ML16340B772

Character of High Frequency Strong Ground Motion.

ML16340B772
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Site:	Diablo Canyon
Issue date:	05/22/1981
From:	Hanks T, Mcguire R ERTEC WESTERN, INC., INTERIOR, DEPT. OF, GEOLOGICAL SURVEY
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ML16340B771	List: ML16340B770 ML16340B772 ML16341B642 ML20004B685
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S THE CHARACTER OF HIGH-FREQUENCY STRONG GROUND NOTION Thomas C. Hanks and Robin K. HcGuire Abstract Analysis of more than 300 horizontal components of ground accelerations written by the San Fernando earthquake, eight other moderate-to-large P

California earthquakes; and seven Orovi lie aftershocks reveal that these acceleration time histories are, to a very good approximation, band-limited, white, Gaussian noise within the S-wave arrival window; the b'and limitation is defined by the spectral corner frequency fo and f the highest passed by the accelerograph or the Earth's attenuation operator, max'requency and the S-wave arrival window is (0 < t-R/o < Td), where R is distance, 8 is shear-wave velocity and T is the faulting duration. An examination of the root-mean-square acceleration (a rms ) characteristics of these records for 0 < t. - R/s < Td in terms of the relation

= 0 85 2 fmax rms 06 pR f where ha is the earthquake stress drop, yields the surprising result that all 16 earthquakes have stress drops, as determined by record values of a rms'ery nearly equal to 100 bars =(+ a factor of 2). The source dependence of 1 U.S. Geological Survey. 345 Hiddlefield Road, Henlo Park, California 94025 Ertec, Inc., 14142 M. 20th Avenue, Golden, Colorado 80401 8$ 529088 S I

h I 1 l

I c I a thus depends solely on the parameter 1/ fo, which increases only as the one-sixth power of seismic moment for constant stress drop earthquakes.

~0 put another way, model and record in agreement within factor of rms are a a 2 approximately 85 percent of the time for da 100 bars and knowledge of 1/Jf .

On the basis that acceleration time histories are finite-duration, band-limited, white, Gaussian noise, for any of which a is fixed by rms to = 100 bars and 1/1/, ee can estimate the peak accelerations (arne) for all of these records with considerable accuracy (50 percent or less). The relation is 2ln 2fmax amax arms

~0 r

wh e e a is defined above. lilith less accuracy, this relation fits the peak acceleration set of Hanks and Johnson '(1976) as well, again with ao = 100 bars. At a fixed, close distance, we determine the magnitude dependence of a to be log a - Oe30 M for 4 < M= M < 6 1/2, remarkably close to that recently determined empirically by Joyner et al (1981) for 5.0 < M < 7.7, their coefficient on M (moment magnitude) being 0.28 + 0.04. In the model presented here, the magnitude-dependence of peak acceleration is a function of faul'ting duration alone; larger earthquakes have larger peak accelerations because they last longer, not because they are intrinsically more powerful.

These well-behaved characteristics of high-frequency strong ground motion also suggest that the stress differences which develop in the course of crustal faulting are comparably well-behaved, both in the average stress release across the characteristic source dimension and in the spectral composition and distribution of stress differences that develop across smaller dimensions'

'li

~ [ I

'arthquake ground motion at periods short compared to the faulting duration has at'tracted the attention of seismologists and engineers alike in recent years, principally through the realization that existing characterizations of high-frequency strong ground motion in terms of causative faulting processes or in terms of potential for structural damage are at best rudimentary. There is considerable seismological interest in how high-frequency ground motion data might be used to infer the character of quasi-static and dynamic stress differences which are associated with crustal faulting and how such inhomogeneities might condition the cause and effect of crustal earthquakes. In a related vein, it is still a matter of vigorous debate whether or not N > 6 crustal earthquakes are organized and coherent enough in the real Earth to materially affect high-frequency amplitudes (for example, peak acceleration) in a limited azimuthal range via source directivity effects (e.g. Brune and Blume, 1981). Finally, there is has been much recent interest in magnitude saturation and how this phenomenom might work for peak acceleration at some fixed, close distance or'might not work.

From an engineering'erspective, it has long been appreciated that peak acceleration does not correlate particularly well with all types of structural damage (Blume, 1979). While'his might be expected from recent studies indicating that peak acceleration depends only weakly on magnitude at close distances (e.g. Hanks and Johnson, 1976; Joyner et al, 1981), it surely must be of concern to both engineers and seismologists that the most widely used measure of high-frequency strong ground motion over the past five decades is not a particularly good measure of real or potential structural damage. But

eg I if peak acceleration isn', what isV While considerable attention has been paid by the engineering comnunity to the n~tion of "effective peak

'..acceleration," it is yet without a "precise definition in physical terms" (Applied Technology Council, 1978, p. 298). Moreover, it seems almost certain that the duration of high-frequency strong ground motion at some fixed level of amplitudes will play an important role in producing structural damage (and that any definition of effective peak acceleration will have to incorporate duration effects), but this problem has received very little quantitative attention.

Although we shall be more precise about the matter shortly, by "high-frequency strong ground motion" we mean. in this study ground acceleration in the frequency band from = 1 Hz up to the highest frequency (f ) passed by the Earth' attenuation operator or the instrumental recording system, with amplitudes sufficiently large to be of engineering concern. Thus, as a formal matter of frequency band, the concept of high-frequency strong ground motion as investigated here is fully S

complementary to that of long-period strong-ground motion (e.g. Hanks, 1975; 1979b). This distinction between high-frequency and long-period strong ground motion runs more deeply, however,'since there is an important deterministic character of long-period strong ground motion, easily discernible in both amplitude and phase, that is generally absent in high-frequency strong ground motion.

Ground motion observations and theoretical modeling are in accord that at source-station distances greater than two source depths, strong ground motion at periods > 1 sec will generally be dominated by surface waves, especially in 1

sedimentary basins where near-surface impedance contrasts dramatically enhance

\

the surface wave contributions (Hanks, 1975, 1976; Heaton and Helmberger,

b l

-I

p $ 1977, 1978; Swanger and Boore, 1978a,b; Kudo, 1978). Indeed, when layered half-space models are a reasonable approximation to the Earth structure for the source-station pair of interest, the observed long-period strong ground

-motion can be synthesized with striking accuracy (e.g., Heaton and Helmberger, 1977; Swanger and Boore, 1978a).

At frequencies of 2 Hz and greater, however, the character of strong ground motion is considerably more complicated, principally because of the

.loss of phase coherence across even very small station separations (Figure 1). At these frequencies, M p 6 earthquakes can be almost arbitrarily complicated, and it is not to be expected that wave propagation effects at such frequencies will be much simpler, given any heterogeneity at

~E all in crustal structur e. In view of the complicated phase inter ference in Figure I, it is safe to say that it will be impossible to deterministically synthesize high-frequency strong ground motion in the time domain (that is, acceleration time histories) except under rather unusual circumstances which, in general, will involve small station-spacing (< 2 km), small (M < 5) simple earthquakes at close (R < 10 km) distances, and uncomplicated geology. Evidently, a realistic characterization of high-frequency strong ground motion wi 11 require one or more stochastic parameters that can account for phase incoherence.

In the context of these matters, we here explore further the efficacy of

--the root-mean-square acceleration (a rms )/faulting duration.(T ) pair as a d

measure of high-frequency strong ground motion for seismologic and engineering purposes, using the recent studies of Hanks (1979a) and McGuire and Hanks (1980) as the Point of deParture. Since a only defined for rms can be some

.interval or "duration" of ground motion and since observed ground motion durations sei'm plainly to be a*function of period and distance, some attention is paid to -this matter in the next (second) section, wherein the model

I

\

', arms estimates of Hanks (1979a) and McGuire and Hanks (1980) are briefly recapitulated. In the third section, the model estimates are exercised against record values for a set of eight moderate-to-large California

. earthquakes and seven aftershocks of the Orovi lie, California, earthquake (Aug 1, )975; ML 5.7). These comparisons, together with those of McGuire and Hanks (1980) for the San Fernando earthquake (Feb. 9, 1971; M =6.4), produce the surprising result that, as measured by their high-frequency excitation, these earthquakes all seem to have stress drops very close to 100 bars (variations of about a factor of 2), despite the variations in their stress drops as determined by conventional means of about a factor of 50.

In the fourth section a comparison of the peak accelerations to the record arms values reveals, also surprisingly, that peak acceleration is as stable a measure of high-frequency strong ground motion as arms. Assuming that far-field, ground accelerations are finite-duration, band-limited, white, Gaussian noise, we are able to estimate the peak acceleration data from the San Fernando earthquake records (McGuire and Hanks, 1980), from the Oroville aftershocks and other California earthquakes studied here, and from the peak acceleration set of Hanks and Johnson (1976) altogether almost 400 values with reasonable accuracy. In the fifth section, we verify directly with frequency-of-occurrence counts that acceleration amplitudes within the time interval Td beginning with the direct shear arrival distribute in a nearly A

Gaussian manner. Finally, we discuss the significant potential the results of this study hold for estimating high-frequency strong ground motion for engineering purposes and for inferring the nature of stress differences that arise in the course of crustal faulting.

"6

~4 V

THE MODEL ESTIYATE OF ar Hanks (1979a) and McGuire and Hanks (1980) have demonstrated how an operation of Parseval's theorem on the Brune (1970, 1971) source model may be used to estimate arms for far-field shear waves in th'e presence of anelastic attenuation. The estimate may be written

= 2 a

rms 2R (2m) 8 e6 6 IlT R

or a

vms

= 2R e6 (2m) 106 2

sa

~R

(

f , f f 'ax max = gs

~R (2) e; when the anelastic attenuation of spectral amplitudes is of the form

-mfR/gB. alternatively, the accelerograph if this f ax may be set equal to the natural frequency of is the controlling factor. In these relations, R is hypocentral distance, g is specific attenuation, e is shear-wave velocity, o is density> f is the spectral corner frequency of the far-field shear radiation, and aa is the earthquake stress drop. R e5 is the radiation pattern in the coordinates of the instrumental axes. In Hanks (1979a) and McGuire and Hanks (1980), the rms-value for shear excitation R

~

0,6 was used, together with the assumption that the energy was equally partitioned into two horizontal components. With these assumptions, the factor of 2R eb~

becomes 0.85, the leading factor of 2 accounting for free-surface amp 1 ification.

It is implicit in these a estimates that, in the absence of anelastic attenuation, the far-field acceleration would be white for f > fo a (and in the presence of anelast,ic attenuation approximately white in the band

I l

o < f< fma). This corresponds to the ~ ", y ~ 2, model of far-'field shear-displacement spectral amplitudes. Hanks (1979a) has argued that the 2 model is the one generally but certainly not always applicable to crustal earthquakes, but, in fact, very little is known in detail about the form and variability of high-frequency source excitation, with the exception of the San Fernando earthquake (Berrill, 1975). Interestingly enough, the construction of synthetic accelerograms on the basis of finite-duration, band-limited, white noise in acceleration, an engineering approach to strong ground motion for more than 30 years (e.g., Housner, 1947, and numerous more recent studies), is consistent with the ~ model in the presence of anelastic attenuation. As it turns out, the results of this study provide substantial additional support for the y = 2 model of high-frequency excitation, although there remains a possibility of a trade-off between it and the earthquake stress drop, a matter to which we will return.

