Logarithmic ScalesLogarithmic Scales

Connecting qualitative observations Connecting qualitative observations with quantitative measurements.with quantitative measurements.

Logarithmic ScalesLogarithmic Scales

““A A logarithmic scalelogarithmic scale is a scale of is a scale of measurement that uses the logarithm of measurement that uses the logarithm of a physical quantity instead of the quantity a physical quantity instead of the quantity itself.” --- From the Wikipedia article on itself.” --- From the Wikipedia article on Logarithmic scales.Logarithmic scales.

Question: Why would anyone want to Question: Why would anyone want to measure the logarithm of quantity? measure the logarithm of quantity?

There are two very good reasons.There are two very good reasons.

Taking in a Large Range Taking in a Large Range of Values. of Values.

XKCD is my very favorite on-line comic XKCD is my very favorite on-line comic strip. strip.

OK… that’s cute, but what does it have to OK… that’s cute, but what does it have to do with logarithms?do with logarithms?

Logarithmic scale of the Universe ----XKCD

Suppose we consider the sequence . . .

Logarithms revisitedLogarithms 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.

Suppose we consider the sequence . . .

Logarithms revisitedLogarithms 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.

Human Perception is Human Perception is Naturally LogarithmicNaturally Logarithmic

In music-In music---frequency of vibration vs. our --frequency of vibration vs. our perception of pitch. Our ears hear equalperception of pitch. Our ears hear equal ratiosratios in frequency as equal in frequency as equal changeschanges in in pitch. pitch. For instance, concert A is 440 Hz in For instance, concert A is 440 Hz in

frequency. The A one octave below concert frequency. The A one octave below concert A is 220 Hz. The A one octave above concert A is 220 Hz. The A one octave above concert A is 880 Hz. A A is 880 Hz. A change in pitch on 1 octavechange in pitch on 1 octave corresponds to a corresponds to a doublingdoubling of the frequency of the frequency. .

Measuring Measuring EarthquakesEarthquakesMore on logarithmic scales and More on logarithmic scales and human perceptionhuman perception

How intense is an How intense is an Earthquake?Earthquake?

The severity of an earthquake can be expressed in The severity of an earthquake can be expressed in terms of both terms of both intensityintensity and and magnitudemagnitude..IntensityIntensity based on observed effects of ground shaking on people, based on observed effects of ground shaking on people,

buildings, and natural features. buildings, and natural features. It varies from place to place within the disturbed region It varies from place to place within the disturbed region

depending on the location of the observer with respect to the depending on the location of the observer with respect to the earthquake epicenter. earthquake epicenter.

MagnitudeMagnitude related to the amount of seismic energy released at the related to the amount of seismic energy released at the

hypocenter of the earthquake. hypocenter of the earthquake. It is based on the amplitude of the earthquake waves recorded It is based on the amplitude of the earthquake waves recorded

on instruments which have a common calibration. on instruments which have a common calibration. The magnitude of an earthquake is thus represented by a The magnitude of an earthquake is thus represented by a

single, instrumentally determined value. single, instrumentally determined value.

Intensity Scales Intensity Scales (qualitative)(qualitative)

Poarid (Italian)– 1627; first attempt to classify Poarid (Italian)– 1627; first attempt to classify earthquakes by intensity in 1627; four levels of intensity earthquakes by intensity in 1627; four levels of intensity use to describe effects of earthquakes experienced at use to describe effects of earthquakes experienced at different towns. different towns.

More attempts made in 18More attempts made in 18thth and 19 and 19thth centuries. centuries. Early 1900’s Mercalli’s scale; 12 levels of intensity.Early 1900’s Mercalli’s scale; 12 levels of intensity. Modified version (1931) published by Wood & Modified version (1931) published by Wood &

Neumann. Known as the Modified Mercalli Intensity Neumann. Known as the Modified Mercalli Intensity Scale of 1931, it has become the standard used by the Scale of 1931, it has become the standard used by the United States engineering seismology community (incl. United States engineering seismology community (incl. National Geophysical Data Center and the U.S. National Geophysical Data Center and the U.S. Geological Survey.)Geological Survey.)

Modified Mercalli ScaleModified Mercalli Scale

The The ModifiedModified Mercalli intensity scaleMercalli intensity scale is is used for measuring the used for measuring the intensityintensity of an of an earthquake.earthquake. Quantifies the effects of an earthquake on Quantifies the effects of an earthquake on

the Earth's surface, humans, objects of the Earth's surface, humans, objects of nature, and man-made structures.nature, and man-made structures.

12 levels, with 1 denoting a weak 12 levels, with 1 denoting a weak earthquake and 12 one that causes almost earthquake and 12 one that causes almost complete destruction. (See handout)complete destruction. (See handout)

USGS Earthquake investigations.USGS Earthquake investigations.

USGS is the Federal agency responsible for collecting earthquake USGS is the Federal agency responsible for collecting earthquake intensity data. intensity data.

Used “Earthquake Report” questionnaire and field investigators to Used “Earthquake Report” questionnaire and field investigators to analyze earthquake damage in serious earthquakes. analyze earthquake damage in serious earthquakes. Questionnaires sent to postmasters, National Weather Service Questionnaires sent to postmasters, National Weather Service

offices, military installations, etc. requesting a report of all offices, military installations, etc. requesting a report of all effects of the earthquake in their area. Newspaper accounts, effects of the earthquake in their area. Newspaper accounts, published scientific reports, and reports provided by published scientific reports, and reports provided by seismology collaborators were also used.seismology collaborators were also used.

Expert observers may investigate and photograph the Expert observers may investigate and photograph the damage.damage.

In the end, a seismologist analyzes each report and assigns In the end, a seismologist analyzes each report and assigns intensities on the basis of the effects at each town;intensities on the basis of the effects at each town;

Intensity (or isoseismal) maps are then constructed for Intensity (or isoseismal) maps are then constructed for earthquakes felt over large areas. earthquakes felt over large areas.

The Richter ScaleThe Richter Scale Magnitude of most earthquakes is measured on the Magnitude of most earthquakes is measured on the Richter Richter

scalescale, invented by Charles F. Richter in 1934. The Richter , invented by Charles F. Richter in 1934. The Richter magnitude is calculated from the amplitude of the largest magnitude is calculated from the amplitude of the largest seismic wave recorded for the earthquake, no matter what seismic wave recorded for the earthquake, no matter what type of wave was the strongest. type of wave was the strongest.

ComparisonsComparisons

Magnitude / Intensity Comparison

Magnitude Typical MaximumModified 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 orhigher

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.

How Does the Richter How Does the Richter Scale Work?Scale Work?

Baseline: the “zero level earthquake” ---Baseline: the “zero level earthquake” ---seismographic reading measures .001 seismographic reading measures .001 millimeters at a distance of 100 km from millimeters at a distance of 100 km from the epicenter.the epicenter.

(Note: All seismographic readings are “adjusted” (Note: All seismographic readings are “adjusted” based on distance so that they represent the based on distance so that they represent the reading one would have gotten had one been reading one would have gotten had one been 100 km from the epicenter.)100 km from the epicenter.)

Richter Magnitude. Richter Magnitude. Set Set xx00=.001 mm on seismograph---zero =.001 mm on seismograph---zero

levellevel Suppose we have a seismograph reading of Suppose we have a seismograph reading of xx mm. We are interested in the mm. We are interested in the ratioratio

Now we define the Richter magnitude of the Now we define the Richter magnitude of the Earthquake by Earthquake by

0

xx

100

( ) logx

M xx