Waveform inversion and time-reversal imaging in attenuative TI media

Transcription

1 Waveform inversion and time-reversal imaging in attenuative TI media Tong Bai 1, Tieyuan Zhu 2, Ilya Tsvankin 1, Xinming Wu 3 1. Colorado School of Mines 2. Penn State University 3. University of Texas at Austin

2 Outline 2 nearly constant-q wave equation (GSLS) waveform inversion with local similarity decoupled constant-q wave equation time-reversal imaging with Q compensation

3 Outline 2 nearly constant-q wave equation (GSLS) waveform inversion with local similarity decoupled constant-q wave equation time-reversal imaging with Q compensation

4 Outline 2 nearly constant-q wave equation (GSLS) waveform inversion with local similarity decoupled constant-q wave equation time-reversal imaging with Q compensation

5 Forward simulation 3 equation of motion: ρ ü i σ ij, j = f i ρ: density u: displacement stress-strain relationship: σ ij Ψ ijkl ε kl = 0 σ: stress ε: strain Ψ: relaxation function f: body force r: memory variable

6 Forward simulation 3 equation of motion: ρ ü i σ ij, j = f i ρ: density u: displacement stress-strain relationship: σ ij C U ijklε kl C ijkl r kl = 0 memory variables: ṙ kl = 1 τ σ (r kl + ε kl ) σ: stress ε: strain Ψ: relaxation function f: body force r: memory variable

7 Thomsen-style parameters (P & SV) 4 A P0 1 2Q P0 = 1 2Q 33 A S0 1 2Q S0 = 1 2Q 55 ε Q Q 33 Q 11 Q 11 δ Q 1 d 2 A P 2A P0 dθ 2 θ=0 Y.Zhu and Tsvankin (2006)

8 P- and SV-waves in TI media 5 Elastic Q P0 = Q S0 = 30 P SV

9 6 P- and SV-waves in TI media Isotropic Q ε Q = 0.6 δ Q = 0 Difference (Iso - TI)

10 7 P- and SV-waves in TI media Isotropic Q ε Q = 0 δ Q = 1.5 Difference (Iso - TI)

11 Outline 8 nearly constant-q wave equation (GSLS) waveform inversion with local similarity decoupled constant-q wave equation time-reversal imaging with Q compensation

12 Adjoint-state method 9 Objective function: F (m) = 1 u(x, t, m) d(x, t) 2 2

13 Adjoint-state method Objective function: F (m) = 1 2 u(x, t, m) d(x, t) 2 Gradients: F T = C ijkl 0 u i x j r kl dt Tarantola (1988); Bai et al. (2017) r: forward-simulated memory variables u : adjoint displacement 9

14 Parameterization: Attenuation coefficients 10 A P0 1 2Q 33 P-wave vertical coefficient A S0 1 2Q 55 S-wave vertical coefficient A Ph (1 + ε Q ) A P0 1 2Q 11 A Pn (1 + δ Q ) A P0 P-wave horizontal coefficient analogous to V nmo

15 BP section 11 A P0 A S0 A Ph A Pn

16 Initial model 12 A P0 A S0 A Ph A Pn

17 Inverted model 13 A P0 A S0 A Ph A Pn

18 BP section 14 A P0 A S0 A Ph A Pn

19 Data fitting (5% error in V P0 &V S0 ) 15 Observed Simulated Difference

20 Inverted model (distorted velocity) 16 A P0 A S0 A Ph A Pn

21 Preconditioning with local similarity 17 F (m) = 1 2 u(x r, t, m) S(x r, t) d(x r, t) 2 S(x r, t): shifting operator from local similarity (Fomel, 2009)

22 Preconditioning with local similarity 17 F (m) = 1 2 u(x r, t, m) S(x r, t) d(x r, t) 2 F (m) m = [ ] T u(xr, t, m) [u(x r, t, m) S(x r, t) d(x r, t)] m S(x r, t): shifting operator from local similarity (Fomel, 2009)

