1. John C. Bancroft and grad students
AVO, migration apertures Fresnel zones, stacking, Q, anisotropy, and fast inversions
John C. Bancroft and grad students
University of Calgary
CREWES 2003
NSERC

2. Grad students Shuang Sun Pavan Elapavuluri Kun Liu (and Hugh Geiger)
John Millar
Xiang Du
Zhihong (Nancy) Cao
Chunyan (Mary) Xiao

3. Outline Differentiators Fresnel zones AVO migration apertures Q
Wavefield extrapolators
Multiples and radon transforms
Inversion

4. Digital differentiator
Accuracy
Speed
Beauty
(1, -1)
1, 0, -1) better
Theoretical shape?

5. Inverse FFT of (j)

6. Derivative assumes: Maximum bandwidth Ideal interpolator
Sinx/x operator
Derivative of sinx/x
SincFn.m

7. What is the basic unit of laryngitis?
Shuang Sun
Stacking specula energy
Fresnel zone for offset and dipping reflectors
Balanced AVO in areas of poor geometry
Caution: model based migration
One hoarsepower
What is the basic unit of laryngitis?

8. What size hole can we see with seismic?

9. What happens at the edge of a reflector?

10. Fresnel zone
Fresnel Size:
frequency
depth
velocity
Claerbout

11. Depends on the size of the Fresnel radius
Only for stacked data!!!

12. Analytic function for zero-offset case (no wavelet)
Solution to the acoustic wave equation for zero-offset, the diffraction response can be written as:
Diffraction operator D0:
te : minimum two-way travel time to the edge of the reflector;
t’=t-te: time measured after onset time te ;
e : angle between the normal to the reflector and raypath of the
minimum travel time to the edge of the reflector.
Berryhill, 1977

13. Diffraction amplitude
Time
Angle

14. Diffraction amplitude

15. Scaled amplitude (evaluate phase)

16. Fresnel zone with Ricker wavelet source
Source signature with dominant frequency 50hz
Reflected signal amplitude

17. Fresnel zone with fixed bandwidth
Frequency from hz
Frequency from hz

18. Reflector smaller than Fresnel zone: width 50m

19. Reflector width: 1000m

20. Amplitude (no wavelet)

21. Diffraction modelling

22. Valid area for AVO ?

23. Comments
Locations for AVO analysis before migration should be at least a Fresnel radius away from the target edge.
The amplitudes will be in error if the target size is smaller than or equal to the Fresnel zone.
Wavelet will contribute to the size of contamination area.
Only zero offset was considered
Should consider a prestack migration to perform AVO analysis.

24. Offset traveltime 2h
DSR eqn.

25. Cheop’s pyramid - DSR eqn.
Hyperbola
CMP at T0
Zero off.
DSR eqn.
Const. Off
CMP gath.

26. Prestack data, flat… 7 Cheop’s .
Specula energy

27. Reflections from dipping event
Reflecting element

28. Prestack data, dipping... Cheop’s p.
Defines exact shape of surface,
specula energy

29. Cheop’s summation: specula energy
Fresnel zones

30. Prestack Fresnel zone horizontal layers

31. P. S. Fresnel zone for dipping event

32. Contours : equal angle of incidence
Offset increases with x
Angle 1

33. Constant offset migration

34. What about dipping specula energy?
Smear of dips
Still OK ???

35. CMP gather

36. Limited aperture EO gather

37. Comments … Prestack specula energy can be identified, Limited aperture
horizontal and dipping
Limited aperture
faster, better SNR, may preserve amplitudes
Optimum size of a super-CMP gather can be defined using Fresnel zones
Should use prestack migration gathers for AVO
Use model based migration with caution

38. Comments Constant offset migration EOM
smears AVO energy (OK)
requires velocity information
EOM
also smears AVO energy (OK)
Stacking within the Fresnel zone may reduce acquisition geometry artifacts

39. What is one millionth of a mouth wash?
Pavan Elapavuluri Q
What is one millionth of a mouth wash?
One microscope

40. Wavelets with varying Q and Time
Build Q wavelet table: [Q, time]
Q: 30, 50, 70, 90, 110, …
T: 0 to 6.0 sec
Cross-correlate (coherence) trace with all wavelets
At a given time, peak corresponds to Q

41. 9 wavelets
T=1000
T=3000
T=5000
Q=90
Q=70
Q=50

42. 3 Spikes on trace Q = 70
T=1000
T=3000
T=5000
Q=90
Q=70
Q=50

43. Noise on trace Q = 70
T=1000
T=3000
T=5000
Q=90
Q=70
Q=50

44. Kun Liu (and Hugh Geiger)
Wavefield extrapolation
What is 2000 pounds of Chinese soup?
Won ton

45. Input

46. 200m depth extrapolation …with adaptive taper

47. What is half a large intestine?
Xiang Du
Finite element method integrated with
Finite difference migration
A semicolon
What is half a large intestine?

48. Comparison: FE-FDM with FX-FDM
Result of FE-FDM
Result of FX-FDM (SU)

49. Trace comparisons

50. Steep oblique model
Velocity model
Seismic section

51. Comparison of methods
Result of FE-FDM
Result of FX-FDM (SU)

52. Runtime and memory requirements
FE-FD 26 secs 2.2 meg
FK-FD 31 secs 1.3 meg.

53. What is one thousand aches?
Zhihong (Nancy) Cao
Chunyan (Mary) Xiao
Tutorials
Review of Multiple suppression techniques
Multiple attenuation using the Radon transform
What is one thousand aches?
A kilohurtz

54. after Kabir and Marfurt, 1999

55. Semblance plot is a Radon transform

56. Lorraine Bloom
David Henley
Larry Lines
The model
and inverse

57. Modelling
process
Inversion expert

58. The
Model
(Not the inverse)

59. The
Inverse

60. The artifacts (of inversion)
Before modelling
After inversion

61. Of course, …
some things are easier to model and invert.

62. Inversion Mathematical process Geophysical terminology
Used in many geophysical applications:
decon., statics, tomography, tau-p, …
Geophysical terminology
Estimation of rock properties
Modelling <-> Inversion

63. What is the ratio of an igloo’s circumference to its diameter?
Millar time
Multigrid inversion
Borrowed from fluid dynamics
Recursive
Variable grid size
Solutions are frequency dependent
Fast
What is the ratio of an igloo’s circumference to its diameter?
Eskimo pi

65. Iterative solution

66. Laplace’s equation

67. Variable grid size

68. Analogous to FFT ???

69. Sorting a 2-D array into a vector

71. Extractions, smoothings and interpolations
S = Solution with one iterative step
33x33
17x17
3x3
9x9
5x5
3x3
9x9
5x5
Pass defects down and errors up

72. Performance

73. Conclusions: Migration apertures speed, SNR, balanced amplitudes
CMP AVO: consider Fresnel zones
Finite element / Finite difference migration
Multiples and Radon transforms
Q and differential operators
Multi grid inversions

74. The end