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UBC Theses and Dissertations

Electromagnetic induction using powerlines Fisk, Lynda Elaine 1984

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ELECTROMAGNETIC INDUCTION USING POWERLINES by LYNDA ELAINE FISK .Sc. B i o l o g y , U n i v e r s i t y Of W e s t e r n O n t a r i o , 1 9 8 1 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department Of G e o p h y s i c s And Astronomy) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Au g u s t 1984 © Lynda E l a i n e F i s k , 1984 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 t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y of B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head of my Department or by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d 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 a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of G e o p h y s i c s And A s t r o n o m y The U n i v e r s i t y of B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : 5 J u l y 1984 ABSTRACT The p o s s i b i l i t y of o b t a i n i n g i n f o r m a t i o n about s u b s u r f a c e 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 p o w e r l i n e s as a source has been i n v e s t i g a t e d . P o w e r l i n e s have been mo d e l l e d as p u r e l y i n d u c t i v e s o u r c e s over a l a y e r e d e a r t h . S o l u t i o n s f o r the f i e l d components produced by an e l e v a t e d s t r a i g h t l i n e source of c u r r e n t over a l a y e r e d e a r t h a re p r e s e n t e d . Exact s o l u t i o n s a re i n the form of a ( i n v e r s e ) F o u r i e r t r a n s f o r m . These e x p r e s s i o n s a re e v a l u a t e d n u m e r i c a l l y u s i n g F a s t F o u r i e r Transform (FFT) a l g o r i t h m s . P r e l i m i n a r y i n t e r p r e t a t i o n s of magnetic f i e l d d a ta g a t h e r e d i n the v i c i n i t y of a p o w e r l i n e have been made u s i n g these e x p r e s s i o n s . Emphasis a t t h i s stage i s on f o r w a r d model r e s u l t s . For the case of a bent l i n e of c u r r e n t the computation of exact s o l u t i o n s becomes c o s t l y , so t h a t approximate s o l u t i o n s a re used employing complex image t h e o r y . Image t h e o r y r e s u l t s agree w i t h exact s o l u t i o n r e s u l t s throughout the q u a s i - s t a t i c range, except at d i s t a n c e s l e s s than the o r d e r of a s k i n depth from the s o u r c e . E f f e c t s due t o l a t e r a l i n h o m o g e n e i t i e s i n a homogeneous h a l f space were modelled w i t h the use of an e l e c t r o l y t i c tank. A c y l i n d r i c a l wave analogue model was used t o study the e f f e c t of a r e c t a n g u l a r s t r u c t u r e embedded i n a homogeneous h a l f space. i i i Inphase and q u a d r a t u r e measurements of the t o t a l f i e l d a r e o b t a i n e d . i v T A B L E O F C O N T E N T S A B S T R A C T i i L I S T O F T A B L E S v i i L I S T O F F I G U R E S v i i i A C K N O W L E D G E M E N T S x i I N T R O D U C T I O N 1 N O T A T I O N 4 C H A P T E R 1. EM I n d u c t i o n i n a L a y e r e d E a r t h b y a S t r a i g h t L i n e o f C u r r e n t . 1.1 F o r m u l a t i o n o f t h e L a y e r e d E a r t h P r o b l e m 6 1 .2 S o l u t i o n f o r a S t r a i g h t L i n e o f S o u r c e . . . 16 1 . 3 R e s p o n s e F u n c t i o n s 19 1 .4 R e s u l t s f o r G o l d R i v e r D a t a 4 5 V CHAPTER 2. EM I n d u c t i o n i n a L a y e r e d E a r t h by a Bent L i n e of C u r r e n t 2.1 A p p r o x i m a t e S o l u t i o n s U s i n g Image T h e o r y 55 2.2 Image S o l u t i o n s f o r T o t a l F i e l d s a b o u t a Bent L i n e S o u r c e 59 2.3 G o l d R i v e r S u r v e y R e s u l t s 61 2.4 'Rule-of-Thumb' f o r I n t e r p r e t a t i o n 72 CHAPTER 3. EM I n d u c t i o n i n a Homogeneous E a r t h w i t h a L a t e r a l I n h o m o g e n e i t y 3.1 Tank Model R e s u l t s 74 3.2 D a t a R e d u c t i o n 88 CONCLUDING REMARKS 96 REFERENCES 98 APPENDIX A. H e r t z P o t e n t i a l s 100 APPENDIX B. R e f l e c t i o n and T r a n s m i s s i o n F i l t e r s 106 APPENDIX C. D i r e c t i o n a l F i l t e r s 110 APPENDIX D. Computational Procedure v i i LIST OF TABLES I . Model Dimensions 77 I I . G e o p h y s i c a l Dimensions 77 v i i i L I S T OF FIGURES 1.1 System geometry 6 1.2 Wavenumber components 8 1.3 N o t a t i o n f o r L a y e r e d E a r t h Model 10 1.4 T r a n s f o r m Method F o r O b t a i n i n g EM R e s p o n s e s 15 1.5 Geometry f o r t h e Case of a S t r a i g h t L i n e o f C u r r e n t . 16 1.6 H a l f Space R e s p o n s e s l o g 1 0 | H ^ ( x , z = 0) | v e r s u s l o g , 0 ( x ) 21 1.7 L a y e r Over a H a l f Space R e s p o n s e s l o g T o | H 2 T ( x , z = 0 ) | v e r s u s l o g , 0 ( x ) 23 1.8 H a l f Space R e s p o n s e s l o g 1 0 | H x ( x , z = 0 ) | v e r s u s l o g 1 0 ( x ) 26 1.9 L a y e r Over a H a l f Space R e s p o n s e s l o g 1 0 | H x ( x , z = 0) | v e r s u s l o g 1 0 ( x ) 28 1.10 N o r m a l i z e d H a l f Space R e s p o n s e s l o g 1 0 ( | H 2 T ( x , z = 0 ) | / | ( x , z = 0 ) | ) v e r s u s l o g , 0 ( x ) 31 1.11 N o r m a l i z e d L a y e r o v e r a H a l f Space R e s p o n s e s l o g 1 0 ( |H^ T(x,z=0) |/|H 2 L U (x,z=0) | ) v e r s u s l o g 1 0 ( x ) .... 33 1.12 N o r m a l i z e d H a l f Space R e s p o n s e s l o g 1 0 ( | H x T ( x , z = 0 ) | / | H x L ( x , z = 0 ) | ) v e r u s l o g 1 0 ( x ) 35 1.13 N o r m a l i z e d L a y e r Over a H a l f Space R e s p o n s e s l o g 1 0 ( | H x T ( x , z = 0 ) | / | H x U ( x , z = 0 ) | ) v e r s u s l o g 1 0 ( x ) 37 1.14 N o r m a l i z e d H a l f Space R e s p o n s e s | H x T ( x , z = 0 ) | / | H Z 1 ( x , z = 0 ) | v e r s u s x 40 i x 1.15 N o r m a l i z e d L a y e r Over a H a l f Space R e s p o n s e s |H x T ( x , z = 0) | / | H S T ( x , z = 0 ) | v e r s u s x 42 1.16 A p p a r e n t R e s i s t i v i t y U s i n g C a g n a i r d ' s R e l a t i o n 44 1.17 L o c a t i o n Map f o r G o l d R i v e r S u r v e y 47 1.18 Model R e s u l t s f o r 60Hz 50 1.19 Model R e s u l t s f o r 420Hz 52 2.1 Image T h e o r y R e s u l t s v e r s u s E x a c t R e s u l t s 57 2.2 Geometry f o r A F i n i t e L e n g t h W i r e 59 2.3 S p a t i a l D i s t r i b u t i o n of M o d e l l e d P o w e r l i n e s 61 2.4 Model R e s u l t s f o r 60Hz F i e l d D a t a 64 2.5 Model R e s u l t s f o r 420Hz F i e l d D a t a 66 2.6 Model R e s u l t s f o r S e c t i o n s 1 and 2 a t 60Hz 69 2.7 Model R e s u l t s f o r S e c t i o n s 1 and 2 a t 420Hz 71 3.1 Model A r r a n g e m e n t and C o - o r d i n a t e System 76 3.2 H^ M o d e l l i n g R e s u l t s f o r a L a t e r a l I n h o m o g e n e i t y .... 80 3.3 H^ M o d e l l i n g R e s u l t s f o r a L a t e r a l I n h o m o g e n e i t y .... 82 3.4 Ey M o d e l l i n g R e s u l t s f o r a L a t e r a l I n h o m o g e n e i t y .... 84 3.5 |H X T (x,z=0) ( x , z = 0) | V e r s u s x f o r a L a t e r a l I n h o m o g e n e i t y 86 3.6 H^. (x, z=0). x R e s u l t s f o r a L a t e r a l I n h o m o g e n e i t y .... 91 3.7 H x (x,z=0).x R e s u l t s f o r a L a t e r a l I n h o m o g e n e i t y .... 93 3.8 E y (x,z=0).x R e s u l t s f o r a L a t e r a l I n h o m o g e n e i t y .... 95 B.I H a l f Space Boundary C o n d i t i o n s 106 B.2 L a y e r e d E a r t h Model 108 D.I Example P r o b l e m : C o a x i a l D i p o l e System 112 D.2 Example P r o b l e m 1 R e s u l t s 116 D.3 Example Problem 2 R e s u l t s D.4 Example Problem 3 R e s u l t s x i A C K N O W L E D G E M E N T S I w o u l d l i k e t o e x p r e s s my t h a n k s t o a l l t h o s e who h a v e h e l p e d me i n o n e way o r a n o t h e r t o c o m p l e t e t h i s t h e s i s . P r i m a r y a m o n g t h e s e p e o p l e i s J u l i a n C a b r e r a w h o s e a d v i c e , e n c o u r a g e m e n t , k n o w l e d g e , a n d i n f i n i t e p a t i e n c e w e r e a n e s s e n t i a l e l e m e n t f r o m b e g i n n i n g t o e n d . F o r h i s h e l p I am p e r m a n e n t l y g r a t e f u l . F o r v a r i o u s a d v i c e , g u i d a n c e a n d a n s w e r s t o q u e r i e s , I am g r a t e f u l t o P r o f e s s o r S l a w s o n , my s u p e r v i s o r . I h a v e a l s o b e e n f o r t u n a t e i n t h e c o l l a b o r a t i o n o f D o u g O l d e n b u r g , who c h e c k e d t h e m a t h e m a t i c s p r e s e n t e d w i t h i n . T o u n n a m e d o t h e r s who a p p e a r e d t o l e n d me a h a n d , my g r a t i t u d e i s e q u a l . 1 INTRODUCTION T h i s t h e s i s i n v e s t i g a t e s and presents p r e l i m i n a r y s t r a t e g i e s f o r the use of powerline r a d i a t i o n as a primary f i e l d f o r e l e c t r o m a g n e t i c (EM) surveys. T h i s study was motivated by the idea that EM f i e l d s surrounding powerlines might be used f o r l a t e r a l p r o f i l i n g surveys i n the search f o r m i n e r a l d e p o s i t s and/or shallow depth sounding surveys. Since powerlines a f f o r d a f r e e , powerful, c o n t r o l l e d m u l t i - f r e q u e n c y source, and the l o g i s t i c s of t r a n s p o r t i n g t r a n s m i t t e r c a b l e and generator are e l i m i n a t e d , t h i s survey method may prove to be c o m p e t i t i v e with more t r a d i t i o n a l EM survey methods. The use of powerlines f o r depth sounding or l a t e r a l p r o f i l i n g i s a unique method. The method i s d i s t i n c t because of the nature of the source, which i s f i x e d i n space and broadcasts at f i x e d f r e q u e n c i e s . U s u a l l y , powerline r a d i a t i o n c o n s t i t u t e s a noise source f o r t r a d i t i o n a l g e o p h y s i c a l surveys, such as m a g n e t o t e l l u r i e s (MT). E s s e n t i a l l y , powerlines c o n s t i t u t e e l e v a t e d l i n e sources of a l t e r n a t i n g c u r r e n t which broadcast at i n t e g r a l m u l t i p l e s of 60 Hertz (Hz). T h i s harmonic enrichment i s due to n o n - l i n e a r i t i e s in the power t r a n s m i s s i o n system as p o i n t e d out by Hayashi et a l . [ 1 ] . Up to 10 harmonics of 60Hz have been measured i n the f i e l d by M c C o l l o r et a l . [ 2 ] . These f r e q u e n c i e s are s i m i l a r to those used in. MT methods, as w e l l as f o r other g e o p h y s i c a l i n d u c t i o n methods which use f r e q u e n c i e s i n the range of 10-5000HZ. 2 T h i s t h e s i s models powerlines as an e l e v a t e d , ungrounded l i n e of a l t e r n a t i n g c u r r e n t . The l i n e i s c o n s i d e r e d ungrounded because r e t u r n c u r r e n t s from the grounded p a r t s of the t r a n s m i s s i o n l i n e are small ( t y p i c a l l y 1-30 Amperes; McCollar et a l . [ 2 ] ) , and act over long d i s t a n c e s of s e v e r a l hundred k i l o m e t e r s between grounding p o i n t s . In order to understand the behavior of EM f i e l d components i n the v i c i n i t y of a powerline, the EM response of the e a r t h due to powerline e x c i t a t i o n must be s o l v e d . T h i s i s known as the forward problem. The usual s t r a t e g y f o r the numerical s o l u t i o n of forward problems c o n s i s t s of assuming the s i m p l e s t e a r t h model, that i s , an e a r t h whose e l e c t r i c a l p r o p e r t i e s vary with depth o n l y . Layered earths are the simplest to model n u m e r i c a l l y . Hence Chapters 1 and 2 present s o l u t i o n s f o r l a y e r e d e a r t h models due to a l i n e source of c u r r e n t . Chapter 1 p r e s e n t s the s o l u t i o n to the EM i n d u c t i o n problem f o r a s t r a i g h t l i n e of c u r r e n t (the s i m p l e s t and most convenient source geometry) over a l a y e r e d e a r t h . Because the wavelengths of the s i g n a l s broadcast by powerlines are long (5000km at 60Hz) r e l a t i v e to the measurement d i s t a n c e s ( t y p i c a l l y 10km) from the source, plane wave r e f l e c t i o n theory i s i n a p p r o p r i a t e f o r the s o l u t i o n of the forward l a y e r e d e a r t h problem. The forward problem i s s o l v e d u s i n g the l i n e a r f i l t e r method of L a j o i e et a l . [ 3 ] . The s o l u t i o n i s i n the form of a ( i n v e r s e ) F o u r i e r 3 e a r t h m o d e l s a r e e v a l u a t e d u s i n g F a s t F o u r i e r T r a n s f o r m ( F F T ) a l g o r i t h m s ; T h e r e s p o n s e o f t h e e a r t h i s c h a r a c t e r i z e d b y a n u m b e r o f r e s p o n s e f u n c t i o n s , t h e a d v a n t a g e s a n d d i s a d v a n t a g e s o f w h i c h a r e d i s c u s s e d . F i n a l l y , d a t a g a t h e r e d i n t h e v i c i n i t y o f a p o w e r l i n e i s i n t e r p r e t e d f o r a l a y e r e d e a r t h m o d e l u s i n g o n e o f t h e r e s p o n s e f u n c t i o n s . C h a p t e r 2 p r e s e n t s a n a p p r o x i m a t e s o l u t i o n t o t h e f o r w a r d p r o b l e m f o r t h e c a s e o f a b e n t l i n e o f c u r r e n t o v e r a l a y e r e d e a r t h . T h i s i s m e a n t t o a c c o m m o d a t e t h e s p a t i a l l y c o m p l e x g e o m e t r i e s w h i c h p o w e r l i n e s a s s u m e i n p r a c t i c e . A n a p p r o x i m a t e s o l u t i o n t o t h e f o r w a r d p r o b l e m i s e m p l o y e d t o r e d u c e c o m p u t a t i o n a l c o s t s a n d t o a v o i d s o m e o f t h e c o m p u t a t i o n a l d i f f i c u l t y p r e s e n t e d b y t h e e x a c t s o l u t i o n m e t h o d u s e d i n C h a p t e r 1. D a t a g a t h e r e d i n t h e v i c i n i t y o f a p o w e r l i n e i s a g a i n i n t e r p r e t e d i n t e r m s o f a l a y e r e d e a r t h m o d e l . C h a p t e r 3 p r e s e n t s a n a l o g u e m o d e l r e s u l t s f o r t h e c a s e o f a s t r a i g h t l i n e o f c u r r e n t w i t h a b u r i e d l a t e r a l i n h o m o g e n e i t y . T h e b e h a v i o r o f t h e f i e l d c o m p o n e n t s i n t h e r e g i o n o f a f i n i t e d i m e n s i o n l a t e r a l i n h o m o g e n e i t y i s o f i n t e r e s t w h e n d e t e c t i o n o f l a t e r a l c h a n g e s i n e l e c t r i c a l p r o p e r t i e s i s d e s i r e d . A g a i n , t h i s m o d e l s t u d y w a s m o t i v a t e d b y t h e i d e a t h a t p o w e r l i n e s m i g b t b e u s e d f o r l a t e r a l p r o f i l i n g s u r v e y s f o r t h e s e a r c h f o r m i n e r a l d e p o s i t s . 