Locating earthquake epicenters
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We start with a couple of quick definitions: focus (and the plural foci – with the “c” pronounced soft) and epicenter.
Remember that a fault is a place in the crust where rocks are broken and forced to move in different directions on either side of the fracture. They are most often
found near plate margins. Before a fault moves the force causing the motion bends the rocks in opposite directions (elastically deforming them) and stores
potential energy, just like energy is stored in a bent ruler.
When the rock breaks the energy spreads away from the fracture by elastically deforming the surrounding rocks (for very long distances) as waves of various sorts.
Faults do not move as a point, but they also do not move smoothly along their entire lengths. When a portion of a fault moves there is some place, a more-or less
small region, that moves farther than everything else. This area of maximum motion is called the focus. The focus might be at Earth’s surface or it might be at
great depth. The deepest ever recorded was some 700km deep beneath South America.
However deep the focus, the point immediately above it is the epicenter. When you see a map of an earthquakes location what you are seeing is the epicenter.
The depth to focus is usually written beside it.
FAULT
EPICENTER
FOCUS
This side moves
away from you.
This side moves
toward you.
This is the record of an earthquake in
Indonesia in 2009 as recorded at GSW’s
seismic station. The three long period
records are shown. The short period record
is not. (Why?)
EPICENTER
FOCUS
Note the yellow dot north of Australia and
the inset map that pinpoints the epicenter
location more specifically. Also note that
the depth (to the focus) is given as 29km.
How do we know the location? That’s what
we aim to understand in this slideshow.
Notice how “noisy” the waves seem on the
real seismograms. We’ll look at very
simplified models in this slideshow and in
lab.
(Graphic by Sam Peavy)
When the waves are generated they begin to move away from the focus
(and epicenter) in every direction. One of those directions will be towards
our seismograph, and we will record the waves. Obviously it will take
some time for them to get to us, and how long that obviously depends on
how far away the earthquake occurred.
All we will “see” is a consequence of the ‘quake – the arrival of the waves
at our location, not the earthquake itself.
*
Nevertheless, we will be able to figure out how far away
the focus was from us, therefore when the ‘quake
occurred, and where the epicenter was.
To do this we have to remember that p-waves are faster than s-waves and examine
the consequences of that fact.
This will be easier to understand if we quantify it, though you won’t see a
calculation like this on a test.
Over short distances the waves move at rates of around 400 km/min (p) and
200km/min (s). Both types speed up with distance, so that after they’ve moved
4000 km the rates are closer to 550 km/min (p) and 300 km/min (s), but the ratio of
the two rates remains constant, so we can do our calculations at the base rate and
get a good idea of what happens overall.
Imagine an epicenter is 400 km away from our seismograph. Our needle will make
its first jiggle in one minute, and that will be when the faster p-wave arrives, moving
at 400 km/min. Those waves will make smaller and smaller jiggles on our needle
over time.
The first s-wave jiggle will take an extra minute to reach us, not arriving for two
minutes, because it is only moving half as fast (200 km/min).
Our seismogram will look something like this (only far more “noisy”, remember):
First p-wave jiggle
“Lag Time”
(s-waves damp)
No jiggles
(p-waves damp)
(direction of recording)
First s-wave jiggle
We will clearly see two sets of jiggles – one from the p-waves and one from the
s-waves. The difference recorded by the seismogram between the times the
two sets first affected our instrument will be the lag time. What is it in this
case? (Return to the previous slide if you don’t remember.)
Since the p-wave arrives at our station in one minute and the s-wave takes
two, the lag time will be one minute.
Notice that we can’t possibly know that it took one minute and two minutes,
we just know that the two first kicks are a minute apart. It could have been 2
minutes and three, three minutes and four, etc.
So what?
Now imagine that an epicenter is 800 km away from our station rather than 400 as in
the previous example. We will use the same rates to do our calculations. Even
though they won’t give us perfectly correct answers they will let us see an
interesting fact.
The p-waves, moving at 400 km/min, will take two minutes to arrive, and our
seismogram will have a first kick that records this. The waves will damp (get smaller)
as time goes by.
