UBC Theses and Dissertations

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UBC Theses and Dissertations

Analysis of geomagnetic depth sounding data Stinson, Kerry James 1981

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ANALYSIS OF GEOMAGNETIC DEPTH SOUNDING DATA by KERRY JAMES STINSON B . S c , S i m o n F r a s e r U n i v e r s i t y , 1973 A T H E S I S SUBMITTED IN P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF S C I E N C E i n THE F ACULTY OF GRADUATE STUDIES (Department o f G e o p h y s i c s a n d A s t r o n o m y ) We a c c e p t t h i s thesis as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE U N I V E R S I T Y OF B R I T I S H COLUMBIA O c t o b e r , 1981 (c) K e r r y J a m e s S t i n s o n , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f (Si^-pA^s^c s QLW</ A sfv o *o i*i<j The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date Oct. X%/% > A b s t r a c t The e l e c t r o m a g n e t i c i n d u c t i o n problem i s n o n - l i n e a r , and t h u s i s v e r y d i f f i c u l t t o s o l v e f o r a l l but the s i m p l e s t symmetries. Because of t h i s , q u a n t i t a t i v e m o d e l l i n g of the c o n d u c t i v i t y s t r u c t u r e from geomagnetic depth sounding d a t a i s e x p e n s i v e and time consuming, and the p o s s i b i l i t y t h a t the anomaly i s produced by c h a n n e l l i n g of r e g i o n a l l y induced c u r r e n t s may i n v a l i d a t e the r e s u l t s . For t h i s reason t r a d i t i o n a l methods of a n a l y s i s are g e n e r a l l y q u a l i t a t i v e i n n a t u r e , w i t h q u a n t i t a t i v e i n f o r m a t i o n e s t i m a t e d on the b a s i s of s i m p l i f i e d models of the anomaly. The t h e o r y and assumptions used i n these t r a d i t i o n a l methods are s t u d i e d i n t h i s t h e s i s , and the range of t h e i r a p p l i c a b i l i t y i s i n v e s t i g a t e d . To a v o i d the c u r r e n t c h a n n e l l i n g c o m p l i c a t i o n , and t o a l s o get a l i n e a r r e l a t i o n between the model and the d a t a , the problem i s r e f o r m u l a t e d w i t h the s u b s u r f a c e c u r r e n t d e n s i t y as the model parameter, r a t h e r than the c o n d u c t i v i t y . The d i s a d v a n t a g e of t h i s f o r m u l a t i o n i s t h a t models t h a t f i t the d a t a a r e v e r y non-unique. The c h a r a c t e r of t h i s non-uniqueness has been e x p l o r e d u s i n g B a c k u s - G i l b e r t a p p r a i s a l , and by the c o n s t r u c t i o n of u n c o n s t r a i n e d models. The r e s u l t s i n d i c a t e t h a t r e a s o n a b l e r e s o l u t i o n of the t r u e model's h o r i z o n t a l f e a t u r e s i s p o s s i b l e , but t h a t v e r t i c a l r e s o l u t i o n w i l l be l a c k i n g . To i c o u n t e r t h i s , the i n f i n i t e range of p o s s i b l e models i s c o n s t r a i n e d by i n t r o d u c i n g e x p e c t e d p h y s i c a l f e a t u r e s of the t r u e model i n t o the model c o n s t r u c t i o n a l g o r i t h m . ' T h i s c o n s t r u c t i o n a l g o r i t h m was t e s t e d u s i n g d a t a g e n e r a t e d from a v a r i e t y of a r t i f i c i a l models, and was s u c c e s s f u l i n r e s o l v i n g b o th the h o r i z o n t a l and v e r t i c a l p o s i t i o n s of the major f e a t u r e s i n a l l of them. The a l g o r i t h m was then used t o d e t e r m i n e the s u b s u r f a c e c u r r e n t s t r u c t u r e f o r r e a l d a t a taken a c r o s s the Cascade anomaly i n Washington S t a t e . T a b l e of C o n t e n t s Page A b s t r a c t i i L i s t of T a b l e s v i i L i s t of F i g u r e s v i i i Acknowledgements x i i i I n t r o d u c t i o n 1 Chapter I : The Source F i e l d 6 1.1 G e n e r a l Nature of Sources 6 1.2 M a t h e m a t i c a l Models of the Source F i e l d 17 Chapter I I : The E l e c t r o m a g n e t i c I n d u c t i o n Problem 26 2.1 G e n e r a l Source F i e l d Over a Homogenous H a l f - S p a c e 26 2.2 U n i f o r m Source F i e l d Over a Two-Dimensional E a r t h : Forward M o d e l l i n g 48 Chapter I I I : T r a d i t i o n a l Methods of G.D.S. A n a l y s i s 58 3.1 The F o r m u l a t i o n and S e p a r a t i o n of the Normal F i e l d ....58 3.2 V i s u a l Methods of A n a l y s i s 74 3.3 The I n d u c t i o n Tensor and I n d u c t i o n Arrows 78 3.4 Q u a n t i t a t i v e Methods Used i n G.D.S 97 V Chapter IV: The C u r r e n t D e n s i t y Model 107 4.1 The C u r r e n t D e n s i t y F o r m u l a t i o n 107 4.2 Uniqueness and B a c k u s - G i l b e r t A p p r a i s a l 113 4.3 C o n s t r u c t i o n : The S m a l l e s t Model 129 4.4 C o n s t r a i n e d Model C o n s t r u c t i o n U s i n g L i n e a r Programming 142 Chapter V: A n a l y s i s of G.D.S. A c r o s s the Cascade Anomaly ....160 C o n c l u s i o n s 188 B i b l i o g r a p h y 192 Appendix A: Maxwell's E q u a t i o n s i n a Conductor 212 Appendix B: C o r r e l a t i o n of P r i c e ' s I n d u c t i o n S o l u t i o n s w i t h Plane Wave S o l u t i o n s 219 Appendix C: The U n i f o r m F i e l d Assumption 229 Appendix D: S e p a r a t i o n of the E x t e r n a l and I n t e r n a l F i e l d s ..234 Appendix E: D e t e r m i n a t i o n of the I n d u c t i o n Tensor Elements ..240 Appendix F: P r o p e r t i e s of C u r r e n t D i s t r i b u t i o n s t h a t Mimic a L i n e C u r r e n t 246 vi Appendix G: A n a l y t i c and Numeric I n t e g r a t i o n s 251 Appendix H: D e t e r m i n i n g the Major A x i s of an E l l i p s o i d 264 v i i L i s t of T a b l e s Page T a b l e 1.1 S k i n Depths f o r a Homogenous E a r t h 8 T a b l e 2.1 Parameter Ranges i n G.D.S 39 T a b l e 2.2 V a l u e s of £ f o r V a r y i n g T, A , w i t h CT= .005 S/m 40 T a b l e 2.3 V a l u e s of ft f o r V a r y i n g T, A , w i t h (5 = .5 S/m 41 T a b l e 2.4 Phase D i f f e r e n c e Between the Anomalous S u r f a c e Magnetic F i e l d and the Normal S u r f a c e E l e c t r i c F i e l d , f o r Both the L i n e C u r r e n t and the Jones-Pascoe R e s u l t s 57 T a b l e 5.1 S t a t i o n P o s i t i o n s ....182 v i i i L i s t of F i g u r e s Page F i g . 1.1 The S t e a d y s t a t e I n t e r a c t i o n Between the E a r t h ' s D i p o l e F i e l d and the S o l a r Wind 10 F i g . 1.2 R e l a t i o n s h i p Between t h e S o l a r Wind-Magnetosphere C o u p l i n g F u n c t i o n and the M a g n e t o s p h e r i c Substorm Index .. •• 13 F i g . 1.3 P o l a r Magnetic Substorm Models 15 F i g . 1.4 Geometry of the Source F i e l d and the E a r t h 18 F i g . 1.5 P l a n e Wave Am p l i t u d e Spectrum of a L i n e C u r r e n t Source 24 F i g . 2.1 The R a t i o s of t h e Complex Magnitudes of the I n d u c i n g and Induced Magnetic F i e l d 37 F i g . 2.2 Three R e p r e s e n t a t i v e Models Used f o r the Jones-Pascoe Forward I n d u c t i o n Program 53 F i g . 2.3 R e s u l t s From the Jones-Pascoe Program f o r the Three Models of F i g . 2.2, a t a P e r i o d of 5 Min 54 F i g . 2.4 R e s u l t s From the Jones-Pascoe Program f o r the Three Models of F i g . 2.2, a t a P e r i o d of 50 Min 55 F i g . 3.1 The Assumed C o n d u c t i v i t y Model of G.D.S 59 i x F i g . 3.2 One D i m e n s i o n a l S u r f a c e A r r a y P e r p e n d i c u l a r t o the S t r i k e of a Two-D i m e n s i o n a l E a r t h 68 F i g . 3.3 Magnetometer L o c a t i o n s and Magnetograms f o r a Substorm i n Aug., 1972 75 F i g . 3.4 R e l a t i o n Between Induced C u r r e n t and I n d u c i n g F i e l d Presuming F i r s t Order I n d u c t i o n Only 77 F i g . 3.5 Contoured A m p l i t u d e and Phase of the Z Component of the Magnetograms of .Fig. 3.3, a t a P e r i o d of 68.3 Min 77 F i g . 3.6 The D i r e c t i o n of the In-Phase I n d u c t i o n Arrow Near a L i n e C u r r e n t Running N o r t h - S o u t h 88 F i g . 3.7 F i r s t Order I n d u c t i o n i n C r o s s i n g C o n d u c t i v e Paths 89 F i g . 3.8 The E f f e c t s of C u r r e n t C h a n n e l l i n g t h r o u g h a C o n d u c t i v e Sphere 92 F i g . 3.9 A V a r i e t y of Two-Dimensional C u r r e n t D e n s i t y Models 99 F i g . 3.10 The V e r t i c a l F i e l d s of the C u r r e n t Models of F i g . 3.9 99 F i g . 4.1 The Presumed Two-Dimensional Model 108 F i g . 4.2 A v e r a g i n g F u n c t i o n s C a l c u l a t e d f o r the Two-Dimensional C u r r e n t D e n s i t y Problem.' 126 X F i g . 4.3 D e t e r m i n a t i o n of the A p p r o p r i a t e W e i g h t i n g F a c t o r t o O f f s e t Geometric Decay 133 F i g . 4.4 L^-Norm S m a l l e s t Model C o n s t r u c t i o n 135 F i g . 4.5 L^-Norm S m a l l e s t Model C o n s t r u c t i o n t o Test H o r i z o n t a l R e s o l u t i o n .138 F i g . 4.6 L2 _Norm S m a l l e s t Model C o n s t r u c t i o n t o Test V e r t i c a l R e s o l u t i o n 140 F i g . 4.7 L,-Norm L i n e a r Programming Model C o n s t r u c t i o n 149 F i g . 4.8 L|-Norm L i n e a r Programming Model C o n s t r u c t i o n t o Te s t H o r i z o n t a l R e s o l u t i o n 152 F i g . 4.9 L| -Norm L i n e a r Programming Model C o n s t r u c t i o n t o Test V e r t i c a l R e s o l u t i o n 153 F i g . 4.10 L|-Norm L i n e a r Programming Model C o n s t r u c t i o n U s i n g V e r t i c a l C u r r e n t Dike 154 F i g . 4.11 Lj-Norm L i n e a r Programming Model C o n s t r u c t i o n U s i n g D i p p i n g C u r r e n t Dike 155 F i g . 4.12 Li-Norm L i n e a r Programming Model C o n s t r u c t i o n U s i n g Jones-Pascoe I n d u c t i o n Program Output f o r a S i n g l e C o n d u c t i v e Path 157 F i g . 4.13 L (-Norm L i n e a r Programming Model C o n s t r u c t i o n U s i n g Jones-Pascoe I n d u c t i o n Program Output f o r Two V e r t i c a l l y S eparated C o n d u c t o r s 158 x i F i g . 5.1 A r r a y of Magnetometers Used by Law et a l t o Study the Cascade Anomaly 161 F i g . 5.2 In Phase and Quadrature Arrows f o r Data from A r r a y of F i g . 5.1 162 F i g . 5.3 The A r r a y of Magnetometers which Measured the Data Used i n t h i s T h e s i s 163 F i g . 5.4 Magnetograms f o r the M a g n e t i c Storm of Feb. 1980 164 F i g . 5.5 Magnetograms f o r the M a g n e t i c Storm of Feb. 1980 165 F i g . 5.6 The A m p l i t u d e S p e c t r a of the Anomalous F i e l d a t ORT 1 68 F i g . 5.7 The A m p l i t u d e S p e c t r a of the Anomalous F i e l d a t MUD 169 F i g . 5.8 The A m p l i t u d e S p e c t r a of the Anomalous F i e l d a t GRE 170 F i g . 5.9 The Degree of P o l a r i z a t i o n Between the Three D i r e c t i o n a l Components of the Anomalous F i e l d 174 F i g . 5.10 The R a t i o of the S m a l l e s t t o the Second S m a l l e s t ' E i g e n v a l u e s of t h e P o l a r i z a t i o n E l l i p s o i d 177 F i g . 5.11 The Induced F i e l d from Two Conductors a t D i f f e r e n t Depths 179 x i i F i g . 5.12 The E s t i m a t e d D i p and S t r i k e of a L i n e C u r r e n t Model 180 F i g . 5.13 Lj-Norm L i n e a r Programming C u r r e n t Models U s i n g the R e a l Data 183 F i g . 5.14 A F i n a l Map of the Cascade Anomaly 187 F i g . B.1 R o t a t i n g t o a New H o r i z o n t a l C o o r d i n a t e System 223 F i g . C.1 The P r o p o g a t i o n Angle of the T r a n s m i t t e d Wave i n t o a H a l f - S p a c e of U n i f o r m C o n d u c t i v i t y 231 F i g . G.1 The P a r a m e t e r i z e d C u r r e n t D e n s i t y Model 261 x i i i Acknowledgements The s t o l i d b r i c k e x t e r i o r of the Geophysics and Astronomy b u i l d i n g a t U.B.C. b e l i e s the warmth and enthusiasm t h a t I have e n c o u n t e r e d w i t h i n . I thank a l l the people of the department f o r the e n j o y a b l y c o n d u c i v e atmosphere t h a t has s u s t a i n e d me d u r i n g t h i s t h e s i s . My s u p e r v i s o r , Doug Oldenburg, d e s e r v e s s p e c i a l mention; h i s i n s i g h t , knowledge, and i n f i n i t e eagerness have been drawn on throughout t h i s t h e s i s " . A l s o worthy of note a r e the many l o n g d i s c u s s i o n s I have . had w i t h my o f f i c e m a t e Tim Scheuer, on a l l a s p e c t s of l i f e and g e o p h y s i c s . I would a l s o l i k e t o thank P e t e r F u l l a g a r f o r h i s s u g g e s t i o n s on the i n v e r s i o n methods. I thank G e r a l d H e n s e l and Dr. John Booker of the U n i v e r s i t y of Washington Geophysics Department f o r k i n d l y s u p p l y i n g me w i t h the Cascade anomaly d a t a . I am immensely g r a t e f u l t o my w i f e T r a c y f o r her support and p a t i e n c e d u r i n g the c o u r s e of t h i s t h e s i s . With l u c k she can tak e the c a n d l e out of the window now. 1 INTRODUCTION G e o m a g n e t i c d e p t h s o u n d i n g ( G . D . S . ) i s one o f t h e many e l e c t r o m a g n e t i c m e t h o d s u s e d t o d e t e r m i n e t h e c o n d u c t i v i t y s t r u c t u r e o f t h e e a r t h . In t h i s p a r t i c u l a r method t h e e l e c t r o m a g n e t i c m e a s u r e m e n t s u s e d a r e t h e s u r f a c e v a l u e s o f t h e t h r e e d i r e c t i o n a l c o m p o n e n t s o f t h e m a g n e t i c f i e l d . G . D . S . i n p r a c t i c e i s f u r t h e r s u b d i v i d e d i n t o p r o b l e m s o f g l o b a l a n d ' l o c a l e x t e n t , w i t h t h e f i r s t e v a l u a t i n g t h e e a r t h ' s g r o s s c o n d u c t i v i t y s t r u c t u r e u s i n g t h e a s s u m p t i o n o f r a d i a l s y m m e t r y , and t h e s e c o n d d e t e r m i n i n g more l o c a l s t r u c t u r e s u s i n g t h e p l a n e e a r t h a s s u m p t i o n . T h i s t h e s i s w i l l be c o n c e r n e d o n l y w i t h t h e l o c a l G . D . S . p r o b l e m , so h e n c e f o r t h , u n l e s s s p e c i f i e d o t h e r w i s e , G . D . S . w i l l r e f e r o n l y t o t h i s s u b d i v i s i o n o f t h e p r o b l e m . In G . D . S . p r o b l e m s , t h e s o u r c e s ( o r p r i m a r y f i e l d s ) f o r t h e e l e c t r o m a g n e t i c i n d u c t i o n i n t o t h e e a r t h have t r a d i t i o n a l l y been n a t u r a l l a r g e s c a l e v a r i a t i o n s i n t h e m a g n e t o s p h e r e . S e v e r a l t y p e s o f n a t u r a l s o u r c e have been u s e d i n t h e p a s t , a s w i l l be o u t l i n e d i n C h a p t e r 2, and t h e o n e s most commonly u s e d f o r G . D . S . s t u d i e s w i l l be d i s c u s s e d i n somewhat more d e t a i l , a l s o i n t h a t c h a p t e r . A l 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 p r o b l e m s a r e i d e n t i c a l i n i n i t i a l f o r m u l a t i o n ; f r o m t h i s p o i n t t h e i n t r a c t a b i l i t y o f t h e g e n e r a l p r o b l e m ha s l e d t o t h e b r a n c h i n g i n t o s e p a r a t e methods 2 a l o n g p a t h s of d i f f e r e n t s i m p l i f y i n g a s s u m p t i o n s . In the m a g n e t o t e l l u r i c method, the assumption i s u s u a l l y t h a t of a one-d i m e n s i o n a l p l a n e e a r t h , w i t h c o n d u c t i v i t y v a r y i n g o n l y w i t h d e p t h . In t h i s o n e - d i m e n s i o n a l c a s e , the f o r w a r d problem of e l e c t r o m a g n e t i c i n d u c t i o n can be s o l v e d q u i c k l y and c h e a p l y , f o r both the p l a n e l a y e r e d c a s e , ( C a g n a i r d , 1953; K e l l e r and F r i s c h n e c h t , 1966) and f o r the c o n t i n u o u s case (Oldenburg, 1979). The i n v e r s e problem i s a l s o q u i t e w e l l s o l v e d , and a v a r i e t y of d i f f e r e n t methods are i n the l i t e r a t u r e ( C a g n a i r d , 1953; Becher and Sharpe, 1969; B a i l e y , 1970; Oldenburg, 1979; F i s c h e r e t a l , 1980). The f o r w a r d problem of the t w o - d i m e n s i o n a l i n d u c t i o n problem i s more d i f f i c u l t and more e x p e n s i v e , a l t h o u g h n u m e r i c a l s o l u t i o n s do e x i s t (Jones and P r i c e , 1970; Jo n e s , 1970; Madden and S w i f t , 1969). There i s a l s o some a m b i g u i t y about the c o r r e c t n e s s of the s o l u t i o n s i n each c a s e , and the f a s t e s t way t o do the problem n u m e r i c a l l y ( P r a u s , 1975). C o r r e s p o n d i n g l y , a l t h o u g h methods f o r i n v e r s i o n of e l e c t r o m a g n e t i c s u r f a c e r e a d i n g s over t w o - d i m e n s i o n a l s t r u c t u r e s do e x i s t , (Jupp and V o z o f f , 1977), they a r e v e r y e x p e n s i v e and t h e i r i t e r a t i v e p r o c e d u r e s can f a i l because of l a c k of convergence (Jupp and V o z o f f ,1977). In g e n e r a l , most model c o n s t r u c t i o n s f o r the two-d i m e n s i o n a l case a r e s i m p l y f o r w a r d m o d e l l i n g a t t e m p t s , w i t h s t a r t i n g models e i t h e r a d j u s t e d i n t e r a c t i v e l y ( P o r a t h e t a l , 1970), or a d j u s t e d randomly i n Monte C a r l o f a s h i o n (Anderssen ,1975; Woods, 1979) u n t i l a s a t i s f a c t o r y f i t w i t h the 3 d a t a i s a c h i e v e d . The t h r e e - d i m e n s i o n a l problem e a s i l y i n v o l v e s an or d e r of magnitude i n c r e a s e i n d i f f i c u l t y over the t w o - d i m e n s i o n a l c a s e . The s o l u t i o n of the f o r w a r d p r oblem i s an enormously e x p e n s i v e p r o p o s i t i o n (Lee e t a l , 1981), and t o t h i s a u t h o r ' s knowledge t h e r e a r e a t t h i s time no wo r k i n g methods f o r t r u e t h r e e -d i m e n s i o n a l i n v e r s i o n . In t h i s t h e s i s , the t h r e e - d i m e n s i o n a l case w i l l not be c o n s i d e r e d . The assumption of v e r t i c a l o n e - d i m e n s i o n a l i t y i s p r o b a b l y v e r y good i n some a r e a s , such as i n the deep ocean away from the r i d g e s (Oldenburg, 1981) or i n se d i m e n t a r y b a s i n s ( V o z o f f , 1972). However, i n many c a s e s the reason an area w i l l be of i n t e r e s t w i l l be because of i t s anomalous e l e c t r o m a g n e t i c r e s p o n s e s , w i t h t h i s anomalous b e h a v i o r i m p l y i n g an u n d e r l y i n g s t r u c t u r e more complex than the one- d i m e n s i o n a l c a s e . I t i s t o handle t h e s e cases t h a t the methods c o n s t i t u t i n g G.D.S. have been d e v i s e d . U n l i k e some e l e c t r o m a g n e t i c i n d u c t i o n methods, such as the g l o b a l G.D.S. problem i n which the assumption of r a d i a l symmetry i s made, or c o n t r o l l e d source s t u d i e s i n which the p r i m a r y f i e l d i s known, t h e r e a r e no u n i v e r s a l l y a p p l i e d a ssumptions i n G.D.S. R a t h e r , the av o i d a n c e of the i n t r a c t a b i l i t y of the g e n e r a l i n d u c t i o n problem i s a t t a i n e d s i m p l y by c o n c e n t r a t i n g on q u a l i t a t i v e r e s u l t s , w i t h l i t t l e emphasis p l a c e d on r i g o r o u s q u a n t i t a t i v e e v a l u a t i o n . The i n t u i t i o n needed t o a r r i v e a t these q u a l i t a t i v e r e s u l t s 4 i s g e n e r a l l y o b t a i n e d by r e v i e w i n g the s o l u t i o n s f o r s i m p l e , but b a s i c , f o r w a r d problems. For t h i s reason , Chapter I I g i v e s the r e s u l t s f o r i n d u c t i o n of a g e n e r a l s o u r c e f i e l d over a homogenous h a l f space ( i n s e c t i o n 2.1) and a l s o g i v e s the r e s u l t s f o r t h r e e i d e a l i z e d t w o - d i m e n s i o n a l s t r u c t u r e s which s h o u l d be q u i t e r e p r e s e n t a t i v e of the range of t w o - d i m e n s i o n a l a n o m a l i e s p o s s i b l e ( i n s e c t i o n 2.2). A r e v i e w of the t r a d i t i o n a l methods used i n geomagnetic depth sounding i s p r e s e n t e d i n Chapter I I I , w i t h many of the d i s c u s s i o n s d r awing upon t h e r e s u l t s of Chapter I I . I t w i l l be seen t h a t b oth the q u a l i t a t i v e and q u a n t i t a t i v e i n f o r m a t i o n o b t a i n e d u s i n g these methods i s c r u c i a l l y dependent on the h o r i z o n t a l v a r i a t i o n s i n the source f i e l d , as w e l l as upon i n d u c t i o n o u t s i d e the measurement a r r a y . Both of these f a c t o r s are d i f f i c u l t t o e s t i m a t e w e l l (Gough, 1973; L i l l e y , 1975). As w e l l , many of the f i n a l q u a l i t a t i v e r e s u l t s o b t a i n e d from t h e s e methods a r e based on the i n t u i t i o n of the i n d i v i d u a l a u t h o r s , where o f t e n i t would be p r e f e r a b l e t o have an automated method t o a v o i d i n d i v i d u a l b i a s e s . To a v o i d these problems, a d i f f e r e n t method of a n a l y s i s of G.D.S. d a t a f o r presumed t w o - d i m e n s i o n a l a n o m a l i e s i s suggested i n Chapter IV. In t h i s approach the c u r r e n t d e n s i t y d i s t r i b u t i o n i n the e a r t h i s t r e a t e d as the d e s i r e d model r a t h e r than the c o n d u c t i v i t y s t r u c t u r e . By f o r m u l a t i n g the problem t o c o n s i d e r o n l y the r e s u l t s of the i n d u c t i o n p r o c e s s , ( t h a t i s , the induced c u r r e n t s i n the e a r t h ) r a t h e r than the t o t a l i n d u c t i o n p r o c e s s , 5 the d i f f i c u l t i e s due t o the n o n - u n i f o r m i t y of the source f i e l d and the n o n - l o c a l i n d u c t i o n e f f e c t s a r e a v o i d e d . As w e l l , the problem i n t h i s f o r m u l a t i o n i s l i n e a r , s i m p l i f y i n g m a n i f e s t l y the i n v e r s i o n p r o c e d u r e . On the o t h e r hand, by i g n o r i n g the p r i m a r y f i e l d a l t o g e t h e r a g r e a t d e a l of m o d e l - l i m i t i n g i n f o r m a t i o n i s thrown away, and so we e xpect the non-uniqueness of our r e s u l t s t o be i n c r e a s e d . In f a c t , u s i n g o n l y the t w o - d i m e n s i o n a l a s s u m p t i o n , the c u r r e n t d e n s i t y f o r m u l a t i o n w i l l be shown t o be h o p e l e s s l y non-unique ( s e c t i o n 4.2 and 4.3). However, by presuming c e r t a i n p h y s i c a l c h a r a c t e r i s t i c s e x p e c t e d of n a t u r a l t w o - d i m e n s i o n a l c u r r e n t s t r u c t u r e s , and i n c o r p o r a t i n g these as c o n s t r a i n t s i n a model c o n s t r u c t i o n r o u t i n e ( s e c t i o n 4.4) t h i s non-uniqueness can be g r e a t l y reduced. In the f i n a l c h a p t e r (Chapter V) of t h i s t h e s i s , the new f o r m u l a t i o n d e s c r i b e d i n Chapter IV i s a p p l i e d t o r e a l data t aken a l o n g a l i n e a r a r r a y i n s o u t h w e s t e r n Washington, over the Cascade Anomaly. Throughout t h i s t h e s i s , the attempt i s made t o p r e s e n t the m a t e r i a l i n as r e a d a b l e a f a s h i o n as p o s s i b l e . For t h i s reason p r o o f s or m a t h e m a t i c a l developments which a r e not c o n s i d e r e d e s s e n t i a l f o r the c o n t i n u i t y of the d i s c u s s i o n are e i t h e r r e f e r e n c e d , i f p o s s i b l e , or r e l e g a t e d t o the Appendix. 6 Chapter I The Source F i e l d I.1 General Nature of Sources The sources u t i l i z e d for geomagnetic depth sounding are the naturally occurring variations in the earth's magnetic f i e l d . These can include the variations in the earth's internal f i e l d as well as the variations and disturbances in the external magnetosphere, but the most commonly employed are the l a t t e r . Because of the unp r e d i c t a b i l i t y of most magnetospheric disturbances, the 'choice' of source to be used i s usually limited to post-procurement editing of data. For t h i s reason most f i e l d experiments e n t a i l continuous recording of magnetic data for long periods, of up to months. The selection of source type w i l l depend on the size and position of the array, as well as upon the depth one wishes to 'see' into the earth. In general, analysis of. the results w i l l be s i m p l i f i e d i f the horizontal wavelengths of the source f i e l d are much larger than the dimensions of the array. Thus, for large arrays (200 km. by 500 km.) the sources used w i l l be the large scale disturbances 7 such as the D s t , or p o l a r magnetic substorms. For s m a l l e r a r r a y s , i t might be p o s s i b l e t o use the more l o c a l i z e d s ource f i e l d s , such as m i c r o p u l s a t i o n s . The d e s i r e d depth of p e n e t r a t i o n would determine the range of i n i t i a l a m p l i t u d e of the source r e q u i r e d , as w e l l as the dominant p e r i o d range i n the s o u r c e . The s k i n depth f o r m u l a f o r a homogenous e a r t h i s : d = f T where T = p e r i o d d = depth a t which f i e l d o r i g i n a l magnitude O = c o n d u c t i v i t y -2. I f a c o n d u c t i v i t y of 10 S/m i s assumed, which i s perhaps a r e a s o n a b l e average f o r c r u s t a l and mantle c o n d u c t i v i t i e s ( B r a c e , 1971), then the approximate s k i n depths can then be e v a l u a t e d f o r the dominant p e r i o d ranges ( G a r l a n d , 1979, pg. 257)) of s e l e c t e d magnetospheric d i s t u r b a n c e s , (see T a b l e 1.1). (1.1.1) has decayed t o 1/e of 8 T a b l e 1.1: S k i n Depths f o r a Homogenous E a r t h M a g n e t o s p h e r i c D i s t u r b a n c e Dominant P e r i o d S k i n Depth (d) L i g h t n i n g -a 10 s. 0.159 km. M i c r o p u l s a t i o n 10 - 1 0 s . 1.59-50.3 km P o l a r Substorm 1 0 1 s . 159 km. Dst 4 10 s. 503 km. D i u r n a l V a r i a t i o n s S o l a r Lunar 24 h r s . 25 h r s . 1480 km. 1540 km. D i u r n a l v a r i a t i o n s , l o n g p e r i o d (up t o one y e a r ) e x t e r n a l v a r i a t i o n s , and even the i n t e r n a l l y o r i g i n e d s e c u l a r v a r i a t i o n s , have been used f o r g l o b a l s t u d i e s of the e a r t h ' s g r o s s r a d i a l c o n d u c t i v i t y s t r u c t u r e (Chapman, 1919; Chapman and P r i c e , 1930; 9 L a h i r i and P r i c e , 1939; R i k i t a k e , 1950; Runcorn, 1955; McDonald, 1957; E c k h a r d t e t a l , 1963; Banks, 1969). However, t h i s t h e s i s d e a l s o n l y w i t h the l o c a l G.D.S. problem, so a t t e n t i o n w i l l be f o c u s s e d m a i n l y on thos e d i s t u r b a n c e s w i t h p e r i o d ranges e n a b l i n g i n v e s t i g a t i o n t o depths of 500 km. A l t h o u g h m i c r o p u l s a t i o n s f i t i n t o t h i s c a t e g o r y they a r e not a common source i n G.D.S. As w e l l , they a r e d i v e r s e and complex i n n a t u r e ( J a c o b s , 1970); f o r thes e r e a s o n s they w i l l not be d i s c u s s e d h e r e . A l l the e x t e r n a l s o u r c e s t o be ' c o n s i d e r e d r e s u l t from d i s t u r b a n c e s i n the s t e a d y - s t a t e i n t e r a c t i o n between the e a r t h ' s d i p o l e f i e l d and the s o l a r wind. In t h i s s t e a d y s t a t e , or q u i e t t i m e , the magnetic f i e l d f r o z e n i n t o the h i g h l y c o n d u c t i n g plasma of the s o l a r wind compresses the e a r t h ' s d i p o l e f i e l d (see F i g . 1 . 1 ) . A shock f r o n t of h i g h l y compressed f i e l d l i n e s marks the boundary between the r e g i o n s of i n f l u e n c e of the i n t e r p l a n e t a r y magnetic f i e l d (IMF) and the e a r t h ' s d i p o l e f i e l d , and a c t s as an e f f e c t i v e ' s h i e l d ' , e x c u d i n g the i o n s of the s o l a r wind from the e a r t h ( N i s h i d a , 1978). However, i t has l o n g been a c c e p t e d t h a t a p o r t i o n of th e s e e n e r g e t i c charged p a r t i c l e s must be f i n d i n g e n t r y i n t o the e a r t h ' s i o n o s p h e r e ( R o s t o k e r , 1972). T h i s was e x p e r i m e n t a l l y c o n f i r m e d by the c o r r e l a t i o n between s a t e l l i t e measurements of the i o n f l u x i n the i o n o s p h e r e and observ e d a c i t i v i t y on the sun ( G a r l a n d , 1979, pg. .253). The mechanism f o r the i o n e n t r y i s s t i l l not w e l l 10 F i g . 1.1 The s t e a d y s t a t e i n t e r a c t i o n between the e a r t h ' s d i p o l e f i e l d and the s o l a r wind. The d o t t e d l i n e i n d i c a t e s t h e magnetopause, i n s i d e which t h e e a r t h ' s f i e l d i s c o n f i n e d . 11 u n d e r s t o o d or agreed upon, but i t has become apparent t h a t the d i r e c t i o n and magnitude of the components of the IMF a r e i m p o r t a n t c o n t r o l l i n g parameters i n the p r o c e s s ( A k a s o f u , 1979). S a t e l l i t e measurements of the IMF and c o i n c i d e n t s u r f a c e r e a d i n g s of the e a r t h ' s f i e l d have i n d i c a t e d t h a t a southward d i r e c t e d IMF i s p r o b a b l y a n e c c e s s a r y ( a l t h o u g h not s u f f i c i e n t ) c o n d i t i o n f o r substorm a c t i v i t y ( R o s t o k e r , 1972). P e r r e a u l t and A k a s o f u q u a n t i f i e d t h i s by showing t h a t an e m p i r i c a l r e l a t i o n c o u l d be found between the IMF and the development of geomagnetic storms ( P e r r e a u l t and A k a s o f u , 1978; A k a s o f u , 1979). The r a t e of energy d i s s i p a t i o n , u ( t ) , was e v a l u a t e d f o r 15 major geomagnetic storms, u s i n g measurements of r i n g c u r r e n t and a u r o r a p a r t i c l e i n j e c t i o n , and e s t i m a t e s of J o u l e d i s s i p a t i o n i n the i o n o s p h e r e . I t was found t h a t t h i s e s t i m a t e d energy d i s s i p a t i o n u ( t ) c o u l d be c l o s e l y d u p l i c a t e d by an e m p i r i c a l l y d e t e r m i n e d f u n c t i o n € ( t ) , dependent o n l y on i n t e r p l a n e t a r y p a r a m e t e r s : € ( t ) s i n ( J o u l e s / s e c ) (1.1.2) w i t h : V ( t ) speed of the s o l a r wind plasma B ( t ) t o t a l magnitude of the IMF 12 0(t) : T a n " ' ( I B y / B z l ) f o r Bz > 0 180 - T a n " 1 ( I B y / B z l ) f o r Bz < 0 J?0 : e s t i m a t e of l i n e a r dimension of the c r o s s - s e c t i o n of the magnetopause (assumed = 7Re) The r i g h t hand s i d e of e g u a t i o n 1.1.2 i s c l o s e l y r e l a t e d t o the P o y n t i n g v e c t o r f l u x of the i n t e r p l a n e t a r y e l e c t r o m a g n e t i c f i e l d , so t h a t 6 ( t ) can be regard e d as the r a t e of energy c o u p l i n g between t h e s o l a r wind and magnetosphere ( A k a s o f u , 1979). I t i s noted t h a t i f the IMF has o n l y a northward component, then the c o u p l i n g s h o u l d be z e r o , and t h a t f o r a c o m p l e t e l y southward d i r e c t e d IMF the c o u p l i n g i s a maximum VB*j£*. The c o r r e l a t i o n of fc(t) w i t h substorm a c t i v i t y (as g i v e n by the substorm index AE) has s u b s e q u e n t l y been found t o be q u i t e good f o r v a l u e s of €.(t) l e s s than 10 J/s ( A k a s o f u , 1979). (see F i g . 1.2). Once i n s i d e the magnetosphere, the b u l k of the s o l a r i o n f l u x i s c a r r i e d t o the m a g n e t o t a i l ( R o s t o k e r , 1972; A k a s o f u , 1979), w i t h a c e r t a i n amount of these charged p a r t i c l e s s p i r a l l i n g a l o n g the e a r t h ' s d i p o l e f i e l d l i n e s . At the p o l e s the convergence of the f i e l d l i n e s r e s u l t s i n the p a r t i c l e ' s r e f l e c t i o n so t h a t they t r a v e l from p o l e t o p o l e (Akasofu and Chapman, 1961). The c e n t r i f u g a l f o r c e due t o the c u r v a t u r e of the f i e l d l i n e s , as w e l l as the inhomogeneity of the e a r t h ' s f i e l d ( A l f e n , 1950,pg.14-23; A k a s o f u and Chapman, 1961) r e s u l t s i n an ea s t w a r d d r i f t of the e l e c t r o n s i n t h e i r p o l e t o p o l e 13 ) F i g . 1.2 T h e r e l a t i o n s h i p b e t w e e n t h e s o l a r w i n d - m a g n e t o s p h e r e c o u p l i n g f u n c t i o n a n d t h e m a g n e t o s p h e r i c s u b s t o r m i n d e x AE f o r a s t o r m i n t h e m o n t h o f J u l y , 1974 ( a f t e r A k a s o f u ( 1 9 7 9 ) ) . 1 4 t r a v e l , and a westward d r i f t of the p r o t o n s . T h i s d r i f t g i v e s the e f f e c t of a r i n g c u r r e n t a t e q u a t o r i a l l a t i t u d e s a t about two or t h r e e e a r t h r a d i i ( A kasofu and Chapman, 1 9 6 1 ) , w i t h the r e s u l t of t h i s r i n g c u r r e n t b e i n g a d i p o l e f i e l d which opposes the e a r t h ' s i n t e r n a l d i p o l e f i e l d . Low v a l u e s of energy i n p u t i n t o the magnetopause ( 6 < 1 0 ' " J/s ) c o r r e s p o n d t o e x t r e m e l y q u i e t c o n d i t i o n s , w i t h 'normal' energy d i s s i p a t i o n p r o c e s s e s i n the magnetosphere m a i n t a i n i n g a steady s t a t e . When the v a l u e of € i n c r e a s e s t o a c r i t i c a l v a l u e of i o " J/s , i t i s suggested t h a t the normal d i s s i p a t i v e modes cannot handle the i n c r e a s e d r a t e of energy b e i n g t r a n s f e r r e d t o . t h e m a g n e t o t a i l ( A k a s o f u , 1 9 7 9 ) . The r e s u l t i s t h a t a p o r t i o n of the c u r r e n t s i n the m a g n e t o t a i l are then d i v e r t e d a l o n g magnetic f i e l d l i n e s t o the p o l a r i o n o s p h e r e . E q u i v a l e n t c u r r e n t systems f o r t h i s p r o c e s s have been suggested by a number of a u t h o r s u s i n g ground based measurements, ( B i r k e l a n d , 1908 ; Bostrom, 1964 ; K i s a b e t h and R o s t o k e r , 1 971 ; K i s a b e t h and R o s t o k e r , 1 9 7 7 ) , w i t h most r e c e n t models ( f o r n o r t h e r n l a t i t u d e s ) f e a t u r i n g f i e l d - a l i g n e d c u r r e n t s ( B i r k e l a n d c u r r e n t s ) c o n n e c t e d by a s t r o n g westward e l e c t r o j e t i n the morning s e c t o r , and B i r k e l a n d c u r r e n t s c o n n e c t e d by a s t r o n g e a s t w a r d e l e c t r o j e t i n the e v e n i n g s e c t o r (see F i g . 1 . 3 ) . T h i s p o s t u l a t e d d i v e r s i o n of m a g n e t o t a i l energy i n t o c u r r e n t s i n the p o l a r i o n o s p h e r e i s g e n e r a l l y a c c e p t e d as the b a s i c model f o r the p o l a r substorm. I t has been noted ( A k a s o f u , 1979) t h a t the s t r o n g c o r r e l a t i o n between the c o u p l i n g r a t e € ( t ) and substorm 15 NOON (b) (a) «- J F i g . 1.3 P o l a r magnetic substorm models: (a) d e t a i l e d model of a u r o r a l zone e l e c t r o j e t s and c o n n e c t e d f i e l d - a l i g n e d B i r k e l a n d c u r r e n t s from R o s t o k e r (1978) (b) m a g n e t o s p h e r i c and f i e l d - a l i g n e d c u r r e n t model from K i s a b e t h ( 1 9 7 5 ) . 16 a c t i v i t y , b o th i n growth and i n decay (see F i g . 