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Inversion of horizontal loop electromagnetic soundings over a stratified earth Fullagar, Peter Kelsham 1981

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INVERSION OF HORIZONTAL LOOP ELECTROMAGNETIC SOUNDINGS OVER A STRATIFIED EARTH by PETER KELSHAM FULLAGAR B.Sc.(Hons), Monash U n i v e r s i t y , 1975 M.Sc., Monash U n i v e r s i t y , 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF. THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of G e o p h y s i c s and Astronomy) We accept t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J a n u a r y , 1981 © P e t e r Kelsham F u l l a g a r , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o lumbia, I agr e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e 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 purposes may be g r a n t e d by the Head o f my Department o r by h i s 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 o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be 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 o f Geophysics and Astronomy The U n i v e r s i t y o f B r i t i s h Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 19th January, 1981 i i ABSTRACT A d e t a i l e d study of e l e c t r o m a g n e t i c i n d u c t i o n i n a sequence of c o n d u c t i v e l a y e r s has been completed f o r the case when the i n d u c i n g f i e l d s are g e n e r a t e d by an a l t e r n a t i n g c u r r e n t i n a h o r i z o n t a l l o o p . The study was undertaken w i t h a view t o the development of a computer program t o p e r f o r m a u t o m a t i c i n v e r s i o n of h o r i z o n t a l l o o p e l e c t r o m a g n e t i c (HLEM) freq u e n c y soundings taken • over h o r i z o n t a l l y s t r a t i f i e d ground. The program c o n s t i t u t e s a new i m p l e m e n t a t i o n of the g e n e r a l approach of Backus and G i l b e r t (1967, 1968, 1970). By means of a l i n e a r i s e d i t e r a t i v e scheme, i t c o n s t r u c t s l a y e r e d c o n d u c t i v i t i e s which s a t i s f y a g i v e n s e t of o b s e r v a t i o n s t o an a c c u r a c y c o n s i s t e n t w i t h the o b s e r v a t i o n a l u n c e r t a i n t i e s . S u b s e q u e n t l y , the non-uniqueness a d m i t t e d by the l i m i t e d amount of d a t a can be a p p r a i s e d by computing averages of the o r i g i n a l c o n s t r u c t e d model and comparing them w i t h averages c o r r e s p o n d i n g t o o t h e r d i s s i m i l a r models which a l s o s a t i s f y the d a t a . In examples the B a c k u s - G i l b e r t averages f a i t h f u l l y r e f l e c t the c h a r a c t e r of the " t r u e " c o n d u c t i v i t y i n r e g i o n s of h i g h c o n d u c t i v i t y , but they a r e of l i m i t e d v a l u e i n d e l i n e a t i n g r e s i s t i v e zones. The program has been a p p l i e d s u c c e s s f u l l y t o the i n v e r s i o n of r e a l d a t a from Grass V a l l e y , Nevada. A uniqueness theorem i s p r e s e n t e d f o r i n v e r s i o n of HLEM frequ e n c y s o u n d i n g s . I t has been proved t h a t an u n l i m i t e d q u a n t i t y of p e r f e c t l y a c c u r a t e HLEM f r e q u e n c y soundings ( a t a f i x e d r e c e i v e r l o c a t i o n ) s u f f i c e s t o c o m p l e t e l y determine the c o n d u c t i v i t y as a f u n c t i o n of d e p t h . T h i s r e s u l t , which i s b e l i e v e d t o be new, enhances the c r e d i b i l i t y of c o n c l u s i o n s based on i n v e r s i o n of HLEM so u n d i n g s . i v TABLE OF CONTENTS Page A b s t r a c t i i L i s t of T a b l e s v i i L i s t of F i g u r e s v i i i Acknowledgements x D e d i c a t i o n x i i I n t r o d u c t i o n 1 Chapter I : The Forward Problem 16 1-1 F o r m u l a t i o n of the Problem 17 1-2 S p e c i f i c a t i o n of the P r i m a r y or Source F i e l d 22 1-3 An I n t e g r a l E q u a t i o n R e p r e s e n t a t i o n f o r the E l e c t r i c F i e l d .... 26 1-4 S p e c i f i c a t i o n of the Responses f o r a L a y e r e d E a r t h .... 28 1-5 A t t e n u a t i o n and R e f l e c t i o n 33 1-6 The I n d u c t i o n Response F u n c t i o n 37 I - 7 C o m p u t a t i o n a l Procedure : 43 Chapter I I : Model C o n s t r u c t i o n 53 I I - 1 F r e c h e t D e r i v a t i v e s and L i n e a r i s e d C o n s t r u c t i o n .... 54 11-2 Computation of the F r e c h e t K e r n e l s 63 11-3 S o l u t i o n of a L i n e a r I n v e r s e Problem i n the Presence of E r r o r s .... 64 Page 11-4 S p e c i a l F e a t u r e s of HLEM.INV 75 ( i ) Data O p t i o n s 75 ( i i ) M o d i f i c a t i o n of Depth P a r t i t i o n 78 ( i i i ) E x p l i c i t Treatment of E r r o r s 79 ( i v ) Compensation f o r C o i l M i s a l i g n m e n t 89 I I - 5 T h e o r e t i c a l Examples 94 Chapter I I I : A p p r a i s a l 116 I I I - l B a c k u s - G i l b e r t A p p r a i s a l f o r L i n e a r I n v e r s e Problems .... 118 111 -2 The R e s o l u t i o n M a t r i x 122 111-3 A p p r a i s a l Techniques f o r N on-Linear I n v e r s e Problems .... 128 111 -4 The Uniqueness of HLEM I n v e r s i o n 131 I I I - 5 A p p r a i s a l i n HLEM.INV 138 ( i ) C o n s t r u c t i o n of a S u i t e of A c c e p t a b l e Models ... 139 ( i i ) B a c k u s - G i l b e r t A v e r a g i n g 158 Chapter IV: I n v e r s i o n of Data from Grass V a l l e y , Nevada .... 184 IV- 1 C o n s t r u c t i o n 186 IV-2 A p p r a i s a l 193 C o n c l u s i o n 210 R e f e r e n c e s 216 V I Page Appendix A: Induced C u r r e n t D i s t r i b u t i o n f o r a P a r t i c u l a r C o n d u c t i v i t y .... 221 Appendix B: E v a l u a t i o n of I n t e g r a l s f o r the D e t e r m i n a t i o n of E, Hr, Hz and the D i s c r e t e F r e c h e t K e r n e l s .... 229 B - l P r i m a r y E l e c t r i c F i e l d I n t e g r a l 229 B-2 P r i m a r y R a d i a l Magnetic F i e l d I n t e g r a l 230 B-3 P r i m a r y V e r t i c a l Magnetic F i e l d I n t e g r a l 231 B-4 I n t e g r a l s f o r D i s c r e t e F r e c h e t K e r n e l s i n the F i r s t L a y e r .... 232 Appendix C: Uniqueness of I n v e r s i o n of HLEM Frequency Soundings .... 235 C - l S l i c h t e r ' s Theorem 237 C-2 B a i l e y ' s Theorem 241 C-3 G e l ' f a n d - L e v i t a n Methods 243 C-4 A Uniqueness Theorem f o r HLEM Frequency Soundings .... 244 Appendix D: A S i m p l i f i e d A n a l y s i s of C o i l M i s a l i g n m e n t E r r o r .... 260 D-l M i s a l i g n m e n t E r r o r f o r Hz Measurements 261 D-2 M i s a l i g n m e n t E r r o r f o r Hr Measurements 262 D-3 M i s a l i g n m e n t E r r o r f o r W a v e - T i l t Measurements 263 V I 1 LIST OF TABLES Page 1.1 Comparison of D i g i t a l L i n e a r F i l t e r i n g and Romberg I n t e g r a t i o n .... 47 2.1 Data f o r Hr C o n s t r u c t i o n 97 2.2 Data f o r Hz C o n s t r u c t i o n 98 2.3 Data f o r W C o n s t r u c t i o n 99 2.4 Parameter L i s t f o r C o n s t r u c t i o n Examples 101 4.1 W a v e - T i l t Data from Grass V a l l e y , Nevada 187 v i i i LIST OF FIGURES Page 1.1 System Geometry 18 1:2 Example of I n d u c t i o n Response F u n c t i o n s 41 1.3 Comparison w i t h Responses C a l c u l a t e d by Ryu et a l . (1970) .... 49 2.1 Examples of Convergence f o r C o n v e n t i o n a l and M o d i f i e d F o r m u l a t i o n s .... 87 2.2 "True" Model f o r T h e o r e t i c a l C o n s t r u c t i o n Examples .... 95 2.3 S t a r t i n g Model f o r T h e o r e t i c a l C o n s t r u c t i o n Examples .... 95 2.4 C o n s t r u c t i o n w i t h Hr Data 102 2.5 C o n s t r u c t i o n w i t h Hz Data 105 2.6 C o n s t r u c t i o n w i t h W Data 108 2.7 C o n s t r u c t i o n w i t h Hr & Hz Data I l l 3.1 C o n s t r u c t i o n from Remote S t a r t i n g Models 133 3.2 C o n s t r u c t i o n of B a s e l i n e Model 142 3.3 E f f e c t of Depth P a r t i t i o n on C o n d u c t i v i t y Bounds .... 148 3.4 Examples of E x t r e m a l Models 151 3.5 | Hr | & J Hz j E x t r e m a l Model Conductances 157 3.6 Examples of Trade-Off Curves 164 3.7 |Hr j & [Hz| Averages, and Assessment of t h e i r U n i v e r s a l i t y .... 169 3.8 j Hr | & 0r Averages, and Assessment of t h e i r U n i v e r s a l i t y .... 172 3.9 Re{Hr,Hz} & Im{Hr,Hz} Averages, and Assessment of t h e i r U n i v e r s a l i t y .... 174 IX Page 3.10 C o m p a r i s o n of A v e r a g e s of t h e " T r u e " Model w i t h B a s e l i n e Model A v e r a g e s .... 177 3.11 | Hr | & j Hz | A v e r a g i n g F u n c t i o n s 179 3.12 Re{Hr,Hz} & Im{Hr,Hz} A v e r a g i n g F u n c t i o n s 182 4.1 Hot S p r i n g s i n N o r t h w e s t e r n Nevada 185 4.2 C o n s t r u c t i o n of Model s a t i s f y i n g G r a s s V a l l e y W a v e - T i l t D a t a 189 4.3 T3-R2 B a s e l i n e Model 195 4.4 E x t r e m a l M o d e l s f o r t h e G r a s s V a l l e y W a v e - T i l t D a t a .... 196 4.5 T3-R2 T r a d e - O f f C u r v e s 202 4.6 A v e r a g e s f o r A c c e p t a b l e G r a s s V a l l e y S t r u c t u r e s , and A s s e s s m e n t of t h e i r U n i v e r s a l i t y .... 203 4.7 A v e r a g i n g F u n c t i o n s f o r G r a s s V a l l e y A p p r a i s a l 205 4.8 C o n d u c t a n c e f o r G r a s s V a l l e y E x t r e m a l M o d e l s 209 A . l COMTAL Images of I n d u c e d C u r r e n t D i s t r i b u t i o n s 222 A.2 GV.STD C o n d u c t i v i t y 227 X ACKNOWLEDGEMENTS T h i s t h e s i s c o u l d n e v e r have been c o m p l e t e d were i t n o t f o r t h e a c t i v e s u p p o r t o f a l a r g e number o f p e o p l e . P r e - e m i n e n t amongst t h e s e i s Doug O l d e n b u r g , my s u p e r v i s o r ; i n a v e r y r e a l s e n s e t h i s d i s s e r t a t i o n i s an o u t g r o w t h o f my a s s o c i a t i o n w i t h Doug, a nd I am g r e a t l y i n d e b t e d t o h i m . I t was he who o r i g i n a l l y s p a r k e d my i n t e r e s t i n i n v e r s e t h e o r y , a n d h i s i n f e c t i o u s e n t h u s i a s m w h i c h r e - k i n d l e d i t on t h o s e o c c a s i o n s when t h e F r o n t i e r o f G e o p h y s i c s seemed l i k e t h e p r o v e r b i a l i m m o v a b l e o b j e c t ! D u r i n g t h e p a s t t h r e e and a h a l f y e a r s I have d e v e l o p e d a g r e a t r e s p e c t and a f f e c t i o n f o r Doug, a nd I hope t o c o n t i n u e t o r e a p t h e b e n e f i t s o f h i s p r o f e s s i o n a l c o u n s e l a n d p e r s o n a l warmth i n t h e y e a r s t o come. T h a n k s a r e due t o G e o r g e B l u m a n , G a r r y C l a r k e , Ron C l o w e s , a n d Bob E l l i s f o r s e r v i n g on my g r a d u a t e a d v i s o r y c o m m i t t e e . I am e s p e c i a l l y g r a t e f u l t o G e o r g e f o r s e v e r a l v e r y h e l p f u l d i s c u s s i o n s , p a r t i c u l a r l y i n r e l a t i o n t o t h e u n i q u e n e s s p r o o f ; h i s a d v i c e a n d e n c o u r a g e m e n t have been g r e a t l y a p p r e c i a t e d . I w i s h t o a c k n o w l e d g e t h e g e n e r o u s a s s i s t a n c e o f M i k e H o v e r s t e n , D r . H.F. M o r r i s o n , a n d t h e r e s t o f t h e E n g i n e e r i n g G e o s c i e n c e g r o u p a t B e r k e l e y , b o t h i n p r o v i d i n g t h e d a t a f r o m G r a s s V a l l e y , a n d i n a l l o w i n g me t o v i s i t t h e i r f i e l d s i t e i n J u l y , 1 978. I h a v e been p a r t i c u l a r l y f o r t u n a t e i n r e c e i v i n g c o n t i n u o u s f i n a n c i a l s u p p o r t i n t h e f o r m o f U.B.C. G r a d u a t e F e l l o w s h i p s ( 1 9 7 7 - 7 9 ) a nd a K i l l a m M e m o r i a l P r e - D o c t o r a l S c h o l a r s h i p ( 1 9 7 9 - 8 0 ) . X 1 I h a v e made a g r e a t many f r i e n d s amongst t h e s t u d e n t s , f a c u l t y , a n d s t a f f a t t h e D e p a r t m e n t o f G e o p h y s i c s and A s t r o n o m y , a l l o f whom ha v e c o n t r i b u t e d t o t h e happy e n v i r o n m e n t i n w h i c h I w o r k e d . F o u r o f my f e l l o w g r a d u a t e s t u d e n t s a r e w o r t h y o f s p e c i a l n o t e : P a t r i c i a M o n ger, who e x p l a i n e d t h e COMTAL s y s t e m t o me ( s e v e r a l t i m e s ) ; Ed W a d d i n g t o n , who was my l o n g - s u f f e r i n g c o m p a n i o n on t h e " n i g h t s h i f t " ; Dan Au, who p r o v i d e d a t w e n t y - f o u r h o u r a i r p o r t s e r v i c e ; a n d Shlomo L e v y , who s p a r e d me t h e d i s g r a c e o f b e i n g t h e w o r s t d a r t s p l a y e r i n t h e d e p a r t m e n t . X I 1 To Merms and The B i g 1 INTRODUCTION The h o r i z o n t a l l o o p e l e c t r o m a g n e t i c (HLEM) method of g e o p h y s i c a l e x p l o r a t i o n i s one of a l a r g e number of e l e c t r o m a g n e t i c (EM) methods used t o d e t e c t s u b t e r r a n e a n c o n d u c t o r s ( K e l l e r and F r i s c h k n e c h t , 1966, pp278-427; Ward, 1966a). The p h y s i c a l p r i n c i p l e u n d e r l y i n g a l l EM methods i s t h a t of i n d u c t i o n : c u r r e n t s a r e induced i n the ground i n the v i c i n i t y of an o s c i l l a t i n g source c u r r e n t . As the name i m p l i e s , the HLEM method i s d i s t i n g u i s h e d by .the n a t u r e of the s o u r c e : an a l t e r n a t i n g c u r r e n t i s c o n f i n e d i n a c i r c u l a r or n e a r - c i r c u l a r c o i l which i s a l i g n e d i n a h o r i z o n t a l p l a n e . Far from the c e n t r e of the c o i l , the i n d u c i n g magnetic f i e l d c l o s e l y resembles t h a t due t o a v e r t i c a l magnetic d i p o l e . T h i s o s c i l l a t i n g " d i p o l e " f i e l d i n d u c e s eddy c u r r e n t s i n a l l c o n d u c t o r s i t e n c o u n t e r s , and the ."secondary" magnetic f i e l d s a s s o c i a t e d w i t h t h e s e induced c u r r e n t s can be d e t e c t e d a t the s u r f a c e w i t h e i t h e r an i n d u c t i o n c o i l or a magnetometer. The c h o i c e of t r a n s m i t t e r f r e q u e n c y , t r a n s m i t t e r - r e c e i v e r (T-R) s e p a r a t i o n , and c o i l r a d i u s i s made w i t h r e f e r e n c e t o the aims and l o g i s t i c s of the p a r t i c u l a r s u r v e y . For example, the g r e a t e r m o b i l i t y of s m a l l c o i l s i s an advantage f o r a i r b o r n e work or i n rough t e r r a i n , but can be enjoyed o n l y a t the expense of a reduced depth of p e n e t r a t i o n a t any p a r t i c u l a r f r e q u e n c y ; on the o t h e r hand, enhanced p e n e t r a t i o n r e s u l t s from an i n c r e a s e i n T-R s e p a r a t i o n w i t h f r e q u e n c y h e l d f i x e d , but a d e t e r i o r a t i o n i n the s i g n a l - t o - n o i s e 2 r a t i o must be a c c e p t e d . The e f f e c t of t r a n s m i t t e r f r e q u e n c y on the d i s t r i b u t i o n of induced c u r r e n t s i s i l l u s t r a t e d d i r e c t l y f o r a c e r t a i n c o n d u c t i v i t y i n Appendix A. Most s i g n i f i c a n t l y , the depth of p e n e t r a t i o n d e c r e a s e s m o n o t o n i c a l l y w i t h i n c r e a s i n g f requency. In v a r i o u s g u i s e s the HLEM method f i n d s a number of a p p l i c a t i o n s , p r i n c i p a l l y i n the f i e l d s of m i n e r a l e x p l o r a t i o n , geothermal (and groundwater) e x p l o r a t i o n , and c i v i l e n g i n e e r i n g . In m i n e r a l e x p l o r a t i o n the t e c h n i q u e may be employed t o l o c a t e m e t a l l i c m i n e r a l s d i r e c t l y , or t o d e l i n e a t e g e o l o g i c a l s t r u c t u r e s i n order t o i d e n t i f y s i t e s f a v o u r a b l e f o r the emplacement of ore (Ward, 1966b, p l 2 2 ; D o l a n , 1967). The t a r g e t i n geothermal e x p l o r a t i o n i s u s u a l l y a body of hot c o n d u c t i v e b r i n e ( T r i p p et a l . , 1978); HLEM d a t a from a geothermal p r o s p e c t i n Nevada a r e i n v e r t e d i n Chapter IV. One p o s s i b l e e n g i n e e r i n g a p p l i c a t