4 N O T A T I O N I n t h i s t h e s i s , t h e e a r t h o c c u p i e s t h e l o w e r h a l f s p a c e a n d a i r o c c u p i e s t h e u p p e r h a l f s p a c e . T h e e a r t h i s a s s u m e d t o b e c o m p r i s e d o f h o m o g e n e o u s , i s o t r o p i c , h o r i z o n t a l l a y e r s . T h e m a g n e t i c p e r m e a b l i l i t y v o f t h e e a r t h i s a s s u m e d t o b e e q u a l t o Uor t h e p e r m e a b l i l i t y o f f r e e s p a c e . S i m i l a r l y , t h e p e r m i t t i v i t y o f t h e e a r t h i s e 0 , t h e p e r m i t t i v i t y o f f r e e s p a c e . M e t e r -k i l o g r a m - s e c o n d u n i t s a r e e m p l o y e d a n d a t i m e d e p e n d a n c e o f e x p ( i d ) t ) i s i m p l i c i t t h r o u g h o u t . T h e z d i r e c t i o n i s p o s i t i v e w i t h i n c r e a s i n g d e p t h . T A B L E O F S Y M B O L S D e f i n i t i o n C a r t e s i a n c o - o r d i n a t e s u n i t v e c t o r s i n x , y , z d i r e c t i o n s w a v e n u m b e r s i n t h e x , y d i r e c t i o n s ( r a d i a n s / m e t e r ) w a v e n u m b e r i n z d i r e c t i o n ( r a d i a n s / m e t e r ) w a v e n u m b e r ( r a d i a n s / m e t e r ) E l e c t r i c f i e l d ( v o l t s / m e t e r ) M a g n e t i c f i e l d ( a m p e r e s . t u r n ) / m e t e r ) S y m b o l x , y , z k ,K. k^=/{ iuu(o+iu>e) } k=y/(k„ 2 + k y 2 + k ^ 2 ) E ( x , y , z ) H ( x , y , z ) f,=9f/9x p a r t i a l d e r i v a t i v e n o t a t i o n f 2=3f/3y f 3 = 9 f / d z F ( k , , k y , z ) = J* J " F ( x , y , z ) e x p { - i ( k x x + k y y ) } d x d y f o r w a r d F o u r i e r t r a n s f o r m OO CO F ( x , y , z ) = l / ( 4 7 r 2 ) / -00 3/9x < > i k x 9/3y <—> i k y 3/3z <——» k a <j 8 = / ( 2 / ( W M 0 O ) ) Mo Co r ( x f y , z ) II(x,y ,z) s u p e r s c r i p t i s u p e r s c r i p t t s u p e r s c r i p t T s u p e r s c r i p t s fF( k y , k y , z )exp{ i ( k x x + k y y) }dk x dky i n v e r s e F o u r i e r t r a n s f o r m d e r i v a t i v e o p e r a t o r s i n s p a t i a l & f r e q u e n c y domains c o n d u c t i v i t y ( S i e m e n s / m e t e r ) a n g u l a r f r e q u e n c y ( r a d i a n s / s e c o n d ) s k i n d e p t h ( m e t e r s ) p e r m e a b i l i t y = 4 7 r E - 7 H e n r i e s / m e t e r p e r m i t t i v i t y = 8 . 8 5 4 E - 1 2 F a r a d s / m e t e r M a g n e t i c H e r t z V e c t o r E l e c t r i c H e r t z V e c t o r i n c i d e n t o r p r i m a r y t r a n s m i t t e d T o t a l s e c o n d a r y o r r e f l e c t e d 6 1.1 F O R M U L A T I O N O F T H E L A Y E R E D E A R T H P R O B L E M C o n s i d e r t h e s y s t e m s h o w n i n F i g u r e 1 . 1 . A c u r r e n t I e x p ( i o ; t ) f l o w s i n a h o r i z o n t a l e l e v a t e d w i r e l o c a t e d a h e i g h t h a b o v e a l a y e r e d e a r t h . F I G U R E 1 . 1 . S Y S T E M G E O M E T R Y T h e e a r t h m o d e l a n d c o - o r d i n a t e s y s t e m . T h e l i n e s o u r c e r e m a i n s a c o n s t a n t h e i g h t h o v e r a s t r a t i f i e d e a r t h m o d e l . P e r m e a b i l i t y n0 a n d p e r m i t t i v i t y e 0 a r e c o n s t a n t s t h r o u g h o u t t h e w h o l e s p a c e . C u r r e n t s a r e i n d u c e d i n t h e c o n d u c t i n g g r o u n d i n t h e v i c i n i t y o f t h e c u r r e n t s o u r c e a c c o r d i n g t o F a r a d a y ' s l a w : V x E = - 3 B / 9 t A l l EM m e t h o d s s e e k t o d e t e r m i n e p r o p e r t i e s o f t h e s u b s u r f a c e b y m e a s u r e m e n t a n d a n a l y s i s o f s e c o n d a r y EM f i e l d s p r o d u c e d b y t h e s e c u r r e n t s i n t h e g r o u n d . T o t a l f i e l d s , c o m p o s e d o f b o t h t h e p r i m a r y a n d s e c o n d a r y f i e l d s , c a n b e m e a s u r e d w i t h a n i n d u c t i o n c o i l o r m a g n e t o m e t e r . A n a l y s i s o f t h e s e f i e l d s r e l i e s o n t h e a b i l i t y t o s o l v e t h e " f o r w a r d p r o b l e m " , t h a t i s , h a v i n g t h e m e a n s t o s o l v e f o r t h e t o t a l f i e l d s g i v e n a s p e c i f i c e a r t h 7 model. The e l e c t r o m a g n e t i c response of a l a y e r e d e a r t h i s a c l a s s i c g e o p h y s i c a l problem which has r e c e i v e d much a t t e n t i o n i n the l i t e r a t u r e ( f o r g e n e r a l t r e a t m e n t s see, f o r example, B r e k h o v s k i k h [ 4 ] ; K e l l e r and F r i s c h k n e c h t [ 5 ] ; P a t r a and M a l l i c k [ 6 ] ; Wait [ 7 ] ) . S o l u t i o n s t o the f o r w a r d EM i n d u c t i o n problem t o a s o u r c e f i e l d p o s s e s s i n g c y l i n d r i c a l symmetry (such as a magnetic d i p o l e ) are u s u a l l y i n the form of a Hankel T r a n s f o r m (HT) ( F r i s c h k n e c h t [8];Ryu et a l . [ 9 ] ; F u l l a g e r [ 1 0 ] ) : F ( r , X ) = J k ( X ) J ( X r ) d X (1.1) 0 where F ( r , X ) i s a c y l i n d r i c a l component of the EM f i e l d , k(X) i s a complex f u n c t i o n i n v o l v i n g the l a y e r e d e a r t h and s o u r c e p a r a m e t e r s , J i s a B e s s e l f u n c t i o n of the f i r s t k i n d (the o r d e r of which depends on the s p a t i a l o r i e n t a t i o n of the d i p o l e ) , and X i s a dummy v a r i a b l e of i n t e g r a t i o n . W i t h t h i s f o r m u l a t i o n , i t i s p r a c t i c a l t o d e a l o n l y w i t h s o u r c e s p o s s e s s i n g c y l i n d r i c a l symmetry. A C a r t e s i a n f o r m u l a t i o n of the f o r w a r d i n d u c t i o n problem has been p r e s e n t e d by Weaver [11],and L a j o i e e t a l . [ 3 ] . In t h i s c a s e , the s o l u t i o n t o the l a y e r e d e a r t h EM i n d u c t i o n problem i s a two d i m e n s i o n a l i n v e r s e F o u r i e r T r a n s f o r m ( F T ) : F S(x,y,z=0) - 1/(47r2)jp jTRF(k x ,k v , z = 0 ) F L ( k . ,k v ,z=0) -OO -00 ' / e x p { i ( k x x + k y ) } d k y d k (1 .2) 8 w h e r e F M x , y , z = 0 ) i s a C a r t e s i a n c o m p o n e n t o f t h e s e c o n d a r y EM f i e l d , R F ( k x , k y , z = 0 ) i s a c o m p l e x f u n c t i o n i n v o l v i n g t h e l a y e r e d e a r t h p a r a m e t e r s , k x , k y a r e p r o j e c t i o n s o f t h e w a v e n u m b e r k ( s e e F i g u r e 1 . 2 ) o n t o t h e x - y p l a n e , a n d F U ( k „ , k y , z = 0 ) i s t h e F T o f a C a r t e s i a n c o m p o n e n t o f t h e i n c i d e n t f i e l d . T h i s f o r m u l a t i o n a l l o w s c o m p l e t e l y a r b i t r a r y s o u r c e g e o m e t r y , a n d i s s o l v e d n u m e r i c a l l y u s i n g d i g i t a l F a s t F o u r i e r T r a n s f o r m ( F F T ) a l g o r i t h m s . z k 2 = / { i u n ( o + i w e 0 ) } It = k„ i+ky j+k a R F I G U R E 1 . 2 . WAVENUMBER C O M P O N E N T S C a r t e s i a n c o m p o n e n t s o f t h e w a v e n u m b e r It. B o t h f o r m u l a t i o n s ( 1 . 1 ) a n d ( 1 . 2 ) r e p r e s e n t a s u p e r p o s i t i o n o f m o d e s . I n t h e c a s e o f ( 1 . 2 ) t h e s e m o d e s a r e c o s i n u s o i d s o f d i f f e r e n t s p a t i a l f r e q u e n c i e s , o r p l a n e w a v e s . F o r t h e p u r p o s e s o f t h i s t h e s i s , w h i c h w i l l c o n s i d e r s o u r c e c u r r e n t s w i t h a r b i t r a r y g e o m e t r y , t h e F o u r i e r T r a n s f o r m ( 1 . 2 ) 9 p r o v i d e s t h e m o s t p r a c t i c a l m e a n s o f c o m p u t i n g t h e e x a c t f o r w a r d EM i n d u c t i o n p r o b l e m . F o u r i e r T r a n s f o r m s i n t h e S o l u t i o n o f t h e F o r w a r d p r o b l e m A n o u t l i n e o f t h e m a t h e m a t i c a l s t e p s n e c e s s a r y f o r o b t a i n i n g e q u a t i o n 1.2 f o l l o w s . A m o r e d e t a i l e d d e r i v a t i o n i s g i v e n i n A p p e n d i x A a n d B . C o n s i d e r a s o u r c e w h i c h r a d i a t e s w i t h a t i m e d e p e n d a n c e o f t h e f o r m e x p ( i c j t ) . I n o r d e r t o s o l v e t h e f o r w a r d EM i n d u c t i o n p r o b l e m f o r u n g r o u n d e d s o u r c e s , t h e e l e c t r i c a n d m a g n e t i c v e c t o r s E a n d "ft, a r e e x p r e s s e d i n t e r m s o f t h e m a g n e t i c H e r t z v e c t o r p o t e n t i a l T . T h e a d v a n t a g e s o f f o r m u l a t i n g t h e p r o b l e m i n t h i s m a n n e r a r e t w o f o l d . . F i r s t , t h e f o u r M a x w e l l e q u a t i o n s a r e r e d u c e d t o t w o e q u a t i o n s . S e c o n d , f o r e a r t h m o d e l s w i t h l a y e r s d e f i n e d b y z = c o n s t a n t , i t i s o n l y n e c e s s a r y t o u s e t h e z c o m p o n e n t o f r ( H a r r i n g t o n [ 1 2 ] ) . T h u s , r w i l l b e t r e a t e d a s a s c a l a r q u a n t i t y . U s i n g p a r t i a l d e r i v a t i v e n o t a t i o n , t h e r e l a t i o n b e t w e e n E , H a n d r i s : E ( x , y , z ) = - T 2 ( x , y , z ) i + r , ( x , y , z ) 3 H ( x , y , z ) = T 3 , ( x , y , z ) i + r 3 2 ( x , y , z ) 5 - ( r , , ( x , y , z ) + r 2 2 ( x , y , z ) ) R (1 .3) I t i s d e m o n s t r a t e d i n A p p e n d i x A t h a t T ( x , y , z ) a n d t h e C a r t e s i a n c o m p o n e n t s o f IS a n d H s a t i s f y t h e H e l m h o l t z e q u a t i o n i n a h o m o g e n e o u s , i s o t r o p i c m e d i u m . C o n s e q u e n t l y , i n a l a y e r e d 10 e a r t h , the H e l m h o l t z e q u a t i o n i s s a t i s f i e d i n each l a y e r i : (v 2-k z 2)r=o (1.4) To s o l v e e q u a t i o n (1.4) s u b j e c t t o the case of a l a y e r e d e a r t h model, one need o n l y s a t i s f y the boundary c o n d i t i o n s i n terms of r ( x , y , z ) . At a boundary d e f i n e d by z=d^ (see F i g u r e 1.3), the s e — + a r e : r(x,y,z=d^)=r(x,y,z=d i <) T 3 (x,y ,z=d~)=r3 (x,y,z=dj) - i . d , 1-1 -Zxd, FIGURE 1.3. NOTATION FOR THE LAYERED EARTH MODEL These e x p r e s s i o n s a r e t r a n s f o r m e d t o the wavenumber domain so t h a t d e r i v a t i v e o p e r a t o r s become m u l t i p l i c a t i v e o p e r a t o r s : T ( k x , k y , z = d [ ) = T ( k x , k y , z = d * ) (1.5) k t T ( k x , ky , z=d ~) = k l + | T ( k x , k y , z=d- L ) (1.6) 11 B y i t e r a t i v e l y s o l v i n g e q u a t i o n s ( 1 . 5 ) a n d ( 1 . 6 ) f o r t h e c a s e o f a l a y e r e d e a r t h i t i s p o s s i b l e t o d e f i n e t r a n s m i s s i o n a n d r e f l e c t i o n c o e f f i c i e n t s i n t h e w a v e n u m b e r d o m a i n . T h i s p r o c e d u r e i s d e s c r i b e d i n m o r e d e t a i l i n A p p e n d i x B . B e c a u s e t h e y a r e d e f i n e d i n t h e w a v e n u m b e r d o m a i n t h e y a r e c a l l e d r e f l e c t i o n a n d t r a n s m i s s i o n f i l t e r s ( R F a n d T F r e s p e c t i v e l y ) . G i v e n a n i n c i d e n t o r p r i m a r y f i e l d c o m p o n e n t a t t h e t o p o f a s t a c k o f l a y e r s , t h e t r a n s m i s s i o n a n d r e f l e c t i o n f i l t e r s w i l l g i v e t h e f i e l d t r a n s m i t t e d t h r o u g h t h e s t a c k , a n d t h e f i e l d r e f l e c t e d f r o m t h e s t a c k , r e s p e c t i v e l y . T h e i n d u c t i o n p r o b l e m i s t h e r e f o r e r e d u c e d t o a s i m p l e f i l t e r i n g o p e r a t i o n i n t h e w a v e n u m b e r d o m a i n . U p w a r d / D o w n w a r d C o n t i n u a t i o n : . U p w a r d a n d d o w n w a r d c o n t i n u a t i o n a r e u s e f u l f o r e v a l u a t i n g f i e l d s a t d i f f e r e n t v a l u e s o f z = c o n s t a n t . A n a n a l y t i c e x p r e s s i o n o f t h e F o u r i e r t r a n s f o r m o f s o m e i n c i d e n t C a r t e s i a n c o m p o n e n t may b e a v a i l a b l e f o r a p l a n e z = c o n s t a n t w h i c h i s n o t t h e s a m e a s t h e m e a s u r i n g p l a n e . U p w a r d o r d o w n w a r d c o n t i n u a t i o n t o t h i s m e a s u r i n g p l a n e i s t h e n r e q u i r e d . C a r e m u s t b e t a k e n s o a s n o t t o c o n t i n u e f i e l d s t h r o u g h p l a n e s z = c o n s t a n t w h i c h c o n t a i n s o u r c e s . U p w a r d a n d d o w n w a r d c o n t i n u a t i o n a r e e a s i l y p e r f o r m e d b y u s e o f t h e o p e r a t o r e x p { - z k } w h i c h i s d e f i n e d b y t r a n s f o r m i n g ( 1 . 