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Variational formulation and finite element type solution for free oscillations of a rotating laterally… Moon, Wooil 1976

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A VARIATIONAL FORMULATION AND FINITE ELEMENT TYPE SOLUTION FOE FHEE OSCILLATIONS OF A ROTATING LATERALLY HETEROGENEOUS EARTH by Kooil Moon B.J.Sc, Univ. Of Toronto, 1970 M.Sc, Columbia University, 1972 .a t h e s i s submitted in p a r t i a l f u l f i l l m e n t of the requirements for the degree of doctor of philosophy IN THE DEPARTMENT OF GEOPHYSICS AND ASTRONOMY WE ACCEPT THIS THESIS AS CONFORMING TO THE REQUIRED STANDARD THE UNIVERSITY OF BRITISH COLUMBIA APRIL,1976 (c) Wooil Moon, 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t ten pe rm i ss ion . Department o f r G a o p h y s i c s and As t ronomy The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2 0 7 5 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1 W S ABSTRACT A v a r i a t i o n a l t y p e , f i n i t e element method i s proposed f o r the s t u d y of normal modes o f a r o t a t i n g , l a t e r a l l y h e t e r o g e n e o u s E a r t h . T h i s method i s very p o w e r f u l and, i n t h e o r y , overcomes a l l t h e l i m i t a t i o n s o f the p e r t u r b a t i o n method. However, the p r e s e n t s i z e of computer memory puts s e v e r e l i m i t a t i o n s on the use o f t h i s methcd. The L a g r a n g i a n e n ergy i n t e g r a l i s d e r i v e d f o r a g e n e r a l c o n f i g u r a t i o n of a p e r f e c t l y e l a s t i c continuum i n b c t h s h p e r i c a l and o b l a t e e l l i p s o i d a l c o o r d i n a t e systems. The assumed s o l u t i o n s f o r t h e d i s p l a c e m e n t s and t h e p e r t u r b a t i o n of the g r a v i t a t i o n a l p o t e n t i a l a r e formed by t e n s o r p r o d u c t s of t h e c u b i c Hermite b a s i s f u n c t i o n s of t h r e e c o o r d i n a t e v a r i a b l e s . The undetermined c o e f f i c i e n t s of t h e s e assumed s o l u t i o n s are s o l v e d by t h e m i n i m i z a t i o n o f the L a g r a n g i a n energy i n t e g r a l by a R a y l e i g h - R i t z t e c h n i g u e . The r e s u l t s o f the n u m e r i c a l s o l u t i o n of t h i s approach show t h a t (1) t h i s a l g o r i t h m i s very e f f i c i e n t and p r o m i s i n g i n one-d i m e n s i o n a l i z e d normal mode problems, (2) t h a t the degenerate f r e q u e n c y of ^\X)x s p l i t s i n t o (2JL+1) components, even f o r t h e n o n - r o t a t i n g , homogeneous s p h e r i c a l E a r t h model, due t o t h e i n t e r p o l a t i o n scheme of the a z i m u t h a l b a s i s f u n c t i o n s and (3) because t h e n u m e r i c a l s p e c t r a l s p l i t t i n g i s v e r y l a r g e , t h e e f f e c t s o f s e l f - g r a v i t a t i o n and r o t a t i o n cannot be examined c l e a r l y i n t h i s s t u d y . However, l a t e r a l h e t e r o g e n e i t i e s which break t h e symmetry o f t h e p h y s i c a l shape of the E a r t h g r e a t l y a f f e c t t h e normal mode s p e c t r a . i i i TABLE OF CONTENTS ABSTRACT . i i LIST OF FIGURES ............................................ V LIST OF TABLES v i i ACKNOWLEDGEMENTS ..................................... . . . . . v i i i 1. INTRODUCTION 1 1.1 Review Of Previous Work ........................... 2 1.2 Review Of Perturbation Solutions For The Effects Of Rotation, E l l i p t i c i t y And Lat e r a l Heterogeneity ..................................... 9 1.3 Scope Of This Discussion .......................... 15 2. VARIATIONAL FORMULATION OF EQUATION OF MOTION .......... 20 2.1 Generalized Equation Of Motion .................... 20 2.2 Boundary Conditions ............................... 24 2.3 C l a s s i c a l Solution Of Normal Mode Problem For The Earth 26 2.4 Lagrangian Integrals Of One-dimensionalizad Normal Mode Problem ............................... 32 2.5 Lagrangian Integral For The Generalized Three-dimensional Problem 34 2.6 Boundary Conditions In Vari a t i o n a l P r i n c i p l e s Of Normal Mode Problems .............................. 40 3. NUMERICAL FORMULATION OF NORMAL MODE SOLUTIONS FOR SIMPLE EARTH MODELS 44 3.1 Vari a t i o n a l Type Finite-element Algorithm With Cubic Hermite Polynomial Basis Functions, 44 i v 3.2 One-dimensional Vari a t i o n a l Solution Of Torsional O s c i l l a t i o n 50 3.3 Three Dimensional Solution For Non-rotating, Non-gravitating, Spherical Simple Earth Model ..... 56 3.4 Effects Of Rotation On The Normal Modes Of Se l f - g r a v i t a t i n g Spherical Earth Model ............ 61 3.5 Ef f e c t s Of Simple Lateral Heterogeneities ......... 65 4. ELLIPSOIDAL EARTH MODEL ................................ 68 4.1 Recent Developments In The Research On The Ef f e c t s Of E l l i p t i c ! t y Of The Earth On The Normal Modes 68 4.2 E l l i p s o i d a l Coordinate System ..................... 70 4.3 Lagrangian Energy Integral For The Oblate Spheroidal Coordinate System ...................... 73 5. NUMERICAL RESULTS .81 5.1 Normal Modes Of A Homogeneous E l a s t i c Sphere With Simple Heterogeneities ....................... 84 5.2 Normal Modes Of Gravitating, Rotating And La t e r a l l y Heterogeneous Sphere ....................104 5.3 E l l i p s o i d a l Earth Model ........108 5.4 Remarks On Computational Error .................... 109 6. CONCLUSIONS ............................................111 BIBLIOGRAPHY AND REFERENCES ....115 APPENDIX A ... .................120 APPENDIX B .124 V LIST OF FIGURES Figure 1. Spherical Coordinate System ....................21 Figure 2. Cubic Hermite Basis Function In ft Normalized Interval Of (r^ * T^ t) • •••••• 49 Figure 3, Cubic Hermite Basis Functions Overlapping In An Interval (rh_, ,r f e + () ....................... 52 Figure 4. The Boundary Condition Hatching In The Energy Matrices C v ] And C T J .................... 52 Figure 5. The Configuration Of Elements With Node Numbers In The Spherical Coordinate System ..... 65 Figure 6, E l l i p s o i d a l Coordinate System .................. 70 Figure 7. Whole Earth Energy Matrix Assembly ............. 83 Figure 8. Displacement Fields Of Normal Modes Of 0T° And 0Sa° For A Sphere ....................... 85 Figure 9. Radial Surface Deformation Fields Of aS* .......92 Figure 10. Displacement Fields U^ Of Mode 0S^ ............. 94 Figure 11. $6 coordinate i n t e r p o l a t i o n scheme for m d i f f e r e n t m of ^ra ............................. 97 Figure 12. Spectral s p l i t t i n g of a S* mode for the models A1, A3, A1A7 and HEMI ................... 101 Figure 13. Spectral s p l i t t i n g of gS™ mode for the models of A1, A3, A1A7 and HEMI ......102 Figure 14, Spectral s p l i t t i n g of mode for the models AT, A3, A1A7 and HEMI 103 Figure 15. Spectral s p l i t t i n g of 0T? mode for the models GO, GW and GWA1 106 Figure 16. Spectral s p l i t t i n g of „S~ mode for the models GO, GW and GWA1 .........................107 v i i LIST OF TABLES Table 1. Theoretical A P r i o r i Error Bounds For The Linear, Cubic Splines And Cubic Hermite Interpolations In One-dimensional Problem ...... 46 Table 2. Comparision Of The Torsional O s c i l l a t i o n Frequencies Found From The One-dimensional V a r i a t i o n a l Solutions And Wiggins(1968) Type Solution .................................. 55 Table 3. Normalised Dimensionless Normal Mode Frequencies For The Models 0,A1, A3, A1A7 And HEMI 99 Table 4. Normalised Dimensionless Norioal Mode Frequencies For The Models 0, GO, GW And GWA1 ....................................105 v i i i ACKNOWLEDGEMENTS I wish to thank my supervisor Dr. R.A.Wiggins f o r suggesting the problem and providing stimulating and e f f i c i e n t guidance whenever required. I thank Prof. G.D.Garland for his encouragements i n the early stages of t h i s research and Dr.R.M.Clowes and Prof. R.D.Russell who provided much help f u l assistance i n the l a t e r stages. I also wish to thank Dr. G.K.C.Clarke f o r reading many chapters of t h i s t h e s i s with encouragement and constructive c r i t i c i s m s . I wish to acknowledge Dr. R.Mathon of the University of Toronto who provided the FORTRAN coding of the QZ-algorithm and Ms. Carol Bird of the computing center at the University of B r i t i s h Columbia who provided help on many occasions. Grateful thanks also go to my fellow graduate students f o r t h e i r good companionships, p a r t i c u l a r l y to Paul G.Somerville and Robert W. Clayton* and f i n a l l y to my wife for her i n f i n i t e patience and love. This research i s supported by NRC operating grants A-8854 and A -7707. 1 CHAPTER 1 INTRODUCTION I n g e n e r a l t h e r e a r e two methods o f d e a l i n g w i t h normal mode d i f f e r e n t i a l e q u a t i o n s o f a dynamic system. I n t h e f i r s t and more o r d i n a r y one, t h e d i f f e r e n t i a l e q u a t i o n s a r e s o l v e d by assuming a m a t h e m a t i c a l l y r e a s o n a b l e s o l u t i o n f o r t h e g i v e n c o o r d i n a t e system and matching t h e boundary c o n d i t i o n s t o s o l v e f o r t h e undetermined c o e f f i c i e n t s . Lamb (1882), and s u b s e q u e n t l y L o v e ( 1 9 1 1 ) , T a k e u c h i (1950) , A l t e r m a n e t a l . (1958) and Backus and G i l b e r t (1967) s o l v e d t h e normal mode problems o f E a r t h by t h i s method. I n t h e second method, which was i n v e n t e d by Lagrange (1736 -1813), and d e v e l o p e d and a p p l i e d by Gauss, Green, and o t h e r s , th e e q u a t i o n s a r e e x p r e s s e d as t h e c o n d i t i o n s t h a t a c e r t a i n q u a d r a t i c f u n c t i o n o f t h e d i f f e r e n t i a l c o e f f i c i e n t s of t h e f i r s t o r d e r , i n t e g r a t e d o v e r the s y s t e m , s h a l l r e t a i n a s t a t i o n a r y v a l u e when s m a l l v a r i a t i o n s a r e imposed on the v a r i a b l e s . I n a d y n a m i c a l s y s t e m , t h i s f u n c t i o n i s t h e k i n e t i c and p o t e n t i a l energy ( g r a v i t a t i o n a l , e l a s t i c , r o t a t i o n a l e t c . ) p e r u n i t volume. I t s i n t e g r a l i s t h e t o t a l energy o f t h e system. When i t s s t a t i o n a r y v a l u e i s known, t h e e q u a t i o n s o f t h e system a re o b t a i n e d a t once by a p p l i c a t i o n o f t h e v a r i a t i o n s . The second p r o c e d u r e i s t h e r e f o r e an easy and s t r a i g h t f o r w a r d one, and has been a p p l i e d t o normal mode problems f o r the E a r t h by R a y l e i g h (1877), S t o n l e y (1926), J e f f r e y s (1935), Backus and G i l b e r t (1967) and Wiggins (1973,1976 ( i n p r e s s ) ) . T h e s c o p e oi this t h e s i s i s p r i n c i p a l l y -.• c o n c e r n e d with t h e 2 numerical v a r i a t i o n a l solution for normal modes of the most general and simple Earth model. Since Lamb(1882) solved the normal mode problem of an e l a s t i c sphere and Benioff(1954) f i r s t suggested the free o s c i l l a t i o n of the Earth from his observation of the Khamchatka earthquake of 4 Nov. 1952, there have been numerous t h e o r e t i c a l solutions of d i f f e r e n t Earth models to interpret exactly the observations of normal modes of Earth. 1.1 Review Of Previous works Lamb (1882) was the f i r s t to examine t h e o r e t i c a l l y the nature of the fundamental modes of vibration of an e l a s t i c sphere by solving the equation of motion with surface boundary conditions. Because of the o r i g i n a l i t y and completeness of h i s paper, i t i s s t i l l referenced in recent text books. He showed how spheroidal and t o r o i d a l o s c i l l a t i o n s can be solved and estimated the gravest fundamental mode period of the Earth as 1hr. 18m. Love (1911) considered a compressible and s e l f - g r a v i t a t i n g Earth and showed that gravity not only acts as a restoring force to shorten the eigenperiods but also causes a large i n i t i a l state of stress throughout the Earth, Hoskin(1920) extended Love*s approach to a r a d i a l l y heterogeneous Earth, taking the e l a s t i c parameters to be simple algebraic functions of radius. Sezawa(1927) considered the propagation of Rayleigh waves on spherical surfaces, which correspond to the high order spheroidal o s c i l l a t i o n s . •, Takeuchi (1950) solved the equation of motion to obtain 3 t i d a l deformations f o r the two Bullen's Earth models{1936,1940). This was the f i r s t time ever an Earth model was used to calcul a t e and to predict a geophysical phenomenon. His computation was performed with a desk calculator and the results were compatible with the observations . Benioff (1954) f i r s t reported the actual observation of the free o s c i l l a t i o n of the Earth after Khamchatka earthquake of 4 Nov. 1952. Following t h i s f i r s t observation, Alterman et al.(1958) made a most comprehensive study of free o s c i l l a t i o n s and earth tides for three Earth models: Bullen's model B and Bullard's models I and I I . , Their r e s u l t s , computed using the f i r s t generation of el e c t r o n i c computers, were i n good agreement with Benioff's observations within the observational error. The normal mode periods of order up to 4 and the Love numbers h, k and are also calculated for various depths of earthguake sources for the Earth models. The possible effect of l a t e r a l heterogeneity of the mantle and the problem of core o s c i l l a t i o n s were also mentioned i n th e i r report. G i l b e r t and MacDonald (1960) considered two Gutenberg and Jeffreys-Bullen Earth models and calculated the free periods of to r o i d a l o s c i l l a t i o n s for various types of earthquake sources. They used an extended method of the Thomson-Haskel matrix approach. They also considered the correspondence between free o s c i l l a t i o n and ray theory. In 1961 Benioff, Press and Mith v e r i f i e d t h e i r previous report of observing the free o s c i l l a t i o n s of the Earth and also reported the s p l i t t i n g of spectral peaks from th e i r analysis of n the s t r a i n and pendulum seismograph recordings made i n C a l i f o r n i a and Peru from the Great Chilean earthquake of 1960 May 22 . Backus and Gi l b e r t (1961) and Pekeris et a l . (1961) simultaneously considered the ef f e c t of Earth's rotation on the normal modes of the Earth, after Benioff et a l . (1961) reported th e i r observation of spectral doublets. Both these authors applied perturbation theory on the assumption that the rotation of the Earth i s much smaller than the degenerate normal mode angular frequencies and they showed that the degenerate normal mode frequency i n the absence of rotation was resolved by small r o t a t i o n a l v e l o c i t i e s into (2JJ+1) modes where J( i s the angular order. Since the longest seismic o s c i l l a t i o n of the Earth has a period of less than an hour th i s theory i s applicable to a l l orders of free o s c i l l a t i o n s . Even though the observed spectral s p l i t t i n g of normal modes dif f e r e d s l i g h t l y depending on the seismograms used, both Backus and Gi l b e r t (1961) and Pekeris et a l . {1961) obtained