# Low-Frequency Expansion Approach for Seismic Data Based on Compressed Sensing in Low SNR
## 4 Application to Seismic Data
### Synthetic Data Example
It can be seen that the low-frequency information shows increasing interference with the increase in the noise. Two low-frequency expansion methods are utilized to calculate the reflection coefficient and compensate the frequency. Figure 8c,e shows the reflection coefficients obtained by the two methods, respectively. It can be seen that the accuracy of the reflection coefficients inversed by the compressive sensing theory is further reduced, and the accuracy of the reflection coefficients inversed by the proposed method is basically unaffected. Comparing the spectrum before (Figure 8b) and after frequency expansion (Figure 8d,f), the low-frequency information based on CS can be better extended before 5 Hz, and the effect is poor at around 10 Hz (pointed by the arrow in Figure 8d). The frequency expansion effect of the proposed method is still not affected by the reduction of SNR; the low-frequency information is extended, and the frequency band is broadened to a certain extent. Figure 7: Comparisons of low-frequency expansion when SNR = 9.41. **(a,b)** Noisy data and their frequency spectrum, (c,d) reflection coefficients estimated by CS and extended spectrum, (**e,f**) reflection coefficients estimated by proposed method and extended spectrum. Figure 9 shows the contrast of the two methods when the SNR is 3.14. Figure 9a,b is the noisy seismic data and their spectrum, respectively. Figure 9c,e is reflection coefficients obtained by the two methods, respectively. It can be seen that the accuracy of the reflection coefficient based on compressed sensing theory is lower, and the accuracy of the reflection coefficient obtained by the proposed method is still unaffected by noise. Comparing the spectrum before and after frequency expansion (Figure 9d,f), the low-frequency information expansion effect based on CS is poor, especially for the low-frequency information between 4 and 10 Hz (indicated by the arrow in Figure 9d), which is not extended. The proposed method is still not affected by the reduction of SNR. Low-frequency information is extended, and the frequency band is broadened to a certain extent. Figure 8: Comparisons of low-frequency expansion when SNR = 5.10. (**a,b**) Noisy data and their frequency spectrum, (**c,d**) reflection coefficients estimated by CS and extended spectrum, (**e,f**) reflection coefficients estimated by proposed method and extended spectrum. To further illustrate the effectiveness of the proposed approach, the Power Similarity method is calculated to prove the superiority of the proposed method. The Power Similarity is described as follows:  
\[Q=\frac{\sum\left(P_{n}-P_{c}\right)}{\sum P_{c}} \tag{14}\]  
where \(P_{c}\) is the extended spectrum of the seismic data without noise, \(P_{n}\) is the extended spectrum of the seismic data with noise, generally, \(P_{c}\), \(P_{n}\in\left[0\ 10\ \mathrm{Hz}\right]\). The smaller \(Q\) is, the greater the similarity is. Figure 10 quantitatively compares the Power Similarity of the two methods under different SNR. It can be seen that the similarity of low-frequency spectrum energy extended by the two methods are gradually decreases with the decrease of SNR. The similarity decline speed of the proposed method is slower, which indicates that the effect of low-frequency expansion is better. Figure 9: Comparisons of low frequency expansion when SNR = 3.14. **(a,b)** Noisy data and their frequency spectrum, (c,d) reflection coefficients estimated by CS and extended spectrum, (e,f) reflection coefficients estimated by proposed method and extended spectrum. In summary, when the SNR of seismic data is high, both methods can accurately estimate the reflection coefficient and effectively extend the low-frequency information. With the decrease in the SNR, the accuracy of the reflection coefficient obtained by the inversion method based on CS gradually reduced. However, accuracy of the reflection coefficient obtained by the proposed method is less affected and can estimate the reflection coefficient more accurately.