A second implicit. matter concerns the duration of motion over which the model or record estimates of arms are to be constructed. For the model estimates of (1) or (2), it is clearly the faulting duration (Td) wh;ch is of interest. Observational experience (e.g., Fletcher et al, 1980) as well as theoretical expectations (e.g., Brune, 1970; Hanks, 1979a) suggest that Td is closely allied with reciprocal fo, so long as estimates of fo unbiased by wave propagation and/or directivity effects are available. As in Hanks (1979a) and McGuire and Hanks (1980), we use here T,- (f,)'n (3) conjunction with (1) and (2).

'8

l p 'E ~

+

The faulting duration, however, is not the same as the ground motion s 5 p

duration of an actual record. quantifying ground motion duration is a tricky bus'iness (HcGuir'e and Barnhard, 1979), principally hecause ground motion is, in general, I'duration an increasing function of period and distance (Figure 2). When this is the case, it is almost always the result of waveguide effects, such as travelling surface waves, that result in delayed arrivals.

Figure 2 illustrates how this works, in the context of how peak ground-motion values can be contaminated when duration effects are not accounted for. The left-hand side of Figure 2, from top to bottom, plots peak displacement, velocity, and acceleration along a profile of stations extending to the southeast of the San Fernando earthquake (Profile 1 of Hanks,. 1975). The curves of the form R

" are not meant to fit the data in any quantitative sense, only to demonstrate that peak displacements decay less rapidly than peak velocities which in turn decay less rapidly than peak accelerations.

This is the expected result; the surprising r'esult, in view of what has been said earlier about the progressive loss of coherence in passing to higher frequencies, is that the peak accelerations decay with no more and seemingly less scatter than do the velocities and displacements. The reason for this is indicated on the right side of Figure 2 where the arrival time of the peak amplitude relative to S! (a high-frequency direct shear arrival, Hanks, 1975) is plotted against distance. The peak accelerations at all distances (32 < R < 139 km) arrive in the 10-second body (S)-wave arrival window, but for the peak velocities and displacements this is only true at the closest distances (R < 43 km). Thus, the latter data sets mix body-wave and surface-wave amplitudes, leading to additional scatter. Since the ordinate values on the right-hand side of Figure 2 are a minimum estimate of ground

motion duration, one may infer that, in general; ground-motion duration increases with period and distance.'evertheless, the faulting duration is, except in very unusual circumstances involving strong directivity effects, a lower bound to the ground motion duration.'Horeover, our experience and Figure 2 suggest that the ground motion duration converges to the faulting duration at high enough frequencies (specifically those of interest in this study) and/or at close enough distances. Since the principal contributions to the rms vvalues a

determined observationally in this study come from high-frequency (f > 2 Hz) acceleration amplitudes, it is our point of view that the faulting duration is a reasonable estimate of the duration of the high-frequency ground motion of interest. In any event, nominal variations in duration estimates (~50 percent) have little effect on the resulting a rms~

at least for the San Fernando earthqauke (tlcCann, 1980).

'inally, equations (1) and (2) are expressions for the far-field radiation, specifically that the incoming radiation arrives as bodily shear waves. The "far-field" assumption in fact involves two conditions, one that the observation distance R be much greater than the maximum wavelengths of interest in the radiated field and the other that R be much greater than the dimension of the source of the radiation. Because a both theoretically rms~

and observationally, depends so heavily on frequencies in excess of 5 Hz, the first condition is not much of a problem. Neither is the second, if we can assume that, for the acceleration amplitudes of interest here, we can simply add up the effects of many small, localized, and isolated bursts to obtain the net effect over Td, equivalently the much larger dimension of the final faulted surface. This, too, seems, reasonable if the acceleration time histories are so white, random, and Gaussian as we find them to be. Even so, 10

I r <

4 we are not verifying the far-field assumption for the analysis presented herein; we are simply making it. One problem remains, that for a site close to a major earthquake, for which R can vary greatly from the closest approach (say, 10 km or less) to the nethermost faulting location (perhaps 50 km or more). In the framework here, this would necessitate "partitioning" such an earthquake into = IO km segments and adding up the effects for appropriate distance and temporal delay. This is easy enough to do in principle, but it does involve some additional assumptions; in the absence of firm

. observational constraints on this problem, we do not here address it in any quantitative way.

Thus, we are investigating, both observationally and theoretically, a rms as a measure of high-frequency strong ground motion across a time window given by the faulting duration and beginning with the direct shear arrival. It shall be left as implicit that the surface wave contributions (that is, the long-period strong ground motion) may be constructed separately so to synthesize ground motion across the entire frequency band of engineering interest. This shall result in a narrow frequency band that will, in effect, be counted twice, but it involves small enough frequencies that its contribution to a rms is safely negligible.

COMPARISOH OF MODEL TO RECORD ESTIMATES OF a rms To compare the model arms estimates with those obtained from the recorded accelerograms, we need to know two parameters of the causative earthquake (da, f ), three properties of the medium (p, 8 and g) hypocentral distance R, and the time window for the record estimate (O <<- R/8 < Td). For p, 8, and g, we assume conventional values of

I ~ + ~

s ~ C 2.7 gm/cm, 3.2 km/sec and 300, respectively, unless otherwise specified.

The. first set of data analyzed in this section is a group of eight California earthquakes, each contributing a small number of significant records to the series "Strong Notion Earthquake Accelerograms" prepared by the Earthquake Engineering Research Laboratory of the California Institute of Technology.

The source parameters of these earthquakes, mostly determined without reference to the strong-motion accelerograms analyzed here, are culled or estimated from published studies; altogether we investigate 25 strong-motion accelerograms in this set (50 horizontal components). The second set of observations are for seven Oroville aftershocks, each writing about ten strong-motion accelerograms at R = 10 km.

A. lhe Larger California Earthquakes Table 1 lists the source parameters of these earthquakes, references to those estimates, and the strong motion accelerograms of these earthquakes analyzed in this study, identified according to the format of the series "Strong Notion Earthquake Accelerograms". These event-record combinations, span a considerable range in source strength (5.3 < M < 7.7; 1 x 10 " < N

< 2 x 10 dyne-cm), in hypocentral distance (12 < R < 130 km) and in stress droP (6 < ~~ < 140 bars). Mhile M .for these earthquakes are probably accurate.to ~ 50 percent, with the possible exception of the Wheeler Ridge earthquake for which only one, single-station estimate is available, the aa estimates are less precise; they are. no more than "consensus" values through the referenced studies. For the Bor rego Nountain earthquake, we have not attempted such a consensus estimate and investigate two end-member models.

12

I ~ ~

r Otherwise, we estimate that the aa estimates are accurate to a factor of 3 or so and the Tdd fo . estimates are accurate to

• 50 percent, again with the possible exception of the Wheeler Ridge earthquake.

Figure 3 compares the ratio of record/model a rms estimates to the earthquake stress drops (Table 1). Here, the "agreement" between model and record is perhaps best described as bad; for example, the record estimate for the Wheeler Ridge earthquake is almost 20 times larger than for the model estimate. Figure 3 does suggest, however, that the magnitude of the mismatch is a strong function of (aa) with an intercept at ao

'= 100 bars for a record/model a s

ratio of l. In Figure 4, then, we blindly recompute the model values on the basis of the given N and ba = 100 bars. This necessitates small changes in f (e.g. through the use of equation ( g) below), and the record estimates are recomputed on the basis of the new Td<s (although the record estimates change only slightly). With a full range scati.er of about. a factor of 6 and thus standard deviation of 4J a 6, t.he model and record estimates are linearly related; Figure 4 suggests that a stress drop slightly greater than 100 bars would "fit" the data slightly better. But why are the published estimates of aa seemingly low, very much so in some cases?. And why do these events, insofar as their high-frequency

-excitation is concerned, seem to exhibit the same stress drop of about 100 bars, + a factor of - 2V Temporarily, we set these issues. aside to present similar comparisons for the seven Orovi lie aftershocks.

13

I B. The Orovi lie Aftershocks

~~

To analyze the Oroville aftershocks, we take the inverse approach to that described previously, since otherwise we would be forced to use stress drops obtained from the same records we wish to investigate with respect to a That is, for each component of motion for each aftershock we estimate na by inverting equation (2). In abbreviated form the result is ho .19R f ar where we have set p = 3.0 gm/cm , e = 3.6 km/sec (Seekins et 'al, 1978) and 2 R<~ 0.85, an assumption we discuss in more detail below; and where ba is in bars, R is in km, and ar (cm/sec ) is the record estimate determined for a duration of acceleration beginning with the S wave and lasting for-(f

( o)

-1 .

Geometric means of the spectral ly determined fo ' for each aftershock (Fletcher et al, 1981) are given in Table 2; fmahas been set equal to 25 Hz, the nominal natural frequency of the St1A-1 recording devices.

Table 2 is constructed in the same way as Table 1 of Seekins and Hanks (1978), which gives peak accelerations for all of the source-station pairs in Table 2. For each source-station pair for which a usable record exists (a positively identified record triggering prior to the S wave), Table 2 gives two values of ha, one for each horizontal component of motion. -

The principal result to be extracted from Table 2 is identical to the one we obtained in the last section: in order to square record and model estimates, stress drops very nearly equal to 100 bars are required. "

The last column of Table 2 gives arithmetic and logarithmic averages of the stress drop estimates for each

1 earthquake, excluding DJR and. D>IR for which strong site resonances exist (Seekins and Hanks, 1978).

Despite the nearly monochromatic (at = 20 Hz) visual appearance of the DMR accelerograms, the ao's obtained from a at DWR are generally in agreement lms with those from other stations. This is certainly not the case at DJR, where the ho's so obtained are considerably greater (by factors of 2 to 4) than the average value. Since estimates from all other stations are much more tightly distributed about the means, something clear ly anomalous is occurring at DJR.

/

At first glance, this would appear to be the strong site resonance at' 6 Hz that always affects the DJR accelerograms, but similar difficulties with velocity, radiated energy, and displacements have also been encountered for the DJR records (e.g. Fletcher et al, 1980; J. L. Boatwright, personal communication). In view of these unusual ground motion characteristics at DJR for the aftershocks, it is probably /not coincidental that very substantial

/

damage was done to the wood-frame residence at this site at the time of the mainshock.

Five of the seven aftershocks have average stress drops between 87 and 101 bars (arithmetic averaging), very constant stress drops indeed by conventional seismological standards (e.g. Hanks, 1977). The other two have average stress drops of about twice as much. In all cases, the standard deviations are only fractions of the average value. Excluding DMR and DJR, Table 2 gives 104 aa estimates, of which only 6 are less than 50 bars and only 10 are greater than 200 bars; all but one of the latter group are associated with the two higher stress drop events 0350 and 2234. The significance of this result is two-fold. First, using wo = 100 bars, one can estimate a rms with the observations within a factor of 2 most of the time (- 85 percent).

I Second~ arms is a remarkably stable measure of high-frequency strong ground motion; the ha estimates in Table 2 are a fairly transparent image of a rms~

fo is the same for any aftershock (and does not vary much across the set) and s.ince R varies only between 10 and 15 km for most source-station pairs.

