P O L A R I S A T I O N O F T H E S - P H A S E O F S E I S M O G R A M S (*) R . STONELEY 11 lias l)een o b s c r v e d b y F. N e u m a n n ( 1 ) , P . B y e r l y ( - ) a n d I. L e l i m a n n ( ! ) t l i a t in s e i s m o g r a n i s of e a r t b q u a k e s r e c o r d e d a l m o - d e r a l e d i s t a n c e s , lite S p h a s e a p p e a r s first as SII, f o l l o w e d s o m e 10- 1 4 s e c o n d s l a t e r b y SV. Miss L e h m a n n a t t r i b u t e s t b i s p b e n o m e n o n l o t h e s m a l l a n g l e t b a t t l i e S — o s c i l l a t i o n s m a k e w i t l i t b e h o r i z o n t a l , w h i l c P r o f . B y e r l y suggests t b a t s o m e k i n d of d o u b l é r e f r a c t i o n m a y b e t h e c a u s e . B o t l i a g r e e t b a t t h e i n t e r r a i d o e s n o t s e e m t o d e p e n d o n e p i c e n t r a l d i s t a n c e . E v i d e n t l y m o r e o b s e r v a t i o n a l e v i d e n c e i s n e e d e d , b u t a p r e l i m i n a - r y e x a m i n a t i o n can b e m a d e of t h e p o s s i b i l i t y t b a t t h e p b e n o m e n o n a r i s e s b e c a u s e t h e o u t e r p a r t of t h e E a r t h ' s m a n t l e is n o t e l a s t i c a l l y i s o t r o p i e . T h e s i m p l e s t h y p o t h e s i s w e can m a k e to c o n f o r m t o t b i s a s s u m p t i o n is p e r h a p s t h a t ali d i r e c t i o n s a t r i g h t a n g l e s to t h e v e r - t i c a l a r e e q u i v a l e n t ; t b i s t y p e of s o l i d is t e r m e d « t r a n s v e r s e l y i s o t r o - p i e ». T h e a s s u m e d t y p e of a n i s o t r o p y is s u f f i c i e n t l v g e n e r a l to g i v e an i n d i c a t i o n of t h e o r d e r of m a g n i t u d e of t h e a n o m a l y to b e ex- p e c t e d . F u r t l i e r , i t is u n l i k e l y t h a t t h e o u t e r l a y e r s a r e v e r y m a r k e d l y a n i s o t r o p i e , e l s e t li is d e p a r t u r e f r o m i s o t r o p y w o u l d p r o b a b l y b a v e b e e n n o t e d l o n g ago. I t s l i o u l d suflìce t b e n , t o t a k e as a s t a n d a r d of c o m p a r i s o n a t r a n s v e r s e l y i s o t r o p i e m a t e r i a l w l i o s e e l a s t i c c o n s t a n t s a r e k n o w n : s u c h a c r y s t a l is b e r v i , t b e c o n s t a n t s of w l i i c l i w e r e d e t e r - m i n e d b y V o i g t ('). T h e s e i s m o l o g i c a l i m p l i c a t i o n s of a t r a n s v e r s e l y i s o t r o p i e m e d i u m b a v e b e e n i n v e s t i g a t e c i , a n d o n e i m p o r t a n t f a c t is t h a t if a l i d i r e c t i o n s p e r p e n d i c u l a r to t h e v e r t i c a l a r e e q u i v a l e n t t b e n ( n e g l e c t i n g t h e c u r - v a t u r e of t h e E a r t h ) t h e v e l o c i l i e s of SII a n d SV a r e d i f f e r e n t , a n d t h e y d e p e n d on tlie a n g l e of i n c i d e n c e of t h e w a v e . T h e d i f i e r e n c e i n t l i e l i m e of p r o p a g a t i o n t h r o u g l i a s u r f a c e l a y e r of t l i i c k n e s s li c a n e a s i l y b e c a l c u l a t e d . L e t suffixes H, V r e f e r l o SII, SV r e s p e c t i v e l y ; l e t c h e t h e w a v e - v e l o c i t y i n a n y m e d i u m a n d i tlie a n g l e of i n c i d e n c e ( * ) L a t r a d u z i o n e in l i n g u a i t a l i a n a d e l l a p r e s e n t e N o t a è r i p o r t a t a a p a g . 