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Earthquake Loc ation Tuton'al: GraphicalApproach and A pp roxim ate Ep icentraiLocationTechniques

Jose Pu jo lCE RI , The U nivers ity of Mem phis

INTRODUCTION

As is we l l k n o wn , th e p r o b le m o f e a r th q u a k e lo c a t io n i s o n e

o f th e m o s t b a s ic in s e i smo lo g y , b u t b e c a u s e o f it s i n h e r e n t

ma th e ma t i c a l c o mp le x i ty i t i s d i f f i c u l t t o p r e s e n t i t i n a wa y

th a t g iv es a r e al i st i c se n s e o f h o w e a r th q u a k e s a r e lo c a te d a n d

i l lu s t r a t e s th e f a c to r s th a t c o n t r ib u te to u n c e r t a in t i e s in th e

so lu t ion . For th is r eason i t i s h igh ly des irab le to be ab le to d is -

c u s s th e mo s t e s s e n t i a l f a c t s r e g a r d in g e a r th q u a k e lo c a t io n

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

l e m. T h i s i s imp o r ta n t wh e n t e a c h in g th e s u b je c t to u n d e r -

g r a d u a te s tu d e n t s , t o E a r th s c i e n c e g r a d u a te s tu d e n t s wh o d o

n o t p la n to b e c o me s e i s mo lo g i s t s , a n d e v e n to h ig h - s c h o o l

sc ience teachers .

On e p o s s ib le wa y o f d o in g th i s is to u s e th e f a ct th a t in a

me d iu m wi th d e p th - d e p e n d e n t v e lo c i t i e s th e t r a v e l t ime s l i e

o n a s u rf a c e o f r e v o lu t io n w i th i ts min im u m a t th e e p ic e n te r .

F o r a h o mo g e n e o u s me d iu m th e s u r f a c e i s a h y p e r b o lo id ,

d e s c r ib e d b y a s imp le e q u a t io n . By p lo t t in g th e s e s u rf a c es f o r

e a r th q u a k e s w i th d i f f e r e n t h y p o c e n t r a l lo c a t io n s a n d o r ig in

t imes i t is poss ib le to prese n t a r ea l is t ic g raphic a l d iscuss ion of

th e mo s t im p o r t a n t a s p e ct s o f e a r th q u a k e lo c a t io n . As s h o w nbelow, th is approach can be used to i l lus tra te is sues such as

th e n o n u n iq u e n e s s o f th e c o m p u te d lo c a t io n s , t h e ef f ec t o f

e r r o rs , a n d th e a d v a n ta g e s o f u s in g c o mb in e d P - a n d S - wa v e

i n f o r m a t i o n .

Af te r th is d iscuss ion i t would a lso be des irab le to in tro-

d u c e t e c h n iq u e s th a t s tu d e n t s c a n u s e to e s t ima te e p ic e n t r a l

lo c a t io n s a f t e r th e y h a v e p ic k e d th e i r o wn a r riv a l time s . M o s t

in t r o d u c to r y s e i s mo lo g y b o o k s d e s c r ib e th e S-P t ime s

me th o d , a n d a l th o u g h i t i s c o n c e p tu a l ly v e r y s imp le , t h e

e x a mp le s g iv e n h e r e d e mo n s t r a t e th a t i t s a p p l i c a t io n i s n o t

s t r a ig h t f o r wa r d .

I n p a r t i c u la r , t h e v e lo c i ty to b e u s e d d e p e n d s o n th e

e v e n t d e p th a n d e p ic e n t r a l d i s t a n c e s . F u r th e r mo r e , Ru f f

( 2 0 0 1 ) n o te d th a t th e S-P t i m e s m e t h o d i s n o t h o w e a r t h -

q u a k e s a r e a c tu a l ly lo c a te d a n d f o r th i s r e a s o n s u g g e s te d th e

u s e o f a me th o d b a s e d o n th e d r a win g o f h y p e r b o la s . T h i s

m e t h o d w a s u s e d b ef o re t h e a d v e n t o f t h e c u r r e n t c o m p u t e r -

b a s e d lo c a t io n p r o g r a ms a n d g e n e r a l ly p e r f o r ms we l l . Bo th

t h e h y p e r b o l a a n d t h e S-P t ime s me th o d s w i l l b e d i s c u s s e d

h e r e , w i th a p p l i c a t io n s to s y n th e t i c a n d a c tu a l d a ta .

An in te r e s t in g b u t u n e x p e c te d r e s u l t is t h a t in s o me c a s es

th e S-P t i m e s m e t h o d c a n b e m o d i f i ed i n s u c h a w a y t h a t t h e

e p ic e n t r a l l o c a t io n s c a n b e d e te r m in e d f a ir ly ac c u r ate ly. T h i s

f a c t c a n b e e x p la in e d in a s e m iq u a n t i t a t iv e w a y a n d w a s v er i -

f i e d w i th a c tu a l d a ta f r o m a n a r e a w i th l a r g e l a t e r a l v e lo c i ty

v a r i a t io n s . As f o r th e h y p e r b o la me th o d , i t i s s h o wn th a t i t

can be used to loca te events with re la t ive ly small e r ror s under

a w id e r a n g e o f c o n d i t io n s , a n d th a t th e h y p e r b o la s c a n b e

c o mp u te r - g e n e r a te d w i th l i t t l e e f f o r t , wh ic h f a c i l i t a t e s th e

a p p l i ca t i o n o f t h e m e t h o d .

THE EARTHQUAKELOCATION PROBLEM: A

GRAPHICAL APPROACH

T h e e a r th q u a k e lo c a t io n p r o b le m c a n b e s t a t e d a s f o l lo ws :

g iv e n a se t o f ar r iv a l t ime s a n d a v e lo c i ty mo d e l , d e te r m in e

th e o r ig in t ime a n d th e c o o r d in a te s o f th e h y p o c e n te r .

I n a c tu a l p r a c t i c e , e a r th q u a k e s a r e lo c a te d a c c o r d in g to

th e f o l lo win g g e n e r a l p r in c ip le s . B e c a u s e th e r e l a t io n b e twe e n

a r r iv a l t ime s a n d h y p o c e n t r a l c o o r d in a te s a n d o r ig in t ime i s

n o t s imp le e v e n f o r th e s imp le s t v e lo ci ty mo d e l ( se e E q u a t io n

2 b e lo w) , e a r th q u a k e s c a n n o t b e lo c a te d in o n e s t e p . W h a t i sd o n e i s to s o lve th e lo c a t io n p r o b le m in a n i t e r a t iv e wa y .

The f i r s t i te ra t ion has these s teps : es t imate in i t ia l va lues

o f th e h y p o c e n t r a l c o o r d in a te s a n d o r ig in t ime , u s e th e s e in i-

t i a l v a lu e s a n d th e v e lo c i ty mo d e l to c o mp u te th e o r e t i c a l

a r r iv a l t ime s , c o mp u te th e i r d i f f e r e n c e s w i th th e o b s e r v e d

t ime s , a n d th e n u s e th e s e d i f fe r e n c es to g e t a n e w e s t ima te o f

th e lo c a t io n a n d o r ig in t ime . E a c h f o l lo win g i t e r a t io n u s e s a s

in i t i a l e s t ima te s th o s e c o mp u te d in th e p r e v io u s i t e r a t io n .

T h i s i t e r a t iv e p r o c e s s s to p s wh e n s o me c o n d i t io n i s me t .

T h e r e a re a n u m b e r o f s t o p p i n g c o n d i t i o n s, b u t i n a n y c as e

w e m e a s u r e t h e q u a l i t y o f t h e c o m p u t e d l o c a ti o n b y t h e r o o t -

me a n - s q u a r e r e s id u a l , d e f in e d a s

' ; :1

(1 )

w h e r e T /~ i s the observ ed a r r iva l t ime for the i th s ta t ion , T[

i s th e c o r r e s p o n d in g th e o r e t i c a l a r r iv a l t ime c o mp u te d in th e

la s t i t e r a t io n , N i s th e n u m b e r o f st a t io n s , a n d th e 4 in th e

S e i s m o l o g ic a l R e s e a r ch L e t t e rs J a n u a r y /F e b r u a r y 2 0 0 4 V o l u m e 7 5 , N u m b e r 1 6 3

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denominator i s related to the fact that there are four

u n k n o w n s ( t h ree h y p o cen t r a l co o rd in a tes an d o r i g in t ime) .