"o<<ovei th<<ec<<d ar estimates, as reflected by the ha estimates, depend only weakly, if at all, on the shear-wave radiation pattern. Table 3 compares, for the 0231 aftershock, the stress drop estimates normalized by the average value aa = 101 bar for each component of motion to the radiation pattern coefficient normalized by <R > = 0.6 for the same component of motion. The coefficient 1.76 simply centers the R ~/<R eIII olti ~> distribution at unity for easier comparison with the aa/aa distribution. According to equations (1) and (2), ha/Za should have an inverse relationship to Rey/<Re~>, since for all ha estimates in Tables 2 and 3 we have used

<R ey> = 0.6. No such correlation exists in Table 3. Moreover, the form of the two distributions is grossly different: while the avoca estimates vary over a total range 'of a factor of 2, varies over ey./<R e6 >

R 2 orders of magnitude. Evidently, the excitation of high-frequency motion is far more isotropic than double-couple point source models allow -- or that scattering mechanisms along the propagation path give this appearance. If this is indeed a source effect, as we tentatively conclude three sections hence, it can only mean that the high-frequency radiation arises from faulting processes far more complicated than a point-source representation. Neither does a to rms appear be closely related to site geology, with the exception of DJR, although the peak acceleration data showed a gross distinction between sedimentary and hard-rock sites (Seekins and Hanks, 1978).

16

I C. Discussion It is a surprising result that the observed high-frequency ground motion of these 15 earthquakes suggests, according, to (1), (2), and (4), that all have stress drops very nearly equal to 100 bars. Our study of the San Fernando earthquake (NcGuire and Hanks, 1980) produced the same result, although we did not appreciate it as such at that time: a stress drop of 100 .-

bars would have fit the data better (and in an absolute sense very well) than the 50 bars we actually used (Figure 4 of NcGuire and Hanks, 1980). Formally at least, all of these stress drops are the average static stress drops determined from the Brune (1970, 1971) scaling, although they have been estimated differently here from what is conventionally done (e.g. Hanks and Wyss, 1972; Thatcher and Hanks, 1973; Tucker and Brune, 1973). The enigma is that, in a set of 16 earthquake stress drops with such a diverse range in magnitude and tectonic setting, there is no reason to expect they would be so nearly the same (10 2

• 0.3

'ars). While the constancy of earthquake stress drops has been a long-recognized seismological phenomenom, it has ordinarily

,been in the context of estimates distributing over two orders of magnitude (e.g. Hanks, 1977); not + a factor of 2. Nore specifically, our results for stress drops differ from those published for the eight earthquakes in Table 1 and for the San Fernando earthquake, often by a considerable margin. When it exists, however, the problem is always that the published stress drops are lower than the ones determined here. Certainly C

it is possible to underestimate stress drops; whether the source dimension is estimated spectrally or from aftershock dimensions, it is far more likely that a value too large will be obtained than one too small, thus leading to underestimated stress drops.

I > v k

~ \ ~

Irrespective, however, of the biasing problems that might affect this set of data, why is it that all of these earthquakes, given their record a values, seem to have such nearly coincident stress drops of 100 bars, as determined'from (1), (2) and (4), no matter what the conventionally determined value? lie do not know the answer to this question, but we suspect that some tradeoff might exist between the stress drop as estimated in the published studies and the high-frequency spectral decay rate (implicit and fixed at i = 2 in (1) and (2)). That is, if so is less than 100 bars and y is less than 2 or if ho is greater than 100 bars and q is greater than 2, it is Possible, at least in PrinciPle, that the same result for rms would a as wa = 100 bars and q = 2.

'Given the available observations, it is premature to press this possible connection between Aa and y too far, but it is potentially of some seismological significance since aa and q a'e generally considered to be independent quantities. nevertheless, there has been, recently, considerable interest in what relations might exist between frequency-magnitude statistics (b-values), so, and q (e.g., Hanks, 1979a; Andrews, 1980), and the results of this section provide at least some justification for further consider ation of these issues. From the Point of view of estimating 'a a

rms as a measure oof strong ground motion for engineering purposes, however, these matters make no real difference. The observed a values fcr the 176 horizontal components s

records considered in this study and the 160 horizontal compared considered in the study of the San Fernando earthquake (HcGuire and Hanks, 1980) are estimated with remarkable accuracy using aa = 100 bars and (1) and (2); most of the observations, with the consistent exception of those at DJR, are within a factor of 2 of this estimate.

18

I I ~

~

' PEAK ACCELERATIONS The value of peak acceleration as a measure of strong ground motion for engineering purposes has come under close scrutiny in recent years, despite the central role it has played in these matters since the first strong motion accelerograms were written almost 50 years ago. In the first place, there is considerable question concerning the reliability of peak acceleration as a measure of damage potential (e.g., Blume, 1979). A second difficulty with peak acceleration is its weak dependence on source strength; at least for potentially damaging earthquakes, peak acceleration at some fixed distance can be expected to "saturate" for exactly the same reasons that the magnitude scales mb, ML, and Ms also saturate (e.g., Hanks, 1979b; Hanks and Kanamori, 1979). Still another difficulty is the large scatter that exists in any reasonably large peak acceleration data set, even'hrough very small ranges in hypocentral distance and earthquake magnitude (e.g., Seekins and Hanks, 1978). Because of these problems, it came as some surprise to us that peak acceleration (a ) correlated so strongly with arms over 0 < t- R/8 < Td at all distances and azimuths for the San Fernando earthquake (NcGuire and Hanks, 1980, Figure 6).

Figure 5 shows a similarly strong correlation between a max an arms for the seven Orovi lie aftershocks, plotted on a linear scale. The strong correlation between a ax and arms and the sharply defined bounding ratios of 2 6 amax/arms e 3 suggest amax is simply the last amplitude point in a Gaussian distribution of acceleration amplitudes, removed from the mean (of zero) by two to three standard deviations. In the next section, we verify directly with frequency-of-occurrence counts that acceleration amplitudes within 0 << R/8 <

Td indeed distribute in a nearly Gaussian manner; here we explore the predictability of peak 'accelerations on this basis.

19

1 I

Apcording to Vanmarcke and Lai (1977), ground acceleration time histories that are stationary, random, and Gaussian in a time interval s will be characterized by a'ax 2s arms

~o (5) 0 where the amax so determined should be exceeded once on the average or, with a probability of e , will not be exceeded during s os

~

here T 0 "the predominant period of the earthquake motion".. In the framework of this study, we equate so with the faulting duration Td for which the record arms values are calculated and for To take the reciprocal of fmax, the highest frequency passed by the attenuation operator or the accelerograph.

Equation (5) then becomes max 21n (6a) mf R rms 0 a

max 2 fmax (6b) n arms f I

Plainly, some highly restrictive assumptions are implicit in (5) and (6), the discussion of which>>e delay in order to investigate whether or not this estimate of amax/a max rms works.

Figure 6 Plots the amax/arms data f'r each of our three PrinciPal data sets. Figure 6a is for the seven Oroville aftershocks; Since virtually all of the observations are written at much the same distance (10 to 15 km), we have plotted observed values of a max /a rms versus f0 for each of the

~ seven earthquakes. The triangles pointing horizontally and connected by the long-dashed line are the values predicted, from equation (6b) with fmax =

"z ~ Th<<ange in f for these seven events is not great, and the range in 20

I ~

the theoretical ama/arms is much less; even so, .with the exception of aftershock 8, the median values of .the observations (connected by the sh6rt-dashed line) track the trend of the theoretical prediction. In every instance, the observed a maxx/arm is within ~ 30 percent of the theoretical prediction.

Typically, however, the median values are lower than the predicted values by 10 to 15 percent; all of this effect can be attributed to our using Volume II (equally spaced data) peak accelerations, which generally are 10 to 20 percent smaller than peak accelerations scaled from Volume I data (unequally spaced, uncorrected for instrument response). Indeed, some care should be exercised in the use of peak accelerations taken from equally. spaced representations of short-duration, high-frequency accelerograms such as those for the Oroville aftershocks; in three cases, all at DMR, the Volume II a max was a factor of 2 or more lower than the Volume I amax (in these cases, the Volume I datum was used).

Similarly, Figure 6b plots a x/ar observations for the San Fernando earthquake as a function of distance. The theoretical prediction is from equation (6a), with g = 300, s = 3.2 km/sec, and f 0.1 Hz (gerrill; Ig75; HcGuire and Hanks, 1980). In this case, the observed ratios're all within

~ 50 percent of the theoretical estimate. Generally speaking, a decreases with increasing distance, in accord with the theoretical estimate, but here the theoretical estimate is slightly less (by 5 to 10 percent) than what might be a median observed value as, a function of distance.

Finally, Figure 6c presents observed and theoretical values of a /a for the other moderate-to-large California earthquakes. For this data set there are large variations in both fo and R (Table 1), so Figure 6c plots the record estimate versus the model estimate (equation (6a)

'21

I ~ < I I ~ ~I evaluated for appropriate f, o% R, g, and g). Again, the observations agree with the theoretical estimate to + 50 percent in every .instance, more of the observations than not being greater than the prediction.

To explore further the predictive value of (6), we estimate the NL > 4 peak acceleration sets at close distances of Hanks and Johnson (1976) and Seekins and Hanks (1978) by combining equations (1) and 6a):

max r)n (7) n pR To evaluate (7), we set wa = 100 bars, consistent with the results of the last section (aa is known for some but not all of the earthquakes represented in Figure 7), and set R = 10 km, nominally consistent with the observations. All magnitudes for these earthquakes are t< except for the two largest which are H . Again for the purposes 'of evaluating (7), we treat the abscissa as moment magnitude M (Hanks and Kanamori, 1979). Thus i'1 is determined by 0

log M - 1.5 M+ 16.0 (8) from which f for each N can be determined by the constant stress drop relation (Hanks and Thatcher, 1972) bo = 100 bars = 8 q7

' fo (9)

~8 The numerical result of these operations is sho~n as the heavy solid curve in Figure 7. He have chosen not to plot (7) for M < 4 (or plot the HL < 4 data of Hanks and Johnson (1976) and Seekins and Hanks (1978)) because as f0 aPProaches fma equation (5), which assumes at least several cycles of 22

~

~ t

ground motion, is no longer valid. Below M = 4, faulting durations need be more than O.l sec or so. Neither have we plotted (7) for M > 6.5, for reasons

. discussed below.

The two light solid curves are an evaluation of the expression of Joyner et al (1981):

log a = 1.76 + 0.28 M log R 0.00255 R + 0.27P (10) where we have set P=O for 50 percent probability of exceedance. The upper curve is evaluated at R = 7.3 km (zero horizontal distance to the epicenter or

'faulting surface); the lower curve is for R =. 10 km. Note that we give the relation of Joyner et al (1981) in cm/sec2, 4

where they give it in units of 980 cm/sec; R is measured in km. The constants in (10) are determined from a regression analysis on peak accelerations recorded at distances out to severa) hundred km for earthquakes with 5.0 < M < 7.7.

Also shown in Figure- 7 is the relation of Donovan (1973) lo9 amax = + 0.25 M 1.52 log (R + 25) evaluated at R = 0 and R = 10 km (light dashed lines). ){ere we have takenM for magnitude, and R is a "closest" distance in km. Donovan (1973) does not detail the range of validity for (ll); we plot it for 5 < M < 7 1/2.

In the range of mutual validity, 5.0 < M < 6.5, the empirical relations of Donovan (1973) and Joyner et al (1981) and the theoretical relation developed hei e are remark ab ly s imi ar; the severa 1 1 dependence s of 1 og a max on M are 23

nearly identical. For our model, the a dependence on

" II 1/3 max 0.45 o (")"'3/2 0 p)q 4.141 (12) 1/3

.R(~<)

works out to be almost exactly

~

1/5 max o (13) for 4 < M < 6.5. With the use of (8), this becomes log amax ,30 M (or tlL) (14)

The coefficient on M is 0.25 for the Donovan {1973) relation and 0.28 + 0.04 for the Joyner et al {1981) relation. Differences in absolute values for log amax of 0.2 to 0.3 units can (and do) arise through various definitions of the "zero" or closest distance, a problem not readily amenable to resolution given the scatter of comparable magnitude in the observations for any M at close distances. 1n view of the limitations of the data set in Figure 7, its natural scatter, and the real variations in R .for the observations that have been suppressed with R = 10 km, all of these relations "fit" the observations about as well as can be expected for 5 < M < 6 1/2.

