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Angle-domain Wave-equation Reflection Traveltime Inversion
Sanzong Zhang, Yi Luo and Gerard Schuster
(1) KAUST, (2) Aramco
1 12
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Outline
Introduction Theory and method Numerical examples Conclusions
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Outline
Introduction Theory and method Numerical examples Conclusions
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Velocity Inversion Methods
Data space
Image space
Ray-based tomography
Full Waveform inversion
Ray-based MVA
Wave-equ. MVA
Inversion
(Tomography)
(MVA)
Wave-equ. Reflection traveltime inversion
Wave-equ. Reflection traveltime inversion
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Problem
-2
e =
Pred. data β Obs. data Model Parameter
π
βπ
βπ
The waveform (image) residual is highly nonlinear with respect to velocity change.
The traveltime misfit function enjoys a somewhat linear relationship with velocity change.
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Angle-domain Wave-equation Reflection Traveltime InversionTraveltime inversion without high-frequency approximation Misfit function somewhat linear with respect to velocity perturbation.Wave-equation inversion less sensitive to amplitude Multi-arrival traveltime inversionBeam-based reflection traveltime inversion
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Outline
Introduction Theory and method Numerical examples Conclusions
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Wave-equation TransmissionTraveltime Inversion
1). Observed data 5 0
Time (s)
4). Smear time delay along wavepath
2). Calculated data 0
Time (s)
5
πππππ
-1.5 1.5 0 Lag time (s)
3). πππππ βπ
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Angle-domain Wave-equationReflection Traveltime Inversion
Suboffset-domain crosscorrelation function : )
ππ :πππππ€πππππππππππ‘ππ πππ‘π
: : time shift
gs
xx-h x+h
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Angle-domain CIG decomposition (slant stack ):
π (π₯ , π§ ,π ,π )=β« π (π₯ ,π§+h tanπ , h ,π|π± π ) hπangle-domain suboffset-domain
Angle-domain crosscorrelation function :
)
Angle-domain Crosscorrelation
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Angle-domain Crosscorrelation: physical meaning
)
Angle-domain crosscorrelation is the crosscorrelationbetween downgoing and upgoing beams with a certain angle. The time delay for multi-arrivals is available in angle-domain crosscorrelation function .
π₯
π§
π)
Local plane wave
ππ₯
π§
) Local plane wave
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Angle-domain Wave-equation Reflection Traveltime Inversion
Objective function: π=12βπ± βπ½ [βπ (π± ,π)]π
Velocity update: (x)= (x) + (x)
Gradient function:πΎπ( x )= β π π
ππ (π± )= ββπ±βπ½
βπ π(βπ)ππ (π± )
Traveltime wavepath
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Traveltime Wavepath
π (π₯ , π§ ,π , βπ )= maxβπ<π <π
π (π₯ , π§ ,π ,π )
π (π₯ , π§ ,π , βπ )=m ππβπ<π <π
π (π₯ ,π§ ,π ,π )
Angle-domain time delay
οΏ½ΜοΏ½ β π=π π (π₯ , π§ ,π ,π)
ππ |π=βπ
=0
Angle-domain connective function
Traveltime wavepath π(βπ)ππ (π₯)=β
π π βπ
ππ (π₯) /π οΏ½ΜοΏ½ βπ
π (βπ )
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Transforming CSG Data Xwell Trans. Data
= +reflection transmission transmission
Src-side Xwell Data
Redatuming data
source
Redatuming source
Observed data Rec-side Xwell Data
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Forward propagate source to trial image points and get downgoing beams
Backward propagate observed reflection data from geophonses to trial image points , and get upgoing beams
Crosscorrelate downgoing beam and upgoing beam, and pick angle-domain time delay
Workflow
βπ
π π½ Smear time dealy along wavepath to
update velocity model
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Introduction Theory and method Numerical examples Simple Salt Model Sigsbee Salt Model Conclusions
