Section 2: The Distribution of Earthquakes

Activity #9: AFTERSHOCK SEQUENCES

Concept: It is possible to quantify properties of an aftershock sequence in such a way that future activity of that sequence can be anticipated by closely examining its beginning.

Materials:

Procedure:

In this activity, you will look at some of the characteristic properties of aftershock sequences, and relate them to variables in two equations: the modified version of Omori's Law, and the Gutenberg-Richter relation for aftershock sequences. You will get a chance to study plots that represent the decay in numbers and the apparent decay in magnitude that aftershocks experience as a sequence progresses. And finally, you will see how the two equations you've been working with can be combined, and how that combined equation can be used.

The activity is divided into three exercises. Each is explained within its own text, so you may begin as soon as you're ready.

Exercise 1 Defining Sequence Properties

There are three main factors that determine the "character" of any aftershock sequence, and each relates to one of the constants in either Omori's Law or the Gutenberg-Richter relation. These three important and independent properties are the magnitude distribution, "productivity", and the rate of decay. Magnitude distribution describes the number of aftershocks relative to magnitude. It is represented by the slope of the Gutenberg-Richter relation, the constant b in that equation. If you've worked through Activity #8, you should already be familiar with this constant. "Productivity" relates to the amount of aftershocks a sequence produces. This property is represented by a in the G-R equation. The rate of decay describes the severity of the decline in the frequency of aftershocks after the mainshock, and is represented by the constant p in Omori's Law.

Before we take a closer look at each property, let's make sure that we've defined exactly which equations we're using. First, the Gutenberg-Richter relation we use for aftershock sequences is slightly different from the equation we defined in Activity #8. Actually, it is more closely related to the original form of the G-R relation, which related the distribution of earthquakes of a variable magnitude M to a fixed magnitude of 8, like this:

log N(M) = a - b(M - 8)

The value of the (negative) slope of this line is still b, but the value of a is not that of the y-axis intercept, as it is in the form log N(M) = a - bM that we derived in Activity #8. Instead, a represents the value of log N(M) when M equals 8, and that part of the equation goes to 0.

As commonly used for aftershock sequences, the equation is similar:

log N(M) = a - b(M - Mm)

where Mm is the magnitude of the mainshock, and M is the variable, representing the magnitudes of aftershocks. Since many people like to make sure the term in parentheses is always positive, you will typically see this equation written with the terms in parentheses switched, and a plus sign in front of b:

log N(M) = a + b(Mm - M)

Realize that this doesn't mean the slope of the line is positive; the variable M has a negative sign in front of it, so fundamentally, nothing has changed.

Hopefully, this bit of playing around with equations hasn't lost you completely -- we still have one more equation to look at: Omori's Law.

On page 22, you saw the simplest version of Omori's Law:

N(t) t-p

More commonly used in scientific studies, however, is a modified version of this law, which in fact more closely resembles the original, empirically-derived model developed by Omori in the 1890s. This modified Omori law describes the rate of decay of an aftershock sequence by equating the number of aftershocks at some time t after a mainshock with the quantity t plus c (a constant) to the negative power of p (another constant), all multiplied by K (a third constant):

N(t) = K * (t + c)-p

We have now established the form of the two equations from which our three aftershock sequence properties -- magnitude distribution (b), "productivity" (a), and the rate of decay (p) -- were taken. Now let's look at how each affects the "character" of an aftershock sequence, starting with the rate of decay, p.

Rate of Decay ( p )

Let's start our look at the significance of the constant that defines the rate of decay (the p value) by graphing a simple model of Omori's Law for three different values of p. This will not only allow you to see what the shape of an exponential decay curve looks like, you will be able to see how that shape changes as the value of p, the rate of decay, changes. You will need either a graphing calculator or similar computer graphing program, or a sheet of regular, linear graph paper to begin.

Scientific studies (notably, the paper cited at the end of this activity) have shown that p values for aftershock sequences in southern California vary between about 0.5 and 1.5, with the mean at a little more than 1.0 (1.08, to be exact). Hence, you will graph the function

N(t) = t-p

for p equal to 0.5, 1.0, and 1.5. If you're using a machine to do the graphing, you need no further instructions. Graph the three functions, study the curves that result, and then answer question 1, below.

If you're graphing by hand, plot about a dozen points per function for t between 0 and 10. (Note what happens at t = 0.) A good set might be: t = 0.1, 0.2, 0.3, 0.5, 0.7, 1.0, 1.3, 1.6, 2.0, 3.0, 5.0, 8.0, and 10.0.

