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Bulletin of the Seismological Society of America, Vol. 80, No. 5, pp. 1205-1231, October 1990

EFFECTS OF CENTROID LOCATION ON DETERMINATION OF EARTHQUAKE MECHANISMS USING LONG-PERIOD

SURFACE WAVES

BY JIAJUN ZHANG AND THORNE LAY

ABSTRACT

Determination of shallow earthquake source mechanisms by inversion of long- period (150 to 300 sec) Rayleigh waves requires epicentral locations with greater accuracy than that provided by routine source locations of the National Earth- quake Information Center (NEIC) and International Seismological Centre (ISC). The effects of epicentral mislocation on such inversions are examined using synthetic calculations as well as actual data for three large Mexican earthquakes. For Rayleigh waves of 150-sec period, an epicentral mislocation of 30 km introduces observed source spectra phase errors of 0.6 radian for stations at opposing azimuths along the source mislocation vector. This is larger than the 0.5-radian azimuthal variation of the phase spectra at the same period for a thrust fault with 15 ° dip and 24-km depth. The typical landward mislocation of routinely determined epicenters of shallow subduction zone earthquakes causes source moment tensor inversions of long-period Rayleigh waves to predict larger fault dip than indicated by teleseismic P-wave first-motion data. For dip-slip earthquakes, inversions of long-period Rayleigh waves that use an erroneous source location in the down-dip or along-strike directions of a nodal plane, overestimate the strike, dip, and slip of that nodal plane. Inversions of strike-slip earthquakes that utilize an erroneous location along the strike of a nodal plane overestimate the slip of that nodal plane, causing the second nodal plane to dip incorrectly in the direction opposite to the mislocation vector. The effects of epicentral mislocation for earthquakes with 45 ° dip-slip fault mechanisms are more severe than for events with other fault mechanisms. Existing earth model propagation corrections do not appear to be sufficiently accurate to routinely determine the optimal surface-wave source location without constraints from body-wave information, unless extensive direct path (R1) data are available or empirical path calibrations are performed. However, independent surface-wave and body-wave solutions can be remarkably consistent when the effects of epicentral mislocation are accounted for. This will allow simultaneous unconstrained body-wave and surface-wave inversions to be performed despite the well known difficulties of extracting the complete moment tensor of shallow sources from fundamental modes.

INTRODUCTION

Accurate determination of earthquake source mechanisms is a fundamental, long- standing task in seismology. The seismic data routinely used to determine source mechanisms of earthquake include P-wave first motions, body waves, surface waves, and free oscillations. Since P-wave first-motion data are usually incomplete, and body waves for large earthquakes are frequently off-scale or of too limited bandwidth to determine static parameters, it is often necessary to use surface waves to determine the source mechanisms. In analyzing long-period surface-wave or free oscillation data, the epicentral parameters, which include the geographical coordinates of the source and the origin time, must be known or determined in the analysis.

1205

1206 J. ZHANG AND T. LAY

Very-long-period surface waves have limited resolution of source location due to strong trade-offs with the source radiation. Thus, body-wave determinations of source location are often assumed, even if it is known that the body-wave locations have significant uncertainty. The combined effects of errors in the standard hypocentral parameters, spatial-temporal finiteness of the sources, and errors in the reference earth model often cause the epicentral location and origin time estimated from first arrival times of P waves to deviate by several tens of kilometers from the point source parameters determined by radiation of long-period seismic waves (Dziewonski et al., 1981).

The purpose of this study is to investigate the effects of epicentral mislocation on determination of earthquake mechanisms using long-period surface waves. This is a critical step in working toward a capability of simultaneously analyzing all seismic wave information to extract a unified model of each seismic source. While some efforts toward this end have been made (e.g., EkstrSm, 1989; Honda and Seno, 1989), many existing procedures are not entirely self-consistent in their treatment of different wave types. The effects of epicentral mislocation on source moment tensor inversions can be analyzed using synthetic seismograms as well as by processing actual data.

In this paper, we first compute the effect on long-period surface-wave inversions of mislocations in along-strike or down-dip directions of a nodal plane for dip-slip andstrike-slip earthquakes using synthetic seismograms. Then we analyze data for three shallow subduction events in central Mexico, exploring the effect of various published locations for these events. The events are the 14 March 1979 Petatlan (Me = 7.6), 25 October 1981 Playa Azul (Ms = 7.3), and 21 September 1985 Michoac~n (M~ = 7.6) earthquakes. These earthquakes occurred in the northwest portion of the Middle America trench (Fig. 1) and ruptured the interface between the North American plate and the subducting Cocos plate near the Orozco fracture zone. For these earthquakes, the sources can be approximated by point sources for the wave periods utilized, allowing the effects of epicentral location on the surface- wave inversions to be isolated. The availability of high-quality digital data and extensive previous work on the body wave locations for these events make it possible to analyze in detail the effects of the epicentral location on determination of earthquake mechanisms for thrust events using the long-period surface waves. Accurate determination of source parameters for shallow thrust events is particularly important for constraining plate convergence rate, direction and dip. In the following, we briefly describe the method, and then proceed to the specific event applications.

INVERSION METHOD

For our surface-wave inversions, we use the method of Zhang and Kanamori (1988b), which is based on the moment tensor inversion approach of Kanamori and Given (1981) as modified to a two-step inversion algorithm following Romanowicz and Guillemant (1984). A point source with step time function can be described by a stress glut rate distribution, F, localized at rc and to:

~ ( ~ , t ) = __M~(~ - ~ ) ~ ( t - t~), (1)

where M is the moment tensor (Backus and Mulcahy, 1976). The moment tensor may be defined with respect to a Cartesian coordinate system (x, y, z) with the

CENTROID LOCATION AND DETERMINATION OF MECHANISMS 1207

Mex ican Ear thquakes

18.5

" . O/ TM t ~0 / - . ~ , , ~ , , ~

18.o ,~ - . , : ~, ,, _$ (I) / ,~

^ , l ' - . , ,~ ,-JFz~ LPR x

"~ 1~ " " ~ ~,~"< "~ JHD + "J 17.5 + ~ " " " ~'~ EMS •

" / F " - , . CMT o ~ - NEIC z~

' 1 / Isc • /

IT % o05 Long i tude (deg)

F](3. I. Various epicenter locations of ( I ) 14 March 1979 Petat lan, (2) 25 October 1981 Playa Azu], and (3) 21 September 1985 Michoac~n earthquakes, from (a) seismic recordings in Mexico (EMS: solid circles), (b) teleseismic P-wave relocations using the JHD method (plusses) (Karen C. McNally, personal comm.), (c) the Harvard CMT method (open circles), (d) routine NEIC preliminary determination of epicenters (NEIC: open triangles), and (e) routine ISC (solid triangles) procedures. The epicenter locations preferred for radiation of long-period Rayleigh waves for the three events analyzed in this study are also shown (LPR: crosses). The dashed and dotted lines, which are orthogonal for each event, depict sequences of epicenter locations used in the inversions of long-period Rayleigh waves. Map of central Mexico (inserted to the upper right corner) showing the epicenters locations of these earthquakes determined by NEIC and local seismic recordings, which occurred in the Middle America trench near the Orozco Fracture Zone (OFZ).

origin at ~ on the polar axis of the sys tem (r, 0, ~). The spec t rum of a se ismogram for angular f requency w for a point source with step t ime function can be wri t ten a s

6

O(z, ~)= Z %A, (2) k ~ l

where ft = Mzz, f2 = M ~ , [3 = M ~ , [4 = Mxz, f5 = Myz, [~ = Mxy represent the six independent componen t s of the m o m e n t tensor. The coefficients, ~k, are Sk(r, #c, to, w)/5(~, #c, w)]~(#c, w), where Sk, /5, and I are the effects of the source exci tat ion corresponding to fk, propagat ion, and ins t rmnen t respectively. The l inear funct ion of the m o m e n t tensor: S = F~=~ ~kfk represents the source spectrum.

The propaga t ion kernel t5 is given by

1 . i - i ~ a O ' ] I1 .] /5(~, ~ , w ) - (sin 0) '/2 e x p ( , a ) e x p t - - - ~ ) e x p t ~ m ~ , ] e x p ~ - - ~ ] , (3)

where a is the initial phase, a the radius of the Ear th , ~ ' the propaga t ion distance, c the phase velocity, U the group velocity, Q the quali ty factor, and m the number of the polar or ant ipodal passages. For large ear thquakes we write the spectra

~(r~, 4, tc, z) = ~(r~, 4, t~, ~)#(¢, e)/~(~), (4)


where V is the source spectrum of a point source with step time function, F the finiteness effect caused by rupture propagation, 0 the azimuth of the station measured from the direction of rupture, and /~ represents the dislocation time function of the point source. We write the source finiteness effect as

P(z, o)~(~) = ,(~, ¢) + i~(z, ¢). (5)

The observed source spectra, S = v + i8, are calculated from the spectra of the surface-wave seismogram U by correcting for instrument response and propagation for a given source location and origin time. We invert the observed source spectra for the source finiteness and moment tensor using a two-step procedure similar to Zhang and Kanamori (1988b), with the exception that source finiteness factors are included in the ith-station block of the kernel in the first-step inversion:

B i

1 1 ] -vRsin 2@, ~ vRcos 2~i, - ~ ~R, -~'Rsin ¢i, -~Rcos ~

1 1 -~'Rsin 2@, ~ ~'Rcos 2@, - ~ ~R, ~Rsin @, ~RCOS @

and the data vector is replaced by

S = [VR(~, h, ~bl), 8R(~, h, ~1), " " , VR(W, h, ~@), ~R(~, h, ~N)] T.

