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Earth and Planetary Science Letters, 45 (1979) 435-444 GEODETIC EVIDENCE FOR RIFTING IN AFAR: A BRITTLE-ELASTIC MODEL OF THE BEHAVIOUR OF THE LITHOSPHERE A. Tarantola, Jean-Claude Ruegg, and Jean-Claude Lépine / u \ u I o r t r r . L U ' r s .u i i q . n r L l v o r . F t u e u n r . d x f s r u r u i r n s r i n r r r o r u . q l t o r r r p i i u r p n P r "r . u . l r s p - o i i : r r l ]l o t u r , x r l s n h r s J f n b s r s . i l r ] . r r j r p . u r r l q o u m q i J ^ f r tl n r t . s . r t l I a t q r . tr r rn s o r l s. r . I J ^ r l . . s o r r i . ) r t r l I I I 0 l u o l l l o r l \ c n n : r s r p( ; 1 _ s ! 6 1p u . t / 6 1 i " | u N . l q s . . u . . r i tr r p . r t r F u ' u r a J U o9.l n s . r a q l urrp IDluauFdxS'z 3 q ri u ' I . ^ i n s r r L l o r s n r r u r o d u r : r r l r r o :ur . n 1 1 t r r u r ^ . r d u o i r ' i i t r u r s i l q n o r l l n r t r i l o dl u r f u n I i s r u ' 1 . l r r l o r u dz z . q i J o n,u scn ruod arro s r u r o d. r o u a u r ur l l r a 6 l S l q . r N l { i r l r . , r r u D r xso. $ i i ^ r n s s | l l J [ 9 ] s F u , . r 1 li . r J r l s n f S / 6 I r i q n r r { o N u ' P . r n ! ' r r u J rr s r u e r a a a u o zl . r r u 3 r 3 q l i r r t ) ! r o x n s s t u r o d. r r N s r ! u . . r l ] i u r / n o l o j s r F r r o t L I a r . r i , r t )u r i x J r u r p o a :r i r s c l . J os u r r t r i q l . u i r o J j . d i r J ^ \ s r u r a L . n t r . r u r i r . l so / i . d n s 3 u . p o . r to . $ l '8uotu,r{Sltue . f r ^ 1 r l r ) jf t n o q Pr a r Nu ' : s r a ^ o ru o D e u r r o J .rpr u o r r a t . r l l o u N . t o ^l u r n s s u^ \ . u E . r oq r r l q . ' t r p L r t - { ! r r L t rl o r c o L Jr i u u r e q r u r L I u as s i n s s uu i d o p u Ps r l i u J 9fI oN d a d l u'i,lnquruo) p u u o r l r o p t . r r t u r l i o d u r u r p . . n p u ' ! u e q rr . i r e l : t n t n r ; c u J o s r u a n ao ^ { rr ! r a \ r s n . r L J l J os . I E n b { t u r . x r l . u e q l . a n s n | ]u r ' 8 1 6 1 - 9 . ' r q r r ! ^ o Nu o p : r n . . o s r s u rJ U r . t o ^ p u . 3 r u r s r ilsu f x o d r u r u Y Isl puoqeN e n b l L t d r r : o arJn r i t s u ti ' l r e ' a u o l r J o q . o r u r d n r n ! s e a . u o T { D . L l tl u r p u n o r n s . a r . . : r . l €s s o r . t S u r t u . r \ e s t u r o dp : t u i u n u o u a l J o t r o ^ \ ] a u. r r ' ! o e l u o r s r . s r d . q d r 'qr r l n i u ' r d u J s e u r f ! n o q . r . l d i L t r i e i u u r n l r o s r s s . . o r dr q r . i f n r s o ] u o 6 a r s [ t ] u r s u . d . : p . q r , l 3 I l p s n b N l q d! p t n l r t s u ll r t t , { q I L 6 t a r L r r s I ' . u r . J r i d ! . . q . 