# Low-Frequency Expansion Approach for Seismic Data Based on Compressed Sensing in Low SNR
## 4 Application to Seismic Data  
### Synthetic Data Example  
The reflection coefficients at 0.1 s, 0.2 s, 0.21 s, 0.3 s, 0.31 s and 0.4 s are defined as 0.8, 0.6, 0.8, 0.4, 0.4 and 0.4, respectively. The sampling point is 400 and the sampling frequency is 1 ms. The synthetic seismogram is obtained by convolution of the reflection coefficient and the Riker wavelet with dominant frequency of 30 Hz, as shown in Figure 4a. The corresponding reflection coefficients and spectrum are shown in Figure 4b,c, respectively. Fourier transform is performed on the composite record to obtain the reflection coefficient of the whole frequency band, and the low-frequency part of the reflection coefficient and the high frequency part of the composite record are taken to form a new spectrum as shown in Figure 5b. Figure 5a is the seismic record after low-frequency expansion. Comparing the results before (Figure 4c) and after frequency expansion (Figure 5b), it can be seen that the  
Figure 3: Low-frequency expansion process with well constraint based on CS. low-frequency energy (0-7 Hz) has been significantly improved from \(-\)29 dB to \(-\)22 dB, especially the energy at 5 Hz is improved obviously. Clearly, the low-frequency expansion method based on the CS theory can extend the low-frequency of the data well. Figure 4: Simulated seismic data. (**a**) Model data, (**b**) reflection coefficient model, (**c**) frequency spectrum of (**a**). Different intensity noise is added to the synthetic seismogram to test the anti-noise ability of this method. The noise data with different SNR (14.72, 9.41, 5.10 and 3.14) are shown in Figures 6a, 7a, 8a and 9a. It can be seen that the effective signal is gradually submerged in the noise with the decrease in SNR. For these noisy seismic records, the sparse inversion method based on compressed sensing and the proposed method are used, respectively. When the SNR of seismic record is 14.72, the reflection coefficients are calculated by the CS method and the well constrained CS method, and then the spectrum of seismic record after frequency expansion is compared. Figure 6a is the noisy seismic record, Figure 6b is the spectrum of the noisy data. Figure 6c,e shows the reflection coefficients obtained by the two methods, respectively. It can be seen that both methods can accurately reverse the reflection coefficient. Comparing the spectrum before (Figure 6b) and after frequency expansion (Figure 6d,f), it can be seen that the low-frequency energy (especially in 0-5 Hz) have been significantly improved from \(-\)50 dB to \(-\)22 dB. It shows that the low-frequency information has been extended and the frequency band has also been broadened to a certain extent. Figure 5: Simulated seismic data after frequency-broadening processing. (**a**) Model data, (**b**) frequency spectrum of (**a**). The SNR is reduced to 9.41 (Figure 7a), and Figure 7b is its spectrogram. It can be seen that the energy of low-frequency energy in Figure 7b is higher than that in Figure 5b. This is because the added noise contains low-frequency components. The noisy data are processed by the low-frequency expansion method based on CS and the proposed method. Figure 7c,e shows the reflection coefficients obtained by the two methods, respectively. It can be seen that accuracy of the reflection coefficient reversed by CS theory is reduced, but the accuracy of reflection coefficient reversed by the proposed method is not affected. Comparing the spectrum before (Figure 7b) and after frequency expansion (Figure 7d,f), it can be seen that both methods still can overcome the noise interference and improve the low-frequency information energy. Figure 6: Comparisons of low frequency expansion when SNR = 14.72. (**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. The SNR continues to reduce to 5.10, Figure 8a,b is the noisy seismic record and its spectrum.