UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Analysis of geomagnetic depth sounding data 1981

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1981_A6_7 S85.pdf
UBC_1981_A6_7 S85.pdf [ 8.51MB ]
UBC_1981_A6_7 S85.pdf
Metadata
JSON: 1.0052943.json
JSON-LD: 1.0052943+ld.json
RDF/XML (Pretty): 1.0052943.xml
RDF/JSON: 1.0052943+rdf.json
Turtle: 1.0052943+rdf-turtle.txt
N-Triples: 1.0052943+rdf-ntriples.txt
Citation
1.0052943.ris

Full Text

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 0 S 4- .53x10 . 1 3 x 1 0 4 220 44 A- 1x10 . 1 0x1 0 S . 1 7 x 1 0 4 . 3 3 x l 0 3 83 1 4 2.8 5x 1 0 3 A- .25x10 .42x1 0 3 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 ' = = ^ [ Ĝ  (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 . 1̂ 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 - Ĝ-.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. Heacock, G e o p h y s i c a l Monograph 4, American G e o p h y s i c a l U n i o n . B r a c e w e l l , R.W., (1965), 'The F o u r i e r T r a n s f o r m and i t s A p p l i c a t i o n s ' , M c G r a w - H i l l , New York. Budden, K., (1961), 'The Wave-Guide Mode Theory of Wave P r o p o g a t i o n ' , P r e n t i c e - H a l l , N . J . C a g n a i r d , L., ( 1 953), G e o p h y s i c s , J_8 , pg. 605-635. ' B a s i c Theory of the M a g n e t o t e l l u r i c Method of G e o p h y s i c a l P r o s p e c t i n g ' Chapman, S., (1919), P h i l . T r a n s . Roy. Soc. London, A218 , pg. 1-118. ' S o l a r and Lunar D i u r n a l V a r i a t i o n s of T e r r e s t r i a l Magnetism' 197 Chapman, S. and P r i c e , A.T., (1930), P h i l . T r a n s . Roy. Soc. London, A229 , pg. 427-460. '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 . I n t e r . , J_ , p g . 2 4 - 3 4 . ' G e o m a g n e t i c V a r i a t i o n s a n d E l e c t r i c a l C o n d u c t i v i t y S t r u c t u r e i n S o u t h - W e s t e r n A u s t r a l i a ' F i s c h e r , G., S c h n e g g , P.-A., a n d P e g u i r o n , M., ( 1 9 8 0 ) , t o be p u b l i s h e d . • 'An A n a l y t i c O n e - D i m e n s i o n a l I n v e r s i o n Scheme' F u l k s , W., ' A d v a n c e d C a l c u l u s ' , J o h n W i l e y a n d S o n s I n c . , U.S.A. F u l l a g a r , P.K., ( 1 9 8 0 ) , Ph.D. T h e s i s . ' I n v e r s i o n o f 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 S o u n d i n g s O v e r a S t r a t i f i e d E a r t h ' G a r l a n d , G.D., ( 1 9 7 9 ) , ' I n t r o d u c t i o n t o G e o p h y s i c s , ( M a n t l e , C o r e a n d C r u s t ) ' , W.B. S a u n d e r s Co., T o r o n t o . 199 Gauss, C.F., (1838), A l l g e m e i n e T h e o r i e des Erdmagnetismus, r e p r i n t e d i n Werke, Band 5, pg. 121-193. Gough, D.I., (1973), Geophys. J . R. a s t r . S o c , 35 , pg. 83-98. 