Stacking-velocity tomography in tilted orthorhombic media

Transcription

1 Stacking-velocity tomography in tilted orthorhombic media Qifan Liu* and Ilya Tsvankin Center for Wave Phenomena, Colorado School of Mines

2 Methodology modes: PP, S 1 S 1, S 2 S 2 (PP + PS = SS) input: NMO ellipses, zero-offset times, reflection time slopes model: homogeneous layers with plane/curved boundaries inversion for: interval medium parameters, interfaces Developed for TTI and ORTH with horizontal symmetry plane (Grechka et al., 2005)

3 NMO ellipse V 2 nmo(α) = W 11 cos 2 α + 2W 12 sin α cos α + W 22 sin 2 α (Grechka and Tsvankin, 1998) single layer: W ij = f (p 1, p 2, q) (p 1, p 2, q): slowness vector of zero-offset ray q(p 1, p 2 ) from Christoffel equation layered model: Dix-type averaging (tracing zero-offset ray)

4 Stacking-velocity tomography (SVT) Input data for each event: τ 0, p 1, p 2 W 11, W 12, W 22

5 Workflow Work-flow trial model (interval parameters) zero-offset ray p 1, p 2,τ 0 reflector effective NMO ellipses W ij misfit in W ij and depths

6 Objective function F(m) = 1 W cal (m) W obs + α 1 z i(m) z j (m)

7 Inversion Gauss-Newton optimization Frechét derivatives feasibility depends on model assumptions

8 Orthorhombic model vector medium tilted orthorhombic (TOR) medium: parameters: (V V P0, V S0, ɛ (1,2), δ (1,2,3), γ (1,2) po, V so, ε 1, δ 1, γ 1, ε 2, δ 2, γ 2, δ 3 ) + (β 1, β 2, β 3 ) x 3 x 1 x 2

9 Tilted model orthorhombic vector (TOR) medium tilted orthorhombic (TOR) medium: parameters: (V V P0, V S0, ɛ (1,2), δ (1,2,3), γ (1,2) po, V so, ε 1, δ 1, γ 1, ε 2, δ 2, γ 2, δ 3, β) 1 +, (β β 21, ββ 32, β 3 )

10 Layered TOR medium interval parameters: V P0, V S0, ɛ (1,2), δ (1,2,3), γ (1,2), β 1, β 2, β 3 interfaces: polynomial coefficients

11 Data vector d(q, Y) = {τ 0,Q (Y), p 1,Q (Y), p 2,Q (Y), W 11,Q (Y), W 12,Q (Y), W 22,Q (Y)} Q = P, S 1, S 2 Y: CMP location

12 Plane dipping reflector ψ φ

13 Data vector for single TOR layer p 1 /p 2 = tan ψ for all three modes only 4 out of 6 in (p 1,Q, p 2,Q ) are independent only one τ 0,Q is independent (Grechka et al., 2005)

14 Feasibility for single TOR layer 14 measurements: τ 0,P p 1,P, p 1,S1, p 1,S2, p 2,P W 11,Q, W 12,Q, W 22,Q 15 parameters: V P0, V S0, ɛ (1,2), δ (1,2,3), γ (1,2), β 1, β 2, β 3, D, φ, ψ

15 Assumption: Reflector coincides with symmetry plane [x 1, x 2 ] β 2 = ψ, β 3 = φ Dip plane deviates from sym. plane δ (3) has no influence on W ij x 3 x 1 x 2

16 Additional assumption: Dip plane coincides with symmetry plane [x 1, x 3 ] β 1 = ψ W ij,q has two independent elements x 3 x 1 x 2

17 Analytic solution strike line: V (1) nmo,p = V P δ (1) V (1) nmo,s1 = V S σ (1) V (1) nmo,s2 = V S2 (V S2 = V S0 ) 1 + 2γ (1) = V S γ (2)

18 Analytic solution dip line: V (2) nmo,p = V P δ (2) / cos φ V (2) nmo,s1 = V S γ (2) / cos φ = V (1) nmo,s2 / cos φ V (2) nmo,s2 = V S σ (2) / cos φ p 1,Q = sin φ/v Q0 V P0, V S1, V S2 D = V P0 τ 0,P

19 P-waves only V (1) nmo,p = V P δ (1) V (2) nmo,p = V P δ (2) /cos φ model vector: V P0, δ (1,2), φ, D data vector: τ 0, p 1, V (1,2) nmo,p need one constraint (e.g., vertical velocity)

20 P-wave inversion (known vertical velocity)

21 Arbitrary dip-plane orientation x 3 x 1 x 2

22 Feasibility β 2 = ψ, β 3 = φ β 3 = φ 14 measurements: τ 0,P, p 1,P, p 1,S1, p 1,S2, p 2,P, W ij,q surements:,p, p 1,S1, p 1,S2, p 2,P, W ij,q 12 parameters: V P0, V S0, ɛ (1,2), δ (1,2), γ (1,2), β 1, D, φ, ψ meters:, ε 1,2, δ 1,2, γ 1,2, D, ψ, φ, β 1

23 Objective function

24 Inversion results

25 Inversion with noise (1% for τ 0, 2% for p 1,2, 2% for V nmo )

26 Inversion with noise (1% for τ 0, 5% for p 1,2, 3% for V nmo )

27 Inversion with noise (1% for τ 0, 5% for p 1,2, 3% for V nmo ) (vertical velocity known)

28 Summary extension of multicomponent SVT to tilted ORTH media single layer: - generally underdetermined - feasible if reflector coincides with symmetry plane - P-waves: need additional constraints

29 Ongoing work layered TOR medium horizontal + dipping reflectors curved interfaces

30 Acknowledgments Vladimir Grechka Andrés Pech A-Team