Introduction to Seismic Imaging

Transcription

1 Introduction to Seismic Imaging Alison Malcolm Department of Earth, Atmospheric and Planetary Sciences MIT August 20, 2010

2 Outline Introduction Why we image the Earth How data are collected Imaging vs inversion Underlying physical model Data Model Imaging methods Kirchhoff One-way methods Reverse-time migration Full-waveform inversion Comparison of Methods

3 Approximate Techniques Kirchhoff Integral technique Related to X-ray CT imaging Generalized Radon Transform Conventionally uses ray theory One-way Based on a paraxial approximation Usually computed with finite differences

4 One-Way Methods Physical Motivation downward continuation imaging condition Claerbout 71, 85 s r

5 One-Way Methods Physical Motivation downward continuation imaging condition r r r r r r r r r r v/2 s s s s s s s s s s

6 One-Way Methods Physical Motivation downward continuation imaging condition Claerbout 71, 85 s r

7 One-Way Methods Physical Motivation downward continuation imaging condition s r s r V(x) V(x)

8 A Data Model δg(s, r, t) = G 0 (r, t t 0, x)v(x) 2 t G 0(x, t 0, s)dxdt 0 X T δg(s, r, ω) = ω 2 G 0 (r, ω, x)v(x)g 0 (x, ω, s)dx X s r G 0 (x, t, s) G 0 (r, t, x) V(x)

9 A Data Model δg(s, r, t) = G 0 (r, t t 0, x)v(x) 2 t G 0(x, t 0, s)dxdt 0 X T δg(s, r, ω) = ω 2 G 0 (r, ω, x)v(x)g 0 (x, ω, s)dx X X ω 2 G (r, ω, x)g (s, ω, x)ṽ(x)dx

10 One-Way Methods Approximating the Wave Equation Idea (1D, c constant): ( 2 x 2 t )u = ( x t )( x + t )u c not constant: (c(x) 2 2 x 2 t )u = (c(x) x t )(c(x) x + t )u c(x)( x c(x)) x u c(x) smooth better approximation

11 One-Way Methods Approximating the Wave Equation Taylor (81), Stolk & de Hoop (05) Multi-dimensional: Lu = f ( ) u z z u = ( 0 1 A 0 )( ) ( ) u 0 + z u f A(z, x, x, t ) = 2 x c(z, x) 2 2 t

12 One-Way Methods Approximating the Wave Equation Taylor (81), Stolk & de Hoop (05) Diagonalize in smooth background ( ) ( )( ) ( ) u+ ib+ 0 u+ f+ z = + u 0 ib u f b ± (ξ, x, ω) = ± ξ 2 c 0 (x) 2 ω 2 B ± FIOs

13 One-Way Methods Approximating the Wave Equation Taylor (81), Stolk & de Hoop (05) Diagonalize in smooth background ( ) ( )( ) u+ ib+ 0 u+ z = + u 0 ib u ( f+ f ) Solution: G 0 = ( ) G+ 0 0 G G 0 propagator in c 0

14 One-Way Methods Approximating the Wave Equation Taylor (81), Stolk & de Hoop (05) Relate u ± to u and z u ( ) ( ) ( ) u+ = 1 (Q+ ) 1 HQ + u u 2 (Q ) 1 HQ z u q ± = [ ( ω c(z, x) Q ± differ in sub-principal parts Q ± ψdos ) 2 ξ 2 ] 1/4

15 One-Way Methods Modeling Data Stolk & de Hoop (05) δg(s, r, ω) ω 2 Q (r)g (r, ω, x)v(x)g + (x, ω, s)q + (s)dxdz X V(x) = 1 4 Q (z, x)δc(z, x)c 0 (z, x) 3 Q + (z, x)

16 One-Way Methods Modeling Data Stolk & de Hoop (05) δg(s, r, ω) ω 2 Q (r)g (r, ω, x)v(x)g + (x, ω, s)q + (s)dxdz X V(x) = 1 4 Q (z, x)δc(z, x)c 0 (z, x) 3 Q + (z, x) Reciprocity: s r r s

17 One-Way Methods Modeling Data Stolk & de Hoop (05) δg(s, r, ω) ω 2 Q (r)g (r, ω, x)v(x)g + (x, ω, s)q + (s)dxdz X V(x) = 1 4 Q (z, x)δc(z, x)c 0 (z, x) 3 Q + (z, x) Drop Qs ω 2 G (0, r, ω, z, x)g (0, s, ω, z, x) X Z E 2 : b(z, r, s) δ(t)b(z, r, s) E 1 : a(z, x) δ(r s)a(z, r+s 2 ) (E 2 E 1 a)(z, x)dzdx

