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**Solid Earth**
An interactive open-access journal of the European Geosciences Union

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- Abstract
- Introduction
- Ocean tide model accuracy assessment using tide gauges
- Impact of ocean tide model errors on OTL displacement
- Kinematic GPS estimation of OTL displacement
- Optimal Green's function for the East China Sea region
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement

SE | Articles | Volume 11, issue 1

Solid Earth, 11, 185–197, 2020

https://doi.org/10.5194/se-11-185-2020

© Author(s) 2020. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/se-11-185-2020

© Author(s) 2020. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Developments in the science and history of tides (OS/ACP/HGSS/NPG/SE...

**Research article**
14 Feb 2020

**Research article** | 14 Feb 2020

Asthenospheric anelasticity effects on ocean tide loading around the East China Sea observed with GPS

^{1}Department of Geomatics Engineering, Minjiang University, Fuzhou, China^{2}School of Engineering, Newcastle University, Newcastle upon Tyne, UK^{3}SEGAL, University of Beira Interior, Covilhã, Portugal

^{1}Department of Geomatics Engineering, Minjiang University, Fuzhou, China^{2}School of Engineering, Newcastle University, Newcastle upon Tyne, UK^{3}SEGAL, University of Beira Interior, Covilhã, Portugal

**Correspondence**: Junjie Wang (wangjunjie.gnss@qq.com)

**Correspondence**: Junjie Wang (wangjunjie.gnss@qq.com)

Abstract

Back to toptop
Anelasticity may decrease the shear modulus of the asthenosphere by 8 %–10 %
at semidiurnal tidal periods compared with the reference 1 s period of
seismological Earth models. We show that such anelastic effects are likely
to be significant for ocean tide loading displacement at the *M*_{2} tidal
period around the East China Sea. By comparison with tide gauge
observations, we establish that from nine selected ocean tide models (DTU10,
EOT11a, FES2014b, GOT4.10c, HAMTIDE11a, NAO99b, NAO99Jb, OSU12, and
TPXO9-Atlas), the regional model NAO99Jb is the most accurate in this
region and that related errors in the predicted *M*_{2} vertical ocean tide
loading displacements will be 0.2–0.5 mm. In contrast, GPS observations on
the Ryukyu Islands (Japan), with an uncertainty of 0.2–0.3 mm, show 90th-percentile discrepancies of 1.3 mm with respect to ocean tide loading
displacements predicted using the purely elastic radial Preliminary
Reference Earth Model (PREM). We show that the use of an anelastic PREM-based Earth
model reduces these 90th-percentile discrepancies to 0.9 mm. Use of an
anelastic radial Earth model consisting of a regional average of the
laterally varying S362ANI model reduces the 90th-percentile to 0.7 mm, which
is of the same order as the sum of the remaining errors due to uncertainties
in the ocean tide model and the GPS observations.

How to cite

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How to cite.

Wang, J., Penna, N. T., Clarke, P. J., and Bos, M. S.: Asthenospheric anelasticity effects on ocean tide loading around the East China Sea observed with GPS, Solid Earth, 11, 185–197, https://doi.org/10.5194/se-11-185-2020, 2020.

1 Introduction

Back to toptop
The periodic redistribution of ocean mass around the Earth's surface due to ocean tides deforms the solid Earth, a phenomenon known as ocean tide loading (OTL). The resulting OTL displacements can reach several centimetres in the vertical component and more than 1 cm in the horizontal components, with the Earth's response to the OTL depending strongly on the material properties within its interior (Farrell, 1972). In the past 2 decades, Global Positioning System (GPS) data analysis techniques have been developed to directly measure OTL displacements with millimetre accuracy and even submillimetre accuracy at some frequencies (e.g. Allinson et al., 2004; Thomas et al., 2007; Yuan et al., 2009; Penna et al., 2015). With parallel substantial advancements in the accuracy of global ocean tide models (Stammer et al., 2014; Ray et al., 2019), comparisons of GPS-observed and predicted (modelled) OTL displacements have several times revealed the deficiencies of using spherically symmetric, non-rotating, elastic, and isotropic (SNREI) Earth models. One of the reasons for these deficiencies is that these models have been derived from seismic data and represent the Earth's elastic properties at a reference period of 1 s but have typically been assumed to be directly applicable at tidal frequencies.

Ito et al. (2009) found the average amplitude ratios between GPS tidal
displacement observations and an Earth tidal model (including OTL and Earth
body tide) across Japan were greater than 1, indicating observational
agreement with inelastic Earth models. Ito and Simons (2011) further
attempted to invert GPS-observed displacements for one-dimensional profiles
of the elastic moduli and density beneath the western United States,
demonstrating the limitations of the Preliminary Reference Earth Model
(PREM) (Dziewonski and Anderson, 1981). Also, Yuan and Chao (2012) and Yuan
et al. (2013) reported continental-scale spatially coherent differences
between GPS-observed and predicted OTL displacements at sites located more
than 150 km inland from the coastline and attributed these differences to
elastic and inelastic deficiencies in the a priori Earth body tide model.
Subsequently, these GPS results were used by Lau et al. (2017) to look for
lateral variations in body tide models of the lower mantle. For western
Europe, Bos et al. (2015) showed that large discrepancies exist between
GPS-observed and modelled OTL displacements, arising from disregarding
anelastic dispersion in the asthenosphere that occurs when the elastic
constants of the Earth model are modified to be applicable at tidal periods.
Such an effect could bring about a reduction of around 8 %–10 % of the shear
modulus in the asthenosphere at tidal frequencies. In addition, Martens et
al. (2016) observed spatial coherence among residual *M*_{2} OTL
displacements across South America, postulating deficiencies in the a priori
SNREI Earth models.

