1. Seismic Resolution of Zero-Phase Wavelets Designing Optimum Zero-Phase Wavelets R. S. Kallweit and L. C. Wood Amoco Houston Division DGTS January 12, 1977 G. Partyka Oct 2006

2. Questions Can different types of zero-phase wavelets be compared in terms of temporal resolution. What wavelet shape is well suited for comparing against? Can we separate the ability of zero-phase wavelets to resolve thin beds from variations in side-lobe tuning effects? What is an optimum wavelet shape? Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

3. What are the Limits of Temporal Resolution? 1.How thin can a bed become and still be resolvable? In other words, when is the measured interval time essentially the same as the true interval time? 2.What are the errors between the true interval times and the measured interval times through thick beds? Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 Review: Rayleigh Ricker Widess G. Partyka Oct 2006

4. Why Focus on Zero-Phase? Non zero-phase wavelets greatly complicate resolution. Zero-phase wavelets simplify resolution: –traces containing zero-phase wavelets will have seismic interfaces located in general at the centers of the peaks and troughs of the trace (neglecting tuning effects and noise). Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 Thick Bed Well Log Thin Bed Well Log G. Partyka Oct 2006

5. What is an Optimum Wavelet Shape to Compare Against? Low Pass Sinc Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 frequency amplitude TimeFrequency time G. Partyka Oct 2006

6. Temporal Resolution of the Low-Pass Sinc Wavelet TbTb TRTR fmfm frequency amplitude T0T0 f4f4 Temporal resolution is established in terms of the maximum frequency temporal resolution:T R = 1 / (3.0)f m = 1 / (1.5)f 4 wavelet breadth:T b = 1 / (1.4)f m = 1 / (0.7)f 4 peak-to-trough:T b / 2 = 1 / (2.8) f m = 1 / (1.4)f 4 1 st zero crossings:T 0 = 1 / 2f m = 1 / f 4 relationship of T b to T R T R = 0.47T b = 0.93T b / 2 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

7. What effect does a low cut have on resolution? Negligible for wavelets with bandpasses of 2-octaves or greater Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 frequency amplitude TimeFrequency time ….but, must have signal in bandpass. G. Partyka Oct 2006

8. Temporal Resolution (T R ) of the Bandpass Sinc Wavelet TbTb TRTR fmfm frequency amplitude T0T0 f4f4 f1f1 f4f4 f 4 sinc f 1 sinc band-pass sinc f m = (f 1 + f 4 ) / 2 f1f1 The band-pass sinc wavelet is the difference between two (f 1 and f 4 ) sinc functions. Effects of the f 1 sinc are negligible for wavelets with band-pass ratios 2-octaves and greater. temporal resolution:T R = 1 / (1.5)f 4 ; 2 octaves (where f 4 / f 1.ge. 4) wavelet breadth:T b = 1 / (0.7)f 4 ; 2 octaves (where f 4 / f 1.ge. 4) peak-to-trough:T b / 2= 1 / (1.4)f 4 ; 2 octaves (where f 4 / f 1.ge. 4) 1 st zero crossings:T 0 = 1 / (2f m ); all octaves relationship of T b to T R :T R = 0.47T b = 0.93T b / 2 ; for sincs.ge. 2 octaves Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

9. Sinc Bandwidth and Temporal Resolution with Constant f max 0 10 20 30 peak-to-trough separation (ms) 0102030 spike separation (ms) 2–64 Hz Sinc (5 octaves) TRTR 0 10 20 30 peak-to-trough separation (ms) 0102030 spike separation (ms) 16–64 Hz Sinc (2 octaves) TRTR Temporal Resolution is the same for all sinc wavelets with 2 octaves or greater bandwidths having the same f max T R = 1 / (1.5)f max Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

10. Non-Binary Complexity 10 15 20 25 peak-to-trough separation (ms) spike separation (ms) 30 0 5 0 15102025305 1.51 1.0 -0.8 -0.6 TRTR 16-to-64 Hz (2-octave) sinc wavelet convolved with alternate polarity spike pairs of unequal amplitude Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 What is the effect on temporal resolution when the amplitude of the second spike of a set of alternate polarity spike pairs is varied? G. Partyka Oct 2006

11. Questions Can different types of zero-phase wavelets be compared in terms of temporal resolution. What wavelet shape is well suited for comparing against? Can we separate the ability of zero-phase wavelets to resolve thin beds from variations in side-lobe tuning effects? What is an optimum wavelet shape? Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

