The importance of the Vp/Vs ratio in determining the error propagation and the resolution in linear AVA inversion

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1 The importance of the Vp/Vs ratio in determining the error propagation and the resolution in linear AVA inversion Mattia Aleardi* & Alfredo Mazzotti University of Pisa Earth Sciences Department GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 1

2 Motivations of the study In linear AVA inversion for deep (hydrocarbon) exploration a background Vp/Vs ratio equal to 2 is frequently assumed However, this ratio can be an underestimation of the true value in case of shallow, undercompacted sediments or in case of overpressured layers In addition to lithology and fluid prediction in hydrocarbon exploration the linear AVA method is also used for: Overpressure prediction Estimating geotechnical properties of shallow marine sediments (for offshore construction activity). In these cases the Vp/Vs ratio is usually much higher than 2 Therefore, we want to investigate the influence of the background Vp/Vs ratio on the final outcomes of linear AVA inversion GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 2

3 Agenda Linear AVA inversion Theoretical aspects Three-term Aki & Richards and two-term Ursenbach & Stewart approximations Sensitivity Analysis varying the background Vp/Vs ratio: Condition number; Eigenvectors in model space; Model resolution matrix (T-SVD); Model covariance matrix (T-SVD); Conclusions GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 3

4 AVA METHOD AVA method PROCESSED SEISMIC CMP CMP DATA AVA AVA EXTRACTION Amplitude Angle AVA AVA INVERSION Vp Vs ρ Vp Vs ρ Amplitude Angle Vp Vs ρ Vp Vs ρ GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 4

5 AVA method The AVA technique is usually based on approximations of the Zoeppritz equations: (cos(θ1) cos(θ2) sin(ϕ2) α 1 2 ρ 2 ρ α 1 ρ cos(2ϕ 2 ) 2 α 2 /ρ 1 α 1 pp sin(2 ϕ 1 α 2 /β 2 ) 2 T 0 sin(2θ 2 ) α pp 2 cos(2ϕ β 2 ) )(R T 2 ps)=( cos(θ1) ) 1 0 α,β andρ : P- and S-wave velocity and density θ 1 : Incidence angle θ 2 and ϕ 2 : Refraction angles for the transmitted P- and S-waves R pp : P-wave reflection coefficient T pp : P-wave transmission coefficient T ps : S-wave transmission coefficient GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 5

6 AVA method The most used approximations are: Three terms Aki and Richards equation: R PP (θ) = R p cos 2 θ 8 γ2 sin 2 θ R s (4 γ 2 sin 2 θ 1)R ρ R p = 1 2 Δ V p V p R s = 1 2 Δ V s V s R ρ = 1 2 Δ ρ ρ γ=(v s1 + V s2 )/(V p1 + V p2 ) Two terms Ursenbach and Stewart equation: R pp (θ) = (1 + 4 γ2 cos 2 θ 1 5 sin 2 θ) R I cos 2 θ 8 γ2 sin 2 θ R J R I = V p2 ρ 2 V p1 ρ 1 V p2 ρ 2 + V p1 ρ 1 R J = V s2 ρ 2 V s1 ρ 1 V s2 ρ 2 + V s1 ρ 1 γ=(v s1 + V s2 )/(V p1 + V p2 ) GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 6

7 Sensitivity Analysis d=g(m) m=g -g d Singular value decomposition of G matrix: G=USV T θ S V T U Eigenvectors in data space Eigenvectors model space Condition Number Model Resolution Matrix Singular values S max /S min R=G -g G Eigenvectors in model space Unit Covariance Matrix Cm=G -g G -g GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 7

8 The Condition Number Seabed cases Aki & Richards Equation Threshold of of Stability Ursenbach & Stewart Equation Deep exploration cases Fixed a threshold of stability (C.N.=2x10²) the problem in case of Vp/Vs=2 becomes stable if we use a two terms parametrization; Conversely, if we consider the seabed case (Vs/Vp close to zero) both parametrizations are unstable; We must apply a method to stabilize the inversion such as the T-SVD method or the Tikhonov regularization. GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 8

9 Sensitivity Analysis: Eigenvectors in model space VP/VS=2 Eigenvectors in in model space VP/VS>>2 I I I I II II GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 9

10 Sensitivity Analysis: Eigenvectors in model space VP/VS=2 Eigenvectors in in model space VP/VS>>2 I I II II T-SVD GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 10

11 Sensitivity Analysis: Eigenvectors in model space VP/VS=2 Eigenvectors in in model space VP/VS>>2 I I II II T-SVD GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 11

12 Sensitivity Analysis: Eigenvectors in model space VP/VS=2 Eigenvectors in in model space VP/VS>>2 I I RI RI RJ RJ RJ RJ RI RI GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 12

13 Sensitivity Analysis: Eigenvectors in model space VP/VS=2 Eigenvectors in in model space VP/VS>>2 I I RI RI RJ RJ RJ RI T-SVD RJ RI GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 13

14 Model resolution matrix (T-SVD) VP/VS=2 VP/VS>>2 GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 14

15 Model covariance matrix (T-SVD) VP/VS=2 VP/VS>>2 GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 15

16 Conclusions The Vp/Vs ratio exerts a strong influence on the stability, resolution and error propagation in linear AVA inversion; From the analysis of the condition number we note that: Vp/Vs equal to 2: the inverse problem becomes stable as we pass from the three- to the two-term approximation. Vp/Vs ratio very high: the inverse problem is ill-conditioned even if a twoterm approximation is considered. In the case of linear AVA inversion with very high Vp/Vs ratios, the application of a regularization method (i.e the T-SVD method) is needed to stabilize the inversion process. GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 16

17 Conclusions The Vp/Vs ratio exerts a strong influence on the stability, resolution and error propagation in linear AVA inversion; From the analysis of the condition number we note that: Vp/Vs equal to 2: the inverse problem becomes stable as we pass from the three- to the two-term approximation. Vp/Vs ratio very high: the inverse problem is ill-conditioned even if a twoterm approximation is considered. In the case of linear AVA inversion with very high Vp/Vs ratios, the application of a regularization method (i.e the T-SVD method) is needed to stabilize the inversion process. The orientation of the eigenvectors in model space shows that: High Vp/Vs ratios: the eigenvectors associated with the Vs-related parameter ( and RJ) span the null-space of the inversion kernel. This fact, combined with the observation of the resolution matrices, highlights that the determination of the Vs contrast (or the S-impedance contrast) becomes a hopelessly non-unique problem in the case of high Vp/Vs values. Increasing the Vp/Vs values the error propagation from data to model space and the cross-talk between and become more and more severe. GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 17

18 Thank you for your attention ANY QUESTIONS? GNGTS Bologna 26 Nov Linear AVA inversion and Vp/Vs ratio 18