1 | r t s . i e ^ r r si D r r s { q d o i tl u J r { J r ( u.rqy 1r'3r:r.rt [r r] s:INrd p u | r r l r q D r Ya q t u : J A r . q { r r p u n o q a r r l . l ; ! r r a r J r t aqr Jo sallo, riirlrrlo^ tEr\r ru\: iqr pLIr(uirsls r : p r i u . p V J o J l n D ) s u ' r d . r u N . r o u J i a r r oI u l t 3 L I suot rI ni r^ llu Iflxr l.:)|d]ir E or relLrls r$ru 3'll p u r t J J tr o r . l v l o s r u o zi u r l t u I . u 3 r q n sI I Ni u o u P s r I ] n o q i l ol o ] | t q n d r d . q r u r u u l . s Y i . q q n o q ) e q t 's3!rprn.q ucrr.lr..qursrp or ar.td rr |:sn i.l plnoqs\.nssrJ rido 3urtns:rrqr q uoo r.tur ru:ru q:h i.pou, c q.ns lrq] Flsr ni s' ]r p& .r.p.qr lt3h ir !'l rus rqrrLu punoa.i E qrr{ i:tou rtntrq.Jlrsqruffrlruaorts\rlt.r.qds.aroJqrloInor4x{lrqIDunF.u3qrJossiFlrnrllrsJrdrrdrdsLI.l '3r'nJjdu.r pur i|idtrt qroq p3rnsraur3ra,{ F0t x t ro !prc IJJ Ttrtl truldrrllo .qr.to 5ur.rtspur ssur srLIrriPnrsor ntr. n rno !.Irrtr JrJ\ slu)0.rnstJLu r!rto.;rsltr. n . r d n r J t r n r . t r t oI ,^r r s s ! r p u r ' t S . t d n . p , u r u ; r r u l os : ) . i b L t l J D . ' r uI r n o q r J o! u r u r J i q d s r pt n u J i i L I r r . (. r t t d : j . o l u , a u . r t r r o u t o ; u r l r d l . l u . n r . l t 5 . r r t d u . r q r r v p u r u D r r J v . q r u 3 J . r r : qI n f u n o q . r q d i u r . h r : q r J o u o q r s I . u U T R qi sNr t t r q ru r r t l v r s . l t r r n o s u r s ! 6 l r r q L u t r ^ o N n p , t n r t r o f n ) ] r J J o r l o s t u l t r r u . r J o d u r l u \ r uori.^ !.s',!al 6r6l 0a ^l.l!.{I.r.r 6L6t I .unLf r:ran nnsnt 3)q1 f au") ^r D t,rrs,i,, l u"l,tl ca nPiJ sutJ ()titi 2q.lt nt dnb^hlJ rt rrrr4 (t6t f7)ePtala4 tunDtrrt 2t arhfnldo,t srrnJ,f )Pu),t 3ta)otutt1 3NI,t:]'l-J t pur SSEittI J I Y]OI_NYXYI Y lXi| IdSOHrt'I3lt-LrtOUnOt VItt{ :lll.t :lO'l:loOll JI LSV'IS-:I'IIIIll{ Y rUYIY NI 3NI rJIU tIU-l Ef,NgCI,'{3JI-LIOO!3 Et_f sr,nPIrJLllaN .q.I ur trruud - rn,tr.rnuv .iurd unj iwqrJtq.J i!r1rr:rrs FrM\I] at rfr Eai (jl6t) sr 1r,r37r1u,11s ajrbliturQloq e l 8 u N urt o r p . r r d d N . { F n o a u . S o u rroELr it r o s u i l i q r l u | . r r re a r o s u . t r r s r r sL r t . u rl g p s l n d r u o iu . i q s e r l . i l o s u r r F 3 3r o Jr o s r c r u r l r l s L r t i u r . q l p u c ' ] r u J r t . p o a ; . q r o t ) r _ r r p r o t rsrleu ! t r r J l r q n i l r r r u rs r r u u o r u rp ! p r ^ r pu - r i qs e qu o r i r a r . q ri u o u u ' r r o J r fr q r J o u o n r t u i s i r . l . r I s u D t u r u r u c l q o o l J 3 p r oo I ' u o ' t c r . r d r i r n i s r uo r p . a i u N . p L ' c! u r u l r - r i . r . J . i u . s o r t .i l t r t o l u . p u e d a p u1r o us I r r r n q s r l n 5 . r . q r j o r r n r . r d r t r r d ( o r I s a o s u o n c l u i s . r d 3 xr I l . n s 'uotrrrrrp pi:11 {r.rrrqrs l|r pL!e pi\aJ .{J.rIq[ lLrrod u r o t . , \ r r f t . r u o t r . r r q r i . J o e o r : a n 1 l l 3 u . . r N l d ssr e p p ! r u . s J r du r u o a r BI . { n s e r r r r r p o . t l o n l n s i r s r l l uorrRrerdlitul € 'le rJ u,.rrpJ:N pu. . I t l . , . r r a . s l op a ' l s q , ] n d . q i l h e r . p . s a q rJ o u o r l ( l u r s i ps r a l d u o r r r o l ' { ( u o , ) e r o rl o j ; i l l : . s ) y p t ' ' N s r u r o d u r e ^ \ r e q u t 0 5 u r L t r . . sr u r i r E uItr r ! r r l