23 Preconditioning with local similarity 17 F (m) = 1 2 u(x r, t, m) S(x r, t) d(x r, t) 2 F (m) m = [ ] T u(xr, t, m) [u(x r, t, m) S(x r, t) d(x r, t)] m S(x r, t): shifting operator from local similarity (Fomel, 2009)

24 Data fitting (local similarity) 18 Shifted Simulated Difference

25 Inverted model (local similarity) 19 A P0 A S0 A Ph A Pn

26 Inverted model (accurate velocity) 20 A P0 A S0 A Ph A Pn

27 Summary (WI) 21 local similarity corrects for time errors

28 Summary (WI) 21 local similarity corrects for time errors velocity errors influence amplitude

29 Summary (WI) 21 local similarity corrects for time errors velocity errors influence amplitude future: WE-based spectral-ratio method

30 Outline 22 nearly constant-q wave equation (GSLS) waveform inversion with local similarity decoupled constant-q wave equation time-reversal imaging with Q compensation

31 Viscoelastic constitutive law 23 σ mn = Ψ mnpq ε pq

32 Viscoelastic constitutive law 23 σ mn = Ψ mnpq ε pq Kjartansson s constant-q model

33 Viscoelastic constitutive law 23 σ mn = Ψ mnpq ε pq Kjartansson s constant-q model Fractional time derivative

34 Viscoelastic constitutive law 23 σ mn = Ψ mnpq ε pq Kjartansson s constant-q model Fractional time derivative

35 Viscoelastic constitutive law 23 σ mn = Ψ mnpq ε pq Kjartansson s constant-q model Fractional time derivative Fractional Laplacian

36 Viscoelastic WE with fractional Laplacian 24 σ ij = Dispersion {}}{ f 1 (C ijkl, γ ijkl, ω 0 ) ( 2 ) γ ijkl ε kl ( ) ( + f 2 C ijkl, γ ijkl, ω ) 2 γ ijkl 1 ε 2 kl 0 }{{ t} Amplitude

37 Viscoelastic WE with fractional Laplacian σ ij = Dispersion {}}{ f 1 (C ijkl, γ ijkl, ω 0 ) ( 2 ) γ ijkl ε kl ( ) ( + f 2 C ijkl, γ ijkl, ω ) 2 γ ijkl 1 ε 2 kl 0 }{{ t} Amplitude γ ij = 1 π tan 1 ( 1 Q ij ), 24

38 Viscoelastic WE with fractional Laplacian σ ij = Dispersion {}}{ f 1 (C ijkl, γ ijkl, ω 0 ) ( 2 ) γ ijkl ε kl ( ) ( + f 2 C ijkl, γ ijkl, ω ) 2 γ ijkl 1 ε 2 kl 0 }{{ t} Amplitude γ ij = 1 π tan 1 ( 1 Q ij ), 0 < γ ij < 0.5 for Q ij > 0 24

39 Implementation 25 finite-differences for time derivative Fourier pseudospectral method for fractional Laplacian

40 Validation test 26

41 Spectral ratios 27 V P0 = 3 km/s V S0 = 1.5 km/s ε = 0 δ = 0 ρ = 2.0 g/m 3 Q P0 = 50 Q S0 = 30 ε Q = 0 δ Q = 0

42 Spectral ratios 28 analytic V P0 = 3 km/s V S0 = 1.5 km/s ε = 0.2 δ = 0.08 ρ = 2.0 g/m 3 Q P0 = 35 Q S0 = 60 ε Q = 0.3 δ Q = 0.6

43 Wavefield comparison V P0 = 2 km/s V S0 = 1 km/s Elastic Viscoelastic ε = 0 δ = 0 ρ = 2.0 g/m 3 Amplitude only 3 4 Dispersion only Q P0 = 20 Q S0 = 20 ε Q = 0.6 δ Q = 1.2

