This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: VOL. 84. NO. BS JOURNAL OF GEOPHYSICAL RESEARCH MAY I0. 1979 Monsieu— b Values and or" Seismic SourceModels: Implications for Tectonic Stress Variations Along Active Crustal Fault Zones and the
Estimation oinghFrequency Strong Ground Motion THOMAS C. HANKS U.S. Geological Survey. Mania Park. California 94025 In this study the tectonic stress along active crustal fault zones is taken to be of the form 50) + Joplx.
y). where 60) is the average tectonic stress at depth y and oeplx. y) is a seismologically observable.
essentially random function of both fault plane coordinates; the stress differences arising in the course of
crustal faulting are derived from Aopfx. y). Empirically known frequency of occurrence statistics.
momentmagnitude relationships. and the constancy ol'earthquake stress drops may be used to infer that
the number of earthquakes Ngfdimension 2r is of tlle form N ~ l/r‘ and that the spectral composition
of Aoptx. y) is of the form I'AepUc)! ~ l/k'. where Ao,(k) is the twodimensional Fourier transform of
Ao,(x. y) expressed in radial wave number k. The 7 = zmodel ofthe farﬁeld shear wave displacement
spectrum is consistent with the spectral composition Magus)! ~ l/k'. provided that the number of
contributions to the spectral representation ofthe radiated ﬁeld at frequency f goes as (k/ka)’. consistent
with the quasistatic frequency of occurrence relation N  l/r': kn is a reference wave number associated
with the reciprocal source dimension. Separately. a variety ofseismologic observations suggests that the y
= 2 model is the one generally. although certainly not always. applicable to the highfrequency spectral
decay of the farﬁeld radiation of earthquakes. in this framework. then. b values near I. the general
validity of the y = 2 model. and the constancy of earthquake stress drops independent of size are all
related to the average spectral composition oi'Aa,(x. y). 13?,(k) ~ l/k’. Should one ofthese change as
a result of premonitory effects leading to failure. as has been speciﬁcally proposed for b vaIUes. it seems
likely that one or all of the other characteristics will change as well from their normative values.
Irrespective ofthese associations. the farﬁeld. highfrequency shear radiation for the 'y = 2 model in the
presence of anelastic attenuation may be interpreted as bandlimited. ﬁnite duration white noise in
acceleration. its rms value. or"... is given by the expression a"... = O.BS[2"'(21r)‘/ 06] (Ao/pRXfmu/fo)".
where _\o is the earthquake stress drop. .9 is density. R is hypocentral distance. I, is the spectral corner
frequency. and I"... is determined by R and speciﬁc attenuation l/Q. For several reasons. one of which is
that it may be estimated in the absence of empirically deﬁned ground motion correlations. on... holds
considerable promise as a measure of highfrequency strong ground motion for engineering purposes. INTRODUCTION Very little is known about the heterogeneities in material
properties and tectonic stress that exist along active cruStal
fault zones. yet such heterogeneities are likely to play a central
role in earthquake mechanics. It is now knmﬂthat crustal
earthquake stress drops. in their average value Ac. are several
tens of bars and that this value is independent of source
strength Over l2 orders of magnitude in seismic moment [e.g..
Aki. 1972: Thatcher ondHonks. 1973; KonomoriondAnderson.
I975; Hanks. I977]. Because earthquakes are generally epi
sodic functions of space and time along even the most well
developed crustal fault zones. it may then be inferred that
stress variations of at least Aer commonly exist along aetive
crustal fault zones (although they might be greatly reduced. if
not eliminated. at the time and place of throughgoing earth
quake faulting. giving rise. for example, to the notably aseis
mic section of the San Andreas fault that broke in the great
earthquake of I857). But because it may also be inferred that
such faults. in general. can be no further away frr_)_n_1 repeated
failure than stress reaccumulation comparable to An. it sceE
plain that tectonic stress heterogeneities of the order of Aer
must play a central role in determining why a particular earth
quake occurs at a particular point in space and time and
therefore in any rational capability that purports to predict
earthquakes. Beyond these truisms. analysis of the nature and extent of
Stress heterogeneities along seismically active faults is com
plicated by important but poorly understood problems. One of This paper is not subject to U.S. copyright. Published in 1979 by the
American Geophysical Union. these is the average tectonic stress operative to cause failure on
the fault zone in the ﬁrst place; whether this value is of the
order of IOO bars (or perhaps somewhat less) or ofthe order of
a kilobar (or perhaps somewhat greater) is as yet unresolved
[e.g.. Hanks. l9‘i7]. 1n the ﬁrst case one may anticipate that
variations in tectonic stress must be ofthe order of 100% ofthe
average value. but in the second case they need only be a small
fraction of the average value (although they could be larger). A second difﬁculty is that variations in the stresses driving
relative motions. in the selfstress [Andrews. I978] resulting
from past faulting episodes (which may be quite nonuniform if
nonuniform faulting displacements are common in the case of
individual earthquakes). and in material properties along the
fault zone all contribute to inhomogeneity in tectonic stresses
along the fault zone of interest. Even if the amplitudewave
length oontent of actual stress variations along faults was known.
which it is not. a difﬁcult problem would remain in separat
ing out the causative processes to which it should be related. Similarly. there is accumulating evidence that the dynamic faulting displacements and associated stress diﬂ'erences can be
highly inhomogeneous in the course of crustal faulting. and it
is natural to suspect (but difﬁcult to prove) that these in
homogeneities arise from variations ofthe preexisting tectonic
stresses across the incipient rupture surface. For both the San
Fernando (for example. Hanks “974] and BoutIron [I9‘lB}.
