Wave-Equation Waveform Inversion for Crosswell Data M. Zhou and Yue Wang Geology and Geophysics Department University of Utah

Outline Theory: Wave Equation Inversion Theory: Wave Equation Inversion 2 Problems: Local Minima & Visco 2 Problems: Local Minima & Visco McElroy Xwell data McElroy Xwell data Conclusions Conclusions

Waveform Inversion  = || P(x,t) – P(x,t) || 2 pred obs v(x,z)  v(x,z) -   v(x,z) P(x,t) pred [P(x,t) – P(x,t) ] pred obs Migration { Waveform Residual { Acooustic, Elastic or Viscoelastic

Why Waveform Inversion? 0 km Depth 0.2 km km Better Resolution & Focusing Blurring Faults LithologyDistortion Gas

Outline Theory: Wave Equation Inversion Theory: Wave Equation Inversion 2 Problems: Local Minima & Visco 2 Problems: Local Minima & Visco McElroy Xwell data McElroy Xwell data Conclusions Conclusions

Why Waveform Inversion is Not Used? Poor Convergence, Local Minima, Cost Localminima   traveltime) v v 1 2   traveltime + 1 st Arrival) v v 1 2 v v 1 2 1 st -Arrival Waveforminversion Waveforminversion Global Minima

km/s Depth (m) 0 90 X (m) Model (Zhou et al., 1995) Model 1.5m X 1.5m grid 18 shots / 36 geophones 60 Hz Ricker wavelet

0 90 X (m) WIF km/s Depth (m) 0 90 X (m) Model 0 90 WT10 Model : Tomograms 0 90 X (m) WIF20 + WI10

X (m) Model WIF20 Time (sec) X (m) WT10 Model 2: Synthetic CSG

Outline Theory: Wave Equation Inversion Theory: Wave Equation Inversion 2 Problems: Local Minima & Visco 2 Problems: Local Minima & Visco McElroy Xwell data McElroy Xwell data Conclusions Conclusions

Problem: Invert Vp & Vs by elastic wave inver. Vp accurate by elastic waveform inversion Vp accurate by elastic waveform inversion Vs inaccurate by elastic waveform inversion Vs inaccurate by elastic waveform inversion Why? Insufficient physics in forward modeler Anisotropy? Attenuation? Biot? Coupling? Source effects? 3-D scattering? Source effects? 3-D scattering? Conjecture: Add attenuation

Waveform Inversion  = || P(x,t) – P(x,t) || 2 pred obs v(x,z)  v(x,z) -   v(x,z) P(x,t) pred [P(x,t) – P(x,t) ] pred obs Migration { Waveform Residual { Viscoelastic Viscoelastic

0 90 Offset (m) Depth (m) P 0 90 Offset (m) S m/s m/s Fault Model

0 Time (s) Depth (m) Elastic Seismogram

0 Time (s) Depth (m) =40 Qp=40Qs=25 Viscoelastic Seismogram

0 90 Offset (m) Depth (m) P 0 90 Offset (m) S m/s m/s A B C Ray Tracing Tomograms

0 90 Offset (m) Depth (m) P 0 90 Offset (m) S m/s m/s A B C A B C D Viscoelastic Tomograms

0 90 Offset (m) Depth (m) P 0 90 Offset (m) S m/s m/s Elastic Tomograms

Outline Theory: Wave Equation Inversion Theory: Wave Equation Inversion 2 Problems: Local Minima & Visco 2 Problems: Local Minima & Visco McElroy Xwell data McElroy Xwell data Conclusions Conclusions

200 Sources * 200 Receivers 180 ft 2700 ft McElroy Geometry 500 ft

0 500 Depth (ft) Time (s) Shot Gather 101

Bandpass Filter Median Filter 3D ==> 2D Borehole Filter, wavelet and radiation pattern Processing

