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Angle-domain Wave-equation
Reflection Traveltime Inversion
1
2
Sanzong Zhang, Yi Luo and Gerard Schuster
(1) KAUST, (2) Aramco
1
Outline




Introduction
Theory and method
Numerical examples
Conclusions
Outline




Introduction
Theory and method
Numerical examples
Conclusions
Velocity Inversion Methods
Data space
(Tomography)
Ray-based tomography
Wave-equ. Reflection
traveltime inversion
Full Waveform inversion
Inversion
Image space
(MVA)
Ray-based MVA
Wave-equ. Reflection
traveltime inversion
Wave-equ. MVA
Problem
 The waveform (image) residual is highly nonlinear
with respect to velocity change.
2
e=
-
∆
Pred. data – Obs. data
∆
Model Parameter
 The traveltime misfit function enjoys a somewhat
linear relationship with velocity change.
Angle-domain Wave-equation
Reflection Traveltime Inversion
 Traveltime inversion without high-frequency
approximation
 Misfit function somewhat linear with respect
to velocity perturbation.
 Wave-equation inversion less sensitive to
amplitude
 Multi-arrival traveltime inversion
 Beam-based reflection traveltime inversion
Outline




Introduction
Theory and method
Numerical examples
Conclusions
Wave-equation Transmission
Traveltime Inversion
1). Observed data 
0
Time (s)
5
2). Calculated data 
0
Time (s)
5
3). 
-1.5

0
Lag time (s)

∆
1.5
4). Smear time delay ∆
along wavepath
Angle-domain Wave-equation
Reflection Traveltime Inversion
Suboffset-domain crosscorrelation function :
 , , ℎ,  =
x  ( − ℎ, ,  + |x ) ( + ℎ, , |x )
s
 ( − ℎ, , |x )
ℎ: 
: time shift
g
 ( + ℎ, , |x )
x-h x x+h
:   
:   
Angle-domain Crosscorrelation
 Angle-domain CIG decomposition (slant stack ):
 , , ,  =
angle-domain
 ,  + ℎ tan  , ℎ,   ℎ
suboffset-domain
 Angle-domain crosscorrelation function :
 , , ,  =
 
ℎ
  − ℎ,  + ℎ tan  ,  +   
 ( + ℎ,  + ℎ tan  , | ) 
Angle-domain Crosscorrelation:
physical meaning
 
  − ℎ,  + ℎ tan  ,   
 ( + ℎ,  + ℎ tan  , | ) 



ℎ

 ( − ℎ,  + ℎ tan  , |  )
Local plane wave

 , , ,  = 0 =

 ( + ℎ,  + ℎ tan  , |  )
Local plane wave
 Angle-domain crosscorrelation is the crosscorrelation
between downgoing and upgoing beams with a certain angle.
 The time delay for multi-arrivals is available in angle
-domain crosscorrelation function .
Angle-domain Wave-equation
Reflection Traveltime Inversion
Objective function:  = 12
∆(, )

Velocity update:


+1 (x)=  (x) +  ∙  (x)
Gradient function:

 (x)= −
=−
()


(∆)
∆
()
Traveltime wavepath
Traveltime Wavepath
 Angle-domain time delay ∆ 
 , , , ∆ = max  , , , 
−<<
 , , , ∆ = m  , , , 
−<<
 Angle-domain connective function