among many such investigations) and the Borrego Mountain
[Burdick and Hellman, 1976; Heoron and Heimberger. [977]
earthquakes. there is considerable evidence that faulting was
initiated with localized but massive faulting with associated
stress diﬂ‘erences not at all representative of those inferred for
the entire faulting process. In a similar manner the larger peak Paper number 930052. 2235 s —.——..—.— w—w ———_ l 2236 accelerations (20.13) at close distances (R a lo km) almost
certainly represent localized. dynamic stress differences many
times greater than the average earthquake stress drop [Hanks
and Johnson. I916]. In a very real sense. of course. the ideas
presented in the studies cited above are simply scaleddown
versions (in spatial dimension and wave period) of the com
plex. multipleevent interpretation of large and great earth
quakes (a number ofsuch investigations are referenced by Das
and Akr‘ [I977]. who present their barrier model of the earth
quake mechanism partially in this context). In any event. considerable interest has developed around
these observations and ideas. for at least two important rea
sons. First. higherquality recordings and more detailed analy
sis ofsuch earthquakes may provide a clearer understanding of
the nature and extent of tectonic stress heterogeneities along
active crustal fault zones. Second. reliable estimates of high
frequency strong ground motion and their use in the aseismic
design of highfrequency structures depend quite strongly on
the nature and extent of these localized dynamic stress diﬁ'er
ences that devel0p in the course of crustal faulting. In this study the problem of tectonic stress variations is
addressed through interpretations. developed herein. of b
value data and the highfrequency spectral characteristics of
the radiated ﬁeld (or1r models) for crustal earthquakes. In fact.
however. the quantities being investigated are stress differ
ences available to be released at the time of faulting. to which
may be added a stress function ofwhich even the average value
is unspeciﬁed in this study and which. in the absence of addi
tional information. is unknown. Following the ideas of An
drews [1978]. one may infer but cannot prOve that this un—
known stress function is intrinsically smooth. arising
fundamentally from largescale stresses driving relative motion
across the fault. and that the actual variations in tectonic
stresses along active crustal fault zones are. to a ﬁrst approxi
mation, reasonably estimated through the ideas developed in
this study. (J VALUES AND EARTHQUAKE STRESS DROPS Hanks [1977] showed that the relations between frequency
of occurrence N ofearthquakes of magnitude 2M. log N = a — bM (1)
between seismic moment Ma and M.
log Mn = cM + d (2) and between source radius r. earthquake stress drop Ao‘. and
M... Mo = 1.5.
N = const/(5)“:J (4) N in (4). consistent with N in (l ). is the number of earthquakes
with dimension Zr. and 3—0 has been used for A0 in (3). It is
empirically knewn that b is generally. but not always. very
nearly equal to II irrespective of the choice of region and time
interval in which earthquakes are counted. Also. c is empiri
cally known to be 1.5 whether local magnitude ML [Thatcher
and Hanks. I973] or surface wave magnitude M. [Kanamorr'
and Anderson. 1975] is used in (2). although serious departures
from (2) with c = .5 begin to develop for M. E: 7]. Equation
(4) implies that if the earthquakes of the counted sample share
the same Ac. as they do on the average for all samples for
which the Au have been determined [e.g.. Hanks. I977]. earth can be combined to obtain, using (2 = l and c = HANKS: FAULT M ECHANJCS quake magnitude frequency of occurrence statistics reduce to a
simple matter of geometrical scaling in terms of the reciprocal
faulting area. Equation (1). however. is also satisﬁed (with a different a
value) by the density distribution of the number of earth
quakes with respect to M. dN/dM [Richren I958. p. 359]. In
this interpretation the density distribution of the number of
earthquakes with respect to r is proportional to r". a result
anticipated in the more complicated but essentially similar
model of Caputo [I976]. To interpret (4). imagine a planar fault surface large in
comparison to any earthquake source dimension of interest.
and a population ofincipient earthquakes to occur upon it: the
earthquake population is characterized by the frequency of
occurrence relation (4) and average stress drops equal to 33
but with scatter about this value comparable to that observed
in the available stress drop data. Before any of the earthquakes
occur. all of the stress differences that will be realized at the
time of occurrence for each and every event exist on the fault
surface in 'potential‘ form; we denote this distribution in both
spatial dimensions as the stress drop potential function Acrp(x.
y). As a matter of convenience. we assign zero mean to .30.,(x.
y) and denote the average shear stress on the fault as 50»). In
the earth. &(y) is the average tectonic stress and is presumably
a function ofdepth: it is not. however. sampled by earthquake
stress drops or any other measure of the radiated ﬁeld of
earthquakes. Within this framework we can expect the stress drop to be
realized across an area of incipient rupture A to be derived
from the rootmeansquare (rms) value of Aa,(x. y) over A.
where we understand. purely as a formality. that only thOSe
regions of mostly positive depot. y) are candidates for rupture.
The meansquare value of Aa,(x. y) across A is mm  Alf taupe. or dx dy (5) Since Aapbr. y) produ es earthquakes which. on the average.
have stress drops ~Aa and satisfy the frequency ofoccurrence
relation (4). (5) must be constant independent ofA. Because of
Rayleigh‘s theorem. ﬂatlxﬁetknrkdk (6) must also be constant. independent ofA. where Audit) is the
twodimensional Fourier transform of Aa,(x. y) expressed in
terms of radial wave number k: wavelength A is 21r/k. It is difﬁcult to be general about the circumstances in which
(6) will be constantfor any A. but we can arrange a special
case by assuming IAa,(k)I ~ k'" and a band limitation for
AepUc) between km... and km". Physically. the k'" dependence
of ]d”a.,(k) can be rationalized on the basis of similarity. and
the band limitation means. for example. that A greater than the
seismogenic depth or less than a grain size do not contribute
materially to Ac,(x. y). New. for n > I and k...“ >> k....... the
constancy of (6) requires that Altman" = const (7) which. dimensionally. can only be arranged by taking it = 2
and km... ~ A“". Physically. this means that the only signiﬁcant spectral con
tributions to (A0,?) occur at A comparable to the source di
mension of incipient rupture. For the It" dependence of
lAa,(k) it is. of course. clear that the shorterwavelength
contributions will be negligible. but for the same reason the HANKS: FAULT MECHANICS o:
'6
2:
ﬂ
2
log frequency
Fig. I. Spectral representation of the cu" and to" source models for two constant stress drop earthquakes observed at the same dis
tance R in a uniform. elastic. isotropic full space. longer A contributions are plainly a problem. But iflt >> (4)“
made a signiﬁcant contribution to (1303‘) across our chosen A.
