## 3 Fault Model Inversion  
We will select a representative and geologically reasonable near surface velocity model to test the feasibility and reliability of the improved travel time tomographic inversion algorithm proposed in this paper. The synthetic data is a three-layer fault model. The selection of algorithms and parameters are as follows:  
1. The maximum signal-to-noise ratio method is used for the first arrival travel time extraction.
2. A second-order MSFM method with the highest accuracy is used for the forward travel time calculation.
3. Selecting the unit matrix as the data weighting matrix.
4. The weighted matrix of the model chooses a composite constraint matrix, which combines the second-order differential Laplacian with the prior information model obtained by the t0 difference method.
5. The dynamic change of regularization factor selection with the iteration process.
6. Sparse linear equations and least square problems (LSQRs) with the highest efficiency and stability for solving a large sparse matrix.
7. The number of iterations is 10.  
The fault model is shown in Figure 9. The lateral distance is 100 m, and the vertical depth is 40 m. There is a steep inclined reverse fault interface between 40 m and 60 m. The interface depth of the first refractive layer increases from 8 m to 16 m, and the interface depth of the second refractive layer increases from 16 m to 20 m. The first layer has a speed of 1500 m/s, the second layer has a speed of 3500 m/s, and the third layer has a speed of 5500 m/s. The size of the grid is 1 m. The observation system refers to shooting and receiving conditions on the surface. The offset distance is 4 m, the shot spacing is 2 m, and the trace spacing is 1 meter. A total of 49 seismic sources and 97 receivers are used.  
Figure 10a is a preliminary interpretation result of the t0 difference method for a real model. In order to avoid the local extremum in the inversion process, Figure 10b is obtained by an inverse distance gradient weighted interpolation of the interval velocity on the interface of the velocity mutation from the t0 difference method. Then, the model is used as an a priori model in the inversion process. Figure 10c is the inversion result. Figure 10d is the corresponding ray path and Figure 11 is the fitting curve of the inversion iteration error. The error curve is used to evaluate the applicability of the model.  
From the inversion results, the layered morphology fitting is very accurate, and the velocity values of each layer are also consistent with those of the real model. Errors mainly occur at the interface of the fault location, and the inversion results show a smooth effect. Steep faults become gentle, which is one of the characteristics of refraction travel time tomography. In addition, the uncertainty of the depth and velocity of the stratum interface increases towards the edge of the model, which is caused by insufficient ray coverage at the edge of the model. From the ray path, the ray tends to gather at the high-speed area and basically glide along the high-speed interface. It can be seen that the depth of the three-layer interface depicted is more accurate. From the error fitting curve, after 10 iterations, the root mean square error converges quickly and drops to a very small level, which shows that the data fitting is very good. In general, the three-layer fault structure has been reconstructed well, which verifies the effectiveness and reliability of the algorithm.  
Figure 10: Inversion results of the fault model. (**a**) t0 results; (**b**) prior models; (**c**) inversion result; (**d**) ray paths.  
Figure 9: Synthetic model.