Scaling Law of Seismic Spectrum

JOURNALOl• GEOPHYSICALR•.s•.Aacn VOL. 72, No. 4 FEBaUAaY 15, 1967

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 different waves at different periods.The •2 model gives a satisfactoryagreement with such observations 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, because,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 autocorrelation 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 autocorrelationfunction 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•. The corresponding/)(•,t) and •(•, t) Fig. 2. Schematic diagram of autocorrelation are alsoshown in Figure 1. Their autocorrelation functions of dislocation velocity and dislocation functionsare shownschematically in Figure 2. accelerationat a given point • on a fault. The dashedlines in thesefigures are for the case in which the dislocation takes the form of a In our first model,we assumethat the temporal ramp function in time. We now construct two autocorrelationfunction of dislocationvelocity earthquake source models by fitting'simple decreasesexponentially with the lag r, that is formulas to the two autocorrelation functions. f_•b(•, t)b(•, tq- r) at - •oe-kr'•l (19) Our secondmodel is the one proposedby O(S.t) Haskell. I-Ie assumes that the autocorrelation r' T function of dislocation acceleration takes the followingform'

•/,,

b (%.t) = (20) We shall assumean identicalspatial correlation f•ction for both models. The correlation >t betweenthe dislocationvelocity at $ and t and II that at $ + v and t' - t + V/v, that is

',, ff.D(e,t)D(e + de • in, cate the degreeof pendency of fa•t propagation.The pers•tency•1 decrease•th

ii the distancev betweenthe two points. FoHo•ng •askell, we shah adopt the functionalform of Fig. 1. Schematicdiagram of dislocationand its e-• •,• for t•s expression,and •so for the co•e- time derivativesat a given point • on a fault. sportingfunction of •(•, 0. SCALING LAW OF SEISMIC SPECTRUM 1221

The above temporal and spatial autocorrela- we have tion functions are expressedin a single form, if %/4krkr •o we write = w DoL (29) k•,kL Inserting this into (25), we get

= •oe-•'"'-•'•-"/" (21) IA½)I

for the first model,and __ wDoL 1+ co_sc 0 2 o•2 {lq- (w/k•)2} (30) for our first model. Since the above function ße -•'•-"" (22) decreasesproportionally to oJ-'-for large oz,we shall call this the 'o>squaremodel.' for the second model. Their Fourier transfo•s On the other hand, the sourcefactor of ampli- are tude spectral density for our secondmodel will decrease proportionally to o•-• for large oJ. 4k•,kL ,•o q•0 is equalto L•D dkrk •,•/8, accordingto Haskell. •(•,•) = /• + (• --•/•)•}(• + • Inserting this into (24) and (28), we obtain: (23)

•(•,•) = /•:•+ (•8k•'kL'q•øa'2 -- •/•)•}(• + • • wDoL (•a) {i-q-(co-so½ •)(•'-•'•)}2•0 2 •/2 {1-•'(•) :•} Using (15) and (23), we may obtain the (31) sourcefactor of amplitude spectral density for our first model as follows' We shall call this the 'o>cube model.'

