source inversion of the 1988 upland, california,

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Bul l e t i n o f t he

Se i smo l o g i c a l So c i e t y o f A me r i c a

Vol. 80 June 1990 No. 3

SOURCE INVERSION OF THE 1988 UPLAND, CALIFORNIA,

EARTHQUAKE: DETERMINATION OF A FAULT PLANE FOR A

SMALL EVENT

BY JIM MORI AND STEPHEN HARTZELL

ABSTRACT

W e e x a m i n e d s h o r t -p e r i o d P w a v e s t o in v e s t i g a t e if w a v e f o r m d a t a c o u l d b e

u s e d t o d e t e r m i n e w h i c h o f t w o n o d a l p l a n e s w a s t h e a c t u a l f a u l t p l a n e f o r a

s m a l l (ML 4 . 6 ) e a r t h q u a k e n e a r U p l a n d , C a l i fo r n i a . W e r e m o v e d p a t h a n d s it e

c o m p l i c a t io n s b y c h o o s i n g a s m a l l a f t e r s h o c k (ML 2 . 7 ) a s a n e m p i r i c a l G r e e n

f u n ct io n . T h e m a in s h o c k P w a v e s w e r e d e c o n v o l v e d b y u s i n g t h e e m p i r ic a l

G r e e n f u n c t i o n t o p r o d u c e s i m p l e f a r - f i e l d d i s p l a c e m e n t p u l s e s . W e u s e d a l e a s t -

s q u a r e s m e t h o d t o i n v e r t t h e s e p u l s e s f o r t h e s l i p d is t r ib u t i o n o n a f i n i te f a u lt .

Bo th n o d a l p la n e s (s t r ik e 1 2 5 ° , d ip 8 5 ° a n d s t r ik e 2 2 1 ° , d ip 4 0 ° ) o f th e f i r s t -

m o t i o n fo c a l m e c h a n i s m w e r e t e s t e d a t v a r io u s r u p t u r e v e l o c i t ie s . T h e s o u t h w e s t

t r e n d i n g f a u l t p l a n e c o n s i s t e n t l y g a v e b e t t e r f it ti n g s o l u t i o n s t h a n t h e s o u t h e a s t -

t r e n d i n g p l a n e , W e d e t e r m i n e d a m o m e n t o f 4 . 2 x 1 0 22 d y n e - c m . T h e r u p t u r e

v e l o c it y , a n d t h u s t h e s o u r c e a r e a c o u l d n o t b e w e l l r e s o l v e d , b u t if w e a s s u m e

a r e a s o n a b l e r u p t u r e v e l o c i t y o f 0 . 8 7 t i m e s t h e s h e a r w a v e v e l o c i t y , w e o b t a i n as o u r c e a r e a o f 0 . 9 7 k m 2 a n d a s t r e s s d r o p o f 3 8 b a r s . C h o i c e o f a s o u t h w e s t -

t r e n d i n g f a u l t p l a n e i s c o n s i s t e n t w i t h t h e t re n d o f th e n e a r b y p o r ti o n o f t h e

T r a n s v e r s e R a n g e s f r o n t a l f a u l t z o n e a n d i n d i c a t e s l e f t - l a t e r a l m o t i o n . T h i s

m e t h o d p r o v i d e s a w a y t o d e t e r m i n e t h e fa u l t p l a n e f o r s m a l l e a r t h q u a k e s t h a t

h a v e n o s u r f a c e r u p t u r e a n d n o o b v i o u s t re n d i n a f t e r s h o c k l o c a t io n s .

INTRODUCTION

A small ea rthqu ake (ML 4.6) occur red near the town of Upland, California, on 26

June 1988. It was located (34 ° 8.09' × 117 ° 42.58') close to th e fronta l fau lt zone

of the Tran sver se Ranges at a depth of 7.9 km. Th is is an area o f relatively low seis-

micity, and little is known about previous historical fault movements (Cramer

and Harrington, 1987; Morton and Matti, 1987; Pechmann, 1987). The location

and well-constrained focal mechanism, obtained from 124 network first motions

(L. Jones, personal comm.), are shown in Figure 1. The routine Southern California

Network locations of about 50 aftershocks cluster in a volume with dimensions of

a few kilometers and, as is typical for events of this size, do not show an obvious

trend that can be identified as the fault plane. We have used waveform data from

the Sout her n California netwo rk to try to resolve which of the two nodal planes

was the fault plane for the earthquake.

The short-period (1 to 8 hz) seismograms for the distances used in this study

(42 to 85 km) are quite complicated; consequently, it is difficult to extract infor-matio n about th e source. One solution to this proble m is to use a smaller earthqu ake

from the same location as an empirical Green function. Deconvolution of the

seismograms of the main event, using the empirical Gr een functions, should elimi-

nate all of the path, site, and i nstru ment effects (Frankel e t a l . , 1986; Li and

Thurber, 1988; Mori and Frankel, 1990). We have used this method to obtain

5o7



508 J. MORI AND S. HARTZELL

54 ° -

l i b o 1 1 7 °

FIG. 1. Focal mechanis m (lower hemisphere) and location of the 1988 Upla nd earthquake. Trianglesrepresent stations which recorded data used in this study.

relatively simple far-field P-wave displacement pulses. We then inverted this data

for slip on a fault plane to test which of the two nodal planes bett er fits the data.

