Earthquakes, log relationships, trig functions tom.h.wilson [email protected] Department of Geology and Geography West Virginia University Morgantown,

Earthquakes, log relationships, trig [email protected] of Geology and GeographyWest Virginia UniversityMorgantown, WVGeology 351 - geomathematics

Objectives for the dayTom 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

Tom Wilson, Department of Geology and Geography

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some interesting seismotectonics @ http://usgsprojects.org/fragment/index.html


http://usgsprojects.org/fragment/download.htmlTom Wilson, Department of Geology and Geography


Oceanic crustal fragment underlies complex sea bottom bathymetryTom 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 seismologyThe Gutenberg- Richter relationship

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


IRIS Seismic Monitorhttp://www.iris.edu/seismon/Tom Wilson, Department of Geology and Geography


Larger number of magnitude 2 and 3s and many fewer M5sTom Wilson, Department of Geology and Geography


Magnitude distributionTom 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 ofMagnitude m and greaterSome worldwide data

m

N/year

5.25

537.03

5.46

389.04

5.7

218.77

5.91

134.89

6.1

91.20

6.39

46.77

6.6

25.70

6.79

16.21

7.07

8.12

7.26

4.67

7.47

2.63

7.7

0.81

7.92

0.66

7.25

2.08

7.48

1.65

7.7

1.09

8.11

0.39

8.38

0.23

8.59

0.15

8.79

0.12

9.07

0.08

9.27

0.04

9.47

0.03

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

What does y represent in this case?What is b?the interceptHere 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 12th, 2010 Haitian magnitude 7.0 earthquake

Shake mapUSGS NEIC

USGS NEIC

Notice the plot axis formatsLimited observations

The seismograph network appears to have been upgraded in 1990.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. Lets work through an example using a magnitude of 7.2

Lets determine N for a magnitude 7.2 quake.

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

LogarithmsTom Wilson, Department of Geology and Geography

Logarithms are based (initially) on powers of 10.We know for example that 100=1,101=10102=100103=1000And negative powers give us10-1=0.110-2=0.0110-3=0.001, etc.


General definition of a logTom Wilson, Department of Geology and Geography

The logarithm of x, denoted log x solves the equation 10log x =x

The logarithm of x is the exponent we have to raise 10 to - to get x.

So log 1000 = 3 since 103 = 1000 &

Log 10y =y since


Some more review examplesTom 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 to get 10, the log is just .

Some other general rules to keep in mind are thatlog (xy)=log x + log y log (x/y)= log x log ylog xn =n log x


andb 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 asleaves 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 () yWhat power do we have to raise the base 10 to, to get 45


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

log10 is referred to as the common logarithmthusloge 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 earlierWhat 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 powerTom 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? (dont forget to put the minus sign in front of b!)-1.53 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 relationshipsmore logs

See http://www.cspg.org/documents/Conventions/Archives/Annual/2012/313_GC2012_Comparing_Energy_Calculations.pdfTom Wilson, Department of Geology and Geography

For applications to microseismic events produced during fracing.


Review: Heres 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 timeTom Wilson, Department of Geology and Geography


Try another: sin(4x)Tom Wilson, Department of Geology and Geography


Graphical sketch problem similar to problem 18What approach could you use to graph this function?Really only need three points: y (x=0), x(y=0) and one other.

XY|Y|077-3.500?

Have a look at the basics.xlsx fileSome 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 constantsJust 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 todayTom 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

*Earthquake focal mechanisms indicate that the upper zone, where most events occur, is in downdip compression, while the lower zone is in downdip extension. This DSZ is located at a depth where the curvature of slab is greatest; at greater depths it unbends into a more planar donfiguration. *Dip slip breachball (left lateral)**0.03 or one every 33 years*Linear so symmetricalCalculate three points: y=0, x=0 and x=2x(at y=0)

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Earthquakes, log relationships, trig functions tom.h.wilson [email protected] Department of Geology and Geography West Virginia University Morgantown,