(, , , )
∆ =
=0

Traveltime wavepath =∆
(∆)
∆ ∆
=−
()
() (∆)
Transforming CSG Data  Xwell Trans. Data
reflection
=
Src-side Xwell Data
transmission
transmission
+
source
Redatuming data
Rec-side Xwell Data
Observed data
Redatuming source
Workflow
 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 angledomain time delay
 Smear time dealy along wavepath to
update velocity model
Outline
 Introduction
 Theory and method
 Numerical examples
Simple Salt Model
Sigsbee Salt Model
 Conclusions
Simple Salt Model
0
0
z (km)
t (s)
(a) True velocity model
0
x (km)
8
5
5
0
(c) Initial Velocity Model
8
0
1
z (km)
z (km)
0
4
x (km)
(d) RTM image
0
x (km)
8
4
0
V(km/s)
4
(b) CSG
x (km)
8
Angle-domain Crosscorrelation
(b) Angle-domain Crosscorrelation
(a) Initial Velocity Model
z (km)
0
4
0
x (km)
8
 , , ∆
(c) Angle-domain Crosscorrelation
∆ = (tan )2
∆
∆: time delay
 , , , ∆
:
curvature
:
reflection angle
Inversion Result
(a) Initial velocity model
z (km)
0
4
5
0
x (km)
(b) Inverted velocity model
Velocity(km/s)
8
0
z (km)
1
4
0
x (km)
8
Inversion Result
(a) RTM image
z (km)
0
4
0
x (km)
(b) RTM image
0
x (km)
8
z (km)
0
4
8
Outline
 Introduction
 Theory and method
 Numerical examples
Simple Salt Model
Sigsbee Salt Model
 Conclusions
Sigsbee Model
(b) Initial velocity model
z(km)
0
z(km)
0
(a) True velocity model
Vinitial = 0.85 Vtrue
6
0
x(km)
(c) RTM image
12
x(km)
4.5
0
Velocity (km/s)
z(km)
6
0
6
0
1.5
x(km)
12
12
Initial Velocity Model
z(km)
0
CIG
0
z(km)
0
x(km)
Semblance
6
-50°

+50°
6
-0.04
-0.2
∆()
0
z(km)
6

12
Crosscorrelation
∆ = (tan )2
0.2
0.04
-50°

+50°
Initial Velocity Model
z(km)
0
CIG
z(km)
0
x(km)
Semblance
12
Crosscorrelation
0
-0.2
∆()
0
z(km)
6
∆ = (tan )2
6
-50°

+50°
6
-0.04

0.2
0.04
-50°

+50°
Initial Velocity Model
z(km)
0
CIG
0
z(km)
0
x(km)
Semblance
6
-50°

+50°
6
-0.04
12
Crosscorrelation
-0.2
∆()
0
z(km)
6

∆ = (tan )2
0.2
0.04
-50°

+50°
Inverted Velocity Model
z(km)
0
CIG
0
z(km)
0
x(km)
Semblance
6
-50°

+50°
6
-0.04
12
Crosscorrelation
-0.2
∆()
0
z(km)
6

0.04
0.2
-50°
∆ = (tan )2

+50°
Inverted Velocity Model
z(km)
0
CIG
6
-50°
-0.2
0
z(km)
0

12
Crosscorrelation
x(km)
Semblance
+50°
6
-0.04
∆()
0
z(km)
6

0.04
0.2
-50°
∆ = (tan )2

+50°
Inverted Velocity Model
z(km)
0
CIG
z(km)
0
x(km)
Semblance
6
-50°

+50°
12
Crosscorrelation
0
-0.2
∆()
0
z(km)
6
6
-0.04

0.04
∆ = (tan )2
0.2
-50°

+50°
RTM Image
(b) RTM image using inverted model
0
z(km)
z(km)
(a) RTM image using initial velocity
0
6
0
6
x(km)
12
0
x(km)
12
Outline




Introduction
Theory and method
Numerical examples
Conclusions
Velocity Inversion Methods
Data space
(Tomography)
Ray-based tomography
Wave-equ. traveltime
inversion
Full Wavform inversion
Inversion
Image space
(MVA)
Ray-based MVA
Wave-equ. traveltime
inversion
Wave-equ. MVA
Angle-domain Wave-equation
Reflection Traveltime Inversion
 Traveltime inversion without high-frequency
approximation
 Misfit function somewhat linear with respect
to velocity perturbation.
 Wave-equation inversion less sensitive to
amplitude
 Multi-arrival traveltime inversion
 Beam-based reflection traveltime inversion
Thank you for your attention

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