Introduction of source finiteness factors in the kernel allow us to weight the error of the first-step inversion as discussed by Zhang and Lay (1989). In general, the source finiteness effects are not known independently, so we vary the source parameters (in this case the duration of a boxcar source time function) to find a best model. This involves inverting the source spectra S to determine the source finiteness model which minimizes the error, ¢, the normalized error of the first-step inversion.

The second step is to search for the best depth and corresponding moment tensor for given excitation functions and the preferred source duration using the solution vectors obtained in the first step. Since the excitation function depends on the depth, the error of the second-step inversion is a function of the assumed depth. For each trial source depth, we calculate the error, p, the normalized error of the second-step inversion. We find the depth that minimizes the error p and determine the moment tensor.

DATA

Table 1 lists the epicentral parameters of the three large Mexican earthquakes studied here, as reported by various investigators and seismological agencies. The parameters determined from local seismic recordings are taken from Gettrust et al. (1981) for the Petatlan earthquake, Havskov et al. (1983) for the Playa Azul earthquake, UNAM Seismology Group (1986) (location), and the monthly listings of the National Earthquake Information Center (NEIC) (origin time) for the Michoac~n earthquake, respectively. Figure 1 shows the various epicentral locations of the three events from (a) seismic recordings in Mexico (EMS), (b) teleseismic P-wave relocations using the joint hypocenter determination method (JHD) (Karen

CENTROID LOCATION AND DETERMINATION OF MECHANISMS

TABLE 1 EPICENTRAL DATA OF THREE LARGE MEXICAN EARTHQUAKES

1209

Event Date UTC Latitude Longitude M~ Reference

Petatlan 14 March 1979 1107:11 17.457°N 101.462°W 7.6 la 1107:16.3 17.813°N 101.276°W lb 1107:11.2 17.76°N 101.30°W lc 1107:12.9 17.562°N 101.459°W ld

Playa Azul 25 October 1981 0322:13 17.750°N 102.250°W 7.3 2a 0322:15.5 18.048°N 102.084°W 2b 0322:16 18.18°N 102.01°W 2c 0322:14.0 18.080°N 102.159°W 2d

Michoac~n 21 September 1985 0137:13 17.618°N 101.815°W 7.6 3a 0137:13.5 17.802°N 101.647°W 3b 0137:15.1 17.81°N 101.69°W 3c

17.585°N 101.972°W 3d

(la) Gettrust e t al. (1981); (2a) Havskov e t al. (1983); (3a) NEIC (origin time) and UNAM Seismology Group (1986) (location); (lb)-(3b) NEIC; (lc)-(3c) ISC; (ld)-(3d) Joint Hypocenter Determination (Karen C. McNally, personal comm.).

C. McNally, personal comm.), (c) long-period body-wave and mantle-wave inversions using the Harvard centroid moment tensor method (CMT) listed in the Monthly Listings of NEIC, (d) routine NEIC locations, and (e) routine International Seismological Centre (ISC) locations. Note the 50-km variation between epicenters for a given event and their tendency to spread along the down-dip direction.

Long-period Rayleigh wave spectra from the Mexican earthquakes were obtained from stations of the International Deployment of Accelerometers (IDA) (Agnew et al., 1976), Global Digital Seismograph Network (GDSN) (Engdahl et al., 1982b), and GEOSCOPE (Romanowicz et al., 1984) networks. The periods used are 150, 175, 200, 225, 256, 275, and 300 sec. The spectra are corrected for propagation phase delay using the laterally heterogeneous earth model M84C obtained by Woodhouse and Dziewonski (1984), and for attenuation using the Q model of Dziewonski and Steim (1982).

EARTH MODEL

Before considering source mislocation effects on surface-wave inversions, it is important to assess the effects of uncertainities in the source velocity structure. As seen from (2), the excitation terms, which depend on source velocity and density structure, are required for determination of source parameters. For the earthquakes studied here, we have ascertained that moment tensor inversions using excitation terms computed for a range of earth models constructed from information on local structure in the source region are stable. As is true for many subduction zone events (Zhang and Kanamori, 1988b) we find the average ocean model of Regan and Anderson (1984) (hereinafter referred to as R-A) is a reasonable choice for compu- tation of Rayleigh wave excitation for the purpose of depth determination.

P-wave velocity models of the crust beneath the central coast of Mexico have been previously obtained from recordings of local seismic networks (Reyes et al., 1979; Vald~s, et al., 1982; Stolte et al., 1986). The model of Stolte et al. (1986) has a 28-km-thick crust with a constant gradient P-wave velocity overlying a half-space with constant velocity; the P-wave velocity at the surface is the same as that at the top of the crust of the PREM earth model (Dziewonski and Anderson, 1981). Detailed models of the shear velocity, density, and attenuation of the crust and


upper mantle in this region are not available. We constructed an ad hoc model, which consists of a crustal layer with the same P-wave velocity as the model of Stolte et al. (1986) and a mantle like that of the 0 to 20 Ma ocean model of Regan and Anderson (1984). The values of S-wave velocity and density at the surface of the crust of this model are taken to be the same as the PREM model, and the values at bottom of the crust are calculated from the values at the bottom of the crust of the PREM model in proportion to the local P-wave structure.

The depth estimate of the earthquakes studied here from moment tensor inversions using this ad hoc model proved to be essentially the same as those obtained using the average ocean model of Regan and Anderson (1984). Because the temper- ature and composition in the upper mantel beneath the coast of Mexico may differ significantly from the average values of the earth's upper mantle, the S-wave velocities in this region may perhaps deviate significantly from those assumed in the ad hoc model. In the following, we only describe results using the excitation functions computed for the average ocean model of Regan and Anderson (1984). We recognize that this results in an unknown bias of all of our depth estimates, probably on the order of _+5 km.

THE EPICENTRAL LOCATION PROBLEM

Long-period surface-wave or free oscillation studies often assume the standard bulletin source location and origin time determined from the arrival times of P waves at teleseismic distances by either the NEIC or ISC (e.g., Silver and Jordan, 1983; Romanowicz and Monfret, 1986; Zhang and Kanamori, 1988]o). The potential problem with this is shown in Figure 1, where the epicenters reported by NEIC and ISC for the Mexican events are seen to be about 25 to 55 km inland from the epicenters determined by local seismic recordings. Epicentral parameters given by ISC are about the same as those from NEIC for the Petatlan and Michoac~n earthquakes, but place the Playa Azul event even further inland than the NEIC epicenter. The epicentral locations obtained by the JHD method are consistent with locations from Mexico recordings (EMS) with the exception of the Playa Azul earthquake. We expect the latter two sets of epicentral locations to be less biased than the ISC and NEIC determinations. Local recordings for the Playa Azul event are sparse, thus in that case the EMS location may be less reliable (Karen C. McNally, personal comm.). Clearly, there is substantial uncertainty with respect to the best a priori source locations to use. We will show that this is of first order importance for the surface-wave inversion procedure.

The uncertainty in location of the Mexican events is probably similar to that for other subduction zone events. Few travel-time observations from the interior of the Pacific basin are available for routine teleseismic determinations of epicenters, and as a result circum-Pacific source locations tend to be least well-constrained in directions perpendicular to the margin. For Mexico, the location instability perpendicular to the margin is accentuated by systematic geographic variations of station delays. For subduction zones with steeply dipping high-velocity slabs such as Japan and the Aleutians, teleseismic locations of shallow epicenters also tend to be biased arcward relative to locally determined locations (Engdahl et al., 1982a).

In their development of simultaneous inversion of long-period body-wave and mantle-wave data for hypocentral parameters and moment tensor, Dziewonski and Woodhouse (1983) pointed out that introduction of a finite dip-slip fault motion may mimic a phase shift produced by a change in the epicentral position of the


point source, or by an equivalent velocity anomaly. In an inversion in which the epicentral coordinates are fixed, uncorrected effects of lateral heterogeneity or mislocation of the source tend to more severely affect the components M= and Mzy. These terms of the moment tensor control dip and are usually the poorest resolved terms for shallow sources, particularly for inversions using fundamental modes (Kanamori and Given, 1981). The CMT procedure, in which the epicentral coordinates are redetermined, is stabilized by the inclusion of long-period body waves, for which the trade-off between the location and mechanism is much less severe than for surface waves. However, the effects of errors in hypocentral parameters and in the reference earth model on the results of inversions of long-period surface waves alone are not thoroughly discussed in the literature, so we will explore this explicitly.