1 1 3 . 4 11. S T O N E L E Y i n t l i a t m e d i u m . If r is t h e r a d i a i d i s t a n c e f r o n t t l i e c e n t r e of t h e E a r t l i , t h e n f o r a n y r a y , i n t h e s t a n d a r d n o t a t i o n , r s i n i dT P ~ c ~~<ÌK S u p p o s e t l i a t a d i s t u r h a n c e o r i g i n a t i n g a t A g e n e r a t e s SH a n d SV w a v e s a r r i v i n g a t B a l o n g p a t h s PQ„B a n d PQVB. L c t I\M a n d QVN h e p e r p e n d i c n l a r t o P „ M a n d Q„ N r e s p e c t i v e l y , w h e r e r e f r a c t i o n is s u p p o s e d t o o c c u r a t t l i e b a s e of a s u r f a c e l a y e r . T l i e n f r o m t h e d i a g r a m P„M = NQn—li ( t a n iv — t a n j„ì s i n i, so t l i a t t h e d i f f e r e n c e of t h e t i m e of t r a n s m i s s i o n of S a l o n g t h e p a t h s PyQy a n d PUQ„ i n t h e l o w e r m a t e r i a l , s u p p o s e d i s o t r o p i e , is 2(/i/c) s i n i ( t a n i v — t a n i„) — 2dTt s a y . A l t e r n a t i v e l y , if 0 is t l i e c e n t r e of t h e E a r t l i a n d R t h e r a d i u s a t t h e d i s c o n t i n u i t y , w h i l e t h e a n g l e Qv OQ„\zd A , t h e n QVQ„ — R. d \ — R(dA/dTi) dT\=c dri\ s i n i, g i v i n g t h e s a n i e r e s u l t as h e f o r e . T l i u s , i n a l i , t h e d i f f e r e n c e ili t h e t i m e s of t r a n s m i s s i o n of SH a n d SV T T 2h 2,1 • - 2h • • U • , • i i „ — 1 y = s e c i„ s e c i v s i n i (tan iY — t a n t„) cH cv c w h e r e c , i r e f e r t o t h e m e d i u m i m n i c d i a t e l y h e l o w t h e j t i n c t i o n . T o s e e t h e t y p e of v a r i a t i o n of c v a n d c „ w i t l i i n c l i n a t i o n 0° t o t h e v e r t i e a l , i t w i l l s u f l ì c e t o w o r k w i t l i t h e d e n s i t y a n d e l a s t i c c o n - s t a n t s of h e r y l , a n d t o s c a l e d o w n t h e r e s u l l s so as t o c o r r e s p o n d t o a v e l o c i t y 3 . 4 0 k m / s e c . i n t h e l a y e r f o r SV a t tì = 0 " a n d i l = 9 0 " . I n l'OI.,\RISATlO'' OF THE: S • l'IBSF. OF SF.iS~IO!;RA~IS 5 the undedying material we take c=4.38 km'see.; thus the modcl eorresponds to a granitic eontinent rCHting on ultrahasic mateI·ial. The I·erluisite fonllulae for C H amI c v are found in thc paper alrearIy cited (H). For a transversely isotropie hody in whieh the stmin.encrgy function W is given Ily 2 W =A(é 11+ e\y}+ C e 2 .. + 2F (eu +f>yy) e .. + +2 (A-21\') e n e n -1- L (e~y"+e2 .. )+ Ne ~~y thc velocity c of a wave eorreHponding to HS is given Ily where n = eOH ù; l=sin ù. For waves of SV type 2, c' . ~Al'+C n'-II (A-L)l'-(C- L)n' l' +41'1'.' l! in whieh J=F +L. Voigt's values, in dynesi cm", are A=2.fi94XI0 I \ C=2.363XI01~j A-2N=0.961XI01~; F=0.661XI0 1\ L=0.6;)3Xl01~; N=O.866XIOI~. From the foregoing formulae, the value8 of CII and Cv, sealed so that they correspond to matel"ial in which C~ll and c~ ,. fOI· i) = 0" are 11.56 X 10 10 , are given in the following tahle: T AnLE I ,')0 CH X 1O"~ c,. X 10-~ O 3.40 3.40 10 3.41 3.48 20 3.46 3.68 30 3.54 I 3.905 40 3.62 4.0,1 50 3.71 4.02 fiO 3.79 3.855 70 3.86 3.64 80 I 3.90 3.47 90 I 3.92 3.'10 1he rIiffel"ing trends of CH amI c,· with U dre evident fi"Om this tahle. In what follows the layer wilI he tl·eated as thin, so that for propa- L E f i • ** 6 R . S T O N E L E Y g a t i o n i n t h i s l a y e r t l i e E a r t l i m a y h o c o n s i d e r e d as p i a n e . T h e n iH a n d ;'v a r e g i v e n b y sin i s i n iH s i n i v c cH c v w h e r e c = 4 . 