I f the m odel veloci t ies were close to the actual ones and

the ar r ival t imes were er ror- f lee, then r m s would be close to

zero , bu t in real i ty the veloci ty model i s on ly approximate

and the data are af fected by errors , and as a consequence r m s

ma y be s ign if ican t ly larger than zero . In som e cases an im por-

tan t source o f er ror i s large lateral veloci ty var iat ions , so that

the 1D veloci ty mod els general ly used for ear thqu ake loca-

t ion are no longer adequate. Note, however ; that the goal o flocat ion programs i s to min imize r m s , not the locat ion er rors ,

wh ich r emain u n k n o wn as l o n g as each ea r t h q u ak e i s l o ca t ed

separately. Therefore, a relatively small value of r m s does no t

always mean a corresponding ly smal l er ror in locat ion . This i s

wh y t h e m o s t r e l iab le d e t e rm in a t i o n o f ea r th q u ak e l o ca ti o n s

req ui r es t h e s imu l t an eo u s d e t e rm in a t i o n o f a 3 D v e lo c it y

mo d e l .

Th i s b r i e f i n t ro d u c t i o n t o t h e ea r t h q u ak e l o ca t i o n p ro b -

lem ignores al l i t s mathemat ical aspects , which involve rather

advanced analysis techniques ( e . g . , Lee and Stewar t , 1981) . I t

i s possib le , however , to gain an u nder s tand ing of some of i ts

essen t ial aspects us ing a g raphical approach . To in t roduce i t

a s su me th a t t h e Ear th can b e r ep resen t ed b y a h o mo g en eo u s

m ed ium and let a and /3 be the P- and S-w ave velocit ies. For

s impl ici ty we wi l l a l so assume that we are deal ing wi th ep i -

cen t ral d is tances that do n o t exceed a few hund reds o f k i lo -

mete r s . Th i s way t h e re i s n o n eed t o b e co n cern ed wi th t h e

Ear th' s cu rv a tu re an d t h e p ro b l em can b e s t a ted i n C ar t es i an

coord inates , w hich wi l l be used here. The o r ig in o f the coor-

d inate system wi l l be some convenien t , arb i t rary po in t .

No w co n s id er an ea r t h q u ak e wi th h y p o cen t r a l co o rd i -

nates (X e, y e , h ) and let To b e t h e o r i g in t ime o f t h e ev en t . Th en

the ar r ival t ime at a s tat ion w i th coord inates (x , y , 0) i s g iven

b y

dy ) - + - g - Vo +

~ ( X X e + ( Y - - Y e +__ )2 )2 h2

5 = a , , 3 (2 )

where d i s the hypocent ral d is tance ( i . e . , t h e d i s t an ce b e tween

th e h y p o cen t e r an d t h e s t a ti o n ) . Eq u a t i o n 2 can b e wr i t t en as

t 2 ( x , y ) ( T -T o ) 2 -- a l ( x 2 + y 2 ) + a 2 x + a 3 y + a 4 (3 )

where t i s t ravel t ime and the coeff icien ts a i ab so rb t h e co n -

s tan t terms in Equat ion 2 (Pu jo l and Smal ley , 1990) . These

two eq uat ions show tha t t (x, y ) can be represen ted b y a hyper-

b o lo id cen t e r ed a t ( x e , Y e ) " Several examples , corresponding to

P- and S-wave t ravel t imes fo r d i f feren t even t dep ths , are

sh o wn in F ig u re 1 . No te t h a t t h e co o rd in a t es o f th e m in i -

m u m o f each h y p erb o lo id co in c id e wi th t h e l o ca t i o n o f t he

ep icen ter .

Al th o u g h t h e h o mo g en eo u s v e lo c i t y mo d e l i s ex t r emely

simple, Equat ion 3 i s impor tan t because i t can be used to

approximate the t ravel - t ime surfaces that are ob tained for

mod els in which the veloci ty var ies wi th de p th . In such a case

the t ravel - t ime surfaces are surfaces o f revo lu t ion a bout a ver-

t ical ax is passing th roug h the ep icen ter . Th is fact was used by

P u jo l an d S mal l ey (1 9 9 0 ) t o d ev e lo p a meth o d t o d e t e rmin e

ep i cen te r s wi th o u t a n y k n o wled g e o f th e v e lo c i ty mo d e l . Th e

basic idea i s to f i t a quadrat ic surface to the observed arr ival

t imes. Because the o r ig in t ime i s no t known, i t i s es t imateddur ing the f i t t ing process . Once the best - f i t hyperbo lo id has

b een d e t e rmin ed , t h e co o rd in a t es o f it s m in im u m are t ak en

as the ep icen t ral coord inates .

Because Equat ion 3 can be used to represen t approx i -

mate ly a r r i v a l t imes i n med ia wi th d ep th -d ep en d en t v e lo c i -

t ies , we can der ive a num be r o f general resu lt s by

considerat ion of f igures generated using Equa t ion 2 . For

example, f rom Figure 1 we see that the d i f ference

d t = t ( x s , Y s ) - t ( X e , Y e ) fo r a fixed posi t ion o f the ep ice n ter an d

given s tat ion coord inates ( x s , Y s ) i s h ig h ly d ep en d en t o n t h e

dep th o f the even ts and the types o f waves. As the dep th

increases, d t decreases . Th is ef fect can be unders tood wi th the

help of Equa t ion 2 , which shows that fo r g iven values o f x

and y , the ef fect o f h on t increases wi th h . As a consequenc e

d t will decrease as h increases because t ( x e , Y e ) = h ~ 5 . Also no te

that fo r a g iven dep th , d t i s larger fo r S waves than for P

waves. To see that, let tp and t s represen t P- and S-wave t ravel

t imes, respectively. Then, ts = ( a / 3 ) t p a n d d t s = ( a / 3 ) d t p . T h e s e

two equal i t ies are val id fo r any veloci ty model fo r which the

ratio a / 3 i s a constan t .

No w l e t u s lo o k a t t h e ea r t h q u ak e l o ca t i o n p ro b l em f ro m

a d i f feren t po in t o f v iew , namely , consider the observed

arr ival t imes as samples o f some arr ival - t ime surface. The n

locat ing an ear thquake i s equ ivalen t to reconst ruct ing a sur-

face f rom a f in i te , general ly smal l, nu m ber o f samples. Theconnect ion between th is v iew and the s tandard one i s as fo l -

lows. Once the hypocent ral locat ion and the o r ig in t ime are

found , one can generate the corresponding arr ival - t ime sur-

face, which can be p lo t ted together wi th the observat ions as

in F igure 1 . Therefore, we can use these surfaces to es tab l i sh

in a qual i ta t ive way what we can expect under d i f feren t ci r -

cu ms tan ces .

Con sider fo r exam ple the ef fect o f sampl ing an d errors in

the ar r ival t imes. Suppose that tw o even ts at dep ths o f 5 and

50 km are recorded by s tat ions corresponding to the do ts in

Figure 1 . Because these s tat ions sample the surfaces wel l , i t

can be expe cted that i f the veloci ty mode l were a fai th fu l rep-

resen tat ion of the actual veloci ties and the d ata were er ror-

f ree, the even ts w ould be wel l located regard less o f the dep th .

How ever , i f the ar r ival t imes are af fected by errors , then i t is

clear that fo r a g iven am ou nt o f er ror, i t s ef fect i s po ten t ial ly

larger fo r the 50 km event , which has a d t mu ch smal l e r t h an

th a t o f t h e 5 k m ev en t.

Nex t we wi l l i n v es t i g a t e wh a t h ap p en s wh en t h e ea r t h -

quakes occu r ou ts ide o f the netw ork . To s impl i fy the p resen-

tat ion i t wi l l be assumed that the ep icen ter i s a lways at the

64 Se ismo log ica lResearchLet te rs Vo lume75, Num ber1 January /February2004

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1 2

1 0 h = 5 k m 1 0 r8

~" 6 ~" 8

" ~ 4 - - -a . ~ 4

2

4 0 , I ~ I 4 0 , l I

O ( k r n ) - 4 0 - 2 0 0 2 0 4 0 0 - 4 0 - 2 0 0 2 0Yx ( k m )1 2

1 0 h = 2 5 k m 1 6 -

8

126

4 ~ ~ r ~ ~ 1 08

2I I I I I

4 0 0 - 4 0 - 2 0 0 2 0 4 0

I

4O

I I I I I4 0 0 - 4 0 - 2 0 0 2 0 4 0

1 6 " 1

1 4 4 h = 5 0 k m 2 0

1 81 2

8 .___ 12

6I I I I I I I

4 0 0 - 4 0 - 2 0 0 2 0 4 0 4 0 0 __ 40 - 2 0 0 2 0 4 0

A F i g u r e 1 . P-wave ( le f t) and S-wave ( r ight) t ravel- time sur faces for ear thquakes at var ious depths h generated using Equat ion 2 w ith T o= 0. All the epicen -

ters are at the or ig in. For the event at 5 km the t ravel t imes for par t icu lar values of x and y are a lso shown (dots) . When. the or ig in t ime is added to those t imes

they represent the ar r iva l t imes that wo uld be recorded by a seismic network.

or i g in o f t he coo r d i na t e s y s t em , t ha t To - 0 , and tha t the s ta-

t i ons i n t he ne t w or k have t he x coo r d i na t e l a r ge r t han o requal to X km. This means tha t the s ta t ion c loses t to the epi -

center wi l l be a t l east X km away f rom i t. To analyze the ef fec t

o f th i s geom e t r y on ea r t hquake l oca t i on w e w il l cons ide r

three events wi th h = 15, 17 , 19 km and X = 20.

