Bulletin of the Seismological Society of America, Vol. 72, No. 6, pp , December 1982

Transcription

1 Bulletin f the Seismlgical Sciety f America, Vl. 72, N. 6, pp , December 1982 ANALYSIS OF THE GROUND ACCELERATIONS RADIATED BY THE 1980 LIVERMORE VALLEY EARTHQUAKES FOR DIRECTIVITY AND DYNAMIC SOURCE CHARACTERISTICS BY JOHN BOATWRIGHT AND DAVID M. BOORE ABSTRACT The strng mtin accelergraph recrdings f the 24 January 1980 main shck and the 27 January 1980 aftershck f the Livermre Valley earthquake sequence are analyzed fr systematic variatins with azimuth r statin lcatin. The variatin f the peak acceleratins with epicentral azimuth is apparently reversed fr the tw events: the main shck acceleratins are larger t the suth, and the aftershck acceleratins are larger t the nrthwest. We eliminate the site effects by frming the rati f the peak acceleratins recrded at the same statin, after crrecting fr the epicentral distance. This analysis indicates that surce direcuvity caused a ttal variatin f a factr f 10 in the peak acceleratins. Cmparisn f this variatin with the spatia! extent f the aftershck sequences suggests that the strng directivity in the radiated acceleratins is the result f unilateral ruptures in bth events. The accelergrams recrded at 10 statins within 35 km f the events were digitized t analyze the azimuthal variatin f the rms acceleratin, the peak velcity, and the radiated energy flux. The variatin f rms acceleratin crrelates almst exactly with the variatin f the peak acceleratins. This crrelatin is analyzed using bth deterministic and stchastic mdels fr the acceleratin wavefrms. The peak velcities, crrected fr epicentral distance, vary with azimuth by a factr f 5 fr bth events, while the radiated energy flux varies by a factr f 30 fr the main shck and 15 fr the aftershck. The peak velcities are strngly crrelated with the radiated energy flux. The radiated seismic energies are estimated t be 2.6 ± 0.9 x 102 dyne-cm fr the main shck and 1.5 ± 0.3 x 1020 dyne-cm fr the aftershck. INTRODUCTION The azimuthal variatin f the radiated seismic energy due t the gemetry f the rupture grwth, r directivity, has lng been recgnized at intermediate perids (several tens f secnds) in teleseismic recrdings f large earthquakes (e.g., Beniff, 1955). The existence f strng directivity effects at shrter perids, especially at perids f engineering interest, is a matter f current debate. Recrdings f grund acceleratins frm several recent earthquakes in Califrnia including the 1971 San Fernand (McGuire and Hanks, 1980), 1979 Cyte Lake (Archuleta, 1979), and Imperial Valley (Swanger et al., 1981) have been interpreted as shwing directivity. Althugh cnvincing, the evidence is nt indisputable. Objectins that have been raised t this interpretatin appeal t cnsideratins f the hetergeneity f the stress release in an extended rupture and systematic effects due t radiatin pattern, lcal sil amplificatin, fundatin-sil interactin, and anelastic attenuatin. While these cmpeting effects cannt pssibly bviate all these interpretatins, they hinder the direct quantificatin f directivity. In this paper, we present what appears t be unambiguus evidence fr directivity in the acceleratins radiated by tw earthquakes which ccurred near Livermre Valley, Califrnia, during January The analysis fr directivity is enhanced by the gd azimuthal cverage, the mderate size f the faults relative t the distance 1843

2 1844 JOHN BOATWRIGHT AND DAVID M. BOORE frm the statins, and the fact that many statins recrded bth events. By cmparing the acceleratins recrded at the same statins, it is pssible t eliminate many f the difficulties assciated with analyzing data frm a single event. The peak acceleratin data were first presented as part f a letter t the editr by Bre and Prcella (1980); this paper presents a mre cmplete analysis f the accelergrams written by these earthquakes. While peak acceleratin is the mst prevalent quantificatin f strng grund mtin, there are ther measurements which may be analyzed t test the directivity in the radiated wave field; in particular, we have analyzed the rms acceleratin, the peak velcity, and the radiated energy flux f the S-wave arrivals at 10 f the clsest statins. The peak velcity and the radiated energy flux (the integral f the square f the grund velcity) are velcity analgs f the peak and rms acceleratin; the energy flux, uncrrected fr attenuatin, is prprtinal t the square f the rms velcity times the signal duratin. This wrk presents the first bservatinal analysis f directivity in these integral measures f strng grund mtin. THE LIVERMORE VALLEY EARTHQUAKES The Livermre Valley earthquakes prvide an excellent pprtunity t study directivity in high-frequency grund mtin. Bth the main shck and the largest aftershck were well recrded by strng mtin accelergraphs (Figure 1). The main shck ccurred n 24 January 1981 at a hypcentral depth f 10.5 km and had a mment f abut dyne-cm. The mment f the largest aftershck, which ccurred n 27 January 1981, at a depth f 15 km, was abut dyne-cm (Blt et al., 1981; J. F. Scheimer, ral cmmunicatin, 1982). Shrt-perid P-wave first mtins indicate that the faulting invlved strike-slip mtin n near-vertical planes. The main shck had a shrt-perid P-wave ndal plane striking 11 west f nrth, while the aftershck had a ndal plane striking 37 west f nrth (Cckerham et al., 1980). Figure 2 shws the epicentral lcatins f the aftershcks which ccurred within 24 hr f the main shck. The aftershck sequence indicates that the main shck rupture prpagatin was primarily t the sutheast. The trend f the aftershck sequence is rtated 25 frm the strike f the nrth-suth ndal plane f the main shck fcal mechanism, but is parallel t the strike f the aftershck ndal plane. As discussed in the next sectin, the peak mtins radiated by the main shck arrive relatively late in the wavefrms. In the analysis fr the directivity, these mtins are assumed t be radiated by the sectin f the fault which strikes t the sutheast; the shrt-perid fcal mechanism is assumed t crrespnd nly t a small initial rupture event. Figure 3 shws the seismicity pattern fr a 5-hr perid n either side f the 27 January aftershck; 2- and 24-hr perids shw similar patterns. The rupture in the aftershck appears t have prpagated t the nrthwest; this directin is inferred frm the cluster f small earthquakes t the nrthwest f the epicenter. As Figure 2 shws, hwever, the regin f suppsed aftershcks f the 27 January event is als the site f aftershck activity fllwing the main shck. These clusters may represent the respnse f an unruptured asperity between the tw events and nt indicate rupture prpagatin t the nrthwest in the 27 January aftershck. THE STRONG MOTION ACCELEROGRAMS A subset f the accelergrams written by the 24 January main shck are shwn in Figure 4, superimpsed n a map f the regin shwing the twn f Livermre and the Sacrament River. The acceleratins pltted t the right f the statin lcatins are the SH cmpnents f the hrizntal mtin. The amplitudes f the