3 8 k m / s e c . a n d (sin i)/c=(dT/d A0)/110.7 S i n c e dT/d A is k n o w n f r o m s e i s m o l o g i c a l t a b l e s , t h e a n g l e i c a n h e c o m p u t e d f o r v a r i o u s e p i c e n t r a l d i s t a n c e s A. T o c o m p l e t e t h e c a l c u l a t i o n s w e first c o n s i d e r SII, a n d p u t {) = in iri T a h l e I. T h u s ( s i n in )/c c a n h e c o m p u t e d as a f u n c t i o n of i , a n d f r o m t h e f o r e g o i n g e q u a t i o n /„ is f o u n d as a f u n c t i o n of A . T h e c a l c u l a t i o n s f o r S F a r e m u d i l o n g e r , b u t t h e g e n e r a l p r i n c i p l e is t h e s a n i e . W e t h u s o b t a i n T a b l e I I . I t w i l l h e s e e n t l i a t t h e d i l f e r e n c e s h e t w e e n /',, a n d i v a r e n o t l a r g e . T h e v a i n e of ( T H — T v ) / 2 / i c a n n o w h e c o m p u t e d d i r e c t l y . T A B L E I I A dTA »H iy i d e g r e e s sec/deg d e g r e e s d e g r e e s d e g r e e s 1 0 2 4 . 4 5 6 . 3 5 8 . 7 7 4 . 5 2 0 1 9 . 4 S d 3 9 . 3 4 5 . 0 5 0 . 0 2 0 1 8 . 7 S r 3 8 . 0 4-3.0 4 7 . 7 3 0 1 5 . 8 3 0 . 0 3 4 . 8 3 8 . 8 4 0 1 5 . 0 2 8 . 5 3 2 . 3 3 6 . 4 5 0 1 3 . 9 2 6 2 9 . 4 3 3 . 5 6 0 1 2 . 8 2 3 . 5 2 6 . 4 3 0 . 5 7 0 1 1 . 8 2 1 . 5 2 3 . 0 2 7 . 8 8 0 1 0 . 5 1 9 . 0 2 0 . 7 2 4 . 5 9 0 9 . 0 1 6 . 2 1 7 . 2 2 0 . 9 1 0 0 9 . 4 1 5 . 0 1 6 . 0 1 9 . 2 S i n c e t h e o b j e c t of t h i s i n v e s t i g a t i o n is l o a s c e r t a i n o n l y t h e o r d e r of m a g n i t u d e of t h e e f f e c t of d e p a r t u r e f r o m i s o t r o p y , i t s u f - f i c e s t o c o n s i d e r t h r e e v a l u e s of ( T i , — T v ) / 2 h . F o r A = 2 0 ° ( S r ) it is 0 . 0 3 9 , f o r A = 5 0 ° i t is 0 . 0 3 5 , a n d f o r A = 8 0 " i t i s 0 . 0 2 5 . T h u s , t h e v a r i a t i o n w i t l i e p i c e n t r a l d i s t a n c e is n o t l a r g e . H o w e v e r , t h e t i i r e d i f - f e r e n c e is q u i t e s m a l l , e.g. f o r / i = 3 3 k m . , a b o u t t h e m a x i m u m p e r - m i s s i b l e t h i c k n e s s of t h e g r a n i t i c l a y e r , T „ — T v a m o u n t s o n l y t o a b o u t 1 . 6 sec. T h u s t h e e x p l a n a t i o n i n t e r m s of d e p a r t u r e f r o m iso- POI.ARISATION OF T H E S - P H A S E OF S E I S M O C R A M S 7 t r o p y i n t h e o u t e r l a y e r s s e e m s to f a i ] , f o r w i t l i a t y p i e a l c r y s t a l l i n e m e d i u m (« a d j u s t e d » t o f i t l l i e v e l o c i t y of Sg), i t w o u l d n e e d a t l i i c k n e s s of a n i s o t r o p i e r o c k of t h e o r d e r of 3 0 0 k m to a c c o u n t f o r a d i f f e r e n c e of 1 4 s e c o n d s in t h e t i m e s of a r r i v a i of SII a n d SV a t a n e p i c e n t r a l