1. Making CMP’s
From chapter 16 “Elements of 3D Seismology” by Chris Liner

2. Outline Convolution and Deconvolution Normal Moveout Dip Moveout
Stacking

3. Outline Convolution and Deconvolution Normal Moveout Dip Moveout
Stacking

4. Convolution means several things:
IS multiplication of a polynomial series
IS a mathematical process
IS filtering

5. Convolution means several things:
IS multiplication of a polynomial series
A * B = C
E.g., A= ]; B = [ ];
C = [ ]

6. Convolutional Model for the Earth
output
input
Reflections in the earth are viewed as equivalent to a convolution process between the earth and the input seismic wavelet.

7. Convolutional Model for the Earth
output
input
SOURCE * Reflection Coefficient = DATA
(input) (earth) (output)
where * stands for convolution

8. Convolutional Model for the Earth
SOURCE * Reflection Coefficient = DATA
(input) (earth) (output)
where * stands for convolution
(MORE REALISTIC)
SOURCE * Reflection Coefficient + noise = DATA
(input) (earth) (output)
s(t) * e(t) n(t) = d(t)

9. s(f,phase) x e(f,phase) + n(f,phase) = d(f,phase)
Convolution in the TIME domain is equivalent to MULTIPLICATION in the FREQUENCY domain
s(t) * e(t) n(t) = d(t)
FFT
FFT
FFT
s(f,phase) x e(f,phase) + n(f,phase) = d(f,phase)
Inverse FFT
d(t)

10. CONVOLUTION as a mathematical operator
signal
has 3 terms (j=3) -1 2
-1/2
earth
Reflection Coefficient
has 4 terms (k=4)
1/4
1/4
1/2
time z
1/2
-1/4
3/4
-1/4
3/4
Reflection Coefficients with depth (m)

11. -1/2 2 1
1/4
1/2
-1/4
3/4 x = +

12. -1/2 2 -1
1/4
1/2
-1/4
3/4 x = +

13. -1/2 2 1
1/4
1/2
-1/4
3/4 x = +

14. -1/2 2 1
1/4
1/4
1/2
-1/4
3/4 x = +

15. -1/2 2 1
1/2 1
1/4
1/2
-1/4
3/4 x = +

16. -1/8 1
-1/4
5/8 x =
1/4
1/2
-1/4
3/4
-1/2 2 1 +

17. -1/4
-1/2
3/4 x =
1/4
1/2
-1/4
3/4
-1/2 2 1 +

18. 1/8
1 1/2
1 5/8 x =
1/4
1/2
-1/4
3/4
-1/2 2 1 +

19. -3/8 x =
1/4
1/2
-1/4
3/4 +
-1/2 2 1

20. x =
1/4
1/2
-1/4
3/4 + -1 2
-1/2

21. MATLAB %convolution a = [0.25 0.5 -0.25 0.75]; b = [1 2 -0.5];
c = conv(a,b)
d = deconv(c,a)
c =
matlab

22. Outline Convolution and Deconvolution Normal Moveout Dip Moveout
Stacking

23. Normal Moveout
Hyperbola: x T

24. Normal Moveout “Overcorrected” Normal Moveout is too largex T
“Overcorrected”
Normal Moveout is too large
Chosen velocity for NMO is too
(a) large (b) small

25. Normal Moveout “Overcorrected” Normal Moveout is too largex T
“Overcorrected”
Normal Moveout is too large
Chosen velocity for NMO is too
(a) large (b) small

26. Normal Moveout “Under corrected” Normal Moveout is too smallx T
“Under corrected”
Normal Moveout is too small
Chosen velocity for NMO is
(a) too large
(b) too small

27. Normal Moveout “Under corrected” Normal Moveout is too smallx T
“Under corrected”
Normal Moveout is too small
Chosen velocity for NMO is
(a) too large
(b) too small

28. Vinterval from Vrms
Dix, 1955

29. Vrms V1 V2
Vrms < Vinterval V3

30. Vinterval from Vrms

31. Primary seismic eventsx T

32. Primary seismic eventsx T

33. Primary seismic eventsx T

34. Primary seismic eventsx T

35. Multiples and Primariesx M1 T M2

36. Conventional NMO before stackingx M1
NMO correction
V=V(depth)
e.g., V=mz + B T M2
“Properly corrected”
Normal Moveout is just right
Chosen velocity for NMO is correct

