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

2 PC 3/20/07 Survey configuration Marine seismic

3 PC 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 PC 3/20/07 GPS Streamer Steering Streamer control - with IRMA (Intrinsic Ranging by Modulated Acoustics) and Q-fin: GPS IRMA range data Bird controller Closing the loop IRMA controller TRINAV TRIACQ

5 PC 3/20/07 velocity Seismic record Seismic “Image” time

6 PC 3/20/07 Complexity Acoustic approximation is often made Ray methods: James Hobro, Chris Chapman, Henrik Bernth

7 PC 3/20/07 (Acoustic) Problem definition Recover inhomogeneous subsurface velocity (density, impedance, …) field from surface measurements Born/Fr é chet Kernel Green function from –Full wave equation –One-way wave equations –Asymptotic ray theory –Maslov –Gaussian beam….x.x (x,y) sr z s: source r: receiver x: scatterer/reflector

8 PC 3/20/07 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 PC 3/20/07 Frequency domain formulation Frequency domain adjoint formulation (Pratt): Forward model: Back-propagation Forward propagation

10 PC 3/20/07 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 f0 f1 f2 …. fn ….. J *Sirgue, L and Pratt, R.G. (2004) “Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies”, Geophysics 69(1), pp *Pratt, R.G. et al (1998, 1999) *Ghattas et al. & Tromp et al

11 PC 3/20/07 Depth Vp Test model Exact velocity modelTraveltime tomography starting model Sensitivities Surface Vp Depth

12 PC 3/20/07 Inversion results f < 20 Hz 1.5Hz 5Hz 16Hz Truth

13 PC 3/20/07 Inversion results Vp vs Depth 1/4 1/2 3/4 Depth Offset

14 PC 3/20/07 Optimization Quasi-Newton, LBFGS –Solve for v p, 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 PC 3/20/07 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 , 2006

16 PC 3/20/07 Multigrid Helmholtz solver - subsalt Wavefield Sigsbee salt velocity model Multiple grids Depth Offset Vp

17 PC 3/20/07 Plane wave synthesis source receiver xs xr X,p,T

18 PC 3/20/07 Ray approximation

19 PC 3/20/07 Results Low frequency starting model –caustics & pseudo-caustics Surface Initial velocity model

20 PC 3/20/07 Densified rays show stability even in such a complicated model; waveforms show back- scattering CJT, 1999 time

21 PC 3/20/07 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.

22 PC 3/20/07 Ray sensitivity equations (1) Hamiltonian system Dynamic ray tracing Paraxial sensitivity

23 PC 3/20/07 Ray tracing for gradients Calculating the kernel Solve ODEs Propagator solves for Sparse automatic differentiation evaluates

24 PC 3/20/07 Fr é chet derivatives Basis functions Regularize over wavepaths Regularized gradient (5Hz)

25 PC 3/20/07 Workflow

26 PC 3/20/07 Measures of resolution Resolution matrix –Posterior covariance –Lanczos solver – Hessian vector products only Gauss Newton Diag(R) Velocity model Depth Offset Depth Vp Offset

27 PC 3/20/07 Resolution of inverse operator

28 PC 3/20/07 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

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