Seismic Trace Interpolation Using Adaptive Prediction Filters

Transcription

1 Seismic Trace Interpolation Using Adaptive Prediction Filters Mostafa Naghizadeh and Mauricio Sacchi Signal Analysis and Imaging Group (SAIG) Department of Physics University of Alberta CSEG 2008 Calgary

2 Outline: Introduction Theory PF interpolation for stationary data (Linear events) Adaptive PF interpolation for non-stationary data (Hyperbolic and Parabolic events) Examples Synthetic data Real data Conclusions

3 Main Goal: Interpolating curved (hyperbolic or parabolic) events and dispersive events using prediction filters

4 Review of reconstruction methods: Signal Processing methods (Spectrum) FX (Prediction Filters) methods (Spitz, 1991 and Porsani, 1999) FK (Masking operators) methods (Gulunay, 2003) Minimum Weighted Norm Interpolation (Liu and Sacchi, 2004) Fourier Reconstruction with Sparse Inversion (Zwartjes, 2005) MSAR Reconstruction (Naghizadeh and Sacchi, 2007) Wave-equation methods (Physics) NMO and DMO operators (Ronen, 1987) Offset and shot continuation (Bagaini and Spagnolini,1999) Seismic data mapping and reconstruction (Stolt, 2002) Migration Operators (Malcolm et al., 2005 and Trad, 2003).

5 Theory and Implementation

6 FX interpolation method (Spitz,1991): Linear events : Stationary signal in the FX domain Stationary signal : Parameterized by prediction filters Prediction filters of the original data at low frequencies = Prediction filters of the interpolated data at high frequencies Prediction filters can estimate the missing samples of the signal

7 Time Shift vs Phase Shift : t X τ time shift To F-X f f f f f n N X f f n : 1, e -i2πf τ, (e -i2πf τ ), (e -i2πf n n 2 n 3 τ ),...

8 Auto-Regressive equivalence of f/2 and f,... e, e, e, e, e, e, : 2 f n n n n n n f -i2 f -i2 f -i2 f -i2 f -i2 f -i2 n ) ( ) ( ) ( ) ( ) ( τ π τ π τ π τ π τ π τ π,... e, e, e, e, e, e, : f n n n n n n f -i2 f -i2 f -i2 f -i2 f -i2 f -i2 n ) ( ) ( ) ( ) ( ) ( τ π τ π τ π τ π τ π τ π

9 Ordinary FX Interpolation

10 Estimation of Prediction Filter (PF)

11 Interpolation using PFs (part 1)

12 Interpolation using PFs (part 2)

13 Interpolation using PFs (part 3)

14 Adaptive FX Interpolation

15 Non-linear events: Non-stationary signal in the FX domain Non-stationary signal : Adaptive prediction filters Currently used methodology: Dividing data into small windows. Very time consuming The length of windows. Our Proposed method in this talk: Computing adaptive local PFs. Exponentially down-weighting the effects of nearby windows. Utilizing Recursive Least Square (RLS) principles to avoid computational loads.

16 Adaptive PFs (as local as possible)

17 Adaptive PFs (using past data with less weight) forgetting factor

18 Least Squares solution of adaptive PFs (Forward)

19 Recursive Least Squares solution of adaptive PFs

20 Initialization of Recursive Least Square solution (backward) The only inversion of the algorithm

21 Interpolation using adaptive PFs (part 1)

22 Interpolation using adaptive PFs (part 2)

23 Interpolation using adaptive PFs (part 3)

24 Interpolation using adaptive PFs (part 4)

25 Synthetic Examples

26 Original synthetic section with linear events

27 Interpolated data using adaptive PFs (λ=1)

28 Interpolated data using adaptive PFs (λ=.25)

29 Twice interpolated data using adaptive PFs (λ=1)

30 Three times interpolated data using adaptive PFs (λ=1)

31 Original synthetic section with hyperbolic events

32 Decimated section with hyperbolic events

33 Original synthetic section with hyperbolic events

34 Decimated section with hyperbolic events

35 Interpolated data using Spitz FX method

36 Interpolated data using adaptive PFs (λ=1)

37 Interpolated data using windowed Spitz method

38 Interpolated data using adaptive PFs (λ=0.15)

39 Residual section (Spitz FX interpolation)

40 Residual section (Adaptive PFs λ=1)

41 Residual section (Windowed Spitz method)

42 Residual section (Adaptive PFs λ=.15)

43 Phase indicator in the FX domain (decimated data)

44 Phase indicator in the FX domain (original data)

45 Phase indicator in the FX domain (interpolated data λ=0.3)

46 Phase indicator in the FX domain (twice interpolated data) λ=0.6

47 Synthetic data with conflicting dips Original Interpolated (λ=0.25)

48 FK domain of data with conflicting dips Original Interpolated (λ=0.25)

49 Synthetic data with conflicting dips (with high frequencies) Original Interpolated (λ=0.25)

50 FK domain of data with conflicting dips (higher frequencies) Original Interpolated (λ=0.25)

51 Real Data Example

52 Shot gather from Gulf of Mexico

53 Decimated shot gather

54 Interpolated using Adaptive PFs (λ=0.55)

55 Original shot gather

56 Residual section

57 Near offset section from Gulf of Mexico

58 Interpolated using adaptive PFs (λ=0.2)

59 Conclusions: Exponentially Weighted Recursive Least Square Algorithm is used to estimate the adaptive prediction filters. Adaptive prediction filters computed at low frequencies are used locally to interpolate the high frequencies. Adaptive F-X interpolation depends on two parameters: operator length and the forgetting factor. The method eliminates the need of selecting window parameters (window size and amount of overlapping between adjacent windows). The proposed adaptive F-X interpolation algorithm is robust under strong changes of curvature and continuity. It can be fully automated and extended to the multidimensional data.

60 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)