# AVO-Friendly Velocity Analysis Based on the High-Resolution PCA-Weighted Semblance
## 2 Method
### High-Resolution AB Semblance with a PCA-Based Weighting Function  
Principal component analysis is a common multivariate analysis method, and it has been widely used in data mining, pattern recognition, machine learning, and image compression [29; 30; 31]. Principal component analysis regroups the original relevant indicators into a new set of unrelated comprehensive indicators. For the NMO-corrected seismic event, if the true scanning velocity is adopted, the NMO-corrected seismic event will be completely flattened, and the data matrix will have a strong spatial coherence. Thus, the first principal component has the absolute advantage compared with other components. If the wrong scanning velocity is adopted, the NMO-corrected seismic event is not flattened, then the first principal component of the data matrix is not overwhelming. We denote the amplitude of NMO-corrected gather as a matrix \(\mathbf{A}\in\mathbb{R}^{N\times M}\), then the covariance matrix of \(\mathbf{A}\) is  
\[\mathbf{C_{A}}\mathbf{=}(\mathbf{A}\mathbf{-}\widetilde{\mathbf{A}})( \mathbf{A}\mathbf{-}\widetilde{\mathbf{A}})^{T}=\sum\limits_{i}\lambda_{i} \mathbf{u}_{i}, \tag{5}\]  
where \(\widetilde{\mathbf{A}}\) is the mean value of matrix \(\mathbf{A}\) and \(\lambda_{i}\) represents the eigenvalue of the (i)-th principal component and \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{k}\). \(\mathbf{u}_{i}\) represents the corresponding eigenvectors of the covariance matrix. If the matrix \(\mathbf{C_{A}}\) is a low-rank matrix or its column vectors have a strong linear coherence with each other, then the first principal component \(\lambda_{1}\) is much larger than the others, and the remains quickly decrease to zero. For example, Figure 1a shows a seismic event at time 0.24 s with the velocity \(v=1950\) m/s. We used a series of scanning velocities to perform NMO corrections, then analyzed the corresponding eigenvalues of the principal components. As shown in Figure 1b, the eigenvalue of the first principal component had the absolute advantage when the true velocity \(v=1950\) m/s was used, and the remaining eigenvalues decreased rapidly. If we used the velocity \(v=1940\) m/s with a slight deviation to perform NMO correction, the corresponding first principal component no longer has the absolute advantage compared with others, indicating that the first principal component is highly sensitive to the variation of scanning velocity. Thus, we define two weighting factors  
\[\beta_{1}=\frac{\lambda_{1}}{\sum\limits_{j=1}^{k}\lambda_{j}}, \tag{6}\]  
and  
\[\beta_{2}=\frac{\lambda_{1}}{\lambda_{2}}, \tag{7}\]  
to improve the resolution of the semblance spectrum. We demonstrated two weighting factors \(\beta_{1}\) and \(\beta_{2}\) with different scanning velocities in the above case (Figure 1b). As shown in Figure 2, the weighting factors \(\beta_{1}\) and \(\beta_{2}\) both had a narrow pulse at true velocity \(v=1950\) m/s. As long as the scanning velocity deviated slightly from the true value, the two weighting factors \(\beta_{1}\) and \(\beta_{2}\) decreased rapidly, indicating that the weighting factors were very sensitive to changes in velocity. Then, we combined factors \(\beta_{1}\) and \(\beta_{2}\) to construct a new weighting function \(w(it,v_{i})\). The new weighting function is defined as  
\[w(it,v_{i})=\frac{\lambda_{1}^{2}}{\lambda_{2}\sum\limits_{j=2}^{k}\lambda_{ j}+\epsilon}, \tag{8}\]  
where \(\epsilon\) is a stability factor for avoiding division by zero. Since \(w(it,v_{i})\) is the product of the two weighting factors \(\beta_{1}\) and \(\beta_{2}\), its sensitivity to velocity is double compared to the single weighting factor. Then, for different scanning velocities \(v_{i}\), the weighting functions \(w(it,v_{i})\) become a vector. \[\mathbf{w}(it)=(w(it,v_{1}),w(it,v_{2}),...,w(it,v_{N})), \tag{9}\]where the vector \(\mathbf{w}(it)\) represents the weights at time \(it\) with a series of scanning velocities. We denote the maximum weight as \(w_{\max}(it)=\max(\mathbf{w}(it))\).