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# Inversion of the density structure of the lithosphere in the North China Craton from GOCE satellite gravity gradient data

- Yu Tian
^{1, 2, 3}Email authorView ORCID ID profile and - Yong Wang
^{1, 2}

**Received:**2 August 2018**Accepted:**20 October 2018**Published:**5 November 2018

## Abstract

## Keywords

- North China Craton
- Density inversion
- Gravity gradient data
- Lithosphere

## Introduction

The lithosphere in the North China Craton (NCC) has suffered destruction since the Phanerozoic. Since the concept of lithospheric thinning in North China was proposed in the early 1990s, research on the destruction of the NCC has engendered increasingly heated debates. Particularly, in recent years, geophysicists, geochemists and geologists in China and elsewhere have conducted extensive research on the destruction of the NCC and achieved a clearer understanding of the deep structure, formation and evolution of the NCC (e.g. Zhao et al. 2001; Kusky and Li 2003; Zhai and Santosh 2011; Sun and Kennett 2017). However, currently, there is still a debate on a number of issues related to the NCC, such as the destruction mechanism, time of occurrence and geodynamic mechanism, based on the researchers’ varying backgrounds and approaches (Wu et al. 2008). As one of the oldest cratons on Earth, the NCC is a notable example of a craton that has undergone modification and destruction. Studying the destruction of the NCC has provided a window for understanding the formation, evolution, stabilization and destruction of paleocontinents.

Currently, there are two mainstream views on what caused the destruction of the NCC, namely delamination (Gao et al. 2009) and thermal erosion (Zhu et al. 2012), both of which involve tectonic deformation and material distribution inside the lithosphere in the NCC. Therefore, it is of vital importance to acquire high-resolution structural data of the lithosphere. As seismic tomography technology advances, numerous researchers have extensively studied the crustal and upper mantle velocity structures in the NCC and its surrounding area and achieved some remarkable results (e.g. Hearn et al. 2004; Zheng et al. 2006; Chen 2009; Huang and Zhao 2009; Zhao et al. 2009; Tian and Zhao 2011). In contrast, there are relatively few studies on the deep underground density structure. The internal density structure of the Earth is the basis for studying the composition and internal structural deformation of the lithospheric crust. Fang (1996) obtained the density distribution in the lithosphere in North China through inversion of Bouguer gravity anomaly data using a constrained least-squares method. Wang et al. (2014) obtained a high-resolution three-dimensional density structure of the lithosphere in the NCC through inversion of Bouguer gravity anomaly data using an algebraic reconstruction inversion method with the seismic travel time as the constraint. Xu et al. (2016) obtained the density of the lithospheric crust as well as the depth of the lithosphere–asthenosphere boundary in the NCC through rapid joint inversion of gravity, geoid and topography data.

In recent years, techniques for measuring gravitational potential fields have evolved from static to dynamic methods, and measurement instruments have also gradually advanced from ground measurement systems to aerial and satellite measurement systems. As the second derivative of the gravitational potential, the gravity gradient can satisfactorily reflect a high-frequency density change and help increase the resolution of the density structure. Full tensor of the gravity gradient can effectively help increase the amount of observation data. Gravity gradient data can reflect the shape and size of an underground anomalous body from various perspectives. While there are many components of the gravity gradient, the inversion solution is non-unique. To constrain the non-unique problem of gravity gradient data inversion, seismic data are introduced, which, together with the multiple components of the gravity gradient, are subjected to a joint inversion to reduce the non-uniqueness of the inversion solution. In this study, based on a high-resolution P-wave velocity structure obtained from seismic tomography, a 3D initial density model is constructed for the lithosphere in the NCC based on the empirical velocity–density conversion relation for North China. The GOCE satellite gravity gradient data are processed. Considering that the anomalous gravity gradient effect is a comprehensive reflection of a myriad of factors and non-unique problem, the anomalous gravity gradient effect caused by the topographic relief at each underground layer interface is eliminated. In addition, by fully considering that the density distribution in the lithosphere is primarily collectively affected by its internal temperature and composition, the anomalous gravity gradient effect caused by the internal temperature of the lithosphere is also eliminated. Separation of the temperature effects from the residual gravity gradient anomalies provides constraints of the density changes due to variations in lithosphere and mantle composition. Thus, the anomalous gravity gradient effect caused by the non-uniform matter density distribution is obtained. Based on the anomalous gravity gradient components obtained, the preconditioned conjugate gradient inversion algorithm was adopted to calculate the horizontal density structure in the NCC. Because the Lagrange empirical parameter in the preconditioned conjugate gradient (PCG) inversion algorithm exhibits uncertainty in data inversion, this algorithm is improved. The distribution of the density structure at various depths in the NCC is obtained through inversion and is analysed and interpreted based on the available geological and geophysical data.