12. Wavelet Shape and Sidelobe Interference Wavelets designed with a vertical or near-vertical high end slope exhibit high frequency sidelobes that can cause significant distortions in reflection amplitudes and associated event character. An alternate wavelet is proposed called the Texas Double in recognition of the primary characteristic being a 2-octave slope on the high frequency side. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

13. Texas Double Wavelets Time Domain: –negligible high frequency sidelobe tuning effects. –maximum peak-to-sidelobe amplitude ratios. Frequency Domain: –vertical or near-vertical low-end slope. –2-octave linear slope on the high-end. Amplitudes are measured using a linear rather than decibel scale. –end frequencies correspond to the highest and lowest recoverable signal frequency components of the recorded data. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

14. Development of High Frequency Side-Lobes REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006

15. Development of High Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

16. Development of High Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 3 octave slope High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

17. Development of High Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 2 octave slope High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

18. Development of High Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 1 octave slope High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

19. Development of High Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 High frequency sidelobes can be attenuated to an insignificant level via a 2-octave or greater high side slope. G. Partyka Oct 2006 Wavelet

20. Development of Low Frequency Side-Lobes REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006

21. Development of Low Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

22. Development of Low Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 4.0 octaves Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

23. Development of Low Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 3.0 octaves Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

24. Development of Low Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 2.4 octaves Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

25. Development of Low Frequency Side-Lobes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 2.0 octaves Low frequency sidelobes are a function of the wavelet’s bandpass. They cannot be reduced beyond what is shown here. G. Partyka Oct 2006 Wavelet

26. High Frequency Held Constant (Klauder Wavelets) REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 G. Partyka Oct 2006

27. High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 4.0 octaves G. Partyka Oct 2006 Wavelet

28. High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 3.0 octaves G. Partyka Oct 2006 Wavelet

29. High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 2.4 octaves G. Partyka Oct 2006 Wavelet

30. High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 2.0 octaves G. Partyka Oct 2006 Wavelet

31. High Frequency Held Constant (Klauder Wavelets) frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 1.4 octaves G. Partyka Oct 2006 Wavelet

32. Decreasing the Low Frequency Slope REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 G. Partyka Oct 2006

33. Decreasing the Low Frequency Slope frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 3 octaves G. Partyka Oct 2006 Wavelet

34. Decreasing the Low Frequency Slope frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 2 octave slope G. Partyka Oct 2006 Wavelet

35. Decreasing the Low Frequency Slope frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 3 octave slope G. Partyka Oct 2006 Wavelet

36. Decreasing the High and Low Frequency Slopes REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 G. Partyka Oct 2006

37. Decreasing the High and Low Frequency Slopes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 3 octave sinc G. Partyka Oct 2006 Wavelet

38. Decreasing the High and Low Frequency Slopes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 G. Partyka Oct 2006 Wavelet

39. Decreasing the High and Low Frequency Slopes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 G. Partyka Oct 2006 Wavelet

40. Decreasing the High and Low Frequency Slopes frequency 100 amplitude 2030 40 5060 REFLECTIVITYIMPEDANCE 0 100 Travel Time (ms) 200 300 50 150 250 0 100 200 300 50 150 250 Temporal Thickness (ms) 05040302010 Temporal Thickness (ms) 50403020100 Texas Double G. Partyka Oct 2006 Wavelet

41. Proposed Standard Equi-Resolution Comparison One of the difficulties involved in trying to compare traces containing different zero-phase wavelets designed over identical bandpasses is the question of what to compare and measure each trace against. It is rather unsatisfactory to compare the traces against one another since there are too many unknowns. A standard comparison is needed. The standard trace proposed is one where the convolving wavelet has the same temporal resolution as the sinc wavelet over a given bandpass but has no sidelobes whatsoever. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

42. Low-Pass Sinc vs Low-Pass Texas Double 0 10 20 30 peak-to-trough separation (ms) 0102030 spike separation (ms) 0–0-62-64 Hz Sinc TRTR 0 10 20 30 peak-to-trough separation (ms) 0102030 spike separation (ms) TRTR Conclusion: Over a given low-pass, temporal resolution of the Texas Double wavelet equals 80% of the temporal resolution of the sinc wavelet. T R = 1 / 1.5f 4 0–0-20-80 Hz Texas Double T R = 1 / 1.2f 4 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

