The procedure is simple to state: guess a location, depth
and origin time; compare the predicted arrival times of the
wave from your guessed location with the observed times at
each station; then move the location a little in the direction
that reduces the difference between the observed and calculated
times. Then repeat this procedure, each time getting closer
to the actual earthquake location and fitting the observed
times a little better. Quit when your adjustments have become
small enough and when the fit to the observed wave arrival
times is close enough.
You can try to fit an earthquake location on the map just
to see how the procedure goes. Note that the earthquake arrives
first on station C, thus C is a good first guess for the location.
Many earthquakes in California occur between 2 and 12 kilometers
depth and we will guess a 6 km. depth. The origin time should
be a few seconds before the time of the wave at the first
station. Let's guess an origin time of 10 seconds, measured
on the same clock that made the time scale at the bottom of
the figure and timed the seismograms. Then we can list the
tentative travel times by subtracting the origin time from
the observed arrival times:
Station |
A |
B |
C |
D |
E |
F |
Observed time |
16.5 |
17.8 |
11.3 |
15.2 |
22.3 |
18.3 |
Tentative travel time |
6.5 |
7.8 |
1.3 |
5.2 |
12.3 |
8.3 |
Note the scale at the left of the figure. It shows travel
times for waves from an earthquake at a depth of 6 kilometers.
The scale starts at 1.3 seconds because the wave reaches the
surface 1.3 seconds after the earthquake origin time. You
can make a tracing of the scale and move the earthquake on
the map until the tentative travel times match the travel
times from the scale. Where do you think the earthquake was?
Are the times for each station systematically early or late,
requiring a shift in the origin time?
To open a window with the earthquake location shown on the
map, CLICK HERE.
The earthquake was near station C. The depth was about 6
km and the origin time was about 10 seconds. (We guessed very
well!) A real magnitude 3.4 earthquake occurred at this location
on April 29, 1992. It was felt by many people who were sitting
or at rest.
Mathematically, the problem is solved by setting up a system
of linear equations, one for each station. The equations express
the difference between the observed arrival times and those
calculated from the previous (or initial) hypocenter, in terms
of small steps in the 3 hypocentral coordinates and the origin
time. We must also have a mathematical model of the crustal
velocities (in kilometers per second) under the seismic network
to calculate the travel times of waves from an earthquake
at a given depth to a station at a given distance. The system
of linear equations is solved by the method of least squares
which minimizes the sum of the squares of the differences
between the observed and calculated arrival times. The process
begins with an initial guessed hypocenter, performs several
hypocentral adjustments each found by a least squares solution
to the equations, and iterates to a hypocenter that best fits
the observed set of wave arrival times at the stations of
the seismic network.
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