGS_PPP JPL_PPP JPL_AMB
1 0.5 0 1.5
North 1 0.5 0 3
Up 2 1 0
M2
S2
N2
K2
K1
O1
P1
Q1
Figure 2. RMS misfits of the GPS tidal residuals Zobs – Zbody 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. [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 with pffiffiffi parameters p ffiffiffi the process noise values of 3mm= h and 0:3mm= h (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 variancecovariance 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 0.5 0.4
East (mm)
East
0.3 0.2 0.1 0.5
North (mm)
1.5
0.4
M2 S2 N2 K2 K1 O1 P1 Q1
0.3 0.2 0.1 0 1.5 1.2
Up (mm)
RMS misfit (mm)
RMS misfit (mm)
RMS misfit (mm)
YUAN ET AL.: TIDAL DISPLACEMENTS DETERMINED BY GPS
0.9 0.6 0.3 0 0
500
1000
1500
2000
2500
3000
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.
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YUAN ET AL.: TIDAL DISPLACEMENTS DETERMINED BY GPS
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.
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.
2.3. Impact of Ambiguity Resolution and Orbit Products [27] Since the PPP technique requires the satellitedependent parameters to remain fixed, any errors in satellite orbits and clocks would propagate directly into the stationspecific 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 Zobs – Zbody 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. [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.)
2622
YUAN ET AL.: TIDAL DISPLACEMENTS DETERMINED BY GPS Table 1. Estimates and Their Formal Errors of the Tidal Geocenter Motions for Eight Principal Tidal Constituents X (mm)
Y (mm)
Z (mm)
Tide
cos
sin
cos
sin
cos
sin
M2 S2 N2 K2 K1 O1 P1 Q1
1.472 0.011 0.587 0.011 0.263 0.002 0.135 0.008 1.797 0.013 1.350 0.004 0.606 0.006 0.250 0.001
0.801 0.011 0.327 0.011 0.199 0.002 0.030 0.008 0.921 0.013 0.242 0.004 0.316 0.006 0.021 0.001
1.110 0.011 0.029 0.011 0.259 0.002 0.014 0.008 0.895 0.013 0.877 0.004 0.292 0.006 0.208 0.001
0.291 0.011 0.289 0.011 0.117 0.003 0.156 0.008 1.768 0.013 0.637 0.004 0.576 0.006 0.042 0.001
1.169 0.014 0.193 0.014 0.283 0.003 0.106 0.010 1.074 0.017 0.183 0.006 0.366 0.008 0.042 0.002
1.461 0.014 0.619 0.014 0.291 0.003 0.205 0.010 4.378 0.017 2.871 0.006 1.437 0.008 0.452 0.002
RMS misfit (mm)
RMS misfit (mm)
RMS misfit (mm)
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 Zobs – Zbody 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. [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.
0.6 East 0.4 0.2 0 0.6 CSR4.0 NAO99b FES2004 TPXO7.2 HAMTIDE11a DTU10 EOT11a GOT4.7
North 0.4 0.2
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 Table 2. WRMS Misfits of the GPS Tidal Residuals Zobs – Zbody 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)
0 1.8 Up 1.2 0.6 0
[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.
Component M2
S2
N2
K2
K1
O1
P1
2623
M2
S2
N2
K2
K1
O1
P1
Q1
East
All 0.37 0.31 0.09 0.21 0.34 0.15 0.17 0.03 Inland 0.32 0.28 0.07 0.20 0.30 0.14 0.13 0.03 Coast 0.40 0.33 0.09 0.22 0.36 0.16 0.19 0.04
North
All 0.36 0.36 0.08 0.27 0.38 0.14 0.22 0.04 Inland 0.37 0.35 0.07 0.28 0.39 0.12 0.23 0.04 Coast 0.35 0.37 0.08 0.27 0.38 0.14 0.22 0.05
Vertical
All 0.69 0.93 0.18 0.76 1.49 0.38 0.52 0.12 Inland 0.31 0.73 0.11 0.77 1.32 0.19 0.43 0.08 Coast 0.85 1.04 0.22 0.76 1.59 0.47 0.57 0.14
Q1
Figure 5. RMS misfits of the GPS tidal displacement estimates Zobs – Zbody 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.
Sites
YUAN ET AL.: TIDAL DISPLACEMENTS DETERMINED BY GPS M2 12
8 1.0 mm, east
6
4
Amplitude (mm)
10
2
0 12
8 1.0 mm, north
6
4
Amplitude (mm)
10
2
0 60
40 2.0 mm, up
30
20
Amplitude (mm)
50
10
0
Figure 6. Spatial distribution of phasor vectors of residuals Zres = Zobs – Zth 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. 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).
3.
Results and Discussions
3.1. GPS Tidal Residuals and Accuracy Assessment [36] The final values of the GPS tidal displacement estimates Zobs – Zbody 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
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YUAN ET AL.: TIDAL DISPLACEMENTS DETERMINED BY GPS O1 5
3
0.5 mm, east 2
Amplitude (mm)
4
1
0 5
3
0.5 mm, north 2
Amplitude (mm)
4
1
0 25
15
1.0 mm, up 10
Amplitude (mm)
20
5
0
Figure 7. Same as Figure 6, but for the O1 tide.
GPS=model RMS misfitj;k ¼
N 1X Zj;k;n 2 N n¼1
!1 =2 (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 2 11=2 N X Zj;k;n C B C B C B n¼1 sj;k;n ¼B C N C B X 1 A @ 2 s n¼1 j;k;n 0
WRMSj;k
(7)
where sj,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. [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)
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Figure 8. Same as Figure 6 but zooming-in on North America (left) and Europe (right). corresponds to the K2 period [King et al., 2005, 2008; Schenewerk et al., 2001; Zhong et al., 2010]. By examining long-running, short-baseline (