4 ) t o t h e w a v e n u m b e r d o m a i n a n d s o l v i n g t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n w h i c h r e s u l t s : { d 2 / d z 2 - k 2 } r = 0 D i r e c t i o n a l F i l t e r s : 12 By a p p l y i n g f i l t e r s d e f i n e d i n the wavenumber domain t o any component of the secondary magnetic f i e l d , i t i s p o s s i b l e t o o b t a i n a l l the o t h e r C a r t e s i a n components of H. The o r i g i n a l f o r m u l a t i o n of t h i s was p r e s e n t e d by S k e e l s [ 1 3 ] , and S k e e l s and Watson [ 1 4 ] . T h i s can be seen by t r a n s f o r m i n g (1.3) t o the wavenumber domain and s o l v i n g f o r F(x,y,z=0) (see Appendix C ) . These f i l t e r s a r e d e f i n e d as f o l l o w s : H x ( k y , k y , z = 0 ) = i k x k H £ ( k , , k y , z = 0 ) / ( k y 2 + k y 2 ) H y ( k x ,k y,z=0) = i k y k H 2 ( k x , k y , z=0)/(k„ 2 + k y 2 ) In summary, the f i r s t s t e p r e q u i r e s t h a t a C a r t e s i a n component of the i n c i d e n t f i e l d , say , be F o u r i e r t r a n s f o r m e d t o k x - k y space. F i n d i n g the t r a n s f o r m of the i n c i d e n t f i e l d on any p l a n e z=constant may be performed n u m e r i c a l l y o r , b e t t e r y e t , a n a l y t i c a l l y . The m a t r i x of t r a n s f o r m e d d a t a may then be downward c o n t i n u e d t o the s u r f a c e z=0, and m u l t i p l i e d w i t h the r e f l e c t i o n f i l t e r R F ( k x , k y , z = 0 ) t o o b t a i n the r e f l e c t e d or secondary f i e l d s a t z=0. T o t a l f i e l d s may be found by r e p l a c i n g the r e f l e c t i o n f i l t e r by [ R F ( k x , k y , z = 0 ) + 1 . 0 ] . Once t h i s has been done, the f i e l d may be upward c o n t i n u e d t o an a r b i t r a r y s u r f a c e z = c o n s t a n t , and then n u m e r i c a l l y i n v e r t e d by FFT t o t h e x-y domain. The essence of t h i s method i s the removal or f i l t e r i n g of s p a t i a l f r e q u e n c i e s from a C a r t e s i a n component of the i n c i d e n t EM f i e l d . Which f r e q u e n c i e s a r e removed depends s t r i c t l y on the 13 e a r t h c o n d u c t i v i t y s t r u c t u r e s u c h t h a t r e s i s t i v e e a r t h m o d e l s a c t a s a l o w p a s s f i l t e r a n d c o n d u c t i v e e a r t h m o d e l s a c t a s a b a n d p a s s f i l t e r . T h e r e s u l t i n g o u t p u t o f s u c h a f i l t e r i n g o p e r a t i o n y i e l d s t h e s e c o n d a r y f i e l d s f r o m t h e p a r t i c u l a r e a r t h m o d e l . B e c a u s e d i s c r e t e F F T ' s a r e u s e d t o i n v e r t b a c k t o t h e x - y d o m a i n , t h e e n t i r e p r o c e d u r e i s s u b j e c t t o t h e u s u a l d i s c r e t e F F T p r o b l e m s o f a l i a s i n g a n d t r u n c a t i o n . E r r o n e o u s p r e d i c t i o n s o f s e c o n d a r y f i e l d s r e s u l t w h e n t h e s e t y p e s o f e r r o r s a r e n o t c o r r e c t e d . A p p e n d i x D d e s c r i b e s t h e c o m p u t a t i o n a l p r o c e d u r e s r e q u i r e d t o a v o i d t h e s e e r r o r s . F i g u r e 1 . 4 p r e s e n t s a b l o c k d i a g r a m t o s u m m a r i z e t h e s t e p s u s e d t o s o l v e t h e EM i n d u c t i o n p r o b l e m . T h i s f o r m u l a t i o n w i l l now b e a p p l i e d t o t h e c a s e o f a s t r a i g h t l i n e o f a l t e r n a t i n g c u r r e n t , l o c a t e d a h e i g h t h a b o v e a l a y e r e d e a r t h . 14 F I G U R E 1 . 4 . T R A N S F O R M M E T H O D F O R O B T A I N I N G E M R E S P O N S E S A b l o c k d i a g r a m r e p r e s e n t a t i o n s h o w i n g t h e s t e p s r e q u i r e d f o r o b t a i n i n g t h e f r e q u e n c y d o m a i n s o l u t i o n d u e t o a n a r b i t r a r y c u r r e n t s o u r c e o v e r a o n e d i m e n s i o n a l e a r t h . FFT METHOD OF OBTAINING SECONDARY H FIELDS Hz (x,y,z h)|Fourl.er Trans(or •analytic 'numerical Z to X s Hx Up (FFT)-* Fi l ter * Cont. Hxl»,y,i-h) Down Cont. k,z>C >Hz" , Up Cont. (FFT)-i Hz(x,y,z=.h) Tl 1 i * i Z to ,Y s Hy Up (FFT)" 1 Filter i Cont. Hy(x,y,i=h) To obtain total fields replace RF by RF*1 16 1.2 SOLUTIONS FOR THE FIELD COMPONENTS ABOUT AN INFINITE ELEVATED STRAIGHT LINE OF CURRENT: lexp(iujt) h Hz •7-7-7- / / / z<+) FIGURE 1.5. GEOMETRY FOR THE CASE OF A STRAIGHT LINE OF CURRENT A l i n e of c u r r e n t I j a t h e i g h t z=-h ( F i g u r e 1.5) has p r i m a r y f i e l d components a t the e a r t h ' s s u r f a c e z=0 g i v e n by: H X L ( x , y , z = 0)=-Ih/{27r(x 2+h 2)} (1.7) H 4 L (x,y,z=0) = Ihx/{27r(x 2+h 2)} (1.8) These e x p r e s s i o n s a r e v a l i d f o r q u a s i - s t a t i c c o n d i t i o n s , t h a t i s , f o r d i s t a n c e s from the s o u r c e which a r e much l e s s than a f r e e space wavelength. S i n c e the f r e e space wavelength of a 60Hz wave i s 5000 km, and measurement d i s t a n c e s a r e on the o r d e r of 10 km, t h i s a p p r o x i m a t i o n i s r e a s o n a b l e . 17 N o t i n g t h a t H x ( x , y , z = 0 ) i s a n e v e n f u n c t i o n o f x a n d H z t ( x , y , z = 0 ) i s a n o d d f u n c t i o n o f x , a n a l y t i c e x p r e s s i o n s f o r t h e F o u r i e r t r a n s f o r m o f ( 1 . 7 ) a n d ( 1 . 8 ) a r e f o u n d i n m o s t t a b l e s o f i n t e g r a l s : r L u ( k x , k z = 0 ) = - I h / ( 2 7 r ) J c o s k y x / ( x 2 + h 2 ) d x j r e x p ( i k y y ) d y = - l / ( 2 7 r ) e x p ( - h k x ) 6 ( k y ) k x >0 Re(h)>0 H u ( k x , k y ,z=0) = i l / ( 2 7 r ) / x s i n k x / ( x 2 + h 2 ) d x / e x p ( i k y y ) d y = 7 - 0 0 - 0 0 ' = i l / ( 2 7 r ) k x e x p ( - h k , ) 8 ( k y ) k x>0 R e ( h ) > 0 T h u s , f o r t h e q u a s i - s t a t i c c a s e , t h e i n c i d e n t c o m p o n e n t s H X L a n d a r e a H i l b e r t p a i r : H X L (k„ , k y , z = 0 ) = - i s g n ( k „ ) H Z L (k„ , k y ,z=0) T h i s r e s u l t may a l s o b e o b t a i n e d b y c o n s i d e r i n g a d i r e c t i o n a l f i l t e r a s s u m i n g q u a s i - s t a t i c c o n d i t i o n s f o r a o n e - d i m e n s i o n a l c a s e . N o t i n g t h a t t h e i n c i d e n t e l e c t r i c v e c t o r h a s o n l y a j c o m p o n e n t , t h e t r a n s f o r m o f E m a y b e f o u n d b y F o u r i e r t r a n s f o r m a t i o n o f V x E = - 3 B / 3 t : £ y L ( k x , k y , z = 0 ) = - i w M O H i 1 ' ( k x , k y ,z=0)/k x = - i a > M 0 I h e x p ( - h k x ) / ( 2 ? r k x ) 6 ( k y ) F o r a s t r a i g h t l i n e s o u r c e t h e d o u b l e i n t e g r a l r e d u c e s t o a s i n g l e i n t e g r a l b e c a u s e o f t h e p r e s e n c e o f t h e d e l t a f u n c t i o n 6 ( k y ) . C o m p u t a t i o n a l l y , t h i s i s c o n v e n i e n t b e c a u s e o n l y o n e -d i m e n s i o n a l t r a n s f o r m s a r e t h e n r e q u i r e d . A f t e r p e r f o r m i n g t h e f i l t e r i n g o p e r a t i o n o n e o b t a i n s t h e 18 f o l l o w i n g t o t a l EM f i e l d s over a l a y e r e d e a r t h : H X T (x,z=0)=-I/(27r)7 ( 1+RF(k x ,z=0) ) e x p ( - h k x ) c o s k y x d k x (1.9) -oo H, (x,z=0) = i l / ( 2 i r ) J ( 1+RF(k x ,z = 0) ) e x p ( - h k x ) s i n k x x d k x (1.10) -oo E y T(x,z=0)=-i£JM OIh/(27r)7( 1 + R F ( k x ,z = 0) ) e x p ( -h k x ) c o s k x x / k x dk x (1.11) The s o l u t i o n s t o ( 1 . 9 ) , (1.10) and (1.11) f o r z<0 may be found by upward c o n t i n u a t i n g the f i e l d s b e f o r e i n v e r t i n g t o the x-y domain. E q u a t i o n s ( 1 . 9 ) , ( 1 . 1 0 ) , and (1.11) do n o t , i n g e n e r a l , p o s s e s s a n a l y t i c s o l u t i o n s . C o n s e q u e n t l y , t h e s e e q u a t i o n s a r e e v a l u a t e d by d i s c r e t e FFT. C o m p u t a t i o n a l d e t a i l s a r e g i v e n i n Appendix D. These e q u a t i o n s have been e v a l u a t e d f o r a v a r i e t y of e a r t h models. The response of a o n e - d i m e n s i o n a l e a r t h model i s of i n t e r e s t s i n c e i n t e r p r e t a t i o n r e l i e s on t h e u n d e r s t a n d i n g of the f o r w a r d problem. The problem t o be a d d r e s s e d next i s how t o c h a r a c t e r i z e the response of an e a r t h model. t o l i n e s ource e x c i t a t i o n . 19 1.3 RESPONSE FUNCTIONS FOR A STRAIGHT LINE OF CURRENT: A response f u n c t i o n i s a parameter chosen t o c h a r a c t e r i z e the response of an e a r t h model t o sour c e e x c i t a t i o n . Response f u n c t i o n s may be used t o c o n s t r u c t models t h a t f i t o b s e r v e d d a t a based on the c h a r a c t e r i z i n g response f u n c t i o n . T h i s p r o c e s s i s known as f o r w a r d m o d e l l i n g or model c o n s t r u c t i o n . E q u a t i o n s (1.9),(1.10) or (1.11) c o u l d be used as response f u n c t i o n s , however, t h e r e a r e d i s a d v a n t a g e s t o t h i s . E q u a t i o n s , ( 1 . 9 ) , ( 1 . 1 0 ) , and (1.11) have complex s o l u t i o n s A+iB, where A i s the inphase and B i s the o u t - o f - p h a s e or q u a d r a t u r e component. In o r d e r t o measure A and B i n ,tbe f i e l d , the phase of the p r i m a r y f i e l d must be m o n i t o r e d and compared t o the phase of the t o t a l f i e l d measured a t the r e c e i v e r . I n p r a c t i c e , the l e a s t c o s t l y and q u i c k e s t survey t h a t can be run i s a magnetic survey which measures the t o t a l magnetic f i e l d a m p l i t u d e s | ( x , z = 0 ) | and | H x T ( x , z = 0 ) | . An i n d u c t i o n c o i l or a f l u x g a t e magnetometer are a l l t h a t a r e r e q u i r e d t o p e r f o r m the s u r v e y . C o n s e q u e n t l y , a c o n v e n i e n t response parameter might be based on | H ^ ( x , z = 0 ) | and/or | H X T ( x , z = 0 ) | : A) |H^. T(x,z=0)| B) |H x T(x,z=6)| F i g u r e 1.6 shows l o g 1 0 |H^ T(x,z=0) | v e r s u s l o g . 1 0 ( x ) iot homogeneous h a l f spaces of d i f f e r i n g r e s i s t i v i t i e s (see i n s e t F i g u r e 1.6). F i g u r e 1.7 shows l o g 1 0 | H ^ (x,z=0)| v e r s u s l o g 1 0 ( x ) f o r a l a y e r over a h a l f space w i t h v a r y i n g r e s i s t i v i t y c o n t r a s t s P 2 / P 1 and s k i n depths i n the f i r s t l a y e r 5,. FIGURE 1.6. HALF SPACE RESPONSE FOR STRAIGHT LINE CURRENT T L o g 1 0 { | H z (x,z=0)|} v e r s u s l o g 1 0 ( x ) Response c u r v e s f o r a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y p due t o a l i n e s o u r c e e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . C u r r e n t a m p l i t u d e i s 100 Amperes. Frequency i s 60Hz. FIGURE 1.7. LAYER OVER A HALF SPACE RESPONSE FOR STRAIGHT LINE CURRENT Log, 0{ IH^1" (x,z=0) |} v e r s u s l o g 1 0 ( x ) Response c u r v e s f o r a l a y e r over a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y c o n t r a s t p 2 / P i and s k i n depths i n the f i r s t l a y e r 5, due t o a l i n e s ource e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . C u r r e n t a m p l i t u d e i s 100 Amperes. Frequency i s 60Hz. zz 24 Data a r e p r e s e n t e d i n t h i s manner t o c l e a r l y show t h a t -r |H-[ (x,z=0)| f a l l s o f f as 1/x near the source as p r e d i c t e d by Wait [ 1 5 ] , and as 1/x 3 f a r from the sou r c e as p r e d i c t e d by M c C o l l a r et a l . [ 2 ] . The t r a n s i t i o n from 1/x t o 1/x 3 depends on the r e s i s t i v i t y of the h a l f space ( i n the case of F i g u r e 1.6) or the r e s i s t i v i t y c o n t r a s t ( i n the case of F i g u r e 1.7) and the f r e q u e n c y of the s o u r c e . The f a r f i e l d b e h a v i o r may be e x p l a i n e d p h y s i c a l l y by c o n s i d e r i n g t h a t secondary f i e l d s a r i s e from an image source l o c a t e d a t a complex depth Q (Wait and S p i e s [ 1 6 ] ) . The t o t a l f i e l d t h e n , i s the sum of the p r i m a r y and secondary f i e l d s as f o l l o w s : For x/|Q|»1: | H j ( x , z = 0 ) | = I / ( 2 T T ) { 1 / X - X / ( X 2 + | Q 2 | ) } ^I/(2w){1/X-(1/x-|Q 2|/xf+|Q«|/x 5 } = I / ( 2 T T ) { | Q 2 | / X 3 + } T h i s e x p l i c i t l y shows the 1/x 3 n a t u r e of |H^T (x,z=0>| f o r s u f f i c i e n t l y l a r g e x. T F i g u r e s 1.8 and 1.9 show r e s p e c t i v e l y , l o g 1 0 | H x (x,z=0)| v e r s u s l o g 1 0 ( x ) f o r a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y , and f o r a l a y e r over a h a l f space w i t h v a r y i n g r e s i s t i v i t y c o n t r a s t s . By t h e same argument as f o r the H z c a s e , t h e f a r f i e l d c h a r a c t e r i s t i c s of H x can be e x p l a i n e d by c o n s i d e r i n g : | H x T ( x , z = 0 ) | S l / ( 2 7 r ) { h / x 2 + | Q | / ( x 2 + | Q 2 | ) } " I / ( 2 T T ) { 1 / X 2 } T h i s shows the 1/x 2 b e h a v i o r of | H x T ( x , z = 0 ) | f o r l a r g e x. FIGURE 1.8. HALF SPACE RESPONSE FOR STRAIGHT LINE CURRENT L o g 1 0 { | H x (x,z=0)|} v e r s u s l o g 1 0 ( x ) Response c u r v e s f o r a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y p due t o a l i n e s o u r c e e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . C u r r e n t a m p l i t u d e i s 100 Amperes. Frequency i s 60Hz. 26 O 4-LU O o ? ( y 3 1 3 W / S 3 y ' 3 d W t i ) 0 I D 0 " l ' f l ( X ) , X H l ) 0 I 0 0 n FIGURE 1.9. LAYER OVER A HALF SPACE RESPONSE FOR STRAIGHT LINE CURRENT L o g 1 0 { | H X T (x,z=0)|} v e r s u s l o g 1 0 ( x ) Response c u r v e s f o r a l a y e r over a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y c o n t r a s t Pi/P\ and s k i n depths i n the f i r s t l a y e r 6, due t o a l i n e s ource e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . C u r r e n t a m p l i t u d e i s 100 Amperes. Frequency i s 60Hz. 29 N o r m a l i z e d Response F u n c t i o n s : E q u a t i o n s , ( 1 . 9 ) , ( 1 . 