QUANTIFICATION OF EARTHQUAKES

H.R. WASON

4.1 Intensity Scale

Towards the end of the nineteenth century, a need was recognized to better define and correlate the macroseismic (directly observable) effects of different earthquakes on humans and their environment. M.S. de Rossi of Italy and F. A. Forel of Switzerland combined their efforts in 1883 to create a numerical scale, that defined ten different discrete levels of severity, or intensity for earthquake effects. Their effort, the Rossi-Forel Scale(RF), was the first of many different intensity scales that have evolved over the years and have proven to be of great importance in the evaluation of historical seismicity and in descriptions of seismic hazard.· Each level, or step, in an intensity scale provides a primarily qualitative description of earthquake effects. These descriptions may include human perceptions such as being awakened from sleep, being unable to stand, or even being subjected to general panic; the effects on buildings and construction such as damage to chimneys, collapse of adobe houses, or loss of service of underground pipelines; and the effects on the natural surroundings such as changes in water-flow from springs, landslides, and the opening of fissures in the ground. Often the choice of scale is a matter of local, that is, national preference. For example, in southern Europe, the 12 level Mercalli-Cancani-Sieberg Intensity Scale (MCS) is used. It relies in part on determining intensity level by the percentage of buildings damages. This has proven useful in a country where there is a relatively uniform type and quality of village construction. The intensity scale used by the Japanese Meteorological Agency (JMA) is an eight level (zero to seven) scale that makes reference to earthquake effects on typical Japanese items such as latticed sliding doors, stone lanterns and wooden houses. The Medvedev-Sponheuer-Karnik Intensity Scale (MSK), a twelve level scale, was developed by central and eastern European scientists and is currently used by many countries outside of that region.

The intensity scale currently used in the United States and India is the

Modified Mercalli Intensity scale (MMI). The Mercalli scale, as modified and revised by Wood and Neumann (1931), is the basis upon which individual scientists, state, and federal agencies evaluate past and present earthquake effects. Full narrative descriptions associated with each intensity level of the Modified Mercalli Intensity Scale can be found elsewhere (e.g. Reiter 1990). Similar to the MSK and MCS scales it consists of twelve intensity levels ranging from I to XII. Roman numerals are used to differentiate intensity from magnitude. Their use serves to emphasize that they define discrete, qualitative levels not given to fractionalization. An increase in intensity describes a more severe effect on what people feel and can observe around them. At MMI I, few, if any, people feel the earthquake. At MMI VII everyone feels it (those inside buildings run outside), some chimneys are damaged and, depending upon the quality of construction and design, building damage ranges from negligible to considerable. At MMI XII, damage is total and the landscape may be permanently distorted. Roughly speaking, the 12 degrees of the macroseismic intensity scale correspond to the following: I An imperceptible disturbance. II A disturbance noticed by only a very few people. III A disturbance perceptible to a number of peoples and sufficiently strong for them to determine the direction and duration. IV A disturbance felt by a number of people indoors. V A disturbance felt by all the inhabitants of the district at night, sleepers are awakened. VI People are sufficiently frightened to leave their houses; slight fall of pebbles and plaster. VII Chimneys fall; cracks develop in the walls of houses. VII Partial destruction of some buildings. IX-XI Severe damage and destruction of buildings XII Total devastation. Figure 1. shows a graphical comparison of the different intensity scales in common use. As can be seen, the MMI and MSK scales are almost the same at all intensity levels.

Fig. 1 A comparison of the seismic intensity scales (after Murphy and O'Brien,

1977; and Richter,1958) 4.1.1 Isoseismals The most important difference between earthquake magnitude and intensity is that earthquake magnitude is determined by quantitatively analyzing instrumental recordings utilizing specific, explicitly defined formulas, while intensity is determined by a largely subjective comparison to a qualitative. narrative scale. In addition, magnitudes defines the strength of the earthquake source while intensity .in almost cases. defines the severity of earthquake effects ata particular location ,regardless of where, or how strong earthquake source is .

