1 Introduction

High resolution static Earth’s gravity field models have been widely used in geodesy, geophysics, geodynamics, glaciology, and oceanography (Luo et al. 2014). Gravity field models and their physical meaning is significant to our understanding of the Earth. Therefore, the evaluation of the precision of the Earth’s gravity field models is meaningful.

Many scholars have studied the precision of different gravity field models in mainland China. For example, Zhang et al. (2009) used GPS-leveling data and 5′ × 5′ surface mean free-air gravity anomalies in mainland China to evaluate the EGM2008 (Pavlis et al. 2008, 2012). They concluded that the precision of EGM2008 height anomalies achieves 20 cm in the study area; and the precision of EGM2008 free-air gravity anomalies is about 10.5 mGal. Liang et al. (2018) constructed the SGG-UGM-1 gravity field model by combining EGM2008 gravity anomaly and GOCE observation data. They found that the precision of gravity disturbance calculated by the SGG-UGM-1 model was comparable to that of EGM2008 and EIGEN-6C4 in the Maowusu region. Li et al. (2014) evaluated the GOCE-only models, GRACE-only models and combined gravity field models by using 2′ × 2′ gravity anomaly grid data of mainland China. Their result showed that the GOCE mission improves the precision of the medium-short wavelength components. Considering the influence of topography, Meng et al. (2017) used the GPS leveling data to evaluate the precision of EGM2008 and EIGEN-6C4 models. Their results showed that the topographic information could compensate for the high-frequency part of the ultra-high degree gravity field model, and this effect is more significant in the big undulating mountainous terrains.

Besides mainland China, several researches have been conducted on the EGM2008 model (e.g., Claessens et al. 2010; Hirt et al. 2010; Kotsakis et al. 2010; Gruber 2009; Morgan and Featherstone 2009). Dawod et al. (2010) has analyzed the performance of EGM2008 using 305 GPS/levelling points in Egypt. The result showed that its precision is 0.23 m. After first-order regression and kriging models processing, the standard deviation decreased to 0.17 m. The accuracies of other gravity field models have also been investigated by previous studies. For example, Goyal et al. (2018) analyzed the performance of 15 recent gravity field models by using 145 GNSS/leveling data. Statistical results indicate that the EGM2008 model is the best with root-mean-square error (RMSE) of 0.28 m, after fitting with the seven-parameter model; GECO performs significantly better results with RMSE of 0.19 m for India. Moreover, Odera (2020) adopted GPS/leveling data and free-air gravity anomaly (FGA) data in Kenya to study the precision of EGM2008, EIGEN-6C4, GECO and SGG-UGM-1 models. The standard deviations (SD) of the differences between observed GPS/levelling geoid data and those from gravity field models show that EIGEN-6C4 is the best for 11.48 cm over Nairobi area. While the best performance of SGG-UGM-1 at 10.00 mGal was found by using 8,690 free-air gravity anomaly data. The study results also indicated that the differences between observations and model values are small in flat areas, and the differences of SD increase with the complexity of the terrain. Odera and Fukuda (2017) used terrestrial free-air gravity anomalies and geometric geoid undulations to evaluate GOCE-based models including: DIR (releases 1, 2, 3, 4, and 5), TIM (releases 1, 2, 3, 4 and 5), SPW (releases 1, 2 and 4) and GOCO (releases 1, 2, 3 and 5). Their study results showed that latest GOCE global gravity field models (releases 4 and 5) do not improve the performance over the earlier released global gravity field models (releases 1, 2, 3) in Japan at degree 150.

