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Disk ray theory in transversely isotropic media Yedlin, Mathew Jacob 1978

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DISK RAY THEORY IN TRANSVERSELY ISOTROPIC MEDIA by Mathew Jacob Y e d l i n B. Sc. (Hons.) , U n i v e r s i t y o f A l b e r t a , 1971 M.Sc, U n i v e r s i t y of Toronto, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Geophysics and Astronomy) accept t h i s t h e s i s as conforming t o the r e q u i r e d standard The U n i v e r s i t y Of B r i t i s h Columbia A p r i l , 1978 Mathew Jacob Yedlin,1978 In presenting th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permission. 0 Department of GEOPHYSICS AND ASTRONOMY The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date APRIL 28, 1978 A b s t r a c t The f i r s t motion approximation has been used t o c a l c u l a t e s y n t h e t i c seisnograms i n t r a n s v e r s e l y i s o t r o p i c , l i n e a r , e l a s t i c media. To achieve t h i s end the equations of motion have been s o l v e d i n a g e o m e t r i c a l o p t i c s regime. Formal l y , t h i s has been accomplished by the use of asymptotic propagator matrices. T h i s formalism i s important, s i n c e t h e phase of the JWKB r e f l e c t i o n c o e f f i c i e n t can be e a s i l y c a l c u l a t e d by c o n s i d e r a t i o n of the r a d i a t i o n c o n d i t i o n . C a l c u l a t i o n of t h i s r e f l e c t i o n c o e f f i c i e n t has shown t h a t the t u r n i n g p o i n t behaviour i s i d e n t i c a l to t h a t obtained f o r an i s o t r o p i c medium. The s i m i l a r i t y of the t u r n i n g p o i n t behaviour i s a d i r e c t consequence of the p h y s i c a l r e s u l t t h a t a t a t u r n i n g p o i n t the phase and group v e l o c i t i e s are i n the same d i r e c t i o n . To understand the r e s u l t s of the f i r s t motion approximation a p p l i e d to a simple upper mantle model, i t i s f i r s t necessary to understand the b a s i c p h y s i c s of t r a n s v e r s e l y i s o t r o p i c media. T h i s has been achieved by examination of the d i s p e r s i o n r e l a t i o n a r i s i n g from Newton*s Laws f o r an e l a s t i c s o l i d . From the d i s p e r s i o n r e l a t i o n , i t has been demonstrated how the Green's Function can be c o n s t r u c t e d using elementary p r o j e c t i v e geometry. Subsequently, the nature of the Green's Fu n c t i o n has been analyzed. The a n a l y s i s of the Green's Function (wave i i i s urface) i s important because i t f a c i l i t a t e s comprehension of any dynamical r e s u l t s . The s y n t h e t i c s e i sinograms were c a l c u l a t e d using ray parameter versus d i s t a n c e curves. These curves were obtained by i n t e g r a t i o n of the ray eguations d e r i v e d form the d i s p e r s i o n r e l a t i o n s . , A Gaussian-Kantorovich method was u t i l i z e d t o perform the r e q u i r e d i n t e g r a t i o n . T h i s hy b r i d i n t e g r a t i o n technique proved to be extremely f a s t and a c c u r a t e . When the r e s u l t i n g p - d e l t a curve was used to c a l c u l a t e the s y n t h e t i c seismogram, the main e f f e c t of the a n i s o t r o p i c model c o n s i d e r e d was a kinematic one - the main a r r i v a l s were e a r l i e r than those f o r an i s o t r o p i c model. i v T a ble of Contents a b s t r a c t • • • - i i Table of Contents • - i v L i s t o f F i g u r e s • • — — v Forward •—• — • — - v i i i Acknowledgements - — x i Chapter I • • *. • 1 Chapter I I — • • •—• • • — 1 3 I n t r o d u c t i o n • 13 S e c t i o n 1. Foundations For The Equations Of Motion —-m Se c t i o n 2. S o l u t i o n s Of The Equations Of Motion -27 Chapter I I I 52 i n t r o d u c t i o n 52 S e c t i o n 1. Bay Theory P a r t I • — 54 S e c t i o n 2. Bay Theory P a r t I I 64 Chapter IV — • : — 8 5 I n t r o d u c t i o n ! — 85 S e c t i o n 1. Development Of The Equations Of Motion — - — 8 7 S e c t i o n 2. D e t a i l s Of The Source — 99 S e c t i o n 3. S o l u t i o n For The Homogeneous System Using A i r y F u n ctions — ~ 107 Chapter V — — --136 I n t r o d u c t i o n 136 S e c t i o n 1, I n t u i t i v e Development Of Disk Ray Theory -138 S e c t i o n 2. Equal Phase Method-JHKB R e f l e c t i o n C o e f f i c i e n t — •— 118 S e c t i o n 3. Equal Phase Method- Implementation 155 S e c t i o n 4. Simple Seismic C a l c u l a t i o n s - - - - — — 1 6 8 Co n c l u s i o n s — : — ; — — •—198 B i b l i o g r a p h y —•* --200 V L i s t Of F i g u r e s F i g u r e Page 1.1 Model f o r o l i v i n e o r i e n t a t i o n near a r i d g e 6 2.1 Geometric c o n s t r u c t i o n of wave s u r f a c e from the slowness s u r f a c e 34 2.2 Slowness and wave s u r f a c e f o r a t r a n s v e r s e l y i s o t r o p i c medium 35 2.3 I l l u s t r a t i o n of the d e f i n i t i o n of pole and p o l a r 36 2.4 C o n s t r u c t i o n of the p o l a r r e c i p r o c a l 38 2.5 P o i n t s on the slowness s u r f a c e s h a r i n g a common tangent 40 2.6 D i f f e r i n g group v e l o c i t i y magnitudes i n the neighborhood o f p o i n t s s h a r i n g a common tangent 41 2.7 I l l u s t r a t i o n o f wave s u r f a c e with s i n g u l a r i t i e s H2 2.8 Parametric r e p r e s e n t a t i o n o f the slowness s u r f a c e 46 5.1 S u i t e o f rays l e a v i n g a shot 139 5.2 A c o u s t i c plane waves i n a quartz c r y s t a l 141 5.3 Disk i n t e r c e p t i n g the su r f a c e i n an a n i s o t r o p i c medium 142 5.4 Simple p - d e l t a curve 145 5.5 P - d e l t a curve f o r compressional waves-no a n i s o t r o p y 173 5.6 P - d e l t a curve f o r guasi-compressional waves-10% a n i s o t r o p y 174 5.7 P- d e l t a curve f o r guasi - c o m p r e s s i o n a l waves-30% a n i s o t r o p y 175 5.8 D i r e c t i v i t y f u n c t i o n {compressional waves) h o r i z o n t a l displacement-no a n i s o t r o p y 176 v i 5.9 D i r e c t i v i t y f u n c t i o n (quasi-compressional waves) h o r i z o n t a l displacement- 10% a n i s o t r o p y 177 5.10 D i r e c t i v i t y f u n c t i o n (quasi-compressional waves) h o r i z o n t a l displacement- 30% a n i s o t r o p y 178 5.11 D i r e c t i v i t y f u n c t i o n (compressional waves) v e r t i c a l displacement-no a n i s o t r o p y 179 5.12 D i r e c t i v i t y f u n c t i o n (quasi-compressional waves) v e r t i c a l displacement-10% a n i s o t r o p y 180 5.13 D i r e c t i v i t y f u n c t i o n (quasi-compressional waves) v e r t i c a l displacement-30% a n i s o t r o p y 181 5.14 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r compressional waves c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5 . 5 182 5.15 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r quasi-compressional waves c a l c u l a t e d u s i n g p - d e l t a curve shown i n f i g u r e 5.6 10% a n i s o t r o p y 183 5.16 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r quasi-compressional waves c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.7 30% a n i s o t r o p y 184 5.17 S y n t h e t i c seismogram of v e r t i c a l displacement f o r compressional waves c a l c u l a t e d u s i n g p-d e l t a curve shown i n f i g u r e 5.5 185 5.18 S y n t h e t i c seismogram of v e r t i c a l displacement f o r q u a s i - c o m p r e s s i o n a l waves c a l c u l a t e d u s i n g p - d e l t a curve shown i n f i g u r e 5.6 10% a n i s o t r o p y 186 5.19 S y n t h e t i c seismogram of v e r t i c a l displacement f o r q u a s i - c o m p r e s s i o n a l waves c a l c u l a t e d u s i n g p - d e l t a curve shown i n f i g u r e 5.6 30% a n i s o t r o p y 187 5.20 P - d e l t a curve f o r quasi-shear waves- 10% a n i s o t r o p y 188 5.21 P - d e l t a curve f o r quasi-shear waves- 30% a n i s o t r o p y 189 v i i 5.22 D i r e c t i v i t y f u n c t i o n {guasi-shear waves) h o r i z o n t a l displacement-10% a n i s o t r o p y 190 5.23 D i r e c t i v i t y f u n c t i o n {guasi-shear waves) v e r t i c a l displacement- 10% a n i s o t r o p y 191 5.24 D i r e c t i v i t y f u n c t i o n {guasi-shear waves) h o r i z o n t a l displacement-30% a n i s o t r o p y 192 5.25 D i r e c t i v i t y f u n c t i o n {guasi-shear waves) v e r t i c a l displacement-30% a n i s o t r o p y 193 5.26 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r guasi-shear waves, c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.20-10% a n i s o t r o p y 194 5.27 S y n t h e t i c seismogram of v e r t i c a l displacement f o r g u a s i - s h e a r waves, c a l c u l a t e d using p-d e l t a curve shown i n f i g u r e 5.20-10% a n i s o t r o p y 195 5.28 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r guasi-shear waves, c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.21-30% a n i s o t r o p y 196 5.29 s y n t h e t i c seismogram of v e r t i c a l displacement f o r guasi-shear waves, c a l c u l a t e d using p-d e l t a curve shown i n f i g u r e 5.21-30% a n i s o t r o p y 197 v i i i f o r w a r d ! A i l s , O b j e c t i v e s , And O r i g i n a l i t y The main t h r u s t of the t h e s i s w i l l be the a n a l y t i c d e r i v a t i o n (using g e o m e t r i c a l o p t i c s ) and numerical c a l c u l a t i o n of simple dis k r a y seismograms i n t r a n s v e r s e l y i s o t r o p i c , v e r t i c a l l y inhomogeneous e l a s t i c media. In Chapter I the main emphasis i s the motivation f o r the study of a n i s o t r o p y . T h i s i s f o l l o w e d i n Chapter I I by the p r e s e n t a t i o n of the b a s i c p h y s i c s of a n i s o t r o p i c media. The o b j e c t i v e i n t h i s c h a p t e r i s the development o f an understanding o f the wave s u r f a c e , the Green*s Function f o r the elastodynamic equations d r i v e n by a p o i n t source. T h i s o b j e c t i v e i s achieved by d e s c r i b i n g i n d e t a i l the geometric c o n s t r u c t i o n s , based on p r o j e c t i v e geometry. The nature of the s i n g u l a r i t i e s o f the wave s u r f a c e are then analyzed a l g e b r a i c a l l y . Chapter I I I uses the p h y s i c a l r e s u l t s of Chapter I I to develop the r e l e v a n t r a y eguations. Two d i f f e r e n t approaches are used to obt a i n these equations. The o b j e c t i v e i n p r e s e n t i n g these methods i s to compare them a l g e b r a i c a l l y , and use the f i n a l r e s u l t i n the c o n s t r u c t i o n of an e f f i c i e n t , a c c u r a t e , r a y t r a c i n g technique. The purpose i n de v e l o p i n g a good r a y - t r a c i n g technique i s t h a t p - d e l t a curves may be c o n s t r u c t e d , and from the s e , the main i x o b j e c t i v e of the t h e s i s nay be r e a l i z e d . Chapter IV i s concerned with the the development of a uni f o r m l y v a l i d s o l u t i o n t o the equations of motion. T h i s o b j e c t i v e i s achieved by u s i n g c l a s s i c asymptotic techniques, and the t u r n i n g p o i n t problem i s s o l v e d using the Langer t r a n s f o r m a t i o n . In Chapter 7, the main o b j e c t i v e o f the t h e s i s i s r e a l i z e d . An i n t u i t i v e d e r i v a t i o n of d i s k ray theory i s presented. T h i s i s f o l l o w e d by an e v a l u a t i o n of the JWKB r e f l e c t i o n c o e f f i c i e n t , which i s then used to o b t a i n the a n a l y t i c form f o r the disk ray theory seismogram. T h i s a n a l y t i c form i s used to compute seismograms f o r simple e a r t h models. In order t o d e l i n e a t e the author*s c o n t r i b u t i o n from t h a t e x i s t i n g i n the l i t e r a t u r e , a l i s t of o r i g i n a l r e s u l t s i s l i s t e d below: 1) a c l e a r e l u c i d a t i o n of the wave s u r f a c e slowness s u r f a c e correspondence has been demonstrated., 2) A d i f f e r e n t i a l geometric proof has been developed t o i l l u s t r a t e how i n f l e c t i o n p o i n t s on the slowness s u r f a c e cause cusps on the wave s u r f a c e ; 3) Gaussian i n t e g r a t i o n has been a p p l i e d to the tau f u n c t i o n ; 4) The method of Kant o r o v i c h has been combined with the Gaussian i n t e g r a t i o n method t o produce an a c c u r a t e , f a s t r a y - i n t e g r a l c a l c u l a t o r ; 5) The source term method o f Takeuchi and S a i t o {1972) has been c a s t i n t o the up and downgoing wave formalism; 6) The method of asymptotic propagators {Chapman 1974b; Hoodhouse 1978) has been extended t o t r a n s v e r s e l y i s o t r o p i c X media, and, the JWKB r e f l e c t i o n c o e f f i c i e n t s have been c a l c u l a t e d f o r such media; 7) fin i n t u i t i v e d e r i v a t i o n f o r disk ray theory i n a n i s o t r o p i c media has been presented; 8} An a n a l y t i c d e r i v a t i o n o f the d i s k ray theory seismogram, i n c l u d i n g source e f f e c t s has been obtained; 9) Seismograms have been c a l c u l a t e d f o r v e r t i c a l and h o r i z o n t a l displacements f o r guasi-shear and q u a s i -compressional waves, i n the case of a poin t e x p l o s i o n . x i Acknoy1edqements Due to the unusual nature of the s u p e r v i s i o n f o r t h i s t h e s i s , the author i n t e r a c t e d with many i n d i v i d u a l s , and r e c e i v e d h e l p f u l s u g g e s t i o n s i n the course of c o u n t l e s s d i s c u s s i o n s . In p a r t i c u l a r , I would l i k e to acknowledge the c o n t r i b u t i o n of Ralph Wiggins, who suggested the p r o j e c t , and e x p l a i n e d some of the i n t u i t i v e a spects of d . r . t . John Woodhouse i n t r o d u c e d me to h i s ideas on asymptotic propagators, and d i s c u s s i o n s with him on the i n t u i t i v e b a s i s f o r d . r . t . i n a n i s o t r o p i c media a l s o proved h e l p f u l . , C h r i s Chapman was k i n d t o send p r e p r i n t s of h i s soon-to-be pu b l i s h e d 1978 s y n t h e t i c seismogram papers, and a l s o to e x p l a i n some of the s u b t l e t i e s of the JWKB method. Jim Varah helped s e t the stage f o r the h y b r i d method of i n t e g r a t i o n developed, by i n t r o d u c i n g the "method of the s u b t r a c t i o n of the s i n g u l a r i t y " i n i t s s i m p l e s t form. Throughout the t h e s i s work, B r i a n Seymour provided encouragement and weekly advi c e on a l l aspects of the work. To him I o f f e r my deepest thanks. Hick M o r t e l l p a r t i c i p a t e d i n many of these weekly d i s c u s s i o n s . I am very g r a t e f u l f o r h i s suggestions. George McMechan a l s o o f f e r e d encouragement, and I had many h e l p f u l d i s c u s s i o n s with him, x i i p a r t i c u l a r l y the d a i l y d i s c u s s i o n s d u r i n g the two Augusts spent at U n i v e r s i t y o f V i c t o r i a . Thank you very such George. To the members of the Geophysics Department, Bob E l l i s , Bon Clowes, Garry C l a r k e , Don R u s s e l l , and my s u p e r v i s o r . Tad O l r y c h , thank you not o n l y f o r the h e l p f u l d i s c u s s i o n s about d i v e r s e t h e s i s t o p i c s , but a l s o thanks f o r i n t r o d u c i n g me to geophysics, and seismology i n p a r t i c u l a r . ft s p e c i a l mention of g r a t i t u d e goes to John G i l l i l a n d , f o r h e l p f u l suggestions when the p r o j e c t was i n i t s i n f a n c y . A l s o a word of thanks to my f e l l o w students i n the department, f o r t h e i r comments and i n t e r e s t i n g d i s c u s s i o n s . I should l i k e to express g r a t i t u d e and acknowledge seme e n l i g h t e n i n g d i s c u s s i o n s with John O r c u t t , George Backus, and S t u a r t Crampin concerning wave propagation i n a n i s o t r o p i c media. * * * * * In an acknowledgement s e c t i o n o f a t h e s i s , i t i s always i n order t o thank the t y p i s t . There was no t y p i s t f o r t h i s t h e s i s . Rather, the author was f o r t u n a t e t o have the a s s i s t a n c e o f the boys at St. Andrew's H a l l i n t y p i n g and producing t h i s t h e s i s . The work production d e a d l i n e c o u l d s. not have been met without the help of John Hester, who x i i i d r a f t e d the equations, and helped i n production matters, and members of the H a l l { S t e f f a n Wegner, John Skinner, Perry W i l l i a m s , Randy Watson, Greg Jaron, Paul James, Ken Rush, Bob Cowin, Dave Simmons, Martin Connolly, P e t e r Cary, Shlcmo Levy, and H i l l o Shaw), S i n c e r e s t thanks a l s o t o Ralph P u d r i t z , who aided and encouraged the f i n a l p r o d u c t i o n of the t h e s i s . The author g r a t e f u l l y acknowledges the f i n a n c i a l support given by the N a t i o n a l Research C o u n c i l of Canada and f i n a n c i a l a s s i s t a n c e of the I n t e r n a t i o n a l N i c k e l Company of Canada. To one and a l l , my s i n c e r e s t merci beaucoup. 1 Chapter. I Since e a r l i e s t times, nan has been i n t e r e s t e d i n the study o f a n i s o t r o p y . As Husgrave (1970) remarks: "Presumably, Stone age man observed and very probably s t u d i e d , the phenomenon of c l e a v a g e -mechanical as w e l l as anatomical!" Impetus f o r the study of a n i s o t r o p y i n geophysics, and i n p a r t i c u l a r seismology, stems not from any romantic impulse, but r a t h e r from p e t r o l o g y . Only 18 years ago. B i r c h {1960) p u b l i s h e d h i s measurements of the v e l o c i t y o f compressional waves i n rocks at a pressure of 10 Kbars, u s i n g pulsed t r a v e l time t e c h n i gues. An a s s o c i a t e , Verma {1960), used p u l s e - i n t e r f e r e n c e methods t o measure the a c o u s t i c wave v e l o c i t y i n high d e n s i t y c r y s t a l s . In p a r t i c u l a r , he found t h a t f o r o l i v i n e , the compressional wave v e l o c i t i e s were as f o l l o w s : 0 \ re c tion * axis D oo] / axis C° to] 1: ex * is [p o {] 9.S7 7. 