1.2) i n d i c a t e s t h a t i t i s the energy c o u p l i n g r a t e which c o n t r o l s a l l phases of the substorm. T h i s i s i n c o n t r a s t t o the p r e v i o u s concept ( f o r example, Rostoker, 1972) of the substorm as s i m p l y an energy u n l o a d i n g d e v i c e a c t i v a t e d when the energy d e n s i t y of the m a g n e t o t a i l reached some c r i t i c a l l i m i t . In a d d i t i o n t o the d i v e r s i o n of c u r r e n t s from the m a g n e t o t a i l t o the p o l a r i o n o s p h e r e , t h e r e i s a l s o l a r g e s c a l e i n j e c t i o n of i o n s i n t o the r i n g c u r r e n t system, w i t h a r e s u l t a n t d e c r e a s e i n the main d i p o l e f i e l d . T h i s e f f e c t and i t s subsequent slow decay c o n s t i t u t e the storm-time d i s t u r b a n c e or D s t . The s o u r c e most commmonly used f o r G.D.S. i s the p o l a r substorm. The main reasons f o r t h i s a r e the s t r e n g t h (hundreds of n a n o t e s l a ) and t h e u n i f o r m i t y ( s c a l e l e n g t h of thousands of k i l o m e t e r s ) of i t s d i s t u r b a n c e f i e l d . In p r a c t i c e t h e r e a r e u s u a l l y more than one o v e r l a p p i n g substorms, w i t h the f i n a l s u p e r p o s i t i o n c o n s t i t u t i n g a p o l a r storm. The e n t i r e storm i s used as the s o u r c e , w i t h g e n e r a l l y no attempt made t o s e p a r a t e the i n d i v i d u a l substorms, or the D s t . I.2 Mathematical Models of the Source F i e l d To f a c i l i t a t e a p h y s i c a l understanding of the i n d u c t i o n process i n the case of complex magnetospheric source f i e l d s , i t i s p r a c t i c a l to mathematically formulate the true source f i e l d as a summation of elementary s o l u t i o n s of the wave equation, and then attempt the p h y s i c a l understanding i n terms of a s i n g l e s o l u t i o n . In the region 0 > z > -h, between the s u r f a c e of the e a r t h and the lowest c u r r e n t element of the source f i e l d (see F i g . 1.4), Maxwell's equations a r e : V-B = 0 (1.2.1) V-D = 0 (1.2.2) VxH = D (as 0*= 0) (1.2.3) 18 F i g . 1.4 Geometry of the s o u r c e f i e l d and the e a r t h 19 V x E = -B (1.2.4) Thus, presuming p = f i 0 , and €=£<> everywhere in the region : > -h, using a time dependei of 1.2.3 and 1.2.4, we arri v e at 0 > z nce of e t W / t , and taking the c u r l V E = 9 aE or V H = %H with: and V-E = 0 (1.2.5) (1.2.6) (1.2.7) (1.2.8) The solution of 1.2.5 (or 1.2.6) results in two independent solutions, corresponding to the TE mode ( E 4 = 0) and the TM mode (H 4 = 0) (Budden, 1961, pg. 13-15,22-30). For the TE mode, the 20 e l e m e n t a r y s o l u t i o n f o r the f r e q u e n c y w i s : E ( x , y , z , t ) = E o{ky,-kx,0}e L k« X e i k » * e ^ V " * (1.2.9) and f o r the TM mode: H ( x , y , z , t ) = H o { k y , - k x , 0 } e ' ^ K e £ K * V ^ V " ' (1.2.10) where i n both c a s e s : ki +k! +k\c, = - <fc0 (1.2.11) Note t h a t the p a r e n t h e s e s {} denote the v e c t o r components. To c o m p l e t e l y d e s c r i b e the p r i m a r y f i e l d of an a r b i t r a r y s ource would r e q u i r e summation of the el e m e n t a r y s o l u t i o n s of both the TE and TM modes over a l l p o s s i b l e k* and k^ v a l u e s . (By v i r t u e of 1.2.11, k 2 a i s not i n d e p e n d e n t ) . For s i m p l i c i t y i n the d i s c u s s i o n , c o n s i d e r a source t h a t i s p r o d u c i n g TE mode waves o n l y . ( I t w i l l be shown i n s e c t i o n 2.1 of Chapter I I t h a t t h i s i s the o n l y mode that need be c o n s i d e r e d f o r i n d u c t i o n i n the e a r t h ) . I n t h i s case the e l e c t r i c f i e l d v e c t o r of the p r i m a r y f i e l d a t any p o i n t (x,y) i s : OO CO CU ) f E ( x , y , z , t ) = e t l A , t ^ A ( k x , k y ) {ky,-kx,0} •oo -co 21 • e * e J J e dkxdky (1.2.12) The n e g a t i v e s i g n f o r the e x p o n e n t i a l i n v o l v i n g the z component i s chosen t o ensure t h a t the waves w i l l always by p r o p a g a t i n g downward; k ^ here i s thus > 0 a l w a y s . The x and y components of E a r e not independent because of 1.2.8, so the spectrum A(kx,ky) (denoted the 'angul a r spectrum' by Booker and Clemmow (1950)) can be found u s i n g the v a l u e s of e i t h e r Ex or Ey over an a r b i t r a r y p l a n e s u r f a c e (Booker and Clemmow, 1950). U s i n g the z = 0 p l a n e , d r o p p i n g the time dependence f o r c o n v e n i e n c e , and c o n s i d e r i n g Ex o n l y , we have: 00 CO Ex(x,y,0) = ^ J A ( k x , k y ) k y e t k * * e ^ * 1 dkxdky ~ 0 0 " 0 0 (1.2.13) T h i s i s a double i n v e r s e f o u r i e r t r a n s f o r m , a l l o w i n g us t o w r i t e : A ( k x,ky) = l / ( 4 T r v k y ) ^ Ex (x , y, 0) e^** e" 3 ^ d x d y -co -co (1.2.14) I t i s apparent t h e n , as shown by Booker and Clemmow (1950) and Wait (1954), t h a t the a n g u l a r s p e c t r a of complex s o u r c e s can be c a l c u l a t e d f o r known v a l u e s of Ex on one p l a n e , and then used f o r the c a l c u l a t i o n of the e l e c t r i c ( or magnetic ) f i e l d a t any o t h e r p o s i t i o n . I t was i n d i c a t e d i n s e c t i o n 1.1 of Chapter I t h a t the e q u i v a l e n t model f o r a substorm would i n some a r e a s 22 resemble a h o r i z o n t a l l i n e c u r r e n t ; the a n g u l a r spectrum f o r t h i s model i s thus of o b v i o u s i n t e r e s t and i s c a l c u l a t e d below. A l i n e c u r r e n t of magnitude I i n the y d i r e c t i o n , o s c i l l a t i n g a t a fr e q u e n c y w, a t a h e i g h t h, has an a s s o c i a t e d e l e c t r i c f i e l d w i t h o n l y a component i n the y d i r e c t i o n (Landau and L i f s c h i t z , 1960, pg.195; W a i t , 1970, pg.23): E y ( x , z ) = - i ( j i e w l ) / ( 2 f r ) K 0 {(<pa)'l[x*- + (z+hf ^ } (1 .2.15) where K 0 i s the m o d i f i e d B e s s e l f u n c t i o n of o r d e r z e r o . E x p r e s s i n g t h i s as a summation of elem e n t a r y s o l u t i o n s on an a r b i t r a r y p l a n e z = c o n s t a n t : CO E y ( x , z ) = j ~ A ( k x ) e " t k x X L d k x (1.2.16) (Note t h a t as t h e r e can be no p l a n e wave p r o p a g a t i o n i n the y d i r e c t i o n , then k x and k ? 0 L a r e no l o n g e r independent i n t h i s c a s e ) . The a m p l i t u d e spectrum i s g i v e n by: CO A(kx) = - ( i H o w I ) / ( 4 1 V l - ) | K 0 { c Pj' l[x 2'+(z+h) 1 ] ' l ] e i ^ K dx -oo = -(iju 0wl ) / [4Tr ( qVkx ) 3 e ( 1 .2.17) (from G r a d s h t y n and R h y s i k , 1965, pg. 736) Thus, the f i n a l e x p r e s s i o n f o r Ey i s : 23 oo E y ( x , z ) = -(ho«I)/(4n)Je-^«+k^'UV^ - 0 0 ( 1 . 2 . 1 8 ) For the l i n e c u r r e n t example, a p l o t of the s u r f a c e v a l u e of the e l e m e n t a r y wave a m p l i t u d e v s . the h o r i z o n t a l w a v elength, 2 IT /kx, i s g i v e n i n F i g . 1.5 f o r a l i n e c u r r e n t of p e r i o d one hour, a t a h e i g h t of 100 km. I t i s i m p o r t a n t t o note from t h i s f i g u r e , t h a t even f o r t h i s v e r y u n c o m p l i c a t e d s o u r c e an i n f i n i t e s u p e r p o s i t i o n of waves of d i f f e r i n g wavenumbers i s produced. For each elementary s o l u t i o n (as denoted by e q u a t i o n 1 . 2 . 9 or 1 . 2 . 1 0 ) the t o t a l h o r i z o n t a l w avelength of the p r i m a r y f i e l d w i l l be g i v e n by: A = 2fT/(kx l + ky 1 ( 1 . 2 . 1 9 ) As seen from the l i n e c u r r e n t example, most r e a l f i e l d s w i l l be composed of an i n f i n i t e number of waves of d i f f e r e n t h o r i z o n t a l w a v e l e n g t h s . In G.D.S. the i m p o r t a n t wavelength w i l l be the minimum v a l u e which s t i l l has a s i g n i f i c a n t a m p l i t u d e a t the s u r f a c e . T h i s wavelength w i l l be the v a l u e of the l a r g e s t s i g n i f i c a n t s p a t i a l n o n - u n i f o r m i t y , and i s commonly c a l l e d the s c a l e l e n g t h . I t w i l l be seen i n Chapter 2 and Chapter 3 t h a t the s c a l e l e n g t h of the p r i m a r y f i e l d and i t s e s t i m a t i o n a re of fundamental importance i n v i r t u a l l y a l l a s p e c t s of c o n v e n t i o n a l G.D.S. a n a l y s i s . The t r a d i t i o n a l range of v a l u e s of X was g i v e n by P r i c e (1962): 24 9.0 FIG. 1.5 Plane wave amplitude spectrum on the surface of the earth from a l i n e current at height 100 km. The log of the amplitude i s plotted against the log of the s p a t i a l wavelength X ( A in km.) at periods: 1 sec. (O) 20 hr. (A) 25 4x10**km. < }\ < 4x10 s km. The maximum v a l u e c o r r e s p o n d s t o the c i r c u m f e r e n c e of the e a r t h , whereas the minimum v a l u e was o b t a i n e d from an e s t i m a t e of the s h a r p d r o p - o f f p o i n t of the a m p l i t u d e spectrum f o r a l i n e c u r r e n t , as shown i n F i g . 1.5. (Note t h a t P r i c e a l s o c o n s i d e r e d a h e i g h t of 100 km. f o r h i s l i n e c u r r e n t , and used f o u r t i m e s t h i s h e i g h t as h i s e s t i m a t e of the c u t - o f f p o i n t ) . Methods of e x p e r i m e n t a l l y e s t i m a t i n g the s c a l e l e n g t h w i l l be d i s c u s s e d i n s e c t i o n two of Chapter I I I . 26 C h a p t e r I I T h e E l e c t r 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 2.1 G e n e r a l S o u r c e F i e l d O v e r a Homogenous H a l f - S p a c e T o i l l u s t r a t e t h e m a j o r f e a t u r e s o f t h e i n d u c t i o n p r o c e s s , i n c l u d i n g n o n - u n i f o r m i t y o f t h e s o u r c e f i e l d , t h e s i m p l i f i e d c a s e o f a n i s o t r o p i c , h o m o g e n o u s e a r t h i s c o n s i d e r e d . A s w i l l be s e e n i n C h a p t e r I I I , t h e r e a l c o n d u c t i v i t y s t r u c t u r e o f t h e e a r t h w i l l o f t e n be m o d e l l e d a s a n o r m a l , one d i m e n s i o n a l s t r u c t u r e c o n t a i n i n g a s m a l l a n o m a l o u s r e g i o n . To be a b l e t o i n t u i t t h e v a r i a t i o n s i n t h e ' n o r m a l ' i n d u c e d f i e l d o f t h e one d i m e n s i o n a l e a r t h due t o t h e i n c l u s i o n o f t h e a n o m a l o u s p o r t i o n , one must o b v i o u s l y f i r s t u n d e r s t a n d f u l l y t h e n o r m a l f i e l d c a s e . I t i s p o s s i b l e t o s o l v e f o r t h e n o r m a l f i e l d f o r b o t h a l a y e r e d e a r t h ( C a g n a i r d , 1953; K e l l e r a n d F r i s c h n e c h t , 1966) a n d a c o n t i n u o u s l y v a r y i n g e a r t h ( O l d e n b u r g , 1 9 7 9 ) , b u t f o r s i m p l i c i t y o n l y t h e h o m o genous e a r t h c a s e w i l l be t r e a t e d h e r e . The e x t e n s i o n o f t h e t h e o r y t o t h e f u l l o n e - d i m e n s i o n a l p r o b l e m i s r e l a t i v e l y s t r a i g h t f o r w a r d , a n d i n a n y c a s e , t h e m a j o r q u a l i t a t i v e p o i n t s o f t h e d i s c u s s i o n w i l l be t h e same f o r b o t h . 27 Much of the development t h a t w i l l be p r e s e n t e d here i s s i m i l a r t o t h a t done by P r i c e (1950,1962). However , s l i g h t l y d i f f e r e n t assumptions w i l l l e a d t o a d i f f e r e n t p h y s i c a l d e s c r i p t i o n of the s o l u t i o n s and s h o u l d p r o v i d e more i n s i g h t i n t o the problem. I t s h o u l d be emphasized t h a t the assumptions t h a t P r i c e makes are not i n v a l i d ; i t i s s i m p l y t h a t the form of h i s s o l u t i o n s l e a d t o a c o n c e p t u a l l y d i f f e r e n t way of v i e w i n g the i n d u c t i o n p r o c e s s . Presume two homogenous h a l f - s p a c e s as i n F i g . 1.4 ( d i s r e g a r d i n g the sou r c e f i e l d ) , w i t h the p o s i t i v e z d i r e c t i o n downwards, and CT as denoted. As shown i n Appendix A, f o r a time dependence of e ^ w ' i n a homogenous medium, M a x w e l l ' s e q u a t i o n s can be put i n the form: V E = <pE ( 2 . 1 . 1 ) where: 9 = iw^o-- vxfji0€.o (2.1.2) As w e l l , E i s n o n - d i v e r g e n t everywhere except a t the boundary between the h a l f - s p a c e s (see Appendix A ) : V-E = 0 (2.1.3) 28 For the range of v a l u e s of p e r i o d range commonly used i n second term i n e q u a t i o n 2.1 comparison t o the f i r s t term, space. In o t h e r words, the c o n d u c t o r w i l l be n e g l i g i b l e U s i n g t h i s and the z e r o conduct have f o r e q u a t i o n s 2.1.1 and 2. 9 = 9a= - w ^ 0 £ o z °" e x p e c t e d i n the e a r t h , and the G.D.S. ( l i s t e d i n T a b l e 2.1), the .2 w i l l be i n s i g n i f i c a n t i n w i t h i n the c o n d u c t i v e lower h a l f d i s p l a c e m e n t c u r r e n t s i n the compared t o the r e a l c u r r e n t s , i v i t y i n the upper h a l f - s p a c e , we 1.2: < 0 (2.1.4) <f> = <f>e = iwju00- z > 0 (2.1.5) U s i n g a s e p a r a t i o n of v a r i a b l e s t o s o l v e e q u a t i o n s 2.1.1 and 2.1.3, we presume a s o l u t i o n of the form: E ( x , y , z ) = Z ( z ) F ( x , y ) e ' ' * u t where: (2.1.6) F ( x , y ) = {Fx,Fy,Fz} (2.1.7) (Note, the time term, eLV°^, w i l l appear i n a l l e q u a t i o n s , so we 29 w i l l drop i t h e n c e f o r t h , r e s u r r e c t i n g i t o n l y i n the f i n a l s o l u t i o n s ) . U s i n g 2.1.6 i n 2.1.1, w i t h a s e p a r a t i o n c o n s t a n t - V* we g e t : ? - ( ^ ) / Z = - V l = ( 0 + y ^ ) / F x (2.1.8) S i m i l a r l y , u s i n g e q u a t i o n 2.1.6 t o s e p a r a t e 2.1.3 w i t h a s e p a r a t i o n c o n s t a n t - e<: ( 4 J E L + l £ i L ) / F 2 = - ( l i ) / E = - * (2.1.9) The s o l u t i o n s f o r the e q u a t i o n s i n v o l v i n g 2(z) i n 2.1.8 and 2.1.9 a r e , r e s p e c t i v e l y : E ( z ) = e i ( ^ ^ ^ a (2.1.10) Z(z) = e * * (2.1.11) 30 O b v i o u s l y t h e n , »<= M S ) 1 * ? ) " 1 (2.1.12) The l e f t hand e q u a t i o n i n 2.1.9 has two p o s s i b l e s o l u t i o n s : S o l u t i o n 1 (2.1.13) and F z = 0 (2.1.14) S o l u t i o n 2 V 1 3x (2.1.15) (2.1.16) where the second s o l u t i o n makes use of the l a s t e x p r e s s i o n i n e q u a t i o n 2.1.8. 31 I t i s n o t e d t h a t s o l u t i o n one r e q u i r e s the 'Z' component of the e l e c t r i c f i e l d t o be z e r o ; t h i s c o r r e s p o n d s t o a TE mode s o l u t i o n ( i e . , E i s i n the boundary p l a n e ) . On the o t h e r hand, upon t a k i n g the c u r l of the e l e c t r i c f i e l d f o r s o l u t i o n 2, i t i s found t h a t the 'Z' component of the magnetic f i e l d i s z e r o , c o r r e s p o n d i n g t o the TM mode s o l u t i o n ( i e . , H i n the boundary p l a n e ) . Thus, i t i s expected t h a t the s e p a r a t i o n of the problem i n t o s o l u t i o n s of Type 1 and 2 s h o u l d be d i r e c t l y r e l a t e d t o the s e p a r a t i o n of p l a n e waves i n t o TE and TM mode components, and t h i s w i l l be seen t o be the c a s e . In i d e n t i c a l f a s h i o n t o P r i c e ' s development, the f i r s t type i s s o l v e d by l e t t i n g : Fx = >JfeQ (2.1.17) Fv = -1 ax (2.1.18) where P(x,y) i s a s c a l a r q u a n t i t y . Thus, a l l terms on the r i g h t hand s i d e of e q u a t i o n 2.1.8 l e a d t o : + J l L + S) 1P = o (2.1.19) 32 The f u n c t i o n P ( x , y ) and the v a l u e of ^ can be shown t o be the same i n both h a l f - s p a c e s . Thus, the f i n a l form of the f i r s t t y p e s o l u t i o n i s : E , ( x , y , z ) . l ^ - $ , 0 > U , e - < ^ \ B , e ( * W ' ^ (2.1.20) f o r z < 0 B,(,,y.,) - { » - » ? . . 0 } C, , - ^ * ^ " % * (2.1.21) f o r z > 0 w i t h P(x,y) s a t i s f y i n g 2 . 1 . 1 9 . (Note t h a t i n 2.1.21 the s o l u t i o n f o r 2 ( 2 ) has o n l y the n e g a t i v e e x p o n e n t i a l term t o a v o i d unbounded s o l u t i o n s a t 2 =*oo ). I t s h o u l d be p o i n t e d out here t h a t each s o l u t i o n c o r r e s p o n d i n g t o a d i f f e r e n t "V (where S) can v a r y between 0 and o o) i s as v a l i d as any o t h e r ; the t o t a l s o l u t i o n w i l l be the summation (or i n t e g r a t i o n ) of a l l t h e s e 'elementary' s o l u t i o n s over the e n t i r e range of . U s i n g Maxwell's e q u a t i o n r e l a t i n g H and the c u r l of E (A.13 i n Appendix A ) , the magnetic f i e l d f o r the f i r s t type s o l u t i o n i s found t o be: 33 H , ( x , y , z ) - i ( ^ + 9 J ' V ( v H -3' ( 2 . 1 . 2 2 ) f o r z < 0 . C i e - ( ^ * * . V " » ( 2 . 1 . 2 3 ) f o r z > 0 A t t h e b o u n d a r y o f t h e h a l f - s p a c e s , z = 0, t h e t a n g e n t i a l c o m p o n e n t s o f E , a n d a l l t h r e e c o m p o n e n t s o f H must be c o n t i n u o u s . T h i s l e a d s t o : B, = -A j • (1 - R ) / ( 1 + R) ( 2 . 1 . 2 4 ) a n d C, = A , • ( 2 R ) / ( 1 + R ) ( 2 . 1 . 2 5 ) w h e r e : R = ( 2 . 1 . 2 6 ) 3 4 To get the same form as P r i c e ( 1 9 5 0 ) , l e t : A, = A, 1 ( V + Va. ) / ( W j u o ) ( 2 . 1 . 2 7 ) B , ' = - B , i ( O l + qv ) ' 7(wji 0) ( 2 . 1 . 2 8 ) The magnetic f i e l d a t the s u r f a c e then becomes: H(x, y,o) = - { ( A / + B l ' ) i L , ( A ; + B ; )VL,( A; - b ; ) v s ^ y , } ( 2 . 1 . 2 9 ) and from 2 . 1 . 2 4 , 2 . 1 . 2 7 and 2 . 1 . 2 8 : B ; = A , 1 ( 1 - R ) / ( 1 + R ) ( 2 . 1 . 3 0 ) From the s i g n of the e x p o n e n t i a l i n e q u a t i o n 2 . 1 . 2 2 (and u s i n g the v a l u e of < P A , (from e q u a t i o n 2 . 1 . 4 ) i t i s c l e a r t h a t A / i s the complex magnitude of the i n d u c i n g or p r i m a r y f i e l d , and B, 1 i s the complex magnitude of the induced or secondary f i e l d . I t w i l l be seen i n F i g . 2 . 1 t h a t the argument of ( 1 - R ) / ( 1 + R ) i s always between 0 and T h i s ensures t h a t i n e q u a t i o n 2 . 1 . 2 9 the magnitude of the h o r i z o n t a l components of the p r i m a r y magnetic f i e l d w i l l always be i n c r e a s e d by the a d d i t i o n of the secondary h o r i z o n t a l f i e l d , whereas the magnitude of the p r i m a r y 35 v e r t i c a l f i e l d w i l l always be dec r e a s e d by the secondary v e r t i c a l f i e l d . S u b s t i t u t i n g 2.1.4 and 2.1.5 i n t o 2.1.26, the v a l u e of R i s : R = [( S>v- w ^ o € e ) / ( iw/* 6<r)] ,' ^ (2.1.31) The s e p a r a t i o n c o n s t a n t , "v , w i l l be shown l a t e r i n t h i s development t o be e q u i v a l e n t t o t h e , t o t a l h o r i z o n t a l s p a t i a l wavenumber, ( k x l + k y 1 ) ' 1 , so t h a t i s d i r e c t l y r e l a t e d t o the s c a l e l e n g t h , "X , d i s c u s s e d i n s e c t i o n 1.2: > = 2TT / s> (2.1.32) Thus, the range of e x p e c t e d - v a l u e s of "0 i s r e a d i l y o b t a i n e d from the range of ?i a r r i v e d a t i n s e c t i o n 1.2; th e s e a r e g i v e n i n Table 2.1, a l o n g w i t h the ranges of the o t h e r v a r i a b l e s i m p o r t a n t i n G.D.S. Even f o r the l a r g e s t e x p e c t e d s c a l e l e n g t h , and f o r a l l p e r i o d s g r e a t e r than 5 seconds, i t i s found t h a t : >)* » w l | u e t 0 (2.1.33) Thus, 2.1.31 can be ap p r o x i m a t e d by: 36 R = I / O + i p )''*-(2.1.34) where: (2.1.35) I f p, i s v e r y l a r g e , the magnitude of R (from e q u a t i o n 2.1.34) w i l l be v e r y s m a l l , so t h a t from e q u a t i o n 2.1.30 |B|/A (| X 1. T h i s c o r r e s p o n d s t o s i g n i f i c a n t i n d u c t i o n of the h o r i z o n t a l components i n t o the e a r t h , and almost t o t a l e x t i n c t i o n of the v e r t i c a l f i e l d . As w e l l , the phase d i f f e r e n c e between A,' and B,' w i l l be near z e r o . I f j3 i s v e r y s m a l l , the magnitude of R w i l l approach 1, so t h a t | B,'/Aj i s n e a r l y z e r o , and B,1 i s n e a r l y ft/2 out of phase w i t h A,'. T h i s i n d i c a t e s near t o t a l r e f l e c t i o n of the i n d u c i n g f i e l d from the s u r f a c e . The complete dependence of B('/Aj on j} i s p l o t t e d i n F i g . 2.1. A median v a l u e of |?> = 1 i s suggested as the v a l u e of B above which t h e r e i s s i g n i f i c a n t i n d u c t i o n i n t o the e a r t h . U s i n g the ranges of S> and w i n Ta b l e 2.1, and t a k i n g c o n d u c t i v i t i e s of .005 S/m ( r e s i s t i v e c r u s t a l r o c k ) and .5 S/m ( c o n d u c t i v e c r u s t a l r o c k ) ( B r a c e , 1971; G a r l a n d , 1979, pg. 277) the range of v a l u e s of JS i s c a l c u l a t e d , and d i s p l a y e d i n T a b l e s 2.2 and 2.3. I t i s c l e a r from the t a b l e s t h a t the s c a l e l e n g t h has a s i z e a b l e e f f e c t on the v a l u e of 8. However, i t must be noted from F i g . 2.1 t h a t f o r a l l v a l u e s of f> g r e a t e r than 100, the magnitude and phase of |BJ/A' | w i l l be a p p r o x i m a t e l y the 37 •4.0 0.0 4.0 LOGJp) 8.0 a.o 2.1 T h e r a t i o o f t h e c o m p l e x m a g n i t u d e s o f t h e i n d u c i n g a n d i n d u c e d m a g n e t i c f i e l d a t t h e s u r f a c e f o r t h e T E mode s o l u t i o n i s B'/A'. G i v e n i s t h e d e p e n d e n c e o f t h i s r a t i o on t h e d i m e n s i o n l e s s p a r a m e t e r p . a ) MOD(B'/A') v s LOG, 0 (fi) b) ARG (B'/A' ) v s L O G I O (p>) 38 same, so t h a t f o r a l l ^  > 10*km. the s u r f a c e v a l u e s w i l l be n e a r l y the same. A l s o from T a b l e s 2.2 and 2.3 i t i s c l e a r t h a t f o r t h e m a j o r i t y of t h e v a l u e s of p e r i o d , c o n d u c t i v i t y , and s c a l e i e n g t h t h e r e w i l l be s i g n i f i c a n t i n d u c t i o n of the p r i m a r y f i e l d i n t o the e a r t h f o r the TE mode. 39 T a b l e 2.1: P a r a m e t e r R a n g e s i n G.D.S. 0" ( c r u s t a l ) .005 S/m - .5 S/m ( s e p a r a t i o n p a r a m e t e r ) 1.57x10 m - 1.57x10 m A ( h o r i z o n t a l s c a l e l e n g t h ) 400 km. - 40000km. T ( p e r i o d ) 1 s . - 7 2 0 0 0 s . ( 2 0 h r s . ) w = 2TT/T -s 8.7x10 Hz. - 6.28 Hz. / T a b l e 2 . 2 : V a l u e s o f p f o r v a r y i n g T , ?k . C = .005S/m T (km. ) 1 0 s . 1m. 5m. 20m. 2 h r . 1 O h r . 4x1 0 4 4 .16x10 . 2 7 X 1 0S 4-.53x10 . 1 3 x 1 0 4 220 44 A-1x10 . 1 0x1 0S . 1 7 x 1 0 4 . 3 3 x l 0 3 83 1 4 2.8 5x 1 0 3 A-.25x10 .42x1 03 83 -21 3.5 .69 1x1 0 3 100 1 7 3.3 .83 . 14 .028 5x1 0 Z 25 4 .2 .83 .21 .035 .0069 T T a b l e 2 . 3 : V a l u e s o f p f o r v a r y i n g T , ?v . 0" = .5S/m T (km. ) 1 Os. 1 m . 5m. 20m. 2 h r . 1 O h r . 4x1 0* % .16x10 • 27X10- 7 .53x10 . 1 3 x 1 0 5 .22x1 0 4400 1x10 . 1 0 x 1 0 1 . 1 7 x 1 0 4 5 .33x10 8300 1 400 280 A 5x10* .25x10 . 42x10* 8300 -21 00 350 69 1 x 1 0l . 1 0x1 0s" 1700 330 83 1 4 2.8 5x10* 2500 420 83 21 3.5 .69 T 42 R e t u r n i n g t o the second type s o l u t i o n , the form of the e l e c t r i c f i e l d s a t i s f y i n g 2.1.8, 2.1.15, and 2.1.16 i s found t o be: E z ( x , y , z ) - { ^ , ^ l ^ ) F z } A,e (2. 1.36) f o r z < 0 »3 (0X+<J>« f o r z > 0 (2.1.37) w i t h Fz s a t i s f y i n g : (2.1.38) In s i m i l a r f a s h i o n t o the case f o r the f i r s t type s o l u t i o n , F z ( x , y ) and are found t o be the same f o r b o t h the n o n c o n d u c t i v e and the c o n d u c t i v e h a l f s paces. As w e l l , each v a l u e of S> between 0 and co merely r e p r e s e n t s one ele m e n t a r y s o l u t i o n w i t h the complete s o l u t i o n a g a i n b e i n g the sum of a l l el e m e n t a r y s o l u t i o n s . A l s o i n analogous f a s h i o n , the c o r r e s p o n d i n g magnetic f i e l d f o r each e l e m e n t a r y s o l u t i o n i s found t o be: 43 (2.1.39) f o r z < 0 Hx<x f y ,z ) - ( i / w ^ ) _ ^ — { l & . F - ^ F 0 } C xe (2.1 .40) f o r z > 0 Again u s i n g the boundary c o n d i t i o n s at z = 0, we g e t : B i - A X { [ i - _ | ? _ R ] / [ I + - | L R ] } T O . Ya (2.1.41) C Z = A R { 2 / [ l + A R ] } (2. 1.42) where R i s the same as d e f i n e d p r e v i o u s l y , i n e q u a t i o n 2.1.26, and: (2.1.43) To o b t a i n the same format as f o r the type one s o l u t i o n , def i n e : 44 (2.1.44) and (2.1 .45) Thus, the magnetic f i e l d a t the s u r f a c e becomes ( u s i n g 2.1.39): H 2 ( x , y , 0 ) = {(k\ + B ^ ) ^ . , - ( A [ + B l ' ) ^ , 0 ) } (2.1.46) w i t h : - A l [ ( - ^ R - D/(-^-R + D ] (2. 1 .47) When the a m p l i t u d e of (A^ + Bj*) i s e v a l u a t e d over the range of v a l u e s of c,w, and-0 g i v e n i n T a b l e 2.1, i t i s found t h a t i n a l l c a s e s : l A i + Bj_' I = \C[\ V- 0 (2.1.48) The l a r g e s t v a l u e of IC^I o c c u r s when the p e r i o d i s 1 second, the c o n d u c t i v i t y i s .005 S/m., and the h o r i z o n t a l wavelength i s 40,000 km., and i s s t i l l o n l y 2.8 x 10 . Thus, f o r the range of parameters of G.D.S., bo t h the t r a n s m i t t e d f i e l d and the t o t a l 4 5 s u r f a c e f i e l d a r e e f f e c t i v e l y z e r o , so t h a t the TM mode w i l l n e i t h e r induce c u r r e n t s w i t h i n t he e a r t h , nor be measureable a t the s u r f a c e . The TM mode can t h e r e f o r e always be i g n o r e d i n G.D.S. The a n a l o g y between the f i r s t t y p e s o l u t i o n and TE mode waves, and the second type s o l u t i o n and TM modes waves i s comp l e t e d i n Appendix B. A l s o i n c l u d e d i s the d e r i v a t i o n of the s t a n d a r d form of the F r e s n e l r e l a t i o n s and S n e l l ' s law. I t i s found t h a t t h e el e m e n t a r y s o l u t i o n s of t h e i n d u c t i o n e q u a t i o n c o r r e s p o n d i n g t o d i f f e r e n t v a l u e s of the s e p a r a t i o n c o n s t a n t , are i n f a c t p l a n e waves of d i f f e r i n g t o t a l h o r i z o n t a l wavenumber. The main p o i n t s of t h i s a n a l o g y a r e g i v e n below. F i r s t type s o l u t i o n (TE Mode) E , ( x , y , z , t ) = {ky,-kx , 0}A, e ^ e ^ ^ ^ e ^ 1 + {ky,-kx , 0}B, e ^ e l V k ' » l e i - t ( 2 . 1 . 4 9 ) H , ( x , y , z , t ) = (-1/wjUe) {( kza ) kx , ( kza ) ky ,-V1} I A i e ^ e J^e e 46 + d / w ^ e ) { ( k z a ) k x , ( k z a ) k y , "v*1} B, e i k * x e t k ^ e i k ^ * e i u j t f o r z < 0 E , ( x , y , z , t ) = { k y , - k x , 0 } C, e * * " e ^ V ^ 6 * e i u > ± H , ( x , y , z , t ) = (1/wjuio) { ( k z e ) k x , ' ( k z e ) k y , - ^ } C, e ^ V S V ' ^ V " ^ f o r z > 0 H e r e , A,, B,, a n d C, o b e y 2.1.24 a n d 2.1.25, S e c o n d t y p e s o l u t i o n (TM Mode) E 1 ( x , y , z , t ) = (1/w € o ) { ( k z a ) k x , ( k z a ) k y , - V v } A, e * e J Je **• e ( 2 . 1 . 5 0 ) ( 2 . 1 . 5 1 ) ( 2 . 1 . 5 2 ) + ( 1 / w O { ( k z a ) k x , ( k z a ) k y , ^ v } B ^ e ^ V ^ V ^ e * " * ( 2 . 1 . 5 3 ) 47 H. 2 ( x , y , z , t ) = { k y , - k x , 0 } A ^ ^ V j V ^ e - { k y , - k x , 0 } B ^ V ^ e ^ * e d l 0 t ( 2 . 1.54) f o r z < 0 E \ ( x , y , z , t ) = ( l / w £ 6 ) { ( k z e ) k x , ( k z e ) ky,->0 V} ( 2 . 1 . 5 5 ) . H 2 U , y , z , t ) = { k y , - k x , 0 } cie:k*Vk:i V : k* V"' ( 2 . 1 . 5 6 ) f o r z > 0 H e r e , A t , B X , a n d C 1 o b e y 2.1.41 a n d 2.1.42. I n t h e s e e l e m e n t a r y s o l u t i o n s , t h e w a v e n u m b e r s a r e shown ( i n A p p e n d i x B) t o be r e l a t e d t o t h e s e p a r a t i o n p a r a m e t e r b y : Ckx* +ky'" = V ( 2 . 1.57) k z a = - i ( W L = - i ( 0 r-w U 0 C T ) " < ( 2 . 1 . 5 8 ) 48 kze = - i ( % + ^ ) ' K = - i ( ^ 'L + i w / u 6 c ) " r (2.1.59) The e q u i v a l e n c e of the el e m e n t a r y waves of Chapter I ( s e c t i o n 1.2) t o the elem e n t a r y s o l u t i o n s of the homogenous e a r t h i n d u c t i o n problem i s thus complete. The s e p a r a t i o n c o n s t a n t i s found t o be r e l a t e d t o the s c a l e l e n g t h d i s c u s s e d i n t he so u r c e f i e l d d e s c r i p t i o n : A = 2 t r / v (2.1.60) A g a i n , the complete s o l u t i o n w i l l be a summation of a l l the ele m e n t a r y s o l u t i o n s , or i n the c o n t i n u o u s c a s e , an i n t e g r a l over a l l h o r i z o n t a l wavenumbers. In t h i s complete s o l u t i o n o n l y the TE mode need be c o n s i d e r e d . 2.2 Uni form Source F i e l d Over a Two-Pimensional E a r t h :  F o r w a r d M o d e l l i n g A common s i m p l i f i c a t i o n i n G.D.S., and one used i n most f o r w a r d m o d e l l i n g a l g o r i t h m s , i s t h a t the s o u r c e f i e l d i s e f f e c t i v e l y h o r i z o n t a l l y u n i f o r m . Thus, the waves of the sour c e f i e l d can be c o n s i d e r e d t o be p r o p o g a t i n g v e r t i c a l l y downward. The l i m i t s of t h i s assumption a r e d i s c u s s e d i n d e t a i l i n Appendix C; i n q u i c k summary, the r e s u l t s i n d i c a t e t h a t f o r 49 p e r i o d s l e s s than 2 h r s . and f o r a r e a s where the source f i e l d s c a l e l e n g t h i s . > 5000 km. (such as a t m i d l a t i t u d e s ) , t h i s a s s umption w i l l be v a l i d . Jones and Pascoe (1971) have embodied t h i s s i m p l i f i c a t i o n i n a f i n i t e d i f f e r e n c e approach t o c a l c u l a t i n g the s u r f a c e magnetic and e l e c t r i c f i e l d s over two d i m e n s i o n a l s t r u c t u r e s , u s i n g the G a u s s - S e i d e l i t e r a t i v e method. T h e i r programs handle s e p a r a t e l y the s i t u a t i o n s where the e l e c t r i c v e c t o r of t h e p r i m a r y f i e l d i s p a r a l l e l t o t h e s t r i k e of t h e s t r u c t u r e ( E-p o l a r i z e d ) or the magnetic v e c t o r of the p r i m a r y f i e l d i s p a r a l l e l t o the s t r i k e ( H - p o l a r i z e d ) . However, i n the case of H p o l a r i z a t i o n , the symmetry a l o n g the s t r i k e of the s t r u c t u r e e nsures t h a t the o n l y non-zero component of the t o t a l magnetic f i e l d a t the s u r f a c e w i l l be the h o r i z o n t a l component p e r p e n d i c u l a r t o t h e s t r i k e . F u r t h e r , f o r t h i s p o l a r i z a t i o n the c o n t i n u i t y of any magnetic f i e l d v e c t o r a c r o s s an i n t e r f a c e g u a r a n t e e s t h a t the v a l u e s of t h i s non-zero magnetic component w i l l be t h e same a t e v e r y s u r f a c e p o s i t i o n (Jones and P r i c e , 1970). Thus, i n G.D.S. the o n l y i n t e r e s t i n g case i s t h a t of E p o l a r i z a t i o n . Because of t h e expense i n v o l v e d i n r u n n i n g the program of Jones and Pascoe ( J - P ) , o n l y a few s e l e c t e d c o n d u c t i v i t y s t r u c t u r e s have been s t u d i e d . I t i s o f t e n c o n s i d e r e d t h a t t h e r e a r e t h r e e major t y p e s of l o c a l i z e d a n o m a l i e s p r e s e n t i n the e a r t h , t h e s e b e i n g (Schmucker, 1970, pg. 78; Oldenburg, 1969 , pg. 117): 5 0 1 ) near s u r f a c e a n o m a l i e s i n the upper c r u s t i n t e r m e d i a t e a n o m a l i e s i n the lower c r u s t and uppermost 2) mantle 3) deep a n o m a l i e s due t o imbalances or u n d u l a t i o n s i n the h i g h c o n d u c t i v i t y l a y e r of the upper mantle The t h r e e models shown i n F i g . 2.2a,b,and c a r e r e p r e s e n t a t i v e of these t h r e e anomaly t y p e s . For each, the J-P f o r w a r d programs have been used t o g e n e r a t e the s u r f a c e v a l u e s due t o an E-p o l a r i z e d i n d u c i n g f i e l d . The "anomalous' f i e l d was then produced by s u b t r a c t i n g from th e s e the 'normal' f i e l d v a l u e s due t o the a n o m a l y - f r e e one d i m e n s i o n a l s t r u c t u r e . The r e s u l t s a r e g i v e n i n F i g . 2.3a,b, and c f o r a p e r i o d of 5 min., and i n F i g . 2.5a,b, and c f o r a p e r i o d of 50 min. I f f i r s t o r d e r i n d u c t i o n e f f e c t s o n l y a r e i m p o r t a n t , t h a t i s , mutual i n d u c t i o n i s i n s i g n i f i c a n t , then the induced c u r r e n t s w i l l be d i r e c t l y p r o p o r t i o n a l t o , and i n phase w i t h , the l o c a l e l e c t r i c f i e l d . Thus, f o r t h e s e l o c a l i z e d c o n d u c t i v e b o d i e s used, the anomalous induced c u r r e n t s would resemble l i n e c u r r e n t s i n phase w i t h the l o c a l e l e c t r i c f i e l d . From e q u a t i o n 2.1.52, the downgoing e l e c t r i c f i e l d i n a h a l f - s p a c e of c o n d u c t i v i t y can be w r i t t e n ( c o n s i d e r i n g o n l y the TE mode): 51 E(x,y,z,t) = {ky,-kx,0} C(kx,ky) e'^e'^e'^V"'4 ( 2 . 2 . 1 ) where from equation 2 . 1 . 5 9 : kze = - i (^,"+iw/Al>ff),'v ( 2 . 2 . 2 ) and from 2 . 1 . 5 7 : S>V= kx1" + ky"*" ( 2 . 2 . 3 ) To allow for v e r t i c a l .incidence, the vector components are normalized by d i v i s i o n withV, giving: - i / \) ( S t A \ © • X + C O S © • n \ E ( x r y , z , t ) = {cos©,-sin©,0} C (©) e J -Ck4,4 Cult . e t c e ( 2 . 2 . 4 ) with: sin© = kx/S) ; cos© = ky/X> ( 2 . 2 . 5 ) T a k i n g the s t r i k e of the two dimensional structure to be in the 'y' d i r e c t i o n , then a v e r t i c a l l y propogating (S) =0) E-polarized wave (sin© =0) of the T E mode ( E % =' 0) w i l l be represented by: 52 E ( z , t ) = {1,0,0} C' e " i k * c * e l u , t (2.2.6) The p o r t i o n of the s p a t i a l e x p o n e n t i a l which c o r r e s p o n d s t o p r o p a g a t i o n i s o b t a i n e d by t a k i n g the r e a l p a r t of kze from e q u a t i o n 2.2.2, so t h a t the phase d i f f e r e n c e between the e l e c t r i c f i e l d a t the s u r f a c e and t h a t a t a depth Z 0 w i l l be g i v e n by: Z e ( i n r a d i a n s ) (2.2.7) I f f i r s t o r d e r i n d u c t i o n o n l y i s s i g n i f i c a n t , the c u r r e n t a l o n g the anomaly and i t s c o r r e s p o n d i n g magnetic f i e l d a t the s u r f a c e w i l l be i n phase w i t h the e l e c t r i c f i e l d a t the depth Z Q. The phase d i f f e r e n c e between the anomalous magnetic f i e l d and the e l e c t r i c f i e l d measured at the s u r f a c e w i l l thus be from e q u a t i o n 2.2.7. To t e s t the a p p r o x i m a t i o n of f i r s t o r d e r i n d u c t i o n the f i e l d of a l i n e c u r r e n t a t the same depth as the anomalous c o n d u c t i v i t y has been superimposed on the r e s u l t s from the fo r w a r d i n d u c t i o n , i n F i g . 2.3a,b, and c, and F i g . 2.4a,b, and c. In a l l c a s e s the magnitude of t h i s c u r r e n t i s n o r m a l i z e d t o the maximum v a l u e of the c a l c u l a t e d h o r i z o n t a l f i e l d from the J -P i n d u c t i o n program. F or both the s h a l l o w and the i n t e r m e d i a t e a n o m a l i e s , the match of the l i n e c u r r e n t and J-P r e s u l t s i s n e a r l y e x a c t f o r the h o r i z o n t a l f i e l d component. T h i s s h o u l d i n t u r n mean t h a t the v e r t i c a l f i e l d match i s as good, because the 53 SURFACE POSITION (KM-) 0 l O O 200 300 400 J — i i i i i SURFACE POSITION (KM) 0 100 200 300 400 o . in X -I Q_ . L U • o in . 1 — 1 1 1 1 1 1 1 fb) SURFACE POSITION (KM1 0 100 200 300 400 J 1 1 I I L o . in 5T. X -h-CL UJ O o in. 0 fl o b (<0 2.2 T h r e e r e p r e s e n t a t i v e m o d e l s u s e d f o r t h e J o n e s - P a s c o e ( 1 9 7 1 ) f o r w a r d i n d u c t i o n p r o g r a m . T h e s i z e o f t h e a n o m a l i e s i s t o s c a l e . a ) S h a l l o w b) M i d c r u s t a l c ) M a n t l e U n d u l a t i o n o r 'bump' 54 100. 0 4 0 0 , 0 JOO.O F i g . 2.3 R e s u l t s f r o m t h e J o n e s - P a s c o e p r o g r a m f o r t h e t h r e e m o d e l s ( a , b , c ) o f F i g . 