1 0 ) , and (1.11) a l l depend on the c u r r e n t a m p l i t u d e I on the l i n e . T h e r e f o r e , knowledge of the v a l u e of I i s r e q u i r e d t o c o n s t r u c t models t h a t f i t o b s e r v e d H z, H x or Ey d a t a . In l i g h t of the s e c o n s i d e r a t i o n s , the f o l l o w i n g response f u n c t i o n s which a r e independent of I were chosen: C) |H 2 T (x,z=0) |/|H a l (x,z=0) | D) | H * T ( x , z = 0 ) | / | H x l ( x , z = 0 ) | e) | H x T ( x , z = 0 ) | / | H 2 T ( x , z = 0 ) | These n o r m a l i z e d response f u n c t i o n s w i l l now be d i s c u s s e d . F i g u r e 1.10 shows l o g 1 0 { | H 2 T ( x , z = 0 ) l / l H ^ (x,z=0)|} v e r s u s l o g 1 0 ( x ) f o r a homogeneous h a l f space and F i g u r e 1.11 shows the same f u n c t i o n f o r a l a y e r over a h a l f space w i t h v a r y i n g r e s i s t i v i t y c o n t r a s t s . F i g u r e 1.12 shows l o g 1 0 { |H x T(x,z=0) |/|H X U (x,z=0) |} v e r s u s l o g 1 0 ( x ) f o r a homogeneous h a l f space. F i g u r e 1.13 shows the same f u n c t i o n f o r a l a y e r over a h a l f space w i t h v a r y i n g r e s i s t i v i t y c o n t r a s t s . A g a i n , the f a r f i e l d b e h a v i o r of t h e s e graphs may be e x p l a i n e d i n the same manner as f o r the u n n o r m a l i z e d response f u n c t i o n s . F i g u r e 1.13 shows the response f u n c t i o n | H x T ( x , z = 0 ) ( x , z = 0 ) | as a f u n c t i o n of x f o r the case of a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y and F i g u r e 1.14 shows the same t h i n g f o r the case of a l a y e r over a h a l f space w i t h v a r y i n g r e s i s t i v i t y c o n t r a s t s . The advantage 30 FIGURE 1.10. NORMALIZED HALF SPACE RESPONSES L o g 1 0 { | H £ T ( x , z = 0 ) | / 1 H 2 L ( x , z = 0 ) | } v e r s u s l o g 1 0 ( x ) Response c u r v e s f o r a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y p due t o a l i n e s o u r c e e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . Frequency i s 60Hz. 31 S S 3 1 N 0 r S N 3 W I Q o CO DC LU X o L D O f ( X ) Z H / I ( X ) ± Z H ' n O I 0 0 1 FIGURE 1.11. NORMALIZED LAYER OVER A HALF SPACE RESPONSES Log, 0{ | H 2 T ( x , z = 0) |/|H 2 L (x,z=0) |} v e r s u s l o g , 1 0 ( x ) Response c u r v e s f o r a l a y e r over a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y c o n t r a s t s P 2 / P 1 and s k i n depths i n the f i r s t l a y e r 5, due t o a l i n e s ource e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . Frequency i s 60Hz. L O G 1 O i l H Z T ( X ) I / H Z ; D I M E N S I O N L E S S ^ ro CD m 1 o o m •+ i o # cn ro cn m 34 FIGURE 1.12. NORMALIZED HALF SPACE RESPONSES L o g 1 0 { | H x T ( x , z = 0 ) 1 / l H / (x,z=0)|} v e r s u s l o g ( x ) Response c u r v e s f o r a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y p due t o a l i n e s o u r c e e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . Frequency i s 60Hz. L O G 1 0 I H X T ( X ) / H X 1 ( X ) } D I M E N S I O N L E S S O I — k o X rn m CO FIGURE 1.13. NORMALIZED LAYER OVER A HALF SPACE RESPONSES L o g 1 0 { | H x T ( x , z = 0 ) | / | H X L ( x , 2 = 0 ) | } v e r s u s l o g 1 0 ( x ) Response c u r v e s f o r a l a y e r o ver a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y c o n t r a s t s P 2 / P 1 and s k i n depths i n the f i r s t l a y e r 5, due t o a l i n e s ource e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . Frequency i s 60Hz. 0 , 3 6 4 E t 0 I X I — I cn cn L U I o X CD z: L U O L D O 0 , 2 7 1 E t O O p 2 /p 1 =.5; 65094 m KI • 60HZ 30A hi 15m 1000m Pi 0 . 3 0 0 E t ( H 0 . 6 0 0 E L O G 1 0 ( X ( M E T E R S ! } 3 8 o f t h i s r e p o n s e f u n c t i o n i s t h a t i t i s l i n e a r w i t h i n a s h o r t d i s t a n c e o f t h e s o u r c e a n d c o n s e q u e n t l y f i t t i n g o b s e r v a t i o n s t o i t i s r e l a t i v e l y s i m p l e . T h i s r e s p o n s e f u n c t i o n i s t h e e a s i e s t t o c o m p u t e i n p r a c t i c e , b e c a u s e no a b s o l u t e c a l i b r a t i o n o f t h e m e a s u r i n g i n s t r u m e n t i s r e q u i r e d . M e a s u r e m e n t s o f | H x T ( x , z = 0 ) | a n d | H 2 T ( x , z = 0 ) | made w i t h t h e same i n s t r u m e n t a r e a l l t h a t a r e r e q u i r e d . I n s p e c t i o n o f A ) , B ) , C ^ , D ) , a n d E ) g i v e no d i r e c t i n f o r m a t i o n a b o u t t h e e a r t h r e s i s t i v i t y s t r u c t u r e . E s t i m a t e s o f l a y e r r e s i s t i v i t i e s a n d t h i c k n e s s e s r e l i e s on c o m p u t e r a n a l y s i s . T h e s e d r a w b a c k s m o t i v a t e d t h e u s e o f t h e a p p a r e n t r e s i s t i v i t y t r a n s f o r m a t i o n u s e d on m a g n e t o t e l l u r i c (MT) d a t a , t h a t i s ( C a g n i a r d [ 1 7 ] ) : F ) p a = l/Mo<4 | E y T ( x , z = 0 ) | / | H x T ( x , z = 0 ) | ] 2 F i g u r e 1 .16 shows a p p a r e n t r e s i s t i v i t y p f c a s a f u n c t i o n o f f r e q u e n c y . F i g u r e 1 . 1 6 ( a ) i s t h e r e s p o n s e f o r a h o m o g e n e o u s h a l f s p a c e w i t h p = l 0 0 o h m . m . A s e x p e c t e d t h e r e s p o n s e i s f r e q u e n c y i n d e p e n d e n t . F i g u r e 1 . 1 6 ( b ) shows t h e d i f f i c u l t i e s w h i c h a r i s e when c o n s i d e r i n g a l a y e r o v e r a h a l f s p a c e . T h e o v e r b u r d e n i s 100m t h i c k w i t h p ,=4000ohm.m, a n d t h e h a l f s p a c e r e s i s t i v i t y . p 2 = 4 0 0 0 0 o h m . m . B e c a u s e o f t h e b a n d l i m i t e d , h i g h f r e q u e n c y c o n t e n t ( r e l a t i v e t o MT) o f s i g n a l s b r o a d c a s t by p o w e r l i n e s ( 6 0 -660Hz m e a s u r e d by M c C o l l o r [ 2 ] ) t h i s m e t h o d d o e s n o t y i e l d u s e f u l i n f o r m a t i o n a b o u t l a y e r e d m e d i a b e c a u s e o f t h e l i m i t e d s k i n d e p t h s a f f o r d e d by t h e s i g n a l . T h i s a p p r o a c h may be u s e f u l i f s u f f i c i e n t l y low f r e q u e n c y s u b h a r m o n i c s c a n be m e a s u r e d t o i n c r e a s e t h e s i g n a l b a n d w i d t h a t t h e low e n d . 39 FIGURE 1.14. NORMALIZED HALF SPACE RESPONSES | H x T ( x , z = 0 ) l / l H j J (x,z=0)| v e r s u s x Response c u r v e s f o r a homogeneous h a l f space of v a r y i n g r e s i s t i v i t y p due t o a l i n e s o u r c e e l e v a t e d 15 meters above the e a r t h ' s s u r f a c e . Frequency i s 60Hz. 0 * 41 FIGURE 1.15 LAYER OVER A HALF SPACE NORMALIZED RESPONSES | H x T ( x , z = 0)|/|H^ T (x,z = 0 ) | v e r s u s x Response c u r v e s f o r a l a y e r over a homogeneous h a l f space w i t h v a r y i n g r e s i s t i v i t y . C u r r e n t h e i g h t i s 15m and f r e q u e n c y i s 60Hz. CO O . LxJ CD CO o +-UJ cn OJ =r ° SS31N0ISN3W r0 I (X) j _ Z H 1 / I (X) j _ X H o I UJ co FIGURE 1.16 APPARENT RESISTIVITY USING CAGNIARD'S RELATION 1.16(a) Apparent r e s i s t i v i t y as a f u n c t i o n of f r e q u e n c y f o r the case of a l i n e s o u r c e 20 meters above a homogeneous h a l f space Of P!=100 ohm.m. The s t a t i o n i s l o c a t e d 5000m from the s o u r c e . 1.16(b) Apparent r e s i s t i v i t y as a f u n c t i o n of f r e q u e n c y f o r the case of a l i n e s o u r c e 20 meters above a l a y e r over a h a l f space. The overbu r d e n r e s i s t i v i t y i s 4000ohm.m, 100m t h i c k , u n d e r l a i n by a h a l f space of r e s i s t i v i t y 40000ohm.m. 0.200E-03 CO CO tu I i 1 h-«- ^ Z O cc CC Cu Cu cc 0.0 (A) 0.500E+OZ O.USOC*03 FREQUENCY (HZ) 45 1.4 RESULTS FOR GOLD RIVER DATA On November 5 1983, a magnetic survey was run east of Go l d R i v e r on Vancouver I s l a n d u s i n g an e x i s t i n g powerline i n the area as a t r a n s m i t t e r . F i g u r e 1.17 shows the l o c a t i o n of the survey, s t a t i o n p o s i t i o n s , and t r a n s m i s s i o n l i n e . In t h i s survey, the x d i r e c t i o n corresponds to the p e r p e n d i c u l a r d i s t a n c e from the t r a n s m i s s i o n l i n e nearest the survey s t a t i o n s . T h i s d i r e c t i o n i s approximately geographic n o r t h . A simple i n d u c t i o n c o i l was used as a r e c e i v e r to make amplitude measurements of the t o t a l h o r i z o n t a l and v e r t i c a l f i e l d components at 60 and 420Hz. To make |H z T(x,z=0)| measurements, the d i p o l e moment of the c o i l r e c e i v e r was o r i e n t e d v e r t i c a l l y . To make h o r i z o n t a l magnetic f i e l d measurements, the r e c e i v e r moment was o r i e n t e d h o r i z o n t a l l y i n a d i r e c t i o n so as to o b t a i n a s i g n a l maximum. The d i r e c t i o n s of these maxima do not n e c e s s a r i l y correspond to the component i n the x d i r e c t i o n , so that the p r o j e c t i o n s of these maxima i n the x d i r e c t i o n were c a l c u l a t e d . The response f u n c t i o n employed t o i n t e r p r e t data c o l l e c t e d was the r a t i o |H X T(x,z=0)|/|H Z T (x,z=0)|. T h i s response f u n c t i o n was chosen because of i t s advantages as o u t l i n e d i n s e c t i o n 1.3. An e a r t h model was generated and i n t e r a c t i v e l y a l t e r e d to produce a reasonable f i t to observed data. G e o l o g i c a l l y , t h i s p a r t of Vancouver I s l a n d i s known to be FIGURE 1.17 LOCATION MAP FOR GOLD RIVER SURVEY L o c a t i o n map showing survey area, t r a n s m i s s i o n l i n e and s t a t i o n l o c a t i o n s . The t r a n s m i s s i o n l i n e i s o u t l i n e d i n ye l l o w . The transformer s t a t i o n i s shown as a green square and the survey l i n e i s shown i n red. Map s c a l e i s approximately 1:500,000. Contour i n t e r v a l i s 100 f e e t . 47 48 c o m p o s e d o f J u r a s s i c age i n t r u s i o n s ( I s l a n d I n t r u s i o n s ) a n d T r i a s s i c a g e v o l c a n i c s ( K a r m u t s e n F o r m a t i o n ) . The I s l a n d I n t r u s i o n s a r e b a t h o l i t h s o f g r a n i t o i d r o c k s ( 3 0 0 < p ( o h m . m ) < l 0 0 0 0 0 ; T e l f o r d [ 1 8 ] ) a n d a r e t h o u g h t t o u n d e r l i e ' a b o u t 1/4 o f t h e i s l a n d ' s s u r f a c e . The K a r m u t s e n v o l c a n i c s (1000<p(ohm.m)<100000 ; T e l f o r d [ 1 8 ] ) a r e a l s o t h o u g h t t o u n d e r l i e a l a r g e p a r t o f t h e i s l a n d ( g e o l o g y f r o m M u l l e r , [ 1 9 ] ) . F i g u r e s 1 .18 a n d 1 .19 show f i e l d d a t a r e s u l t s s u p e r i m p o s e d on m o d e l l e d r e s u l t s f o r 60 a n d 4 2 0 H z . An e a r t h m o d e l . w h i c h s i m u l t a n e o u s l y f i t s t h e 60 a n d 420Hz d a t a i s shown i n t h e i n s e t s t o t h e s e f i g u r e s . The e a r t h m o d e l p r e d i c t s a r e s i s t i v e u p p e r l a y e r o f r e s i s t i v i t y p i=11OOOohm.m w h i c h i s 1400m t h i c k u n d e r l a i n by a c o n d u c t i v e h a l f s p a c e w i t h p 2 = 1 2 0 0 o h m . m . F i g u r e 1 .19 shows d e v i a t i o n s f r o m t h e s t r a i g h t l i n e t r e n d p r e d i c t e d by t h e f o r w a r d m o d e l l e d r e s u l t s . T h i s may be due t o a l a t e r a l i n h o m o g e n e i t y b e t w e e n s t a t i o n s x=2000m a n d x=3200m. S i m i l a r d e v i a t i o n s d o n o t a p p e a r i n t h e 60Hz d a t a s u g g e s t i n g t h a t t h e i n h o m o g e n e i t y may be s h a l l o w . T a n k m o d e l r e s u l t s ( s e c t i o n 3 . 1 ) i n d i c a t e t h a t t h e r a t i o | H x T ( x , z = 0 ) ( x , z = 0 ) | i n c r e a s e s s i g n i f i c a n t l y a t t h e e d g e s o f a c o n d u c t o r b u r i e d i n a r e s i s t i v e m e d i u m , a n d d e c r e a s e s o v e r t h e c e n t e r o f t h e a n o m a l y . B e c a u s e t h e r a t i o | H x T ( x , z = 0 ) | / | H £ T ( x , z = 0 ) | i n F i g u r e 1 .19 d e c r e a s e s b e t w e e n x=2000 a n d x=3200m, t h i s m i g h t i n d i c a t e a b r o a d c o n d u c t i v e z o n e b e t w e e n t h e s e s t a t i o n s . T h i s b e h a v i o r c a n be s e e n by d r a w i n g a s m o o t h l i n e t h r o u g h t h e d a t a . R e s p o n s e c u r v e s w i l l be a f f e c t e d by l a t e r a l , l o c a l i z e d F I G U R E 1 . 1 8 M O D E L R E S U L T S F O R 6 0 H z O b s e r v e d d a t a a p p e a r s a s a n a s t e r i s k a n d t h e t h e o r e t i c a l p r e d i c t i o n a p p e a r s a s a s o l i d l i n e . L i n e h e i g h t w a s m o d e l l e d a s 15 m e t e r s . 0 . 1 0 i E + 0 1 CO CO LU K i ^ nr. o \ *-* r—- co X z JT. ^ 27. a 0 . 9 3 0 E " 0 1 J 4 4 1 J 1 4 i \ V 4-1 60 H Z h=15m 140 3m p=l.lxio 4cim I p=1.2Xl03Om * data — theory 4 1- 4 4 4 ^-4 1-0 . 6 0 0 E + 0 3 X ( M E T E R S ) 0 . 5 0 0 E + O 4 cn o FIGURE 1.19 MODEL RESULTS FOR 420Hz Observed data appears as an a s t e r i s k and the t h e o r e t i c a l p r e d i c t i o n appears as a s o l i d l i n e . L i n e height was modelled as 15 meters. X ( M E T E R S ) 53 i n h o m o g e n e i t i e s c a u s i n g a s c a t t e r i n g of o b s e r v e d d a t a . C o n s e q u e n t l y , i n f o r m a t i o n about h o r i z o n t a l l a y e r i n g w i l l be p e r t u r b e d . Q u a n t i t a t i v e i n t e r p r e t a t i o n may be s u b j e c t t o u n c e r t a i n t y and may be i m p o s s i b l e i n some c a s e s . I t may be r e a s o n a b l e t o t r y t o overcome i n t e r p r e t a t i o n problems by smoothing f i e l d d a ta but how t h i s a f f e c t s the i n f o r m a t i o n c o n t e n t c o n c e r n i n g l a y e r s i s unknown. In t h i s c a s e the s c a t t e r i n g was smoothed by f i t t i n g a s t r a i g h t l i n e which m i n i m i z e d the square of the d i f f e r e n c e between o b s e r v a t i o n s and t h e o r e t i c a l r e s u l t s f o r b o t h the 60 and 420Hz d a t a s i m u l t a n e o u s l y . Assuming l a y e r e d e a r t h models i n g e o l o g i c a l l y complex a r e a s i s h a z a r d o u s . I t makes sense t o t r y and produce l a y e r e d e a r t h models from a r e a s where h o r i z o n t a l l a y e r i n g a s sumptions a r e l e s s l i k e l y t o be v i o l a t e d . The model g e n e r a t e d i s f e l t , a t l e a s t , t o i n d i c a t e the r e d u c t i o n of r e s i s t i v i t y a t d e p t h . Reduced r e s i s t i v i t y a t depth i s thought t o be t y p i c a l of t e c t o n i c e n v i r o n m e n t s . I t s h o u l d be noted t h a t t h e model p r e s e n t e d r e p r e s e n t s o n l y one p o s s i b l e model t h a t f i t s t h e d a t a t o a l e a s t squares c r i t e r i o n . Other models, f o r example, a t h r e e l a y e r model, may f i t t he d a t a e q u a l l y w e l l . A q u i s i t i o n of m u l t i f r e q u e n c y d a t a w i l l h e l p c o n s t r a i n the number of o n e - d i m e n s i o n a l models which w i l l f i t the d a t a . 