Toward SystemLevel Earthquake Science

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Turning Points

X(t), TpX(p), phase delay?pX(p)
v
z
T(p), phase delay?T(p)-?/2

• Constructive interference when
• phase along ray and at bounce points
• satisfies
• ?pX(p)?T(p)-?/2-n2???i.e.
• n2??/2 / T(p)-pX(p) n2??/2 / ?(p)
• Dispersion curve! N0 is fundamental mode

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Phase and group velocity

T(p), delay?T(p)-?/2
v
z
X(t), TpX(p), delay?pX(p)

C1/p phase velocity UX(p)/T(p) group velocity
d?/dk
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tanh?(1/?12-p2)1/2 ?2(p2-1/??2)1/2/ ?1(1/??2
-p2)1/2? c1/p is the phase velocity, varying
between ?1 and ?2Every value of c produces
multiple values of ? (or T)Smallest value of ?
defines the fundamental mode,second smallest
the first higher mode (over-tone).
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?????1,?1

?2,?2
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tanh?(1/?12-p2)1/2 ?2(p2-1/??2)1/2/ ?1(1/??2
-p2)1/2?
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Rayleigh Waves(2p2-1/?2)2-4p2(p2-1/?2)1/2
(p2-1/?2)1/2 0 the Rayleigh function (P and
SV co-exist at surface of a halfspace)
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Dispersion u(x,t)cos(?1t-k1x)cos(?2t-k2x) ?1
?-??, ?2 ??? k1k-?k, k2k?k u(x,t)cos(?t-
??t-kx?kx)cos(?t??t-kx-?kx) cos(?t-kx)-(??t-
?kx)cos(?t-kx)(??t-?kx) 2cos(?t-kx)cos(?kx-
??t) Signal of average frequency ? whose
amplitude is modulated by longer period wave of
frequency ?? Ud?/dk group velocity

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Surface Waves

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Recall the momentum equation???2ui/?t2
?j?ijfi , 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)
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Helpful to define a concept that separates the
source from the wave propagationui(x,t)G f
Gij(x,tx0,t0)fj(x0,t0)f force vectorG
Greens function response to a small
sourceLinear equationDisplacement from any
body force can be computed as the superposition
of individual point sources
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Force CouplesForces must occur in opposing
directions to conserve momentum
D no net torque
double couple D net
torque no net torque
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9 Force Couples Mij (the moment tensor), 6
different (MijMji). Mfd
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
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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

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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
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Because of ambiguity MijMji 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
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The moment tensor can be diagonalized from its
eigenvalues and eigenvectors, and rotation to
another coordinate system. For example The
principal axes are rotated 45 degrees w/r to the
original system. The new x and y axes are called
the tenson (T, maximum compression) and pressure
(P, maximum tension) axes, respectively.
0 M0 0 M M0 0 0 0 0 0

M0 0 0 M 0 -M0 0 0 0 0

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Far-field P-wave displacement for double-couple
point source In spherical coordinates
x3/rcos?, x1/rsin? cos?, xi/rri uP(1/4???3)
sin2? cos? (1/r) ?M0(t-r/????t r


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Far-field S-wave displacement for double-couple
point source uS(x,t)(1/4???3)(cos2?cos??-cos?
sin??)(1/r) ?M0(t-r/????t


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• Earthquake focal mechanism determination from
  first P motion (assuming double-couple model)
• Only vertical component instruments 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)

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• Earthquake focal mechanism determination from
  first P motion (assuming double-couple model)
• Focal sphere is shaded in compressional
  quadrants, generating beach ball
• Tension axis in the middle of shaded region
• Pressure axis in the middle of unshaded region
• Only represents stress tensor if slip on plane
  of max shear
• Normal faulting white with black edges
• Reverse faulting black with white edges
• Strike-slip cross pattern

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Far-field pulse shapes Earthquake rupture
doesnt 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 areaM0
?Save A

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Directivity Haskell source model
M(t) ?M(t)?t
0 tr
0 tr
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Directivity Haskell source model (Vr
0.7-0.9?)
Vr L
rupture
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Far-field displacement is the convolution of two
boxcar functions, one with width ?r and one with
width ?d


