appraising earthquake hypocenter location errors

8/11/2019 Appraising Earthquake Hypocenter Location Errors

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Bul le t in o f the Se i smolog ica l Soc ie ty o f Am er ica Vo l. 76 No . 6 p p . 1699-1717 Decem ber 1986

A P P R A I S I N G E A R T H Q U A K E H Y P O C E N T E R L O C AT I O N E R R O R S : AC O M P L E T E , P R A C T I C A L A P P R O A C H F O R S IN G L E - E V E N T

L O C AT I O N S

BY G RY L. PAVLIS

BSTR CT

F o r c o n v e n t i o n a l s i n g l e - e v e n t , n o n l i n e a r , l e a s t - s q u a r e s h y p o c e n t r a l e s t i m a t e s ,I s h o w t h a t th e t o t a l e rr o r i s e x p r e s s i b l e a s a l in e a r c o m b i n a t i o n o f t h r e e t e r m s :1 ) m e a s u r e m e n t e rr o r; 2 ) m o d e l in g e r r o r s c a u s e d b y in a d e q u a c y o f t h e t r a v el -

t i m e t a b l e s ; a n d 3 ) a n o n l i n e a r t e r m . E r r o r s in c a l c u l a ti n g t r a v e l - t i m e p a r t ia ld e r i v a t i v e s a r e s h o w n t o h a v e n o e f f e c t , p r o v i d e d a s t a b l e s o lu t io n c a n b e f o u n d .T h i s is in c o n t r a s t t o l in e a r p r o b l e m s w h e r e e r r o r s i n c a l c u l a t in g m a t r i x e l e m e n t sc a n d i s to r t th e s o l u ti o n d r a s t ic a l l y. T h e e r r o r a p p r a i s a l t e c h n i q u e d e v e l o p e d h e r ee x a m i n e s e a c h o f t h e t h re e e r r o r t e r m s i n d e p e n d e n tl y . T h e f ir s t c a n b e a n a l y z e db y s t a n d a r d c o n f i d e n c e e l l ip s o i d s w i t h c r it ic a l v a l u e s b a s e d o n m e a s u r e m e n te r r o r s t a t is t ic s . T h e s e c o n d c a n c a u s e c o n v e n t i o n a l e r r o r e ll ip s o i d c a l c u l a ti o n st h a t d e r i v e a c r i ti c a l v a l u e f r o m a n e s t i m a t e b a s e d o n r m s r e s i d u a l s , t o g i v em i s l e a d i n g r e s u l t s . I i n t r o d u c e a n a l te r n a t i v e e x t r e m a l b o u n d p r o c e d u r e f o ra p p r a i s i n g s u c h e r r o r s . T r a v e l - t i m e m o d e l i n g e r r o r s a r e b o u n d e d a s t h e p r o d u c to f r a y a r c l e n g th a n d a n e s t i m a t e o f t h e n o m i n a l s c a l e o f s l o w n e s s e r r o r s a l o n gt h e r a y p a t h . T h e s e a r e u s e d t o d e r i v e a n u p p e r b o u n d o n s y s t e m a t i c e r r o r s ine a c h h y p o c e n t r a l c o o r d i n a t e b a s e d o n a n o v e l b o u n d i n g c r i te r i a . F in a l ly, I s h o wt h a t , f o r e r r o r s o f a r e a s o n a b l e s c a l e , t h e n o n l i n e a r e r r o r t e r m c a n b e e s t i m a t e da d e q u a t e l y u s i n g a s e c o n d - o r d e r a p p r o x i m a t i o n . G i v e n a n u p p e r b o u n d o n t h et o t a l l o c a ti o n e r r o r , b o u n d s o n t h e t r a v e l - ti m e e r r o r in d u c e d b y n o n l in e a r i t y c a nb e c a l c u l a t e d f r o m t h e s p e c t r a l n o r m o f t h e H e s s i a n f o r e a c h m e a s u r e d a r r iv a lt i m e . T h e s y s t e m a t i c e r r o r s in e a c h h y p o c e n t r a l c o o r d i n a t e d u e t o n o n l i n e a r it yc a n t h e n b e b o u n d e d u s i n g th e s a m e c r i t e ri a u s e d f o r c o n s t r u c t in g m o d e l i n ge r r o r b o u n d s . T h i s o v e r a l l p r o c e d u r e i s c o m p l e t e b e c a u s e i t a l l o w s o n e t oi n d e p e n d e n t l y a p p r a i s e t h e r e l a t i v e i m p o r t a n c e o f a l l s o u r c e s o f h y p o c e n t r a le r r o r s. I t i s p r a c t i c a l b e c a u s e t h e r e q u i r e d c o m p u t a t i o n a l e f f o r t is s m a l l.

INTRODUCTION

S o m e o f t h e m o s t s i g n if ic a n t a d v a n c e s i n e a r t h s c ie n c e t h a t c a n b e a t t r ib u t e d t o

s e is m o l o g y h a v e r e s u l t e d f r o m i n t e r p r e ta t i o n s o f s p a ti a l p a t t e r n s o f e a r t h q u a k eh y p o c e n t e r s . F o r l a rg e s c a le o b s e r v a t i o n s , s u c h a s t h e f a c t t h a t e a r t h q u a k e s o c c u rm a i n l y a l o n g p l a t e b o u n d a r i e s , t h e i s su e o f e a r t h q u a k e l o c a t i o n e rr o r s i s o f m i n o ri m p o r t a n c e . A s w e t r y t o e x t r a c t m o r e i n f o r m a t i o n f r o m s e i s m i c i t y p a t t e r n s ,h o w e v e r , w e s o o n f a c e t h e i s su e o f h o w p r e c i s e e a r t h q u a k e l o c a t i o n s r e a l ly a r e.

I t is w e l l k n o w n t h a t e a r t h q u a k e l o c a t i o n s a r e s u b j e c t t o t w o q u i t e d i ff e r e n t t y p e so f e r ro r s : r e la t iv e lo ca t io n sca t t e r an d sy s t em a t i c b i a se s e .g ., Do u g la s , 1 9 67 ; Dew ey,1 97 1, 1 97 2; J o r d a n a n d S v e r d m p , 1 9 8 1) . T h e f i r st is c a u s e d b y m e a s u r e m e n t e r r o r sa n d s h o r t ra n g e f l u c t u a t i o n s i n t h e s e is m i c v e l o c i t y s t r u c t u r e o f t h e e a r t h L e a v e r ,1 9 84 ) a n d c a n p r o p e r l y b e t r e a t e d f r o m a p u r e l y s t a t i s t i c a l p o i n t o f v i ew. T h e l a t t e r

i s c a u s e d l a rg e ly b y i n a d e q u a t e k n o w l e d g e o f e v e n t h e l o n g w a v e l e n g t h v e l o c i tys t r u c t u r e o f t h e e a r t h e .g ., J o r d a n a n d S v e r d r u p , 1 9 81 ; P a v l i s a n d H o k a n s o n , 1 9 85 ).I a rg u e b e l o w t h a t i t i s n o t p r o p e r t o c o n s i d e r s u c h e r r o r s a s r e s u l ti n g f r o m a z e r om e a n r a n d o m p r o c e s s . C o n v e n t i o n a l e r r o r e ll ip s o i d s F l in n , 1 9 6 5 ) i m p l i c i tl y m a k et h i s a s s u m p t i o n J o r d a n a n d S v e r d r u p , 1 9 8 1 ), w h i c h I s h o w h e r e c a n y i e ld v e r ym i s l e a d i n g r e s u lt s . I n th i s p a p e r , I i n t ro d u c e a n e w, a l te r n a t i v e m e t h o d f o r b o u n d i n g

1699


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1 7 0 0 G A R Y L . P AV L I S

s y s t e m a t i c m i s lo c a t io n e r r o rs b a s e d o n a c o m p o n e n t - w i s e b o u n d i n g t h e o r y. U n l i k es im p l e m a t r i x v e c t o r n o r m b o u n d i n g c r it er ia , t h e b o u n d s c o n s t r u c t e d b y t h is n e wa p p r o a c h d o n o t t e n d t o b e o v e r l y p e s s i m i s t i c , p r o v i d e d o n e c a n s p e c i f y a r e a s o n a b l eu p p e r b o u n d o n t h e s c al e o f s l o w n e s s a n o m a l i e s w i t h in t h e n e t w o r k .

