# Accelerating High-Resolution Seismic Imaging by Using Deep Learning
## 2 Methods
### End-to-End Learning with Small Patches
\tag{2}\]  
Here, \(s(x,t)\) is the scaled amplitude of each patch, and \(max\) and \(min\) denote the positive maximum and negative minimum amplitudes, respectively. \(f(x,t)\) denotes the original image patches. When the original data is greater than or equal to 0, \(max\) is used as the numerator; otherwise, minus \(min\) is divided. After the amplitude balance, the data range of each patch is normalized to \([-1,+1]\). As shown in Figure 7, the \(PSTM\) block represents the balanced conventional PSTM result patches, and the \(QPSTM\) block denotes the balanced QPSTM patches. The \(PSTM\) patch was fed to the network as the input. After the feature encoding and decoding, the network predicted the output, denoted as \(PPSTM\) in the figure. It should be noted that the \(PPSTM\) had the same size as that of the \(PSTM\) and \(QPSTM\) patches. In the training phase, the difference (shown as \(DIFF\) in figure) was obtained element-wise in the patch. Furthermore, we evaluated the loss, a scalar that represents the difference between the predicted and the ground truth result, by using the mean absolute error (MAE), which is defined as  
\[Loss=\sum_{i=1}^{nx}\sum_{t\
eq 1}^{nt}|p(x,t)-q(x,t)|, \tag{3}\]  
where \(p(x,t)\) is the predicted patch with the filtering effect of the network from the original PSTM imaging patch, and \(q(x,t)\) represents the QPSTM imaging patches. In this case, we use the MAE instead of the mean square error (MSE) as the loss function, because the MSE can smear the different phases of the seismic events, thereby mitigating the gradient difference. In a DL framework, the obtained loss is back-propagated to the network to update the network's hyper-parameters and make the network more suitable for the training data set. Among the several optimization methods that can be used to train the network, in this study, the widely used Adam optimizer [46] was employed owing to its computational efficiency and ability to obtain the adaptive estimates of the lower-order moments. The prediction phase is easier to understand than the training procedure. Figure 7 illustrates the prediction and training phases together. In the prediction phase, only the PSTM patches are input to a well-trained network. In our framework, the training is conducted, and the scale is applied for every patch (as defined in Formula (2)). However, the objective is to attain the high-resolution images. To this end, in the prediction phase, when scaling the PSTM patch, we memorize \(max,min\) at the time. After feeding the patches into the network and obtaining the predicted patches, \(max,min\) is eliminated from these patches. Consequently, each predicted patch has a comparable energy intensity and a constant scale factor. In addition, Figure 5 and Table 1 indicate that the last layer of the network is an up-sampling convolution layer. It is difficult to reduce the boundary artifacts induced by the size extension. To suppress these artifacts, we retain only the inner half of the predicted patch. In particular,we predict the patch with a size of \(64\times 128\) pixels; however, we only select the inner \(32\times 64\) pixels as the meaningful result. We choose the overlapped patches as the input and obtain the overlapped output. After cropping and merging, the final result can be obtained. In the **end-to-end** learning, the framework is mapped from one image profile to another image profile with some features emphasized. The features are automatically learned from the given data during the training process. The feature extraction and representation are hidden in the network's training/prediction procedure. In computer vision science, the image is treated as a whole input; however, in geophysics, especially in exploration seismology, the image is characterized by high complexity and diversity. Therefore, we proposed the use of a small patch-to-patch learning method, which is also an image-to-image model. In all the aspects, we achieved end-to-end learning.