making cmp’s
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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
E.g., A= 0.25 + 0.5 -0.25 0.75]; B = [1 2 -0.5];
0 1 2 30.25 0.5 0.25 0.75A z z z z 0 1 22 0.5B z z z
A * B = C
C = [0.2500 1.0000 0.6250 0 1.6250 -0.3750]
Convolutional Model for the Earthinput
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 Earthinput
output
SOURCE * Reflection Coefficient = DATA(input) (earth)
(output)
where * stands for convolution
Convolutional Model for the Earth
(MORE REALISTIC)
SOURCE * Reflection Coefficient = DATA(input) (earth)
(output)
where * stands for convolution
SOURCE * Reflection Coefficient + noise = DATA(input) (earth)
(output)
s(t) * e(t) + n(t) = d(t)
Convolution Convolution in the TIME TIME domain is equivalent to MULTIPLICATIONMULTIPLICATION in in
the FREQUENCYFREQUENCY domain
s(t) * e(t) + n(t) = d(t)
s(f,phase) x e(f,phase) + n(f,phase) = d(f,phase)
FFT FFT FFT
Inverse FFT
d(t)
CONVOLUTION as a mathematical operator
0
j
j j k kkD s e
2
-1/2
-1
z
Reflection Coefficients with depth (m)
1/4
1/2
-1/4
3/4
1/41/2
-1/43/4
Reflection Coefficient
signalsignalhas 3 terms (j=3)has 3 terms (j=3)
earthearth has 4 terms (k=4)has 4 terms (k=4)
time
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
x
x
x
x
x
=
=
=
=
=
0
0
0
0
0
0
+
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
x
x
x
x
x
=
=
=
=
=
=
0
0
0
0
0
0
0
+
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
x
x
x
x
x
x
x
=
=
=
=
=
=
=
0
0
0
0
0
0
0
0
+
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
x
x
x
x
x
x
x
x
=
=
=
=
=
=
=
=
0
0
0
1/4
0
0
0
0
1/4
+
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
x
x
x
x
x
x
x
x
x
=
=
=
=
=
=
=
=
=
0
0
0
1/2
1/2
0
0
0
0
1
+
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
x
x
x
x
x
x
x
x
x
x
=
=
=
=
=
=
=
=
=
=
0
0
0
-1/8
1
-1/4
0
0
0
0
5/8
+
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
x
x
x
x
x
x
x
x
x
x
=
=
=
=
=
=
=
=
=
=
0
0
0
0
-1/4
-1/2
3/4
0
0
0
0
+
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
x
x
x
x
x
x
x
x
=
=
=
=
=
=
=
=
0
0
0
1/8
1 1/2
0
0
0
1 5/8
+
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
x
x
x
x
x
x
x
=
=
=
=
=
=
=
0
0
0
-3/8
0
0
0
-3/8
+
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
x
x
x
x
x
x
=
=
=
=
=
=
0
0
0
0
0
0
0
+
c = 0.2500 1.0000 0.6250 0 1.6250 -0.3750
%convolutiona = [0.25 0.5 -0.25 0.75]; b = [1 2 -0.5];c = conv(a,b)d = deconv(c,a)
2 3 40.25 0.5 0.25 0.75a z z z z
MATLAB
2 32 5b z z z
matlab
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
Normal Moveout
22 2
0 2
xT T
V
22
0 0 02( ) ( )
xT x T x T T T
V
x
T
Hyperbola:
Normal Moveoutx
T
“Overcorrected”
Normal Moveout is too large
Chosen velocity for NMO is too
(a) large (b) small
Normal Moveoutx
T
“Overcorrected”
Normal Moveout is too large
Chosen velocity for NMO is too
(a) large (b) smallsmall
Normal Moveoutx
T
“Under corrected”
Normal Moveout is too small
Chosen velocity for NMO is
(a) too large
(b) too small
Normal Moveoutx
T
“Under corrected”Normal Moveout is too small
Chosen velocity for NMO is
(a) too largetoo large
(b) too small
Vinterval from Vrms
122 2
1 1interval
1
n n n n
n n
V t V tV
t t
Dix, 1955
2i i
RMSi
V tV
t
Vrms
V1
V2
V3
Vrms < Vinterval
Vinterval from Vrms
Vrms T Vinterval from Vrms ViViT VRMS from V interval1500 0 01500 0.2 1500 450000 15002000 1 2106.537443 4000000 20003000 2 3741.657387 18000000 3000
SUM 3.2 22450000
Primary seismic eventsx
T
x
T
Primary seismic events
x
T
Primary seismic events
x
T
Primary seismic events
Multiples and Primariesx
TM1
M2
Conventional NMO before stackingx
TNMO correction
V=V(depth)
e.g., V=mz + B
M1
M2
“Properly corrected”
Normal Moveout is just right Chosen velocity for NMO is correct
Over-correction (e.g. 80% Vnmo)
x
TNMO correction
V=V(depth)
e.g., V=0.8(mz + B)
M1
M2
x
TM1
M2
f-k filtering before stacking (Ryu)
x
TNMO correction
V=V(depth)
e.g., V=0.8(mz + B)
M1
M2
x
T
M2
Correct back to 100% NMO
x
TNMO correction
V=V(depth)
e.g., V=(mz + B)
M1
M2
x
TM1
M2
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
Dip Moveout (DMO)
How do we move out a dipping reflector in our data set?
