Transcript PPT - KAUST
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=
-
∆𝜏
Pred. data – Obs. data
∆𝜏
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 𝑝𝑜𝑏𝑠
0
Time (s)
5
2). Calculated data 𝑝𝑐𝑎𝑙𝑐
0
Time (s)
5
3). 𝑝𝑜𝑏𝑠
-1.5
0
Lag time (s)
𝑝𝑐𝑎𝑙𝑐
∆𝜏
1.5
4). Smear time delay ∆𝜏
along wavepath
Angle-domain Wave-equation
Reflection Traveltime Inversion
Suboffset-domain crosscorrelation function :
𝑓 𝑥, 𝑧, ℎ, 𝜏 =
𝑑x𝑠 𝑝𝑓 (𝑥 − ℎ, 𝑧, 𝑡 + 𝜏|x𝑠 )𝑝𝑏 (𝑥 + ℎ, 𝑧, 𝑡|x𝑠 )𝑑𝑡
s
𝑝𝑓 (𝑥 − ℎ, 𝑧, 𝑡|x𝑠 )
ℎ: 𝑠𝑢𝑏𝑜𝑓𝑓𝑠𝑒𝑡
𝜏: time shift
g
𝑝𝑏 (𝑥 + ℎ, 𝑧, 𝑡|x𝑠 )
x-h x x+h
𝑝𝑓: 𝑓𝑜𝑤𝑎𝑟𝑑 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑒𝑑 𝑑𝑎𝑡𝑎
𝑝𝑏: 𝑏𝑎𝑐𝑘𝑤𝑎𝑟𝑑 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑒𝑑 𝑑𝑎𝑡𝑎
Angle-domain Crosscorrelation
Angle-domain CIG decomposition (slant stack ):
𝑓 𝑥, 𝑧, 𝜃, 𝜏 =
angle-domain
𝑓 𝑥, 𝑧 + ℎ tan 𝜃 , ℎ, 𝜏 𝐱𝑠 𝑑ℎ
suboffset-domain
Angle-domain crosscorrelation function :
𝑓 𝑥, 𝑧, 𝜃, 𝜏 =
𝑑𝐱 𝑠
𝑑ℎ
𝑝𝑓 𝑥 − ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡 + 𝜏 𝐱 𝑠
𝑝𝑏 (𝑥 + ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡|𝐱𝑠 ) 𝑑𝑡
Angle-domain Crosscorrelation:
physical meaning
𝑑𝐱 𝑠
𝑝𝑓 𝑥 − ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡 𝐱 𝑠
𝑝𝑏 (𝑥 + ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡|𝐱𝑠 ) 𝑑𝑡
𝑧
𝑧
𝜃
𝑑ℎ
𝑥
𝑝𝑓 (𝑥 − ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡|𝐱 𝑠 )
Local plane wave
𝜃
𝑓 𝑥, 𝑧, 𝜃, 𝜏 = 0 =
𝑥
𝑝𝑏 (𝑥 + ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡|𝐱 𝑠 )
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: 𝜀 = 12
∆𝜏(𝐱, 𝜃)
𝐱
Velocity update:
𝟐
𝜽
𝑐𝑘+1 (x)= 𝑐𝑘 (x) + 𝛼𝑘 ∙ 𝛾𝑘 (x)
Gradient function:
𝜕𝜀
𝛾𝑘 (x)= −
=−
𝜕𝑐(𝐱)
𝐱
𝜽
𝜕(∆𝜏)
∆𝜏
𝜕𝑐(𝐱)
Traveltime wavepath
Traveltime Wavepath
Angle-domain time delay ∆𝜏 𝜃
𝑓 𝑥, 𝑧, 𝜃, ∆𝜏 = max 𝑓 𝑥, 𝑧, 𝜃, 𝜏
−𝑇<𝜏<𝑇
𝑓 𝑥, 𝑧, 𝜃, ∆𝜏 = m𝑖𝑛 𝑓 𝑥, 𝑧, 𝜃, 𝜏
−𝑇<𝜏<𝑇
Angle-domain connective function
𝜕𝑓(𝑥, 𝑧, 𝜃, 𝜏)
𝑓∆𝜏 =
=0
𝜕𝜏
Traveltime wavepath 𝜏=∆𝜏
𝜕(∆𝜏)
𝜕𝑓∆𝜏 𝜕𝑓∆𝜏
=−
𝜕𝑐(𝑥)
𝜕𝑐(𝑥) 𝜕(∆𝜏)
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
𝑓 𝑧, 𝜃, ∆𝜏
(c) Angle-domain Crosscorrelation
∆𝜏 = 𝛼(tan 𝜃)2
∆𝝉
∆𝜏: time delay
𝑓 𝑥, 𝑧, 𝜃, ∆𝜏
𝛼:
curvature
𝜃:
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°
𝜃
+50°
6
-0.04
-0.2
∆𝜏(𝑠)
0
z(km)
6
𝛼
12
Crosscorrelation
∆𝜏 = 𝛼(tan 𝜃)2
0.2
0.04
-50°
𝜃
+50°
Initial Velocity Model
z(km)
0
CIG
z(km)
0
x(km)
Semblance
12
Crosscorrelation
0
-0.2
∆𝜏(𝑠)
0
z(km)
6
∆𝜏 = 𝛼(tan 𝜃)2
6
-50°
𝜃
+50°
6
-0.04
𝛼
0.2
0.04
-50°
𝜃
+50°
Initial Velocity Model
z(km)
0
CIG
0
z(km)
0
x(km)
Semblance
6
-50°
𝜃
+50°
6
-0.04
12
Crosscorrelation
-0.2
∆𝜏(𝑠)
0
z(km)
6
𝛼
∆𝜏 = 𝛼(tan 𝜃)2
0.2
0.04
-50°
𝜃
+50°
Inverted Velocity Model
z(km)
0
CIG
0
z(km)
0
x(km)
Semblance
6
-50°
𝜃
+50°
6
-0.04
12
Crosscorrelation
-0.2
∆𝜏(𝑠)
0
z(km)
6
𝛼
0.04
0.2
-50°
∆𝜏 = 𝛼(tan 𝜃)2
𝜃
+50°
Inverted Velocity Model
z(km)
0
CIG
6
-50°
-0.2
0
z(km)
0
𝜃
12
Crosscorrelation
x(km)
Semblance
+50°
6
-0.04
∆𝜏(𝑠)
0
z(km)
6
𝛼
0.04
0.2
-50°
∆𝜏 = 𝛼(tan 𝜃)2
𝜃
+50°
Inverted Velocity Model
z(km)
0
CIG
z(km)
0
x(km)
Semblance
6
-50°
𝜃
+50°
12
Crosscorrelation
0
-0.2
∆𝜏(𝑠)
0
z(km)
6
6
-0.04
𝛼
0.04
∆𝜏 = 𝛼(tan 𝜃)2
0.2
-50°
𝜃
+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