Displacement of an epicenter introduces variations of the propagation phase delay and attenuation, which are determined by the propagation kernel in (4). This is manifested as variations of the amplitude and phase of the observed (propagation corrected) source spectra. For a displacement of the epicentral location, 51, the variation of phase corrections of the observed spectra is - ~ cos(¢ - ¢~)5//c, where ¢ and ¢~ are the angles measured counter clockwise from a reference direction to the station and to 61, respectively. For convenience, we chose the reference direction to parallel the down-dip direction. The relative variation of the amplitude of the observed source spectra is very small, on the order of 61 cos(¢ - ¢l)/a.

The systematic, azimuthally varying errors in the phase spectra caused by mislocation of the source can be much greater than the azimuthal variation of the phase spectra expected for a shallow, dip-slip fault. Figure 2 shows, for Rayleigh waves of periods from 40 to 400 sec, the relative phase change for stations on opposite sides of the epicenter due to a shift of the source location by 30 km along with the azimuthal variation of the phase spectra of dip-slip earthquakes at 24-km depth with dip angles of 15 ° and 30 ° computed for the PREM and R-A models. For

Rayleigh Wave Phase Variation (Dip-Slip Earthquake) 3.0 ~ , j

"', I \ ~ Phase Variation (30 km Epicenter Mislocation,PREM)

2.5 ' ,/ \ Dip Depth=24 km \.. X (deg)

r- ",t ' ~ \ ' \ . . . . 15 PREM D~

• .~\'~ \ " . X - - - 30 PREM 2.c

.o ~ "%. "" \ - - - 30 R-A 1 . . = "~. \~'~.. " ' . \

" t -

> q)

~ 1 . 0 , \ \ ~ "~ t -

13_ ~ . ~ .

o go ~o ~o ~o~o1'oo ~oo aoo 40o Period (s)

FIG. 2. Relative phase change of Rayleigh waves with periods from 40 to 400 sec for stations at opposite azimuths from the source due to a shift of the source location by 30 km (solid line) and azimuthal variations of the phase spectra of dip-slip earthquakes at 24-km depth with dip angles of 15 ° and 30 ° computed for the PREM and R-A models (dashed lines).


150-sec period, the shift of the source location introduces a relative phase change of as much as 2~ 51/c ~- 0.6 radian. For dip-slip earthquakes, the phase spectrum varies as tan-l[2 cot 25 QR (1) sin ( ~ ] / ( S R (1) - - PR (1) cos 2~b)] (Kanamori and Stewart, 1976). Since QR (~) is less than SR (~) and PR (~) at shallow depths, the azimuthal variation of the phase spectrum is small. For example, for a period of 150 sec Q(1), Sn (1), and PR (1) at 24-km depth are about -0.4, 3.9, and 2.2, respectively, for the PREM earth model (Dziewonski and Anderson, 1981); the azimuthal variation of the phase spectrum of 150-sec period for a thrust fault with 15°-dip at 24-km depth is less than 0.46 radian. For a fault with a dip close to 45 °, the imaginary parts of the source spectra become very small, and the errors in the phase spectra caused by the mislocation dominate the observed radiation pattern of the source spectra. Figure 2 shows that the azimuthal variation of the phase spectrum for the PREM model is much smaller than for the R-A model, which suggests that results of moment tensor inversions using excitations computed for the former model are more sensitive to epicentral mislocations.

The attenuation corrected amplitude spectra are less sensitive to the shift of the epicenter than the phase spectra. For shallow thrust earthquakes, the amplitude spectrum of Rayleigh waves varies approximately as ~ M0sin PR(~)COS 2~b). Therefore, the seismic moment and dip of the fault plane trade-off directly in the amplitude spectrum, so resolution of the mechanism is primarily dependent on the phase, but as shown in Figure 2 mislocation effects can closely mimic source mechanism effects over a broad frequency range.

The effects of uncertainty in epicentral location on source moment tensor inversions can be explored using synthetic seismograms. We computed synthetic Rayleigh wave seismograms for the PREM and R-A models for stations with various azimuths at an arc distance of 90 ° from the epicenter for dip-slip and strike-slip point sources with step time-functions. We then inverted the spectra of the synthetic seismograms for the source moment tensor assuming an epicenter mislocated at an arc distance of 0.05 ° from the true epicenter. The source moment tensors obtained from inversions are essentially pure double couples.

Figure 3 to 6 show the effects of epicentral mislocation for dip-slip earthquakes. Inversions of long-period Rayleigh waves that utilize an erroneous location in the down-dip direction of a nodal plane overestimate the dip of that nodal plane (Fig. 3); effects on the strike and slip of this nodal plane are much smaller. Inversions that utilize an erroneous location in the along-strike direction of a nodal plane overestimate the dip (Fig. 4), strike (Fig. 5), and slip (Fig. 6) of that nodal plane. Errors of the derived dip, strike, and slip of the nodal plane are larger for earthquakes with a fault dip of 45 ° than for those with other dip angles, for shallow earthquakes than for deep earthquakes, and for the PREM model than for the R-A model.

Figure 7 shows the effects of epicentral mislocation for strike-slip earthquakes. Inversions that utilize an erroneous location in the along-strike direction of the first nodal plane overestimate the slip of this nodal plane, causing the second nodal plane to dip incorrectly in the direction opposite to the mislocation vector. Relative to the slip of the first nodal plane and dip of the second nodal plane, variations of the dip and strike of the first nodal plane and strike and slip of the second nodal plane are very small.

Although quantitative estimates of the errors in moment tensor inversions can be computed using such numerical procedures, it is useful to consider the theoretical basis for the effects of mislocations on source moment tensor inversions. Because of the importance of accuracy of large subduction zone earthquake mechanisms, we


Effect of Mis loca t ion (0.05°down-dip) 80 . . . . n i , ,

True Dip PREMDepth (kin)

(deg) 24 40 24 R-A 4O Dip-Sl ip Ear thquake 70 45 • - . e e - . o & . - . ~ & . - . ~ . . . e

35 e- - e e - - o ~ - ,& ~ - .A . , - " " : " 25 =--..--o e . - - . -o ,=-----,L & . - - - a . . o ' ' " " " - " "0

60 15 = = o ..o A A a. .a . . . . . . . . . . . . . • - - .

. . . . - . • . . . . . . . - O ° ~ j j

- . 4 1 ~

..o 50 . . . . . . " " . . . . . . . . " ' ~ . . . . . . ": : " ' : " . . . . . . . . . - i : 'Y ' - - • : ~

o :_: : : : - : - i ; -; : : : : : : : : : >

- -~k . . . . . A , - - - - ~ k - - A ' " - - - ~ . . . . . & '

a:,. 4 0 - - - ~ ~ _ _ - - - : - - : ~ : - - - = : : : : ~ : : = : : ~_ : . . .= . .= - -~ - : : ~ : : : - _

Q " . - o

30 . . . . . . . . . . . . . . ~ _

20 ~. .~

- - i | - - I I i - - I i H ~ " . . . . ,

1 ;0 200 250 300

Per iod (s)

FIG. 3. Effects of mislocation in the down-dip direction for dip-slip earthquakes. Synthetic seismograms are computed for point sources at depths of 24 and 40 km with various fault dips of 15 °, 25 °, 35 °, and 45 ° with the earth models PREM and R-A. Apparent dips of the fault plane are obtained assuming an epicenter mislocated at an arc distance of 0.05 ° from the true epicenter by moment tensor inversions of Rayleigh wave spectra of periods between 150 to 300 sec of the synthetic seismograms.

60

50

Ef fect of Mis loca t ion (0.05°along-str ike)

Depth (km i True Dipl(deg) ' I ,

PREM 4 • .45..o 0- 35_, ,-2S--, = 15 . Dip-Sl ip Ear thquake 40 e - - o e- - e a - - - - o o

R-.,,, 420, t . . . , t ~, ..-, ' , ~ ~ -" " "

+ . . . . . . . &===== == ~== . -= === =~ ==== : = ; ; ; ~ k ; ; ' ; ; ~ ; ; ; ; ; ; ; ~

8 ~ ~v 40

D

¢:D

g 3o <

20

. . . . - - O - - - - - - : - - - - 0 . . . . •

200 250 300

Per iod (s)

FIG. 4. Effects of mislocation in the along-strike direction for dip-slip earthquakes. Synthetic seismograms are the same as for Figure 3. Apparent dip of the fault plane is obtained from inversions at various periods assuming an epicenter mislocated at an arc distance of 0.05 ° along the strike direction.

f ocus t h e d i s c u s s i o n b e l o w o n s h a l l o w , t h r u s t e a r t h q u a k e s , t h o u g h t h e e q u a t i o n s a r e

t o t a l l y g e n e r a l . T h e s o u r c e s p e c t r u m , V = I VI e x p [ i ~ ] , is a f u n c t i o n o f t h e m o m e n t t e n s o r

o f t h e s o u r c e a n d t h e e x c i t a t i o n f u n c t i o n s , w h i c h d e p e n d o n t h e s o u r c e d e p t h . T h e s e i s m i c m o m e n t t e n s o r is o f t e n d e c o m p o s e d i n t o s i m p l e r m e c h a n i s m s a n d

1214 J . Z H A N G A N D T. L A Y

Effect of Mislocation (0.05=along-strike)

2 5 ~True Dip Depth (kin) I d PREM R-A

~" ( e g ) 24 40 24 40 r 45 • - - • o . . o • - - • A . . ,~ r~;~ ~,;~,,~,.,~,,~, = ~,~ t. ,_ar t .~uar , e • ~" I - 3 5 e * ~ • ~ - - O & - - ~ & - - ~ , ' "

201 - 25 = - - - - ~ ~ - - - - o , . - - - , ~ - - - - - ~ - I - 1 5 : : 0 0 J- • ~, .~, . . i t ~ e - -

/ I 'True Strike 0 • "

~ 1 5 N ft.)

r a 10

<

° ° J I I

"True Slip 90 ~

. . . . . ;.::;--" - - !

