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  1. Recall the momentum equation:∂2ui/∂t2 = ∂jij+fi , where fi is the body force termAn earthquake source is usually considered slip on a surface (displacement discontinuity), not a body forceFortunately, it can be shown that a distribution of body forces exists, which produces the equivalent slip (equivalent body forces) 

  2. Helpful to define a concept that separates the source from the wave propagation:ui(x,t)=G * f = Gij(x,t;x0,t0)fj(x0,t0)f = force vectorG = Green’s function = response to a ‘small’ sourceLinear equationDisplacement from any body force can be computed as the superposition of individual point sources 

  3. Force Couples:Forces must occur in opposing directions to conserve momentum D no net torque double couple: D net torque no net torque 

  4. 9 Force Couples Mij (the moment tensor), 6 different (Mij=Mji). |M|=fd M11 M12 M13 Good approximation for distant M= M21 M22 M23 earthquakes due to a point source M31 M32 M33 Larger earthquakes can be modeled as sum of point sources

  5. ui(x,t)=G * f = Gij(x,t;x0,t0)fj(x0,t0)Displacement from a force couple can be computed asui(x,t) = Gij(x,t;x0,t0)fj(x0,t0)-Gij(x,t;x0-xk,t0)fj(x0,t0) = [∂Gij(x,t;x0,t0) /∂x0 ] fj dwhere the force vectors are separated a distance d in the xk directionui(x,t) = [∂Gij(x,t;x0,t0) /∂x0 ] Mjk(x0,t0) ^ ^ 

  6. Description of earthquakes using moment tensors: Parameters: strike , dip , rake  Right-lateral =180o, left-lateral =0o,=90 reverse, =-90 normal faulting Strike, dip, rake, slip define the focal mechanism 0 M0 0 Example: vertical right-lateral al M= M0 0 0 M0=DA scalar seismic momen 0 0 0 

  7. Description of earthquakes using mome Parameters: strike , dip , rake  Vertical fault, right-lateral =180o Vertical fault, right-lateral =0o Strike, dip, rake, slip define the focal m 0 M0 0 Example: vertical right-lateral along x M= M0 0 0 M0=DA scalar seismic moment (Nm) 0 0 0 

  8. Because of ambiguity Mij=Mji two fault planes are consistent with a double-couple model: the primary fault plane, and the auxillary fault plane (model for both generates same far-field displacements). Distinguishing between the two requires further (geological) information

  9. Far-field P-wave displacement for double-couple point source: uPi(x,t)=(1/43) (xixjxk/r3)-(1/r) ∂Mjk(t-r/t r2=x12+ x22 + x32 For the fault in the (x1,x2) plane, motion in x1 direction, M13=M31=M0 and: uPi(x,t)=(1/23) (xix1x3/r3)-(1/r) ∂Mj(t-r/t In spherical coordinates: x3/r=cos, x1/r=sin cos, xi/r=ri uP=(1/43) sin2 cos (1/r) ∂M0(t-r/t r ^ ^

  10. Far-field S-wave displacement for double-couple point source: uSi(x,t)=[(ij-ij)k]/(1/43)(1/r) ∂Mjk(t-r/t, i = xi/r r2=x12+ x22 + x32 For the fault in the (x1,x2) plane, motion in x1 direction, M13=M31=M0 and: uS(x,t)=(1/43)(cos2cos-cossin)(1/r) ∂M0(t-r/t ^ ^

  11. Earthquake focal mechanism determination from first P motion (assuming double-couple model): • Only vertical component instruments needed • No amplitude calibration needed • Initial P motion easily determined (up or down) • Up: ray left the source in compressional quadrant • Down: ray left source in dilatational quadrant • Plotted on focal sphere (lower hemisphere) • Allows division of focal sphere into compressional/dilatational quadrants • Focal mechanism is then found from two orthogonal planes (projections on the focal sphere)

  12. Earthquake focal mechanism determination from first P motion (assuming double-couple model): • Focal sphere is shaded in compressional quadrants, generating ‘beach ball’ • Normal faulting: white with black edges • Reverse faulting: black with white edges • Strike-slip: cross pattern

  13. Far-field pulse shapes: Earthquake rupture doesn’t occur instantaneously, thus we need a time dependent moment tensor M(t) Near-field displacement is permanent Far-field displacement (proportional to ∂M/t) is transient (no permanent displacement after the wave passes): uSi(x,t)=[(ij-ij)k]/(1/43)-(1/r) ∂Mjk(t-r/t area=M0= Save A

  14. Directivity: Haskell source model Point source: amplitude will vary with azimuth but rise time is constant Larger events: integrating over point sources M(t) ∂M(t)∂t 0 tr 0 tr

  15. Directivity: Haskell source model (Vr ~ 0.7-0.9) Rupture toward you at end of fault: d = - L/ + L/Vr (last arrival rupture pulse L/Vr -first arrival P wave, L/ Rupture away from you at end of fault: d = L/ + L/Vr (last arrival L/Vr (time of rupture to the end of the fault) + L/ (time of the P waves generated by the last rupture instant at L/Vr) - first arrival 0s) Vr L rupture

  16. Far-field displacement is the convolution of two boxcar functions, one with width r and one with width d:

  17. Stress Drop = average difference between stress on fault before and after the earthquake. t2)t1dS A is fault area Assume long skinny fault (w<<L) with average displacement Dave and slip in the direction of L. Strain is then  = Dave/w, and we have Dave/w In general: CDave/L where L is a characteristic rupture dimension, C is a nin-dimensional constant that depends on rupture geometry Infinite long strike-slip fault: L=w/2, C=2/ S

  18. Earthquake magnitude Most related to maximum amplitudes in seismograms. Local Magnitude (ML): Richter, 1930ies Noticed similar decay rate of log10A (displacement) versus distance Defined distance-independent magnitude estimate by subtracting a log10A for reference event recorded on a Wood-Anderson seismograph at the same distance ML=log10A(in 10-6m)-log10 A0(in 10-6m) =log10A(in 10-6m)+2.56log10 dist (in km) -1.67 for 10<dist<600km only

  19. Earthquake magnitude Body wave magnitude (mb): (used for global seismology) mb=log10(A/T)+Q(h,) T is dominant period of the measured waves (usually P, 1s) Q is an empirical function of distance  and depth h (details versus amplitude versus range) Surface wave magnitude (Ms): (used for global seismology, typically using Rayleigh waves on vertical components) Ms=log10(A/T)+1.66 log10 + 3.3 = log10A20+1.66 log10 + 2.0 (shallow events only)

  20. Earthquake magnitude Saturation problem motivated the moment magnitude Mw Mw=2/3 log10M0-10.7 (M0 moment in dyne-cm, 107dyne cm=1Nm) = Mw=2/3 log10M0-6.1 (M0 moment in Nm) Scaling derived so Mw is in agreement with Ms for small events More physical property, does not saturate for large events

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