peat8002 - seismology lecture 13: earthquake magnitudes...

PEAT8002 - SEISMOLOGYLecture 13: Earthquake magnitudes and

moment

Nick Rawlinson

Research School of Earth SciencesAustralian National University

Earthquake magnitudes and momentIntroduction

In the last two lectures, the effects of the source ruptureprocess on the pattern of radiated seismic energy wasdiscussed.However, even before earthquake mechanisms werestudied, the priority of seismologists, after locating anearthquake, was to quantify their size, both for scientificpurposes and hazard assessment.The first measure introduced was the magnitude, which isbased on the amplitude of the emanating waves recordedon a seismogram.The idea is that the wave amplitude reflects the earthquakesize once the amplitudes are corrected for the decreasewith distance due to geometric spreading and attenuation.

Earthquake magnitudes and momentIntroduction

Magnitude scales thus have the general form:

M = log[

AT

]+ F (h,∆) + C

where A is the amplitude of the signal, T is its dominantperiod, F is a correction for the variation of amplitude withthe earthquake’s depth h and angular distance ∆ from theseismometer, and C is a regional scaling factor.Magnitude scales are logarithmic, so an increase in oneunit e.g. from 5 to 6, indicates a ten-fold increase inseismic wave amplitude.Note that since a log10 scale is used, magnitudes can benegative for very small displacements. For example, amagnitude -1 earthquake might correspond to a hammerblow.

Earthquake magnitudes and momentRichter magnitude

The concept of earthquake magnitude was introduced byCharles Richter in 1935 for southern Californiaearthquakes.He originally defined earthquake magnitude as thelogarithm (to the base 10) of maximum amplitudemeasured in microns on the record of a standard torsionseismograph with a pendulum period of 0.8 s,magnification of 2800, and damping factor 0.8, located at adistance of 100 km from the epicenter.This standard instrument, known after its designer as theWood-Anderson seismometer, consists of a small coppercylinder attached to a vertical metal fiber. The restoringforce is supplied by tension in the fiber.


The picture on the leftshows a short periodWood-Anderson torsionseismometer.The instrument as a wholeis sensitive to horizontalmotions, which aredetected via light reflectedfrom a small mirror in thecylinder.The magnitude scaledevised by Richter is nowreferred to as the localmagnitude ML.


In practice, the scale requires different calibration curvesfor regions such as stable continental interiors, ascompared to the Southern California region for which thescale was originally defined.This is because the attenuation of seismic waves withdistance can be very different for different geologicalprovinces.The magnitude of the largest arrival (often the S-wave) ismeasured and corrected for the distance between sourceand receiver, given by the P- and S-wave differential arrivaltimes. The scale for Southern California is defined by:

ML = log A + 2.76 log ∆− 2.48


Richter magnitudes in their original form are no longerused because they only apply to Southern California andthe Wood-Anderson seismograph is now rarely used forrecording the seismic wavefield.However, local magnitudes are sometimes still reported,because many buildings have resonant frequencies near 1Hz, which is close to that of a Wood-Andersonseismograph. Therefore, ML is often a good indicator of thepotential for structural damage.A number of different global and local magnitude scaleshave been produced since the original formulation ofRichter. Some of these will now be discussed.

Earthquake magnitudes and momentBody and surface wave magnitudes

For global studies of teleseismic events, the two primarymagnitude scales that have traditionally been used are thebody wave magnitude, mb and the surface wavemagnitude, Ms.mb is measured from the early portion of the body wavetrain, usually that associated with the P-wave, and isdefined as:

mb = log[

AT

]+ Q(h,∆)

In this case, A is the ground motion amplitude in micronsafter the effects of the seismometer are removed, T is thewave period in seconds, and Q is an empirical term that isa function of angular distance and focal depth.Q can be derived as a global average or for a specificregion.


The plot below shows an estimate of the Q-factor for bodywave magnitude mb derived from earthquakes in the Tongaregion.


Measurements of mb depend on the seismometer usedand the portion of the wave train measured.Common US practice is to use the first 5 s of the record,and periods less than 3 s (usually about 1 s), oninstruments with a peak response of about 1 s.mb is usually measured out to a distance of 100◦, beyondwhich core diffraction has a complicated effect on theamplitude.The surface wave magnitude, Ms, is measured using thelargest amplitude (zero to peak) of the arriving surfacewaves. Gutenberg and Richter first devised a scale forteleseismic surface waves in 1936.


