Section 2: The Distribution of Earthquakes

Activity #8: THE GUTENBERG-RICHTER RELATION

Materials:

• the graphing template
  
• writing instruments (colored pencils, pens, etc.)
  
• paper and/or a calculator for making simple calculations
  
• (optional) a calculator capable of computing logarithms
  
• a straight edge of some kind (ruler, etc.)
  
• example of completed graph (see Exercise 1, below)
  

Procedure:

Beno Gutenberg and Charles Richter were two of the pioneers of modern seismology; each contributed greatly to the development of the field as a modern, quantitative science. In the 1930s, as instrumental recording of earthquakes was becoming a reality in many areas of the world, these two scientists 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 over, say, 10 years, 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 activity consists of two exercises designed to familiarize you with the "Gutenberg-Richter relation", as the pattern described by these early seismologists came to be known. The exercises are outlined below. Each has its own set of instructions and review questions. Work through each as directed within the exercise itself.

In the first exercise you will be given a set of data to graph. Once you have determined which kind of graph to use and have plotted the data, it will be up to you to figure out the equation that describes the Gutenberg-Richter relation. Then you will plot another set of data from a different time and region, and be asked to compare the two.

In the second exercise, you'll use the SCEC Data Center earthquake catalog search to construct your own data set of a year's worth of recent southern California seismicity. You will then plot this and compare it to the other sets of data previously graphed.

As each exercise is self-explanatory, you may begin when ready.

Exercise 1 Southern California vs. the World

Your introduction to Gutenberg-Richter plots will be a relatively easy one. The data will be provided; you only need to determine what sort of graph to make, and then plot the pre-made data sets. You'll start with a set of data from southern California, then plot worldwide totals of earthquakes against this and compare.

For southern California, the data set below was compiled according to the following guidelines:

• Start date: January 1, 1987; End date: December 31, 1996.
  
• "Southern California" is defined as that area between 32° and 36.25° North latitude, and 114.75° and 121° West longitude.
  
• Only earthquakes of magnitude 2.5 and greater were used, because the Southern California Seismic Network's (SCSN) database is almost certainly incomplete below this magnitude for the area, as outlined above.
  

Earthquake Numbers in Southern California, 1987 through 1996

Before you can do any graphing, you'll need to decide what type of graphing scale to use. Choose a simple x-y plot, with magnitude M as the x-axis and number of earthquakes greater than magnitude M as the y-axis.

Note that the x-axis data, the magnitudes, are very much linear in scale, increasing in half-unit steps. However, look how greatly our y-axis numbers change -- we'll need to plot the number 1 and the number 13590 on the same graph! If we used a proportional linear scale for each axis, the y-axis would be huge, while the x-axis would be miniscule!

But note that numbers we want to plot on the y-axis jump about a factor of ten for every unit in magnitude increase. This suggests that we could use a y-axis based on powers of 10, or a logarithmic scale, while we use a linear scale for the x-axis.

Hence, we can plot this data set on a logarithmic-linear graph. Print out the log-linear graphing template if you haven't already. You are now ready to begin making your first graph of the data set above. Do so now. Remember to pay attention to the scale for each axis, but don't worry too much about making your points exact. When you've finished plotting the data, work through the questions below.

1. In roughly what form did the points you plotted fall? (What shape or pattern?) Or are they completely random, with no form at all?

2. You should see a roughly linear arrangement of points. Using a straight edge, draw a single line that best represents the set of points you've plotted. That line does not need to run through the center, or even touch, all of the points in your set.
   

You now have a line that represents the data you graphed -- the numbers of earthquakes with respect to magnitude over 10 years in southern California (1987-1996). You are ready to describe the Gutenberg-Richter relation just as the two of them did, decades ago.

The equation for a line on a simple x-y plot is y = bx + a, where a is the y-intercept and b represents the slope of the line. To keep b positive at all times, we can think of the above equation as true for positive-sloping lines (going up as you move left-to-right), and the equation y = a - bx as true for negative-sloping lines.