then it is most likely that (Aarp') across a larger A' would also
be =(KEY: that is. in such an eventuality the rupture ofA’
would be the event of interest. having incorporated in its
rupture the smaller area A. In other words. limiting the rup
ture area to some A must mean that A >> A‘" cannot contrib
ute signiﬁcantly to (Aap’) across A: otherwise. a larger area
would rupture. As such. the frequency of occurrence relation
(4) may be written as N ~ 0A.)" ~ (It/kn)“. where A is the
wavelength corresponding to any earthquake source dimen
sion of interest and A. is some reference source dimension. In this context. then, a spectral composition of depot. y)
of the form Iaupur)! ~ It" will guarantee constant stress
drop earthquakes independent of the size of the rupture sur
face and that the frequency of occurrence relation will be
satisﬁed. It is worth emphasizing. however. that this represen
tation can well be nonunique and need not be correct. even
though a diﬁ'erent representation that will guarantee the con
stancy of(6) for any A is not obvious. We shall ﬁnd. however.
that the dynamic ﬁeld radiated by earthquakes in the case of
the 7 = 2 model is consistent with lEap(k) ~ k" and
provides separate support for this representation. In supposing that these ideas are relevant to currently active
crustal fault zones. some additional points should be made.
First. stress drops both higher and lower than lit—a will occur
with certain probabilities determined by the distribution of
Aarp(x. y) about its rms value. Existing stress drop data are
mostly in the range of several bars to several hundred bars.
allmving for likely biasing to lower values in the case of many
of the smaller earthquakes [Thatcher and Hanks. I973; Hanks.
1977]. These determinations suggest a log normal distribution
about a logarithmic mean of approximately 30 bars (K anamorl
and Anderson [[975] have suggested for = 60 bars on the basis
of arithmetic averaging). one logarithmic standard deviation
being about 0.5. Thus while_t'lle areaindependent rms value of Aa,(x. y) is determined by Aa‘ a: 30 bars. it may vary. at least ‘ occasionally. to several hundred bars. It is interesting that this
latter value is approximately the same as the variation about
the mean of the frictional strength of common crustal rocks at
constant pressure and temperature [Byerlee. 1978]. Second. active crustal fault zones are plainly not inﬁnite in
both spatial dimensions. For those earthquakes with fault
length L sufficiently greater than fault width w a: h/sin 6 2237 (where h is the seismogenic depth and 6 is the fault dip). the
twodimensional character of the fault surface collapses essen
tially to one. and it can be expected that the ideas presented
above will no longer hold. For it 2: l5 km and a vertical
transform fault. one may estimate roughly that this will occur
when L 2 30 km or. equivalently. when M. 2 6). In particular.
(3) then takes the form M. = k'Acer‘ (a) MoreOver. (2) with c = [.5 begins to fail at slightly larger M..
7—7). Finally. as is well known. M, becomes an increasingly
poorer measure of source strength for M.I 2 10" dyn cm. or
M, 2 7! [e.g.. Kanamorr'. I977]. As such. present uncertainties
in estimating both c and 'magnitude' at large magnitude pre
clude. at the present time. an extension of these results to the
more nearly onedimensional character of large and great
earthquakes. But these difficulties in no way change the argu
ments given ab0ve for M. S 6) earthquakes for which r S, w. EVIDENCE son AND INTERPRETATION or
THE w"SOURCE MODEL In spectral form the farﬁeld radiation emanating from
simple seismic source models [e.g.. Aki. I967; Brune. I970.
l97l] is generally characterized by a longperiod level 9., pro
portional to M... a corner frequency f. proportional to r". and
a highfrequency spectral decay of the form U/f°)"' (in the
fOIIOwing discussion. frequency is denoted byj'in hertz rather
than to in radians per second). The corner frequency f0. funda
mentally. is closely allied with the reciprocal duration of fault
ing T..", but it is well known that several ‘faulting durations'
can be deﬁned. in particular those associated with the fault length. fault width. and the rise time of a propagating dis/ placement discontinuity. Depending on the faulting geometry
and rise time characteristics. the associated corner frequencies
can be well separated. leading to more complicated high
frequency spectral amplitude decay (that is. 7 is a function of
frequency). Moreover. by making the displacement disconti
nuity a smooth enough function of time. 7 can become arbi
trarily large at high enough frequencies. Whether in fact a
generally applicable source representation of highfrequency
spectral characteristics exists within the inﬁnity of possibilities
is as yet theoretically controversial and observationally unre
solved. More as a matter ofconvenience than a matter ofhard
fact. highfrequency spectral characteristics of seismic sources
are generally discussed in terms of f2. and In related by the
constant stress drop assumption (9J3 = const in the context
of the {tnf. relations of Hank: and Thatcher [l972]) and 7 = 2
(the «iisquare model) or 7 = 3 (the wcube model. in the
terminology ofAlrr' [1967]). Figure l schematically illustrates the 7 = 2 and 7 = J
seismic source models in terms of two idealized farﬁeld shear
wave displacement spectral amplitudes at the same distance R.
In both the 7 = 2 and 7 = 3 cases the two earthquakes have
been assigned the same Aer. so the corner frequencies lie on a
line of slope 3 in these loglog plots. In both cases the larger
event (event I) has fl. and Mo 3 orders of magnitude larger
than the smaller event (event 2). and f3” is ID times smaller
than f."'. At frequencies greater than fol". spectral amplitudes are IO
times greater for event I than for event 2 in the 7 = 2 case but
are the same in the 7 = 3 models. How do we interpret these
models in terms of timedomain amplitudes. recognizing that
Tu'“ 2: IOTd‘“? Figure 2 presents the extreme interpretations.