All of these relations, however, either do or would underestimate the Hanks and Johnson (1976) data for 4 < M < 5, by factors of 2 to 3. Our theoretical expression agrees well with the the HL > 4 Oroville aftershock mean-v>><< amax data, however, and we suspect that these are the more compelling observational constraints. The data set of Hanks and Johnson (1976) was compiled simply from the largest peak accelerations then known at R -" 10 km, without knowledge of what might be an appropriate mean value at M < 5. Nevertheless, there remains a possibility that the log a max dependence on M (4 g M g 6 1/2) at close distances might be some~hat less than 24

~ ~

that given either by our model or the empirical relations of Donovan (1973) and Joyner et al (1981).

Of greater concern is the M dependence of a at close distances for max M> 6 1/2.. In the M > 6 1/2 and R g 10 km space, the observations are very sparse, and both the assumed functional forms of the empirical studies and our theoretical model are, of uncertain validity. Nevertheless at M = 6.5 and R = 10 km,we are impressed by the convergence of observational results, theoretical expectation, and popular consensus: peak accelerations in the vicinity of 1/2 g (+50 percent) will be the norm for M= 6 1/2 earthquakes at close distances, not the exception (e.g. Donovan, 1973; Hanks,and Johnson, 1976; Boore et al, 1980; Campbell et al, 1980; Joyner et al, 1981; and the results of this study). But,what may we expect for greater M'P In the model presented here, all the magnitude dependence of peak acceleration is a function of duration alone: In view of the reasonable match of the observations to theoretical expectation, at least for 4 < M < 6 1/2, we conclude that larger earthquakes have larger peak accelerations because they last 'longer, not, becuase they are intrinsically more powerful, an hypothesis advanced previously by Hanks and Johnson (1976) without the benefit of equation (7). Thus, at close disi.ances, if the duration of ~stree est motion cannot increase beyond a certain value (perhaps, for reference, 10 to 20 seconds), then neither can the peak acceleration, if we can always count on finding it in the strongest motion portion of the record.

'1 Specfically, larger magnitude earthquakes will be associated with fault lengths that become progressively larger than R = 10 km,'hich then would be the distance of closest. approach. Fven though the faulting duration of such an event will continue to increase, the a at any "close" site over that duration will decrease with increasing duratio~, because of the greater distances the 25

radiation from the outlying faulting segments must travel to that site. But we do not expect amax to decrease for such events if it is caused by faulting on the closest segment,.taking it to be the source of the strongest motion On the other hand,ama need not increase by much above " 1/2 g for M> 6 1/2 at close distances, and we expect it will not, the linear increase in log a - above M = 6.5 assumed in the empirical re,lations of Donovan (1973) and Joyner et al (1981) notwithstanding.

THE GAUSSIAN CHARACTER OF ACCELERATION TINE HISTORIES

'he results of the two previous sections suggest that there is considerable evidence that the high-frequency strong ground motion that arrives as band-limited (fo direct

<< shear waves is finite-duration (0 fmax) white, Gaussia'n th<<ma/arms ratios estimated on this basis with a variety of noise.

< t-R/8 <

The agreement Td),

between observations, while far from perfect, nevertheless provides a stringent test of the assumptions implicit in equations (5), (6), and (7), given any scatter at all in the observations not explicitly accounted for in the model and uncertainties in the parameters used in the model.. In particular, we have estimated to ~ 50 percent, most of the time, the last amplitude point (a max in almost 400 series of several hundred to more than a thousand acceleration amplitudes, depending on the record. It is unlikely that this coincidence would materialize if any of the above characterizations were seriously in error, although we have previously noted the potential trade offs between stress drops and high-frequency spectral decay rates. In this section, we verify the Gaussian character directly with frequency-of-occurrence plots of the amplitude data.

26

I I

I SI that acceleration amplitudes within the

~ ~

Figues 8a and 8b demonstrate shear-wave arrival windo~ (0 < t-R/'s < Td) distribute in a very nearly Gaussian manner. Figure 8a shows these amplitude distr ibutions plotted on normal probability paper for one component of motion from each of our three principal sets of data, those for the San Fernanado earthquake, the other moderate-to-large California earthquakes, and the Orovi lie aftershocks. The abscissa in each case is acceleration amplitude normalized to 'a rms for that

, record. In Figure 8b are similar results for all of the San Fernando earthquake records analyzed by NcGuire and Hanks (1980), amplitudes again being normajized to a for each record of the set. Similar data for the rms entire Orovi lie aftershock data set (Table 2) and the other moderate-to-large California earthquakes (Table 1) overlap the symbols in Figure 8b to the point where the three sets could not be distinguished at the given symbol size; they are not shown, here.'oth individually and collectively, then, these figures demonstrate that acceleration amplitudes in the shear->>ave arrival window are distributed in a manner only trifling different from Gaussian, but this should come as no great surprise, in view of our ability to estimate so well the last amplitude point (the peak acceleration) under the Gaussian assumption.

SUNNARY ANO CONCLUSIONS From an analysis of more than 300 components of ground acceleration of 16 earthquakes of magnitude 4.0 < N < 7.7 at distances of 10 g R g 100 km in this study and HcGuire and Hanks (1980), high-frequency strong ground motion is, to a very good approximation, band-limited (fo < f < fmax ) > finite duration (0 < t- R/8 < Td), white, Gaussian noise. Its rms value is fixed by the 27

~ ~ ~

single eartlnluaae source parameter 1/p (or eouiva]ent]Z aio ), since r all of these. events ~~ has been est imated from ~eco~d est imates o rms

'to be nearly equal to 100 bar's. Although we have made no formal statistical analysis of the matter, with these parameters and equal'ly simplistic wave propagation effects, one can model estimates of a rms that agree within a factor of 2 of the record estimates about 85 percent of the time. These results are of great significance for the rational construction of synthetic accelerograms, at least for H < 6 1/2 earthquakes.

On this same basis, one can predict peak accelerations with comparable accuracy. The magnitude-dependence of peak acceleration determined empirically by Joyner et al (1981) for 5.0 < M < 7.7 is remarkably similar to that determined by equations (7) and (8) for 4.0 g Mg 6 1/2, the magnitude coefficient being 0.28-in the first case and Oa30 in the second. This

=magnituae dependence of peak acceleration is a function of faulting duration alone; at a fixed, close distance larger earthquakes have larger peak accelerations only because they last longer, not because they are intr insically more powerful. Even from just a seismological point of view, then, faulting duration is an important ground motion parameter. Should "effective peak acceleration" continue to want for a "precise definition in physical terms," we recommend that it be a rms for the time interva 0 < t R/8 < Td.

Seismologically, these results are no less fascinating. Of considerable significance is the result that the earthquake stress drops as determined by a are very nearly equal to 100 bars; estimates for any of the 16 earthquakes of interest here vary over + a factor of 2, not 2 orders of

'28

~ ~

~ v ~

magnitude. While all of these are California earthquakes, they occurred in a variety of tectonic settings and spanned a considerable range in source strength; had past experience in. stress-drop determinations been the measure of expecta'ion, there would be no reason why these events would have so nearly the same stress drops. The results reported here, however, are the only available stress droPs determined by a . "'nd they may be more akin to dynamic stress drop than an average static stress drop, in'iew of the high-frequency character of the motion from which they are derived. Only further study of a larger class of earthquakes will reveal whether or not a wider ra~ge of arms-determined stress drops will materialize and whether or not the offsetting relation between ho and y, as postulated earlier, might exist. Similarly, Table 2 suggests that a is a remarkably and rms surprisingly stable measure of high-frequency ground motion, no matter what purpose to which it is put; further study of this phenomenom is also warranted.

While scattering mechanisms undoubtedly contribute some component to the chaotic but stochastically well-behaved character of high-frequency strong ground motion, we believe for two reasons that these properties are fundamentally due to properties of the source, not path. First, there is no distance-dependence of the scatter in the ratio for max /a rms a

Fernando earthquake; the acceleration time histories are as white, Gaussian, and random at 20 km as at 120 km, although the bandwidth narrows with increasing distance. Secondly, in the case of the 0103 aftershock of the Orovi lie earthquake, Boatwright (1981) was able to identify and locate 5 sub-events within the - 0.7 second faulting duration. This complexity is plainly a function of the source, not path; indeed, had there been much phase incoherence at all along the propagation paths, it is unlikely that this sub-event analysis would have been possible.

29

~ ~

~l ' '

On this matter as wel l, further study is plainly indicated, but the results of this study suggest that the stress differences that develop in the course of crustal faulting, giving rise to the acceleration time histories studied here, are also very well-behaved, both in the average stress release through the characteristic source volume and in the character of stress differences that develop across smaller dimensions. In particular, given the white, random, and Gaussian character of acceleration time histories for

, 0 < t-R/8 < Td, the I linear relation of Hanks and Johnson (1976) between dynamic stress differences for localized and isolated events in the source region and acceleration amplitudes at R should allow for a straightforward mapping of the latter into the former. These results should be of value in the further development of the relationships that might exist between frequency-of-occurrence statistics (b-values), average earthquake stress drops (ao) and rate of high-frequency spectral decay in the far-field (y) and how any or all of them might be related to the distribution and spectral composition of quasi-static stress differences existing along active crustal fault zones (e.g. Nur, 1978; Hanks, 1979a; Andrews, 1980; von Seggern, 1980).

Finally, it seems almost certain that inhomogeneities in the quasistatic I tectonic stress field that gives rise to the earthquake or inhomogeneities in stress differences that develop dynamically in the course of crustal faulting or most-likely both effects are an intrinsic characteristic of crustal earthquakes. M = 4 earthquakes appear to be as complicated at periods = 0. 1 sec as M = 6 earthquakes are at periods = 1 sec and M = 8 earthquakes are at periods = 10 sec. Indeed, earthquake "complexity" may be a self-similar charcteristic of crustal earthquakes. In view of the heterogeneous morphology 30

4 of active crustal fault zones, so readily apparent in their en-echelon, locally discontinuous character as:observed both geologically (Wallace, 1968;

,Tchalenko, lgi0) and seismically (Faton, 1970; Bakun, 1980), this "complexity" character of crustal earthqual:es is probably to be expected.

31

I s

Ref erences B,J, ~1950). A stochastic fault model, 1. si.ai.ic case. J. ~Geo I Res.. 85, 3867-3877,.

Applied Technology Council (1978). Tentative provisions for the development of seismic regulations for buildings, Hational Bureau of Standards Special Publication 510, pp. 506.

Bakun, H.H., R.tl. Stewart, C.G. Bufe, and S.N. Harks (1980) Implication of seismicity for failure of a section of the San Andreas fault. Bull.

Seism. Soc. Am. 70, 185-201.

Boatwright, J. L. (1981) quasi-dynamic models of simple earthquakes:

application to an aftershock of the 1975 Oroville, California earthquake, Bull. Seism. Soc. Am., 71, 69-94..

Berrill, J.B. (1975). A study of high-frequency strong ground motion from the San Fernando, earthquake, Ph.D. Thesis, California Institute of Technology, Pasadena.

Blume, J.A. (1979). On instrumental versus effective acceleration, and design coefficients, Proceedings of the 2nd U.S. Hational Conference on E tl E gi i 9, 868-882, E ~E Institute, Berkeley.