5 ~ _ : _ . . j . . . . _ ~ - - ' ~ - - - " ~ - - - - ~ - - . 7 . 2 = ~ . ~ ~ ~ ~-_ _-7_- . . . . - - _---~_-~_~- ~ - - ~ -

. . . . , . . . . , . . . .

200 250 3 0 0

Per iod (s)

FIG. 5. Effects of mislocation in the along-strike direction for dip-slip earthquakes. Synthetic seismograms are the same as for Figure 3. Apparent strike of the fault plane (true strike = 0 °, slip = 90 °) is obtained from inversions at various periods assuming an epicenter mislocated at an arc distance of 0.05 ° along the strike direction (0 ° in azimuth).

Ef fec t o f M is loca t ion (0.05=along-str ike)

. . . . ' ' ' ' - - ; " .., .s~ True Dip PREMDepth (kin) R-A

130 (deg) 24 40 24 40 Dip Sl ip E a r t h q u a k e 45 e - - e e - - o • - - , t A - - , ~ - ~ •

2 5 e - - . - . - - e Q.----...O & . - - - - . - • ,~-.-----,~

15 = : o o • " ~ " , ~ " • ' ° " 7 ]

true Strike 0 . " x . - J .J 1 2 0 - T m e S , i p 90 - " ~ f - ~ ' ' ' " J I

. <

11o

Q. < 12- " ~ . - j ~ - C . 3 - ~ -

=. .-~ lOO ~ . _ _ _ : - ~- ._-_Z.-~ ~ _ _ _..:8=.~ :..-_--8"

-~--- _--...~ --~ .---.--- __.. __.~ ~-,-~-.-.-.--.~--~ ,..-.--,-~ -~-----~. -._.--:_--_:--:_~--.-.--.~- ~ . . . . . . - .

9~)~ ' ' ' ' I , , , , I , , , , - - 0 2 0 0 2 5 0 3 0 0

Per iod (s)

FIG. 6. Effects of mislocation in the along-strike direction for dip-slip earthquakes. Synthetic seismograms are the same as for Figure 3. Apparent slip of the fault plane (true strike = 0 °, slip -- 90 °) is obtained from inversions at various periods assuming an epicenter mislocated at an arc distance of 0.05 ° along the strike direction (0 ° in azimuth).

c h a r a c t e r i z e d b y s e v e r a l p a r a m e t e r s : s e i s m i c m o m e n t Mo a n d f a u l t m e c h a n i s m (~b I, 5, a n d X) o f t h e b e s t d o u b l e c o u p l e o f t h e m o m e n t t e n s o r , a n d t h e r a t i o E =

I )'m:,/)'max I, w h e r e ~min a n d hm~x are , in a b s o l u t e v a l u e , t h e s m a l l e s t a n d l a r g e s t e i g e n v a l u e s , r e s p e c t i v e l y .

F o l l o w i n g Z h a n g a n d K a n a m o r i (1988b) , we w r i t e t h e r e a l (v) a n d i m a g i n a r y (~)

C E N T R O I D L O C A T I O N A N D D E T E R M I N A T I O N O F M E C H A N I S M S 1215

Ef fec t o f M i s l o c a t i o n ( 0 . 0 5 ° a l o n g - s t r i k e NP1 )

3 0 De'-thp ' ' P R E M ' ' R - A ' ' ' ,,O /

( km) Dip(NP2) Slip(NP1)Dip(NP2) Slip(NP1) ~-~"

20 2, : . . . . . . : - - : : . . . . . : = - - :

~ : - - = = = 2 : . . . . • . . . . ~ - - ~ : : : - - - ~ - - - -

D : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : . . . . . . . . ~ . . . . . . ~ . . . . . . . . A

o - 1 0 L . . . . . . . . . * . . . . . . . o o

. _ - * o .

(a) (b) . . . . . . . . . > True NP1NP2 NP1 NP2 " ' " ~ `o .

- oo -20 Strike 0 270 0 270 " ~ . °

Dip 90 90 90 90 Str ike-Slip Ear thquake " - . Slip 0 180 180 0

.310~ 0 , , ; , t ~ , , ; I , , , , 2 0 0 2 5 0 3 0 0

P e r i o d (s)

FIG. 7. Effects of mislocation in the along-strike direction for strike-slip earthquakes. Synthetic seismograms are computed for vertical strike-slip faults with: (a) first nodal plane (NP1): Cf = 0 °, ~ = 90 °, h = 0°; second nodal plane (NP2): ¢~ = 270 °, ~ = 90 °, X = 180; and with (b) NPI: ~f = 0 °, & = 90 °,

= 180°; NP2: Cf - 270 °, 5 = 90 °, ~ = 0 °. The apparent dip (NP2) and slip (NP1) are obtained from inversions at various periods assuming an epicenter mislocated at an arc distance of 0.05 ° along the strike direction of NP1. The difference between the apparent value and true value of the dip (NP2) (or slip (NP1)) gives the variation of the dip (NP2) (or slip (NP1)).

p a r t s o f t h e s p e c t r u m ~? as

1 1 v = - -~ f2(PR(1)COS 2¢ + SR (1)) + ~/3(PR(nCOS 2¢ -- SR (1)) --/6PR(1)sin 2¢ (6a)

} = AQR(~)cos q~ +/sQR(~)s in ¢ (6b)

w h e r e for a doub le coup le t h e m o m e n t t e n s o r c o m p o n e n t s fi c an be w r i t t e n as

f2 = 2 M 0 ( - c o s X s in 5 s in 2¢f + s in ;~ s in 5 cos 5 cos 2¢ s - s in X s in 5 cos 5);

f3 = 2Mo(cos }, s in 5 s in 2¢ s - s in ~ s in 5 cos 5 cos 2¢ t - s in X s in ~ cos 5);

f4 = M0(s in }, cos 25 s in q~r + cos }, cos 5 cos ¢~);

f5 = M 0 ( - c o s ~ cos 5 s in Cr + s in ~ cos 25 cos ¢~);

/6 = M o ( - c o s X s in 5 cos 2¢ i - s in ~ s in 5 cos ~ s in 2¢s).

S ince c h a n g e in t h e e p i c e n t r a l l o c a t i o n of t h e p o i n t source i n t r o d u c e s v a r i a t i o n s of t he o b s e r v e d source spec t r a , t h e r e is a c o r r e s p o n d i n g v a r i a t i o n of t he m o m e n t t e n s o r a n d c e n t r o i d d e p t h o b t a i n e d f rom each inve r s ion . F o r t h e m o m e n t t e n s o r of a doub le couple , t h e a m p l i t u d e , I~ ' l = (v 2 + ~2)1/2, d e p e n d s on b o t h Mo a n d m e c h a n i s m , whi le t h e p h a s e , ~ = t a n - l ( ~ / v ) , d e p e n d s on ly on t h e m e c h a n i s m . S ince t h e a m p l i t u d e a n d p h a s e a re n o n l i n e a r f u n c t i o n s of t h e m o m e n t t enso r , for a ge ne ra l m o m e n t t e n s o r t h e r e l a t i o n b e t w e e n t h e p e r t u r b a t i o n s of t he p a r a m e t e r s e, M0, ¢s, 5, a n d X a n d t h e p e r t u r b a t i o n s of t h e a m p l i t u d e a n d p h a s e is complex . A v a r i a t i o n


of the observed amplitude or phase will, in general, lead to perturbations of all six components of the moment tensor, and therefore, to e, Mo, el, 6, and X. In order to illustrate the relation, we compare the first order terms of these perturbations.

For a shallow dip-slip earthquake, from (6), the first azimuthal terms of 0~/0¢s, 09/06, and 0~/0X vary as - s in 45 sin ¢, -cos ¢, and sin 6 sin ¢, respectively. These terms vanish at the free surface. For 01 V 12/0¢I, 01 V 12/05, 01 V 12/0X, and O I V I2/O(Mo 2) the first azimuthal terms vary as -Mo2sin 2¢ ml 2, Mo2cos 2¢ sin 45, M02sin 2¢ m2, and cos 2¢ m32 respectively, where mi (i = 1, 2, 3) are functions of 6.

For a shift of the epicentral location in the dip direction, the observed source spectra vary as -cos ¢ for phase and remain about the same for amplitudes. Inversions of the observed spectra give results with an increase of the dip and corresponding decrease of the seismic moment; on the other hand, ¢I and X remain about the same, since the first order terms of the variations of the phase and amplitude due to changes of Cr and X are not compatible with that due to the location shift.