exactly the same th e o r e t i c a l r e s u l t s . They also estimated the r e l a t i v e amplitudes of the m=(2#+1) modes for i l = 2 and | = 3 to compare t h e i r r e s u l t s with the observations. MacDonald and Ness (1961) reviewed the published normal mode observations and considered the e f f e c t s of rotation by using a perturbation method. They also concluded that rotation removes the degeneracy and r e s u l t s i n a s p l i t t i n g of spectral peaks of order I into (2JJ + 1) multiplets. Furthermore they predicted that rotation couples the t o r o i d a l and spheroidal o s c i l l a t i o n s . Sato and Usami(1962) numerically solved the c h a r a c t e r i s t i c equation for the o s c i l l a t i o n of a homogeneous e l a s t i c sphere as 5 a b a s i c data f o r t h e i r l a t e r s t u d y o f l a t e r a l l y h eterogenous and s p h e r o i d a l E a r t h models. J o b e r t (1962) c o n s i d e r e d t h e o r e t i c a l seismograms f o r t o r s i o n a l o s c i l l a t i o n s u s i n g Gutenberg's c o n t i n e n t a l and Dorman's 8009 o c e a n i c E a r t h models. Usami and S a t o (1962) c o n s i d e r e d t o r s i o n a l f r e e o s c i l l a t i o n s of a homogeneous e l a s t i c s p h e r o i d . They c o n c l u d e d t h a t e l l i p t i c i t y c a uses t h e mode t o s p l i t i n t o (£+1) m u l t i p l e t s , w h i l e t h e E a r t h ' s r o t a t i o n causes i t t o s p l i t i n t o (2J1 + 1) m u l t i p l e t s . A lsop(1963) c a l c u l a t e d the p e r i o d s o f s p h e r o i d a l o s c i l l a t i o n s n u m e r i c a l l y f o r f o u r E a r t h models and a l s o c o n s i d e r e d the phase and group v e l o c i t i e s f o r the f i r s t two s h e a r modes. Usami, S a t o , and Landisman (1965) and Landisman e t a l . (1970) c o n s i d e r e d the t h e o r e t i c a l seismograms f o r both s i m p l i f i e d and r e a l i s t i c E a r t h models and examined t h e r e l a t i o n s between f r e e o s c i l l a t i o n s and t r a v e l l i n g waves. They a l s o s t u d i e d the e g u i v a l e n c e of mantle and c r u s t a l R a y l e i g h waves t o t h e s p h e r o i d a l f r e e o s c i l l a t i o n s and t h e e g u i v a l e n c e o f G-waves and Love waves t o t o r o i d a l f r e e o s c i l l a t i o n s . S a i t o ( 1 9 6 7 ) c o n s i d e r e d t h e r a d i a t i o n p a t t e r n s of s u r f a c e waves and f r e e o s c i l l a t i o n s f o r v e r t i c a l l y heterogeneous e l a s t i c media w i t h a r b i t r a r y s o u r c e s . Backus and G i l b e r t (1967) d e s c r i b e d a g e o p h y s i c a l i n v e r s i o n t e c h n i q u e f o r e x p l o r i n g the c o l l e c t i o n of a l l E a r t h models which f i t g i v e n g r o s s E a r t h d a t a , They a p p l i e d t h e t h e o r y t o t h e normal modes of t h e e l a s t i c - g r a v i t a t i o n a l o s c i l l a t i o n o f t h e 6 Earth by u t i l i z i n g Rayleigh's v a r i a t i o n a l p r i n c i p l e . This important work not only opened a new f i e l d of geophysical inversion, but also became a stepping stone for the l a t e r studies of normal modes by Dahlen(1968,1972 etc.) and Luh(1973). Wiggins (1968) considered Birch's Earth model and computed t e r r e s t r i a l v a r i a t i o n a l tables of normal mode periods, r o t a t i o n a l s p l i t t i n g parameters and energy functions. He also computed the overtone eigenfunctions f o r several orders to make the table more complete. Dahlen (1968) considered the normal modes of a slowly rot a t i n g e l l i p t i c a l Earth. Using Rayleigh's v a r i a t i o n a l p r i n c i p l e (Backus and Gil b e r t (1967)) he computed approximate normal mode eigenfreguencies for slowly rotating and s l i g h t l y aspherical and anisotropic Earth models. He found that f o r an ar b i t r a r y spheroidal or t o r o i d a l multiplet, the ce n t r a l member (m=0) of the multiplet i s s h i f t e d s l i g h t l y i n frequency and that the other members of the multiplet are s p l i t apart asymmetrically by the eff e c t s of the Earth's rotation and e l l i p t i c i t y . , He also reported that f o r the angular order numbers of 10<^<25 the fundamental spheroidal eigenfreguency i s very nearly equal-to the fundamental t o r o i d a l eigenfreguency and i t would be necessary or at least more convenient to use a quasi-degenerate perturbation theory to examine the ef f e c t s of rota t i o n and e l l i p t i c i t y on these multiplets. Later(1969) he examined more cl o s e l y these modes (10<i<25) and aS, and fS 3 . He found that some of the normal modes of a slowly r o t a t i n g , s l i g h t l y e l l i p t i c a l Earth model may consist of about half spheroidal type motion and half t o r o i d a l type motion. In 7 conclusion, he conveniently divided the normal modes into two categories. The f i r s t category included a l l those normal modes for which i t i s s u f f i c i e n t to neglect the e f f e c t of any l a t e r a l heterogeneity i n the Earth with respect to e l l i p t i c i t y . Every normal mode in t h i s category can be characterized by a single value of i . The second category includes a l l those normal modes for which i t i s not possible to neglect the l a t e r a l heterogeneity. At l e a s t a l l the fundamental normal mode multiplets with angular order greater than 4=22 are included i n t h i s category. He said i t i s not possible to characterize normal modes i n t h i s second class by a single value of m. Dahlen did a very thorough job on the perturbation solution f o r the e f f e c t s of the rotation and e l l i p t i c i t y of Earth. Zharkov and Lyubimov(1970a) used the perturbation method to construct a theory of t o r s i o n a l o s c i l l a t i o n s of the Earth i n a weakly s p h e r i c a l l y asymmetric Earth model. They expanded the small parameter increments and (shear modulus and density) in terms of spherical harmonics and determined the dependence of the v a r i a t i o n of 4tf on 5M and oj*. , They estimated AH without using an e l e c t r o n i c computer. Later(1970b) they applied the same technique to develop a theory of spheroidal vibrations of the Earth f o r a weakly spherically asymmetric Earth model and concluded, as Dahlen (1968) did, that the spreading of degenerate frequencies caused by the Earth's rotation i s of order 1/^  and decreases rapidly with increasing X • Eor large H the spreading i s determined mostly by the l a t e r a l heterogeneity of Earth. Madariaga(1972) considered the eff e c t s of l a t e r a l 8 heterogeneities on the t o r s i o n a l free o s c i l l a t i o n s by means of the theory of perturbation of degenerate eigenfreguencies of a s p h e r i c a l l y averaged Earth. Like Zharkov and Lyubimov(1970a) he expanded the l a t e r a l heterogeneities i n a series of spherical harmonics and deduced the selection rules by which the expansions of l a t e r a l heterogeneity s e l e c t i v e l y couples the eigenf unctions. Later, Hadariaga and .Ski (1972) applied the above solution to actual Earth models with continent-ocean and plate tectonic heterogeneity. They obtained multiplet widths l e s s than 1% f o r £ =4 0. This i s much less than some observed multiplet widths of 1.7$ of the degenerate freguency(Smith(1966) ). They also discussed some of the d i f f i c u l t i e s i n the inversion of the s p l i t spectral peaks. Recently Luh(1973) applied Rayleigh's v a r i a t i o n a l p r i n c i p l e to calculate the f i r s t order corrections to the eigenfreguencies of a s p h e r i c a l l y symmetric, non-rotating and e l a s t i c Earth model due to the l a t e r a l variations i n the r e a l Earth. His work i s an extension of Madariaga*s (1972) re s u l t to the remaining problem of guasi-degeneracy of the normal modes for a l a t e r a l l y heterogeneous Earth. Later Luh{1974) extended his research to a rotating l a t e r a l l y heterogeneous Earth model and obtained somewhat d i f f e r e n t computed values from those of Hadariaga(1972). In a q u a l i t a t i v e nature t h e i r r e s u l t s agreed. He concluded that, when Jjj i s small, second order r o t a t i o n a l corrections may become comparable to f i r s t - o r d e r corrections due to the l a t e r a l perturbations and, as higher-order corrections are too formidable to solve, he recommended a v a r i a t i o n a l method. 9 Wiggins(1973,1976) applied an one dimensional f i n i t e element type Bayleigh-Eitz method with cubic Hermite polynomials to develop an algorithm to solve f o r normal mode periods, p a r t i a l derivatives and r o t a t i o n a l s p l i t t i n g parameters. However he did not consider the e f f e c t s of e l l i p t i c i t y nor l a t e r a l heterogeneity. Buland and Gil b e r t (1975, personal communication) have reported they are applying a v a r i a t i o n a l method (1-D f i n i t e element method) to solve higher order normal mode problems. Smylie(1975, personal communication) i s also considering t h i s v a r i a t i o n a l method to solve core o s c i l l a t i o n problem for rotating r a d i a l l y heterogeneous Earth models. Dahlen (1976) studied the ef f e c t s of l a t e r a l heterogeneity, r o t a t i o n , and hydrostatic e l l i p t i c i t y of the Earth on the normal modes. Even though he expressed great uncertainties due to the treatement of the hydrostatic e l l i p t i c i t y , he concluded that there must exis t s i g n i f i c a n t l a t e r a l heterogeneities i n the Earth's lower mantle and that the l a t e r a l heterogeneity i n the upper mantle associated with the differences i n structure beneath oceans and continents must persist to depths of at least a few hundred kilometers. 10 1.2 Review Of Perturbation Methods For The Effects Of Rotation, E l l i p t i c i t y , And Lateral Heterogeneity. The normal mode problem can be solved by the separation of the-equations of motion, i f the material properties and parameters vary only with radius, and retain the o r i g i n a l s p h e r i c a l symmetry. Boundary conditions have to be s a t i s f i e d at a l l i n t e r n a l boundaries as well as at the free surface boundary. However, i f the material properties and parameters vary i n such a way that the spherical symmetry of the system breaks down, the conventional method of solution cannot be applied to the system. The variations of material properties and parameters i n the Earth are small and the c l a s s i c a l perturbations have been applied to f i n d the e f f e c t s of these s p h e r i c a l l y unsymmetric material properties on the normal mode eigenfunctions of a sp h e r i c a l l y symmetric Earth model. As b r i e f l y surveyed in the previous section, Backus and Gil b e r t (1961) and Pekeris e t . a l . (1961) solved f o r the e f f e c t of rotation on the normal mode periods of Earth by a perturbation method assuming that the Earth's rotation i s much smaller than the unperturbed normal mode frequency of Earth. They expanded the perturbed normal mode frequency by where U^J*- i s the frequency of the unperturbed mode, SL i s the angular velocity of rotation of Earth, and KLdx i s the non-degenerate perturbed normal mode frequency. They calculated the ef f e c t s of rotation correct to second order in JI and derived the s p l i t t i n g parameters correct to zercth order i n (^). 11 Dahlen (1968) applied the perturbation of parameters i n Rayleigh's v a r i a t i o n a l p r i n c i p l e to investigate a rotating e l l i p s o i d a l Earth model. He obtained angular frequency multiplets of rotating e l l i p s o i d a l Earth model, correct to f i r s t order in £{y) , the e l l i p t i c i t y of Earth, and correct to second order in ; - ^ T ==. \ + + + where the superscript r indicates a parameter depending only on the rotation SL while the superscript e indicates a parameter depending on the e l l i p t i c i t y . For „ ^ , „"(i » v i P i £ » and , see Dahlen(1 968, Appendices a and B). Backus and G i l b e r t (1961) and Dahlen (1968) both concluded that for the lower order fundamental normal modes, at le a s t $<25, the Earth's rotation and e l l i p t i c i t y are the~dominant perturbing e f f e c t s . . They showed that the degeneracy of any multiplet „S A or „T A i s in general completely removed f o r the rotating e l l i p t i c Earth. They also showed that the f i r s t order e f f e c t of e l l i p t i c i t y and the second order e f f e c t of rotation not only act to s h i f t the entire multiplet but also cause the s p l i t t i n g of a multiplet.to be asymmetrical. Zharkov and Lyubimov (1970a,b) , Madariaga{1972) and Madariaga and Aki (1972) applied the c l a s s i c a l perturbation theory to study the effects of l a t e r a l inhomogeneity on the 12 normal mode solutions of Earth. They expanded the perturbation of the l a t e r a l heterogeneity in spherical harmonics; bo s •5-1 e— s and used the Wigner-Eckart theorem to simplify the complicated i n t e g r a l s and to compute the coupling c o e f f i c i e n t s between the normal modes and the expansion of the l a t e r a l heterogeneities. They showed that the expansion of the l a t e r a l heterogeneities i n spher i c a l harmonics permitted the separation of the r a d i a l and angular i n t e g r a l s of the scattering c o e f f i c i e n t s . From the angular integrals^ they derived a selection rule which controls the coupling between the degenerate eigenfunctions and the l a t e r a l heterogeneities. Following Dahlen (1 968,1969) , Luh(1973) expanded the l a t e r a l heterogeneity as 55. ( r . Q. f > ) = § 5 ^ M ( r ) YLM where ^ i s J, K, and for j=1, ... #4. He then substituted t h i s parameter perturbation i n t o the f i r s t order perturbation eguation to solve for the e f f e c t s of l a t e r a l heterogeneity. He used the Wigner-Eckart theorem to simplify many complicated angular integrals involving the s t r a i n deviator tensor to more basic and manageable in t e g r a l s . Later Luh(1974) included rotation and e l l i p t i c i t y to obtain the normal mode 13 solutions of a more general Earth model. For the case where the e l l i p t i c i t y f(r) i s caused by Earth's rotation and the Earth i s i n hydrostatic equilibrium, as Dahlen(1968) assumed, the perturbation i s represented by The e l l i p t i c i t y p e r t u r b a t i o n changes the l o c a t i o n s of i n t e r n a l jump d i s c o n t i n u i t i e s i n (j , «r ,J* ). So a s p h e r i c a l s u r f a c e a t r=a becomes a s u r f a c e of an e l l i p s o i d with I He then d i v i d e d the Earth's s u r f a c e i n t o d i f f e r e n t subareas, to * « s r e p r e s e n t l a t e r a l inhomogeneities, beneath which ( J , £ ) are s f u n c t i o n s of r a d i u s o n l y . A s u r f a c e f u n c t i o n f i s i n t r o d u c e d t o d e s c r i b e each sub-area, * , , ( l OM the S'th ^ui>are<i ' I O otherwise ) LM where ^ and ^ denotes j , fC and of the r e g i o n and of the E a r t h model f o r j=1,2,3 r e s p e c t i v e l y . Then the t o t a l p e r t u r b a t i o n used by Luh(1974) i s , b e s i d e s r o t a t i o n , then 14 where Luh's Earth model HOD508 (obtained by inverting 508 gross Earth data) with the above perturbation representation can be said to resemble the r e a l Earth more closely than any other Earth models studied before. From the perturbation solution of the ordinary degenerate normal modes the following conclusions can be reached: 1) the ef f e c t of Earth's rotation i s i n s i g n i f i c a n t whenever i^^j i s small, 2) the ef f e c t of l a t e r a l heterogeneity on low n and low X i s likewise i n s i g n i f i c a n t and 3) the e l l i p t i c i t y corrections cannot be neglected at high j£ modes. For the cl o s e l y coupled modes the quasi-degenerate solutions indicate that 1) the energy of each quasi-degenerate l e v e l i s nearly evenly divided between two multiplets when t h e i r combined ordinary degenerate l e v e l s are crowded, 2) the strength of coupling i s not related to the proximity of the unperturbed frequencies, but depends greately on the s i z e of the matrix elements that render coupling, 3) f o r cl o s e l y coupled cases the differences between the ordinary degenerate perturbation s p l i t t i n g and those of guasi-degenerate l i n e s are highly s t r i k i n g as predicted, 4) the C o r i o l i s coupling of the form ^ - ^.Tj f or small n and n' (n,n'<£5) i s very strong and 5) most importantly, the usual mode designation loses i t ' s 15 meaning as well as i t ' s i d e n t i t y when the coupling i s strong. ; 1.3 Scope Of Discussion As reviewed i n the previous sections, the Earth models i n normal mode studies have been improved greatly since Lamb (1882) f i r s t calculated the free o s c i l l a t i o n periods of a homogeneous s o l i d sphere. The application of the perturbation methods during the l a s t decade also helped greatly i n developing a more r e a l i s t i c understanding of the normal mode phenomenon. In summary, the most up to date and r e a l i s t i c Earth model we have considered i n our t h e o r e t i c a l study of free o s c i l l a t i o n s i s , 1) slowly rotating (—-- <<^  1), >w Si 2) a s l i g h t l y oblate e l l i p s o i d a l (e (a) ^  1/298. 