o ^ \ r . q r J o u o r r . r r ) ' ! o . r L r n r u r u i resp u Ni A Jp L r rl s l u r o d u a e a l r q u r t z S u r L F r i r' r i u r O t o r r r d I r r r u ! . t t l o u o n . u r r o l t I Fuo'$rrxc.ir.t. !r.rsuoruip srlns.r.s.ril ' u 3 r n o q Nu r s q r t a . i r n r l . . r u s r s r pi q i , r o r o l r t p f f u . r s €o r I r l i r ^ , { a q , ' J i L I r n i osrl u r u r r r . s . i u r J io a l : q } ; u r l c s u r . l r r r . i L U o rsJn l u i ^ . r d r o r o l r r t r n r u l l u i r $ . 1 i i u l p i . r x . r o u ' s . r . u r d . r r s r pr s i r l r r F n o q l l Y 6 1 6 I r l . r . l \ p x r 8 1 6 I r r q u r i { o N r r : ) l ^ \ r . qp 3 l n t r r os n r . u l i t o t r l I]uurstrrot InssJ s.nurdJrrslp rtrros sssllrlrrl.\ 'u:l l u l r u u o ! ^ u a l 0 . ! u . u J D o : ; 1 r | !] \ o r l s . r n t l J r q r j o r r ' ] dj . t r o l t r r l l l L t l jilrrrrlpu dstu Lrrlr:lsrrnollit l 3!:l Iz] iatl.rrS uo{ rtroz rJr rrqqnoqC lusv arl J. sltnrJ r^!rr umu.'0 ,on.zu ,o9"zv ,08.1I t\\- , . . : , " , f, - l N -l( ,1li TAALT ] C o n r t . . t t r r . dv n r l F o \ o f d , \ i n . e : b . r v . r n 1 9 i I r n d l 9 ? 8 Sssnr.nt Dillmrclnl 00: 0.08 ,0.:l 019 A! BC 0ll Blt NL CD CE DE DH DI Lf II l:n" I\1 HI IIL 0.18 ,.008 -0 t7 0:l 0.t5 0!6 007 022 058 0.1I s.:nr.nr llQ H1' ]N IQ JL JS LS LT IIN \IR NQ NR 0.01 2.12 0.11 005 -0.01 0.01 il.5l ST -417 0.15 0.20 0.07 0.02 0.1l 0 0,1 Tv UV 0.03 QU QV laultnrS. l U . ! r u r e dc o r l t t r c t i o n isn r h e n o r r l ) . r s lz o n e !r r c l 3 r g e rl h ! 0 t h o J cn r t h c s o u t h i v c \ol n e s T h ^ i s p r o b e t r l yd L r et o t h c g e o l r e l r i c r c] o l l f i S U r l r r i oonf l h e n e l i v o r k$ h e r e t h c I r o r i h w e s t c rpno i n t s! r e c l o s c r r . , r ] | ( .I rr ,. . u , r l . r r \ .| . . t o t h e s h o r tw r v e l c n g t ho f t h e d e f d n r x i i o . f h c i r o J i _ c o u l de x l c t l y a t c o u n tI o r l h e d e ' o m a t i o n o b s c e d . l l r e . o m r u t r t i o i h a sb c c nm r d e 3 sf o l i o w s :F f o m t h c L i e i j n i l t u no i t h e s r r r i | t e n s o r! s e g m e not f l e n g t t r / i s d e i o r i .c d b y ! u n i i n { n s r r a i n€ t o I l e n g l h1 + j / e i v e nb y : . lr e,,u't' e x t c n s i o r ds l f t i n r o u g h l )p . r t c d i c r l r r r o l h c I r u l t d " . ' i . , . l , . r ' . . l r ' . r. l ( J _ ^ l l , r r ,r i N c o n t r r c l r o ni n ! t p . o \ L n r t . l y r h c s r f ) . , l i t . . r i o r . T h i i isr v.ry i rtorlrrr polntT e c t o n i co b s e N a t i o . i\f l h c i . n . r f l o . r o l l h c l t v r l l c y s l o $ t h r t b r i r l l ed c i o a n r t i o n( o i l l ) ' nr ) o r J f r l i x u l t sa . d o p e nf i s s u r st r ! n d i n gN \ v - S En. ' t h c g e n c n ld i r e c t i o no f t h e r i l i ) o i c u r si D s i d cr r r r r e r n o t n r o r c! l t a | 3 ( x 3 5 ) k n r w i d eh t h e e m e r : c dp r r l . 'Ih. l h r r t h i sb r i t l l c d . f o i i r r l r o e x r . n ! t \ !ssunrpticn ' . . 