Influence of Ocean Tidal Loading on Continuous Gravity Observations in Eastern China

Global and regional ocean tide models are used to analyze the gravity effect of ocean tidal loading (OTL) for gravity stations in East China. The accuracies of OTL correction results for 21 gravity stations in East China are evaluated. The global ocean tide model is the most effective for the OTL correction of inland gravity stations (up to 90%) but is less effective for coastal gravity stations (only 60%). Considering regional ocean tide models, the applicability of OTL correction increased to 80% in coastal gravity stations. Based on the root sum square (RSS) method, among 16 combination models, the optimal combination model for OTL correction is the global model FES2014b by combing the regional model OSU.Chinasea.2010 (F14O). The RSS, which has reached 7.1 nms−2, is the minimum of the 16 combination models. Simulating with the F14O in the China Sea and adjacent areas, the gravity amplitude of the OTL is about 10 nms−2 in inland areas and > 50 nms−2 along the coastline. Especially in Southeastern China coastal areas and the southwestern coastal areas of the Korean Peninsula, the gravity amplitude of the OTL reaches about 80 nms−2. Moreover, the OTL changes drastically possibly owing to coastal topography. The results of this study provide a reference for selecting ocean tide models for high-precision analysis of continuous gravity observations in East China.


Introduction
In high-precision tidal gravity observations on the Earth's surface, ocean tide is an important source of disturbance. In coastal areas, the gravity effect of ocean tidal loading (OTL) can reach several micro-Gal (1 lGal = 10 nms -2 ), and OTL displacement can reach [ 10 cm. Therefore, the gravity effect of OTL cannot be ignored when the gravity tide is analyzed at coastal stations (Yang et al., 2016). Both ocean tides and solid earth tides are gravitational interactions with the moon and sun. Therefore, they are coupled together with the same frequency. It is difficult to remove the gravity effect of OTL from the solid earth tide by filtering methods. Thus, a convolution method with the ocean tide model and Green's function, first proposed by Farrell (1972), has been used to calculate the gravity effect of OTL. Recently, with improvements in the resolution and coverage of modern satellite altimetry and in the accuracy of tide gauge stations, many new global ocean tide models have been developed. Models with the highest accuracy and widest application include SCHWIDERSKI, NAOglobal.99b, FES2004, GOT4.7, DTU10, OSU.TPXO72.2010, EOT11a, HAMTIDE11a, FES2014b, and EOT20 (Schwiderski, 1980a(Schwiderski, , 1980bLe Provost et al., 1994;Matsumoto et al., 2000;Lyard et al., 2006;Cheng and Andersen, 2011;Fok, 2012;Savcenko and Bosch, 2012;Ray, 2013;Taguchi et al., 2014;Lyard et al., 2021;Har-Davis et al., 2021).
Many global ocean tide models have been constructed with modern satellite altimetry and tide gauge sea level data. However, owing to the complex local coastline geometry, regional seafloor topography in coastal areas, and the different data sources of the models, issues remain in assessing the gravity effect of the OTL by global ocean tide models in coastal areas (Hinderer et al., 2020;Yang et al., 2016). Numerous studies have aimed to improve the OTL simulated method and identify optimal regional tide models. Ocean tide models have been evaluated by the gravity residual vector before and after OTL correction in gravity stations, the ocean tide amplitude between the tide gauges and models, and the root sum square of the ocean tide models (Hinderer et al., 2020;Penna et al., 2008;Stammer et al., 2014;Sun et al., 2006;Tan et al., 2021;Wei et al., 2021). Many superconducting and spring gravimeters have been installed for researching the gravity effect of OTL in gravity stations. The relative error of the OTL has achieved 0.3% by superconducting gravity stations (Sun et al., 1998;Spiridonov and Vinogradova, 2020;Wei et al., 2021). The selection of the global and regional ocean tide models, the elevations of the gravity stations, and the precision and number of the gravimeters are the most important factors for evaluating the gravity effect of the OTL (Hinderer et al., 2020;Sun et al., 2006). In the coastal areas of China, there is only a minor difference in OTL correction among different global ocean tide models. The FES (Finite Element Solution) and NAO (National Astronomical Observatory of Japan) models are considered the best ocean tide models. In terms of regional models, NAOregional.99Jb and OSU.Chinasea.2010 can be used (Li et al., 2012;Sun et al., 2006;Yang et al., 2016). However, all the discussions have limitations and non-uniqueness. For example, some of the discussions have used a single gravimeter. Others only used a few gravimeters in a small region of China. There are few universal results, and they are based on different gravimeter types. The influence of gravity stations' elevation has been ignored in the OTL evaluation.
In this study, aiming to evaluate the accuracy of the gravity data in 21 gravity stations with 4 different types of gravimeters, 8 global, 2 regional, and 16 combination ocean tide models were used to investigate the gravity effect of the OTL in East China. To identify the optimal ocean tide models, all the ocean tide models have been assessed with the ratio of the OTL correction, root mean square (RMS), and root sum square (RSS) methods by considering the influence of the stations' elevation and position. Then, the spatial distribution characteristics of the OTL amplitude have been analyzed in East China by the optimal ocean tide model. The optimal ocean tide model will be provided for studying the gravity effect of the OTL by high-precision tidal gravity observation in East China.  (Table 1).