3) , 3) r a d i a l l y heterogeneous with d i s c o n t i n u i t i e s i n J 5 , K a n d a t arbi t r a r y depths (including l i q u i d and s o l i d cores) g • • 4) s l i g h t l y l a t e r a l l y heterogeneous at the Earth's surface ( continent and ocean l a t e r a l heterogeneity and plate tectonic l a t e r a l heterogeneity up to depths of 400km) , 5) and s l i g h t l y l a t e r a l l y heterogeneous at the core-mantle boundary. If the normal mode eigenfrequencies and eigenfunctions f o r the above Earth model are calculated by parameter perturbation technique using Eayleigh's v a r i a t i o n a l p r i n c i p l e (Luh (1974)) , the following results can be summarized: 1) For the case of a s p h e r i c a l , non-rotating, e l a s t i c and i s o t r o p i c Earth model, the (2^+1) spheroidal modes 16 have a degenerate frequency of A while the (2 +1) t o r o i d a l modes t T * have the degenerate T f reguency of . 2) The degeneracy of any multiplet „ S e or T^?. i s completely removed by the perturbing e f f e c t s of rotation, e l l i p t i c i t y or l a t e r a l heterogeneities. 3) For the range of angular order number from about £-10 to about X=25 , the fundamental spheroidal eigenfreguency 0w/ i s very nearly equal to the fundamental t o r o i d a l eigenfreguency 0WA. In such cases, the ordinary degenerate perturbation method i s not appropriate and i t becomes necessary to use a guasi-deqenerate perturbation theory to examine the ef f e c t s of rotation, e l l i p t i c i t y and l a t e r a l heterogeneity on the multiplets. U) For the lower order (small I ) fundamental modes, the Earth's rotation and e l l i p t i c i t y are the dominant perturbing e f f e c t s . Every normal mode in t h i s category can be characterized by a simple m; to zeroth order every such mode i s either spheroidal .S^" or t o r o i d a l or mixed ^ S ^ - j J , or „ s / - v S 4 ~ , depending on the quasi-degeneracy. 5) For the higher order normal modes, (angular order greater than about ^=22) the effects of l a t e r a l heterogeneity i s the most dominant perturbation. In general, i t i s not possible to characterize the normal modes by a single value of m in t h i s case, sever, the following d i f f i c u l t i e s and l i m i t a t i o n s are 17 also encountered: 1) Hhen two multiplets nearly coincide in frequency, ordinary degenerate perturbation methods must be replaced by f i r s t order quasi-degenerate perturbation methods, introducing mathematical and computational complications. 2) When the eff e c t s of rot a t i o n becomes dominant, the f i r s t order perturbation calculations become i n s u f f i c i e n t and the higher order terms must be considered. This would introduce a great deal of horrendous algebraic manipulations and Luh (1974) recommended the use of a v a r i a t i o n a l method for such 3) If the condition =^ = << 1.0 cannot be s a t i s f i e d as in the study of core o s c i l l a t i o n s , where vtOje i s comparable to (Crossley (1975) ) , the above perturbation formalism f a i l s . 4) The sel e c t i o n rule (Zharkov and Lyubimov(1970a,b) , Hadariaga(1972) and Luh (1973)) indicates that only the even angular orders of the l a t e r a l heterogeneity perturbation contribute to the s p l i t t i n g within a multiplet. The effects of the odd angular orders i n the-perturbation are not completely understood either t h e o r e t i c a l l y or observationally. 5) fts the overtone number n of a mode increases, the r a d i a l eigenfunction penetrates deeper into the Earth's i n t e r i o r and deep l a t e r a l heterogeneities become important. Evidence of the l a t e r a l 18 heterogeneities below the upper mantle are reported by Davies et a l . (1972) , Kanasewich et a l . (1973) and Jordan et a l . (1974). The l a t e r a l heterogeneities reported are rather arb i t r a r y with respect to the Earth's surface l a t e r a l heterogeneity. Therefore, the l a t e r a l heterogeneity assumed at the core-mantle boundary, as considered by Luh(1974), i s not most appropriate because he r e s t r i c t e d the l a t e r a l heterogeneities at the core-mantle boundary to be a projection of the free surface l a t e r a l heterogeneities, Most recently Dahlen (1976) reported from his research that there exists a s i g n i f i c a n t l a t e r a l heterogeneity in the Earth's lower mantle and that the Earth's surface l a t e r a l heterogeneities must extend at least a few hundred kilometers. He reached these two conclusions because his most up-to-date perturbation scheme, including the e f f e c t s of Earth's surface l a t e r a l heterogeneity, rotation and the hydrostatic e l l i p t i c i t y , could not interprete the observed normal mode data very s a t i s f a c t o r i l y . 6) The usual mode designation and i t ' s i d e n t i t y lose t h e i r meaning for the cases of strongly coupled modes, and the interpretation of the normal modes by the modes of the sphe r i c a l l y summetric, non-rotating, i s o t r o p i c Earth model i s not proper. These d i f f i c u l t i e s and l i m i t a t i o n s are, i n general, due to complicated mathematical and computational reasons, and also to the assumptions made for the application of the perturbation 19 schema. In the following, a new approach to solving the normal mode problem i s proposed to overcome -at least i n theory- a l l the above mentioned d i f f i c u l t i e s and to develop a t r u l y general Earth model; This new v a r i a t i o n a l type f i n i t e element method has i t s own d i f f i c u l t i e s (as w i l l be explained i n Chapter 3 and Chapter 6) but i t i s p o t e n t i a l l y more powerful and has fewer l i m i t a t i o n s than the above surveyed perturbation methods. In chapter 2 , after a b r i e f review of the t h e o r e t i c a l solution of normal modes, the one dimensional v a r i a t i o n a l Lagrangian i n t e g r a l form of the eguations of motion i s derived. Then the derivation i s extended to a three dimensional v a r i a t i o n a l Lagrangian in t e g r a l in a spherical coordinate system. In Chapter 3, the Lagrangian i n t e g r a l s obtained i n Chapter 2 are transformed into matrix forms f o r the numerical computations i n Chapter 5. Chapter 4 explains how the e l l i p s o i d a l Earth may be treated i n the v a r i a t i o n a l solution. Computational r e s u l t s and error analyses are included i n Chapter 5 , and the conclusions are presented in Chapter 6. 20 CHAPTER 2 VARIATIONAL FORMULATION OF EQUATION OF MOTION In t h i s chapter, 1) the c l a s s i c a l normal mode period equations, which w i l l be used for test purposes, 2) the Lagrangian inte g r a l s for the one dimensional case and 3) the extension to a general three dimensional case are derived f o r the numerical c a l c u l a t i o n s i n the next chapter. The Earth model assumed i s spherical,; perfectly e l a s t i c , s e l f - g r a v i t a t i n g and r a d i a l l y heterogeneous. The convenient extensions for l a t e r a l heterogenieties and rotation are shown. 2.1 Generalized Equation Of Motion. Let the Earth model consists of a s e l f - g r a v i t a t i n g , p e r f e c t l y e l a s t i c continuum occupying an arbi t r a r y bounded volume V with surface ^V. Assume further that i t has a steady angular velocity of rotation -T2_ about i t s centre of mass, that jV(r) denotes the density, T„(v) denotes i n i t i a l s t a t i c stress f i e l d , and ^ e(r) i s the g r a v i t a t i o n a l potential of the body occupying the volume V. If we assume a small time-dependent p a r t i c l e displacement f i e l d u(r,t) for a small disturbance, t h i s displacement w i l l change both the volume V(t) and the surface ^V(t) of the given continuum and w i l l be accompanied by perturbations j'){r,t) i n the density, ^ ) ( r , t ) i n the g r a v i t a t i o n a l p o t e n t i a l , and T e ( r , t ) i n the stress tensor at a f i x e d location. Figure 1. shows the notations and the Earth model in s p h e r i c a l coordinate system. Figure 1 Spherical Coordinate System with variables used in this thesis. 22 The usual lagrangian convention i s to l a b e l a material p a r t i c l e by i t s location r at time t=0. Then Te (r, t)+T E (r,t) i s the value of the stress tensor at r i n space at time t . , To f i r s t - o r d e r in u, the incremental stress T(r,t) i s T = TE + l i - v T o ^ and from c o n s t i t u t i v e r e l a t i o n s where [J^MJL i s the l i n e a r isentropic e l a s t i c c o e f f i c i e n t (fourth-order tensor) . In general CAJUA which i s the l i n e a r e l a s t i c c o e f f i c i e n t tensor with twenty one independent components, i . e . , Gy** = ( * - | > J & * + / * ( • + *) (o where X ( r ) i s i s e n t r o p i c bulk modulus and yM(r) i s the is e n t r o p i c r i g i d i t y , then, TA.- i s and for the s p e c i a l case of T 0 (r) being a purely hydrostatic stress To O) ==. - PoCrj l 0 (*) and 23 Then the most general and exact equations governing the motion about the equilibrium configuration are (after Dahlen (1972)) v.(T + T0 -Cf.*fOv(_€.*-«p+i;) + f W where f (r,t) i s an external body force per unit volume, - j ^ - i s the Lagrangian time derivative, and In (7) the equilibrium condition for the i n i t i a l steady state i s C<0 and the r o t a t i o n a l potential due to the c e n t r i p e t a l acceleration i s For the given eguations of motion (7) and (8), i f we neglect the second order terms and subtract equation (9) from equation (7), we obtain 24 Now from the l i n e a r i z e d continuity equation we have These r e l a t i o n s w i l l l a t e r ce extended furthur to derive the Lagrangian inte g r a l s for the normal modes of the Earth. 2.2 Boundary Conditions Biot(1965) showed how the v a r i a t i o n a l p r i n c i p l e and the boundary conditions may be deduced from general s t r a i n energy considerations. The approach shown below i s s i m i l a r to his derivation, but the general outline follows the approach by Dahlen (1972) . Consider an a r b i t r a r y small simply connected surface element dA centered on an a r b i t r a r y material p a r t i c l e r located on the undeformed surface <>V at time t=0. Let the unit outward normal to dA be "h (r) and denote the corresponding unit outward -A normal to the deformed surface <?V(t) by -n (r,t) on dA(t). To f i r s t - o r d e r i n u, the n(r,t)dA(t) on the deformed surface (t) i s related to n(r)dA on the corresponding undeformed surface £V by the r e l a t i o n (Biot(1965), Dahlen (1972)) , 25 0S) The d i l a t i o n terra in (15) arises from the stretching of the surface element dA (t) and the other term arises from the deflection of the normal n ( r , t ) . Then, to f i r s t order, T^ - i n eguation (4 ) w i l l become and, i n the s p e c i a l case of a hydrostatic i n i t i a l stress T(r)= -p o(r)I, The eguation of motion (14) w i l l be, from (17) and (1) , And since dA i s an a r b i t r a r y element on the undeformed surface >V, and since n(r)»T(r) i s already continuous, the l i n e a r i z e d surface t r a c t i o n continuity i s equivalent to n(r)»T(r,t) being continuous at a l l points r of 3>V. Then we have the boundary conditions at a l l points r of the undeformed boundary surface av, and of any i n t e r n a l material discontinuity surfaces, i . e . , 26 1) u (r,t) continuous v J A y\ ->- . 2) n(r).T(r>t) continuous and n (r) • T (r ,t) =0 on 9-1 (?<0 3) (r,t) continuous (*0 4) n (r)- <7^t(r,t) + U^G {„ (r) n (r). u (r,t) continuous (zO These are the very general boundary conditions for the s o l i d continuum case., I f there i s a l i q u i d - s o l i d boundary the boundary condition (19) of u(r,t) i s not necessarily continuous. y< — Instead only n-u(r,t) i s continuous and the shear stress vanishes across the boundary. Also, i n the Alterman et a l . (1959) and Wiggins(1968) type s o l u t i o n , the re g u l a r i t y of the integrated displacement solution becomes an important boundary condition at the centre of the Earth. 2.3 C l a s s i c a l Solution Of Normal Mode Problem Of Earth. From the very general equations of motion (18), we w i l l derive the solutions obtained by Alterman et a l . (1959) to test the present v a r i a t i o n a l solution. In V, i f a p a r t i c l e at a point r i n the deformed state was at r-u i n the equilibrium, the density at a point r i n the deformed state w i l l be expressed by j c r ; = M F - M J - r ° ( p ) v ^ = f c - v - (p. M ) ( * O to the f i r s t order of small quantities. Then, we have f, = - (f.