Bulletin of the Seismological Society of America, Vol. 72, No. 6, … · 2017-02-27 · Bulletin of the Seismological Society of America, Vol. 72, No. 6, pp. 2063-2080, December 1982

Bulletin of the Seismological Society of America, Vol. 72, No. 6, pp. 2063-2080, December 1982

EFFECTS OF LATERAL HETEROGENEITY AND SOURCE PROCESS TIME ON THE LINEAR MOMENT TENSOR INVERSION OF

LONG-PERIOD RAYLEIGH WAVES

BY ICHIRO NAKANISHI AND HIROO KANAMORI

ABSTRACT

Spectra of mantle Rayleigh waves recorded on the IDA network are inverted to determine the seismic moment tensor of the 9 June 1980 California-Mexico Border earthquake, the 29 July 1980 Vanuatu Islands earthquake, and the 9 July 1980 Santa Cruz Islands earthquake. Examinations are made to correct for phase velocity lateral heterogeneity and source process time. A simple regionalization (stable continent, tectonic region, young ocean, and old ocean) im- proves results of the linear inversion, but the phase correction is not large enough, and the discontinuous change of phase velocity across the boundary between regions causes an artificial rapid change in apparent phase velocity as a function of azimuth. Therefore, a more detailed but gradual representation of the lateral heterogeneity is desirable to correct for the propagation effect in the linear inversion. A source process time which consists of the nondirectional part of the apparent duration of main faulting and the delay time of the faulting from initial break is estimated by a phase analysis of Rayleigh waves. The source process time is generally proportional to the seismic moment on the log-log scale, but some earthquakes deviate considerably from the general relation. Therefore, the measurement of the source process time should be made before the moment tensor inversion. The phase errors introduced by the lateral heterogeneity and source process time may cause a bias to low scalar moment.

INTRODUCTION

The importance of phase velocity data in the linear inversion for the seismic moment tensor is widely known, and efforts have been made to correct for the lateral heterogeneity in the moment tensor inversion of short-period (shorter than 100 sec) Rayleigh waves (Aki and Patton, 1978; Patton, 1980; Tr6hu et al., 1981; Romanowicz, 1981). On the other hand, long-period surface waves and free oscillations have been analyzed by using laterally homogeneous phase velocities or eigen- periods (Gilbert and Dziewonski, 1975; Mendiguren and Aki, 1978; Dziewonski et al., 1981; Kanamori and Given, 1981, 1982). This is because the average great-circle phase velocities are accurately measured with a standard deviation of about 0.3 per cent in the period range from 200 to 300 sec, and the average free oscillation periods at longer periods are even more accurately measured. The error of initial phase due to an error in the applied phase velocity is approximately 3~ = 2~rr3C/TC 2, where T is the period, r the propagation distance, and C the phase velocity. If we consider R2 observed at a minor arc distance of 90 ° and assume T, C, and 8C/C to be equal to 250 sec, 4.917 km/sec, and 0.3%, respectively, we have 8~ = 0.46 radian. The value of 0.3 per cent, however, is a representative standard deviation for one-circuit measurements. Actual variations in phase velocities along individual source-receiver paths may be larger than that value.

The systematic azimuthal variation of 0.5 radian or more causes a bias in the moment tensor inversion of shallow-focus earthquakes. The coefficients of the moment tensor elements Myz and Mxz in the imaginary part of the linear equation [equation (2) of Kanamori and Given, 1981; equation (1) of this paper] are very small at long periods (200 to 300 sec) for shallow events (Kanamori and Given, 1981).

2o63

2064 ICHIR0 NAKANISHI AND HIRO0 KANAMORI

Thus , t h e a z i m u t h a l v a r i a t i o n of t he in i t ia l phase (or t h e i m a g i n a r y p a r t of sur face-

w a v e spec t ra ) caused by t h e l a t e r a l h e t e r o g e n e i t y l eads to a b iased e s t i m a t e o f Myz

and Mxz. T h e s e e l e m e n t s are s o m e t i m e s c o n s t r a i n e d to be equa l to zero in t h e

invers ion . W e s h o w l a t e r tha t , w i t h t he se cons t ra in t s , t he l a t e r a l h e t e r o g e n e i t y o f

p h a s e v e l o c i t y causes a bias to low sca la r m o m e n t .

A n o t h e r p r o b l e m we discuss is an ef fec t of t h e f in i te d u r a t i o n of t h e source p rocess

on t h e m o m e n t t e n s o r invers ion . K a n a m o r i and G i v e n {1982) e s t i m a t e d t h e source

p roces s t i m e on t h e bas is of t h e su r f ace -wave m a g n i t u d e . F u r u m o t o and N a k a n i s h i

(1982), app ly ing a p h a s e ana lys i s o f l ong -pe r iod sur face waves , h a v e s h o w n t h a t t h e

se i smic m o m e n t is p r o p o r t i o n a l to t he cube of t h e source p rocess t i m e for l ow-ang le

TABLE 1

LIST OF 1980 EARTHQUAKES USED*

No. Date Time Latitude Longitude Depth Ms Region z ~ No. (m d) (h m s) (o) (o) (kin) (sec) (see) of

Obser- vations

1 1 1 16 42 40.0 38.815N 27.78OW 10 6.7 Azores 18 - - 1 2 1 2 20 58 44.2 5.984N 126.188E 63 - - Mindanao 30 21 2 3 2 7 10 49 16.0 54.158S 158.890E 10 6.5 MacquarieIs. 21 17 5 4 2 23 5 51 3.2 43.530N 146.753E 44 7.0 KurfleIs. 27 34 5 5 2 27 21 17 20.2 6.017S 150.189E 53 6.6 New Britain 28 4 2 6 3 8 22 12 10.3 22.673S 171.357E 38 6.7 Loyalty Is. 41 27 8 8 6 9 3 28 18.9 32.220N 114.985W 5 6.4 Cal-Mex Border 18 9 3

10 6 18 17 14 54.5 9.475N 126.657E 54 6.8 Mindanao 25 12 7 11 6 25 23 18 20.4 5.233S 151.686E 49 6.5 New Britain 49 - - 1 12 7 8 23 19 19.8 12.410S 166.381E 33 7.5 Santa Cruz'Is. 50 21 7 13 7 9 20 56 53.2 12.689S 166.004E 33 6.7 Santa Cruz Is. 36 20 8 14 7 14 16 15 1.7 29.273S 177.154W 49 6.6 Kermadec 32 10 3 15 7 17 19 42 23.2 12.525S 165.916E 33 7.9 Santa Cruz Is. 83 12 8 16 7 29 3 11 56.3 13.101S 166.338E 48 6.7 VanuatuIs. 14 19 6 17 7 29 14 58 40.8 29.598N 81.092E 18 6.5 Nepal 14 16 5 18 9 26 15 20 37.1 3.225S 142.237E 33 6.5 Papua 16 13 7 19 10 10 12 25 23.5 36.195N 1.354E 10 7.3 Algeria 30 21 6 20 10 24 3 25 34.4 21.989S 170.165E 33 6.7 Loyalty Is. 24 12 7 21 10 25 7 0 7.9 21.982S 170.025E 33 6.7 Loyalty Is. 34 12 4 22 10 25 11 0 5.1 21.890S 169.853E 33 7.2 Loyalty Is. 45 20 8 24 11 8 10 27 34.0 41.117N 124.253W 19 7.2 N. Calif. 34 23 6 25 11 11 10 36 58.2 51.422S 28.796E 10 6.7 S. Africa 17 8 4 26 11 23 18 34 53.8 40.914N 15.366E 10 6.9 Italy 41 16 10 27 12 17 16 21 58.8 49.479N 129.496W 10 6.8 Vancouver Is. 12 23 3 28 12 31 10 32 11.0 46.060N 151.453E 33 6.5 Kurile Is. 15 11 2

