Some remarks on the leading order imaging series Depth imaging of reflection data from a layered acoustic medium with an unknown velocity model Simon A. Shaw University of Houston M-OSRP Annual Meeting April 20th, 2005

Outline Motivation, objectives Background 1-D acoustic inverse series A leading order imaging series Analysis Conclusions

Motivation Current migration theory requires correct velocity model to give correct location Current methods for deriving the velocity model can be inadequate, especially for complex media More reservoirs beneath complex geology

Objective (of this research) To improve our ability to accurately locate reflectors, especially in areas where the velocity model is difficult to estimate

Objective (of this talk) To review the 1D constant density variable velocity acoustic leading order imaging series algorithm To better understand what it is what it does how it does it

Outline Motivation, objectives Background 1-D acoustic inverse series A leading order imaging series Analysis Conclusions

Background and review Inverse scattering series-based algorithms: Do not require any a priori subsurface information – no velocity model, no event picking, no moveout assumptions … Require the source wavelet Allow / require the data to get involved in their own processing. They are data-driven.

Background and review Scattering Theory is Perturbation Theory Relates Actual Field, G Reference Field, G0 difference between Actual Medium and Reference Medium properties, V

Actual Field = Reference Field + Scattered Field Background and review Actual Medium Reference Medium Perturbation L – L0 = V Actual Field = Reference Field + Scattered Field G G0 S

Actual Field = Reference Field + Scattered Field Background and review Actual Medium Reference Medium Perturbation L – L0 = V Actual Field = Reference Field + Scattered Field G G0 S Forward Problem

Actual Field = Reference Field + Scattered Field Background and review Actual Medium Reference Medium Perturbation L – L0 = V Inverse Problem Actual Field = Reference Field + Scattered Field G G0 S

Background and review Inverse Problem Measured Scattered Field Perturbation D V Solution for V is an infinite series in the data, D

Reference medium is never updated Inverse scattering series V = V1 + V2 + V3 + … Inverse Series is solution for V in terms of G0 and measured G Linear 2nd Order 3rd Order Nonlinear…. Data must multiply itself Reference medium is never updated

Linear inverse scattering V ≈ V1 Inverse Born approximation for V Linear Nonlinear…. Data must multiply itself Assumes G ≈ G0

Iterative linear G0 towards G Linear Linear Linear Repeated linear inverse Linear Linear Linear Updates G0 towards G E.g., velocity model updating

Reference medium is never updated Inverse scattering series Inverse Series is solution for V in terms of G0 and measured G Reference medium is never updated V = V1 + V2 + V3 + … Linear 2nd Order 3rd Order Multiplication: data events communicate with each other Nonlinear…. Data must multiply itself

Measured Scattered Field Background and review Seismic Inverse Problem Measured Scattered Field Moses (1956) Razavy (1975) Weglein et al. (1981) Stolt and Jacobs (1980) Perturbation V D Free-surface multiple removal Internal multiple removal Imaging in space Target identification

Imaging using the inverse series (Increasing realism) Production algorithm Embryonic concepts Prototype algorithm Analysis and testing Generalization Idea Non-linear, wavefield at depth Pattern, isolate subseries 2D, elastic, variable background Task separation Taylor series Risk time

Outline Motivation, objectives Background 1-D acoustic inverse series A leading order imaging series Analysis Conclusions

1D acoustic inverse problem

1D Earth, 3D wave propagation

Inverse scattering series Linear 2nd Order 3rd Order Nonlinear…. Data must multiply itself

1D acoustic inverse series

Outline Motivation, objectives Background 1-D acoustic inverse series A leading order imaging series Analysis Conclusions

where a1 is the data imaged with the constant reference velocity Leading order imaging series 1-D with offset where a1 is the data imaged with the constant reference velocity

Leading order imaging series 1-D/1-D

Leading order imaging series

Leading order imaging series

Leading order imaging series

Leading order imaging series

Leading order imaging series

Leading order imaging series

Leading order imaging series

Leading order imaging series

Communication with all deeper events Depth imaging without the velocity model z

+ Taylor Series at each mislocated interface Depth imaging without the velocity model + First term: image with the reference velocity Sum of second and higher terms Sum of imaging series

Outline Motivation, objectives Background 1-D acoustic inverse series A leading order imaging series Analysis Conclusions

How does it work? z A Shift: Taylor Series for: Wrong depth Correct depth

A cascaded series Shift = zb = correct depth zb zb = wrong depth zb = correct depth Shift = zb zb Coefficients hold the key to the correct depth (Depth correction) (Wrong depth, “time”) (Data amplitude) Required information resides in data’s amplitudes and travel times

Higher order imaging contributions (Innanen, 2005) Leading order shift 4R1 functions of p z Higher order imaging contributions (Innanen, 2005)

Reference/Actual velocity contrast: 10% Leading order imaging: 3 terms correction a1 Reference/Actual velocity contrast: 10%

Reference/Actual velocity contrast: 10% Sum 3 leading order terms correction a1 Reference/Actual velocity contrast: 10%

Reference/Actual velocity contrast: 10% Sum 5 leading order terms correction a1 Reference/Actual velocity contrast: 10%

Leading order imaging series

Closed Form (After R.G. Keys)

Convergence properties Converges for finite kz and Converges faster for low kz small

Depth errors ? When will Actual depth of reflector Depth of reflector predicted by reference (in a1) Depth of reflector predicted by LOIS

Depth errors, condition when Actual depth of reflector Depth of reflector predicted by reference (in a1) Reference vertical slowness Actual vertical slowness

Analytic example Two interfaces: Transmission coefficient is satisfied for prestack data independent of Reflector depths, (za, zb) Velocity, (z0, z1)

Random noise

Random noise

Coherent noise (internal multiples) Residual (internal) multiples have two effects: They will be imaged They will impact the location at which primaries are imaged

Coherent noise (internal multiples) Data travel times Image depths Residual internal multiple Residual internal multiple

Coherent noise (internal multiples) Data travel times Image depths Internal multiple Internal multiple

Coherent noise (internal multiples) Data travel times Image depths Residual internal multiple Residual internal multiple

Coherent noise (internal multiples) Data travel times Image depths Internal multiple Internal multiple

Outline Motivation, objectives Background 1-D acoustic inverse series A leading order imaging series Analysis Conclusions

Conclusions Inverse scattering series has ability to image reflectors in space without requiring or solving for the actual velocity An imaging subseries has been isolated that is an improvement over conventional imaging with the reference and has good convergence properties Numerical tests: leading order imaging series algorithm does not require zero frequency to provide benefit

Conclusions To improve imaging series accuracy and lessen low frequency dependence Keep contrasts smaller (use best background velocity estimation) Record lower frequencies: industry trend is to recording lower frequencies Results encouraging therefore 2D, 3D and elastic generalizations progressing

Acknowledgments Drs. Art Weglein, Ken Matson, Doug Foster, Hua-Wei Zhou and Stuart Hall Craig Cooper, Hugh Rowlett, Rob Habiger Bob Keys, Dennis Corrigan M-OSRP colleagues – Kris Innanen, Bogdan Nita, Fang Liu, Haiyan Zhang, Jingfeng Zhang, Einar Otnes, Adriana Ramirez M-OSRP sponsors, especially BP and ConocoPhillips