ACSDFEEWQ

1. Logarithmic Scales
Connecting qualitative observations
with quantitative measurements.

2. Logarithmic Scales
 “A logarithmic scale is a scale of
measurement that uses the logarithm of
a physical quantity instead of the quantity
itself.” --- From the Wikipedia article on
Logarithmic scales.
 Question: Why would anyone want to
measure the logarithm of quantity?
There are two very good reasons.

3. Taking in a Large Range of
Values.
XKCD is my very favorite on-line comic
strip.
OK… that’s cute, but what does it have to
do with logarithms?
Logarithmic scale of the Universe
----XKCD

4. Suppose we consider the sequence . . .
Logarithms revisited
1,10,100,1000,10000,100000,1000000,
What happens if the we the logarithm (base 10) of each
term? We get . . .
0,1, 2, 3, 4, 5, 6,
If we start with a sequence in which the ratio from one
term to the next is constant and we take logarithms, then
the resulting sequence will have terms that are evenly
spaced.

5. Suppose we consider the sequence . . .
Logarithms revisited
1,10,100,1000,10000,100000,1000000,
What happens if the we the logarithm (base 10) of each
term? We get . . .
0,1, 2, 3, 4, 5, 6,
More generally, if we start with a very spread out
sequence that gets more and more spread out as we go
along, taking logarithms, compresses the spread . . . This
is how the XKCD cartoon allows us to take in the whole
universe at a glance.

6. Human Perception is
Naturally Logarithmic
 In music---frequency of vibration vs. our
perception of pitch. Our ears hear equal
ratios in frequency as equal changes in
pitch.
 For instance, concert A is 440 Hz in
frequency. The A one octave below concert
A is 220 Hz. The A one octave above
concert A is 880 Hz. A change in pitch on
1 octave corresponds to a doubling of the
frequency.

7. Measuring
Earthquakes
More on logarithmic scales and
human perception

8. How intense is an
Earthquake?
The severity of an earthquake can be expressed in
terms of both intensity and magnitude.
 Intensity
 based on observed effects of ground shaking on people,
buildings, and natural features.
 It varies from place to place within the disturbed region
depending on the location of the observer with respect to the
earthquake epicenter.
 Magnitude
 related to the amount of seismic energy released at the
hypocenter of the earthquake.
 It is based on the amplitude of the earthquake waves recorded
on instruments which have a common calibration.
 The magnitude of an earthquake is thus represented by a
single, instrumentally determined value.

9. Intensity Scales
(qualitative)
 Poarid (Italian)– 1627; first attempt to classify
earthquakes by intensity in 1627; four levels of intensity
use to describe effects of earthquakes experienced at
different towns.
 More attempts made in 18th and 19th centuries.
 Early 1900’s Mercalli’s scale; 12 levels of intensity.
 Modified version (1931) published by Wood &
Neumann. Known as the Modified Mercalli Intensity
Scale of 1931, it has become the standard used by the
United States engineering seismology community (incl.
National Geophysical Data Center and the U.S.
Geological Survey.)

10. Modified Mercalli Scale
 The Modified Mercalli intensity scale is
used for measuring the intensity of an
earthquake.
 Quantifies the effects of an earthquake on
the Earth's surface, humans, objects of
nature, and man-made structures.
 12 levels, with 1 denoting a weak
earthquake and 12 one that causes almost
complete destruction. (See handout)

11. USGS Earthquake investigations.
 USGS is the Federal agency responsible for collecting earthquake
intensity data.
 Used “Earthquake Report” questionnaire and field investigators to
analyze earthquake damage in serious earthquakes.
 Questionnaires sent to postmasters, National Weather Service
offices, military installations, etc. requesting a report of all
effects of the earthquake in their area. Newspaper accounts,
published scientific reports, and reports provided by
seismology collaborators were also used.
 Expert observers may investigate and photograph the damage.
 In the end, a seismologist analyzes each report and assigns
intensities on the basis of the effects at each town;
 Intensity (or isoseismal) maps are then constructed for
earthquakes felt over large areas.

12. The Richter Scale
 Magnitude of most earthquakes is measured on the
Richter scale, invented by Charles F. Richter in 1934.
The Richter magnitude is calculated from the amplitude
of the largest seismic wave recorded for the
earthquake, no matter what type of wave was the
strongest.

13. Comparisons
Magnitude / Intensity Comparison
Magnitude Typical Maximum
Modified Mercalli Intensity
1.0 - 3.0 I
3.0 - 3.9 II - III
4.0 - 4.9 IV - V
5.0 - 5.9 VI - VII
6.0 - 6.9 VII - IX
7.0 and
higher
VIII or
higher
However, beware that such
comparisons can be dicey, as
damage depends on a lot more
than the intensity of the
Earthquake, such as the population
density and the quality of
construction materials used.

14. How Does the Richter
Scale Work?
 Baseline: the “zero level earthquake” ---
seismographic reading measures .001
millimeters at a distance of 100 km from
the epicenter.
(Note: All seismographic readings are “adjusted”
based on distance so that they represent the
reading one would have gotten had one been
100 km from the epicenter.)

15. Richter Magnitude.
 Set x0=.001 mm on seismograph---zero
level
 Suppose we have a seismograph reading of
x mm. We are interested in the ratio
 Now we define the Richter magnitude of the
Earthquake by
0
x
x
10
0
( ) log
x
M x
x
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