54 CHAPTER 2 : EM INDUCTION IN A L A Y E R E D EARTH BY A BENT L I N E OF CURRENT T h e C a r t e s i a n f o r m u l a t i o n ( e q u a t i o n 1 .2) i s i d e a l l y s u i t e d t o t h e c a s e o f a b e n t l i n e o f c u r r e n t b e c a u s e o f t h e c o m p l e x g e o m e t r y a s s u m e d by t h e s o u r c e . H o w e v e r , t h i s f o r m u l a t i o n i s c o m p u t a t i o n a l l y e x p e n s i v e when l a r g e two d i m e n s i o n a l m a t r i c e s a r e e m p l o y e d . T o c o n s t r u c t e a r t h m o d e l s w h i c h f i t o b s e r v a t i o n s , t h e f o r w a r d p r o b l e m must be s o l v e d many t i m e s . T h e r e f o r e , c o m p u t a t i o n a l e x p e n s e i s o f p r i m a r y c o n s i d e r a t i o n . T h e C a r t e s i a n f o r m u l a t i o n i s e x p e n s i v e f o r t h e c a s e o f a b e n t l i n e o f c u r r e n t f o r a number o f r e a s o n s . F i r s t , t h e i n c i d e n t f i e l d d o e s n o t h a v e an a n a l y t i c t r a n s f o r m on a n y p l a n e z = c o n s t a n t b e c a u s e o f t h e c o m p l e x g e o m e t r y a s s u m e d by t h e s o u r c e . T h e r e f o r e , t h e i n c i d e n t f i e l d must be t r a n s f o r m e d n u m e r i c a l l y . F o r l a r g e m a t r i c e s t h i s i s e x p e n s i v e . S e c o n d , t h e i n v e r s i o n o f l a r g e two d i m e n s i o n a l m a t r i c e s w i l l be n e c e s s a r y t o o b t a i n t h e s o l u t i o n i n t h e x - y d o m a i n . S o l u t i o n s f o r a s t r a i g h t l i n e o f c u r r e n t r e q u i r e 4096 p o i n t t r a n s f o r m s ( A p p e n d i x D ) , t h e r e f o r e , one c a n e x p e c t a t l e a s t t h i s number o f p o i n t s f o r one o f t h e m a t r i c e ' s d i m e n s i o n s . B e c a u s e o f t h e c o m p u t a t i o n a l e x p e n s e s e e n w i t h t h i s m e t h o d an a p p r o x i m a t e s o l u t i o n t o t h e EM i n d u c t i o n p r o b l e m f o r a l a y e r e d e a r t h was a d o p t e d . 55 2.1 A p p r o x i m a t e S o l u t i o n s U s i n g Image T h e o r y : I t h a s b e e n known f o r some y e a r s t h a t t h e s e c o n d a r y f i e l d s p r o d u c e d by a t i m e h a r m o n i c s o u r c e may be a p p r o x i m a t e d by a v e r s i o n o f image t h e o r y ( B a n n i s t e r [ 2 0 ] , W a i t a n d S p i e s [ 1 6 ] ; Thomson a n d Weaver [ 2 1 ] ) . T h e e s s e n c e o f t h e image t h e o r y t e c h n i q u e i s t o a p p r o x i m a t e e q u a t i o n 1.2 by a s e r i e s e x p a n s i o n a s s u m i n g q u a s i - s t a t i c c o n d i t i o n s . T h i s e x p a n s i o n r e v e a l s t h a t s e c o n d a r y f i e l d s may be w e l l a p p r o x i m a t e d by l o c a t i n g an image o f t h e s o u r c e a t a c o m p l e x d e p t h Q : Q = h + F ( 1 - i ) where h i s t h e e l e v a t i o n o f t h e l i n e s o u r c e , F i s a c o r r e c t i o n f a c t o r w h i c h a c c o u n t s f o r t h e p r e s e n c e o f l a y e r s i n t h e e a r t h m o d e l a n d i a c c o u n t s f o r q u a d r a t u r e o r 90 d e g r e e p h a s e s h i f t s o f t h e s e c o n d a r y f i e l d r e l a t i v e t o t h e p r i m a r y . T h e r e a s o n s f o r a d o p t i n g t h i s p a r t i c u l a r f o r m u l a t i o n a r e t w o f o l d . F i r s t , i t p r o v i d e s a s i m p l e p h y s i c a l p i c t u r e a n d s o l u t i o n w i t h o u t t h e e x p e n s e o f s o l v i n g the* more i n v o l v e d p a r t i a l d i f f e r e n t i a l e q u a t i o n s s u b j e c t t o t h e u s u a l b o u n d a r y c o n d i t i o n s a n d , s e c o n d , i t p r o v i d e s a c h e c k on t h e a c c u r a c y o f e x a c t s o l u t i o n s w h i c h a r e more c o m p u t a t i o n a l l y e x p e n s i v e . Image t h e o r y a p p r o x i m a t i o n s a r e o b s e r v e d t o be s t r i c t l y v a l i d f o r f i e l d p o i n t s o b s e r v e d a t d i s t a n c e s on t h e o r d e r o f a s k i n d e p t h f r o m t h e s o u r c e ( B a n n i s t e r [ 2 0 ] ) . F i g u r e 2.1 shows s e c o n d a r y i n p h a s e a n d q u a d r a t u r e H- a n d H f i e l d s c o m p u t e d f o r t h e c a s e o f FIGURE 2.1. IMAGE THEORY RESULTS VERSES EXACT RESULTS Image t h e o r y r e s u l t s f o r a h a l f space of 10000 ohm.m a r e shown as a s t e r i s k s . E x a c t s o l u t i o n r e s u l t s appear as a s o l i d l i n e . L i n e h e i g h t i s 15m, fr e q u e n c y i s 60Hz, and c u r r e n t a m p l i t u d e i s 100 Amps. (A) Shows the inphase and q u a d r a t u r e components of secondary H x. (B) Shows the inphase and q u a d r a t u r e components of secondary H 2. • o . j n t - 0 7 . X c £ u w u i x CC w X e „ w 2 * - O . i m - O V * J 1 • 4 1 • • • 1*0 X M H E T E R S ) l.!OZC*fll X ( M E T E R S ) I.IOZC*OI O . t l l C - I S WJ X X. K UJ u CC CC S OC E CC x 3 « a X ( M E T E R S ) 0.10ZE»t( O.ZflC-OV M X K U l W S i £ s cc e CC x - Q . S 1 X C - 0 I X ( M E T E R S ) o.toze*06 58 an i n f i n i t e s t r a i g h t l i n e o f c u r r e n t o v e r a ha- l f s p a c e o f r e s i s t i v i t y p= 10OOOohm.m a t 6 0 H z . T h e s k i n d e p t h 5=]/{2/{uu0o)} i n t h i s c a s e i s a b o u t 6 . 5 k m . T h e f i g u r e c o m p a r e s e x a c t s o l u t i o n r e s u l t s w i t h image t h e o r y r e s u l t s . T h e g o o d a g r e e m e n t b e t w e e n image t h e o r y a n d e x a c t s o l u t i o n r e s u l t s , a l o n g w i t h t h e c o m p u t a t i o n a l s a v i n g s p r o m p t e d an e x a m i n a t i o n o f image t h e o r y r e s u l t s f o r t h e c a s e o f a b e n t l i n e s o u r c e o f c u r r e n t o v e r a l a y e r e d e a r t h . 59 2.2 Image S o l u t i o n s f o r T o t a l F i e l d s about a Bent L i n e Source: The f i e l d s about a bent l i n e source may 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 f i e l d s due to f i n i t e l e n g t h s t r a i g h t wire elements i n the q u a s i - s t a t i c c a se. The e x p r e s s i o n f o r the i n c i d e n t f i e l d about a f i n i t e l e n g t h wire element i s found i n most t e x t s : |H L(x,y,z=0) |=I ( s i n B j - s i n B , ) / ( 4 7 r ( d 2 + h 2 ) Y P(x,y,z»0) FIGURE 2 .2 GEOMETRY FOR A FINITE LENGTH WIRE Plan view showing the angles B 0,B,, and B 2. The wire i s between p o i n t s A and B and i s e l e v a t e d a height h above z=0. The p e r p e n d i c u l a r d i s t a n c e i n the x-y plane between P(x,y,z=0) and the wire element i s d. For a s i n g l e wire element then (see F i g u r e 2 . 2 ) : H i L(x,y,z= 0 ) = I ( s i n B 2 - s i n B , )d / ( 4 7 r(d 2+h 2)) H x l(x,y,z=0)=I ( s i n B 2 - s i n B , ) c o s ( B 0 )h/(47r(d 2+h 2 )) The t o t a l f i e l d s (sum of the primary and secondary) a r e : H 1 T(x,y,z=0)=I ( s i n B 2 - s i n B , ) d / ( 4 7 r ) { l / ( d 2 + h 2 ) - 1 / ( d 2 + Q 2 ) } H» T(x,y,z=0) = I ( s i n B 2 - s i n B , ) c o s ( B 0 ) / ( 4 i r ) {h/(d 2+h 2 )+Q/(d 2+Q 2)} These simple e x p r e s s i o n s were used to compute the response 60 f u n c t i o n |HX (x,y,z=0)|/|H 2 (x,y,z=0)| f o r a s u p e r p o s i t i o n of f i n i t e wire elements. 61 2 . 3 G o l d R i v e r S u r v e y R e s u l t s T h i s s e c t i o n p r e s e n t s f o r w a r d m o d e l r e s u l t s f o r t h e G o l d R i v e r S u r v e y d a t a . The r e s p o n s e f u n c t i o n e m p l o y e d t o i n t e r p r e t d a t a was t h e r a t i o | H x T ( x , y , z = 0 ) | / I H ^ ( x , y , z = 0 ) | b e c a u s e o f t h e a d v a n t a g e s a f f o r d e d by t h i s r e s p o n s e f u n c t i o n a s o u t l i n e d i n S e c t i o n 1 . 3 . T h e s p a t i a l d i s t r i b u t i o n o f f i n i t e l e n g t h w i r e s e m p l o y e d . a p p e a r s i n F i g u r e 2 . 3 . T h i s c h o i c e o f g e o m e t r y was made t o a p p r o x i m a t e t h e g e o m e t r y o f t h e t r a n s m i s s i o n l i n e u s e d f o r t h e G o l d R i v e r 1983 s u r v e y . T F IGURE 2 . 3 S P A T I A L D I S T R I B U T I O N OF F I N I T E WIRES T h e s p a t i a l d i s t r i b u t i o n o f f i n i t e l e n g t h w i r e s e m p l o y e d t o a p p r o x i m a t e t h e t r a n s m i s s i o n l i n e u s e d f o r t h e G o l d R i v e r s u r v e y . T h e two s e c t i o n s o f l i n e (1 ) a n d (2) a r e j o i n e d by a t r a n s f o r m e r s t a t i o n . T h e s u r v e y l i n e i s shown by a d a s h e d l i n e . I t s h o u l d be n o t e d t h a t t h e a m p l i t u d e s a n d p h a s e s o f t h e c u r r e n t s on t h e l i n e s shown i n F i g u r e 2 . 3 a r e u n k n o w n . I t was t h e r e f o r e a s s u m e d t h a t t h e c u r r e n t s were e q u i a m p l i t u d e a n d e q u i p h a s e . T h i s may n o t be t h e c a s e b e c a u s e of, t h e p r e s e n c e o f t h e t r a n s f o r m e r s t a t i o n a t G o l d R i v e r ( s e e F i g u r e 1 .17 a n d 2 . 3 ) . 62 F i g u r e s 2 . 4 a n d 2 . 5 show f i e l d d a t a r e s u l t s s u p e r i m p o s e d on m o d e l l e d r e s u l t s f o r 60 a n d 420Hz u s i n g o n l y s e c t i o n (1) o f t h e l i n e . One s e c t i o n o f l i n e was u s e d t o b e g i n m o d e l c o n s t r u c t i o n b e c a u s e o f t h e g r e a t d i f f i c u l t y e n c o u n t e r e d i n c o n s t r u c t i n g m o d e l s u s i n g b o t h s e c t i o n s (1) a n d ( 2 ) . M o d e l r e s u l t s u s i n g b o t h s e c t i o n s w i l l be d e s c r i b e d s h o r t l y . An e a r t h m o d e l w h i c h s i m u l t a n e o u s l y f i t s b o t h 60 a n d 420Hz d a t a f o r t h e c a s e o f one s e c t i o n o f l i n e i s shown i n t h e i n s e t s t o t h e s e f i g u r e s . T h e e a r t h m o d e l p r e d i c t s a r e s i s t i v e u p p e r l a y e r o f r e s i s t i v i t y p , = l 0 0 0 0 o h m . m ,1000m t h i c k , u n d e r l a i n by a c o n d u c t i v e h a l f s p a c e o f r e s i s t i v i t y p 2 = l 1 1 O o h m . m . T h i s a g r e e s w e l l w i t h t h e e a r t h m o d e l p r e d i c t e d i n S e c t i o n 1 . 4 . When b o t h s e c t i o n s (1) a n d (2) o f t h e t r a n s m i s s i o n l i n e a r e c o n s i d e r e d , p r o b l e m s a r i s e i n m a t c h i n g o b s e r v e d r e s u l t s w i t h p r e d i c t e d r e s u l t s . B o t h F i g u r e s 2 . 6 a n d 2 . 7 show a p o o r f i t t o t h e m o d e l l e d r e s u l t s . T h i s may be d u e t o p r o b l e m s i n a s s u m i n g e q u i p h a s e a n d e q u i a m p l i t u d e c u r r e n t s on t h e two s e c t i o n s o f l i n e . F u t u r e m e a s u r e m e n t s s h o u l d i n c l u d e p h a s e a n d a m p l i t u d e m e a s u r e m e n t s o f t h e c u r r e n t s on t r a n s m i s s i o n l i n e s e c t i o n s when a t r a n s f o r m e r s t a t i o n i s known t o be p r e s e n t . T h e e a r t h m o d e l shown i n t h e i n s e t s o f F i g u r e s 2 . 6 a n d 2 . 7 show r e s i s t i v e o v e r b u r d e n p ,=12000ohm.m w h i c h i s 1300m t h i c k u n d e r l a i n by a c o n d u c t i v e h a l f s p a c e o f r e s i s t i v i t y p 2 =1OOOohm.m. F o r F i g u r e s 2 . 4 , 2 . 5 , 2 . 6 a n d 2 . 7 , l a c k o f f i t w i l l a r i s e b e c a u s e t h e a p p r o x i m a t e s o l u t i o n e m p l o y e d h e r e i s s t r i c t l y v a l i d f o r f i e l d FIGURE 2.4 MODEL RESULTS FOR 60Hz DATA Model r e s u l t s showing f i e l d data ( a s t e r i s k s ) superimposed on modelled r e s u l t s ( s o l i d l i n e ) f o r 60Hz. T h i s model uses only s e c t i o n ( l ) of the l i n e shown i n F i g u r e 2.3. 64 FIGURE 2.5 MODEL RESULTS FOR 420Hz DATA Model r e s u l t s showing f i e l d data ( a s t e r i s k s ) superimposed on modelled r e s u l t s ( s o l i d l i n e ) f o r 420Hz. T h i s model uses only s e c t i o n (1) of the l i n e shown i n F i g u r e 2.3. 