'The I n t e r p r e t a t i o n of Magnetometer A r r a y S t u d i e s ' G r a d s h t y n , I.S. and R h y s i k , I.M., (1965), 'Tables of I n t e g r a l s , S e r i e s and P r o d u c t s , Academic P r e s s , New York. H e n s e l , G., (1981), P r i v a t e c o r r e s p o n d e n c e . J a c o b s , J.A., (1970), 'Geomagnetic M i c r o p u l s a t i o n s ' , S p r i n g e r - V e r l a g , New York. J o n e s , F.W., (1970), Geophys. J . R. a s t r . S o c , 22 , pg. 17-28. ' E l e c t r o m a g n e t i c I n d u c t i o n i n a N o n - H o r i z o n t a l l y S t r a t i f i e d Two-Layered Conductor' 200 J o n e s , F.W., ( 1 9 7 1 ) , P h y s . E a r t h P l a n e t . I n t e r . , 4 , p g . 417- 424. ' E l e c t r o m a g n e t i c I n d u c t i o n i n a T w o - D i m e n s i o n a l M o d e l o f a n A s y m m e t r i c T w o - L a y e r e d C o n d u c t o r ' J o n e s , F.W. a n d P a s c o e , L . J . , ( 1 9 7 1 ) , G e o p h y s . J . R. a s t r . S o c , 24 , p g . 3-30. 'A G e n e r a l C o m p u t e r P r o g r a m t o D e t e r m i n e t h e P e r t u r b a t i o n o f A l t e r n a t i n g E l e c t r i c C u r r e n t s i n a T w o - D i m e n s i o n a l M o d e l o f a R e g i o n o f U n i f o r m C o n d u c t i v i t y w i t h a n Embedded I n h o m o g e n e i t y ' J o n e s , F.W. a n d P r i c e , A . T ., ( 1 9 7 0 ) , G e o p h y s . J . R. a s t r . S o c , 2_0 , p g . 317-334 . 'The P e r t u r b a t i o n s o f A l t e r n a t i n g G e o m a g n e t i c F i e l d s by C o n d u c t i v i t y A n o m a l i e s ' J u p p , D.L.B. a n d V o z o f f , K., ( 1 9 7 7 ) , G e o p h y s . J . R. a s t r . S o c , 50 , p g . 333-352 ' T w o - D i m e n s i o n a l M a g n e t o t e l l u r i c I n v e r s i o n ' . 201 K e l l e r , G.V. and Frischnecht, F.C., ( 1 9 6 6 ) , ' E l e c t r i c a l Methods in Geophysical Prospecting', Pergamon Press, New York. Kisabeth, J.L., ( 1 9 7 5 ) , Phys. Earth Planet. Inter., J_0 , pg. 2 4 1 - 2 4 9 . 'Substorm Fields In and Near the Auroral Zone' Kisabeth, J.L. and Rostoker, G., ( 1 9 7 1 ) , J. Geophys. Res., ]_6 , pg. 6815. 'Development of the Polar E l e c t r o j e t During Polar Magnetic Substorms' Kisabeth, J.L. and Rostoker, G., ( 1 9 7 7 ) , Geophys. J. R. astr . S o c , 49 , pg. 6 5 5 - 6 7 4 . 'Modelling of Three-Dimensional Current Systems Associated with Magnetospheric'Substorms' Kuckes, A.F., ( 1 9 7 3 ) , Geophys. J. R. astr. S o c , 32 , pg. 1 1 9 - 131. 'Relations Between E l e c t r i c a l Conductivity of a Mantle and Fluctuating Magnetic F i e l d s ' 202 L a h i r i , B.N. a n d P r i c e , A.T., ( 1 9 3 9 ) , P h i l . T r a n s . Roy. S o c . L o n d o n , A237 , p g . 5 0 9 - 5 4 0 . ' ' 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 N o n - U n i f o r m C o n d u c t o r s a n d t h e D e t e r m i n a t i o n o f t h e C o n d u c t i v i t y o f t h e E a r t h f r o m T e r r e s t r i a l M a g n e t i c V a r i a t i o n s ' L a n d a u , L.D. a n d L i f s c h i t z , E.M., ( i 9 6 0 ) , ' E l e c t r o d y n a m i c s o f C o n t i n u o u s M e d i a ' , P e r g a m o n P r e s s , New Y o r k . Law, L.K., A u l d , D.R., a n d B o o k e r , J.R., ( 1 9 8 0 ) , J . G e o p h y s . R e s . , 85 , p g . 