i o n i s the d e t e r m i n a t i o n of p e r m a f r o s t t h i c k n e s s ; t h i s i s a c c o m p l i s h e d i n d i r e c t l y v i a d e t e c t i o n of the u n f r o z e n ( r e l a t i v e l y c o n d u c t i v e ) ground which u n d e r l i e s the r e s i s t i v e p e r m a f r o s t l a y e r ( H o e k s t r a , 1978). In c o n t r a s t t o the r e c e n t r a p i d advances i n i n s t r u m e n t a t i o n , development of v i a b l e i n v e r s i o n r o u t i n e s f o r EM data has proceeded v e r y g r a d u a l l y . The slow p r o g r e s s i s due p a r t l y t o the heavy emphasis p l a c e d on s e i s m i c methods by the o i l i n d u s t r y i n the p a s t , and p a r t l y t o the i n h e r e n t c o m p l e x i t y of e l e c t r o m a g n e t i c phenomena. To i l l u s t r a t e the second p o i n t , i t may be r e c a l l e d t h a t d i s p e r s i o n and a t t e n u a t i o n can o f t e n be d i s r e g a r d e d i n s e i s m o l o g y , but must always be c o n s i d e r e d i n EM problems. The r e l a t i v e n e g l e c t of EM methods i s r e f l e c t e d i n the 3 f a c t t h a t much of p r e s e n t - d a y EM i n t e r p r e t a t i o n i s p e r f o r m e d by c u r v e - m a t c h i n g , whereas s e i s m i c i n t e r p r e t a t i o n i s h i g h l y a u t o m a t e d . T h i s t h e s i s p r o j e c t was t h e r e f o r e u n d e r t a k e n i n r e s p o n s e t o a p e r c e i v e d need f o r i m p r o v e d i n v e r s i o n p r o c e d u r e s f o r EM d a t a i n g e n e r a l , and f o r HLEM d a t a i n p a r t i c u l a r . A c c o r d i n g l y , a FORTRAN a l g o r i t h m has been d e v e l o p e d t o p e r f o r m a u t o m a t i c i n v e r s i o n of HLEM s o u n d i n g s i n t h e s i m p l e s t c a s e of i n t e r e s t , namely when t h e e a r t h i s c o m p r i s e d of a s e q u e n c e of f l a t - l y i n g l a y e r s . T h i s s i m p l i f i e d " g e o l o g y " w i l l be i m p l i c i t l y assumed i n a l l s u b s e q u e n t d i s c u s s i o n . I t i s w e l l t o examine t h e w e a k n e s s e s of t h e c u r v e - m a t c h i n g method i n o r d e r t o a p p r e c i a t e t h e e x t e n t t o w h i c h t h e new program, HLEM.INV, r e p r e s e n t s an improvement. C u r v e - m a t c h i n g can be t e d i o u s and t i m e - c o n s u m i n g , and r a p i d l y becomes i m p r a c t i c a l as t h e number of l a y e r s i n c r e a s e s . Thus F r i s c h k n e c h t (1967) and Vanyan e t a l . (1967) p r o v i d e c u r v e s f o r no more t h a n f o u r l a y e r s . F o r more c o m p l i c a t e d l a y e r e d s t r u c t u r e s c u r v e - m a t c h i n g i s v i r t u a l l y u n w o r k a b l e . T h e s e l i m i t a t i o n s a r e n o t s h a r e d by HLEM.INV b e c a u s e e s s e n t i a l l y no r e s t r i c t i o n i s imposed on t h e number o f l a y e r s w h i c h can be c o n s i d e r e d . A l s o , c u r v e - m a t c h i n g p r o v i d e s a t b e s t one p o s s i b l e model c o m p a t i b l e w i t h t h e o b s e r v a t i o n s , whereas t h e r e a r e i n g e n e r a l an i n f i n i t e number o f e q u a l l y a c c e p t a b l e m o d e l s ; c o n s e q u e n t l y , i t i s u n j u s t i f i a b l e t o base c o n c l u s i o n s r e g a r d i n g t h e n a t u r e o f t h e t r u e c o n d u c t i v i t y on t h e a p p e a r a n c e of a p a r t i c u l a r model w h i c h s a t i s f i e s t h e d a t a . C u r v e - m a t c h i n g c a n n o t p r o v i d e any i n d i c a t i o n o f t h e e x t e n t t o w h i c h a p a r t i c u l a r a c c e p t a b l e model i s r e p r e s e n t a t i v e of a l l models s a t i s f y i n g t h e d a t a . 4 HLEM.INV may be r e g a r d e d as a new i m p l e m e n t a t i o n of the g e n e r a l approach of Backus and G i l b e r t (1967, 1968, 1970), and as such i s a c l o s e r e l a t i v e of the r e s i s t i v i t y and m a g n e t o t e l l u r i c (MT) i n v e r s i o n r o u t i n e s w r i t t e n by Oldenburg (1978, 1979). C e n t r a l t o the B a c k u s - G i l b e r t approach i s the c l e a r d i s t i n c t i o n drawn between c o n s t r u c t i o n of one or more models which s a t i s f y the d a t a , and a p p r a i s a l of such a c c e p t a b l e models i n an e f f o r t t o i d e n t i f y the f e a t u r e s common t o a l l a c c e p t a b l e models. With r e s p e c t t o t h i s t e r m i n o l o g y , c u r v e - m a t c h i n g may be c a t e g o r i s e d as a c o n s t r u c t i o n p r o c e d u r e , w i t h no p r o v i s i o n f o r a p p r a i s a l . The r e c o g n i t i o n of the importance of a p p r a i s a l a l s o s e t s HLEM.INV a p a r t from the HLEM i n v e r s i o n a l g o r i t h m w r i t t e n by Glenn e t a l . (1973). T h e i r program i s based on a g e n e r a l i s e d l i n e a r i n v e r s i o n approach ( W i g g i n s , 1972), and i s e s s e n t i a l l y a c o n s t r u c t i o n r o u t i n e o n l y . The reduced emphasis a c c o r d e d t o a p p r a i s a l by Glenn et a l . can be a t t r i b u t e d i n p a r t t o a p h i l o s o p h i c a l d i f f e r e n c e : they appear t o r e g a r d the model parameters as r e a l p h y s i c a l v a r i a b l e s , and t o v i s u a l i s e i n v e r s i o n as the p r o c e d u r e whereby the parameters are a s s i g n e d v a l u e s which resemble t h e i r " t r u e " v a l u e s t o a degree d e t e r m i n e d by the q u a l i t y of the d a t a ; i n B a c k u s - G i l b e r t i n v e r s i o n , on the o t h e r hand, any model or s e t of parameters i s r e g a r d e d as a m a t h e m a t i c a l a b s t r a c t i o n which may or may not bear any d i r e c t r e l a t i o n t o the a c t u a l s t r u c t u r e of the ground. The s t a n d p o i n t of Glenn e t a l . i s v a l i d o n l y i f two r a t h e r s t r i n g e n t c o n d i t i o n s a r e met: ( i ) the r e a l , e a r t h must ( t o a v e r y good a p p r o x i m a t i o n ) be c o m p r i s e d of the s p e c i f i e d number of l a y e r s ; ( i i ) t h e r e must be o n l y one s t r u c t u r e w i t h t h a t number 5 of l a y e r s which s a t i s f i e s the d a t a . In c o n t r a s t , HLEM.INV has been w r i t t e n on the presumption t h a t n o t h i n g i s known about the c o n d u c t i v i t y 'a p r i o r i ' ( e x c e pt t h a t i t may be m e a n i n g f u l l y r e p r e s e n t e d as a f u n c t i o n of depth o n l y ) , and t h a t the d a t a c o n s t i t u t e the o n l y source of e n l i g h t e n m e n t . On t h i s b a s i s , one set of c o n d u c t i v i t i e s and t h i c k n e s s e s which reproduces the o b s e r v a t i o n s i s c o n s i d e r e d no more i n d i c a t i v e of the t r u e v a r i a t i o n of c o n d u c t i v i t y w i t h depth than any o t h e r s e t which p r o v i d e s an e q u a l l y c l o s e f i t t o the d a t a . T h e r e f o r e , i n kee p i n g w i t h the B a c k u s - G i l b e r t a pproach, HLEM.INV i n c l u d e s b oth c o n s t r u c t i o n and a p p r a i s a l s u b r o u t i n e s . Another i m p o r t a n t d i f f e r e n c e between HLEM.INV and the Glenn et a l . a l g o r i t h m i s t h a t the former has been d e s i g n e d from the o u t s e t t o i n v e r t n o i s y d a t a , and so o f f e r s a complete t r e a t m e n t of e r r o r s ( i n c l u d i n g c o r r e l a t e d e r r o r s ) , whereas Glenn e t a l . d e f e r a comprehensive e r r o r a n a l y s i s . In t h i s r e s p e c t a l s o the i n f l u e n c e of Backus and G i l b e r t i s a p p r e c i a b l e . I t has been e s t a b l i s h e d t h a t r e c o g n i t i o n of the importance of a p p r a i s a l i s the p r i n c i p a l d i s t i n c t i o n between HLEM.INV and p r e - e x i s t i n g a i d s f o r i n v e r s i o n of HLEM so u n d i n g s . C o n s e q u e n t l y the t o p i c of a p p r a i s a l bears f u r t h e r a m p l i f i c a t i o n , e s p e c i a l l y s i n c e i t i s o n l y i n the c o n t e x t of a p p r a i s a l t h a t the s i g n i f i c a n c e of the uniqueness theorem i n Appendix C can be a p p r e c i a t e d . B r o a d l y s p e a k i n g , the aim of a p p r a i s a l i s t o come t o terms w i t h the non-uniqueness of the i n v e r s e problem. Non-uniqueness a r i s e s i n HLEM i n v e r s i o n because a l i m i t e d number of i n a c c u r a t e measurements of magnetic f i e l d on the s u r f a c e do not s u f f i c e t o c o m p l e t e l y s p e c i f y the c o n d u c t i v i t y , even when i t b i s assumed t o be a f u n c t i o n of depth o n l y . N e v e r t h e l e s s , the data do c o n s t r a i n the c o n d u c t i v i t y t o some e x t e n t , and the e s s e n t i a l problem i n a p p r a i s a l i s t o summarise those c o n s t r a i n t s i n p h y s i c a l l y - m e a n i n g f u l terms. To t h i s end, Backus and G i l b e r t (1970) show t h a t i t i s p o s s i b l e t o det e r m i n e "averages" of an a c c e p t a b l e model, and t h a t these averages a re common t o a l l a c c e p t a b l e models which c l o s e l y resemble the known model. However, the t o t a l i t y of a c c e p t a b l e models may i n c l u d e g r e a t l y d i f f e r e n t s t r u c t u r e s ; o n l y i f i t can be demonstrated t h a t a l l p o s s i b l e a c c e p t a b l e models g i v e r i s e t o the same averages can the averages o b t a i n e d from a p a r t i c u l a r a c c e p t a b l e model be c o n s i d e r e d c h a r a c t e r i s t i c of the t r u e s t r u c t u r e . T h e r e f o r e i t i s not w o r t h w h i l e t o compute B a c k u s - G i l b e r t averages u n l e s s t h e r e i s a r e a s o n a b l e l i k e l i h o o d t h a t a l l a c c e p t a b l e models w i l l i n f a c t e x h i b i t s i m i l a r i t i e s , a t l e a s t i n c e r t a i n depth ranges. For i n v e r s i o n of HLEM fr e q u e n c y soundings over a s t r a t i f i e d e a r t h the l i k e l i h o o d of t h i s e v e n t u a l i t y i s g r e a t l y i n c r e a s e d by the e s t a b l i s h m e n t of the uniqueness of the HLEM i n v e r s e problem i n Appendix C. I t i s proved t h e r e t h a t an u n l i m i t e d q u a n t i t y of ( a c c u r a t e ) HLEM freq u e n c y soundings s u f f i c e s t o c o m p l e t e l y s p e c i f y the c o n d u c t i v i t y as a f u n c t i o n of depth; t h i s i s thought t o be the f i r s t such p r o o f f o r HLEM fr e q u e n c y s o u n d i n g s . The theorem enhances the v i a b i l i t y of B a c k u s - G i l b e r t a v e r a g i n g by r e n d e r i n g i t more l i k e l y t h a t a l l a c c e p t a b l e models w i l l resemble one a n o t h e r . The remainder of the t h e s i s i s c o m p r i s e d of f o u r c h a p t e r s , a c o n c l u s i o n , and f o u r a p p e n d i c e s . The c o n t e n t s of each of these components w i l l be d e s c r i b e d b r i e f l y below. The l o g i c a l / development of the t h e s i s p a r a l l e l s the o v e r a l l approach t o the i n v e r s e problem, as w i t n e s s e d by the t i t l e s of the f i r s t t h r e e c h a p t e r s : "The Forward problem", "Model C o n s t r u c t i o n " , and " A p p r a i s a l " . The uniqueness proof i s p r e s e n t e d i n Appendix C i n o r d e r t o p r e s e r v e the l o g i c a l c o n t i n u i t y . The s e p a r a t i o n of the p r o o f from the main body of the t h e s i s s h o u l d not be c o n s t r u e d t o s i g n i f y t h a t the theorem i s of secondary i m p o r t a n c e . The f i r s t c h a p t e r concerns the s o l u t i o n of the f o r w a r d problem, i . e . the development of the c a p a c i t y t o compute the responses of any l a y e r e d c o n d u c t i v i t y f o r a g i v e n l o o p r a d i u s , s ource f r e q u e n c y , and T-R s e p a r a t i o n . The f o r m u l a t i o n of the problem g i v e n i n 1-1 (the f i r s t s e c t i o n ) i s i d e n t i c a l t o t h a t of Ryu et a l . (1970), but the s o l u t i o n i s s i m p l i f i e d t h r ough the use of a Green's f u n c t i o n . A n a l y t i c e x p r e s s i o n s f o r the source f i e l d s i n a i r are d e r i v e d i n 1-2 and Appendix B; t o the best of t h i s a u t h o r ' s knowledge, the i n d u c i n g f i e l d s have not p r e v i o u s l y been w r i t t e n i n c l o s e d form. In 1-3 the well-known r e c u r s i o n f o r m u l a f o r the i n p u t impedance of any l a y e r e d c o n d u c t i v i t y i s d e r i v e d ; the importance of the i n p u t impedance d e r i v e s from the f a c t t h a t i t appears i n formulae from which the o b s e r v a b l e magnetic f i e l d s a re computed by Hankel t r a n s f o r m a t i o n . The need f o r a scheme t o p e r f o r m e f f i c i e n t n u m e r i c a l e v a l u a t i o n of the Hankel t r a n s f o r m i n t e g r a l s poses a n o n - t r i v i a l problem i n n u m e r i c a l a n a l y s i s ; e f f i c i e n c y i s e s s e n t i a l because a l a r g e number of such i n t e g r a l s must be computed d u r i n g each i n v e r s i o n run . The c o m p u t a t i o n a l p r o c e d u r e employed i n HLEM.INV i s o u t l i n e d i n 1-7. As per Ryu et a l . (1970, p864), the f i r s t s t e p i s the d e c o m p o s i t i o n of each i n t e g r a l i n t o two i n t e g r a l s 8 r e p r e s e n t i n g the p r i m a r y and secondary components of the magnetic f i e l d . The p r i m a r y i n t e g r a l s can be e v a l u a t e d i m m e d i a t e l y u s i n g the a n a l y t i c f o r m u l a e a l r e a d y mentioned, but the secondary f i e l d i n t e g r a l s must be computed n u m e r i c a l l y . In HLEM.INV the d i g i t a l l i n e a r f i l t e r i n g of Koefoed et a l . (1972) has been a p p l i e d t o the s e i n t e g r a l s . The d i g i t a l f i l t e r i n g i s much f a s t e r (and t h e r e f o r e cheaper) than c o n v e n t i o n a l q u a d r a t u r e r o u t i n e s ; moreover, comparison w i t h the Romberg a l g o r i t h m i n 1-7 i n d i c a t e s t h a t the l i n e a r f i l t e r i n g i s s u f f i c i e n t l y a c c u r a t e f o r g e o p h y s i c a l p u r p o s e s . The* t o p i c s p r e s e n t e d i n s e c t i o n s 1-3, 1-5, and 1-6 a r e not e s s e n t i a l t o an u n d e r s t a n d i n g of the f o r w a r d problem s u b r o u t i n e s i n HLEM.INV, but are i n c l u d e d t o p r o v i d e i n s i g h t i n t o the HLEM problem and t o s e r v e as r e f e r e n c e m a t e r i a l i n l a t e r s e c t i o n s of the t h e s i s . For example, the i n d u c t i o n response f u n c t i o n d e f i n e d i n 1-6 f e a t u r e s p r o m i n e n t l y i n the uniqueness pr o o f i n Appendix C. Chapter I I i s an e x p o s i t i o n of the c o n s t r u c t i o n p r o c e d u r e employed i n HLEM.INV. The pr o c e d u r e adopted i n HLEM.INV i s b a s i c a l l y the same as t h a t used by Glenn et a l . (1973), t o the e x t e n t t h a t c o n s t r u c t i o n i s l i n e a r i s e d and proceeds by i t e r a t i v e improvement of a s t a r t i n g model. D u r i n g each i t e r a t i o n the aim i s t o compute s m a l l changes i n the c o n d u c t i v i t y which when added to the s t a r t i n g model w i l l produce a new model f o r which the t h e o r e t i c a l responses bear a c l o s e r resemblance t o the o b s e r v a t i o n s . The d e s i r e d c o n d u c t i v i t y p e r t u r b a t i o n s c o n s t i t u t e the unknowns i n a system of l i n e a r e q u a t i o n s . The c o e f f i c i e n t m a t r i x i s a s e t of d e r i v a t i v e s of i n d i v i d u a l o b s e r v a t i o n s w i t h r e s p e c t t o i n d i v i d u a l parameters ( c o n d u c t i v i t i e s ) . These 9 d e r i v a t i v e s a r e s p e c i f i c t o a p a r t i c u l a r model, and must c o n s e q u e n t l y be re-computed f o r each i t e r a t i o n . I t i s i n t h i s sense t h a t the problem i s n o n - l i n e a r ; were the d e r i v a t i v e s model-independent, an a c c e p t a b l e model c o u l d be o b t a i n e d v i a the s o l u t