Contour lines of equal intensity in a map ,are called isoseismals. These isoseismals bound areas within which the predominant intensity is the same. There may be locations of similar intensity outside the isoseismal and higher or lower intensities may be included within its bounds Isoseismal or intensity maps are drawn' from observed intensities with lines separating regions of different degrees on a map. The first such map seems to have been drawn by P.N. C. Egen for the earthquake of· 1828 in the Netherlands. using his own scale of six degrees (Davison. 1927). Intensity maps, despite

their lack of precision. are a very important means of establishing distributions of ground vibration levels due to earthquakes. These maps have a great deal of information regarding the extent and intensity of shaking of the ground, and the response of buildings and other structures. From the point of view of intensity. the size of an earthquake depends not only on its maximum value but also on the extents of the areas with various degrees of intensity.

The distribution of intensity on the Earth's surface shown on isoseismal maps depends not only on the size of an earthquake but also on its focal depth and the attenuation of the shaking of the ground with distance. If the epicentral intensity is Io. the intensity I at a certain distance ∆., can be expressed by

(4.1)

where h is the focal depth, a is a coefficient related to geometric spreading, and b is related to the anelastic attenuation. Thus, from the intensity map of an earthquake we can estimate its size given by Imax or IO, the macroseismic epicenter. depth of focus, and values of the coefficients a and b, which give information on how intensities are attenuated in the region near the epicenter(the near field).

Despite the lack of precision, the information provided by isoseismal maps is very important and complementary to that provided by analysis of instrumentally recorded seismic waves. For historical earthquakes, this is the only information available. Even for recent earthquakes. this information is very important, especially from the point of view of engineering. The study of earthquakes does not end with the analysis of the shaking of the ground but rather extends also to consideration of damage to buildings and the responses of persons affected by them.

4.2 Magnitude Scales The best way to quantify the size of an earthquake is to determine its seismic

moment, Mo, and the shape of the overall source spectrum. This can be done by

recovering the source time function from either body or surface waves, but this

requires relatively complete modeling of the waveform in question. It is desirable

to have a measure of earthquake size that is much simpler to make, for example,

using the amplitude of a single seismic phase, such as the P wave. Unfortunately,

as we have seen, the amplitude and waveform character of a far-field P or S wave

are proportional to the moment rate; thus different fault dislocation histories with

the same seismic moment can produce very different amplitude signals. Further,

the effects of the time function will depend on the frequency band of observation,

and thus the amplitude of various phases will vary greatly from instrument to

instrument. These limitations aside, measurements based on wave amplitude are

still very useful because of their simplicity and because high-frequency shaking in

a narrow frequency band is often responsible for damage from earthquakes.

The concept of earthquake magnitude, a relative-size scale based on measurements

of seismic phase amplitudes, was developed by K. Wadati and C. Richter in the

1930s, over 30 year before the first seismic moment was calculated in 1964.

Magnitude scales are based on two simple assumptions.

The first is that given the same source-receiver geometry and two earthquakes of

different size, the "larger" event will on average produce larger-amplitude arrivals.

The second is that the amplitudes of arrivals behave in a "predictable" fashion. In

other words, the effects of geometric spreading and attenuation are known in

a statistical fashion. The general form of all magnitude scales is given by

M = log(A/T) + f(∆,h) + Cs + Cr (4.2)

where A is the ground displacement of the phase on which the amplitude scale is

based, T is the period of the signal, f is a correction for epicentral distance (∆) and

focal depth (h), Cs is a correction for the sitting of a station (e.g., variability in

amplification due to rock type), and Cr is a source region correction. The

logarithmic scale is used because the seismic wave amplitudes of earthquakes vary

enormously. A unit increase in magnitude corresponds to a 10-fold increase in

amplitude of ground displacement. Magnitudes are obtained from multiple

stations to overcome amplitude biases caused by radiation pattern, directivity and

anomalous path properties. Four basic magnitude scales are in use today: ML, mb,

Ms and Mw.

4.2.1 Local Magnitude (ML)

The first seismic magnitude scale was developed by C. Richter in early

1930s and was motivated by his desire to issue the first catalogue of California

earthquakes. This catalogue contained several hundred events, whose size ranged

from barely perceptible to large, and Richter felt that an earthquake description

must include some objective size measurement to assess its significance. Richter

observed that the logarithm of maximum ground motion decayed with

distance along parallel curves for many earthquakes. All the observations were

from the same type of seismometer, a simple Wood-Anderson torsion instrument.