Data Pre-processing, Tide Analysis, and Calibration
The continuous gravity data are processed by Tsoft and ETERNA 3.4 (Van Camp and Vauterin, 2005;Wenzel, 1996). The original second sampling gravity data in gravimeters are downsampled with the average filter being 1-min sampling time series. The removal and recovery method is used to remove gaps, steps, and spikes from the 1-min sampled time series. Next, the time series is corrected by the calibration factors (Hwang et al., 2009;Wei et al., 2021). Then, the 1-min sampling time series is decimated to hourly data and analyzed with co-located atmospheric pressure by ETERNA 3.4. The length of the observation data, calibration factors, mean errors (r) of tidal parameters, and atmosphere admittance of the gravity stations are listed in Table 1. From Table 1, the mean errors of the tidal parameters in the diurnal wave are larger than in the semidiurnal wave. The mean errors of M2, which is the most important indicator of the quality for tidal gravity observation, is approximately ± 4 -8 9 10 -4 for PET and GPH and approximately ± 3 9 10 -4 for SG in XZ. The accuracy of GPH or PET, which is close to that of SG, can be provided for OTL research. Although the M2 accuracy of DZW is only 1/3 or 1/4 of SG, the results are still acceptable (Wei et al., 2021). In addition, the mean errors of O1 and P1 in PET, GPH, and DZW are two to five times higher than that of the SG. The accuracy acquired for only 1 year (Table 1) is lower than the result obtained by Wei (2021), who reported an accuracy of approximately ± 20 9 10 -4 for 2-3 years. Accuracy was related to the length of gravity data (Hwang et al., 2009). The atmosphere admittances in Table 1 are consistent with the theoretical values, which are in the range of -3.8 to -2.2 nms -2 /hPa (Ducarme et al., 2002). Considering the accuracy and range of the atmosphere admittance of these spring relative gravimeters, the tidal observation data could be used to analyze the gravity effect of the OTL in East China.

Ocean Tide Models
The construction of the global ocean tide model is usually divided into two approaches: (1) the empirical approach based on the direct analysis of the satellite altimetry data and (2) the modeling approach   (Cheng & Andersen, 2011;Fok, 2012;Le Provost et al., 1994;Lyard et al., 2006Lyard et al., , 2021Matsumoto et al., 2000;Ray, 2013;Savcenko and Bosch, 2012;Taguchi et al., 2014). Two regional models are NAOregional.99Jb from the National Astronomical Observatory (NAO, Japan) and OSU.Chinasea.2010 from the Oregon SU (Oregon State University, USA) (Egbert and Erofeeva, 2002;Matsumoto et al., 2000). The information on these models is summarized in Table 2.
Among the global ocean tide models, the latest FES2014b has the highest spatial resolution (1/16°) and the most tidal waves (34). It contains 9 nonlinear, 7 long-period, and 18 linear diurnal and semidiurnal wave groups. The regional ocean tide models cover the area near the Chinese mainland (Egbert and Erofeeva, 2002;Matsumoto et al., 2000). Besides the single model, the combination models (Global ? Region) are also evaluated.