ASSUMPTION OF SIMILARITY

w¾/ 4k•,kL •o The straightforwardway of testing the earthquake sourcemodels proposed above is to com- pare the predicted spectrum directly with the [k•2+/.COS 0 •)2(,•2 ]1/2 + (.1,)2)1/2observed one. For this purpose, however, we must know about such effects of the propaga- To determine the value of •o, we put •o = 0 in tion medium as dissipationand complex inter- (10). Then ferenees on the seismic spectrum for a wide frequency range. Although such knowledgehas been accumulating, especially for long-period waves [ef. Press, 1964], it does not yet satis- (26) factorily cover the frequency range required for the presentstudy. = w As mentioned in the preceding section, we Comparingthe aboveequation with (25), we get will remove this difficulty by comparingseismic waves having a common propagational path but coming from earthquakesof different sizes. %/4krkL•o= w (27) Further, in order to specify an earthquakeby a singlesource parameter, 'magnitude,'we must If we define an average dislocationby reduce to one the number of parameters ap- pearingin (30) and (31) by assumingthat they Oo= Z (28) are related to each other in some manner. 1222 KEIITI AKI The simplest of such assumptionsmay be that large and small earthquakes are similar y(t)-- w phenomena.If any two earthquakesare geo- (32) metrically similar, the fault width w is propor- I tional to the length L. If they are physically '" 42 similar, all the nondimensionalproducts formed by the sourceparameters will be the same.The averagedislocation Do will be proportionalto L and, consequently,to w. This implies that if an where•oo is given by the equation earthquakeis a Starr fracture, the pre-existing stress or strength is constant and independent t - --(d$/dco).... (33) of sourcesize [Tsuboi, 1956]. Since the wave If this approximationis valid, the trace ampli- velocity is practically independent of source tude of waves with frequency • read directly and may be consideredconstant for our present on the record will be proportionalto the spec- purpose,all the quantitieshaving the dimension tral density]Y(•)I. The quantityd•/&o" in of velocity must also be constant and inde- (32) is the sum of a propagation term and a pendent of source size. Thus, the similarity sourceterm. Since the propagationterm is pro- assumptionsimply that the rupture velocity v portional to the travel distance, the source is a constantand that all the quantities having term may be neglectedat long distances.Thus, the dimensionof time, suchas k• -• and (vkL)-•, we may assume that the trace amplitude of are proportionalto L. surface waves with period of 20 sec is equal For simplicity, we shall further assumethat to the amplitude spectral density of waves with cos 0 -- 0 and that vkL -- k•. A value of k• that period, except for a factor that is inde- greater than vk• may be a more realisticchoice, pendent of the sourcesize. The validity of this becausek• -• is related to the time required for assumptionis confirmedby comparingthe ratio formation of fracture across the fault width, of traceamplitudes of Love waveswith a cer- whereas (vk•) -• is related to the time required tain period from two aftershocksof the Kern for propagationof fracture along the length of County earthquake with the ratio of amplitude the fault. We shall examine later the case in spectral densities at that period obtained by which 10 vk• -- kr. Essentiallythe same result the Fourier analysismethod. Both ratios agree as when vk• -- kr will be obtained,except for well. the value of k• correspondingto a sourcesize. Thus, the dependenceof amplitude spectral density,IA(•)I, on the magnitudeM• will be SCALING L•w OF SEISMIC SPECTRUM suchthat log IA(o•)] at the periodof 20 secis Under the assumptionsdescribed in the pre- equal to M• plus a constant.In other words, cedingsection, we can expressthe sourcefactor two spectrum curves corresponding to two of amplitude spectraldensity as a function of earthquakesizes differing by M• -- 1.0 will be L, % and several nondimensionalconstants. separated by 1.0 along the ordinate at the Taking L as a parameter,we shall obtain a periodof 20 sec,if the curveIA(•)I is drawn group of curves of spectral density, each of on a logarithmic scale. Figures 3 and 4 shows which correspondsto an earthquakeof a certain such groups of curves for the o•-squareand size. In order to find which curve corresponds •-cube models,respectively. to a given earthquakesize, we must have a scale The curves shown in each of these charts to measure size. The most convenient scale for have an identical shape. The frequency that our purposeis the surfacewave magnitude scale, characterizesthe shape of the curve, such as definedby Gutenbergand Richter [1936]. This k•, is proportionalto L -•, and the spectralden- magnitude,designated as Ms, is proportionalto sity at • -- k• is proportionalto L ', as can be the logarithm of amplitude of teleseismicsur- found from (30) and (31) under the assump- face waves with period of about 20 sec. Since tion of similarity. Therefore, the points cor- at this period the waves are usually well dis- respondingto the characteristicfrequency lie persed,we may expressthe wave train y(t) by on a straight line with gradient 3, as shownby the stationary phaseapproximation, as follows: dashedlines in Figures 3 and 4. As mentioned SCALING LAW OF SEISMIC SPECTRUM 1223 PERIOD IN SEC nitudes. The magnitude of earthquakesstudied 0.1 0.2 0.5 I 2 5 I0 20 50 I00 200 500 I000 by him coversthe range 4.5 to 8. After several trials, we choosethe absolutevalue of magnitude that gives the best agreement between

