The Basics of
Earthquake Location
William Menke
Lamont-Doherty Earth Observatory
Columbia University
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
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!
Arrival Time ≠Travel Time
Q: a car arrived in town after traveling for an
half an hour at sixty miles an hour. Where
did it start?
A. Thirty miles away
Q: a car arrived in town at half past one,
traveling at sixty miles an hour. Where and
when did it start?
A. Are you crazy?
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!
Suppose you
contour arrival time
on surface of earth
Earthquake’s
(x,y) is center
of bulls-eye
but what about
its depth?
Deep
Shallow
Since origin
time unknown
we have not
marked it on
time axis
Earthquake’s
depth related to
curvature of
arrival time at
origin
Fundamental data:
arrival time tpi of waves
from earthquake p to station i
Wave could be either P
wave or S wave. Both are
used.
Fundamental Relationship
Arrival Time = Origin Time + Travel Time
tpi = tp + Tpi
ray
earthquake p
with origin time tp
Traveltime Tpi along ray
connecting earthquake p
with station q can be
calculated using ray theory
Locating an earthquake
requires knowing the
earth’s seismic velocity structure
accurately
so that traveltime can be calculated
between stations and hypothetical
hypocenters
example
velocity
structure
(Iceland)
in this case
assumed to
vary only
with depth
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)
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
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.
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 dt=dx/v. But a change
perpendicular to the ray has no effect.
This is Geiger’s Principle, and illustrated in
the next slide.
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
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.
but with any method, a key
question is …
What can go wrong?
here are some possibilities …
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.
Bad Station Geometry …
But P and S waves from each of
two stations won’t do it, because
there is a left-right ambiguity
earthquake here?
station 1
station 2
or here?
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
shallow and late
deep and early
Depending upon ray geometry,
this trade-off can also involve
depth and origin time
Recently, a new earthquake
location method has been
developed
that instead of locating a single
earthquake on the basis of its
arrival times (as above)
locates groups of earthquakes on
the basis of the difference
in their arrival times
This method is often called the
Double-Difference Method
the following figures illustrate its
power
Earthquakes in
Long Valley
Caldera,
California
located with
traveltimes
Note
amorphous
clouds of
earthquakes, no
evidence of
faults.
Courtesty of
Felix
Walhhauser,
LDEO
Earthquakes in
Long Valley
Caldera,
California
located with the
doubledifference
method
Note many
earthquakes fall
on lines, so
there is clear
evidence of
faults!
Courtesty of
Felix
Walhhauser,
LDEO
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
But that is another story …