Bos et al. (2015) showed the feasibility of representing the behaviour of
the asthenosphere across an absorption band from seismic to tidal
frequencies by a constant quality factor *Q*, which provides a rough
transformation to account for the anelastic dispersion effect. Hence, it can
be postulated that the asthenosphere should always produce ∼8.5 % OTL displacement discrepancies with respect to a purely elastic
PREM-based Earth model, not only in western Europe where Bos et al. (2015) demonstrated this effect but also all over the world. However, these
discrepancies will not be equally observable in all localities, either
because ocean tide amplitudes are too small within the 50–250 km distance
range from the analysis point that samples asthenospheric behaviour or
because regional uncertainties in ocean tide models are too large to be able
to attribute any observed discrepancy to the Earth model. To identify
regions where the findings of Bos et al. (2015) are testable, we have
examined the global distribution of a “detectability ratio”. This is defined
as the ratio between the elastic–anelastic OTL displacement discrepancy
(taken to be the difference between OTL predicted using a purely elastic
PREM Green's function, as described in Sect. 3, and that using Bos et
al.'s (2015) anelastic S362ANI(*M*_{2}) Green's function) as the numerator and the
combination of expected GPS observational and ocean tide model related
errors as the denominator. For the latter, the ocean tide model related
error is characterized as the standard deviation (SD) of the predicted
elastic OTL displacements at each location, using each of the DTU10, EOT11a,
FES2014b, GOT4.10c, HAMTIDE11a, NAO99b, OSU12, and TPXO9-Atlas numerical
ocean tide models (see Table 1 for references). The GPS observational error
is assigned a SD of 0.3 mm following Penna et al. (2015), which assumes
that at least 2.5 years of continuous GPS data will be available.

Figure 1a shows a global 1∕8^{∘} grid of detectability ratio for the
*M*_{2} vertical OTL displacement, which is unfavourable (less than 1) for
most inland and deep ocean regions. Many of the areas where it exceeds 1,
such as off the coasts of southern Greenland, eastern Africa, and western
Central America, are poorly sampled with continuously operating GPS
networks. However, the East China Sea (ECS) region exhibits a favourable
combination of large OTL displacements and fairly consistent ocean tide
models across much of it, so the detectability ratio here exceeds 3
across a wide area and contains a healthy distribution of long-running GPS
sites (Fig. 1b shows the 102 GPS sites used). Accordingly, we have
selected this as a suitable region for an independent test of Bos et al.'s (2015) conclusions. A further attraction of this region for the testing of
Earth models is that its position overlying a subduction zone means that it
represents a very different tectonic setting to the mature passive margin in
western Europe studied by Bos et al. (2015).

Figure 1c shows the predicted *M*_{2} vertical OTL displacements across the
ECS region using the FES2014b ocean tide model (Carrère et al., 2016)
and an elastic PREM Green's function. It can be seen that the *M*_{2}
vertical OTL displacement amplitudes are as large as 20–25 mm around the
Ryukyu Islands and on the southeast coast of China, so the anelastic OTL
displacement discrepancies would be expected to be about 2 mm and therefore
detectable using GPS. Overall, the accuracy of recent ocean tide models is
believed to be good, e.g. Stammer et al. (2014) show sub-centimetre *M*_{2}
root mean square (RMS) agreement between bottom pressure observations and
seven recent models in the deep oceans globally and additionally, the
FES2014b model has been suggested as providing a clear advancement in global
ocean tide modelling (Ray et al., 2019). However, the fact that the tides in
the ECS are large and complex owing to the irregular geometry of the basin
(Lefèvre et al., 2000) implies that careful evaluation of the ocean tide
models is still necessary in this region to ascertain the optimal model and
thus minimize the effect of errors in ocean tide models on the OTL
predictions.

In this paper, we first assess the accuracy of a selection of up-to-date
ocean tide models in the ECS and quantify their contribution to the
predicted OTL error budget. We then describe the kinematic GPS analysis
approach for obtaining the observed OTL displacements. Finally, we examine
the evidence of asthenospheric anelasticity effects in the ECS region based
on the GPS-observed OTL displacements. We consider the *M*_{2} constituent
and the vertical component of OTL displacement, as these are dominant in the
ECS region.

2 Ocean tide model accuracy assessment using tide gauges

Back to toptop
A prerequisite for using GPS measurements of OTL displacement for
evaluating the Earth's interior material properties is that the impact of
ocean tide model errors on the predicted OTL displacement is understood and
found to be near negligible. Therefore, we first evaluate the quality of
ocean tide models in the ECS region (considered throughout this paper as
116 to 133^{∘} E in longitude and 23 to
42^{∘} N in latitude) by assessing their consistency with each
other and by comparing them with tide gauge observations.