## Data processing

### Initial density model data processing

The collected P-wave velocity values were converted to density values using Eq. (1). Figure 1b–g shows the density values converted from the P-wave velocity values. According to the P-wave tomography data of the study area (Tian et al. 2009), values of 2840 kg/m^{3} (10 km), 2980 kg/m^{3} (25 km), 3320 kg/m^{3} (42 km), 3320 kg/m^{3} (60 km), 3320 kg/m^{3} (80 km), and 3330 kg/m^{3} (120 km) were used as the absolute density values of the layers at different depths. The differences between converted density and absolute density represent the density perturbations. We adopted the density perturbations as the initial values for density inversion.

The high-frequency signals induced by the geological characteristics of the shallow and middle layers are more significant in the gravity gradient data. In the previous research (Wang et al. 2014), there has been uncertainty to some extent at the depth of 150 km for the gravity data inversion. Accordingly, the bottom layer of the inversion model is determined to be at the depth of 120 km. Combined with the initial value for density inversion velocity perturbations, an initial inversion model was established for the inverse calculation based on the initial value. The inversion model is divided into six layers in the depth direction with the lower boundary at depths of 10 km, 25 km, 42 km, 60 km, 80 km and 120 km. In the horizontal direction, a 0.25° × 0.25° mesh with equidistantly spaced nodes is generated on each of the top four layers, and a 0.5° × 0.5° mesh with equidistantly spaced nodes is generated on each of the bottom two layers. This model partition method not only increases the abnormal effect of the deep blocks at the surface but also effectively reduces the number of parameters of the inversion model.

### Preprocess of the GOCE data

As GOCE satellite gravity gradiometer consists of three-axis accelerometers, two of the axis are high sensitive and the other one is not so sensitive. In the GOCE data, the precisions of the original gravity gradient observation components \({\varvec{V}}_{xy}\) and \({\varvec{V}}_{yz}\) are lower than those of the high-precision tensors \({\varvec{V}}_{xx}\), \({\varvec{V}}_{xz}\), \({\varvec{V}}_{yy}\) and \({\varvec{V}}_{zz}\) (Rummel et al. 2011; Yi et al. 2013). To use the gravity gradient data to detect anomalies, four high-accuracy anomalous gravity gradient components (\({\varvec{T}}_{xx}\), \({\varvec{T}}_{xz}\), \({\varvec{T}}_{yy}\), \({\varvec{T}}_{zz}\)) provided by the GOCE satellite can be used to study regional internal structures and obtain more realistic and reliable structural models of the Earth (e.g. Bouman et al. 2015; Fecher et al. 2015; Li et al. 2017).

Obtaining more accurate mass and density anomalies especially helps with the geometrical interpretation, as they allow all of the local extrema to be better localized. Although the data signal becomes stronger and the noise is amplified, this problem can be resolved by improving the signal-to-noise ratio. We did not select spherical harmonic data because they are often regularized with respect to potential but not measured gradients.

The statistic difference value between gravity gradient after downward continuation and the 220th-order model of the EGM2008

Parameters | \({\text{d}}{\varvec{T}}_{xx}\) | \({\text{d}}{\varvec{T}}_{xz}\) | \({\text{d}}{\varvec{T}}_{yy}\) | \({\text{d}}{\varvec{T}}_{zz}\) |
---|---|---|---|---|

Mean | − 0.013 | − 0.09 | − 0.030 | 0.035 |

Sigma | 0.33 | 0.42 | 0.64 | 0.71 |

### The topographic correction of the gravity gradient data

The gravity gradient tensors are not sensitive to the density variation vertically, and the lithospheric horizontal density variation is what we really concerned about. Therefore, we have to reduce the other known gravity gradient anomalies that can be determined by the other reliable measures. As the layer interface undulation data can be determined by the seismic wave velocity data, the topographical data determined by the other methods can meet the higher resolution, which are more reliable. Therefore, we adopted these established interface models (CRUST 1.0, ETOPO1) and reduced the corresponding gravity gradient anomaly effect from the corrected gravity gradient data.