DATA AND EMPIRICAL GREEN FUNCTION DECONVOLUTION

Dat a used in this st udy are from thre e stations (LJB, RAY, SIL) of the Sout hern

California Network and from the Pasadena (PAS) Very Broad Band instrument

(Fig. 1). The three short-period network stations, which have high-and low-gain

channels with a combined dynamic range of about 72 db, and the large dynamic

range (140 db) Pasa dena instr ument , allowed on-scale recording of both the ML 4.6

main shock and the ML 2.7 aftersho ck that was used as the empirical Gre en function.

The Nyquist frequency for the PAS ins tru men t is 8 hz; consequently, all of the data

used in this study were low-pass filtered, using a thir d-ord er Butte rwo rth filter with

a corner at 8 hz.

In choosing an event to use as an empirical Green function, we looked for

aftershocks that had waveforms that were most similar to the main shock, and so

would indicate similar focal mechanisms and locations. The particular aftershock

that was used as the empirical Green function was chosen because it was the

smallest well-recorded aftershock that had waveforms similar to the main shock.

The smallest aftershock was chosen in order to minimize the source contribution

in the waveform. We found that the epicenters of the main shock and aftershock

were within 250 m of each other when they were relocated using the same set of

stations. This is less than half a wavelength of the highest freq uency (8 hz) used in

the data, so that, initially, all of the propag ation effects should be the same for the

two earthquakes. However, the rupture extent for the main shock was estimated to

be 1 to 2 kin, so that this assumption may not be as valid for later parts of the

waveform.

The deconvolution of the main shock, using the empirical Green function, was



S O U R C E I N V E R S I O N O F T H E 1988 U P L A N D E A R T H Q U A K E 509

done by dividing the Fourier Transfo rm of the main shock by the transform of the

aftershock (Mori and Frankel, 1990). A water-level criterion (set at 0.1 per cent of

the maximum amplitude) was used to fill up the "holes" in the spectrum of the

aftershock, in order to stabilize the deconvolution against division by small values

in the transform of the empirical Green function. The main shock and aftershockwaveforms and the resu ltan t deconvolutions are shown in Figure 2. All four of the

stations show relatively simple deconvolved pulse shapes, inte rpreted as being the

far-field P waves with all of the effects of the path , site, and inst rumen t re~.~oved.

In the inversion, we used only the pulse shape data, so the areas under the

displacement pulses were normalized to give the same moment at each station.

Before normalizing the waveforms, we calculated the mome nt for the main shock

using the following expression from Boatwright (1980):

DM o = 4 ~ [ p ( X o ) p ( x ) c ( x ) ] l/ 2 C ( X o ? / 2 - F 5~

where (Xo) = 2.80 and (x) = 2.28 g/cm 3 are the densi ties at the source and receiver,

respectively; c ( x o ) = 6.56 and c ( x ) = 3.40 km/sec are the velocities at the source and

receiver, respectively; D is the epicentral distance; F is the radiation pattern which

also includes a free surface correction at the receiver; and ~ is the area under the

PAS

148691

7 5 8

L J B

3 6 8 0 8

381

9 5 3 6 5

2 2 0

SI L

0 . 0 0

F IG . 2 . O n t h e l e f t a r e p ai r s o f w a v e f o r m d a t a s h o w i n g t h e m a i n s h o c k (top) a n d t h e a f t e r s h o c k u s e da s t h e e m p i r i c a l G r e e n f u n c t i o n (bottom). T h e n u m b e r s o n t h e l ef t a re t h e m a x i m u m a m p l i t u d es i nd i g it a l c o u n t s o f e a c h t r ac e . T h e w a v e f o r m s o n t h e r i g h t a r e t h e f a r - fi e ld d i s p l a c e m e n t p u l s e s r e s u l t i n gf r o m d e c o n v o l u t i o n o f t h e m a i n s h o c k u s i n g t h e a f t e r s h o c k . T h e t i m e s c a l e is in s e c o n d s .