To date, no single ocean tide model has been demonstrated as optimal in all
regions of the world (Stammer et al., 2014; Ray et al., 2019), so we
selected eight recent global models (DTU10, EOT11a, FES2014b, GOT4.10c, HAMTIDE11a,
NAO99b, OSU12, TPXO9-Atlas) and one regional model (NAO99Jb) for the
quality assessment. The key features of the models are listed in Table 1.
All models, except for GOT4.10c, directly assimilate TOPEX/Poseidon (T/P)
altimeter data plus, for some of the models, data from one or more of the
ERS-1/2, Geosat Follow-On (GFO), Jason-1/2, Envisat, and ICESat altimetry
satellites, as well as tide gauge data. FES2014b, HAMTIDE11a, NAO99b, and
TPXO9 are barotropic data-assimilative models. DTU10 and EOT11a are both
based on an empirical correction to the global hydrodynamic tide model
FES2004 (Lyard et al., 2006), while the a priori model for GOT4.10c is a
collection of global and regional models blended at mutual boundaries. OSU12
is a purely empirical model determined by an analysis of multi-mission
satellite altimeter measurements. TPXO9-Atlas is obtained by combining the
base global TPXO9 and local solutions for all coastal areas including around
Antarctica and the Arctic Ocean. The regional model, NAO99Jb, covers the
area from 110 to 165^{∘} E in longitude and from
20 to 65^{∘} N in latitude, including the whole area
of our considered ECS region, and assimilates more local tide gauge data
than do the other models.

^{a} T/P, TOPEX/Poseidon; GFO, Geosat Follow-on; TG, tide gauge.
^{b} E, empirical adjustment to an adopted a priori model; H, assimilation
into a barotropic hydrodynamic model.

To evaluate the consistency among the different ocean tide models for the
dominant *M*_{2} constituent, all models were bilinearly interpolated onto
a common 1∕16^{∘} grid across the ECS region, and the SDs of the
phasor differences from the mean were computed per grid point using Eq. (2) of Stammer et al. (2014) and are shown in Fig. 2. It can be seen that
away from the coastlines, all models are quite similar with the SDs no more
than 1–2 cm, which likely arises because they have more or less assimilated
the same altimeter data, albeit over different durations. However, closer to
the coast large intermodel discrepancies arise, especially in the Seto
Inland Sea and near the coast of eastern China and the western Korean Peninsula, where the
SD exceeds 30 cm in places. To check if the large discrepancies are caused
by the older models, we considered the three most recent models (FES2014b,
GOT4.10c, and TPXO9-Atlas) and computed the differences per pair of
FES2014b-GOT4.10c, FES2014b-(TPXO9-Atlas), and GOT4.10c-(TPXO9-Atlas).
However, similar patterns and size of errors as in Fig. 2 were obtained
with the modern model difference pairs. The only changes were that the
intermodel differences for the more modern models tend to tail off slightly
more rapidly on moving away from the coast of eastern China.

To ascertain which models are the causes of the large SDs in some subareas and to assess their accuracy, we compared each model with observations from 75 coastal tide gauges (58 from the Japan Oceanographic Data Center and 17 from the University of Hawaii Sea Level Center) in the ECS region, as shown in Fig. 2. Unfortunately no tide gauge data are currently available within the Korea subarea. Using the UTide package (Codiga, 2011), the tidal constants observed at these locations were deduced from hourly sea level time series spanning 4 to 69 years, with a median time-series length of 26 years. For time series shorter than 18.6 years, we applied nodal corrections during the harmonic tidal analysis (Foreman et al., 2009). The observed tidal constants are listed in Table S1 in the Supplement.

In order to investigate in detail the problematic coastal areas of eastern
China, the western Korean Peninsula, and the Seto Inland Sea, the region is divided into the
separate subareas shown in Fig. 2, basically in accordance with the zones
of intermodel discrepancy. Moreover, for the sake of describing the ocean
tide model errors as precisely as possible in the next section, the subarea
denoted as Kyushu is further divided. The *M*_{2} phasor difference between
each model and each tide gauge was computed, and the RMS of these
differences per model for all tide gauges in each subarea is listed in
Table 2.

For eastern China, FES2014b and NAO99Jb perform quite well (RMS of
10–12 cm), whereas DTU10 and EOT11a are the worst models (RMS of 47–59 cm).
This could be explained by the fact that the FES2004 model, on which DTU10
and EOT11a are both based, has several grossly incorrect tidal values in
this area owing to the insufficient satellite altimetry data available at the
time. Such problems with the earlier set of finite element solution (FES) ocean tide models were also
seen from tidal gravity observations in Wuhan, China (Baker and Bos, 2003),
near this subarea. RMS agreements of better than 4 cm between tide gauge
observations and each of the models are obtained for the Ryukyu Islands
subarea, except for TPXO9-Atlas. This is despite TPXO9-Atlas having the
finest resolution among the models (1∕30^{∘}), whereas the coarser
(1∕2^{∘}) GOT4.10c and NAO99b models have better than 4 cm RMS
agreement. Around the island of Kyushu, the observations compare
consistently well with FES2014b and NAO99Jb (RMS lower than 4 cm), while the
comparisons are poor for DTU10, EOT11a, HAMTIDE11a, OSU12, and TPXO9-Atlas
along the west coast of Kyushu and for GOT4.10c and NAO99b along the north
coast of Kyushu. NAO99Jb exhibits the best agreement with the observations
in the Ariake Sea and Seto Inland Sea, which is expected as it assimilates
data from 219 local tide gauges (Matsumoto et al., 2000). This also results
in NAO99Jb being more accurate than NAO99b in most parts of the ECS region.
However, the agreement between NAO99Jb and the tide gauges is no better than
the other models in the Kanmon Straits, because the tide gauges there were
installed in 2011, after the release of NAO99Jb, and hence none of their
data have been assimilated. Nonetheless, NAO99Jb is the most accurate ocean
tide model in the ECS region as a whole.

