1. Larry Braile, Purdue University
The Five-Slinky Model – Waves in all directions and illustration of how seismologists use many P-wave arrival times to find the epicenter or hypocenter of an earthquake
Attach to wood with
screws and washer
Half length
slinky (hold
second half
in hand)
Double slinky
(taped together)
Larry Braile, Purdue University
web.ics.purdue.edu/~braile
This PPT

2. 1. Demonstration of seismic wave propagation using a slinky: See instructions for using a slinky to demonstrate wave propagation for P, S, Rayleigh, and Love waves at: This web page also provides instructions for making the 5-slinky model (see photo on previous page) and using the 5-slinky model to demonstrate waves traveling in “all” directions and the different travel times of wave propagation to different distances.
2. The 5-slinky model can also be used to illustrate the method that seismologists use to locate earthquakes (x, y, z, and origin time), mostly from many P-wave arrival times. To illustrate the earthquake location method, consider that the travel time from the source (person causing a sudden disturbance on the five slinky wood block) to one of the “recorders” (the people holding the ends of the slinky) arrives too early (for that distance; as compared to well-documented travel time data, such as for travel times in the Earth). That would mean that the “earthquake” would be farther away than the current distance from the source (epicenter) to that seismograph (end of the slinky) indicating that to better determine the location of the epicenter from this station, the trial epicenter needs to be moved farther away from that station. With multiple stations and a directed trial and error approach, one can find the optimum location. A graphic example is provided below – notes relate to the 5-slinky demo, and the quantitative example of trial epicenters (directed trial and error search method) on the slides below.

3. How to locate an earthquake with P wave arrivals –
similar to the 5-slinky demo
When earthquake location is discussed in introductory books, the S-minus-P method is almost always presented. Although the S-minus-P method is a relatively simple concept and can be used to locate earthquakes, it is very rarely used by seismologists because it is less accurate, usually used for only 3-4 seismograms, and, generally, cannot be used to accurately determine the depth of an earthquake. Almost all earthquakes are located by an optimization (or directed search) method that can easily be illustrated with the 5-slinky model. Instructions for using the 5-slinky model to demonstrate a directed trial and error search and a quantitative example of the search method are presented in the following slides (this will take some time and repetition to fully understand!).
Example of P and S wave arrivals from an earthquake.

4. * We start with a Trial Epicenter Trial Epicenter 1
Scale for measuring distances.
20 km
North
Similar arrangement of source (wood block with attached slinkys, star) and seismographs (people holding the ends of five slinkys, triangles) in the 5-slinky model demonstration of seismic wave propagation. 3
We start with a Trial
Epicenter 4
Trial
Epicenter 1
(create P wave here)
Or person holding the end of a slinky in the 5-slinky model demo.
Because we know the locations of the trial epicenter and the 5 seismograph stations, we can calculate (or measure on a map as shown here) the distance (D) from the trial epicenter to each station. Then, we can calculate the travel time (T) from the trial epicenter to each station using a well
Distance 2
Note time to stations * 1 5
Seismograph
Station
known Earth velocity model. For our simple example, we’ll assume the velocity is a constant 6 km/s (homogeneous Earth model). Then, the travel times are equal to T = D/V or T = D/6. The results are shown in the Table on the left. Now, the only unknown (for this trial epicenter) is the Origin Time of the earthquake. That can be determined using the method shown on the next slide. (This is a simplified example where the velocity is a constant and the epicenter is at the surface. For “more real Earth” situations [variable seismic velocity model, earthquake at depth] the same method works but the calculations are more complex.)
Dist. (km) Trav. Time (s)

5. For any trial epicenter (this is Trial
Epicenter 1), we can calculate the distance
(D, in km) between the trial epicenter and the
seismograph recording stations. Then, we
assume an Earth velocity model. In this
example, we use a simple flat Earth model with
a velocity (v) of 6 km/s. Using the distances
and the Earth velocity, we can calculate the
Travel time (T) from the trial epicenter to each
station [T(s) = D(km)/v(km/s)]. Plotting a
graph of the D and T data and drawing a “best fit
line” through the data points yields the graph
shown above. The equation shown is a best fit line.
The coefficient is 1/6 where 6 is the velocity in km/s. The second coefficient is very close to zero seconds and is the relative origin time of the simulated earthquake. Using the actual arrival times at the stations, the complete origin time (hours, minutes seconds) can be calculated. In our example, we have used an origin time of 00:00:00. We can then use the observed arrival times at stations and the expected arrival times for the trial epicenter to see if the trial epicenter is correct. Usually, the trial epicenter will not be consistent with the actual source location. However, the differences between the observed arrival times at stations and the expected arrival times based on the trial epicenter location can indicate which direction to move the trial epicenter to produce a better match of observed and predicted times. This can be demonstrated with a directed “trial and error” approach as illustrated below, and with the 5 slinky model demonstration. This approach is actually very similar in concept, to the sophisticated computer program (using an optimization method) that seismologists use to locate earthquakes.
In this example, we have assumed an origin time of 00:00:00. If we did not know the origin time (as is the usual situation) we would plot the arrival times
and this point would be the origin time (and
the second coefficient above).