3 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1845 acceleratins are scaled t crrect fr the gemetrical spreading using the hypcentral distances listed in Table 2. The fault plane inferred frm the aftershck distributin is drawn as a dark wavy line. The directivity in the radiated acceleratins is clearly demnstrated by the difference in amplitude between statin DVD t the suth and statins ANT and WCS t the nrthwest. This variatin with azimuth is reversed in Figure 5, which shws the accelergrams written at the same statins by the 27 January aftershck. The acceleratins radiated by the aftershck are larger at ANT and WCS than at DVD while they are apprximately the same fr the statins t the east and west. :58" ANTe OCL '\ ~1~i~ ~4-80 IDPP FR.~.~, ~E "}~J-ZT- 80 TRY x ~ lz~ \\ IADV D - "-,~t// 57. 5,2 SJT \\ ~VR X~'., KILOMETERS I I I 3 I '~ I FIG. 1. Lcatin map, adapted frm McJunkin and Ragsdale (1980). Epicenters f the 24 and 27 January events (Cckerham et al., 1980) are shwn by stars. Permanent and temprary accelergraph statins shwn by three- and tw-letter designatins, respectively (McJunkin and Ragsdale, 1980; Switzer et al., 1981). Temprary statins were installed after the 24 January event. Structure type indicated by symbl: circle, building less than three stries; square, building taller than tw stries; triangle, dam abutment r te. In additin t the strng difference in the azimuthal variatin f the amplitudes radiated by the tw earthquakes, there is a significant difference in the character f the acceleratin wavefrms. The peak acceleratins in the main shck accelergrams arrive relatively late in the wavefrms, whereas the peak acceleratins in the aftershck accelergrams ccur as the secnd pulse. This difference is assumed t reflect the cmplexity f the rupture prcess f the main shck relative t that f the aftershck. In the main shck, the initial mtins f the accelergrams are

4 1846 JOHN BOATWRIGHT AND DAVID M. BOORE radiated by an initial rupture event whse fcal mechanism is given by the shrtperid P-wave radiatin pattern. The peak mtins in the accelergrams are assciated with rupture n the sutheast extensin f the aftershck distributin. The peak mtins in the wavefrms radiated by the 27 January event are interpreted 24 HRS. FOLLOWING 1-24 EVENT I I I 3750'N I- ; I : 57 40' N 1 1 I 'W 'W FIG. 2. Aftershcks fr 24 hr fllwing the 24 January event, with magnitude shwn by symbl size ( fr ML :> 2.0, O fr ML > 4.0). Lcatins f the 24 and 27 January events are indicated by crsses. 5 HRS. PRIOR TO 1-27 EVENT 5 HRS. AFTER 1-27 EVENT I I i I I I 3750'N O,:pO 0% ~0 i_zr 0,-27 ce) 3740'N I I I I I I 121%50' W ' W 'W ' W FIG. 3. Aftershcks fr 5-hr perids befre and after 27 January event (shwn by largest symbl). as examples f stpping phases, i.e., the acceleratin pulses radiated by the deceleratin f a rupture frnt (Savage, 1965; Madariaga, 1977), as their plarities are reversed with respect t the first mtin f the wavefrms. Befre analyzing these acceleratins, it is useful t cnsider sme f the site

5 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1847 effects in the data by cmparing accelergrams frm statins which are near the same azimuth frm the epicenters f the events. In Figure 6, we have pltted the SH cmpnents f the acceleratin recrded at statins DVD and VLR, which are separated by 1 km. Althugh the relative amplitudes are similar, the site respnse,,,i,,~,,l,''i... I... ANT 3800 ' WCS -- I...,,,t I sec \ \ llrrlrl H l+liii,~,,llll,i,,,,i,,l DPP _ :5745, LIVERMORE mllr~,,ll,~,,,~+,~llrllrjll+ll, ASE E_ 0 5 ~ i i i I t3 KM E ' I ll - ~ r,,~lr,,;,,,,~]~*~... I... DVD... I... I... tf,,l i MSJ ' ' ' FIG. 4. Selected SH accelergrams f the 24 January main shck superimpsed n a map f the Livermre area, shwing the directivity in peak mtins and relative cmplexity f the main shck accelergrams. The acceleratin scales have been adjusted t cmpensate fr the expected gemetrical spreading; the scales are 100 cm/sec 2 between the large tick marks. f the tw statins differs strngly. The Del Valle Dam te has a strng resnance at 8 Hz, and the Veterans Administratin Hspital resnates at abut 0.4 Hz (R. B. Matthiesen, ral cmmunicatin, 1980). Because these resnant frequencies are smewhat utside the peak frequency f the grund acceleratin at these statins,

6 1848 JOHN BOATWRIGHT AND DAVID M. BOORE the different respnses have relatively little effect n the measurements f the peak and rms acceleratin, ther than t slightly amplify the mtins at DVD fr the 24 January main shck. Figure 7 shws a mre extreme example f differential site effects, using the accelergrams written at three statins whse azimuths frm the epicenters are within 25 f each ther. Statins SRE and SRM are less than 5 km frm each 38"00' ~I''''L'''IB,'''I'JlIIJIIIIII ANT wcs \\.k~,!ie I L,~,,g;,I LIVERMORE \ DPP -- i745 ' ASE iiiil:~,,[~,~,r,l,,i,~,,i,,,,i DVD 0 5 I i I I I I KM I i730 ' 12200' M S J ' ' FIG. 5. Selected SH accelergrams f the 27 January aftershck. The acceleratin scales have been adjusted as in Figure 4. Nte the strng amplificatin t the nrthwest, relative t the 24 January accelergrams. ther, yet their accelergrams are remarkably different. The Eastman Kdak Building has a strng site amplificatin at 1.5 Hz, as the large acceleratin pulses in bth recrds indicate, while the San Ramn Fire Statin appears t have a brad spectral hle arund 1 Hz. Because the acceleratin radiated by the main shck is strngest at these frequencies, the differences in the wavefrms and in the peak and