1 - , . r . : rr .' .' d I j C. r r." l s u p p o r r eb d y s . i s n r i cd r r u ,i . c -l o c a t i . n o t r h . n i r ' r r s h o c kr n d p o n c , i s ' rs c i s n ) i c i tLy Sl . I n o d d i l i o ni o t h c d i s t ! n c en r e c $ r c r l . n i s t, h u r e l c r . l l i n !o i r l i n e c r o s s i n r! h e s u b l c r i 3Il ' l r z o ' L e [ 7 ] s h o w su r o u g h$ b s i d . n c co f i r b o u tl 0 c r ns t l o n s l ) c o r r . l ! 1 e dw i t h l h e t e c t o r r i co b s c r v r l i o no f n o n l x l l1l T h c w i d t h o f I h c b r i t i l e r r e a i s s t n r l i . rt h r t h e d i n r c n s i o nosf t h e c e n t r { lt r i s n g l . s\ ' e c ! n c o n . l u ( l ' ! h , r tt w o o f f o s i t e m e c h . . i c x lb . h a v i o ! r sh a v e o . c u r c d : 3 n e r l c r r s i o nl i m i t ! C t o t h c b r i t t l . T o n e . I ' l , c . , r . , r i o , , r r J . I | " | o | c| | J \ | | I n c a s u r eodr t h e l r i r n S l c si l ] t h c c c n t r u l r r t s l N ' r l ' l b . s m r l l e rt l u n t h e r c t u . r le x r e n s i o n r t l e b r i ! t l ' l w o f r c ! s c r n b e P o l n r e do u l : l h e c o h . r c t c " f w h 3 r e ; i s t h . L r n j tv c c t o ri n t h c d i r e c l i o no f t h ' r e g r n c n ! r h c o b s e r v esdl r a i nf i e l d r s i t i s s h o w ni n F i 8 .: . 3 n d F o r r h o , i u o n t l ls e g n r c f t : t h e c i e a rp h c n o r l ] c n ooni P o s r s e i : i r i . c o t r l c l i o n , n o ! e x p e . t . dl t r i o r i i n a d i n e n s i l cr i f t T o n e (2j + eyy(u')' c,,(u'). + 2e"{"t ;T | e s ed r l r s L r g g ctsot ! s t h a t I m c c h r n j s nor i c h s t i c s l r e t c | i r gf o l l o w e db y s u d d . nb r . l k i n ! r l r d ' 1 ! \ t i c K n o w i i g l h e n r e a s u . cdd! t r A r y r n d t l ' e v a l u c s l e b o u d s i n r i h rl o t h o s .i n l t o . l u c e db v R e r d[ 9 1 ," 3nd !Y Ior enchsidc of a singlctriansle,we obtain f o r t h e i l . r p r c l r t i . i n o f t h d 1 L ) 0 6S r n f i r r n c i i c o a s."-sler of lltrc€ cqurlions i,r th. irnknowns€jx. rllr Pl 'u' "no cr-rl I |,L . . T h u s .r n e h s t i cr e b o u n dm . . h r r r i s n it s f l o r t r e d T h u s .t h e d i r e c l i c nc n d m 3 g n i t u d .o f p i n c i ! 3 1 t o i n l e r p r . t r t et h e s ed a t r : l l . f o r e a c r i s j sJ 8 i r ' r r h o r i z o n t r ls t r a i na r : d e t . n n j n e db l r c l a s s i c aeli g e n ' Io r e g i o riri r n r c c r e u r r p h t e b o u n d r v i s $ b n l r d r r ' I l _ i r " u ,:' ' ,'" The re$rlts of this fomlutstion rre sholvnin eLi\ir' c l r i f to ai l ) . P h t . s r n d . o r r r e q u c n l l !d ' i o ' I i B c lr{'i t l r c r e J c i r c s \ ! h c n s l r r i n theexler!sionrl rllv. T h e . e s o r s u r o u n d i n st h e ] x i r l s r r i l s h o w s3 n '1rrr3qrJ. rDd t.La.qns rql 'n u$oqs.r. uoudnr: tunsl] aiu r L r t u 1 r , , . \ i J u . n u\ r T ' r r n . ' " " . . , , - t ' - o r , , " . . . . '; " q . v a i n ; r J . q rl o t F t h { o t . r l r u . x $ . q \ . l . r s . r i i o r ; F t r . r . P u D l s . q r r a r r r l p r s r o t t l l r J o q r i u r l i q . L3 r u a ! ' : n r b ' t r , D, -D o r.,'. -,r ,:. I r!i F.