3 Impact of ocean tide model errors on OTL displacement

Back to toptop
In this section we assess the impact of ocean tide model errors on the
predicted OTL displacements, which is needed to ensure the confident
geophysical interpretation of the GPS-observed OTL displacement residuals
considered thereafter. For a particular tidal constituent, the OTL
displacement *u* at a point ** r** on the Earth's surface may be computed (predicted)
with the following convolution integral (Farrell, 1972):

$$\begin{array}{}\text{(1)}& u\left(\mathit{r}\right)=\underset{\mathrm{\Omega}}{\int}\mathit{\rho}G\left(\left|\mathit{r}-{\mathit{r}}^{\prime}\right|\right)Z\left({\mathit{r}}^{\prime}\right)\mathrm{d}\mathrm{\Omega},\end{array}$$

where Ω represents the global water areas, *ρ* is the density of
seawater, *G* is a Green's function that describes the displacement at ** r** from a
unit point load, and

$$\begin{array}{}\text{(2)}& u\left(\mathit{r}\right)=\sum _{\mathrm{\Omega}}\mathit{\rho}{Z}_{i}{G}_{i},\end{array}$$

where *G*_{i} here is the integrated Green's function for the *i*th element of
Ω, as per Agnew (1997), and the tidal heights *Z*_{i} are represented
over Ω by inputting a global ocean tide model.

Bos et al. (2015) took the SD of predicted OTL displacements computed per
point for a set of ocean tide models as the error contribution of the ocean
tide models in western Europe, assuming that there were no systematic biases
shared by the models. However, we have shown in Sect. 2 that for the ECS
region, the SD among the models is not always a good indicator of their
accuracy. To check this, *M*_{2} vertical OTL displacements were computed
for a 1∕8^{∘} grid across the ECS region for each of the nine ocean
tide models (NAO99Jb was augmented globally outside its boundary by
FES2014b) using the SPOTL (NLOADF) software version 3.3.0.2 (Agnew, 1997). A
Green's function computed based on the isotropic, purely elastic version of
PREM was input (as for all elastic PREM-generated results in this paper) and
is provided in Table S3 in the Supplement. As the GPS sites considered in
this study are on land, the upper 3 km water layer in PREM was replaced with
the density and elastic properties from the underlying rock layer. The OTL
displacement SDs among the models per point are shown in Fig. 3a, and it
can be seen that the distribution of the SDs is similar to those shown for
the ocean tide models in Fig. 2, with large SDs of up to 2.5 mm arising
around eastern China, the western Korean Peninsula, and the Seto Inland Sea. However, as
shown in Sect. 2, these large SDs arise from large errors in some (but
not all) of the nine ocean tide models, and NAO99Jb was shown to be the most
accurate model across the ECS region. Therefore, it is unreasonable to use
the intermodel SD as an indicator of OTL displacement accuracy for all of
the ECS region. Instead, we now present an approach which allows us to
quantify (to the first order) the resulting OTL displacement prediction error
individually for a particular ocean tide model.

Assuming the ocean is divided into *k* specified water areas Ω_{k}
(e.g as per Table 2) and that the ocean tide model error magnitude per
area is *δ*_{k}, the corresponding OTL accuracy *δ**u*_{k} is

$$\begin{array}{}\text{(3)}& \mathit{\delta}{u}_{k}\left(\mathit{r}\right)=\sum _{{\mathrm{\Omega}}_{k}}\mathit{\rho}{\mathit{\delta}}_{k}{G}_{i}.\end{array}$$

Then, assuming no correlation between each of the *k* areas, the total OTL
displacement prediction error may be computed as

$$\begin{array}{}\text{(4)}& \mathit{\delta}u\left(\mathit{r}\right)=\sqrt{\sum \mathit{\delta}{u}_{k}^{\mathrm{2}}\left(\mathit{r}\right)}.\end{array}$$

Note that in practice there are likely to be negative correlations between adjacent water areas, which will result in the error estimates from Eq. (4) being too large (conservative).

To evaluate the OTL error using Eq. (4) for NAO99Jb, the most accurate ocean tide model in the ECS region, we define the ocean tide model errors for the separate subareas (as per Fig. 2) as the RMS difference between NAO99Jb and the tide gauge observations within the subarea (Table 2). For the Korean subarea, although no tide gauge data source is available, the error of NAO99Jb for Korea can be estimated as the mean value of the RMS of the areas around Kyushu excluding the Kanmon Straits, considering the fact that NAO99Jb also assimilated the tide gauge data around the Korean Peninsula. The “other water areas” (comprising the central ECS subarea and all other global water areas not named in Fig. 2) are either open oceans or narrow coastal areas that are far from the ECS. To be conservative, a slightly larger value of 0.7 cm is chosen as the RMS error of NAO99Jb and its complement of FES2014b for these areas, in accordance with the largest RMS model differences of 0.66 cm for deep oceans inferred by Stammer et al. (2014).