5 2 9 7 - 5 3 0 2 . 'A G e o m a g n e t i c V a r i a t i o n A n o m a l y C o i n c i d e n t 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 ' L e e , K.H., P r i d m o r e , D.F., a n d M o r r i s o n , H.F., ( 1 9 8 1 ) , G e o p h y s i c s , 46 , p g . 7 9 6 - 8 0 5 . 'A H y b r i d T h r e e - D i m e n s i o n a l E l e c t r o m a g n e t i c M o d e l l i n g Scheme' 203- L e v y , S. a n d F u l l a g a r , P.K., ( 1 9 8 1 ) , G e o p h y s i c s , £ 6 , p g . 1235- 1 2 4 3 . ' R e c o n s t r u c t i o n o f a S p a r s e S p i k e T r a i n f r o m a P o r t i o n o f i t s S p e c t r u m a n d A p p l i c a t i o n t o H i g h R e s o l u t i o n D e c o n v o l u t i o n ' L i g h t h i l l , M.J., ( 1 9 5 8 ) , ' I n t r o d u c t i o n t o F o u r i e r A n a l y s i s a n d G e n e r a l i z e d F u n c t i o n s ' , C a m b r i d g e a t t h e U n i v e r s i t y P r e s s . L i l l e y , F.E.M., ( 1 974 ), P h y s . E a r t h P l a n e t . I n t e r . , 8 , pg.. 301- 316. ' A n a l y s i s o f t h e G e o m a g n e t i c I n d u c t i o n T e n s o r ' L i l l e y , F.E.M., ( 1 9 7 5 ) , P h y s . E a r t h 2 3 1 - 2 4 0 . ' M a g n e t o m e t e r A r r a y S t u d i e s : A o f O b s e r v e d F i e l d s ' P l a n e t . I n t e r . , H) , p g . R e v i e w o f t h e I n t e r p r e t a t i o n L i l l e y , F.E.M. a n d B e n n e t t , D . 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 . 9-14 ' L i n e a r R e l a t i o n s h i p s i n G e o m a g n e t i c V a r i a t i o n S t u d i e s ' 204 M c D o n a l d , K.L. , ( 1 9 5 7 ) , J . G e o p h y s . R e s . , 62 , p g . 1 1 7 - 1 4 1 . ' P e n e t r a t i o n o f t h e G e o m a g n e t i c S e c u l a r F i e l d T h r o u g h a M a n t l e w i t h a V a r i a b l e C o n d u c t i v i t y ' M adden, T.R. a n d S w i f t J r . , C M . , ( 1 9 6 9 ) , I n : 'The E a r t h ' s C r u s t a n d U p p e r M a n t l e ' , P . J . H a r t ( e d . ) , Am. G e o p h y s . M o n o g r a p h 13, p g . 4 6 9 . ' M a g n e t o t e l l u r i c S t u d i e s o f t h e E l e c t r i c a l C o n d u c t i v i t y S t r u c t u r e o f t h e C r u s t a n d U p p e r M a n t l e ' M a r e s c h a l , M., ( 1 9 8 1 ) , t o be p u b l i s h e d . ' S o u r c e E f f e c t s a n d 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 S o u n d i n g D a t a a t S u b - A u r o r a l L a t i t u d e s ' N i s h i d a , A., ( 1 9 7 8 ) , ' G e o m a g n e t i c D i a g n o s i s o f t h e M a g n e t o s p h e r e ' , S p r i n g e r - V e r l a g , New Y o r k . O l d e n b u r g , D.W., ( 1 9 6 9 ) , M S c . T h e s i s ' S e p a r a t i o n o f M a g n e t i c S u b s t o r m F i e l d s ' 205 O l d e n b u r g , D.W., (1976), Geophys. J . R. a s t r . S o c , £4 , pg. 413-431. ' C a l c u l a t i o n of F o u r i e r T r a n s f o r m s by the B a c k u s - G i l b e r t Method' Old e n b u r g , D.W., (1979), G e o p h y s i c s , 44 , pg. 1218-1244. 