-rrl . 1 r , p i . ,ttl iu L ' ' d \ . . ' llrr. roJ r.ruit urr4s.qrlo srntrr pu. suorltr.nplrdhuld a ii'.] m.l 0t urErls or *l* e r n s s r ,u o d o r o I t n E ]l P U . r r o u p a l e ^t r c e u ^{o|l P^el 8l6L ^oN o< 439 b r i r t l . r h . . s h o l d( r b o u l t . 6 X i 0 a j n o l | r c ! s ! , r \ s c n r 3 l lr . e l , l e r ) , r l i ew c : r k . s zr o . e , i i r l r c! r n c r o o r o i , , ,.' . . ) I . . . r . - .a . r ' . :, r r u r ! ! r g ,I ' s u n D g ,! n d n r j . c t n o f n j r s n ! ( d r k . s ) ! r lheresultuie { r t r c ni l s s u r e s 4. A nuD'.ric.lnrod.l l n o r d e rt o r e s rL t r j jh v p o t t r e s iosf e t 3 s r i c i t vr.a h o r n . q e n e o u rsf.lf i n i t . . t i r o . r i i r n e n s i o nhaotr i T o n r 3 L D o u e it r ? sa s s ' n e d _i v i t h i 0 : 7 p o n s o l ] . r! r l i o . . I h e n r o d c r \ s u t t o s e di n i t j x l t ! u n n r e r s c dr J r dr f c q ! i t i b , n u n r .T h o r e h t j y ed r i f l o f r h c , ^ r r b r a n! n d A i r i Q n p l a t e si s s r n u l a t e db y ! n u D i ! \ i r l s t r . r j nr 3 . h i , r g t h e t h r e s l l o l d\ r l u e j u s t b c n ) r el h e c n s i s T . h e c . j s i 5i t s e t l r s r l r . l ' s ' m u l s t e db v t t r e( j p r n j n go f s o r x ef i s s L r r c 5 . r l I l u . - r o . rJ r I I b ,, r e - . . Jb . r t r , I r ' [ . . i i , . ] . 'r , J I , , . I r. a n d s c ' ! n i c d a r a .T h e m i . n n r ! n ! n b c r o f f i l s u r e s n e c . s s s r tvo l l t ! d e r l u a t e l yt l t d r l ! h r s b e e nf o u r r d e q u r l t o 2 b y u s i n gl r i i r l n r o d e h . T h e r n o l . l c o n r r i n sr . n f a r a r r c r e r st:h e , a t L t ea n u o I | e n t a l i o io r l h e L n r i a l s l r r i I t i . t d b c f o r et h e c n l l sl r w o f ! r s n r e t e r sa) n d l t r ec o o . d i i [ l e so l l h e endf-onts olthe nvo fissurcs(eiglrt tr.ametc6) (see r r g .j l . 'l h e s t l a i n si n o u r r s . o a t i r n e n s i o n c t , I l ohdter rl e b e e nc o d r I u r e du s j n gr h e d l s p t ! c c m e ndri s c o r , r u r L ! n e r h o d I 1 0 . 1I I . l n t h i s t r c t | o d r h ec x ! . r s r r e $r n d d r s f h c e n r e . l i e l d i i I l i n c r r t yc i r s i i ct i o m o g e f e o u s n e d i u m! r e C c r i v e du, n d c rp t r n es t r r h c o d i t i o nb y ' n c r n so i t h c P l t k o v i r c hf u f c r i D i s I r : j , t i o r nr t r e d r s F l i c c m e ndrr s c o n l i n u i tey x i s l i n qi n a i t c g m e nqrr r h D o l hn o n n r l r n d l r n g c n t i r l .n o n _ 7 j r oc o n s t r n rv j l u e s . T . .