The co r r e s pond i ng P - w ave a r r i va l - t i m e s u r f ace s ( F i gu r e

2A) are c lose to each other , a l though they are c lear ly d i s t in-

guishable . The t ime di f ferences be tween adjacent sur faces

range roughly between 0 .1 and 0 .2 s in absolute va lue , and

t he r e f o r e i t w ou l d s eem t ha t w hen l oca t i ng t he even t s i t

w ou l d be pos s ib l e t o r ecove r t he t h r ee s e ts o f hypocen t r a l

coordinates and or ig in t imes cor rec t ly . This i s not necessar i ly

the case, however. In fact , as Figure 2B shows, i t is possible to

m od i f y t he o r i g i n t i m es and ep i cen t ra l l oca t i ons o f tw o o f theeven ts i n s uch a w ay t ha t t he t h r ee s u r f ace s becom e a l m os t

in d i s t inguish able .

To gene r a t e F i gu r e 2B , 0 .87 km and 1 .74 km w er e added

to the e picent ra l co ordin ate x~ of the events a t 17 km and

19 km, and 0 .05 s and 0 .10 s , respect ive ly , were subt rac ted

f r om t he o r i g i n t i m e TO. In th i s case most of the t ime di f fer -

ences ( in absolute va lue) be tween adjacent sur faces are l ess

than 0 .05 s , wi th the l a rges t d i f ference less than 0 .08 s .

To s ee t he connec t i on be t w een t he s e r e s u l t s and t he

ea r t hquake l oca t i on p r ob l em , a s s um e t ha t an even t i s l oca t edus i ng s am pl e s f r om one o f t he s ur f ace s s how n i n F i gu r e 2A

and t ha t t he r e is som e a m o un t o f e rr o r i n t he p i ck i ng o f the

ar r iva l t imes . I f the d i f ferences Ti~ - T[ t ha t r em a i n a ft e r t he

even t ha s been l oca t ed a r e s i m i l a r i n m agn i t ude t o t he t i m e

di f ferences be tween the three sur faces in Figure 2B, then the

com pu t ed hypocen t r a l l oca t i ons and o r i g i n t i m e cou l d be any

of the t h r ee com bi na t i ons ( o r s om e o t he r ) t ha t w e r e u s ed to

genera te the sur faces . In prac tice , when locat ing on e of these

events one may get d i f ferent resul t s depending on the in i t i a l

va l ue s o f hypocen t r a l coo r d i na t e s and To us ed and / o r on t he

m a t hem a t i ca l t e chn i ques u s ed t o upda t e t he i n i t i a l e s t i m a t e s

a t e ach i t e r a ti on . The r e f o r e , F i gu r e 2B s how s t ha t u nde r t hos e

s am pl i ng cond i t i ons t he r e w i l l be a t r ade - o f f be t w een hypo-cen t r a l dep t h and ep i cen t r a l l oca t i on and o r i g i n t i m e . O f

course , the s i tua t ion wi l l be even worse when the ve loci ty

m ode l i s no t w e l l kno w n o r w hen 1D m ode l s a r e g r os sl y i nad -

equate , as i s the case in subduct ion zones . In the l a t t e r case

t he t r ave l - ti m e s u r face s no l onge r have t he az i m u t ha l s ym m e-

t ry seen in Figure 1 and there wi l l not be a sur face of revolu-

t ion tha t wi l l f i t a l l the observat ions per fec t ly , even when the

data are f ree f rom er rors .

Se ismo log ica l Research Le t te r s January /February2004 Vo lum e 75 , Number1 65

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( A )1 2

1 0

.E_ 8

n 6 f 4 0

4 0 2 0 0 - 2 0 - 4 0 -

y ( k m )

(B )1 2

1 0

a_ 6

f / f

4 0 2 0 0

~ 0 4 ~, , . , . L

- 2 0 - 4 0 - 4 0

= 1 2

1 0 4 0

/ / f /

4 0 2 0 0 - 2 0 - 4 0 - 4 0

A F igu re 2. (A) Three P-wave ar r iva l-t ime sur faces for events wi th h = 15,

17, 19 km and To= O. In all ca ses the ep icentral coordinates xe and Ye are

equal to zero and the values of xare equa l to 20 km or larger. (B) Sim ilar to

the surfaces in (A) after xe and To were changed to 0.87 km and -0.0 5 s for

h= 17 km, and to 1.74 km and -0.10 s for h= 19 km. For h= 15 km theparameters remained unchan ged. (C) S-wave ar r iva l- time sur faces cor re-

spon ding to the pa rameters used for the surfaces in 2B. Note that in 2B the

three sur faces are a lmost ind ist inguishable, whi le in th is f igure that is not

the case, particular ly for po ints close to the epicenter.

So far we have referred to P-wave t imes. What i s the

effect o f add ing S-wave t imes? I t i s very s ign i f ican t because

th ey ad d co mp le t e ly n ew in fo rmat io n t o t h e p ro b l em. Th i s

can b e seen wi th t h e h e lp o f F ig u re 2 C , wh ich w as o b t a in ed

by sh i f t ing the o r ig inal S-wave surfaces (no t shown) as the P-

wave surfaces were, wi th the resu l t that the S-wave surfaces

have no t moved toward a s ing le surface, par t icu lar ly fo r the

poin ts closest to the ep icen ter . Therefo re, i f the d i f ferences

TwO - Tic are s imi lar in magni tude to the t ime d i f ferences

seen in F igure 2B bu t smal ler than in F igure 2C, add ing S-

wave in format ion wi l l help const rain the even t locat ions .

There fore, i f avai lab le , S-wave data sh ould be used whe n

locat ing ear thquakes .

I t mus t be no ted , however , that p ick ing S-wave arrivals i s

no t as s imple as fo r the P waves fo r two reasons. F i rs t , when

only ver t ical -c om pone nt record ings are avai lab le i t i s possib le

to m i s id en ti fy l a rg e -amp l i tu d e co n v er t ed wav es ( e . g . , S-to-P)

as p r ima ry S waves. This i s l ikely to happ en in areas such as

the Mississ ipp i embayment , where low-veloci ty mater ials

over l ie h igh-veloci ty rocks ( e . g . , Pujo l e t a l . , 1 9 9 8 ) . Wh en

three-component record ings are avai lab le th is misiden t i f ica-

t ion prob lem does no t ar ise , bu t accurate iden t i f icat ion o f the

S-wave arrival may be d i f f icu lt because som et imes the S wave-

forms are compl icated , thus requ i r ing considerab le exper i -

ence on the pa r t o f the person p ick ing the ar r ival times.

Arr ivals f rom o ther waves can also be used to help con-s t rain the hypocent ral locat ions as long as thei r correspond-

ing t ime surfaces have geometr ic p roper t ies considerab ly

d i f feren t f rom those of the P-wave surfaces . I f the hypo cent ral

d is tances are no t too large, i t may no t be possib le to f ind or i -

g in t ime and/or dep th and ep icen t ral sh i f t s that wi l l s imul ta-

neously merge t ravel - t ime surfaces o f in t r ins ical ly d i f feren t

types . A good example i s the P head (or ref racted) waves,

which propagate along a layer boundary . For the s imple case

of a layer over a hal f-space, the t ravel t ime for an ev en t located

wi th in the layer is g iven by

-

1 ~ / ( x Xe + ( Y - - Y e +_ _ ) 2 ) 2 ( 2 H _ h ) 1

0 5 2 0 5 1 0 5 2

2(4 )

( e . g . , Lee and Stewar t , 1981), whe re H is layer th ickness , a 1

and a 2 are the wave veloci ties in the layer and in the hal f -

space, respectively, a 2 > a 1, an d the othe r s ym bols are as in

Eq u a t i o n 2 . Eq u a t i o n 4 can b e wr i t t en as

, ( x , y ) - D ( x , y ) + b ( 5 )

whe re a = 1 / a 2 , d i s the ep icen t ral d is tance (g iven b y thesquare roo t ) , and b i s a constan t equal to the second term on

the r igh t -hand s ide. Equat ion 5 represen ts a s t raigh t l ine in

the var iab les t and d , and the corresponding t ravel - t ime sur-

face i s a t runcate d cone w i th the ver t ical ax is passing th rou gh

the ep icen ter . The cone i s t runcated because the head waves

ex is t on ly fo r d is tances d that exceed a cr i t ical d is tance that

depends on H, h , and the two veloci t ies . In add i t ion , the head

waves can be d is t ingu ished f rom the d i rect P waves because of

thei r lower f requency conten t . These d i f ferences between the

two t y p es o f wav es sh o u ld b e k ep t i n m in d w h en t h e h y p er -

bo la method , d iscussed below, i s app l ied .