The spatial resolution of the four preprocessed GOCE gravitational gradient grid tensors (\({\varvec{T}}_{xx}\), \({\varvec{T}}_{xz}\), \({\varvec{T}}_{yy}\), \({\varvec{T}}_{zz}\)) at a noise-free satellite altitude is 10 arc-min (http://goce.kma.zcu.cz/), the spatial resolution of the ETOPO1 is 1′ × 1′, and the spatial resolution of the CRUST1 is 1° × 1°. As the spatial resolutions of the three models are different, the downloaded data are used with their original spatial resolution for the downward continuation and forward-modelling calculation, which makes the calculated results more reliable. We homogenized the same spatial resolution of 0.5° × 0.5° before calculating the remaining gravity gradient. For the higher spatial resolution of the data (preprocessed GOCE gravitational gradient grid data, topography data ETOPO1), we extracted data from the calculated results (Fig. 2b, d). For the gravity gradient effect caused by the underground layer interface (CRUST1.0), we used the common kriging interpolation method to obtain the data needed for this calculation.

As the North China Craton is such a large area, it is more suitable to use the tesseroids instead of the planar prisms for forward modelling. We used the tesseroids as the modelling units under the spherical coordinates. We adopted the software named “Tesseroids” (Uieda et al. 2016), which helps us to implement the forward-modelling calculations (https://tesseroids.readthedocs.io/en/latest/). The software “Tesseroids” can be used to convert the tesseroids in the spherical coordinates into the planar prisms in the Cartesian coordinates, with the converted planar prisms can be used, then the software can implement the forward-modelling calculations under the Cartesian coordinates which corresponds into the forward modelling calculations.

In addition, 1′ × 1′ ETOPO1 mesh topography data (Amante and Eakins 2009) were also acquired. The whole NCC was segmented to 1′ × 1′ mesh points. The longitude and latitude coordinates of the mesh points were used as the length and width boundaries of the upright cuboid unit, the centre of the mesh as the cuboid centre, and the topographic height and geoid as the cuboid upper and lower boundaries, respectively. A density of 2 670 kg/m^{3} was used for the part of the upright cuboid unit above sea level, and a density of 1 030 kg/m^{3} was used for the part below sea level. Using the prism unit-based forward-modelling method (Nagy et al. 2000), the reduction in the gravity gradient at a height of 10 km generated by the topographic mass was calculated. To reduce the effects of high-frequency topographic errors, the result from forward modelling of the topographic mass was filtered based on the previous research results (Jekeli and Rapp 1980). The anomalous gravity gradient effect in this area caused by the topographic masses was thus obtained (Fig. 2c). As there are too many figures of the four components, we only display the figures of one component \({\varvec{T}}_{zz}\) in the initial data processing.

### The underground layer interface correction and uncertainty analysis

^{3}(Laske et al. 2013), respectively; when calculating the gravity gradient effect resulting from the relief at the Moho interface, the average depth of the Moho was set to 35 km, and the density difference between the crust and mantle was set to − 420 kg/m

^{3}(Laske et al. 2013). The values of the two mean depths are the average values of the sediment and Moho depths from CRUST 1.0. We calculated and adopted the mean density difference value of every point between the mean depth and the corresponding layer (sediment and Moho) from CRUST 1.0.

*R*

_{s}represents the fluctuation of each average gravity gradient caused by the uncertainty of the sedimentary layer depth compared with the total remaining gravity gradient.

*R*

_{m}represents the fluctuation of average gravity gradient caused by the uncertainty of the Moho layer depth compared with the total remaining gravity gradient. The percentage of

*R*

_{s}and

*R*

_{m}are within the reasonable range compared with the effect of uncertainties on residual lithosphere and upper mantle gravity by the parameter tests (Herceg et al. 2015). The effect of uncertainty layer depth on the inversion results is analysed in Section

*Anti*-

*noise ability test of PCG algorithm*.

Sedimentary layer interface correction and uncertainty analysis of the layer depth

Gravity gradient tensors | Maximum of gravity gradient tensors by sedimentary layer ( | Minimum of gravity gradient tensors by sedimentary layer ( | Maximum uncertainty of gravity gradient tensors by sedimentary layer ( | Minimum uncertainty of gravity gradient tensors by sedimentary ( | Mean uncertainty of gravity gradient tensors by sedimentary layer ( | Resultant change in gravity gradient tensors |
---|---|---|---|---|---|---|

\({\varvec{T}}_{xx}\) | 6.4258 | − 1.7022 | 0.7810 | − 1.2032 | 0.038 | − 1.35 |

\({\varvec{T}}_{xz}\) | 0.8060 | − 0.5912 | 0.1908 | − 0.2594 | − 0.006 | 0.32 |

\({\varvec{T}}_{yy}\) | 2.9595 | − 1.8646 | 1.2278 | − 1.4064 | 0.057 | − 1.88 |

\({\varvec{T}}_{zz}\) | 1.6786 | − 7.3107 | 1.5172 | − 1.5437 | − 0.148 | − 2.18 |