7 3 2 H i s r e s u l t s were important, i n t h a t a c o n c l u s i v e demonstration was provided t h a t some of the minerals comprising the e a r t h were a n i s o t r o p i c , i n s o f a r as t h e i r a c o u s t i c wave v e l o c i t i e s were concerned. F u r t h e r s t i m u l u s to i n v e s t i g a t e was s u p p l i e d by the f i r s t of many ocean r e f r a c t i o n experiments performed by R a i t t (1963) and h i s co-workers. R a i t t ' s i n i t i a l data were used by Hess (1961) t o d e r i v e the f i r s t of many models of the oceanic upper mantle. Hess (1961) combined the r e s u l t s o f R a i t t and Verma and a r r i v e d at a p r e l i m i n a r y model. s The sub-Moho compressional wave v e l o c i t y , o b tained by R a i t t , was determined by shooting along d i f f e r e n t azimuths i n the neighbourhood of Mendocino f r a c t u r e zone. The maximum v e l o c i t y (8.7 km/sec.) was measured p a r a l l e l t o the f r a c t u r e zone and the minimum v e l o c i t y (8.0 km/sec.) was p e r p e n d i c u l a r to the f r a c t u r e zone. Hess reasoned, using Verma's data, that the b-axis of o l i v i n e (which d e f i n e s the (0,1,0) plane) should a l i g n p e r p e n d i c u l a r t o the f r a c t u r e zcne, and the a - a x i s should a l i g n p a r a l l e l to i t . T h i s would be achieved by the (0,1,0) plane g l i d i n g along the plane of shear. For such a model the compressional v e l o c i t y p a r a l l e l t o the f r a c t u r e zone would be 9.87 km/sec. and p e r p e n d i c u l a r to i t 7.73 km/sec. although the p h y s i c s o f Hess's model appears c o r r e c t , the numerical v a l u e s are c o n s i d e r a b l y o f f ( F u c h s 1977). fi year l a t e r Backus (1965) again looked a t R a i t t * s data and very e l e g a n t l y d e r i v e d an eguation r e l a t i n g Pn v e l o c i t y 3 to the azimuth of t h e r e f r a c t i o n p r o f i l e . The e q u a t i o n . was de r i v e d by a p p l i c a t i o n of formal p e r t u r b a t i o n theory to the elastodynamic wave equation, when the medium was " m i l d l y a n i s o t r o p i c " ( l e s s than 10%). The p e r t u r b a t i o n t o the wave v e l o c i t y was then expressed as a F o u r i e r s e r i e s i n the azimuth angle. As Backus demonstrated, (1.1-1) proved to be very u s e f u l in f i t t i n g the measured v e l o c i t y as a f u n c t i o n of the azimuth angle t o o b t a i n 5 e l a s t i c parameters. Backus a l s o o btained c o r r e c t i o n terms to (1.1-1) i n the case t h a t the c u r v a t u r e of the upper mantle was s i g n i f i c a n t . In c o n c l u s i o n , he pointed out that Hess's model based on alignment of o l i v i n e c r y s t a l s c o u l d have an a l t e r n a t i v e , t h a t i s , the a n i s o t r o p y could be due to a s t a t i c s t r e s s p a t t e r n i n an i s o t r o p i c m a t e r i a l . Three years l a t e r , C h r i s t e n s e n and Crosson (1968) p o s t u l a t e d t h a t an o l i v i n e - r i c h upper mantle m a t e r i a l moving v e r t i c a l l y i n the presence of f r a c t u r e zones, would be t r a n s v e r s e l y i s o t r o p i c , with the a x i s of symmetry i n c l i n e d to the v e r t i c a l . From a study o f u l t r a m a f i c r o c k s , the p r e f e r r e d o r i e n t a t i o n of the b- a x i s o f o l i v i n e was pe r p e n d i c u l a r to s c h i s t o s i t y or banding, Dunite (90% + E Cos 0 + \- S i r s V<p (1. 1-1) o l i v i n e ) , i n the above o r i e n t a t i o n , was found t o behave l i k e a t r a n s v e r s e l y i s o t r o p i c medium f o r compressional waves. As w e l l , i t was expected t h a t i n the case of a t r a n s v e r s e l y i s o t r o p i c medium, the maximum compressional wave v e l o c i t y should be p e r p e n d i c u l a r to the symmetry a x i s . The year 1969 provided a f u r t h e r wealth o f i n f o r m a t i o n about the oceanic upper mantle. fiaitt and Shor (1969) obtained more data i n the n o r t h e a s t P a c i f i c . , For i n t e r p r e t a t i o n , a modified time term method was used to i n c l u d e azimuthal a n i s o t r o p y . T h i s was achieved by i n c o r p o r a t i n g Backus* eguation (1.1-1) i n the time term method of Berry and west (1966). They found t h a t the d i f f e r e n c e i n v e l o c i t y (0.3 km/sec.) p a r a l l e l and p e r p e n d i c u l a r to f r a c t u r e zones was not as l a r g e as t h a t p r e d i c t e d by Hess. F r a n c i s (1969) used the r e s u l t s of S a i t t and Shor to o f f e r an a l t e r n a t i v e to the model of Hess. His p r i n c i p a l o b j e c t i o n t o Hess's model was t h a t t r a n s l a t i o n a l g l i d i n g of the (0,1,0) f a c e of the c r y s t a l s was, f i r s t l y , i m p o s s i b l e , s i n c e the creep of u l t r a m a f i c s should be i n s i g n i f i c a n t away from the r i d g e , and secondly, i n c o n c e i v a b l e , s i n c e the d e f o r m a t i o n a l s t r a i n r e g u i r e d would have to extend a l a r g e d i s t a n c e away from the r i d g e . F u r t h e r , F r a n c i s , c i t i n g R a yleigh (1968), s t a t e d that f o r o l i v i n e , the predominant g l i d e mechanism was (0,k,l) [1,0,0]}, not n e c c e s s a r i l y (0,1,0) [1,0,0], As a r e s u l t , the model a r r i v e d at to 5 e x p l a i n the data o f R a i t t was t h a t v e l o c i t y shear, g r e a t e s t under the r i d g e due to c o n v e c t i o n c u r r e n t s , would cause s l i p with (0,k,l) [1,0,0] th e major mechanism.. In the s i m p l e s t case, the v e l o c i t y shear o r i e n t s the o l i v i n e so t h a t (0,1,0) plane l i e s i n the plane of flow and the d i r e c t i o n of flow i s the (1,0,0) d i r e c t i o n . Such a model ( i l l u s t r a t e d i n F i g . 1) e x p l a i n s the low compressional wave v e l o c i t y p a r a l l e l to the r i d g e and the high compressional wave v e l o c i t y p e r p e n d i c u a l a r to the r i d g e , although the a n i s o t r o p y f a c t o r i s too l a r g e , 20% (Fuchs 1977). F i n a l l y , another model f o r the oceanic upper mantle was developed by Crosson and C h r i s t e n s e n (1969) who used Backus* (1965) f o r m u l a t i o n p o s t u l a t i n g the e x i s t e n c e o f a t i l t e d , t r a n s v e r s e l y i s o t r o p i c upper mantle. The data of R a i t t (1963) f i t t e d p r e v i o u s l y by Backus were r e f i t t e d u s i n g the t i l t e d t r a n s v e r s e l y i s o t r o p i c model. The f i t s obtained using the t r a n s v e r s e l y i s o t r o p i c model are i n e x c e l l e n t agreement with Backus* f i t u s i n g general a n i s o t r o p y . Since 1969, a number of r e f r a c t i o n surveys have been undertaken with a view to check and r e i n f o r c e the i n i t i a l d i s c o v e r y of R a i t t . Keen and T r a m o n t i n i (1970), found that i n the A t l a n t i c Ocean, the Pn v e l o c i t y was 7.9 km/sec, with an a n i s o t r o p y f a c t o r o f 8%. Again t h e formula of Backus was used t o f i t the data, but a l s o the p o s s i b i l i t y t h a t c u r v a t u r e or l a t e r a l v a r i a t i o n s a f f e c t e d the r e s u l t was examined. I t was found t h a t n e i t h e r of these e f f e c t s c o u l d 6 DEPTH km 50 f TENSION AND IGNEOUS ACTIVITY 0 + DISTANCE km 50 ; :C>RVJ5T-; STAGNANT ZONE V BASALT \ DISCHARGE.' A A A A A A A C C f " b s (after Francis(l969)) F i g u r e 1.1 Hodel f o r o l i v i n e o r i e n t a t i o n near the r i d g e with the a - a x i s , which d e f i n e s the d i r e c t i o n of the f a s t e s t p-wave v e l o c i t y p e r p e n d i c u l a r t o the r i d g e s t r i k e . A f t e r F r a n c i s (1 969) . e x p l a i n the observed a n i s o t r o p y s i n c e i t was not p o s s i b l e t h a t i n the area of the survey, a l a t e r a l v a r i a t i o n of 165? co u l d occur. For c u r v a t u r e to a f f e c t the r e s u l t s , i t was necessary t h a t a v a r i a t i o n o f 7 km i n the depth o f the Moho 7 d i s c o n t i n u i t y e x i s t i n the r e g i o n of the survey. F o l l o w i n g t h i s survey, Keen and B a r r e t t (1971) estimated oceanic upper mantle a n i s o t r o p y i n the P a c i f i c , and found t h a t the d i r e c t i o n of maximum compressional wave v e l o c i t y at an azimuth of 107°, was not e x a c t l y the d i r e c t i o n of s t r i k e of the nearest f r a c t u r e zone, but only approximately so. Again, a value of 8% a n i s o t r o p y was o b t a i n e d , with a mean value of the compressional wave v e l o c i t y computed as 8.07 km/sec. Host r e c e n t l y a s e i s m i c r e f r a c t i o n p r o f i l e p a r a l l e l to and p e r p e n d i c u l a r to the E x p l o r e r Ridge has been completed. The Pn v e l o c i t i e s d e r i ved by Clowes and Malecek (1976) were 7.3 km/sec. p a r a l l e l t o the r i d g e and 7.85 km/sec. p e r p e n d i c u l a r t o the r i d g e with an a n i s o t r o p y of 75E. Dpon c o n s i d e r a t i o n of the data presented above, i t appears t h a t f o r the o c e a n i c upper mantle, an i n t r i n s i c a n i s o t r o p y of 1% t o 8% i s o b t a i n e d , with the d i r e c t i o n of maximum v e l o c i t y p a r a l l e l t o the f r a c t u r e zones. However, why should the a n i s o t r o p y be r e s t r i c t e d t o oceanic upper mantle? As Fuchs (1977) p o i n t s out, the temperatures (800 to 1000°C) and pressures (10 kbar.) are higher at the c o n t i n e n t a l c r u s t - m a n t l e boundary than at the oceanic c r u s t -mantle boundary. T h e r e f o r e , the higher temperatures and pressures are more l i k e l y to generate the p r e f e r r e d o r i e n t a t i o n of o l i v i n e , held r e s p o n s i b l e f o r t h e azimuthal a n i s o t r o p y observed. The e x i s t e n c e of a n i s o t r o p y i n the c o n t i n e n t a l 8 l i t h o s p h e r e had been suggested as e a r l y as 1966 by Crampin. However, i t was not u n t i l the work of Bamford (1973,1976) t h a t the e x i s t e n c e of l a r g e s c a l e c o n t i n e n t a l a n i s o t r o p y was observed. Bamford used data from the F e d e r a l Republic of Germany r e f r a c t i o n p r o f i l e s , and used a modified time term method t o i n t e r p r e t the data. From the l a t e s t a n a l y s i s (1977), which i n c l u d e d r e s u l t s from the 1972 Rhinegraben experiment, Bamford o b t a i n e d an a n i s o t r o p y of 6% to 1%, with the d i r e c t i o n of maximum Pn v e l o c i t y N20° E. What i s i n t e r e s t i n g about the r e s u l t i s that the magnitude of the a n i s o t r o p y i s equal t o th a t deduced from oceanic upper mantle i n v e s t i g a t i o n s . However, the d i r e c t i o n of maximum c r u s t a l Pn v e l o c i t y d i f f e r s by 20° from the oceanic r e s u l t s , i n d i c a t i n g a s l i g h t v a r i a t i o n i n the p o s s i b l e a n i s o t r o p i e s of the oceanic upper mantle and the c r u s t a l upper mantle. 9 P e t r o l o g i c a l Evidence f o r A n i s o t r o p y The i m p l i c a t i o n of the s e i s m i c experiments d e s c r i b e d i n the p r e v i o u s s e c t i o n i s that the sub-Moho i s a n i s o t r o p i c . To r e i n f o r c e the s e i s m o l o g i c a l evidence f o r the e x i s t e n c e of a n i s o t r o p y i t i s e s s e n t i a l to c o n s i d e r p e t r o l o g i c a l work which supports the s e i s m i c r e s u l t s . A f t e r V e n a ' s (1960) measurement of the compressional and shear wave v e l o c i t i e s i n o l i v i n e , the next s i g n i f i c a n t work on the s u b j e c t was by B a l e i g h (1968), who s t u d i e d the mechanisms of p l a s t i c deformation o f o l i v i n e . His a n a l y s i s of the s l i p system f o r o l i v i n e (0,k,l) [1,0,0] was l a t e r used by F r a n c i s (1969) i n c o n s t r u c t i n g h i s model of oceanic upper mantle. More p e t r o l o g i c a l r e s u l t s on the deformation of o l i v i n e were obtained by Ave'Lallement and C a r t e r (1970) They r e c r y s t a l l i z e d o l i v i n e at temperatures g r e a t e r than 1050°C, at pressures of 3 kbar, f o r a s t r a i n r a t e of 10—3/sec. E x t r a p o l a t i o n of the r e s u l t s t o s t r a i n r a t e s and temperatures t y p i c a l of o c e a n i c l i t h o s p h e r e ( 1 0 _ * * / s e c . and 500°C) showed t h a t the b a x i s (0,1,0) o f the new c r y s t a l s a l i g n e d i n the d i r e c t i o n o f maximum compressive s t r e s s . Thus, Ave L'allement and c a r t e r deduced t h a t the above mechanism accounts f o r the o r i e n t a t i o n of o l i v i n e c r y s t a l s i n oceanic l i t h o s p h e r e . , However, as Fuchs (1977) has pointed out, the proposed mechanism could not e x p l a i n the 10 s e i s m i c r e s u l t s f o r two reasons: A) The azimuthal v e l o c i t y dependence would occur only near the r i d g e a x i s where the temperature i s high (500°C); B) The b a x i s would not a l i g n p e r p e n d i c u l a r to the plane of shear as suggested, s i n c e the p r i n c i p a l a x i s i s a t 45° to the plane of shear. The r e s u l t s of Ave«Lallement and C a r t e r however, have been used by some authors ( C h r i s t e n s e n , 1971) to e x p l a i n the o r i e n t a t i o n of the o l i v i n e f a b r i c i n the rocks from the Twin S i s t e r s i n Washington. Fu r t h e r p e t r o l o g i c s t u d i e s have been c a r r i e d out by P e s e l n i c k et a l (1974) i n the Ivr e a zone of the I t a l i a n A l p s . T h i s zone i s of p a r t i c u l a r i n t e r e s t , s i n c e i t may be the boundary o f the mantle and deep c r u s t . Again the o l i v i n e was found t o have a p r e f e r r e d o r i e n t a t i o n . The measured v e l o c i t y a n i s o t r o p y was 7%, and was independent of pressure. The d i r e c t i o n o f minimum v e l o c i t y was found to be pe r p e n d i c u l a r to the f o l i a t i o n plane, and the d i r e c t i o n of maximum v e l o c i t y was p a r a l l e l to the f o l i a t i o n plane. P e s e l n i c k et a l favored F r a n c i s 1 (1969) i n t e r p r e t a t i o n f o r the gen e r a t i o n of ocean upper mantle a n i s o t r o p y , with the p r i n c i p a l s l i p mechanism, as d e s c r i b e d by R a l e i g h , being r e s p o n s i b l e f o r p r e f e r r e d o r i e n t a t i o n o f o l i v i n e . , Within the l a s t year, two a r t i c l e s have appeared which e m p h a t i c a l l y r e q u i r e i n t r i n s i c a n i s o t r o p y of the upper mantle to e x p l a i n the s e i s m i c r e s u l t s . Since the v e l o c i t y 11 and a n i s o t r o p y of e c l o g i t e i s too smal l t o account f o r that observed, B o t t i n g a and A l l e g r e (1976) have deduced t h a t p e r i d o t i t e , not e c l o g i t e , must be the main c o n s t i t u e n t f o r sub-Moho a n i s o t r o p y . Green and Liebermann (1976) s t a t e the f o l l o w i n g : " I t appears to us u n l i k e l y t h a t u n c e r t a i n t i e s i n knowledge or e r r o r i n p r e d i c t i o n of the e l a s t i c p r o p e r t i e s of minerals and t h e i r P,T d e r i v a t i v e s are s u f f i c i e n t t o permit v a l u e s of Vp>8.5 km/sec i n p e t r o l o g i c a l l y reasonable rock types under upper mantle c o n d i t i o n s , without appeal to s e i s m i c a n i s o t r o p y " . 12 Other Forms of A nisotropy The p r e v i o u s s e c t i o n s were concerned with the r o l e of i n t r i n s i c a n i s o t r o p y i n the c o n s t i t u e n t s of the upper mantle. However, t h e r e are other types o f a n i s o t r o p y which have s i g n i f i c a n c e i n other s t r a t a of the e a r t h . Such a n i s o t r o p y can be generated i n two ways: 1) p r e - s t r a i n s t a t i c s t r a i n f i e l d ; and 2) t h i n l a y e r s . Many g e o p h y s i c i s t s have been concerned with the t h i n l a y e r problem which i s important i n e x p l o r a t i o n seismology. As e a r l y as 1950, Thomson i n v e s t i g a t e d the p e r i o d i c , i s o t r o p i c two l a y e r e d medium, c o n s i s t i n g of f i n e l a y e r s , u s i n g the now- famous propagator matrices. Dhrig and Van Melle <1955) noted t h a t f i n e s t r a t i f i c a t i o n s o f a l t e r n a t i n g bands of limestone and s h a l e , or sand and s h a l e c o u l d be regarded as smoothly a n i s o t r o p i c i f the dominant wavelength was much g r e a t e r than the bed t h i c k n e s s . Potsma (1955) q u a n t i f i e d the above i n t u i t i v e r e s u l t s by c a l c u l a t i o n of the e f f e c t i v e e l a s t i c c o n s t a n t s o f a p e r i o d i c i s o t r o p i c two-l a y e r e d medium. However, the r e s u l t s are s p e c i f i c only f c r p e r i o d i c v a r i a t i o n s of the l a y e r p r o p e r t i e s . Backus (1962) has given a more g e n e r a l treatment of the f i n e l y l a y e r e d i s o t r o p i c medium, which can be modelled as a smooth, t r a n s v e r s e l y i s o t r o p i c medium. 13 CHAPTER II Introduction In t h i s chapter the basic elastodynamics of anisotropic e l a s t i c media w i l l be presented. By examination of symmetry properties of the media, the most general equations of motion can be reduced to simpler and simpler forms. By substitution of plane wave solutions into the elastodynamic wave equations, dispersion r e l a t i o n s may be derived. These dispersion r e l a t i o n s can be used to construct the Green's function for the par t i c u l a r e l a s t i c material under consideration., This method of construction i s based on pri n c i p l e s of elementary projective qeometry, which was at i t s peak i n the 19th century. (Klein 1939; Poncelet 1822; Briot 6 Bouquet 1896) The geometric re s u l t s of t h i s chapter are important, since they form the physical basis for the derivation of the ray equations to be considered Chapter I I I . These ray equations provide the basis f o r the construction of p-delta curves, which may then be used for the ca l c u l a t i o n of synthetic seismograms. 1 4 Section 1 Foundations for the Eguations of Motion. The eguations of motion are derived on the assumption that the e l a s t i c material i s Hookean (Love 1945). That i s , the free energy about an equilibrium configuration i s expressed as a Taylor's expansion i n the s t r a i n (Musgrave 1970; Landau-Lifschitz 1959) as follows: where A F=free i n t e r n a l energy due to an e l a s t i c deformation, tij i s the s t r a i n tensor and © refers to the undeformed state of the s o l i d . , since work must be done to deform the system A F ^ O ,and the r i g h t hand side of (2.1-1) must be a posi t i v e d e f i n i t e quadratic form. The work done i n deforminq an e l a s t i c body, equal to the change in free energy, i s : W = | < j e ' j (2.1-2) By comparison of (2.1-1) and (2.1-2) i t i s evident that: (2. 1-3) where 6ij i s the s t r e s s t e n s o r . Eguation (2.1-3) i s a c o n s t i t u t i v e one r e l a t i n g s t r e s s to s t r a i n i n a l i n e a r e l a s t i c medium. That i s . (2. 