The Basics of Earthquake Location

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The Basics of Earthquake Location

• William Menke
• Lamont-Doherty Earth Observatory
• Columbia University

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• The basic data in earthquake location is
• Arrival Time, t
• The time of day that a wave from the earthquake
  arrives at a seismograph station

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• The distinction between
• Arrival Time time of day something arrives
• And
• Travel Time the length of time spent traveling
• Is very important in earthquake location!

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Arrival Time ?Travel TimeQ a car arrived in
town after traveling for an half an hour at sixty
miles an hour. Where did it start? A. Thirty
miles awayQ a car arrived in town at half past
one, traveling at sixty miles an hour. Where
andwhen did it start? A. Are you crazy?
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An earthquake location has 4 Parameters x, y
(epicenter) z (depth) t (origin
time)Together, (x, y, z) are called the
hypocenter. The fact that origin time is an
unknown adds complexity to the earthquake
location problem!
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Suppose you contour arrival timeon surface of
earth
Earthquakes (x,y) is center of bulls-eye
but what about its depth?
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Since origin time unknownwe have not marked it
on time axis
Deep
Earthquakes depth related to curvature of
arrival time at origin
Shallow
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Fundamental dataarrival time tpi of wavesfrom
earthquake p to station i
Wave could be either P wave or S wave. Both are
used.
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Fundamental RelationshipArrival Time Origin
Time Travel Timetpi tp Tpi
Traveltime Tpi along ray connecting earthquake p
with station q can be calculated using ray theory
ray
earthquake p with origin time tp
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Locating an earthquakerequires knowing
theearths seismic velocity structureaccuratel
yso that traveltime can be calculatedbetween
stations and hypothetical hypocenters
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examplevelocitystructure(Iceland)in this
caseassumed to vary only with depth
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• Basic Principle
• Best estimates of the hypocentral parameters and
  origin time are the ones that best predict the
  arrival times at all the stations.
• Usually, best predicts means minimizing
• the least-squares prediction error, E
• Ep Si tpiobserved tpipredicted 2
• where tpipredicted tppredicted
  Tpipredicted
• and where Tpipredicted depends on (xp, yp, zp)

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• The mathematical problem is to find the
• hypocentral parameters,
• xppredicted(xp, yp, zp)predicted
• and origin time,
• tppredicted
• that give the best fit
• (which is to say, minimize the error)
• But the problem is that the traveltime varies in
  a complicated, non-linear way with the
  hypocentral parameters, xppredicted

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• The usual solution is to use an iterative method
• Step 1 Guess a set of hypocentral parameters,
  h(xp, yp, zp, tp) (xp, , tp) and use it to
  predict the traveltime
• Step 2 Determine how much the arrival time would
  change if the guess were changed by a small
  amount, dh (dx, dt).
• Step 3 Use that information to attempt to find a
  slightly different h that reduces the error, E.
• Do steps 2 and 3 over and over again, hoping that
  eventually the error will become acceptably
  small.

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• It turns out that Step 2 is incredibly easy.
• A small change in origin time, dt, simply shifts
  the arrival times by the same amount, dt dt.
• The effect of a small change in location depends
  on the direction of the shift. A change dx along
  the ray direction shifts the time by dtdx/v.
  But a change perpendicular to the ray has no
  effect. This is Geigers Principle, and
  illustrated in the next slide.

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• Step 3 is pretty easy too. The trick is to
  realize that the equation that says the observed
  and predicted traveltimes are equal is now linear
  in the unknowns
• tpiobs tpipre tp Tpipre
• tpguess dt Tpipre(xpguess) (t/v)?dx
• Or by moving two terms to the left
• tpiobs - Tpipre(xpguess) - tpguess dt (t/v)?dx

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• The methodology for solving a linear equation in
  the least-squares sense is very well known. It
  requires some tedious matrix algebra, so we wont
  discuss it here. But is routine.

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but with any method, a key question is

• What can go wrong?

here are some possibilities
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• Too few data

Since there are four unknowns,you must have at
least four arrival time measurements. Any fewer,
and you cannot locate the earthquake.
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• Bad Station Geometry

But P and S waves from each of two stations wont
do it, because there is a left-right ambiguity
earthquake here?
station 1
station 2
or here?
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• Another poor geometry

When the stations are all to one side of the
stations, the rays all leave the source in
roughly the same direction and location trades
off with origin time
station 2
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shallow and late
deep and early
Depending upon ray geometry, this trade-off can
also involve depth and origin time
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Recently, a new earthquake location method has
been developedthat instead of locating a single
earthquake on the basis of its arrival times (as
above)locates groups of earthquakes on the
basis of the differencein their arrival times
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This method is often called theDouble-Difference
Methodthe following figures illustrate its
power
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Earthquakes in Long Valley Caldera, California
located with traveltimes Note amorphous clouds of
earthquakes, no evidence of faults.
Courtesty of Felix Walhhauser, LDEO
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Earthquakes in Long Valley Caldera, California
located with the double-difference method Note
many earthquakes fall on lines, so there is clear
evidence of faults!
Courtesty of Felix Walhhauser, LDEO
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The basic data in the double-difference method is
the differential arrival time between two
different earthquakes observed at the same
station Dtpqi tpi - tqi
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But that is another story