Lab 3

Finding an Epicenter from Seismograms

 

Times and Speeds for P and S Waves Yield Distance to the Epicenter

 

Seismograms are pieces of paper on which a wavy line represents vibrations over time. Seismograms record vibrations as a function of time, not distance, but we can obtain the distance to an epicenter from seismograms, by the fact that P and S earthquake waves travel at different speeds.

 

To show mathematically how this is possible, see the Appendix where the equation we will use in this Lab exercise is derived.

 

In this laboratory exercise, you determine the location of an earthquake’s epicenter using data from seismograms.

Method of Calculating Distance Using a Seismogram

 

The Appendix yields an equation that gives the time of arrival t_p of an earthquake’s P wave. That time is related to P and S wave speeds and to the interval of time between arrivals of the P and S waves. From such information, the method of determining the distance d from the seismic measuring station to the epicenter is as follows:

 

Directly from the seismogram we obtain the difference in time of arrival of the P and S waves, Δt. (Here, the Greek capital letter “delta” (that is, Δ) is used to represent the words “difference in” because “delta” and “difference” both begin with the letter “d”.)

 

We insert the value of Δt into the following equation to obtain t_p :

t_p = (V_s * Δt) / (V_p – V_s) ,

[this equation is the one derived in the Appendix]

 

In the above equation for t_p we use as values for V_s and V_p the following –

 

P and S waves have been found to have the following average speeds:

 

V_p = 6.01 km/sec

V_s = 4.1 km/sec .

 

Then, using t_p we calculate distance d from

 

d = V_p * t_p  . (This is the familiar equation of distance = speed X time.)

 

Example of Calculating Distance from a Value of Δt :

 

To illustrate, we will put in some values for the items on the right-hand side of the equation for t_p. We use the values for V_p and V_s from above. Let’s say as an example that Δt determined from the seismogram is found to be Δt = 200 sec. Then,

 

t_p = (4.1 km/sec * 200 sec) / (6.01 km/sec – 4.1 km/sec) =  429.3 sec . In this equation note the division sign and the minus sign.

 

Then, the distance d is :

 

d = V_p * t_p = 6.01 km/sec * 429.3 sec = 2580.2 km .

 

Using a Seismogram to Obtain Δt and Thence to Obtain Distance to the Epicenter

 

From a seismogram, we must find the time interval Δt. We then use the Δt to find the distance from the seismic station to the epicenter.

 

Let the points on the seismogram when the seismic waves arrive be labeled as follows:

 

Point A = arrival time of P-wave , and

Point B = arrival time of S-wave .

 

Figure 1. Fictitious seismogram labeled to show points A and B.

 

We need to find the time interval Δt  between points A and B from the seismogram. We will do this by a simple technique in MS Word (although in this computer age it is routine for the time interval to be computed automatically). The seismogram may not provide the time interval directly, and we may have to figure it out from the time axis on the graph paper.

 

Here is how –

 

On the seismogram find the time axis (usually found beneath the actual trace of the seismic wave). In an actual seismogram, there will always be a time axis, and it will usually be labeled with times. Measure the linear distance between two marks on the time axis for which the times are given. Let’s say that two of the time marks on the axis are for 0 sec and 300 sec (i.e., 5 minutes). If upon measuring the distance on the seismogram between those two time marks, the distance might turn out to be 1.5 inches, then the “time scale” (i.e., the “scale factor”) for the seismogram is 300 sec per 1.5 inches, or 300 sec / 1.5 in = 200 sec/in. The numbers here are simply an example. In our case below, where we use Figure 3, the time scale has been pre-set at 24 sec between tic marks. You will only have to measure the distance in inches between tic marks to obtain the time scale in seconds per inch (sec/in).

 

(Normally we would use millimeters or centimeters (the Metric system) instead of inches, because most scientists use the Metric system, but here we will use inches because MS Word makes measurements in inches.)

 

Now measure the linear distance in inches between points A and B. The procedure for making these measurements in MS Word is explained below in step-by-step fashion. Let’s say as an example that the linear distance is 1 inch. Then, by using the imaginary time scale of 200 sec/inch, the Δt is

 

Δt = 1 inch x 200 sec/inch = 200 sec. 