1-4) The q u a n t i t y Cij K £ i s the f o u r t h order s t i f f n e s s t e n s o r with 81 components. S i n c e £ c j ^ ^ a n d , and (2.1-5) the f o l l o w i n g symmetry c o n d i t i o n s are v a l i d ; C~WS c t > * ; (2.1-6) Hence, C i i J K £ t n e f o u r t h order s t i f f n e s s tensor with 81 components can be reduced t o 21 independent components. Fu r t h e r r e d u c t i o n o f the number of components i s achieved c o n s i d e r a t i o n of g r e a t e r symmetries f o r the p a r t i c u l a r medium i n v o l v e d . Newton's laws f o r an e l a s t i c s o l i d i n the absence of body f o r c e s become 17 (2. 1 - 7 ) Eguation (2.1-7) r e l a t e s the mass times a c c e l e r a t i o n ( i n C a r t e s i a n c o - o r d i n a t e s ) to the divergence of the s t r e s s t e n s o r . That the a c c e l e r a t i o n i s r e l a t e d to the divergence of the s t r e s s t e n s o r i s d e r i v e d by many authors (Love 1945; B u l l e n 1965; J e f f r e y s 1959) The study of eguation (2.1-7) w i l l be the f o c a l p o i n t i n t h i s chapter. Of course source terms w i l l have t o be i n c l u d e d i n any t r a n s i e n t pulse propagation problem (Takeuchi S S a i t o 1972). In most s e i s m o l o g i c a l problems i t i s n e i t h e r e s s e n t i a l nor p o s s i b l e t o c o n s i d e r a l l 21 components of CL^KH • Media u s u a l l y under c o n s i d e r a t i o n have some degree o f symmetry. T h i s symmetry may be e x p l o i t e d and the number of components of the e l a s t i c tensor reduced. These manipulations are s i m p l i f i e d i f the b a s i c concepts of group theory are i n t r o d u c e d . A group G i s d e f i n e d t o c o n s i s t of a s e t of elements &,B,C,... with an o p e r a t i o n of computation ( m u l t i p l i c a t i o n ) such t h a t the f o l l o w i n g axioms are s a t i s f i e d : 18 1) C l o s u r e - For any A,B, members o f G, AB, i s a l s o a member of Gj 2) A s s o c i a t i v e Rule - For any A,B,C, members o f G (AB) C=A (BC) =ABC; 3) I d e n t i t y - There e x i s t s an element of G known as I so t h a t f o r a l l A, members of G, AI=IA =A. 4) Inverse - For every element A of G there e x i s t s an i n v e r s e A - J , a l s o i n G, such that A - 1A=AA _ 1=I. To demonstrate the u t i l i t y of the above a b s t r a c t paradigm, c o n s i d e r the elements of the Group a s s o c i a t e d with the c o u n t e r c l o c k w i s e r o t a t i o n of a sguare. R o t a t i o n s of 0, 90* 180, 270 degrees, can be represented as complex numbers C l i L ^ ^1CL/^- # T h e group m u l t i p l i c a t i o n t a b l e i s g iven below. i 1 i QJirh *<* 1 1 1 (2.1-8) Here the i d e n t i t y element i s 1, and the i n v e r s e elements are obvious from the t a b l e . An i n t e r e s t i n g point t o note i s th a t f o r the given m u l t i p l i c a t i o n r u l e , the element of G, e x p ( i # V 2 ) , can generate the other elements of the group. I t i s known as the generator of the group. Since any group can be def i n e d i n terras of i t s s e t of generators, i t i s s u f f i c i e n t to c o n s i d e r group o p e r a t i o n s i n v o l v i n g o n l y these generators. 19 In the study of the v a r i o u s symmetries of the e l a s t i c tensor the p a r t i c u l a r t r a n s f o r m a t i o n of i n t e r e s t i s known as a poin t t r a n s f o r m a t i o n . These t r a n s f o r m a t i o n s leave a t l e a s t one p o i n t of the e l a s t i c medium f i x e d , and c o n s i s t of r o t a t i o n s and r e f l e c t i o n s . The complete d e s c r i p t i o n of a l l the symmetries of a p a r t i c u l a r e l a s t i c medium can be represented by t h i s group, whose elements d e f i n e the p a r t i c u l a r symmetry o p e r a t i o n s which are allowed. As shown above f o r the simple case o f the r o t a t i o n of the sguare, not a l l the elements of the group need be con s i d e r e d . Only the generators o f the group need be i n v o l v e d i n any c o - o r d i n a t e manipulation. A p p l i c a t i o n of the above methodology i s bes t i l l u s t r a t e d by an example. F i r s t , CcjicSL c a n fee r e w r i t t e n as a second order t e n s o r i f the f o l l o w i n g i n d e x i n g scheme i s used (Daley & Hron 1977; Musgrave 1970): Old Index P a i r New S i n g l e Index 11 1 22 2 33 3 23,32 31,13 5 12,21 6 In t h i s scheme the new s t r e s s - s t r a i n r e l a t i o n would be r e w r i t t e n as: (2.1-9) with the s t r e s s <5\ 11 33 6 xi (2. 1-10) and the s t r a i n e = i i (2. 1-11) Now f o r a p a r t i c u l a r symmetry, the a p p l i c a t i o n of a p a r t i c u l a r p o i n t t r a n s f o r m a t i o n to generate a new e l a s t i c tensor may be w r i t t e n as (ftuld 1973) 21 d p * (2.1-12) where (2. 1-13) fis the transformation has at least one fixed point, (2.1-12) must hold i d e n t i c a l l y . For a n o n t r i v i a l example, consider an e l a s t i c medium with monoclinic symmetry. The simple r e f l e c t i o n matrix which represents the generator of t h i s point group i s (2. 1-14) For t h i s mirror r e f l e c t i o n , the matrix used in (2.1-12) i s 22 0 (2.1-15) a p p l y i n g the symmetry c o n d i t i o n (Auld 1973), i t i s found t h a t Hence, 8 of the e l a s t i c c o n s t a n t s v a n i s h . T h i s l e a v e s 13 e l a s t i c constants. However, th e r e i s freedom to choose the x-y c o - o r d i n a t e s i n the r e f l e c t i o n plane. T h i s reduces the degrees of freedom by one, so o n l y 12 c o n s t a n t s d e s c r i b e a monoclinic c r y s t a l medium. A p p l i c a t i o n o f t h i s method f o r the case of media with hexagonal symmetry shows t h a t (Landau and L i f s c h i t z 1959) t h e r e are o n l y 5 independent e l a s t i c constants. (2. 1-16) 23 I n t u i t i v e Arguments The use of group theory i n determining the number of independent e l a s t i c c o n s t a n t s f o r a p a r t i c u l a r c r y s t a l i s j p r e c i s e , but other means are a v a i l a b l e to study the e l a s t i c t e n s o r f o r media with d i f f e r e n t symmetries. As Feynman (1966) s t a t e s : " There i s a branch o f mathematics c a l l e d * group t h e o r y 1 t h a t d e a l s with such s u b j e c t s , but u s u a l l y you can f i g u r e out what you want with common sense." The s i m p l e s t e l a s t i c m a t e r i a l to c o n s i d e r i s one that i s homogeneous and i s o t r o p i c . Hhen t h i s m a t e r i a l i s s t r e t c h e d by a f o r c e F, the amount of e x t e n s i o n Al i s r e l a t e d to the f o r c e by Young's Modulus f r ' Y A J t ' < 2 ' 1 - 1 7 > As w e l l the bar c o n t r a c t s p e r p e n d i c u l a r l y t o the d i r e c t i o n of s t r e t c h . The change i n width per u n i t width i s r e l a t e d by Poisson's r a t i o to the change i n l e n g t h per u n i t l e n g t h . 24 W (2. 1-18) The above s t r e t c h i n g experiment can be done i n any o r i e n t a t i o n , s i n c e the medium i s i s o t r o p i c . Hence o n l y two e l a s t i c c o n s t a n t s a r e needed to d e s c r i b e a homogeneous, i s o t r o p i c medium. The next type of e l a s t i c m a t e r i a l t o be c o n s i d e r e d i s one resembling plywood i n g e o m e t r i c a l s t r u c t u r e . I t i s composed of i n f i n i t e s i m a l l y t h i n laminae. Each lamina can be viewed as i s o t r o p i c and hence i s symmetric about the z a x i s . To determine the minimum number of e l a s t i c c o n s t a n t s f o r t h i s medium, two experiments s u f f i c e . Apply an e x t e n s i o n normal to the laminae 25 There are l a t e r a l c o n t r a c t i o n s equal and p a r a l l e l t o the planes of the laminae. For t h i s case a lounges flodulus and Poisson r a t i o can be measured. The amount of e x t e n s i o n i n t h i s case i s determined by the weakest lamina, and the l a t e r a l c o n t r a c t i o n f~w i s the same along the x and y axes. Now repeat the experiment with F p a r a l l e l to the laminae. The p h y s i c a l s i t u a t i o n i s d i f f e r e n t , s i n c e i t i s the s t r o n g e s t lamina which determines the amount of e x t e n s i o n . There are two Poisson r a t i o s , because l a t e r a l c o n t r a c t i o n p e r p e n d i c u l a r to the a p p l i e d f o r c e F i s both p a r a l l e l t o and p e r p e n d i c u l a r to the symmetry a x i s of the laminae. These two experiments show t h a t t h i s m a t e r i a l (known as t r a n s v e r s e l y i s o t r o p i c ) has f i v e e l a s t i c c o nstants. Backus 26 (1962) showed how one could c a l c u l a t e the e l a s t i c c o n s t a n t s of a t r a n s v e r s e l y i s o t r o p i c medium, which i s d y n a m i c a l l y e q u i v a l e n t t o a f i n e l y l a y e r e d i s o t r o p i c medium by the a p p l i c a t i o n of an analagous argument. A l s o , i t i s important to note t h a t the t r a n s v e r s e l y i s o t r o p i c medium i s e q u i v a l e n t to a medium with hexaqonal symmetry, s i n c e i t s e l a s t i c p r o p e r t i e s can be d e s c r i b e d i n terms of f i v e e l a s t i c c o n s t a n t s . A more com p l i c a t e d type of e l a s t i c medium can be now examined. I t can be viewed as f i n e l y - l a y e r e d , but i n three orthoqonal d i r e c t i o n s . In t h i s case an e x t e n s i o n a l o n g each of the x, y, and z axes would allow measurement of a Young*s Modulus and two Poisson r a t i o s . T h i s g i v e s a t o t a l of nine e l a s t i c c o n s t a n t s . Such an e l a s t i c medium i s c a l l e d o r t h o t r o p i c . O r t h o t r o p i c m a t e r i a l s are e s p e c i a l l y important as they are c o n s i d e r e d to be among the main c o n s t i t u e n t s o f the upper mantle. (Green S Liebermann (1976) ) 27 S e c t i o n 2 - S o l u t i o n s of the Eguations of Motion Rs seen i n S e c t i o n 1, the eguations of motion with no body f o r c e s , are w r i t t e n ( i n C a r t e s i a n c o - o r d i n a t e s ) as (2. 2-1) In s o l v i n g such a system of d i f f e r e n t i a l equations, the s a l i e n t f e a t u r e s can be obtained by us i n g plane waves as t r i a l s o l u t i o n s . L e t u.- A - L e f u , C p * V * - i J C2.2-2) where A ^ i s the amplitude, i s the slowness (the summation convention i s i m p l i e d on a p p r o p r i a t e repeated i n d i c e s ) . S u b s t i t u t i o n of (2.2-2) i n t o (2.2-1) g i v e s 28 Equation (2.2-3) has a s o l u t i o n when the determinant of the c o e f f i c i e n t s of A ^  vanishes. Thus the s o l u t i o n t o (2.2-1) reduces t o the s o l u t i o n of an eigenvalue problem. That i s . da(fV P j P AJ=,0 ( 2 - 2 - , ) Equation (2.2-4) i s a s e x t i c i n the components of p. I t a l s o can be viewed as a c u b i c i n p t ^ j • T h i s c u b i c has t h r e e r o o t s , each r o o t d e s c r i b i n g a g u a d r a t i c form i n the components of p. Thus each r o o t of (2.2-4) r e p r e s e n t s a qu a d r i c s u r f a c e known as the slowness s u r f a c e . There i s a corresponding e i g e n v e c t o r f o r each ro o t . The ei g e n v e c t o r represents the displacement a s s o c i a t e d with the p a r t i c u l a r r o o t . The e x i s t e n c e of three r o o t s corresponds t o three d i f f e r e n t p o l a r i z a t i o n s o f displacement. These p o l a r i z a t i o n s are s t i l l orthogonal but are not pure mode v i b r a t i o n s . (Musgrave (1970)) That i s , they do not a l i g n e i t h e r p a r a l l e l to or p e r p e n d i c u l a r t o the wave propagation, 29 as do P or S waves propagating i n an i s o t r o p i c medium. For each p o l a r i z a t i o n o f the motion, the corresponding g u a d r i c s u r f a c e p r o v i d e s a c o n s t r a i n t on the components of p i . e . on the plane waves which are allowed to propagate. I t i s known t h a t the most g e n e r a l s o l u t i o n of equation (2.2-1) i s an envelope of a l l plane waves, which.are allowed by the r o o t s of (2.2-4) (Kraut 1963; Duff 1975; Courant & H i l b e r t 1966). T h i s envelope i s known as the wave s u r f a c e (Kraut 1963; Musqrave 1970) . 30 C o n s t r u c t i o n o f Have Sur f a c e - A n a l y t i c Treatment The a n a l y t i c c o n s t r u c t i o n of the wave s u r f a c e can be obt a i n e d by d i f f e r e n t i a t i n g the phase f u n c t i o n (chosen f o r convenience at t=1) with r e s p e c t t o the parameters (the components of t h e slowness) s u b j e c t t o t h e c o n s t r a i n t t h a t the slowness v e c t o r s t r a c e out a three sheeted slowness s u r f a c e (Fowler 1929; L i g h t h i l l 1960; Duff 1975; Musgrave 1970). L e t the eguation o f t h e j t h sheet of the slowness s u r f a c e be G (p) =0.0. The phase f u n c t i o n i s p^x^-1 = 0 . T o f i n d the envelope o f a l l plane waves the method of Lagrange m u l t i p l e r s i s used, and then i s e l i m i n a t e d from the two eguations: (2.2-5) and The c o - o r d i n a t e s of the envelope are (2.2-6) 31 PI Sp K (2.2-7) Equation (2.2-7) shows that the c o - o r d i n a t e s X« of the wave s u r f a c e H are r e l a t e d to the normals of the slowness s u r f a c e S. (Note t h a t j r e f e r s qo the j»th sheet o f the slowness s u r f a c e . ) The above c a l c u l a t i o n i s i d e n t i c a l t o the t r a d i t i o n a l s t a t i o n a r y phase approach. That i s , i t i s r e q u i r e d t o f i n d the s t a t i o n a r y p o i n t s of the phase where are the components of the wave number. The angular frequency,U / % i s r e l a t e d t o the wave number by the d i s p e r s i o n r e l a t i o n o b t a i n e d by s o l v i n q f o r the j t h eigenvalue, using an eigen v a l u e eguation i d e n t i c a l t o (2.2-4), T h e r e f o r e , the s t a t i o n a r i t y c o n d i t i o n i s 32 - O or ii^H a V ^ * I ^ J L / (2.2-9) But "bur^ are the components of the group velocity. Eguation (2.2-9), which i s simply the equation of the rays for a homogeneous anisotropic medium, demonstrates that the co-ordinates of the wave surface at one second can be obtained by c a l c u l a t i n g the group v e l o c i t y for a l l angles defined by the allowable wave vectors. In general the vector ^ur/SU£, i s not p a r a l l e l to . This implies that i n a homogeneous anisotropic medium, the slowness, and hence the phase ve l o c i t y , are not i n the same dir e c t i o n as the group velocity. A l l the physical p e c u l i a r i t i e s of waves t r a v e l l i n g i n such media are derivable from that simple r e s u l t . 33 C o n s t r u c t i o n of Save S u r f a c e - Geometrical D e r i v a t i o n The shape of the wave s u r f a c e may be obtained by a geo m e t r i c a l argument. In the l a s t s e c t i o n , i t has been shown t h a t the components of the group v e l o c i t y generate the wave s u r f a c e a t t=1 second. S i n c e the d i r e c t i o n of the group v e l o c i t y i s gi v e n by the normal t o the slowness s u r f a c e , the slowness corresponding t o the group v e l o c i t y (the group slowness) may be e a s i l y determined. I t s r e c i p r o c a l i s the group v e l o c i t y . Por the case o f a slowness curve (slowness s u r f a c e f o r a two dimensional problem i n the X-Z p l a n e ) , the geometric c o n s t r u c t i o n proceeds as f o l l o w s ( F i g 2.1): 1) For the slowness v e c t o r OB at an a n g l e d , f i n d the corresponding u n i t normal v e c t o r $ ; the components of IT are the d i r e c t i o n a l c o s i n e s f o r the l i n e ES; 2) Find the group slowness, corresponding to the d i r e c t i o n of n. I t s magnitude i s the pe r p e n d i c u l a r d i s t a n c e from the o r i g i n to the l i n e BS and i t s d i r e c t i o n i s given by the angle (p , which i s a r c t a n ( n y / n ^ ) ; 3) Take the r e c i p r o c a l of the magnitude of the group slowness vector OC. The v e c t o r OD a t the same angle as OC i s the group v e l o c i t y corresponding t o the slowness v e c t o r OB. I f steps one, two, and t h r e e are repeated f o r a l l angles & , the two dimensional wave s u r f a c e can be t r a c e d out. The complete slowness s u r f a c e and wave s u r f a c e f o r a t r a n s v e r s e l y i s o t r o p i c m a t e r i a l are i l l u s t r a t e d i n Px Figure 2.1 Geometric c o n s t r u c t i o n o f wave s u r f a c e from slowness s u r f a c e F i g u r e 2.2 • The geometric c o n s t r u c t i o n d e s c r i b e d above has i t s o r i g i n i n p r o j e c t i v e geometry ( K l e i n 1939; Poncelet 1822; Napoleoni 1977, pe r s o n a l communication; Courant S H i l b e r t 1966; B r i o t S Bouguet 1896) . In the terms of p r o j e c t i v e 35 Figure 2.2 Slowness and wave s u r f a c e f o r a t r a n s v e r s e l y i s o t r o p i c medium. P o l a r r e c i p r o c a l c o n s t r u c t i o n was used t o c o n s t r u c t the wave s u r f a c e . geometry, the slowness s u r f a c e and wave s u r f a c e are known as po l a r r e c i p r o c a l s with r e s p e c t t o the u n i t sphere (a c i r c l e i n two dimensions). To see t h a t , the f o l l o w i n g d e f i n i t i o n s (for the two dimensional case) are i n order; 1) " I f P,Q are p o i n t s on the diameter AB of a c i r c l e such t h a t AB:PQ i s harmonic and i f QB i s drawn p a r a l l e l t o the tangent of A then QB i s c a l l e d the p o l a r o f P with r e s p e c t to the c i r c l e and P i s c a l l e d the pole o f QB." (Fig 2.3 ) . I f 0 i s the ce n t e r of the c i r c l e , i t f o l l o w s at once t h a t OP*OQ = r a d i u s sguared." ( D u r e l l 1947). 2) A s u r f a c e W , c o n s i s t i n g of a l o c u s of poles whose p o l a r s with r e s p e c t to the u n i t c i r c l e are the tangents t o a s u r f a c e S, i s s a i d t o be the p o l a r r e c i p r o c a l o f S with r e s p e c t t o the u n i t c i r c l e . F i g u r e 2.3 Diagram i l l u s t r a t i n g d e f i n i t i o n of pole and p o l a r . To see t h a t d e f i n i t i o n (2) d e s c r i b e s the r e l a t i o n of the wave s u r f a c e t o the slowness s u r f a c e , i t i s s u f f i c i e n t to show t h a t the c o n s t r u c t i o n i m p l i e d by (2) i s i d e n t i c a l to the one d e s c r i b e d p r e v i o u s l y . F i r s t imbed a s u r f a c e S i n s i d e the u n i t c i r c l e as f o l l o w s . Then Draw a tangent at P. T h i s tangent c u t s the c i r c l e a t R and S Construct the pole D of BS . Wow by symmetry, the l i n e OD b i s e c t s the angle RDS, and i n t e r s e c t s l i n e RS a t Q i n a r i g h t angle. Hence, by d e f i n i t i o n (1) {Napoleoni 1977,personal communication) OQ.OD = 1 or OD i s (1/OQ). The above c o n s t r u c t i o n , t h e r e f o r e , i s i d e n t i c a l to t h a t 38 d e s c r i b e d u s i n g arguments of group v e l o c i t y . In three Figure 2.4 C o n s t r u c t i o n of the p o l a r r e c i p r o c a l s u r f a c e dimensions, analogous d e f i n i t i o n s h o l d . To o b t a i n these a n a l o g i e s simply s u b s t i t u t e the words p o l a r planes, tangent planes, and spheres f o r p o l a r l i n e s , tangents, and c i r c l e s . 