Frequency dependent AVO analysis

Transcription

1 Frequency dependent AVO analysis of P-, S-, and C-wave elastic and anelastic reflection data Kris Innanen CREWES 23 rd Annual Sponsor s Meeting, Banff AB, Nov 30-Dec 2, 2011

2 Outline 1. Geophysical setting: elastic & anelastic targets 2. Formulation 3. Results: expansions truncated at 2 nd order 4. Predictions & observations: I: elastic: R PS and target V P ; importance of V P /V S = 2 II: anelastic: mode conversions from Q S / Q P contrasts III: anelastic: R PS and R SS inversion; Q 1 R/ ω 5. Conclusions

3 Geophysical setting θ R PS (θ, ω) R PP (θ, ω) R SS (φ, ω) φ R SP (φ, ω) V P0,V S0, ρ 0 V P0,V S0, ρ 0 V P1,V S1, ρ 1,Q P1,Q S1 V P1,V S1, ρ 1,Q P1,Q S1 Elastic overburden, anelastic target θ φ : P-wave angle of incidence : S-wave angle of incidence

4 Theoretical tools 1. Knott-Zoeppritz: exact equations for R PP, R PS, etc. precise amplitudes regardless of angle (for plane waves) inaccessible: geophysical interpretation difficult 2. Aki-Richards approximation for R PP, R PS, etc. highly accessible to geophysical interpretation inaccurate RE. large contrasts, influence of parameters

5 Example: R PS and target V P 1. Knott-Zoeppritz: exact equations for R PP, R PS A = ρ 1 /ρ 0 B = V S0 /V P0 C = V P1 /V P0 D = B(V S1 /V S0 ) X = sin θ S Y = (1 2Y 2 X 2 ) X S B CX S D S 1 BX S C DX 2B 2 XS 1 BS B 2AD 2 XS C ADS D S B 2B 2 XS B ACS D 2AD 2 XS D S Y = 1 Y 2 X 2 R P R S T P = T S X S 1 2BS 1 X S B Question: what influence does a ΔV P have on R PS? Answer: I haven t the foggiest idea. Inaccessible to analysis

6 Example: R PS and target V P 2. Aki-Richards: approximate form for R PS : R PS γ tan φ 2 [ (1 + δ) ρ ρ +2δ V S V S ] where δ = 2 sin2 θ γ 2 +2 cos θ cos φ γ Question: what influence does a ΔV P have on R PS? Answer: easy no effect. Highly accessible to analysis

7 Example: R PS and target V P P I S R φ 0 θ 0 V S0 = 1500m/s ρ 0 =2.0gm/cc V P0 = 3000gm/cc V S1 = 2300m/s ρ 1 =2.5gm/cc V P1A V P1B = 4000gm/cc = 4500gm/cc V P1C = 5000gm/cc / 0 1 %" %" %",)&, -. %" %" %" %# %# %# " # $ &'()*+ " # $ &'()*+ " # $ &'()*+ Highly accessible to analysis Wrong for certain angles/contrasts

8 Formulation 1. Matrix forms for Knott-Zoeppritz 2. Adjust for finite Q P, Q S in target medium 3. All contrasts 5 dimensionless perturbations: a VP =1 V P 2 0 VP 2 a VS =1 V S VS 2 1 a ρ =1 ρ 0 a QP = 1 a QS = 1 ρ 1 Q P1 Q S1 4. Expand R PP, R PS, R SP, R SS in orders of each & sinθ 5. Truncate at 1 st or 2 nd order 1 st order ~ Aki-Richards approximation with incidence angle 2 nd order ~ low-moderate angle, low-large contrast 3 rd order and higher ~ negligible