* NEIS origin times, locations, and surface wave magnitudes are used.

t h r u s t e a r t h q u a k e s a long d e e p sea t r enches . F u r u m o t o and N a k a n i s h i also h a v e

s h o w n t h a t t h e e m p i r i c a l r e l a t i on is no t as c lear for no rma l , s t r ike-s l ip , or d e e p

e a r t h q u a k e s , and t h a t s o m e e v e n t s dev i a t e cons ide r ab ly f r o m t h e gene ra l r e la t ion .

Thus , t he b l ind use of t h e e m p i r i c a l r e l a t i o n b e t w e e n the e a r t h q u a k e size and the

source p roces s t i m e m a y cause e r ro rs in t he m o m e n t t e n s o r invers ion . T h e ana lys i s

m e t h o d u s e d by F u r u m o t o and N a k a n i s h i (1982), w h i c h was sugges t ed by F u r u m o t o

(1979), does n o t n e e d t h e i n f o r m a t i o n a b o u t t he e a r t h q u a k e size for e s t i m a t i n g a source p rocess t ime. W e use t he m e t h o d to co r r ec t for t h e source p roces s t i m e in t he

m o m e n t t e n s o r invers ion .

K a n a m o r i and G i v e n (1982) d e t e r m i n e d m o m e n t t enso r s for 25 la rge sha l low

e a r t h q u a k e s in 1980 us ing the l inea r i nve r s ion m e t h o d desc r ibed in K a n a m o r i and

PROPAGATION AND SOURCE FINITENESS EFFECTS ON MOMENT TENSOR 2065

Given (1981). We adopt the same linear inversion method for long-period surface waves. We use path-dependent phase velocities calculated from regionalized phase velocities instead of the path-independent phase velocities used by Kanamori and Given to derive the initial phase at the source. We use the source process time measured directly by a phase analysis of Rayleigh waves instead of the magnitude- based source process time.

DATA

We used data from 25 large shal]ow earthquakes with surface-wave magnitude greater than about 6.5 which occurred in 1980. We performed the linear moment tensor inversion for the California-Mexico Border, the Vanuatu Islands, and the

9 0 °

6 0 °

3 0 °

N

0 o

S

3 0 °

6 0 °

G

90 OO 6 0 ° 120 ° E 180 ° W 120 ° 6 0 ° 0 °

FIG. 1. Great-circle paths connecting the California-Mexico Border earthquake of 9 June 1980 and the IDA stations. The solid and open circles represent the epicenter and the antipode, respectively. The solid triangles indicate the IDA stations. The regionalization is taken from Dziewonski and Steim (1982}. -, stable continent and shelf; +, tectonically active region; I , ocean floor younger than 38 m.y.; blank, ocean floor older than 38 m.y. The grid size is 5 ° × 5 °. This regionalization is based on the work of Mauk (1977}.

Santa Cruz Islands earthquakes, and we measured the source process times for all the earthquakes. The source parameters determined for the earthquakes by the National Earthquake Information Service (NEIS) are listed in Table 1. We used the vertical component of Rayleigh waves recorded on the IDA network (Agnew et al., 1976). For the correction for the instrument response, we used the transfer functions and the constants provided by the IDA project team at the Institute of Geophysics and Planetary Physics, University of California, San Diego.

We used the regionalized phase velocities derived by Dziewonski and Steim (1982). They performed a pure-path decomposition of great-circle phase velocities, group velocities, and Q determined through a waveform inversion of mantle Rayleigh waves recorded at the IDA stations using the simplified Mauk's (1977) regionalization. Dziewonski and Steim divide the Earth's surface into stable continents, tectonic

2066 ICHIR0 NAKANISHI AND HIR00 KANAMORI

r eg ions i n c l u d i n g i s l and arcs , o c e a n f loors y o u n g e r t h a n 38 m.y. , a n d o c e a n f loors o l d e r t h a n 38 m.y . F i g u r e 1 shows t h e i r r e g i o n a l i z a t i o n a d o p t e d in th i s s t udy . T h e a d o p t i o n o f r e g i o n a l i z e d p h a s e ve loc i t i e s d e t e r m i n e d b y a p u r e - p a t h d e c o m p o s i t i o n is b a s e d on a n a s s u m p t i o n t h a t a p u r e - p a t h m e t h o d is a good f i r s t - o r d e r a p p r o x i - m a t i o n for s t u d y i n g t h e m a n t l e s t r uc tu r e . N a k a n i s h i a n d A n d e r s o n (1982) h a v e s h o w n t h a t t h e w o r l d w i d e d i s t r i b u t i o n of g roup v e l o c i t y of m a n t l e R a y l e i g h w a v e s d e t e r m i n e d b y t h e s p h e r i c a l h a r m o n i c i nve r s ion is c o n s i s t e n t w i t h r e su l t s of p r e v i o u s p u r e - p a t h s tud ies , a l t h o u g h t h e y h a v e o b s e r v e d s o m e d e v i a t i o n s f r o m t h e p u r e - p a t h r e su l t s .

MOMENT TENSOR INVERSION USING REGIONALIZED PHASE VELOCITIES

W e a s s u m e t h a t a su r face w a v e p r o p a g a t e s a long a g r ea t - c i r c l e p a t h c o n n e c t i n g a sou rce a n d a rece ive r , a n d t h a t t h e a p p a r e n t p h a s e s lowness is e x p r e s s e d as a p a t h

TABLE 2 RESULTS OF LINEAR MOMENT TENSOR INVERSION USING LATERALLY HOMOGENEOUS OR

REGIONALIZED PHASE VELOCITY CALIFORNIA-MExICO BORDER EARTHQUAKE (EVENT 8)

Phase velocity Homogeneous Regionalized Period (sec) 256.0* 196.92 256.0 196.92 Depth (km) t 16.0 16.0 16.0 16.0 T (sec) 15o0 15.0 15.0 15.0 Mxy$ 0.78 _ 0.84 0.22 +_ 0.92 0.87 ± 0.63 -0.05 _ 0.74 Myy - Mxx~ 8.82 _ 1.72 5.37 ± 1.88 10.99 _ 1.28 8.29 _ 1.50 Myy + Mx~$ 1.31 ± 0.69 0.16 ± 0.79 1.16 _ 0.51 0.26 ± 0.63

Major double couple M0:~ 5.13 2.78 6.14 4.27 61 (deg)§ 90.0 90.0 90.0 90.0 }tl (deg)§ 180.0 180.0 180.0 180.0 q~l (deg)§ 140.0 137.4 139.5 134.7 82 (deg)§ 90.0 90.0 90.0 90.0 ~2 (deg)§ 0.0 0.0 0.0 0.0 q~2 (deg)§ -130.0 -132.6 -130.5 -135.3

Second double couple (%)¶ 25.6 5.8 18.9 6.1

rms error (cm.sec) 0.042 0.056 0.031 0.044

* The result for a period of 256.0 sec using the laterally homogeneous phase velocity is taken from Kanamori and Given (1982).

t A source depth of 16 km assumed in Kanamori and Given (1982) is adopted in this study. Unit of seismic moment is 1025 dyne. cm.

