The Richter Scale1 Earthquake Senior Mathematics B Exponential and Logarithmic Functions

The Richter Scale2 The magnitude of an earthquake is a measure of the energy released by it. It is usually measured on the Richter Scale, which is logarithmic, and each unit represents a tenfold increase in amplitude of the measured waves and nearly a 30-fold increase in energy. Thus, in a magnitude 6 earthquake, for example, the movement of the ground is 10 times greater and the energy released about 30 times more than in an earthquake with a magnitude of 5. Generally, an earthquake of magnitude 6 or more will cause major damage to buildings near its epicentre.

The Richter Scale3 Richter defined an earthquake that caused his seismograph to move sufficiently to produce a 1 mm amplitude in the seismogram's tracing as magnitude 4. If the amplitude of the trace was 10 mm, it was magnitude 5. A 100 mm amplitude was produced by an earthquake defined as magnitude 6, and so on.

The Richter Scale4 Richter magnitude Amplitude on Seismograph 11 micron = metres 41 mm (= 10 0 mm) = 10 3 microns 510 mm (= 10 1 mm) = 10 4 microns 6100 mm (= 10 2 mm) = 10 5 microns

The Richter Scale5 (a)All other factors being equal, determine: 1)The Richter Scale magnitude of an Earthquake which caused Richter’s seismograph to record a trace of amplitude 60 mm. 2)The amplitude of the trace recorded on Richter’s seismograph resulting from an Earthquake of magnitude 8.5. (b)Assign a meaning to an Earthquake of magnitude zero on the Richter Scale.

The Richter Scale6 (a) All other factors being equal, determine: (1) The Richter Scale magnitude of an Earthquake which caused Richter’s seismograph to record a trace of amplitude 60 mm; (2) The amplitude of the trace recorded on Richter’s seismograph resulting from an Earthquake of magnitude 8.5.

The Richter Scale7 (b) Assign a meaning to an Earthquake of magnitude zero on the Richter Scale. If 0 = log 10 (Amplitude in mm) + 4, then Amplitude in mm = Thus Richter assigned a magnitude of 1 to an Earthquake which produced an amplitude of 1 micron on his seismograph, an Earthquake of magnitude zero would correspond to one producing an amplitude on Richter’s seismograph of 0.1 microns i.e mm or metres.

The Richter Scale8 Earthquake Effects Earthquake Severity Richter Earthquake Magnitudes Effects Less than 3.5 Generally not felt, but recorded Often felt, but rarely causes damage. Under 6.0 At most slight damage to well-designed buildings. Can cause major damage to poorly constructed buildings over small regions Can be destructive in areas up to about 100 kilometres across where people live Major earthquake. Can cause serious damage over larger areas. 8 or greater Great earthquake. Can cause serious damage in areas several hundred kilometres across.

The Richter Scale9 Earthquake Effects Descriptor Richter magnitudes Earthquake Effects MicroLess than 2.0Microearthquakes, not felt. Very minor Generally not felt, but recorded. Minor Often felt, but rarely causes damage. Light Noticeable shaking of indoor items, rattling noises. Significant damage unlikely. Moderate Can cause major damage to poorly constructed buildings over small regions. At most slight damage to well-designed buildings. Strong Can be destructive in areas up to about 100 miles (160 kilometres) across in populated areas. Major Can cause serious damage over larger areas. Great Can cause serious damage in areas several hundred miles across. Rare great9.0 or greaterDevastating in areas several thousand miles across.

The Richter Scale10 Earthquake Effects (a) In it is reported, “The first reported earthquake in Australia was felt at Port Jackson (Sydney) in June 1788, when Governor Phillip reported ‘The 22nd of this month (June) we had a slight shock of an earthquake; it did not last more than 2 or 3 seconds. I felt the ground shake under me, and heard a noise that came from the southward, which I at first took for the report of guns fired at a great distance’.” Estimate the magnitude (measured on the Richter Scale) of Governor Arthur Phillip’s earthquake, commenting on the likely accuracy of your estimate. Give any assumptions made.

The Richter Scale11 Estimate the magnitude (measured on the Richter Scale) of Governor Arthur Phillip’s earthquake, commenting on the likely accuracy of your estimate. Governor Phillip’s Earthquake was probably of magnitude about 4.0 (in range 3.5 to 4.5) on the Richter Scale. I assume he would have mentioned any damage to buildings in his report if there was any (none mentioned, so assume no damage), and assume epicentre was not too far distant (it is likely that the epicentre was to the south where the noise came from. Since this was within earshot, it could not have been a very great distance from him). The Earthquake may have been greater than 4.5 if the epicentre was a large distance away.