Indeed, high-resolution ground gravity observations data in China are not or seldomly used in the derivation of the current ultra-high resolution global gravity field models (GGMs). Thus, the precision of these models in China is usually lower than that of other regions of the world (Yang et al. 2012). Although many scholars have evaluated different gravity field models for their applicability in China, evaluating the precision of ultra-high gravity field models in mainland China with ground gravity data is very few. Furthermore, few studies adopt the initial ground gravity observations during the assessment process, and instead, grid data are often used. Therefore, this study presents an external precision (Erol et al. 2020; Huang and Kotsakis 2009) evaluation of current ultra-high degree GGMs, including: EGM2008, EIGEN-6C4, GECO, SGG-UGM-1, XGM2019e_2159, XGM2019e and SGG-UGM-2, using observed ground gravity data of the Sichuan Basin and the southeast margin of the Tibetan Plateau. Moreover, the internal error estimate (Gruber 2004; Wu et al. 2021) was calculated. In Sect. 2, the study area and data sets are introduced. Section 3 provides a detailed description of the research method. Section 4 gives the results and analysis, and a discussion is conducted in Sect. 5. Finally, the conclusions are given in Sect. 6.

2 Study area and data

2.1 Study area

The study area comprises the southeast margin of the Qinghai-Tibet Plateau and the western region of the Sichuan Basin (see Fig. 1), including the Bayan Har plate, the South China block, and the Sichuan-Yunnan block, where the topography is high in the west and low in the east (Wang et al. 2007). These regions were chosen because they host both mountains and plains, allowing analysis of the precision of gravity field models in different terrains.

Fig. 1
figure 1

Study area and gravity datasets distribution. Black points represent the positions of the gravity observations, and the yellow rectangles show the locations of the two subregions of the study area

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2.2 Gravity field models

With the implementation of altimetry satellite missions and the development of satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG) technologies, the construction of the Earth’s gravity field model has entered a new era of development. The ultra-high degree gravity field models have made great progress by combining various types of data. For example, the highest degree and order has risen from 360 in the 1990s to 5540 at present. In this study, we select the high-resolution gravity field models published on the International Center for Global Earth Models (ICGEM) website (http://icgem.gfz-potsdam.de/tom_longtime) in recent years for comparative evaluation. Table 1 provides an overview about the models. In order to facilitate comparison, this study evaluates the models EGM2008, EIGEN-6C4, GECO, XGM2019e_2159 and SGG-UGM-2 with maximum degree of 2190, and SGG-UGM-1 with maximum degree of 2159 in Sect. 4. Section 5 discusses XGM2019e, which has a maximum degree of 5540.

Table 1 Global gravitational models used in this study
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2.3 Ground gravity data

In the Sichuan Basin and the southeast margin of the Tibetan Plateau, the ground gravity measurement data are obtained from flow Gravity/GPS combined measurement. Based on the topography of the study area, ground gravity datasets were divided into two subranges, A and B (Fig. 1). A is located in the Qinghai-Tibetan Plateau and B is the Sichuan Basin. There are 603 data points in area A where the minimum, maximum and average elevations are 704 m, 4234 m and 2129.1 m, respectively. The number of data points in region B is 302, where the minimum, maximum and average elevations are 292.5 m, 781.1 m, and 460.4 m, respectively.

The ground gravity measurement data came from three sources. First, profiles AA and BB data of the Qinghai-Tibet Plateau region were measured in 2009 (Zhang et al. 2014). The precision of gravity is 0.082 mGal and the spacing between consecutive measurement points is under 1 km. Second, data were collected from 302 stations at the Sichuan Basin, which were measured in 2012 (Fu et al. 2014). The accuracies are 0.02 mGal and 5 cm for gravity and GPS position (elevation) data, respectively. And measurement points were spaced at about 2–5 km apart. Finally, profiles CC and DD data of the Qinghai-Tibet Plateau region were measured in 2013 (Yang et al. 2015), with a measurement precision of 0.02 mGal and the spacing between gravity stations was 2.5 km.

After obtaining the raw gravity data of the observation network, the free-air gravity anomalies are derived by using the normal reduction, the earth tide reduction and the height reduction. Here, the data processing was done in a uniform system. Therefore, the gravity anomalies data are derived in the WGS84 reference frame.