M o s t c o n v e n t i o n a l e r r o r e s t i m a t e s a r e b a s e d o n a l i n e a r a p p r o x i m a t i o n t o a s e to f n o n l i n e a r e q u a t i o n s [ te c h n i q u e s d i s c u s s e d b y Ta r a n t o l a a n d Va l e t te (1 9 82 ) a n dR o w l e t t a n d F o r s y t h ( 1 9 8 4 ) a r e e x c e p t i o n s ] . T h e a n a l y s i s I g i v e h e r e i n d i c a t e s t h a tn o n l i n e a r i t y c a n b e v i e w e d a s a d i f f e r e n t t y p e o f s y s t e m a t i c b i a s w h o s e e f f e c t isv i r t u a l l y i n s e p a r a b l e f r o m t h a t d u e t o m o d e l i n g e r r o rs . O n t h e p o s i t i v e s id e , h o w e v e r ,c a l c u l a t io n s p r e s e n t e d h e r e s u g g e s t t h e i n f l u e n c e o f n o n l i n e a r i t y i s p r o b a b l y s m a l lf o r m o s t l o c a t i o n s a n d i s l a rg e l y c o n t r o l l e d b y m o d e l i n g e r r o r b i a s e s. I n a n y c a s e , itis s h o w n t h a t s u c h e r ro r s c a n b e b o u n d e d u s in g a s e c o n d - o r d e r a p p r o x i m a t i o n a n da c o m p o n e n t - w i s e b o u n d i n g t h e o r y s i m i la r t o t h a t u s e d f o r b o u n d i n g t h e i n f l u e n c eo f m o d e l i n g e r r o rs .

T h e n e t r e s u l t o f t h i s p a p e r i s a p r o c e d u r e f o r a p p r a i s i n g l o c a t i o n e r r o r s t h a t Ic l a im i s c o m p l e t e a n d p r a c t ic a l . I t i s c o m p l e t e b e c a u s e i t g iv e s r e l a t iv e l y i n d e p e n d e n tm e a s u r e s f o r a ll t h r e e s o u r c e s o f h y p o c e n t r a l e r ro r : ( 1 ) m e a s u r e m e n t e r ro r s ; ( 2)v e l o c i t y m o d e l e rr o r s ; a n d (3 ) n o n l i n e a r i t y. T h e p r o c e d u r e i s p r a c t i c a l b e c a u s e t h ee r r o r e s t i m a t e s d e s c r i b e d a r e r e la t i v e l y e a s y t o c a lc u l a t e a n d r e q u i r e f e w a s s u m p -t io n s . T h i s i s i n c o n t r a s t t o t h e c o m p l e t e e r ro r a n a l y s i s o f Ta r a n t o l a a n d V a l e t t e( 19 8 2 ) w h i c h is c o m p l e t e b u t n o t n e c e s s a r i l y p r a c t i c a l b e c a u s e i t is c o m p u t a t i o n a l l yd e m a n d i n g a n d r e q u i r e s p r i o r i a s s u m p t i o n s a b o u t t h e f o r m o f t h e u n c e r t a i n t y i nt h e v e l o c i t y s t r u c t u r e o f t h e e a r t h ( T h u r b e r , 1 9 86 ).

FUNDAMENTAL EQUATIONS AND DEFINITIONS

T h e d a t a a l w a y s u s e d f o r e a r t h q u a k e l o c a t i o n s a re a s e t o f m a r r i v a l ti m e sm e a s u r e d f r o m o n e o r m o r e p h a s e s r e c o r d e d b y a n e t w o r k o f s e i sm o m e t e rs . I w il ld e n o t e t h i s v e c t o r o f m e a s u r e d a r r i v a l t i m e s a s ( t ) ~ R ~ . I t is r e la t e d t o r e a l i t y a sfo l lo ws

t ) = t + r l + ~ 1 )

w h e r e

t - - t r u e t r av e l - t im e v ec to r ;r = o r ig in t im e (1 E R m d e n o te s a v ec to r o f a l l o n es ) ; an d

= m e a s u r e m e n t e r r o r.

W i t h a c t u a l d a t a , t is u n k n o w a b l e . I n s t e a d , w e m u s t a l w a y s r e l y o n a m a t h e m a t i c a lm o d e l o f t c a l c u l a t e d f r o m s o m e e s t i m a t e o f t h e e a r t h ' s s e i s m i c v e l o c i t y s t r u c t u r e .I use tmodol( ) to sym bo l ize th is ve c to r o f t rave l t im es , tmoael i s a fun c t io n o f thee s t i m a t e d sp t i l c o o r d i n a t e s o f t h e h y p o c e n t e r , . A c o m p l e t e s p e c i f ic a t i o n o f t h eh y p o c e n t e r , o f c o u r s e , a l so r e q u i r e s a n e s t i m a t e o f r, ~ , a s w e ll . F o r c o n v e n i e n c e ,t h i s 4 - v e c t o r w il l b e s y m b o l i z e d a s /~ . W i t h t h e s e d e f i n i ti o n s , I d e f i n e th e r e s i d u a lf u n c t i o n

r ] ~ ) ---- t ) - - t m o d e l ; ~ ) - - T 1 . (2)

U s i n g e q u a t i o n ( 1) , e q u a t i o n ( 2) b e c o m e s

(3)


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PPR ISING E RTHQU KE HYPOCENTER LOC TION ERRORS 1701

w he re emodel X) = t - - t~odel X) a n d A r = r - ~ .

A l l c o m p u t e r i z e d e a r t h q u a k e l o c a ti o n m e t h o d s m i n i m i z eII r ( /~ ) I I , where I[ IId e n o t e s a v e c t o r n o r m . T h e d e t a i l s o f h o w t h i s p r o c e e d s v a r y g r e a t l y, b u t a l l m e t h o d sa r e t h e s a m e i n o n e r e s p e c t. A l l a r e i t e r a t i v e p r o c e d u r e s t h a t c a l c u l a t e a s e ri e s o f

c o r r e c t i o n s , 5 h h , t o e s t i m a t e h a s

/~ = h o + ~ 5 h h ( 4 )k= l

w h e r e niter i s t h e t o t a l n u m b e r o f c o r r e c t i o n s r e q u i r e d t o o b t a i n a s t a b l e s o l u t i o nf r o m a n i n i t i a l g u e ss , ho , o f t h e h y p o c e n t e r . B y s t a b l e I m e a n s p e c i fi c a ll y t h a t t h es e q u e n c e 5 hk i s c o n v e r g e n t s u c h t h a t

I I I I < 5 )

w h e r e e i s s o m e a p p r o p r i a t e l y s m a l l n u m b e r. T h e 5 hk a r e a l w a y s c a l c u l a t e d a s

5hk = A~-~ P /~k-1) 6)

w i t h f tk-1 = ho + F,~---~ 5 h l . A ~-I ER xm i s a g e n e r a l i z e d in v e r se . W i t h t h e e x c e p t i o no f t h e n o n l i n e a r m e t h o d d e s c r i b e d b y T h u r b e r 1 98 5) , A ~_~ i s a l w a y s c a l c u l a t e dd i r e c tl y f r o m t h e m a t r i x o f p a r t i a l d e r i v a ti v e s z[ w i t h c o m p o n e n t s

0X , j = t m o e ,) .

