Logarithms and the Richter Scale
Overview:
Quantifying the magnitude of earthquakes is an application of logarithms. Earthquake intensity can vary significantly so the logarithm of intensity is used to compare magnitude of differing intensities.
Richter scale magnitude is the base ten logarithm of earthquake intensity divided by a standard intensity
The basic equation of magnitude and intensity is:
M = log (I/S), where:
• M is the magnitude of an earth quake
• I is the intensity of the earthquake measured by the amplitude of a wave form on a seismograph reading taken 100 km from the epicenter.
• S is the intensity of a “standard” (and very weak) earthquake, whose amplitude on a seismograph is one micron, or 10^-4 cm.
About magnitude and intensity
With the Richter scale, earthquake magnitude is a computed value based on an empirical measured value, which is the intensity of an earthquake. Intensity is the wave height on a seismograph that records earth movement during an earthquake.
The Richter scale is a scale of magnitude values. The seismograph reports intensity. Intensity is the amplitude of a graph on a seismometer that records ground movement during an earthquake. Intensity is a measured quantity that is recorded by an instrument. Magnitude by comparison is a computed value.
More about the equations
The basic equation for magnitude and intensity is
M = log (I/S) or magnitude is the base ten logarithm of intensity over standard.
Applying some algebra and rules of logarithms, Intensity can be isolated
M = log (I/S)
10^M = 10^(Log(I/S))
10^M = I/S
I = (10^M) * S
If M = log(I/S) and I = S then M = log (I/S) = log (1) = 0 because 10^0 = 1. This means that the magnitude of a ‘small’ earthquake is 0 on the Richter scale
Richter scale problems
If the intensity of an earthquake is given then the magnitude can be computed.
If the magnitude of an earthquake is given then the intensity can be computed.
One unknown can be solved with one equation, and two unknowns can be solved with two equations
Conversion:
The most important equation to know is that magnitude is the base ten logarithm of intensity, and that intensity is the wave height on a seismograph recording movement. The expression is:
M = Log(I/S)
Using algebra and rules of logarithms, Intensity can be isolated and expressed in terms of magnitude.
10^M = 10^(Log(I/S))
10^M = I/S
If I=80,500,000 and S=1 then M=Log(80,500,000) = 7.9
If I=48,275,000 and S=1 then M=Log(48,275,000) = 7.7
If I=251,200 and S=1 then M=Log(251,200) = 5.4
If M = 8.6 then I = 10^8.6 = 398,107,171
If M = 6.7 then I = 10^6.7 = 5,011,872
If M = 7.7 then I = 10^7.7 = 50,118,723
Example
If one earthquake is 20x the intensity of another, how much larger is the magnitude on the Richter scale?
M = log(I/S) = log(20I2/I2) = log(20) = 1.3
Example
SFO earthquake in 1906 was magnitude 8.3 on the Richter scale
Loma Prieta earthquake in 1989 was magnitude 7.1 on the Richter scale
How many more times intense was the 1906 earthquake?
8.3 = log(I1/s) and 7.1 = log(I2/S)
This works if S = 1.
Objective is to compute log((I2/S)/(I1/S)) = log(I2/I1) =
log(I2) - log(I1/S) = 8.3 - 7.1 = 1.2
log(I2/I1) = 1.2 exponentiate both sides
10^(I2/I1) = 10^1.2
I2/I1 = 10^1.2 = 15.8
Intensity is 15.8x
Example
SFO 1906 magnitude = 8.3, Japan 1906 = magnitude 4.9
How many times more intense was SFO 1906?
Magnitude is the base 10 log of intensity divided by a constant S. M = log(I/S). Assume S = 1 (It is actually 10^-4)
1). 8.3 = log(I1/S) and
2). 4.9 = log(I2/S)
Objective is to compute I1/I2
Subtract 2 from 1:
8.3 - 4.9 = log(I1/S) - log(I2/S) = log((I1/S)/(I2/S)) = Log(I1/I2)
3.4 = log(I1/I2)
10^3.4 = 10^(log(I1/I2)) = I1/I2 = 2,511
Example
Northridge 1994 magnitude 6.8. Call this M1
Kobe 1995 magnitude 7.2. Call this M2
How many times more intense was Kobe? In other words find I2/I1.
