Earthquakes, log relationships, trig functions Geology 351 - Geomath Earthquakes, log relationships, trig functions tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV

Objectives for the day Explore the use of frequency of earthquake occurrence and magnitude relations in seismology Learn to use the frequency magnitude model to estimate recurrence intervals for earthquakes of specified magnitude and greater. Learn how to express exponential functions in logarithmic form (and logarithmic functions in exponential form). Review graphical representations of trig functions and absolute value of simple algebraic expressions Tom Wilson, Department of Geology and Geography

Problems Tom Wilson, Department of Geology and Geography

Earthquake Seismology Application The Gutenberg-Richter Relation Are small earthquakes much more common than large ones? Is there a relationship between frequency of occurrence and magnitude? Fortunately, the answer to this question is yes, but is there a relationship between the size of an earthquake and the number of such earthquakes?

World seismicity in the last 7 days (preceding January 22nd) Tom Wilson, Department of Geology and Geography

Another site being phased in If you change this to 2.5+ you only get about 220 Tom Wilson, Department of Geology and Geography

Larger number of magnitude 2 and 3’s and many fewer M5’s Tom Wilson, Department of Geology and Geography

Number of earthquake of magnitude m and greater (y axis) versus magnitude (x axis) Total number for the week Tom Wilson, Department of Geology and Geography

When in doubt -collect data. Observational data for earthquake magnitude (m) and frequency (N, number of earthquakes per year (worldwide) with magnitude greater than m) ? Number of earthquakes per year of Magnitude m and greater What would this plot look like if we plotted the log of N versus m?

Number of earthquakes per year of Magnitude m and greater On log scale Number of earthquakes per year of Magnitude m and greater Looks almost like a straight line. Recall the formula for a straight line?

What does y represent in this case? What is b? the intercept

-b is the slope and c is the intercept. The Gutenberg-Richter Relationship or frequency-magnitude relationship -b is the slope and c is the intercept.

January 12th Haitian magnitude 7.0 earthquake

Shake map USGS NEIC

USGS NEIC

Notice the plot axis formats

Low magnitude seismicity The seismograph network appears to have been upgraded in 1990.

In the last 110 years there have been 9 magnitude 7 and greater earthquakes in the region

Look at problem 19 (see additional group problems)

With what frequency should we expect magnitude 7 With what frequency should we expect magnitude 7.2 earthquakes in the Haiti area?

Substitute 7.2 for m and solve for N.

How do you solve for N? What is N? Let’s discuss logarithms for a few minutes and come back to this later. 0.03 or one every 33 years

In the last 110 years there have been 9 magnitude 7 and greater earthquakes in the region

Thus, given y =100, we know that x must be equal to 2. Logarithms The allometric or exponential functions are in the form and b and 10 are the bases. These are constants and we can define any other number in terms of these constants raised to a certain power. Given any number y, we can express y as 10 raised to some power x Thus, given y =100, we know that x must be equal to 2.

By definition, we also say that x is the log of y, and can write So the powers of the base are logs. “log” can be thought of as an operator like x (multiplication) and  which yields a certain result. Unless otherwise noted, the operator “log” is assumed to represent log base 10. So when asked what is We assume that we are asking for x such that

Sometimes you will see specific reference to the base and the question is written as leaves no room for doubt that we are specifically interested in the log for a base of 10. One of the confusing things about logarithms is the word itself. What does it mean? You might read log10 y to say -”What is the power that 10 must be raised to to get y?” How about this operator? -

The power of base 10 that yields () y What power do we have to raise the base 10 to, to get 45 Tom Wilson, Department of Geology and Geography

We’ve already worked with three bases: 2, 10 and e We’ve already worked with three bases: 2, 10 and e. Whatever the base, the logging operation is the same. How do we find these powers?

In general, or Try the following on your own

log10 is referred to as the common logarithm thus loge or ln is referred to as the natural logarithm. All other bases are usually specified by a subscript on the log, e.g.

Return to the problem developed earlier Where N, in this case, is the number of earthquakes of magnitude 7.2 and greater per year that occur in this area. What is N? You have the power! Call on your base!

Base 10 to the power Since Take another example: given b = 1.25 and c=7, how often can a magnitude 8 and greater earthquake be expected? Tom Wilson, Department of Geology and Geography

The Richter magnitude scale determines the magnitude of shallow earthquakes from surface waves according to the following equation where T is the period in seconds, A the maximum amplitude of ground motion in m (10-6 meters) and  is the epicentral distance in degrees between the earthquake and the observation point. More logs!

Some in-class problems for discussion (see handout) e.g. Worksheet – pbs 16 & 17: sin(nx) … and basics.xls

Finish up work on these in-class problem. Individually show your work. Tom Wilson, Department of Geology and Geography

Let’s try sin(4x) Tom Wilson, Department of Geology and Geography

Graphical sketch problem similar to problem 18 What approach could you use to graph this function? X Y |Y| 7 -3.5 ? Linear so symmetrical Calculate three points: y=0, x=0 and x=2x(at y=0) Really only need three points: y (x=0), x(y=0) and one other.

Have a look at the basics.xlsx file Some of the worksheets are interactive allowing you to get answers to specific questions. Plots are automatically adjusted to display the effect of changing variables and constants Just be sure you can do it on your own!

Spend the remainder of the class working on Discussion group problems Spend the remainder of the class working on Discussion group problems. The one below is all that will be due today Tom Wilson, Department of Geology and Geography

Warm-up problems 1-19 will be due NEXT Tuesday. Tom Wilson, Department of Geology and Geography

Next week we will spend some time working with Excel. Tom Wilson, Department of Geology and Geography

For Next Time Look at additional group problems handed out today and bring questions to class on Tuesday Look over problems 2.11 through 2.13 Continue your reading We examine the solutions to 2.11 and 2.13 using Excel next week.