Matrix forms for the Knott-Zoeppritz equations

Transcription

1 Kris Innanen ABSTRACT In this note we derive convenient matrix forms for the Knott-Zoeppritz equations. The attempt is to recapture results used (quoted not derived) by Levin and Keys and to extend these to include the case of an incident S-wave. In addition to being straightforward to solve and analyze these forms are used in several other papers in this year s CREWES report. INTRODUCTION Our aim is to derive solutions to the Zoeppritz equations for an incident P-wave in a form similar to that quoted by Levin and Keys (Levin 986; Keys 989) and to extend these to include the results for an incident S-wave. These forms have proven convenient both for the purposes of extension to anelastic media and direct inversion (Innanen ); in this report they will be used in discussions on reflectivity decomposition of elastic reflection coefficients AVF analysis of anelastic reflection of shear and converted waves and in an implementation of the poroelastic AVO theory due to Russell et al. (). BASIC EQUATIONS We take as our starting point the relations on pg. 4 of Aki and Richards (). The assumption of continuity of displacement and traction across an elastic boundary has led to four equations relating all relevant displacement amplitudes above and below the boundary. In our terminology (see Figure ) these equations relate etc. as follows: sin θ ( + ) + cos φ ( + S R ) = sin φ ( + ) + cos φ ( + ) (a) cos θ ( ) sin φ ( S R ) = cos φ ( ) sin φ ( ) (b) ρ (VS /V P ) sin θ cos θ ( ) ρ V S (V S /VP ) sin θ (SI S R ) (c) =ρ (VS /V P ) sin θ cos θ ( ) + ρ V S (V S /VP ) sin θ (ST ) ρ V P (V S /VP ) sin θ (PI ) ρ (VS /V P ) sin θ cos φ ( + ) (d) =ρ V P (V S /VP ) sin θ (PT + ) ρ (VS /V P ) sin θ cos φ ( + ). We consider several ratios of elastic parameters: A ρ ρ B V S V P B V P V S C V P V P D V S V P E V P V S F V S V S. () INCIDENT P-WAVE: R PP AND R PS We now limit the number of waves incident on the boundary to one: a P-wave incident from above (see Figure ). That is we let = P I = =. (3) CREWES Research Report Volume 3 ()

2 Innanen SR φ θ θ V P V S ρ V P V S ρ φ FIG.. Displacement amplitudes associated with the Knott-Zoeppritz equations. We define the displacement coefficients associated with the remaining reflected and transmitted waves to be R PP R PS S R T PP T PS. (4) We will parametrize the coefficients in terms of the P-wave angle of incidence in the upper medium θ. So beyond substituting the forms in equation (4) into the relations in equations () we further transform all angle-dependent factors to expressions in sin θ using Snell s law particularly cos θ = sin θ sin φ = V S sin θ sin θ = V P sin θ V P V P cos θ = V P sin θ VP and cos φ = V S sin θ VP (5) or setting X sin θ Finally for convenience we define cos θ = X sin φ = BX sin θ = CX cos θ = C X and cos φ = D X. (6) Γ j (X) j X Γ j (X) j X. After substitution of equations () (7) equations () can be expressed in matrix form as R PP P R PS T PP = b P (8) T PS (7) CREWES Research Report Volume 3 ()

3 where P X Γ B (X) CX Γ D (X) Γ (X) BX Γ C (X) DX B XΓ (X) BΓ B (X) AD XΓ C (X) ADΓ D (X) Γ B (X) B XΓ B (X) ACΓ D (X) AD XΓ D (X) and b P X Γ (X) B XΓ (X) Γ B (X). Forming auxiliary matrices P P and P S by replacing the first and second columns of P with b P respectively we may use Cramer s rule to determine the displacement reflection coefficients: R PP (θ ) = detp P detp We thus recover the forms used by Levin and Keys. R PS(θ ) = detp S detp. (9) S R φ θ θ φ V P V S ρ V P V S ρ FIG.. Displacement amplitudes given an incident P-wave. INCIDENT S-WAVE: R SP AND R SS The extension for an incident S-wave is carried out similarly. As illustrated in Figure 3 we disallow all incident fields except an S-wave from above by setting and we define displacement coefficients = P I = = () R SS S R R SP T SS T SP. () CREWES Research Report Volume 3 () 3

4 Innanen This time we parametrize in terms of sin φ i.e. the sine of the S-wave incidence angle. Using equation (7) the other trigonometric functions are expressible in terms of Y sin φ as cos φ = Γ (Y ) sin θ = B Y sin θ = EY sin φ = F Y cos θ = Γ E (Y ) cos φ = Γ D (Y ) and cos θ = Γ B (Y ). () After substitution of equations () () equations () can also be expressed in matrix form as R SS S R SP T SS = b S (3) T SP where and S Γ (Y ) B Y Γ F (Y ) EY Y Γ B (Y ) F Y Γ E (Y ) Γ (Y ) Y Γ B (Y ) AF Γ F (Y ) AF Y Γ E (Y ) Y Γ (Y ) B Γ (Y ) AF Y Γ F (Y ) AEΓ F (Y ) b S Γ (Y ) Y Γ (Y ) Y Γ (Y ). Forming auxiliary matrices S P and S S by replacing the first and second columns of S with b S respectively we again use Cramer s rule to determine the displacement reflection coefficients: R SS (φ ) = dets S dets R SP(φ ) = dets P dets. (4) Thus are generated the incident-s counterparts to the incident-p solutions of Levin and Keys. 4 CREWES Research Report Volume 3 ()

5 SR φ θ θ φ V P V S ρ V P V S ρ FIG. 3. Displacement amplitudes given an incident S-wave. CONCLUSIONS We derive matrix forms for the Knott-Zoeppritz equations. The attempt is to recapture results of Levin and Keys show how they are arrived at and to extend these to include the case of an incident S-wave. We have made particular use of these forms in several other papers in this year s CREWES report. ACKNOWLEDGMENTS The sponsors of CREWES and NSERC are gratefully acknowledged for their support. REFERENCES Aki K. and Richards P. G. Quantitative Seismology: University Science Books nd edn. Innanen K. A. Inversion of the seismic AVF/AVA signatures of highly attenuative targets: Geophysics 76 No. R R. Keys R. G. 989 Polarity reversals in reflections from layered media: Geophysics Levin F. K. 986 When reflection coefficients are zero: Geophysics Russell B. R. Gray D. and Hampson D. P. Linearized AVO and poroelasticity: Geophysics 76 No. 3 C9 C9. CREWES Research Report Volume 3 () 5