37. Over-correction (e.g. 80% Vnmo)x x M1 M1
NMO correction
V=V(depth)
e.g., V=0.8(mz + B) T T M2 M2

38. f-k filtering before stacking (Ryu)x x M1
NMO correction
V=V(depth)
e.g., V=0.8(mz + B) T T M2 M2

39. Correct back to 100% NMO x x M1 M1 NMO correction V=V(depth)
e.g., V=(mz + B) T T M2 M2

40. Outline Convolution and Deconvolution Normal Moveout Dip Moveout
Stacking

41. Outline Convolution and Deconvolution Normal Moveout Dip Moveout
Stacking

42. How do we move out a dipping reflector in our data set?
Dip Moveout (DMO)
(Ch. 19; p )
How do we move out a dipping reflector in our data set? m
Offset (m)
TWTT (s) z

43. For a dipping reflector:
Dip Moveout
A dipping reflector:
appears to be faster
its apex may not be centered
Offset (m)
For a dipping reflector:
Vapparent = V/cos dip
TWTT (s)
e.g., V=2600 m/s
Dip=45 degrees,
Vapparent = 3675m/s

44. CONFLICTING DIPS Different dips CAN NOT
be NMO’d correctly at the same time
Offset (m)
TWTT (s)
3675 m/s
2600 m/s
Vrms for dipping
reflector too low &
overcorrects
Vrms for dipping
reflector is correct but
undercorrects horizontal reflector

45. DMO Theoretical Background (Yilmaz, p.335)
(Levin,1971)
is layer dip
“NMO”

46. DMO Theoretical Background (Yilmaz, p.335)
(Levin,1971)
“DMO”

47. Three properties of DMO
“NMO”
“DMO”
(1) DMO effect at 0 offset = ?
(2) As the dip increases DMO (a) increases (B) decreases
(3) As velocity increases DMO (a) increases (B) decreases

48. Three properties of DMO
“NMO”
“DMO”
(1) DMO effect at 0 offset = 0
(2) As the dip increases DMO (a) increases (B) decreases
(3) As velocity increases DMO (a) increases (B) decreases

49. aka “Pre-stack partical migration”
Application of DMO
aka “Pre-stack partical migration”
(1) DMO after NMO (applied to CDP/CMP data)
but before stacking
DMO is applied to Common-Offset Data
Is equivalent to migration of stacked data
Works best if velocity is constant

50. DMO Implementation before stack -I
Offset (m)
(1) NMO using background Vrms
TWTT (s)

51. DMO Implementation before stack -II
Reorder as COS data -II
Offset (m)
TWTT (s)
NMO (s)

52. DMO Implementation before stack -III
f-k COS data -II
X is fixed k
NMO (s) f
NMO (s)

53. f-k COS data -II
X is fixed k
NMO (s) f
NMO (s)

54. f-k COS data -II
X is fixed k
NMO (s) f
NMO (s)

55. Outline Convolution and Deconvolution Normal Moveout Dip Moveout
Stacking

56. NMO stretching T0 V1 V2
“NMO Stretching”

57. NMO stretching V1 T0 V2
“NMO Stretching”
V1<V2

58. NMO stretching V1 V1<V2 NMO “stretch” = “linear strain” V2
Linear strain (%) = final length-original length
original length
X 100 (%)

59. NMO stretching original length = final length = V1 V1<V2 V2
X 100 (%)
X 100 (%)

60. “function of function rule”
NMO stretching
X 100 (%)
Assuming, V1=V2:
X 100 (%)
Where,
“function of function rule”

61. NMO stretching
So that…

62. stretching for T=2s,V1=V2=1500 m/s
Green line assumes
V1=V2
Blue line is for general case,
where V1, V2 can be different
and delT0=0.1s (this case: V1=V2)
Matlab code
X 100 (%)

63. Stacking + + =

64. Stacking improves S/N ratio+ + =

65. Semblance Analysis X + + =
Twtt (s)
“Semblance”

66. Semblance Analysis X V + + = V1 V2
Twtt (s) V3
Peak energy