Using Eq. (4) and inputting the NAO99Jb RMS errors per subarea, the
*M*_{2} vertical OTL displacement errors at each point of a 1∕8^{∘}
grid were computed and are shown in Fig. 3b. It can be seen that the
largest errors of 1–2 mm are for the points falling within the eastern China
subarea, but these can be explained by the NAO99Jb model having a fairly
large assumed RMS error of 11.7 cm for this subarea, and this has the
largest influence on the OTL displacement there. This is, however, likely very
conservative and results in errors for much of the eastern China subarea
that are too large, because it can be seen from Figs. 2 and 3 that it is
only very close to the coast where large intermodel discrepancies arise.
Away from the coast much of the intermodel ocean tide agreements for the
eastern China subarea are about 2 cm. For the rest of the ECS region,
notably where most of the GPS sites are located, the OTL errors arising from
NAO99Jb model RMS errors are no more than ∼0.5 mm, even for
sites on the east of Kyushu where the intermodel OTL SDs are large
(∼2.5 mm).

To provide a more detailed indication of the influence on the OTL of the NAO99Jb ocean tide model errors from each of the defined subareas, three GPS sites (0487, 0706, and 1094) are considered, located on the east and west sides of Kyushu and on the Ryukyu Islands, respectively (Fig. 3b). The contribution of each subarea to both the OTL displacement and its accompanying error are shown in Table 3, which provides further clarification that the local ocean tides are the principal contributor to the OTL displacements, as well as the OTL errors. The large effect from the “other water areas” is mainly due to their vast area, although most of this is far from our study area and will have no impact on regional comparison of Earth models. The Kanmon Straits and eastern China, where NAO99Jb performs relatively poorly, have little effect on the OTL displacements at these sites, with contributions to the OTL amplitude and error of only 1.0–1.5 mm and less than 0.1 mm, respectively. Furthermore, the effect of the ocean tide model errors from these two subareas is no more than 0.13 mm for all three sites. These computations were repeated for all the GPS sites, and only three of the 102 GPS sites had a total OTL prediction error greater than 0.5 mm. It can therefore be concluded that the OTL displacements computed using the NAO99Jb ocean model are suitable for investigating possible anelasticity effects in the ECS region.

4 Kinematic GPS estimation of OTL displacement

Back to toptop
Using the NASA GNSS-Inferred Positioning System (GIPSY) software in kinematic precise point positioning (PPP) mode, Penna et al. (2015) showed for sites in western Europe with at least 2.5 years of GPS data (4 years recommended) that vertical OTL displacements may be estimated with a precision of about 0.2–0.4 mm. We apply the same approach for GPS sites in the ECS region. In order to assess the accuracy and precision of the OTL displacements, particularly to check that the tuned coordinate and tropospheric delay process noise values for western Europe are applicable for the ECS region, we insert an artificial harmonic displacement per GPS site. We then assess how well it is recovered from the kinematic PPP GPS processing, as per Penna et al. (2015) but in the coordinate time series used for the final OTL displacement estimation rather than as a preliminary investigation step.

All available continuous GPS data in the ECS region were collated for the window 2013.0–2017.0, with the distribution of the 102 sites used shown in Fig. 1. These comprised 96 sites from the GPS Earth Observation Network (GEONET), which all had at least 95 % data availability throughout the 4-year window considered and are located mainly on the Ryukyu Islands and Kyushu. We also collated data from six International GNSS Service (IGS) sites in China and South Korea, although two sites (SHAO and YONS) only had 2.5 years of data. On the Ryukyu Islands and along the coast of Kyushu, the sites exhibit detectability ratios of greater than 1, with the median value being 2.1, although close to the Seto Island Sea the ratio reduces to less than 1. The data spans of at least 2.5 and typically 4 years are sufficient to separate the different major tidal constituents robustly according to the Rayleigh criterion.

