Reconstruction of irregularly sampled and aliased data with linear prediction filters

Transcription

1 Reconstruction o irregularly sampled and aliased data with linear prediction ilters Mostaa Naghizadeh and Mauricio Sacchi Signal Analysis and Imaging Group (SAIG) Department o Physics University o Alberta 23 September 2007 SEG Annual Meeting

2 Outline: Review o reconstruction methods Algorithm Band-Limited Minimum Weighted Norm Interpolation Multi-Step Auto-Regressive reconstruction Examples Synthetic data (2D & 3D) Real data (3D) Conclusions

3 Review o reconstruction methods: Wave-equation methods NMO and DMO operators (Ronen, 1987) Oset and shot continuation (Bagaini and Spagnolini,1999) Seismic data mapping and reconstruction (Stolt, 2002) Migration Operators (Malcolm et al., 2005 and Trad, 2003). Signal Processing methods FX (Prediction Filters) methods (Spitz, 1991 and Porsani, 1999) Band-Limited Fourier Reconstruction (Duijndam et al., 1999) FK (Masking operators) methods (Gulunay, 2003) Minimum Weighted Norm Interpolation (Liu and Sacchi, 2004) Fourier Reconstruction with Sparse Inversion (Zwartjes, 2005)

4 Limitations o available methods: FX (Prediction Filters) method (Spitz, 1991 and Porsani, 1999). Only applicable or regularly sampled data. Band-Limited Fourier Reconstruction (Duijndam et al., 1999). Not valid or the aliased part o data. Minimum Weighted Norm Interpolation (Liu and Sacchi, 2004) and Fourier Reconstruction with Sparse Inversion (Zwartjes, 2005). Unable to eliminate the high amplitude artiacts in spectrum.

5 Overall Strategy: Our methodology to overcome the limitations: Reconstructing a part o data which is band-limited (Low requencies) using Band-Limited Minimum Weighted Norm Interpolation (BLMWNI). Leads to regularly sampled data in the selected band o data. Extracting Prediction Filters rom regularly sampled part. Prediction ilters o all requencies can be computed rom small band o requencies that we reconstructed them above. Reconstruction using Prediction Filters. Prediction ilters can be used as regularization term to reconstruct the missing samples at all requencies.

6 Band-Limited Minimum Weighted Norm Interpolation (BLMWNI)

7 Band-Limited Minimum Weighted Norm Interpolation (BLMWNI) Analogy Between the time-space and the requency-wavenumber domain k x t

8 Band-Limited assumption in the -k domain: n Data in FK domain : Desired to be computed : Known and orced to be zero

9 BLMWNI : Missing Samples : Available Samples Spatial Domain Wavenumber Domain FFT IFFT Data in FX domain Data in FK domain IFFT Final reconstructed data in FX domain F : Fourier Operator : Band-Limiting and Weighting Operator G : Sampling Operator

10 Multi-Step Auto-Regressive reconstruction

11 Auto-Regression Representing current sample as linear combination o past and uture samples (prediction) x 1, x 2, x 3, x 4, x 5, x 6, x 7,, x N. a 1 x 1 + a 2 x 2 + a 3 x 3 = x 4 a 1 x 2 + a 2 x 3 + a 3 x 4 = x 5 Prediction Filter a 1,a 2,a 3 a 1 x 3 + a 2 x 4 + a 3 x 5 = x 6 Prediction step Spitz (1991) proves that AR operators are able to predict linear seismic events in FX domain.

12 Time Shit vs Phase Shit : t X τ time shit To F-X n N X n : 1, e -i2π τ, (e -i2π τ ), (e -i2π n n 2 n 3 τ ),...

13 Proo o Multi-Step Auto-Regression,... e, e, e, e, e, e, : 3 n n n n n n -i2 -i2 -i2 -i2 -i2 -i2 n ) ( ) ( ) ( ) ( ) ( τ π τ π τ π τ π τ π τ π,... e, e, e, e, e, e, : 2 n n n n n n -i2 -i2 -i2 -i2 -i2 -i2 n ) ( ) ( ) ( ) ( ) ( τ π τ π τ π τ π τ π τ π,... e, e, e, e, e, e, : n n n n n n -i2 -i2 -i2 -i2 -i2 -i2 n ) ( ) ( ) ( ) ( ) ( τ π τ π τ π τ π τ π τ π