d i s t a n e e of 80". W h i l e o n e c a n n o t e n t i r e l y r i d e o u t t l i e p o s s i h i l i t y of a p r e f e r r e d o r i e n t a t i o n in t l i e o u t e r c r y s t a l l i n e p a r t of t h e m a n t l e ( r o u g l i l y t h e p a r t a b o v e t h e 20" d i s c o n t i n u i t y ) it is t o h e r e m e m h e r e d t h a t t h e m o d e l c h o s e n is o n e in w l i i c h t h e a x e s of s y m - m e t r y of ali t h e c o n s t i t u e n t c r y s t a l s a r e r a d i a i . A c e o r d i n g l y , u n l e s s d e c i s i v e e v i d e n c e of a n i s o t r o p y i n t l i e E a r t l i ' s c r u s t is f o r t h e o m i n g f r o m o t h e r s o u r c e s , t l i i s w i l l n o t h e a n a d e q u a t e e x p l a n a t i o n of t h e o h s e r v e d l i m e d i f f e r e n c e T „ — T v . P r o v i s i o n a l l y , t h e n , M i s s L e h m a n n ' s e x p l a n a t i o n s e e m s t h e m o r e l i k e l y . Comunicazione presentala al Convegno per la costituzione della F.S.E., tenutosi a Verona dal 12 al 16 settembre 1950 (v. « Annali di Geofìsica », voi. I l i , pag. 579). SUMMARY In earthquakes recorded at moderate distances it lias been ohserved that S phase appears first as SII, folloived some 10 to 14 seconds later by SV. The object of tliis paper is to try to decide ichether doublé refraction is likely to be the explanation of tliis jìlwnomenon. A simpie model to consider ivould be a « transversely isotropie » material, symmetrical about the radiai direction. Formulae for the velocities of SII and S V waves are available; tliese velocities depend on the angle that the ray makes ivi t li the norma!. It is unlikely that the Eartli could be as markedly anisotropie as the minerai beryl, which is transversely isotropie; aceordingly, this material, of ivhich the fi ve clastic constants are knoivn is taken as an extreme example, and the velocities of SH and SV for different angles of incidence are « scaled down » so as to match the velocity of distortional ivaves in granite. It is then possible to calcitiate the difference in the timo taken by ivaves from one point of the surface of the Earth to anotlier point on the surface according as the S wave in the surface layer is of SH or SV type. It is found that, even in this extreme case, a layer of anisotropie rock some 30 km thick tvould account only for a time difference of 8 R . S T O N E L E Y about 1 y, seconds. Thus, if the Earth ivere as strongly anisotropie as beryl (which is unlikelyj oue would need tlie layer of preferred orien- tatimi to extend to a depth of about 300 km. This is difficult to believe, and thus no great credence can be attaclied to ari explanation in ternis of doublé refraction. B I B L I O G R A P H Y ( ! ) F . NEUMANN, Bull. Seis. Soc. Amer. 20, 1 5 - 2 0 ( 1 9 3 0 ) . ( 2 ) P . BYERLY, Bull. Seis. Soc. Amer. 28, 1 2 ( 1 9 3 8 ) . ( 3 ) I . LEUMANN, Pubi. Bur. Cent. Séism. N . 1 2 , 1 9 ( 1 9 3 5 ) . ( ' ) W . V O I C T , Lehrbuck der Kristaliphysik ( 1 9 2 8 ) . ( s ) R . STONELEY, Geophys. Supp. 5, 3 4 3 , 1 9 4 9 , e.g. K . E . B u l l e n , T h e o r e t i c a l S e i s m o l o g y , p . 1 0 9 . ( f i ) R . STONELEY, LOC. cit. e q u a t i o n s ( 1 1 ) o ( 1 1 a ) .