Here. for purposes of illustration we have taken fol“ = 0.05 2238 cl IV\,I o. d)
N22“; Fig. 2. Time domain interpretation of the w" and at" source
models: (a) to" model when Is radiation arrives continuously across
T, (2 s on the lefthand side and 20 s on the righthand side. of which
only 10 s are shown in the ﬁgure). (1)) to" model when ls radiation
arrives in the ﬁrst 15 interval. (c) or" model when ls radiation arrives
continuously across T... and (d) to" model when ls radiation arrives
in the ﬁrst 15 interval. Relative ls amplitudes are given in two groups
of four. one for the to" model interpretation and one for the to"
model interpretation: the choice of I in the upper lefthand corner of
each square is arbitrary. Hz. ﬁg" = 0.5 Hz. Td‘“ = 20 sec. and Tam = 2 sec. and we are
investigating possible interpretations of 15 time domain am
plitudes. those used in determining m... Figure 2a is the interpretation for the 7 = 2 earthquakes if
the 15 energy arrives more or less continuously over the
complete faulting duration. In this case. ls spectral ampli
tudes for the larger event are 10 times greater than for the
smaller event. but the 15 time domain amplitudes are the same
for both events—they have the same In... [fall the 15 energy
arrives at the same time. however. the 15 time domain ampli
tudes and m, of the larger event are 10 times larger (Figure 215). For the 7 = 3 earthquakes. ls spectral amplitudes must be
the same. In Figure 2c this is achieved in a manner analogous
to that in Figure 2a. but now 15 time domain amplitudes for
event l are 10 times smaller than for event 2'. that is. m. must
decrease with M... Figure 2d is the analogue to Figure Zb; here
ls time domain amplitudes for the two earthquakes are the same; they have the sam, m...
The interpretation in Ft ur c is certainly unacceptable: m, does not systematically decrease with increasing Mo. Neither.
homdoes m. increase beyond m. 2: 6)7. and the inter
pretation WWfsWnrmw. Z
10" dyn cm. One‘s preference for the interpretation in Figure
2a or 2d and thus one‘s preference for 7 = 2 or 7 = 3 seismic
source models then depends on whether one believes that all
(or most) of the 15 energy arrives more or less continuously
through T, (T, > I s) or arrives more or less impulsively in a
~ls window (and in the case ofma. the ﬁrst one or two such
windows) no matter what the value of T... It is appropriate to
recall now that both possibilities are extreme. and grossly
simpliﬁed. interpretations: the truth. in most cases. should lie
somewhere in between. Even so. when T, >>  s in the case of
the larger earthquakes (M. 2: 6)). it is clear that Figure 2d is
much more the exception than the rule. as almost all short
period seismograms of large and great earthquakes reveal.
Thus I conclude. as Aki‘ [1967] did more than a decade ago.
that nthM. data support they = 2 model. in the interpretation
of Figure 20. as the one generally (but certainly not always)
applicable to the representation of highfrequency spectral
characteristics. With the assumptions that (l) fault propagation in both
coordinates ofthe fault plane is equally phase coherent and (2)
the source displacement time function is a propagating ramp HANKS: FAULT M ECHANICS of ﬁnite duration (with singular particle accelerations). Geller
[1976] followed Haskell [1964] to obtain 7 = 3 at high fre—
quencies. His justiﬁcation of this model with existing m.M.
data is not correct. however. because he assumed that m,, and
M. faithfully represent spectral amplitudes at I and 20s
periods. respectively. across the entire range of magnitudes
observable at teleseismic distances. Geller [1976] notes that ‘it
is not exactly correct' to do this; quite generally. it is not at all
correct to do this. except for the smaller earthquakes (M 2 5)
for which lo 2 1 Hz. In the latter case. both m. and M. become
longperiod measurements. proportional to H... but then. of
course. m.M. data carry no information at all about high
frequency spectral characteristics of earthquake sources. There are. in addition. several other observations that are in
general accord with the highfrequency spectral charaCteristics
of the w" model. First. the difference of a factor of 20 in the
maximum m. of ~7.0 and maximum M. of ~83 is 'exac11y'
predicted by the7 = 2 model (because ofthc period shift in the
amplitude measurement from I to 20 5) provided that m, 2
M. at a: 7. 1n the ‘latest' form of the linear relations between m.
and M.. m. = M. at 6.75 [Ricliter. 1958. p. 348]. Second. the
same arguments used above tojustify they = 2model in terms
of ms and M. data. and the upper limit to each. may also be
used to explain why peak acceleration data at a ﬁxed. close
distance are such a weak function of magnitude. especially
abOve ML a til5 [Hanks and Johnson. 1976]. Third. the highfrequency spectral characteristics of the San
Fernando earthquake are very well known [Benill. 1975]. even
at frequencies 2 orders of magnitude greater than I. = 0.1 Hz
[Wyss and Hanks. 1972]. because ofthe large number ofstrong
motion accelerograms that recorded this earthquake at local
distances. With allowance for radiation pattern effects and
anelastic attenuation. the 7 = 2 model ofBrune [1970. I971].
parameterized by Mo = 10" dyn cm and r = 10 km. is the
simplest possible interpretation of the data. although more
complicated interpretations are possible and. in view of the
highly inhomogeneous character of faulting for this earth
quake. perhaps warranted. Finally. a great number of spectral determinations have
been made in the course ofnumerous source parameter stud
ies. although the great bulk of these are singlestation measure
ments (in addition to those cited by Hanks [1977]. see also
Tn'funac [19720. b]. Johnson and McEur‘lly [I974]. Balctm er al.
[1976]. and Hamel! and Brune [1977]). Of the three parame
ters n... f... and 7. 7 is almost always the least well determined.
Even so. 7 = 2 is the value most often recovered. although the
same observations clearly demonstrate that 7 is not precisely
2. or even particularly close 01 it. for each and every earth
quake. Still and all. the several sets of observations summa
rized in this section leave little alternative to the conclusion
that the 7 = 2 model is the one generally. if certainly not
always. operative. Figure 3 presents the acceleration spectral amplitudes. in the
presence of anelastic attenuation for the two 7 = 2 events
whose displacement spectral amplitudes are given in Figure I.
