UBC Theses and Dissertations

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UBC Theses and Dissertations

Inversion of magnetotelluric impedances from above young lithosphere Whittall, Kenneth Patrick 1982

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INVERSION FROM OF MAGNETOTELLURIC IMPEDANCES ABOVE YOUNG LITHOSPHERE by KENNETH PATRICK WHITTALL S c . , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n \ THE FACULTY OF GRADUATE STUDIES Department of G e o p h y s i c s and Astronomy We a c c e p t t h i s t h e s i s as 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 THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1982 © K e n n e t h P a t r i c k W h i t t a l l , 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of 6&ofHY&IC% MID A&rgQA)Q*l/ The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date S<g*PT 3 Q , <<7gZ i i A b s t r a c t Ocean b o t t o m magnetometer d a t a f r o m a s i t e on t h e P a c i f i c p l a t e above 3 my o l d l i t h o s p h e r e a r e i n v e r t e d f o r e l e c t r i c a l c o n d u c t i v i t y as a f u n c t i o n o f d e p t h . M a g n e t o t e l l u r i c i m pedances a r e c a l c u l a t e d by t h e v e r t i c a l g r a d i e n t method u s i n g t h e f i e l d s a t t h e OBM i n c o n j u c t i o n w i t h t h o s e measured a t t h e V i c t o r i a G e o m agnetic O b s e r v a t o r y . The a p p r o x i m a t i o n s i n v o l v e d a r e examined. Winnowing c r i t e r i a a r e p r o p o s e d w h i c h i s o l a t e t h o s e impedances c o m p a t i b l e w i t h a l l t h e model and s o u r c e f i e l d a s s u m p t i o n s . T h e s e t h e n d e f i n e t h e b e s t p o s s i b l e d a t a s e t . A number of i n v e r s i o n a l g o r i t h m s a r e a p p l i e d t o t h e d a t a and a wide r a n g e of a c c e p t a b l e c o n d u c t i v i t y p r o f i l e s a r e c o n s t r u c t e d . A l l p r o f i l e s e x h i b i t a u n i f o r m , r e l a t i v e l y h i g h c o n d u c t i v i t y of a b o u t 0.2 S/m from t h e s u r f a c e down t o a d e p t h of 100 km. E x a c t and a p p r o x i m a t e bounds on t h e c o n d u c t a n c e a r e c a l c u l a t e d i n an e f f o r t t o q u a n t i f y t h e n o n - u n i q u e n e s s of t h e d i v e r s e c o n d u c t i v i t y m o d e l s . P r o f i l e s w i t h a minimum of s t r u c t u r e a r e used t o c a l c u l a t e t h e p a r t i a l m e l t i n g and t e m p e r a t u r e v a r i a t i o n s b e n e a t h t h e 3 my o l d s i t e . A l l r e s u l t s a r e compared w i t h t h r e e o t h e r m a g n e t o t e l l u r i c a n a l y s e s above 1, 30 and 72 my o l d l i t h o s p h e r e . The 3 my o l d datum i s d i s c o r d a n t and does n o t f i t t h e t r e n d s i n t e r p r e t e d from the o t h e r t h r e e s t u d i e s . i i i T a b l e o f C o n t e n t s A b s t r a c t ... , . . i i L i s t o f F i g u r e s . ...v Acknowledgements . v i C h a p t e r 1 The Ocean Bottom M a g n e t o t e l l u r i c Method ..1 1.1 I n t r o d u c t i o n 1 1.2 The E x p e r i m e n t , 4 C h a p t e r 2 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 Method .....1.2 2.1 MT F i e l d s i n a One D i m e n s i o n a l E a r t h ...12 2.2 MT F i e l d s i n a Two D i m e n s i o n a l E a r t h 19 2.3 D e r i v a t i o n o f t h e Impedance T e n s o r f o r 1-D M o d e l s ...23 2.4 Impedance T e n s o r I n v a r i a n t s and Symmetries 26 2.5 Impedances from M a g n e t i c F i e l d s A l o n e 27 2.6 P l a n e Wave A s s u m p t i o n 33 2.7 S p h e r i c i t y of t h e E a r t h 35 C h a p t e r 3 D a t a R e d u c t i o n 36 3.1 T a k i n g T r a n s f o r m s 36 3.2 S c a l a r R a t i o E s t i m a t e s 38 3.3 T e n s o r R a t i o E s t i m a t e s 42 3.4 Winnowing R a t i o E s t i m a t e s 45 3.4.1 Ocean F i l t e r 45 3.4.2 V e r t i c a l F i e l d 46 3.4.3 R a t i o e r r o r s 46 3.4.4 D i s p e r s i o n R e l a t i o n s and I n e q u a l i t i e s ...47 3.5 OBM D a t a R e d u c t i o n 50 C h a p t e r 4 I n v e r s i o n of Impedances f o r C o n d u c t i v i t y 54 4.1 B e s t F i t t i n g M o d e l s 56 i v 4.2 L a y e r e d M o d e l s .56 4.2.1 A L a y e r o v e r a H a l f s p a c e . 56 4.2.2 C o n s t a n t nah2 L a y e r s 57 4.3 Smooth M o d e l s ...60 4.3.1 L i n e a r i z e d I n v e r s i o n .60 4.3.2 G e l ' f a n d - L e v i t a n I n v e r s i o n 61 4.4 C o n d u c t a n c e Bounds 61 C h a p t e r 5 C o n c l u s i o n s 66 5.1 OBM R e s u l t s Compared w i t h O t h e r S t u d i e s 66 5.2 S u g g e s t i o n s f o r F u t u r e Work 74 R e f e r e n c e s 78 V L i s t of F i g u r e s F i g u r e 1 E x p e r i m e n t l o c a t i o n map 6 F i g u r e 2 F o u r d a y s of OBM d a t a 7 F i g u r e 3 F o u r d a y s of V I C d a t a .....9 F i g u r e 4 C o m p l e t e OBM h o r i z o n t a l D c h a n n e l ..1.0 F i g u r e 5 One d i m e n s i o n a l model 13 F i g u r e 6 Two d i m e n s i o n a l model 20 F i g u r e 7 Ocean f i l t e r 31 F i g u r e 8 Winnowing c r i t e r i a ...48 F i g u r e 9 F i n a l a p p a r e n t r e s i s t i v i t i e s and p h a s e s 53 F i g u r e 10 OBM c o n d u c t i v i t y m o d e l s 58 F i g u r e 11 OBM c o n d u c t a n c e bounds 64 F i g u r e 12 Minimum s t r u c t u r e c o n d u c t i v i t y models 67 F i g u r e 13 OBM p a r t i a l m e l t i n g and t e m p e r a t u r e c u r v e s 70 F i g u r e 14 JDF, CAL, NCP p a r t i a l m e l t i n g and t e m p e r a t u r e c u r v e s 72 v i A cknowledgements I t h a n k D r . Doug O l d e n b u r g f o r h i s e b u l l i e n t s u p p o r t t h r o u g h o u t t h e t e n u r e of t h i s p r o j e c t . H i s m o r n i n g c h a l k t a l k s were a s r e v i v i f y i n g a s a 50 m i l e b i c y c l e r i d e and when t h e d u s t s e t t l e d an a p h o r i s m o r two was u s u a l l y f o u n d . As w e l l , I thank D r . L a w r i e Law f o r h i s months of h a r d work s p e n t i n a c q u i r i n g , and h i s g e n e r o s i t y i n s u p p l y i n g , t h e s u p e r b q u a l i t y magnetometer d a t a . I t h a n k K e r r y S t i n s o n , Wang J i a Y i n g and Don P l e n d e r l e i t h f o r many i n t e r e s t i n g d i s c u s s i o n s and s u g g e s t i o n s , t h e s p i r i t s of w h i c h a r e i n c a r n a t e i n t h i s t h e s i s . Thanks a r e a l s o due a l l t h e o t h e r p e o p l e of t h e d e p a r t m e n t who make work h e r e v e r y p l e a s a n t i n d e e d . 1 CHAPTER 1 THE OCEAN BOTTOM MAGNETOTELLURIC METHOD 1.1 I n t r o d u c t i o n E l e c t r o m a g n e t i c f i e l d s a r e i n d u c e d i n t h e e a r t h by a v a r i e t y o f n a t u r a l s o u r c e s . The w o r l d w i d e o c c u r r e n c e of l i g h t n i n g s t o r m s g e n e r a t e s h i g h f r e q u e n c y o s c i l l a t i o n s g r e a t e r t h a n a b o u t 1 Hz. Below 1 Hz, 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 m a g n e t i c f i e l d c a u s e v a r i a t i o n s o v e r a wide band of p e r i o d s r a n g i n g f r o m 1 t o 600 s. A t l o n g e r p e r i o d s ( m i n u t e s t o d a y s ) m a g n e t i c s t o r m s r e s u l t i n g from t h e i n t e r a c t i o n o f t h e s o l a r w ind p l a s m a w i t h t h e m a g n e t o s p h e r e d o m i n a t e t h e s p e c t r u m . A r i n g c u r r e n t forms a t an a l t i t u d e o f from 2 t o 10 e a r t h r a d i i a s o p p o s i t e l y c h a r g e d p a r t i c l e s s p i r a l a l o n g m a g n e t i c f i e l d l i n e s i n o p p o s i t e d i r e c t i o n s . The q u i e t day s o l a r v a r i a t i o n (Sq) d e r i v e s from a l a r g e , s t a b l e c u r r e n t s y s t e m c o n c e n t r a t e d i n t h e i o n o s p h e r e n e a r t h e m a g n e t i c p o l e s . T h i s o c c u r s m a i n l y i n r e s p o n s e t o s o l a r h e a t i n g w i t h p e r t u r b a t i o n s p e r p e t r a t e d by t h e s o l a r w i n d . Movements o f t h e h i g h l y c o n d u c t i n g o c e a n g e n e r a t e a d d i t i o n a l f i e l d s w h i c h a r e i m p o r t a n t f o r t h i s p a r t i c u l a r o cean bottom e x p e r i m e n t . S u r f a c e and i n t e r n a l waves, t u r b u l e n c e , t i d e s and c u r r e n t s a l l c o n t r i b u t e t o 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 o b s e r v e d on t h e ocean f l o o r . L a r s e n ( 1 9 7 3 ) , F i l l o u x (1973) and Cox e t a l (1970) d i s c u s s b o t h i o n o s p h e r i c and o c e a n i c s o u r c e s and t h e i r r e l e v a n c e t o e l e c t r o m a g n e t i c i n d u c t i o n a t s e a . The e l e c t r o m a g n e t i c (EM) waves g e n e r a t e d by a l l s o u r c e s r e f l e c t from and r e f r a c t i n t o t h e e a r t h . The r e f r a c t e d waves d i f f u s e v e r t i c a l l y downward f o r a l m o s t any a n g l e o f i n c i d e n c e b e c a u s e of t h e h i g h a i r / e a r t h c o n d u c t i v i t y c o n t r a s t . In 2 g e n e r a l , t h e e a r t h a t t e n u a t e s s h o r t p e r i o d o s c i l l a t i o n s more r a p i d l y t h a n l o n g p e r i o d ones and c o n d u c t o r s a t t e n u a t e t h e f i e l d s more t h a n r e s i s t i v e l a y e r s . A measure o f t h e d e c a y a s a f u n c t i o n of f r e q u e n c y i s t h e m a g n e t o t e l l u r i c impedance Z(u) - E(u)/H(u) where E ( u ) and H(o) a r e o r t h o g o n a l f r e q u e n c y domain e l e c t r i c and m a g n e t i c f i e l d s , r e s p e c t i v e l y ( C a g n i a r d , 1953). I f , o v e r an i s o t r o p i c , h o r i z o n t a l l y l a y e r e d (one d i m e n s i o n a l ) e a r t h , t h e impedance i s known p r e c i s e l y a t e v e r y f r e q u e n c y t h e n t h e c o n d u c t i v i t y - d e p t h p r o f i l e , e(z), i s d e t e r m i n e d e x a c t l y ( B a i l e y , 1970). Thus, t h e v a r i a t i o n of p e n e t r a t i o n o f EM waves as c o d e d by Z(u) i s t h e key t o u n l o c k i n g 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 a one d i m e n s i o n a l e a r t h . Over a more complex two d i m e n s i o n a l e a r t h (<y(z,y) say) t h e impedance becomes a 2x2 t e n s o r r e l a t i n g t h e h o r i z o n t a l e l e c t r i c and m a g n e t i c f i e l d s ( s e e s e c t i o n 2 . 2 ) . T h i s i s b e c a u s e c e r t a i n 2-D and 3-D models may c a u s e c u r r e n t s t o f l o w p a r a l l e l as w e l l as p e r p e n d i c u l a r t o a p r i m a r y i n d u c i n g m a g n e t i c f i e l d . The t e n s o r f o r m u l a t i o n i s a d v a n t a g e o u s s i n c e i n t h e p r i n c i p a l a x e s frame of a 2-D s t r u c t u r e t h e impedances a l o n g and a c r o s s s t r i k e do not depend on t h e s o u r c e f i e l d p o l a r i z a t i o n . S e c t i o n 2.4 d e s c r i b e s r o t a t i o n p r o p e r t i e s of t h e t e n s o r w h i c h d e t e r m i n e t h e p r i n c i p a l a x e s of t h e 2-D s t r u c t u r e and d i s t i n g u i s h between 2-D and t h e v e r y much more d i f f i c u l t 3-D models ( V o z o f f , 1972). U n f o r t u n a t e l y , f o r p r a c t i c a l a p p l i c a t i o n s , c o n s t r a i n t s imposed upon eiz) by t h e MT method a r e not t o o r e s t r i c t i v e . The r e s o l v i n g power of MT d a t a and t h e u n i q u e n e s s of 3 c o n s t r u c t e d models a r e two p r o b l e m s w h i c h have as y e t no r i g o r o u s s o l u t i o n s . Q u a l i t a t i v e l y , i t i s known t h a t h i g h r e s o l u t i o n i s a c h i e v e d f o r c o n d u c t i v e z o n e s where t h e f i e l d s a r e r a p i d l y a t t e n u a t e d . R e s i s t i v e l a y e r s a r e more p o o r l y d e f i n e d . F u r t h e r m o r e , below a c e r t a i n d e p t h e v e n a l o n g p e r i o d datum i s i n s e n s i t i v e t o h i g h c o n d u c t i v i t y s i n c e t h e f i e l d s h ave d e c a y e d t o v e r y s m a l l a m p l i t u d e s ( P a r k e r , 1982). The s e c o n d p r o b l e m o f model u n i q u e n e s s o c c u r s b e c a u s e we can o n l y s e c u r e a f i n i t e number of i n a c c u r a t e MT r e s p o n s e s . U s i n g t h e s e d a t a , any i n v e r s i o n a l g o r i t h m can r e t u r n o n l y a few of t h e i n f i n i t e l y many c o n d u c t i v i t y p r o f i l e s w h i c h f i t t h e d a t a e q u a l l y w e l l . W i t h i n a p a r t i c u l a r r o u t i n e t h e r e may be p a r a m e t e r s t o v a r y w h i c h r e s u l t i n s l i g h t l y d i f f e r e n t m o d e l s . However, a d i f f e r e n t i n v e r s i o n scheme i s l i k e l y 'to r e t u r n m odels w h i c h a r e g l o b a l l y d i s t i n c t and w h i c h c a n n o t be f o u n d by t h e o t h e r a l g o r i t h m ( s e e C h a p t e r 4 ) . Thus, n o n - u n i q u e n e s s p r o h i b i t s o v e r - i n t e r p r e t a t i o n of m i n u t e s t r u c t u r e s v i s i b l e on a s i n g l e <r(z) p r o f i l e ( P a r k e r e t a l , 1983). I n f e r r i n g what s t r u c t u r e s a r e common t o a l l models w h i c h f i t t h e d a t a i s d i f f i c u l t f o r n o n - l i n e a r i n v e r s e p r o b l e m s s u c h as MT. R a t h e r t h a n d e a l i n g w i t h a p r o f u s i o n of c(z) c u r v e s t h e r e may be more p r o f i t i n q u a n t i f y i n g t h e bounds on t h e a v e r a g e c o n d u c t i v i t y between two d e p t h s o r t h e range of c o n d u c t a n c e models a l l o w e d by t h e d a t a ( O l d e n b u r g , 1983). The c o n d u c t a n c e i s s i m p l y t h e i n t e g r a l of t h e c o n d u c t i v i t y from t h e s u r f a c e down t o a c e r t a i n d e p t h . The g l o b a l bounds on t h e c o n d u c t a n c e p r e s e n t e d i n s e c t i o n 4.4 g i v e a measure of t h e r e s o l u t i o n of t h e d a t a as a f u n c t i o n o f d e p t h . 4 E l e c t r i c a l c o n d u c t i v i t y i s an i m p o r t a n t p r o p e r t y s i n c e i t i s r e l a t e d t o o t h e r g e o p h y s i c a l q u a n t i t i e s s u c h as t e m p e r a t u r e , p a r t i a l m e l t i n g and p e t r o l o g y . T h e s e , i n t u r n , p r o v i d e c o n s t r a i n t s on s t r u c t u r a l and dynamic m o d e l s o f t h e e a r t h ' s c r u s t and m a n t l e . R e s u l t s f r o m o c e a n b o t t o m i n s t r u m e n t s p l a c e d above 1, 30 and 72 my o l d l i t h o s p h e r e by Law and G r e e n h o u s e (1981) and F i l l o u x (1967,1977) v a r y w i t h age i n a way t h a t i s c o n s i s t e n t w i t h t h e t h e o r i e s o f s e a - f l o o r s p r e a d i n g . D a t a from t h e s e t h r e e s i t e s were i n v e r t e d and compared by O l d e n b u r g (1981) t o r e v e a l a c o r r e l a t i o n between c o n d u c t i v i t y and l i t h o s p h e r i c age a l s o n o t e d by F i l l o u x ( 1 9 8 0 ) . The p a t t e r n r e p o r t e d by O l d e n b u r g (1981) was t h a t f o r i n c r e a s i n g age t h e d e p t h t o a c o n d u c t i v e r e g i o n i n c r e a s e d f r o m 70 t o 120 t o 180 km, f o r t h e s e d a t a s e t s . The c o n d u c t i v i t y h i g h s a r e w e l l c o r r e l a t e d w i t h t h e s e i s m i c low v e l o c i t y zone and may be r e p r e s e n t a t i v e of a p a r t i a l l y m o l t e n a s t h e n o s p h e r e . R e s u l t s f r o m t h e new s t u d y above 3 my o l d l i t h o s p h e r e d i s r u p t t h i s c o h e r e n t p i c t u r e somewhat by p e r s i s t e n t l y showing a f a i r l y u n i f o r m h i g h c o n d u c t i v i t y zone from t h e s u r f a c e down t o a b o u t 100 km d e p t h . S u b s e q u e n t a n a l y s i s o f t h e p r e v i o u s t h r e e d a t a s e t s u s i n g d i f f e r e n t i n v e r s i o n schemes has shown the o b s e r v e d t r e n d s a r e not as s t r o n g as i n i t i a l l y b e l i e v e d . But even a l l o w i n g f o r t h e h i g h d e g r e e of n o n - u n i q u e n e s s i n h e r e n t i n t h e MT method i t a p p e a r s t h a t t h e 3 my datum i s s t r i k i n g l y d i s c o r d a n t ( s e e s e c t i o n 5 . 