z
m Offset (m)
TWTT (s)
(Ch. 19; p.365-375)
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
Offset (m)
TWTT (s)
Vrms for dipping reflector too low &
overcorrects
Vrms for dipping reflector is correct but
undercorrects horizontal reflector
3675 m/s
2600 m/s
CONFLICTING DIPS Different dips CAN NOT
be NMO’d correctly at the same time
DMO Theoretical Background (Yilmaz, p.335)
2 22 2
0 2
cos( )
xT x T
V
(Levin,1971)
22 2 2
0 2( ) (1 sin )
xT x T
V
2 2sin cos 1
2 22 2 2
0 2 2( ) sin
x xT x T
V V
“NMO”
is layer dip
DMO Theoretical Background (Yilmaz, p.335)
2 22 2
0 2
cos( )
xT x T
V
(Levin,1971)
22 2 2
0 2( ) (1 sin )
xT x T
V
2 2sin cos 1
2 22 2 2
0 2 2( ) sin
x xT x T
V V
“DMO”
2 22 2 2
0 2 2( ) sin
x xT x T
V V
“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
Three properties of DMO
2 22 2 2
0 2 2( ) sin
x xT x T
V V
“DMO”“NMO”
(1) DMO effect at 0 offset = 00
(2) As the dip increasesincreases DMO (a) increasesincreases (B) decreases
(3) As velocity increasesincreases DMO (a) increases (B) decreasesdecreases
Three properties of DMO
Application of DMOaka “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
2 22 2 2
0 2 2( ) sin
x xT x T
V V
(1) NMO using
background Vrms
Offset (m)
TW
TT (
s)
22 2 2
0 2( ) sin
xT x T
V
Reorder as COS data -II
2 22 2 2
0 2 2( ) sin
x xT x T
V V
Offset (m)
TW
TT (
s)
2 22 2 2
02 2( ) sin
x xT x T
V V
NM
O (
s)
DMO Implementation before stack -II
f-k COS data -II
NM
O (
s)
X is fixed
f
k
NM
O (
s)
DMO Implementation before stack -III
f-k COS data -II N
MO
(s)
X is fixed
f
k
NM
O (
s)
f-k COS data -II N
MO
(s)
X is fixed
f
k
NM
O (
s)
Outline
•Convolution and Deconvolution
•Normal Moveout
•Dip Moveout
•Stacking
NMO stretching
V1
V2
T0
“NMO Stretching”
NMO stretching
V1
V2
T0
“NMO Stretching”
V1<V2
NMO stretching
V1
V2
V1<V2
0 0T T0T 1T
1 1T TNMO “stretch” = “linear strain”
Linear strain (%) = final length-original length
original length
X 100 (%)
NMO stretching
V1
V2
V1<V2
0 0T T0T 1T
1 1T T
X 100 (%)
original length = 1T final length = 0T
NMO “stretch” = 0 1
1
T TT
X 100 (%)0
1
1TT
0T
NMO stretching
X 100 (%)0
1
1TT
220 2
0 0 0
( )x
d TdT TVdT dT T
12 22
0 0 2
12
2x
T TV
12 22
0 0 2
xT T
V
12 2
2 20
1 1x
T V
X 100 (%)
Where,
“function of function rule”
Assuming, V1=V2:
NMO stretching
12 22
0 20
0
xT
VdTdT T
12 2
2 20
1x
T V
So that…
X 100 (%)0
1
1TT
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
Stacking
+ + =
+ + =
Stacking improves S/N ratio
+ =
Semblance Analysis
22
1 1 2
22
1 1 2
22
1 1 2
“Semblance”
+
22
3 33
2 2 2
X
Tw
tt (
s)
+ =
Semblance Analysis
+
X
Tw
tt (
s)
V3
V1
V2
V
Peak energy
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