Moving at only 200 km/min the s-wave will not travel the necessary 800 km and
draw a jiggle on our seismogram for four minutes!
Maybe you already see where this is going.
The seismogram will look like this (without the red parts, of course):
Just for practice, identify the parts indicated.
p-wave first arrival
s-wave first arrival
Lag time
What will the lag time be in this case?
Because one epicenter is farther away, the s-wave will take proportionally even longer
to arrive at the seismograph and jiggle the pen – two minutes of lag time(4 minutes – 2
minutes) rather than one minute.
This is important because, as you should be able to convince yourself, for a given
distance there is one and only one possible lag time! Another distance will have yet
another lag time.
Seismogram at 400 km
1 minutes
2 minutes
Seismogram at 800 km
What would the lag time be for a seismogram at 1200 km?
Therefore, the lag time is all we need to
calculate the distance to the epicenter!
We do this with a nifty little program called a Jeffries-Bullen (or J-B) calculator, which you will see in lab.
Let’s imagine that our seismic station has recorded an earthquake whose squiggles look like this. We read
the seismogram from left to right, starting at the top left corner. The flat line just below the top was what
the pen drew some time before the event – follow it to the right edge, then drop down to the second line
and read left to right again. Nothing interesting happens until we get to the sixth line, and then we the
jiggles begin. This is the arrival of the first p-wave. (Why?) This is also the start time, or time 0 (zero) of
our record.
Notice that it is at the left end of the series of p-wave kicks – the first indication we have of the event.
“Time 0” – arrival of the first p-wave.
We need to estimate the lag time between the two waves, rounded to the nearest
minute. To do this we simply count off minute markers from 0 at the first p-wave
kick to the first s-wave kick (red arrow). In this example the s-wave arrived in a
little less than 5 minutes. If we insert (just in our heads, usually) approximations of
quarter minutes between 4 and 5 (blue lines) we see that the s-wave arrival is just
above and pretty close to the 4¾ mark, so we’ll round our time to 4.8 minutes.
Notice that we have probably just introduced a small error into our eventual result.
Time 0
1
2
3
4
5
… and we enter that value into our data table
Lag Time (min)
Distance (km)
Americus
__4.8___
_________
Madison
________
_________
Los Angeles
________
_________
Now repeat the process for the Madison, WI seismogram. This one does not have
the minutes indicated, but does have the quarter minutes in the correct spot –
between three and four minutes after the first p-wave in this case. We see that the
first s-wave kick came just before the 3¾ mark and so round the lag time to 3.7
minutes.
… and we enter that value into our data table
Lag Time (min)
Distance (km)
Americus
__4.8___
_________
Madison
__3.7___
_________
Los Angeles
________
_________
Finally, we do the same for the L.A. seismogram. The first s-wave arrived
more than two but not three minutes after the p-wave. Its first kick is closer
to the 3 minute mark than the 2¾ mark so we round to 2.9 minutes.
… and we enter that value into our data table
Lag Time (min)
Distance (km)
Americus
__4.8___
_________
Madison
__3.7___
_________
Los Angeles
__2.9___
_________
The graphic below was screen-copied from the J-B calculator program after sliding the red
bar to a lag time (S-P difference) of about 4.8 minutes. (Remember that we’ve already
rounded our value and don’t worry about the extra hundredth.) The epicenter distance,
and, remember, the only possible epicenter distance that coincides with a lag time of 4.81
min is 2923 km. We’ll round this to 2925.
… and we enter that value into our data table
Lag Time (min)
Distance (km)
Americus
__4.8___
__2925__
Madison
__3.7___
_________
Los Angeles
__2.9___
_________
So what we know at this point is only how far from
Americus the epicenter was. That doesn’t tell us where
it was, but it does indicate a lot of places that it was
not.
If we repeat the process for the 3.7 minute lag time recorded at the Madison, WI station we find that the epicenter
was ~1950 km away from Madison, rounding to the nearest 25. Again, it doesn’t tell us where the ‘quake occurred,
but it does tell us a lot of places where it didn’t, and you can probably predict where that leads: many of the places
that this seismogram says were not the epicenter are different places that what the Americus Seismogram said, so we
have ruled out even more possible locations. In fact, we have ruled out all but two, as you will soon see.