Bolt,, B. A. (1978). The local magnitude NL of the Kern County earthquake of

\

July 21, 1952, Bull. Seism. Soc. Am. 68, 513-515.

Boore, D. M., ll. B. Joyner, A. A. Oliver III, and R. A. Page (1980). Peak accelerai:ion, velocity, and displacement from strong-motion records, Bull.

Seism. Soc. Am. 70, 305-321.

Brune, J.N. (1970). Tectonic stress and the spectra of seismic shear waves from earthquakes, J. ~Geoph s. Res. 75, 4997-5009.

Brune, J.N. (1971). Correction, J. ~Goo h s. Res. 76, 5002.

.32

\'

> C

a 4

~ s ~

Joyner, W.B., D.M. Boore, and'R.L. Porcella (1981). Peak horizontal acceleration and velocity from strong motion records including records

~~

.from the 1979 Imperial Yal ley, California, earthquake, U.S. Geol. ~Surve Open-f i le Report 81-365.

Kanamori, H. and D.L. Anderson (1975). Theoretical basis of some empirical relations in seismology, Bull. Seism. Soc. Am. 65, 1073-1096.

Kanamori, H. and P.C. Jennings (1978). Determination of local magnitude, ML, from strong-motion accelerograms,- Bull. Seism. Soc. Am. 68, 471-485.

Kudo, K. (1978). The contribution of Love waves to strong ground motions, Proceedings of the Second International Conference on Microzonation for Safer Construction Research and Applications, 765-776.

McCann, M. W. (1980). rms acceleration and duration of strong ground motion,

~Oe t. Civ. Enr19. Rept. No. 46, Stanford University.

McGuire, R.K. and T.P. Barnhard (1979). Four, definitions of strong motion duration: their predictability and utility for seismic. hazard analysis, U.S. Geol. ~Surve Open-file Report 79-1515.

McGuire, R.K. and T.C. Hanks (1980). RMS accelerations and spectral amplitudes of strong ground motion during the san Fernando, California earthquake, Bull. Seism. Soc. Am. 70, 1907-1919.

Bur, A. (1978). tionuniform friction as a physical basis for earthquake mechanics, Pure Appl. Geophysics 116, 964-989.

Schnabel, P. B. and H. B. Seed (1973). Accelerations in rock for earthquakes in the western United States, Bull. Seism. Soc. Am. 63, 501-516.

Seekins, L.C. and T.C. Hanks (1978). Strong motion accelerograms of the Orovi'lie aftershocks and peak acceleration data, Bull. Seism. Soc. Am. 68, v

677-690.

.35

~ l ~

4 Seekins, L.C., D.P. Hill, and T.C. Hanks (1978). Shear-wave velocity structure near Orovi lie, California, Bull. Seism. Soc. Am. 68, 691-698.

Swanger, H.J. and .D.M. Boore (1978a). Simulation of strong-motion displacements using surface-wave modal superposition, Bull. Seism. Soc.

Am. 68, 907-922.

Swanger, H.J. and D.N. Boore (1978a). Importance of surface waves in strong ground motion in the period range 1 to 10 seconds, Proceedings of the Second International Conference on Microzonation for Safer Construction Research and Application, 1447-1457.

Tchalenko, J.S. (1970). Similarities between shear zones of different magnitudes, Geol. Soc. Am. Bull. 81, 1625-1640.

Thatcher, H. and T.C. Hanks (1973). Source parameters of sourthern California earthquakes, J. ~Geoph s. Res. 78,'8547-8575.

Tocher, D., (1959). Seismic history of the San Francisco region, San Francisco Earthquakes of March 1957, ~aif Div ~Min ~~ $ n c. ~R 57.

Tr ifunac; lf. D. (1976). Preliminary analysis of peaks of strong earthquake ground motion-dependence of peaks on earthquake magnitude, epicentral distance, and recording site conditions, Bull. ~Scrim. ~S. Am. $ 6, 189-219.

Trifunac, H.O. and J.N. Brune (1970). Complexity of energy release during the Imperial Yal1 ay, California, earthquake of 1940, Bull. Seism. Soc. Am. 60, 137-].60.

Trifunac, M.O. and F.E. Udwadia.(1975). Parkfield, California, earthquake of June'27, 1966: a three-dimensional moving dislocation, Bull. Seism. Soc.

Am. 64, 511-533.

I I

Tucker, B.E. and J.H. Brune (1973). Seismograms, S-wave spectra, and source parameters for aftershocks of San Fernando earthquake, in San Fernando,

" California Earl:hquake of ~februar 9, 1971, Yol. III (Geological and Geophysical Studies),69-122, U.S. Dept. Comnerce Vanmarcke, E.H. and S.P. Lai (1977). Strong-motion duration of earthquakes,

~ge t. of Civil ~En . Pub. No. R77-16, Massachusetts Institute of Technology, Cambridge.

von Seggern, D. (1980). A random stress model for seismicity statistics and earthquake prediction, ~geoph s. Res. Lett. 70, 637-640.

Wallace, R.E. (1968). Notes on stream channels offset by the San Andreas fault, Southern Coast Ranges, California, Proceedings of .Conference on Geologic Problems of San Andreas Fault System, Stanford Univ. Publ. Geol.

Sci. 11, 6-21.

37

l 4

)

Table 1 8arthpuake SOurCe ParameterS and Strorrg-Notion Acceler ogr ams Earthauake Mo aa Td~ fo-1 Accelerograms References date; nagnftude dyne-cm bars sec lO S-arrival R (CIT) Sec kn f.prrg BeaCh 2 1/2 x 1025 10 10 8021 1.0 50 Thatcher and Ha~ks (1973)

Har. 11, 1933; Hf 6.3 V314 1.0 55 Ka~amprf and Arrderso< (1975)

V315 0. 5 20 Ieperfaf Valley 3 1026 50 12 AO011 0.7 20 Tr ifunac and Brune (1970)

Hay lgr 1940; Hse7 lr Hf a6o4 Kern County 2 x 1027 60 20 A003 11. 7 130 Kaoamori and Arrderson (1975)

July 21, 1952; Ms~7.7. Hi~7.2 A004 2.7 40 Hanks ( 1976, 1979a)

A005 6.2 90 Kananprf and Je~~fngs (1978)

A006 8.5 120 Bolt (1978)

Hheeler Rfdge 3 x 1024 8031 4.0 40 Tt atcher a~d Hanks (1973)

Jarr. 12, 1954L 5 9 Sarr Francfsco2 x 1024 50 1.7 A013 0.6 20 Tocher (1957)

Har. 22, 1957; HL 5.3 A014 1.1 20 A015 1.0 15 A016 0.5 20 AO17 0.3 30 Parkffelo - 3 1/2 x 1025 17 8033 1.6 20 Kanamprf arrd Anderson (1975)

June 28, 1966; Hs 6.4, Hf 5.5 8034 4.6 20 Tr ffunac arrd fJdrradfa (1975) 8035 1.2 25 8036 1.9 25 8037 1.1( ) 30 Borrego Hovntafn3 7 x 1025 A 17 10 A019 6.6 60 Hanks and Hyss (1972)

Apr 9, 1968; Hsi6.7, HL 6.4 8 140 5 A020 100 Karramprf arrd Arrderson (1975)

Bvrdick and Hellma~ (1975)

Heatorr aud Helmberger (lp77) lyt le Cree'k4 l x 1024 30 M334 0.2 12 Sept 12. 1970; Mt~5.4 M335 0.0 18 Hanks et al (1975)

M338 0.0 - 25 Analysis of this record performed rrfth p 2.5 ~ gm/cm3.~ ei2.5

~ km/sec r arrd Q 200 Hp deternfrred frOm log Hp 1 5 Ht + 16 0'p determined f'rom aftershoc'k dfme~sforr r 2km Model A according to irrterpretatiorrs of Hao'ks aud 'Myss (1972) and Karramorf and Arrdersorr (1975); Model 8 according to Bvrdfc'k aud Hellmarr (1o76) 4 Hp and fp frerr unpvbffshed dai.a of T. C. Hanks

I I ~ I

Table 2 arms - determined Stress Drop Estimates (bars) for Seven Oroville Aftershoc'ks Oate Origin Mt fo 2 3 OAP 1 4 5 ESH OMC OJR 7 OMR 6 ~

8 Hr min (H~) N9ON N85M N90N H90E N35N SOOE NSOE H24M N90E H90N SOOE 555E N9OM H20M ar it>

SOOE 505M SOVE NOOE 555M H9OE NOOE 566N 7iOOE SOON N9OE H35E SQOE 57Q'X log Aug 3 0103(A) 4.6 1.4 . 42 63 129 83 110 156 107 125 97 38 27 114 103 135 143 117 91 87 Aug 3 0247(8) 4.1 1.9. 96 113 133 54 101 59 53 90 45 83 97 106 71 122 37 86 Aug 6 0350(F) 4.7 1.3 121 123 185 186 96 167 171 248 344 304 173 124 293 279 180 138 160 171 119 887 144 164 Aug 8 0700(K) 4.9 1.6 83 105 72 68 78 88 65 178 65 87 56 178 82 73 138 200 62 83 95 82 Aug 16 0548(P) . 4.0 3.0 45 164 58 70 107 106 190 70 101 123 53 97 59 312 98 105 74 41 228 56 94 114 91 85 Sep 26 0231(T) 4.0 2.2 88 91 77 146 89 95 358 130 S 5 101 74 110 104 153 89 81 258 83 98 Sep 27 2234(0) 4.6 '.2 320 176 175 204 178 142 144 474 184 278 155 195 324 209 132 161 235 998 231 127 164 187 Exclud>ng OJR and OMR Denotes a record thai. did not clear ly trigger on or before 5-wave arrival

r

~ I ~ ~ ~

Table 3 Comparison of .Hormalized arms determined stress drop estimates to normalized shear-wave radiation patterns for the Orovi lie aftershoc'k 0231 ..

Structure Component ba/ha 1. 76 RgjII/(Roy >

OHC H24W 0. 94 n. 56 S66M 0. 80 0. 96 C06 S55E l. 29 1. 03 H35E 0. 83 2. 40 H90E 0. 88 1. 57 HOOF 0.88 0. 3A SOOE l. <5 9. 16 H90E 1. 51 1. 10 CD4 H35W 0.76 0.41 SSSH 1. 03 '1. 08 OAP H90W 0. 87 1.5n SOOE 0. 73 n. 024 C01 H90E 0. 90 l. 18 ROOE 1.09 Q. 43

l 4

Figure Captions 1, Transverse components of ground acceleration of the San Fernando earthquake for 15 seconds beginning with the S-wave arrival at the Seismo,logical Laboratory (top) and the Athenaeum (bottom) of the California Institute of Technology in Pasadena; these sites are 36 and 40 km, respectively, from the hypocenter.

2. Peak displacements, velocities, and accelerations for the San Fernando earthquake along Profile 1 of Hanks (1975) are plotted from top to bottom on the left side of the figure. 1'he crosses are the more transverse component of horizontal motion, the circles the more radial component.

The arrival times of these phases with respect to S! (Hanks, 1975) are plotted on the right side of the figure. Note that the peak accelerations at all distances arrive within 10 seconds of S!.

3. The ratio of record/model ar for each source-station pair of'able 1 is plotted against earthquake stress drop.

4. ComPa~ison of ~eco~d ar s to model arms for the source-station Pgirs of Table 1, (and Figure 3), when all model estimates are calculated with bv = 100 bars.