18 One-Way Methods Imaging Claerbout (78) Stolk & de Hoop (06) downward continuation imaging condition s r

19 One-Way Methods Imaging Claerbout (78) Stolk & de Hoop (06) downward continuation ˆd(z, s, r, t) = H(z, 0) Q 1 d(s, r, t) (H(z, z 0 ))(s, r, t, s 0, r 0 ) = (G (z, z 0 ))(r, t, r 0 ) (G (z, z 0 ))(s, t, s 0 ) H is an FIO H H is ΨDO imaging condition

20 One-Way Methods Imaging Claerbout (78) Stolk & de Hoop (06) downward continuation imaging condition V(z, x) ˆd(z, x, x, 0) V(x) = 1 4 Q (z)δc(z, x)c 0 (z, x) 3 Q + (z) Recall Q are ΨDOs & H H is ΨDO We have again located the singularities

21 One-Way Methods Example 0 lateral position (km) depth (km) 5

22 Approximate Techniques Kirchhoff Integral technique Related to X-ray CT imaging Generalized Radon Transform Conventionally uses ray theory One-way Based on a paraxial approximation Usually computed with finite differences

23 Exact Techniques Reverse-time migration (RTM) Run wave-equation backward Usually computed with finite differences No approximations (to the acoustic, isotropic, linearized wave-equation, for smooth media assuming single scattering) Full-waveform inversion (FWI) Iterative method to match the entire waveform Gives smooth part of velocity model

24 Reverse-Time Migration Modeling Data δg(s, r, t) = G 0 (r, t t 0, x)v(x) 2 t G 0(x, t 0, s)dxdt 0 X T δg(s, r, ω) = ω 2 G 0 (r, ω, x)v(x)g 0 (x, ω, s)dx X s r G 0 (x, t, s) G 0 (r, t, x) V(x)

25 Reverse-Time Migration Modeling Data δg(s, r, t) = G 0 (r, t t 0, x)v(x) 2 t G 0(x, t 0, s)dxdt 0 X T δg(s, r, ω) = ω 2 G 0 (r, ω, x)v(x)g 0 (x, ω, s)dx X F : δc δg is again an FIO

26 Reverse-Time Migration Forming an image δg(s, r, ω) = X F : δc δg F : δg δc ω 2 G 0 (r, ω, x)v(x)g 0 (x, ω, s)dx is again an FIO is again an FIO F F is ΨDO for complete coverage Stolk et al. (09) no direct rays (transversal intersection) no source-side caustics c 0 C

27 Reverse-Time Migration Forming an Image Procedure: Whitmore (83), Loewenthal & Mufti (83), Baysal et al (83) back propagate in time imaging condition G 0 (x, t, s) s r G 0 (r, t, x) V(x)

28 Reverse-Time Migration an Adjoint State Method Lailly (83,84), Tarantola (84,86,87) Symes (09) For a fixed source, s, (c 2 2 t 2 )q(x, t; s) = δg(r, t; s)δ(x r)dr R s q(, t, ) = 0 for t > T receivers act as sources, reversed in time Im(x) = 2 q(x, t; s) 2 c 2 t (x) G 0(x, t, s)dtds

29 Reverse-Time Migration Example Liu et al (07)

30 Reverse-Time Migration Example Liu et al (07)

31 Reverse-Time Migration Example Liu et al (07)

32 Exact Techniques Reverse-time migration (RTM) Run wave-equation backward Usually computed with finite differences No approximations (to the acoustic, isotropic, linearized wave-equation, for smooth media assuming single scattering) Full-waveform inversion (FWI) Iterative method to match the entire waveform Gives smooth part of velocity model

33 Full-waveform Inversion Problem Setup Tarantola (87), Virieux & Operto (09) Recall our initial formulation: Lu := ( 2 1 c 2 2 t )u = f u = 0 t < 0 z u z=0 = 0 FWI attempts to solve for c directly given u,f no splitting of c but band limited data smooth solution

34 Full-waveform Inversion Problem Setup Tarantola (87), Virieux & Operto (09) Recall our initial formulation: Lu := ( 2 1 c 2 2 t )u = f Compute: min c Lu d L2 ((S,R) [0,T])

35 Full-waveform Inversion Some Issues Symes (09) Compute: min c Lu d L2 ((S,R) [0,T]) Objective function appears non-convex Can we restrict the domain of models so that it is? e.g. c min < c < c max What space must c be in for objective function to be differentiable? Data are finite bandwidth we cannot resolve structures on all scales

36 Full-waveform Inversion Work-around 1: Partial Linearization Symes (09) Idea: Extend δc(x) from X to ˆX Example: δc(x) δĉ(x, h) = δ(h)δc(x) s x h/2 x r form extended image here x + h/2 form image here Claerbout (85) suggested this extension

37 Full-waveform Inversion Work-around 1: Partial Linearization Symes (09) Idea: Extend δc(x) from X to ˆX Example: δc(x) δĉ(x, h) = δ(h)δc(x) Now extend to F[δc] δg ˆF[δĉ(x, h)] δg s.t. ˆF is invertible (I ˆF ˆF is smoothing) Find c 0 s.t. supp (δĉ(x, h)) {(x, h) h = 0} Stolk & de Hoop (2001), Stolk et al (2005) show invertibility

