Exercise 4 - Fractal analysis & estimating recurrence intervals of earthquakes

Part 1 - Estimation of fractal dimension of a coast line.

The initial part of this will be done in class. The class will generate length values for a coast line segment using different size measuring elements (rulers). Take these values, and plot and regress them appropriately so that you can judge whether a fractal relationship exists, and if so, estimate the fractal dimension. To do this plot the log of the resulting length as y and the log of the ruler as x. Print the plot and add a less than 100 word explanation of what information it conveys (unless I indicate otherwise). For comparison the fractal dimension of the west coast of Britain is 1.25.

Part 2 - Estimation of earthquake recurrence intervals

Obtain earthquake data from a relevant web source (or other data source) from an area of interest over a certain time span. You should have several hundred data points to work with in order to establish a good plot. Below are the details of how you can go from web site to earthquake data in an Excel spreadsheet in less than 10 minutes.

Getting the data from the United States Geological Survey earthquake site and into Excel.

Transforming the data.

EQ data
EQ size
#>Eqsize
#>Eqsize/years of record
log of #>Eqsize per year
5.4
1.3
265
10.1923
1.0083
4.9
1.4
265
10.1923
1.0083
4.8
1.5
264
10.1538
1.0066
4.7
1.6
264
10.1538
1.0066
4.6
1.7
264
10.1538
1.0066
4.6
1.8
262
10.0769
1.0033
4.6
1.9
259
9.9615
0.9983
4.6
2
254
9.7692
0.9899
4.5
2.1
245
9.4231
0.9742
4.4
2.2
237
9.1154
0.9598
4.2
2.3
225
8.6538
0.9372
4.1
2.4
217
8.3462
0.9215
4.1
2.5
206
7.9231
0.8989
3.9
2.6
189
7.2692
0.8615
3.9
2.7
171
6.5769
0.8180
3.9
2.8
146
5.6154
0.7494
3.9
2.9
127
4.8846
0.6888
3.9
3
99
3.8077
0.5807
3.8
3.1
89
3.4231
0.5344
3.7
3.2
79
3.0385
0.4827
3.7
3.3
62
2.3846
0.3774
3.7
3.4
39
1.5000
0.1761
3.7
3.5
32
1.2308
0.0902
3.7
3.6
25
0.9615
-0.0170
3.7
3.7
19
0.7308
-0.1362
3.6
3.8
18
0.6923
-0.1597
3.6
3.9
13
0.5000
-0.3010
3.6
4
13
0.5000
-0.3010
3.6
4.1
11
0.4231
-0.3736
3.6
4.2
10
0.3846
-0.4150
3.6
4.3
10
0.3846
-0.4150
3.6
4.4
9
0.3462
-0.4607
3.5
4.5
8
0.3077
-0.5119
3.5
4.6
4
0.1538
-0.8129
3.5
4.7
3
0.1154
-0.9379
3.5
4.8
2
0.0769
-1.1139
3.5
4.9
1
0.0385
-1.4150
3.5
5
3.5
5.1

Example of the upper portion of a table of calculations created for Iceland data (n=266 total). The longer list of earthquake data is truncated at the bottom in this example (there were more values that are not shown here).

Plotting and regressing the data: Plot earthquake size as x (remember it is already a log scale measure) vs. log cumulative frequency per year in a scatter plot. What you are plotting is the yearly number of earthquakes greater than a given size. Add a trend line. Print out the plot (our save as a file to insert in Word). To get the best fit line equation, under the Data option chose Data analysis and then choose Regression, or highlight your chart, and use the Add Trendline option. Print out the table or plot of results.

Here is the raw plot for the Iceland data. The Y axis is the log of yearly frequency so that 0 represents 1 year and -1 would represent 10 years, and -2 would represent 100 years.

 

Repeat values at the high end. If you have a lower number of data points are a major outlier, then it is common you end up with a series of identical values. These will plot as a horizontal step in your data. Basically, if you have a size 7 earthquake as your largest and a size 5 as your next largest, and your earthquake size increment for plotting purposes is .2 or the like, then for size 5.2 you have just one earthquake, and for 5.4 you have just one earthquake, and so on up until you reach 7. For your plot and best fit line you should delete all but the highest repeat value, and then make your plot and create your best fit line.

Is there a bias on the basis of earthquake size in your data? If you look at your plot (or the plot above) it is likely that at lower earthquake sizes the line formed by your scatter plot flattens out parallel to the x axis and departs from the expected linear relationship. Why? What determines how many of the much smaller earthquakes one sees in an area? The sensitivity of your seismic detection array does, and this is variable from location to location. The smaller earthquakes aren't picked up consistently. With this as a rationale, you can then look at the fractal relationship only for the range of earthquakes that will be consistently picked up by the array. Remember that you will also be extrapolating to estimate the larger earthquakes. Try excluding from your data all earthquakes less than magnitude 4 (or whatever magnitude at which you start to see the departure), and repeat the above steps to calculate a best fit trend line. With the trend line in hand you can then extrapolate to estimate the size of a larger, rarer earthquake.

Example of Iceland data where only the earthquakes greater than 3 are considered. It suggest that every year you expect to see a richter magnitude 3.6 or so. What might be the size of the 100 year earthquake (frequency of -2 on this scale). The earthquake risk in Iceland is much lower than other active plate boundaries (according to this view/analysis) - why might that be? Note the consistent departures from the best fit line. There is a concept of bifractal relationship where there are "two parts". Such a more sophisticated analysis might lead to a better estimate of larger EQ frequency.

Report format: Your report should include the following components.

We will compare values from the different areas assigned.

Some FAQs that might help:

How do you get your fractal dimension? To calculate the fractal dimension you take the absolute value of your slope on the size yearly frequency map and multiply by 2 (see Turcotte reference for details).

How do you compute the recurrence interval for a given size earthquake once you have the best fit line? Take the magnitude of earthquake size you want to compute the recurrence interval for, and insert it into the x position in your best-fit linear equation and solve for y. Remember that your x axis is the log of the # of earthquakes of a given size or larger per year. You need to both take the inverse log, and change the units from per year to years. The following formula will do this - 1/(10^y). 10^y is basically the inverse log. You can use the power function in Excel to make this computation.

What does the fractal dimension mean in term of the physics of earthquakes?? This is a good question to ask. What follows is only a partial answer. Obviously as the slope decreases, for the same intercept value, the recurrence interval decreases. This means more of the seismic energy is released by larger earthquakes relative to shallower earthquakes than for a a earthquake distribution with a higher fractal dimension. The other parameter is the slope intercept. For our plots this is the size earthquake which has a recurrence interval of 1 year. The higher that is the more seismic energy that is being released. However, what is going on mechanics wise during earthquake events that produces a higher slope and fractal dimension?? Turcotte et al. 2002 (www.pnas.org/content/99/suppl.1/2530.full.pdf ) describe a cascade model that may give some insight. How easily does any initial given slip event ripple along the fault to create a given size earthquake? Perhaps conjure up the term cascadability. This might be related to how tightly or loosely coupled one section of the fault is to another and how stress is transmitted from one element to another, which is perhaps fundamentally tied to the spatial distribution and variability of frictional parameters along the fault.


Copyright by Harmon D. Maher Jr.. This material may be used for non-profit educational purposes if proper attribution is given. Otherwise please contact Harmon D. Maher Jr.