5. ama vs arms for the seven Orovi lie aftershocks (Table 2). In order to present this data on linear scales, four data pairs with a > 11O

~

rms

'I cm/sec are not included. Otherwise, there is one point for each source/station component'air in Table 2. Circles are bedrock sites, crosses are sedimentary sites, according to the classific'ation of Seekins and Hanks (1978).

f' )

6. Record and model estimates of a max

a. Oroville aftershocks, as a function of f

b. San Fernando earthquake, as a function of R

c. Other moderate-to-large California earthquakes, observed ratio as a function of theoretical ratio 7 The l'1L > 4 peak acceleration data at R = 10 km of Hanks and Johnson (1976) (solid symbols) and the Orovi lie aftershock mean-value peak accelerations (Seekins and Hanks, 1978) (open and half-filled circles for sediment and bedrock sites, respectively). The peak acceleration-magnitude relations at fixed close distances of Donovan (1973) and Joyner et al (1981),

labeled JBP, are as shown for two choices of R. The numerical expression of the model developed in this study is labeled as such.

8. Acceleration amplitude frequency-of-occurrence counts, normalized to the record arms over 0 < t-R/g < Td

a. individual components from the three principal data sets, the San Fernando earthquake, the other moderate-to-large California IV earthquakes, and the Oroville aftershocks

b. all records for the San Fernando earthquake used by NcGuire and Hanks C

(1980).

I

~ ~

1 'I ~

TOMS

I I P I

~ ~

O E so E6

a. 20 E4p R I~2 0 O

Ip ~2O x ~ 0 O x 0 x x O 0 CL p C:

20 i E 4P 0 00 R o Io x x O 20 0 x x a Y 0 0 cu 0 p a 500 CL

~EO 0 O

uE 200 Cl 4Q E

oO IOO n 20 0 L 0x 0 0 x 0 x

x t) Q p 50 Io I 50 0 50 IOO l50 CL Distance, km

S RECORD / MODEL ARMS V(HEELER RIDGE

~ LONG BEACH

.~ PARKFIELD, BORREGO tIT. (4)

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SCALING LAWS FOR LARGE EARTHQUAKES CONSE(FENCES FOR PHYSICAL MODELS Christopher H. Schol2; Depar tment o f Geol ogi cal S ci ence s and Lamont-Doher ty Geological Obs ervatory of Columbia University, Palisades, New York 10964 ABSTRACZ It is F

observed that. the mean slip in large ear thquakes correlates linearly with fault length L and is not related to fault width, W. If we interpret this in terms of an elastic model, it implies that static stress drop increases with aspect ratio (L/W) . We also observe a tendency, particularly for strike-slip earthquakes, for aspect ratio> and hence static stress drop> to increase with seismic moment. Dynamic models of rupture of a rectangular fault in an elastic medium show that the final slip should be controlled by the fault width and scale with the dynamic stress drop. The only way these models can be reconciled, with the obser-vations is if dynamic stress drop correlates with fault length so that it is also nearly proportional to aspect ratio- This could only happen if fault length is determined by the dynamic stress drop. There are several serious objections to this, which lead us to suspect that these models may be poor representations of large earthquakes ~ Firstly, it conflicts with the observations for small earthquakes (modeled as circular sources) that stress drop is nearly constant and independent of source radius- Secondly,

A I I

xt conf lxcts with the observation that fault length is often determined by rupture zones of previous earthquakes or tectonic complications'e speculate that the boundary condition at the base of the fault, that slip is zero> is unrealistic because that edge is in a ductile region at the ase of the seismogenic layer. In a model in which slip is not so con-strained at the base of the fault nor at the top (the fxee surface), such that no healing wave originates from these edges, final slip would be deterau.ned by fault length. The observations would then be interpreted as meanxng that the static and dynamic stress drops of laxge earthquakes are nearly constant. These two alternatives predict very different scaling of the dynamics of large earthquakes ~ The width-dependent model predicts that average particle velocities are larger for long ruptures but the rise tame wall be the same as in a shorter event of the same width. The length-dependent model predicts the oppos i te.

I INTRODUCT I&V, "A central problem in earthquake seismology has been to find scaling laws that relate the static parameters such as slip and stress drop to the dimensions of the rupture and to understand these relationships in terms of the dynamic parameters, the most fundamental of which are rupture velocity and dynamic stress drop.

In doing so> it is essential to distinguish between small earthquakes and large earthquakes. Tectonic earthquakes nucleate and are bounded within a region of the earth between the surface and a depth h,0 the seismogenic layer. The seismogenic depth, h,0 depends on the tectonic environment but in a given region the maximum width of an earthquake occurring on a fault of dip 6 is W 0

~ h o

/sin 6 ~ We will define a small earthquake as one with a source radius r < W 0

/2 and a large earthquake as one in which r > W 0

/2. Thus a small earthquake can be represented as a circular source in an elastic medium, whereas a large earthquake is more suitably treated as a rectangular rupture with one edge at the free sur face.

It has been repeatedly demonstrated (e.g., Aki> 1972; Thatcher and Hanks, 1973; Hanks, 1977) that the stress drops of small earthquakes are nearly constant and independent of source dimensions. This result, when interpreted with dynamic models of finite circular ruptures ( Madariaga, 1976; Archuleta, 1976; Das, 1980 ), simply means that the dynamic stress drop is constant.

If the same were true for large earthquakes, the dynamic models of rectangular faulting in an elastic medium (Day, 1979; Archuleta and Day>

1980; Das, 1981) would predict that the mean slip is a linear function of fault width. In

'J the next section we will show that this prediction is not borne out by the observations ~ Mhat is observed instead is that slip correlates linearly with fault length. The principal point of this paper xs to discuss the consequences of that observation for the physics of large earthquakes.

I OB SERVATI(NS For small. earthquakes, using the definition of seismic moment, M 0,

and the relationship Aa ~ ~r 7Ã 16 where r is source radius, u is mean slip, and 'Aa is stress drop. If stress drop is constant> the relationship between M and fault area, A> is

~

0 o

=( 16'3/2 )A3/2 7

Large earthquakes, however,, are more nearly rectangular ruptures of width W and length L and in this case, for an elastic model in which slip is res trict ed to be wi thin W, Aa Cyu (2)

W where C is a geometrical constant-If stress drop were constant, we would expect to find that M

o 4WC (3)

In Figure 1 we show a plot of log LW vs. log M, for the large 0

interplate thrust and strike-slip earthquakes from the data set of Sykes

and Quittmeyer (1981) . These observations are listed in Table 1. The data for each type of earthquake define a line, but with a slope less than one, indicating that stress drop systematically" increases with moment ~ The 4

offset between the data for the strike-slip and thrust events is also an important feature that we will discuss later.

These data indicate that u is not simply related to W and that her is not constant for large earthquakes'n the contrary, many workers (e.g ~ ,

Bonilla and Buchanan, 1970; Slemmons, 1977 ) have argued that u correlates

/

with L, and recently Sykes and gui ttmeyer (1981) have argued that the correlation is linear. Plots of u vs. L on linear scales are shown in Figures 2 and 3 for strike-slip and thrust earthquakes, respectively.

Zn view of the usual uncertainties in the estimates of u and L, and any naturally occurring variations in dynamic stress drop (with which slip should be expected to scale), the correlation between u and L is fairly strong. We fit it with a straight line with an intercept at the origin u ~aL (4) and find that a = 2 x 10 for the thrust events and 1.25 x 10 for the strike-slip events. At least for the strike-slip events> slip is clearly not dependent on width because the widths of all the events in Figure 2 are between 10-15 km> i.e. > they are essentially the same.

4 Prom this observation we would then expect that-M 0

paL W (5)

S which is confirmed in Figure 4. For reference, the line drawn through the data has a slope of one.

Since 3q L2W:A3/2(L)

W and since the aspect ratio L/W varies only by a factor of about 20 in the data-set, we would have found a good correlation between M and A 3/2 as

~

did Aki (1972) and Kanamori and Anderson (1975) had we plotted log A vs.

log M .

0 The question is not whether M correlates better with L W than 0

with A3/2 ~ The issue of concern is that Kanamori and Anderson's inter-pretation of their correlation as meaning that stress drop is constant is only true if L/W is constant, because from (2) and (4), we have

~ L Aa Cp~ (6)

That L/W is a constant is an explicitly s tated assumption of Aki (1967, 1972) and Kanamori and Anderson (1975)> and although Abe (1975 ) and Geller (1976) attempted to observationally justify this assumption, it is not generally true. In Figures 5 and 6 we plot Aa vs. L/W for the two types of earthquakes. The correlation between them is very clear for the strike-slip events, and less so for the thrust events, for which there is a much smaller variation of aspect ratio. That L/W does not have a large vari-ation for the thrust events seems to simply result from the fact that the seismogenic width of subduction zones, W is about 100 km, so that only 0,

extremely large events can achieve high values of aspect ratio.

Ve can now understand why stress drop increases systematically with M

0>

as shown in Figure 1. The width of large strike-slip earthquakes is

C 8

limited by the'eismogenic depth to W:0 15 km so that they grow princi-pally in the L direction. This results in a systematic increase in L/W, and hence Acr, with M . The subduction zone thrust earthquakes have dif-ferent widths but L increases faster than W with increasing moment, pro-ducking the same result, i-e ~ > Aa increases with L/W or M ~ The offset between the data for thrust and strike-slip events in Pigure 1 occurs simply because the widths of the thrust events are much greater than those of the strike-slip events' strike-slip event must have a much greater aspect ratio, and hence stress drop, than a thrust event of the same moment-

PHYSICAL CONSEQJENCES The principal feature of the observations that we wish to explain is the correlation between slip and fault length. It is a surprising obser-vation because intuition would first lead one to expect slip to depend on width, yet this is not observed. This intuition is re-inforced by the results of dynamic models of rectangular faults in an elastic medium (Day, 1979; Archuleta and Day, 1980; Das, 1981). These models show that slip is controlled by the width of the fault and that it scales with dynamic stress drop.

The situation is illustrated in Figure 7, which shows surface slip along the Sault for two representative strike-slip earthquakes. These earthquakes have essentially the same width, and differ only in length- If the dynamic stress drop were the same for these two earthquakes, then according to the theory, the Ft. Tejon earthquake would be the equivalent of six Mudurnu earthquakes placed end to en'learly that is not the case.

If the dynamic, elastic models are correct representations of earth-quakes, then the only way they can be reconciled with the observations is if dynamic stress drop correlates with aspect ratio. Since the width of strike-slip events is nearly constant, and the width varies much less than length for the thrust events, 'this would be approximately true if dynamic stress drop correlates linearly with fault length. The only way this can happen without violating causality is if fault length is determined by dynamic stress drop. This is not an entirely unphysical proposition, because dynamic stress drop determines the stress intensity factor, which

l 10 is important. in fracture growth..It is not obviously apparent'> however, why L should increase linearly with ha dt the dynamic stress drop.

~

There are several major objections to this interpretation- The first is that we have to assume that for large earthquakes Aa d

determines the rupture length, which 'directly contradicts the observations for small

'arthquakes. Although stress drop appears to increase with source radius over a limited -ange in sane data sets (Aki, 1980), it shows no obvious variation with source radius over a very broad range (Hanks> 1977 ). We can offer no reasonable explanation for why large earthquakes should behave differently than small earthquakes in this important respect.

A second objection is that this assumption conflicts with the prin" cipal observations that led to the concept of seismic gaps: that the length of large earthquakes is often controlled by the rupture zones of previous earthquakes or by structural features . transverse to the fault zone. Of course, one could soften the original assumption to: 6a d

deter" manes the length unless the rupture encounters a rupture zone of a previous earthquake or a transverse feature. The rejoinder is that if the latter were as common as is thought, it would have the effect of destroying the correlation between u and L that is observed.