The Moho layer interface correction and uncertainty analysis of the layer depth

Gravity gradient tensors | Maximum of gravity gradient tensors by Moho layer ( | Minimum of gravity gradient tensors by Moho layer ( | Maximum uncertainty of gravity gradient tensors by Moho layer ( | Minimum uncertainty of gravity gradient tensors by Moho layer ( | Mean uncertainty of gravity gradient tensors by Moho layer ( | Resultant change in gravity gradient tensors |
---|---|---|---|---|---|---|

\({\varvec{T}}_{xx}\) | 10.099 | − 4.924 | 4.862 | − 2.468 | − 0.036 | 1.28 |

\({\varvec{T}}_{xz}\) | 4.514 | − 8.584 | 1.9071 | − 2.758 | 0.044 | − 2.35 |

\({\varvec{T}}_{yy}\) | 5.805 | − 6.260 | 2.4803 | − 3.250 | 0.062 | − 2.04 |

\({\varvec{T}}_{zz}\) | 8.916 | − 14.587 | 3.7330 | − 3.5636 | − 0.158 | − 2.33 |

### The gravity gradient effect correction of the mantle

*R*is the Earth’s radius;

*Z*is the buried depth of the field source; and

*n*is the order of the spherical harmonic function. The long-wavelength gravity gradient anomalies obtained from calculations using the second- to 44th-order spherical harmonic coefficients (Li et al. 2017) were deducted. The interfacial relief-corrected anomalous gravity gradient component at the height of 10 km was obtained by subtracting the topographic relief-corrected anomalous gravity gradient component by the gravity gradient anomalies resulting from the relief at the Moho and sedimentary layer interfaces as well as the long-wavelength gravity gradient effect at the same height, as shown in Fig. 4.

### Gravity gradient effect caused by the temperature variations in the lithosphere

Previous research results (Kaban et al. 2003, 2010; Mooney and Kaban 2010) show that the cause of density variations in the lithosphere is multifaceted and that the density distribution in the lithosphere is primarily affected by its internal temperature and composition. And the results by Herceg et al. (2015) show that the residual mantle gravity anomalies are mainly caused by a heterogeneous density distribution in the lithosphere and uppermost mantle, which depends on variations in lithosphere mantle temperature and composition. Separation of the temperature effects from the residual gravity anomalies provides constraints of the density changes due to variations in lithosphere and mantle composition. Lithospheric thinning in North China not only reflects the variation of the thickness, but also accompanies the change of the substance properties and the transformation of the thermal state. Both mainstream views, namely delamination (Gao et al. 2009) and thermal erosion (Zhu et al. 2012), would induce inhomogeneous density distribution of the lithosphere, but the characteristics of the density distribution induced by different mechanisms are various. For further discussions of the inhomogeneous density distribution caused by the component in the lithosphere, the gravity gradient effect caused by the temperature variations in the lithosphere should be separated.

While in previous studies on the study area, the non-uniform density of the lithosphere resulting from temperature variations and composition was not distinguished, and the gravity gradient effect caused by the temperature variations in the lithosphere was not differentiated; consequently, the density obtained through inversion contained the gravity gradient effect caused by the temperature variations in the lithosphere. In this study, we first collected 0.5° × 0.5° temperature data of the study area. Two groups of temperature data were used. The temperature data of the area above a depth of 70 km were obtained using the steady-state heat conduction equation with the temperature at the depth of 70 km as the boundary condition (An and Shi 2007). The temperature data of the area between the depths of 70 km and 120 km were obtained using the Euler state equation based on S-wave tomography data (Yang et al. 2013).

## Methodology

### Inversion method

The inversion of the gravity gradient tensor is used to solve linear equations. Because the unknown variable m is much larger than the sampled data vector, the inversion is an underdetermined problem and the solution is not unique. Proper restrictions are needed for the objective function in order to narrow the solution area. We seek an objective function that enables us to account for prior information; the constructed objective function aligns with both the observations and additional geophysical constraints. Therefore, in the geophysical inversion theories, objective functions often consist of a data fitting function and a model objective function (Constable et al. 1987). The inversion problem is thus transformed to seek for a model m, which satisfies the data fitting function and minimizes the objective function.

*i*th observation.

*R*is the sum of the squared first-order derivatives of the three components of the model vector m along the

*x*,

*y*and

*z*directions.