The variation of dip caused by the shift of the source location in the dip direction depends on the depth of the earthquake. The variations of the first azimuthal terms of c)~lc)~[, 0~/05, and 0~/0h are proportional to the excitation coefficient QR (1), which vanishes at the free-surface and increases rapidly with depth. Therefore the variation of the dip for deep events is smaller than for shallow events.

For a shift of the epicentral location along the strike direction (with the use of Aki and Richards (1981) convention), the variation of the observed source phase goes as - s in ¢. Because variations of ¢I and X cause the phase to also vary as sin ¢, the inversions result in moment tensor solutions with large variations of Cs and X for various source locations along the strike direction.

In the following, we use a spatial search approach to examine the effects of the source coordinate variations on the moment tensors that are obtained for the Mexico earthquakes. For each earthquake, we calculated the long-period source spectra of Rayleigh waves by correcting for attenuation and propagation phase delay assuming various point source or centroid locations and using optimized finite source durations. For each assumed location we invert for a moment tensor, to assess the resolution and trade-offs with source location.

1. 14 March 1979 Petatlan Earthquake

We first consider sources distributed along the 28 ° azimuthal trend, along which the various body-wave epicenters are distributed (Fig. 1). This direction is perpendicular to the Pacific margin, thus is the most likely direction for epicentral bias. Figures 8a and b show results of first-step inversions, and Figures 8c and d results of second-step inversions, respectively, for a sequence of source locations distributed along a line 150 km long with the southwest end located 50 km seaward from the EMS location. In the inversions, we adopted the EMS origin time (Gettrust et al., 1981). Differences in origin time can directly be absorbed into the source duration estimate and thus have only second-order affects on these results.

The first-step inversion is to isolate the source finiteness effect and is made for the observed Rayleigh wave spectra of a given period. For each assumed source location, we obtained the minimum of the first-step errors, a, (Fig. 8a) and corresponding apparent duration (Fig. 8b) from inversions at a given period. For all periods, the variation of a is very small, and the apparent durations are essentially independent of the assumed source location. The duration estimates do depend on

C E N T R O I D LOCATION A N D D E T E R M I N A T I O N OF M E C H A N I S M S 1217

1979 Petatlan

~.4

.6 . . . . , . . . . , . . . . I 70 i

1

.5' - ~.6s- ~. ~ !.:,-T,,l:~..~..t..~,,~.:.~. ! ~

• ~ - o . . ~ - ~ - ~ - ~ 6 0

.... o . ~ , . . . ~ . o . o . o , . . . ~ o . o . ,

- :.:.*:=:l~:l:~:~:.*:l~:l~:~l:l- 7

(a) . . . . . . . . . . . . . . 50

.1"2 40

1

/

.10 / 30 : g

/ x:: /

X ~ 2 0 o8,, /

"" (c) ! . . . .

% o so loo %

. . . . I . . . . I . . . .

. X . )(- -X- ~¢-'M-~ -

- . 4 . -X- -X--X-.K" ~--X" 2

- ÷ - + - ~ . + ~ " 3 A - , ~ - ~ , - A - .

A.~,-b,~-~,L-,~-A" 7 . l , l - i l * . , l l . i . I F . l - , . l . . l l . . l l . l ~ . I F o l l . . l - . l -

'-" o - @ . ~ - -@-.@- o oo - o -@--@-~,-

- 0 - 0 - 1 ~ 1 " 5

,~,4 . o . , . - o - D - m . ~ , m - - ~ "m" -

(b) . . . . t . . . . t . . . .

. . . . t . . . . t . . . .

\ ',..

,.

\ ,.. x :./'~"

. . . . , . . . . io SW ~ Distance (kin)

L I d

NE

FIG. 8. Results of inversions for various source locations for the Petatlan earthquake using the origin time given by Gettrust et al. (1981). The sources are distributed along a line trending a 28 ° direction (dashed line in Fig. 1) with distances measured in kilometers from the EMS location (solid circle 1 in Fig. 1). (a) The minimum errors, o, in the first step inversion for determination of apparent duration for every assumed source location at periods of (1) 150, (2) 175, (3) 200, (4) 225, (5) 256, (6) 275, and (7) 300 sec. (b) Apparent durations determined for each source location at various periods. (c) The minimum errors, p, in the second step inversion in determination of source moment tensor. A point source with a duration of 64 sec is used for all inversions. The range of preferred source location of radiation of long- period Rayleigh waves is shown with a bar. (d) The depth for every source location, where the error, p, is minimized. The minimum error, p, and corresponding depth from inversions with epicentral parameters (location and origin time) given by NEIC are also shown in (c) and (d) (crosses).

t h e r e f e rence e a r t h m o d e l ( Z h a n g a n d K a n a m o r i , 1988a), i n a d e q u a c i e s o f w h i c h a c c o u n t for t h e s c a t t e r b e t w e e n d i f f e r e n t f r equenc ie s (Fig. 8b). F o r a source a t t he

E M S loca t ion , we o b t a i n e d a n ave rage 64-sec d u r a t i o n for t h e p e r i o d s of 256 a n d 275 sec. I n t h e fo l lowing, we d e t e r m i n e t h e source d e p t h a n d m o m e n t t e n s o r u s ing a p o i n t source m o d e l w i t h 64-sec d u r a t i o n for a l l a s s u m e d source loca t ions .

T h e s e c o n d - s t e p i n v e r s i o n is to d e t e r m i n e t h e d e p t h a n d source m o m e n t t ensor . F o r eve ry a s s u m e d source loca t ion , we so lved for t h e m o m e n t t e n s o r b y l ea s t squa re s a n d f i nd t h e d e p t h t h a t m i n i m i z e s t h e s e c o n d - s t e p e r ror , p. T h e r e su l t s for a source a t t h e E M S l o c a t i o n a re s h o w n in F i g u r e s 9a a n d b, wh ich i l l u s t r a t e t h e p r o c e d u r e u n d e r l y i n g c ruc ia l a s p e c t s of t h e s e c o n d - s t e p i n v e r s i o n in F i g u r e s 8c a n d d. T h e e r ro r p is m i n i m i z e d a t a d e p t h o f 23 k m (Fig. 9a), a n d t h e c o r r e s p o n d i n g m o m e n t t e n s o r s o l u t i o n is e s s e n t i a l l y a p u r e doub le couple . T h e s t r ike , e l , s l ip, X, a n d dip,


March 14, 1979 Petatlan .4 ' I ' I ' I ' I '

,3

1:).2

.1

(a) (b) O ; o , , , , , , , , , 2 0 3 0 4 0 5 0 6 0

Depth (km) FIG. 9. Results for the Peta t lan earthquake. (a) Residual versus depth curve for the moment tensor

inversion of Rayleigh waves for EMS epicentral parameters (Gettrust et al., 1981). The error, p, is minimized at a 23-kin depth. (b) The source mechanism of the bes t double couple (shaded area) obtained for the 23-km depth is compared with the fault plane solution obtained by Chael and Stewart (1982) (short dashed lines: strike Cf = 293 °, dip ~ = 14 °, slip X = 87 °) from P-wave first motion and IDA surface- wave data and with the best double couple of the Harvard CMT solution (long dashed lines: ¢[ = 307 °, 5 = 17 °, }, = 104°).

6, associated with the best double couple of the moment tensor (following the decomposition used by Dziewonski and Woodhouse, 1983) are 295 °, 85 °, and 12 °, respectively. This mechanism is very consistent with the best double couple of the Harvard CMT solution and the fault plane solution obtained by Chael and Stewart (1982) from P-wave first-motion and IDA surface-wave data (Fig. 9b). This is a very encouraging result, for it is obtained entirely from fundamental mode Rayleigh waves, with no a priori constraints other than the choice of location. The earth model propagation corrections over the spectral range of 150 to 300 sec are apparently adequate to overcome the instability of determining accurate dip for a shallow thrust event using fundamental modes. Now we examine the effect of using a less reliable location.

Figures 8c and d show the minimum error of the second-step inversion, p, and corresponding depth, at which the errors are minimized, for each position along the 28 ° trend. As discussed below, each solution has a different source mechanism. For a source 10 km to the southwest of the EMS location, the p is locally minimized at a depth of 26 km. For the origin time and epicentral location reported by NEIC, respectively 5 sec later and 45 km northeast of the EMS solution, inversions give a 53-sec apparent duration and a 21-km depth. The corresponding total error, p, is about 25 percent larger than that for a source 10 km to the southwest of the EMS location.

Figure 10 shows, for various source locations along the 28 ° trends, the phase of the observed source spectra of 150-sec and 256-sec periods and the predicted source spectra calculated using the moment tensor solutions for each location. As the source shifts along the dip direction (NE), the calculated source phase delay


1

o

-1

-2

Petatlan 03/14/1979

l o k ~ . s w . . . . . . . . . . . . . . . . . . . . . . , . . . . I - . . _ . = , . • - - , ~ = =

" . . . . , . . . . . " . . . . . . • , • • , •

ol lOi.m NE . . . . . . . . . . . . . . . . . . . , .i I- --'•- • J~ • m. • • m••=

( - - ~ • ~ k J L • = = • • maD

-2~" " • ~ • • • -~L . . . . . . . . . . . " . . . . . . . " . . . . . . . . . . . . .

ol 30k NE . . . . . . . . . . . . . . . . ,

. 2 ~ = ~ | ~ • - • ~mm •

~_ .~L . . . . . . . . . . . • . . . . . . • . . . . . . . . . . . .

ol 50kroNE . . . . . . . . . . . . . . . . . . . . . . .

