Full Waveform Inversion algorithm using Common Scatter Points gathersusing Common Scatter Points gathers
Hassan Khaniani*, John C. Bancroft and Gary F. Margrave2011 CREWES Annual Sponsors Meeting
December,02,2011
1
Outline
• Review of Full Waveform InversionReview of Full Waveform Inversion• PSTM Full Waveform Inversion algorithm
l• Examples • Discussions• Conclusions
2
Review
• Nonlinear geophysical optimization process
• Born in mid 1980s due to pioneering work of p gLailly(1983) & Tarantola (1984,1986,1988), well developed in more than two decades
• Forward modeling tool including wave equation modeling
• Inverse problem can be solved with gradient based linear iterative methods
3
Theory of PSDM FWI0 4000
(m)
0
5003500
4000
dept
h (
10002500
3000
21
PSDM FWI
lateral position (m) 4600 4800 5000 5200 54001500 2000
fwI in
, 2
1min ( , )2
sx xu x tφ δ=
1
m)
0
5003500
4000fwI in
3
1( , ) ( ) ( ( , ))k t f t kbsk
x z dt P u x tv
γ δ= ∂ ∂
( ) ( ) ( )
Tarantola, 1984, Vigh, 2008
dept
h (m
10002500
30001( , ) ( , ) ( , )k k k kv x z v x z x zα γ+ = −
Sh d i l i ilateral position (m)
4600 4800 5000 5200 54001500 2000Shot records simulation using
solution of wave equationPSDM
4
R S
Prestack Time FWI pt
h (m
)
0
500
3000
3500
4000
me
(s)
0
0.5
3000
3500
4000
me
(s)
0
0.50.1
R
lateral position (m)
dep
4600 4800 5000 5200 5400
1000
1500 2000
2500
lateral position (m)
tim
4600 4800 5000 5200 5400
1
1.5 2000
2500
lateral position (m)
tim
4600 4800 5000 5200 5400
1
1.5 0
0.05Depth Time Rc
Forward Modeling
time
Forward Modeling
Migration/Inversion
DSR equation
Lateral position
X= Distance between scatterpoint and S/R CMP
h=Half S/R offset 5
CSP gather
CSP gather
Offset domain Equivalent Offset domain
τ
X= Distance between scatterpoint and S/R CMP
2 2 2 2
2 2
( X h ) ( X h )t( S ,G )4 v 4 v
τ τ+ −= + + +2
2 e2h( )v
,t S G τ +
=
X Distance between scatterpoint and S/R CMP
h=Half S/R offset
PSTM FWI Inversion
21 1( , ) ncR x Lτ ∂= − ( , )4 ( , )c x
v xτ
τ τ ∂
21 1 2 ( )v o vδ δ= − +near
izatio
n
2 2 3 ( )( ) ( , ) ( , )
o vv v v x v x
δδ τ τ
= ++
22 4hδ
lin
22
3
4( ', )( , , ) ( ) '.
( ', )
e
e
hv x tvu x h t K s d dh dx
v x
δ τδ τ τ
τ
= − = × ∗
( )γ τ7
( , )xγ τ
Gradient ComputationPSTM vs PSDM
True di
initial True gradientinitial Velocity gradientVelocityVelocitygradientVelocity
1z
1τ
dept
h
time
2z2τ
PSDM FWI PSTM FWIInnanen, 2011, Geophysical inversion II: seismic inversion:U of Calgary, unpublished course notes. 8
Algorithm of PSTM FWICalculate ( x, )γ τ
startingv ( x, )τ
Scan for optimum No
αrmsv ( x, )τ
Update v( x, )τcalculate small
enough?
u( x,t )δu( x,t )δ
( , )
?
