# Low-Frequency Expansion Approach for Seismic Data Based on Compressed Sensing in Low SNR
## 4 Application to Seismic Data
### Field Data Example  
The field data consists of 100 traces, and the sampling interval is 2 ms. The low frequency expansion method based on CS and the proposed method are used to extend the low-frequency.  
Figure 11a,b shows the actual single piece of trace data and its spectrum. It can be seen that the energy of the seismic record is relatively weak at about 5 Hz. The low-frequency expansion method based on CS can effectively extend the low-frequency information, and the energy of the signal about 5 Hz is improved to a certain extent. The proposed method can effectively extend the low-frequency information of 0-10 Hz as a whole.  
Due to the limited energy of the extended low-frequency information, it is not easy to show in the full frequency section, so the seismic records are displayed by low frequency filtering. Figure 12a shows the filtering results of the original seismic profile at 2-10 Hz. Figure 12b,c shows the low-frequency expansion results processed by the method based on CS and the proposed method. Through comparative analysis, the following points can be seen:  
1. The seismic event at position A in the original section is discontinuous, so it is difficult to judge the basic trend of the formation. After the expansion of the two methods, energy of the seismic event at this position is significantly enhanced, and the basic trend of the formation is clear.
2. The seismic event at B in the original section are also discontinuous, and the continuity has been improved to a certain extend after CS low-frequency expansion. In the proposed method, the inversion process of the reflection coefficient is constrained by the well data, and the continuity of the seismic event after expansion is substantially improved.
3. The seismic event at the position C is not obvious. After expansion processing of CS method, the low-frequency energy is improved, but the continuity is still poor. After the expansion processing of the proposed method, the continuity of the event is significantly improved, and the effect of the low-frequency expansion is the best.  
Figure 10: Comparison of power spectrum similarity under different SNR.  
To highlight the effectiveness of the proposed approach, the segment B is taken out and shown in Figure 13. As shown in Figure 13a, the continuity of the seismic event between traces 25 and 34 in the profile is poor. After expansion processing of CS method, the energy of the seismic event between trace 25 and 34 is improved, and the continuity of the seismic event is enhanced. The proposed method can further improve the energy of the seismic event between traces 25 and 34, so as to better reflect the trend of the formation.  
Figure 11: Comparisons of low-frequency expansion analysis for actual single trace record. (**a**) Noisy data, (**b**) frequency spectrum of original data, (**c**) data processed by CS, (**d**) frequency spectrum of (**c**), (**e**) data processed by the proposed method, (**f**) frequency spectrum of (**e**).  
Similarly, 1-3 Hz information is taken from the original section (as shown in Figure 14), where a, b and c are the 1-3 Hz information in the original seismic section of the work area, the low-frequency expansion result based on CS and the low-frequency expansion result of the proposed method. The three seismic events framed in the figure have poor continuity. After using the CS method, the energy of low-frequency information is enhanced, and the continuity of the event is improved. After using the proposed method, the energy of low-frequency information is further enhanced, and the resolution has better promotion.  
Figure 12: Filtering profile with 2–10 Hz of actual data before and after low frequency expansion. (**a**) Original data, (**b**) processed data by CS, (**c**) processed data by the proposed method.  
Figure 13: Partial enlargement effect at position B in 2–10 Hz filtering profile before and after low-frequency expansion. (**a**) Original data, (**b**) processed data by CS, (**c**) processed data by the proposed method.