*x*,

*y*and

*z*directions, respectively.

Because the gravity data do not have fixed depth resolutions and kernel functions decay fast with depth, the inversion results close to ground surface cannot adequately represent the accurate location of the anomaly (Li and Oldenburg 1998). To overcome the tendency of density to concentrate at the surface, a depth weighting function was added to the model objective function to counteract the natural decay of the kernel function. Therefore, the depth weighting function (Eq. 12) is included in the model objective function so that the kernel function can accurately present the weights of anomaly at various depths.

We selected the four well-observed components with the same spatial resolution \({\varvec{T}}_{xx}\),\({\varvec{T}}_{xz}\),\({\varvec{T}}_{yy}\) and \({\varvec{T}}_{zz}\) as the observed data, which implies the Jacobian matrix \({\varvec{A}} = \left[ \begin{array}{c} {\varvec{G}}_{xx} \hfill \\ {\varvec{G}}_{xz} \hfill \\ {\varvec{G}}_{yy} \hfill \\ {\varvec{G}}_{zz} \hfill \\ \sqrt \mu {\varvec{W}}_{i} \hfill \\ \end{array} \right]\) and \({\varvec{b}} = \left[ \begin{array}{c} {\varvec{T}}_{\text{xx}} \hfill \\ {\varvec{T}}_{\text{xz}} \hfill \\ {\varvec{T}}_{\text{yy}} \hfill \\ {\varvec{T}}_{\text{zz}} \hfill \\ 0 \hfill \\ \end{array} \right]\).

The PCG method is an iterative method that provides linear or nonlinear solutions. It is mainly used to solve for equations such as \(\phi = \frac{1}{2}{\varvec{m}}^{\text{T}} {\varvec{Am}} - {\varvec{m}}^{\text{T}} {\varvec{b}}\) whose solution is \({\varvec{m}} = {\varvec{A}}^{ - 1} {\varvec{b}}\), equivalent to Eq. (18). This algorithm takes the gradient of initial point as the initial conjugate direction and seeks for the minimum of the objective function along this direction to improve stability. Following the computing process of the preconditioned conjugate gradient inversion, model \({\varvec{m}}\) is solved. The fitting error is used as a criterion for the iteration. If the model \({\varvec{m}}\) satisfies the predetermined difference \(\beta\), the iteration ceases. Details regarding the PCG algorithm can be found in Pilkington (1997, 2009) and Li and Yang (2011).

*N*is the number of observations, \({\varvec{d}}_{i}^{\text{obs}}\) is the measured data, and \({\varvec{d}}_{i}^{\text{fwd}}\) is the theoretical response from inversion models. Figure 9 shows the flow chart of the procedure of the PCG inversion algorithm.

### Anti-noise ability test of PCG algorithm

The uncertainty of the gravity gradient effect due to the total uncertainty sedimentary and Moho interface depth is the − 0.07% (\({\varvec{T}}_{xx}\)), − 2.03% (\({\varvec{T}}_{xz}\)), − 3.92% (\({\varvec{T}}_{yy}\)), − 4.51% (\({\varvec{T}}_{zz}\)), respectively. To ensure the reliability of the inversion results, we designed two sets of complex models (M1 and M2), and the 8% Gaussian noise was added to the four gravity gradient components in the synthetic model tests. To test the feasibility of the inversion method for several anomalies, model M1 consists of four anomaly blocks with different sizes, depths and densities. To demonstrate the effectiveness of the inversion method for large blocks, we designed two large blocks with density anomalies in Model M2.

Conditions and results for two different synthetic inversion models

Observation (T | Iteration | Maximum value of results (10 | Minimum value of results (10 | Mean value of results (10 | |
---|---|---|---|---|---|

Model 1 |
T
| 5 | 1.0649 | 0.0831 | 0.489 |

T | 6 | 1.1134 | 0.0956 | 0.512 | |

Model 2 |
T
| 6 | 0.8527 | 0.0654 | 0.445 |

T | 8 | 0.9033 | 0.0872 | 0.478 |

## Results

^{−2}. As shown in Fig. 11, after 19 outer iterations, the curve of defined RMS misfit tends to be horizontal, the variation is more and more slight between the values of defined RMS misfit, an defined RMS misfit of 3.5 × 10

^{−3}was obtained, at which point the inverse iterative computation was complete.

## Discussion

### Characteristics of the distribution of density anomalies in the eastern NCC

A notable spatial distribution pattern can be found for density anomalies in the whole eastern NCC, which adjoins to Bohai Bay. This is in agreement with the theory (Tian and Zhao 2011) that this area has suffered significant intense destruction.

