Imaging with Multiply Scattered Waves: Removing Artifacts

Transcription

1 Imaging with Multiply Scattered Waves: Removing Artifacts Alison Malcolm Massachusetts Institute of Technology Collaborators: M. de Hoop, Purdue University B. Ursin, Norwegian University of Science and Technology

2 Subsalt imaging 0 lateral position (km) depth (km) 5

3 Subsalt imaging 0 lateral position (km) depth (km) 5

4 Outline Overview of nonlinearity Standard Reverse-time imaging One-way approximate method Interferometric improvements

5 Nonlinearity c 0 (x) 2 2 t u 0(x, t) u 0 (x, t) = 0 L 0 u 0 = 0 c(x) 2 2 t u(x, t) u(x, t) = 0 Lu = 0 V = L L 0 = (c(x) 2 c 0 (x) 2 ) 2 t c 0 (x) C δc(x) oscillatory

6 Nonlinearity Lippmann-Schwinger: L 0 δu = Vu Born Approximation: L 0 δu = Vu 0 c 0 (x) 2 2 t u 0(x, t) u 0 (x, t) = 0 L 0 u 0 = 0 c(x) 2 2 t u(x, t) u(x, t) = 0 Lu = 0 V = L L 0 = (c(x) 2 c 0 (x) 2 ) 2 t

7 Outline Overview of nonlinearity Standard Reverse-time imaging One-way approximate method Interferometric improvements

8 Reverse-Time Migration d(s, r, t) = Ω { G(s, x, t t 0 ) V(x)G(x, r, t 0 )dxdt 0 }

9 Reverse-Time Migration I(x) = S x,t G(x, s, ω) x,t d (x, s, ω)dsdω Separation of Scales δc(x) = c 0 (x) + δc(x) c 0 C δc(x) oscillatory Include part of δc in c 0

10 Jones et al I(x) = Reverse-Time Migration S x,t G(x, s, ω) x,t d (x, s, ω)dsdω

11 I(x) = Reverse-Time Migration S x,t G(x, s, ω) x,t d (x, s, ω)dsdω 0 lateral position (km) depth (km) 5 Top Balder Ekofisk Top Chalk Top Hod Base Chalk m/s

12 Outline Overview of nonlinearity Standard Reverse-time imaging One-way approximate method Interferometric improvements

13 One-Way Migration Lu = f ( ) u z z u = ( )( ) ( ) 0 1 u 0 + A 0 z u f A(z, x, x, t ) = 2 x c(z, x) 2 2 t Taylor (81), Stolk & de Hoop (05)

14 One-way Migration Diagonalize (parabolic approx.) 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

15 Artifacts Series setup Solution: z U = BU + F L 0D G := (I z B)G = Iδ ( ) G+ 0 G = 0 G G propagator in c 0

16 Artifacts Series setup Bremmer Series (Bremmer 51, de Hoop 96) Born Series (Lippmann & Schwinger 50) L 0D δu = VU (I GV) δu = G(VU 0 ),

17 Artifacts Series setup Bremmer Series (Bremmer 51, de Hoop 96) Born Series (Lippmann & Schwinger 50) δu 1 = G(VU 0 ) δu m = G(VδU m 1 ) m = 2,..., M δu M δu m m=1

18 Artifacts Series setup δu = G(VU 0 ) + G(VG(VU 0 )) +... Set V = j V j equate orders G(V 1 U 0 ) = δu G(V 2 U 0 ) = G(V 1 G(V 1 U 0 )) G(V 3 U 0 ) = G(V 1 G(V 1 G(V 1 U 0 ))) e.g. Moskow (2008), Weglein (2003), AM & de Hoop (2005)

19 Nullspace series setup G(V 1 U 0 ) = δu G(V 2 U 0 ) = G(V 1 G(V 1 U 0 )) V 1 = V 1 + V 1 + V 1 G(V 2 U 0 ) = G(V 1 G(V 1 U 0)) G(V 1 G(V 1 U 0)) G(V 1 G(V 1 U 0)) G(V 1 G(V 1 U 0)) AM, Ursin, de Hoop (2009)

20 Nullspace series setup G(V 1 U 0 ) = δu G(V 2 U 0 ) = G(V 1 G(V 1 U 0 )) V 1 = V 1 + V 1 + V 1 G(V 1 G(V 1 U 0)) + G(V 1 G(V 1 U 0)) + G(V 1 U 0) = δd Similar to Cheney & Bonneau (2004)

