## 3 Results
### Equivalent Source Inversion
#### 3.1.2 Checkerboard Test  
Before the inversion of the surface time-varying gravity field, it is necessary to examine the source resolution capacity of the gravity monitoring network in the study area. To this end we applied the checkerboard test, which is commonly used in seismic tomography studies [49]. As we aimed to conduct equivalent source inversion for crustal apparent density, we only tested the resolution in the horizontal directions. We assigned alternatively distributed sources of positive and negative anomalies of the same size in the study area, measured the theoretical gravity anomalies at the surface monitoring stations by forward gravity modeling, and then used the inversion method to solve the source parameters and compare the results with the designed values to evaluate the differences in spatial resolution caused by the uneven distribution of the existing monitoring stations [50]. We designed discretized source bodies with a size of 0.5\({}^{\circ}\)\(\times\) 0.5\({}^{\circ}\), a burial depth of 10 km, an apparent density of \(\pm\)1.0 kg/m\({}^{3}\), and a thickness of 1 km to examine the resolution capacity of the gravity monitoring network in the middle segment of the North Tianshan Mountains with the hypothetical equivalent source model [51]. The range of surface gravity anomalies obtained by forward modeling was about \(\pm\)30 uGal.  
Figure 3 shows the theoretical gravity anomalies at the ground surface associated with a checkerboard model with a resolution of 0.5\({}^{\circ}\)\(\times\) 0.5\({}^{\circ}\), as well as the spatial distribution of the monitoring stations in the study area. The locations of the stations are marked by grey rectangles, and the color of the rectangular patches indicates the measured gravity anomalies at the ground surface. The monitoring stations are unevenly distributed in the study area, and the employed resolution was chosen after considering the source resolution capacity of the stations, the optimal design and layout of the monitoring network, and the uncertainty of the source signals [52]. Then, the theoretical gravity anomalies with a checkerboard pattern were inverted to obtain the crustal apparent density using the equivalent source inversion method. The results are presented in Figure 4, where the grey rectangles mark the locations of the surface monitoring stations. The obtained source parameters can provide a reliable reference for the gravity variations in different regions. The figure shows that the predetermined source parameters cannot be accurately recovered in regions far away from the monitoring stations, and that there are certain areas that are not covered by the monitoring network. At the edges of the covered area, the inverted source body densities therefore show transition features, mainly manifested as the constraints of spatial smoothing regularization. The test shows good resolution in the area covered by the monitoring network, and the results were representative of the surface gravity changes induced by the regional crustal material migration. This indicates that the network's  
resolution would not have significant impacts on the subsequent inversion of the actual mobile gravity data.