It. is worth giving a specific example. If we compare the 1966 Park-field earthquake (L ~ 30 km, u ~ 30 cm, W ~ 15 km) and the 1906 San Fran-cisco earthquake (L ~ 450 km, u ~ 450 cm, 'W ~ 10 km) we need to explain the difference in u by a difference in had of about a factor of 15. Since the correlation between u and L is also good in these examples, we also need to argue that bod determined L in these cases ~ On the other hand, it can be argued that the length of the 1966 earthquake was determined by the length of the gap between the rupture zone of the 1857 earthquake (or the

J 11 fault offset near Cholame) and the southern end of the creeping section of the San Andreas faul t. S imila rly, the 1906 ear th quake filled the gap between the northern end of the creeping section at San Juan Bautista and the end of the fault at Cape Mendocino. If our argument that Dad deter-mines L is true, then these latter observations are coincidences ~ Almost identical arguments can be made for many of the other earthquakes in our data set.

The third point is less an objection than a surprising consequence of this interpretation. The Hoei earthquake of 1707 ruptured about 500 km of the Nankai trough in Japan (Ando> 1975; Shimazaki and Nakata, 1980) . 'The sam'e plate boundary was ruptured twice subsequently, in two sets of delayed multiple events, the Ansei I and II events of 1854, and the Tonankai and Nankaido events of 1944 and 1946. In support of a time"predictable model of earthquake recurrence, Shimazaki and Nakata argued that the greater recurrence time between the, first two sequences (147 years) and the second (91 years) is because the slip (and stress drop) were greater in 1707 than in either 1854 or 1946, the greater uplift at Muroto Point in 1707 (1.8 m) than in 1856 (1.2 m) or 1946 (1.15 m) being the evidence The reason why this should happen is readily explained by the correlation between u and L.

Thus the ratio of fault length of the Hoei and Ansei II earthquakes, 500 km/300 km ~ 1 ~ 7 can explain the ratio of uplift at Muroto Point, 1.8/1.2 ~ 1.5 and recurrence time, 147/91 1 .6.

However, if this is interpreted as being due to a difference in dynamic stress drop, then one has to argue that a significant change in dynamic stress drop (50X) can occur on the same fault zone between succes-sive earthquakes'ne could argue that this could occur because the slip P in one earthquake might change the relative position of asperities on the

n 12 fault. However, since the slip xn an earthquake is about 10 L, this 4

would mean that the gross frictional properties of the fault are controlled by asperities of dimensions on the order of 10 or less of the rupture dimensions. Since there will be a very large number of such small fea-tures> the average change between successive earthquakes would more likely

~ be expected to be negligible ~

In the above discuss on we have created enough doubt about the appli-cability of the dynamic rectangular models to consider that they may be failing, in sane fundamental way, to properly describe the physics of large ear th quakes .

For a rupture propagating at a constant rupture velocity, v> the slip, for both circular and rectangular faults, is very close to [Day, 1979; Das, 1980, 1981l u(x,y,t) u (t 2 ~

( 2+ 2)

V

)

)/2

~ l/2

2) 2 V

< t < th (7)

~ where x and y are measured relative to the point of rupture initiation.

Equation (7) is the self-similar solution of Kostrov (1964). The asymp-totic particle velocity, u which scales the slip is, (Kostrov, 1964; Dahlen, 1974)

~

u 0

~ K had 8 (8) where K is a function of rupture velocity.

Qhen the rupture reaches its final perimeter and stops, a healing wave propagates back into the rupture, arriving at time t . por or t > ]'

t h sap decelerates and comes to a halt. The healing wave is not the stopping phase, which is a wave radiated in all directions from the tip of a

t 11

13 stopping crack (Savage, 1965). A stopping phase cannot physically stop the slip in these models. because such a wave will lose energy with distance f

whereas the results of the models are independent of dimension. A healing wave must be interpreted as a wave that propagates into the interior of the rupture in an analogous way, and for analogous physical reasons, as the

'topping of cars on a highway propagates up the s tream of traf fic.

Causality restricts it to travel at a velocity slower than a stopping phase. Thus Madariaga (1976, p. 648') observed, "It appears as if a

'healing'ave propagates inward from the edge of the fault some time after the P and S stopping phases."

Since slip is terminated by the healing wave, the rise time and final slip at any point on the fault is determined by the distance to the nearest boundary (Day, 1979; Das, 1981). Therefore. it is easy to see why mean slip on a rectangular fault should be controlled by the fault width.

A healing wave is the result of the boundary condition that u ~ 0 at the edges of the fault. If the models are poor representations of large spaces'he earthquakes, the most likely problem is that these boundary conditions are unrealistic. The models are of rectangular faults embedded in an elastic whole boundary condition u ~ 0 is imposed on all edges of the fault and healing waves thus propagate from each edge. Since large earth-quakes rupture the free surface, slip is unconstrained there and a healing wave will not propagate from that edge. However even if an elastic half-space model were available, we would still expect slip to be width-dependent since it would be controlled by the healing wave from the base of the fault.

In large earthquakes the base of the fault is at the bottom of the seismogenic layer. A plausible explanation for the seismogenic depth is

I 1'4 that it is the result o f a brittle-ductile transition . Thus a large earthquake cannot propagate to greater depth because the energy at the crack t ip is dissipated in plastic deformation . A more re al is t ic model then may bb one in which the base of the fault is in a pl as t ic , rather than elastic, region and therefore the condition u ~ 0 is no longer valid at that edge .

Qe illustrate in Figure 8 the difference between an elastic model and an elastic-plastic model . The most s igni fican t difference is that in the elastic-plastic model (Figure Sb ) slip at the base o f the fault may be allowed to be greater than zero as a result of plastic deformation in a zone surrounding the rupture tip . This is simply the equivalent> in shear ,

'f the blunting of a crack tip that occurs in tensile crack propagation in ductile materials. The plastic deformation around the base of the fault smooths out the stress singularity associated with finite slip there> and will continue as long as slip continues ~ This may have the effect of inhibiting a healing wave from originating at the baseg and if healing waves propagate only from the ends of the fault, slip and rise time will depend on fault length, not width ~

No model is available with these boundary conditions but we can approximate one. If we make'he approximation that slip stops abruptly 4

with the arrival of the healing wave, then the final slip on che fault will be, from (7),

2+ 2) 1/2 u(xy) ~ u (t ( (9) o h 2 )

v which we can calculate. This is a 'quasidynamic'odel (Boatwright, 1980),

i.e-, a kinematic model that simulates a dynamic model.

15 It can readily be shown for the circular case that (9) yields final slip values that are everywhere within 5X of that of the dynamic numerical models of Madariaga (1976) and Das (1980), and Day (1979) has shown that (9), when properly truncated, also yields a very good approximation to final slip in his rectangular models. We use it to simulate an elastic-plastic half-space model by simply assuming that no healing wave propa-"'ates from either the top or bottom of the fault.

The procedure we use is very similar to that used by Day (1979, pp. 23-26), and simply involves the calculation of th. We assumed v ~ 0.98, for which the corresponding value of K is 0.81 (Dahlen> 1974),

and that the velocity of the healing wave is v38. In Figure 9 we show slip

.at the surface as a function of distance from the center of the fault for a bilateral case with L/W = 4. The mean slip is found to scale as L

u~-2ibad (10) so this model would lead to the interpretation that the linear correlation between u and L that is observed means that the dynamic stress drop for large interplate earthquakes is approximately constant. Equating (10) with (4) we obtain hc d

= 12 bars and 7.5 bars for thrust and strike"slip earthquakes, respectively. Returning to Figure 4, the line drawn through the data is the prediction of this model for Aa d

= 10 bars. Furthermore, in this model, where slip is unconstrained at top and bottom, static stress drop will also be a function of fault length, since the scale length that determines the strain change will be the fault length. The observation made earlier that ha is a function of aspect ratio is due to the incorrect use of equation (2) to calculate it. According to this model, Ao is also approximately constant for these earthquakes ~

DISCUSSION The observation that slip increases with fault length in large earthquakes poses severe consequences when viewed in the light of dynamic rupture models- In conventional dynamic models (W models), slip is deter-mined by fault width, rather than length. These models can only be recon-ciled with the observations if it is assumed that the dynamic stress drop determines the fault length, and the several major objections to this possibility were detailed earlier. With different assumptions concerning the boundary conditions at the base of the fault, it may be possible 'to construct a dynamic model in which slip depends on fault length (L model) ~

This model avoids the objections raised to the W model but is based on a speculative, although not entirely ad hoc, assumption concerning the boundary cond i t ions .

Furthermorey severe constraints are placed on L models from the geo-detic data obtained for the 1906 San Francisco earthquake. The simplest form of L model is one in which slip is totally unconstrained at the base of the fault. If this were the casey strain release would extend out to distances comparable to fault length, rather than depth, but as Brune (1974) has pointed out> the strain release in 1906 was concentrated within a few tens 'of km from the fault. From angle changes in the Pt. Arena triangulation network fangle g from Thatcher (1975, Fig. 4) ] one can

-5 estimate a straxn drop of 8 x 10 within 12 km of the fault, a figure somewhat more consistent with a W model than an L model. Thus if L models ara relevant, they must be models in which slip is only ~artiall con-strained at the base of the fault- In the absence of numerical modeling of

17 this type, one can't tell if this type of model will result in L scaling or

~ hybrid scaling intermediate to the L and W extremes.

These L and W models represent, in many respects, opposite extremes concerning the mechanism of large earthquakes and so it is useful to discuss the contrasting way in which they scale. For earthquakes in which L < 2W~ the models are indistinguishable in their gross manifestations ~

In Figure 10 we schematically show a comparison between an earthquake of dimensions about L = 2W and one of the same width but about 15 times longer. Specifically, this might be a comparison of the 1966 Parkfield

'earthquake> say, and the 1906 San Francisco earthquake.

On the left of the figure we show a snapshot of slip on the fault during the smaller earthquake ~ We only show the part that is actually slipping during the snapshot. We also show the time history of slip at some representative point. For simplicity, it is simply shown as a ramp with a rise time, tR. On the right is shown the predictions of the two models for the longer earthquake ~

In a bilateral case, as shown, the W model predicts that the slipping portion of the fault splits into two patches of length ~ W that propagate away from each other at a velocity 2v as .they sweep over the fault surface.

Since the rise time tR = W/28, remains the same but the slip is fifteen times greater, the dynamic stress drop, and hence particle velocity, must be fifteen times greater.