21 One-Way Imaging Algorithm forward propagate source backward propagate data imaging condition s r Claerbout (71,85)

22 Imaging Algorithms source side u s (z, x, ω) = (G + (z, 0))(x, ω, s) receiver side u Σs (z, x, ω) = dr(g (z, 0))(r, ω, x)d(s, r, ω) Σ s

23 Imaging Algorithms source side u s (z, x, ω) = (G + (z, 0))(x, ω, s) receiver side u Σs (z, x, ω) = dr(g (z, 0))(r, ω, x)d(s, r, ω) Σ s V 1 (z, x) ds dωu s (z, x, ω)u Σ s (z, x, ω)ω 2

24 Nullspace series setup G(V 1 U 0 ) = δu G(V 2 U 0 ) = G(V 1 G(V 1 U 0 )) V 1 = V 1 + V 1 + V 1 G(V 1 G(V 1 U 0)) + G(V 1 G(V 1 U 0)) + G(V 1 U 0) = δd Similar to Cheney & Bonneau (2004)

25 Algorithm multiples propagate down store W form image, I V 1 W I propagate up W I form image r s

26 Algorithm multiples propagate down store W form image, I V 1 W I propagate up W I form image r r s

27 One-Way Example 0 lateral position (km) depth (km) 2 4

28 Nullspace Examples 0 lateral position (km) depth (km)

29 Nullspace Examples 0 receiver position (km) time (s) 2 3 4

30 Nullspace Examples 0 receiver position (km) time (s) 2 3 4

31 Nullspace Examples 0 lateral position (km) depth (km) 1 2 3

32 Nullspace Examples 0 lateral position (km) depth (km) 1 2 3

33 Nullspace Examples 0.5 lateral position (km) lateral position (km) depth (km) depth (km) V 1 V 1 V 1 V 1 V 1 V 1

34 Comparing Models depth (km) depth (km) depth (km)

35 Outline Overview of nonlinearity Standard Reverse-time imaging One-way approximate method Interferometric improvements

36 Interferometry r 1 r 2 s H(r 1, r 2, ω) S G(s, r 1, ω)g (s, r 2, ω)ds Vasconcelos (2007)

37 Interferometry r 1 r 2 s H(r 1, r 2, ω) S G(s, r 1, ω)g (s, r 2, ω)ds Vasconcelos (2007)

38 Interferometry H(r 1, r 2, ω) = C Assuming: Far-field ν G iωg Radiation condition S G (s, r 1, ω) ν G(s, r 2, ω) ν G (s, r 1, ω)g(s, r 2, ω) Homogeneous model outside S, S circle H(r 1, r 2, ω) G(s, r 1, ω)g (s, r 2, ω)ds S

39 Interferometry and RTM s x d Born (x, ω, s) = Ω y x H Σ s 0 (x, ω, x )V(x )G 0 (x, ω, s)dx Stolk et al. (2009) x

40 Interferometry and RTM r 1 r 2 s d Born (x, ω, s) = Ω H Σ s 0 (x, ω, x )V(x )G 0 (x, ω, s)dx Stolk et al. (2009)

41 Interferometry and RTM Standard imaging: d Born (x, ω, s) = G 0 (x, ω, x )V(x )G 0 (x, ω, s)dx Ω Ω G 0 (s, ω, y)d Born (y, ω, s)ds V(y) Interferometry from migration: d Born (x, ω, s) = H Σ s 0 (x, ω, x )V(x )G 0 (x, ω, s)dx Ω

42 Interferometry and RTM s x Ω y x G 0 (s, ω, x )d Born (x, ω, s)ds x = H Σ s 0 (x, ω, x )V(x ) Esmersoy (1988), van Manen (2006), Thorbecke (2007) Vasconcelos et al. (2009)

43 Interferometry and RTM s x s x y x x x x similar to extended images of e.g. Symes (2008), Vasconcelos (2009)

44 Summary Standard Reverse-time imaging ignores separation of scales One-way approximate method respects separation of scales requires multiple passes Interferometric improvements respects separation of scales single pass artifacts?

45 Acknowledgements Ivan Vasconcelos & Ian Jones GX Technologies Total

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