… and we enter that value into our data table
Lag Time (min)
Distance (km)
Americus
__4.8___
__2925__
Madison
__3.7___
__1950__
Los Angeles
__2.9___
_________
Finally, the J-B graph tells us that the 2.9 minute lag time at L.A., CA corresponds to a
distance of about 1375 km (rounding to the nearest 25). Again, this information rules
out many more potential candidates for the location of the epicenter. In fact, we have
narrowed the number to one possibility, which must be correct.
… and we enter that value into our data table
Lag Time (min)
Distance (km)
Americus
__4.8___
__2925__
Madison
__3.7___
__1950__
Los Angeles
__2.9___
__1375__
The final step is to represent our findings
graphically and make it obvious what we have
discovered.
As is very often the case in geology, a map is what
we use as our starting point.
The next step involves graphically illustrating where the epicenter can and cannot be relative to each seismograph. All the information
we have that relates to this is a distance away. You should infer therefore that the map scale is going to be part of our thinking. Let’s
have a look at it.
Notice that there’s actually more
detail that the numbers provided suggest.
For one thing, we have blocks that are 500 km
long as well as the 1000’s given. More
importantly, at the left end we have small
blocks that are 100 km long. These will help
our accuracy a lot!
500
1500
2500
3500
400
300
200
100
We start by spreading our compass so that the pin and the lead are the distance apart that our seismogram indicates – for Americus, about 2925 km. We want to do
this as precisely as we can. Our immediate thought is to put the pin on the scale zero and the lead out near 3000, but we can do it more accurately the other way
because of the 100 km divisions in the first 500km. So instead, we put the pin at 3000 and open the compass to the first 100 km mark (3000-100=2900). Then we open
it a little more – until it is about ¼ the way from 100 to zero. That gives us a fair estimate of the last 25 km.
Now put the pin of the compass as close as you can to the center of the block at Americus and draw a circle (or as much of one as will fit on the paper)
around it. Be careful not to let the compass arms spread or compress. The radius of this circle will be 2925 km, scaled, of course, to the map.
Think for a minute – what do we have? Each and every point on this circle lies 2925 km away from Americus. Furthermore, these are the
only points on Earth that are that distance from Americus. Any point inside the circle is closer; any point outside it is too far away. Thus,
this circle illustrates every possible location for the epicenter, and rules out many, many other possibilities!
Now do the same for the Madison, WI seismograph. The distance is about 1950 km, a little less than 2000, so pin the compass at 2000 and
open it to put the pencil about half way (50 km) between the zero and the one hundred mark. 2000-50=1950.
Draw a circle around Madison with that radius. The points on this circle are all the points on Earth the right distance from Madison.
R=1950 km
Remember Venn diagrams in your early math classes? The intersection of two sets is the
set which includes members from both sets.
INTERSECTION
The set of all points the
right distance from
Americus.
The set of all points the
right distance from
Madison.
on our map, the set of all points, every last one of them, that are the right distance
from Americus are represented by a circle. The same is true for Madison.
The epicenter of the Earthquake must be in the intersection of these two sets,
right?
The two circles on our map intersect at only two points – one in eastern Canada and one in the western USA. One of these must
be the epicenter. (You can probably decide which by looking at the lag time for the LA seismogram, but we’ll finish the exercise
anyway).
Once we add the circle (r=1350 km) for the LA seismograph station we pinpoint the location exactly. The epicenter has to be at the
intersection of all three circles – that is, the intersections of the three sets of points that are the right distance from all three stations! In this
case it is in southwestern Wyoming.
R=1350 km
What about locating the focus? The idea is exactly the same, but is a little more
complex geometrically than locating an epicenter. Once the epicenter is known then
appropriate stations (close to the epicenter) are chosen. The distance “circles” are
drawn vertically rather than horizontally, and their intersection is the focus.
FAULT
Of course this is usually a 3-dimensional problem so the “circles” are really spheres
drawn around the three stations. The spheres will intersect at the focus.
FOCUS
This side moves
away from you.
This side moves
toward you.