tO-SQUARE MODEL ' theory and observation.The valuesassigned to the curves in Figures 3 and 4 are determined in this manner, and the correspondingtheo- / retical spectral ratios are shown in Figure 5, together with the observed ratios given by Berckhemer.

LOVE WAVES FROM,Two CALIFORNIA SHOCKS

1 The applicability of the theoretical curves of spectral densitiesobtained in the preceding sectionis tested by the use of recordsof Love waves from two aftershocks of the Kern f-- 7.0 County, California, earthquake of 1952. The epicentersof these two earthquakesare within ,/ 8.5 severalmiles of each other, accordingto Richter [1955], and they show identical first motion patterns, according to Bdth and Richter

PERIOD IN SEC / , 0.5 I 2 õ I0 20 50 I00 200 500 I000 5(300

03-CUBE MODEL

M,defined _ 8.o

FREQUENCY IN C/S

, Fig. 3. I)eper•der•ceof amplitude spectra] den- '"' sity of earthquake magnitude M, for the •-square /f •--- 7.5 model. • . before,the spacingof curvesfor differentearth- quake magnitudesis determinedby the deft- 6.5 nition of M,. The definitionalone, however, cannotgive the absolutevalue of magnitude correspondingto eachcurve. If we know the absolute value for one of •,•'Y' ,.o the curves,the values for the rest are de- terminedfrom the definitionof M•. First we 4.5 adopta trial valueof magnitudefor one of the curvesand assignmagnitude values to other curves according to the definition. Then we can find the ratio of spectral densitiesfor

two different magnitudesas a function of fre- 2,0 1.0 0.5 02 0.1 0.05 0•2 0.010005 OJ:)02 0.0006

quency or period. This ratio is comparedwith FREQUENCY IN C/S the observedone given by Berckhemer [1962]. Fig. 4. Dependence.of amplitude spectral den- letsdata include six sets of twoearthquakes sity on earthquake magnitude M, for the e-cube with the same epicenter but of different mag- model. 1224 KEIITI AKI

AI /A2! ß IOO 8/6,5 .. AI/A21 8/7,5 80 - 6 ß / 5 60 4

40 /

! 20' ß

I I I I '-m I I I I i • ' -- T i0 20 30 40 50 s IO 20 30 40 50 60 70 S

AI/A2• AI/A 21 7,4/6,5 50 7,5 / 6,5

20 20

I I ] • •T , , ] , , , t IO 20 30 40 I0 20 30 S

Ai/A2 AI/A21 300

3 200

IOO 6,2 / 5,7

, I I I • =T I I I I 5 IO 15 20 S 5 IO 15 20 S

Fig. 5. Comparisonof theoretical and observedspectral ratio, plotted against period, for pairs of earthquakeshaving nearly the same epicenterbut different size. Observedvalues are reproducedfrom Berclchemer[1962]. The numbers shown for each pair are the earthquake magnitude for the pair. Solid line denotes•-square model; dashedline denotes•-cube model.

