Download - Making CMP’s
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)
1/41/2-1/43/4
1/41/2
-1/43/4
Reflection Coefficient
signalsignal has 3 terms (j=3)has 3 terms (j=3)
earthearth has 4 terms (k=4)has 4 terms (k=4)
time
000
1/41/2-1/43/4000
000
-1/2210000
xxxxx
=====
000000
+
000
1/41/2-1/43/4000
000
-1/22-10000
xxxxx
======
0000000
+
000
1/41/2-1/43/4000
000
-1/2210000
xxxxxxx
=======
00000000
+
000
1/41/2-1/43/4000
000
-1/2210000
xxxxxxxx
========
000
1/40000
1/4
+
000
1/41/2-1/43/4000
000
-1/2210000
xxxxxxxxx
=========
000
1/21/200001
+
000
1/41/2-1/43/4000
000
-1/2210000
xxxxxxxxxx
==========
000
-1/81
-1/40000
5/8
+
000
1/41/2-1/43/4000
000
-1/2210000
xxxxxxxxxx
==========
0000
-1/4-1/23/40000
+
000
1/41/2-1/43/4000
000
-1/2210000
xxxxxxxx
========
000
1/81 1/2
000
1 5/8
+
000
1/41/2-1/43/4000
000
-1/2210000
xxxxxxx
=======
000
-3/8000
-3/8
+
000
1/41/2-1/43/4000
000-12
-1/20000
xxxxxx
======
0000000
+
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 TV
22
0 0 02( ) ( ) xT x T x T T TV
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 tVt 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
T NMO correctionV=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
T NMO correctionV=V(depth)
e.g., V=0.8(mz + B)
M1
M2
x
TM1
M2
f-k filtering before stacking (Ryu)x
T NMO correctionV=V(depth)
e.g., V=0.8(mz + B)
M1
M2
x
T
M2
Correct back to 100% NMOx
T NMO correctionV=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 MoveoutA 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 TV
(Levin,1971)
22 2 2
0 2( ) (1 sin )xT x TV
2 2sin cos 1
2 22 2 2
0 2 2( ) sinx xT x TV V
“NMO”
is layer dip
DMO Theoretical Background (Yilmaz, p.335)
2 22 2
0 2
cos( ) xT x TV
(Levin,1971)
22 2 2
0 2( ) (1 sin )xT x TV
2 2sin cos 1
2 22 2 2
0 2 2( ) sinx xT x TV V
“DMO”
2 22 2 2
0 2 2( ) sinx xT x TV 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( ) sinx xT x TV 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( ) sinx xT x TV V
(1) NMO using
background Vrms
Offset (m)
TWTT
(s)
22 2 2
0 2( ) sinxT x TV
Reorder as COS data -II
2 22 2 2
0 2 2( ) sinx xT x TV V
Offset (m)
TWTT
(s)
2 22 2 2
02 2( ) sinx xT x TV V
NMO
(s)
DMO Implementation before stack -II
f-k COS data -II
NMO
(s)
X is fixed
f
k
NMO
(s)
DMO Implementation before stack -III
f-k COS data -II NM
O (s
)
X is fixed
f
k
NMO
(s)
f-k COS data -II NM
O (s
)
X is fixed
f
k
NMO
(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
V2V1<V2
0 0T T0T 1T
1 1T TNMO “stretch” = “linear strain”
Linear strain (%) = final length-original length original length
X 100 (%)
NMO stretching
V1
V2V1<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
11T
T
0T
NMO stretching
X 100 (%)0
11T
T
220 2
0 0 0
( )xd TdT TVdT dT T
12 22
0 0 2122
xT TV
12 22
0 0 2xT TV
12 2
2 20
1 1xT V
X 100 (%)
Where,
“function of function rule”
Assuming, V1=V2:
NMO stretching1
2 220 2
0
0
xTVdT
dT T
12 2
2 20
1 xT V
So that…
X 100 (%)0
11T
T
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
221 1 2
221 1 2
221 1 2
“Semblance”
+
223 33
2 2 2
X
Twtt
( s)
+ =
Semblance Analysis
+
X
Twtt
( s)
V3
V1
V2
V
Peak energy