Gravity and GPS data used in this study were obtained from in situ observations conducted in 2014 and 2017 (Wang et al. 2018). Normal correction, free-air correction, Bouguer correction, and terrain correction were performed on the adjusted gravity data to obtain complete BGA (Fig. 2). GPS measurements are used in both the free-air correction and Bouguer correction. The high accuracy and resolution GDEMV2 model, released by the Chinese Academy of Science (http://www.gscloud.cn), and the valid method proposed by Fu et al. (2014) are used in the terrain correction. According to isostasy theory, BGAs are typically low in high-altitude areas due to the existence of the mountain root, whereas those in ocean or basin areas are high. The altitude of the Longxi Block (west) is higher than that of the Ordos Block (east). The mean BGAs in the western part are − 216.7 mGal, − 231.3 mGal, and − 193.5 mGal for each profile (P1–P3), whereas those in the eastern part are − 185.3 mGal, − 167.9 mGal, and − 170.4 mGal, respectively; thus, the gravity results conform to the theory. The Liupan Shan is located in the transitional zone between high and low BGAs. However, low BGAs do not correspond to the highest altitudes in the Liupan Shan (Fig. 2); i.e., the three lowest BGAs are observed to the west of the Liupan Shan, indicating that these abnormal results occur throughout the entire region. This phenomenon is clearest in profile P2 across the middle of the Liupan Shan, where the offset between the peak altitude in the Liupan Shan and low BGA values is 34 km.
In order to evaluate the cause of this offset, we inverted the crustal density structure. For avoiding the multiple solutions of gravity inversion, joint inversion of gravity and seismology was performed. The inversion procedure was as follows:
- 1.
Take the seismic velocity result of a profile as the initial model and convert velocity to density according to empirical formulas.
- 2.
Calculate the modeled BGAs of the initial model using the two-dimensional body method (Talwani et al. 1959) and compare them with the observed BGAs. If the misfit exceeds the threshold, then perform step 3; otherwise, proceed to step 4.
- 3.
Invert the density structure using the non-linear least squares algorithm (Marquardt 1963) with constraints of the measurements, then repeat step 2.
- 4.
Accept the model.
There are infinitely many models that reduce the misfit to a desired value in gravity inversion. To find a particular model, an efficient and popular method is to require that the inverted model is close to a reference model (Li and Oldenburg 1998). We adjusted the model slightly according to the BGAs once the initial model was determined. Therefore, the initial model was crucial (see Additional file 1). In this study, we transferred the velocity structure of profile DD’ from Bao et al. (2013), located in the center of the study area, to the density structure using Eq. 1, which was obtained from the relationship between velocity and density in the CRUST1.0 model (Laske et al. 2013):
$$\rho = 0.0562 \times v_{\text{s}}^{2} + 0.0472 \times v_{\text{s}} + 1.9265,$$
(1)
where \(\rho\) is the density and \(v_{\text{s}}\) is the S-wave velocity. The density structure required further processing but was not used directly as the initial model because it was located 2° from the profile P2. We assumed that the structures of two profiles located 0.1° apart were similar; therefore, the density structure of a profile could be used as the initial model for another profile located 0.1° apart. Through this “profile propagation” method, we gradually propagated to profiles P2 and P3 from profile DD’. The initial model of profile P3 could not be obtained by this method because P3 does not trend in the EW direction. Instead, it was interpolated using the inverted density structures of other profiles.
Overall, folds in the shallow crustal structure correspond to fluctuations in BGAs, whereas variations in the deep crustal structure correspond to larger trend of BGAs. Misfits between the observed BGAs and modeled BGAs (Fig. 2) for profiles P1, P2, and P3 are 1.7 mGal, 1.9 mGal, and 3.2 mGal, respectively. The misfit reflects the reliability of the inversion result and generally does not exceed 5 mGal (Zhang et al. 2017). The inverted structure has a conspicuous vertically layered feature, similar to the initial model. The Moho of all three profiles are deep in the west and shallow in the east, indicating that the crust gradually thins from the Tibetan Plateau to the interior. The Moho result is constant and there is no mountain root beneath the Liupan Shan; these results are identical to those of Bao et al. (2013). In this study, the interface between the third and fourth layer is regarded as the interface between the upper and lower crust. In profile P2, the interface between the upper and lower crust, which corresponds to low BGAs, is notably concave to the west of the Liupan Shan. This concave shape was also reported in a previous seismic study (Zhou et al. 2000). The shallow low-density layer inside the Ordos Block is very thick, corresponding to the thick sedimentary layer in the Loess Plateau. The inversion results are close to those of Zheng et al. (2010) and in good correspondence with CRUST 1.0.