# Greens functions - solution for earthʼs surface movement due to a slipping patch at depth

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1 Greens functions - solution for earthʼs surface movement due to a slipping patch at depth Need to know slip distribution along the fault as a function of time No closed-form solution for Greens functions (Gnp). Makes use of full-space solution and image solution (recall Segallʼs 2D dislocation static solutions) u n (x,t) v j u i (ξ, τ) Σ u n (x,t) u i (ξ, τ)

2 Full-space solution: displacement at a point due to a dislocation r near- and intermediate field terms far-field terms (P and S wave)

3 near- and intermediate field terms near field and intermediate field displacement is proportional to moment. It results in permanent ground displacement. Drops off as 1/r 4 and 1/r 2. far-field terms (P and S wave) far field displacement is proportional to moment rate, not moment. It goes to 0 because moment rate goes to 0. far field displacement decreases as 1/r far field equations apply if λ >> 2L2 r dislocation patch is normal to the plotted plane φ is 0 in the plotted plane θ =0 nodal plane P wave θ =3π/2 nodal plane θ = π/2 nodal plane θ = π nodal plane

4 dislocation patch is normal to the plotted plane φ is 0 in the plotted plane θ =0 S wave θ = π/2 θ =3π/2 α = 3β so the S wave amplitude is about 5 times bigger than the P wave θ = π p amplitude of the Fourier transform of a one-sided pulse at very low frequencies equals the area under the pulse (e.g. EQ 11). Thus it s proportional to moment. p 945 -also, corner frequency is inversely proportional to the duration of the pulse, so dimension of the source can be obtained (Brune EQ from last time): r = 2.34β 2πf c

5 σ = 7 M o 16 r 3 Finite source models - page 947 needed when observer is close to fault and for permanent displacement (near field), and to understand directivity effect (near and far field) Brune (1970) self-similar crack: as slip grows, crack dimension grows. Haskell (1964) slip on a rectangular fault idealized as a propagating ramp (carpet wrinkle analogy) Duration of slip is far shorter than duration of the quake Demonstrated by Tom Heaton, Kinematic vs. dynamic rupture models

6 This looks sort of like the Haskell propagating ramp. Wald and Heaton, 1994

7 Directivity effects Propagation effects (on strong motion) scattering and attenuation loss of high frequencies ( fmax ) loss goes as exp πκf κ is bigger in sed basins (fig. 16) sedimentary basins larger amplitudes, longer duration, surface waves generated at edge of basin (strong, late arrivals) focusing of seismic energy (Kobe; Northridge/Santa Monica) nonlinear elasticity near the surface soft sediments in top few hundred m amplify shaking because wave propagation is slowed down - nonlinearity decreases seis veloc at high diff stress, making lower frequencies resonate, + damping shaking more. Page 950

8 Propagation effects (on strong motion) sedimentary basins larger amplitudes, longer duration, also - high frequencies are lost to attenuation example from sites on rock and in sed basin: 1985 Mexico City earthquake Page 950 Topographic Effects topography: topo wavelength close to S wave wavelength leads to amplification of shaking at ridge tops Recent example - Haiti (Susan Hough) Hotel Montana (at the top of a ridge)

9 Topographic Effects λ = V f 2000 ms 1 10 Hz 200 m Theoretical Models for estimation of strong shaking Stochastic: band-limited white noise in frequency domain One important approach to generating synthetic seismograms was proposed by Boore (1983). In this approach, one generates a completely random time series, applies a shape function in the time domain so that the time series has a reasonable envelope of amplitude with time, transforms to the frequency domain, modifies the Fourier amplitude spectrum to match the amplitude and shape of a spectral model, and then transforms back to the time domain to obtain the synthetic seismogram. The representation theorem is thus applied in the frequency domain Synthetic Greens Functions: Derived from full space solution, enforcing traction-free surface Generally available for 1-D (layered) elastic structure (with all terms) JFʼs term project Empirical Greens Functions: Greens functions are seissmic records of small earthquakes Assume focal mechanism is same as for larger future quake Records are not always available for desired source-receiver pairs

10 To model strong motion: at each snapshot, you need slip and slip rate on each discrete patch of the fault Wald and Heaton, 1994

11

12

13

14 Figure S1

15 σ = 7 16 M o r 3 Figure S2 r = 2.34β 2πf c Figure S3

16 Figure S4 Figure S5

17 Figure S6 k relates slip (offset) to the shear stress change Equation for stiffness k for a small, elliptical crack is: k = G (1 ν)l dx1 G ν L = shear modulus = Poissonʼs ratio = length of slipping area slipping elliptical patch of fault length L Earthquake machine: k is the spring stiffness dx2 elastic rock with properties G and ν Earth: k is the elastic force due to offset along crack (length L)

18 k< σ n(b a) D c G (1 ν)l > σ n(b a) < D c k G ν L (b a) σ n = normal stress D c = crack stiffness = shear modulus = Poissonʼs ratio = length of slipping area = friction weakening parameter = friction weakening distance L> D c G (1 ν)(b a)σ n This tells us that the slipping patch of fault must be bigger than a critical size to go unstable, even for a velocity weakening fault L> D c G (1 ν)(b a)σ n This tells us that the slipping patch of fault must be bigger than a critical size to go unstable, even for a velocity weakening fault Typical values in the seismogenic zone: Dc = 10 microns (based on experiments) larger?? (based on minimum measured earthquake size) G = 30 GPa (b-a) = around 10-2 eff normal stress = 2e8 Pa at 10 km depth L > 0.15 m at 10 km, 1.5 m at 1 km. in mines: 15 m patch of slip measured (M 0) in mine at 1 km depth This implies a finite nucleation time.

19 Nucleation time: shallow crust: of the order of a month to a year? 12 hours is too short (no correlation with Earth tides) tremor and volcanic quakes are triggered by passing surface waves - so nucleation times must be very fast. on the Moon - nucleation is very fast (is it even friction?)

20 Asperity is being loaded by creep on adjoining parts of fault that are assumed to be velocity-weakening (unstable) but not creeping over a large enough area to go unstable (creep rate accelerates with time as theory predicts) slowly creeping fault surface locked patch, 300 m diameter elastic rock with properties G and ν L> D c G (1 ν)(b a)σ n Asperity L is small locally low normal stress due to high pore pressure? locally large (b-a)? patch surrounded by velocitystrengthening material?

21 Slow earthquakes Ide et al., 2007 Ide et al., 2007 K. Creager, University of Washington.

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