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**Understanding Earthquake Mechanics**: Learn the basics behind earthquakes, including moment tensors and focal mechanis

Explore the fundamental concepts of earthquakes, such as force couples, moment tensors, and faulting mechanisms. Understand how displacement is related to body forces using mathematical equations and models. Discover how earthquake focal mechanisms are determined and differentiated based on geological data. Gain insights into wave propagation and the effects of seismic forces.

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**Understanding Earthquake Mechanics**: Learn the basics behind earthquakes, including moment tensors and focal mechanis

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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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