AIRCurrents
Measuring Earthquake
Magnitude
01.2008
Editor’s note: One of the many challenges facing the catastrophe modeler
is reconciling the magnitudes of historical earthquakes that were recorded
using a wide variety of different magnitude scales. Indeed, different sources
will publish different magnitudes for the same event. For the lay person, it
can be pretty confusing. In this article, AIR’s Dr. Mehrdad Mahdyiar helps
us make sense of it all. He discusses the history of the various scales, their
advantages and limitations, and why the scientific community seems finally
to have settled on one: the moment magnitude scale.
by Dr. Mehrdad Mahdyiar, Director of Earthquake Hazard, Research and
Modeling, AIR Worldwide
When an earthquake occurs, the first thing people want
to know is its magnitude and location—the two most
important pieces of information that characterize both the
event and its damage potential. Several magnitude scales
are currently in use. However, the one that the lay-person is
perhaps most familiar with is long-lived Richter scale. Indeed,
the popular press frequently, but mistakenly, labels reported
earthquake magnitudes as Richter magnitudes. From a
scientific point of view many advances have taken place
since Richter proposed his scale in 1935.
Recognition of the various magnitude scales used in
historical catalogs and by seismologists today, and the
proper normalization across them is critical to an appropriate
assessment of seismic risk. In this article we discuss the
scales that are used in the scientific literature and draw
the reader’s attention to the one that is today most widely
accepted as an objective and physically sound measure of
earthquake magnitude.
Interpreting the Historical Record
Historical earthquake catalogs are the compilation of
the time, geographic locations, and magnitudes of past
earthquakes. This information provides seismologists
with valuable insight into regional seismicity for hazard
analysis. However, a glance through any historical catalog
quickly reveals the difficulties seismologists face. Under the
magnitude heading will be found terms like ML, mb, mB, MS,
and MW, among others. Each represents a different approach
to describing the size, or strength, of earthquakes.
Earthquakes are complex phenomena, and many different
characteristics can be used to measure their strength. In
general, an earthquake can be characterized either by a set
of parameters that describes what happened underground,
at the source, or by parameters that define its effects on the
surface. Examples of the first category include rupture length
and width, the average relative displacement across the
fault surface, and the amount of stress relaxation that the
rupture area experiences after the earthquake. Examples of
the second category are ground displacement, velocity and
acceleration, and damage to buildings and infrastructure.
With the exception of the moment magnitude scale, MW,
all magnitude scales are based on parameters in the second
category. After all, measuring the effects of an earthquake
on the surface is much simpler than trying to quantify
what happened underground—sometimes many miles
underground. Each of these magnitude scales in the second
category attempt to measure the strength of earthquakes
by quantifying the different types of seismic waves they
generate.
AIRCurrents
01.08|Measuring Earthquake Magnitude
by Dr. Mehrdad Mahdyiar
Early Magnitude Scales
Richter Magnitude
The first seismometer in North America was installed in 1897
at the Lick Astronomical Observatory on Mount Hamilton
near San Jose, California. By 1935, when Charles Richter—a
young physicist turning seismologist at the California
Institute of Technology—developed the famous Richter
magnitude scale, a number of instruments invented by Harry
O. Wood and J.A. Anderson were already in operation in
different parts of the state.
Richter wanted to publish the first earthquake catalog for
California and felt the need to assign some sort of metric
to earthquakes to quantify their strength. Following an
earlier 1931 study by Kiyoo Wadati in Japan, he plotted the
peak ground motions (PGM) of few recorded earthquakes
versus distance to see if he could identify a pattern. He
and a colleague Hugo Gutenberg observed similarities
between earthquakes in the rate of decay of PGM with
distance. However, since PGM itself could be quite different
from one earthquake to another, they plotted PGM on a
logarithmic scale and constructed an empirical relationship
that statistically captured the observed pattern of PGM
decay with distance. As a result of their research, the Richter
magnitude scale ML, which stands for local magnitude, was
defined as:
ML = log (A) – log (A0)
where A is the recorded PGM on the Wood-Anderson
seismometer and A0 is the PGM of a reference calibrating
earthquake of “zero magnitude.” Richter arbitrarily defined
a zero magnitude earthquake as one that creates a 1/1000
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1
0
-1
log (A, A0)
Before the invention and deployment of seismometers—
instruments that measure and record motions of the
ground—the only available information on earthquakes
came from reports of the damage they caused. Not
surprisingly, therefore, the first scales were developed
by quantifying the damage and scaling earthquakes
accordingly. Two such scales were the Rossi-Forel and
Mercalli intensity scales that were developed in 1880 and
1902, respectively. The history of the first earthquake
instrument goes back to the Chinese scholar Zhang Heng, in
about 132 A.D. However, the first modern instrument that
was able to record ground motions as a function of time
was developed by John Milne in the early 1890s while a
professor of mining and geology at Tokyo’s Imperial College
of Engineering.