6. * Test the Trial Epicenter by calculating predicted arrival times
Scale for measuring distances.
20 km
North
Note differences between the observed and predicted arrival times. The goal is to move the trial epicenter in a reasonable way to reduce the differences in arrival times, to obtain the best epicenter. 3
Test the Trial Epicenter by calculating predicted arrival times 4
The trial epicenter is not accurate (as would be expected for our first guess). The observed arrival time differences can be used to improve the epicenter location. The differences indicate that we need to move the epicenter. It makes sense to focus on stations with the largest differences.
Trial
Epicenter 1
Distance 2 * 1 5
Seismograph
Station
Compare
Dist. (km) Trav. Time (s) Obs. Arrival T Pred. Arrival T Difference (s)
:00: :00: (move farther)
:00: :00:
:00: :00: (move closer)
:00: :00: (move closer)
:00: :00:

7. * * Test Trial Epicenter 2 Trial Epicenter 2
Scale for measuring distances.
20 km
North
Note that the 5-slinky model can be used to illustrate this correction to the epicenter by moving the wood block and leaving the “seismographs” fixed. 3
Test Trial Epicenter 2 4
Trial
Epicenter 2
Moved closer to 3
Distance
Moved closer to 4 * 2
Note that moving the trial epicenter from location 1 to location 2 reduced the observed versus predicted arrival times, so epicenter 2 is a significant improvement; but we will need one or more additional trial epicenters.
Moved farther from 1 * 1 5
Seismograph
Station
Trial
Epicenter 1
Compare
Differences smaller than for Epi. 1
Dist. (km) Trav. Time (s) Obs. Arrival T Pred. Arrival T Difference (s)
:00: :00:
:00: :00: (move closer)
:00: :00: (move closer)
:00: :00:
:00: :00: (move farther)

8. Scale for measuring distances.
20 km
North 3
If there are still differences in the observed versus predicted times, we may need to consider the depth (hypocenter) of the earthquake.
Epicenter 3 results in good match between Observed and Predicted times 4
Final
Epicenter
Moved closer to 3
Distance * *
Trial
Epicenter 2
With only 3 trials in our directed search (because of the small number of stations, and a little luck), we have reduced the observed versus predicted time differences to zero, so we have the final and accurate epicenter. With more stations and real data, many trial epicenters will be required but the procedure will be the same. 2
Moved closer to 2 *
Moved farther from 5 1 5
Seismograph
Station
Trial
Epicenter 1
Dist. (km) Trav. Time (s) Obs. Arrival T Pred. Arrival T Difference
:00: :00:
:00: :00:
:00: :00:
:00: :00:
:00: :00:
Differences smaller than
for Epi. 2

9. Scale for measuring distances.
20 km
North 3
The 5-slinky model can also be used to illustrate the effect of depth by moving the wood block down, leaving the “seismographs” fixed; increases travel times. 4
Final
Epicenter
Moved closer to 3
As stated earlier, because of the large number of stations usually available for earthquakes, this location process is usually performed with an optimization computer program. Noisy data, variable velocity models, and the fact that earthquakes occur at different depths (so we need the hypocenter), make the computer solution necessary. However, the procedure is very similar to that illustrated here and can be illustrated with the 5-slinky model!
Distance * * 2
Moved closer to 2 *
Moved farther from 5 1 5
Seismograph
Station
Dist. (km) Trav. Time (s) Obs. Arrival T Pred. Arrival T Difference
:00: :00:
:00: :00:
:00: :00:
:00: :00:
:00: :00:

10. The EqLocate program (written by Alan Jones from a concept by L
The EqLocate program (written by Alan Jones from a concept by L. Braile) for locating earthquakes using seismograms is a directed search method that works for real seismograms, a real Earth velocity model, and earthquakes at depth.
The EqLocate program (Alan Jones, Windows) can be downloaded from:
A tutorial for EqLocate (including the images here and below) can be found at:
Flowchart for the EqLocate program. Steps 1, 2, 3, 7, and 8 are performed by the user. Steps 4, 5 and 6 are performed by the program by calculations and adjusting the map display and data displayed in the map and seismogram windows. Step 8, connected by the long, curved arrows, represents multiple selections of the trial epicenter (iteration) to improve the fit until an optimum location is found.

11. Example of using the EqLocate program to locate an earthquake using recorded seismograms
Seismograms, Observed arrival times (“picks”; downward red lines), theoretical arrival times (predicted/calculated for the hypocenter shown at right; upward blue lines; after several adjustments to the trial epicenter), and calculated S-wave arrival times (upward green lines) for the southern Indiana earthquake.
Stations (red triangles; black when calculated distance is the same as actual distance) and trial epicenters (colored dots) for the June 18, 2002 southern Indiana earthquake. The small error ellipse and small RMS error indicate that a reasonably accurate location has been determined.

12. More on seismic waves and the slinky:
Example of P and S wave arrivals from an earthquake:
EqLocate program:
EqLocate Tutorial:
This PPT