-Uj (13) and i f we neglect the rotation of the Earth and the external 27 body force per unit volume f, the equation of motion (18), to f i r s t order, becomes exactly i d e n t i c a l to the ones given by aiterman et a l . (1959) and Takeuchi and Saito(1972) This can also be written as = f , V < $ ( - v (J\>u) + ( V . f l7 , V . ( J ? , V . ( S f J t For the t o r s i o n a l o s c i l l a t i o n , the assumed solution of the form Mr UG = o where Yj (&,^ >) denotes a spherical harmonics of order Jl » I t i s substituted into equation (25) and we have which can be rewritten as two f i r s t - o r d e r d i f f e r e n t i a l equations, i . e . . <*l ~~ T ^  + /* 3 28 where The equations i n (28) are solved numerically by the University of B r i t i s h Columbia Computing Center subprogram DIFSY i n Chapter 3 f o r comparison with the v a r i a t i o n a l solutions. The boundary conditions to be s a t i s f i e d i n these cases are Y= a. where r=a i s the free-surface and r=b i s the core-mantle boundary. For the non-rotating, perfectly spherical Earth model, the spheroidal o s c i l l a t i o n s are not coupled with the t o r s i o n a l o s c i l l a t i o n s and are i d e n t i f i e d with the following assumed solutions 0 ' < f J ^ i * * ( 3 0 J The spheroidal o s c i l l a t i o n s involve d i l a t i o n and cause perturbations i n the g r a v i t a t i o n a l potential c j^(r,t) , i . e . , 29 The non-vanishing d i l a t i o n requires (H) for the stress terms. The boundary condition f o r requires the continuity of at any surface of discontinuity and the sixth function i s defined as v-0*3 Then the following six f i r s t - o r d e r d i f f e r e n t i a l equations represent the spheroidally o s c i l l a t i n g normal modes when they were solved simultaneously, i . e . , il« ' S A T n > 3-/ 30 fa- + f - f r 4v 4-t 6\ f< r •91 - - U < + 0 » > I n normal mode r e s e a r c h , AIterman e t a l , (1959) and , l a t e r , many o t h e r s s o l v e d t h i s system o f f i r s t - o r d e r d i f f e r e n t i a l e q u a t i o n s o r s l i g h t l y m o d i f i e d v e r s i o n s w i t h t h e a p p r o p r i a t e boundary c o n d i t i o n s and t h e r e g u l a r i t y of t h e i n t e g r a t e d s o l u t i o n s a t the c e n t r e o f E a r t h . T h i s has been t h e most p o p u l a r method t o s o l v e t h e normal mode problem because o f i t s r e l a t i v e s i m p l i c i t y . The above system o f s i x f i r s t - o r d e r d i f f e r e n t i a l e q u a t i o n s has t o be 31 modified i n the l i q u i d outer core because of the vanishing of the shear modulus and the changes of the boundary conditions., Now i f we assume the Earth model i s made of a set of uniform layers, as we assume i n the actual numerical integration, then within each layer -^L=0 and equation (27) for t o r o i d a l o s c i l l a t i o n s reduce to which i s , in f a c t , a spherical Bessel d i f f e r e n t i a l equation with pa r t i c u l a r solutions, . J » t * 0 = ^ ^ 0 ' C*J where Then, for the given boundary condition that the shear s t r e s s vanishes at the free surface and at the core-mantle boundary, the c h a r a c t e r i s t i c equation i s obtained as which leads to the recurrence r e l a t i o n U-0 J , . , . - J^ +i O<0 = o (*D This i s a part of the c h a r a c t e r i s t i c equations solved by Sato et al.(1962) to obtain the basic data on the free o s c i l l a t i o n of a homogeneous e l a s t i c sphere. The numerical solution of t h i s eguation i s used in the next chapter f o r test purposes. 32 2.4 Lagrangian Integrals Of The One-dimensional Normal Hode Problem. Now we w i l l derive the Lagrangian inte g r a l s f o r the t o r s i o n a l and spheroidal o s c i l l a t i o n s by the inner product approach rather than the energy method. For t o r s i o n a l o s c i l l a t i o n , we have from equation (27), If we l e t xlj, be the vector space consisting of a l l twice continuously d i f f e r e n t i a b l e vector f i e l d s y (r) defined throughout the undeformed Earth model, for any two y(r) and y(r) of , an inner product on can be defined as - ' a where * denotes the complex conjugate. Now take the inner product of the equation (27) with another a r b i t r a r y vector y,(r) of the inner product space which, a f t e r a few algebraic steps, can be written 33 This i s the Lagrangian i n t e g r a l for the t o r s i o n a l o s c i l l a t i o n s solved i n Chapter 3 as a test of the method before the three dimensional formulation i s begun. From the boundary condition that shear stresses at the free surface and at the core-mantle boundary vanish, the l a s t term i n the eguation (41) i s i d e n t i c a l l y zero. The l a s t term of equation (41) i s , i n f a c t , the natural boundary condition i n the v a r i a t i o n a l solution. The numerical solution of equation (41) i s carried out by using the cubic Hermite polynomial basis as w i l l be shown f u l l y i n Chapter 3. By a s i m i l a r derivation, the Lagrangian i n t e g r a l f o r the spheroidal o s c i l l a t i o n i s 34 <K 1 +=9 •2. In the above expression, the terms with i\) are associated with the k i n e t i c energy of the system and terms without 60 are associated with potential ( e l a s t i c and gravitational) energy. 2.5 Lagrangian Integral For The Generalized Three-dimensional Problem In general there are two ways of deriving Lagrangian energy i n t e g r a l s for a given dynamic system. One way i s to s t a r t with the d e f i n i t i o n of k i n e t i c and potential energy terms of the p a r t i c l e s i n the motion. The other way i s by forming an inner 35 product of the equation of motion and an arbit r a r y solution vector defined i n the same vector or functional space as the equation of motion. we w i l l follow the inner product approach since i t i s more simple and convenient mathematically. Define the Fourier transforms of the various f i r s t order terms, u (r,t) , $ 0(r,t) , ^ (r,t) , T (r,t) and f ( r , t ) , JO LHJ>X etc. And, f o r a fixed frequency 60 , l e t Ay be the vector space consisting of a l l twice continuously d i f f e r e n t i a b l e vector f i e l d s u (r,tw ) defined throughout the undeformed Earth model. Then f o r any two u(r,w ) and u(r,«o ) of ^ , define an inner product on ^ as V The Fourier transformed l i n e a r i z e d equation of motion i s f Now by taking the inner product of equation (45) with u(r , t j ) of the inner product space yd 3 F i r s t in the l e f t hand side of equation (46), consider the evaluation of the volume i n t e g r a l defining the b i l i n e a r 36 f u n c t i o n a l , ?</{u',u) as. ( f 7 j Then an application of Gauss's theorem(Backus and G i l b e r t 1967) The po t e n t i a l energy b i l i n e a r functional becomes v 0 , / - ^ *Vx Now from Green's f i r s t i d e n t i t y the g r a v i t a t i o n a l perturbation term can be written as (*>) * -I V Now and the Fourier transformed continuity eguation and Poisson's eguation are By substituting (52) , (53) and (54) into (51) we fi n d a n o t h e r a p p l i c a t i o n o f G a u s s ' s theorem V 2>V + gives • r* u 4 1 / and 38 V \U *i [$,' ^ (s6) By adding (49) and (56) , + •r In t h i s eguation the f i r s t i n t e g r a l i s the e l a s t i c potential energy; the second and t h i r d integrals are the perturbed g r a v i t a t i o n a l potential energy and the fourth i n t e g r a l i s the i n i t i a l g r a v i t a t i o n a l energy plus the r o t a t i o n a l potential energy. The l a s t surface i n t e g r a l i s the boundary i n t e g r a l term for the surfaces of d i s c o n t i n u i t i e s . Therefore, the Lagrangian 39 i n t e g r a l of the system i s , from equations (49) and (57) V V 4v J 5v The f i r s t v a r i a t i o n of the above Laqrangian i n t e g r a l »ith respect to the complex displacement u(r) i s zero, to the f i r s t -order i n Vu (r). This i s the precisely the form of Rayleigh's v a r i a t i o n a l p r i n c i p l e . ' •-: =•* C i ; , „ i t can be seen that a l l energy integ r a l s in the Lagranqian are b i l i n e a r functionals and are Hermitian symmetric. Consequently, the 40 Lagrangian i n t e g r a l derived above i s an e l a s t i c g r a v i t a t i o n a l normal mode problem and, by application of the p r i n c i p l e of leas t action, the sum of the b i l i n e a r functionals i s i d e n t i c a l l y zero when evaluated at any u(r,t) which i s an eigenfunction of the Earth model V with eigenfreguency tO . This means that the above derived Lagrangian i n t e g r a l can be interpreted as a pr i n c i p l e of conservation of energy for the e l a s t i c -g r a v i t a t i o n a l mechanical system V for normal modes. The above Lagrangian i n t e g r a l i s solved by r e s t r i c t i n g u(r,t) to be r e a l , i n which case the C o r i o l i s term i n t e g r a l vanishes. The numerical aspects of the solution are considered in d e t a i l i n Chapter 3. 2.6 Boundary Conditions In Variational P r i n c i p l e s Of Normal Mode Problems. In general, there are two e s s e n t i a l l y d i f f e r e n t types of boundary conditions, natural and geometric, in the va r i a t i o n a l formulations of boundary value problems(Courant et a l . (1970)). For a given functional I T (or I Sp or I J t > ) , (where u and u are the f i r s t and second derivatives of u with respect to r) the variation of I j , i s . 41 St r <U4r -t-2 ? o (*<0 and, since the v a r i a t i o n ^u i s a r b i t r a r y , to s a t i s f y the condition we have the conditions. '.IE o r ( « 0 Condition equation (62) i s the Euler equation which i s s a t i s f i e d by the energy functionals of the system. Expressions (63) and (64) are c a l l e d the natural boundary conditions. The natural boundary conditions s a t i s f i e d at the free surface or at the s o l i d - l i q u i d boundary are', i n general, c a l l e d free boundary conditions. 42 For the one-dimensionalized t o r s i o n a l o s c i l l a t i o n problem, we have, from the Lagrangian i n t e g r a l (41), which i s automatically s a t i s f i e d from eguation (30) at the free surface and at the core-mantle boundary, because of the vanishing of the shear stresses at these boundries (from section 2,2 and section 2.3) Since, i n the numerical solution i n Chapter 3, the minimization of the Lagrangian automatically imposes the natural boundary conditions we do not have to d i r e c t l y impose the boundary conditions at the free surface and at the core-mantle boundary. For the generalized three-dimensional case, we have, from eguation (58) , This natural boundary condition i s s a t i s f i e d at the free surface and at the core-mantle boundary as i n the t o r s i o n a l o s c i l l a t i o n case. From eguations (20) and (22) 43 t-a.c. and the natural boundary condition ( 6 6 ) i s i d e n t i c a l l y s a t i s f i e d and zero. If these natural boundary conditions were not s a t i s f i e d we must impose the condition that the geometric boundaries s a t i s f y O g o In the process of solving the Lagrangian i n t e g r a l s , with some appropriate basis functions i n each coordinate d i r e c t i o n , we encounter i n t e r n a l boundary conditions at each nodal point and nodal surface. These i n t e r n a l boundary conditions are s a t i s f i e d by simply matching the displacements and the stress terms at a l l nodal points or nodal surfaces. This i s very straight-foward i n the f i n i t e element type solution process. 44 CHAPTER 3 NUMERICAL FORMULATION OF NORMAL MODES OF SIMPLE EARTH MODELS In the v a r i a t i o n a l type finite-element method, we assume a t r i a l solution consisting of appropriately chosen piecewise, continuously d i f f e r e n t i a b l e polynomial basis functions and minimize the Lagrangian energy functional with the proper boundary conditions. From a mathematical point of view, t h i s i s an extension of the Rayleigh-Ritz-Galerkin technigue. The basic idea of t h i s method i s an old one, f i r s t proposed by Ritz(1900-1938) in the 1930's, but the use of the piecewise basis functions i n the solution i s more recent (Birkhoff et a l . (1966)). 3.1 V a r i a t i o n a l Type Finite-element Algorithm Hith Cubic Hermite Polynomial Basis Functions. As shown i n Chapter 2, we have a Lagrangian energy i n t e g r a l I T (or I S f or I, p) from eguations (41), (42) and (58) where T(r,u,u) i s the kin e t i c energy functional and V(r,u,Ti) i s the potential energy functional in the Fourier transformed domain. Now, i f the integrand F (r,u,u) s a t i s f i e s Euler's equation (62) and the natural boundary conditions (63) and (64) are s a t i s f i e d , then there i s an approximate solution of the form 45 *^ = 2 a * f M ' O where "j tit^ j fe , denotes N l i n e a r basis functions i n the piecewise continuous, twice continuously d i f f e r e n t i a b l e function space, PC , and u (r) £ PC . The t r i a l solution -u(r) minimizes the Lagrangian energy functional I T (or I S p or lt„ ) , i n the l i m i t . Then we have I T (or Iit> or lSj> ) i s minimum i f , from equations (59) , (62) and (63) , which gives a generalized eigenvalue problem of type {OJ. - *ZTl\ C*3 - o where elements of the matrices £v ~] and [_T 3 w i l l be given l a t e r for the d i f f e r e n t cases. This generalized eigenvalue problem i s solved by the QZ-algorithm (Moler and Stewart (1973)), which i s b r i e f l y described in Appendix A. For the basis functions which w i l l be used in the t r i a l solution (62), we have several choices. The most freguently used basis functions i n the finite-element method are l i n e a r functions, quadratic functions, splines( cubic, B-spline and L-spline) and cubic Hermite polynomials. There are higher order basis functions but the high cost of the numerical implementation i s often compromised by the accuracy of the 46 desired solution. When Wiggins (1973) solved the normal mode problem for the Earth by the Bayleigh-Ritz scheme he used Hermite polynomials as basis functions. Linear functions are most often used i n engineering problems but due to the large error bounds and the nature of the coordinate system used i n the present problem, they are not suitable. The a p r i o r i error bounds are given i n Table 1 for the three most often used basis functions. Table 1. Theoretical a p r i o r i error bounds for the l i n e a r , cubic spline and cubic Hermite interpolations i n one-dimensional problem. Interpolation functions j a p r i o r i error bounds l i n e a r i n t e r p o l a t i o n cubic spline interpolation cubic Hermite int e r p o l a t i o n 3?+ Cubic Hermite polynomial basis functions are used i n the present numerical scheme, because they have the smallest a p r i o r i error bounds among the three shown in Table 1 , and because they are piecewise continuous and twice continuously differe.ntia.ble. , They have computational s i m p l i c i t y and s a t i s f y the necessary continuity conditions f o r stresses at designated boundaries. Cubic Hermite int e r p o l a t i o n i n one-dimension i s defined as 47 given Ay>z 0=r,<r,<... <^<r^<r^ ( ... < r ^ = 1 . 0 where H(^r) i s a 2 (n+1)-dimensional vector space, and the cubic Hermite basis, h f e(r^, i s unique i n H (Ar) such that h ^ r ^ 0 * * , Hk, |£N + 1, and Dh^{re) = 0 , Uk, J U N + 1, and h'^r^O, 1*k,i^N+1, and Dh^ (rp = (fj^ , 1^k, JE6K +1. The shape of the cubic flermite basis function i s shewn in the Figure 2 fox an arbitrary i n t e r v a l ( r 4 , r t ). The d e f i n i t i o n of the cubic Hertnite basis function i s (from Schultz (1973)) -t- \ o 4 r 4 * u - J -t r,8 \ O A — r ^ — -f 3 -2 0 B> —<LJ^L life r ^ ^ , , for 1 £ k £ N , and L (y) - 2. + — n>^r^ho C and (Rt - r * - ( ) J ( K - r Q l rid. - Q o f o r 1 4. k i H, and i f * £: f ^ T 6 O O -o o ^ r - rx, 49 F i g u r e 2 C u b i c H e r m i t e B a s i s f u n c t i o n i n a n o r m a l i z e d i n t e r v a l o f ( rk> r»u.