i o n of a s i n g l e s e t of l i n e a r e q u a t i o n s . Owing t o the presence of e r r o r s i n the d a t a , the p e r t u r b a t i o n e q u a t i o n s must be s o l v e d w i t h some c a r e . The most s e r i o u s e f f e c t s of n o i s e can be a v o i d e d by never demanding an exact f i t t o the r e s i d u a l s . The e q u a t i o n s are s o l v e d t o an a c c u r a c y c o n s i s t e n t w i t h the u n c e r t a i n t i e s i n the d a t a u s i n g the s p e c t r a l e x p a n s i o n t e c h n i q u e d e s c r i b e d by P a r k e r (1977a), by means of which i t i s p o s s i b l e t o e x p r e s s the e x a c t s o l u t i o n as a l i n e a r c o m b i n a t i o n of o r t h o n o r m a l e i g e n v e c t o r s . Lanczos (1961) has shown t h a t the most p o o r l y d e t e r m i n e d components of the e i g e n v e c t o r e x p a n s i o n a r e those a s s o c i a t e d w i t h the s m a l l e s t e i g e n v a l u e s . T h e r e f o r e , the e i g e n v e c t o r s c o r r e s p o n d i n g t o the s m a l l e s t e i g e n v a l u e s can be d i s c a r d e d u n t i l the d e s i r e d r e s i d u a l s a r e reproduced t o a p r e c i s i o n c o n s i s t e n t w i t h t h e i r u n c e r t a i n t i e s . Thus the g r e a t advantage of the s p e c t r a l d e c o m p o s i t i o n method l i e s i n the f a c t t h a t i t p e r m i t s i d e n t i f i c a t i o n (and hence removal) of the l e a s t r e l i a b l e components of the s o l u t i o n . Chapter I I i s d i v i d e d i n t o f i v e s e c t i o n s . In the f i r s t the F r e c h e t k e r n e l s f o r the HLEM problem a r e d e r i v e d from the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n , and the l i n e a r i s e d p e r t u r b a t i o n problem i s f o r m u l a t e d . The F r e c h e t k e r n e l s a r e f u n c t i o n s which r e l a t e a s m a l l change i n c o n d u c t i v i t y t o the r e s u l t i n g change i n an o b s e r v a t i o n v i a an i n n e r p r o d u c t , and as such are fundamental 1 U t o t h e l i n e a r i s a t i o n . The k e r n e l s a r e s p e c i f i c t o a p a r t i c u l a r c o n d u c t i v i t y , and t h e r e i s a d i f f e r e n t k e r n e l f o r e a c h o b s e r v a t i o n . The n u m e r i c a l c o m p u t a t i o n o f t h e k e r n e l s i s d i s c u s s e d i n t h e s e c o n d s e c t i o n , w i t h some d e t a i l s r e l e g a t e d t o A p p e n d i x B. The t h i r d s e c t i o n i s an o u t l i n e o f t h e s p e c t r a l d e c o m p o s i t i o n method f o r t h e s o l u t i o n o f t h e p e r t u r b a t i o n e q u a t i o n s i n t h e p r e s e n c e o f n o i s e . I n t h e f i f t h s e c t i o n t h e c o n s t r u c t i o n a l g o r i t h m i s t e s t e d on s i x t e e n d i f f e r e n t t h e o r e t i c a l e x a m p l e s , f o r w h i c h t h e d a t a were c o n t a m i n a t e d w i t h a p p r o x i m a t e l y 2% n o i s e . The p r o g r a m p r o d u c e d an a c c e p t a b l e m o d e l i n a r e l a t i v e l y s m a l l number of i t e r a t i o n s i n e a c h c a s e . I n I I - 4 , f o u r s p e c i a l f e a t u r e s o f HLEM.INV a r e d e s c r i b e d : ( i ) d a t a o p t i o n s ; ( i i ) m o d i f i c a t i o n o f t h e d e p t h p a r t i t i o n ; ( i i i ) e x p l i c i t t r e a t m e n t o f e r r o r s ; a n d ( i v ) c o m p e n s a t i o n f o r c o i l m i s a l i g n m e n t e r r o r . The f i r s t o f t h e s e s u b - s e c t i o n s s i m p l y d o c u m e n t s t h e s i x t e e n d i f f e r e n t f o r m s o f i n p u t d a t a w h i c h c a n be i n v e r t e d by t h e p r o g r a m , e.g. a m p l i t u d e o f r a d i a l m a g n e t i c f i e l d , o r r e a l and i m a g i n a r y p a r t s o f v e r t i c a l m a g n e t i c f i e l d . The s e c o n d c o n c e r n s t h e manner i n w h i c h t h e number o f l a y e r s i s i n c r e a s e d , f r o m a r e l a t i v e l y s m a l l number i n i t i a l l y t o a p r e s c r i b e d maximum, a s c o n s t r u c t i o n p r o c e e d s . The p r i n c i p a l r e a s o n f o r t h e i n c l u s i o n o f t h i s c a p a b i l i t y i s economy, b e c a u s e b e g i n n i n g w i t h a l a r g e number o f l a y e r s a d d s t o t h e c o s t w i t h o u t i m p r o v i n g t h e r a t e o f c o n v e r g e n c e ; i n some c a s e s e x t r a l a y e r s a c t u a l l y r e t a r d c o n v e r g e n c e . The t h i r d s p e c i a l f e a t u r e i s a s u b r o u t i n e t o s o l v e m o d i f i e d p e r t u r b a t i o n e q u a t i o n s i n w h i c h t h e e r r o r s a r e i n c l u d e d e x p l i c i t l y . When so f o r m u l a t e d , t h e p r o b l e m i s s t a b i l i s e d i n much t h e same way a s i t w o u l d be by t h e method 11 of r i d g e r e g r e s s i o n , but w i t h o u t the i n t r o d u c t i o n of a d d i t i o n a l unknown parameters such as "k" i n e q u a t i o n (10) of Inman (1975). The f o u r t h n o v e l f e a t u r e of HLEM.INV i s a s i m p l i f i e d scheme which goes some way towards compensating f o r the d e l e t e r i o u s e f f e c t s of m i s a l i g n m e n t of the r e c e i v e r and t r a n s m i t t e r ( S i n h a , 1980). The mi s a l i g n m e n t a n g l e i s m o d e l l e d as the r e a l i s a t i o n of a Normal random v a r i a b l e w i t h s m a l l s t a n d a r d d e v i a t i o n , and the r e s u l t i n g i n c r e a s e i n v a r i a n c e f o r each k i n d of datum i s computed u s i n g formulae d e r i v e d i n Appendix D. When m i s a l i g n m e n t i s t r e a t e d i n t h i s way, the e r r o r s i n the da t a a r e no l o n g e r independent. However, the system of e q u a t i o n s may be t r a n s f o r m e d i n t o an e q u i v a l e n t system w i t h independent e r r o r s by e x p l o i t i n g the s p e c t r a l d e c o m p o s i t i o n of the c o v a r i a n c e m a t r i x ( G i l b e r t , 1971) . The purpose of the t h i r d c h a p t e r i s t o d e s c r i b e and demonstrate the a p p r a i s a l r o u t i n e s i n c o r p o r a t e d i n t o HLEM.INV. As mentioned p r e v i o u s l y , a p p r a i s a l i s an e s s e n t i a l phase i n any comprehensive i n v e r s i o n procedure because t h e r e a r e u s u a l l y an i n f i n i t e number of models which s a t i s f y a s e t of o b s e r v a t i o n s e q u a l l y w e l l . The aim of a p p r a i s a l i s t h e r e f o r e t o i d e n t i f y the f e a t u r e s which a re common t o a l l such a c c e p t a b l e models. B a c k u s - G i l b e r t a v e r a g i n g , d e s c r i b e d i n 111-1, i s the mains t a y of the HLEM.INV a p p r a i s a l scheme. Backus and G i l b e r t (1968) d e f i n e averages of the, model as i n n e r p r o d u c t s of the model w i t h a v e r a g i n g f u n c t i o n s . The a v e r a g i n g f u n c t i o n s may be v i s u a l i s e d as the "windows" through which the da t a enable the e a r t h t o be viewed. These f u n c t i o n s a r e d e f i n e d as l i n e a r c o m b i n a t i o n s of F r e c h e t k e r n e l s , w i t h the r e s u l t ( i n l i n e a r problems) t h a t the 12 averages themselves a r e l i n e a r c o m b i n a t i o n s of the d a t a . C o n s e q u e n t l y , any a c c e p t a b l e model w i l l g i v e r i s e t o the same a v e r a g e s , and i n t h i s sense the averages are " u n i v e r s a l " i n a l i n e a r i n v e r s e problem. S i n c e the d a t a are c o n t a m i n a t e d w i t h e r r o r s i n any r e a l i n v e r s i o n , the averages a r e n e c e s s a r i l y i n a c c u r a t e . Backus and G i l b e r t show t h a t the l e a s t a c c u r a t e averages at any depth are a s s o c i a t e d w i t h the most d e l t a - l i k e a v e r a g i n g f u n c t i o n s which can be c o n s t r u c t e d a t t h a t depth. Put more s u c c i n c t l y , they demonstrate t h a t r e s o l u t i o n and a c c u r a c y a r e m u t u a l l y a n t a g o n i s t i c a t t r i b u t e s of an a v e r a g e . Thus i t i s o f t e n d e s i r a b l e t o s a c r i f i c e some r e s o l u t i o n , i . e . t o a c c e p t an a v e r a g i n g f u n c t i o n w i t h a s l i g h t l y broader peak, i n o r d e r t o o b t a i n an average w i t h an a c c e p t a b l y s m a l l v a r i a n c e . Backus and G i l b e r t i n t r o d u c e a parameter t o c o n t r o l the " t r a d e - o f f " between r e s o l u t i o n and a c c u r a c y , thus p e r m i t t i n g g r e a t f l e x i b i l i t y i n the c h o i c e of a v e r a g i n g f u n c t i o n t o s u i t . a p a r t i c u l a r a p p l i c a t i o n . T h i s i s i n c o n t r a s t t o the use of the r e s o l u t i o n m a t r i x f o r a p p r a i s a l , a d v o c a t e d by the proponents of c o n v e n t i o n a l g e n e r a l i s e d l i n e a r i n v e r s i o n , e.g. Wiggins (1972), because (as shown i n 111-2) the rows of the r e s o l u t i o n m a t r i x c o r r e s p o n d t o the most d e l t a - l i k e a v e r a g i n g f u n c t i o n s ; the r e s u l t i n g parameter e s t i m a t e s (averages) a r e t h e r e f o r e n e c e s s a r i l y the l e a s t r e l i a b l e . The a d a p t a t i o n of B a c k u s - G i l b e r t a v e r a g i n g f o r a n o n - l i n e a r i n v e r s e problem, such as the HLEM problem, i s d e s c r i b e d i n 111 — 3. I t i s shown t h e r e t h a t i n the n o n - l i n e a r case the B a c k u s - G i l b e r t averages a r e u n i v e r s a l o n l y i f a l l a c c e p t a b l e 13 models a r e s u f f i c i e n t l y s i m i l a r t o one a n o t h e r . As remarked e a r l i e r and a m p l i f i e d i n 111-4, the uniqueness theorem i n c r e a s e s the l i k e l i h o o d t h a t the averages f o r a p a r t i c u l a r a c c e p t a b l e model w i l l p e r t a i n t o the t r u e s t r u c t u r e , but i n the f i n a l a n a l y s i s the o n l y way t o a s s e s s u n i v e r s a l i t y of averages i s by d i r e c t comparison of averages a s s o c i a t e d w i t h d i f f e r e n t a c c e p t a b l e models. In HLEM.INV t h e r e are s u b r o u t i n e s which per f o r m such a comparison f o r a l i m i t e d number of a c c e p t a b l e models. In p r i n c i p l e , the assessment of u n i v e r s a l i t y i s i d e n t i c a l t o t h a t conducted by Oldenburg (1979), but the c h o i c e of models i n HLEM.INV i s s i g n i f i c a n t because an e f f o r t has been made t o f i n d models which a r e q u a l i t a t i v e l y v e r y d i f f e r e n t from one a n o t h e r i n o r d e r t o o b t a i n as r e p r e s e n t a t i v e a sample as p o s s i b l e of the t o t a l i t y of a c c e p t a b l e models. E x p l i c i t l y , s e v e r a l a c c e p t a b l e models a r e g e n e r a t e d by m a x i m i s i n g and m i n i m i s i n g the c o n d u c t i v i t i e s of each of the l a y e r s i n t u r n , s u b j e c t always t o the d a t a c o n s t r a i n t s . T h i s i s i l l u s t r a t e d i n I I I — 5 ( i ) where e x t r e m a l models a r e computed from some of the models c o n s t r u c t e d i n I I - 5 . The com p u t a t i o n of the e x t r e m a l models proceeds i t e r a t i v e l y , u s i n g a l i n e a r programming f o r m u l a t i o n s i m i l a r t o t h a t proposed by Kennett and N o l e t (1978). Once the e x t r e m a l models have been d e t e r m i n e d , t h e i r a verages w i t h r e s p e c t t o the same a v e r a g i n g f u n c t i o n s a re computed. In I I I - 5 ( i i ) , a v erages c o r r e s p o n d i n g t o d i f f e r e n t a c c e p t a b l e models are d e t e r m i n e d i n t h i s way f o r some of the t h e o r e t i c a l examples i n i " I - 5 . Only when an average i s reproduced by a l l the a v a i l a b l e a c c e p t a b l e models i s i t t e n t a t i v e l y r e g a r d e d as c h a r a c t e r i s t i c of the t r u e s t r u c t u r e . In p r a c t i c e 14 good correspondence between averages has been observed i n zones of r e l a t i v e l y h i g h c o n d u c t i v i t y , c o n s i s t e n t w i t h the f a c t t h a t HLEM soundings are s e n s i t i v e t o c o n c e n t r a t i o n s of c u r r e n t i n the ground. C o n v e r s e l y , the averages tend t o be u n r e l i a b l e i n r e s i s t i v e zones. The HLEM.INV a p p r a i s a l scheme c o n s t i t u t e s an improvement on the c o n v e n t i o n a l B a c k u s - G i l b e r t approach o n l y i n the sense t h a t i t i s b e t t e r adapted t o expose the l i m i t a t i o n s of a v e r a g i n g , and hence t o a v o i d i t s p i t f a l l s . HLEM.INV a p p r a i s a l i s thus more c o n s e r v a t i v e than a s t r a i g h t f o r w a r d a p p l i c a t i o n of B a c k u s - G i l b e r t a v e r a g i n g t o a s i n g l e a c c e p t a b l e model. N e v e r t h e l e s s , the s i m i l a r i t y between a l l known a c c e p t a b l e models ( n o t w i t h s t a n d i n g a c o n c e r t e d e f f o r t t o c o n s t r u c t g l o b a l l y d i s t i n c t models) does not a l t e r the f a c t t h a t i t i s i m p o s s i b l e t o prove t h a t a l l a c c e p t a b l e models resemble one a n o t h e r : i n a b i l i t y t o f i n d d i s t i n c t models can always be d i s m i s s e d as symptomatic of d e f i c i e n c i e s i n the a v a i l a b l e c o n s t r u c t i o n t e c h n i q u e s . Hence the fundamental problem i n n o n - l i n e a r i n v e r s i o n remains s t u b b o r n l y u n r e s o l v e d . In the f o u r t h c h a p t e r HLEM.INV i s a p p l i e d t o r e a l d a t a from Grass V a l l e y , Nevada. C o i l m i s a l i g n m e n t i s taken i n t o account and the a l g o r i t h m performs q u i t e s a t i s f a c t o r i l y : an a c c e p t a b l e model i s c o n s t r u c t e d i n ten i t e r a t i o n s from a u n i f o r m h a l f - s p a c e s t a r t i n g model. A f u l l HLEM.INV a p p r a i s a l r e v e a l s ' the presence of a r a p i d i n c r e a s e i n average c o n d u c t i v i t y , from a p p r o x i m a t e l y 0.1 S/m a t the s u r f a c e t o over 0.4 S/m near 800m. In view of the success of HLEM.INV i n i n v e r t i n g b o th t h e o r e t i c a l and r e a l d a t a , i t i s c o n c l u d e d t h a t HLEM.INV i s a l b p o t e n t i a l l y v a l u a b l e a i d i n the i n t e r p r e t a t i o n of HLEM fr e q u e n c y soundings over a s t r a t i f i e d e a r t h . The program can be adapted f o r i n v e r s i o n of time-domain HLEM soundings and, a f t e r somewhat more s u b s t a n t i a l m o d i f i c a t i o n , f o r i n v e r s i o n of HLEM geo m e t r i c s o u n d i n g s . F i n a l l y , the c r e d i b i l i t y of c o n c l u s i o n s based on i n v e r s i o n of HLEM soundings i s enhanced by the d e m o n s t r a t i o n here t h a t the r e l e v a n t m a t h e m a t i c a l i n v e r s e problem has a unique s o l u t i o n . l b C h a p t e r I THE FORWARD PROBLEM An e s s e n t i a l p r e - r e q u i s i t e f o r any i n v e r s i o n scheme i s 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 " , i . e . t o d e t e r m i n e t h e r e s p o n s e of a p a r t i c u l a r m o d e l . The f o r w a r d p r o b l e m f o r HLEM s o u n d i n g s o v e r a s t r a t i f i e d e a r t h i s f o r m u l a t e d i n t h e f i r s t s e c t i o n o f t h i s c h a p t e r , and i s examined from d i f f e r e n t p e r s p e c t i v e s i n s u b s e q u e n t s e c t i o n s . The f o r w a r d p r o b l e m must be s o l v e d r e p e a t e d l y i n an i t e r a t i v e p rogram l i k e HLEM.INV, so c o m p u t a t i o n a l e f f i c i e n c y has been an i m p o r t a n t c o n s i d e r a t i o n i n th e d e v e l o p m e n t of t h e a l g o r i t h m . In p a r t i c u l a r , t h e need f o r e f f i c i e n c y m o t i v a t e d t h e d e c i s i o n t o r e s t r i c t a t t e n t i o n t o l a y e r e d m o d e l s , t h e s e f a c i l i t a t i n g c o m p u t a t i o n t h r o u g h t h e ag e n c y o f a r e c u r s i o n r e l a t i o n , a s d e s c r i b e d i n 1-4 below. E x p l i c i t d i s c u s s i o n of c o m p u t a t i o n a l d e t a i l s i s d e f e r r e d u n t i l t h e f i n a l s e c t i o n , 1-7. D i f f e r e n t a s p e c t s of t h e p r o b l e m a r e examined i n 1-2,3,5,6. The s e c o n d s e c t i o n p e r t a i n s t o t h e s o u r c e f i e l d g e n e r a t e d by a c u r r e n t l o o p , w h i l e an i n t e g r a l e q u a t i o n f o r t h e e l e c t r i c f i e l d i s d e r i v e d i n t h e t h i r d . C e r t a i n a s p e c t s of t h e i n t e r a c t i o n o f t h e e l e c t r o m a g n e t i c f i e l d s w i t h c o n d u c t i v e s t r a t a a r e e l u c i d a t e d i n 1-5, and an i n d u c t i o n r e s p o n s e f u n c t i o n f o r a l a y e r e d e a r t h e x c i t e d by a c y l i n d r i c a l l y - s y m m e t r i c s o u r c e f i e l d i s d e f i n e d i n 1-6. A l t h o u g h n o t n e c e s s a r y i n s o f a r as an a p p r e c i a t i o n of HLEM.INV i s c o n c e r n e d , t h e s e s e c t i o n s p r o v i d e a d d i t i o n a l i n s i g h t 17 into the physical basis of the method, and f a c i l i t a t e the l o g i c a l development of the remainder of the thesis. 