Elastic wave-equation migration for laterally varying isotropic and HTI media. Richard A. Bale and Gary F. Margrave

Transcription

1 Elastic wave-equation migration for laterally varying isotropic and HTI media Richard A. Bale and Gary F. Margrave a

2 Outline Introduction Theory Elastic wavefield extrapolation Extension to laterally heterogeneous media Migration imaging condition Examples Elastic HTI modeled data Marmousi 2: elastic OBC modeled data Conclusions

3 Introduction Drawbacks to scalar extrapolation for elastic migration: Neglects mode conversions Fails to keep track of polarization changes Difficult to fully account for anisotropic effects, in particular shear wave splitting (birefringence) for HTI media

4 VTI and HTI: decks of Cards VTI: Vertical symmetry axis E.g. Shales y HTI: Horizontal symmetry axis φ x Strong (fast) direction Weak (slow) direction E.g. Fractured carbonates

5 Variation of Polarization with Slowness: HTI

6 Introduction Standard processing of birefringent shear waves: Assumes vertical incidence waves Neglects variation of shear wave polarization with propagation angle Neglects changes in velocity, (and time delay) with propagation angle Often neglect variations of symmetry axis with depth

7 Outline Introduction Theory Elastic wavefield extrapolation Extension to laterally heterogeneous media Migration imaging condition Examples Elastic HTI modeled data Marmousi 2: elastic OBC modeled data Conclusions

8 Theory First order, 6x6 form of elastic wave-equation: b Ab z = iω A = 6x6 fundamental elasticity matrix depends on horizontal slowness p, frequency ω and elastic constants u = displacement vector τ = vertical traction vector u b = τ

9 Theory Diagonalize: v Λv z = iω b = Dv v = v v U D v U = up-going wave-mode vector v D = down-going wave-mode vector Λ = diagonal matrix of eigenvalues (vert. slowness) D = eigenvector matrix (from polarizations) Solution: v ( ) ( ) ( ) 0 z ( i z z = e ωλ v z ) i U ( ) ( ) D z z0 z = e ωλ v z UD 0UD 0

10 V(z) Extrapolation v( p,, 1 )= z ω n + e iωλ n ( z ) (,, ω ) n+ 1 zn v p z n p : horizontal slowness z n : n th depth level v : wave-mode vector in k-ω domain ( k= pω ) Λ n : diagonal matrix of eigenvalues (vert. slowness)

11 V(z) Extrapolation recomposition extrapolation decomposition b ( ) iωλ ( z ) 1,, ω (,, ω ) n n+ 1 zn p z = D e D b p z n+ 1 n n n p : horizontal slowness z n : n th depth level v : wave-mode vector in k-ω domain ( k= pω ) Λ n : diagonal matrix of vertical slowness (P, S1, S2) b : displacement-stress vector in k-ω domain D n : eigenvector matrix (from polarizations)

12 V(z) Extrapolation z n z n+1 continuity b n bn+1 b n+1 decomposition 1 D n Dn recomposition + v n v n+1 e phase shift iωλ n ( z z ) n+1 n

13 V(x,z) Extrapolation Operator ( ) ( ) ( ) ( ) ( ) dp e z p p x p x p x z x px i n n n n n PSPI ω ω ω π ω ω + =,,,,,, 2,, 1 1 b D E D b Lateral Dependence Fourier Transform: ( ) ( ) dx e z x z p px i n n = ω ω,ω,,, b b Extrapolation: ( ) ( )( ) [ ] n n n n z z p x i p x = +1, exp,, Λ E ω ω

14 PSPI Elastic Extrapolation z D(x 1 ), Λ(x 1 ) D(x 2 ), Λ(x 2 ) D(x 3 ), Λ(x 3 ) z+ z v P v S1 v S2 φ :

15 Spatial Interpolation: PSPI Standard PSPI Extrapolate with N reference velocities Interpolate based on actual velocity at each spatial position Isotropic elastic case: dependence on V P and V S is (almost) separable cost N Vp + N Vs OK HTI elastic case: non-separable dependence on 6 parameters cost (N Vp N Vs )(N ε N δ N γ )N φ BAD!

