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Making CMP’s
From chapter 16 “Elements of 3D Seismology” by
Chris Liner
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
Convolution means several things:
•IS multiplication of a polynomial series
•IS a mathematical process
•IS filtering
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]
A  0.25z  0.5z  0.25z  0.75z
0
1
2
B  z  2 z  0.5z
0
1
2
3
Convolutional Model for the Earth
input
output
Reflections in the earth are viewed as equivalent to a
convolution process between the earth and the input
seismic wavelet.
Convolutional Model for the Earth
output
input
SOURCE * Reflection Coefficient = DATA
(input)
(earth)
(output)
where * stands for convolution
Convolutional Model for the Earth
SOURCE * Reflection Coefficient = DATA
(input)
(earth)
(output)
where * stands for convolution
(MORE REALISTIC)
SOURCE * Reflection Coefficient + noise = DATA
(input)
s(t)
(earth)
*
e(t)
(output)
+ n(t)
=
d(t)
Convolution in the TIME domain is
equivalent to MULTIPLICATION in
the FREQUENCY domain
s(t)
FFT
*
e(t)
+ n(t)
FFT
FFT
s(f,phase) x e(f,phase) + n(f,phase)
=
d(t)
= d(f,phase)
Inverse FFT
d(t)
CONVOLUTION as a mathematical
operator
signal
-1
has 3 terms (j=3)
D j  k0 s j k ek
j
2
-1/2
earth has 4 terms (k=4)
z
Reflection Coefficient
1/4
1/4
1/2
-1/4
3/4
Reflection Coefficients with depth (m)
-1/4
1/2
3/4
time
0
-1/2
2
0 x 1
=
0
0 x 0
=
0
0 x 0
=
0 +
1/4 x 0
=
0
1/2 x 0
=
0
-1/4
3/4
0
0
0
0
0
0
-1/2
0 x 2
=
0
0 x -1
=
0
0 x 0
=
0 +
1/4 x 0
=
0
1/2 x 0
=
0
0
=
0
-1/4
3/4
0
0
0
0
0
0
0
0 x -1/2 =
0
0 x 2
=
0
0 x 1
=
0 +
1/4 x 0
=
0
1/2 x 0
=
0
-1/4 x 0
=
0
3/4 x 0
=
0
0
0
0
0
0
0
0 x 0
=
0
0 x -1/2 =
0
0 x 2
0 +
=
1/4 x 1
= 1/4
1/2 x 0
=
0
-1/4 x 0
=
0
3/4 x 0
=
0
0 x 0
=
0
0
0
1/4
0
0 x 0
=
0
0 x 0
=
0
0 x -1/2 =
0 +
1/4 x 2
= 1/2
1/2 x 1
= 1/2
-1/4 x 0
=
0
3/4 x 0
=
0
0 x 0
=
0
0 x 0
=
0
0
1
0 x 0
=
0
0 x 0
=
0
0 x 0
=
0 +
1/4 x -1/2 = -1/8
1/2 x 2
=
1
-1/4 x 1
= -1/4
3/4 x 0
=
0
0 x 0
=
0
0 x 0
=
0
0 x 0
=
0
5/8
0 x
=
0
0 x 0
=
0
0 x 0
=
0 +
1/4 x 0
=
0
1/2 x -1/2 = -1/4
-1/4 x 2
= -1/2
3/4 x 1
= 3/4
0 x 0
=
0
0 x 0
=
0
0 x 0
=
0
0
0
0
0
0 x 0
=
0 +
1/4 x 0
=
0
1/2 x 0
=
0
-1/4 x -1/2 =
3/4 x 2
1/8
= 1 1/2
0 x 1
=
0
0 x 0
=
0
0 x 0
=
0
0
0
1 5/8
0
0
+
0
1/4 x 0
=
0
1/2 x 0
=
0
-1/4 x 0
=
0
3/4 x -1/2 = -3/8
0 x 2
=
0
0 x 1
=
0
0 x 0
=
0
0
0
-3/8
0
0
+
0
1/4
1/2 x 0
=
0
-1/4 x 0
=
0
3/4 x 0
=
0
0 x -1
=
0
0 x 2
=
0
0 x -1/2 =
0
0
0
0
MATLAB
%convolution
a = [0.25 0.5 -0.25 0.75]; b = [1 2 -0.5];
c = conv(a,b)
d = deconv(c,a)
a  0.25 z  0.5 z 2  0.25 z 3  0.75 z 4
b  z  2 z 2  5z3
c = 0.2500
1.0000
0.6250
0
1.6250 -0.3750
matlab
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
Normal Moveout
Hyperbola:
2
x
2
2
T  T0  2
V
x
T
2
x
T ( x)  T ( x)  T0  T02  2  T0
V
Normal Moveout
x
T
“Overcorrected”
Normal Moveout is too large
Chosen velocity for NMO is too
(a) large (b) small
Normal Moveout
x
T
“Overcorrected”
Normal Moveout is too large
Chosen velocity for NMO is too
(a) large (b) small
Normal Moveout
x
T
“Under corrected”
Normal Moveout is too small
Chosen velocity for NMO is
(a) too large
(b) too small
Normal Moveout
x
T
“Under corrected”
Normal Moveout is too small
Chosen velocity for NMO is
(a) too large
(b) too small
Vinterval from Vrms
VRMS 
2
V
  i ti 
t 
i
Vinterval
V t  V t