Full details of the GPS data processing strategy used are provided in Penna
et al. (2015); in summary it is as follows. Daily, 30 h, kinematic PPP
GPS solutions were generated for each site using GIPSY version 6.4 software
with Jet Propulsion Laboratory (JPL) reprocessed version 2.1 fiducial
satellite orbits, Earth orientation parameters, and 30 s satellite clocks
held fixed in the IGb08 reference frame. A priori hydrostatic and wet zenith
tropospheric delays from the European Centre for Medium-Range Weather
Forecasts reanalysis products were used, with residual zenith tropospheric
delays estimated every 5 min (applying a process noise of 0.1 mm s${}^{-\mathrm{1}/\mathrm{2}}$), together with north–south
and east–west tropospheric gradients. The VMF1 gridded mapping function was
used with an elevation cut-off angle of 10^{∘}, and corrections were
applied for solid Earth and pole tides according to the International Earth Rotation Service (IERS) Conventions
2010 (Petit and Luzum, 2010), along with IGS satellite and receiver antenna
phase centre variation corrections. Ambiguities were fixed to integers
according to the approach of Bertiger et al. (2010). Receiver coordinates
were estimated every 5 min, with a coordinate process noise of
3.2 mm s${}^{-\mathrm{1}/\mathrm{2}}$ applied.
OTL displacement was modelled using the IERS Conventions (2010) HARDISP.F routine, based on amplitudes and phase lags generated using the NLOADF
software with the NAO99Jb model (augmented in the rest of the world with the
FES2014b model) and a PREM elastic Green's function, computed in the centre
of mass of the solid Earth and oceans (CM) frame to be compatible with the
JPL orbits. In each daily solution, an artificial 13.96 h harmonic signal
of 3.0 mm amplitude was introduced in each of the east, north, and vertical
components, with the phase referenced to zero defined at the GPS timeframe epoch
J2000, and hence the GPS harmonic estimation capability with the
aforementioned GIPSY processing settings assessed. The value of 13.96 h was chosen as
the period of this displacement following Penna et al. (2015), as it is
approximately in the semidiurnal band but is distinct from the main tidal
harmonics, so it will not be contaminated by geophysical signals.

The estimated coordinates at 5 min resolution within the central 24 h of
the daily 30 h kinematic PPP GPS solutions (which ran from 21:00 UTC the
previous day to 03:00 UTC the next day) were averaged in nonoverlapping, 30 min
bins then concatenated to form coordinate time series. Harmonic analysis
was then undertaken using UTide to estimate the residual *M*_{2} vertical
OTL displacement signal per site, and also a 13.96 h harmonic was
estimated to assess how well the introduced 3.0 mm amplitude artificial
signal could be recovered. The resulting UTide formal errors were
0.1–0.2 mm.

The *M*_{2} vertical OTL residual phasors extracted from the harmonic
analysis are listed in Table S2 in the Supplement and shown in Fig. 4, along with the artificial 13.96 h harmonic signal residual phasors. It can
be seen from Fig. 4 that on the Ryukyu Islands and in the west coastal
area of Kyushu the *M*_{2} vertical OTL GPS-observed minus model
discrepancies (residuals) can reach over 1.5 mm, corresponding to about
7 % of the total loading signal. The typical magnitudes of phasor
differences between the recovered and original artificial 13.96 h
harmonic signals are 0.2–0.3 mm, providing an indication of the accuracy
level of our GPS-observed *M*_{2} vertical OTL displacements and indicating
that the optimal process noise values found for western Europe by Penna et
al. (2015) are also applicable to the ECS region. Since the ocean tide error
of NAO99Jb maps to only an error of 0.2–0.5 mm for the predicted *M*_{2}
vertical OTL displacement values across the Ryukyu Islands and Kyushu
(Fig. 3b), it can be concluded that the 1.5 mm discrepancies must be
dominated by errors in the elastic PREM Green's function.

5 Optimal Green's function for the East China Sea region

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As Green's functions essentially depend on the material properties of the
adopted Earth models, an improvement of the agreement between GPS-observed
and predicted OTL values (reduction in the observational residuals) could be
expected by modifying the Earth models, and the representation of the
asthenosphere has been demonstrated to be especially important (Bos et al.,
2015). So far we have used Green's functions computed from isotropic, purely
elastic PREM, and we first consider whether the more recent elastic S362ANI
Earth model (Kustowski et al., 2008), which is a transversely isotropic
seismic tomographic model for the mantle, results in a reduction in the
residuals. This model provides horizontal and vertical shear velocities
(transversely isotropic) on a regular longitude–latitude grid for various
depths. For each depth layer between longitudes 122 and
133^{∘} E and latitudes 23 and 35^{∘} N, we
computed averaged shear velocities, which were used to compute the load Love
numbers following Bos and Scherneck (2013), together with the density and
compressional velocities of the STW105 model that was also developed by
Kustowski et al. (2008). It can be seen from Table 4 that using the elastic
S362ANI Green's function reduces the overall RMS of the residuals by about
0.1 mm compared to the elastic PREM Green's function (and similarly for the
maximum and 90th-percentile values), which could be explained by its use of
the regional mean shear velocity.

We next considered whether using Green's functions with the anelastic
dispersion effect in each of PREM and S362ANI results in reductions in the
residuals. The elastic properties for these Earth models have been derived
from seismic observations and are valid at the reference period of 1 s. To
include the anelastic dispersion effect, the values of the shear modulus
were converted from a period of 1 s to the period of the *M*_{2} harmonic
using the relation formula given by eq. (9.66) in Dahlen and Tromp (1998)
with a constant absorption band, as described by Bos et al. (2015). The bulk
modulus has a much higher-quality factor *Q* and is assumed not to be affected.
After modifying the shear modulus, the load Love numbers were computed as
described in Bos and Scherneck (2013), and the respective anelastic Green's
functions will be hereafter referred to as PREM_M2 and
S362ANI_M2 (these, together with all the Green's functions
used in this section, are listed in Tables S3 and S4 in the Supplement). It
can be seen from Table 4 that use of PREM_M2 and
S362ANI_M2 reduces the overall RMS of the residuals from
∼0.5 to ∼0.4 mm. However, if only the GPS
sites on the Ryukyu Islands are considered, the RMS residual is reduced from
∼0.7 mm with elastic PREM and elastic S362ANI, to
∼0.5 mm with PREM_M2 and to ∼0.4 mm with S362ANI_M2. The respective 90th-percentile
residual values reduce from 1.3 mm with PREM to 0.9 mm with
PREM_M2 and to ∼0.7 mm with
S362ANI_M2. This reduction across the Ryukyu Islands when
using S362ANI_M2 instead of PREM can be clearly seen in
Fig. 5. However, one can observe that the residual phasors for
S362ANI_M2 still show some correlations along the Ryukyu
Islands, which might be due to the tectonic setting of the subduction zone.