1 ' . 0 0 2 ' . 0 0 3 ' . 0 0



5 1 0 J . M O R I A N D S . HARTZELL

TABLE 1

MOMENT ESTIMATE

Distance U MomentStation Azimuth F

(km) (cm sec) (dyne-cm)

PA S 42.5 272 ° 0.88 6.23 x 10 4 1.94 x 1022

LJB 52.1 346° -1 .04 7 . 5 8 x 1 0 4 2.46 x 1022

RAY 83.5 97 ° 0.68 8.57 X 10 4 6.75 x 1022SIL 84.7 73 ° 0.65 6.69 x 10 4 5.59 x 1022

Ave rage 4.19 x 1022

d i s p l a c e m e n t p u l se . T h e r e s u lt s , s h o w n i n T a b l e 1, g iv e a n a v e r a g e v a l u e o f

4 .2 x 1022 d y n e - c m f o r t h e f o u r s t a ti o n s . T h i s v a l u e f o r t h e t o t a l m o m e n t , t o g e t h e r

w i t h a n a s s u m e d v a l u e o f 4 .0 x 1 01 1 d y n e / c m f o r t h e r i g i d it y , w a s u s e d t o s c a le t h e

r e s u l t s o f t h e i n v e r s i o n f o r sl ip d i s t r i b u t i o n .

INVERSION METHOD

W e f o l l o w e d H a r t z e l l a n d H e a t o n ( 19 8 6 ) in c a l c u l a ti n g t h e i n v e r s i o n o f t h e

w a v e f o r m s f o r th e d i s t r i b u t e d s li p o n t h e f a u l t p l a n e . I n t h i s m e t h o d , t h e f a u l t p l a n e

is d i v i d e d in t o d i s c r e t e s u b f a u l ts , a n d s y n t h e t i c s e i s m o g r a m s a r e c a l c u l a te d a t e a c h

s t a t i o n f o r a u n i t a m o u n t o f s li p o n e a c h s u b f a u lt . A s y n t h e t i c w a v e f o r m a t e a c h

s t a t i o n c a n t h e n b e f o r m e d f r o m a l i n e a r c o m b i n a t i o n o f t h e s u b f a u l t s e i s m o g r a m s .

A l i n e a r l e a s t- s q u a r e s i n v e r s i o n , w h i c h m i n i m i z e s t h e e r r o r b e t w e e n t h e s y n t h e t i c s

a n d t h e d a t a , i s u s e d t o s o l v e fo r th e a m o u n t s o f s li p o n e a c h o f t h e s u b f a u l ts .

I n t h is s t u d y , th e p r o g r a m Q U A K E 7 b y P . S p u d i c h w a s u se d t o c a lc u l at e t h e

s y n t h e t i c s e i s m o g r a m s f o r e a c h s u b f a u l t ( S p u d i c h a n d F r a z e r , 1 9 84 ). T h e f a u l t

p l a n e w a s d i v i d e d i n t o t r i a n g u l a r s h a p e d s u b f a u lt s , w h i c h e n a b l e d u s to m a k e a

s i m p l e a n d f a s t c a l c u l a t i o n f o r t h e s o u r c e - t i m e f u n c t i o n a s t h e r u p t u r e s w e e p s a c ro s s

a s u b f a u l t a r e a . F o r a ll o f t h e f a u l t m o d e l s, a c o n s t a n t r u p t u r e v e l o c i t y a n d a

c o n s t a n t r i s e t i m e o f 0. 0 6 s e c w e r e a s s u m e d . F i g u r e 3 s h o w s t h e 1 1 b y 1 1 g r i d o f

t r i a n g l e s , 0 . 3 6 b y 0 . 6 0 k m i n s iz e , t h a t w e u s e d f o r a l l o f t h e i n v e r s i o n s . T h e g r i d is

c e n t e r e d a r o u n d t h e n e t w o r k l o c a ti o n a n d o r i e n t e d to m a t c h o n e o f t h e n o d a l p l a n e s

o f t h e f i r s t - m o t i o n m e c h a n i s m . T h e p l a n e s t r i k in g s o u t h e a s t a t 1 2 5 ° a n d d i p p i n g

8 5 ° t o t h e s o u t h w e s t w a s t r i e d f i r s t ( p l a n e 1 ) , t h e n t h e p l a n e s t r i k i n g s o u t h w e s t a t

2 2 1 ° a n d d i p p i n g 4 0 ° t o t h e n o r t h w e s t ( p l a n e 2 ) w a s u s e d .

B e c a u s e t h e d a t a w e a r e m o d e l i n g a r e a s s u m e d t o b e t h e f a r - fi e l d d i s p l a c e m e n t

p u l s e s h a p e s , a l l t h a t is n e e d e d f o r e a c h s u b f a u l t s y n t h e t i c is th e s o u r c e - t i m e

f u n c t i o n g e n e r a t e d b y t h a t s u b f a u lt . A l l o f t h e s y n t h e t i c s o u r c e - t i m e f u n c t i o n s w e r e

s i m p l e u n i p o l a r t r i a n g l e s , n o t c o m p l i c a t e d w a v e f o r m s , s o t h i s i n v e r s i o n w a s q u i t e

s t a b l e . A p o s i t i v i t y c o n s t r a i n t ( L a w s o n a n d H a n s o n , 1 9 7 4 ) w a s p l a c e d o n t h e

s o l u t io n , b u t n o d a m p i n g o r m i n i m i z a t i o n c o n s t r a i n t s w e r e n e c e s s a r y . T h e i n v e r s i o n

w a s d e s i g n e d t o r e s o lv e t h e s h a p e o f t h e a z i m u t h a l l y d e p e n d e n t w a v e f o r m s , c o n s e -

q u e n t l y a ll o f t h e d i s p l a c e m e n t s p u l s e s w e r e n o r m a l i z e d to t h e a v e r a g e m o m e n t o f

4 .2 x 1 02 2 d y n e - c m , a n d t h e f o c a l m e c h a n i s m w a s k e p t f ix e d .