A,4 = l h4 = r ) .

j = 1 , 2 , 3 ) , a nd

7)

T h e r e a r e e s s e n t i a l l y a s m a n y v a r i a t i o n s i n h o w A h * i s c a l c u l a t e d a s t h e r e a r el o c a t i o n p r o g r a m s . F o r t u n a t e l y, w e d o n o t h a v e t o p r e s e n t a d i f f e r e n t e r r o r a n a l y s i sf o r e v e r y p o s s i b i l i ty t h a t e x i s ts . T h e r e a s o n i s t h a t e v e r y e x i s t i n g l o c a t i o n m e t h o dc a n b e e q u a t e d t o a w e i g h t e d le a s t - s q u a r e s p r o b l e m . F o r o u r p u r p o s e s , t h i s m e a n ss p e c i f i c a l l y t h a t t h e r e e x i s t s a p o s i t i v e d e f i n i t e m a t r i x , W E R mm f o r w h i c h t h es o l u t i o n to t h e u s u a l e q u a t i o n s o f c o n d i t i o n

= 8 )

s a t i s f i e s

5 h = A r < ~ ( 9 )

w h e r e

A = W ~ I , r = W ~ , 1 0)

i s a s in e q u a t i o n 5 ), a n d A + d e n o t e s t h e p s e u d o i n v e r s e o f A e .g ., L a w s o n a n dH a n s o n , 1 9 74 , p p . 3 6- 4 0 ). E q u a t i o n 9 ) m e a n s t h a t / ~ i s a s ta b l e s o l u t io n in t h el e a s t - s q u a r e s s e n s e t o t h e w e i g h t e d e q u a t i o n s o f c o n d i t i o n ,W A 6 h = W i .W i t hs t a n d a r d l e a s t -s q u a r e s p r o c e d u r e s li k e H Y P O 7 1 L e e a n d L a h r , 1 97 5) , t h e c o r re -


4/19

1 7 0 2 GARY L PAVLIS

s p o n d e n c e i s o b v i o u s . W i t h p r o c e d u r e s u s i n g d a m p e d l e a s t s q u a r e s (e .g ., H e r r m a n n ,1 97 9) o r th e r e c e n t l y p u b l i s h e d a p p l i c a ti o n o f N e w t o n ' s m e t h o d b y T h u r b e r ( 19 8 5) ,t h e c o r r e s p o n d e n c e i s n o t s o o b v i o u s . H o w e v e r , e q u a t i o n ( 9) s ti ll h o l d s b e c a u s e t h ep u r p o s e o f b o t h a l g o r it h m s is to p r o m o t e c o n v e r g e n c e o f t h e s e q u e n c e d e f i n e d in

eq u a t io n (4 ). B o t h seek a so lu t io n th a t m in im ize s I] r II=

[ ] T W 2] ] 1 /2 .A s l o n g a st h e l o c a t i o n i s w e l l c o n s t r a i n e d , b o t h m e t h o d s w i ll c o n v e rg e t o th e s a m e s o l u t i o n a sa n e q u i v a l e n t l e a s t -s q u a r e s p r o c e d u r e. F u r t h e r m o r e , e v e n a p r o c e d u r e w h i c h s e e k sto m inim ize L1 ( I] ? ]01 = ~ ?=1 ] r~ I ) can be ca st in th i s fo rm (A nd ers on , 1982) . W ea re a ls o fo r ce d , o n t h e o t h e r h a n d , t o m a k e t h r e e f u n d a m e n t a l a s s u m p t i o n s .

1 . h i s we l l co n s t r a in ed . T h a t i s, we a ss u m e A + i s n o t s in g u la r.2 . ]~ i s th e g loba l m in im um of 0] r ]1, no t a loc a l m in im a.3 . ]~ i s n o t e n o r m o u s l y d i f f e r e n t f r o m t h e t r u e h y p o c e n t e r , h .

T h e c o n s e q u e n c e s o f n o s . 2 a n d 3 a re i d e n t ic a l , b u t t h e w a y t h e y c a n a r i s e isd i f fe r e n t . W h a t I m e a n b y " e n o r m o u s l y d i f f e re n t " i s s t a t e d s p e c i fi c a ll y b e l o w.

A N A LY S I S O F H Y P O C E N T R A L E R R O R S

Second-order t h e o r y. / t i n e v i t a b l y c o n t a i n s e r r o r s t h a t a r i s e f r o m a n u m b e r o ff a c to r s . To e x a m i n e t h is , e x p a n d e a c h c o m p o n e n t o f t h e r e s id u a l v e c t o r i n a Ta y l o rse r i e s a s f o l lo ws (Lee an d S tew ar t , 1 9 8 1 , p . 1 2 4 )

4

~i(/~ + 5 h) = ~,(h ) + A,jSh j + fii (11 )J=l

w h e r e Av is a s d e f in e d i n e q u a t i o n ( 7 ), a n d n i i s t h e s u m o f a ll s e c o n d a n d h i g h e ro r d e r t e rm s o f t h e e x p a n s i o n . T h e m e q u a t i o n s o f ( 11 ) c a n b e s u m m a r i z e d i n m a t r ixf o r m a s

f ( h + 5 h ) = ? ( /~ ) + A Sh + f i ( 1 2 )

w h e r e t h e c o r r e s p o n d e n c e o f d i f fe r e n t te r m s is o b v i ou s . Tw o f a c ts a b o u t A a n d in e e d t o b e e m p h a s i z e d : ( 1) t h e y a r e c a l c u l a t e d f r o m a m a t h e m a t i c a l m o d e l o f r e a li ty,a n d ( 2) t h e y a r e t h e m s e l v e s n o n l i n e a r f u n c t i o n s o f h . I n e q u a t i o n ( 1 2) , b o t h a r ee v a l u a t e d a t / ~ .

W e s h a ll p u t e q u a t i o n ( 12 ) t o u s e b y a s s u m i n g 5 h is t h e d i ff e re n c e b e t w e e n h a n drea l i t y ( i.e ., h = / t + 5 h ) . I f we th en ap p ly th e we ig h t in g m a t r i ce s d e f in ed in

e q u a t i o n s ( 8) t o ( 10 ), w e o b t a i n t h e s i m i l a r f o r m

r = ~ A S h 13)

w h e r e n = W ~ . r a n d ~ a r e a s h o r t h a n d u s e d to d e n o t er h ) a n d r ( / t ) , r e s p e c t i v e l y.F r o m e q u a t i o n ( 9 ) , w e s e e

A+ r - 5h + A+ n = A+ ~ < ~, 1 4 )

s i n c e w e r e q u i r eA+ A = I . H ow ev er, f rom eq ua t ion (5 ) , II A +~ ]] ~ 0 . Hen ce , us ing

e q u a t i o n ( 3) , e q u a t i o n ( 14 ) c a n b e w r i t t e n a s

5 h ~ A + emod~l + n + e ) (15)

wi th in th e to l e r an c e p a r a m e te r sp ec i f i ed b y e. (N o te emode~ i s ev a lu a te d a t h , n o tt l . )


5/19

A P P R A I S I N G E A RT H Q U A K E H Y P O C E N T E R L O C AT IO N E R R O R S 17 3

A c o n v e n t i o n a l a n a l y s i s wo u l d a s s u m e n ~ 0 , y i e l d in g c la s s ic a l r e s u l ts i n v o l v i n ge r r o r e l li p s o id s ( F l i n n , 1 9 65 ). On e o f t h e m a i n p o i n t s o f t h i s p a p e r, h o we v e r, i s t oc o n s i d e r t h e i m p o r t a n c e o f n . To d o s o, we n e e d a m e t h o d f o r c a l c u l a ti n g i t.Ap p l i c a ti o n o f e q u a t i o n ( 5. 68 ) o f L e e a n d S t e w a r t ( 19 8 1 ) t o e q u a t i o n ( 11 ) y i e l d s

5~ = 5 h T H ~ S h . . . (16)

wh e r e H , is t h e H e s s i a n m a t r i x . A n a l y t i c f o r m s f o r H i fo r a c o n s t a n t v e l o c i t ym e d i u m a n d a t wo - l a y e r m o d e l a r e g iv e n in a r e c e n t p a p e r b y T h u r b e r ( 19 8 5 ). I u s eT h u r b e r ' s r e s u l t s b e l o w t o a p p r o x i m a t e 5 i t o s e c o n d o r d e r a s

~ ~ l S h T H i S h = (n2),. (17)

E q u a t i o n ( 1 5 ) t h e n b e c o m e s

~ h ~ A + e m o d e l + n 2 - b e ) . 18)

T h e r e a r e t wo l e v el s o f a p p r o x i m a t i o n i n e q u a t i o n ( 18 ): ( 1) t h e c o n v e rg e n c e c r i te r i o ne, a n d (2 ) t h e s e c o n d - o r d e r a p p r o x i m a t i o n o f n . ( T h e l a t t e r i s e m p h a s i z e d w i t h t h e2 s u b s c r i p t o n n .)