1 ) . 1.2 The E x p e r i m e n t S e v e r a l o c e a n bottom magnetometers (OBM's) were d e p l o y e d 5 o f f V a n c o u v e r I s l a n d by L.K.Law d u r i n g t h e summer o f 1980. Law and G r e e n h o u s e (1981) d e s c r i b e t h e i n s t r u m e n t d e s i g n . The one u s e d f o r t h i s s t u d y was f u r t h e s t f r o m t h e c o a s t i n a b o u t 2760m of w a ter and above 3 my o l d l i t h o s p h e r e ( s e e F i g u r e 1 ) . I t r e c o r d e d t h r e e o r t h o g o n a l components of t h e m a g n e t i c f i e l d f rom 23 J u l y t o 26 A u g u s t 1980. The s e n s o r s were f l u x g a t e m a gnetometers a c c u r a t e t o 1 nT. E v e r y m i n u t e t h e component v a l u e s were r e c o r d e d on d i g i t a l t a p e . T i l t m e t e r s w i t h i n t h e i n s t r u m e n t p a c k a g e showed t h e OBM's h o r i z o n t a l a x e s t o be a b o u t one d e g r e e from l e v e l so no c o r r e c t i o n was made t o t h e r e c o r d e d f i e l d v a l u e s . However, a c o r r e c t i o n f o r t h e o r i e n t a t i o n of t h e h o r i z o n t a l a x e s was made t o a l i g n them w i t h g e o m a g n e t i c n o r t h (H) and e a s t ( D ) . The r e l a t i v e a m p l i t u d e of t h e h o r i z o n t a l components d u r i n g m a g n e t i c a l l y q u i e t t i m e s d e t e r m i n e d t h i s r o t a t i o n a n g l e as 20.6° c l o c k w i s e . An example of 4 d a y s of r o t a t e d OBM d a t a i s shown i n F i g u r e 2. The two h o r i z o n t a l c h a n n e l s (H and D) a r e p l o t t e d below t h e v e r t i c a l c h a n n e l ( Z ) . The t i m e on t h e h o r i z o n t a l a x i s i s i n h o u r s s i n c e 0701 23 J u l y 1980 UT. The r e c o r d i s d o m i n a t e d by t h e d i u r n a l Sq v a r i a t i o n w i t h peak t o peak a m p l i t u d e s o f up t o 100 nT on t h e D component. S u p e r i m p o s e d on t h e s e r e g u l a r o s c i l l a t i o n s i s a sudden commencement m a g n e t i c s t o r m b e g i n n i n g n e a r hour 52. R e c o r d s from t h e V i c t o r i a G e o m agnetic O b s e r v a t o r y (VIC) a r e r e q u i r e d t o c a l c u l a t e t h e s e a - f l o o r impedance ( s e e s e c t i o n 2 . 5 ) . T h e r e were two gaps i n t h e VIC d a t a w h i c h t o t a l l e d 9 h o u r s . T h e s e m i s s i n g h o u r s a r e f i l l e d by a s i g n a l most s i m i l a r t o a l l 33 d a i l y r e c o r d s . T h a t common s i g n a l was f o u n d u s i n g 6 F i g u r e 1 . L o c a t i o n map f o r t h e ocean bottom magnetometer (OBM) and t h e V i c t o r i a G eomagnetic O b s e r v a t o r y ( V I C ) . B a t h y m e t r i c c o n t o u r s a r e i n m e t e r s . The arrow n e a r t h e OBM p o i n t s t o w a r d s t h e n o r t h m a g n e t i c p o l e . 7 24 56 88 TIME (HOURS] 120 F i g u r e 2. Four days o f t h e t h r e e component OBM d a t a . D i s g e o m a g n e t i c e a s t , H i s g e o m a g n e t i c n o r t h and Z i s v e r t i c a l l y down. 8 t h e method d e s c r i b e d by O l d e n b u r g , S c h e u e r and L e v y ( 1 9 8 2 ) . E a c h d a i l y t r a c e i s decomposed i n t o a l i n e a r c o m b i n a t i o n o f t h e same o r t h o n o r m a l b a s i s f u n c t i o n s . The common s i g n a l i s t h e n t h a t l i n e a r c o m b i n a t i o n o f b a s i s f u n c t i o n s w h i c h m i n i m i z e s t h e e r r o r between i t and a l l t h e r e c o r d e d d a t a . The m a g n i t u d e of a c o e f f i c i e n t i n t h e l i n e a r c o m b i n a t i o n i n d i c a t e s t h e r e l a t i v e i m p o r t a n c e of t h e a s s o c i a t e d b a s i s f u n c t i o n . S m a l l c o e f f i c i e n t s have b a s i s f u n c t i o n s w h i c h a r e w e a k l y c o r r e l a t e d w i t h t h e d a i l y t r a c e s and may c o n t a i n s i g n i f i c a n t amounts of n o i s e . L a r g e c o e f f i c i e n t s m u l t i p l y b a s i s f u n c t i o n s w h i c h d o m i n a t e e v e r y t r a c e and have l a r g e s i g n a l t o n o i s e r a t i o s . F o r t h i s a p p l i c a t i o n , one c o e f f i c i e n t i n t h e l i n e a r c o m b i n a t i o n i s much l a r g e r t h a n a l l t h e o t h e r s combined. The common s i g n a l i s c o n s t r u c t e d from t h e p r i n c i p a l component b a s i s f u n c t i o n c o r r e s p o n d i n g t o t h i s l a r g e s t c o e f f i c i e n t . Note t h a t i f a l l b a s i s f u n c t i o n s a r e u s e d t h e n t h e method r e d u c e s t o a s i m p l e a v e r a g e of t h e d a i l y r e c o r d s . The a p p r o p r i a t e p a r t s of t h e p r i n c i p a l component were i n s e r t e d i n t o t h e two d a t a gaps w i t h a minimum of o f f s e t a t t h e e n d p o i n t s . T r a c e s r e c o r d e d a t VIC c o r r e s p o n d i n g t o t h o s e of F i g u r e 2 a r e shown i n F i g u r e 3. The r e d u c e d h i g h f r e q u e n c y c o n t e n t a t OBM r e l a t i v e t o VIC i s c a u s e d by t h e s t r o n g l y c o n d u c t i n g , h i g h l y a t t e n u a t i n g o c e a n l a y e r ( s e e F i g u r e 7 ) . An example o f one c h a n n e l of t h e e n t i r e 33 day OBM d a t a s e t i s g i v e n i n F i g u r e 4. E a c h s u b g r a p h i s one day l o n g , i s i n d i v i d u a l l y n o r m a l i z e d , and b e g i n s a t a t i m e near l o c a l m i d n i g h t . M a g n e t i c s t o r m s d i s t u r b t h e r e c o r d o v e r d a y s 2 t o 5 and 24 t o 27. 9 2 4 56 88 TIME [HOURS) 1 2 0 F i g u r e 3. F o u r d a y s of d a t a from V I C . Y i s g e o m a g n e t i c e a s t , X i s g e o m a g n e t i c n o r t h and V i s v e r t i c a l l y down. 10 F i g u r e 4. The c o m p l e t e 33 day r e c o r d o f t h e OBM h o r i z o n t a l D c h a n n e l . E a c h day i s i n d i v i d u a l l y n o r m a l i z e d and b e g i n s a b o u t 1.5 h o u r s b e f o r e l o c a l m i d n i g h t . 12 CHAPTER 2 THEORY OF THE MAGNETOTELLURIC METHOD T h i s c h a p t e r c o n t a i n s t h e o r y r e l e v a n t t o t h e MT method i n g e n e r a l and t o t h i s o c e a n b o t t o m a p p l i c a t i o n i n p a r t i c u l a r . M a x w e l l ' s e q u a t i o n s a r e u s e d t o d e r i v e t h e MT e q u a t i o n s w h i c h g o v e r n t h e d i f f u s i o n o f h a r m o n i c e l e c t r o m a g n e t i c waves i n t o t h e c o n d u c t i n g e a r t h . In s e c t i o n 2.1 and 2.3 t h e impedance a t t h e s u r f a c e o f a s i m p l e l a y e r e d e a r t h i s d e r i v e d . I t s v a l u e i s o b t a i n e d f r o m a r e c u r s i o n f o r m u l a i n terms of t h e impedances o r r e f l e c t i o n c o e f f i c i e n t s a t l o w e r l e v e l s . In a two d i m e n s i o n a l e a r t h a 2x2 impedance t e n s o r r e l a t e s t h e h o r i z o n t a l e l e c t r i c and m a g n e t i c f i e l d s ( s e c t i o n 2 . 2 ) . The t e n s o r f o r m u l a t i o n i s more s t a b l e s i n c e t h e p r i n c i p a l impedances d e r i v e d from i t do not depend on t h e s e n s o r o r i e n t a t i o n o r on t h e t o t a l f i e l d p o l a r i z a t i o n . The r o t a t i o n a l p r o p e r t i e s of t h e t e n s o r o u t l i n e d i n s e c t i o n 2.4 g i v e t h e p r i n c i p a l d i r e c t i o n s of a 2-D e a r t h and d i s t i n g u i s h between t h i s and a more complex 3-D e a r t h s t r u c t u r e . The f i l t e r i n g e f f e c t of t h e h i g h l y c o n d u c t i n g ocean on t h e d i f f u s i n g f i e l d s i s c o n s i d e r e d and e x p l o i t e d t o c a l c u l a t e s e a - f l o o r impedances from m a g n e t i c f i e l d s a l o n e ( s e c t i o n 2 . 5 ) . L a s t l y , i n s e c t i o n s 2.6 and 2.7, t h e p l a n e wave and f l a t e a r t h a p p r o x i m a t i o n s a r e examined and f o u n d t o be v a l i d f o r t h e p e r i o d r a n g e u s e d i n t h i s s t u d y . 2.1 MT F i e l d s i n a One D i m e n s i o n a l E a r t h When t h e e a r t h model i s r e s t r i c t e d t o be composed of a s t a c k of u n i f o r m , h o r i z o n t a l , f l a t l a y e r s t h e n i t i s termed one d i m e n s i o n a l s i n c e t h e c o n d u c t i v i t y v a r i e s o n l y w i t h d e p t h . 13 Z J -1 zn-l z l ayer 1 a , l ayer 2 l a y e r j l a y e r n i ° i a 2 y 2 o-j V j °"n * n A l B l A ? B ? A j - 1 B j - i A n - 1 B n - i h, h 2 > y n F i g u r e 5. The one d i m e n s i o n a l model w i t h n o t a t i o n u s e d i n t h i s t h e s i s . W i t h i n l a y e r j o f t h i c k n e s s hj t h e c o n d u c t i v i t y i s CJ , t h e p a r a m e t e r = JT>Tutf7 and and Bj a r e t h e a m p l i t u d e s o f t h e downgoing and u p g o i n g waves, r e s p e c t i v e l y , a t t h e l e v e l ZJ . L a y e r n i s a h a l f s p a c e of i n f i n i t e t h i c k n e s s . B n_, i s e q u a l t o z e r o . 14 An example o f s u c h an e a r t h i s i l l u s t r a t e d i n F i g u r e 5. I f t h e i n c i d e n t EM f i e l d s a r e p l a n e waves and t h e y a r e r e f r a c t e d t o t h e v e r t i c a l t h e n s i m p l e d i f f u s i o n e q u a t i o n s p i l o t t h e i r v o y a g e i n t o t h e e a r t h . F u r t h e r m o r e , t h e impedance Z(o) m easured a t t h e s u r f a c e i s c o m p a c t l y e x p r e s s e d a s a r e c u r r e n c e r e l a t i o n i n t e r m s of t h e l a y e r p a r a m e t e r s . We b e g i n w i t h t h e u b i q u i t o u s M a x w e l l e q u a t i o n s c u r l E = - M d H / H c u r l H = <y(z)E •+ c^E/c^t d i v E = 0 d i v H = 0 (2.1) where E i s t h e e l e c t r i c f i e l d v e c t o r , H i s t h e m a g n e t i c f i e l d v e c t o r , n i s t h e m a g n e t i c p e r m e a b i l i t y (assumed c o n s t a n t f o r a l l l a y e r s and e q u a l t o t h e f r e e s p a c e v a l u e y o = 4 i r x l 0 ~ 7 H/m), e i s t h e d i e l e c t r i c c o n s t a n t ( e q u a l t o 8.86x10" 1 2 F/m), and * ( z ) i s t h e c o n d u c t i v i t y . A s s u m i n g t h e f i e l d s a r e q u a s i -p e r i o d i c , t h a t i s , E ~ e x p ( + i u t ) and H ~ e x p ( + i o t ) , and a l l t r a n s i e n t s have d e c a y e d away, we c a n w r i t e t h e f i r s t two e q u a t i o n s i n (2.1) as c u r l E = - i j x u H c u r l H = <r(z)E + i e o E (2.2) B e c a u s e t h e p r o d u c t eo i s v e r y s m a l l compared t o e(z) ( K e l l e r and F r i s c h k n e c h t 1966, p212) t h e s e c o n d t e r m i n t h e l a s t e q u a t i o n may be d r o p p e d . M o r e o v e r , t h e f i e l d s i n a 1-D e a r t h do not change w i t h c o o r d i n a t e s x or y b e c a u s e of t h e v e r t i c a l l y d i f f u s i n g p l a n e wave a s s u m p t i o n . T h e r e f o r e , d e r i v a t i v e s w i t h r e s p e c t t o x and y v a n i s h and e q u a t i o n s (2.2) become 15 dEy/dz = +i»/oHx dHy/dz = -tfEx dEx/dz = - i ^ u H y dHx/dz = +<rEy 0 = Hz 0 = Ez (2.3) D i f f e r e n t i a l e q u a t i o n s f o r e a c h h o r i z o n t a l component of E and H a r e f o u n d from e q u a t i o n s (2.3) t o be d 2 E x / d z 2 = r 2 E x d 2 H x / d z 2 = r 2 H x d 2 E y / d z 2 = r 2 E y d 2 H y / d z 2 = y 2 H y (2.4) where r 2 = i c u t f ( z ) . S o l u t i o n s t o t h e s e d i f f u s i o n e q u a t i o n s a l l have t h e same form w i t h i n l a y e r s where r 2 i s c o n s t a n t . F o r a h a l f s p a ce o f c o n s t a n t c o n d u c t i v i t y e0, a t y p i c a l s o l u t i o n i s - r z + i o t • + r z + i o t E x ( z ) = A e + B e where r= ( 1 +i ) ( »o<t0/2.)* = ( 1 +i ) fi and A and B a r e c o n s t a n t s . The s o l u t i o n i s a l i n e a r c o m b i n a t i o n of two i n d e p e n d e n t d e c a y i n g waves, one t r a v e l l i n g down ( t h e f i r s t term) and t h e o t h e r t r a v e l l i n g up ( t h e s e c o n d t e r m ) . F o r a h a l f s p a c e t h e r e i s no u p g o i n g wave s i n c e t h e r e a r e no s o u r c e s w i t h i n t h e e a r t h and t h e r e a r e no h o r i z o n s t o r e f l e c t t h e downgoing wave back t o w a r d s t h e s u r f a c e . T h e r e f o r e , i n t h e l i n e a r c o m b i n a t i o n a bove, t h e c o e f f i c i e n t B=0. The r e m a i n i n g term i s -fiz i(ut-£z) -fiz - i f i z A e e = E x ( 0 ) e e w h i c h r e p r e s e n t s a wave d e c a y i n g as i t t r a v e l s down i n t o t h e h a l f s p a c e . The r a t e o f a t t e n u a t i o n i s g o v e r n e d by t h e p a r a m e t e r fi. At a d e p t h z = l / p t h e wave has d e c a y e d t o 1/e o f i t s s u r f a c e a m p l i t u d e E x ( 0 ) . T r a d i t i o n a l l y , 6=1/0 i s c a l l e d t h e s k i n d e p t h . F o r a g i v e n h a l f s p a c e c o n d u c t i v i t y a0, 6 i s s m a l l f o r h i g h f r e q u e n c i e s and l a r g e f o r low f r e q u e n c i e s . 16 Hence, p r o g r e s s i v e l y , l o n g e r p e r i o d s sample p r o g r e s s i v e l y d e e p e r s t r u c t u r e s . A l t e r n a t i v e l y , f o r a f i x e d f r e q u e n c y o, 6 i s s m a l l e r f o r l a r g e r c o n d u c t i v i t i e s . T h i s i m p l i e s r a p i d f i e l d a t t e n u a t i o n w i t h i n c o n d u c t i n g l a y e r s . F o r a f i x e d t i m e t , t h e p h a s e of t h e e l e c t r i c f i e l d c h a n g e s by 2ir r a d i a n s from i t s s u r f a c e v a l u e when 0z = 2ir o r z=2ir6. T h i s d i s t a n c e i s a measure of t h e w a v e l e n g t h of t h e d i f f u s i n g f i e l d . W i t h i n l a y e r j o f a more complex, l a y e r e d 1-D e a r t h we w r i t e f o r Zj>z>z^_, + i u t + y ; ( z ; - z ) + i o t - r ; ( z ; - z ) E x ^ ( z ) = A j e 4 4 +.B^e * * (2.5) where r j = (1 + i') ( w e ^ / 2 ) 1 = 0 + i ) p j , and A^ and Bj a r e t h e a m p l i t u d e s of t h e downgoing and u p g o i n g waves r e s p e c t i v e l y as t h e y c r o s s t h e l e v e l z^. The c o r r e s p o n d i n g s o l u t i o n f o r H y ^ ( z ) i s f o u n d from e q u a t i o n s (2.3) and (2.5) as 7; +it)t + 7 ; ( Z ; ~z) + i U t ~ y : ( Z : ~ Z ) Hy;Cz) = [A;e * * - B,e * * ] (2.6) * i»io * 4 a g a i n f o r z^>z>z^_, . The q u a n t i t y of i n t e r e s t i s t h e impedance Z x y ( z ) w h i c h i s t h e r a t i o o f t h e e l e c t r i c t o t h e m a g n e t i c f i e l d and as s u c h i t i s i n d e p e n d e n t of c h a n g e s i n t h e i n c i d e n t f i e l d s t r e n g t h . y ; (z: -z) - y : ( Z ; -z) E x ; ( z ) i n u hie 4 + B;e A 4 Z x y ( z ) = — - — = 1 i (2.7) H y ^ ( z ) y- ri(z\-z) -rAzi-z) 4 « A j e 4 * - B^e 4 4 To f i n d t h e impedance a t t h e s u r f a c e of a 1-D e a r t h , Z x y ( O ) , Aj and B^ must be c a l c u l a t e d . V a l u e s f o r t h e s e c o n s t a n t s a r e f o u n d by c o n s i d e r i n g Zxy a t two l e v e l s w i t h i n t h e same y 2 = c o n s t a n t l a y e r . The i d e n t i t y (A^/B^ )^ =exp( In (A^/B^ ) *) i s u s e f u l i n p r o d u c i n g 1 7 lptt ^ Z x y ( z ) = --- c o t h [ ( l n ( A i / B : ) * + r ; ( z ; - z ) ] w i t h i n l a y e r j . A t t h e b o t t o m of t h i s l a y e r z=z^ and Z x y ( z ^ ) = — c o t h [ ( l n ( A ^ / B | ) z ] At t h e t o p o f t h e j ' t h l a y e r z = z a n d i f " x. Z x y ( z ^ _ , ) = — c o t h [ l n ( A j / B ^ ) * + h • ] where hj=z^-z^__! i s t h e t h i c k n e s s o f t h e j ' t h l a y e r . C o m b i n i n g t h e s e l a s t two e q u a t i o n s g i v e s i ^ u / r i Z x y ( z ; ) ^ Z x y ( z / _ , ) = c o t M c o t h - 1 / - 1 A - j + r- h • ] (2.8) • r j \ i v u / 1 * T h i s shows t h a t t h e impedance a t t h e t o p of a l a y e r may be w r i t t e n i n terms of t h e impedance a t t h e bo t t o m o f t h a t l a y e r . F u r t h e r m o r e , s i n c e t h e h o r i z o n t a l components of E and H a r e c o n t i n u o u s a c r o s s l a y e r b o u n d a r i e s , Zxy i s c o n t i n u o u s . So Zxy(O) i s f o u n d by s t e p p i n g up r e c u r s i v e l y • t h r o u g h t h e model l a y e r s b e g i n n i n g w i t h t h e b o u n d a r y c o n d i t i o n a t t h e t o p of t h e b a s a l h a l f s p a c e Zxy(z„_, ) = i p o / r n (2.9) s i n c e i n e q u a t i o n (2.8) h h = ° ° and c o t h ( » ) = 1. F o r a l a y e r o v e r a h a l f s p a c e l e t z^ - i=z o = 0 and z^ = z,=h, i n e q u a t i o n ( 2 . 