7 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1849 a 1/24 K 1 I DVD cm sec... I... l... llrllrllll,+l... VLR z~ =17.6km = 185 sec b 1/27 DVD l L cm sec 2... r... I... I... P... I... sec VLR A=14.5km ~=197 FIO. 6. (a) SH accelergrams f the 24 January main shck recrded at statins DVD and VLR. The scales are adjusted as in Figures 4 and 5. Distance and azimuth frm the epicenter are indicated belw the accelergrams. (b) SH accelergrams written by the 27 January aftershck at the same statins. rms acceleratin measurements are prnunced. The accelergrams recrded at statin A3E are shwn as a cntrl fr the cmparisn f SRE and SRM, althugh A3E has a site resnance at abut 6 Hz. The accelergrams written by the aftershck at SRE and A3E are remarkably similar in shape. Because f the site amplificatin,

8 1850 JOHN BOATWRIGHT AND DAVID M. BOORE 0 W n" ~:: t i < J~ W 0:: 5~ O I I I i

9 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1851 hwever, the accelergrams recrded at SRE are nt used t estimate surce parameters. PEAK GROUND ACCELERATIONS Despite these prnunced site effects, because f the large number f statins which recrded bth f the events, this data set is f cnsiderable use fr studying Statin TABLE 1 PEAK ACCELERATION DATA 24 January Event 27 January Event Structure Peak Grund Type* d~" d (kin) ~ Acceleratin~ (km) ) (g) A3E I ANT BSD CAP CDT CRB DPP DVD ECO GWJ HSU HVR KMC MSJ PHS RCC SJT SRE SRM TIB 1 N Recrd TRY VLR WCS WVC CL 1 Statin Nt Installed FR 1 Statin Nt Installed MT 1 Statin Nt Installed Peak Grund Acceleratin (g) N Recrd N Recrd N Recrd N Recrd * Statin identificatin, crdinates, and structure type (1 = buildings less than r equal t tw stries, 2 = buildings larger than tw stries, 3 = dam te r abutment) frm Switzer et al. (1981). CL, FR, and MT are temprary statins installed after the 24 January event (McJunkin and Ragsdale, 1980). t Clsest hrizntal distance and azimuth frm surce t statin, assuming faults extending frm N, W t N, W and N, W t N, W fr the 24 and 27 January events, respectively. Fault lcatins based n earthquake lcatins by Cckerham et a/. (1980). $ Peak grund acceleratin, largest f the tw hrizntal cmpnents scaled frm riginal recrds by R. L. PrceUa. directivity. In this sectin, the peak grund acceleratins are analyzed t estimate the directivity in each event. The peak mtin (PGA) is defined as the largest acceleratin f the tw hrizntal cmpnents; the values were scaled ff the riginal recrds by R. L. Prcella (Table 1) and have uncertainties f abut _ g. In rder t islate azimuthal variatin f peak mtins fr each event, the effects f

10 1852 JOHN BOATWRIGHT AND DAVID M. BOORE attentin and gemetrical spreading must be remved. We have divided the bserved peak acceleratins by the peak acceleratins predicted frm the attenuatin curves f Jyner and Bre (1981). lg PGA = M - lg r r r = (d ) 1/2, (1) I i ~ i I 1 i i t I I I I I I I I I STRUCTURE <:3 STORIES 1-24 AV/B=O.9/ ~, Q STRUCTURE _> 3 STORIES /., ~ ~ DAM TOE OR hbutment e~ 0 I - _lo i L,, i I L I t 1 L L [ ~ I I, I AZIMUTH L I,, I I i I t I ~ t ~ i I t, _,-27 /0,8~ "0._~ i -I L [ I J l I I I I f ~ ~ t I r I I 0 IO0, AZIMUTH FIG. 8. The lgarithms f the bserved peak acceleratin divided by the predicted peak acceleratin, pltted against azimuth frm surce t receiver. The predicted mtins are estimated frm equatin (1), using mment magnitudes f 5.8 and 5.5 fr the 24 and 27 January events. The curves superimpsed n the data are the theretical predictins determined frm equatin (2) adjusted vertically t the data. where M is mment magnitude f the earthquake and d is the clsest distance t the surface prjectin f the fault rupture in kilmeters. The crrected peak acceleratins are pltted in Figure 8; they shw markedly different variatins with azimuth fr the tw events. The azimuthal variatin fr the peak acceleratins recrded in the basements f large and small structures is apprximately the same, but the tw sets f data differ frm ne anther by a cnstant factr. This is

11 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1853 cnsistent with analyses f the acceleratins frm ther earthquakes, especially the 1971 San Fernand event. The peak acceleratins recrded in the basements f tall buildings are less than thse recrded in small structures: Bre et al. (1980) find the difference t be 0.2 lg units fr the San Fernand data. Cnsidering this effect f structure size, the estimates f the ttal azimuthal variatin f the peak acceleratins yields a factr f 8 fr the main shck (0.9 lg units) and a factr f 5 fr the aftershck (0.7 lg units). A critical cnsideratin in this analysis is the effect f the radiatin patterns. It is nt pssible t discern the SH ndes, predicted at 188 and 278, in the measurements f peak acceleratin shwn in Figure 8. Althugh this difficulty des nt rule ut the pssibility that the radiatin patterns influence the peak acceleratins, the dense sampling f statin azimuths frm 180 t 300 indicates that the radiatin patterns are significantly bscured. The radiatin patterns are assumed t be bscured by tw effects: the scattering f bdy waves at high frequencies and the cmplexity f the dynamic rupture prcess. Nte als that the peak hrizntal S- wave mtin is measured withut regard t whether it is SH r SV mtin. As discussed earlier, it is difficult t estimate directivity fr single events: apparent azlmuthal variatins can be caused by carse sampling f radiatin patterns, by using imprper distance crrectins, r by azimuthally dependent gelgy and structure-size distributins. T eliminate these effects, we have analyzed the rati f peak acceleratins at thse statins which recrded bth events. Because the epicentral distances are similar, this rati is relatively insensitive t the attenuatin relatin used t crrect fr the epicentral distance. The results pltted in Figure 9 shw a marked azimuthal dependence, with a ttal variatin f a factr f 30 (1.5 lg units). Because we are using the rati f the peak acceleratins at these statins, this variatin can nly result frm the directivity in the tw events. If the dminant mtin f the main shck ccurred n the same fault plane as the aftershck, as suggested by the aftershck sequence fllwing the main shck, this rati als eliminates any bias which might result frm the radiatin patterns. It is imprtant t analyze the azimuthal variatins using theretical predictins f directivity. As a simple mdel fr the variatin f the peak grund acceleratin, we cnsider the directivity functin, Au DS(~) = [1 - ~ cs ~]-~ (2) (Madariaga, 1977; Batwright, 1982) where h v is the change f rupture velcity assciated with the radiatin f the acceleratin pulse, fl is the shear wave velcity at the surce, and is the angle between the directin f rupture and the takeff directin f the ray. This equatin is exactly applicable nly fr the peak acceleratins measured frm the acceleratin pulses radiated by the same faulting event. In calculating the theretical curves, the takeff angles f the rays are assumed t be hrizntal; ~ is taken as the angle between the rupture directin, 0r, and the azimuth t the statin, 0- The rupture directins were assumed t be hrizntal and aligned with the sutheastward extensin f the distributin f aftershcks fr the main shck (Or = 143 ) and with the nrthwest (0r = 323 ) fr the aftershck. The theretical curves calculated fr A v = 0.7fi, 0.8fl, and 0.9fi are pltted n Figures 8 and 9. The curves in Figure 8 were adjusted vertically t the small structure data.

