1. 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

2. Outline Motivation Theory of WQ Numerical Examples Limitations
Predicted
Observed
𝑓 𝑜𝑏𝑠
𝑓 𝑐𝑎𝑙𝑐
Motivation
Theory of WQ
Numerical Examples
Synthetic Data Examples
Field Data Example
Limitations
Conclusions

3. Outline Motivation Theory of WQ Numerical Examples Limitations
Synthetic Data Examples
Field Data Example
Limitations
Conclusions

4. Motivation for Q Compensation
Offshore Brunei (Gamar et al., 2015)

5. Motivation for Q compensation
North Sea (Valenciano and Chemingui, 2012)

6. Motivation for Q Compensation
Offshore Brazil (Zhou et al., 2011)

7. Motivation for Q Compensation
Problem: FWI Q(x,y,z) not robust
Solution: Skeletonized Inversion for Q
Predicted
Observed
Frequency (Hz)
Amp. Spectrum
Predicted
Observed
Time Δf
𝜖= 1 2 𝑠 𝑟 Δ𝑓 2 e=
Y. Quan & Jerry Harris, 1997, Seismic attenuation tomography using the frequency shift method

8. Outline Theory of WQ Motivation Numerical Examples Limitations
Synthetic Data Examples
Field Data Example
Limitations
Conclusions

9. FWI vs Skeletal Inversion FWI vs Skeletal Inversion
True Q Model Q
Observed Traces
vs Predicted Traces
200
d(t) 2
time
e=||dpred - dobs ||2 vs Model
Z (km) 80
FWI gets stuck in local minima e 4
local minima 40 1 2 3
X (km)
Model

10. FWI vs Skeletal Inversion FWI vs Skeletal Inversion
Observed vs Predicted Spectra
True Q Model Q
Skeletal data = Peak Frequency
200
D(f) 2
fpred 80
fobs
Frequency (Hz)
Z (km)
e=||fpred - fobs ||2 vs Model
Skeletal inversion = rapid convergence 4 e
global minima 40 1 2 3
X (km)
Model

11. Similarities with Wave-equation Traveltime Inversion
Wave-equation traveltime tomography (Luo and Schuster, 1991; Woodward 1992)
Properties
Wave-equation Q tomography (Dutta and Schuster, 2016)
𝜖= 1 2 𝑠 𝑟 Δ𝜏 𝒙 𝑟 , 𝒙 𝑠 2
Misfit function:
𝜖= 1 2 𝑠 𝑟 Δ𝑓 𝒙 𝑟 , 𝒙 𝑠 2 Δ𝜏
Predicted
Observed Δf
𝜕𝜖 𝜕𝑐(𝒙) =− 𝑠 𝑟 𝜕Δ𝜏 𝜕𝑐 𝒙 Δ𝜏( 𝒙 𝑟 , 𝒙 𝑠 )
Gradient:
𝜕𝜖 𝜕𝑄(𝒙) =− 𝑠 𝑟 𝜕Δ𝑓 𝜕𝑄 𝒙 Δ𝑓( 𝒙 𝑟 , 𝒙 𝑠 )

12. Wave-equation Q Tomography
There are 3 steps in WQ:
1) Misfit function 𝜖:
𝜖= 1 2 𝑠 𝑟 Δ𝑓 𝒙 𝑟 , 𝒙 𝑠 2
Δ𝑓= 𝑓 𝑐𝑎𝑙𝑐 ( 𝒙 𝑟 , 𝒙 𝑠 )− 𝑓 𝑜𝑏𝑠 ( 𝒙 𝒓 , 𝒙 𝑠 )
𝜕𝜖 𝜕𝑄(𝒙) =− 𝑠 𝑟 𝜕Δ𝑓 𝜕𝑄 𝒙 Δ𝑓( 𝒙 𝑟 , 𝒙 𝑠 ) Δ𝑓
2) Frechet Derivative : df/dQ =
We know dP/dQ from wave equation
3) Gradient: Q(k+1) = Q(k) - a 𝜕𝜖 𝜕𝑄 .
Smear frequency-shift residuals along wavepaths
Wave-equation Q tomography (Dutta and Schuster, 2016)

13. Viscoacoustic Wave Equation
SLS Model
Time-domain visco-acoustic wave equation:
𝜕𝑃 𝜕𝑡 +𝐾 𝜏+1 𝛻⋅𝒗 + 𝑟 𝑝 =𝑓( 𝒙 𝑠 ,𝑡)
𝜕𝒗 𝜕𝑡 + 1 𝜌 𝛻𝑃=0
𝜕 𝑟 𝑝 𝜕𝑡 + 1 𝜏 𝜎 𝑟 𝑝 +𝜏𝐾 𝛻⋅𝒗 =0
𝑃= Pressure
𝒗= Particle velocity
𝑟 𝑝 = Memory variable
𝜏 𝜖 , 𝜏 𝜎 = Strain/Stress relaxation times
𝜏 𝜎 = 𝑄 2 − 1 𝑄 𝜔
𝜏 𝜖 = 𝑄 𝑄 𝜔
= 2 𝑄 1 𝑄 𝑄 2
𝜏= 𝜏 𝜖 𝜏 𝜎 −1
𝑓= Point-source function

14. Outline Numerical Examples Motivation Theory of WQ
Synthetic Data Examples
Field Data Example
Limitations
Conclusions

15. Synthetic Example True Q Model Acquisition 60 sources 200 receivers
Predicted
Observed
𝑓 𝑜𝑏𝑠
𝑓 𝑐𝑎𝑙𝑐
200 2
Z (km) 80
Acquisition
60 sources
200 receivers
𝑓 𝑝𝑒𝑎𝑘 = 15 Hz 4 40 1 2 3
X (km)