As residuals at the ∼0.7 mm level remain after accounting for
anelasticity effects with the regional S362ANI_M2 model, we
also tested the optimality of the Green's function by computing a range of
Green's functions based on different asthenosphere depths and values of *Q*.
For the density and compressional velocity, S362ANI only provides global
mean profiles. In our work, the asthenosphere is defined a priori to be
between depths of 80 and 220 km with a *Q* of 70 as per Kustowski et al. (2008). Following a similar method to Bos et al. (2015), we vary the depths
of the top (D1) and bottom (D2) of the asthenosphere of S362ANI and the
amount of anelastic dispersion (*Q*) in this layer. For each combination of
these three parameters, a new Green's function was computed via the load
Love number formulation. While computing the load Love numbers, we
transformed the shear modulus from the reference period (1 s) to *M*_{2} as
described above. The *Q* value in the other layers is at least twice that of
the asthenosphere, so the frequency dependence will be smaller, but to be
consistent the elastic properties were also transformed to the period of
harmonic *M*_{2}. However, these *Q* values were not varied in our inversion.
New Green's functions were then derived and used to predict the *M*_{2}
vertical OTL values using the NAO99Jb ocean tide model. This transformation
produces complex-valued shear moduli and therefore complex-valued Green's
functions, but the imaginary part is less than 5 % of the real part (Bos et
al., 2015) and can be neglected. The optimal Green's function was
considered to be that which minimized the sum of the squared misfits between
the observed and predicted OTL phasor values using all the GPS sites. It was
obtained when *Q* was 90 (corresponding to a reduction of the shear modulus of
about 7.6 % at the *M*_{2} period), and the estimated values of D1 and D2
were 40 and 220 km, respectively, implying an asthenosphere extending to
shallower depths than its original definition for this region in S362ANI. It
can be seen from Table 4 that the residuals statistics with this
mod_S362ANI_M2 Green's function are
practically identical to those using S362ANI_M2 (and although
not shown, very similar patterns for the residuals arise as in Fig. 5),
which confirms the large influence of the asthenosphere. In terms of
practical usage for the region, S362ANI_M2 with its
correction of S362ANI for anelastic dispersion provides a simple way to
improve predicted OTL displacements instead of performing the complex
numerical optimization scheme each time. As a further simple practical
implementation for the region, the global PREM_M2 leads to
almost comparable results as S362ANI_M2.

As a further check on the explanation of the observed and model
discrepancies which we have attributed to asthenospheric anelasticity, we
used the CARGA programme (Bos and Baker, 2005) to compute the effect of
varying seawater density on the *M*_{2} vertical OTL displacements for the
102 GPS sites. First we computed the OTL displacements using a constant
global density of seawater of 1030 kg m^{−3}. Then we recomputed the OTL
displacements using the spatially varying (0.25^{∘} × 0.25^{∘}) mean column seawater density from the World Ocean Atlas
(Boyer et al., 2013) and found the mean change in *M*_{2} vertical
amplitude at our GPS sites was 0.03 mm (maximum difference of 0.16 mm). We
then also corrected the mean seawater density per column for
compressibility according to Ray (2013) and found that the mean change in
*M*_{2} vertical OTL displacement amplitude increased to 0.11 mm (maximum
difference of 0.37 mm). Whilst such magnitude differences now have the
potential to be detectable by geodetic observations, they are too small to
explain our observed 1.5 mm discrepancies.

6 Conclusions

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By introducing the detectability ratio for the asthenospheric anelasticity
effects and considering the distribution of the available GPS sites, the ECS
region was selected as a potential area to observe the anelastic dispersion
in the asthenosphere. Using an intercomparison of eight recent global models
(DTU10, EOT11a, FES2014b, GOT4.10c, HAMTIDE11a, NAO99b, OSU12, TPXO9-Atlas)
and one regional (NAO99Jb) model and a validation with tide gauges, NAO99Jb
has been demonstrated to be the most accurate tide model in the region. In
the open sea areas NAO99Jb could be slightly worse than the other ocean tide
models, due to the assimilation of more satellite altimetry data in the
latter, but this does not outweigh the benefits of forcing the NAO99Jb model
to fit a large number of tide gauge observations. We quantified the impact
of the errors in NAO99Jb on the predicted OTL values, based on the RMS
difference between NAO99Jb and the tide gauge observations. Compared to the
approach of using the SD of predicted OTL displacements as the error
contribution of the ocean tide models, this method can allow for systematic
biases shared by the models, so the outputs are more conservative. For the
GPS sites located in Japan, the errors in NAO99Jb result in *M*_{2} vertical
OTL displacement errors of 0.2–0.5 mm.