-1L = - - • • • ~ • • == • • • = = • = . 2L-" " ~ . ' l " - ' . '= ; ~ - - .

1 . . . . , . . . . . , . . . . . , . . . . . , . . . . , . . . . . 0 1 O k • SW

- 2 - • •

====================================== 0 1 O k • NE

• I • = • • • • • • "1 , i r a ' ' • ~ • •• IN • m

" 30 km'NE . . . . . . . . . . . . . . . . . . . . . . . . . .

50 km'NE . . . . . . . . . . . . . . . . . . . . . . . . . .

dLm n • • arena,d• . _mUm • • _ m=.nn nlm = ~ N N i ~ l m m n ~ ~ - r

" 70 km'NE . . . . . . . . . . . . . . . . . . . . . . . . . . . mm mmmm# . ~ m • • • m row.

0 60 120 180 240 300 360 0 60 120 180 240 300 360

T=150 s Azimuth (deg) (a) T=256 s Azimuth (deg) (b)

FIG. 10. Phase angles of observed propagation corrected source spectra of 150-sec (a) and 256-sec (b) periods for source locations, distributed along the 28 ° azimuthal trend shown in Figure 1, 10 km SW, 10 km NE, 30 km NE, 50 km NE, and 70 km NE from the EMS epicenter. Solid lines indicate theoretical values calculated for the moment tensor solution obtained at each source location.

increases for stations to the NE and decreases for stations to the SW. For the source 10 km to the SW of the EMS location, the phase radiation pattern yields a shallow thrust-fault mechanism; for the source 70 km to the NE, however, it requires a dip-slip, reverse-fault mechanism. Since the wave number, w/c, for long periods is smaller thanfor shorter periods, the variation of source phase due to the change of the source location decreases as periods get longer (compare Figure 10b with 10a).

Figure 11 shows the amplitude radiation pattern of 150-sec period. Compared with the phase radiation pattern, the amplitude radiation patterns vary only slightly, because the spreading and attenuation corrections are very small.

Figures 12a to c show, for each source location in Figure 8, the results of the moment tensor inversion for two depths: the variable depth which minimizes the inversion error, p, (shown in Fig. 8d) and for a fixed depth of 23 km. For all of the assumed source locations, the moment tensor solutions are essentially pure double couples, with the parameter e being less than 0.08 for inversions with variable depth and less than 0.05 for inversions with fixed depth (Fig. 12a). Figure 12b shows the strike, dip, and slip for both nodal planes of the best double couple of the moment tensor solutions. For all assumed source locations, the best double couples represent predominantly dip-slip fault motion with a northwesterly striking nodal plane, and the variation of the strike and slip is less than 20 ° . However, for sources 30 to 70 km to the northeast of the EMS location, the dip of the northwesterly striking plane changes from about 15 ° to 75 ° , and the seismic moment of the best double couple varies by a factor of 2 for inversions with variable depth and by a factor of 5 for inversions with fixed depth (Fig. 12c). The northwesterly striking plane of the solution, obtained from inversions for a source depth of 21 km with the epicentral


Petatlan 03/14/1979 T=150 s

" , . . . . . . i 0 S w . . . . . . . . . . . . . . . . . . .

• • Q

~d ,

5 ~" " i . . . . . . . i 0 I~m NE . . . . . . . . . . . . . . . . . . .

• 0 0 • • •

~ o

ff ~ • 0 0 0 • •

< o i. . . . . . . Sd NE . . . . . . . . . . . . . . . . . . . .

• • u

0 . . . . . . . . . . ' ' ' ' ' ' ' ' ' '

. . . . . . . . . . . . . . . . . . . . t

0 0 60 120 180 240 300 360 Azimuth (deg)

FIG. 11. Amplitudes of the observed spectra of 150-sec period for the same source locations as in Figure 10. Solid lines indicate theoretical values calculated for the moment tensor solution obtained at each source location.

parameters given by the NEIC, has a dip about 40 °, inconsistent with the first- motion mechanism (Fig. 12d). We feel that this is the result of an incorrect assumed location rather than intrinsic instability of the dip determination from surface- wave inversion.

Figures 12b and c show that the variation of dip correlates well with the seismic moment. For the inversions with fixed depth, the sources to the SW of the EMS location have smaller dips and larger seismic moments than the sources to the NE. For shallow thrust events, the amplitude and phase of the excitation of Rayleigh waves are governed to the first order by Mosin 25 and cot 25, respectively (see Kanamori and Stewart, 1976, equation A-8). The variations of the dip and seismic moment shown in Figure 12 are consistent with rapidly varying phase spectra and slowly varying amplitudes spectra as the source shifts along the down-dip direction.

Because of errors in the reference earth model and the source finiteness model, which result in the residual scatter in Figure 10 to 12, it is not very stable to determine the earthquake location from surface-wave data alone. However, the surface-wave data may be useful to constrain the centroid of the long-period radiation to within a few tens of kilometers in the down-dip direction. Figure 8c shows that, allowing for a variation of error of the inversions of about 10 percent, the preferred long-period centroid is within 30-km distance of the EMS location. The source mechanisms obtained for locations near the EMS location are stable and consistent with the P-wave first motions. This also holds for the JHD location,


1 9 7 9 Petatlan • 1C . . . . , . . . . , . . . .

310

1 2 0 - " - "~- "~- -~- - - - - , - ~- _,~ s t r i k e 2

1 0 0 ~ 1 _ - - - e - - o - . - o - - o - - o - 4 _

" O . . . .

80 _ ~ ~ c~ dip 2 - ~ i , ¢ - .

" 6 0

• 40 . ~,

E dip 1 . " .

• . • Chael and S t e w a r t [ 1 9 8 2 ]

. . . . ~ . . . . ' . . . . Steeply deeping plane: • • • i . . . . i . . . .

s t r i k e = 1 1 6 d i p = 7 6

i i i i I i i i i I i i i 1

0 5o 100 " T SW Distance (km) NE

FIG. 12. Results of inversions for the source locations of the Petat lan event used in Figure 8. (a) The parameter +( I Xmln/Xma= I ) Of the moment tensor solution• (b) The dip (solid circles), strike (crosses), and slip (open circles) of the northwesterly striking nodal plane (symbols with solid lines) and auxiliary plane (symbols with dashed lines) of the best double couple of the moment tensor solution obtained for each source location. The symbols without lines show the results of the northwesterly striking plane for inversions with the depth fixed at 27 km for all source locations• (c) The seismic moment of the best double couples in the unit of 102°Nm; (d) The focal mechanism determined by Chael and Stewart (1982) from P-wave first motion and IDA surface-wave data (solid lines). A lower hemisphere, equal area projection is shown with solid circles indicating compression, open circles dilatation, and crosses nodal character. Corresponding values for the moment tensor solution for the NEIC epicentral parameters are indicated by triangles in (a), (b) (northwesterly striking plane only), and (c), and by dashed lines (best double couple) in (d). The arrow indicates the source location where p is minimized (Fig. 8c).

which is about 10 km NE of the EMS location• On the other hand, inversions for the sources close to the NEIC location result in larger residual error and source mechanisms incompatible with P-wave f irs t-motion data.

How about mislocations along the strike of the arc? We now consider various source locations distr ibuted along the 118 ° azimuthal t rend (Fig. 1), which is almost parallel to the direction of fault strike (NW). Figure 13 shows results for sources distr ibuted along the southeas t -nor thwest line centered at the EMS location. The inversion errors, p, are minimized 30 km to the N W of the EMS location (Fig. 13a). For all locations, the est imates of the centroid depth are almost the same (Fig. 13b); the parameter E is ~less than 0.12 and minimized for a source 10 km to the SE of the EMS location (Fig. 13c); the dip of the best double couple is about the same, however the slip and strike vary from 50 ° to 130 ° and from 280 ° to 320 °, respectively (Fig. 13d). The seismic moment of the best double couple does not vary much (Fig. 13d), which is compatible with the small variat ion of the dip.

Figure 14 shows the phase of observed and predicted source spectra of 150-sec period• For a shift of the source location in the direction of strike, the observed phase delay increases for stations to the N W and decreases for stat ions to the SE.


197! K . • • i . . . .

.10 E"

e % '.... 2C .....

"% .10 '-,, (a) '

~, ,o° .06 " "."-,r" 3 2 0 t ~ ; . . ~ ~ • , . . . . j E.05

2 8 0 " " '~ ' - " -=

Petatlan 3C . . . . ,

• . ~ . .

""4, ,,e"-9 "° %e...e.=4...e.o.et "°

(b)

. . . . | . . . .

~ x ~ = ¢ ~¢" 0 . . . . I . . . .

"o ~ ~ " ~ " ~ "x~ 'x~ 3.