3. Is the slope of your graphed line positive or negative?

4. Your graph has a y-axis that is logarithmic. Thus, a negative-sloping line on this graph would be described as log y = a - bx. Indeed, instead of calling it the y-axis, think of it as a function N of the magnitude, M, which itself can be substituted for x (since the x-axis is magnitude). Make these substitutions in the equation given above. What do you get?
   

You now have a mathematical expression that represents the Gutenberg-Richter relation, the correlation between the magnitude of earthquakes and their relative numbers. It should look something like

Had you come up with this 70 years earlier, this expression might have been named after you!

But do all data sets of earthquakes counted according to magnitude plot in this same linear manner? And even if they are all linear, does the slope of different sets vary significantly?

To begin to answer these questions, let's plot another set of data -- this time, the average values of an entire year's worth of worldwide seismicity. Using that same piece of graph paper you used to plot the southern California data set, plot the set of data given in the table below, then answer the questions that follow.

Average Worldwide Seismicity Totals for a Single Year

5. Did this set of data plot as a line, too? If you haven't already, draw a line that is the best fit for these new data points.

6. How does the slope of this new line (worldwide seismicity) compare with that of the southern California data?
   

Use a ruler to actually measure the slope of each line you graphed. Pick any segment of each line and sketch out a right triangle, with the legs parallel to the axes, and the line itself forming the hypotenuse. Measure the height of the vertical side of the triangle and divide this by the length of the horizontal side. Your answer will be the slope of the line, otherwise known as the b value.

Another way to find the b value is to note the value of N(M) at each of two points along the line, exactly one magnitude unit (i.e. in the x-direction) apart. Divide the larger number (the point on the left) by the smaller number (the point on the right), and then take the logarithm of this quotient. That answer is the b value of this line.

7. Using whichever method you prefer, what numerical values do you get for the b value of each line?
   

Now that you've completed your own graphs, feel free to take a look at this example of a finished graph for the same data sets. How does this compare with yours?

As it turns out, when Gutenberg-Richter plots are made for various data sets all over the world, most end up having a b value very close to 1, usually slightly less. This basic relation seems to be a universal property of seismicity.

Exercise 2 A Year of Your Life

In this exercise, you will again be making a Gutenberg-Richter plot of southern California seismicity. This time, however, you will not be provided a pre-formed data set; you must retrieve the data yourself! Fortunately, you will use the SCEC Data Center catalog search to help you accomplish this.

Since you already know the basics of what is required to plot a Gutenberg-Richter relation, let's concentrate on using the catalog search of the SCEC Data Center Earthquake Hypocenter and Phase Database. Either by opening a new browser window, or simply following the link, and returning after you get a first impression, go there now. Then read through the basics of how to search the database, below. (If you already know how to use the catalog search, you may skip the bracketted section below.)

How to Use the SCEC Data Center Catalog Search

The catalog search will allow you to obtain a list of all the earthquakes in the SCEC Data Center Database that meet certain parameters you can define. Those parameters include:
• Catalog and Format: For this exercise, we will use only the default values for these parameters.

• Start and End Dates: To narrow your search to a range between two specific dates. The database extends back only as far as August 1, 1983, and usually is complete up to as recent as five days before today's date. The date format is: YYYYMMDDhhmmss, where YYYY=year (all four digits), MM=month, DD=day, hh=hour, mm=minute, and ss=second. You should only bother with the first eight digits (leaving the last six as zero) unless you're doing a very specific search.

• Maximum and Minimum Magnitude: To select only earthquakes within a specific "size" range. A maximum magnitude of 9.0 will ensure no upper "cut-off". A minimum of 0.0 is as low as you will need to go.

• Maximum and Minimum Depth (km): To select only earthquakes within a defined range of depths, in kilometers. Generally, there's no need to change these defaults.

• Southern and Northern Latitude: You will only be able to search within a rectangular area -- these let you define its top (north) and bottom (south) edges. You may wish to consult a map to define these more precisely.

• East and West Longitude: As above, but for the left (west) and right (east) edges of the rectangle.

• Type of Event: L, the default value, stands for "local". In this activity, you should always use this default value.

When the parameters are set the way you want them, click the button labelled "Submit request" to send your search request to the database. Then click the button marked "Continue submission" in the pop-up alert window that should appear when you make your request. Depending on the speed of your connection and the breadth of your search, a list of hypocenters matching your specified parameters will appear within several seconds. For the most lengthy searches, it may take a minute or more.

