Sampling functions and sparse reconstruction methods

Transcription

1 Sampling functions and sparse reconstruction methods Mostafa Naghizadeh and Mauricio Sacchi Signal Analysis and Imaging Group Department of Physics, University of Alberta EAGE 2008 Rome, Italy

2 Outlines: Motivations Sparse Reconstruction Sparseness imposing norms Minimum Weighted Norm Interpolation Sampling operators (irregular vs. regular) Synthetic examples 1D, 2D and 3D examples Real data example (3D) Conclusion

3 Motivations I: Main Goal: Analyzing the effects of various sampling operators on sparse reconstruction methods

4 Sparse Reconstruction (Minimum Weighted Norm Interpolation)

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

6 MWNI : Missing Samples : Available Samples Spatial Domain FFT Wavenumber Domain IFFT Data in FX domain Data in FK domain IFFT Final reconstructed data in FX domain F : Fourier Operator : Weighting Operator G : Sampling Operator

7 Sampling Operators (Irregular Vs. Regular)

8 Non-uniform sampling and its spectra (Continuous signal) Sampling function Spectra of Sampling function Doing some math tricks

9 Non-uniform sampling operator for discrete signals Original signal Sampled signal Sampling operator Convolution

10 Spectrum of regular sampling operators with various decimations (from top to bottom: decimation factors equal to 1,2,3,4 and 5)

11 Spectrum of random sampling operators with 50% missing samples

12 Spectrum of random sampling operators with 90% missing samples

13 Spectrum of sampling operators with Gaps (from top to bottom: big gaps to small gaps)

14 Synthetic Examples

15 1D example: random sampling Original Missing Reconstructed

16 1D example: regularly missing samples (every other traces) Missing Reconstructed

17 1D example: random sampling from regularly missing signal (every other trace) Missing Reconstructed

18 1D example: random sampling from regularly missing signal (decimation factor equal to 3) Missing Reconstructed

19 1D example: random sampling (with the distance between two consecutive samples is always bigger than 4). Missing Reconstructed

20 1D example: Error Vs. Percentage of available samples (for random sampling operators)

21 2D example: Irregular sampling T-X Domain Original Missing MWNI-reconstructed

22 2D example: Irregular sampling F-K Domain Original Missing MWNI-reconstructed

23 2D example: Regular sampling T-X Domain Regularly missing traces MWNI-Reconstructed

24 2D example: Regular sampling F-K Domain Regularly missing traces MWNI-Reconstructed

25 2D example: Gap example T-X Domain Original Missing MWNI-reconstructed

26 2D example: Gap example F-K Domain Original Missing MWNI-reconstructed

27 Original 3D data with linear events

28 Randomly missing 90% of 3D data

29 MWNI result for randomly missing 90% of 3D data

30 Regularly missing 50% of 3D data

31 MWNI result for regularly missing 50% of 3D data

32 Original 3D data with mild hyperbolic events

33 Randomly missing 75% of original data

34 Reconstructed using MWNI

35 Original 3D data

36 3D Gulunay s f-k interpolation

37 91 traces per shot 19 shot gathers from the Gulf of Mexico

38 4 shots of the original shots

39 Regularly decimated shots with factor of 4

40 Non-zero traces of the regularly decimated data

41 MWNI reconstruction of the regularly decimated data

42 Original data

43 Randomly eliminating 75% of the traces

44 Non-zero traces of the randomly eliminated data

45 Sparse reconstruction of the randomly eliminated data

46 Original data

47 Conclusions: Random sampling causes low amplitude noises in the spectrum domain and therefore sparse reconstruction methods perform successfully (if signal is Sparse). With the regular sampling, the resulted artifacts are as high amplitude as the original spectrum of signal. Hence imposing sparseness constraints can not recover missing samples. Sparse reconstruction can reconstruct big gaps to an acceptable level of accuracy. MSAR reconstruction can overcome the failure of sparse reconstruction methods for regularly or almostregularly sampled data (Our next talk).

48 Acknowledgments SAIG Sponsors: BP Encana ENI SPA AGIP Division ConocoPhillips Divesco Fugro Hydro Statoil CGGVeritas GEDCO Landmark Graphics Faculty of Science, University of Alberta Natural Sciences and Engineering Research Council of Canada (NSERC)

49

50

51 Regularized Inversion (quadratic cost function): Ill-posed linear system G : Inverse DFT Cost function Solution Minimum Norm Solution This solution is more familiar

52 Regularized Sparse Inversion: Cost function Solution Cauchy norm : Less damping (Faithful to available data) : More damping (Constrained to model data (zero))