# Development of a Deep Neural Network Model for the
## 2 Data and Method
### Data Used for Model Training and Testing
#### 2.1.2 Synthetic Data Generations  
Our synthetic data, that include both training data and true data, were constructed through forward modeling with our randomly generated velocity models. In generating these models, we adhered to fundamental geological principles. Firstly, P-wave velocities typically ranged from 1500 m/s to 3200 m/s, a range established by the site engineers' past experiences. Secondly, velocities generally increased with greater depth. To create our velocity models, we started with an initial vertical velocity profile model and introduced random variations. The model was divided into 30 depth layers, covering a depth range of 3000 m and each layer was assigned with a constant velocity initially. Vertically, the velocities ranged from 2000 m/s to 3000 m/s, increasing linearly from the bottom layer to the top. For each layer, we added a random variation of 50 m/s to simulate natural heterogeneity (Figure 2a). Horizontally, the model spans 8000 by 12,000 m, structured as a 30 by 30 grid. We employed two methods to introduce variance into a plane with constant velocity, which was calculated based on the previously discussed vertical velocity profile. Both methods start with this constant velocity plane that is defined by the vertical velocity profile discussed in the aforementioned text. For the first method, we used the following equation to add variance to our velocities on the horizontal plane:  
\[V(x,y)=a_{1}\sin\left(a_{2}x\right)+a_{3}\cos(a_{4}y)+a_{5}\]  
where \(x\) and \(y\) are the horizontal coordinates, \(a_{i(i=1-5)}\) are random variables. We either start with \(a_{1}=60\), \(a_{2}=0.25\), \(a_{3}=50\), \(a_{4}=0.15\), \(a_{5}=0.20\), or we randomly select six points and add random variables to their velocities; for the remaining grid points on the horizontal plane, we calculate velocities using inverse distance weighting. Both methods use 180 parameters (30 vertical layers and each layer has 6 parameters) to characterize our velocity model. These methods ensure that our forward modeling process incorporates realistic horizontal velocity variations, enhancing the accuracy and applicability of our synthetic data for training seismic inversion models. This approach ensures that the forward modeling process realistically reflects the geological characteristics of the study area, thereby enhancing the reliability and applicability of the synthetic data for training the seismic inversion models. With the generated velocity models, we hypothetically placed earthquake sources within the model and then used ray-tracing methods to calculate the P-wave arrival times at each receiver. These arrival times and earthquake locations were subsequently used as the training data. For the models, the event locations were randomized. For each model, 4000 events were generated. We employed two methods. First, events happening in locations that were spatially continuous; the coordinates of the first earthquake were (X, Y, Z). For the second earthquake, we adjusted the coordinates by adding random values to X, Y, and Z (i.e., X + random (\(-0.5\), \(0.5\))). The same approach was also applied to the Y and Z coordinates. This method creates a series of events that are roughly connected, simulating real-world scenarios where earthquakes predominantly occur along the same fault plane. We hereby denote this as event generation methods I. Or, when we have entirely randomly scattered events, we denote this as event generation methods II. Figure 2: (a) Vertical velocity profile and vertical direction line graph on the x-z plane at Y = 0; (b) Horizontal velocity profile on the x–y plane at Z = 0 (triangles denote points of abrupt velocity change). Ray tracing is utilized in both linear and nonlinear inversion methods to model the paths taken by seismic waves as they travel through the Earth. Ray-tracing algorithms simulate how these paths bend and twist in response to varying subsurface properties, which is essential for accurately predicting the travel times and amplitudes of seismic waves.