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Seismology – Lecture 2 Normal modes and surface waves

Seismology – Lecture 2 Normal modes and surface waves. Barbara Romanowicz Univ. of California, Berkeley. From Stein and Wysession, 2003. Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland. Surface waves. SS. S. P. Shallow earthquake. From Stein and Wysession, 2003. one hour.

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Seismology – Lecture 2 Normal modes and surface waves

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  1. Seismology – Lecture 2Normal modes and surface waves Barbara Romanowicz Univ. of California, Berkeley CIDER Summer 2010 - KITP

  2. From Stein and Wysession, 2003 CIDER Summer 2010 - KITP

  3. Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland Surface waves SS S P

  4. Shallow earthquake CIDER Summer 2010 - KITP From Stein and Wysession, 2003 one hour

  5. Direction of propagation along the earth’s surface T Z L

  6. Surface waves • Arise from interaction of body waves with free surface. • Energy confined near the surface • Rayleigh waves: interference between P and SV waves – exist because of free surface • Love waves: interference of multiple S reflections. Require increase of velocity with depth • Surface waves are dispersive: velocity depends on frequency (group and phase velocity) • Most of the long period energy (>30 s) radiated from earthquakes propagates as surface waves CIDER Summer 2010 - KITP

  7. After Park et al, 2005 CIDER Summer 2010 - KITP

  8. Free oscillations CIDER Summer 2010 - KITP

  9. CIDER Summer 2010 - KITP

  10. Free Oscillations (Standing Waves) The k’th free oscillation satisfies: In the frequency domain: SNREI model; Solutions of the form k = (l,m,n) CIDER Summer 2010 - KITP

  11. Free Oscillations In a Spherical, Non-Rotating, Elastic and Isotropic Earth model, the k’th free oscillation can be described as: l = angular order; m = azimuthal order; n = radial order k = (l,m,n) “singlet” Degeneracy: (l,n): “multiplet” = 2l+1 “singlets ” with the same eigenfrequency nwl

  12. Spheroidal modes : Vertical & Radial component Toroidal modes : Transverse component overtones Fundamental mode n=0 n=1 nTl l : angular order, horizontal nodal planes n : overtone number, vertical nodes CIDER Summer 2010 - KITP

  13. n=0 nSl Spheroidal modes

  14. Spatial shapes:

  15. Depth sensitivity kernels of earth’s normal modes

  16. Sumatra Andaman earthquake 12/26/04 M 9.3 0S2 0S3 0S0 20.9’ dr=0.05m 53.9’ 3S1 2S2 1S3 0S4 44.2’ 0S5 1S2 2S1 0T2 0T3 0T4

  17. Rotation, ellipticity, 3D heterogeneity removes the degeneracy: • -> For each (n, l) there are 2l+1 singlets with different frequencies

  18. 0S2 0S3 2l+1=5 2l+1=7

  19. mode 0S3 7 singlets

  20. Geographical sensitivity kernel K0(q,f) 0S3 0S45

  21. Mode frequency shifts Δω SNREI-> ωo frequency Frequency shift depends only on the average structure along the vertical plane containing the source and the receiver weighted by the depth sensitivity of the mode considered:

  22. P(θ,Φ) S R Masters et al., 1982

  23. Data Model Anomalous splitting of core sensitive modes

  24. Mantle mode Core mode

  25. Seismograms by mode summation  Mode Completeness: Depends on source excitation f  Orthonormality (L is an adjoint operator): * Denotes complex conjugate

  26. Normal mode summation – 1D A : excitation w : eigen-frequency Q : Quality factor ( attenuation ) CIDER Summer 2010 - KITP

  27. Spheroidal modes : Vertical & Radial component Toroidal modes : Transverse component n=0 n=1 nTl l: angular order, horizontal nodal planes n : overtone number, vertical nodes CIDER Summer 2010 - KITP

  28. CIDER Summer 2010 - KITP

  29. Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland Surface waves SS S P

  30. Standing waves and travelling waves Ak ---- linear combination of moment tensor elements and spherical harmonics Ylm When l is large (short wavelengths): Replace x=a Δ, where Δ is angular distance and x linear distance along the earth’s surface Jeans’ formula : ka = l + 1/2

  31. Hence: Plane waves propagating in opposite directions

  32. -> Replace discrete sum over l by continuous sum over frequency (Poisson’s formula): With k=k(ω) (dispersion) Phase velocity: S is slowly varying with ω ; The main contribution to the integral is when the phase is stationary:

  33. S is slowly varying with ω ; The main contribution to the integral is when the phase is stationary: For some frequency ωs The energy associated with a particular group centered on ωs travels with the group velocity:

  34. Rayleigh phase velocity maps Period = 50 s Period = 100 s Reference: G. Masters – CIDER 2008

  35. Group velocity maps Period = 50 s Period = 100 s Reference: G. Masters CIDER 2008

  36. Importance of overtones for constraining structure in the transition zone overtones n=1 n=2 n=0: fundamental mode

  37. Overtones By including overtones, we can see into the transition zone and the top of the lower mantle. from Ritsema et al, 2004

  38. 120 km Fundamental Mode Surface waves 325 km 600 km Body waves 1100 km Overtone surface waves 1600 km 2100 km 2800 km Ritsema et al., 2004

  39. Anisotropy • In general elastic properties of a material vary with orientation • Anisotropy causes seismic waves to propagate at different speeds • in different directions • If they have different polarizations

  40. Types of anisotropy • General anisotropic model: 21 independent elements of the elastic tensor cijkl • Long period waveforms sensitive to a subset (13) of which only a small number can be resolved • Radial anisotropy • Azimuthal anisotropy CIDER Summer 2010 - KITP

  41. Radial Anisotropy Montagner and Nataf, 1986

  42. Radial (polarization) Anisotropy • “Love/Rayleigh wave discrepancy” • Vertical axis of symmetry • A=r Vph2, • C=r Vpv2, • F, • L= r Vsv2, • N= r Vsh2 (Love, 1911) • Long period S waveforms can only resolve • L , N • => x = (Vsh/Vsv) 2 • dln x =2(dln Vsh – dlnVsv)

  43. Azimuthal anisotropy • Horizontal axis of symmetry • Described in terms of y, azimuth with respect to the symmetry axis in the horizontal plane • 6 Terms in 2y (B,G,H) and 2 terms in 4y (E) • Cos 2y -> Bc,Gc, Hc • Sin 2y -> Bs,Gs, Hs • Cos 4y-> Ec • Sin 4y -> Es • In general, long period waveforms can resolve Gc and Gs

  44. Montagner and Anderson, 1989

  45. x   y Axis of symmetry z • Vectorial tomography: • Combination radial/azimuthal (Montagner and Nataf, 1986): • Radial anisotropy with arbitrary axis orientation (cf olivine crystals oriented in “flow”) – orthotropic medium • L,N, Y, Q CIDER Summer 2010 - KITP

  46. x = (Vsh/Vsv)2 Isotropic velocity Radial Anisotropy Azimuthal anisotropy Montagner, 2002

  47. Depth= 100 km Pacific ocean radial anisotropy: Vsh > Vsv Ekstrom and Dziewonski, 1997 Montagner, 2002

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