[1958]. The Richter magnitude(M•,; localscale are obtained by Berckhemer'smethod, in which for southernCalifornia) of one of them is 6.1, the ratio is obtained between the corresponding and that of the other is 5.8. The difference of peaks of waves by directly reading amplitudes 0.3 correspondsto the maximum amplitude on the record. As shown in Figure 7, the cor- ratio of 2 on the record of the standard Wood- respondenceof peaks and troughsbetween the Andersonseismograph. two earthquakesis excellent,and there is no The amplituderatios of Love wavesfrom the difficulty in obtaining such ratios. Figure 8 two earthquakesobserved at Weston, Ottawa, showsthe ratio of the amplitude spectral den- and ResoluteBay are shownin Figure 6. They sity obtained by the Fourier analysismethod. SCALING LAW OF SEISMIC SPECTRUM 1225 samespectral density at short periodsfor the o,so,u'r .,¾ I . two earthquakesand doesnot explain the ob- o•O ß WESTON | •'Ms 0.85 servation. A more general comparisonof the local magnitudescale M•, and the surfacewave mag- o7 _ nitude scaleM, is difficult. The maximumamplitude recordedby the Wood-Andersonseismo- graph would not be directly proportional to the amplitude spectral density at any fixed period,because the signalduration and prevail- // I •3 model ing period may change with the earthquake I0 20 $0 sourcesize. The spectral ratio may be nearly PERIOD IN SEC equalto the maximumamplitude ratio for such Fig. 6. Comparisonof theoreticaland observed earthquakeswith small differencein magnitude spectral ratio for two aftershocksof the Kern as studiedin the presentsection, but the equal- County, California, earthquakeof 1952.Observed ity cannothold for larger magnitudedifference. ratios are obtained from trace amplitude. Further, the empiricalrelation between M•, and M,, with which the theoreticalrelation is to be

There is no, significantdifference between the compared,has not yet been stabilized[Richter, results obtained by the two methods,justify- 1958]. ing the simpleprocedure used by Berckhemer. The theoretical curves of spectral density RELATION BETWEEN ms AND M, ratio for an earthquake pair with magnitudes M, around 6.0 which best fit the observations On the other hand, the relation between the are shown in these figures. It is remarkable magnitude scale mB, defined as the logarithm that the observedratio is about 7 at the period of amplitude of teleseismicbody waves, and of 20 sec; in other words,the differencein M, M, has been well establishedempirically by between the two earthquakesis 0.85, about 3 Gutenberg and Richter [1965a]. Further, it is times larger than the differencein M•, obtained well known that the usual record of teleseismic by Richter. Our u-squaremodel explainsthis body waves, obtainedby a standard short-pe- fact satisfactorily,because M•, must have been riod seismographsuch as Benioff's, shows a measuredon waves with periods of less than 1 rather narrow spectral band around I c/s. sec, and this model predicts a spectral density Therefore, we may correlate the amplitude of ratio of about 2 at these periods.On the other body waveswith the spectraldensity at I c/s. hand, the u-cube model predicts nearly the If our signal is a finite portion of a Gaussian

o)-• o)

b)•./

•1 MINUTE•

IEW IN S INS RESOLUTE BAY WESTON OTTAWA Fig. 7. Love wavesfrom the Kern County aftershocks(no. 194 above,no. 141 below; num- bersassigned by Richter [1955]) recordedat Ottawa,Resolute, and Weston. 1226 KEIITI AKI

o RESOLUTEBAY M, for the to-squareand .to-cubemodel. The ß WESTON • Ms-----0.85 shaded area indicatesthe range between the i- 10 n OTTAWA above-mentioned two extreme cases of the de- o pendenceof spectraldensity on signalduration. The theoretical curve for the to-cube model does o ' not agree with the empirical one given by •. 6 Gutenbergand Richter. On the other hand, the agreementis excellentfor the to-squaremodel, ,•o4 • M•.=Q3/, / except for smallermagnitudes. Looking at the o originaldata from whichGutenberg and Richter o 2 -•'• • model derived their empirical formula, we find that •a modeI .... the theoretical curve based on the to-square 0 5 I0 15 20 25 model better explainsthe observationsat small PERIOD IN SEC magnitudesthan the empirical curve as shown in Figure 10. This result stronglysupports the Fig. 8. Comparison of theoretical and observed spectral ratio for two aftershocks of the Kern applicabilityof the scalelaw of seismicspec- County, California, earthquake of 1952. Observed trum derived from the to-squaremodel on the spectral ratios are obtained by Fourier analysis. assumptionof similarity.