When you've finished plotting points, sketch in the best smooth curve that intersects all these points. Try and make your curves different colors to keep them distinct, and make sure you note which curve relates to which value of p. Study your graph, and use it answer the question below.

  1. Which of these three functions decayed (approached N(t) = 0) the fastest? Which showed the slowest decay?

  2. Since you graphed the simple version of Omori's Law, you didn't have to worry about the constants c or K. Looking at your graphs, explain why the constant c in the modified Omori law (in southern California sequences, c averages 0.05, for t in days) is necessary to relate this function to a real-world process. Would it ever make sense to talk about aftershocks at time t if t were less than zero?

Now that you've made your own model graphs of Omori's Law, you'll have the chance to look at two graphs of actual aftershock data from southern California sequences.

Follow this link to a page of two plots of seismic activity from southern California aftershocks sequences. Each figure shows the number of aftershocks per day during that sequence, plotted against the number of days after each mainshock. These "days" are not calendar days, but are 24-hour periods, divided at the time of day the mainshock originally occurred. The two sequences shown here are both from moderate earthquakes that occurred in 1986. The first in the North Palm Springs sequence, which followed a M 5.6 mainshock. The other is the Oceanside sequence, which occurred offshore after a M 5.4 mainshock. Both generated a great many aftershocks. The plots show how the number of these aftershocks (per day) fell off with time as an exponential decay curve, like the ones you graphed before. Since these are from the real world, however, they are not as "pretty" as a simple mathematically-defined curve. Try and overlook the minor variations and see the overall curve of each.

Study these plots briefly for similarities and differences between them. Note that the vertical scale for the first is twice that of the second, and that each plot has a vertical discontinuity because of the very large number of aftershocks that occurred within the first 24-hour period after the mainshock.

  1. Can you tell which sequence has a higher p value (i.e. which decays faster)?

If not, that's OK. Try this method to make it more evident: Look at the range of values for the number of aftershocks, N(t), in the span between days 10 and 20 for each sequence. Find a rough average value for N(t) during this span. For each sequence, divide the number of aftershocks from the first day (the peak value) by this average. Compare your results. The larger quotient belongs to the sequence with the faster decay -- the higher p value.

  1. Following the process above, can you now make a better guess as to which sequence has the higher p value? Which do you think it is?

If you can determine the value of p early in an aftershock sequence, you can use this to anticipate the number of aftershocks that will occur at a later date, by using the modified Omori law to solve for N(t). You can even anticipate how long a sequence will last (at what time t the rate of aftershocks will drop to below the background seismicity rate). To do these things, however, you must first determine the rate of decay (p)!

Each of the questions below gives you the chance to solve one of these problems, starting with how to find the rate of decay (p). You will need a calculator, and knowledge about solving problems involving logarithms and exponent variables. Remember that log yx = x * log y. For each question, use the modified Omori law, and assume that c is always equal to 0.05, for t in days. Also, assume that these aftershock sequences follow a perfect decay curve (smooth, with no "bumps").

  1. Find the rate of decay (the value of p) for a sequence with 57 aftershocks on the first day following the mainshock, and only 3 aftershocks on day 10 (t = 10). Is this a relatively high, low, or average rate of decay for a southern California aftershock sequence?

  2. If an aftershock sequence has the rate of decay p = 0.7, and 900 aftershocks occur in the first 24 hours after the mainshock, how many aftershocks could be expected to occur of the 30th day of the sequence? What if this same sequence, instead, had p = 1.4? Then how many aftershocks would you expect on day 30?

  3. A total of 250 earthquakes (aftershocks) occur within the aftershock zone of a magnitude 5.7 mainshock on the first day of an "average southern California aftershock sequence" (p = 1.08). If the background seismicity rate in the aftershock zone was 3 per day before the mainshock struck, for about how many days will this aftershock sequence last?

Magnitude Distribution ( b )

If an aftershock sequence generates 32 earthquakes greater than magnitude 4, how many of those will be greater than magnitude 5? To answer that kind of question, knowledge about the magnitude distribution, or b value, of an aftershock sequence is essential.

If you've worked through Activity #8, you've seen what a plot of magnitude distribution looks like. That's essentially all the Gutenberg-Richter relation is -- a set of linear points that show the magnitude distribution of a seismic data set; the slope of the line is negative b.

To study b values of aftershock sequences, you will plot two sets of data from the same aftershock sequences you studied in the section above on rate of decay, North Palm Springs and Oceanside, both of which occurred in 1986.