39 P r o p e r t i e s of the Wave Surface Although the d e r i v a t i o n of the wave s u r f a c e from the slowness s u r f a c e i s based on a simple g e o m e t r i c a l c o n s t r u c t i o n , the s u r f a c e s t h a t r e s u l t sometimes have p e c u l i a r f e a t u r e s . From a mathematical viewpoint, these p e c u l i a r i t i e s are the s i n g u l a r p o i n t s o f the wave s u r f a c e . There are two types of s i n g u l a r i t i e s : cusps and double p o i n t s . I t i s e a s i e r t o see how the two types of s i n g u l a r i t i e s a r i s e i f a plane s e c t i o n of the slowness s u r f a c e i s considered. The f i r s t type o f s i n g u l a r i t y to be i n v e s t i g a t e d w i l l be a double point. <Af, B', fig.2.7) In Figure 2.5, p o i n t s A and B of the slowness s u r f a c e (one guadrant shown) share the same tangent l i n e RS with normal n. Hence, the corresponding group v e l o c i t i e s must be i n the same d i r e c t i o n . By the c o n s t r u c t i o n shown i n f i g u r e 2.4 the magnitude of the cor r e s p o n d i n g group v e l o c i t i e s i s the r e c i p r o c a l of the l e n g t h OP. T h i s proves t h a t the p o i n t s A and B map onto the same point of the wave s u r f a c e . In the thr e e dimensional case, a cone of slowness v e c t o r s would share the same tangent plane and map onto one point of the wave s u r f a c e . T h i s phenomenon i s known as i n t e r n a l c o n i c a l r e f r a c t i o n (Landau and L i f s h i t z 1960). From F i g 2.5, i t i s c l e a r t h a t at a neighboring p o i n t o f A, D, and at a 40 Figure 2.5 P o i n t s ft and B on slowness s u r f a c e s h a r i n g a common tangent, r e s u l t i n g i n a double poi n t on the wave s u r f a c e . neighboring point o f B, F, the d i r e c t i o n s of the group v e l o c i t y are the same, but the magnitudes are d i f f e r e n t . T h i s i s i l l u s t r a t e d i n F i g . 2.6, where the two tangent l i n e s XZ and 07 are p a r a l l e l but the p e r p e n d i c u l a r d i s t a n c e Figure 2.6 D i f f e r i n g group v e l o c i t y magnitudes i n the neighborhood of p o i n t s A and B which share a common tangent. with i t than p o i n t D. The wave s u r f a c e f o r F i g . 2 .5 i s shown i n F i g . 2.7 with the group v e l o c i t i e s of the corresponding slownesses l a b e l l e d w i t h dashes. The point A* Y Figure 2.7 I l l u s t r a t i o n of wave s u r f a c e with s i n g u l a r i t i e s corresponding t o slowness s u r f a c e o f f i g u r e 2.5 or B* (which i s i d e n t i c a l ) i s the double point and here th wave s u r f a c e i n t e r s e c t s i t s e l f . In F i g . 2.7 the other s i n g u l a r i t y mentioned p r e v i o u s l y i s a l s o present, the cusp The cusp a r i s e s because of an i n f l e c t i o n p o i n t on the slowness curve (or s u r f a c e ) . At the i n f l e c t i o n p o i n t , 43 (tf, * ti i° f i g . 2.5) the tangent becomes s t a t i o n a r y . The pole of the corresponding tangent must a l s o become s t a t i o n a r y . T h i s p o l e , which r e p r e s e n t s the group v e l o c i t y at the i n f l e c t i o n p o i n t of the slowness curve, i s the t i p of cusp df,*,X'» i n F i g . 2.7). I f the curve i n F i g . 2.7 i s given p a r a m e t r i c a l l y by X ( t ) , and Y ( t ) , then the s t a t i o n a r i t y c o n d i t i o n d e s c r i b e d above i s simply J (2.2-10) From c l a s s i c a l d i f f e r e n t i a l geometry, (Fowler 1929; E i s e n h a r t 1968) eguation (2.2-10), i s the known c o n d i t i o n f o r a s i n g u l a r p o i n t o f a curve. The a d d i t i o n a l c o n s t r a i n t t h a t i s s u f f i c i e n t to prove that the s i n g u l a r point i s indeed a cusp. Let the slowness s u r f a c e be given p a r a m e t r i c a l l y as 44 r = f ( t ) (2.2-12) Then x ( t ) - f (t) cos (t) y (t)=f (t) s i n ( t ) (2.2-13) where x (t) , y ( t ) are the c o - o r d i n a t e s o f a p o i n t on the slowness curve f o r any given t . a t an i n f l e c t i o n p o i n t of the slowness s u r f a c e , 4 V = O (2. 2-14) or p a r a m e t r i c a l l y , X N / - j / x a O (2. 2-15) Using equation (2.2-13) f o r the d e f i n i t i o n of x ( t ) and y ( t ) , i t i s easy to show t h a t eguation (2.2-15) becomes •f Z-ft » 0 (2.2-16) Equation (2.2-16) i s the c o n d i t i o n f o r an i n f l e c t i o n p o i n t , when a curve i s given i n p o l a r c o - o r d i n a t e s (Fowler 1929). For the given slowness curve, the corresponding wave s u r f a c e ( i n two dimensions) i s 45 t (2.2-17) where X (t) and Y (t) are the c o - o r d i n a t e s of a p o i n t on the wave s u r f a c e and pitt) * T A N > " ' ^ -1 j ( F i g . 2.8) Eguation (2.2-17) can be s i m p l i f i e d t o X l t ) = J_U Cost + i s i n t ) ( Z (2.2-18) Now compute X(t) and Y ( t ) : 46 <jLi)' -J± C-fcoxt + *Smt] t j J ' J c ^ W 2 f S f « ^ +-fcos-tl (2.2-20) 47 S i m p l i f y i n g : \(r>* -5 .nA If-ti + 2 ^ J (2.2-21) Cost [ f r ' - l f 4 2 ^ 1 However, at t h e i n f l e c t i o n p o i n t , a t x ( t ) , y ( t ) , (by equation (2.2-16)). T h i s proves that a t the p o i n t of the wave s u r f a c e c o r r e s p o n d i n g t o x ( t ) , y ( t ) , X(p)s^(t)-0. I t remains to show that the second d e r i v a t i v e i s non-zero at t = t . To do t h i s i t i s necessary to compute /({.) •  Now +y{{)-0 only i f , at t = t . d.U2--f( + 2^) = 0 (2.2-22) or To prove t h a t eguation (2.2-22) cannot h o l d , i t i s necessary to r e v e r t to the slowness s u r f a c e . For the slowness s u r f a c e do* j£0. P a r a m e t r i c a l l y , ai? 48 (2. 2-23) At t = t equation (2.2-23) becomes (2.2-24) A as xy - yx vanishes i d e n t i c a l l y at t=t. I f the parametric equations of the slowness s u r f a c e (2.2-13) are used then Thus (2.2-22) cannot be s a t i s f i e d , and indeed the s i n g u l a r p o i n t i s a c u s p i d a l one. Another way of proving the c o n d i t i o n f o r the e x i s t e n c e of the cusp i s based on a c o n s t r u c t i o n given here i n two dimensions i n v o l v i n g the v e l o c i t y curve (or s u r f a c e ) . Suppose that the phase v e l o c i t y i s s p e c i f i e d as a f u n c t i o n of angle (2.2-25) 49 C-cjCe) (2.2-26) That i s , f o r each angle & a plane wave with u n i t normal k i s propagated and i t s phase v e l o c i t y c i s measured. To f i n d the group v e l o c i t y , r e w r i t e (2.2-26) as UJrKcjt©) (2.2-27) Follo w i n g Backus (1970), the group v e l o c i t y i s obtained from (2.2-27) by d i f f e r e n t i a t i o n , u s i n g the g r a d i e n t o p e r a t o r i n po l a r c o - o r d i n a t e s . The components o f the group v e l o c i t y i n the 0 and k d i r e c t i o n s are given as S^g'16' (2.2-28, From (2,2-28) a p h y s i c a l p i c t u r e of the group v e l o c i t y can be obtained. The group v e l o c i t y i s a v e c t o r whose k component i s the phase v e l o c i t y , and whose 0 component i s Thus the angle the group v e l o c i t y makes with the phase v e l o c i t y i s T 7 W ~ l C^'te^/g^) t T h i s proves the important r e s u l t noted by many authors. (Landau 5 L i f s c h i t z 1960; witham 1974). I t i s c l e a r from f i g u r e 2.8 that the group v e l o c i t y leads or l a g s the phase v e l o c i t y by the angle ^ . 50 Suppose that i t leads the phase v e l o c i t y of t h e plane wave propagating at an angle 0 .* Then with r e s p e c t to 0=0, the angle the group v e l o c i t y makes i s As long as dp/de i s >0,the group v e l o c i t y w i l l l ead the phase v e l o c i t y . But when ckr/<£e<0 , i t w i l l l a g . T h e r e f o r e , the c r i t i c a l point i s at ^ / d e = 0 (2.2-30) From (2.2-29), i t f o l l o w s t h a t ^ Ce)+9l©>'0 (2.2-31) at the c r i t i c a l p o i n t . The above c o n d i t i o n f o r a c u s p i d a l point i n the wave s u r f a c e has been obtained by wxtham{1974) and Potsma(1955) u s i n g a s l i g h t l y d i f f e r e n t argument. Husgrave (1957) has a l s o given proofs o f the e x i s t e n c e of c u s p i d a l p o i n t s , but h i s arguments are somewhat d i f f e r e n t from those used here. He has a l s o d e r i v e d the c o n d i t i o n s (2.2-29) *Note: t i n f i g u r e 2.8 i s e g u i v a l e n t t o 0 , the v a r i a b l e which r e l a t e s phase v e l o c i t y to the angle o f propagation 51 t h a t the e l a s t i c c o n s t a n t s must obey i n order that c u s p i d a l edges e x i s t i n the three dimensional problem. A d e t a i l e d study of the wave s u r f a c e , which i s the Green's Function f o r a p o i n t source i n homogeneous, a n i s o t r o p i c media i s important, s i n c e the q u a l i t a t i v e f e a t u r e s of waves propagating i n such media can r e a d i l y be d e r i v e d . The s i n g u l a r p o i n t s correspond to the f o c u s s i n g of energy. Thus i n computing seismograms i n a n i s o t r o p i c media, i t i s expected that these f o c u s s i n g e f f e c t s should appear i f the cusps i n the wave s u r f a c e are l a r g e enough. 52 CHAPTER I I I I n t r o d u c t i o n Following the d i s c u s s i o n of the b a s i c p h y s i c s of waves propagating i n a n i s o t r o p i c media, i t i s a s t r a i g h t f o r w a r d matter t o i n v e s t i g a t e the k i n e m a t i c s of these waves. In p a r t i c u l a r , a t t e n t i o n w i l l be focussed on t r a n s v e r s e l y i s o t r o p i c media . As i n d i c a t e d i n Chapter I and I I , the main emphasis o f t h i s t h e s i s w i l l be the a n a l y t i c d e r i v a t i o n and numerical c a l c u l a t i o n o f s y n t h e t i c seismograms i n t r a n s v e r s e l y i s o t r o p i c media. The study of the kinematics o f plane waves i s u s u a l l y approached v i a asymptotic ray theory (Cerveny 1972), or by the c o n s i d e r a t i o n o f the jumps i n dynamical g u a n t i t i e s across the wavefronts {Vlaar 1968). As w i l l be shown i n t h i s chapter the two methods are completely e g u i v a l e n t , although d e t a i l s d i f f e r . Once the ray theory has been developed, p - d e l t a curves (Wiggins 1973; B u l l e n 1965; McHechan 1976) w i l l be c o n s t r u c t e d . These p - d e l t a curves w i l l then be u t i l i z e d d i r e c t l y i n the next chapter f o r the c o n s t r u c t i o n of s y n t h e t i c seismograms . A v i s u a l p r e s e n t a t i o n of the r a y s i s e a s i l y o b t a i n a b l e , once part of the numerical schemes f o r i n t e g r a t i o n of the ray equations have been d e r i v e d . A new hyb r i d scheme o f i n t e g r a t i o n w i l l be developed, based on work of Gauss, Kantorovich (1934), Krylov (1962), ana Chapman (1971). 54 Ray Theory Part I The methods o f ray theory have been used i n o p t i c s s i n c e Newton's time. Not u n t i l Hamilton's d e r i v a t i o n of the p r i n c i p l e of l e a s t a c t i o n was a connec t i o n made between o p t i c s and mechanics. I t i s p r e c i s e l y t h i s connection which i s u t i l i z e d so e f f e c t i v e l y i n seismology. F o l l o w i n g the d e r i v a t i o n presented i n Methods-of• Mathematical P h y s i c s V o l I I {Courant S H i l b e r t 1966) many Russian s e i s m o l o g i s t s (Alekseyev (1961); Y e l i s e y e v n i n (1964) ; Babich 1961) have s u c c e s s f u l l y a p p l i e d the ray method i n d e t a i l and Cerveny (1972) has a p p l i e d the method t o ge n e r a l a n i s o t r o p i c media. The s p e c i f i c case of t r a n s v e r s e i s o t r o p y has been considered by Daley and Hron (1977). A summary of t h e i r b a s i c r e s u l t s w i l l be o u t l i n e d below. The eguations of motion to be con s i d e r e d were d e r i v e d i n Chapter I I : (3.1-1) The b a s i c assumption of the ray method i s that 55 5*; 1 (no summation here) where A i s the wavelength under c o n s i d e r a t i o n . With t h i s condition, i t i s p o s s i b l e to express the displacements i n terms of an asymptotic, expansion : v ) (3.1-2) where n wx = v e c t o r amplitude T ~ phase o f the wavefront CAJ = angular frequency. S u b s t i t u t i o n of (3.1-2) i n t o (3.1-1) and c o n s i d e r a t i o n of the terms i n v o l v i n g i-*J y i e l d s the f o l l o w i n g eigenvalue problem: 56 (3.1-3) The s u p e r s c r i p t o r e f e r s to the z e r o t h order term of the asymptotic s e r i e s (3.1-2). Eguation (3.1-3) i s i d e n t i c a l to the eigenvalue problem posed i n the p r e v i o u s chapter. I f the e l a s t i c tensor i s s c a l e d by the d e n s i t y , eguation(3.1-3) becomes; Eguation (3.1-4) has three e i g e n v a l u e s whose numerical values, when s e t equal to one, d e f i n e three o r t h o g o n a l e i g e n v e c t o r s , the p o l a r i z a t i o n s o f p a r t i c l e motion. In p a r t i c u l a r , i f the eigenproblem a s s o c i a t e d with the matrix (3. 1-4) where P 57 then, the c o n d i t i o n G=1, corresponds to the eguation (3. 1-4) above. I t i s i n s t r u c t i v e t o determine the e i g e n v a l u e s a s s o c i a t e d with (3.1-5), and s e t Gj =1. m order t h a t a s o l u t i o n of (3.1-5) e x i s t s (3.1-6) Since each Gj i s a f u n c t i o n o f the pt- , a p a r t i a l d i f f e r e n t i a l eguation f o r the wavefront has been obtained: (3.1-7) Eguation (3.1 - 7) can be s o l v e d by the methods of c h a r a c t e r i s t i c s (Courant S H i l b e r t 1966) . The s o l u t i o n i s d e f i n e d by the eguations 58 as er-as >PI (3. 1-8) where s i s a parameter along the ray. I f eguation ( 3 . 1 - 8 ) i s to be r e c a s t i n terms of T , the phase of the wavefront, an a d d i t i o n a l r e s u l t must be u t i l i z e d . T h i s r e s u l t i s obtained by a p p l i c a t i o n of E u l e r ' s theorem on homogeneous f u n c t i o n s to the e i g e n v a l u e s & m . That i s . a P.-ov-(3.1-9) or as can be seen from (3.1-2), the phase of t h e wavefront i s d efined by: 59 t - t C X j ) (3.1-10) Dpon d i f f e r e n t i a t i o n of (3.1-10) with r e s p e c t t o t , the r e s u l t obtained i s n d t (3.1-11) A combination of (3.1-8), (3.1-9) and (3.1-11) changes the form of the ray equations. Expressed i n terms of the phase of the wavefront T , eguations (3.1-8) become 2 3^: (3.1-12) The above t h e o r e t i c a l d e r i v a t i o n may be e l u c i d a t e d i f the example of t r a n s v e r s e i s o t r o p y i s considered. The e l a s t i c tensor (Cerveny & Psencik 1972) f o r a t r a n s v e r s e l y i s o t r o p i c medium i s 60 a 13 o iS {3. 1-13) The ray w i l l l i e e n t i r e l y i n the (>dy X}) plane i f the source l i e s in the plane and the inhomogeneity depends only on X3. With these r e s t r i c t i o n s , substitution of (3.1-13) into (3.1-16) y i e l d s the following determinantal eguation (Daley and Hron 1977); d e l 0 o (3. 1-14) The roots are given by solving the cubic 61 ( p : v B V * X ( P f F J A ^ O - ( R V R V ^ J o. 1-15) One obvious root i s = ?>\ir ?j*ss The other two roots are determined by the quadratics (3.1-16) The solutions to these quadratics are 62 2 2 <3. 1-17) I f &t and r e s p e c t i v e l y are s e t equal t o one, then each eguation i n (3. 1-17) i s an a l g e b r a i c r e l a t i o n between the h o r i z o n t a l and v e r t i c a l slownesses f o r compressional and shear waves r e s p e c t i v e l y . The ray equations d e s c r i b i n g the ray t r a j e c t o r y may now be d e r i v e d using (3. 1-7) and (3. 1-12): f o r example, the t r a j e c t o r i e s f o r v e r t i c a l l y p o l a r i z e d quasi-compressional waves a^e 2 JVZ-4L dp, 1_ U-L) (3. 1-18) _ I The two equations i n (3.1-18) may be combined i n t o a s i n g l e ray eguation 63 (3. 1-19) Eguation (3.1-19) i s i d e n t i c a l t o t h a t i n f e r r e d from Cerveny and Psencik (1972) . 64 Ray Theory Part IT another approach t o the ray t r a j e c t o r y c a l c u l a t i o n s , i n v o l v i n g c o n s i d e r a t i o n s o f the d i s c o n t i n u i t i e s of the p a r t i c l e v e l o c i t y a c r o s s a s u r f a c e t r a v e l l i n g through -space, has been developed by Vla a r (1968). His approach t o ray theory w i l l be d e r i v e d below f o r v e r t i c a l l y inhomogeneous, t r a n s v e r s e l y i s o t r o p i c media. C y l i n d r i c a l c o o r d i n a t e s w i l l be u t i l i z e d , so t h a t the c y l i n d r i c a l symmetry of the medium may be e x p l o i t e d . As w e l l , the f i n a l a l g e b r a i c r e s u l t s w i l l be proven i d e n t i c a l t o those obtained i n Part I. As V l a a r ' s n o t a t i o n w i l l be used throughout, the f o l l o w i n g d e f i n i t i o n s are i n order: a) the wavefront w i l l be denoted by /""*; b) A f i e l d g u a n t i t y f , before the wavefront has passed. H i l l be designated by f ~ . a f t e r the wavefront has passed, the f i e l d q u a n t i t y w i l l be s i g n i f i e d by £•; c) The jump i n the f i e l d g u a n t i t y f i s shown as {£*) = {£*)-(£-), with both f+ and f - being evaluated a t the a p p r o p r i a t e time and l o c a t i o n d) The p o s i t i o n of the wavefront at time t can be w r i t t e n as Immediately some r e s u l t s may be d e r i v e d from the above. Since J 65 then at in/ ~ Ut dt dtj <3.2-1) As s e l l , on the wavefront ^ ( ^ © , 0 ~t - O D i f f e r e n t i a t i o n of t h i s r e l a t i o n y i e l d s flV-Vl (3.2-2) Equation (3.2-2), analogous to (3.1-11) i n the l a s t s e c t i o n , has the f o l l o w i n g p h y s i c a l i n t e r p r e t a t i o n . The g r a d i e n t of V i s normal to the wavefront and i t s components are the components of the slowness vect o r . Hence, the v e l o c i t y of the wavefront, normal to i t s e l f , has a magnitude of . I f the u n i t normal t o the wavefront i s denoted by 66 |r\j , then the components o f the slowness are given by p. * n-.WW 8 J O L Equation (3.2-1) can be r e w r i t t e n as I VI so** =' (3. 