earthquakes, log relationships, trig functions

Earthquakes, log relationships, trig functions tom.h.wilson

[email protected]

Department of Geology and Geography West Virginia University

Morgantown, WV

Objectives for the day

Tom Wilson, Department of Geology and Geography

• 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

Problems


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)


Another site being phased in


If you change this to 2.5+ you only get about 220

Larger number of magnitude 2 and 3’s and many fewer M5’s


Number of earthquake of magnitude m and greater (y axis) versus magnitude (x axis)


Total number for the week

5 6 7 8 9 10

Richter Magnitude

0

100

200

300

400

500

600

Num

ber o

f ear

thqu

akes

per

yea

r

m N/year5.25 537.035.46 389.045.7 218.775.91 134.896.1 91.206.39 46.776.6 25.706.79 16.217.07 8.127.26 4.677.47 2.637.7 0.817.92 0.667.25 2.087.48 1.657.7 1.098.11 0.398.38 0.238.59 0.158.79 0.129.07 0.089.27 0.049.47 0.03

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

magnitude greater than m)

What would this plot look like if we plotted the log of N versus m?

Num

ber o

f ear

thqu

akes

per

yea

r of

M

agni

tude

m a

nd g

reat

er

0.01

0.1

1

10

100

1000

Num

ber o

f ear

thqu

akes

per

yea

r

5 6 7 8 9 10

Richter Magnitude

Looks almost like a straight line. Recall the formula for a

straight line?

On

log

scal

e N

umbe

r of e

arth

quak

es p

er y

ear

of

Mag

nitu

de m

and

gre

ater

bmxy +=

0.01

0.1

1

10

100

1000

Num

ber o

f ear

thqu

akes

per

yea

r

5 6 7 8 9 10

Richter Magnitude

What does y represent in this case?

Ny log=

What is b?

the intercept

5 6 7 8 9 10

Richter Magnitude

0.01

0.1

1

10

100

1000

Num

ber o

f ear

thqu

akes

per

yea

r

cbmN +−=log

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

Magnitude2 3 4 5 6 7 8

N (p

er y

ear -

mag

nitu

de m

and

hig

her)

0.01

0.1

1

10

100

Gutenberg Richter (frequency magnitude) plot

Haiti (1973-2010) Magnitude 2 and higher

log( )N bm c= − +Notice the plot axis formats

Year1975 1980 1985 1990 1995 2000 2005 2010

Mag

nitu

de

2

3

4

5

6

7Earthquake Occurrence 1973- present (Haiti and surroundings)

The seismograph network appears to have been upgraded in 1990.

Low magnitude seismicity

Year1920 1940 1960 1980 2000

Mag

nitu

de

6.0

6.5

7.0

7.5

8.0Large Earthquakes Haiti Region (last century)

In the last 110 years there have been 9 magnitude 7

and greater earthquakes in the region

Magnitude2 3 4 5 6 7

Log 1

0 N

-2

-1

0

1

2

3Frequency (log10N) Magnitude Plot (Haitian Region)

Look at problem 19 (see additional group problems)

Magnitude

2 3 4 5 6 7

Log

10 N

-2

-1

0

1

2


logN=-0.935 m + 5.21

With what frequency should we expect magnitude 7.2

earthquakes in the Haiti area?

log 0.935 5.21log 0.935(7.2) 5.21log 1.52

N mNN

= − += − += −

Magnitude

2 3 4 5 6 7

Log

10 N

-2

-1

0

1

2


logN=-0.935 m + 5.21

Substitute 7.2 for m and solve for N.

log 0.935 5.21log 0.935(7.2) 5.21log 1.52

N mNN

= − += − += −

How do you solve for N?

What is N?

Let’s discuss logarithms for a few minutes and come back to this later.

Year1920 1940 1960 1980 2000

Mag

nitu

de

6.0

6.5

7.0

7.5

8.0Large Earthquakes Haiti Region (last century)

In the last 110 years there have been 9 magnitude 7

and greater earthquakes in the region

Logarithms The allometric or exponential functions are in the form

cxaby = and

cxay 10=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.

xyei 10 .. =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.

xy 10=

By definition, we also say that x is the log of y, and can write

( ) xy x == 10loglogSo 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

45y where,log =yWe assume that we are asking for x such that

4510 =x

Sometimes you will see specific reference to the base and the question is written as

45y where,log10 =yy10log 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? -

ypow →10


ypow →10

The power of base 10 that yields (→) y

653.1log10 =y 1.65310 45=

10 45 = pow →

10 45 = 1.653pow →

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.

5log 10 asks what is the power that 5 must be raised to, to get 10.

? 10log5 =

How do we find these powers?

5log10log 10log

10

105 =

431.1699.01 10log5 ==

105 431.1 =thus

In general, basenumber

base10

10log

)(log number) some(log =

or

ba

b10

10log

)(log alog =

Try the following on your own

?)3(log)7(log 7log

10

103 ==

8log8

21log7

7log4

10So log is often written as log, with no subscript

log10 is referred to as the common logarithm

ln. asten often writ is log e

2.079 ln8 8log ==e

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.

etc. ,logor og 25l

log 0.935 5.21log 0.935(7.2) 5.21log 1.52

N mNN

= − += − += −

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


log 1.52N = −Since

?10N =

Take another example: given b = 1.25 and c=7, how often can a magnitude 8 and greater earthquake be expected?

The Richter magnitude scale determines the magnitude of shallow earthquakes from surface waves according to the following equation

3.3log66.1log10 +∆+=TAm

where T is the period in seconds, A the maximum amplitude of ground motion in µm (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)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 45 90 135 180 225 270 315 360

)

… and basics.xls

Finish up work on these in-class problem. Individually show your work.


Let’s try sin(4x)


Graphical sketch problem similar to problem 18

What approach could you use to graph this function?

X Y |Y| 0 7 7

-3.5 0 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. The one below is all that

will be due today



• Warm-up problems 1-19 will be due NEXT Tuesday.

Next week we will spend some time working with Excel.


• 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.

earthquakes, log relationships, trig functions

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