Synthetic data

In this section I discuss the results obtained by applying the scheme to three synthetic data. Figure a and b show a synthetic data set which contains three linear events and its spectrum, respectively. From the slant stack of the data, the dip spectrum shown in Figure c was obtained by cuttting the noisy low amplitude portion followed by smoothing after picking the maximum absolute values along the $\tau$ for each ray parameter. The prediction-error filter is generated by putting zeros along the dips picked and its spectrum, shown in Figure d. The interpolated result, Figure e, shows quite well aligned linear events and the spectrum after interpolation, Figure f, shows no more aliasing.

For testing the applicability to noisy data, random noise with the amplitude of .2 times of signal amplitude was added to the synthetic data set shown in Figure a. Figure a and b show such a synthetic data and its spectrum. After interpolation, Figure e shows a good interpolation except that noises are also interpolated along the picked dips. Figure f shows the spectrum after interpolation and it shows clearly the effect to the noises. But this artifact does not cause a serious problem because the noise level is relatively low and noises are only locally interpolated.

 

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Figure 1 (a) A synthetic data set with three dipping events (b) the spectrum of the synthetic data set (c) The dips picked from slant stack (d) the spectrum of the prediction-error filters (e) The interlaced data set (f) the spectrum of the interlaced data set

 

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Figure 2 (a) A synthetic data set with three dipping events and additional noise (b) the spectrum of the synthetic data set (c) The dips picked from slant stack (d) the spectrum of the prediction-error filters (e) The interlaced data set (f) the spectrum of the interlaced data set

Figure a shows a synthetic data set which contains several linear events with some background noise and 50 percent of the traces are randomly missing. For this case, again, the dip spectrum picks three dominant dips as in the interlacing case. Since there are fewer constraints in the region of more missing traces, the convergence of this model is a little bit slower than that of the interlacing case. The result of interpolation, Figure e, shows quite good interpolation along the given dips except that the noise region tends to line up along the dip. But the amplitude of the noise is relatively low and does not affect the spectrum too much.

 

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Figure 3 (a) A synthetic data set with randomly missing traces ( 50 percent of total traces are missing ) (b) the spectrum of the synthetic data set (c) The dips picked from slant stack (d) the spectrum of the filter simulated (e) The interpolated data set (f) the spectrum of the interpolated data set