# Seismic Periodic Noise Attenuation Based on Sparse Representation Using a Noise Dictionary
## 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 11 th 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.  
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 11 th 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 4: Seismic signal on 11 th trace. (**a**) Signal on the 11 th trace in time domain and (**b**) its amplitude spectrum.  
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.  
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 7: De-noising result on the 11 th trace. (**a**) De-noised signal and (**b**) eliminated noise on the 11 th 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**).  
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._