 

Because the distance numbers are fictitious and were made up, this time was deliberately arranged to be the same value of Δt that we used in the example earlier.

 

And from Δt we had calculated t_p and d as follows (repeating what we did before):

 

t_p = (V_s * Δt) / (V_p – V_s) = (4.1 km/sec * 200 sec) / (6.01 km/sec – 4.1 km/sec) = 429.3 sec .

 

Then, the distance d to the epicenter is:

 

d = V_p * t_p = 6.01 km/sec * 429.3 sec = 2,580.2 km .

 

Using Distance Values from Three Seismograms to Determine the Location of an Epicenter

 

On a map, the distance of an epicenter from a seismic station can be represented as a circle. The seismogram provides the distance, but does not provide the direction to the epicenter. With only one seismogram, the epicenter could be anywhere around the circle whose radius is the distance determined for the epicenter. With two seismograms, however, there are two points of intersection of the circles around the seismic stations where the epicenter could be located. Even better, three distances from three different seismograms will enable the determination of the epicenter’s exact location by a three-way intersection.

 

The diagram in Figure 2 shows the situation graphically.

 

Figure 2. Exact Epicenter Location Requires 3 Distance Circles.

 

The Lab Exercise

From one seismogram, you will determine the distance from the seismic station’s location to the epicenter of an earthquake. This will provide for your drawing one circle on a map. But you need two more circles to solve for the location of the epicenter.

You will be given the S-P time intervals Δt from two other seismograms, located at other stations, relieving you from having to find the values of Δt yourself from two more seismograms. From those provided values of Δt you will calculate the distances of the two stations. These distances will provide for your drawing two more circles on the map. Then you will determine the exact location of the epicenter from the intersection of the three circles.

As a refresher, see Figure 1 above, a fictitious seismogram fully labeled to show points A and B and the time axis.

The actual seismogram to be used for this lab is shown below in Figure 3. Notice that it contains three traces, one trace for each of the three directions of motion in three-dimensional space. The two horizontal traces emphasize S waves (shear waves, side-to-side), while the vertical trace de-emphasizes S waves and emphasizes longitudinal waves.

Although Figure 3 shows an actual seismogram, it was not obtained from the location labeled as Albuquerque, and furthermore the time axis has been altered. For this lab, the time scale on this seismogram has been arbitrarily set at 24 sec for the distance between two tic marks. (You will have to measure the distance in inches between tic marks to determine the time scale in sec per inch). These changes were made for convenience to suit this lab exercise.

From the seismogram shown in Figure 3 (using the method described above but presented step-by-step below), you will determine the S-P time interval Δt between the first arrivals of P and S waves at the Albuquerque seismograph station. Then you will use the Δt to obtain tp and ultimately the distance d. But first, enter the value of Δt you obtain into the column for Δt in Table 1 below. If you need more help to obtain Δt from the seismogram, look at the details in the next section.

Details of Procedure for Measuring Δt on the Seismogram

To obtain an accurate measure in seconds of the S-P time interval Δt, you will have to measure the distance between the P and S waves on the seismogram, and compare that distance with the time scale at the bottom of the seismogram. Here is the step-by-step procedure:

Procedure for measuring the S-P time interval Δt (presuming use of MS Word) —

1– Arrange for display of the drawing toolbar (in MS Word 2003), or the portion of the “ribbon” in MS Word 2007, or other means on other computers, so that you can select an icon for drawing lines and circles. Note: If you have a Macintosh computer the procedures below will be slightly different.
2– Click on line icon
3– Position the cursor on the seismogram anywhere on the P-line (red), depress left mouse button, move cursor to create and extend a line horizontally to the S-line (green). This is our Δt interval.
4– Measure the line length by —

a– Position cursor on line
b– Right-click mouse button
c– Choose Format Auto Shape (or Size)
d– Choose Size
e– Read Width for the Δt interval line length in inches (the Height should be 0.0 inches if your line is actually horizontal)

 

Compare the Δt line length just obtained with the time scale at the bottom of the seismogram. For example, draw a line that extends over 10 units of the scale (where a unit is the distance between a pair of tic marks). Measure the length of that line, and then divide that length by 10 to find the distance for one scale unit (the distance between two tic marks). That distance represents 24 sec.