§ 8, dip; h, slip; q~, strike. ¶ Ratio of the moment of the minor double couple to that of the major double couple.

i n t e g r a l o f t h e p h a s e s lowness of t h e s u b r e g i o n a long t h e g r e a t - c i r c l e p a t h . T h e fo l lowing d e s c r i b e s t h e r e s u l t for e a c h e a r t h q u a k e .

California-Mexico border earthquake (event 8). F i g u r e 1 s h o w s t h e l o c a t i o n s of t h e e p i c e n t e r a n d t h e I D A s t a t i o n s on a m a p w i t h t h e r e g i o n a l i z a t i o n of D z i e w o n s k i a n d S t e i m (1982). A l so s h o w n in t h e f igure is t h e g r e a t - c i r c l e p a t h for e a c h s t a t ion . A l t h o u g h t h e s t a t i o n s t o t a l on ly 10, t h e a z i m u t h a l c o v e r a g e is r e a s o n a b l y good, as can be s een in F i g u r e 1. W e h a v e p e r f o r m e d t h e m o m e n t t e n s o r i n v e r s i o n o f R a y l e i g h w a v e s p e c t r a a t p e r i o d s o f 256 a n d 197 sec us ing b o t h t h e l a t e r a l l y h o m o g e n e o u s a n d t h e r e g i o n a l i z e d p h a s e ve loc i t ies . S ince t h e e a r t h q u a k e is l o c a t e d a t a sha l l ow

P R O P A G A T I O N A N D S O U R C E F I N I T E N E S S E F F E C T S O N M O M E N T T E N S O R 2067

d e p t h ( - 5 km) , t h e e l e m e n t s Myz a n d Mxz of t h e m o m e n t t e n s o r c a n n o t be

d e t e r m i n e d wel l f r o m l o n g - p e r i o d R a y l e i g h w a v e s ( K a n a m o r i a n d Given , 1981). T h u s , we c o n s t r a i n e d Myz a n d M=z to b e e q u a l to zero a n d d e t e r m i n e d Mxy, Myy - M ~ , a n d Myy + M~x. H e r e t h e source is a s s u m e d to b e f ree o f v o l u m e c h a n g e (Mx~ + Myy + M~z - 0). T h e s e c o n s t r a i n t h e s o l u t i o n to be e i t h e r p u r e s t r i ke s l ip on a

Lateral ly

I _(a) x

x ~ +3-

0

_, '

I I 0 60 120

¢- o

+2

o +l e-

Homogeneous C

x I I I x I

T = 2 5 6 . 0 sec x x -

x x X

x x x

x

J xESK, R 2 •

CxMO, R2 CMR O - xGAR, R 2 I I GIAR' R~ ~

4 ~ 0 0 0

•

-2

+1

C ~ 1

180 240 300 Azimuth, deg

Regionalized C

1 T - - T r 1

T= 2 5 6 . 0 sec o o 1 0 0

o b R2 OESK, R2 G~R,_C

1 I I J - I - ESK, R 3

o i ,60 0 60 120 180 240 300 :560

Azimuth, deg

£3 o.J

E (D

o~

E <

0.010

( c )

1/ ,

6O

I I I I

T = 2 5 6 . 0 sec x x

"~" ) x

I 120 180 240 300 360

Azimuth, deg

FIG. 2. Phase and amplitude spectra of the California-Mexico Border earthquake at a period of 256.0 sec as functions of azimuth. The correction for the source finiteness is made in the phase spectra by using a source process time of 15.0 sec. The source phase of ¢r/4 is subtracted from the phase spectra. Amplitude lobes are unequal in azimuthal extent, yet phase patterns show four equal quadrants. This is due to the minor double couple of 45 ° dip-slip fault, which has an initial phase of ir/4 independent of azimuth. (a) The crosses indicate the phases derived by using the laterally homogeneous phase velocity. The solid line is the phase calculated for the model determined by the inversion. (b) The open circles represent the phases derived by using the regionalized phase velocities of Dziewonski and Steim (1982). The solid line is the phase calculated for the model determined by the inversion. (c) The crosses indicate the amplitudes derived by using the laterally homogeneous group velocity and Q. The solid and dashed lines represent the amplitudes calculated for the models determined in the inversions by using the regionalized and laterally homogeneous phase velocities, respectively.

v e r t i c a l p l a n e or p u r e d ip s l ip on a p l a n e d i p p i n g 45 ° ( M e n d i g u r e n , 1977; K a n a m o r i a n d Given , 1981). A sou rce p r o c e s s t i m e of 15 sec u s e d in K a n a m o r i a n d G i v e n (1982) is a d o p t e d in o u r i n v e r s i o n in o r d e r to c o r r e c t for t h e n o n d i r e c t i o n a l p a r t of t h e p h a s e d e l a y due to t h e source p rocess . A l t h o u g h th i s v a l u e was e s t i m a t e d f rom M s , i t m a y be a p p r o p r i a t e in t h i s case, as wil l be s h o w n in t h e n e x t sec t ion . F i g u r e s 2, 3, a n d T a b l e 2 show t h e r e s u l t s o f t h e inve r s ion . T h e s o l u t i o n o b t a i n e d b y

2068 ICHIRO NAKANISHI AND HIROO KANAMORI

Kanamori and Given (1982) is used for the case of the laterally homogeneous phase velocity and a period of 256 sec. We determined three other solutions.

