# A Denoising Method for Seismic Data Based on SVD and Deep Learning
## 2 Method  
### Singular Value Decomposition (SVD)  
The SVD can perform principal component decomposition and is widely used in seismic data noise reduction. The complete singular value decomposition formula is:  
\[\mathrm{D_{m\times n}}\ =\ \mathrm{USV^{T}}\ =\ \sum_{K\ =\ 1}^{R}u_{k}\sigma_{k}v_{k}^{T} \tag{1}\]  
The D matrix has m sampling points for each channel and \(n\) channels. R denotes the rank of the matrix D, U the orthogonal matrix of m \(\times\) m, V the orthogonal matrix of n \(\times\) n, S the diagonal matrix of m \(\times\) n; \(\sigma_{k}\) is singular value and is arranged in order from highest to lowest in the S-matrix. The left singular vector (LSV) is designated by u\({}_{k}\) and, the RSV by v\({}_{k}\). See Appendix A for details.  
\[\mathrm{D_{m\times n}}\ =\ \mathrm{USV^{T}}\ =\ \sum_{K\ =\ 1}^{R}u_{k}\sigma_{k}v_{k}^{T}\ =\ u_{1}\sigma_{1}v_{1}^{T}+u_{2}\sigma_{2}v_{2}^{T}+ \ldots+u_{R}\sigma_{R}v_{R}^{T} \tag{2}\]  
The matrix D decomposes into R submatrices. Since the seismic signal is continuous in space, the noise is randomly distributed. The higher singular value often reflects the effective signal. The lower singular value generally represents the noise. As a result, the singular value can be used to distinguish between the noise and the effective signal.  
\[\mathrm{D_{m\times n}}\ =\ \mathrm{USV^{T}}\ =\ \sum_{K\ =\ 1}^{r}u_{k}\sigma_{k}v_{k}^{T}\ (1\leq r\leq R) \tag{3}\]  
K is frequently adjusted to 1 or 2 when reconstructing seismic data. The reconstruction error of seismic data is the sum of the matrix corresponding to the remaining singular values. By using SVD, it is discovered that the LSV and RSV of 2D seismic data indicate various physical meanings. The RSV indicates the lateral variation trend of the amplitude of the seismic lineups, whereas the LSV represents the seismic wavelet. The first left singular vector typically represents the normalized wavelet of the seismic signal to the seismic data after SVD. At the same time, the noise is represented by the following columns of LSV. Similarly, the first right singular vector indicates the amplitude variation of the effective signal, whereas the variations of the remaining RSV are more random and represent noise [12; 16].  
However, the seismic signal becomes significantly disturbed by intense noise. Figure 1a shows intense amplitude noise in the 2 nd and 6 th seismic data. The seismic data are decomposed by SVD, as shown in Figure 1b. The first LSV represents noise, and the second represents the effective signal. The traditional singular value method cannot denoise seismic data in this case. In Figure 1c, the first RSV correspondingly represents the amplitude change of seismic data, in which sizeable abrupt change represents intense amplitude noise. By contrast, the absolute value of the second RSV generally shows an increasing trend, which confirms that seismic signal amplitude gradually increases.  
The RSV characteristics corresponding to noise and effective signal are very different. Therefore, we could distinguish the effective signal and noise based on the RSV characteristics in addition to the singular value. As a result, flexible control over the rank selection of seismic data reconstruction was possible. The goal of intelligent noise suppression was finally accomplished.  
Figure 1: _Cont._