^tL ^1 Jor r ) o " .n . r - . N o D t l n r c di r y i r n e r rc o n r b i n x r j ooni r t r ee t e J n c n I r r y B c c a u sler e a r ed e ! t r n gw i t h s t r . l n sa . d n o r w r i n s l r e s s ersh. e r e s u t r rsr e l n d e t e n d e nor l i h . v r l i t r t a k . ' r f o r l h e Y o u r r q ln o d u l u s . T h c c o n p u r r i j o no f r r i ! t , r 1 j , l . t sb y t l u sm c u r o . l , r e l o r t o r r ! t t s t l i r t ! t p . o . r c ht h c o b s c n , e d strri,, l i e l . 1N e v c J h r l e l e sssj n , c ei h . n d r r b e ro f i n d e p e n o c n l o l r s e n e dd r r a t \ , : l 3 ) i s l u g c r r h anrhcnur)rber o . I lra r'. - .., J.,.r.1 , r. r t / . 0 1 ^ . '" | .t0," I crrc,r:r..e, c l r . u l l r er h c b . s i n o d e i t h ] r r i t s r h e d!r, r i ! l . ( ! ) l h e J n o n c l t u nb c i o r eL h c . , n n . r n c & j r , \ ! u p . r r n r e r J { . l h e v r t u e sr n d o r i e n l a l i o n oi rh. unir\irl rc!1onxlnr)D snnuLlhg dre rclrlive drilt of ihe Afuu[n a r d A l r i c r r p h t . \ . a h ) T h € m o d . i l u s t r r r r rl h c . . $ . l N o 'i .' ,r. 1 . d r o . , - , i n r , . f , . \i . r r , l . s , r - r r . r.!rltnrS irnrr rbis oprnij)-!!re dedur!d I.rorntnc dirlirence b B $ v . c nn i i . s l ! ) x n d ( b ) .I h e . i ! h l c o $ { t j n t r r . s . fo r c r o u r p o i n r r , 1R , , C r r d , D , ! n d t t r cl v o p t u a n . t c r so i I j g .. l r , a r c t h ep a r x D e r c ro f l h c n o d . l . p o i s s o . ,rsr r i o n l ] \ q r . q u : l l ( ) 0 . 1 t , b o u r d r r i e Isd s u p p . s e d v c r yl a r t r o m l ] j € r r s L u $ .! D , , n . r r , i n d d r r c i h c , n m r t ! n d r r n t e n r i r l d b p b c c e . t ! r n d s t r t s c sn n p o s . dr j b o r n J ! r y c o . d j ! o n s . 5. Remlts of the iNeNion T h e i n v e 6 i o n ! l g o r i t h n u s c ! l i I t t r i ss t u d y i s d . cribedjn thc Attendix. Originrl d!rr for Lh. ijtreNon a . r l " . r , J . : r I r ' r 1 . r r ' . . . . i , , r . . . . \ ' | rn _ r g e o d c l i cn e t o f l E p o i n i s . I n s t c l do l u b i t r r r y c h o o s j f g( l r e s s3 3 i n a e t i r d c n t d r r r . w e l l r k ef o r ! I e i . ! . r s i o n r o u l i n cr t r . t 5 i r e l ! r i v ed i i p l i c c l n . n r sb c r w e e nr l l I ) o i n L si n, o r J e rl o f o r r l t p o i n r si r t h c t r r ) i i d e t I c s J d r !w c i g h r i l r g w c h r v . r s u ' n c d ! s t ! n d : r l dc r r o ro f 0 . 0 5 n r f d t f i e s cd r t ! . T h . s r ! n ! n ! n r o d d f o r l h e i D r c 6 i o no . r r g t h e h s t t r , ] l m o d . l p r e v i o u s loi b i i i n . d - I i r r . s i n p r r r n r e t e r i5o r t h i ss r r i i g d r o d . 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U r u , l rrso u i r r s t u . u a l o r u l B r u o z . u o r l r ' r q ru { o r { ss x t S L 6 t rii.a1.q puNI P U N 9 1 6 I ' i ] l u , l i x . r t u ' l - i , , l Ns h - I rr. u l l t u l sr ' ^ l f u o r a s [9r] 8 ! 6 r r ; q u r a i . N u r p a n J r s qxoi l t a q s r r -.t{ F ( ' r r 1 x . 