16. Synthetic Example True Q Model WQ Tomogram Q Q 200 200 2 Z (km) 80 80
𝑓 𝑜𝑏𝑠
𝑓 𝑐𝑎𝑙 Δ𝑓
Predicted
Observed
Synthetic Example
True Q Model
WQ Tomogram Q Q
200
200 2
Z (km) 80 80 4 40 40 1 2 3
X (km) 1 2 3
X (km)

17. Synthetic Example True Q Model WQ Tomogram Q 10000 0.5 Z (km) 1.5 20 Q
Observed
Synthetic Example
Predicted
True Q Model Q
10000
0.5
Z (km)
1.5 20
WQ Tomogram
0.5
1.5 4 8 12
Z (km)
X (km) 20
10000 Q

18. Standard RTM 1 2
Z (km) 4 8 12
X (km)

19. Standard LSRTM 1 2
Z (km) 4 8 12
X (km)

20. Q LSRTM 1 2
Z (km) 4 8 12
X (km)

21. Standard RTM 1 2
Z (km) 4 8 12
X (km)

22. Outline Numerical Examples Motivation Theory of WQ Field Data Example
Synthetic Data Examples
Field Data Example
Limitations
Conclusions

23. Crosswell Field Data 183 m 9 m 9 m 3 m 3 m 305 m 293 m Reflector
96 receivers
98 sources
Data Sampling: ¼ ms
Total Record Length: s

24. Crosswell Field Data Velocity Tomogram Q Tomogram Q 50 100 150 X (m)30 40 60 70
km/s
2.1
Z (m)
100
200
300
1.9
1.7
1.5 50
100
150
X (m)

25. Predicted vs Observed Peak Frequencies4 8 12 Hz 50
100
150
200
Source Index
100
200
300
400
Receiver Index

26. Crosswell Field Data Standard Migration Q-PSDM Z (m) 100 200 300 50
150
X (m)
Standard Migration 50
100
150
X (m)
Q-PSDM

27. Crosswell Field Data Standard Migration Q-PSDM Z (m) 100 200 300 50
150
X (m) 50
100
150
X (m)

28. Outline Conclusions Conclusions Motivation Theory of WQ
Numerical Examples
Synthetic Data Examples
Field Data Example
Limitations
Conclusions
Motivation
Theory of WQ
Numerical Examples
Synthetic Data Examples
Field Data Example
Limitations
Conclusions

29. Limitations Low-Intermediate Q resolution
Velocity-Q ambiguity: Q  time delays
Sequential Q and V inversion, or possibly simultaneous Q+V inversion

30. Outline Conclusions Motivation Theory of WQ Numerical Examples
Synthetic Data Examples
Field Data Example
Limitations
Conclusions

31. Backpropagated weighted residual
Conclusions
A novel wave-equation Q tomography method is presented.
Predicted
Observed
𝑓 𝑜𝑏𝑠
𝑓 𝑐𝑎𝑙𝑐
𝜖= 1 2 𝑠 𝑟 Δ𝑓 𝒙 𝑟 , 𝒙 𝑠 2
Δ𝑓= 𝑓 𝑐𝑎𝑙𝑐 ( 𝒙 𝑟 , 𝒙 𝑠 )− 𝑓 𝑜𝑏𝑠 ( 𝒙 𝒓 , 𝒙 𝑠 )
≈ 𝑠 𝑟 ∫𝑑𝑡 𝛻⋅𝒗(𝒙,𝑡; 𝒙 𝑠 ) 𝑔 𝒙 𝑟 ,−𝑡;𝒙,0 ∗𝑃 𝒙 𝑟 ,𝑡; 𝒙 𝑠 𝑜𝑏𝑠 Δ𝑓( 𝒙 𝑟 , 𝒙 𝑠 )
Gradient:
Source
Backpropagated weighted residual
∫𝛼 𝑑𝑙=Δ𝑓

32. Conclusions Inverted Q tomograms ⇒ Improvements in imaging.
Standard Migration 1 2
Z (km) 4 8 12
X (km)

33. Conclusions Inverted Q tomograms ⇒ Improvements in imaging. Q-PSDM 1 2
Z (km) 4 8 12
X (km)

34. Conclusions Inverted Q tomograms ⇒ Improvements in imaging.
Z (m)
100
200
300 50
150
X (m)
Standard Migration
Q-PSDM

35. Limitations Low-Intermediate Q resolution Velocity-Q ambiguity
Sequential Q and V inversion, or possibly simultaneous Q+V inversion

36. Acknowledgements SEG for providing this platform.
Sponsors of the CSIM consortium.
Exxon for the Friendswood data.
KAUST Supercomputing Laboratory and IT Research Computing Group.

37. Motivation for Q Compensation
Problem: Q distorts amplitude and phase of propagating waves.
Q=1000
Q=40
Q=20 2 4 8
X (km)
Z (km)
𝑓 𝑝𝑒𝑎𝑘 =20 Hz

38. Motivation for Q Compensation
Problem: Q distorts amplitude and phase of propagating waves.
Q=1000
Q=40
Q=20 2 4 8
X (km)
Z (km)
𝑓 𝑝𝑒𝑎𝑘 =20 Hz

39. Motivation for Q Compensation
Problem: Q distorts amplitude and phase of propagating waves.
Q=1000
Q=40
Q=20 2 4 8
X (km)
Z (km)
𝑓 𝑝𝑒𝑎𝑘 =20 Hz