 

Divide the line length for the Δt interval by the distance for one scale unit, and then multiply by 24 sec to find the total number of seconds in the Δt interval. Write that number in the column for Δt in Table 1.

 

 

 

Figure 3. Actual seismogram, but from a fictitious seismic station, and with the time scale altered for convenience in this lab exercise. An orange line has been drawn between the beginnings of the P and S waves. Because MS Word does not allow highly accurate positioning, the orange line does not exactly line up with the red and green lines. This deficiency in MS Word will mean that results in this lab will not be exact. Macintosh computers may give more accurate results.

 

Table 1. Epicenter Distances from Three Seismic Stations.
Station Name	S-P time interval Δt (sec)	tp (sec)	Distance to epicenter (km)	Radial distance on map (in)
Albuquerque
Boise	79.58
Sacramento	30.96

Details of Procedure for Obtaining t_p

Then, determine the value of t_p by means of the equation yielding t_p —

t_p = (V_s * Δt) / (V_p – V_s) ,

 

In the above equation for t_p , remember that the values for V_s and V_p are the following –

 

V_p = 6.01 km/sec

V_s = 4.1 km/sec .

 

Put the resulting value of t_p in the column for t_p  in Table 1.

Details of Procedure for Obtaining d

Now, use the equation below yielding d from t_p to determine the distance d of the Albuquerque seismograph station from the earthquake’s epicenter, and enter that distance into the column for d in Table 1. Using t_p we calculate distance d from

 

d = V_p * t_p .

The S-P time intervals (Δt ) from two other fictitious seismograph stations are also listed in Table I: Boise and Sacramento. (Figure 4 is a map showing the location of Albuquerque and the other two seismic stations of interest.) Use the above equations to complete Table 1 for Boise and Sacramento. Determine their t_p values and distances d from the earthquake’s epicenter, using their figures for Δt, and enter their t_p values and distances d into Table 1.

           

Figure 4. Western United States, showing the location of the fictitious seismic stations in Albuquerque, Boise, and Sacramento. For explanations of the circles see the text.

Details of Procedure for Obtaining Radii of the Circles

What remains is to figure out the radii of the circles you will draw on the map in Figure 4. For your lab report, download a copy of the map in Figure 4 from the Blackboard module under Lab 4, and paste it into an MS Word document where you will compose your lab report. Make the map big on the paper.

To draw the circles on the map in Figure 4, you are going to have to figure out how large they should be. You already know the actual ground distances (on the surface of the Earth) from the next to last column in Table 1. But you must calculate how large the circles are when those ground distances are represented on the map. You must convert ground distances from the next to last column in Table 1 into radii of circles to be drawn on the map (last column of Table 1).

This calculation of the radii of the circles requires that you determine the scale factor for the map— On the map find the scale bar in the lower left corner. Its full length represents a ground distance of 400 km on the Earth itself. Measure the length of the scale bar with MS Word, using the size-measuring procedure presented earlier. It might be close to 1 inch or 30 mm. (This number will vary depending on how large the map is on your computer screen. On your computer screen, or on a hard copy printout, you can vary the size of the map, and so we cannot say what the length of the scale bar is in your case; only you can). If the length is 1 inch, then the scale factor is 400 km / I inch, or 400 km per inch (each inch on the map would represent 400 km on the Earth). You will obtain a scale factor that is probably different from this value.

Once you have the value of your map’s scale factor, divide it into each of the ground distances in Table 1 and put the results in the last column. These results are the radii of the circles you must draw on the map (not the diameters, but the radii).

In the above procedure the units of measurement will work out perfectly as follows —

ground distance (km) divided by scale factor (km/inch)  = map distance (inches)

Details of Procedure for Drawing Circles

Now, the next step is to draw circular arcs (partial circles, or whole circles if they will fit) on the map in Figure 4 using radii from column 5. Hint: the circle for Albuquerque will be too large to fit on the map entirely.