Below are the two sets of data you will need. Each represents a year of aftershock activity. The two sets span slightly different magnitude ranges. Oceanside aftershocks were offshore, away from many seismic instruments, and so below magnitude 2.0, the catalog of this sequence is certainly incomplete (indeed, you may not want to put much weight on the data point for magnitude 2.0, either). Also, the largest aftershocks of the North Palm Springs mainshock were larger than those of the Oceanside mainshock.

You will need to print out at least one sheet of log-linear graph paper. Though it might be best to put both plots on a single sheet of graph paper, the two data sets will overlap, and this could be confusing, especially if you don't have different colors with which to plot them.

Go ahead and begin plotting as soon as you're ready. Then continue with the rest of the instructions below.

North Palm Springs (1986),
M 5.6 mainshock
 Oceanside (1986),
M 5.4 mainshock

Magnitude (M)
 Number of
Aftershocks,
M or larger
1.8
 1300
2.0
 862
2.2
 572
2.4
 396
2.6
 255
2.8
 167
3.0
 95
3.2
 60
3.4
 38
3.6
 20
3.8
 12
4.0
 12
4.2
 6
4.4
 5
4.6
 1

Magnitude (M)
 Number of
Aftershocks,
M or larger
1.8
 -
2.0
 1513
2.2
 929
2.4
 505
2.6
 281
2.8
 174
3.0
 113
3.2
 60
3.4
 37
3.6
 24
3.8
 10
4.0
 6
4.2
 3
4.4
 -
4.6
 -

Once you have completed plotting the points on your G-R aftershock graph(s), use a straight edge to draw lines that best fit each data set. Draw the lines so that they extend somewhat beyond the set of points you have.

  1. Just from looking at the lines, can you say which sequence has the higher b value (the steeper slope)?

(Review from Activity #8:) Find the numerical value of b for each line, using one of the following methods:

  1. Having found numerical answers for each line's b value, which sequence has the larger value?

  2. Values for b in southern California aftershock sequences range from about 1.4 down to 0.5, with a mean value of 0.91. How do the b values of these two sequences compare to that range of values? Are they relatively high, relatively low, or about average?

Though mainshocks are typically the earthquakes most people remember years later, large aftershocks can do a lot of damage, and result in injuries or even deaths. This is especially true when the mainshock damages but does not destroy most of the buildings in an area. Since the structures remain standing after the mainshock, people may assume they are safe for occupation, and stay indoors. But when a moderately strong aftershock shakes a weakened building, that can be the final push that triggers the building's collapse. This kind of scenario was seen following the 1952 Kern County earthquake (M 7.5), when a moderate aftershock that struck a full month after the mainshock caused severe damage, multiple injuries, and two deaths in Bakersfield.

  1. All other things being equal, which kind of aftershock sequence would bring a greater risk of secondary damage like that described above: one with a low b value, or one with a high b value?

Productivity (a)

Save the graph of the two Gutenberg-Richter plots you made in the section above. You will need them for this section.

The value of a in the Gutenberg-Richter equation for aftershocks:

log N(M) = a + b(Mm - M)

can be thought of as representing the relative "producing power" of the aftershock sequence, and it is dependent upon the magnitude of the mainshock. A high value of a means that an aftershock sequence produced a great many aftershocks, given the size of its mainshock. A low value of a means that aftershock output was small, for the size of the mainshock that produced them.

The easiest way to find the value of a for a sequence is to simply project the line of that sequence's G-R plot far enough to the right of the graph that it intersects the magnitude of the mainshock. Take your already completed graph of the magnitude distribution for the North Palm Springs and Oceanside aftershock sequences and do this, now.

Once that line intersects the magnitude of the mainshock, read the value of N(M) for that point of intersection. Now, find the logarithm of that number, and you have the a value of that sequence! Simple enough?

Perform this operation on each of your graphed sequences. Then work through the questions below.

  1. Which sequence has the higher a value? (Remember, since a is negative, the higher value will have a lower absolute value.) This is the more "productive" aftershock sequence.

  2. Values for a in southern California aftershock sequences vary from as high as almost -0.5 to lower than -3.0, with an average of -1.67. Are these two sequences particularly "productive", about normal, or "unproductive", relative to the mean?

  3. Could an aftershock sequence have an a value greater than zero? If somehow you found this to be true in the early stages of a sequence of earthquakes, what might you conclude about the "mainshock", and what might you expect to occur in the near future?

Exercise 2 Sequence vs. Sequence

In this exercise, you will study some plots of activity from actual aftershock sequences in southern California. Each sequence is represented by two different graphs. One shows the number of aftershocks per day versus the number of days since the mainshock, and the other shows the magnitude of the largest aftershock of the sequence each day, for every day that there was at least one recorded aftershock.