2.2, a t a p e r i o d o f 5 m i n . T h e s o l i d l i n e r e p r e s e n t s t h e J - P r e s u l t s , a n d — • © — - i s t h e s u p e r i m p o s e d f i e l d o f a l i n e c u r r e n t a t t h e d e p t h o f t h e a n o m a l o u s c o n d u c t i v i t y . T h e m a g n i t u d e o f t h e l i n e c u r r e n t i s n o r m a l i z e d t o t h e maximum v a l u e o f t h e Bx c o m p o n e n t . 55 Bx Bz 8 0 , 0 1 SO.0 200 0 3 2 0 . 0 100 0 3 0 SO 0 160.0 240 .0 3 2 0 . 0 4 0 0 . 0 4 0 0 . 0 X X i g . 2.4 R e s u l t s f r o m t h e J o n e s - P a s c o e p r o g r a m f o r t h e t h r e e m o d e l s (a.,b,c) o f F i g . 2.2, a t a p e r i o d o f 50 m i n . T h e s o l i d l i n e r e p r e s e n t s t h e J - P r e s u l t s , a n d e i s t h e s u p e r i m p o s e d f i e l d o f a l i n e c u r r e n t a t t h e d e p t h o f t h e a n o m a l o u s c o n d u c t i v i t y . T h e m a g n i t u d e o f t h e l i n e c u r r e n t i s n o r m a l i z e d t o t h e maximum v a l u e o f t h e Bx c o m p o n e n t . 5 6 v e r t i c a l f i e l d i s always t h e n e g a t i v e H i l b e r t t r a n s f o r m of the h o r i z o n t a l f i e l d f o r a two d i m e n s i o n a l s t r u c t u r e (see Appendix D). However, the v e r t i c a l components a r e not n e a r l y as s i m i l a r as t h e i r h o r i z o n t a l c o u n t e r p a r t s . I t i s s u g g e s t e d t h a t t h i s i n d i c a t e s some minor f a i l i n g i n the J-P program, of e i t h e r i n h e r e n t n a t u r e , or due t o the g r i d s e l e c t i o n by the a u t h o r . The matches f o r the t h i r d t y p e anomaly a r e q u i t e good f o r t h e h o r i z o n t a l component, but a r e v e r y poor f o r the v e r t i c a l component. In t h i s c a s e , however, the o r i g i n a l a ssumption of a l o c a l i z e d anomaly i s not t r u e , so t h a t mutual i n d u c t i o n i s p o s s i b l e w i t h i n the h i g h c o n d u c t i v i t y zone beneath the u n d u l a t i o n . Thus, the f i t of the l i n e c u r r e n t f i e l d i s not e x p e c t e d t o be as good i n t h i s c a s e . To f u r t h e r check th e v a l i d i t y of the l i n e c u r r e n t a p p r o x i m a t i o n the phase d i f f e r e n c e <j> between the anomalous f i e l d and the s u r f a c e e l e c t r i c f i e l d has been c a l c u l a t e d from e q u a t i o n 2.2.7, and a l s o d i r e c t l y from the J-P r e s u l t s (see T a b l e 2.4). The v a l u e s f o r the s h a l l o w and m i d c r u s t a l anomalies a r e c o m p a r a t i v e i n each example. For the mantle u n d u l a t i o n t h e r e i s a l a r g e d i f f e r e n c e between the two e s t i m a t e s of phase, and t h i s i s a g a i n i n d i c a t i v e t h a t i n t h i s case mutual i n d u c t i o n i s i m p o r t a n t . 57 T a b l e 2.4: Phase D i f f e r e n c e Between the Anomalous S u r f a c e M a g n e t i c F i e l d and t h e Normal S u r f a c e E l e c t r i c F i e l d , f o r Both the L i n e C u r r e n t and the Jones-Pascoe R e s u l t s . Anomaly Type Depth P e r i o d Phase eq.2.2.7 Phase J-P S h a l l o w 9 km. 5 min. 1 .87° 1.74° M i d c r u s t a l 35.5 km. 5 min. 7.38° 5.3° Man t l e 'Bump' 140 km. 5 min. 29.10° 21.0° S h a l l o w 9 km. 50 min. .59° .46° M i d c r u s t a l 35.5 km. 50 min. 2.33° 2.44° Mantl e 'Bump' 140 km. 50 min. 9 . 1 9 ° 35.93° In summary, f o r l o c a l i z e d a n o m a l i e s i n a host m a t e r i a l of co m p a r a t i v e low c o n d u c t i v i t y , the J-P fo r w a r d m o d e l l i n g r e s u l t s suggest t h a t t h e l i n e c u r r e n t a p p r o x i m a t i o n i s i n f a c t v a l i d , i n d i c a t i n g t h a t the f i r s t o r d e r s i m p l i f i c a t i o n f o r such a n o m a l i e s i s a p p r o p r i a t e . 58 Chapter I I I T r a d i t i o n a l Methods of G.D.S. A n a l y s i s 3.1 The F o r m u l a t i o n and S e p a r a t i o n of the Normal F i e l d G.D.S. has been used f o r one d i m e n s i o n a l as w e l l as two and t h r e e d i m e n s i o n a l sounding (Schmucker, 1970; Kuckes, 1973; L i l l e y , 1975; Woods, 1979), but the one d i m e n s i o n a l problem w i l l not be d i s c u s s e d h e r e . The g e n e r a l presumption made i n G.D.S. a n a l y s i s , as o r i g i n a l l y s uggested by Schmucker (1970), i s t h a t the e a r t h model i s b a s i c a l l y p l a n e l a y e r e d , w i t h o n l y r e l a t i v e l y s m a l l anomalous v a r i a t i o n s of c o n d u c t i v i t y (as i n d i c a t e d i n F i g . 3.1a): O j ( x , y , z ) = C^"+C*(x,y ,z) (3.1.1) The magnetic f i e l d w i t h i n the j l a y e r , H j ( x , y , z ) can always be w r i t t e n : 59 F i g . 3.1 P o s s i b l e C o n d u c t i v i t y M o d e l s ( a ) T h e a s s u m e d c o n d u c t i v i t y m o d e l o f G.D.S. i s b a s i c a l l y p l a n e l a y e r e d , w i t h o n l y s m a l l l o c a l i z e d r e g i o n s o f a n o m a l o u s c o n d u c t i v i t y . (b) A c o n d u c t i v i t y m o d e l t h a t i s p l a n e l a y e r e d e x c e p t f o r a l a r g e d i s c o n t i n u o u s ' s t e p ' . ( c ) The a b u t m e n t o f two d i f f e r e n t l a y e r e d s t r u c t u r e s ( a s a t t h e l a n d - s e a b o u n d a r y ) . . I n b o t h (b) a n d ( c ) t h e c o n c e p t o f a n o r m a l f i e l d b r e a k s down i f t h e a r r a y i s c l o s e t o , o r s p a n s t h e d i s c o n t i n u i t y . 60 F i g . 3 . 1 AIR EARTH CO 00 AIR cr,' EARTH.-9}" 61 H j ( x , y , z ) =Hj(x,y,z) + H^(x,y,z) (3.1.2) where Hj i s the 'normal' f i e l d which would e x i s t i n the absence -i Ok. _i r of the anomalous c o n d u c t i v i t y , and Hj i s the v a r i a t i o n from Hj due t o the anomaly. The g e n e r a l i n d u c t i o n e q u a t i o n i s e q u a t i o n A.19 from Appendix A: (3.1.3) I n s e r t i n g e q u a t i o n s 3.1.1 and 3.1.2 i n t o 3.1.3, and u s i n g e q u a t i o n A.31 from Appendix A f o r the normal f i e l d i n a r e g i o n of c o n s t a n t c o n d u c t i v i t y , the anomalous f i e l d Hj can be e x p r e s s e d as a f u n c t i o n of the normal f i e l d : 7V^ + -feT" x ( V x H>) - iw^OjH* = iv|-.<?HT " | ^ x ( § x Hp J (3.1.4) T h i s e q u a t i o n , a l t h o u g h never used d i r e c t l y , forms the b a s i s f o r most of the methods t h a t attempt t o r e l a t e the d e s i g n a t e d normal f i e l d t o the anomalous f i e l d i n some s t a t i s t i c a l manner, and then from the e n s u i n g r e l a t i o n s e s t i m a t e the anomalous s t r u c t u r e . In c a s e s where the e x t e n t of the anomalous v a r i a t i o n i s v e r y l a r g e , the concept of a normal l a y e r e d s t r u c t u r e w i l l not be a p p l i c a b l e , and the d e f i n e d 62 concept of a normal f i e l d w i l l break down. Examples of t h i s a re when the measuring a r r a y spans a l a r g e s u b s u r f a c e c o n d u c t i v i t y s t e p , or i s i n the v i c i n i t y of the abutment of two d i f f e r e n t l a y e r e d s t r u c t u r e s (as i n the case of the e a r t h - o c e a n boundary), (see F i g . 3.1b,c) To a v o i d t h e s e problems of d e f i n i t i o n L i l l e y (1974) has suggested t h a t the t o t a l f i e l d be f o r m u l a t e d i n terms of i n t e r n a l and e x t e r n a l p a r t s r a t h e r than anomalous and normal p a r t s . Banks (1979) has a l s o c r i t i c i z e d Schmucker's approach on t h e grounds t h a t the e a r t h ' s c r u s t i s f a r t o o l o c a l l y h eterogenous f o r a normal f i e l d t o e v e r t r u l y e x i s t . In view of the marked s i m i l a r i t y of magnetograms i n g e n e r a l over an a r r a y , t h i s i s p r o b a b l y too h a r s h an i n d i c t m e n t ; however, the e x i s t e n c e of a t r u e normal f i e l d s h o u l d always be viewed w i t h some r e s e r v a t i o n s . I f the concept of the measured f i e l d b e i n g composed of normal and anomalous components i s a c c e p t e d , i t s u s e f u l n e s s w i l l t hen be dependent on the a b i l i t y t o s e p a r a t e t h e s e two components. S i m i l a r l y , i f L i l l e y ' s s u g g e s t i o n t o use i n t e r n a l and e x t e r n a l components i s adopted, then i t w i l l be r e q u i r e d t h a t t h e s e two components be s e p a r a b l e from the t o t a l measured f i e l d . As i n d i c a t e d i n T a b l e s 2.2 and 2.3 and F i g . 2.1, the i n d u c e d v e r t i c a l component over a c o n d u c t i v e h a l f - s p a c e almost c o m p l e t e l y c a n c e l s out the v e r t i c a l component of the i n d u c i n g f i e l d f o r n e a r l y a l l v a l u e s of p e r i o d , c o n d u c t i v i t y , and s c a l e l e n g t h e x p e c t e d i n G.D.S. Except f o r a c o m b i n a t i o n of l o n g p e r i o d (> 2 h r . ) , s h o r t s c a l e l e n g t h (< 5000 km.), and low 63 c o n d u c t i v i t y (< .005 S/m.), the v e r t i c a l component of the normal f i e l d w i l l be v e r y s m a l l . Thus, i n anomalous r e g i o n s , the s e p a r a t i o n i s o f t e n e f f e c t i v e l y a l r e a d y done f o r t h i s component, as v i r t u a l l y a l l of the measured v a l u e w i l l be anomalous, and i n t e r n a l . T h i s forms the b a s i s f o r the v i s u a l methods d e s c r i b e d i n s e c t i o n 3.2 of t h i s c h a p t e r , which use the u n s e p a r a t e d d a t a f o r p r e l i m i n a r y a n a l y s i s . To s e p a r a t e the normal and anomalous f i e l d s , two methods suggested by Schmucker (1970) a r e commonly employed. These methods d i f f e r o n l y i n t h e i r manner of s e p a r a t i o n of the h o r i z o n t a l components. In the f i r s t method a s t a t i o n presumed t o be d i s t a n t from the anomalous r e g i o n (as would be i n d i c a t e d by the v i s u a l methods of s e c t i o n 3.2) i s s e l e c t e d as the r e f e r e n c e s t a t i o n . I f the s c a l e l e n g t h of the observed f i e l d i s much l a r g e r than the a r r a y s i z e , then the h o r i z o n t a l components H and D of the f i e l d {H,D,Z} measured a t t h i s s i t e w i l l s e r v e as the normal f i e l d . (Note t h a t H i s the component of the f i e l d i n the d i r e c t i o n of magnetic n o r t h , D i s the component i n the d i r e c t i o n of magnetic e a s t , and Z i s the v e r t i c a l component, where the p o s i t i v e d i r e c t i o n of Z i s downwards). In the case t h a t t h i s c o n d i t i o n i s not met, one can use the r e g i o n a l h o r i z o n t a l d e r i v a t i v e s of the f i e l d s t o c a l c u l a t e the f i r s t o r d e r c o r r e c t i o n s a t any p o s i t i o n (x,y) (where the p o s i t i o n (0,0) i s the r e f e r e n c e s i t e , 'x' i s toward magnetic n o r t h , and 'y' i s toward magnetic e a s t ) : 64 H M ( x , y ) = H(0,0) + ( " ^ ) . x + ( ^ ) . y ^ x j -(3.1.5) D N ( x , y ) = D(0,0) + ( ^ L ) - x + (I&»).y 3x * » j (3.1.6) From e q u a t i o n s 2.1.20 and 2.1.36 i n Chapter I I , and the v a l u e s of |B /A I g i v e n , the v e r t i c a l e l e c t r i c f i e l d a t the s u r f a c e s h o u l d always be n e a r l y z e r o . Thus, the v e r t i c a l component of the c u r l of the magnetic f i e l d s h o u l d be near z e r o , a l l o w i n g a check of the v a l u e s of ^ U i / a x . and ^Hv/aij : l i i i i. - 22*. = i w e c E , v o ^ ^ ^ x ° * (3.1.7) The second method used t o s e p a r a t e the normal and anomalous h o r i z o n t a l f i e l d s t a k e s advantage o f " t h e p resumption t h a t the s c a l e l e n g t h of the i n d u c i n g f i e l d s h o u l d be much l a r g e r than the s p a t i a l wavelengths of the induced anomalous f i e l d . Thus, s p a t i a l smoothing of the measured h o r i z o n t a l components over the a r r a y a r e a s h o u l d d e f i n e the normal f i e l d . To d e f i n e the normal v e r t i c a l f i e l d , Schmucker uses the s p a t i a l d e r i v a t i v e s of the normal h o r i z o n t a l f i e l d . From e q u a t i o n s 2.1.19 and 2.2.23 of Chapter I I i t can be shown t h a t a t any f r e q u e n c y the v a l u e of the v e r t i c a l magnetic f i e l d i n the homogenous e a r t h case i s l i n e a r l y r e l a t e d t o the d e r i v a t i v e s of 65 the h o r i z o n t a l components by a frequency dependent c o n s t a n t , C: 1 = c ( + ^ M ) (3.1.8) where: C = l/ (^+9 e ) ' / z (3.1.9) In the p l a n e l a y e r e d case the r e l a t i o n i s the same as i n e q u a t i o n 3.1.8, but now C i s a more complex, fre q u e n c y dependent f u n c t i o n of the c o n d u c t i v i t y s t r u c t u r e (Schmucker, 1970, pg.15). (I n both c a s e s C i s a measure of the s k i n depth of p e n e t r a t i o n ) . The v a l u e of C i s c a l c u l a t e d from some g i v e n model of the one d i m e n s i o n a l c o n d u c t i v i t y s t r u c t u r e . U s i n g t h i s , and e s t i m a t e s of the h o r i z o n t a l f i e l d g r a d i e n t s , i t i s then p o s s i b l e t o e v a l u a t e the v a l u e of the v e r t i c a l normal f i e l d , Z^. In p r a c t i c e , t h i s method of d e t e r m i n i n g Z^ i s not v e r y r e l i a b l e f o r a number of r e a s o n s . The d i f f i c u l t i e s i n a c c u r a t e l y d e t e r m i n i n g the one d i m e n s i o n a l s t r u c t u r e and the h o r i z o n t a l f i e l d g r a d i e n t s c o n s t i t u t e the f i r s t o b v i o u s problem w i t h t h i s method. More fund a m e n t a l , however, a r e the problems i n c u r r e d because of the f r e q u e n c y dependence of C, and the time v a r i a t i o n s of the h o r i z o n t a l g r a d i e n t s . I f t h e v a l u e of Z N i s c a l c u l a t e d i n the f r e q u e n c y domain, then t h e time v a r i a t i o n s of the g r a d i e n t s cannot be taken i n t o a c c o u n t , whereas i n the time domain the f r e q u e n c y dependence of C cannot be i n t r o d u c e d (Schmucker, 1970, 66 pg.16). To a v o i d t h e s e d i f f i c u l t i e s , i t i s t h i s a u t h o r ' s o p i n i o n t h a t t h e normal v e r t i c a l f i e l d Z N i s j u s t as r e l i a b l y d e t e r m i n e d by a s s i g n i n g t o i t the v a l u e of Z N a t the r e f e r e n c e s i t e . Whatever the method used t o d e t e r m i n e Z N, i t s s p a t i a l v a r i a t i o n can s a f e l y be i g n o r e d . T h i s f o l l o w s from e q u a t i o n 3.1.8, as the g r a d i e n t s of Z w w i l l now be second o r d e r c o r r e c t i o n s w i t h r e s p e c t t o H N and D w. Once a normal f i e l d { H ^ D ^ Z ^ } has been d e f i n e d , the anomalous f i e l d { H A ( x , y ) , D A ( x , y ) , Z f t ( x , y ) } a t any s t a t i o n p o s i t i o n can be c a l c u l a t e d from the measured f i e l d , { H ( x , y ) , D ( x , y ) , Z ( x , y ) } : H A ( x , y ) = H(x,y) - H w (3.1.10) D f l ( x , y ) = D ( x , y ) - D N I 7 (3.1.11) Z A ( x , y ) = Z(x,y) - Z N (3.1.12) The problem of s e p a r a t i n g the i n t e r n a l and e x t e r n a l f i e l d s was f i r s t s o l v e d by Gauss (1839) f o r a s p h e r i c a l e a r t h . For the f l a t e a r t h c a s e , t h e s e p a r a t i o n method has been d e r i v e d i n a 67 v a r i e t y of ways ( V e s t i n e , 1941; S i e b e r t and K e r t z , 1957; Weaver, 1963)). The d e r i v a t i o n of t h e s e p a r a t i o n formulae f o r a two d i m e n s i o n a l s t r u c t u r e which i s g i v e n i n Appendix C i s taken m a i n l y from Weaver (1963). C o n s i d e r a c o n t i n u o u s one d i m e n s i o n a l a r r a y r u n n i n g e a s t - w e s t p e r p e n d i c u l a r t o the n o r t h - s o u t h s t r i k e of a two d i m e n s i o n a l c u r r e n t d e n s i t y s t r u c t u r e (see F i g . 3.2). By symmetry c o n s i d e r a t i o n s t h e o n l y non-zero components of the magnetic f i e l d w i l l be D and Z. The r e l a t i o n s between the i n t e r n a l and e x t e r n a l components of D and Z (as d e r i v e d i n Appendix C) a r e : K(D X) = - z r (3.1 .13) K(D E) = Z E (3.1.14) K ( Z X ) = D x (3.1.15) K ( Z C ) = -D £ (3.1.16) where t h e o p e r a t o r K i s the H i l b e r t t r a n s f o r m : 68 AIR F i g . 3.2 One d i m e n s i o n a l s u r f a c e a r r a y p e r p e n d i c u l a r t o the s t r i k e of a t w o - d i m e n s i o n a l e a r t h . The c u r r e n t d e n s i t i e s t r a v e l i n t o (or out of ) the page. 69 oo r K ( A ( x ) ) = -1/1Y j A(u) / ( x - u ) du "* (3.1.17) w i t h d e n o t i n g the p r i n c i p a l v a l u e of the i n t e g r a l . The measured d a t a i s : D(x) = D r ( x ) + D £ ( x ) Z(x) = Z x ( x ) + Z E ( x ) Thus, combining 3.1.13 - 3.1 . 19 , we get D x ( x ) = {D(x) + K [ Z ( x ) ] } / 2 (3.1.18) (3.1.19) (3.1.20) Z j ( x ) = {Z(x) - K [ D ( x ) ] } / 2 (3.1.21) The i n t r i n s i c s h o r t c o m i n g here l i e s i n a fundamental p r o p e r t y of the H i l b e r t t r a n s f o r m . Because the denominator of the i n t e g r a n d i n 3.1.17 i s an odd f u n c t i o n , the H i l b e r t t r a n s f o r m of a c o n s t a n t i s z e r o . Thus, f o r the normal f i e l d , t h i s s e p a r a t i o n t e c h n i q u e w i l l never be a b l e t o s e p a r a t e the i n t e r n a l and e x t e r n a l p o r t i o n s i f the i n d u c i n g f i e l d i s u n i f o r m , 70 t h a t i s , i f i t c o n s i s t s of waves p r o p a g a t i n g v e r t i c a l l y downward ( c o r r e s p o n d i n g t o an i n f i n i t e s c a l e l e n g t h ) . As seen i n s e c t i o n 2.1 (arid as d i s c u s s e d by Weaver ( 1 9 7 3 ) ) , t h i s i s an i n t r i n s i c p r o p e r t y . In the case of a u n i f o r m f i e l d the r a t i o of the secondary t o the p r i m a r y f i e l d a m p l i t u d e can have any v a l u e between 0 and 1, and t h e r e i s no way t o d i s t i n g u i s h from measurements what the t r u e v a l u e i s . Thus, t h e r e w i l l be problems s e p a r a t i n g the i n t e r n a l and e x t e r n a l p o r t i o n s of the normal f i e l d i f the s c a l e l e n g t h of the i n d u c i n g f i e l d i s g r e a t e r than the d i m e n s i o n s of the a r r a y . I t i s c l e a r from the above t h a t e s t i m a t i o n of the s c a l e l e n g t h from the s u r f a c e r e a d i n g s w i l l be i m p o r t a n t . In a l a y e r e d s t r u c t u r e , the c r o s s i n g of a p l a n e wave i n t o the next l a y e r w i l l never r e s u l t i n a change of the o r i g i n a l h o r i z o n t a l wavenumbers f o r the r e f l e c t e d or r e f r a c t e d wave (as seen f o r the e a r t h - a i r i n t e r f a c e i n s e c t i o n 2.1, a l s o , see Panofsky and P h i l l i p s (1962, pg. 196). Thus, the s c a l e l e n g t h s h o u l d be c a l c u l a b l e from e i t h e r the s e p a r a t e d normal f i e l d , or the s e p a r a t e d e x t e r n a l f i e l d . In t h e case of the homogenous e a r t h , the i n d u c i n g f i e l d ( Hp) c o n s i s t i n g of o n l y a s i n g l e TE mode wave may be found by a p p l y i n g e q u a t i o n 1.2.4 t o 1.2.9: H p ( x , y , 0 ) = H 0 { - [ k z k x ] , - [ k z k y ] , [ k x 1 + k y ' ] }e t^ , < e l k 3 ^ (3.1.22) T h i s would c o r r e s p o n d t o the s e p a r a t e d e x t e r n a l f i e l d . U s i n g e q u a t i o n 2.1.29 the r e s u l t a n t t o t a l s u r f a c e f i e l d (H) w i l l be: 71 H(x,y,0) = H 0 { [ - k z k x ( l + T ) ] , - [ k z k y ( 1 + T ) ] , [ ( k x 1 + k y l ) d - T ) ] } e l k x*e (3.1.23) where T i s the r a t i o of in d u c e d t o i n d u c i n g complex a m p l i t u d e s from e q u a t i o n 2.1.30. H c o r r e s p o n d s t o the s e p a r a t e d normal f i e l d . The true' s c a l e l e n g t h i s g i v e n by: A = 2TT/(kx l+ky V 1 (3.1.24) U s i n g the h o r i z o n t a l components of e i t h e r the s e p a r a t e d normal or e x t e r n a l f i e l d s , an e s t i m a t e of A i s o b t a i n e d from: > = ia(H Z +DVV[i (^L+^- ) ] (3.1.25) where i t i s noted t h a t H 1, D a a r e H-H, D-D, and not |H|a,|D|*". The e x p r e s s i o n f o r the p l a n e l a y e r e d case i s i d e n t i c a l t o e q u a t i o n 3.1.25, where the c o n s t a n t , C i n t h i s case i s a complex f u n c t i o n of the l a y e r e d s t r u c t u r e . In both c a s e s C i s a measure of t he depth of p e n e t r a t i o n of the f i e l d . I f the i n d u c i n g f i e l d c o n s i s t s of many waves of d i f f e r i n g s p a t i a l w a v e l e n g t h s , the n o n - l i n e a r i t y of e q u a t i o n 3.1.25 w i t h r e s p e c t t o H and D w i l l r e s u l t i n an i n c o r r e c t v a l u e f o r A . However, i t i s ex p e c t e d t h a t the v a l u e w i l l s t i l l be a r e a s o n a b l e e s t i m a t e of the s m a l l e s t s i g n i f i c a n t h o r i z o n t a l w a v e l e n g t h . I t s h o u l d be noted t h a t as the complex form of the wave s o l u t i o n has been used i n e q u a t i o n 3.1.22 and 3.1.23 t h a t e v a l u a t i o n of X by t h i s method would n e c c e s s a r i l y be done i n 72 the F o u r i e r t r a n s f o r m f r e q u e n c y domain. A s i m i l a r e s t i m a t o r has been used which i s a p p l i e d t o the time domain v a l u e s of the h o r i z o n t a l normal or e x t e r n a l f i e l d ( P o r a t h et a l , 1971): V = 21YF / | V F | (3.1.26) F i s the t o t a l h o r i z o n t a l f i e l d measured a t the s p a t i a l p o s i t i o n -a of the maximum g r a d i e n t of F, and |v"F| i s the magnitude of the maximum g r a d i e n t . T h i s e s t i m a t o r w i l l be i n c o r r e c t except under v e r y f o r t u i t o u s c i r c u m s t a n c e s . C o n s i d e r the e x p r e s s i o n f o r the complex f i e l d i n e q u a t i o n 3.1.23. Because the e s t i m a t o r uses r e a l time domain v a l u e s , we use o n l y the r e a l p a r t from t h i s e x p r e s s i o n . R o t a t i n g i n t o a new c o o r d i n a t e frame so t h a t t h e r e i s o n l y one h o r i z o n t a l component F a l o n g ^, the v a l u e of F w i l l be: F = F Q k H c o s ( k H ) (3.1.27) where: F c = -H^kz-Realf 1+T] (3.1.28) and k H i s the t o t a l h o r i z o n t a l wavenumber. U s i n g t h i s t o e v a l u a t e ~)\ i n e q u a t i o n 3.1.26, we g e t : 73 = (2 T T / k H ) c o t ( k „ 1j ) ( 3 . 1 . 2 9 ) = A c o t ( k H ^ ) ( 3 . 1 . 3 0 ) T h u s , t h e e s t i m a t e o f t h e v a l u e w i l l be i n c o r r e c t by t h e f a c t o r c o t ( k H ^ ) , and as t h i s c a n have any v a l u e be tween 0 and CO, t h e p o s s i b l e f l u c t u a t i o n s o f V f r o m t h e t r u e v a l u e X c o u l d be h u g e . I t w o u l d be p o s s i b l e t o u se t h i s e s t i m a t o r i f t h e g r a d i e n t a n d t o t a l f i e l d v a l u e were t a k e n a t p o s i t i o n s a q u a r t e r s c a l e l e n g t h a p a r t . H o w e v e r , a s i t i s t h e s c a l e l e n g t h t h a t i s b e i n g d e t e r m i n e d t h i s i s no t a v e r y p r a c t i c a l s u g g e s t i o n . F o r b o t h of t h e s e methods t h e e s t i m a t o r s a r e a p p l i e d t o e i t h e r t h e s e p a r a t e d n o r m a l o r s e p a r a t e d e x t e r n a l f i e l d s , so t h a t e r r o r s i n t h e s e p a r a t i o n s c o u l d r e s u l t i n e r r o r s i n t h e s c a l e l e n g t h v a l u e . H o w e v e r , b o t h methods o f s e p a r a t i o n o f t h e n o r m a l and a n o m a l o u s f i e l d s w i l l t e n d t o e r r o r by n o t i n c l u d i n g e n o u g h s m a l l s p a t i a l w a v e l e n g t h s , w h e r e a s t h e s e p a r a t i o n o f t h e i n t e r n a l and e x t e r n a l f i e l d s i s u n a b l e t o s e p a r a t e t h e l o n g s p a t i a l w a v e l e n g t h s . T h u s , e s t i m a t e s o f A made u s i n g b o t h k i n d s o f s e p a r a t e d f i e l d s s h o u l d p r o v i d e bounds on t h e v a l u e o f t h e s c a l e l e n g t h . 74 3.2 V i s u a l Methods of A n a l y s i s As was mentioned i n the p r e v i o u s s e c t i o n , because the i n d u c e d v e r t i c a l f i e l d over a normal l a y e r e d e a r t h always opposes the i n d u c i n g v e r t i c a l f i e l d , the anomalous a m p l i t u d e w i l l g e n e r a l l y be much l a r g e r than the normal a m p l i t u d e . The v e r t i c a l component i s t h e r e f o r e e f f e c t i v e l y a l r e a d y s e p a r a t e d , by v i r t u e of i t s b e i n g m a i n l y anomalous and i n t e r n a l . T h i s o f f e r s the r a t i o n a l e f o r d o i n g i n i t i a l p r e l i m i n a r y a n a l y s i s b e f o r e s e p a r a t i o n , by u s i n g d i s p l a y s of the d a t a i t s e l f . For example, c o n s i d e r the case of a b u r i e d l i n e a r f e a t u r e w i t h enhanced c o n d u c t i v i t y r e l a t i v e t o the host r o c k . As seen i n s e c t i o n 2.2 of Chapter I I , the anomalous magnetic f i e l d a t the s u r f a c e w i l l mimic the f i e l d of a l i n e c u r r e n t . The v e r t i c a l f i e l d w i l l thus undergo a r e v e r s a l i n s i g n a l o n g a p r o f i l e a t r i g h t a n g l e s t o the s t r i k e of the l i n e a r f e a t u r e , w i t h the z e r o f i e l d p o i n t d i r e c t l y over t o p of i t . As seen i n F i g . 3.3, t h i s a l l o w s p r e l i m i n a r y t r a c i n g of l i n e a r f e a t u r e s ; c l e a r l y a c o n d u c t o r runs r o u g h l y n o r t h - s o u t h between s t a t i o n s CHU and RAW, WIC and CUS, and REE and BAK, i n the s o u t h e r n p a r t of the a r r a y . A nother p o s s i b i l i t y i s t h a t the time v a r i a t i o n s of the v e r t i c a l component of the magnetic f i e l d w i l l be c l o s e l y c o r r e l a t e d w i t h one of the h o r i z o n t a l f i e l d components. T h i s i s seen i n F i g . 3.3, i n which , f o r example, the Z components of s t a t i o n s REE and WIC a r e c l o s e l y c o r r e l a t e d w i t h t h e i r 75 F i g . 3.3 (a) Magnetometer l o c a t i o n s f o r the magnetograms of ( b ) . The a r r a y c o n s i s t s of 8 l i n e s t r e n d i n g E a s t - West, which a r e numbered from N o r t h t o South. (b) Magnetograms f o r a substorm of August, 1972, from the s o u t h e r n p a r t of the a r r a y ( a f t e r A l a b i e t a l , 1975). 76 a s s o c i a t e d Y components. The c o r r e l a t i o n i n d i c a t e s t h a t i t i s the p r i m a r y f i e l d of t h a t h o r i z o n t a l component which i s the dominant so u r c e of i n d u c t i o n i n t h e anomaly. Presume f i r s t o r d e r i n d u c t i o n o n l y , t h a t i s , the secondary e f f e c t s of mutual i n d u c t i o n by the induced f i e l d a r e i g n o r e d . The anomalous c u r r e n t w i l l then be p e r p e n d i c u l a r t o the c o r r e l a t e d magnetic f i e l d component, as i l l u s t r a t e d i n F i g . 3.4. In accordance w i t h the r e s u l t s of s e c t i o n 2.2 of Chapter I I , the anomalous c o n d u c t i v i t y must then t r e n d i n t h i s same d i r e c t i o n . For the anomaly of F i g . 3.3, t h i s t r e n d w i l l t h e r e f o r e be i n the n o r t h -s o u t h d i r e c t i o n , which i s i n agreement w i t h the r e s u l t of the Z r e v e r s a l s . An enhanced use of the u n s e p a r a t e d d a t a t a k e s advantage of the f e a t u r e noted i n Chapter I I i n both s e c t i o n s 1 and 2, t h a t the induced f i e l d due t o a c o n d u c t i v e r e g i o n w i l l be more c l o s e l y i n phase w i t h the i n d u c i n g f i e l d s the h i g h e r i t s c o n d u c t i v i t y i s . Thus, by t a k i n g t e m p o r a l F o u r i e r t r a n s f o r m s of the v e r t i c a l component at e v e r y s t a t i o n i n a two d i m e n s i o n a l a r r a y , and c o n t o u r i n g s e p a r a t e l y the a m p l i t u d e and phase r e s u l t s , one w i l l c l e a r l y see the p a t h s of l a r g e h i g h l y c o n d u c t i v e a n o m a l i e s by the c l u s t e r i n g of the c o n t o u r s i n the phase diagram. In F i g . 3 . 5 , the p a t h of the N o r t h American C e n t r a l P l a i n s anomaly through Saskatchewan and i n t o the U.S.A. i s c l e a r l y d e l i n e a t e d i n the c o n t o u r p l o t of the phase of the Z component ( A l a b i et a l , 1975). 77 F i g . 3.4 A plan view of an anomalous current in the earth i s shown. Presuming only f i r s t order induction, the induced current in a l o c a l i z e d conductor w i l l be perpendicular to the magnetic vector of the inducing f i e l d . F i g . 3.5 The contoured amplitude and phase of the magnetograms of Fi g . 3.3b, at a period of 68.3 min. 78 3.3 The I n d u c t i o n Tensor and I n d u c t i o n Arrows Presume t h a t the normal f i e l d {H N,D N,Z N} has been d e f i n e d , so t h a t the anomalous f i e l d {H^,D^,Zft} can be s e p a r a t e d from the t o t a l f i e l d {H,D,Z}. The methods of a n a l y s i s u s i n g i n d u c t i o n arrows a r e then based on the assumption t h a t f o r any frequency w, the anomalous f i e l d components a r e r e l a t e d t o the normal f i e l d components by the i n d u c t i o n t e n s o r , I : The elements of I may be complex, t o a l l o w f o r p o s s i b l e phase d i f f e r e n c e s between the anomalous and normal f i e l d s . O b v i o u s l y t h i s r e l a t i o n can always be d e f i n e d a t each time t , s i n c e t h e r e a r e o n l y t h r e e e q u a t i o n s f o r n i n e unknowns. U n f o r t u n a t e l y , t h i s a d m i t s the p o s s i b i l i t y t h a t any scheme used t o compute the t e n s o r elements w i l l r e s u l t i n the v a l u e s b e i n g (3.3.1) where: I = (3.3.2) 79 time dependent. If the induction tensor elements can fluctuate with time t h i s indicates the tensor i s a function of the inducing f i e l d as well as the underlying conductivity. The ensuing problem of trying to separate the two influences in the tensor w i l l severely l i m i t i t s usefulness in resolving the earth's structure. Thus, the tensor must be reasonably independent of the inducing f i e l d , and thus of time, to be of value. If the source f i e l d at each frequency w was comprised of only one elementary electromagnetic • wave, characterized by wavenumbers (kx,ky,kz), then obviously the l i n e a r i t y of equation 3 . 1 . 4 ensures that the values of the induction tensor w i l l be constant,even with time varying amplitudes. For a single wave, and a given conductivity structure, the re l a t i o n of 3.3.1 would be: values of the normal f i e l d components for an inducing wave of unit amplitude at wavenumbers (kx,ky). A(kx,ky) i s the actual time varying amplitude of the inducing wave, and I'(kx,ky) i s the induction tensor for the wavenumbers (kx,ky). The values of the elements in I'(kx,ky) w i l l depend on the interaction of the / i ' (kx,ky)A(kx,ky,t) / H w(kx,ky) D^(kx,ky) V Z*N(kx,ky) ( 3 . 3 . 3 ) The vector components {H N(kx,ky),D N(kx,ky),Z N(kx,ky)} are the 80 transmitted wave with the anomaly, and so w i l l be dependent on the s p a t i a l c h a r a c t e r i s t i c s of the wave as given by (kx,ky). Thus, I'(kx,ky) w i l l be d i f f e r e n t for each (kx,ky). Because the normal f i e l d of a wave at a plane boundary (incident plus reflected) w i l l always have the same magnitude as the transmitted wave, the actual magnitude of the inducing wave does not enter into I'(kx,ky). In the case of more than one wave in the source f i e l d the t o t a l anomalous f i e l d w i l l be the integral over a l l possible wavenumbers: CO CD where now A(kx,ky) i s an amplitude density in wavenumber space. The t o t a l normal f i e l d i s given by: Thus, to be able to represent the t o t a l anomalous f i e l d as given in equation 3 . 3 . 4 , by an expression of the form of equation 3.3.1 in which I i s source ( and thus time), invariant requires that: ( 3 . 3 . 4 ) ( ( 3 . 3 . 5 ) 81 00 c o -oc -co = j ^ [ l - I *(kx,ky) ] A ( k x , k y , t ) ( H w(kx,kyAdkxdky ) w ( k x , k y ) J ! N ( k x , k y ) / (3.3.6) Because t h e v a l u e s of A ( k x , k y , t ) a r e a r b i t r a r y , t h i s can o n l y be s a t i s f i e d i f f o r a l l ( k x , k y ) : I = I 1 ( k x , k y ) (3.3.7) As was p o i n t e d out e a r l i e r , t h i s i s not i n g e n e r a l t r u e . Thus t h e r e i s n o t h i n g t h a t r e q u i r e s a p r i o r i t h a t the i n d u c t i o n t e n s o r I r e l a t i n g t h e normal and anomalous f i e l d s i n 3.3.1 w i l l be source independent. A v a r i a t i o n of e q u a t i o n 3.3.1 was suggested by D r a g e r t (1973, pg. 41 - 4 2 ) . T a k i n g the s p a t i a l F o u r i e r t r a n s f o r m s of the a r r a y measurements of both the normal and anomalous f i e l d s a t a p a r t i c u l a r f r e q u e n c y , w, one would o b t a i n t h e i r r e s p e c t i v e complex a m p l i t u d e s a t each p o s s i b l e p a i r of wavenumbers kx,ky. The a s s e r t i o n was then t h a t the r e l a t i o n of 3.3.1 would be t r u e when c o n s i d e r e d i n d e p e n d e n t l y a t each wavenumber p a i r (meaning t h a t I , the i n d u c t i o n t e n s o r , would be a f u n c t i o n of k x , k y ) . The s u g g e s t i o n i s not c o r r e c t , because a s i n g l e e l e m e n t a r y wave of a r b i t r a r y wavenumbers kxo,kyo, can g e n e r a t e an e n t i r e h o r i z o n t a l spectrum of waves i n the anomalous f i e l d . C o n s i d e r the example, t r e a t e d i n s e c t i o n . 