1-1 Formulation of the Problem: Following Ryu et a l . ( 1 9 7 0 ) , consider the system depicted in F i g . 1 . 1 . An o s c i l l a t i n g current, Ie , flows in a c i r c u l a r loop of radius a, situated height h above a sequence of homogeneous layers. The axis of the loop i s v e r t i c a l , and i s coincident with the z-axis of a c y l i n d r i c a l coordinate system; the positive z-direction is taken to be downwards. Conductivity, cr , and d i e l e c t r i c constant, S , are assumed to be i s o t r o p i c , and permeability is assigned i t s free-space value throughout. The v e r t i c a l and r a d i a l magnetic f i e l d s , Hz and Hr respectively, are measured on the surface (z=0) at a distance R from the axis of the c o i l (r=R). As implied by the harmonic time dependence of the current, a frequency domain formulation has been adopted. The a x i a l symmetry shared by the source and the layered ground guarantees that the azimuthal magnetic f i e l d , H$, i s i d e n t i c a l l y zero, and that E g is the only non-zero e l e c t r i c f i e l d component, i.e. I = ( 0 , E e , 0 ) H = (Hr ,0,Hz) Maxwe11's equations in the frequency domain for the three non-zero components reduce to Te i w t c Z i p h a . R -z 2 z 3 t h, e2,<r2 Z j Zj-1 T z F i g . 1.1: System Geometry - B E iw/*Hr = — (1.1) 3z iwyUHz = - r " 1 — ( r E ) (1.2) ' dr 3Hr d H z _ — - — 3 = (iw£ + cr ) E + I<S(r-a)<5 (z+h) (1.3) d z Br where E S E g . E l i m i n a t i n g Hr and Hz, t h e g o v e r n i n g PDE f o r E(F,z,w) r e v e a l e d t o be V 2 E - E / r 2 + {vyx £(z)-iw /u.C^(z) ]E = i w/* I (w )6 ( r - a )cf( z+h) ( 1 . The d i m e n s i o n l e s s form of (1.4) i s y 2 E - E / r 2 + [ y 2 £ ( z ) - i ^ 2 £ T ( z ) ]E = J ( r - a ) J ( z + h ) (1.5) where E ( r , z , w ) = E( r , z , w)/(w/* I ) (1.6) y2 = R 2 w ^ £ ( 0 ) (1.7) /3 2 = R 2w /ctc5 :(0) (1.8) £ ( z ) = £(z)/£(0) (1.9) Cr(z) = cr(z)/<r(0) (1.10) 20 and where a l l l e n g t h s have been s c a l e d w i t h r e s p e c t t o the t r a n s m i t t e r - r e c e i v e r (T-R) s e p a r a t i o n , R. In terms of these d i m e n s i o n l e s s v a r i a b l e s , (1.1) and (1.2) become &E Hr = - i — (1.11) 3z Hz = i r " x — (rE) " (1.12) dr where H = HR/l . (1.13) The s o l u t i o n of (1.5) can be o b t a i n e d most r e a d i l y by a p p l i c a t i o n of a Hankel t r a n s f o r m . The Hankel t r a n s f o r m ( o r d e r 1) of a f u n c t i o n f ( r ) i s d e f i n e d by oo C { 1 ( f ( r ) } = f ( A ) = £f ( r ) J ( ( A r ) r . d r (1.14) o The c o r r e s p o n d i n g i n v e r s e t r a n s f o r m from f ( A ) t o f ( r ) i s i d e n t i c a l i n form, w i t h the r o l e s of A and r i n t e r c h a n g e d . A p p l i c a t i o n of t h i s t r a n s f o r m t o (1.5) has the e f f e c t of s e p a r a t i n g the r - and z-dependence. The z-dependence of the H a n k e l - t r a n s f o r m e d e l e c t r i c f i e l d , E ( A,z,w), i s c o d i f i e d i n the f o l l o w i n g ODE: E" - [ A 2 - ^ 2£(z) + i/5 2 <r(z)]E = iaJ ( ( A a) 6(z+h) (1.15) where a prime denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o z. The 21 be r e c o v e r e d from (1.16) P h y s i c a l l y , the e l e c t r i c f i e l d , and hence the magnetic f i e l d v i a (1.11) and ( 1 . 1 2 ) , i s seen t o be the s u p e r p o s i t i o n of an i n f i n i t e number of "modes", a s s o c i a t e d w i t h d i f f e r e n t v a l u e s of A . Each mode c o n s i s t s of a p l a n e wave, f o r which the r a d i a l v a r i a t i o n of a m p l i t u d e i s governed by the B e s s e l f u n c t i o n , J(( A r) . In p a r t i c u l a r , the a m p l i t u d e of modes a s s o c i a t e d w i t h v e r y s m a l l v a l u e s of A i s almost u n i f o r m w i t h r e s p e c t t o r , and when A=0 the DE (1.15) i s i d e n t i c a l t o the fundamental e q u a t i o n f o r the m a g n e t o t e l l u r i c (MT) method ( C a g n i a r d , 1953, p612). For a complete f o r m u l a t i o n of the problem, i t remains o n l y t o s p e c i f y the boundary c o n d i t i o n s . From Maxwell's e q u a t i o n s i t f o l l o w s d i r e c t l y ( J a c k s o n , 1975, ppl7-20) t h a t E, Hr, and Hz must be c o n t i n u o u s everywhere i f s u r f a c e c u r r e n t s a r e d i s a l l o w e d . Thus i f E + ( r ) and E ~ ( r ) denote the e l e c t r i c f i e l d s on e i t h e r s i d e of an i n t e r f a c e , then E + ( r ) = E " ( r ) Vr whence, a f t e r Hankel t r a n s f o r m a t i o n , E + ( A ) = E " ( A ) (1.17) measurable e l e c t r i c f i e l d , E ( r , z , w ) , may E ( A f Z»w) by i n v e r s e Hankel t r a n s f o r m a t i o n : oo E(r,z,w) = j*E( A ,z,w) A J,( A r ) d A o The c o n t i n u i t y c o n d i t i o n f o r Hr t r a n s f e r s i n t o the Hankel 22 domain i n l i k e f a s h i o n . C o n t i n u i t y of Hz i s a s s u r e d by c o n t i n u i t y of E, v i a ( 1 . 1 2 ) . •1-2 S p e c i f i c a t i o n of the P r i m a r y or Source F i e l d : In a i r , (1.15) i s an inhomogeneous l i n e a r ODE w i t h c o n s t a n t coef f i c i e n t s : w i t h £ c d e n o t i n g the f r e e - s p a c e p e r m i t t i v i t y , and the c o n d u c t i v i t y of a i r . Ryu e t a l . (1970) s o l v e f o r the e l e c t r i c f i e l d i n a i r u s i n g F o u r i e r t r a n s f o r m s . The s o l u t i o n may be o b t a i n e d more r e a d i l y , however, by e x p l o i t i n g t he f a c t t h a t (1.18) i s the d e f i n i n g e q u a t i o n f o r a Green's f u n c t i o n . P r o c e e d i n g i n the manner d e s c r i b e d by Friedman (1956, p p l 6 2 - 1 6 3 ) , the zone above the ground (z < 0) i s d i v i d e d i n t o two i n t e r v a l s , z > -h and z < -h, i n both of which the e l e c t r i c f i e l d must s a t i s f y the homogeneous form of ( 1 . 1 8 ) . Thus, E" - u^E = i a J ( A a)<f(z+h) (1.18) where u I = A 2 " T£o + i / * 2 ^ (1.19) E ( A , z , w ) = k^A.e"' 2 + B0e~u'z] k< [ C0 e -U0Z 0 > z > -h z < -h (1.20) where A0 , B0 , Cc , Dc , k >, and k < are unknown c o n s t a n t s . The 16 f i e l d must decay t o z e r o w i t h d i s t a n c e from the c o i l ; t h e r e f o r e Dff=0, and w i t h o u t l o s s of g e n e r a l i t y the c o n s t a n t Z0 may be a s s i g n e d the v a l u e one. In or d e r t o ensure the c o n t i n u i t y of E at z=-h, the f o l l o w i n g c h o i c e of k > and k< i s a p p r o p r i a t e : k^ . = e 5 + B - e kK = A 0 e  B,  S u b s t i t u t i o n of the p r e s c r i b e d v a l u e s f o r C0 , D0 , k>, and k^ i n t o (1.20) y i e l d s E(A,z,w) = e _ U' h[A„e"' 2 + B„ e'"'* ] X ( z+h) + .... (1.21) .... eUoZ [h0eU'h+ BeeU'h JX(-z-h) where !)-C(y) i s the H e a v i s i d e s t e p f u n c t i o n . D i f f e r e n t i a t i n g (1.21) t w i c e w i t h r e s p e c t t o z and s u b s t i t u t i n g i n t o ( 1 . 1 8 ) , i t may be shown t h a t E" - u„2E = -2u,B e<f(z+h) (1.22) Comparing (1,22) and ( 1 . 1 8 ) , the v a l u e of B0 may be d e t e r m i n e d : B c = - i a j r ( A a ) / 2 u e (1.23) T h i s r e s u l t i s c o n s i s t e n t w i t h e q u a t i o n (10) of M o r r i s o n e t a l . 24 (1969) . P h y s i c a l l y , |B 0(A,a)| i s the a m p l i t u d e f u n c t i o n f o r the modes emanating from the c o i l . The c u r r e n t l o o p i s seen t o r a d i a t e a t almost a l l v a l u e s of A, as e x p e c t e d f o r a l o c a l i s e d s o u r c e . The dominant modes c o r r e s p o n d t o the A - v a l u e s near the maxima and minima of J(( Aa). The p r i n c i p a l lobe of J(( A a) a t t a i n s i t s maximum when A — 1.8/a = Ap, say, where the mode c o r r e s p o n d i n g t o Ap c o u l d be regard e d as the " p r i n c i p a l mode". I f the l o o p r a d i u s were i n c r e a s e d , Ap would d e c r e a s e and the p r i n c i p a l mode would bear a c l o s e r resemblance t o a u n i f o r m p l a n e wave, i n keeping w i t h p h y s i c a l i n t u i t i o n . I t may be observed t h a t t h e r e i s an i n f i n i t e number of modes which a r e not gene r a t e d by a p a r t i c u l a r l o o p , c o r r e s p o n d i n g t o the z e r o s of J(( A a) . The c o n s t a n t , A 0, i n (1.21) r e p r e s e n t s the a m p l i t u d e of the secondary ("upgoing") e l e c t r i c f i e l d g e n e r a t e d by i n t e r a c t i o n of the source f i e l d w i t h the s t r a t i f i e d ground. The d e t e r m i n a t i o n of A0 i s the t o p i c of s e c t i o n 1-4. For the p r e s e n t , suppose t h a t t h e r e i s o n l y a i r beneath the c o i l . Then t h e r e a r e no induced c u r r e n t s (ho=0), and (1.21) reduces t o the d e f i n i n g r e l a t i o n f o r the source f i e l d , E0 : Eo(\,z,w) = -iaJ(( A a ) e x p [ - u j z+h| ]/2uc (1.24) The presence of u0 i n the exponents i n (1.24) i n d i c a t e s v i a (1.19) t h a t A c o n t r o l s g e o m e t r i c a l a t t e n u a t i o n , the large-A modes d e c a y i n g v e r y r a p i d l y w i t h d i s t a n c e from the p l a n e of the c o i l , z=-h. The measurable source f i e l d may be d e t e r m i n e d by i n v e r s e Hankel t r a n s f o r m a t i o n of ( 1 . 2 4 ) . Hence, r e c a l l i n g ( 1 . 1 6 ) , oo EJ[r,z,w) = - i a ^ e x p [ - u j z + h | ]J (( Aa) A Jf( Ar)dA/2u«, (1.25) An a n a l y t i c e x p r e s s i o n e x i s t s f o r t h i s i n t e g r a l i f u0 i s r e p l a c e d by A. T h i s a p p r o x i m a t i o n c o r r e s p o n d s p h y s i c a l l y t o the n e g l e c t of d i s p l a c e m e n t c u r r e n t s i n a i r , and i s p e r f e c t l y a c c e p t a b l e i n most c a s e s of g e o p h y s i c a l r e l e v a n c e because the term "tf2&0 i n (1.19) becomes a p p r e c i a b l e o n l y a t v e r y h i g h f r e q u e n c i e s , e.g. f o r R=30m, %2 So = 1 when w <a 1 0 7 . M a t h e m a t i c a l l y , the v a l u e of u i s dominated by y2S0 as A — ^ 0 ; however, the c o n t r i b u t i o n t o the i n t e g r a l (1.24) f o r such s m a l l v a l u e s of A i s c o m p l e t e l y n e g l i g i b l e , e s p e c i a l l y i n view of the f a c t t h a t J(( A a) —>0 as A—> 0. T h e r e f o r e , t o an e x c e l l e n t a p p r o x i m a t i o n , (1.25) may be e v a l u a t e d as a s p e c i a l case of the f o l l o w i n g f o r m u l a , p r o v i d e d by Watson (1945, p389): OO \e~X*Jn(\ a ) J n ( A r)dA = Qn.i(s)/(7Tv/aT) (1.26) e where s = { X 2 + a2 + r 2 ) / 2 a r (1.27) and where Q ^ x ) denotes the Legendre f u n c t i o n of the second k i n d , o r d e r m. Hence, comparing (1.25) and ( 1 . 2 6 ) , the d i m e n s i o n l e s s p r i m a r y f i e l d may be r e p r e s e n t e d as f o l l o w s E ( r , z ) = -Wa/r Q.(s)/2rr (1.28) 0 4 lb w i t h j[ = \z+h\ i n (1.27). T h i s i s b e l i e v e d t o be the f i r s t time t h a t a c l o s e d form e x p r e s s i o n has been o b t a i n e d f o r the i n d u c i n g f i e l d g e n e r a t e d by a h o r i z o n t a l l o o p c u r r e n t . Legendre f u n c t i o n s of the second k i n d can be e x p r e s s e d i n terms of h y p e r g e o m e t r i c f u n c t i o n s (Gradshteyn and R y z h i k , 1965, pl016) and are t h e r e f o r e amenable t o r a p i d n u m e r i c a l e v a l u a t i o n (see Appendix B ) . Consequently the f o r m u l a (1.28) and i t s d e r i v a t i v e s are of p r a c t i c a l v a l u e , and they a r e used t o f a c i l i t a t e a c c u r a t e e v a l u a t i o n of Hankel t r a n s f o r m i n t e g r a l s , as d e s c r i b e d i n 1-7. 1-3 An I n t e g r a l E q u a t i o n R e p r e s e n t a t i o n f o r the E l e c t r i c F i e l d The Green's f u n c t i o n d e r i v e d above p e r m i t s a s t r a i g h t f o r w a r d d e t e r m i n a t i o n of the i n t e g r a l e q u a t i o n which must be s a t i s f i e d by E(A,z,w). The DE (1.15) may be r e - w r i t t e n i n the form where J ( A,z,w) = { i ^ j - [ 8 ( z ) - 6 , ] + cr (z) }E ( A , z , w) (1.30) denotes the net c u r r e n t ( i n c l u d i n g d i s p l a c e m e n t c u r r e n t ) , and E" - u^E = iy32J + i s J ( z + h ) (1.29) where S( A ,a) = aJ,( A a) (1.31) r e p r e s e n t s the e f f e c t i v e source c u r r e n t a m p l i t u d e a t each A. The geometry of the system i s such t h a t c u r r e n t f l o w i s everywhere r e s t r i c t e d t o h o r i z o n t a l c i r c u l a r paths c e n t r e d on the a x i s of the c o i l . T h e r e f o r e , the secondary e l e c t r i c f i e l d a t any p o i n t may be v i s u a l i s e d as a s u p e r p o s i t i o n of the f i e l d s i nduced by an i n f i n i t e number of c u r r e n t l o o p s . For a c u r r e n t l o o p of u n i t a m p l i t u d e a t depth z=t, i t f o l l o w s from (1.24) t h a t the e l e c t r i c f i e l d decays e x p o n e n t i a l l y w i t h d i s t a n c e a c c o r d i n g t o E^ (A,z,w) = -e x p ( - u j z - t | )/2u # The net e l e c t r i c f i e l d i s composed of the p r i m a r y f i e l d p l u s c o n t r i b u t i o n s from a l l the c u r r e n t elements i n the ground. Hence, oo E(A,z,w) = Etf(A,z,w) + iyj2 ^ J ( A , t , w ) E t ( A ,z,w)dt (1.32) o T h i s e x p r e s s i o n may a l s o be o b t a i n e d i n a p u r e l y f o r m a l manner by m u l t i p l y i n g (1.29) by the Green's f u n c t i o n and i n t e g r a t i n g by p a r t s , as per Friedman (1956, pp238-239). By e x t e n s i o n of the s u p e r p o s i t i o n p r i n c i p l e u n d e r l y i n g the d e r i v a t i o n of ( 1 . 3 2 ) , i t f o l l o w s t h a t the e l e c t r i c f i e l d a t any depth may be m e a n i n g f u l l y decomposed i n t o " i n t e r n a l " and " e x t e r n a l " components, ^ and £, due t o c u r r e n t s r e s p e c t i v e l y below and above the depth i n q u e s t i o n ; t h i s f e a t u r e has been e x p l o i t e d by B a i l e y (1970, p l 8 7 ) i n h i s uniqueness pr o o f f o r the 28 geomagnetic i n d u c t i o n problem. Thus from (1.32) i( A,z,w) £ ( A , z , w ) = Efl( A , z , w ) i y 3 2 j e x p [ - u j z - t ) ]J ( A ,t,w)dt/2u„ (1.34) o At the s u r f a c e the e x t e r n a l component i s j u s t the s o u r c e f i e l d , E0 , w h i l e the i n t e r n a l component c o r r e s p o n d s t o the induced f i e l d . 1-4 S p e c i f i c a t i o n of the Responses f o r a L a y e r e d E a r t h : The net e l e c t r i c f i e l d a t the s u r f a c e i s composed of an up-going wave emanating from the ground and a down-going wave from the s o u r c e . With the a m p l i t u d e of the down-going wave s p e c i f i e d by ( 1 . 2 3 ) , i t remains t o determine the a m p l i t u d e , h0 , of the up-going wave. The f i e l d s i n a l a y e r e d E a r t h may be computed u s i n g a w e l l known r e c u r s i o n r e l a t i o n from t r a n s m i s s i o n l i n e t h e o r y ( S c h e l k u n o f f , 1943, p l 9 7 ) . A d e r i v a t i o n of the r e c u r s i o n r e l a t i o n i s s k e t c h e d i n t h i s s e c t i o n , and some i n c i d e n t a l r e s u l t s a re d e r i v e d f o r l a t e r use. Le t £ j, OJ, and hj denote r e s p e c t i v e l y the d i e l e c t r i c c o n s t a n t , c o n d u c t i v i t y , and t h i c k n e s s of the j t h l a y e r , l o c a t e d between depths Z j and zj+i , as shown i n F i g . 