The relative size of events is calculated by comparison to a reference event:

(4.3)

where A is the recorded trace amplitude for a given earthquake at a given distance

as written by standard type of instrument, and A0 is that for a particular earthquake

selected as standard. The magnitude is thus a number characteristic of the

earthquake and independent of the location of the recording stations.

Three arbitrary choices enter into the definition of M :

(1) the use of a particular type of seismometer with specific constant,

(2) the use of ordinary logarithms to the base 10,

(3) selection of the standard shock whose amplitudes are represented by A0. This

standard shock has also been called the zero shock, since, if A=A0, M=0. This

clearly does not mean “no earthquake”; a small earthquake might conceivably

record with amplitudes smaller than those of the standard shock, which would give

it a negative magnitude. The zero level was intentionally chosen low enough to

make the magnitudes of the smallest recorded earthquake positive.

Richter chose his reference earthquake ,with ML= 0, as being that which would

theoretically produce a maximum trace, or record (Ao), of 0.001 mm ( one

micron) on a seismogram from the then standard Wood-Anderson seismograph,

100 km from the epicenter.

By using the reference event to define a curve, we can rewrite (4.3) as

ML=log A + 2.76 log ∆ - 2.48 (4.4)

At first glance this equation is not in the form of (4.2), but Richter made a number

of restrictions that can be factored out of (4.2). First, all the instruments used were

narrowband and identical, and thus the maximum amplitude phase was always of a

single dominant period, T. Second, all the seismicity was shallow (less than 15 km

deep), and the travel paths were confined to southern California. Thus the

corrections for regional dependence and focal depth are approximately constant,

and equation (4.4) is actually a particular subset of (4.2).

Earthquakes with ML < 3.0 are called microearthquakes and are rarely felt.

Table 22-1 Logarithms· of the Amplitudes Ao (in millimeters) with which a Standard Torsion Seismometer (T0 = 0.8, V = 2800, h = 0.8) Should Register an Earthquake of Magnitude Zero

∆ (km) .

-logA0 ∆

(km) -logA0 ∆

(km) log A0

0 1.4 150 3.3 390 4.4 5 1.4 160 3.3 400 4.5 10 1.5 170 3.4 410 4.5 15 1.6 180 3.4 420 4.5 20 1.7 190 3.5 430 4.6 25 1.9 200 3.5 440 4.6 30 2.1 210 3.6 450 4.6 35 2.3 220 3.65 460 4.6 40 2.4 230 3.7 470 4.7 45 2.5 240 3.7 480 4.7 50 2.6 250 3.8 490 4.7 55 2.7 260 3.8 500 4.7 60 2.8 270 3.9 510 4.8 65 2.8 280 3.9 520 4.8 70 2.8 290 4.0 530 4.8 80 2.9 300 4.0 540 4.8 85 2.9 310 4.1 550 4.8 90 3.0 320 4.1 560 4.9 95 3.0 330 4.2 570 4.9 100 3.0 340 4.2 580 4.9 110 3.1 350 4.3 590 4.9 120 3.1 360 4.3 600 4.9

*Since A0is less than 1, its logarithm is negative, and the table shows values for -log A0.

The smallest events that are recorded have magnitudes less than zero, and the largest ML recorded is about 7 ( max. magnitude recorded is 9.5), which gives seven orders of magnitude in ground displacement. In practice, ML is usually a measure of the regional-distance S wave. The magnitude for each of the horizontal seismometers is averaged in a least -squares sense to give an ML, for a given station. The values of ML , from each station are averaged to give the "magnitude". ML, may vary considerably from station to station, due not only to station corrections but also to variability in the radiation pattern.

The seismograph used as standard for magnitude determinations of local shocks is a simple type with torsion suspension of the mass developed by H. O. Wood and J. Anderson (or its equivalent). The farther the earthquake source is from the seismograph, the smaller the amplitude of the seismic wave, just as a light appears dimmer as the observing distance from the source increases. Because earthquake sources.

are located at all distances from seismographic stations, Richter further developed a method of making allowance for this attenuation with epicentral distance when calculating the Richter magnitude of an earthquake.

