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Analytical and analogue methods of studying electromagnetic variations at the earth's surface 1967

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The Uni v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of HARRY W... DOS SO B.A., The Univ e r s i t y of B r i t i s h Columbia,"1955 M.Sc, The Uni v e r s i t y of B r i t i s h Columbia, 1957 WEDNESDAY, MAY 3, 1967 AT 10:30 A.M. IN ROOM 206, CHEMICAL ENGINEERING BUILDING COMMITTEE IN CHARGE Chairman: B. N. Moyls External Examiner: D. I. Gough Unive r s i t y of Alberta Edmonton, Alberta R. W. Burling R. M. E l l i s J. A. Jacobs M. W. Ovenden R. D. Russell T. Watanabe Research Supervisor: J. A. Jacobs ANALYTICAL AND ANALOGUE METHODS OF STUDYING • ELECTROMAGNETIC VARIATIONS AT THE EARTH'S SURFACE ABSTRACT The research described in this thesis deals with both mathematical ..and analogue, .methods of studying electromagnetic.variations at the earth's surface. F i e l d components .are studied for frequencies in the range 10 ^ - 10^ cps and for earth c o n d u c t i v i t i e s in the range 10" 1 6 - 10" 1 0 emu. Expressions are ..developed for the .electric . and .mag- netic . . f i e l d components .at the surface and within the upper layer of a...hori.zontalLy ..stratified..flat. conducting earth in the f i e l d of incident plane waves. Amplitudes and phase angles are obtained for various frequencies, angles of incidence, layer thicknesses, depths, and con- d u c t i v i t i e s . As an extension of this problem, expressions for a multilayer earth (n layers) are devel- oped and evaluated. Each of several thick layers i s divided into a s u f f i c i e n t number of sublayers, with changing conductivity, to represent, to a good approxi- mation, a continuous change in conductivity. The conductivity d i s t r i b u t i o n s used are of i n t e r e s t i n geo- physics. The r e s u l t s for the plane wave model, indicate that the amplitudes and phase, angles are strongly affected by the conductivity structure. The e l e c t r i c and magnetic f i e l d s at the surface of a f l a t homogeneous conducting earth i n the near f i e l d of an o s c i l l a t i n g l i n e current are. studied. The equations for the amplitudes and phase angles developed by Law and Fannin (1961) are used for the c a l c u l a t i o n s . Amplitudes and phase angles are obtained for various frequencies, c o n d u c t i v i t i e s , source heights, and locations with respect to the overhead current. The r e s u l t s indicate that the v e r t i c a l to h o r i z o n t a l magnetic f i e l d r a t i o s are i n the range of experimentally observed, values. An analogue model su i t a b l e for studying the behaviour of the natural geomagnetic and t e l l u r i c f i e l d variations- for various geological structures was constructed. Two types -of f i e l d sources were used - an o s c i l l a t i n g sheet current and an o s c i l l a t i n g l i n e current. Amplitudes and phase angles for the horizontal e l e c t r i c , horizontal magnetic, and v e r t i c a l magnetic f i e l d components are obtained and discussed for various geological structures including a f l a t layered earth, c y l i n d r i c a l bodies embedded in the surface layer, v e r t i c a l f a u l t s and dykes, sea mounts and conducting domes, c o a s t l i n e structures (sea-land interface and an upwelling in a high- conductivity zone within the mantle), and islands in an ocean channel. The r e s u l t s obtained for the c o a s t l i n e structures and islands in an ocean channel tend to support the proposed structures suggested by various workers (Schmucker 1964, Lambert and Caner 1965, Lokken and Maclure 1966.) in describing the experimentally observed coastal magnetic f i e l d anomalies. The analogue model constructed and used for this work r e a d i l y lends i t s e l f to studying a wide range of geological structures for a v a r i e t y of source f i e l d s in addition to those used here „ GRADUATE STUDIES F i e l d of Study: Geomagnetism Advanced Geophysics P r i n c i p l e s of Earth Science J . A. Jacobs W. F. Slawson Geomagnetism and Aeronomy T. Watanabe Plasma Waves T. Watanabe PUBLICATIONS 6 short .pedagogical a r t i c l e s (not l i s t e d below) 7 P a c i f i c Naval Laboratory Reports (not l i s t e d below) Singh, P.P., Dosso, H.W., and G r i f f i t h s , G.M., 1959. The energies and r e l a t i v e pair production cross sections for Zn&5 and Na22. .gamma rays. Can. J. Phys. _37, 1055. Dosso, H.W., 1962. The magnetic f i e l d at the surface of a s t r a t i f i e d f l a t conductor in the f i e l d of plane waves with a p p l i c a t i o n to geophysics. Can. J. Phys. 40, 1583. Dosso, H.W., 1965. The e l e c t r i c and magnetic f i e l d s i n a s t r a t i f i e d f l a t conductor for incident plane waves. Can. J. Phys. 43, 898. Dosso, H.W., 1966. A plane wave analogue model for stud} ing electromagnetic v a r i a t i o n s . Can. J. Phys. 44, 68. Dosso, H.W., 1966. A multi-layer conducting earth i n the f i e l d of plane waves. Can. J. Phys. 44, 81. Dosso, H.W., 1966. Further r e s u l t s for a multi-layer conducting earth i n the f i e l d of plane waves. Can. J. Phys. 44, 1197. Dosso, H.W., 1966. Analogue model measurements for electromagnetic v a r i a t i o n s near v e r t i c a l f a u l t s and dykes. Can. J. Earth S c i . 3, 287. Dosso, H.W., 1966. The e l e c t r i c and magnetic f i e l d s at' the surface of a f l a t conducting earth i n the near f i e l d of an o s c i l l a t i n g l i n e current. Can. J. Phys. 44, 1923. Dosso, H.W., 1966. Analogue model measurements for electromagnetic variations near a c o a s t l i n e . Can. J. Earth S c i . 3, 917. ( i ) A N A L Y T I C A L A N D A N A L O G U E M E T H O D S O F S T U D Y I N G E L E C T R O M A G N E T I C V A R I A T I O N S A T T H E E A R T H ' S S U R F A C E b y H A R R Y W I L L I A M D O S S O B . A . ( H o n s . ) , 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 , 1955 M . S c , 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 , 1957 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n t h e D e p a r t m e n t o f G E O P H Y S I C S W e a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E 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 A p r i l , 1967 ! ( c ) H a r r y W i l l i a m D o s s o 1 9 6 7 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t die 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 reference and study. 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 extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted t y the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or 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 allowed without my w r i t t e n p e r m i s s i o n . Department of ' G e o p h y s i c s The U n i v e r s i t y of B r i t i s h Col-umbia Vancouver 8, Canada Date May, < 1967 ( i i ) A B S T R A C T T h i s t h e s i s d e a l s w i t h b o t h m a t h e m a t i c a l a n d a n a l o g u e m o d e l s f o r s t u d y i n g e l e c t r o m a g n e t i c v a r i a t i o n s a t t h e e a r t h ' s s u r f a c e . T h e f i e l d c o m p o n e n t s a r e s t u d i e d f o r f r e q u e n c i e s i n t h e r a n g e 10 t o ^0J c y c l e s / s e c - a n d f o r e a r t h c o n d u c t i v i t i e s i n t h e r a n g e 1 0 " ^ t o 10 "^ 0 e m u . E x p r e s s i o n s a r e d e v e l o p e d f o r t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s a t t h e s u r f a e e a n d w i t h i n t h e u p p e r l a y e r o f a h o r i z o n t a l l y s t r a t i f i e d f l a t c o n d u c t i n g e a r t h i n t h e f i e l d o f i n c i d e n t p l a n e w a v e s . E x t e n s i v e r e s u l t s o f ; a m p l i t u d e s a n d p h a s e a n g l e s a r e o b t a i n e d f o r v a r i o u s f r e - q u e n c i e s , a n g l e s o f i n c i d e n c e , l a y e r t h i c k n e s s e s , d e p t h s , a n d c o n d u c t i v i t i e s . A s a n e x t e n s i o n o f t h i s p r o b l e m , e x - p r e s s i o n s f o r a m u l t i l a y e r e a r t h ( n l a y e r s ) a r e d e v e l o p e d a n d e v a l u a t e d . E a c h ' o f s e v e r a l t h i c k l a y e r s i s d i v i d e d i n t o a s u f f i c i e n t n u m b e r o f s u b l a y e r s , w i t h c h a n g i n g c o n d u c t i v i t y , t o r e p r e s e n t t o a g o o d a p p r o x i m a t i o n a c o n t i n u o u s c h a n g e i n c o n d u c t i v i t y . T h e c o n d u c t i v i t y d i s t r i b u t i o n s u s e d a r e o f i n t e r e s t i n g e o p h y s i c s . T h e - r e s u l t s f o r t h e p l a n e w a v e m o d e l i n d i c a t e t h a t t h e a m p l i t u d e s a n d p h a s e a n g l e s a r e s t r o n g l y a f f e c t e d b y t h e c o n d u c t i v i t y s t r u c t u r e . T h e e l e c t r i c a n d m a g n e t i c f i e l d s a t t h e s u r f a c e o f a f l a t h o m o g e n e o u s c o n d u c t i n g e a r t h i n t h e n e a r f i e l d o f a n o s c i l l a t i n g l i n e c u r r e n t a r e s t u d i e d . T h e e q u a t i o n s f o r t h e a m p l i t u d e s a n d p h a s e a n g l e s d e v e l o p e d b y L a w a n d F a n n i n (1961) a r e u s e d f o r t h e c a l c u l a t i o n s . E x t e n s i v e r e s u l t s o f ( i l l ) amplitudes and phase angles are obtained f o r various f r e - quencies, conductivities, source heights, and locations with respect to the overhead current. The results indicate that the v e r t i c a l to horizontal magnetic f i e l d r a t i o s are i n the range of experimentally observed values. - An analogue model suitable f o r studying the behavior of the natural geomagnetic and t e l l u r i c f i e l d variations f o r various geological structures was constructed. The two types of f i e l d sources used were an o s c i l l a t i n g sheet current and an o s c i l l a t i n g l i n e current. Extensive measurements of amplitudes and phase angles f o r the horizontal e l e c t r i c , the horizontal magnetic, and the v e r t i c a l magnetic f i e l d compon- ents are obtained and discussed f o r various geological structures including a f l a t layered earth, c y l i n d r i c a l bodies embedded i n the surface layer, v e r t i c a l f a u l t s and dykes, sea mounts and conducting domes, coastline structures (sea- land interface and an upwelling i n a high-conductivity zone within the mantle), and islands i n an ocean channel. The results obtained f o r the coastline structures and islands i n an ocean channel tend to support the proposed structures suggested by various workers (Schmucker 196̂ -, Lambert and Caner 1965? Lokken and Maclure 1966) i n describing the experi- mentally observed coastal magnetic f i e l d anomalies. The • analogue model constructed and used f o r this work re a d i l y lends i t s e l f to studying a wide range of geological s t r u c t - ures f o r a v a r i e t y of source f i e l d s i n addition to the ones used here. ( i v ) Page ABSTRACT i i L I ST OF ILLUSTRATIONS v i LIST- OF TABLES x v i ACKNOWLEDGEMENTS x v i i CHAPTER 1 INTRODUCTION 1 i 1.1 H i s t o r i c a l Rev iew 1.2 O b j e c t o f t he T h e s i s 15 CHABTER 2 MATHEMATICAL MODELS 18 2.1 A S i m p l e L a y e r e d C o n d u c t i n g E a r t h i n 18 t h e F i e l d o f P l a n e Waves 2.1.1 I n t r o d u c t i o n 18 2.1.2 M a t h e m a t i c a l A n a l y s i s 19 2.1.3 D i s c u s s i o n o f R e s u l t s 28 2.2 A Complex L a y e r e d C o n d u c t i n g E a r t h i n $h t h e F i e l d o f P l a n e Waves 2.2.1 I n t r o d u c t i o n 5*+ 2.2.2 M a t h e m a t i c a l A n a l y s i s 55 2.2.3 D i s c u s s i o n o f R e s u l t s 58 2.3 A Homogeneous C o n d u c t i n g E a r t h i n 68 t he F i e l d o f a L i n e C u r r e n t 2.3.1 I n t r o d u c t i o n 68 2.3.2 • M a t h e m a t i c a l A n a l y s i s 72 2.3.3 D i s c u s s i o n o f R e s u l t s 72 (v) Page CHAPTER 3 ANALOGUE MODELS 87 3.1 S hee t C u r r e n t Sou rce 87 3.1=1 I n t r o d u c t i o n 87 3.1.2 M a t h e m a t i c a l A n a l y s i s 89 3.1.3 Mode l D e s c r i p t i o n and 91 • Measurement Techn i que s 3 . 1 A D i s c u s s i o n o f R e s u l t s 99 3.2 L i n e C u r r e n t Sou r ce 17*f 3.2.1 I n t r o d u c t i o n '['Jh 3.2.2 M a t h e m a t i c a l A n a l y s i s 175" 3.2.3 • Mode l D e s c r i p t i o n 175 3.2.!+ D i s c u s s i o n o f R e s u l t s 176 CHAPTER h • SUMMARY AND CONCLUSIONS 202 BIBLIOGRAPHY 2 0 6 ( v i ) L I ST OF ILLUSTRATIONS F i g u r e 1. Mode l u s ed i n the p l a n e wave p r o b l e m . 2. The a m p l i t u d e s ( a ) - ( c ) and t he phase a n g l e s (d) as a f u n c t i o n o f 6 f o r f=1 c y c l e / s e c , (1) o=10~ 1 1 , (2) 1 0 ~ 1 \ (3) 1 0 ~ 1 ^ and (If) 1 0 ~ 1 6 emu. 3. The r a t i o R as a f u n c t i o n o f f r e q u e n c y f o r o=10"'1^ emu, and (1) 9=5, (2) 10, (3) 25, (k) k5, (5) 65, and (6) 85°. k. S k i n d e p t h as a f u n c t i o n o f f r e q u e n c y f o r a range o f c o n d u c t i v i t i e s . 5. The a m p l i t u d e s H x , H y 5 H z ? and the r a t i o R as f u n c t i o n s o f © f o r o2=1 0~1 o^=10" 1 6 emu, f=1 c y c l e / s e c , and d 2 = .1 o5 cm. 6. • The dependence o f H z on t he a n g l e o f i n c i d e n c e and l a y e r t h i c k n e s s f o r o 2 = 1 0 ~ 1 \ ciy=^Q~°^^ e m U ' and f=1 c y c l e / s e c . 7. H z as a f u n c t i o n o f f and d 2 f o r $=50 p, ( a ) - d 2 = 1 0 ~ 1 \ - d 3=.l'0"" 1 6,- (b) o 2 = 1 0 ~ 1 6 , d 3 =10~ l I f emu. 8. < E z as a f u n c t i o n o f f f o r 6=1+5P, 0 2 =10~ 1 1 , o ^ i O " 1 6 emu, (1) d 2=10 ? (2) 103-,- (3) 10-, • (lf>- 10^; o =10~ 1 5 , (5) d 2 * 1 0 3 (6) 10 3 , (7) IO**", (8) 10?; o = 1 0 - ™ , ' (9) d 2=10, ( 1 0 ) M 0 3 , (11). 10^, (12). 10^; d 3 = 1 0 ~ 1 3 , (13) d 2=10, (1^) 10 3 , (15) 10^, (16) 10^ cm. 9- H z as a f u n c t i o n o f f f o r 9=>+5p, o 2 =10"" 1 6 , o 3 =10~ 1 1 emu, (-1) d 2 =10 3 , ' (2) 10^, (3) 1 0 6 , (h) 10?; o 3 = 1 0 " 1 3 (5) d 9 =10 3 , (6) 1 0 5 5 (7) 10 6 , (8) 1 0 7 ; o ,=10~^ , (9) d 0 =10 3 3 (10) 10^, (11) 10 6 , (12) 107 cm. 10. H z as a f u n c t i o n o f d 2 f o r f=1 , 16 c y c l e s / s e c and 6=i+5 P, (A) o 9 = 1 0 ~ " , d . = 1 0 ~ 1 6 emu, (B) d o = 1 0 ~ 1 6 . -11 ' ~> d 3 =10 emu. ( v i i ) F i g u r e Page 11. 0Z as a f u n c t i o n o f d 2 f o r f=1 c y c l e / s e c , ©= k 5° 5 -+3 ( A ) d 2 = 1 0 ~ 1 1 , d 3 = 1 0 - 1 ° , 1 0 ~ 1 ^ , 1 0 " l l f ? 1 0 ~ 1 3 emu, (B) d 2 = 1 0 " 1 6 , d 3 = 1 0 ~ 1 1 5 1 0 ~ 1 3 , 1 0 ~ 1 \ 1 0 " 1 5 emu. 12. The a m p l i t u d e s (a) and the phase a n g l e s (b) as k6 1 1 1U- a f u n c t i o n o f z f o r d 2 = L x 1 0 , d 3 = 1 0 emu, d 2 =2x10 l f cm, d y 0 0 ? -~ = 1 c v c l e / s e c ) a n d ©=-+5°. 13. H z as a f u n c t i o n o f f f o r d 2 = 1 0 3 , d̂ =1 . 6x10^ cm, if 7 z=0, and ©= k 5°. 1*f. H „ as a f u n c t i o n o f f f o r d o = 1 0 ~ 1 1 , d . = 1 0 ~ 1 l f , 1*8 z - . 1 6 > o ^ L 3 0^=10 emu, z=0, 6=^5 , d^=1:.:6x10 cm, (1) d 2 =0, (2) 1 0 2 , (3) 1 0 3 , (h) 10^ ; and d 3 =8x10^ cm, (5) d 2 =0, (6) 10, (7) 1 0 2 , (8) ^o\, (9) 1oV (10) 10^ cm. 15- H_ (a) and Q0 (b) as a f u n c t i o n o f z f o r f=1 59 c y c l e / s e c , ©=^5°? d 2 =10 , d^=2x10 cm, and the f o l l o w i n g d 2 , C o , d ^ : (1) 1 0 " 1 1 , 10~1 \ 1 0 " 1 6 (2) 1 0 " 1 1 , 1 0 " 1 ^ , 1 0 " 1 1 , ' (3) - 1 0 " 1 1 , ' 1 0 ~ 1 6 , 1 0 " l I f , ' ih) 1 0 " 1 6 , i o ~ 1 l + , i o " 1 1 , (5) 1 0 - 1 6 , 1 0 ~ 1 V , ' 1 0 - 1 6 , - 1V - 1 6 10' -16,- 10' - 1 L 10 , 10' 10 , 10' (6) i o ~ 1 6 , i o ~ 1 1 , 1 0 - 1 V ( 7 ) i o ~ 1 1 , i o " 1 \ ' f o " 1 l f , ' (8) 1 0 ~ 1 1 , 1 0 ~ 1 6 , 1 0 - 1 6 , (9) 1 0 - 1 6 , 0 " l I f , 1 ( T 1 \ (10) 1 0 ~ 1 6 , 1 0 ~ 1 1 , 1 0 ~ 1 1 emu. 16. H_ (a) and <* (b) as a f u n c t i o n o f d 0 f o r z=0. 52 d - j - d ^ l O cm, f=1 c y c l e / s e c , ©=̂ -5 , and t he f o l l o w i n g d p , c<35 d^s d 2 =10 d j=10 , CD d 3 = r o - 1 5 3 ( 2 ) i o ~ 1 \ (3) i c H 3 , (k ) 1 0 - 1 2 , (5) 1 0 ~ 1 1 ; and d 2 = 1 0 ~ 1 1 , 0^=10 " 1 6 , ' (6) d 3 = l ' 0 ~ 1 - 6 , ' (7) (8) 1 0 " l I + ? (9) 1 0 ~ 1 3 , (10) 1 0 - 1 2 emu. 17. H (a) and <*> (b.) a s - a f u n c t i o n o f d Q f o r z = d Q , 53 LL n d^-d 9 =10 cm, f=1 c y c l e / s e c , ©=i+5 ? and t h e ~> - 16 -11 f o l l o w i n g d p , d 3 , d ^ : d 2 =10 , d^=10 , (1) d 3 = 1 0 " 1 5 emu, (2) 1 0 " l I f , (3) 1 0 ~ 1 3 , (^) 1 0 ~ 1 2 , (5) 1 0 " 1 1 ; a n d d 2 = 1 0 ~ 1 1 , d ^ l O " 1 6 , (6) d 3 = 1 0 ~ 1 6 , (7) 1 0 ~ 1 5 , (8) 1 0 ~ l L f , (9) 1 0 ~ 1 3 , (10) 1 0 - 1 2 emu. ( v i i i ) F i g u r e 18. Mode l u s ed i n the c a l c u l a t i o n s f o r a complex l a y e r e d e a r t h . 19. Complex l a y e r e d c o n d u c t o r . 20. " Mode l u s e d i n c a l c u l a t i n g ' e q u i v a l e n t ' c o n d u c t i v i t i e s . 21. Mode l u s e d i n t he l i n e c u r r e n t p r o b l e m . 22. H y (a ) and 0 y (b) as f u n c t i o n s o f y f o r 0=1 O * " 1 6 emu, h=2x107 cm, and (1) f=10~3, (2) 1'0"2, (3) 10"1, \k) 1, and (5) 10 c y c l e s / s e c . 1 6 23. - H z (a ) and •• (b) as• f u n c t i o n s o f y f o r d=10' emu, h=2x10? cm, and (1) f = 1 0 _ 3 , (2) 10^2, (3) 10~ 1 , Xk) 1 , a n d (5) 10- c y c l e s / s e c - . 2k. E • (a ) and ̂ x (b) as f u n c t i o n s o f y f o r o=10"16 emu, h=2x10? cm, and (1) f=10~3, (2) 10~2, (3) 10"1, ik) 1, and (5) 10 c y c l e s / s e c . 25. H y (a ) and H z (b) as f u n c t i o n s o f y f o r d=10"11 emu, h=2x107 cm, and (1) f=10~ 3 , (2) 10~2, (3) 10"1, (!+) 1, and (5) 10 c y c l e s / s e c . 26 . E x (a ) and 0̂ , 0Z, ^ x (b) as f u n c t i o n s o f y f o r o=10~11 emu, h= 2x107 cm, and (1) f=10~3, (2) 10 r 2, (3) 10~ 1 , Xh) 1, and (5) 10 c y c l e s / s e c . 27. H. (a ) and H_ (b) ats f u n c t i o n s o f y f o r f=10""1 J ' 7 16 c y c l e s / s e c , h=2x10' cm, and (1) o=10 , (2) 10"15, (3) 10"™, ( i f ) 10"13, (5) 10"12, (6) 10r1\, and (7) 10~ 1 0 emu. 28. E x (a ) and 0y, 0 Z (b) as f u n c t i o n s o f y f o r f=10~1 c y c l e s / s e c , h=2x107 cm, and (1) o = i 0 ~ 1 6 , (2) 10~1^ (3) 1 ( T 1 \ (h) l O ^ 1 3 , (5) 10"12, (6) 10"11, and (7) 1'0"10 emu. 29. H ( a ) , H_ ( b ) 5 and 0_ ( c ) as a f u n c t i o n o f y f o r y -1 . - 1 6 . • 7 f=10 c y c l e s / s e c , o=10 emu, and (1) h=10', (2) 2x107, (3) 3x107, and (>+) lfx107 cm. ( i x ) F i g u r e Page 30. H z as a f u n c t i o n o f f r e q u e n c y f o r h=10' 7 cm, 85 (a) 0=1 CT16,' (b) d = 1 0 " 1 1 emu, and (1) y =10 7 , (2) 2 x 1 0 7 , (3) L x 1 0 7 , and (1+) 8 x 1 0 7 cm. 31. D iag ram o f the ana l ogue m o d e l . 93 Diagram o f t he mode l l a y e r e d c o n d u c t o r . 93 33. The e l e c t r i c and m a g n e t i c f i e l d d e t e c t o r s . 95 3k. B l o c k d i a g r a m o f t he mode l measurements s y s tem. 98 35 • B_ (a) and H (b) f o r a v e r t i c a l g r a p h i t e c y l i n d e r 105 f o r f=3x10 c y c l e s / s e c , d^=1 cm, and t r a v e r s e s a long- (1) y=0, (2) R, (3) R/2, (k) 3R/2 cm. 36. E (a) and H (b) f o r a v e r t i c a l g r a p h i t e c y l i n d e r 106 LL J f o r f=6x10 c y c l e s / s e c , d^=1 cm, and t r a v e r s e s a l o n g (1) y=0,- (2) y=R, (3) y=R/2, (k) y=3R/2 cm. 37. ^x~<Py  a v e r t i c a l g r a p h i t e c y l i n d e r f o r d^=1 108 cm, f =3x10 l + c y c l e s / s e c ( a ) - ( b ) , ^ = 6 x 1 0 ^ c y c l e s / s e c ( c ) - ( d ) , and (1) y=0. (2) R, (3) R/2, and (k) 3 R / 2 cm. 38. E v f o r t r a v e r s e s a l o n g (a) y=0, (b) y=R, H. f o r 108 x y t r a v e r s e s a l o n g ( c ) y=0$ (d) y=R, f o r a v e r t i c a l LL g r a p h i t e c y l i n d e r f o r f=3x10 c y c l e s / s e c , and (1) d ^ l , (2) 8 .6 , (3) 16.2-, and (>+) 23>.8 cm. h • 39- E v (a) f o r f=3x10 c y c l e s / s e c , f o r ( b ) - ( c ) 109 3x10 c y c l e s / s e c , ( d ) - ( e ) 6x10 c y c l e s / s e c , f o r a v e r t i c a l c o n c r e t e c y l i n d e r f o r d^=1 cm and t r a v e r s e s a l o n g (1) y=0, (2) R ? (3) R/2, and ( h ) 3R/2. h O . E „ (a) and H (b) f o r a h o r i z o n t a l g r a p h i t e c y l i n d e r 111 x y and E „ ( c ) f o r a h o r i z o n t a l c o n c r e t e c y l i n d e r f o r f=3x1 CP" c y c l e s / s e c ' , d^=1 cm, and t r a v e r s e s a l o n g y=0. V l . H y , H z ( a ) , ' E x ( b ) , \ |/ x - 0 y ( c ) , 0 z - 0 y (d) and E x / H y 11"+ (e) f o r t he H p o l a r i z a t i o n f o r a g r a p h i t e dyke w i t h w=23 cm. ( x ) F i g u r e Page 'te. Hy, H z ( a ) , E x ( b ) , t x - 0 y ( c ) E ^ H y , H y / H z ( d ) , 116 and, 0„?-frr (e ) f o r t he E p o l a r i z a t i o n f o r a g r a p h i t e Z 7 dyke w i t h w = 2 3 cm. k 3 . E Y (a ) f o r t h e H p o l a r i z a t i o n , H- , H , (b) and 1 1 8 x . i y z E ( c ) f o r t he E p o l a r i z a t i o n f o r a g r a p h i t e dyke w i t h w = 2 9 cm, and ( 1 ) d = 1 , ( 2 ) 6, ( 3 ) 1 1 , and (if) 16 cm. k I f . E x (a ) f o r t he H p o l a r i z a t i o n w i t h d = 5 cm f o r 1 2 0 ( 1 ) w = 5 , ( 2 ) 1 0 , ( 3 ) 1 5 , ( L ) 2 0 , ( 5 ) 2 5 , (6) 3 0 , and ( 7 ) 100 cm; H y , H z (b) and E x ( c ) f o r the E p o l a r i z a t i o n w i t h d = 2 9 cm, f o r ( 1 ) w = 0 . 2 , ( 2 ) 1 , and ( 3 ) 5 cm f o r a g r a p h i t e dyke . i f5 . • Hy, H z ( a ) , E x ( b ) , H z /Hy ( c ) ^ x - $ y ( d ) , E ^ H y ( e ) , 1 2 2 and 4> f o r t he H p o l a r i z a t i o n f o r a g r a p h i t e y c y l i n d e r w i t h d = 7 . 6 cm, and R = 3 0 o k cm f o r t r a v e r s e s a l o n g (1) y = 0 , ( 2 ) R / 2 , ( 3 ) R cm. I f6 . Hy, H z ( a ) , E x ( b ) ? H z / H y ( c ) , V H y ( d ) ' V 0 y ( e ) 1 2 3 and 0 - # „ f o r the E p o l a r i z a t i o n f o r a g r a p h i t e £ 7 c y l i n d e r w i t h d = 7 . 6 cm and R = 3 0 . L i - cm f o r t r a v e r s e s a l o n g ( - 1 ) x = 0 , ( 2 ) R / 2 , and ( 3 ) R cm. If7-. H , H_, E , H /H , and E /H f o r the E p o l a r i z a t i o n 1 2 5 7 x' • " 7 X 1 y f o r a c y l i n d r i c a l g r a p h i t e s h e l l 2 cm t h i c k w i t h d = 7 . 6 cm and R = 3 0 . i + cm f o r t r a v e r s e s a l o n g ( 1 ) x = 0 , and ( 2 ) x = R / 2 cm. If8. H , H , E_, H / H . and E /H. f o r the E p o l a r i z a t i o n 126 y z x " z y x y f o r a c y l i n d r i c a l g r a p h i t e s h e l l 2 cm t h i c k w i t h d = 7 o 6 cm and R = 3 0 „ l f cm f o r t r a v e r s e s a l o n g (1) x=R-1 , ( 2 ) x=R cm. ! + 9 . The a m p l i t u d e s and phase a n g l e s f o r t r a v e r s e s o v e r 1 3 3 an i n v e r t e d t r u n c a t e d cone ; f o r the: E p o l a r i z a t i o n . and f r e q u e n c i e s (a) 3x10 , (b) 10 c y c l e s / s e c . ( x i ) F i g u r e 50; T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a n i n v e r t e d t r u n c a t e d c o n e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x103, ( b ) 103 c y c l e s / s e c . 51• T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a n i n v e r t e d t r u n c a t e d c o n e f o r t h e H p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x10 ( b ) 10- c y c l e s / s e c . 52. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a w e d g e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s • ( a ) 3x10̂ ,' ( b ) 10^ c y c l e s / s e c . 53. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a w e d g e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x1035 ( b ) 103 c y c l e s / s e c * . 5k. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a w e d g e f o r t h e H p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x10^, ( b ) 103 c y c l e s / s e c . 55- • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - a t i o n a n d f=3x10 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e (1), t h e b l o c k a l o n e (2). 56. • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - LL a t i o n a n d f=10 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e (-1), • t h e b l o c k a l o n e (2).. 57- T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - a t i o n a n d f=3x103 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e (1) ? t h e b l o c k a l o n e (2). 58. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - a t i o n a n d f=103 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e ( I ) , 1 t h e b l o c k a l o n e (2). ( x i i ) F i g u r e Page 59 • The a m p l i t u d e s and phase a n g l e s f o r t he E p o l a r i z - 151 a t i o n f o r t r a v e r s e s o v e r t he wedge w i t h the wedge edge d i r e c t i y o v e r t he b l o c k edge and f r e q u e n c i e s (a) 3x1 oV (b) I O 3 c y c l e s / s e c * . 60. The a m p l i t u d e s and phase a n g l e s f o r t he E p o l a r i z - 153 a t i o n f o r t r a v e r s e s o v e r t he wedge w i t h t he b l o c k edge e x t e n d i n g 10 cm beyond the wedge edge and f r e q u e n c i e s (a) 3x10 s (b) 10° c y c l e s / s e c . 61 . The a m p l i t u d e s and phase a n g l e s f o r the H p o l a r i z - 155 a t i o n f o r t r a v e r s e s o v e r the wedge e x t e n d i n g 10 cm beyond the b l o c k edge and f r e q u e n c i e s (a) 3x10 , (b) 1 0 3 c y c l e s / s e c . 62 . The a m p l i t u d e s and phase a n g l e s f o r t he H p o l a r i z - 157 a t i o n f o r t r a v e r s e s o v e r t he wedge w i t h t he wedge edge d i r e c t l y o v e r t he b l o c k edge and f r e q u e n c i e s . ( a ) 3x10^, (b) 1 0 3 c y c l e s / s e c . 63. The a m p l i t u d e s and phase a n g l e s f o r t he H p o l a r i z - 158 a t i o n f o r t r a v e r s e s o v e r t he wedge w i t h the b l o c k edge e x t e n d i n g 10 cm beyond t he wedge edge and f r e q u e n c i e s (a) 3x10 s (b) 10-3 c y c l e s / s e c . 6^-. Map o f Texada I s l a n d and t he S t r a i t o f G e o r g i a . 160 65. The a m p l i t u d e s and phase a n g l e s f o r the E p o l a r i z - 163 a t i o n f o r (a) t r a v e r s e 0 and (b) t r a v e r s e 1„ 66. The a m p l i t u d e s and phase a n g l e s f o r the E p o l a r i z - 16*+ a t i o n f o r (a) t r a v e r s e 3 and (b) t r a v e r s e 5« 67. The a m p l i t u d e s and phase a n g l e s f o r t he H p o l a r i z - 166 a t i o n f o r (a) t r a v e r s e 0 and (b) t r a v e r s e 1„ 68 . The a m p l i t u d e s and phase a n g l e s f o r t he H p o l a r i z - 167 a t i o n f o r (a) t r a v e r s e 3 and (b) t r a v e r s e 5. 69. • The a m p l i t u d e s and phase a n g l e s f o r t r a v e r s e s i n 170 t h e y d i r e c t i o n o v e r a g r a p h i t e cone f o r ) 1 ) 1 f r e q u e n c i e s (a) 3x10 ?- (b) 10 c y c l e s / s e c . ( x i i i ) F i g u r e P a g e 70. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s i n 172 , t h e x d i r e c t i o n o v e r a g r a p h i t e c o n e f o r f r e q u e n c i e s ( a ) 3x10 , ( b ) 10 c y c l e s / s e c . 71. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s i n 173 t h e y d i r e c t i o n o v e r a c o n c r e t e c o n e f o r f r e q u e n c i e s ( a ) 3x10 ( b ) 10 J c y c l e s / s e c . 72. T h e a m p l i t u d e s f o r t r a v e r s e s i n t h e y d i r e c t i o n 178 f o r a n o v e r h e a d o s c i l l a t i n g l i n e c u r r e n t . 73. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 179 c u r r e n t a n d t r a v e r s e s i n t h e y d i r e c t i o n o v e r a n i n v e r t e d t r u n c a t e d g r a p h i t e c o n e f o r k k f r e q u e n c i e s ( a ) 3x10 5 ( b ) 10 c y c l e s / s e c . 7*+. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 180 c u r r e n t a n d t r a v e r s e s i n t h e y d i r e c t i o n o v e r a n i n v e r t e d t r u n c a t e d g r a p h i t e c o n e f o r f r e q u e n c i e s ( a ) 3x10 3 , ( b ) 10^ c y c l e s / s e c - . 7 5 - T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 182 c u r r e n t a n d t r a v e r s e s i n t h e x d i r e c t i o n o v e r a n i n v e r t e d t r u n c a t e d g r a p h i t e c o n e f o r k O f r e q u e n c i e s ( a ) 3x10 ? ( b ) 10^ c y c l e s / s e c . 76. - T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 18U- c u r r e n t a n d t r a v e r s e s o v e r a w e d g e f o r t h e E k p o l a r i z a t i o n a n d f r e q u e n c i e s - - ( a ) 3x10 5 k ( b ) 10 c y c l e s / s e c o 77« T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 185 c u r r e n t a n d t r a v e r s e s o v e r a w e d g e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x10 3 , ( b ) 10 3 c y c l e s / s e c . 78. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 186 c u r r e n t a n d t r a v e r s e s o v e r a w e d g e f o r t h e H p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x10^, ( b ) 10 3 c y c l e s / s e c . ( x i v ) F i g u r e P a g e 79- T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - 188 L. a t i o n a n d a l i n e c u r r e n t w i t h f=3x10 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e (1), t h e b l o c k a l o n e (2). 80. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - 189 LL a t i o n a n d a l i n e c u r r e n t w i t h f=10 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d Ob) t h e w e d g e a l o n e (1), t h e b l o c k a l o n e (2). 81. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - 190 a t i o n a n d a l i n e c u r r e n t w i t h f=3x10J c y c l e s / s e c - f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e ( 1 ) , t h e b l o c k a l o n e (2).> • 82. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - 191 a t i o n a n d a l i n e c u r r e n t w i t h f=103 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e (1) , t h e b l o c k a l o n e (2). 83. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 193 c u r r e n t a n d t h e E p o l a r i z a t i o n f o r t r a v e r s e s o v e r t h e w e d g e w i t h t h e w e d g e e d g e d i r e c t l y o v e r t h e b l o c k e d g e a n d f r e q u e n c i e s ( a ) 3x1 o \ ( b ) 10^ c y c l e s / s e c . 8̂ . ' T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 19^ c u r r e n t a n d t h e E p o l a r i z a t i o n f o r t r a v e r s e s o v e r t h e w e d g e w i t h t h e b l o c k e d g e e x t e n d i n g 10 c m LL b e y o n d t h e w e d g e e d g e a n d f r e q u e n c i e s ( a ) 3x10 , ( b ) 103 c y c l e s / s e c . 85. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e 196 c u r r e n t a n d t h e H p o l a r i z a t i o n f o r t r a v e r s e s o v e r ( xv ) F i g u r e Page a wedge e x t e n d i n g 10 cm beyond t he b l o c k edge and f r e q u e n c i e s (a) 3x1o\ (b) 103 c y c l e s / s e c . 86. The a m p l i t u d e s and phase a n g l e s f o r a l i n e 197 c u r r e n t and the H p o l a r i z a t i o n f o r t r a v e r s e s o v e r the wedge w i t h t he wedge edge d i r e c t l y o v e r t he b l o c k edge and f r e q u e n c i e s (a) 3x10 , (b) 10J c y c l e s / s e c . 87. The a m p l i t u d e s and phase a n g l e s f o r a l i n e 198 c u r r e n t and the H p o l a r i z a t i o n f o r t r a v e r s e s o v e r t he wedge w i t h t he b l o c k edge e x t e n d i n g 10 cm beyond t he wedge edge and f r e q u e n c i e s (a ) 3x10^, • (b) 103 c y c l e s / s e c . 88. The a m p l i t u d e s and phase a n g l e s f o r the E p o l a r i z - 200 a t i o n and a l i n e c u r r e n t w i t h f=3x10lf c y c l e s / s e c : f o r t r a v e r s e s o v e r t he wedge w i t h t h e b l o c k edge e x t e n d i n g 10 cm beyond the wedge edge f o r wedge edge p o s i t i o n s (a) y = r-5? (b) y = -̂ 0 cm. ( x v i ) L IST OF TABLES T a b l e Page I. The e f f e c t o f v a r y i n g the number o f s u b l a y e r s . 61 I I . C o n d u c t i v i t y i n c r e a s i n g w i t h d e p t h . 62 I I I . C o n d u c t i v i t y d e c r e a s i n g w i t h d e p t h . 63 IV.. C o n d u c t i v i t y d e c r e a s i n g w i t h d e p t h be l ow a 65 c o n d u c t i n g s e a . V . C o n d u c t i v i t y i n c r e a s i n g w i t h depth,, 69 V I . C o n d u c t i v i t y d e c r e a s i n g be low a u n i f o r m 70 c o n d u c t i n g s e a . V I I . M o d e l d i m e n s i o n s . 103 V I I I . G e o p h y s i c a l d i m e n s i o n s . 103 ( x v i i ) A C K N O W L E D G E M E N T S I t i s a p l e a s u r e t o a c k n o w l e d g e t h e h e l p f u l d i s - c u s s i o n s w i t h D r . J o A . J a c o b s a n d D r . T . W a t a n a b e o n v a r i o u s a s p e c t s o f t h i s r e s e a r c h . I w i s h t o t h a n k D r . J . A . J a c o b s f o r h i s c a r e f u l r e v i e w i n g o f t h e m a n u s c r i p t . I a m a l s o i n d e b t e d t o D r . T . W a t a n a b e a n d D r . R . M . E l l i s f o r r e v i e w i n g p a r t s o f t h e m a n u s c r i p t . I w o u l d a l s o l i k e t o a c k n o w l e d g e u s e f u l d i s c u s s i o n s w i t h D r . - J . E i L o k k e n , D r . S . Z . M a c k , D r . K . C . M a c l u r e , a n d D r . J . T . W e a v e r o f t h e P a c i f i c N a v a l L a b o r a t o r y , E s q u i m a l t , B . C , a n d M r . B . C a n e r o f t h e V i c t o r i a M a g n e t i c O b s e r v a t o r y I h a v e g r e a t l y a p p r e c i a t e d t h e e n c o u r a g e m e n t o f m y w i f e , - M a r t h a , t h r o u g h o u t t h e t i m e o f t h i s r e s e a r c h . I w i s h t o t h a n k M r s . M . G e l l i n g f o r h e r c a r e f u l w o r k i n t y p i n g t h i s t h e s i s . T h i s r e s e a r c h w a s s u p p o r t e d b y g r a n t s f r o m t h e D e f e n c e R e s e a r c h B o a r d o f C a n a d a a n d t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a . 1 C h a p t e r 1. I N T R O D U C T I O N 1.1 H i s t o r i c a l R e v i e w T h e n a t u r a l l y o c c u r r i n g e l e c t r o m a g n e t i c v a r i a t i o n s o b - s e r v e d a t t h e s u r f a c e o f t h e e a r t h a r e o f c o n t i n u i n g i n t e r e s t i n g e o m a g n e t i c a n d m a g n e t o t e l l u r i c s t u d i e s , , I t i s w e l l k n o w n t h a t t h e c o n d u c t i v i t y s t r u c t u r e o f t h e e a r t h , a s w e l l a s t h e s o u r c e , p l a y s a n i m p o r t a n t r o l e i n a f f e c t i n g t h e n a t u r e o f t h e s e v a r i a t i o n s . A n i n c r e a s e d u n d e r s t a n d i n g o f t h i s n a t u r a l e l e c t r o m a g n e t i c b a c k g r o u n d l e a d s t o a b e t t e r u n d e r s t a n d i n g o f t h e n a t u r e o f t h e s o u r c e o f t h e s e v a r i a t i o n s a n d t h e s u b s u r f a c e s t r u c t u r e o f t h e e a r t h . T h e p o s s i b i l i t y o f o b t a i n i n g i n f o r m a t i o n o n t h e d i s t r i - b u t i o n o f e l e c t r i c a l c o n d u c t i v i t y w i t h i n t h e e a r t h , f r o m m e a s u r e m e n t s o f t h e m a g n e t i c v a r i a t i o n s a t t h e e a r t h ' s s u r f a c e , w a s c o n s i d e r e d a s e a r l y a s 1889 b y S c h u s t e r . S i n c e t h a t t i m e m a n y w o r k e r s h a v e i n v e s t i g a t e d t h e e f f e c t s o f v a r i o u s s o u r c e f i e l d s a n d c o n d u c t i v i t y d i s t r i b u t i o n s b o t h f o r s p h e r i c a l a n d f l a t s t r u c t u r e s . S o m e o f t h e e a r l y g e n e r a l a n a l y s i s w a s d o n e b y C h a p m a n (1919)? C h a p m a n a n d W h i t e h e a d (1922), C h a p m a n a n d P r i c e (193O), P r i c e (1929, 1930), a n d L a h i r i a n d P r i c e (1939). B e g i n n i n g i n t h e 1920 ! s c o n s i d e r a b l e a t t e n t i o n w a s g i v e n t o t h e u s e o f t h e t e l l u r i c m e t h o d f o r g e o l o g i c a l e x p l o r - a t i o n . T h i s m e t h o d r e q u i r e s a r e f e r e n c e s t a t i o n u s e d i n c o n j u n c t i o n w i t h a m o v a b l e s t a t i o n . P o t e n t i a l g r a d i e n t s a r e d e t e r m i n e d a t t h e s u r f a c e b y m e a s u r i n g p o t e n t i a l d i f f e r e n c e s b e t w e e n p a i r s o f e l e c t r o d e s e m b e d d e d i n t h e e a r t h . B o t h K a t o a n d K i k u c h i (1950) a n d T i k h o n o v (1950) s u g g e s t e d t h a t t h e 2 e l e c t r i c a l c h a r a c t e r i s t i c s o f t h e e a r t h ' s c r u s t a l s t r u c t u r e c o u l d b e d e t e r m i n e d b y t h e c o m b i n e d a n a l y s i s o f g e o m a g n e t i c a n d t e l l u r i c f i e l d v a r i a t i o n s . C a g n i a r d (1953) o u t l i n e d t h e t h e o r y o f t h i s s o - c a l l e d m a g n e t o t e l l u r i c m e t h o d f o r p l a n e w a v e s i n c i d e n t o n a h o r i z o n t a l l y s t r a t i f i e d c o n d u c t o r . W a i t (195 k ) a n d P r i c e (1962) h a v e d i s c u s s e d t h e l i m i t a t i o n s i n C a g n i a r d ' s r e s u l t s w i t h r e s p e c t t o t h e d i m e n s i o n s o f t h e s o u r c e f i e l d . C a g n i a r d ' s t h e o r y r e q u i r e s t h a t t h e f i e l d s b e e s s e n t i a l l y c o n s t a n t f r o m t h e p o i n t o f o b s e r v a t i o n t o a h o r i z o n t a l d i s t a n c e o f t h e o r d e r o f a " s k i n d e p t h " i n t h e u n i f o r m c o n d u c t o r o f i n f i n i t e d e p t h . T h e m a g n e t o t e l l u r i c m e t h o d c o n s i s t s o f s i m u l t a n e o u s m e a s u r e m e n t s o f t h e h o r i z - o n t a l e l e c t r i c a n d h o r i z o n t a l m a g n e t i c f i e l d c o m p o n e n t s o f t h e n a t u r a l e l e c t r o m a g n e t i c f i e l d a t t h e e a r t h ' s s u r f a c e . T h e e l e c t r i c f i e l d i s m e a s u r e d a s a p o t e n t i a l g r a d i e n t b e t w e e n t w o p o i n t s o n t h e s u r f a c e o f t h e e a r t h . • T h e m a g - n e t i c f i e l d a l s o i s m e a s u r e d a t t h e e a r t h ' s s u r f a c e . T h e r a t i o o f t h e e l e c t r i c f i e l d t o t h e m a g n e t i c f i e l d a s a f u n c t i o n o f f r e q u e n c y p r o v i d e s i n f o r m a t i o n o n t h e c o n d u c t - i v i t y d i s t r i b u t i o n i n h o r i z o n t a l l a y e r s . T h e p h a s e d i f f e r - e n c e b e t w e e n t h e e l e c t r i c a n d m a g n e t i c f i e l d s i s a l s o d i a g n o s t i c o f t h e s t r u c t u r e . T i k h o n o v a n d L i p s k a i a (1952) 5 R i k i t a k e (1951)? a n d L i p s k a i a - (1953) h a v e t r e a t e d v a r i o u s a s p e c t s o f t h e s i g n i f i c a n c e o f a m p l i t u d e r a t i o s a n d p h a s e d i f f e r e n c e s a s r e l a t e d t o t h e s u b s u r f a c e s t r u c t u r e . T h e m a g n e t o t e l l u r i c m e t h o d h a s b e e n u s e d w i t h v a r y i n g d e g r e e s o f s u c c e s s a s a m e t h o d o f e x p l o r i n g t h e s u b s u r f a c e n a t u r e 3 o f t h e e a r t h o T h e w i d e r a n g e o f f r e q u e n c i e s p r o v i d e d b y n a t u r e m a k e p o s s i b l e s o m e s t u d y o f t h e e a r t h ' s s t r u c t u r e e v e n t o g r e a t d e p t h s . A t t h e p r e s e n t t i m e a n o t h e r m e t h o d m o r e c o m m o n l y u s e d i n s t u d y i n g t h e p r o b l e m o f s o u r c e f i e l d s a n d t h e c o n d u c t i v i t y d i s t r i b u t i o n i n t h e e a r t h i s b a s e d o n t h e a n a l y s i s o f t h e t h r e e m a g n e t i c f i e l d c o m p o n e n t s o f t h e n a t u r a l e l e c t r o m a g n e t i c v a r i a t i o n s . T h i s m e t h o d e m p l o y s m e a s u r e m e n t s f r o m s e v e r a l s t a t i o n s t o g i v e a s p a t i a l d i s t r i b u t i o n . T h e r e l a t i v e a m p l i - t u d e s o f t h e h o r i z o n t a l a n d v e r t i c a l f i e l d c o m p o n e n t s a n d t h e p h a s e d i f f e r e n c e s a r e d i a g n o s t i c o f b o t h t h e c o n d u c t i v i t y d i s t r i b u t i o n w i t h i n t h e e a r t h a n d t h e s o u r c e f i e l d s . T h i s m e t h o d i s m o r e a p p r o p r i a t e t h a n t h e m a g n e t o t e l l u r i c m e t h o d f o r s t u d y i n g c e r t a i n p r o b l e m s w h e r e a l a r g e h o r i z o n t a l g r a d i e n t i n c o n d u c t i v i t y e x i s t s o r w h e r e d i s c o n t i n u i t i e s i n t h e h o r i z - o n t a l d i r e c t i o n o c c u r ( c o a s t l i n e , f a u l t s , a n d d y k e s ) . A m o n g t h e e l e c t r o m a g n e t i c v a r i a t i o n s o b s e r v e d a n d s t u d i e d w i d e l y a t p r e s e n t a r e t he w e l l k n o w n t y p e c a l l e d " m i c r o p u l s a t i o n s l , . T h e t e r m g e o m a g n e t i c m i c r o p u l s a t i o n s r e f e r s t o f l u c t u a t i o n s i n t h e e a r t h ' s m a g n e t i c f i e l d w h i l e t h e a s s o c i a t e d e l e c t r i c f i e l d v a r i a t i o n s m a y be d e s c r i b e d a s m i c r o - p u l s a t i o n s i n t h e e a r t h ' s e l e c t r i c f i e l d . T h e p e r i o d s o f t h e s e p u l s a t i o n s a r e i n the r a n g e f r o m a b o u t 0.1 s e c o n d t o 10 m i n u t e s ( f r e q u e n c y r a n g e f r o m a b o u t 10 c y c l e s / s e c t o JOT-* c y c l e s / s e c ) . T h e g e o m a g n e t i c m i c r o p u l s a t i o n s h a v e i n the p a s t c o m m o n l y b e e n c l a s s i f i e d i n t o t h r e e m a i n c a t e g o r i e s : P c , P t , a n d P g . T h e P c ( c o n t i n u o u s p u l s a t i o n s . ) m i c r o p u l s a t i o n s a r e a h c o n t i n u o u s s e r i e s o f g e o m a g n e t i c v a r i a t i o n s o f s m a l l a m p l i t u d e , ( m u c h l e s s t h a n 1 y , 1 y = 1 0 " ^ g a u s s ) a n d l a s t i n g a s l o n g a s s e v e r a l h o u r s . T h e r a n g e o f p e r i o d s f o r P c ' s t e n d s t o b e b e t w e e n 1 0 a n d 6 0 s e c ( r o u g h l y i n t h e f r e q u e n c y r a n g e 0 . 