The Richter Scale12 Earthquake Effects (b) Comment on the damage done by the Newcastle earthquake on 28 December, 1989 which measured 5.6 on the Richter Scale. What would have been expected from a category 5.6 quake? What actual damage was done? Why the discrepancy? View NBN news video clip here. It is at:

The Richter Scale13 Comment on the damage done by the Newcastle earthquake on 28 December, 1989 which measured 5.6 on the Richter Scale. One would have expected major damage to poorly constructed buildings over small regions, at most slight damage to well- designed buildings. The extensive damage to buildings and other structures resulted from an underlying thin layer of alluvium allowing the shaking to cause greater damage than that expected for a relatively small magnitude earthquake.

The Richter Scale14 Earthquake Effects (c) The Chinese Earthquake in 1556 killed the most people, but did not rate as high on the Richter Scale as the Chilean Earthquake in Comment ….?

The Richter Scale15 The Chinese Earthquake in 1556 killed the most people, but did not rate as high on the Richter Scale as the Chilean Earthquake in The Chinese Earthquake hit an area where most of the population at the time lived in artificial caves in loess cliffs, many of which collapsed during the disaster. The Richter Scale is based upon the magnitude of the seismic energy released by the Earthquake, which does not necessarily correspond to the actual damage done or people killed.

The Richter Scale16 Richter Magnitudes “The Richter Scale is a measurement using the numbers 1 to 12 to express the magnitude of earthquakes.” You can’t believe everything you read on the internet! The “teaching treasure” quoted above does not seem to agree with other sources listed: Please check out the next three slides.

The Richter Scale17 Richter Magnitudes Source: The largest recorded earthquake in the world was a magnitude 9.5 in Chile on May 22, When the Chilean earthquake occurred in 1960, seismographs recorded seismic waves that travelled all around the Earth. These seismic waves shook the entire earth for many days!

The Richter Scale18 Richter Magnitudes Source: There is no beginning nor end to this scale. However, rock mechanics seems to preclude earthquakes smaller than about -1 or larger than about 9.5.

The Richter Scale19 Richter Magnitudes Source: The magnitude of (or energy released by) an earthquake is recorded by a seismograph using the Richter Scale. There is no upper limit to this scale as there is no upper limit to the amount of energy an earthquake might release. The most severe earthquakes recorded so far have not exceeded 9.5 on the Richter Scale.

The Richter Scale20 Richter Magnitudes (a) Explain the range of values (minimum to maximum) possible on the Richter Scale.

The Richter Scale21 Explain the range of values (minimum to maximum) possible on the Richter Scale. The Richter Scale is an open- ended one, with no minimum or maximum. In 1935, Richter’s seismograph could not detect an amplitude of less than one micron (corresponding to a magnitude 1 quake). Modern equipment can detect lesser amplitudes, and thus zero and negative Richter magnitudes are possible. The largest recorded quake was of magnitude 9.5. We hope not to see larger quakes as the devastation would be enormous, with a magnitude 12 quake virtually destroying the Earth.

The Richter Scale22 Richter Magnitudes (b) Combine information from tables available on the internet to form a new table for an Australian audience under the headings Richter Magnitude Approximate TNT for Seismic Energy Yield Example giving all measures in metric units and giving the Richter Magnitude to most closely match the TNT equivalent of the given example.

The Richter Scale23 Richter Magnitudes Source: Richter TNT for Seismic Example Magnitude Energy Yield (approx) ounces Breaking a rock on a lab table pounds Large Blast at a Construction Site pounds ton Large Quarry or Mine Blast tons tons tons tons Small Nuclear Weapon tons Average Tornado (total energy) tons tons Little Skull Mtn., NV Quake, million tons Double Spring Flat, NV Quake, million tons Northridge, CA Quake, million tons Hyogo-Ken Nanbu, Japan Quake, 1995; Largest Thermonuclear Weapon million tons Landers, CA Quake, billion tons San Francisco, CA Quake, billion tons Anchorage, AK Quake, billion tons Chilean Quake, trillion tons (San-Andreas type fault circling Earth) trillion tons (Fault Earth in half through centre, OR Earth's daily receipt of solar energy)

The Richter Scale24 Richter Magnitudes

The Richter Scale25 Richter Magnitudes

The Richter Scale26 Richter Magnitudes

The Richter Scale27 Is there a relationship between the Richter Magnitude and TNT Equivalent Energy Yield?