E q u a t i o n ( 18 ) is t h e f o c a l p o i n t o f t h i s p a p e r. T h e f i rs t s t e p i s c l e a rl y to i n v e s t i g a t et h e l i m i t s o f t h i s a p p r o x i m a t i o n . T h i s i s d o n e i n t h e f o l lo wi n g s e c t i o n u s in g c o m p u t e rs imu la t i ons .

C o m p u t e r s i m u l a t i o n s . I n t h i s s t u d y, I c h o s e t o c o n s i d e r t h e l o c a t i o n p r e c i s io n o f

e a r t h q u a k e s i n th e v i c i n i ty o f t h e r u p t u r e z o n e o f t h e 1 9 8 4 M o rg a n H i ll , C a l i fo r n i a,e a r t h q u a k e ( C o c k e r h a m a n d E a t o n , 1 9 84 ). A c ru d e a p p r o x i m a t i o n t o th e v e l o c i tys t r u c t u r e i n t h i s a r e a w a s u s e d t o g e n e r a t e t r a v e l t i m e s f o r a s e r ie s o f s y n t h e t i ce v e n t s. T h i s v e l o c it y m o d e l c o n s i s t e d o f t w o c o n s t a n t v e l o c it y q u a r te r - s p a c e s j o i n e da l o n g a v e r ti c a l p l a n e s t r ik i n g n o r t h 3 1 .5 we s t a n d p a s s i n g t h r o u g h t h e p o i n t3716 ' no r th l a t i t ude by 121o40 , w es t l ong i tude (F igu re 1 ). Syn th e t i c a r r i va l t imesf o r e v e r y s t a t io n i n t h e U . S . Ge o l o g ic a l S u r v e y C e n t r a l C a l i f o r n ia N e t w o r k( C A L N E T ) a n d a ll U n i v e r s i t y o f C a l i fo r n i a B e r k e l e y s t a t io n s w i t h in 1 00 k m o f t h isp o i n t ( 1 2 1 s t a t i o n s ) w e r e c a lc u l a t e d f r o m t h i s m o d e l f o r t h e s e t o f e v e n t s s h o w n i nF i g u r e 1 u s i n g t wo d i f f e r e n t q u a r t e r- s p a c e m o d e l s . Ve l o c i t i e s f o r t h e s e t wo m o d e l s

a r e g i v e n i n Ta b l e 1. T h e m e a s u r e m e n t e r r o r v e c t o r i n e q u a t i o n ( 1) wa s s i m u l a t e db y u s i n g a r a n d o m n u m b e r g e n e r a t o r to p r o d u c e r a n d o m s a m p l e s f ro m a n o r m a ld i s t r i b u t i o n w i t h z e ro m e a n a n d a v a r ia n c e o f 0 .0 5 se c . T h e s a m e ~ wa s t h e n a d d e dt o e a c h s y n t h e t i c e v e n t a r r iv a l t i m e v e c t o r. T h e a d v a n t a g e o f t h i s i s t h a t i t a l lo wsc o m p a r i s o n o f e r r o r s i n d u c e d b y a s a f u n c t i o n o f p o s i t io n . On th e o t h e r h a n d , i tg i ve s a p ri v i le g e d p o s i t i o n t o a s e t o f r a n d o m n u m b e r s . Ho we v e r, r e p e t i t i o n o f t h e s er e s u l ts w i t h d i f f e r e n t r a n d o m v e c t o r s in d i c a t e s t h e r e s u l t s a r e n o t v e r y d e p e n d e n to n t h e e x a c t c h o i c e o f ~ , a n d t h e o n e p r e s e n t e d h e r e i s r e p r e s e n t a t i v e .

A l l e v e n t s we r e l o c a t e d u s i n g a s i m p l e , d a m p e d l e a s t - s q u a r e s p r o c e d u r e s i m i l a rt o t h a t d e s c r ib e d b y H e r r m a n n ( 19 7 9) , u s i n g tr a v e l t im e s c a l c u la t e d f ro m a c o n s t a n t

v e l o c i t y m e d i u m wi t h a v e l o c i ty o f 5 .6 k m / s e c . F i g u r e s 2 a n d 3 s h o w t h e r e s u l t i n gl o c a t i o n e s t i m a t e s . A l l l o c a t i o n e s t i m a t e s s h o w a n e a s t wa r d b i a s c a u s e d b y a p p r o x -i m a t i n g t h e q u a r t e r- s p a c e m o d e l w i t h a c o n s t a n t v e l o c i t y m e d i u m . T h e s c a le o f t h i sb i a s i s ~ 1 0 k m f o r m o d e l A e v e n t s a n d ~ 2 k m f o r m o d e l B e v e n t s . F i g u r e 4 c a n b eu s e d to e x a m i n e t h e v a l i d it y o f t h e s e c o n d - o r d e r a p p r o x i m a t i o n i n th e c o n t e x t o ft h e s e t wo d i f f e r e n t e rr o r s c a le s . T h i s i s s u m m a r i z e d h e r e b y e x a m i n i n g o n l y t h e


6/19

1 7 0 4 G A RY L . PAV L I S

3 720 '

3 71 0 '

CMH CAOz~J A' ~ ~'MHC / /

~ cs c ~ _ ..~ ~ j

JST ~A~O ~o ~/o~ C C O ~0 5 Km \

? J A L . , , ' ,' ,' ' , ' , ' , , , C A D ; , , , ~ . . . . , ,

121 50 ' 121 40 ' 121 30 '

A Ao Booooo

o o o o o o o o o o o o o o o o o o oooooooo

E

. t= -10f:k{D

- 2 0

1 0 2 0 3 0Distance km )

F IG . 1 . A c t u a l l o c a ti o n o f s y n t h e t i c e v e n t s e x a m i n e d i n t h i s s tu d y. ( A ) M a p v i e w. ( B ) C r o s s - s e c ti o n .S t a t i o n l o c a t i o n s a r e m a r k e d w i t h t r i a n g l e s a n d t h e i r t h r e e c h a r a c te r n a m e c o d e . T h e c r o s s - s e c t io n A Ai s a s s h o w n i n t h e m a p v i e w. T h e d i a g o n a l l i n e r u n n i n g f r o m n o r t h w e s t t o s o u t h e a s t m a r k s t h e b o u n d a r yof the quar te r - s paces u s ed to genera te the s yn the t i c da ta a r r iva l t im es (T ab le 1 ) .

T B L E 1

S Y N T H E T I C Q U A RT E R - S PA C E M O D E L PA R A M E T E R S

Southwest NortheastMod el Quarter Space Quarter Space

Velocity Velocityk m / s e c ) k m / s e c )

A 5.0 6 .2B 5 .5 5 .7

n o r m o f t h e f i r s t - a n d s e c o n d - o r d e r p r e d i c t i o n e r r o r

E ~ = ]]~ h - A + ( e~ o d ~ l + e ) ] ]

a n d

E 2 - - ]] 5 h - A + ( em o d ~ + n 2 + e ) ] l .

T h e 4 - v e c t o r n o r m i s c a l c u l a t e d f o r a n y 4 - v e c t o r a a s

II a = [a~ 2 + a22+ a3 2 + v o 2 a4 2 ] 1 /2 ( 1 9 )


7/19

APPRAISING EARTHQUAKE HYPOCENTER LOCATION ERRORS

213720'

3710'-

~ M H C ~ ~

S T

0 5 Km \Z, AL i t t t t ~CAD

. . . . . . . ' ' : . . . . . . . . .

121 50' 121 40 ' 121 30'

1705

A

x:: -1 0

121

t`

i

-2o0 10 20 30Distance (km)

F[G. 2. Estimated locations of synthetic events from model A using 5.6 km/sec constant velocitymedium. For comparison, the map view A) and cross-section B) frames are identical to those shown inFigure 1. Estima ted locations are at t he ce nters of crossing lines which are the projecti ons of the majoraxes of the conventional 95 per cent confidence ellipsoids. Critical values for the se ellipsoids are basedon an F statisti c as originally advocated by Flin n 1966).

where v0 is the const ant velocity medium velocity. Vo is used to convert origin timeerrors into an equivalent length scale so all components of a are at least measuredin the same units.

