# Seismic Periodic Noise Attenuation Based on Sparse Representation Using a Noise Dictionary
## 4 Field Example  
The proposed method was tested using field data. The field data are a land seismic shot recorded with a 2 ms sample rate and a 40 m geophone interval, as shown in Figure 10a. The data contain periodic noise which may be caused by power lines, engine operation and other interferences. The noise is so strong that it affects the quality of the subsequent processes, especially for the first 12 traces. As shown in Figure 10c, we cannot see any waves after the time 1.5 s. There are three horizontal bands around the frequencies of 7 Hz, 14 Hz and 50 Hz in the f-k domain (Figure 10b). The periodic noise at the frequency of approximately 14 Hz, highlighted in Figure 10b, is the weakest, and the periodic noise at the frequency of 50 Hz is the strongest.  
We use the ambient noise marked by a red rectangle in Figure 10a to construct the noise dictionary. The estimated waveform is shown in Figure 11. The result of noise attenuation by the proposed method is shown in Figure 12. The periodic noise is attenuated to a large degree. The horizontal bands of the f-k spectrum are almost eliminated, except for weak residual noise of about 7 Hz frequency. In addition, there are no spectral notches in the f-k spectrum (Figure 12b).  
The notch filtering method is applied to the field data for a comparison. The narrow stop band of the notch filter is the frequency range \([f_{0}-\Delta f,f_{0}+\Delta f]\), where \(f_{0}\) is the noise frequency and \(2\times\Delta f\) is the noise bandwidth. The values of \(f_{0}\) for the three noise bands are set to 7 Hz, 14 Hz and 50 Hz, respectively. The noise bandwidth is estimated to be 2 Hz and \(\Delta f=1\)Hz is set. The f-k spectra of the filtering result are shown in Figure 13. The horizontal bands around the frequencies of 7 Hz, 14 Hz and 50 Hz are separated from the seismic data. However, the seismic waves are also eliminated at those frequencies. This causes spectral notches of seismic waves (Figure 13a). To further show the effectiveness of our method, the spectra of the single-trace de-noised signals by the two methods are compared in Figure 14. For the notch filtering method, the spectral notch causes amplitude loss of seismic events around the frequencies of 7 Hz, 14 Hz and 50 Hz. However, amplitude loss does not occur using our proposed method. The eliminated noise (Figure 15a) is stationary. This is consistent with the characteristics of periodic noise caused by the power lines or engine operation. For comparison, the noise eliminated by the notch filtering method is not stationary because it contains seismic signals.  
Figure 9: Amplitude spectra of the (**a**) noisy signal, (**b**) eliminated noise and (**c**) de-noised signal. The de-noised signal is shown as a red line and the theoretical signal as a black line in (**c**).  
Figure 11: Waveform estimated from the ambient noise.  
Figure 10: Field data. (**a**) Field data in offset-time domain, (**b**) its f-k spectrum and (**c**) the signal on the 8 th trace, where the red rectangle marks the ambient noise and the red arrow points to the 14 Hz weak noise.  
Figure 12: Result of noise attenuation by our method: (**a**) de-noised data, (**b**) its f-k spectrum and (**c**) the f-k spectrum of eliminated noise.  
Figure 14: Amplitude spectra of the de-noised signals on the 8 th trace by the proposed method (the red solid line) and the notch filtering method (the blue dash-dotted line).  
Figure 13: Spectra of (**a**) the de-noising data and (**b**) the eliminated noise by the notch filtering method.