1.1. In t h i s l a y e r , (1.15) may be s i m p l i f i e d t o E" - u?E = 0 Zj ^ z ^ Z j + , (1.35) 2 9 where u.2 = \ 2 ~ %2£j + ty2"] (1.36) The g e n e r a l s o l u t i o n of (1.35) i s a l i n e a r c o m b i n a t i o n of two i n d e p e n d e n t s o l u t i o n s , r e p r e s e n t i n g u p - g o i n g and down-going p l a n e waves: E ( A , z , w ) = Aj exp[ Uj( Z - Z J ) ] + Bj exp[-u.( Z - Z J ) ] (1.37) I n t r o d u c i n g s u b s c r i p t e d q u a n t i t i e s , Ej and Hrj , t o d e n o t e t h e v a l u e s of t h e e l e c t r i c and r a d i a l m a g n e t i c f i e l d s a t t h e j t h i n t e r f a c e , i t f o l l o w s from (1.37) and (1.11) t h a t Ej = E ( A , zj , w ) = Aj + B j (1.38) Hrj = H r ( A , Z j , w ) = -iu.(Aj - Bj ) S i m i l a r l y , a t t h e ( j + l ) t h i n t e r f a c e , Ej*, = hf + Bj* (1.39) Hr j + ( = - i u / A j ^ - Bf) where Aj* = A j e x p ( u j h j ) (1.40) = Bj e x p ( - U j hj ) The r e c u r s i o n r e l a t i o n now t o be d e r i v e d f a c i l i t a t e s 30 d e t e r m i n a t i o n of the i n p u t impedance a t any i n t e r f a c e i n terms of the e l e c t r i c a l p r o p e r t i e s and t h i c k n e s s e s of the u n d e r l y i n g l a y e r s . The i n p u t impedance, Z J, a t the j t h i n t e r f a c e i s d e f i n e d as the r a t i o of e l e c t r i c t o r a d i a l magnetic f i e l d a t t h a t i n t e r f a c e , i . e . Z} = E j / H r j (1.41) In accordance w i t h the numbering i n F i g . 1.1, the i n p u t impedance a t the s u r f a c e (the f i r s t i n t e r f a c e ) i s denoted by Z 1. The immediate o b j e c t i v e i s t o d e r i v e an e x p r e s s i o n f o r ZJ (the i n p u t impedance a t the j t h i n t e r f a c e ) , i n terms of Z and the i n t r i n s i c impedance, Zj , of the i n t e r v e n i n g ( j t h ) l a y e r . The i n t r i n s i c impedance of any l a y e r i s d e f i n e d as the r a t i o of e l e c t r i c f i e l d t o r a d i a l magnetic f i e l d f o r a down-going wave; thus Zj = - i / u ; (1.42) S u b s t i t u t i n g from (1.39) and (1.42) i n t o ( 1 . 4 1 ) , Z J +' = -Z.(Af + Bf)/(kf - Bf) (1.43) whence, r e c a l l i n g ( 1 . 4 0 ) , Aj = Bj exp(-2u/ hj ) (Z J +' - Z j ) / ( Z J + ' + Zj ) (1.44) E q u i v a l e n t l y , u s i n g (1.38) and ( 1 . 4 2 ) , 31 ZJ = -Zj(Aj + Bj )/(Aj - Bj ) . (1.45) S u b s t i t u t i n g (1.44) i n t o (1.45) and r e a r r a n g i n g , the d e s i r e d r e l a t i o n r e s u l t s : j Z J + ' + Zj tanh(u: h; ) Z = Zj (1.46) Zj + Z t a n h ( u j hj ) T h i s d e r i v a t i o n i s e s s e n t i a l l y the same as t h a t p r e s e n t e d by Wait (1953, p418). The form of (1.46) i s c o n d u c i v e t o r a p i d c o m p u t a t i o n of the i n p u t impedance at the s u r f a c e f o r any g i v e n set of l a y e r s . Once Z 1 has been d e t e r m i n e d i n t h i s manner, A0 can be computed from (1.44) and ( 1 . 2 3 ) : - i a A0= — J,( A a)exp(-2u,h) ( Z 1 - Z t f ) / ( Z X + Z0 ) (1.47) The f o r w a r d problem may now be c o n s i d e r e d s o l v e d . I t i s c o n v e n i e n t a t t h i s p o i n t t o d e r i v e a r e c u r s i o n f o r m u l a f o r the c o e f f i c i e n t s {Bj : j = l , N L } . Such a f o r m u l a f a c i l i t a t e s d e t e r m i n a t i o n of the f i e l d s a t a l l i n t e r f a c e s , as r e q u i r e d i n I I - 2 . The d e s i r e d f o r m u l a r e s u l t s when two d i f f e r e n t e x p r e s s i o n s f o r Ej+r a r e equated. The f i r s t f o l l o w s i m m e d i a t e l y from (1.39) and ( 1 . 4 4 ) : E j +, = B j * ( l + Af/B*) = 2B j*Z J +' / ( Z J + ' + Zj ) (1.48) 32 To d e r i v e the second, observe from (1.45) t h a t Aj = Bj(Z J - Z ; ) / ( Z J + Zj ) (1.49) Hence, u s i n g ( 1 . 3 8 ) , Ej+i = 2Bj +, Z^'/iZ*** + Zj+i ) (1.50) E q u a t i n g (1.48) and ( 1 . 5 0 ) , the c o n n e c t i o n between Bj and B j - i i s r e v e a l e d : Bj = Bj*_,(ZJ + Zj ) / ( Z J + Zj-, ) (1.51) In the same p r e p a r a t o r y v e i n , i t i s advantageous t o determine here the a s y m p t o t i c l i m i t s of E, f o r l a r g e w and l a r g e A, because they s e r v e as checks on the a s y m p t o t i c s o l u t i o n d e r i v e d i n Appendix C. An e x p r e s s i o n f o r the s u r f i c i a l e l e c t r i c f i e l d may be o b t a i n e d as a s p e c i a l c a s e of ( 1 . 4 8 ) : - i a E = J(( A a ) e x p ( - u t f h ) Z 1 / ( Z 1 + Z0 ) . (1.52) where (1.23) has been i n v o k e d . T h i s e x p r e s s i o n i s e q u i v a l e n t t o e q u a t i o n (20) of M o r r i s o n e t a l . (1969). C o n s i d e r f i r s t the h i g h f r e q u e n c y l i m i t . As w —> , i t f o l l o w s from ( 1 . 3 6 ) , ( 1 . 7 ) , and (1.8) t h a t U j hj — > <x> and hence t h a t t a n h ( u j h y ) — > 1, f o r a l l j=l,2,....,NL where NL i s the number of l a y e r s . T h e r e f o r e , from ( 1 . 4 6 ) , Z 1 — > Z, , i . e . the 33 response of the ground i s c o m p l e t e l y d e t e r m i n e d by the e l e c t r i c a l p r o p e r t i e s of the f i r s t l a y e r . S u b s t i t u t i n g t h i s a s y m p t o t i c v a l u e f o r Z 1 i n t o ( 1 . 5 2 ) , and r e c a l l i n g ( 1 . 4 2 ) , i t i s c o n c l u d e d t h a t ' E, — > - i a J ( A a ) exp( -uah)/( u, + uc ) as w >• «* (1.53) Thus, i f d i s p l a c e m e n t c u r r e n t s are i g n o r e d , the e l e c t r i c f i e l d a t the s u r f a c e approaches z e r o as w a t e x t r e m e l y h i g h f r e q u e n c i e s . In the l i m i t of l a r g e A , on the o t h e r hand, Z 1 —> Z0 —>• -i/A as per (1.42) and (1.36),- so from (1.52) - i a . . E. — > J.(Aa)e as A — > °° (1.54) 2 A 1-5 A t t e n u a t i o n and R e f l e c t i o n I t i s p o s s i b l e t o a c q u i r e a degree of p h y s i c a l i n s i g h t by c o n s i d e r a t i o n of ( 1 . 4 6 ) . I t has a l r e a d y been ob s e r v e d i n the d e r i v a t i o n of (1.53) t h a t Z J > Zj as R e l u j h j } > •* (1.55) In t h i s l i m i t the ( j + l ) t h and deeper l a y e r s do not i n f l u e n c e the impedance a t the j t h i n t e r f a c e , nor a t any s h a l l o w e r i n t e r f a c e s . In p a r t i c u l a r , the responses a t the s u r f a c e w i l l be u n a f f e c t e d by the e l e c t r i c a l p r o p e r t i e s of the l a y e r s u n d e r l y i n g the j t h l a y e r , t h i s l a y e r a c t i n g as an " e l e c t r o m a g n e t i c s h i e l d " . More 34 p r e c i s e l y , Z m Zj i m p l i e s t h a t the f i e l d s decay almost t o z e r o as they pass through the j t h l a y e r , i . e . t h a t the t h i c k n e s s of t h a t l a y e r i s much g r e a t e r than i t s s k i n - d e p t h . The ( d i m e n s i o n l e s s ) s k i n - d e p t h , dj , of the j t h l a y e r i s a f u n c t i o n of both X and w, and i s d e f i n e d by <$j(A,w) = 1/Re{uj} = l/(<*jcos0/) (1.56) where oCj = /(A 2 - 12 £j ) 2 + /3 4 2 (1.57) $j = O.Stan- 1 { ji2oj/{ X2 - y 2£j )} (1.58) Hence, Re { UJ hj } = hj /c*j and e f f e c t i v e s h i e l d i n g o c c u r s when hj /6j » 1. I t f o l l o w s from (1.56-58) t h a t 6j d e c r e a s e s m o n o t o n i c a l l y as e i t h e r A or w i n c r e a s e s . At any g i v e n f r e q u e n c y , t h e r e f o r e , the s k i n - d e p t h i s maximum when A=0, the maximum g i v e n by max{ 6j ( A ,w)} = 4 ( 0 ' w ) = [ Sf + fi* <7j2 c o s ( t a n " 1 { / 3 2 ^ / ( y'Sj)})]'1 (1.59) I f i n a d d i t i o n /Z2<?j » K2£j> (1.59) reduces t o the f a m i l i a r s k i n - d e p t h f o r m u l a f o r u n i f o r m p l a n e waves ( J a c k s o n , 1975, p298). T h e r e f o r e the s t a n d a r d s k i n - d e p t h f o r m u l a i s u s e f u l i n the c o n t e x t of HLEM soundings o n l y i n s o f a r as i t p r o v i d e s an 3b upper l i m i t on the p e n e t r a t i o n ; i n f a c t t h i s upper l i m i t i s never a t t a i n e d because the source does not ge n e r a t e the A = 0 mode, as a t t e s t e d by ( 1 . 2 3 ) . In a d d i t i o n t o i t s s u p e r i o r p e n e t r a t i o n , the \=0 mode i s a l s o c h a r a c t e r i s e d by the s h o r t e s t w avelength and s l o w e s t phase speed. These f u r t h e r p r o p e r t i e s may be deduced from the f o l l o w i n g formulae f o r wav e l e n g t h , l j , and phase speed, V J , i n the j t h l a y e r : l j = 2 T T / ( < * j S i n e ; ) ( 1 . 6 0 ) VJ = w/(<*jsin0j ) ( 1 . 6 1 ) The h i g h l y d i s p e r s i v e c h a r a c t e r of the d i s t u r b a n c e , e v i d e n c e d by ( 1 . 6 1 ) , ( 1 . 5 7 ) , and ( 1 . 5 8 ) , u n d e r l i e s i n t e r p r e t a t i o n of time-domain EM r e c o r d s : e a r l y a r r i v a l s c o r r e s p o n d t o h i g h f r e q u e n c i e s and s h a l l o w p e n e t r a t i o n , w h i l e l a t e r a r r i v a l s a r e c h a r a c t e r i s e d by low f r e q u e n c i e s and deeper p e n e t r a t i o n . T h i s d e s c r i p t i o n i s somewhat o v e r - s i m p l i f i e d f o r HLEM soundings because modal d i s p e r s i o n i s a l s o a f a c t o r . In the f o r e g o i n g d i s c u s s i o n , c o n d u c t i v e zones have been c h a r a c t e r i s e d as zones of a t t e n u a t i o n . A l t e r n a t i v e l y , from the p e r s p e c t i v e of p r o p a g a t i n g waves, c o n d u c t o r s may be r e g a r d e d as " r e f l e c t o r s " . In p a r t i c u l a r , the r a t i o ^ j + i of r e f l e c t e d t o i n c i d e n t a m p l i t u d e at the ( j + l ) t h i n t e r f a c e i s , from ( 1 . 4 4 ) and ( 1 . 4 0 ) , f>j+t = A J / B . * = (Z - Z j ) / ( Z " + Z j ) ( 1 . 6 2 ) I f Z a Z j + , , t h e r e i s no up-going wave i n the ( j + l ) t h l a y e r ; i n t h i s case pj+t i s a c o n v e n t i o n a l r e f l e c t i o n c o e f f i c i e n t , and (1.62) s i m p l i f i e s t o (>i+i = (uj - U j> , ) / ( U J + uj+i ) (1.63) I t may be obser v e d t h a t ^j+i -1 when | UJ+, | >> 1, s i g n i f y i n g complete r e f l e c t i o n w i t h a 180° phase change; r e c a l l i n g ( 1 . 3 6 ) , a l l modes w i l l on t h i s b a s i s s u f f e r t o t a l r e f l e c t i o n i f , f o r example, f$20j >> 1. A h i g h l y c o n d u c t i v e zone t h e r e f o r e a c t s l i k e an i m p e r f e c t t w o - s i d e d m i r r o r , not o n l y s h i e l d i n g the u n d e r l y i n g l a y e r s from the source f i e l d s , but a l s o hampering the upward p r o p a g a t i o n of any secondary f i e l d s from those deeper l a y e r s . In c o n t r a s t , r e f l e c t i o n s from r e s i s t i v e zones a r e n e c e s s a r i l y l e s s pronounced, and r e s i s t o r s may be c o n s i d e r e d almost " t r a n s p a r e n t " . These g e n e r a l comments are c o n s i s t e n t w i t h the o b s e r v a t i o n s of Vanyan (1967, p l 7 6 ) and W e i d e l t (1972, p282) t o the e f f e c t t h a t i n t e r p r e t a t i o n of EM soundings i s l e a s t ambiguous when the c o n d u c t i v i t y i n c r e a s e s m o n o t o n i c a l l y w i t h d e p t h . In p r i n c i p l e , s t r o n g r e f l e c t i o n s may a r i s e from a marked c o n t r a s t i n d i e l e c t r i c c o n s t a n t between l a y e r s . However, i n most g e o p h y s i c a l s i t u a t i o n s the v a r i a t i o n i n c o n d u c t i v i t y i s much more extreme than t h a t i n d i e l e c t r i c c o n s t a n t (Grant and West, 1965, p469). Water might c o n s t i t u t e a p o s s i b l e " d i e l e c t r i c t a r g e t " i n some c i r c u m s t a n c e s , i t s d i e l e c t r i c c o n s t a n t b e i n g about 80 t i m e s t h a t of f r e e space ( K e l l e r and F r i s c h k n e c h t , 37 1966, p54) . 1-6 The I n d u c t i o n Response F u n c t i o n A l l i n f e r e n c e s c o n c e r n i n g the e l e c t r i c a l p r o p e r t i e s of the ground must be based e x c l u s i v e l y on the secondary f i e l d s . A re-arrangement of the m a t h e m a t i c a l e x p r e s s i o n s f o r these induced f i e l d s l e a d s t o the d e f i n i t i o n of a s p e c t r a l response f u n c t i o n f o r the l a y e r e d e a r t h , a p p r o p r i a t e f o r any c i r c u l a r l y symmetric s o u r c e . Let AE denote the H a n k e l - t r a n s f o r m e d secondary e l e c t r i c f i e l d , i . e . A E ( A , Z , W ) = E ( X » Z f W ) " Ej :A,z,w) (1.64) The o b s e r v a b l e secondary magnetic f i e l d s a t the s u r f a c e may be r e l a t e d t o AE ( A,0,w) u s i n g ( 1 . 1 6 ) , ( 1 . 1 1 ) , ( 1 . 1 2 ) , and the i d e n t i t y r " 1 ! — [ r J ( A r ) ] = A j f l ( A r ) , (1.65) dr 1 0 Hence, oo A H r ( r , 0 , w ) = - i f A E ' ( \ , 0 , w ) A J,( A r ) d A (1.66) o oo Hz(r,0,w) = i ^ A E ( A,0,W ) \ 2 J O ( A r ) d A (1.67) o A p h y s i c a l l y more i l l u m i n a t i n g statement of t h e s e t r a n s f o r m 3d r e l a t i o n s i s f u r n i s h e d by the schematic e q u a t i o n oo A H m ( r , 0 , w ) = ^K( X,w; £ ,<r)B*( A ,a,h)£ m ( A , r ) d A (1.68) o where K i s the response f u n c t i o n f o r the ground, B* i s the modal a m p l i t u d e f u n c t i o n on the s u r f a c e (z=0), and P£ i s a " f i l t e r f u n c t i o n " . Each of thes e q u a n t i t i e s w i l l now be d e f i n e d e x p l i c i t l y . The response f u n c t i o n , K , i s d e f i n e d by K ( A,w; e,<T) = A E ( A , 0 , W ) / B * (1.69) where, r e c a l l i n g (1.23) and ( 1 . 4 0 ) , B * = — J , ( A a ) e " U # h (1.70) P h y s i c a l l y s p e a k i n g , the a m p l i t u d e spectrum, Bff , of the modes ge n e r a t e d by the c o i l i s m o d i f i e d by d i f f e r e n t i a l a t t e n u a t i o n over the d i s t a n c e h from the c o i l t o the s u r f a c e . Thus B^ i s the e f f e c t i v e source spectrum. The d i v i s i o n of A E by B^ * i n (1.69) t h e r e f o r e removes the e f f e c t of the s o u r c e , r e n d e r i n g K c h a r a c t e r i s t i c of the e l e c t r i c a l p r o p e r t i e s of the ground a t g i v e n v a l u e s of A and w. A c c o r d i n g t o ( 1 . 6 8 ) , both magnetic components may be ex p r e s s e d as moments of the same response f u n c t i o n . T h i s i s a p h y s i c a l l y p l a u s i b l e r e p r e s e n t a t i o n , s i n c e A H r and A H z a r e d i f f e r e n t m a n i f e s t a t i o n s of the same induced c u r r e n t d i s t r i b u t i o n . The v a l i d i t y of the r e p r e s e n t a t i o n w i l l now be 3 9 e s t a b l i s h e d m a t h e m a t i c a l l y . E l i m i n a t i o n o f A J f r o m ( 1 . 3 9 ) y i e l d s U J E J > ' - EJV, = 2 H » B ; ( 1 - 7 1 ) w h e r e ( 1 . 1 1 ) s h o u l d b e r e c a l l e d . A s p e c i a l c a s e o f t h i s e q u a t i o n ( c o r r e s p o n d i n g t o j = 0 ) r e l a t e s s u r f i c i a l v a l u e s o f E a n d E ' ; e x p r e s s e d i n t e r m s o f s e c o n d a r y f i e l d s u s i n g ( 1 . 6 4 ) , t h i s r e l a t i o n b e c o m e s u # AE - AE; = 2ucB* - uCEC( A , 0 , w ) + E ; ( A , 0 , W ) ( 1 . 7 2 ) I t f o l l o w s f r o m ( 1 . 2 4 ) a n d ( 1 . 7 0 ) t h a t t h e p r i m a r y f i e l d a t t h e s u r f a c e a d o p t s t h e v a l u e s E J A , 0 , w ) = B * ( 1 . 7 3 ) E ; ( A , 0 , w ) = -u0B* ( 1 . 7 4 ) S u b s t i t u t i o n o f ( 1 . 7 3 ) a n d ( 1 . 7 4 ) i n t o ( 1 . 7 2 ) y i e l d s t h e s i m p l e e q u a t i o n A B I = u , A E ' | n ( 1 . 7 5 ) T h u s t h e r e s p o n s e f u n c t i o n d e f i n e d i n ( 1 . 6 9 ) a p p l i e s t o b o t h r a d i a l a n d v e r t i c a l m a g n e t i c f i e l d s , p r o v i d e d o n l y t h a t a f a c t o r u0 i s a b s o r b e d i n t o t h e f i l t e r f u n c t i o n f o r A H r . T h e r e f o r e t h e H a n k e l k e r n e l s , o r f i l t e r f u n c t i o n s , i n ( 1 . 6 8 ) a r e d e f i n e d b y 4U *0 A,D = i \2J0( A r) (1.76) #,( A ,r) = -i Au0J,( Ar) (1.77) i s the a p p r o p r i a t e k e r n e l f o r A H z , and 9C{ f o r AUr. The g e o m e t r i c a l o r t h o g o n a l i t y of Hr and Hz i s r e f l e c t e d i n the mat h e m a t i c a l o r t h o g o n a l i t y of J f and J0 . T y p i c a l l y the response f u n c t i o n a t a p a r t i c u l a r f r e q u e n c y v a r i e s smoothly w i t h \, as w i t n e s s e d by F i g . 