## 3 Synthetic Data Test  
This paper assumes that the velocity is known, and the proposed method is applied on a layered model to test its viability and efficiency. Velocities, densities, and Q values in each layer are listed in Table 1. A shot gather is generated by ray tracing with a Ricker wavelet of 60 Hz peak frequency. The velocity can be estimated by tomographic inversion or refraction method in field data. Tomographic inversion is divided into two categories: diving wave tomography and waveform tomography [46]. Theoretically, the waveform tomography inversion algorithm has higher velocity inversion accuracy, but the method is more dependent on the initial model and seismic low-frequency data, so its application is limited. Research on diving wave tomography inversion has shifted from mid-deep models to near-surface models in recent years, gradually weakening the dependence on stratigraphy and moving toward a data-driven direction. Refractive wave method includes the wave front method, delay time method, intercept time method, t0 difference method, and generalized reciprocity method. The principle of the method is to calculate the layer thickness and velocity of each layer from the travel time data of direct and refracted waves [47]. The constant Q model is used for velocity dispersion and attenuation calculation of seismic wave (Figure 2). Figure 3 shows the calculated attenuated travel time by using the proposed method and the conventional attenuated travel time tomography method. The proposed method produced more accurate attenuated travel time results. With the increase of the offset distance, the conventional method will gradually increase the attenuated travel time, and compared with the proposed method, the error in the attenuated travel time is large. For the proposed method, we can see that when the offset distance is about 20, the error between it and the true value is the smallest, and as the offset distance increases, the error gradually becomes larger, and when the offset distance reaches the maximum, the error between it and the true values also reaches the maximum. The error between the proposed method and the true value is within an acceptable range, so this method is considered to be reliable. Then the Q value is estimated by using Equation (4), and the initial Q value is set to 100. The Q value results are shown in Figure 4. The conventional attenuated travel time tomography method contains error in attenuated travel time and results in a bias of Q value results, especially for the shallow layer. In the first layer, the error in the Q value results is small by the conventional attenuated travel time tomography method. Equivalent Q values are calculated in the first layer by horizontally selecting the seismic reflection waveform, so \begin{table}
\begin{tabular}{l l l l l l l} \hline  & **Depth** & **Density** & **Vs** & **Vp** & & \\  & **/m** & **/(kg\(\cdot\)m\({}^{-3}\))** & **/(m\(\cdot\)s\({}^{-1}\))** & **/(m\(\cdot\)s\({}^{-1}\))** & **True Q** & **Inverted Q** \\ \hline
1 & 200 & 2095.2 & 1200 & 2100 & 50 & 49.991 457 \\
2 & 200–400 & 2143.3 & 1300 & 2500 & 100 & 100.001 44 \\
3 & 400–600 & 2188.5 & 1400 & 2700 & 200 & 199.972 46 \\
4 & 600–800 & 2290.6 & 1500 & 3000 & 300 & \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for layered model. Figure 2: Synthetic shot gather of the layered model. The move out of the traces are corrected. there is no additional influence from the overburden. The overburden effect comes from the attenuation effect of different travel paths in the overburden of the reflections from A1 and from A2 seen in Figure 1. This cumulative attenuation effect cannot be eliminated and leads to errors in Q estimation in the conventional LSR method. While in the deeper layers, the Q value is calculated by longitudinally selecting the seismic reflection waveform. Especially the second layer contains more effects from the overburden and thus leads to error in the conventional attenuated travel time tomography method.