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 GutenbergRichter 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 GutenbergRichter 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 GutenbergRichter 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 GR 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 GutenbergRichter 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 GR relation, which related the distribution of earthquakes of a variable magnitude M to a fixed magnitude of 8, like this:
The value of the (negative) slope of this line is still b, but the value of a is not that of the yaxis 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:
where M_{m} 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:
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
On page 22, you saw the simplest version of Omori's Law:
More commonly used in scientific studies, however, is a modified version of this law, which in fact more closely resembles the original, empiricallyderived 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):
We have now established the form of the two equations from which
our three aftershock sequence
Let's start our look at the significance of the constant that
defines the rate of decay (the
Scientific studies (notably, the paper cited
at the end of this activity) have shown that
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
If you're graphing by hand, plot about a dozen points per function
for t between 0 and 10. (Note what happens at
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.
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 24hour 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 mathematicallydefined 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 24hour period after the mainshock.
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
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
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
If you've worked through Activity #8,
you've seen what a plot of magnitude distribution looks like.
That's essentially all the GutenbergRichter relation
To study
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 loglinear 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 




Once you have completed plotting the points on your GR 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.
(Review from Activity #8:) Find the numerical value of b for each line, using one of the following methods:
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
Save the graph of the two GutenbergRichter plots you made in the section above. You will need them for this section.
The value of a in the GutenbergRichter equation for aftershocks:
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 GR 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.
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.
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?
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 loglinear graph paper, preferably the same one on which you made your previous plots. Draw a bestfit 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.
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 
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.
Now, make one last GR 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.
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 
Exercise 3 It's Two Equations in One!
Now that you have been introduced to the GutenbergRichter relation for aftershocks:
which gives the frequency of aftershocks
greater than or equal to
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
First, we will need to convert the GutenbergRichter relation to a nonlogarithmic form, to match the modified Omori law. That's pretty simple; just take the GR relation we have:
and raise each side to the power of 10, so that the lefthand side is
no longer
Now that the GR equation is in this form, all you need to do is simply combine the two independent equations into one function with two variables:
...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
Reasenberg, Paul A. and L.M. Jones (1989). Earthquake Hazard After a Mainshock in California, Science, 243, 1173  1176.