Calculation of the Gravity Effect of the Ocean Tidal Loading
The OTL correction was divided into two steps. First, the OTL was calculated by the convolutions between Green's functions and the tidal height of global ocean tide models (Farrell 1972). There are many types of Green's functions, including displacement, gravity, tilt deformation, and so on. In the study, the mass loading Green's function is selected to simulate the gravity effect of the OTL. The gravity effect of the OTL can be calculated with Eq. (1): where Dg e is the gravity variation caused by the direct attraction of seawater to the station, Dg l is indirect gravity variation from OTL, G is Newton's gravity constant, q w is the density of seawater, R is the mean radius of the Earth, h(u, k) is the tidal height at the loading point (u is latitude and k is longitude), w is Earth's spherical distance, u = cosw, p = (R ? H)/R, H is the elevation of the measuring point, ds = R 2 cosududk, and K is the mass loading Green's function (Farrell, 1972).
Considering the complex local coastline geometry, the corresponding spatial region in the global ocean tide model has been replaced by the regional ocean tide model (Wang & Zhou, 2020). Equation (1) can be rewritten to Eq. (2): where u 0 is the truncation angle of the near and far regions, h l is the tidal height of the regional model, and h g is the tidal height of the global model.
The SPOTL software package is used to calculate the OTL (Agnew, 1997(Agnew, , 2012. From Eqs. (1) and (2), the OTL is divided into the direct attraction by seawater and the indirect effect by the vertical deformation. The elevation is another important factor for the direct attraction by seawater (Hwang and Huang, 2012). The M2 OTL amplitude is simulated with different elevations (from 0 to 500 m) in Fig. 2. Figure 2 shows that the M2 OTL amplitude of the wave increases gradually with elevation. If the elevation increases by 10 m, the M2 OTL amplitude also increases by 0.15 nms -2 . If the elevation reaches 500 m, the M2 OTL amplitude changes from 37 to 45.5 nms -2 , and the effect on M2 OTL amplitude would be 8.5 nms -2 in XZ. In East China, when the elevation of the gravity stations changes from 3 to 880 m, the effect on M2 OTL amplitude changes  (Table 3). It is comparable to the accuracy (10 nms -2 ) of PET or gPhone, and larger than the accuracy (1 nms -2 ) of SG, such as at MDJ, TA, and CBS. Therefore, the station elevation is estimated by the DEM of the Shuttle Radar Topographic Mission (SRTM: https://srtm.csi.cgiar. org/srtmdata/), which can provide even more accurate and unified elevation data.

Quantitative Evaluation of the Ocean Tide Models
The second step is the OTL correction. In this study, this method has been used for correcting the observed tidal vector.

Corrected and Uncorrected Residual Vector by Ocean Tide Models
The uncorrected residual vector B (B, b) and the corrected residual vector X (X, n), as given in Eqs.
XðX; nÞ ¼ BðB; bÞ À LðL; kÞ ð 4Þ where A is the pressure-corrected tidal vector, B is the residual vector, vectors B and A are not corrected by the ocean tide models, and X is the corrected tidal vector, RES B is the norm of cumulative uncorrected residual gravity vectors, and RES X is the norm of cumulative corrected residual gravity vectors. d 0 and A 0 are the amplitude parameter and theoretical amplitude of the modeled tidal vector A 0 by the Dehant, Ducarme, and Wahr (DDW) model, respectively. DDW is a hydrostatic inelastic body tide model calculated from the PREM Earth model (Dehant et al., 1999). d and a are the observed amplitude parameter and phase of the observed tidal vector A, which is the pressure-corrected tidal vector. B and b are the amplitude and phase of the residual vector B, which is the pressure-corrected residual vector. L and k are the amplitude and phase of the OTL vector L, which can be integrated with the ocean tide models in the gravity station. X and n are the amplitude and phase of the corrected tidal vector X, and m is the index of tidal waves. From Eq. (4), the RES B should theoretically be larger than the RES X . If the noise level of the observation system is small, the RES X should  converge to the accuracy of the gravimeters. Considering the rule of the OTL in China, the OTL in coastal stations should be larger than that in inland gravity stations. All of the RES B must agree on the rule.

Root Mean Square (RMS) and Root Sum Square (RSS)
There are two methods to identify the optimal ocean tide model: the so-called ratio (R) of the OTL correction method based on the RES B and RES X in Eq. (7) and the root mean square (RMS) and the root sum square (RSS) method based on the amplitude parameter, amplitude, and phase between the tidal gravity observation and the ocean tide model. R is an important indicator for checking the applicability of the OTL correction in gravity stations. The larger the R, the better the applicability of the OTL correction is. At the same time, the RMS has been used for evaluating each tidal wave of the ocean tide model in Eq. (8). The RSS can directly evaluate the agreement of the OTL between the ocean tide model and tidal gravity observation in East China (Li et al., 2012) in Eq. (9). The smaller the RSS, the more suitable the ocean tide model is for the OTL correction in that region. The specific expressions of R, RMS, and RSS are as follows: where R is the normalized residual; d 0 , A 0 , and u 0 are the theoretical amplitude parameters, amplitude, and phase (0°) of the DDW model, respectively. d m and u m are the amplitude parameters and phase of the measured data after atmospheric and OTL correction, respectively. K is the number of gravity stations, n is the serial number of gravity stations, M is the number of tidal wave divisions, and m is the index of tidal waves.