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Source Spectra Fourier transform of a boxcar with
width and amplitude 1 ?exp(i?t)dt sinc(?/2)
FB(t) FB(t/?r)?r sinc(??r/2)
FB(t) FB(t/?r) B(t/?d) ?r
?d sinc(??r/2) sinc(??d/2) A(??????? g Mo
sinc(??r/2)sinc(??d/2)
0.5 -0.5
?r 2?/?r
?
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Stress Drop ??? average difference between
stress on fault before and after the
earthquake. ??????????????????t2)???t1??dS

S
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???CMo/aL3 Circular fault with radius r
???7Mo/16r3 Generally stress drops between 10-100
bars (1-10MPa) Interplate earthquakes have
smaller stress drops than intraplate earthquakes,
in general (intraplate faults are
stronger). Absolute level of stress NOT
constrained!
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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 MLlog10A(in 10-6m)-log10 A0(in 10-6m)
log10A(in 10-6m)2.56log10 dist (in km)
-1.67 for 10 42
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Earthquake magnitude Body wave magnitude (mb)
(used for global seismology) mblog10(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) Mslog10(A/T)1.66
log10 ? 3.3 log10A201.66 log10 ? 2.0
(shallow events only)
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Earthquake magnitude Calibration of all scales
not possible due to differences in frequency of
the waves involved Corner frequency typically
moves to lower frequency for larger events -
magnitude linear below corner frequency, but not
above

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Earthquake magnitude Saturation problem
motivated the moment magnitude Mw Mw2/3
log10M0-10.7 (M0 moment in dyne-cm, 107dyne
cm1Nm) Mw2/3 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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Earthquake magnitude Intensity scale Measures
damage to structures Often used is Mercalli
I-XII Can be used to examine historic earthquakes
without seismic records For example, eastern
US.
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Intensity Observed Effects I Not felt at all II
Felt only by a few individuals, indoors and?at
rest, usually on upper floors of tall
buildings. III Felt indoors by many persons, but
not necessarily recognized as an earthquake.
Chandeliers and hanging plants swing.


IV Felt both indoors
and out. Feels like the vibration caused by a
heavy truck or train passing. Windows rattle. V
Strong enough to awaken sleeping persons. Small
objects knocked off shelves. Beverages may splash
out of cups or glasses on tables. VI Perceptible
to everyone. May cause public fright. Pictures
fall off walls. Weak masonry cracks. Some plaster
may fall from ceilings. VII Difficult to stand
upright. Ornamental masonry falls from buildings.
Waves may be seen in ponds and swimming
pools. VIII Mass panic may occur. Chimneys, smoke
stacks and water towers may lean and fall.
Unsecured frame houses slide off foundations. IX
Panic is general. Heavy damage to masonry
structures and to?underground pipes. Large cracks
open in ground.X Many buildings collapse. Water
splashes over riverbanks. XI-XII Virtually total
destruction.
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Radiated Energy Log10Es (ergs)
5.82.4mb11.81.5Ms Es(Ms7)/Es(Ms6)32! Es(Ms7
)/Es(Ms5)1,000!
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Magnitude power laws and b-values Log10Na-bMs
(Gutenberg Richter relationship) N number of
events with magnitude Ms ?M b (the slope) is
called the b value, generally 0.8-1.2 b1 -
number of earthquakes increses by 10 for each
drop in magnitude From energy release, many small
events do not release significant amounts of
energy
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Heat flow paradox Total released energy
radiated energy frictional heat EEsEf Es?E
(? is the seismic efficiency) EDaveA?ave
(?ave average stress on fault, not stress
drop!) EM0?ave/? since DaveMo/?A Es?
M0?ave/? ??Es/M0?ave 0.5??/?ave since Es
??M0/2? Ef DaveA?f (?f is frictional shear
stress) ?f0.85?n, ?n?n200MPa (Byerlees law)
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• Heat flow paradox
• Even using lower estimate from (2), the predicted
  heat flow from the San Andreas fault is 10 x
  more than observed
• Assuming ?50-170MPa, ??3MPa, ?1-3 (low!)
• Shear stress much lower than Byerlee suggests?
• Stress measurements near SAF - average ?
  10-20MPa
• (weak fault)
• How come slip occurs so easily despite low shear
  stress compared to large normal stresses -
  fluids, fault opening, gouge?