>-50 3.2 One-dimensional Variational Solution Of Torsional O s c i l l a t i o n Is a f i r s t step to the three dimensional generalized v a r i a t i o n a l solution of normal modes of the Earth, the one dimensional t o r s i o n a l mode problem i s solved with cubic Hermite polynomial basis functions. From Chapter 2, we have the lagrangian energy functional (41) for the to r s i o n a l o s c i l l a t i o n , i . e . , If we assume a t r i a l solution cf the form (66), f t 1 for the N layers in the s o l i d mantle region between the core-mantle boundary r and the free-surface boundary a, then from the r e l a t i o n (64), we have 51 and , v r y The i n t e g r a t i o n l i m i t s are from r b to , but, i n a c t u a l p r a c t i s e , i t i s performed from r +., to r ^ + ( f o r the k'th matrix element. T h i s g i v e s the same r e s u l t s i n c e f o r the k'th matrix element the b a s i s f u n c t i o n s are zero o u t s i d e the i n t e r v a l (r^_ ( ,r f e + )) . The b a s i s f u n c t i o n s a l s o o v e r l a p each other i n t h i s i n t e g r a t i o n i n t e r v a l as shown i n the F i g u r e 3 . From the nature of the piecewise c o n t i n u i t y and the o v e r l a p p i n g of the b a s i s f u n c t i o n s i n the a c t u a l i n t e g r a t i o n i n t e r v a l , the matrix eigenvalue problem. { CO - ^ CO \ c o =-o i n v o l v e s symmetric block t r i d i a g o n a l matrices Cv ~) and £ T J . The elements c f the matrices [ [ v^and f T ^ a r e 1 u and Mi \ , >(*-0(e+O C\/ i f ^ f Iff' iAi k<i , Mi \ The matching of boundary c o n d i t i o n s , as g r a p h i c a l l y shown i n 5 2 Figure 3, re s u l t s in the overlapping of the block matrices in f V ] and [JT\]• The schematic example i s shown i n Figure 4. An elegant and detailed treatment cf boundary conditions i n one dimensional v a r i a t i o n a l problem i s given by Wiggins (1976). -I T-1C4 1 Figure 3 The Cubic Hermite Basis functions overlapping in an arbitrary integration interval ( r ^ , ,r f c t () . The integrations i n the one dimensional problem are done a n a l y t i c a l l y and thus exactly, which i s not possible f o r the more complicated problems. O Figure 4 The boundary condition matching i n the energy matrices [v] and [ T ] . 54 As a test problem, the normal modes of torsional o s c i l l a t i o n for a simple spherical Earth model with homogeneous, i s o t r o p i c one-layered mantle i s solved hy two methods; the Wiggins (1968) solution technigue and the present v a r i a t i o n a l solution with two d i f f e r e n t nodal points. For the better understanding of errors, the v a r i a t i o n a l solutions are f i r s t integrated with N=2 and then with N=10. The radius of the Earth i s normalised such that the core-mantle boundary i s at 0.5454 and the free surface i s at 1.0. The density and the shear modulus are also normalised to 1.0. Then the eguations in (28) (Wiggins (1968)) are integrated using subroutine DIFSY of the University of B r i t i s h Columbia Computing Center, with the error control set to EPS = 1.0x10~7. For the v a r i a t i o n a l solutions, the mantle i s divided into two and ten layers giving t o t a l r a d i a l nodal points 3 and 11 respectively including the boundary nodes. The matrices C v i j 3 and L~Ti j J are calculated, obtaining blcck-tri-diagonal matrices C v > s and £ T J ' S , then the generalized eigenvalue problem i s -n solved by the QZ-algorithm with EPS=1.0x10 55 Table 2. Cemparision of the to.rsic.nal free o s c i l l a t i o n frequencies found from the one dimensional v a r i a t i o n a l solutions and llterman et a l . (1959) type solution. Toroidal Mode 2 0 1 2 3 4 5 0 1 2 3 4 \ 0 1 2 3 4 5 0 1 2 3 0 1 2 0 1 z 3 10 20 I 30 I 40 0 0 0 0 Var i a t i o n a l Type Solution (2-layer) 5.8079 66.5671 209.9236 472.3389 1480.8981 2143.7765 14.3167 79.274G 221.5732 483.5133 1494.0799 25.3522 96.2925 237.2335 498.4935 1511.6362 2167.9302 38.7643 117.5661 257.0167 517.3521 54.4473 142.9385 281.0638 540. 18 37 1559.4941 72.3378 172. 1567 309.5371 567. 1080 138.9997 500.0233 1072.3134 1854.6703 Variational Type Solution < 10-layer) 38.7627 117.5462 256.8044 493.8476 54.4446 142.9118 280.6191 516.7423 850.1181 72.3336 172.1249 3C9.2009 543.6046 138.9891 499.8205 1070.8317 1849.5201 Alterman et a l . Type Solution (DIFSY) 5.8075 "* 66.5525 209.6486 448.3534 782.6623 1212.5294 14.3156 79.2161 221.3256 459.6725 793.8447 25.3504 96.2688 237.0064 474.8070 808.7754 1238.4827 38.7616 117.534 3 256.7892 493.7950 54.4433 142.8970 280.7970 516.6863 849.9599 72.3325 172.1076 309. 1710 54 3.5441 138.9875 499.8184 1070.8239 1849.5105 56 The r e s u l t s o f the above t h r e e s o l u t i o n p r o c e d u r e s a r e t a b u l a t e d i n T a b l e 2. The d e t a i l e d e r r o r a n a l y s i s w i l l be t r e a t e d w i t h t h e t h r e e d i m e n s i o n a l s o l u t i o n r e s u l t s i n the Chapter 5 . However, we can e a s i l y see t h a t , from the r e s u l t s i n T a b l e 2, the e r r o r i n c r e a s e s g r e a t l y as n i n c r e a s e s f o r the both c a s e s of N=2 and N= 10. The a b s o l u t e e r r o r a l s o i n c r e a s e s as 1 i n c r e a s e s . E c t h r e s u l t s were e x p e c t e d , but the e r r o r i n c r e a s e w i t h n i n t h e cases o f N=2 (number of l a y e r s ) m e r i t s s p e c i a l a t t e n t i o n at t h i s s t a g e , because, i n the t h r e e d i m e n s i o n a l s o l u t i o n , the n o d a l p o i n t s i n each c o o r d i n a t e axes a r e l i m i t e d by computer c o r e s t o r a g e . 3.3 Three D i m e n s i o n a l S o l u t i o n For N o n - r o t a t i n g , N o n - g r a v i t a t i n g , S p h e r i c a l Simple E a r t h Model From the p r e v i o u s c h a p t e r , we have t h e f o l l o w i n g l a g r a n g i a n i n t e g r a l f o r the n o n - r o t a t i n g , n o n - g r a v i t a t i n g , i s o t r o p i c , s p h e r i c a l E a r t h u c d e l , , ( e q u a t i o n s (46) and (49) ) V ff f ~- ^ 1 For t h i s s i m p l e E a r t h model, the L a g r a n g i a n energy i n t e g r a l i n c l u d e s o n l y the k i n e t i c energy term, the e l a s t i c p o t e n t i a l energy t e r m , and the s u r f a c e boundary i n t e g r a l term. As i n t h e one d i m e n s i o n a l c a s e , any s o l u t i o n u which m i n i m i z e s t h i s L a g r a n g i a n i n t e g r a l w i l l be an e i g e n - s o l u t i c n o f t h e system. To a p p l y a t r i a l s o l u t i o n c o n s i s t i n g of the c u b i c Hermite 57 polynomial basis i n a spherical coordinate, we can rewrite the equation (72) i n matrix form, i . e . . V where the superscript j>.c. means 'piecewise continuous*, and r O o r o O (superscript f means the transpose of a matrix) and 58 t A A o o o A A+v* A o o o A A A*y* o o O O o A o o 0 O o O A o o O o O / A ( 7 0 The l a s t term i n eguation (72) vanishes on the application of surface boundary condition. In a three dimensional, spherical coordinate system, we introduce a t r i a l solution consisting of tensor products of three coordinate basis functions s i m i l a r to the ones introduced i n the previous section. le l e t H (? S p h ) be H ( ? 5 P 0 = H (4r) ©H C ^ 0 D | i O f ) where The H(A r), H(Ag)/ and H(^>) are the cubic Hermite polynomial basis functions in r , © and & coordinate directions and each 59 forms a 2(11+1), 2(M + 1) and 2 (L +1)-dimensional vector space. Then we w i l l have an i n t e r p o l a t i o n scheme with H(3 y^) i n 8 (N+1) (M+1) (L+1)-dimensional piecewise t r i c u b i c polynomial v e c t o r space, where (N+1), (M+1) and (L+1) are the number of nodes i n each c o o r d i n a t e d i r e c t i o n . Then se w i l l have a t r i a l s o l u t i o n of form M =CH]CO = O - i r where c*s are the undetermined c o e f f i c i e n t s and On the s u b s t i t u t i o n of (74), (75) , (76) and (78) i n t o (73), we have the Lagrangian i n t e g r a l 60 For the minimum of I,„ , g i v e s , a g a i n , a g e n e r a l i z e d e i g e n v a l u e problem of type o which i s s o l v e d by t h e Q Z - a l g o r i t h m as mentioned above. The boundary c o n d i t i o n s are s a t i s f i e d at each n o d a l s u r f a c e by matching t h e d i s p l a c e m e n t b a s i s f u n c t i o n s as w e l l as t h e i r p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o each c o o r d i n a t e v a r i a b l e . The b a s i c p r i n c i p l e o f matching the d i s p l a c e m e n t s and t h e i r p a r t i a l d e r i v a t i v e s i s e x a c t l y the same as i n the one d i m e n s i o n a l problem, o n l y more c o m p l i c a t e d and t e d i o u s . For the n o r m a l i z e d s i m p l e E a r t h model w i t h a=1.0, A=1.0 andylA =1.0, t h e r e s u l t s are t a b u l a t e d i n Table 3 w i t h the r e s u l t s c f the c l a s s i c a l s o l u t i o n s from e g u a t i o n (39). I n the v a r i a t i o n a l s o l u t i o n t h e E a r t h model has 2 nodes, 3 nodes and 4 nodes r e s p e c t i v e l y a l o n g r , & and (j> a x i s i n the s p h e r i c a l c o o r d i n a t e . The 168x168 m a t r i c e s [V"3 a n d f I D a r s symmetric and banded w i t h bandwidth 144. I n the t h r e e d i m e n s i o n a l v a r i a t i o n a l s o l u t i o n , the e i g e n v a l u e s a re c a l c u l a t e d a l l a t once w i t h o u t any 61 discrimination. fls shown i n the previous section, the eigenvalues for the overtones with higher n w i l l depart considerably from the true values but the eigenvalues for snsall n and X modes are expected to be reasonably close to the upper l i m i t of true values. Cne d i f f i c u l t y in the three dimensional solution i s to i d e n t i f y each of the modes as they are characterized i n the conventional solution. If the rotation or other perturbing effects are included i n the solution, the i d e n t i f i c a t i o n w i l l become even more d i f f i c u l t than i n the present simple case due to coupling e f f e c t s . 3.4 Effects of Rotation In The Normal Hodes Of Sel f - g r a v i t a t i n g Spherical Earth Hcdel If we add s e l f - g r a v i t a t i o n and rotation, we w i l l have the general lagrangian energy i n t e g r a l , from eguation (58) , I, f V: where V V 62 I, C ; JL" A vA ^ f« " j >«> 3 + V v In the above eguation, the l a s t term i s i d e n t i c a l l y zero upon the application of the boundary conditions. If ae include the s e l f - g r a v i t a t i o n we have an extra unknown besides the displacement f i e l d s . The general unknown vector u w i l l , again, be interpolated by cubic Hermite basis functions. Then the "u i s O o 4 , 1 . - - i * I J which reduces to eguation (74) during the integration of k i n e t i c energy terms. The matrix [VDJ w i l l , now, be 63 O O \ Y Y *& O o ( Y 0 0 O 1 o o _L a- & -r) o o For the is o t r o p i c materials [ E J i s same as before., The matrix form of the energy i n t e g r a l s for the kinetic and e l a s t i c s t r a i n terms i s the same as before( equation (80) ). The gr a v i t a t i o n a l and the r o t a t i o n a l potential terms w i l l be 41/ R u ] T where and 1_ < ? O h l \^ O rVa-T© <9 a- a- <p> \ ? O 0 0 0 o function s only cf r, in t h i s case Y a n d T h e g r a v i t a t i o n a l p e r t u r b a t i o n i n t e g r a l t e r m s c a n a l s o w r i t t e n i n m a t r i x f o r m a s f o l l o w s 4V f-T V T 0 o o \ 5-0 o o r^*"? -> o 0 o o 0 o a o 0 o o o o «3 r * a n d 65 V 4v^jr^] o o o o I o o o o o o o o 1 o During the e v a l u a t i o n of Cv J # most of the volume i n t e g r a l s sere e v a l u a t e d a n a l y t i c a l l y cr by s e r i e s expansion. However, some s u r f i c i a l p a r t of the volume i n t e g r a l c o u l d not be c a r r i e d out by t h i s simple method and a numerical scheme was used. The i n t e g r a t i o n schemes using the s e r i e s expansion and the numerical methods are e x p l a i n e d i n appendix fl. In gen e r a l the e v a l u a t i o n of the volume i n t e g r a l s r e g u i r i n g numerical treatment i s very c o s t l y as expected and makes the whole a l g o r i t h m very expensive. 3.5 E f f e c t s Of Simple l a t e r a l H e t e r o g e n i e t i e s In the p e r t u r b a t i o n technigue, .-, l a t e r a l h e t e r o g e n e i t y was a p p l i e d as a s e r i e s expansion of s u r f a c e s p h e r i c a l harmonics. In the v a r i a t i o n a l type f i n i t e element method, the l a t e r a l h e t e r o g e n e i t i e s , which may be represented by /„(r, <9,<£) , or are a p p l i e d during the e v a l u a t i o n of volume i n t e g r a l f o r each elements. 66 Figure 5 The configuration of elements with node numbers. Therefore, the constants ^oi^*B,j>), \(t,&,p) or ^iT,$,<f>) may have d i s c o n t i n u i t i e s across any nodal surfaces without disturbing the i n t e r n a l boundary conditions for the displacement, stress cr g r a v i t a t i o n a l perturbation. A general element block i s shown in Figure 5 . However i n present research the elementary block i s more simple with 4 nodes; r = ( 0 , 1 ) , $ = (0,-E-) and ^>=(0,S_). i n the present c a l c u l a t i o n the constants ^(r,#,^>), A(r,0,^>) and /*{x,$,<p) are assumed to be stepwise constant in each element and have d i s c o n t i n u i t i e s at the nodal surfaces i f they were d i f f e r e n t i n each block. A l l other computations are i d e n t i c a l to those in the previous section. A spline i n t e r p o l a t i o n scheme using cubic Hermite polynomial can be computer-programed for the i n t e r p o l a t i o n cf 67 f o ( r , b,j>) , X (£*&*(f>) and / M ( r , p , ^ ) # bu t t h i s would make the c o m p l i c a t e d program e v e r more ho r rendous t o hand le and was not i m p l e m e n t e d . 68 CHAPTER 4 ELLIPSOIDAL EARTH MODEL After the epoch-making Chilean earthguake of May 22 1960, the s p l i t t i n g cf spectral peaks were f i r s t interpreted as the effe c t of the rotation of the Earth by Pekeris et a l . (1961), Backus and Gilbert (1961) and MacDcnald and Ness (1961). Another cause, proposed by Osami and Stc(1962), to explain the s p l i t t i n g was the e l l i p t i c i t y of Earth. They solved the eguation of motion i n a spheroidal coordinate system to explain that the to r s i o n a l o s c i l l a t i o n m T X mode s p l i t s into (j£ -1-1) modes by the e l l i p t i c i t y of the Earth. Later Caputo(1963) applied the perturbation technigue to examine the effects of the e l l i p t i c i t y on the normal modes. The more conveniently treated application of perturbation technigue i s developed by Backus and Gil b e r t (1967) and Dahlen (1968), and the eff e c t s of the e l l i p t i c i t y of the Earth are extensively studied by Dahlen (1969) using t h i s scheme. 