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Multi-Scale Seismic Velocity Inversion Methods and Applications

Explore the advancements in seismic inversion methods for recovering subsurface velocity structures using frequency domain formulations, full wave equations, and multi-scale approaches. This technical paper discusses the use of Maslov waveforms, ray sensitivity equations, regularization techniques, and closure constraints in the process. Gain insights into generating accurate velocity models through efficient waveform inversion strategies.

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Multi-Scale Seismic Velocity Inversion Methods and Applications

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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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