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Aspects of seismic inversion

Aspects of seismic inversion. Paul Childs *Schlumberger Cambridge Research With acknowledgements to: Colin Thomson*, ZhongMin Song † , Phil Kitchenside † Henk Keers* † Schlumberger WesternGeco, Gatwick HOP, Newton Institute, June 19 th 2007. Survey configuration. Marine seismic.

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Aspects of seismic inversion

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  1. Aspects of seismic inversion Paul Childs *Schlumberger Cambridge Research With acknowledgements to: Colin Thomson*, ZhongMin Song †, Phil Kitchenside † Henk Keers* † Schlumberger WesternGeco, Gatwick HOP, Newton Institute, June 19th 2007

  2. Survey configuration Marine seismic

  3. Spectral interference notches from receiver-side free-surface ghost; O/U receivers allow for up/down separation, hence ghost removal; flatter, broader spectrum shows Earth structure better in seismic sections (=> better “attribute analysis”).

  4. GPS Bird controller TRINAV Closing the loop TRIACQ StreamerSteering • Streamer control - with IRMA (Intrinsic Ranging by Modulated Acoustics) and Q-fin: GPS IRMA range data IRMA controller

  5. velocity Seismic record time time Seismic “Image”

  6. Complexity • Acoustic approximation is often made Ray methods: James Hobro, ChrisChapman, Henrik Bernth

  7. Recover inhomogeneous subsurface velocity (density, impedance, …) field from surface measurements Born/Fréchet Kernel Green function from Full wave equation One-way wave equations Asymptotic ray theory Maslov Gaussian beam… (x,y) s r .x (Acoustic) Problem definition z s: source r: receiver x: scatterer/reflector

  8. Fréchet kernels for multi-scale waveform inversion • Full wave equation inversion • Acoustic wave equation • Frequency domain • Helmholtz equation • Multigrid solver • Multiscale approach • Ray modeling • Turning waves • Maslov asymptotic approx. • Sensitivity, resolution & influence

  9. Frequency domain formulation • Frequency domain adjoint formulation (Pratt): • Forward model: Forward propagation Back-propagation

  10. f0 f1 f2 …. fn ….. Multi-scale approach* • Low frequency => large wavelength => large basin of convergence • Multiscale continuation • Solve for low frequency ~3 Hz • Increase frequencies incrementally • Use last [subsurface velocity] as initial guess for new frequency J *Sirgue, L and Pratt, R.G. (2004) “Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies”, Geophysics 69(1), pp.231-248 *Pratt, R.G. et al (1998, 1999) *Ghattas et al. & Tromp et al

  11. Exact velocity model Vp Vp Surface Depth Surface Depth Surface Depth Surface Surface Test model • Traveltime tomography starting model Surface Surface • Sensitivities

  12. Inversion results f < 20 Hz Vp@ 1.5Hz Vp@ 5Hz Vp@ 16Hz Vp@ Truth

  13. Inversion results Vp vs Depth Depth Offset 1/2 3/4 1/4

  14. Optimization • Quasi-Newton, LBFGS • Solve for vp, ρ, source wavelet,… • Project constraints • L2, H2 + TV regularization • Gauss-Newton + line search • Constrained by modelling cost • Multiple right hand sides • Direct solver (SuperLU/MUMPS) • Multigrid solvers

  15. Multigrid Preconditioner (Erlangga et al, 2006)* • H: Helmholtz • Indefinite • Not coercive • Non-local • C: Complex shifted Laplace, improved spectral properties • Preconditioner for H is C solved by Multigrid • *Y.A. Erlangga and C.W. Oosterlee and C. Vuik (2006). • “A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems”, • SIAM J. Sci. Comput.,27, pp. 1471-1492, 2006

  16. Multigrid Helmholtz solver - subsalt Multiple grids Vp Offset Depth Wavefield Sigsbee salt velocitymodel

  17. X,p,T xr xs Plane wave synthesis receiver source

  18. Results Initial velocity model • Low frequency starting model • caustics & pseudo-caustics Surface

  19. Densified rays show stability even in such a complicated model; waveforms show back- scattering time time time CJT, 1999

  20. Maslov* waveforms • Integral over plane waves • Sensitivity • Asymptotic theory *C.H.Chapman & R.Drummond, “Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory”, Bull. Seism. Soc. America, vol 72, no. 6, pp.S277-S317, 1982.

  21. Ray sensitivity equations (1) • Hamiltonian system • Dynamic ray tracing • Paraxial sensitivity

  22. Ray tracing for gradients • Calculating the kernel • Solve ODEs • Propagator solves for • Sparse automatic differentiation evaluates

  23. Basis functions Regularize over wavepaths Regularized gradient (5Hz) Fréchet derivatives

  24. Gauss Newton Measures of resolution Offset • Resolution matrix • Posterior covariance • Lanczos solver • Hessian vector products only Diag(R) Depth Velocity model Vp Depth Offset

  25. Resolution of inverse operator

  26. Closure • Frequency domain finite difference (FD) full waveform inversion • regularization • multi-scale optimization • Full wave equation FD(FE, SEM…) may be too detailed • reduced physics for forward models • Which approximations inform the inverse solution ? • Are Maslov waveforms effective for turning ray waveform inversion ? • Uncertainty estimates

  27. End Comments ?

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