S-PTIMES EPICENTRALLOCATION METHOD

Th i s meth o d i s co mmo n ly d i scu ssed i n i n t ro d u c to ry b o o k s ,

bu t because the (s imple) der ivat ion of the equat ion that i s the

basis o f the m etho d i s general ly no t p rov ided , i t is g iven here.

Co n s id er ag a in a h o m o g en eo u s h a l f- sp ace wi th P - an d S -

wave veloci t ies a and /3 and let d be the d is tance between the

h y p o cen t e r an d a g iv en s t a t io n . Th en

66 Se ismo log ica l Research Le t te r s Vo lum e75 , Number1 January /Februa ry2004

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d d- - - ; ts = - - ( 6 )tp-a

a n d

t - P - I - ~ - l l d - -a l ( - ~ - -a l l d( 7 )

so th a t

- V ( , s - , , )

w i t h

a (9)v - ( a / / / ~ ) - 1

W i th th e s im p le v e lo c i ty m o d e l d i sc u sse d h ere , t h e h y p o -

c e n te r w i l l b e a t so m e p o in t o n th e su r fa c e o f t h e h e m isp h e rew i th th e c e n te r a t t h e s t a t io n a n d r a d iu s e q u a l t o d . I f tw o s t a -

t io n s h a v in g h y p o c e n t ra l d i s t a n c e s e q u a l t o d 1 a n d d 2 (n o t

e q u a l t o d l ) a r e c o n s id e re d , t h e re w i l l b e tw o in t e r se c t in g

h e m isp h e re s a n d th e h y p o c e n te r w i l l b e a p o in t a lo n g th e i r

l ine o f in te rsec t ion . I f a th i rd s ta t io n is ava i lab le , the in te rsec -

t io n o f t h e th re e h e m isp h e re s w i l l b e th e h y p o c e n te r . T h e se

c o n s id e ra t io n s a re th e b a s is o f t h e S - P t im e s e p ic e n t r a l l o c a -

t io n m e th o d , w h ic h i s a p p l i e d a s fo l lo w s . F o r e a c h s t a t io n

c o m p u te t~ - t p c o n v e r t i t t o d i s t a n c e m u l t ip ly in g b y a v e lo c -

i ty v th a t i s a p p ro p r i a t e fo r t h e a re a o f i n te re s t , a n d d ra w a c ir -

c l e o n a m a p u s in g th a t d i s t a n c e a s th e r a d iu s . Wh e n se v era l

s ta t ions a re used , the c i rc le s a re expec ted to over lap in some

c o m m o n a re a su r ro u n d in g th e e p ic e n te r . T o se e th i s c o n s id e r

tw o s t a t io n s , w h ic h fo r s im p l i c i ty w i l l b e a s su m e d to b e

a l ig n e d w i th th e e p ic e n te r (F ig u re 3 ) . T h e p ro je c t io n o f t h etw o h e m isp h e re s o n th e su r fa c e a re tw o c i r c l es c e n te re d a t t h e

s t a t io n s . C le a rly , t h e e p ic e n te r i s n o t a t t h e in t e r se c t io n o f t h e

tw o c i r c l e s , a l th o u g h i t w i l l b e c o m e c lo se r t o i t a s t h e d e p th

decreases . I f a th i rd s ta t io n is ava ilab le , the posi t i on o f the

th ird c i rc le is genera l ly dep ic ted as in F igure 3B.

T h e m e th o d w a s t e s t e d in i t i a l ly w i th a c tu a l d a t a

r e c o r d e d i n t h e N e w M a d r i d s e is m i c z on e b y s t a t io n s o f t h e

P A N D A p o r t a b l e n e t w o r k a n d w i t h s y n t h e t i c d a t a g e n e r a t e d

u s i n g s t at i o n l o c a t io n s f r o m t h e s a m e n e t w o r k a n d t w o v e l o c-

i ty m o d e l s . O n e w a s th e m o d e l u se d fo r t h e s t a n d a rd lo c a t io n

o f th e e ve n t s . T h e m o d e l h a s a n u p p e r l a y e r 0 . 6 5 k m th i c k

w i th lo w v a lu e s o f a a n d /3 ( e q u a l t o 1 . 8 a n d 0 . 6 k m /s ,

r e sp e c t iv e ly ) u n d e r l a in b y h ig h - v e lo c i ty ro c k s ( a e q u a l t o

5 . 9 5 k m /s ) (P u jo l e t a l . , 1 9 9 7 ) . T h e r a t io a / ~ i s equa l to 1 .73

for a l l the laye rs excep t the f i rs t one .

T h e s e c o n d m o d e l i s a v a r i a t io n o f t h e p re v io u s o n e , w i th

th e sa m e l a y e r b o u n d a r i e s , w i th o u t t h e lo w v e lo c i t i e s i n th e

( A )

(1 )

$ 1 E S 2

( B ) Y

> X

A F i g u r e 3 . G e o m e t ry f o r t h e S - P t im e s m e t h o d fo r th e a p p r o x im a t e d e t e r m in a t io n o f e p i c e n te r s . ( A ) d1 a n d d 2 a r e t h e d i s t a n c e s i n E q u a t i o n 8 f o r tw o d i ff e r e n t

s t a t i o n s ( in d i c a t e d b y $ 1 a n d 8 2 ) . T h e h y p o c e n t e r ( H ) l ie s a t t h e i n t e rs e c t io n o f th e t w o h e m is p h e r e s . T o s i m p l i fy t h e f i g u r e i t i s a s s u m e d t h a t th e e p i c e n t e r (E )

a n d t h e t w o s t a t i o n s a r e a l ig n e d . ( B ) M a p v i e w . T h e s o l id c i rc l e s a r e t h e i n t e rs e c t io n s w i t h t h e s u r f a c e o f th e h e m i s p h e r e s i n 3 A . N o t e t h a t t h e e p i c e n t e r i s n o t

a t t h e i n t e r s e c t io n o f th e c i r c le s . T h e d a s h e d c ir c le i s t h e p r o j e c t io n o f a h e m is p h e r e c o r r e s p o n d i n g t o s o m e o t h e r s t a ti o n .

S e i s m o l o g i c a l R e s e a r c h L e t t e r s J a n u a r y / F e b r u a r y 2 0 0 4 V o l u m e 7 5 , N u m b e r 1 6 7

8/7/2019 Pujol 2004


km

14 0

1 2 0

10 0

80

60

40

20

v l = 8 . 0 0 k m / s " h = 6 . 0 k m

- El.,

\

l,,

- \ ,, . . . . .

4 / - . . . ,/ , %

[] / 1 \

"~ - " / i /

\ . . . p ,

.... _ , , / / ' 0 , , ) ~ . ~ " " ". . . J \

- _ : ._ ,_. -,--, [] .\

I-I/ 'I -/ i

/ / " \ '

_ . . . . . . . ~- - ' - , , ,, [] ~ /, ,

' ?4' L /\ - -.

--- .,. /

" " v . . . . . . i

0 2'0 4'0 -6'0 8 0 10 0

km

140

1 2 0

v z = 8 . 0 0 k m / s "

10 0

80

6 0 -

4 0 -

2 0 -

120 140km 0

h = 1 8. 0 k m

\

\,

\,

\

'/

/

'1 /

[ ] i . //

]/"

lo o

\,

_ . . . . ~ /.

..< ~./" ~ ~' :

/ / \

/ ........ . , I

D " 1 I I

/ / i " .w- - / . . . . . . ~ -

. , ; /

"- . /, I {~ i",, i- ....,

. . .. . . - - - :~ . i . . . . . /_--< " [] ,,,

[ ] t / " , ,,

' z\ , " /

/

/

. / . . t \- ~ . . . . . . . . 9

, \ /

\2

2'0 ,l 'o a'o l z i O km

km

14 0

1 2 0

i 0 0

v = 6.50 kin /s; h = 6.0 km

'; i i

', /

',, , /:

",, / ./

", / / .-'1 2 0

-', / . ./

/ , / ' . /

,, :' / . ,:

, r-i / / .- " 10 0 -

, / / .--

', /: / ..l"

-. . /

80 "--.. ,, :. / // ..--

, / / .

[]. k, /' / . I

60 "'" ' '. . . . . "

40 . . . . / , , . , % . - - _ _

/ /V I :. : ',',, "%. - - ~ _

/ / .': ; , , / i t '", "~'(~'-.,/

2 0 . / , . i 7 ',,, ' . , D , \ \ " - . . 2 0 -

/'/ / ~ \,x ~ ~-

# i ,' : i , " i I

0 20 4'0 6'0 8'0 i 00 120 I "40 k m

km

140 -

8 0 -

6 0 -

4 0 -

v = 6.50 kin/s;

.

El .