2-3) where V n = magnitude o f the normal v e l o c i t y fv// = magnitude of ray v e l o c i t y oC = angle between ray v e l o c i t y v e c t o r and u n i t wavefront normal. From (3.2,1-3) and (3) a l g e b r a i c r e l a t i o n s between the jumps i n s t r e s s and i n p a r t i c l e v e l o c i t y may be d e r i v e d . Since U-i - 0, the jump i n the p a r t i c l e v e l o c i t y i s z ero, i t f o l l o w s t h a t ^ ^"/^C^ ~0 . On a p p l i c a t i o n of t h i s r e s u l t to equation (3.2-1) , the ensuing equation i s obtained: 67 (3.2-4) OLL; - - | (Va L f | |V r |coio< at where cos 0 0 i s the same as i n (3.2-3). S u b s t i t u t i o n o f V n f o r IVrj CoS oC and noting t h a t / (V*i)*l s 1^*] i f n i s def i n e d as the u n i t normal t o the wavefront, y i e l d s the f i n a l r e s u l t F u r t h e r r e d u c t i o n o f (3.2-5) i s e s s e n t i a l i n i n f e r r i n g the f i n a l a l g e b r a i c eguation r e l a t i n g jumps i n p a r t i c l e v e l o c i t y across the wavefront t o the corresponding jumps i n s t r e s s e s . T h i s r e d u c t i o n i s achieved by m u l t i p l y i n g (3.2-5) by the components of 1^  , n ^  and u s i n g the f a c t t h a t (3.2-5) (3.2-5) becomes Then, i n component form eguation 68 (3. 2-6) Equation (3.2-6) r e l a t e s the jump i n the j t h component of the g r a d i e n t of p a r t i c l e displacement t o the jump i n p a r t i c l e v e l o c i t y a c r o s s the wavefront. T h i s equation i s the d e s i r e d one r e q u i r e d f o r c a l c u l a t i o n of ray t r a j e c t o r i e s s i n c e i t i s known that the c h a r a c t e r i s t i c s are c a r r i e r s of jumps i n dynamical q u a n t i t i e s . However, another r e l a t i o n i s needed before the jumps i n the s t r e s s a c r o s s the wavefront can be r e l a t e d to the p a r t i c l e * s v e l o c i t y . D e t a i l s of the d e r i v a t i o n are given i n V l a a r ' s paper (1968). The r e q u i r e d equation which r e l a t e s the d i s c o n t i n u i t y i n s t r e s s a c r o s s the wavefront, to the jump i n momentum i s Combination of equations (3.2-6) and (3.2-7) w i l l r e s u l t i n an eigenvalue problem s i m i l a r t o those d i s c u s s e d p r e v i o u s l y . As b e f ore, a system of p a r t i a l d i f f e r e n t i a l equations a s s o c i a t e d with the e i g e n v a l u e problem can be e l u c i d a t e d . 69 The s o l u t i o n of these equations y i e l d s the ray t r a j e c t o r i e s . The ray t r a j e c t o r i e s , obtained i n d i r e c t l y from equations (3.2-6) and (3.2_7) are of course i d e n t i c a l t o those c a l c u l a t e d e a r l i e r . To demonstrate the eq u i v a l e n c e of the two approaches, the example of a v e r t i c a l l y inhomogeneous, t r a n s v e r s e l y i s o t r o p i c medium w i l l aqain be examined. In t h i s f o l l o w i n g example, the c y l i n d r i c a l symmetry of the medium i s e x p l o i t e d . Since the medium i s c y l i n d r i c a l l y symmetric, a l l 0 d e r i v a t i v e s are zero. The s t r a i n s are given as 2 U i * r 2 *a <9o " r (3. 2-8) Since the jumps of the t r a c t i o n s a c r o s s the wavefront are needed i n eguation (3.2-7), i t i s necessary t o o b t a i n the jumps i n the s t r e s s e s . Where t h e d i s c o n t i n u i t i e s i n d e r i v a t i v e s are r e q u i r e d , equation (3.2-6) i s used. The r e f o r e , the r e q u i r e d jumps i n s t r a i n are 70 (3. 2-9) 2 where pr^ P& are the components o f the slowness. ( u r ) * (^if j (k-e) a r e t h e 1 u n ,P s i n p a r t i c l e v e l o c i t y . The corresponding jumps i n the t r a c t i o n are obtained by using the c o n s t i t u t i v e r e l a t i o n s r e l a t i n g s t r a i n and s t r e s s (eguation (3.2-9)). These are: ^:*C11C-pr(ur)')4.C1,C-pi(ua)B) - C l i ( - p rCa rr ) ^ C 33 ( - p 2 ( a / } where Cc j = ^ c j f w i t h Aj d e f i n e d p r e v i o u s l y , (see 3.1 - 4 . ) S u b s t i t u t i o n o f (3.2-10) i n t o (3.2-7) and use of the f a c t s t h a t p; - and fd - O ( s i n c e A = O ) y i e l d s 71 — - o {3.2-1 1) Equation (3.2-11) has a s o l u t i o n i f the determinant of the c o e f f i c i e n t matrix vanishes. The r e s u l t a n t c u b i c i n 2- z-p f Pg. , i s i d e n t i c a l to t h a t d e r i v e d i n Part I , eguation (3.1-15). To prove t h i s e q uivalence i n t r o d u c e again the s c a l e d e l a s t i c c o n s t a n t s ^ 6 J = ^ j / f • Opon computing the determinant and s e t t i n g i t to z e r o , the f o l l o w i n g c u b i c i s obtained. ("Pr V ?l 0{(-prlA(I - p*A^ + l)(-pe1A33-p^^4 I ) - pjpfl*,,*^ ] (3. 2-12) Equation (3.2-12) becomes eguation (3.2-15) of Part I i f p r and p^. are i d e n t i f i e d with p, and . Since eguation 72 (3.2-12) i s a c u b i c r e l a t i o n between pr and p f c , three s o l u t i o n s of Pft as a f u n c t i o n of pr can be o b t a i n e d , each of which corresponds to a p a r t i c u l a r p o l a r i z a t i o n of p a r t i c l e motion. One obvious s o l u t i o n of (3.2-12) i s P , - / 4 _ ( I - P ; A B ) T h i s s o l u t i o n can be i d e n t i f i e d with h o r i z o n t a l l y p o l a r i z e d shear waves. The other two s o l u t i o n s a r e : 2 a. W h e r e b- ( A 6tO 1 p , 1 -A 3 3 (p rA , r 0 (3.2-13) In equation (3.2-13) the outer *•» and »-* s i g n s corresponds t o up or downgoing waves , while the inner s i g n can be i d e n t i f i e d with guasi-shear waves, and the i n n e r ,~* s i g n can be i d e n t i f i e d with guasi-compressional wavesj I f eguation 3.2-13 i s w r i t t e n as p^- p r, l )* F L(p 0p t) ' O , then the corresponding ray eguations are (Vlaar 1968): 73 <*S d p * (3. 2-14) as The eguation necessary to compute the ray t r a j e c t o r y i s where the s u b s c r i p t i r e f e r s to the two p o s s i b l e p o l a r i z a t i o n s of motion. Equation (3.2-15) i s i d e n t i c a l t o the ray equations o b t a i n e d e a r l i e r f o r a p l a n a r , v e r t i c a l l y inhcmogeneous, t r a n s v e r s e l y i s o t r o p i c medium. Since the medium i s c y l i n d r i c a l l y symmetric, simply r e p l a c e r by x. 5 r i — L All 2 p , 3 P r " 2JJ[ S p r (3.2-15) 74 C a l c u l a t i o n of t - d e l t a and p - d e l t a curves. Theory The r e s u l t s of the p r e v i o u s s e c t i o n allow the c a l c u l a t i o n of t - d e l t a curves and p - d e l t a curves which have found such broad a p p l i c a t i o n i n the c a l c u l a t i o n of s y n t h e t i c seismograms. I n t e g r a t i o n of the eguation H - ~ —L- - ^ i - (3.3-1) given i n the p revious s e c t i o n , w i l l determine the d i s t a n c e to the t u r n i n g p o i n t f o r a given ray parameter ( h o r i z o n t a l slowness) i n a p a r t i c u l a r v e r t i c a l l y inhomogeneous, t r a n s v e r s e l y i s o t r o p i c medium. The t r a v e l time can be obtained from the r e l a t i o n (3.3-2) where 75 cm d t = 2.p>- o^k/Zp - p z + p^cV_ d l - (3.3-3) Note t h a t (3.3-3) can be immediately i n t e g r a t e d t o g i v e the one way t r a v e l time as t=JV?~a!^ +-pr  (3.3-4) The eguations which w i l l be used to c a l c u l a t e both p - d e l t a and t - d e l t a curves are obtained by m u l t i p l y i n g the r i g h t hand s i d e of (3.3-1) and (3.3-3) by (3.3-2), s i n c e the m a t e r i a l i s l a t e r a l l y homogeneous. In order that the c a l c u l a t i o n s of p - d e l t a and t - d e l t a curves proceed a c c u r a t e l y and e f f i c i e n t l y , a h y b r i d computational scheme based on the work of Chapman (1971) and Kantorovich (1934) i s u t i l i z e d . Kantorovich's method i s used to s u b t r a c t out the s i n g u l a r i t y which can be i n t e g r a t e d a n a l y t i c a l l y . The remainder of the i n t e g r a l i s i n t e g r a t e d using a fancy Gaussian i n t e g r a t i o n scheme d e s c r i b e d i n AE£I22ia^te C a l c u l a t i o n of I n t e g r a l s . Chapman's usage of the fancy Gaussian method i s extended, i n that the technique 76 i s a p p l i e d a l s o to the i n t e g r a l s with sguare r o o t s i n g u l a r i t i e s . The h y b r i d technigue d e s c r i b e d above i s necessary due to the s i n g u l a r behaviour of the ray i n t e g r a l s near the t u r n i n g p o i n t . For example, i n the case of an i s o t r o p i c , v e r t i c a l l y inhomogeneous medium, the ray i n t e g r a l s a re where t{p) i s the two-way v e r t i c a l delay time. The i n t e g r a n d i n (3.3-«ib) has an i n t e g r a b l e s i n g u l a r i t y and o s i n c e can be seen by a change of v a r i a b l e f o r example. (3. 3-5) o I f W$) i s monotonic and smooth M/Jiu, w i l l be smooth, and the behavior o f the i n t e g r a l w i l l be dominated by the 77 s i n g u l a r i t y at u = o, (corresponding to the t u r n i n g p o i n t ) . I f the f u n c t i o n i s more complicated, an a r t i f i c e due to Kantorovich (1934) may be used. Suppose the i n t e g r a l i n q u e s t i o n i s of the form I = J ^ ( n x ) ) c | x (3.3-6) a. where some composite f u n c t i o n . A l s o suppose that at x = a, " 7 / f x j - O and as a r e s u l t Q$(o) d i v e r g e s , i . e . $60 c o u l d be ^rz . The procedure i s t o expand ^Cx) about x = a, to as many terms as d e s i r e d . For example, In (3.3-6), then, add and s u b t r a c t t o o b t a i n 13J (p(f'icL%x^)eixMJ(fl^x))-^(^.ft]Wx. (3.3-8) Since cj^ u s u a l l y i s a simple f u n c t i o n l i k e vTx" , the f i r s t term i n (3.3-8) can be i n t e g r a t e d a n a l y t i c a l l y . The second term i s r e g u l a r about the s i n g u l a r p o i n t , at l e a s t t o f i r s t order i n x., Hence, a numerical method, such as Gaussian i n t e g r a t i o n may be employed. 78 The above technique may nos be a p p l i e d t o the p a r t i c u l a r ray i n t e g r a l necessary f o r the c a l c u l a t i o n of p -d e l t a curves i n t r a n s v e r s e l y i s o t r o p i c , v e r t i c a l l y inhomogeneous media. R e c a l l t h a t where = the depth at the t u r n i n g point and f = b t y ^ - ^ f l C L r I cW>e + (3.3-9.1) p r = ray parameter A>3ij Aff, An ace the reduced e l a s t i c c o n s t a n t s ( f/j/p etc. ) • At the t u r n i n g p o i n t , "f 6 - O . I f the wave being considered i s guasi-compressional, then ^i-O a t ^ £ with the corresponding ray parameter chosen to be | V c \J~An Conversely, i f the wave i s a guasi-shear wave, the correspondinq ray parameter i s pr- '^{A^J * I n e q u a t i o n (3.3-9) , the in t e g r a n d may be r e w r i t t e n as 79 I (3. 3-10) The f u n c t i o n i n (3.3-10) i s now a composite f u n c t i o n of the form mentioned p r e v i o u s l y with Adherence t o the p r e s c r i p t i o n o u t l i n e d p r e v i o u s l y J U n e c e s s i t a t e s c a l c u l a t i o n of . A f t e r l e n g t h y a l g e b r a , i t may be shown t h a t a t the t u r n i n g p o i n t 61 7? (3.3-11) 80 Expansion of ^ about the turning point yields w o - . C t b U - 2 j = a U - O ( 3 . 3 - 1 2 ) W h e r e u. = C ? b Substitution of ( 3 . 3 - 1 2 ) into ( 3 . 3 - 9 ) , and use of ( 3 . 3 - 8 ) yields 81 The f i r s t term i n (3.3-13) may be integrated a n a l y t i c a l l y . &s mentioned previously, the second term, which i s not singular, can be integrated numerically using a Gaussian method. However, since the integrand i n the second term i n (3.3-13) s t i l l behaves l i k e ' " * > i s convenient to recast the integrand in the form This i s f i r s t accomplished by a simple change of variables \/zz i. -"2 » Then, l e t fifes1- ~ J 1 / • T n e integrand i n IT?) the second terra in (3.3-13) i s written as _J I (3.3-14) Eguation (3.3-U) i s i d e n t i c a l to 82 1 _ / / - iW - | j * h i * ) (3.3-15) where hW>* AV_ _ _ L (3.3-16) To integrate an i n t e g r a l of the form T = Lbisd -cAV (3.3-17) a Gaussian method i s used. The method i s s l i g h t l y more elaborate than usual, and the polynomials which are orthogonal on (0,1) with respect to the weight function rj^ are closely related to the Legendre polynomials, which are orthogonal on the i n t e r v a l £ -1,1 ] with respect to a unit weight function. The polynomials are (Krylov 1962): 83 where (%) are the Legendre polynomials of even order. Given the above formula, the i n t e g r a l (3.3-17) can be c a l c u l a t e d n u m e r i c a l l y as J /vT~ M I (3.3-18) where are the squares of p o s i t i v e r o o t s of the Legendre polynomials and Af/ are weights o b t a i n a b l e d i r e c t l y from the weights used i n o r d i n a r y Gaussian i n t e g r a t i o n . U s u a l l y yC^ , and are obtained d i r e c t l y from t a b l e s . The numerical a l g o r i t h m e l u c i d a t e d above, a h y b r i d of techniques of Kantorovich and Krylov, can a l s o be used i n the c a l c u l a t i o n of i n t e g r a l s of the form 1 = Js/^tfdx. (3.3-19) Such i n t e g r a l s a r i s e i n the c a l c u l a t i o n of ~t(p) , the v e r t i c a l delay time. P - d e l t a curves c a l c u l a t e d u sing the 84 above method are presented in Chapter V along with the calcu l a t i o n of synthetic seismograms. 85 CHAPTER IV S o l u t i o n s Of The Eguations Of Motion The r e s u l t s i n Chapter I I I were derived so t h a t the k i n e m a t i c s of the wave propagation c o u l d be e x p l a i n e d i n terms of r a y s , p - d e l t a and t - d e l t a curves. These r e s u l t s w i l l be u t i l i z e d i n Chapter V, which i s concerned with the seismogram c a l c u l a t i o n . I t i s f i r s t necessary t o s o l v e the r e l e v a n t eguations of motion and i n c l u d e the source terms. The s o l u t i o n s of the equations of motion, i n the elastodynamic case have been known f o r some time. (Love 1945). Most o f t e n the s o l u t i o n i s obtained i n terms of plane waves. (Musgrave 1970). In the case of v e r t i c a l l y inhomogeneous media, the h o r i z o n t a l c o - o r d i n a t e s are F o u r i e r transformed, and the r e s u l t a n t equations are a s e t of coupled o r d i n a r y d i f f e r e n t i a l eguations, to be s o l v e d as a f u n c t i o n of z. (Chapman 1977b, gives an e x c e l l e n t review of the method.) These coupled equations are solved using e i t h e r a numerical scheme, ( G i l b e r t & Backus 1966) or an asymptotic method, which i n c o r p o r a t e s the wave kinematics d e s c r i b e d i n Chapter I I I . In the s o l u t i o n of these coupled e g u a t i o n s , the h o r i z o n t a l slowness, p, and the angular frequency w, appear 86 as parameters. The s o l u t i o n i s then i n v e r s e transformed t o ob t a i n displacements and s t r e s s e s as a f u n c t i o n o f ( r , z , t ) . I t i s the e v a l u a t i o n of the i n v e r s e transform to which much a t t e n t i o n has been p a i d v i z . Chapman (1974a), Wiggins (1976), Helmberger (1968), Fuchs and Muller (1971), Helmberger and Wiggins (1974). , Very r e c e n t l y , i t has been discov e r e d t h a t an adequate approximation t o the f i r s t motion represented on the seismogram can be obtained using the equal phase method (Chapman 1976a) or e g u i v a l e n t l y d i s k ray theory (Wiggins 1976). Chapter V w i l l be concerned with the a p p l i c a t i o n of the above approximation to v e r t i c a l l y inhomogeneous, t r a n s v e r s e l y i s o t r o p i c , e l a s t i c media. 87 S e c t i o n 1. Development of the Eguations of Hotion When the earth may be regarded as v e r t i c a l l y inhomogeneous, t r a n s v e r s e l y i s o t r o p i c , and e l a s t i c i n i t s dynamic response to an a p p l i e d f o r c e , the eguations of motion and s t r e s s - s t r a i n r e l a t i o n s can be r e a d i l y d e r i v e d . In order to i n c o r p o r a t e the source d e s c r i p t i o n i n t o the f o r m u l a t i o n , i t i s expedient to use the formalism developed by Takeuchi and S a i t o (1972). In the case of a f l a t earth and the v e r t i c a l symmetry a x i s possessed by a t r a n s v e r s e l y i s o t r o p i c medium, v e c t o r c y l i n d r i c a l harmonics are employed. Then the displacement and s t r e s s e s can be expanded as f o l l o w s : (4.1-1) where 6 =• s t r e s s v e c t o r i n the z d i r e c t i o n u = displacement v e c t o r and £7(*-,©)- l^r)e^6 K i r Kr <4.1-2 with er } <f"@ £•£ the u n i t v e c t o r s i n the r , 0 , z d i r e c t i o n s . The c y l i n d r i c a l harmonics i > K ,5,. , T K are s i m p l i f i e d whe the r e i s a v e r t i c a l symmetry a x i s s i n c e ^ (m=o) , n They become: 89 (4. 1-3) The c y l i n d r i c a l harmonic expansion (4.1-1) can be a p p l i e d to the eguations of motion and the c o n s t i t u t i v e r e l a t i o n s as o u t l i n e d i n the paragraphs below. Con t i n u i n g with the formalism of Takeuchi and S a i t o (1972) , the s t r a i n - d i s p l a c e m e n t r e l a t i o n s are given as e U r where = i j t h component of s t r a i n Uj = j t h component of displacement. The above s t r a i n - d i s p l a c e m e n t r e l a t i o n s are s u b s t i t u t e d i n t o 90 the c o n s t i t u t i v e r e l a t i o n s as o u t l i n e d below: r dr r d £ wh'ere 6ij - i j t h component of s t r e s s ( i = z , G, z; j = r , Q , z ) , and A, C, F, N,L are the 5 e l a s t i c parameters d e s c r i b i n g a t r a n s v e r s e l y i s o t r o p i c medium. Then eguations (4.1-5) are s u b s t i t u t e d i n t o the eguations of motion below (1.1-6) to obt a i n eguations (4.1-7). The equations of motion are 91 ^ 4 K„ dr r 7 (4. 1-6) ^ r i t r where r = d e n s i t y fp # f@, fg. = components of the body f o r c e s and, 61^ - i j t h components of s t r e s s . S u b s t i t u t i o n of (4.1-5) i n t o (4.1-6) r e s u l t s i n the f o l l o w i n g set o f e g u a t i o n s : 92 A J L f l 1 (rur) J * F i _ flu*\ + ^ + / 3 ^ c / » u r (4. 1-7) Equation s e t (4.1-7) and the c o n s t i t u t i v e r e l a t i o n s <5rr= A _ L M r \ £ r } - 2 M u r f F S u , r d r r £2; = F I l K r ^ l j (4.1-8) \r.3r / * z completely d e f i n e the d i f f e r e n t i a l system to be s o l v e d . Before a p p l y i n g the c y l i n d r i c a l harmonic expansion i t i s u s e f u l t o r e w r i t e the two s e t s o f eguations (4.1-7), and (4. 1-8) such t h a t a l l z d e r i v a t i v e s appear on the l e f t hand 93 s i d e and the r i g h t hand s i d e has only ^ o p e r a t o r s and body f o r c e components. B r i t t e n i n t h i s manner, the complete system i s *Z ' L T 7 (4.1-9-1) (1.1-9-2) T C [ r 3 r J + C (4.1-9-3) H i - / O ^ - l M r O " / 3 ^ (4.1-9-4) 94 ^2 S r l r a r J (4.1-9-6) The s i x equations above separate i n t o two systems, a f o u r t h order system,and a second order system. These are u c r j r {it. 1-10) >2 irSr 1 and 95 hi c h? (4.1-11) Eguation s e t (4.1-10) corresponds to the d i f f e r e n t i a l system d e s c r i b i n g P-SV waves i n an i s o t r o p i c medium, while the second order system (4.1-11) i s analogous t o t h a t d e s c r i b i n g SH propagation. In the ensuing d i s c u s s i o n only the "P-SV system, eguation s e t (4.1-10), w i l l be c o n s i d e r e d . Now, the formalism developed e a r l i e r may be a p p l i e d to equation s e t (4.1-10). Only one t r i c k i s used and t h a t i s the f a c t t h a t the operator^-p [^,r^."