Remember: M = log(I/S).
M2 = 7.2 = log(I2/S)
M1 = 6.8 = log(I1/S)
Subtract M2 from M1:
7.2 - 6.8 = log(I2/S) - log(I1/S) = log((I2/S)/(I1/S)) = log(I2/I1) = 0.4
0.4 = log(I2/I1) Exponentiate both sides
10^0.4 = 10^(log(I2/I1))
10^0.4 = 2.51
10^(log(I2/I1)) = I2/I1
I2/I1 = 2.51
Kobe was 2.51x intense
Example
Alaska 1964 magnitude 8.6. Call this M1
SFO 1906 magnitude 8.3. Call this M2
How many times more intense was Alaska 1964?
M1 = Log(I1/S)
8.6 = Log(I1/S)
M2 = Log(I2/S)
8.3 = Log(I2/S)
Subtract M1 from M2:
8.6 - 8.3 = Log(I1/S) - Log(I2/S)
0.3 = Log(I1/I2)
10^0.3 = 10^(Log(I1/I2))
1.995 = I1/I2
Alaska 1964 was about 2x more intense
Example
If magnitude 1 is 7.3 and magnitude 2 is 4.8 what is the ratio of I1/I2?
7.3 = Log(I1/S)
4.8 = Log(I2/S)
Subtract M2 from M1:
7.3 - 4.8 = Log(I1/S) - Log(I2/S) = Log(I1/I2)
2.5 = Log(I1/I2) Exponentiate both sides
10^2.5 = 10^(Log(I1/I2)) = I1/I2
10^2.5 = 316.2
I1/I2 = 316.2
Example
The SFO earthquake in 1906 was of an estimated 8.3 magnitude on the Richter scale.
The Colombia-Ecuador earthquake had 4 times the intensity.
What is the magnitude of the Colombia earthquake?
SFO: M = Log(I/S) = 8.3
Colombia:
M = Log(4I/S) = Log(4) + Log(I/S) note that Log(I/S)=8.3
M= Log(4) + 8.3 Base ten log of 4 = 0.602
M = 0.602 + 8.3 ~=8.9
Colombia earth quake was magnitude 8.9
Example
M1: Landers earthquake was magnitude 7.3
M2: Northridge in 1994 was magnitude 6.7
How many times more powerful (intense, I) was Landers
M1 = Log(I1/S) = 7.3
7.3 = Log(I1/S)
10^7.3 = 10^(Log(I1/S)) Exponentiate both sides
10^7.3 = I/S Simplify
M2 = Log(I2/S) = 6.7
6.7 = Log(I2/S)
10^6.7 = 10^(Log(I2/S)) Exponentiate both sides
10^6.7 = Log(I2/S)
Divide M1 by M2:
10^7.3 / 10^6.7 = (I1/S) / (I2/S) = I1/I2
10^(7.3) - 10^(6.7) = 10^(7.3 - 6.7) = 10^(0.6) = 3.98
I1/I2 = 3.98 meaning Landers was 3.98x intense
Example
If an earthquake had magnitude 6.5 then compute the magnitude of an earthquake that had 35x the intensity
M1 = 6.5 = Log(I/S) M2 = Log(35I/S)
6.5 = Log(I/S)
10^6.5 = 10^(Log(I/S))
10^6.5 = I/S
I = 10^6.5/S
M2 = Log(35I/S)
M2 = Log(35*10^6.5/S/S)
M2 = Log(35*10^6.5)
M2 ~= 8.04
Magnitude of earthquake 35 times the intensity of a magnitude 6.5 earthquake is 8.0
Example
M1 = 1985 earthquake in Mexico of magnitude 8.1
M2 = 1976 earthquake in China of 1.26x intensity.
Compute the magnitude of M2
M1 = Log(I1/S) M2 = Log(1.26I/S)
8.1 = Log(I1/S)
Subtract M1 from M2:
M2 - M1 = Log(1.26I1/S) - Log(I/S)
M2 - 8.1 = Log((1.26I/S) / (I/S)) Simplify
M2 - 8.1 = Log(1.26)
M2 = Log(1.26) + 8.1 Log(1.26) ~=0.1
M2 ~= 8.2
Summarize this: Compare magnitude and intensity differences