# Signal Separation Based on Knowledge Representation
## 2 Theory
### Knowledge Representation
#### 2.2.1 Observed Knowledge Representation  
Vertical seismic profiling (VSP) is a technique used in seismic exploration to record seismic signals at different depths within a borehole. VSP signals comprise the transmission wavefield, which can be utilized for calculating velocity models, and the reflection wavefield, which is often migrated to obtain subsurface images for geological interpretation [35; 36]. Effectively separating upgoing and downgoing waves is fundamental and critical, as it directly impacts the quality of the final images. VSP signals comprise both the upgoing and downgoing wavefields. The objective of wavefield separation is to accurately extract these wavefields from complete seismic signals.  
Taking the separation of upgoing and downgoing waves as an example, the distinct gradient disparities observed in the image enable human observers to identify the differences between upgoing and downgoing waves. The result of knowledge representation is thus the gradient information of different target source signals. [PERSON] et al. [37] provided a  
Figure 1: The three scenarios via knowledge representation framework. Utilizing human cognition to represent knowledge inherent in the data, such as distinguishing between intertwined upward and downward lines based on gradient disparities, offers a method for knowledge representation. For complex data beyond direct human characterization, but adhering to certain physical principles, knowledge representation takes the form of specific mathematical constraints, such as partial differential equations. Lastly, in scenarios with numerous associated data samples relevant to the target, knowledge representation involves classifying and partitioning data samples based on their distinct features.  
method and approach for effectively extracting gradient features in the problem of separating upgoing and downgoing waves, along with a detailed explanation of the upgoing and downgoing wave separation problem. Seismic record signals can be divided into upgoing and downgoing waves based on the number of wave reflections, which can be represented as  
\[\mathbf{X}=\mathbf{S}_{u}+\mathbf{S}_{d} \tag{3}\]  
Here, \(\mathbf{S}_{u}\) represents the upgoing wave signal, and \(\mathbf{S}_{d}\) represents the downgoing wave signal.  
As shown in Figure 2, experts can observe the characteristic differences in the mixed signal image of upgoing and downgoing waves to extract and record the angles \(\alpha_{i}\) and \(\beta_{i}\) between the upgoing and downgoing waves and the image axis, achieving knowledge representation of the target signal based on human experiential knowledge. Therefore, we can easily obtain the knowledge representation results of upgoing and downgoing waves as follows:  
\[\mathbf{K_{u}}=\{\alpha_{i}|1\leq i\leq q,1\leq q\} \tag{4}\]  
\[\mathbf{K_{d}}=\{\beta_{i}|1\leq i\leq p,1\leq p\}\]  
Here, \(q\) is an empirical parameter representing the number of angles formed by the selected upgoing wave segments with the axis, while \(p\) is an empirical parameter representing the number of angles formed by the selected downgoing wave segments with the axis.  
Figure 2: Knowledge representation based on human cognition is employed to characterize observed signals. The figure illustrates the mixed wavefield recordings observed in VSP, which consist of both upgoing and downgoing wavefield signals. Through human observation, the angles of the upgoing and downgoing waves are extracted as knowledge derived from the observed signals.