There is a notable high-density anomalous body A1 in the North China Basin at a depth of 10 km (Fig. 12a) in the eastern NCC. The amplitude of the anomalies is significantly smaller at the top of the lower crust and the upper mantle (at depths of 25 km and 42 km, respectively). This anomalous body (Fig. 12b, c) is mainly distributed along the Beijing–Baoding line. This finding matches the Tangshan–Xingtai seismic zone and is in good agreement with the results concerning the P-wave velocity derived from a study conducted by Huang and Zhao (2004).

Density anomalies vary significantly between the two sides of the north–north-east-trending Tancheng–Lujiang fault zone in the eastern NCC. The anomalous body A2 on the west side of the fault zone exhibits notable characteristics, whereas the block on the east side of the fault zone exhibits no significant density anomalies. At the depths of 10–42 km (Fig. 12a–c), the anomalous body A2 on the west side exhibits significant low-density anomalies. At larger depths of 42–60 km, the maximum amplitude of the density anomalies in the anomalous body A2 on the west side decreases, while the body A3 exhibits no density variations. At the depth of 60 km, the density change of the anomalous body A2 and A3 is relatively gentle, and both the anomalous body A2 on the west side and the anomalous body A3 on the east side exhibit low-density anomalies, but the density characteristic of the body A3 is not evident as the A2 obviously. At depths of 80–120 km, the anomalous body A2 on the west side transitions from a low-density anomalous body to a high-density anomalous body, whereas the A3 still exhibits no obvious density variations. These results show that there is always a significant difference in the density distribution between the blocks on the two sides of the Tancheng–Lujiang fault zone and that the Tancheng–Lujiang fault zone has already penetrated the lithosphere.

In Bohai Bay (area A4), at depths of 42–120 km, particularly in the upper mantle, density anomalies are distributed along the Tancheng–Lujiang fault zone, low-density anomalies are more common, and the maximum amplitude of the density anomaly variations is large. Previous research results (Teng et al. 1997; Su et al. 2009) show that Bohai Bay experiences the combined action of the extension of the main fault at the eastern boundary of the Tancheng–Lujiang fault zone and potential plumes forming in the upper mantle. The gravity gradient effect caused by the upper mantle temperature has already been eliminated; however, density anomalies are still notably distributed in this area, and the maximum value of the low-density anomalies can reach − 50 kg/m^{3} at a depth of 120 km (Fig. 12e). These results show that density anomalies in Bohai Bay are primarily caused by an extension of the main fault at the eastern boundary of the Tancheng–Lujiang fault zone.

Notable low-density anomalies can be observed at a depth of 10 km (Fig. 12a) in the transition area A5 between the Yanshan block and the Songliao Basin. As the depth increases, continuous high-density anomalies can be found in the upper mantle. These density anomalies may be a result of the western extension of the high-density mantle lithosphere beneath the Songliao Basin towards beneath the Yanshan orogenic belt (Tian and Zhao 2011).

Compared with the central and western NCC, there are no wide areas of the density anomalies but a notable spatial distribution pattern in the whole eastern NCC, which indicates that this area has suffered different degrees destruction. Furthermore, comparing Fig. 12e, f with Fig. 6a, b which has been marked on the figures, we find that the notable density distribution exists with the high-temperature distribution. Combined with our results and previous research results, Teng et al. (1997); Su et al. (2009) show that there are obvious plumes forming in the upper mantle in the eastern NCC, which means that the high temperature of the lithosphere and mantle in the eastern NCC is accompanied with the non-uniform density of the lithosphere resulting from the composition.

### Characteristics of the distribution of density anomalies in the central NCC

The central NCC, composed of the central transition zone, mainly exhibits notable low-density anomalies. The anomalous bodies in this area exhibit significant segment-wise characteristics. The Taihangshan tectonic zone, with an overall north-east–south-west-trending strike direction, can be divided into three blocks, namely the southern block B1, the central block B2 and the northern block B3. At the same depth, the magnitude and distribution of anomalies both vary between these three blocks. At depths of 25–120 km, the left side of the southern block B1 adjoins to the positive-density anomalies in the western NCC, the central block B2 adjoins to the high-density anomalies in the eastern NCC and the northern block B3 adjoins to the high-density anomaly zone in the North China Basin in the east and adjoins to the Yinshan–Yanshan block in the west, forming a large low-density anomaly zone. The maximum values of the low-density anomalies in the southern block B1 and the central block B2 are more significant.