This was developed more extensively by Gutenberg(1945), who devised the formula

Ms = log A + 1.656 log ∆ + 1.818

where A is the horizontal component of the maximumground displacement (in microns) due to surface waveswith periods of 20 s.Many formulae for Ms have been proposed since that ofGutenberg. Vanek (1962) devised the formula:

Ms = log[

AT

]max

+ 1.66 log ∆ + 3.3

which has been officially adopted by the InternationalAssociation for Seismology and Physics of the Earth’sInterior (IASPEI).


The (A/T )max term is the maximum of all A/T values ofthe wave group on a record.If Rayleigh waves with a period of 20s are used, whichoften have the largest amplitudes, the above expressionreduces to

Ms = log A20 + 1.66 log ∆ + 2.0

noting that in general log[X/Y ] = log X − log Y , andlog 20 = 1.3.As in the expression for mb, A is the ground motionamplitude in microns after the effects of the seismometerhave been removed.

Earthquake magnitudes and momentLimitations

As measures of earthquake size, magnitudes have twomajor advantages. First, they are directly measured fromseismograms without sophisticated signal processing.Second, the estimates they yield are intuitively meaningful(magnitude 5 is moderate, magnitude 6 is strong etc.).However, magnitudes also have several related limitations.First, they are totally empirical, and thus have no directconnection to the physics of the earthquake. Theequations used are not even dimensionally correct (A/T isnot dimensionless, yet the logarithm of this quantity is stilltaken).A second problem is with consistency; magnitudeestimates vary noticeably with azimuth, due partly to thesource radiation pattern.


Different magnitude scales also yield different values, andbody and surface wave magnitudes do not correctly reflectthe size of large earthquakes.This is demonstrated in the table below, which showssignificant discrepancies between mb and Ms

Loma Prieta, 1989

Truckee, 1966

San Fernando, 1971

San francisco, 1906

Alaska, 1964

Chile, 1960

Earthquake Fault area (km )(lengthxwidth)

10x10

20x14

40x15

450x10

500x300

800x200

Averagedislocation (m)

0.3

1.4

1.7

4

7

21

Surface wavemagnitude

5.9

6.6

7.1

7.8

8.4

8.3

Body wavemagnitude

mb

5.4

6.2

6.2

6.2

Moment(dyn−cm)

1.2x10

3.0x10

5.4x10

5.2x10

2.4x10

8.3x1024

26

26

27

29

30

MomentMagnitude

5.9

6.7

6.9

7.8

9.1

9.5

Ms

M0

Mw

2


The earthquakes with moments greater than the SanFernando earthquake all have mb = 6.2, even as themoment increases by a factor of 20,000.Similarly, the earthquakes larger than the San Franciscoearthquake have Ms ≈ 8.3, even as the moment increasesby a factor of 400.This effect, called magnitude saturation, is a generalphenomenon for approximately mb > 6.2 and Ms > 8.3.In the above table, the 1906 San Francisco earthquakeapproximately represents the maximum size of continentaltransform earthquakes.However, the Alaska and Chilean earthquakes had muchlarger rupture areas because they occurred on shallowdipping subduction thrust interfaces.


Faults associated with subducting slabs can have widths of100s of km on which strain can build up and eventually bereleased seismically.The larger fault dimensions give rise to greater slip, so thecombined effects of larger fault area and more slip causethe largest earthquakes to occur at subduction zonesrather than transforms.Uncertainties in earthquake magnitude estimates haveseveral sources. First, the Earth’s seismic structureexhibits significant lateral variations, particularly at shallowdepth.Estimation techniques used to compute magnitudes varyover time (hence pre-1964 earthquakes do not have mbvalues).


Different techniques (body wave, surface wave, geodetic,geological) can yield contrasting estimates of magnitude.Fault dimensions and dislocations are average values forquantities that can vary significantly along the fault.All of these effects are understandable given thatamplitudes depend on the scalar moment, the azimuth ofthe seismometer relative to the fault geometry, the distancefrom the source, and the source depth.In addition, because the source time function has a finiteduration, depending on fault dimensions and rise time, theamplitudes vary with frequency.