\\ h a s the eigenvalue transformed s e t of equations i s then d e r i v e d by s u b s t i t u t i o n of (4.1-1) (with m=0) i n t o equation s e t (4.1-10) . Thus the f o l l o w i n g system i s o b t a i n e d , when i t i s a p p l i e d t o X 0 fur prececal 1 k=wp) . The 96 hi L T r " ^u-t~ J i u?p H r +• J_5"i^ a* c c | £ r t = - pvo* u.r- ( A-J?)(-u^pO u r-1. cop <5; _ ^ f r (4.1-12) where are the c o e f f i c i e n t s of K and i n {U, 1-1a) ^2*' b'ri are t h e c o e f f i c i e n t s of R K and SK i n (4.1-lb) and w = angular freguency p = h o r i z o n t a l slowness k = wp = wavenumber. A l t e r n a t i v e l y U r } U.^ , o r i j 6^^ can be viewed as the B e s s e l transforms of LLr , , o V i , ^ * T h i s i s not s t r i c t l y c o r r e c t , as can be seen by examination of SK which i s accompanied by the d i f f e r e n t i a l o p e r a t o r . The d e t a i l s a f of the source terms w i l l be d e s c r i b e d below, but f i r s t i t i s i n s t r u c t i v e to c a s t [H. 1-12) as the matrix d i f f e r e n t i a l eguation 97 d 2. ' u u. 1 2i [A] o o -7, - L i ) p O C L o O (4. 1-13) \ I C —f L J p/c Equation (4.1-13) can be a l t e r e d so t h a t w does not appear at a l l i n [&J. That equation i s ? 7L 0 0 0 7c 0 - 0 P p 0 99 Se c t i o n 2. D e t a i l s of the Source The power of the v e c t o r c y l i n d r i c a l harmonic expansion was perhaps not too ev i d e n t i n the l a s t s e c t i o n . One c o u l d ask "Why not use B e s s e l transforms d i r e c t l y ? " The answer i s given i n t h i s s e c t i o n . A vector c y l i n d r i c a l harmonic decomposition i s the best one f o r d e s c r i b i n g the most general types of sources. An expansion i d e n t i c a l to t h a t of (4.1-1) can be w r i t t e n f o r a p o i n t f o r c e of magnitude G a c t i n g at (fo) ®*, 0^ and p o i n t i n g i n the "m" d i r e c t i o n , i . e . , i f (4.2-1) where then a i r where 100 o o -ZlT and s i m i l a r l y L ©-- £fc In order to o b t a i n more g e n e r a l sources, only the s u p e r p o s i t i o n of p o i n t sources a c t i n g i n d i f f e r e n t d i r e c t i o n s need be c o n s i d e r e d . Suppose there e x i s t s a p o i n t f o r c e of magnitude G a c t i n g i n the d i r e c t i o n "m" separated by from a f o r c e a c t i n g i n the d i r e c t i o n then 101 xuA. I (4.2-3) where G = magnitude of the f o r c e Co = j = j t h expansion c o e f f i c i e n t of the f o r c e i . e . c o e f f i c i e n t n m r» -r-rn of /fx ,3K O R ' K i Q t h e expansion (a. 1-1). In (4.2-3), which r e p r e s e n t s a couple f o r c e , now m u l t i p l y and d i v i d e the r i g h t hand s i d e by/frjand l e t S V , - ? 0 . Then, l e t t i n g GJffojhe the moiaent of the c o u p l e f o r c e i t f o l l o w s t h a t — - M. , _ ~ r (4.2-4) where <"o, 0o i £ o are the source c o - o r d i n a t e s and 102 r~j = j t h component i n the c y l i n d r i c a l harmonic expansion of the f o r c e as given i n eguation (4.2-2) The term on the r i g h t hand s i d e o f eguation (4.2-4) may be w r i t t e n as a d i r e c t i o n a l d e r i v a t i v e , y i e l d i n g Pi - e G\6 15 (4.2-5) or i f u n i t moment i s c o n s i d e r e d . where 103 application of the above results to equations (4.1-14) of the l a s t section i s straightforward since a l l that i s needed i s substitution of Fr, Fj into (4.2-6) dropping _ L ur it" the e x p l i c i t time dependence ^ and then substituting into (4.1-14). For s i m p l i c i t y expressions 1 w i l l be evaluated for f~ ("„ =0 . since only the zeroth order term in the c y l i n d r i c a l harmonic expansion i s being considered J>- a n <* p£ simplify considerably. Thus from (4.2-2) (4. 2-7) e--ea and e-e0 Then substitution of (4.2-7) into (4.2-6) res u l t s i n 104 e * e ° (4.2-8) The f o l l o w i n g f a c t s are then used t o s i m p l i f y the e x p r e s s i o n s i n (4.2-8): 105 0) I U r ) U) J 0(Kr) r-r 0=o (3) V? r S ^ i_Km r SCl -V and j _ ^ f r -* r 0 2 y r 3 I o (4. 2-9) Z T h i s r e s u l t s i n 2 (1.2-10) where: 1) the e x p l i c i t harmonic time dependence has been removed; 2) i n d i c a t e s d i f f e r e n t i a t i o n with r e s p e c t to the argument 106 of the d e l t a f u n c t i o n ; 3) n r i V\& , 0 ^ d e f i n e d i r e c t i o n of the coupl e ; 4) nor , m& , r r ) e d e f i n e d i r e c t i o n of the f o r c e ; I n s e r t i o n of (4.2-10) i n t o the source term i n eguation (4.1-14) y i e l d s F = O (4.2-11) A d d i t i o n of t h r e e o r t h o g o n a l couples together produces an e x p l o s i o n source given by / o 0 (4.2-12) where k=wp. I t i s eguation (4.2-12) which w i l l be used as a source i n a l l seisaogram c a l c u l a t i o n s i n Chapter V. 107 S e c t i o n 3. S o l u t i o n o f the Homogeneous System Using a i r y F u n c tions Consider the homogeneous d i f f e r e n t i a l system obtained by dropping the f o r c e term i n (4.1-14) A fundamental matrix s a t i s f i e s (4.3-1) ( G i l b e r t S Backus 1966), i . e. (4. 3-2) In order to s o l v e f o r the fundamental matrix i t i s i n s t r u c t i v e and u s e f u l t o use the asymptotic formalism developed by Wasow (1965), Chapman (1974b), and Woodhouse (1977) F i r s t r e c a l l t h a t 108 rA'3- ? 0 \ c o 0 Vc 0 0 I ° P p ° / So that a l g e b r a i c c a l c u l a t i o n s nay be l a t e r s i m p l i f i e d l e t z = p (no r e l a t i o n t o depth v a r i a b l e z) (4. 3-4) w = 1/L Y = pF/C t = 1/C A. X = p - (A - F /C)p I t i s e a s i e s t to d e s c r i b e the complicated s e t of c a l c u l a t i o n s which f o l l o w i n a manner analogous t o a cooking r e c i p e . Step 1: F i n d a Block Diagonal Transformation f o r (4.3-2). I t i s d e s i r e d to f i n d a t r a n s f o r m a t i o n which puts the matrix i n (4. 3-2) i n block d i a g o n a l form. The 109 eigenvalues of each block w i l l be the square o f the eigenvalues i n the o r i q i n a l matrix (WJ (Woodhouse 1977, p e r s o n a l communication). The o r i g i n a l e i q envalues o f £q J are the up and downgoing v e r t i c a l wave numbers. Thus f o r some matrix lR~\ = ( r , , £ ^ , £ 4 ) l e t o 1 O \ <V o (4. 3-5) or Frcm (4.3 - 5 ) , i t i s seen t h a t [fl] [/)] [ff] = Required Block Diaqonal Matrix To f i n d (X) i t i s r e g u i s i t e t h a t the e i q e n v e c t o r s o f the 9L el-matrix £^"J be found. The matrix [A] i s 110 o O O (4.3-6) The eigenvalues of (4.3-6) are and ^ £ (represented from here as/-4<2 and Lb£) , the v e r t i c a l wave numbers squared for quasi-compressional and guasi-shear waves. The eigenvector matrix f f l j i s © -£<(tztw;y) O O ° £3!^* til -L&z) (4.3-7) and i t s inverse i s 111 O ^jC wx+y? - LBZ) - t ^ U * + ^ y ) O (4 .3-8) The n o r m a l i z a t i o n f a c t o r s have been chosen to s a t i s f y energy f l u x c o n d i t i o n s , which w i l l be d e s c r i b e d l a t e r . So, step 1 i s completed, s i n c e the r e g u i r e d t r a n s f o r m a t i o n has been found, A p p l i c a t i o n of the tr a n s f o r m a t i o n jj^]=[n'] [ / ^ r e s u l t s i n (4.3-2) becoming cl i or 112 JLl (4.3-10) The reason f o r the second term i s t h a t the parameters of density and s t i f f n e s s do depend on the depth. Th e r e f o r e ASjQfO. Note, however, t h a t i f high f r e g u e n c i e s are c o n s i d e r e d , t h e f i r s t term on the r i g h t hand s i d e of (4.3-10) i s dominant. Hence, (4.3-10) i s an asymptotic expansion and can be w r i t t e n as AtH'3 *sT K K H ' ) (4.3-11) where s = -iw 113 a n * [nil« IR-T'CAIUI « o I LA2 o o o \ © I LB2 O/ and Step 2: F i n d a Block Diagonal T r a n s f o r m a t i o n f o r a l l orders i n (4.3-11) In order to implement t h i s step, i t i s important t o r e a l i z e t h a t W" 4 C*J has the same form a s f j f l j . T h i s a t t r a n s f o r m a t i o n allows s e p a r a t i o n of the block d i a g o n a l components from the non block d i a g o n a l components of Follow i n g Chapman (1974b) a t r a n s f o r m a t i o n i s c o n s t r u c t e d such t h a t 114 [H' l 'WlH*] ( ' , • 3 - , 2 , with the c o n d i t i o n s t h a t (1) 4 L H ' i , . sY LH l] (2) Now t o f i r s t order i n s, s u b s t i t u t i o n o f (4.3-12) i n t o (4.3-11) g i v e s ate] IH1} f Lelifc!! = sY tn)] U&ILH']) « U c U S (4.3-13) S u b s t i t u t i o n o f c o n d i t i o n s (1) and (2) i n t o (4.3-13), t a k i n g only powers up to S y i e l d s 115 s (4.3-14) C o n s i d e r a t i o n of c o e f f i c i e n t s of s and 1 i n (4.3-14) r e g u i r e s t h a t : and (4.3-15) c U From a) i t i s evident that and from b) Now the s t r u c t u r e s of the above matrices are known, and eguations such as (4.3-16) can be sol v e d r e c u r s i v e l y (Wasow 1965). In p a r t i c u l a r \ L A X 0 V o o IBZ 0 J LM',1» [R3 ' ' d C E ] e l l o ° 3.4 ^ o o V 9* - 9 j where ^ i j i s the i j t h element of the matrix As Hasow (1965) and Chapman (1974b) p o i n t out, the matrices have an " a n t i " b l o c k - d i a g o n a l s t r u c t u r e so t h a t recurrence formulae may be used to c a l c u l a t e the elements, The s a l i e n t f e a t u r e s o f the above c a l c u l a t i o n are that 117 and that the t r a c e o f the bl o c k s i s z e r o , i . e . J i H - a n d 3*3+3«/w=0- T n e f i r s t terms i n the tran s f o r m a t i o n have now been found, and the system (4.3-11) may be w r i t t e n as OO (4. 3-18) where the main concern i s with the f i r s t two terms i n (4.3-18). Since the [MRj are a l l block d i a g o n a l , the fo u r t h order system i n (4.3-18) decouples i n t o two f i r s t order systems. S y m b o l i c a l l y t h i s d e c o u p l i n g i s w r i t t e n as tn1] - L K " ] © K 1 where 0 denotes the d i r e c t sum. In the system under c o n s i d e r a t i o n 118 K 1 * / o \ \ Ln1/] -to \\ \LA2 O J \l$2 o 1 <4. 3-19) Step 3: Convert each o f the decoupled 2 by 2 systems i n t o the A i r y Eguation and s o l v e u s i n g Hankel F u n c t i o n s The c a l c u l a t i o n s o u t l i n e d below w i l l be done f o r o n l y one of the two by two systems, the guasi-compressional one. C o n s i d e r a t i o n of the f i r s t two terms i n the asymptotic s e r i e s on the r i g h t hand s i d e o f (4.3-18) y i e l d s the d e s i r e d 2 by 2 system of d i f f e r e n t i a l eguations: (4. 3-20) where the s u p e r s c r i p t ^ i n d i c a t e s t h a t the -? * <3- system corresponding to guasi-compressional waves i s c o n s i d e r e d and 119 0 I L A I 0 ( a . 3 - 2 1 ) To convert (4.3-20) i n t o an a i r y eguation l e t (4. 3-22) <x*\d let V\ 2/3 where w = angular frequency n^ = \ J L a i = v e r t i c a l wave number z<* = t u r n i n g p o i n t lHt*fo*,p)=0 ) z = l e v e l at which s o l u t i o n i s c o n s i d e r e d S u b s t i t u t i o n of the above t r a n s f o r m a t i o n i n t o (4.3-20) r e s u l t s i n 120 SUJ£3 = ^ p j ^ C c r , C r t 3 C c ] j [ H a " ] or with 4 k and . In order to e v a l u a t e the r i g h t hand s i d e of (4.3-24), the expressions f o r [c] and ^ £ j must be determined, £ c ] i s chosen so that the f i r s t term on the r i g h t hand s i d e of (4.3-24) i s ^ £j f / - / 3 j , a n d j ^ j can be e v a l u a t e d frcm the expression f o r h (see 4.3-22). Then, 121 and where cX=t = v e r t i c a l wave number w = angular frequency (a.3-25a) (4.3-25b) I n s e r t i o n of the above two ex p r e s s i o n s i n t o (4.3-24) r e s u l t s i n 122 «n l VN o ; (4. 3-26) From (4.3-26) i t i s c l e a r t h a t the f i r s t matrix i s the c o e f f i c i e n t of u** i n an asymptotic expansion and the second matrix on the r i g h t hand s i d e has powers of w i n i t from the terms. T h e r e f o r e a f u r t h e r t r a n s f o r m a t i o n may be used (Basow 1965; Chapman 1974b) t o s o l v e (4.3-26). I t i s given as (4.3-27) r 4*1 and|_H J s a t i s f i e s the eguation (4.3-28) with the £DJ< J obtained by s u b s t i t u t i o n of (4. 3-27) i n t o (4.3-26). Again a set of r e c u r r e n c e r e l a t i o n s i s the outcome. These a r e : 123 - - L C I e».3-29> with r ^ T » - CD:-3 - y trfiL^ Wasow shows t h a t a system such as (4,3-29) has a s o l u t i o n i f and only i f (4.3-30) Such being the case, r e l a t i o n s (4.3-30) can be used t o f i n d the required£p^J« The p r i n c i p a l concern here i s to o b t a i n [jD^J* the f i r s t term i n the t r a n s f o r m a t i o n (4.3-27). As can be seen from (4.3-29), s i n c e [S^^  = ^ # the most q e n e r a l s o l u t i o n of t ^ o ^ J i s (4.3-31) To determine g, and g a the t r a c e r e l a t i o n s (4.3-30) are used f o r j j ^ J. F i r s t ^ i t i s necessary to c a l c u l a t e ^ f ^ which i s done from the equation 124 [ft* Co;]' ' CKKl£l (4.3-32) S u b s t i t u t i o n of[M ? * J ,[0*], and[o?J i n t o (4.3-32) and use of (4.3-26) r e s u l t s i n (4.3-33) Taking the t r a c e of^c", J a n d s e t t i n g i t equal t o zero y i e l d s or 125 29,' « -/M«* (4.3-34) which has the s o l u t i o n (assuming (h (z )) = 1) 3« " ( h ' / i O ^ , ^ (4.3-35) or an i d e n t i c a l a n a l y s i s shows that i n (4.3-31) must be zero i n order t h a t [p*J be r e g u l a r at 2 = 2 ^ , the t u r n i n g p o i n t . Therefore 126 What i s l e f t now i s the s o l u t i o n of (4.3-28). Step 4: The solution of Eguation (4.3-28). In order t o s o l v e eguation (4.3-28) c o n s i d e r f i r s t a column of£~H*/. C a l l i t / . Then (4.3-28) becomes ^ S "* ( K O ) ~ U , K C R * * ^ ^ 3 " 3 7 > d y From (4.3-37) i t i s observed t h a t 4 t f cLK Eguation (4.3-38) has the s o l u t i o n (4.3-38) 127 where-T, and-f^ are Hankel f u n c t i o n s of the 1/3 order and f i r s t and second k i n d r e s p e c t i v e l y . The c o n s t a n t s C, and C^ may be a r b i t r a r i l y chosen and, henceC, and Cj, w i l l be picked to be a phase f a c t o r times other constants which w i l l be absorbed as the c a l c u l a t i o n proceeds. Therefore WHC,H;,^"') • < ^ ( K V 3-391 To f i n d the other independent s o l u t i o n s , the d i f f e r e n t i a l equation and the p r o p e r t i e s o f the Hankel f u n c t i o n s are used. Thus {4.3-40) i' 2 A p p l i c a t i o n of formulas i n (4.1-27) i n Abramowitz and Stegun (1965) f o r d e r i v a t i v e s o f the Hankel f u n c t i o n s e.g. And of the i d e n t i t y 128 to (4.3-40) r e s u l t s i n Choice of C, a ft"''* and c=e™''/'!- together »ith the r e l a t i o n V^^y, y i e l d s T herefore the columns o f / / / * j ! r e (4.3-41) Step 5: Complete s o l u t i o n o f the decoupled 2x2 and 4x4 systems The complete s o l u t i o n of the 2x2 system f o r q u a s i -compressional waves i s obtained by m u l t i p l y i n g a l l the 129 t r a n s f o r m a t i o n s ana t h e s o l u t i o n (4.3-41) t o g e t h e r . or combining r e s u l t s of equations (4.3-25) (b) , (4.3-36) and (4.3-41) , the fundamental matrix s o l u t i o n of the 2x2 system i s The constant 6, can be absorbed i n t o the d e f i n i t i o n of C 3/V and C-^described p r e v i o u s l y . A l s o , V> can be w r i t t e n as (4.3-43) where w •= angular freguency R e c a l l a l s o t h a t the argument of the Hankel f u n c t i o n s i s T h e r e f o r e , the fundamental s o l u t i o n (4.3-42) becomes (4.3-44) A completely analagous s o l u t i o n e x i s t s f o r the 2x2 system r e l a t e d to guasi-shear waves. That i s 2 - ^ H ^ e ^ n j f y u ^ Thus, the complete s o l u t i o n f o r the fundamental matrix, u s i n g only the f i r s t term i n the [.8] t r a n s f o r m a t i o n ([#J=.I( i n eguation (4.3-12)), i s [Hi = £-rLCH*!l © C r f * l ] (4.3-45) The columns of£rV] are given below column 1 IL rjTu>otfct u> - i i 1 - COLA 2) H;3C^ Q J ' (-/* wy V £ . - BtAi) e^ H'^ C C J Q J column 2 r YA _L_ TfcoCL column 3 132 r>/> L 2. w>c +yt ^ 62)e^H;C OX},,) column 4 Asymptotic expansion o f the Hankel f u n c t i o n s u s i n g the r e l a t i o n s {4.2-3) ana (4.2-4) (Abramowitz & and Stegun 1965) allows the fundamental matrix^"/-/J to be w r i t t e n i n a more coamon form. 133 where [ H ] - M C A ' (4.3-46) L A ! o O and with = e p t w - - t *2 - - w L A 2 L A ( t z + w y ) LAt^t f yi -LAZ) L$(ti + wy) fc-w< - "tLZZ -wy1 -txe + y*2 - y/-B2 135 ' w h e r e * . . The £ 5 have been chosen as n o r m a l i z a t i o n s with r e s p e c t t o u n i t energy f l u x (see Chapman 1973 £ B i o t 1957). Both forms of the fundamental matrix ^H^ a na" the source vect o r obtained i n S e c t i o n 2, w i l l be used i n the next chapter, i n the c o n s t r u c t i o n of s y n t h e t i c seismograms. 136 CHAPTER V C a l c u l a t i o n of S y n t h e t i c Seismograms I n t r o d u c t i o n In t h i s f i n a l chapter, the basic r e s u l t s of the l a s t t h ree chapters w i l l be i n c o r p o r a t e d . These r e s u l t s have been obtained with a view t o c o n s t r u c t i n g s y n t h e t i c seismograms i n t r a n s v e r s e l y i s o t r o p i c media, s i n c e the apparatus of ray kinematics has been developed p r e v i o u s l y , the seismograms w i l l be c a l c u l a t e d on t h a t b a s i s . Chopra (1958) has e s t a b l i s h e d the connection between ray theory and s a d d l e p o i n t methods i n elastodynamics. He s t a t e s that Bromwich was the f i r s t t o d e r i v e a ray s e r i e s , and s i n c e then the correspondence t o ge o m e t r i c a l o p t i c s -r e f l e c t i o n and r e f r a c t i o n has been w e l l - g u a l i f i e d . But, with the advent of f a s t e r computers, more elegant t e c h n i q u e s such as Haskell-Thomson propagator ma t r i c e s , o r g e n e r a l i z e d ray theory have been employed t o s o l v e the l i n e a r elastodynamic wave eguations o c c u r r i n g i n seismology. I t i s i n t e r e s t i n g t o note that developments i n s y n t h e t i c seismology have come f u l l - c i r c l e , s i n c e the s a d d l e p o i n t method has again come i n t o vogue, 60 years a f t e r Bromwich*s work. 137 The new saddlepoint technique has been c a l l e d disk ray theory (Wiggins 1976) or the equal phase method (Chapman 1976a). These two approaches were presented simultaneously, and provide a f i r s t motion approximation to the motion. Both methods of developing this approximation w i l l be described in the following sections. As well, some seismograms for simple models w i l l be calculated with a view to comparing the r e s u l t s for i s o t r o p i c and transversely i s o t r o p i c media. 