THE MATHEMATICS OF EARTHQUAKES:
Introduction:
About 55 times per day, there is an earthquake somewhere on our planet,
approximately 20,000 every year. 1 Earthquakes are usually sudden and are the
cause of thousands of deaths and numerous casualties. Luckily, technologies are
evolving, and with the help of maths, we are able to study such disasters.
The investigation will explore the mathematics of earthquakes. I will first investigate
different types of scales used for measuring the earthquake’s magnitude or relative
size as well: the logarithmic scale, the Richter scale and the moment scale. The
logarithmic scale is a nonlinear scale often used when analyzing a large range of
quantities.2 Instead of increasing in equal increments, each interval is increased by
a factor of the base of the logarithm.3
In addition , I will then investigate a mathematical concept that will allow us to find
the origin point by establishing a connection between three other points, called
triangulation. When seismic data is collected from at least three different locations, it
can be used to determine the epicenter by where it intersects. 4
In order to test our knowledge, we will use such scales, formulas and triangulation
method to find the point of origin of one of the worst earthquakes in history, the 2010
earthquake in Bio-Bio, Chile.
The aim of this investigation is to study a particular aspect of earthquakes focusing
on two specific aspects: 1) the mathematics behind measuring earthquake intensity
2) the mathematics behind the point of origin.
1 [ CITATION USG20 \l 3082 ]
2 [ CITATION Jor17 \l 3082 ]
3 [ CITATION Jor17 \l 3082 ]
4 [ CITATION QUE16 \l 3082 ]
,EARTHQUAKES AND SCALES:
To refresh your memory of school math, logs are just another way of writing
exponential equations, one that allows you to separate the exponent on one side of
the equation.5 Logarithmic scales are used for things that come in an extremely wide
variety of sizes.6 Earthquakes are just one example: a 9.0 earthquake releases a
trillion times as much energy as a 1.0 earthquake. 7 If we used a linear scale, we
would have to use giant numbers to describe earthquakes; for example, a 6.0
magnitude earthquake would be a 1,000,000,000 size earthquake (magnitude of its
effects) , and a 9.0 magnitude earthquake would be described as a
1,000,000,000,000 size earthquake.8 Logarithmic scales allow us to measure things
that are orders of magnitude in difference. 9If using a linear scale, either all the
smaller quakes would be lumped together or the large quakes would be far off the
charts.10 A logarithmic scale allows us to list and plot them all accordingly. 11
Other type of data that could be represented in a logarithmic scale rather than a
linear scale could be the, unfortunately, large number of worldwide deaths due to
COVID-19, bacterial growth or even in when it comes to human hearing, our ear
responds logarithmically to sound.12
The use of logarithmic scales to measure the severity of earthquakes used to be
done through the Richter scale. The Richter scale is a logarithmic scale used to
measure the "size" (intensity) of an earthquake.13 It defines the magnitude of the
earthquake to be the logarithm of the ratio of the the amplitude of the seismic wave
to an arbitrary amplitude.14 We can calculate the magnitude of an earthquake using
the following equation:
M=log(I / IN)
Here the variables represent:
5 [ CITATION Nao12 \l 3082 ]
6 [ CITATION Red15 \l 3082 ]
7 [ CITATION Red15 \l 3082 ]
8 [ CITATION Red15 \l 3082 ]
9 [ CITATION Red15 \l 3082 ]
10 [ CITATION Red15 \l 3082 ]
11 [ CITATION Red15 \l 3082 ]
12 [ CITATION DrI18 \l 3082 ]
13 [ CITATION Chr20 \l 3082 ]
14 [ CITATION Chr20 \l 3082 ]
, M the magnitude of the earthquake
I the amplitude of the seismic wave or the intensity
IN the arbitrary amplitude or arbitrary intensity
For instance, in the case of the earthquake I will study later on in this study, ‘I’ would
15
be 2.34 m and the ‘IN’ would be 8.8 Mww. 2.34m is the displacement of the particle
motions.16 Likewise, 8.8 Mww is an arbitrary scale based on observations of
phenomena such as: the type and extent of damage, whether sleeping people were
woken… 17
M=8.8
I= 2.34m
IN= 8.8 Mww.
M=log(I / IN)
8.8=log( 2..8 )
8.8= -0.57526681474
(Mr.Browne I think I did this wrong I don’t really know how to do it..)
In 1979, the American seismologist, Tom Hanks, and the Japanese seismologist,
Hiroo Kanamori, introduced the Moment Magnitude Scale (MW) the ultimate
18
successor to the Richter scale. The moment magnitude scale is used by
19
seismologists to compare the energy released by earthquakes. Calculations of an
earthquake’s size using the moment magnitude scale are tied to an earthquake’s
seismic moment (M0) rather than to the amplitudes of seismic waves recorded by
seismographs.20 The moment magnitude scale is the only scale capable of reliably
measuring the magnitudes of the largest, most destructive earthquakes because it is
15 [ CITATION Wik202 \l 3082 ]
16 [CITATION Sei201 \l 3082 ]
17 [ CITATION Sei20 \l 3082 ]
18 [ CITATION Sci201 \l 3082 ]
19 [ CITATION Sci201 \l 3082 ]
20 [ CITATION Joh20 \l 3082 ]
Introduction:
About 55 times per day, there is an earthquake somewhere on our planet,
approximately 20,000 every year. 1 Earthquakes are usually sudden and are the
cause of thousands of deaths and numerous casualties. Luckily, technologies are
evolving, and with the help of maths, we are able to study such disasters.