The first circle should, of course, be centered on Albuquerque, representing the distance from Albuquerque to the earthquake’s epicenter. Initially, let’s just get the size correct. If you are working on a hard copy map, use a compasses to draw the arc, with tines spread enough to yield the correct radius for each circle. But you can use MS Word to draw and size the circles. On the computer screen, you will have to adjust the sizes of circles until the circle size shows the correct radius as measured in inches (last column in Table 1).

Circles can be created by means of MS Word drawing features. The procedure is as follows —

– Arrange for display of the MS Word drawing toolbar, or other means, for drawing lines and circles
– Click on ellipse icon
– Position cursor on map (anywhere, because later the circle’s location will be adjusted), depress the shift key, depress left mouse button, move cursor to create a perfect circle (if you forget to depress the shift key the figure drawn will be an ellipse rather than a circle). The circle’s size does not matter yet; size will be adjusted below.
– Initially the circle will be opaque white. For convenience later, make it transparent or partially transparent by the following procedure (see Figure 4 for illustration of both opaque and partially transparent circles) —

– Position cursor inside circle
– Right-click mouse button
– Choose Format Auto Shape
– Choose Colors and Lines
– Adjust transparency (try 60-70%)

Details of Procedure for Centering Circles over the Stations

 

– Now draw two lines like cross hairs to enable proper location of the circle over Albuquerque —

– Draw one line horizontally across the map and intersecting Albuquerque’s location (the red square); draw another line vertically and also intersecting Albuquerque’s red square. These lines will be used like cross hairs to center the circle over the red square.
– Position the cursor inside the circle, depress the left mouse button, and move the circle to center it roughly over the red square.
– To position the circle exactly, release the button. Then click inside the circle – this will create tiny marker circles on the circle’s perimeter. Move the circle until those tiny marker circles are aligned on the horizontal and vertical lines used as cross hairs over the red square.

 

Details of Procedure for Adjusting the Size of Each Circle

 

Now, adjust the circle size —

– Position cursor inside the circle
– Right-click mouse button
– Choose Format Auto Shape (or Size)
– Choose Size
– Put a check mark inside the tiny box labeled Lock Aspect Ratio (this will force any change in the vertical dimension to be matched by the same change in the horizontal dimension, and vice versa)

– Adjust the number in one or the other dimension boxes to be twice the distance in the last column of Table 1 (because that distance is a radius, but Size in MS Word measures diameter, not radius)

– The result will be a circle of the right size, positioned over Albuquerque’s red square. Some of the circle will be off the page in the case of Albuquerque.

In the above way, draw the arc or full circle for Albuquerque. This arc shows the possible locations for the epicenter based on arrival times at Albuquerque.  Then add two more arcs or circles, one for each of the other two seismographic stations (Boise and Sacramento), positioning the center of these other two circles in the correct locations for each of those two additional seismographic stations.

The three arcs should intersect at one point, or close to one point. Errors in the construction of this exercise by the lab instructor (map scales, map reproduction, and so on), plus any errors in your arithmetic and/or graphical methods, will combine to produce some total error in the determination of the epicenter. This total error will show up as a failure of the three arcs to intersect at a single point on the map.

N.B. Surprise! — It is possible for the three circles to be sized incorrectly and still intersect. Thus, intersection does not guarantee that you have done your arithmetic and graphing correctly. But gross failure to intersect means absolutely that you have made a large error somewhere.

In addition to the three arcs or circles on the map in your lab report (put a copy of the map into the report), pinpoint the location of the earthquake’s epicenter on the map and label it with the letter E. Estimate by eye, from the map labels, the latitude and longitude of the location E. This is the final step for completing the exercises of this lab – to specify the actual location of E in terms of latitude and longitude. Express the result as degrees north latitude and degrees west longitude; as an example, the location could be 24 N, 145 W. (If you omit N and W, your answer is incomplete and will be wrong.)

Lab Report

Write your lab report in the required style as prescribed in the Lab Report Format document – include all sections, properly labeled. Provide a brief Introduction, a Methods section, and then a Results section. Finally, write an Abstract – position it at the top, immediately under the title and your name (both centered) and before the Introduction. The Abstract summarizes the entire lab exercise in a few sentences (including the location of the epicenter – always state major results in the Abstract).