'One-Dimensional I n v e r s i o n of N a t u r a l Source M a g n e t o t e l l u r i c O b s e r v a t i o n s ' Oldenburg, D.W., (1981), Geophys. J . R. a s t r . S o c , (65 , pg. 359-394. ' C o n d u c t i v i t y S t r u c t u r e of Oceanic Upper M a n t l e Beneath the P a c i f i c P l a t e ' P a n o f s k y , W.K.H. and P h i l l i p s , M., (1962), ' C l a s s i c a l E l e c t r i c i t y and Magnetism', Addison-Wesley P u b l . Co., R e a d i n g , U.S.A. P a r k e r , R.L., (1977), Ann. Rev. E a r t h P l a n e t . S c i . , 5 , pg. 35- 64. ' U n d e r s t a n d i n g I n v e r s e Theory' 206 P a r k i n s o n , W.D., (1959), Geophys. J . R. a s t r . S o c , 2 , pg. 1- ' D i r e c t i o n s of R a p i d Geomagnetic F l u c t u a t i o n s ' P a r k i n s o n , W.D., (1962), Geophys. J . R. a s t r . S o c , 6 , pg. 441- 449. 'The I n f l u e n c e of C o n t i n e n t s and Oceans on Geomagnetic V a r i a t i o n s ' P e r r e a u l t , P. and A k a s o f u , S - I . , (1978), Geophys. J . R. a s t r . Soc . , 5_4 , pg . 547 . P o r a t h , H., Oldenburg, D.W., and Gough, D.I., (1970) Geophys. J . R. a s t r . S o c , J_9 , pg. 237-260. ' S e p a r a t i o n of Magnetic V a r i a t i o n F i e l d s and C o n d u c t i v e S t r u c t u r e s i n the Western U n i t e d S t a t e s ' P o r a t h , H . , Gough, D.I., and C a m f i e l d , P.A., (1971), Geophys. J . R. a s t r . S o c , 20 , pg. 387-398. 'Conductive S t r u c t u r e s i n the North-Western U n i t e d S t a t e s and South-West Canada' 207 P r a u s , 0 . , ( 1 9 7 5 ) , P h y s . E a r t h P l a n e t . I n t e r . , _1_0 , p g . 2 6 2 - 2 7 0 . ' N u m e r i c a l a n d A n a l o g u e M o d e l l i n g o f I n d u c t i o n E f f e c t s i n L a t e r a l l y N o n - u n i f o r m C o n d u c t o r s ' P r i c e , A . T . , ( 1 9 5 0 ) , Q u a r t . J . M e c h . A p p l . M a t h . , 3 , p g . 3 8 5 - 4 1 0 . ' E l e c t r o m a g n e t i c I n d u c t i o n i n a S e m i - I n f i n i t e C o n d u c t o r w i t h a P l a n e B o u n d a r y * P r i c e , A . T., ( 1 9 6 2 ) , J . G e o p h y s . R e s . , , 67 , p g . 1 9 0 7 - 1 9 1 8 . 'The T h e o r y o f M a g n e t o - T e l l u r i c M e t h o d s when t h e S o u r c e F i e l d i s C o n s i d e r e d ' R i k i t a k e , T. , ( 1 9 5 0 ) , B u l l . E a r t h q u a k e R e s . I n s t . , T o k y o U n i v e r s i t y , 28 , p g . 4 5 - 1 0 0 , 2 1 9 - 2 6 2 , 2 6 3 - 2 8 3 . ' E l e c t r o m a g n e t i c I n d u c t i o n w i t h i n t h e E a r t h a n d i t s R e l a t i o n t o t h e E l e c t r i c a l S t a t e o f t h e E a r t h ' s I n t e r i o r ' R o s t o k e r , G., ( 1 9 7 2 ) , R e v . G e o p h y s . S p a c e P h y s . , J_0 , p g . 157 - 2 1 1 . ' P o l a r M a g n e t i c S u b s t o r m s ' 208 R o s t o k e r , G., ( 1 9 7 8 ) , J . Geomagn. G e o e l e c t r . , 30 , p g . 