## 3 Methodology
### Loss Function  
To utilize knowledge representations for guiding wave separation, this study modeled them as a loss function, \(L\), which includes the reconstruction, representation, gradient, and cosine losses. The specific loss function is defined as follows:  
\[L=L_{rec}+\lambda_{1}L_{rep}+\lambda_{2}L_{grad}+\lambda_{3}L_{cos}, \tag{12}\]  
where \(L_{rec}\) represents the reconstruction loss that measures the difference between the original and reconstructed wavefields, \(L_{rep}\) represents the representation loss, which measures the difference between the predicted and physics-knowledge-represented wavefields, \(L_{grad}\) represents the gradient loss obtained from the morphological knowledge representation, emphasizing the wavefield continuity, and \(L_{cos}\) represents the cosine loss obtained from the knowledge representation of the difference in the propagation direction,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Layer**} & \multicolumn{2}{c}{**Output Channels**} & \multicolumn{2}{c}{**Kernel Size**} & \multicolumn{1}{c}{**Stride**} & \multicolumn{1}{c}{**Activation**} & \multicolumn{1}{c}{**Output Shape**} \\ \hline \multicolumn{1}{c}{Input} & 1 & - & - & - & 1600 \(\times\) 100 \(\times\) 1 \\ \hline \multirow{5}{*}{Encoder Layer} & Conv2D & 32 & \(3\times 3\) & \(1\times 1\) & tanh & 1600 \(\times\) 100 \(\times\) 32 \\ \cline{2-7}  & MaxPool & - & \(2\times 2\) & \(2\times 2\) & - & 800 \(\times\) 50 \(\times\) 32 \\ \cline{2-7}  & Conv2D & 64 & \(3\times 3\) & \(1\times 1\) & tanh & 800 \(\times\) 50 \(\times\) 64 \\ \cline{2-7}  & Conv2D & 32 & \(3\times 3\) & \(1\times 1\) & tanh & 800 \(\times\) 50 \(\times\) 32 \\ \cline{2-7}  & MaxPool & - & \(2\times 2\) & \(2\times 2\) & - & 400 \(\times\) 25 \(\times\) 32 \\ \hline \multirow{5}{*}{Decoder Layer} & Conv2D & 32 & \(3\times 3\) & \(1\times 1\) & tanh & 400 \(\times\) 25 \(\times\) 32 \\ \cline{2-7}  & UnSampling2D & - & \(2\times 2\) & - & - & 800 \(\times\) 50 \(\times\) 32 \\ \cline{2-7}  & Conv2D & 64 & \(3\times 3\) & \(1\times 1\) & tanh & 800 \(\times\) 50 \(\times\) 64 \\ \cline{2-7}  & Conv2D & 32 & \(3\times 3\) & \(1\times 1\) & tanh & 800 \(\times\) 50 \(\times\) 32 \\ \cline{2-7}  & UnSampling2D & - & \(2\times 2\) & - & - & 1600 \(\times\) 100 \(\times\) 32 \\ \cline{2-7}  & Conv2D & 1 & \(3\times 3\) & \(1\times 1\) & tanh & 1600 \(\times\) 100 \(\times\) 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Structure and Parameter Configuration of Each Layer in the Net-U (net1) Convolutional Autoencoder.  
Figure 7: Workflow of the proposed Knowledge-Guided Convolutional Autoencoder (KCCAE) for unsupervised VSP wavefield separation. The diagram illustrates the process where original VSP data are input into the KCCAE. The model’s dual autoencoder networks, Net-U and Net-D, then predict the upgoing and downgoing wavefields, respectively. This separation is guided by a composite loss function derived from physical, morphological, and directional difference knowledge representations, in addition to a reconstruction loss.  
used to reduce the aliasing phenomena in wavefield separation. Next, we describe the principles of each loss function and their roles in wavefield separation.