In the frequency band I. S f S In... acceleration spectral
amplitudes are constant. I...“ being determined by setting the
argument of the exponent in the expression e—IlR/QB equal to
I. Then one interpretation. again nonunique. of Figure 3 is
that the corresponding acceleration time histories are band
limited (I. S f 5 Ln“). ﬁnite duration (0 S r — R/B S T.)
white noise. The whiteness arises from the constant spectral
amplitudes equal to 901'.” in the band I. S f S In“... The
randomness has simply been assumed. but in view of the
generally chaotic nature of strong motion accelerograms for HANKS: FAULT MECHANICS M ~ 5 earthquakes at R :5 [0 km. in ﬁnite time windows and
frequency bands. this assumption does not seem unreasonable.
indeed. the idea that ground acceleration time histories can be
treated as bandlimited. ﬁnite duration white noise has been
the basis for considerable work in the analysis of existing
accelerograms and in the computation of synthetic accelero
grams for more than 30 years in the engineering community
le.g.. Housner. [947; Hudson. 1956; Bycroﬂ. [960; Housnerand
Jennings. l964; Jennings at £11.. I968]. Hanks and Johnson [1976] developed the following relation
between the amplitude ii of any acceleration pulse at R and the
dynamic stress diﬁ'erence a... giving rise to it in the source
region:  _ L a u p R (9)
where p is density. In this framework the interpretation ofthe
7 = 2 model given above in terms of bandlimited. ﬁnite
duration white noise in ground acceleration translates directly
into a white. random distribution ofdynamlc stress differences
(at wavelengths less than r). but only with respect to the
essentially onedimensional conﬁguration of Figure 3. That is.
loosely speaking. the abscissa of Figure 3 is a onedimensional
wavelength spectrum of a twodimensionalfaulting process.
N0w. if the radiated ﬁeld is drawn from IAo,,(k)[ ~ (Ir/kn)"
and ifthe number ofcontributions to the spectral representa
tion of the radiated ﬁeld at frequency f. where I ~ It, is
proportional to (Ir/kn)“. as suggested by the quasi—static fre
quency of occurrence relation (4), then the wave number spec
trum of the radiated dynamic stress diﬁ'erences will be constant
for k 2 l/r. Through (9) this implies a white acceleration
spectrum and thus the 7 = 2 model of the farﬁeld shear
lisplacement spectrum. DISCUSSION The consistency of the spectral composition inferred for
Aa,(x. y) and the 7 = 2 model of the highfrequency ra
diated ﬁeld is notable. in view of the grossly differing time
and dimension scales associated with them individually (as
long as several decades and hundreds of kilometers in the case
of Aap(x. y) and as short as fractions of seconds and tens of
meters in the case of the 7 = 2 model for small earthquakes).
Because of the variety of uncertainties and unknowns associ
ated with this coincidence. it is probably premature to make
too much of it or reach too far for its physical signiﬁcance.
Even so. it would follow quite naturally if the tectonic stress
along active crustal fault zones was of the form 602) + output.
y). with the previously described characteristics for each. if this is the case. 1: values different from i might be accom
panied by 7 diﬁ‘erent from 2 for those earthquakes of the “ l
'5 I3’
to
O
2
log frequency
Fig. 3. Acceleration amplitude spectra at R for the or“ earth quakes of Figure 2. with attenuation explicitly shown. 2239 counted sample. in particular. when it < i. there is a relative
excess of larger earthquakes to smaller earthquakes. which
may be interpreted in terms of lfoplkﬂ deﬁcient in short
wavelength amplitudes relative to a normative k" depen
dence; then 7 > 2 would be expected if the dynamic Stress
diﬁ‘erences arise from the same tectonic stress ﬁelds. Presently
available data. unfortunately. are not suited for a critical ex
amination of this hypothesis principally because of the poor
control on 7. Correlations of welldetermined bvalue and 7 data may be
particularly important to devel0p in view of the suggestion
that b decreases prior to larger earthquakes and is therefore a
possible means of earthquake prediction [Schoiz eI ai.. 1973;
Wyss and Lee. 1973: Rikirake, l975]. although the data are
hardly conclusive on this matter [Lahr and Pomeroy. 1970].
With respect to the ideas presented here. several points are
worth making about this possibility. First. ifb = l in a certain
region Over a long enough period of time. then b values esti
mated over a shorter period of time that explicitly excludes a
larger earthquake (i.e.. the one to be predicted) will be natu
rally biased to values that are greater than one. not less than
one. Thus those areas with b < l in a time intervaljust before
an earthquake larger than any member of the set counted to
determine in are especially interesting. Second. as discussed
previously. one interpretation of b < l is that IA?,(k)[ is
relatively deﬁcient in short~wavelength amplitudes; the devel
opment of longerwavelength stress concentrations would
seem to be a nautral prelude to the occurrence of a larger
earthquake. Third, ifb < l is accompanied by 7 > 2 for those
earthquakes that are counted to deﬁne b. one may proceed on
an earthquake~byearthquake basis rather than waiting the
much longer period of time for enough earthquakes to yield a
welldetermined 17. Another possibility. and an important one.
is that b diﬁ'erent from I may be accompanied by earthquake
stress drops that are not independent of source size. More
generally. if changes in any one of the normative situations of
b values near 1. the general validity of the 7 = 2 model. the
constancy of earthquake stress drops independent of size. and
A~ap(k)l ~ it" occur as a result of processes premonitory to
largerscale faulting. it seems reasonable that at least one and
perhaps all of the other phenomena will change as well. Finally. it is worth noting that if. as in the interpretation
here. the frequency of oceurrence statistics. or b values. are
gOverned by the spectral composition of A0,,(x. y). then the
Overall rate of seismicity. or a value. is presumably controlled
by Edy). At least along major plate margins. 50:) increases
slowly on a time scale of hundreds of years until the area of
interest is ruptured by throughgoing faulting. at which time
61y) precipitously decreases by the earthquake stress drop. it is
well known that the San Francisco Bay area. to aconsiderable
distance away from the San Andreas fault. was considerably
more seismic at the M 2, 6 level in the ~70 years prior to the
1906 earthquake than it has been in the ~70 years since
[Tother. 1959]. excluding the immediate aftershock sequence.