1.2. In F i g . 1.2a the a m p l i t u d e of the response f u n c t i o n f o r a u n i f o r m h a l f - s p a c e (cf=0.1 S/m) i s p o r t r a y e d f o r a n g u l a r f r e q u e n c y 10 r a d / s e c . I t i s apparent t h a t o n l y the small-A modes g i v e r i s e t o a p p r e c i a b l e induced c u r r e n t s ; a t l a r g e r v a l u e s of A the g e o m e t r i c a l a t t e n u a t i o n c o m p l e t e l y dominates the response. F i g . 1.2b d e p i c t s |K| a t the same frequ e n c y f o r a t w o - l a y e r c o n d u c t i v i t y f o r which Of = 0.1 S/m, h = 18.75m, <Z[ = 3 S/m. The d i f f e r e n c e s between the two f i g u r e s a r e due e n t i r e l y t o the i n f l u e n c e of the c o n d u c t i v e "basement" i n the second model. The p r i n c i p a l d i f f e r e n c e i s the e x t e n s i o n of the range of A - v a l u e s over which i n d u c t i o n i s a p p r e c i a b l e ; as a r e s u l t of the c o n d u c t i v e zone a t de p t h , the- second model responds t o modes which do not e x c i t e the h a l f - s p a c e a t a l l . The enhanced response of the second model a t l a r g e r v a l u e s of \ can be a t t r i b u t e d t o r e f l e c t i o n a t the upper s u r f a c e of the c o n d u c t o r ; as per (1.63) and ( 1 . 3 6 ) , the r e f l e c t i o n c o e f f i c i e n t ^ does not approach z e r o u n t i l A2 » /3 2 °l • 41 F i g u r e 1.2 Examples of I n d u c t i o n Response F u n c t i o n s (a) Response f u n c t i o n f o r u n i f o r m h a l f space (<~f - 0.1 S/m) f o r a n g u l a r frequency 10 r a d / s e c . (b) Response f u n c t i o n f o r a t w o - l a y e r model (of = n . l S'/m, h, = 18.75m, e~l = 3 S/m) a t a n g u l a r f r e q u e n c y 10 r a d / s e c . The e f f e c t of the c o n d u c t o r i s t o extend the range of A f o r which a p p r e c i a b l e i n d u c t i o n may o c c u r . 42 Half-space response function a;=1 Orad/sec -5 0 5 F i g . 1.2a Two-layer response function co=J\ Orad/sec -5 0 5 ln(X) F i g . 1.2b 43 1-7 C o m p u t a t i o n a l Procedure The e v a l u a t i o n of Hankel t r a n s f o r m i n t e g r a l s l i k e (1.66) and (1.67) c o n s t i t u t e s the major c o m p u t a t i o n a l h u r d l e i n the s o l u t i o n of the f o r w a r d problem. An e s s e n t i a l p r e r e q u i s i t e i s the c a l c u l a t i o n of the i n p u t impedance Z 1 ( o r , e q u i v a l e n t l y , the response f u n c t i o n K) at many, s t r i c t l y a l l , v a l u e s of A. T h i s r equirement prompted the d e c i s i o n t o c o n f i n e the study t o l a y e r e d models, f o r which the r e c u r s i o n r e l a t i o n i s a v a i l a b l e , r a t h e r than t o a l l o w the e l e c t r i c a l p r o p e r t i e s t o v a r y c o n t i n u o u s l y , i n which case the d e t e r m i n a t i o n of Z 1 would i n v o l v e r e p e a t e d n u m e r i c a l i n t e g r a t i o n of ( 1 . 1 5 ) . The r e s t r i c t i o n t o l a y e r e d models does not compromise the u t i l i t y of the a l g o r i t h m , because as many l a y e r s as a r e n e c e s s a r y t o f i t the d a t a can be i n t r o d u c e d . In the l i m i t as the number of l a y e r s becomes l a r g e the r e p r e s e n t a t i o n i s e s s e n t i a l l y c o n t i n u o u s because any f u n c t i o n i s e n t e r e d i n t o a d i g i t a l computer as a f i n i t e number of sampled v a l u e s . S u b s t i t u t i n g (1.52) i n t o ( 1 . 1 6 ) , i t may be observed t h a t the e v a l u a t i o n of E(r,0,w) i n v o l v e s the i n t e g r a t i o n of a p r o d u c t of B e s s e l f u n c t i o n s : 7 21 E(r,0,w) = - i a \ exp(-u,h)J( Aa)J.( A r) A dA/u, (1.78) The i n t e g r a n d i n (1.78) i s t h e r e f o r e h i g h l y o s c i l l a t o r y ; moreover, f o r h=0 ( i . e . f o r a c o i l on the grou n d ) , the o s c i l l a t i o n s a r e a t t e n u a t e d weakly as ,A • In o r d e r t o mute the o s c i l l a t i o n s , Ryu et a l . (1970) i n v o k e d the a s y m p t o t i c r e l a t i o n (1.54) and r e - w r o t e (1.78) as the sum of two i n t e g r a l s : 44 E(r,O fw) = f e"*'' J,( A a)J f( A r ) d A -oo ia (1.79) £[E(A,0,w) - e" A h Jt( Aa)/2A ] A J,( A r)dA A l t h o u g h Ryu et a l . appear t o adopt t h i s approach p u r e l y f o r reasons of n u m e r i c a l e x p e d i e n c y , i t i s apparent from (1.64) and (1.25) t h a t (1.79) r e p r e s e n t s a p h y s i c a l d e c o m p o s i t i o n of the o b s e r v e d f i e l d i n t o p r i m a r y and secondary components. Ryu et a l . e v a l u a t e both i n t e g r a l s i n (1.79) n u m e r i c a l l y ; however, i f d i s p l a c e m e n t c u r r e n t s i n a i r a r e n e g l i g i b l e , the f i r s t i n t e g r a l may be e v a l u a t e d e x a c t l y u s i n g ( 1 . 2 8 ) . The second i n t e g r a l i n (1.79) ( r e p r e s e n t i n g the secondary f i e l d ) has been e v a l u a t e d i n t h i s s tudy u s i n g the d i g i t a l l i n e a r f i l t e r i n g method of Koefoed et a l . (1972), r e c e n t l y g e n e r a l i s e d by Johansen and Sorensen (1979). B r i e f l y , the l i n e a r f i l t e r i n g t e c h n i q u e d e r i v e s from the c o o r d i n a t e t r a n s f o r m a t i o n r = exp(x) A = exp(-y) which c o n v e r t s a Hankel t r a n s f o r m i n t e g r a l i n t o a c o n v o l u t i o n i n t e g r a l : oo T ( r ) = [ c ( A ) J ( A r)dA a oo T(x) = e~* j " c ( y ) e x p ( x - y ) J ( e * " y )dy (1.80) o The f u n c t i o n C(y) i s sampled M+1 times a t i n t e r v a l s of Ay, 45 and i s s u b s e q u e n t l y approximated as a sum of s i n e [ s i n ( v ) / v ] f u n c t i o n s : C(y> - 2 l c(y w)sinc[k(y-y r v,) ] (1.81) m s o where k = 7 T / A y . S u b s t i t u t i n g f o r C(y) i n (1.80) and t a k i n g the F o u r i e r t r a n s f o r m , the e f f e c t of the s i n e f u n c t i o n s i s condensed i n t o a s i n g l e m u l t i p l i c a t i v e f a c t o r , A y . e x p ( - i ^ y w ) . N o t i n g t h a t the N y q u i s t a n g u l a r f r e q u e n c y f o r the s a m p l i n g i s k, a f i 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 a t i o n y i e l d s A y 2TT e * T ( x ) * — ^ 21 f F ( ^ ) e x P( i ^ 7 m ) d i ) C ( y m ) (1.82) where 77 = x - y (1.83) 'm m and where F ( j £ ) i s the F o u r i e r t r a n s f o r m of e 5 J ( e s ) . T h i s r e s u l t c o r r e s p o n d s t o e q u a t i o n (20) of Koefoed et a l . , and may be ex p r e s s e d s c h e m a t i c a l l y as M T ( r ) * r - x 2 _ W m C ( y m ) (1.84) A y c where W = \F( i ) e x p ( - i i j n ) d / (1.85) m so k 2TC j£ »w A c c o r d i n g t o (1 . 8 4 ) , the t r a n s f o r m i n t e g r a l (1.80) may be approxi m a t e d as a weighted sum of f u n c t i o n e v a l u a t i o n s , {C(y ):m=0,M}. The a c c u r a c y of t h i s method i s dependent 46 p r i m a r i l y on the c h o i c e of A y ; c l e a r l y t h e r e i s a t r a d e - o f f between a c c u r a c y and c o m p u t a t i o n a l l a b o u r , and Koefoed et a l . choose A y = l n l 0 / 1 0 on e m p i r i c a l grounds. For Hankel t r a n s f o r m s w i t h r e s p e c t t o Je and J, , Koefoed et a l . p r o v i d e s i x - f i g u r e w e i g h t s at r e s p e c t i v e l y s i x t y - o n e and f o r t y - s e v e n v a l u e s of y. Were the s e t o prove i n a d e q u a t e , i t would be p o s s i b l e t o use the e i g h t - f i g u r e w e i g h t s computed at over two hundred v a l u e s of y by Anderson (1977), though t h i s o p t i o n would n e c e s s a r i l y i n c r e a s e the computing c o s t s . The a c c u r a c y a t t a i n a b l e w i t h the Koefoed et a l . w e i g h t s f o r the Ryu et a l . (1970) #4 t w o - l a y e r model has been a s s e s s e d by comparison w i t h v a l u e s o b t a i n e d u s i n g the Romberg a l g o r i t h m ( A c t o n , 1970, p p l 0 6 - 1 0 7 ) , and the r e s u l t s a re summarised i n T a b l e 1.1. Convergence of the Romberg scheme was assumed when s u c c e s s i v e e s t i m a t e s d i f f e r e d by l e s s than 0.1%, t h i s c r i t e r i o n b e i n g a p p l i e d i n d e p e n d e n t l y t o the r e a l and i m a g i n a r y p a r t s . In e s t i m a t i n g the a b s o l u t e a c c u r a c y of the e s t i m a t e of the i n v e r s e Hankel t r a n s f o r m so computed, some a l l o w a n c e must be made f o r the f a c t t h a t the range of i n t e g r a t i o n i s f i n i t e i n p r a c t i c e . The magnitude of t h i s e f f e c t was e s t i m a t e d by i n c r e a s i n g the range u n t i l the d i f f e r e n c e between s u c c e s s i v e v a l u e s was 0.1% or l e s s . Hence, the a c c u r a c y of the Romberg e v a l u a t i o n s i s p r o b a b l y of the o r d e r of 0.2%. The g r e a t e s t d i s c r e p a n c y between the Koefoed et a l . and Romberg v a l u e s i s a p p r o x i m a t e l y 0.06% i n |Hz| when w = 10 r a d / s e c * T h e r e f o r e the d i g i t a l f i l t e r i n g i s a c c u r a t e t o a p p r o x i m a t e l y 0.2% a l s o i n the cases c o n s i d e r e d . E r r o r s of t h i s magnitude a r e *An e r r o r of 0.1% i n phase i s i n t e r p r e t e d t o mean ±0.36°. 47 Ta b l e 1.1 Comparison of D i g i t a l L i n e a r F i l t e r i n g and Romberg I n t e g r a t i o n T h e o r e t i c a l responses f o r Ryu et a l . t w o - l a y e r #4 c a l c u l a t e d t w i c e t o compare two q u a d r a t u r e methods: the d i g i t a l l i n e a r f i l t e r i n g of Koefoed et a l . (1972) and i n t e g r a t i o n u s i n g the Romberg a l g o r i t h m . The f i e l d a m p l i t u d e v a l u e s a re e n t e r e d as l o g of t h e i r MKS v a l u e s . Phases a r e i n d e g r e e s . The symbol * i s used t o i n d i c a t e t h a t the range of i n t e g r a t i o n i s e"' < A < e 6 , w h i l e # r e l a t e s t o the i n t e r v a l e" 7 < A < e 7 . Ang.Freq. 0.1 1.0 10.0 rad/sec -8.8054 -7.2261 -6.2683 Koefoed Romberg l o g | E | -8.8054* -7.2261* -6.2687* -5.0547 -4.0163 -4.3286 Koefoed Romberg l o g i n r | -5.0545* ^4.0163* -4.3286# -3.9775 -4.0265 -5.0102 Koefoed Romberg xog1HZ| -3.9775# -4.0266# -5.0070# 266.03 240.42 176.95 Koefoed Romberg v e 266.03* 240.42* 176.94* 9r 78.38 5.63 -44.34 Koefoed Romberg 78.39* 5.62* -44.35# Bz 183.73 158.79 85.51 Koefoed 183.73# | 158.79# | 85.53# Romberg 48 a s m a l l f r a c t i o n of t h e e x p e c t e d " n o i s e " i n most g e o p h y s i c a l a p p l i c a t i o n s . J u s t as f o r t h e e l e c t r i c f i e l d , t h e H a n k e l t r a n s f o r m i n t e g r a l s f o r H r ( r ,0,w) and Hz(r,0,w) can be decomposed i n t o s e p a r a t e i n t e g r a l s r e p r e s e n t i n g p r i m a r y and s e c o n d a r y components. As shown i n A p p e n d i x B, t h e p r i m a r y f i e l d i n t e g r a l s a dmit c l o s e d form e x p r e s s i o n s d e r i v e d from (1.28) by d i f f e r e n t i a t i o n w i t h r e s p e c t t o h and r i n a c c o r d a n c e w i t h (1.11) and (1.12) r e s p e c t i v e l y . The s e c o n d a r y f i e l d i n t e g r a l s a r e a l w a y s e v a l u a t e d by means of l i n e a r d i g i t a l f i l t e r i n g . R e s p o n s e s computed i n t h e manner o u t l i n e d above a r e compared i n F i g . 1.3 w i t h v a l u e s i n t e r p o l a t e d f r o m t h e p u b l i s h e d f i g u r e s of Ryu e t a l . ( 1 9 7 0 ) . The o v e r a l l agreement i s v e r y s a t i s f a c t o r y , t h o u g h t h e r e i s some s u g g e s t i o n t h a t t h e c o r r e s p o n d e n c e i s somewhat p o o r e r a t h i g h f r e q u e n c i e s ( l a r g e fi ) . In view of t h e a b s o l u t e a c c u r a c y d e m o n s t r a t e d i n T a b l e 1.1, t h e d i s c r e p a n c i e s a r e a t t r i b u t e d p r i m a r i l y t o i n t e r p o l a t i o n e r r o r s . D i s p l a c e m e n t c u r r e n t s a r e a p o s s i b l e c o n t r i b u t o r y f a c t o r t o l a r g e r d i f f e r e n c e s a t h i g h e r f r e q u e n c i e s , s i n c e f r e e s p a c e p e r m i t t i v i t y i s a s s i g n e d t o t h e g r o u n d i n t h i s s t u d y whereas Ryu e t a l . a p p e a r t o d i s r e g a r d d i s p l a c e m e n t c u r r e n t s c o m p l e t e l y ; however, t h i s e f f e c t i s not s u f f i c i e n t t o a c c o u n t f o r t h e o b s e r v e d d i f f e r e n c e s by i t s e l f b e c a u s e even when fi =10, /32<?f OC 700y2£o. 49 F i g u r e 1.3 Comparison w i t h Responses C a l c u l a t e d by Ryu e t . a l . (1970) The v a l u e s denoted "x" were i n t e r p o l a t e d from the graphs p u b l i s h e d by Ryu et a l . , w h i l e s o l i d l i n e s connect the v a l u e s computed i n t h i s s t u d y . A l l v a l u e s p e r t a i n to Ryu et a l . Two-layer model #4: = 0.01 S/m, h, = 18.75m, <?£ = 0.3 S/m. The independent v a r i a b l e , yg, v a r i e s as /w, as per ( 1 . 8 ) . (a) E l e c t r i c f i e l d a m p l i t u d e s . (b) E l e c t r i c f i e l d phases. (c) R a d i a l magnetic f i e l d a m p l i t u d e s . (d) R a d i a l magnetic f i e l d phases. (e) V e r t i c a l magnetic f i e l d a m p l i t u d e s , ( f ) V e r t i c a l magnetic f i e l d phases. 50 Ryu et al . #4 a = 1 0 m R = 1 0 0 m C D o Ryu et al . #4 a = 1 0 m R = 1 0 0 m 270 240 h 210 Ob 180 h 150 -1 .0 Fig. 1.3b 51 Ryu et al . #4 a = 1 0 m R = 1 0 0 m - 5 . 5 1 1 1 1 1 ' 1 i I - 1 . 0 - 0 . 5 0 .0 0 . 5 1.0 F i g . 1.3c Ryu et al . #4 a = 1 0 m R = 1 0 0 m -60 1 ' 1 1 1 i i • I - 1 . 0 - 0 . 5 0.0 0.5 1.0 logO?) F i g . 1.3d Ryu et a l . #4 a = 1 0 m R = 1 0 0 m ~3 . 5 I l 1 r- 1 1 1 1 Ryu et al . #4 a = 1 0 m R = 1 0 0 m 210 i -180 150 -120 -90 -60 ' 1 1 1 1 ' 1 1 I - 1 . 0 - 0 . 5 0.0 0.5 1.0 F i g . 1.3f 53 Chapter I I MODEL CONSTRUCTION The a b i l i t y t o c o n s t r u c t a model t o s a t i s f y a g i v e n s e t of o b s e r v a t i o n s i s an i m p o r t a n t a t t r i b u t e of any i n v e r s i o n scheme. The main purpose of t h i s c h a p t e r i s t o d e s c r i b e the i t e r a t i v e c o n s t r u c t i o n a l g o r i t h m i n c o r p o r a t e d i n t o the program HLEM.INV. The t h e o r e t i c a l b a s i s f o r l i n e a r i s e d c o n s t r u c t i o n i s w e l l known (Backus and G i l b e r t , 1967, 1968; P a r k e r , 1977a), and i s summarised below f o r the sake of completeness and, more e s p e c i a l l y , t o p r o v i d e background f o r subsequent d e s c r i p t i o n of i n n o v a t i o n s i n HLEM.INV. The c h a p t e r i s d i v i d e d i n t o f i v e s e c t i o n s . In the f i r s t s e c t i o n the c o n s t r u c t i o n problem i s f o r m u l a t e d as a system of l i n e a r e q u a t i o n s f o r a c o n d u c t i v i t y p e r t u r b a t i o n . U n d e r l y i n g t h i s f o r m u l a t i o n a re the F r e c h e t d e r i v a t i v e s f o r the HLEM problem, r e l a t i n g s m a l l changes i n c o n d u c t i v i t y t o the a s s o c i a t e d changes i n the r e s p o n s e s . The comp u t a t i o n of the F r e c h e t d e r i v a t i v e s f o r HLEM i n v e r s i o n i s d i s c u s s e d i n the second s e c t i o n , and the method of s o l u t i o n of the l i n e a r p e r t u r b a t i o n e q u a t i o n s i n the presence of e r r o r s i s o u t l i n e d i n the t h i r d . In the f o u r t h s e c t i o n c e r t a i n s p e c i a l f e a t u r e s of the HLEM.INV c o n s t r u c t i o n a l g o r i t h m a r e d e s c r i b e d , p r i o r t o a d e m o n s t r a t i o n of the performance of the program on t h e o r e t i c a l examples i n the f i n a l s e c t i o n . 