43. Equivalent Temporal Resolution: Ormsby to Low-Pass Sinc fsfs frequency amplitude f4f4 f3f3 0.5 0.6 0.7 f s /f 4 0.8 0.9 1.0 00.10.20.30.40.50.60.70.80.91.0 f 3 / f 4 Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

44. Is it worth giving up the 20% loss in Temporal Resolution? Can the benefits associated with attenuating high- frequency sidelobes outweigh the 20% loss in temporal resolution? Well-log based comparisons, suggest that they can. …as long as that 20% is not critical to the required imaging. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

45. Well-Log Comparison raw rc 00-00-20-8000-00-62-6408-09-62-6408-09-16-6408-09-20-80 Reflectivity raw 00-00-20-8000-00-62-6408-09-62-6408-09-16-6408-09-20-80 Raw InputDesired Standard High Frequency Tuning Effects Only Negligible EffectSinc WaveletTexas Double G. Partyka Oct 2006

46. Well-Log Comparison - 3 Octaves raw rc 00-00-20-8000-00-62-6408-09-62-6408-09-16-6408-09-20-80 raw layering 00-00-20-8000-00-62-6408-09-62-6408-09-16-6408-09-20-80 LayeringReflectivity G. Partyka Oct 2006

47. Well-Log Comparison - 2 Octaves raw rc 00-00-20-8000-00-62-6416-17-62-6416-17-18-6416-17-20-80 raw layering 00-00-20-8000-00-62-6416-17-62-6416-17-18-6416-17-20-80 LayeringReflectivity G. Partyka Oct 2006

48. Gradually Increasing Frequency Content to Examine Tuning Effects When filters change in a linear and gradual manner, we would hope that the traces would do likewise. Unfortunately, sidelobe interference gets in the way. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006

49. Well-Log Example - Layering Sinc Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw Layering 6-10-096-100 6-10-066-070 6-10-036-040 Raw Layering 6-10-096-100 6-10-066-070 6-10-036-040

50. Well-Log Example - Layering 10 Hz High-Cut Slope Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw Layering 6-10-090-100 6-10-060-070 6-10-030-040 Raw Layering 6-10-090-100 6-10-060-070 6-10-030-040

51. Well-Log Example - Layering Texas Double Wavelets – high frequency side varies 52 to 112 Hz; low frequency held constant Raw Layering 6-10-028-112 6-10-021-082 6-10-013-052 Raw Layering 6-10-028-112 6-10-021-082 6-10-013-052

52. Well-Log Example - Layering Texas Double Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw Layering 6-10-025-100 6-10-020-070 6-10-011-040 Raw Layering 6-10-025-100 6-10-020-070 6-10-011-040

53. Well-Log Example - Layering Sinc Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw Layering 6-10-096-100 6-10-066-070 6-10-036-040 Raw Layering 6-10-096-100 6-10-066-070 6-10-036-040

54. Well-Log Example - Reflectivity Sinc Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw RC 6-10-096-100 6-10-066-070 6-10-036-040 Raw RC 6-10-096-100 6-10-066-070 6-10-036-040

55. Well-Log Example - Reflectivity 10 Hz High-Cut Slope Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant 6-10-090-100 6-10-060-070 6-10-030-040 6-10-090-100 6-10-060-070 6-10-030-040 Raw RC

56. Well-Log Example - Reflectivity Texas Double Wavelets – high frequency side varies 52 to 112 Hz; low frequency held constant 6-10-028-112 6-10-021-082 6-10-013-052 6-10-028-112 6-10-021-082 6-10-013-052 Raw RC

57. Well-Log Example - Reflectivity Texas Double Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant 6-10-025-100 6-10-020-070 6-10-011-040 6-10-025-100 6-10-020-070 6-10-011-040 Raw RC

58. Well-Log Example - Reflectivity Sinc Wavelets – high frequency side varies 40 to 100 Hz; low frequency held constant Raw RC 6-10-096-100 6-10-066-070 6-10-036-040 Raw RC 6-10-096-100 6-10-066-070 6-10-036-040

59. Texas Double in Practice One may implement the Texas Double on real data by first running an amplitude whitening program followed by a 2-octave high-side slope Ormsby filter. The Texas Double design criteria should not be a goal of data acquisition. It is of utmost importance that the signal-to-noise ratio of the high- frequency components be as large as possible, and therefore filtering process such as the Texas Double should occur in data processing and not in data acquisition. Seismic Resolution of Zero-Phase Wavelets, R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977 G. Partyka Oct 2006