and it seems reasonable that a stress drop of approximately
[00 bars along the San Andreas fault at the time ofthe earth
quake played a central role in the greatly reduced seismicity
rate. At the same time. however. this situation underscores the
potential ambiguity at all wavelengths 2}: between the long
wavelength character of A0,,(x. y) and 50) variable along the
fault length. HIGHFREQUENCY STRONG GROUND MOTION Whether or not .5 values and 7 are related through a com
mon origin in a tectonic stress of the form t'r(y) + Acr,(x. y) 2240 along active crustal fault zones, as discussed in the last section.
the general validity of the 7 = 2 model has important implica
tions for new approaches to the estimation of highfrequency
strong ground motion for aseismic design purposes. One possi
bility that suggests itself immediately is developed below. in
comparison with the existing approach. The alternate point of
view. that strong motion accelerograms written at close dis
tances (R :2 10 km) for potentially damaging earthquakes are
important data for investigating in more detail the validity of
the 7 = 2 model. shall be left as being implicit. Since the ﬁrst strong motion accelerograms were written
more than 40 years ago. peak acceleration has been the most
commonly used single index of strong ground motion. It has.
however. been known for some time that peak acceleration
need not be. and too often cannot be. a uniformly valid mea—
sure of strong ground motion over the entire frequency band
and amplitude range of engineering interest. The very charac
ter of the peak acceleration datum as a shortperiod. time
domain amplitude measurement is the principal reason for two
important limitations on its value as a measure of strong
ground motion. First. for M 2 5 earthquakes at close dis
tances. taken here as a rough threshold of potentially dam
aging ground motion. the period of this phase is much shorter
than the faulting duration. Thus the peak acceleration simply
cannot measure gross source properties of potentially dam
aging and destructive earthquakes. even if such data may. in a
large enough set of observations. indicate limiting conditions
on the failure process in very localized regions of the fault
surface. Second. this same characteristic of the peak accelera
tion datum makes precise corrections for wave propagation
eﬁects. including anelastic attenuation and elastic scattering.
impossible except under very unusual conditions. Both of
these problems. but especially the second. are in turn respon
sible for the notoriously large scatter in peak acceleration data.
even through very small variations of magnitude. distance. and
site conditions. It is this last problem that limits the utility of
peak acceleration even as a measure of highfrequency strong
ground motion. These diﬁiculties in interpreting. manipulating. and extrapo
lating peak acceleration data are widely acknowledged by
engineers and seismologists alike. and recently acquired peak
acceleration data for 3 S M S 5 earthquakes at R 2 l0 krn
have accentuated them [Hanks and Johnson. l976; Seekr'rrs and
Hanks. 1978]. But if peak acceleration is not a reliable mea sure of highfrequency strong ground motion. as is gener
ally agreed to be the case. then what is? One such measure that is almost certainly better is the rms
acceleration. a...... First. since the time integral of the square
ground acceleration is proportional to the work per unit mass
done on a set of linear. viscously damped. singledegreeof
freedom oscillators with natural frequencies between 0 and m
[A rias. I970]. a"... is then of considerable engineering impor
tance (to the extent that actual structures may be approxi
mated by such oscillators) with respect to the design capabili
ties of the rate of dissipation of this energy. Second. as a
broadband integral measure. it almost certainly will be a more
stable measure of highfrequency strong ground motion than
individual highfrequency time domain amplitude measure
ments. Finally. as described below. an... can be directly related
to a very few parameters ofthe earthquake source and source—
station propagation path and thus can be estimated in the
absence of strong ground motion observations or empirical
correlations derived from them. HANKS: FAULT MECHANICS The analysis begins with Parseval‘s theorem. EMUI'dIzl—rfalﬂwll’dw where a(r) is the acceleration time history and &(w) is its
Fourier amplitude spectrum. For 5(a)) we take the y = 2 model
of Figure 3 and note that for large earthquakes at close dis
tances. I...“ >> I... so that contributions to the righthand side
of (IO) for f S 1'. are small. We further assume that the
signiﬁcant motion is conﬁned to the shear wave arrival win
dow 0 S r — R/B < T, and anelastic attenuation cuts the
spectral amplitudes off sharply at f a: fm... Then (10) may be
written (10) [g i‘1(!)'arr = '22,; I’m" ﬁ(w)'dw (1') The rms acceleration is III
an“.II a(r)l’dr: (12)
Equations (ii) and (12). together with
5(a)) = Quiztrfo)’ In S f 5 fm... (l3)
and the approximation
f9 = 1/7} (14)
result in
arm. = 2"‘(21r)”flofo’(fm../fn)"’ (l5) for I...“ >> f... Finally. for the Brune {l970. l9'll] scaling. AU = [Hanks and Thatcher. l972]. which. upon substitution in (15).
gives (17) = 2tr2(21r22 AU (Eyre
a"... 0.85 106 FR 1.0 The factor of0.85 introduced in (l7) accounts for free surface
ampliﬁcation of SH waves (2.0). vectorial partition onto two
horizontal components of equal amplitude (1/2“). and the rms value of the shear wave radiation pattern (0.6) [Thatcher
and Hanks. l973]. Table 1 compares am. values estimated from ([7) with
observed. whole record values for the San Fernando earth
quake at Pacoima Dam and the Kern County earthquake at
Taft and with 'observed’ values corrected for (record length/
T“)m to estimate the a..." value that occurs in the time inter
val of the S wave arrival plus T... For the San Fernando
earthquake the 'observed' value is 70% greater than the esti
mated value at Pacoima Dam; in the case of the Kern County
earthquake at Taft the ‘observed‘ value is only 30% greater
than the estimated value. By conventional seismological stan
dards in estimating highfrequency amplitudes. this agreement
is remarkable. These comparisons are. on the one hand. encouraging with
respect to the use ofam as a measure of highfrequency strong
ground motion and. on the other hand. further evidence for
the general validity of the 7 = 2 model. in both respects.