44 Wavefield comparison V P0 = 2 km/s V S0 = 1 km/s Elastic Viscoelastic ε = 0 δ = 0 ρ = 2.0 g/m 3 Amplitude only 3 4 Dispersion only Q P0 = 20 Q S0 = 20 ε Q = 0.6 δ Q = 1.2

45 Wavefield comparison V P0 = 2 km/s V S0 = 1 km/s Elastic Viscoelastic ε = 0 δ = 0 ρ = 2.0 g/m 3 Amplitude only 3 4 Dispersion only Q P0 = 20 Q S0 = 20 ε Q = 0.6 δ Q = 1.2

46 Wavefield comparison 30 Elastic Amplitude only 1 2 Viscoelastic Dispersion only 3 4 V P0 = 2 km/s V S0 = 1 km/s ε = 0 δ = 0 ρ = 2.0 g/m 3 Q P0 = 20 Q S0 = 20 ε Q = 0.6 δ Q = 1.2

47 Wavefield comparison 31 V P0 = 2 km/s V S0 = 1 km/s ε = 0 δ = 0 ρ = 2.0 g/m 3 Q P0 = 20 Q S0 = 20 ε Q = 0.6 δ Q = 1.2

48 Trace comparison Elastic Phase only Amplitude only Viscoelastic 32

49 Trace comparison Elastic Phase only Amplitude only Viscoelastic 32

50 Outline 33 nearly constant-q wave equation (GSLS) waveform inversion with local similarity decoupled constant-q wave equation time-reversal imaging with Q compensation

51 Time reversal (TR) with attenuation 34 T.Zhu (2014)

52 Time reversal (TR) with attenuation 34 T.Zhu (2014)

53 Forward simulation σ ij = Dispersion {}}{ f 1 (C ijkl, γ ijkl, ω 0 ) ( 2 ) γ ijkl ε kl ( ) ( + f 2 C ijkl, γ ijkl, ω ) 2 γ ijkl 1 ε 2 kl 0 }{{ t} Amplitude γ ij = 1 π tan 1 ( 1 Q ij ) 35

54 Backward simulation σ ij = Dispersion {}}{ f 1 (C ijkl, γ ijkl, ω 0 ) ( 2 ) γ ijkl ε kl ( ) ( f 2 C ijkl, γ ijkl, ω ) 2 γ ijkl 1 ε 2 kl 0 }{{ t} Amplitude γ ij = 1 π tan 1 ( 1 Q ij ) 35

55 TR imaging for microseismic sources 36 origin time Q P0 = 30 Q S0 = 20 ε Q = 0.6 δ Q = 0.4

56 Simulated data 37 Elastic

57 Simulated data 37 Elastic Viscoelastic

58 Simulated data Elastic Viscoelastic (noisy) 37

59 Source localization by TR IC = max u x Reference (noise-free) 38

60 Source localization by TR IC = max u x Reference (noise-free) 38

61 Source localization by TR IC = max u x Reference (noise-free) No Q-comp 38

62 Source localization by TR IC = max u x Reference (noise-free) No Q-comp Isotropic Q-comp 38

63 Source localization by TR IC = max u x Reference (noise-free) No Q-comp Isotropic Q-comp 38

64 Source localization by TR IC = max u x Reference (noise-free) No Q-comp Isotropic Q-comp Anisotropic Q-comp 38

65 Source localization by TR IC = max u x Reference (noise-free) No Q-comp Isotropic Q-comp Anisotropic Q-comp 38

66 Summary (TR) 39 decoupled constant-q propagator for TI media

67 Summary (TR) 39 decoupled constant-q propagator for TI media Q-compensated TR imaging for microseismic

68 Summary (TR) 39 decoupled constant-q propagator for TI media Q-compensated TR imaging for microseismic future: Q-compensated RTM

69 Acknowledgments 40 Oscar Jarillo Michel Ivan Lim Chen Ning