Making CMP

1
Making CMPs
From chapter 16 Elements of 3D Seismology by
Chris Liner
2
Outline

• Convolution and Deconvolution
• Normal Moveout
• Dip Moveout
• Stacking

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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

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Convolution means several things

• IS multiplication of a polynomial series

A B C
E.g., A 0.25 0.5 -0.25 0.75 B 1 2 -0.5
C 0.2500 1.0000 0.6250 0
1.6250 -0.3750
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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.
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Convolutional Model for the Earth
output
input
SOURCE Reflection Coefficient DATA
(input) (earth)
(output) where stands for convolution
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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)
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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)
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CONVOLUTION as a mathematical operator
signal
has 3 terms (j3)
-1
2
-1/2
earth
Reflection Coefficient
has 4 terms (k4)
1/4
1/4
time
1/2
z
1/2
-1/4
3/4
-1/4
3/4
Reflection Coefficients with depth (m)
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0 0 0 -1/2 2 1 0 0 0 0
0 0 0 0 0 0
x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0

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0 0 0 -1/2 2 -1 0 0 0 0
0 0 0 0 0 0 0
x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0

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0 0 0 -1/2 2 1 0 0 0 0
0 0 0 0 0 0 0 0
x x x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0

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0 0 0 -1/2 2 1 0 0 0 0
0 0 0 1/4 0 0 0 0 1/4
x x x x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0

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0 0 0 1/2 1/2 0 0 0 0 1
0 0 0 -1/2 2 1 0 0 0 0
x x x x x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0

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0 0 0 -1/8 1 -1/4 0 0 0 0 5/8
x x x x x x x x x x

0 0 0 -1/2 2 1 0 0 0 0
0 0 0 1/4 1/2 -1/4 3/4 0 0 0

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0 0 0 0 -1/4 -1/2 3/4 0 0 0 0
x x x x x x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0
0 0 0 -1/2 2 1 0 0 0 0

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0 0 0 1/8 1 1/2 0 0 0 1 5/8
x x x x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0
0 0 0 -1/2 2 1 0 0 0 0

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0 0 0 -3/8 0 0 0 -3/8
x x x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0

0 0 0 -1/2 2 1 0 0 0 0
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0 0 0 0 0 0 0
x x x x x x

0 0 0 1/4 1/2 -1/4 3/4 0 0 0

0 0 0 -1 2 -1/2 0 0 0 0
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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 0.2500 1.0000 0.6250 0
1.6250 -0.3750
matlab
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Outline

• Convolution and Deconvolution
• Normal Moveout
• Dip Moveout
• Stacking

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Normal Moveout
Hyperbola
x
T
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Normal Moveout
x
T
Overcorrected
Normal Moveout is too large
Chosen velocity for NMO is too (a) large (b)
small
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Normal Moveout
x
T
Overcorrected
Normal Moveout is too large
Chosen velocity for NMO is too (a) large (b)
small
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Normal Moveout
x
T
Under corrected
Normal Moveout is too small
Chosen velocity for NMO is (a) too large (b) too
small
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Normal Moveout
x
T
Under corrected
Normal Moveout is too small
Chosen velocity for NMO is (a) too large (b) too
small
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Vinterval from Vrms
Dix, 1955
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Vrms
V1
V2
Vrms lt Vinterval
V3
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Vinterval from Vrms
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Primary seismic events
x
T
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Primary seismic events
x
T
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Primary seismic events
x
T
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Primary seismic events
x
T
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Multiples and Primaries
x
M1
T
M2
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Conventional NMO before stacking
x
M1
NMO correction VV(depth) e.g., Vmz B
T
M2
Properly corrected
Normal Moveout is just right
Chosen velocity for NMO is correct
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Over-correction (e.g. 80 Vnmo)
x
x
M1
M1
NMO correction VV(depth) e.g., V0.8(mz B)
T
T
M2
M2
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f-k filtering before stacking (Ryu)
x
x
M1
NMO correction VV(depth) e.g., V0.8(mz B)
T
T
M2
M2
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Correct back to 100 NMO
x
x
M1
M1
NMO correction VV(depth) e.g., V(mz B)
T
T
M2
M2
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Outline

• Convolution and Deconvolution
• Normal Moveout
• Dip Moveout
• Stacking

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Outline

• Convolution and Deconvolution
• Normal Moveout
• Dip Moveout
• Stacking

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Dip Moveout (DMO)
(Ch. 19 p.365-375)
How do we move out a dipping reflector in our
data set?
m
Offset (m)
TWTT (s)
z
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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., V2600 m/s
Dip45 degrees, Vapparent 3675m/s
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CONFLICTING DIPS Different dips CAN NOT be NMOd
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
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DMO Theoretical Background (Yilmaz, p.335)
(Levin,1971)
is layer dip
NMO
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DMO Theoretical Background (Yilmaz, p.335)
(Levin,1971)
DMO
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Three properties of DMO
DMO
NMO
(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
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Three properties of DMO
DMO
NMO
(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
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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

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DMO Implementation before stack -I
Offset (m)
(1) NMO using background Vrms
TWTT (s)
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DMO Implementation before stack -II
Reorder as COS data -II
Offset (m)
TWTT (s)
NMO (s)
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DMO Implementation before stack -III
f-k COS data -II
X is fixed
k
NMO (s)
f
NMO (s)
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f-k COS data -II
X is fixed
k
NMO (s)
f
NMO (s)
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f-k COS data -II
X is fixed
k
NMO (s)
f
NMO (s)
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Outline

• Convolution and Deconvolution
• Normal Moveout
• Dip Moveout
• Stacking

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NMO stretching
T0
V1
V2
NMO Stretching
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NMO stretching
V1
T0
V2
NMO Stretching
V1ltV2
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NMO stretching
V1
V1ltV2
NMO stretch linear strain
V2
Linear strain () final length-original length
original length
X 100 ()
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NMO stretching
original length
final length
V1
V1ltV2
V2
X 100 ()
NMO stretch
X 100 ()
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NMO stretching
X 100 ()
Assuming, V1V2
X 100 ()
Where,
function of function rule
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NMO stretching
So that
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stretching for T2s,V1V21500 m/s
Green line assumes V1V2
Blue line is for general case, where V1, V2 can
be different and delT00.1s (this case V1V2)
Matlab code
X 100 ()
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Stacking



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Stacking improves S/N ratio



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Semblance Analysis
X



Twtt (s)
Semblance
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Semblance Analysis
X
V



V1
V2
Twtt (s)
V3
Peak energy