ML, in its original form is rarely used today because Wood-Anderson

torsion instruments are uncommon and, of course, because most earthquakes do

not occur in southern California. However, ML remains a very important

magnitude scale because it was the first widely used "size measure," and all other

magnitude scales are tied to ML. Further, ML is a very useful scale for engineering.

Many structures have natural periods close to that of a Wood-Anderson instrument

(0.8 s), and the extent of earthquake damage is closely related to ML.

130 3.2 370 4.3 140 3.2 380 4.4

4.2.2 Body-Wave Magnitude (mb)

Although the local magnitude is useful, the limitations imposed by instrument

type and distance range make it impractical for global characterization of

earthquake size. Beyond regional distances, where direct P becomes a distinct

phase, it is convenient to define a magnitude based on the amplitude of the P

wave, which is termed mb. This magnitude is based on the first few cycles of the

P-wave arrival and is given by

mb= log(A/T) + Q(h,∆) (4.5)

where A is the actual ground-motion amplitude in micrometers and T is the

corresponding period in seconds. The reason for using the first few swings of the

P wave is that the effects of radiation pattern and depth phases can result in a

complicated waveform signature. In practice, the period at which mb is usually

determined is 1s (the WWSSN and many regional network, short-period

instruments have a "peaked" response near 1 Hz). It is not unusual to have scatter

of the order of ±0.3 for individual mb measurements for a given event, requiring

extensive averaging. Occasionally, long-period instruments are used to determine

body-wave magnitude for periods from 5 to 15s, and these are usually referred to

as mB. When mB is measured, it is usually for the largest body wave (P, PP, etc.).

The correction for distance and depth Q (h.∆) is determined empirically. The

following figure shows values for Q(h,∆); note that the corrections are fairly

uniform beyond 30° but are complex at upper-mantle distances. This reflects the

complexity of the body waves in this epicentral distance range. The correction

dramatically decreases at 20° because the upper-mantle triplications result in very

large amplitude arrivals.

4.2.3 Surface-Wave Magnitude(Ms)

Beyond about 600 km the long-period seismograms of shallow earthquakes

are dominated by surface waves, usually with a period of approximately 20s. The

amplitude of these waves depends on distance differently than the amplitude of

body waves, and surface-wave amplitudes are strongly affected by the source

depth. Deep earthquakes do not generate much surface-wave amplitude, and thus

there is no appropriate correction for source depth. The equation for surface-wave

magnitude is given by

MS = log A20 + 1.66 log ∆ + 2.0 ..……… .(4.6)

where A20 is the amplitude of the 20-s-period surface wave in micrometers. In

general, the amplitude of the Rayleigh wave on the vertical component is used in

equation (4.6).

Both MS and mb were designed to be as compatible as possible with ML; thus

at times all three magnitudes give the same value for an earthquake.

Unfortunately, this is rarely the case. In these magnitudes we are making a

frequency-dependent measurement of amplitudes at about 1.2, 1.0, and 0.05 Hz

for ML, mb, and Ms, respectively. If we consider the source spectrum of a seismic

pulse, it is easy to see that only for small earthquakes (very short fault lengths)

with corner frequencies well above 1 Hz will the amplitude be the same for all

three frequencies. For earthquakes above a certain size, the frequency at which

we measure mb will be located on the ω-2 decay slope, and thus all earthquakes

above this size will have a constant mb. This is called magnitude saturation.

Figure 2 illustrates one model of the variation of source spectra for different size

earthquakes, and it is apparent that mb begins to saturate at approximately mb =

5.5 and is fully saturated by mb = 6.0, whereas Ms does not saturate until

approximately Ms = 7.25 and is fully saturated by Ms = 8.0. There are many

examples of reported mb, larger than 6.0, which implies that these particular

source spectra are not valid for all events. It is clear that ML begins to saturate at

about magnitude 6.5. It is desirable to have a magnitude measure that does not

suffer from this saturation deficiency.

Fig. 2. Spectra for different sized earthquakes and the relationship of these

spectra to the frequencies at which surface wave magnitude and body wave magnitude are determined.