Outline
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Simple Salt Model
04
0
8
(a) True velocity model
x (km)
z (k
m)
0 8 x (km)
0
5
(b) CSG
t (s
)1
5
V(km/s)
04
0
8
(c) Initial Velocity Model
x (km)
z (k
m)
04
0
8
(d) RTM image
x (km)
z (k
m)
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βπ
ππ½
04
0
8
(a) Initial Velocity Model
x (km)
z (k
m)
Angle-domain Crosscorrelation(b) Angle-domain Crosscorrelation
(c) Angle-domain Crosscorrelation
βπ=πΌ( tan π)2
βπ :πΌ :π :
time delay
curvature reflection angle
βπ
π½π
π (π§ ,π , βπ )
π (π₯ , π§ ,π ,βπ )
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Inversion Result
04
0
8
(a) Initial velocity model
x (km)
z (k
m)
0
4
(b) Inverted velocity model
z (k
m)
0 8 x (km)
1
5
Velocity(km/s)
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Inversion Result
0
4
(b) RTM image
z (k
m)
0 8 x (km)
04
0
8
(a) RTM image
x (km)
z (k
m)
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Introduction Theory and method Numerical examples Simple Salt Model Sigsbee Salt Model Conclusions
Outline
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Sigsbee Model
Vinitial = 0.85 Vtrue
0
60 12
z(km
)
x(km)
0
60 12
z(km
)
x(km)
1.5
4.5
Velocity (km/s)
(a) True velocity model (b) Initial velocity model
0
60 12
z(km
)
x(km)
(c) RTM image
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Initial Velocity Model0
60 12
z(km
)
x(km)
0
6-50Β° +50Β°
CIG
πΌ-0.04 0.04
z(km
)
Crosscorrelation
-50Β° +50Β°
0
6
z(km
)
βπ=πΌ( tan π)2
π π
Semblance
βπ(π )
-0.2
0.2
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Initial Velocity Model0
60 12
z(km
)
x(km)
0
6-50Β° +50Β°
CIG
πΌ-0.04 0.04
z(km
)
Crosscorrelation
-50Β° +50Β°
0
6
z(km
)
π
Semblance
π
βπ=πΌ( tan π)2
-0.2
0.2
βπ(π )
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Initial Velocity Model0
60 12
z(km
)
x(km)
0
6-50Β° +50Β°
CIG
πΌ-0.04 0.04
z(km
)
Crosscorrelation
-50Β° +50Β°
0
6
z(km
)
π π
βπ=πΌ( tan π)2
Semblance
βπ(π )
0.2
-0.2
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Inverted Velocity Model0
60 12
z(km
)
x(km)
0
6-50Β° +50Β°
CIG
πΌ-0.04 0.04
z(km
)
Crosscorrelation
-50Β° +50Β°
0
6
z(km
)Semblance
π π
βπ(π )
-0.2
0.2
βπ=πΌ( tan π)2
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Inverted Velocity Model0
60 12
z(km
)
x(km)
0
6-50Β° +50Β°
CIG
πΌ-0.04 0.04
z(km
)
Crosscorrelation
-50Β° +50Β°
0
6
z(km
)Semblance
π π
-0.2
0.2
βπ(π )
βπ=πΌ( tan π)2
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Inverted Velocity Model0
60 12
z(km
)
x(km)
0
6-50Β° +50Β°
CIG
πΌ-0.04 0.04
z(km
)
Crosscorrelation
-50Β° +50Β°
0
6
z(km
)Semblance
π π
βπ(π )
-0.2
0.2
βπ=πΌ( tan π)2
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RTM Image
0
6
0 12
z(km
)
x(km)
(a) RTM image using initial velocity
0
6
0 12
z(km
)
x(km)
(b) RTM image using inverted model
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Outline
Introduction Theory and method Numerical examples Conclusions
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Velocity Inversion Methods
Data space
Image space
Ray-based tomography
Full Wavform inversion
Ray-based MVA
Wave-equ. MVA
Inversion
(Tomography)
(MVA)
Wave-equ. traveltime inversion
Wave-equ. traveltime inversion
![Page 32: Angle-domain Wave-equation Reflection Traveltime Inversion](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816475550346895dd65aa8/html5/thumbnails/32.jpg)
Angle-domain Wave-equation Reflection Traveltime InversionTraveltime inversion without high-frequency approximation Misfit function somewhat linear with respect to velocity perturbation.Wave-equation inversion less sensitive to amplitude Multi-arrival traveltime inversionBeam-based reflection traveltime inversion
![Page 33: Angle-domain Wave-equation Reflection Traveltime Inversion](https://reader036.vdocuments.us/reader036/viewer/2022062310/56816475550346895dd65aa8/html5/thumbnails/33.jpg)
Thank you for your attention