8 ) . Then iMtf / Z,r A Zxy(O) = c o t h [ n h , + c o t h " 1 j ] (2.10) r 1 \ i >io / where Z , =Zxy (z ,) = i//u/r 2 • K e l l e r and F r i s c h k n e c h t (1966) and Kaufman and K e l l e r ( 1 9 8 1 ) , f o r example, p r e s e n t t h e s i m p l e e x t e n s i o n of t h i s f o r m u l a f o r any number of l a y e r s o v e r a h a l f s p a c e . 18 A n o t h e r u s e f u l e q u a t i o n f o r Zxy(O) i s o b t a i n e d by r e c a s t i n g e q u a t i o n (2.8) i n terms o f a c o n t i n u e d f r a c t i o n o f r e f l e c t i o n c o e f f i c i e n t s . T h i s r e p r e s e n t a t i o n i s c e n t r a l t o t h e t h r e e i n v e r s i o n programmes o f P a r k e r and Whaler (1981) and t h a t o f F i s c h e r e t a l (1 9 8 1 ) . S i n c e t h e impedance i s c o n t i n u o u s a c r o s s a l a y e r i n t e r f a c e we use e q u a t i o n (2.7) t o w r i t e i f t o A ; . | + , B ; - i z r ( z i - « ) = z i ( z r ' , = ;-; A l l + r ; h ; - r ; h : inu A^e 1 *+ Bj-e * 11 i A^e 4 4 - B^e ^ ^  where h^=z^-z,-_, . T h i s i s r e w r i t t e n a s ') "\ "Y +r\hi ~r\h\ where t h e r e f l e c t i o n c o e f f i c i e n t R . . PB..Z.B (2.„, The r a t i o o f u p g o i n g t o downgoing a m p l i t u d e s i s + r i h i " r i h \ Bi: ,_ . Vli^!______:_!ie A i _ , + r i h i ~r'\hi M u l t i p l y i n g t o p and bot t o m by R q - o ^ a n c ^ a d d i n g (1-1) t o t h e num e r a t o r g i v e s , a f t e r a n o t h e r few s t e p s , + 2r< h, = 1/R^.,M + i " i (2.12) + 2r-hJ e 2 • A V , V I + 2 19 w h i c h i s t h e d e s i r e d c o n t i n u e d f r a c t i o n a t l a s t . From e q u a t i o n ( 2 . 7 ) , w i t h i n l a y e r j ~2y: AZ i i . u 1 + B ; / A J e * Z x y ( z ) = i — i — yj, -2y; Az /• 1 " V A - j e 1 where Az=z^-z. The bou n d a r y c o n d i t i o n i s t h a t B h_,=0 s i n c e t h e r e i s no r e f l e c t e d wave from t h e b o t t o m l e s s b a s a l h a l f s p a c e . The f i n a l s u r f a c e impedance i s " 2 r , h , i nu 1 + B y/A! e Zxy (0) = — (2.13) r i - 2 n h , 1 " B,/A, e F o r a u n i f o r m h a l f s p a c e B ^ O , h,= <=o and Zxy ( 0 ) = i »a/r 1 wh i c h a g r e e s w i t h e q u a t i o n ( 2 . 9 ) . F o r a l a y e r o v e r a h a l f s p a c e B 2=0 and f r o m e q u a t i o n (2.12) B i 1 1 - 1 / R 2 1 2 r r , ~ r2 A! R12 1/R12 r1 + r 2 Hence ~ 2 r i h 1 i >JU 1 + R , 2 e Zxy ( 0 ) = (2.14) r i ~ 2 r i h , 1 - R 1 2 e The f o r e g o i n g d i s c u s s i o n from e q u a t i o n (2.5) onwards c a l c u l a t e d Zxy from Ex and Hy. T h e r e i s an e q u i v a l e n t d e v e l o p m e n t f o r Zyx d e r i v e d f r o m Ey and Hx. In f a c t Zyx=-Zxy f o r t h e 1-D model assumed h e r e . 2.2 MT F i e l d s i n a Two D i m e n s i o n a l E a r t h A h y p o t h e t i c a l 2-D model i s p r e s e n t e d i n F i g u r e 6. In t h e s t r i k e d i r e c t i o n x t h e r e i s no s t r u c t u r a l v a r i a t i o n . Hence, F i g u r e 6. A two d i m e n s i o n a l e a r t h w i t h s t r i k e d i r e c t i o n 21 f o r v e r t i c a l l y d i f f u s i n g p l a n e waves d e r i v a t i v e s w i t h r e s p e c t t o x v a n i s h . The M a x w e l l e q u a t i o n s (2.2) become "bEz/e>Y ~ ^Ey/b;z = - i ^ u H x 3Ex/S>z = - i y o H y - c>Ex/c>y = -ifiuliz S)Hz/^y - "£>HyA>z = <rEx ; ~bHx/hz = tfEy - bHx/2>y = <rEz (2.15) U s u a l l y t h e i n c i d e n t p r i m a r y f i e l d w i l l be e l l i p t i c a l l y p o l a r i z e d i n t h e x-y p l a n e and w i l l i n d u c e complex s e c o n d a r y f i e l d s b e c a u s e o f t h e l a t e r a l i n h o m o g e n e i t i e s . A l i n e a r impedance t e n s o r , Z, r e l a t i n g t h e t o t a l e l e c t r i c and m a g n e t i c f i e l d s a d m i t s a l l p o s s i b l e i n f l u e n c e s between components. The l i n e a r i t y r e s u l t s from t h e l i n e a r M a x w e l l e q u a t i o n s . Thus, we assume E = Z H or (Ex\ I Z1 Z2 Z3\ /Hx Ey = Z4 Z5 Z6 Hy VEz/ \ Z7 Z8 Z9/ \Hz/ S i m i l a r l y , we p o s t u l a t e an a d m i t t a n c e t e n s o r Y where H = Y E. S i m p l i f i c a t i o n s r e s u l t from d e c o m p o s i n g th e p r i m a r y f i e l d i n t o two i n d e p e n d e n t p o l a r i z e d f i e l d s . The f i r s t , E - p o l a r i z a t i o n , has t h e e l e c t r i c f i e l d v e c t o r a l o n g t h e s t r i k e d i r e c t i o n . Thus, E=(Ex,0,0) and H=(0,Hy,Hz). The s e c o n d , H - p o l a r i z a t i o n , has t h e m a g n e t i c f i e l d v e c t o r p a r a l l e l t o s t r i k e w i t h E = ( 0 , E y , E z ) and H=(Hx,0,0). T h e s e two p o l a r i z a t i o n s a r e i m p o r t a n t b e c a u s e a p r i m a r y f i e l d i n e i t h e r s t a t e i n d u c e s a s e c o n d a r y f i e l d i n t h a t same s t a t e v i a e q u a t i o n s ( 2 . 1 5 ) . T h e r e f o r e , t h e t o t a l f i e l d , p r i m a r y p l u s s e c o n d a r y , decomposes i n t o E- and H - p o l a r i z a t i o n s . The c o r r e s p o n d i n g impedance r e l a t i o n s h i p s a r e 22 E - p o l H - p o l . lEx) Z2 0 = # . Z5 \ ol Z8 ( °\ / 0 • • Ey = Z4 - • • l E z / Z7 • • 'Hy ^Hz-/Hx' 0 I 0 where .. i n d i c a t e s an u n c o n s t r a i n e d e l e m e n t . The 3x3 Z t e n s o r c o l l a p s e s t o a 2x2 form b e c a u s e of t h e f o l l o w i n g a r g u m e n t s . A c c o r d i n g t o t h e a d m i t t a n c e t e n s o r , Hz i s ' a l i n e a r c o m b i n a t i o n of Ex, E y , and E z . But from Z t h e s e f i e l d components a r e , i n t u r n , a l i n e a r c o m b i n a t i o n of Hx, Hy, and Hz. So we w r i t e Hz = A Hx + B Hy where A and B a r e complex c o n s t a n t s . F o r E - p o l a r i z a t i o n Hx=0 and so Hz=BHy. W i t h Hz p r o p o r t i o n a l t o Hy t h e t h i r d column of Z i s i n c o r p o r a t e d i n t o t h e s e c o n d column. The r e s u l t i n g 3x2 m a t r i x now has i t s s e c o n d and t h i r d rows l i n e a r l y d e pendent so one row i s d i s c a r d e d . Hence, E - p o l (Ex < 0, ... Zxy . . . 0 / ' °1 .Hy / A s i m i l a r argument f o r H - p o l a r i z a t i o n y i e l d s H - p o l ( 0\ Ey/ 0 Zyx . . . / (Hx 1 I 0, To c o m p l e t e l y s p e c i f y a l l f o u r e l e m e n t s of t h e 2x2 impedance t e n s o r t h e t o t a l f i e l d must be composed of EM waves of b o t h p o l a r i z a t i o n s . In g e n e r a l t h e n , /Ex\ / 0 Zxy\/Hx\ = ( 2 ' 1 6 ) \Ey/ lZyx 0 / \Hy/ In r e f e r e n c e frames w h i c h do n o t have a x e s p a r a l l e l and p e r p e n d i c u l a r t o s t r i k e t h e d i a g o n a l e l e m e n t s a r e n o n - z e r o (see s e c t i o n 2.4 f o r r o t a t i o n p r o p e r t i e s of t h e impedance 23 t e n s o r ) . 2.3 D e r i v a t i o n of t h e Impedance T e n s o r f o r 1-D M o d e l s The h e u r i s t i c a rguments f o r a 2x2 Z g i v e n i n t h e p r e c e e d i n g s e c t i o n a r e s u p p o r t e d by a more r i g o r o u s d e v e l o p m e n t f o l l o w i n g L o e w e n t h a l and L a n d i s m a n ( 1 9 7 3 ) . T h e s e a u t h o r s d e r i v e d f r o m f i r s t p r i n c i p l e s t h e impedance t e n s o r f o r an a n i s o t r o p i c l a y e r e d medium. In t h i s s e c t i o n i t i s shown ( f o r 1-D i s o t r o p i c m o d e l s ) t h a t i f a 2x2 Z i s assumed t h e n u s i n g j u s t t h e e l e c t r i c and m a g n e t i c f i e l d s r e a s o n a b l e e x p r e s s i o n s f o r e a c h e l e m e n t a r e p o s s i b l e . E q u a t i o n s (2.5) and (2.6) a l o n g w i t h s i m i l a r e x p r e s s i o n s f o r Ey and Hx a r e put i n m a t r i x form and t h e e x p ( + i o t ) f a c t o r i s d r o p p e d . Thus, + 7{ AZ 0 . , / ~y: AZ 0 \ y. E x j ( z ) \ le J E y ^ ( z ) / \ 0 e ^ /UjyJ [ 0 e 7 ^ ) + y- Az. / H x ; ( z ) \ / 0 -e » WEtx' i U l +y:AZ 4 n (E\X\ 1 (2.17) U y : ( z ) . i 1 n where A z = z i - z , z^>z>z- , and A: and B- have been r e p l a c e d by i i 0 J o E +^ and E"^ w h i c h a r e t h e downgoing and u p g o i n g e l e c t r i c f i e l d a m p l i t u d e s , r e s p e c t i v e l y , a t t h e l e v e l Zy Note t h a t i f E and H a r e b o t h w r i t t e n as a 2x2 t e n s o r t i m e s a common v e c t o r C t h e n a 2x2 impedance t e n s o r Z i s f o u n d from t h e f o l l o w i n g . L e t E = W, C and H = (y/i»io) W2 C where W, and W2 a r e 2x2 t e n s o r s as y e t unknown. R e w r i t e t h e s e c o n d e x p r e s s i o n as C = ( i ^ u / y ) W 2 _ 1 H and s u b s t i t u t e i n t o 24 t h e f i r s t e x p r e s s i o n E = ( i u u / r ) W, W2'1 H T h e r e f o r e , i *io 2 = W, W2" 1 [ (.2.19) •y So t h e impedance t e n s o r i s s i m p l y a c o m b i n a t i o n o f t h e t e n s o r s w h i c h p r e m u l t i p l y t h e common v e c t o r , C. In s e a r c h of t h i s common v e c t o r we f i r s t r e c a l l t h a t t h e impedance i s c o n t i n u o u s a c r o s s an i n t e r f a c e so we w r i t e S ^ ) . | 1 . l ( « i ) H 1 ( » i ) S u b s t i t u t e e q u a t i o n s (2.17) and (2.18) i n t o t h i s l a s t e x p r e s s i o n and n o t e t h a t Az = 0 E ; ( z J ) = I E +. + I E". - 1 1 = ~ i = ~ \ .Ii l | i U where I_ i s t h e 2x2 i d e n t i t y m a t r i x and / 0 -1 \ ' \ i o j I f we e x p r e s s E". i n terms of E +. t h e n t h e p r o b l e m i s s o l v e d w i t h C = E +-. In f a c t , from t h e e q u a t i o n f o r E - ( z ; ) above o i A E- = (I - 2A" 1 ) E +. (2.20) where A = Z ( Z : ) G -Ii + I T h e r e f o r e , E ^ ( z ^ ) = 2 (I_ - A" 1 ) E +^ H . (z • ) = G A" 1 E + . ' 3 i ^  ~ \ 25 and f r o m e q u a t i o n (2.19) i IIU Z • ( Z . ) = — ( i - A ' 1 ) [G A ' 1 ] " 1 . . ' r i The impedance w i t h i n t h e j ' t h l a y e r i s f o u n d by s u b s t i t u t i n g E"^ f r om e q u a t i o n (2.20) i n t o e q u a t i o n s (2.17) and ( 2 . 1 8 ) . + 7;AZ ~ 7 ; A z E - ( z ) = [e » £ + e » U - 2A" 1 ) ] E + - = W, E +. * — A 1 y • +y •AZ — 7 • AZ 7 • H . ( z ) = — - G [e 1 I ~ e ^ (£ - 2A" 1 ) ] E + = — 1 W2 E +. i >/ u i i j i u i where A z = Z j - z . Hence, i ti u Z • ( z ) = — W, W2" 1 Zj>z>z^_, ri Example (1) A H a l f s p a c e im I 0 1 \ Z,(0) = — 7 , V - i 0/ w h i c h a g r e e s w i t h t h e p r e v i o u s r e s u l t s o f Zxy=-Zyx, Zxx=Zyy=0 and e q u a t i o n ( 2 . 9 ) . Example (2) One L a y e r o v e r a H a l f s p a c e +7,h, ~ y i h , W, = (e + R, 2 e ) I +7ih, - r , h , W2 = (e - R 1 2 e ) G and -271h, i ttu 1 + R1 2 e / 0 1 \ Z,(0) = r, - 2 n h , 1-1 0/ 1 - R, 2e T h i s i s a g e n e r a l i z a t i o n o f e q u a t i o n (2.14) w i t h R 1 2 d e f i n e d as i n e q u a t i o n ( 2 . 1 1 ) . 26 2.4 Impedance T e n s o r I n v a r i a n t s and Symmetries Once a l l t h e t e n s o r e l e m e n t s a r e e s t i m a t e d (by t e c h n i q u e s d e s c r i b e d i n s e c t i o n s 3.2 and 3.3) i t i s i m p o r t a n t t o s e l e c t o n l y t h o s e p e r i o d s a t w h i c h t h e t e n s o r i s c o n s i s t e n t w i t h c e r t a i n t h e o r e t i c a l c o n s t r a i n t s . When t h e c o n s t r a i n t s a r e v i o l a t e d i t i n d i c a t e s t h e p r e s e n c e o f l a r g e amounts o f n o i s e o r a breakdown i n one o r more of t h e s o u r c e and model a s s u m p t i o n s . O t h e r c r i t e r i a f o r c h o o s i n g p e r i o d s c o m p a t i b l e w i t h t h e a s s u m p t i o n s a r e p r e s e n t e d i n s e c t i o n 3.4. Many o f t h e t e n s o r p r o p e r t i e s a r i s e from an a r b i t r a r y r o t a t i o n of t h e c o o r d i n a t e a x e s a b o u t t h e v e r t i c a l . In a r o t a t e d frame t h e impedance t e n s o r i s Z' = T Z T T where t h e s u p e r s c r i p t T i n d i c a t e s a t r a n s p o s e and c o s e s i n e - s i n e c o s e i s t h e m a t r i x w h i c h r o t a t e s t h e x-y a x e s of F i g u r e 5 or 6 c l o c k w i s e by e>0. F o r any s u c h s i m i l a r i t y t r a n s f o r m a t i o n t r ( Z ) = t r ( Z ' ) d e t ( Z) = d e t ( Z ' ) Zxy-Zyx = Z'xy-Z'yx (2.21) where t h e t r a c e t r { Z ) = Zxx + Zyy, and d e t ( Z ) i s t h e d e t e r m i n a n t of t h e m a t r i x . We have a l r e a d y seen i n s e c t i o n s 2.1 and 2.3 t h a t f o r 1-D m o d els Zxx=Zyy=0 and Zxy=-Zyx. S e c t i o n 2.2 showed t h a t i n t h e p r i n c i p a l a x e s frame of 2-D models Zxx=Zyy=0, and so f o r one or two d i m e n s i o n a l models tr(£)=0 i n any r e f e r e n c e frame. D u r i n g t h e c o u r s e of a c o o r d i n a t e r o t a t i o n t h r o u g h 180° t h e impedance t e n s o r e l e m e n t s t r a v e l a l o n g e l l i p s e s i n t h e 27 complex p l a n e ( E g g e r s , 1982). The d i a g o n a l e l e m e n t s s h o u l d d i s p l a y two minima as t h e c o o r d i n a t e a x e s c o i n c i d e w i t h t h e p r i n c i p a l d i r e c t i o n s of t h e 2-D e a r t h . The o f f - d i a g o n a l e l e m e n t s s h o u l d have j u s t a s i n g l e minimum. V o z o f f (1972) i. 1 shows t h a t t h e p r i n c i p a l a x e s o r i e n t a t i o n o v e r :2-D s t r u c t u r e s i s f o u n d d i r e c t l y by m a x i m i z i n g |Zxy j 2 + | Z yx| 2 w i t h r e s p e c t t o 9. T h i s same a n g l e a l s o m i n i m i z e s |Zxx| 2 + | Z y y | 2 . A c o m b i n a t i o n of t h e f i r s t and t h i r d i n v a r i a n t s i n e q u a t i o n (2.21) g i v e s a n o t h e r r o t a t i o n a l i n v a r i a n t c a l l e d t h e skew | Zxx + Zyy| S, = |Zxy — Zyx| C o m b i n i n g t h e f i r s t and s e c o n d i n v a r i a n t s g i v e s Zxx + Zyy S 2 = ZxxZyy - ZxyZyx T h e s e have been u s e d t o a s s e s s t h e 2-D a s s u m p t i o n . ( V o z o f f , 1972) s i n c e S,=S 2=0 f o r any 1-D o r 2-D e a r t h . A n o t h e r symmetry p r o p e r t y d i s c u s s e d by F i s c h e r (1975) i s t h a t f o r 2-D models w i t h a v e r t i c a l p l a n e of symmetry A r g ( Z x x ) = A r g ( Z y y ) = A r g ( Z x y + Z y x ) where A r g i s t h e p r i n c i p a l v a l u e of t h e argument of t h e complex impedance. He s t a t e s t h a t t h i s r e l a t i o n s h i p as w e l l as S,=0 must h o l d above t h e s e t y p e s o f s t r u c t u r e s . 