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Synthetic data tests

To choose reference filter windows and time-varying filter windows, we analyzed the characteristics of our TVMF formulation using synthetic data. Referring to the velocity model (Table 1), note the synthetic common-shot record with a dominant frequency of 40 Hz shown in Figure 1a. A 5%-density white spike noise was then added to the model. Three different noise peaks (noise amplitude is half, one, and two times the maximum value of the reflections, and the corresponding noise intensity is 1/2, 1, and 2) were chosen to separately test the TVMF. Only noise with twice the maximum value of the reflected wave is displayed in Figure 1b. Energy attenuation has not been taken into account, nor has spherical spreading or AVO. Here, the signal has been drowned out by noise.

Stratum thickness (m) 500 1739 2069  
Stratum velocity (m/s) 2500 3162 4138 4500

Table 1. Velocity model

Definitions of signal energy (equation 5), noise energy (equation 6), and SNR (equation 7) were used to analyze characteristics of the reference median filter. We chose the stationary MF with different filter-window lengths of from 3 to 19 points, comparing various signal energies (Figure 2a), noise energies (Figure 2b), and the SNR's (Figure 2c) after filtering. To meet the assumptions of the SNR model, we chose five traces near the zero offset, where seismic events are approximately invariant across the five traces. Note that in the figures, curves of the three noise intensities (1/2, 1 and 2) appear to have similar tendencies. Both signal energy and noise energy decrease as filter-window length increases, but the SNR displays a wiggle shape. The energy levels of signals are stable after filter-window length reaches 11 points, but the energy levels of noise are stable after filter-window length reaches 15 points. And because the curves show a different rate of descent, the SNR has a different tendency. It reaches a peak at the 5-point filter window and decreases afterward because signal energy attenuates faster than noise energy. The SNR then reaches the minimum near 11 points, where the signal energy is stable, and next, the SNR improves again because the noise energy still decreases. Finally, the SNR reaches stability, however, the signal has been barely damaged.

In the noisy model (Figure 1b), the SNR is -10.7 dB. After stationary MF filtering, even the minimum SNR remains much larger than -10.7 dB, illustrating that all stationary median filters can improve the SNR in the noisy model. We can select large filter windows of reference median filters to keep noise energy to a minimum when threshold $T$ is well able to separate signal from noise, but at the same time we can define time-varying filter windows on the basis of reference filter windows so that the reference filter window will be limited. Three basic principles should be satisfied:

1. The reference filter window can be chosen as a large value to separate signal from noise when noise energy is small.

2. Time-varying filter windows at signal positions should be small values to protect the signal when removing noise.

3. Time-varying filter windows at noise positions should be large values to attenuate noise energy.

In Figure 2, reference filter window $C$ should be larger than the stable signal point (11 points), and time-varying filter windows $C-\gamma$ and $C-\delta$ at signal positions should be in the range of from 5 to 7 points in order to preserve signal energy. Time-varying filter windows $C+\alpha$ and $C+\beta$ should be limited in range from 11 to 13 points in order to attenuate noise energy and save calculation time. To meet all principles, $C$ can be 11 points, with $\alpha=2$, $\beta=0$, $\gamma=4$, and $\delta=6$. After more tests on synthetic and real data it became clear that these filter parameter choices work for most real data.

model noise0
Figure 1.
Synthetic model (a) and white-noise model (b).
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es en snr
Figure 2.
Comparison of different noise levels (Intensity (1/2, 1, and 2) is the amplitude ratio between noise and reflections). Signal energy (a), noise energy (b), and SNR (c).
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We used the TVMF, with defined reference and time-varying filter windows, to process the noise model (Figure 1b), the result of which is shown in Figure 3a. When assumptions can be met, the SNR estimation using the stack method can be used to compare results. After TVMF processing, the SNR is 19.7 dB. To compare, we also used the low-pass filter for preserving the signal in the dominant frequency band (Figure 3b, the SNR is -7.4 dB). After TVMF processing, white spike noise attenuated well, but the result after low-pass filtering became a signal with band-limited noise, and a great deal of low-frequency noise remained. Next, we also used the TVMF to process the band-limited noise model (Figure 3b). SNR analysis shows that the parameters of TVMF only change a little, so the same parameters can be used for processing. The TVMF cannot eliminate all band-limited noise, like white noise, but the energy of the noise has been degraded. At the same time, the TVMF introduces a few high-frequency noise components having low energy, but these can be easily removed by using a high-cut filter. Figure 4a shows the result after TVMF and high-cut filtering; the SNR is 0.5 dB. The low-pass filter does nothing about band-limited noise. The 11-point stationary MF (Figure 4b, the SNR is -8.4 dB) can be used to compare with the TVMF. The stationary MF can remove most of the noise, but useful information is also destroyed. After comparing the result of the TVMF with those of the stationary MF and the band-pass filter, we conclude that the TVMF is superior when processing random, spike-like noise.

We can compare results of using different methods by analyzing their spectra as well. We chose spectra of the trace at a distance of 4 km, corresponding to the pertinent parts of Figures 1, 3, and 4. Results in Figure 5 show that spectral values of random noise are larger than those of the signal at every frequency and that reflected waves have been masked by random noise (Figure 5b). The low-pass filter can eliminate noise in the high-frequency band, but it does nothing in the dominant-frequency band (Figure 5c). After the band-limited model is processed by the TVMF using an 11-point reference filter window, most spectral components in the dominant-frequency band can be recovered. An additional high-cut filter was used to remove high-frequency noise introduced by the TVMF (Figure 5d). The corresponding time profile is shown in Figure 4a.

w11tvmf noise1
Figure 3.
Denoised result after 11-point TVMF filtering (a) and denoised result (band-limited noise model) after low-pass filtering (b).
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cb11tvmf c11mf
Figure 4.
Denoised band-limited noise model using different methods (the result of Figure 3b was input to these tests); 11-point TVMF followed by a high-cut filter (a) and 11-point stationary MF (b).
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smodel snoise0 snoise1 scb11tvmf
Figure 5.
Comparison of amplitude spectra (trace at distance of 4 km) before and after processing. Amplitude spectra in Figure 1a (a), corresponding spectra in Figure 1b (b), corresponding spectra in Figure 3b (c), and corresponding spectra in Figure 4a (d).
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Given the result of model filtering, the TVMF can easily remove spike-like noise, especially noise having a white spectrum. When the TVMF is used to attenuate band-limited, spike-like noise, its filter ability decreases, although it can still work better than other common methods.

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Next: Processing of field data Up: Liu etc.: 1-D time-varying Previous: 1-D time-varying median filter