138 S e c t i o n 1 - I n t u i t i v e Development of Disk Ray Theory Wiggins (1976) has d e s c r i b e d a method f o r computing seismograms, based on e a r l i e r r e s u l t s (Wiggins and Madrid 1974). The p h y s i c a l b a s i s f o r computing the seismograms i s t h a t the amplitude i s p r o p o r t i o n a l to the change i n the ray parameter p. T h e r e f o r e , given a p- d e l t a curve f o r a continuous v e l o c i t y depth model, t h e steps to compute the seismogram a r e : 1) E s t a b l i s h t h e p o s i t i o n and time o f the geo m e t r i c a l a r r i v a l at the r e c e i v e r p o s i t i o n ; 2) F i n d the phase a r r i v a l times f o r a l l r a y s l e a v i n g the source and a r r i v i n g at the r e c e i v e r p o s i t i o n ; Bays other than the g e o m e t r i c a l ray a r r i v a l w i l l be delayed i n time with r e s p e c t to i t , a m a n i f e s t a t i o n of Fermat's p r i n c i p l e ; 3) At some time delay r e l a t i v e t o the main geometric a r r i v a l begin computation of dp such t h a t one d i g i t i z a t i o n u n i t of delay time i s used up; 4) H u l t i p l y t h e |dp| by a d i r e c t i v i t y f a c t o r , g e n e r a l l y dependent on p; 5) a f t e r completing 3 and H f o r the p - d e l t a curve, convolve the seismogram with an i n v e r s e o p e r a t o r which removes the 1 / s g r t ( t ) behaviour of the seismogram. The above methodology w i l l now be a p p l i e d i n d e t a i l t o the t r a n s v e r s e l y i s o t r o p i c medium. Consider a s u i t e of r a y s , l e a v i n g the source, as i n Fig.5.1. as can be seen, each ray has an a s s o c i a t e d plane wave f r o n t , which i s not orthogonal t o the ray t r a j e c t o r y . 139 Figu r e 5 . 1 - S u i t e of rays l e a v i n g shot Each of the plane waves (disks) w i l l a r r i v e at the r e c e i v e r but delayed with r e s p e c t t o the main a r r i v a l which i s ray 2 i n the f i g u r e . There may be, however, an o b j e c t i o n to using such a ray r e p r e s e n t a t i o n , as both L i g h t h i l l (1960) and Crampin (1977) p o i n t out. The t h e s i s of t h e i r argument. 140 which i s c o r r e c t , i s t h a t i n a n i s o t r o p i c media a plane wave f r o n t cannot propagate, s i n c e energy must be s u p p l i e d p a r a l l e l t o the wavefront. But, s i n c e the source under c o n s i d e r a t i o n i s a p o i n t source, energy i s s u p p l i e d i n a l l d i r e c t i o n s , as shown i n chapter IV. Fu r t h e r evidence f o r the v a l i d i t y of plane wave models has been provided by the work of Staudt and Cook (1967) (see F i g . 5 . 2 ) . An e m i t t i n g transducer placed beside a gu a r t z c r y s t a l has been used t o generate plane wave f r o n t s , which are the v e r t i c a l l i n e s i n the p i c t u r e . The d i r e c t i o n of energy flow, the ray d i r e c t i o n , i s i n d i c a t e d by the white band i n the f i g u r e , which i s at 45 degrees t o the plane wave f r o n t s . The white band i s i n d i c a t i v e of the d i r e c t i o n o f energy flow, s i n c e t h i s flow a f f e c t s the o p t i c a l parameter 6 ,which a f f e c t s the l i g h t s c a t t e r e d by the c r y s t a l . From the f i g u r e , i t i s c l e a r t h a t energy has been s u p p l i e d p a r a l l e l t o the wavefront. T h i s energy p a r a l l e l to the wavefront a r i s e s from the f a c t t h a t there are edge e f f e c t s from the e m i t t i n g transducer, i . e . a plane wave i s a purely mathematical g e n e r a l i z a t i o n , a l b e i t a u s e f u l one. Having e s t a b l i s h e d the v a l i d i t y of the d i s k s u p e r p o s i t i o n model, i t i s necessary to f i n d the phase delay time f o r a l l d i s k s a r r i v i n g a t the r e c e i v e r . A simple g e o m e t r i c a l argument i s i l l u s t r a t e d below. From Chapter 3, the a r r i v a l time of the disk a s s o c i a t e d with ray B i s p )<0 ^l~(Py) ' T k i s d i s k i n t e r s e c t s the po i n t X^,the F i g u r e 5 . 2 - A c o u s t i c plane waves i n a quartz c r y s t a l ( A f t e r Staudt and Cook 1967) q e c a e t r i c a l ray a r r i v a l d i s t a n c e at soae time e a r l i e r than ?3 ^ +~t~^p3.) • T h e r e f o r e , t i a e aust be s u b t r a c t e d f r o a P3 + ttPl) '  ThlS t i a e iS g i v e n by 3 142 F i g u r e 5.3 - Disk i n t e r c e p t i n g s u r f a c e i n a n i s o t r o p i c medium A t c AXft. V r (5.1-1) where \jr =ray v e l o c i t y 143 = d i s t a n c e from the node A t o the p o s i t i o n n3 By the law of s i n e s (5. 1-2) Cos e The time t o be s u b t r a c t e d i s c a l c u l a t e d as A i c ~^H^z 5in(TT/a-C^f«)XVK«] (5.1-3) But V rCose = Vp , the phase v e l o c i t y , and Sln(%'{ite)) - CoiW + e.) . Therefore v/ * 3 L (5.1-4) From the wave surface-slowness s u r f a c e c o n s t r u c t i o n of Chapter 2^  CoS(iie)_ n f the ray parameter of ray . Thus, the a r r i v a l time of the d i s k a s s o c i a t e d with ray ftj at X- X0 i s 144 or *• pA^-uPj) (5. 1-5) Eguation (5.1-5), i d e n t i c a l t o that obtained i n the i s o t r o p i c s i t u a t i o n , i s used to c a l c u l a t e the JdpJ a s s o c i a t e d with the phase delay (Fig 5.4). A p p l i c a t i o n o f equation (5.1-5) to the simple p - d e l t a curve i n Fig.5.4 demonstrates t h a t the r e g u i r e d phase delay of r a y f t j , with r e s p e c t to the g e o m e t r i c a l rayR^ , i s where p.d. i s the phase delay. In F i g . 5.4 p.d. i s the area of the t r i a n g l e OUS. I f the slope of the p - d e l t a curve i s s, then 145 Figure 5. « - Simple p - d e l t a curve 146 (5. 1-7) r S Sp, {IS it) 'k where dp = 1-p dt=p.d. S=slope of P-Xcurve curve. As S i g g i n s p o i n t s out, the amplitude i s given by F C p ^ l i p l (5,1-8) where F(Px)= d i r e c t i v i t y at the p o i n t P=P^. To c o n s t r u c t the seismogram then, i t i s necessary to compute the amount of £p f o r the phase delays o f o ft- where n i s an i n t e g e r and i s the d i g i t i z a t i o n i n t e r v a l . For the nth p o s i t i o n i n the seismogram, a p p l i c a t i o n o f eguation (5.1-7) g i v e s where n i s the i n t e g e r t i m i n g index r e l a t i v e t o the main a r r i v a l . There are two important p o i n t s to note about (5.1-9). F i r s t , the amplitude i s given as 147 F(pj|6pJ (5.1-10) Second, the seismogram w i l l have a shape l i k e <- s i n c e \Tr) - \l o-1 i s a f i n i t e - d i f f e r e n c e r e p r e s e n t a t i o n of the d e r i v a t i v e of "t , T h e r e f o r e a f t e r the c a l c u l a t i o n o f a l l Ah^S has been completed, the seismogram must be convolved with an i n v e r s e operator which removes the u dependence. The n e c e s s i t y of such an operator w i l l be proved i n S e c t i o n 3. I t i s important t o note t h a t f o r computational purposes, Wiggins (1976), has given a 3-point r e c u r s i o n formula f o r the operator. 1 4 8 S e c t i o n 2 The egual phase method JHKB R e f l e c t i o n C o e f f i c i e n t In the preceding s e c t i o n , an i n t u i t i v e approach to the f i r s t motion approximation i n t r a n s v e r s e l y i s o t r o p i c media was presented. Here, an a l t e r n a t i v e r e p r e s e n t a t i o n i s developed, using the fundamental matrix s o l u t i o n s obtained i n chapter IV. . To begin with a s o l u t i o n of the homogeneous system o f eguations, Eguation (4. 1-14) must be made v a l i d a t the t u r n i n g p o i n t . In order to do t h i s , the method o f Chapman (1977a) w i l l be used. In the case of "compressional" waves propagating, the c o e f f i c i e n t s r e p r e s e n t i n g the up and downgoing components must be chosen i n a s p e c i a l way. That i s , the s o l u t i o n to the homogeneous system, V [ H i /ce ce" o \ O (5.2-1) where V = s o l u t i o n vector = fundamental matrix (H) (4.3-45) C = amplitude of up and downgoing p-waves must obey the r a d i a t i o n c o n d i t i o n t h a t 2 -*-oo " The choice of z~?-<>o i n s t e a d of the u s u a l +«o i s achieved by changing c o - o r d i n a t e s , so t h a t the r e f e r e n c e p o i n t i s the 149 t u r n i n g p o i n t of the r a y . So the o b j e c t of the e x e r c i s e i s to f i n d the value o f c i n (5.2-1). P r i o r t o t h i s , however, some p r e l i m i n a r y a n a l y s i s needs t o be done on the r e p r e s e n t a t i o n of t h e Hankel f u n c t i o n s appearing in£nj. From Abramowitz and Stegun (1965) where (5. 2-2) Below the t u r n i n g p o i n t u i s n e g a t i v e . T h e r e f o r e , -u i s p o s i t i v e , and s i n c e 13(. l~ U* J ^  £ J , f o r l a r g e values of (-u), i t i s r e g u i r e d t h a t the q u a n t i t i e s i n (5.2-1) be chosen so t h a t the '-?c -> do not appear i n the f i n a l s o l u t i o n f o r \£ , T h i s i s f u r t h e r i l l u s t r a t e d below. C o n s i d e r a t i o n of the f i r s t component of V w i l l be s u f f i c i e n t to c a l c u l a t e 0 ^ . S u b s t i t u t i o n of the fundamental matrix values i n t o (5.2-1) r e s u l t s i n the f i r s t component of \/ being c a l c u l a t e d as 150 y , * b e (5.2-3) where w I n s e r t i o n of asymptotic forms of the Hankel f u n c t i o n s , i n t o ( ' $ " . , 2 - 3 ) # n e g l e c t i n g the Ac(-u) w n i . C h a r e bounded r e s u l t s i n ] (5.2-4) In order t h a t the term i n the b r a c k e t s vanish, i t i s r e g u i r e d that -/JI - « 12.) [b >2 / Or 2* * HI ^JI O r * 0 ~2L 3 ~ 4 The value of oC has been determined so t h a t the s o l u t i o n 151 (5.2-1) i s v a l i d a t the t u r n i n g p o i n t , where both up and downgoing waves e x i s t . In order t o c a l c u l a t e a r e f l e c t i o n c o e f f i c i e n t ( i n c i d e n t g u a s i - c o a p r e s s i o n a l wave), eguation (5.2-1) i s p r e a u l t i p l i e d by [N ] the i n v e r s e matrix of e i g e n v e c t o r s (see Chapman 1977b). T h i s r e s o l v e s the p h y s i c a l v a r i a b l e s , which are the components of V^ , i n t o up and downgoing wave components. The f i r s t component of jVJ V w i l l be the r e f l e c t i o n c o e f f i c i e n t and the second component w i l l r e p r e s e n t the i n c i d e n t wave ( r e c a l l t h a t the o r i g i n has been s h i f t e d to the t u r n i n g p o i n t ) . Furthermore, s i n c e the s o l u t i o n i s t o be examined f a r away from the t u r n i n g p o i n t , 0"G oay be w r i t t e n as Thus, c a l c u l a t i o n o f t h e v e c t o r , r e p r e s e n t i n g a l l up and downgoing waves above the t u r n i n g p o i n t y i e l d s the f o l l o w i n g : 152 (5.2-5) where i?pp »s t" />g r e f l e c t i o n c o e f f i c i e n t f o r i i n c i d e n t "p" wave; Rip = i n c i d e n t p-wave amplitude; Bps = r e f l e c t e d w s " wave converted from p; and B i s = i n c i d e n t " s " wave. The important r e s u l t t o note about (5.2-5) i s t h a t there i s no converted quasi-compressional wave due to the " r e f l e c t i o n " at the t u r n i n q p o i n t . To a c t u a l l y o b t a i n the value of Bpp, the s u b s t i t u t i o n o f i n t o (5.2-5) y i e l d s 153 ,-AV/i (5. 2-6) In (5.2-6), i f a u n i t i n c i d e n t amplitude i s co n s i d e r e d , Equation (5.2-7) i s the famous JWKB r e f l e c t i o n c o e f f i c i e n t mentioned by Chapman (1976b) and Ri c h a r d s (1973). I t can be used t o c o n s t r u c t seismograms as Chapman has demonstrated. However, as Chapman (1977, p e r s o n a l communication) has pointed out, the r e s u l t (5.2-7) i s not completely general. I t does not obey the r e l a t i o n c * e (5. 2-7) which must be t r u e , i f the f u n c t i o n d e s i r e d , a f t e r an inv e r s e F o u r i e r transform i s performed, i s to be r e a l , c o r r e c t r e f l e c t i o n c o e f f i c i e n t i s The SynL^e.*^0" (5.2-8) An analogous c o e f f i c i e n t e x i s t s f o r g u a s i - shear waves: 55 With the above r e f l e c t i o n c o e f f i c i e n t s , the f i r s t motion approximation may be obtained i n a s t r a i g h t f o r w a r d manner. 155 S e c t i o n 3 - The Equal Phase Method - Implementation With the c a l c u l a t i o n o f the r e f l e c t i o n c o e f f i c i e n t , the c o n s t r u c t i o n of the seismogram proceeds as i n d i c a t e d below. F i r s t , the inhomogeneous matrix eguation (4. 1-14) must be s o l v e d , and then, the s o l u t i o n i s s u b s t i t u t e d i n t o the c y l i n d r i c a l harmonic expansion eguation (4.1-1). The s o l u t i o n of t h e inhomogeneous matrix eguation i s given as (Chapman 1976b) where i s the fundamental matrix a n d ) ^ / s ) i s the source term (4.2-12). As 2 : , i s a r b i t r a r y , the f i r s t term can be neglected. The second term i n s i d e the parentheses i n (5.3-1) re p r e s e n t s the decomposition of the source vector i n t o i t s up and downgoing wave components. S u b s t i t u t i o n of the asymptotic form of the fundamental m a t r i x / / v j (4.3-46) i n t o (5.3-1), and i n t e g r a t i o n a c r o s s the point of d i s c o n t i n u i t y , t0, y i e l d s 156 ( 5 . 3 - 2 ) where ur = angular freguency Ufa) = h e a v i s i d e step f u n c t i o n = i j t h component of [Nl i n (4.3-46) ( Note: z i n ^ A/^  i s not the depth but the ray parameter -(see (4.3-4)) The f i r s t element o f Sji , i s the amplitude c a r r i e d by a downgoing guasi-compressional wave, and the t h i r d element i s the amplitude c a r r i e d by a downgoing guasi-shear wave. In an i s o t r o p i c medium £ 3 \ and £iff are i d e n t i c a l l y z e r o , s i n c e the e x p l o s i o n source term (4.2-12) generates no shear waves. Thus, i n (5.3-2), the f i r s t obvious d i f f e r e n c e between i s o t r o p i c and t r a n s v e r s e l y i s o t r o p i c media becomes s e l f -e v i d e n t . With an e x p l i c i t e xpression f o r the source, the s o l u t i o n to (4. 1-14) f o r the v e r t i c a l and h o r i z o n t a l displacements becomes = source term * r e f l e c t i o n c o e f f i c i e n t * a p p r o p r i a t e 157 eig e n v e c t o r components (5.3-3) where $ = v e c t o r of h o r i z o n t a l and v e r t i c a l displacements T h i s w i l l be i l l u s t r a t e d below f o r the case of downgoing guasi-compressional waves. From (5.3-2), the source term generating downgoing guasi-compressional waves i s The r e f l e c t i o n c o e f f i c i e n t i s given as The r e l e v a n t e i g e n v e c t o r components are IV * € U t w - i i 1 - wLA2) Thus, the s o l u t i o n of eguation (4.1-14) f o r the components r e p r e s e n t i n g v e r t i c a l and h o r i z o n t a l displacements i s 158 (5. 3-5) where V , - c ur 7, . ' V x =• - ur/a. Vj = transformed horizontal displacement Vx = transformed v e r t i c a l displacement (see (4.1-13), (4.1-14)) -tip) = a = the v e r t i c a l delay time Expression of V ' in terms of the elements of V r e s u l t s i n (5.3 - 5 becoming - sg n (y*) e^^U „ Stt (5.3-6) 2 / 0 f / t L A N 3-1 The problem i s almost solved.' Hhat i s necessary now i s to insert Vj and into the c y l i n d r i c a l harmonic expansion, and do the integrations required. Before proceeding with those calculations, i t i s i n s t r u c t i v e to note some properties of Bessel functions and t h e i r r e l a t i o n s to Hankel functions. 159 The appearance of Bessel functions i n the harmonic expansion, or a l t e r n a t i v e l y , the Bessel transform occurs i n t h i s elastodynamic problem due to c y l i n d r i c a l symmetry. For example, from (4.1-1) (neglecting constants and the time dependence) As Chapman (1977a) has noted, i t i s expedient to convert integ r a l s appearing above to ones where uTthe angular freguency i s r e a l and k=wp. Substitution of k=wp into the o (5.3-7) i n t e g r a l f o r W r e s u l t s in (5.3-8) Substitution of the rela t i o n s 160 T0(u>p03 H C^topOt- HQCu>pr) _ i n t o (5.3-8) r e s u l t s i n . . , . „ , ~ (5.3-9) - _L <-ozJ^  p wcp) H0(u>pr)dCp only i f uT > 0 andWCp) i s even i n p. For a l l r e a l , \V »J_ I tol u>J p w ( p ) H^Cwpr^ clp (5.3-10) The contour i s chosen to s a t i s f y the r a d i a t i o n c o n d i t i o n (Chapman 1977a) . A s i m i l a r a n a l y s i s can be a p p l i e d to the i n t e g r a l f o r u through use of the r e l a t i o n 161 1 dJ0Lmf) - - I,Cu>pr) Then the expr e s s i o n f o r u becomes (5.3-11) only i f Ct(p) i s odd i n p. E q u a t i ons (5.3-10) and (5.3-11) w i l l now be used below i n the c a l c u l a t i o n of the i n v e r s e transform (or e v a l u a t i o n o f the c y l i n d r i c a l harmonic expansion) . In doing the f i n a l i n v e r s e transform, i t must be r e c a l l e d t h a t the r e s u l t s have been d e r i v e d f o r a plane wave of angular freguency U/ . T h e r e f o r e , eguations (5.3-6) must be m u l t i p l i e d by the source time f u n c t i o n transform,~T(urJ. Then the complete s o l u t i o n f o r U . (the h o r i z o n t a l displacement) i s given as 162 U * V, -J_ fe" 4 W Cu>|u>|T<u»^u> . |-H (CwprX-53niu)e<u, t (p ;M1(SupdLp 8ITZJ J —oo — o o (5.3-12) and 8lf (5.3-13) where (5.3-6),(5.3-10) and (5,3-11) have been used and the i n v e r s e time transform has been performed as w e l l . In order to render (5.3-12) and (5.3-13) i n a form s u i t a b l e f o r c a l c u l a t i o n , the asymptotic forms of the Hankel f u n c t i o n s are used: 163 V toprir u>prrr These expansions are s u b s t i t u t e d i n t o e xpressions (5.3-12) and (5.3-13), and t h i s f a c t i m p l i e s t h a t the c u r v a t u r e of the wavefront i s neglected. Thus, the e x p r e s s i o n s f o r V/ and become (5.3-14) and (5. 3-15) To f u r t h e r s i m p l i f y the above expressions the f o l l o w i n g " t r i c k s " are used: 1) I u r l = (cor) 2) £ i s w r i t t e n as (-',) e-3) the r e s u l t i n g (-1) i s absorbed i n t o the tO"" i n t e g r a l ; 164 4) the convolution theorem i s used as well as the following inverse transforms: t ' Spirt" These " t r i c k s " result i n (5.3-16) and (5. 3-17) where Qf^f+rfp); the phase time of a l l plane waves at the position r. Heaviside step function. With the c a l c u l a t i o n at the present state, i t i s time to invoke the egual phase method. This i s a mathematical ' statement of the physics developed i n Section 1, in which the disk algorithm was explained. The phase 0 i s expanded about the equal phase points pj (Chapman 1976b). These P j are the ray parameters of rays which, at the receiver position r, have equal phase delays r e l a t i v e to the main a r r i v a l . The contributions from each set of equal phase ray parameters are summed and occur in the seismogram at the 1 6 5 a p p r o p r i a t e time. T h i s corresponds e x a c t l y to the method explained i n S e c t i o n 1 . Note a l s o t h a t , i n S e c t i o n 1 , an o perator was d e s c r i b e d , which had t o be convolved with the seismogram. I f one of the o p e r a t o r s i s c o n s i d e r e d with the term i n ( 5 . 