2.5 Impedances from M a g n e t i c F i e l d s A l o n e The OBM u s e d f o r t h i s s t u d y r e c o r d e d o n l y m a g n e t i c f i e l d components so t h a t more i n f o r m a t i o n i s r e q u i r e d t o c a l c u l a t e t h e MT i m p e dances. T h a t i n f o r m a t i o n ( w i t h a few a p p r o x i m a t i o n s ) i s p r o v i d e d by t h e r e c o r d s of t h e V i c t o r i a 28 G e o m a g n e t i c O b s e r v a t o r y ( V I C ) . We assume t h a t VIC i s on t h e oc e a n s u r f a c e d i r e c t l y above OBM and use t h e v e r t i c a l g r a d i e n t method d e s c r i b e d by Schmucker ( 1 9 7 0 ) , P o e h l s and von H e r z e n (1976) and Law and G r e e n h o u s e ( 1 9 8 1 ) . To f i n d t h e impedance on t h e s e a - f l o o r u s i n g t h i s t e c h n i q u e we need t o assume 1) t h e f i e l d s a t t h e s u r f a c e above OBM a r e a d e q u a t e l y r e p r e s e n t e d by t h e f i e l d s a t VIC, 2) t h e r e a r e no m a g n e t i c f i e l d s i n d u c e d by t h e homogeneous, i s o t r o p i c o cean and 3) t h e s t r u c t u r e below OBM and VIC i s 1-D. The f i r s t a s s u m p t i o n , w h i c h t r a n s p o r t s VIC t o above OBM, r e q u i r e s t h a t t h e s o u r c e f i e l d s r e m a i n u n i f o r m o v e r t h e 350 km s e p a r a t i n g OBM from V I C . In a d d i t i o n , t h e d i f f e r e n c e between t h e o c e a n i c and c o n t i n e n t a l e n v i r o n m e n t s must n o t a f f e c t t h e f i e l d s s i g n i f i c a n t l y . C o m p a r i s o n of r e c o r d s from s t a t i o n s s e p a r a t e d by s e v e r a l h u n d r e d s of k i l o m e t e r s r e v e a l s t h a t t h e f i e l d s a r e i n d e e d q u i t e u n i f o r m o v e r t h e s e d i s t a n c e s . Chave e t a l (1981) g i v e an a p p r o x i m a t e e x p r e s s i o n f o r t h e s o u r c e f i e l d h o r i z o n t a l s c a l e l e n g t h w h i c h f o r t h e p e r i o d s c o n s i d e r e d h e r e i s much l a r g e r t h a n t h e OBM—VIC s e p a r a t i o n . Law and Gr e e n h o u s e (1981) a r g u e t h a t t h e e f f e c t s of c h a n g i n g s o u r c e ,and c o n d u c t i v i t y s t r u c t u r e s c o u n t e r a c t e a c h o t h e r . The s e c o n d a s s u m p t i o n of an i m m o b i l e , u n i f o r m ocean i s t e n a b l e a t p e r i o d s o t h e r t h a n t i d a l p e r i o d s . L a r s e n (1968) and Cox e t a l (1970) d i s c u s s EM f i e l d s i n d u c e d by o c e a n i c m o t i o n s . T h e r e a r e s e v e r a l t i d a l s p e c t r a l l i n e s c l u s t e r e d a r o u n d p e r i o d s of 12 and 24 h o u r s w h i c h a r e a v o i d e d i n t h i s a n a l y s i s . The t h i r d a s s u m p t i o n , t h a t of a 1-D s t r u c t u r e , i s n e v e r t r u e but i t must be made i n o r d e r t o d e r i v e an impedance from 29 m a g n e t i c f i e l d s a l o n e . A l l t h e i n v e r s i o n t e c h n i q u e s u s e d h e r e r e q u i r e a 1-D e a r t h anyway, so t h e a p p r o x i m a t i o n i s i n v o k e d s o o n e r r a t h e r t h a n l a t e r . The m a g n e t i c f i e l d s measured a t t h e two ocean l e v e l s a r e , from e q u a t i o n (2.6) (exp( + i o t ) , H - p o l a r i z a t i o n ) . , r i + r i Z i " r i Z , +i-ot Hy(0) = --- (A,e - B,e ) e i MO y, + i u t H y ( z , ) = --- (A, - B,) e i ttu •i-w i t h . r 1 = ( 1 +i ) ( cu<f/2) x , *=3.3 S/m (an a v e r a g e f o r t h e ocean) and z 1=2760 m. S o l v i n g f o r t h e unknown c o n s t a n t s y i e l d s - i o t - r , z , inu e (Hy(0) - H y ( z , ) e ) A i = ___ _ r 1 2 s i n h r 1 z , -i«t +r1z, i ^ u e (Hy(0) - H y ( z , ) e ) B i = — _ r i 2 s i n h r 1 z , The s e a - f l o o r impedance from e q u a t i o n (2.7) i s i i/u A , + B , Zxy (z , ) = r , A , - B , T h e r e f o r e i f i u c o s h n z , - 1 /Ry Z x y ( z ! ) = (2.22) r i s i n h r 1 z ! where R y = H y ( z , ) / H y ( 0 ) . F o r E - p o l a r i z a t i o n and e x p ( + i o t ) i f i u c o s h n z , - 1/Rx Z y x ( z , ) = + (2.23) r1 s i n h r 1z ! where R x = H x ( z , ) / H x ( 0 ) . F o r 1-D m o d els we e x p e c t Zxy=-Zyx. Comparing e q u a t i o n s (2.22) and (2.23) we see t h a t t h i s i s t r u e 30 s i n c e t h e h o r i z o n t a l m a g n e t i c components Hx and Hy a r e a f f e c t e d i n e x a c t l y t h e same way o v e r 1-D m o d e ls as t h e y d i f f u s e f r o m t h e s e a - s u r f a c e t o t h e s e a - f l o o r . Thus, R=Rx=Ry. Be c a u s e of t h e f i l t e r i n g e f f e c t o f t h e o c e a n t h e OBM f i e l d s a r e a t t e n u a t e d and phase l a g g e d w i t h r e s p e c t t o V I C . T h i s c o n s t r a i n s |R| t o be l e s s t h a n u n i t y and t h e phase of R, 4>R, t o be l e s s t h a n z e r o d e g r e e s . F i g u r e 7 shows |R| and $R f o r an o c e a n w i t h <y=3.3 S/m and 2^=2760 m above a r a n g e o f e a r t h s t r u c t u r e s . As t h e h a l f s p a c e e a r t h model becomes l e s s c o n d u c t i v e t h e a t t e n u a t i o n of t h e f i e l d s between s e a - s u r f a c e and s e a - f l o o r i n c r e a s e s ( c u r v e s a t h r o u g h f ) . T h i s i s due t o a d e s t r u c t i v e i n t e r f e r e n c e of t h e downgoing wave w i t h t h e u p g o i n g wave r e f l e c t e d from t h e s e a b o t t o m . The r e f l e c t i o n c o e f f i c i e n t 1*7 - r«7 R 1 2 = ~ J C 1' + a p p r o a c h e s u n i t y as <r2 a p p r o a c h e s z e r o and so t h e n e t f i e l d i n t h e o cean i s v e r y s m a l l . As t h e e a r t h c o n d u c t i v i t y i n c r e a s e s t o w a r d s t h a t of t h e o c e a n , R 1 2 a p p r o a c h e s z e r o and |R| f a l l s o f f w i t h p e r i o d a c c o r d i n g t o t h e s i m p l e s k i n d e p t h d e c a y expi-z^/6) where 6= (2/»ue, )*". F o r e2>cy t h e r e f l e c t i o n c o e f f i c i e n t i s n e g a t i v e and t h e i n c i d e n t and r e f l e c t e d waves i n t e r f e r e c o n s t r u c t i v e l y t o keep |R| n e a r u n i t y o v e r a b r o a d range o f p e r i o d s . Two t h r e e l a y e r m o d e ls ( c u r v e s g and h) have more v a r i a b l e |R| c u r v e s but t h e y s t i l l l i e w i t h i n i n t u i t i v e bounds and a p p r o a c h t h e h a l f s p a c e c u r v e s a t s h o r t and l o n g p e r i o d s . In c o n t r a s t , t h e <t>R f u n c t i o n s f o r t h e m u l t i l a y e r e a r t h models p l o t t e d i n F i g u r e 7B a r e more s e n s i t i v e t o t h e 31 F i g u r e 7. The o c e a n f i l t e r . The m a g n i t u d e o f t h e r a t i o R o f a s e a - f l o o r m a g n e t i c f i e l d component t o t h e c o r r e s p o n d i n g s e a - s u r f a c e m a g n e t i c component i s shown i n A. The phase of t h i s r a t i o i s shown i n B. A l l c u r v e s a r e c a l c u l a t e d a s s u m i n g an ocean t h i c k n e s s o f 2670 m and c o n d u c t i v i t y <r = 3.3 S/m. C u r v e s a t h r o u g h f a r e d e r i v e d f r o m s u b o c e a n i c h a l f s p a c e models of c o n d u c t i v i t y 10, 1, 0.1, 0.01, 0.001 and 0.0001 S/m, r e s p e c t i v e l y . C u r v e g r e s u l t s from a two l a y e r o v e r a h a l f s p a c e s u b o c e a n i c model w i t h <y n = 0 . 0 1 , « 2 = 0.001, <r3 = 0.1 S/m, h 1 = h 2 = 50 km. C u r v e h i s from a s i m i l a r model w i t h « 1 =0.001, * 2 = 0.1, <y 3 = 0 . 0 1 S/m, h 1 = h 2 = 50 km. 32 1 0 " 2 I O " 1 1 0 ° 1 0 1 1 0 2 1 0 3 PERIOD (HR) •33 d e t a i l s of l a y e r i n g , t h e g e n e r a l t r e n d , however, i s t h a t t h e p hase l a g i n c r e a s e s as t h e s u b o c e a n i c c o n d u c t i v i t y d e c r e a s e s . 2.6 P l a n e Wave A s s u m p t i o n In t h i s s e c t i o n we c o n s i d e r t h e v a l i d i t y o f t h e p l a n e wave a p p r o x i m a t i o n t o t h e s o u r c e f i e l d s . The most p r o b a b l e o r i g i n o f t h e f i e l d s i s i n t h e h i g h l y c o n d u c t i n g E - l a y e r of t h e i o n o s p h e r e a t an a l t i t u d e of a b o u t 110 km. The c u r r e n t s a r e not u n i f o r m l y d i s t r i b u t e d a r o u n d t h e e a r t h but c o n c e n t r a t e d i n a u r o r a l and e q u a t o r i a l e l e c t r o j e t s . One model t h a t t r i e s t o a c c o u n t f o r n o n - u n i f o r m s o u r c e f i e l d s i s t h a t of a l i n e c u r r e n t a t some h e i g h t h above a 1-D e a r t h (Hermance and P e l t i e r , 1970). Such a c u r r e n t f l o w i n g i n t h e y d i r e c t i o n l e a d s t o a g e n e r a l i z a t i o n of one of e q u a t i o n s (2.4) as V E y / ^ x 2 + }>2Ey/bz2 = i„u«(z)Ey Assume E y ( x , z ) = X ( x ) Z ( z ) and use s e p a r a t i o n of v a r i a b l e s t o o b t a i n X ( x ) = A cos(mx) + B s i n ( m x ) Z ( z ) = C e x p ( k z ) + D e x p ( - k z ) where -m2 i s t h e s e p a r a t i o n c o n s t a n t , k2 = ( i i J o c ( z ) + m 2) and A, B, C, and D a r e c o n s t a n t s . The e l e c t r i c f i e l d i s E y ( x , z ) = [Acos(mx) + B s i n ( m x ) ] [ C e x p ( k z ) + D e x p ( - k z ) ] (2.24) The h o r i z o n t a l m a g n e t i c f i e l d Hy i s d e r i v e d from t h i s u s i n g one of t h e M a x w e l l e q u a t i o n s ( 2 . 2 ) . Note t h a t f o r p l a n e waves t h e h o r i z o n t a l wavenumber m=0 and e q u a t i o n (2.24) r e d u c e s t o an e x p r e s s i o n s i m i l a r t o t h o s e i n s e c t i o n 2.1. The t o t a l f i e l d a t t h e e a r t h ' s s u r f a c e i s formed by an a p p r o p r i a t e s u p e r p o s i t i o n o f s o l u t i o n s f o r a l l wavenumbers 0 < m < ° ° . I f t h e 34 s o u r c e model i s g e n e r a l i z e d from a l i n e c u r r e n t t o a s h e e t c u r r e n t t h e n P e l t i e r and Hermance (1971) show how a l i n e a r c o m b i n a t i o n o f l i n e c u r r e n t s o l u t i o n s i s made t o r e p r e s e n t t h e s h e e t c u r r e n t d i s t r i b u t i o n . T h e s e a u t h o r s d e r i v e i n t e g r a l e q u a t i o n s f o r t h e f i e l d s due t o a s h e e t c u r r e n t w i t h a G a u s s i a n d i s t r i b u t i o n 110 km above a 1-D e a r t h . The s t a n d a r d d e v i a t i o n of t h e d i s t r i b u t i o n i s t a k e n as 240 km. They f i n d t h a t t h e agreement between n o n - u n i f o r m and p l a n e wave s o u r c e impedances i s a f f e c t e d by o b s e r v e r p o s i t i o n and e a r t h c o n d u c t i v i t y . We d i s c u s s t h e s e two p a r a m e t e r s i n t u r n . D i r e c t l y b e n e a t h t h e G a u s s i a n e l e c t r o j e t t h e agreement i s good, b e i n g g e n e r a l l y w i t h i n e x p e r i m e n t a l e r r o r u n t i l p e r i o d s g r e a t e r t h a n s e v e r a l h o u r s . The p l a n e wave a p p r o x i m a t i o n becomes worse as t h e o b s e r v e r moves l a t e r a l l y from under t h e e l e c t r o . j e t . A t a h o r i z o n t a l d i s t a n c e of 2.5 t o 3 s t a n d a r d d e v i a t i o n s t h e m i s f i t r e a c h e s a maximum. S t i l l l a r g e r d i s t a n c e s r e v e r s e t h i s t r e n d and improve t h e agreement once a g a i n . The OBM s i t e u s e d i n t h i s s t u d y was l o c a t e d i n t h e mid-l a t i t u d e s ( g e o m a g n e t i c c o o r d i n a t e s ) p e r h a p s 2000 km from t h e c o r e of a u r o r a l e l e c t r o j e t s and even f u r t h e r f r o m e q u a t o r i a l e l e c t r o j e t s . T h e r e f o r e , on t h e b a s i s of p o s i t i o n , t h e p l a n e wave a p p r o x i m a t i o n i s v a l i d . The s e c o n d f a c t o r i n f l u e n c i n g t h e a c c u r a c y of t h e p l a n e wave a p p r o x i m a t i o n t o t h e G a u s s i a n e l e c t r o j e t s o u r c e i s t h e e a r t h c o n d u c t i v i t y . B e n e a t h t h i s e x t e n d e d s o u r c e t h e r e w i l l be l a r g e d e v i a t i o n s from p l a n e wave t h e o r y when t h e e a r t h i s v e r y r e s i s t i v e s u c h as i s f o u n d i n c o n t i n e n t a l a r e a s . But t h e e a r t h i s e x p e c t e d t o become v e r y c o n d u c t i v e a t l a r g e d e p t h s b e c a u s e 35 of t h e h i g h t e m p e r a t u r e s t h e r e . P e l t i e r and Hermance (1971) show t h a t f o r t h e s e t y p e s o f m o d e l s t h e agreement i s much i m p r o v e d . F o r a c t i v e , o c e a n i c a r e a s s u c h as i n o u r s t u d y t h e h i g h t e m p e r a t u r e s and hence h i g h c o n d u c t i v i t i e s a r e c l o s e t o t h e s u r f a c e . T h e r e f o r e , on t h e b a s i s o f e a r t h c o n d u c t i v i t y , t h e p l a n e wave a p p r o x i m a t i o n i s s t i l l r e a s o n a b l y v a l i d . 2.7 S p h e r i c i t y of t h e E a r t h P l a n e waves p r o p a g a t i n g v e r t i c a l l y downwards i n a 1-D e a r t h behave d i f f e r e n t l y i f t h e one d i m e n s i o n i s c h a n g e d from a C a r t e s i a n d e p t h z t o a s p h e r i c a l p o l a r r a d i u s r . A d i s c u s s i o n o f t h e f l a t e a r t h a p p r o x i m a t i o n i s g i v e n by Kaufman and K e l l e r , 1981, C h a p t e r 6. They show t h a t s p h e r i c i t y becomes i m p o r t a n t as t h e w a v e l e n g t h X., or s k i n d e p t h 6, of t h e f i e l d a p p r o a c h e s t h e r a d i u s of t h e e a r t h , a. Thus t h e impedances of t h e f l a t and s p h e r i c a l e a r t h a r e v e r y c l o s e when a/6 > 10 where 6= J~/>T/j*tr'. C o m b i n i n g t h e s e a p p r o x i m a t e r e l a t i o n s g i v e s pT < 445 f o r p e r i o d s T i n h o u r s and an e a r t h of c o n s t a n t r e s i s t i v i t y p n-m. P e r i o d s up t o 10 h o u r s a r e a c c e p t a b l e i f t h e major f i e l d a t t e n u a t i o n o c c u r s i n a r e g i o n w i t h p o f t h e o r d e r of 45 n-m. A r i g o r o u s a n a l y s i s o f t h e c o n v e r s i o n from a s p h e r i c a l t o a f l a t e a r t h i s g i v e n by W e i d e l t ( 1 9 7 2 ) . F o r t h e p e r i o d s c o n s i d e r e d i n t h i s s t u d y an e a r t h f l a t t e n i n g t r a n s f o r m a t i o n i s n o t r e q u i r e d . 36 CHAPTER 3 DATA REDUCTION The methods o f t r a n s f o r m i n g t h e d a t a t o t h e f r e q u e n c y domain a r e examined i n . s e c t i o n 3.1. Once i n t h e f r e q u e n c y domain t h e s c a l a r and t e n s o r t r a n s f e r f u n c t i o n s between t h e OBM and VIC h o r i z o n t a l f i e l d s a r e c a l c u l a t e d ( s e c t i o n s 3.2 and 3 . 3 ) . G r e a t c a r e i s t a k e n t o compute e r r o r b a r s t h a t have a f i r m s t a t i s t i c a l b a s i s . Winnowing c r i t e r i a a r e p r o p o s e d i n s e c t i o n 3.4 t o f a c i l i t a t e s e l e c t i o n o f t h e b e s t p o s s i b l e d a t a s e t . F i n a l l y , some d e t a i l s o f t h e OBM a n a l y s i s a r e p r e s e n t e d ( s e c t i o n 3 . 5 ) . 