T h e i n v e r s i o n p r o c e d u r e s o l v e d t h e f o l l o w i n g e q u a t i o n f o r t h e v e c t o r x t h a t

c o n t a i n s t h e u n k n o w n v a l u e s o f s li p f o r e a c h o f t h e 1 21 s u b f a u lt s :

A x = B

T h e A m a t r i x ( 30 0 b y 1 21 p t s .) c o n t a i n s t h e s y n t h e t i c s e i s m o g r a m s f o r t h e

1 21 s u b f a u l t s a t e a c h o f t h e f o u r s t a t i o n s , s a m p l e d a t 5 0 s a m p l e s / s e c f o r 1 . 5 s ec .



STRIKE i ° - " 1S O U R C E I N V E R S I O N O F T H E 1988 U P L A N D E A R T H Q U A K E 511

0 ' . 0 0 1 ' .0 0 2 ' .0 0 3 ~ . 0 0

KM

F IG . 3 . S u b f a u l t g r i d u s e d f o r t h e i n v e r s i o n o n b o t h f a u l t p l a n e s . T h e f i ll e d c ir c le i s t h e h y p o e e n t e r .

The B vector contains 1.5 sec of data sampled at 50 samples/sec for the four

stations (300 pts).

Th e i nversion was carried out assuming various ruptur e velocities from 2.6 to 4.6

km/sec (corresponding to 0.70 to 1.2 times the S-wave velocity, assuming a P- to

S-wave velocity ratio o f 1.7) on the southea st-str iking nodal plane, plane 1, and

then on the southwest-striking nodal plane, plane 2. The solution for each of the

cases was evaluated by using the variance (2),

a2 = ((A x) - B)2/N

N is the numb er of degrees of freedom,

N = nd ata - nsol - 1,

where ndata is the fixed numb er of data poi nts in B and nsol is the n umber of

subfaults tha t had nonzero slips. Th e value of nsol varies because of the po sitivity

constraint, which solves only for those subfaults which it determines has non-

negative slip. Comparisons of the variances show which models produce the syn-

thetics that best match the data.

D E T E R M I N A T I O N O F F A U L T P L A N E

The variances for a range of rupt ure velocities on the two nodal planes are shown

in Figure 4. In all cases, except for the slowest rupture velocity of 2.6 km/sec, plane

2 gives a better fit to the data. The variance reduction using plane 2 compared to

plane 1, is 2 to 12 per cent for the slower rupture velocities and 20 to 30 per cent

for the higher ru pture velocities, indicating a significant imp rove ment of the fit to

the data for plane 2.



0.020

0.015

J . M O R I A N D S . H A R T Z E L L

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< 0.010

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0.005Plane 2

512

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

2 . 5 3 . 5 4 . 5

RUPTURE VELOCITY (KM/SEC)

F I G, 4 . V a r i a n c e s o f t h e s o l u t i o n s f o r a r a n g e o f r u p t u r e v e l o c i t i e s o n t h e t w o n o d a l p l a n e s . T h e

a s s u m e d S - w a v e v e l o c i t y i n t h e s o u r c e r e g i o n i s 3 .8 k m / s e c .

F i g u r e 5 s h o w s t h e s l ip s o l u t i o n s o b t a i n e d f o r s e v e r a l r u p t u r e v e l o c i t ie s o n p l a n e s1 a n d 2 . W h e n p l a n e 1 is u s e d t h e r u p t u r e s p r e a d s o u t r a t h e r s y m m e t r i c a l ly f r o m

t h e h y p o c e n t e r . F o r t h e p r e f e r r e d p l a n e 2 , t h e r u p t u r e p r o p a g a t e s m o r e u p d i p a n d

t o w a r d t h e s o u t h w e s t . T h e u p d i p r u p t u r e p r o p a g a t i o n o b t a i n e d f o r p l a n e 2 i s

c o n s i s t e n t w i t h t h e o b s e r v a t i o n t h a t m a n y e a r t h q u a k e s i n i t i a t e n e a r t h e b o t t o m o f

t h e s o u r c e a r e a a n d r u p t u r e u p w a r d t o w a r d t h e s u r f a c e ( S i b s o n , 1 9 8 2 ) .