6 7 - 1 0 7 . ' E l e c t r i c a n d M a g n e t i c F i e l d s i n t h e I o n o s p h e r e a n d M a g n e t o s p h e r e ' R u n c o r n , S.K., ( 1 9 5 5 ) , T r a n s . Am. G e o p h y s . U n i o n , 3_6 , p g . 191- 1 98. 'The 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 E a r t h ' s M a n t l e ' R y s h i k , I.M. a n d G r a d s t e i n , I . S . , ( 1 9 6 3 ) , ' T a b l e s o f S e r i e s , P r o d u c t s , a n d I n t e g r a l s ' , V e b D e u t s h e r V e r l a g D e r W i s s e n s c h a f t e n , B e r l i n , 1. Samson J . C , ( 1 9 7 7 ) , G e o p h y s . J . R. a s t r . S o c . , 5J_ , p g . 583- 6 0 3 . ' M a t r i x a n d S t o k e s V e c t o r R e p e r e s e n t a t i o n s o f D e t e c t o r s f o r P o l a r i z e d W a v e f o r m s : T h e o r y , w i t h Some A p p l i c a t i o n s t o T e l e s e i s m i c Waves' S c h m u c k e r , U., ( 1 9 7 0 ) , ' A n o m a l i e s o f G e o m a g n e t i c V a r i a t i o n s i n t h e S o u t h w e s t e r n U n i t e d S t a t e s ' , M o n o g r a p h , S c r i p p s I n s t . O c e a n o g r a p h y 209 S i e b e r t , M. und K e r t z , W., (1957), Zur Z e r l e g u n g e i n e s L o k a l e n Erdmagnetishen F e l d e s i n A u s s e r n and I n n e r n A n t e i l ' , Nachr. Akad. Wiss. G o t t i n g e n , I I . M a t h - P h y s i k . K1., pg. 87-112. S o k o l k n i k o f f , I.S., (1951), 'Tensor A n a l y s i s , Theory and A p p l i c a t i o n s ' , John W i l e y and Sons, I n c . , New York. S t r a n g , G., (1976), ' L i n e a r A l g e b r a and i t s A p p l i c a t i o n s ' , Academic P r e s s , New York. T e l f o r d , W.M., G e l d a r t , L.P., S h e r i f f , R.E., Keys, D.A., (1976), ' A p p l i e d G e o p h y s i c s ' , Cambridge U n i v e r s i t y P r e s s . V e s t i n e , E.H., (1941), I . T e r r e s t . Magnetism At m o s p h e r i c E l e c , 46 , pg. 27-41. V o z o f f , K., (1972), G e o p h y s i c s , 37 , pg. 98-141. 'The M a g n e t o t e l l u r i c Method i n the E x p l o r a t i o n of Sedimentary B a s i n s ' 210 W a i t , J.R., ( 1 954), G e o p h y s i c s , J_9 , pg. 281-289. 'The R e l a t i o n Between T e l l u r i c C u r r e n t s and the E a r t h ' s Magnetic F i e l d ' W a i t , J.R., (1970), ' E l e c t r o m a g n e t i c Waves i n S t r a t i f i e d Media', Pergamon P r e s s , T o r o n t o . Weaver, J.T., ( 1963), 3. Geophys., 29 ,'pg. 29-36. 'On the S e p a r a t i o n of L o c a l Geomagnetic F i e l d s i n t o E x t e r n a l and I n t e r n a l P a r t s ' Weaver, J.T., (1973), Phys. E a r t h P l a n e t . 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 îi£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 D̂ /( 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 .

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
China 23 8
Bolivia 7 0
United States 2 0
France 2 0
Japan 2 0
India 1 0
City Views Downloads
Beijing 23 0
La Paz 7 0
Unknown 2 6
Tokyo 2 0
Mountain View 1 0
Ahmedabad 1 0
Ashburn 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items