In the Results section, include Figure 3 from this document showing the seismogram, and the map of Figure 4 (obtained from Blackboard), showing the three circles you have drawn and the pinpointed location of the epicenter. (It is possible to grab Figure 3 from this document — right-click it and choose Copy, and then Paste the figure into your lab report developed in MS Word. However, a better copy of the map is on Blackboard.)

Include a copy of Table 1, completely filled in with data.

If you worked with the map in hard copy print form, then you will have to scan the map to get an electronic copy for inclusion in this report. However, you probably worked with the map on your computer screen right inside your developing lab report. (If not, then copy your map (with the three circles and designation of E) by right-clicking on the image and choosing Copy, and then Paste the map image into your lab report that you are developing in MS Word.)

State the latitude and longitude of the epicenter. Repeating what was said above, express the result as degrees north latitude and degrees west longitude; as an example, the location could be 24 N, 145 W.

Properly label the seismogram and map with figure legends as prescribed in the Lab Report Format document (legends should be placed beneath the figures).

In the text of the report, refer to the figures and to Table 1. For example, you may have text discussing the circles on the map, and therefore you would put in one of the sentences the following — “(see Figure 1).” For convenience, you may put the figures and table at the end of your report, instead of positioning them within the text (within the text would be the common arrangement in scientific literature). To position figures within the text requires some familiarity with MS Word and the techniques are tricky at times.

The total length of text in your report (not counting any space given to figures and Table 1) can be as little as 1-2 pages single space or a little more.

If you need help in manipulating the figures and the table, to get them into your report, you may find help in the document “Graphing Helps,” on Blackboard under Assignments/Before You Begin.

Appendix: Derivation of the Equation for t_p

 

Using simple algebra, we will obtain an equation to be used with data from a seismogram.

 

P and S waves have been found to have the following average speeds:

 

V_p = 6.01 km/sec

V_s = 4.1 km/sec

 

These equations yield graphs of distance versus time with the form of straight lines (v = d/t for distance d and time t), and therefore these equations are called linear equations. In actuality, distance-time graphs for earthquake waves are not straight lines, but slightly curved lines, as seen in slides 11-13 of the Power Point presentation for Week 3. But the linear equations here will be used in this lab as an approximation to reality.

 

Because speed equals distance divided by time, we obtain distance from speed multiplied by time, or d = V x t. This is true for both types of waves; we must use the correct speed and time for each type of wave.

 

The P waves have higher speed but therefore shorter travel times, from the location of the earthquake to the location of the seismic measuring station, while the S waves have lower speed and correspondingly longer travel times. But the distance traveled is the same with both types of waves. Hence we can write

 

d_p = d_s where d_p is the symbol for distance for the P wave, and d_s is the symbol for distance for the S wave.

 

Substituting d = V * t into both sides and using appropriate subscripts,

 

V_p * t_p = V_s * t_s .

 

Now,  we know from seismograms that t_s is longer than t_p by some time interval, so we can write t_s = t_p + <time interval between P and S waves>. Let’s substitute the symbol Δt in place of <time interval between P and S waves>, where Δ is a symbol for the Greek letter “delta” meaning “difference”; so Δt stands for the difference in time of arrival of P and S waves. Thus, using the symbol Δt,

t_s = t_p + Δt .

 

Then,

 

V_p * t_p = V_s * (t_p + Δt) .

 

Now we solve for t_p in this equation by taking the following steps :

 

V_p * t = V_s * t + V_s * Δt .

 

Collecting terms,

 

(V_p – V_s) t = V_s * Δt .

 

Finally, the solution for t_p is :

 

t_p = (V_s * Δt) / (V_p – V_s) .

 

Notice that t_p depends only speeds that we already know (the values above), and the difference in time of arrival of the P and S waves, Δt, which we measure directly from the seismogram.

 

Once we have t_p, we will use it to get the distance to the epicenter by using

 

d = V_p * t_p .

 

 

An alternative method for using Δt to obtain distance d is to employ the S-T graphs such as found on the Power Point presentation.  No matter which method is used (the equation for t_p or an S-T graph), we have to get Δt in order to find d.