Internal error estimate
The geoid error degree variance and cumulative geoid height error of the six gravity field models are calculated by Eqs. (1) and (2), and the results are shown in Fig. 2.
According to Fig. 2(a), the precision of geoid of all models is in millimeter level, and the overall best performance is produced by SGG-UGM-2. The maximum values of geoid error degree variance of EGM2008, EIGEN-6C4, SGG-UGM-1, GECO, SGG-UGM-2 and XGM2019e_2159 are at degree and order 108, 360, 210, 220, 210 and 300, respectively. EGM2008 exhibits larger differences with other models in the range of 80 ~ 200 degrees, which may be caused by the fact that the GOCE satellite data is not used in EGM2008 (Pavlis et al. 2012). The GOCE mission provides great improvements in detecting the low to medium wavelength gravity signals by introducing SGG (Wan et al. 2012; Wan and Yu 2013).
GECO, SGG-UGM-1 and SGG-UGM-2 all adopt EGM2008 as a reference model in the construction process, and thus these three models are compared. The degree variance of GECO’s geoid errors is smaller than that of EGM2008 below degree 280 due to the usage of GOCE data. Above 280, the geoid error degree variances of EGM2008 and GECO are almost the same, which may be related to the fact that GECO uses the same ground and altimetry data as EGM2008 (Gilardoni et al. 2016). Both SGG-UGM-1 and SGG-UGM-2 contain GOCE data, but SGG-UGM-2 uses the newly derived marine gravity anomalies and ITSG-Grace2018 model data (Liang et al. 2020), which leads to better performance of SGG-UGM-2 than SGG-GM-1.
The degree variances of EIGEN-6C4 geoid errors are larger than that of GECO and EGM2008 for degrees 260–370. Outside this degree range, EIGEN-6C4 (Förste et al. 2014) shows better performance than GECO and EGM2008. This is because the EIGEN-6C4 uses LAGEOS data from 1985 to 2010, GRACE RL03 GRGS data from 2003 to 2012, GOCE-SGG data, and DTU10 data with 2′ resolution, all of which improve its precision.The XGM2019e_2159 gravity field model has a jump after d/o 719 due to the different data sources used in the derivation of XGM2019e_2159 above degree 719 (Zingerle et al. 2020).
It can be seen from Fig. 2b that the cumulative geoid height errors of all gravity field models become horizontal after degree 300, with SGG-UGM-2 having the highest precision. At other degrees, the precision of gravity field models have a decreasing sequence as SGG-UGM-1 > XGM2019e_2159 > EIGEN-6C4 > GECO > EGM2008. EGM2008 has the maximum cumulative error, and its error reaches 8.2 cm at degree2190. The cumulative errors of EIGEN-6C4, GECO, SGG-UGM-1, SGG-UGM-2, and XGM2019e_2159 up to their maximum degree are 3.4, 4.2, 2.7, 1.9, and 3.1 cm respectively, which are much smaller than that of EGM2008.
This study shows that different datasets reflect different information when deriving gravity field models, and GRACE can provide high-precision medium-to-long wavelength gravity field information, whereas GOCE provides more accurate and richer information about medium-short wavelength signals of the Earth's gravity field compared to the GRACE (Yi et al. 2013). Furthermore, the precision of the gravity field model is greatly improved by combining multi-source gravity data.
Evaluation of GGMs using independent data
We use an independently observed gravity dataset to further assess the precision of gravity field models in this section. Different truncation degrees (Gruber and Willberg 2019) are chosen to calculate the gravity anomaly difference in the plateau region and the Sichuan Basin according to Eqs. (3) and (5). The statistics of these differences are listed in Tables 2 and 3, and the error standard deviations of the studied gravity field models are shown in the Figs. 3 and 4.
From the contents of Table 2 and Fig. 3, the differences between the gravity anomalies calculated by EGM2008, EIGEN-6C4, SGG-UGM-1, SGG-UGM-2 and XGM2019e_2159 gravity field models and the measured gravity data gradually increase within d/o 360. It may be related to the influence of topography undulation in the study area. Above degree 360, the GECO model performs the best, the XGM2019e_2159 model has the lowest precision. The three models that use the same GOCE data, i.e., EIGEN-6C4, SGG-UGM-1, and SGG-UGM-2, have similar precision slightly better than EGM2008. When calculated to the maximum degree of all models: EGM2008 (d/o2190), EIGEN-6C4 (d/o2190), GECO (d/o2190), SGG-UGM-1(d/o2159), SGG-UGM-2 (d/o2190), and XGM2019e_2159 (d/o2190), the accuracies respectively are 39.063, 38.333, 37.080, 38.383, 38.321, and 40.229 mGal. Wang et al. (2017) concluded that the standard deviation of the EGM2008 difference is 41.9 mGal in this area, which is consistent with the results of this study.