1 t o — 2 1 . 6 x 1 0 c y c l e s / s e c ) . . T h e P t ( p u l s a t i o n t r a i n ) m i c r o p u l s a t i o n s a r e s e p a r a t e s e r i e s o f d a m p e d o s c i l l a t i o n s . T h e r a n g e o f p e r i o d s o f t h e s e o s c i l l a t i o n s t e n d s t o b e b e t w e e n *+0 a n d 1 0 0 s e c ( r o u g h l y i n t h e f r e q u e n c y r a n g e 2.5 x 1 0 t o 1 0 c y c l e s / s e c ) . T h e a m p l i - t u d e s o f t h e P t m i c r o p u l s a t i o n s a r e i n g e n e r a l l a r g e r t h a n t h o s e o f t h e P c t y p e b u t a r e s t i l l l e s s t h a n l y * T h e P g ( g i a n t p u l s a t i o n s ) m i c r o p u l s a t i o n s a r e a s e r i e s o f o s c i l l a t i o n s o f m u c h l a r g e r a m p l i t u d e s t h a n t h o s e o f t h e P c a n d P t t y p e s ( 1 O y o r g r e a t e r ) . T h e p e r i o d s o f t h e P g t y p e t e n d t o b e m u c h l o n g e r ( 1 t o 1 0 m i n u t e s , o r r o u g h l y i n t h e —2 ' — ̂ f r e q u e n c y r a n g e 1 0 t o . 1 0 ° c y c l e s / s e c ) , t h a n t h o s e o f t h e o t h e r t w o t y p e s . J a c o b s a n d W e s t p h a l (1963) a n d J a c o b s (196*t) h a v e r e c e n t l y m a d e e x t e n s i v e r e v i e w s o f t h e s t a t e o f k n o w l e d g e o f g e o m a g n e t i c m i c r o p u l s a t i o n s . I n t h e s e w o r k s , a s w e l l a s i n o t h e r s [ e . g . M a t s u s h i t a ( 1 9 6 3 ) ] 9 i t w a s p o i n t e d o u t t h a t t h e c l a s s i f i c a t i o n d i v i d i n g m i c r o p u l s a t i o n s i n t o t h r e e c a t e g o r i e s , P c , P t , a n d P g , w a s t o o b r o a d . I n f o r m a t i o n c o l l e c t e d d u r i n g t h e I n t e r n a t i o n a l G e o p h y s i c a l Y e a r ( I . G . Y o ) i n d i c a t e d t h a t t h e r e w e r e m a n y m o r e t y p e s o f v a r i a t i o n s t h a n t h e P c , P t , a n d P g c l a s s i f i c a t i o n i n d i c a t e d . A s a : r e s u l t o f t h i s b r o a d c l a s s i f i c a t i o n a g r e a t v a r i e t y o f n o n - u n i f o r m n o t a t i o n s a n d 5 d e f i n i t i o n s d e v e l o p e d f o r t h e d i f f e r e n t p h e n o m e n a . A n a t t e m p t a t f o r m a l i z i n g t h e s y m b o l i s m a n d n o t a t i o n w i t h t h e v i e w o f e n c o u r a g i n g m o r e c o n s i s t e n t u s a g e w a s m a d e b y a n a d h o c c o m m i t t e e o f t h e I n t e r n a t i o n a l A s s o c i a t i o n o f G e o m a g n e t i s m a n d A e r o n o m y ( I . A . G ' . A . ) a t t h e I n t e r n a t i o n a l U n i o n o f G e o d e s y a n d G e o p h y s i c s ( I . U . G . G . ) g e n e r a l a s s e m b l y h e l d a t B e r k e l e y i n A u g u s t , 1963. T h e r e s u l t i n g c l a s s i f i c a t i o n a n d n o t a t i o n o f g e o m a g n e t i c m i c r o p u l s a t i o n s w h i c h w a s a d o p t e d h a s b e e n d i s c u s s e d b y J a c o b s e t a l (196*+)° G e o m a g n e t i c m i c r o p u l s a t i o n s w e r e c l a s s i f i e d i n t o t w o m a i n c a t e g o r i e s ; P c - ( c o n t i n u o u s m i c r o p u l s a t i o n s 1 ) a n d P i ( i r r e g u l a r m i c r o p u l s a t i o n s ) . T h e s u b d i v i s i o n s i n t h e s e t w o c a t e g o r i e s a r e a s f o l l o w s : • f ( s e c " 1 ) P i T ( s e c ) f ( s e c " 1 ) P C T ( s e c ) P c 1 0.2-5 P c 2 5-10 P c 3 10-1+5 P c h 1+5-1 50 P c 5 150-600 5-0.2 P i 1 1 -1+0 1-0.025 0.2-0.1 P i 2 1+0-150 0 . 0 2 5 - 0 . 0 0 6 0.1-0.022 M i c r o p u l s a t i o n s o b s e r v e d a t t h e e a r t h ' s s u r f a c e a r e b e l i e v e d t o o r i g i n a t e o u t s i d e t h e e a r t h , t h e s o u r c e o f t h e e n e r g y b e i n g t h e s u n . I t i s n o w c o m m o n l y b e l i e v e d t h a t t h e i n t e r a c t i o n o f t h e e n e r g y , e m i t t e d f r o m t h e s u n , w i t h t h e c h a r g e d p a r t i c l e s o f t h e i o n o s p h e r e a n d t h e f a r r e a c h i n g e a r t h ' s m a g n e t i c f i e l d , r e s u l t s i n i o n o s p h e r i c e l e c t r i c c u r - r e n t s , c a u s i n g e l e c t r o m a g n e t i c d i s t u r b a n c e s o b s e r v a b l e a t t h e e a r t h ' s s u r f a c e . S t u d i e s h a v e b e e n m a d e b y v a r i o u s w o r k e r s t o d e t e r m i n e t h e p o s i t i o n s i n t h e i o n o s p h e r e o f s u c h c u r r e n t s y s t e m s a n d t o d e t e r m i n e t h e r e l a t i o n s h i p b e t w e e n m a g n e t i c f i e l d v a r i a t i o n s a t t h e e a r t h ' s s u r f a c e a n d i o n o s p h e r i c 6 c u r r e n t s y s t e m s ( e . g . J a c o b s a n d S i n n o 1960 ) , H a s e g a w a 1960, Z m u d a (1960? C a m p b e l l a n d R e e s 1961, B o m k e 1962, a n d m a n y o t h e r s ) . T h e v a r i a t i o n s o b s e r v e d a t t h e e a r t h ' s s u r f a c e w i l l d e p e n d b o t h o h t h e n a t u r e o f t h e s o u r c e ( c u r r e n t d i s t r i - b u t i o n i n t h e i o n o s p h e r e ) a n d o n t h e e l e c t r i c a n d m a g n e t i c p r o p e r t i e s o f t h e e a r t h . T h e s e v a r i a t i o n s a r e s t u d i e d i n a n a t t e m p t t o i n v e s t i g a t e b o t h t h e n a t u r e o f t h e s o u r c e a n d t h a t o f t h e i n t e r i o r o f t h e e a r t h . I n r e c e n t y e a r s m u c h w o r k h a s b e e n d o n e i n a n f t t e m p t t o e x p l a i n g e o m a g n e t i c m i c r o p u l s a t i o n s i n t e r m s o f h y d r o m a g n e t i c w a v e s ( e . g v , J a c o b s a n d W a t a n a b e 1962, J a c o b s a n d W e s t p h a l 1 96*+, - m a n y r e f e r e n c e s a r e g i v e n i n t h e s e p a p e r s ) . I t i s b e l i e v e d t h a t h y d r o m a g n e t i c w a v e s , o r i g i n - a t i n g i n o u t e r s p a c e a n u m b e r o f e a r t h r a d i i d i s t a n t f r o m t h e e a r t h , p r o p a g a t e t o p o l a r r e g i o n s a l o n g m a g n e t i c f i e l d l i n e s . ( T h e h y d r o m a g n e t i c w a v e s , u n a b l e - t o p r o p a g a t e t h r o u g h t h e i o n o s p h e r e a n d a t m o s p h e r e o f t h e e a r t h , a r e t r a n s f o r m e d i n t o e l e c t r o m a g n e t i c w a v e s a s t h e y p a s s i n t o t h e i o n o s p h e r e . T h e s e e l e c t r o m a g n e t i c w a v e s t r a v e l t o t h e e a r t h ' s s u r f a c e w h e r e t h e y a r e - o b s e r v e d a s g e o m a g n e t i c m i c r o p u l s a t i o n s . I t i s - a l s o b e l i e v e d t h a t h y d r o m a g n e t i c w a v e s e n t e r i n g t h e i o n o s p h e r e w i l l i n p a r t b e r e f l e c t e d a s w e l l a s g i v i n g r i s e t o s e c o n d a r y w a v e s ( m o d i f i e d A l f v e n w a v e s ) w h i c h d o n o t p r o p a g a t e a l o n g - f l i p i d l i n e s , b u t t r a v e l o b l i q u e l y t o , o r p e r p e n d i c u l a r t o t h e f i e l d l i n e s t o l o w e r l a t i t u d e s . T h e s e w a v e s a l s o r e s u l t i n e l e c t r o m a g n e t i c w a v e s w h i c h a r e o b s e r v e d a s g e o m a g n e t i c m i c r o p u l s a t i o n s o v e r t h e e a r t h ' s s u r f a c e . 7 T h e a n a l y s i s o f m i c r o p u l s a t i o n : m e a s u r e m e n t s i s i n g e n e r a l c o m p l i c a t e d b y t h e f a c t t h a t b o t h t h e n a t u r e o f t h e s o u r c e o f t h e v a r i a t i o n s a n d t h e c o n d u c t i v i t y d i s t r i b u t i o n o f t h e e a r t h ' s i n t e r i o r a r e i n m a n y r e s p e c t s n o t s u f f i c i e n t l y w e l l u n d e r s t o o d t o a l l o w a s p e c t s o f t h e v a r i a t i o n s t o b e a t t r i b u t e d t o e i t h e r t h e n a t u r e o f t h e s o u r c e o r t o t h e i n h o m o g e n e i t i e s i n t h e c o n d u c t i v i t y s t r u c t u r e o f t h e e a r t h - W i t h t h i s p r o b l e m i n m i n d m a n y w o r k e r s h a v e s t u d i e d m a t h e m a t i c a l m o d e l s f o r s p e c i a l c a s e s a s s u m i n g s p e c i f i c f i e l d s o u r c e s a n d c o n d u c t i v i t y d i s t r i b u t i o n s . : C o m p a r i s o n s o f t h e r e s u l t s f o r t h e s e s p e c i a l c a s e s w i t h a c t u a l f i e l d m e a s u r e m e n t s y i e l d s o m e i n f o r m a t i o n o n p l a u s i b l e s o u r c e f i e l d s a n d c o n d u c t i v i t y s t r u c t u r e s . T h e m a t h e m a t i c a l s t u d i e s o f s o u r c e f i e l d s u s u a l l y i n - v o l v e d a s i m p l i f i e d e a r t h s t r u c t u r e c o n s i s t i n g o f a f l a t c o n - d u c t i n g e a r t h o f i n f i n i t e d e p t h o r a h o r i z o n t a l l y l a y e r e d c o n d u c t i n g e a r t h . S e v e r a l t y p e s o f s o u r c e s s t u d i e d b y m a n y w o r k e r s a r e : p l a n e w a v e s o u r c e s - ( r e q u i r i n g a v e r y d i s t a n t s o u r c e o r a v e r y l a r g e o s c i l l a t i n g c u r r e n t s h e e t ) , d i p o l e s , a n d l i n e c u r r e n t s . - F o r s i m p l i c i t y , t h e a n a l y s i s o f f i e l d m e a s u r e m e n t s i s o f t e n b a s e d o n t h e a s s u m p t i o n o f a p l a n e w a v e s o u r c e f i e l d . C a g n i a r d ' s - (1953) m u c h u s e d t h e o r y o f t h e m a g n e t o t e l l u r i c m e t h o d i s b a s e d o n t h e p l a n e w a v e a s s u m p t i o n . R e f i n e m e n t s t o C a g n i a r d ' s t h e o r y a n d t h e l i m i t a t i o n s o f t h e p l a n e w a v e a s s u m p t i o n h a v e b e e n d i s c u s s e d b y v a r i o u s w o r k e r s ( e . g . W a i t 195 k ? P r i c e 1962). W a i t (1962 a ) h a s r e v i e w e d t h e s t a t e o f k n o w l e d g e o f m a g n e t o t e l l u r i c f i e l d s a n d t h e m a g n e t o t e l l u r i c 8 m e t h o d o H e i n c l u d e d a l a r g e n u m b e r o f l a y e r e d e a r t h i n t e r - p r e t a t i o n c u r v e s . I n a r e c e n t p a p e r W a t a n a b e (196k) d i s c u s s e d t h e c h o i c e o f b o u n d a r y c o n d i t i o n s u s e d i n t h e m a g n e t o t e l l u r i c m e t h o d . S r i v a s t a v a (196-2) h a s m a d e a d e t a i l e d e x a m i n a t i o n o f t h e c o n d i t i o n s f o r t h e v a l i d i t y o f t h e m a g n e t o t e l l u r i c m e t h o d . A s p a r t o f t h i s s t u d y , h e a n a l y z e d , b y v a r i o u s m e t h o d s , f i e l d m e a s u r e m e n t s o f m a g n e t o t e l l u r i c f i e l d s f r o m a n u m b e r o f s t a t i o n s i n C e n t r a l A l b e r t a . E l l i s (196k) s t u d i e d t h e e a r t h ' s n a t u r a l e l e c t r o m a g n e t i c f i e l d i n t h e f r e q u e n c y r a n g e 0.001 t o 5 c y c l e s / s e c , w i t h t h e p u r p o s e o f a p p l y i n g e l e c t r o m a g n e t i c f i e l d m e a s u r e m e n t s t o s t u d y i n g t h e c o n d u c t i v e i t y o f t h e e a r t h a t d e p t h . H e u s e d p o w e r s p e c t r u m t e c h n i q u e s i n a n a l y z i n g t h e f i e l d m e a s u r e m e n t s . I n r e c e n t y e a r s c o n s i d e r - a b l e a t t e n t i o n h a s b e e n g i v e n t o u s i n g t h e v e r t i c a l a n d h o r i r - z o n t a l m a g n e t i c f i e l d m e a s u r e m e n t s f o r a n a l y s i s a s w e l l a s u s i n g - t h e m a g n e t o t e l l u r i c m e t h o d . T h e a s s u m p t i o n t h a t t h e m a g n e t i c - v a r i a t i o n s m a y b e d e s c r i b e d a p p r o x i m a t e l y i n t e r m s o f p l a n e w a v e s i n c i d e n t o n t h e e a r t h r e s u l t s i n t h e p r e d i c t i o n o f a v e r y s m a l l v e r t i c a l c o m p o n e n t c o m p a r e d w i t h t h e h o r i z o n t a l c o m p o n e n t . C a l c u l a t i o n s c a r r i e d o u t b y D o s s o a n d L o k k e n (1961) f o r p l a n e w a v e s i n c i d e n t o n a f l a t h o m o g e n e o u s c o n d u c t i n g e a r t h f o r v a r y i n g a n g l e s o f i n c i d e n c e a n d a w i d e r a n g e o f f r e q u e n c i e s a n d c o n d u c t i v i t i e s i n d i c a t e t h a t t h e m a g n i t u d e o f t h e v e r t i c a l c o m p o n e n t s h o u l d i n d e e d b e s m a l l . M e a s u r e m e n t s m a d e b y t h e P a c i f i c N a v a l L a b o r a t o r y a n d o t h e r s i n d i c a t e t h a t t h e a m p l i - t u d e r a t i o o f v e r t i c a l t o h o r i z o n t a l c o m p o n e n t s i s a b o u t 103 t i m e s g r e a t e r t h a n t h a t p r e d i c t e d b y t h e p l a n e w a v e m o d e l . 9 One e x p e c t s t he v e r t i c a l component t o be h i g h l y dependent on i n h o m o g e n e i t i e s and s t r a t i f i c a t i o n p l a n e s w i t h i n t he e a r t h ' s s t r u c t u r e . Measurements made n e a r c o a s t a l r e g i o n s , where a l a r g e c o n d u c t i v i t y g r a d i e n t and deep l y i n g i n h o m o g e n e i t i e s e x i s t , y i e l d l a r g e a m p l i t u d e s o f the v e r t i c a l component when compared w i t h v e r t i c a l components a t s t a t i o n s i n l a n d where a u n i f o r m g e o l o g i c a l s t r u c t u r e i s l i k e l y ( D u f f u s e t a l 1959? 1960, 1^62, P a r k i n s o n 1959, 1962, Shand e t a l 1959, C h r i s - t o f f e l e t a l 1961, Schmucker 1961*, S r i v a s t a v a and J acob s 196^, Lamber t and Caner 1965)» The v e r t i c a l t o h o r i z o n t a l magne t i c f i e l d r a t i o , however , even f o r i n l a n d s t a t i o n s where u n i f o r m h o r i z o n t a l l a y e r s a r e l i k e l y , i s much l a r g e r t h a n p r e d i c t e d by the p l a n e wave mode l . The p l a n e wave model does , however , r e a d i l y l e n d i t s e l f t o s t u d y i n g ma the - m a t i c a l l y t he e f f e c t o f i n h o m o g e n e i t i e s i n t he c o n d u c t i n g s u b s t r a t u m . T h i s a s p e c t o f t he p l a n e wave mode l w i l l be t r e a t e d l a t e r i n t h i s work . A v a r i e t y o f d i p o l e f i e l d s f o r a f l a t homogeneous c o n d u c t i n g e a r t h o r f o r a h o r i z o n t a l l y s t r a t i f i e d e a r t h have been t r e a t e d and d i s c u s s e d i n the l i t e r a t u r e ( e . g . Wo l f 19^6, W a i t 1951, 1958, Quon 1963, Weaver 1965). W a i t (1962 a) s t u d i e d t he r e spon se o f an o s c i l l a t i n g magne t i c d i p o l e o v e r a f l a t e a r t h w i t h the c o n d u c t i v i t y v a r y i n g e x p o n e n t i a l l y w i t h d e p t h . Quon O963) gave an e x t e n s i v e t r e a t m e n t f o r d i p o l e s o u r c e s o v e r a c o n d u c t i n g e a r t h . - He d e a l t w i t h b o t h v e r t i c a l and h o r i z o n t a l e l e c t r i c and magne t i c d i p o l e s o v e r a two l a y e r c o n d u c t i n g f l a t e a r t h . Weaver (1965) t r e a t e d t he 10 p rob l em o f the f i e l d o f a magne t i c d i p o l e s i t u a t e d i n the uppe r l a y e r o f a s e m i - i n f i n i t e two l a y e r e a r t h . T h i s has a p p l i c a t i o n t o a d i p o l e s i t u a t e d i n t he s e a . E a r l i e r t h e o r e t - i c a l i n v e s t i g a t i o n s o f d i p o l e s o u r c e s s i t u a t e d i n a s e m i - i n f i n i t e c o n d u c t i n g medium have been c a r r i e d ou t by v a r i o u s p e o p l e ( e . g . L i e n 1953, W a i t 1953, W a i t and C a m p b e l l 1953). I n a r e c e n t p a p e r W a i t (1966) s t u d i e d the e l e c t r o m a g n e t i c f i e l d s o f a d i p o l e o v e r a homogeneous a n i s t r o p i c h a l f s p a c e . He d e v e l o p e d e x p r e s s i o n s f o r the e l e c t r o m a g n e t i c f i e l d s f o r b o t h v e r t i c a l and h o r i z o n t a l d i p o l e s o u r c e s . The g e n e r a l t h e o r y o f e l e c t r o m a g n e t i c i n d u c t i o n i n a s e m i - i n f i n i t e homogeneous c o n d u c t o r was t r e a t e d by P r i c e (1950). As an i l l u s t r a t i v e example, he examined t he s p e c i a l ca se o f an o s c i l l a t i n g l i n e c u r r e n t . I n a more r e c e n t work ( P r i c e 1962) he s t u d i e d the e f f e c t o f a g e n e r a l s ou r ce o f f i n i t e d i m e n s i o n s . S r i v a s t a v a (1965) o u t l i n e d a method o f i n t e r p r e t a t i o n o f m a g n e t o t e l l u r i c d a t a when the s o u r c e f i e l d i s c o n s i d e r e d . He n o t e s t h a t phase a n g l e s a r e v e r y s i g n i f i c a n t i n d e t e r m i n i n g t he c o n d u c t i v i t y d i s t r i b u t i o n s f r o m measurements made a t t he e a r t h ' s s u r f a c e . Weaver ex tended the work o f P r i c e (1950, 1962) f o r a u n i f o r m c o n d u c t o r t o a t w o - l a y e r c o n d u c t o r . He o b t a i n e d g e n e r a l s o l u t i o n s f o r t he magne t i c f i e l d components i n the uppe r l a y e r o f a t w o - l a y e r c o n d u c t o r . The s p e c i a l ca se o f an o s c i l l a t i n g l i n e c u r r e n t above a c o n - d u c t i n g e a r t h has been t r e a t e d by Law and F a n n i n (1961). They c a r r i e d o u t c a l c u l a t i o n s f o r a p l a n e c o n d u c t i n g e a r t h i n the n e a r f i e l d o f a l i n e s ou r ce s i t u a t e d a t a h e i g h t o f 2 x 1 0 y 11 meters o b t a i n i n g v e r t i c a l to h o r i z o n t a l magnetic f i e l d r a t i o s ranging from 0 to 0.5 f o r an angular frequency of 0.3 radians/sec. These r e s u l t s are i n the range of experiment- a l l y observed valu e s . Weaver (1961) has extended t h i s l i n e c u r r e n t model to apply to more d i s t a n t p o i n t s on the earth's surface by c o n s i d e r i n g a c y l i n d r i c a l e a r t h i n the near f i e l d of a r a d i a t i n g l i n e c u r r e n t . The values he obtained compare f a v o r a b l y w i t h those of Law and Fannin (1961) i n the r e g i o n where the plane e a r t h assumption i s v a l i d . R e sults f o r the l i n e c u r r e n t source f o r a wide range of frequencies^ conduct- i v i t i e s , and source heights w i l l be t r e a t e d l a t e r i n t h i s work. The e f f e c t of the gross f e a t u r e s of the e a r t h on the electromagnetic v a r i a t i o n s a t the earth's surface f o r a given assumed source f i e l d can be s t u d i e d by assuming a simple con- d u c t i v i t y s t r u c t u r e f o r the ea r t h as was discussed i n the preceding paragraphs. However, the earth's c r u s t i s probably more d i s t i n g u i s h e d by i r r e g u l a r i t i e s than by r e g u l a r i t i e s * Although l a y e r e d s t r u c t u r e s do e x i s t , other s t r u c t u r e s of con s i d e r a b l e i n t e r e s t i n geophysics a t the present time are those having d i s c o n t i n u i t i e s or c o n d u c t i v i t y gradients i n the h o r i z o n t a l d i r e c t i o n such as v e r t i c a l f a u l t s , dykes, and e a r t h - sea i n t e r f a c e s . Some a n a l y t i c a l work i n v o l v i n g v e r t i c a l f a u l t s t r u c t u r e s and s i m p l i f i e d source f i e l d s has been done. . d ' E r c e v i l l e and Kunetz (1962) have s t u d i e d the case of the magnetic f i e l d everywhere p a r a l l e l to the t r a c e of the f a u l t * Rankin (1962) has extended the work of d ' E r c e v i l l e and Kunetz to the case of a dyke d i s c o n t i n u i t y . To study the enhancement 12 o f t h e v e r t i c a l m a g n e t i c f i e l d c o m p o n e n t i n c o a s t a l r e g i o n s , W e a v e r (1963) c o n s i d e r e d t h e c a s e o f t h e m a g n e t i c f i e l d e v e r y - w h e r e n o r m a l t o t h e t r a c e o f a v e r t i c a l f a u l t r e p r e s e n t i n g a c o a s t a l b o u n d a r y s e p a r a t i n g s e a a n d l a n d . B o t h s e a a n d l a n d w e r e t a k e n t o b e i n f i n i t e i n d e p t h o n e i t h e r s i d e o f t h e v e r t i c a l i n t e r f a c e . I n t h e a n a l y s i s h e a s s u m e d t h a t t h e h o r i - z o n t a l m a g n e t i c f i e l d w a s c o n s t a n t a c r o s s t h e t r a c e o f t h e f a u l t . • T h i s - ' e n h a n c e m e n t o f t h e v e r t i c a l m a g n e t i c f i e l d c o m - p o n e n t h a s b e e n t h e s u b j e c t o f v a r i o u s t h e o r e t i c a l i n v e s t i g - a t i o n s ( f o r e x a m p l e , R i k i t a k e 1961, w h e r e e a r l i e r r e f e r e n c e s a r e a l s o q u o t e d ) . S c h m u c k e r (196k) h a s d e s c r i b e d c e r t a i n c o a s t a l a n o m a l i e s i n t h e m a g n e t i c f i e l d c o m p o n e n t s i n t e r m s o f t h e e d g e e f f e c t o f t h e o c e a n a n d i n h o m o g e n e i t i e s i n t h e d e e p c o n d u c t i v i t y s t r u c t u r e i n c l u d i n g a n u p w e l l i n g i n a h i g h c o n - d u c t i v i t y z o n e w i t h i n t h e m a n t l e . L a m b e r t a n d C a n e r (1965) a l s o s u g g e s t t h a t t h e i r o b s e r v e d e n h a n c e m e n t o f t h e v e r t i c a l m a g n e t i c f i e l d n e a r t h e c o a s t l i n e f o r t h e l o w e r f r e q u e n c i e s i s n o t d u e t o t h e s e a b u t r a t h e r d u e t o i n h o m o g e n e i t i e s a t m u c h g r e a t e r d e p t h s . F o r h i g h e r f r e q u e n c i e s ( p e r i o d s l e s s t h a n 10 m i n . ) t h e c o n d u c t i v i t y c o n t r a s t o f t h e l a n d - s e a i n t e r - f a c e w a s f o u n d t o b e i m p o r t a n t . T h e c o a s t l i n e p r o b l e m , a s w e l l a s o t h e r g e o l o g i c a l p r o b l e m s i n v o l v i n g s u c h s t r u c t u r e s a s f a u l t s a n d d y k e s , c a n b e a r m u c h m o r e s t u d y . - O n e m e t h o d o f s t u d y i n g t h e s e p r o b l e m s i s b y t h e u s e o f a s c a l e d m o d e l e m p l o y i n g t h e w e l l k n o w n p r i n c i p l e o f s i m i l i t u d e ( S t r a t t o n 19k1)• T h i s m e t h o d i s p a r t i c u l a r l y u s e f u l f o r p r o b l e m s w h i c h d o n o t r e a d i l y y i e l d t o 13 a n a l y t i c a l s o l u t i o n . • H u b h e r t (1937) p r e s e n t e d t h e t h e o r y o f s e a l e d m o d e l s a s a p p l i e d t o g e o l o g i c a l s t r u c t u r e s a n d t h e m a g n e t o t e l l u r i c s c a l i n g f a c t o r s w e r e t r e a t e d b y C a g n i a r d (1953). T h e p r o b l e m s o f a c o n d u c t i n g s p h e r e i n a u n i f o r m f i e l d a n d i n t h e f i e l d o f a d i p o l e s o u r c e h a v e b e e n s t u d i e d b o t h t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y w i t h t h e a i d o f a s c a l e d m o d e l ( e . g . M a r c h 1953? S l i c h t e r 1932). S l i c h t e r (1932) c o n d u c t e d a m o d e l s t u d y o n a l a r g e s p h e r e i n a n a p p l i e d u n i f o r m f i e l d . T h e m o d e l c o n s i s t e d o f a h o l l o w s p h e r e c o a t e d w i t h t i n f o i l . T h e s o u r c e f r e q u e n c i e s h e u s e d w e r e s u c h t h a t t h e t i n f o i l w a s s e v e r a l s k i n d e p t h s t h i c k , , a n d h e n c e t h e s p h e r e b e h a v e d l i k e a s o l i d c o n d u c t i n g s p h e r e i n w h i c h c u r r e n t s f l o w o n l y n e a r t h e s u r f a c e . T h e f i e l d s o u r c e ( t r a n s m i t t e r ) w a s s u f f i c i e n t l y f a r f r o m t h e s p h e r e t o j u s t i f y t h e a s s u m p t i o n o f a u n i f o r m f i e l d . S i n c e e x a c t s o l u t i o n s f o r a c o n d u c t i n g s p h e r e i m m e r s e d i n a n o n - c o n d u c t i n g m e d i u m c a n b e o b t a i n e d a n a l y t i c a l l y ( W a i t 1953)? " t h e m o d e l m e a s u r e m e n t s f o r t h i s g e o m e t r y c a n r e a d i l y b e c h e c k e d a g a i n s t t h e o r e t i c a l v a l u e s . • T h e g r e a t e s t v a l u e o f s c a l e m o d e l s i s b e i n g a b l e t o s t u d y t h e e l e c t r o m a g n e t i c r e s p o n s e o f b o d i e s w i t h g e o m e t r i e s w h i c h a r e n o t a m e n a b l e t o a n a l y t i c a l s o l u t i o n . T o s i m p l i f y t h e t h e o r e t i c a l a n a l y s i s o f t h e p r o b l e m o f a c o n d u c t i n g s p h e r e t h e u s u a l a s s u m p t i o n i s t h a t t h e s u r r o u n d i n g m e d i u m i s n o n - - c o n d u c t i n g . T h e r e s p o n s e o f c o n d u c t i n g b o d i e s i s s i g n i f i c a n t l y m o d i f i e d w h e n t h e s u r r o u n d i n g m e d i u m i s c o n d u c t i n g . T h e m o d e l d i s c u s - s e d l a t e r i n t h i s w o r k , i s s u i t a b l e f o r s t u d y i n g t h i s p r o b l e m . T h e p r o b l e m o f t h e r e s p o n s e o f a s e m i . - i n f i n i t e p l a n e Ik h a s b e e n s t u d i e d b y W e s t ( 1 9 6 0 ) . H e u s e d a l a r g e s t a i n l e s s s t e e l s h e e t t o r e p r e s e n t a h a l f V p l a n e . T h e s h e e t , i n t h e h o r i - z o n t a l p o s i t i o n c o u l d r e p r e s e n t a c o n d u c t i n g l a y e r b u t t i n g a g a i n s t a n o n - c o n d u c t i n g l a y e r , o r a c o n d u c t i n g l a y e r o v e r a n o n - c o n d u c t i n g m e d i u m . I n a v e r t i c a l p o s i t i o n , o r i n a n i n c l i n e d p o s i t i o n i t c o u l d r e p r e s e n t a c o n d u c t i n g d y k e w i t h n o n - c o n d u c t i n g m e d i a o n e i t h e r s i d e . - T h e s o u r c e f i e l d s w e r e p r o v i d e d b y h o r i z o n t a l a n d v e r t i c a l l o o p s y s t e m s . D o u l o f f ( 1 9 6 1 ) s t u d i e d t h e r e s p o n s e o f a d i s c i n a d i p o l e f i e l d u s i n g a s c a l e d m o d e l . H e s t u d i e d v e r t i c a l a n d h o r i z o n t a l d i s c s o f v a r i o u s d i a m e t e r s s i t u a t e d i n t h e f i e l d o f a n o v e r h e a d h o r i z o n t a l l o o p c u r r e n t . T h e r e s p o n s e w a s v e r y s i m i l a r t o t h a t o f a s e m i r - i n f i n i t e p l a n e s t u d i e d b y W e s t ( 1 9 6 0 ) . Y o s t ( 1 9 5 2 ) a n d Y o s t e t a l ( 1 9 5 2 ) h a v e s t u d i e d t h e o r e t - i c a l l y a n d w i t h t h e a i d o f a s c a l e d m o d e l , t r a n s i e n t s g e n e r a t e d i n a l a y e r e d c o n d u c t i n g e a r t h . ; T h e y u s e d s h e e t s o f v a r i o u s m e t a l s t o r e p r e s e n t c o n d u c t i n g l a y e r s i n t h e e a r t h . T h e e x c i t - i n g f i e l d s w e r e s u p p l i e d b y c u r r e n t p u l s e s i n a h o r i z o n t a l l o o p o v e r t h e s t a c k s o f m e t a l s h e e t s . T h e r e s p o n s e s i g n a l t o g e t h e r w i t h t i m i n g m a r k e r s w e r e d i s p l a y e d o n a n o s c i l l o s c o p e , a n d t h e r e f l e c t e d f i e l d s , a s a f u n c t i o n o f t i m e f o l l o w i n g t h e c u r r e n t p u l s e , w e r e s t u d i e d . T h e w o r k o f O r s i n g e r a n d V a n N o s t r a n d ( 1 9 5 k ) w a s a s e q u e l t o t h a t o f Y o s t ( 1 9 5 2 ) a n d Y o s t e t a l ( 1 9 5 2 ) , i n t h a t i t e m p l o y e d a m o d e l l i k e t h e o n e d e s c r i b e d b y t h e l a t t e r i n m a k i n g f i e l d m e a s u r e m e n t s t o d e t e r m i n e d e p t h s i n a l a y e r e d 15 earth. The metal sheets i n the model were adjusted by t r i a l and e r r o r to give, as nearly as possible, the same signal as that obtained from the f u l l scale f i e l d measurements. In this way an analogue of the layered earth was constructed. Their results f o r measurements at various locations agreed with subsequent d r i l l hole measurements to within an accuracy of 5%o Roden (196^) has described a model f o r investigating the- e f f e c t of i r r e g u l a r i t i e s i n the shape of the coastline on magnetic diurnal v a r i a t i o n s . • He used a t h i n copper sheet to represent the ocean. He concluded that the ocean edge should cause observable enhancement of the magnetic f i e l d f o r v a r i a t - ions with periods as long as 2h hours. Rankin; (19,60) and Rankin et a l (1965) described a magnetotelluric model consisting e s s e n t i a l l y of an o s c i l l a t i n g l i n e current over a conducting s a l t water solution, which represented the upper layer of a model earth. Some' magneto- t e l l u r i c model measurements were obtained f o r f a u l t s and dykes. A model somewhat si m i l a r to Rankin's (1960),; but designed to provide considerably more information, i s disr- cussed l a t e r i n this work. This model i s p a r t i c u l a r l y s u i t - able f o r studying some in t e r e s t i n g problems of conducting s-tructures surrounded by media of lower conductivity. 1 .2 Object of the -Thesis There are four objectives of t h i s thesis. The f i r s t objective i s to develop and evaluate mathematical expressions 16 f o r t h e e l e c t r o m a g n e t i c f i e l d c o m p o n e n t s b o t h a t t h e s u r f a c e a n d w i t h i n t h e u p p e r l a y e r o f a h o r i z o n t a l l y s t r a t i f i e d c o n - d u c t i n g e a r t h u s i n g a p l a n e w a v e s o u r c e m o d e l . S i m i l a r w o r k s i n t h e l i t e r a t u r e t e n d t o d e a l w i t h s p e c i a l a s p e c t s o f t h i s p r o b l e m s u c h a s n o r m a l i n c i d e n t w a v e s a n d r a t i o s o f c o m p o n e n t s . I n t h i s w o r k g e n e r a l e x p r e s s i o n s f o r t h e i n d i v - i d u a l c o m p o n e n t s w i l l b e o b t a i n e d a n d e v a l u a t e d f o r a w i d e r a n g e o f f r e q u e n c i e s , a n g l e s o f i n c i d e n c e , l a y e r t h i c k n e s s e s , d e p t h s , a n d c o n d u c t i v i t i e s . F r o m t h e c a l c u l a t e d v a l u e s , i n f o r m a t i o n o n t h e v a l i d i t y o f t h e a s s u m p t i o n o f a p l a n e w a v e m o d e l f o r n a t u r a l l y o c c u r r i n g e l e c t r o m a g n e t i c v a r i a t i o n s c a n b e o b t a i n e d . I n a d d i t i o n , t h e e f f e c t o f h o m o g e n e o u s a n d i n h o m o g e n e o u s c o n d u c t i n g l a y e r s c a n b e a s s e s s e d . T h e s e c o n d o b j e c t i v e i s t o s t u d y t h e p r o b l e m o f a p l a n e h o m o g e n e o u s c o n d u c t i n g e a r t h i n t h e n e a r f i e l d o f a n o s c i l l a t i n g l i n e c u r r e n t b y e v a l u a t i n g t h e c o m p o n e n t s ( a m p l i - t u d e s a n d p h a s e a n g l e s ) f o r a w i d e r a n g e o f f r e q u e n c i e s , c o n d u c t i v i t i e s , s o u r c e h e i g h t s , a n d l o c a t i o n s w i t h r e s p e c t t o t h e o v e r h e a d c u r r e n t . A g a i n , i n f o r m a t i o n o n t h e v a l i d i t y o f t h e a s s u m p t i o n o f a l i n e c u r r e n t m o d e l f o r n a t u r a l l y o c c u r r i n g e l e c t r o m a g n e t i c v a r i a t i o n s a t t h e e a r t h ' s s u r f a c e c a n b e a s s e s s e d . T h e r e s u l t s f o r t h e p l a n e w a v e m o d e l a n d t h o s e f o r t h e l i n e c u r r e n t m o d e l f o r s o m e s p e c i a l c a s e s c a n ; b e c o m p a r e d . T h e t h i r d o b j e c t i v e i s t o d e s i g n a n d c o n s t r u c t a n a n a l o g u e m o d e l ( i n c l u d i n g s u i t a b l y s c a l e d g e o l o g i c a l s t r u c t - u r e s ) s u i t a b l e f o r s t u d y i n g e l e c t r o m a g n e t i c v a r i a t i o n s . T h e 17 measurements w i l l i n c l u d e t h e a m p l i t u d e s and phase a n g l e s o f the v e r t i c a l magne t i c f i e l d as w e l l as t h o s e o f the h o r i z o n t a l e l e c t r i c and magne t i c f i e l d components f o r b o t h s h e e t c u r r e n t and l i n e c u r r e n t s o u r c e s . The f o u r t h o b j e c t i v e i s t o u se t h i s ana l ogue mode l t o s t u d y a v a r i e t y o f g e o p h y s i c a l p rob lems i n v o l v i n g t y p i c a l g e o l o g i c a l s t r u c t u r e s . The r e s u l t s s h o u l d p r o v i d e new i n f o r m a t i o n on s e v e r a l p rob lems i n c l u d i n g t he c o a s t l i n e p r o b l e m . 18 C h a p t e r 2 . M A T H E M A T I C A L M O D E L S 2 .1 A S i m p l e L a y e r e d C o n d u c t i n g E a r t h i n t h e F i e l d o f P l a n e W a v e s 2 . V . 1 I n t r o d u c t i o n A c o m p a r i s o n o f t h e a m p l i t u d e s a n d p h a s e s o f t h e n a t u r a l g e o m a g n e t i c a n d t e l l u r i c f i e l d v a r i a t i o n s p r o v i d e s a m e t h o d f o r o b t a i n i n g i n f o r m a t i o n o n t h e e a r t h ' s c r u s t a l s t r u c t u r e . T h e b e h a v i o r o f t h e n a t u r a l l y o c c u r r i n g e l e c t r o - m a g n e t i c v a r i a t i o n s o b s e r v e d a t t h e e a r t h ' s s u r f a c e i s i n g e n e r a l d e t e r m i n e d b y t h e n a t u r e o f t h e s o u r c e f i e l d s a n d b y t h e d i s t r i b u t i o n o f e l e c t r i c a l c o n d u c t i v i t y w i t h i n t h e e a r t h . A t p r e s e n t t h e r e a r e t w o m e t h o d s c o m m o n l y u s e d t o s t u d y t h e c o n d u c t i v i t y s t r u c t u r e o f t h e e a r t h . O n e i s b a s e d o n t h e a n a l y s i s o f t h e t h r e e m a g n e t i c f i e l d c o m p o n e n t s o f t h e n a t u r a l e l e c t r o m a g n e t i c v a r i a t i o n s . T h e o t h e r i n v o l v e s t h e h o r i z o n t a l e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s a n d t h e i r f r e q u e n c y d e p e n d e n c e . T h e a n a l y s i s o f f i e l d m e a s u r e m e n t s f o r t h e a b o v e m e t h o d s n o r m a l l y a s s u m e s h o r i z o n t a l l a y e r s o f u n i f o r m c o n d u c t i v i t y a n d a u n i f o r m i n d u c i n g f i e l d . T h i s p a r t o f t h e p r e s e n t w o r k d e a l s w i t h t h e e l e c t r i c a n d m a g n e t i c f i e l d s a t t h e s u r f a c e a n d i n t h e u p p e r l a y e r o f a s t r a t i f i e d f l a t e a r t h i n t h e f i e l d o f p l a n e w a v e s . E x - p r e s s i o n s f o r t h e a m p l i t u d e a n d p h a s e o f t h e c o m p o n e n t s o f t h e e l e c t r i c a n d m a g n e t i c f i e l d s a r e o b t a i n e d a n d e v a l u a t e d f o r v a r i o u s f r e q u e n c i e s , a n g l e s o f i n c i d e n c e , l a y e r t h i c k - n e s s e s , d e p t h s , a n d c o n d u c t i v i t i e s . T h e c o n d u c t i v i t i e s 19 d = 1 0 ~ 1 0 t o 1 0 " 1 6 e m u a n d t h e f r e q u e n c i e s f = 1 C f 3 t o 10* c y c l e s / s e c s t u d i e d a r e o f ' i n t e r e s t i n g e o p h y s i c s . 2.1.2 - M a t h e m a t i c a l A n a l y s i s C o n s i d e r a p l a n e e l e c t r o m a g n e t i c w a v e h a r m o n i c i n t i m e , t r a v e l i n g i n f r e e s p a c e ( m e d i u m 1 ) , a n d i n c i d e n t u p o n t h e s u r f a c e o f a s t r a t i f i e d f l a t c o n d u c t o r . • T h e c o n d u c t o r c o n s i s t s o f t h r e e f l a t , h o m o g e n e o u s , i s o t r o p i c l a y e r s ( m e d i a 2, 3, k ) , w i t h t h e t h i r d l a y e r i n f i n i t e i n d e t f t h . T h e f o u r m e d i a c a n b e c h a r a c t e r i z e d b y t h e c o n s t a n t s e ^ , d j ( j = 1? 2, 3, I f ) , w h e r e e , pc, d a r e t h e d i e l e c t r i c c o n s t a n t , m a g n e t i c p e r m e a b i l i t y , a n d c o n d u c t i v i t y r e s p e c t i v e l y . T h e m a g n e t i c p e r m e a b i l i t y o f e a c h o f t h e c o n d u c t o r s i s a s s u m e d t o b e t h e s a m e a s t h a t o f f r e e s p a c e . A l t h o u g h t h e G a u s s i a n s y s t e m o f u n i t s w i l l b e u s e d ( c o n d u c t i v i t y e x p r e s s e d i n e m u ) i n t h e m a t h e m a t i c a l a n a l y s i s , t h e g r a p h i c a l r e s u l t s w i l l b e e x p r e s s e d i n u n i t s w h i c h a r e u s e d m o r e c o m m o n l y i n g e o p h y s i c s . T h e m a g n e t i c f i e l d w i l l b e g i v e n i n g a u s s w h i l e t h e e l e c t r i c f i e l d w i l l b e g i v e n i n v o l t s / m e t e r . I n e a c h m e d i u m M a x w e l l ' s e q u a t i o n s m u s t b e s a t i s f i e d . , (2) • • V x H - i < W d c 2 E + e § f ) = 0, (3) • - V.E = 0 , ( I f ) V.H = 0. T h e r i g h t - h a r i d s i d e o f e q u a t i o n (3) i s s e t e q u a l t o z e r o s i n c e 20 w e a r e c o n s i d e r i n g c h a r g e - f r e e m e d i a . I t i s e a s i l y s h o w n t h a t t h e e l e c t r i c a n d m a g n e t i c f i e l d v e c t o r s f o r t h e i n c i d e n t p l a n e w a v e i n m e d i u m 1 s a t i s f y i n g t h e s e e q u a t i o n s c a n b e e x p r e s s e d a s (5) = E Q e x p i d L j i i ^ . r + c o t ) , (6) H 1 1 = H Q e x p K ^ n ^ . r + c o t ) , w h e r e E „ a n d H a r e t h e a m p l i t u d e s o f t h e i n c i d e n t e l e c t r i c o o a n d m a g n e t i c f i e l d v e c t o r s r e s p e c t i v e l y , i = ^-1 , co i s t h e .A a n g u l a r f r e q u e n c y , n ^ ^ i s t h e u n i t v e c t o r i n t h e d i r e c t i o n o f p r o p a g a t i o n , r i s t h e p o s i t i o n v e c t o r f r o m a f i x e d o r i g i n , (7) k . j = — ij jj„j e 1 co - i ^ + T r ^ c co , 10 a n d c i s t h e v e l o c i t y o f l i g h t i n f r e e s p a c e (3 x 10 c m / s e c ) . T h e e l e c t r i c a n d m a g n e t i c f i e l d v e c t o r s ( S t r a t t o n , 1 9 ^ 1 ) a r e r e l a t e d b y T h e f i e l d v e c t o r s f o r t h e r e f l e c t e d w a v e h a v e the s a m e f o r m a s (5) a n d ( 6 ) . S i n c e a l l f i e l d v e c t o r s w i l l c o n t a i n t h e t i m e v a r i a t i o n e x p ( i c o t ) , t h i s f a c t o r w i l l b e s u p p r e s s e d i n t h e f o l l o w i n g d e v e l o p m e n t . L e t u s c o n s i d e r a r i g h t - h a n d e d c o - o r d i n a t e s y s t e m a s s h o w n i n F i g . 1 w i t h t h e o r i g i n a t t h e i n t e r f a c e o f m e d i a 1 a n d 2, a n d w i t h t h e c o n d u c t i n g l a y e r s p a r a l l e l t o t h e x y p l a n e , T h e p l a n e o f i n c i d e n c e i s t h e y z p l a n e . T h e r e s u l t a n t f i e l d v e c t o r s i n e a c h m e d i u m a r e t h e 2 1 F i g u r e 1 . < M o d e l u s e d i n t h e p l a n e w a v e p r o b l e m . 