16 Spatial Interpolation: PSPAW Phase shift plus adaptive windowing Windows ( molecules ) constructed from elementary small windows ( atoms ) c.f. Scalar adaptive method (Grossman et al., 2002) 1. Compute phase slowness for P, S1, S2 modes as a function of lateral position and phase angle 2. For each molecule, atom acceptance based on: Maximum phase error over slownesses Maximum variation of HTI symmetry axis 3. Begin new molecule if either criteria are violated Cost # Windows (usually OK)

17 Adaptive Windowing: Phase slowness for HTI to Isotropic transition model

18 Imaging Condition Forward extrapolated source wavefield: Backward extrapolated w D = ( D D D w w ) T wp S1 S 2 ( U U U v ) T receiver wavefield: U = vp vs1 vs 2 P-wave S1-wave S2-wave P-P Image I w v D U P P P P = D 2 wp dω P-S1 Image I w v D U P S1 P S1 = D 2 wp dω etc.

19 Outline Introduction Theory Elastic wavefield extrapolation Extension to laterally heterogeneous media Migration imaging condition Examples Elastic HTI modeled data Marmousi 2: elastic OBC modeled data Conclusions

20 Isotropic Model

21 Isotropic Model

22 Isotropic Data P-P Image (PSPAW)

23 Isotropic Data P-S Image (PSPAW)

24 AVO on Flat Reflector from Migration of Single Shot PP PS

25 HTI: S1=-45, S2=45 HTI Model Iso: S1 =SH=90, S2 =SV=0 Symmetry axis inline NOTE: In following images, we (arbitrarily) assign SH mode to S1, and SV to S2, for isotropic layers.

26 HTI Data P-P Image (PSPAW)

27 HTI Data P-S1 Image (PSPAW) Migrated with true HTI model

28 HTI Data P-S1 Image (PSPAW) Incorrect Imaging of isotropic interface (no SH mode should exist) Migrated with isotropic model

29 HTI Model P-S2 Image (PSPAW) Migrated with true HTI model

30 HTI Data P-S2 Image (PSPAW) Focussing degrades Migrated with isotropic model

31 The Marmousi-2 Elastic OBC Model From Martin, Marfurt and Larsen, Marmousi-2: an updated model for the investigation of AVO in structurally complex areas, SEG 2002 Original (acoustic) Marmousi model

32 Marmousi-2 Mid-section: P-Impedance Water Layer

33 Marmousi -2 Mid-Section: PP Image (PSPI) Water layer multiples Diffraction Noise

34 Marmousi-2 Mid-section: S-Impedance Water Layer: I S =0

35 Marmousi -2 Mid-Section: PS Image (PSPI) Water layer Multiple?

36 Marmousi-2 Shallow: I P Water Wet Sand Gas Charged Sand

37 Marmousi -2 Shallow: PP Image

38 Marmousi -2 Shallow: PS Image

39 Marmousi-2 Shallow: I S Water Wet Sand Gas Charged Sand

40 Conclusions Developed elastic wave-equation migration applicable to HTI anisotropy AVO response compares well to Zoeppritz for flat reflector under isotropic layer Two PSPI-type algorithms for spatial variations Standard PSPI for isotropic cases PSPAW for HTI HTI migration focuses S1 and S2 images - isotropic migration fails to Marmousi tests demonstrate: Multiples and aliased noise are problematic Imaging in structural area: PP better than PS Shallow resolution of PS better than PP Fluid lithology discrimination

41 Acknowledgements Sponsors of CREWES Sponsors of POTSI: Pseudodifferential Operator Theory and Seismic Imaging MITACS: Mathematics of Information Technology and Complex Systems PIMS: Pacific Institute of the Mathematical Sciences NSERC: Natural Sciences and Engineering Research Council of Canada Allied Geophysical Laboratory, University of Houston for permission to use the Marmousi II data Prof. Robert Wiley Gary Martin (GX Technology) Kevin Hall, CREWES