5 €, $p t a u. r n s s ui r q l i l s r u : t r o r r d ' us s es { t l r o d d n ss u o \ r J ro $ I 8 t 6 r r . q o r . r o NJ o s r s u r. r l S u n p 3 r . t u r j o o ru o r r r u r o J i pp . r n s F u n l o r t s i q l t E L j l p J u n s s ra . \ . L.la t r 6 r f n : 1 1 6 I u s i a r i q t r u r o j J.d i^\ sio!ot.rnsE!(.rlrto.li ou Lf,lnoqrN ' l sI r l l s J u . p ! n o q a r r l di u r r . r : . ' r r q r u r r r r t . k r q l l n Jo s r r r r J o r d I f . ) ' u c r { r . ui r 1 1u o u o u r u o ] u ' s a r 6 s r u o z i i u l l . r Li . L J r .o r p . ] r , t i r . , u r r; q u f r t r t r ' t . 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J .llu 3 s r r r s r r . u N ' F r ' r f r ui q r p 1 r 0t l r p f : r ^ r . v l o r r l r r l r r { I t a h I 6 L ' L * n l , n !. . r ; i ' ! i t q U . u r p . l \ r l s ) t n s i l l - a . i ^ ! Ju ! t . s n u o r l r r r t i p\ n o r ^ a r dr L l t r t l r r i t ; u r u r ' { . e . r o J s r n t N ru c . u ' . r 3 s , o \ u i r u r c r r s ' ; r r t u r u { o ' l s p l a t - l u r r r r s! ' l r t . ) n J x r o r . r r q e . \ i t i p o u rI . r u s r r l ro r t o q t . u r ' { r r n ! n u o r s r pt u ! u I . l L l s ' f J r l t r u r i t d . l y ' u r e r r ( sa : I r s . u . p Y . l o , r t n 3i q r j u s r t n u ! , j o l s u . r l i u r l r r r i r t \ a L I q r r ^ \r u i ( ! . J r i D u ! s r ' { . n t \ i t J t o ' u I . r ) r Uo ! r ( o r - r N c ! f J o ! n l | \ ] l l r r d J t : ^ r \ a $ p u t ( ; r l ! r L i i s l r l u p a L { irq p . u r ( ! 1 r . p i t r o o d s r u r r r r s I r ! o i l i i J l t t . j o L , o r r I ] ] r ' . u o i L t0. it. ,x 6 s I c ) 1 I r n . 1 r l i . o ru r r q ! ' r ' l l t i r I r l r i l r r u q ' i I r s r ] . ! q r r o J .ntfr .Lll l,! srolf tr. N s ;soI I r..u \.rl iurod : l p ! r ( r r t r . q q n o ( 1 5I t r u r P r r g r o ls r ' N o 0 ! _rl l u ' { u r r r i r u r ; u o iu , I I : . I o - i r n \ n r p r o . . ) s . t l LN o f : I s (r i j . r L r / ' r\ r t ( t i i u u ; s r r l ) . U u n s s ul r ' ! i r q N s p . i r N q o r q r - r os r \ | i r l r q r L \ s ! l t . ) t r c - rq ] r l \ i u o l u ' l j f - r ou n s s u ! r o s r r s r o l( i . r r n l o \l . ) r a ^ u r J ' { l W . r h r r ) k !c o 1 1 . r S ! . fso r l i c l p J " ld i s . u s n ) n ,r n l l | r r l i . r L h r l j - F I i . 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I . d e x f r . s i o l l o i t h c ! o r r r i x . c . n r x t r i x1 . r r ! c r ) ' s i i n i h ! i r l c r s c i ) r o b l c mT. h i s r sd u c I o 1 1 r f. r . ( l l r r t r f ( l , c i , r i r i d f r o , l . l r r i s n o l t \ t l i r i t l l i . L o i - l u c . di n t h . d e f i i i t i o n o f I h r n r n r i n r i r ! r gl u L r c t i o .X ?( . q u r r r o n . r - l ) . l h . n l | c s e c o n dt . n n i n t h . r i g l r t - h t ' nsdn l r o l e q u J t i o n( , \ - ' l ) i s n r i $ i n gr n d $ r c s i r f r ) l i l l c r t i osr l e r d i n gt o e ( t u . r l i o(nA - 5 )c l on o t l s k c p h c e . T h e " r c s o l L r l i ornl r t n x " [ ] l l i l i l l o $ 5 t o s r u d ) - . b ) J . . .