66 O SS31N0 J 9N3WI 0 = Z ' X) Z H I / I ( 0 = Z - X) XH 67 p o i n t s on the o r d e r of a s k i n d e pth away from the s o u r c e . Data shown here i s w i t h i n the range of a s k i n d e p t h . The p r e l i m i n a r y i n t e r p r e t a t i o n s made i n t h i s s e c t i o n p r o v i d e a d d i t i o n a l e v i d e n c e which s u p p o r t s the i d e a of reduced r e s i s t i v i t y a t depth i n t h i s a r e a . FIGURE 2.6 MODEL RESULTS FOR SECTIONS 1 AND 2 AT 60Hz Model r e s u l t s showing f i e l d d a t a ( a s t e r i s k s ) superimposed on m o d e l l e d r e s u l t s ( s o l i d l i n e ) f o r 60Hz. T h i s model uses s e c t i o n s (1) and (2) of the l i n e shown i n F i g u r e 2.3. Poor agreement between o b s e r v e d d a t a and t h e o r e t i c a l d a t a may be due t o problems i n assuming e q u i a m p l i t u d e and equiphase c u r r e n t s on th e s e two s e c t i o n s of l i n e . o + r— o o i L U o CO CD CO CC LU h— UJ X = Z ' X ) S S 3 1 N 0 I S N 3 W I Q Z H 1/ I ( 0 = Z - X ) X H FIGURE 2.7 MODEL RESULTS FOR SECTIONS 1 AND 2 AT 420Hz Model r e s u l t s showing f i e l d d a t a ( a s t e r i s k s ) superimposed on m o d e l l e d r e s u l t s ( s o l i d l i n e ) f o r 420Hz. T h i s model uses s e c t i o n s (1) and (2) of the l i n e shown i n F i g u r e 2.3. Poor agreement between o b s e r v e d d a t a and t h e o r e t i c a l d a t a may be due t o problems i n assuming e q u i a m p l i t u d e and equiphase c u r r e n t s on t h e s e two s e c t i o n s of l i n e . !HX ( X f Z = 0 ) | / | H Z ( X , Z = D I M E N S I O N L E S S CO cn m + o 4= U 72 2.4 IMAGE SOLUTIONS FOR A STRAIGHT LINE OF CURRENT: RULE OF THUMB FOR INTERPRETATION Image t h e o r y r e s u l t s f o r a s t r a i g h t l i n e of c u r r e n t over a homogeneous h a l f space have been used t o d e v e l o p a r u l e of thumb f o r i n t e r p r e t a t i o n . E s s e n t i a l l y , the s l o p e of |H^(x,z=0) |/|H 2 J(x,z=0) | v e r s u s x p l o t s a r e used t o i n v e r t f o r an apparent r e s i s t i v i t y . A development of t h i s r u l e f o l l o w s . The t o t a l magnetic f i e l d s about a s t r a i g h t l i n e of c u r r e n t f o r a homogeneous e a r t h a r e : H x T(x,z=0)=-l{h/(x 2+h 2)+Q/(x 2+Q 2)}/27r H 2 T ( x , z = 0 ) = l{x/(x 2+h 2)-x/(x 2+Q/ 2)}/27r where Q = 6 ( l - i ) + h where 6=half space s k i n d e pth So t h a t : |H x T(x,z=0) |/|H 3 L T(x,z=0) |=x/( |Q|-h)+h( |Q 2|+h)/{x( |Q|+h)(|Q|-hH (2.1) For l a r g e v a l u e s of x, e q u a t i o n 2.1 becomes: | H x T ( x , z = 0 ) | / | H a ( x , z = 0)|=x/(| Q|-h) The s l o p e (S) of the l i n e i s then S-1/(|Q|-h) So (l/S+h) 2=25 2+2h5+h 2 T h i s i s a q u a d r a t i c i n 5 and has s o l u t i o n : 5= - h V { 2 ( l/S 2+2h/S)+h 2} T h i s f o r m u l a can be used t o g i v e an e s t i m a t e f o r the h a l f space r e s i s t i v i t y a t a s i n g l e f r e q u e n c y : pCi=un062/2 73 T h i s e q u a t i o n w i l l be u s e f u l f o r on s i t e i n t e r p r e t a t i o n and w i l l a l l o w r a p i d c a l c u l a t i o n of an a p p a r e n t ha-lf space r e s i s t i v i t y a t a g i v e n f r e q u e n c y . 74 CHAPTER 3: EM INDUCTION IN A HOMOGENEOUS EARTH WITH A L A T E R A L INHOMOGENEITY 3.1 TANK MODEL R E S U L T S T h e p r o b l e m o f l a t e r a l e l e c t r i c a l i n h o m o g e n e i t i e s i s one o f c o n s i d e r a b l e i n t e r e s t i n g e o p h y s i c s . T h e f e a t u r e o f i n t e r e s t i s t h e b e h a v i o r o f EM f i e l d c o m p o n e n t s i n t h e n e i g h b o u r h o o d o f a b o d y embedded i n t h e e a r t h . One m e t h o d o f s t u d y i n g t h i s p r o b l e m , i s by t h e u s e o f a s c a l e d a n a l o g u e m o d e l m a k i n g u s e o f t h e p r i n c i p l e o f s i m i l i t u d e ( S t r a t t o n [ 2 2 ] ) . T h e u s e f u l n e s s o f a n a l o g u e m o d e l s i s d e m o n s t r a t e d f o r p r o b l e m s i n w h i c h n u m e r i c a l s o l u t i o n s d o n o t e x i s t o r a r e c o m p u t a t i o n a l l y e x p e n s i v e . An e l e c t r i c a l i n h o m o g e n e i t y i n t h e v i c i n i t y o f a p o w e r l i n e i s one s u c h p r o b l e m . T h e m a t h e m a t i c a l d e v e l o p m e n t o f t h e m o d e l p r o b l e m i s d e s c r i b e d i n many t e x t s ( S t r a t t o n [ 2 2 ] ; K e l l e r a n d F r i s c h k n e c h t [ 5 ] ) a n d w i l l n o t be g i v e n h e r e . T h e m o d e l l i n g e q u a t i o n i s : E q u a t i o n (3 .1) g i v e s t h e r e l a t i o n s h i p b e t w e e n c o n d u c t i v i t i e s ( o ) f r e q u e n c i e s ( f ) a n d l e n g t h s ( l ) f o r t h e g e o p h y s i c a l ( s u b s c r i p t g.) a n d t h e m o d e l ( s u b s c r i p t m) s i t u a t i o n s s o t h a t t h e r a t i o s o f t h e f i e l d c o m p o n e n t s a n d p h a s e d i f f e r e n c e s a r e t h e same f o r b o t h p r o b l e m s . T h i s s e c t i o n d e s c r i b e s r e s u l t s o b t a i n e d w i t h a n a n a l o g u e m o d e l . A d e t a i l e d d e s c r i p t i o n o f t h e m o d e l l i n g s y s t e m a n d F I G U R E 3 . 1 . M O D E L A R R A N G E M E N T A N D C O - O R D I N A T E S Y S T E M T h e g e o p h y s i c a l p r o b l e m m o d e l l e d i s t h a t o f a s t r a i g h t l i n e s o u r c e i n t h e v i c i n i t y o f a c o n d u c t i v e b u r i e d b l o c k i n r e s i s t i v e h o s t r o c k . S Y S T E M G E O M E T R Y 35cm 77 i n s t r u m e n t a t i o n may be f o u n d i n D o s s o [ 2 3 ] . T h e c o - o r d i n a t e s y s t e m a n d m o d e l a r r a n g e m e n t e m p l o y e d i s shown i n F i g u r e 3 . 1 . The g e o p h y s i c a l p r o b l e m m o d e l l e d i s t h a t o f a c y l i n d r i c a l wave f i e l d i n c i d e n t on a r e c t a n g u l a r i n h o m o g e n e i t y b u r i e d w i t h i n a h o m o g e n e o u s e a r t h . T h e s o u r c e 13. e f f e c t i v e l y e x t e n d s i n f i n i t e l y i n t h e y d i r e c t i o n . T h e m o d e l l i n g t a n k was n o t i n f i n i t e i n e x t e n t b u t t h e m o d e l d i m e n s i o n s a r e s u f f i c i e n t l y s m a l l a s t o m i n i m i z e e d g e e f f e c t s due t o t h e t a n k w a l l s . I n p h a s e a n d q u a d r a t u r e EM f i e l d c o m p o n e n t s f o r a r e c t a n g u l a r s t r u c t u r e i m m e r s e d i n a c o n d u c t i n g e l e c t r o l y t i c t a n k were o b t a i n e d . T h e m o d e l d i m e n s i o n s a n d c o r r e s p o n d i n g , g e o p h y s i c a l d i m e n s i o n s a r e g i v e n i n T a b l e s I a n d I I . T h e s c a l i n g f a c t o r s i n v o l v e d s a t i s f y ( 3 . 1 ) . In t h e t a b l e s , 1 , , 1 2 , 1 3 , o , , o 2 , f , d , a n d h a r e r e s p e c t i v e l y : t h e l e n g t h o f t h e b l o c k i n t h e x d i r e c t i o n , b l o c k l e n g t h i n t h e y d i r e c t i o n , b l o c k l e n g t h i n t h e z d i r e c t i o n , t h e c o n d u c t i v i t y o f t h e s a l t s o l u t i o n i n t h e t a n k , t h e c o n d u c t i v i t y o f t h e g r a p h i t e b l o c k , t h e f r e q u e n c y , t h e d e p t h o f s a l t s o l u t i o n o v e r t h e b l o c k , a n d t h e e l e v a t i o n o f t h e l i n e s o u r c e o v e r t h e t a n k ' s f l u i d s u r f a c e . T a b l e I M o d e l D i m e n s i o n s f ( H z ) l , ( m ) l 2 ( m ) l 3 ( m ) a , ( S / m ) a 2 ( S / m ) d(m) h(m) 3 7 3 3 . 7 . 3 5 . 3 0 5 . 0 3 5 2 1 . 120000 . .001 . 0 0 3 2 T a b l e I I G e o p h y s i c a l D i m e n s i o n s f ( H z ) l , ( m ) l 2 ( m ) l 3 ( m ) a , ( S / m ) 6 0 . 2 1 0 0 . 1830 . 2 1 0 . . 0 0 0 0 3 6 3 o 2 ( S / m ) . 2 0 d(m) 6 . h(m) 1 9 . 2 7 8 T h e m o d e l r e s u l t s w i l l now be d i s c u s s e d i n d e t a i l . F i g u r e s 3 . 2 , 3 . 3 , a n d 3 . 4 show r e s p e c t i v e l y t h e b e h a v i o r o f H a , H x , a n d Ey a s a f u n c t i o n o f p o s i t i o n x . T h e g e n e r a l b e h a v i o r o f H 2 , H x , a n d Ey c a n be e x p l a i n e d by r e p l a c i n g t h e c o n d u c t i v e b l o c k by a c u r r e n t d i s t r i b u t i o n w h i c h a p p r o x i m a t e s t h e t r u e e d d y c u r r e n t d i s t r i b u t i o n w i t h i n t h e b l o c k . T h e f i e l d s p r o d u c e d by t h e s e c u r r e n t s a r e t h e s e c o n d a r y f i e l d s due t o t h e b l o c k a l o n e . In t h i s c a s e t h e d i s t r i b u t i o n o f e d d y c u r r e n t s i n t h e b l o c k c a n be a p p r o x i m a t e d by a l o o p s o u r c e o f c u r r e n t . T h i s i s i n t u i t i v e l y r e a s o n a b l e s i n c e c u r r e n t t e n d s t o c o n c e n t r a t e a l o n g e d g e s , a n d t h e e d g e s o f t h e b l o c k f o r m a r e c t a n g u l a r l o o p . T h e s u p e r p o s i t i o n o f t h e f i e l d s due t o a r e c t a n g u l a r l o o p o f c u r r e n t a n d t h e f i e l d s a b o u t a l i n e s o u r c e o f c u r r e n t a p p r o x i m a t e t h e b e h a v i o r o f H 2 , H x , a n d Ey shown i n t h e f i g u r e s . The i n p h a s e c o m p o n e n t i n a l l t h r e e c a s e s i s l a r g e r t h a n t h e q u a d r a t u r e s i n c e i t c o n t a i n s t h e p r i m a r y f i e l d . B o t h H x a n d show r a p i d v a r i a t i o n i n t h e v i c i n i t y o f t h e e d g e s o f t h e a n o m a l y . T h i s c h a n g e w o u l d be g r e a t e r i f t h e d e p t h o f s o l u t i o n o v e r t h e b l o c k were d e c r e a s e d . T h e r e s p o n s e f u n c t i o n | H x T ( x , z = 0 ) | / | H - J ( x , z = 0 ) | i s o f i n t e r e s t i n t h e v i c i n i t y o f a l a t e r a l i n h o m o g e n e i t y s i n c e t h i s r e s p o n s e f u n c t i o n h a s b e e n u s e d t o i n t e r p r e t d a t a i n S e c t i o n 1 . 4 . L a r g e d e v i a t i o n s f r o m t h e l i n e a r t r e n d o f t h i s r e s p o n s e f u n c t i o n w i l l i n d i c a t e a l a t e r a l i n h o m o g e n e i t y . F i g u r e 3 . 5 shows | H x T ( x , z = 0 ) l / j H ^ 1 ( x , z = 0 ) | a s a f u n c t i o n o f x . T h e r e s p o n s e i s a s y m m e t r i c w i t h t h e e d g e s o f t h e b l o c k c l e a r l y d e l i n e a t e d by two 7 9 F I G U R E 3 . 2 . E M M O D E L L I N G R E S U L T S F O R A L A T E R A L I N H O M O G E N E I T Y T a n k m o d e l r e s u l t s s h o w i n g i n p h a s e a n d q u a d r a t u r e H z r e s p o n s e s f o r a r e c t a n g u l a r b l o c k b u r i e d i n a h o m o g e n e o u s e n v i r o n m e n t . 80 F I G U R E 3.3. E M M O D E L L I N G R E S U L T S F O R A L A T E R A L I N H O M O G E N E I T Y T a n k m o d e l r e s u l t s s h o w i n g i n p h a s e a n d q u a d r a t u r e H x r e s p o n s e s f o r a r e c t a n g u l a r b l o c k b u r i e d i n a h o m o g e n e o u s e n v i r o n m e n t . 8 2 .8 6 • ELECTROMAGNETIC MODELLING Source: Line current Structure: Rectangular Block X=35cm ; Y=30cm ; Z=3.5cm Frequency: 3733.7 HZ Xcm h=3.2mm € ^ W//////////////M[3.5 cm -35cm-z F I G U R E 3 . 4 . EM M O D E L L I N G R E S U L T S F O R A L A T E R A L I N H O M O G E N E I T Y T a n k m o d e l r e s u l t s s h o w i n g i n p h a s e a n d q u a d r a t u r e E y r e s p o n s e s f o r a r e c t a n g u l a r b l o c k b u r i e d i n a h o m o g e n e o u s e n v i r o n m e n t . 84 ELECTROMAGNETIC MODELLING Source: Line current 35cm » z F I G U R E 3.5. EM M O D E L L I N G R E S U L T S F O R A L A T E R A L I N H O M O G E N E I T Y T a n k m o d e l r e s u l t s s h o w i n g |H X ^" ( x , z = 0 ) l/ lH^ 7 ( x , z = 0 ) | r e s p o n s e f o r a r e c t a n g u l a r b l o c k b u r i e d i n a h o m o g e n e o u s e n v i r o n m e n t . 86 ELECTROMAGNETIC MODELLING Source: Line current h= 3.2 mm -0 • 4 35cm • 87 p e a k s . T h e p e a k n e a r e s t t h e s o u r c e i s s m a l l e r t h a n t h e s e c o n d p e a k . T h i s i s p r o b a b l y d u e t o t h e l i n e a r i n c r e a s e o f t h e r e s p o n s e f u n c t i o n w i t h i n c r e a s i n g x ( r e c a l l F i g u r e s 1 . 1 6 a n d 1 . 1 7 ) . 88 3.2 D a t a R e d u c t i o n The e f f e c t s o f t h e p r i m a r y f i e l d a p p e a r t o g i v e a 1/x d e c a y t o t h e r e s p o n s e s H 2, H x , and Ey. T h i s i s most a p p a r e n t i n t h e c a s e o f t h e v e r t i c a l component H j . In an a t t e m p t t o remove t h e " s o u r c e e f f e c t " , r e s u l t s were m u l t i p l i e d by a w e i g h t i n g f u n c t i o n w h i c h i s p r o p o r t i o n a l t o t h e i n v e r s e of t h e p r i m a r y f i e l d a t t h a t p o i n t . F o r s m a l l v a l u e s o f h, t h i s w e i g h t i n g f u n c t i o n i s j u s t x. T h e s e r e s u l t s a r e p r e s e n t e d i n F i g u r e s 3.6, 3.7, and 3.8 s h o w i n g H^.x, Hy.x and Ey.x r e s p e c t i v e l y . S o u r c e " c o r r e c t e d " H 2 and H x d e l i n e a t e t h e edges o f t h e anomaly b e s t . The s o u r c e c o r r e c t e d p l o t s f o r E y a r e much more complex, and i t i s h a r d e r t o d i s c e r n t h e anomaly edges by l o o k i n g a t t h e d a t a a l o n e . F i g u r e s 3.6 and 3.7 s t i l l show a d e c a y w i t h i n c r e a s i n g x i n t h e s o u r c e c o r r e c t e d p l o t s . T h i s m ight be a c c o u n t e d f o r by t h e e f f e c t o f t h e h a l f s p a c e on t h e f i e l d components. S i n c e t h e EM i n d u c t i o n p r o b l e m i s n o n - l i n e a r , however, one c a n n o t c o r r e c t t h i s t r e n d by s i m p l y s u b t r a c t i n g t h e h a l f s p a c e r e s p o n s e f rom t h e d a t a . A n o t h e r e x p l a n a t i o n f o r t h e p r e s e n c e o f t h e d e c a y i n s o u r c e c o r r e c t e d p l o t s may be edge e f f e c t s . The s o u r c e c o r r e c t e d r e s u l t s s u g g e s t a s t r a t e g y f o r p e r f o r m i n g l a t e r a l p r o f i l i n g s u r v e y s u s i n g p o w e r l i n e s ( b e n t o r s t r a i g h t ) as a p r i m a r y s o u r c e . B e c a u s e m a g n e t i c s u r v e y s a r e c h e a p and y i e l d t h e b e s t r e s u l t s i n terms o f d i s c e r n i n g t h e e d ges of a b u r i e d anomaly, a f l u x g a t e s u r v e y run o v e r a g r i d i s 89 a l i k e l y c a n d i d a t e f o r o r e b o d y p r o s p e c t i n g . M e a s u r e m e n t s o f | H x ~ ^ ( x , z = 0 ) | a n d | H ^ T ( x , z = 0 ) | w i l l y i e l d t h e s a m e s h a p e d c u r v e s d e l i n e a t i n g t h e a p p r o x i m a t e l o c a t i o n o f t h e e d g e s a n d c e n t e r o f a n y s h a l l o w c o n d u c t i v e b o d y w i t h s u f f i c i e n t c o n d u c t i v i t y c o n t r a s t . D a t a r e d u c t i o n w o u l d c o n s i s t o f c o r r e c t i n g f o r t h e s o u r c e e f f e c t b y m u l t i p l y i n g b y t h e a p p r o p r i a t e w e i g h t i n g f u n c t i o n W ( x , y , z = c o n s t a n t ) w h i c h w o u l d b e t h e i n v e r s e o f t h e i n c i d e n t f i e l d c o m p o n e n t H Z L o r H X L a t t h a t p o i n t ( x , y , z = c o n s t a n t ) . T h i s m e t h o d c o u l d b e o f i n t e r e s t i n m i n e r a l e x p l o r a t i o n s i n c e t h e p r e s e n c e o f a p o w e r l i n e a f f o r d s a f r e e , p o w e r f u l , c o n t r o l l e d f r e q u e n c y s o u r c e , a n d t h e l o g i s t i c s o f t r a n s p o r t i n g t r a n s m i t t e r c a b l e a r e e l i m i n a t e d . F I G U R E 3 .6. EM M O D E L L I N G R E S U L T S F O R A L A T E R A L I N H O M O G E N E I T Y T a n k m o d e l r e s u l t s s h o w i n g i n p h a s e a n d q u a d r a t u r e ( H ^ ) . x r e s p o n s e s f o r a r e c t a n g u l a r b l o c k b u r i e d i n a h o m o g e n e o u s e n v i r o n m e n t . 