Seismic data reconstruction via weighted nuclear-norm minimization

Figures & data

Figure 1. (a) Original seismic data $X_1$. (b) Corrupted data with 50% missing traces.

Figure 2. Reconstructions for $X_1$ under $\mathit{\delta }=0$. (a)–(d): Reconstructed 100th single traces via IST, LMaFit, WSST and WISD, respectively. Left-half part is the 100th trace comparison, while the right-half part is the corresponding reconstruction error comparison.

Table 1. The SNR and CPU times of reconstructions for $X_1$ by four different algorithms with three different noise levels.

Noise	SNR and CPU times for $X_1$
level	IST(CPU)	LMaFit(CPU)	WSST(CPU)	WISD(CPU)
$\mathit{\delta }=0$	14.4704(33.78s)	15.7683(4.08s)	17.2115(9.55s)	17.1280(2.44s)
$\mathit{\delta }=0.01$	13.5495(27.51s)	14.0634(2.82s)	15.4221(1.92s)	15.4270(1.39s)
$\mathit{\delta }=0.03$	11.3439(22.85s)	11.0226(2.82s)	12.3375(1.05s)	12.3395(0.67s)

Table 2. The SNR and CPU times of reconstructions for $X_2$ by four different algorithms with three different sampling ratios.

Sampling	SNR and CPU times for $X_2$
ratio (%)	IST(CPU)	LMaFit(CPU)	WSST(CPU)	WISD(CPU)
25	11.3900(251.4s)	15.5493(0.83s)	16.5086(15.32s)	16.5438(13.12s)
35	15.2804(77.28s)	15.5753(1.97s)	17.7437(24.34s)	17.7888(12.18s)
50	17.4351(34.73s)	17.6134(0.34s)	18.9050(11.14s)	18.9372(8.88s)

Table 3. The SNR and CPU times of reconstructions for $X_3$ by four different algorithms with three different sampling ratios.

Sampling	SNR and CPU times for $X_3$
ratio (%)	IST(CPU)	LMaFit(CPU)	WSST(CPU)	WISD(CPU)
25	9.8132(95.69s)	11.0437(1.02s)	11.9217(47.28s)	11.9793(18.00s)
35	11.5813(59.10s)	12.0613(1.09s)	13.1103(47.58s)	13.2361(17.35s)
50	13.7402(34.71s)	14.7511(1.19s)	16.7312(20.34s)	16.7332(18.64s)

Figure 3. The SNR change of reconstructions by four different algorithms for $X_1$ as the sampling ratio increases from 20 to 90%.

Figure 4. Left column: original data $X_2$ (a) and $X_3$ (c). Right column: (b) and (d) are the corresponding corrupted data with 50% traces missing for $X_2$ and $X_3$, respectively.

Figure 5. Reconstructions for $X_2$. (a)–(d): Reconstructed eighth single traces via IST, LMaFit, WSST and WISD, respectively. Left-half part is the eighth trace comparison, while the right-half part is the corresponding reconstruction error comparison.

Figure 6. Reconstructions for $X_3$. (a)–(d): Reconstructed central single traces via IST, LMaFit, WSST and WISD, respectively. Left-half part is the central trace comparison, while the right-half part is the corresponding reconstruction error comparison.

Figure 7. The SNR change of reconstructions by four different algorithms ((a) for $X_2$ and (b) for $X_3$) as the sampling ratio increases from 20 to 90%.