2.2 of Chapter I I , of a s i n g l e downgoing wave i m p i n g i n g on an e a r t h t h a t i s homogenous ex c e p t f o r a 82 b u r i e d c y l i n d e r of h i g h r e l a t i v e c o n d u c t i v i t y . As was seen, the anomalous f i e l d of t h i s example t o f i r s t o r d e r resembled t h a t of a b u r i e d l i n e c u r r e n t . I t was shown i n s e c t i o n 1.2 of Chapter I t h a t a l i n e c u r r e n t has energy a t an i n f i n i t e number of wavenumber v a l u e s , so t h a t the s p a t i a l t r a n s f o r m of the anomalous f i e l d would be non-zero a t an i n f i n i t e number of wavenumber v a l u e s . On t h e o t h e r hand, as was shown i n s e c t i o n 2.1 of Chapter I I , the normal f i e l d due t o a s i n g l e i m p i n g i n g wave w i l l o n l y have one non-zero wavenumber component, a t the wavenumbers, kxo,kyo, of the o r i g i n a l i n d u c i n g wave. Thus, t o r e l a t e the anomalous f i e l d components a t each wavenumber p a i r kx,ky would be i m p o s s i b l e a t a l l v a l u e s f o r which the normal f i e l d v a l u e was z e r o and the anomalous f i e l d was n o t . I t has been shown t h e n , t h a t the r e l a t i o n of e q u a t i o n 3.3.1 i s not u n i v e r s a l l y t r u e . However, i t has been proposed t h a t f o r a r r a y s a t l a t i t u d e s mid-way between the p o l a r a r e a s and the e q u a t o r , the source f i e l d w i l l be e f f e c t i v e l y h o r i z o n t a l l y u n i f o r m f o r the ranges of parameters used i n G.D.S. ( L i l l e y and B e n n e t t , 1973; Banks, 1973), w i t h e s t i m a t e s of the s c a l e l e n g t h b e i n g on the o r d e r of 5000 t o 1 0 , 0 0 0 km. (Gough, 1973; Banks, 1973; L i l l e y , 1973; M a r e s c h a l , 1981). Under the s e c o n d i t i o n s i t i s c l a i m e d t h a t the i n d u c t i o n t e n s o r w i l l be independent of the sourc e v a r i a t i o n s . C o n s i d e r the e f f e c t on the i n d u c t i o n t e n s o r f o r m u l a t i o n i n the u n i f o r m f i e l d s i t u a t i o n . In the l i m i t as the kx and ky v a l u e s of a wave go t o z e r o , the d i r e c t i o n and mode (TM or TE) of the wave no l o n g e r s e r v e t o d i s t i n g u i s h t he 83 d i r e c t i o n of the e l e c t r i c and magnetic f i e l d s i n the h o r i z o n t a l p l a n e . For a v e r t i c a l l y downgoing wave t h e r e i s no d i s t i n c t i o n between the TE and TM modes, and the o n l y wavenumber i s k z , y e t t h e r e a r e an i n f i n i t e number of p o s s i b l e r o t a t i o n s of the wave components about the d i r e c t i o n n o r m al. I t i s found, however, t h a t any v e r t i c a l wave can be decomposed i n t o a sum of two waves: one w i t h i t s e l e c t r i c v e c t o r a l o n g some g i v e n h o r i z o n t a l d i r e c t i o n (E p o l a r i z e d ) , the o t h e r w i t h i t s magnetic v e c t o r a l o n g t h i s same d i r e c t i o n (H p o l a r i z e d ) (Jones, 1971). Thus, the u n i f o r m f i e l d case s t i l l i n v o l v e s two wave t y p e s w i t h the consequent r e l a t i o n between the anomalous and normal f i e l d now b e i n g : (3.3.8) where the s u b s c r i p t s E and M r e f e r t o E p o l a r i z e d and H p o l a r i z e d r e s p e c t i v e l y . E q u a t i o n 3.3.8 has been s i m p l i f i e d because of the z e r o v a l u e s of D w e and H N h due t o our c h o i c e of r e f e r e n c e axes, and because of the z e r o v a l u e of the normal v e r t i c a l f i e l d i n t h e u n i f o r m f i e l d c a s e . A l s o from t h i s , we have the t o t a l normal f i e l d b e i n g g i v e n by { HNE' DNM'0}- T n u s , we can r e w r i t e 3,3.8 a s : 84 / Hft Here, the superscripts E and M indicate the o r i g i n a l tensor that the elements are from. Thus, in the case of a perfectly horizontally uniform f i e l d the induction tensor w i l l be independent of the source f i e l d , and should thus contain retrievable information about the underlying conductivity structure. What needs to be determined then, i s whether a s u f f i c i e n t degree of uniformity for induction tensor invariance i s attained at scale lengths in the range 5000 - 10,000 km. This is considered in d e t a i l in Appendix C. The discussion indicates that for the scale lengths suggested for midlatitudes •, and for periods greater than 2 hrs., the.values of I'(kx,ky) w i l l in fact be nearly i d e n t i c a l . Thus, the rel a t i o n of equation 3.3.1 w i l l be v a l i d for these parameter ranges. It should be noted that i t i s because the tensor relates the normal f i e l d to the anomalous f i e l d that the magnitude of the transmitted wave need not be considered. This i s not true of the similar relation between the external and internal f i e l d s . If these portions of the surface f i e l d are related by a tensor in i d e n t i c a l fashion to that in equation 3.3.1 (as done by L i l l e y (1974)), the s i g n i f i c a n t changes in the r a t i o of the transmitted wave to the external f i e l d at a l l values of (kx,ky) w i l l always result in a 85 t e n s o r t h a t i s time dependent. C o n s i d e r t h a t the i n d u c t i o n t e n s o r i s i n f a c t independent of t i m e . One can then d e f i n e t h e r e l a t i o n between t h e normal and anomalous f i e l d a t each f r e q u e n c y w as i n e q u a t i o n 3.3.1, except f o r an u n c o r r e l a t e d ' n o i s e ' term: (3.3.10) The e s t i m a t e d v a l u e s of the elements of I w i l l be those which m i n i m i z e the power of the u n c o r r e l a t e d terms (Schmucker, 1970, pg. 20 - 2 1 ) . By m i n i m i z i n g the power of each component of the n o i s e v e c t o r w i t h r e s p e c t t o the r e a l and i m a g i n a r y p a r t s of each of the t e n s o r e lements, t h r e e independent s e t s of l i n e a r e q u a t i o n s a r e o b t a i n e d , one f o r each'column of I (see Appendix D f o r the complete method). As an example, the s e t of e q u a t i o n s f o r t h e t h i r d column i s : S%«-*KJ J (3.3.11) where: 8 6 S = / S H w H w S ^ j H N S?MHM f ( 3 . 3 . 1 2 ) i s t h e m a t r i x of auto and c r o s s powers. The power of two s i g n a l s A,B of l e n g t h To i s d e f i n e d a s : Sfl B(w) = A(w)B*(w)/ To ( 3 . 3 . 1 3 ) The most im p o r t a n t terms of the i n d u c t i o n t e n s o r a r e those of the bottom row, c o r r e s p o n d i n g t o the r e l a t i o n of t o the normal f i e l d : ( 3 . 3 . 1 4 ) Presuming t h a t the c o n t r i b u t i o n due t o i n d u c t i o n by the normal v e r t i c a l component w i l l i n g e n e r a l be v e r y s m a l l , we have: ( 3 . 3 . 1 5 ) For each f r e q u e n c y w, two i n d u c t i o n arrows can be d e f i n e d , c o r r e s p o n d i n g t o the p o r t i o n of Z^ which i s in-phase w i t h and 87 U(w) = - { x - R e a l [ C j H ( w ) ] + y - R e a l [ C ^ ( w ) ]} (3.3.16) and the p o r t i o n which i s out of phase w i t h H N and D N: V(w) = {x•Imag[C f r H(w)] + y•Imag[C*&(w)]} (3.3.17) where x i s t h e u n i t v e c t o r i n the d i r e c t i o n of magnetic n o r t h and y i s the u n i t v e c t o r i n the d i r e c t i o n of magnetic e a s t . The n e g a t i v e s i g n on the l e f t hand s i d e of e q u a t i o n 3.3.16 i s commonly i n t r o d u c e d t o m a i n t a i n • the c o n v e n t i o n s e t by P a r k i n s o n ' s o r i g i n a l development of i n d u c t i o n arrows ( P a r k i n s o n , 1959, 1962). A l s o , i n t h e case of a l o c a l i z e d c o n d u c t i v e p a t h , i f f i r s t o r d e r i n d u c t i o n o n l y i s s i g n i f i c a n t , the s i g n c o n v e n t i o n w i l l ensure t h a t the in-phase i n d u c t i o n arrow U w i l l p o i n t towards t h i s c o n d u c t i v e anomaly. T h i s i s i l l u s t r a t e d i n F i g . 3.6 f o r a h o r i z o n t a l l i n e c u r r e n t r u n n i n g n o r t h - s o u t h a t depth z c . I t i s g e n e r a l l y presumed t h a t the in-phase i n d u c t i o n arrow at a f r e q u e n c y w w i l l p o i n t towards nearby c u r r e n t c o n c e n t r a t i o n s induced a t t h a t f r e q u e n c y , as seemingly i n d i c a t e d i n F i g . 3.6. (Banks, 1973; Beamish, 1977; G a r l a n d , 1979, pg. 270). However, i t i s not c l e a r t h a t t h i s can be a c c e p t e d i n g e n e r a l . C o n s i d e r the h o r i z o n t a l c o n d u c t i v e anomaly shown i n F i g . 3.7. T h i s anomaly i s not o f f e r e d as a p o s s i b l e r e a l e a r t h model, but s i m p l y as a t e s t of the g e n e r a l i t y of the i n d u c t i o n arrow's p r o p e r t i e s . U s i n g the f i r s t o r d e r i n d u c t i o n 88 H Polarized Hn z a < 0 C£1> < 0 E ^Polarized Ed ~*"6 n C 2 B » C l H Z ^ > 0 c « >o -« • A X F i g . 3.6 The d i r e c t i o n of the in-p h a s e i n d u c t i o n arrow near a l i n e c u r r e n t r u n n i n g N o r t h - South. 89 'n c, >0 >0 J&= rC|0| Drj JH=rC2aaHn C,<0 C2<0 a 2 — •.V X F i g . 3 .7 F i r s t o r d e r i n d u c t i o n i n c r o s s i n g c o n d u c t i v e p a t h s . The r a t i o s of c u r r e n t s i n t h e two c o n d u c t o r s w i l l depend on b o t h t h e c o n d u c t i v i t y and t h e magnitude of each component of t h e i n d u c i n g f i e l d . However, t h e c o e f f i c i e n t s of the i n d u c t i o n t e n s o r w i l l depend o n l y on t h e c o n d u c t i v i t y . 90 a p p r o x i m a t i o n throughout ( t h a t i s , i g n o r i n g s e l f - i n d u c t i o n ) , we w i l l have the H component of the normal f i e l d i n d u c i n g c u r r e n t i n the eas t - w e s t d i r e c t i o n and the D component of the normal f i e l d i n d u c i n g c u r r e n t i n the n o r t h - s o u t h d i r e c t i o n . The f i n a l r e l a t i o n f o r Z A i n terms of H W and D^, a t any s u r f a c e p o s i t i o n (x,y) i s : Z f t(x,y,w) = r (w) • [ C 2 ( x , y ) C T Z H n + C ( ( x , y ) G , D W ] ( 3 . 3 . 1 8 ) C, and Cj. a re s p a t i a l l y dependent c o n s t a n t s , w i t h t h e i r s i g n as i n d i c a t e d i n F i g . 3 . 7 , and r(w) i s a fr e q u e n c y dependent complex c o n s t a n t r e l a t i n g the normal components H ^ D ^ t o t h e i r a s s o c i a t e d e l e c t r i c f i e l d s a t the depth of the anomalous c o n d u c t i v i t i e s . Thus, the in-phase i n d u c t i o n arrow i n t h i s example i s : U = - { [ r ( w ) < r l c 1 ( x f y ) ] x + [ r (w)tr, C, (x,y) ]y} ( 3 . 3 . 1 9 ) The d i r e c t i o n of U w i l l depend on the p o s i t i o n of the s t a t i o n t h r o u g h the v a l u e s of C, and . More s i g n i f i c a n t l y , the d i r e c t i o n of U w i l l depend on the c o m p a r a t i v e v a l u e s of cr, and C"2, r a t h e r than on the r e l a t i v e s t r e n g t h s of the c u r r e n t s i n each c o n d u c t o r . I f the s t a t i o n was s i t u a t e d such t h a t |C,| and \CZ\ were e q u a l (which would be t r u e a l o n g the b i s e c t o r s of the a x e s ) , then f o r <"i>>C"i, U would p o i n t towards the c o n d u c t o r l y i n g a l o n g the y a x i s , and f o r < ^ i » C T 2 f U would p o i n t towards 91 the c o n d u c t o r l y i n g a l o n g the x a x i s . The r e l a t i v e a m p l i t u d e s of the c u r r e n t d e n s i t i e s i n t h e s e c a s e s c o u l d t a k e on any v a l u e , depending on the a m p l i t u d e s of H w and . T h i s c l e a r l y i l l u s t r a t e s t h a t the i n d u c t i o n a rrows do not n e c e s s a r i l y p o i n t towards c u r r e n t c o n c e n t r a t i o n s , but r a t h e r , from the examples c o n s i d e r e d , they p o i n t g e n e r a l l y towards the a r e a of h i g h c o m p a r a t i v e c o n d u c t i v i t y . Up t o t h i s p o i n t , o n l y the case where l o c a l i n d u c t i o n e f f e c t s were dominant i n the p r o d u c t i o n of the anomalous f i e l d has been c o n s i d e r e d , so t h a t t h e r e was a n e c c e s s a r y c o r r e l a t i o n between the l o c a l l y measured normal and anomalous f i e l d s . Another p o s s i b i l i t y f o r anomalous b e h a v i o r i s c u r r e n t c o n c e n t r a t i o n , or c h a n n e l l i n g (Whitham and Andersen, 1965; Dyck and G a r l a n d , 1969; Gough, 1973), i n which a u n i f o r m c u r r e n t f l o w due t o l a r g e s c a l e r e g i o n a l i n d u c t i o n i s c h a n n e l l e d t h r o u g h a l o c a l i z e d h i g h c o n d u c t i v i t y zone (see F i g . 3.8). The anomalous magnetic f i e l d i n the c h a n n e l l e d c u r r e n t case does not depend on l o c a l i n d u c t i o n , t h u s , i t i s not i m m e d i a t e l y c l e a r how the c a l c u l a t e d v a l u e s of the i n d u c t i o n t e n s o r , and t h e r e f o r e , the i n d u c t i o n arrow, w i l l r e l a t e t o the p o s i t i o n and o r i e n t a t i o n of the p e r t u r b i n g c o n d u c t i v e body. In i g n o r i n g i n d u c t i o n e f f e c t s , we reduce the problem t o one of q u a s i - s t a t i c d i r e c t c u r r e n t (DC) f l o w . The p e r t i n e n t M a x w e l l ' s e q u a t i o n s , a f t e r e l i m i n a t i o n of the time d e r i v a t i v e terms, a r e (from e q u a t i o n s A.12, A.13, A.14 and A.27 i n Appendix A ) : 92 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i i i i i i i i i / f f i i > « » \ u ~ K 1 ' 1 1 ::: :f Y : : : , , , ^ ; : ; I I / / / i i i / i i i i i i i i i i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Cb) F i g . 3.8 The e f f e c t s of c u r r e n t c h a n n e l l i n g t h rough a c o n d u c t i v e s p h e r e . The arrows g i v e the magnitude and d i r e c t i o n of the e l e c t r i c f i e l d o u t s i d e the sp h e r e , f o r the two o r t h o g o n a l d i r e c t i o n s of the r e g i o n a l c u r r e n t f l o w : (a) r e g i o n a l c u r r e n t f l o w s E a s t - West (b) r e g i o n a l c u r r e n t f l o w s N o r t h - South 93 V-E = 0 (3.3.20) V-H = 0 (3.3.21 ) VxE = 0 (3.3.22) VxH = (TE = J (3.3.23) The c u r l - f r e e n a t u r e of the e l e c t r i c f i e l d e n a b l e s us t o w r i t e the e l e c t r i c f i e l d as the g r a d i e n t of a s c a l a r p o t e n t i a l : E = -VV (3.3.24) which when c o u p l e d w i t h e q u a t i o n 3.3.20 g i v e s \ us L a p l a c e ' s equat i o n : (3.3.25) In t h i s q u a s i - s t a t i c c a s e , we a l s o have from e q u a t i o n A.20 i n Appendix A the boundary c o n d i t i o n e n s u r i n g the c o n t i n u i t y of the 94 normal component of c u r r e n t f l o w a c r o s s a boundary: V - ( f l T E ) = V- J = 0 (3.3.26) E q u a t i o n s 3.3.24, 3.3.25 and 3.3.26 govern the DC problem. In the f o l l o w i n g d i s c u s s i o n i t w i l l be presumed f o r s i m p l i c i t y t h a t the source f i e l d i s p e r f e c t l y h o r i z o n t a l l y u n i f o r m . A l l the arguments p e r t a i n i n g t o the v a l i d i t y of t h i s a ssumption i n p r a c t i c e and i t s e f f e c t on the i n v a r i a n c e of the i n d u c t i o n t e n s o r w i l l a p p l y here as w e l l , but w i l l not be d i s c u s s e d . We w i l l a l s o presume f i r s t o r d e r i n d u c t i o n e f f e c t s o n l y , so t h a t the n o r t h - s o u t h c u r r e n t s w i l l be p r o p o r t i o n a l t o D^ o n l y , and the e a st-west c u r r e n t s w i l l be p r o p o r t i o n a l t o o n l y . The l i n e a r i t y of the e q u a t i o n s i n v o l v e d a l l o w s us t o c o n s i d e r s e p a r a t e l y the c a s e s i n w h i c h the r e g i o n a l c u r r e n t i s o n l y n o r t h - s o u t h , or o n l y e a s t - w e s t . Any o t h e r d i r e c t i o n of r e g i o n a l c u r r e n t f l o w can a l w a y s be c o n s i d e r e d as a s u p e r p o s i t i o n of t h e s e , w i t h the r e s u l t a n t c u r r e n t v e c t o r a t any p o i n t b e i n g the sum of the v e c t o r s c o r r e s p o n d i n g t o each of the two s o l u t i o n s . In F i g . 3.8, a sphere of r a d i u s 'a' and c o n d u c t i v i t y imbedded i n a h o s t rock of lower c o n d u c t i v i t y <3", s e r v e s as the model. E q u a t i o n s 3.3.25 and 3.3.26 can be s o l v e d i n s p h e r i c a l c o o r d i n a t e s , f o r the boundary c o n d i t i o n of a u n i f o r m e l e c t r i c -a f i e l d E 0 a t l a r g e d i s t a n c e s away from the s p here. The symmetry of the problem e n s u r e s t h a t t h e s o l u t i o n f o r one d i r e c t i o n of 9 5 t h i s r e g i o n a l e l e c t r i c f i e l d can be s i m p l y r o t a t e d by 1"T/2 t o g i v e the s o l u t i o n f o r the o r t h o g o n a l d i r e c t i o n . As an example, the s o l u t i o n f o r the p o t e n t i a l o u t s i d e t h e sphere f o r the u n i f o r m f i e l d i n the 'x' d i r e c t i o n i s ( T e l f o r d e t a l , 1976, pg. 647-649): V ( x , y ) = - E 0 x { l - [ a 3 ( b - D ] / [ ( b + 2 ) ( x X + y l ) V l ] } (3.3.27) where 'b' i s the c o n d u c t i v i t y r a t i o 0"2/cT, . From e q u a t i o n 3.3.35, the d i r e c t i o n a l d e r i v a t i v e s of the • p o t e n t i a l g i v e us the components of the e l e c t r i c v e c t o r a t any p o i n t o u t s i d e the sphere. For the u n i f o r m f i e l d i n the 'x' d i r e c t i o n : E x ( x , y ) = E Q{1 + [ a i ( b - l ) ( 2 x l - y 2 ) ] / [ ( b + 2 ) ( x 1 + y I f / l ] } (3.3.28) E,j(x,y) = E 0 { [ a S ( b - 1 ) ( 3 x y ) ] / [ ( b + 2 ) ( x l + y l ) S / i ] } (3.3.29) The e l e c t r i c f i e l d v e c t o r s f o r each of t h e two u n i f o r m f i e l d d i r e c t i o n s a r e shown i n F i g . 3.8. I t i s c l e a r t h a t the c u r r e n t c o n c e n t r a t i o n e f f e c t s on the o u t s i d e of the sphere a r e most pronounced i n the r e g i o n s a l o n g the a x i s of symmetry p a r a l l e l t o t h e d i r e c t i o n of the r e g i o n a l c u r r e n t f l o w ( r e g i o n s 96 1 and 3 ) , and are a t a minimum i n the r e g i o n s a l o n g the r a d i a l l i n e o r t h o g o n a l t o t h i s ( r e g i o n s 2 and 4 ) . Thus, i n r e g i o n s 1 and 3 the anomalous v e r t i c a l .magnetic f i e l d w i l l be most dependent on D ^ f i n r e g i o n s 2 and 4 i t w i l l be most dependent on H N, and i n the i n t e r m e d i a t e r e g i o n s t h e r e w i l l be a c o n t i n u o u s g r a d i n g , w i t h the TT/4 b i s e c t o r s marking the l i n e s of e q u a l dependences. Thus, from t h i s q u a l i t i t a v e d i s c u s s i o n , the i n -phase i n d u c t i o n arrows w i l l always p o i n t g e n e r a l l y towards the sphere. In f a c t , symmetry w i l l r e q u i r e t h a t the arrows w i l l a lways p o i n t towards the v e r y c e n t e r . In q u i c k summary of the main p o i n t s r e g a r d i n g the i n d u c t i o n t e n s o r and the i n d u c t i o n arrow we have: (1) R e l a t i n g the anomalous magnetic f i e l d v e c t o r t o the normal f i e l d v e c t o r v i a the i n d u c t i o n t e n s o r i s not u n i v e r s a l l y c o r r e c t . (2) In the case of a h o r i z o n t a l l y u n i f o r m f i e l d the i n d u c t i o n t e n s o r w i l l c o r r e c t l y r e l a t e the anomalous and normal f i e l d s , t o f i r s t o r d e r . The i n d u c t i o n arrows w i l l p o i n t i n g e n e r a l towards r e g i o n s of anomalously h i g h c o n d u c t i v i t y , which w i l l not n e c c e s s a r i l y be the r e g i o n s of h i g h e s t c u r r e n t c o n c e n t r a t i o n . The degree of u n i f o r m i t y r e q u i r e d f o r the i n v a r i a n c e of the t e n s o r w i l l be a t t a i n e d a t v a l u e s of the s c a l e l e n g t h encountered i n p r a c t i c e a t m i d l a t i t u d e s , f o r p e r i o d s g r e a t e r than ~ 2 h r s . (3) The u n c e r t a i n t y of t h e i n v a r i a n c e of the i n d u c t i o n t e n s o r 97 w i l l a p p l y t o the case of c u r r e n t c h a n n e l l i n g as w e l l . However, i f the f i e l d i s u n i f o r m , s u c h as a t m i d l a t i t u d e s , the i n d u c t i o n a rrows w i l l p o i n t a l s o toward the anomalous h i g h c o n d u c t i v i t y zone. 3.4- Q u a n t i t a t i v e Methods Used i n G.D.S. Q u a n t i t a t i v e m o d e l l i n g of the e a r t h ' s c o n d u c t i v i t y f o r a r e a s c o n t a i n i n g presumed anomalous s t r u c t u r e has r e c e i v e d much l e s s emphasis i n the 'past than t h e c o r r e s p o n d i n g q u a l i t a t i v e m o d e l l i n g . The t r u e i n v e r s e problem of f i n d i n g a model t h a t f i t s the d a t a i n e i t h e r a 'one-shot', or an i t e r a t i v e p r o d e d u r e , has not been s o l v e d t o t h i s a u t h o r ' s knowledge. The o t h e r approach t o t h i s i s t o s e l e c t a p o s s i b l e model, e i t h e r based on q u a l i t a t i v e i n f o r m a t i o n and i n t u i t i o n , or s i m p l y p i c k e d randomly, and then use f o r w a r d m o d e l l i n g r o u t i n e s t o see i f i t f i t s the d a t a . I f model s e l e c t i o n i s done i n t e r a c t i v e l y w i t h the fo r w a r d m o d e l l i n g then the model can be a d j u s t e d u n t i l a good f i t i s o b t a i n e d ( P o r a t h e t a l , 1970; Gough, 1973; ; D r a g e r t , 1973, pg.94-96). However, the f i n a l model i s always non-unique, so t h a t i t i s v e r y p o s s i b l e t h a t the b i a s e s of d i f f e r e n t i n d i v i d u a l s d o i n g the model p e r t u r b a t i o n s c o u l d r e s u l t i n v e r y d i f f e r e n t f i n a l s o l u t i o n s . As w e l l , i t has been found i n some cas e s t h a t a model t o f i t the d a t a cannot be found (Whitham and 98 Andersen, 1965; P o r a t h e t a l , 1971). (The suggested reason f o r t h i s was t h a t the anomalous f i e l d was due t o c u r r e n t c h a n n e l l i n g , so t h a t the a r e a s c o n s i d e r e d i n the model s t u d i e s were s i m p l y not l a r g e enough t o encompass the t r u e model.) The a l t e r n a t i v e t o the i n t e r a c t i v e approach would be the 'Monte C a r l o ' method, i n which models a r e s e l e c t e d randomly f o r subsequent t r i a l i n the f o r w a r d m o d e l l i n g r o u t i n e (Cochrane and Hyndman, 1970). For the two d i m e n s i o n a l and t h r e e d i m e n s i o n a l c a s e s t h i s would c l e a r l y be p r o h i b i t i v e l y e x p e n s i v e . As w e l l , i t i s not c l e a r t h a t f o r a r b i t r a r i l y • complex models t h a t the e x i s t i n g f o r w a r d m o d e l l i n g r o u t i n e s w i l l g i v e a c c u r a t e r e s u l t s . Thus, e i t h e r the models must be made s i m p l e r (meaning they no l o n g e r a r e c o m p l e t e l y a r b i t r a r y ) or the f o r w a r d model r e s u l t s must be c o n s i d e r e d q u e s t i o n a b l e . To a v o i d t h e s e d i f f i c u l t i e s , and y e t s t i l l o b t a i n e s t i m a t e s of some of the b a s i c q u a n t i t a t i v e v a l u e s , such as the depth and l a t e r a l e x t e n t of the anomalous c o n d u c t i v i t i e s , c e r t a i n a p p r o x i mate measures have been d e v i s e d . I f the v i s u a l d a t a d i s p l a y s and i n d u c t i o n arrows c o n s i s t e n t l y i n d i c a t e a l o n g narrow anomaly, then i t i s common t o model the i n t e r n a l source as a l i n e c u r r e n t . Presume t h a t the c u r r e n t , of magnitude I , i s p a r a l l e l t o the e a r t h ' s s u r f a c e and i n the n e g a t i v e 'y' d i r e c t i o n , and t h a t t h e measuring a r r a y i 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 the c u r r e n t (see F i g . 3.9). Then, by Ampere's law, the magnitudes of the d i r e c t i o n a l components Bx,Bz a t any p o s i t i o n 'x' a l o n g the a r r a y w i l l be g i v e n by: 99 3.9 A v a r i e t y of d i f f e r e n t t w o - d i m e n s i o n a l c u r r e n t d e n s i t y models, w i t h t h e c u r r e n t s t r a v e l l i n g i n t o or out of the page. In a l l c a s e s the s t a t i o n a r r a y i s a t the s u r f a c e , p e r p e n d i c u l a r t o the s t r i k e of the c u r r e n t s . The d i f f e r e n t models a r e i n d i c a t e d by: (a) <> (b) LU (c) X (d) X 3.10 The dependence of the 'peak t o -peak' w i d t h of the Bz component, f o r the c u r r e n t d e n s i t y models of F i g . 3.9. The w i d t h ' s a r e seen t o v a r y w i t h the depth of the anomaly as w e l l as w i t h i t s l a t e r a l e x t e n t . 100 Fig. 3.9 Fig.3.10 87.5 100.0 Station Position 101 B x ( x ) = j i 0 I z 0 / { 2 f T [ ( x - x 0 ) * +z*]} (3.4.1) B z ( x ) = j u 0 I ( x - x o ) / { 2 f T [ ( x - x 0 ) + z * ] } (3.4.2) where ( x 0 , z 0 ) i s the p o s i t i o n of the l i n e c u r r e n t (see F i g . 3.10 fo r sample p l o t s of Bx(x) and B z ( x ) ) . The depth of the l i n e c u r r e n t can be found from e i t h e r the d i s t a n c e between the p o i n t a t which t he Bx f i e l d i s a maximum, ' ( x o , 0 ) , and the p o i n t s a t which i t i s h a l f the maximum v a l u e ( x i / , 0 ) : (3.4.3) or from the s e p a r a t i o n between the p o s i t i o n s (x.^,0), (x_,0) of the p o s i t i v e and n e g a t i v e peaks of the Bz component: z o = 1 / 2 | X _ - X J (3.4.4) I t i s sometimes c l a i m e d t h a t t h i s depth r e p r e s e n t s the maximum depth p o s s i b l e t o the anomalous c u r r e n t , and t h a t any ot h e r c u r r e n t d i s t r i b u t i o n g i v i n g the same d a t a v a l u e s on the s u r f a c e must be s h a l l o w e r . T h i s statement i s not s t r i c t l y c o r r e c t , as i s shown i n the f o l l o w i n g argument. C o n s i d e r a d i s t r i b u t i o n of c u r r e n t s a l o n g a sheet p a r a l l e l 102 t o t he e a r t h ' s s u r f a c e , a t a depth Z | . I f the c u r r e n t s i n t h i s sheet a r e c o n s t r a i n e d t o f l o w o n l y i n the p o s i t i v e or n e g a t i v e 'y' d i r e c t i o n , w i t h t h e i r magnitudes not v a r y i n g w i t h 'y', th e n a l i n e a l c u r r e n t d e n s i t y j ( x ) i s s u f f i c i e n t t o d e s c r i b e them. I t w i l l be shown t h a t f o r z ( < z 0 i t i s always p o s s i b l e t o f i n d a j ( x ) such t h a t a t the s u r f a c e t h e f i e l d of t h e l i n e c u r r e n t a t ( x 0 , z 0 ) i s c o m p l e t e l y d u p l i c a t e d . Without l o s s of g e n e r a l i t y we can s i m p l i f y the a l g e b r a of the problem by s e t t i n g the • c o o r d i n a t e o r i g i n on the s u r f a c e d i r e c t l y above the l i n e c u r r e n t . Thus, the f i e l d components of the o r i g i n a l l i n e c u r r e n t reduce t o : Bx(x) = u 0 I z 0 / [ 2 1 T ( z 0 + x j ) ] (3.4.5) B z ( x ) = p 0 I x / [ 2 T T (Zo+xJ-) ] (3.4.6) i i The s u r f a c e f i e l d s , Bx , Bz , due t o the sheet d i s t r i b u t i o n of c u r r e n t s a r e o b t a i n e d u s i n g the B i o t - S a v a r t law, and a r e : CO Bx' (x) = (/L»o/2*nr) ^ { [ j ( u ) z , ] / [ ( x - u ) l + z l 1 ] } d u ~°° (3.4.7) 103 c o B z \ x ) = (^o/2TT ) f { [ j ( u ) ( x - u ) ] / [ < x - u ) l + z j " ] } d u (3.4.8) Thus, f o r the f i e l d s of the sheet d i s t r i b u t i o n of c u r r e n t t o mimic t h a t of the o r i g i n a l l i n e c u r r e n t r e q u i r e s t h a t : oo y [ j ( u ) z ] / [ ( x - u ) 2 + z ^ ] d u = I z 0 / ( z * + x l ) ~°° (3.4.9) and CO ^ [ j ( u ) ( x - u ) ] / [ ( x - u ) l + z | l ] d u = l x / ( z * + x r ) ~°° (3.4.10) T a k i n g the s p a t i a l F o u r i e r t r a n s f o r m of both s i d e s of e q u a t i o n 3.4.9 and 3.4.10 (where the F o u r i e r t r a n s f o r m , 3~ , i s d e f i n e d as i n e q u a t i o n C.5 i n Appendix C ) , n o t i n g t h a t the l e f t hand s i d e i n b o th cases i s a c o n v o l u t i o n , and then a p p l y i n g the F a l t u n g (or c o n v o l u t i o n ) theorem, we a r r i v e a t : J [ j ( x ) ] . J [z , / ( x * + z . 1 ) ] = I [ z 0 / (z 0 l+x v) ] (3.4.11) 3 tjU)]-} [x/(x1+z,1')] = l J [ x / ( z f l V ) ] (3.4.12) U s i n g the g e n e r a l F o u r i e r t r a n s f o r m e x p r e s s i o n s (from Appendix C, e q u a t i o n s C.3 and C.4): 104 } [ a / ( x 2 + a 1 ) ] = T r e " 1 ! 1 " (3.4.13) J [ x / U ^ + a 1 )] = - T f i e ' ^ s g n ^ ) (3.4.14) both e q u a t i o n s 3.4.11 and 3.4.12 r e s u l t i n : (3.4,15) where: (3.4.16) T a k i n g the i n v e r s e F o u r i e r t r a n s f o r m of e q u a t i o n 3.4.15 we a r r i v e a t the f i n a l r e s u l t : j ( x ) = I<$V[TT( J^+x 1)] (3.4.17) Thus, we can d u p l i c a t e the f i e l d due t o any l i n e c u r r e n t w i t h a d i s t r i b u t i o n of c u r r e n t on a sheet s i t u a t e d between the l i n e c u r r e n t and the s u r f a c e . C o r r e s p o n d i n g l y , t h i s s u g g e s t s t h a t i f such a sheet c u r r e n t was p r e s e n t a t the same t i m e as the l i n e c u r r e n t , but w i t h i t s d i r e c t i o n of f l o w r e v e r s e d , then the f i e l d a t a t the s u r f a c e would be n u l l . Thus, i t i s c o n c e i v a b l e 105 t h a t c u r r e n t d i s t r i b u t i o n s c o u l d e x i s t of any magnitude and d e p t h , but which i n t o t a l g i v e r i s e t o no s u r f a c e r e a d i n g s . I f such an ' a n n i h i l a t o r * d i s t r i b u t i o n was p r e s e n t , then c u r r e n t s c o u l d c o n c e i v a b l y be f l o w i n g u n d e t e c t e d a t g r e a t e r depths than t h a t of the o r i g i n a l l i n e c u r r e n t . I t i s a l s o shown i n Appendix F t h a t even when a l l the c u r r e n t s i n any model a r e r e s t r i c t e d t o b e i n g i n phase, (so t h a t ' a n n i h i l a t o r ' d i s t r i b u t i o n s a r e i m p o s s i b l e ) t h e r e can s t i l l be c u r r e n t s i n the model t h a t a re deeper than the l i n e c u r r e n t , w i t h the model d u p l i c a t i n g the s u r f a c e d a t a of the l i n e c u r r e n t . The c o r r e c t statement about the p o s s i b l e depth of the anomalous c u r r e n t s i s t h a t i f the s u r f a c e f i e l d r e a d i n g s can be d u p l i c a t e d by a l i n e c u r r e n t a t a c e r t a i n d e p t h , then t h e r e must be some c u r r e n t a t t h a t depth or s h a l l o w e r . T h i s statement i s proved i n Appendix F. Another method of e s t i m a t i n g the depth of an anomaly i s by c o n s i d e r i n g the s m a l l e s t p e r i o d a t which the anomalous f i e l d i s s t i l l s i g n i f i c a n t , and then u s i n g t h i s f r e q u e n c y i n a s k i n depth c a l c u l a t i o n . One problem i n v o l v e d w i t h t h i s method i s t h a t of e s t i m a t i n g the one d i m e n s i o n a l c o n d u c t i v i t y s t r u c t u r e f o r use i n the s k i n depth c a l c u l a t i o n . The o t h e r problem a r i s e s because of the p o s s i b i l i t y of c h a n n e l l e d c u r r e n t s . I t i s v e r y p o s s i b l e t h a t the c h a n n e l l i n g of the r e g i o n a l c u r r e n t system w i l l not be s t r i c t l y h o r i z o n t a l , so t h a t t h e c h a n n e l l e d c u r r e n t under t h e a r r a y w i l l be a t a d i f f e r e n t d e pth than t h a t a t which the r e g i o n a l system was i n d u c e d . The dominant p e r i o d s of the anomalous , f i e l d s i n such a case would t h u s be i n d i c a t i n g the 106 d e p th of the r e g i o n a l c u r r e n t system, and not t h a t of the c h a n n e l l e d c u r r e n t s . E s t i m a t e s of the l a t e r a l w i d t h of the anomaly might be made from the s p a t i a l e x t e n t of the anomalous f i e l d , as i n d i c a t e d on the c o n t o u r maps of the v e r t i c a l f i e l d ' s phase, or on c o n t o u r maps of the ampitude of the h o r i z o n t a l anomalous f i e l d . However, the c a l c u l a t e d f i e l d s (see F i g . 3.10) f o r the d i f f e r e n t c u r r e n t d e n s i t y models of F i g . 3.9 i l l u s t r a t e t h a t the i n d i c a t e d w i d t h i s a f u n c t i o n of d e p t h , as w e l l as the a c t u a l l a t e r a l e x t e n t . Thus, the t r u e w i d t h may be d i f f i c u l t t o e s t i m a t e i n t h i s manner, p a r t i c u l a r l y f o r deep a n o m a l i e s . A method used t o c a l c u l a t e the e x t e n t of anomalous b o d i e s has been t o s e l e c t a s t a r t i n g model t h a t q u a l i t a t i v e l y f i t s the d a t a , and then p e r t u r b the s i z e of the anomaly u n t i l a best f i t w i t h the da t a i s a t t a i n e d ( P o r a t h e t a l , 1970). As d i s c u s s e d p r e v i o u s l y , t h i s type of i n t e r a c t i v e m o d e l l i n g i s open t o problems of human b i a s , as w e l l as b e i n g e x p e n s i v e t o p e r f o r m . In summary of t h i s s e c t i o n , i t i s c l e a r t h a t the i n t r a c t a b i l i t y of the i n d u c t i o n problem f o r two and t h r e e d i m e n s i o n a l models s e v e r l y l i m i t s the e x t e n t of q u a n t i t a t i v e a n a l y s i s p o s s i b l e . The methods and e s t i m a t o r s used have been shown t o be of q u e s t i o n a b l e v a l i d i t y . 1 07 C h a p t e r I V The C u r r e n t D e n s i t y M o d e l 4. 1 T h e C u r r e n t D e n s i t y F o r m u l a t i o n D e t e r m i n i n g t h e e a r t h ' s c o n d u c t i v i t y a s a f u n c t i o n o f s p a t i a l p o s i t i o n i s t h e - u l t i m a t e g o a l i n G.D.S. a n a l y s i s . H o w e v e r , f o r r e a s o n s t o be o u t l i n e d s h o r t l y , a more u s e f u l a p p r o a c h i n many c a s e s i s t o c o n s i d e r t h e c u r r e n t d e n s i t y a s t h e m o d e l p a r a m e t e r t o be d e t e r m i n e d . I n a n i n i t i a l s i m p l i f i c a t i o n , o n l y two d i m e n s i o n a l m o d e l s w i l l be a l l o w e d , w i t h a l l c u r r e n t f l o w i n g o n l y i n t h e 'y' d i r e c t i o n . A l t h o u g h t h i s two d i m e n s i o n a l c o n s t r a i n t w i l l o b v i o u s l y n o t be a p p l i c a b l e i n a l l r e a l e x a m p l e s , a s u f f i c i e n t l y l a r g e number o f e l o n g a t e d a n o m a l i e s i n G.D.S. d o a p p e a r t o h a v e t h i s d e g r e e o f s y m m e t r y , m a k i n g i t a r e a s o n a b l e i n i t i a l s i m p l i f i c a t i o n ( G o u g h , 1 9 7 3 ) . T h e symmetry a s s u m p t i o n s e n s u r e t h a t a l i n e a r a r r a y p e r p e n d i c u l a r t o t h e s t r i k e o f t h e s t r u c t u r e i s s u f f i c i e n t t o m e a s u r e a l l n o n -r e d u n d a n t i n f o r m a t i o n a v a i l a b l e a t t h e e a r t h ' s s u r f a c e ( s e e F i g . 4 . 1 ) . I t w i l l be p r e s u m e d t h a t t h e s e p a r a t i o n o f t h e a n o m a l o u s a n d n o r m a l f i e l d s m e a s u r e d a l o n g t h i s a r r a y h a s b e e n p e r f o r m e d , 108 AIR S t a t i o n s - v ^ —•—x X *"* s\ /\ 7* X X X EARTH Densities F i g . 4.1 The presumed t w o - d i m e n s i o n a l model. The c o n d u c t i v i t y i s i n v a r i a n t i n the Y d i r e c t i o n (out of the p a g e ) , and a l l c u r r e n t s a r e assumed t o f l o w i n t h i s d i r e c t i o n . A l i n e a r a r r a y of magnetometers p e r p e n d i c u l a r t o the s t r i k e of th« model i s s u f f i c i e n t t o r e c o r d a l l non-redundant i n f o r m a t i o n a v a i l a b l e . 1 0 9 s o t h a t h e n c e f o r t h a l l d a t a w i l l be c o n s i d e r e d t o be p u r e l y a n o m a l o u s a n d i n t e r n a l . T h e g e n e r a l e x p r e s s i o n u s e d t o c a l c u l a t e t h e m a g n e t i c f i e l d d u e t o t h e c u r r e n t i n a g i v e n v o l u m e V i s t h e B i o t - S a v a r t law ( P a n o f s k y a n d P h i l l i p s , 1962, p g . 1 2 5 ) : where r i s t h e d i f f e r e n c e v e c t o r b e t w e e n t h e p o s i t i o n o f t h e ( 4 . 1 . 1 ) l i n e ^ c u r r e n t , ( x * , y ' , z ' ) a n d t h e p o s i t i o n a t w h i c h B i s m e a s u r e d , ( x , y , z ) : r = x, ( x - x 1 ) + X;, ( y - y ( ) + x ^ z - z 1 ) ( 4 . 1 . 2 ) T h e c o m p o n e n t s o f B ( x , y , z ) a r e t h u s : B x ( x , y , z ) = u D / 4 T r y [ [ { [ ( z - z 1 ) J 3 - ( y - y ' ) j ? ] / [ ( x - x 1 ) ^ V + ( y - y ' )"" + ( z - z ' )x 3 d x ' d y ' d z ( 4 . 1 . 3 ) B y ( x , y , z ) = ^ o / 4 r r [ [ f { [ ( x - x 1 ) j 4 - ( z - z ' ) j x ]/[ (x-x ' V 110 + ( y - y ) + ( z - z ) J dx dy d z (4.1.4) B z ( x , y , z ) = ^ o / 4 t v ] ' ( [ { [ ( y - y , ) j x - ( x - x , ) j,, ] / [ (x-x' ) Z V + ( y - y ' ) z + ( z - z ' J 7" ] d x ' d y ' d z ' ( 4 . 1 . 5 ) F o r t h e s i m p l i f i e d c u r r e n t d e n s i t y s y s t e m c o n s i d e r e d , j w i l l be i n t h e 'y' d i r e c t i o n o n l y , a n d i t s a m p l i t u d e w i l l n o t v a r y i n t h a t d i r e c t i o n . The v a l u e s o f B ( x , y , z ) w i l l be t a k e n o n l y a t z = 0. U s i n g t h e s e s i m p l i f i c a t i o n s , a n d i n t e g r a t i n g f r o m y =-cr> t o y =+co , t h e c o m p o n e n t s o f B ( x , y , 0 ) become: B x ( x , y , 0 ) = / i o / 2 T T ^ { [ - z ' j ^ ] / [ ( x - x 1 f + z , Z - ] } d x ' d z ' S ( 4 . 1 . 6 ) B y ( x , y , 0 ) = 0 ( 4 . 1 . 