22 c o m p l e x v e c t o r s u m s o f t h e f i e l d v e c t o r s o f t h e i n c i d e n t a n d r e f l e c t e d p l a n e w a v e s , i . e . (9) = + H \ 2 = l \ e x p C i k^n^ . r ) + B ̂  e x p C i k^n^ . r ) , a n d (10) = E ^ + E j 2 = C j e x P ( i k - j n - j - | - r ) + D - j e x p ( i k ^ n ^ . r ) , -* ' \ w h e r e ky By Cy D.. ( j = 1, 2, 3, M-) a r e c o m p l e x v e c t o r s a n d k j i s d e f i n e d i n t h e s a m e w a y a s k ^ [ e q u a t i o n (7 ) ] . I n e q u a t i o n s (9) a n d (10) = H Q , C j = E Q , a s d e f i n e d i n e q u a t i o n s (5) a n d (6), a n d B ^ a n d a r e z e r o s i n c e m e d i u m h i s o f i n f i n i t e d e p t h . B o t h t h e e l e c t r i c a n d t h e m a g n e t i c f i e l d v e c t o r s i n e a c h m e d i u m c a n b e r e s o l v e d i n t o a c o m p o n e n t n o r m a l t o t h e p l a n e o f i n c i d e n c e a n d a c o m p o n e n t p a r a l l e l t o t h e p l a n e o f i n c i d e n c e ( S t r a t t o n , 19 k 1) . I n c i d e n t M a g n e t i c F i e l d N o r m a l t o t h e P l a n e o f I n c i d e n c e I n o u r c o - o r d i n a t e s y s t e m t h i s p o l a r i z a t i o n g i v e s t h e x c o m p o n e n t o f t h e m a g n e t i c f i e l d a n d t h e y a n d z c o m p o n e n t s o f t h e e l e c t r i c f i e l d . L e t t h e s i n g l e p r i m e s r e p r e s e n t t h e c o m p o n e n t s n o r m a l t o t h e p l a n e o f i n c i d e n c e a n d t h e d o u b l e p r i m e s r e p r e s e n t t h e c o m p o n e n t s p a r a l l e l t o t h e p l a n e o f i n - c i d e n c e . U s i n g t h e n o t a t i o n (11) a . , = k ^ n ^ . r = k . , ( y s i n 9^ + z c o s 9 . , ) , . = k . n . p . r = k . ( y s i n 9 . - z c o s 9 . ) ? t h e x c o m p o n e n t s o f t h e c o m p l e x m a g n e t i c f i e l d v e c t o r s a n d t h e y a n d z c o m p o n e n t s o f t h e c o m p l e x e l e c t r i c f i e l d v e c t o r s i n t he f o u r med ia become (13) ( H j ) x = A j e x p ( i o j ) - + e x p ( i P j ) , ( 1 k ) ( E j ) = [- C*j expC ia j . ) + D'j e x p ( i P j ) ] cos 9^, (15) ( E j ) z ' = [C'j e x p ( i a ^ ) + D*j e x p ( i P j ) ] s i n Qy The complex c o n s t a n t s i n e q u a t i o n s (13) t o (15) c an be d e t e r - mined u s i n g t h e boundary c o n d i t i o n s t h a t t h e t a n g e n t i a l components o f t h e e l e c t r i c and magne t i c f i e l d s a r e c o n t i n u o u s , A p p l y i n g t h e s e boundary c o n d i t i o n s a t z = d ^ , we f i n d t h a t (16) where (17) B ' = m«A» , khCos9., - k - , c o s 9 ^ I f we l e t (18) (19) t h e n a t z = d, (20) where (21) A t z = 0 (22) Y-| = e x p ( i 2 k 3 d 2 c o s 9 3 > + m^ 5 y 2 = e x p ( i 2 k 3 d 2 c o s 9 3 ) - m^ , B 2 = m 2 A 2 , m 2 = Y ^ k ^ c o s 9 2 - Y 2 ^ 2 c o s ^ 3 Y ^ k 3 c o s 9 2 + k 0 c o s 9 . Y 2 K 2 e x p ( i 2 k 2 d 2 c o s 9 2 ) 2h w h e r e f k 2 c o s © ^ Cl+mp - k , c o s © 2 ( 1 - m 2 ) (23) m, - — — - — • — k 2 c o s © ^ (1+m2) + k^cos© 2 ( -1 - m 2 ) I t f o l l o w s t h a t ( 2 1 + ) T h e n u s i n g e q u a t i o n (13) f o r j = 2 , ( 2 5 ) ( H 2 ) x = A ] [T^TJ [ e x p ( i a 2 ) + m ^ e x p ( i p 2 ) ] A f t e r d e t e r m i n i n g C 2 a n d D 2 , t h e e l e c t r i c f i e l d c o m p o n e n t s u s i n g e q u a t i o n s (1*+) a n d (15) a r e g i v e n b y ( 2 6 ) ( E (27) ( E 2 ) z = C /l-m'\ 2 ) y = -C" [j~^J L " e x P ( ± a 2 ^ ~ m 2 e x P ( i P 2 ^ C O S 61 » / l t i j \ k 2 1 r [ e x p ( i a 9 ) + n u e x p ( i B 9 ) ] -5- s i n e1 V 1 + m 2 / k 2 I n c i d e n t M a g n e t i c F i e l d P a r a l l e l t o t h e P l a n e o f I n c i d e n c e T h i s p o l a r i z a t i o n g i v e s t h e y a n d z c o m p o n e n t s o f t h e m a g n e t i c f i e l d a n d t h e x c o m p o n e n t o f t h e e l e c t r i c f i e l d . T h e e q u a t i o n s f o r t h e s e c o m p o n e n t s i n t h e f o u r m e d i a c a n b e w r i t t e n a s (28) ( H j ) = [ A ' j e x p C i a j ) + B ' j e x p ( i P j ) ] c o s 6., (29) ( R \ . ) z = [ - A ' j e x p ( i a j ) + B ' j e x p ( i P j ) ] s i n 6 j , (30) ( E j ) x = C j e x p ( i a j ) - D j e x p ( i P j ) . 25 I n a d d i t i o n t o t h e b o u n d a r y c o n d i t i o n s p r e v i o u s l y s t a t e d w e h a v e t h a t f o r c o n s t a n t u. t h e n o r m a l c o m p o n e n t o f t h e m a g n e t i c f i e l d i s c o n t i n u o u s . A g a i n a p p l y i n g t h e b o u n d a r y c o n d i t i o n s w e f i n d t h a t a t z = ( 3 D w h e r e (32) I f w e l e t (33) - _ ; M m3 = B 3 = m 3 A 3 , s i n Q ^ c o s Q ^ - s i n 9 ^ c o s Q ^ s i n 6 3 c o s © k + 3^9^0039.^ e x p ( i 2 k 3 d 3 c o s © 3 ) . Y 3 = e x p ( i 2 k 3 d 2 c o s 9 3 > + m 3 , ( 3 k ) t h e n a t z = d , (35) Yh. = e x p ( i 2 k 3 d 2 c o s 9 3 ) - m 3 - I I 1 1 , 1 ! B 2 = m 2A 2 , (36) i t m 2 = Y 3 s i n 9 2 c o s 9 v Y k s i n 9 3 c o s 9 2 Y 3 s i n 9 2 c o s 9 3 + Y h . s i n ® 3 C 0 S ® 2 e x p ( i 2 k 2 d 2 c o s 9 2 ) A t z = 0 (37) w h e r e (38) _ M II , It B 1 = m 1A 1 , sin9.j cos9 2 (1+m 2) - sin92Cos9.| (1-mp s i n 9 1 cos92(1+m2') + 31^20039.! (1-m2) ' I t f o l l o w s t h a t 26 / 1 + m " \ c o s e . H e n c e u s i n g e q u a t i o n s (28) a n d (29) f o r j = 2, (MO) ( H 2 ) y = A! , 1 (j^n-j [ e x p ( i a 2 ) + e x p ( i B 2 ) ] c o s e 1 , / 1 - m " \ (M-1) ( H 2 ) z = - A " I [ e x p ( i a 2 ) - m 2 e x p ( i B 2 ) ] s i n e l A f t e r d e t e r m i n i n g C 2 a n d D 2 t h e e l e c t r i c f i e l d c o m p o n e n t u s i n g e q u a t i o n (30) r e d u c e s t o (M-2) ( E 2 ) x = c j 1 - m " \ HT [ e x p ( i a 9 ) - mlX e x p ( i B 9 ) ] 1 - m 2 y ^ ^ ^ E q u a t i o n s (25), (M-0) a n d (M-1) a r e t h e c o m p l e x c o m p o n - e n t s o f t h e v a r y i n g m a g n e t i c f i e l d a n d e q u a t i o n s ( 2 6 ) , (27) a n d (M-2) a r e t h e c o m p l e x c o m p o n e n t s o f t h e v a r y i n g e l e c t r i c f i e l d i n t h e u p p e r c o n d u c t i n g l a y e r f o r p l a n e e l e c t r o m a g n e t i c w a v e s i n t h e y z p l a n e . T h e A m p l i t u d e a n d P h a s e o f t h e x , y , z C o m p o n e n t s U s i n g t h e r e l a t i o n s h i p s ( S t r a t t o n , 19M-1) (M-3) k ^ s i n e 1 = k 2 s i n © 2 = k ^ s i n ©^ = k ^ s i n © ^ , 27 ( k 6) k^COS© k = -Ji^ - k 2 s i n 2 e 3 ) , ©2? a n d ^1+ m a y b e e l i m i n a t e d f r o m m j , m " 5 a n d m ^ E a c h o f t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s m a y b e e x p r e s s e d i n t e r m s o f t h e m o d u l u s a n d a r g u m e n t o f t h e c o m p l e x q u a n t i t i e s . S i n c e t h e a m p l i t u d e s H Q a n d E Q a r e d e s c r i p t i v e o f t h e i n c i d e n t w a v e s a n d h e n c e a r b i t r a r y a s f a r a s c o m p u t a t i o n s a r e c o n c e r n e d , i t i s c o n v e n i e n t t o e x p r e s s e q u a t i o n s (25) ? C+0), a n d ( k 1 ) a s ( L 7 ) ( k 8 ) < H 2>x ( H 2 > x | H ' " H 1 O 0 < H 2 V ( H 2 ) y | H " " H " o 0 < H 2 > Z •« e x p ( i 0 v ) = H V exp ( i < o x ) , x H " o e x p ( i < j > ) = H e x p ( i < ? > ) , a n d e q u a t i o n s (*+2)y ( 2 6 ) , a n d (27) a s (50) (51)« (52)- ( E 2 > x E ' " E 1 o o ( B 2 M E" " E" 0 0 < E2> z < E 2> Z | e x p ( i | x ) = E x e x p d ^ ) x r x ' "S e x p ( i \ | r ) = E y e x p ( i \ j r • ) , e x p ( i \ | r ) = E e x p ( i ^ ) , w h e r e t h e m o d u l u s i s t h e a m p l i t u d e o f t h e c o m p o n e n t a n d t h e 28 a r g u m e n t i s t h e p h a s e a n g l e . E x p r e s s e d i n t h i s f o r m . . , t h e a m p l i t u d e s a r e d i m e n s i o n l e s s a n d t h e p h a s e a n g l e s a r e m e a s u r e d r e l a t i v e t o t h e i n c i d e n t f i e l d c o m p o n e n t s . E q u a t i o n s ( V 7 ) t o (52) g i v e t h e f i e l d c o m p o n e n t s i n t h e u p p e r c o n d u c t i n g l a y e r a n d a r e i n a c o n v e n i e n t f o r m f o r c a l c u l a t i n g t h e e l e c t r o m a g - n e t i c f i e l d c o m p o n e n t s a t v a r i o u s d e p t h s i n a c o n d u c t i n g s e a . A l s o , s i n c e a l l t h e c o m p o n e n t s e x c e p t t h e v e r t i c a l e l e c t r i c f i e l d c o m p o n e n t a r e c o n t i n u o u s a t t h e f r e e s p a c e - e a r t h i n t e r - f a c e , t h e s e e q u a t i o n s a r e s u i t a b l e f o r c a l c u l a t i n g t h e c o m p o n e n t s a t t h e e a r t h ' s s u r f a c e ( z = 0 ) . T h e v e r t i c a l e l e c t r i c f i e l d c o m p o n e n t a t t h e e a r t h ' s s u r f a c e i s i n g e n e r a l n o t o f i n t e r e s t i n g e o p h y s i c s . T h e s p e c i a l c a s e s o f ( 1 ) a u n i f o r m c o n d u c t i n g e a r t h , o f i n f i n i t e d e p t h , a n d (2) a s i n g l e c o n d u c t i n g l a y e r o v e r l y i n g a : : h o m o g e n e o u s c o n d u c t i n g e a r t h o f i n f i n i t e d e p t h a r e o b t a i n e d b y s e t t i n g ( 1 ) - ° ° s a n d (2) d - = o o . R e s u l t s f o r t h e s e v a r i o u s c a s e s a r e d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n ( 2 . 1 . 3 ) . 2.1.3 D i s c u s s i o n o f R e s u l t s A c o m p u t e r p r o g r a m m e w a s w r i t t e n t o e v a l u a t e t h e a m p l i - t u d e s a n d p h a s e a n g l e s o f t h e c o m p o n e n t s a s g i v e n b y e q u a t i o n s ( k7) t o (51)" E x t e n s i v e c o m p u t a t i o n s w e r e c a r r i e d o u t f o r v a r i o u s f r e q u e n c i e s , a n g l e s o f i n c i d e n c e , l a y e r t h i c k n e s s e s , d e p t h s , a n d c o n d u c t i v i t i e s . T h e f r e q u e n c i e s a n d c o n d u c t i v i t i e s s t u d i e d w e r e r e s p e c t i v e l y f - 1 0 •J t o \0J c y c l e s / s e c a n d 0 = 1 0 " " t o 1 0 " e m u . T h e f r e q u e n c i e s a r e w i t h i n t h e m i c r o - p u l s a t i o n ; a n d l o w e r f r e q u e n c y r a n g e o f t h e e a r t h ' s n a t u r a l e l e c t r o m a g n e t i c f i e l d v a r i a t i o n s . 29 I n t h e c o m p u t a t i o n s t h e v a l u e s (j, = J J , , e = e Q f o r a l l m e d i a a n d g - 0 f o r f r e e s p a c e w e r e u s e d . T h e d i e l e c t r i c c o n s t a n t , e , w a s n o t v a r i e d s i n c e i t i s n o t i m p o r t a n t i n t h i s r a n g e o f f r e q u e n c i e s a n d c o n d u c t i v i t i e s ( D o s s o a n d L o k k e n , 1961). • F o r f r e q u e n c i e s g r e a t e r t h a n 10 c y c l e s / s e c ti d o e s b e c o m e i m p o r t a n t , p a r t i c u l a r l y i f i t h a s a l a r g e v a l u e a s i n t h e c a s e o f w a t e r . ( a ) H o m o g e n e o u s C o n d u c t i n g E a r t h o f I n f i n i t e D e p t h I n o r d e r t o o b t a i n s o m e g e n e r a l i n f o r m a t i o n o n t h e b e h a v i o r o f t h e f i e l d c o m p o n e n t s f o r a p l a n e w a v e s o u r c e , t h e s i m p l i f i e d p r o b l e m o f a f l a t h o m o g e n e o u s c o n d u c t i n g e a r t h o f i n f i n i t e d e p t h w a s s t u d i e d f i r s t . F o r t h i s p r o b l e m d 2 = co w a s u s e d i n e q u a t i o n s (*+7) t o (51)° A l t h o u g h c o m p u t a t i o n s w e r e c a r r i e d o u t f o r t h e e n t i r e f r e q u e n c y a n d c o n d u c t i v i t y r a n g e s o f i n t e r e s t i n t h i s w o r k ( f = 1 0~ 3 t o 10 3 c y c l e s / s e c , 1 f\ 11 d = 10 t o 10 e m u ) , o n l y a f e w t y p i c a l r e s u l t s w i l l b e s h o w n . H e r e , a s i n a l l t h e f o l l o w i n g r e s u l t s , t h e e l e c t r i c f i e l d r e l a t i v e a m p l i t u d e s a r e g i v e n i n M K S u n i t s ( v o l t s / m e t e r ) w h i l e t h e m a g n e t i c f i e l d r e l a t i v e a m p l i t u d e s a r e g i v e n i n g a u s s i a n u n i t s ( g a u s s ) . T h e c o n v e r s i o n f a c t o r f o r c o n v e r t i n g t h e e l e c t r i c f i e l d g a u s s i a n u n i t t o t h e M K S u n i t i s c/10 , w h e r e c = 3 x 10 . T h e c o n v e r s i o n f a c t o r ( c 10" ) f o r t h e e l e c t r i c f i e l d i s s h o w n o n t h e g r a p h s . F i g u r e s 2 ( a ) - ( d ) s h o w h o w t h e a m p l i t u d e s a n d p h a s e a n g l e s o f t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s a t t h e s u r f a c e o f t h e c o n d u c t o r d e p e n d o n t h e a n g l e o f i n c i d e n c e f o r a p l a n e w a v e s o u r c e o f f r e q u e n c y 1 c y c l e / s e c f o r t h e c o n d u c t - 30 F i g u r e 2. T h e a m p l i t u d e s ( a ) - ( c ) a n d t h e p h a s e a n g l e s ( d ) a f u n c t i o n o f G f o r f=1 c y c l e / s e c - , (1) o = 1 0 " 1 1 , (2) 10~1 (3) 1 0 " 1 5 , a n d (1+) 1 0 " 1 6 e m u . 31 i v i t i e s 1C f 1 1 , 1 (T1^, 10"1^? a n d 10~16 e m u . A l t h o u g h t h e v e r t i c a l e l e c t r i c f i e l d c o m p o n e n t w a s c o m p u t e d i t i s n o t s h o w n . F o r t h e p r o b l e m s t u d i e d h e r e , E a n d E a r e s i m p l y r e l a t e d , w i t h E „ P E „ c o s 6 , a l t h o u g h f r o m e q u a t i o n s (26) a n d (h2) t h e x • y r e l a t i o n s h i p a p p e a r s m o r e c o m p l i c a t e d . H . f o r e a c h c o n d u c t - Z ' i v i t y , h a s i t s m a x i m u m v a l u e f o r a n a n g l e o f i n c i d e n c e o f a p p r o x i m a t e l y L 5 ° . F r o m c u r v e s (1) a n d (3), a n d (2) a n d (•+) i n F i g . 2 ( b ) i t c a n b e s e e n t h a t H _ v a r i e s a s d ~ " ^ . T h i s z w a s f o u n d t o b e t h e c a s e f o r t h e f r e q u e n c y r a n g e f = 10~3 t o 103 c y c l e s / s e c . A s c a n b e s e e n f r o m e q u a t i o n s (M-1) a n d (M-2) E x a n d H z a r e a l s o s i m p l y r e l a t e d . H x , H , a n d a l l t h e p h a s e a n g l e s d o n o t d e p e n d o n t h e c o n d u c t i v i t y o r f r e q u e n c y . T h e a n g u l a r d e p e n d e n c e f o r a l l t h e c o m p o n e n t s a n d p h a s e a n g l e s h a s t h e s a m e f o r m f o r a l l o t h e r f r e q u e n c i e s a n d c o n d u c t i v i t i e s e o n s i a e r e d . I n s t u d y i n g g e o m a g n e t i c v a r i a t i o n s , e s p e c i a l l y i n t h e m i c r o p u l s a t i o n r r g n g a , i t i s s o m e t i m e s a s s u m e d t h a t t h e v a r i a t - i o n s c a n b e e x p l a i n e d i n t e r m s o f p l a n e w a v e s i n c i d e n t o n t h e e a r t h . I f t h i s a s s u m p t i o n w e r e v a l i d , m e a s u r e m e n t s m a d e i n r e g i o n s w h e r e a u n i f o r m g e o l o g i c a l s t r u c t u r e i s l i k e l y s h o u l d c o m p a r e f a v o u r a b l y w i t h v a l u e s p r e d i c t e d b y a p l a n e w a v e m o d e l . I n f i e l d m e a s u r e m e n t s , i t i s c o m m o n t o s t u d y t h e v e r t i c a l t o h o r i z o n t a l m a g n e t i c f i e l d r a t i o . M e a s u r e m e n t s m a d e a t i n l a n d l o c a t i o n s s u c h a s R a l s t o n , A l b e r t a , w h e r e a u n i f o r m g e o l o g i c a l s t r u c t u r e i s c o n s i d e r e d l i k e l y , y i e l d r a t i o s u p t o a b o u t 0.1 ( C h r i s t o f f e l e t a l , 1961). A m p l i t u d e r a t i o s R = H ( H 2 + H 2 ) " " ^ z x y -1 % a s a f u n c t i o n o f f r e q u e n c y f o r a n e a r t h c o n d u c t i v i t y d = 10 y 32 - e m u a n d v a r i o u s a n g l e s o f i n c i d e n c e a r e s h o w n i n P i g . 3- A t 1 c y c l e / s e c t h e r a t i o i s i n t h e n e i g h b o r h o o d o f 10~ k f o r t h e v a r i o u s a n g l e s o f i n c i d e n c e . C o m p a r i n g t h i s r a t i o w i t h t y p i c a l f i e l d m e a s u r e m e n t s s u g g e s t s t h a t t h e o b s e r v e d a m p l i t u d e r a t i o s c a n n o t b e e x p l a i n e d o n t h e b a s i s o f h o r i z o n t a l l a y e r e d s t r u c t - u r e a n d t h e a s s u m p t i o n o f a p l a n e w a v e m o d e l . I t i s a p p a r e n t f r o m F i g . 3> t h a t f o r t h e p l a n e w a v e m o d e l H z v a r i e s a s t h e s q u a r e r o o t o f t h e f r e q u e n c y . F i e l d m e a s u r e m e n t s m a d e b y C h r i s t o f f e l e t a l (1961) a n d o t h e r s i n d i c a t e t h a t t h e r a t i o , i n g e n e r a l , d e c r e a s e s w i t h f r e q u e n c y . ( b ) O n e C o n d u c t i n g L a y e r O v e r l y i n g a : H o m o g e n e o u s C o n d u c t i n g E a r t h o f I n f i n i t e D e p t h A l t h o u g h t h e p l a n e w a v e m o d e l p r e d i c t s a m u c h s m a l l e r v e r t i c a l - t o - h o r i z o n t a l m a g n e t i c f i e l d r a t i o t h a n i s o b s e r v e d i n a c t u a l f i e l d m e a s u r e m e n t s , i t i s o f i n t e r e s t t o u s e t h i s m o d e l t o s t u d y t h e e f f e c t o f i n h o m o g e n e i t i e s i n t h e c o n d u c t i n g s u b s t r a t u m . O n e w o u l d e x p e c t t h a t t h e e f f e c t o f i n h o m o g e n - e i t i e s i n c o n d u c t i v i t y f o r a p l a n e w a v e s o u r c e w o u l d b e s i m i l a r f o r o t h e r f i e l d s o u r c e s s u c h a s l i n e c u r r e n t s . T h e p l a n e w a v e m o d e l r e a d i l y l e n d s i t s e l f t o s t u d y i n g l a y e r e d c o n d u c t o r s . T h e e f f e c t o f a c o n d u c t i n g h o r i z o n t a l l a y e r o f v a r y i n g t h i c k n e s s a n d c o n d u c t i v i t y o v e r l y i n g a h o m o g e n e o u s c o n d u c t i n g e a r t h o f i n f i n i t e d e p t h i s s t u d i e d h e r e . F o r t h i s p r o b l e m = oo w a s u s e d i n e q u a t i o n s ( k 7 ) t o (51)* T h e f i e l d c o m p o n e n t s a n d p h a s e a n g l e s w e r e c o m p u t e d f o r t h e s a m e f r e - q u e n c i e s a n d c o n d u c t i v i t i e s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . S i n c e f r e q u e n t r e f e r e n c e t o " s k i n d e p t h " ( t h e - a m p l i t u d e F i g u r e 3. T h e r a t i o R a s a f u n c t i o n o f f r e q u e n c y f o r o=10~ 1 3 e m u , a n d (1) 9 = 5 , ( 2 ) 1 0 , (3) 2 5 5 ( k ) k 5 , ( 5 ) 6 5 s a n d ( 6 ) 8 5 ° . 3 k o f a w a v e i s a t t e n u a t e d b y a f a c t o r 1 / e = 0.368 i n a s k i n d e p t h ) w i l l b e m a d e t h r o u g h o u t t h i s w o r k , a g r a p h s h o w i n g s k i n d e p t h a s a f u n c t i o n o f f r e q u e n c y f o r v a r i o u s c o n d u c t i v i t i e s i s g i v e n i n F i g . h. F i g u r e 5 s h o w s h o w t h e a m p l i t u d e s o f t h e t h r e e c o m p o n - e n t s o f t h e m a g n e t i c f i e l d a n d t h e r a t i o R o f t h e v e r t i c a l t o t h e h o r i z o n t a l c o m p o n e n t s d e p e n d o n t h e a n g l e o f i n c i d e n c e o f t h e p l a n e w a v e s . T h e a n g u l a r d e p e n d e n c e w a s f o u n d t o b e t h e s a m e f o r a l l f r e q u e n c i e s , l a y e r t h i c k n e s s e s a n d c o n d u c t i v - i t i e s . S i n c e a n d E w a r e s i m p l y r e l a t e d t o H _ t h e y a r e n o t x y z s h o w n h e r e . H a n d H , f o u n d t o b e t h e s a m e f o r a l l c o n d u c t - x y i v i t i e s , f r e q u e n c i e s a n d l a y e r t h i c k n e s s e s , w i l l n o t b e s h o w n i n t h e f o l l o w i n g f i g u r e s i n t h i s s e c t i o n . T h e p h a s e a n g l e s w i l l n o t b e t r e a t e d i n d e t a i l i n t h i s s e c t i o n . W e n o t e a g a i n , t h a t t h e r a t i o R , i s v e r y s m a l l . S i n c e t h e h o r i z o n t a l c o m p o n e n t s a r e u n a f f e c t e d b y c h a n g e s i n l a y e r t h i c k n e s s , c o n - d u c t i v i t y , a n d f r e q u e n c y , H z r a t h e r t h a n t h e r a t i o R i s s t u d i e d i n t h e f o l l o w i n g f i g u r e s i n t h i s s e c t i o n . A s c a n b e s e e n f r o m F i g . 6, t h e m a g n i t u d e o f t h e v e r t - i c a l c o m p o n e n t i s s t r o n g l y d e p e n d e n t o n t h e d e p t h a t w h i c h t h e c o n d u c t i v i t y d i s c o n t i n u i t y o c c u r s . T h e c o n d u c t i n g l a y e r h a s a c o n d u c t i v i t y 0 = 1 0 e m u c o r r e s p o n d i n g t o t h a t o f f r e s h —16 w a t e r w h i l e t h e r e g i o n b e l o w h a s a c o n d u c t i v i t y a - 10" e m u c o r r e s p o n d i n g t o t h a t c f d r y e a r t h . T h e d e p e n d e n c e o f H _ o n t h e f r e q u e n c y o v e r a l a r g e f r e q u e n c y r a n g e f c r v a r i o u s l a y e r t h i c k n e s s e s i s b r o u g h t o u t i n F i g . 7. T h e l a y e r t h i c k n e s s i s a s t r o n g f a c t o r i n d e t e r - 35 ' « l I i I— 10"2 I 102 104 f CYCLES/SEC F i g u r e h. • S k i n d e p t h a s a f u n c t i o n o f f r e q u e n c y f o r a r a n g e o f c o n d u c t i v i t i e s . 36 9 DEGREES F i g u r e 5. T h e a m p l i t u d e s H , H , H z , a n d t h e r a t i o R a s f u n c t i o n s o f Q f o r d 2 = - 1 0 ~ ' l f , d 3 = 1 0 ~ 1 6 e m u , f = 1 c y c l e / s e c , a n d d o = 1 0 ^ c m . 37 H, 20 40 60 0 DEGREES 80 T h e d e p e n d e n c e o f - H z o n t h e a n g l e o f i n c i d e n c e 0 - ^ = 1 0 " ^ e m u , a n d F i g u r e 6 a n d l a y e r t h i c k n e s s f o r f = 1 c y c l e / s e c . f CYCLES/SEC F i g u r e 7 . H a s a f u n c t i o n o f f a n d ( b ) O 2 = 1CH 6 3 ' q . = 1 0 " 1 l f e m u . f CYCLES / SEC d 2 f o r 3= 5 0 ° , ( a ) c 2 = 1 0 ~ l I f , < j . =10 " 1 6 , 39 m i n i n g t h e f r e q u e n c y d e p e n d e n c e . F o r a s i n g l e l a y e r o f i n f i n i t e d e p t h ( d 2 = OP) i n F i g s . 7 ( a ) a n d ( b ) H z v a r i e s a s t h e s q u a r e r o o t o f t h e f r e q u e n c y . F i g u r e s 8 a n d 9 s h o w t h e m o r e d e t a i l e d d e p e n d e n c e o f R" z o n f r e q u e n c y , l a y e r t h i c k n e s s , a n d c o m b i n a t - i o n o f c o n d u c t i v i t i e s . I t i s e v i d e n t f r o m F i g s . 8 a n d 9 t h a t t h e c o n d u c t i v i t y o f t h e s u b s t r a t u m h a s a n i m p o r t a n t e f f e c t f o r s h a l l o w s u r f a c e l a y e r s . I t i s a l s o s h o w n t h a t i f t h e s u r f a c e l a y e r h a s t h e g r e a t e r c o n d u c t i v i t y , t h e n f o r s o m e c a s e s H z d o e s n o t v a r y w i t h f r e q u e n c y . T h i s i s b r o u g h t o u t i n c u r v e s 2, 3, 7, a n d 11 i n F i g . 8. F i g u r e 9 s h o w s t h a t H z d o e s v a r y w i t h f r e q u e n c y f o r a l l d e p t h s c o n s i d e r e d i f t h e s u r f a c e l a y e r h a s t h e s m a l l e r c o n d u c t i v i t y . T h e l a y e r d e p t h a n d t h e c o n d u c t - i v i t y c o m b i n a t i o n s v e r y s t r o n g l y a f f e c t b o t h t h e a m p l i t u d e o f t h e v e r t i c a l c o m p o n e n t a n d i t s f r e q u e n c y d e p e n d e n c e . G r e a t e r d e t a i l o f t h e d e p e n d e n c e o f H o n l a y e r t h i c k n e s s f o r f r e - z q u e n c i e s 1 , h a n d 1 6 c y c l e s / s e c i s p r o v i d e d i n F i g . 1 0 . 1 i "1 A C u r v e s A s h o w t h a t , f o r c 2 = 1 0 e m u a n d c , = 1 0 " e m u a n d ^ h s u r f a c e l a y e r t h i c k n e s s e s r a n g i n g f r o m I O - 1 c m t o 1 0 c m , H _ z h a s t h e s a m e v a l u e f o r e a c h f r e q u e n c y i n t h e r a n g e 1 t o 16 c y c l e s / s e c , w h e r e a s c u r v e s B s h o w t h e f r e q u e n c y d e p e n d e n c e t o b e q u i t e d i f f e r e n t i f t h e c o n d u c t i v i t i e s o f t h e s u r f a c e l a y e r a n d t h e s u b s t r a t u m , a r e i n t e r c h a n g e d . F o r l a r g e d 2 , H z a p p r o a c h e s t h e v a l u e s o b t a i n e d f o r a s i n g l e c o n d u c t i n g l a y e r o f i n f i n i t e d e p t h . F i g u r e 11 s h o w s h o w t h e p h a s e <j>„ d e p e n d s o n t h e s u r - z f a c e l a y e r t h i c k n e s s f o r a v a r i e t y o f c o n d u c t i v i t y c o m b i n a t - i o n s f o r a f r e q u e n c y o f 1 c y c l e / s e c . F o r s m a l l a n d l a r g e d 9 ? ho f CYCLES / SEC F i g u r e 8. H z as a f u n c t i o n o f f f o r e ^ 0 , d 2 =10~ 1 1 ? o 3 =10~ 1 6 emu,- (1) d 2=10, (2) 10 3 , (3) 1 o \ (h) 105$ d 3 =10 " 1 ^ , (5) d 2 =10, (6) 10 3 , (7.) 10^", (8) 1 0 * ; d 3 =10~ l I f , (9) d 2 =10, (10) 10 3 , (11) 10**, (12) 10^5 d 3 = 1 0 " 1 3 , (13)> d2=10,- (1*+)> 1o3, (15)' 10^, (16) 10* cm. k1 f CYCLES / SEC F i g u r e 9. • H z a s a f u n c t i o n o f f f o r e= k5° s o 2 = 1 0 " l b s 0 3 =10" 1 1 e m u , (1) d 2 = 1 0 3 . (2) 10^ s (3) 10 6 , (h) 10 7 ; O 3 =10~ 1 3 , - (5) d 2=103, (6) 10 5 , (7) 10 6 3 (8) 10 7 ; 0 3=10~ 1^,. (9) d 2 = 1 0 3 s (10) 10^, (11) 10 6 , (12) 10 7 c m . F i g u r e 10. • H as a f u n c t i o n o f dp f o r f =1 , L , 16 c y c l e s / s e c and e= k5°, (A) o 2 = 1 0 ~ 1 1 , o 3 =10~^ 6 emu, (B) o 2 = 1 0 ~ 1 6 , 0^=10 -^ emu. ^3 d 0 cm F i g u r e 11 „ <p„ a s a f u n c t i o n o f cL-, f o r f=1 c y c l e / s e c , ©= (A) d 2=1C~ 1*, d- =1(T16, 10~ 1*, 10~ l l + ? 10~ 1 3 e m u , ( B ) d 2 = 1 C f 1 6 , d.=10" 1 1, 10~ 1 3, K T 1 ^ , 10~ 1* e m u . 0_ h a s a v a l u e o f 225° . T h i s a g r e e s w i t h t h e v a l u e o f 0„ f o r Zs Zi a s i n g l e l a y e r o f i n f i n i t e d e p t h . 0 d e p a r t s g r e a t l y f r o m 225° w h e n t h e r e a r e t w o c o n d u c t i n g l a y e r s a n d t h e s u r f a c e l a y e r i s l e s s t h a n a s k i n d e p t h t h i c k ( s e e F i g . k). T h e p h a s e s h i f t s t o v a l u e s g r e a t e r t h a n 225° i f G*2 a n d t o ! V a l u e s , s m a l l e r t h a n 225° i f d 2 > d y C u r v e s A a n d B s h o w t h a t t h e g r e a t e r t h e d i f f e r e n c e b e t w e e n d 2 a n d d^ t h e g r e a t e r t h e m a g n i - t u d e o f t h e m a x i m u m p h a s e c h a n g e a n d t h e g r e a t e r t h e r a n g e f o r w h i c h t h i s p h a s e c h a n g e o c c u r s . I t w a s f o u n d t h a t a l t h o u g h t h e m a g n i t u d e o f t h e m a x i m u m p h a s e c h a n g e f o r a g i v e n c o n d u c t i v i t y c o m b i n a t i o n i s i n d e p e n d e n t o f t h e f r e q u e n c y , t h e l a y e r t h i c k - n e s s r e q u i r e d f o r m a x i m u m p h a s e c h a n g e d o e s d e p e n d o n f r e q u e n c y . ( c ) T w o C o n d u c t i n g L a y e r s Q v e r l y i f a g a n H o m o g e n e o u s C o n d u c t i n g B a r t h o f I n f i n i t e B e p t h T o f u r t h e r s t u d y t h e e f f e c t o f i n h o m o g e n e i t i e s i n c o n d u c t i v i t y , t h e p r o b l e m o f t w o l a y e r s o v e r l y i n g a ; i h o m o g e n - e o u s c o n d u c t i n g e a r t h o f i n f i n i t e d e p t h i s t r e a t e d h e r e . T h e e l e c t r o m a g n e t i c f i e l d s w e r e c o m p u t e d f o r p o i n t s i n t h e u p p e r c o n d u c t i n g l a y e r a s w e l l a s a t t h e s u r f a c e . T h e a m p l i t u d e s a n d p h a s e a n g l e s i n t h e u p p e r c o n d u c t i n g l a y e r a r e o f p a r t i c - u l a r i n t e r e s t w h e n t h e u p p e r c o n d u c t i n g l a y e r i s a n . o c e a n . P h y s i c a l m e a s u r e m e n t s o f m a g n e t i c f i e l d v a r i a t i o n s i n t h e o c e a n a r e b e i n g m a d e b y v a r i o u s l a b o r a t o r i e s ( e . g . P a c i f i c N a v a l L a b o r a t o r y ) a n d s o m e i n f o r m a t i o n o n t h e a t t e n u a t i o n , a n d p h a s e c h a n g e s a s a f u n c t i o n o f d e p t h i n t h e s e a i s o f i n t e r e s t . A l t h o u g h e m p h a s i s i s p l a c e d o n , t h e c a s e w h e r e t h e u p p e r l a y e r h a s t h e c o n d u c t i v i t y o f s e a w a t e r , o t h e r c o n d u c t i v i t i e s a r e M-5 a l s o c o n s i d e r e d . S i n c e t h e a n g u l a r d e p e n d e n c e i s t h e s a m e f o r a l l c a s e s c o n s i d e r e d h e r e , t h e a n g l e o f i n c i d e n c e i s a r b i t r a r - i l y t a k e n a s L+5°» T h e a t t e n u a t i o n t h a t t a k e s p l a c e w i t h d e p t h i n t h e c o n d u c t i n g l a y e r i s s h o w n i n F i g . 1 2 ( a ) a n d t h e d e p e n d e n c e o f t h e p h a s e a n g l e s o n d e p t h i s s h o w n i n F i g . 1 2 ( b ) . T h e a t t e n - u a t i o n w i t h d e p t h f o r e a c h o f H a n d H i s c o n s i d e r a b l y x y g r e a t e r t h a n f o r E v , E . a n d H g = A l t h o u g h t h e s u r f a c e v a l u e s x y z o f H a n d H a r e t h e s a m e f o r a l l c o n d u c t i v i t i e s a n d f r e q u e n c i e s c o n s i d e r e d , t h e a t t e n u a t i o n w i t h d e p t h d o e s d e p e n d o n t h e f r e - q u e n c y a n d t h e c o n d u c t i v i t y o f t h e u p p e r l a y e r . H , E , a n d E „ d e p e n d o n t h e c o n d u c t i v i t y a n d t h i c k n e s s e s o f e a c h l a y e r a s w e l l a s d e p e n d o n t h e f r e q u e n c y . T h e s e c o m p o n e n t s a r e d i a g - n o s t i c o f t h e e n t i r e c o n d u c t i v i t y s t r u c t u r e , w h i l e H x a n d E y a r e n o t . S i n c e E „ a n d E „ a r e s i m p l y r e l a t e d t o H , o n l y H i s s h o w n i n t h e r e m a i n i n g f i g u r e s i n t h i s s e c t i o n . T h e e f f e c t o f t h e l o w e r s e m i - i n f i n i t e c o n d u c t i n g l a y e r o n H z a t t h e s u r f a c e o v e r a s m a l l f r e q u e n c y r a n g e 1 t o 32 c y c l e s / s e c i s s h o w n i n F i g . 13" I n t h i s c a s e t h e u p p e r t w o l a y e r s a r e , i n f a c t , v e r y s h a l l o w i n t e r m s o f s k i n d e p t h s a n d h e n c e t h e l o w e r s e m i - i n f i n i t e l a y e r p l a y s t h e d o m i n a n t r o l e . T h e c o n d u c t i v i t i e s a n d l a y e r t h i c k n e s s a r e t h o s e w h i c h m i g h t b e p o s s i b l e f o r l a n d m a s s e s o n t h e e a r t h . T h e d e p e n d e n c e o f H z a t t h e s u r f a c e o n f r e q u e n c y o v e r a l a r g e f r e q u e n c y r a n g e 1 0 J t o 1 0 c y c l e s / s e c f o r v a r i o u s l a y e r t h i c k n e s s e s i s b r o u g h t o u t i n F i g . 1 1 * ' . T h e c o n d u c t i v i t y o f t h e u p p e r l a y e r i s -11 dp = 10 e m u , t h e c o n d u c t i v i t y o f s e a w a t e r . C u r v e s 1 a n d - r Z cm Z cm F i g u r e 1 2 . T h e a m p l i t u d e s ( a ) a n d t h e p h a s e a n g l e s ( b ) a s a f u n c t i o n o f z f o r d 2 = k x 1 ( T 1 1 , o 3 = 1 0 " l i + e m u 3 d 2 = 2 x 1 Q l + c m , d =QO? f = i c y c l e / s e c , a n d - e = l + 5 ° . k 7 F i g u r e 13. • H z a s a f u n c t i o n o f f f o r d 2 = 1 0 - 3 , d ^ = 1 . 6 x 1 0 c m , z = 0 , a n d 0 = k 5 ° . J+8 F i g u r e 1 L . H z a s a f u n c t i o n o f f f o r d 2 = 1 0 , ^ = 1 0 d ^ l O " 1 6 e m u , z = 0 , 0=M - 5 ° , d 3 = 1 . 6 x 1 0 ^ c m , ( 1 ) d 2 = 0 , (2) 1 0 2 (3) 1 0 3 , (h) 1 0 ^ ; a n d d 3 = 8 x 1 0 * c m , (5) d 2 = 0 , ( 6 ) 1 0 , (7) 1 0 2 , ( 8 ) 1 0 3 , ( 9 ) 1 0 \ ( 1 0 ) 1 0 * c m . h9 10 v a r y a p p r o x i m a t e l y a s t h e s q u a r e r o o t o f t h e f r e q u e n c y f o r p a r t o f t h e f r e q u e n c y r a n g e . F o r c u r v e 1, t h e s u r f a c e l a y e r h a s z e r o t h i c k n e s s , w h i l e t h e s e c o n d l a y e r i s s h a l l o w c o m p a r e d w i t h t h e s k i n d e p t h f o r f r e q u e n c i e s b e l o w 1 c y c l e / s e c , a n d h e n c e t h e s t r u c t u r e i s v e r y n e a r l y t h a t o f a s i n g l e s e m i - i n f i n i t e l a y e r w i t h c o n d u c t i v i t y e q u a l t o t h a t o f d ^ . F o r c u r v e 10, t h e s u r f a c e l a y e r i s m u c h d e e p e r t h a n a s k i n d e p t h f o r f r e q u e n c i e s a b o v e 1 c y c l e / s e c , a n d h e n c e t h e s t r u c t u r e a g a i n b e h a v e s a s a s i n g l e s e m i - i n f i n i t e l a y e r w i t h c o n d u c t i v i t y e q u a l t o t h a t o f d^» F o r c u r v e 10, t h e s u r f a c e l a y e r i s m u c h d e e p e r t h a n a s k i n d e p t h f o r f r e q u e n c i e s a b o v e 1 c y c l e / s e c , a n d h e n c e t h e s t r u c t u r e a g a i n b e h a v e s a s a s i n g l e s e m i - i n f i n i t e l a y e r w i t h c o n d u c t i v i t y e q u a l t o t h a t o f t h e s u r f a c e l a y e r . A s c a n b e s e e n f r o m t h e c u r v e s o f F i g . 1 k , t h e f r e q u e n c y d e p e n d - e n c e f o r a t h r e e - l a y e r c o n d u c t i n g s t r u c t u r e i s , i n g e n e r a l , n o t a s i m p l e o n e . I t i s e v i d e n t f r o m F i g s . 1 5 ( a ) a n d ( b ) t h a t , f o r a s h a l l o w s u r f a c e l a y e r , t h e c o n d u c t i v i t y s t r u c t u r e h a s a n i m - p o r t a n t e f f e c t . T h e c o m b i n e d t h i c k n e s s o f t h e u p p e r t w o l a y e r s i s l e s s t h a n a s k i n d e p t h f o r a s i n g l e l a y e r o f t h e l o w e r c o n - d u c t i v i t y f o r a l l t h e s t r u c t u r e s i n d i c a t e d . T h a t t h e l o w e r s e m i - i n f i n i t e l a y e r i s I m p o r t a n t f o r s h a l l o w s u r f a c e l a y e r s i s p a r t i c u l a r l y e v i d e n t w h e n c o m p a r i n g c u r v e s 1 a n d 2 w i t h 7 5 3 w i t h 8, i f a n d . 5 w i t h 9 5 a n d 6 w i t h 10. A l t h o u g h t h e a m p l i t u d e a n d p h a s e a n g l e s a r e h i g h l y d e p e n d e n t o n t h e s t r u c t u r e f o r a l l c a s e s s h o w n . , v e r y l i t t l e a t t e n u a t i o n a s a f u n c t i o n o f d e p t h i n t h e s h a l l o w s u r f a c e l a y e r i s i n d i c a t e d . C u r v e s 2, a n d 10, icr3 - H, r4 10 ,-5 10 F i g u r e 15° H 1,8 3,7 4 -lO- taj 270 250- to LU LU cr o LU Q 230- 210- I03 Z c m 10 10 (b) I03 Z c m d 0 = 2 x 1 0 10 1 4 \ 10 ( a ) a n d <j>2 ( b ) a s a f u n c t i o n o f z f o r f = 1 c y c l e / s e c 9 = 1 + 5 ° d c m , a n d t h e f o l l o w i n g o 9 , d _ , 0 , .. (1) 1 0 10' 10' •1M =16 ! 5 ( 3 ) 10 1 0 ~ 1 6 , ( 6 ) 1 0 " 1 6 10 - 1 6 ' ( 9 ) 10 - 1 6 , 10 , '" 1 1 S 1<T1\ 1 0 - 1 6 , ( 2 ) 10" 10-1H-,- ( L l ) 1 0 - l O 5 1 0 - l l + ? 1 0 - H 5 ( 5 ) 1 0-16 1 0 ~ \ ( 7 ) 1 0 " 1 1 , 1 0 - 1 ^ , 1 0 " 1 \ ( 8 ) 1 0 " 1 1 , 1 0 ~ 1 \ ( 1 0 ) 1 0 " 1 6 , 10- 1 1, 1 0 ~ 1 1 emu. 51 w h i c h r e f e r t o s t r u c t u r e w i t h t h e l o w e r s e m i - i n f i n i t e l a y e r —11 h i g h l y c o n d u c t i n g ( 1 0 e m u ) ? s h o w s o m e a t t e n u a t i o n . T h e s p e c i a l c a s e o f c o n s t a n t t h i c k n e s s o f t h e s e c o n d l a y e r i s s h o w n i n F i g s . 16 a n d 17. F i g u r e s 1 6 ( a ) a n d ( b ) g i v e t h e a m p l i t u d e a n d p h a s e a n g l e a t t h e s u r f a c e a s a f u n c t - i o n o f s u r f a c e - l a y e r t h i c k n e s s , w h i l e F i g s . 1 7 ( a ) a n d ( b ) g i v e t h e a m p l i t u d e a n d p h a s e a n g l e a t t h e i n t e r f a c e o f t h e f i r s t a n d s e c o n d l a y e r ( z = d^) f o r t h e s a m e s t r u c t u r e . I t c a n b e s e e n f r o m c u r v e s 1 t o 5 i n F i g s . 1 6 a n d 17 t h a t t h e s e c o n d l a y e r h a s l i t t l e e f f e c t w h e n i t i s m u c h l e s s t h a n a s k i n d e p t h t h i c k ; h o w e v e r , i t s p r e s e n c e i s i m p o r t a n t , a s i s s e e n f r o m c u r v e s 6 t o 1 0 , w h e n t h e s u r f a c e l a y e r i s s h a l l o w a n d h i g h l y — 11 c o n d u c t i n g ( s e a w a t e r , 1 0 e m u ) . F i g u r e s 1 6 ( a ) a n d ( b ) a r e a l m o s t i d e n t i c a l t o F i g s , ^^•(a.) a n d ( b ) f o r s u r f a c e - l a y e r d e p t h s m u c h l e s s t h a n a s k i n d e p t h , b u t t h e y d i f f e r r a d i c a l l y f o r s u r f a c e - l a y e r d e p t h s g r e a t e r t h a n a s k i n d e p t h . W e h a v e s e e n e a r l i e r t h a t t h e a t t e n u a t i o n i s s m a l l f o r d e p t h s l e s s t h a n a s k i n d e p t h a n d h e n c e t h e s i m i l a r i t y f o r s h a l l o w s u r f a c e l a y e r s i s e x p e c t e d . O n t h e b a s i s o f r e s u l t s d i s c u s s e d i n t h i s s e c t i o n , i t i s s e e n t h a t f o r t h e p l a n e w a v e m o d e l t h e a m p l i t u d e s a n d p h a s e a n g l e s o f t h e v e r t i c a l m a g n e t i c f i e l d c o m p o n e n t s a n d t h e h o r i - z o n t a l e l e c t r i c f i e l d c o m p o n e n t s b o t h a t t h e s u r f a c e a n d i n t h e u p p e r c o n d u c t i n g l a y e r a r e s t r o n g l y d e p e n d e n t - o n t h e s t r u c t u r e o f t h e c o n d u c t o r . I t i s o f i n t e r e s t a l s o t o n o t e t h a t , a l t h o u g h t h e s e c o m p o n e n t s a r e r a t h e r s e n s i t i v e t o s t r u c t u r e , t h e a t t e n u a t i o n w i t h d e p t h i n t h e s u r f a c e l a y e r f o r Q 11 w O Q -J- n O N o cm o H (D \ w (D CD O N O O I - ON VJT. <D N «•» II U) IV) o s—-II ^ ^ _>. p o p p .1.00 p* O N — 1 - & - f c t ESI - p- -̂v (D » >>—' C O Hj N - ' v-^ O O —* H ! O O • • s vn. —1 H* , -° cm CO ^ Q ^ - r ro w Hj O c+ O 0 O — 1 Q. i O U ) — I ! - O ro Q ^ -T P i \ 0 w |V) o o I O O N -*• I I II O J ->• - J . o -* — 1 ON-* ^ a, o p - r 1 ^ & 1  p, - 1 IV) o. o II O IV) I —1 I II ->• o - 1 - 1 - r o -° » o 2 B H3 <£z DEGREES ro ro 2£  5h t h e s e c o m p o n e n t s i s r a t h e r s m a l l . 2.2 A C o m p l e x L a y e r e d C o n d u c t i n g E a r t h i n t h e F i e l d o f P l a n e W a v e s 2.2.1 I n t r o d u c t i o n F o r s i m p l i c i t y , t h e a n a l y s i s o f f i e l d m e a s u r e m e n t s i s o f t e n b a s e d o n t h e a s s u m p t i o n o f s e v e r a l h o m o g e n e o u s c o n d u c t - i n g l a y e r s . H o w e v e r , i t i s o f i n t e r e s t t o s t u d y t h e e f f e c t o f i n h o m o g e n e o u s c o n d u c t i n g l a y e r s , s i n c e i n s o m e c a s e s t h e s e m a y b e m o r e r e a l i s t i c c o n d u c t i v i t y d i s t r i b u t i o n s . A p p l y i n g t h e m a g n e t o t e l l u r i c m e t h o d o f a n a l y s i s t o m e a s u r e m e n t s m a d e a t M e a n o o k , A l b e r t a , N i b l e t t a n d S a y n - W i t t g e n s t e i n (1960) h a v e o b t a i n e d r e s u l t s i n d i c a t i n g a d e c r e a s e i n c o n d u c t i v i t y w i t h d e p t h . R e l a t i v e l y n e a r t h e s u r f a c e t h e c o n d u c t i v i t y c o u l d b e e x p e c t e d t o d e c r e a s e r a p i d l y w i t h d e p t h b e c a u s e t h e r o c k m a t e r i a l b e c o m e s p r o g r e s s i v e l y d r i e r , w h e r e a s a t g r e a t d e p t h s t h e i n c r e a s i n g t e m p e r a t u r e s h o u l d t e n d t o i n c r e a s e t h e c o n d u c t - i v i t y . T h i s s e c t i o n o f t h e p r e s e n t w o r k d e a l s w i t h t h e e l e c - . t r i e a n d m a g n e t i c f i e l d s a t t h e s u r f a c e o f a m u l t i l a y e r c o n d u c t i n g s t r u c t u r e t h a t r e p r e s e n t s 3 t o a g o o d a p p r o x i m a t i o n , a n i n h o m o g e n e o u s c o n d u c t o r w h o s e c o n d u c t i v i t y i s d e s c r i b e d b y a c o n t i n u o u s f u n c t i o n . T h e c o n d u c t i v i t y f u n c t i o n s a n d f r e - q u e n c i e s c o n s i d e r e d a r e o f i n t e r e s t i n g e o p h y s i c s . I n s t u d y i n g t h e r e s p o n s e o f a n o s c i l l a t i n g m a g n e t i c d i p o l e o v e r a c o n t i n u o u s l y s t r a t i f i e d f l a t e a r t h . W a i t - (1962 b ) c o n s i d e r e d a c o n d u c t i n g e a r t h w h o s e c o n d u c t i v i t y v a r i e d e x p o n - e n t i a l l y w i t h d e p t h . T w o f u n c t i o n s c o n s i d e r e d b y h i m . a r e u s e d 55 h e r e ; t h e s e h a v e t h e f o r m d ( z ) = d Q e x p ( - a z ) a n d d ( z ) = . 0 [1 + k z e x p ( - a z ) ] , w h e r e a a n d k a r e p o s i t i v e c o n s t a n t s w i t h d i m e n s i o n s c m ' „ A n o t h e r f u n c t i o n d i s c u s s e d i n a r e v i e w a r t i c l e b y W a i t ( 1 9 6 2 a ) h a s t h e f o r m o ( z ) = d Q(1 + k z ) " " P , w h e r e k a n d p a r e p o s i t i v e c o n s t a n t s . O t h e r f u n c t i o n s c o n s i d e r e d i n t h e p r e s e n t w o r k a r e o ( z ) - d Q e x p ( a z ) s d ( z ) • = c [1 + k z e x p ( a z ) ] , d ( z ) = d 0(1 + k z ) , a n d d ( z ) = d Q ( 1 + k z ) . 