Random sampling: new insights into the reconstruction of coarsely-sampled wavefields

Numerical results

In the seismic context, the effect of coarse sampling in the $f$ -$k$ domain is illustrated in Fig. 3. Figs. 3(a) and 3(d) show densely-sampled data and the corresponding amplitude spectrum. Figs. 3(b) and 3(e) show regularly sub-sampled data and the corresponding amplitude spectrum. Finally, Figs. 3(c) and 3(f) show randomly sub-sampled data and the corresponding amplitude spectrum. Note how random sampling creates incoherent noise across the spectrum.

Although Fourier does not provide the sparsest representation for seismic data, there exists successful interpolation algorithms that solve Eq. 3 with $\tensor{S}:=\tensor{F}$ (see e.g. Zwartjes and Hindriks, 2001; Xu et al., 2005). We use the algorithm called curvelet reconstruction with sparsity-promoting inversion (Hennenfent and Herrmann, 2005; Herrmann, 2005; Hennenfent and Herrmann, 2006; Herrmann and Hennenfent, 2007) since curvelets provide a sparser representation for seismic data than Fourier (see e.g. Candes et al., 2006; Hennenfent and Herrmannn, 2006). In this case, $\tensor{S}$ is defined as the curvelet transform (Candes et al., 2006, and references therein). The incoherent noise generated by random sampling remains incoherent in the curvelet domain since curvelets are strictly localized in the $f$ -$k$ domain. Fig. 4(a) and 4(b) show the interpolation results for the data of Figs. 3(b) and 3(c), respectively. The signal-to-reconstruction-error ratios are 6.92 dB for regular sub-sampling and 13.78 dB for random sub-sampling. For the same number of receivers, coarse random sampling leads to a much better reconstruction than coarse regular sampling. When a minimum velocity constraint is imposed during the reconstruction process, the same conclusion holds although the difference is reduced.

data_12p5m,data_subREG,data_subIRREG,fkdata_12p5m,fkdata_subREG,fkdata_subIRREG
Figure 3. Seismic data and their corresponding spectrum. Densely-sampled data (a) and corresponding amplitude spectrum (d). Data regularly sampled below Nyquist rate (b) and corresponding amplitude spectrum (e) with strong aliasing beyond 25 Hz. Data randomly sampled at the same sub-Nyquist rate as (b) and corresponding amplitude spectrum (f) corrupted by broadband noise.

crsi_subREG,crsi_subIRREG,fkcrsi_subREG,fkcrsi_subIRREG
Figure 4. Synthetic seismic data reconstruction using 2-D curvelet reconstruction with sparsity-promoting inversion. Interpolation result - SNR = 6.9 dB - (a) and corresponding amplitude spectrum (c) given data of Fig. 3(b). Interpolation result - SNR = 13.78 dB - (b) and corresponding amplitude spectrum given data of Fig. 3(c). For the same number of receivers, coarse random sampling leads to a much better reconstruction than coarse regular sampling.

Random sampling: new insights into the reconstruction of coarsely-sampled wavefields