We note the following points from the comparison of the solutions. First, the lateral variations of phase velocity cause underestimates of the seismic moment. The source spectra for shallow earthquakes should be almost real (i.e., phase is equal to 0 or ~r) after the corrections for the instrument, polar phase shift, propagation effect, and a constant source phase of ~r/4. The observed phase, however,

+5 (a)

+ 4 - -

+ 3 -

c + 2 --

~ -I-I - -

0 c- O

La te ra l l y Homogeneous C Reg iona l i zed C

I I I [ x I I x T : 1 9 6 . 9 2 s e c

X X X

X X X X

m X ' m X

X X

(b)

0

I I I I T=196.92 sec o o o

O o O o o ~ 0 - - - I

l I io o

OGAR, R 2

oCMO, R2 CMO

b xESK, R3 cMO'R2x x xESK, R 2 GAR

GAR, Rg R,.,]

- 2 - - EoSK , R,] I I I I i I I I l

0 60 120 180 240 300 360 0 60 120 180 240 Azimuth, deg Azimuth, deg

~ I I I x I T=196.92 sec

X (uuU~.l x x x _ u~ X

Q; / X ,~, "

t < f

X

0"010 60 120 180 240 300 360 Azimuth, deg

OESK, R 2

C M O

GAR R3C

I 300 ~60

FIG. 3. Phase and amplitude spectra of the California-Mexico Border earthquake at a period of 196.92 sec as functions of azimuth. The correction for the source finiteness is made in the phase spectra by using a source process time of 15.0 sec. The source phase of ~r/4 is subtracted from the phase spectra. The symbol notation is the same as in Figure 2.

deviates significantly from 0 or ~, as Figures 2 and 3 show. If we let 30 be the deviation of the phase from 0 or ~r, the real part of the spectrum is proportional to cos 3~. Since the moment tensor components M~, Mxx, Myy are linearly related to the real part of the spectrum [e.g., equation (7) of Kanamori and Given, 1981], they are reduced by a factor of cos 3~. Thus, if we assume Myz = Mx~ -- 0, and amplitude perturbations are well-behaved in a statistical sense or are small, a nonzero 30 results in an underestimate of the seismic moment. The bias to low moment is


reduced by the application of the regionalized phase velocities. Second, the simple regionalization reduces the standard deviations of the moment tensor elements even at long periods, although the geometry of the source mechanism is not significantly changed. This reduction in the standard deviation is mainly due to the increase in the real part of the source spectrum (and corresponding decrease in the imaginary part) attained by the correction for the lateral heterogeneity. The component Myy - Mxx shows the largest change, as can be seen in Table 2. The change is 1.70o and 1.95o for 256 sec and for 197 sec, respectively, where o is the standard deviation of the component. Third, there still remains a systematic difference of the observed initial phases from those calculated for the models obtained by using the regionalized phase velocities. This difference in the phase spectra is a source of the bias to low

+ 2 ~ , , , | I (a) Laterally Homogeneous C I +" / x T=256 secl

I I- x GUA, R 2 T= 15 sec : I

L X xx , x GUA, rx> l 0 I ~ o x J xJ

I ~ - / ] I I ] I /

+2 , , , T : 2 5 6 sec (blo Regionalized C ' 3} CL + i - oGUA, R z T : 15 sec_J

GUA, R 0 oo:1 O o - O o : o

- - I I I ]

~, 0.5 (C) T:25 '6 sec: x v = 15 sec

c~ X X

/ \ ,,/ x\ o x A,R

t- \ t' Y ~O.l ' , I, ~_ E ×

F , ,% R,, , , 0 6 0 120 180 2 4 0 5 0 0 5 6 0

Azimuth, deg

FIG. 4. Phase and amplitude spectra of the Vanuatu Islands earthquake at a period of 256.0 see as functions of azimuth. The correction for the source fmiteness is made in the phase spectra by using a source process time of 15.0 sec. The source phase of ~r/4 is subtracted from the phase spectra. Data points for GUA are shown in the figure. The dashed and solid lines in (c) represent the amplitudes calculated for the models determined in the inversion by using the phase data in (a) and (b), respectively. The other symbol notations are the same as in Figure 2.

amplitude in the spectra calculated for the solutions obtained by the regionalized inversion. Finally, the effects of the lateral heterogeneity are stronger for 197 sec than for 256 sec. rms (8@) is 0.54 and 0.99 radian for 256 and 197 sec, respectively, after the heterogeneity correction, and the ratio of cos [rms (8@)] -- 0.64. This value is in good agreement with the ratio of Mo (0.69) between these periods (Table 2).

Vanuatu Islands earthquake (event 16). We inverted the Rayleigh wave spectra at a period of 256 sec. Since the earthquake is located at a depth of 48 km (Table 1), we have determined Mxy, Myy - Mxx, Myy + Mxx, Myz, and Mxz with the assumption Mxx + Myy + Mzz = O. Here we assume a depth of 43 km for the point source in order to compare our result with that of Kanamori and Given {1982). We assume a source process time of 15 sec used by them. We shall show later that this value is appropriate for this earthquake. Figure 4 and Table 3 show the result of the


inversion. The solution determined by Kanamori and Given (1982) is used here for the case of the laterally homogeneous phase ve]ocity.

A large oblique-slip component (hi -- 61 ° or h2 = 116 °) in the solution of the homogeneous inversion has almost vanished in that of the regionalized inversion (hl = 92 ° or )~2 = 88°). The components Myz and Mxz show large changes as compared with the other components of the solution, although the standard deviations are ]arge for My~ and M~z (Table 3). This result shows the sensitivity of My~ and Mx~ to the lateral heterogeneity. The moment suffers from the underestimate caused by the lateral heterogeneity, although it is small.

TABLE 3 RESULTS OF LINEAR MOMENT TENSOR INVERSION

USING LATERALLY HOMOGENEOUS OR REGIONALIZED PHASE VELOCITY

VANUATU ISLANDS EARTHQUAKE (EVENT 16)

Phase velocity Homogeneous* Regionalized Period (sec) 256.0 256.0 Depth (km)t 43.0 43.0 r (sec) 15.0 15.0 Mxy$ 0.58 _+ 0.18 0.66 ± 0.15 Myy - Mxx~ -1.67 +_ 0.57 -1.75 _ 0.46 Myy + Mx~ -1.69 ± 0.19 -1.75 ± 0.15 Myz$ -0.37 ± 0.78 0.16 ± 0.64 Mxz$ -0.57 + 0.72 -0.01 + 0.58

Major double couple M0~ 1.94 1.98 6, (deg)§ 45.6 47.4 ~1 (deg)§ 61.0 91.9 ~, (deg)§ 143.5 162.8 62 (deg)§ 51.4 42.6 ~2 (deg)§ 116.3 88.0 ~2 (deg)§ 1.9 -19.9

Second double couple (%)¶

rms error (cm.sec)

3.6 11.3

0.076 0.062

* The result for the laterally homogeneous phase velocity is taken from Kanamori and Given (1982}.

t A source depth of 43 km assumed in Kanamori and Given (1982) is adopted in this study.

Unit of seismic moment is 1026 dyne. cm. § 8, dip; )~, slip; ~, strike. ¶ Ratio of the moment of the minor double couple to

that of the major double couple.