For model A, the first-order errors are extremely large. This indicates that in thepresence of such a large bias, conventional first-order analysis techniques can beexpected to give very misleading results. Inc lusion of the second-order term improvesthe picture by a factor of about 4, but E2 is not negligibly small. Ee is still almosttwo orders of magnitude larger than the convergence parameter e which was 10 mfor all tests. On the other hand, for model B, Ee is generally of the same size as e.The only exceptions are the three shallowest sources along the vertical arm of thejack-shaped test patter n. This failure is to be expected, however, because the d epthsof these three events are poorly constrained and required auxiliary constraints toobtain a stable solution. This is not accounted for in this theory. Furthermore,Figure 2 shows that these events are located very close to station CCO. Since thesecond-order term varies as R -1, where R is the hypocen tral distance from a givenstat ion Thurber, 1985), one expects the thi rd-or der term to va ry as R -2. Thus,higher order terms would only be important for very short hypocentral distances,which is consis tent with these results.

I conclude from the results presented in Figure 4 that the second-order approx-


8/19


9/19

PPR ISING E RTHQU KE HYPOCENTER LOC TION ERRORS 17 7

ANALYZING DIFFERENT SOURCES OF HYPOCENTRAL ERROR

In t roduct ion . Having seen the limitations of the approximation given by equation18), we can approach the practical problem of what to do with this result. The

problem is that with real dat a the vectors emodel, n, and e are fun dam ent all y

unknown. Our only information about them is provided by a projection of theresidual vector, ( I - AA +) i . Unfortunately, the above analysis shows that ~ is asum of all three error terms and is evaluated at the wrong place in space-time. Thebasic idea here is to use auxiliary information to appraise the relative importanceof each term. This provides a valuable error appraisal tool to provide a morecomplete and realistic appraisal of location uncertainties. I now consider methodsfor estimating each of these terms. The order in which they are considered issignificant.

Measu rem en t e r ro r t e rm . Of the three terms in equation 18), e is unique in th atit is the only truly statistical quantity. Thus, although e may be unknowable, we

can assume we know someth ing about its statistics [see Freedman 1966) or Leaver1984) for examples of part icula rly careful studies]. For impulsive arrivals measured

from analog records, the are appro ximately normally distributed with zero meanBuland, 1976). For emergent arrivals, the e, tend to have a distribution skewed

toward positive numbe rs due to a tenden cy to pick weak arrivals too late Anderson,1982). Finally , with newer compu ter pick ing methods, the distribution of e~ may besomewhat complicated, but at least the gross details of the distribution are knownAllen, 1982; Leaver, 1984). In any case, if we know something about the p robability

densi ty func tion of e, it is a sta ndard exercise in regression analysis see Flinn,1965; Hoel, 1971; Jordan and Sverdrup, 1981; and their associated references) that

the errors induced by e can be appraised by examination of confidence ellipsoids.For the hypocente r location problem, this amoun ts to outlining an ellipsoidal regionin space-time characterized by

h - / t ) ~ C ~ - ~ h - ~ )


10/19

17 8 GARY L PAVLIS

where F, 4 , m - 4) is an F statist ic with 4 and m - 4 degrees of freedom for thecritical level ~ Flinn, 1965). Thi s almost always leads to error estima tes tha t areconsiderably larger than those based on measurement errors alone. The conven-tional wisdom is that this is justified, as it is one way to account for what I have

called emode~ and n . I hope to convince the reader in th e following sections tha t theconventional wisdom is wrong and a d ifferent analysis is necessary.M o d e l e r r o r t e r m . Unlike e, it is not really proper to consider emodo~ as a statistical

quantity. The failure of the statistical model to predict systematic biases in hypo-center location estimates is well doc umented e.g., Bolt e t a l . 1968; Brown and Lee,1971; Morrison e t a l . 1976; Uhrham mer , 1981). Nonethe less, i t is useful to considerresults from the synthetic data tests considered above as they demonstrate at leastpart of the reason for the failure of the statistical model to correctly predict errorsdue to emodel.

Figure 2 shows the principle axes of the 95 per cent confidence ellipsoids defined

by the st and ard F statisti c given in 22). None of the ellipsoids enclose the truelocations of any of the events examined here. Th is failure is not due to measur ementerrors as the error induced by e is less than 30 m for every event. The bestexplanation is that the application of equation 22) ignores a well-known caveat ofthe properties of the F distribution summarized in the following quote from anintroduc tory statistics book

Unfortu nately, the preceding test F test) is not reliable if X a nd Y do not possessnormal distributions. Ju st as in the case of applying the x 2 distribution to tes t asingle variance, the F distribution may be highly unreliable if X and Y possessdistributions whose fourth moments are considerably larger than for a normal

dis tribut ion Hoel, 1971, p. 273).For the case considered here, that turns out to be precisely the problem. Figure 5shows the distr ibution of individual components of ~, emodel, e, and n for all modelB events. Model A results are simila r except the time scale is roughly an order ofmagnitude larger. These are not shown for the sake of brevity.) If one did not lookcarefully at the residuals the only informat ion we have in real life), one might beled to think the residuals were Gaussian a nd jump to the conclusion tha t e~odel wasalso Gaussian. In reality, we see the componen ts of emod~l have a d istribution t ha tis basically bimodal due to the fact th at the data were generated from a two quarter-space model. A bimodal distribution is an example of a distribution with a large

fourth-order moment. This is obscured in the distribution of ~ by the added influenceof e and n, causing the distribution of ~ to be nearly Gaussian. Nonetheless, thefourth-order moment of the distribution of ~ was apparently sufficiently large tomake the F distribution error ellipsoids unreliable.

A statistician would say that the error ellipsoid problem described above wascaused by the violation of our assump tion t hat the residuals were Gaussian, and weshould take steps to account for the actual distribution of ~. I would argue thatwould accomplish little and misses the basic point. T ha t is, there are two kinds oferrors in hypo center estimates: 1) systematic biases and 2) scatt er in the relativelocation o f events. Th e former is dom ina ted by emod~l. The latte r is largely controlledby e a nd shor t spatial wavelength structu res in emod~l, but relative location scatte rcan also be induced by in teraction with e~odel thr oug h variat ions in effective arraygeometry caused by arrivals not being recorded at all stat ions e.g., Pavlis andHokanson, 1985). I assert that these two kinds of errors should be appraiseddifferently. Relative location errors can and should be appraised by error ellipsoidsbased on known properties of the measurement error statistics. Systematic biases


11/19

A P P R A IS IN G E A RT H Q U A K E H Y P O C E N T E R L O C AT IO N E R R O R S 1 7 0 9

1.0

0 8 -

(D0 0 6 -~DU -

> ~ 0 4 -

n 0 , 2 -

l- 0 3 - 0 '1

R e s i d u a ls M e a s u r e m e n t E r ro r

011 03

1 0 -

0 8 -

0 6 -

0 4 -

0 2 -

O 0- 0 3 - 0 1 0 1 0 3

1 0

o > , 0 8 -

0 6 -u_

~ 0 4 -

~ 0 2 -

O 0- 0 3 - 0 1

Modeling rror1 0

0 6

0 4

O 2

, 0 00 1 3 - 0 3

E r r o r ( s e c )

Second Order rrors

- 0 ' 1 0 '1 0 3E r r o r ( s e c )

FIG. 5. Freq uency distributions of different travel-tim e error terms from m odel B syn thetic d ata.Distributions w ere derived from all 53 e vents (5413 total travel times). E ach plo t is normalized to peakfrequency to facilitate comparisons. For the second-order, nonlinear errors, only two points fell outsidein the first three bins ( in terval 0 .01 sec). T he tw o extremes are beyond the dy namic range of the plot

and occur at t ime s ~0.0 4 and 0 .06, respectively.

s h o u l d b e t r e a t e d s e p a ra t e ly . I n o w a d v a n c e a n e w m e t h o d o l o g y f o r b o u n d i n gs y s t e m a t i c m i s l o c a t i o n s .