2.4 Terrain data

Surface elevation is needed for terrain correction (TC) in the evaluation of gravity precision. The Shuttle Radar Topography Mission (SRTM) was a joint project of the National Aeronautics and Space Administration (NASA) and the National Geospatial-Intelligence Agency (NGA). It provides elevation data with three resolutions, i.e., 1, 3 and 30 arcsec (Yang et al. 2011) referred to as SRTM1, SRTM3 and SRTM30, respectively. SRTM1 covers only the continental United States, while SRTM3 and SRTM30 cover the whole world. SRTM3 data (https://srtm.csi.cgiar.org/download/) is most widely used and thus is also used in this study.

3 Methods

3.1 Internal error estimate

The precision is expressed in terms of geoid error degree variance and cumulative geoid height errors as follows (Erol et al. 2020; Pail et al. 2011):

$$\mathop \sigma \nolimits_{n} = R \times \sqrt {\sum\limits_{m = 0}^{n} {\left[ {\mathop \sigma \nolimits_{nm}^{2} (c) + \mathop \sigma \nolimits_{nm}^{2} (s)} \right]} }$$
(1)
$$\mathop \sigma \nolimits_{N} = R \times \sqrt {\sum\limits_{n = 2}^{n} {\sum\limits_{m = 0}^{n} {\left[ {\mathop \sigma \nolimits_{nm}^{2} (c) + \mathop \sigma \nolimits_{nm}^{2} (s)} \right]} } }$$
(2)

where \({\sigma }_{n}\) is the degree variance of geoid errors; \({\sigma }_{N}\) is the cumulative geoid height errors; and \(R\) is the Earth’s average radius. \(n\) and \(m\) are the degree and order of the spherical harmonic coefficients, respectively. \({\sigma }_{nm}\left(c\right)\) and \({\sigma }_{nm}\left(s\right)\) are the standard deviation of fully normalized spherical harmonic coefficients.

3.2 External precision assessment

The validation of GGMs was based on the comparisons between gravity anomalies obtained from GGMs and the corresponding ones determined from the ground gravity data. The gravity anomaly, \(\mathop {\Delta g}\nolimits_{GGM}\), obtained from the GGM is as follows (Barthelmes 2013):

$$\Delta g_{GGM} = \frac{GM}{{\mathop r\nolimits^{2} }}\sum\limits_{n = 2}^{{\mathop n\nolimits_{\max } }} {(n - 1)(\frac{a}{r})^{n} \sum\limits_{m = 0}^{n} {(\mathop {\overline{C} }\nolimits_{nm} } } \cos m\lambda + \mathop {\overline{S} }\nolimits_{nm} \sin m\lambda )\mathop {\overline{P} }\nolimits_{nm} (\cos \theta )$$
(3)

where \(GM\) is the geocentric gravitational constant; \(r\) is the distance to the geocentric; \(n\) and \(m\) are, respectively, the degree and order of the spherical harmonic coefficients. \({n}_{max}\) is the maximum degree of a gravity field model; \(a\) is the semi-major axis of the reference ellipsoid. \({\overline{C} }_{nm}\) and \({\overline{S} }_{nm}\) are fully normalized spherical harmonic coefficients; \(\theta\) and \(\lambda\) are geocentric co-latitude and longitude, respectively. \({\overline{P} }_{nm}(\mathrm{cos}\theta )\) is the fully normalized associated Legendre functions for degree \(n\) and order \(m\).

The fully normalized spherical harmonic coefficients of different gravity field models (EGM2008, EIGEN-6C4, GECO, SGG-UGM-1, XGM2019e_2159, XGM2019e and SGG-UGM-2) are derived for different ellipsoids, and have their own geocentric gravitational constant GM and the semi-major axis of the reference ellipsoid a. The ground gravity anomalies data are derived in the WGS84 reference frame. According to the Eq. (4) of He et al. (2017), the fully normalized spherical harmonic coefficients of all gravity field models are converted to values under the WGS84 reference frame. Finally, the modeled gravity data and the ground gravimetry data are unified under the same reference frame.

