#### 4.1. Data Reduction and Weighting

The InSAR observations shown in

Figure 2 consist of millions of pixels, and it is impractical to use all of them to invert the model. To reduce the number of points and to improve the computational efficiency, several methods can be used to downsample the original interferograms, including uniform sampling [

39], quadtree sampling [

40], resolution-based quadtree sampling [

41] and equation-based sampling [

42]. We adopted the equation-based sampling approach, which can distinctively sample near-field points according to a deformation gradient and prevent the far-field noises based on the difference in Green’s function [

42]. The location-dependent look vectors of each observation were then calculated considering the differences of incidence and azimuth angle based on the precise orbit data and local topography. Finally, 502 and 568 observations were obtained from the Sentinel-1A tracks T056A and T136D interferograms, respectively, and 416 observations were obtained from the ALOS-2 track P160A interferogram.

Because three different datasets were used for modeling, an appropriate determination of the weighting of each dataset was required during inversions. Although the empirical errors of the three interferograms derived from the 1-D covariance function (

Section 3.2) can be used to weight the three datasets directly, Xu

et al. [

12] suggested that the HVCE method has various advantages over other methods, especially when the number of redundant observations is relatively large in a joint inversion. In this study, the HVCE method was employed to weight the three different observations to derive the coseismic slip distribution of the Pishan earthquake. In the HVCE method, the relative weight ratio is obtained according to the posterior misfit information of each dataset. During inversion, the starting value for the relative weight ratio among the datasets was given as 0.131:1:0.174 based on the variances (standard deviations) of the three interferograms. After 5 iterations, the variances of unit weight became almost uniform for all three datasets, and the relative weight ratio converged to 0.344:1.000:0.495.

#### 4.2. Finite Fault Slip Model

The source parameters for the 2015 Pishan event were modeled through joint inversions of the ascending and descending Sentinel-1A and the ascending ALOS-2 data using the analytical solutions of a rectangular dislocation in a homogeneous, elastic half-space [

43]. During inversion, a two-step strategy was employed to solve the fault parameters of the one-segment model. In step one, a nonlinear inversion was used to determine a set of model parameters (fault geometry) by minimizing the misfit under the assumption of a uniform slip on a rectangular fault; in step two, a linear inversion was used to estimate the slip distribution on the modeled fault plane.

In this study, a hybrid minimization algorithm basing on multi-peak particle swarm optimization (M-PSO) [

44] was used to invert nine fault parameters, including location, strike, dip, length, depth, width and slip, by minimizing the misfits between the observed and the model predictions, assuming a Poisson ratio of 0.25 and a shear modulus of 3.32 × 10

^{10} N/m

^{2} [

45]. A Monte Carlo bootstrap simulation technique [

46] was used to estimate the uncertainties of the fault parameters. Model solutions from 100 simulations perturbed with noise from the statistical properties based on previous 1-D covariance functions were used to estimate the standard deviation from their distributions.

Table 2 shows all fault parameters and their errors. In general, the errors are relatively small. The inversion results indicate that the rupture fault is a blind thrust fault dipping approximately 24° to the southwest. The inverted geodetic moment magnitude is Mw 6.4, which is consistent with the GCMT result and slightly lower than that of CENC, but larger than the value from USGS.

Under the determined fault geometry, the slip on the rupture plane shows a linear relationship with surface displacements based on classic linear-elastic dislocation theory. Fixing the fault geometry for the optimal fault plane determined in the uniform slip modeling enables the fault length and width to be extended to 36 km along the strike and 40 km along the down-dip direction, respectively, and then discretized into 1440 patches with a size of 1 km by 1 km. Then, a constrained least squares method is used to solve the objection function:

where

G is Green’s matrix relating unit slip at the patches to the predicted displacement,

$s$ is the slip vector on each patch,

d is the observed InSAR LOS displacement,

L is the second-order finite difference approximation of the Laplacian operator used to avoid unphysical oscillating slip distribution, and

k^{2} is the smoothing factor.