3.1 T a k i n g T r a n s f o r m s A c r i t i c a l s t e p i n t h e a n a l y s i s i s t o t r a n s f o r m t h e h i g h q u a l i t y d a t a o f , f o r example, F i g u r e 4 t o e q u a l l y h i g h q u a l i t y f r e q u e n c y domain d a t a . The d a i l y r e c o r d s a r e d o m i n a t e d by t h e Sq v a r i a t i o n . I t s e f f e c t s a r e most p r o n o u n c e d n e a r l o c a l noon and a r e m i n i m a l n e a r l o c a l m i d n i g h t when t h e i n d u c e d c u r r e n t s a r e d e c a y i n g v e r y g r a d u a l l y . The Sq s i g n a l imposes c o n s t r a i n t s on t h e r e c o r d l e n g t h s e l e c t e d f o r t r a n s f o r m a t i o n t o t h e f r e q u e n c y domain. S i n c e t h e f a s t F o u r i e r t r a n s f o r m assumes a p e r i o d i c e x t e n s i o n t h e r e s h o u l d be no d i s c o n t i n u i t y between t h e i n i t i a l and f i n a l t i m e domain f i e l d v a l u e s . I f t h e r e i s a jump, s p u r i o u s f r e q u e n c i e s w i l l l e a k i n t o t h e t r a n s f o r m domain. To m i n i m i z e t h e o f f s e t we b r e a k t h e r e c o r d i n t o m u l t i p l e s of 24 h o u r s as i n F i g u r e 4. T h e r e a r e two a d v a n t a g e s of s e g m e n t i n g t h e d a t a i n t h i s way. F i r s t l y , t h e t i d a l s p e c t r u m has many l i n e s c l u s t e r e d a r o u n d 12 and 24 h o u r s . The e n e r g y i n t h e s e l i n e s c a n s p r e a d o ut i n t o a d j a c e n t p e r i o d s i f 37 t h e r e c o r d l e n g t h i s n o t an e x a c t m u l t i p l e of t h e t i d a l p e r i o d . D a t a segments w h i c h a r e m u l t i p l e s o f 24 h o u r s w i l l h e l p m i n i m i z e t h e s p r e a d i n g . A b e t t e r p l a n would be t o n o t c h o u t a l l t h e known t i d a l l i n e s o r t o remove them by a l e a s t s q u a r e s f i t t o t h e i r known p e r i o d s . S e c o n d l y , w e l l - d e t e r m i n e d F o u r i e r c o e f f i c i e n t s , w i t h many d e g r e e s of f r e e d o m , a r e f o u n d by s t a c k i n g t h e segment t r a n s f o r m s a s w e l l as s m o o t h i n g them w i t h a s p e c t r a l window. The n e x t s t e p i n t h e a n a l y s i s s e e k s t o remove any l o n g p e r i o d t r e n d s f r o m t h e d a t a and a l s o t o match t h e e n d p o i n t s t o r e d u c e t r a n s f o r m r i n g i n g . One a l t e r n a t i v e i s t o remove t h e mean and t a p e r w i t h a c o s i n e b e l l t y p i c a l l y of h a l f w i d t h t e n p e r c e n t o f t h e s e r i e s l e n g t h . T h i s method f a i l s t o d e a l w i t h l o n g p e r i o d t r e n d s . F u r t h e r m o r e , t a p e r i n g c o r r e s p o n d s t o a f r e q u e n c y domain c o n v o l u t i o n and any b r o a d s m o o t h i n g o f t h e s t e e p 1/f n a t u r a l e l e c t r o m a g n e t i c s p e c t r u m w i l l o v e r e m p h a s i z e l o n g e r , p e r i o d s . A s e c o n d a l t e r n a t i v e i s t o remove a l e a s t s q u a r e s l i n e a r t r e n d and t a p e r . B e c a u se of t h e Sq waveform, s u c h a t r e n d r e m o v a l o f t e n l e a v e s t h e s e r i e s w i t h a l a r g e r e n d p o i n t o f f s e t t h a n i n i t i a l l y p r e s e n t . The t h i r d a l t e r n a t i v e i s t h e one u s e d i n t h i s a n a l y s i s . A ramp between t h e f i r s t s e r i e s p o i n t and t h e l a s t i s s u b t r a c t e d . T h i s a n n i h i l a t e s a l l w r a p a r o u n d o f f s e t s w h i l e a t t h e same t i m e removes a r e a s o n a b l e e s t i m a t e of t h e l o n g p e r i o d t r e n d . The method i s f a u l t y f o r segments w h i c h have r a p i d c h a n g e s n e a r t h e i r ends b u t b e c a u s e t h e y a l l a r e c h o s e n t o b e g i n n e a r l o c a l m i d n i g h t t h i s i s n o t a p r o b l e m . Most ramps have s l o p e s l e s s t h a n 1 n T / h r . A f t e r t h e t r a n s f o r m s a r e s t a c k e d t h e y a r e smoothed w i t h a 38 P a r z e n window. P r o p e r t i e s o f s p e c t r a l windows a r e f u l l y d i s c u s s e d by J e n k i n s and W a t t s ( 1 9 6 8 ) . The P a r z e n window i s p r e f e r r e d o v e r t h a t of B a r t l e t t o r Tukey b e c a u s e i t has s m a l l s i d e l o b e s and g i v e s n o n - n e g a t i v e s p e c t r a l e s t i m a t e s . The d e g r e e s o f f r e e d o m ( d o f ) f o r t h i s window a r e 3.71T/M where T i s t h e s e r i e s d u r a t i o n and M i s t h e t i m e c o r r e s p o n d i n g t o t h e maximum l a g of t h e t i m e domain window. E a c h F o u r i e r c o e f f i c i e n t b e g i n s w i t h 2 d o f , one e a c h f o r i t s r e a l and i m a g i n a r y p a r t . S m o o t h i n g w i t h a P a r z e n window g i v e s e a c h e s t i m a t e 3.71T/M d o f . S t a c k i n g n segments r e s u l t s i n a t o t a l o f 3.71nT/M d o f . Many d e g r e e s of f r e e d o m r e d u c e t h e s p e c t r a l e s t i m a t e e r r o r b ut i n c r e a s e t h e b i a s . A b a l a n c e must be f o u n d between t h e s e two c o m p e t i n g e f f e c t s t o g i v e a c c u r a t e impedances c l o s e t o t h e i r t r u e v a l u e s . D e t a i l s of t h e c h o i c e s made i n t h i s s t u d y a r e p r e s e n t e d i n s e c t i o n 3.5. 3.2 S c a l a r R a t i o E s t i m a t e s As m e n t i o n e d i n s e c t i o n 2.5 t h e OBM r e c o r d e d o n l y m a g n e t i c f i e l d components. However, by t a k i n g t h e r a t i o o f t h e OBM f i e l d s t o t h o s e measured a t VIC a p p r o x i m a t e MT impedances a r e f o u n d v i a t h e v e r t i c a l g r a d i e n t method. W i t h o u t e l e c t r i c f i e l d s , t h e n , we must put our e f f o r t i n t o f i n d i n g s t a b l e e s t i m a t e s f o r t h e once removed r a t i o R r a t h e r t h a n f i n d i n g t h e s e a - f l o o r impedance Z ( z , ) d i r e c t l y . E q u a t i o n s (2.22) and (2.23) a r e t h e n u s e d t o g i v e t h e impedance. R a t i o s a r e f o u n d from e i t h e r s c a l a r models s u c h as c o n s i d e r e d i n t h i s s e c t i o n or from t h e t e n s o r models of s e c t i o n 3.3. The s c a l a r model i s 39 D ( o ) = R ( o ) Y ( o ) + e 1 ( u ) H(o) = R(o) X ( o ) + e 2 ( o ) where D(o) and H(o) a r e t h e h o r i z o n t a l f i e l d s a t OBM, Y(u) and X(o ) a r e t h e c o r r e s p o n d i n g f i e l d s a t VIC, e ^ o ) and e 2 ( u ) a r e w h i t e n o i s e e r r o r s a nd R(o) i s t h e t r a n s f e r f u n c t i o n t o be f o u n d . A s t r a i g h t f o r w a r d e s t i m a t e of R = D/Y s u f f e r s s e v e r e l y from w i d e l y v a r y i n g F o u r i e r c o e f f i c i e n t s and t h e e r r o r s e,. A b e t t e r e s t i m a t e i s f o u n d by a s s u m i n g R i s c o n s t a n t o v e r a band of a d j a c e n t p e r i o d s and from day t o day a t a s i n g l e p e r i o d . Then t h e r e a r e many e q u a t i o n s f o r a s i n g l e complex number R. S e v e r a l s o l u t i o n s a r e p o s s i b l e . F o r i n s t a n c e , c o n s i d e r i n g o n l y t h e e a s t - w e s t f i e l d s , we w r i t e <DD*> <YY*> <DD*> <DY*> R 1 = ^ 2 = I ^ 3 I <YD*> <YY*> <D> R„ = R 5 = <D/Y> <Y> where < > i n d i c a t e s an a v e r a g e o v e r d a y s a n d / o r p e r i o d s a nd * i n d i c a t e s a complex c o n j u g a t e . R a t i o s R u and R 5 have been t e s t e d on s y n t h e t i c examples and a r e not as s t a b l e as t h e o t h e r s . They y i e l d e s t i m a t e s of R f u r t h e r f r o m t h e t r u e v a l u e ( b i a s ) and undergo much more v a r i a t i o n . R, and R 2 a r e c o m p l i m e n t a r y s i n c e t h e f i r s t o v e r e s t i m a t e s and t h e s e c o n d u n d e r e s t i m a t e s R i n t h e p r e s e n c e of i n c o h e r e n t n o i s e . To show t h i s w r i t e D = D $ + D„ Y = Y s + .Y„ where t h e s u b s c r i p t s r e f e r s t o t h e t r u e s i g n a l and t h e s u b s c r i p t n r e f e r s t o n o i s e . Then 40 <DD*> = <(D 4 + D h ) ( D * + D*)> = <D SD*> + <D„D*> s i n c e <DsD*>=0 f o r i n c o h e r e n t n o i s e D „ . S i m i l a r l y , <YY*> = <Y SY*> •+ <Y„Y*> and <DY*> = <D SY*> a s s u m i n g D„ and Y„ a r e a l s o i n c o h e r e n t . T h e r e f o r e <DD*> <D,D*> + <D„D*> <D SD?> <D hD*> R i = = = + ] <YD*> ^ <Y SD*> <Y SD*> <D SD*> and <DY*> <D 5Y*> <D SY*> <Y nY*> R z = = = t l +  <YY*> <Y SY*> + <Y nY*> <Y SY*> <Y SY*> Thus, D n o i s e power w i l l b i a s upwards and Y n o i s e power w i l l b i a s R 2 downwards. Some s o r t o f a v e r a g e of R, and R 2 would r e d u c e t h e b i a s i n t h e e s t i m a t e . R 3 p r o v i d e s t h a t a v e r a g e s i n c e i t i s t h e g e o m e t r i c mean of R, and R 2. The phase of R 3 i s o b t a i n e d from t h e c r o s s s p e c t r u m <DY*> w h i c h i s t h e phase of b o t h and R 2. The e r r o r s i n t h e s c a l a r r a t i o e s t i m a t e s a r e f o u n d u s i n g t h e t h e o r y of f r e q u e n c y domain r e s p o n s e f u n c t i o n s d e s c r i b e d by J e n k i n s and W a t t s , 1968, p429 and Bendat and P i e r s o l , 1971, p l 9 9 . These a u t h o r s show how t h e smoothed power s p e c t r u m of th e n o i s e may be w r i t t e n a p p r o x i m a t e l y as < e e*> = <ee*> + <YY*> |R-R|2 where <ee*> i s t h e smoothed power s p e c t r u m of t h e r e s i d u a l n o i s e £ = D - RY and R = <DY*>/<YY*> i s t h e e s t i m a t e o f R wh i c h m i n i m i z e s t h e l 2 - n o r m o f e. I f t h e s m o o t h i n g g e n e r a t e s n d e g r e e s of f r e e d o m 41 and e'e'* r e p r e s e n t s t h e t r u e n o i s e power t h e n n < e e * > n<£e*> n<YY*> . = _ .+ I R-R | 2 e ' e ' * e'e'* e'e** i s d i s t r i b u t e d a s c h i - s q u a r e d w i t h n d e g r e e s of freedom,. Hence, by t h e a d d i t i v e p r o p e r t y o f x 2 v a r i a b l e s t h e two s t a t i s t i c a l l y i n d e p e n d e n t terms on t h e r i g h t hand s i d e of t h e p r e v i o u s e q u a t i o n a r e d i s t r i b u t e d a s x 2 w i t h d e g r e e s o f f r e e d o m t h a t sum t o n. The f i r s t t e r m on t h e r i g h t i s o n l y d i f f e r e n t f r o m t h e t e r m on t h e l e f t i n t h a t i t i s b a s e d on e i n s t e a d of e. Two a d d i t i o n a l c o n s t r a i n t s a r e p l a c e d on £, one e a c h by t h e r e a l and i m a g i n a r y p a r t of R. T h e r e f o r e , t h e f i r s t t e r m on t h e r i g h t i s d i s t r i b u t e d a s x 2 - 2 . T h i s l e a v e s t h e s e c o n d term on t h e r i g h t d i s t r i b u t e d as x 2 2 . The F-d i s t r i b u t i o n g o v e r n s t h e r a t i o of two x 2 v a r i a b l e s w i t h ' x 2 , / n ! x 2 2 / n 2 d i s t r i b u t e d a s F n l , h 2 . T h e r e f o r e , n - 2 <YY*> | R - R | 2 2 i s d i s t r i b u t e d a s F 2 , h _ 2 . S i n c e = < ( D - R Y ) ( D * - R * Y * ) > = <DD*> (1 - r2DY) where t h e s q u a r e d c o h e r e n c e <DY*><YD*> y 2 D y =  <DD*><YY*> we w r i t e n - 2 <YY*> | R - R | 2 2 <DD*> ( 1 - T 2 D Y ) 42 i s d i s t r i b u t e d a s F 2 , n _ 2 . The (1-6) c o n f i d e n c e i n t e r v a l f o r th e t r a n s f e r f u n c t i o n R i s t h e n 2 <DD*> |R-R|2 < — (l-r 2DY) f 2 , h _ 2 ( 1 ~ 6 ) = k 2 n-2 <YY*> where f 2 , „ _ ; 2 (1-6) i s t h e 100(1-6) p e r c e n t a g e p o i n t o f an F-d i s t r i b u t i o n w i t h 2 and n-2 d e g r e e s o f fre e d o m . I n d i v i d u a l |R| and ^R c o n f i d e n c e i n t e r v a l s a r e a p p r o x i m a t e d by ( J e n k i n s and W a t t s , 1968, p434) I | S | " IRI I * k s i n | *R - *R| < k/|R| . 3.3 T e n s o r R a t i o E s t i m a t e s The s c a l a r r a t i o model assumes t h a t t h e r e i s no c o r r e l a t i o n between D and X or H and Y. T h i s i s ne v e r q u i t e t r u e . The t e n s o r r a t i o model a l l o w s more g e n e r a l f i e l d r e l a t i o n s h i p s of t h e form DU) = R n ( u ) YU) + R 1 2 U ) X(u) + e,(o) H(u) = R 2 1 U ) Y(o) + R 2 2 U ) X(u) + e 2 U ) (3.1) C o n s i d e r o n l y t h e f i r s t e q u a t i o n i n v o l v i n g t h e e a s t - w e s t OBM f i e l d . A ssuming t h a t t h e t r a n s f e r f u n c t i o n s a r e c o n s t a n t o v e r a d j a c e n t p e r i o d s and from day t o day a t a s i n g l e p e r i o d we w r i t e D^  = R,,Y^ + R, 2X^ + e^ j=1,...N d a y s and/or p e r i o d s . We need t h e many d a y s and p e r i o d s t o d e t e r m i n e t h e two unknown p a r a m e t e r s R,^ and R 1 2 b e c a u s e o t h e r w i s e t h e r e i s j u s t one e q u a t i o n i n two unknowns. T h e r e f o r e , t h i s i s now an o v e r d e t e r m i n e d p a r a m e t r i c l i n e a r i n v e r s e p r o b l e m . In m a t r i x form t h e p r e c e e d i n g e q u a t i o n 43 becomes D = A R + e where D=(D,,D 2,...D N) , R = ( R 1 1 f R 1 2 ) , e = ( e , , e 2 , — e „ ) and / ? i X, \ A = Y 2 X 2 Uj X*/ We want o u r p a r a m e t e r R t o m i n i m i z e t h e l 2 - n o r m e r r o r | | e | | 2 . L e t f d e n o t e t h e complex c o n j u g a t e t r a n s p o s e . | | e | | 2 = e f e = (D-AR)f(D-AR) = DtD - R f A t D - DfAR + Rf A t A R The minimum o c c u r s when AfAR = A f D . These a r e c a l l e d t h e no r m a l e q u a t i o n s . The c o r r e s p o n d i n g l e a s t s q u a r e s p a r a m e t e r e s t i m a t e i s R = ( A t A ) " 1 A t D w h i c h i s a l s o known i n . i n v e r s e t h e o r y a s t h e s m a l l e s t model. W r i t t e n i n f u l l t h e normal e q u a t i o n s a r e <YY*> <XY*>^ / <YX*> <KK*>) \ R 1 2 , \ \ and t h e p a r a m e t e r e s t i m a t e s a r e <DY*><XX*>-<DX*><XY*> R, 1 = <YY*><XX*>-<YX*><XY*> <YY*><DX*>-<YX*><DY*> ^1 2 = <YY*><XX*>—<YX*><XY*> F o r t h e n o r t h - s o u t h OBM f i e l d we have <HY*><XX*>-<HX*><XY*> R21 = <YY*><XX*>-<YX*><XY*> <YY*><HX*>—<YX*><HY*> R 2 2 = <YY*xXX*>-<YX*><XY*> (3.2) (3.3) 44 C o n f i d e n c e i n t e r v a l s f o r t h e p a i r of p a r a m e t e r s R,, and R 1 2 o r R 2 2 and R 2, can be f o u n d u s i n g t h e same method as f o r s c a l a r r a t i o s . However, t h i s t e c h n i q u e s o l v e s f o r t h e j o i n t c o n f i d e n c e l i m i t s o f a l l t h e p a r a m e t e r s . Pedersen' (1982) t a k e s a d i f f e r e n t a p p r o a c h and d e r i v e s c o n f i d e n c e i n t e r v a l s f o r e a c h p a r a m e t e r s e p a r a t e l y . He shows how s m a l l d e v i a t i o n s from t h e l e a s t s q u a r e s s o l u t i o n R r e s u l t i n a change i n t h e v a l u e o f Q=||e|| 2 from i t s minimum Q 0. The r a t i o ( n - 4 ) ( Q - Q 0 ) / Q 0 i s d i s t r i b u t e d as F , , h _ 4 . The c o n f i d e n c e i n t e r v a l s f o r t h e r e a l and i m a g i n a r y p a r t s of R,, a r e e q u a l and e q u a l t o 1 1-r 2D.YX <DD*> s q r t [ f i , t i - « d - 6 ) ] = k, (3.4) n-4 1 - r 2 Y X <YY*> and f o r t h e l i m i t s on t h e r e a l and i m a g i n a r y p a r t s of R 1 2 we have 1 1-r 2D.YX <DD*> s q r t [ f , , „ _ , ( 1 - 6 ) ] = k 2 (3.5) n-4 1 - r 2 Y X <XX*> where f 1 , h _ i t ( 1 - 6 ) i s t h e 100(1-6) p e r c e n t a g e p o i n t of t h e F-d i s t r i b u t i o n w i t h 1 and n-4 d e g r e e s of freedom, r 2 Y X i s t h e s q u a r e d c o h e r e n c e of t h e VIC h o r i z o n t a l f i e l d and <YY*>|<DX*>|2+<XX*>|<DY*>|2-2Re[<XY*><DX*><YD*>] r 2 D . Y X = <DD*>(<YY*><XX*>-|<YX*>| 2) i s t h e m u l t i p l e s q u a r e d c o h e r e n c e of t h e OBM D c h a n n e l on t h e h o r i z o n t a l VIC c h a n n e l s . A p p r o x i m a t e c o n f i d e n c e i n t e r v a l s f o r t h e m a g n i t u d e and phase a r e I |Ri1 I " | R i i I I - k i I |R 1 2| " | R i 2 | I * k 2 sin|#R,, - «R,,| < k,/|R,,| s i n | * R , 2 - *R, 2| ^ k 2 / | R 1 2 | 45 w i t h s i m i l a r e x p r e s s i o n s f o r t h e c o n f i d e n c e l i m i t s o f R 2 2 and R2 1 • 3.4 Winnowing R a t i o E s t i m a t e s C a l c u l a t i n g s c a l a r o r t e n s o r r a t i o s a s d e s c r i b e d i n t h e p r e c e e d i n g two s e c t i o n s g e n e r a t e s s e v e r a l h u n d r e d complex numbers a l l v y i n g f o r t h e a t t e n t i o n o f t h e i n t e r p r e t e r and h o p i n g f o r p r o m o t i o n t o s e a - f l o o r impedances f o r use i n model c o n s t r u c t i o n . O n l y a few r a t i o e s t i m a t e s a r e r e q u i r e d a t p e r i o d s s p a c e d r o u g h l y l o g a r i t h m i c a l l y . T h i s s e c t i o n d e s c r i b e s some c r i t e r i a u s e d t o e x t r a c t t h e b e s t p o s s i b l e d a t a s e t w h i c h must be c o m p a t i b l e w i t h t h e a s s u m p t i o n s made c o n c e r n i n g t h e s o u r c e f i e l d s and t h e e a r t h . An i n t e r a c t i v e computer programme was w r i t t e n t o a p p l y any c o m b i n a t i o n o f t h e winnowing c r i t e r i a w i t h any r e j e c t i o n l e v e l t o t h e v a s t number of R ( u ) . The a l g o r i t h m a c t s l i k e t h e s i e v e of E r a s t o t h e n e s t o a l l o w a few p r i m e c a n d i d a t e s t o f a l l t h r o u g h t h e mesh and be u s e d f o r model c o n s t r u c t i o n . 