yes
End
9
Synthetic Example
0 20 2
21 shot records
)
0.2
0.4
0.6
0.8
)
0.2
0.4
0.6
0.8
Tim
e (s 1
1.2
1.4
1 6
Tim
e (s 1
1.2
1.4
1 6
CMP0 500 1000 1500 2000 2500 3000 3500
1.6
1.8
2
1.6
1.8
2
true velocity in time initial velocity in time
10
Synthetic Example
0 20 2
)
0.2
0.4
0.6
0.8
)
0.2
0.4
0.6
0.8
Tim
e (s
1
1.2
1.4
1 6
Tim
e (s 1
1.2
1.4
1 6
CMP3500 4000 4500 5000 5500 6000 6500
1.6
1.8
2
1.6
1.8
2
true velocity in time updated velocityIteration 5
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Iteration 5
Synthetic Example
0 20 2
)
0.2
0.4
0.6
0.8
)
0.2
0.4
0.6
0.8
Tim
e (s
1
1.2
1.4
1 6
Tim
e (s 1
1.2
1.4
1 6
CMP3500 4000 4500 5000 5500 6000 6500
1.6
1.8
2
1.6
1.8
2
true velocity in time updated velocityIteration 15
12
Iteration 15
Synthetic Example
0 20 2
)
0.2
0.4
0.6
0.8
)
0.2
0.4
0.6
0.8
Tim
e (s
1
1.2
1.4
1 6
Tim
e (s 1
1.2
1.4
1 6
CMP3500 4000 4500 5000 5500 6000 6500
1.6
1.8
2
1.6
1.8
2
true velocity in time updated velocityIteration 25
13
Iteration 25
Synthetic Example
0 20 2
)
0.2
0.4
0.6
0.8
)
0.2
0.4
0.6
0.8
Tim
e (s
1
1.2
1.4
1 6
Tim
e (s 1
1.2
1.4
1 6
CMP3500 4000 4500 5000 5500 6000 6500
1.6
1.8
2
1.6
1.8
2
true velocity in time updated velocityIteration 30
14
Iteration 30
Synthetic Example
0 20 2
)
0.2
0.4
0.6
0.8
)
0.2
0.4
0.6
0.8
Tim
e (s
1
1.2
1.4
1 6
Tim
e (s 1
1.2
1.4
1 6
CMP3500 4000 4500 5000 5500 6000 6500
1.6
1.8
2
1.6
1.8
2
true velocity in time updated velocityIteration 35
15
Iteration 35
Synthetic Example
0 20 2
)
0.2
0.4
0.6
0.8
)
0.2
0.4
0.6
0.8
Tim
e (s
1
1.2
1.4
1 6
Tim
e (s 1
1.2
1.4
1 6
CMP3500 4000 4500 5000 5500 6000 6500
1.6
1.8
2
1.6
1.8
2
true velocity in time updated velocityIteration 40
16
Iteration 40
Synthetic Example
0 20 2
)
0.2
0.4
0.6
0.8)
0.2
0.4
0.6
0.8
Tim
e (s
)
1
1.2
1.4
Tim
e (s 1
1.2
1.4
1 6
CMP3500 4000 4500 5000 5500 6000 6500
1.6
1.8
2
1.6
1.8
2
true velocity in time updated velocityIteration 55
17
Iteration 55
Synthetic Example
0true velocity45 it ti d t d l it
0.5
45 iterations updated velocitystarting model
1
time
(s)
1.5
1500 2000 2500 3000 3500 4000 45002velocity (m/s)
true l itvelocity
18
E l 2Example 2Hussar current results
19
Hussar example
iteration# 0
initial velocitySW NWiteration# 0
0.2
0 4 4000
4500
5000SW NWTi
me
(s)
0.4
0.6
0.83000
3500
T
1
1.2
1500
2000
2500
500 1000 1500 2000 2500 3000
1.4
1000
1500
20
Hussar example
iteration# 1
updated velocitySW NWiteration# 1
0.2
0 4 4000
4500
5000SW NWTi
me
(s)
0.4
0.6
0.83000
3500
T
1
1.2
1500
2000
2500
CMP
500 1000 1500 2000 2500 3000
1.4
1000
1500
21
Hussar example
iteration# 8
updated velocitySW NWiteration# 8
0.2
0 4 4000
4500
5000SW NWTi
me
(s)
0.4
0.6
0.83000
3500
T
1
1.2
1500
2000
2500
CMP
500 1000 1500 2000 2500 3000
1.4
1000
1500
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Hussar example
updated velocityiteration# 70SW NW
0.2
0 4 4000
4500
5000iteration# 70
0.2
0 4 4000
4500
5000SW NWTi
me
(s)
0.4
0.6
0.83000
3500
Tim
e (s
)
0.4
0.6
0.83000
3500
T
1
1.2
1500
2000
2500
T
1
1.2
1500
2000
2500
CMP
500 1000 1500 2000 2500 3000
1.4
1000
1500
CMP
500 1000 1500 2000 2500 3000
1.4
1000
1500
23
Hussar example
iteration# 0 5000
iteration# 70 5000
me
(s)
0.2
0.4
0.6
0 83000
3500
4000
4500
me
(s)
0.2
0.4
0.6
0 83000
3500
4000
4500
Tim 0.8
1
1.2
1.41500
2000
2500Ti
m 0.8
1
1.2
1.41500
2000
2500
CMP
500 1000 1500 2000 2500 3000
1000
CMP
500 1000 1500 2000 2500 3000
1000
24
Hussar example
γ(x,t) sonic log 1435initial velocity
0.1 0.1
initial velocitysonic velocity
me
(s) 0.2
0 3
0.2
0 3
Tim 0.3
0.4
0.3
0.4
CMP500 1000 1500
0.52000 3000 4000
0.5
l it ( / )CMP velocity (m/s)
25Gredient function In time
Shot records comparison
real data
26
Shot records comparison
predicted data
27
Discussions
real data
28
Comments
• Converted waveConverted wave • Anisotropic modeling /inversion
29
Conclusions
Developed a PSTM FWI algorithms for velocityDeveloped a PSTM FWI algorithms for velocity inversion.Use CSP gathers for initial velocity modelUse CSP gathers for initial velocity model.Not accurate for complex structures because
f i i i i f d d iof using time migration forward and inverse process.Faster.
30
Acknowledgments
• CREWES Sponsors for supports• Dr. Kristopher Innanen• Naser Yousefzadeh• Marcus Wilson• Marcus Wilson• Ben Wards
THANK YOU !THANK YOU !31
hyperbola tilt
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