12 1854 JOHN BOATWRIGHT AND DAVID M. BOORE Althugh it is nt pssible t chse between the changes f rupture velcity, the agreement f the theretical curves with the main shck data is surprisingly gd. Because f the increased scatter f the data, hwever, the fit f the theretical curves t the aftershck data is less cnvincing. In Figure 9, the theretical curves are cmpared t the ratis f the peak acceleratins: the agreement is remarkable in that n free parameters were varied t btain it. Using nly the small-structure and dam site data, the change f rupture velcity appears t be greater than 0.7ft. The takeff angles f the shear waves recrded at statins 10 t 35 km frm the epicenters range frm 40 t 10 abve hrizntal (R. Cckerham, written cmmunicatin, 1981}. Fr a purely hrizntal rupture, this range f takeff angles weakens the expected directivity by apprximately 10 per cent. The directivity wuld als be weakened if the directins f the ruptures were nt purely hrizntal. These tw cnsideratins indicate that the theretical curves calculated using ~ -- ~ - (~r represent upper bunds fr the expected directivity. Therefre, A v = 0.7fi is a strng lwer bund fr the change f rupture velcity. L ' ' ' ' ' ' ' _,,.. ~ i 0 I AVERAGE AZIMUTH FIG. 9. The lgarithms f the rati f peak acceleratins frm the 24 and 27 January events crrected fr distance with theretical curves determined fr varius changes f rupture velcity superimpsed. In rder t cnsider the variatin f peak acceleratin which may result frm differential site effects r radiatin patterns, we analyze the distributin f the lgarithms f the prducts, at each statin, f the crrected peak acceleratins. Assuming that the directivity in the tw events is exactly reversed in azimuth and equal in amplitude, the standard deviatin f this distributin is equal t twice the variatin which may be attributed t site effects. This prcedure estimates the ttal peak t peak variatin expected frm site effects and radiatin patterns as a factr f 3 (half a lg unit). RMS ACCELERATIONS We have analyzed the rms acceleratins f the SH cmpnents f the shear waves radiated by the main shck and the largest aftershck using a subset f the statins which recrded the tw events. The statins chsen represent as cmplete an azimuthal distributin as pssible; they als include mst f the strng mtin statins within 35 km f the tw events.

13 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1855 While mstly SH mtin, the shear waves are nt perfectly plarized. Because the SH cmpnent is always the largest cmpnent f mtin, hwever, this analysis is cnsistent with the previus analysis f the peak grund acceleratin. The measurements f a,n~ and the duratins used t calculate them are shwn in Table 2. The chice f the interval ver which the square f the acceleratin is averaged was made in a subjective fashin, where the direct S wave is enclsed as clsely as pssible. As discussed by McCann and Bre (1982), this technique, althugh subjective, is ~vell cnditined; varying the duratin has little effect n the rms estimates. T islate the variatin f these measurements with azimuth, we multiply the rms acceleratin by the hypcentral distance, t crrect fr the gemetrical spreading. The resulting estimates are then averaged t btain the mean Ra... and the lgarithm f Rarm~/Rarms is calculated. These lgarithms are pltted in Figure 10, alng with the lgarithms f the bserved peak acceleratins divided by the predicted peak acceleratins fr the same statins. The crrected arm~ values track the crrected peak acceleratin values well: the crrected values are mre than 25 per cent frm each ther at nly fur f the statins. Statin TABLE 2 RMS ACCELERATION DATA 24 January Event 27 January Event R * r arm~ fma* ~" 0 R ~" ar,~ h a (kin) (sec) (cm/sec 2) (Hz) (bars) (km) (sec) (cm/sec 2) (bars) A3E ANT DPP DVD MSJ SRE SRM VLR WCS FR Statin Nt Installed * Hypcentral distance cmputed frm nearest epicentral distances listed in Table I and the hypcentral depths f 10.5 km fr the main shck and 15 km fr the aftershck. The crrelatin is smewhat strnger than the crrelatin presented by Hanks and McGuire {1981) fr the accelergrams written by the 1975 Orville aftershck sequence. If we use this crrelatin t extraplate the variatin f the crrected peak acceleratins t the crrected rms acceleratins, we then estimate a directivity effect f a factr f 10 (peak t peak) fr measurements f rms acceleratin frm a strngly unilateral event. As demnstrated by Batwright (1982), the rms acceleratin may be used t estimate the rms dynamic stress drp averaged ver the rupture area. In general, the relatin between the rms acceleratin and the rms dynamic stress drp depends n the shape f the acceleratin spectrum. Fr acceleratin spectra which are flat between the crner frequency,/c, and a high-frequency limit, fmax, we can use the relatin, -1/2 p ~2 ~fmax 2 R-'~rms A. = 1.13 \ f (3a)