Santa Cruz Islands earthquake (event 13). T h i s e a r t h q u a k e is loca ted ve ry close to the V a n u a t u I s l ands e a r t h q u a k e ana lyzed above (Tab le 1). T h e seismic m o m e n t s d e t e r m i n e d by K a n a m o r i a n d G i v e n (1982) are a l m o s t the s ame (Vanua tu , 0.19 × 1027 d y n e . c m ; S a n t a Cruz, 0.17 × 1027 d y n e . c m ) . T h e i nve r s ion for t he V a n u a t u

I s l ands e a r t h q u a k e sugges ted t h a t a source process t ime of a b o u t 15 sec e s t i m a t e d f rom Ms m i g h t be app rop r i a t e for (he S a n t a Cruz I s l and ea r thquake . We a d o p t e d a source process t ime of 12 sec a n d a source d e p t h of 43 k m used in K a n a m o r i a n d G iven (1982) in order to see a n y change a t t a i n e d by cor rec t ing for the l a te ra l he te rogene i ty . We d e t e r m i n e d Mxy, Myy - Mxx, Myy + Mx~, Myz, and Mxz wi th the


a s s u m p t i o n M= + Myy + Mzz = 0. F igure 5 a, b, a n d d) a n d T a b l e 4 show the resu l t s

of the invers ion . T h e so lu t ion of K a n a m o r i a n d G i v e n (1982) is used for the case of

the l a t e ra l ly h o m o g e n e o u s phase veloci ty. T h e r e d u c t i o n of the s t a n d a r d dev ia t ions of the so lu t ion is small , as T a b l e 4

shows, a n d the obse rved phase spec t ra exh ib i t a sys t ema t i c de lay w i th respec t to

+21 (a)' +i I SUR' R2

(b)'

*~/SURo , R,

Late'ral ly Homog'eneou's C I ~U~, R 2 SUR, R 3 T=256 sec/

x T= 12 sec -]

X ~ X × X X x X

~ l ~ i x l I

T=256 sec = Reg iona l ized C a GU~ R 2 r= 12 sec J

R z SUR o, R 3 !

-I

+2 (C) I Regionalized C I T=2516 sec I GU 'e R "r = 36 sac /

+l-SUR, R 2 " , , SU i, R, GUA, R,I

Th ~ / , , • / 0 " i , , , • V, . .~..-"---~

, "I -~. I I I I

: (d)×' ' ' o ~ 0.5 - T=256 sec

F: FSUR, R 2 x SUR, R 3 x

o.r I# i x

do Jzo ,8o " ' s6o Azimuth , deg

FIG. 5. Phase and amplitude spectra of the Santa Cruz Islands earthquake (event 13) at a period of 256.0 sec as functions of azimuth. The correction for the source fmiteness is made in the phase spectra of (a), (b), and (c) by using a source process time of 12.0, 12.0, and 36.0 sec, respectively. The source phase of 9/4 is subtracted from the phase spectra. Data points for SUR and GUA are shown in the figure. (a) The crosses indicate the phases derived by using the laterally homogeneous phase velocity. The solid line is the phase calculated for the model determined by the inversion. (b) The open circles represent the phases derived by using the regionalized phase velocities of Dziewonski and Steim {1982) and the source process time of 12.0 sec. The solid line is the phase calculated for.the model determined by the inversion. (c) The solid circles indicate the phase derived by using the regionalized phase velocities and the source process time of 36.0 sec. The solid line is the phase calculated for the solution. The arrows attached to the solid circles for SUR indicate the reduction of the phase advance attained by using • =-46 sec for SUR in the inversion. (d) The crosses indicate the amplitudes derived by using the laterally homogeneous group velocity and Q. The dashed, dotted, and solid fines represent the amplitudes calculated for the models determined in the inversions by using the phase data in (a), (b), and (c), respectively.

those ca l cu la t ed for the so lu t i on except for s t a t ions S U R a n d G U A (Figure 5b). S ince th is e a r t h q u a k e is loca ted ve ry close to the V a n u a t u I s l ands e a r t hqua ke , it is diff icult to a t t r i b u t e th is de lay to i n a p p r o p r i a t e r eg iona l i za t ion of the Ea r th . A n or igin t ime er ror could exp la in the sy s t ema t i c pha se delay, b u t th i s r equ i r e s a n e r ro r of a b o u t 30 sec. W e of ten exper ience mi s loca t i on of e a r t h q u a k e s in i s l and arc regions.

2 0 7 2 ICHIRO NAKANISHI AND IIIROO KANAMORI

T h e e p i c e n t r a l m i s l o c a t i o n e r r o r c ause s a p h a s e d i f f e rence of t h e f o r m 60 = ~ S L cos O/C, w h e r e w is t h e a n g u l a r f r equency , 3 L t h e d i s t a n c e b e t w e e n t h e m i s l o c a t e d a n d t r u e ep i cen t e r , O t h e a z i m u t h of a s t a t i o n m e a s u r e d f rom t h e l ine c o n n e c t i n g t h e m i s l o c a t e d e p i c e n t e r to t h e t r u e ep icen te r , a n d C t h e p h a s e v e l o c i t y {Pa t ton a n d Aki , 1979). T h e e p i c e n t r a l m i s loca t i on , howeve r , does n o t e x p l a i n t h e o b s e r v e d a l m o s t c o n s t a n t p h a s e d e l a y (F igu re 5b). T h e a b o v e d i s cus s ion sugges t s t h a t a n a n o m a l o u s l y long source p r o c e s s is t h e on ly p o s s i b i l i t y for t h e e x p l a n a t i o n o f t h e s y s t e m a t i c p h a s e de lay . T h i s p o s s i b i l i t y is e x a m i n e d in t h e fo l lowing sec t ions .

TABLE 4 RESULTS OF LINEAR MOMENT TENSOR INVERSION USING LATERALLY

HOMOGENEOUS OR REGIONALIZED PHASE VELOCITY SANTA CRUZ ISLANDS EARTHQUAKE (EVENT 13)

Phase velocity Homogeneous* Regionalized Regionalized Period (sec) 256.0 256.0 256.0 Depth (km)t 43.0 43.0 43.0 z (sec) 12.0 12.0 36.0 Mxy$ 0.54 ± 0.19 0.53 _ 0.17 0.57 _ 0.16 Myy - Mxx:~ -1.42 +_ 0.51 -1.59 _+ 0.46 -1.79 _+ 0.41 Myy + Mx~; -1,62 _ 0.18 -1.67 + 0.17 -1.78 _ 0.15 Myz$ -0.02 _+ 0.71 0.16 ± 0.64 0.30 ± 0.57 Mxz$ -0.14 _ 0.75 0.53 ± 0.69 0.15 ± 0.61

Major double couple Mo:~ 1.70 1.84 1.97 ~1 (deg)§ 44.8 47.8 41.7 },1 (deg)§ 97.0 65.5 78.4 ~1 (deg)§ -13.8 -34.0 -24.1 32 (deg)§ 45.6 47.6 49.3 X2 (deg)§ 83.1 114.5 100.2 62 (deg)§ 156.4 180.1 171.3

Second double couple (%)¶

rms error (cm.sec)

4.08 2.79 7.15

0.092 0.083 0.074

* The result for the laterally homogeneous phase velocity is taken from Kanamori and Given (1982).

t A source depth of 43 km assumed in Kanamori and Given (1982) is adopted in this study.

Unit of seismic moment is 102~ dyne. cm. § 5, dip; 2,, slip; q~, strike. ¶ Ratio of the moment of the minor double couple to that of the

major double couple.