14 Multi-Step Auto-Regression Step size Maximum possible step size Number o prediction ilters at each requency

15 Computing Prediction Filters (One-Step) 0 Spatial axis min Normalized requency axis max -1=0 0.5

16 0 Computing Prediction Filters (Two-Step) Spatial axis min Normalized requency axis 2*min max 2*max -1=0 0.5

17 Computing Prediction Filters (Three-Step) 0 Spatial axis min Normalized requency axis 3*min max -1=0 3*max 0.5

18 Computing Prediction Filters (Four-Step) 0 Spatial axis min Normalized requency axis max 4*min -1=0 0.5

19 Prediction Filters Computing Prediction Filters (Multi-Step) Averaging PF s 0Normalized requency axis min max Spatial axis 0.5

20 Estimation o 1D Prediction ilter

21 Estimation o 2D Prediction ilter (3x3)

22 Computing missing samples using prediction ilters : Missing Samples : Available Samples : Estimated Prediction Filter * =

23 2D Synthetic Examples

24 T-X Domain Randomly missing traces BL-MWNI MSAR

25 F-K Domain Randomly missing traces BL-MWNI MSAR

26 T-X Domain Original Regularly missing traces

27 T-X Domain BL-MWNI MSAR

28 F-K Domain Original Regularly missing traces

29 F-K Domain BL-MWNI MSAR

30 Number o prediction ilters at each requency

31 3D Synthetic Examples

32 Original 3D cube

33 Data with missing traces

34 Reconstructed data using MWNI

35 Original data

36 Reconstructed data using MSAR

37 Data with missing traces

38 Reconstructed data using MWNI

39 Original data

40 Reconstructed data using MSAR

41 Number o prediction ilters at each requency

42 2D & 3D Real data Examples

43 A near oset section rom Gul o Mexico Original section Section with missing traces

44 Near oset section rom Gul o Mexico MWNI MSAR

45 Reconstructed band o data using BLMWNI t-x domain -k domain

46 Output o C program or inormation o MSAR ******************************************************* Inormation o MSAR plan Dimensionality: 3D Maximum jumping step: 7 --> 7 Dimension o original data: n1=900 n2=91 n3=19 Dimension o MWNI plan: n1=8192 n2=256 n3=64 MWNI-applied requencies: min=0.025 nmin=205 max=0.070 nmax=574 Band-Limitation o MWNI-applied requencies : bmin2=0.400 bmin3=0.150 bmax2=0.950 bmax3=0.250 Maximum user-deined MSAR requency: maxp=0.250 nmaxp=257 Size o Prediction Filters: np2=5 np3=3 *******************************************************

47 Cube o shots rom Gul o Mexico Trace number Time [s] Original

48 MSAR reconstruction (every other shot) Trace number Time [s] MSAR Reconstructed

49 MWNI o cube o shots rom Gul o Mexico Trace number Time [s] MWNI Reconstructed

50 Zoom o MWNI 3.0 Trace number Time [s] MWNI Reconstructed

51 Zoom o the original cube o shots rom Gul o Mexico 3.0 Trace number Time [s] original

52 Zoom o MSAR reconstruction (every other shot) 3.0 Trace number Time [s] MSAR Reconstructed

53 Original MWNI MSAR

54 Conclusions: Band-limiting approaches can reconstruct some part o data (low requencies) with good resolution. The prediction ilters o all data can be computed rom a band o data using Multi-Step AR. Reconstruction o data using prediction ilters avoids ampliying high amplitude artiacts in spectra. MSAR can be used in various areas o seismic data processing like noise attenuation etc. Extending MSAR to the higher dimensions is straightorward.

55 Acknowledgments SAIG Sponsors: CGGVeritas ConocoPhillips Encana ENI SPA AGIP Divesco Fugro Norsk Hydro Statoil Faculty o Science, University o Alberta Natural Sciences and Engineering Research Council o Canada (NSERC)