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.

• Use a map to find the longitude and latitude at the center of an area you are interested in.
• Enter the database and use the circular search option for the global data set and request the screen output option. Use the USGS/NEIC database for 1973 to present. Enter the appropriate longitude and latitude values. Choose a radius for your search area. For an active tectonic area a radius of 500 km will likely result in several hundred earthquakes. Make sure you get at least 50 earthquakes to work with. Ask it to conduct the search for you.
• When the results have returned from the USGS data base, highlight and copy into the clipboard the data from the resulting table. Note that you will have a lot of extraneous information you don't need.
• Save a copy of the page as a text file so you can print it and in case you need to access it later.
• Open up an Excel sheet and transfer the data into it. Note that it might stick an entire line of data in one cell. You should end with N rows of text data with the magnitude values you want embedded in the middle of an individual cell entry. If it does this under the Data option in Excel select the Text to columns operation. This will detect the spaces and insert the various text columns into different Excel columns. This is like magic and saves a lot of time. This trick may be very useful at other times.
• Clean up your Excel spreadsheet by deleting columns/rows you don't need. You might do this on a copy of the original file, so that if something goes wrong you can restart with that, instead of having to retrieve the information all over again. The two columns you should keep are the year, and magnitude. Also delete rows without magnitude data.
• You are ready to proceed with the rest of the analysis.
• I have completed all the above steps with a Mac.

Transforming the data.

• You can first sort the data by earthquake size to start with just to see what is going on.
• The next step is to compute how many of the earthquakes in your data set are greater than a given size. You can sit there and manually count, but there are much faster and more reliable ways to accomplish this in Excel . We will describe one way, 1, and mention three others in passing (2-4)). Remember that the end product you want is the number of earthquakes in your data set greater than a given size, for an array of different sizes.
• Method 1) Using the COUNTIF function (simplest and preferred):
• This function works well but you have to be careful with your syntax. Given an array it counts the number of cells in that array that meet criteria you specify.
• An example of the formula you would insert could look like: =COUNTIF(A\$2:A\$314, ">" & B2)
• A\$2:A\$314 represents the data array you are operating on.
• ">" & B2 is the criteria and basically can be read as - values greater than whatever is in cell B2. In this case B2 should be your earthquake magnitude size. You should therefore have an array that lists the earthquake sizes you are going to plot up on your x axis. With such an array you can then drag-copy the formula down.
• The result of the entire formula entry should be the number of earthquakes in your data array larger than what ever value is in B2.
• The dollar signs are added simply so that those cell references do not advance when the formula is copied down.
• Remember to use the help option in the bottom left of the function box to help learn more about how a function works.
• Alternate method 2) Using the FREQ function. This is how I used to do it on older versions of excel, but it no longer seems to produce simple frequencies when you copy and drag the formula down. You will need to create a bin array. It can work better to start with higher numbers and go to lower.
• Alternate method 3) You can also use the HISTOGRAM routine in the data analysis option in excel (the one you used to make your histograms two labs ago).
• Alternate method 4) If you hate the FREQ function, you can also simply use the row number of your sorted earthquake size number to extract the cumulative frequency. If you have your earthquake size column sorted in descending order, then the row number tells you how many values are greater than a given value (usually you will subtract 1 from the row number to take into account the column headings). You can then extract the information by hand into an x,y array that you can then continue to work on to create the plot you need.
• One more transformation can help. The frequency you have is of earthquakes > than size m for x number of years, where x is the length of your record. It will make the end product much more useful, and allow for better comparisons) if you transform this into a frequency per year by simply dividing the cumulative frequency by the number of years of record your data set covers.
• The final transformation is to take the log of your cumulative frequency of earthquakes > than size m per year. Print out (or insert in Word document file) a copy of the your results, along the lines of the chart below. You are then set to make your plot, regress a line and see how good a fit exists.
• An example exists below of the what your Excel sheet might look like is below.
 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.

• Area earthquake data taken from.
• Sample size.
• First page of list of data values.
• Source of data.
• Print out of column of Excel calculations (first page).
• Print out of plot with linear trend line fits.
• Print out of regression parameters.
• Discussion of following:
• What is the "goodness of the fit" of the data to the line. Note any distinctive departures and perhaps comment on why?
• Report on the fractal dimension. Compare it to the 1.78 value for southern California given by Turcotte.
• What will be the recurrence interval for an earthquake of magnitude 7?
• evaluate whether this makes sense given the tectonic setting.

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.