7 ) B z ( x , y , 0 ) = / V 2 t t ^ { [ ( x ' - x ) j l / I U-x^+z^Hdx'dz' S (4.1.8) w h e r e S i s t h e a r e a o f i n t e r e s t o u t s i d e o f w h i c h w i l l be p r e s u m e d t o be z e r o . I n f u t u r e , a s B x ( x , y , 0 ) a n d B z ( x , y , 0 ) do 111 n o t d e p e n d on y o r z , t h e y w i l l s i m p l y be r e f e r r e d t o a s B x ( x ) and B z ( x ) . The m a j o r a d v a n t a g e o f i n v e r t i n g t h e d a t a t o f i n d a c u r r e n t d e n s i t y m o d e l r a t h e r t h a n a c o n d u c t i v i t y m o d e l i s t h e a v o i d a n c e o f t h e i n t r a c t a b i l i t y o f t h e n o n - l i n e a r i n d u c t i o n e q u a t i o n . In t h e new f o r m u l a t i o n t h e m o d e l p a r a m e t e r j ( x , z ) i s l i n e a r l y r e l a t e d t o t h e d a t a a s shown i n e q u a t i o n s 4 . 1 . 6 and 4 . 1 . 8 . The f o r w a r d p r o b l e m e x p r e s s e d i n t h e s e e q u a t i o n s i s e a s i l y and c h e a p l y s o l v e d , a s i s shown i n s e c t i o n 4 .3 o f t h i s c h a p t e r . As w e l l , t h e s o l u t i o n o f t h e g e n e r a l l i n e a r i n v e r s e p r o b l e m i s v e r y w e l l u n d e r s t o o d , w i t h a v a r i e t y o f me thods and t e c h n i q u e s a v a i l a b l e i n t h e l i t e r a t u r e ( B a c k u s and G i l b e r t , 1 9 6 7 , 1 9 6 8 ; W i g g i n s , 1972; O l d e n b u r g , 1976; P a r k e r , 1 9 7 7 ) . A n o t h e r a d v a n t a g e t o u s i n g a c u r r e n t d e n s i t y mode l i s t h a t i t s v e r y f o r m u l a t i o n a v o i d s t h e p r o b l e m o f c u r r e n t c h a n n e l l i n g . The r e g i o n o f i n t e r e s t o f t h e m o d e l c a n be d e f i n e d p r e c i s e l y by u s i n g t h e g e o m e t r i c d e c a y o f t h e m a g n e t i c f i e l d away f r o m a l i n e c u r r e n t t o d e t e r m i n e t h e b o u n d a r i e s b e y o n d w h i c h c u r r e n t s w i l l h a v e n e g l i g i b l e e f f e c t a t t h e m e a s u r i n g s t a t i o n s . The p r o b l e m o f h a v i n g t o s o l v e o v e r a l a r g e and p o o r l y d e t e r m i n e d r e g i o n o f i n t e r e s t t o e n s u r e t h a t t h e e f f e c t s o f c h a n n e l l i n g o f t h e r e g i o n a l c u r r e n t f l o w a r e i n c l u d e d ( P o r a t h e t a l , 1971 ; G o u g h , 1973) w i l l s i m p l y n o t be e n c o u n t e r e d . The i n h e r e n t d i s a d v a n t a g e i n u s i n g t h e c u r r e n t d e n s i t y m o d e l i s i t s n o n - u n i q u e n e s s . I t ha s been p o i n t e d ou t e a r l i e r i n s e c t i o n 3.4 o f C h a p t e r I I I t h a t t h e f i e l d due t o a l i n e c u r r e n t 1 12 c a n be d u p l i c a t e d by an i n f i n i t e number o f d i f f e r e n t c u r r e n t d i s t r i b u t i o n s . As w e l l , i t i s p o s s i b l e f o r ' a n n i h i l a t o r d i s t r i b u t i o n s ' t o e x i s t w h i c h w i l l h a v e n o n - z e r o c u r r e n t d e n s i t y v a l u e s , b u t w h i c h w i l l n o t g i v e r i s e t o any m e a s u r a b l e s u r f a c e m a g n e t i c f i e l d . As f o r m u l a t e d , t h e c u r r e n t d e n s i t y i n v e r s e p r o b l e m c a n o n l y u t i l i z e t h e i n t e r n a l l y o r i g i n e d p o r t i o n o f t h e m e a s u r e d m a g n e t i c f i e l d , e f f e c t i v e l y i g n o r i n g t h e e x t e r n a l f i e l d r e q u i r e d t o i n d u c e t h e i n t e r n a l c u r r e n t s y s t e m . T h e d i s r e g a r d i n g o f t h i s i n f o r m a t i o n e x p l a i n s t h e d i f f e r e n c e i n u n i q u e n e s s d i f f i c u l t i e s b e t w e e n t h e c o n d u c t i v i t y a n d c u r r e n t d e n s i t y m o d e l a p p r o a c h e s . Th e c u r r e n t d e n s i t y m o d e l h a s b e e n u s e d b e f o r e ( B a n k s , 1979; Woods, 1979 ) , b u t t h e u n i q u e n e s s p r o b l e m h a s l e d t h e a u t h o r s t o c o n s t r a i n t h e c u r r e n t m o d e l s t o a n i n f i n i t e s m a l l y t h i n s h e e t a t some c o n s t a n t d e p t h . T h e t h i n s h e e t f o r m u l a t i o n w i l l p r o b a b l y g i v e a g o o d i n d i c a t i o n o f t h e h o r i z o n t a l p o s i t i o n o f t h e a c t u a l e a r t h c u r r e n t s , b u t i t c a n n o t g i v e a n y i n f o r m a t i o n a s t o t h e i r d e p t h s . As w e l l , t h e d e p t h o f t h e t h i n s h e e t must be c h o s e n w i t h some c a u t i o n , f o r a s shown i n A p p e n d i x F, i f t h e t h i n s h e e t i s d e e p e r t h a n t h e a c t u a l c u r r e n t , t h e n no c u r r e n t d i s t r i b u t i o n a l o n g i t w i l l be p o s s i b l e t h a t f i t s t h e d a t a . The c u r r e n t d e n s i t y i n v e r s e p r o b l e m l e f t s i m p l y a s o r i g i n a l l y s t a t e d i s p r o b a b l y h o p e l e s s l y n o n - u n i q u e . T h i s w i l l be i l l u s t r a t e d i n s e c t i o n 4.2 by B a c k u s - G i l b e r t t y p e a p p r a i s a l , a n d i n s e c t i o n 4.3 by t h e c o n s t r u c t i o n o f a v a r i e t y o f d i s s i m i l a r m o d e l s , a l l u n l i k e t h e o r i g i n a l m o d e l , b u t a l l o f 113 which match the d a t a . However, by imposing c e r t a i n e x p e c t e d p h y s i c a l f e a t u r e s of t h e e a r t h model as c o n s t r a i n t s i n the c o n s t r u c t i o n p r o c e s s , i t i s p o s s i b l e t o g r e a t l y r e s t r i c t the range of a l l o w a b l e models. I t w i l l be shown i n s e c t i o n 4.4 t h a t ' t r u e ' models which match th e s e p h y s i c a l r e q u i r e m e n t s are c l o s e l y r e c o v e r e d i n t h i s c o n s t r a i n e d c o n s t r u c t i o n . 4.2 Unigueness and B a c k u s - G i l b e r t A p p r a i s a l The statement t h a t a p a r t i c u l a r i n v e r s e problem i s non-unique i s r a t h e r n e b u l o u s , as t h e r e are v a r y i n g ' t y p e s ' and 'degrees' of non-uniqueness. In c e r t a i n problems the uniqueness of the d a t a s e t produced by a p a r t i c u l a r model can be p r o v e d , such as i n the g l o b a l G.D.S. problem ( B a i l e y , 1970) or i n h o r i z o n t a l l o o p e l e c t r o m a g n e t i c sounding over a s t r a t i f i e d e a r t h ( F u l l a g a r , 1981). However, i n both c a s e s , t o o b t a i n the unique model r e q u i r e s an i n f i n i t e amount of p e r f e c t l y a c c u r a t e d a t a . As t h i s c o n d i t i o n can never be reached i n p r a c t i c e , t h e r e w i l l a l w ays be a c e r t a i n range of a c c e p t a b l e v a l u e s f o r the model, even though t h e o r e t i c a l l y t h e r e i s a one t o one correspondence between models and d a t a s e t s . The non-uniqueness i n c u r r e d i n the above-mentioned cases would be e x p e c t e d t o be ' l e s s ' than t h a t i n c a s e s such as t h e c u r r e n t d e n s i t y problem, i n which an i n f i n i t e amount of p e r f e c t d a t a would s t i l l not guarantee a 1 1 4 unique model upon i n v e r s i o n . As w e l l , even though the t o t a l model may not be u n i q u e , t h e r e may be p o r t i o n s of i t , or f e a t u r e s i n i t , which a r e r e q u i r e d i n a l l models t h a t f i t the d a t a . Backus and G i l b e r t (1967,1968,1970) have o u t l i n e d the a p p r a i s a l p r o c e d u r e t o q u a n t i f y t h e s e d i f f e r e n t a s p e c t s of non-u n i q u e n e s s . In a l i n e a r problem, the model and d a t a can always be r e l a t e d by a Fredholm i n t e g r a l e q u a t i o n of the f i r s t k i n d ( P a r k e r , ( 1 9 7 7 ) : where: e- : i s the i datum g- : i s the k e r n e l f u n c t i o n a s s o c i a t e d w i t h the i t k datum m : i s the model R : i s the r e g i o n of i n t e r e s t o u t s i d e which the model i s c o n s i d e r e d t o be z e r o ( i n t h i s case R i s a s u r f a c e ) C o n s i d e r t h a t t h e r e a r e N d a t a , w i t h N a s s o c i a t e d l i n e a r l y independent k e r n e l f u n c t i o n s . A f u n c t i o n , d e s i g n a t e d an a v e r a g i n g f u n c t i o n , i s now c o n s t r u c t e d from a l i n e a r c o m b i n a t i o n of the k e r n e l f u n c t i o n s : (4.2.1) 115 N A ( r , T 0 ) = £4i(r0 ) g L ( r ) ( 4 . 2 . 2 ) s u c h t h a t t h e f u n c t i o n A i s u n i m o d u l a r ( i e . i t h a s u n i t a r e a ) a n d i s a s ' c l o s e ' a s p o s s i b l e t o a D i r a c d e l t a f u n c t i o n a t some d e s i g n a t e d p o s i t i o n r 0 . T h e m e a s u r e o f c l o s e n e s s t o a d e l t a f u n c t i o n c a n be d e f i n e d i n a number o f d i f f e r e n t ways ( B a c k u s a n d G i l b e r t , 1970; O l d e n b u r g , 1 9 7 6 ) ; t h e m e a s u r e u s e d h e r e i s t h e f i r s t D i r i c h l e t c r i t e r i o n ( O l d e n b u r g , 1 9 7 6 ) : S ( r 0 ) = £ $ [ A ( r , r 0 ) - <f( r-tQ ) ]*ds R ( 4 . 2 . 3 ) T h e a v e r a g e o f a m o d e l a t t h e p o s i t i o n r Q i s d e f i n e d a s : < m ( r 0 ) > = J r m ( r ) A ( r , r ^ ) d s R ( 4 . 2 . 4 ) I f t h e a v e r a g i n g f u n c t i o n A ( r , r c ) was e x a c t l y a D i r a c d e l t a f u n c t i o n , t h e n : < m ( r 0 ) > = $ } m ( r ) £ ( r , r 0 ) d s = m ( r 0 ) ( 4 . 2 . 5 ) —k s o t h a t t h e v a l u e o f t h e m o d e l a t t h e p o s i t i o n r c w o u l d h a v e b e e n e x a c t l y r e c o v e r e d . I n a n y e v e n t , i n d e p e n d e n t o f t h e f o r m o f A ( r , r ^ , ) , t h e v a l u e s o f t h e a v e r a g e s < m ( r 0 ) > a r e u n i q u e t o t h e p r o b l e m , a s t h e y a r e d e p e n d e n t o n l y on t h e v a l u e s o f t h e c o e f f i c i e n t s a n d on t h e d a t a : 1 1 6 <m(r^)> = f ^ m ( r ) A ( r ( r 6 ) d s = $ £ m ( r ) £ o < . ( r 0 ) g - ( r ) d s M = ^ ^ ( r e ) [ f [ m ( r " ) g - ( r ) d s ] = ( r o)e-( 4 . 2 . 6 ) T h u s , t h e a v e r a g e s must be t h e same f o r a l l m o d e l s , i n c l u d i n g t h e t r u e m o d e l , a n d t h e r e f o r e c o m p l e t e l y c o d i f y o u r u n i q u e i n f o r m a t i o n a b o u t t h e p r o b l e m . I f t h e a v e r a g i n g f u n c t i o n f o r a p o s i t i o n r c i s v e r y c l o s e t o a d e l t a f u n c t i o n a t t h a t p o s i t i o n , t h e n i t s u n i t a r e a e n s u r e s t h a t t h e m o d e l a v e r a g e < m ( r c ) > i s p r o b a b l y v e r y c l o s e t o t h e t r u e . m o d e l v a l u e a t t h a t p o i n t . As w e l l , a s a l l m o d e l s h a v e t h e same u n i q u e v a l u e o f t h e a v e r a g e t h e n t h e r a n g e o f m o d e l v a l u e s a t t h i s p o s i t i o n w i l l be n e c c e s s a r i l y l i m i t e d . By c o n s t r u c t i n g a v e r a g i n g f u n c t i o n s a t a v a r i e t y o f p o s i t i o n s i n t h e r e g i o n o f i n t e r e s t R, one c a n q u i c k l y d e t e r m i n e t h o s e p o s i t i o n s a t w h i c h a l l m o d e l s w i l l be s i m i l a r , a n d t h o s e a t w h i c h t h e r a n g e o f p o s s i b l e m o d e l v a l u e s w i l l be l e s s c o n s t r a i n e d . —X I n p r a c t i c e , b e c a u s e t h e a v e r a g e s < m ( r & ) > w i l l a l w a y s be c a l c u l a t e d a s a l i n e a r c o m b i n a t i o n o f t h e d a t a ( a s i n e q u a t i o n 4 . 2 . 6 ) , t h e e r r o r s i n t h e d a t a w i l l i n t r o d u c e e r r o r i n t o t h e a v e r a g e s . L e t t h e o b s e r v e d v a l u e o f t h e d a t a be e ^ , t h e t r u e v a l u e , e;* , a n d t h e e r r o r i n t h e o b s e r v e d v a l u e , <fet, s u c h t h a t : 117 (4.2.7) S i m i l a r l y , we w i l l have f o r the averages: <m(r D)> = <m(r0 )>* + £<m(r^)> (4.2.8) The assumption w i l l be made that the e r r o r s are Gaussian, so that the expected value and c o v a r i a n c e of the e r r o r terms are: E[<fe-] = 0 (4.2.9) COVfcfe- J e j ] = E[cf e Lfe - ] (4.2.10) We w i l l a l s o presume the e r r o r s are u n c o r r e l a t e d so that the c o v a r i a n c e matrix, C, reduces to a d i a g o n a l matrix, with the d i a g o n a l elements being given by the square of the standard e r r o r 0"c of each datum e^: Using 4.2.7 i n equation 4.2.6, we have: (4.2.11) 118 < m ( r D ) > = £ ^ - e L KJ hi i - l t= i = < m < r 0 )> + £*:«re<. (4.2.12) T h u s N <f<m(r 0 )> = ^oiLfe^ (4.2.13) T h e e x p e c t e d v a l u e o f t h e e r r o r i n t h e a v e r a g e i s : E [ <m ( r0 ) > ] = E [ <f <m ( ) > ] = E [ f e : ) = 0 T h e v a r i a n c e o f t h e a v e r a g e t h e n b e c o m e s V A R [ < m ( r 0 )>] = VAR[<f<m( r 0 ) > ] (4.2.14) 1 19 N N = E[{£*JeL)(£o<j£eS)] M N/ M N « - x , J 5 ' i=, ( 4 . 2 . 1 5 ) The f u t u r e a l g e b r a o f t h e a v e r a g i n g f u n c t i o n d e t e r m i n a t i o n w i l l be g r e a t l y s i m p l i f i e d i f we now r e f o r m u l a t e t h e p r o b l e m by n o r m a l i z i n g w i t h r e s p e c t t o t h e s t a n d a r d e r r o r s , c?c : ec(r) = e ' ( r ) / ^ = f $ { [ m ( r )g{.(r) ] / o i }ds = ^  m(r ) G C ( r ) d s R ( 4 . 2 . 1 6 ) V A R [ < m ( r c ) > ] = ^<^-VAR[ e^ ( r ) ] (4.2.17.) B a c k u s a n d G i l b e r t ( 1 9 7 0 ) show t h a t r e s o l u t i o n ( a s q u a n t i f i e d by an y s u i t a b l e ' d e l t a - n e s s ' c r i t e r i o n ) a n d a c c u r a c y a r e m u t u a l l y a n t a g o n i s t i c p r o p e r t i e s , s o t h a t a n y i m p r o v e m e n t i n one n e c c e s s a r i l y d e g r a d e s t h e o t h e r . T h u s , i n e a c h a p p l i c a t i o n o ne must c h o o s e t h e c o e f f i c e n t s °<i s o a s t o o b t a i n t h e op t i m u m ' t r a d e - o f f b e t w e e n r e s o l u t i o n a n d a c c u r a c y . T o a l l o w a 1 20 c o n t i n u o u s l y v a r i a b l e t r a d e - o f f the parameter © i s introduced. The c h o i c e of G ,between 0 and TT/2, w i l l determine the r e l a t i v e emphasis on r e s o l u t i o n or accuracy, when the c o e f f i c i e n t s are c a l c u l a t e d by minimizing the f o l l o w i n g o b j e c t i v e f u n c t i o n with respect to each o<L: Y ( r 0 ) = c o s © S ( r ^ ) + s i n 9 VAR[<m(r0 )>] (4.2.18) Backus and G i l b e r t (1970) prove that the averaging f u n c t i o n obtained i n t h i s way (where S would be any s u i t a b l e 'closeness' c r i t e r i o n ) w i l l have the lowest p o s s i b l e value of the varia n c e for a given value of the r e s o l u t i o n . The t r a d e - o f f curve of r e s o l u t i o n and va r i a n c e as a f u n c t i o n of & thus obtained w i l l be the optimal curve for the problem. A value of © = 0 w i l l s t r i c t l y minimize the 'closeYiess' of A ( r , r 0 ) to a d e l t a -f u n c t i o n , and the values of °<L determined w i l l r e s u l t in maximum r e s o l u t i o n , but with a consequent maximum in the v a r i a n c e . On the other hand, 9 = Tr/2 w i l l ensure that the vari-ance i s a minimum, but now with r e s o l u t i o n a l s o at i t s minimum. We add the unimodular c o n d i t i o n to the problem by the method of Lagrange m u l t i p l i e r s , so that our o b j e c t i v e f u n c t i o n becomes: 121 V ( r 0 ) = 00565 (4) + s i n £ v A R [ < m ( r 0 )>] + 2B [ 1 ( r ) d s ] R ( 4 . 2 . 1 9 ) w i t h t h e v a l u e s o f °(i s a t i s f y i n g : 1 ^ G i ( r ) ] d s = £ * < L ^ G - ( r ) d s ( 4 . 2 . 2 0 ) M i n i m i z i n g w i t h r e s p e c t t o e a c h g i v e s : Vfy3c<;= 0 = c o s© ( } S ( r e ) / X L ) + sin6 ( V*<C){VAR[S<in(r 0' )>]} - 2 | a j y [ } A ( r f r 0 )/M,)ds R ( 4 . 2 . 2 1 ) S u b s t i t u t i n g t h e e x p r e s s i o n s f o r S ( r D ) , V A R [ £ < m ( r e ) > ] , a n d A ( r , r 0 ) f r o m e q u a t i o n s 4.2.3, 4 . 2 . 1 5 , a n d 4.2.2 i n t o e q u a t i o n 4.2.19 we o b t a i n f o r e a c h o(; : ' -M cos0{ [£<*j f ^ G j ( r ) G ^ ( r ) d s ] - G L ( r 0 ) } + s i n e - * : - p ^ G ^ r J d s = 0 R ( 4 . 2 . 2 2 ) To u s e s i m p l e r m a t r i x n o t a t i o n , l e t : 122 ^ G 0 ( r ) G L ( r ) d s = Pj = R (4.2.23) be an element of the NxN inner product matrix P , and: ^ G L ( r ) d s = rjL (4.2.24) —* ~i be an element of the 1xN v e c t o r U, and G ^ ( r 0 ) and °<i be elements of the 1xN v e c t o r s G 0 , and <\ r e s p e c t i v e l y . Thus, c o n s i d e r i n g the m i n i m i z a t i o n with respect to each of the <<i's as i n 4.2.21 the f i n a l matrix r e l a t i o n i s : cos0[^-P-G o] + sin9-<* - pu = 0 (4.2.25) The unimodular c o n s t r a i n t i s now: _» -» o( • U = 1 (4.2.26) The inner product matrix, P , i s o b v i o u s l y symmetric, and i s a l s o p o s i t i v e d e f i n i t e : 1 i P- x T =^^X- P; X = i i J f ( G L ( r ) x l ) ( G J ( r ) X j )ds L j R 123 = ff [<CGi ( r ) x - X^Gj ( r ) X j ) ] d s R. L j = ( ^ G L ( r ) x L ) 2 ds > 0 (4 . 2 . 2 7 ) Thus, r can be d i a g o n a l i z e d ( P a r k e r , 1 9 7 7 ) : r = R A R T ( 4 . 2 . 2 8 ) where R and R are o r t h o g o n a l NxN m a t r i c e s such t h a t : R- = R T ( 4 . 2 . 2 9 ) and A i s an NxN d i a g o n a l m a t r i x c o n t a i n i n g the e i g e n v a l u e s of P , where a l l e i g e n v a l u e s a r e g r e a t e r than z e r o . U s i n g t h i s e x p a n s i o n , m u l t i p l y i n g both s i d e s by R and u t i l i z i n g e q u a t i o n 4 . 2 . 2 9 , 4 . 2 . 2 5 becomes: c o s 9 [ £ - R - A - G • R] + s i n 6 * R - B U -R = 0 ( 4 . 2 . 3 0 ) The m a t r i x P can be c o n s i d e r e d t o both ' r o t a t e ' and ' s t r e t c h ' t h e components of an a r b i t r a r y v e c t o r ; the d e c o m p o s i t i o n of P i n e q u a t i o n 4 . 2 . 2 7 s e p a r a t e s t h e s e two o p e r a t i o n s , w i t h the d i a g o n a l m a t r i x A b e i n g pure ' s t r e t c h i n g ' , and R and R T b e i n g pure r o t a t i o n s . We denote the r e s u l t a n t v e c t o r s of the r o t a t i o n of <*,G0, and U by R a s : 124 e< = oC- R ( 4 . 2 . 3 1 ) ( 4 . 2 . 3 2 ) U = U-R ( 4 . 2 . 3 3 ) U s i n g t h e s e r e l a t i o n s a n d c o l l e c t i n g t e r m s i n e q u a t i o n 4.2.30, t h e r o t a t e d c o e f f i c i e n t s f o r t h e a v e r a g i n g f u n c t i o n a r e f o u n d t o b e : c< = [BU + c o s © G 0 ] - D ( 4 . 2 . 3 4 ) w h e r e D i s t h e d i a g o n a l m a t r i x : D = c o s 6 A + s i n G l ( ( 4 . 2 . 3 5 ) R o t a t i n g two v e c t o r s w i l l n o t c h a n g e t h e i r i n n e r p r o d u c t , s o t h e u n i m o d u l a r c o n d i t i o n o f e q u a t i o n 4.2.26 i s n o t c h a n g e d : 125 oC . U = << • TJ = 1 ( 4 . 2 . 3 6 ) T h i s e q u a t i o n a l l o w s t h e v a l u e o f t h e L a g r a n g e m u l t i p l i e r |3> t o be c a l c u l a t e d : p = [ 1 -cos6c?0- D '• U ] / [ U - D-1- U] ( 4 . 2 . 3 7 ) T h e v a l u e s o f t h e a v e r a g i n g f u n c t i o n c o e f f i c i e n t s a r e t h e n f o u n d by r o t a t i n g e q u a t i o n 4 . 2 . 3 4 w i t h t h e m a t r i x R T : c< = c<- R = tpU + c o s S S 0 ]D • R r ( 4 . 2 . 3 8 ) T h e u t i l i t y o f t h e d i a g o n a l i z a t i o n a p p r o a c h i s now a p p a r e n t , a s t h e d e c o m p o s i t i o n o f t h e i n n e r p r o d u c t m a t r i x n e e d be done o n l y o n c e f o r a l l v a l u e s o f © a n d r ^ , w h e r e a s o t h e r w i s e a m a t r i x i n v e r s i o n w o u l d be r e q u i r e d f o r e a c h d i f f e r e n t v a l u e . I n t h e two d i m e n s i o n a l c u r r e ' n t d e n s i t y p r o b l e m t h e two f o r m s o f t h e k e r n e l s r e l a t i n g t h e d a t a a n d t h e m o d e l f o r d i f f e r e n t s t a t i o n p o s i t i o n s a r e g i v e n i n e q u a t i o n s 4 . 1 . 6 a n d 4 . 1 . 8 . F o r a g i v e n a r r a y o f s t a t i o n s o v e r a r e g i o n o f i n t e r e s t R, t h e c a l c u l a t i o n o f t h e a v e r a g i n g f u n c t i o n a t a p o s i t i o n ( x 0 , z o ) a n d a n y © p r o c e e d s i n t h e same manner a s j u s t o u t l i n e d . T h e e x a c t m e t h o d s u s e d t o c a l c u l a t e t h e i n t e g r a l s n e c c e s s a r y f o r d e t e r m i n a t i o n o f t h e e l e m e n t s o f t h e i n n e r p r o d u c t m a t r i x P a n d t h e e l e m e n t s o f t h e v e c t o r U a r e g i v e n i n A p p e n d i x G. D i s p l a y e d i n F i g . 4 .2 a r e t h e a v e r a g i n g f u n c t i o n s c a l c u l a t e d i n t h e a b o v e 1 26 F i g . 4.2 A v e r a g i n g f u n c t i o n s c a l c u l a t e d f o r the t w o - d i m e n s i o n a l c u r r e n t d e n s i t y problem. The a v e r a g i n g f u n c t i o n s have been c a l c u l a t e d a t maximum r e s o l u t i o n ( Q = 0) on the t r a d e - o f f c u r v e , and a r e thus as ' c l o s e ' as p o s s i b l e t o a D i r a c d e l t a f u n c t i o n a t the p o i n t s : (a) (1,25) (b) (10,25) Fin. 4 . 2 1 27 (a) AVERAGING FUNCTION XLOC ~ 25 ZLOC ~ ] 6 = 0 . 0 0 0 Cb) AVERAGING FUNCTION XLOC = 25 ZLOC = 10 9 = o .ooo 128 manner f o r two p o s i t i o n s i n t h e r e g i o n o f i n t e r e s t R. B o t h a v e r a g i n g f u n c t i o n s h a v e b e e n c a l c u l a t e d a t t h e h i g h e s t r e s o l u t i o n , 0 = 0 . I t i s n o t e d i m m e d i a t e l y t h a t t h e s e f u n c t i o n s a r e v e r y s p r e a d o u t , p a r t i c u l a r l y w i t h r e s p e c t t o d e p t h , z , a n d t h u s show v e r y l i t t l e r e s e m b l a n c e t o t h e d e l t a f u n c t i o n t h e y a r e t o e m u l a t e . The r e s o l u t i o n a t a n y o t h e r v a l u e o f © w o u l d be e v e n w o r s e . I t i s a l s o c l e a r t h a t t h e p e a k o f t h e a v e r a g i n g f u n c t i o n i s much more l o c a l i z e d i n t h e 'x' d i r e c t i o n t h a n i n t h e ' z ' d i r e c t i o n , i n d i c a t i n g t h a t t h e h o r i z o n t a l r e s o l u t i o n o f t h e t r u e m o d e l ' s f e a t u r e s w i l l be s u p e r i o r t o t h e r e s o l u t i o n i n d e p t h . A more c r i t i c a l f e a t u r e o f t h e a v e r a g i n g f u n c t i o n s i s t h e s h i f t o f t h e p e a k o f t h e f u n c t i o n t o w a r d t h e s u r f a c e away f r o m i t s c a l l e d f o r p o i n t , t h i s b e i n g p a r t i c u l a r l y n o t i c e a b l e i n F i g . 4.2b. T h i s l a r g e b i a s i n t h e p o s i t i o n o f t h e a v e r a g i n g f u n c t i o n s c l e a r l y i n v a l i d a t e s t h e c o n c e p t o f t h e a v e r a g e a s b e i n g a moment o f t h e m o d e l a r o u n d t h e d e s i r e d p o i n t . T h u s , i n t h e s e c a s e s , t h e a c t u a l a v e r a g e s w o u l d h o l d l i t t l e m e a n i n g . A"s w e l l , i t f u r t h e r p o i n t s o u t t h e e x t r e m e l a c k o f d e p t h r e s o l u t i o n . The b i a s i s due t o t h e d e c r e a s e i n t h e r e q u i r e d m a g n i t u d e o f c u r r e n t s i n m o d e l s f i t t i n g t h e d a t a w h i c h a r e c o n c e n t r a t e d c l o s e r t o t h e s u r f a c e . T h i s i s a f u n d a m e n t a l p r o p e r t y o f t h i s c u r r e n t d e n s i t y f o r m u l a t i o n , a n d a s s u c h , i s u n a v o i d a b l e t h r o u g h c h a n g e s s u c h a s a r r a y d e s i g n , e t c . T h e c o m p u t e d a v e r a g i n g f u n c t i o n s f o r t h e c u r r e n t d e n s i t y p r o b l e m h a v e t h u s shown t h a t t h e n o n - u n i q u e n e s s o f t h e p r o b l e m i s i n d e e d s e v e r e . H o w e v e r , a l t h o u g h t h e r e s o l u t i o n o f t h e t r u e 1.29 m o d e l ' s f e a t u r e s w i t h r e s p e c t t o d e p t h i s i n d i c a t e d t o be e x t r e m e l y p o o r , t h e r e s h o u l d be f a i r h o r i z o n t a l r e s o l u t i o n . 4 . 3 C o n s t r u c t i o n : T h e S m a l l e s t M o d e l A c u s t o m a r y f i r s t m o d e l t o c o n s t r u c t when i n v e r t i n g d a t a i s one w i t h minimum s t r u c t u r e . H e r e , t h a t f i r s t m o d e l w i l l be t h e s m a l l e s t m o d e l , i n a l e a s t s q u a r e s s e n s e , w h i c h f i t s t h e d a t a . T h e o b j e c t i v e f u n c t i o n t o be m i n i m i z e d i s : (£(J) = ^ | J ( x ' , z ' ) | Z d x ' d z ' ( 4 . 3 . 1 ) To e n s u r e t h a t t h e c u r r e n t d e n s i t y m o d e l J f i t s t h e d a t a , t h e d a t a e q u a t i o n s a r e a d d e d t o t h e o b j e c t i v e f u n c t i o n v i a t h e L a g r a n g e m u l t i p l i e r s , pi: <f) ( J ) = J J | J (x ' , z ' ) | d x ' d z ' R N + 2 ^ ^ [e. - JJG- (X ' , Z ' ) J ( X * , Z * ) d x ' d z ' <--' R ( 4 . 3 . 2 ) H o w e v e r , t h e s m a l l e s t m o d e l f o u n d by m i n i m i z i n g Cj^(J) i n e q u a t i o n 4 . 3 . 2 w o u l d n o t be p h y s i c a l l y r e a s o n a b l e , a s t h e ' s m a l l e s t ' c r i t e r i o n w o u l d e n s u r e t h a t a l l t h e c u r r e n t s w o u l d c o n g r e g a t e i n t h e u p p e r m o s t p o r t i o n o f t h e r e g i o n R. To o f f s e t 130 t h i s e f f e c t , a w e i g h t i n g f u n c t i o n , W ( x , z ) i s i n t r o d u c e d i n t o t h e o b j e c t i v e f u n c t i o n : <^(J) = ^ W(x* , z ' ) | J ( x ' , z ' ) | d x ' d z ' + 2 ^ |3- [ e L - G ^ ( x ' , z ' ) j ( x ' , z ' ) d x ' d z ' ] ( 4 . 3 . 3 ) T h e e x a c t f o r m o f t h e w e i g h t i n g w i l l be d e a l t w i t h l a t e r . M i n i m i z i n g t h e o b j e c t i v e f u n c t i o n w i t h r e s p e c t t o an a r b i t r a r y i n f i n i t e s m a l p e r t u r b a t i o n o f t h e m o d e l g i v e s : (f>(J+£j) - ( J ) = 0 = 2 Jfw(x' , z' ) J ( x ' , z ' )«fj(x' , z ' ) d x ' d z ' -2 £ B ' ^ j d ' J ( x ' , z' ) G C ( x ' , z' ) d x ' d z ' * ( 4 . 3 . 4 ) As c f j i s a r b i t r a r y , t h i s r e q u i r e s t h a t : J ( x , z ) = £ )?• G: ( X , Z ) / W ( X , Z ) ( 4 . 3 . 5 ) H o w e v e r , t h e m o d e l must a l s o s a t i s f y t h e d a t a , s o t h a t f o r a l l v a l u e s o f ' j ' : 131 ej = J ( x ' ,z' )Gj (x' ,z' )dx'dz' N r r = £ B- )\ [Gj (x' , 2 ' ) G c ( x ' , 2 ' ) / W ( x ' , 2 ' ) ] d x ' d 2 ' (4.3.6) or i n m a t r i x form: (4.3.7) where: R l ' = = ^ [ G^  (x ' , 2 ' )G^ (x ' , z ' ) /W(x ' , 2 1 ) ]dx'dz' (4.3.8) and p i s the v e c t o r c o n t a i n i n g the Lagrange m u l t i p l i e r s , fi . 1^  i s symmetric and p o s i t i v e d e f i n i t e , so t h a t P ' can be d i a g o n a l i 2 e d : e = B • R - A R T I A . " V <"V-(4.3.9) Thus, the v a l u e s of fi a r e e a s i l y found: B = e • R A 'R (4.3.10) a l l o w i n g the model J ( x , 2 ) t o be c a l c u l a t e d from e q u a t i o n 4.3.5. The r e q u i r e d form of the w e i g h t i n g f u n c t i o n W (x rz) must now be d e t e r m i n e d . C o n s i d e r two l i n e c u r r e n t s , I, , I a , of e q u a l magnitude, but a t d i f f e r e n t p o s i t i o n s , ( x , , z , ) and ( x 2 , z 2 ) 1 32 r e s p e c t i v e l y (see F i g . 4.3). The r a t i o of the Bx component of the f i r s t c u r r e n t t o t h a t of the second a t a s t a t i o n ( x o , 0 ) i s : Bx,/Bx 2 = { [ ( x 1 - x 0 ) 2 + z 1 1 n / z^.U . / K x . - x ^ + z , 1 ]} (4.3.11) T h i s r a t i o c o u l d be made u n i t y by m u l t i p l y i n g the c u r r e n t s I, , I 2 v X by the weight f a c t o r s W, , and Wj : W* (x , z ) = zi/[ (x- - x 0 ) l +zL* ] (4.3.12) where i = 1 or 2. F o l l o w i n g the same r e a s o n i n g , t he w e i g h t i n g f u n c t i o n f o r the Bz component i s found t o be: W L*(x*,z) = ( x - - x 0 ) / [ ( x L - x e ) l + Z i * ] (4.3.13) Thus, the w e i g h t i n g f u n c t i o n i s d i f f e r e n t f o r the two components. As w e l l , f o r each s t a t i o n p o s i t i o n x D the w e i g h t i n g f u n c t i o n s would change. C l e a r l y t h e r e i s no u n i v e r s a l w e i g h t i n g f u n c t i o n which w i l l a p p l y i n a l l c a s e s . For t h i s reason an approximate w e i g h t i n g f u n c t i o n t o o f f s e t the depth e f f e c t s o n l y i s s u g g e s t e d : W(x , z) = z * (4.3.14) where 'z' i s the depth of the l i n e c u r r e n t , and i s some w e i g h t i n g f a c t o r . At l a r g e v a l u e s of the s t a t i o n p o s i t i o n x 0 , 133 F i g . 4.3 D e t e r m i n a t i o n o f t h e a p p r o p r i a t e w e i g h t i n g f a c t o r t o o f f s e t t h e g e o m e t r i c d e c a y i n t h e m a g n e t i c f i e l d away f r o m a l i n e c u r r e n t . 134 the a s y m p t o t i c v a l u e s of the weghting f u n c t i o n s of e q u a t i o n s 4.3.5 and 4.3.6 a r e p r o p o r t i o n a l t o 1/z and 1/z 1 r e s p e c t i v e l y , so t h a t a r e a s o n a b l e v a l u e of o< i n e q u a t i o n 4.3.14 s h o u l d be between 1 and 2. The n u m e r i c a l i n t e g r a t i o n t e c h n i q u e used t o o b t a i n the elements of P f o r the a v e r a g i n g f u n c t i o n s of the p r e v i o u s s e c t i o n i s e a s i l y adapted t o i n c l u d e the v a r i a b l e w e i g h t i n g f u n c t i o n i n e q u a t i o n 4.3.14 (see Appendix G ) , so t h a t the c a l c u l a t i o n of the s m a l l e s t models f o r a g i v e n d a t a set r e q u i r e s l i t t l e e x t r a programming. The a r t i f i c i a l d a t a t o be used was g e n e r a t e d from p a r a m e t e r i z e d c u r r e n t d e n s i t y models (see F i g . G.1). The f i e l d components, Bx,Bz, a t any s t a t i o n p o s i t i o n due t o a r e c t a n g u l a r g r i d element of c o n s t a n t c u r r e n t can be c a l c u l a t e d i n c l o s e d form (see Appendix G). Because of the l i n e a r i t y of the e q u a t i o n s i n v o l v e d , the c o n t r i b u t i o n s t o the s u r f a c e f i e l d from each g r i d element can then be summed, g i v i n g the r e s u l t a n t f i e l d due t o the c u r r e n t model. U s i n g these d a t a s e t s , ' s m a l l e s t ' models were then c o n s t r u c t e d u s i n g the o u t l i n e d method, f o r d i f f e r e n t v a l u e s of the weight f a c t o r , <X . The f i r s t ' t r u e ' model used t o generate a r t i f i c i a l d a t a was t h a t shown i n F i g . 4.4a. The models c o n s t r u c t e d u s i n g t h i s d a t a , f o r w e i g h t i n g s of o( = 0, c< = 1, and <K = 2, are i n F i g . 4.4b,c,and d r e s p e c t i v e l y . In the unweighted case (°< = 0 ) , the c u r r e n t s , as e x p e c t e d , a r e c o n c e n t r a t e d a t the s u r f a c e . However, the c a l c u l a t e d model does i n d i c a t e the p r o p e r h o r i z o n t a l p o s i t i o n of the o r i g i n a l model. In the second and t h i r d examples 135 F i g . 4.4 L^-Norm s m a l l e s t model c o n s t r u c t i o n . The v a r i o u s models are : (a) the true model (b) model c o n s t r u c t e d at *< = 0. (c) model c o n s t r u c t e d at °< = 1 . (d) model c o n s t r u c t e d at <<= 2. co 1 37 the w e i g h t i n g s a re i n c r e a s e d from oi = 1 t o «< = 2, and the c u r r e n t s i n t h e models i n each case a r e 'pushed' deeper. U n f o r t u n a t e l y , the models s t i l l show l i t t l e resemblance t o the t r u e model, a l t h o u g h they a g a i n have a c c u r a t e l y i n d i c a t e d the ' t r u e ' h o r i z o n t a l p o s i t i o n . As w e l l , t h e r e i s c l e a r l y no i n d i c a t i o n i n the c a l c u l a t e d models of the depth a t which the t r u e model might be. To f u r t h e r check the apparent a b i l i t y t o h o r i z o n t a l l y r e s o l v e f e a t u r e s , the model i n F i g . 4.5a was used as the t r u e model, w i t h the c a l c u l a t e d models f o r the d i f f e r e n t v a l u e s of °i b e i n g i n F i g . 4.5b,c,and d. Once a g a i n the unweighted model i s c o n c e n t r a t e d a t the s u r f a c e . I t i s noteworthy i n t h i s case t h a t the h o r i z o n t a l f e a t u r e s of the model a r e q u i t e p o o r l y r e s o l v e d , which suggest t h a t the sheet c u r r e n t models used by Banks (1979) arid Woods (1979) might run i n t o s i m i l a r d i f f i c u l t i e s . The weighted model c o n s t r u c t i o n s f o r t h i s example q u i t e c l e a r l y show the h o r i z o n t a l p o s i t i o n s of the t r u e model c u r r e n t s , but a g a i n g i v e no i n d i c a t i o n of the depth. The f i n a l model c o n s t r u c t i o n was a f u r t h e r t e s t of the presumed l a c k of v e r t i c a l r e s o l u t i o n . The t r u e model i s shown i n F i g . 4.6a, and the weighted models a r e i n F i g . 4.6b,c,and d. The r e s u l t s c o n f i r m the p r e v i o u s c o n c l u s i o n s , as the two l i n e c u r r e n t s a r e not r e s o l v e d a t any w e i g h t i n g v a l u e . The weighted s m a l l e s t model c o n s t r u c t i o n s done here have s u b s t a n t i a t e d the p r e d i c t i o n s made from the B a c k u s - G i l b e r t a p p r a i s a l . The g e n e r a l h o r i z o n t a l f e a t u r e s of the t r u e model a r e a l l r e p r o d u c e d t o some e x t e n t i n the c a l c u l a t e d model, whereas 138 F i g . 4 .5 L^-Norm smallest model construction. The various models are: (a) the true model (b) model constructed at <* = 0. (c) model constructed at <* = 1 . o (d) model constructed at <* = 2. 139 140 F i g . 4.6 L £-Norm s m a l l e s t model c o n s t r u c t i o n . The v a r i o u s models a r e : (a) the true model (b) model c o n s t r u c t e d at «< = 0. (c) model c o n s t r u c t e d at °C = 1 . (d) model c o n s t r u c t e d at = 2. G O 142 the v e r t i c a l f e a t u r e s a r e not r e s o l v e d a t a l l . 4.4 C o n s t r a i n e d Model C o n s t r u c t i o n s U s i n g L i n e a r Programming I t has been shown i n the p r e v i o u s s e c t i o n s t h a t the s u r f a c e data do not l i m i t the range of p o s s i b l e models enough t o g i v e v e r t i c a l r e s o l u t i o n of the t r u e model's f e a t u r e s . To f u r t h e r r e s t r i c t the range of p e r m i s s i b l e models, i t i s s u g g e s t e d t h a t e x p e c t e d p h y s i c a l f e a t u r e s of the model be i n c o r p o r a t e d , or f a v o u r e d , i n the model p o h s t r u c t i o n . In a g r e a t number of c a s e s the a nomalies which a t t r a c t a t t e n t i o n i n G.D.S. a r e those which appear t o be v e r y l o c a l i z e d , as t h e s e suggest i n t e r e s t i n g g e o p h y s i c a l s t r u c t u r e s such as geothermal h o t s p o t s , a n c i e n t c r a t o n b o u n d a r i e s , f a u l t zones, e t c . T h i s l o c a l i z e d t y pe of model c o n s i s t i n g o n l y of a few l a r g e model elements w i l l not be f a v o u r e d by the l e a s t - s q u a r e s L 2-norm used i n the s m a l l e s t model c a l c u l a t i o n , which i s c l e a r from the f l a t t e n e d , spread-out models c a l c u l a t e d i n the p r e v i o u s s e c t i o n . The norm which most f a v o u r s c o n s t r u c t i o n of the d e s i r e d s p a r s e , l o c a l i z e d models i s the L,-norm (Levy and F u l l a g a r , 1981). A n o t h e r p h y s i c a l f e a t u r e i s t h a t no c u r r e n t s i n the model w i l l be e x p e c t e d t o be more than IT/2 d i f f e r e n t i n phase. From e q u a t i o n B.9 i n Appendix B, the complex v e r t i c a l wavenumber i n a homogenous e a r t h i s : 143 = i ( ^ + i w ^ c r ) " 1 -(4.4.1) where i s the h o r i z o n t a l wavenumber, 2rt"/;\ (as d e f i n e d i n s e c t i o n 2.1 of Chapter I I ) . The r e a l p a r t of k J e w i l l d e f i n e the o s c i l l a t o r y or ' t r a v e l l i n g ' n a t u r e of the wave, so t h a t the depth at which the wave has t r a v e l l e d a q u a r t e r wavelength from the s u r f a c e i s g i v e n by: z'/4 = ™ / 2 {REAL[i(^> v+ ivft^f1]} (4.4.2) The v a l u e of z y + w i l l i n c r e a s e w i t h i n c r e a s i n g "v , so i t w i l l a lways be t r u e t h a t : z, 4 > IT/2 (2/w/U 0 C T )" 1 = 1 . 