RELATIONBETWEEN FAULT LENGTH AND M, noise, the amplitude spectral density will be FOR THE (o-SQUAREMODEL proportional to the square root of the signal duration. On the other hand, if the signal is a In derivingthe scalinglaw of seismicspec- finite portion of a coherent sinusoidaloscilla- trum we assumed that the characteristic fre- tion, the spectral density will be proportional quency k• is proportional to L -'. We can check to the signal duration. We may assumethat this assumptionagainst geological or geodetic the actual seismic signal has an intermediate observationson an earthquakefault of known nature betweenthe above two extremes.Then, magnitudeM,. The value of k• for a givenM, we may write the spectral density at I e/s as is found from the theoretical curves for the follows: to-squaremodel shownin Figure 3. Then k•-' shouldbe proportionalto L, if the assumption A(1) = c0nstX A= X tø"•z'ø (34) of similarity holds.Figure 11 showsthe rela- whereA., is the maximumtrace amplitudeand t is the signalduration. me Accordingto Gutenbergand Richter [1956b], 8 , , , log t is related to m• by the empiricalformula •)- squa

log t = --1.9 -]- 0.4m• (35) •-cube model • Insertingthis equationinto (34), we obtain 7 log A(1) = cons•-]- (1.2-,• 1.4)ms (36) From this equation and the charts of spectral density curves given in Figures 3 and 4, we 6 can obtain the theoretical relation between m• 'ms= 0,63Ms + 2,5 / (Gutenberg-Richter. 1956) and M, on the basis of the to-squareand to-cube models.The constantin (36) is determinedin such a way that m• and M, agree at 6.75, in accordance with the Gutenberg-Richter em- 5 , 5 6 7 8 Ms pirical formula Fig. 9. Theoretical relation between ms and mB= 6.75 + 0.6a(M,- 6.75) (37) M, based upon the •-square and •o-cubemodels, as comparedwith the Gutenberg-Richterempiri- Figure 9 showsthe relation betweenrns and cal formula. SCALING LAW OF SEISMIC SPECTRUM 1227

Ms-Ms f[ [ I I I [ • I I • [ [ I ] i [ [ 1.21• 0 f,•)- squar e mocl el Ms FROMSURFACE WAVES - , ' [ o• M•Me =0.4(Ms-7) A KERNCOUNTY, 195Z .Sr•• o o 0 AVERAGES

.4- • -• o••oo

2- • • • ••o• o• o ' o •••• o o •o o o 0 - • ooo o o •m• o

-.4- o o• •6- o

ß•8- o - [ t i I [ i I I I I I I I I I I I 5.0 6.0 7.0 8.o Ms Fig. 10. Theoretical relation between ms and M, based upon the •-square model, as compared with that observedby Gutenberg and Richter [1965a].

tion betweenTo -- 2,rk•-• and L for the earth- • To sec quake fault given in Toeher'slist [Tocher, I0 2.0 50 I00 200 500 I000 1960]. A linear relationshipholds between L