$$\mathop {\overline{C} }\nolimits_{nm} = \frac{GM}{{GM_{1} }}\mathop {\left( {\frac{a}{{a_{1} }}} \right)}\nolimits^{n} \mathop C\nolimits_{nm} {,}\mathop {\overline{S} }\nolimits_{nm} = \frac{GM}{{GM_{1} }}\mathop {\left( {\frac{a}{{a_{1} }}} \right)}\nolimits^{n} \mathop S\nolimits_{nm}$$
(4)

where \(GM\) is the geocentric gravitational constant of different gravity field models, a is the semi-major axis of the reference ellipsoid of different gravity field models. \({C}_{nm}\) and \({S}_{nm}\) are fully normalized spherical harmonic coefficients of different gravity field models. \(G{M}_{1}\) is the geocentric gravitational constant \(3.986004418\times {10}^{14}{m}^{3}/{s}^{2}\) of WGS84, \({a}_{1}\) is the semi-major axis \(6378137m\) of WGS84. \({\overline{C} }_{nm}\) and \({\overline{S} }_{nm}\) are fully normalized spherical harmonic coefficients of different gravity field models under WGS84 reference frame.

Differences between the observed gravity anomalies \(\mathop {\Delta g}\nolimits_{real}\) and the corresponding gravity anomalies \(\mathop {\Delta g}\nolimits_{GGM}\) obtained from gravity field models were determined as follows:

$$\Delta g = \mathop {\Delta g}\nolimits_{real} - \mathop {\Delta g}\nolimits_{GGM}$$
(5)

Considering the terrain varies largely in the study area, it is necessary to apply the terrain correction for the free-air gravity anomalies. Scholars have done a lot of research on terrain correction (Kane 1962; Forsberg 1984, 1985; Leaman 1998; Omang and Forsberg 2000; Garcia-Abdeslem and Martin-Atienza 2001; Fullea et al. 2008). Fullea et al. (2008) developed a Fortran 90 program FA2BOUG for terrain correction, which divides the data around the calculation point into three zones, an inner zone, an intermediate zone and a distant zone. Three zones are divided in the following way: when the distance d of the calculation point from the flow point is 20 < d ≤ 167 km, the corresponding zone is considered as the distant zone; when 1 < d ≤ 20 km, it is considered as the intermediate zone; when d < 1 km, it is considered as the inner zone. This study sets different grid lengths in different areas. The step length of the distant zone is 4 km, and the step length of the intermediate zone is 2 km. In the inner zone, only one grid is involved, so there is no step length definition. In this study, the FA2BOUG method is adopted to conduct the terrain correction using SRTM3 data.

Considering the terrain correction c (Omang and Forsberg 2000), based on Eqs. (3) and (5), the gravity anomalies \(\delta g\) used for assessment are expressed as:

$$\delta g = \mathop {\Delta g}\nolimits_{real} - \mathop {\Delta g}\nolimits_{GGM} + c$$
(6)

4 Results and analysis

4.1 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.

Fig. 2
figure 2

Results of the internal error estimate

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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.

4.2 Evaluation of GGMs using independent data

4.2.1 Initial results

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.

Table 2 The statistics of the differences between the gravity anomaly derived from gravity field models and the measured data in plateau area (unit: mGal)
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Table 3 The statistics of the differences between the gravity anomaly derived from gravity field models and the measured data in Sichuan Basin (unit: mGal)
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Fig. 3
figure 3

The standard deviation of differences in the plateau area

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Fig. 4
figure 4

The standard deviation of differences in Sichuan Basin

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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.

4.2.2 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.

Table 4 The statistics of the differences between the gravity anomaly derived from gravity field models and the measured data after terrain correction in plateau area (unit: mGal)
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Table 5 The statistics of the differences between the gravity anomaly derived from gravity field models and the measured data after terrain correction in Sichuan Basin (unit: mGal)
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Table 6 The precision improvements obtained by terrain correction for each gravity field model (unit: mGal)
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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.