T h e s l i p d i s t r i b u t i o n o n t h e p r e f e r r e d p l a n e 2 i s w h a t o n e m i g h t q u a l i t a t iv e l y

e x p e c t f r o m l o o k in g a t t h e w a v e f o r m s i n F i g u re 2. T h e m a i n f e a t u r e t o n o t e i s t h a t

P A S a n d L J B l o o k s i m i la r i n s h a p e a n d R A Y a n d S I L l o o k s im i la r , w i t h t h e P A S /

L J B p a i r b e i n g s l i g h t l y b r o a d e r t h a n t h e R A Y / S I L p a i r . T h i s s u g g e s t s s o m e

d i r e c t iv i t y t o w a r d t h e e a s t , w i t h t h e r u p t u r e m o v i n g e i t h e r s o u t h e a s t o n t h es o u t h e a s t - t r e n d i n g p l a n e ( p l a ne 1 ) o r u p d i p o n t h e s o u t h w e s t - t r e n d i n g p l a n e

(p l ane 2 ) .

I t a p p e a r s t h a t t h e r e a s o n t h e i n v e r s io n r e s u l t s f a v o r o n e f a u l t p l a n e o v e r t h e

o t h e r is al so d u e t o th e s i m il a ri ty o f t h e P A S / L J B w a v e f o r m s a n d t h e S I L / R A Y

w a v e f o r m s . F o r t h e f a v o r e d f a u l t p l a n e 2 , t h e t a k e - o f f a n g le s f r o m t h e f a u l t n o r m a l

a r e r e l a t i v e l y s i m i l a r f o r P A S ( 13 3 °) a n d L J B ( 1 16 ° ), s o i t i s n o t d i f f i c u l t f o r t h e

i n v e r s i o n t o f i n d s o l u t i o n s w h i c h p r o d u c e w a v e f o r m s t h a t a r e s i m i l a r a t P A S a n d

L J B . F o r f a u l t p l a n e 1 , t h e t a k e - o f f a n g l e s a r e q u i t e d i f f e r e n t t o P A S ( 12 3 °) a n d

L J B ( 5 0 ° ) , c o n s e q u e n t l y t h e i n v e r s i o n d o e s n o t d o a s w e l l i n m a k i n g s i m i l a r

s y n t h e t i c w a v e f o r m s a t P A S a n d L J B . T h e s a m e r e a s o n i n g a p p l i e s f o r t h e s i m i l a r

w a v e f o r m s a t S I L a n d R A Y , w h e r e t h e t a k e - o f f a n g le s a r e s i m i la r f o r p l a n e 2

(61 ° , 53 °) bu t s l igh t ly gre ate r fo r p la ne 1 (39 ° , 62°) .

T h u s , t h e f a u l t -p l a n e d e t e r m i n a t i o n d e p e n d s o n t h e d i f fe r e n c e b e t w e e n P A S /

L J B w a v e f o r m s a n d t h e R A Y / S I L w a v e f o r m s . E v e n t a k i n g i n t o a c c o u n t t h e n o i s e

l e v e l s e e n i n t h e w a v e f o r m s , t h e r e a p p e a r s t o b e a d i s t i n c t d i f f e r e n c e i n s h a p e



S O U R C E I N V E R S I O N O F TH E 1988 U P L A N D E A R T H Q U A K E

NW SE NW qqF NW

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P L A N E 1

3 0 (

2 . 8 KM/SEC3 . 4 K M / S E C 4 . 0 K M / S E C P L A N E 2

0 . 0 1 . 0 2 . 0 3 . 0

KM

F IG . 5 . S o l u t i o n s o f s li p d i s t r i b u t i o n f o r t h r e e r u p t u r e v e l o c it ie s o n e a c h o f t h e t w o n o d a l p l a n e s . T h ev e r t i c a l p i n i n t h e m i d d l e o f t h e g r i d m a r k s t h e h y p o c e n t e r .

between the PAS/LJB waveforms and the RAY/SIL waveforms, that can be used

for the fault plane determination.

R U P T U R E V E L O C I T Y

For both planes, the variances become smaller with higher rupture velocities.

This result indicates tha t the best-fitting solution is for the highest rupture velocity

tested, 4.6 km/sec, implying a transonic rupture velocity of 1.2 times the shear-

wave velocity. However, we believe this result is just a consequence of the poor

resolution we have on the ru ptur e velocity. Because all the stations are far from thesource, it is difficult to measure the directivity effects needed to obtain a rupture

velocity. There is a direct relation between the rupture velocity and the lateral

dimensions of the source. Assuming higher rup ture velocities, the source dimension

increases, given the fixed pulse duration observed in the data. When the source

dimension increases, more subfaults (variables) contribute to the solution, conse-

quently the solution can be more complicated and can better fit the data. Thus, the

reduction of the variance for higher rupture velocities is believed to be simply a

result of solving the pr oblem with more variables. Th e numb er of nonzero values in

the immediate region of the hyp oce nter increased from 10 for 2.6 km/se c to 22 for

4.6 km/sec. The increasing complexity of the solution with increasing rupturevelocity, can be seen in the solutions on plan e 1 of Figure 5.