2 . 2 . 2 M a t h e m a t i c a l A n a l y s i s T h e a n a l y t i c a l e x p r e s s i o n s f o r t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s f o r t h i s m u l t i l a y e r c o n d u c t o r ( n - 1 l a y e r s ) f o l l o w s i m p l y a s a n e x t e n s i o n o f t h e t h r e e - l a y e r c o n d u c t o r p r o b l e m t r e a t e d e a r l i e r ( S e c t i o n 2 . 1 . 2 ) . I n s o l v i n g M a x w e l l ' s e q u a t i o n s i n t u r n f o r f o u r a n d t h e n f o r f i v e c o n d u c t i n g l a y e r s a r e c u r r e n c e r e l a t i o n b e c o m e s e v i d e n t a n d t h e e x p r e s s i o n s f o r t h e c a s e o f n m e d i a a r e o b t a i n e d . S i n c e t h e m a t h e m a t i c a l a n a l y s i s o f t h i s p r o b l e m f o l l o w s t h e s a m e p r o c e d u r e a s w a s u s e d i n 2 . 1 . 2 , t h e m a t h e m a t i c s w i l l n o t b e g i v e n i n d e t a i l h e r e . W e a r e a g a i n c o n s i d e r i n g a p l a n e e l e c t r o m a g n e t i c w a v e h a r m o n i c i n t i m e w i t h a n g u l a r f r e q u e n c y CD , t r a v e l - i n g i n f r e e s p a c e ( m e d i u m 1 ) a n d i n c i d e n t u p o n t h e s u r f a c e o f a s t r a t i f i e d f l a t c o n d u c t o r a s s h o w n i n F i g . 18. T h e c o n d u c t o r c o n s i s t s o f n - 1 f l a t , h o m o g e n e o u s , i s o t r o p i c l a y e r s ( m e d i a 2 , 3 ? - - s n ) , w i t h t h e l a s t l a y e r i n f i n i t e i n d e p t h . T h e n m e d i a a r e c h a r a c t e r i z e d b y t h e c o n s t a n t s e^s | j , • , d . ( j =• 1 , - - ? n ) . T h e m a g n e t i c p e r m e a b i l i t y o f 56 o f e a c h c o n d u c t i n g l a y e r i s a s s u m e d t o b e t h a t o f f r e e s p a c e * F o l l o w i n g t h e m e t h o d u s e d e a r l i e r ( S e c t i o n 2.1.2) a n d u s i n g t h e s i m p l i f i e d e x p r e s s i o n f o r t h e p r o p a g a t i o n c o n s t a n t (53) k \ . = c ^ i ^ 2 ~ i W ^ d ^ c o ) * = ( - i W L t j d v t o ) ^ f o r j = 2 , 3, - - , n - , t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n - e n t s i n t h e u p p e r c o n d u c t i n g l a y e r c a n b e e x p r e s s e d a s : (5 k ) ( E 2 ) x = [ e x p [ - z v 2 ( 1 + i ) ] - - m£ e x p [ z v 2 ( 1 + i ) ] J . ( 5 5 ) ( ^ y = - o ( ^ 4 ) | e x p [ - z v 2 ( 1 + i ) ] - m 2 e x p [ z v 2 ( 1 + i ) ]j c o s 6 1 , (56)> ( E 2 ) Z = E Q H ^ I j ^ e x p [ - z v 2 ( 1 + i ) ] + m ^ e x p { z v 2 ( U i ) ] j ^ s i n 6,, , ( 57K ( H 2 ) x • = jexp [ - zv 2 ( 1 + i ) ] + m 2 e x p [ z v 2 ( 1 + i ) f j , / W ' \ . (58) ( H 2 ) y = H ^ j ^ ^ J | e x p [ - z v 2 ( 1 + i ) ] + m 2 ' exp" [zv 2 <1 -+ i ) ] j c o s © 1 , ( 5 9 ) « - ( H 2 ) z > => - • H j^^Jjexp[-zv 2 ( 1 + i ) ] - m ^ exp[zv2(1 + i)]J s i n e 1 , w h e r e k ^ = ^•(LL^S^CO 2 - - i W j j , ^ c 2 c o ) " ^ = ^ J JJ,̂  e ^ s i n c e t h e c o n - d u c t i v i t y o f f r e e s p a c e i s z e r o , v 2 = 2u J d 2 f -, i = J -1 , a n d f i s t h e f r e q u e n c y . • T h e t i m e v a r i a t i o n e x p ( i c o t ) c o m m o n t o a l l c o m p o n e n t s i s n o t i n c l u d e d . T h e a p p r o x i m a t i o n i n d i c a t e d 57 i n e q u a t i o n (53) i s v a l i d f o r t h e f r e q u e n c i e s a n d c o n d u c t i v i t - p i e s c o n s i d e r e d i n t h i s w o r k s i n c e [ i - e - c o i s o r d e r s o f m a g n i t u d e J V 2 s m a l l e r t h a n W ^ . o . c co f o r a l l c a s e s . F o r t h e f r e q u e n c y a n d J J c o n d u c t i v i t y r a n g e s o f i n t e r e s t i n t h i s w o r k t h e v a r i o u s t e r m s i n e q u a t i o n s (5 L ) t o (59) a r e g i v e n b y : ( 6 0 ) t u2(-1<-i-) ( I r t-mpcos ©.j - (l-mvp m1 ^"u 2 (1 - i ) (1+m^)cos © 1 +• (1-m£) (61-)- u 2 ( 1 - i ) ( H - m 2 ' ) - O - m " ) c o s © 1 m 1 = u 2 ( 1 - i ) ( 1 + m » ) ( 1 -m») c o s ©., ' ( 6 2 ) - m " = m ' = I J ( a ^ ^ m ! + 1 ) ^ ^ ( q ^ r m ! + 1 X y o 7 e x p [ - 2 d . v . ( 1 + i ) ] J J f o r j = 2 , 3 5 n-1 , w i t h m 1 ! = m ! = 0 f o r j = n . ( 6 3 ) a J + 1 = e X p [ - 2 d . v . + 1 ( l + i ) ] ( 6 l f ) a n d (65) u 2 = c ^ ( o 2 / f ) S i n c e t h e a m p l i t u d e s E Q a n d H Q a r e d e s c r i p t i v e o f t h e i n c i d e n t w a v e s a n d h e n c e a r b i t r a r y a s f a r a s c o m p u t a t i o n s a r e c o n c e r n e d , i t i s a g a i n c o n v e n i e n t t o e x p r e s s e q u a t i o n s (5*+) " to (59) i n t h e f o r m g i v e n e a r l i e r b y e q u a t i o n s C+7) t o (52). 58 2.2..3 D i s c u s s i o n o f R e s u l t s N u m e r i c a l v a l u e s f o r t h e a m p l i t u d e s a n d p h a s e s o f t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s a t t h e s u r f a c e f o r v a r i o u s c o n d u c t i v i t y p r o f i l e s w e r e c o m p u t e d . T h e c o n d u c t i v i t - i e s a n d f r e q u e n c i e s c o n s i d e r e d a r e t h e s a m e a s t h o s e u s e d p r e v i o u s l y : d = 1 0 ~ 1 1 t o 1 0 " 1 6 e m u a n d f = 1 0 " - ^ 1 f a n d 1 0 3 c y c l e s / s e c . T h e a n g l e o f i n c i d e n c e i s M-5°. T h e c o n d u c t o r , c o n s i s t i n g o f n-1 l a y e r s , i s d i v i d e d i n t o s e v e r a l p r o f i l e s , w i t h a d i f f e r e n t c o n d u c t i v i t y f u n c t i o n f o r e a c h , p r o f i l e ^ a s s h o w n i n F i g . 1 9 . T h e s u b l a y e r s w i t h i n a p r o f i l e a r e e q u a l l y s p a c e d . T h e c o n d u c t i v i t y o f t h e f i r s t l a y e r i n e a c h p r o f i l e h a s t h e v a l u e d Q g i v e n f o r t h e c o n d u c t i v i t y f u n c t i o n i n t h a t p r o f i l e . F o r e a c h c a s e s t u d i e d , t h e r e s u l t s o b t a i n e d a r e c o m p a r e d w i t h v a l u e s f o r o n e - l a y e r h o m o g e n e o u s p r o f i l e s . T h e r e s u l t s f o r t h e v a r i o u s c o n d u c t i v i t y s t r u c t u r e s s t u d i e d a r e g i v e n i n t a b u l a r f o r m . T o d e t e r m i n e h o w m a n y s u b - l a y e r s a r e r e q u i r e d i n a p r o f i l e f o r t h e c o n d u c t o r t o b e h a v e i n a w a y s i m i l a r t o a c o n d u c t o r w h o s e c o n d u c t i v i t y * a s a f u n c t i o n o f d e p t h , v a r i e s i n a c o n t i n u o u s f a s h i o n * c a l c u l a t i o n s f o r o n e s p e c i a l c a s e w e r e c a r r i e d o u t f o r a v a r y i n g n u m b e r o f l a y e r s ( i . e . N 1 = 1 , 1 0 , 1 0 0 , 300, 500, 700, 1 , 0 0 0 ) . F o r t h i s c a s e t h e f i r s t p r o f i l e , w i t h D^ = 1 0 y c m , h a s N ^ c o n d u c t i n g l a y e r s w i t h c o n d u c t i v i t i e s d e t e r m i n e d b y t h e f u n c t i o n o = d Q —11 e x p ( - a z ) , w h e r e d =• 1 0 e m u , a n d a i s c h o s e n s u c h t h a t - 1 6 1 t h e c o n d u c t i v i t y i s 1 0 e m u a t z = 1 0 y c m . T h e s e c o n d p r o - f i l e c o n s i s t s o f a s i n g l e c o n d u c t i n g l a y e r o f i n f i n i t e d e p t h — 1 6 w i t h a c o n d u c t i v i t y o f 1 0 e m u . T h e r e s u l t s a r e g i v e n i n 59 INCIDENT WAVE MEDIUM I (free space) REFLECTED WAVE X MEDIUM 2 (conductor) MEDIUM 3 (conductor) 1 d n - | l I \ * \ t w MEDIUM N (conductor • \ of infinite depth) Z F i g u r e 18. M o d e l u s e d i n t h e c a l c u l a t i o n s f o r a c o m p l e x l a y e r e d e a r t h . free space 0*2 I d 4 Profile I N, sub-layers crAz) Profile II N 2 sub-layers <rbU) Profile TXT N 3 sub-layers (z) d n r D=oo <r0 F i g u r e 19- • C o m p l e x l a y e r e d c o n d u c t o r . 6 0 T a b l e I . I t c a n b e s e e n t h a t v e r y l i t t l e c h a n g e t a k e s p l a c e b e y o n d 300 l a y e r s f o r t h i s p a r t i c u l a r c a s e . A l l t h e f o l l o w i n g c a l c u l a t i o n s h a v e b e e n c a r r i e d o u t f o r 300 l a y e r s . A l t h o u g h a l l t h e c o m p o n e n t s a n d p h a s e a n g l e s w e r e c o m - p u t e d , o n l y H _ a n d 0 _ f o r t h e v a r i o u s c a s e s s t u d i e d a r e t r e a t e d b e l o w . T h e v a l u e s o f H a n d H a r e c o n s t a n t , E a n d x y x E _ . a r e s i m p l y r e l a t e d t o H _ , a n d E _ i s v e r y s m a l l a n d n o t o f y z Zi m u c h i n t e r e s t e x p e r i m e n t a l l y . T h e e f f e c t o n H a n d Q o f t h e c o n d u c t i v i t y i n c r e a s i n g Zi Zi w i t h d e p t h f o r v a r i o u s c o n d u c t i v i t y d i s t r i b u t i o n s i s s h o w n i n T a b l e I I . T h e s t r u c t u r e c o n s i s t s o f a s u r f a c e l a y e r 10 7 c m t h i c k d i v i d e d i n t o 300 l a y e r s o f e q u a l t h i c k n e s s e s a n d a s e m i - i n f i n i t e s u b s t r a t u m . T h e c o n s t a n t s , k a n d a , u s e d i n t h e c o n d u c t i v i t y f u n c t i o n s f o r N o s . 1 - k a r e c h o s e n t o b e s u c h -1 ^ t h a t t h e c o n d u c t i v i t y i n c r e a s e s f r o m 10-- e m u a t t h e s u r f a c e -11 7 t o 10 e m u a t a d e p t h o f 10' c m . F o r p u r p o s e s o f c o m p a r i s o n , t h e r e s u l t s f o r a h o m o g e n e o u s c o n d u c t i n g s u r f a c e l a y e r w i t h -1 5 c o n d u c t i v i t y 10 y e m u a n d a s e m i - i n f i n i t e s u b s t r a t u m w i t h —11 c o n d u c t i v i t y 10 e m u a r e g i v e n i n N o . 6 . T h e c o n s t a n t s , k a n d a , u s e d i n N o . 5 a r e s u c h t h a t t h e c o n d u c t i v i t y r e a c h e s a m a x i m u m a t 5 x 10^ c m a n d t h e n r e t u r n s t o 10~ 1 ^ e m u a ^ ^Q7 CNIO A g a i n f o r p u r p o s e s o f c o m p a r i s o n , t h e r e s u l t s f o r a h o m o g e n e o u s -1 "5 m e d i u m w i t h a c o n d u c t i v i t y o f 10 y e m u a r e g i v e n i n N o . 7° F r o m t h e r e s u l t s l i s t e d i n T a b l e I I , i t i s r e a d i l y s e e n t h a t t h e c o n d u c t i v i t y v a r i a t i o n h a s a m a r k e d e f f e c t o n b o t h t h e a m p l i t u d e a n d p h a s e a n g l e . T a b l e I I I s h o w s t h e e f f e c t o f d e c r e a s i n g c o n d u c t i v i t y 61 T A B L E I T h e e f f e c t o f v a r y i n g t h e n u m b e r o f s u b l a y e r s . N1 E x ( 1 0 " 6 c ) E y ( 1 0 - 6 c ) * x H y H z 1 0 * x y * x 0° < 1 1 .03x10~* 1.1+5xio~* 2.00 1 M 7.27x1O"6 M-6.6 2 2 6 . 6 ° 0 0 2 2 6 . 6 ° 10 2.56x10"* 3 . 6 2 x 1 0 ~ * 11 1 .81X10"6 3.81 183.1° 11 183.1° 100 >+.05x10"* 5.73x1'0"* n it 2.87x10"6 2.3^ 182„3° 11 182.3° 300 h.21x10"* 5.95x10"* 11 2.98x10~6 2.28 182.3° 11 11 182.30 500 M-.2M-X10~* 6.00x10~* n 11 3.00x10~6 2.27 182.3° 11 ti 182.3° 700 ^.25x10"* 6.02x10"* n 11 3.01x10~6 2.27 182.3° 11 182.3° 1000 M - .26x1 0"* 6.03x10~* ti 3.02x1O"6 2.27 182.3° • 182.3° D 1 = 10 y c m ? D 2 = °°> d a = 1 0 ~ 1 1 e x p ( - a z ) 9 a = 1 .15xio~ l I f s O-K =• 10" e m u . 62 T A B L E I I C o n d u c t i v i t y i n c r e a s i n g w i t h d e p t h f = 10" 3 s e c " ' ' f = 1 s e c -1 f = 103 s e c " ^ N o ' H z 0° H v z z 0° ^ z H z ^ z 1 1 .86x10"6 259.2° lf.7kx10"L|' 2̂ -0.8° 2.28x10~2 22^.7° 2 6.75x10~7 2^0.1° 7.00x10'"̂  2lf1 A ° 9.95x1O"3 250.1° 3 1 .05x10~6 2^7.5° 1.85X10~L 2^7.0° 2.19x10"2 230.1+° h 7-30x10"8 2lf0.1° 1.05x10"^ 253.7° 7.05x1O"3 267.0° 5 2.35x10"8 2̂ -3.9° 7A5X10"6 2 6 6 . h ° 7.00x1O"3 268.8° 6 2.27x10~6 265 0 6° 7 = 65x10"k 223.k° 2.31x10~2 223.7° 7 2-36x10~5 225.00 7- k5xio" k 225.0° 2.31x10"2 223.70 T A B L E I I ( c o n t i n u e d ) D1 D2 ( c m ) ( c m ) N1 c ( e m u ) o b ( e m u ) 10 7 oo 300 10"1^ e x p ( a z ) , o=9-21x10~7 1 0- 1 1 » 11 11 10~1^ ( 1 + k z ) , k=1.00x10 -3 tt » 11 11 10~1^ ( 1 + k z 2 ) , k=1 .00x10~10 it » 11 11 10" 1^[1+kz e x p ( a z ) ] . k=1.00, a =6.91x10~7 n it 11 1 0 _ 1 ? [ 1+ k z e x p ( - a z ) ] , k=k0.0, a=1.98x10" .. 1 1<T1* 10-11 „ t. ! 1(T1* io" 1* 63 T A B L E I I I C o n d u c t i v i t y d e c r e a s i n g w i t h d e p t h ,. f = 10" •3 s e c " 1 f = 1 s e c -1 f = 10 3 s e c " 1 N o . H z ^ z H z 0° ^ z H z 0° ^ z 1 2.82x10 r •7 203 A ° 7 0 95x10" 6 22^.8° 2.36x1o" k 225.0° 2 5-65x10" •6 189.8° 7.00x10" 6 229.1° 2.36x1o" k 225.0° 3 2.36x10" •5 225.0° 7.50x10" L 2 2 k „ 9 ° 2.30x1o" 2 223.6° k 2.36x10" •7 225.0° 7-95x10" 6 225.0° 2.36x1o~ k 225.0° T A B L E I I I ( c o n t i n u e d ) D1 D 2 ( c m ) ( c m ) N1 o a ( e m u ) d b ( e m u ) 10 7 o o 300 '10" •11 e x p ( - a z ) ? a=9° 21x10" 7 II II tt 10" ° 1 1 ( 1 + k z ) " 1 , k=1 . 00x10" 7 tt it it ti 10" ° 1 1 ( 1 + k z ) " 2 ; . k=9» 90x10" 6 it it it 1 10" 11 tt 6h a s a f u n c t i o n o f d e p t h f o r t h r e e d i f f e r e n t c o n d u c t i v i t y d i s t r i - b u t i o n s . T h e s t r u c t u r e a g a i n c o n s i s t s o f a s u r f a c e l a y e r 10^ c m t h i c k d i v i d e d i n t o 300 l a y e r s a n d a s e m i - i n f i n i t e s u b s t r a t - u m . T h e c o n s t a n t s k a n d a a r e c h o s e n t o b e s u c h t h a t t h e - 1 1 -1 5 c o n d u c t i v i t y d e c r e a s e s f r o m 10 e m u a t t h e s u r f a c e t o 10 ' 7 e m u a t a d e p t h o f 10 c m . T h e c o n d u c t i v i t y o f t h e h o m o g e n e o u s -11 s u b s t r a t u m i s 10 e m u . A g a i n t h e a m p l i t u d e a n d p h a s e a r e s t r o n g l y d e p e n d e n t o n b o t h t h e c o n d u c t i v i t y d i s t r i b u t i o n a n d t h e f r e q u e n c y . R e s u l t s f o r a h o m o g e n e o u s c o n d u c t i n g s u r f a c e l a y e r w i t h c o n d u c t i v i t y t h e s a m e a s t h a t o f t h e s u r f a c e o f t h e m u l t i l a y e r s t r u c t u r e a r e g i v e n i n T a b l e I I I , N o . k. T h e v a l u e s d i f f e r b y a s m u c h a s s e v e r a l o r d e r s o f m a g n i t u d e . T a b l e I V d e a l s w i t h a l a y e r o f d e c r e a s i n g c o n d u c t i v i t y b e l o w a h o m o g e n e o u s c o n d u c t i n g l a y e r . T h e h o m o g e n e o u s s u r f a c e l a y e r i s c h o s e n t o h a v e a c o n d u c t i v i t y e q u a l t o t h a t o f s e a w a t e r . T h e l a y e r b e l o w i s d i v i d e d i n t o 300 l a y e r s , t h e c o n - d u c t i v i t y o f e a c h b e i n g d e t e r m i n e d b y t h e c o n d u c t i v i t y f u n c t i o n g i v e n i n T a b l e I V . • A g a i n a a n d k a r e c h o s e n t o -11' ' h b e s u c h t h a t t h e c o n d u c t i v i t y d e c r e a s e s f r o m 10 e m u a t 10 - 1 5 - 6 c m t o 10 y e m u a t 10 c m . T h e s e m i - i n f i n i t e s u b s t r a t u m h a s a - 1 5 c o n d u c t i v i t y o f 10 y e m u . T h e h o m o g e n e o u s c o n d u c t i n g s e a ; h a s t h e e f f e c t o f m a k i n g H _ l e s s s e n s i t i v e t o t h e s t r u c t u r e b e l o w . I n c o m p a r i n g H z f o r N o s . -1-3 w i t h t h e r e s u l t s f o r N o . i t i s s e e n t h a t f o r a f r e q u e n c y o f 103 c y c l e s / s e c t h e a m p l i t u d e i s n o t a t a l l a f f e c t e d b y t h e s t r u c t u r e b e l o w t h e s e a . T h i s i s d u e t o t h e f a c t t h a t t h e s k i n d e p t h f o r t h i s f r e q u e n c y i s l e s s t h a n 10 c m i n s e a w a t e r . 65 T A B L E I V C o n d u c t i v i t y d e c r e a s i n g w i t h d e p t h b e l o w a u n i f o r m c o n d u c t i n g s e a f = 10" •3' s e c " ^ f = 1 s e c r 1 f = 103 -1 s e c N o • H z 0° *z H z 0° ^ z H z 0° 1 2.09x10" •6 183.9° 7 «55X1 0" >5 221.9° 2.36x1 o - l f 225.0° 2 1.15x1 o" •5 200.1° 1 .90x10" 181+.10 2.36x1 o _ l + 225.0° 3 8» 65x10" •7 195.1° 1 .25x10" •5 188.9° 2.36x1 o - I f 225.0° h 2.77x10" •7 195=1° 7.55x1o" •6 225.0° 2.36x1 o"^ 225.0° T A B L E I V ( c o n t i n u e d ) D1 D2 D^ ( c m ) ( c m ) ( c m ) N1 g a ( e m u ) d b ( e m u ) ° c ( e m u ) 10^ 106 00 300 10~11 1 0~11 e x p [ - a Cz-D.,) ] ,a=9.303x10"6 10~1* » "• " " " 10~11 [1+k(z -D 1 ) ] ~ 1 ,k=1 .01x10"2 " 11 11 u 10~ 1 1[1+k(z -D 1 ) ]~ 2 ,k=1.00x10"^ II II I! II 0" 6 6 O n t h e b a s i s o f r e s u l t s o b t a i n e d , i t i s s e e n t h a t t h e a m p l i t u d e s a n d p h a s e s o f t h e v a r y i n g e l e c t r o m a g n e t i c f i e l d c o m p o n e n t s a r e s t r o n g l y a f f e c t e d b y t h e i n h o m o g e n e i t y o f t h e c o n d u c t i n g m e d i u m ' . ; T h e c o n d u c t i v i t y d i s t r i b u t i o n s s t u d i e d a r e o n e s t h a t m i g h t b e p l a u s i b l e i n d e s c r i b i n g t h e c o n d u c t - i v i t y o f p a r t s o f t h e e a r t h ' s c r u s t a l s t r u c t u r e . A n o t h e r a s p e c t o f t h e i n h o m o g e n e o u s c o n d u c t i n g e a r t h p r o b l e m o f i n t e r e s t h e r e i s t h e r e p l a c i n g o f t h e i n h o m o g e n - e o u s c o n d u c t o r s b y ' e q u i v a l e n t ' h o m o g e n e o u s c o n d u c t o r s . T h e p r o b l e m c o n s i s t e d o f d e t e r m i n i n g t h e c o n d u c t i v i t y o f t h e h o m o - g e n e o u s l a y e r s f o r w h i c h t h e a m p l i t u d e s o f t h e f i e l d c o m p o n - e n t s w e r e t h e s a m e a s f o r t h e i n h o m o g e n e o u s l a y e r s o f t h e s a m e t h i c k n e s s . - T h e c o n d u c t i v i t i e s w e r e d e t e r m i n e d b y u s i n g e q u a t i o n s - ( V 7 ) t o (51) f o r t h e f i e l d c o m p o n e n t s f o r a t h r e e - l a y e r m o d e l a n d c o m p u t i n g t h e a m p l i t u d e s a n d p h a s e s w h i l e i n c r e m e n t i n g t h e c o n d u c t i v i t y i n s m a l l s t e p s . T h e c o o r d i n a t e s y s t e m u s e d f o r t h e i n h o m o g e n e o u s a n d t h e h o m o g e n e o u s c o n d u c t o r m o d e l s i s s h o w n i n F i g . 20. T h e p r i m e d q u a n t i t i e s i n : F i g . - 20 a n d i n t h e f o l l o w i n g d i s c u s s i o n r e f e r t o t h e h o m o g e n e o u s c o n d u c t o r c a s e w h i l e t h e u n p r i m e d q u a n t i t i e s r e f e r t o t h e m u l t i l a y e r c a s e . • A l t h o u g h a l l t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s w e r e c o m p u t e d , o n l y t h e z c o m p o n e n t o f t h e m a g n e t i c f i e l d i s d i s c u s s e d b e l o w . • T h e r e s u l t s f o r t h e v a r i o u s c o n d u c t i v i t y s t r u c t u r e s s t u d i e d f o r f r e q u e n c i e s f = 10~ 3 S i , a n d 1.03 c y c l e s / s e c - a r e g i v e n i n t a b u l a r f o r m . T h e a m p l i t u d e s , p h a s e a n g l e s , c o n d u c t i v i t i e s a n d d e p t h s a r e r e p r e s e n t e d b y H , 0-, o a n d D r e s p e c t i v e l y . *1L J4> r Profile I N| sub- layers I t Profile H N 2 sub- layers a : (Z) I D=a> F i g u r e 20. M o d e l - u s e d i n c a l c u l a t i n g Layer I - D,' L a y e r IT r J Laye r HI D = a> e q u i v a l e n t ' c o n d u c t i v i t i e s . 68 T a b l e V d e a l s w i t h a n i n h o m o g e n e o u s s u r f a c e l a y e r w i t h c o n d u c t i v i t y i n c r e a s i n g w i t h d e p t h a n d a s e m i - i n f i n i t e s u b s t r a t u m . R e s u l t s a r e g i v e n f o r z = 0, = 107, D 2 = ° ° , -11 o b - 10 e m u , N 1 = 300 l a y e r s , a n d N 2 - 1 l a y e r f o r f i v e d i f f e r e n t c o n d u c t i v i t y d i s t r i b u t i o n s . T a b l e V I d e a l s w i t h a n i n h o m o g e n e o u s s u r f a c e l a y e r w i t h c o n d u c t i v i t y d e c r e a s i n g w i t h d e p t h b e l o w a u n i f o r m c o n d u c t i n g s e a a n d a s e m i - i n f i n i t e s u b s t r a t u m . T h e c a l c u l a t - i o n s w e r e c a r r i e d o u t f o r z = , = 10 c m , D 2 = 10 c m , -11 -1^ = ocy o & = 10 e m u , cQ - 10 J e m u , N 1 = 1 l a y e r , a n d N 2 = 300 l a y e r s . S i n c e t h e s h a l l o w c o n d u c t i n g s e a a n d t h e i n h o m o g e n e o u s c o n d u c t i n g l a y e r a r e e a c h v e r y m u c h l e s s t h a n a s k i n d e p t h t h i c k f o r f = 10 - 3 c y c l e s / s e c , t h e s e m i - i n f i n i t e s u b s t r a t u m p l a y s t h e d o m i n a n t pa l '© a n d h e n c e t h e p h a s e a n g l e s 4>„ a n d 0 ' a r e a l m o s t i d e n t i c a l . F o r f = 103 c y c l e s / s e c t h e c o n d u c t i n g s e a i s s u f f i c i e n t l y t h i c k t o a c t a s a s h i e l d f o r t h e c o n d u c t i v i t y s t r u c t u r e b e l o w a n d a g a i n t h e p h a s e a n g l e s a r e a l m o s t i d e n t i c a l . • I t i s a p p a r e n t f r o m t h e r e s u l t s g i v e n i n T a b l e s V a n d V I t h a t a l t h o u g h i t i s p o s s i b l e t o o b t a i n t h e s a m e a m p l i t u d e s f o r t w o q u i t e d i f f e r e n t c o n d u c t i v i t y s t r u c t u r e s t h e p h a s e a n g l e s a r e i n g e n e r a l q u i t e d i f f e r e n t . 2.3 A H o m o g e n e o u s C o n d u c t i n g B a r t h i n t h e N e a r F i e l d o f a L i n e C u r r e n t 2.3.1 I n t r o d u c t i o n A l t h o u g h t h e p l a n e w a v e m o d e l r e a d i l y l e n d s i t s e l f t o T A B L E V C o n d u c t i v i t y increasing, w i t h depth N o - ( s e c - 1 - ) H z d a(emu) d -'•{••emu) cl 3 10 1 .88x10~6 M^M-xtO^ 3 2:28x10~2 6.75x1O"7 7.00x10"* •3 9«95-x10~3 10° 1 10 10~ 3 1 1 10 : 10 1 10 r 3 ,3 10~ 3 1 10 ,3 259.2 2I+3.O 10" 1*exp(az) 5 a=9«21x10~7 2MO.-8 225,0 " 22M-.7 223.7 " 2M0.1 22h.h 10~1*(1+kz), k=1.00x10~3 2Vl A 225.0 250.1 22h.h 11 II 10" 3 1.05x10"6 2M-7.5 225.h 10" 1*(1+kz 2) ? k=1.00x10" 1.85x10"^ 2M7.O 225.0 10 u 2.19x10"2 230.h 223.8 7.30x10" 1.05x10' •8 =5 1 .60x10' •13 2A5x10~1* •15 1 .02x10 1 .23x10 1 .13x10' 5.50x10' -12 •13 >15 5.67x10~13 •15 1 .62x10" 1 .11 x10" 7.05x10~3 2.35x10~8 7.1+5x10~6 2M-0.1 225.0 10" 1 *[1+kz exp(az)] 5k=1 .00;a=6.91xiO" 7 1 .OM-xlO"10 5.09x10~12 1 .10x10 ' 253»7 225.0 267.0 225»6 11 it 2M-3.9 225.0 10"1*[1+kz exp(-az)] ̂ =^0.0^=1-98X10"6! .01x10"9 -11 7.00x10 r3 266.k 225.0 288.8 22M-.6 it ti 1 .00x10 1 .12x10 .-A T A B L E V I C o n d u c t i v i t y d e c r e a s i n g b e l o w a u n i f o r m c o n d u c t i n g s e a N o ' ( s e c " 1 ) H z vz K° d l 3 ( e m u ) d £ ( e m u ) 10 _ 3 1 1 103 2.09x10"6 6.30x10"6 k. L0x10' 7 ' 183.9 210.2 225.0 185-3 213.3 225.0 10" " 1 1 e x p [ - a ( z - D 1 ) • ] , it 11 a=9-30x10"*6 -1 2 1 .11 x1 0 ' * 9-27x10"12 1.00x1O"11 10""3 2 1 103 1.15x10"^ 1.89x10"^ k.38x10~7 200.1 180.0 225.6 200.2 193.5 225.0 10* ' 1 1 [ 1 + k ( z - D 1 ) ] - 1 , 11 it k=1 .01x1 O"*2 3.96x1 o - 1 l f 1.99x10~13 1 .00x10-11 10"3 3 1 103 8.65x10"*6 1 .23x1 o~5 k.kOxl0"7 195.1 181.5 225.1 195.3 203.8 225.0 10" * 1 1 [ 1 + k ( z - D 1 ) ] - 2 , tt 11 k=1 .00x10"^ 1 .1 9x10"13 1.30x10"12 1.00x1O-11 71 s t u d y i n g a n a l y t i c a l l y t h e e f f e c t o f c e r t a i n i n h o m o g e n e i t i e s i n t h e c o n d u c t i v i t y s t r u c t u r e , i t c a n n o t i n g e n e r a l a c c o u n t f o r t h e m a g n i t u d e o f t h e a m p l i t u d e r a t i o s o f t h e o b s e r v e d v e r t i c a l t o h o r i z o n t a l m a g n e t i c f i e l d . A s d i s c u s s e d e a r l i e r , t h e p l a n e w a v e m o d e l p r e d i c t s a m p l i t u d e r a t i o s s e v e r a l o r d e r s o f m a g n i t u d e s m a l l e r t h a n t h o s e o b s e r v e d i n f i e l d m e a s u r e m e n t s i n t h e c a s e o f t h e r e a l e a r t h . I t i s o f i n t e r e s t a s w e l l t o c o n s i d e r f i e l d s o t h e r t h a n a p l a n e w a v e f i e l d . P r i c e (1962.) s t u d i e d t h e e f f e c t o f a g e n e r a l s o u r c e o f f i n i t e d i m e n s i o n s o v e r a s e m i - i n f i n i t e h o m o g e n e o u s c o n d u c t - i n g e a r t h . • T h e s p e c i a l c a s e o f a s e m i - i n f i n i t e f l a t c o n d u c t - i n g e a r t h i n t h e n e a r f i e l d o f a n o s c i l l a t i n g l i n e c u r r e n t h a s b e e n t r e a t e d i n d e t a i l b y L a w a n d F a n n i n (1961). T h e y d e v e l o p e d t h e e q u a t i o n s f o r t h e f i e l d c o m p o n e n t s a n d o b t a i n e d 7 r e s u l t s f o r a s o u r c e h e i g h t o f 2 x 10' c m , a f r e q u e n c y o f 0.3 r a d i a n s / s e c , a n d a c o n d u c t i v i t y o f 1 .25 x 10 J e m u . T h e i r r e s u l t s d e m o n s t r a t e d t h a t f o r a f r e q u e n c y o f 0.3 r a d i a n s / s e c - (1 .9' c y c l e s / s e c ) t h e v e r t i c a l m a g n e t i c f i e l d c o m p o n e n t c o u l d h a v e a m a g n i t u d e r a n g i n g f r o m 0 t o 0.5 o f t h a t o f t h e h o r i z o n t a l f i e l d . T h e s e r e s u l t s a r e i n t h e r a n g e o f e x p e r i m e n t a l l y o b s e r v e d v a l u e s . T h i s s e c t i o n d e a l s w i t h t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s f o r a n i o n o s p h e r i c l i n e c u r r e n t a n d a f l a t c o n d u c t i n g e a r t h f o r a w i d e r a n g e o f f r e q u e n c i e s , s o u r c e h e i g h t s , a n d c o n d u c t i v i t i e s . 72 2.3.2 M a t h e m a t i c a l A n a l y s i s T h e m a t h e m a t i c a l a n a l y s i s c a r r i e d o u t b y L a w a n d F a n n i n (1961) f o r t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s a s s o c i a t e d w i t h a h o r i z o n t a l o s c i l l a t i n g l i n e c u r r e n t a b o v e a h o m o g e n e o u s f l a t e a r t h w i l l n o t b e r e p r o d u c e d h e r e . T h e i r s o l u t i o n ; , u s e d i n t h e p r e s e n t c a l c u l a t i o n s , e x p r e s s t h e f i e l d c o m p o n e n t s i n t e r m s o f H a n k e l , B e s s e l a n d S t r u v e f u n c t i o n s . S i n c e t h e i r f i n a l e q u a t i o n s a r e n e c e s s a r i l y r a t h e r l e n g t h y a n d w o u l d r e q u i r e n u m e r o u s d e f i n i n g e q u a t i o n s t h e y a r e n o t r e p r o d - u c e d h e r e . • A c o m p u t e r p r o g r a m w a s w r i t t e n t o e v a l u a t e t h e a m p l i t u d e s a n d p h a s e a n g l e s o f t h e c o m p o n e n t s f o r t h e f r e - q u e n c i e s , c o n d u c t i v i t i e s a n d s o u r c e h e i g h t s o f i n t e r e s t i n t h e p r e s e n t w o r k . T h e c o o r d i n a t e s y s t e m u s e d i s s h o w n i n F i g . 21 . T h e c u r r e n t i s a s s u m e d t o v a r y h a r m o n i c a l l y w i t h t i m e a n d i s e x p r e s s e d a s I e x p ( i c o t ) . I t i s a g a i n c o n v e n i e n t t o e x p r e s s t h e h o r i z o n t a l e l e c t r i c , t h e h o r i z o n t a l m a g n e t i c , a n d t h e v e r t i c a l m a g n e t i c f i e l d c o m p o n e n t s i n t h e d i m e n s i o n l e s s f o r m u s e d i n e q u a t i o n s ( L 7) t o (52), w h e r e t h e m o d u l i E x , R y , a n d H a r e t h e r e l a t i v e a m p l i t u d e s o f t h e c o m p o n e n t s f o r a c u r r e n t Zi o f c o n s t a n t a m p l i t u d e a n d t h e a r g u m e n t s ty , 0 , a n d 0 a r e t h e x y z p h a s e a n g l e s r e l a t i v e t o t h e i r r e s p e c t i v e p h a s e s a t t h e s o u r c e . 2 . 3 . 3 D i s c u s s i o n o f R e s u l t s N u m e r i c a l v a l u e s f o r t h e a m p l i t u d e s a n d p h a s e s o f t h e f i e l d c o m p o n e n t s f o r a r a n g e o f p o i n t s a l o n g t h e s u r f a c e o f t h e c o n d u c t i n g e a r t h w e r e c o m p u t e d f o r a w i d e r a n g e o f i o n o - s p h e r i c l i n e c u r r e n t h e i g h t s ( h =•• 107 t o h x 107 c m ) , c u r r e n t f r e q u e n c i e s ( f ~ 1CT3 t o 10 c y c l e s / s e c ) , a n d e a r t h c o n d u c t - 73 LINE SOURCE ( x , o , - h ) IMAGE SOURCE ( x , o , h ) F i g u r e 21 . M o d e l u s e d i n t h e l i n e c u r r e n t p r o b l e m . 7 k i v i t i e s ( d = 10" t o 10 e m u ) . H e r e a s i n a l l t h e e a r l i e r g r a p h i c a l r e s u l t s , i n o r d e r t o f a c i l i t a t e c o m p a r i s o n s w i t h t h e m o d e l m e a s u r e m e n t s d i s c u s s e d i n C h a p t e r 3, t h e n u m e r i c a l v a l u e s f o r t h e e l e c t r i c f i e l d a m p l i t u d e s a r e g i v e n i n u n i t s o f v o l t s / m w h i l e t h o s e f o r t h e m a g n e t i c f i e l d a m p l i t u d e s a r e i n u n i t s o f g a u s s . F i g u r e s 22, 23, a n d 2k s h o w t h e a m p l i t u d e s H , H z a n d E v , a n d t h e p h a s e s * , 0_, a n d \k f o r t h e f r e q u e n c i e s f = 1 0 " 3 , x y z 1 x -2 -1 10 Tj:. 10 , 1, a n d 10 c y c l e s / s e c a s a f u n c t i o n o f y . H e r e y i s t h e d i s t a n c e a l o n g t h e e a r t h ' s s u r f a c e f r o m t h e p o s i t i o n v e r t i c a l l y b e l o w t h e l i n e s o u r c e a s s h o w n i n F i g . - 2 1 . T h e s o u r c e h e i g h t i s h = 2 x 10^ c m a n d t h e c o n d u c t i v i t y o f t h e —16 e a r t h i s 10" e m u . I t i s a p p a r e n t t h a t f o r a c o n d u c t i v i t y o f —16 10" e m u H a n d E a r e s t r o n g l y d e p e n d e n t o n f r e q u e n c y w h i l e H „ i s l e s s s e n s i t i v e . S i n c e H i s m u c h m o r e a f f e c t e d b y a y z J c h a n g e i n f r e q u e n c y t h a n i s H , t h e r a t i o o f t h e v e r t i c a l t o t h e h o r i z o n t a l m a g n e t i c f i e l d c o m p o n e n t s w i l l a l s o b e s t r o n g l y d e p e n d e n t o n f r e q u e n c y , w i t h t h e r a t i o d e c r e a s i n g a s t h e f r e q u e n c y i n c r e a s e s . - T h i s 1 d i f f e r s f r o m t h e p l a n e w a v e m o d e l r e s u l t s d i s c u s s e d e a r l i e r w h e r e i t w a s f o u n d t h a t t h e r a t i o i s a n i n c r e a s i n g f u n c t i o n o f f r e q u e n c y . F i g u r e s 25 a n d 26 d e a l w i t h t h e c a s e w h e r e t h e c o n d u c t - —16 i n g e a r t h ( d =• 10" e m u ) h a s b e e n r e p l a c e d b y a c o n d u c t i n g —11 s e a ( d - = 1 0 e m u ) o f i n f i n i t e d e p t h . I n c o m p a r i n g t h e s e a m p l i t u d e s a n d p h a s e a n g l e s w i t h t h o s e o f F i g s . 22, 23 a n d 2k, i t c a n b e s e e n t h a t t h e f r e q u e n c y d e p e n d e n c e i s v e r y d i f f e r e n t i n t h e t w o c a s e s . F o r f r e q u e n c i e s u p t o 1 c y c l e / s e c H i s 10 10° h b X 10 0 8 12 16 Y ( l0 7 cm) F i g u r e 2 2 . H _ ( a ) a n d cp ( b ) a s f u n c t i o n s o f y f o r d=1Cr I D e m u , h = 2 x 1 0 8 12 16 Y ( I 0 7 cm) - 1 6 20 24 7̂ ( 1 ) f = 1 0 = i - n - 3 ? ' ( 2 ) i o " 2 , "(3) 1 0 r 1 , ( k ) 1 , a n d (5) 10 c y c l e s / s e c . c m , a n d F i g u r e 2 3 . H z ( a ) a n d 4- ( b ) a s f u n c t i o n s o f y f o r 0=10" 1 6 e m u , h=2x10 7 c m a n d (1) f = i O ~ 3 , (-2) 1G~ 2 r -<3) 1G" 1, (if) 1, a n d (5) 10 c y c l e s / s e c . F i g u r e 2k. • E x ( a ) a n d ^ x ( b ) a s f u n c t i o n s o f y f o r 0=10 e m u , h=2x10' c m , a n d (1) f=10~3, (2) 10~2, (3) 10~1 , (k) 1 , a n d (5) 1-0 c y c l e s / s e c . 8 12 16 Y (I07cm) 8 12 16 Y(l07cm) 24 Figure 25. H y (a) and R"z (b) as functions of y for o=10~11 emu, h=2x107 cm, and (1) f=10-35 <2) 10"2, ( 3 ) 10"1, <*+)-1, and (5) 10 cycles/sec. -<1 CO IO"T— 1—L 1 1 140 120 co UJ UJ tr o UJ 4 0 2 0 I I » l • I 3 , 4 , 5 = 2 = 3 .4 I • I 5 (b) ' • I • 1 0 8 12 16 Y ( I 0 7 c m ) 2 0 2 4 8 12 16 Y (10 7 cm) ^ 1 1 2 0 2 4 7̂ vO F i g u r e 2 6 . E x ( a ) - a n d xf> , <J>Z, ^ x ( b ) a s f u n c t i o n s o f y f o r 0 = 1 0 e m u , h = 2 x 1 0 ' c m , ° , (3) 1 ( T 1 , (k) 1 , a n d (5) 1 0 c y c l e s / s e c . a n d ( 1 ) f = 1 0 J , ( 2 ) 1 0 " " 8 0 i n d e p e n d e n t o f f r e q u e n c y [ c u r v e s o f F i g . 2 . 5 ( a ) ] , H_ z -1/2 1 / 2 v a r i e s a s f , w h i l e E v a r i e s a s f ' . A s c a n b e s e e n f r o m F i g s . 2 5 ( a ) a n d ( b ) , t h e r a t i o o f H t o H i s m u c h s m a l l e r ( l e s s t h a n 0.05 f o r a l l v a l u e s o f y ) f o r t h e c o n d u c t i n g s e a t h a n i t w a s f o r t h e c o n d u c t i n g e a r t h ( a s l a r g e a s 0.5K T h e p h a s e a n g l e s [ F i g . 2 6 ( b ) ] f o r a l l c o m p o n e n t s s h o w v e r y l i t t l e c h a n g e a s a f u n c t i o n o f f r e q u e n c y a n d p o s i t i o n : i n t h e c a s e o f t h e c o n d u c t i n g s e a . T h e e f f e c t o f t h e c o n d u c t i v i t y o f t h e e a r t h i s b r o u g h t o u t i n g r e a t e r d e t a i l i n F i g s . 27 a n d 2.8. R e s u l t s a r e s h o w n f o r t h e c o n d u c t i v i t y r a n g e o* = 1 0 " ^ t o l O " " 1 0 e m u f o r a 7 s o u r c e h e i g h t o f 2 x 10- c m a n d a f r e q u e n c y o f 0.1 c y c l e s / s e c . A s s e e n f r o m F i g . 27 ( a ) , R V i s n o t v e r y s t r o n g l y a f f e c t e d b y a c h a n g e i n c o n d u c t i v i t y . H a n d E ^ [ F i g s . 27 (b) a n d 2 8 ( a ) ] Z J \ . d o s h o w l a r g e c h a n g e s a s a f u n c t i o n o f c o n d u c t i v i t y , b o t h d e c r e a s i n g a s t h e c o n d u c t i v i t y i n c r e a s e s . F o r c o n d u c t i v i t i e s —1 h- 7 g r e a t e r t h a n 10 e m u a n d x g r e a t e r t h a n 2 x 10' c m b o t h R~ z -1 / 2 a n d E v a r y a p p r o x i m a t e l y a s d • . F r o m t h e s e r e s u l t s a n d f r o m t h o s e o f F i g s . 25 (b) a n d 2 6 ( a ) i t c a n b e s e e n t h a t f o r —11 a c o n d u c t i n g s e a (d = 10 e m u ) a n d a s u f f i c i e n t l y l o w f r e - - 1 / 2 q u e n c y H _ v a r i e s a p p r o x i m a t e l y as (df) • w h i l e E . v . v a r i e s Z A. 1/2 1 a p p r o x i m a t e l y a s f ' d~ . T h e e f f e c t o f t h e s o u r c e h e i g h t o n t h e m a g n e t i c f i e l d s i s t r e a t e d i n F i g , , 29- R e s u l t s a r e g i v e n f o r f = 0 . 1 c y c l e s / s e c , 0 = 1 0 " e m u a n d s o u r c e h e i g h t s h = 1 0 ' , 2 x 1 0 ' , 3 x 1 0 ^ , a n d h x 1 0 ^ c m . . H , a l t h o u g h n o t s t r o n g l y d e p e n d e n t o n t h e s o u r c e h e i g h t , i n c r e a s e s w i t h i n c r e a s i n g s o u r c e h e i g h t 10' 10" ro i o 10 — ^ ^ S v ; (a) . I . I . I . I . I . 2 0 2 4 0 4 8 12 16 2 0 2 4 0 4 8 12 16 Y ( , ° 7 c m > Y ( l 0 7 c m ) F i g u r e 27 H Y U ) a n d H , (b) as f u n c t i o n s o f j f o r f = l <T 1 c y c l e s / s e c , h = 2 X 1 0 ? C m and U - 1 0 - 1 6 , ( 2 ) 1 0 - 1 5 , ( 3) 1 0 ^ \ (!>) 1 0 ^ 3 , ( 5 ) 1 ( r 1 2 ( f i ) 1 0-11 (7) 10 l u emu. F i g u r e 28._ • ( a ) a n d c/y , <£z ( b ) a s f u n c t i o n s o f y f o r f=10~' c y c l e s / s e c , h=2xl0 7 c m a n d (1.) 0=10-16, (2) 10~ 1 5, (3) 10" l k, (>f) 10" 1 3, (5) 10" 1 2, (6.) 10" 1 1, a n d (7) 10" 1 0 e m u . 0 4 8 12 16 2 0 24 0 4 8 12 16 2 0 24 Y ( l 0 7 c m ) Y ( l 0 7 c m ) F i g u r e 29- • H y ( a ) , - Kz ( b ) , a n d 0- ( c ) a s a f u n c t i o n o f y f o r f=10 c y c l e s / s e c , d = i C ~ 1 6 e m u , a n d (1) h=10^, (2) 2x107, (3) 3x107, a n d (h) ^fx107 c m . 8>f f o r p o s i t i o n s x g r e a t e r t h a n h x 107 c m . F o r s m a l l y ( l e s s t h a n k x 107 c m ) , H z d e c r e a s e s r a p i d l y a s t h e s o u r c e h e i g h t 7 i n c r e a s e s . A t a h o r i z o n t a l d i s t a n c e o f a p p r o x i m a t e l y 12 x 10' c m , H z i s i n d e p e n d e n t o f t h e s o u r c e h e i g h t ; f o r l a r g e r v a l u e s o f y , R" z i n c r e a s e s w i t h i n c r e a s i n g s o u r c e h e i g h t * A l t h o u g h E x ' ^ x ? a n d ^ y w e r e c o m p u t e d t h e y a r e n o t s h o w n h e r e . T o f u r t h e r s h o w t h e f r e q u e n c y d e p e n d e n c e , F i g . 30 s h o w s h o w E L v a r i e s a s a f u n c t i o n o f f r e q u e n c y f o r d r y e a r t h z 1 f\ — 1 1 (d = 10 e m u ) a n d s e a w a t e r (d = 10 e m u ) , f o r s e v e r a l v a l u e s o f y . I t i s e v i d e n t f r o m F i g s . 30 (a) a n d ( b ) t h a t f o r 1 f\ — 1 / ? d = 10 e m u H _ v a r i e s a p p r o x i m a t e l y a s f ' f o r t h e h i g h e r z f r e q u e n c i e s w h i l e b e i n g a l m o s t i n d e p e n d e n t o f f r e q u e n c y f o r 11 t h e l o w e r f r e q u e n c i e s . F o r a = 10 e m u , h o w e v e r , F i g . 30 (b) - 1 /2 s h o w s t h a t H z v a r i e s a s f ' f o r t h e e n t i r e f r e q u e n c y r a n g e c o n s i d e r e d . O n t h e b a s i s o f r e s u l t s o b t a i n e d f o r t h e a m p l i t u d e s a n d p h a s e a n g l e s i t c a n b e s e e n t h a t a l t h o u g h t h e t h r e e f i e l d c o m p o n e n t s a r e a l l s t r o n g l y d e p e n d e n t o n t h e h o r i z o n t a l d i s t - a n c e f r o m t h e l i n e s o u r c e , t h e h o r i z o n t a l m a g n e t i c f i e l d c o m p o n e n t i s n o t v e r y s e n s i t i v e t o f r e q u e n c y , c o n d u c t i v i t y , o r s o u r c e h e i g h t . T h e v e r t i c a l m a g n e t i c f i e l d a n d t h e h o r i z o n t a l e l e c t r i c f i e l d a r e i n g e n e r a l s t r o n g l y d e p e n d e n t o n f r e q u e n c y , c o n d u c t i v i t y a n d s o u r c e h e i g h t , a s w e l l a s d e p e n d e n t o n t h e l o c a t i o n w i t h r e s p e c t t o t h e s o u r c e . T h e r a t i o s E ^ H y a n d H z / H y a r e h e n c e q u i t e s e n s i t i v e t o c h a n g e s i n f r e q u e n c y * c o n d u c t i v i t y , a n d s o u r c e l o c a t i o n . I f o n e a s s u m e s o s c i l l a t i n g l i n e c u r r e n t s i n t h e f C Y C L E S / S E C f C Y C L E S / S E C F i g u r e 30.- H a s a f u n c t i o n o f f r e q u e n c y f o r h=107 c m , ' ( a ) o=i0" 1^, • ( b > 0=10"11 e m u , a n d (1) y=107,- (2) 2x1O7, (3) ^ x l O 7 , a n d (>+-) 8x107 c m . 86 i o n o s p h e r e as a s o u r c e o f the e l e c t r o m a g n e t i c v a r i a t i o n s on the e a r t h , t h e n measurements o f t he f i e l d components made a t t h e e a r t h ' s s u r f a c e s h o u l d be dependent on the s o u r c e f r e - quency , the s o u r c e p o s i t i o n , and the c o n d u c t i v i t y o f t he e a r t h . I n t he ca se o f t he r e a l e a r t h i n h o m o g e n e i t i e s i n t he c o n d u c t i v i t y s t r u c t u r e wou ld c o n s i d e r a b l y m o d i f y the a m p l i - t ude s and phases o f t h e f i e l d components . 87 C h a p t e r 3- A N A L O G U E M O D E L S 3 - 1 S h e e t C u r r e n t S o u r c e .3... 1... 