' H

//

i

//

/

/

:i

i ,/ /

/ /"

/ I/"/ i / /

h = 1 8 .0 k m

/

/

//

// / -"

9 /

,/ / -

i/ ../ "

/ ./

// . /

/

/

/

/

/

/ . . ._ __

\, 1:/' / ..

- _ _ _ ~ . . , . L , ' ~ , I , . " _

/ / 5 ,/,' ': ',, " '- %

, .> ,,'I '~ "',. ~"k --.

0 : " / " ' ' "

0 2'0 4'0 6'0 8'0 i 00

i

1 2 0 1 4 0 k m

, i F igure 4. Top:Results of the S-P time s method when applied to synthetic data for two events with depths h equal to 6 and 18 km . The squares represent

stations and v1 is the velocity used to generate the circles u sing Equation 8. The actual epicenter is indicated by the circled dot. Bottom.Results of the hy perbola

method for the synthetic data used in the top plots. Because there are five stations, the num ber of hyperbolas is ten (se e Equation A-IO ). The veloc ity vwas

chosen such that the hyperbo las intersect at a com mon point.

upper layer and with a/fl alternating between 1.6 and 1.8.

This second model was introduced to make sure that the

results obtained were not a consequence of a constant velocity

ratio. The two models will be referred to as models I and II,

respectively. The first four examples considered below corre-

spond to synthetic data and constitute a small subset of all the

tests, but they represent a good summary of the resultsobtained using either synthetic or actual data.

To apply the method one should choose a value for the

velocity v in Equation 8. A rule of thumb is that v is equal to

8 for most crustal earthquakes (Lay and Wallace, 1995), but

this value is not always appropriate. In fact, it was noted that

the velocity to be used depends on the distance of the event

to the stations and on the event depth, which means that

before locating events with this method using data from a

given network it will be necessary to test it with known loca-

tions to establish the velocity (or velocities) to be used and the

general performance of the metho d.

Let us consider first two events at depths of 6 and 18 km

with the same epicenter surrounded by the five stations used

to locate them (Figure 4). For this test velocity model II was

used to generate the synthetic data. With a velocity of 8 km/sthe epicenter is near the center o f the area where all the circles

overlap. Reducing the velocity reduces the overlap area, thus

reducing the error made when estimating the epicentral loca-

tion. For a velocity of 7.4 km/s the epicenter is still within the

overlap area. For 7 km/s the area is significantly reduced and

close to the epicenters, but no longer within the overlap area.

Of course, when the epicentral location is unknown it is not

possible to determine the best velocity to be used.

68 Seismological Research Letters Volume 75, Number1 January/February20 04

8/7/2019 Pujol 2004


k m Vl= 7 .10 km /s" h = 0 .1 kmv2= 8 .15

140 t

12 0

. . . . . 7 ; - : ; : ~ - . = 0 - - ~ ~ i b ' - - . .' ~ <I .~- ~(';.... - ....... '::~7<<<'--'~\~\ "-

80 9J+ ~" ---. . . . . -~\- " \ \ .....

/ /, / \ , ,,,,\ ', t\\ \60 \,

" ["" i ~[ ':7

\, I"-1 [] 1 ,I ,/ ; ,i '' l

\ [] / / " ' ' i '0 i ~ / i/ i '

4 '0 ' '0 20 60 80 100 120 14 0k m

k m

140 -

120 -

10 0

80

60

4 0

20

v l= 8 .0 0 km /s" h = 2 0 .0 kmv2 = 9 ~ ~ Z - - ...........

....,x\, \

'7>, ','~\",.,

.,, ,,,,

t, "

!'

yl[]

[] /

[] ,/

2'0 . . . . .0 40 60 80 I00 120 i Okm

5.95v = Km/~'

k m

14 0

12 0

100

80

60

40

20

\

\

t,

\

\,

\, \

\ \

\ \

\ ',

\ .

\',,

h

/'

,,,,'

,,,'

,,

//

,,

/' ,

,,'

,;,'

,,

"' 7'

'/ i/' ,)

~--- /.), , , / , , I

t," ,"1 i' t/

- I , / "(,: []

E3 / '\,

0.1 km,: ' /' ;IW

I'l/ / "l,;' .,", / / i /,//,,,/, ,,

/ ; i , , . ,/,//,,~r.......

7r/

/

/

km

14 0

12 0

10 0

/

/

80

" 60/

/

40

20

v = 7 .10 km/s"

.... " '7 .....\ ,),!~ i

\ \ \tl '

h = 20.0 k m!

/ ,

, // , ,. '

/'

/

//

//

/ " , -

/ ' , ,

/ ,-

/' . ,"

/

//'

'~ 'iili",' i

'" k ' ' // -""

"?4 .i,\,,,,,7,./.-

\',,", I-I/

I-1/ \ ,

i i i

40 60 80

/

f./

2 'o ' 6 'o 8 'o ' ' 6 ' '0 40 100 1 0 14 0k m 0 20 1 0 120 14 0k m

i Figure 5. Top:Results of the S- Ptim es metho d when applied to two sets of synthetic data. The velocity v~ was used to generate the dashed circles under

the condition that all of them passed close to the known epicentral location. The velocity v2 was used to generate the solid circles and is equ al to the inverse

of the slope of the corresponding ts- tpvs. epicentral distance curve. To draw these curves the epicentral location m ust be known. S ee the text for details. Other

symbols as in Figure 4. Bottom. Results of the hyperbola method for the synthetic data used in the top plots. For the events at 20 km depth two of the arrival

times correspond to head waves and three to direct waves. In this case it is not possib le to find a common intersection point for all the hyperbolas.

If the epicenter is outside of the netw ork the overlap area

is not well defined and the selection of the velocity is not

straightforwar d. Figure 5 shows t he results for a very shallow

event (e.g., an explosion, h = 0.1 km) and for a deeper one

(h = 20 km). The arrival times were computed using velocity

model I. The dashed circles were drawn using Equation 8

with velocities v 1 equal to 7.1 and 8.0 km/s chosen in such a

way that all of them passed through points close to the

know n epicentral locations. If those locations were not

known, choosing the appropriate velocities would be diffi-

cult. Interestingly, by trial and error it was found that there is

a velocity (indicated with v2) for which the circles intersect at

a point almost perfectly. However, the intersection points are

about 11 and 18 km away from the epicenters, and the value

of v2 depends on the event.

The fact that it is possible to find a velocity for which the

circles intersect at a point was observed for all the other test

cases involving events in similar distance ranges (and even

smaller) and was investigated empirically. Two things were

noted. First, in each test the t s - t p vs. epicentral distance

curve (time-distance curve, for short) is a straight line. Sec-

ond, the velocity required for the intersection of the circles is

equal to the inverse of the slope s of the time-distance curve.

For the event at 0.1 km depth all the arrivals correspond to

head waves propagating along the bottom of the first layer,

and in this case the time-distance curve is a straight line. In

fact, from Equations 4 and 5 we see that

Seismological Research Letters Janu ary/F ebru ary20 04 Volume 75, Num ber1 69

8/7/2019 Pujol 2004


o r

(11)

wh ere c = b s - b p i s the t ime in tercep t . Eq uat io n 11 can be

wr i t t en as

D - v ( t s - t p ) - v c (12)

wi th v as i n Eq u a t i o n 9 . Eq u a t i o n 1 2 can b e i n t e rp re t ed i n

two ways. F i rs t , i t i s the e quat io n of a s t raigh t l ine in the t im e

and d is tance var iab les . I f we use Eq uat io n 9 wi th a = 5 .95

( i . e . , the veloci ty o f the head w aves) and a/ /3 = 1 .73 we get

v = 8 .15 , w hich i s equal to the value o f v2 used in F igure 5 .

Second , Equ at ion 12 is a l so the equ at ion of a ci rcle on the

(x, y ) p lane w hen D is f ixed . Therefore , un l ike the s i tuat iond esc r i b ed b y Eq u a t i o n 8 , wh en Eq u a t i o n 1 2 ap p l i es t h e c i r -

cles wi l l in tersect a t a po in t on the surface, in agreement wi th

the resu l t s shown in F igure 5 . In add i t ion , f rom Equat ion 11

we see that the t ime that must be used to d raw the ci rcles i s

actual ly ts - t p - c , n o t t , - t p . Becau se fo r co mmo n v a lu es o f

a/ /3 the value of c wi l l be posi t ive, the ef fect o f ignor ing i t i s

t o en l a rge t h e r ad ii o f t h e c i rc l es b y a co n s t an t am o u n t eq u a l

to v c . This exp lains why the po in ts where the ci rcles in tersect

i s a t a larger d is tance f ro m the s tat ions tha n the actual ep icen-

ter locat ion . For the even t at 0 .1 km depth the correct ion v c

i s equal to 10 .7 km, and when i t i s used to d raw the ci rcles

they in tersect a t the t rue ep icen t ral locat ion .