MEASUREMENT OF SOURCE PROCESS TIME

Method. W e m e a s u r e t h e source p r o c e s s t i m e b y a p h a s e a n a l y s i s of l o n g - p e r i o d R a y l e i g h w a v e s ( F u r u m o t o , 1979; F u r u m o t o a n d N a k a n i s h i , 1982) in o r d e r to e s t i m a t e t h e p h a s e sh i f t due to t h e f in i te sou rce p rocess .

T h e F o u r i e r t r a n s f o r m of t h e v e r t i c a l c o m p o n e n t of R a y l e i g h w a v e s a t p o i n t P (r, 8, q~) due to a s t e p - f u n c t i o n p o i n t sou rce is e x p r e s s e d as fo l lows in t e r m s o f a f r e q u e n c y - i n d e p e n d e n t m o m e n t t e n s o r ( Mxx, Myy, Mzz, Mxz, Myz, Mxz) ( K a n a m o r i a n d Given , 1981),

PROPAGATION AND SOURCE F I N I T E N E S S E F F E C T S ON MOMENT TENSOR 2073

CR(CO,~)- (sin~-~" exp t~- • exp - - i - - • exp t--~-

X (-PR(~)[Mxysin 2~ - ½(Myy - M~x)eOs 2~k]

1 (2NR(1) _ SR(1))(Mxx + Myy) 1 (SR(1) 4" NRO))Mzz + -6

+ iQR(~)(Myzsin ~ + M=cos 0)}, (1)

where PR (1), SR (1), NR (~), and QR (1) are the excitation functions given by Kanamori and Stewart (1976) and Kanamori and Given (1981), ¢o is the angular frequency, R the radius of the Earth, A the propagation distance, and m the number of the polar passages. For large events, the corrections for the source fmiteness are necessary in the spectrum (1). The finiteness is approximated by a function defined by Ben- Menahem (1961), for unilateral faulting,

sin X F(~, O) - exp(-iX) (2)

X

and

coL oJL X = - - cos O, (3)

2V 2C

where L is the fault length, V the rupture velocity, C the phase velocity, and O [not to be confused with the colatitude 8 in equation (1)] the azimuth of the station measured from the rupture direction. Fukao and Furumoto (1975) reported a delay of the main faulting from the initial break for the Tokachi-Oki earthquake of 16 May 1968. The effect of the delay is expressed as

/) (o~) = exp(--io~TD), (4)

where TD is the delay time of the main faulting. If there is an error in the origin time, the error may be included in TD. Thus, the observed spectrum of the vertical component of Rayleigh waves is generally approximated as follows,

? . ( ~ , ¢~, o ) = ¢ . ( ~ , ¢)) • P ( ~ , o ) • / 5 ( ~ ) • i ( ~ ) , (5)

where _T(o~) is the instrument response. Here we use the phase of the spectrum (5). We consider the case that we have

spectra of R2n+l, R2m, and R2~+a from an earthquake observed at a station. As equation (1) shows, the geometrical part of the spectrum [( } in equation (1)] of R2m is the complex conjugate of that of R2n+l. Therefore, the phase spectra of R2~+1 and R2m are expressed by

and

toRA2n+I 2n~r (I}2n+l = 4 C2n+l e T + (I)~- X(O) - 0~VD -- eP~t (6)

7r o~Rh2m 2 m - 1 (I)2~ ----

4 C2m 2 - - ~ - ¢ ~ - X ( O + ~r) - ¢ 0 ~ n - - ( I ) ~ t , ( 7 )


where O~ is the phase of ( } in equation (1) for R2n+l, and (I)~t is the phase delay due to the seismograph. From equations (6) and (7) we have

~r 752~+1 52m\ - ~ R / ~ + - - . + \

+ ( n + m - (8)

Here we define a sum of the duration of the main faulting (L/V) and the delay time of the faulting (~n) as ~,

L r = ~ + 2~n. (9)

This is the source time defined by Furumoto (1979). In this paper, we call • the source process time instead of the source time. As equations (6), (7), and (9) show, the nondirectional part of the phase delay due to the source process is

~L ~0~ - - + ~0~'D -- (10) 2V 2

We determine the source process time T and the phase delay ~v/2 by a phase analysis of R2n+l, R2m, and R2~+~ in the following way (Furumoto and Nakanishi, 1982). We write the phase delay due to the wave propagation in equation (8) as

\C2n+~ C2~] ~°R[--~v + + 2~R¢

= ( n + m ) - - , (11) Ca

where 1/CG and 1/CR~ is the average phase slowness for the great circle and the minor arc, respectively. 2~rRw/CG in equation (11) is obtained from the phase difference between R2~+1 and R2n+3,

2~rR 02n+1 - 02n+3 = ~ o - - ~r. (12)

Ca

From equations (8), (9), (11), and (12) we have

02n+1 "~ (ID2m "4- (n + m)((I)2n+l -- 02n+s) + 20i~t = --~0~'. (13)

We obtain the following equation for R2n, R2m+l, and R2n+2 in the same way

02n + 02.~+1 + (n + m ) ( O 2 n - - 02n+2) + 2(I)inst ---- --~'r. (14)

Thus, we can determine • and wT/2 by using equation (13) or (14), and then we can correct for the nondirectional part of the phase delay due to the finite source process exactly.


For the horizontal components of R and G waves, we have

O2~+1 + O2m + (n + m ) ( O 2 n + l -- O2n+3) "4- 2Oinst -t- ~ ---- --~OT (15)

for

(R2n+l , R2m, R2n+3) o r (V2n+l , V2m, V2n+3),

and

O2n + O2m+l + (n + m)(O2n - O2n+2) "~- 2Oinst "~- ~r = --a~T (16)

for

(R2n, R2m+I, R2n+2) or (V2n, V2rn+l, V2n+2).

We use equations (13) and (14) in the measurement of T. Equations (13), (14), (15), and (16) have an ambiguity of 2N~, where N is an arbitrary integer. This means that there is an ambiguity of N times the period in T. Results of Furumoto and Nakanishi (1982) show that earthquakes with M0 less than 102s dyne .cm have T shorter than 100 sec. We analyze the earthquakes whose seismic moments are less than 102s dyne. cm at periods from 200 to 300 sec. Therefore, the ambiguity in the phase spectrum does not cause problems in our study.

Measurement. We measured the source process times for 25 large shallow earthquakes in 1980 using the method described above. Furumoto and Nakanishi (1982) measured T for great and large earthquakes with M0 greater than 1027 dyne. cm using mainly data from the WWSSN network. We extend the moment range down to 5 × 1025 dyne. cm by using the digital data from the IDA network.