4.2.4 Moment Magnitude(Mw)

Given the limitations of the various magnitude scales, it would be beneficial if a

single unified magnitude scale applicable to earthquakes of all sizes, depths, and

locations could be developed. Such a magnitude scale has been developed and it

stems from the concept of seismic moment. Geologically, seismic moment (Mo) is

a description of the extent of deformation at the earthquake source. It is simply

defined as:

MO = μ A D (4.7)

where μ is the rigidity modulus (resistance to shearing motion), A is the fault

rupture area and D is the average dislocation or relative movement (slip) between

the opposite sides of the fault. Seismologically, moment is measured at epicentral

distances much larger than, and using wavelengths much longer than, the

dimensions of the earthquake fault rupture. It can readily be determined from

seismograms by techniques that utilize long-period seismic waves or the long-

period (low frequency) end of the spectrum (Bullen and Bolt. 1985).

The advantages of seismic moment over body or surface wave amplitudes used

in other magnitude measurements is:

i) that moment is directly related to the size of the earthquake source

and, specifically to quantities which are often measurable in the

field.

ii) In addition, because it is determined from very long period seismic

waves it can be used to quantify even the largest earthquake.

iii) It does not rely upon a single wave or wave type which may or

may not be observed depending on earthquake depth or epicentral

distance.

iv) Its disadvantage is that, in contrast to other parameters it cannot be

measured directly from the seismogram without some additional

analysis.

For convenience, and in order to be consistent with past practice, Kanamori (1977)

and Hanks and Kanamori (1979) devised a moment magnitude scale (Mw or M) as

Mw = (Log Mo -16.05)/1.5 (4.8)

or

MW = log 10 M0 / 1.5 – 10.7 (4.9)

where M0 is the seismic moment of the earthquake in dyne-cm. The moment

magnitude has the advantage (as a measure of size in earthquakes) that it does not

saturate at the top of the scale and has a sounder theoretical basis than Ms and mb .

4.2.5 Duration/Coda Magnitude

Recently, many local and regional networks have been using magnitude

scales based on the total duration of the measured seismic wave train rather than

the amplitude of a specific wave of wave type. which may be "clipped" or not

recorded completely. This wave train consists of the usual body and surface waves

followed by a seemingly disorganized collection of seismic waves that eventually

disappear into the background noise. This latter part of the wave train is called the

coda and is believed to be made up of waves arriving at a seismograph station after

being scattered or reflected off lateral variations in the earth's structure. Aki in

1969 developed a coda magnitude (MC) that could be obtained from those

characteristics. The duration magnitude ( MD ) which is based on the total

duration of the earthquake, can be used to describe small earthquakes that are often

of more interest to seismologists than engineers. Duration magnitudes (MD) or coda

magnitudes (Mc) are very useful in measuring local earthquakes because they are

not dependent on any single seismic wave and can be used when peak amplitudes

go off-scale and cannot be recorded at a seismograph station.

4.3 Comparison of Different Magnitude Scales

When initially developed, these magnitude scales were considered to be equivalent; in other words, earthquakes of all sizes were thought to radiate fixed proportions of energy at different periods. But it turns out that larger earthquakes, which have larger rupture surfaces, systematically radiate more long-period energy. Thus, for very large earthquakes, body-wave magnitudes badly underestimate true earthquake size; the maximum body-wave magnitudes are about 6.5 - 6.8. In fact, the surface-wave magnitudes underestimate the size of very large earthquakes; the maximum observed values are about 8.3 - 8.7.

It is important to note how the different magnitude scales saturate, or stop

increasing with increasing earthquake size or moment. This occurs because each

magnitude scale, except moment magnitude, is determined using a seismic wave

of a particular period and wavelength. Seismic waves, whose wavelengths are

much smaller than the earthquake source, do not increase in amplitude as the

earthquake source size, moment, and energy release increase. Thus mb, which

uses P waves of about one second period and less than 10 km wavelength cannot

really reflect the energy release or deformation from faults whose rupture

dimension is tens of kilometers or greater. mb saturates at about magnitude 6.5.