5C

-~-~-: : d i p ~ (e) -05 'o . . . . . . . . . . . . . . . . s o

SE Distance (kin) NW

FIG. 13. Results of inversions for the Petatlan earthquake for various source locations distributed along a line centered at the EMS location trending a 298 ° direction (dotted line in Fig. 1). (a) The minimum errors, p, obtained for various source locations; (b) the depth, where p is minimized; (c) the parameter e; (d) the dip, strike, and slip of nodal planes of the best double couple of the moment tensor solution (see Fig. 12 for classification of symbols); (e) the seismic moment of the best double couple.

Source locations to the N W of the EMS location, result in left-lateral, oblique thrus t fault mechanisms, while locations to the SE result in right-lateral, oblique thrust fault mechanisms. Since the inversion errors for source locations to the SE are much larger than the errors for sources to , the NW, the latter are preferred. However, the source mechanism for a source 30 km to the NW, where the error is minimized, has a large oblique component , and is inconsistent with the P-wave first-motion data. We consider this to be due to errors in the reference earth model and source finiteness model. This indicates tha t an inversion for location based solely on reduction of surface-wave spectral variance may not give a satisfactory location or mechanism.

2. 25 October 1981 Playa Azul Earthquake

The 25 October 1981 Playa Azul earthquake has a smaller magnitude (Ms = 7.3) than the Petat lan and Michoac~n earthquakes. In contrast with the Petat lan earthquake, the J H D location is about 40 km to the north of the EMS location for the Playa Azul earthquake. The NEIC location is about 10 km to the east of the J H D location. Figure 15 shows inversion results for sources distributed along a line 100 km long trending N27°E direction from the EMS location. We used the EMS


Petatlan 03/14/1979 T=150 s

0] 40 krn SE i _ • • m ~ i L • •

-2~ ~ • - -

- ? ~ ~ , ' : : : : : : : : : ' : : ' , ' : : : : : : : : : : : - 0 ], 20 krn SE

4 - • -

g 0 1 0 . . . . . . . . . ,

, . • [ r _ . . . % . . . • _21- • • ~ • ., - :::::::::::::::::::::::::::::::::::::::::::,

I 20 krn NW 0 i

- 1 ~ = • I I • •

-2~ - • ~ , - - ~ !- . . . . . . . . . . . . . , . . . . . . . - . . . . . . . . . . . . . -

o1 4 k;.'NW . . . . . . . . . . . . . . . . . . . . . . . . . . .

-2I" _ " ~ J "

-3~ ' . . . . . . . . . . ' . . . . . . ; ." . . . . . . . ' . . . . 0 60 120 180 240 300 360

Azimuth (deg)

FIG. 14. Phase angles of the observed source spectra of 150-sec period for the source locations used in Figure 13:40 km SE, 20 km SE, 0 kin, 20 km NW, 40 km NW from the EMS location for the Petatlan earthquake.

origin t ime (Havskov et al., 1983). The inversions again give similar results for source durat ion for all source locations used; and the first-step errors, o, do not vary much (Fig. 15a). For a source at the EMS location, we obtained a 50-sec duration. We used a point source with 50-sec durat ion to determine the moment tensor solutions for all source locations with the results being shown in Figures 15b to f. The second-step inversion error, p, is minimized for a source 50 km to the NE of the EMS location (Fig. 15b), which corresponds to a depth of 37 km (Fig. 15c). The variat ion of dip caused by the shift of the source location is smaller than for the Peta t lan earthquake, which may suggest tha t the Playa Azul ear thquake is deeper than the Pe ta t lan earthquake, consis tent with the greater depth determinations found in our inversions.

Figure 16 shows the inversion results for Playa Azul sources distr ibuted along a line paralleling the t rench direction (Fig. 1). The error, p, is minimized N W of the line through the body-wave locations, but the curve has a broad minimum (Fig. 16a); the est imates of the depth are about the same (Fig. 16b). The overall variat ion of slip is about 30 ° less than tha t for the Peta t lan earthquake, suggesting tha t the source depth and dip of the fault plane of the Playa Azul ear thquake are different from tha t of the Peta t lan earthquake.

The focal mechanism of the Playa Azul ear thquake obtained by LeFevre and McNally (1985) from surface waves and P-wave first-motion data has a steeply


1981 P laya Azul

" ' ' " ' - - . - - . - - - . - - - . . . . . . . . . . . . . . . ¢P

.5 P % ," [ 7 .... . • .lo - . , , , , ,.." t

. . . . . . 0 , t . . . . : . . ?!1 O" . 4 I - 3 - " ~ ^

I O ,.- "~-~ . . . . . . ~--~-..~.. /

. . . . . . "~--o--,o--,e-.-¢.....~ " . t

/ <a) o I .... "'"*"t . 2 i . . . . ' . . . . 3 0 " . . . . ' . . . . "

300 . . . . , . . . . . 08 1 . . . . , . . . .

280 ~ " 0 6 ~ ' , x (d )

~I00 "° °" ; " °"(' " '°" " 0 2 [ ~ : l i . .

<

.x.i 10 . . . . ' . . . . 6 . . . . . . . . . . '

50 100 " 0 50 100

SW Dis tance (km) NE

FIG. 15. Results of inversions for the Playa Azul earthquake for various source locations distributed along a line originating at the EMS location and trending a 27 ° direction (dashed line in Fig. 1). (a) The minimum error, ~, obtained at various periods; (b) the minimum error, p; (c) the depth, where p is minimized; (d) the parameter E; (e) the dip, strike, and slip of nodal planes of the best double couple of the moment tensor solutions (see Fig. 12 for classification of symbols); (f) the seismic moment of the best double couple. Corresponding values for the moment tensor solution for the NEIC epicentral parameters are indicated by isolated symbols in (b), (c), and (e) (northwesterly striking plane).

dipping nodal plane with a strike of 121 ° and dip of 79 ° (the dip is constrained by P-wave first-motion data). The strike and dip of the steeply dipping nodal plane obtained from the long period Rayleigh waves are 100 ° to 130 ° and 63 ° to 64 °, respectively, for the source locations assumed. The dip is about 15 ° smaller than that obtained from P-wave first-motion data, which may either reflect complexity of the source process, or inaccuracies of the earth models used in both studies. The strike of the nodal plane obtained for the source where p is minimized is about 14 ° less than in the body-wave mechanism, while better agreement is obtained for the best location along the down-dip direction. The slight NW bias of the best locations for the Petatlan and Playa Azul events is probably the result of inaccurate aspherical earth corrections.

3. 21 September 1985 Michoacdn Earthquake

The 21 September 1985 Michoacfin earthquake is the largest aftershock of the 19 September Michoacfin main shock. The P-wave first-motion data constrain the dip


o12 . . . . , . . . . "

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2 1 0 0

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1981 P laya Azu l

.90

.-e% ~ . . - $ - - . 9 "

(b) . . . . i . . . .

. . . . I . . . .

(e) I . . . . . . . . . . . . . ,

. . . . o so % 0 ; 50 SE Distance (kin) NW

FIG. 16. Results of inversions for the Playa Azul earthquake for various source locations distributed along a line trending a 297 ° direction (dotted line in Fig. 1) and centered 50 km to the NE of the EMS location. (a) The minimum error; p; (b) the depth, where p is minimized; (c) the parameter e; (d) the dip, strike, and slip of nodal planes of the best double couple of the moment tensor solutions (see Fig. 12 for classification of symbols); (e) the seismic moment of the best double couple.

and strike of the steeply dipping nodal plane to be about 81 ° and 127 ° , respectively (Eissler et al., 1986). The NEIC and ISC locations lie between the epicenters of the Playa Azul and Peta t lan earthquakes. The epicentral locations of EMS and J H D are about 30 to 40 km southwest of the NEIS and ISC locations, as was the case for the Petat lan earthquake (Fig. 1). The J H D location for the Michoc~n earthquake shown in Figure 1 was determined using teleseismic P-wave arrivals reported by NEIC.

Figure 17 shows the results of inversions for sources distributed along a line 150 km long trending 27 ° in azimuth. The EMS location is on the line 50 km from its southwest end. For all the source locations, the variation of the first-step error, o, is only a few percent (Fig. 17a). The source duration is about 48 sec for the source at the EMS location. We used a point source with 48-sec duration to determine the moment tensor solutions. The second-step error, p, is minimized for a source 10 km to the NE of the EMS location (Fig. 17b), which corresponds to a depth of 28 km (Fig. 17c). The moment tensor solution for the NEIC location has a ~ of 0.24,


1985 M i c h o a c a n a f te rshock

. 6 / . . . . . . ' . . . . 1 ' ' " ' ' I .18~ . . . . , . . . . , . . . . j

1 T ....................... "---¢ "st., .'" 4 4p1,[-;', ""

f- ......... 72 ............ 4.12 I. . . . . . . . . . . . x. . . . . . , . . . / !b). a .4 ;--,:,, ,-=',~:::'*:z:z-,,-z.~.~.t / . . . . , . . . . , . . . .

oh. ¢0)

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2 . . . . . ' . . . . ' . . . . / 20 / . . . . ~ . . . . , . . . . • : . : . : . .L ... : . : . . , . . . . . 41 . . . . , . . . . , . . . . |

2 8 ° ~ - ~ - " s'tiik-e- ~ .3 i- ~ (d)~(d)

15o~ . ~ , - ] e . I ~ p.15 ~ ~ ~ .~

Jo U -050 0 50 1 O0 1 O0

S W D is tance (kin) N E

FIG. 17. Results of inversions for the Michoac~n aftershock for various source locations distributed along a line trending a 27 ° direction and centered at the EMS location (Fig. 1). (a) The minimum error, ~, obtained at various periods; (b) the minimum error, p; (c) the depth, where p is minimized; (d) the parameter e; (e) the dip, strike, and slip of nodal planes of the best double couple of the moment tensor solutions (see Fig. 12 for classification of symbols); (f) the seismic moment of the best double couple. Corresponding values for the moment tensor solution for the NEIC epicentral parameters are indicated by isolated symbols in (b), (c), and (e) (northwesterly striking plane).

suggesting a large non-double couple component of the moment tensor, but locations to the southwest have much smaller e.

