# Three-Dimensional Velocity Field Interpolation Based on Attention Mechanism
## 3 Experiments
### Synthetic Data  
Since it is impossible to obtain the real three-dimensional velocity field data for the entire work area, we need to evaluate the network's interpolation results through synthetic data. Synthetic data can be generated according to specific patterns and rules, ensuring that all variables are controllable during the testing process while also reducing the complexity of network pretraining. After generating synthetic data for the entire work area, we simulate sparse data by sampling and inputting them into the network, and then compare the residuals of the interpolation results and synthetic data. By using synthetic data, we can test and verify the algorithms of this paper under known geological conditions. Figure 4a shows synthesized three-dimensional velocity field data. Subsequently, sampling of the synthesized simulated three-dimensional velocity field is conducted, as shown in Figure 4b. We use the sampling results as actual observed data. Moreover, we can set the sampling rate to evaluate the network's three-dimensional interpolation capability by comparing different existing data volumes. This is then applied to subsequent real data. To compare the effects of basic 3 DUnet and the constructed JointA 3 DUnet network's three-dimensional interpolation, we use both networks to interpolate the sampled synthetic data. To control the singularity of the dependent variables, the inputs, loss functions, and training methods of the two networks are the same, and the input data are a randomly sampled 10% of the synthetic data. The obtained three-dimensional velocity field is shown in Figure 5. From Figure 5, neither 3 DUnet nor JointA 3 DUnet has been able to achieve an ideal interpolation effect. In the areas with layer undulations and low values, the interpolation results of both networks are uneven. The reason for this phenomenon is that the receptive field of the convolution kernel is too small, and the network cannot learn useful features in positions with large data loss. Therefore, this paper introduces dilated convolution to increase the receptive field of the convolution kernel. In the later experiment, we chose the expansion rate as 1,2,5, and the interpolation results are shown in Figure 6. From the figure, the continuity of the undulating layers has been improved. However, the interpolation effect of 3 DUnet in the low-value areas is still not ideal after combining with dilated convolution. The network JointA 3 DUnet in this paper can overcome this problem better after combining with dilated convolution. As shown in Figure 7, we conducted experiments on a synthetic dataset. We visualized the overall relative error, and observation point relative error for the interpolation results obtained from both methods as well as after adding dilated convolution. From Figure 7, it can be observed that JointA 3 DUnet outperforms the traditional 3 DUnet in terms of observation point relative error and overall relative error. Compared with 3 DUnet, the JointA 3 DUnet without adding dilated convolution reduces the overall relative error by 22.31%, and the relative error of observation points by 28%. After adding dilated convolution, the errors of both JointA 3 DUnet and 3 DUnet decrease, but the error of JointA 3 DUnet is still the smallest. Compared with 3 DUnet, the overall relative error of JointA 3 DUnet decreases by 22.77%, and the relative error of observation points decreases by 21.43%. Therefore, the interpolation results of JointA 3 DUnet are more accurate and better than those of 3 DUnet. Although the observation point relative error of 3 DUnet is also lower, the overall effect is not ideal. This is due to the training method used. The loss function defined in this paper fits the speed data of the observation point positions. As the number of training times increases, even the traditional 3 DUnet can fit the observation points. However, the missing parts of the non-observation points need to be generated by network interpolation.