3.4.1 Ocean F i l t e r As d e m o n s t r a t e d i n s e c t i o n 2.5 and F i g u r e 7 t h e ocean f i l t e r a t t e n u a t e s and phase l a g s OBM f i e l d s w i t h r e s p e c t t o t h o s e a t V I C . These e f f e c t s r e q u i r e |R|<1 and * R < 0 ° . F o r p e r i o d s T<0.1 h r t h e a t t e n u a t i o n i s s e v e r e and so e s t i m a t e s may f a l l p r e y t o n o i s e . A l s o t h e s e p e r i o d s a r e l i a b l e t o be i n f l u e n c e d by n e a r - s u r f a c e i n h o m o g e n e i t i e s . Hence, a s h o r t p e r i o d c u t o f f o f T=0.1 hr i s e s t a b l i s h e d . W i t h T>0.1 hr and p l a u s i b l e s u b o c e a n i c models a f u r t h e r rough l i m i t on t h e phase 46 of R i s s e t as ) -90° < *R < 0° 3.4.2 V e r t i c a l F i e l d F o r a 1-D e a r t h and p l a n e i n c i d e n t EM waves t h e v e r t i c a l f i e l d i s i d e n t i c a l l y z e r o . Over 2-D models t h e v e r t i c a l f i e l d i s a l i n e a r c o m b i n a t i o n of t h e h o r i z o n t a l f i e l d s . To a v o i d c o n t r a d i c t i n g our 1-D a s s u m p t i o n we s e l e c t p e r i o d s a t w h i c h t h e v e r t i c a l f i e l d i s r e l a t i v e l y s m a l l and i n c o h e r e n t 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 f i e l d s . T h a t i s , we r e q u i r e <ZZ*> <ZZ*> « 1 « 1 and 7 2Z.DH « 1 <DD*> <HH*> where Z i s t h e v e r t i c a l OBM f i e l d and y 2Z.DH i s t h e m u l t i p l e s q u a r e d c o h e r e n c e of Z on D and H. These c o n d i t i o n s a r e s i m i l a r t o r e q u i r i n g t h a t t h e t i p p e r , as d e s c r i b e d by V o z o f f ( 1 9 7 2 ) , be s m a l l and h i g h l y v a r i a b l e . 3.4.3 R a t i o E r r o r s To c o n s t r a i n t h e c o n d u c t i v i t y m o d e ls as much as p o s s i b l e we s e l e c t r a t i o e s t i m a t e s w i t h s m a l l e r r o r s . The c o n f i d e n c e l i m i t f o r m u l a e f o r t e n s o r r a t i o s , e q u a t i o n s (3.4) and ( 3 . 5 ) , show t h a t s m a l l r a t i o e r r o r s r e s u l t from r 2 D . Y X and y 2H.YX near u n i t y . H i g h c o h e r e n c i e s o c c u r when t h e model i n e q u a t i o n (3.1) i s a good d e s c r i p t i o n o f t h e t r u e s i t u a t i o n a t a p a r t i c u l a r p e r i o d w i t h v e r y l i t t l e r e s i d u a l i n c o h e r e n t n o i s e . In a d d i t i o n , r 2 Y X must be s m a l l . I f i t i s not t h e n n ot o n l y do th e e r r o r s become l a r g e but t h e d e n o m i n a t o r i n t h e e x p r e s s i o n s f o r R*; i n e q u a t i o n s (3.2) and (3.3) a p p r o a c h e s z e r o and l e a d s 47 t o v e r y b i a s e d e s t i m a t e s . 3.4.4 D i s p e r s i o n R e l a t i o n s and I n e q u a l i t i e s W e i d e l t (1972) and F i s c h e r and Schnegg (1980) d e s c r i b e d i s p e r s i o n r e l a t i o n s and i n e q u a l i t i e s between a p p a r e n t r e s i s t i v i t i e s and p h a s e s w h i c h must be s a t i s f i e d o v e r 1-D m o d e l s . These c o n s t r a i n t s a r e not i m m e d i a t e l y a p p l i c a b l e s i n c e t h e y r e q u i r e a l a r g e number of a c c u r a t e l y known r e s p o n s e s . T h i s i s d e f i n i t e l y n o t t h e c a s e f o r p r a c t i c a l MT d a t a . L i n e a r i n v e r s e t h e o r y w h i c h i s d e s i g n e d t o o p e r a t e w i t h a f i n i t e number of i n a c c u r a t e d a t a c o u l d be u s e d t o p r e d i c t a phase c u r v e from a more r e l i a b l y d e t e r m i n e d a p p a r e n t r e s i s t i v i t y c u r v e or v i c e v e r s a . T h i s a p p r o a c h was not u s e d h e r e . F i g u r e 8 shows some o f t h e winnowing c r i t e r i a as f u n c t i o n s of p e r i o d . G r a p h A i s t h e s q u a r e d c o h e r e n c y of t h e VIC h o r i z o n t a l f i e l d , r 2 Y X . At p e r i o d s n e a r 10 h o u r s t h e c u r v e s u r g e s up t o very, h i g h v a l u e s . R a t i o s c a l c u l a t e d near h e r e w i l l be h e a v i l y b i a s e d upwards and w i l l have l a r g e e r r o r s . G r a p h B i s t h e m u l t i p l e s q u a r e d c o h e r e n c e of an OBM c h a n n e l on t h e VIC h o r i z o n t a l c h a n n e l s , r 2D.YX. T h i s c u r v e s h o u l d be near u n i t y . Two deep n o t c h e s a t one h a l f and one hour p e r i o d p l u s a f a l l o f f a t v e r y s h o r t p e r i o d s i n d i c a t e r e g i o n s t o a v o i d . G r a p h C p l o t s t h e r e l a t i v e m a g n i t u d e of t h e OBM v e r t i c a l f i e l d <ZZ*>/<DD*> and s u g g e s t s a r a t h e r narrow range of a c c e p t a b l e p e r i o d s . G r a p h D shows t h e c o h e r e n c y between t h e OBM v e r t i c a l and h o r i z o n t a l c h a n n e l s , r 2 Z . D H . A g a i n p e r i o d s n e a r one h a l f , one and t e n h o u r s s h o u l d be a v o i d e d . 48 F i g u r e 8. Winnowing c r i t e r i a as f u n c t i o n s of p e r i o d . The s q u a r e d c o h e r e n c e o f t h e h o r i z o n t a l VIC f i e l d i s shown i n A. The m u l t i p l e s q u a r e d c o h e r e n c e of t h e OBM D c h a n n e l on t h e VIC h o r i z o n t a l c h a n n e l s i s shown i n B. The m a g n i t u d e of t h e v e r t i c a l OBM f i e l d r e l a t i v e t o one of i t s h o r i z o n t a l f i e l d s i s shown i n C. The m u l t i p l e s q u a r e d c o h e r e n c y between t h e OBM v e r t i c a l and h o r i z o n t a l c h a n n e l s i s shown i n D. A l l c u r v e s a r e c a l c u l a t e d f r o m s p e c t r a w i t h 136 d e g r e e s of f r e e d o m . ] .0 I 1 1—i i i I 111 1 1—I i M 111 r 1 . 0 I 1 — I — i i i i 111 1 1—i i i i 111 i PERIOD (HR) 50 3.5 OBM D a t a R e d u c t i o n T h i s s e c t i o n d e s c r i b e s some s p e c i f i c d e t a i l s of t h e OBM d a t a a n a l y s i s up t o t h e s e l e c t i o n o f a p p a r e n t r e s i s t i v i t y and p h a s e d a t a f o r use i n model c o n s t r u c t i o n . F i r s t o f a l l , t h e r e l a t i v e m a g n i t u d e s o f t h e OBM h o r i z o n t a l f i e l d components d u r i n g m a g n e t i c a l l y q u i e t t i m e s a r e u s e d t o r o t a t e t h e d a t a i n t o t h e g e o m a g n e t i c c o o r d i n a t e s y s t e m ( t h e same as a t V I C ) . The r o t a t i o n a n g l e i s 20.6° c l o c k w i s e . F o l l o w i n g t h e d i s c u s s i o n o f s e c t i o n 3.1 t h e t o t a l r e c o r d i s b r o k e n up i n t o m u l t i p l e s o f 24 h o u r s . S e v e r a l d i f f e r e n t d a t a g r o u p i n g s a r e u s e d . Many d e g r e e s o f f r e e d o m a r e p o s s i b l e when 33 one-day segments a r e s t a c k e d and smoothed. However, a b e t t e r s e l e c t i o n o f l o n g e r p e r i o d F o u r i e r c o e f f i c i e n t s w h i c h a v o i d t h e major t i d a l h a r m o n i c s a t 4, 6, 8, 12 and 24 hr i s p r o v i d e d by s t a c k i n g segments o f , s a y , 72 h o u r s d u r a t i o n . F o r t h i s s t u d y 31 one-day, 16 two-day, 11 t h r e e - d a y , 8 f o u r - d a y , 4 e i g h t - d a y and 3 e l e v e n - d a y n o n - o v e r l a p p i n g segments a r e s e l e c t e d f o r a n a l y s i s . The t r a n s f o r m s w i t h i n e a c h g r o u p a r e s t a c k e d and smoothed by a P a r z e n window w i t h T/M=2.5, 2.5, 3.3, 4, 5 and 5 r e s p e c t i v e l y . The r e s u l t i n g d e g r e e s of fr e e d o m a r e 287, 148, 136, 119, 74 and 56 r e s p e c t i v e l y . Smoothed t e n s o r r a t i o s c a l c u l a t e d from t h e s e d i f f e r e n t d a t a g r o u p i n g s a r e s i m i l a r . P e r i o d s where t h e y d i s a g r e e d r a m a t i c a l l y a r e a v o i d e d . R a t i o e s t i m a t e s a t s h o r t e r p e r i o d s a r e s e l e c t e d f r o m t h e one-day s t a c k s i n c e i t has t h e most d e g r e e s o f f r e e d o m . Above T=3 h r t h e c h o i c e becomes v e r y l i m i t e d so e s t i m a t e s a r e s e l e c t e d from t h e two-day g r o u p i n g , and so on. A l l c h o i c e s a r e 51 made on t h e b a s i s of t h e winnowing c r i t e r i a o u t l i n e d i n s e c t i o n 3.4. A t p e r i o d s g r e a t e r t h a n 7 h r r a t i o s f r o m any g r o u p i n g have d i f f i c u l t y p a s s i n g t h e winnowing t e s t s . When s t a n d a r d s a r e r e l a x e d impedances d e r i v e d from t h e s e l o n g e r p e r i o d r e s p o n s e s , p a r t i c u l a r l y t h e p h a s e s , a r e b a d l y m i s f i t by t h e m o d e l l i n g r o u t i n e s . Hence, t h e maximum p e r i o d i s r e l u c t a n t l y t a k e n t o be T=5.5 h o u r s . The t e n s o r r a t i o e s t i m a t e s a r e r o t a t e d i n t o t h e r e f e r e n c e frame w h i c h m a x i m i z e s t h e d i a g o n a l e l e m e n t s (R,, and R 2 2 ) a t t h e e x p e n s e of t h e o f f - d i a g o n a l e l e m e n t s ( R 1 2 and R 2 1 ) f o l l o w i n g t h e p r o c e d u r e of E v e r e t t and Hyndman (1967) and Banks ( 1 9 7 3 ) . C o n f i d e n c e l i m i t s f o u n d f o r t h e u n r o t a t e d d i a g o n a l e l e m e n t s a r e us e d f o r t h e r o t a t e d d i a g o n a l e l e m e n t s . Impedances a r e c a l c u l a t e d u s i n g e q u a t i o n ( 2 . 2 2 ) . The e r r o r s i n t h e m a g n i t u d e and phase of t h e impedance a r e d e r i v e d f r o m a b r u t e f o r c e method. The r a t i o s R a r e p e r t u r b e d by G a u s s i a n n o i s e and t h e v a r i a n c e o f t h e r e s u l t i n g (assumed G a u s s i a n ) d i s t r i b u t i o n of impedances i s c a l c u l a t e d . The e a s t - w e s t OBM f i e l d g i v e s impedances Zxy w h i c h p a s s t h e winnowing c r i t e r i a and a r e s u c c e s s f u l l y m o d e l l e d by a l l th e 1-D i n v e r s i o n schemes. On t h e o t h e r hand, t h e n o r t h - s o u t h OBM f i e l d p r o d u c e s u n s t a b l e Zyx e s t i m a t e s w h i c h a r e not w e l l f i t . Hence, o n l y Zxy i s u s e d f o r model c o n s t r u c t i o n . I t was e x p e c t e d t h a t t h e r e v e r s e would be t r u e s i n c e Zyx v a l u e s a r e b a s e d on m a g n e t i c f i e l d s p e r p e n d i c u l a r t o t h e s t r i k e of t h e v e r y two d i m e n s i o n a l c o n t i n e n t a l s l o p e . E l e c t r i c f i e l d s a r e t h e n a l o n g s t r i k e and t h e a p p a r e n t r e s i s t i v i t i e s a r e l e s s 52 s u s c e p t i b l e t o e r r o r s ( V o z o f f , 1972). However, a l l a p p a r e n t r e s i s t i v i t i e s a r e v e r y low whether t h e y a r e d e r i v e d f r o m Zxy or Zyx o r t h e s c a l a r t r a n s f e r f u n c t i o n a n a l y s i s . The f i n a l d a t a w i t h one s t a n d a r d d e v i a t i o n e r r o r b a r s a r e p r e s e n t e d i n F i g u r e 9. The a p p a r e n t r e s i s t i v i t i e s of F i g u r e 9A a r e c a l c u l a t e d f r o m pa= | Zxy | 2/i"» • They a l l a r e s u r p r i s i n g l y s m a l l , l y i n g between 5 and 10 n-m (*=0.2 and 0.1 S/m) . D i f f e r e n t c o n v e n t i o n s and MT r e s p o n s e f u n c t i o n s r e q u i r e t h e p h a s e s t o be i n d i f f e r e n t q u a d r a n t s . F o r c o n v e n i e n c e , t h e y a r e p l o t t e d i n F i g u r e 9B i n t h e f i r s t q u a d r a n t w h i c h i s e q u i v a l e n t t o t h e l e a d of t h e e l e c t r i c o v e r t h e m a g n e t i c f i e l d . A p h a s e l e s s t h a n 45° i m p l i e s a more r e s i s t i v e s u b s t r a t u m w h i l e a phase g r e a t e r t h a n 45° s u g g e s t s a more c o n d u c t i v e s u b s t r a t u m . 53 20 CL o 10 cn 0 1 1—I—I I I I I I 1 1—I—I I I I I to - {$ * M J i i i i i i i 10 10° to • i i i i i 11 101 90" 60' LU CO c x C L 30' 0" i 1—i—i i i 111 1 1—i—i i i 11 i i I I I I I I I I 1 10 -1 10° PERIOD (HR) j i i i i 111 1 0 ' F i g u r e 9. The f i n a l data c a l c u l a t e d from the v e r t i c a l g r a d i e n t method impedances. Apparent r e s i s t i v i t i e s are shown i n A while the phases are shown i n B. The phase can l i e i n any quadrant and i s p l o t t e d here i n the f i r s t , f o r convenience. 54 CHAPTER 4 INVERSION OF IMPEDANCES FOR CONDUCTIVIY The l i m i t e d number of i m p r e c i s e d a t a shown i n F i g u r e 9 do not u n i q u e l y s p e c i f y a p a r t i c u l a r c o n d u c t i v i t y v e r s u s d e p t h p r o f i l e . N o n - u n i q u e n e s s i s a d i f f i c u l t y common t o a l l g e o p h y s i c a l d a t a s e t s . I t i s e s p e c i a l l y a c u t e f o r t h e non-l i n e a r MT a n a l y s i s s i n c e no t h e o r y y e t e x i s t s t o a s s e s s t h e n o n - u n i q u e n e s s i n a s i m i l a r manner t h a t u n i q u e B a c k u s - G i l b e r t a v e r a g e s q u a n t i f y l i n e a r i n v e r s e p r o b l e m s ( B a c k u s and G i l b e r t , 1970). One way o f g r a p p l i n g w i t h n o n - u n i q u e n e s s i s t o use t h e d a t a t o c o n s t r u c t some o f t h e i n f i n i t e l y many a c c e p t a b l e c o n d u c t i v i t y m o d e l s . By u s i n g d i f f e r e n t i n v e r s i o n a l g o r i t h m s i t i s hoped t h a t most d a r k c o r n e r s o f model s p a c e w i l l be i l l u m i n a t e d and t h a t no o t h e r s i g n i f i c a n t l y d i f f e r e n t models w i l l e x i s t w h i c h s a t i s f y t h e d a t a t o w i t h i n t h e i r e r r o r s . One i n v e r s i o n r o u t i n e ( s e c t i o n 4.1) c o n s t r u c t s models composed of d e l t a f u n c t i o n s of c o n d u c t a n c e . Such models a r e g e o p h y s i c a l l y u n r e a s o n a b l e b u t f i t t h e d a t a more p r e c i s e l y t h a n any o t h e r model t y p e . I f t h i s minimum m i s f i t i s s t i l l u n a c c e p t a b l y l a r g e t h e n t h e d a t a s e t i s i n c o m p a t i b l e w i t h t h e 1-D a s s u m p t i o n and new r e s p o n s e s must be f o u n d . T h i s programme a c t s as a g l o b a l winnowing c r i t e r i o n w h i c h r e j e c t s e n t i r e d a t a s e t s . In a manner s i m i l a r t o o t h e r i n v e r s i o n r o u t i n e s i t can a l s o i d e n t i f y s p u r i o u s , b a d l y d e t e r m i n e d r e s p o n s e s even t h o u g h t h e m i s f i t e r r o r i s d i s t r i b u t e d t h r o u g h o u t t h e d a t a . F o u r i n v e r s i o n r o u t i n e s u s e d h e r e g e n e r a t e i n t u i t i v e l y s a t i s f y i n g s i m p l e m o d e l s . S i m p l e i s d e f i n e d as e i t h e r a s m a l l number of c o n s t a n t c o n d u c t i v i t y l a y e r s ( s e c t i o n 4.2) or a s m o o t h l y v a r y i n g c o n t i n u o u s c o n d u c t i v i t y f u n c t i o n ( s e c t i o n 55 4 . 3 ) . The f i r s t d e f i n i t i o n i s e x p e c t e d t o a p p l y n e a r t h e s u r f a c e where d i s c o n t i n u i t i e s a r e more l i k e l y t o o c c u r . The s e c o n d d e f i n i t i o n i s more a p p l i c a b l e t o t h e deep e a r t h s t r u c t u r e where c h a n g e s a r e more g r a d u a l . The f i n a l r o u t i n e ( s e c t i o n 4.4) a t t e m p t s t o q u a n t i f y t h e n o n - u n i q u e n e s s a l l o w e d by t h e d a t a by f i n d i n g m o d e ls w h i c h m a x i m i z e o r m i n i m i z e t h e i n t e g r a t e d c o n d u c t i v i t y between s p e c i f i e d d e p t h s . Thus, g l o b a l bounds on t h e a v e r a g e c o n d u c t i v i t y o r on t h e c o n d u c t a n c e a r e c a l c u l a t e d . The m i s f i t c r i t e r i o n we use i s b a s e d on t h e c h i - s q u a r e d d i s t r i b u t i o n x 2 = f_ I C J - C U ^ I V S J i where c^ i s a dummy v a r i a b l e r e p r e s e n t i n g t h e measured r e s p o n s e a t Uy c ^ u ^ ^ s t n e r e s p o n s e c a l c u l a t e d from t h e c o n s t r u c t e d model, s^ i s t h e s t a n d a r d d e v i a t i o n of c^ and N i s t w i c e t h e number of p e r i o d s s i n c e a t e a c h p e r i o d we can have c^ i n r e a l and i m a g i n a r y p a r t s or as m a g n i t u d e s and p h a s e s . In p r a c t i c e , t h e d i f f e r e n c e i n t h e x 2 m i s f i t i s s m a l l f o r d i f f e r e n t r e p r e s e n t a t i o n s of t h e MT r e s p o n s e (complex impedances, a p p a r e n t r e s i s t i v i t i e s and p h a s e s ) . We make t h e h y p o t h e s i s t h a t t h e c o n s t r u c t e d model a c c u r a t e l y r e p r e s e n t s t h e r e a l e a r t h s t r u c t u r e . Even when t h i s a s s u m p t i o n i s t r u e x 2 d o e s n o t e q u a l z e r o b e c a u s e o f random e r r o r s i n t h e d a t a , c j . In f a c t , t h e e x p e c t e d v a l u e of x 2 i s N. L a r g e v a l u e s of x 2>>N or s m a l l v a l u e s of x 2<<N have a low p r o b a b i l i t y of r e s u l t i n g m e r e l y from random m i s f i t e r r o r s t o t h e d a t a p o i n t s . T h e r e f o r e , f o r x 2 i n t h e s e r a n g e s we must r e j e c t our o r i g i n a l h y p o t h e s i s of model and t r u e e a r t h e q u i v a l e n c e . Hence, we 56 d e f i n e a c c e p t a b l e models as t h o s e w h i c h have x 2 w i t h i n t h e ( a p p r o x i m a t e ) 95 p e r c e n t c o n f i d e n c e i n t e r v a l N - 2 / 2N1 < x 2 < N + 2 J2N 1 4.1 B e s t F i t t i n g M o d e l s ; P a r k e r (1980) and P a r k e r and Whaler (1981) d e s c r i b e how D + models c o n s i s t i n g o f d e l t a f u n c t i o n s o f c o n d u c t a n c e m i n i m i z e t h e m i s f i t t o t h e d a t a . D* models of t h e OBM d a t a of 12 a p p a r e n t r e s i s t i v i t i e s and 12 p h a s e s have x 2 e q u a l t o a b o u t 20. Hence t h e d a t a s e t i s c o m p a t i b l e w i t h t h e 1-D a s s u m p t i o n and more g e o p h y s i c a l l y r e a s o n a b l e models e x i s t w h i c h f i t t h e d a t a . 