AVO, migration apertures Fresnel zones, stacking, Q, anisotropy, and fast inversions

Transcription

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

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

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

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

5 Inverse FFT of (j!)

6 Derivative assumes: Maximum bandwidth Ideal interpolator Sinx/x operator Derivative of sinx/x d sin( x) x dx = cos x x sin x 2 x SincFn.m 6

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

8 What size hole can we see with seismic? 8

9 What happens at the edge of a reflector? 9

10 Fresnel zone Fresnel zone Fresnel Size: frequency depth velocity v k t R = 2 f Claerbout 10

11 Depends on the size of the Fresnel radius R = v k t 2 f 1 k 2 Only for stacked data!!! 11

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: p Diffraction operator D 0 : () t D () t U ( t ) n ( t) = f 0 t e t e D cosθ t d, arctan π ' ( t + t e ) dt 2 ' e e 0 ( t θ e ) = 2 ' : minimum two-way travel time to the edge of the reflector; ' ( + 2 e ) ' t t t t e sin θ e t =t-t e : time measured after onset time t e ; θ e : angle between the normal to the reflector and raypath of the minimum travel time to the edge of the reflector. Berryhill,

13 Diffraction amplitude Amplitude Time Angle 13

14 Diffraction amplitude 14

15 Scaled amplitude (evaluate phase) 15

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

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

18 Reflector smaller than Fresnel zone: width 50m 18

19 Reflector width: 1000m 19

20 Amplitude (no wavelet) 20

21 Diffraction modelling 21

22 Valid area for AVO? 22

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. 23

24 Offset traveltime CSP 2h 2h t z T ( ) ( ) 2 2 x+ h 0 T x h T = V 4 V DSR eqn. 24

25 Cheop s pyramid - DSR eqn. Offset h Hyperbola 0-2 CMP at T 0 Zero off. -1 Time t -2 DSR eqn. Const. Off CMP gath Displacement x

26 Prestack data, flat 7 Cheop s. h h x x t t Specula energy 26

27 Reflections from dipping event h CMP h Midpoint? h = 0 h = 0 # " Offset reflection moves up dip " Reflecting Scatterpoint element 27

28 Prestack data, dipping... Cheop s p. h h x x t Reflection t # Reflector Reflector " T = T x cosβ 0 2 VRMS Defines exact shape of surface, specula energy 28

29 Cheop s summation: specula energy h x t Fresnel zones 29

30 Prestack Fresnel zone horizontal layers S R Zero offset Offset Fresnel zone 30

31 P. S. Fresnel zone for dipping event S R Zero offset Fresnel zone Offset 31

32 Contours : equal angle of incidence Angle 2 Scatterpoint Angle 1 Offset increases with x Scatterpoint 32

33 Constant offset migration 33

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

35 CMP gather 35

36 Limited aperture EO gather 36

37 Comments 1. Prestack specula energy can be identified, horizontal and dipping 2. Limited aperture faster, better SNR, may preserve amplitudes 3. Optimum size of a super-cmp gather can be defined using Fresnel zones 4. Should use prestack migration gathers for AVO 5. Use model based migration with caution 37

38 Comments 6. Constant offset migration smears AVO energy (OK) requires velocity information 7. EOM also smears AVO energy (OK) 8. Stacking within the Fresnel zone may reduce acquisition geometry artifacts 38

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

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 40

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

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

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

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

45 Input 45

46 200m depth extrapolation with adaptive taper 46

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

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

49 Trace comparisons Amplitude (3) FKFD (2) FEDM (1) Source Time(s) 49

50 Steep oblique model Velocity model Seismic section 50

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

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

53 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 53

54 after Kabir and Marfurt,

55 Semblance plot is a Radon transform 55

56 Larry Lines Lorraine Bloom David Henley 56

57 Modelling process Inversion expert 57

58 The Model (Not the inverse) 58

59 The Inverse 59

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

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

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

63 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 63

64 ax + bx + cx = y lx + mx + nx = y px + qx + rx = y a b c x y i m n x = y p q r x y x= 1 A y ( T ) 1 x= A A A y T Ax = y 64

65 ax + bx + cx = y lx + mx + nx = y px + qx + rx = y x a + l + 0 = y + y + y bx + mx + qx + cx + nx + rx ( ) ( ) x b+ m+ q = y + y + y ax + lx + px + cx + nx + rx ( ) ( ) x l+ m+ n = y + y + y ax + lx + px + bx + mx + qx ( ) ( ) Iterative solution 65

66 Laplace s equation 66

67 Variable grid size 67

68 Analogous to FFT??? 68

69 Sorting a 2-D array into a vector 69

70 70

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

72 Performance 72

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 73

74 The end 74

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