I’ve upload a file with all questions some queations are easy and some queastions may need to use excel or graph.
Student Name
Student #
Homework #2: Earthquakes (100 pts)
Richter-Gutenberg Relationship
This activity is modified from an activity by the SCEC: www.data.scec.org/Module/s2act08.html. The entire homework
must be completed in Excel!
Earthquakes behave a lot like floods: big ones are a lot rarer than small ones, and there’s a predictive relationship
between magnitude and likelihood. The likelihood of an event like an earthquake can be described either as a
frequency—events per unit of time—or as an exceedance interval, which is the average time between successive events
of a given size or larger. The exceedance interval is related to the probability through the relationship,
1
𝑃
where P is the probability of an event happening in any given time period, and is a number that ranges from 0 [zero
chance] to 1 [sure thing]. For example, if an earthquake of M = 5 has an exceedance interval of 20 years, then the
probability P in any given year of an earthquake of M ≥ 5 is 1/20 = 0.05 (also expressed as 5%). Make sure you
understand these relationships before you continue. (This should be a review, but if you didn’t get it with streams, make
sure you get it now!)
𝐸𝐼 =
For earthquakes, a plot of magnitude M vs. the log of the exceedance interval (EI) or frequency (𝑓) is usually linear. The
slope of this line, called the “b value”, gives an indication of the characteristic seismicity rate in an area, and allows
seismologists to make predictions about the frequency or EI of large, rare earthquakes that may not have occurred for
hundreds or even thousands of years.
This relationship was first demonstrated by Beno Gutenberg and Charles Richter, two of the pioneers of modern
seismology. In the 1930s, as instrumental recording of earthquakes was becoming a reality in many areas of the world,
Gutenberg and Richter described a pattern in the seismic data that related the number of earthquakes in a given area
(or around the entire world) over a fixed period of time to the magnitude of those earthquakes. Using Richter’s recently
developed magnitude scale and the newest instrumental records, they found that the number of earthquakes greater
than magnitude 6 that would occur in a given area per year was proportional to the number of earthquakes greater than
magnitude 5 in that area, which was proportional to the number greater than magnitude 4, and so on. This is just
another way of saying that the log(𝑓) or log(EI) vs. M slope was linear.
This part of the homework consists of an exercise designed to familiarize you with the “Gutenberg-Richter relation”, as
the pattern described by these early seismologists came to be known. You will first construct a Gutenberg-Richter plot of
worldwide data, determining the average b-value for earthquakes on Earth. You will then consider two other datasets,
one for Southern California and one for the New Madrid area of Missouri, and make conclusions from them based on
the same type of analysis.
The World
Your first task is to analyze all recorded Earthquakes on Earth over a decade. The data are shown in the table below,
which shows the number of all earthquakes worldwide from the decade spanning 1/1/2000-12/31/2009 that exceeded
various magnitudes. For example, the table shows that there were 5,163 earthquakes of magnitude 5.5 or larger in the
decade starting January 1st, 2000. Enter these data into a spreadsheet exactly as shown.
While you could plot the total number of earthquakes, it’s best to normalize it to a frequency. Calculate the annual
frequency 𝑓 (number/year) in the third column and enter the data into your table, remembering that you have 10 years
of data. I’ve done the first one for you. Now plot column A (magnitude M) vs. column C (𝑓) with a scatterplot. Your graph
should look like this:
Highlight the y-axis and change it to logarithmic; this should linearize your data so you see this:
Add an “exponential” trendline to your data and project it back 4.5 periods so it spans the entire x-axis. You should now
see this:
You can now make the plot more presentable by making the following changes:
•
•
•
•
•
Select the y-axis and “Add Minor Gridlines”
Set the limits of the x-axis to 0.0 and 9.0
Label the x-axis “Magnitude M” and the y-axis “𝑓 = Number/year”
Label the chart “The World”
Choose to display the trendline equation on the chart (you may also need to reposition the equation to make it
easier to read)
Your graph should now look like this:
STOP! If your graph doesn’t look like this, seek help before you go on.
The b-value is the negative slope on this diagram; since the actual slope is always negative (there are fewer large
earthquakes), the negative sign ensures the b-value will always be positive. It can be calculated from the expression:
𝑛
𝑏=−
ln(10)
where n is the number that multiplies x in the exponent in the equation for the slope. For example, in the equation for
the slope above, n = -2.373. (Why this goofy formula? It’s because Excel plots the data as log(M) but calculates the
formula in terms of ln(M), requiring a conversion from ln → log.)
The equation of the line can be used to calculate the predicted frequency of earthquakes (Num/year) for any specified
magnitude M. For example, use the equation to calculate:
The worldwide frequency of earthquakes with M ≥ 5
(5 pts)
The worldwide frequency of earthquakes with M ≥ 1
(5 pts)
The worldwide frequency of earthquakes with M ≥ -2
(5 pts)
You’ll want to use scientific notation to express some of these numbers.
Southern California
You will now repeat these steps for Southern California earthquakes, but you will first extract the data yourself from the
earthquake database. Earthquake data for southern California are compiled by the Southern California Seismic Network
(SCSN) and housed at the Southern California Earthquake Data Center. Go to http://service.scedc.caltech.edu/eqcatalogs/date_mag_loc.php to choose earthquakes for a 19-year period starting January 1st, 2000 and ending on
December 31, 2018. Follow these instructions:
•
•
•
•
Leave the output format as “SCEDC”
Set the start date as year = 2000, month = 01, day = 01, hour = 00, min = 00, and sec = 00.
Set the end date as year = 2018, month = 12, day = 31, hour = 23, min = 59, sec = 59.