9 Results (anelastic P-P example) R PP (θ 0, ω) =R (1) PP (θ 0, ω)+r (2) PP (θ 0, ω)+..., The output is the set of Δs for the expansion: R (1) PP (θ 0, ω) = p a VP + s a VS + ρ a ρ + Qp a QP + Qs a QS R (2) PP (θ 0, ω) = pp a 2 VP + ps a VP a VS + pρ a VP a ρ + pqp a VP a QP + pqs a VP a QS + ss a 2 VS + sρ a VS a ρ + sqp a VS a QP + sqs a VS a QS + ρρ a 2 ρ + ρqp a ρ a QP + ρqs a ρ a QS + QpQp a 2 QP + QsQp a QP a QS + QsQs a 2 QS

10 Results (anelastic P-P example) Interpretability & ease of use. No target anelasticity? Q P contrast only? R (1) PP (θ 0, ω) = p a VP + s a VS + ρ a ρ + Qp a QP + Qs a QS R (2) PP (θ 0, ω) = pp a 2 VP + ps a VP a VS + pρ a VP a ρ + pqp a VP a QP + pqs a VP a QS + ss a 2 VS + sρ a VS a ρ + sqp a VS a QP + sqs a VS a QS + ρρ a 2 ρ + ρqp a ρ a QP + ρqs a ρ a QS + QpQp a 2 QP + QsQp a QP a QS + QsQs a 2 QS

11 Results (anelastic P-P example) Interpretability & ease of use. No target anelasticity? Q P contrast only? R (1) PP (θ 0, ω) = p a VP + s a VS + ρ a ρ + Qp a QP + Qs a QS R (2) PP (θ 0, ω) = pp a 2 VP + ps a VP a VS + pρ a VP a ρ + pqp a VP a QP + pqs a VP a QS + ss a 2 VS + sρ a VS a ρ + sqp a VS a QP + sqs a VS a QS + ρρ a 2 ρ + ρqp a ρ a QP + ρqs a ρ a QS + QpQp a 2 QP + QsQp a QP a QS + QsQs a 2 QS

12 Results (anelastic P-P example) Interpretability & ease of use. No target anelasticity? Q P contrast only? = 0 (Lines et al., 2011) R (1) PP (θ 0, ω) = p a VP + s a VS + ρ a ρ + Qp a QP + Qs a QS R (2) PP (θ 0, ω) = pp a 2 VP + ps a VP a VS + pρ a VP a ρ + pqp a VP a QP + pqs a VP a QS + ss a 2 VS + sρ a VS a ρ + sqp a VS a QP + sqs a VS a QS + ρρ a 2 ρ + ρqp a ρ a QP + ρqs a ρ a QS + QpQp a 2 QP + QsQp a QP a QS + QsQs a 2 QS

13 Observation I: R PS and target V P R PS (θ 0 )=Λ s a VS + Λ ρ a ρ +Λ ps a VP a VS + Λ pρ a VP a ρ + Λ ss a 2 VS + Λ sρ a VS a ρ + Λ ρρ a 2 ρ +... Linear: no influence of V P Nonlinear: coupling of V S, ρ with V P %" %" %" &'(& )* %" %" %" %# %# %# " # $ " # $ " # $ Linear &'(& )* %, %+ %+, %, %+ %+, %, %+ %+, %" + " (-.'/0 %" + " (-.'/0 %" + " (-.'/0

14 Observation I: R PS and target V P R PS (θ 0 )=Λ s a VS + Λ ρ a ρ +Λ ps a VP a VS + Λ pρ a VP a ρ + Λ ss a 2 VS + Λ sρ a VS a ρ + Λ ρρ a 2 ρ +... Linear: no influence of V P Nonlinear: coupling of V S, ρ with V P %" %" %" &'(& )* %" %" %" %# %# %# " # $ " # $ " # $ Nonlinear &'(& )* %, %+ %+, %, %+ %+, %, %+ %+, %" + " (-.'/0 %" + " (-.'/0 %" + " (-.'/0

15 Observation II: V P /V S = 2 ALL 2 nd order coupling involving ρ have coefficients Δ proportional to ρ[ ] 1 2 ( VS0 V P0 ) A V P -V S ratio of 2 has an inherent mathematical importance for the Zoeppritz equations, originating from elastodynamics + BCs alone.