3 - 1 6 ) or ( 5 . 3 - 1 7 ) , then t h i s time v t f u n c t i o n a. (Htm i s e x a c t l y the one d e s c r i b e d i n S e c t i o n 1. , For the c o n t r i b u t i o n t o the seismogram at the j t h egual phase p o i n t , Q(f>) i s expanded about p~ fj ( 5 . 3 - 1 8 ) Use of ( 5 . 3 - 1 8 ) i n t o ( 5 . 3 - 1 6 ) and ( 5 . 3 - 1 7 ) r e s u l t s i n ( 5 . 3 - 1 9 ) To evaluate the double i n t e g r a l i n ( 5 . 3 - 1 9 ) two s t e p s are taken: 1 ) The l i m i t s on the uf i n t e g r a l are r e p l a c e d >&) t o (oj < o J and t h i s g i v e s twice the r e a l p a r t , i . e . , 166 2) The (A/*i n t e g r a l i s evaluated and the p - i n t e g r a l i s evaluated. Dse of the a r t i f i c e s i n e x p l i c i t l y doing step (2) i s t h a t : and f hcp)S(e/(o.Xp-p,))dp =r WtP> J \0'lf)\ = smooth f u n c t i o n of p implementation of (1), (2), (3), and (4) y i e l d s t he f i n a l r e s u l t [ a ^ _L_ d I Tit)) * d. /_HU1\ .it 1, p'kl w„\ _ J (5.3-20) Equation (5.3-20) i s i d e n t i c a l i n a l l r e s p e c t s to t h a t obtained by Chapman (1976b), the only d i f f e r e n c e being the source f a c t o r Sit and the e i g e n v e c t o r components^/ andA^/. A completely analogous e x p r e s s i o n e x i s t s f o r g u a s i -shear waves, the d i f f e r e n c e s being t h a t the phase time i s 167 that a p p r o p r i a t e to the wave-type and t h a t the new source term and e i g e n v e c t o r components are: Eguations (5.3-20) and (5.3-21) w i l l be used t o generate seisffiograms f o r very simple t r a n s v e r s e l y i s o t r o p i c media, as d e t a i l e d i n S e c t i o n 4. 168 S e c t i o n 4 -Simple Seismic C a l c u l a t i o n s Given the complicated t h e o r e t i c a l d e r i v a t i o n of the preceding s e c t i o n s , can any simple p r e d i c t i o n s be made f o r s e i s m i c waves i n the case where ray theory i s v a l i d ? The answer i s a d e f i n i t e yes. For a l l the c a l c u l a t i o n s of s y n t h e t i c seismograms i n t h i s s e c t i o n , the v e l o c i t y model chosen was one c h a r a c t e r i s t i c of the upper mantle, that i s , v= 8.1 +.0027*z. Three s e t s of c a l c u l a t i o n s were done-no a n i s o t r o p y , 10% a n i s o t r o p y , and 30% a n i s o t r o p y . The a n i s o t r o p y i n the model was achieved by i n c r e a s i n g the s t i f f n e s s C by the a p p r o p r i a t e amounts. I n c r e a s i n g C would e f f e c t i v e l y i n c r e a s e the v e r t i c a l v e l o c i t y . For s i m p l i c i t y , i n the f o l l o w i n g d i s c u s s i o n , the model without a n i s o t r o p y w i l l be r e f e r r e d t o as Hodel 1, those with 10 and 30% a n i s o t r o p y as Models 2 and 3 r e s p e c t i v e l y . In f i g u r e s 5.5, 5.6, 5.7 p - d e l t a curves f o r compressional waves are p l o t t e d f o r Models 1, 2, and 3. The rays a r r i v e at e a r l i e r times and s m a l l e r e p i c e n t r a l d i s t a n c e s with i n c r e a s i n g a n i s o t r o p y . For example, i n f i g u r e 5.5, the e p i c e n t r a l d i s t a n c e f o r p=.1 sec./km. i s 4400 km., while i n f i g u r e 5.7, f o r the same p, the d i s t a n c e i s 3800 km. a l s o i t i s c l e a r from the f i g u r e s , t h a t apart from kinematic e f f e c t s , the a n i s o t r o p y has not a f f e c t e d the 169 basic shape of the p-delta curve. Before commenting on the synthetic seismograms calculated from the p-delta curves, i t i s h e l p f u l to consider the d i r e c t i v i t y functions used i n the d. r. t. calculations. The d i r e c t i v i t y function i s taken to mean the product of the source function, the appropriate eigenvector component, and the p**1/2 which arises from the expansion of Bessel function. The d i r e c t i v i t y function assigns an i n i t i a l "amplitude" to each ray. (Figures 5.8, 5.9, and 5.10 show the horizontal d i r e c t i v i t y functions for guasi-compressional waves.) I t i s immediately clear that the anisotropy has had l i t t l e e f f e c t i n changing the shape or magnitude of the d i r e c t i v i t y function for a given ray parameter p. In figures 5.11, 5.12, and 5.13 the d i r e c t i v i t i e s for the v e r t i c a l displacement are i l l u s t r a t e d . Again, the increasing anisotropy has not s i g n i f i c a n t l y changed the shape of the d i r e c t i v i t y as a function of p, although the magnitude, in the case of Hodel 3, has been reduced by about The most important point to note i s that the anisotropy has not introduced any extra bumps on the d i r e c t i v i t y curve. Given the above r e s u l t s , i t i s expected that the main effect of the anisotropy would simply be the advance i n time of the main a r r i v a l , as the amount of anisotropy i s increased. The seismogram for Models 1, 2, and 3 (horizontal displacement) are shown i n figures 5.14, 5,15, 170 and 5.16, while those for v e r t i c a l displacement are shown i n figures 5.17, 5.18, and 5.19. I t i s clear that the observation of the kinematic effect mentioned above can be seen by comparison of figures 5.11 and 5, 16. For example, in figure 5.14, at an epicentral distance of 3100 km., the main pulse arrives at 57 sec. while i n figure 5.16 i t arrives at 56 sec. The d i r e c t i v i t y used to construct the seismogram does not a f f e c t the pulse for either horizontal or v e r t i c a l displacement, since i t i s a smooth function and i s not rapidly changing for values of p i n the neighbourhood of .105 sec./km., which corresponds to a delta of 3200 km. It has been demonstrated that for guasi-compressional ("p") waves, the ef f e c t of anisotropy i s mainly a kinematic one. What happens i n the case of guasi-shear waves? Comparison of p-delta curves shows that for greater anisotropy the epicentral distances are reduced (compare figure 5.21 with 5.20). Hence, the same kinematic e f f e c t s , mentioned e a r l i e r also a f f e c t guasi-shear waves for the pa r t i c u l a r models chosen. However, examination of the d i r e c t i v i t y curves for horizontal and v e r t i c a l displacements (figures 5.22 and 5.23 for Model 2 and figures 5.24 and 5.25 for Model 3), shows that the value of the d i r e c t i v i t y function i s much less for guasi-shear waves than for guasi-compressional waves. This i s to be expected since i n an i s o t r o p i c medium, the d i r e c t i v i t y function for an explosion source i s zero for shear waves. The d i r e c t i v i t y values are 171 somewhat g r e a t e r f o r Model 3 than Model 2, s i n c e l a r g e r amounts of a n i s o t r o p y generate a g r e a t e r guasi-shear component. The r e l e v a n t seismograms f o r gu a s i - s h e a r waves are shown i n f i g u r e s 5.26 - 5.29. Although the seismograms f o r Model 2 are c a l c u l a t e d at d i f f e r e n t e p i c e n t r a l d i s t a n c e s than those i n Model 3, i t i s s t i l l c l e a r that the a r r i v a l s f o r Model 3 are e a r l i e r than those f o r Model 2 due to the d i f f e r i n g p- d e l t a curves (see f i g u r e s 5.20 and 5.21). I t i s evident from the seismograms that the v e r t i c a l components are phase s h i f t e d 180 degrees r e l a t i v e to the h o r i z o n t a l components. Again due to the smooth, s l o w l y v a r y i n g d i r e c t i v i t i e s , t h e p u l s e shapes are not a f f e c t e d by the a n i s o t r o p y . Given a l l the evidence presented thus f a r , i t i s p o s s i b l e t o answer the question posed at the beginning of t h i s s e c t i o n . I t has been demonstrated t h a t f o r the simple models used i n the c a l c u l a t i o n s , the e f f e c t of the a n i s o t r o p y i s a kinematic one-the main a r r i v a l s are simply advanced i n time. The shape of the main a r r i v a l i s not a f f e c t e d by a smooth, s l o w l y v a r y i n g d i r e c t i v i t y f u n c t i o n . T h i s s i t u a t i o n c o u l d be a l t e r e d i f a d i f f e r e n t range of e p i c e n t r a l d i s t a n c e s i s considered, or the d i r e c t i v i t y e f f e c t s due to f r e e s u r f a c e are i n c l u d e d . To keep the r e s u l t s as simple as p o s s i b l e , no s u r f a c e c o n v e r s i o n c o e f f i c i e n t was in t r o d u c e d i n the c a l c u l a t i o n . , T h i s may, of 172 course, be obtained from the e i g e n v e c t o r s c a l c u l a t e d i n Chapter IV, but the algebra i s extremely complicated. Since the e f f e c t s of a s m a l l amount of a n i s o t r o p y can be detected using the dis k ray method (equal phase method), i t i s a va l u a b l e adjunct t o other s y n t h e t i c seismogram c a l c u l a t i o n techniques which have been a p p l i e d to a n i s o t r o p i c media ( K e i t h 6 Crampin 1977; Daley S Hron 1977). _! 1 1 1 1 1 1 0.0 80.0 160.0 240.0 320.0 400.0 , 480 DELTA (KM) ISOTROPIC MODEL1 PWflVES (X101 ) Figure 5.5 P- d e l t a curve f o r i s o t r o p i c Model 1 f o r compressional waves. Mantle model was used (v=(8.1 + .0027 *z) km. /sec.) 174 Figure 5.6 P-delta curve f o r a n i s o t r o p i c Model 2 f o r g u a s i -compressional waves. Anisotropy f a c t o r i s 10%. 175 in o~l 1 1 1 1 1 1 0.0 80.0 160.0 240.0 320.0 400.0 , 480.D DELTA (KM) ANISOTROPIC M0DEL3 PWAVE5 (X10 j ) Figure 5.7 P-delta curve f o r a n i s o t r o p i c Model 3 f o r g u a s i -compressional waves. Anisotropy f a c t o r i s 301. 176 177 F i g u r e 5,9 D i r e c t i v i t y f u n c t i o n (guasi-compressional waves) f o r h o r i z o n t a l displacement f o r a n i s o t r o p i c Model 2. 10% a n i s o t r o p y 178 i o O 'CM —.m CvJ -H in \ ° L J , LU cn ID >--o LU<x) »—i •-0.85 0.93 P(SEC/KM.JPVflVES 1.01 1.09 ANISOTROPIC HORIZONTAL 1.25 DISP. (X10 - i ' F i g u r e 5.10 D i r e c t i v i t y f u n c t i o n (guasi-corapressional waves) f o r h o r i z o n t a l displacement f o r a n i s o t r o p i c Model 3. 30% a n i s o t r o p y 179 F i g u r e 5.11 D i r e c t i v i t y f u n c t i o n (compressional waves) f o r v e r t i c a l displacement f o r i s o t r o p i c Model 1. 1 8 0 CM cn CO i o o , -^CO in \ ° o LU cn > 03 ID o ' 1 1 1 1 1 1 0.B5 0.93 1.01 1.09 1.17 1.25 1.33 P(SEC/KM.)PWRVES ANISOTROPIC VERTICAL DISP. CX10' 1 ) F i g u r e 5.12 D i r e c t i v i t y f u n c t i o n {guasi-compressional waves) f o r v e r t i c a l displacement f o r a n i s o t r o p i c Hodel 2. 101 a n i s o t r o p y 181 • cn i o O L O 2)5 C J L U CO 0 . 8 5 0 . 9 3 1.0J 1.09 1.17 P(SEC/KM.JPVflVES ANISOTROPIC VERTICAL 1.25 , 1.3: DI5P (X10"1 J f i g u r e 5.13 D i r e c t i v i t y f u n c t i o n (guasi-compressional waves) f o r v e r t i c a l displacement f o r a n i s o t r o p i c Model 3. 30% a n i s o t r o p y 182 a a SH 1 1 1 1 1 1 1 r — i 1 1 — i 1 1 1 3000.0 3040.0 3080.0 3120.0 3160.0 3200.0 3239.99 3279.99 DELTA (KM) ISQMODEL P-WAVES HORIZONTAL DISP (ISW=1) Figure 5.14 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r compressional waves c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.5 183 cn. U3 in. to in. to to in in. in r 3000.0 DELTA (KM) T T T T "I 1 3040.0 3080.0 3120.0 3160.0 3200.0 3239.99 3279.99 ANIS0MQDEL2 P-WAVES HORIZONTAL DISP (ISW=1) F i g u r e 5.15 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r guasi-compressional waves c a l c u l a t e d u sing p - d e l t a curve shown i n f i g u r e 5.6., 10% a n i s o t r o p y 184 cn_J LO LUr-_| CO10 oSS-y U 3 CE f— LU • L J -LU C O O 21 a r—. i n i n . i n — r 3000.0 DELTA IKMJ — i 1 1 1 1 1— 3040.0 3080.0 3120.0 ANIS0M0DEL3 P-WAVE5 —i 1 1 1 1 1 1 1 3160.0 3200.0 3239.99 3279.99 HORIZONTAL DISP U5W=1) F i g u r e 5.16 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r guasi-compressional waves c a l c u l a t e d u sing p - d e l t a curve shown i n f i g u r e 5.7. 30% a n i s o t r o p y 185 o cn ir m . to o a-o r~. U in in 3000.0 DELTA (KM) T T n 1 1 r 3040.0 3080.0 3120.0 3160.0 3200.0 J50M0DEL P-WAVES VERTICAL DISP — i r -3239.99 (ISW=2) "I 1 3273.99 F i g u r e 5,17 S y n t h e t i c f o r compressional waves seismogram o f v e r t i c a l displacement c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.5 186 LO LU to to o o id (X LU a i LJ LU CO LU in. LO tn. LO a ID in in in 3000 .0 DELTA (KM) T T 1 i 1 1 1 r 3040.0 3080.0 3120.0 3160.0 3200.0 3239.99 3279.99 ANIS0M0DEL2 P-WAVE5 VERTICAL DI5P (ISV=2) F i g u r e 5.18 S y n t h e t i c seismogram o f v e r t i c a l displacement f o r guasi-compressional waves c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.6 10% a n i s o t r o p y . 187 U3 LUr-. CO"3 o S i m . v * t o CE I— ' 5 G -LU coo LUirH r- -4 in in. in —r 3000.0 DELTA (KM) T T 1 1 1 1" 3040.0 3080.0 3120.0 ANIS0M0DEL3 P-WAVES -1 1 1 1— 3J60.0 3200.0 VERTICAL DISP T T T 3239.99 3279.99 (ISW^2) F i g u r e 5.19 S y n t h e t i c seismogram o f v e r t i c a l displacement f o r guasi-compressional waves c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.7.30% ani s o t r o p y . 188 Figure 5.20 P-delta curve for anisotropic Model 2 f o r quasi shear waves. Mantle model was used (velocity used for guasi- compressional waves divided by 1.732) 10% anisotropy. 189 —4 ro* oo CD ro" LO ID 0.0 50.0 DELTA (KM) 100 0 150.0 200.0 250.0 ANISOTROPIC M0DEL3 SWAVES (X101 ) i 300. Figure 5.21 P-delta curve f o r a n i s o t r o p i c Model 3 f o r g u a s i -shear naves. Mantle model was used ( v e l o c i t y used f o r g u a s i - compressional waves d i v i d e d by 1.732) 30% a n i s o t r o p y . 190 CN o i—«. m i a O X 1.6 1.68 1.76 1.B4 1.92 2.0 2.08 P(SEC/KM.JSWflVES ANISOTROPIC HORIZONTAL DISP. (XIO^ ) F i g u r e 5.22 D i r e c t i v i t y f u n c t i o n (guasi-shear waves) f o r h o r i z o n t a l displacement using Model 2. 10% a n i s o t r o p y 191 CM o _ | _ 1 1 1 1 1 I 1.6 1.58 1.76 1.84 1.92 2 . 0 . 2 . P(SEC/KM.JSWRVES ANISOTROPIC VERTICAL DISP. CX10"1 ) Figure 5.23 D i r e c t i v i t y function (guasi-shear aaves) for v e r t i c a l displacement using Hodel 2. 10% anisotropy F i g u r e 5.24 D i r e c t i v i t y f u n c t i o n (guasi-shear waves) f o r h o r i z o n t a l displacement u s i n g Model 3. 3051 a n i s o t r o p y 193 F i g u r e 5.25 D i r e c t i v i t y f u n c t i o n (guasi-shear waves) f o r v e r t i c a l displacement u s i n g Model 3. 30% a n i s o t r o p y 194 o as CM ' CJ LU cn a T" CM" O 00 a CM co a . CM CC LU a I G LU CO CM a . C M to cn. at. a cn. 2300.0 DELTA IKM) T T 1 — I J— 2340.0 2380.0 2420.0 ANISQM0DEL2 S-WAVES T T T T T 1 1 1 2460.0 2500.0 2540.0 2579 99 HORIZONTAL D I S P ( I S W = 3 ) F i g u r e 5.26 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r guasi-shear waves, c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.20. 10% a n i s o t r o p y . 195 00 C E a i C J L U CO L U z : a to C M ' a XT-CM ' L U CO —' CM a . CM a . CM cn. M" cn. a a - 4 2300.0 DELTA (KM) T T T T ~ l 1 1 1 2340.0 2380.0 2420.0 2460.0 2500.0 2540.0 2?79.99 ANISQMQDEL2 S-WAVES VERTICAL DISP CISW=4) F i g u r e 5.27 S y n t h e t i c seismogram of v e r t i c a l displacement f o r guasi-shear waves, c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.20. 10% a n i s o t r o p y . 196 03 i n . C O 03 • 12 in cn J I d 2 LUo CO^ —•'oi. UJ—• a Q m T T T 1700.0 DELTA (KM) — i 1 1 1 1 1 1 1 r 1740 0 1 1 8 0 . 0 1 8 2 0 . 0 1 8 6 0 . 0 1 9 0 0 . 0 1940 .DT 1 9 8 0 . 0 ANI50M0DEL3 5-WAVE5 HORIZONTAL DISP USV=3) Figure 5.28 S y n t h e t i c seismogram of h o r i z o n t a l displacement f o r guasi-shear waves, c a l c u l a t e d using p - d e l t a curve shown i n f i g u r e 5.21. 30% a n i s o t r o p y . 197 03 ID. CJLO. LU"-* CQ ^3 in. 00 • CE LUQJ I G • LUo SR. U J -o a m. T T T T 1 1 1 1 1 1 1700.0 1740.0 1780.0 1820.0 1860.0 J900.0 194D.0 198D.0 DELTA (KM) ANIS0M0DEL3 5-WAVE5 VERTICAL DISP (ISW=4) Figure 5.29 Synthetic seismogram of v e r t i c a l displacement for guasi-shear waves, calculated using p-delta curve shown in figure 5.21. 30% anisotropy. 198 Conclusions For f a s t , cheap, and a c c u r a t e s y n t h e t i c seismogram c a l c u l a t i o n s , i n the g e o m e t r i c a l o p t i c s l i m i t , d . r . t . provides an important complement to e x i s t i n g methods. Since the geometrical o p t i c s l i m i t i s used, i t i s necessary t o o b t a i n the a p p r o p r i a t e JWKB r e f l e c t i o n c o e f f i c i e n t . T h i s has been achieved by u s i n g the Langer t r a n s f o r m a t i o n and the r a d i a t i o n c o n d i t i o n a t i n f i n i t y . For the simple cases o f anisotropy c o n s i d e r e d , the main e f f e c t seen on the seismograms i s a kinematic one. The main a r r i v a l s f o r the a n i s o t r o p i c model were advanced i n time, as compared with those f o r the i s o t r o p i c model. As i n the i s o t r o p i c case, care must be taken i n d e l i m i t i n g the range of ray parameters e n t e r i n g i n the s e i s m i c c a l c u l a t i o n . In p a r t i c u l a r , s i n c e the d . r . t . a l g o r i t h m operates i n the f a r f i e l d , i t i s necessary that w p r » 1 . T h i s i s a r e s t r i c t i o n on both the freguency content of the waves which are allowed to propagate and on the range o f e p i c e n t r a l d i s t a n c e s . F u r t h e r s t u d i e s of the a p p l i c a t i o n s of d . r . t . to more general forms of a n i s o t r o p y are p r o f i t a b l e avenues of f u t u r e r e s e a r c h . T h i s can be achieved by performing a t h i r d F o u r i e r transform over angle and a p p l y i n g the necessary asymptotic methods to s o l v e the i n i t i a l value problem. 199 Currently, the author i s investigating more complex transversely i s o t r o p i c models, in which the wave surface for guasi-shear waves has a t r i p l i c a t i o n . It i s expected that focussing e f f e c t s w i l l appear i n the seismogram {cf.Chapter II) at appropriate epicentral distances. The possible extensions of the methods used in t h i s thesis show that indeed in the geometrical optics, f a r - f i e l d l i a i t , d.r.t or the-equal phase method i s as powerful as Wiggins and Chapman have indicated. 200 B i b l i o g r a p h y abramowitz, H. , & Stegun, I . A., 1965. Hau^feook of mathematical f u n c t i o n s , Dover P u b l i c a t i o n s i n c . , New York,., alekseyev, a . C., Babich, V. a . , & G e l c h i n s k i y , B. Y . , 1961. 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