Leave the latitudes and longitudes alone; they are set to define an area that encompasses all of southern
California.
• Leave everything else unchanged except the minimum magnitude. This you will increase systematically from
3.5 in 0.5 increments to the magnitude where no earthquakes were recorded.
Start by entering a minimum magnitude of 3.5 and Submit Request. A long list of earthquakes will appear. However, you
are only interested in the number of earthquakes with a magnitude 3.5 or larger that happened during this interval, and
this number is listed at the very bottom of the list. Be patient. If you don’t see 1488 you’ve done something wrong!
Enter 1488 in cell B2, the cumulative number of earthquakes greater than M = 3.5 in the 19-year period starting January
1st, 2000.
Now systematically increase the minimum magnitude in 0.5 magnitude increments to fill in the other rows in column B.
Stop when the record you don’t retrieve any earthquakes.
Once again calculate the number of earthquakes per-year (column C) and reproduce your Richter-Gutenberg plot for
Southern California using the exact same setting as for the world. Only plot the magnitudes you have data for. Forecast
the trendline backwards and forwards to cover the range from M = 0 to M = 9. Label your plot “Southern California”.
Once again, calculate the b-value.
→ Print out your annotated Southern California log(𝒇) vs. M graph with the exponential trendline and
attach it to the end of this report. (15 pts)
Calculate the b value for the Southern California data and write it below (two significant figures):
b-value for Southern California:
(10 pts)
Use the trendline via its equation to estimate the number of smaller and larger earthquakes that occur each year in
Southern California.
M ≥ 1 earthquakes (Num/Year)
(5 pts)
M ≥ 7 earthquakes (Num/Year)
(5 pts)
When the frequency is less than one, it’s essentially an estimate of the probability in any given year. You should know
how to relate probability to exceedance interval. Estimate the exceedance interval for earthquakes with a magnitude of
7 or larger in Southern California. Give your answer to two significant figures.
EI, M ≥ 7 earthquakes (y)
(5 pts)
New Madrid Fault Zone, Missouri
You will now repeat these steps for the New Madrid Fault Zone of Missouri, where earthquake data are compiled by the
University of Memphis and housed at: http://folkworm.ceri.memphis.edu/catalogs/html/cat_nm.html.
Once, again, extract earthquake data for the 30-year period starting January 1st, 1989, as follows:
•
•
•
•
Set the start date as year = 1989, month = 01, day = 01, hour = 00, min = 00, and sec = 00.
Set the end date as year = 2018, month = 12, day = 31, hour = 23, min = 59, sec = 59.
Leave the latitudes and longitudes alone.
Leave everything else unchanged except the minimum magnitude. This you will increase systematically from
2.5 to the magnitude where no earthquakes were recorded.
Clicking “Begin Search” will extract the requested data. The number of earthquakes meeting the criteria will be shown at
the top of the page; for example, for M ≥ 2.5, you will see “795 out of 12819 events selected”. This means that there
were 795 earthquakes larger than M = 2.5, and this is the number you will enter into your spreadsheet. Continue
extracting earthquakes until you don’t retrieve any earthquakes. Then calculate the per-year earthquake frequency and
create a plot for the New Madrid earthquakes identical to the one you generated for the southern California
earthquakes, with the trendline extended from M = 0 to M = 9.
→ Print out your New Madrid log(𝒇) vs. M graph with the exponential trendline and attach it to the end of
this report. (15 pts)
Calculate the b value for the New Madrid zone and write it below (two significant figures):
b-value for New Madrid earthquake zone:
(10 pts)
Use the trendline via its equation to estimate the number of smaller and larger earthquakes that occur each year in
Southern California.
M ≥ 1 earthquakes (Num/Year)
(5 pts)
M ≥ 7 earthquakes (Num/Year)
(5 pts)
Fill in the following table which estimates the exceedance interval for earthquakes that exceed various magnitudes in
the New Madrid earthquake zone. Give your answer to two significant figures. Use units of years unless the EI < 1; in
that case use months. (1 pts each, 6 pts total)
M≥
EI – specify units
3
(2 pts)
4
(2 pts)
5
(2 pts)
6
(2 pts)
7
(2 pts)
Applying What You’ve Learned
OK, now let’s apply what you’ve learned from this activity. Remember, the b-value is the (negative) slope on a diagram
of log(frequency) vs. Magnitude. Summarize the b-values you’ve calculated so far:
World
Southern California
New Madrid Zone
“Average” (1 sig fig)
b-value
You should see that all the b values are similar, and can be approximated as a single “average” value—that’s a very
simple number. It is sometimes known as the “loneliest number”. You only need one significant figure; write it into the
table!
Now let’s say you’ve been appointed director of the Outer Transylvanian Seismological Institute. You have very limited
earthquake data at your disposal, but according to records you know that there were 20 earthquakes with M ≥ 4 in the
last year. Based on this single observation, estimate the exceedance interval in years for an earthquake with M ≥ 7!
EI, M ≥ 7 earthquakes (y)
(10 pts)
This might seem like an impossible task, but it’s within reach if you assume a Gutenberg-Richter relationship and assume
the “typical” value of the slope, b. So imagine what the Gutenberg-Richter plot would look like:
Notice I haven’t used a logarithmic scale, but that shouldn’t matter—you should be able to make connections between
log(𝑓) and 𝑓. (For example, if log(𝑓) = 2.0, then 𝑓 = 102 = 100). The Gutenberg-Richter line must go through the single
data point and have a slope equal to the “average” slope you identified above. You can thus draw the line!
Draw it, and then make inferences about log(𝑓) and thus EI for M ≥ 7.