Skeletonized Wave-equation
Inversion for Q
Gaurav Dutta and Gerard T. Schuster*
Department of Earth Science & Engineering
King Abdullah University of Science and Technology
October 18, 2016
Outline
Predicted
Observed
• Motivation
• Theory of WQ
• Numerical Examples
 Synthetic Data Examples
 Field Data Example
• Limitations
• Conclusions
𝑓𝑜𝑏𝑠 𝑓𝑐𝑎𝑙𝑐
• Motivation
Outline
• Theory of WQ
• Numerical Examples
 Synthetic Data Examples
 Field Data Example
• Limitations
• Conclusions
Motivation for Q Compensation
Offshore Brunei (Gamar et al., 2015)
Motivation for Q compensation
North Sea (Valenciano and Chemingui, 2012)
Motivation for Q Compensation
Offshore Brazil (Zhou et al., 2011)
Motivation for Q Compensation
Problem: FWI Q(x,y,z) not robust
Solution: Skeletonized Inversion for Q
Δf
Amp. Spectrum
Time
Predicted Observed
Predicted
Observed
e=
1
𝜖=
2
𝑠
Δ𝑓
𝑟
Frequency (Hz)
Y. Quan & Jerry Harris, 1997, Seismic attenuation tomography using the frequency shift method
2
• Motivation
Outline
• Theory of WQ
• Numerical Examples
 Synthetic Data Examples
 Field Data Example
• Limitations
• Conclusions
FWI vs Skeletal Inversion
True Q Model
Q
Observed Traces vs Predicted Traces
200
d(t)
time
2
Z (km)
e=||dpred - dobs ||2 vs Model
80
FWI gets stuck in local minima
e
4
40
1
2
X (km)
3
local minima
Model
FWI vs Skeletal Inversion
True Q Model
Observed vs Predicted Spectra
Q
Skeletal data = Peak
Frequency
200
D(f)
2
Z (km)
fpred
80
fobs
Frequency (Hz)
e=||fpred - fobs ||2 vs Model
4
e
Skeletal inversion = rapid convergence
global minima
40
1
2
X (km)
3
Model
Similarities with Wave-equation Traveltime Inversion
Properties
Wave-equation traveltime
tomography (Luo and Schuster,
Wave-equation Q tomography
(Dutta and Schuster, 2016)
1991; Woodward 1992)
Misfit function:
1
𝜖=
2
Δ𝜏 𝒙𝑟 , 𝒙𝑠
𝑠
2
𝑟
1
𝜖=
2
Δ𝑓 𝒙𝑟 , 𝒙𝑠
𝑠
𝑟
Predicted
Observed
Δ𝜏
Gradient:
𝜕𝜖
=−
𝜕𝑐(𝒙)
𝑠
𝑟
𝜕Δ𝜏
Δ𝜏(𝒙𝑟 , 𝒙𝑠 )
𝜕𝑐 𝒙
2
Δf
𝜕𝜖
=−
𝜕𝑄(𝒙)
𝑠
𝑟
𝜕Δ𝑓
Δ𝑓(𝒙𝑟 , 𝒙𝑠 )
𝜕𝑄 𝒙
Wave-equation Q Tomography
There are 3 steps in WQ:
1
𝜖=
2
1) Misfit function 𝜖:
𝜕𝜖
=−
𝜕𝑄(𝒙)
𝑠
𝑟
𝜕Δ𝑓
Δ𝑓(𝒙𝑟 , 𝒙𝑠 )
𝜕𝑄 𝒙
3) Gradient:
=
Q(k)
𝑠
2
𝑟
Δ𝑓Δ𝑓
= 𝑓𝑐𝑎𝑙𝑐 (𝒙𝑟 , 𝒙𝑠 ) − 𝑓𝑜𝑏𝑠 (𝒙𝒓 , 𝒙𝑠 )
2) Frechet Derivative : df/dQ =
Q(k+1)
Δ𝑓 𝒙𝑟 , 𝒙𝑠
-
We know dP/dQ from wave equation
𝜕𝜖
a .
