Full-waveform inversion using seislet regularization

Encoded data test

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Figure 4. Real part of 6 Hz frequency component data after first dynamic phase encoding. (a) First supershot containing shots with index $i$ that satisfies: $i\%4=1$ and $1 \leq i \leq 32$ ; (b) second supershot ($i\%4=2$ ); (c) third supershot ($i\%4=3$ ); (d) fourth supershot ($i\%4=0$ ).

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Figure 5. FWI results of encoded data. (a) Standard FWI without phase encoding; (b) standard FWI with dynamic phase encoding; (c) FWI using seislet regularization with dynamic phase encoding; (d) normalized model error versus iteration for different inversions: (a) (fine dot), (b) (large dash) and (c) (solid line).

To reduce the computational burden, one well-known technique is to build supershots by assembling several sources, which can reduce the computational cost by a factor roughly equal to the number of sources assembled together (Romero et al., 2000; Morton and Ober, 1998). However, the source-combination technique may introduce crosstalk noise arising from the interference of waves generated at spatially adjacent sources. To remove the unwanted crosstalk noise, the blended data is typically paired with phase encoding strategies; among them, dynamic phase encoding (Ben-Hadj-Ali et al., 2011; Baumstein et al., 2011; Krebs et al., 2009) is a particularly effective approach. Dynamic phase encoding in frequency-domain FWI involves changing encoding code and building a new encoded superset at each iteration of each frequency inversion (Ben-Hadj-Ali et al., 2011).

In this test, we combine every 8 equidistant shots in one supershot, creating 4 supershots in total. The first supershot contains shots with indices: 1, 5, 9, 13, 17, 21, 25 and 29. We perform three inversions. The first inversion is the standard FWI with the blended data without phase encoding, which means that data are directly blended together without any designed codes. The second inversion is the standard FWI with the dynamic phase encoding technique, and at each iteration, a new encoding code is generated to build a new super dataset. The encoding code used in this study is the simple phase function $e^{i\gamma}$ , where $\gamma$ is a random number in $[0, 2\pi]$ . Figure 4 shows the real part of the 4 encoded supershots at the first iteration of the 6 Hz frequency data inversion. From each supershot, we can roughly observe 8 large peaks and troughs, and their locations correspond to that of the original sources. The third inversion inverts the dynamic phase encoded data by using FWI with seislet regularization. We set the soft thresholding parameter in the seislet regularization empirically to be 18%, meaning that 82% smaller seislet coefficients get removed at each iteration.

Figure 5 shows the final results of the three inversions. All the results were obtained after 80 iterations. The result of standard FWI without phase encoding contains visible crosstalk artifacts, and the model is blurred by noise. Dynamic phase encoding can effectively suppress some of the artifacts, but there are some remaining artifacts. As shown by Figure 5c, seislet regularization leads to a noise-free and high-resolution model. Because this is a synthetic data inversion test, we can also display the evolution of model misfits. As shown in Figure 5d, FWI with seislet regularization has a faster model convergence rate.

Full-waveform inversion using seislet regularization