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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
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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 .
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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
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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
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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.
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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
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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
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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-
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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
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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
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, # .
I
I /
I I //
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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