54 11 —1 F r e c h e t Der i v a t i v e s and L i n e a r i s e d C o n s t r u c t i o n In g e o p h y s i c a l i n v e r s e problems the aim i s t o d e t e r m i n e , as f a r as t h i s i s p o s s i b l e , the s p a t i a l d i s t r i b u t i o n of one or more p h y s i c a l p r o p e r t i e s f o r some p o r t i o n of the e a r t h . The o b s e r v a t i o n s s e r v e as i n d i r e c t i n d i c a t o r s of the d i s t r i b u t i o n of a p h y s i c a l p r o p e r t y of i n t e r e s t . They may o r i g i n a t e p a s s i v e l y as the s i g n a t u r e of an i n t r i n s i c c h a r a c t e r i s t i c of the ground, i n the manner t h a t d e n s i t y v a r i a t i o n s produce g r a v i t y a n o m a l i e s ; a l t e r n a t i v e l y , they may r e p r e s e n t responses t o some form of e x c i t a t i o n , as i n EM p r o s p e c t i n g or s e i s m o l o g y . The t r u e v a r i a t i o n of any p h y s i c a l p r o p e r t y w i t h i n the e a r t h i s e x c e e d i n g l y complex i n g e n e r a l , and i t i s n e c e s s a r y t o i n t r o d u c e s i m p l i f y i n g a ssumptions b e f o r e a t r a c t a b l e m a t h e m a t i c a l problem can be f o r m u l a t e d . In t h i s s t u d y , f o r example, the c o n d u c t i v i t y of the ground i s deemed t o be i s o t r o p i c and a f u n c t i o n of depth o n l y . Once a r e s t r i c t e d s e t of s t r u c t u r e s has been p r e s c r i b e d , the immediate g o a l i s t o f i n d elements of t h a t s e t which c o n s t i t u t e a c c e p t a b l e m a t h e m a t i c a l "models" of the ground, i . e . t o f i n d models which s a t i s f y the o b s e r v a t i o n s . Thus the e s s e n t i a l m a t h e m a t i c a l problem i s t o d i s t i l l i n f o r m a t i o n c o n c e r n i n g the n a t u r e of a c c e p t a b l e models from the a v a i l a b l e measurements. Whether or not c o n c l u s i o n s based on t h e o r e t i c a l models a r e r e l e v a n t t o the t r u e e a r t h w i l l depend on the s u i t a b i l i t y of the s i m p l i f y i n g a s s u m p t i o n s . In o r d e r t o c o n s t r u c t a model s a t i s f y i n g a g i v e n s e t of o b s e r v a t i o n s i t i s e s s e n t i a l t o know how the o b s e r v a t i o n s a r e i n f l u e n c e d by the p h y s i c a l p r o p e r t y i n q u e s t i o n . In the f o r m u l a t i o n of Backus and G i l b e r t (1968, p l 7 2 ) , c e r t a i n k e r n e l 55 f u n c t i o n s c o n s t i t u t e the l i n k between the model and measurable r e s p o n s e s . More e x p l i c i t l y , a r e s p o n s e , c, i s r e p r e s e n t e d as an i n n e r p r o d u c t of a k e r n e l , G ( z ) , w i t h a model, m(z), i . e . = ^ G ( z ) m ( z ) d z The k e r n e l i s s p e c i f i c t o the p a r t i c u l a r r e s p o n s e , and i s t h e r e f o r e a f u n c t i o n of system p a r a m e t e r s , such as o p e r a t i n g f r e q u e n c y or s o u r c e - r e c e i v e r s e p a r a t i o n . Hence N r e s p o n s e s , {cj : j=l,N}, are a s s o c i a t e d w i t h N d i f f e r e n t k e r n e l s , {G.(z): j = l , N } : Cj = j*Gj(z)m(z)dz j = l , N (2.1) A d i s t i n c t i o n i s drawn between l i n e a r and n o n - l i n e a r i n v e r s e problems: f o r n o n - l i n e a r problems the k e r n e l s a r e c h a r a c t e r i s t i c of a p a r t i c u l a r model, whereas f o r l i n e a r problems the k e r n e l s are "model independent". I f (2.1) i s regarded as the d e f i n i n g r e l a t i o n f o r a l i n e a r i n v e r s e problem, the c o r r e s p o n d i n g r e l a t i o n f o r a n o n - l i n e a r i n v e r s e problem i s C j = ^Gj(z;m)m(z)dz j = l , N (2.2) To c l a r i f y the d i s t i n c t i o n , o b s e r v e t h a t the i n v e r s i o n of g r a v i t y d a t a i s l i n e a r because each " p o i n t mass" i n the c a u s a t i v e body i s weighted s o l e l y a c c o r d i n g t o the i n v e r s e square of i t s d i s t a n c e from the d e t e c t o r , independent of the s p a t i a l d i s t r i b u t i o n of d e n s i t y . E l e c t r o m a g n e t i c (EM) methods, 56 on the o t h e r hand, g i v e r i s e t o n o n - l i n e a r i n v e r s e problems i f the d i s t r i b u t i o n of c o n d u c t i v i t y i s sought because the s i g n a t u r e of any s t r u c t u r e a t depth i s always a t t e n u a t e d by any o v e r l y i n g c o n d u c t o r s , t h i s d i s s i p a t i o n b e i n g i n a d d i t i o n t o p u r e l y g e o m e t r i c a l decay. In p r i n c i p l e , such c o u p l i n g endows EM methods w i t h powers of s t r u c t u r a l d i s c r i m i n a t i o n s u p e r i o r t o those t y p i c a l of g r a v i t y and m a g n e t i c s ; t h i s statement i s borne out by the uniqueness theorem proved i n Appendix C below. In p r a c t i c e , however, the advantage of e x t r a s e n s i t i v i t y i s somewhat o f f s e t by the accompanying i n t e r p r e t a t i o n a l d i f f i c u l t i e s . The n o n - l i n e a r i t y of the h o r i z o n t a l l o o p problem i s most r e a d i l y demonstrated u s i n g the i n t e g r a l e q u a t i o n f o r m u l a t i o n p r e s e n t e d i n 1-3. For s i m p l i c i t y of e x p o s i t i o n , and because c o n d u c t i v e zones are of paramount i n t e r e s t i n t h i s s t u d y , the d i e l e c t r i c c o n s t a n t of the ground w i l l be assumed e q u a l t o the f r e e space p e r m i t t i v i t y , and o n l y v a r i a t i o n s i n c o n d u c t i v i t y w i l l be c o n s i d e r e d h e r e a f t e r . On t h i s b a s i s , the c u r r e n t d e n s i t y at any p o i n t i s due e n t i r e l y t o c o n d u c t i o n , and so from ( 1 . 3 2 ) , ( 1 . 3 0 ) , and (1.64) the secondary e l e c t r i c f i e l d a t the s u r f a c e may be w r i t t e n as A E ( A , 0 , w ) = -iji2 ^*exp(-u < >z)E( A , z , w )cr( z ) dz/2u„ (2.3) e I n v o k i n g (1.75) and p e r f o r m i n g i n v e r s e Hankel t r a n s f o r m a t i o n of (2.3) i n accordance w i t h (1.66) and ( 1 . 6 7 ) , the secondary magnetic f i e l d s may be e x p r e s s e d i n the form (2.2) : 57 \ A H m ( r , 0 , W j ) = fr ( r, z , w: ; <r )cr{ z )dz m=0,1 (2.4) o where the k e r n e l s , i f ] , are d e f i n e d by £ < r f z , w j ) = (-1) m (2.5) o w i t h m=0 f o r A H z and m=l f o r A H r as i n 1-6. S i n c e the e l e c t r i c f i e l d i s n e c e s s a r i l y a f f e c t e d by the c o n d u c t i v i t y , i t i s c l e a r t h a t the k e r n e l s a r e "model dependent" and hence t h a t the HLEM problem i s n o n - l i n e a r . For a l i n e a r i n v e r s e problem, c o n s t r u c t i o n of one or more models s a t i s f y i n g a g i v e n s et of o b s e r v a t i o n s does not u s u a l l y pose any major d i f f i c u l t i e s because i t i s n e c e s s a r y o n l y t o s o l v e a system of l i n e a r e q u a t i o n s . In n o n - l i n e a r problems, however, c o n s t r u c t i o n of an a c c e p t a b l e model i s n o r m a l l y a c o n s i d e r a b l e u n d e r t a k i n g , and the o n e - d i m e n s i o n a l HLEM problem i s not an e x c e p t i o n t o t h i s r u l e . The d i r e c t i n v e r s i o n f o r m u l a d e r i v e d i n Appendix C, f o r example, i s u n s u i t a b l e f o r r o u t i n e i m p l e m e n t a t i o n w i t h r e a l d a t a owing t o i t s s e n s i t i v i t y t o e r r o r s ; t h i s i s a common l i m i t a t i o n of a n a l y t i c a l i n v e r s i o n f o rmulae ( W e i d e l t , 1972, p285; B a i l e y , 1973, p239; Oldenburg and Samson, 1979, p934). A more " r o b u s t " , l i n e a r i s e d c o n s t r u c t i o n p r o c e d u r e was proposed by Backus and G i l b e r t (1967, p256), and has been a p p l i e d s u b s e q u e n t l y t o a number of d i f f e r e n t problems w i t h s u c c e s s ( P a r k e r , 1970; Johnson and G i l b e r t , 1972; Oldenburg, 1978, 1979); i t i s t h i s method which has been i n c o r p o r a t e d i n t o HLEM.INV. The p h i l o s o p h y u n d e r l y i n g B a c k u s - G i l b e r t c o n s t r u c t i o n i s 58 i t e r a t i v e improvement of a s t a r t i n g model v i a the s o l u t i o n of a s u c c e s s i o n of l i n e a r problems, and as such i s not new (Anderssen, 1975, p292). Backus and G i l b e r t have adapted the method of l i n e a r i s a t i o n f o r g e o p h y s i c a l a p p l i c a t i o n s , c h i e f l y t h r o ugh the p o p u l a r i s a t i o n of the concept of a F r e c h e t d e r i v a t i v e (Dunford and S c h w a r t z , 1958, p92). F r e c h e t d e r i v a t i v e s r e l a t e a s m a l l change, 6m, i n a model t o s m a l l changes [£cj : j=l,N} i n the r e s p o n s e s ; d e n o t i n g by {D.(z;m): j = l,N} the F r e c h e t k e r n e l s f o r the h y p o t h e t i c a l problem i n ( 2 . 2 ) , the concept i s c o d i f i e d i n the e q u a t i o n ensures t h a t the o b s e r v a t i o n s v a r y c o n t i n u o u s l y w i t h changes i n the model, as i l l u s t r a t e d by Backus and G i l b e r t (1967, pp249-250). Most g e o p h y s i c a l d a t a f u n c t i o n a l s a r e F r e c h e t d i f f e r e n t i a b l e , though Woodhouse (1976) has i d e n t i f i e d an e x c e p t i o n i n the c o n t e x t of f r e e o s c i l l a t i o n s . The F r e c h e t d i f f e r e n t i a b i l i t y of the o n e - d i m e n s i o n a l EM i n d u c t i o n problem has been e s t a b l i s h e d by P a r k e r (1977b). T h e r e f o r e the B a c k u s - G i l b e r t c o n s t r u c t i o n i s a p p r o p r i a t e i n the p r e s e n t c o n t e x t . The F r e c h e t d e r i v a t i v e s f o r the HLEM problem a r e d e r i v e d i m m e d i a t e l y below. Suppose t h a t E —> E + cfE = E + when crcr + Scr ; then from (2.6) where \&m\ i s the L 2-norm of <Jm(z). F r e c h e t d i f f e r e n t i a b i l i t y (1.15) E" - ( A 2 - y2S + i | 3 2 c r ) c f E = ifi2E*<S<r (2.7) 59 As p e r B i r k h o f f a n d R o t a ( 1 9 6 9 , p 4 2 ) , E i s an i n t e g r a t i n g f a c t o r f o r ( 2 . 7 ) , whence oo EJE' - E'<5E = - i / 3 2 J EE*6<r dy ( 2 . 8 ) z where t h e b o u n d a r y c o n d i t i o n E | ^ = 0 h a s been a p p l i e d . From ( 1 . 7 5 ) a n d ( 1 . 6 4 ) i t f o l l o w s t h a t <*E'(A,0,w) = (AE + -AE)| = u i ^ E + ' -AE ' ) I = uA<*E(A,0,w) ( 2 . 9 ) T h e r e f o r e , [EJE' - E'<5E]| = [ ( u „ E - E')<JE]| B u t f r o m ( 1 . 7 1 ) , [ueE - E ' ] | r s < ? = 2u,B* where B£ i s t h e e f f e c t i v e s o u r c e s p e c t r u m d e f i n e d i n ( 1 . 7 0 ) . H e n c e , t h e l e f t h a nd s i d e o f ( 2 . 8 ) may be s i m p l i f i e d c o n s i d e r a b l y a t z=0, t o y i e l d c?E(A,0,w) = CE( A , Z , W ) E + ( A ,z,w)<f<r(z)dz ( 2 . 1 0 ) 2u„B* i T h i s r e s u l t h a s been o b t a i n e d p r e v i o u s l y ( T i k h o n o v , 1 9 6 5 , p210; P a r k e r , 1977b, p 5 4 5 ) , a n d may be r e c o g n i s e d a s a more g e n e r a l v e r s i o n o f ( 2 . 3 ) . A f t e r H a n k e l - t r a n s f o r m a t i o n , ( 2 . 1 0 ) 60 e x p r e s s e s e x a c t l y t h e c h a n g e i n H a n k e l - t r a n s f o r m e d e l e c t r i c f i e l d due t o a c h a n g e i n c o n d u c t i v i t y . A s . s u c h i t i s n o t v e r y u s e f u l f o r i n v e r s i o n , b e c a u s e E + ( A,z,w) i s unknown. However, i f o n l y " s m a l l " c h a n g e s i n cr" a r e a l l o w e d , - i A2 0 0 J E ( A,0,W ) ~ (E 2 ( A ,z,w ) < £ r(z)dz ( 2 . 1 1 ) T h i s l i n e a r i s e d e q u a t i o n becomes e x a c t i n t h e l i m i t a s | | c f c r | | — ^ 0. The c o r r e s p o n d i n g e x p r e s s i o n f o r d E ' ( A , 0 , w ) f o l l o w s r e a d i l y u s i n g ( 2 . 9 ) . I f s o u n d i n g s a r e t a k e n a t N f r e q u e n c i e s , {WJ : j = l , N } , N e q u a t i o n s o f t h e f o r m ( 2 . 6 ) r e s u l t ; s p e c i f i c a l l y 00 <$Ej = 6e{ X , 0 , WJ ) us ^Dej( A ,z;cr )6<r ( z ) d z ( 2 . 1 2 ) where De.( A , z; C" ) i s t h e H a n k e l - t r a n s f ormed F r e c h e t k e r n e l f o r J e l e c t r i c f i e l d a t a n g u l a r f r e q u e n c y w: , d e f i n e d by -i&2 De ; ( A,z;£T) = E 2 ( A,z,w:) ( 2 . 1 3 ) ' 2u.B* R e c a l l i n g ( 2 . 9 ) , t h e c o r r e s p o n d i n g F r e c h e t k e r n e l s f o r H a n k e l - t r a n s f o r m e d Hr d a t a a r e g i v e n by D r . ( A , z ; c r ) = u dDe ( A,z ; < r ) ( 2 . 1 4 ) * l The F r e c h e t k e r n e l s f o r m e a s u r a b l e e l e c t r i c a n d m a g n e t i c f i e l d c o m p o n e n t s may be d e t e r m i n e d f r o m ( 2 . 1 3 ) a n d ( 2 . 1 4 ) by 61 i n v e r s e Hankel t r a n s f o r m a t i o n . Hence, u s i n g ( 1 . 1 6 ) , ( 1 . 6 6 ) , and ( 1 . 6 7 ) , CO De(r,z;<r) = ^De ( A , z; cr-) A J,( A r ) d A o oo D r ( r , z ; < r ) = - i j" u eDe ( A , z ; cr )\ J,( A r ) d A ^ (2.15) • o oo Dz ( r , z ; cr ) = i De ( A , z ; cr ) \2 Jtf( A r ) d A For p i e c e - w i s e c o n s t a n t models, the i n t e g r a t i o n over z i n (2.12) can be r e p l a c e d by a summation w i t h o u t a p p r o x i m a t i o n , and the elements of a m a t r i x , of d i s c r e t e F r e c h e t k e r n e l s can be d e f i n e d u s i n g ( 2 . 1 3 ) , ( 2 . 1 4 ) , and ( 2 . 1 5 ) : oo 2k+» ^ j k = ( - D m i ^ 2 m J m ( A r ) u ^ d A ^ D e j( A ,z; cr ) d z / h k (2 .16) 2 k where h f c = z k H - zk i s the t h i c k n e s s of the k t h l a y e r , as per F i g . 1 . 1 . The d i v i s i o n by h i s e f f e c t e d so t h a t A has the p h y s i c a l d i m e n s i o n s of a F r e c h e t k e r n e l ; t h i s c h o i c e of d e r i v a t i v e m a t r i x f a c i l i t a t e s comparison w i t h the c o n t i n u o u s f o r m u l a t i o n of Backus and G i l b e r t . In terms of the d i s c r e t e k e r n e l s , changes i n observ e d magnetic f i e l d s may be e x p r e s s e d as f o l l o w s : NL <fH m(r,0,w ) ~ 2l A 7 k h k c f ^ (2.17) where NL i s the number of l a y e r s . E q u a t i o n (2.17) i s the d i s c r e t e analogue of ( 2 . 6 ) . 62 C o n s t r u c t i o n commences w i t h the s e l e c t i o n of a s t a r t i n g model, and the computation of the c o r r e s p o n d i n g responses and F r e c h e t k e r n e l s . The r e s i d u a l s between the t h e o r e t i c a l responses and the o b s e r v a t i o n s are then d e t e r m i n e d , and a l i n e a r system of the form (2.17) i s s o l v e d f o r cfcr-. The p e r t u r b a t i o n i s added t o the o r i g i n a l c o n d u c t i v i t y , and the m o d i f i e d c o n d u c t i v i t y i s then t r e a t e d as the " s t a r t i n g model" f o r the next i t e r a t i o n . The p r o c e s s i s r e p e a t e d u n t i l a model i s o b t a i n e d s a t i s f y i n g the d a t a t o some p r e - d e t e r m i n e d a c c u r a c y . S a b a t i e r (1974) and Anderssen (1975) have e x p r e s s e d m i s g i v i n g s about t h i s l i n e a r i s e d c o n s t r u c t i o n p r o c e d u r e . T h e i r two p r i n c i p a l c o n c e r n s are ( i ) t h a t convergence of the i t e r a t i v e scheme i s not a s s u r e d , and ( i i ) t h a t the n o n - l i n e a r problem may admit more than one s o l u t i o n . The second o b s e r v a t i o n r a i s e s the s c e p t r e of non-uniqueness, which i s c o n s i d e r e d more f u l l y i n the next c h a p t e r . I t s u f f i c e s here t o p o i n t out t h a t uniqueness i s an i m p o r t a n t c o n s i d e r a t i o n i n any g e o p h y s i c a l i n v e r s i o n scheme: the e x t e n t . t o which a f i n i t e s e t of n o i s y d a t a s p e c i f y a p a r t i c u l a r s t r u c t u r e i s always u n c e r t a i n , independent of the method of model c o n s t r u c t i o n . The b e s t , a l b e i t i n c o n c l u s i v e , response t o the f i r s t p o i n t i s p r o v i d e d by the demonstrated "good b e h a v i o u r " of the B a c k u s - G i l b e r t c o n s t r u c t i o n p r o c e d u r e whenever the d e s i r e d model i s - n o t r e q u i r e d t o f i t n o i s y d a t a e x a c t l y . Indeed, ,in o n e - d i m e n s i o n a l r e s i s t i v i t y and MT i n v e r s i o n , r a p i d convergence has been a c h i e v e d w i t h l i n e a r i s e d c o n s t r u c t i o n u s i n g s t a r t i n g models t o t a l l y d i f f e r e n t from any s u c c e s s f u l model (Oldenburg, 1978, 1979); such f e a t s s u r p a s s r i g o r o u s m a t h e m a t i c a l e x p e c t a t i o n s of performance. I t i s 6 3 i n t e r e s t i n g t o s p e c u l a t e t h a t the reason f o r the s t a b i l i t y o bserved by Oldenburg may l i e i n the f a c t t h a t uniqueness theorems e x i s t f o r the problems he c o n s i d e r e d (Langer, 1933; B a i l e y , 1970). 