/ and To, if earthquakessmaller than magnitude / 5OO 6.5 are excluded. In the case of small earthquakes,the surfaceevidence may not revealthe / o / true fault lengthat the earthquakefocus. There 2OO / is alsosuch an ambiguitywith the magnitudeof o small earthquakesthat the magnitudegiven in I00 o Tocher'slist may or may not be taken as M,. Consideringthese facts, we may concludefrom oo •/ o o Figure 11 that geologicaldata do not exclude / ,,/'•'"---L: CONST.XT O the assumptionof similarity. The characteristictime To for a given M, as z 20 o/p/o shownin Figure 11 may seema little too large. / It is possibleto reducethis value without af- / •o D / fecting significantlythe conclusionsobtained above. If we assume that k• -- 10 vk•. instead of k• = vkL, and if we determinea set of spec- o tral density curves using the data of Berck- hemer and others as given above,we find that the value of To becomes about one-third that o given in Figure 11. The agreementbetween t I theory and observationis as goodas that shown 6 6.5 7 7.5 8 ---• Ms in Figures 5, 6, and.8, and we find again that the to-squaremodel explains the relation be- Fig. 11. Relation betweenthe length of earthquake fault measuredby geologicalor geodetic tween ms and M, and that the to-cube model means and the characteristic time of the earth- does not. quake determined from its magnitude on the basis of the •-square model. A linear relation EFFICIENCY OF SEISMIC RADIATION between them supportsthe assumptionof similarity. Let us now examine the efficiencyof seismic 1228 KEIITI AKI energy radiation, which must be independent tributed to a difference in the assumed source of earthquakesource size if the similarity con- model. As mentioned before, the model of dition holdsstrictly. We definethe efficiency Berckhemer,if interpretedby dislocationtheory, as the ratio of the energyradiated in the form is the one in which the dislocation is constant of seismicwaves to the elasticenergy released and independentof sourcesize, but the disloca- by the formationof an earthquakefault. If an tion in our modelis proportionalto the linear earthquakeis a Starr fracture [Starr, 1928], dimension of the source. the fault-releasedelastic energy is proportional DEPARTURE i•ROM SIMILARITY to DolL. Under the assumptionof similarity, this energy will be proportionalto L 3. If we As mentionedbefore, the assumptionof simi- know the energyfor a certainvalue of Ms, we larity implies a constant stress drop in all can determinethe value for any M8 from the earthquakes.If the stressdrop differs for two scalinglaw givenin the precedingsection. As- earthquakes,our scalinglaw will not apply. If sumingthat log E is 23.7 for M, ---- 7.5 from the stress drop varies systematicallywith re- the resultof the writer'sstudy on the Niigata spect to such environmental factors as focal earthquake[Aki, 1966], we get the released depth, orientation of fault plane, and crust- strain energyfor variousM, as shownin Table mantle structure, we may construct different 1. The energy radiated in the form of seismic scaling laws for different environments. Such wavesis evaluatedby the Gutenberg-Richter a study of the seismicspectrum may eventually formula reveal the distributionof stressdrop or strength of material in the earth's crust and mantle. For log E = 11.4 + 1.5M, (38) sucha study, however,we shall need more pre- and is also shownin Table I together with its cise measurements of spectrum over wider ratio to the strain energy.The ratio definitely ranges of frequency than are now available, as increases with decreasing magnitude. Thus, well as detailed knowledgeof the propagation starting with the assumptionof similarity, we factor of the spectrum. have endedby denyingit. Even with the present limited knowledgeof It is, however,not impossiblethat a further the propagation factor, however, we may refinement of the magnitude-energyrelation demonstrate remarkable differences in stress (38) may eventually support the assumption drops betweensome earthquakesby the use of of similarity with regard to the radiation ef- long-period surface waves. The earthquakesto ficiency, because (38) is based upon several