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

Outline ½ Space Rayleigh waves Love waves 2-Layer medium

½ Space Solution to Elastic Wave Equation PP PS P Sin a /V = sin a /V = sin a /V P PS PP Boundary Tractions @ 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 )

½ Space Solution to Elastic Wave Equation PP PS P Sin a /V = sin a /V = sin a /V P PS PP Boundary Tractions @ 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 Tractions @ z=0: TP + TPP + TPS =(tzx , tzy ,tzz )=(0,0,0) Two eqns, two unknowns

½ Space Solution to Elastic Wave Equation PP 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. 0.92cs

Outline ½ Space Rayleigh waves Love waves 2-Layer medium

½ Space Solution to Elastic Wave Equation

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.

Retrograde ellipitical

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

½ Space Solution to Elastic Wave Equation

½ Space Solution to Elastic Wave Equation

Outline ½ Space Rayleigh waves Love waves 2-Layer medium

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.

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

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

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

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

Outline ½ Space Rayleigh waves Love waves 2-Layer medium

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

Case Histories for PP and PS Reflections

North Sea Gas Chimney

North Sea PP & PS

GOM PP/PS Sections

Reservoir Identification

GOM Sub-Salt PP/PS

Methane Case Histories for PP and PS Reflections