A simple way to improve PP and PS AVO approximations. Chuck Ursenbach CREWES Sponsors Meeting Thursday, December 1, 2005

Transcription

1 A simple way to improve PP and PS AVO approximations Chuck Ursenbach CREWES Sponsors Meeting Thursday, December 1, 2005

2 Overview Notes on spherical-wave modeling Reflectivity Explorer observations Theoretical Explanations Improving AVO theories

3 Sphericalwave Reflection Coefficients θ i = 0 θ i = 15 R spherical PP 0 ( θ ) i p RPP( pw ) n( p; θi) dp ξ θ i = 45 θ i = 85 p = sinθ

4 Efficient Explorer calculations Assume that wavelet is of form fn n ( ω) = ω exp( s ω ) Then ω-integration can be analytic This form is similar to Ricker wavelet ( ) 2 2 fricker ( ω) = ω exp ω/ ω 0 fn n ( ω) = ω exp( n ω/ ω ) ω 0 is the maximum frequency for both 0

5 Wavelet comparison ω 0

6 Spherical R PP for Ormsby and n=4 wavelets

7 Representation of Ricker wavelet (note range of axes)

8 Ursenbach, Haase, and Downton, Improvements and verifications for the Spherical Zoeppritz Explorer

9 Reflection of spherical waves in VTI media Posters Ursenbach & Haase Generalized reflections from point sources in a two-layer VTI medium: theory Haase & Ursenbach Spherical-wave AVO-modelling in elastic VTImedia Anelasticity and spherical-wave AVO-modelling in VTI-media

10 Overview Notes on spherical-wave modeling Reflectivity Explorer observations Theoretical Explanations Improving AVO theories

11 Aki-Richards Approximation A-R R 2 2 RPP = Rρ + 4γ sin θ(2 R R ), 2 β + ρ cos θ ( ) R = R + R + R 4 γ (2 R + R ) sin θ + sin θ tan θr Shuey PP ρ β ρ A+ B + C sin θ sin θ tan θ, Shuey-like 3 RPS = γ Rρ + 2 γ(2 Rβ + Rρ) sin θ + O(sin θ) A + ( ) A-R RPS = γ tanϕ Rρ + 2γ cos θ ϕ (2 Rβ + Rρ), = ( 1+ 2) / 2, = ( 1+ 2) / 2 β1+ β2 θ θ θ ϕ ϕ ϕ 3 S sin θ O(sin θ). R R R β ρ = 2 β = 2β ρ = 2ρ γ = + 1 2

12 A further approximation Shuey (1985) also suggested substituting θ 1 for θ as an approximation. What behavior does this give?

13 θ vs. θ 1 approximations R R R β ρ γ = = = = = β = β ρ = ρ

14 θ vs. θ 1 approximations R PP R PS

15 θ vs. θ 1 approximations a) γ new = 0.3 b) R β new = R β c) R β new = 0.1R β

16 Effect of θ θ 1 True linear behavior: no critical point R PS more accurate in 0 < θ 1 < 30 range R PP more accurate if o γ >.35 o R β > R, same sign R PP less accurate if o γ <.3 o R β small, or opposite sign to R

17 Overview Notes on spherical-wave modeling Reflectivity Explorer observations Theoretical Explanations Improving AVO theories

18 Why is θ 1 better at low angles? Differences disappear for R = 0 (θ = θ 1 ) Used MAPLE to linearize R exact PP, R exact PS in R β, R ρ Find coefficient of sine-powers: R RPP( θ1) = A+ Bsin θ1 +, B= [ R 4 γ (2 Rβ + Rρ)] 1 R R θ A B θ B R γ R R R PP( ) = + sin +, = [ 4 (2 β + ρ)](1 ) 1+ 2R 2 (1 R ) R ( θ ) = A sin θ +, A = [ R + 2 γ(2 R + R )] PS 1 S 1 R ( θ) = A sin θ +, A = [ R + 2 γ (2 R + R )](1 R ) PS S S S ρ β ρ ρ β ρ 1 R

19 Alternate expression for B Used MAPLE to linearize R PP exact in R ρ only 16 2 (1 + R ) γ R (nonlinear in β; ρ = 0) = ( 8 γ β) + 1 R 1 R θ1 B R R R R 3 2 β Set R = R, = 1/2 β γ Then B θ 1 = R θ B = R (1 R ) Shuey B = R 2

20 Overview Notes on spherical-wave modeling Reflectivity Explorer observations Theoretical Explanations Improving AVO theories

21 A better expression? Note that sinθ 1 = sinθ 1 R 1+ R tan sin θ (1 R ) 2 2 θ sinθ = sin θ(1 ) R Substitute 1 in the initial gradients of the sinθ 1 expressions. This should give better behavior at low angles and a critical point.

22 New approximation R PP R PS

23 New approximation

24 The Smith-Gidlow approximation S-G 1 R RPP = R + 4γ sin θ(2 R R ) 2 β + 4 cos θ 4 The Fatti approximation Fatti RI 2 2 ( R ) PP = 8γ sin θr 4 sin tan 2 J + γ θ θ R ρ cos θ

25 Conclusions The Aki-Richards expression has been compared using both θ and θ 1 as the dependent variable The expression in terms of θ is best near the critical point The expression in terms of θ 1 is best at low angles for R PS and certain regions of R PP The quality of the θ 1 expression has been justified by theoretical analysis

26 Conclusions A new version of the Aki-Richards approximation is given in which sinθ is multiplied by (1 R ) An estimate of R is already required to obtain θ, so this requires no new information The new expression is more accurate for a wider range of low angles and has a correctly located critical point.