4.1 Recent Development In Research On The Effects Of The E l l i p t i c i t y Of The Earth On The Normal Modes. Most recently Dahlen (1975) applied the normal mode perturbation theory i n correcting great c i r c u l a r surface wave phase velocity measurements for the ef f e c t s of the rotation and the hydrostatic e l l i p t i c i t y of the Earth. He estimated the apparent great c i r c u l a r path length as 69 where a i s the mean radius of the Earth, Q i s the co-latitude and %i(T) and ^ ( T ) are respectively the normal mode multiplet r o t a t i o n a l and e l l i p t i c a l s p l i t t i n g parameters of the Earth. He calculated the s p l i t t i n g parameters and X * ( T ) f ° r t h e t w 0 Earth models 1066A and 1066E (Dziewcnski and G i l b e r t (1972)) for the period range 150-300s. lit a period of 300s, ^Ljji) for Eayleigh waves on the model 1066B with two sharp upper mantle d i s c o n t i n u i t i e s { 420Km and 67CKm ) deviates from the asymtctic l i m i t of e(a)/6.0 by more than a factor of 4. (see Figure 1 of Dziewonski and Sai l o r (1976)) This stunning result was received with a doubtful mind. In a search for a possible explanation of Dahlen's (1975) r e s u l t s , Dziewcnski and Sailor(1976) found that the major contribution to the e l l i p t i c i t y s p l i t t i n g parameter comes from the treatment of the in t e r n a l boundary conditions at the sharp d i s c o n t i n u i t i e s in the upper mantle and they concluded that the re s u l t i s erroneous. Hoodhcuse (1976) proved a n a l y t i c a l l y that this result i s indeed improperly calculated and found the cause to be the wrong application of the Eayleigh's p r i n c i p l e . Backus and Gilb e r t (1976) and Dahlen (1968, 1969 and 1975) used, in the parameter perturbation technigue, a Eayleigh's p r i n c i p l e applied to the lagrangian for the upper mantle d i s c o n t i n u i t i e s ; where cvUv i s the displacement vector of the appropriate 70 boundary, &p i s the surface of d i s c o n t i n u i t i e s and ["L^t denotes the d i s c o n t i n u i t i e s i n the Lagrangian L, across ftp with positive contribution from the side of kp corresponding to ot p o s i t i v e . woodhouse{1916) showed rigorously that Bayleigh's p r i n c i p l e must be applied to the Hamiltonisn instead of to the Lagrangian for the i n t e r n a l d i s c o n t i n u i t i e s , i . e . , At the external boundary, the integrand of eguation (89) reduces to the same as eguation {88), because -f=f ^ vanishes and eguation (88) produces correct r e s u l t . The s p l i t t i n g parameters are calculated with t h i s new formula by Dzieaonski and Sa i l o r (1976) and they are shown in their Figure 1 ; t h e i r r e s u l t s compare cl o s e l y to the results by the conventional method. The s p l i t t i n g parameters computed by Dahlen(1976) for the combined effects cf the rotation, e l l i p t i c i t y and l a t e r a l heterogeneity are weakly affected by the above mentioned error but they have to be corrected accordingly as well as his previous results (Dahlen 1968,1969 ). In reference to t h i s new development, Dahlen(1S76 , p92) s p e c i f i c a l l y points cut that there i s s t i l l great uncertainty in the c a l c u l a t i o n of the e f f e c t s of the e l l i p t i c i t y on the normal modes. 4.2 E l l i p s o i d a l Coordinate System To derive a lagrangian energy functional for the oblate e l l i p s o i d a l Earth model, we consider an oblate e l l i p s o i d a l coordinate system, assuming the centre of the Earth to be the o r i g i n and the polar axis of the Earth to be the axis of r o t a t i o n . Figure 6 E l l i p s o i d a l Coordinate System. 72 i e then have the following r e l a t i o n between the Cartesian (x,y,z), spherical {r, & , cj>) and the oblate spheroidal ( ^ , ^ ,^) coordinates where c i s the distance between the o r i g i n and the focus. The contours of constant ^ form a group of spheroids. The spheroid with ^ = i s the surface of the Earth and ^ =0 i s a c i r c u l a r disc. The polar axis i s represented by '*| = 0 , and the contours of constant are a group of hyperboloids of one sheet whose f o c i are given by A and A . The condition fj = z£- gives the eguatorial plane excluding the c i r c u l a r disc AA*. Thus the equatorial and polar r a d i i are where y„ i s the value of ^ on the surface of the Earth. For the numerical values, and <* — = — <E. (ex.) 7 3 The variable <^6 represents the longitudinal coordinate, as i n the previous spherical coordinate, varying from 0 to 2 X radians. If we l e t Q ~$ o j )> —> &° c^Wl q — > (9 fa*) under the condition that C C<S^t ^ — V we obtain a formula which holds i n the case of a spherical Earth. 4 . 3 Lagrangian Energy Integral For The Oblate Spheroidal Coordinate System In a generalized c u r v i l i n e a r orthogonal coordinate system the s t r a i n r elations are given as follows ( Takeuchi and Saito (1972)) , 4i« > f ^ where the coordinate transformation scale factors h^, h^ and h^ are of the form 74 and e q u a t i o n (SO) g i v e s Now the components o f s t r a i n , i n o b l a t e s p h e r o i d a l c o o r d i n a t e s , a r e where t h e s u b s c r i p t ' e ' means e l l i p s o i d a l c o o r d i n a t e and \ Ax 0 L_ _A\ 0 I > CD n J D O i ° -ty rJL 1 o 0 ^ n ^ 4 w . f o . Q * _ ^ t y l 0 75 In the generalized form of Hooke's law, i t i s assumed that each of the six components of stress i s a l i n e a r function of a l l the components of s t r a i n . Also for an i s o t r o p i c medium, i t i s independent of the choice cf coordinates and a i l transformations from one set of coordinates to another one are invariant. Thus we have from eguation (76) X O O O A. O O O A A**/A O 0 o o o o O o o o o O M o o o O 0 o M we used in the Chapter 3 also have to transformed into the new oblate spheroidal coordinate system, i . e . , from eguation (77), where .... , m ' t v ] 76 and h^, and h ^  are coordinate transformation scale factors and h(^), h«(jf)# ... are defined in Chapter 3 by eguations (67) and (68) . The solution vector (]ucJ w i l l then be, as defined i n eguation (78), [ C e ] = M f C e ] ( Qi where h ,h , ... are -Re, - i ' U ) Then, again from eguation (58), the general Lagrangian energy functional in an oblate spheroidal figure with volume V and the boundary 5V i s I e = \/e - * Te 0°O where 77 I/O zi r 7 1 r ) J ( for this i n t e g r a l evaluation only, <j5,c in |uej i s set to zero ) where ^ i | C C e ] T [ H e ] T [ D e ] T [ £ , ] [ P e ] [ H t ] [ C e ] V ^ H H ^ f I T and I ^ = ffi J { C « < f [ De J [ Me] * [ U e ] T [ D e 3 ] [ Ue] $ 4 V Ve In a generalized orthogonal coordinate system, the gradient, divergence and Laplacian are defined as I 3 f 78 > ( h i and the ^De,J , [l>e tJ and [De s i n the e g u a t i o n (101) a r e , from e g u a t i o n (10 2) , and O © O o o o 0 o o O o o o _ l 3-o 6 o o o o *f Q o O o D o o _ L X 0°o Now the g r a v i t a t i o n a l p o t e n t i a l , , at e g u i l i b r i u m s t a t e , s a t i s f i e s and i n our c a l c u l a t i o n 79 e where B i s the distance to the observation point from the centre of mass, and, at the surface cf an oblate e l l i p s o i d a l Earth, E becomes The potential due to the c e n t r i p e t a l acceleration of the oblate spheroidal Earth, rotating with angular velocity XL about i t s centre of mass i s Now i f we substitute a l l the relations into eguation (101) and perform the volume integrations and minimize the Lagrangian I e with respect to (^CeJ^we again have a generalized matrix eigenvalue problem of the same form as eguation (65) Again t h i s eguation can be solved for the eigenvalues and eigenvectors by using the QZ-algorithm. (Appendix A) In eguation (101), the matrices resulting from the integration of each i n d i v i d u a l elements can be assembled, as described in Chapter 3 , by matching the displacements and t h e i r derivatives along the nodal points and nodal surfaces. For the boundaries which reguire the matching of stress components the matrices can be conveniently transformed as described in d e t a i l by Wiggins (1976). In the cblate spheroidal Earth model, we have 8 0 a c i r c u l a r surface with radius C at the centre of Earth instead cf a point. The regularity and boundary conditions at t h i s central disc are s a t i s f i e d by just matching displacements and the i r derivatives (or stresses) along t h i s surface for every element in contact with i t . Most of the boundary treatment w i l l be same as i n the spherical Earth model, but, i f we have a spherical boundary at some radius inside the e l l i p s o i d a l Earth, then we w i l l have a very complicated boundary matching because, in the oblate spheroidal Earth model, the = ^  gives a spheroidal surface rather than a spherical surface. In such cases, we can represent the spherical boundary i n the spheroidal coordinate variables r e s u l t i n g in an extra matrix transformation and compute the boundary terms for matching. Or we can represent the gap between the spherical surface and the oblate spheroidal surface with same a^=a as an extra element and proceed the solution. ."'u »p* .:o i u i : t \.z.L-a .-:,-3.c" - » a. 3 ? -81} i a a e ; ( i : i „o ; , 81 CHAPTEB KOHEBICAI BESULTS In t h i s chapter, the numerical formulations of the v a r i a t i o n a l normal mode solutions, described i n the previous chapters, are computed for the simple three dimensional Earth models. The Earth models and parameters are normalized into dimensionless variables. Since the Earth models considered i n t h i s research are perfectly e l a s t i c spheres, the average values of the density J , e l a s t i c shear modulus yw , e l a s t i c bulk modulus , s-wave ve l o c i t y v* and p-wave ve l o c i t y v p are defined i n i t i a l l y as 0V r ^ f rMr and The average normalizing values for these parameters are, f o r Bullen's Earth model j3 , 82 and the average Earth's radius i s taken as X •= IK The normalizing factors for the gra v i t a t i o n a l constants and time are After the normalized dimensionless parameters are substituted into the energy i n t e g r a l s , they are evaluated for each element sectionalized by the nodal points in the three coordinates. This matrix has a dimension of 96x96 for the perfectly e l a s t i c , non-rotating, non-gravitating sphere and 128x128 for the gravitating and/or rotating Earth model. SL 83 o i S Y M M E T R I C ! I I I .0 - — - - - h • Apc-vBpc |Alc< - e i C+^A^c+B i c i A c c i - p c c t c c c | ' I 1- r - • - t' - • - ( L . - - J Qzr + p\y iPl^tHl^ Q I C £Gf- Pld + ( c i ? t ^ | z . \ C*-z t Pll + \ O ! i ; Qrc + Hfc i \ * % p 4 - K f p J Figure 7 Whole Earth energy matrix assembly. 84 The l e t t e r s 1 c* represent the elements of the matrix involving the integration of the basis functions which include the o r i g i n (center cf the Earth) and the l e t t e r s 'p' indicate the i n c l u s i o n of the basis functions which have nodes on the polar axes. The matrix blocks with numbers are the only general point energy matrix elements. If the Earth consisted of eight egual elements with o r i g i n , as shown in Figure 5, and each block had names such as A,E,C, ... , H the t o t a l energy matrix would, after boundary matching, look l i k e the one in Figure 7. The r e s u l t i n g matrix w i l l be 168x168 with half bandwidth of 144 for the non-rotating, non-gravitating Earth model and 224x224 with half bandwidth 192 for the gravitating and/or rotating Earth models. Once the potential and k i n e t i c energy matrices are assembled the eigenvalues are found by the QZ-algorithm described in Appendix A. 5.1 Normal Modes Of A Non-gravitating Homogeneous E l a s t i c Sphere With Simple Heterogeneities The dimensionless normal mode freguencies were calculated for the homogeneous e l a s t i c spherical Earth model with elements as shown in Figure 7. The lower order eigenvalues are given in Table 3. For the Earth models without gravitation and r o t a t i o n , we have 168 eigenvalues. 85 Figure 8 Surface displacement fields of the normal modes of eT^ and 0S//.'The displacement fields plotted in the upper half of page are the ones computed from the variational solution and the ones in the lower half of the page are the theoretical displacement fields. AXIS LATITUDE 1.9 GCQCCC 133IJCCCC3C3 '3C33:;-J3 33C3333'j\J3 i eccccc3D3C3 3-*j'3;333333c.:333333;33 33 t C3 3ccc-cc3C3C<:cc33 3;.oo : :3 ivO:c:'j J30 I C C O O C C O C I : C C C C C C C O C : C ' : : : C C ' : C : : ; : ; C 3 i ccococccc tccccc:c;cc:c:::ccccJ i C030oc30jc:ccccc3c3c3 3:3:c3::o3:cc3i o C C 3 C C C 3 3C cc c c C C C C U J ' J G C C 3.>v333333;j3 j ccjacccscccccc.cc': 3 3 3 3333333:03033331 —C33 33333.) 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I & ! t I 0 I ti -1 6 O.M • C0»Ufi'ON9*\CE (£TV((H P«INTE^ !»MS01J CCWlCU* UVtCl s - -I.coco TO -0.(OOOO » - -C.4CCO0 10 -C.4CCC0 0 - -O./O'.'CO 10 e . : > > i « c - i i I - 0.2CCC0 10 c.tccco 2 - o.toooo 10 0.(OOOO > • t.oooo 10 I..VOJ f o r S° 92 Figure 9 Radial surface displacement fields of 0S^ . The plot in the upper half of the page is from the variational solution and the one in the lower half i s the theoretical displacement f i e l d . 93 i s I o Ii C O.M I T U O . t U l l J . l c o o o o -ooooo 00000 o o c c o 00000 uuooo 00000 00000 o o c c o c o o c o O O O O O — UMTUOf J . ' 2.2 i . i t — i m u l u u l u l t u l l u u u -i m m m m a u i H i u u i l l l l l U l l l l l l l l l l l l l l l l l U I l u i i i u i i u u i i i i u i n i i i i i i u i i u n m m i i i i i . i i m M u m i i i i i i i i i m i i i i i i i i i i i i u n i i i i m u u i u i i i u u i I u i i u u m i u u i i i m i i m i I i i n I i I : I i i I 111 n u i i iin1111 I l l l l l l l ' . 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I 1 0000 1 1 • — - - 0 0 * -I - 000 I I I oo • r I 000 I I 0000 I I C C C C O I C O O C C C C C C C C C O C C C O O — C O O C G C C : ^ -I o o o c c c c o o c c c c c : c c c c c : : c c c c c : c c c : c c c o o • I OOOQCCCOOCCCCCCCCCGCCCCC.iOCCCCCCOCt l l l l l l l l l l l l l l l l l l l l l l l 1 i m i i u u u i u u u i l l l I l l U U U U l I 0 C 0 O 0 0 I 0 O 0 O C O I C 0 O O O O I C U 0 C 0 C 1 0 0 0 0 C C C I CCOCOOOt OCCOCOOt C C C C C 0 O I — . — 0 0 0 0 0 0 0 * I C C C C C 0 0 I I C 0 C 0 0 C C I 1 0 0 0 0 0 0 0 1 I OOOOOCCCt I 0 C C C 0 C C 0 I I 0 0 C 0 C C 0 0 I 1 0 C C C C C C C C I I 0 C U 0 0 0 0 0 0 I I 0 C C C C C C C C I -OOOOOOOOOO* 0 0 C O C O 0 O O 0 I 1 C C O C C C C C C C C I I ooooooacccooi I COOOOCCCCCCCI t O O C 0 0 O C C 0 C C C C C I I GOOCGOOOCCOCOCOl I O C O O O C C C C L C C C C O C O O I OCaonCCCCCCCOGOCCCCCCCI cccccccccccocccococooccccocccccccooocccoocooi t(ii\*i'^n'i^'"i" •"l*'llloC''CUtJ'jCCljijO J'J'JilLdCCuCICuCOCO I ooSSSe""'3Soc««2=c-J--':? 1 y . - - . . . . - . ^ . wJi.^-Ji..i.L--L, ^----C-COCO^JCOL-^lK-JCOCCCJOGOCOOCCOQOOO I cooooGcco ; c .ccO ' : c cGococo : ; cccocccccccccc :cccccccccoco 1 C0000CC VOGCGZOoOCCoococ;cocooccccoocccocccoooooooo I cccccccccQc:ccccc:ccc:cccccccccjoccccuoooooooooo I I ooocoococoooccoococccococccocccccccccccccooo I I OCCCCCGCOOC-CJCCCCJUUOOOCaOCUCJOCCCOCOC I ' •—CCC0CCCCOCCCCCCCCCOCCOO0OGOCOCCC0OC — • OOOOOOOOOOOOO'JCCGOCCCOOCCCCCCCOOOC I CC0C0 CC C CC0 00 0 I o o o c c c : c c c c c c c c c c c c c c : : c : c c c c o o o • ccococcooocccccccccoccccccoooo •I ccocccccccccccccccocccoccooaa I ooooococooccccccccccccccccoco I ' ccccccccocccccccooccocooccoo I ocooccccccccccccccccccccooo I ooococococcoccoacoooccccco I c c c c c c c c c c c c c c c c c c c c c c c c o o • I ooocoocoocccccccccccccccco 1 ccocccccoccccccccccocccooo t coccocccccccccccccccccaco i • OOOGOCJUGOCOGGaOOOCOCOOOCOt —• I C0 0CCC 0CJC 0OC O 0J0GJO0 0 OOOOCOCCCCCCCCCGGCCCCCCCCCCCCOCC 1 OQCCG0COO0CCCOCCC0CC0GOOOCC0CC I , CCCCCCCCCCCCOCCCCCCCOGOCCCOCO I-GCCCCCCCCCCOCCCCCCCCCCOCCCCC I cacocoooooocooocoucuooaccoo / I cccccccccccocccccccooooccco 1 looocooocccocccococooocccao t • . 