There is a maximum output length cut-off of 10,000 lines. Should you exceed this, or even come anywhere close to it, it is recommended that you narrow your search to make your list of output more manageable.

To get acquainted with the catalog search, try a very basic sample search to start with: leave all the parameters at their default value (use the "Reset" button if you need to), except for "Minimum magnitude". Change this to "5.0", and submit the request. The list you'll see represents potentially damaging earthquakes that have occurred in southern California since August 1983.

Now that you've seen how the search works, it's time to use it toward the goal of this exercise -- constructing a custom set of data to plot on a Gutenberg-Richter graph!

The data set you'll be making for this exercise will span exactly one year. To "personalize" this set of data (especially if you're working on this exercise in parallel with others), you should use your two most recent birthdays as the start and end dates. Thus, if it's October 2000 and you were born on March 18th, you would use "19990318000000" as your start date, and "20000318000000" as your end date.

You should keep this set comparable to the previous southern California data set we graphed -- indeed, you are encouraged to plot this set on the same piece of graph paper. To do this, you need to choose the same boundaries, in terms of longitude and latitude. The values you should choose are "32.0" and "36.25" for the Southern and Northern latitudes, respectively; for East and West longitude, use "-114.75" and "-121.0". Keep the default values for the depth range and, of course, for "Type of Event".

Lastly, we come to minimum and maximum magnitude. These values will be crucial for constructing a proper data set. You should probably build your data set by making a table, similar to those you saw in Exercise 1, above. Divide your table into narrow magnitude ranges, going no lower than M 2.5 (consider using 0.2 or 0.5 intervals). Then search the entire year for each magnitude range. Be careful not to "double up"! That is, don't search for magnitudes 3.0 to 3.5, and then for magnitudes 3.5 to 4.0. You would count all magnitude 3.5 earthquakes twice! When you receive the search output, count the number of lines, and record that as the number of earthquakes in that magnitude range. Repeat this until your table is complete, then add up the cumulative totals -- these are what you should plot. When your table is complete, start plotting this data on your log-linear graph paper. You can, and probably should, plot this data on the same piece of paper as the other data sets, but if it gets "crowded", feel free to start with a fresh piece of graph paper.

When you have completed graphing the data from your table of earthquake counts versus magnitude, work through the questions below.

1. Did the data set you created using the catalog search "work"? That is, does it plot as a roughly linear set of points on a logarithmic graph?

2. If you haven't already, draw a line to best fit your set of graph points. Then measure the slope of this line. What is the b value of this data set (to the nearest tenth)?

3. How does the slope (b value) of this line compare to the slope of the data for 10 years of seismic activity in southern California? How does it compare to the b value of the data for a year of worldwide seismicity?
   

Ignore b values now, and just look at the earthquake numbers for each of the three data sets you've graphed. You have one year of seismic activity worldwide, one year of activity in southern California only, and ten years of activity in southern California as your data sets. This gives us an excellent opportunity to make some comparisons.

4. Multiply the earthquake counts from the single year of southern California activity by 10. Are these new totals larger than, smaller than, or the same as the totals from the 10-year southern California data set (1987 through 1996)? If we assume that the 10-year period we plotted was average, then what does this say about the year for which you gathered data? Or perhaps "your year" was average; what would that mean for the 10-year period? (Bearing in mind that the M 6.6 Superstition Hills, M 7.3 Landers, and M 6.7 Northridge earthquakes struck during the 10 years in question, which case seems more likely?)

5. Compare the numbers from a year of activity in southern California to a year of worldwide seismicity (you may need to project the one-year southern California line forward to higher magnitudes). What is the ratio, roughly, of the number of worldwide events to that of those earthquakes of the same magnitude in southern California, over the course of a year?

6. (Optional) Calculate the size of the area we call "southern California" (as defined with longitude and latitude, above), in square miles, and compare this to the surface area of the entire Earth. Given this ratio, and the one you found in question #5, above, does southern California have an average, above-average, or below-average rate of seismicity, compared to the mean for the entire world?
   

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