91 h=3.2mm Q -ELECTROMAGNETIC MODELLING Source: Line Source Structure: Rectangular Block X=35cm Y=30.5cm Z=3.5cm Frequency: 3733.7 HZ (inphase Hz }-x X cm 3.5 cm •35 cm-z 92 F I G U R E 3.7. EM M O D E L L I N G R E S U L T S F O R A L A T E R A L I N H O M O G E N E I T Y T a n k m o d e l r e s u l t s s h o w i n g i n p h a s e a n d q u a d r a t u r e ( H x K x r e s p o n s e s f o r a r e c t a n g u l a r b l o c k b u r i e d i n a h o m o g e n e o u s e n v i r o n m e n t . * 35cm * F I G U R E 3.8. EM M O D E L L I N G R E S U L T S F O R A L A T E R A L I N H O M O G E N E I T Y T a n k m o d e l r e s u l t s s h o w i n g i n p h a s e a n d q u a d r a t u r e ( E y ) . x r e s p o n s e s f o r a r e c t a n g u l a r b l o c k b u r i e d i n a h o m o g e n e o u s e n v i r o n m e n t . 95 35cm 96 CONCLUDING REMARKS As an outgrowth of the work presented here, some recommendations f o r f u r t h e r work w i l l be presented. In regards to survey d e s i g n , i t i s recommended that measurements be made on a chained r e g u l a r g r i d . At present, measurements are made by d r i v i n g along a v a i l a b l e l o g g i n g roads using the v e h i c l e ' s odometer to r e c o r d d i s t a n c e s . Measurements should be made as c l o s e to the l i n e (0-10km) as access w i l l permit, s i n c e i t i s here that the s i g n a l to noise r a t i o w i l l be g r e a t e s t . T h i s s t r a t e g y assumes that the response f u n c t i o n to be used in the i n t e r p r e t a t i o n w i l l be | H x T ( x , y , z = 0 ) l / l H ^ i X f Y , z = 0 ) | . T h i s f u n c t i o n has the advantage of becoming l i n e a r w i t h i n a short d i s t a n c e (0-500m) of the t r a n s m i t t e r (powerline) so that model c o n s t r u c t i o n w i l l become an e x e r c i s e i n l i n e a r r e g r e s s i o n and a n a l y s i s of v a r i a n c e . Large d e v i a t i o n s of observed |HX (x,y,z=0)|/|H Z (x,y,z=0)| from a l i n e a r t rend w i l l s i g n a l l a t e r a l inhomogeneities. When making measurements of h o r i z o n t a l magnetic f i e l d components, i t i s recommended that the o r i e n t a t i o n of the d i p o l e moment of the r e c e i v o r be o r i e n t e d i n a constant d i r e c t i o n . T h i s d i r e c t i o n may be p u r e l y a r b i t r a r y . T h i s w i l l e l i m i n a t e the need fo r data r e d u c t i o n and w i l l t h e r e f o r e allow the data to be p l o t t e d immediately. T h i s w i l l allow an estimate of an apparent h a l f space r e s i s t i v i t y to be made i n the f i e l d . In a d d i t i o n to making an |H y^(x,y,z=0)| reading, the 97 which these are generated w i l l p r ovide the much needed bandwidth i n c r e a s e at the low frequency end of the primary s i g n a l which w i l l be r e q u i r e d to improve the depth of p e n e t r a t i o n f o r deeper soundings. A l i n e a r t rend w i l l s i g n a l l a t e r a l inhomogeneities. In areas where transformer s t a t i o n s are known to be pre s e n t , i t w i l l be necessary to measure c u r r e n t amplitudes and phases on e i t h e r s i d e of the s t a t i o n . Although the phase i n primary and secondary windings of a transformer are not expected to change i n balanced step-up or step-down transformers, c e r t a i n t ransformers ,namely, the Y-A transformer, possess phase changes. In the case of the Y-A transformer, the v o l t a g e on the secondary lags 30 degrees behind the v o l t a g e s on the primary. T h i s i s , a p p a r e n t l y , the p r e f e r r e d c o n n e c t i o n i n the U.S. and Canada (Dommel [ 2 4 ] ) . There are other transformers which produce 90 or , 150 degree phase s h i f t s , the l a t t e r being p r e f e r r e d i n some European c o u n t r i e s . 98 REFERENCES [I] K. Hayashi,T. Oguti, T. Watanabe, K. Tsuruda, S. Kokubun and R.E. H o r i t a , "Power Harmonic R a d i a t i o n Enhancement During the Sudden Commencement of a Magnetic Storm", Nature, v o l . 275, pp.627-629, Oct. 1978. [2] D.C. M c C o l l o r , T. Watanabe, W.F. Slawson, and R.M. S h i e r , "An E.M. 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M u l l e r , "Geology of Vancouver I s l a n d , F i e l d T r i p 7 Guidebook; G e o l o g i c a l A s s o c i a t i o n of Canada, M i n e r o l o g i c a l A s s o c i a t i o n of Canada J o i n t Meeting,1977. [20] P.R. B a n n i s t e r , "Summary of Image Theory Expr e s s i o n s f o r the Q u a s i - S t a t i c F i e l d s of Antennas at or Above the Ear t h ' s Surface", Proc. IEEE, v o l . 67., no. 7, pp. 1001-1008, J u l y 1979. [21] D.J. Thomson and J.T. Weaver, "The Complex Image Approximation f o r Induction i n a M u l t i - l a y e r e d E a r t h " , J . Geophys. Res. V o l . 80, pp. 123-129, 1975. [22] J.A. S t r a t t o n , E l e c t r o m a g n e t i c Theory, New York: McGraw-H i l l , 1941. [23] H.W. Dosso, "A Plane-Wave Analogue Model f o r Studying Electromagnetic V a r i a t i o n s " , Can. J . Phys., vol.44, pp.67-80, 1966. [24] H.W. Dommel, Notes on Power Systems A n a l y s i s , Dept. Of E l e c t r i c a l E n g i n e e r i n g , The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, 1975. 100 A P P E N D I X A : M A G N E T I C AND E L E C T R I C H E R T Z V E C T O R P O T E N T I A L S F o l l o w i n g W e a v e r [ 1 1 ] , t h e f o r w a r d 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 i s f o r m u l a t e d i n t e r m s o f t h e m a g n e t i c H e r t z v e c t o r p o t e n t i a l r a n d t h e e l e c t r i c H e r t z v e c t o r p o t e n t i a l n. D e r i v a t i o n o f E a n d H i n t e r m s o f II a n d r f o l l o w s . 1. EM F i e l d E q u a t i o n s ( M K S ) T h e b e h a v i o r o f a n EM f i e l d w i t h i n a u n i f o r m , i s o t r o p i c a n d s o u r c e f r e e m e d i u m o f c o n d u c t i v i t y a , p e r m e a b i l i t y n0, a n d p e r m i t i v i t y e 0 i s g o v e r n e d b y : V x E = - i c J M 0 H ( A . 1 ) VxH=Mo ( o+icje Q ) E ( A . 2 ) I n f u t u r e l e t a=u0(a+iue0) a n d f o r q u a s i - s t a t i c c o n d i t i o n s a=n0o D e f i n i n g t h e u s u a l v e c t o r p o t e n t i a l A s u c h t h a t : V x A = B ( A . 3 ) a n d s u b s t i t u t i n g i n t o ( 1 ) o n e o b t a i n s : V x ( E + i c j A ) = 0 S i n c e a n y q u a n t i t y w i t h v a n i s h i n g c u r l m a y b e w r i t t e n g r a d i e n t o f s o m e s c a l a r p o t e n t i a l <)> we may w r i t e : E = - i w A - V 0 a s t h e ( A . 4 ) 101 By s u b s t i t u t i n g (A.3) and (A.4) i n t o : VxH=oE V.D=0 one o b t a i n s coupled equations i n A and <f> which reduces Maxwell's equations to two equations: VxM 0VxA=a (-V</>-icJA) V. e0 (-V0-icjA) = O (One performs t h i s s u b s t i t u t i o n because A and <p must s a t i s f y the remaining Maxwell equations V.D=0 and V.B=0) Uncoupling of these equations i s u s u a l l y achieved by invoking a Lorentz c o n d i t i o n : V.A+ac>=0 (A.5) then 0 and the components of A s a t i s f y (by s u b s t i t u t i o n of (A.5) i n t o (A.3) and (A.4)): {V 2-k a 2}<*(x,y,z) = 0 (A.6) such that k-, 2 = io)u0 ( a+ia>e0 ) Equation (A.5) i s a u t o m a t i c a l l y s a t i s f i e d i f : A=aTT+Vxf (A. 7) (A.8) 102 where n i s the E l e c t r i c Hertz v e c t o r p o t e n t i a l r i s the Magnetic Hertz v e c t o r p o t e n t i a l These p o t e n t i a l s are g e n e r a l i z a t i o n s of a f u n c t i o n i n t r o d u c e d for the EM f i e l d of an o s c i l l a t i n g d i p o l e by H. H e r t z . S u b s t i t u t i n g (A.8) i n (A.6) one o b t a i n s : { v 2 - k z 2 } n = o S i m i l a r l y , { V 2-k 2 2}f=0 The r e l a t i o n between E and 1 to r and n i s found by s u b s t i t u t i o n of (A.7) and (A.8) i n t o : B=VxA E=-V0-3A/dt to o b t a i n B=Vx[an+Vxf] =aVxH+VxVxr (A.9) E=-VxVxn-icJVxf (A. 10) The i n d u c t i o n theorem i s used to s i m p l i f y (A.9) and (A.10) above fo r the case of a l a y e r e d e a r t h . The theorem s t a t e s that secondary f i e l d s induced i n any region may be represented uniquely by a d i s t r i b u t i o n of the normal component of f and n 103 o v e r t h e b o u n d a r y d e f i n i n g t h e r e g i o n ( L a j o i e [ 3 ] ; H a r r i n g t o n [ 1 2 ] ) . F o r a h o r i z o n t a l l y l a y e r e d e a r t h , i t i s e a s y t o see t h a t o n l y t h e z components of r,Tt a r e r e q u i r e d . T h e r e f o r e , f o r t h e r e m a i n d e r o f t h e d i s c u s s i o n , n and T w i l l be t h e z component o f n, r and w i l l be t r e a t e d a s s c a l a r f u n c t i o n s . T h u s , (A.9) and (A.10) b e c o m e : ( u s i n g p a r t i a l d e r i v a t i v e n o t a t i o n ) E ( x , y , z ) = ( n 3 1 ( x , y , z ) - i c j r 2 ( x , y ,z) ) i + ( n 3 2 ( x ^ z j + i u r , ( x , y , z ) ) j -(II, , ( x , y , z ) + r 2 2 ( x , y , z ) ) R (A.11) B ( x , y , z ) = (aII 2 ( x , y , z ) + T 3 , ( x , y , z ) ) i + (-aIIi ( x , y , z ) + r 3 2 ( x , y , z ) ) j - ( T , , ( x , y , z ) + r 2 2 ( x , y , z ) ) R (A.12) E q u a t i o n s ( A . 1 1 ) and (A.12) may be e x p r e s s e d i n t h e wavenumber domain t r a n s f o r m i n g t h e d e r i v a t i v e o p e r a t o r s t o m u l t i p l i c a t i o n s a c c o r d i n g t o : 3/3x * > i k x 3/3y <t > i k y The 3/3z o p e r a t o r i n t h e wavenumber domain a r i s e s by c o n s i d e r i n g t h e F o u r i e r t r a n s f o r m of t h e H e l m h o l t z e q u a t i o n i n any l a y e r i : { V 2 - k^ 2 } r ( x , y , z ) = 0 w h i c h y i e l d s an o r d i n a r y d i f f e r e n t i a l e q u a t i o n w i t h s o l u t i o n s of upward/downward t r a v e l l i n g p o t e n t i a l s : F ( k x , k y , z ) = C e x p ( k z ) + D e x p ( - k z ) where C,D a r e c o n s t a n t s and k=v/(k^ 2 + k 2 + k^ 2 ) 104 F o r z p o s i t i v e d o w n , we r e q u i r e D=0 f o r t h e u p w a r d c o n t i n u a t i o n o f f i e l d s : r ( k x , k y , z ) = C e x p ( k z ) S o l v i n g f o r C , l e t z = 0 r ( k y , k y , z ) = T ( k x , k y , 0 ) e x p ( k z ) ( A . 1 3 ) t h u s 9 / 9 z h a s a t r a n s f o r m k E q u a t i o n ( A . 1 3 ) d e f i n e s t h e u p w a r d c o n t i n u a t i o n o p e r a t o r . I f we a r e t o e m p l o y t h e H e r t z v e c t o r s i n s t e a d o f t h e c o m p o n e n t s o f E a n d B , b o u n d a r y c o n d i t i o n s i n t e r m s o f t h e s e v e c t o r s n e e d b e k n o w n . T h e u s u a l b o u n d a r y c o n d i t i o n s a t a n i n t e r f a c e a r e : 1) c o n t i n u i t y o f 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 H 2 ) c o n t i n u i t y o f n o r m a l c o m p o n e n t s o f *B a n d D BOUNDARY C O N D I T I O N S A T AN I N T E R F A C E z = d L A p p l y i n g c o n d i t i o n s 1) a n d 2 ) a b o v e t o e q u a t i o n s ( A . 1 1 ) a n d ( A . 1 2 ) , a n d e x p r e s s i n g t h e m i n t e r m s o f t h e H e r t z p o t e n t i a l s i n s t e a d o f E a n d H , W e a v e r [ 1 2 ] s h o w s t h a t : c o n t i n u i t y o f t a n g e n t i a l E r e q u i r e s t h a t : n 3 ( x , y , z = d - L )=n 3 ( x , y , z = d ^ ) c o n t i n u i t y o f t a n g e n t i a l ~H: r 3 ( x , y , z = d ^ ) = r 3 ( x , y , z = d - L ) c o n t i n u i t y o f n o r m a l B i w ; i 0 r ( x , y , z = d - ^ ) = i c j ^ 0 r ( x , y , z = d L ) ( A . 14) M o r e c o m p l e t e d e r i v a t i o n s o f t h e a b o v e a r e g i v e n i n L a j o i e [ 4 ] 105 and Weaver [ 1 2 ] . SIMPLIFICATION FOR UNGROUNDED SOURCES I t may be d e m o n s t r a t e d by t h e e q u i v a l e n c e p r i n c i p l e t h a t as f a r as i n d u c t i o n i n t h e e a r t h (z>0) i s c o n c e r n e d , any s o u r c e c u r r e n t d i s t r i b u t i o n i n t h e r e g i o n z<0 may be r e p r e s e n t e d u n i q u e l y by an e q u i v a l e n t s u r f a c e c u r r e n t i n t h e p l a n e z=0. The b o u n d a r y c o n d i t i o n a t z=0 i s t h e n ( H a r r i n g t o n [ 1 3 ] ) : n x [ H , ( x , y , z = 0 + ) ] - n x [ H 2 ( x , y , z = 0 ) ] = J s ( x , y , z = 0 ) where J s i s t h e s u r f a c e c u r r e n t d e n s i t y H, i s t h e m a g n e t i c f i e l d i n r e g i o n 1 H 2 i s t h e m a g n e t i c f i e l d i n r e g i o n 2 L a j o i e [4] shows t h a t n, t h e e l e c t r i c H e r t z v e c t o r p o t e n t i a l i s a s s o c i a t e d w i t h s u r f a c e c u r r e n t s a s s o c i a t e d w i t h t h e g r o u n d e d p a r t s o f t h e s o u r c e c u r r e n t . T h i s means t h a t f o r an u n g r o u n d e d s o u r c e n=0. T hus, (A.11) and (A.12) s i m p l i f y t o : E ( x ,y , z ) =-i<jr 2 (x ,y , z ) l + i ^ r , (x ,y , z ) j 1*(x,y,z)=r 3 , ( x , y , z ) i + r 3 2 (x,y,z)5 - { T , , ( x , y , z ) + r 2 2 ( x , y , z ) } R 106 APPENDIX B: REFLECTION AND TRANSMISSION FILTERS F o r wavenumber domain s o l u t i o n s t o t h e f o r w a r d EM i n d u c t i o n p r o b l e m , r e f l e c t i o n and t r a n s m i s s i o n FILTERS a r e d e f i n e d . T h e s e a r e t h e u s u a l r e f l e c t i o n and t r a n s m i s s i o n c o - e f f i c i e n t s f o r p l a n e waves, w h i c h a r e d e f i n e d i n t h e k „ - k y domain r a t h e r t h a n t h e s p a t i a l x-y domain. G i v e n th e p r i m a r y o r i n c i d e n t H e r t z p o t e n t i a l a t t h e t o p o f a s t a c k of l a y e r s , t h e t r a n s m i s s i o n and r e f l e c t i o n f i l t e r s w i l l g i v e t h e H e r t z p o t e n t i a l t r a n s m i t t e d t h r o u g h t h e s t a c k , and t h e p o t e n t i a l r e f l e c t e d from t h e s t a c k . (1) Homogeneous H a l f Space R e f l e c t i o n and T r a n s m i s s i o n F i l t e r s : F o r a p u r e l y i n d u c t i v e s o u r c e , o n l y t h e m a g n e t i c H e r t z p o t e n t i a l i s r e q u i r e d . B e g i n by c o n s i d e r i n g t h e b o u n d a r y c o n d i t i o n s w h i c h must be s a t i s f i e d . 