Image gather reconstruction using StOMP

Phase encoding

Phase encoding is another powerful technique to reduce data size. The basic idea of phase encoding is to sum several independent experiments together each scaled by a different multiplier. This multiplier could be something as sophisticated as a Gold code (Gold, 1967) or as simple as scaling the different experiments by -1, 0, or 1. The idea is that the combining of the different experiments will not, or will minimally, affect the final model. This ``magic'' is achieved because the encoded experiments only interact slightly when applying the operator to produce the model or through inversion techniques that attempt to undo the encoding. Examples of the first approach in geophysics can be seen in phase encoded migrations, such as plane wave migration (Whitmore, 1995), or more general phase encoded migrations(Shan, 2008). Examples of using phase encoding in inversion can be seen velocity estimation (Guerra, 2010) did inverse imaging (Leader and Almomin, 2012; Tang, 2011).

For the correlation gather construction problem, I attempted to encode multiple different correlations into each data sample. In terms of operators, we can think of the subsampling of correlations as applying a subsampling operator $\bf S$ to all possible correlations $\bf d$ . In the phase encoded case, we are going to add another operator $\bf P$ that first sums a number of different correlations together and then subsamples them leaving a new dataset $\bf S \bf R \bf d$ . This in turn changes the operator $\bf L$ in algorithm 1 to $\bf P \bf W$ . For this test, I combined 20 different correlations in a random pattern to form each data point. Figure 8 shows the same three offset and angle gathers seen in Figures 4 and 6. Note the noticeably better job recovering the deeper portion of subsurface offsets. Figure 9 shows the angle gathers from the fully sampled correlation gathers and the subsampled, phase encoded gathers muted to the believable angle range. The gathers with the notable exception of more lower frequency noise in the recovered gathers.

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Figure 8. A, B, and C show the recovered subsurface offset gathers using compressive sensing as shown in plot A, B and C of Figure 4 and 6. D, E, and F show the corresponding angle gathers. Note the noticeable improvement in A, B, and C compared to the data shown in Figure 6.

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Figure 9. A comparison of the angle gathers after muting of the fully sampled correlation gathers (A, B, and C) and the phase encoded, subsampled correlation gathers (D, E, and F).

Image gather reconstruction using StOMP