# Low-Frequency Expansion Approach for Seismic Data Based on Compressed Sensing in Low SNR
## 3 Low Frequency Expansion
### Basic Theory  
Suppose \(w\) is seismic wavelet, \(r\) is underground reflection coefficient, \(g\) is noisy seismic data, and \(n\) is random noise; \(g\) can be described as:  
\[g=w*r+n \tag{7}\]  
Formula (7) is transformed to the frequency domain by Fourier transform:  
\[G=WR+N=WFr+N \tag{8}\]  
where \(F\) is the basis function of Fourier transform. \(G\), \(W\), \(R\) and \(N\) are sparse representations of \(g\), \(w\), \(r\) and \(n\) in the frequency domain.  
The reflection coefficient \(r\) in convolution model is sparse in time domain, which conforms to the precondition of CS theory. According to CS theory, the L\({}_{1}\) norm optimization problem for solving the underdetermined linear equation in Equation (8) is as follows:  
\[J=\frac{1}{2}\|WR-G\|_{2}^{2}+\lambda\|r\|_{1}^{1}=\frac{1}{2}\|WFr-G\|_{2}^{2} +\lambda\|r\|_{1}^{1} \tag{9}\]  
where \(\|\bullet\|_{2}^{2}\) is the error function of L2 norm constraint; \(W\) is the measurement matrix constructed by seismic wavelet \(w\), \(W_{ij}=w_{i-\tau_{j}+1}\), \(\tau_{j}\) represents the reflection coefficient location; \(F\) is the sparse representation matrix; \(\|\bullet\|_{1}^{1}\) is the exact solution of L\({}_{1}\) norm constraint; \(\lambda\) is a regularization parameter, which is used to adjust the weights of the two norms to ensure the balance between sparsity and data fidelity. The larger the \(\lambda\) is, the larger the weight of the L\({}_{1}\) norm is, and the sparser the reflection coefficient is. The first term of Equation (9) is constrained by L\({}_{2}\) norm and converges to complete seismic data in the process of solution. The second term of Equation (9) is constrained by the L\({}_{1}\) norm to obtain more sparse reflection coefficient \(r\).  
Based on the obtained sparse reflection coefficient \(\widetilde{r}\), the full band spectrum \(\widetilde{R}\) can be expressed as:  
\[\widetilde{R}=F\widetilde{r} \tag{10}\]  
\(\widetilde{R}\) is the spectrum of the seismic reflection coefficient in the full-frequency band obtained from limited bandwidth seismic data. Assuming that the original data spectrum is \(G\), the frequency spectrum after compensation is \(\widetilde{G}\). The specific method of using \(\widetilde{R}\) to expand the low frequency information in seismic data is as follows:  
1. Select the point where \(\widetilde{R}\) and \(G\) are equal;
2. The spectrum value in \(\widetilde{R}\) before the point and the spectrum value in \(G\) after the point are extracted and combined into the spectrum after low-frequency expansion; the formula is described as follows: \[\widetilde{G}=L(\widetilde{R})+H(G)\] (11) where \(L\) and \(H\) are the operators to extract the low-frequency and high-frequency spectrum values, respectively.
3. The seismic data after low-frequency expansion can be obtained by inverse Fourier transform of \(\widetilde{G}\). The low-frequency expansion process based on CS is shown in Figure 2.  
Note that in Equation (8), the sampling matrix \(\mathbf{W}\) formed by the Fourier transform of \(w\) is not a completely random function, so the missing frequency information of the seismic data is also recovered to a certain extent.