4 ( i ) r U y ,k y ,z = 0 ) = r ( k x ,k y ,z = 0) ( i i ) k- LT(k x ,k y ,z=0) = k L + ) T ( k x , k y ,z=0+) ZrO | _ _ r1 z FIGURE B.1 HALF SPACE BOUNDARY CONDITIONS So t h a t : 107 r " k ( k x ,k y , z = 0 ) = r L ( k y ,k y , z = 0 ) - T S ( k y ,k y ,z=0) (B. 1) k L T ( k y ,k y fz=0) = kL_| T U ( k x ,k y ,z=0) + kL_, r ( k x , k y , z = 0 ) (B.2) (B.1) and (B.2). a r e s o l v e d s i m u l t a n e o u s l y c o n s i d e r i n g t h a t F U ( k x , k y , z = 0) i s known ( t h e i n c i d e n t H e r t z p o t e n t i a l a t z = 0 may be d e r i v e d f r o m t h e known s o u r c e - c u r r e n t d i s t r i b u t i o n ) . S o l v i n g (1) and (2) one o b t a i n s : r S ( k y ,k y,z=0) = [kL_, - k j j / f k ^ , + k L ] . r L ( k x , k y , z=0) R e f l e c t i o n F i l t e r f o r 1/2 s p a c e rl(kx ,ky ,z=o) = [2k L_, ]/[k L_, +k L ] . r L ( k x ,ky,z=o) V — ' T r a n s m i s s i o n F i l t e r f o r 1/2 s p a c e S i n c e t h e EM f i e l d s a r e d e r i v a b l e form H e r t z p o t e n t i a l s by t h e use of l i n e a r o p e r a t o r s i n t h e wavenumber domain, t h e r e f l e c t i o n f i l t e r may be a p p l i e d d i r e c t l y t o t h e F o u r i e r t r a n s f o r m o f any i n c i d e n t f i e l d component i e . H £ S ( k x ,z = 0)=RF(k y , k y , z = 0) .H £ L ( k x ,ky ,z = 0) 108 ( i i ) R e f l e c t i o n F i l t e r f o r A L a y e r e d E a r t h : B e c a u s e t h e t r a n s m i s s i o n f i l t e r i s n o t u s e f u l f o r c o m p u t a t i o n o f s e c o n d a r y o r r e f l e c t e d f i e l d s , o n l y t h e r e f l e c t i o n f i l t e r w i l l b e c o n s i d e r e d . A g a i n , u s e t h e b o u n d a r y c o n d i t i o n s p r e v i o u s l y g i v e n , a n d c o n s i d e r t h e t o t a l m a g n e t i c H e r t z v e c t o r r a t t h e b o u n d a r y d ^ : r ( k x , k y , d c_,)=r ( k y , k y , d l _ , ) + r * ( k x ,k , d L _ , ) (B.3a) r 3 T ( k x ,k d H ) = - r 3 L ( k x , k y , d L _ , ) + r 3 S (k x , k y , d - _ , ) ( B.3b) F I G U R E B.2 L A Y E R E D E A R T H BOUNDARY C O N D I T I O N S = R F ( k x , k y , d L ) r L ( k y , k y , d ^ ) e x p ( - 2 k L t L ) ( B . 4 ) 109 S u b s t i t u t i n g ( B . 4 ) i n t o ( B . 3 a ) : T L T ( k y , k y , d L _ , ) = T ( k y , k y , d j _ , ) [ 1 + R F ( k x , k y , d L ) e x p ( - 2 k L t L ) ] ( B . 5 ) S u b s t i t u t e ( B . 4 ) i n t o ( B . 3 b ) : T 3 T ( k x , k y r d L _ , )= - T 3 L ( k x , k y , d L . , ) [ 1+RF ( k y , k y , d L ) e x p ( -2 k L t L ) ] ( B . 6 ) S u b s t i t u t i n g ( B . 5 ) , ( B . 6 ) , ( B . 3 a ) a n d ( B . 3 b ) i n t o t h e b o u n d a r y c o n d i t i o n s , a n d e l i m i n a t i n g r L( k^ , ky , d [ _ | ) o n e o b t a i n s : R F ( k x , k y , d j _ , ) = [ ( 1 - D ) + R E ( 1+D) ] / [ ( 1 + D ) + R E ( 1 - D ) ] w h e r e R E = R F ( k x , k y , d ^ ) e x p ( - 2 k ^ t ^ ) D » k t / ( k L . , ) T h i s f o r m u l a t i o n e x p r e s s e s t h e R F a t z = d j _ | a s a f u n c t i o n o f t h e f i l t e r a t z = d ^ . T h i s a l l o w s t h e R F a t t h e t o p o f a s t a c k o f l a y e r s t o b e c o m p u t e d s e p a r a t e l y i n a s u b r o u t i n e . 1 10 A P P E N D I X C : D I R E C T I O N A L F I L T E R S B y a p p l y i n g d i r e c t i o n a l f i l t e r s t o 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 , i t i s p o s s i b l e t o o b t a i n a l l o t h e r C a r t e s i a n c o m p o n e n t s o f H . T h i s w i l l b e d e r i v e d b e l o w . A s s h o w n i n A p p e n d i x A , t h e z c o m p o n e n t o f r ( x , y , z ) s a t i s f i e s a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n i n t h e w a v e n u m b e r d o m a i n w h i c h h a s u p / d o w n t r a v e l l i n g s o l u t i o n s : { d 2 / d z 2 - k 2 } r ( x , y , z ) = 0 ( C . 1 ) F o r u p w a r d t r a v e l l i n g p o t e n t i a l s : T ( k x , k y , z ) = r ( k x , k y , z = 0 ) e x p ( - k z ) F r o m A p p e n d i x A we h a v e H s ( k x , k y , z ) = ( k x 2 + k y 2 ) T ( k v , k y , z = 0 ) e x p ( - k z ) T ( k x , k y , z = 0 ) = H i ( k x , k y , z = 0 ) / ( k x 2 + k y 2 ) ( C . 2 ) T h e o t h e r c o m p o n e n t s o f H a r e g i v e n b y s u b s t i t u t i o n o f (2) i n t o H „ ( k ^ , k y , z = 0 ) = i k x k r ( k y , k y , z = 0 ) = i k x k / ( k x 2 + k y 2 ) H J L ( k x , ky ,z=0) 111 H y ( k y , k y ,z = 0 ) = i k y k r ( k y , k y ,z = 0 ) = i k y k / ( k y 2 + k y 2 ) H 2 ( k x , k y , z = 0 ) 1 12 APPENDIX D: COMPUTATIONAL PROCEDURE Because d i s c r e t e FFT's are u t i l i z e d to so l v e the exact forward problem, the procedure i s su b j e c t to the usual d i s c r e t e FFT problems. I n c o r r e c t sampling i n the s p a t i a l domain r e s u l t s i n erroneous p r e d i c t i o n s f o r secondary f i e l d components. T h i s s e c t i o n d e s c r i b e s the p r e c a u t i o n s which must be taken to insu r e that secondary f i e l d s are p r e d i c t e d c o r r e c t l y . The case of a c o a x i a l system w i l l be used as an example s i n c e FFT r e s u l t s can be compared with Hankel Transform (HT) r e s u l t s f o r the same problem. Earth Model h=.60m i r . r / / 7 f r f r t*^ r^g r r 7 t r-7 C , = C o d , = . 6 0 » Mi = Mo 1 / / 1 t 1 / II l i t 6 = 3. Om Cj = C 0 M2 = Mo o 2 = . 10o, FIGURE D.1. EXAMPLE PROBLEM: COAXIAL DIPOLE SYSTEM The modelled c o a x i a l system: A l a y e r over a h a l f space with l a y e r t h i c k n e s s of .60m, s k i n depth 5=3.Om, p e r m e a b i l i t y and p e r m i t t i v i t y are the fr e e space v a l u e s . E l e v a t i o n of t r a n s m i t t e r and r e c e i v o r i s .6m. A C a r t e s i a n a n a l y t i c s o l u t i o n f o r the secondary magnetic f i e l d 1 1 3 H x d u e t o t h e s y s t e m i l l u s t r a t e d i n F i g u r e D . 1 i s : ( L a j o i e e t a l . [ 3 ] ) ' H x (k„ , k y , z = - h ) = 2 7 r m k x 2 e x p ( - 2 p h ) R F ( k y , k y , z = 0 ) / p ( D . 1 ) w h e r e p = / ( k x 2 + k y 2 ) 00 00 H x ( x , y , z = - h ) = 27rm/ J " k 2 e x p ( - 2 p h ) R F ( k x , k y , z = 0 ) -oo -co * e x p { i ( k x x + k y y ) } / p d k y d k y ( D . 2 ) T h e c y l i n d r i c a l s o l u t i o n f o r t h e s a m e p r o b l e m i s i n t h e f o r m o f a H a n k e l t r a n s f o r m ( K e l l e r a n d F r i s c h k n e c h t [ 5 ] ) : H x ( x , y , z = - h ) = M { ( 1 - y 2 / x 2 ) ( p / 5 ) T 2 ( A , B ) - T 0 ( A , B ) } ( D . 3 . ) w h e r e M = m x 2 / { 4 7 r 6 3 ( x 2 + y 2 ) } m = d i p o l e m o m e n t T 2 ( A , B ) = / R ( D , X ) X e x p ( - X A ) J , ( X B ) d X T 0 ( A , B ) = / R ( D , X ) X 2 e x p ( - X A ) J 0 ( X B ) d X R ( D , X ) = 1 - { 2 X ( U + V ) + ( U - V ) e x p ( - U D ) } / { ( U + X ) ( U + V ) - ( U ~ X ) ( U -V ) e x p ( - U D ) } U=y/(X 2 + i 2 ) V = v / ( X 2 + i k 2 ) k = o 2 / o , A = ( z + h ) / 5 B = p / 5 D = 2 d / 6 J 0 = B e s s e l f u n c t i o n o f f i r s t k i n d , o r d e r 0 1 14 J , = B e s s e l f u n c t i o n of f i r s t k i n d , o r d e r 1 X = v a r i a b l e o f i n t e g r a t i o n EXAMPLE 1: E q u a t i o n ( l ) was computed u s i n g Ax=Ay=1.0m on a 256 by 256 g r i d and i n v e r t e d t o t h e s p a t i a l domain by F F T . F i g u r e s D.2A and D.2B i l l u s t r a t e t h e r e a l ( i n p h a s e ) and i m a g i n a r y ( q u a d r a t u r e ) p a r t s o f (1) as a f u n c t i o n o f k x o n l y , p l o t t e d up t o t h e N y q u i s t f r e q u e n c y s u c h t h a t : k.x = ir/Ax=.314E+0l N o t i c e t h a t t h e s p e c t r a d e c a y s m o o t h l y t o z e r o by N y q u i s t and t h a t t h e r e i s no t r u n c a t i o n a t t h e h i g h f r e q u e n c i e s . Thus, t h e i n v e r s e t r a n s f o r m a g r e e s w e l l w i t h r e s u l t s p r e d i c t e d by t h e H a n k e l t r a n s f o r m f o r m u l a t i o n s e e n i n F i g u r e s D.2C and D.2D. EXAMPLE 2: In t h i s c a s e e q u a t i o n (1) was e v a l u a t e d u s i n g Ax=Ay=0.30m on a 128 by 128 g r i d . F i g u r e s D.3A and D.3B i l l u s t r a t e t h e r e a l ( i n p h a s e ) and i m a g i n a r y ( q u a d r a t u r e ) p a r t s of ( l ) i n t h e f r e q u e n c y domain as a f u n c t i o n o f k x o n l y , p l o t t e d up t o N y q u i s t = ! 0 . 5 r a d i a n s / m e t e r . B o t h s p e c t r a i n d i c a t e t h a t t h e s a m p l i n g has been i n a d e q u a t e . T h i s a r i s e s b e c a u s e , i n t h e x-y domain, o n l y a s m a l l p o r t i o n o f t h e f i e l d has been samp l e d s i n c e b o t h t h e r e c o r d l e n g t h and s a m p l i n g i n t e r v a l have been r e d u c e d . C o n s e q u e n t l y , i n 3C and 3D t h e p r e d i c t e d f i e l d s p o s s e s s a low f r e q u e n c y F I G U R E D . 2 . E X A M P L E 1: I N P H A S E AND Q U A D R A T U R E HX S E C O N O H R T Q U A D R A T U R E HX S E C O N D A R Y Q U A D R A T U R E o 9 U F I G U R E D . 3 . E X A M P L E 2 : I N P H A S E AND Q U A D R A T U R E HX SECONDARY INPHASE i (RHPEHE/METERI HX SECONDARY INPHASE (flMPEHESl X SECONDARY QUADRATURE HX SECONDARY QUADRATU . ^ - r ^ o , • (flMPEHESI 0 1 I 119 C o s i n e - l i k e waveform and a r e s u b s t a n t i a l l y d i f f e r e n t from t h e HT r e s u l t s . EXAMPLE 3: H e r e e q u a t i o n (1) was e v a l u a t e d u s i n g Ax=Ay=3.0m on a 256 by 256 g r i d . F i g u r e s D.4A and D.4B i n d i c a t e t h a t a l i a s i n g has o c c u r e d s i n c e t h e s p e c t r a do n o t d e c a y s m o o t h l y t o z e r o by N y q u i s t . A l i a s i n g e r r o r s i n t h i s c a s e have b o o s t e d t h e h i g h f r e q u e n c y c o n t e n t a s seen i n F i g u r e s D.4C and D.4D. T h e s e e xamples a l e r t t h e u s e r t o e x e r c i s e c a u t i o n when u s i n g t h e FFT f o r m u l a t i o n . E a c h t i m e a f o r w a r d p r o b l e m i s t o be s o l v e d , t h e s p e c t r a must be c h e c k e d t o see t h a t s a m p l i n g i n t h e s p a t i a l x-y domain i s c o r r e c t . A l i a s i n g i s c o n t r o l l e d by Ax and Ay. T r u n c a t i o n of t h e f i e l d i s c o n t r o l l e d by t h e d i s t a n c e f r o m t h e s o u r c e a t w h i c h t h e s e c o n d a r y f i e l d d e c a y s t o n e a r z e r o v a l u e s . The number o f p o i n t s i n t h e x and y d i r e c t i o n a r e d e t e r m i n e d by a b a l a n c e between t h e t h e d i s t a n c e s i n t h e x and y d i r e c t i o n s , and t h e s a m p l i n g i n t e r v a l s i n t h e s e d i r e c t i o n s . A l i a s i n g e r r o r s o c c u r i f Ax o r Ay a r e t o o l a r g e . A l t h o u g h one c a n e a s i l y use l a r g e Ax and Ay v a l u e s t o e l i m i n a t e t r u n c a t i o n i n t h e x-y domain, t h e r e s u l t may be t o s e v e r l y a l i a s t h e sampled f i e l d i n t h e k y - k v domain. F I G U R E D . 4 . E X A M P L E 3: I N P H A S E A N D Q U A D R A T U R E HX HX S E C O N D A R Y I N P H A S E i (AMPERE/METER) m i o o i» m i o X o m —i m 3 CO u m • o to • > / • 1 * •n X n H • H • o HX S E C O N D A R Y I N P H A S E • (AMPERES) 1 22 D E T A I L S ON T H E C O M P U T A T I O N F O R T H E C A S E O F A S T R A I G H T L I N E O F C U R R E N T T h e c o m p u t a t i o n o f t o t a l f i e l d s f o r t h e c a s e o f a s t r a i g h t l i n e o f c u r r e n t e m p l o y e d 4096 p o i n t t r a n s f o r m s u s i n g a A x o f 5.0 m e t e r s . T h i s c h o i c e w a s c h e c k e d f o r a l i a s i n g e r r o r s b y i n v e r t i n g t h e e x p r e s s i o n H x L ( k x , k / , z = 0 ) = - I h / ( 2 i r ) { e x p ( - h k y ) } t o t h e x - y d o m a i n a n d c o m p a r i n g t h e i n v e r t e d r e s u l t w i t h t h e e x p r e s s i o n f o r t h e i n c i d e n t f i e l d : H X L ( x , z = 0 ) = - I / 2 7 r { h / ( x 2 + h 2 ) } T h e c o m p u t a t i o n o f s e c o n d a r y f i e l d s f o r t h e c a s e o f a s t r a i g h t l i n e o f c u r r e n t e m p l o y e d 4096 p o i n t t r a n s f o r m s u s i n g , a Ax o f b e t w e e n 500 a n d 1000 m e t e r s d e p e n d i n g o n t h e e a r t h m o d e l . E a c h t i m e t h e f o r w a r d p r o b l e m f o r t h e s e c o n d a r y f i e l d s w a s c o m p u t e d , a m p l i t u d e s p e c t r a f o r t h e c o m p o n e n t s H , a n d H w e r e c h e c k e d a s p e r E x a m p l e s 1, 2, a n d 3 o f t h i s a p p e n d i x . 

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