57 <f (4.4.3) where tT i s the s k i n d e p t h . Thus, the induced c u r r e n t s t h a t a r e more than out of phase w i t h those a t the s u r f a c e w i l l be i n s i g n i f i c a n t , as they a r e deeper than the s k i n d e pth of the i n d u c i n g wave. ( I t i s noted t h a t i n the case of c u r r e n t s t h a t are c h a n n e l l e d l a r g e d i s t a n c e s v e r t i c a l l y upward, t h i s argument w i l l break down.) To r e i t e r a t e , the two e x p e c t e d p h y s i c a l f e a t u r e s of the model a r e : 1. The c u r r e n t model w i l l be s p a r s e , t h a t i s , i t w i l l 144 c o n s i s t of v e r y l o c a l i z e d c u r r e n t e l e m e n t s . 2. No c u r r e n t s i n the model w i l l be more than fr/2 out of phase. Both of these c o n s t r a i n i n g f e a t u r e s a r e e a s i l y a p p l i e d u s i n g l i n e a r programming, w i t h an L t-norm o b j e c t i v e f u n c t i o n t o be m i n i m i z e d . The c u r r e n t d e n s i t y model i s p a r a m e t e r i z e d i n t o an N x M g r i d of r e c t a n g u l a r e l e m e n t s , w i t h each h a v i n g a c o n s t a n t c u r r e n t w i t h i n them (see F i g . G . 1 ) . The c o n t r i b u t i o n t o the s u r f a c e f i e l d a t ( x k , 0 ) , due t o a c u r r e n t J : J i n the g r i d element ( i , j ) , can be c a l c u l a t e d (see Appendix G ) , and i s l i n e a r l y p r o p o r t i o n a l t o J ; ; : Presume now t h a t the c u r r e n t s i n a l l g r i d elements a r e i n phase. ( The p o s s i b i l i t y of phase d i f f e r e n c e s w i l l be c o n s i d e r e d l a t e r ) . The l i n e a r i t y of the c u r r e n t d e n s i t y e q u a t i o n s then a l l o w the t o t a l f i e l d a t any s u r f a c e p o s i t i o n ( x k , 0 ) to be c a l c u l a t e d from the sum of each g r i d c o n t r i b u t i o n : B M '<*k> (4.4.4) (4.4.5) 145 N tt B * ( x k ) = i $ J. • ki{ (x. ) (4.4.6) (4.4.7) C o n s i d e r i n g a l l t h e s u r f a c e s t a t i o n s ( x^ : k = 1 f D and both d i r e c t i o n a l components r e s u l t s i n an L x (2NM) s e t of l i n e a r e q u a t i o n s , which i n most c a s e s w i l l be underdetermined ( t h a t i s , 2NM > L ) . A model t o f i t t h e s e e q u a t i o n s can be found u s i n g l i n e a r programming. T h i s model w i l l be the s m a l l e s t model i n the L,-norm sense, i f the model m i n i m i z e s the o b j e c t i v e f u n c t i o n : <£(j) £ I J C , i (4.4.8) As w e l l , because the L, -norm r e s u l t s i n s p a r s e r , more l o c a l i z e d models than the l e a s t - s q u a r e s norm (L^-norm), the c a l c u l a t e d model w i l l be i n accordance w i t h the f i r s t e x p e c t e d p h y s i c a l f e a t u r e . L i n e a r programming s o l v e s the underdetermined problem w i t h o n l y p o s i t i v e v a l u e s of the v a r i a b l e s a l l o w e d , which i s why the f o r m u l a t i o n t o t h i s p o i n t has presumed the g r i d element c u r r e n t s a r e a l l i n phase. The d e s i r e d c o n d i t i o n was a much weaker one; t h a t t h e r e be no c u r r e n t s i n the model more than tr/2 out of phase. Presume now t h a t the c u r r e n t s i n each g r i d element ( i , j ) a t each f r e q u e n c y 'w' a r e not i n phase, w i t h t h e i r complex time 146 domain r e p r e s e n t a t i o n b e i n g : ( t ) - | J y | . « « « * * ^ (4.4.9) and the c o r r e s p o n d i n g f r e q u e n c y domain r e p r e s e n t a t i o n b e i n g : J i j (w) = | J i j | cos + i | J i j | sin©^ (4.4.10) f o r w > 0. I f no c u r r e n t s are more than Tr/2 out of phase, then i t i s ensured t h a t a minimum v a l u e of <9cj can be found such t h a t [ ©:J - G^-.rv] > 0 a l w a y s . Phase s h i f t a l l the c u r r e n t s by t h i s minimum v a l u e of B : ( t ) = Jcj ( t ) e " 1 8 ^ = |Jij I e J (4.4.11) In the freq u e n c y domain the i m a g i n a r y and r e a l components of each c u r r e n t element ( i , j ) a r e : J i j (w) = | c o s ( ©Lj -<9~In) + i | J ^ | s i n ( ©cj -(4.4.12) Thus, as C O S ( S ; J and s i n ( G^-tSU,'*.) a r e guaranteed t o be g r e a t e r than z e r o , then the v a l u e s of the r e a l and i m a g i n a r y p a r t s of a l l c u r r e n t s (w) w i l l be g r e a t e r than z e r o . The l i n e a r programming approach as o u t l i n e d can then be a p p l i e d 147 s e p a r a t e l y t o the r e a l and im a g i n a r y p o r t i o n s of the phase-s h i f t e d d a t a , , and the c o n s t r u c t e d models w i l l be i n accordance w i t h the second e x p e c t e d p h y s i c a l f e a t u r e . The f i n a l r e a l and im a g i n a r y p o r t i o n s a r e then recombined t o g i v e the a m p l i t u d e and phase of each model element. The o r i g i n a l p h a s e - s h i f t of the data i s s u b t r a c t e d from the phase v a l u e s t o g i v e the f i n a l phase r e p r e s e n t a t i o n of the model. U s i n g the unweighted L (-norm ' s m a l l e s t ' c r i t e r i a as i n e q u a t i o n 4 . 5 . 8 w i l l r e s u l t i n the same u n p h y s i c a l s h a l l o w models as i n the p r e v i o u s l e a s t - s q u a r e s c o n s t r u c t i o n . To o f f s e t t h i s the weight f u n c t i o n W(z) i s a g a i n i n c l u d e d i n the o b j e c t i v e f u n c t i o n : As w e l l , i n p r a c t i c e t h e r e w i l l be some e r r o r i n the d a t a , so to f i t the d a t a e q u a t i o n s of 4 . 5 . 6 and 4 . 5 . 7 e x a c t l y would be o v e r f i t t i n g t he d a t a . T h i s i s han d l e d by expanding each e q u a t i o n i n t o two i n e q u a l i t y c o n s t r a i n t s . For example, e q u a t i o n 4 . 5 . 6 becomes: ( 4 . 4 . 1 3 ) B K ( x k ) + * \ ( x k ) >t£ J ^ A * j ( x k ) ( 4 . 4 . 1 4 ) 1 48 B K ( x k ) - ^ B x ( x k ) < £ £ J L - A C i ( x k ) (4.4.15) where cTB)C(xk) i s the e s t i m a t e d s t a n d a r d d e v i a t i o n i n B x ( x ^ ) . The model would then be r e q u i r e d t o f i t the da t a o n l y t o w i t h i n the s t a n d a r d d e v i a t i o n of each data p o i n t . T h i s completes the f o r m u l a t i o n of the -norm l i n e a r programming c o n s t r u c t i o n . The l i n e a r programming c o n s t r u c t i o n was a p p l i e d t o the same s u i t e of t r u e models as used p r e v i o u s l y . Because the method of c a l c u l a t i n g the data from the p a r a m e t e r i z e d t r u e model i s i d e n t i c a l t o the method used t o c a l c u l a t e the c o n t r i b u t i o n of each g r i d element i n the c o n s t r u c t i o n , the g r i d meshes used i n each were made d i s s i m i l a r , w i t h the g r i d s i z e i n the data c a l c u l a t i o n b e i n g 20 x 20, and t h a t i n the c o n s t r u c t i o n , 16 x 16. The f i r s t s e t of t r u e and c a l c u l a t e d models are shown i n F i g . 4.7. In a l l c a s e s , o n l y the a m p l i t u d e c u r r e n t d e n s i t y model i s g i v e n . In F i g . 4.7, the c a l c u l a t e d models a r e c o n s t r u c t e d a t v a r i o u s v a l u e s of the weight f a c t o r c< , f o r s u r f a c e f i e l d s c a l c u l a t e d u s i n g the t r u e model. G a u s s i a n w h i t e n o i s e w i t h a s t a n d a r d d e v i a t i o n of 10% of the maximum f i e l d v a l u e was added t o the d a t a . Because of the r a p i d g e o m e t r i c decay i n the f i e l d of the anomalous c u r r e n t , t h i s n o i s e l e v e l w i l l be v e r y h i g h a t s t a t i o n s t o e i t h e r s i d e of the anomaly, and i s thus u n r e a s o n a b l y l a r g e . The unweighted model i s c o n c e n t r a t e d near the s u r f a c e as e x p e c t e d . However, f o r b o t h of the w e i g h t e d c o n s t r u c t i o n s (o< = 1, = 2) t h e c a l c u l a t e d models a r e v e r y s i m i l a r t o the t r u e F i g . 4 . 7 L j - N o r m l i n e a r p r o g r a m m i n g m o d e l c o n s t r u c t i o n . 10% n o i s e was a d d e d t o t h e t r u e m o d e l d a t a . ' T h e v a r i o u s m o d e l s a r e : ( a ) t h e t r u e m o d e l (b) m o d e l c o n s t r u c t e d a t = 0 . ( c ) m o d e l c o n s t r u c t e d a t °( - 1 . ( d ) m o d e l c o n s t r u c t e d a t o< = 2. o 151 m o d e l , w i t h t h e h o r i z o n t a l a n d t h e v e r t i c a l f e a t u r e s r e c o v e r e d . When t h e g r i d s i z e i s t a k e n i n t o a c c o u n t , t h e t o t a l c u r r e n t i n t h e l a r g e p e a k i s a l s o f o u n d t o be c l o s e t o t h a t o f t h e t r u e m o d e l . I t i s an i m p o r t a n t p o i n t t h a t b o t h v a l u e s o f t h e w e i g h t i n g p r o d u c e s i m i l a r r e s u l t s , a s t h i s r e m o v e s t h e p r o b l e m o f d e t e r m i n i n g t h e ' b e s t ' v a l u e o f o ( f o r e a c h d a t a s e t t o be i n v e r t e d . I n f a c t , v a l u e s a s l a r g e a s 10 were t r i e d f o r ^< , a n d c a u s e d l i t t l e d i f f e r e n c e i n t h e c o n s t r u c t e d m o d e l , i n d i c a t i n g g r e a t s t a b i l i t y w i t h r e s p e c t t o t h e w e i g h t i n g . I n a l l s u b s e q u e n t m o d e l s t h i s s t a b i l i t y i s t a k e n a d v a n t a g e o f , w i t h t h e v a l u e o f °< f i x e d a t 1. The h o r i z o n t a l a n d v e r t i c a l r e s o l u t i o n o f t h i s c o n s t r u c t i o n w e r e t h e n c h e c k e d u s i n g d a t a f r o m t h e t r u e m o d e l s o f 4.8a a n d 4.9a, t o w h i c h 1 % n o i s e was a d d e d . The c o n s t r u c t i o n was d one u s i n g a w e i g h t i n g o f oC = 1, a n d t h e r e s u l t s a r e i n F i g . 4.8b a n d 4.9b. T h e r e s o l v i n g a b i l i t y i n b o t h t h e h o r i z o n t a l a n d t h e v e r t i c a l d i r e c t i o n i s e v i d e n t . The t r u e m o d e l s u s e d t o t h i s p o i n t h a v e a l l m a t c h e d t h e f i r s t p r e s u m e d p h y s i c a l f e a t u r e , i n w h i c h t h e t r u e m o d e l was a s s u m e d t o be s p a r s e , w i t h o n l y a few l o c a l i z e d c u r r e n t e l e m e n t s . T o c h e c k t h e l i m i t o f t h e L ( - n o r m c o n s t r u c t i o n w i t h r e s p e c t t o l e s s l o c a l i z e d t r u e m o d e l s , t h e m o d e l s o f F i g . 4.10a a n d 4.11a w e r e u s e d t o g e n e r a t e t h e s u r f a c e d a t a . The c o n s t r u c t e d m o d e l s i n F i g . 4.10b a n d F i g . 4.11b do show t h e t e n d e n c y o f . t h e I ^ - n o r m t o o v e r - l o c a l i z e t h e c u r r e n t e l e m e n t s . H o w e v e r , t h e d a t a s t i l l s u p p l y e n o u g h i n f o r m a t i o n s o t h a t t h e 1 52 fil 4 . 8 L i n e a r p r o g r a m m i n g m o d e l c o n s t r u c t i o n t o t e s t h o r i z o n t a l r e s o l u t i o n . T h e c o n s t r u c t e d m o d e l was c a l c u l a t e d w i t h a w e i g h t i n g o f <X = 1. 1 % n o i s e was a d d e d t o t h e t r u e m o d e l d a t a . ( a ) t h e t r u e m o d e l ( b ) t h e c o n s t r u c t e d m o d e l 1 53 F i g . 4.9 L i n e a r programming model c o n s t r u c t i o n t o t e s t v e r t i c a l r e s o l u t i o n . The c o n s t r u c t e d model was c a l c u l a t e d w i t h a w e i g h t i n g of °< = 1 . 1% n o i s e was added t o the t r u e model d a t a . (a) the t r u e model (b) the c o n s t r u c t e d model 154 F i g . 4 . 1 0 L i n e a r p r o g r a m m i n g m o d e l c o n s t r u c t i o n t o t e s t t h e a l g o r i t h m w i t h l e s s l o c a l i z e d t r u e m o d e l s , i n t h i s c a s e a v e r t i c a l c u r r e n t d i k e . T h e w e i g h t i n g was o< = 1 , a n d 1% n o i s e was a d d e d t o t h e t r u e m o d e l d a t a . ( a ) t h e t r u e m o d e l ( b ) t h e c o n s t r u c t e d m o d e l 155 F i g . 4.11 Linear programming construction to test the algorithm with less l o c a l i z e d true models, in t h i s case a dipping dike. The weighting was c< = 1, and 1% noise was added to the true model data. (a) the true model (b) the constructed model 156 m a i n f e a t u r e s o f b o t h d i k e - l i k e a n o m a l i e s a r e r e c o v e r e d , i n c l u d i n g t h e i r d i p . I n summary, t h i s l i n e a r p r o g r a m m i n g m o d e l l i n g a p p r o a c h i s f o u n d t o be s t a b l e w i t h r e s p e c t t o b o t h t h e w e i g h t i n g f u n c t i o n a n d t o r e a s o n a b l e a m o u n t s o f n o i s e i n t h e d a t a . I t h a s g o o d v e r t i c a l a s w e l l a s h o r i z o n t a l r e s o l u t i o n when t h e t r u e m o d e l s c o n s i s t o f a s m a l l number o f l o c a l i z e d m o d e l e l e m e n t s . E v e n f o r d i k e - l i k e a n o m a l i e s w h i c h a r e l o c a l i z e d o n l y i n one d i r e c t i o n , t h e c o n s t r u c t i o n s t i l l r e c o v e r s t h e m a j o r f e a t u r e s o f t h e t r u e m o d e l . The d a t a s e t s u s e d t o t e s t t h e l i n e a r p r o g r a m m i n g c o n s t r u c t i o n a l g o r i t h m h a v e t o t h i s p o i n t b e e n g e n e r a t e d f r o m c u r r e n t d e n s i t y m o d e l s , w i t h a l l c u r r e n t e l e m e n t s i n t h e s e m o d e l s b e i n g i n p h a s e . A more r e a l i s t i c t e s t i s t o u s e a s t h e s t a r t i n g d a t a t h e o u t p u t f r o m t h e J o n e s ' f o r w a r d i n d u c t i o n p r o g r a m , w h e r e now t h e t r u e ' m o d e l s w o u l d be c o n d u c t i v i t y m o d e l s . T h i s w i l l i n p a r t i c u l a r c h e c k t h e c o n s t r u c t i o n a l g o r i t h m ' s a b i l i t y t o h a n d l e d a t a c o n t a i n i n g v a r i o u s p h a s e s h i f t s , a n d w i l l a l s o t e s t t h e c o r r e l a t i o n b e t w e e n t h e c o n s t r u c t e d c u r r e n t s t r u c t u r e a n d t h e o r i g i n a l c o n d u c t i v i t y m o d e l . T h e f i r s t c o n d u c t i v i t y m o d e l u s e d t o g e n e r a t e d a t a was t h a t i n F i g . 4 .12a, c o n s i s t i n g o f a l o c a l i z e d c o n d u c t o r i n a l a y e r e d s t r u c t u r e , w i t h a h o r i z o n t a l p o s i t i o n o f 189 km. a n d a d e p t h o f 35 km. B e c a u s e o f t h e i n c o n s i s t e n c y b e t w e e n t h e two d i r e c t i o n a l c o m p o n e n t s o f t h e o u t p u t i n J o n e s ' p r o g r a m ( a s n o t e d i n s e c t i o n 2.2 o f C h a p t e r I I ) , o n l y t h e Bx c o m p o n e n t h a s b e e n u s e d f o r 157 SURFACE POSITION (KMi O (OO 200 300 400 F i g . 4.12 L i n e a r programming model c o n s t r u c t i o n u s i n g as s t a r t i n g d a t a t h e o u t p u t from t h e Jones-Pascoe i n d u c t i o n program. The models a r e : (a) t h e t r u e c o n d u c t i v i t y model (b) t h e c o n s t r u c t e d c u r r e n t d e n s i t y model 158 S U R F A C E P O S I T I O N (KM) O 100 200 300 400 >99, 'yp-F i g . 4.13 L i n e a r programming model c o n s t r u c t i o n u s i n g as s t a r t i n g d a t a t h e o u t p u t from the Jones-Pascoe i n d u c t i o n program. The t r u e c o n d u c t i v i t y model i n t h i s case c o n s i s t s of two v e r t i c a l l y d i s p l a c e d c o n d u c t o r s . (a) the t r u e c o n d u c t i v i t y model (b) the c o n s t r u c t e d c u r r e n t d e n s i t y model 159 t h e s e i n v e r s i o n s . T h e c o n s t r u c t e d a m p l i t u d e c u r r e n t d e n s i t y m o d e l i s shown i n F i g . 4.12b. The b a s i c s t r u c t u r e o f t h i s c u r r e n t d e n s i t y m o d e l i s a c l o s e r e p r o d u c t i o n o f t h e o r i g i n a l c o n d u c t i v i t y m o d e l , a n d t h e d e p t h ( 3 4 . 6 km.) a n d h o r i z o n t a l p o s i t i o n (188.1 km) a l s o m a t c h t h a t o f t h e o r i g i n a l m o d e l . A s e c o n d c o n d u c t i v i t y m o d e l was u s e d t o c h e c k t h e v e r t i c a l r e s o l u t i o n , w i t h two l o c a l i z e d c o n d u c t o r s a t 9 a n d 35.5 km. d e p t h , w i t h b o t h a t a h o r i z o n t a l p o s i t i o n o f 189 km. ( s e e F i g . 4 . 1 3 a ) . A g a i n , t h e c o n s t r u c t e d c u r r e n t m o d e l ( i n F i g . 4.13b) h a s r e s o l v e d t h e m a j o r f e a t u r e s o f t h e c o n d u c t i v i t y m o d e l , a n d h a s a c c u r a t e l y d e l i n e a t e d t h e h o r i z o n t a l p o s i t i o n (188 km.) o f t h e o r i g i n a l c o n d u c t o r s . As w e l l , a r e a s o n a b l e e s t i m a t e o f t h e d e p t h s o f t h e c o n d u c t o r s ( 8 . 3 km., 30.2 km.) h a s b e e n made. C h a p t e r V A n a l y s i s o f G.D.S. A c r o s s t h e C a s c a d e A n o m a l y G.D.S. d a t a f r o m an a r r a y o f m a g n e t o m e t e r s s p a n n i n g t h e C a s c a d e a n o m a l y o f W a s h i n g t o n S t a t e (Law e t a l , 1980) h a s b e e n o b t a i n e d f r o m t h e U n i v e r s i t y o f W a s h i n g t o n ' s G e o p h y s i c s S e c t i o n . The C a s c a d e a n o m a l y was o r i g i n a l l y d e t e c t e d by a l i n e a r a r r a y o f s t a t i o n s i n s o u t h w e s t e r n W a s h i n g t o n ( s e e F i g . 5.1) w i t h t h e i n d u c t i o n a r r o w r e s p o n s e s i n d i c a t i n g a l o c a l i z e d n o r t h - s o u t h c o n d u c t o r b e t w e e n t h e s t a t i o n s KOS a n d WHI ( s e e F i g . 5 . 2 ) . The c o i n c i d e n c e o f t h i s c o n d u c t i v e p a t h w i t h t h e C a s c a d e v o l c a n i c b e l t g e n e r a t e d s i g n i f i c a n t i n t e r e s t , p r o m p t i n g f u r t h e r i n v e s t i g a t i o n s t o t r a c e t h e c o u r s e o f t h e c o n d u c t i v e p a t h t o t h e n o r t h a n d t o t h e s o u t h . One s u c h i n v e s t i g a t i o n r e c o r d e d t h e s u r f a c e m a g n e t i c f i e l d o f a p o l a r m a g n e t i c s t o r m i n F e b r u a r y o f 1980, a t t h e a r r a y s i t e s i n d i c a t e d i n F i g . 5.3. T h e d a t a f r o m t h i s e v e n t ( s e e F i g . 5.4 a n d 5.5) i s u s e d h e r e t o d e m o n s t r a t e t h e L , - n o r m l i n e a r c o n s t r u c t i o n p r o g r a m m i n g r o u t i n e d e s c r i b e d i n s e c t i o n 4 o f C h a p t e r I V . The d i s c u s s i o n s o f a p p r a i s a l a n d c o n s t r u c t i o n i n C h a p t e r I V were a l l ' b a s e d on t h e p r e s u m p t i o n s t h a t t h e n o r m a l a n d a n o m a l o u s 161 F i g . 5.1 The a r r a y of magnetometers used by Law e t a l (1980) t o s t u d y the Cascade anomaly of Washington S t a t e . 162 F i g . 5.2 In-phase and q u a d r a t u r e i n d u c t i o n arrows f o r 30,300, and 3000s. ( A f t e r Law et a l , 1980). 1 6 3 F i g . 5.3 T h e a r r a y o f m a g n e t o m e t e r s w h i c h m e a s u r e d t h e d a t a u s e d i n t h i s t h e s i s ( a f t e r H e n s e l , 1 9 8 0 ) . T h e a p p r o x i m a t e c o n t i n u a t i o n o f t h e C a s c a d e a n o m a l y ( H e n s e l , 1980) i s m a r k e d a s a d a s h e d l i n e . T h e s t u d y a r e a i s i n d i c a t e d by a r e c t a n g l e i n F i g . 5.1. 1 64 (a) . 5.4 M a g n e t o g r a m s f o r t h e m a g n e t i c s t o r m o f F e b r u a r y , 1980. T h e m e a s u r i n g s t a t i o n s a r e : ( a ) N I S ( b ) ORT 1 65 ( A ) F i g . 5.5 M a g n e t o g r a m s f o r t h e m a g n e t i c s t o r m o f F e b r u a r y , 1980. T h e m e a s u r i n g s t a t i o n s a r e : ( a ) MUD ( b ) GRE f i e l d s h a d a l r e a d y b e e n s e p a r a t e d , a n d t h a t t h e s u b s u r f a c e s t r u c t u r e was two d i m e n s i o n a l . T h e l i n e a r p r o g r a m m i n g c o n s t r u c t i o n f u r t h e r a s s u m e d t h a t t h e i n t e r n a l c u r r e n t s p r o d u c i n g t h e a n o m a l o u s f i e l d were v e r y l o c a l i z e d a n d t h a t n o n e o f t h e s e c u r r e n t s w e r e more t h a n 'TT/2 d i f f e r e n t i n p h a s e . T h u s , b e f o r e a n y c o n s t r u c t i o n o f m o d e l s i s d o n e , t h e s e p a r a t i o n o f t h e t o t a l f i e l d i n t o i t s n o r m a l a n d a n o m a l o u s p o r t i o n s must be p e r f o r m e d , a n d t h e n a l l t h e a s s u m p t i o n s must be c h e c k e d u s i n g t h e a n o m a l o u s f i e l d v a l u e s . The n o r m a l f i e l d d e f i n e d i n C h a p t e r I I I was t h e t o t a l s u r f a c e f i e l d t h a t w o u l d be i n d u c e d o v e r t h e r e g i o n a l one d i m e n s i o n a l s t r u c t u r e . H o w e v e r , t h e s t a t i o n s i n t h i s a r r a y a r e a l l v e r y c l o s e t o t h e l a n d - s e a b o u n d a r y , so t h a t t h e g e o m a g n e t i c c o a s t e f f e c t ( P a r k i n s o n , 1959; E v e r e t t a n d Hyndman, 1967) w i l l be p r o n o u n c e d a t a l l o f them. A l t h o u g h i t i s o f i n t e r e s t a n d i s o f t e n s t u d i e d , t h e c o a s t e f f e c t i n t h i s e x p e r i m e n t s i m p l y o b s c u r e s t h e d e s i r e d r e s p o n s e f r o m t h e i n l a n d C a s c a d e c o n d u c t o r . F o r t h i s r e a s o n i t i s a d v a n t a g e o u s t o i n c l u d e t h e a n o m a l o u s f i e l d o f t h e c o a s t e f f e c t i n t h e d e f i n e d n o r m a l f i e l d , s o t h a t u p on s u b t r a c t i n g t h i s n o r m a l f i e l d f r o m t h e t o t a l f i e l d o n l y t h e a n o m a l o u s p o r t i o n due t o t h e C a s c a d e c o n d u c t o r r e m a i n s . Law e t a l ( 1 9 8 0 ) h a v e d o n e t h i s u s i n g t h e c a l c u l a t e d r e s p o n s e o f t h e c o a s t e f f e c t a t e a c h i n l a n d s t a t i o n . ( E v e r e t t a n d Hyndman, 1 9 6 7 ) . H o w e v e r , f o r t h e a r r a y s u p p l y i n g t h e d a t a f o r t h i s t h e s i s t h e c o a s t e f f e c t w i l l be c o m p l i c a t e d by t h e p r o x i m i t y o f t h e P u g e t S o u n d . A s w e l l , t h e two o u t e r m o s t s t a t i o n s i n t h i s a r r a y , NIS 167 a n d GRE, a r e o n l y s e p a r a t e d l o n g i t u d i n a l l y by a b o u t 80 km. B e c a u s e o f t h e s e p o i n t s , t h e s i m p l i f i c a t i o n w i l l be u s e d h e r e t h a t t h e c o a s t e f f e c t i s c o n s t a n t a c r o s s t h e a r r a y . The s t r o n g s i m i l a r i t y b e t w e e n t h e m a g n e t o g r a m s a t t h e s t a t i o n s w h i c h a r e m o s t d i s t a n t f r o m t h e p r e s u m e d l o c a t i o n o f t h e C a s c a d e a n o m a l y (NIS a n d O R T ) , i n d i c a t e t h a t t h i s s i m p l i f i c a t i o n i s r e a s o n a b l e . T h e f i e l d a t t h e o u t e r m o s t s t a t i o n , N I S , i s d e s i g n a t e d t h e n o r m a l f i e l d . T h e a n o m a l o u s f i e l d a t t h e r e m a i n i n g t h r e e s t a t i o n s i s t h e n c a l c u l a t e d a s t h e d i f f e r e n c e b e t w e e n t h e i n d i v i d u a l m e a s u r e d f i e l d s a n d t h i s n o r m a l f i e l d , a n d b o t h t h e o n e - d i m e n s i o n a l s t r u c t u r e r e s p o n s e a s w e l l a s t h e c o a s t e f f e c t s h o u l d be r e m o v e d . Th e f i r s t s t e p i n t h e a n a l y s i s o f t h e a n o m a l o u s f i e l d ( h e n c e f o r t h c a l l e d s i m p l y t h e f i e l d ) a t t h e s t a t i o n s ORT, MUD, a n d GRE w i l l be t h e c a l c u l a t i o n o f t h e s p e c t r a f o r e a c h d i r e c t i o n a l c o m p o n e n t . T h e f i r s t 23.75 h o u r s o f t h e d i g i t i z e d m a g n e t o g r a m s w e r e u s e d , w h i c h was 4096 p o i n t s a t a 20 s e c o n d d i g i t i z i n g i n t e r v a l . T h e mean a n d l i n e a r t r e n d were r e m o v e d f r o m e a c h s i g n a l a n d t h e F a s t F o u r i e r T r a n s f o r m was a p p l i e d . T h e a m p l i t u d e s p e c t r a a r e g i v e n i n F i g . 5.6, 5.7, a n d 5.8. T h e o r i g i n a l m e a s u r e m e n t o f t h e m a g n e t i c f i e l d was d i g i t a l a l s o , w i t h e a c h c o u n t r e p r e s e n t i n g .25 n a n o t e s l a ( n t . ) . T h u s , a n i n t u i t i v e ' t h r e s h h o l d l e v e l ' , b e l o w w h i c h t h e a m p l i t u d e s p e c t r a m i g h t be c o n s i d e r e d t o be m e r e l y n o i s e , w o u l d be .125 n t . ( T h e h a l f c o u n t v a l u e i s t a k e n b e c a u s e t h e r e i s an e q u a l a m p l i t u d e o f s i g n a l a t t h e c o r r e s p o n d i n g n e g a t i v e f r e q u e n c y ) . I f t h i s l e v e l 168 H SPECTRUM . D SPECTRUM F i g . 5.6 The amplitude s p e c t r a of the anomalous f i e l d at ORT-(a) H (b) D (c) Z 169 H SPECTRUM D SPECTRUM C O F i g . 5.7 The amplitude s p e c t r a of the anomalous f i e l d at MUD-(a) H (b) D (c) Z H SPECTRUM D SPECTRUM Z SPECTRUM The amplitude s p e c t r a of the anomalous f i e l d at GRE (a) H (b) D (c) Z 171 i s used, the m e a n i n g f u l range f o r most of the s p e c t r a would o n l y i n c l u d e p e r i o d s g r e a t e r than ~1 hour. However, the s t r i c t use of t h i s ' t h r e s h h o l d ' l e v e l i s not a p p r o p r i a t e , because the i n d u c i n g f i e l d s a r e more c o r r e c t l y m o d e l l e d as a sum of f r e q u e n c y components w i t h time v a r y i n g a m p l i t u d e s ; b ( t ) = £ a • ( t ) c o s ( w - t ) (5.1) r a t h e r than as a sum of f r e q u e n c y components w i t h c o n s t a n t a m p l i t u d e , as m o d e l l e d by the F o u r i e r t r a n s f o r m . The time v a r i a t i o n s of the a m p l i t u d e , a j ^ t ) , w i l l s p r e a d the energy of each f r e q u e n c y component of the s i g n a l over a range of f r e q u e n c i e s i n the F o u r i e r t r a n s f o r m r e p r e s e n t a t i o n . As an example, f o r a s i g n a l b ( t ) as i n e q u a t i o n 5.1, but w i t h a s i n g l e f r e q u e n c y component, w 0 , i t s f r e q u e n c y spectrum w i l l be: B(w) = [ A c ( w ) / 2 ] «> [<f(w-w 0)] + [ A Q ( w ) / 2 ] ® [cf(w+w c)] (5.2) w i t h : A6Cw) = J [ a c ( t ) ] (5.3) Thus, the t h r e s h o l d l e v e l can p r o b a b l y be taken t o be much lower than h a l f the count r a t e . The p o s s i b i l i t y t h a t the model of e q u a t i o n 5.1 i s more p h y s i c a l l y r e p r e s e n t a t i v e of the source s u g g e s t s t h a t complex d e m o d u l a t i o n (Banks, 1975) may be a more 172 r e a s o n a b l e approach t o d e t e r m i n i n g s p e c t r a than the F o u r i e r t r a n s f o r m approach, but i t has not been used h e r e . The assumptions embodied i n the c o n s t r u c t i o n r o u t i n e of s e c t i o n 4 i n Chapter IV can now be checked. A measure of the degree of p o l a r i z a t i o n of the magnetic f i e l d v e c t o r a t any f r e q u e n c y 'w' can be o b t a i n e d from (Samson, 1977): p(w) = [ n T r ( S Z ) - ( T r S ? ] / [ ( n - 1 ) ( T r S ) ^ ] (5.4) where p(w) i s t h e degree of p o l a r i z a t i o n a t 'w', 'n' i s the number of v e c t o r components, and S(w) i s the smoothed s p e c t r a l m a t r i x of the magnetic v e c t o r . The unsmoothed s p e c t r a l m a t r i x i s g i v e n by: S' (w) w i t h S'^ -g, (w) d e f i n e d a s : S^fc (w) = A(w)B*"(w) (5.6) (A(w) and B(w) a r e the complex magnitudes of the s p e c t r a f o r each v e c t o r component). The v a l u e of the degree of p o l a r i z a t i o n can v a r y between 0 and 1, c o r r e s p o n d i n g t o c o m p l e t e l y random S^ (w) S i D (w) S^ (w) \S4 H(w) S', (w) S ^ ( w ) , (5.5) f o r two a r b i t r a r y v e c t o r components A and B b e i n g 1 73 p o l a r i z a t i o n a n d t o a c o m p l e t e l y p o l a r i z e d s i g n a l . U s i n g t h e s p e c t r a p r e v i o u s l y d e t e r m i n e d , t h e s p e c t r a l m a t r i x S'(w) was c a l c u l a t e d , a n d s u b s e q u e n t l y s m o o t h e d w i t h a 3 p o i n t f i l t e r (.25,.5,.25) t o g i v e S ( w ) : R e a l [ S A B ( w ) ] = R e a l [ S ' A E (w) ] ® (.25,.5,.25) ( 5 . 7 ) I m a g [ S f t B ( w ) ] = Imag[S' (qg w(w) ] ® . (.25,.5,.25) ( 5 . 8 ) T h e s m o o t h i n g o f t h e s p e c t r a l m a t r i x b e f o r e t h e d e g r e e o f p o l a r i z a t i o n i s c a l c u l a t e d i s b a s e d on t h e i m p l i c i t a s s u m p t i o n t h a t t h e f i e l d i s more c o r r e c t l y m o d e l l e d by e q u a t i o n 5 . 1 , r a t h e r t h a n by t h e F o u r i e r t r a n s f o r m e x p r e s s i o n . I f t h e u n s m o o t h e d s p e c t r a l m a t r i x was u s e d , t h e d e g r e e o f p o l a r i z a t i o n f r o m e q u a t i o n 5.4 w o u l d be 1 a t e v e r y f r e q u e n c y , i n a c c o r d a n c e w i t h t h e c o n s t a n t a m p l i t u d e a n d c o n s t a n t p h a s e a s s u m p t i o n o f t h e F o u r i e r t r a n s f o r m m o d e l . T h e v a l u e s o f p(w) a r e p l o t t e d i n F i g . 5.9 f o r t h e s t a t i o n s ORT, MUD, a n d GRE. T h e m a g n e t i c v e c t o r a t a l l s t a t i o n s i s s t r o n g l y p o l a r i z e d f o r n e a r l y a l l f r e q u e n c i e s , w i t h t h e v e c t o r a t MUD a n d ORT b e i n g a l m o s t c o m p l e t e l y p o l a r i z e d . T h i s a l l o w s us t o a p p r o x i m a t e t h e f r e q u e n c y c o m p o n e n t s o f t h e m a g n e t i c v e c t o r a t e a c h s t a t i o n p o s i t i o n 'x' by a c o m p l e t e l y p o l a r i z e d s i g n a l ( B o r n a n d W o l f , 1975, p g . 3 2 ) : 174 Ca) P E R I O D ( M I N . ) 50 25 16 7 4- r — - I I I M O.OO CYC.I F.C/MIN 10 _L_ ft) «•• i .n • • M O . H | .Qg CYCLES/ MIN. • •I t . l l CO 50 I PERIOD (MIN.) z? 'V v> CYCLES/MIN. F i g . 5.9 T h e d e g r e e o f p o l a r i z a t i o n b e t w e e n t h e t h r e e d i r e c t i o n a l c o m p o n e n t s o f t h e a n o m a l o u s f i e l d a s a f u n c t i o n o f p e r i o d . ( a ) ORT (b) MUD ( c ) GRE 175 H(x,w,t) = HQ(x,w.)- [cos(wt+ <}>H(x,w) ) ] (5.9) D(x,w,t) = D e(x,w) • [ c o s ( w t + <pt(x,w) ) ] (5.10) Z(x,w,t) = Z Q(x,w) • [cos(wt+<r\(x,w) ) ] (5.11) As i n d i c a t e d , H 0, D e, Z 0 , 9 H r a n d 9 * a r e a l l independent of t i m e . The v a r y i n g p o s i t i o n of the v e c t o r d e s c r i b e d by these t h r e e components w i l l t r a c e out the s u r f a c e of an e l l i p s o i d , w i t h p r i n c i p l e axes h a v i n g l e n g t h s A,B, and C, and d i r e c t i o n s the p o l a r i z a t i o n of the v e c t o r . The p r o c e d u r e t o o b t a i n the p r i n c i p l e axes l e n g t h s and d i r e c t i o n s i s d e t a i l e d i n Appendix H. B a s i c a l l y , the t h r e e component e q u a t i o n s of 5.8 - 5.10 a r e c o n v e r t e d t o a q u a d r i c e q u a t i o n r e p r e s e n t i n g an e l l i p s o i d , which i s then s i m p l i f i e d by d i a g o n a l i z i n g the m a t r i x of c o e f f i c i e n t s . The r e l a t i v e l e n g t h s of the p r i n c i p l e components A,B, and C a r e then g i v e n by the e i g e n v a l u e s (Xi.) of the c o e f f i c i e n t m a t r i x , which a r e the non-zero elements of the d i a g o n a l m a t r i x : . These parameters c o m p l e t e l y d e t e r m i n e the n a t u r e of 176 A i / > ; ' -(5.12) B «< 1/-X,,"1 (5.13) C « 1 / W 1 (5.14) The d i r e c t i o n of each p r i n c i p l e a x i s i s g i v e n by the e i g e n v e c t o r a s s o c i a t e d w i t h i t s e i g e n v a l u e . The s p e c t r a f o r each s t a t i o n were smoothed u s i n g the t h r e e p o i n t f i l t e r (.25,.5,.25),the e i g e n v a l u e s were d e t e r m i n e d , and the r a t i o s of the s m a l l e s t t o second l a r g e s t e i g e n v a l u e s were c a l c u l a t e d a t each f r e q u e n c y . The r e s u l t s g i v e n f o r the p e r i o d s from 20 min. - 4 h r s . (see F i g . 5.10 ) i n d i c a t e t h a t a t both ORT and MUD the anomalous f i e l d v e c t o r i s e f f e c t i v e l y l i n e a r l y p o l a r i z e d . The h i g h degree of p o l a r i z a t i o n found i n the anomalous f i e l d a t a l l t h r e e s t a t i o n s , and i n p a r t i c u l a r ORT and MUD, c o u l d be due o n l y t o e i t h e r a h i g h l y p o l a r i z e d i n d u c i n g f i e l d , or t o a d i r e c t i o n of c o n s t a n t symmetry i n the c o n d u c t i v e s t r u c t u r e . From the o r i g i n a l magnetograms i n F i g . 5.4 and 5.5 i t appears t h a t the i n d u c i n g f i e l d i s i n f a c t p o l a r i z e d , a t l e a s t f o r the l o n g p e r i o d s . However, even f o r a l i n e a r l y p o l a r i z e d 177 PERIOD (MIN) Ca) e.i i n Cb) CYCLES . 'M IN . CO o.D li oi CYCI.ES/MIN. O.I a ii F i g . 5.10 The r a t i o of the s m a l l e s t t o the second s m a l l e s t e i g e n v a l u e s from t h e m a t r i x of c o e f f i c i e n t s of the p o l a r i z a t i o n e l l i p s o i d , as a f u n c t i o n of p e r i o d . (a) ORT ( b ) MUD (c) GRE 178 i n d u c i n g f i e l d , i f t h e c o n d u c t i v i t y s t r u c t u r e i s t h r e e -d i m e n s i o n a l t h e a n o m a l o u s f i e l d w i l l n o t be e x p e c t e d t o be l i n e a r l y p o l a r i z e d , b e c a u s e o f t h e p r o b a b l e p h a s e s h i f t s b e t w e e n t h e d i r e c t i o n a l c o m p o n e n t s . T h u s , i t i s f e l t t h a t t h e l i n e a r p o l a r i z a t i o n a t ORT a n d MUD i n d i c a t e s t h a t t h e s t r u c t u r e h a s a d i r e c t i o n o f c o n s t a n t s y m m e t r y . F u r t h e r t o t h i s , a s shown i n F i g . 5 . 1 1 , t h e f a c t t h a t two s t a t i o n s h a v e a l i n e a r l y p o l a r i z e d s i g n a l s u g g e s t s t h a t t h e c u r r e n t must be l o c a l i z e d a t a s i n g l e d e p t h , a s c u r r e n t s a t d i f f e r e n t d e p t h s w o u l d i n t r o d u c e a p h a s e s h i f t b e t w e e n t h e d i r e c t i o n a l c o m p o n e n t s a t a t l e a s t one s t a t i o n . I f t h e f i e l d i s p r e s u m e d t o be due t o a l i n e c u r r e n t , t h e v e c t o r d i r e c t i o n s o f t h e l o n g e s t p r i n c i p l e a x i s a t any two s t a t i o n s must b o t h be p e r p e n d i c u l a r t o t h e l i n e c u r r e n t , a n d a l s o must be p e r p e n d i c u l a r t o t h e n o r m a l s t o t h e l i n e c u r r e n t w h i c h i n t e r s e c t t h e s t a t i o n s . T h i s w o u l d c o m p l e t e l y d e t e r m i n e t h e l o c a t i o n o f t h e l i n e c u r r e n t , i n c l u d i n g t h e d i p a n d s t r i k e , i f t h e v e c t o r d i r e c t i o n s were p e r f e c t l y a c c u r a t e . U s i n g t h e d i r e c t i o n s o f t h e l a r g e s t p r i n c i p l e a x i s f r o m t h e a s s o c i a t e d e i g e n v e c t o r s , t h e d i p a n d s t r i k e o f t h e c o n j e c t u r e d l i n e c u r r e n t was c a l c u l a t e d u s i n g t h e ORT a n d MUD e i g e n v e c t o r s . T h e mean v a l u e s a n d s t a n d a r d d e v i a t i o n o f t h e d i p a n d s t r i k e o f t h e p r o p o s e d l i n e c u r r e n t p a t h w e r e f o u n d f r o m t h e 1 h r . - 20 h r . p e r i o d r a n g e t o be ( s e e F i g . 5 . 1 2 ) : D I P = - 1 . 0 0 1 ° + 7 . 