Equat ion 12 i s val id fo r head waves in arb i t rary layered

models as long as they come f rom the same layer . For those

waves the t ravel t ime vs . d is tance equat ion i s s imi lar to Equa-

t ion 5 , wi th a equal to the inverse o f the veloci ty o f the waves

an d b a co n s t an t t h a t d ep en d s o n t h e d ep th o f th e ev en t an d

on layer th icknesses and veloci t ies ( e . g . , Lee and Stewar t ,

1981) .

For the even t at 20 km dep th th ree o f the ar rivals corre-

sp o n d t o d i r ec t wav es an d two t o h ead wav es , b u t t h e t ime-

d is tance curve i s essen t ial ly a s t raigh t l ine. Therefore, we can

assum e that these waves are f ic t it ious head waves fo r which an

eq u a t i o n s imi l a r t o Eq u a t i o n 1 2 ap p li es wi th ap p ro p r i a t e v a l -

u es o f v an d c , wh ich mu s t b e d e t e rm in ed f ro m th e o b se rva-t i o n s . Th en wh a t we sa id i n t h e p reced in g p arag rap h ap p l i es

to th is case. The veloci ty v2 used in F igure 5 fo r the 20-k m -

dep th eve n t is equal to 1 / s . I t s large value (9 .34 km/s) i s ind ic-

at ive o f a deeper even t . T he P-wave veloci t ies in the layer that

con tains the even t and in the under ly ing layer are 6 .6 and

7 .3 k m/ s , r e sp ec ti v ely . W h e n u s in g Eq u a t i o n 9 wi th t h ese

two veloci t ies and a / 3 = 1 .73 we get v equal to 9 and 10 km /

s , wh ich b rack e t v2 . On t h e o th e r h an d , v c is equal to

1 7 .6 k m, an d wh en t h i s amo u n t i s su b t r ac t ed f ro m th e r ad i i

the ci rcles in tersect a t the t rue ep icen t ral locat ion .

To tes t the semiquant i ta t ive exp lanat ion g iven above, the

meth o d was ap p l i ed t o d a t a f ro m th e An d ean fo re l an d i n

Argent ina, where large lateral veloci ty var iat ions ex is t (Pu jo l ,

1992) . The s tat ions and even ts used (F igure 6 ) were selected

to create a worst -case scenar io . The P- an d S-wave s tat ion cor-

rect ion pai rs fo r the two s tat ions o n the r igh t o f the p lo ts are

( -0 .6 4 s , -0 .9 7 s ) an d ( -0 .8 2 s , - 1 .2 3 s ), r e sp ec ti v ely . F o r t h e

lef tmo st s tat ion the corre spon ding pai r i s (0 .63 s , 1 .52 s) ,wh i l e fo r t h e r emain in g s t a t i o n s t h e P -wav e co r r ec t i o n s a r e

b e t w e e n - 0 . 1 9 s a n d - 0 . 0 1 s an d t h e S - w av e co rr ec ti on s

b e tween 0 .1 2 s an d 0 .4 4 s . Th ese co r r ec t i o n s were co mp u ted

u s in g t h e j o in t h y p o cen t r a l d e t e rmin a t i o n ( JHD) t ech n iq u e .

For these even ts the ts - tp vs. d is tance curves fo r the s tat ions

of F igure 6 are approximately l inear . Because of the large

range of the correct ions , and p roba bly p ick ing errors also, i t

i s no t p ossib le to f ind a s ing le veloci ty fo r which al l the ci rcles

in tersect . However , when the veloci ty v2 and the values

d r = v c der ived f rom the t ime-d is tance curves are used , the

ci rcles in tersect a t po in ts close to the ep icen ters d eterm ined as

par t o f a convent iona l s ing le-even t locat ion . I f the even t loca-

t i o n s were n o t k n o wn i t wo u ld n o t b e p o ss ib l e t o g en era t e

t ime-d is tance curves , bu t us ing a s imple t r ia l -and-error

appro ach i t was possib le to f ind values v2 and d r fo r which al l

the ci rcles in tersect in a smal l area (F igure 6 ) . I f a po in t in that

area were chosen as the ep icen ter , then i t cou ld be used to

generate a t ravel - t ime curve f rom which new values o f z , ,2 and

d r could be der ived . Therefore, i t would be possib le to d raw

new ci rcles whose in tersect ions would be closer to the actual

ep icen t ral locat ion .

HYPERBOLA EPICENTRALLOCATION M ETHOD

To in t ro d u ce t h i s meth o d co n s id e r t h e fo l l o win g s i t u a t i o n .An ex p lo s io n o ccu r s i n a ch emica l p l an t an d two se i smo lo -

g i st s wh o h ear i t immed ia t e ly l o o k a t t h e i r ch ro n o mete r s an d

record the t imes. The seismologis ts are in d i f feren t par ts o f

t h e t o wn ; o n e o f t h em ca ll s th e o th e r a n d t h e two sh are t h e ir

i n fo rmat io n . W h at can t h ey say ab o u t t h e l o ca t i o n o f t h e

explosion? Let t I and r be the tw o record ed t im es an d d 1 and

d 2 the d is tances f rom the e xp losion to the po in ts w ere i t was

reco rd ed . O f co u rse , t h e d i s tan ces a r e n o t k n o wn , b u t t h e

se i smo lo g i st s k n o w th a t

4 = g ( t l - T o); 4 = g ( t 2 - T o ) ( 1 3 )

wh ere V i s t h e v e lo c it y o f so u n d an d TO is the o r ig in t ime.

Subtract ing the two d is tances g ives

d 1 - d 2 = V ( t 1 - t 2 ) ( 1 4 )

Th e r i g h t si d e o f Eq u a t i o n 1 4 is k n o w n . Th ere fo re , t h e ex p lo -

s ion occurre d along a curve tha t sati sf ies the fo l low ing condi-

t ion : The d i f ference in d is tances f rom two f ixed po in ts ( the

70 SeismologicalResearchLet ters Volume75,Number1 January/February2004

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= 1 3 . 5 k m / s " h = 1 0 7 k m ;V2k m

14 0

12 0

I0 0

80

60

40

20

d r = 1 0 7 k mk m

14 0

12 0

D 1 0 0

- [] 80....\

, '", , ..... (30

i ...... 4o

[] \ !] "\ \ i

~ ~ .~ / , i o4'0 . . . .0 2 0 6 0 8 0 l O 0 1 2 0 1 4 0 k m

v e = 1 4 . 3 k m / s " h = 1 0 5 k m ; d r = l l 7 k m

/

\.\\

.{ []

0

\\'\'"', ,,,

i

20 4'0 60 80 I O0 120 140 km

k m14 0

12 0

1 0 0

8 0

6 0

4 0

2 0

v 2 = 1 3 . 0 k m / s ; h = 1 0 7 k m ; d r = l O 0 k m

//

H

[]

i

0 2 0 4 0 6 0 8 0 10 0 1 2 0 1 4 0 k m

k m

14 0

12 0

1 0 0

80

60

40

20

0

v 2 = 1 3 . 0 k m / s " h = 1 0 5 k m ; d r = l O 0 k m

/

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 k m

, i F igure 6. Top.Results of the S-Ptim es method for actual data from two events recorded in the Andean foreland in Argentina, where large lateral velocity

variations exist. The circles were obtained using Equation 12 with v2 = vand dr = vcdetermined from the slope and intercept of the ts- tpvs. epicentral distance

curves. To draw these curves the epicentral location must be known. See the text for details. Other symbols as in Figure 4. Bottom. Similar to the top plots for

the same two events with v2 and dr determined by trial and error to force all the circles to intersect in the vicinity of a common point.

recording sites) to any point on the curve is constant. This is

the definition of a hyperbola. Although a hyperbola has two

branches, it would be easy to determ ine whic h one is relevant

because one of the two recorded times will generally be

smaller than the other. If they were the same, then the explo-

sion would be along a straight line perpendicular to the line

joining the two recording sites. Once the appropriate branchhas been identified, it can be drawn on a map. A person driv-

ing along the hyperbola would eventually find the explosion

site. If the explosion was recorded at three sites, then the cor-

responding times could be combined to generate three hyper-

bolas, and their common intersection would be the location

of the explosion.

We can apply these ideas to the problem of earthquake

location. Consider again a hom ogene ous med ium and an

earthquake recorded at a number of stations. Any two sta-

tions and the hypocenter define a plane on which we can

apply the results discussed for the explosion. For each station

pair there are thus a plane through the hypocenter and an

associated hyperbola that satisfies an equation similar to

Equation 14 with v replaced by a, and with the hypocenter

located at the point were all the hyperbolas meet (there maybe other points were some hyperbolas meet).

If the hyperbolas are projected onto the surface of the

Earth their projections will also be hyperbolas (although with

different equations) and their intersection at depth will

project onto the event epicenter. If the event depth is much

smaller than the epicentral distances a common velocity

likely will make all the surface hyperbolas intersect near a

com mo n point, but this is unlikely to happen for smaller epi-

Seismological Research Letters January/Fe bruary200 4 Volume 75, Numbe r1 71

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cen t ral d is tances , in which case there wi l l be an area around

the ep icen ter where al l the hyperbo las become close to each

other .

