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

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

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

4. ½ Space Solution to Elastic Wave EquationPP PS P
Sin a /V = sin a /V = sin a /V P PS PP
Boundary z=0:
TP + TPP + TPS =(tzx , tzy ,tzz )=(0,0,0) a
2 unknowns, 2 eqns constraint -> Rpp, Rps
Step 3. Express PS displacements as Y potentials: (u,0,w)= (dY / dz, 0, -dY /dx)
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-w t) + Rpp ei(xKx-zKz-w t) : Y = RPS ei(xKx-zKz-wt)
Boundary z=0:
TP + TPP + TPS =(tzx , tzy ,tzz )=(0,0,0)
Two eqns, two unknowns

5. ½ Space Solution to Elastic Wave EquationPP PS P
Sin a /V = sin a /V = sin a /V P PS PP SS SP S a a
incident P reflected PP : reflected PS
Step 5. F = ei(xKx+zKz-w t) + Rpp ei(xKx-zKz-w t) : Y = RPS ei(xKx-zKz-wt)
kz = sqrt(w2/cp2 - kx2 ) if kx < w /cp
kz = i sqrt(kx2 - w2/cp2 ) if kx > w /cp
Rayleigh Waves propagate along surface and attenuate with depth at vel cs

6. Outline
½ Space
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

11. Dan Russell animations – Rayleigh wave
Note, amplitude diminishes
With depth exponentially
Animation courtesy of Dr. Dan Russell, Kettering University

12. ½ Space Solution to Elastic Wave Equation

13. ½ Space Solution to Elastic Wave Equation

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

15. 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.

16. 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

17. 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

18. 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

19. 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

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

21. 4 unknowns, 4 eqns constraint -> Rpp, Rps, Tpp, Tps
Sin a /V = sin a /V = sin a /V = sin a /V PP PS P S
4 unknowns, 4 eqns constraint -> Rpp, Rps, Tpp, Tps

25. Case Histories for PP and PS Reflections

28. North Sea Gas Chimney

29. North Sea PP & PS

30. GOM PP/PS Sections

31. Reservoir Identification

33. GOM Sub-Salt PP/PS

34. Methane Case Histories for PP and PS Reflections