Earthquakes, log relationships, trig functions tom.h.wilson Department of Geology and Geography West Virginia University Morgantown, WV Geology geomathematics

Objectives for the day Tom Wilson, Department of Geology and Geography Explore the use of earthquake frequency-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

related materials that may be of interest Tom Wilson, Department of Geology and Geography some interesting

Tom Wilson, Department of Geology and Geography

Oceanic crustal fragment underlies complex sea bottom bathymetry Tom Wilson, Department of Geology and Geography

Life over a subducting oceanic place zone can be exciting Tom Wilson, Department of Geology and Geography

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? A useful log relationship in seismology The Gutenberg- Richter relationship

World seismicity – Jan 9 to jan16, 2014 Tom Wilson, Department of Geology and Geography

IRIS Seismic Monitor 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

Magnitude distribution Tom Wilson, Department of Geology and Geography

Observational data for earthquake magnitude (m) and frequency (N, number of earthquakes per year (worldwide) with 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 Some worldwide data

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

What does y represent in this case? What is b? the intercept Here is our formula for a straight line …

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

January 12 th, 2010 Haitian magnitude 7.0 earthquake

Shake map USGS NEIC

Notice the plot axis formats Limited observations

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

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

Magnitude 7 earthquakes are predicted from this relationship to occur about once every 20 years. Let’s work through an example using a magnitude of 7.2

Let’s determine N for a magnitude 7.2 quake.

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

Logarithms Tom Wilson, Department of Geology and Geography Logarithms are based (initially) on powers of 10. We know for example that 10 0 =1, 10 1 = = =1000 And negative powers give us = = =0.001, etc.

General definition of a log Tom Wilson, Department of Geology and Geography The logarithm of x, denoted log x solves the equation 10 log x =x The logarithm of x is the exponent we have to raise 10 to - to get x. So log 1000 = 3 since 10 3 = 1000 & Log 10 y =y since

Some more review examples Tom Wilson, Department of Geology and Geography What is log  10? We rewrite this as log (10) 1/2. Since we have to raise 10 to the power ½ to get  10, the log is just ½. Some other general rules to keep in mind are that log (xy)=log x + log y log (x/y)= log x – log y log x n =n log x

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. Take a look at exponential (allometric) functions

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 log 10 y to say - ”What is the power that 10 must be raised to to get y?” How about this operator? -

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

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

log 10 is referred to as the common logarithm thus log e 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 What is N? Where N, in this case, is the number of earthquakes of magnitude 7.2 and greater per year that occur in this area. You have the power! Call on your base!

Base 10 to the power Tom Wilson, Department of Geology and Geography Since Take another example: given b = 1.25 and c=7, how often can a magnitude 8 and greater earthquake be expected? (don’t forget to put the minus sign in front of b!) is the power you have to raise 10 to to get N. log N = ….

What energy is released by a magnitude 4 earthquake? A magnitude 5? More logs and exponents! Seismic energy-magnitude relationships more logs

See 2/313_GC2012_Comparing_Energy_Calculations.pdf Tom Wilson, Department of Geology and Geography For applications to microseismic events produced during frac’ing.

Review: Here’s a problem similar to the inclass problem from last time. (see handout) e.g. Worksheet – pbs 16 & 17: sin(nx) … and basics.xls

A review of the problems from last time Tom Wilson, Department of Geology and Geography

Try another: 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? XY|Y| ? 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. The one below is all that will be due today Tom Wilson, Department of Geology and Geography

Warm-up problems 1-20 will be due next Tuesday. Bring any remaining questions to class on Thursday

In the next class, we will spend some time working with Excel. Tom Wilson, Department of Geology and Geography

Hand in group problems before leaving today Look over problems 2.11 through 2.13 Continue your reading We examine the solutions to 2.11 and 2.13 using Excel next time. Next Time