0 184 Offset (ft) Depth (ft) P Offset (ft) S B A C D ft/s ft/s Elastic Tomograms

2700 V Depth (ft) 3150 Receiver Well P-Velocity Profile: Tomo vs Sonic Log

2850 Depth (ft) 3050 V Source Well S-Velocity Profile: Tomo vs Sonic Log

Bandpass Filter Median Filter 3D ==> 2D Borehole filter, Wavelet and Src Rad. Qp and Qs by Harris’ Redshift Method Processing

Freq. ~ Time Covariance of Source Spectrum Appr. Qp Extract All Direct P Waves Estimate Q from 1 st Arrivals

Freq. vs Time Freq. vs Time Time (s) Freq. (hz)

B A C D Offset (ft) Depth (ft) P Offset (ft) S ft/s ft/s Viscoelastic Tomograms

2850 Depth (ft) 3050 V Source Well S-Velocity Profile: ViscoTomo vs Sonic Log

2850 Depth (ft) 3050 V Source Well S-Velocity Profile: ElasticTomo vs Sonic Log

0 Offset (ft) Depth (ft) A B Viscoelastic S-Tomogram

0 Offset (ft) Depth (ft) A B Elastic S-Tomogram

0 Offset (ft) Depth (ft) Visco. 184 C D Poisson Ratio

0 Offset (ft) Depth (ft) Elastic 184 C D Poisson Ratio

Conclusions 4 Iteration, Pressure Data, Visco.WTW4 Iteration, Pressure Data, Visco.WTW Visco better than ElasticVisco better than Elastic Issues:Issues: Q(z), multicomponent, anisotropy Q(z), multicomponent, anisotropy CPU Time CPU Time Intrinsic atten. vs Scattering atten.? Intrinsic atten. vs Scattering atten.? Scale of Q vs Velocity? Scale of Q vs Velocity? Need more Physics such as Anisotropy? Need more Physics such as Anisotropy?

B A C D Visco. WTW Tomograms Offset (ft) Depth (ft) P Offset (ft) S ft/s ft/s

S Tomogram Comparison 0 Offset (ft) Depth (ft) Visco. 184 C D

S Tomogram Comparison 0 Offset (ft) Depth (ft) Elastic 184 C D

Conclusions Visco.WTW High-resolution P- and S-velocity ! YES

Conclusions Visco.WTW Visco.WTW Porosity, Lithology, AVO YES

Acknowledgments Acknowledgments We are grateful for the support of :  the 1997 members of University of Utah Tomography and Modeling/Migration Consortium

Viscoelastic CSP Gather Depth (ft) Time (s)

Evaluate Q Values Appr. Qp + Synthetic Modeling Test Qp Qs

CSP Gather Depth (ft) Time (s)

Viscoelastic CSP Gather Depth (ft) Time (s)

Elastic CSP Gather Depth (ft) Time (s)

Viscoelastic Waveform Inversion Depth (ft) Time (s) Depth (ft) Time (s) Depth (ft) Time (s) Elastic Viscolastic CSG101

P-Velocity Profiles : P-sonic Logs : Elastic P-velocity Profiles : Visco. P-velocity Profiles

S-Velocity Profiles 2700 V Depth (ft) 3150 V Receiver Well Source Well

S-Velocity Profiles : S-sonic Logs : Elastic S-velocity Profiles : Visco. S-velocity Profiles

Elastic CSP Gather Depth (ft) Time (s)

Viscoelastic CSP Gather Depth (ft) Time (s) Qp=80Qs=50

Outline Motivation+Theory WI Motivation+Theory WI Examples Examples Xwell Synthetic Xwell Synthetic CDP Synthetic : Chirp CDP Synthetic : Chirp CDP Gulf of Mexico CDP Gulf of Mexico Conclusions Conclusions

Synthetic CDP Model (FD Acoustic) Depth (m) Distance (m) (m/s) 40 m Synthetic CDP Model (100 shot gathers, 100 receivers/gather)