Figure 18 shows the results of inversions for sources distr ibuted along a line tha t is centered at the EMS location paralleling the t rench direction (Fig. 1). The first- step error, p, is minimized near the EMS location {Fig. 18a); the estimates of the depth are about the same for the source locations to the SE of the EMS location (Fig. 18b). For the source about 30 to 40 km to the SE of the EMS location, the moment tensor solutions has the smallest parameter E (Fig. 18d); the dip and strike of the steeply dipping nodal plane are about 72 ° and 114 ° to 120 ° , respectively; and the seismic moment of the best double couple is 2.70 to 2.75 x 102°Nm. This solution is most compatible with the body-wave mechanism and gives a location close to the CMT location.


. . . . I

.13

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: ." (a) % . % l . . . . i o"

o12 . . . . ' . . . .

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1985 Michoacan aftershock . . . . . . . .

35

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. . . . . . . . . . . . . . ~ . . . . (b) . . . . I , , , ,

(c)

. . . . g . . . . 0 5O krn) NW

FIG. 18. Results of inversions for the Michoachn aftershock for various source locations distributed along a line trending 297 ° direction (dotted line in Fig. 1) and centered at the EMS location. (a) The minimum error, p; (b) the depth, where p is minimized; (c) the parameter e; (d) the dip, strike, and slip of nodal planes of the best double couple of the moment tensor solutions (see Fig. 6 for classification of symbols); (e) the seismic moment of the best double couple.

DISCUSSION AND CONCLUSION

The azimuthal pattern of propagation corrections for observed phase spectra of long-period Rayleigh waves depends strongly on the assumed epicentral location. An error of only 30 km in the epicenter may thus cause significant biases in source inversions for shallow thrust earthquakes. The biases include errors in the strike, dip, and slip of the nodal planes, size of the non-double couple component of the source moment tensor, depth of the centroid, and seismic moment.

Determination of source mechanisms for shallow, dip-slip earthquakes from long- period (150 to 300 sec) Rayleigh waves thus requires source locations with greater accuracy than the routine NEIC or ISC locations, even if source finiteness is not a concern. Epicentral locations with high accuracy can be obtained from local seismic recordings or from teleseismic arrivals using procedures such as the JHD method, b u t i t is s t i l l n e c e s s a r y to e x p l o r e t h e ef fec ts o f u n c e r t a i n sou rce l o c a t i o n i n t h e s u r f a c e - w a v e a n a l y s i s , p a r t i c u l a r l y g i v e n t h e l i m i t e d a c c u r a c y of e x i s t i n g e a r t h

mode l s .

1228 J . Z H A N G A N D T. LAY

The effects of velocity anomalies, source mislocation, and rupture propagation all contribute to the observed phase spectra. For Mexican earthquakes, large scale velocity anomalies associated with the East Pacific Rise to the south and the North America Craton to the north produce strong azimuthal variation of propagation phase delay. However, in constrast with the effects of epicentral mislocation, the effects of lateral heterogeneity are in general nonlinear functions of frequency, varying rapidly with azimuth; thus we speculate that these effects may be largely corrected by existing earth models or averaged out by using many stations at different azimuths and ranges. For a source with unilateral rupture propagation, if ~[' represents the direction and length of the rupture, the observed phase spectra vary as ~0 cos(¢ - -¢ l , )~ I ' / 2c , which is equivalent to the effects of a epicentral mislocation 5l = - ~ I ' / 2 . However, the epicentral mislocation does not produce the amplitude variations that arise from destructive interference during rupture propagation. For inversions of long-period surface waves from earthquakes sources with significant finite ruptures, the directivity must be taken into account in the source finiteness model along with any mislocation effects.

Ideally, surface-wave inversions would include the source location as an unknown to be determined in the inversion. Following Dziewonski et al. (1981), we expand the spectrum of the ith record with respect to small perturbations in the source coordinates (re, 0c, Co), origin time to, and moment tensor fk:

6

~ _ ~(o) = b~Sr¢ + c~50~ + d~Sc~ + e~Stc + Y~ ~kSfk , (7) k ~ l

where ~'i (°) is the theoretical spectrum calculated for the starting values of the source coordinates and origin time and the initial estimate of the moment tensor, and bi, ci, di, and e~ are the kernel functions for corresponding parameters. Equation (7) can be solved iteratively. An alternative approach to the solution of the equation is to perform a parameter space seach (Dziewonski and Woodhouse, 1983). Un- less path corrections and source finiteness models are very accurate, epicentral parameters and the source moment tensor determined from long-period surface waves alone are subject to large uncertainties. Analyses of the three large earthquakes in Central Mexico indicates that caution is needed when using variance reduction estimates to determine an optimal source location from long- period Rayleigh waves.

A more stable procedure is to invoke consistency with other wave types, as was done by our preference for locations giving mechanisms consistent with body-wave and tectonic observations. Even allowing for this, the resolution of source location with Rayleigh waves alone is comparable to that of routine epicentral determinations by teleseismic P waves such as performed by NEIC and ISC. We believe that the JHD locations of these earthquakes are more accurate than the locations given by other teleseismic methods. Using the JHD epicenters for the Petatlan and Playa Azul earthquakes and a position 30 to 40 km to the SE of the JHD location for the Michoacfin earthquake, the source mechanisms obtained from inversions of long- period Rayleigh waves were found to be consistent with the first-motion data for these earthquakes. These locations are each about 30 km to the SE of the locations where the Rayleigh-wave inversion errors are minimized for these events. For these Mexican earthquakes, the epicentral locations inferred from long-period Rayleigh


waves actually show less bias than the 25 to 55 km inland bias of the standard epicentral locations reported by NEIC and ISC. It is, of course, possible that the long-period mechanisms, which average the entire source process, should differ from both first motion and body waveform derived mechanisms. Simultaneous inversion methods should address whether the small additional variance reduction achieved by separately optimizing the surface-wave solutions actually can resolve a distinct point source representation.

For the Petatlan, Playa Azul, and Michoacfin earthquakes, the errors, p, of the Rayleigh-wave inversions are minimized at source locations within 30 to 40 km from the JHD locations and are about 0.06, 0.09, and 0.12, respectively; the variations of the error for source locations within 30 to 40 km from the JHD locations are 10, 7, and 8 percent for sources locations along the dip direction, and 10, 4, and 5 percent for source locations along the strike direction, respectively. This may indicate that the source location of radiation of long-period Rayleigh waves is best resolved for the Petatlan earthquake.

The epicenter of the Playa Azul earthquake determined from JHD method or inferred from long-period Rayleigh waves is about 30 to 40 km landward relative to the locations of the Petatlan and Michoacfin earthquakes. If these earthquakes all occurred on a 15 ° dipping planar interface, the Playa Azul event occurred 10 km deeper, as indicated by the inversions. The absolute baseline of all of our depths must be viewed as uncertain by ___5 km, due to the poorly known source velocity structure; however, it appears that the relative depths are much better constrained. The depth difference can, of course, also be attributed to distortion of the plate interface near the Orozco fracture zone or location of the Playa Azul earthquake in the upper plate. Accurate determination of depths and epicentral locations of these large earthquakes is essential to resolution of these speculations. This analysis clearly indicates the need for careful consideration of source location in routine long-period surface-wave analysis. As more on-scale R1 and G1 data is collected by very broadband systems, location accuracy will greatly improve and joint body and surface wave procedures can proceed on a less ad hoc basis than at present.

ACKNOWLEDGMENTS

The data used in this study were made available by courtesy of the IDA project team at the Institute of Geophysics and Planetary Physics, University of California, San Diego, the GEOSCOPE project team at the Insti tut de Physique du Globe de Paris, and GDSN network personnel of USGS at Denver, Colorado. We thank Karen C. McNally for providing information on the locally determined source epicenters and the results of the JHD relocation. J im Dewey provided a thorough review of an earlier version of this manuscript. This work was supported by NSF grant EAR-8451715, USGS grant14-08- 0001-G1843, and by the W. M. Keck Foundation. Contribution number 100 of the Institute of Tectonics and C. F. Richter Seismological Laboratory, University of California, Santa Cruz.

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INSTITUTE OF TECTONICS C. R. RICHTER SEISMOLOGICAL LABORATORY UNIVERSITY OF CALIFORNIA SANTA CRUZ, CALIFORNIA 95064

Manuscript received 14 November 1989

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