h0wevcr. further examination of existing data is required. and
strong motion accelerograms at R S 10 km are a particularly
valuable set of observations for these analyses. s HANKS: FAULT MECHANICS 2241 TABLE 1. Comparisons of Estimated and 'Observed‘ a"... Values
San Fernando Kern County
Earthquake Earthquake
at Pacoima Darn at Taft
Act. bars 50' 60'
1'... Hz 0.1? 0.051
R. km ~10 ~40
Imus Hz ~30§ ~8§
0...... Cm/s'
Estimated I40 30
Observed I I 120.110 26.27
‘Observed' 240 39 'Konamon' and Anderson [1975]. lHanks and Wyss [1972]. Estimated for L = 50 km. r = L/2. and f. = 2345/2". §From rf.....R/Q.B = I With .6 = 3.2 km/s and Q = 300. IIFor both horizontal components. from Brody er of. [1971] for
Pacoima Dam and Hudson and Brady [1969] for Taft. SUMMARY AND CONCLUSIONS The nature and magnitude of variations in the deviatoric
stresses existing along active crustal fault zones are central to a
full understanding of the cause and eﬂ'ect of earthquakes. but
these variations are at the present time only poorly under
stood. These stresses may be written in the form 60:) + Ao.,(x.
y). Where EU) is the average tectonic stress at depth y and
Ao,(x. y) is the seismologically observable stress drop poten
tial function. The constancy of earthquake stress drops inde
pendent ofsource dimension suggests that the spectral compo
sition oon,(x. y) is ofthe form lAEAR)! ~ It". Independent
support for this representation exists in the general validity of
the 7 = 2 model of the far~ﬁcd shear wave displacement
spectrum. under the reasonable assumptions that the radiated
ﬁeld of earthquakes is also drawn from the stress differences of
Ao,(x. y) and that the number of contributions to the radiated
ﬁeld at frequency {goes as (it/kn)“. consistent with the quasi
static frequency ofoccurrence relation N ~ l/r’. Separately. a
variety of seismologic observations suggests that the 7 = 2
model is the one generally. although certainly not always.
applicable to the highfrequency spectral decay of the farﬁeld
radiation of earthquakes. That the constancy of earthquake stress drops. 1; values near
I. and the general validity of the 7 = 2 model may all be
related to the same spectral composition of AGAX. y) is a
notable result. although there is as yet considerable uncer
tainty. especially in an observational sense. in relating these
phenomena to a common physical origin. namely. Iﬁptkﬂ ~
k". If. however. these phenomena indeed share a common
explanation in a tectonic stress of :70») + Aop(x. y) existing
along active crustal fault zones. where AopLx. y) has spectral
composition lAj‘ErAkH ~ It". a possible consequence is that 1:
values diﬂ'erent from 1 would be accompanied by 7 different
from 2 for those earthquakes counted to determine 12. Another
is that changes in 1; values from 1 may be accompanied by
earthquake stress drops different from 5 and/or different
from the normative area independence. Irrespective of these possible associations. the 7 = 2 model
in the presence of anelastic attenuation suggests that high
frequency strong ground motion has a straightforward inter
pretation as bandlimited. ﬁnite duration white noise in accel
eration. An estimate of a"... is easily constructed from this
interpretation. and for several reasons it appears to be of
potential importance as a measure of highfrequency strong ground motion for aseismic design purposes. Alternatively.
these same ideas. together with strong motion accelerograms
written at close distances for potentially damaging earth quakes. may be used forinvestigating in more detail the 7 = 2
model. Acknowledgments. I have enjoyed the critical remarks of D. J.
Andrews. A. McGarr. and W. Thatcher in developing the ideas ofthis
manuscript and useful conversations with M. Caputo regarding fre
quency ofoccurrence statistics. I deeply appreciate the eﬂ'orts of E. A.
Flinn in his capacity of editor and an unknown associate editor in
having me evaluate critically an original. and erroneous. guess that the
spectral composition of ought. y) was white. a guess written into the
work of Hanks (1977) without consequence. Carol Sullivan patiently
typed this manuscript several times. This research was supported in
part by National Science Foundation grant ENV76—81816. Pub
lication approved by the Director. U.S. Geological Survey. REFERENCES Aki. K.. Scaling law of Seismic spectrum. J. Geophys. Res. 72. 1212—
1231. I967. Aki. K.. Earthquake mechanism. Terronophyst‘rs. 13.423446. 1972. Andrews, D. J.. Coupling of energy between tectonic procuses and
earthquakes. J. Geophys. Res. 33. 22592264. 1978. Arias. A.. A measure of earthquake intensity. in Seismic Design for
Nuclear Power Plants. edited by R. J. Hanson. MIT Press. Cam
bridge. Mass.. 1970. Bakun. W. H.. C. G. Bufe. and R. M. Stewart. Body—wave spectra of
central California earthquakes. Bull. Seismol. Soc. Amer.. 66. 363
384. 1976. Berrill. J. B.. A study of highfrequency strong ground motion from
the San Fernando earthquake. PhD. thesis. Calif. Inst. ofTechnol..
Pasadena. I975. Bouchon. M.. A dynamic source model for the San Fernando earth
quake. Butl. Seismol. Sac. Amer,, 68. 15551576. 1978.
Brady. A. G.. D. E. Hudson.and M. J. Trifunac. Earthquake Accelero grams. vol. I.part C. Earthquake Engineering Research Laboratory.
California institute of Technology. Pasadena. I971. Brune. J. N.. Tectonic stress and the spectra of seismic shear waves. J.
Geophys. Res” 75. 49975009. 1970. Brune. J. N.. Correction. J. GeOphys. Res. 76. 5002. 1971.
