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

From Stein and Wysession, 2003 CIDER Summer KITP

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

From Stein and Wysession, 2003 Shallow earthquake CIDER Summer KITP one hour

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

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 KITP

After Park et al, 2005 CIDER Summer KITP

Free oscillations CIDER Summer KITP

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

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

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 KITP Fundamental mode overtones

Spheroidal modes n=0 nSlnSl

Spatial shapes:

Depth sensitivity kernels of earth’s normal modes

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

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

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

mode 0 S 3 7 singlets

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

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->

S R P(θ,Φ) Masters et al., 1982

Anomalous splitting of core sensitive modes Data Model

Mantle mode Core mode

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

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

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 KITP

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

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

Hence: Plane waves propagating in opposite directions

-> 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:

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 :

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

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

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

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

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

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

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 KITP

Montagner and Nataf, 1986 Radial Anisotropy

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 )

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

Montagner and Anderson, 1989

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 KITP

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

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

Gung et al., 2003

Marone and Romanowicz, 2007 Absolute Plate Motion

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

Layer 1 thickness Mid-continental rift zone Trans Hudson Orogen

“Finite frequency” effects CIDER Summer KITP

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

Panning et al., 2009

Waveform tomography

observed synthetic Waveform Tomography