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P-SV Waves and Solution to Elastic Wave Equation for a ½ Space and 2-Layer Medium and Reflection Coefficients. ½ Space. Outline. Rayleigh waves. Love waves. 2-Layer medium. P. PS. Sin a / V = sin a /V = sin a /V. P. P. PS. PP. PS. P. PP. PS. Boundary Tractions @ z=0:

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  1. P-SV Waves and Solution to Elastic Wave Equation for a ½ Space and 2-Layer Medium and Reflection Coefficients

  2. ½ Space Outline • Rayleigh waves • Love waves • 2-Layer medium

  3. P PS Sin a /V = sin a /V = sin a /V P P PS PP PS P PP PS Boundary Tractions @ z=0: TP + TPP + TPS =(tzx, tzy ,tzz )=(0,0,0) a ½ Space Solution to Elastic Wave Equation 2 unknowns, 2 eqns constraint -> Rpp, Rps (reflec. coeffs) 0 (dF/dx, 0, dF/z) Step 1. Express P displacements as f potentials: (u,0,w)= (df/dx, 0, df/dz) Hooke’s law tzx=2mezx Step 2. Express P stress as f potentials: tzx = m(du/dz+dw/dx) = 2md2f /dzdx Hooke’s law tzz=le+2mezz tzz = l(du/dx+dw/dz) +2mdw/dz = tzz = l(d2f / dx2+ d2f / dz2) + 2m d2f /dz2 TP + TPP = (md2f / dxdz, 0, l(d2f / dx2+ d2f / dz2) + 2m d2f /dz2 )

  4. P PS Sin a /V = sin a /V = sin a /V P P PS PP PS P PP Boundary Tractions @ z=0: TP + TPP + TPS =(tzx, tzy ,tzz )=(0,0,0) a ½ Space Solution to Elastic Wave Equation 2 unknowns, 2 eqns constraint -> Rpp, Rps Step 3. Express PS displacements as Y potentials: (u,0,w)= (dY / dz, 0, -dY/dx) -(m(d2 X /dz2 - d2 X /dx2 ), ?) Step 4. Express PS stress as Y potentials: TPS = -(d2Y / dx2 – d2Y/dz2, 0, d2Y/dxdz) incident P + reflected PP : reflected PS Step 5. F = ei(xKx+zKz-wt) + Rppei(xKx-zKz-w t) : Y = RPSei(xKx-zKz-wt) Boundary Tractions @ z=0: TP + TPP + TPS =(tzx, tzy ,tzz )=(0,0,0) Two eqns, two unknowns

  5. P PS Sin a /V = sin a /V = sin a /V P P PS PP SP S P SS PP PS a a ½ Space Solution to Elastic Wave Equation Only with incident S at critical angle a do we get horiz. Traveling Rayleigh waves incidentS + reflectedSS : reflected SP Step 5. F = ei(xKx+zKz-wt) + Rppei(xKx-zKz-w t) : Y = RPSei(xKx-zKz-wt) SS SP kz= sqrt(w2/cp2 - kx2 ) if kx < w /cp “i*ikzz= -kz z ” causes attenuation in depth so Rayleigh waves only propagate along surface kz = isqrt(kx2 - w2/cp2 ) if kx > w /cp Rayleigh Waves propagate along surface and attenuate with depth at vel. 0.92cs

  6. ½ Space Outline • Rayleigh waves • Love waves • 2-Layer medium

  7. ½ Space Solution to Elastic Wave Equation

  8. Rayleigh Wave (R-Wave) Animation Deformation propagates. Particle motion consists of elliptical motions (generally retrograde elliptical) in the vertical plane and parallel to the direction of propagation. Amplitude decreases with depth. Material returns to its original shape after wave passes.

  9. Retrograde ellipitical

  10. Dan Russell animations – Rayleigh wave Note, amplitude diminishes With depth exponentially Animation courtesy of Dr. Dan Russell, Kettering University http://www.kettering.edu/~drussell/demos.html

  11. ½ Space Solution to Elastic Wave Equation

  12. ½ Space Solution to Elastic Wave Equation

  13. ½ Space Outline • Rayleigh waves • Love waves • 2-Layer medium

  14. Love Wave (L-Wave) Animation Deformation propagates. Particle motion consists of alternating transverse motions. Particle motion is horizontal and perpendicular to the direction of propagation (transverse). To aid in seeing that the particle motion is purely horizontal, focus on the Y axis (red line) as the wave propagates through it. Amplitude decreases with depth. Material returns to its original shape after wave passes.

  15. Love Waves • Love waves, resulting from interacting SH waves in a layered medium • Love waves cannot exist in a half-space • Consider up-going & down-going SH-waves in layer and in the halfspace 16

  16. Love Waves • Love waves, resulting from interacting SH waves in a layered medium • As with Rayleigh waves, Love waves have to satisfy the conditions of (a) energy trapped near the interface (b) a traction-free free surface

  17. Love Waves Love waves, resulting from interacting SH waves in a layered medium • Combining the boundary conditions at the interface, we obtain • Dividing the 2nd by the 1st expression provides a particularly important equation: • This dispersion relation gives the apparent velocity cx as a function of kx or ω • Waves of different frequency (period) travel at different speed 18

  18. Love Waves Love waves, resulting from interacting SH waves in a layered medium • Love waves occur because incoming waves (with some wavenumber kx) are “trapped” within the surface layer • Think about constructive interference between incoming and reflected waves • This happens only if the waves come in at the right angles of incidence (wavenumber kx), which thus constitute so called “modes” of the solution • Rewrite the Love-wave dispersion relation (DR) in terms of cx, kx, and ω • Because the square roots must be real, cx is bounded: β1 < cx < β2 19

  19. ½ Space Outline • Rayleigh waves • Love waves • 2-Layer medium

  20. S P Sin a /V = sin a /V = sin a /V = sin a /V P S PS PP PP PS 4 unknowns, 4 eqns constraint -> Rpp, Rps, Tpp, Tps

  21. Case Histories for PP and PSReflections

  22. North Sea Gas Chimney

  23. North Sea PP & PS

  24. GOM PP/PS Sections

  25. Reservoir Identification

  26. GOM Sub-Salt PP/PS

  27. Methane Case Histories for PP and PS Reflections

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