We also tried to resolve the ruptu re velocity using the Akaike inf ormat ion

criterion, AIC (Akaike, 1974), which is a statistical test used to compare models

with differe nt number s of parame ters. However, the results were similar to the

variance and did not show any preferential rupture velocity. Both statistical tests




did show a very weak local minimum at a rupture velocity of 3.3 km/sec. For lack

of any other evidence, we have simply assumed this reasonable value (which

corresponds to 0.87 times the shear-wave velocity) for our prefer red rupture velocity.

RESULTS

The solution of slip distribution for the rupture velocity of 3.3 km/sec on plane 2

is shown in Figure 6. An est imate of the unce rta inty in the slip solution is given by

the diagonal elements of the model covariance matrix multiplied by some estimate

of the error in the data. We used the stand ard deviation in the absolute moment

determination from the four stations (Table 1) as an estimate of the data error.

The resulting uncertaint ies in the slip of each subfault are shown on Figure 6. The

main area of slip near the hypocenter is significantly larger than the error estimates,

indicating a reliable part of the solution, whereas the smaller amounts of slip on

the downdip edge of the grid are within the unce rta inty estimates and not considered

a resolvable part of the slip solution.Assuming that the earthquake ruptured the area of the nine subfaults near the

epicenter, the source area is 0.97 km 2. The static stress drop is given by Kanam ori

and Anderson (1975)

A a = C ~ ( D / L )

where C is a geometrical factor near 1.0, tL is the rigidity, D is the average slip, and

L is a fault dimension. Using C -- 1.0, ~ = 4.0 x 1011 dyne /c m2, an average value of

slip for the nine subfaults, (D = 9.1 cm) and a fault dimension equal to the square

root of the area of the nine subfaul ts (L = 0.99 km), gives a stress drop of 38 bars.Using the solutions for the other rupture velocities gave stress drops of 11 to

37 bars, with the slower rupture velocities having generally smaller rupture areas

and thus higher stress drops.

..50

2 0

10

0

NE SW NE SW/

5 .5 KM /SEC U N C ERTAIN TY

6 . o 1 ' .o i . o i . o

KM

FIG. 6. (Left) Preferred solution for the slip distribution using a rupture velocity of 3,2 km/sec onplane 2. (R igh t ) Uncertainties in the solution estimated from the model covariance matrix and thestandard deviation of the absolute moment estimate.



SOURCE INVERSION OF THE 1988 UPLAND EARTHQUAKE 515

I I

I I I

A ,o I L ,

[ i R A Y

( ) . 0 ( ~ . 5 1 ' . 0 1 ' . 5 i . OS E C

FIG. 7. Modelsynthetics dottedlines) for a rupture velocityof S.2 km/sec on plane 2 and deconvolveddisplacement waveforms (solid lines) used in the inversion.

In Figure 7, the synthetic seismograms for the preferred solution are shown,

together with the far-field displacement data that were used in the inversion. The

synthetic seismograms from Figure 7 were also convolved with the empirical Green

functions and are compared with the original low-pass filtered data in Figure 8. In

both figures, the synthet ics match the data reasonably well for all the stations, even

including the high-frequency features.

DISCUSSION

The method described above provides a way of determi ning fault planes for smallevents that have neither surface rupture nor aftershock locations which clearly

define a fault plane. Using a small event as an empirical Green function has the

advantage of easily correcting the waveforms for the complicated path and site

effects. Extremely detailed knowledge of the velocity structure would have been

necessary to construct synthetic Green functions to model an earthquake of this

size in the 1 to 8 hz frequency band. Further, the simple displacement pulses which

resulted from the deconvolution are easy to invert for fault slip. The inversion is

much more stable than inverting complicated seismograms which include all the

path and site effects.

With good azimuthal station coverage, fault-plane determination should be pos-

sible for most str ike-slip events with source dimensions of a few kilometers, because

the directivity effects due to the finiteness of the source would be significantly

different for the two nodal planes. Dip-slip events may be more difficult, because

the azimuthal dependence of the directivity effects would be much harder to see for

two nodal planes th at have a similar strike. Dip-slip events would probably require




P A S , R A Y

. _ 2 8 0 0 / ' , , 4 8 9 5 5 , , , , , o ,

. . . . . . . . . . . . . , . . . . . . . . . . . , ~ , , , ' , t ' , 4 " -

I ' , . " " , / . . . .6 0 4 9 ' , j ' 5 6 8 0 i r i , i , ~ ,

L J B S I L/

1 6 o g 8 o ,~" , ' ~ f ,8 7 2 8 5 ,;,, ;'~\~

/

. . . . . ' , I \ ;1 , ,.. . . . . . . . . . . - . ,, ," , , . ._ - . . . . . ~ . , ~ , - , ; -

1 4 8 6 9 1 " ~ J l ' \ ' i _ l ~ 9 5 5 6 3 i , ,

0 . 0 0 1 . 0 0 . 0 0 . 0 0 4 . 0 0

S E C

FIG. 8. Model synthetics from Figure 8 convo]vedwith the empirical Green functions (dotted lines)and the original low-pass filtered data (solid lines).