T o b o u n d em oa ~l, w e n e e d t o r e c o g n i z e i t s r e l a t io n t o t h e s e i s m i c v e l o c i t y s t r u c t u r eo f t h e e a r t h . A s s u m i n g r a y t h e o r y a n d g i v e n t h e d e f i n i t i o n o f em od~l [ e q u a t i o n ( 3) ] ,i t f o l l o w s t h a t

e m o d e l ) z= ~ t r u e s U m o d e l st ~ ~ j F modeL

| U t r u e - - U r n o d e l ) sr t ~odel 23)

w h e r e u tru~ s t h e a c t u a l s l o w n e s s a s a f u n c t i o n o f p o s i t i o n w i t h i n t h e e a r t h a n d F f f i s t h e a c t u a l r a y p a t h c o r r e s p o n d i n g t o t h e p h a s e d e f i n e d a s a r r iv a l i. S i m i l a r l y ,F, m d~ i s th e r a y p a t h c o r r e s p o n d i n g t o t h e s a m e a r r iv a l , b u t b a s e d o n a m o d e l o ft h e s l o w n e s s f u n c t i o n , U ~nodel. I n t h e e a r t h ,r ~ t r u ea n d F, rede1 c a n b e e x p e c t e d t o n o td i ff e r e n o r m o u s l y a n d th e ~ i n (2 3 ) is r e a s o n a b l y s a t is f i ed . S i n c e r o c k s l o w n e s s i s

a l w a y s f in i te , th e i n t e g r a l i n e q u a t i o n ( 23 ) c a n b e b o u n d e d . T h i s y i e l d s t h e b o u n d

I (~modo~), [ =< ~ , A u (2 4)

w h e r e A u i s a n u p p e r b o u n d o n I U tru e - - Um ode~ a n d ~ , i s t h e r a y p a t h l e n g t h , fd sa l o n g F, m del. T h u s , e a c h c o m p o n e n t o f em odel c a n b e b o u n d e d i n t e r m s o f a c o m m o n


12/19


13/19


14/19

7 2 G A RY L . PAV L I S

wh ere II/-/~ I[ im p l i e s th e L2 o r sp e c t r a l n o rm o f H , b ec au se th e seco n d -o rd e r t e rm ineq u a t io n (1 7 ) is q u a d ra t i c . T h e sca l e, p , i n eq u a t io n (2 6 ) in v o lv es o n ly th e sp a t i a lc o o r d i n a t e s , a s t h e H e s s i a n d o e s n o t d e p e n d o n t h e o r ig i n ti m e ( T h u r b e r , 1 9 85 ).W e c a n t h e n a g a in a p p l y t h e t h e o r e m i n t h e A p p e n d i x t o b o u n d t h e e rr o r, 5 h~ , d u e

t o n o n l i n e a r i t y a s

II (Sh ~), II < lp 2 ~ I A ~ III ~ II. (2 8)2=1

Th ese b o u n d s , l i k e th o s e u sed fo r em od el, a r e b a sed o n a sca l e f ac to r, p , an d a se t o fv a r i ab le f ac to r s , [ ]/ /J II, wh ich v a ry w id e ly fo r d i f f e r en t a r r iv a l s . T h e sca l e f ac to r i se s p e c i a ll y c r u c i a l i n th i s c a s e , h o w e v e r , a s t h e c a l c u l a t e d b o u n d s d e p e n d o n p 2.T h e r e f o r e , b e f o r e p r o c e e d i n g , i t is n e c e s s a r y t o d i s c u s s h o w p c a n b e e s t i m a t e d .

N o r m a l l y, i t ca n b e e x p e c t e d t h a t t h e t o t a l e r r o r i s d o m i n a t e d b y t h e e f f e c ts o f

emoted. In th is case , th e m os t o bv iou s es t i m ate o f p is ]]~hm odel I]w h e r e ~hmo el i s t h ev e c t o r o f b o u n d s i n e q u a t i o n ( 25 ). T h i s w i ll c e r ta i n l y y i e l d a s e t o f a b s o l u t e b o u n d s .U n f o r t u n a t e l y, t h e r e s u l t s w i ll t e n d t o b e o v e r l y p e s s i m i s t ic . A g r a p h i c i l lu s t r a t i o no f t h i s is s h o w n i n F i g u r e 7. T h e s e t o f b o u n d s l a b e l e d c w e r e c o n s t r u c t e d t h i sw a y. F ig u r e 7 s h o w s o n l y r e s u l t s f o r t h e y c o o r d i n a t e o f t h e s e r ie s o f s y n t h e t i ce v e n t s p e r p e n d i c u l a r t o t h e s t r ik e o f t h e f a u l t s h o w n i n F i g u r e 1, b u t t h e s e r e s u l t sa r e r e p r e s e n t a t i v e . T h e b o u n d s a r e s o l a rg e t h a t t h e y a r e n o t o n l y u se l e s s, b u t a l s ov e r y m i s le a d i n g . T h i s o c c u r s b e c a u s e w e a r e u s i n g t h e s q u a r e o f o n e b o u n d t oc o n s t r u c t a n o t h e r , w h i c h i n e v i t a b l y g e t s t o o p e s s im i s t i c . I e x p e r i m e n t e d w i t he s t im a t i n g p fr o m r m s r e s id u a ls , b u t t h i s h a d t h e o p p o s i t e p r o b l e m . W e s a w a b o v e

t h a t e r r o r e ll ip s o i d s s c a l e d u s i n g r m s u n d e r e s t i m a t e d t h e s y s t e m a t i c b ia s in t h e s el o c a ti o n s . F o r t h e s a m e r e a s o n , a n y e s t i m a t e s o f p d e r i v e d f ro m r m s a c c o r d i n g t oa n y c o n v e n t i o n a l r e c i p e w i ll u n d e r e s t i m a t e p le a d in g t o a n u n d e r e s t i m a t e o f t h eb o u n d s d u e t o n .

A n e s t i m a t e f o r p t h a t p r o v e d a d e q u a t e , a l t h o u g h s t il l n o t e n t i r e l y s a ti s fy i n g , w a sa c r u d e g u e s s o n e m i g h t b e a b l e t o m a k e w i t h t h e r e a l d a t a t h a t t h e s e s y n t h e t i cd a t a m o d e l s re p r e s e n t . T h a t i s, a ll e v e n t s a r e o f f s e t p e r p e n d i c u l a r t o t h e f a u l t b ye p i c e n t r a l d i s t a n c e s o f ~ 9 a n d ~ 1 . 5 k m f o r m o d e l s A a n d B , r e s p e c ti v e l y. I f w ep r e s u m e t h i s i s r e p r e s e n t a t i v e f o r o n e d e g r e e o f f r e e d o m o f t h e s o l u t i o n , t h e n ar e a s o n a b l e g u e s s f o r t h e s c a le p i s 9 ~ f3 k m a n d 1 .5 ~ k m f o r m o d e l s A a n d B ,

r e s p e c t iv e l y, s in c e 5 x h a s t h r e e d e g r e e s o f f r e e d o m . U s i n g t h e s e v a l u e s r e s u l t s i nt h e b o u n d s s h o w n i n F i g u r e 7 l a b e l e d b . T h e s e a r e s ti ll s o m e w h a t o v e r l y p e s s i-m i s t ic , b e i n g a p p r o x i m a t e l y a n o r d e r o f m a g n i t u d e l a rg e r t h a n t h e a c t u a l e r ro r. T h i si s l a rg e l y c a u s e d b y t h e f a c t t h a t t h e s e e s t i m a t e s o f p a r e n o t p e r f e c t e i th e r. I d e a lv a l u e s o f p a re v a l u e s o f ~ 1 0 a n d - 1 . 5 k m f o r m o d e l s A a n d B , r e s p e c ti v e l y. H e n c e ,i f w e k n e w p a p r i o r i w e w o u l d b e a b le t o r e d u c e th e b o u n d s s h o w n i n F ig u r e 7 b ya f a c t o r o f 3 . I n t h a t c a se , th e s e b o u n d s w o u l d d o a r e m a r k a b l y g o o d j o b o f m e a s u r in gt h e p o t e n t i a l i m p a c t o f n o n l i n e a r i t y i n t h e s o l u t io n . I t is i m p o r t a n t t o r e c o g n iz et h a t t h i s i s a m a j o r a d v a n t a g e o f t h e b o u n d i n g c r i t e r i o n b a s e d o n e q u a t i o n ( 28 ).T h a t is , p r o v i d e d p ca n b e e s t i m a t e d a c c u r a t e l y b y s o m e a u x i l ia r y m e a n s , t h e b o u n d s