Effect of Global Ocean Tide Models
To identify the optimal global ocean tide model for the gravity effect of the OTL in East China, B and X are calculated by the eight global ocean tide models (Table 2), DDW global body tide model, and gravity data (Eqs. (3), (4)). By accumulating B and X (Eqs. (5), (6)), RES B and RES X are compared with different station positions (coastal gravity stations and inland gravity stations) in Fig. 3. Then, the optimal ocean tide models have been identified by the R in eight ocean tide models. This is shown in Fig. 4.
In Fig. 3a, the RES B is about 20-40 nms -2 (blue dotted line a) in inland gravity stations. The RES B of coastal gravity stations (60-120 nms -2 , blue dotted line c) is generally 2-4 times more than that of inland gravity stations in Fig. 3b. The relationship is consistent with the rule that the closer to the ocean, the larger the gravity effect of the OTL is.
Corrected by the eight global ocean tide models, except for DL, the RES X s of the others are about 20-40 nms -2 (red dotted line b, e, and red solid line d). However, some of the RES X s are \ 10 nms -2 , approaching the observation accuracy (10 nms -2 ) of PET or gPhone. For coastal gravity stations, the RES X of stations QD, WZ, XZ, DL, and XM (red solid line d, 50 nms -2 ) is two times larger than that of the other station (red dotted line e, 20 nms -2 ), which are far from the Chinese coastline.
When corrected by global ocean tide models, the RES B is reduced in most gravity stations (Fig. 3). Except for LY, there are about 10 to 20 nm -2 OTL corrections in inland gravity stations. Meanwhile, except for QD, DL, and XM, there are about 20 to 80 nm -2 OTL corrections in coastal gravity stations. The difference in OTL corrections between inland Vol. 180, (2023) Influence of Ocean Tidal Loading on Continuous 941 and coastal gravity stations is in agreement with the OTL rule in East China. In Fig. 4, the R ranged from 20 to 90% for inland and coastal gravity stations (Fig. 4). However, there are only 20%-40% in the gravity stations closer to the coastline (such as QD, DL, and XM). Therefore, the global ocean tide models are available for correcting in inland gravity stations and partially in coastal gravity stations, but are not effective enough in the coastal gravity stations, which are even closer to the coastline.

Effects of Regional Ocean Tide Models
The amplitude of the ocean tide models changes greatly along the coastline of China, owing to the complex topography. Many global ocean tide models have been constructed with modern satellite altimetry and tide gauge sea level data. However, relatively few Chinese tide gauge sea level data and more precise topography data are assimilated in the global models. Therefore, the regional ocean tide models, which contain regional tide gauge sea and even higher resolution coastline data, should be included. Two regional ocean tide models (Table 2) are used to calculate the OTL for all gravity stations. RES B and RES X are calculated for different stations' positions in Fig. 5. R should also be considered as the important indicator for choosing the optimal one in Fig. 6. From Fig. 5a, there is only \ 20 nms -2 OTL correction in inland gravity stations. In coastal gravity stations (such as ZJ, SZ, ZZ, QZ, PT, and FZ), the RES B is about 20-80 nms -2 (Fig. 5b). Corrected by OSU.Chinasea.2010, the RES X has Figure 5 Comparison of cumulative uncorrected and corrected residual gravity amplitudes by regional ocean tide in inland a and coastal b gravity stations Figure 6 OTL correction ratio of regional ocean tide in inland a and coastal b gravity stations Vol. 180, (2023) Influence of Ocean Tidal Loading on Continuous 943 achieved comparable results to the global models at some coastal gravity stations (Figs. 3b and 5b). The law obtained from Fig. 5 also can be acquired from Fig. 6. The R for inland gravity stations ranges from 20 to 90% (Fig. 6a). However, the R for coastal gravity stations is [ 50% (Fig. 6b). Except for some stations (QD, WZ, XZ, DL, and XM), which are located closer to the coastline, the OTL correction of the OSU.Chinasea.2010 model is better than the NAOregional.99Jb for coastal gravity stations. Thus, different regional ocean tide models have distinct behaviors and can only be used to correct some coastal gravity stations. If the stations are located nearest the coastline, the regional ocean models perhaps may not be used directly.