Th e h y p erb o l a meth o d was u sed b y M o h o ro v i e id (1 9 1 5 )

and was considered a p ract ical ep icen t ral locat ion method

u n t i l th e ad v en t o f t h e cu r r en t co mp u te r -b ased meth o d s . F o r

ex amp le , Ben -M en ah em an d Bf i t h (1 9 6 0 ) imp lemen ted an

an a lyt ica l meth o d wh o se g eo met r i c co u n t e rp ar t i s t h e h y p er -

b o l a meth o d , an d H u seb y e ( ca . 1 9 6 5 ) d ev e lo p ed a g rap h ica l

me th o d b ased o n t h e u se o f co mp u te r -g en era t ed se ts o fh y p erb o l as . Here we t es t t h e meth o d wi th t h e d a t a u sed t o

test the S - P t imes meth o d . Th e co n s t ru c t i o n o f t h e h y p erb o -

las is descr ibed in the App endix . Th e select ion of the veloci ty

was gu ided by the condi t ion that the hyperbo las in tersect a t a

co mmo n p o in t . To av o id b i as , t h e t ru e ep i cen t r a l l o ca t i o n

was as su med as u n k n o w n an d was sh o wn o n ly a f te r t h e v elo c -

i ty fo r a g iven even t was selected . For the f ive s tat ions used in

the tes ts there are ten possib le d i f feren t hyperbo las (see Equ a-

t ion A-10) , a l l o f wh ich w ere used .

F o r t h e two ev en t s wi th in t h e n e two rk (F igu re 4) t h e co r -

r esp o n d in g h y p erb o l as h av e a co m m o n p o in t o f i n t e rsec t i on ,

which co incides wi th the ep icen t ral locat ion , a l though in

so me o th er t e s ts i t was fo u n d t h a t i n s t ead o f a co mm o n p o in t

there i s a relatively smal l area where every hyperbo la in tersects

at leas t ano ther one, wi th the ep icen ter roughly at the cen ter

of the area. This s i tuat ion was fo und for s tat ion d is t r ibu t ions

l ike that shown in F igure 4 .

The hyperbo las fo r the even t at 0 .1 km depth in F igure 5

in tersect a t a po in t a lmost co inciden t wi th the ep icen ter fo r a

veloci ty o f 5 .95 km/s . This p recis ion in the veloci ty may appear

excessive, bu t a ch ange in veloc ity as small as 0.15 km/s is

enough to p roduce some hyperbo las that are considerab ly far

f rom the common in tersect ion . As no ted in the p rev ious sec-

tion, for this event all the arrivals correspond to head waves

refracted at the bo t to m of the f i rst layer and t ravel wi th a veloc-i ty o f 5 .95 km /s .

Th e even t at 20 km dep th in F igure 5 is in teres t ing

because i t shows that in som e cases it i s no t possib le to m ake

al l the hyperb o las in tersect in the v icin i ty o f a po in t . A lso in

th is case a smal l change (0 .1 km /s) in the veloci ty used has an

appreciab le ef fect on the hyperbo las . The reason for the fai l -

u re o f the m eth od i s that th ree o f the arr ivals corre spond to

d i rect waves and two to head waves, so that the basic hypoth-

esis o f the method that a l l the waves are o f the same type i s

v io lated . I f th is s i tuat ion occurs w i th an a ctual even t , one

should inspect the seismograms and select the s tat ion pai rs

t h a t i n c lu d e s imi la r t y p es o f wav es . W h en o n ly t h e t h r ee s t a -

t i on s w i th d i r ec t wav es were u sed , an o th er t y p e o f p ro b l em

occurred , namely , two qu i te d i f feren t in tersect ion po in ts , cor-

respon ding to close values o f velocity , were foun d . The refore,

in th is case the m eth od does no t g ive a rel iab le locat ion , bu t a

posi t ive aspect o f i t is that th is fact can be es tab l i shed f rom

the resu l t s ob tained .

Th e meth o d was a l so t e s t ed wi th t h e d a t a f ro m th e

An dea n foreland . In th is case, us ing a veloci ty o f about

10 .5 km/s , m ost o f the hyperbo las in tersect in the v icin i ty o f

a po in t that i s abou t 20 km to the sou theast o f the ep icen ters

sh o wn in F ig u re 6 . Th e r e l a t i o n b e tween t h e ep i cen t e r an d

the in tersect ion po in t i s s imi lar to that seen in F igure 5 fo r the

ev en t a t 2 0 k m. Us in g smal l e r n u mb er s o f h y p erb o l as d o es

not improve the resu l t s s ign i f ican t ly .

CONCLUSIONS

Th i s t u to r i a l sh ows t h a t t h e ea r t h q u ak e l o ca t i o n p ro b l em can

be in t roduced using a s imple g raphical approach that a l lows areal is t ic d iscussion of quest ions such as the t ra de-off between

depth and or ig in t ime and/or ep icen t ral locat ion , the ef fect o f

errors , and the advantages o f us ing S-wave or head-wave

arr ivals in add i t ion to the t rad i t ional P-wave arr ivals . Also

d iscussed are two approximate ep icen t ral locat ion methods:

t h e p o p u l a r S - P t imes meth o d , an d t h e o th e r b ased o n t h e

co n s t ru c t i o n o f h y p erb o l as an d t h a t r eq u i res o n ly P -wav e

arrival t imes.

Th e analysis o f syn thet ic an d actual data helpe d es tab l ish

g en era l co n d i t i o n s u n d er wh ich t h e two meth o d s can b e

expected to p rov ide reasonably good ep icen t ral es t imates . For

th e S - P t imes me th o d i t was fo u n d t h a t fo r ev ent s o u t s id e o f

the network i t i s somet imes possib le to locate the ep icen ter

wi th considerab le accuracy . For even ts ins ide the network the

h y p erb o l a m eth o d was fo u n d t o p e r fo rm v ery we l l. Bo th

methods requ i re veloci ty factors to conver t t imes to d is tances ,

and to ob tain the best resu l t s i t i s necessary to do a t r ia l -and-

error search for the best velocit ies. For this reason i t is conve-

n ien t to be ab le to generate the ci rcles and hyperbo las wi th a

computer , par t icu lar ly the lat ter , which are d i f f icu l t to d raw

manual ly . B y com puter iz ing the p rocess i t is possib le to locate

ep icen ters w i th er rors o f a few k i lome ters , par t icu lar ly i f the

two methods are used together .

In su mmary , t h ese meth o d s p ro v id e g o o d t each in g t o o l s

and produce ep icen t ral locat ions that are adequate fo rpro jects that requ i re on ly approximate resu l t s . Therefore,

t h ey can b e u sed as p a r t o f a co mp reh en s iv e p ro g ram o f ea r th -

q u ak e ed u ca t i o n i n v o lv in g u n d erg rad u a t es o r h ig h - sch o o l

s tudents in teres ted in p ro jects such as the U.S . Educat ional

Seismology Network in i t ia t ive (h t tp : / /www.ind iana.edu /

~usesn/about .h tmt) . E l

ACKNOWLEDGMENTS

Th i s w o rk was su p p o r t ed b y t h e S t a t e o f Ten n essee C en t e r s o f

Ex ce l l en ce P ro g ram. CERI co n t r i b u t i o n No . 4 6 7 . I t h an k S .

Ho u g h an d an an o n y mo u s r ev i ewer fo r t h e i r co n s t ru c t i v e

c o m m e n t s .

REFERENCES

Ben-Menahem, A. and M . B fith (1960). A m ethod for determination ofepicenters of near earthquakes, Geafis ica Pura e Applicata 46 ,37-46.

Husebye, E. (ca. 1965). A rap id, graphical m ethod for epicenter loca-tion (unpublished ), Seismological Institute , Uppsa la, Sweden.

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Jo h n so n , R . , E K i o k em ei s t e r , an d E . W o l k (1 9 7 8 ) . Calc ulus w i th Ana-lytic Geom etry, A l l y n an d B aco n .

Lay, T. and T . Wal lace (1995 ). M ode rn G lobal Se ismology , A cad em i c

Press.

Lee , W . a n d S . S t ew ar t (1 9 8 1 ) . Pr inc ip le s and Appl ic a t ions o fMicroearthquake Networks , A cad em i c P ress .

Mo h o ro v i e i d , A . (1 9 1 5 ). D i e b es t i m m u n g d es ep i zen t ru m e i nes n ah b e -

bens, Gerl. Beitr. z. Geophys. 1 4 , 1 9 9 -2 0 5 .

Pujol , J . (1992). Jo in t hypocentral locat ion in media wi th la teral veloc-

i ty varia t ions and in terpre tat ion o f the sta t ion correct ions, Physicsof the Ea rth an d P lanetary Inter iors 7 5 , 7 - 2 4 .

