# Single-Shot Time-Lapse Target-Oriented Velocity Inversion Using Machine Learning
## 2 Methodology
### Machine Learning Training and Testing
A series of convolutional layers with increasing dilations is at the core of the TCN architecture, widely used for sequence modeling; the difference here is that we combine this concept with the standard convolutions across traces. The next layer is a global max pooling layer that retains only the maximum value of each feature map outputted by the convolutional layers in order to reduce dimensionality and constrain the neural network complexity. With this pooling, only the strongest patterns make it through the rest of the network, and it also introduces some invariance to subtle changes in the input [25], resulting in better generalization capabilities. The pooling outputs are submitted to a dropout layer with a 2% rate, and then to a fully connected layer of 1111 neurons. The next layer then applies a linear shifting and scaling to the 0-100 range, which is the interval of variation in the anomaly velocities. By performing this, we force the data in the previous layers to be within the [\(-\)1, 1] interval for a better conditioning of the error gradient during backpropagation, increasing the stability and convergence of training. Finally, the data are reshaped as an 11 \(\times\) 101 matrix to form the predicted velocity anomaly \(\Delta v\). The proposed network architecture has parameters of about \(2.39\times 10^{5}\). For training, we use Adam optimizer with the mean squared error (MSE) loss function, an adaptively decreasing learning rate (starts at \(10^{-3}\), reduces by 80% whenever validation loss does not improve for 25 consecutive epochs), batch size 32, and early stopping as the training stopping criterion. Dropout and early stopping are regularization measures to reduce the overfitting issue [25; 26]. Figure 5 shows the MSE loss curves computed for the training and the validation sets. The loss function indicated in the figure shows a good convergence during the neural network training and validation processes. The curves for the training and validation subsets are very close to one another, an evidence that the training converged and that the network was not overfitting. To compare the accuracy of the inverted velocity reservoir anomaly with the true anomaly, we employ the Structural Similarity Index Measure (SSIM) [27], which is a scalar metric that quantifies the perceptual similarity between images. SSIM varies in the [0, 1]  
Figure 4: Proposed neural network architecture. The input of the ML is the seismic time-lapse difference within the reservoir time window and the output is the velocity anomaly of the target region. Figure 5: Loss curves computed on the training and validation subsets during the training of the neural network employed for velocity inversion. The loss function is calculated with velocities in the scale of the 4D anomalies (0–100 m/s range). range; 1 indicates that the compared images have good similarity, and 0 means that they are very different. In addition, we employ a mean average error (MAE) to quantify uncertainty in our analysis. The MAE estimates the pixel-to-pixel difference between the ML-estimated velocity anomaly and the real anomaly.