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Reflection Coefficients For a downward travelling P wave, for the most general case: u x x z u z z x Where the first term on the RHS is the P-wave displacement component and the second term is the shear-wave displacement component Reflection Coefficients and where both shear stress, 2 2 (2 ) xz 2 xz x 2 z 2 1 2 and as well as normal stress is continuous across the boundary: 2 2 2 2 (2 ) zz 2 xz z Reflection Coefficients When all these conditions are met and for the special case of normal incident conditions, we have that Zoeppritz’s equations are: 2V P 1V P 2 1 R \ / I 2 I1 2V P 1V P P P 2 1 I 2 I1 T \ P P \ 2 I1 I 2 I1 2 1V P 1 2V P 1V P 2 1 On occasions these equations will not add up to what you might expect…! Reflection Coefficients T R \ P P / I I 2 I1 2 1 \ \ I 2 I1 I 2 I1 P P I I 2 I1 2 1 I 2 I1 Reflection Coefficients T R \ P P / I I 2 I1 2 1 \ \ I 2 I1 I 2 I1 P P I I 2 I1 2 1 I 2 I1 I I 2 1 I 2 I1 Reflection Coefficients T R \ P P / I I 2 I1 2 1 \ \ I 2 I1 I 2 I1 P P I I 2 I1 2 1 I 2 I1 I I 2 1 I 2 I1 1 Reflection Coefficients T R \ P P / I I 2 I1 2 1 \ \ I 2 I1 I 2 I1 P P I I 2 I1 2 1 I 2 I1 I I 2 1 I 2 I1 1 T R \ P P / 1 \ P P \ Reflection Coefficients T R \ P P / I I 2 I1 2 1 \ \ I 2 I1 I 2 I1 P P I I 2 I1 2 1 I 2 I1 I I 2 1 I 2 I1 1 T R \ P P / 1 \ P P \ Reflection Coefficients What happens when we have a complete reflection with a 180 degree phase shift, as we might have when a ray in water travels upward toward a free surface and reflects completely at the interface? Reflection Coefficients What happens when we have a complete reflection with a 180 degree phase shift, as we might have when a ray in water travels upward toward a free surface and reflects completely at the interface? We know that in this case: 1 R \ P P / Reflection Coefficients What happens when we have a complete reflection with a 180 degree phase shift, as we might have when a ray in water travels upward toward a free surface and reflects completely at the interface? We know that in this case: 1 R \ P P But, T R \ P P What must: / / 1 \ P P \ ? T \ P P \ Reflection Coefficients What happens when we have a complete reflection with a 180 degree phase shift, as we might have when a ray in water travels upward toward a free surface and reflects completely at the interface? We know that in this case: 1 R \ P P But, T R \ P P So, / / 1 \ P P \ \ \ 2 T P P Reflection Coefficients Briefly, how to consider displacements at interfaces using potentials, when mode conversion occurs: Z- layer 1 X+ layer 2 In layer 1, just above the boundary, at the point of incidence: P u x x \ \ S z P P P x z \ / \ S / Z+ Reflection Coefficients Briefly, how to consider displacements at interfaces using potentials, when mode conversion occurs: Z- layer 1 X+ layer 2 In layer 2, just below the boundary, at the point of incidence: u x \ P P z \ \ P S x \ Z+ Reflection Coefficients So, if we consider that (1) stresses as well as (2) displacements are the same at the point of incidence whether we are in the top or bottom layer the following must hold true so that (3) Snell’s Law holds true: u u , x x u z u z zz zz zx zx Reflection Coefficients We get the general case of all the different types of reflection and transmission (refraction or not) coefficients at all angles of incidence : sin \ P cos \ P sin 2 \ P cos 2 \ P cos R \ \ P P R \ / P S T \ \ P P T \ \ P S Reflection Coefficients Variation of Amplitude with angle (“AVA”) for the fluid-over-fluid case (NO SHEAR WAVES) I 2 cos I1 1 R ( ) I 2 cos I1 1 V 2 sin V1 V 2 sin V1 2 2 (Liner, 2004; Eq. 3.29, p.68; ~Ikelle and Amundsen, 2005, p. 94) Reflection Coefficients What occurs at and beyond the critical angle? V V 1 2 c sin c sin 2 sin 1 V V 1 2 Reflection Coefficients FLUID-FLUID case What occurs at the critical angle? I 2 cos I1 1 R ( ) I 2 cos I1 1 V 2 sin V1 V 2 sin V1 2 2 (Liner, 2004; Eq. 3.29, p.68; ~Ikelle and Amundsen, 2005, p.94) Reflection Coefficients Reflection Coefficients at all angles: preand post-critical Case: Rho: 2.2 /1.8 V: 1800/2500 Matlab Code NOTES: #1 Reflection Coefficients At the critical angle, the real portion of the RC goes to 1. But, beyond it drops. This does not mean that the energy is dropping. Remember that the RC is complex and has two terms. For an estimation of energy you would need to look at the square of the amplitude. To calculate the amplitude we include both the imaginary and real portions of the RC. NOTES: #2 Reflection Coefficients For the critical ray, amplitude is maximum (=1) at critical angle. Post-critical angles also have a maximum amplitude because all the energy is coming back as a reflected wave and no energy is getting into the lower layer NOTES: #3 Reflection Coefficients Post-critical angle rays will experience a phase shift, that is the shape of the signal will change. Energy Coefficients \ P P E For the energy coefficients at normal incidence : T R We saw that for reflection coefficients : E \ / P P / 1 \ P P \ R \ / P P 2V P 2 T \ / \ \ P P 1V P P P 1 Energy Coefficients \ P P E For the energy coefficients at normal incidence : T R We saw that for reflection coefficients : E / \ / P P 1 \ P P \ R \ / P P 2V P 2 T \ / \ \ P P 1V P P P 1 The sum of the energy is expected to be conserved across the boundary E P \ E \ / P P E \ / P S E \ \ P P E \ \ P S Amplitude versus Offset (AVO) Zoeppritz’s equations can be simplied if we assume that the following ratios are much smaller than 1: VS VP VP average VS average average For example, the change in velocities across a boundary is very small when compared to the average velocities across the boundary; in other words when velocity variations occur in small increments across boundaries… This is the ASSUMPTION Amplitude versus Offset (AVO) If the changes across boundaries are relatively small, then we can make a lot of approximations to simplify the reflection and transmission coefficients: R \ / P P 2 2 V VP 1 z 1 S sin 2 2 i 2 VP average 2 z average V P average