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 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

Outline Motivation Theory of WQ Numerical Examples Limitations 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 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

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

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

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

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 )

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)

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 = 1+ 1 𝑄 2 − 1 𝑄 𝜔 𝜏 𝜖 = 1+ 1 𝑄 2 + 1 𝑄 𝜔 = 2 𝑄 1 𝑄 + 1+ 1 𝑄 2 𝜏= 𝜏 𝜖 𝜏 𝜎 −1 𝑓= Point-source function

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

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)

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)

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

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

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

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

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

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

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: 0.375 s

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)

Predicted vs Observed Peak Frequencies 4 8 12 Hz 50 100 150 200 Source Index 100 200 300 400 Receiver Index

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

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

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

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

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

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

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

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

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

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 2 4 8 X (km) Z (km) 𝑓 𝑝𝑒𝑎𝑘 =20 Hz

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

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