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ML = log(A) - log (A0)
(.001 mm) -3
Zero magnitude
reference earthquake
-4
-5
0
100
200
300
400
500
600
distance, km
mm PGM at 100 km distance on the Wood-Anderson
seismometer in southern California. Figure 1 shows the plot
of the PGM used by Richter for constructing the magnitude
scale. The ML scale is a logarithmic scale in the sense that a
unit increase in magnitude represents a tenfold increase in
PGM.
A reconstruction of Richter’s plot of peak ground motions on
the Wood-Anderson seismometer for selected earthquakes
in Southern California. The dotted line represents the
reference earthquake—one that creates PGA of 1/1000 mm
at a distance of 100 km. The information on this plot was
used to construct the famous Richter magnitude scale.
Interestingly, the sensitivity of seismometers to different
kinds of earthquakes depends on the resonant oscillation
period of the mass and spring that comprises the recording
instrument itself. The recording instrument within the
Wood-Anderson seismometer, for example, has a natural
period of 0.8 seconds. However, it is the very 0.8 s natural
period of the Wood-Anderson instrument that limits its
sensitivity to the long-period seismic waves produced by
very large (or deep, or distant) earthquakes. As a result, the
Richter magnitude scale “saturates”, preventing the proper
measurement of the magnitude of large earthquakes.1
The concept behind the Richter magnitude scale was
widely adopted in many parts of the world. One of its
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AIRCurrents
01.08|Measuring Earthquake Magnitude
by Dr. Mehrdad Mahdyiar
main attractions was that the natural period of the WoodAnderson seismometer, at 0.8 s, is close to the natural
period of many buildings, i.e. about 1 s, and is thus is a
good measure of building response. Presently, however, no
Wood-Anderson instrument remains in operation and ML
in its original form is no longer reported. In cases where ML
magnitudes are still calculated the analyses are based on
the simulation of Wood Anderson seismograms by other
recording instruments.
Beyond Richter
Charles Richter developed his magnitude scale in California
using recorded PGM of shallow, crustal California
earthquakes and a distance range of up to 600 km. Beyond
600 km, earthquake ground motions are dominated by
surface waves with a dominant period of about 20 seconds.
The formation and the propagation of surface waves
through the Earth’s crust are different than those of the
body waves, that is, the so-called P waves and S waves that
are the first set of wave trains that reach a seismic station.
The amplitudes of the surface waves are very sensitive to the
depth of the earthquake. For example, deep earthquakes
do not create large-amplitude surface waves. In 1945,
Gutenberg proposed two new magnitude scales to address
the issue of scaling distant shallow and deep earthquakes.
The surface wave magnitude scale, MS, was introduced
to measure the size of distant shallow earthquakes based
on the amplitude of 20-second period surface waves. The
body wave magnitude scale was introduced to measure the
size of distant shallow or deep earthquakes based on the
amplitude of direct or reflected P waves. When the recording
instrument itself is characterized by a short natural period—
about 1 second—the body wave scale is denoted as mb and
when the instrument is a long period one—about 5 to 15
seconds—the scale is denoted as mB.
In practice, the magnitude of an earthquake is estimated as
the statistical average of a set of recorded magnitudes at
many different stations. Often there are large variations in
the magnitude estimates, reflecting the variations in seismic
radiation in different directions, the regional geological
complexities along the source-to-station propagation path,
and the local soil conditions that may amplify or de-amplify
the ground motions.
The Modern Magnitude Scale
As has been discussed, each of the magnitude scales
discussed so far measure the strength of an earthquake by
scaling its different types of ground motions with different
dominant period. This makes all of these magnitude scales
period, or frequency, dependent. This takes us back to the
saturation problem. That is, different magnitude scales
saturate, or stop increasing with increasing earthquake size.
Generally, the shorter the dominant period of the ground
motions used to develop the magnitude scale, the sooner
it saturates with increasing the magnitude. For example,
ML saturates sooner than MS since it scales an earthquake’s
magnitude with ground motions having a dominant period
of about 0.8 s compared to 20 s dominant period for MS.