1 .. I n t r o d u c t i o n T h e b e h a v i o r o f t h e n a t u r a l l y o c c u r r i n g e l e c t r o m a g r n e t i c v a r i a t i o n s o b s e r v e d a t t h e e a r t h ' s s u r f a c e i s i n g e n e r a l d e t e r m i n e d b y t h e n a t u r e o f t h e s o u r c e f i e l d a n d b y t h e d i s t r i b u t i o n o f t h e e l e c t r i c a l c o n d u c t i v i t y w i t h i n t h e e a r t h . I n C h a p t e r 2 w e c o n s i d e r e d t w o t y p e s o f f i e l d s o u r c e s , a p l a n e w a v e s o u r c e r e q u i r i n g a v e r y d i s t a n t s o u r c e o r a v e r y l a r g e c u r r e n t s h e e t , , a n d a l i n e c u r r e n t s o u r c e » T h e r e s u l t s o b t a i n e d i n d i c a t e d t h a t t h e v a r y i n g f i e l d s o b s e r v e d a t t h e e a r t h ' s s u r f a c e s h o u l d b e q u i t e d i f f e r e n t f o r t h e t w o s o u r c e s . I n c o n s i d e r i n g h o r i z o n t a l c o n d u c t i n g l a y e r s , i t w a s a l s o f o u n d t h a t t h e d i s c o n t i n u i t i e s i n c o n d u c t i v i t y i n t h e v e r t i c a l d i r e c t i o n a l s o h a d a n i m p o r t a n t e f f e c t o n t h e f i e l d s o b s e r v e d a t t h e e a r t h ' s s u r f a c e . • T h e p r o b l e m o f d i s c o n t i n u i t i e s i n c o n d u c t i v i t y i n t h e h o r i z o n t a l d i r e c t i o n d o e s n o t s u b m i t a s r e a d i l y t o m a t h e m a t i c a l t r e a t m e n t . T h i s c h a p t e r d e a l s w i t h a n a n a l o g u e m e t h o d u s i n g a m o d e l f o r s t u d y i n g t h e p r o b l e m o f c o n d u c t i v i t y d i s c o n t i n u i t i e s i n g e n e r a l f o r s h e e t c u r r e n t (3°1) a n d l i n e c u r r e n t (3.2) s o u r c e s . T h e p r o b l e m o f h o r i z o n t a l c o n d u c t i v i t y d i s c o n t i n u i t i e s i s o n e o f c o n s i d e r a b l e i n t e r e s t i n g e o p h y s i c s a t t h e p r e s e n t t i m e . A m o n g t h e f e a t u r e s o f i n t e r e s t i s t h e b e h a v i o r o f t h e e l e c t r o m a g n e t i c f i e l d c o m p o n e n t s i n t h e n e i g h b o r h o o d o f a 88 c o a s t l i n e a s w e l l a s i n t h e n e i g h b o r h o o d o f v e r t i c a l f a u l t s a n d d y k e s i n t h e e a r t h ' s c r u s t . F o r s i m p l i c i t y , t h e a n a l y s i s i s c o m m o n l y b a s e d o n t h e a s s u m p t i o n o f p l a n e w a v e s o r a u n i f o r m i n d u c i n g f i e l d . T h e c a s e o f t h e m a g n e t i c f i e l d p a r a l l e l t o t h e t r a c e o f a v e r ^ t i _ c a l f a u l t w a s s t u d i e d b y d ' E r c e v i l l e a n d K u n e t z ( 1 9 6 2 ) . o I t w a s e x t e n d e d t o t h e c a s e o f a d y k e d i s - c o n t i n u i t y b y R a n k i n ( 1 9 6 2 ) . W e a v e r ( 1 9 6 2 ) c o n s i d e r e d t h e c a s e o f t h e m a g n e t i c f i e l d n o r m a l t o t h e t r a c e o f t h e f a u l t . A l t h o u g h s o m e w h a t d i f f e r e n t r e s u l t s w o u l d b e e x p e c t e d f o r t h e r e a l c a s e o f a s h e l v i n g o c e a n f l o o r a n d l a n d i n t e r f a c e , h i s r e s u l t s i n d i c a t e t h a t t h e a m p l i t u d e o f t h e v e r t i c a l m a g - n e t i c f i e l d . c o m p o n e n t i n c r e a s e s s h a r p l y a s t h e d i s c o n t i n u i t y i s a p p r o a c h e d , a n d s u p p o r t t h e o b s e r v e d e n h a n c e m e n t o f t h i s c o m p o n e n t i n t h e n e i g h b o r h o o d o f c o a s t l i n e s ( L a m b e r t a n d C a n e r 4 9 6 5 ) . O n e m e a n s o f s t u d y i n g p r o b l e m s s u c h a s t h e " c o a s t e f f e c t " i s w i t h a s c a l e d m o d e l u s i n g t h e w e l l - k n o w n p r i n c i p l e s o f s i m i l i t u d e ( S t r a t t o n 1 9 ^ 1 ) » M a g n e t o t e l l u r i c s c a l i n g f a c t o r s w e r e t r e a t e d b y C a g n i a r d ( 1 9 5 3 ) ° R a n k i n e t a l ( 1 9 6 5 ) h a v e d e s c r i b e d a m a g n e t o t e l l u r i c m o d e l c o n s i s t i n g o f a n o s c i l l a t i n g l i n e c u r r e n t o v e r a c o n d u c t i n g s a l t s o l u t i o n w h i c h r e p r e s e n t e d t h e u p p e r l a y e r o f t h e e a r t h . • T h e y d i s c u s s e d m e a s u r e m e n t s o f t h e m a g n e t o t e l l u r i c r a t i o ( h o r i z o n t a l e l e c t r i c t o h o r i z o n t a l m a g n e t i c f i e l d ) f o r a l i n e c u r r e n t a b o v e a t w o - l a y e r m o d e l e a r t h . T h e m o d e l u s e d i n . t h e p r e s e n t w o r k w a s d e s i g n e d t o s t u d y t h e v e r t i c a l t o h o r i z o n t a l 89 m a g n e t i c f i e l d r a t i o a s w e l l a s t h e m a g n e t o t e l l u r i c f i e l d r a t i o f o r v a r i o u s f i e l d s o u r c e s a n d a w i d e r a n g e o f s u i t a b l y s c a l e d g e o l o g i c a l s t r u c t u r e s . T h e a m p l i t u d e s a n d p h a s e a n g l e s o f t h e h o r i z o n t a l e l e c t r i c , t h e h o r i z o n t a l m a g n e t i c , a n d t h e v e r t i c a l m a g n e t i c f i e l d c o m p o n e n t s f o r t h e v a r i o u s p o l a r i z - a t i o n s , a n g l e s o f i n c i d e n c e , a n d s o u r c e f i e l d s a r e m e a s u r e d . T h e p r e s e n t s e c t i o n (3.1) d e a l s w i t h t h e m o d e l e a r t h s t r u c t - u r e s i n t h e n e a r f i e l d o f a s h e e t c u r r e n t o f l a r g e d i m e n s i o n s . F o r a s h e e t c u r r e n t o f s u f f i c i e n t l y l a r g e e x t e n t , t h e m a g n e t o - t e l l u r i c r a t i o s a n d t h e p h a s e a n g l e s s h o u l d a p p r o x i m a t e t h e v a l u e s e x p e c t e d o n t h e b a s i s o f a p l a n e w a v e f i e l d a s s u m p t i o n . 3.1.2 M a t h e m a t i c a l A n a l y s i s T h e m a t h e m a t i c a l d e v e l o p m e n t f o r t h e s c a l i n g f a c t o r s i n t h i s m o d e l p r o b l e m i s b a s e d o n t h e w e l l - k n o w n p r i n c i p l e o f s i m i l i t u d e ( S t r a t t o n 1 9VI). C o n s i d e r M a x w e l l ' s f i e l d e q u a t i o n s a s g i v e n e a r l i e r [ e q u a t i o n s (1) a n d (2)]: (66) V x E« + £ | f ' = 0 , (67) V x H ' - J §f̂ - - W d c E ' = 0 c 0 1 T h e s e e q u a t i o n s m a y b e e x p r e s s e d i n d i m e n s i o n l e s s f o r m b y m a k i n g t h e s u b s t i t u t i o n s (68) E ' = e Q E , H ' = h _ H , (69) P = H o V e = e o K e ' d = ° o S ' 90 (70) • d = d Q D , t = t Q T , w h e r e B y H, K ^ , K g 5 S , D a n d T a r e d i m e n s i o n l e s s q u a n t i t i e s , a n d e Q , h Q , L L q , e o , d 0 ? d Q a n d t a r e t h e u n i t q u a n t i t i e s o f t h e e l e c t r i c f i e l d , m a g n e t i c f i e l d , m a g n e t i c p e r m e a b i l i t y , d i e l e c t r i c c o n s t a n t , c o n d u c t i v i t y , l e n g t h , a n d t i m e r e s p e c t - i v e l y . E q u a t i o n s (66) a n d (67) t h e n b e c o m e (71) x E + a | f = 0 (72) V x H - p | f - T E = 0 where (73). - - ^ f r ) , O V 0 / >=^f fe and (75) . *f = Wcd Qd 0S (h^)° T h e s o l u t i o n s t o e q u a t i o n s (71) a n d (72) w i l l b e i n v a r i a n t u n d e r a c h a n g e i n s c a l e i f t h e d i m e n s i o n l e s s q u a n t i t i e s <st, p , a n d y a r e i n v a r i a n t . • H e n c e a n a c t u a l g e o p h y s i c a l e l e c t r o m a g - n e t i c b o u n d a r y v a l u e p r o b l e m c a n b e r e p r e s e n t e d b y a s c a l e d m o d e l . I f w e d i r e c t o u r a t t e n t i o n t o p r o b l e m s w h e r e e a n d p, h a v e t h e f r e e s p a c e v a l u e s , a n d e l i m i n a t e e o / h 0 f r o m e q u a t - 91 i o n s (73) t o (75)? t h e n t h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s f o r i n v a r i a n c e u n d e r a c h a n g e i n s c a l e a r e t h a t d f = c o n s t a n t , a n d d o = c o n s t a n t , o r (76) o f d = c o n s t a n t , where f i s t h e f r e q u e n c y . I f w e n o w l e t p r i m e d " s y m b o l s refer t o a c t u a l g e o p h y s i c a l d i m e n s i o n s a n d u n p r i m e d s y m b o l s refer t o m o d e l d i m e n s i o n s , t h e n e q u a t i o n (11) c a n b e e x - p r e s s e d a s (77) o f d 2 = o ' f ' d ' 2 . I f c o n d i t i o n (72) i s s a t i s f i e d , t h e n t h e g e o p h y s i c a l p r o b l e m a n d i t s s c a l e d m o d e l w i l l h a v e t h e s a m e s o l u t i o n s . 3 . 1 . 3 M o d e l d e s c r i p t i o n a n d m e a s u r e m e n t t e c h n i q u e T h e e x p e r i m e n t a l a r r a n g e m e n t o f t h e m o d e l i s s h o w n i n F i g . 31. T h e c u r r e n t s h e e t w a s p r o v i d e d b y c u r r e n t f l o w i n g i n a n a r r a y o f h o r i z o n t a l b r a s s w i r e s o f 1 c m s e p a r a t i o n h e l d o n a f r a m e h>8,8 m l o n g a n d 2 . M + m w i d e . ' T h e p a r a l l e l w i r e s w e r e a t t a c h e d t o h e a v y b r a s s b u s b a r s a t t h e e n d s o f t h e f r a m e . C o p p e r b u s b a r s a t t a c h e d t o t h e b r a s s b u s b a r s e x - t e n d e d a b o u t 2 . ^ m b e y o n d t h e e n d s o f t h e f r a m e a n d t h e n c o m p l e t e d t h e c i r c u i t w i t h t h e r e t u r n c o n d u c t o r a b o u t 7 .6 u f r o m t h e w i r e a r r a y . < T h e f r a m e w a s s u p p o r t e d i n s u c h a w a y t h a t i t c o u l d b e r a i s e d o r l o w e r e d r e a d i l y a s w e l l a s t i l t e d to a l t e r t h e a n g l e w i t h r e s p e c t t o t h e h o r i z o n t a l . T h e p o w e r w a s p r o v i d e d b y a 1 5C>-W M c i n t o s h p o w e r a m p l i f i e r c o n n e c t e d 92 d i r e c t l y to the loop as shown i n F i g . 31 . ' The current i n the loop was maximized by adding a capacitor to make a resonant c i r c u i t as shown i n F i g . - 31. The current was monitored by measuring the voltage across a r e s i s t i v e element of the loop. For the measurements described i n t h i s work, the current sheet was kept In a horizontal p o s i t i o n 1.25 m above the surface of the s a l t s olution i n a plywood tank 2.¥+ m by 1.68 m and O.76 m deep. - To minimize e f f e c t s of the concrete f l o o r and the earth below, the bottom of the tank was l i n e d with a 5-cm layer of graphite i n the form of machined rods of square cross section. • For measurements made l a t e r i n this work the ends of the tank perpendicular to the d i r e c t i o n of the e l e c t r i c f i e l d of the source were l i n e d with graphite. This reduced the edge e f f e c t s , ' This*point w i l l be treated l a t e r i n t h i s work. The s a l t solution over the graphite layer i n the tank represented the upper conducting layer i n the r e a l earth problem. - Figure 32 shows the layered conductor with the s a l t water (conductivity ov,) forming the upper layer and the graphite l i n i n g (conductivity d'̂ ) at' the bottom of the tank forming the lower layer. The detector c a r r i e r , to which the f i e l d detectors were attached, consisted of a l u c i t e plate 100 cm long, 15 cm wide and 2 cm thick. The c a r r i e r , though free to move along the r i g i d laminated plywood beam above the tank, was secured to the beam by means of a l u c i t e r a i l glued to the beam. The r a i l , with a T-shaped cross section, mated with a s l o t of the same cross section extending the f u l l length of the c a r r i e r . F i g u r e 32. D i a g r a m o f t h e m o d e l l a y e r e d c o n d u c t o r . 9^ S m a l l n y l o n r o l l e r s a t t a c h e d t o t h e c a r r i e r a l l o w e d i t t o m o v e a l o n g t h e b e a m w i t h l i t t l e f r i c t i o n , , • P o s i t i o n i n g h o l e s i n t h e b e a m a n d i n t h e c a r r i e r a l l o w e d t h e c a r r i e r t o b e p o s i t i o n e d a t f i x e d 1 - c m i n t e r v a l s o v e r t h e f u l l l e n g t h o f t h e t a n k . S i n c e t h e b e a m c o u l d b e m o v e d t o a n y l o c a t i o n p a r a l l e l t o e i t h e r t h e X o r Y a x i s ( F i g < . • 3-1), i t w a s p o s s i b l e t o l o c a t e t h e c a r r i e r a t a n y d e s i r e d p o i n t a t t h e s u r f a c e o f t h e s a l t s o l u t i o n i n t h e t a n k . . A p u l l e y a n d b e l t s y s t e m , d r i v e n b y a g e a r e d m o t o r m o u n t e d o n o n e e n d o f t h e b e a m , a l l o w e d c o n t i n - u o u s a u t o m a t i c r e c o r d i n g o n a n X - Y p l o t t e r f o r t r a v e r s e s a c r o s s t h e t a n k . T h e c a r r i e r w a s d r a w n a l o n g a t a s p e e d s u i t a b l e f o r t h e K - Y p l o t t e r u s e d . A . c o m p l e t e t r a v e r s e r e q u i r e d e i g h t m i n u t e s . T h e X - i n p u t t o t h e X - Y p l o t t e r w a s . p r o v i d e d b y a b r a s s s l i d e r a t t a c h e d t o t h e c a r r i e r a n d m a k i n g a s l i d i n g c o n t a c t w i t h t h e r e s i s t a n c e w i r e f a s t e n e d t o t h e b e a m . • A 6 - v o l t s t o r a g e b a t t e r y p r o v i d e d t h e d . c . p o t e n t i a l g r a d i e n t a l o n g t h e r e s i s t a n c e w i r e . T h e e l e c t r i c a n d m a g n e t i c f i e l d d e t e c t o r s , e a c h m o u n t e d o n t h e e n d o f s e p a r a t e l u c i t e t u b e s , w e r e s e c u r e d t o t h e c a r r i e r . T o p o s i t i o n t h e d e t e c t o r , i t w a s s i m p l y i n s e r t e d i n t h e h o l e t h r o u g h t h e p l a t e a n d s e c u r e d a n d h e l d v e r t i c a l l y b y l u c i t e s e t s c r e w s . T h e h e i g h t o f t h e d e t e c t o r w a s r e a d i l y a d j u s t e d t o a l l o w f o r d i f f e r e n t w a t e r d e p t h s i n t h e t a n k . T h e h o r i z o n t a l m a g n e t i c f i e l d d e t e c t o r s h o w n i n F i g . 33 c o n s i s t e d o f a p a i r o f t w i n c o i l s f r o m a G15^ O L E . • p h o n o - g r a p h c a r t r i d g e m o u n t e d s i d e b y s i d e i n t h e s e a l e d e n d o f a l u c i t e t u b e 1.91 c m i n d i a m e t e r a n d 35 c m l o n g . - E a c h c o i l 95 DIFFERENTIAL AMPLIF IER I TWIN COILS T V o u t V - UNWANTED S I G N A L ( NOISE, C A B L E PICKUP, ETC. ) V o u t tt < v 2 " v l ) I 1 V 2 = V + v2 Lrr - DIFFERENTIAL A M P L I F I E R V, = V+v , - - 1 J ELECTRIC PROBE F i g u r e - 3 3 . - T h e e l e c t r i c a n d m a g n e t i c f i e l d d e t e c t o r s . 96 was 0 . 6 ? cm l o n g and 0.57 cm i n d i a m e t e r . - To remove unwanted s i g n a l s f r o m the d e t e c t o r s and c o n n e c t i n g l e a d s , t he two c o i l s were c o n n e c t e d i n s e r i e s i n a way t o p r o v i d e a s u i t a b l e i n p u t t o a d i f f e r e n t i a l a m p l i f i e r . The l e a d f r o m the o u t e r l a y e r o f each c o i l was c o n n e c t e d t o t he s h i e l d o f a t w o - c o n d u c t o r l o w p i c k - u p c a b l e ( I n t e r - 8 Weave P e r f e c t i o n M i c a c a b l e ) and t he o t h e r ends o f the c o i l s were c o n n e c t e d , by means o f t he two c o n d u c t o r s o f t he c a b l e , t o the d i f f e r e n t i a l i n p u t o f t h e l o w n o i s e b a t t e r y o p e r a t e d P r i n c e t o n A p p l i e d R e s e a r c h mode l CR^f d i f f e r e n t i a l a m p l i f i e r . A 70 cm c a b l e c o n n e c t e d t he d e t e c t o r t o t he a m p l i f i e r w h i c h was mounted on one end o f the d e t e c t o r c a r r i e r . • The a m p l i f i e r t hu s moved t o g e t h e r w i t h t he p robe w h i l e t r a v e r s e s were made a c r o s s t he t a n k . ' The p r e s e n c e o f the a m p l i f i e r had a n e g l i g i b l e e f f e c t on t h e f i e l d a t t he s u r f a c e o f the s a l t s o l u t i o n . • A l t h o u g h t he a m p l i f i e r was s i t u a t e d i n a r e l a t i v e l y i n t e n s e e l e c t r o - magne t i c f i e l d , i t s w e l l d e s i g n e d s h i e l d i n g made i t q u i t e i n s e n s i t i v e t o the f i e l d . • The o u t p u t f r om the a m p l i f i e r was c o n n e c t e d by a 9 m t w o - c o n d u c t o r s h i e l d e d c a b l e t o t he d i f f e r - e n t i a l i n p u t o f a T e k t r o n i x 502 o s c i l l o s c o p e where i t was f u r t h e r a m p l i f i e d . - The a m p l i t u d e o f the o u t p u t f r om the o s c i l l o s c o p e was measured by an A . C . ; H e w l e t t - P a c k a r d 3l+U-0r-A DVM and t he phase a n g l e was measured by an Ad -Yu 52^-A3 - d i g i t a l phase computer whose o u t p u t was d i s p l a y e d on a H e w l e t t - P a c k a r d 3¥+0-A DVM. • A r e f e r e n c e s i g n a l o f f i x e d phase a n g l e was p r o v i d e d by a s m a l l c o i l s i t u a t e d a t a f i x e d p o s i t i o n n e a r t he f i e l d s o u r c e . • The a m p l i t u d e s and phase 97 a n g l e s d i s p l a y e d on the mete r s c o u l d be r e c o r d e d f o r each p o i n t , o r a l t e r n a t i v e l y an ana l ogue r e c o r d i n g f o r c omp le te t r a v e r s e s c o u l d be o b t a i n e d u s i n g a H e w l e t t - P a c k a r d 580-A d i g i t a l t o ana l o gue c o n v e r t e r and a M o s e l y M o d e l 3 X -Y r e c o r d e r . The i n s t r u m e n t a r rangement i s shown i n F i g , 3 k » The v e r t i c a l m a g n e t i c f i e l d d e t e c t o r , shown i n F i g . 33", was c o n s t r u c t e d i n t he same way as the h o r i z o n t a l magne t i c f i e l d d e t e c t o r , e x c e p t t h a t t he t w i n c o i l s were o r i e n t e d w i t h t h e i r axes i n the v e r t i c a l d i r e c t i o n . B o t h magne t i c f i e l d d e t e c t o r s were c a l i b r a t e d f o r the f r e q u e n c y range o f -1-30 k-c/sec w i t h a mode l 350 B e l l Gaus smete r . The c a l i b r a t i o n was e x t r a p o l a t e d t o 60 k c / s e c . The a ve rage h o r i z o n t a l e l e c t r i c f i e l d a l o n g t he s u r - f a c e o f t he s a l t s o l u t i o n was d e t e r m i n e d by mea su r i n g t h e v o l t a g e d i f f e r e n c e between p o i n t s ] cm a p a r t . • To remove s i g n a l s common t o b o t h p robes and to m i n i m i z e the e f f e c t o f s i g n a l s i n d u c e d i n the l e a d s , the d e t e c t o r was d e s i g n e d t o p r o v i d e a s u i t a b l e i n p u t t o t he d i f f e r e n t i a l a m p l i f i e r . The • d e t e c t o r c o n s i s t e d o f t h r e e p robes mounted i n the s e a l e d end o f a l u c i t e t ube (1.91)cm d i a m e t e r and 35 cm l o n g ) w i t h the p o i n t s j u s t p r o t r u d i n g t h r o u g h the s e a l e d ends o f the tube and mak ing c o n t a c t w i t h the s u r f a c e o f the s a l t s o l u t i o n . • The two o u t e r p r o b e s , 1 A 8 cm a p a r t , were j o i n e d t o t h e two l e a d s o f t he s h i e l d e d c a b l e ( I n t e r - 8 Weave) c o n n e c t e d t o t he d i f f e r - e n t i a l a m p l i f i e r , w h i l e t h e t h i r d p r o b e , a t t h e m i d p o i n t between t he o u t e r p r o b e s , was c o n n e c t e d t o t he c a b l e s h i e l d . Thus ' t h e t h i r d p robe p r o v i d e d a common r e f e r e n c e p o t e n t i a l f o r D U A L - B E A M SCOPE TEKTRONIX 5 0 2 AMPLITUDE PHASE ANGLE DIGITAL P H A S E COMPUTER A D - Y U 5 2 4 A 2 DIGITAL V O L T M E T E R HP 3 4 4 0 A ( A C ) DIGITAL VOLTMETER HP 3 4 4 0 A (D C) DIFF. AMPLIFIER! PA.R.-CR4A PHASE REFERENCE D E T E C T O R X - Y RECORDER MOSELEY MODEL N0 .3 DIGITAL-ANALOG I C O N V E R T E R HP 5 8 0 A i - g u r e • B l o c k - d i a g r a m o f t h e m o d e l m e a s u r e m e n t s s y s t e m . 99 t h e t w o o u t e r p r o b e s , , a n d t h i s a r r a n g e m e n t p r e s e n t e d a s u i t - a b l e i n p u t t o t h e d i f f e r e n t i a l a m p l i f i e r . • T h e a m p l i t u d e s a n d p h a s e a n g l e s w e r e m e a s u r e d i n t h e s a m e w a y a s d e s c r i b e d f o r t h e m a g n e t i c f i e l d s i g n a l s . 3 . 1 . h D i s c u s s i o n o f R e s u l t s T h e c o o r d i n a t e s y s t e m u s e d i n t h e d i s c u s s i o n i s s h o w n i n F i g . 3 2 i - T h e g e o p h y s i c a l p r o b l e m b e i n g m o d e l l e d h e r e i s t h e o n e o f a f l a t h o r i z o n t a l s t r a t i f i e d e a r t h , e x - t e n d i n g i n f i n i t e l y i n t h e X a n d Y d i r e c t i o n s , i n t h e f i e l d o f l a r g e i o n o s p h e r i c s h e e t c u r r e n t . I n t h e m o d e l , t h e u p p e r l a y e r w i t h c o n d u c t i v i t y O g i s t h e s a l t s o l u t i o n a n d t h e l a y e r w i t h c o n d u c t i v i t y o ^ i s a l a y e r o f g r a p h i t e l i n i n g t h e b o t t o m o f t h e t a n k . ; T h e s e c o n d l a y e r i s o f s u f f i c i e n t l y h i g h c o n - d u c t i v i t y t h a t t h e e l e c t r o m a g n e t i c f i e l d d o e s n o t p e n e t r a t e t o r e g i o n s b e l o w t h i s l a y e r . • T h e m o d e l i s n o t i n f i n i t e i n e x t e n t a n d h e n c e t h e p r o b l e m c a n n o t b e m o d e l l e d e x a c t l y . H o w e v e r , i f t h e m o d e l d i m e n s i o n s a r e s u f f i c i e n t l y l a r g e , t h e e n d e f f e c t s w i l l b e s m a l l . - T h e v a l i d i t y o f t h e m o d e l w a s t e s t e d b y c o m p a r i n g t h e m e a s u r e d v a l u e s o f E a n d H w i t h c a l c u l a t e d v a l u e s u s i n g x y e q u a t i o n s (U-8) a n d (50) f o r p l a n e w a v e s i n c i d e n t o n a t w o - l a y e r e d s y s t e m o f t h e s a m e c o n d u c t i v i t y a s t h e m o d e l b u t i n f i n i t e i n t h e X a n d Y d i r e c t i o n s . - I t w a s a s s u m e d t h a t t h e c u r r e n t s h e e t w a s o f s u f f i c i e n t l y l a r g e d i m e n s i o n s t h a t t h e h o r i z o n t a l e l e c t r i c a n d t h e h o r i z o n t a l m a g n e t i c f i e l d c o m p o n - e n t s ' . s h o u l d h a v e v a l u e s p r e d i c t e d f o r a p l a n e w a v e s o u r c e . 100 O n l y t h e r e l a t i v e a m p l i t u d e s can be o b t a i n e d by c a l c u l a t i o n , s i n c e t h e a m p l i t u d e o f t he i n c i d e n t f i e l d i s an unknown. S i n c e H„ f r o m the mode l measurements was q u i t e i n s e n s i t i v e t o edge e f f e c t s , i t was, u s ed as a b a s i s f o r compar ing t he c a l - c u l a t e d v a l u e s o f t he f i e l d components w i t h t he measured v a l u e s . - S o l u t i o n s t o M a x w e l l ' s e q u a t i o n s f o r p l a n e waves i n c i d e n t on a h o r i z o n t a l l y s t r a t i f i e d c o n d u c t o r p r o v i d e a r e l a t i o n s h i p between E and H . • The v a l u e o f H as measured tr jf f o r t he mode l can t h e n be u sed t o p r e d i c t a v a l u e f o r E . I t was f o u n d t h a t t he v a l u e p r e d i c t e d f o r E„ a g r e e d v e r y c l o s e l y w i t h t he v a l u e d e t e r m i n e d f o r E f r o m the model measurements f o r p o i n t s w e l l removed f r om the edges o f t he . h t a n k , • The v a l u e s a g r eed to w i t h i n 2% a t ' 3 x 1 0 c y c l e s / s e c . k and t o w i t h i n h % a t ' 6 x 1 0 c y c l e s / s e c . The phase a n g l e measurements a l s o a g r e e d t o w i t h i n a few p e r c e n t . • D e t a i l e d measurements were made o f b o t h E and H . I t was f ound t h a t H-__ was e s s e n t i a l l y c o n s t a n t o v e r t he e n t i r e s u r f a c e a r e a , w h i l e E v a r i e d l e s s t h a n -2% o v e r a c e n t r a l r e g i o n 130 cm l o n g and 50 cm w ide when t he t ank was s i t u a t e d c e n t r a l l y be l ow t h e c u r r e n t s h e e t w i t h t he l o n g d i m e n s i o n i n t he d i r - e c t i o n o f t h e e l e c t r i c f i e l d . The b r o k e n l i n e s i n a l l t he g r a p h i c a l r e s u l t s t o f o l l o w i n d i c a t e the measured v a l u e s o f t he a m p l i t u d e s and phase a n g l e s f o r u n i f o r m h o r i z o n t a l c o n - d u c t i n g l a y e r s . I t w i l l be n o t e d t h a t t h e e l e c t r i c f i e l d i n t h e X d i r e c t i o n d e c r e a s e s r a p i d l y n e a r t he edge o f t h e t a n k . The edge e f f e c t s were g r e a t l y r educed i n some o f t he l a t e r work , by l i n i n g the end o f t he t ank w i t h g r a p h i t e r o d s . I n 101 the early measurements -(Figs „ 35 to *f8) the ends of the tank were not l i n e d with graphite. - The v e r t i c a l magnetic f i e l d H„ was also studied i n great d e t a i l f o r uniform conducting layers. Various types of scaled structures, representing geological structures such as f a u l t s , dykes, sea coasts, etc., were placed i n the s a l t solution. • Departure of the amplitudes and phase angles from the o r i g i n a l values can he attributed to the structure introduced. • The materials used f o r the various structures were graphite and concrete. The conductivity of graphite was taken to be 1.2 x 10""^ emu, while that of concrete was assumed to be \CT^§ emuu ' The conductivity of the s a l t solution was measured to be 2.1 x l b ~ 1 < ^ emu. • The various types of graphite s t r u c t - ures used i n th i s work were machined from large graphite blocks and sheets. • The scaling factors involved s a t i s f y equation (77)'. The conductivities of the model structures f o r a l l measure- ments are 10^ greater than those f o r the geophysical problem. -10 The s a l t solution,of conductivity 2.1 x 10 emu 5represents _ i < a layer of earth having a conductivity of 2.1 x 10 y emu, while the graphite structure,of conductivity 1.2 x 10"^ emu, -11 ' corresponds to sea water of conductivity 1.2 x 10 em. Several sets of scaling factors f o r the frequency and l i n e a r dimensions were used depending on the p a r t i c u l a r problem studied. I t i s convenient to express the f i e l d components as E„\ exp(\iir,* !IL e x p(i* ), and H exp(ic/> -), where the modulus 102 i s t he a m p l i t u d e o f t he component and t he argument i s t he phase a n g l e . T h i s n o t a t i o n i s c o n s i s t e n t w i t h t h a t u sed i n C h a p t e r 2 . The symbols d , R, W, and a w i l l be u s ed t o r e p r e s e n t , r e s p e c t i v e l y , t he d e p t h , r a d i u s , w i d t h , and c o n - d u c t i v i t y o f t he mode l s t r u c t u r e , w h i l e d ^ , d 2 , c 2 , a n d °3 w i l l r e p r e s e n t , r e s p e c t i v e l y , t he d e p t h o f w a t e r o v e r t h e s t r u c t u r e , t he t o t a l d e p t h o f w a t e r i n the t a n k , t he c o n - d u c t i v i t y o f t he s a l t s o l u t i o n , and the c o n d u c t i v i t y o f t he g r a p h i t e l a y e r . The mode l r e s u l t s w i l l now be d i s c u s s e d i n d e t a i l . F o r d i s c u s s i o n pu rpo se s t he mode l measurements f o r t he s h e e t c u r r e n t s o u r c e a r e d i v i d e d i n t o t he f o l l o w i n g g r o u p s : • (a ) c y l i n d r i c a l s t r u c t u r e s , (b) f a u l t and dyke s t r u c t u r e s , ' (<s) c o a s t l i n e s t r u c t u r e s , (d) i s l a n d s t r u c t u r e s , and (e) seamount and c o n d u c t i n g dome s t r u c t u r e s . I n (a) t he h o r i z o n t a l f i e l d components a r e t r e a t e d f o r t r a v e r s e s p a r - a l l e l t o t h e e l e c t r i c f i e l d o f t he s o u r c e , w h i l e i n ( b ) , (c),> ( d ) , and (e) t he v e r t i c a l magne t i c f i e l d component and t he h o r i z o n t a l f i e l d components f o r t r a v e r s e s b o t h p a r a l l e l and p e r p e n d i c u l a r t o t he e l e c t r i c f i e l d o f t h e s o u r c e a r e p r e s e n t e d . (a) C y l i n d r i c a l . S t r u c t u r e s The mode l d i m e n s i o n s i n v o l v e d and t he c o r r e s p o n d i n g g e o p h y s i c a l d i m e n s i o n s u sed i n F i g s . 35 t o *f0 a r e summar ized i n T a b l e s V I I and V I I I r e s p e c t i v e l y . The s t r u c t u r e s s t u d i e d i n F i g s . 35 t o *+0 a r e c y l i n d e r s o f v a r i o u s l e n g t h s w i t h a d i a m e t e r o f 30.h cm. The r e s u l t s T A B L E V I I M o d e l d i m e n s i o n s C a s e f ( s e c " 1 ) d 1 ( c m ) d 2 ( c m ) R ( c m ) d ( e m u ) d 2 ( e m u ) d ^ ( e m u ) T w o l a y e r s C y l i n d e r 3x1 b \ 6 x 1 0 ^ it. it 31 .5 - - 2 . 1 x 1 0 " 1 0 1 " 30.1+ -i 0 2 x i O " 6 ? 1 0 ~ 1 6 ••- 1 . 2 x 1 0 " 6 ti T A B L E V I I I G e o p h y s i c a l d i m e n s i o n s C a s e f ' ( s e c " 1 ) d ^ c m ) d-^Ccm.) R ' ( c m ) ^ > d ' ( e m u ' ) d ^ e m u ) d ^ ( e m u ) T w o l a y e r s C y l i n d e r 0 . 3 : , 0 . 6 - it « 1 0 5 3 . 1 5 X 1 0 6 - - • . - - 2 . 1 x 1 0 ~ 1 ^ 3 . 0 ^ + x l O 6 1 ^ x l O " " 1 1 , 1 0 ~ 2 1 1 . 2 1 x 1 0 " " 1 1 10V shown i n F i g s , 35 to 39 are f o r v e r t i c a l cylinders immersed i n the s a l t s o l u t i o n with the axis of the cylinder i n each case along the z-axis of the coordinate system shown i n F i g . 32. Figure kO deals with cylinders having t h e i r axes p a r a l l e l to the y-axis. The depth of the s a l t solution used f o r these measurements was 3 1 ° 5 cmo ' The two model frequencies used k k were 3 x 1 0 cycles/sec and 6 x 1 0 cycles/sec. Figure 35 shows the behavior of E x and Hy as a function of p o s i t i o n f o r f = 3 x 10^ cycles/sec f o r the case of d^ =1 cm over a v e r t i c a l graphite cylinder. • For y = 0 and y = R / 2 , E__ changes very rapi d l y i n the neighborhood of the edge of the cylinder. < The change i n E„ would be even greater i f the depth of water over the cylinder were decreased. This' structure on a geophysical scale would correspond to a deep c i r c u l a r sea surrounded by earth with a v e r t i c a l earth-sea' in t e r f a c e . The value of E„ d i r e c t l y over the cylinder at y = 0 agrees to within a few percent of ."the: J value calculated f o r plane waves using Maxwell's equations f o r the case of 1 cm of s a l t solution over a semi^-infinite layer of graphite. H also shows some change i n the neighbor- hood of the int e r f a c e . Unlike E x , H y does not become constant d i r e c t l y over the cylinder f o r y = 0. k Figure 36 shows the behavior of E x and H y f o r f = 1 0 cycles/sec^and the same structure as f o r F i g . 35° The curves are very s i m i l a r except that H shows s l i g h t l y more change. Measurements made f o r a wide range of frequencies (10 to 10 y cycles/sec), but not shown here, indicated that the graphite tn o b x 0 CM X ° T — ' — i — 1 — i — ' — i — 1 — i — 1 — i — 1 — i — i — i — 1 — i — » — i — 1 — r • i • i * - 2 R - ^ -i I i I i I i I i I i L J J L ( b ) -100 - 8 0 - 6 0 - 4 0 - 2 0 0 20 4 0 6 0 80 100 X cm F i g u r e 35- E v ( a ) a n d H ( b ) f o r f=3x10 c y c l e s / s e c , d =̂1 (1) y=0, (2) R , (3) R/2, (1+) f o r a v e r t i c a l g r a p h i t e c y l i n d e r c m , a n d t r a v e r s e s a l o n g 3R/2 c m . 1 0 $ 3 2 > 1° CD ~x3 LU A ~ 2 N |\ / V - i — i — I — i — i — i — i 1—i—I—i—I—i—I—i—I 1—I—'—I—i 1—r -100 -80 -60 -40 -20 0 20 40 60 80 100 X c m 3 2 </> Z3 O s° id x fr-2R-* 1 — i — | — i — | — i — | — i — | — i — | — i — | — i — | — i — | — ' — | — > — | — r 2 - _L *-2R—* (b) J i_ l i I i _ -100 -80 -60 -40 -20 0 20 40 60 80 100 X c m F i g u r e 36; E y ( a ) a n d H v ( b ) f o r a v e r t i c a l g r a p h i t e c y l i n d e r f o r f = 6 x 1 0 c y c l e s / s e c , d ^ 1 c m , a n d t r a v e r s e s a l o n g ( 1 ) y = 0 , (2) y = R , (3) y = R / 2 , (If) y=3R/2 c m . 10? cylinders were not e n t i r e l y homogeneous. At some frequencies the shape of the curves i n the neighborhood of the cylinder edge changed s l i g h t l y i f the cylinder was rotated to a new p o s i t i o n . Figure 37 displays the difference i n phase of the two components of the f i e l d studied. ^ x — <p • i s the phase d i f f e r - ence between the horizontal e l e c t r i c and the horizontal magnetic f i e l d . Figures37(a) and (b) show the behavior of the phase angle f o r the case described in: F i g , 35? while F i g s . 37(c) and (d) refer to the case described 3)ji F i g . 36. I t can be seen that the phase difference, undergoes a much greater change f o r f = 6 x 10 cycles/sec than f o r f = 3 x 10 c y c l e s / s e c -The dependence of E„ and H,_ ,on the depth of the s a l t x y solution over the graphite cylinder f o r y = 0 and y = R at 1 f = 3 x 10^ cycles/sec i s brought out i n F i g . 38. I t can be seen that changing the depth of the s a l t solution over the -cylinder greatly affects' E i n the neighborhood of the cylinder. - Figure 39 deals with the case of a v e r t i c a l concrete cylinder submerged i n the s a l t solution with d^ = 1 cmu Figure 39(a) shows E as a function of p o s i t i o n f o r traverses -A-over the cylinder. Measurements of E were also made f o r a -X. k frequency of 6 x 1 0 cycles/sec^ but since they d i f f e r e d very l i t t l e from measurements f o r 3 x 1 0 cycles/sec the results are not shown* • Since H was not at a l l affected by the . concrete cylinder, the results f o r H are not shown. •- The 108 CO LU UJ or LU o 'E > b x CO If) x LU in 3 O O X CM (a) (b) I i I i i i l i i i 1 i i i 1 i 1 i l i 37 (d) j i | i | i | i | i | i j i i • i • i . 0 20 40 60 80 0 20 40 60 80 X cm ' J (a) 1 i I i l i i • I i 1 ^ H y l i 1 i 1 i i > i i l*R-»l U Hy (c) 38 (0 1 1 1 1 1 1 1 1 1 1 1 1 1 . 1 1 1 1 1 . 0 20 40 60 80 0 20 40 60 80 X cm F i g u r e 3 7 . - t x " ° % f o r a v e r t i c a l g r a p h i t e c y l i n d e r f o r d ^ l cm, f=3x10 L f c y c l e s / s e c ( a ) - ( b ) , f=6x10^" c y c l e s / s e c ( c ) - ( - d ) , . and (1) y=0,< ( 2 ) R, ( 3 ) R/2, and (1+) 3R/2 cm. F i g u r e 3 8 . E x f o r t r a v e r s e s a l o n g (a) y=0, (b) y=R, H y f o r . t r a v e r s e s a l o n g ( c ) y=0, (d) y=R, f o r a v e r t i c a l g r a p h i t e c y l i n d e r f o r f =3x10 l f c y c l e s / s e c , and (1) d l = 1 , ( 2 ) 8 . 6 , ( 3 ) 16.2, and A) 2 3 . 8 cm. 109 -, 1 1 1 1 1 1 1 j 1 1 1 | 1 1 1 1 1 1 r 0 2 0 4 0 6 0 8 0 0 2 0 4 0 6 0 8 0 X cm F i g u r e 39. • ( a ) f o r f=3x10^" c y c l e s / s e c , - tyx-<Py f o r ( b ) - ( c ) 3x10k c y c l e s / s e c , • ( d ) - ( e ) 6x10 c y c l e s / s e c , f o r a v e r t i c a l c o n c r e t e c y l i n d e r f o r d =̂1 c m a n d t r a v e r s e s a l o n g (1) y=0, (2) R , - (3) R/2 3 a n d {k) 3R/2. 110 c o n d u c t i v i t y o f c o n c r e t e i s s u f f i c i e n t l y l o w t h a t t h e d e p t h o f c o n c r e t e i n v o l v e d h e r e i s a v e r y s m a l l f r a c t i o n o f a s k i n d e p t h . : The c o n c r e t e a c t s e s s e n t i a l l y l i k e a l a r g e c a v i t y i n t h e c o n d u c t i n g s o l u t i o n , and t h e enhancement o f E o v e r t he c y l i n d e r i s due t o t he l a r g e c o n d u c t i v i t y c o n t r a s t . H i s n o t shown s i n c e i t i s n o t a t a l l a f f e c t e d by t he c o n c r e t e c y l i n d e r . T h i s s t r u c t u r e on a g e o p h y s i c a l s c a l e wou ld c o r r e s p o n d t o a body o f v e r y l o w c o n d u c t i v i t y embedded i n t he e a r t h . • F i g s.39 ( b ) and ( c ) show the phase d i f f e r e n c e ty - $ f o r f = 3 x 10 k c y c l e s / s e c w h i l e F i g s .39 ( d ) and (e) x y show t h e phase d i f f e r e n c e f o r f = 6 x 10 c y c l e s / s e c . As was t he ca se f o r t he phase d i f f e r e n c e s f o r t r a v e r s e s o v e r t h e g r a p h i t e c y l i n d e r , i t i s e v i d e n t f r om F i g s .39 ( b ) - (e ) t h a t t h e phase d i f f e r e n c e tyK - <p undergoes a much g r e a t e r L. h change f o r f = 6 x 10 c y c l e s / s e c t h a n i t does f o r f = 3 x 10 c y c l e s / s e c . ' I n compar i ng F i g s L 37 and 39, i t i s i n t e r e s t i n g t o n o t e t h a t t he phase d i f f e r e n c e s i n t he n e i g h b o r h o o d o f t he g r a p h i t e c y l i n d e r a r e q u i t e d i f f e r e n t f r o m the phase d i f f e r - ences i n t he n e i g h b o r h o o d o f t h e c o n c r e t e c y l i n d e r . • F i g u r e s k 0 ( a ) and (b) d e a l w i t h t he ca se o f t he g r a p h i t e c y l i n d e r i+5 cm l o n g p o s i t i o n e d w i t h i t s a x i s p a r a l l e l LL t o t he y - a x i s w i t h d^ = 1 cm and f = 2 x 10 c y c l e s / s e c . T h i s s t r u c t u r e on a g e o p h y s i c a l s c a l e c o u l d c o r r e s p o n d t o a h i g h l y c o n d u c t i n g c y l i n d r i c a l body l o c a t e d n e a r t he s u r f a c e o f t he e a r t h . H i s v e r y l i t t l e a f f e c t e d by the g r a p h i t e v c y l i n d e r , whereas E d e c r e a s e s r a p i d l y as t he c y l i n d e r i s a p p r o a c h e d . • F i g u r e MX-c) i n d i c a t e s a v e r y l a r g e enhancement 111 1 — i — I — i — l — i — I — i — I — i — | — ' — I — 1 — I — 1 — I — ' I 1 -100 -80 -60 -40 -20 0 20 40 60 80 100 X cm -100 -80 -60 -40 -20 0 20 40 60 80 100 X cm F i g u r e 1+0. • E • (a) a n d H • ( b ) f o r a h o r i z o n t a l g r a p h i t e y c y l - i n d e r - - a n d - E • - ( c ) - f o r a h o r i z o n t a l c o n c r e t e c y l i n d e r f o r LL X f = 3 x 1 0 c y c l e s / s e c , d-.,=1 c m , a n d t r a v e r s e s a l o n g y = 0 . 112 of: E „ when the h o r i z o n t a l g r a p h i t e c y l i n d e r i s r e p l a c e d by a .A. c o n c r e t e c y l i n d e r . - H i s n o t shown, s i n c e i t i s a g a i n n o t a t a l l s e n s i t i v e t o t h i s s t r u c t u r e . Measurements were made a l s o f o r d-.j = 0.1 cm. • I n t h i s c a s e E x d i r e c t l y o v e r t he c y l i n d e r i n c r e a s e d by a f a c t o r o f k above t he v a l u e o b t a i n e d f o r d\j = 1 cm. On t he b a s i s o f t he r e s u l t s d i s c u s s e d so f a r , i t i s a p p a r e n t t h a t i f t h e g r a p h i t e and c o n c r e t e c y l i n d e r s repres-;- e n t e d c y l i n d r i c a l b o d i e s -embedded i n t h e e a r t h , b o t h E and \|rx -, -ty i n t h e n e i g h b o r h o o d o f t he c y l i n d e r s wou ld be v e r y d i f f e r e n t f o r a h i g h l y c o n d u c t i n g c y l i n d e r f o r a h i g h l y r e s i s t i v e c y l i n d e r . I t i s a l s o e v i d e n t t h a t t he d e p t h be l ow t h e s u r f a c e o f , t he e a r t h wou ld have an i m p o r t a n t e f f e c t on the v a l u e o f E a t t he s u r f a c e . -X. Ob) F a u l t and Dyke S t r u c t u r e s I n (a ) o n l y t he h o r i z o n t a l f i e l d components f o r t r a - v e r s e s p a r a l l e l t o the e l e c t r i c f i e l d o f t he s o u r c e were c o n s i d e r e d . I n (b) t he v e r t i c a l magne t i c f i e l d component as w e l l as t h e h o r i z o n t a l f i e l d component a r e i n c l u d e d f o r t r a - v e r s e s b o t h p a r a l l e l ( i n t he x d i r e c t i o n ) t o and p e r p e n d i c - u l a r ( i n t he y d i r e c t i o n ) t o the e l e c t r i c f i e l d o f t he s o u r c e , thus p e r m i t t i n g t he s t u d y o f b o t h p o l a r i z a t i o n s f o r the v e r t i c a l f a u l t and dyke p r o b l e m s . - F o r a l l measurements t he model f r e q u e n c y was 3 x 1 0 c y c l e s / s e c , the d e p t h o f t h e s a l t s o l u t i o n o v e r the submerged s t r u c t u r e a p p r o x i m a t e l y 0 . 0 5 cm, and t h e t o t a l d e p t h o f the s a l t s o l u t i o n i n the t ank i s 61 cm. 113 The s c a l i n g f a c t o r s i n v o l v e d a g a i n s a t i s f y e q u a t i o n (77)? w i t h t he c o n d u c t i v i t y and t h e f r e q u e n c y each a f a c t o r o f 1Cr g r e a t e r , and t h e l i n e a r d i m e n s i o n s a f a c t o r o f 1Cr s m a l l e r , than- t h o s e a " f o r t he g e o p h y s i c a l p r o b l e m , ' The symbols d, ' R, and W a g a i n r e p r e s e n t , r e s p e c t i v e l y , t he d e p t h , r a d i u s and w i d t h o f t he g r a p h i t e s t r u c t u r e . The mode l measurements f o r t he i n c i d e n t magne t i c f i e l d p a r a l l e l t o the dyke d i s c o n t i n u i t y -(H p o l a r i z a t i o n ) a r e d e s c r i b e d i n F i g - . h \ . • The g r a p h i t e dyke i s 61 cm deep and 23 cm w i d e . - I t s l e n g t h , 102 cm, i s s u f f i c i e n t l y g r e a t t h a t - i t can be c o n s i d e r e d i n f i n i t e l y l o n g f o r t he c o r r e s p o n d i n g g e o p h y s i c a l d i m e n s i o n . - As can be seen f r o m F i g . L 1 ( a ) , H„ shows a s m a l l g r a d u a l i n c r e a s e as t he dyke i s approached-* H i s v e r y s m a l l , b u t does become measu rab l e n e a r t he i n t e r - f a c e s . ' I t c an be seen f r o m F i g . » M ( b ) t h a t E changes v e r y l i t t l e up t o abou t 1 cm f r o m the d i s c o n t i n u i t i e s and t h e n d e c r e a s e s r a p i d l y t o v e r y s m a l l v a l u e s d i r e c t l y o v e r t he d yke . - I t i s r a t h e r i n t e r e s t i n g t o n o t e t h a t f o r t h i s deep dyke , i t s p r e s e n c e i s n o t d e t e c t a b l e on t he b a s i s o f e l e c t r i c f i e l d measurement u n t i l p o i n t s v e r y n e a r t he d i s c o n t i n u i t y a r e r e a c h e d . - F i g u r e s M ( c ) and (d) show t h e b e h a v i o r o f t he pha se d i f f e r e n c e s , ^ - A • and <$> • cp . • I t i s a p p a r e n t t h a t x . y ^ y a change i n phase a n g l e i s o b s e r v a b l e a t more d i s t a n t p o i n t s where t h e r e i s no o b s e r v a b l e change i n E„» » The r a t i o 1 E / H , x x y w h i c h i s t he q u a n t i t y u s u a l l y c o n s i d e r e d i n m a g n e t o t e l l u r i c a n a l y s e s , i s shown i n F i g . ' M (-e). We n o t e t h a t E ^ R y does show some change a t p o i n t s q u i t e remote f r o m the i n t e r f a c e . 