Table 3 and Fig. 4 show the results of the Sichuan Basin, where the precision of EGM2008 model is higher than the other models up to degree 360 by about 1 ~ 4 mGal. At truncation degrees between 720 and 1440, the XGM2019e_2159 model has the highest precision, EGM2008 is second and GECO has the lowest precision. The precision of EGM2008 is highest and XGM2019e_2159 is second at degrees 2159 and 2190. One can observe that SGG-UGM-1 has a slight advantage compared to SGG-UGM-2. It indicates that the new data in SGG-UGM-2 do not improve its precision in Sichuan Basin. Up to the highest degree of each model, i.e., d/o2190 for EGM2008, EIGEN-6C4, GECO, SGG-UGM-2, XGM2019e_2159 and d/o2159 for SGG-UGM-1, the standard deviations of the differences are: 7.202, 10.260, 10.723, 9.595,9.912, and 7.553 mGal, respectively.
Comparing Figs. 3 and 4, it is easily found that the differences shown in Fig. 3 firstly increase and then decrease but it is not this case in Fig. 4. This should be due to the fact that in the Tibet Plateau region, the gravity signals contain much more medium and short wavelength signals than those in Sichuan Basin and thus the truncation errors are largely different in the two regions. Therefore, in order to present gravity signals which high precision, higher degrees of gravity field model are needed in Tibet Plateau compared to Sichuan Basin.
Considering terrain correction
Taking into consideration the large terrain undulation in the study area, topographic undulation can cause changes in the short-wavelength signals, which have an impact on the high-frequency part of the Earth's gravity field. Theoretically, the terrestrial gravity measurements contain the full spectral information of the gravity field. However, the gravity field models are represented by finite spherical harmonic expansion (Godah and Krynski 2015) and thus contain insufficient high-frequency information. Local areas, especially areas with complex topographic changes, cannot be well approximated with the gravity field models. Therefore, it is necessary to adopt the terrain data to represent the short-wave component as a supplement. This study adopts SRTM data for a terrain correction to the ground measured gravity data in Qinghai-Tibet Plateau and Sichuan Basin. It should be noted that, the terrain correction only provides the gravity signals beyond the maximum of the gravity field models, and thus we performed high-pass filter (Smith and Sandwell 1994) on the initial terrain correction c. In order to correspond to the maximum degree of all gravity filed models: EGM2008 (d/o2190), EIGEN-6C4 (d/o2190), GECO (d/o2190), XGM2019e_2159 (d/o2190), SGG-UGM-2 (d/o2190), and SGG-UGM-1 (d/o2159), the truncation wavelengths are 20,000/2190 km and 20,000/2159 km, respectively.
Based on Eq. (6), new statistics are obtained. Tables 4 and 5 present the statistics for the plateau region and Sichuan Basin respectively. Table 6 shows the precision improvements obtained by terrain corrections.
In the plateau area, comparing Tables 2 and 4, it is shown that the terrain correction on the model precision assessment results of each model contributed 8 mGal. After terrain correction, the resultant precisions at the maximum degree of all models—EGM2008 (d/o2190), EIGEN-6C4 (d/o2190), GECO (d/o2190), SGG-UGM-1 (d/o2159), SGG-UGM-2 (d/o2190), and XGM2019e_2159 (d/o2190)—are: 31.900, 30.082, 28.907, 30.454, 30.924, and 31.396 mGal, respectively. It is shown in Table 6 that the terrain correction has a 22% contribution to the precision correction of GECO and XGM2019e_2159 models. For other models, the contribution exceeds 18%.
According to Tables 3 and 5, the terrain correction only affects about 1 mGal on the precision assessment results of all gravity field models. After terrain correction, EGM2008 yielded the best performance in this region with a precision of 6.648 mGal. EIGEN-6C4 was the most improved with a precision of 0.908 mGal. Due to the flat topography in the Sichuan Basin, terrain correction is negligible in this region.
The above results show that the precision of the studied gravity field models presents great differences in the study area. In the plateau region, the precision of GECO is the best, and the precision of the XGM2019e_2159 model is the lowest before terrain correction, whereas EGM2008 is the lowest performer after terrain correction. In the Sichuan Basin, EGM2008 and GECO models showed opposite significant changes, and the precision of EGM2008 model was the highest irrespective of terrain correction, while the GECO model performed poorly. The results indicate that the effect of topography on gravity must be considered in big undulating terrains, while the effect can be neglected when the topography is relatively flat.