Full Waveform Inversion algorithm using Common Scatter ... · Full Waveform Inversion algorithm...

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

11

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

22

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

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