11 k * b X in IS M 'E b X. O CD x LU 0 8 6 4 H. (b) i 1 i 1 i 1 i 1 — r - 1 - ! — 1 i 1 i 1 i 1 i • i -100 -80 -60 -40 -20 0 20 40 60 80 100 X cm co S 70 Q •6-̂ 60 1.0 -5*0.8 E > 0.6! cn S 0.4 LU KJ-C- 160" J i I i L 20 40 60 80 J i L J i I i_ 20 40 60 80 (e) — i — i — | — i — | — i — | — i — | T -^ -? - j — i — i — i — i — i — | — i — i — -100 -80 -60 -40 -20 0 20 40 60 80 100 X cm • H y , - H z ( - a ) y E x ( b ) , < ( ^ y ^ H y ( d ) a n d - E ^ H y ( e ) f o r t h e H - p o l a r i z a t i o n f o r a g r a p h i t e d y k e w i t h w=23 c m . 115 T h i s i s d u e t o t h e f a c t t h a t EL u n d e r g o e s a g r a d u a l i n c r e a s e e v e n f o r p o i n t s w e l l r e m o v e d f r o m t h e i n t e r f a c e . F i g u r e k2 d e a l s w i t h t h e i n c i d e n t e l e c t r i c f i e l d p a r a l l e l t o t h e d y k e d i s c o n t i n u i t y ( E p o l a r i z a t i o n ) , w i t h m e a s u r e m e n t s m a d e f o r t r a v e r s e s a l o n g t h e y - a x i s . I t i s e v i d e n t f r o m F i g . k 2 ( a ) t h a t b o t h EL a n d EL s h o w a l a r g e y z c h a n g e i n t h e n e i g h b o r h o o d o f t h e d i s c o n t i n u i t y . O f p a r t i c - u l a r i n t e r e s t i s t h e o b s e r v a t i o n t h a t H u n d e r g o e s a l a r g e i n c r e a s e r i g h t a t t h e i n t e r f a c e , i n c r e a s i n g b y a b o u t a f a c t o r o f 2.5 o v e r t h e r e f e r e n c e v a l u e f o r t h e c a s e o f a h o m o g e n e o u s c o n d u c t o r . T h e b e h a v i o r o f E , a s s h o w n i n F i g . k 2 ( b ) , i s q u i t e d i f f e r e n t f o r t h i s p o l a r i z a t i o n f r o m t h a t f o r t h e H p o l a r i z a t i o n s h o w n i n F i g . V l ( b ) . E u n d e r g o e s a f a i r l y .X. g r a d u a l d e c r e a s e , b e g i n n i n g a t p o i n t s w e l l r e m o v e d f r o m t h e i n t e r f a c e , a n d r e a c h e s v e r y s m a l l v a l u e s d i r e c t l y o v e r t h e d y k e . A g a i n , i n t h e g e o p h y s i c a l p r o b l e m , t h e r a t i o s E /EL T h e r a t i o s a r e s h o w n i n F i g . ^-2 (d ) . O f s p e c i a l i n t e r e s t i s c o n d u c t i n g l a y e r s , r e a c h e s a m a x i m u m v a l u e o f a p p r o x i m a t e l y u n i t y a t a b o u t 2 c m (2x10^ c m i n t h e g e o p h y s i c a l p r o b l e m ) f r o m t h e d y k e d i s c o n t i n u i t y . T h i s r a t i o r i s e s s t e e p l y a s t h e i n t e r f a c e i s a p p r o a c h e d a n d f a l l s o f f v e r y r a p i d l y o v e r t h e d y k e . I f t h e m e a s u r e m e n t s w e r e m a d e f o r a f a u l t , r a t h e r t h a n f o r a d y k e , t h e r a t i o w o u l d b e s o m e w h a t d i f f e r e n t s i n c e EL., a s s h o w n i n F i g . ' k 2 ( a ) , w o u l d f a l l t o l o w e r v a l u e s o v e r t h e g r a p h i t e . T h e m a x i m u m v a l u e o f H /EL w o u l d , h o w e v e r , h a v e t h a n t h e a m p l i t u d e s t h e m s e l v e s . t h e f a c t t h a t t h e r a t i o EL/EL , w h i c h i s v e r y s m a l l f o r u n i f o r m z y 116 _ "Oh x N b > 4 X 2U TE 0 b 4 x o (0 UJ H 4 T—1—r (a) i — • — r (b) — i — i — | — i — | — i — i — i — | — i — i — i — i — i — i — i — i — i — i — • — i — -100 -80 -60 -40 -20 0 20 40 60 80 100 Y cm ->—I— 1—I— 20 40 60 80 0 Y cm T — | 1 1—1—|— 20 40 60 80 F i g u r e hZ. H y ? H z ( a ) , , E x ( b ) , ̂ x - 0 y ( c ) , E x / H y 5 < EJ/EZ ( d ) , a n d 0 z - 0 y ( e ) f o r t h e E p o l a r i z a t i o n f o r a g r a p h i t e d y k e w i t h w=23 c m . 117 a p p r o x i m a t e l y t h e s a m e v a l u e a s i t h a s f o r t h e d y k e . T h i s w o u l d s u g g e s t t h a t f o r t h e f r e q u e n c y c o n s i d e r e d (0.3 c y c l e s / s e e ) t h e m a x i m u m v a l u e o f H z / H y , f o r m e a s u r e m e n t s a t a v e r t i c a l e a r t h - s e a ' i n t e r f a c e , w o u l d b e a p p r o x i m a t e l y 1 . - T h e p h a s e d i f f e r e n c e s , ty„ — 0 V a n d <j> • - fc, a r e s h o w n i n F i g s . k 2 ( c ) x <y y a n d ( e ) . - $ z - - 0 y u n d e r g o e s a p a r t i c u l a r l y l a r g e c h a n g e a n d s h o w s t h e e f f e c t s o f t h e d y k e a t p o i n t s f a r r e m o v e d f r o m t h e d y k e . T o s t u d y t h e e f f e c t t h a t t h e d e p t h o f t h e d y k e h a s o n t h e f i e l d c o m p o n e n t s , m e a s u r e m e n t s w e r e m a d e f o r b o t h t h e H a n d t h e E p o l a r i z a t i o n s f o r d y k e s o f c o n s t a n t w i d t h (29 c m ) f o r f o u r d i f f e r e n t d y k e d e p t h s . - T h e n u m b e r e d c u r v e s i n F i g . k-3 c o r r e s p o n d t o t h e d e p t h s ( 1 ) 1 c m , (2) 6 c m , (3) 11 c m , a n d ( k > 1 6 c m . F o r t h e H p o l a r i z a t i o n [ F i g . k 3 ( a ) ] t h e m a x i m u m v a l u e o f E n e a r t h e d y k e d i s c o n t i n u i t y i s s t r o n g l y .A. d e p e n d e n t o n t h e d e p t h o f t h e c o n d u c t i n g d y k e . • A s t h e d y k e b e c o m e s d e e p e r , t h e s h a p e o f t h e c u r v e f o r E a p p r o a c h e s t h a t -A. s h o w n i n F i g . k 1 ( b ) . F o r a n i n f i n i t e l y d e e p d y k e , - E „ w o u l d s h o w n o i n c r e a s e a t a l l a s t h e d y k e i s a p p r o a c h e d , b u t w o u l d f a l l o f f s h a r p l y a t t h e i n t e r f a c e . H y a n d H z f o r t h i s c a s e a r e n o t s h o w n s i n c e t h e y s h o w e d n e g l i g i b l e c h a n g e a s t h e d y k e w a s t r a v e r s e d . • F i g u r e s k 3 ( b ) a n d ( c ) d e a l w i t h t h e E p o l a r - i z a t i o n f o r t r a v e r s e s m a d e a l o n g t h e y - a x i s a c r o s s t h e d y k e s p o s i t i o n e d p a r a l l e l t o t h e y - a x i s . I n F i g . k 3 ( b ) t h e m a x i m u m v a l u e s o f H a n d H i n c r e a s e w i t h i n c r e a s i n g d y k e d e p t h . E , J Zt A a s s h o w n i n F i g . ^ k 3 ( - c ) , i s a l s o a f f e c t e d b y t h e d y k e d e p t h , b u t t h e c h a n g e s a r e n o t a s g r e a t a s t h e y a r e f o r t h e H p o l a r - 118 'E 12 > X o CD CO LU - _ I " 2 E > -—w—» s X (a) -I00 -80 -60 -40 -20 0 20 40 60 80 100 X c m _ IO <r 9 8 x m e 6 N >% x < O X o to >< LU I '' I (b) i- • i ' / / / / / / * ? / / / / - T — i — | — i — | — i — | — I — | — i — | — i — I — r — \ — r — y -100 -80 -60 -40 -20 0 20 40 60 80 100 Y c m F i g u r e k 3 . E • (a) f o r t h e H p o l a r i z a t i o n , H__f H (b ) and E „ •A. J ( c ) f o r t h e E p o l a r i z a t i o n f o r a g r a p h i t e dyke w i t h w=29 cm. and (-1) d=1, (2.) ( 3 ) 11, and (K) 16 cm. 119 i z a t i o n s h o w n i n F i g . k 3 ( a ) . C u r v e h i s n o t s h o w n i n F i g s . k 3 ( b ) a n d ( c ) s i n c e i t c o i n c i d e d a l m o s t e x a c t l y w i t h c u r v e 3« T o s t u d y t h e b e h a v i o r o f t h e f i e l d c o m p o n e n t s a s a f u n c t i o n o f d y k e w i d t h , t h e d e p t h o f t h e d y k e w a s k e p t c o n s t a n t a n d t h e w i d t h v a r i e d . F i g u r e ¥ f ( a ) d e a l s w i t h t h e H p o l a r i z a t i o n f o r a d y k e d e p t h o f 5 c m a n d w i d t h s o f ( 1 ) 5 c m , ( 2 ) 10 c m , ( 3 ) 1 5 c m , (h) 20 c m , ( 5 ) 2 5 c m , ( 6 ) 30 c m , a n d ( 7 ) 100 c m . T h e r e s u l t s i n d i c a t e t h a t t h e m a x i m u m v a l u e o f E i n t h e n e i g h b o r h o o d o f a s h a l l o w d y k e i n t e r f a c e i s d e p e n d e n t o n t h e w i d t h o f t h e d y k e . T h e s t r u c t u r e f o r c u r v e 7? o n t h e g e o p h y s i c a l s c a l e , w o u l d c o r - r e s p o n d t o a v e r t i c a l f a u l t 5 x 1 0 ^ c m d e e p w i t h a c o n d u c t - i v i t y c o n t r a s t o f a p p r o x i m a t e l y 10 a t t h e i n t e r f a c e . T h e s i d e s o f t h e d y k e a r e e s s e n t i a l l y i n f i n i t e l y f a r a p a r t a n d t h e f i e l d a t o n e i n t e r f a c e i s n o t a f f e c t e d b y t h e s e c o n d i n t e r - f a c e . F i g u r e s k l f ( b ) a n d ( c ) d e a l w i t h t h e E p o l a r i z a t i o n f o r a d y k e d e p t h o f 29 c m a n d w i d t h s o f ( 1 ) 0 . 2 c m , ( 2 ) 1 c m , a n d ( 3 ) 5 c m . I t c a n b e s e e n f r o m F i g . M+Cb) t h a t t h e v a l u e s o f H y a n d Ez i n t h e n e i g h b o r h o o d o f t h e d y k e d i s c o n t i n u i t y v a r y w i t h d y k e w i d t h . C u r v e s 2 a n d 3 f o r H h a v e a l m o s t t h e s a m e m a x i m u m v a l u e s , e v e n t h o u g h t h e d y k e w i d t h s a r e v e r y d i f f e r - e n t . • O n e w o u l d e x p e c t t h a t a s t h e d y k e w i d t h i s f u r t h e r i n c r e a s e d , t h e c u r v e f o r H w o u l d s h o w a c o n s i d e r a b l e d e c r e a s e d i r e c t l y o v e r t h e d y k e , t h e m a x i m u m v a l u e w o u l d d e c r e a s e , a n d t h e c u r v e w o u l d h a v e a s h a p e s i m i l a r t o t h o s e o f F i g . k 3 ( b ) * H _ f o r t h e t h r e e d y k e w i d t h s c o n s i d e r e d d o e s d e c r e a s e t o v e r y s m a l l v a l u e s d i r e c t l y o v e r t h e d y k e . T h e e l e c t r i c f i e l d , a s 120 1 — i — • — i — • — i — 1 — r • i • i - 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 C X i — " — r ' i • i ' i ) 2 0 4 0 6 0 8 0 100 c m >- •>- 3 - J 2 - | 1- ' J ; — 1 — i — | — i — i — i — i — i — I — 1 — l 1 I •' I 1 i ' i ' i i . i . i . " ^ ^ ^ — i — l — CP I O X If) X o X o <x> CD X u - 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 Y c m 4 0 6 0 8 0 100 F i g u r e k L K E x ( a ) f o r t h e H p o l a r i z a t i o n w i t h d=5 c m f o r (1)w=5,- (-2)10, (3)15, (k)20y (5)25, (6)30, a n d (7)100 c m ; H y , H z ( b ) a n d E x ( c ) f o r t h e E p o l a r i z a t i o n w i t h d=29 c m , f o r (-1)w=0.2,' (2)1, a n d (3)5 c m f o r a g r a p h i t e d y k e . 12.1 s h o w n i n P i g . M+(-c •)•,'• a l s o c h a n g e s w i t h d y k e w i d t h . ' W e s h o u l d p o i n t o u t t h a t - f o r t h e v e r y n a r r o w d y k e ( c u r v e -1) s t h e c u r v e w o u l d p r o b a b l y - h a v e a d i f f e r e n t s h a p e i f a s m a l l e r p r o b e s e p a r a t i o n f o r t h e e l e c t r i c f i e l d d e t e c t o r w e r e u s e d c - T h e • m i n i m u m v a l u e s f o r c u r v e s 1 a n d 2 w o u l d p r o b a b l y c o r r e s p o n d m o r e c l o s e l y t o t h e m i n i m u m v a l u e f o r c u r v e 3 . • ' F i g u r e s - k 5 a n d h6 s h o w t h e r e s u l t s f o r m e a s u r e m e n t s a c r o s s a v e r t i c a l g r a p h i t e c y l i n d e r 30»h c m i n d i a m e t e r a n d 7.6 c m i n h e i g h t 0 T h e u p p e r s u r f a c e o f t h e c y l i n d e r h a s i t s c e n t e r a t t h e ' o r i g i n o f o u r c o o r d i n a t e s y s t e m . F i g u r e *+5 d e a l s w i t h t r a v e r s e s a l o n g (-1) y = Qr (2) y = R/23. a n d (3) y =- R , w h e r e R i s t h e r a d i u s o f t h e c y l i n d e r . A l t h o u g h R y a n d H s h o w e d v e r y l i t t l e c h a n g e f o r a s h a l l o w d y k e f o r t h e H p o l a r i z a t i o n , i t c a n b e s e e n f r o m F i g * ^ ( a ) t h a t f o r a c y l i n d e r t h e y d o s h o w s o m e - c h a n g e i n t h e n e i g h b o r h o o d o f t h e d i s c o n t i n u i t y . - T h e b e h a v i o r o f t h e e l e c t r i c f i e l d i s q u i t e d i f f e r e n t f o r t h e c y l i n d e r t h a n t h a t f o r a d y k e o f a p p r o x - ? - i m a t e l y e q u a l d e p t h . ' T h i s c a n b e s e e n b y c o m p a r i n g c u r v e 1 o f Figo' ^ 5 ( h ) w i t h , c u r v e s 2 and 3 o f F i g . - k30a)»' T h e m a x i m u m v a l u e o f E v f o r c u r v e 1 o f f i g . k5(b) i s a p p r o x i m a t e l y 13.5 w h i l e t h e ' m a x i m u m v a l u e o f E v f o r c u r v e s 2 a n d 3 o f Fig.' .X. ^3(a) i s a b o u t 80 ' C u r v e 1 f o r F i g . *+5(c) i s . n o t s h o w n s i n c e H v / H ; r r e m a i n s e s s e n t i a l l y z e r o a l o n g y = 0. • T h e p h a s e d i f f e r s z y e n c e s a n d E ^ / H y a r e g i v e n i n F i g s • „ k5Ca)' - ( c ) . T h e r e s u l t s f o r t r a v e r s e s a l o n g t h e x d i r e c t i o n a r e g i v e n i n F i g . L6. I n c o m p a r i n g t h e c u r v e s f o r H a n d H o f y ^ F i g . k 6 ( a ) w i t h t h o s e l a b e l l e d c u r v e 3 i n F i g . k 3 ( b ) . $ i t i s 122 r i i i » i i [ » 1 i • i • i i i i i i 0 20 40 60 80 0 20 40 60 80 X cm F i g u r e >+5, H y - j - H - ( a ) , E x • ( b ) . - H z / H y - ( e v H j ^ y ( d ) , E ^ H y (e)...» a n d cfr -c& . f o r t h e H p o l a r i z a t i o n f o r a g r a p h i t e c y l i n d e r w i t h d = y . 6 c m , a n d R=30 , . 1 + c m f o r t r a v e r s e s a l o n g (1) y = 0 , (2) R/2 5 <3) R c m . 123 0 20 40 60 80 0 20 40 60 80 Y c m 0 20 40 60 80 0 20 40 60 80 Y c m F i g u r e h6. . h y H z ( a ) , E x (b), H z / H y ( c ) , E x / H y ( d ) , ( e ) a n d 4 > z - 0 y f o r t h e E p o l a r i z a t i o n f o r a g r a p h i t e c y l i n d e r w i t h d=7 .6 c m a n d R = 3 0 . k c m f o r t r a v e r s e s a l o n g (1) x=0,- (-2)' R / 2 , a n d (3) R c m . 12 k e v i d e n t t h a t t h e e n h a n c e m e n t o f t h e m a g n e t i c f i e l d c o m p o n e n t s f o r t h e d y k e d i s c o n t i n u i t y i s c o n s i d e r a b l y g r e a t e r t h a n f o r t h e c y l i n d e r d i s c o n t i n u i t y . ' C u r v e 1 o f F i g . k 6 ( b ) i s v e r y s i m i l a r t o c u r v e 2 o f F i g . ' k 3 ( c ) , i n d i c a t i n g t h a t t h e b e h a v i o r o f E i s e s s e n t i a l l y t h e s a m e f o r t h e c y l i n d e r a s f o r t h e JL • • d y k e . • T h e r a t i o H z ' / H b e c o m e s a p p r e c i a b l e , - r e a c h i n g a m a x - i m u m v a l u e o f O . 3 5 n e a r t h e c y l i n d e r e d g e f o r a t r a v e r s e a l o n g x = 0. T h e E Y / H c u r v e s a r e v e r y s i m i l a r t o t h e E x • y x c u r v e s . - T h e p h a s e d i f f e r e n c e s a r e s h o w n i n F i g s . k 6 ( e ) a n d ( f ) . T h e v a l u e o f 0 •• . - 0 T T s h o w s v e r y l a r g e d e p a r t u r e s f r o m - z y t h e v a l u e s f o r t h e c a s e o f a h o m o g e n e o u s c o n d u c t o r i n d i c a t e d b y t h e b r o k e n l i n e . M e a s u r e m e n t s f o r a- c y l i n d r i c a l s h e l l 2 c m t h i c k , a n d o f t h e s a m e d i a m e t e r a n d h e i g h t a s t h e c y l i n d e r u s e d i n F i g s . k 5 a n d k 6 , a r e d e s c r i b e d i n F i g s . k 7 a n d k 8 . O n t h e g e o p h y s - i c a l " s c a l e t h i s s t r u c t u r e c o u l d c o r r e s p o n d t o a c o n d u c t i n g r i n g d y k e 2 x 1 0 ; c m i n w i d t h a n d 7'6 x 10^ c m i n d e p t h . R e s u l t s f o r t r a v e r s e s a l o n g t h e x d i r e c t i o n a r e n o t s h o w n s i n c e t h e y w e r e f o u n d t o b e a l m o s t t h e s a m e a s t h o s e f o r t h e s o l i d g r a p h i t e c y l i n d e r . < F i g u r e k 7 s h o w s t h e r e s u l t s f o r t r a v e r s e s a l o n g ( 1 ) x = 0 a n d (2) x = R / 2 , w h i l e F i g . k 8 s h o w s t h e r e s u l t s f o r (3) x =• R T 1 c m a n d ( k ) x = R . F i g - u r e s k 7 ( a - ) - a n d ( c ) i n d i c a t e t h a t t h e b e h a v i o r o f H a n d H i s q u i t e d i f f e r e n t f o r t h i s c y l i n d r i c a l s h e l l t h a n f o r t h e s o l i d c y l i n d e r . T h e c u r v e s f o r H a n d H a r e q u i t e s i m i l a r t o t h o s e f o r n a r r o w d y k e s d i s c u s s e d e a r l i e r . T h e r a t i o H z / H y a t t a i n s a m a x i m u m v a l u e o f a p p r o x i m a t e l y 0.5- 1 T h e b e h a v i o r 125 g x in N X_ >% X O X o CD x LU 8 6 4 2 0 6 4 2 Hy I' Hy - A 1 • —' 2 Hz / t- — " ~ t * i • i • H—* _ 777WA I 1 ' 1 _ f x \ ' 1 1 i • H—> 1 1 1 1 I Ex 1/ "2 20 40 60 80 0 Y c m 20 40 60 80 1— 1 — i — 1 — I — ' — I — 1 — l — » " 0 20 40 60 80 0 20 40 60 .80 Y c m F i g u r e k7'. H y 5 H z r E x , - H z / H y , a n d E x / H y f o r t h e E p o l a r i z a t - i o n f o r a c y l i n d r i c a l g r a p h i t e s h e l l 2 c m t h i c k w i t h d=7„6 c m a n d R = 3 0 A c m f o r t r a v e r s e s a l o n g (1 )x=0, a n d ( 2 ) x = R / 2 c m . 1 2 6 x If) ^ 2 4 - - 10 I E > 8 <? o x 6 o o> to 4 LU X . 1 Hy " ^ ^ - 2 H„ " H- • tway/i - * - R * Hz • ^ j - 1 i i • i V i 1 i 1 l E x 1 1 ' 1 I 1 I 1 2 0 4 0 6 0 8 0 0 2 0 4 0 6 0 8 0 Y c m 1—i—i—•—|—i—i—1—r r 0 2 0 4 0 6 0 8 0 0 Y c m 1—1 I 1 I 1 I 1 2 0 4 0 6 0 8 0 F i g u r e h&. < H , ; R" . E . H z / H , a n d E / H f o r t h e E p o l a r i z a t -y & x y x y i o n f o r a c y l i n d r i c a l g r a p h i t e s h e l l 2 c m t h i c k w i t h 6=7.6 c m a n d R=30.L c m f o r t r a v e r s e s a l o n g ( 1 ) x = R < - 1 ? ( 2 ) x = R c m . 127 o f E ^ R y i s v e r y s i m i l a r t o t h a t f o r t h e s o l i d c y l i n d e r e x c e p t t h a t t h e r a t i o i s l a r g e r i n s i d e t h e c y l i n d r i c a l s h e l l t h a n i t w a s d i r e c t l y o v e r t h e s o l i d c y l i n d e r . • T h e a m p l i t u d e s a n d a m p l i t u d e r a t i o s s h o w n i n F i g . k8 f o r y = R a r e v e r y s i m i l a r t o t h o s e f o r t h e s o l i d c y l i n d e r . F r o m t h e s e r e s u l t s , i t w o u l d a p p e a r t h a t a l a r g e H / H r a t i o f o r t h e E p o l a r i z a t i o n s h o u l d b e e x p e c t e d a t t h e i n t e r f a c e o f a d e e p v e r t i c a l f a u l t , a n d t h a t t h e r a t i o c o u l d b e a s l a r g e a s 1 f o r f r e q u e n c i e s i n t h e n e i g h b o r h o o d o f 0.3 c y c l e s / s e c . I f o n e a p p r o x i m a t e s t h e e a r t h - s e a i n t e r f a c e b y a v e r t i c a l f a u l t , t h e r a t i o s h o u l d b e c o n s i d e r a b l y l e s s s i n c e t h e d e p t h o f t h e s e a i s r e l a t i v e l y s h a l l o w c o m p a r e d w i t h a s k i n d e p t h o n d r y l a n d . T h e m o d e l r e s u l t s i n d i c a t e t h a t t h e m a x i m u m v a l u e o f t h e r a t i o s h o u l d b e m u c h s m a l l e r f o r s h a l l o w f a u l t s t h a n f o r d e e p f a u l t s . ' F u r t h e r m o r e , t h e r e a l e a r t h - s e a i n t e r f a c e i s a s l o p i n g i n t e r f a c e r a t h e r t h a n a v e r t i c a l i n t e r f a c e , a n d t h i s w o u l d r e s u l t i n a s t i l l s m a l l e r r a t i o . T h e m o d e l m e a s u r e m e n t s f o r t h e c y l i n d e r a l s o i n d i c a t e t h a t t h e c u r v a t u r e o f t h e c o a s t l i n e w o u l d a f f e c t t h e f i e l d s . •  I n t h e n e x t s e c t i o n ( c ) , t y p i c a l e a r t h - s e a - i n t e r f a c e s a r e s t u d i e d u s i n g m a c h i n e d g r a p h i t e w e d g e s a n d t r u n c a t e d c o n e s t o r e p r e s r - e n t t h e s h e l v i n g o c e a n f l o o r . M e a s u r e m e n t s r e p o r t e d b y L a m b e r t a n d C a n e r (1965) i n d i c a t e a m p l i t u d e r a t i o s , , H z / H y , f o r t h e w e s t c o a s t o f V a n c o u v e r I s l a n d t o b e a p p r o x i m a t e l y 0.3 f o r f r e q u e n c i e s i n t h e n e i g h b o r h o o d o f 0.01 c y c l e s / s e c - . I n t h e i r w o r k , t h e ' r e c o r d i n g s t a t i o n , a l t h o u g h l o c a t e d a t t h e c o a s t , w a s a c o n s i d e r a b l e d i s t a n c e f r o m t h e c o n t i n e n t a l s h e l f , . 128 a n d h e n c e w e l l r e m o v e d f r o m t h e s t e e p e r e a r t h - s e a i n t e r f a c e . O n e c o u l d e x p e c t a s o m e w h a t l a r g e r r a t i o n e a r e r t h e c o n t i n - e n t a l s h e l f . • T h e m o d e l r e s u l t s i n d i c a t e a l s o t h a t t h e p o l a r i z a t i o n o f t h e f i e l d w i t h r e s p e c t t o t h e d i s c o n t i n u i t y p l a y s a n i m p o r t a n t r o l e i n d e t e r m i n i n g t h e b e h a v i o r o f t h e e l e c t r o m a g - n e t i c v a r i a t i o n s . • O f i n t e r e s t i s t h e f a c t t h a t t h e m o d e l r e s u l t s i n d i c a t e t h a t H y i s n o t c o n s t a n t f o r t h e p o l a r i z a t i o n o f t h e e l e c t r i c f i e l d p a r a l l e l t o t h e f a u l t . ' W e a v e r (1962), i n h i s a n a l y t i c a l t r e a t m e n t o f t h e v e r t i c a l f a u l t p r o b l e m , m a d e t h e ' a s s u m p t i o n t h a t H w a s c o n s t a n t . T h e m a x i m u m v a l u e h e o b t a i n e d f o r t h e r a t i o H v / H T _ w a s 2.6L a t t h e i n t e r f a c e . . z y A l t h o u g h h i s v a l u e i s s o m e w h a t h i g h e r t h a n t h e m o d e l m e a s u r e - m e n t s i n d i c a t e , t h e n a t u r e o f h i s c o m p u t e d c u r v e i s v e r y s i m i l a r t o t h e r e s u l t s o b t a i n e d f r o m t h e m o d e l e x p e r i m e n t . • I n g e n e r a l , t h e b e h a v i o r o f t h e e l e c t r i c f i e l d i s f o u n d t o b e v e r y s e n s i t i v e t o c h a n g e s i n s h a l l o w s t r u c t u r e f o r t h e H p o l a r i z a t i o n a n d r a t h e r i n s e n s i t i v e t o c h a n g e s i n s h a l l o w s t r u c t u r e f o r t h e E p o l a r i z a t i o n . > T h e b e h a v i o r o f t h e m a g - n e t i c f i e l d s , o n t h e o t h e r h a n d ? i s m o r e s e n s i t i v e t o d i s - c o n t i n u i t i e s f o r t h e E p o l a r i z a t i o n t h a n f o r t h e H p o l a r i z - a t i o n . ( c ) - C o a s t l i n e S t r u e t u r e s A m o n g t h e p r o b l e m s o f c o n t i n u i n g , i n t e r e s t i n g e o p h y s i c s i s t h e c o a s t l i n e a n o m a l y . ' E x p e r i m e n t a l r e s u l t s - ( L a m b e r t a n d C a n e r 1965) i n d i c a t e t h a t t h e e n h a n c e m e n t o f t h e v e r t i c a l 129 m a g n e t i c f i e l d c o m p o n e n t n e a r a n e a r t h - s e a i n t e r f a c e c a n n o t b e a c c o u n t e d f o r o n t h e b a s i s o f t h e c o n d u c t i n g s e a a l o n e . L a m b e r t a n d C a n e r (1965) s u g g e s t t h a t t h e o b s e r v e d e n h a n c e - m e n t n e a r t h e c o a s t l i n e f o r t h e l o w e r f r e q u e n c i e s i s n o t d u e t o t h e e a r t h - s e a - i n t e r f a c e b u t r a t h e r t o i n h o m o g e n e i t i e s a t m u c h g r e a t e r d e p t h s , n a m e l y , t o t h e u p w e l l i n g o f a h i g h l y c o n d u c t i n g z o n e w i t h i n t h e m a n t l e o n t h e s e a s i d e o f t h e i n t e r f a c e . • S c h m u c k e r (196k) h a s d e s c r i b e d t h e C a l i f o r n i a c o a s t a l a n o m a l y a s b e i n g d u e t o t h e e d g e e f f e c t o f t h e P a c i f i c O c e a n a n d i t s e f f e c t " u p o n i n d u c t i o n w i t h i n t h e h i g h l y c o n d u c t i n g r e g i o n s o f t h e u p p e r m a n t l e . 1 H e a t t r i b u t e s s o m e i n l a n d a n o m a l i e s t o i n h o m o g e n e i t i e s i n t h e d e e p c o n - d u c t i v i t y s t r u c t u r e s , s o m e t o t h e d i f f e r e n c e s i n c o n d u c t i v - i t i e s i n t h e s u r f a c e l a y e r s , a n d s o m e - t o c o m b i n a t i o n s o f s u r f a c e , i n t e r m e d i a t e , a n d d e e p l y i n g c o n d u c t i v i t y i n h o m o - g e n e i t i e s ' . ' F o r t h e d e e p l y i n g i n h o m o g e n e i t i e s h e c o n s i d e r s a s t e p m o d e l r e p r e s e n t i n g t h e u p w e l l i n g o f a h i g h l y c o n d u c t i n g z o n e w i t h i n t h e m a n t l e . T h i s , s e c t i o n d e a l s w i t h a m o d e l s t u d y o f e a r t h - s e a i n t e r f a c e s - ( s l o p i n g i n t e r f a c e s ) , s c a l e d m a n t l e s t r u c t u r e s , a n d c o m b i n a t i o n s o f t h e s e . - A s w a s d i s c u s s e d e a r l i e r , t o m i n i m i z e t h e e f f e c t s o f t h e f l o o r a n d e a r t h b e l o w , t h e b o t t o m o f t h e t a n k w a s l i n e d w i t h a 5 - c m l a y e r o f g r a p h i t e i n t h e f o r m o f m a c h i n e d r o d s o f s q u a r e c r o s s s e c t i o n . - I n t h e s a m e m a n n e r , i t w a s f o u n d t h a t t h e e d g e e f f e c t s c o u l d b e r e d u c e d b y l i n i n g t h e v e r t i c a l e n d s o f t h e t a n k ( p e r p e n d i c u l a r t o t h e e l e c t r i c f i e l d o f t h e s o u r c e ) w i t h g r a p h i t e t o f o r m a g r a p h i t e 130 w a l l . T h i s h a s t h e e f f e c t o f m a k i n g t h e t a n k s e e m l a r g e r i n e x t e n t . T h e e l e c t r i c f i e l d i n t h e x - d i r e c t i o n i s i u n i f o r m n o w r i g h t t o t h e g r a p h i t e w a l l , w h e r e a s w i t h o u t i t t h e e l e c t r i c f i e l d b e g i n s t o d e c r e a s e 30 c m f r o m t h e e d g e a n d f a l l s o f f r a p i d l y a s i t i s a p p r o a c h e d . • I n t h e e a r l i e r w o r k t h e t a n k w a s l o c a t e d s y m m e t r i c - a l l y b e l o w t h e s o u r c e ( s h e e t c u r r e n t ) w i t h t h e l o n g d i m e n s i o n p a r a l l e l t o t h e e l e c t r i c f i e l d o f t h e s o u r c e . • F o r t h i s p o s i t i o n t h e v e r t i c a l m a g n e t i c f i e l d f o r t r a v e r s e s i n t h e y - d i r e c t i o n h a d a m i n i m u m v a l u e a t p o i n t s h a l f w a y a c r o s s t h e t a n k , i n c r e a s i n g s y m m e t r i c a l l y o n e i t h e r s i d e o f t h e c e n t r a l l i n e . I n p a s s i n g t h r o u g h t h i s m i n i m u m t h e p h a s e a n g l e o f t h e v e r t i c a l m a g n e t i c f i e l d c h a n g e s b y 180°. T h i s p h a s e c h a n g e i s t r o u b l e s o m e w h e n m a k i n g m e a s u r e m e n t s f o r n o n - s y m m e t r i c a l s t r u c t u r e s s u c h a s w e d g e s o r l a r g e s t r u c t u r e s l o c a t e d n e a r t h e c e n t e r o f t h e t a n k , a n d i t i s d e s i r a b l e n o t t o l o c a t e t h e m o d e l s t r u c t u r e s t o o n e a r t h i s p h a s e r e v e r s a l r e g i o n . I n o r d e r t o h a v e a p p r e c i a b l e s p a c e f o r s o m e o f t h e l a r g e s t r u c t - u r e s a n d a t t h e s a m e t i m e a v o i d i n g t h e p h a s e r e v e r s a l r e g i o n , t h e t a n k w a s t u r n e d w i t h t h e l o n g d i m e n s i o n p e r p e n d i c u l a r t o t h e e l e c t r i c f i e l d , a n d s e t o f f c e n t e r w i t h r e s p e c t t o t h e s o u r c e s o t h a t t h e m i n i m u m i n t h e v e r t i c a l m a g n e t i c f i e l d a p p e a r e d n e a r o n e e n d o f t h e t a n k . - I n t h i s w a y a r e g i o n o f t h e t a n k a b o u t 2 m b y 1.7 m w a s a v a i l a b l e f o r m a k i n g m e a s u r e ^ - m e n t s . G r a p h i t e s t r u c t u r e s i n t h e f o r m o f w e d g e s a n d a t r u n - c a t e d c o n e p l a c e d i n t h e s a l t s o l u t i o n w e r e u s e d t o r e p r e s e n t 131 o c e a n s . • L a r g e g r a p h i t e b l o c k s p l a c e d a t t h e a p p r o p r i a t e d e p t h i n t h e t a n k w e r e u s e d t o r e p r e s e n t t h e u p w e l l i n g o f h i g h l y c o n d u c t i n g z o n e s w i t h i n t h e e a r t h ' s m a n t l e . T h e s c a l i n g f a c t o r s i n v o l v e d s a t i s f y e q u a t i o n (77)• T h e c o n d u c t i v i t i e s o f t h e m o d e l s t r u c t u r e s f o r a l l m e a s u r e - m e n t s a r e t a k e n t o b e 1 C r g r e a t e r t h a n t h o s e f o r t h e g e o - p h y s i c a l p r o b l e m . T w o s e t s o f s c a l i n g f a c t o r s f o r t h e l i n e a r d i m e n s i o n s a n d f r e q u e n c y w e r e u s e d . ' F i g u r e s k9 t o 5k? d e a l - i n g w i t h t h e e a r t h - s e a i n t e r f a c e , i n v o l v e s c a l i n g f a c t o r s d / d ' = 1/10^ a n d f / f = 10̂ /1 . T h e m o d e l f r e q u e n c i e s u s e d , LL LL 0 "3 f = 3 x 10 , 1 0 , 3 x 1 0 ° , a n d 10J c y c l e s / s e c , c o r r e s p o n d t o f r e q u e n c i e s o f 0.3, 0.1, 0.03, a n d 0.01 c y c l e s / s e c i n t h e g e o p h y s i c a l p r o b l e m . • F i g u r e s 55 t o 6 , § , d e a l i n g w i t h t h e c o a s t l i n e - m a n t l e p r o b l e m , i n v o l v e s c a l i n g f a c t o r s d / d ' = 1/(5 x 10^) a n d f / f = 1 oVo .O k . F o r t h i s c a s e t h e m o d e l f r e q u e n c i e s u s e d c o r r e s p o n d t o f r e q u e n c i e s o f 0.012, 0.00k, 0.0012, a n d 0.000L c y c l e s / s e c i n t h e g e o p h y s i c a l p r o b l e m . • F i g u r e s k9 t o 51 d e a l w i t h a n i n v e r t e d t r u n c a t e d g r a p h i t e c o n e w i t h i t s b a s e a t t h e s u r f a c e o f t h e s a l t s o l u t - i o n . - T h e d i a m e t e r o f t h e b a s e o f t h e c o n e w a s 30A c m , a n d t h e d i a m e t e r o f t h e t r u n c a t e d e n d 5 c m . T h e t h i c k n e s s o f t h e g r a p h i t e f r o m t h e b a s e t o t h e t r u n c a t e d e n d w a s 2.5 c m . T h e s t r u c t u r e r e p r e s e n t s a c i r c u l a r s e a 30.h k m i n d i a m e t e r , s l o p i n g f r o m z e r o d e p t h a t t h e s h o r e t o 2.5 k m a t a d i s t a n c e o f 12.7 k m f r o m t h e s h o r e . • M e a s u r e m e n t s w e r e m a d e b o t h f o r t r a v e r s e s p a r a l l e l a n d p e r p e n d i c u l a r t o t h e e l e c t r i c f i e l d 132 LL LL o f t h e s o u r c e . T h e f r e q u e n c i e s u s e d w e r e f = 3 x 10 , 10 , 3 x 10^ a n d 10-̂  c y c l e s / s e c . - F i g u r e s k 9 ( a ) a n d (b) s h o w t h e Li. LL a m p l i t u d e s a n d p h a s e a n g l e s f o r 3 x 1 0 a n d 10 c y c l e s / s e c r e s p e c t i v e l y , w h i l e F i g s . 50 (a) a n d ( b ) s h o w t h e m e a s u r e m e n t s f o r 3 x 10^ a n d 10^ c y c l e s / s e c r e s p e c t i v e l y . B o t h F i g . L9 a n d F i g . 50 d e a l w i t h t r a v e r s e s p e r p e n d i c u l a r t o t h e e l e c t r i c f i e l d - o f t h e s o u r c e . - I t i s a p p a r e n t f r o m F i g s . k9 a n d 50 t h a t t h e n o n - u n i f o r m i t y o f t h e s o u r c e f i e l d d o e s a f f e c t t h e b e h a v i o r o f t h e f i e l d c o m p o n e n t s i n t h e n e i g h b o r h o o d o f t h e e d g e o f t h e c o n e . - H , f o r e x a m p l e , s h o w s a d e c r e a s e o n t h e s i d e o f d e c r e a s i n g h o r i z o n t a l f i e l d a n d a n e n h a n c e m e n t o n t h e s i d e o f t h e i n c r e a s i n g f i e l d . T h e a s y m m e t r y o f t h e p h a s e c u r v e s i s - a l s o a t t r i b u t e d t o t h e n o n - u n i f o r m i t y o f t h e s o u r c e f i e l d . - T o f u r t h e r e x a m i n e t h e e f f e c t o f n o n - u n i f o r m i t y o f t h e i n d u c i n g f i e l d , t h e t r u n c a t e d c o n e w a s p l a c e d a t v a r i o u s l o c a t i o n s i n t h e t a n k . < W h e n t h e c o n e w a s l o c a t e d a t t h e c e n t r a l p o s i t i o n , w i t h t h e h o r i z o n t a l i n d u c i n g f i e l d d e c r e a s - i n g o n e i t h e r s i d e , H- s h o w e d a d e c r e a s e o n b o t h s i d e s n e a r t h e c o n e . A l t h o u g h t h e g e n e r a l s h a p e o f R " z , o n t h e o t h e r h a n d , d i d n o t c h a n g e w i t h p o s i t i o n , t h e e n h a n c e m e n t n e a r t h e e d g e s o f t h e c o n e d i d d e p e n d s o m e w h a t o n p o s i t i o n . T h i s i n d i c a t e s t h a t t h e n a t u r e o f t h e s o u r c e i s i m p o r t a n t . ' A s s e e n f r o m F i g s . L9 a n d 50,' E „ u n d e r g o e s a l a r g e c h a n g e n e a r t h e i n t e r f a c e f o r e a c h f r e q u e n c y s t u d i e d . T h e c h a n g e i n p h a s e a n g l e x l r f o r t h e e l e c t r i c f i e l d i n c r e a s e s a s t h e f r e q u e n c y d e c r e a s e s w h e r e a s c h a n g e s i n b o t h $ T r a n d 0„ d e c r e a s e a s t h e y z f r e q u e n c y d e c r e a s e s . • O n t h e b a s i s o f t h e r e s u l t s g i v e n i n 133 F i g u r e k9. • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a n i n v e r t e d t r u n c a t e d c o n e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3-X-10"1", ( b ) 10^ c y c l e s / s e c . 13^ -p-,—|—i—|—.—|—i |—i — i — r - 1 — | — i — i — i — j i—f— 20 4 0 6 0 8 0 100 w 2 0 4 0 60 80 100 Y c m F i g u r e 50. • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a n i n v e r t e d t r u n c a t e d c o n e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x10^, ( b ) 10^ c y c l e s / s e c . 135 F i g s . k9 a n d 50 i t i s a p p a r e n t t h a t a s f a r a s m a g n e t i c f i e l d a m p l i t u d e s a r e c o n c e r n e d , t h e p r e s e n c e o f t h e t r u n c a t e d g r a p h i t e c o n e i s n o t o b s e r v a b l e u n t i l p o i n t s v e r y n e a r t h e i n t e r f a c e a r e r e a c h e d , a n d t h a t f o r f r e q u e n c i e s a s l o w a s 103 c y c l e s / s e c t h e c o n e i s h a r d l y o b s e r v a b l e a t a l l . E a n d i r „ , h o w e v e r , u n d e r g o l a r g e c h a n g e s a t l o w f r e q u e n c i e s . T h e r e s u l t s i n d i c a t e t h a t a c i r c u l a r s e a 30.k k m i n d i a m e t e r a n d s l o p i n g t o a d e p t h o f 2.5 k m w o u l d h a v e v e r y l i t t l e e f f e c t o n H a n d H f o r e l e c t r o m a g n e t i c f i e l d f r e q u e n c i e s i n t h e n e i g h - b o r h o o d o f 0.01 c y c l e s / s e c , a l t h o u g h i t - w o u l d h a v e a c o n s i d e r - a b l e e f f e c t f o r f r e q u e n c i e s o f 0.1 c y c l e s / s e c a n d h i g h e r . F i g u r e 51 g i v e s t h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s p a r a l l e l t o t h e e l e c t r i c f i e l d o f t h e s o u r c e . F i g u r e LL 51(a) g i v e s t h e r e s u l t s f o r 3 x 10 c y c l e s / s e c a n d F i g . 51(b) f o r 10J c y c l e s / s e c . M e a s u r e m e n t s w e r e a l s o m a d e f o r 10 a n d 3 x 103 c y c l e s / s e c b u t a r e n o t s h o w n h e r e . H s h o w s a s m a l l 7 LL i n c r e a s e o v e r t h e c o n e f o r a f r e q u e n c y o f 3 x 10 c y c l e s / s e c , b u t n o c h a n g e f o r a f r e q u e n c y o f 103 c y c l e s / s e c . H _ u n d e r - g o e s a g r a d u a l i n c r e a s e a s t h e i n t e r f a c e i s a p p r o a c h e d , t h e n d e c r e a s e s o v e r t h e c o n e . • F o r a f r e q u e n c y o f 103 c y c l e s / s e c , H i s h a r d l y a f f e c t e d b y t h e p r e s e n c e o f t h e c o n e . T h e e n - Zi. h a n c e m e n t o f E r i g h t a t t h e i n t e r f a c e i n c r e a s e s a s t h e .A. f r e q u e n c y d e c r e a s e s . - B o t h QZ a n d ty , i n t h e n e i g h b o r h o o d o f t h e t r u n c a t e d c o n e , a r e h i g h l y f r e q u e n c y d e p e n d e n t . • T h e c h a n g e i n <t>_ d e c r e a s e s w i t h d e c r e a s e i n f r e q u e n c y w h i l e t h e c h a n g e i n ty i n c r e a s e s w i t h a d e c r e a s e i n f r e q u e n c y . T h e r e s u l t s o f F i g . 51 a s w e l l a s t h o s e o f F i g s . k9 a n d 50 136 10 'O X CD o X 00 OJ ' E <*> 'o X 00 '> LU 16 H V 200 i—1—i—1—i—•—r 8 0 4 0 0 - 40 - i — I — i — i — r 20 4 0 6 0 80 I - M - E - ^UaJS* (a) X c m i I i — | — i — | — i | — i — r — 20 4 0 6 0 80 100 rji M 'Q 2 x S I xT" N 'O x 16 CD ro 5 - 1 2 e o X O *"">< LU 8 E »» H, i—<—r • i •—r 2 0 0 - 160 120 i — | — — i — r * H — I — i — I — i 20 4 0 6 0 8 0 100 E ^Uaf (b) X c m | , | • I | - 2 0 4 0 6 0 8 0 100 F i g u r e 51 »• •' The a m p l i t u d e s and phase a n g l e s f o r t r a v e r s e s o v e r an I n v e r t e d t r u n c a t e d cone f o r t h e H p o l a r i z a t i o n k 3 a n d f r e q u e n c i e s (a) 3x10 (b) 1 0 J c y c l e s / s e c . 137 indicate that the v e r t i c a l magnetic f i e l d i s l i t t l e affected by the presence of this model sea f o r frequencies of 0.01 cycles/sec, but i s considerably affected at higher frequencies. The horizontal magnetic f i e l d on the other hand, f o r traverses along a diameter i n the d i r e c t i o n of the e l e c t r i c f i e l d , i s e s s e n t i a l l y unaffected f o r the frequency range studied. • The horizontal e l e c t r i c f i e l d i s much affected by the presence of the model sea at a l l frequencies. - To further study the e f f e c t of the earth-sea i n t e r - face a graphite wedge sloping from 0.1 cm to 2.5 cm i n a horizontal distance of 50 cm with a constant thickness beyond this was used to represent the shelving ocean f l o o r . ' The • width of the wedge was 60 cm, - This' wedge, with the plane surface at the surface of the s a l t solution and the sloping surface submerged, was a scaled model of the ocean on the west coast of Vancouver Island (Toflno area). The sloping part of the wedge represented the continental slope which begins about 25 km from the shore and f a l l s from a depth of abouti.*0.1 km to 2,5 km i n a horizontal distance of about 50 km. The depth of the ocean beyond the continental slope i s about 2.5 km. The graphite wedge, which i s a model of the ocean, did not include the f i r s t 25 km of the ocean (con- t i n e n t a l s h e l f ) . A model of the continental shelf would require a very thin sheet of graphite (representing the very shallow water) extending 25 cm beyond the sharp edge of the wedge. 138 M e a s u r e m e n t s o f t h e n a t u r a l m a g n e t i c f i e l d v a r i a t i o n s a t T o f i n o h a v e b e e n m a d e b y L a m b e r t a n d C a n e r 01 965)• T o f i n o i s n e a r t h e c o a s t a n d h e n c e t h e d i s t a n c e t o t h e c o n t i n e n t a l s l o p e i s a b o u t 25 k m . T h e i r m e a s u r e m e n t s i n d i c a t e d a c o n s i d - e r a b l e e n h a n c e m e n t o f t h e v e r t i c a l t o h o r i z o n t a l m a g n e t i c f i e l d r a t i o a t T o f i n o a b o v e t h e v a l u e f u r t h e r i n l a n d . M o d e l m e a s u r e m e n t s c a n a i d i n s t u d y i n g t h e e f f e c t o f a s e a - l a n d i n t e r f a c e . F i g u r e s 52, 53 a n d 5^ d e a l w i t h m e a s u r e m e n t s m a d e f o r t r a v e r s e s o v e r t h e w e d g e . I n t h e g e o p h y s i c a l p r o b l e m t h i s c o r r e s p o n d s t o m e a s u r e m e n t s m a d e o n l a n d a n d s e a a l o n g a l i n e p e r p e n d i c u l a r t o t h e c o a s t l i n e . F i g u r e s 52 a n d 53 d e a l w i t h t h e E p o l a r i z a t i o n ( w e d g e e d g e p a r a l l e l t o t h e e l e c t r i c f i e l d o f t h e s o u r c e ) , a n d F i g . 5 k w i t h t h e H p o l a r i z a t i o n ( w e d g e e d g e p e r p e n d i c u l a r t o t h e e l e c t r i c f i e l d o f t h e s o u r c e ) . F i g u r e s 5 2 ( a ) , ( b ) , 5 3 ( a ) , a n d ( b ) g i v e t h e r e s u l t s r e s p e c t - i v e l y f o r f r e q u e n c i e s 3 x 1 0 , 10 , 3 x 10 J c y c l e s / s e c . I t i s a p p a r e n t t h a t t h e e l e c t r i c a n d m a g n e t i c f i e l d s a r e a f f e c t e d LL b y t h e g r a p h i t e w e d g e . F o r a f r e q u e n c y o f 3 x 10 c y c l e s / s e c , H y a n d H z s h o w c o n s i d e r a b l e c h a n g e n e a r t h e w e d g e w i t h H y d e c r e a s i n g a s t h e w e d g e i s a p p r o a c h e d a n d t h e n i n c r e a s i n g s l i g h t l y o v e r t h e w e d g e . A s w a s p o i n t e d o u t e a r l i e r ( i n t h e d i s c u s s i o n o f F i g s . k 9 a n d 50) t h e b e h a v i o r o f H n e a r t h e w e d g e e d g e i s s o m e w h a t d e p e n d e n t o n t h e n o n - u n i f o r m i t y o f t h e s o u r c e f i e l d . H z i n c r e a s e s s h a r p l y a s t h e w e d g e i s a p p r o a c h e d t h e n f a l l s r a p i d l y t o v e r y s m a l l v a l u e s o v e r t h e w e d g e . F o r l o c a t i o n s b e y o n d 10 c m f r o m t h e w e d g e e d g e H_ _ a n d H _ s h o w n o y z 139 - i — T — | — i — | — i — | — i — | 1 — | — i — i — i — | — i — | — i — r 20 4 0 6 0 8 0 100 v y 20 4 0 60 80 100 Y cm -i—•—i— 1 —i—•—i—•—i—• — • — i — ' — i — 1 — i — 1 — r 20 4 0 6 0 80 100 20 4 0 6 0 8 0 100 Y c m F i g u r e 52- T h e . a m p l i t u d e s a n d p h a s e - a n g l e s f o r t r a v e r s e s o v e r a w e d g e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s U'. LL h ^. Ca)' 3x10 , ( b ) 1 (T c y c l e s / s e c . 