16 Prediction: mode conversions from Q S contrasts θ R PS (θ, ω) Lines, Wong, Sondergeld, Treitel & others: P-P reflections from contrasts in Q only. V P0,V S0, ρ 0 Mode conversions: R V P0,V S0, ρ 0,Q P1,Q PS, R SP? S1 Λ Qp (θ 0, ω) =0 Λ QpQp (θ 0, ω) =0 Λ QsQs (θ 0, ω) =0 R PS (θ 0, ω) =Λ Qp a QP + Λ Qs a QS + Λ QpQp a 2 QP + Λ QpQs a QP a QS + Λ QsQs a 2 QS Λ Qs (θ 0, ω) =2F S (ω) ( VS0 V P0 ) sin θ 0 Λ QpQs (θ 0, ω) = ( VS0 V P0 ) F P (ω)f S (ω) sin θ 0

17 Prediction: mode conversions from Q S contrasts A Q S contrast alone generates a mode conversion The amplitude of the Q S mode conversion is proportional to A Q P contrast alone generates no mode conversion However, Q P contrasts will influence, to 2 nd order, the amplitudes of mode conversions due to Q S contrasts, with a strength proportional to ( ω log ω p ( ω log ) ω s ) ( ω log ω s ( VS0 ) V P0 ) sin θ 0 ( VS0 V P0 ) sin θ 0

18 Linear anelastic inversion Frequency rate of change of R PP, R PS, R SS ( π Q 1 S 2 V P 0 ω ) V S0 sin θ ω R PS(θ, ω) Q 1 S Q 1 P (2π ω) ω R PP(0, ω) [2π ω (1 7 sin2 φ) 1 ] ω R SS(φ, ω) ))(/'*0'12345(14'**313'52 $ 6(((+("(& )) (7("((. %" #" &'(& )) %# %#" %$ # " %$" " # #" $ *(+,-. " # #" $ *(+,-.

19 Conclusions 1. Accuracy v. interpretability scalable analysis 2. 2 nd order influences for low angles, large contrasts 3. Predictions & observations: I: elastic: R PS and target V P ; importance of V P /V S = 2 II: anelastic: mode conversions from Q S / Q P contrasts III: anelastic: R PS and R SS inversion; Q R/ ω 4. Practical frequency dependent AVO and/or AVF analysis? Next talk.

20 Acknowledgments CREWES project sponsors & researchers NSERC

21 Linear anelastic inversion R PP (θ), R PS (θ) in the elastic limit (-) and for 2 frequencies (-,-) & */. '( *2. 0,)0 11 0,)0 11 '( '( & " # $ % )*+,-. '4( '4 '$( '$ '(( '( *5. '#( ( & &( " )*+,-. 0,)0 13 0,)0 13 '( & " # $ % )*+,-. '& '" '6 *+. f = 3Hz f = 20Hz '# ( & &( " )*+,-. V P0 V P1 V S0 V S1 = 2000m/s = 5000m/s = 1500m/s = 3500m/s ρ 0 =2.0gm/cc ρ 1 =4.0gm/cc Q P1 = 30 Q S1 =5

22 Linear anelastic inversion DATA = [ R PP (θ fixed,ω 1 ), R PP (θ fixed,ω 2 ), R PP (θ fixed,ω 3 ) ] %$ '.+ %$ '6+ %# %#,)&, -- %",)&, -- %" %" " # $ &'()*+ "2 ':+ %" " # $ &'()*+ "2 '(+ Exact Recovered -) " 2 -) " 2 2 /0 /1 2 /0 /1 From R PP AVF: target Q P recovered, Q S not

23 Linear anelastic inversion DATA = [ R PS (θ fixed,ω 1 ), R PS (θ fixed,ω 3 ) ] )1- )9- %# %#.+(. /0 %$ %'.+(. /0 %$ %' %& %" " # #" $ ()*+,- )=- %& %" " # #" $ ()*+,- )*- Exact Recovered 5 5 / :;<4 & $ / :;<4 & $ $ $ From R PS AVF: target Q S recovered, Q P not