As equations (13) and (14) show, we need no corrections or assumptions except for the instrument response and assumptions in deriving these equations (e.g., unilateral faulting and great-circle propagation of surface waves) in determining T. We should obtain almost the same value of T for every station and every period if the surface waves propagate along the great-circle path connecting the source and the receiver, if the apparent slowness is expressed as an average of the phase slowness of the subregion along the great-circle path [equation (11)], and if the correction for the instrument is correct. We calculated T in the period range from 200 to 300 sec for each earthquake-station pair. Since the calculation requires a trio of Rayleigh waves, the number of stations where T can be measured is limited compared to those which can be used in the moment tensor inversion, especially for relatively small earthquakes. Figure 6 shows ¢ measured for the Santa Cruz Islands earthquake of 17 July (event 15), the largest earthquake in our data set. As the figure shows, T scatters considerably except at periods around 260 sec. Our experience in the moment tensor inversion of long-period Rayleigh waves also suggests that we obtain the most stable results from the spectra at those periods. We use the average of T at a period of 256 sec. Table 1 lists the average r measured for the 25 earthquakes, the standard deviations AT, and the numbers of observations.

Figure 7 shows ¢ of three earthquakes, for which we have performed the moment tensor inversion using the regionalized phase velocities. Figure 7, a and b, show that average r of the California-Mexico Border and the Vanuatu Islands earthquakes are

2076 ICI-IIRO NAKANISttI AND }tIROO KANAMORI

close to the source process t ime of 15 sec, which is a ssumed in the m o m e n t tensor inversion for the two ear thquakes . On the o ther hand, the San ta Cruz Is lands ea r thquake (event 13) (Figure 7c) has an average ~ of 36 sec three t imes longer t han the source process t ime of 12 sec adopted in the m o m e n t tensor inversion for this ear thquake.

REDETERMINATION OF MOMENT TENSOR FOR THE SANTA CRUZ ISLANDS EARTHQUAKE (EVENT 13)

T h e m e a s u r e m e n t of T in the above section suggests t ha t the source process t ime of 12 sec is not appropr ia te for the San ta Cruz Is lands ea r thquake (event 13). Thus, we invert the Rayle igh wave spect ra of this ea r thquake again using the source process t ime of 36 sec de te rmined above. Figure 5 (c and d) and Tab le 4 give the result of the inversion. As Figure 5c shows, the sys temat ic phase delay has a lmos t disappeared. T h e reduct ion of the d iscrepancy be tween the observed and the calculated phase spect ra increases the scalar m o m e n t of the solution. M0 of the th i rd solution (= 1.97 x 10 ~° dyne . cm) (Table 4) is a lmos t the same as tha t of the second solution of the Vanua tu Islands ea r thquake (= 1.98 x 102° dyne . cm) (Table 3).

250 . . . . . I . . . . . Santa Cruz Is. T = 8 5 s e c l a tT=256sec

200 / . - . . . . i -TWO/RZ,R3,R4r / ~ ', FNNA(RS,R4,RS)

150 / ~ i SEY(R2,R3,R4)- " ~ ""-. ', . , /ONO(R2,P,5,R4)

~Loo ,~.-... ~ ?ESK(R2,R3,R4r • fEIO(R2,R3,R4)

50 ~ i -~---~-GA R(R 2, R3,R4)- l

} I I I I i I I ~ { I 210 220 230 240 250 260 270 2 0 290

Per iod, sec

FIO. 6. ¢ calculated for the Santa Cruz Islands earthquake (event 15). The solid circles indicate the period points at which the source process times are calculated. Trio of Rayleigh waves used in the calculation is shown in parentheses in the figure. ~ is an average of ¢ at a period of 256.0 sec.

T h e phase spec t ra observed a t S U R and GUA show large phase advances (Figure 5c). We measured ¢ f rom the Rayle igh waves recorded at SUR. T is negat ive in the period range f rom 200 to 300 sec. • = - 4 6 sec a t a per iod of 256 sec. Str ict ly T = ( -46 + 256 N) sec, where N is an a rb i t ra ry integer. T h e negat ive T is consis tent wi th the phase advance in the observed spectra . We adop ted the source process t ime of 36 sec for all the stations. I f we use T = - 4 6 sec for S U R in the inversion, we obtain the reduct ion of the phase advance by o~(36 + 46)sec/2 = 1.0 radian for SUR. This value is a lmost the same as the differences of the observed phase spec t ra be tween S U R and the other normal stations, as indicated by arrows in Figure 5c. Therefore , the phase advances for S U R m a y suggest t h a t the clock is slow at the station, if we assume N = 0. Since GUA is s i tuated nea r the t rough of the ampl i tude spect ra (Figure 5d), only R2 and R8 are seen on the seismogram. R1 is dis tor ted due to the shor t epicentral distance (h = 33°). Thus , we cannot measure T for GUA. T h e phase spectra f rom the Vanua tu Is lands ea r thquake also show the phase advances for GUA (Figure 4, a and b). We de te rmined a four th solution excluding the da ta f rom S U R and GUA, bu t the solution is a lmost the same as the th i rd solution.

The above discussion suggests t h a t ¢ of equat ion (9) should be generalized to


include a clock error for an anomalous station

• ' = T + 2~c, (17)

where Tc is a lead or a lag of the clock. We can correct for the clock error by using T' as a special source process t ime for the station. Therefore, we do not need to exclude the station from the m o m e n t tensor inversion. Th i s procedure m a y be useful when we use data f rom a sparse network, such as the I D A network.

DISCUSSION

Applicability of regionalized phase velocities. The simple regionalized phase velocities of Dziewonski and Steim (1982) reduce the s tandard deviat ions of the

220 240 260 28O + l o o ( a ) ' ' ' - ' ; ' ' '

/ CaI-MexBorder T=18sec ? IT=256sec 1

+SOITwO(R1,R2,R3) . . . . - - __ i 1 I /

+100 (b) Vanuatu Is. ~=14sec ot =256sec

+50 0M0(R2,R3,R4)-.... " - ~ • II

CMO(R 1, R2,R3)-...... " NNA<R1,R2,R3)----,- ~ . ~ ~ -

0 NNA(R2,R3,R4)~S ~ - _ ~ _

P F O ( R 2 , R S , R 4 ) J ~

-50 -BDF(R2 R3 R4) j , , , , , ,

i i L , ' , I ,

(C) Santa Cruz Is. ~=56sec atT=256sec +100 GAR(RI,R2,R3~ !

CMO(RI,R2,R3)\\ ', +50 - S E Y ( R i , R 2 , R ~ ) ) ~ . ~ ~ ~

ESK(R2,R3,R4)~"~-~_ I ~ ~ ~ P F O ( R 2 , R 3 , R 4 ) _ _ ~ _ . , j ~ i ~ i

0 NNA(R2,R3,R4)~ r - , BDF(R2,R3,R4) I /

-50 PFO(RI,R2,R3Y, , , ~ , , 220 250 240 250 260 270 280

Period, sec Fro. 7. ¢ calculated for three earthquakes whose moment tensor inversions are made in this study.