 tn  tn1
2
n n
Dix, 1955
2
n 1 n 1



1
2
Vrms
V1
V2
Vrms < Vinterval
V3
Vinterval from Vrms
Vrms
1500
1500
2000
3000
SUM
T
0
0.2
1
2
3.2
Vinterval from Vrms
1500
2106.537443
3741.657387
ViViT
VRMS from V interval
0
450000
1500
4000000
2000
18000000
3000
22450000
Primary seismic events
x
T
Primary seismic events
x
T
Primary seismic events
x
T
Primary seismic events
x
T
Multiples and Primaries
x
T
M1
M2
Conventional NMO before stacking
x
T
M1
NMO correction
V=V(depth)
e.g., V=mz + B
M2
“Properly corrected”
Normal Moveout is just right
Chosen velocity for NMO is correct
Over-correction (e.g. 80% Vnmo)
x
T
M1
NMO correction
V=V(depth)
M2
e.g., V=0.8(mz + B)
f-k filtering before stacking (Ryu)
x
T
M1
NMO correction
V=V(depth)
M2
e.g., V=0.8(mz + B)
Correct back to 100% NMO
x
T
M1
NMO correction
V=V(depth)
M2
e.g., V=(mz + B)
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
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
Dip Moveout
A dipping reflector:
• appears to be faster
•its apex may not be centered
Offset (m)
TWTT (s)
For a dipping reflector:
Vapparent = V/cos dip
e.g., V=2600 m/s
Dip=45 degrees,
Vapparent = 3675m/s
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
DMO Theoretical Background
(Yilmaz, p.335)
2
2
x
cos

2
2
T ( x)  T0 
V2

sin 2   cos 2   1
2
x
T 2 ( x)  T02  2 (1  sin 2  )
V
2
2
x
x
T 2 ( x)  T02  2  2 sin 2 
V
V
“NMO”
(Levin,1971)
is layer dip
DMO Theoretical Background
(Yilmaz, p.335)
2
2
x
cos

2
2
T ( x)  T0 
V2
sin 2   cos 2   1
2
x
T 2 ( x)  T02  2 (1  sin 2  )
V
2
2
x
x
T 2 ( x)  T02  2  2 sin 2 
V
V
“DMO”
(Levin,1971)
Three properties of DMO
2
2
x
x
T ( x)  T  2  2 sin 2 
V
V
2
2
0
“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
Three properties of DMO
2
2
x
x
T ( x)  T  2  2 sin 2 
V
V
2
2
0
“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
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
DMO Implementation before stack -I
TWTT (s)
Offset (m)
(1) NMO
using
background
Vrms
2
2
x
x
T 2 ( x)  T02  2  2 sin 2 
V
V
2
x
T 2 ( x)  T02  2 sin 2 
V
DMO Implementation before stack -II
Reorder as COS data -II
NMO (s)
TWTT (s)
Offset (m)
2
2
x
x
T 2 ( x)  T02  2  2 sin 2 
V
V
2
2
x
x
T 2 ( x)  2  T02  2 sin 2 
V
V
DMO Implementation before stack -III
f-k COS data -II
X is fixed
NMO (s)
k
NMO (s)
f
f-k COS data -II
X is fixed
NMO (s)
k
NMO (s)
f
f-k COS data -II
X is fixed
NMO (s)
k
NMO (s)
f
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
NMO stretching
T0
V1
V2
“NMO Stretching”
NMO stretching
T0
V1
V2
“NMO Stretching”
V1<V2
NMO stretching
T0
T1
T0  T0
V1<V2
V1
T1  T1
V2
NMO “stretch” = “linear strain”
Linear strain (%) = final length-original length
original length
X
100 (%)
NMO stretching
original length = T1
T0
T1
T0  T0
V1<V2
final length =
T0
V1
T1  T1
V2
T0  T1
T1
 T0 

 1
 T1 
NMO “stretch” =
X
100 (%)
X
100 (%)
T0
NMO stretching
 T0 

 1
 T1 
X
100 (%)
Assuming, V1=V2: 1

x2 2
  1  2 2   1 X 100 (%)
2
 T0 V 
x
d (T02  2 )
dT
T
V
Where,


dT0
dT0
T0
1

 2 x  2
 T0  T0  2 
V 
1


1  2 x2  2
 2T0  T0  2 
2
V 
2
“function of function rule”
NMO stretching
So that…
1
x2 2
 2
 T0  2 
V 
dT0 

dT
T0
1
2

x2
 1  2 2 
 T0 V 
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
 T0 

 1
 T1 
X
100 (%)
Stacking
+
+
=
Stacking improves S/N ratio
+
+
=
Semblance Analysis
X
+
+
=
1   1  2
2
2
1   1  2
2
2
“Semblance”
1   1  2
2
2
3   3 
2
222
2
3
Semblance Analysis
X
+
V
+
=
V1
V2
V3
Peak energy

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