3 0 ' 1 1 8 °' / ~ , . _ 4 5 ' 3 0 ' 1 5 'I I

L. I I~ o io 2o KM

1 5 '

3 4 ' O '

Fro. 9. Map showing inferred left-lateral motion for the small southwest trending section connectingthe Cucamonga and Sierra Madre th rust faults. Fault locations and displacements are taken from Cramerand Harrington (1987).

one or more statio ns close to the source with up ward t ake- off angles, in order to

resolve the effects caused by the d ifference in the dip angles of the nodal planes.

The size of earthq uakes tha t can be studied in this man ner is somewhat l imited.

The event has to be large enough so tha t th e finiteness of the source can be observed

in the fre quenc y ba nd of the data. I f the source is too large, however, the d econvo-



S O U R C E I N V E R S I O N O F T H E 1988 U P L A N D E A R T H Q U A K E 517

lution of the waveforms using a smaller earthquake will not work, because the small

event would not be an appropriate Green function for the entire areal extent of the

larger event. For short-period network data in the frequency band of 1 to 20 hz,

magnitude 4 to 5 earthquakes are suitable.

The inversion results for the 1988 Upland ear thquake indicate tha t the southwest-trending plane was the fault plane. This implies that the slip was left-lateral on a

southwest-trending plane dipping toward the northwest. This orientation is con-

sistent with the trend of a nearby portion of the Transverse Ranges frontal fault

zone, as it bends to the southwest connecting the Cucamonga fault with the Sierra

Madre Fault. From focal mechanisms of a few small earthquakes in the region,

Cramer and Harr ington (1987) inferred tha t slip on this portion of the fault was

left-lateral strike-slip, similar to the movement we described in this study (Fig. 9).

Strike-slip faulting on this southwes t-trending portion of the fault and the adjacent

thrus t faulting on the east-west-trending portions on either side is consistent with

the general north-south convergence across the Transverse ranges (Morton andYerkes, 1987).

C O N C L U S I O N S

By using a small aftershock as an empirical Green function, complicated path

and site effects can be deconvolved from P waveforms recorded at regional distances.

This process results in far-field source-time functions with a high-frequency reso-

lution of up to 8 hz. This frequency range is higher than can be modeled using only

synthetics with our present knowledge of velocity structures. The deconvolved

waveforms were inverted for the slip distr ibution on a discretized fault plane, testing

the two nodal planes of the focal mechanis m at various rupture velocities. Becausethe data fit the southwest-trending nodal plane better than the southeast-tr ending

nodal plane for almost all of the rupture velocities, we believe that this approach

enabled us to conclude that the southwest trending plane was the fault plane for

this earthquake.

A C K N O W L E D G M E N T S

W e b e n e f i t e d f r o m t h e h e l p f u l re v i e w s o f A . F r a n k e l , T . H e a t o n , W . E l l s w o r t h , A . S n o k e , a n d

D . B o o r e. L . J o n e s p r o v i d e d l o c a t i o n s a n d f o c al m e c h a n i s m s f r o m S o u t h e r n C a l i f o r n i a N e t w o r k .

R E F E R E N C E S

A k a i k e , H . ( 1 9 7 4 ) . A n e w l o o k a t t h e s t a t i s t i c a l m o d e l i d e n t i f i c a t i o n , I E E E T r a n s . A u t o m . C o n t r o l

A C - 1 9 , 7 1 6 - 7 2 3 .

B o a t w r i g h t , J . ( 1 98 0 ). A s p e c t r a l t h e o r y f o r c i r c u l a r s e is m i c s o u r c es : s i m p l e e s t i m a t e s o f s o u r c e d i m e n s i o n ,

d y n a m i c s t r e s s d r o p s , a n d r a d i a t e d e n e r g y , B u l l . S e i s m . S o c . A m . 7 0 , 1 - 2 8 .

C r a m e r , C . H . a n d J . M . H a r r i n g t o n ( 1 98 7 ). S e i s m i c i ty a n d t e c t o n i c s o f t h e C u c a m o n g a f a u l t a n d t h e

e a s t e r n S a n G r a b r i e l M o u n t a i n s , S a n B e r n a r d i n o C o u n t y , I n R e c e n t r e v e r s e f a u l t i n g i n t h e T r a n s v e r s e

Ranges , Cal i forn ia , U.S . Geol . Surv . Pro fe ss . Paper 1339 , 7 - 2 6 .