To start off, just compare the first two sequences, North Palm Springs (1986) and Oceanside (1986). More specifically, look only at the plots showing the magnitude of the largest aftershock per day. (You've already looked at their rates of decay, anyway.) The two plots should look reasonably similar. Each shows only one gap, a day when there were no aftershocks, in the first 120 days. Each gap is near day 110. But there should be at least one evident difference in the trends of each plot.

  1. What is that difference?

  2. You've already found the b value of each sequence, in Exercise 1. Can you see the connection between the relative b values of each sequence and the difference in their magnitude trends?

  3. You've also compared the values of p and a for each sequence. Do you think either of these may be influencing this trend? Why or why not?

Now, study the plots for the Whittier Narrows sequence (1987). This aftershock sequence was spawned by a larger mainshock than the others we've seen in this activity. Look at both its activity per day (the vertical scale is the same as it is for Oceanside), and at the plot of largest magnitude per day. How does it compare to the North Palm Springs and Oceanside sequences?

  1. Would you guess that the a value of the Whittier Narrows aftershock sequence was larger, smaller, or about the same as the a values of the two previous sequences?

  2. How do you think the p value of this sequence compares to each of the others? Try using the method introduced in Exercise 1 to evaluate this.

Below is a data set of aftershock distribution by magnitude for the Whittier Narrows sequence. As you did with the other sequences, plot this set of points on a piece of log-linear graph paper, preferably the same one on which you made your previous plots. Draw a best-fit line for these points, and make sure to project it all the way to magnitude 5.9, the size of the Whittier Narrows mainshock, so that you can find a value for a.

Whittier Narrows (1987) sequence, M 5.9 mainshock


Magnitude (M)
 Number of Aftershocks,
M or greater
1.8
 204
2.0
 150
2.2
 117
2.4
 84
2.6
 60
2.8
 47
3.0
 40
3.2
 25
3.4
 17
3.6
 13
3.8
 7
4.0
 7
4.2
 5
4.4
 5
4.6
 5
4.8
 1
5.0
 1
5.2
 1

  1. What values do you find for a and b for the Whittier Narrows sequence? How do they compare to those for the two previous sequences graphed, and to the average values for southern California aftershock sequences?

Finally, look at the graphs of activity that followed the magnitude 5.7 Mojave earthquake of 1992. Note that the vertical scale on the upper plot is radically different than it was for our other sequences.

  1. In a word, describe the output of this sequence, relative to its mainshock magnitude. Take a guess at what its a value might be.

Now, make one last G-R plot (on the same sheet of paper, if possible) for the data below. Do your best to draw a line representing the magnitude distribution, and to project it all the way out to magnitude 5.7.

Mojave (1992) sequence, M 5.7 mainshock


Magnitude (M)
 Number of Aftershocks,
M or greater
1.4
 24
1.6
 14
1.8
 12
2.0
 7
2.2
 7
2.4
 6
2.6
 2

  1. What values of a and b did you obtain for the very weak sequence that followed the Mojave earthquake?

Exercise 3 It's Two Equations in One!

Now that you have been introduced to the Gutenberg-Richter relation for aftershocks:

log N(M) = a + b(Mm - M)

which gives the frequency of aftershocks greater than or equal to magnitude M after a mainshock of magnitude Mm, and are also familiar with the modified version of Omori's Law:

N(t) = K * (t + c)-p

which expresses the rate of aftershocks at time t after the mainshock, it is possible to combine the two in such a way that we can find the rate of aftershocks of magnitude M or greater at time t after a mainshock of magnitude Mm.

First, we will need to convert the Gutenberg-Richter relation to a non-logarithmic form, to match the modified Omori law. That's pretty simple; just take the G-R relation we have:

log N(M) = a + b(Mm - M)

and raise each side to the power of 10, so that the left-hand side is no longer log N(M), but simply N(M):

N(M) = 10a + b(Mm - M)

Now that the G-R equation is in this form, all you need to do is simply combine the two independent equations into one function with two variables:

Rate(t, M) = 10a + b(Mm - M) * (t + c)-p

...and you're done!

It is possible to turn this equation into an equation that can be solved to find the probability that one or more aftershocks in a given magnitude range will occur within a specified time range. This allows seismologists to make probability forecasts for aftershock sequence activity; for example, at one week after a magnitude 6 mainshock, you might hear that "there is a 20% chance of a magnitude 5 or larger aftershock in the next 30 days". Since using this equation involves solving a double integral, we won't bother going into it here. This might make an excellent independent research project! For a more in-depth look at this topic, see the following article:

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