3D anisotropic phase shift operators

Robert J. Ferguson and Gary F. Margrave

U of C CREWES

Introduction

• Through Rayleigh-Sommerfeld modelling and imaging we can:

– Control up ↑ and down ↓ and mode conversion:– Cause reflection and transmission.– Ensure high frequency stability and reduced runtime.

• Extrapolate wavefields with phaseshift operators one planewave at atime.

• Handle heterogeneous media with Gabor windows [Margrave et al., 2002].

• Anisotropy?

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Phaseshift operators in TTI media

• Phaseshift operators are defined relative to the recording surface.

• TTI media are defined relative to the axis of symmetry.

• To reconcile, build operators in the TI coordinate system, map to thedata coordinates, then apply.

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Planewave vs. angle, parameters vs. coefficients

• Monochromatic spectrum ϕ is phaseshifted to a new depth ∆z accordingto

ϕ∆z = ϕ ei ω q ∆z. (1)

• In TI media q is [Daley and Hron, 1977]:

q → q (pI, C11, C12, C13, C33, C44) . (2)

• Alternatively, q is [Thomsen, 1986]:

q → q (θI, α0, β0, ε, δ∗, γ) . (3)

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• For RS, we want to use planewaves, but α0, β0, ε, δ∗, and γ are ubiquitous

- reconcile notation [Ferguson and Sen, 2004] so:

q → q (pI, α0, β0, ε, δ∗, γ) . (4)

– Matlab’s symbolic math toolbox helps.

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What is pI in a TTI medium?

• For TTI, pI = sin θIv

where θI is the angle between planewave unit normalp and normal a to the symmetry plane.

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−0.20

0.20.4

0.60.8

11.2

0

0.5

1

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x2 (m)

x1 (m)

p

a

x 3 (m

)

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• Use Snell’s Law to eliminate raytracing:

pI =sin θI

v= |p × a| |p| . (5)

• Recall, p1 = kxω

, p2 =ky

ω, and p3 = 1

v

√

1 − (v p1)2− (v p2)

2, so given

α0, β0, ε, δ∗, and γ, qα, qβSHand qβSV

are computable for TTI media.

• Note, q are referenced to a - map q (a → p) to extrapolate ϕ (p) soinvoke Snell’s Law again so that

q2

r = q2 + p2

I − p2

1 − p2

2, (6)

is the general form for the three modes.

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Branch points

• For VTI and HTI, the evanescent region is defined.

• TTI is more complicated, and branchpoints must be determined, forexample, set qβSH

= 0 to get [Bale, 2006]

1 = β2

0

[

p2

1 + p2

2

]

[2 γ + 1] , (7)

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−8 −6 −4 −2 0 2 4 6

x 10−4

0

1

2

3

4

5

6

7

8x 10

−4

qβSH

branch points

p1 (s m−1)

q (s

m−

1 )

ℜℑBP

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Example: P-waves

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Example: SV-waves

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Example: SH-waves

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Conclusions and future work

• 3D phaseshift operators for homogeneous TTI media:

– P, SV, and SH.

• Need group direction impulses for SV and SH.

• Compute branch points for P and SV.

• Rather than map qpI→ qp then extrapolate ϕp, extrapolate ϕpI

thenmap ϕ (pI) → ϕ (p).

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Acknowledgements

• CREWES sponsors and staff.

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References

[Bale, 2006] Bale, R. A., 2006, Elastic wave-equation depth migration ofseismic data for isotropic and azimuthally anisotropic media: PhD thesis,University of Calgary.

[Daley and Hron, 1977] Daley, P. F. and F. Hron, 1977, Reflection andtransmission coefficients for transversely isotropic media: Bulletin of theSeismological Society of America, 56, 87–94.

[Ferguson and Sen, 2004] Ferguson, R. J. and M. K. Sen, 2004, Estimatingthe elastic parameters of anisotropic media using a joint inversion ofp-wave and sv-wave traveltime error: Geophysical Prospecting, 52, 547–558.

[Margrave et al., 2002] Margrave, G., M. Lamoureux, J. Grossman, and V.Iliescu, 2002, Gabor deconvolution of seismic data for source waveform

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and Q correction: 72nd Ann. Internat. Mtg, 2190–2193, Soc. of Expl.Geophys.

[Thomsen, 1986] Thomsen, L., 1986, Weak elastic anisotropy: Geophysics,51, 1954–1966. Discussion in GEO-53-04-0558-0560 with reply by author.

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