**3. Synthetic Example**

A synthetic dataset consisting of 21 traces is shown in Figure 2 with a 1 ms sampling rate and a 10 m geophone interval. Multitoned noises of 40 Hz and 50 Hz are incorporated to the noise-free data. In addition, a small amount of white Gaussian noise is added. Figure 3 shows the contaminated seismic dataset and its frequency–wavenumber (f-k) spectrum. The reflections are hardly identifiable owing to the strong periodic noise. The contaminations in the seismic data appear as two horizontal bands at 40 and 50 Hz in the f-k domain. The signal on the 11th trace (Figure 4a) shows the influence of multitoned noise clearly. Its amplitude spectrum (Figure 4b) also shows high spikes at frequencies of 40 Hz and 50 Hz.

**Figure 2.** Synthetic seismic data.

**Figure 3.** Synthetic contaminated seismic data. (**a**) Contaminated seismic data in offset-time domain and (**b**) its frequency-wavenumber (f-k) spectrum where the red rectangle marks the ambient noise we used.

**Figure 4.** Seismic signal on 11th trace. (**a**) Signal on the 11th trace in time domain and (**b**) its amplitude spectrum.

The first 0.4 s of data is dominated by periodic noise marked by a red rectangle in Figure 3a. In addition, it does not consist of seismic waves. Therefore, it is chosen to construct the noise dictionary. The waveform estimated from the chosen data is shown in Figure 5. After sparse representation, the result of noise attenuation is shown in Figure 6. The main noise is attenuated and reflections can be clearly seen in Figure 6a. The eliminated noise is periodic noise as shown in Figure 6b. Their spectra show that the multitoned noise is attenuated and no seismic waves are eliminated. The de-noising result on the 11th trace and the corresponding amplitude spectra are shown in Figure 7. The de-noised seismic signal is close to the theoretical signal in both time sequence and amplitude spectrum, except for weak white Gaussian noise.

**Figure 5.** Noise waveform estimated by the proposed method.

**Figure 6.** De-noising result of the synthetic seismic data by the proposed method. (**a**) De-noised data and (**b**) eliminated noise by the proposed method; and (**c**,**d**) their corresponding f-k spectra.

**Figure 7.** De-noising result on the 11th trace. (**a**) De-noised signal and (**b**) eliminated noise on the 11th trace; and (**c**,**d**) their amplitude spectra. The red line corresponds to the de-noised signal and the black line corresponds to the theoretical signal in (**a**,**c**).

A synthetic signal is shown in Figure 8a, where the main noise is stationary and the waveform is recurring except for weak white Gaussian noise. The eliminated noise and de-noised signal are shown in Figure 8b,c, respectively. Their amplitude spectra are shown in Figure 9. Both the time sequence and amplitude spectrum of the de-noised signal are consistent with the theoretical signal. Because the amplitude spectrum is complex, it is hard to estimate its amplitudes, frequencies and phases, which are important for model-based approaches [8,9]. Therefore, the proposed method can effectively attenuate the multitoned noise and even periodic noise with a complex waveform but not influence the seismic events.

**Figure 8.** De-noising result of the synthetic noisy signal. (**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 9.** *Cont*.

**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**).

### **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*<sup>0</sup> − Δ*f* , *f*<sup>0</sup> + Δ*f* ], where *f*<sup>0</sup> is the noise frequency and 2 × Δ*f* is the noise bandwidth. The values of *f*<sup>0</sup> 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 Δ*f* = 1Hz 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 10.** Field data. (**a**) Field data in offset-time domain, (**b**) its f-k spectrum and (**c**) the signal on the 8th trace, where the red rectangle marks the ambient noise and the red arrow points to the 14 Hz weak noise.

**Figure 11.** Waveform estimated from the ambient 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 13.** Spectra of (**a**) the de-noising data and (**b**) the eliminated noise by the notch filtering method.

**Figure 14.** Amplitude spectra of the de-noised signals on the 8th trace by the proposed method (the red solid line) and the notch filtering method (the blue dash-dotted line).

**Figure 15.** Eliminated noise by (**a**) the proposed method and (**b**) the notch filtering method.