We then estimated the *M*_{2} vertical OTL displacements for 102 sites
around the ECS using GPS with typical accuracy of 0.2–0.3 mm. On the Ryukyu
Islands and in the west coastal area of Kyushu, the discrepancies between
GPS-observed and predicted values can reach over 1.5 mm (1.3 mm 90th
percentile) when using the NAO99Jb tide model and the purely elastic PREM
Green's function. The discrepancies cannot be explained by the sum of the
remaining errors due to ocean tide models and the uncertainty in the GPS
observations themselves or by the small change in elastic parameters that
results from using a regional average of the elastic S362ANI model in place
of PREM. However, modelling of the anelastic dispersion effect using the *Q*
values, which lowers the shear modulus by about 8 % in the asthenosphere,
reduces the 90th-percentile discrepancies to 0.9 and 0.7 mm for PREM and
S362ANI, respectively. We estimated a regionally optimal Green's function by
varying the depth and thickness of the asthenosphere of the S362ANI Earth
model and its *Q* values, but this resulted in essentially no further reduction
in the discrepancies.

This paper has confirmed the importance of considering the asthenospheric anelasticity effects observed by Bos et al. (2015). It is necessary to incorporate dissipative effects for the Green's functions based on seismic Earth models; use of elastic parameters at 1 s period is insufficient. The PREM_M2 Green's function is near-optimal for the ECS region and western Europe, and it represents a sensible compromise with global applicability, so it is therefore a pragmatic choice for OTL prediction in geodetic analysis. For sites in areas where the detectability ratio exceeds 1 as shown in Fig. 1a or where the highest accuracy is demanded, a regional anelastic Green's function calculated directly from a laterally varying Earth model such as S362ANI should be considered.

Data availability

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Data availability.

GPS data were obtained from the Crustal Dynamics Data Information System, one of the International GNSS Service data centres (ftp://cddis.nasa.gov/gnss/data/daily/, Crustal Dynamics Data Information System, 2016) and by request from the GPS Earth Observation Network (GEONET) of the Geospatial Information Authority of Japan (GSI) (http://datahouse1.gsi.go.jp/terras/terras_english.html, GSI, 2015). The ocean tide models used were those provided in the SPOTL software version 3.3.0.2 distribution (https://igppweb.ucsd.edu/~agnew/Spotl/spotlmain.html, Agnew, 2013), except that FES2014b was obtained from https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/global-tide-fes.html (LEGOS, NOVELTIS, and CLS, 2016), OSU12 was obtained from https://geodesy.geology.ohio-state.edu/oceantides/OSU12v1.0/readme1st.dat (Fok et al., 2012), TPXO9-Atlas was obtained from http://volkov.oce.orst.edu/tides/tpxo9_atlas.html (Egbert and Erofeeva, 2019), and GOT4.10c was provided by its author Richard Ray from the NASA Goddard Space Flight Center via personal communication (2016). The Earth models were obtained from http://ds.iris.edu/ds/products/emc-referencemodels/ (IRIS DMC, 2011); JPL orbits clocks were obtained from https://sideshow.jpl.nasa.gov/pub/JPL_GPS_Products_IGb08/ (JPL, 2019); tide gauge data was obtained from the Japan Oceanographic Data Centre (https://www.jodc.go.jp/jodcweb/JDOSS/index.html, JODC, 2018) and the University of Hawaii Sea Level Center (ftp://ftp.soest.hawaii.edu/uhslc/rqds, Caldwell et al., 2015).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/se-11-185-2020-supplement.

Author contributions

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Author contributions.

PJC, NTP, and JW devised the study. NTP undertook GIPSY processing of the GPS data. JW carried out the analysis of the tide gauge observations, ocean tide loading displacements, and GPS coordinate time series under the supervision of NTP and PJC, and MSB computed the elastic and anelastic Green's functions. All authors contributed to the discussion of the results and writing of the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Developments in the science and history of tides (OS/ACP/HGSS/NPG/SE inter-journal SI)”. It is not associated with a conference.

Acknowledgements

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Acknowledgements.

All data providers listed in the Data Availability statement are thanked, as well as Duncan Agnew, Daniel Codiga, and NASA JPL for providing the SPOTL, UTide, and GIPSY software packages, respectively. The figures were generated using the Generic Mapping Tools software (Wessel et al., 2013). Constructive reviews and comments by Duncan Agnew, Richard Ray, Philip Woodworth, and an anonymous reviewer are appreciated.

Financial support

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Financial support.

This research has been supported by the Chinese Scholarship Council (grant no. 201606710069) and the UK Natural Environment Research Council (grant no. NE/R010234/1).

Review statement

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Review statement.

This paper was edited by Philip Woodworth and reviewed by Duncan Agnew, Richard Ray, and one anonymous referee.

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Short summary

Changes in the Earth's elastic strength at increasing timescales of deformation affect predictions of its response to the shifting weight of the oceans caused by tides. We show that these changes are detectable using GPS and must be accounted for but that 3-D or locally-tuned models of the Earth's behaviour around the East China Sea provide only slightly better predictions than a simpler model which varies only with depth. Use of this model worldwide will improve precise positioning by GPS.

Changes in the Earth's elastic strength at increasing timescales of deformation affect...

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