1U0 i—1—i—1—i—'—i—•—i—1 —•—i—1—i—1—i—'—i—1—r 20 4 0 60 80 100 20 4 0 6 0 8 0 100 Y cm -I 1 1 1 1 1 1 1 1 T 1 20 4 0 6 0 8 0 100 / / % "~ ~ Ls' / ~ ~~ - Ty H (b) Y c m 20 4 0 60 8 0 100 F i g u r e 53- " T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a w e d g e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x10 3 , ( h ) 10 3 c y c l e s / s e c . 1 k1 20 4 0 60 8 0 I00 20 4 0 60 80 I00 X cm F i g u r e 5*+. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s o v e r a w e d g e f o r t h e H p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3*1o\ (b) 10 3 c y c l e s / s e c . -|!+2 appreciable change due to the wedge. However, the phase angle 0 i s affected by the wedge at locations as f a r as 30 cm from the wedge edge. • For the H p o l a r i z a t i o n , as shown i n F i g . 5k5 Hy. again increases over the wedge. For a frequency of 3 x 10 cycles/sec H y i s affected by the wedge at locations as f a r as 30 cm from the edge. H z begins to increase at a distance s l i g h t l y greater than 10 cm from the edge, reaches a maximum about 1 cm beyond the edge, and then f a l l s to very small values at a distance of about 15 cm beyond the edge. E begins to r i s e r a p i d l y 20 cm from the edge, reaches a maximum value d i r e c t l y over the edge of the wedge, and then f a l l s to very small values 1 cm beyond the edge. The phase angles <* and ^ x are e s s e n t i a l l y unaffected by the wedge as the edge i s approached, they then change rapidly r i g h t at the', edge. ; As can be seen from F i g s . 52 and 53 the magnetic f i e l d components are l e s s affected by the wedge as the frequency decreases. At 10̂  cycles/sec, which corresponds to 0.01 cycles/sec i n the geophysical scale, the wedge has only a small e f f e c t on H or H_. I t i s .apparent from._the—results Tor this model sea-land interface that the. enhancement of the amplitude r a t i o s of the v e r t i c a l to horizontal magnetic f i e l d observed i n coastal interface alone. ycles/seC' the regions cannot be attributed to the earth-seal For frequencies i n the neighborhood of 0.01 c sea should have only a small e f f e c t f o r a l l locations. At higher frequencies the sea-earth interface should have a greater e f f e c t on H y and H z but only f o r locations near the continental slope. • In most cases locations on land would be 1k3 t o o - d i s t a n t f r o m t h e c o n t i n e n t a l s l o p e f o r H „ a n d H „ t o b e y • z m u c h a f f e c t e d . • S i g n i f i c a n t p h a s e s h i f t s i n c o u l d h o w e v e r b e e x p e c t e d a t r e l a t i v e l y l a r g e d i s t a n c e s f r o m t h e c o n t i n e n t a l s l o p e ( F i g . 52 a ) 0 T o s t u d y t h e s e a - c o a s t p r o b l e m f u r t h e r t h e e f f e c t o f t h e u p w e l l i n g o f a c o n d u c t i n g z o n e w i t h i n t h e m a n t l e u n d e r t h e o c e a n w a s m o d e l l e d „ - T h e u p w e l l i n g o f h i g h l y c o n d u c t i n g u p p e r m a n t l e m a t e r i a l s h o u l d b e c o m e p a r t i c u l a r l y i m p o r t a n t f o r t h e l o w e r f r e q u e n c i e s w h i c h p e n e t r a t e t h e o c e a n a n d t h e e a r t h ' s c r u s t . - I t i s w e l l k n o w n t h a t t h e e a r t h ' s c r u s t i s m u c h t h i n n e r u n d e r t h e o c e a n t h a n u n d e r t h e c o n t i n e n t s . • H i g h l y c o n d u c t i n g z o n e s i n t h e m a n t l e m a y h e n c e b e m u c h n e a r e r t h e e a r t h 1 S ' s u r f a c e f o r r e g i o n s u n d e r t h e o c e a n s a n d n e a r c o a s t - l i n e s . • S c h m u c k e r (196 K) h a s d e s c r i b e d c o a s t a l a n o m a l i e s i n t h e m a g n e t i c f i e l d v a r i a t i o n s a s b e i n g d u e t o t h e e d g e e f f e c t o f t h e s e a a n d t h e u p w e l l i n g o f c o n d u c t i n g m a t e r i a l i n t h e m a n t l e i n t h e f o r m o f a s t e p . I n t h e p r e s e n t s t u d y t h e u p w e l l i n g m a n t l e w a s m o d e l l e d u s i n g l a r g e g r a p h i t e b l o c k s a t a s u i t a b l e d i s t a n c e b e l o w t h e s u r f a c e o f t h e s a l t s o l u t i o n . • A g r a p h i t e w e d g e r e p r e s e n t i n g t h e s e a w a s s u s p e n d e d a t t h e s u r f a c e . ; T h e g r a p h i t e w e d g e w a s p l a c e d i n v a r i o u s p o s i t i o n s w i t h r e s p e c t t o t h e g r a p h i t e s t e p b e l o w . S i n c e i n t h i s c o a s t l i n e - m a n t l e p r o b l e m i t i s o f i n t ^ e r e s t t o s t u d y t h e b e h a v i o r o f f r e q u e n c i e s l o w e r t h a n t h o s e d i s c u s s e d i n t h e w e d g e m e a s u r e m e n t s , t h e s c a l i n g f a e t o r s w e r e m o d i f i e d . ; T h e l i n e a r d i m e n s i o n s , a n d t h e f r e q u e n c y s e a l i n g f a c t o r s u s e d w e r e r e s p e c t i v e l y d / d ; f • f 4 ^ ^ ^ ~ 4 # ^ - - a n d 1¥+ f / f 1 = 10^/O.O k . • Wi th - t h e s e f a c t o r s t he wedge u s ed h e r e t o r e p r e s e n t t he sea was one f i f t h t he s i z e o f t he one u s ed e a r l i e r . - The wedge s l o p e d t o 0.5 cm t h i c k n e s s i n a h o r i z o n t a l d i s t a n c e o f 10 cm and had a c o n s t a n t t h i c k n e s s o f 0.5 cm beyond. - The 0.5 cm t h i c k n e s s o f g r a p h i t e r e p r e s e n t e d t he 2.5 km ocean d e p t h . L a r g e g r a p h i t e b l o c k s (1 f o o t cubes ) were s t a c k e d t o r e p r e s e n t c o n d u c t i n g zones w i t h i n the m a n t l e . The t op f l a t s u r f a c e o f t he a s semb led g r a p h i t e b l o c k was 3 cm be l ow t he s u r f a c e o f the s a l t s o l u t i o n . T h i s d e p t h r e p r e s e n t s a d e p t h o f 15 km i n the g e o p h y s i c a l p r o b l e m . The 1 d e p t h o f w a t e r between t he uppe r s u r f a c e o f t he c o n d u c t i n g b l o c k and t he l o w e r s u r f a c e o f t he f l a t p a r t o f t he wedge was 2.5 cm. The s t e p - h e i g h t was 61 cm. The model f r e - k k o o quene-ies u s e d 3 x 10 , 10 , 3 x 10 , and TO-* c y c l e s / s e e c o r r e s p o n d t o f r e q u e n c i e s o f 0.012, 0.00k, 0.0012, and 0.000*+ c y c l e s / s e c i n t he g e o p h y s i c a l p r o b l e m . The s h e e t c u r r e n t was k e p t a t t he o r i g i n a l p o s i t i o n , 1.25 m above t he . s u r f a c e o f the s a l t s o l u t i o n . F i g u r e 55(a) shows t he a m p l i t u d e s and phase a n g l e s f o r t r a v e r s e s o v e r the wedge w i t h t he wedge and b l o c k edges p a r a l l e l t o t he e l e c t r i c f i e l d o f t he s o u r c e (E p o l a r i z a t i o n ) k f o r a mode l f r e q u e n c y o f 3 x 10 c y c l e s / s e c . The edge o f t he wedge ex tend s 10 cm beyond t he b l o c k edge. I n t he g eo - p h y s i c a l p r ob l em t h i s wou ld c o r r e s p o n d t o t he ocean e x t e n d i n g 50 km beyond t he edge o f t he c o n d u c t i n g m a n t l e . •  F i g u r e 55(D) g i v e s t he r e s u l t s f o r the wedge (-1) and the b l o c k (2) s e p a r a t e l y . - I n compar i ng t he c u r v e s o f F i g . 55(a) w i t h t ho se i k 5 20 4 0 6 0 8 0 I00 20 4 0 6 0 8 0 I00 Y cm i—j—r—r—i—i—i—j—»—j—i • — i — i — i — j — i — r — B — i — > — j — 20 4 0 60 80 100 K / 20 4 0 60 80 100 F i g u r e -55. ; T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z a t i o n a n d f=3x1cA c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) - t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d Cb) t h e w e d g e a l o n e (-1), t h e b l o c k a l o n e (-2). 14-6 o f F i g . 5 5 ( b ) t h e r e l a t i v e i m p o r t a n c e o f t h e w e d g e a n d o f t h e b l o c k c a n b e a s s e s s e d - I t i s a p p a r e n t t h a t f o r a m o d e l f r e ? * LL q u e n c y o f 3 * 1 0 c y c l e s / s e c b o t h t h e w e d g e a n d t h e b l o c k s t r u c t u r e p l a y a n i m p o r t a n t r o l e i n i n f l u e n c i n g t h e b e h a v i o r o f t h e f i e l d c o m p o n e n t s a n d t h e p h a s e a n g l e s . T h i s w o u l d s u g g e s t t h a t f o r f r e q u e n c i e s i n t h e n e i g h b o r h o o d o f 0 . 0 1 ' c y c l e s / s e c b o t h t h e e a r t h - s e a i n t e r f a c e a n d t h e u p w e l l i n g m a n t l e c o n d u c t i n g z o n e s b e l o w h a v e a n i m p o r t a n t e f f e c t o n t h e e l e c t r o m a g n e t i c v a r i a t i o n s n e a r a c o a s t l i n e . • F i g u r e s 56 5 57? a n d 58 d e a l w i t h t h e s a m e p r o b l e m a s F i g . 55 e x c e p t t h a t t h e m o d e l f r e q u e n c i e s h a v e b e e n c h a n g e d LL O O t o 1 0 , 3 x 1 0 ~ % a n d 1 0 J c y c l e s / s e c r e s p e c t i v e l y . T h e s e f r e - q u e n c i e s c o r r e s p o n d t o h x , l O " ^ , 1 . 2 x 1 0 " ^ , a n d k- x 1 0 ~ L f c y c l e s / s e c r e s p e c t i v e l y i n t h e g e o p h y s i c a l p r o b l e m . W e s e e t h a t a t t h e l o w e r f r e q u e n c i e s H y a n d H z a r e n o t a f f e c t e d a p p r e c i a b l y b y t h e w e d g e b u t a r e s t i l l a f f e c t e d b y t h e b l o c k s t r u c t u r e . E o n . t h e o t h e r h a n d i s s t r o n g l y a f f e c t e d b y b o t h t h e w e d g e a n d t h e b l o c k , f o r a l l f r e q u e n c i e s . A l l t h e p h a s e a n g l e s a r e a f f e c t e d s o m e w h a t b y b o t h s t r u c t u r e s , ty u n d e r - g o e s a v e r y l a r g e r a p i d c h a n g e n e a r t h e w e d g e e d g e . T h e p h a s e a n g l e <p_ a l s o u n d e r g o e s l a r g e c h a n g e s a t t h e h i g h e r f r e q u e n c i e s a n d i s a f f e c t e d a t p o i n t s m u c h m o r e d i s t a n t f r o m , t h e s t r u c t u r e t h a n i s t h e c a s e f o r ty . F r o m t h e r e s u l t s o f F i g s . 55 t o 58 w e c a n c o n c l u d e t h a t f o r f r e q u e n c i e s i n t h e —2 n e i g h b o r h o o d o f 1 0 c y c l e s / s e c , b o t h t h e e a r t h - s e a i n t e r - f a c e a n d t h e u p w e l l i n g c o n d u c t i n g z o n e s h o u l d b e i m p o r t a n t i n d e t e r m i n i n g t h e b e h a v i o r o f t h e n a t u r a l l y o c c u r r i n g 1 L7 2 0 4 0 6 0 8 0 I00 2 0 4 0 6 0 8 0 I00 Y cm H- -I 1 1 1 1 1 1 1 T — ] — - T ' 1 1 1 J T j~ 2 0 4 0 6 0 8 0 100 K, 2 0 4 0 6 0 8 0 Y cm F i g u r e 56. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E LL p o l a r i z a t i o n a n d f=10 c y c l e s / s e c f o r t r a v e r s e s o v e r ' ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e ( 1 ) , t h e b l o c k a l o n e ( 2 ) . • 1k8 -I 1 1 1 1 1 1 1 1 T 2 0 4 0 6 0 8 0 I00 20 Y cm l — • — i — > — i — 1 — i — ' — r — - ~ 4 0 6 0 8 0 100 2 0 4'o 6 b 8b ibo v 2b " 4 0 ' 6 0 ' s'o' ido Y cm F i g u r e 57' T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z a t i o n a n d f=3x10-3 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e ( 1 ) , t h e b l o c k a l o n e ( 2 ) . 1k9 T — | 1 1 1 1 1 1 I 1 1 ' 1 1 1 1 1 1 1 1 1 1 20 4 0 60 80 100 „ 20 4 0 60 80 100 Y cm F i g u r e 58. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z a t i o n a n d f=103 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e (1), t h e b l o c k a l o n e (2). 1 5 0 m a g n e t i c v a r i a t i o n s , w h i l e f o r f r e q u e n c i e s n e a r k x 1 0 c y c l e s / s e c o n l y t h e c o n d u c t i n g z o n e s w i t h i n t h e m a n t l e s h o u l d h a v e a n i m p o r t a n t e f f e c t . T h e h o r i z o n t a l e l e c t r i c f i e l d s h o u l d b e a f f e c t e d b y b o t h t h e s e a a n d t h e u p w e l l i n g c o n d u c t i n g m a n t l e f o r t h e w h o l e f r e q u e n c y r a n g e s t u d i e d . • T o f u r t h e r e x a m i n e t h e f r e q u e n c y d e p e n d e n c e f o r t h e c o m b i n e d m o d e l s t r u c t u r e ( w e d g e a n d b l o c k ) i n > F i g s . 55 t o 58 , c o n s i d e r a g a i n F i g s . 5 5 ( a ) , 5 6 ( a ) , 5 7 ( a ) a n d 5 8 ( a ) . A t t h e h i g h e r f r e - q u e n c i e s b o t h H „ a n d H a r e c o n s i d e r a b l y a f f e c t e d b y t h e y 2 s t r u c t u r e o n l y f o r r e g i o n s n e a r t h e e d g e o f t h e b l o c k a n d w e d g e . H z i n c r e a s e s a s t h e w e d g e i s a p p r o a c h e d , r e a c h i n g a m a x i m u m d i r e c t l y o v e r t h e w e d g e e d g e , t h e n f a l l s t o v e r y s m a l l v a l u e s b e y o n d t h e b l o c k e d g e . - H b e g i n s t o r i s e a t t h e w e d g e e d g e , r e a c h i n g a m a x i m u m d i r e c t l y o v e r t h e b l o c k e d g e , t h e n d e c r e a s e s s l i g h t l y a n d b e c o m e s c o n s t a n t a t p o i n t s w e l l r e m o v e d f r o m t h e b l o c k e d g e . • B„ s h o w s - a g r a d u a l d e c r e a s e a s t h e w e d g e i s a p p r o a c h e d . • T h e p h a s e a n g l e s \Lr a n d 0 a r e n o t x y m u c h a f f e c t e d b y t h i s s t r u c t u r e b u t 0 , s h o w s l a r g e c h a n g e s Zs e v e n f o r p o i n t s w e l l r e m o v e d f r o m t h e e d g e o f t h e s t r u c t u r e . I t i s a p p a r e n t f r o m F i g s . 55 t o 58 t h a t a t t h e h i g h e r f r e r - q u e n c i e s t h e w e d g e h a s c o n s i d e r a b l e e f f e c t o n I L _ a n d H_ b u t y z a t t h e l o w e r f r e q u e n c i e s t h e b l o c k s t r u c t u r e b e c o m e s i n c r e a s - i n g l y i m p o r t a n t . - T o s t u d y h o w t h e r e l a t i v e p o s i t i o n s o f t h e w e d g e a n d t h e b l o c k a f f e c t t h e f i e l d c o m p o n e n t s t h e p o s i t i o n o f t h e w e d g e w a s c h a n g e d . ; T h e r e s u l t s i n F i g . - 59 a r e f o r t h e w e d g e e d g e d i r e c t l y a b o v e t h e b l o c k e d g e , w h i l e t h e r e s u l t s i n F i g . 6 0 151 20 4 0 60 80 I00 20 40 60 80 I00 Ycm F i g u r e 59v • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E . p o l a r i z a t i o n f o r t r a v e r s e s o v e r t h e w e d g e w i t h t h e w e d g e e d g e d i r e c t l y o v e r t h e b l o c k e d g e a n d f r e q u e n c i e s ( a ) 3 x 1 c \ ( b ) 1 c y c l e s / s e c . 152 are f o r the block edge extending 10 cm beyond the wedge edge. In the r e a l earth problem the structure f o r F i g . 59 would represent the s i t u a t i o n where the edge of the mantle conduct- ing zone \wsfisr d i r e c t l y below the shoreline, while F i g . 60 would represent the case where the conducting mantle-step occurred 50 km inland. Figure 59(a) deals with a model fre*> 1+ quency of 3 x 10 cycles/sec and F i g . 59(b) deals with a frequency of 10̂  cycles/sec. Measurements were also made f o r frequencies of 10 and 3 x 10J cycles/sec but are not shown. The changes i n the amplitudes and phase angles f o r these l a t t e r frequencies were intermediate to those results shown In F i g s . 59(a) and (b). Figure 59(a) shows that H z begins to increase at distances as g£eat as 20 cm from t h i §dg§ of the structure. For t h i geophysical problem, this i n d i e a t i s that fo r friqueaeies i n t h i neighborhood of 0.01 eyele§/§ee, H g begins to shew enhanced values 100 km from the shoreline, then f a l l s to very small values over the sea f o r regions beyond the continental slope. Figure 59(b) shows that f o r frequencies as . -Li low as H- x 10 i cycles/sec both Hy and H z are l i t t l e affected by the coastline-mantle structure u n t i l points d i r e c t l y over the structure are reached. In the geophysical problem H and H„ f o r the structure modelled i n F i g . 59(b), should be l i t t l e affected by the presence of the ocean and the upwelling con- ducting zones of the mantle f o r points on land near the coast. From* the results shown i n Fig.' 60 i t i s apparent that the coast e f f e c t f o r H z w i l l be observed much farth e r inland (as f a r as 1 50 km) f o r frequencies i n the neighborhood 153 20 4 0 60 80 I00 w 20 4 0 6 0 80 I00 cm 20 4 0 6 0 8 0 I00 20 4 0 6 0 8 0 DO Y c m F i g u r e 60. > The a m p l i t u d e s and phase a n g l e s f o r the E p o l a r i z a t i o n f o r t r a v e r s e s o v e r the wedge w i t h t he b l o c k edge e x t e n d i n g 10 cm beyond the wedge edge and f r e q u e n c i e s (a) 3x1 O^,' (b) <1()3 c y c l e s / s e c . i 5 k o f - 0 . 01 c y c l e s / s e c i f t he u p w e l l i n g m a n t l e c o n d u c t i n g zone ex tend s w e l l beyond t he s h o r e l i n e . A t l o w e r f r e q u e n c i e s i n t he n e i g h b o r h o o d o f h x 1 0 " ^ c y c l e s / s e c , H z wou ld b e g i n t o d e c r e a s e a t a p p r o x i m a t e l y 50 km f r o m the sho re ( d i r e c t l y o v e r t he m a n t l e stefc) and m a i n t a i n a c o n s t a n t v a l u e a t p o i n t s o v e r t h e sea beyond 50 km f r o m the s h o r e . * A t t h i s f r e q u e n c y t he sea has v e r y l i t t l e i n f l u e n c e , bu t the, u p w e l l i n g m a n t l e c o n - d u c t i n g zone i s i m p o r t a n t . A c ompa r i s on o f F i g s . 5 5 ( a ) , 5 9 ( a ) , ' a n d 60 (a ) shows t h a t 0 Z i s q u i t e s e n s i t i v e t o changes i n the r e l a t i v e p o s i t i o n s o f t he wedge and b l o c k . • I t i s e v i d e n t f r o m F i g s . 55 t o 60 t h a t t he r e l a t i v e p o s i t i o n s o f t h e c o n d u c t i n g s t e p and t he ocean c o a s t have an i m p o r t a n t e f f e c t on t he b e h a v i o r o f t he a m p l i t u d e s and phase a n g l e s o f t he components . F i g u r e s 61 , 62, and 63 d e a l w i t h t he H p o l a r i z a t i o n (wedge edge p a r a l l e l t o t he h o r i z o n t a l magne t i c f i e l d o f t he s o u r c e ) . F i g u r e 61 shows t he r e s u l t s f o r model f r e q u e n c i e s LL O 3 x 10 and 1 0 J c y c l e s / s e c f o r the wedge e x t e n d i n g 10 cm beyond t he b l o c k edge. EL undergoes a g r a d u a l i n c r e a s e f o r b o t h f r e q u e n c i e s as t he s t r u c t u r e i s app roached w h i l e EL shows z a s l i g h t enhancement n e a r t he wedge edge and t h e n d e c r e a s e s t o s m a l l e r v a l u e s o v e r t he wedge. ' E i n c r e a s e s v e r y r a p i d l y as the wedge i s app roached t h e n f a l l s e x t r e m e l y s h a r p l y t o v e r y s m a l l v a l u e s o v e r t he wedge. 1 The change i n the phase a n g l e ty- n e a r t h e wedge i n c r e a s e s as t he f r e q u e n c y d e c r e a s e s . The phase a n g l e 0__ i s n o t a t a l l a f f e c t e d by the s t r u c t u r e . I n compar i ng A i n F i g s . 61 (a ) and (b) we see t h a t i t s b e h a v i o r z 1 5 5 WM7M (a) T - 1 — i — 1 — r 2 0 4 0 60 80 100 v 20 40 60 80 100 X cm _ ' 'V • 1 1 1 • l i 1 i J. 4 LU EZZZZZzZZZZ Î W7777777A i * i 1 i * i 1 i 1 2 0 4 0 60 80 100 - i — i — i — | — (b) i— 1—i—>—r~ 4 0 60 80 100 X cm F i g u r e 6 1 . T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e H p o l a r i z - a t i o n f o r t r a v e r s e s o v e r t h e w e d g e e x t e n d i n g 1 0 c m b e y o n d t h e b l o c k e d g e a n d f r e q u e n c i e s ( a ) 3 x 1 0 ^ , ( b ) 1 0 - ^ c y c l e s / s e c . 156 near the structure i s frequency dependent. • Figure 62 shows the results f o r the wedge edge d i r e c t l y over the mantle edge while F i g . 63 deals with the block edge extending 10 cm beyond the wedge edge. The behavior of- H y, H z, 0 y and QZ i s very s i m i l a r to that indicated i n F i g . 61. The behavior of - E ,, JL however, i s quite s e n s i t i v e to the r e l a t i v e positions of the wedge edge and block edge. Results were also obtained f o r model frequencies of 10 and 3 x IO-* cycles/sec but are not shown since they indicated no large departure from the results f o r the other frequencies already discussed. In general,, the results f o r the H p o l a r i z a t i o n indicate that the magnetic f i e l d components undergo only small changes as the structure i s approached while the horizontal e l e c t r i c f i e l d i s very sensitive to the conductivity structure. Measurements were also made f o r the top f l a t surface of the block structure 1.5 cm below the surface of the s a l t solution. - In the geophysical problem, t h i s depth represents a-depth of 7°5 km« Since a l l the results f o r this case were almost i d e n t i c a l to those f o r the 3 cm depth discussed i n d e t a i l i n t h i s work, they are not shown. The results of t h i s work on model coastline s t r u c t - ures indicate that, i n general, f o r frequencies i n the neighborhood of 0.01 cycles/sec' the sloping sea-earth i n t e r - face should have an important e f f e c t on the H"z/Hy r a t i o very near the coast but that an upwelling highly conducting mantle structure near the coast would become increasingly more important at decreasing frequencies. I t appears that the 157 cn E> 2 CO I 2* ' o X oo 16 CvJ 5tl2 c? 8 O s 4 LU K H, 2 0 0 160 t7 / M / LS r r m m l ' I ' I ' 120 f 8 0 f 4 0 0 - 4 0 I ' I ' i — i — r 20 4 0 60 80 100 1777777} (a) X c m i — i — i — i — i — i — i — i — > — i — 20 4 0 60 80 100 3* b x 2 ID x N I b x cr> ro |6 € IE > h e X O 4 U J " y • 1 — 1 — r ' i i • • i i • • mrnn (b) T—•—n—i ' " i " i i i 1 i i i i i i • — i — r — r 20 40 60 80 100 v 20 40 60 80 100 X c m F i g u r e 62„ The a m p l i t u d e s and phase a n g l e s f o r t h e H p o l a r i z a t i o n f o r t r a v e r s e s o v e r t he wedge w i t h t he wedge edge d i r e c t l y o v e r t he b l o c k edge and f r e q u e n c i e s (a ) 3x1 o \ (<b) 1C-3 c y c l e s / s e c . 158 ro & 2 cn 10 O co 16 E <*> 8 o x „ oq 4 x LU H% 2 0 0 160 1 I 1 1 ' i — 1 — i — i — i — » '20 V I I MS U77rrm 80 <f> 4 0 0 - 4 0 T — 1 — I ' 20 4 0 6 0 8 0 100 (a) — 1 — I — I — 1 — I — I — I — • — I — 20 4 0 60 8 0 DO X cm 1—1—r 6 0 8 0 100 X cm l — ' — i — 1 — r - - — r 20 4 0 6 0 8 0 100 F i g u r e 63. • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e H p o l a r i z a t i o n f o r t r a v e r s e s o v e r t h e w e d g e w i t h t h e b l o c k e d g e e x t e n d i n g 10 c m b e y o n d t h e w e d g e e d g e a n d f r e q u e n c i e s ( a ) 3x1o\ ( b ) 10^ c y c l e s / s e c . 159 e x p e r i m e n t a l l y o b s e r v e d e n h a n c e m e n t o f t h e H z / H y r a t i o a t f r e l a t i v e l y l a r g e d i s t a n c e s f r o m t h e c o a s t c o u l d b e a t t r i b u t e d , i n p a r t , t o t h e u p w e l l i n g o f a c o n d u c t i n g z o n e i n t h e m a n t l e . T h e r e s u l t s o b t a i n e d h e r e s u p p o r t t h e m o d e l u s e d b y S c h m u c k e r (196*+) i n d e s c r i b i n g c o a s t a l a n o m a l i e s i n t h e m a g n e t i c f i e l d v a r i a t i o n s ; • I n a d d i t i o n , t h e m o d e l r e s u l t s i n d i c a t e t h a t m e a s u r e m e n t s o f t h e H y / H z r a t i o o b t a i n e d b y L a m b e r t a n d C a n e r (1965) c o u l d b e e x p l a i n e d , a s t h e y s u g g e s t , b y a n u p w e l l i n g c o n d u c t i n g z o n e w i t h i n t h e m a n t l e . (d) • Ialand S t r u e t u r e s Of s o m e e x p e r i m e n t a l i n t e r e s t i s t h e p r o b l e m o f s m a l l i s l a n d s i n a w a t e r c h a n n e l , s u c h a s a r e f o u n d i n t h e S t r a i t o f G e o r g i a w h i c h i s b o u n d e d b y t h e B r i t i s h C o l u m b i a M a i n l a n d o n t h e o n e s i d e a n d V a n c o u v e r I s l a n d o n t h e o t h e r s i d e . T h e d i m e n s i o n s o f t h i s w a t e r c h a n n e l a r e s u c h t h a t a s c a l e d m o d e l c a n r e a d i l y b e m a d e . T h e s e c t i o n o f t h e S t r a i t o f G e o r g i a a r e a c h o s e n f o r t h i s m o d e l s t u d y i n c l u d e s T e x a d a I s l a n d . F i g u r e 6h s h o w s t h e r e g i o n t h a t w a s m o d e l l e d . - T h i s r e g i o n w a s s t u d i e d u s i n g t h e p r e s e n t m o d e l t e c h n i q u e a s a r e s u l t o f c o l l a b o r a t i o n w i t h s c i e n t i s t s a t t h e P a c i f i c N a v a l L a b o r a t o r y a t E s q u i m a l t , B . C . T h e m o d e l m e a s u r e m e n t s w e r e m a d e j u s t p r i o r t o f i e l d m e a s u r e - m e n t s m a d e b y t h e P a c i f i c N a v a l L a b o r a t o r y i n M a y 1966. T h e y u s e d b u o y e d m a g n e t o m e t e r s t o m a k e m a g n e t i c f i e l d m e a s u r e m e n t s a t v a r i o u s p o i n t s i n t h e v i c i n i t y o f T e x a d a I s l a n d ; t h e i r i n s t r u m e n t s r e c o r d e d t h e t o t a l f i e l d v a r i a t i o n s a n d n o t t h e VANCOUVER ISLAND F i g u r e 61+„ - M a p o f T e x a d a I s l a n d a n d t h e S t r a i t o f G e o r g i a . 161 c o m p o n e n t s . S i m u l t a n e o u s m e a s u r e m e n t s w e r e m a d e a t t h e i r r e f e r e n c e s t a t i o n n e a r S u r p r i s e M o u n t a i n (1000 f e e t a b o v e s e a l e v e l ) o n T e x a d a I s l a n d . T h e m o d e l o f t h e s e a a n d i s l a n d s r e q u i r e d a l a r g e s h e e t o f g r a p h i t e 120 c m l o n g , 3 L c m w i d e , a n d O.k- c m t h i c k . T h i s l a r g e s h e e t w a s c o n s t r u c t e d b y c e m e n t i n g f o u r s m a l l e r s h e e t s t o g e t h e r e n d t o e n d . • A n e l e c t r i c a l l y c o n d u c t i n g e p o x y c e m e n t o f a p p r o x i m a t e l y t h e s a m e c o n d u c t i v i t y a s g r a p h i t e w a s u s e d . T h e s e e p o x y j o i n t s i n t h e s h e e t p r e s e n t e d n o s e r i o u s p r o b l e m s i n t h e m o d e l m e a s u r e m e n t s . S u i t a b l y s h a p e d h o l e s , t o r e p r e s e n t T e x a d a I s l a n d a n d t h e o t h e r s m a l l i s l a n d s s h o w n i n F i g . 6 k , w e r e c u t i n t h e g r a p h i t e s h e e t . T h e g r a p h i t e s h e e t w a s f u r t h e r m a c h i n e d t o t h e a p p r o p r i a t e t h i c k n e s s t o r e p r e s e n t t h e c o a s t s a n d t h e v a r y i n g d e p t h o f t h e o c e a n . T h i s m a c h i n i n g w a s d o n e u s i n g a s m a l l l o w s p e e d c i r c u l a r s a n d e r . T h e s c a l i n g f a c t o r s u s e d h e r e a r e t h e s a m e a s t h o s e u s e d i n ( a ) a n d ( b ) i n t h i s c h a p t e r , w i t h 1 c m i n t h e m o d e l c o r r e s - p o n d i n g t o 10^ c m o n t h e g e o p h y s i c a l s c a l e . T h e g r a p h i t e s h e e t w a s s u s p e n d e d a t t h e s u r f a c e o f t h e s a l t s o l u t i o n i n t h e t a n k . A t h i n f i l m o f s o l u t i o n c o v e r e d t h e s h e e t . M e a s u r e m e n t s o f t h e e l e c t r i c a n d m a g n e t i c f i e l d s f o r b o t h t h e E a n d H p o l a r i z a t i o n s w e r e m a d e f o r t h e t r a v e r s e s l a b e l l e d 0 t o 7 i n F i g . 6 k . T h e m o d e l f r e q u e n c y o f 10^ c y c l e s / s e c c o r r e s p o n d s t o 0.1 c y c l e s / s e c o n t h e g e o p h y s i c a l s c a l e . F i g u r e s 65 a n d 66 s h o w t h e E p o l a r i z a t i o n ( t r a v e r s e s n o r m a l t o t h e e l e c t r i c f i e l d o f t h e s o u r c e ) w h i l e F i g s . 67 a n d 68 s h o w t h e H p o l a r i z a t i o n ( t r a v e r s e s p a r a l l e l t o t h e e l e c t r i c 162 f i e l d - o f t he s o u r c e ) . In- F i g s . 65 t o 68 t he p o s i t i o n o f the g r a p h i t e s h e e t i s i n d i c a t e d w i t h V-.I. i n d i c a t i n g t he Vancouve r I s l a n d s i d e and B v C. i n d i c a t i n g t he M a i n l a n d s i d e . The a m p l i t u d e s and phases f o r t r a v e r s e s 0 and 1 a r e shown i n F i g s 6 5 ( a ) and (b ) r e s p e c t i v e l y . - T r a v e r s e 0 pa s se s t h r o u g h the r e f e r e n c e s t a t i o n p o s i t i o n ( S u r p r i s e Moun t a i n ) u s ed by t h e P a c i f i c N a v a l L a b o r a t o r y i n t h e i r f i e l d m e a s u r e - ment s . F i g u r e 65 shows t h a t EL i s enhanced on b o t h t h e V-.I. ' and- B. C. s i d e s as the c o a s t i s a p p r o a c h e d , f a l l s t o s m a l l e r v a l u e s o v e r t he s h a l l o w s e a , and t h e n i n c r e a s e s s l i g h t l y as Texada I s l a n d i s a p p r o a c h e d . - H shows enhancement o v e r t he sea n e a r t he two c o a s t s . ; E undergoes a l a r g e i n c r e a s e as Texada I s l a n d i s c r o s s e d . B o t h 0 Z and undergo a s u b s t a n t - i a l change as the I s l a n d i s t r a v e r s e d . The r e s u l t s f o r t r a v e r s e s 3 and 5 a r e shown i n F i g s . 66(a) and (b) r e s p e c t i v e l y . - The r e s u l t s f o r t r a v e r s e 3 a r e v e r y s i m i l a r t o t ho se o f t r a v e r s e 0 w i t h t he e x c e p t i o n s t h a t E undergoes a s m a l l e r change as> Texada I s l a n d i s c r o s s e d and - JL • \k a ; l a r g e r change as t h e B. j CL c o a s t l i n e i s c r o s s e d . - T r a - • JL v e r s e 5 ( F i g . 66 b ) does n o t pass o v e r Texada I s l a n d , b u t c r o s s e s a much s m a l l e r i s l a n d . Harwood I s l and . . I t i s tfLn^fps- e s t i n g t h a t Hy and H z behave a l m o s t i n t he same manner as f o r t he t r a v e r s e s o v e r Texada I s l a n d . Th i s - i n d i c a t e s t h a t t h e i r b e h a v i o r i s a f f e c t e d more by the c o a s t s t h a n by t he p r e s e n c e o f t he i s l a n d s . E i s r a t h e r d i f f e r e n t , i t now shows JL enhancement i n c r o s s i n g Harwood I s l a n d and i s l i t t l e a f f e c t e d by Texada I s l a n d . The phase c u r v e s emphas ize the change i n 163 F i g - t i r e 6-5 „ T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z a t i o n f o r - ( a ) t r a v e r s e 0 a n d ( b ) t r a v e r s e 1 » 16>+ 20 4 0 6 0 8 0 100 v 2 0 4 0 6 0 8 0 100 Y cm F i g u r e 66 . * The a m p l i t u d e s and phase a n g l e s f o r t h e E p o l a r i z a t i o n f o r (a) t r a v e r s e 3 and (b) t r a v e r s e 5° 165 s t r u c t u r e a n d a r e s i g n i f i c a n t l y d i f f e r e n t f o r t r a v e r s e s w h i c h c r o s s T e x a d a I s l a n d a n d t h o s e w h i c h d o n o t . T h e a m p l i t u d e s a n d p h a s e s f o r t h e H p o l a r i z a t i o n f o r t r a v e r s e s 0. 1, 3? a n d 5 a r e s h o w n i n F i g s . 6 7 ( a ) , ( b ) , 6 8 ( a ) a n d ( b ) r e s p e c t i v e l y . I t i s a p p a r e n t f r o m t h e s e f i g u r e s t h a t b o t h H a n d H „ a r e v e r y l i t t l e a f f e c t e d b y t h e s t r u c t u r e f o r y * t h i s p o l a r i z a t i o n o T h e p h a s e a n g l e $ z d o e s u n d e r g o a c h a n g e a s l a r g e a s ^ - O 0 . T h e c u r v e f o r t r a v e r s e 5 ( F i g . 68 b ) i s c o n s i d e r a b l y d i f f e r e n t f r o m t h e c u r v e s f o r t r a v e r s e s w h i c h d o n o t c r o s s T e x a d a I s l a n d . I n c o n t r a s t t o t h e R y a n d H z c u r v e s , E i s v e r y s e n s i t i v e t o t h e s t r u c t u r e . I t r i s e s v e r y s h a r p l y a s t h e c o a s t s a r e a p p r o a c h e d , f a l l s t o v e r y s m a l l v a l u e s o v e r t h e s e a a n d i n c r e a s e s a g a i n a s t h e s h o r e s o f t h e i s l a n d s a r e c r o s s e d . T h e p h a s e a n g l e ty i s a l s o v e r y s e n s i t i v e t o t h e c o a s t l i n e s a n d s h o r e s o f t h e i s l a n d s . • A n u n u s u a l l y l a r g e c h a n g e i n ty t a k e s p l a c e i n t r a v e r s i n g H a r w o o d I s l a n d ( t r a v e r s e 5)° M e a s u r e m e n t s w e r e m a d e a l s o f o r t h e o t h e r t r a v e r s e s i n d i c a t e d i n F i g . * 6k b u t a r e n o t s h o w n h e r e , s i n c e t h e m a i n f e a t u r e s o f a l l t h e m e a s u r e m e n t s m a d e a r e c o v e r e d b y t h e f o u r t r a v e r s e s d i s c u s s e d . T h e m o d e l m e a s u r e m e n t s i n d i c a t e t h a t f o r t h e g e o p h y s i c - a l p r o b l e m a p p r e c i a b l e c h a n g e s i n b o t h H a n d H _ c o u l d b e y z e x p e c t e d n e a r b o t h c o a s t l i n e s o n l y f o r t h e E p o l a r i z a t i o n . T h e e l e c t r i c f i e l d E „ f o r t h e E p o l a r i z a t i o n s h o u l d s h o w a s h a r p d e c r e a s e a s t h e c o a s t l i n e s a r e a p p r o a c h e d a n d t h e n f a l l v e r y s h a r p l y o v e r t h e o c e a n , s h o w i n g s o m e e n h a n c e m e n t n e a r 166 - i — i — ' — i — 1 — i — i — i — 1 — r 20 4 0 6 0 8 0 100 X cm l — • — i — 1 — i — 1 — i — • — r 20 4 0 6 0 8 0 100 V9 2 2 fsl X * 16 ' o E > B J C J — HVI. H Y H, 200 160 i • i i i — > — r — 120 Y 80 4 0 0 - 4 0 ' | i i i | i i ' j i 2 0 4 0 60 8 0 100 - BJOT HVI. * (b) X cm — i — i — | — i — | — i — | — i — | — 20 4 0 6 0 8 0 100 F i g u r - e 67. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e H p o l a r i z a t i o n f o r ( a ) t r a v e r s e 0 a n d ( b ) t r a v e r s e 1. 167 F i g u r e 6 8 * T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e H p o l a r i z a t i o n f o r ( a ) t r a v e r s e 3 a n d ( b ) t r a v e r s e 5. 168 i s l a n d s . I t s h o u l d r i s e s t e e p l y as t he c o a s t i s app roached f o r the ' H- p o l a r i z a t i o n o B o t h <* and \l/ s h o u l d show cons ide r ? - z 1 x a b l e change' f o r b b t h p o l a r i z a t i o n s „ The f i e l d measurements made i n t h e Texada I s l a n d r e g i o n ( Lokken and M a c l u r e , 1966) i n d i c a t e s m a l l changes i n t he a m p l i t u d e o f t h e r e s u l t a n t mag- n e t i c f i e l d v a r i a t i o n s and changes i n phase a n g l e somewhat l a r g e r t h a n t h o s e p r e d i c t e d by t he model measurements . - Much o f t h e m i c r o p u l s a t i o n s a c t i v i t y t h e y r e c o r d e d was a t a f r e - quency l o w e r t h a n O d c y c l e s / s e c • One wou ld e x p e c t f a u l t s and a p o s s i b l e u p w e l l i n g o f m a n t l e c o n d u c t i n g zones i n t he c o a s t a l r e g i o n , and such s t r u c t u r e wou ld g r e a t l y a f f e c t b o t h t h e v e r t i c a l m a g n e t i c f i e l d a m p l i t u d e and phase a n g l e ( see t h e r e s u l t s i n the ' p r e v i o u s s e c t i o n s ) . .The f a u l t s and u p - w e l l i n g o f t h e c o n d u c t i n g zones w i t h i n t he m a n t l e wou ld become i n c r e a s i n g l y more I m p o r t a n t f o r l o w e r f r e q u e n c i e s . I n g e n e r a l i t appea r s t h a t t h e r e s u l t s o b t a i n e d by t he P a c i f i c N a v a l L a b o r a t o r y can be e x p l a i n e d i n terms o f the combined e f f e c t o f t he e a r t h - s e a - I n t e r f a c e a t t h e c o a s t l i n e s and i s l a n d s , , and i n h o m o g e n e i t i e s i n the f o r m o f f a u l t s and u p w e l l i n g o f t he m a n t l e c o n d u c t i n g zones a t g r e a t e r d e p t h s . (e) Seamount and -Conduc t i n g -Dome' S t r u c t u r e s I n t h i s s e c t i o n measurements f o r model s t r u c t u r e s r e p r e s e n t i n g seamounts and c o n d u c t i n g domes a r e t r e a t e d b r i e f l y . E l e v a t i o n s o f t he ocean bo t tom v a r y f r o m submar ine mounta in s t o s m a l l i s l a n d s . > S i n c e seamounts- (submerged v o l - c a n i c h i l l s o r moun ta i n s ) w i l l have a c o n d u c t i v i t y l o w e r t h a n 169 t h a t o f s e a w a t e r , c o n d u c t i v i t y g r a d i e n t s w i l l e x i s t a n d a n o m a l i e s i n t h e n a t u r a l l y o c c u r r i n g e l e c t r o m a g n e t i c v a r i a t - i o n s m e a s u r e d a t t h e o c e a n s u r f a c e m i g h t b e e x p e c t e d . S i m i l a r l y , c o n d u c t i n g d o m e s ( m i n e r a l o r s a l t d o m e s ) b e l o w t h e s u r f a c e o f l a n d w o u l d g i v e r i s e t o c o n d u c t i v i t y g r a d i e n t s , a n d a n o m a l i e s i n t h e e l e c t r o m a g n e t i c v a r i a t i o n s a t t h e e a r t h ' s s u r f a c e . n e a r t h e c o n d u c t i n g d o m e s c o u l d b e e x p e c t e d . • I n t h i s w o r k a g r a p h i t e c o n e 30A c m h i g h a n d 30.k c m i n d i a m e t e r a t t h e b a s e w a s u s e d t o r e p r e s e n t a c o n d u c t i n g d o m e . T h i s c o n e w a s s e t o n a g r a p h i t e c y l i n d e r 3 k c m h i g h a n d o f t h e s a m e d i a m e t e r a s t h e b a s e o f t h e c o n e . T h e a p e x o f t h e c o n e w a s l e v e l w i t h t h e s u r f a c e o f t h e w a t e r i n t h e t a n k . T h e g r a p h i t e c o n e w o u l d r e p r e s e n t a c o n d u c t i n g d o m e 30A k m h i g h a n d 30.*+ k m i n d i a m e t e r a t t h e b a s e . T h e s a m e g e o m e t r y w a s u s e d f o r t h e s e a m o u n t s t r u c t u r e . T h e m a t e r i a l u s e d w a s c o n c r e t e . T h e c o n d u c t i v i t y o f c o n c r e t e i s a b o u t 10^ l o w e r t h a n t h a t o f t h e c o n c e n t r a t e d s a l t s o l u t i o n . F i g u r e 69 s h o w s t r a v e r s e s o v e r t h e g r a p h i t e c o n e i n t h e d i r e c t i o n p a r a l l e l po t h e h o r i z o n t a l m a g n e t i c f i e l d o f t h e s o u r c e . F i g u r e 6 9 ( a ) g i v e s t h e r e s u l t s f o r a m o d e l f r e - q u e n c y o f 3 x 10 c y c l e s / s e c w h i l e 6 9 ( b ) s h o w s t h e r e s u l t s - LL f o r a f r e q u e n c y o f 10 c y c l e s / s e c . A t t h e s e f r e q u e n c i e s t h e a m p l i t u d e s o f t h e t h r e e c o m p o n e n t s a r e a f f e c t e d b y t h e p r e s e n c e o f t h e c o n e . T h e p h a s e a n g l e o f t h e v e r t i c a l c o m p o n - e n t o f t h e m a g n e t i c f i e l d i s a p p r e c i a b l y i n f l u e n c e d b y t h e g r a p h i t e c o n e . B o t h -ty a n d 0 a r e e s s e n t i a l l y u n a f f e c t e d . y M e a s u r e m e n t s w e r e a l s o m a d e f o r m o d e l f r e q u e n c i e s o f 3 x 10^ 170 F i g u r e 69, T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s i n t h e - y d i r e c t i o n o v e r a g r a p h i t e c o n e f o r f r e q u e n c i e s ( a ) 3x10% ( b ) 10^ c y c l e s / s e c . 171 and 10^ c y c l e s / s e c bu t t he r e s u l t s a r e n o t p r e s e n t e d h e r e . B o t h H and H show much l e s s change a t t h e s e l o w e r f r e q u e n c -y z i e s . Measurements f o r t r a v e r s e s p a r a l l e l t o t he h o r i z o n t a l k e l e c t r i c f i e l d o f t he s o u r c e f o r model f r e q u e n c i e s o f 3 x 10 and 10 c y c l e s / s e c a r e shown i n F i g s . 70(a) and (b) r e s p e c t - i v e l y . These r e s u l t s i n d i c a t e t h a t H_ i s n o t a t a l l a f f e c t e d by the cone f o r t h i s , p o l a r i z a t i o n . H i s enhanced o v e r t h e cone , w h i l e E f a l l s v e r y s h a r p l y and r eache s a minimum JL d i r e c t l y o v e r t he cone . - The phase a n g l e s a r e n o t n o t i c e a b l y a f f e c t e d by the g r a p h i t e cone . - Measurements f o r t he m o d e l l e d seamount f o r t r a v e r s e s p a r a l l e l t o t he magne t i c f i e l d o f t he s o u r c e , and f o r mode l LL 0 f r e q u e n c i e s o f 3 x 10 and 10J c y c l e s / s e c r e s p e c t i v e l y , a r e g i v e n i n F i g s . 