𝜕𝑄
Smear frequency-shift
residuals along wavepaths
Viscoacoustic Wave Equation
SLS Model
Time-domain visco-acoustic wave equation:
𝜕𝑃
𝑃 = Pressure
+ 𝐾 𝜏 + 1 𝛻 ⋅ 𝒗 + 𝑟𝑝 = 𝑓(𝒙𝑠 , 𝑡)
𝒗 = Particle velocity
𝜕𝑡
𝑟𝑝 = Memory variable
𝜕𝒗 1
𝜏𝜖 , 𝜏𝜎 = Strain/Stress
+ 𝛻𝑃 = 0
𝜕𝑡 𝜌
relaxation times
1
𝜏𝜎 =
𝜔
1
𝜏𝜖 =
1
1+ 2 −𝑄
𝑄
1
1+ 2 +𝑄
𝑄
𝜔
𝑓 = Point-source function
𝜕𝑟𝑝 1
+
𝑟𝑝 + 𝜏𝐾 𝛻 ⋅ 𝒗
𝜕𝑡 𝜏𝜎
=0
𝜏𝜖
2 1
1
𝜏 = −1 =
+ 1+ 2
𝜏𝜎
𝑄 𝑄
𝑄
• Motivation
Outline
• Theory of WQ
• Numerical Examples
 Synthetic Data Examples
 Field Data Example
• Limitations
• Conclusions
Synthetic Example
True Q Model
Q
Predicted
Observed
200
2
Z (km)
𝑓𝑜𝑏𝑠 𝑓𝑐𝑎𝑙𝑐
80
•
•
•
4
40
1
2
X (km)
3
Acquisition
60 sources
200 receivers
𝑓𝑝𝑒𝑎𝑘 = 15 Hz
Δ𝑓
Predicted
Observed
Synthetic Example
True Q Model
𝑓𝑜𝑏𝑠
WQ Tomogram
Q
𝑓𝑐𝑎𝑙
Q
200
200
80
80
40
40
Z (km)
2
4
1
2
X (km)
3
1
2
X (km)
3
Observed
Predicted
Synthetic Example
True Q Model
Q
10000
Z (km)
0.5
1.5
WQ Tomogram
Q
20
10000
Z (km)
0.5
1.5
4
20
X (km)
8
12
Standard RTM
Z (km)
1
2
4
X (km)
8
12
Standard LSRTM
Z (km)
1
2
4
X (km)
8
12
Q LSRTM
Z (km)
1
2
4
X (km)
8
12
Standard RTM
Z (km)
1
2
4
X (km)
8
12
• Motivation
Outline
• Theory of WQ
• Numerical Examples
 Synthetic Data Examples
 Field Data Example
• Limitations
• Conclusions
Crosswell Field Data
183 m
9m
9m
3m
305 m
3m
293 m
Reflector
96 receivers
98
sources
Data Sampling: ¼ ms
Total Record Length: 0.375 s
Crosswell Field Data
Velocity Tomogram
Q Tomogram
km/s
Q
70
2.1
60
100
Z (m)
1.9
50
200
1.7
40
1.5
300
30
50
100
X (m)
150
50
100
X (m)
150
Predicted vs Observed Peak Frequencies
Hz
12
Source Index
50
100
8
150
4
200
100
200
Receiver Index
300
400
Crosswell Field Data
Standard Migration
Q-PSDM
Z (m)
100
200
300
50
100
X (m)
150
50
100
X (m)
150
Crosswell Field Data
Standard Migration
Q-PSDM
Z (m)
100
200
300
50
100
X (m)
150
50
100
X (m)
150
Outline
• Motivation
• Theory of WQ
• Numerical Examples
 Synthetic Data Examples
 Field Data Example
• Limitations
• Conclusions
Limitations
• Low-Intermediate Q resolution
• Velocity-Q ambiguity: Q  time delays
• Sequential Q and V inversion, or possibly
simultaneous Q+V inversion
• Motivation
Outline
• Theory of WQ
• Numerical Examples
 Synthetic Data Examples
 Field Data Example
• Limitations
• Conclusions
Conclusions
•
A novel wave-equation Q tomography method is presented.
Predicted
Observed
1
𝜖=
2
Δ𝑓 𝒙𝑟 , 𝒙𝑠
𝑠
2
𝑟
Δ𝑓 = 𝑓𝑐𝑎𝑙𝑐 (𝒙𝑟 , 𝒙𝑠 ) − 𝑓𝑜𝑏𝑠 (𝒙𝒓 , 𝒙𝑠 )
𝑓𝑜𝑏𝑠 𝑓𝑐𝑎𝑙𝑐
•
Gradient: ≈
∫ 𝑑𝑡 𝛻 ⋅ 𝒗(𝒙, 𝑡; 𝒙𝑠 ) 𝑔 𝒙𝑟 , −𝑡; 𝒙, 0 ∗ 𝑃 𝒙𝑟 , 𝑡; 𝒙𝑠
𝑠
𝑟
Source
𝑜𝑏𝑠 Δ𝑓(𝒙𝑟 , 𝒙𝑠 )
Backpropagated weighted residual
∫ 𝛼 𝑑𝑙 = Δ𝑓
Conclusions
•
Inverted Q tomograms ⇒ Improvements in imaging.
Standard Migration
Z (km)
1
2
4
X (km)
8
12
Conclusions
•
Inverted Q tomograms ⇒ Improvements in imaging.
Q-PSDM
Z (km)
1
2
4
X (km)
8
12
•
Conclusions
Inverted Q tomograms ⇒ Improvements in imaging.
Standard Migration
Q-PSDM
Z (m)
100
200
300
50
100
X (m)
150
50
100
X (m)
150
Limitations
• Low-Intermediate Q resolution
• Velocity-Q ambiguity
• Sequential Q and V inversion, or possibly
simultaneous Q+V inversion
Acknowledgements
• SEG for providing this platform.
• Sponsors of the CSIM consortium.
• Exxon for the Friendswood data.
• KAUST Supercomputing Laboratory and IT Research
Computing Group.
Motivation for Q Compensation
Problem: Q distorts amplitude and phase of propagating waves.
Q=1000
Q=40
Q=20
Z (km)
2
4
4
X (km)
8
4
X (km)
8
𝑓𝑝𝑒𝑎𝑘 = 20 Hz
4
X (km)
8
Motivation for Q Compensation
Problem: Q distorts amplitude and phase of propagating waves.
Q=1000
Q=40
Q=20
Z (km)
2
4
4
X (km)
8
4
X (km)
8
𝑓𝑝𝑒𝑎𝑘 = 20 Hz
4
X (km)
8
Motivation for Q Compensation
Problem: Q distorts amplitude and phase of propagating waves.
Q=1000
Q=40
Q=20
Z (km)
2
4
4
X (km)
8
4
X (km)
8
𝑓𝑝𝑒𝑎𝑘 = 20 Hz
4
X (km)
8