14 1856 JOHN BOATWRIGHT AND DAVID M. BOORE (Batwright, 1982). This spectral mdel is apprpriate fr the acceleratins radiated by the main shck. Fr acceleratin spectra where fmax < 5f0 r which fall ff as ~-1 abve the crner frequency, this relatin may be rewritten as, 2p fi2 ha = - - = Ra... (3b) 3 ~ ~ vhv I I I I I I E I I i I W I [] O O Ib [] _ W -I.J r~ W W w [k (.9...i 1-24 I i i i r I I I I PEAK RMS ACCEL ACCEL. Structure < 3 stries Q 0 Structure _>5 stries [] Dam te r abutment A I i i _,~ ~ ~ - O- -- (/3 ~ ~_ 0 O 0 [] --0 O -- - n-27 -'6-' k k ',i, I.i I L ~, I k -I AZIMUTH Fr. 10. Cmparisn f crrected peak grund acceleratin and rms acceleratin fr the 10 statins at which the rms acceleratin was measured. The rms acceleratins are crrected fr hypcentral distance and divided by the mean, R-a... The slid symbls are the crrected peak acceleratins; the pen symbls are the crrected rms acceleratins. This spectral mdel is apprpriate fr the aftershck. Here p = 2.8 gm/cm 3 and fl km/sec are the density and the shear wave velcity, and we assume v = 0.75fi is the average rupture velcity, &v = 0.85fi is the change f rupture velcity assciated with the radiated acceleratins, and ~ 8 = 0.77 is the average highfrequency radiatin pattern fr SH waves which takeff between 10 and 40 frm the hrizntal frm a strike-slip fault (see Batwright, 1982, Figures 1 and 2), multiplied by tw t accunt fr the free-surface. The effect f the near-surface density, p' = 2.5 gm/cm ~, and shear wave velcity, fl' = 1.5 km/sec, n the acceleratin amplitudes is apprximated by the factr (p'fi'/p fi)1/2 = 0.64 (Aki and

15 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1857 Richards, 1980). Using the estimates f R, a,~, and f,,,~ listed in Table 2, with/0 = 0.7 Hz fr the main shck and 0.9 Hz fr the aftershck, then gives estimates f the rms dynamic stress drp fa ffi 152 +_ 29 bars fr the main shck and 173 _+ 26 bars fr the aftershck. The estimate f the dynamic stress drp f the aftershck can be tested against the initial slpe technique prpsed by Batwright (1980), where = p -. In this equatin, ~ = ~ sin 8, where ~ is the angle between the takeff dh'ecti~n f the ray and the nrmal t the fault surface. The measurements f the initial slpe ~f the velcity wavefrms, -~, are listed in Table 3 alng with the resulting estimates f the dynamic stress drp. The radiatin pattern crrectin, F ~ , is the average f the abslute value f the SH radiatin pattern ver the range f the Statin TABLE 3 DYNAI~[(C STRF_~S DROP DATA R 8 (t) aa (kin) (cn,jsec2) (ba~) A3E ANT DPP MSJ SRE WCS FIR 15.6 (D takeff m~gles, multiplied by tw t crrect fr the free surface. The measurements f the initial slpe were'crrected fr attenuatin by subtracting t*/2 (= T/2Q where T is the travel time and Q = 200) frm the duratin ver which the slpe was estimated. The resulting estimate f the dynamic stress drp is bars, slightly less than the estimate f the rms dynamic stress drp determined frm the rms acceleratins. These estimates f the dynamic,stress drp are smewhat larger than the a~,,, stress drps determined by Hanks'and MeG.trite (1981)fr.seven mderate aftershcks f the 1975 Orville, Califrnia, earthquake, whse a,= stress drps ranged frm 90 t 170 bars. The depths f the aftershcks studied ranged frm 6 t 11 km, and they had predminantly nrmal faulting mechanisms. As the arm~ stress drps returned by Hanksand McGuire's (1981) inversin underestimate the rms dynamic stress drps by abut 20 per cent (Batwright, 1982), the estimates f stress drp frm these tw ~different tectnic envirnments appear t be remarkably, similar. THEORETICAL RELATIONS BETWEEN PEAK AND RMS ACCELERATION The crrelatin between the peak acceleratins and the rms acceleratins shwn in Figure 10 suggests cmparing these measurements t predictins frm theretical

16 1858 JOHN BOATWRIGHT AND DAVID M. BOORE mdels f the acceleratin wavefrms. We will cnsider tw mdels: the deterministic mdel prpsed by Batwright (1982) and the stchastic mdel similar t the mdel f Hanks and McGuire (1981). Frm equatin (6) f Batwright (1982), the peak acceleratin radiated by a subevent f radius r' and dynamic stress drp ha' may be written as, am~. = AV (5) # R ~ where ~ is the width f the acceleratin pulse, ~ar~x is the area under the acceleratin pulse, and # is the rigidity. Dividing the area f the acceleratin pulse by the rms acceleratin and assuming that the dynamic stress drp f the strngest subevent (ha') is greater than r equal t the rms dynamic stress drp f the verall event (ha), equatins (3b) and (5) determine an upper bund fr the radius f the subevent which radiated the peak acceleratin ~amax 2 ha' r' m arms 3 ha V" (6) This relatin has the same dimensins as equatin (25) f McGarr (1.981) which relates the asperity radius t the rati f the peak velcity t the peak acceleratin. Because it uses the area f the peak acceleratin pulse rather than the peak acceleratin, hwever, it is nearly independent f the attenuatin. Nte that the use f equatin (3b) presumes that the acceleratin spectra are peaked and fallff as ~--1 abve the cmer frequency. Figure 11 Shws the measurements made n a number f the accelergrams written by the 27 January aftershck. The measurements are listed in Table 4. The peak acceleratins were calculated frm the rtated SH cmpnents rather than by using the values listed in Table 1. If the rms acceleratins were calculated frm the cmpnents n which the peak acceleratins were riginally measured, the results wuld be identical. The nly anmalus pulse area is the pulse area f the main shck accelergram recrded at SRE. As shwn in Figure 7, this accelergram has an unusually brad pulse which is nt seen n the statins which are near the same azimuth frm the event. If we mit this statin, we btain r'/v ~ 0.56 _ 0.04 sec fr the main shck and r'/v ~= _ 0.04 sec fr the aftershck. The distributin f the ratis f pulse area t arm~ is well cnstrained fr bth events. If the dynamic stress drp f the subevent is apprximately equal t the rms dynamic stress drp f the whle event, equatin (6) can be used t estimate the radius f the subevent. Assuming that the average rupture velcity is 0.75fl where the shear wave velcity at depthis 3.3 km/sec, we btain 1.5 +_ 0.1 km fr the radius f ne aftershck. This estimate f surce size is slightly smaller than the spatial extent f the aftershck cluster which fllwed this event. Fr the mainshck, equatin (6) gives an upper bund f km fr the strngest subevent. We cannt identify a specific sectin f the aftershck distributin which might crrespnd t this subevent. Equatin (6) is a deterministic analg f the stchastic relatin derived by Hanks and McGuire (1981), which relates the peak acceleratin t the rms acceleratin by assuming that the grund acceleratin in a bdy wave arrival is finite duratin and