be comparedhere are the Niigata earthquake simplifiedassumptions. of June 16, 1964, and the Parkfield (California) It should be noted here that Bdth and Duda earthquakeof June28, 1966. [1964], usingBerckhemer's result [Berckhemer, The stress drop in the Niigata earthquake 1962], reached an entirely different conclusion was obtainedby the following procedure[Aki, on the radiation effciency.They found that the 1966]. The geometry of fault movement was efficiencyincreases with increasingmagnitude determined from the radiation patterns of P of the earthquake.This differencemay be at- waves,$ waves[Hirasawa, 1966], and G waves. The spectral density of displacementdue to G waves was estimated for periods of 50 to 200 TABLE 1. ReleasedStrain Energy, Seismic Wave Energy and Efficiencyof see, corrected for dissipation and geometric Seismic Radiation spreading,and comparedwith the theoretical excitation function [Haskell, 1964; Ben-Mena- log E,,t log E,,,* Ew[E,t hem and Harkrider, 1964] correspondingto a sourceof that geometry. From this comparison 8.5 26.7 24.2 0.003 we estimatedthe• product of rigidity g, area $ 8.0 25.2 23.4 0.016 of fault surface, and average dislocationAu, 7.5 23.7 22.7 0.10 which correspondsto the moment Mo of the 7.0 22.5 21.9 0.25 6.5 21.6 21.2 0.40 component couple of the equivalent doublet [Maruyama, 1963; Burridgeand Knopof],1964; * log E,. = 11.4 q- 1.5 M.. Haskell, 1964]. The value of Mo (-- 1• AuS) SCALING LAW OF SEISMIC SPECTRUM 1229 for the Niigata earthquake was 3 X 10• dynes Parkfield earthquake. Consideringthat the ef- cm. All the near field evidence (echo-sound- feet of finite size was significantfor the Niigata ing survey,aftershock epicenters, and Tsunami earthquake (about a factor of % at a period source area) indicated a fault length, L, of of 70 see) but probably not for the Parkfield about 100 kin. The focal depths of the main earthquake,we estimate the ratio of the source shock and aftershocksindicated a fault width, moment Mo for the Parkfield earthquaketo that w, of about 20 kin. Assumingthat p -- 3.7 X for the Niigata earthquake as 1/250. Thus, we 10• dynes cm-•, correspondingto a shear ve- get a moment value of about 1 x 10• dynes locity of 3.6 kin/see and density of 2.85 g/cm•, for the Parkfieldearthquake. we obtainedthe value of the averagedislocation Using the same rigidity value as for the as 400 cm by inserting the valuesof L, w, and Niigata earthquake and the observedvalues of p into the equationMo -- 1• Au ß Lw. This fault length and dislocationmentioned before, value agreeswell with those observedby echo- we get a fault width of about 13 km from the soundingsurveys made just before and after above value of moment. This value of fault the earthquake[Mogi et al., 1965]. Finally, the width gives us an extremely low estimate of stress drop was estimated as about 125 bars strain release.Since Knopoff's fracture model with the aid of Starr'stheory [Starr, 1928]. is more appropriate for a strike slip than Now, let us comparethe Niigata earthquake Starr's, we estimate the strain releaseby the with the Parkfield earthquake. The Parkfield formula e -- Au/2w [Knopo#, 1958]. We get earthquaketook place right on the San Andreas a value of • of 2 X 10-6 and a corresponding fault near Cholame and Parkfield. The PDE stressdrop of about 0.7 bar, which is indeeda card of the Coast and GeodeticSurvey reports remarkably low value. Even if there is an order the epicenter as (35.9øN, 120.5øW), and the of magnitudeerror in estimatingthe value of origin time as 04:26:12.4 GCT, June 28, 1966. moment,the stressdrop is still severalbars. The magnitudeis 5.8, 5.5, and 6• as given by As mentionedbefore, if the stress drop. is the Pasadena,Berkeley, and Palisadesstations, differentbetween two earthquakes,the scaling respectively.According to a personalcommuni- law derivedin the presentpaper will not apply cation from Clarence R. Allen and Stewart W. to them. We found some indication of violation Smith of the CaliforniaInstitute of Technology, of the scaling law when we compared the the near field measurements revealed a strike Parkfieldearthquake with oneof the aftershocks slip fault