Overall, the Linfen–Weihe graben block is characterized by high-velocity anomalies alternating with low-velocity anomalies. Block B4 in the northern Linfen–Weihe Graben is mainly distributed in the Datong volcanic zone. Density anomalies in block B4 undergo significant changes with increasing depth. At a depth of 25 km (Fig. 12b), block B4 exhibits high-density anomalies. However, as the depth increases from 42 to 120 km, notable low-density anomalies can be found in block B4. This suggests that the Datong volcano is no longer active (Tian et al. 2009), but low-density anomalies are still distributed beneath this area.

At depths of 60–120 km, block B5 in the central NCC is surrounded by the low-density anomalies in the central Taihangshan block B2 and the northern Linfen–Weihe graben block B4. However, the high-density anomalies in block B5 become increasingly prominent as the depth increases and adjoin to the high-density anomaly zone in the North China Basin in the eastern NCC at the depth of 80 km (Fig. 12e). This indicates that the destruction of the lithosphere in the NCC may have already extended from the North China Basin to the central NCC.

At depths of 10–25 km, notable low-density anomalies can be found in the Yinshan–Yanshan block B6, and they are surrounded by high-density anomalies. These low-density anomalies are distributed in a relatively shallow zone along an orogenic belt and are mainly affected by the granitic bodies distributed along the orogenic belt (Hou et al. 2014). In addition, as the depth increases to 60 km (Fig. 12d), the aforementioned low-density anomalies adjoin to block B3 in the northern section of the Taihangshan tectonic zone and block B4 at the junction of the Linfen graben and the Weihe graben, forming a large low-density anomaly zone. Compared to previous findings (Wang et al. 2014), in this study, low-density anomalous blocks are more widely distributed in this area. The whole central NCC block is located in an orogenic belt, where the mantle temperature is relatively high (An and Shi 2007). This indicates that this area is significantly affected by both temperature and composition. A high heat-flow environment is present in this area, resulting in the upwelling of hot material in the deep asthenosphere, which modifies the lithospheric mantle.

### Characteristics of the distribution of density anomalies in the western NCC

Tectonic deformation and intense magmatic activity have occurred in the Ordos area since the Mesozoic–Cenozoic, resulting in the formation of a special tectonic framework in Ordos and its surrounding area (Zhang et al. 2011). The Ordos area is surrounded by two rift valleys, namely the Yinchuan–Hetao rift valley and the Shaanxi–Shanxi rift valley. Overall, the Ordos area displays as a stable and solid tectonic unit.

At a depth of 42 km (Fig. 12c), the central Ordos block C1 exhibits high-density anomalies, whereas the northern Ordos block exhibits no notable anomalies. Based on the receiver function (Tian et al. 2011), the central Ordos block is 41 km in thickness, and the northern Ordos block is 45 km in thickness. Based on this information, it is inferred that the central and southern areas of the Ordos are in the lithospheric mantle at a depth of 42 km. At depths of 60–120 km, high-density anomalies in the central Ordos block C1 transition to low-density anomalies, which are surrounded by high-density anomalies in the southern and northern areas of Ordos. The central Ordos block adjoins to the Yinshan orogenic belt in the north and the Qinling orogenic belt in the south. Because the characteristics of the distribution of density anomalies are consistent with those of the surface structure, it is inferred that the characteristics of the distribution of density anomalies on the south and north sides of Ordos are affected by the deep dynamic processes in the orogenic belts on the two sides.

The Qilian block is composed of block C2 in the south-east and block C3 in the north-west. Overall, a part of the Qilian block in the crust exhibits notable low-density anomalies. The upper boundary of this area corresponds to the Haiyuan fault zone. The direction of density anomalies in this area is basically consistent with the strike direction of the surface fault zone. At the junction of the Qilian block and the Ordos block, the distribution of density anomalies exhibits a significant abrupt change—from low-density anomalies to high-density anomalies. Generally, the anomalies in the Qilian block are north-west–south-east-trending, which is a behaviour that is closely related to the regional tectonic framework—the movement of the Tibetan Plateau changes from northward to eastward upon being blocked by the stable Ordos block and Alxa block. At a depth of 60 km (Fig. 12d), block C2 in the north-east exhibits continuous high-density anomalies; in addition, as the depth increases, the distribution of high-density anomalies becomes increasingly prominent, and the amplitude of the high-density anomalies increases. In comparison, block C3 in the north-west still exhibits notable low-density anomalies, in which the amplitude decreases as the depth increases. The seismic sounding results show that the Moho depth in the Qilian block is less than 60 km (Tian and Zhao 2011). In addition, the tomography results show that intense low-velocity anomalies at the bottom of the crust transition to intense high-velocity anomalies at the top of the Moho (Li et al. 2002). High-density anomalies found in block C2 in the south-east suggest that the block enters the lithospheric mantle at this depth.