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Since the size of the energy matrix i s very large, t h i s plot i t s e l f i s very costly and, in Figures 8, 9 and 10, the surface eigenfunctions are plotted for u r, u e and u^ , components of oT° and oS° , the ur for 0S* and the u^ component of 0S* to show the r e l a t i v e surface displacements. In these figures, the upper diagrams are the displacements from the v a r i a t i o n a l solution and the lower ones are the displacements from the t h e o r e t i c a l spherical harmonic solutions. The models have 2,3 and 4 nodes along r a d i a l , cc-l a t i t u d i n a l and azimuthal coordinates, which i s very crude. The eigenfunction plots of the t h e o r e t i c a l solutions and the solutions from these crude models naturally have minor differences but the general outlines of the modes are e a s i l y i d e n t i f i e d . One fact we have to notice i s that as X increase the eigenfuction plot from the v a r i a t i o n a l solution shows a gradual phase s h i f t i n both the azimuthal d i r e c t i o n and also the c o - l a t i t u d i n a l d i r e c t i o n (Figure 10). One of the most s i g n i f i c a n t features of the v a r i a t i o n a l numerical solution i s that the degenerate eigenf reguency »>U3^ s p l i t s into (2m+1) spectral components as shown in Figure 12. This f a c t can e a s i l y be explained by the interpolation scheme of the h { cj>) for the azimuthal component of the eigenvibration. 97 Figure 11 <f> -coordinate interpolation scheme for „Y^with different xa values. 98 As shown i n Figure 11, the interpolation function h If) and h' {<p ) only depend on the number of nodes i n the c/> coordinate and t h i s set of cubic Hermite basis functions, in the i n t e r v a l O L cj> 2 , interpolate a l l (2m+1) j> variations of each eigenfunction and, depending on the accuracy of int e r p o l a t i o n for each of the (2m+1) modes, the degenerate normal mode frequency s p l i t s into (2m+1) l e v e l s . The degree of s p l i t t i n g i s not uniform and, to explain t h i s , a very detailed error analysis i s necessary. However, the ^tig. frequency i s not affected either by t h i s numerical spectral s p l i t t i n g or by the e f f e c t s of other perturbations studied. To examine the ef f e c t s of the l a t e r a l heterogeneity, four simple models are considered; O The earth as a sphere without any perturbations. f\ L The j> , /» and A. for the block A (Figure 7) are 1-35 greater than the rest of the sphere. A 3> -•-• The j> , JA and A- for the block A are 2 % greater. A1A7 ""• T i i e i 3 l o c } c A a n d G have f r /* and A values 1 35 higher than the rest of the model. HEM X. The upper hemisphere has j» , /* and A. values 1 % higher than those of the lower hemisphere. In these types of crude l a t e r a l heterogeneities, we can handle only large scale perturbations physically and the j>, /w and 7\. were varied a l l together by 1 % and up to 2 %. 99 T a b l e 3 N o r m a l i s e d D i m e n s i o n l e s s Normal Mode F r e q u e n c i e s For The Models 0,A1,A3,A1A7 And HEMI. T h e o . S o l . f o r Unpert.Model 2.501 (JO -t 2.47532 2.50129 2.52173 2.54039 2.5627 8 A1 2.50106 2.51329 2.52172 2.53641 2.56386 A3 +• 2.50405 2.51628 2.52270 2.54238 2.56691 A1A7 2.50C07 2.51830 2.52165 2.52848 2.56799 HEMI 2.48048 2.50030 2.51676 2.54503 2.56510 2. 814 W o - 2 - I 2.75133 2.76808 2.78408 2.80064 2.81711 2.75361 2.77130 2.78448 2.79708 2.81759 2.74869 2.77229 2.78483 2.79607 2.82859 2.75661 2.77033 2.78409 2.79679 2.8264 8 2.74712 2.78481 2.78954 2.79059 2.82911 3. 865 * > = - J - 2 - I o I 3 3.87275 3.88723 3.89506 3.87747 3.88721 3. 89237 3.87034 3.88750 3.89810 3.87035 3.87191 3.88117 3.89121 4.04 4 -m=.-3 j 3. 97555| 3. 97523| 3. 976221 3. 97424J 3. 96725 - 2 j 3. 97808} 3. 97800| 3. 98099J 3. 97701| 3. 97425 — I j 3. 98082J 3. 98197| 3. 98308! 3. 98172J 3. 97702 0 I 3. 98792J 3. 987551 3. 98973| 3. 98746! 3. 99005 I I 3. 99409I 3. 994931 3. 99786J 3. 99384! 3. 99694 2 t 3. 99689| 3. 99693J 3. 99992J 3. 99595J 3. 99896 3 I 3. 99907\ 3. 999131 4. 004061 3. 99807J 4. 00689 I n the r e a l E a r t h , t h e most p r o b a b l e l a t e r a l h e t e r o g e n e i t y c f the above s i z e s c a l e would be much l e s s t h a n 1 %, and the l a t e r a l h e t e r o g e n e i t i e s c o n s i d e r e d i n a l l the p r e v i o u s p e r t u r b a t i o n t e c h n i q u e s a r e l e s s t h a n 1 %. 100 The normalised dimensionless normal mode frequencies for these models and the theoretical solution are given i n Table 3. The t h e o r e t i c a l solutions are obtained by solving the ch a r a c t e r i s t i c equations for the t o r s i o n a l (eguation (39)) and spheroidal o s c i l l a t i o n freguencies were obtained by the routine by Higgins (1976). As predicted i n Chapter 3, from the results of the one dimensional solution, the errors increase greatly as £ or YI increases and conseguently only the lower order t o r s i o n a l and spheroidal o s c i l l a t i o n s are l i s t e d i n Table 3. The spectral s p l i t t i n g of the above l i s t e d modes are graphically shown i n figures 12 ( 0 s J ), 13 ( 0S~) and 14 l0T™), From these Figures, the following facts can be observed ( the l i n e s with a dot on the l e f t hand side are the zeroth degree multiplet l i n e s ) ; (1) The numerical spectral s p l i t t i n g of the model *0* i s much larger than expected, reaching up to 4 % of the degenerate freguencies and „i>)^  (the v e r t i c a l l i n e s represent 1 % ) . (2) The ef f e c t s of the l a t e r a l heterogeneities, of the models considered, seem much smaller than the numerical s p l i t t i n g . (3) In »S2 and 0S™ modes, the l a t e r a l heterogeneities of types A3 and HEMI have the most e f f e c t s on the normal modes. The zeroth degree mode frequency, as well as the m'th degree spectral l i n e s , s h i f t up to 0.3%, in both models. (4) In t o r s i o n a l o s c i l l a t i o n , the most spectral s h i f t occurs for the model A3 and greatly reduced e f f e c t s are exhibited for the model HEMI. For 0T* mode, some of the azimuthal s p l i t t i n g freguencies with d i f f e r e n t m are missing or uni d e n t i f i a b l e . 101 0 A1 A3 A1 A7 HEMI Figure 12 Spectral s p l i t t i n g of 0S^"mode for the Earth models Al, A3, A1A7 and HEMI. The lines with a dot on the l e f t hand side are the zeroth degree multiplet lines and the others are for m= —2,-1, 1 and 2. The v e r t i c a l line represents a relative magnitude 1% of degenerate frequency. 102 A1 A3 AT A7 HEMI Figure 13- Spectral s p l i t t i n g of „S" mode for the Earth models Al, A3, A1A7 and HEMI. The lines with a dot on the l e f t side are the zeroth degree multiplet lines and the other lines are for m = -3,-2,-1, 1, 2 and 3. The v e r t i c a l line represents a relative magnitude 1% of the degenerate frequency. 103 I o B x A1 A3 A1 A7 HEMI Figure 14 Spectral s p l i t t i n g of eT^nmode for the models A l , A3, A1A7 and HEMI. The lines with a dot on the l e f t side are the zeroth degree multiplet lines and the other lines represent m = -2,-1, 1 and 2. The v e r t i c a l line represents a relative magnitude 1% of the degenerate frequency. 104 5.2 Normal nodes Of Gravitating, Rotating And Lat e r a l l y Heterogeneous Sphere As a next step, s e l f - g r a v i t a t i o n and rotation are added into our model. The angular velocity of rotation and gr a v i t a t i o n a l constant are normalised as mentioned at the beginning of t h i s chapter. The values are J?_=0. 03857 G =0.1290x10 The models considered are i $ 0 .... only the s e l f - g r a v i t a t i o n i s added to the previous model ' 0 • c^ vV — the normalised J"2_ i s added to the model G° (•ifW/l 1 t ^ i e ^ a t e r a l heterogeneity of type Al i s combined with the model GW The normalised dimensionless freguencies are tabulated in Table 4 for these models and the spectral s p l i t t i n g diagrams are shown in Figures 15 and 16 for the modes aT~ and •. As shown in these f i g u r e s , the e f f e c t s of the s e l f - g r a v i t a t i o n and rotation are very small. The numerical spectral s p l i t t i n g i s dominant and even the combined rotation and the l a t e r a l heterogeneity e f f e c t s on the s p l i t t i n g are n e g l i g i b l e . From previous r e s u l t s of s i m i l a r research using perturbation techniques, the e f f e c t s of l a t e r a l heterogeneities on the normal modes are expected to be small and the ef f e c t s of the r o t a t i o n a l potential would be large for the lower order modes. 105 Table 4 Normalised Dimensionless Normal Mode frequencies For The Models 0, GO, GW and GWA1. Thee.Sol.for Unpert.Hodel -+ GO GW G«A1 2.501 -) I 2 2 .47532 2 .50129 2 .52273 2 .54039 2 .56278 2 .47893 2.50491 2 .52335 2.54201 2 .56440 2.46371 2 .50107 2 .52383 2.54281 2 .56049 2 .46735 2 .50111 2 .52388 2 .54299 2 .55711 2 .706 CO 7V)=-2 2 .75133 2 .76808 2 .78408 2 .80064 2 .81771 2.75523 2.77296 2.78451 2 .80327 2 .81783 2 .75028 2 .77613 2 .78446 2.79571 2 .82159 2 .74865 2 .77434 2 .78455 2 .79613 2 .82178 - 2 3.865 3.87275 3 .88723 3 .89506 3. 87438 3.88 886 3.89 668 3.87133 3 .87328 3 .89025 3.89668 3.91256 3.87113 3.874425 3 .88906 3 .89670 3 .89810 3.981 Ws-3 | 3 . 97555 | 3 .977174 3 . 97465 | 3 . 97368 - 2 | 3 . 97804 | 3 .979713 3 . 9791 11 3 . 97828 — 1 I 3 . 980821 3.98245I 3 . 98277} 3 . 98122 o 1 3 . 98792J 3 . 9 8 7 8 0 | 3 .98901J 3 . 98815 1 1 3 . 99409| 3.995121 3 . 99493 | 3 . 99516 2 1 3 . 99689J 3.998521 3 . 99890 | 3 . 99949 3> 3 . 99907| 4.000891 4 . 00069 | 4 .00571 J; 0 GO GW GWA1 Figure 15 Spectral s p l i t t i n g of 0Tj >mode for the models GO, GW and GWA1. The lines with a dot on the l e f t side are the zeroth degree multiplet lines and the other lines represent m = -2,-1, 1 and 2. The ver t i c a l line represents a relative magnitude 1% of the degenerate frequency. 107 0 GO GW GWA1 Figure 16 Spectral s p l i t t i n g of 0S~ mode for the models GO, GW and GWA1. The lines with a dot on the l e f t side are the zeroth degree multiplet lines and the other lines represent m = - 2 , -1, 1 and 2. The v e r t i c a l line represents a relative magnitude 1% of the degenerate frequency. 108 However the r e s u l t s of thi s research indicate that the e f f e c t s of s e l f - g r a v i t a t i o n and rotation are very small r e l a t i v e to the numerical spectral s p l i t t i n g and no reasonable conclusions can be obtained. The extreme l a t e r a l heterogeneities of types A3 and HEMI , however, seem to af f e c t the normal mode freguencies greatly even with large numerical spectral s p l i t t i n g . Thus the r e a l Earth must be well symmetrically balanced in the J , /A and A. d i s t r i b u t i o n and the degree of l a t e r a l heterogeneities (<-1.035) considered in the perturbation techniques are well j u s t i f i e d from a physically r e a l i s t i c viewpoint. 5.3 E l l i p s o i d a l Earth Model The e l l i p s o i d a l Earth models can also be studied by exactly the same approach as above and the uncertainties created by the perturbation solutions can be cleared up. However, the test runs of the integration routines c a l c u l a t i n g the energy matrices which involved the hyperbolic functions, were extremely expensive and after a b r i e f search for more economical algorithms, the e l l i p s o i d a l Earth model study was l e f t f o r future i n v e s t i g a t i o n . An integration method, which might provide a solution f o r t h i s p a r t i c u l a r case, was proposed by Higgins(1975,personal communication). In t h i s method a number of exact numerical integrations are carried out as a function of radius on a regularly spaced grid i n 6 and <j> . The intermediate regions are then interpolated to complete the desired integration for the given block. This idea seems most appropriate but i t requires a great deal of t r i a l computing runs and numerical tests and i t was not applied. 109 5.4 Remarks On Computational Error During the problem formulation stage, a group of available basis functions were studied and, from the a p r i o r i error c r i t e r i a ( and the convenient numerical manipulability ), cubic Hermite polynomial basis functions were chosen for t h i s research. The numerical computations are a l l performed i n double precision and consequently the re s u l t s are very accurate. The eigenvalue evaluation routines are supposed to have a minimum -2 accuracy of 10 for the normal matrices of size 50x50. The matrices t r i e d i n t h i s research are much larger than 50x50, but the re s u l t s are expected to be reasonably accurate even though no guantitative test was made. According to Birkoff et a l . (1966) the accuracy of the eigenvalue i s 0{h 6), (h= (x^-x-J ) , and the accuracy of the eigenf unction i s 0{h* ) for the one-dimensional eigenvalue problem. The r e s u l t s of the one-dimensional t o r s i o n a l o s c i l l a t i o n problem solved i n t h i s research (Table 2) indicate t h i s v a r i a t i o n a l method with cubic Hermite polynomial basis i s very accurate and powerful, even i n the extreme case of 2 layer models. Errors increase rapidly as Z and n increase. In the three-dimensional v a r i a t i o n a l s o l u t i o n , the error bounds are expected to be very large and t h i s i s shown i n the results of t h i s chapter. There i s no t h e o r e t i c a l estimation of error bounds for the three dimensional f i n i t e element method and we can only compare the numerical r e s u l t s with known solutions. 110 Even though no attempt i s made t o e s t i m a t e the e r r o r s • q u a n t i t a t i v e l y , the r e s u l t s i n T a b l e 3 and 4 i n d i c a t e the maximum e r r o r s i n the e i g e n f r e g u e n c i e s of the modes w i t h o r d e r s 2 and 3, r e a c h up t o 14 % o f the t h e o r e t i c a l degenerate s o l u t i o n s . T h i s w i l l i n c r e a s e r a p i d l y as the o r d e r X and v\ i n c r e a s e . 111 CH AP,XE8 6 CONCLUSIONS The task cf developing computational technigues for ca l c u l a t i n g the free o s c i l l a t i o n s of a r e a l i s t i c Earth model has been carried out recently by a number of authors. Their approach has been the application cf perturbation technigues to the c l a s s i c a l solution of the normal mode problem f o r a sp h e r i c a l l y symmetric, non-rotating, i s o t r o p i c and r a d i a l l y heterogeneous Earth model. However, there remain several problems which cannot be solved by the conventional perturbation technigues. In thi s t h e s i s , a v a r i a t i o n a l type f i n i t e element solution of the free o s c i l l a t i o n s of Earth i s proposed. The work c l e a r l y demonstrates that t h i s technigue i s very powerful and e f f i c i e n t in one-dimensional problems and i s also feasible for three-dimensional normal mode problems. In p r i n c i p l e , i t overcomes a l l the d i f f i c u l t i e s associated with the use of perturbation methods. F i r s t l y , the va r i a t i o n a l approach allows the e f f e c t s of the rotation cf the Earth to be included without r e s t r i c t i o n . In the perturbation method, one has to assume that • — - « . 