1 1 7 ° 179 AIR Stations A , / H, ^ F k X . B < I = W EARTH X \ \ lei<P> Is = ir » > F i g . 5 . 1 1 Two l i n e a l c o n d u c t o r s a r e shown a t d i f f e r e n t d e p t h s . A s shown i n s e c t i o n 2.2 o f C h a p t e r I I , t h e r e s u l t a n t f i e l d f r o m e a c h o f t h e s e c o n d u c t o r s c a n be m o d e l l e d a s a l i n e c u r r e n t , a s shown a t s t a t i o n s A a n d B. A s t h e c o n d u c t o r s a r e a t d i f f e r e n t d e p t h s , t h e r e w i l l be a p h a s e d i f f e r e n c e b e t w e e n t h e i r r e s p e c t i v e f i e l d s , s o t h a t b o t h s t a t i o n s c a n n o t h a v e a l i n e a r l y p o l a r i z e d s i g n a l . 180 D I P 50 Period (min.) 25 16.7 12.5 10 0 . 1 2 S T R I K E o Cycles/mm. 0.]<! F i g . 5.12 T h e e s t i m a t e d d i p a n d s t r i k e t h a t a l i n e c u r r e n t w o u l d n e e d t o m a t c h t h e f i e l d d i r e c t i o n o f t h e l a r g e s t p r i n c i p a l a x i s o f t h e p o l a r i z a t i o n e l l i p s o i d a t s t a t i o n s ORT a n d MUD. 181 STRIKE = -15.029° + 5.825° where the d i p i s downward i n the d i r e c t i o n of the s t r i k e , and the s t r i k e i s measured c l o c k w i s e from n o r t h . The d i p i s s m a l l enough t o be i g n o r e d , t h e r e b y a l l o w i n g the use of the two-d i m e n s i o n a l a p p r o x i m a t i o n . The assumptions of Chapter IV have been v a l i d a t e d f o r t h i s d a t a s e t , except f o r the assumption of the maximum phase d i f f e r e n c e . The check of the phase d i f f e r e n c e s was i n c o r p o r a t e d i n the m o d e l l i n g r o u t i n e i t s e l f , and i t was found t h a t the assumption was not v i o l a t e d . The r e m a i n i n g s t e p i s t o r o t a t e the v e c t o r d a t a and s t a t i o n p o s i t i o n s t o a new c o o r d i n a t e system a l i g n e d a l o n g the e s t i m a t e d s t r i k e of the s t r u c t u r e , u s i n g the s t a t i o n NIS as the o r i g i n of the new c o o r d i n a t e system. The e r r o r i n the an g l e of the s t r i k e w i l l i n t r o d u c e e r r o r i n t o both the s t a t i o n p o s i t i o n , and the r o t a t e d magnetic d a t a , w i t h the r e s u l t a n t e r r o r i n the magnetic f i e l d v a l u e s e a s i l y i n c o r p o r a t e d i n t o the l i n e a r programming method. There i s no easy way t o i n c o r p o r a t e the e r r o r i n the s t a t i o n p o s i t i o n i n t o the l i n e a r programming, but i t s r e l a t i v e v a l u e w i l l be s m a l l so i t has s i m p l y been i g n o r e d . The f i n a l s t a t i o n p o s i t i o n s (x) a r e g i v e n i n T a b l e 5.1. 182 T a b l e 5.1: S t a t i o n P o s i t i o n s S t a t i o n P o s i t i o n NIS 0.0 km. ORT 33.2 km. MUD 57.7 km. GRE 80.9 km. U s i n g the o n e - d i m e n s i o n a l s t a t i o n p o s i t i o n s ( x ) , the component of the r o t a t e d f i e l d p e r p e n d i c u l a r t o the proposed s t r i k e , and the v e r t i c a l component, the c o n s t r u c t i o n of models was performed u s i n g the l i n e a r programming method of Ch. IV. For the l o n g e r p e r i o d s (1 - 4 h r s . ) the c o n s t r u c t e d models c o n s i s t e n t l y r e q u i r e a l o c a l i z e d c u r r e n t i n the depth range 11 -21 km., a t a s t a t i o n p o s i t i o n , x = 50.6 + 2.8 km. (see F i g . 5.13a,b,and c ) . V a r y i n g the s i z e of the r e g i o n of i n t e r e s t , or the ' t i g h t n e s s ' of f i t t o the d a t a can r e s u l t i n the removal or appearance of the c u r r e n t s a t the edges of the r e g i o n , but never 183 F i g . 5.13 L i n e a r p r o g r a m m i n g c u r r e n t m o d e l s c a l c u l a t e d u s i n g t h e r e a l d a t a a t a v a r i e t y o f p e r i o d s . T h e p e r i o d s o f e a c h m o d e l , w i t h t h e i n d i c a t e d p o s i t i o n o f t h e c e n t r a l l i n e c u r r e n t , a r e : ( a ) 4 h r . D e p t h 16.9 km. S t a t i o n P o s i t i o n ... 46.5 km. (b ) 2 h r . D e p t h 14.1 km. S t a t i o n P o s i t i o n . . . 5 0 . 7 km. "(c) 1 h r . D e p t h 21.0 km. S t a t i o n P o s i t i o n ... 52.5 km. (d ) 20 m i n . D e p t h 11.0 km. S t a t i o n P o s i t i o n ... 52.5 km. F i g . 5 . . I T 4 hr 2 hr 1 h r 20 min. 185 r e s u l t s i n the d i s a p p e a r a n c e of t h i s c e n t r a l c u r r e n t . T h i s i n d i c a t e s t h a t the o t h e r c u r r e n t elements a r e s t r i c t l y a r t i f a c t s of t h e n o i s e and the m o d e l l i n g p a r a m e t e r s , whereas the c e n t r a l c u r r e n t i s r e q u i r e d t o s a t i s f y the d a t a . I t i s noted t h a t the d e pth of t h i s c u r r e n t (15.7 ± 4.2 km.) i s i n good agreement w i t h the v a l u e of 1 7 + 8 km. e s t i m a t e d f o r the s o u t h e r n s e c t i o n of the Cascade anomaly, u s i n g a b e s t f i t t i n g l i n e c u r r e n t model f o r a p e r i o d of 1 h r . (Law e t a l , 1980). For the l i n e a r programming models c o n s t r u c t e d a t h i g h e r f r e q u e n c i e s , the c e n t r a l c u r r e n t , a l t h o u g h s t i l l e v i d e n t , becomes l e s s and l e s s s i g n i f i c a n t ( F i g . 5.13d). For p e r i o d s l e s s than 15 min. i t was not always p o s s i b l e t o f i n d a model t h a t matched the d a t a , i n d i c a t i n g t h a t e i t h e r the t w o - d i m e n s i o n a l symmetry assumption was no l o n g e r v a l i d a t t h e s e p e r i o d s , or t h a t the c u r r e n t s were perhaps t r a v e l l i n g i n a d i f f e r e n t d i r e c t i o n . The i n d u c t i o n arrow r e s u l t s of Law e t a l (see F i g . 5.2) a r e i n a ccordance w i t h t h i s , as they i n d i c a t e the d i s a p p e a r a n c e of the n o r t h - s o u t h c u r r e n t s between KOS and WHI f o r p e r i o d s l e s s than ~1 h r . The appearance of the Cascade anomaly c u r r e n t s o n l y f o r the l o n g e r p e r i o d s ( 30 min. - 4 h r s . ) i s a t odds w i t h i t s s h a l l o w d e p t h . As i n d i c a t e d i n T a b l e 1.1, the s k i n depths f o r these p e r i o d s a t a c o n d u c t i v i t y of .01 S/m (which i s v e r y h i g h f o r s h a l l o w c r u s t a l r o c k s ) a r e on the o r d e r of hundreds of k i l o m e t e r s . T h i s s u g g e s t s t h a t the c u r r e n t s a r e p r o b a b l y due t o c h a n n e l l i n g of a r e g i o n a l c u r r e n t system, and a r e not due t o 186 l o c a l i n d u c t i o n . The proposed d i r e c t i o n of the segment of the Cascade anomaly s t u d i e d i n t h i s t h e s i s i s shown i n F i g . 5.14, a l o n g w i t h the s o u t h e r n s e c t i o n from Law e t a l , and an approximate n o r t h e r n s e c t i o n from H e n s e l ( 1 9 8 1 ) . The i m p l i c a t i o n of t h i s 'complete' p a t h i s t h a t the Cascade anomaly c u r r e n t s a r e s i m p l y b e i n g c h a n n e l l e d a l o n g some r e l a t i v e l y c o n d u c t i v e path i n t o the Puget Sound. T h i s appears t o r u l e out the p o s s i b i l i t y of a c o n d u c t i v e c o n d u i t c o n n e c t i n g the c h a i n of Cascade v o l c a n o e s . 187 F i g . 5.14 A - f i n a l map o f t h e C a s c a d e a n o m a l y s h o w i n g t h e a p p r o x i m a t e p o s i t i o n s o f p o r t i o n s a s e s t i m a t e d b y : ( a ) Law e t a l ( 1 9 8 0 ) » » • » — ( b ) T h i s t h e s i s — I — i — i — i — ( c ) H e n s e l (1981) 188 CONCLUSIONS A d e t a i l e d i n v e s t i g a t i o n h a s b e e n made o f t h e a s s u m p t i o n s a n d l i m i t a t i o n s i n h e r e n t i n t h e t r a d i t i o n a l m e t h o d s u s e d i n G.D.S. T h e a p p r o x i m a t i o n o f a n i n d u c t i o n t e n s o r l i n e a r l y r e l a t i n g t h e n o r m a l a n d a n o m a l o u s f i e l d s i s f o u n d t o be p r o b a b l y v a l i d a t m i d l a t i t u d e s , f o r p e r i o d s g r e a t e r t h a n ~ 2 h r s . H o w e v e r , a s s h o r t e r p e r i o d s a r e u s e d , o r i n r e g i o n s w h e r e t h e i n d u c i n g f i e l d c a n n o t be c o n s i d e r e d h o r i z o n t a l l y u n i f o r m , i t s v a l i d i t y w i l l d e g r a d e , a n d u l t i m a t e l y w i l l f a i l . T h e i n d u c t i o n a r r o w s d e r i v e d f r o m t h e i n d u c t i o n t e n s o r w i l l s h a r e t h e same l i m i t a t i o n s . As w e l l , i t h a s b e e n shown u s i n g v e r y s i m p l e e x a m p l e s t h a t t h e a r r o w s w i l l n o t n e c c e s s a r i l y p o i n t t o w a r d s c u r r e n t c o n c e n t r a t i o n s , b u t r a t h e r w i l l p o i n t t o w a r d s h i g h r e l a t i v e c o n d u c t i v i t i e s , f o r b o t h i n d u c e d a n d c h a n n e l l e d t y p e s o f a n o m a l i e s . T h e v a r i o u s q u a n t i t a t i v e m e a s u r e s commonly e m p l o y e d i n G.D.S. t o d e t e r m i n e d e p t h s o f c u r r e n t s , l a t e r a l e x t e n t s o f a n o m a l i e s , a n d s c a l e l e n g t h s , h a v e b e e n shown t o be o f l i m i t e d u s e f u l n e s s , a n d i n f a c t , o ne e s t i m a t o r f o r t h e s c a l e l e n g t h ( P o r a t h e t a l , 1971) i s r e v e a l e d t o be e r r o n e o u s . I t i s a l s o c o n c l u d e d t h a t q u a n t i t a t i v e m o d e l l i n g o f t h e c o n d u c t i v i t y s t r u c t u r e a t t h i s s t a g e i n i t s d e v e l o p m e n t i s n o t a l w a y s t h e mos t p r a c t i c a l way t o p r o c e e d , a s t h e m o d e l l i n g i s e x p e n s i v e a n d 189 t i m e c o n s u m i n g b e c a u s e o f t h e n o n - l i n e a r i t y o f t h e i n d u c t i o n f o r m u l a t i o n . A s w e l l , b e c a u s e o f t h e p o s s i b i l i t y t h a t t h e a n o m a l y i s due t o c h a n n e l l i n g o f r e g i o n a l l y i n d u c e d c u r r e n t s y s t e m s , t h e e x t e n t o f t h e r e g i o n o v e r w h i c h t h e m o d e l l i n g i s t o be d o n e i s a l w a y s i n d o u b t . T h e f a i l u r e o f m o d e l l i n g e f f o r t s i n c e r t a i n s t u d i e s h a s b e e n b l a m e d on t h i s i n a b i l i t y t o f i n d t h e p r o p e r r e g i o n o f i n t e r e s t ( W h i t h a m a n d A n d e r s e n , 1965; P o r a t h e t a l , 1 9 7 1 ) . T o a v o i d t h e s e d i f f i c u l t i e s a n d t o a l s o p u t t h e p r o b l e m i n l i n e a r f o r m , i t i s s u g g e s t e d i n C h a p t e r IV t h a t t h e a n o m a l y be m o d e l l e d i n t e r m s o f c u r r e n t d e n s i t y r a t h e r t h a n c o n d u c t i v i t y , w i t h an i n i t i a l s i m p l i f i c a t i o n o f t w o - d i m e n s i o n a l i t y . The r e g i o n o f i n t e r e s t o f t h e m o d e l i s e a s i l y d e t e r m i n e d i n t h i s f o r m u l a t i o n , a n d t h e l i n e a r i t y a l l o w s f o r f a s t a n d i n e x p e n s i v e c o m p u t a t i o n s f o r b o t h t h e f o r w a r d a n d i n v e r s e p r o b l e m s . The m a j o r d i s a d v a n t a g e o f t h e c u r r e n t d e n s i t y a p p r o a c h i s t h e n o n -u n i q u e n e s s t h a t i s i n h e r e n t i n t h i s a p p r o a c h . T h e e x t e n t a n d t y p e o f t h i s n o n - u n i q u e n e s s i s e x p l o r e d u s i n g t h e a v e r a g i n g f u n c t i o n s o f B a c k u s - G i l b e r t a p p r a i s a l ( 1 9 6 7 , 1 9 6 8 , 1 9 7 0 ) . I t i s f o u n d t h a t t h e s u r f a c e d a t a w i l l c o n s t r a i n t h e r a n g e o f p o s s i b l e m o d e l s s u c h t h a t r e s o l u t i o n o f t h e m a j o r h o r i z o n t a l f e a t u r e s o f t h e t r u e m o d e l w i l l be a p p a r e n t i n a l l c o n s t r u c t e d m o d e l s . H o w e v e r , t h e a v e r a g i n g f u n c t i o n s i n d i c a t e t h a t c o n s t r a i n i n g t h e m o d e l c o n s t r u c t i o n w i t h t h e d a t a a l o n e w i l l r e s u l t i n no r e s o l u t i o n o f t h e t r u e m o d e l ' s v e r t i c a l f e a t u r e s . T h e s e c o n c l u s i o n s were c o n f i r m e d by t h e c o n s t r u c t i o n o f t h e 190 L^-norm w e i g h t e d s m a l l e s t models, w i t h the d i f f e r e n t models t h a t f i t the d a t a a l l c o r r e c t l y i n d i c a t i n g the h o r i z o n t a l p o s i t i o n s of the c u r r e n t elements i n the t r u e model, but w i t h none of the models g i v i n g an a c c u r a t e v e r t i c a l placement. To overcome t h i s u n i q u e n e s s d i f f i c u l t y , c e r t a i n e x p e c t e d p h y s i c a l f e a t u r e s of the t r u e model were i n c o r p o r a t e d or f a v o u r e d i n the model c o n s t r u c t i o n . I t was suggested t h a t many anomalies would be l o c a l i z e d and would be s p a r s e l y d i s t r i b u t e d , and so a w e i g h t e d L,-norm o b j e c t i v e f u n c t i o n was i n t r o d u c e d f o r the c o n s t r u c t i o n of a p a r a m e t e r i z e d c u r r e n t d e n s i t y model u s i n g l i n e a r programming. The c u r r e n t s were a l s o presumed t o be due m a i n l y t o f i r s t o rder i n d u c t i o n , so t h a t no s i g n i f i c a n t c u r r e n t s i n the model would be more than Tr/2 d i f f e r e n t i n phase. With these c o n s t r a i n t s and model f e a t u r e s i n c o r p o r a t e d i n the l i n e a r programming c o n s t r u c t i o n , i t was found t h a t both v e r t i c a l and h o r i z o n t a l r e s o l u t i o n of the f e a t u r e s of l o c a l i z e d , s p a r s e l y d i s t r i b u t e d t r u e models was now p o s s i b l e . T h i s a b i l i t y of the c o n s t r u c t i o n a l g o r i t h m t o r e c o v e r the major f e a t u r e s of the t r u e model was found t o be s t a b l e w i t h r e s p e c t t o the w e i g h t i n g f a c t o r ( f o r << £ 1 ) , which a l l o w e d t h i s f a c t o r t o be f i x e d a t << = 1 f o r a l l subsequent m o d e l l i n g . As w e l l , the s u c c e s s of the a l g o r i t h m p e r s i s t e d i n the p r e s e n c e of r e a s o n a b l e amounts of n o i s e (up t o 10% of the maximum d a t a v a l u e f o r s i m p l e models; and 1-2% of the maximum data v a l u e f o r more complex m o d e l s ) . Even when d a t a was i n v e r t e d u s i n g as t r u e models c u r r e n t d e n s i t y c o n f i g u r a t i o n s t h a t were l o c a l i z e d o n l y i n one d i r e c t i o n , the 191 r e s u l t a n t c o n s t r u c t e d m o d e l s were s t i l l a g o o d r e p l i c a t i o n o f t h e t r u e m o d e l . I n t h e f i n a l c h a p t e r t h e l i n e a r p r o g r a m m i n g c o n s t r u c t i o n was a p p l i e d t o r e a l d a t a m e a s u r e d a t s t a t i o n s c r o s s i n g t h e C a s c a d e a n o m a l y i n W a s h i n g t o n S t a t e . T he d a t a was c h e c k e d , a n d f o u n d t o be i n g o o d a c c o r d w i t h t h e a s s u m p t i o n o f two-d i m e n s i o n a l i t y . As w e l l , t h e n e a r l y l i n e a r p o l a r i z a t i o n o f t h e s i g n a l a t two s t a t i o n s i n d i c a t e d t h a t t h e a n o m a l y p r o b a b l y c o n s i s t e d o f o n l y a s i n g l e l o c a l i z e d c u r r e n t . A f t e r r o t a t i n g t h e d a t a a n d s t a t i o n p o s i t i o n s i n t o t h e p r o p e r r e f e r e n c e f r a m e , c u r r e n t d e n s i t y a m p l i t u d e m o d e l s were c o n s t r u c t e d o v e r a r a n g e o f p e r i o d s f r o m 20 m i n . t o 4 h r s . a n d i t was f o u n d t h a t a l l m o d e l s r e q u i r e d a l o c a l i z e d c u r r e n t n e a r t h e s t a t i o n MUD, a t a d e p t h o f f r o m 1 1 - 2 1 km. The mean v a l u e o f d e p t h o f 15.7 + 4.2 km. f r o m a l l t h e m o d e l s i s i n g o o d a g r e e m e n t w i t h t h e e s t i m a t e d d e p t h o f Law e t a l ( 1 9 8 0 ) f o r t h e s o u t h e r n p o r t i o n o f t h e a n o m a l y . T h e t h r e e s u g g e s t e d s e g m e n t s f r o m Law e t a l ( 1 9 8 0 ) , t h i s t h e s i s , a n d H e n s e l ( 1 9 8 1 ) show g o o d c o n t i n u i t y , a n d i n d i c a t e t h a t t h e c u r r e n t p a t h d o e s n o t e x t e n d t o t h e n o r t h e r n C a s c a d e v o l c a n o e s . T h e s h a l l o w n e s s o f t h e s i g n i f i c a n t c u r r e n t s i n t h e m o d e l s i n c o m p a r i s o n w i t h t h e s k i n d e p t h s a t t h e i r r e s p e c t i v e p e r i o d s i n d i c a t e s t h a t t h e a n o m a l y i s p r o b a b l y due t o t h e c h a n n e l l i n g o f r e g i o n a l c u r r e n t s t h r o u g h a l o c a l h i g h c o n d u c t i v i t y f e a t u r e . 1 92 BIBLIOGRAPHY A k a s o f u , S - I . , ( 1 9 7 9 ) , ' D y n a m i c s o f t h e M a g n e t o s p h e r e ' , A k a s o f u ( e d . ) , D. R e i d e l P u b l . Co., p g . 4 4 7 - 4 6 0 . 'What i s a M a g n e t o s p h e r i c S u b s t o r m ' A k a s o f u , S - I . a n d Chapman, S., ( 1 9 6 1 ) , J . G e o p h y s . R e s . , 66 , p g . 1 3 2 1 - 1 3 5 0 . 'The R i n g C u r r e n t , G e o m a g n e t i c D i s t u r b a n c e s , a n d t h e Van A l l e n R a d i a t i o n B e l t s ' A l a b i , A.O., C a m f i e l d , P.A., a n d G o u g h , D . I . , ( 1 9 7 5 ) , G e o p h y s . J . R. a s t r . S o c , 43 , p g . 81 5-833. 'The N o r t h A m e r i c a n C e n t r a l P l a i n s C o n d u c t i v i t y A n o m a l y ' A l f e n , H., ( 1 9 5 0 ) , ' C o s m i c a l E l e c t r o d y n a m i c s ' , O x f o r d U n i v e r s i t y P r e s s , O x f o r d . A n d e r s s e n , R.S., ( 1975), Phys. E a r t h P l a n e t . I n t e r . , J_0 , pg. 292-298. 'On the I n v e r s i o n of G l o b 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 Data' Backus, G. and G i l b e r t , F., (1967), Geophys. J . R. a s t r . S o c , J_3 , pg. 247-276. 'Numerical A p p l i c a t i o n s of a Formalism f o r G e o p h y s i c a l I n v e r s e Problems' Backus, G. and G i l b e r t , F., (1968), Geophys. J . R. a s t r . S o c , 1 6 , pg. 169-205. 'The R e s o l v i n g Power of Gross E a r t h Data' Backus, G. and G i l b e r t , F., ( 1 9 7 0 ) , P h i l . T r a n s . Roy. S o c , A266 , pg. 123-192. 'Uniqueness i n the I n v e r s i o n of I n a c c u r a t e Gross E a r t h Data' B a i l e y , R.C., (1970), P r o c . Roy. Soc. Lond., A315 , pg. 185-194. ' I n v e r s i o n of the Geomagnetic I n d u c t i o n Problem' 194 B a n k s , R . J . , ( 1 9 6 9 ) , G e o p h y s . J . R. a s t r . S o c , _T7 , p g . 457-487. ' G e o m a g n e t i c V a r i a t i o n s a n d t h e E l e c t r i c a l C o n d u c t i v i t y o f t h e U p p e r M a n t l e ' B a n k s , R . J . , ( 1 9 7 3 ) , P h y s . E a r t h P l a n e t . I n t e r . , 7 , p g . 339-348. ' D a t a P r o c e s s i n g a n d 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 ' B a n k s , R . J . , ( 1 9 7 5 ) , G e o p h y s . J . R. a s t r . S o c , 43^ , p g . 8 3 - 1 0 1 . 'Complex D e m o d u l a t i o n o f G e o m a g n e t i c D a t a a n d t h e E s t i m a t i o n o f T r a n s f e r F u n c t i o n s ' B a n k s , R . J . , ( 1 9 7 9 ) , G e o p h y s . J . R. a s t r . S o c , 56 , p g . 139-157. 'The U s e o f E q u i v a l e n t C u r r e n t S y s t e m s i n t h e I n t e r p r e t a t i o n o f G e o m a g n e t i c D e e p S o u n d i n g D a t a ' 195 B e a m i s h , D., ( 1 9 7 7 ) , G e o p h y s . J . R. a s t r . S o c , 50 , p g . 311-332. 'The M a p p i n g o f I n d u c e d C u r r e n t s A r o u n d t h e K e n y a R i f t : A C o m p a r i s o n o f T e c h n i q u e s ' B e c h e r , W.D. a n d S h a r p e , C.B., ( 1 9 6 9 ) , R a d i o S c i e n c e , 4 , p g . 1 0 8 9 - 1 0 9 4 . 'A S y n t h e s i s A p p r o a c h t o M a g n e t o t e l l u r i c E x p l o r a t i o n ' B i r k e l a n d , K., ( 1 9 0 8 ) , 'The N o r w e g i a n A u r o r a P o l a r i s E x p e d i t i o n 1 9 0 2 -1903, V o l . 1 , S e c t i o n 1 , A s c h h o u g , C h r i s t i a n i a . B o o k e r , H.G. a n d Clemmow, P.C., ( 1 9 5 0 ) , P r o c I . E . E . , 97 , P a r t I I I , p g . 11-17. 'The C o n c e p t o f a n A n g u l a r S p e c t r u m o f P l a n e Waves, a n d i t s R e l a t i o n t o t h a t o f P o l a r D i a g r a m a n d A p e r a t u r e D i s t r i b u t i o n ' B o r n , M. a n d W o l f , E . , ( 1 9 7 5 ) , ' P r i n c i p l e s o f O p t i c s ' , P e r g a m o n P r e s s , T o r o n t o . 196 Bostrom, R., ( 1 9 6 4 ) , J . Geophys. Res., 69 , pg. 4 9 8 3 . 'A Model of the A u r o r a l E l e c t r o j e t s ' B r a c e , W.F., (1971), 'The S t r u c t u r e and P h y s i c a l P r o p e r t i e s of the E a r t h ' s C r u s t ' , ed. J.G. 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'The E l e c t r i c a l M a gnetic S t a t e of the I n t e r i o r of the E a r t h as I n f e r r e d from T e r r e s t r i a l M a g n e t i c V a r i a t i o n s ' Cochrane, N.A. and Hyndman, R.D., ( 1 9 7 0 ) , Can. J . E a r t h S c i . , 6 , pg. 1208-1218. D r a g e r t , H., (1973), Ph.D. T h e s i s . 'Broad-band Geomagnetic Depth-Sounding A l o n g an Anomalous P r o f i l e i n the Canadian C o r d i l l e r a ' Dyck, A.V. and G a r l a n d , G.D., (1969), Can. J . E a r t h S c i . , 6 , pg. 513-516. 'A C o n d u c t i v i t y Model f o r C e r t a i n F e a t u r e s of the A l e r t Anomaly i n Geomagnetic V a r i a t i o n s ' E c k h a r d t , D., L a r n e r , K., Madden, T., (1963), J . Geophys. Res., 68 , pg. 6279-6286. ' L o n g - P e r i o d M a g n e t i c F l u c t u a t i o n s and M a n t l e E l e c t r i c a l C o n d u c t i v i t y E s t i m a t e s ' 198 E v e r e t t , J . E . a n d Hyndman, R.D., ( 1 9 6 7 ) , P h y s . E a r t h P l a n e t . 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I n t e r . , 7 , pg. 266-281 . ' I n d u c t i o n i n a L a y e r e d P l a n e E a r t h by Un i f o r m and Non-Un i f o r m Source F i e l d s ' W i g g i n s , R.A., (1972), Rev. Geophys. Space Phys., H) , pg. 251-285. 'The G e n e r a l L i n e a r I n v e r s e Problem: I m p l i c a t i o n of S u r f a c e Waves and Free O s c i l l a t i o n s f o r E a r t h S t r u c t u r e ' 21 1 Woods, D . V . , ( 1 9 7 9 ) , Ph.D. T h e s i s . ' G e o m a g n e t i c D e p t h S o u n d i n g S t u d i e s i n C e n t r a l A u s t r a l i a ' 212 Appendix A Maxwell's Equations i n a Conductor Maxwell's equations i n t h e i r g eneral form a r e : V-D = V - B = 0 V X E = - £ £ . (A.1 ) (A.2) (A.3) VxH = j + IB. (A.4 ) I f we presume that a l l time dependences are of the form e L U j t and that 6 , ^ . are equal to € 0 , f*o everywhere, then the equations s i m p l i f y t o : 213 V - D = (A.5) V-B = 0 (A.6) V x E = - i w B (A.7) V xH = J. + iwD (A.8) E m p l o y i n g t h e c o n s t i t u t i v e r e l a t i o n s f o r an i s o t r o p i c m e d i u m : 5* = c r E (A.9) D = € Q E (A.10) B = fi0H (A.11) 214 we a r r i v e a t t h e s t a r t i n g f o r m o f M a x w e l l ' s e q u a t i o n s f o r a l l p r o b l e m s i n t h i s t h e s i s : V - E = _/>/€o ( A . 1 2 ) V-H = 0 (A . 1 3 ) V x E = 1WI ( A . 1 4 ) V xH = ( <y + i w t o ) E ( A . 1 5 ) A l l i n d u c t i o n p r o b l e m s t h a t w i l l be d e a l t w i t h i n t h i s t h e s i s w i l l c o n s i s t o f a n o n - c o n d u c t i v e h a l f - s p a c e ( a i r ) a n d a c o n d u c t i v e h a l f - s p a c e ( e a r t h ) w i t h t h e s o u r c e i n t h e n o n -c o n d u c t i v e h a l f - s p a c e a t a d i s t a n c e f r o m t h e b o u n d a r y p l a n e ( a s shown i n F i g . 1 . 5 ) . T h e Z a x i s w i l l a l w a y s be p e r p e n d i c u l a r t o t h e b o u n d a r y , w i t h t h e p o s i t i v e d i r e c t i o n downward. W i t h i n t h e e a r t h , t h e l o w e s t c o n d u c t i v i t y e x p e c t e d ( ~ 10 5 S/m) i s s t i l l much g r e a t e r t h a n t h e v a l u e o f w€ 0 c o r r e s p o n d i n g v 215 t o t h e h i g h e s t f r e q u e n c y u s e d i n G.D.S. ( t h e f r e q u e n c y r a n g e i s shown i n T a b l e 2 . 1 ) : = 1 0 ~ 5 S/m » w ^ K € 0 = 5 . 6 x 1 0 ~ l Z S/m ( A.16) T h u s , w i t h i n t h e e a r t h we c a n a l w a y s n e g l e c t t h e d i s p l a c e m e n t c u r r e n t t e r m , s o t h a t A.15 b e c o m e s : V x H = G E f o r z > 0 ( A . 1 7 ) T a k i n g t h e c u r l o f b o t h s i d e s o f e q u a t i o n A.17 a n d u t i l i z i n g e q u a t i o n A.13 on t h e l e f t h a n d s i d e o f t h e s u b s e q u e n t e q u a t i o n , we a r r i v e a t : V l H = -7t7 x E + cr ( $ x E ) ( A . 1 8 ) Upon r e p l a c i n g E f r o m A . 1 7 , a n d V x E f r o m A . 1 4 , we h a v e : v"*H = - ^ p X ( v x H ) + i w j u ^ H ( A . 1 9 ) T h i s i s t h e g e n e r a l e q u a t i o n f o r t h e m a g n e t i c f i e l d i n an i n h o m o g e n o u s e a r t h . The c o r r e s p o n d i n g r e s u l t f o r t h e e l e c t r i c f i e l d i s o b t a i n e d i n a n a l o g o u s f a s h i o n by t a k i n g t h e c u r l o f b o t h s i d e s o f A.14 a n d t h e n s i m p l i f y i n g w i t h A.12 a n d A . 1 7 : 216 v^E = i w ^ f f E + (V/> ) / € 0 ( A . 2 0 ) G o i n g b a c k t o e q u a t i o n A . 1 5 , t a k i n g t h e d i v e r g e n c e o f b o t h s i d e s ( a n d n o t i n g t h a t t h e d i v e r g e n c e o f t h e c u r l o f a v e c t o r i s a l w a y s z e r o ) , i t w i l l a l w a y s be t r u e t h a t : "V • [ ( Cr + i w € 0 ) E ] = 0 ( A . 2 1 ) T h u s , by a p p l i c a t i o n o f t h e d i v e r g e n c e t h e o r e m , t h e v e r t i c a l c o m p o n e n t o f t h e v e c t o r ( O" + i w d e ) E - i s c o n t i n u o u s a c r o s s t h e b o u n d a r y p l a n e s e p a r a t i n g t h e h a l f s p a c e s , s o t h a t a t t h e b o u n d a r y : (OkJr + i w € o ) E 4 ( 0 ; ^ = (CWn, + i w € 0 ) E M c a , ^ ( A . 2 2 ) U s i n g A.16 a n d t h e z e r o c o n d u c t i v i t y o f t h e a i r , we h a v e : ( A . 2 3 ) A g a i n f r o m A . 1 6 , i t i s a p p a r e n t t h a t a t t h e b o u n d a r y p l a n e o f t h e two h a l f - s p a c e s , t h e v e r t i c a l e l e c t r i c f i e l d i n t h e e a r t h a n d c o n s e q u e n t l y t h e v e r t i c a l c u r r e n t s , w i l l be i n s i g n i f i c a n t . I f 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 a f t e r t h e b o u n d a r y , t h e n by s y m metry c o n s i d e r a t i o n s , E-j a n d J t i n t h e e a r t h w i l l a l w a y s be i n s i g n i f i c a n t , r e g a r d l e s s o f t h e s o u r c e t y p e . C o n s i d e r i n g e q u a t i o n A.21 a g a i n , e x p a n d i n g t h e t e r m s , 2 1 7 u t i l i s i n g A . 1 2 , a n d f i n a l l y c o n v e r t i n g ' iw' b a c k t o ^ , l e a d s t o : 1|. -,5a-E, ( A . 2 4 ) I f O" v a r i e s o n l y i n t h e Z d i r e c t i o n , a n d E j f r o m t h e p r e v i o u s d i s c u s s i o n i s i n s i g n i f i c a n t i n t h e e a r t h , t h e n t h e s o u r c e t e r m f o r c h a r g e c r e a t i o n on t h e r i g h t h a n d s i d e o f A. 2 4 i s n e g l i g i b l e , l e a d i n g t o t h e e q u a t i o n g o v e r n i n g t h e r a t e o f d e c a y o f e x i s t i n g f r e e c h a r g e : = 0 ( A . 2 5 ) T h e s o l u t i o n o f t h i s i s : ( A . 2 6 ) E v e n f o r t h e most r e s i s t i v e r o c k ( O" = 1 0 ~ ? S/m) t h e h a l f - l i f e f o r f r e e c h a r g e i s o n l y ~ 1 0 s . E f f e c t i v e l y t h e n , t h e r e w i l l be no f r e e c h a r g e i n t h e e a r t h i f cr = c r ( z ) o n l y ( e x c e p t a t t h e b o u n d a r y ) . As w e l l t h e r e c a n be no f r e e c h a r g e i n t h e n o n -c o n d u c t i n g h a l f - s p a c e . T h u s , e v e r y w h e r e e x c e p t a t t h e b o u n d a r y : V ' E = 0 ( A . 2 7 ) 218 S i m p l i f i c a t i o n o f A.20 i s now p o s s i b l e b e c a u s e o f t h e l a c k o f f r e e c h a r g e , l e a v i n g : V 1 E = <p E ( A . 2 8 ) w h e r e : 9 = -y>^aafic ( A . 2 9 ) z < 0 ( a i r ) 9 = i w / u 0 c r ( A . 3 0 ) z > 0 ( e a r t h ) A g a i n , t h e s e r e s u l t s a r e f o r t h e c a s e w h e r e G" i s a f u n c t i o n o f d e p t h o n l y . I f we now f u r t h e r s t i p u l a t e t h a t t h e v a r i a t i o n o f cr w i t h d e p t h o c c u r s i n a l a y e r e d f a s h i o n , w i t h <3~(z) c o n s t a n t w i t h i n e a c h l a y e r ( g i v i n g VO~ = 0 w i t h i n e a c h l a y e r ) , t h e n A. 19 s i m p l i f i e s t o : ( A . 3 1 ) w i t h i n t h e n ^ l a y e r , w h e r e 9*. i s t h e v a l u e o f ^ i n t h a t l a y e r . 219 A p p e n d i x B C o r r e l a t i o n o f P r i c e ' s I n d u c t i o n S o l u t i o n s w i t h P l a n e Wave As s e e n i n s e c t i o n 2.1 o f C h a p t e r I I , t h e r e q u i r e d n o n -d i v e r g e n c e o f t h e e l e c t r i c f i e l d i n t h e homogenous e a r t h i n d u c t i o n p r o b l e m c a n be s a t i s f i e d i n two ways, l e a d i n g t o two i n d e p e n d e n t t y p e s o f s o l u t i o n s . S o l u t i o n s o f t h e f i r s t t y p e w i l l be c o r r e l a t e d h e r e w i t h waves w i t h a t r a n s v e r s e e l e c t r i c f i e l d ( T E ) a n d t h e s e c o n d t y p e w i l l be c o r r e l a t e d w i t h waves w i t h a t r a n s v e r s e m a g n e t i c f i e l d ( T M ) . T y p e one s o l u t i o n s The f i r s t t y p e s o l u t i o n s f o r t h e e l e c t r i c f i e l d a r e g i v e n by e q u a t i o n s 2.1.20 a n d 2.1.21 i n C h a p t e r I I : S o l u t i o n s E . (x , y , z ) = -CS>* + q O " x * + B (B.1 ) z < 0 220 E , < x , y , z ) = [ C , e - ^ l + ^ V ' l a i ( B . 2 ) z > 0 w i t h P ( x , y ) s a t i s f y i n g ( f r o m 2 . 1 . 1 9 ) : + ILZ. + S^P = 0 "ax 1 ^ V J ' -( B . 3 ) a n d w i t h A,,B,, a n d C 4 s a t i s f y i n g ( f r o m 2 . 1 . 2 4 , 2 . 1 . 2 5 , a n d 2. 1 . 2 6 ) : B, = - A | • [ ( 1 - R ) / ( 1 + R ) ] C, = A ,[2R/( 1+R) ] w h e r e : R = [( ^ +?a>/ ( ^ +9e)] T h e e l e m e n t a r y s o l u t i o n t o B.3 i s : p ( x , y ) = eik** e U 3 * w i t h : (B.4) ( B . 5 ) ( B . 6 ) ( B . 7 ) 221 kx 2 + k y Z = (B.8 ) L e t t i n g : k z e = -i ( + <j>e j ' ' 2 " ( B . 9 ) a n d : k z a = -i ( 0* + <Va. ( B . 1 0 ) a n d u s i n g B.7, t h e n t h e e l e m e n t a r y s o l u t i o n s B.1 a n d B.2 t a k e on s t a n d a r d p l a n e wave f o r m a t ( w h e r e i t i s n o t e d t h a t t h e t i m e d e p e n d e n c e h a s b e e n r e s u r r e c t e d ) : E , ( x , y , z ) = i { k y , - k x , 0 } [ A , e ~ a * * e ^V^U e i u , t + B, e i k i ^ eCk*K e l k 3 U i u l t ( B . 1 1 ) z < 0 ^ / \ • (, , «•> „ k j _ ^ <-kKX i k ¥ w c oo t E , ( x , y , z ) = i { k y , - k x , 0 } C, e K e e He ( B . 1 2 ) z > 0 To s i m p l i f y t h e s e r e s u l t s we r o t a t e by a n a n g l e -o\ a b o u t t h e Z a x i s t o a new h o r i z o n t a l c o o r d i n a t e s y s t e m ( f , Y) ) ( s e e 222 F i g . B.1) s u c h t h a t t h e e l e c t r i c v e c t o r i s o n l y a l o n g |? , a n d h o r i z o n t a l p r o p o g a t i o n i s o n l y i n t h e d i r e c t i o n . T h e e q u a t i o n f o r t h e e l e c t r i c f i e l d b e c o m e s : E , ( ^ , r> f 2 ) «i{S> ,0,0) [A, e ^ 2 e ' ^ e C w t + B, e *°- e L e z < 0 (B.13) E , ( ^ , Y £ , z ) = i { " 9 , 0 , 0 } C | e~ c k*V^ e i u j t ( B . 1 4 ) z > 0 I n t e r m s o f t h e i n c i d e n t a n g l e 9" a n d t h e t r a n s m i t t e d a n g l e © ' (wh e r e t h e a n g l e s may be c o m p l e x ) : ka s i n © = \) (B.15) ka c o s & = k z a (B.16) f o r z < 0 w h e r e : 223 F i g . B.1 R o t a t i n g b y a n a n g l e - o< a b o u t t h e Z a x i s t o a new h o r i z o n t a l c o o r d i n a t e s y s t e m . T h e h o r i z o n t a l d i r e c t i o n o f propagation i s now p u r e l y i n t h e r? d i r e c t i o n . 224 ka = ( S)1- + k z a * ) i s t h e t o t a l wavenumber i n t h e a i r . A l s o : ke s i n = V) ( B . 1 7 ) ( B . 1 8 ) i ke c o s c? = k z e f o r z > 0 w h e r e : ( B . 1 9 ) ke = ( V*1" + k z e * ) ( B . 2 0 ) i s t h e t o t a l wavenumber i n t h e e a r t h . T h e e q u i v a l e n c e o f i n t h e e a r t h a n d t h e a i r , c o m b i n e d w i t h B.15 a n d B.18, r e s u l t s i n S n e l l ' s law ( P a n o f s k y a n d P h i l l i p s , 1962, p g . 1 9 6 - 1 9 7 ) : k a / k e = s i n © / s i n © ( B . 2 1 ) A l s o , e q u a t i n g t h e f i r s t t e r m i n B.13 w i t h t h e i n c i d e n t wave a m p l i t u d e E i , a n d t h e s e c o n d t e r m w i t h t h e r e f l e c t e d wave a m p l i t u d e E r , we h a v e : 225 E r / E i - B,/A, = [ ( - ^ i ) " 1 - i ] (B.22) Upon s u b s t i t u t i o n using B.9, B.10, B.16 and B.19, we a r r i v e at the usual F r e s n e l r e l a t i o n f o r the e l e c t r i c f i e l d f o r the TE mode (Panofsky and P h i l l i p s , 1962, pg.198): E r / E i = [ c o s © - (ke/ka)cos©']/[cos© + (ke/ka)cos ©'] (B.23) In s i m i l a r f a s h i o n we have f o r the t r a n s m i t t e d wave amplitude, Et : E t / E i = C,/At - 2 ' ( )''V[1 + (*l±*S^)'i] (B.24) which a f t e r s u b s t i t u t i o n becomes the second F r e s n e l r e l a t i o n f o r the TE mode (Panofsky and P h i l l i p s , 1962, pg.198): E t / E i = 2 cos B/[ (ke/ka)cos©' + cos© ] (B.25) Type two s o l u t i o n s The second type s o l u t i o n s f o r the magnetic f i e l d are given by equations 2.1.39 and 2.1.40 i n Chapter I I : H i ( X r y . z ) = (i/wp0) ^ {1E*,-Mi,,0} [A 226 (B.26) f o r z < o st<*.y.*> • < i / v . ) s ^ 3 I l i t i a . . - 2 a . o ) c , . - " 1 ^ " * (B.27) f o r z > 0 w i t h F E ( x , y , z ) s a t i s f y i n g ( f r o m 2 . 1 . 3 8 ) : ^ 1 * a n d w i t h A 2 , B Z , C 2 s a t i s f y i n g ( f r o m 2.1.41 a n d 2.1.42) To. Ya. q = A 2 - 2 / [ i + ( - J 5 - ) ! * ] A g a i n , t h e e l e m e n t a r y s o l u t i o n t o B.28 i s : F a ( x , y ) = e i k * K e i k S l w i t h : (B.28) (B.29) (B.30) (B.31 ) 227 k x 2 , + k y * = ^ ( B . 3 2 ) U s i n g t h e d e f i n i t i o n s f o r k z e a n d k z a a s i n B.9 a n d B.10, a n d s u b s t i t u t i n g t h e r e s u l t o f B.31 i n t o B.26 a n d B.27 we o b t a i n t h e s t a n d a r d p l a n e wave f o r m a t f o r t h e m a g n e t i c f i e l d i n t h e t y p e two c a s e : Hz(x,y,z) = - ( i / w « 0 ) - 7 ^ — { k y , - k x , 0 } >[hze e l-'e e ~ B z e e 3 3 e e ] (B.33) f o r z < 0 H t ( x , y , z ) = - ( i / W / i 0 ) - j ^ - { k y , - k x , 0 } C z e : t " x e ' ^ e i k ^ 1 e 1 w t (B.34) f o r z > 0 R o t a t i n g i n t o t h e new c o o r d i n a t e s y s t e m ) , i n i d e n t i c a l f a s h i o n t o t h a t d o n e f o r t h e t y p e o n e s o l u t i o n , t h e m a g n e t i c f i e l d b e c o m e s : 228 •B2e f e e ] f o r z < 0 (B . 3 5 ) H z ( f ,7 ,z) = - ( i / w ^ - - ^ - { S > , 0 , 0 } qe^V^V^ ( B . 3 6 ) f o r z > 0 • F o l l o w i n g t h e same p r o c e d u r e a s f o r t h e t y p e o ne s o l u t i o n , S n e l l ' s law i s a g a i n r e c o v e r e d , a n d a l s o t h e F r e s n e l r e l a t i o n s f o r t h e TM mode ( P a n o f s k y a n d P h i l l i p s , 1962, p g . 1 9 8 ) : H r / H i = - B a / A ^ = [ ( k e / k a ) c o s © - c o s ©' ] / [ ( k e / k a ) c o s © + cos6 ] ( B . 