Earthquake magnitudes and momentMoment magnitude

The moment magnitude Mw has several advantagescompared to the magnitude estimates discussed so far. Itis defined by:

Mw =log M0

1.5− 10.73

where M0 is in dyn-cm (1 dyn=10−5 N).Recall that the seismic moment was defined by:

M0 = µDS

where µ is the shear modulus of the rocks involved in theearthquake, S is the rupture area, and D is the averagedisplacement or slip on S.

Earthquake magnitudes and momentMoment magnitude

The main benefit of Mw is that the magnitude is directlytied to earthquake source processes that do not saturate.It turns out that Mw is comparable to Ms until Ms saturatesat about 8.2.The largest recorded earthquake, the 1960 Chile event,had Mw = 9.5. Moment magnitude has become thecommon measure of the magnitude of large earthquakes.Estimation of M0 and hence Mw requires more analysis ofseismograms compared to mb and Ms. However,semi-automated programs like the Harvard CMT projectnow regularly compute moment magnitudes for mostearthquakes larger than about Mw = 5.

Earthquake magnitudes and momentMagnitude saturation

The reason for Ms and mb saturating at large magnitudescan be traced back to the source spectrum of anearthquake. The plot below shows a theoretical sourcespectrum.

TR and TDcorrespond tothe ruptureand rise timesrespectively.fc is referredto as thecornerfrequency.


The above source spectrum assumes that the source timefunction is the convolution of two “boxcar" functions,corresponding to the finite length of the fault and the finiterise time of the faulting at any point.It turns out that the spectral amplitude (obtained byapplying the Fourier transform) of the source signal isgiven by:

|A(ω)| = M0

∣∣∣∣sin(ωTR/2ωTR/2

∣∣∣∣ ∣∣∣∣sin(ωTD/2ωTD/2

∣∣∣∣If the logarithm of both sides is now taken, then:

log A(ω) = log M0 + log[sinc(ωTR/2)] + log[sinc(ωTD/2)]

noting that in general log XY = log X + log Y , andsincX = 1 if X = 0 but otherwise sincX = (sinX )/X .


A useful approximation for sincX is that sincX ≈ 1 forX < 1 and sincX ≈ 1/X for X > 1.This explains why the theoretical source spectrum in theprevious diagram has three linear segments. The flatsegment extending to zero frequency gives M0.The corner frequencies are given by 2/TR and 2/TD.It can be shown that for earthquakes with a shear velocityof about β = 4 km/s, the rupture and rise times can begiven by the approximate expressions TR = 0.35L andTD = 0.1Lf 1/2. In these expressions, L is the width of arectangular fault and f is the ratio of width to length.


The simple analysis provided above shows why mb and Msdiffer, and why both magnitude scales saturate.As the fault length increases, the seismic moment, rupturetime, and rise time increase. Thus, the corner frequenciesmove to the left (i.e. to lower frequencies).The moment M0 determines the zero frequency level,which rises as the earthquake becomes larger.However, the surface wave magnitude Ms is usuallymeasured at a period of 20 s, and so depends on thespectral amplitude at this period.As the the moment increases, a period of 20 s willeventually lie to the right of the first corner frequency, atwhich point Ms will no longer increase at the same rate asM0.


As the moment increasesstill further, 20 s willeventually lie to the right ofthe second cornerfrequency, which results inMs saturating at about 8.2.A similar effect occurs forbody waves, but becausemb is usually estimatedfrom waves with a periodof about 1 s, it saturates atlower moment.


In this example,Ms is plottedagainst M0 andfault area (S).Saturation clearlyoccurs in bothplots.Open and closedcircles denoteintraplate andinterplateearthquakes,respectively.

Earthquake magnitudes and momentIRIS

If you visit the Incorporated Research Institutions forSeismology (IRIS) webpage http://www.iris.edu, you canaccess a large resource of current and past earthquakedata and analyses.

Earthquake magnitudes and momentIRIS

You will find that magnitude estimates of one kind oranother are used in the description of most earthquakes,and form an integral part of the event information that isattached to most seismic datasets that are distributed.If you look up an earthquake on the IRIS website, you maynotice that definitions of magnitude exist other than theones covered in this lecture.One of these is Md , the duration magnitude. This is basedon the duration of shaking as measured by the time decayof the amplitude of the seismogram. This is sometimesdone when the dynamic range of the recording instrumentmakes it impossible to measure peak amplitudes.

peat8002 - seismology lecture 13: earthquake magnitudes...

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