p r o v i d e d b y ( 2 8 ) c a n b e e x p e c t e d t o b e r e a s o n a b l e . T h i s i s a g a i n i n c o n t r a s t t os i m p l e r s c h e m e s I f i r s t t r i e d u s i n g s t a n d a r d b o u n d i n g m e t h o d s b a s e d o n m a t r i xno rm s ( i.e. , 5hn II =< A + JR II n ]1 for an y pai r o f co ns is te n t m at r ix an d ve ct or no rm s,p r o v i d e d ]l n ]] i s a n u p p e r b o u n d o n t h e t r u e n o r m o f n ) . T h e s e a l w a y s g a v e t e r r ib l er e s u l t s e v e n w h e n p w a s c h o s e n e x a c t l y.


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APPRAISING EARTHQUAKE HYPOCENTER LOCATION ERRORS 1713

30 O0

20 00-

10 00-

o O 0

LU

--10 O0

2 0 O 0

-30 00

0 60-

0 40-

0.20 -

O.O0LU

--0.20-

- 0 4 0

0 60-

b

_ _ _ _ _ A

I I i I I I t ] l

-7 -5 -3 -1 1 3 5 7 9

b

b

1 1 o 5 7

Distance from Ongm km)

FIG. 7. Example of nonlin ear error bounding estimates. A) Model A and B) model B. Results shownare for the east-west epicentral coordinate for the line of events oriented perpendicular to the strike

northeast to southwest) in Figure 1, which are representative. The b s c i s s is as described in Figure 4.The actual n onlinea r errors are plotted as discrete point s as triangles labeled a. The bounds construct edfrom the extr emal bounds shown in Figure 6 are those labeled as c, Bounds cons tructed from the guessdescribed in the text are those labeled as b. Note the extreme difference in the scale between results formodels A and B.

DISCUSSION AND CONCLUSIONS

The first major result of this paper is equation (18). It is significant that this wasnot the result I originally expected when I began this study. My original inte nt wasto consider the impact of the fact th at the matrix of partial derivatives [A definedby equations (7) a nd (10)] was calculated from a model of the earth s velocitystructure in the same way the travel time is. It is well known that errors incalculating coefficients of a matrix cause errors in solutions of linear equations.Such errors are commonly appraised by matrix peturbation analysis methods (forleast-squares problems, see, e.g., Lawson and Hanson, 1974, pp. 41-52) to boundthe possible influence of computational errors in real computers with a finiteprecision. The analysis leading to equation (18) shows that even though A is not

calculated perfectly, it makes little difference as long as a stable solution can befound. The basic reason is that in a linear problem, A is fixed; here A is variable.We try to minimize I[ r [] by a sequence of steps given in equat ion (6). Each s tep isa linear one designed to minimize the norm N r A S h J] based on the current valuesof r and A, which vary from step to step. The net result is tha t when the solution


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1714 GARY L. PAVLIS

converges, the only relevant measure of A is the curr ent one. Hence, the fact thatit may be wrong due to inadequacies in calculating ray takeoff angles is irrelevant.

Equa tion 18) shows th at hypoce ntra l errors are a composite of three terms: 1)meas uremen t error; 2) wha t I have called modeling errors; and 3) a non linear

term. Of these, I claim only the f irst shou ld really be viewed as a statis tical quanti ty.This term can be appraised with confidence ellipsoids, but the scale of the processshould be determine d from indep endent studies of measur ement errors such as thatby Fre edman 1966) or Leaver 1984). Scaling confidence ellipsoids by rms followingFlin n 1965) is a dangerous step tha t can produce misleading results as shown bythe results given in Figure 2. This occurred because the synthetic examples studiedhere model a feature of most real data. Th at is, the errors are domin ated by modelingerrors. In this paper, I introduced an alternat ive means for appraising the influenceof modeling errors using a component-wise bounding criteria based on a theoremproven in the Appendix. These bounds are based on arrival-dependent ray arc

lengths and a common scale factor, Au, which is a bound on the average slownessalong a given ray segment. The result is a parallelepiped-shaped bounding regionfor each ear thquake location whose relative dimensions are fixed and whose absolutescale depends linearly on Au.

The third source of earthquake location errors, nonlinearity, is almost alwaysignored in con ventional location procedures. T he analysis presented here indicatesnonlinearity can be viewed as an additional component of the systematic locationbias superimposed on top of that caused by modeling errors. Evidence from thesynthetic examples presented here suggests that such errors can be approximatedto adequate precision for any reasonable location estimate using a second-order

approximation. Using a second-order approxi mation an d a component-wise bound-ing procedure similar to that used for modeling errors, the expected size of thenonlinear error can be bounded. These bounds are constructed from the spectralnorm o f the Hessian for each arrival Thurber, 1985) an d a common scale factor, p,which is a guess of an upper bound on the total hypocentral error for that event.Again, these bounds form a parallelepiped-shaped region of space whose relativedimensions are fixed. In this case, however, the scale of the bounding region isdetermined by the square of the scale factor p. Consequently, the bound s e stimatedthis way are reasonable only when p is chosen reasonably.

How to best estimate th e scale factors Au and p used in the bound ing procedures

described here is an open question. Au is probably best esti mated by nonparame tricstatis tical methods Efron and Gong, 1983) or from a ran dom media viewpoint asadvoca ted by Leaver 1985). Th e best approach for p in many cases is probably tomake a reasonable guess of its size based on other information. For example, if agroup of earthquakes appear to be systema tically offset a distance d from a map pedfault trace, p could be estimated from d as described above. Lacking such informa-tion, p may be esti mated from some measure based on rms residuals or the extremalbounds on systematic biases described here. In doing so, one must recognize,however, that the former may und eresti mate p, and th e latter will always overesti-mate p. In any case, one should recognize the main advantage of a set of bounds

based on a common scale factor. If one decides the original scale factors used tocalculate the bounds were in error, recalculating them based on the revised scale istrivial.

I would claim that one of the most significant facts about the error appraisaltechniques described here is their practicality. The error estimates are easy tocalculate, and the results are easy to understand at a glance. Furthermore, imple-


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A P P R A I S I N G E A RT H Q U A K E H Y P O C E N T E R L O C AT IO N E R R O R S 1 7 1 5

m e n t i n g t h e m r e q u i r e s o n l y m i n o r m o d i f i c a t i o n s t o m o s t l o c a t i o n p r o g r a m s . R e -s c a l i n g c o n f i d e n c e e l l i p s e s t o r e f l e c t o n l y m e a s u r e m e n t e r r o r s i s t r i v i a l a n d i sa l r e a d y a n o p t i o n i n a t l e a s t o n e c o m m o n l o c a t i o n p r o g r a m I a m a w a r e o f