In the L model, the rupture sweeps out over the fault as an expanding patch, with slip continuing within its boundaries until after the final dimensions are reached. In that model, the dynamic stress drop and par-ticle velocities are the same as in the smaller event, but the rise time, L/28 is much longer ~

A 18 In terms of predicting the strong ground motions for a 1906 size earthquake> say, from observed ground motions for a 1966 size earthquake>

the difference between the W and L model is critical'he W model would predict Chat the average particle velocities would be much higher and the duration would be about the same. The L model would predict nearly the opposite-Suppose we start with a square rupture of width W 0

and consider how peak particle velocity, u,P and the asymptotic particle velocity> u 0

increase for ruptures of greater length. For a square rupture with dynamic S

stress drop, had, the maximum value of u P

and the asymptotic value u 0

will be

~ S u cc aa 0 p

and

~ S S had 0

Using the W model., for a rupture of width W 0

and length L > W 0,

the stress drop will have to be greater by the ratio W

ha~

S W

~

d so that u ~ha d S.LW r o

P 0

.l 19 and (l2)

~ W S L u Qc d W 0 0 For the L model, stress drop is the same but the scale length that deter-mines the maximum peak velocity becomes L rather than W, so that

~ L S u o: d P

and (13)

~ L S do d u o:

0 Ceaparing (12) and (13), the two models. differ in the ratios U

P and u

L 0

W u0 So that with a W model, from (12), both peak and asymptotic velocities for a 1906 type earthquake would be about 15 times greater than for the Park-field earthquake ~ For the L model, from (13), the peak velocities would at maximum be about 8 5 greater for a 1906 than a 1966 event, but the asymptotic value would be the same.,

4 20 These remarks, of course, apply only to the simple case of a smoothly propagating rupture- Any heterogeneity will produce local high frequency variations in the velocities ~ However, they serve to point out the impor-tance of determining if large earthquakes are better described by an L model or W model or by some intermediate case, if such can exist ~

I 21 ACKNOWLEDGEMENTS My attempts at trying to understand the consequences of slip corre-lating with fault length had a rather long gestation period, during which the author benefitted from discussions with T. Hanks, J. Boatwright, P.

Richards, S. Das, S. Day> and R- Madariaga ~ Most of the work was done while the author was a visitor at the Department of Earth Sciences, Uni-versity of Cambridge, and a Green Scholar at the Institute of Geophysics and Planetary Physics, University of California, San Diego. Both are

,thanked for their support and hospitality. The work was supported by National Science Foundation grant=EAR 80&7426 and National Aeronautics and Space Administration grant NGR 33-008-146. I thank P. Richards an'd L.

\

Sykes for critical reviews ~ Lamont-Doherty Geological Observatory con-tribution no. 0000.

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Archuleta, R. J. ( 1976) . Experimental and numerical three dimensional simulations of strike-slip earthquakes, Ph.D. thesis> Univ. of Cal i f- > San D iego ~

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20, 671-689.

Ando, M. ( 1975) ~ Source mechanisms and tectonic significance of his-toric earthquakes along the Nankai trough, Japan, Tectono h sics, 27, 119-140. Rcpt' Boatwright, J. (1980). A spectral theory for circular seismic sources; simple estimates of source dimension, dynamic stress drop, and radiated seismic energy, Bull. Seismol ~ Soc. Amer., 70, 1-27.

Bonilla, M. G., and J. M. Buchanan (1970) ~ Interim report on worldwide historic surface faulting, U ~ S. Geol. Surv. Open-Pile , Wash-ington, D.C.

I 6

23.

Brune, J. N. (1974) ~ Current status of understanding quasi-permanent fields associated with earthquakes, EOS Trans. AGU, 55, 820-827.

Dahlen,. F. A. (1974) ~ On the ratio of P-wave to S>>wave corner frequen-cies for shallow earthquake sources, Bull. Seismol. Soc. Amer., 64, 1159-1180.

Das, S. ( 1980) . A numerical method for determination of source-time functions for general three-dimensional rupture propagation, ~Geo h s.

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Das, S. (1979) ~ Three-dimensional spontaneous rupture propagation and implications for the earthquake source mechanism, Geo h s. J. R.

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Day, S. (1979). Three-dimensional finite difference simulation of fault dynamics, Final Rept., NAS2-10459, "

71 pp., Systems, Science and Software, La Jo'llap Calif.

Geller, R. J. (1976) . Scaling relations for ear thquake source para-meters and magnitudes, Bull. Seismol. Soc. Amer., 66, 1501-1523.

Hanks, T. C. (1977) ~ Earthquake stress drops, ambient tectonic stress, and stresses that drive plate motions, Pure A 1- Geo h s ~, 115, 441-458.

Kanamori, H., and D. L. Anderson (1975) ~ Theoretical basis of some Amer' empirical laws of seismology, Bull. Seismol. Soc. Am., 65, 1073-1096.

Kostrov, B. V. (1964) ~ Selfsimilar problems of propagation of shear cracks, J. A 1 ~ Math. Mech., 28, 1077-1087.

Madariaga, R. (1976). Dynamics of an expanding circular fault, Bull.

Seismol. Soc. , 66, 639-666.

Savage, J. C. (1965). The stopping phase on seismograms, Bull. Seismol.

Soc. Amer., 55, 47-58.

g A 24 Shimazaki, K., and T. Nakata (1980) ~ Time"predictable recurrence model for large earthquakes, Geo h s. Res. Lett., 7, 279-282.

Sich, K. (1978).. Slip along the San Andreas fault associated with the great 1857 earthquake, Bull. Seis. Soc. Amer., 68, 1421-1448.

Slemmons, D. B. (1977) . State of the art for assessing earthquake hazards in the United States, Faults and earthquake magnitudes, U .S .

Army Eng- Waterway Exp. Sta., Vicksburg, Hiss., pp. 229, 1977.

Sykes, L. R., and R. C. Quittmeyer (1981). Repeat times of great earthquakes along simple plate boundaries, Third 'maurice Ewin S osium on Earth uake Prediction, 4, edited by D. W. Simpson and P.

G. Richards, AGU, Washington, D.C.

Thatcher, W. (1975). Strain accumulation and release mechanism of the 1906 San Francisco earthquake, J. Geo h s. Res., 80, 4862%872.

Thatcher, W., and T. Hanks (1973) ~ Source parameters of southern Cali-fornia ear th quakes, J. Geo h s. Res., 78, 8547-8576.

TABLE 1 PARAMETERS OF LARGE IHTERPLATE EARTHQUAKES (AVERAGED FROM SYKES AND QUITTMEYER (1981))

Ho L Ae Ho. Date Location 1027 dyne-cm km km, L/tY bars Strike-Sli Earth uakes

1. 10 3ul 1958 SE Alaska 4.3 350 12 29 325 26

2. 9 3an 1857 S. California 7 380 12 32 465 36 3~ 18 Apr 1906 San Francisco 450 10 45 450 19 Hay 1940 Imperial Va., Ca. 0.23 60 10 125 13

5. 27 3un 1966 Parkfield, Calif. 0.03 37 10 4 . 30 4

6. 9 Apr 1968 Borrego Htn, Ca. 0.08 37 12 3 25 3

7. 15 Oct 1979 Imperial Va., Ca. 0.03 30 10 3 30 4

8. 4 Feb 1976 Guatemala 2.6 270 15 18 150 9

.9. 16 Oct 1974 Gibbs F. Z. 0.45 75 12 170 14

10. 26 Dec 1939 Ercincan, Turkey 4.5 350 15 23 285 18

11. 20 Dec 1942 Erbaa Hiksar, Turkey 0.35 70 15 5 .112 8

12. 1 Feb 1944 Gerede-.Bolu, Turkey 2.4 190 15 275 18

13. 18 Har 1953 Gonen-Yenice, Turkey 0.73 58 15 280 21

14. 22 3ul 1967 Hudurnu, Turkey 0.36 80 15 100 7 Thrust Earth uakes

15. 6 Hov 1958 Etorofu, Kuriles 44 150 70 2.1 "

840 37

16. 13 Oct 1963 Eruppu, Kuriles 67 275 110 2.5 445 12

17. 16 tlay 1968 Tokachi-oki, 3apan 28 150 105 1.4 355 10

18. 11 Aug 1969 Shikotan, Kuriles 22 230 105 2.2 180 5

19. 17 3un 1973 Hemuro-oki, 3apan 6.7 90 105 .0.86 140 5

20. 4 Hov 1952 Kamchatka 350 450 175 2.6 890 14

21. 28 tlar 1964 Prince Wm Sound, Alaska 820 750 180 4.2 1215 18

22. 4 .Feb 1965 Rat Island, Aleutians 125 650 80 8.1 480 10

23. 10 pan 1973 Colima, Mexico 3 85 65 1.3 110 5

24. 29 ttov 1978 Oaxaco, Mexico 3 80 70 1.1 110 5

25. 22 tlay 1960 S. Chile 2000 1000 210 4.8 1900 21

26. 17 Oct 1966 C. Peru 20 80 140 0.6 360 12

S FIGURE CAPTIONS Figure 1. Plot of log LM vs- log M 0

for the large intraplate earthquakes from the data set of Sykes and Quittmeyer (1981) . The lines of slope 1 are constant stress drop lines, assuming C ~ 0.6 for the thrust events, and 0.3 for the strike-slip events ~

Figure 2. A plot of mean slip,.u, vs. fault length for the strike-slip events. The line drawn through the data has a slope of 1.25 x 10 Numbers are references to Table 1 ~

Figure 3. The same as Figure 2, for the thrust events. The slope of the line is 2 x 10 Figure 4. A plot of log L2 R vs ~ log M . The line drawn through the data slope of 1, has a quakes. for reference.

Figure 5. Stress drop plotted vs ~ aspect ratio for the strike-slip ear th Figure 6. Stress drop vs. aspect ratio for the thrust earthquakes.

Event 22 is an oblique slip event for which stress drop was calculated based only on the dip slip component and is hence underestimated'vent 15 is an anomalously deep event in the Kuriles (Sykes and oui t tm eye r, 1 981 ) .

I 27

.Figure 7. Schematic representation of two models of large earthquakes.

In A, it is represented by rupture in an elastic half-space. The boundary condition at the base of the rupture is u ~ 0. In B, the rupture penetrates a ductile region. At the base u > 0, which is accommodated by plastic deformation in a zone surrounding the rupture tap.

Figure 8. Surface slip as a function of distance along the .fault plane for two representative strike-slip earthquakes of similar width but different depth- Data for the, Mudurnu earthquake is from Ambraseys (1969) and for the Ft. Tejon earthquake from Sich (1978) ~

Figure 9. Dimensionless slip, u's. length, L', at the free surface from the center to the end of the fault. The model is a quasidynamic one that simulates a dynamic model with boundary conditions similar to those shown in Figure 7b, as described in the text. The normal-ization relations are u ~ Wu'nd Ao L ~ WL' The case shown is bilateral with aspect ratio 4~

Figure 10. A schematic diagram to illustrate the contrasting way in which a model in which width determines the slip (W model ) scales with length as compared to a length dependent model (L model) ~

I 9 THRUST

((

STRIKE SLIP o~

6 Q~

o o

/g QQ ~6 2I E

O O

20 r~

o~

o~r l

l9 y,'P l8 /~6 I'7 27, 28 Log Mo, 'dyne-cm Figure 1

2000 THRUST 25 l500 E

O }000 20 500 22 17 26 19 23 24 0 200 400 600 800 1000 L, km Pigure 3

P 500 ST R l KE S Ll P 400 300 200 IOO

/~<5, 6, 7'OO'00 500 400 500 L, km Figure 2

I 22, tO O

20 19 18 26 27 28 Log Mo, dyne-cm Figure 4

'1

PQ 30 th L

O J3 b

.20 lo l0 20 40 50 LIVl Figure 5

4

~ '

40

~s 50

~ 20 b"

IO 9 22 L/W Figure 6

BRITTLE DUCTILE ~ZONE OF PLASTIC DEFORMATION B

Figure 7

N

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<P

MUDURNU, I967 X

I 50km FT. TEJON, I857 200 500 Pigure 8

C Figure 9

Figure 10

UNITED STATES OF AMERICA NUCLEAR REGULATORY COMMISSION BEFORE THE ATOMIC SAFETY AND LICENSING APPEAL BOARD

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In the Matter of: )

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PACIF IC GAS & ELECTRIC ) Docket Nos. 50-275 O.L.

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(Diablo Canyon Nuclear )

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