Combined Effects of Global and Regional Tide Models
From Sects. 3.1 and 3.2, there are different advantages of the global and regional ocean tide models. Thus, the RES B and RES X have been simulated by 16 combination models (shown in Table 4), and the Rs also have been calculated with different ocean tide models and station positions (Fig. 7).
From Figs. 3a and 7a, the RES X corrected by the combination models is similar to that by global models for inland gravity stations. Nevertheless, for some coastal gravity stations (such as DL, QD, and XZ), the RES X corrected by the combination models is less than that of global ones. All of the RES X was corrected by adding the OSU.Chinasea.2010 has been less than that corrected by NAOregional.99Jb for coastal gravity stations (Fig. 6b). For example, in QD, the RES X corrected by NAOregional.99Jb fell from 40 nms -2 to 10 nms -2 . For XZ, the RES X has decreased from 10 nms -2 to 5 nms -2 . Thus, the performance of the combination models is better than that of the global ones in coastal gravity stations.

Evaluation of the Ocean Tide Model Correction with RMS and RSS
For choosing an available ocean tide model in East China, the 16 combination models (Table 5) have been evaluated by Eqs. (8), (9).
From Table 5, the RMS including EOT11a, FES2004, and FES2014b is less than that including the other global models in each tidal wave. The minimum RSS is 7.136 nms -2 , which is calculated by F14O. The FES2014b ? OSU.Chinasea.2010 model represents the optimal ocean tide model in East China.

The OTL by F14O in East China
By using the FES2014b ? OSU.Chinasea.2010 model and the SRTM digital elevation model Table 4 Combinations of global tide models and regional tide models Global ocean tide model  (Agnew, 1997(Agnew, , 2012 in Fig. 8. The OTL amplitudes of O1, K1, N2, and S2 are in the range of 1 to 40 nms -2 (Fig. 8a, c, d, f). The OTL amplitude of P1 is 1 to 10 nms -2 (Fig. 8b). The OTL amplitude of M2, which is in the range of 1 to 100 nms -2 , is the largest one in Fig. 8e. It has changed in the bay area dramatically, e.g., in the Taiwan Strait, Bohai Bay, and Qiongzhou Strait (about 80 nms -2 ). The OTL amplitude has decreased to \ 10 nms -2 in mainland China and the island of Taiwan. In coastal areas, the OTL amplitude has reached about 50 nms -2 , which is larger than the accuracy of many gravimeters and cannot be ignored in coastal gravity observations of coastal stations. The OTL amplitude in the southeastern coastal area is larger than those in the northern coastal area. The feature may be attributed to the Taiwan Strait, in which the ocean tide waves are reflected, and the ocean tidal force interferes (similar to the interfered and reflected light). The OTL amplitude seems to be increased in the Strait, and a similar phenomenon is also present in the southwestern coastal area of the Korean Peninsula. A similar distribution characteristic is found in other tidal waves (Fig. 8b, c, d, e, f).

Figure 7
Comparison of RES B , RES X , and R corrected by combination ocean tide models in inland and coastal gravity stations

Conclusion
In this study, eight global tide models, two regional tide models, and their combination models were evaluated for the OTL gravity effect of 21 gravity stations in East China. The conclusions are as follows.
(1) The global ocean tide models have provided about 10-20 nms -2 OTL correction for inland gravity stations and about 20-60 nms -2 OTL correction for coastal gravity stations. The R of the OTL correction is about 20%-90% by the global ocean tide model in East China. However, compared with about 40% for inland gravity stations and some coastal gravity, the global ocean tide models only provide about 20% for gravity stations that are closer to the coastline. In summary, global ocean tide models are not effective enough to correct the OTL for some coastal gravity stations.
(2) The OTL correction for the regional tide models is 10 nms -2 for inland gravity stations and 20 nms -2 for coastal gravity stations. The regional ocean tide models provided [ 50% the R of the OTL correction for some coastal gravity stations. This can be attributed to their greater precision and higher resolution of the coastline and models. However, it still not effective enough to correct inland gravity stations and the gravity stations closer to the coastline than the global ones. It cannot be used for OTL correction in East China directly. The corrected applicability of the OSU.Chinasea.2010 model is better than that of NAOregional.99Jb for coastal gravity stations. (3) By adding the regional ocean tide model to the global one, the RES X decreases by about 5 nms -2 for inland gravity stations and about 10 nms -2 in coastal gravity stations. These combination models, which have provided [ 40% (R) of the OTL correction for inland gravity stations and 60% for coastal gravity stations, are effective enough to correct the OTL for all coastal gravity stations. Based on the RMS and RSS of the 16 combination models, the FES2014b ? OSU.Chinasea.2010