H Y P O I N V E R S E , K l e in , 1 9 78 ). C a l c u la t in g u n s c a l e d m o d e l a n d n o n l i n e a r b o u n d s

r e q u i r e s o n l y a m i n o r m o d i f i c a t i o n t o a n y p r o g r a m t h a t e x p l i c it l y c a l c u l a te s A +.O n e t h e n o n l y h a s t o c a lc u l a te t h e v e c t o r o f c o m p o n e n t b o u n d s d e f i n e d i n e q u a t io n s2 4 ) a n d 2 7 ) . T h e m o d e l e r r o r b o u n d e s t i m a t e r e q u i r e s t h e c a l c u l a t io n o f r a y a rc

l e n g th s . F o r l o c a l n e t w o r k s , t h i s c a l c u l a t i o n c a n p r o b a b l y b e a p p r o x i m a t e d a d e -q u a t e l y a s t h e t o t a l s o u r c e r e c e i v e r sp a t i a l s e p a r a t i o n . F o r t e l e s e i s m i c l o c a ti o n s , i tw o u l d p r o b a b l y r e q u i r e a t a b l e . T h e n o n l i n e a r e r r o r b o u n d c a l c u l a t io n r e q u i r e sc a l c u l a t io n o f t h e s p e c t r a l n o r m o f t h e H e s s i a n m a t r i x f o r e a c h a r ri v a l. A n a l y t i cf o r m s a r e p r e s e n t l y k n o w n o n l y f o r c o n s t a n t v e l o c i t y m e d i a a n d a la y e r o v e r a h a l f-s p a c e m o d e l T h u r b e r , 1 9 8 5 ). T h e s e s e c o n d - o r d e r d e r i v a t i v e s c o u l d p r e s u m a b l y b ec a l c u l a t e d n u m e r i c a l l y f o r a n y a r b i t r a r y m o d e l , b u t I e x p e c t t h a t i s p r o b a b l y

u n n e c e s s a r y i n m o s t c a s es . T h u r b e r p o i n t s o u t t h a t t h e H e s s i a n g i v e s a m e a s u r e o fl o c a l w a v e f r o n t c u r v a t u r e . M y e x p e r i e n c e f r o m t h i s w o r k i s t h a t t h i s c a u s e s t h en o n l i n e a r e r ro r s t o b e d o m i n a t e d b y t h e o n e o r tw o n e a r e s t s t a t i o n s t o t h e s o u r cew h e r e t h e w a v e f r o n t c u r v a t u r e i s l a rg e s t. A t n e a r b y s t a t io n s , t h e w a v e f r o n t s w i lln o t d i f fe r t h a t d r a m a t i c a l l y f r o m a c o n s t a n t v e l o c i t y m e d i u m . T h e r e f o r e , I s u s p e c tt h e u s e o f H e s s i a n s a p p r o p r i a te t o c o n s t a n t v e l o c it y m e d i a w o u l d n o r m a l l y g iv er easo n ab le r e su l t s .

F i n a ll y, it is i m p o r t a n t t o s t r e s s th e t w o l i m i t a t io n s o f t h e t e c h n i q u e s d i s c u s s e dh e r e . F i r s t, t h e e r r o r a n a l y s i s p r e s e n t e d h e r e i s b a s e d o n t h e f u n d a m e n t a l a s s u m p t i o nt h a t t h e l o c a t i o n e s t i m a t e i s w e l l c o n s t r a i n e d . T h a t i s, I a s s u m e d n o a u x i li a r y

c o n s t r a i n t l ik e f ix i n g t h e d e p t h i s n e c e s s a r y t o o b t a i n a s t a b l e s o l u ti o n . I n t h e c a s eo f p o o r l y c o n s t r a in e d l o c at io n s , o n e p r o b a b l y s h o u l d r e s o r t to t h e t e c h n i q u e s o fTa r a n t o l a a n d Va l e t t e 1 9 8 2 ) o r R o w l e t t a n d F o r s y t h 1 9 8 4 ). S e c o n d , t h i s e r r o ra n a l y s i s i s w h a t m i g h t b e c a l le d a s i n g l e - e v e n t th e o r y. T h a t i s, e a c h e v e n t i s t r e a t e da s a n i n d e p e n d e n t e n t i t y, w h i c h is a n u n s t a t e d , i m p l ic i t a s s u m p t i o n i n m o s t r o u t i n el o c a ti o n s . I n f u t u r e w o r k , I h o p e t o e x t e n d t h e s e i d e a s t o m e a s u r e s o f r e l a ti v el o c a t i o n p r e c i s i o n o f a s e t o f e a r t h q u a k e l o c a ti o n s . T h a t p r o b l e m is c o m p l i c a t e d b yt h e f a c t t h a t d i f f e r e n t e a r t h q u a k e s w i ll , i n g e n e ra l , b e re c o r d e d b y d i f f e r e n t s t a t i o n sw i t h d i f f e r e n t l e v e ls o f p r e c i s i o n f o r c o r r e s p o n d i n g a r ri v a ls . H o w t h i s d a t a h e t e r o -g e n e i t y i n t e r a c t s w i t h t h e b i a s i n g i n f l u e n c e o f m o d e l i n g e r ro r s i s a n o n t r i v ia l

q u es t io n .A C K N O W L E D G M E N T S

Sp e c i a l th a n k s a r e d u e t o An n e T u r n e r f o r h a n d l i n g t h e s e e mi n g l y e n d l e s s g e n e r a ti o n s o f t h is p a p e rt h a t e me rg e d f ro m t h e wo r d p r o c e s s o r a n d t o K i m So wd e r fo r t u r n i n g my c r u d e c o mp u t e r g r a p h i c s i n towo r k s o f a r t. I t h a n k Ro b e r t Uh r h a m me r a n d C l i f f Th u r b e r f o r p r o v i d i n g v e r y c o n s tr u c t i v e c o mme n t s .I t h a n k Ro b e r t Uh r h a m me r e s p e c i a l ly f o r su g g e s ti n g th e p o s s i b i li t ie s o f n o n p a r a me t r i c s t a t i st i c a lme t h o d s . Th i s wo r k wa s sp o n s o r e d b y t h e Na t i o n a l Sc i e n c e Fo u n d a t i o n u n d e r Gr a n t EAR8 3 - 1 9 4 1 8 a n dby the U.S. G eo log ica l Su rve y unde r IN T E R 14-8 -1 -G1084 .

R E F E R E N C E S

Al len , R. (1982). A u tom at ic phase p icke r s : the i r p re s en t u se a nd fu tu re p rospec t s ,Bu l l . Se i sm. So c . Am.

7 2 , 2 2 5 - 2 4 2 .Ande rson , K. R. (1982) . Robus t ea r thq uake loca t ion us ing M-e s t ima tes ,P h y s . E a r t h P l a n e t . I n t e r i o r s30 , 119-130 .

Bo l t , B. A., C. Lomni tz , a nd T. V. M cEv i l ly (1968). Se i smo log ica l ev idence on the t ec ton ic s o f cen t ra la n d n o r t h e r n Ca l i f o rn i a a n d t h e M e n d o c i n o e s c a r p me n t ,Bu l l . Se i sm. So c . Am.58, 1725-1768.

Brown , R. D. , J r. and W . H. K. Lee (1971) . Ac t ive fau l t s and p re l im ina ry ea r thqu ake ep icen te r s (1969-1970) in the sou the rn pa r t o f the Sa n Fran c i sco Bay reg ion ,U.S. Geo l . Su rv. M ap M F 3 0 7 .


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779-790.

Thurber, C. H. (1986). Analysis methods for kinematic data from local earthquakes, Rev. Geophys. 24(in press).Uhrhammer, R. A. (1981). Northern and central California earthquakes, 1979, in United States Earth-

quakes 1979 C. W. Stover, Editor, U.S. Geol. Survey, Golden, Colorado, and National OceanicAtmospheric Administration, Boulder, Colorado, 106-108.

DEPARTMENT OF GEOLOGYINDIANA UNIVERSITYBLOOMINGTON, INDIANA 47405

Manuscript received 25 March 1986

PPENDIX

The com pone n t - wi s e e r r o r bound i ng c r i t e r ia deve loped in t h is pape r dependsupon the fo l lowing theore m. The proof i s ana log ous to the c ons i s tenc y proof for theL1 and L~ mat r ix norms g iven in Stewar t (1973, p . 179) . The in te res ted readershould compare the two to see why com pone nt - wis e bou ndi ng leads to a l esspess imis t i c bond than a s imple one based on L1.


19/19

APPRAISING EARTHQUAKE HYPOCENTER LOCATION ERRORS 7 7

T h e o r e m : G i v e n A E R m ~ a n d a v e c t o r x E R n w h o s e c o m p o n e n t s c a n b eb o u n d e d a s

[x j [ =-< bj, A 1)

t h e n t h e c o m p o n e n t s o f t h e v e c t o r y E R m d e f i n e d asy = A x a r e b o u n d e d b y

ly~l

appraising earthquake hypocenter location errors

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