The symbol notation is the same as in Figure 6. (a) California-Mexico Border earthquake. (b) Vanuatu Islands earthquake. (c) Santa Cruz Islands earthquake (event 13).

solution and the bias to low moment in the linear moment tensor inversion of long- period Rayleigh waves. Problems, however, remain for the correction for the lateral heterogeneity. The observed initial phases of the California-Mexico Border earthquake show a systematic phase delay from the calculated initial phases for stations CMO and GAR at 256 sec {Figure 2b). As Figure 1 shows, the surface waves from the California-Mexico Border earthquake observed at these stations have long paths over the tectonically active regions, such as the Southeast Pacific, western North America, and the young area in the Indian Ocean. Therfore, the systematic phase delay may suggest that the Rayleigh wave phase velocities in these regions are slower than the pure-path phase velocities for the young ocean and the tectonic region derived by Dziewonski and Steim (1982). The long paths over the low velocity

2078 ICHIR0 NAKANISI-II AND I-IIR00 KANAMORI

in the Southeast Pacific may cause a systematic deviation of the observed initial phases, and eventually lead to a bias to low moment. We cannot avoid this bias by improving azimuthal coverage of stations, for instance using the data from the SRO network.

For station ESK, the correction makes the fit worse (Figure 2b). This deterioration may be an artifact caused by the simple regionalization used in this study. The apparent phase velocities calculated for Re and R3 at ESK from the regionalized phase velocities are 0.0103 and 0.0091 km/sec higher than the value (4.9622 km/sec) used as the laterally homogeneous phase velocity. Figure 1 shows that the path fraction of the young ocean or the tectonic region is very small along the great-circle path connecting the earthquake and ESK, but the ray path runs very close to the young oceans of the Southeast Pacific and the Indian Ocean to the south of Australia. Lateral variations of mantle structure in the oceanic region, for instance, the thickness of the lithosphere, may be considered to be gradual. Therefore, the phase velocities for Re and R3 observed at ESK may be much slower than those

600 _ L I I I ] I I I [ 1 1 1 1 ] I I I l l l i ] j I ~ ~ l ~ r , , ] ~ I I ] l l l v I , , j i - B o Thrust Fault ~ . . I - ' f o - - x Normal Faulf ~ - - '~ Strike Slip A ~ _

~, 100 ~ - " Deep Event o ~ - ° ' ' 9 " ~ ° / -

p - - o ~ G'~o M0= 2.5 x l O ' ~ , / - x 0 x @ O@ 0 _ ~ % ~ e x z~ • (Furumoto Ond

-0~.. 0''~ ~ x 0 - Nokonishi, 1982) S- x

JO JJJ l l i j J f l J J J [ J J J l l J l l l I I I I [ l l l l J I I J J J l l l J 10 26 10 27 1028 1029 10 30

MO, dyne. cm

FIG. 8. Relation between M0 and ¢ obtained by Furumoto and Nakanishi (1982) and this study. The open circle, cross, open triangle, and solid circle indicate the thrust fault, normal fault, strike-slip fault, and deep event, respectively. The solid line is an empirical relation between M0 and ¢ obtained for the low-angle thrust earthquakes by Furumoto and Nakanishi (1982). The results for events 1 and 11 are not shown in the figure because only one estimate of • was available for each event (Table 1). M0 determined by Kanamori and Given (1982) is used for the data points of mobtained in this study.

predicted from the regionalized phase velocities of Dziewonski and Steim. Further- more, the propagation of long-period surface waves is influenced by not only the medium properties just beneath the ray path but also those of the regions adjacent to the ray path (Woodhouse and Girnius, 1982). Therefore, if we adopt ray theory in treating the propagation of long-period surface waves in the linear moment tensor inversion, we should either divide the Earth's surface into a large number of regions or use another representation for the distribution of phase velocity on the Earth's surface, such as a spherical harmonic expansion, in order to make the lateral variations of phase velocity gradual, and reduce the artifacts caused by a rough regionalization.

Source process time in the moment tensor inversion. The Santa Cruz Islands earthquake (event 13) has a much slower source process than the Vanuatu Islands earthquake, although they have almost the same seismic moment. This suggests a limitation in the use of an empirical relation between the earthquake size (magnitude or seismic moment) and the source process time for the correction for the finite source process in the moment tensor inversion. Figure 8 shows a relation between the seismic moment and the source process time obtained by Furumoto and


Nakanishi (1982) and this study. The figure exhibits a linear relation between log M0 and log v with a gradient of } (Furumoto and Nakanishi, 1982). This figure, however, also shows that the source process time varies significantly from earthquake to earthquake even for earthquakes with the same seismic moment. Therefore, a more accurate estimate of the phase shift due to the finite source process should be made before the moment tensor inversion. The phase analysis used in this paper presents a method for that purpose.

Finally, we note two problems in the use of r in the moment tensor inversion. First, T represents the nondirectional part of the apparent duration of the source process. The apparent source process time is

L L TA = V -- C c o s O -~- 20"D

L = ~" - ~ cos O. (18)

The directional part of the phase shift is

~L ~¢, = ~ cos O. (19)

If we take T(period) -- 256 sec, C = 4.9622 km/sec, and L = 100 km, we have 8~ -- 0.25 cos 0 radian. As Figures 4b and 5c show, this phase shift is comparable to the differences of the observed phases from those calculated for the solution obtained from the regionalized inversion. Thus, ~ may dominate the observed initial phases, if the fault length is much longer than 100 km. For such a large earthquake, the correction by using T is not sufficient to determine an accurate moment tensor solution. Kanamori and Given {1981) present a method for that case. Second, our interpretation of T assumes the unilateral propagation of rupture. This assumption may be an oversimplification. However, it is difficult to determine details of the mode of the rupture propagation from the analysis of surface-wave data. We assume the unilateral fault in this paper.

CONCLUSIONS

The simple regionalized phase velocities {stable continent, tectonic region, young ocean, and old ocean) improve the linear inversion of mantle Rayleigh waves. However, there still remains a systematic deviation of the observed phases from those calculated for the solution. A more detailed but smooth phase velocity representation is desirable in order to correct for the propagation effect in the inversion.

Some earthquakes deviate considerably from the general relation between the source process time and the seismic moment. We suggest measuring the source process time before the moment tensor inversion for the purpose of correcting for the nondirectional part of the phase shift due to the finite source process.

The phase errors due to the lateral heterogeneity and source process time may lead to systematic underestimation of the scalar moment in the linear moment tensor inversion.

ACKNOWLEDGMENTS

The IDA data used in this study were made available to us by courtesy of the IDA project team at the Institute of Geophysics and Planetary Physics, University of California, San Diego. We would like to thank Jeff Given for reviewing the manuscript. We thank Adam Dziewonski and Joseph Steim for a


preprint of their paper. We appreciate suggestions made by Howard Pat ton as a reviewer of the manuscript. We thank Laszlo Lenches for drafting the figures. We thank Lynn Finger for typing the manuscript. This research was supported by the U.S. Geological Survey Contracts 14-08-0001-19755 and 14-08-0001-19265 and the National Science Foundation Grant EAR-811-6023.

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SEISMOLOGICAL LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 CONTRIBUTION NO. 3759

Manuscript received 14 April 1982

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Bulletin of the Seismological Society of America, Vol. 72, No. 6, … · 2017-02-27 · Bulletin of the Seismological Society of America, Vol. 72, No. 6, pp. 2063-2080, December 1982