Imaging using compressive sensing

Compressibility

There is significant literature on compressing seismic data. Relatively low compression ratios are achievable by compressing a trace. Significantly higher compression ratios are achieved by multi-dimensional approaches. Generally, the best results have used either multi-dimensional wavelets (Mallat, 1999) or its successor curvelets (Candes and Donoho, 1999). Villasenor et al. (1996) showed that compression ratios of 100:1 were achievable by compressing a 4-D volume ( $t,h_x,h_y,s$ ). Further, Villasenor et al. (1996) states that the header information was the limiting factor in achieving even higher ratios.

Sub-surface offset gathers potentially represent an even higher, up to six, dimensional problem. To test compressibility, I used a 4-D volume ($x,y,z,h_x$ ) of dimensions (32,32,400,64). Figure 2 shows one of these sub-surface offset gathers and its neighbors, note the similarly. Following Villasenor et al. (1996), I chose the 9/7 bi-orthonormal transform(Antonini et al., 1992) used in JPEG compression. Figure 3 shows the resulting transform space and a histogram of the absolute values. I then used several different thresholds (throwing away 90%, 95%, 98%, and 99% of the data in the wavelet domain). Figure 4 shows the result of transforming this thresholding volume back into the space-domain. The resulting images are near-perfect at 95% and potentially acceptable at 98%. This translates into an acceptable compression ratio of approximately 30:1.

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Figure 2. Five sub-surface offset gathers. B and C are one midpoint in X before and after A. E is one midpoint in Y before A. Note the spatial similarity, which lends itself to compression.

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Figure 3. Panel A shows the wavelet domain representation of the 4-D volume used in this experiment. Panel B shows a zoom into portion of the wavelet domain. Panel C shows a histogram of the wavelet domain values. Note how the vast majority of the values are nearly zero.

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Figure 4. The result of zeroing the smallest values of the wavelet domain representation shown in Figure 3A. All four panels show the same sub-surface offset gather shown in Figure 2. A shows the result of clipping 90% of the values; B, 95%; C, 98%; and D,99%. Note how the reconstructed gather is nearly identical up to a 98% clip.

We get a more interesting result if we look at the compressibility of angle gathers. Figure 5 shows the same locations as Figure 2 but in the angle domain. Figure 6 shows the resulting wavelet domain space and a histogram of its absolute values. Note an even greater clustering around 0. Figure 7 shows the result of thresholding the wavelet domain at 95%, 98%, 99%, 99.5%. Note how we see near perfect reconstruction of the angle gather up to 99% and potential up to 99.5%. This is a 100:1 to 200:1 compression ratio (which translates into 20 to 50x reduction in sub-surface offsets).

Framing the compression in terms of angle gathers requires a change in formulation. The $\bf L$ in equation 2 now becomes chain of two operators. The first translates from sub-surface offset to angle. The second is multi-dimensional wavelet transform.

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Figure 5. Four angle gathers at the same locations seen in Figure 2.

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Figure 6. Panel A shows the wavelet domain representation of the 4-D volume used in this experiment transformed into the angle domain. Panel B shows a poriton of the wavelet domain. Panel C shows a histogram of the wavelet domain values. Note how the vast majority of the values are nearly zero.

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Figure 7. The result of zeroing the smallest values of the wavelet domain representation shown in Figure 3A. All four panels show the same sub-surface offset gather shown in Figure 5. A shows the result of clipping 95% of the values; B, 98%; C, 99%; and D,99.5%. Note how the reconstructed gather is nearly identical up to a 99% clip.

Imaging using compressive sensing