F r a n k e l , A . , J . F l e t c h e r , F . V e r n o n , L . H a a r , J . B e r g e r , T . H a n k s , a n d J . B r u n e ( 1 9 8 6 ) . R u p t u r e

c h a r a c t e r i s t i c s a n d t o m o g r a p h i c s o u r c e i m a g i n g o f M L - 3 e a r t h q u a k e s n e a r A n z a , S o u t h e r n

C a l i f o r n i a , J . Geophys . Res . 9 1 , 1 2 6 3 3 - 1 2 6 5 0 .

H a r t z e l l , S . a n d T . H e a t o n ( 19 8 6 ). I n v e r s i o n o f s t ro n g g r o u n d m o t i o n a n d t e l e s e i s m i c w a v e f o r m d a t a f o r

t h e f a u l t r u p t u r e h i s t o r y o f t h e 1 9 79 I m p e r i a l V a ll e y, C a l i f o r n ia , e a r t h q u a k e , B u l l . S e i s m . S o c . A m .

7 6 , 6 4 9 -6 7 4 .

K a n a m o r i , H . a n d D . L . A n d e r s o n ( 1 9 7 5 ) . T h e o r e t i c a l b a s i s o f s o m e e m p i r i c a l r e l a t i o n s i n s e i s m o l o g y ,

B u l l . S e i s m . S o c . A m . 6 5 , 1 0 7 3 - 1 0 9 5 .

L a w s o n , C . L . a n d R . J . H a n s o n ( 1 97 4 ). S o l v i n g L e a s t S q u a r e s P r o b le m s , P r e n t i c e - H a l l , I n c ., E n g l e w o o d

C l i ff s , N e w J e r s e y , 3 4 0 p p .



518 J . M O R I A N D S . H A R T Z E L L

L i , Y . a n d C . H . T h u r b e r ( 1 98 8 ). S o u r c e p r o p e r t i e s o f t w o m i c r o e a r t h q u a k e s a t K i l a u e a V o l c a n o , H a w a i i ,

B u l l . S e i s m . S o c . A m . 7 8 , 1 1 2 3 - 1 1 3 2 .

M o r i , J . a n d A . F r a n k e l ( 1 99 0 ). S o u r c e p a r a m e t e r s f o r e a r t h q u a k e s a s s o c i a t e d w i t h t h e 1 9 86 N o r t h P a l m

S p r i n g s , C a l i f o r n i a , e a r t h q u a k e d e t e r m i n e d u s i n g e m p i r i c a l G r e e n f u n c t i o n s , B u l l . S e i s m . S o c . A m .

8 0 , 2 7 8 - 2 9 5 .

M o r t o n , D . M . a n d J . C . M a t t i ( 19 8 7 ). T h e C u c a m o n g a f a u l t z o n e : g e o lo g ic a l s e t t i n g a n d Q u a t e r n a r yh i s t o r y , I n R e c e n t r e v e r s e f a u l t i n g i n t h e T r a n s v e r s e R a n g e s , C a l i fo r n ia , U . S . G e o l. S u r v . P r o f e s s .

P a p e r 1 3 3 9, 1 7 9 - 2 0 3 .

M o r t o n , D . M . a n d R . F . Y e r k e s ( 19 8 7 ). I n t r o d u c t i o n , I n R e c e n t r e v e r s e f a u l t i n g i n t h e T r a n s v e r s e R a n g e s ,

Ca l i f o rn i a , U .S . Geo l . Surv . Pro f ess . Paper 1339 , 1 - 5 .

P e c h m a n n , J . C . (1 9 8 7) . T e c t o n i c i m p l i c a t i o n s o f s m a l l e a r t h q u a k e s i n t h e c e n t r a l T r a n s v e r s e r a n g e s , I n

R e c e n t r e v e r s e f a u l t i n g i n t h e T r a n s v e r s e R a n g e s , C a l i fo r n ia , U . S . G e ol S u r v . P r o f e s s . P a p e r 1 3 3 9 ,

1 7 9 - 2 0 3 .

S i b s o n , R . H . ( 1 9 8 2 ) . F a u l t z o n e m o d e l s , h e a t f l o w a n d t h e d e p t h d i s t r i b u t i o n o f e a r t h q u a k e s i n t h e

c o n t i n e n t a l c r u s t o f th e U n i t e d S t a t e s , B u l l . S e i s m . S o c . A m . 7 2 , 1 5 1 - 1 6 3 .

S p u d i c h , P . a n d L . N . F r a z e r ( 1 98 4 ). U s e o f r a y t h e o r y t o c a l c u l a t e h i g h - f r e q u e n c y r a d i a t i o n f r o m

e a r t h q u a k e s o u r c e s h a v i n g s p a t i a l ly v a r i a b l e r u p t u r e v e l o c i ty a n d s t r e s s d r o p , B u l l . S e i s m . S o c . A m .

7 4 , 2 0 6 1 - 2 0 8 2 .

U. S. GEOLOGICAL SURVEY

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