Synthetic Example

First, we applied the method to a 2D synthetic prestack dataset (Figure 1a), which was created from a 2D slice of the benchmark French model (French, 1974). The reflector with round dome and corners creates reflection events along midpoint and offset axes. The inflection points of the reflector lead to traveltime triplications at some offsets. Then we removed 70% of randomly selected traces (Figure 1b) from the input data (Figure 1a). We directly used NMO equation 3 to analyze data with missing traces because the dataset is limited in a near-offset range. The scanned parameters were obtained and the results are shown in Figure 2a. The velocity spectra of NMO from data show that the picked velocities are reasonable because velocity scan is less sensitive to missing traces. Next, we calculated the velocity-dependent (VD) slopes using the NMO velocities. The VD slopes calculated from equation 5 with $S=1$ provided accurate results (Figure 2b). Data interpolation with the help of modified Bregman iteration (equations 13 and 14) recovers missing traces as long as seismic data are sufficiently sparse in the transform domain. To demonstrate the superior sparseness of the generalized VD-seislet coefficients, we compare the proposed method with Fourier projection onto convex sets (POCS) interpolation method (Abma and Kabir, 2006a). The results of the interpolation are shown in Figure 3a after carefully selecting parameters. 50 iterations were selected as comparable computational cost as the modified Bregman iteration. Some artificial events were created because Fourier transform cannot provide a naturally sparse domain for curved events. The interpolated error is also slightly large at the locations of the gaps (Figure 4a). The interpolated results can only be partially improved by cutting data into overlapping windows, unless the events display an ideal plane-wave pattern in each window. The generalized VD-seislet transform provides a much sparser domain for reflections. Therefore, the modified Bregman iteration successfully interpolates missing traces (Figure 3b). The number of iteration for the proposed method is 20, which is less than that of Fourier POCS. The difference between the interpolated result and Figure 1a also shows that the proposed method provides reasonable results, in which parts of diffraction events are lost (Figure 4b).

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Figure 1. Synthetic prestack data from the benchmark French model (a) and data with 70% traces removed (b).

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Figure 2. Velocity spectra by using NMO equation (a) and local slopes calculated by equation 5 with $S=1$.

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Figure 3. Interpolated results by using different methods. Fourier POCS method (a) and the proposed method (b).

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Figure 4. The difference between interpolated results and synthetic prestack data. Fourier POCS method (a) and the proposed method (b).

The second example is shown in Figure 5a. The amplitudes of the four reflection events show variation because of anisotropy. To evaluate missing trace interpolation (Figure 5b), we removed 70% of randomly selected traces from the input data (Figure 5a). The variation of amplitude and curvature makes it difficult to recover the missing traces. The interpolated results are shown in Figure 6b, which uses the generalized velocity-dependent (VD)-seislet transform and the modified Bregman iteration with 99% percentile-based soft thresholding. The number of iteration is 20. In the interpolated result, it was visually difficult to distinguish the missing trace locations, which indicates successful interpolation. For comparison, we applied the Fourier POCS method to interpolate the missing traces (Figure 6a). The data were divided into seven patches with 43% overlap along the space axis and 200 iterations were run. The POCS method produces a reasonable result after carefully selecting parameters, but some artificial events were still present becausethe uniformed patching windows cannot guarantee the assumption of stationary plane waves.

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Figure 5. Synthetic data of VTI media (a) and data with 70% traces removed (b).

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Figure 6. Interpolated results by using different methods. Fourier POCS method with patching (a) and the proposed method (b).