4.2 L a y e r e d M o d e l s 4.2.1 A L a y e r o v e r a H a l f s p a c e F i s c h e r e t a l (1981) d e s c r i b e a t e c h n i q u e f o r c o n s t r u c t i n g l a y e r e d c o n d u c t i v i t y m o d e l s . The method i s b a s i c a l l y a r e p e a t e d a p p l i c a t i o n o f t h e l a y e r o v e r a h a l f s p a c e impedance f o r m u l a g i v e n by e q u a t i o n ( 2 . 1 4 ) . The s u r f a c e c o n d u c t i v i t y i s a v a r i a b l e p a r a m e t e r . Once i t i s f i x e d t h e s h o r t e s t p e r i o d datum i s u s e d t o f i n d t h e t h i c k n e s s of t h e s u r f a c e l a y e r and t h e u n d e r l y i n g c o n d u c t i v i t y a s s u m i n g t h e s e c o n d l a y e r i s a h a l f s p a c e . The n e x t l o n g e s t p e r i o d d e t e r m i n e s t h e t h i c k n e s s of t h e s e c o n d l a y e r and t h e c o n d u c t i v i t y o f t h e t h i r d , and so on. I f a p e r i o d has a s k i n d e p t h w h i c h i s not g r e a t e r t h a n some f a c t o r t i m e s t h e model d e p t h so f a r c o n s t r u c t e d t h e n i t i s d e l e t e d and t h e n e x t l o n g e s t p e r i o d t r i e d . T h i s t e s t a s s u r e s t h a t t h e f i e l d has 57 p e n e t r a t e d s i g n i f i c a n t l y i n t o t h e u n d e r l y i n g s t r u c t u r e and p r e v e n t s g e n e r a t i o n o f i n s i g n i f i c a n t t h i n l a y e r s . A l l p e r i o d s a r e u s e d , however, i n c a l c u l a t i n g t h e x 2 m i s f i t of t h e model t o t h e d a t a . F o u r s u c h a c c e p t a b l e F models a r e shown i n F i g u r e 10A. W h i l e t h e d e t a i l s o f t h e o v e r l a p p i n g l i n e s a r e o b s c u r e d t h e g e n e r a l t r e n d i s c l e a r . S u r f a c e c o n d u c t i v i t i e s a r e a b o u t 0.1 t o 0.2 S/m and a more r e s i s t i v e zone i s i n d i c a t e d between 50 and 100 km d e p t h . Below t h i s p o i n t t h e models v a r y a g r e a t d e a l . T h i s f a c t as w e l l as o t h e r a rguments s u g g e s t t h a t t h e d a t a c a n n o t r e s o l v e s t r u c t u r e s below a b o u t 120 km. 4.2.2 C o n s t a n t nch2 L a y e r s P a r k e r (1980) and P a r k e r and Whaler (1981) d i s c u s s what t h e y t e r m H + models w h i c h have c o n s t a n t d 2 = */tfh 2 l a y e r s . At t h e s u r f a c e of s u c h a model the MT r e s p o n s e s may be w r i t t e n i n terms of a p a r t i a l f r a c t i o n e x p a n s i o n . P a r a m e t e r s of t h e e x p a n s i o n a r e f o u n d w h i c h m i n i m i z e t h e x 2 m i s f i t f o r a g i v e n d 2 . The l a y e r c o n d u c t i v i t i e s and t h i c k n e s s e s a r e t h e n f o u n d by r e c u r s i o n t h r o u g h an e q u i v a l e n t c o n t i n u e d f r a c t i o n r e p r e s e n t a t i o n , s i m i l a r i n form t o e q u a t i o n ( 2 . 1 2 ) . S m a l l v a l u e s of d 2 p r o d u c e v e r y t h i n , v e r y h i g h c o n d u c t i v i t y l a y e r s . In f a c t , as d 2 a p p r o a c h e s z e r o t h e x 2 m i s f i t d r o p s t o t h e minimum f o u n d by D + and t h e l a y e r s become l i k e d e l t a f u n c t i o n s . More r e a s o n a b l e v a l u e s f o r d 2 y i e l d m o d e l s w i t h t h i c k e r l a y e r s of more u n i f o r m c o n d u c t i v i t y . F i v e H + models a r e shown i n F i g u r e 10B. As w i t h t h e F m o d e l s , t h e s u r f a c e c o n d u c t i v i t y i s between 0.1 and 0.2 S/m and a r e s i s t i v e zone i s i n d i c a t e d between 50 and 100 km d e p t h . At d e p t h s below 120 58 F i g u r e 10. A c c e p t a b l e c o n d u c t i v i t y models b a s e d on t h e d a t a of F i g u r e 9. F and H* l a y e r e d models a r e shown i n A and B, r e s p e c t i v e l y . S and C 2 + smooth models a r e shown i n C and D, r e s p e c t i v e l y . Some c u r v e s f i t t h e d a t a more p r e c i s e l y t h a n o t h e r s but a l l have a x 2 m i s f i t l e s s t h a n 38 w h i c h i s t h e upper l i m i t of t h e 95 p e r c e n t c o n f i d e n c e i n t e r v a l . S J G M R ( S / M ) S I G M A ( S / M ) 6S 60 km t h e c o n d u c t i v i t y v a r i e s by two o r d e r s o f m a g n i t u d e and i s n o t t o be b e l i e v e d . 4.3 Smooth M o d e l s 4.3.1 L i n e a r i z e d I n v e r s i o n O l d e n b u r g (1979) d e s c r i b e s a method of c o n s t r u c t i n g smooth c o n d u c t i v i t y p r o f i l e s . The n o n - l i n e a r MT e q u a t i o n i s t r a n s f o r m e d i n t o a l i n e a r e q u a t i o n r e l a t i n g s m a l l c h a n g e s i n t h e model t o s m a l l c h a n g e s i n t h e r e s p o n s e s . Smooth p e r t u r b a t i o n s t o a s t a r t i n g model a r e f o u n d w h i c h r e d u c e t h e x 2 m i s f i t between the measured r e s p o n s e s and t h o s e c a l c u l a t e d from t h e new p e r t u r b e d model. The p r o c e s s i s r e p e a t e d u n t i l c o n v e r g e n c e i s a t t a i n e d , u s u a l l y w i t h i n 5 i t e r a t i o n s . T h r e e S models a r e shown i n F i g u r e 10C. A g a i n , s u r f a c e c o n d u c t i v i t i e s a r e n e a r 0.1 t o 0.2 S/m.. T h e r e i s some i n d i c a t i o n of a narrow r e g i o n of i n c r e a s e d c o n d u c t i v i t y n e a r 10 km not p r e s e n t on models from o t h e r a l g o r i t h m s . The r e s i s t i v e zone between 50 and 100 km i s s t i l l p r o m i n e n t . Below 120 km e a c h p r o f i l e r e t u r n s t o t h e c o n d u c t i v i t y of t h e h a l f s p a c e s t a r t i n g model t h a t g e n e r a t e d i t . Thus, t h e d a t a a r e p o w e r l e s s t o a l t e r t h e s t r u c t u r e below a b o u t 120 km. F u r t h e r c o n f i r m a t i o n t h a t t h e h i g h c o n d u c t i v i t i e s a t t h e s e d e p t h s a r e n o t j u s t i f i e d by t h e d a t a i s p r o v i d e d by t h e f o l l o w i n g t e s t . O l d e n b u r g ' s a l g o r i t h m can c o n s t r u c t models from o n l y a p p a r e n t r e s i s t i v i t i e s — p h a s e s a r e not r e q u i r e d . S e v e n t e e n a p p a r e n t r e s i s t i v i t i e s a t p e r i o d s r a n g i n g from 0.2 t o 27 h o u r s a r e i n v e r t e d . W i t h o u t t h e l o n g p e r i o d p h a s e s models a r e f o u n d w h i c h f i t t h e d a t a . A l l t h e models have n e a r l y u n i f o r m c o n d u c t i v i t i e s of 0.07 S/m w i t h no 61 h i g h l y c o n d u c t i v e r e g i o n s a t d e p t h . T h e r e f o r e , we do not b e l i e v e s t r u c t u r e below 120 km b a s e d on j u s t t h e o r i g i n a l 12 complex d a t a . 4.3.2 G e l ' f a n d - L e v i t a n I n v e r s i o n I ; P a r k e r (1980) and P a r k e r and Whaler (1981) d i s c u s s an i n v e r s i o n scheme b a s e d on t h e t e c h n i q u e s o f G e l ' f a n d and L e v i t a n ( 1 9 5 5 ) . As f o r H + models t h e r e s p o n s e s a r e w r i t t e n as a p a r t i a l f r a c t i o n e x p a n s i o n . The s p e c t r a l f u n c t i o n o f t h i s e x p a n s i o n i s m a n i p u l a t e d by t h e G e l ' f a n d - L e v i t a n method t o g i v e a n o t h e r f u n c t i o n w h i c h i s p a r t of a S c h r o d i n g e r e q u a t i o n . T h i s does not seem v e r y p r o g r e s s i v e u n t i l i t i s d i s c o v e r e d t h a t t h e MT e q u a t i o n can be t r a n s f o r m e d i n t o a S c h r o d i n g e r e q u a t i o n . In t h i s r a t h e r i n v o l v e d f a s h i o n , i n f i n i t e l y d i f f e r e n t i a b l e c o n d u c t i v i t y p r o f i l e s a r e c o n s t r u c t e d . The s u r f a c e c o n d u c t i v i t y , e0, i s t h e p a r a m e t e r f o r t h i s a l g o r i t h m . L a r g e v a l u e s of c0 p r o d u c e s m a l l x 2 m i s f i t s and models w h i c h r e s e m b l e smoothed d e l t a f u n c t i o n s . S m a l l v a l u e s o f e0 i n c r e a s e t h e m i s f i t e r r o r and g i v e more u n i f o r m c o n d u c t i v i t y p r o f i l e s . F o u r s u c h C 2 + p r o f i l e s a r e shown i n F i g u r e 10D. V e r y l i t t l e v a r i a t i o n from <r0 i s p r e s e n t i n t h e t o p 100 km. The c u r v e s d i v e r g e somewhat nea r 150 km but t h i s has been shown t o be b e yond t h e d e p t h of r e s o l u t i o n of t h e d a t a . 4.4 C o n d u c t a n c e Bounds To summarize a l l t h e models from t h e f o u r i n v e r s i o n a l g o r i t h m s t h e c o n d u c t a n c e c u r v e s a r e c a l c u l a t e d . The c o n d u c t a n c e a t a d e p t h z i s s i m p l y t h e i n t e g r a l of t h e 62 c o n d u c t i v i t y from t h e s u r f a c e t o l e v e l z. A l l s i x t e e n c o n d u c t i v i t y p r o f i l e s p r e s e n t e d i n F i g u r e 10 a r e i n t e g r a t e d i n t h i s way. A t any p a r t i c u l a r d e p t h t h e maximum and minimum c o n d u c t a n c e i s n o t e d and t h e r e s u l t i n g bound c u r v e s a r e p l o t t e d i n F i g u r e 11. T r u e , g l o b a l bounds w h i c h m a x i m i z e and m i n i m i z e t h e c o n d u c t a n c e a t s e v e r a l d e p t h s a r e a l s o shown. T h e s e p o i n t s a r e f o u n d u s i n g t h e l i n e a r programming t e c h n i q u e s d e s c r i b e d by O l d e n b u r g ( 1 9 8 3 ) . As e x p e c t e d t h e y l i e o u t s i d e t h e a p p r o x i m a t e c u r v e s c a l c u l a t e d on t h e b a s i s o f j u s t 16 d i f f e r e n t m o d e l s . The a p p r o x i m a t i o n t o t h e t r u e bounds i n F i g u r e 11 i s good down t o a d e p t h of ab o u t 80 km. The agreement i m p l i e s t h a t t h e c o n d u c t a n c e s of t h e c o n s t r u c t e d m o d e l s a r e r e p r e s e n t a t i v e of t h e t r u e d i v e r s i t y of c o n d u c t a n c e s p e r m i t t e d by t h e d a t a i n t h i s r e g i o n . The a p p r o x i m a t i o n b r e a k s down a t g r e a t e r d e p t h s where t h e c o n d u c t i v i t y p r o f i l e s , d i s s i m i l a r t h o u g h t h e y a r e , f a i l t o a d e q u a t e l y e x p l o r e t h e s p a c e of a c c e p t a b l e c o n d u c t a n c e s . A minimum bound a t a d e p t h of 115 km was not f o u n d due t o c o n v e r g e n c e p r o b l e m s . I n d i c a t i o n s were, however, t h a t t h e bound p o i n t d i d n o t l i e c l o s e t o t h e a p p r o x i m a t e c u r v e . The c o n d u c t a n c e bound c u r v e s d e r i v e d from e i t h e r method i n d i c a t e w e l l and p o o r l y d e f i n e d r e g i o n s . Where t h e bounds a r e c l o s e , t h e d a t a have t h e power t o c o n s t r a i n t h e t o t a l c o n d u c t a n c e down t o t h a t d e p t h . Wide bounds, however, i n d i c a t e where t h e d a t a have l e s s i n f l u e n c e and admit l a r g e c o n d u c t i v i t y and c o n d u c t a n c e v a r i a t i o n s . The OBM c o n d u c t a n c e bounds a r e narrow down t o a b o u t 50 km. Between 50 and 120 km t h e y g r a d u a l l y i n c r e a s e a s r e s o l u t i o n d e c r e a s e s ( n o t e t h e 63 l o g a r i t h m i c s c a l e ) . U n d e r l y i n g , e x t r e m e l y wide bounds c o n f i r m t h e e a r l i e r p r o p o s i t i o n t h a t r e s o l u t i o n i s l o s t below 120 km. 64 F i g u r e 1 1 . C o n d u c t a n c e bounds. A p p r o x i m a t e upper and l o w e r bounds on t h e c o n d u c t a n c e c a l c u l a t e d f r o m t h e 16 a c c e p t a b l e m o dels i n F i g u r e 10 a r e p l o t t e d as l i n e s . T r u e , g l o b a l bounds a t s e v e r a l d e p t h s a r e p l o t t e d as s q u a r e s . CONDUCTANCE (S) 66 CHAPTER 5 CONCLUSIONS 5.1 OBM R e s u l t s Compared w i t h O t h e r S t u d i e s T h e r e a r e a g r o w i n g number o f . o c e a n b o t t o m MT a n a l y s e s f o r s i t e s on t h e P a c i f i c p l a t e . T h r e e i n p a r t i c u l a r have been amalgamated i n t o a p i c t u r e c o n s i s t e n t w i t h t h e o r i e s o f s e a -f l o o r s p r e a d i n g . O l d e n b u r g (1981) combined r e s u l t s f r o m s t a t i o n s JDF, CAL and NCP above 1, 30 and 72 my o l d l i t h o s p h e r e , r e s p e c t i v e l y . S e v e r a l ^ i n t e r e s t i n g t r e n d s were o b s e r v e d . F i r s t l y , t h e d i s t i n c t c o n d u c t i v e zone v i s i b l e i n models f o r a l l t h r e e s i t e s began a t d e e p e r d e p t h s as t h e l i t h o s p h e r i c age i n c r e a s e d . The s h a r p r i s e i n c o n d u c t i v i t y was i n t e r p r e t e d a s t h e bo u n d a r y between t h e c o l d , r e s i s t i v e l i t h o s p h e r e and t h e h o t , c o n d u c t i v e a s t h e n o s p h e r e . An i n c r e a s e i n d e p t h w i t h age i s e x p e c t e d s i n c e t h e l i t h o s p h e r e t h i c k e n s w i t h age. S e c o n d l y , a p a r t i a l m e l t i n g i n t e r p r e t a t i o n of t h e c o n d u c t i v i t y h i g h showed t h e m e l t f r a c t i o n d e c r e a s e d w i t h i n c r e a s i n g age and was c o r r e l a t e d w i t h t h e s e i s m i c low v e l o c i t y zone i n t e r p r e t e d from R a y l e i g h wave d i s p e r s i o n d a t a . T h e s e r e l a t i o n s h i p s s u g g e s t t h a t t h e c o n d u c t i v e zone i s t h e c o r e of a p a r t i a l l y m o l t e n a s t h e n o s p h e r e . F i n a l l y , t e m p e r a t u r e s , t o o , g e n e r a l l y d e c r e a s e d w i t h i n c r e a s i n g age i n a c c o r d a n c e w i t h a t h i c k e n i n g l i t h o s p h e r e . A new a n a l y s i s by P a r k e r , O l d e n b u r g and W h i t t a l l (1983) c o n f i r m s t h a t t h e s e t r e n d s e x i s t but t h a t t h e y a r e n o t as s t r o n g as i n i t i a l l y b e l i e v e d due t o t h e s u r p r i s i n g l y l a r g e n o n - u n i q u e n e s s o f t h e MT method. T h e r e f o r e , an a d d i t i o n a l a n a l y s i s on a n o t h e r age l i t h o s p h e r e would be most h e l p f u l i n r a t i f y i n g or r e f u t i n g t h e g e n e r a l p a t t e r n o u t l i n e d a b o v e . T h i s was one of t h e 67 F i g u r e 12. Minimum s t r u c t u r e c o n d u c t i v i t y models f o r f o u r P a c i f i c p l a t e o c e a n b o t t o m s t a t i o n s . From A t h r o u g h D t h e s i t e s were above 1, 3, 30 and 72 my o l d l i t h o s p h e r e . The OBM m o d e l s of 12B a r e c l e a r l y d i s t i n c t from t h o s e o f t h e o t h e r s i t e s . S I G M A ( S / M ) S I G M A ( S / M ) i CO I ro O i O O o