Post on 02-Jan-2022
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
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
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
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
Number of earthquake of magnitude m and greater (y axis) versus magnitude (x axis)
Tom Wilson, Department of Geology and Geography
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.
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
3Frequency (log10N) Magnitude Plot (Haitian Region)
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
3Frequency (log10N) Magnitude Plot (Haitian Region)
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
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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.
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| 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
Tom Wilson, Department of Geology and Geography
Next week we will spend some time working with Excel.
Tom Wilson, Department of Geology and Geography