38 Full-waveform Inversion Work-around 2: Separation of Scales Pratt (99), Virieux (09) min c Lu d L2 ((S,R) [0,T]) not convex c initial c true must be small Solve for G(ω) from ω min to ω max No proof optimization converges

39 Full-waveform Inversion Work-around 2: Separation of Scales, example Virieux (09)

40 Outline Introduction Why we image the Earth How data are collected Imaging vs inversion Underlying physical model Data Model Imaging methods Kirchhoff One-way methods Reverse-time migration Full-waveform inversion Comparison of Methods

41 Comparison of Methods Kirchhoff vs One-way Fehler & Huang (02) Kirchhoff

42 Comparison of Methods Kirchhoff vs One-way Fehler & Huang (02) Phase-shift

43 Comparison of Methods Kirchhoff vs One-way Fehler & Huang (02) Split-step

44 Comparison of Methods Kirchhoff vs RTM Zhu & Lines (98)

45 Kirchhoff Comparison of Methods Kirchhoff vs RTM Zhu & Lines (98)

46 Comparison of Methods Kirchhoff vs RTM Zhu & Lines (98) Reverse-time

47 Model Comparison of Methods One-Way vs RTM Farmer (06)

48 One-way Comparison of Methods One-Way vs RTM Farmer (06)

49 Comparison of Methods One-Way vs RTM Farmer (06) Reverse-time

50 One-way Comparison of Methods One-Way vs RTM Farmer (06)

51 Comparison of Methods One-Way vs RTM Farmer (06) Reverse-time

52 Kirchhoff Comparison of Methods Real Data Farmer 2006

53 One-way Comparison of Methods Real Data Farmer 2006

54 Comparison of Methods Real Data Farmer 2006 Reverse-time

55 Summary of Methods Kirchhoff requires a lot of rays (or a simple model) One-way doesn t handle turning waves RTM separation of up/down-going waves is challenging c 0 is key FWI optimization doesn t convergence from an arbitrary starting model... and we never spoke about the source or multiple-scattering...

56 Edip Baysal, Dan D. Kosloff, and John W. C. Sherwood. Reverse time migration. Geophysics, 48(11): , J. F. Claerbout. Imaging the Earth s Interior. Blackwell Sci. Publ., Jon F. Claerbout. Toward a unified theory of reflector mapping. Geophysics, 36(3): , P. A. Farmer. Reverse time migration: Pushing beyond wave equation. EAGE Annual Meeting, Michael C. Fehler and Lianjie Huang. Modern imaging using seismic reflection data. Annual Review of Earth and Planetary Sciences, 30(1): , P. Lailly. The Seismic Inverse Problem as a Sequence of Before-Stack Migrations, pages Conference on Inverse Scattering: Theory and Applications. SIAM, Philadelphia, P. Lailly. Migration Methods: Partial but Efficient Solutions to the Seismic Inverse Problem. Inverse Problems on Acoustic and Elastic Waves. SIAM, Philadelphia, Faqi Liu, Guanquan Zhang, Scott A. Morton, and Jacques P. Leveille. Reverse-time migration using one-way wavefield imaging condition. SEG Technical Program Expanded Abstracts, 26(1): , Dan Loewenthal and Irshad R. Mufti. Reversed time migration in spatial frequency domain. Geophysics, 48(5): , 1983.

57 C. C. Stolk and M. V. de Hoop. Modeling of seismic data in the downward continuation approach. SIAM J. Appl. Math., 65(4): , C. C. Stolk and M. V. de Hoop. Seismic inverse scattering in the downward continuation approach. Wave Motion, 43(7): , Christiaan C. Stolk, Maarten V. de Hoop, and William W. Symes. Kinematics of shot-geophone migration. Geophysics, 74(6):WCA19 WCA34, W. W. Symes. The seismic reflection inverse problem. Inverse Problems, 25:123008, A. Tarantola. Inverse Problem Theory. Elsevier, 3 edition, Albert Tarantola. Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49(8): , Albert Tarantola. A strategy for nonlinear elastic inversion of seismic reflection data. Geophysics, 51(10): , M. E. Taylor. Pseudodifferential Operators. Princton University Press, Princton, NJ, J. Virieux and S. Operto. An overview of full-waveform inversion in exploration geophysics. Geophysics, 74(6):WCC1 WCC26, 2009.

58 N. D. Whitmore. Iterative depth migration by backward time propagation. In Expanded Abstracts, page S10.1. Society of Exploration Geophysicists, Jinming Zhu and Larry R. Lines. Comparison of kirchhoff and reverse-time migration methods with applications to prestack depth imaging of complex structures. Geophysics, 63(4): , 1998.

59 [10, 11, 2] [17, 12, 8] [3, 19, 9] [1, 6, 7] [15, 16, 14] [13, 5, 20] [4, 18]