1. Seismology – Lecture 2 Normal 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. P S SS Surface waves Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland

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

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

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

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

9. Free oscillations CIDER Summer 2010 - KITP

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

12. 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 n  l

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

14. Spheroidal modes n=0 nSlnSl

15. Spatial shapes:

16. Depth sensitivity kernels of earth’s normal modes

17. 53.9’ 44.2’ 20.9’  r=0.05m 0T20T2 2S12S1 0S30S3 0S20S2 0T40T4 1S21S2 0S50S5 0S00S0 0S40S4 3S12S21S33S12S21S3 0T30T3 Sumatra Andaman earthquake 12/26/04 M 9.3

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

19. 0S20S3 2l+1=52l+1=7

20. mode 0 S 3 7 singlets

21. Geographical sensitivity kernel K 0 (  ) 0 S 45 0S30S3

22. ωoωo Δω 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: Mode frequency shifts SNREI->

23. S R P(θ,Φ) Masters et al., 1982

24. Anomalous splitting of core sensitive modes Data Model

25. Mantle mode Core mode

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

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

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

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

31. Standing waves and travelling waves A k ----  linear combination of moment tensor elements and spherical harmonics Y l m 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

32. Hence: Plane waves propagating in opposite directions

33. -> 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:

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

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

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

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

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

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

42. 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

43. Types of anisotropy General anisotropic model: 21 independent elements of the elastic tensor c ijkl 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

44. Montagner and Nataf, 1986 Radial Anisotropy

45. Radial (polarization) Anisotropy “Love/Rayleigh wave discrepancy” –Vertical axis of symmetry A=  V ph 2, C=  V pv 2, F, L=  V sv 2, N=  V sh 2 (Love, 1911) –Long period S waveforms can only resolve L, N =>  = (V sh /V sv ) 2  ln  =2(  ln V sh –  lnV sv )

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

47. Montagner and Anderson, 1989

48. 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, ,  x y z   Axis of symmetry CIDER Summer 2010 - KITP

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

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

51. Gung et al., 2003

52. Marone and Romanowicz, 2007 Absolute Plate Motion

53. Continuous lines: % Fo (Mg) from Griffin et al. 2004 Grey: Fo%93 black: Fo%92 Yuan and Romanowicz, in press

54. Layer 1 thickness Mid-continental rift zone Trans Hudson Orogen

55. “Finite frequency” effects CIDER Summer 2010 - KITP

56. Structure sensitivity kernels: path average approximation (PAVA) versus Finite Frequency (“Born”) kernels S R M S R M PAVA 2D Phase kernels

57. Panning et al., 2009

58. Waveform tomography

60. observed synthetic Waveform Tomography