71(a) and ( b ) . N e i t h e r H„ n o r H a r e a t a l l y z a f f e c t e d by t he c o n c r e t e cone . - E now undergoes a s ha rp JL enhancement d i r e c t l y o v e r t he cone . The phase a n g l e s a r e a g a i n n o t a p p r e c i a b l y a f f e c t e d by the cone . Measurements f o r t r a v e r s e s i n the x - d i r e c t i o n a r e n o t shown, s i n c e o n l y E was s e n s i t i v e t o t he s t r u c t u r e . JL • The r e s u l t s o f F i g s . 69-71 s ugge s t t h a t f o r f r e q u e n c - i e s i n t he n e i g h b o r h o o d o f 0 . 3 c y c l e s / s e c , a n o m a l i e s i n t he e l e c t r i c and magne t i c f i e l d , though i n g e n e r a l s m a l l , s h o u l d e x i s t a t t he e a r t h ' s s u r f a c e n e a r submerged c o n d u c t i n g domes, w h i l e a n o m a l i e s i n the e l e c t r i c f i e l d o n l y , a r e i n d i c a t e d f o r measurements a t t he s u r f a c e o f t h e sea n e a r seamounts o r s u b - merged s m a l l i s l a n d s . • I n h o m o g e n e i t i e s i n the c o n d u c t i v i t y d i s t r i b u t i o n i n seamount r e g i o n s wou ld l i k e l y be p r e s e n t a t 1 72 10 • o 2 x CD CO I cn 'O 00 cvi x 3 > 2 CM o X , oo 1 LU H z 1 6 0 : ^ 1 « i 1 i 1 i 1 i 1 i 1 120 r 8 0 E 7 r ~ T ™ ° E • - 4 0 (a) 2 0 4 0 6 0 8 0 1 0 0 2 0 4 0 6 0 8 0 1 0 0 X cm cn ro '° 2 x 09 CD cn ro ' o x - m 4 pS >» 3 3 2 ' o X I - LU E * H - 1 1 1 1 1 1 1 1 1 1 L 2 0 0 160 120 8 0 x r 4 0 0 • 4 0 - * T x 7 (b) "i I " I i l — ' — l — ' — I — ' ' — i — | — i — | — i — | — i — | — i . 2 0 4 0 6 0 8 0 100 w 2 0 4 0 6 0 8 0 1 0 0 X cm F i g u r e 70. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s i n t h e x d i r e c t i o n o v e r a g r a p h i t e c o n e f o r f r e q u e n c i e s ( a ) 3x10^, ( h ) 10^ c y c l e s / s e c . 173 ' o X OJ oo 8 OJ T e 4 o co 2 UJ . — — ^ ^ y ^ 2 0 0 H z I60 — i o n H • l | 1 | l | i | l | l£U r 8 0 4>° M O H — * ^ — * 4 0 7 (a) i i i j i j • i i i i > i • i • i < i > i 2 0 4 0 6 0 8 0 100 w 2 0 4 0 6 0 8 0 100 Y cm — i - ^ . 2 0 0 H p " " 160 H • 7 rX -i i i | i i i | i i \dU r 8 0 1 *° A H • - 4 0 (b) 1 I 1 1 1 " 1 I " I 20 4 0 6 0 8 0 100 2 0 4 0 6 0 8 0 100 Y cm v° 2 X  c CO 12 | "3» CM ' o o> 8 j f e Is > ' o X o F i g u r e 71. • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a v e r s e s i n t h e y d i r e c t i o n o v e r a c o n c r e t e c o n e , f o r f r e q u e n c i e s | a ) 3x1 o\ ( b ) 10^ c y c l e s / s e c . 1 7 K greater depths, i n the form of f a u l t s and upwellings of con- ducting zones i n the mantle, and hence substantial anomalies could be expected. Measurements were also made with the current sheet i n c l i n e d at 1 0 ° and 2 0 ° with respect to the horizontal f o r the seamount and conducting dome structures as well as f o r some of the structures discussed e a r l i e r . Since the behavior of the components near the structures was not appreciably altered by changing the angle of incidence, none of the results are shown. 3 . 2 Line Current Source 3 . 2 . 1 Introduction In section 2 . 3 the problem of a homogeneous conducting earth i n the near f i e l d of an o s c i l l a t i n g l i n e current was studied. - In comparing the results f o r the l i n e current source with those f o r the plane wave source i t i s found that although the horizontal e l e c t r i c and the horizontal magnetic f i e l d s f o r the two sources are very similar, the v e r t i c a l magnetic f i e l d components were quite d i f f e r e n t . The r a t i o H„/H„ f o r the z y plane wave source i s several orders of magnitude smaller than values observed experimentally at the earth's surface, while the r a t i o of the components f o r the l i n e current source i s within the range of experimentally observed values. Again, as f o r the plane wave source, the problem of d i s c o n t i n u i t i e s i n conductivity In the horizontal d i r e c t i o n does not r e a d i l y submit to mathematical '^rsajtmeht. 1 7 5 This section deals with an analogue method f o r study- ing problems of conductivity d i s c o n t i n u i t i e s f o r an overhead o s c i l l a t i n g l i n e current„ Several of the problems studied f o r the sheet current source w i l l be studied f o r the l i n e current source as well. 3 . 2 . - 2 • Ma themsdkcal • • Ahaly s i s • The mathematical development of the scaling factors f o r the model problem given i n section 3 o 1 » 2 makes no assumption on the type of source f i e l d , and hence i s d i r e c t l y applicable to a l i n e current model asiwell* • The necessary and- s u f f i c i e n t condition f o r invariance under a change i n scale -Is again given by equation (77) = 3 . 2.3 • Mod-el -Descri-T>tion- and^-Mea-surement Technique • The experimental arrangement i s the same as that discussed f o r the sheet current source with the exception that the array of wires used f o r the sheet current was re^ placed by a single copper rod 3/8 inch i n diameter. < This copper rod was located at the same height ( U 2 5 m) as the current sheet source above the surface of the s a l t solution. The tank was again set o f f center with respect to the over^- head l i n e source i n order that the minimum i n H z should occur near one end of the tank* - The minimum H_ appears d i r e c t l y below the l i n e current, and the phase 4>z reverses at t h i s point. • As was pointed out e a r l i e r ^ this phase change i s troublesome when making measurements over structure i f the structure i s located too near this phase reversal region* 176 A i l the measurements discussed i n the following sections were made f o r structures well removed from this phase reversal region. 3 . 2 . k Discussion- of Results The coordinate system used i n the discussion here i s the same as that f o r the sheet current model (Fig. 3 2 ) . The geophysical problem being modelled here i s that of a f l a t h orizontal earth i n the f i e l d of an ionospheric l i n e current. The s a l t solution i n the tank represented a layer of a homo- geneous f l a t earth. • The v a l i d i t y of the model was tested by comparing the measured values of the amplitudes with calculated values f o r a semi-infinite, f l a t conducting earth. The agreement with the calculated values was not quite as good as i n the oase of the current sheet source. - At 3-x 10 cycles/sec the r a t i o B^Hy f o r the model measurements agreed to within 6$ with the com- puted values f o r points d i r e c t l y below the l i n e source. The calculations assume a homogeneous conductor of i n f i n i t e depth extending i n f i n i t e l y i n the horizontal directions while the model has a layer of s a l t solution with a layer of graphite below. Also the edge effects cannot be included i n the c a l - culations. The v e r t i c a l magnetic f i e l d as measured by the model at points well removed from the p o s i t i o n d i r e c t l y below the source, i s considerably larger than that predicted by the c a l c u l a t i o n s . • The edge effects i n part account f o r t h i s . The return loop f o r the l i n e current source, which was l o c - 177 ated about 7 m from the edge of the tank, also increases the v e r t i c a l magnetic f i e l d . In general, however, the shape of the curves giving the amplitudes as a function of p o s i t i o n agree well with the calculated ones. Figure 72 gives the model measurements of the amplitudes f o r traverses normal to the d i r e c t i o n of the l i n e current f o r a frequency of 3 x 10 cycles/sec and a water depth of 6h,h cm. Although various structures were studied'using the l i n e current model, only coastline structures w i l l be d i s - cussed i n d e t a i l here. In order to f a c i l i t a t e comparison with measurements made f o r the sheet current source, the structures were located at the same positions as those f o r the sheet current source. Coastline Structures Measurements f o r the inverted truncated graphite cone are shown i n F i g s . 73 to 75. The scaling factors involved are the same as those used f o r the truncated cone i n the f i e l d of the sheet current discussed e a r l i e r (Figs. L 9 to 51)• The only change i s that the sheet current has been replaced by a l i n e current. Figures 73(a) and (b) show the amplitudes h h and phase angles f o r 3 x 1 0 and 10 cycles/sec respectively, while F i g s . 7Ma) and (b) show the measurements f o r 3 x 10^ and 10^ cycles/sec respectively. Both F i g . 73 and F i g . 7 k deal with traverses perpendicular to the e l e c t r i c f i e l d of the source. As was the case f o r F i g s . h9 and 50, the non- uniformity of the source f i e l d again also a f f e c t s the -\1 co F i g u r e 7 2 . T h e a m p l i t u d e s f o r t r a v e r s e s i n t h e y d i r e c t i o n f o r a n o v e r h e a d o s c i l l a t i n g l i n e c u r r e n t . 179 - I — | 1 — | — i | I 1 — i — I — r 2 0 4 0 6 0 8 0 I00 Y cm 20 4 0 6 0 8 0 I00 T — I — i — | — i — i — i — i — i — | — r 2 0 4 0 6 0 8 0 I00 Y cm 2 0 ' 4 0 6 0 8 0 I00 F i g u r e 73. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r r e n t a n d t r a v e r s e s i n t h e y d i r e c t i o n o v e r a n i n v e r t e d t r u n c a t e d g r a p h i t e c o n e f o r f r e q u e n c i e s ! ( a ) 3x10L|'? ( b ) 10^ c y c l e s / s e c . 180 i — i — i — i — " — i — i — i — i — i — i 1 — i — i — i — i — i — j — i — i — i — i — 2 0 4 0 6 0 8 0 100 w 20 4 0 6 0 8 0 100 Y cm 2 0 4 0 6 0 80 100 2 0 4 0 6 0 8 0 100 Y cm F i g u r e 7 L- T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r r e n t a n d t r a v e r s e s i n t h e y d i r e c t i o n o v e r a n i n v e r t e d t r u n c a t e d g r a p h i t e c o n e f o r f r e q u e n c i e s ( a ) 3x10 3, ( b ) 10^ c y c l e s / s e c . . - 181 behavior of the f i e l d components i n the neighborhood of the edge of the. cone. EL,, as before, shows a decrease on the side of the decreasing horizontal f i e l d and an enhancement on the side of the increasing f i e l d . In comparing F i g s . 73 and 7k with F i g s . k9 and 50 we see that E L and H E undergo some- y z what lar g e r changes near the cone edges f o r the l i n e current source than i s indicated f o r the sheet current source. The r a t i o H_/H„ i s approximately unity near the cone edge on the Z y side of the decreasing horizontal f i e l d f o r the l i n e current source while the r a t i o i s much les s than unity f o r the sheet current source. * The phase angles 0, . and <j>_ are very s i m i l a r V z f o r the two sources, while the phase angle tyx undergoes greater changes f o r the l i n e current source than i t does f o r the sheet current source. Figure 75 gives the amplitudes and phase angles f o r traverses p a r a l l e l to the e l e c t r i c f i e l d of the source. Figure 75(&) gives the results f o r 3 x 1 0 cycles/sec and F i g . 75(b) those f o r 10^ cycles/sec. Comparing F i g . 75 with F i g . 51 we see that the behavior of the amplitudes and phase angles f o r the two f i e l d sources i s very s i m i l a r . The r a t i o Hz/H i s again nearer unity f o r the l i n e current source. For a frequency of 3 x 10 cycles/sec, ty undergoes lar g e r changes f o r the l i n e current source, whereas f o r a frequency of 10^ cycles/sec, tyx undergoes a larger change f o r the sheet current source. To study further the e f f e c t of a sloping earth-sea interface, measurements were made f o r the graphite wedge i n 182 X cm F i g u r e 75. • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r r e n t a n d t r a v e r s e s i n t h e x d i r e c t i o n o v e r a n i n v e r t e d t r u n c a t e d g r a p h i t e c o n e f o r f r e q u e n c i e s ( a ) 3x1-o\' ( b ) 1C-3 c y c l e s / s e c . 183 the f i e l d of the l i n e current. The sealing factors and the physical arrangement were the same as those used f o r the sheet current source (Pigs. 52 to 5 k )• Figures 76 and 77 show the E p o l a r i z a t i o n (wedge edge p a r a l l e l to the e l e c t r i c f i e l d of the source), and' F i g . 78 shows the H p o l a r i z a t i o n (wedge edge perpendicular to the e l e c t r i c f i e l d of the source). - Figures 76(a), (b), 77(a), and (b) give the results respectively f o r frequencies 3 x 1 o \ 1 0 k , 3 x 10^, and 10^ cycles/sec. For the H p o l a r i z a t i o n . F i g s . 78(a) and (b) show the results LL 0 respectively f o r frequencies 3 x 10 and 10 J cycles/sec. In > comparing the amplitudes and phase angles near the wedge edge fo r the overhead l i n e current source (Figs. 52 to 5 L ) with those f o r the overhead sheet current source (Figs. 76 to 78) we see that f o r thi s structure also the behavior is very s i m i l a r f o r the two f i e l d sources, the main difference again being that the r a t i o Hz/Hy i s much larger f o r the l i n e cur- rent source than f o r the sheet current source. Smaller Hz/Hy ra t i o s would be obtained f o r the l i n e current problem i f the wedge edge were situated nearer to the source ( h o r i - zontally) where the v e r t i c a l f i e l d of the source i s smaller. These results indicate that f o r certa i n locations an overhead l i n e current source should r e s u l t i n a larg e r Hz/Hy, r a t i o near sea-land interfaces than would be the case f o r a large sheet current. The e f f e c t of the upwelling of a conducting zone within the earth's mantle under the ocean was studied f o r the l i n e current model using the wedge and block structure F i g u r e 76. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r - r e n t a n d t r a v e r s e s o v e r a w e d g e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x1O^, ( b ) 10^ c y c l e s / s e c . 185 i — i — i — i — i — i — i — i — i — i — i 1 — i — i — i — i — i — i — i — i — i — r — 2 0 4 0 6 0 8 0 100 2 0 4 0 6 0 8 0 100 Y cm F i g u r e 77 o < T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r - r e n t a n d t r a v e r s e s o v e r a w e d g e f o r t h e E p o l a r i z a t i o n a n d f r e q u e n c i e s ( a ) 3x10 3, ( b ) 10 3 c y c l e s / s e c . 186 F i g u r e 78. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r - r e n t a n d t r a v e r s e s o v e r a w e d g e f o r t h e H p o l a r i z a t i o n a n d f r e q u e n c i e s - ( a ) 3x1 C - \ ( b ) 1Cp c y c l e s / s e c . 187 d e s c r i b e d e a r l i e r f o r t h e s h e e t c u r r e n t p r o b l e m . T h e s c a l i n g f a c t o r s w e r e t h e s a m e a s t h o s e u s e d e a r l i e r ( F i g s . 55 t o 6 3 ) . F i g u r e 7 9 ( a ) s h o w s t h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t r a - v e r s e s o v e r t h e w e d g e w i t h t h e w e d g e a n d b l o c k e d g e s - p a r a l l e l t o t h e e l e c t r i c f i e l d o f t h e s o u r c e ( E p o l a r i z a t i o n ) f o r a m o d e l f r e q u e n c y o f 3 x 10 c y c l e s / s e c . T h e e d g e o f t h e w e d g e e x t e n d s 10 c m b e y o n d t h e b l o c k e d g e . • T h i s i s a m o d e l o f t h e o c e a n e x t e n d i n g 50 k m b e y o n d t h e c o n d u c t i n g m a n t l e e d g e . - I n o r d e r t o o b t a i n s o m e i n d i c a t i o n o f t h e r e l a t i v e i m p o r t a n c e o f t h e w e d g e a n d t h e b l o c k , ' F i g . 7 9 ( b ) g i v e s t h e r e s u l t s f o r t h e w e d g e (1) a n d t h e b l o c k (2) s e p a r a t e l y . Pig-- u r e s 80, 81 a n d 82 d e a l w i t h t h e s a m e p r o b l e m a s F i g . ' 7 9 e x c e p t t h a t t h e l#0del f r e q u e n c i e s a r e 10^, 3 x 10^, a n d 10^ c y c l e s / s e c r e s p e c t i v e l y . - A s i n t h e c a s e o f t h e s h e e t c u r r e n t problem, we see that f o r the lower frequencies- Hy and H z are n o t a f f e c t e d a p p r e c i a b l y b y t h e w e d g e b u t a r e s t i l l s o m e w h a t a f f e c t e d b y t h e b l o c k s t r u c t u r e . • E „ i s s t r o n g l y a f f e c t e d b y b o t h t h e w e d g e a n d t h e b l o c k f o r a l l f r e q u e n c i e s s t u d i e d . I f w e c o m p a r e F i g s - . 79, 80, 81 , a n d 82 ( l i n e c u r r e n t s o u r c e ) w i t h P i g s . 55, 56, 57? a n d 58 ( s h e e t c u r r e n t s o u r c e ) r e s p e c t - i v e l y , i t i s a p p a r e n t t h a t t h e e n h a n c e m e n t o f H y , a n d H z o v e r t h e s t r u c t u r e e d g e s i s , i n g e n e r a l , m u c h g r e a t e r f o r t h e s h e e t c u r r e n t s o u r c e t h a n i t i s f o r t h e l i n e c u r r e n t s o u r c e . T h e b l o c k e d g e i n p a r t i c u l a r , s e e m s t o h a v e a v e r y d i f f e r e n t e f f e c t o n H , r a n d H _ f o r t h e l i n e c u r r e n t s o u r c e t h a n i t h a d y z f o r . t h e s h e e t c u r r e n t s o u r c e . > H f o r t h e s h e e t c u r r e n t z s o u r c e [ c u r v e s l a b e l l e d ( 2 ) f o r H i n F i g s . 55 t o 58] u n d e r - Zi 188 o IN O X 00 cvi i 3 > CM x 00 UJ ^ > ^ H y I 2 0 - . r ^ ^ ^ ^ 8 0 —r—|—1—i r - i i | i | i 4 0 '- *0 ^ - 8 0 H * \ A + 1 / ' 1/ / /07 / IS ' (/ / / 0" / ^ IS I I I ! "1—1 1 1 1 1 ' 2 0 40 60 8 0 100 Y cm 2 0 4 0 60 80 100 • — < — i — 1 — i — r 2 0 4 0 6 0 8 0 1 0 0 v 2 0 4 0 6 0 8 0 100 Y cm F i g u r e 79-4 - T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - a t i o n a n d a l i n e c u r r e n t w i t h f = 3x10^ c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e ( 1 ) , t h e b l o c k a l o n e ( 2 ) . 189 20 4 0 6 0 8 0 I00 v 2 0 4 0 6 0 8 0 I00 Y cm 20 4 0 6 0 8 0 I00 v 20 4 0 6 0 8 0 I00 Y cm F i g u r e 80. • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - LL a t i o n a n d a l i n e c u r r e n t w i t h f = 10 c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e ( 4 ) , t h e b l o c k a l o n e ( 2 ) . 190 x o 1 I I <Tl I IT U7V72 Ex ^-40 I > i 1 l— 1—I— 1—i—' 20 40 60 80 100 \l t i crl I IS (a) Y cm - i — i — | — i — | — i — i — i — | — 20 40 60 80 100 ' 2b ' 4b ' 60 ' 8b ' 160' Y cm - i — 1 — i — 1 — i — • — i — 1 — r 20 40 60 80 100 F i g u r e 81. ' T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - a t i o n .... a n d . a l i n e c u r r e n t w i t h f=3x103 c y c l e s / s e c f o r t r a v e r s e s o v e r ( - a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d (>b) t h e w e d g e a l o n e (1), t h e b l o c k a l o n e (2). 191 'O x 2 ID X 3 4 £ 2 x o 5. K V M t k I | " l | ' T l i | | | | | | | 1 20 40 60 80 OO y / / 7 I I. T7TW77[ Y cm i i i j i i i i • i . 20 40 60 80 100 20 4 0 60 80 100 Y cm i — • — i — 1 — i — • — i — • — r 20 4 0 6 0 8 0 DO F i g u r e 8 2 . • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z - a t i o n a n d a l i n e c u r r e n t w i t h f = 10^ c y c l e s / s e c f o r t r a v e r s e s o v e r ( a ) t h e w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e , a n d ( b ) t h e w e d g e a l o n e (1), t h e b l o c k a l o n e ( 2 ) . 192 goes a s ha rp i n c r e a s e d i r e c t l y o v e r the b l o c k edge w h i c h i s n o t t h e ca se f o r the, l i n e c u r r e n t source<» D i r e c t l y o v e r t he wedge e d g e ' H i s enhanced f o r b o t h f i e l d sources , , b u t t h e enhancement f o r the s h e e t c u r r e n t s o u r c e i s g r e a t e r . - H , t o o , shows l e s s change n e a r the s t r u c t u r e edges f o r the l i n e c u r r e n t s o u r c e t h a n i t does f o r the s h e e t c u r r e n t s o u r c e . • The r e s u l t i n g a m p l i t u d e s f o r t h e combined s t r u c t u r e s [wedge and b l o c k , F i g s . 7 9 ( a ) , 8 0 ( a ) , 81 (a ) and 8 2 ( a ) ] a r e hence c o n s i d - e r a b l y d i f f e r e n t f o r t he two s o u r c e s . A n o t h e r d i f f e r e n c e i s t h a t t he a m p l i t u d e r a t i o s o f t he v e r t i c a l t o h o r i z o n t a l f i e l d s f o r t he l i n e c u r r e n t s o u r c e r e a c h v a l u e s g r e a t e r t h a n u n i t y •{Figs. 7 9 ( a ) ? 80 (a ) and 8 2 ( a ) ] whereas t he r a t i o s a r e l e s s t h a n u n i t y f o r a l l p o i n t s a l o n g t he t r a v e r s e s f o r t he s hee t c u r r e n t s o u r c e . • The phase a n g l e <j>_ t end s t o undergo l a r g e r changes and the phase a n g l e \jr x s m a l l e r changes f o r t h e s h e e t c u r r e n t s ou r ce t h a n f o r the l i n e c u r r e n t s o u r c e . • To s t u d y how the r e l a t i v e p o s i t i o n s o f t he wedge and b l o c k a f f e c t t he f i e l d components f o r a l i n e c u r r e n t s o u r c e , the p o s i t i o n o f t he wedge was changed (as was done f o r t he s hee t c u r r e n t s o u r c e , F i g s . 59 t o 6 0 ) . The r e s u l t s i n F i g . 83 a r e f o r t he wedge edge d i r e c t l y o v e r t he b l o c k edge, w h i l e t he r e s u l t s i n F i g . oh a r e f o r t h e b l o c k edge e x t e n d i n g 10 cm beyond t h e wedge edge. - F i g u r e s 83(a) and 8 k ( a ) a r e f o r a model f r e q u e n c y o f 3 x 10^ c y c l e s / s e c w h i l e F i g s . 83(h) and 8 kCb) show : t h e r e s u l t s f o r a f r e q u e n c y o f 10^ c y c l e s / s e c . I n compar ing t h e r e s u l t s o f F i g s . 83 and oh ( l i n e c u r r e n t s o u r c e ) w i t h t h e r e s u l t s o f F i g s . 59 and 60 ( s h e e t c u r r e n t s o u r c e ) we 193 F i g u r e 83. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r r e n t a n d t h e E p o l a r i z a t i o n f o r t r a v e r s e s o o v e r t h e w e d g e w i t h t h e w e d g e e d g e d i r e c t l y o v e r t h e b l o c k e d g e a n d f r e q u e n c i e s ( a ) 3x10 , ( b ) 10 J c y c l e s / s e c . 1 9 k > 2 9 x CO UJ H » T7T77A ' 20 ' 4*0 ' 6'0' 80 ' l6o' V/*///\ (a) ' i ' j — • — i — • — i — • I 20 40 60 80 100 Y cm ' O X «> ro Cn CM 'o CD 4 ro* ' E > ro 'o x o UJ H mrrn x  N s -40 - > — i — 1 — i — 1 — i — 1 — i — 1 — i — 20 40 60 80 100 1/ / v r w r n (b) — I — • — i — • — i — 1 — i — ' — r ~ 20 40 60 80 100 Y cm F i g u r e 8̂1-̂  • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r r e n t a n d t h e E - p o l a r i z a t i o n f o r t r a v e r s e s o v e r t h e w e d g e w i t h t h e b l o c k e d g e e x t e n d i n g 10 c m b e y o n d t h e w e d g e e d g e a n d f r e q u e n c i e s ( a ) 3x10 ,• ( b ) 10 J c y c l e s / s e c . 195 a g a i n see t h a t a l t h o u g h t h e r e i s a c o n s i d e r a b l e s i m i l a r i t y i n t he c u r v e s f o r t he two s o u r c e f i e l d s , t he enhancement of- H and H_ o v e r t h e s t r u c t u r e edges i s c o n s i d e r a b l y s m a l l e r f o r t h e l i n e c u r r e n t s ou r ce t h a n f o r t h e s h e e t c u r r e n t s o u r c e . The g e n e r a l o b s e r v a t i o n s w i t h r e s p e c t t o f r e q u e n c y dependence d i s c u s s e d i n t he r e s u l t s f o r the s h e e t c u r r e n t p r ob l em a p p l y t o t he l i n e c u r r e n t p r ob l em as w e l l . • F i g u r e s 85, 86, and 87 show the H p o l a r i z a t i o n (wedge edge p a r a l l e l t o t he h o r i z o n t a l m a g n e t i c f i e l d o f t he s o u r c e ) . - F i g u r e 85 g i v e s the r e s u l t s f o r t he wedge edge e x t e n d i n g 10 cm beyond t he b l o c k edge. ' F i g . 86 t h o se f o r the wedge edge d i r e c t l y o v e r t he b l o c k edge, and F i g . 87 t h o se f o r t h e b l o c k edge e x t e n d i n g 10 cm beyond the wedge edge* The 1 model f r e q u e n c i e s u s e d f o r t h e measurements shown i n F i g s . 85 t o 87 were 3 x 10 and 1 0 J c y c l e s / s e c . I n compar ing F i g s . 85 t o 87 ( l i n e c u r r e n t s o u r c e ) w i t h F i g s . 61 t o 63 ( s h e e t c u r - r e n t s o u r c e ) , we see t h a t t he b e h a v i o r o f Hy. and H z i s a l m o s t i d e n t i c a l i n t he two c a s e s ; E , however , d i r e c t l y o v e r t h e wedge edge, i s about 30$ l a r g e r f o r t he s h e e t c u r r e n t s o u r c e t h a n f o r t he l i n e c u r r e n t s o u r c e . • The H /H ' r a t i o s , though l e s s t h a n u n i t y f o r a l l p o s i t i o n s and f r e q u e n c i e s f o r t h e l i n e c u r r e n t s o u r c e , a r e l a r g e r t h a n t ho se f o r t he s h e e t c u r - r e n t s o u r c e . The b e h a v i o r o f t he phase a n g l e s i s v e r y s i m i l a r f o r t h e two s o u r c e f i e l d s , w i t h the e x c e p t i o n t h a t ty LL f o r a f r e q u e n c y o f 3 x 10 c y c l e s / s e c , undergoes much g r e a t e r changes d i r e c t l y o v e r the s t r u c t u r e edges f o r the l i n e c u r r e n t s o u r c e t h a n f o r the s h e e t c u r r e n t s o u r c e . ' As was f ound f o r 196 'o X cvi 'o cvi 5l2 8 > CM 'O X 00 _ 2 0 0 / V * . I60 : y - H2 - 120 .0 — i — | — I - T - l | ' 1 ' 1 E + 8 0 i° *x ' > / / /a / / S 9 4 0 V ~" 4> 7 0 E * - 4 0 • S / / rr/ / 1/ ^ _ Ex_ M M/I (a) • • .* 2 0 4 0 6 0 80 100 X cm 20 4 0 6 0 8 0 100 2 0 4 0 6 0 80 100 X cm 20 ' 4 0 ' 6 0 ' 8 0 100 F i g u r e 85. T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r r e n t a n d t h e H p o l a r i z a t i o n f o r t r a v e r s e s o v e r a w e d g e e x t e n d i n g 10 c m b e y o n d t h e b l o c k e d g e a n d f r e q u e n c i e s ( a ) 3-x10 Ob) 10 3 c y c l e s / s e c . 197 ro 1 ^ si I ro ' g 'E CJ"* 8 g CD 4 UJ H. H , 1 — r E- " i — 1 — i — ' — i — 1 — r 2 0 0 160 120 T E T Z m U T 4>° 8 0 4 0 0 - 4 0 >/ A / / I - - 7 (a) t — | 1 1 1 1 1 2 0 4 0 6 0 80 100 i—i—i—r T—i—r X cm 2 0 4 0 6 0 8 0 100 CM 'O x 2 CD ro CM o X <T> 16 ro X s 4 - Hz ' 1 i i i i uT (b) I * I — r 2 0 4 0 6 0 8 0 100 T — | — i — | — i — | — i — | — i — r — 2 0 4 0 6 0 8 0 100 cm F i g u r e ' 86-. • T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r r e n t a n d t h e H p o l a r i z a t i o n f o r t r a v e r s e s o v e r t h e w e d g e w i t h , t h e w e d g e e d g e d i r e c t l y o v e r t h e b l o c k e d g e a n d f r e q u e n c i e s ( a ) 3 x 1 O ^ , ( b ) 1 0 ^ c y c l e s / s e c . 198 CP CD OJ *~lsl H, 2 0 0 _ I 6 0 CP io ' o co 16 CO 5. I2 w 8 o co 4 VP7T7\ I • i • i i I 2 0 8 0 4 0 UJ - 4 0 'XT i 1/ / g 7 > * F 7 W 7 1 (a) —T ' 1 ' I 1 1 ' 1 1 2 0 4 0 6 0 8 0 100 X cm i — 1 — i — 1 — i — 1 — i — 1 — i — 2 0 4 0 6 0 8 0 100 2 0 " 4 0 ' 6 0 ' 8 0 ' l6o „ 2 0 ' 4 0 ' 6 0 8 0 100 X c m F i g u r e 87. > T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r a l i n e c u r - r e n t a n d t h e H p o l a r i z a t i o n f o r t r a v e r s e s o v e r t h e w e d g e w i t h t h e b l o c k e d g e e x t e n d i n g 10 c m b e y o n d t h e w e d g e e d g e LL O a n d f r e q u e n c i e s ( a ) 3x10 , ( b ) 100 c y c l e s / s e c . 199 the sheet current source, i t i s apparent that the r e l a t i v e positions of the wedge and block f o r the overhead l i n e current source a f f e c t the behavior of the amplitudes and phases i n the neighborhood of the structures. In order to compare the measurements f o r the two f i e l d sources (sheet current and l i n e current) i d e n t i c a l structures and locations i n the tank were used f o r both problems — the only difference was a change i n source f i e l d . However, some tests were made to study how the f i e l d components were affected by changing the structure position, In p a r t i c u l a r , observations were made on the e f f e c t which a change i n the horizontal distance to the overhead l i n e current had on the behavior of the components. Figure 8*+(a) (discussed e a r l i e r ) shows the results f o r the wedge edge located at y = k 5 cm (a - horizontal distance of 95 cm from the p o s i t i o n d i r e c t l y below the l i n e current) f o r a frequency of 3 x 10 cycles/sec-. Figure 88(a) shows the results f o r the same structure but with the wedge edge at y » +5 cm (*+5 cm from the p o s i t i o n d i r e c t l y below the l i n e current), w h i l e F i g . 88(b) shows the results when the wedge edge i s d i r e c t l y below the l i n e current. < H„ and p a r t i c u l a r l y H_ show much increased en-y z hancement d i r e c t l y over the edges of the structure as the structure i s moved nearer to the l o c a t i o n d i r e c t l y below the l i n e current. Figure 8 k(a) shows that the enhancement of H„ i s approximately the same over the wedge edge as i t i s over the block edge, while F i g . 88(b) shows a much larger enhance- ment over the block edge than i t does over the wedge edge. 200 "O X o cvi c n f 'O X oo CM > ' 2 x 00 UJ V / M S mrm 0 40 inrm (a) - i — , — i — | — i — i — i — i — • — i — » • i — i — i — i — « — i — i — i — • — r ~ -20 0 20 40 60 v n r n -20 0 20 40 60 Y cm T — • —r • 60 -40 -20 Y cm F i g u r e 88. > T h e a m p l i t u d e s a n d p h a s e a n g l e s f o r t h e E p o l a r i z a t i o n a n d a l i n e c u r r e n t w i t h f=3x10k c y c l e s / s e c f o r t r a v e r s e s o v e r t h e w e d g e w i t h t h e b l o c k e d g e e x t e n d i n g 10 c m b e y o n d t h e w e d g e e d g e f o r w e d g e e d g e p o s i t i o n s ( a ) y = -5, ' ( b ) y = ~h0 c m . 201 The phase angle </>_ also shows considerable change. • For the Z v e r t i c a l source height used here, the wedge tends to shie l d somewhat the block edge from the c y l i n d r i c a l source f i e l d f o r the arrangement of F i g * 88(h) 9 while th i s i s not the case when the structure i s nearer the point d i r e c t l y below the source. • The-, change i n the enhancement over the wedge edge i s much l e s s s r e l a t i v e l y , than i s the change d i r e c t l y over the block edge.- < The behavior of both H z and Ry s when the structure i s near the point d i r e c t l y below the l i n e current source, i s very s i m i l a r to that observed e a r l i e r f o r the overhead sheet current source [e.g.'Pig. 60(a)]» • I t i s apparent from' F i g s « 8kCa)9 88(a)5 and- (••&), that the h o r i - zontal p o s i t i o n r e l a t i v e to the overhead l i n e current i s a very important fa c t o r i n determining the behavior of the oag«* no-tic f i e l d component! u 202 C h a p t e r k » S U M M A R Y A N D C O N C L U S I O N S T h e p r e s e n t r e s e a r c h c o n t r i b u t e s t o t h e u n d e r s t a n d i n g o f t h e e f f e c t t h a t c e r t a i n i o n o s p h e r i c c u r r e n t s o u r c e s a n d e l e c t r i c a l c o n d u c t i v i t y d i s t r i b u t i o n s i n t h e e a r t h ' s c r u s t h a v e o n t h e b e h a v i o r o f t h e n a t u r a l l y o c c u r r i n g e l e c t r o m a g - n e t i c v a r i a t i o n s o b s e r v e d a t t h e e a r t h ' s s u r f a c e . B o t h m a t h e m a t i c a l a n d a n a l o g u e m o d e l s f o r s e v e r a l s o u r c e f i e l d s a n d a v a r i e t y o f c o n d u c t i v i t y d i s t r i b u t i o n s h a v e b e e n t r e a t e d i n d e t a i l . T h e f r e q u e n c i e s (10"°̂  t o 10̂  c y c l e s / s e c ) a n d t h e c o n d u c t i v i t i e s (10 t o 10 e m u ) u s e d a r e o f s p e c i a l i n t e r e s t i n g e o p h y s i c s . E x p r e s s i o n s h a v e b e e n d e v e l o p e d f o r t h e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s a t t h e s u r f a c e a n d w i t h i n t h e u p p e r l a y e r o f a h o r i z o n t a l l y s t r a t i f i e d f l a t c o n d u c t i n g e a r t h i n t h e f i e l d o f i n c i d e n t p l a n e w a v e s . E x t e n s i v e r e s u l t s o f a m p l i t u d e s a n d p h a s e a n g l e s h a v e b e e n o b t a i n e d f o r v a r i o u s f r e q u e n c i e s , a n g l e s o f i n c i d e n c e , l a y e r t h i c k n e s s e s , d e p t h s , a n d c o n d u c t i v i t i e s f o r a f l a t c o n d u c t i n g e a r t h c o n s i s t i n g o f o n e , t w o a n d t h r e e u n i f o r m c o n d u c t i n g l a y e r s . I t i s e v i d e n t f r o m t h e r e s u l t s o b t a i n e d , t h a t a l t h o u g h c h a n g e s i n a h o r i - z o n t a l l y l a y e r e d c o n d u c t i n g e a r t h s t r o n g l y a f f e c t t h e a m p l i - t u d e s a n d p h a s e a n g l e s a t t h e e a r t h ' s s u r f a c e , t h e r e l a t i v e l y l a r g e r a t i o H _ / H _ _ o b s e r v e d w h e r e a l a y e r e d g e o l o g i c a l z y s t r u c t u r e i s l i k e l y , c a n n o t b e e x p l a i n e d i n t e r m s o f a p l a n e w a v e s o u r c e f i e l d . S i n c e i n s t u d y i n g g e o m a g n e t i c v a r i a t i o n s a p l a n e w a v e s o u r c e f i e l d i s f r e q u e n t l y a s s u m e d , t h e r e s u l t s 2 0 3 d i s c u s s e d i n - t h i s w o r k a r e o f i n t e r e s t s i n c e t h e y g i v e s o m e i n d i c a t i o n o f t h e e f f e c t t h e c o n d u c t i v i t y d i s t r i b u t i o n i n a l a y e r e d e a r t h w o u l d h a v e . • A s a n e x t e n s i o n t o t h e p r o b l e m o f a c o n d u c t i n g e a r t h c o n s i s t i n g o f s e v e r a l u n i f o r m l a y e r s , e x p r e s s i o n s h a v e b e e n d e v e l o p e d 1 a n d . e v a l u a t e d f o r a m u l t i l a y e r e a r t h ( n l a y e r s ) . E a c h o f s e v e r a l t h i c k l a y e r s w a s d i v i d e d i n t o a s u f f i c i e n t n u m b e r o f s u b l a y e r s , w i t h c h a n g i n g c o n d u c t i v i t y , t o r e p r e s e n t t o a g o o d a p p r o x i m a t i o n a c o n t i n u o u s c h a n g e i n c o n d u c t i v i t y . S e v e r a l c o n d u c t i v i t y f u n c t i o n s o f i n t e r e s t i n g e o p h y s i c s w e r e t r e a t e d i n d e t a i l . T h e r e s u l t s f o r t h i s m u l t i l a y e r e d e a r t h p r o b l e m ^ w h e n c o m p a r e d w i t h t h e r e s u l t s f o r t w o a n d t h r e e l a y e r s , a l l o w a n a s s e s s m e n t a n d c o m p a r i s o n o f t h e e f f e c t i n - h o m o g e n e o u s a n d h o m o g e n e o u s c o n d u c t i n g l a y e r s h a v e o n t h e r e s u l t i n g f i e l d s a t t h e s u r f a c e o f a c o n d u c t i n g e a r t h . A s e c o n d s o u r c e f i e l d w a s s t u d i e d a n a l y t i c a l l y , i n t h e p r o b l e m o f a p l a n e h o m o g e n e o u s c o n d u c t i n g e a r t h i n t h e n e a r f i e l d o f a n o s c i l l a t i n g l i n e c u r r e n t , b y e v a l u a t i n g t h e c o m p o n e n t s - ( a m p l i t u d e s a n d p h a s e a n g l e s ) f o r a w i d e r a n g e o f f r e q u e n c i e s , c o n d u c t i v i t i e s , s o u r c e h e i g h t s , a n d l o c a t i o n s w i t h r e s p e c t t o t h e o v e r h e a d c u r r e n t . - T h e r e s u l t s f o r t h i s l i n e c u r r e n t m o d e l y i e l d r a t i o s o f t h e a m p l i t u d e s o f t h e v e r t i c a l t o h o r i z o n t a l m a g n e t i c f i e l d s i n t h e r a n g e o f e x - p e r i m e n t a l l y o b s e r v e d , v a l u e s . - F r o m t h e d e t a i l e d r e s u l t s o b t a i n e d , i t i s e v i d e n t t h a t i f a s o u r c e o f t h e e l e c t r o m a g - n e t i c v a r i a t i o n s w e r e a n i o n o s p h e r i c l i n e c u r r e n t , t h e n t h e f i e l d c o m p o n e n t s a t t h e e a r t h ' s s u r f a c e s h o u l d b e s t r o n g l y 20^ dependent on- t h e ; s o u r c e f r e q u e n c y , the s o u r c e p o s i t i o n , and t h e c o n d u c t i v i t y o f t he e a r t h . I n compar ing t h e s e r e s u l t s w i t h t ho se f o r t he p l a n e wave s o u r c e , i t i s c l e a r t h a t t h e f i e l d s a t t he e a r t h ' s s u r f a c e s h o u l d be q u i t e d i f f e r e n t f o r the- two sources , 1 • An ana l ogue mode l s u i t a b l e f o r s t u d y i n g t he b e h a v i o r o f n a t u r a l - geomagnet i c and t e l l u r i c f i e l d v a r i a t i o n s f o r v a r i o u s g e o l o g i c a l s t r u c t u r e s was d e s i g n e d and c o n s t r u c t e d . The two t y p e s o f f i e l d s o u r c e s u s e d were an o s c i l l a t i n g s h e e t c u r r e n t and an o s c i l l a t i n g l i n e c u r r e n t , • The model was d e s i g n e d i n s u ch a way' t h a t o t h e r s o u r c e s , s uch as e l e c t r i c and magne t i c d i p o l e s , c o u l d be u s ed as w e l l * . • D i p o l e s o u r c e s w i l l be s t u d i e d as a c o n t i n u a t i o n o f t h i s p r e s e n t work . E x t e n s i v e measurements o f a m p l i t u d e s and phase a n g l e s have been t r e a t e d f o r v a r i o u s g e o l o g i c a l s t r u c t u r e s i n c l u d i n g a- f l a t l a y e r e d e a r t h , c y l i n d r i c a l c o n d u c t i n g b o d i e s embedded i n t he s u r f a c e l a y e r , v e r t i c a l f a u l t s and d y k e s , sea mounts and c o n d u c t i n g domes, c o a s t l i n e s t r u c t u r e s ( s e a - l a n d i n t e r n - f a c e s and an u p w e l l i n g i n a h i g h - c o n d u c t i n g zone w i t h i n t he m a n t l e ) , and i s l a n d s i n an ocean c h a n n e l 0 ! Measurements f o r the- two. s o u r c e s ( s h e e t c u r r e n t and. l i n e c u r r e n t ) were compared and some s i g n i f i c a n t d i f f e r e n c e s i n t he a m p l i t u d e s and phases f o r t h e two s o u r c e s found,, i n d i c a t i n g t h a t t h e s o u r c e f i e l d d i s t r i b u t i o n i n the g e o p h y s i c a l p r ob l em p l a y s an i m p o r t a n t r o l e i n d e t e r m i n i n g t he b e h a v i o r o f t he f i e l d components a t t h e e a r t h * s s u r f a c e 0 ' Measurements f o r t h e l i n e c u r r e n t s o u r c e d e m o n s t r a t e d t h a t t he b e h a v i o r o f t he f i e l d v a r i a t i o n s 2 0 5 n e a r d i s c o n t i n u i t i e s i s s t r o n g l y d e p e n d e n t o n t h e d i s t a n c e t o t h e o v e r h e a d l i n e c u r r e n t . T h e r e s u l t s o b t a i n e d f o r t h e c o a s t l i n e s t r u c t u r e s a n d i s l a n d s i n a n o c e a n c h a n n e l t e n d t o s u p p o r t t h e s t r u c t u r e s p o s t u l a t e d b y v a r i o u s w o r k e r s i n e x - p l a i n i n g e x p e r i m e n t a l l y o b s e r v e d c o a s t a l m a g n e t i c f i e l d . ' ' i Q m a l i e s 0 I n p a r t i c u l a r , i t w a s f o u n d t h a t a , m o d e l s t r u c t u r e ; o - s e n t i n g a n u p w e l l i n g o f a h i g h l y c o n d u c t i n g z o n e i n t h e e a r t h ' s m a n t l e w o u l d h a v e a s i g n i f i c a n t e f f e c t o n t h e e l e c t r o m a g n e t i c v a r i a t i o n s n e a r c o a s t l i n e s . - A n a l o g u e m e t h o d s e m p l o y i n g s u i t a b l y s c a l e d m o d e l s a r e o f c o n s i d e r a b l e i n t e r e s t i n s t u d y i n g g e o p h y s i c a l p r o b l e m s w h i c h d o n o t r e a d i l y s u b m i t t o a n a l y t i c a l s o l u t i o n . - T h e • r e s u l t s o b t a i n e d a n d d i s c u s s e d d e m o n s t r a t e t h a t t h e a n a l o g u e m o d e l u s e d i n t h i s w o r k r e a d i l y l e n d s i t s e l f t o s t u d y i n g a w i d e r a n g e o f g e o l o g i c a l s t r u c t u r e s f o r a v a r i e t y o f s o u r c e f i e l d s . 206 B I B L I O G R A P H Y B o m k e , H . 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T h e E l e c t r i c a n d M a g n e t i c S t a t e o f t h e I n t e r i o r o f t h e E a r t h a s i n f e r r e d f r o m • T e r r e s t r i a l M a g n e t i c V a r i a t i o n s . P h i l o s o p h i c a l T r a n s - a c t i o n s o f t h e R o y a l S o c i e t y , L o n d o n , 22£, ^-27. C h a p m a n , S . a n d W h i t e h e a d , T . T . 1922. T h e I n f l u e n c e o f E l e c t r i c a l l y C o n d u c t i n g M a t e r i a l w i t h i n t h e E a r t h o n V a r i o u s P h e n o m e n a o f T e r r e s t r i a l M a g n e t i s m . T r a n s a c t - i o n s o f t h e C a m b r i d g e P h i l o s o p h i c a l S o c i e t y , 22, ^63. C h r i s t o f f e l , D . A . , J a c o b s , J . A . , J o l l e y , E . S . , K i n n e a r , J . K . , a n d S h a n d , J . A . 1961. • T h e F r a s e r D e l t a E x p e r i - m e n t o f I960. P a c i f i c N a v a l L a b o r a t o r y R e p o r t 61 -5. d ' E r c e v i l l e , I . - a n d K u n e t z , G . 1962. 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