The Alxa block C4 exhibits continuous notable high-density anomalies from the crust to the mantle, indicating that this area is a stable tectonic unit. Part of the Qaidam block C5 in the crust exhibits low-density anomalies. At a depth of 42 km (Fig. 12c), the lower boundary of the density anomaly zone corresponds to the Riyueshan fault, and the direction of the density anomalies is basically consistent with the strike direction of the surface fault zone. At a depth of 60 km (Fig. 12d), this zone is characterized by low-density anomalies alternating with high-density anomalies that adjoin to the high- and low-density anomalies in the Qilian block. The northern Bayan Har block C6, which adjoins to the Qaidam block C5, exhibits intense low-density anomalies. At depths of 80–120 km, the direction of the density anomalies is no longer consistent with the strike direction of the fault zone. The Qaidam block C5 mainly exhibits high-density anomalies, whereas the low-density anomalies found in the northern Bayan Har block C6 become increasingly prominent as the depth increases.

Considered the whole central and western NCC, the destruction of the lithosphere in the NCC has extended from the North China Basin only to the regional area of the central NCC. Meanwhile, referred to the temperature variation in this area (Fig. 6a, b), the temperature of the lithosphere and mantle in western NCC is relatively low. These shallow responses to the deep characteristics show that the whole main region of the central and western NCC has not been destroyed yet the overall NCC still maintains stable attributes.

## Conclusions

The GOCE data were processed with several corrections to obtain the remaining gravity gradient components, and the non-uniform density distribution in the lithosphere related to the internal temperature variations was assessed. The effect of the gravity gradient on the density variation in the lithospheric mantle due to temperature was separated from that due to composition. The data corrections were completed by analysing the uncertainty related to the models used.

The eastern NCC exhibits notable regional characteristics, the distribution of density anomalies varies significantly within this area and the results show that there is always a significant difference in the density distribution between the blocks on the two sides of the Tancheng–Lujiang fault zone. The Tancheng–Luyan fault zone has already penetrated the lithosphere and has extended to Bohai Bay.

The central NCC, composed of the central transition zone, mainly exhibits notable low-density anomalies. The anomalous bodies in this area exhibit significant segment-wise characteristics. The whole central NCC block is located in an orogenic belt, where the mantle temperature is relatively high; the results show that this area is significantly affected by both temperature and composition. A high heat-flow environment exists in this area, causing the upwelling of hot material in the deep asthenosphere, which modifies the lithospheric mantle.

In the western NCC area, the Ordos block and the Alxa block appear collectively as a stable and solid tectonic unit. The directions of the density anomalies in this part of the Qilian block and the Qaidam block in the crust are basically consistent with the strike directions of the surface fault zones corresponding to the upper and lower boundaries, respectively. It is inferred that the characteristics of the distribution of density anomalies on the south and north sides of Ordos are affected by the deep dynamic processes in the orogenic belts on the two sides. The anomalous density body in the Qilian block is north-west–south-east trending.

## Declarations

### Authors’ contributions

YW provided the initial idea of this study, YT processed and designed all the data and experiments; then, YT applied the processed data and method to the North China Craton, YT and YW analysed the results, and YT wrote the paper. All authors read and approved the final manuscript.

### Acknowledgements

We are so grateful to two reviewers for providing valuable and constructive suggestions that improved this paper. We are grateful to Prof. Josef Sebera in Kiel University and Astronomical Institute (CAS) for providing the initial gravity gradiometry data. We are grateful to Prof. Michael S. Zhdanov in University of Utah for providing constructive comments on this paper. We are grateful to Dr. Mo Tao in University of Utah for providing a lot of help when we prepared this paper. We are also grateful to Prof. Meijian An in the Institute of Geomechanics (CAGS) for providing the temperature data in the NCC.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

The preprocessed GOCE gravitational gradient grid data at noise-free satellite altitude under spherical coordinates can be downloaded from the http://goce.kma.zcu.cz/.

### Funding

This work was supported by the R&D of Key Instruments and Technologies for Deep Resources Prospecting (the National R&D Projects for Key Scientific Instruments) under Grant No. ZDYZ2012-1-04.

### Publisher’s Note

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## Authors’ Affiliations

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