A more scientifically sound measure of the strength of an
earthquake is the energy it radiates. However, since energy
is calculated based on the ground motion it would have the
same saturation problem as the magnitudes. Nevertheless,
a number of empirical relationships have been developed
to correlate seismic energy with different magnitude scales.
For example, Gutenberg and Richter derived the following
relationship for estimating energy from MS:
log (E) = 11.8 + 1.5 * MS
where E is in ergs. It is interesting to note that a unit increase
in magnitude translates to a 31.6 times increase in radiated
seismic energy.
In 1966, Keiiti Aki, a prominent Japanese seismologist,
formulated the concept of seismic moment, M0,as a measure
of an earthquake source strength. Aki defined seismic
moment as the product of (1) an earthquake’s displacement
averaged over its rupture area, (2) the size of its rupture
area, and (3) the shear modulus (a metric for quantifying the
strength of materials) of the rocks near its rupture surface.
In related research, B.V. Kostrov showed in 1974 that the
radiated seismic energy is proportional to the average stress
drop and average displacement over the fault surface.
In 1977, Hiroo Kanamori, using Aki’s seismic moment
concept and Kostrov’s seismic energy-moment relationship,
introduced a new magnitude scale, called moment
magnitude MW. This new magnitude scale has been
resoundingly embraced by the seismological community
primarily because it is based on a source parameter (M0) that
is not frequency dependent, solving the saturation problem.
Thus earthquakes like the magnitude 9.1-9.3 December
2004 Indonesia quake, which could not have been measured
with any accuracy on the Richter scale, can today be
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AIRCurrents
01.08|Measuring Earthquake Magnitude
by Dr. Mehrdad Mahdyiar
quantified with significant reliability—though even here
there remains some uncertainty over the exact magnitude
of an earthquake this size and thus the range of magnitude
typically cited for this event.
Presently, virtually all seismological institutes worldwide
report the magnitudes of earthquakes in terms of moment
magnitude, MW. Having said that, it is interesting to note
that in Japan, where since 1926 the responsibility for
publishing earthquake parameters has fallen to the Japan
Meteorological Agency (JMA), earthquake magnitudes are
measured using the JMA magnitude scale, which is based on
the maximum ground displacement (for large earthquakes)
and/or velocity (for smaller events). Studies have shown that
the difference between MJMA and MW is less significant for
the subduction type earthquakes as compared to the crustal
earthquakes. Recently, because of the ubiquitous use of MW,
the JMA has begun issuing magnitude estimates using both
scales.
In Conclusion
In this article we presented a brief review of the history
and the general concept behind the major earthquake
magnitude scales. As discussed, during the second half
of the last century different magnitude scales have been
devised as seismometers became available to record different
types of waves forms. The motivation for developing these
different scales was the very complexity of the earthquake
phenomenon, which cannot be characterized and scaled
by a single parameter, such as the peak ground motion it
generates. In search for a universal parameter to define
earthquake strength, seismologists introduced the concept
of seismic moment and moment magnitude. MW is a
stable magnitude that does not saturate, since seismic
moment, the basic underlying parameter, is based on source
parameters that are not frequency dependent.
However, while the introduction of MW represents a major
step in unifying the process of measuring the strength of
earthquakes, MW alone cannot capture the many different
characteristics of an earthquake, particularly as it relates to
damage potential at different sites. We need also to know
the stress drop at the source, the velocity by which the
displacement on the fault surface reaches maximum, the
detail of the fault slip distribution over the rupture area, and
the detail of the regional and local site effects, among other
things. These are all important parameters that, in one way
or another, characterize the complex ground motion signals
that are recorded at each of the seismic stations affected by
an earthquake. In other words, the damage potential from
earthquakes of similar MW can be very different because
of these other complexities. That is why, in a catastrophe
model, earthquake magnitude is only one of many
parameters needed to reliably estimate damage and loss.
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1 Mathematically, there is no upper or lower bound for ML. Very sensitive instruments can record very low levels of ground
motion that translate to negative magnitudes. However, because the Wood Anderson seismometer is sensitive to the ground
motions with a period of 0.8 seconds, the ML upper bound is limited by the radiation intensity of the seismic waves of this
period range. This limitation is known as the magnitude saturation problem.
©2008 AIR Worldwide Corporation. All rights reserved. 4