Burdick. L. J.. and G. R. Mellman. Inversion of the body waves from the Borrego Mountain earthquake to the source mechanism. Bu”.
Seisnrot'. Soc. Amer.. 66. 14851499. 1976. Bycroft. G. N.. White noise representation of earthquakes. J. Eng.
Mach. Dt'u. Amer. Soc. Ct‘ut‘f Eng. 86. 116. 1960. Byerlee. J.. Friction ofrocks. Pure Appl. Geophys. 116,615626. I973. Caputo. M.. Mode! and observed seismicin represented in a two
dimensional space. Ann. Geoﬂs.. 29. 277—288. 1976. Das. 5.. and K. Aki. Fault plane with barriers: A versatile earthquake
model. J. Geophys. Res. 82. 56535670. 1977. Geller. R. J.. Scaling relations for earthquake source parameters and
magnitudes. Bull. Seismol'. Soc. Amer.. 66. 1501—1523. 1976. Hanks. T. C.. The faulting mechanism of the San Fernando earth
quake. J. Geophys. Res. 79, 1215—1229. 1974. Hanks. T. C.. Earthquake stress drops. ambient tectonic stresses. and
stresses that drive plate motions. Pure Appl. Geophys. H5. 441458.
1977. Hanks. T. C.. and D. A. Johnson. Geophysical assessment of peak
accelerations. Bull. Seismol. Soc. Amer.. 66. 959963. 1976. Hanks. T. C.. and W. R. Thatcher. A graphical representation of
seismic source parameters. J. Geophys. Res. 77. 4393—4405. 1972. Hartzell. S. H.. and J. N. Brune. Source parameters for the January
1975 Brawleylmperial Valley earthquake swarm. Pure App}.
Geophys, 115. 333—355. 1977. Haskell. N. A.. Total energy and energy spectral density of elastic
wave radiation from propagating faults. Bull. Seismol. Soc. Amer..
54. 1811—1841. 1964. Heaton. T. H.. and D. V. Helmberger. A study ofthe strong ground
motion of the Borrego Mountain. California. earthquake. Bu”.
Seismof. Soc. Amer., 67, 315330. 1977. Housner. G. W.. Characteristics of strongmotion earthquakes. Bull.
Ser'smol. Soc. Amen. 37. 1931. 1947. Housner. G. W.. and P. C. Jennings. Generation of artiﬁcial earth
quakes. J. Eng. Merit. Div. Amer. Soc. Civil Eng. 90. 113150. 1964. Hudson. D. E.. Response spectrum techniques in engineering seis 2242 mology. in World Conference on Earthquake Engineering. Earth
quake Engineering Research Institute. Berkeley. Calif. I956. Hudson. D. E.. and A. 0. Brady. Strong Motion Earthquake
Acceleragroms. vol. 2. part A. Earthquake Engineering Research
Laboratory. California Institute of Technology. Pasadena. I969. Jennings. P. C..G. W. Housner. and N. C. Tsai. St'mulatedEarthquake
Motions. Earthquake Engineering Research Laboratory. California
Institute of Technology. Pasadena. Calif.. I968. Johnson. L. R.. and T. V. McEviIIy. Nearﬁeld observations and
source parameters of central California earthquakes. BUN. Seismal.
Soc. Amer.. 64. [3551886. I974. KanamOri. H.. The energy release in great earthquakes. J. Geophys.
Res.. 82, 298I~2981. I971. Kanamori. H.. and D. L. Anderson. Theoretical basis oI‘some empiri
cal relations in seismology. Bull. Seismot'. Soc. Amer.. 65. I073
I096. I975. Lahr. .I.. and P. W. Pomeroy. The I'oreshockal'tershock sequence of
the March 20. 1966 earthquake in the Republic of Congo. Bull.
Seismot’. Soc. Amer.. 60. l2451258. I970. Richter. C. F.. Elementary Seismology. 768 pp.. W. H. Freeman. San
Francisco. Calil'.. I953. Rikitake. T.. Earthquake precursors. Eu”. Setsmol. Soc. Amer.. 65.
II33—II62. I975. Scholz. C. H.. L. R. Sykes. and Y. P. Aggarwa]. Earthquake predic
tion: A physical basis. Science. I8]. 803810. I973. HANKS: FAULT M ECIIANICS Seekins. L. C.. and T. C. Hanks. Strong motion accelerogtams ofthe
0r0ville aftershocks and peak acceleration data. Butt. Seismal. Soc.
Amer.. 68. 677689. I978. Thatcher. W.. and T. C. Hanks. Source parameters of southern Cah
fornia earthquakes. J. Geophys. Res. 73. 3547—8576. I973. Tocher. D.. Seismic history of the San Francisco region. San Fran
cisco Earthquakes of March 1957. Can’t]? Div. Mines Geol. Spec.
Rep.. 57. I959. Trifunac. M. D.. Stress estimates for the San Fernando earthquake of
February 9. I97]: Main event and thirteen altershocks. Butt. Sets
mol. Soc. Amer.. 62. 72l750. 1972a. Trifunac. M. D.. Tectonic stress and the source mechanism of the
Imperial Valley. California. earthquake of I940. Butt. Set‘smal. Soc.
Amen. 62. I281I302. I972b. Wyss. M.. and T. C. Hanks. The source parameters of the San Fer
nando earthquake inferred from teIeseismic body waves. Bull. Sets
mol. Soc. Amer., 62. SIN602. I972. Wyss. M.. and W. H. K. Lee. Time variations of the average earth
quake magnitude in central California. in Proceedings ofthe Confer
enee on Tectonic Problems of the San Andreas Fault System. Stan
I'ord University Publications. Stanford. Calil.. I973. (Received May 4. I978;
revised November 3. I973:
accepted January I. I919.) ...
View
Full
Document
This note was uploaded on 07/23/2008 for the course GEOSC 508 taught by Professor Marone during the Spring '07 term at Pennsylvania State University, University Park.
 Spring '07
 MARONE

Click to edit the document details