The Richter Scale28 Richter Magnitude Approximate TNT for Seismic Energy Yield Example gramsBreaking a rock on a lab table KgHand grenade Construction site blast KgWWII conventional bombs 2.01 metric tonlate WWII conventional bombs metric tonsWWII blockbuster bomb metric tonsMassive Ordnance Air Blast bomb metric tonsChelyabinsk nuclear accident, kilotonSmall atomic bomb kilotonAverage tornado (total energy) kilotonNagasaki atomic bomb kilotonLittle Skull Mtn., NV Quake, megatonDouble Spring Flat, NV Quake, approx. 5 megatonsNorthridge quake, megatonsLargest thermonuclear weapon 7.5approx. 160 megatonsLanders, CA Quake, approx. 1 gigatonSan Francisco, CA Quake, approx. 5 gigatonAnchorage, AK Quake, approx. 30 gigaton2004 Indian Ocean earthquake teratonestimate for a 10 km rocky bolide impacting at 25 km/s

The Richter Scale29 Earthquake Energies

The Richter Scale30 Verify the general rule of thumb used by geologists that a thirty- fold increase in the energy or intensity of an earthquake is represented by an additive increase of one in the Richter Scale. Let M 0 = 0.67 log 10 (0.37E 0 ) For 30 times this energy, M = 0.67 log 10 (0.37E 0 x 30) = 0.67 (log 10 (0.37E 0 ) + log 10 (30)) = 0.67 log 10 (0.37E 0 ) log 10 (30) = M … = M 0 + 1

The Richter Scale31 Verify the general rule of thumb used by geologists that a thirty- fold increase in the energy or intensity of an earthquake is represented by an additive increase of one in the Richter Scale. or 0.67 log 10 (0.37E) = M – 1.46 log 10 (0.37E) = (M – 1.46)/ E = 10 (M – 1.46)/0.67 E = 10 M/0.67 / ( /0.67 x 0.37) E = 10 M/0.67 / … For Magnitude (M + 1), E = 10 (M+1)/0.67 / E = 10 M/0.67 x 10 1/0.67 / E = (Energy for magnitude M) x 10 1/0.67 E = (Energy for magnitude M) x or approx. a thirty-fold increase in energy.

The Richter Scale32 Earthquake Energies

The Richter Scale33 From Energy in Kilowatt-hours to Magnitude on the Richter Scale: M = 0.67 log 10 (0.37E) For E = 3.68 x 10 10, M = 0.67 log 10 (0.37 x 3.68 x ) = = 8.25 (approx) For E = 2.8 x 10 9, M = 0.67 log 10 (0.37 x 2.8 x 10 9 ) = = 7.5 (approx) For E = 4.08 x 10 6, M = 0.67 log 10 (0.37 x 4.08 x 10 6 ) = = 5.6 (approx)

The Richter Scale34 From Magnitude on the Richter Scale to Energy in Kilowatt-hours: For M = 8.1, E = / 0.67 / = = 2.2 x Kwh (approx) For M = 7.6, E = / 0.67 / = = 3.9 x 10 9 Kwh (approx)

The Richter Scale35 Table 3: Earthquake Energies Earthquake Magnitude (Richter Scale) Energy in Kilowatt-hours 1906 San Francisco, USA x Alaska x Papua New Guinea x Tangshan, China x Newcastle, Australia x 10 6

The Richter Scale36 Earthquake Energies

The Richter Scale37 The logarithmic relation: log(E) = M, with E in Joules, becomes log(E x 3.6 x 10 6 ) = M, with E in Kilowatt-hours This becomes log(3.6E) + 6 = M or1.44M = log(3.6E) M = (log (0.36E) ) / 1.44 M = 0.69 log (0.36E) Which is only very roughly similar to M = 0.67 log(0.37E) For M = 7.5, log(E) = x 7.5 = Therefore E = = Joules = Kwh = 3 x 10 9 Kwh, which is close to the value 2.8 x 10 9 Kwh, from the table in part (b).

The Richter Scale38 Even though the 2 given equations are not identical, they seem to yield similar results. Magnitude (Richter scale) Energy in kilowatt- hours Energy in Joules M from log(E) = M M = (log(E) – 5.24) / x x x x x x x x x x

The Richter Scale39 Frequency of Occurrence of Earthquakes.

The Richter Scale40 Frequency of Occurrence of Earthquakes MNlog N

The Richter Scale41 Frequency of Occurrence of Earthquakes

The Richter Scale42 Frequency of Occurrence of Earthquakes (a) For x (M) = 8, y (log N) = , or N = 2i.e. approx. 2 category 8 Earthquakes per year. (b) For x (M) = 9, y (log N) = , or N = 0.32 i.e. approx. 1 category 9 Earthquake every 3 years. (This illustrates the danger of extrapolating from given data, as indicates only one earthquake of magnitude 8.0 – 8.9 is expected per year, and one of magnitude 9.0 or greater every 20 years.)

The Richter Scale43 References / Sources: