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Scaling Law of Seismic Spectrum JOURNALOl• GEOPHYSICALR•.s•.Aacn VOL. 72, No. 4 FEBaUAaY 15, 1967 Scaling Law of Seismic Spectrum K•,ii•i Axi Department o• Geology and Geophysics MassachusettsInstitute oi Technology, Cambridge The dependenceof the amplitude spectrumof seismicwaves on sourcesize is investigated on the basis of two dislocation models of an earthquake source.One of the models (by N. Haskell) is called the o•8 model, and the other, called the •2 model, is constructedby fitting an exponentially decaying function to the autocorrelation function of the dislocation velocity. The number of source parameters is reduced to one by the assumptionof similarity. We found that the most convenientparameter for our purposeis the magnitude M,, defined for surface waves with period of 20 sec. Spectral density curves are determined for given M,. Comparison of the theoretical curves with observationsis made in two different ways. The observedratios of the spectra of seismicwaves with the same propagation path but from earthquakesof different sizesare comparedwith the correspondingtheoretical ratios, thereby eliminating the effect of propagation on the spectrum. The other method is to check the theory with the empirical relation between different magnitude scales defined for differ- ent waves at different periods.The •2 model gives a satisfactoryagreement with such obser- vations on the assumptionof similarity, but the •a model does not. We find, however, some indicationsof departure from similarity. The efficiencyof seismicradiation seemsto increase with decreasingmagnitude if the Gutenberg-Richtermagnitude-energy relation is valid. The assumptionof similarity implies a constantstress drop independentof sourcesize. A prelimi- nary study of Love waves from the Parkfield earthquake of June 28, 1966, shows that the stress drop at the source of this earthquake is lower than the normal value (around 100 bars) by about 2 orders of magnitude. INTRODUCTION longer-periodwaves are generated.In the early Elaborate studies have been made in recent days of seismologyin Japan, much attention years to find seismicsource parameters such as was given to the presenceof large long-period fault length, rupture velocity, and stressdrop motion in P waves from large local earthquakes at the earthquakesource from the spectrumof [cf. Matuzawa, 1964]. Analysesof seismicwaves seismicwaves. Except for the geometricparam- by Jones[1938], Honda and Ito [1939], Gt•ten- eters obtained from fault plane studies,how- berg and Richter [1942], Byerly [1947], Kanai ever, the magnitudeis the only physicalparam- et al. [1953], Asada [1953], Aki [1956], Kasa- eter that specifiesmost earthquakes.A gap hara [1957], Matumoto [1960], and others exists between the two approachescurrently have shownthat the period of the spectralpeak used,one basedon the use of spectrumand the for P waves,S waves, surfacewaves, and even other on amplitude.The purposeof the present for coda waves increaseswith earthquake mag- paper is to fill this gap by finding a first nitude. approximation to the relation between seismic The most convincingevidence for the greater spectrumand magnitudeof earthquakeson the efficiency of generating long-period waves by basisof somedislocation models of earthquake larger earthquakesis probably given by Berck- sources.For this purpose we must reduce to hemer [1962]. He comparedseismograms ob- one the number of parametersspecifying a tained at a station from two earthquakesof the dislocation.We shall make this reductionby same epicenter but of different size. We shall assumingthat large and small earthquakes reproducehis result later. satisfy a similarity condition. The magnitudeof an earthquakeis definedas The relation betweenseismic spectrum and a logarithm of amplitude of a certain kind of earthquakemagnitude is not a new problem. seismicwave recordedby a certain type of It has been well known that the greater the band-limited seismograph.If there is such a size of an earthquake, the more efficiently size effect on seismic spectra as mentioned 1217 1218 KEIITI AKI above,the unit of magnitudeobtained from one 1 kind of wave recordedby one type of seismo- u0 - 4•rbrcos 20 sin • graph may not correspondto that obtained from another kind of wave recorded on another instrument.In fact, Gutenbergand Richter ßw ,t -- r- •/cøs [1956a] discovereda discrepancybetween the where a and b are the velocities of P and S magnitudescale based upon short-period body waves,respectively. The aboveexpressions have wavesand that basedupon long-periodsurface the foliowingcommon form: waves. It will be shownthat the theoreticalscaling V= P(r, O,,i:,,a, b) law of the seismicspectrum derived from a dislocationmodel of the earthquakesource ßw 15 ,t (3) satisfactorilyexplains the above-mentionedob- ½ servations. where c is the appropriatewave velocity.In terms of the Fourier transform,the aboveform THEORETICALMODELS OF THE EARTHQUAKE can be written as SOURCE U(o•)= P(r, O,'i:', a, b)A(co) (4) FollowingHaskell [1966], we define a dislo- where cation functionD(•, t) which is the displace- ment discontinuityacross a fault plane at a point • and time t. The fault plane extends = u(O,at along the /• axis, and D(•, t) is consideredas the average dislocationover the width w of the = w e dt fault. Taking the starting point of the fault at A(•o) the originof the (x, y, z) coordinates,and the • axis along the x axis, we assumethat the ß ,t -r--cøsO fault endsat • = L and the surroundingme- dium is infinite, isotropic,and homogeneous.If the medim is dissipative,the equationcor- Introducingpolar coordinates(r, O, •,) by the respondhgto (4) will be relation U(oo)= Pit, O,•, a, b, oo,Q(½o)]. A(½o) (6) x -'r cosO where Q(•o) is the dissipationcoefficient. The above expressionshows that .wecan isolatethe y = r sin 0 cos•v (1) propagationterm P((o) whichdoes not, except z = rsin 0sin•v for the directionof fault propagation,include the fault motionparameters. Mathematically, the displacementcomponents of P and $ waves sucha simpleisolation is not permittedfor an at long distances,corresponding to a sourceof arbitrary heterogeneousmedium. Practically, longitudinalshear fault [Haskell, 1964] for however,this separationof propagationfactor example, can be written as from sourcefactor may be permitted,at least as a goodfirst approximation.In this paperwe shall be concernedonly with the sourcefactor U,.-- 4•rbr sin20 sin • A((o), which can be calculatedfrom the disloca- tionD (•, t) accordingto (5). For comparisonwith observationswe shall a 'wfo ,t--r - cos0) use seismicwaves observed at a given station fromdistant earthquakes of the same epicenter, V• cos 0 cos qo the samefocal depth, and the samefault plane solution,but of differentmagnitude. The ratio of the Fourier transforms of two such seismo- ßWfo r- b cos grams may be directly comparedwith the SCALING LAW OF SEISMIC SPECTRUM 1219 theoretical ratio for the source factor A(o•), becausethe propagation factor P((o) may be •(•, •) = •l•. ff_• b(•, 0 ff• B(•, canceled in the observed ratio. This ingenious method comes from Berckhemer [1962]. His ße •(•+')-•½•+•) dk &od• dt theoreticalmodel, however, seems unrealistic, be- cause,from the point of dislocationtheory, his 4•-: ' model impliesthat the amount of dislocationis I ff:•B(k,o•)B(--k--o•) constant,independent of the size of the fault. ße dk &o Following the generalline of approachtaken by Haskell [1966],we introduce the autocorrela- lB(k,o.,)l: tionfunction •(•/, •) of •(•, t)' 4w2 ße dk &o Comparingthis formula with (9), we get the (7) well-known relation Putting the Fourier transform of •k(•, •') as •(k, o•)= lB(k,o•)[ •' (14) •(k, o•),we get Finally, we get from (9), (13), and (14) the relationbetween the amplitudespectral density IA(•o)]and the Fouriertransform of the auto- = ff •)e-'•+'•" d•d. (8)correlation function' •(,, •) = • •(k,w)e' •-'•"• dk (9) IA(w)["= w•[(w cosO)/c, w] (15) On the other hand, A(w) can be re•tten by Thus,the sourcefactor ]A(co)] of the amplitude changing the order of integration and putting spectral density is expressedin terms of the t' = t --(r -- • cosO)/c in (5) as follows' autocorrelationof dislocationvelocity /)(•, t). We followed Haskell [1966] in deriving the expressionsabove. Haskell, however,calculated A(•)= we-'"• fff• b(e, the energyspectral density from the autocorre- lation function $(7, •) of dislocationacceleration .e-•.•,+•.••o• •/• dt• d• (10) /•(•, o. In the above e•ression the integration fi•ts •e extendedto i•ty by putting•(•, t) = 0 for • ( 0 and L ( •. Putt•g the Fourier trans- formof •(•, t) asB(k, w), weobtain (16) The Fouriertransform $(k, w) of t•s function s(•,•) = ff• b(e,t)e-'"'*'• dtde isrelated to •(k, w)simply by (•) $(•, •) = • f(•, •) (•7) 12 ff•• •-• Thus, we obt•n 2 Then we have from (10) and (11) •(•)1• = • $[(• •o•0)/•, •] 0s) ' ,w (12) As shownabove, the amplitudespectral den- c sity of seis•c wavescan be expressedin te• and of the autocogelationfunction of •(•, t) or that of •(•, t). The autocogelationfunction of •(•, t) canbe detersnedif the absolu•value I.a(•)[: = w•' c , •o (13) of the Fouriertramform of •(•, t) is •ven. There •e an i•te number of space-time On the other hand, we get from (7) and (11) f•ctio• that •ve a commonspectrM deity 1220 KEIITI AKI but have different phases.By specifyingan autocorrelation,therefore, we are considering an infinite group of space-timefunctions. The model based upon the autocorrelationfunction // is different from the deterministic one in this //////x% respect and may be called 'statistical,' as was doneby Haskell [1966]. Sincethe earthquakeis essentiallya transient -T 0 T phenomenon,however, the autocorrelationfunc- tion introduced here cannot be treated in the same manner as the one for the stationarytime I! series.The following figures will schematically i]lustrate what form may be expectedfor the autocorrelation function for the dislocation processat an earthquake source. Let the dis- -T _ " _• T locationstart at • - 0 and propagatealong the ) z: • axis with a constantvelocity v; then the dis- "/ •, •, locationat • will be zero for t • •/v and will II II Ii take a constantvalue D0(•) for t • T q- •/v. Ii Ii Figure I showsa schematicpicture of D(•, t) at a given•.
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