associatedwith this earthquake,its of the Kern Countyearthquake. length being about 38 km and its offset about The magnitude of the Parkfield earthquake 5 cm. given by local stationsis 5.5 (Berkeley) ,• 5.8 G2 waves from this earthquake are clearly (Pasadena). The surface wave magnitude Ms recordedby long-period seismographsat Reso- of this earthquake, calculated from the Love lute (A = 40ø) and at Ottawa (A = 35ø). The wave amplitude at a period of 20 see recorded peak-to-peak amplitudes on the records at a at Ottawa, is 6•. This value agrees with the period of 70 see are a little over 1 mm at both magnitudegiven by the Palisadesstation. stations.This correspondsto a spectraldensity On the other hand, the magnitude of num- of ground displacementof about 0.04 em see at ber 141 aftershock[Richter, 1955] of the Kern that period. County earthquake is 6.1. Ms for this earth- The G2 waves from the Niigata earthquake quake, calculatedalso from Love wave ampli- at, the epicentral distanceof 35ø to 40ø show a tude at a period of 20 seerecorded at Ottawa, spectraldensity of about 1.6 em seeat a period is 6.2. of 70 see for a certain radiation azimuth. If the Since the variability of seismicamplitudes is Niigata earthquakesource is a strike slip fault very large, it is dangerousto draw any conclu- like the Parkfield earthquake, and if we ob- sions from measurements at a few stations. served G waves in the direction of maximum However, the magnitude values above suggest radiation,we would expecta spectraldensity of that the spectral density for the Parkfield about 5 em see at a period of 70 see for G2 earthquake may be greater than that for the waves at A -- 35 ø • 40 ø. This value is about Kern County aftershock at long periods, and 125 times as large as that observed from the smaller at short periods.If so, the two spec- 1230 KEIITI AKI trum curvesmust crosseach other, violating the Berckhemer,H., Die Ausdehnungder Bruchfiiche scalinglaw. This result is expectedif the stress i'm Erdbebenherdund ihr Einfiussauf das seis- drop in the Parkfield earthquakeis lower than mische Wellenspektrum, Getlands• Beitr. Geo- phys., 71, 5-26, 1962. that in the Kern County aftershock.The reduc- Burridge, R., and L. Knopoff, Body force equiva- tion of stressdrop is equivalentto the reduction lents. for seismic dislocations, Bull. $eismol. of Do in (30), and it will shift the spectrum $oc. Am., 54, 1875-1888,1964. curves in Figure 3 downward parallel to the Byefly, P., The periods of local earthquake waves in central California, Bull. $eismol. $oc. Am., ordinate,causing an intersectionwith the origi- 37, 291-298, 1947. nal curve in the manner described above. Gutenberg, B., and C. F. Richter, On seismic As we have seenabove, there is a possibility waves, Getlands Beitr. Geophys., 47, 73-131, that the stressdrop in an earthquakemay vary 1936. greatly accordingto its geologicalenvironment. Gutenberg,B., and C. F. Richter, Earthquake magnitude, intensity, energy, and acceleration, We shall probablyhave to assigndifferent scal- Bull. $eismol. $oc. Am., 32, 163-191, 1942. ing lawsto differentenvironments. This implies Gutenberg, B., and C. F. Richter, Earthquake that a single parameter,such as magnitude, magnitude, intensity, energy, and acceleration, cannotdescribe an earthquakeeven as a rough 2, Bull $eismol. $oc. Am., 46, 105-145, 1956a. measure.The measurementof seismicspectral Gutenberg, B., and C. F. Richter, Magnitude and density rather than amplitude will becomein- energy of earthquakes, Ann. Geof•s. Rome, 9, 1-15, 1956b. creasingly important. To understand the ob- Haskell, N., Total energy and energy spectral servedspectrum in terms of the physicsof the density of elastic wave radiation from propa- earthquakesource, however, we shall have to gating faults, Bull. $eismol. $oc. Am., 54, 1811- knowmore about the effectof the propagation 1842, 1964. mediaon the spectrumthan we do now. Haskell, N., Total energy and energy spectral density of elastic wave radiation from propa- Acknowledgments. I should like to thank Dr. gating faults, 2, A statistical sourcemodel, Bull. J. H. 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