5 Discussion

According to the above section, the maximum degree of evaluated gravity field model is only at 2190. However, the XGM2019e model has been up to degree 5540. This section further analyzes the precision of XGM2019e beyond 2190. The results from different truncation degrees, i.e., 2159, 2190, 3000, 4000, 5000, 5480, and 5540, are computed. The statistics of the plateau area and the Sichuan Basin are presented in Tables 7 and 8, respectively. The result of the geoid error degree variance is shown in Fig. 5.

Table 7 The statistics of the differences between the gravity anomaly derived from XGM2019e with different truncated degrees and the measured data in plateau area without terrain correction (unit: mGal)
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Table 8 The statistics of the differences between the gravity anomaly derived from XGM2019e with different truncated degrees and the measured data in Sichuan Basin without terrain correction (unit: mGal)
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Fig. 5
figure 5

Result of the geoid error degree variance

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According to Tables 2 and 7, XGM2019e and XGM2019e_2159 have the same precision at degree 2159. This is because these two models have the same spherical harmonic coefficients for degrees 0 ~ 2159. However, when truncating to 2190, the XGM2019e_2159 model performs a little better than XGM2019e. For XGM2019e, the model precision improves with an increase in the truncation degree, and the precision at d/o5540 is 31.484 mGal.

It can be seen from Table 8 that the precision of XGM2019e does not always increase with an increase in the truncation degree after 2159. The model precision is 17.705 and 12.634 mGal up to 5000 and 5540, respectively. However, the precision is 12.011 mGal up to degree 4000. This trend may be related to the result of internal error estimate (see Fig. 5). Comparing the results of Tables 7 and 8, a higher degree of the gravity field model presents more information of the actual gravity field in the plateau region. However, in flat topography areas, the contribution of ultra-high degrees of gravity field model to the description of Earth’s gravity field is not obvious.

6 Conclusions

In this study, accuracies of six recent ultra-high-degree gravity field models, including EGM2008, EIGEN-6C4, GECO, SGG-UGM-1, XGM2019e_2159 and SGG-UGM-2, are assessed in terms of internal and external precision. The internal error is evaluated by the degree variance and cumulative values of geoid errors, whereas the external precision is evaluated using ground gravity measured data in two different terrains located in the southeast margin of the Qinghai-Tibet Plateau and the western region of the Sichuan Basin. The results show that the geoid error degree variance of all models is at millimeter magnitude, and the precision of the cumulative geoid height error is at centimeter magnitude. GOCE improves the precision of medium to high frequency information in the Earth’s gravity field; using more and newer gravity data, the derived gravity field model has better internal error precision.

It should be noted that gravity field models studied in this study are all global models. The model with the best internal error precision does not always have the highest precision in all regions, and thus, comparisons using independent gravity data are needed. In this study, GECO has the best precision in Qinghai-Tibet Plateau, and the XGM2019e_2159 model has the lowest precision. The three models (EIGEN-6C4, SGG-UGM-1, and SGG-UGM-2) which all use GOCE data have similar precision and are slightly better than EGM2008. In the Sichuan Basin, EGM2008 performs the best, XGM2019e_2159 is the next best, EIGEN-6C4, SGG-UGM-1, and SGG-UGM-2 models have similar precision, and the GECO model performs the lowest. After considering terrain correction, it was observed that the terrain correction contributes a precision improvement of about 8 mGal for all models in the plateau area, with the GECO model still producing the best precision of 28.907 mGal. In the Sichuan Basin area, the terrain correction has a much lower influence on the precision; and the magnitude of the improvement is only about 1 mGal. This means that in the big undulating mountainous terrains, the effect of terrain on gravity must be considered, while when the terrain is relatively flat, the effect of terrain on gravity can be ignored.

Furthermore, a detailed analysis of XGM2019e up to degree 5540 was conducted. This study shows that a higher degree, the gravity field model presents better precision in the plateau region. However, in the flat topography areas, the contribution of the ultra-high degree gravity field model to the Earth's gravity field is not obvious.