The investigation will explore the mathematics of earthquakes. I will first investigate
different types of scales used for measuring the earthquake’s magnitude or relative
size as well: the logarithmic scale, the Richter scale and the moment scale. The
logarithmic scale is a nonlinear scale often used when analyzing a large range of
quantities.2 Instead of increasing in equal increments, each interval is increased by
a factor of the base of the logarithm.3
In addition , I will then investigate a mathematical concept that will allow us to find
the origin point by establishing a connection between three other points, called
triangulation. When seismic data is collected from at least three different locations, it
can be used to determine the epicenter by where it intersects. 4
In order to test our knowledge, we will use such scales, formulas and triangulation
method to find the point of origin of one of the worst earthquakes in history, the 2010
earthquake in Bio-Bio, Chile.
The aim of this investigation is to study a particular aspect of earthquakes focusing
on two specific aspects: 1) the mathematics behind measuring earthquake intensity
2) the mathematics behind the point of origin.
1 [ CITATION USG20 \l 3082 ]
2 [ CITATION Jor17 \l 3082 ]
3 [ CITATION Jor17 \l 3082 ]
4 [ CITATION QUE16 \l 3082 ]
,EARTHQUAKES AND SCALES:
To refresh your memory of school math, logs are just another way of writing
exponential equations, one that allows you to separate the exponent on one side of
the equation.5 Logarithmic scales are used for things that come in an extremely wide
variety of sizes.6 Earthquakes are just one example: a 9.0 earthquake releases a
trillion times as much energy as a 1.0 earthquake. 7 If we used a linear scale, we
would have to use giant numbers to describe earthquakes; for example, a 6.0
magnitude earthquake would be a 1,000,000,000 size earthquake (magnitude of its
effects) , and a 9.0 magnitude earthquake would be described as a
1,000,000,000,000 size earthquake.8 Logarithmic scales allow us to measure things
that are orders of magnitude in difference. 9If using a linear scale, either all the
smaller quakes would be lumped together or the large quakes would be far off the
charts.10 A logarithmic scale allows us to list and plot them all accordingly. 11
Other type of data that could be represented in a logarithmic scale rather than a
linear scale could be the, unfortunately, large number of worldwide deaths due to
COVID-19, bacterial growth or even in when it comes to human hearing, our ear
responds logarithmically to sound.12
The use of logarithmic scales to measure the severity of earthquakes used to be
done through the Richter scale. The Richter scale is a logarithmic scale used to
measure the "size" (intensity) of an earthquake.13 It defines the magnitude of the
earthquake to be the logarithm of the ratio of the the amplitude of the seismic wave
to an arbitrary amplitude.14 We can calculate the magnitude of an earthquake using
the following equation:
M=log(I / IN)
Here the variables represent:
5 [ CITATION Nao12 \l 3082 ]
6 [ CITATION Red15 \l 3082 ]
7 [ CITATION Red15 \l 3082 ]
8 [ CITATION Red15 \l 3082 ]
9 [ CITATION Red15 \l 3082 ]
10 [ CITATION Red15 \l 3082 ]
11 [ CITATION Red15 \l 3082 ]
12 [ CITATION DrI18 \l 3082 ]
13 [ CITATION Chr20 \l 3082 ]
14 [ CITATION Chr20 \l 3082 ]
, M the magnitude of the earthquake
I the amplitude of the seismic wave or the intensity
IN the arbitrary amplitude or arbitrary intensity
For instance, in the case of the earthquake I will study later on in this study, ‘I’ would
15
be 2.34 m and the ‘IN’ would be 8.8 Mww. 2.34m is the displacement of the particle
motions.16 Likewise, 8.8 Mww is an arbitrary scale based on observations of
phenomena such as: the type and extent of damage, whether sleeping people were
woken… 17
M=8.8
I= 2.34m
IN= 8.8 Mww.
M=log(I / IN)
8.8=log( 2..8 )
8.8= -0.57526681474
(Mr.Browne I think I did this wrong I don’t really know how to do it..)
In 1979, the American seismologist, Tom Hanks, and the Japanese seismologist,
Hiroo Kanamori, introduced the Moment Magnitude Scale (MW) the ultimate
18
successor to the Richter scale. The moment magnitude scale is used by
19
seismologists to compare the energy released by earthquakes. Calculations of an
earthquake’s size using the moment magnitude scale are tied to an earthquake’s
seismic moment (M0) rather than to the amplitudes of seismic waves recorded by
seismographs.20 The moment magnitude scale is the only scale capable of reliably
measuring the magnitudes of the largest, most destructive earthquakes because it is
15 [ CITATION Wik202 \l 3082 ]
16 [CITATION Sei201 \l 3082 ]
17 [ CITATION Sei20 \l 3082 ]
18 [ CITATION Sci201 \l 3082 ]
19 [ CITATION Sci201 \l 3082 ]
20 [ CITATION Joh20 \l 3082 ]