I LO I o I i 11 mil 1—i 111 mi—1/1)1 I I I I I I I o m —I o o O LTJ o o ' l l l I I I I ! | I I I I I III 1 L o o IV) o o S I G M A ( S / M ) S I G M A ( S / M ) o I CO I O O o I co i I i 1111 j 1—I I I 11 M | 1—I I IIIII a m o o o o I I I I i ;in | Yi | | | in ' I i I m i O o o o 89 69 m o t i v a t i o n s f o r r e c o r d i n g t h e OBM d a t a on 3 my o l d l i t h o s p h e r e , an age i n t e r m e d i a t e between JDF and CAL. M o d e l s c o n s t r u c t e d by d i f f e r e n t i n v e r s i o n schemes f o r a l l f o u r s t a t i o n s a r e p l o t t e d i n F i g u r e 12. In t h e hope of c a p t u r i n g t h e e s s e n t i a l e a r t h f e a t u r e s , m o d els w i t h a minimum of s t r u c t u r e a r e s e l e c t e d f r o m t h e i n f i n i t y of a c c e p t a b l e p r o f i l e s . C u r v e s f o r JDF, CAL and NCP ( F i g u r e s 12A, 12C, 12D) a l l have t h e same b a s i c f o r m . They a r e q u i t e r e s i s t i v e n e a r t h e s u r f a c e and have a c o n d u c t i v e zone b e g i n n i n g between 40 and 80 km. Below t h i s d e p t h some models i n d i c a t e a d e c r e a s e i n c o n d u c t i v i t y w h i l e o t h e r s m a i n t a i n a f a i r l y c o n s t a n t l e v e l . The OBM p r o f i l e s a r e q u i t e d i s t i n c t ( F i g u r e 12B). They a l l have a r o u g h l y u n i f o r m c o n d u c t i v i t y c e n t e r e d on a b o u t 0.2 S/m u n t i l 120 km d e p t h where r e s o l u t i o n i s l o s t . T h e r e a r e no major c o n d u c t i v e or r e s i s t i v e z o n e s s u c h as a r e f o u n d f o r t h e o t h e r t h r e e s i t e s . To f i n d t h e m e l t f r a c t i o n and t e m p e r a t u r e p r o f i l e s we use t h e p y r o l i t e p l u s 0.1 p e r c e n t water p e t r o l o g y of Ringwood (1975) as w e l l as h i s f ( T , P ) f u n c t i o n w h i c h r e l a t e s t h e m e l t f r a c t i o n , f , t o t h e t e m p e r a t u r e , T, and p r e s s u r e , P. One f u r t h e r e q u a t i o n w h i c h e x p r e s s e s t h e minimum f i n terms of t h e m e l t , s o l i d and b u l k c o n d u c t i v i t i e s i s b o r r o w e d from S h a n k l a n d and Waff ( 1 9 7 7 ) . R e s u l t s f o r t h e OBM d a t a a r e p r e s e n t e d i n F i g u r e 13. Below t h e f i r s t few k i l o m e t e r s o f s e d i m e n t s where t h e p y r o l i t e p e t r o l o g y i s e x p e c t e d t o h o l d t h e m e l t f r a c t i o n i s a b o u t 0.12 ( F i g u r e 13B). T h i s d r o p s f a i r l y s m o o t h l y t o 0.03 by 120 km as l e s s and l e s s m e l t i s r e q u i r e d t o a c c o u n t f o r t h e o b s e r v e d b u l k c o n d u c t i v i t y . One model g i v e s an e x t r e m e l y h i g h 70 F i g u r e 13. A p a r t i a l m e l t i n g and t e m p e r a t u r e i n t e r p r e t a t i o n o f t h e OBM minimum s t r u c t u r e m o d e l s . The minimum s t r u c t u r e models from t h e f o u r i n v e r s i o n r o u t i n e s a r e p l o t t e d i n A. The minimum p e r c e n t a g e s of m e l t r e q u i r e d t o a c c o u n t f o r t h e o b s e r v e d b u l k r o c k c o n d u c t i v i t i e s a r e shown i n B. The c o r r e s p o n d i n g t e m p e r a t u r e p r o f i l e s a r e shown i n C. 72 F i g u r e 14. P a r t i a l m e l t i n g and temperature i n t e r p r e t a t i o n s f o r the JDF data are given in the top row, panels A, B and C. The corresponding r e s u l t s f o r CAL are shown in the middle row, panels D, E and F. R e s u l t s f o r NCP are shown i n the bottom row, panels G, H and I. SIGMA (S/M) SIGMA (S/M) SIGMA (S/M) 1 74 f a t 200 km but t h i s i s beyond t h e d a t a r e s o l u t i o n . t h e t e m p e r a t u r e c u r v e s i n F i g u r e 13C t r e n d l i n e a r l y f r o m 1200°C ne a r t h e s u r f a c e t o 1400°C a t 120 km. The a l m o s t c o n s t a n t c o n d u c t i v i t y OBM models o f F i g u r e 13A g e n e r a t e q u i t e f e a t u r e l e s s m e l t f r a c t i o n and t e m p e r a t u r e p r o f i l e s . In c o n t r a s t , r e s u l t s from t h e o t h e r t h r e e s i t e s , shown i n F i g u r e 14, a r e more s t r u c t u r e d . T h e r e , t h e v a r y i n g c o n d u c t i v i t i e s p r o d u c e m e l t f r a c t i o n s ^ 5 p e r c e n t w i t h i n d e f i n i t e z o n e s and t e m p e r a t u r e c u r v e s w h i c h a r e g e n e r a l l y r e d u c e d from OBM l e v e l s . A f u l l g e o l o g i c i n t e r p r e t a t i o n w h i c h draws t o g e t h e r r e s u l t s from a l l f o u r d a t a s e t s i s beyond t h e scope of t h i s t h e s i s . A c o h e r e n t a n a l y s i s of t h e t h r e e p r e v i o u s d a t a s e t s has a l r e a d y been made by O l d e n b u r g ( 1 9 8 1 ) . The d i s c o r d a n t , h i g h c o n d u c t i v i t i e s below t h e OBM s i t e may be e x p l a i n e d , i n p a r t , by t h e 7 km of c o n d u c t i v e s e d i m e n t s (ff=0.3 t o 1.0 S/m) i n t h e r e g i o n (L.K. Law, p e r s o n n a l c o m m u n i c a t i o n ) . One a d d i t i o n a l s u r v e y t a k e n above 10 my l i t h o s p h e r e i n August 1982 has y e t t o be a n a l y s e d and may s h i f t t h e i n t e r p r e t a t i o n b a l a n c e t o c o n f i r m t h e OBM s i t e s t r u c t u r e or c e r t i f y i t as anomalous. Hence, u n t i l t h i s a n a l y s i s i s c o m p l e t e i t i s p r e m a t u r e and p e r h a p s unwise t o go t h r o u g h g r e a t c o n t o r t i o n s t o accomodate t h e a p p a r e n t l y anomalous OBM r e s u l t s . 5.2 S u g g e s t i o n s f o r F u t u r e Work Ocean b o t t o m measurements of t h e e l e c t r i c as w e l l as t h e m a g n e t i c f i e l d o b v i a t e t h e major p r o b l e m s w i t h t h i s a n a l y s i s . No l o n g e r i s t h e once removed r a t i o R t h e q u a n t i t y i n demand 7 5 s i n c e t h e s e a - f l o o r impedance i s c a l c u l a t e d d i r e c t l y from E ( o ) and H ( .o ) . A l l o f t h e a p p r o x i m a t i o n s a c c o m p a n y i n g t h e v e r t i c a l g r a d i e n t method, s u c h as 1-D s t r u c t u r e and t h e t r a n s p o r t of VI C t o above OBM, a r e c i r c u m v e n t e d . T h u s , one l i n k i n t h e c h a i n o f a p p r o x i m a t i o n s and a s s u m p t i o n s i s d e l e t e d and we a r e c l o s e r t o a d i r e c t measure of t h e e a r t h ' s c o n d u c t i v i t y . M o r e o v e r , t h e impedance t e n s o r r o t a t i o n p r o p e r t i e s a r e a v a i l a b l e t o d i s t i n g u i s h between one, two and t h r e e d i m e n s i o n a l e a r t h s t r u c t u r e s e n a b l i n g a p p r o p r i a t e i n v e r s i o n a l g o r i t h m s t o be s e l e c t e d . C omplete f i v e component (Ex,Ey,Hx,Hy,Hz) o c e a n b o t t o m MT i n s t r u m e n t s e x i s t and have been u s e d s u c c e s s f u l l y . U n f o r t u n a t e l y , t h e one d e p l o y e d f o r t h i s e x p e r i m e n t ended up a few d a y s l a t e r on t h e W a s h i n g t o n c o a s t l o o k i n g as i f i t had been mauled by a t r a w l e r . C h o o s i n g a more 'normal' l a n d s t a t i o n f o r r e p l a c e m e n t of s e a - s u r f a c e f i e l d s i s a d v i s a b l e f a i l i n g a s u c c e s s f u l d e p l o y m e n t of a f i v e component i n s t r u m e n t . VIC i s s u r r o u n d e d by a complex p a t t e r n o f s t r a i t s and i n l e t s w h i c h must make i t s r e c o r d s somewhat an o m a l o u s . O t h e r g e o m a g n e t i c o b s e r v a t o r i e s l o c a t e d n e a r t h e c o a s t of Oregon, f o r example, may y i e l d fewer s p u r i o u s r a t i o e s t i m a t e s . Long d a t a s e r i e s of p e r h a p s two o r t h r e e months a r e e s s e n t i a l f o r many r e a s o n s . Such e x t e n d e d r e c o r d s c o n s i d e r a b l y i n c r e a s e t h e a v a i l a b l e d e g r e e s of f r e e d o m . S t a c k i n g and s m o o t h i n g t h e n g i v e s s t a b l e impedance v a l u e s w i t h s m a l l e r e r r o r b a r s . The e s t i m a t e s a r e not o n l y s t a b l e s t a t i s t i c a l l y b u t p h y s i c a l l y a s w e l l s i n c e t h e y a r e b a s e d on a w i d e r r a n g e of s o u r c e f i e l d p o l a r i z a t i o n s and a n g l e s of a p p r o a c h . Such 76 a c c u r a t e d a t a i s i m p o r t a n t s i n c e i t l i m i t s t h e v a s t non-u n i q u e n e s s of a c c e p t a b l e c o n d u c t i v i t y m o d e l s . Long t i m e s e r i e s a l s o have more c h a n c e o f p r o d u c i n g l o n g p e r i o d r e s p o n s e s w h i c h p a s s t h e winnowing c r i t e r i a and hence p r o v i d e r e s o l u t i o n down t o s e v e r a l h u n d r e d k i l o m e t e r s . i ,: ";• The r e s o l u t i o n p r o v i d e d by MT r e s p o n s e s and t h e u n i q u e n e s s of c o n s t r u c t e d m o d els a r e a c u t e p r o b l e m s . I n t e r e s t i n g f e a t u r e s p r e s e n t on one model a r e a b s e n t from a n o t h e r . I t i s c r i t i c a l t o f i n d a way of q u a n t i f y i n g t h e r e s o l u t i o n and i n f e r r i n g j u s t what s t r u c t u r e t h e d a t a t r u l y r e q u i r e . P a r k e r (1982) p r o v i d e s an a n a l y t i c method o f f i n d i n g t h e maximum d e p t h w h i c h t h e d a t a c an r e s o l v e . O l d e n b u r g (1983) c a l c u l a t e s bounds on t h e c o n d u c t a n c e down t o a c e r t a i n d e p t h or on t h e a v e r a g e c o n d u c t i v i t y w i t h i n a p a r t i c u l a r r e g i o n . A c o m p l e t e a p p r a i s a l t h e o r y f o r n o n - l i n e a r i n v e r s e p r o b l e m s has y e t t o be u n v e i l e d . I f t h e impedances i n d i c a t e a. two or t h r e e d i m e n s i o n a l s t r u c t u r e t h e n t h e i n t e r p r e t a t i o n s h o u l d be made i n terms of t h e s e t y p e s of m o d e l s . The p r o c e s s i s not so e a s y , however. Two d i m e n s i o n a l i n v e r s i o n programmes e x i s t , but t h e non-u n i q u e n e s s must be even worse t h a n t h e v e r y c o n s i d e r a b l e 1-D n o n - u n i q u e n e s s . As f o r t h e 1-D c a s e , some e f f o r t must be made t o a s s e s s t h e v a r i a b i l i t y o f p o s s i b l e 2-D m o d e l s . One way t o l i m i t a c c e p t a b l e c o n d u c t i v i t y p r o f i l e s i s t o i n c o r p o r a t e r e s u l t s from other, g e o p h y s i c a l s u r v e y s . S t r u c t u r e s d e l i n e a t e d by t h e d i f f e r e n t methods h e l p t o c o n s t r a i n t h e p o s s i b l e e a r t h models and d e f i n e a most p r o b a b l e one. G r a v i t y d a t a c o u l d be u s e d a l t h o u g h i t i s a v e r y n o n - u n i q u e t e c h n i q u e 77 i t s e l f and t h e c o n n e c t i o n between d e n s i t y and c o n d u c t i v i t y i s n o t p r e c i s e . Deep s e i s m i c s o u n d i n g s down t o t h e Moho and below c o u l d i d e n t i f y major g e o l o g i c u n i t s w h i c h c o r r e l a t e w i t h c o n d u c t i v i t y a n o m a l i e s . Heat f l o w i s l i n k e d w i t h c o n d u c t i v i t y s i n c e l a r g e v a l u e s of e i t h e r a r i s e f r o m e l e v a t e d t e m p e r a t u r e s . H i g h h e a t f l o w measurements i n t h e OBM a r e a w o u l d r e i n f o r c e t h e c a s e f o r t h e i n t e r p r e t e d h i g h c o n d u c t i v i t i e s . F i n a l l y , t h e n a t u r a l s o u r c e MT method s u f f e r s from an i n c o m p l e t e knowledge of t h e s o u r c e mechanisms and t h e f i e l d s t h e y g e n e r a t e . Many t e c h n i q u e s must be u s e d t o e n s u r e t h a t t h e impedances a r e not c a l c u l a t e d on t h e b a s i s o f p a t h o l o g i c a l s o u r c e s . T e s t s of t h e v a l i d i t y of t h e p l a n e wave a s s u m p t i o n s t i l l depend on s i m p l i f i e d s o u r c e m o d e l s . T h e o r e t i c a l and p r a c t i c a l work on a r t i f i c i a l c o n t r o l l e d s o u r c e s f o r MT s h o u l d p r o v i d e a s u b s t a n t i a l improvement i n measured impedances and a b e t t e r u n d e r s t a n d i n g of t h e e a r t h ' s e l e c t r i c a l c o n d u c t i v i t y . 78 REFERENCES B a c k u s , G. and G i l b e r t , F., 1970. U n i q u e n e s s i n t h e i n v e r s i o n o f i n a c c u r a t e g r o s s e a r t h d a t a , T r a n s . P h i l . Roy. S o c , A266, 123-192. J j B a i l e y , R . C , 1970. I n v e r s i o n of t h e g e o m a g n e t i c i n d u c t i o n p r o b l e m , P r o c . Roy. S o c . London, A315, 185-194. Banks, R . J . , 1973. D a t a p r o c e s s i n g and i n t e r p r e t a t i o n i n g e o m a g n e t i c deep s o u n d i n g , Phys. E a r t h P l a n e t . I n t . , 7, 339-348. B e n d a t , J . S . and P i e r s o l , A.G., 1971. Random d a t a : a n a l y s i s and measurement p r o c e d u r e s , John W i l e y and Sons, New Y o r k . C a g n i a r d , L., 1953. B a s i c t h e o r y of t h e m a g n e t o t e l l u r i c method of g e o p h y s i c a l p r o s p e c t i n g , G e o p h y s i c s , 18, 605-635. Chave, A.D., Von H e r z e n , R.P., P o e h l s , K.A. and Cox, C S , 1981. E l e c t r o m a g n e t i c i n d u c t i o n f i e l d s i n t h e deep ocean n o r t h e a s t of H a w a i i : i m p l i c a t i o n s f o r m a n t l e c o n d u c t i v i t y and s o u r c e f i e l d s , Geophys. J . R. A s t r . S o c., 66, 379-406. Cox, C S . , F i l l o u x , J.H. and L a r s e n , J . C , 1970. E l e c t r o m a g n e t i c s t u d i e s o f o c e a n c u r r e n t s and e l e c t r i c a l c o n d u c t i v i t y below t h e o c e a n f l o o r , i n The Sea, 4, 637-693, ed. A.E. M a x w e l l , John W i l e y and Sons, New Y o r k . 79 E g g e r s , D.E., 1982. An e i g e n s t a t e f o r m u l a t i o n o f t h e m a g n e t o t e l l u r i c impedance t e n s o r , G e o p h y s i c s , 47, 1204-1214. E v e r e t t , J . E . and Hyndman, R.D., 1967. G e o m a g n e t i c v a r i a t i o n s and e l e c t r i c a l c o n d u c t i v i t y s t r u c t u r e i n s o u t h w e s t e r n A u s t r a l i a , P h y s . E a r t h P l a n e t . I n t . , 1., 24-34. F i l l o u x , J.H., 1967. O c e a n i c e l e c t r i c c u r r e n t s , g e o m a g n e t i c v a r i a t i o n s and t h e deep e l e c t r i c a l c o n d u c t i v i t y s t r u c t u r e of t h e o c e a n - c o n t i n e n t t r a n s i t i o n o f c e n t r a l C a l i f o r n i a , Ph.D. t h e s i s , U n i v e r s i t y o f C a l i f o r n i a , San D i e g o . F i l l o u x , J.H., 1973. T e c h n i q u e s and i n s t r u m e n t a t i o n f o r s t u d y o f n a t u r a l e l e c t r o m a g n e t i c i n d u c t i o n a t s e a , P h y s . E a r t h P l a n e t . 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K e l l e r , G.V. and F r i s c h k n e c h t , F.C., 1966. E l e c t r i c a l methods i n g e o p h y s i c a l p r o s p e c t i n g , Pergamon P r e s s , O x f o r d . L a r s e n , J.C., 1968. E l e c t r i c and m a g n e t i c f i e l d s i n d u c e d by deep s e a t i d e s , Geophys. J . R. A s t r . S o c , 1 6, 47-70. L a r s e n , J . C . , 1973. An i n t r o d u c t i o n t o e l e c t r o m a g n e t i c 81 i n d u c t i o n i n t h e o c e a n , Phys. E a r t h P l a n e t . I n t . , 7, 389-398. Law, L.K. and G r e e n h o u s e , J . P . , 1981. G e o m a g n e t i c v a r i a t i o n s o u n d i n g of t h e a s t h e n o s p h e r e b e n e a t h t h e J u a n de F u c a r i d g e , J . G eophys. Res., 86, 967-978. L o e w e n t h a l , D. and L a n d i s m a n , M., 1973. T h e o r y f o r m a g n e t o t e l l u r i c o b s e r v a t i o n s on t h e s u r f a c e o f a l a y e r e d a n i s o t r o p i c h a l f s p a c e , Geophys. J . R. A s t r . 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