Similarly MS. which uses surface waves of about 20 seconds period and 80 km

wavelength, cannot really reflect the energy release or deformation from faults

whose rupture dimension is many hundreds of kilometers long. Ms saturates at

about magnitude 8.5. Except for MS less than about magnitude 5.5, all the

magnitude scales approach, and become approximately equal to, moment

magnitude below their respective saturation points. For engineering purpose

where the catalogues are used for the estimation of hazard or potential of the

seismogenic sources, the conversion of different scales of magnitude is required

for homogenization of magnitude in its size on same scale. Fig. 3 shows the

relationship of the different magnitude scales.

Fig. 3 The relation among various magnitude scales.

Saturation explains the observation that earthquakes of obviously different sizes

and energy releases often have the same magnitude. The 1906 San Francisco

earthquake and the 1960 Chile earthquake both have estimated surface wave

magnitudes of about 8.3. Yet, while the 1906 earthquake rupture was confined to

a long, narrow fault segment believed to be about 5800 square km in area, the

1960 earthquake, the largest in this century, was associated with a fault rupture

some 35 times greater in area, equivalent in size to about one half of the whole

state of California. When the moment magnitude is computed, it turns out that the

1906 earthquake is "only" about magnitude eight while the 1960 earthquake has a

moment magnitude of 9.5. Similarly, with the above definitions, the 1964 great

Alaskan earthquake has the estimated values: MS=8.4, M0 =820 X 1027 dyne- cm,

MW =9.2.

For most moderate shallow focus damaging earthquakes, it is sufficient for

engineering purposes to take ML, MS and MW to be roughly the same.

4.4 Magnitude and Intensity Correlation.

Because of the subjective nature of intensity determination, the discrete character of the intensity scale, and a poor correlation of intensity with specific earthquake source characteristics and recorded ground motion, methods have been developed to derive earthquake magnitude from earthquake intensity data. Once an earthquake magnitude is estimated, some of the physical insight and quantitative techniques developed from the evaluation of instrumentally recorded earthquakes may be used. For example, a well calibrated magnitude estimate of a pre-instrumental earthquake allows a comparison with recent, excellently recorded earthquakes of the same source strength. This can be done without worrying about the bias introduced by the large variations in intensity due to different geologic and soil conditions. Simple methods to convert intensity data into magnitude have been around almost

as long as the magnitude scale itself. The original method for converting intensity to magnitude relied upon assuming a correlation between magnitude and maximum (Imax) or, more likely, epicentral (I0) intensity. Gutenberg and Richter (1956), for example, found that local magnitude (ML could be related to I0 by: ML = 2/3(IO) + 1 (4.10)

where the symbol M indicates that it is an estimated magnitude rather than a measured one. Such a relationship would associate an epicentral intensity (MMI) of VI with a magnitude of 5.0, IX with 7.0 and any unit change in the epicentral intensity with a concurrent change of 0.67 in the associated magnitude. Equation (4.10) was derived by comparing measured earthquake magnitudes in southern California with epicentral intensities. Because different magnitude scales rely upon different seismic waves and may not be equivalent and intensity estimation may vary from region to region, the relation between ML and I0 in southern California is not necessarily applicable to other magnitude scales and other regions. Richter believed that the intensity as associated with any particular location should be based on the mode of the observations rather than the single highest one. In any case, correlations between epicentral intensity and magnitude are not particularly accurate means for estimating magnitude. Table. 1 depicts approximate correlation of magnitude, maximum intensity and mean epicentral distance of perception as foundfor California.

Table.I Approximate Correlation of Magnitude, Maximum Intensity and Mean Epicentral Distance of Perception as Found for California (After Richter)

Magnitude 2 3 4 5 6 7 8 Maximum Intensity I-II III V VI/VII VII/VIII IX/X XI Radius(km) 0 15 80 150 220 400 600

4.5 Seismic Energy and Magnitude Although the magnitude scales discussed above give a means of comparing earthquakes, the total size of an earthquake is best represented by the seismic moment, M0. An alternative measure of earthquake size would he energy released. We calculate energy by considering the history of a particle as it responds to a transient seismic wave field. As a wave passes, the particle, which has a potential energy, will have a velocity and thus a kinetic energy. The sum of the potential and kinetic energies integrated over time will yield the work, or the energy expended. As an example, consider a seismic station situated directly above a monochromatic source of seismic energy. The displacement of the ground at the seismic station will be given by