3 7 ) H t / H i = Cz/hi = [ 2 ( k e / k a ) c o s 6 ] / [ ( k e / k a ) c o s © + c o s S ' ] (B.38) 229 A p p e n d i x C T h e U n i f o r m F i e l d A s s u m p t i o n A common a s s u m p t i o n i n b o t h G.D.S. a n d m a g n e t o t e l l u r i c s i s t h a t t h e e l e c t r o m a g n e t i c p l a n e waves c o m p r i s i n g t h e p r i m a r y f i e l d c a n a l l be c o n s i d e r e d t o be p r o p a g a t i n g v e r t i c a l l y d ownward, s o t h a t t h e f i e l d i s h o r i z o n t a l l y u n i f o r m . T h i s a s s u m p t i o n i n h e r e n t l y i m p l i e s t h a t a l l waves o f s i g n i f i c a n t a m p l i t u d e i n t h e s o u r c e f i e l d must a d h e r e t o t h r e e c o n d i t i o n s : (1) T h e a n g l e f r o m t h e v e r t i c a l o f t h e d i r e c t i o n o f p r o p a g a t i o n o f t h e wave t r a n s m i t t e d i n t o t h e e a r t h i s n e a r z e r o . (2) The h o r i z o n t a l w a v e l e n g t h o f t h e t r a n s m i t t e d wave i s much g r e a t e r t h a n t h e l a t e r a l e x t e n t o f a n y a n o m a l y , s o t h a t t h e r e w i l l be l i t t l e h o r i z o n t a l v a r i a t i o n i n t h e i n d u c i n g f i e l d a c r o s s t h e a n o m a l y . (3) T h e c o m p l e x m a g n i t u d e o f t h e t r a n s m i t t e d wave ( f o r a n o r m a l i z e d i n c i d e n t wave) i s t h e same f o r a l l w a v e s . 230 F r o m e q u a t i o n 2.1.51 o r 2.1.52, a n d u s i n g e q u a t i o n 2.1.57, t h e d o w n g o i n g e l e c t r o m a g n e t i c wave i n t h e e a r t h c a n be e x p r e s s e d a s : -» - i\>e - i f e u i c u t F = F 0 e i e e (C . 1 ) where F i s t h e t o t a l h o r i z o n t a l d i s t a n c e a n d F c a n be e i t h e r t h e e l e c t r i c o r m a g n e t i c v e c t o r . E q u a t i o n C.1 c a n be r e w r i t t e n a s a p r o d u c t o f d e c a y i n g a n d p r o p o g a t i n g p a r t s o f t h e wave: ( C . 2 ) N o t e t h a t f r o m e q u a t i o n 2.1.59 t h e i m a g i n a r y p a r t o f k % e i s a l w a y s n e g a t i v e , s o t h a t e I d o g s -n f a c f c r e p r e s e n t A d e c a y w i t h i n c r e a s i n g d e p t h . T h e p r o p o g a t i o n i n t h e P d i r e c t i o n i s r e p r e s e n t e d by e L 5 , a n d t h a t i n t h e z d i r e c t i o n by - t R c c J C k ^ H , _ e , s o t h a t t h e a n g l e o f p r o p o g a t i o n , m e a s u r e d f r o m t h e v e r t i c a l i s g i v e n b y : 9 = Tan"* [v»/-Real(kze) ] = Tan" 1 [ V / i m a g ( 1 + i p ) ] (C. 3 ) w i t h t h e d e f i n i t i o n o f "p t h e same a s i n e q u a t i o n 2.1.35. T h e d e p e n d e n c e o f B on p i s p l o t t e d i n F i g . C . 1 . I t i s s e e n t h a t f o r p > 100, a l l waves p r o p o g a t e v i r t u a l l y v e r t i c a l l y downward. U s i n g t h e e x p e c t e d p h y s i c a l r a n g e o f g i v e n i n T a b l e s 2.2 a n d 2.3, a n d i n t e r p r e t i n g f o r a m i d r a n g e c o n d u c t i v i t y o f .05 S/m, i t 2 3 1 a rsi. (N CD UJ Q a .to CLZ~ I— L J X a oo a a 1 .0 ^ r 3.0 5.0 LOG rBETfl) 7.0 9.0 T h e p r o p a g a t i o n a n g l e f r o m t h e v e r t i c a l o f t h e t r a n s m i t t e d wave i n t o a h a l f - s p a c e o f u n i f o r m c o n d u c t i v i t y . T h e a n g l e i s p l o t t e d a s a f u n c t i o n o f t h e d i m e n s i o n l e s s p a r a m e t e r , jg . 232 i s s e e n t h a t f o r l a r g e v a l u e s o f t h e h o r i z o n t a l w a v e l e n g t h ( ?\ > 3 5x10 km.) t h i s m a g n i t u d e o f B i s a t t a i n e d f o r a l l p e r i o d s l e s s t h a n ~ 2 h o u r s . T h u s , w i t h i n t h e s e p a r a m e t e r b o u n d a r i e s t h e f i r s t c o n d i t i o n w i l l be s a t i s f i e d . A s w e l l , t h e l a r g e h o r i z o n t a l w a v e l e n g t h s r e q u i r e d w i l l c e r t a i n l y g u a r a n t e e t h a t t h e s e c o n d c o n d i t i o n i s a l s o s a t i s f i e d . T h e r e m a i n i n g c o n d i t i o n , i f met, w i l l e n s u r e t h a t t h e m a g n i t u d e o f t h e a n o m a l o u s r e s p o n s e w i l l be t h e same f o r a l l s i g n i f i c a n t w a v e s i n t h e s o u r c e f i e l d . T h e r e l a t i o n s b e t w e e n t h e c o m p l e x m a g n i t u d e s o f t h e i n c i d e n t ' ( A ) , r e f l e c t e d ( B ) , a n d t r a n s m i t t e d (C) waves a r e d e f i n e d i n e q u a t i o n s 2.1.24 - 2.1.30, w i t h t h e r e s u l t s b e i n g : B = f ( B )• A ( C . 4 ) C = A + B = [ 1 + f(p ) ] A ( C . 5 ) T h e a m p l i t u d e o f f ( B ) i s p l o t t e d i n F i g . 2 . l a . B e t w e e n B = 100 a n d B - » o o , |f(p> )| v a r i e s by a f a c t o r o f 2, s o t h a t c o n d i t i o n (3) i s n o t s a t i s f i e d f o r t h i s r a n g e o f p . H o w e v e r , i n some m e t h o d s o f a n a l y s i s u s e d i n G.D.S., w h i c h w i l l be d i s c u s s e d i n C h a p t e r I I I , one i s c o n c e r n e d o n l y w i t h t h e r e l a t i o n b e t w e e n t h e ' n o r m a l s u r f a c e f i e l d , a n d t h e a d d i t i o n t o t h e ' n o r m a l ' s u r f a c e 233 f i e l d due t o an i n t e r i o r a n o m a l o u s r e g i o n , ( t h a t i s , t h e a n o m a l o u s f i e l d ) . The ' n o r m a l ' f i e l d i s t h e t o t a l s u r f a c e f i e l d t h a t w o u l d e x i s t i n t h e a b s e n c e o f a n y a n o m a l o u s v a r i a t i o n s f r o m a one d i m e n s i o n 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 t h i s c a s e o n l y t h e r a t i o o f C t o (A + B) must r e m a i n c o n s t a n t f o r a l l w aves t o a c t a l i k e , a n d t h i s i s a l w a y s g u a r a n t e e d by t h e c o n t i n u i t y o f t h e m a g n e t i c f i e l d a c r o s s a b o u n d a r y , a s e x p r e s s e d i n e q u a t i o n C . 4 . T h u s , i n t h i s c a s e t h e t h i r d c o n d i t i o n i s n o t r e q u i r e d , s o t h a t t h e u n i f o r m f i e l d a s s u m p t i o n i s v a l i d f o r t h e p a r a m e t e r r a n g e s s a t i s f y i n g t h e f i r s t two c o n d i t i o n s . I t s h o u l d be e m p h a s i z e d t h a t t h i s w o r k s o n l y f o r m e t h o d s u s i n g t h e r e l a t i o n b e t w e e n t h e t o t a l ' n o r m a l ' s u r f a c e f i e l d a n d t h e a n o m a l o u s s u r f a c e f i e l d . I t w i l l n o t be t r u e f o r m e t h o d s w h i c h r e l a t e t h e m a g n i t u d e o f t h e i n c i d e n t f i e l d t o t h e a n o m a l o u s f i e l d , a s t h a t w i l l r e q u i r e t h a t c o n d i t i o n (3) be f u l f i l l e d . 234 Appendix D S e p a r a t i o n of the E x t e r n a l and I n t e r n a l F i e l d s In i l l u s t r a t i o n of the more g e n e r a l s e p a r a t i o n f o r m u l a s d e r i v e d by S i e b e r t and K e r t z (1957) and Weaver (1963), the s i m p l i f i e d case of a two d i m e n s i o n a l e a r t h w i l l be c o n s i d e r e d . Presume t h a t n e i t h e r the i n d u c i n g f i e l d nor the e a r t h s t r u c t u r e v a r i e s i n the 'y' d i r e c t i o n (see F i g . 4.1), and a l s o t h a t the s t a t i o n a r r a y i s alon g " the s u r f a c e a t r i g h t a n g l e s t o the y a x i s . Thus, the magnetic f i e l d w i l l o n l y have d i r e c t i o n a l components X £ , X j and Z C , Z T i n the 'x' and 'z' d i r e c t i o n s r e s p e c t i v e l y . The s u b s c r i p t s 'E' and ' I ' w i l l denote whether the f i e l d component i s of e x t e r n a l or i n t e r n a l o r i g i n . Let a s i n g l e e x t e r n a l l i n e c u r r e n t , I , a t the p o s i t i o n x=0,z=-H, be the sour c e of the f i e l d s a t the s u r f a c e . The measured f i e l d a t any a r r a y p o s i t i o n x, from Ampere's law w i l l be: X c ( x ) = (/i 0I/2Tr)- [H/(H l+x*)] (D.1 ) Z E ( x ) = -(/i 0I/2Tr) [ x / ( H l + x 1 ) ] 2 3 5 We w i l l n e e d t o know t h e g e n e r a l ( R y s h i k a n d G r a d s t e i n , 1 9 6 3 , p g . } [a/(x*+al>] =TTe"^ l a (D.2) F o u r i e r t r a n s f o r m e x p r e s s i o n s 250) : (D.3) } [ x / ( x l + a 1 ) ] = - f r i e ~ ^ l a s g n ( ^ ) (D.4) w h e r e t h e F o u r i e r t r a n s f o r m , J , o f a f u n c t i o n o f x, f ( x ) , i s d e f i n e d h e r e t o b e : oo } [ f ( x ) ] = | * f ( x ) e ' ^ X (D.5) a n d t h e ' s g n ' f u n c t i o n i s 1 o r -1 d e p e n d i n g on t h e s i g n o f i t s a r g u m e n t . U s i n g t h e s e , t h e s p a t i a l F o u r i e r t r a n s f o r m o f t h e m e a s u r e d f i e l d s a l o n g t h e s u r f a c e w i l l b e : J [ X E ( x ) ] = (jUeI/2fr) [ T r e ~ ' f l H ] (D.6) } [ Z E ( x ) ] = (ji0I/2n) [TTeWsgnflj)] T h u s : (D.7) 236 } [ Z E ( x ) ] / J [ X E ( x ) ] = i s g n ( ^ ) = jf [-1/(tTx)] (D.8) (from L i g h t h i l l , 1958, pg. 43). We can r e w r i t e equation D.8: 3 [ Z E ( x ) ] = J [ - l / ( t f x ) ] J [ X E ( x ) ] (D.9) Then, using the F a l t u n g , or c o n v o l u t i o n theorem,(Bracewell, 1965, pg.25) we can r e w r i t e the r e l a t i o n between X E and Z E in the frequency domain as a c o n v o l u t i o n in the time domain: CO Z E ( x ) = f X E ( x * ) • 1 / [ T T ( x - x ' ) ] d x ' -co (D.10) The c o n v o l u t i o n with the f u n c t i o n [-1/(TTx)] i s the H i l b e r t transform ( B r a c e w e l l , 1965, pg. 267). Denoting t h i s o p e r a t i o n by the symbol 'K', D.10 becomes: Z E ( x ) = K[X E(x) ] (D.11) Noting that the r e c i p r o c a l of sgn(^) i s s t i l l s g n ( ^ ) , we c o u l d a l s o have r e w r i t t e n equation D.8 i n the form: J r x £ ( x ) ] = which r e s u l t s i n : - i s g n ( ^ ) J [ Z E ( x ) ] (D.12) 237 X E ( x ) = - K [ Z £ ( x ) ] (D.13) The l i n e a r i t y o f A m p e r e ' s law a l l o w s u s t o e x t e n d t h i s r e s u l t f o r one e x t e r n a l l i n e c u r r e n t , t o t h e g e n e r a l c a s e o f an a r b i t r a r y number a n d d i s t r i b u t i o n o f e x t e r n a l l i n e c u r r e n t s . C o n s i d e r now t h a t t h e s o u r c e i s a l i n e c u r r e n t a t a d e p t h H b e l o w t h e s u r f a c e . F r o m A m p e r e ' s law t h e f i e l d X j , Z-^  a t t h e s u r f a c e w i l l b e : X j U ) = - ( / i 0 I / 2 t r ) [ H / ( x * + H x ) ] (D.14) Z j ( x ) = - ( ^ 0 I / 2 1 T ) [ x / ( x 1 + H 1 ) ] (D.15) F o l l o w i n g t h e same a r g u m e n t s a s b e f o r e , t h e s e e q u a t i o n s l e a d t o : X I ( x ) = K [ Z x ( x ) ] (D.16) Z 1 ( x ) = - K [ X x ( x ) ] (D.17) A g a i n , t h i s r e s u l t c a n be e x t e n d e d t o an a r b i t r a r y d i s t r i b u t i o n o f i n t e r n a l c u r r e n t s . 238 T h e t o t a l f i e l d s m e a s u r e d a t t h e s u r f a c e w i l l b e : X ( x ) = X-j-(x) + X E ( x ) (D.18) Z(x) = Z x(x) + Z E(x) (D.19) T a k i n g t h e H i l b e r t t r a n s f o r m o f t h e m e a s u r e d t o t a l d a t a g i v e s : K [ x ( x ) ] = K[XJ(X)] + K [ X E ( x ) ] = -Z x(x) + Z E(x) (D.20) a n d : K[Z(x)] = K[Z x(x)] +K[ZE(x)] = X X ( x ) - X £ ( x ) (D.21) T h u s , t h e p o r t i o n o f t h e m e a s u r e d f i e l d t h a t i s due t o i n t e r n a l s o u r c e s c a n be s e p a r a t e d : X T ( x ) = (K[Z(x)] + X ( x ) } / 2 (D.22) 239 Zj - U ) = ( Z ( x ) - K [ X ( x ) ] } / 2 (D.23) I t s h o u l d be n o t e d t h a t t h e H i l b e r t t r a n s f o r m o f a c o n s t a n t g i v e s a z e r o r e s u l t / s o t h a t i n t h e c a s e o f a c o n s t a n t f i e l d o f e i t h e r i n t e r n a l o r e x t e r n a l o r i g i n , e q u a t i o n s D.22 a n d D.23 w i l l n o t be a b l e t o s e p a r a t e t h e c o m p o n e n t s , a n d w i l l m e r e l y d i v i d e t h e t o t a l f i e l d i n t o e q u a l p a r t s o f e a c h c o m p o n e n t . 240 A p p e n d i x E D e t e r m i n a t i o n o f t h e I n d u c t i o n T e n s o r E l e m e n t s The f o l l o w i n g m e t h o d o f o b t a i n i n g v a l u e s f o r t h e i n d u c t i o n t e n s o r e l e m e n t s i s due t o S c h m u c k e r ( 1 9 7 0 ) . O t h e r m e t h o d s h a v e b e e n s u g g e s t e d by E v e r e t t a n d Hyndman ( 1 9 6 7 ) , a n d Woods ( 1 9 7 9 , p g . 5 3 ) . The a n o m a l o u s a n d n o r m a l f i e l d s a t a n y f r e q u e n c y 'w' a r e c o n s i d e r e d r e l a t e d by t h e i n d u c t i o n t e n s o r , b u t w i t h t h e p o s s i b i l i t y t h a t a p o r t i o n o f t h e a n o m a l o u s f i e l d c a n n o t be c o r r e l a t e d w i t h t h e f i e l d : / CHH r D A = Cl>D ZA / V c * u H, ( E . 1 ) T h e v a l u e s o f t h e t e n s o r e l e m e n t s a r e c o m p l e x , w i t h t h e r e a l a n d i m a g i n a r y p a r t s r e p r e s e n t e d h e r e by s u p e r s c r i p t I o r R a s shown: C = C R + i r x HH HH H H ( E . 2 ) T h e a u t o a n d c r o s s p o w e r s o f two s i g n a l s a ( t ) a n d b ( t ) o f e q u a l l e n g t h T c w i l l be d e f i n e d a s : 241 S f l B = A ( w ) B * ( w ) / T 0 ( E . 3 ) w h e r e A(w) a n d B(w) a r e t h e F o u r i e r t r a n s f o r m s o f a ( t ) a n d b ( t ) r e s p e c t i v e l y , a n d t h e s t a r '*' d e n o t e s t h e c o m p l e x c o n j u g a t e . The d e s i r e d v a l u e s o f t h e t e n s o r a r e t h o s e w h i c h m i n i m i z e t h e u n c o r r e l a t e d t e r m s , Dr a n d . C o n s i d e r t h e l a s t a n o m a l o u s v e c t o r c o m p o n e n t : z f l " C * H H N + CfcD d N + Z N +Z f (E.4) The v a l u e s o f Zc a n d Zj a r e t h e n : a n d : Z £ = "(CJH + i c \ H ) . H N - (C.J 0 + i C ^ ) - D - ( C ^ + i C ^ )• Z w + z A Z* = - i c J H ) . H j - ( C ^ - i c J D ) . D J ( E . 5 ) ( C 4 % - i C ^ ) - Z N + Z A (E.6)' w h e r e we h a v e e x p a n d e d t h e t e n s o r t e r m s i n t o b o t h r e a l a n d i m a g i n a r y t e r m s . Z^, a n d t h u s t h e a u t o p o w e r o f Z r , i s d e p e n d e n t on s i x i n d e p e n d e n t v a r i a b l e s , a s s e e n i n E.6, s o t h a t t h e minimum m u s t o c c u r when t h e d e r i v a t i v e o f i s z e r o w i t h r e s p e c t t o e a c h o f them: 242 i^i£lL = 0 ( E . 7 ) ( E . 8 ) S^L?£ = 0 ; *Sit}f = 0 ( E . 9 ) U s i n g t h e c h a i n r u l e ( w h e r e 'u' i s a n y o f t h e i n d e p e n d e n t v a r i a b l e s ) we h a v e : tlmL= ( Z r / T C ) ^ L + ( Z r / T c ) l ^ -} U, ( E . 1 0 ) E v a l u a t i n g t h i s u s i n g e q u a t i o n s E.5 a n d E.6 f o r e a c h o f t h e c a s e s , s i x s e p a r a t e e q u a t i o n s a r e o b t a i n e d : (E.11 ) "Z*H M + 1S H* = 0 ( E . 1 2 ) 243 4 D N + V D * = 0 ( E . 1 3 ) -Z*DK + Z f D * = 0 ( E . 1 4 ) zJzN + Z f Z* = 0 ( E . 1 5 ) - Z * Z N + ZfZl = 0 ( E . 1 6 ) F o r e a c h o f t h e p a i r s (E.11 a n d E . 1 2 ) , ( E . 1 3 a n d E . 1 4 ) , a n d ( E . 1 5 a n d E . 1 6 ) t h e o n l y p o s s i b l e s o l u t i o n s a r e e i t h e r t h a t H N , D N , a n d Z N a r e z e r o , w h i c h m a x i m i z e s Z^, o r t h a t : 4 H * = z r H * = 0 ' H * * ° ( E . 1 7 ) Z* Dw = z r O j = 0 , D w * 0 ( E . 1 8 ) 244 Z £ Z M - % Z * « 0 , Z N * 0 ( E . 1 9 ) T h i s i s i n f a c t a r e s t a t e m e n t o f t h e o r i g i n a l p r o b l e m , a s e a c h o f E . 1 7 , E . 1 8 a n d E.19 i s m e r e l y s t a t i n g t h a t Z i s u n c o r r e l a t e d w i t h t h e n o r m a l f i e l d , w i t h : S H „ ^ = o ; s ^ H w = o ( E . 2 0 ) ( E . 2 1 ) I n s e r t i n g t h e e x p r e s s i o n s f o r lg e a c h o f t h e a b o v e e q u a t i o n s , e q u a t i o n s : C * t f S H i o H w + C * D S D N H N + C 2 * ( E . 2 2 ) if. a n d Z c f r o m E.5 a n d E.6 i n t o we a r r i v e a t t h e s e t o f l i n e a r 2N H* = Stk H w ( E . 2 3 ) 245 ( E . 2 4 ) ( E . 2 5 ) w h i c h a l l o w u s t o c a l c u l a t e t h e v a l u e s o f t h e t h r e e i n d u c t i o n t e n s o r e l e m e n t s : f 2H ( E . 2 6 ) w h e r e S i s g i v e n b y : S = S H M H N ST>MH- S 2 « / H W u 2w DM Sr\*\„ S D « i N Siw ?k ) , ( E . 2 7 ) T h e v a l u e s o f t h e i n d u c t i o n t e n s o r e l e m e n t s f o r t h e o t h e r two rows w o u l d p r o c e e d i n i d e n t i c a l f a s h i o n . 246 A p p e n d i x F P r o p e r t i e s o f C u r r e n t D i s t r i b u t i o n s t h a t M i m i c a L i n e C u r r e n t P r e s u m e t h a t a p a r t i c u l a r t w o - d i m e n s i o n a l c u r r e n t d i s t r i b u t i o n t h a t f i t s t h e o b s e r v e d s u r f a c e r e a d i n g s Bx,By a l o n g a l i n e a r a r r a y a t a f r e q u e n c y w, i s a l i n e c u r r e n t a t a d e p t h z Q , a s i n F i g . 3.7. T h e a b i l i t y t o f i t t h e d a t a w i t h t h i s l i n e c u r r e n t m o d e l t h e n i m p o s e s c e r t a i n c o n s t r a i n t s on a l l o t h e r p o s s i b l e m o d e l s t h a t f i t t h e d a t a . As many a c t u a l a n o m a l i e s h a v e s u r f a c e m a g n e t i c f i e l d s t h a t c l o s e l y r e s e m b l e t h o s e o f a s i n g l e l i n e c u r r e n t , t h e s e m o d e l c o n s t r a i n t s a r e o f o b v i o u s i n t e r e s t . C o n s i d e r a c u r r e n t d i s t r i b u t i o n , j ( x , z ) , w h i c h i s n o n - z e r o b e t w e e n z=0, a n d z = c o . F o r t h i s d i s t r i b u t i o n t o d u p l i c a t e t h e s u r f a c e f i e l d d ue t o t h e l i n e c u r r e n t , t h e v a l u e s o f Bx a n d Bz m u st m a t c h a t a l l s u r f a c e p o s i t i o n s 'x' a l o n g t h e a r r a y . B e c a u s e o f t h e l i n e a r r e l a t i o n s h i p b e t w e e n t h e two c o m p o n e n t s ( a s shown i n A p p e n d i x D ) , i t i s s u f f i c i e n t t o c o n s i d e r o n l y one o f t h e c o m p o n e n t s . U s i n g t h e Bx c o m p o n e n t , a s u i t a b l e m o d e l j ( x , z ) w i l l t h u s s a t i s f y : 247 ( U e / 2 f r ) l z 6 / ( x l + z e 1 - ) OO OD = ( f . / 2 t t ) ^ {[ j ( u , v ) - v ] / [ ( x - u ) z + v * ] } d u d v ( F . 1 ) N o t i n g t h e c o n v o l u t i o n a l f o r m o f t h e r i g h t h a n d s i d e we c a n r e w r i t e e q u a t i o n F.1 a s : CX) I z 0 / ( x l + z l 0 ) = ^ [ j ( x , v ) ® ( v / ( x i + v " 1 ) ) ] dv ( F . 2 ) T a k i n g t h e F o u r i e r t r a n s f o r m w i t h r e s p e c t t o 'x' o f b o t h s i d e s o f F.2 g i v e s : ITT e " ' ! U o = ^ [ J ( ^ , v ) TT e 3 ] dv ( F . 3 ) w h e r e : CO - t e a d u J(<£,v) = ^ j ( u , v ) e ~L ^ -00 E q u a t i o n F.3 s i m p l i f i e s t o ( F . 4 ) oo I e lV*' = ^ j ( ^ v ) e dv ( F . 5 ) C o n s i d e r now a m o d e l t h a t h a s no c u r r e n t s a t , o r s h a l l o w e r t h a n t h e l i n e c u r r e n t d e p t h , t h a t i s , j ( x , z ) i s n o n - z e r o o n l y b e t w e e n z 0 + € a n d oo , w h e r e € i s a n a r b i t r a r i l y s m a l l number. U s i n g t h i s , m o v i n g e J ° t o t h e r i g h t h a n d s i d e i n F . 5 , a n d 248 i n t r o d u c i n g a c h a n g e o f v a r i a b l e s , v = ( v - z ) , e q u a t i o n F.5 b e c o m e s : co I e ( F . 6 ) T h e u p p e r l i m i t o f i n t e g r a t i o n i n F.7 must a c t u a l l y be f i n i t e b e c a u s e o f t h e f i n i t e ' d e p t h ' o f t h e e a r t h ; d e n o t e t h i s f i n i t e l i m i t by 'R'. Now, by t h e mean v a l u e t h e o r e m ( F u l k s , 1969, p g . 1 2 5 ) a v a l u e o f t h e i n t e g r a n d w i l l be a b l e t o be f o u n d a t some p o s i t i o n v w i t h i n t h e r a n g e o f i n t e g r a t i o n s u c h t h a t : T h i s must h o l d f o r a l l v a l u e s o f ^, w h i c h means t h a t i n t h e l i m i t a s ^ g o e s t o co , J ( ^ , v ) must i n c r e a s e m o n o t o n i c a l l y t o i n f i n i t y . T h u s , t h e r e q u i r e m e n t t h a t t h e e n t i r e c u r r e n t d i s t r i b u t i o n be a t a g r e a t e r d e p t h t h a n t h a t o f t h e l i n e c u r r e n t r e s u l t s i n p h y s i c a l l y u n t e n a b l e m o d e l s . I n o t h e r w o r d s , a l l r e a l i z a b l e m o d e l s t h a t f i t t h e d a t a must h a v e some c u r r e n t a t , o r s h a l l o w e r t h a n t h e l i n e c u r r e n t d e p t h . R e t u r n t o t h e g e n e r a l p r o b l e m a g a i n , w h e r e t h e c u r r e n t d i s t r i b u t i o n t h a t w i l l m i m i c t h e l i n e c u r r e n t c a n be n o n - z e r o a n y w h e r e b e t w e e n z = 0 a n d z= co. T h e r e q u i r e d r e l a t i o n b e t w e e n t h i s c u r r e n t d i s t r i b u t i o n a n d t h e l i n e c u r r e n t i s t h e n g i v e n by e q u a t i o n F . 5 . I f t h e c u r r e n t d i s t r i b u t i o n J ( ^ , v ) h a s e l e m e n t s w h i c h a r e n o t i n p h a s e w i t h t h e o r i g i n a l l i n e c u r r e n t , t h e n t h a t - i r i u -v) e 5 ( F . 7 ) 249 m o d e l w i l l h a v e b o t h r e a l a n d i m a g i n a r y p o r t i o n s , J R ( ^ , v ) , J j ( ^ , v ) r e s p e c t i v e l y . D e s i g n a t i n g t h e l i n e c u r r e n t a s t h e z e r o p h a s e p o s i t i o n , t h e n I w i l l be p u r e r e a l . T h i s l e a d s t o e q u a t i o n F.5 b e c o m i n g : ^l0= I J R ( ^ V ) e ^ d v ° ( F . 8 ) a n d , CD 0 o = $ J x ( ^ , v ) e~ l5 , t rdv ( F . 9 ) I n t h e c a s e o f F . 9 , a s e ^ ' ^ i s a l w a y s g r e a t e r t h a n z e r o , t h e n e c c e s s a r y z e r o v a l u e o f t h e t o t a l i n t e g r a l r e q u i r e s t h a t e i t h e r J r ( ^ , v ) i s e v e r y w h e r e z e r o , o r t h a t J j . ( ^ , v ) h a s e l e m e n t s o f b o t h n e g a t i v e a n d p o s i t i v e s i g n . I f we now r e s t r i c t o u r r a n g e o f p o s s i b l e m o d e l s t o i n c l u d e o n l y t h o s e where t h e c u r r e n t e l e m e n t s i n t h e m o d e l a r e d i f f e r e n t i n p h a s e by l e s s t h a n TV/2, t h e n t h e o n l y way t o s a t i s f y e q u a t i o n F.9 w i l l be t o h a v e J x ( ^ , v ) z e r o e v e r y w h e r e . T h u s , a p p l y i n g t h i s c o n s t r a i n t f o r c e s a l l c u r r e n t e l e m e n t s t o be i n p h a s e w i t h t h e o r i g i n a l l i n e c u r r e n t . A l t h o u g h t h i s p r e c l u d e s t h e p o s s i b i l i t y o f ' a n n i h i l a t o r ' d i s t r i b u t i o n s i n t h e m o d e l , i t d o e s n o t p r e c l u d e c u r r e n t s a t d e p t h s g r e a t e r t h a n t h a t o f t h e l i n e c u r r e n t . An e x a m p l e o f s u c h a d i s t r i b u t i o n i s o n e c o n s i s t i n g o f a l i n e c u r r e n t o f m a g n i t u d e I 0 / 4 a t a d e p t h 2 z 0 , p l u s a l i n e a l c u r r e n t d e n s i t y : 250 j ( x ) = I 0 { ( z 0 - 2 l ) / [ x l + ( z e - z , r" ] - ( 2 z e - z , ) / 4 [ x 2 + ( 2 z 0 - z , f ]} ( F . 1 0 ) a l o n g a d e p t h z , , w h e r e z, < z Q . I n a c a s e s u c h a s t h a t g i v e n , w h e r e a l l t h e c u r r e n t s a r e i n p h a s e a n d t h e c u r r e n t d i s t r i b u t i o n c o n t a i n s o n l y one l i n e c u r r e n t I a t a d e p t h z, g r e a t e r t h a n z c , t h e n i t must be t r u e t h a t : z o x o ^ M i ( F . 1 1 ) T h i s i s r e q u i r e d t o e n s u r e t h a t t h e X c o m p o n e n t o f t h e o r i g i n a l l i n e c u r r e n t f i e l d i s a l w a y s g r e a t e r t h a n o r e q u a l t o t h e X c o m p o n e n t o f a n y p o r t i o n o f t h e m i m i c k i n g c u r r e n t d i s t r i b u t i o n a t a s y m p t o t i c a l l y l a r g e v a l u e s o f x. 251 A p p e n d i x G A n a l y t i c a n d N u m e r i c I n t e q r a t i o n s I n s e c t i o n 2 o f C h a p t e r I V t h e i n n e r p r o d u c t s o f t h e c u r r e n t d e n s i t y k e r n e l s a r e t o be c o m p u t e d . The k e r n e l s c o r r e s p o n d t o t h e c o n t r i b u t i o n t o t h e s u r f a c e m a g n e t i c f i e l d a t s t a t i o n p o s i t i o n ( x , 0 ) , due t o a c u r r e n t d e n s i t y a t t h e p o s i t i o n ( x ' , z ' ) . T h e r e w i l l be two t y p e s o f k e r n e l s , c o r r e s p o n d i n g t o t h e v e r t i c a l ( z ) a n d h o r i z o n t a l ( x ) c o m p o n e n t s o f t h e f i e l d ( a s g i v e n i n e q u a t i o n s 4.1.6 a n d 4 . 1 . 8 ) : G x ( x , x ' , z * ) = /V2TT {-z'/[ ( x - x ' ) 2 + z * 2 - ] } (G.1) G a ( x , x ' , z ' ) = /V2ir { ( x ' - x ) / [ ( x - x ' f + z ' 1 ] } ) (G.2) T h i s a l l o w s f o r s i x t y p e s o f i n n e r p r o d u c t i n t e g r a l s : ' [ j k = ^ G x ( x j ,x' , z ' ) G x ( x k , x ' , z ' ) d x ' d z ' S 252 /*o%rr a ^ z' 2- d x ' d z ' [ (X. -X' ) l + z ' X ] [ (X. -X' J^+Z'1"] s J k (G.3) w i t h e i t h e r x^ H x^, o r XJ = x ^ . = K G* ( xj , x ' , z ' } G 4 ( x k , x ' , z ' ) d x ' d z S ^oV4TTl f f ( x ' - x k ) ( x ' - x ; )y I (XJ -x' ) l + z - v x.i ) d x ' d z '  ] [ ( x t - x « )* + z « * ] s (G.4) w i t h e i t h e r x ; * x, , o r x: = x, . I j k = J j G 4 ( x k ,x' ,z* ) G K(XJ ,x' ,z') dx'dz' 5 = ^v4Tr i rr - z ' - ( x ' - x k ) dx'dz'  J J [ (XJ -x' )«-+z'»- ] [ ( x k-x ' )*-+z ' *• ] s (G.5) w i t h e i t h e r xj \ x^, o r XJ = x^. Th e r a n g e S w i l l be d e s i g n a t e d A < x' < B; €: < z' < D, where 6 must a l w a y s be g r e a t e r t h a n z e r o . U s i n g s t a n d a r d i n t e g r a t i n g t e c h n i q u e s , t h e f i r s t i n t e g r a t i o n o f t h e s e i n n e r p r o d u c t s c a n be c a r r i e d o u t w i t h r e s p e c t t o e i t h e r o f t h e v a r i a b l e s , r e s u l t i n g i n t h e f o l l o w i n g a n a l y t i c f o r m s . 253 F i r s t i n t e g r a t i o n w i t h r e s p e c t t o z : r ] k = ^o/4ir* $ dx' { ( x ' - x k ) / [ ( x , - x k ) i - ( x ' - x j ) z ] A -{Tan" 1 [ D / ( x ' - x k ) ]-Tan~ l [€./( x ' - x k ) ]} ( x ' - X j ) / [ ( x ' - x j ) 1 - ( x ' - x k ) 1 ] • { T a n - 1 [ D / ( x ' - x j ) ]-Tan"' [ t / ( x ' - x j ) ] } } (G.6) x: * x u 3 • P j k = P o / s r r 1 ^ dx' { i / ( x ' - x k ) * ~ . - I . [ T a n ( D / ( x ' - x k ) ) - T a n ' ( € / ( x ' - x k ) ) ] + ( € / [ ( x ' - x k ) Z + - D/[ ( x ' - x k ) i + D 1 ] ) } (G.7) Xj = x k B r$k = poV4ft r _f dx' { ( x ' - x j )/[ (X'-XJ ^ - ( X ' - X L . ) 2 - ] * {Tan" 1 [ D / ( x ' - x k ) ] - T a n " 1 [ € / ( x ' - x k ) ] } ( x ' - x k ) / [ (X'-XJ ) i - ( x ' - x k ) 2 ] 2 5 4 - i • { T a n [ € / ( x ' - x j ) ]-Tan 1 D^ /( x '-XJ ) ]}} ( G . 8 ) X; * x. Pj k = ^o l/8Tr l £ dx' { 1 / ( x ' - x k ) [ T a n ' 1 ( D / ( x ' - x k ) ) - Tan"' ( V ( x ' - x K ) ) ] + (D/[ ( x ' - x k ) 2 + D ] - € / [ ( x ' - x k ) * + £*])} ( G.9) x j = x k 3 P rf/8T['L § ( x ' - x k ) / [ ( x , - x ^ . ) i - ( x ' - x k ) z ] * l n { [ ( x ' - x j )* + 6 x i r ( x ' - x O * +D*-])dx' (J ( x ' - x ; ) l + D M [ ( x ' - x k ) 1 + 6*]} ( G . 1 0 ) X j * x k f ] k = MoV8irl 5" ( x ' - x k ) 1 dx • d / [ ( x ' - x k ) 1 + € l ] - l / [ ( x ' - x k ) l + D 1 ] } ( G . 1 1 ) x j = x k I n a l l c a s e s , t h e r e s u l t s f o r XJ \ x k c a n be shown t o h a v e a s t h e i r l i m i t i n g v a l u e s t h e XJ = x k e x p r e s s i o n s , when x: x k « 255 F i r s t i n t e g r a t i o n with respect to x 1 £al f z ' l d z ' . l n i 4tt t J 8(z* 1- +d'"). d I [ (XK -A) 1 +z 1 ][(xi -B )*" +z [<x K -B) t+z M[(x- -A) 1 +z 2 Tan' / [ (x«-A) (XJ -A)-Z'"M z' ^ [ z ' ( x K - A ) + z ' ( X J - A ) ] -Tan (XK-B) (xi - B ) - z ' * 1 z* (x k-B)+z' (XJ -B) ] X J * x k (G.12) P ; Jk £l_ C d z ' f (XK 8rr v J l [ z ' t + € -A) (XK-B) ( x k - A ) 1 ] [ z '*• + (x, -B) z ] + J_ Tan z' - i |(x^-A)j - i_Tan"' |(xR-B)jj (G.13) n hi 4fT J_ In 8d [ ( x ; - A ) x + Z ' * ] [ ( x * - B ) > 2 ' M [ ( x k - A ) l + z ' 1 ] [ ( x j - B ) z + z , x ] 256 + J _ 4z' Tan"' /[ ( x k - A ) ( x j - A ) - z ' 1 ] ( [ z ' ( x k - A ) + z ' ( x - - A ) ] - T a n " 1 [ ( x n - B ) ( x j - B ) - z ' 1 1 [ z ' ( x k - B ) + z ' ( X > i -B) ] 1 /d I n 8 ( z ' 1 +d* [ (xk-A ) z + z ' M [ J x i z J i l l z ^ . ] 1 [(xk-B) l+z*s-][(xj-A)-'-+z , ;L] 2d_* z' T a n "1 / [ ( x k - A ) ( x j - A l - z ^ ] 1 [z' ( x K - A ) + z ' (XJ -A) ] T a n " 1 / [ (XK-[z K - B ) (xi - B ) - z ' , • 1 ) ] \ \ ' ( x K - B ) + z ' ( X J - B ) )J\ j (G.14) dz' 1 T a n (xk-A1 - ( x k - A ) |z' z' [z"- + ( x k - A ) ' - ] JI_Tan"' IXK-BI I Z 7 ( x k - B ) , [z , J- + ( x k - B ) l ] J (G.15) 257 r, Ho 4TP ' f z ' d z ' f ^ [ T a n " ' ([ ( x * - A - d )Z - d * + z ' z ] ^ J [ 4 z ' d L \ 2 z T d J - T a n " 1 ([ ( x K - B - d f - d 2 z ' d ^ ) ] d C_l I n f[ (XK-A^+Z' 1 ][ (xo - B ) * + z , a " ^  8 ( z " - + d x ) \ d \ [ ( x k - B ) l + z , i ] [ ( x j - A ) 1 ' + z , J - ] y + 2_ z' T a n -I /[ (X'K-A) ( x i - A ) - z ' r 3 U z - (x. -k A ) + z ' ( X j - A ) ] - T a n /[ ( x k-B) (x j - B ) - z ' t - ] \1) U z ' ( x k - B ) + z ' (x- -B) (G .16 ) x j * x k 8vr [ ( x ^ - B ^ + z ' 1 ] [ ( x k - A ) l + z " - ] (G .17 ) 258 d = < x k - x j ) / 2 (G.18) The i n t e g r a t i o n w i t h r e s p e c t t o t h e f i n a l v a r i a b l e f o r e i t h e r o f t h e two p o s s i b i l i t i e s must be d o n e n u m e r i c a l l y , due t o t h e ' T a n - 1 ' a n d ' I n ' t e r m s i n t h e i n t e g r a n d . T h e m e t h o d o f n u m e r i c a l i n t e g r a t i o n t h a t was u s e d i n v o l v e d s a m p l i n g t h e i n t e g r a n d f u n c t i o n b e t w e e n t h e l i m i t s o f i n t e g r a t i o n , a n d t h e n f i t t i n g a c u b i c s p l i n e t o t h e s e v a l u e s . T h e c u b i c s p l i n e e s t i m a t e s t h e f i r s t , s e c o n d a n d t h i r d d e r i v a t i v e s o f t h e i n t e g r a n d c u r v e a t e v e r y s a m p l e d p o i n t , s o t h a t t h e i n t e g r a l c a n be r e a d i l y e s t i m a t e d : T h e s e c t i o n s o f t h e i n t e g r a n d c u r v e w h i c h c h a n g e d r a p i d l y , a n d t h u s were d i f f i c u l t t o s p l i n e , w e r e l o c a t e d a n d s a m p l e d on a much f i n e r b a s i s . T h e i n t e g r a t e d r e s u l t s o f t h e s p l i n i n g t e c h n i q u e w e r e c o m p a r e d w i t h r e s u l t s u s i n g v a r i o u s U.B.C. c o m p u t e r l i b r a r y i n t e g r a t i n g r o u t i n e s , a n d we r e a l w a y s w i t h i n . 0 0 0 1 . A l s o , t h e n u m e r i c a l i n t e g r a t i o n was d o n e u s i n g t h e i n t e g r a n d s f r o m b o t h t h e f i r s t i n t e g r a t i o n w i t h r e s p e c t t o 'x' a n d w i t h r e s p e c t t o ' z ' , w i t h t h e r e s u l t s a g r e e i n g w i t h i n .0001. The p a r t i c u l a r u t i l i t y o f i n t e g r a t i n g w i t h r e s p e c t t o 'x' f i r s t , i s t h a t i t a l l o w s t h e s i m p l e i n t r o d u c t i o n o f t h e w e i g h t i n g f ( x ' ) d x ' = D f ( x ) + D a f ' ( x ) + D3- f ' ' (x ) + D*- f ' ' ' ( x ) 2! 3! 4! (G.19) 259 f a c t o r , z , i n t o t h e i n t e g r a n d f o r t h e w e i g h t e d s m a l l e s t m o d e l c o n s t r u c t i o n o f C h a p t e r I V . Th e s e c o n d r e s u l t n e e d e d f o r t h e c a l c u l a t i o n o f t h e a v e r a g i n g f u n c t i o n s i n S e c t i o n 2 o f C h a p t e r I V i s t h e i n t e g r a l o f e a c h o f t h e k e r n e l s o v e r t h e r e g i o n o f i n t e r e s t . T h e r e a r e o n l y two p o s s i b i l i t i e s , a n d b o t h h a v e a n a l y t i c s o l u t i o n s : B T> 1 U k = ^ G K U k , x ' , z ' ) d x ' d z ' A £ = f ( x k - B ) . l n / [ D * + ( B - X K f ]-\ 2TT{ 2 U f e l + ( B - x k ) 1 ] / (XK-A), I n f [ D H t A - x . , ) 1 ] 2 \ [ € z - + ( A - x k ) i ] D'Tan" 1 /XR-B\ - €.Tan~' D« Tan" 1 (G.20) B T> G s ( x k , x ' , z ' ) d x ' d z * A € 260 t 2TT - ( l [ D I (.2 . In \PX +a V - £ In /ilV €»• + a l / J w h e r e ; + a [T a n " ' ( 6 / a ) - T a n " 1 ( D / a ) ] - b [ T a n " ' ( € / b ) - T a n ' 1 (D/b) ] ^ (G.21 ) x k - A (G.22) b = x k - B (G.23) F o r b o t h t h e f o r w a r d m o d e l l i n g r o u t i n e a n d t h e l i n e a r p r o g r a m m i n g c o n s t r u c t i o n , t h e c o n t r i b u t i o n t o t h e s u r f a c e f i e l d f r o m a r e c t a n g l e o f c o n s t a n t c u r r e n t d e n s i t y i s r e q u i r e d ( s e e F i g . G . 1 ) . T h e c o n t r i b u t i o n s f o r e a c h c o m p o n e n t a t a s t a t i o n p o s i t i o n 1 X j _ ' a r e ; - z ' d x ' d z ' [ (x, -x' )*+z"- ] = ( / X . / 2 T T ) J j k l ^ ( G.24) 261 ^-Surface J., J 2 i P a r a m e t e r i z e d c u r r e n t d e n s i t y m o d e l . T h e c u r r e n t d e n s i t y i s c o n s t a n t w i t h r e s p e c t t o p o s i t i o n w i t h i n e a c h g r i d e l e m e n t . 262 BzU,; ,0) = ^ k 2rr ( x ' - x i ) dx'dz' J [ (X; -X* )*• + Z " - ] = (/lo/21t ) J A I j ' k The v a l u e s of I:. and I;, a r e : Jk Jk (G.25) \lk Tan - x j - Tan X t _ " z k t i Tan xc -x '• k + t - Tan - i xc k - H X i - X ; l n [ZK 1 + (XC -x-,'-)2- ] + ( x , - x j ) ] k+( xc. -x J t l In [zkli + (xi -x;Vi )* 3 [ z - + ( x £ - x j t , ) M (G.26) < x t ~ x j + i H T a n Y z U i ] - .Tan"'/ z V \ ] \x. - x ' ' Iv - V • I X C X jV ly 263 (x- -X- ' )[Tan" \X; - X : + zW, . In / [ ( x t - x ' j V , ) * + z & 3 \ 2 ^ [ ( x t - x ] V + z ^ ] / + z_k . In A ( x i - x l )» + z l x ] \ 2 \ K x t - X j V , ) l + z ^ ] J (G.27) 264 Appendix H Determining the Major Axis of an E l l i p s o i d Let the linear transformation of a three dimensional vector x be represented by a real valued matrix A: y = A x (H.1) The square of the length of the new vector, y, i s : y Ty = (Ax) TAx = x T(A TA)x = x Bx (H.2) The matrix B w i l l be real and symmetric. The endpoints of a l l the position vectors x for which the squared length of y is a constant w i l l map out a quadric surface (Sokolnikoff, 1951; Strang, 1976), with the governing equation being: y T y = C 265 = (x, ,xz,xz) / b„ b a b n \ / x, b i , b 2 7 . b 2 * j I xj, a 3 x b " b " b V \ X V (H.3) F o r C > 0 t h e q u a d r i c s u r f a c e i s a n e l l i p s o i d . F r o m e q u a t i o n H.2, t h e m a t r i x JB w i l l be p o s i t i v e d e f i n i t e a s w e l l a s s y m m e t r i c , w h i c h e n s u r e s t h a t we c a n a l w a y s d e c o m p o s e B: B = S A S T (H.4) w h e r e S i s an o r t h o g o n a l m a t r i x : s - = s T (H.5) a n d A i s a d i a g o n a l m a t r i x , wiilv. a l l v a l u e s o f t h e d i a g o n a l g r e a t e r t h a n z e r o : A (H.6) T h u s , e q u a t i o n H.3 b e c o m e s : C = x T ( S A S T ) x 2 6 6 = ( x T S ) - A ' ( S T x ) (H.7) D e f i n i n g a new c o o r d i n a t e s y s t e m b y : I - s T i ( H . 8 ) ( w h e r e t h e l e n g t h o f £ i s g u a r a n t e e d t o be t h e same a s t h a t o f x by t h e o r t h o g o n a l i t y o f S) t h e n e q u a t i o n H.7 r e d u c e s t o : (H.9) T h u s , i n t h e r o t a t e d s y s t e m , t h e e q u a t i o n o f t h e e l l i p s o i d b e c o m e s : c = + + ( H . 1 0 ) s o t h a t t h e u n i t v e c t o r s , ^ l i e a l o n g t h e p r i n c i p a l a x i s o f t h e e l l i p s o i d . T he maximum l e n g t h s o f t h e e l l i p s o i d i n e a c h o f t h e a x i s d i r e c t i o n s ( u s i n g t h e f a c t t h a t } V , A l f a n d a r e a l l g r e a t e r t h a n z e r o ) a r e : ^( max = (C/ >vj) 1 ( H . 1 1 ) 267 ^ m * * = (C/\)' / 2-(H.12) ^mA . K = (C/^)''* (H.13) T h u s , a m e a s u r e o f t h e r e l a t i v e l e n g t h s o f t h e p r i n c i p a l a x i s o f t h e e l l i p s o i d i s g i v e n by t h e r a t i o s o f t h e s q u a r e r o o t s o f t h e e i g e n v a l u e s : (H.14) ^ • j m a x ^ i (H.15) As w e l l , t h e d i r e c t i o n s o f t h e p r i n c i p a l a x i s o f t h e e l l i p s o i d i n t h e o r i g i n a l s p a c e a r e g i v e n b y t h e row s ( o r e i g e n v e c t o r s ) o f S: (H.16) 268 (H.17) ^•i _ ^ S l S ' S * 3 ' S 3 3 ) (H.18) w h e r e t h e o r t h o g o n a l i t y o f S e n s u r e s t h e u n i t l e n g t h o f t h e e i g e n v e c t o r s . 

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