P u j ol , J . , R . H er rm a n n , S . -C . Ch i u , an d J . -M. C h i u (1 9 9 8 ) . Co n -s t ra i n ed j o i n t l o ca ti o n o f N ew Ma d r i d se i sm i c zo n e ea r th q u ak es ,

Seismological Research Letters 6 9 , 5 6 - 6 8 .

P u j ol , J . , A . Jo h n s t o n , J . -M. Ch i u , an d Y . -T . Y an g (1 9 9 7 ) . Re f i n em en to f t h ru s t fau l t i n g m o d e l s fo r t h e cen t ra l N ew Ma d r i d se i sm i c zon e ,

Engineer ing Geology 46, 281-298.Pujol, J . and R. S malley (199 0). A preliminary earthquake location

method based on a hyperbolic approximation to travel times, Bu l -

le t in o f the Seismological Society o f Amer ica 80, 1,629-1,642.Ruff, L. (20 01). How to locate earthquakes, Seismological Research Let-

ters 72, 197.

C E R I

T h e U n i v er s it y o f M e m p h i s

M e m p h i s , T N 3 8 1 5 2

p u j o l@ c e r i . m e m p h i s . e d u

AppendixConst ruc t ion of the Hyp erbol as

Re f e r t o F i g u r e A- 1 . Th e e a s t a n d n o r t h d i r e c t i o n s a r e u s e d t o

d e f i n e a lo c a l Ca r t e s i a n c o o r d i n a t e s y s t e m ( x, y ) , w i t h t h e s t a -t i o n c o o r d i n a t e s r e f e r r i n g t o t h i s s y s t e m. Th e p o i n t s A a n d B

r e p r e s e n t s t a t i o n s a n d a r e t h e f o c i o f th e h y p e r b o l a . Fo r a r b i -

t r a r y l o c a t i o n s o f t h e f o c i t h e e q u a t i o n o f t h e h y p e r b o l a i s

c o mp l i c a t e d . Fo r t h i s r e a s o n , b e f o r e c o n s t r u c t i n g t h e h y p e r -

b o l as t h e f o l l o w i n g t r a n s f o r m a t i o n s a re a s s u m e d t o h a v e b e e n

p e r f o r me d ( i n p r a c t i c e t h e y a r e n o t p e r f o r me d ) . F i r s t t h e o r i -

g i n O i s t r a n s l a te d t o O ; w h i c h i s a p o i n t e q u i d i s t a n t f r o m A

a n d B . T h e n t h e a x es a re r o t a t e d i n s u c h a w a y t h a t t h e n e w x

a x is is a l o n g a l i n e t h r o u g h A a n d B . Th e r e f o r e , t h e n e w y i s

p e r p e n d i c u l a r t o t h a t l i n e . Le t x ' a n d y ' i n d i c a t e t h e n e w a x e s .

I n t h i s s y s t e m t h e c o o r d i n a t e s o f A a n d B wi l l b e wr i t t e n a s

( -c , 0) an d (c, 0) , respect ive ly , w hic h m ean s th a t c i s ha l f oft h e d i s t a n c e b e t we e n t h e s t a t i o n s . Le t

I P A I - I P B I - + 2 a ( A- 1 )

O p e r a t i n g a n d i s o la t i n g t h e s q u a re r o o t o n t h e r i g h t s i de g iv e s

1 1 . , / , 2- - 2 c x ' - 1 - + _ - ~ e 2 + y ' . ( A- 4 )a a

S q u a r i n g a g a i n a n d a d d i t i o n a l s i m p l e o p e r a t i o n s g i ve

p2 p2x y

a 2 b 2= 1 ; b 2 -- C2 -- a 2 (A-5)

( a ft e r J o h n s o n e t a l . , 1978).

U s i n g E q u a t i o n A - 5 w r i t t e n a s

I / x , 2 / _~ . a 2 ) 1 ( A - 6 )

wh e r e t h e v e r t i c a l b a r s i n d i c a t e t h e d i s t a n c e b e t we e n p a i r s o f

p o i n t s , P i s a g e n e r i c p o i n t w i t h c o o r d i n a t e s ( x ', y ' ) , a n d 2 a i s

a c o n s t a n t e q u a l t o t h e r i g h t s i de o f E q u a t i o n 1 4 w i t h a n

a p p r o p r i a t e v e l o c i t y . I n c o m p o n e n t f o r m E q u a t i o n A - 1

b e c o m e s

~ /(X t + 6 ) 2 + y , 2 _ ~ / ( X t _ g ) 2 + y ,2 --_+2 a . (A-2)

N o w l e t d = x ' + c a n d e -- x ' - c , mo v e t h e s e c o n d t e r m o n t h e

l e f t o f Eq u a t i o n A - 2 t o t h e r i g h t , d i v i d e b o t h s i d es b y 2 a , a n d

s q u a r e t h e m. Th i s g i v e s

1 y , 2 1 y , 21 ( d 2 + y t 2 ) _ l + _ ( # 2 4 ) + _ ~ e 2 + . ( A -3 )4 a 2 4 a 2 - a

we c a n g e n e r a t e p a i r s ( x ' , y ' ) f o r x ' > 0 . Th e n t h e s e v a l u e s c a n

b e u s e d t o g e n e r a t e t h e p a i r s ( x ', - y ' ) . Th e s e o p e r a t i o n s g e n e r -

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

u e s o f x ' , wh i c h i s t h e d e s i r e d b r a n c h i f a > 0 . I f a < 0 u s e t h e

p o i n t s ( - x ' , y ' ) .

O n c e t h e p o i n t s o n t h e a p p r o p r i a t e b r a n c h o f t h e h y p e r -

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

o r i g i n a l c o o r d i n a t e s y s t e m. Th i s i s d o n e a s f o l l o ws . Le t ( Xo , ,Y o ' ) b e t h e c o o r d i n a t e s o f th e p o i n t O ' i n t h e o r i g in a l s y s te m ,

c o m p u t e d u s i n g

( X l + X 2 ) ( Y l + Y 2 )X o ' = ~ ; Y o ' - ~ (A-7)

2 2

w h e r e ( X l , Yl) and (x2 , Y2) are the coo rdin ates of the two s ta-

t i o n s . Le t 0 b e t h e a n g l e b e t we e n t h e s e g me n t A B a n d t h e x

Seismolog ica l Research Le t te rs Janua ry /February2004 Vo lum e75, Num ber1 73

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ax is . Th en t h e p o i n t s (x, y ) t h a t b e l o n g t o t h e h y p erb o l a a r e

o b t a i n ed u s i n g

x - cos Ox" + s in Oy" + x o, (A-8)

a n d

y - - s i n O x ' + c o s O y ' + Y o " (A-9)

Final ly le t us der ive the number o f d i f feren t hyperbo las

t h a t can b e g en era t ed wh en t h e re a re Ns t a t i o n s . F o r ex amp l e ,

le t N be 5 and label the s ta t ions 1 , 2 , 3 , 4 , 5 . The possib le

t wo - s t a t i o n co mb i n a t i o n s a r e

5-4, 5-3, 5-2, 5-1 (4)

4-3 , 4 -2 , 4 -1 (3 )

3 - 2 , 3 - 1 ( 2 )

2-1 (1)

wh ere t h e n u mb er s i n p a ren t h eses i n d i ca t e t h e n u mb er o f

co mb i n a t i o n s . Th i s sh o ws t h a t t h e t o t a l n u mb er i s eq u a l t o

1 + 2 + 3 + 4 - 10 . By ex tension we see that fo r N s tat ions

t h e n u m b er o f co mb i n a t i o n s , a n d t h u s h y p erb o l as , i s g i v en b y

X-1 1 N ( N - 1 ) ( A - I O )M - I + 2 + 3 + ' " + ( N - 1 ) - Z i - 2i=1

IPAI- IPBI = I Q A I - I Q BI

I R A I - I R BI = I S A I - IS B I

Nor th

Y

P

\

\

\

\

\

, # .

I

I /

I I //

/

I /

I /

I /

I /

Q !

B

~XEa s t

A F i gu r e A - 1 . Geom et r y fo r the de r i va t ion o f t he equa t i on o f t he hyper bo l a . T he xand yaxes cons t i tu te a loca l C ar tes i an coor d i na te sys tem ; A and B i nd ica te

stat ion locat ions. The two b ranche s of the hyp erbola sat is fy the equ al i t ies in the u pp er lef t corner . To construc t the hype rbolas thei r eq uat ions are der ived in a

r o ta ted coor d i na te sys tem cen te r ed a t the p o i n t O ' ( equ i d is tan t f rom A and B) w i th t he x ax i s a li gned w i th A and B .

7 4 S e i s m o l o g i c a l R e s e a r c h L e t t e r s V o l u m e 7 5 , N u m b e r 1 J a n u a r y / F e b r u a r y 2 0 0 4

pujol 2004

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