Traveltime Tomogram Depth (m) Distance (m) (m/s)

Multigrid for half comp. time Depth (m) Distance (m) (m/s)

Chevron Gulf of Mexico Seismic Line Courtesy of Alan Leeds Shots: 990 Channel: 180 Shot spacing: 25 m Receiver: 25 m Sample: 4 ms Length: 8 sec. Offset 173 – 4648 m

(m/s) CDP NUMBER Depth (m) Traveltime Tomogram Waveform

Waveform vs. Traveltime (m/s) CDP NUMBER Depth (m) WaveformTraveltime Gas?

Amplitude Vs. Offset Log10 Amplitude Log10 Offset (m) Waveform Data Traveltime

600 (msec.) 1200 Stacked Section (Waveform vs. Traveltime)

Conclusions Robust to initial model for Xwell. Robust to initial model for Xwell. High resolution tomograms. High resolution tomograms. WI vs. Traveltime Inversion: WIF + WI vs. WI: Stable for Models Tested. Stable for Models Tested. Sensitive to initial model. Sensitive to initial model. Will this approach work for Reflections? Will this approach work for Reflections?

Acknowledgements I am grateful for the financial I am grateful for the financial support from the members of support from the members of the UTAM consortium. the UTAM consortium.

Waveform Tomogram Depth (m) Distance (m) (m/s)

2D Synthetic Data (Blind test) Courtesy of Konstantin Osypov Shots: 401 Channel: 241 Shot spacing: 50 m Receiver: 25 m Sample: 4 ms Length: 2 sec. Offset: – 3000 m

Traveltime Tomogram m/s Horizontal distance (km) Depth (km)

Amplitude 0.1.WIF20 Amplitude Time (s) WT10 Model 2: One Trace Examples ObjectiveTheoryConclusionsMotivation

Waveform Tomogram m/s Horizontal distance (km) Depth (km)

600 (msec.) 1200 Stacked Section (Waveform vs. Traveltime) WaveformTraveltime

600 (msec.) 1200

Summary Traveltime Inversion Traveltime Inversion Waveform Inversion Waveform Inversion slow, sensitive to initial model high resolution fast, insensitive to initial model low resolution

Model 1: Model km/s 050 X (m) Depth (m) Model 1m X 1m grid 41 shots/geophones 200 Hz Ricker wavelet

Model 1: Tomograms 050 X (m) WI km/s Depth (m) Model 050 X (m) Tomo50

050 WIF30 + WI km/s 050 X (m) Depth (m) Model 050 X (m) WIF30 Model 1: Tomograms

Time (sec) X (m) Tomo50 Model WIF30 + WI Model 1: Synthetic CSG

RMS Waveform Residuals Number of Iterations Model 1: Residuals

Outline Motivation+Theory WI Motivation+Theory WI Examples Examples Xwell Synthetic Xwell Synthetic CDP Synthetic : Chirp CDP Synthetic : Chirp CDP Gulf of Mexico CDP Gulf of Mexico Conclusions Conclusions

Visco. Wave Equation Initial Velocity Model Vp, Vs, Qp, Qs Synthetic Seismograms Residuals = ||Syn. - Obs|| 2

u Perturbation of Lame parameters: Gradient Optimization f : from forward wavefield b: from adjoint wavefield

Spectrum Covariance Frequency (hz) Amplitude 1 1/e

 Problem & Methodology  Synthetic Data Example  Field Data Example  Conclusion and Discussion Outline

Full 2-D viscoelastic wave equationFull 2-D viscoelastic wave equation with memory variables. Standard with memory variables. Standard model spring-dashpot Qp and Qs model spring-dashpot Qp and Qs 2-D to 3-D conversion2-D to 3-D conversion Apply borehole transfer functionApply borehole transfer function Invert src waveletInvert src wavelet Invert src radiation patternInvert src radiation patternMethodology