Hanks1979 - VOL. 84. NO. BS JOURNAL OF GEOPHYSICAL RESEARCH...

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Unformatted text preview: VOL. 84. NO. BS JOURNAL OF GEOPHYSICAL RESEARCH MAY I0. 1979 Monsieu— b Values and or" Seismic Source-Models: Implications for Tectonic Stress Variations Along Active Crustal Fault Zones and the Estimation oingh-Frequency 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. moment-magnitude 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 two-dimensional Fourier transform of Ao,(x. y) expressed in radial wave number k. The 7 = zmodel ofthe far-field 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 quasi-static 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 high-frequency spectral decay of the far-field 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 far-field. high-frequency shear radiation for the 'y = 2 model in the presence of anelastic attenuation may be interpreted as band-limited. 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 high-frequency 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 self-stress [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 amplitude-wave- 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 Bout-Iron [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 scaled-down versions (in spatial dimension and wave period) of the com- plex. multiple-event 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. higher-quality 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 high-frequency 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 high-frequency 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 large-scale 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 root-mean-square (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 mean-square 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 two-dimensional 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 shorter-wavelength 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 area-independent 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 two-dimensional 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 one-dimensional 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 far-field radiation emanating from simple seismic source models [e.g.. Aki. I967; Brune. I970. l97l] is generally characterized by a long-period level 9., pro- portional to M... a corner frequency f. proportional to r". and a high-frequency 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 high-frequency 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. high-frequency 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 {tn-f. relations of Hank: and Thatcher [l972]) and 7 = 2 (the «ii-square model) or 7 = 3 (the w-cube model. in the terminology ofAlrr' [1967]). Figure l schematically illustrates the 7 = 2 and 7 = J seismic source models in terms of two idealized far-field 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 log-log 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 time-domain 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 I-s radiation arrives continuously across T, (2 s on the left-hand side and 20 s on the right-hand side. of which only 10 s are shown in the figure). (1)) to" model when l-s radiation arrives in the first 1-5 interval. (c) or" model when l-s radiation arrives continuously across T... and (d) to" model when l-s radiation arrives in the first 1-5 interval. Relative l-s 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 left-hand 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 1-5 time domain am- plitudes. those used in determining m... Figure 2a is the interpretation for the 7 = 2 earthquakes if the 1-5 energy arrives more or less continuously over the complete faulting duration. In this case. l-s spectral ampli- tudes for the larger event are 10 times greater than for the smaller event. but the 1-5 time domain amplitudes are the same for both events—they have the same In... [fall the 1-5 energy arrives at the same time. however. the 1-5 time domain ampli- tudes and m, of the larger event are 10 times larger (Figure 215). For the 7 = 3 earthquakes. l-s spectral amplitudes must be the same. In Figure 2c this is achieved in a manner analogous to that in Figure 2a. but now 1-5 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 l-s 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 WWfsWn-rmw. 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 1-5 energy arrives more or less continuously through T, (T, > I s) or arrives more or less impulsively in a ~l-s 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 nth-M. data support they = 2 model. in the interpretation of Figure 20. as the one generally (but certainly not always) applicable to the representation of high-frequency 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 20-s 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 long-period 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 high-frequency 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 [Ric-liter. 1958. p. 348]. Second. the same arguments used above tojustify the-y = 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 til-5 [Hanks and Johnson. 1976]. Third. the high-frequency spectral characteristics of the San Fernando earthquake are very well known [Ben-ill. 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 single-station 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 band-limited. 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 band-limited. 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 one-dimensional configuration of Figure 3. That is. loosely speaking. the abscissa of Figure 3 is a one-dimensional wavelength spectrum of a two-dimensionalfaulting 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 far-field shear lisplacement spectrum. DISCUSSION The consistency of the spectral composition inferred for Aa,(x. y) and the 7 = 2 model of the high-frequency 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 well-determined b-value 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 longer-wavelength 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~by-earthquake basis rather than waiting the much longer period of time for enough earthquakes to yield a well-determined 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 larger-scale 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 [Tot-her. 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 5-0) variable along the fault length. HIGH-FREQUENCY 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 high-frequency 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 short-period. 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 high-frequency 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 high-frequency 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. single-degree-of- 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 high-frequency strong ground motion than individual high-frequency 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'dI-zl—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 right-hand 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 high-frequency amplitudes. this agreement is remarkable. These comparisons are. on the one hand. encouraging with respect to the use ofam as a measure of high-frequency 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 lA-EAR)! ~ It". Independent support for this representation exists in the general validity of the 7 = 2 model of the far~fic|d 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 high-frequency spectral decay of the far-field 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 band-limited. 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 high-frequency 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. 2259-2264. 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 high-frequency 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. 1555-1576. 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. 4997-5009. 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. 1485-1499. 1976. Bycroft. G. N.. White noise representation of earthquakes. J. Eng. Mach. Dt'u. Amer. Soc. Ct‘ut‘f Eng. 86. 1-16. 1960. Byerlee. J.. Friction ofrocks. Pure Appl. Geophys. 116,615-626. 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. 5653-5670. 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. 441-458. 1977. Hanks. T. C.. and D. A. Johnson. Geophysical assessment of peak accelerations. Bull. Seismol. Soc. Amer.. 66. 959-963. 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 Brawley-lmperial 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, 315-330. 1977. Housner. G. W.. Characteristics of strong-motion earthquakes. Bull. Ser'smol. Soc. Amen. 37. 19-31. 1947. Housner. G. W.. and P. C. Jennings. Generation of artificial earth- quakes. J. Eng. Merit. Div. Amer. Soc. Civil Eng. 90. 113-150. 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. Near-field observations and source parameters of central California earthquakes. BUN. Seismal. Soc. Amer.. 64. [355-1886. 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'oreshock-al'tershock sequence of the March 20. 1966 earthquake in the Republic of Congo. Bull. Seismot’. Soc. Amer.. 60. l245-1258. 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]. 803-810. 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. 677-689. 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. 72l-750. 1972a. Trifunac. M. D.. Tectonic stress and the source mechanism of the Imperial Valley. California. earthquake of I940. Butt. Set‘smal. Soc. Amen. 62. I281-I302. 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. SIN-602. 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.) ...
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