\ \ ' f i l r r o r..r. | \r.,, i n t r o d u c .$ h r t r v en J r n ct h c i i d e t ! ! n n r r ! ' o n l'i'^ (A-6) Fromcqu3tions fA l) rnd (A.6).ncoblrjnsi r(e) (A r) , = cov(,I,) cov It c; be sho$n {l'xl: 0 <,D""< 1 (^-8) a n d i r i s c l c l r f r o m e q u r r i o n( , 4 - 6 )t h r t i i r " " ' s c l o s e t o z c r c ,t h c t r r 3 n r e t e r4 r ot a k e sa v r h . i r r d . ' , 3 n d c n t o f i h e c o r c s p o n d l u gl 3 n n l e t c r p d o f r h c i l r i l i 3 l | -l ..., ,lt Lv r', ' r'. i c : , . .' . . . I ' \ " 1 , l " " d e ( . r n u r i r t t ' p a r 3 m c r c rI.i r o a i s . l d s e Odis. well i o u n i l y , t h e t u r n r c l . r O d s t r o n g l yd e p e n l so r ! h e p a r r n r . r e r9 a o f t h . i t i ! l n r o , l e l rt h c t r r x r i . l . r ' t ' " I n f r . t , t h c p r u b l e n\r{ e s h r l l t r . . l i s n o l ! ( r ! . l i l ] e ! r t r o l , l c r nC l i s t l a c e m e r trsr e n o t l i u c r r l r r e l r t ' d l o o u r D a ' 3 n e t . r s )b, u ! b . c l u s . o u r i r r r l i r lt r o d e l d is nor too lir liorD the finrl solution rf', wc crn, rs it i s c l u s r i c r l l yd o n c ,w o r k i n ! | e i l A l . r d e r r l f r o x ' matior ("lrr.rrir.Lt!robl.nr").B! e\prnd'n: rnr 'lrvlols s c r i . so n c o b l r i n s : C o v ( , r =, ) [ A r ' C o v - ' ( ] ) A r C o v - t ( f ) l - r ,(,!) = co(,1,) cov-!(rj o -/lt)l e = C o ( , I , ) A r C o v - ' ( -) , L L (^ t) (1.10) (A.ll) $ l . r ' , , - l , , r . - ' ,r ( ' ^ i l J r ' r r ' J : r ' r ' r n r o d c l, ( 1 v i I ht h e n o r l i , r . r r ! 1 i r . . lp ' o b l c t r r )r.n d (A4) 16 .l-t o|t F r o me q u l r i o ni A , i ) i t c l n b c s h o w nr h a ! l h j s c o v r . i - lalL(,r)'\ ('\-[) 'l9I i a l { 6 a 6 t r) ! i o s a r . r t \ v I I r ^ q d o a ! )u o 6 r r n ur .r i u ' t 0 r$ . u r n l r r u n -uoui{tolri o} fr.p uo!.i rlo rn' rLLtiu.\:Fr|a o €t 'a6l t z r ( 0 i 6 I )9 9 a uopuo-t Jos )l sllJf xotrqd..rEp rtlrf. isorrrilen.rur t o u o r r : 1 u 1i , l i ! r s r u . n f r d n r r i q t r c f p d r s n r j . r g c a a 'tlr. 69I (1196|)Il ros'u.rlrv f s{,td!r.ra).rrDprtrr'l !rr.ql! s\.f ri{od:uht$ r I I s n : i r . r l J . l t x D J I' '(916t) tZ '18 r:)l r\rtdoJJ r !.rtrnLl rlr! p..luroq s.nu p^Dr. J 1 \ r Ui n N n , i u f , n o s r i s r r J p n . . . r r r L r o u ,r { r h o r r ^ I r u . 6 u r u i r p r r { ! i . , u r r ; . u r t r l o 3 c - r :tlt f , \ \ p u f r : l v t t 0 z luolr . r r r . i i d u , ) r i l . a - J J or I F o r . r t r u o f q ) r L r I i . r l l r . r l t J o uorrruur.rrp.rtl or lorflrJa. slr pu'r ruliqolt is.riur ! . i .r.\ .., t, d .a J ') I !$icrd v .rrr.)t .) r.trpJDql! | ,rt.rurjNt \: 6I .!0t f6l ( 9 1 6 r ) I t s r r ! i L ! r i . u o r r i . ll ] ! r , r : r rn r s t , i r t r r l j n r n i . , u r r jrp rfl tr.rd rriltD.). \\ |] pur uon:l! :j.)t );pu.rd 1 I 8l '8tll 6 a 0 t 1 1 l i 6 l r t 8s a Us f t t d o r r I g l a t 9 i 6 t p u D l - t l l u o \ u r r r ' r p u . . . 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