17 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1859 bandiimited. Their equatin is the first term f the relatin, Y a~a~ -- [2 ln(2n)] 1/2 + [2 ln(2n)] 1/2' (7) ANT WGS A3E Fr. 11. The measureme~ts f pulse width and peak acceleratia fr three f the aeceier,.~g~.ams written by the 24 January aftershck, shw~ reiative t the rma acceiera~ien level fr each pulse. sec Statin TABLE 4 PEAK ACCELERATION PULSE DATA 24 January Event 27 January Event (cm/sec 2) (see) (cm/sec) (cm/sec ~) (sec) (cm/sec) A3E ANT DPP DVD MSJ SRE SRM VLR WCS FR Statin Nt Installed which is itself asympttic t an exact expressin given by Cartw~ight and Lnguet- Higgins (1956). In equatin (7), ~, is Euler's cnstant ( ), and N is the average number f psitive zer crssings in the duratin f strng grund shaking, ~. N may be estimated frm zerth and secnd mments f the energy spectrum (a mre cmplete analysis f the seismlgical applicatins f the results frm randm

18 1860 JOHN BOATWRIGHT AND DAVID M. BOORE vibratin thery is in preparatin). Using, v = 1.,5 and 1.1 sec fr.the main shck and the~ aftershck, respectively, and estimating, N frm a spectral shape which apprximates the data spectra, gi.ves am~x arm, fr the:main shck and a~ = 2.4a~, fr the aftershck. Averaging the ratis f the peak and rms acceleratins measured frm the accelergrams fr each event gives a~, -- (2.5 =t= 0.1)a,,, fr bth events. Cnsidering the strng directivity in these acceleratins, the agreement between the statistical thery and the bservatins is remarkable. PEAK VELOCITY AND RADIATED ENERGY FLUX It is imprtant t cnsider hw the measurements f peak velcity and radiated energy flux vary with azimuths,_ and whether they are as strngly crrelated as the peak and rms acceleratin. The, peak velcities, measured frm the integrals f the SH accelergrams analyzed in tlm last tw sectins, are listed in Table 4. They were crrected fr epicentral distance using the relatin, lg Vmax M - lg r r S r = (d ) 1/2 (8) f Jyner and Bre (1981). In this relatin, S takes a value f 1 fr sil sites and 0 fr rck sites. Statins ANT, MSJ, S1RE, and SRM were assumed t be sil sites; the ther free-field and large structure statins were assumed t be rck sites. The crrected peak velcities are pltted as a functin f azimuth in Figure 12. While nt identical t the variatin f the crrected peak and rms acceleratins shwn in Figure 10, the variatin with azimuth is similar. Excluding the statins SRM and SRE, which are severely cntaminated by their strng site respnses, the ttal variatin is estimated t be a factr f 5 (0.7 lg units) fr bth events. The radiated energy flux in the S waves is calculated by integrating the square f" the grund velcity ver time r frequency, after crrecting fr the attenuatin. This crrectin is determined using Parseval's relatin, e = d2(~)e ' t" d~ = pfli*, (9) ~J where p and fl are the density and the shear wave velcity at the receiver, and I* is the integral f the square f the grund velcity after crrecting, fr attenuatin. The necessary measurements are cmpiled in Table 4. Bth the uncrrected and crrected integrals f the square f the grund velcity (I and I*) are listed t shw hw the attenuatin crrectin affects the estimate f the radiated energy flux. The statin estimates f the ttal radiated, energy are shwn in the last clumn; these estimates are btained frm tb.e relatin, ~,=2~ ~ ~, (10) (RandaU, 1973; Batwright, 1980). Here the radiatin pattern, ~ , is the rms SH radiatin pattern averaged ver the range f.takeff angles and multiplied by tw t accunt fr the free surface. The estimates f the radiated energy, cmpiled in Table 5, vary ver a factr f 30 fr the main shck and 15 fr the aftershck. Because this estimate is determined frm a squared measure f the grund mtin,

19 i ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1861 I '' '1' ' I! ' ' A 0-- OlD - 0 hi I-- (.9 J C:I W ~> rn O CD O _J -I I q, 1-24 J i i I, i, I,, I, I, I ', ~ I, i ~O~T~ ~ ~,u Structure < 5 stries 0 Structure _>3 stries [] Dam te r abutment A 0 O 0 [] -2 2 X _2 >- r wlw Z ~- r 0-- ~- * 0 -I(- i 1-27,,, L I I, I ~ I I L I I AZIMUTH I -2 FIG. 12. Crrected peak velcity and radiated energy estimates pltted against azimuth fr 10 statins. The statins SRE (at 229 and 256 ) and SRM (at 253 and 274 ) have anmalus site respnses. The pen symbls are the crrected radiated energy estimates; the slid symbls are the crrected peak velcities. Nte the strng crrelatin fr the 27 January data. Statin TABLE 5 PEAK VELOCITY AND ENERGY FLUX DATA 24 January Event 27 January Event Vmax I I* E~ Vmax I I* Es (cm/sec) (cm2/sec) (1020 dyne-cm) (cm/sec) (cm2/sec) (102 dyne-cm) A3E ANT DPP DVD MSJ SRE SRM VLR WCS FR Statin Nt Installed