1.0, a t iWji condition which cannot be s a t i s f i e d i n the study of core o s c i l l a t i o n s with very long periods. Secondly, the use of the va r i a t i o n a l technigue t h e o r e t i c a l l y removes the r e s t r i c t i o n that l a t e r a l heterogeneities i n the lower mantle must be projections of those introduced near the surface. The perturbation method reguires that inhomcgeneities be confined to near surface depths (in the order of a few hundred kilometers). Hhile the approach presented 112 here does not include such r e s t r i c t i o n s i n i t s theoretical development, present computational f a c i l i t i e s do not allow i t s application to r e a l i s t i c models of Earth's heterogeneities. Thirdly, uncertainties in the ca l c u l a t i o n of the e f f e c t s of the Earth's e l l i p t i c i t y on the normal modes can he removed by the use of the v a r i a t i o n a l solution formulation. Recently published work by a number of authors who used perturbation technigues has presented some c o n f l i c t i n g r e s u l t s for the e f f e c t s cf e l l i p t i c i t y . Theoretically t h i s c o n f l i c t can be resolved but, again, the computational f a c i l i t i e s l i m i t the application of the v a r i a t i o n a l type f i n i t e element method with cubic Hermite polynomial basis functions. In the t h e s i s , v a r i a t i o n a l solutions for several simple Earth models are presented. Eased cn these r e s u l t s , which are given i n Chapter 5, the following conclusions can be made: (1) In the v a r i a t i o n a l type f i n i t e element solution cf normal modes, the degenerate normal mode freguency ^ u)^ s p l i t s into (2m + 1) components even when there are no perturbations of r o t a t i o n , or l a t e r a l heterogeneities. This i s a result of the c h a r a c t e r i s t i c s of the numerical solution scheme, (2) In general the numerical spectral s p l i t t i n g of the degenerate freguency i s large enough for the crude Earth models considered i n t h i s research that i t obscure the e f f e c t s cf rotation and small l a t e r a l heterogeneities, (3) S e l f - g r a v i t a t i o n seems to have a very small e f f e c t on the normal mode spectra of the models considered, <4) Several l a t e r a l heterogeneities (;>1.03? but < 2,0%), which break the physical symmetry of the Earth model, s h i f t the spectra in general up to 0.4?? including the zeroth degree 113 freguency. Thus t h i s research provides a t h e o r e t i c a l basis f o r eventually obtaining more precise knowledge about the e f f e c t s of rot a t i o n , e l l i p t i c i t y and l a t e r a l heterogeneities of Earth cn the normal modes. However, computation cf r e a l i s t i c r e s u l t s i s not possible now. Either we must await the development of a new generation of computers, or a new computer algorithm must be formulated such that i t would greatly reduce the number of unknowns yet retain the c a p a b i l i t i e s of handling r e a l i s t i c Earth models. However, within the confines of current computing c a p a b i l i t i e s , some immediate extensions of the three-dimensional version cf the present research are possible: 1) S p l i t t i n g parameters can be calculated as a function of -/2_ as SZ becomes large. 2) The spectral s p l i t t i n g can be calculated for Earth models that contain only odd (or even) angular orders of l a t e r a l heterogeneities. This w i l l v e r i f y the r e s u l t s from the perturbation technigue using selection rules (Dahlen 1968,196 9, and Luh 1973,1974). 3) A fa s t integration program may be implimented for the normal mode- solutions of e l l i p s o i d a l Earth models. 4) a new application of the present technigue to the r o t a t i o n a l dynamic of Earth may be developed. As well, the one-dimensional version of the formulation may be applied to core o s c i l l a t i o n s and Earth tide problems with better e f f i c i e n c y and accuracy than any other conventional numerical technigues. 114 In conclusion, the va r i a t i o n a l type f i n i t e element algorithm using cubic Hermite basis functions i s a powerful and e f f i c i e n t method to study the normal modes cf Earth. 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The Theory of Sound vol. I, MacMillan, London. Saito,M., 1967. Excitation of Free O s c i l l a t i o n s and Surface Haves by a Point Source i n a V e r t i c a l l y Heterogeneous Earth, J. Geophys. Res., vol.72, no.14, 3689-3699. Sato,Y. And Usami,T«, 1962. Basic Study on the O s c i l l a t i o n of a Homogeneous E l a s t i c Sphere, Geophys. Magazine, 31, 15-62. Schultz,M.H., 1973. Saline Analysis , Prentice-Hall, New Jersey. Sezawa,K. 1927. On the Propagation of Rayleigh Haves cn Spherical Surfaces, B u l l . Earthg. Res. Inst., 2, 21-28. 118 S l i c h t e r , ! . B . , 1967. S p h e r i c a l o s c i l l a t i o n s o f t h e E a r t h , Geophys. J . R. a s t r . S o c , 14, 171-177. Takeuchi,H., 1950. On the E a r t h T i d e o f the C o m p r e s s i b l e E a r t h of V a r i a b l e D e n s i t y and E l a s t i c i t y , T r a n s . Amer. Geophys. On., 31, 651-689. Takeuch i , H . And S a i t o , f f . , 1972. S e i s m i c S u r f a c e Haves, Methods i n C o m p u t a t i o n a l P h y s i c s , ed. By B.A.Bolt e t . a l . , Academic P r e s s , New York, 217-295. Usami,T. And Sato,Y., 1962. T o r s i o n a l O s c i l l a t i o n of a Homogeneous E l a s t i c S p h e r o i d , B u l l . Seism. Soc. Am., v o l . 5 2 , no. 3, 169-484. 0 s a m i , t . , Sato,Y. And landisman,M., 1965. T h e o r e t i c a l Seismograms of s p h e r o i d a l Type on the S u r f a c e o f a Heterogeneous S p h e r i c a l E a r t h , B u l l . E a r t h g . Res. I n s t . , v o l . 4 3 , 461-660. Van Ioan,C.F., 1S75. A G e n e r a l M a t r i x E i g e n v a l u e A l g o r i t h m , SIAM J . Numer. A n a l y s i s , 12, 819-834. , Ward,B.C., 1975. The Combination S h i f t Q Z - a l g o r i t h m , SIAM J . Numer. A n a l y s i s , 12, 835-853. Wiggins,R.A., 1968. T e r r e s t r i a l V a r i a t i o n a l T a b l e s f o r the P e r i o d s and A t t e n u a t i o n of the Free O s c i l l a t i o n s . Phys. E a r t h . P l a n e t . I n t e r i o r s , v o l . 1 , 201-266. Wiggins,R.A., 1973. High Speed Computation of S u r f a c e Waves and Free O s c i l l a t i o n s , A b s t r a c t s c f Conf. Seism. Soc, Am,, , E o u l d e r , p.30. Wiggins,R.A., 1976. A F a s t , New C o m p u t a t i o n a l A l g o r i t h m f o r Free O s c i l l a t i o n s and S u r f a c e Waves. Geophys. J . R. a s t r . S o c , ( i n p r e s s ) . Wiggins,B.A., 1976. Comment on 'The C o r r e c t i o n of G r e a t C i r c u l a r S u r f a c e Wave Phase V e l o c i t y Measurements f o r the B o t a t i o n and E l l i p t i c i t y of the E a r t h * by F.A.Dahlen, Geophys.J.R. a s t r . S o c , ( i n p r e s s ) . W i l k i n s o n , J.H. And R e i n s c h , C. , 1971. l i n e a r A l g e b r a , S p r i n g e r - V e r l a g , Nev York. Woodhouse,J.H., 1976. On R a y l e i g h ' s P r i n c i p l e , Geophys.J.R. a s t r . S o c , ( i n p r e s s ) . Zharkov,B.N. And lyubimov,V.H., 1970a. T o r s i o n a l O s c i l l a t i o n s of a S p h e r i c a l l y A s y m m e t r i c a l Model of the E a r t h , E u l l . ( I z v . ) Akad. S c i . U.S.S.R., E a r t h P h y s i c s , 2, 71-76. Zharkov,B.N. And Lyubimov,V.M., 1970b. Theory of S p h e r o i d a l V i b r a t i o n s f o r a S p h e r i c a l l y Asymmetric Model of the E a r t h , E u l l , ( I z v . ) Skad. S c i . O.S.S.R., E a r t h P h y s i c s , 10, 613-618. 120 APPENDIX A Qz-algorithm for The Solution Of Ac= ABC. Recently there has teen great progress in the development of new algorithms to solve generalized matrix eigenvalue problems of the form Ac=A.Bc (Wilkinson (1965), Wilkinson and Reinsch (1971), Moler and Stewart (1973), Van Loan(1975) and Ward (1975)). However, the A and B matrices are symmetric and banded i n our case and the algorithm developed by Moler and Stewart(1973) i s chosen over other available algorithms to solve the eigenvalues and eigenvectors, because of i t s generality, s t a b i l i t y and accuracy s i t h reasonable e f f i c i e n c y . I f the matrices A and B are rectangular the VZ-algorithm developed by Van Loan (1975) can be used. 1). In the f i r s t step E i s reduced to upper triangular form, one column at a time, by Householder row transformations. These transformations are simultaneously applied to A. A i s then reduced to upper Hessenberg form by row rotations while maintaining the triangular form of B. This i s possible i f the elements of A are annihilated in a precise order, columnwise from the bottom, as for example when n=5 X K X X X X X X X X t o X X X X X X X ( 0 (13 it) A X 121 Each r o t a t i o n a n n i h i l a t i n g an element of A i n t r o d u c e s one non-zero element on the subdiagonal of B. T h i s element i s then a n n i h i l a t e d by a column r o t a t i o n which when a p p l i e d to A does not d i s t u r b any of the newly i n t r o d u c e d zeroes. By accumulating these r o t a t i o n s i n a dummy space T, f u l l i n f o r m a t i o n i s saved f o r l a t e r use i n back t r a n s f o r m i n g the e i g e n v e c t o r s of the reduced system i n t o those of the o r i g i n a l system. Once A i s upper Hessenberg and B i s upper t r i a n g u l a r form, an i t e r a t i v e technigue of the e x p l i c i t QZ-algorithm i s used to reduce A to upper t r i a n g u l a r form while maintaining the - i t r i a n g u l a r i t y o f B. l e t C be C=AB , then apply QE-a l g o r i t h m with s h i f t <T. Then Q i s determined as an ort h o g o n a l t r a n s f o r m a t i o n such t h a t the matrix E=Q <C - IT I) i s upper t r i a n g u l a r . The next i t e r a t e C i s d e f i n e d as C* =EQT + <r I=QCQT and i s known to be upper Hessenberg form. I f we s e t A =QAZ B' = QBZ where Z i s any u n i t a r y matrix, then / i - l T T - i r ' A B = QAZZ B Q = QAB Q =C I f matrix Z i s chosen sc t h a t B i s upper t r i a n g u l a r form, A'=C'B' i s upper Hessenberg form. In the process of -i i t e r a t i o n , we can avoid c a l c u l a t i n g C=AB e x p l i c i t l y . Then we have the i t e r a t i o n scheme Af+, =QV-Ap-Z 122 During the i t e r a t i o n , B stays as an upper triangular matrix and a seguency of upper Hessenberg matrices i s formed for A which converges to a guasi-triangular matrix, that i s , an upper Hessenberg matrix ahose eigenvalues are the eigenvalues of 1x1 or 2x2 p r i n c i p a l submatrices. The origin s h i f t s at each i t e r a t i o n are the eigenvalues of the loaest 2x2 p r i n c i p a l minor. Whenever a lowest 1x1 cr 2x2 p r i n c i p a l sub-matrix f i n a l l y s p l i t s from the rest of the matrix, the algorithm proceeds with the remaining submatrix. This process i s continued u n t i l the matrix has s p l i t completely into submatrices of order 1 or 2. . 3). The next step i s to check 1x1 and 2x2 p r i n c i p a l submatrices. The eigenvalues of 1x1 problems are the r a t i o s of the corresponding elements of A and B. The eigenvalues cf 2x2 problems might be calculated as the roots of a guadratic eguation, and may be complex even for r e a l A and B. However, in t h i s Q:Z-algorithm the 2x2 p r i n c i p a l submarrices are furthur reduced to 1x1 i f possible and eigenvalues are derived. The computer program (FORTRAN IV), kindly provided by Dr. R.aathon, Dept. Of Computer Science, University of Toronto, consists of four subroutines f o r each of the sections described above and one for the back-substitution to obtain the generalized eigenvectors. This FORTRAN program has been tested before by others and found to have good s t a b i l i t y and accuracy. I t has been tested again nith EVPOWR (Power method) of the University of B r i t i s h Columbia Computing Center for a simple one 123 dimensional t o r s i o n a l o s c i l l a t i o n problem. The test r e s u l t s show the program i s very good i n e f f i c i e n c y and accuracy f o r the moderate size matrices of A and B, but becomes very expensive for large matrices as expected. The more detailed aspects of the technigue can be found in the paper by Holer and Stewart (197 3) . 124 APPENDIX E I n t e g r a t i o n Methods Dsed In The N u m e r i c a l Computations The i n t e g r a t i o n s a l o n g r a d i a l c o o r d i n a t e a r e performed a n a l y t i c a l l y on t h e d i f f e r e n t c o m b i n a t i o n s o f c u b i c Hermite c o u l d be i n t e r p o l a t e d u s i n g t h e same c u b i c Hermite p o l y n o m i a l b a s i s f u n c t i o n s i f they are f u n c t i o n s of r , but i n t h e p r e s e n t c a s e s they are assumed c o n s t a n t s between nodes { o r n o d a l s u r f a c e s ) w i t h d i s c o n t i n u i t i e s at t h e nodes ( o r a t the n o d a l s u r f a c e s ) . T h i s a n a l y t i c a l scheme i s t h e most d e s i r a b l e f e a t u r e of the p r e s e n t b a s i s f u n c t i o n s because of i t s a c c u r a c y and economy. However, f o r the i n t e g r a t i o n of the a z i m u t h a l and co-l a t i t u d e v a r i a b l e s i t i s c o m p l i c a t e d , Again f o r s i m p l i c i t y , the m a t e r i a l c o n s t a n t s J , and A . are assumed t o be c o n s t a n t s between each i n t e g r a t i o n i n t e r v a l and thus pose no problems. The t r i g o n o m e t r i c f u n c t i o n s i n t r o d u c e d by the use o f a s p h e r i c a l c o o r d i n a t e system make some of the i n t e g r a t i o n d i f f i c u l t . I n t e g r a t i o n s of the form p o l y n o m i a l b a s i s f u n c t i o n s . The m a t e r i a l c o n s t a n t s J , / t and A . a r e computed by an a n a l y t i c approach. The r e m a i n i n g i n t e g r a t i o n s cannot be computed a n a l y t i c a l l y so t h a t and 125 are expanded i n series involving Bernoulli numbers (Gradshteyn C f 2 / , l * l < TL) and Byzhik (1965)) i . e . , ^ zfc K During the numerical evaluations of these i n t e g r a l s , each term with k>i i n the expansion i s added to the sums up to k=i u n t i l the sums up to k=i asymptotically approach the true values and the {k=i+1)*th term i s ne g l i g i b l e compared to the sums. The ~i largest term (k=i+1) neglected has an absolute magnitude of 10 or l e s s . This scheme i s much more economical than available numerical integration methods. The s i n g u l a r i t i e s involved with cotQ or -^ZTQ a r e approached as close as 0=1.0x10 and the int e g r a l s are well approximated. In most cases, the above method stably and accurately approximates the i n t e g r a l s , but for the cases when convergence i s not good a numerical integration method, SQUANK of the University of B r i t i s h Columbia Computing Center , i s used. The double precision version of SQUANK i s used as a back-up subroutine for greater accuracy. The algorithm of SQUANK i s based on Simpson's integration r u l e applied over i n t e r v a l s with varying width. For more detailed description of SQUANK, the reader i s referred to the write-up for SQUANK at the University of B r i t i s h Columbia Computing Centre. 

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