Previous studies (e.g., [

44,

47,

48]) indicate that the fault geometry (especially the dip angle) derived from the uniform slip model is not optimal for the distribution slip model. In this study, the grid search method proposed by Feng

et al. [

44] is used to determine the optimal dip and smoothing factor simultaneously. In their method, a log function

$f\left(\delta ,{\kappa}^{2}\right)=\mathrm{log}\left(\psi +\xi \right)$ is used, where

$\delta $ is the dip angle,

ψ is the model roughness and

ξ is the residuals. The optimal dip angle and smoothing factor derived from the log function are 25° and 2.5°, respectively (

Figure 4). The dip angle is slightly different from the nonlinear inversion results.

**Figure 4.**
Plot of the log-function of the sum of root-mean-square error and model roughness for a range of models with different smoothing factors (

k^{2}) and dip angles. Red cross is the global minimum, which is chosen for the inversion presented in

Figure 5.

**Figure 4.**
Plot of the log-function of the sum of root-mean-square error and model roughness for a range of models with different smoothing factors (

k^{2}) and dip angles. Red cross is the global minimum, which is chosen for the inversion presented in

Figure 5.

The preferred coseismic slip distribution and uncertainties from joint inversion of the ascending and descending Sentinel-1A and the ascending ALOS-2 observations are shown in

Figure 5. The rupture fault is dominated by thrust motion with a slight strike-slip component. The total released geodetic moment is approximately 6.1 × 10

^{18} N m (Mw 6.5), which is in agreement with the CENC result but larger than those of GCMT and USGS. The predominant slip occurred on the fault with a peak magnitude of 0.89 m at a depth of 9–14 km (width of 21–33 km). In addition to the main slip, the upper layer 0–7 km is found to have a near-uniform slip with a value of ~0.2 m and with 20% of the total moment, but there is no finding of surface rupture from field investigation organized by Institute of Geology, China Earthquake Administration. The shallower asperity corresponds to an Mw 6.0 sub-event (compared with the maximum aftershock of ML 5.0) with rupture close to the surface, although there is a relatively large uncertainty in the same area. The average errors of slip are ~5 cm, except for the area close to the surface, where the error increases to 7 cm, which can be attributed to reduced measurement constraints.

**Figure 5.**
The 1 km × 1 km finite fault model of the 2015 Pishan earthquake (**a**), standard deviations in slip from the Monte Carlo estimation with 100 perturbed datasets (**b**) and the sum of moment release along strike distance (**c**).

**Figure 5.**
The 1 km × 1 km finite fault model of the 2015 Pishan earthquake (**a**), standard deviations in slip from the Monte Carlo estimation with 100 perturbed datasets (**b**) and the sum of moment release along strike distance (**c**).

Figure 6 shows the simulated interferograms and residuals from our best fitting slip model. It is clear that the general patterns of both Sentinel-1A and ALOS-2 observations can be sufficiently explained by the distributed slip model. There are no notable residual fringes in the surrounding region of the seismogenic fault. The standard deviations between the InSAR observations and the simulated LOS displacements are 5.1 mm, 2.4 mm and 4.2 mm for tracks P056A, T136D and T160A, respectively, which are close to the noise levels of these InSAR observations. The correlation coefficient between the observations and predictions is 99.4%.

**Figure 6.**
Modeled interferograms for the ascending Sentinel-1A track T056A (**a**), the descending Sentinel-1A track T136D (**b**) and the ascending ALOS-2 track P160A (**c**); and their residuals (**d**–**f**) with the distributed slip model. The white line is the top boundary of the distributed slip model, and the black dashed lines represent the surface projections of the modeled fault.

**Figure 6.**
Modeled interferograms for the ascending Sentinel-1A track T056A (**a**), the descending Sentinel-1A track T136D (**b**) and the ascending ALOS-2 track P160A (**c**); and their residuals (**d**–**f**) with the distributed slip model. The white line is the top boundary of the distributed slip model, and the black dashed lines represent the surface projections of the modeled fault.