20 1862 JOHN BOATWRIGHT AND DAVID M. BOORE the square rts f these factrs (5.5 and 4) shuld be cmpared t the variatin f the peak velcity. Such a strng variatin can lead t substantial errrs in estimates f the radiated energy fr unilateral ruptures. The average radiated seismic energies are estimated as 2.6 +_ dyne-cm fr the main shck and 1.5 +_ dyne-cm fr the aftershck by calculating the arithmetic average f the statin estimates. As the rati f the mments f the events calculated by Blt et al. (1981) was abut 3, the aftershck appears t have radiated energy mre efficiently than the main shck. This discrepancy may result in part frm the statin distributin; the aftershck ruptured tward the largest cncentratin f statins (t the nrthwest) whereas the main shck ruptured away frm these statins. The crrelatin between the peak velcities and radiated energies shwn in Figure 12 is as strng as the crrelatin between the peak and rms acceleratins, particularly fr the 27 January aftershck recrdings. SRM is the nly anmalus statin: the crrected radiated energy is apprximately 50 per cent less than the crrected peak velcity. The azimuthal variatin f these measurements appears t be abut twthirds the variatin f the peak acceleratins. Cmparisn f Figure 12 t Figure 10 indicates that the velcity measurements d nt crrelate with the acceleratin measurements as well as they crrelate with each ther. The stchastic relatin f the peak velcity t the rms velcity is similar t the relatin between the peak and rms acceleratin. They differ nly in the number f psitive zer crssings. N is smaller fr velcity pulse shapes than fr acceleratin pulse shapes because velcity spectra have a lwer dminant frequency than acceleratin spectra. Fr the velcity spectra frm these tw events, N is small enugh t require using the exact expressin t which equatin (7) is asympttic. This expressin predicts that the rati f peak velcity t the rms velcity is bunded as 2.3 => Vmax/Vrm~ >= 1.7, fr velcity spectra which fallff between c-1 and c-2 abve the cmer frequency. Using the values f Vmax and I listed in Table 5 and the relatin, t determine averages f the ratis f the peak velcity t the rms velcity, gives Vmax ---- (1.9 +_ 0.1)V~8 fr the main shck and Vmax ( )Vrms fr the aftershek. As nted previusly, the velcity spectra frm the main shck rllff as ~0-1 ver a narrw frequency band while the spectra frm the aftershck rllff as ~0-2 abve the crner frequency. The bservatins are in excellent agreement with the theretical predictins. DISCUSSION Direetivity in these earthquakes was als bserved in lnger perid mtins, including Wd-Andersn seismgraphs and bradband recrders (Blt et al., 1981; Schechter, 1981). Figure 13, adapted frm Schechter (1981), cmpares the Wd- Andersn seismgrams recrded at Arcata and Santa Barbara. The azimuth frm the main shck t Santa Barbara is within 10 f the assumed directin f rupture; the difference in amplitude between SBC and ARC is abut a factr f 10. Fr the aftershck, hwever, the difference is less prnunced. The lw-magnificatin, Wd-Andersn equivalent at Berkeley gave magnitudes f 5.5 and 5.9 fr the first and secnd events, respectively. If the magnitudes had been based n peak acceleratins, the theretical curve in Figure 9 predicts that the difference between first and secnd events wuld have been 0.5 t 0.7 units, depending n the change f

21 ANALYSIS OF GROUND ACCELERATIONS FOR DIRECTIVITY 1863 rupture velcity. The bserved difference f 0.4 crrespnds t the expected difference in peak velcity rather than peak acceleratin. The relative amplitudes f the bradband recrdings at Berkeley are reversed with respect t the Wd-Andersn amplitudes, with the first event having much mre lng-perid mtin (B. Schechter, written cmmunicatin, 1981). This behavir reflects the relative size f the mments f the tw events. The strng directivity in the peak acceleratins radiated by these earthquakes implies that bth events had unilateral ruptures. In equatin (5), the peak acceleratin depends linearly n the asperity radius and the dynamic stress drp. Fr an incherent rupture in which the dynamic stress drp is apprximately cnstant ver the cmpnent subevents, the directivity in the radiated peak acceleratins must be similarly reflected in the relative asymmetry f the rupture prcess. Either the peak acceleratins were radiated by an extremely large and unilateral subevent, r all the larger subevents ruptured in the same directin. A similar argument can be made 1/ N-S ARC E-W - ~ - ~, ~ ~ =330,~=392km 11 E-W SBC N-S ~<~4~\ =153 A=419km m [A_IJIIilIIIII]LLJEIIIIJlI!Iilll [ LllllIJIIL[l!iJrJJJ!!~lJJ!lJ~LIjlL i 2 1 k TIME (secnds) FIG. 13. Wd-Andersn seismgrams f the 24 January main shck and the 27 January aftershck recrded at Arcata and Santa Barbara (reprinted frm Schechter, 1981). cncerning the directivity in the rms acceleratin measurements. Because the rms acceleratin is the mean f the square f the acceleratin, it is mst sensitive t the strngest subevent. Fr cmplex events, then, bth the peak and rms acceleratin shuld be generally insensitive t directivity, except in the case f a simple event r a purely unilateral rupture. The fact that the peak and rms acceleratins shw such strng directivity implies that the rupture prcesses f these earthquakes were unilateral. In the analysis f these acceleratins, we have emphasized the difference between the average rupture velcity, v, and the change in rupture velcity, Av, which cntrls the high-frequency radiatin. The average rupture velcity is strngly cnditined by the cmplexity and gemetry f the rupture grwth. Fr simple ruptures, it may apprach the terminal rupture velcities fr antiplane and inplane cracks, i.e., the shear and Rayleigh wave speeds. Fr cmplex ruptures, hwever, it will be substantially slwer. In cntrast, the radiative change in rupture velcity is a prperty f the crack tip, r the set f crack tips making up a cmplex rupture