X= A cos (2πt/T) (4.11) where A is the amplitude of the wave of period T. The ground velocity is given by υ = - (2πA/T) sin (2πt/T) (4.12) The kinetic energy of a unit mass at a recording station is just given by 1/2ρυ2. If we average this over one complete cycle, we obtain the kinetic energy density:

(4.14)

Note that the energy density is proportional to A2, which is the expected result. Since the mean potential and kinetic energies are equal, we can write E = 2e. If we integrate over the spherical wave front to correct for geometric spreading, we obtain an equation of the form

E = F(r,ρ,c)(A/T)2 (4.15)

where r is the distance traveled, p is the density, and c is the velocity of the wave type. This equation can be recast in a form similar to the general equation for magnitude scales:

log E = log F(r,ρ,c) + 2 log(A/T) (4.16)

Thus it is possible to relate energy to magnitude if F(r, p ,c) is known. Gutenberg and Richter found empirical relationship for mb, and Ms:

log E = 5.8+2.4mb (4.17)

log E = 11.8+1.5 Ms (4.18)

Obviously, the energy calculation in equations. (4.17) and (4.18) suffers from all the problems of the magnitude determination. In particular, since mb, saturates, the estimate of E using (4.17) for any earthquake larger than about magnitude 6.5 is probably low. Equation (4.18) is fairly robust, since Ms does not saturate until very large magnitudes are reached. Equation (4.18) gives an interesting insight into the tremendous range of earthquake size. The difference between the energy released an Ms = 6.0 and an Ms= 7.0 earthquake is a factor of 101.5, or - 32. In other words, the seismic energy released in a magnitude 7.0 earthquake is over 30 times greater than that released in a magnitude 6.0 earthquake, and it is three orders of magnitude greater than that released in a Ms = 5.0 earthquake.

It is also possible to relate seismic moment to the seismic energy. Kostrov (1974) showed that the radiated seismic energy is proportional to the stress drop:

(4.19)

or, rearranging terms using the definition of Mo,

(4.20)

We can use this expression to relate M0 to magnitude through (4.18). If we assume that stress drop is constant and equal to about 30 bars, this yields the relation

log M0 = 1.5 Ms + 16.1 (4.21)

This equation gives a simple way to relate magnitude to seismic moment and, in fact can be used to define a new magnitude scale, MW called the moment magnitude: Mw = (log MO/1.5) - 10.73 (4.22) This scale, derived by Kanamori (1977), is tied to Ms but will not saturate because Mo does not saturate. Generally, determination of Mo is much more complicated than magnitude measurement, although modern seismic analyses are routinely providing Mo for all global events larger than Mw = 5.0. The largest earthquake recorded this century was the 1960 Chilean earthquake, with Mw = 9.5.

4.6 Intensity and Acceleration

With the availability of instrumental data on ground acceleration in regions of different intensities for numerous earthquakes, empirical relations between intensity and ground acceleration have been developed. One such relation is:

log a = I/3 – ½

where a is the acceleration in cm/sec2 and I is the Modified Mercalli intensity. If one lets I = 1 .5, represent the limit of perceptibility between intensities I and II, log a = 0 or a = 1 cm/sec2

Various lines evidence point to this as the level of shaking ordinarily perceptible to persons. If one lets I = 7.5, log a = 2 or a = 100 cm/sec2 = 0.1g approximately. Thus the seismic intensity VII to VIII will approximately represent a ground acceleration of 10% g. This is the acceleration commonly accepted by engineers as that which damages ordinary structures not designed to be earthquake resistant. Through such estimates it is possible to convert isoseismal maps into iso-acceleration and iso-force maps for engineering applications.

Ground shaking may cause sliding of objects if the horizontal component of ground acceleration is able to exert a force larger than the inertia of the object. Observations on sliding, rocking and overturning of objects with measurements of dimensions of the object, its weight and friction, could allow a quantitative estimate of ground acceleration. This would be particularly useful in regions where instrumental data on ground acceleration are not recorded.