22 1864 JOHN BOATWRIGHT AND DAVID M. BOORE prcess. It shuld be clse t the terminal rupture velcity, regardless f the cmplexity f the rupture prcess. The lwer bund f Av = 0.7fi determined frm the directivity in,the radiated peak acceleratins is substantially slwer than these terminal velcities. The range f takeff angles and the pssibility f a subhrizntal rupture directin may have weakened the bserved directivity, hwever. This lwer bund is cmmensurate with the estimate f Av = 0.80 _ 0.07fi, btained by Batwright (1982) frm a cmparisn f the high-frequency cntent f the P and S waves radiated by 10 small earthquakes in Mnticell, Suth Carlina. CONCLUSIONS The peak hrizntal acceleratins, the rms acceleratins, the peak velcities, and the energy flux radiated by the main shck and the largest aftershck f the 1980 Livermre Valley earthquake sequence shw clear evidence f directivity, bth fr each event individually and in the rati f the mtins at cmmn recrding sites. The azimuthal Variatin f the acceleratins, crrected fr the gemetrical spreading and attenuatin, was as much as a factr f 10. A theretical mdel fr the directivity in the acceleratins fits the data well. While the radiative change f rupture velcity cannt be determined uniquely frm the data, it appears t supprt the lwer bund f hv = 0.7ft. Althugh we have established the significant impact that directivity had n the S waves in accelergrams frm the 1980 Livermre Valley earthquakes, we make n claim fr being the first t establish cnclusively directivity at high frequencies. Fr example, Bakun et al. (1978) shwed clear directivity effects in a study f P waves frm tw small magnitude earthquakes (M = 3.0 and 2.0) in central Califrnia. Hwever, this analysis is the first t clearly demnstrate directivity in the highfrequency acceleratins which are f interest t seismic engineers. Mrever, the measurements analyzed in this study, i.e., peak acceleratin, rms acceleratin, peak velcity, and radiated energy flux, are f mre imprtance t seismic engineering than the duratin and frequency f zer crssing measurements analyzed by Bakun et al. (1978). The crrelatins between the peak and rms acceleratins, and the peak velcities and the radiated energy flux imply that the measurements f peak acceleratin and peak velcity made n a large number f accelergrams and seismgrams can be used t determine surce parameters such as the dynamic stress drp and the radiated seismic energy if the signal duratins are als measured. In particular, the relatins between peak velcity, rms velcity, signal duratin, and radiated energy flux might be used t supplant the Gutenberg-Richter magnitude-energy relatinship (Richter, 1958), if bradband instruments are used. While it is pssible that a future earthquake culd exhibit a mre extreme directivity in its radiated wave field, it is suggested that this data set be used as an upper bund fr the expected effect f directivity. Fr peak and rms acceleratin, the maximum ttal variatin is a factr f 10, r a factr f 3 amplificatin f the mean. Fr peak velcity, the ttal variatin is a factr f 5, while the energy flux can vary by a factr f 30. Finally, the ttal variatin expected frm differential site effects, including sil respnse and structural resnances, appears t be a factr f 3. ACKNOWLEDGMENTS We thank Rb Cckerham fr the aftershck lcatins which are shwn in Figures 2 and 3, Rn Prcella fr the peak acceleratin measurements, and Bruce Schechter fr use f the Wd-Andersn recrdings shwn in Figure 13. This paper has been critically reviewed by Art McGarr, Jim Scheimer, and Paul Spudich, whse cmments have helped us t imprve it significantly.

23 ANALYSIS OF GROUND ACCELERATIONS FOR DiI~.ECTIVITY 1865 ~t~eferences Aki~ K and P. G. Richards (1980). Quanlitative Seismlgy: Thery and Methds, W. H. Freeman, San FrancL~c. Archuleta, R. J. (1979). Rupture prpagatin effects in the Cyte Lake earthquake (ab.~;tract), EOS, Trans. Am. Gephys. Unin 60, 8.0. Bakun, W. H., R. M. Stewart, and C. G. Bufe (1978). Direetivity in the high-frequency radiatin f small earthquakes, Bull. Sei.sm. Sc. Am. 68, Beniff, H. (1955). Mechanism and strain characteristics f the White Wlf fault as indicated by the aftershek sequence, Earthquakes in Kern Cunty, Califrnia during 1955 (G. B. Oakeshtt. Editr), Calif. Div. Mines Bull. 171, Batwright, J. (1980). A spectral thery fr circular seismic.surces: simple estimates f surce dimensin, dynamic stress drp and radiated seismic energy, Bull. Seism. Sc. Am. 70, Batwright, J. (1982). A dynamic mdel fr far-field acceleratin, Bull. Seism. Sc. Am. 72, Blt, B. A., T. V. McEvilly, and R. A. Uhrhammer (1981). The Livermre Valley, Califrnia, sequence f January 1980, Bull. Seism. Sc. Am. 71, Bre, D. M. and R. L. Prcella (1980). Peak acceleratin frm strng-mtin recrds: a pstscript, Bull. Seism. Sc. Am. 70, Bre, D. M., W. B. Jyner, A. A. Oliver, I[I, and R. A. Page (1980). Peak acceleratin, velcity, and displacement frm strng-mtin recrds, Bull. Seism. Sc. Am. 70, Cartwright, D. E. and M. S. Lnguet-Higgins (1956). The statistical distributin f the maxima f a randm functin, Pre. Ry. Sc. Lnd., Ser. A 237, Cckerham, R. S., F. W. Lester, and W. L. Ellswrth (19~). A preliminary reprt n the Livermre Valley earthquake sequence January 24-February 26, 1980, U.S. Gel. Surv., Open-File Rept Hanks, T. C. and R. K. McGuire (1981). The character f high-frequency strng grund mtin, Bull. Seism. Sc. Am. 71, Jyner, W. B. and D. M. Bre (1981). Peak hrizntal acceleratin and velcity frm strng-mtin recrds including recrds frm the 1979 Imperial Valley, Califrnia, earthquake, Bull. Sei.sm. Sc. Am. 71, Madariaga, R. (1977). High-frequency radiatin frm crack (stress-drp) mdels f faulting, Gephs. J. 51, McCann, M. and D. M. Bre (1982). Variability in grund mtins: rt mean square acceleratins and peak acceleratin fr the 1971 San Fernand, Califrnia, earthquake, Bull. Seism. Sc. Am. (in press). McGarr, A. (1981)., Analysis f peak grund mtin in terms f a mdel f inhmgeneus faulting, J. Gephys. Res. 86, McGuire, R. K. and T. C. Hanks (1980). rms acceleratins and spectral amplitudes f strng grund mtin during the San Fernand, Califrnia, earthquake, Bull. Seism. Sc. Am. 70, McJunkin, R. D. and J. T. Ragsdale (1980). Strng-mtin recrds frm the Livermre earthquake f 24 and 26 January 1980, Calif. Div. Mines and Gelgy Preliminary Reprt 28. Randall, M. J. (1973). The spectral thery f seismic surces, Bull. Seism. Sc. Am. 63, Richter, C. A. (1958). Elementary Seismlgy, W. H. Freeman, San Francisc, 768 pp. Savage, J. C. (1965). The stpping phase n seismgrams, Bull. Sei~sm. Sc. Am. 58, Schechter, B. (1981). Surce parameters and directivity f the Livermre earthquakes f 1980 (abstract), Earthquake Ntes 52, 83. Swanger, H. J., S. M. Day, J. R. Murphy, and R. Guzman (1981). State-f-the-Art study cncerning nearfield earthquake grund mtin, U.S. Nuclear Reg. Cmm. NUREG/CR-1978.,~;witzer, J., D. Jhnsn, R. Maley, and R. Matthiesen (1981). Western hemisphere strng-mtin accelergraph statin list-1980, U.S. Gel. Bury., Open-File Rept [J.S. GEOLOGICAL SURVEY OFFICE OF EARTHQUAKE STUDIES 345 MIDDLEFIELD ROAD MENLO PARK, CALIFORNIA Manuscript received 19 March 1982