11-2 Computat ion of the F r e c h e t K e r n e l s The i n t e g r a l w i t h r e s p e c t t o z i n (2.16) may be e v a l u a t e d u s i n g a for m u l a d e r i v e d below. Observe from (1.35) t h a t i n the k t h l a y e r zk.+i 2k+t *k+i ^ E 2 d z = jEE"dz/u£ = { [ E E ' ] 2 k * ' - j"(E' ) 2 d z } / u 2 (2.18) 2 k 2k Z k 2k Adding ^E 2dz t o both s i d e s , and deducing from (1.37) t h a t E 2 - ( E ' / u f c ) 2 = 4A f eB f e ( 2 . 1 9 ) , i t f o l l o w s t h a t C zk+i l E 2 d z = 0.5u" 2[EE'] + 2A kB kh f e (2.20) *k Thus the computation of the d i s c r e t e k e r n e l s r e q u i r e s l i t t l e n u m e r i c a l l a b o u r over and above t h a t a l r e a d y r e q u i r e d f o r d e t e r m i n a t i o n of the s u r f i c i a l i n p u t impedance, Z 1, u s i n g ( 1 . 4 6 ) . In HLEM.INV the i n p u t impedance a t each i n t e r f a c e i s s t o r e d d u r i n g the computation of Z 1, and the i n t e r f a c i a l v a l u e s of E and E' r e q u i r e d by (2.20) a r e computed from ( 1 . 3 8 ) , ( 1 . 4 4 ) , and the r e c u r s i o n r e l a t i o n ( 1 . 5 1 ) , r e p e a t e d here f o r 64 c o n v e n i e n c e : Z + Z»-t Here B^(A ,w) i s the a m p l i t u d e of the down-going wave i n the n t h l a y e r f o r mode A, Z n i s the i n t r i n s i c impedance of t h a t l a y e r , and Z n i s the i n p u t impedance a t the n t h i n t e r f a c e . U s i n g ( 2 . 2 1 ) , the i n t e r f a c i a l v a l u e s of E and E' a r e d e t e r m i n e d r e c u r s i v e l y a t p r o g r e s s i v e l y g r e a t e r d e p t h s , b e g i n n i n g w i t h t h e i r known v a l u e s a t the s u r f a c e . W ith one e x c e p t i o n , a l l the Hankel t r a n s f o r m i n t e g r a l s a r i s i n g from s u b s t i t u t i o n of (2.20) i n (2.16) a r e s u i t a b l e f o r e v a l u a t i o n u s i n g the l i n e a r d i g i t a l f i l t e r i n g of Koefoed e t a l . (1972). The e x c e p t i o n i s the i n t e g r a l i n v o l v i n g E E ' \ z s 0 , f o r which the i n t e g r a n d decays s l o w l y as A —> <** when h=0, c f . (1 . 7 8 ) . As i n 1-7, the d i f f i c u l t y i s overcome by s u b t r a c t i n g the large-A l i m i t from the i n t e g r a n d , thus i n d u c i n g s u f f i c i e n t l y r a p i d decay w i t h A t o a l l o w a c c u r a t e n u m e r i c a l e v a l u a t i o n . The i n t e g r a l of the remainder, i . e . of the large-A l i m i t i t s e l f , may be d e t e r m i n e d u s i n g s t a n d a r d f o r m u l a e . The d e t a i l s a re p r e s e n t e d i n Appendix B. 11-3 S o l u t i o n of a L i n e a r I n v e r s e Problem i n the Presence  of E r r o r s D u r i n g each i t e r a t i o n of the c o n s t r u c t i o n p r o c e d u r e i t i s n e c e s s a r y t o s o l v e a system of l i n e a r e q u a t i o n s of the form (2.17) f o r the d e s i r e d p e r t u r b a t i o n , Scr. Lanczos (1961) has 65 d e s c r i b e d how s m a l l e r r o r s i n o b s e r v a t i o n s can cause d e b i l i t a t i n g l y l a r g e e r r o r s i n the s o l u t i o n of a l i n e a r system. S i n c e e r r o r s a re an u n a v o i d a b l e component i n a l l g e o p h y s i c a l measurements, i t i s e s s e n t i a l t o mute t h e i r i n f l u e n c e i n any i n v e r s i o n p r o c e d u r e . The b a s i c s t r a t e g y i s s i m p l y t o s o l v e the e q u a t i o n s i m p e r f e c t l y , t o a p r e c i s i o n c o n s i s t e n t w i t h the exp e c t e d l e v e l of e r r o r i n the o b s e r v a t i o n s . S t a t i s t i c a l c r i t e r i a a r e employed t o c o n t r o l the a c c u r a c y of f i t d u r i n g each i t e r a t i o n . G l o b a l convergence and the n o t i o n of an " a c c e p t a b l e model" a r e a l s o d e f i n e d i n terms of t h e s e c r i t e r i a . The m a t h e m a t i c a l model f o r the e a r t h i s n e c e s s a r i l y h i g h l y s i m p l i f i e d , and the d e v i a t i o n s of the t r u e e a r t h from the assumed form w i l l i n t r o d u c e e r r o r s , i n t o the i n v e r s i o n . These e r r o r s a r i s e e n t i r e l y from the d e f i c i e n c i e s of the t h e o r e t i c a l model, and a r e a d d i t i o n a l t o the u s u a l o b s e r v a t i o n a l u n c e r t a i n t i e s which have d i r e c t p h y s i c a l c a u s e s , e.g. magnetospheric n o i s e i n HLEM so u n d i n g s . In p r a c t i c e the two d i s t i n c t c o n t r i b u t i o n s t o the e r r o r cannot be s e p a r a t e d , and w i l l be t r e a t e d t o g e t h e r i n what f o l l o w s . I t w i l l be assumed t h a t the net e r r o r s a r e r e a l i s a t i o n s of Normal random v a r i a b l e s , { Q j : j = l , N } , w i t h mean z e r o and v a r i a n c e { q j 2 : j = l , N } . In o r d e r t o a s s e s s whether or not the f i t p r o v i d e d by an approximate s o l u t i o n i s s t a t i s t i c a l l y c o n s i s t e n t , i t w i l l be n e c e s s a r y t o d e f i n e "measures of m i s f i t " . The measures used i n HLEM.INV a r e the squared m i s f i t , x 2 , and the L -norm m i s f i t , x 1 . I f o i s a v e c t o r of o b s e r v a t i o n s and c* a v e c t o r of t h e o r e t i c a l r e s p o n s e s , x 1 and x 2 a r e d e f i n e d by 66 (2.22) (2.23) where ij = (oj - c; )/q- (2.24) S i n c e the t h e o r e t i c a l responses a r e known e x a c t l y , the j t h r e s i d u a l , y^. , has the same e r r o r as the j t h o b s e r v a t i o n , oj . N o r m a l i s a t i o n w i t h r e s p e c t t o s t a n d a r d d e v i a t i o n ensures t h a t x 1 and x 2 are not. dominated by c o n t r i b u t i o n s from the n u m e r i c a l l y l a r g e d a t a . The m i s f i t s may be regard e d as r e a l i s a t i o n s of the random v a r i a b l e s X 1 and X 2, where N ^ N X 2 = S I S . 2 (2.26) w i t h Qj =.Qj/qj (2.27) I t i s now p o s s i b l e t o d e f i n e the term " s t a t i s t i c a l l y c o n s i s t e n t f i t " w i t h r e s p e c t t o the p r o b a b i l i t y d i s t r i b u t i o n s of the random v a r i a b l e s X 1 and X 2. I f the e r r o r s a r e independent, X 2 i s d i s t r i b u t e d as a c h i - s q u a r e d v a r i a b l e w i t h N degrees of freedom ( J e n k i n s and Watts, 1969, p79). T h e r e f o r e , £[X 2] = N (2.28) V a r [ X 2 ] = 2N (2.29) 67 The p r o b a b i l i t y d i s t r i b u t i o n o f X 1 i s not so f a m i l i a r , but i t s moments may be computed n o n e t h e l e s s , a s w i l l now be d e m o n s t r a t e d . By i n s p e c t i o n of t h e i d e n t i t y i Q j / q j I = yspq] A i t may be c o n c l u d e d t h a t |Q.| i s t h e s q u a r e r o o t of a c h i - s q u a r e d v a r i a b l e w i t h one d e g r e e o f f r e e d o m . Hence, as p e r J e n k i n s and W a t t s (1969, p71 and p 7 9 ) , oo •CC I Qj I 3 = { e x p ( - y / 2 ) d y / / 2 7 T ^Jl/n o T h e r e f o r e , r e c a l l i n g ( 2 . 2 5 ) , 8'X 1 ] = J2/TI N (2.30) When t h e e r r o r s a r e i n d e p e n d e n t V a r ' X 1 ] can be computed i n an a n a l o g o u s manner. S p e c i f i c a l l y , OC V a r [ | Q ; | ] = { ( v T ~fi/n ) 2 e x p ( - y / 2 ) d y / y / 2 n y = 1 - ^ o whence V a r f X 1 ] = ( 1 - 2 / T T ) N (2.31) The f o r m u l a e (2.30) and (2.31) have been d e r i v e d i n d e p e n d e n t l y by P a r k e r and McNutt ( 1 9 8 0 ) . The f i t t o t h e d a t a i s c o n s i d e r e d " s t a t i s t i c a l l y c o n s i s t e n t " w i t h r e s p e c t t o x 1 or x 2 r e s p e c t i v e l y when 68 N - N S D > / ( ) N < x 1 < N + N S D V / ( 1 - 2 / T T ) N ( 2 . 3 2 ) N - N S D ^ 2 N < x 2 < N + N S D / 2 N ( 2 . 3 3 ) where NSD i s a p a r a m e t e r s u p p l i e d by t h e u s e r . F o r i n d e p e n d e n t e r r o r s t h e C e n t r a l L i m i t Theorem d e c r e e s t h a t X 1 and X 2 a r e a l m o s t Normal random v a r i a b l e s f o r " l a r g e " N; i n t h i s c o n n e c t i o n , t e n may be r e g a r d e d as l a r g e ( s e e F i g . 3.9, J e n k i n s and W a t t s , 1 9 6 9 ) . Thus when N i s l a r g e t h e f a m i l i a r G a u s s i a n c o n f i d e n c e l i m i t s may be i n v o k e d t o g u i d e t h e c h o i c e of NSD. I t has been a r g u e d t h a t t h e p e r t u r b a t i o n e q u a t i o n s s h o u l d not be s a t i s i f i e d e x a c t l y , and t h e a p p r o p r i a t e l e v e l of m i s f i t has been s p e c i f i e d . I t r e m a i n s now t o d e s c r i b e t h e method o f ( a p p r o x i m a t e ) s o l u t i o n . A s p e c t r a l e x p a n s i o n method i s employed i n t h e c o n s t r u c t i o n s u b r o u t i n e s of HLEM.INV ( P a r k e r , 1977a, p p 4 6 - 5 4 ) . The p r i n c i p a l a t t r a c t i o n of t h e s p e c t r a l e x p a n s i o n a p p r o a c h i s t h a t i t e n a b l e s t h e most u n r e l i a b l e components of t h e s o l u t i o n t o be r e c o g n i s e d and s u b s e q u e n t l y d i s c a r d e d , as shown below. The a i m o f c o n s t r u c t i o n i s t o f i n d a model s a t i s f y i n g t h e N o b s e r v a t i o n s , o. L e t "c be t h e r e s p o n s e s of a s t a r t i n g model cr-( r e p r e s e n t e d h e r e as a v e c t o r of NL l a y e r c o n d u c t i v i t i e s ) , and l e t D be t h e r e l e v a n t m a t r i x o f F r e c h e t d e r i v a t i v e s . In so* l i n e a r i s e d c o n s t r u c t i o n a new model cr1 = cr-+ der w h i c h f i t s t h e d a t a more c l o s e l y i s d e t e r m i n e d v i a t h e s o l u t i o n of a s y s t e m o f l i n e a r e q u a t i o n s of t h e form 69 NL where D;k = D j k /q. and where 77. i s d e f i n e d i n (2 . 2 4 ) . S c a l i n g the e q u a t i o n s w i t h r e s p e c t t o the s t a n d a r d d e v i a t i o n s {q} i s c o n v e n i e n t i n view of the d e f i n i t i o n s (2.22-23), and a l s o reduces the system t o d i m e n s i o n l e s s form. When s o l v i n g (2.34) the p e r t u r b a t i o n i s assumed t o be a l i n e a r c o m b i n a t i o n of the k e r n e l s : = D T £ (2.36) f o r some v e c t o r of c o e f f i c i e n t s , b. The p e r t u r b a t i o n of the form (2.36) i s the s m a l l e s t s o l u t i o n ( i n the l e a s t squares s e n s e ) . I t i s i n t u i t i v e l y r e a s o n a b l e t h a t the s m a l l e s t s o l u t i o n b e l o n g s t o the subspace spanned by the F r e c h e t k e r n e l s because any component of which i s o r t h o g o n a l t o t h a t subspace w i l l i n c r e a s e ll^^ll w i t h o u t a f f e c t i n g c o m p l i a n c e w i t h the o b s e r v a t i o n s (Hoffman and Kunze, 1971, p284). S u b s t i t u t i n g (2.36) i n t o ( 2 . 3 4 ) , the p e r t u r b a t i o n e q u a t i o n s become ?J = Jib (2.37) where _/2 i s the i n n e r p r o d u c t m a t r i x , w i t h elements d e f i n e d by NL -^ jh-£A *Ak (2-38) j = l , N (2.34) (2.35) 70 The symmetry o f J2. g u a r a n t e e s t h a t i t a d m i t s a s p e c t r a l d e c o m p o s i t i o n of t h e form J 2 = U A 2 U T (2.39) where A i s a d i a g o n a l m a t r i x of e i g e n v a l u e s , and where U i s an NxN o r t h o g o n a l m a t r i x of " d a t a e i g e n v e c t o r s " . These v e c t o r s c o n s t i t u t e a b a s i s f o r t h e N - d i m e n s i o n a l " d a t a s p a c e " . S i m i l a r l y , i t i s p o s s i b l e t o d e f i n e a r e l a t e d s e t of o r t h o n o r m a l b a s i s v e c t o r s f o r t h e N L - d i m e n s i o n a l model s p a c e . The b a s i s v e c t o r s f o r model s p a c e c o m p r i s e t h e rows of t h e NxNL m a t r i x , where "VP = A _ 1 U T D (2.40) The b a s i s v e c t o r s i n c o r r e s p o n d t o t h e " p a r a m e t e r e i g e n v e c t o r s " of W i g g i n s (1972, p 2 5 6 ) . The o r t h o n o r m a l i t y o f t h e b a s i s i s c o d i f i e d i n t h e e q u a t i o n ^ H f = I (2.41) where H i s a d i a g o n a l m a t r i x of l a y e r t h i c k n e s s e s and where I i s t h e NxN i d e n t i t y m a t r i x . In a c c o r d a n c e w i t h (2.36) and ( 2 . 4 0 ) , t h e d e s i r e d s o l u t i o n o f (2.34) a d m i t s t h e e x p a n s i o n Sir = -ysc (2.42) 7 1 w h e r e oc = A u T b ( 2 . 4 3 ) M u l t i p l y i n g ( 2 . 3 7 ) f r o m t h e l e f t b y A _ 1 y T , a n d r e c a l l i n g ( 2 . 3 9 ) a n d ( 2 . 4 3 ) , oc m a y b e e x p r e s s e d i n t e r m s o f t h e d i m e n s i o n l e s s r e s i d u a l s : oi = A - 1 U T ^ ( 2 . 4 4 ) I t m a y b e s u r m i s e d b y i n s p e c t i o n o f ( 2 . 4 4 ) t h a t a n y e r r o r s i n o ( a n d h e n c e *J£ v i a ( 2 . 2 4 ) ) w i l l b e g r e a t l y m a g n i f i e d b y s m a l l e i g e n v a l u e s . M a t h e m a t i c a l l y s p e a k i n g , s m a l l e i g e n v a l u e s a r i s e w h e n t h e k e r n e l s a r e n e a r l y l i n e a r l y d e p e n d e n t ; p h y s i c a l l y , t h e i r o c c u r r e n c e m a y b e i n t e r p r e t e d a s a n i n d i c a t i o n o f a c e r t a i n r e d u n d a n c y i n t h e m e a s u r e m e n t s . A l t h o u g h r e d u n d a n c y c a n b e r e d u c e d b y t h o u g h t f u l s u r v e y d e s i g n ( G l e n n a n d W a r d , 1 9 7 6 , p l 2 1 1 ) , i t c a n n o t b e c o m p l e t e l y e l i m i n a t e d . T h e u b i q u i t y o f r e d u n d a n c y i s u n d e r s c o r e d b y t h e o b s e r v a t i o n o f B a c k u s a n d G i l b e r t ( 1 9 6 7 , p 2 5 1 ) t h a t i t i s u s u a l f o r t h e k e r n e l s t o b e " a l m o s t , b u t n o t q u i t e , l i n e a r l y d e p e n d e n t " . T h e c l o s e c o n n e c t i o n b e t w e e n s m a l l e i g e n v a l u e s a n d t h e p r o p a g a t i o n o f e r r o r s i s f u r t h e r i l l u s t r a t e d b y c o n s i d e r a t i o n o f t h e v a r i a n c e s o f t h e c o m p o n e n t s o f oc . B y t h e r u l e f o r c o m p o s i t i o n o f v a r i a n c e f o r i n d e p e n d e n t r a n d o m v a r i a b l e s , i t f o l l o w s f r o m ( 2 . 4 4 ) t h a t N A Var[<x/] = A j 2 2 l ( U j i )2Var[Qj] = \j2 ( 2 . 4 5 ) w h e r e t h e u n i m o d u l a r i t y o f t h e e i g e n v e c t o r s i n U h a s b e e n 72 i n v o k e d , and where i t s h o u l d be r e c a l l e d from (2.27) t h a t each A random v a r i a b l e Q- has u n i t v a r i a n c e . Thus, the r e l i a b i l i t y of each c o e f f i c i e n t i n the o r t h o n o r m a l e x p a n s i o n (2.42) d e t e r i o r a t e s i n v e r s e l y w i t h the magnitude of the c o r r e s p o n d i n g e i g e n v a l u e . In c o n j u n c t i o n w i t h the o b s e r v a t i o n t h a t the c h a r a c t e r i s t i c "wavelength" of the b a s i s f u n c t i o n s i s s h o r t e r f o r the s m a l l e r e i g e n v a l u e s ( W i g g i n s , 1972, p258; P a r k e r , 1977a, p54), (2.45) s e r v e s as an i l l u s t r a t i o n of the t e n e t t h a t s m a l l s c a l e s t r u c t u r e may be d i s c e r n e d o n l y a t the expense of a c c u r a c y . C o n s e q u e n t l y , l o n g wavelength f e a t u r e s i n d " c r a r e d e t e r m i n e d w i t h r e a s o n a b l e c e r t a i n t y , whereas the (sometimes w i l d ) o s c i l l a t i o n s which a r e the l e g a c y of s m a l l e i g e n v a l u e s can u s u a l l y be d i s r e g a r d e d , on b oth p h y s i c a l and t h e o r e t i c a l grounds. The s m a l l e i g e n v a l u e s a r e a l s o the most p o o r l y d e t e r m i n e d n u m e r i c a l l y , and can sometimes be d i s c a r d e d on c o m p u t a t i o n a l grounds a l o n e . I t i s t h e r e f o r e u n d e s i r a b l e t o r e t a i n the terms a s s o c i a t e d w i t h the s m a l l e s t e i g e n v a l u e s i n the e x p a n s i o n (2.42) because the s e t erms obscure the p h y s i c a l sense of the s o l u t i o n . A l l terms must be i n c l u d e d i f an e x a c t s o l u t i o n i s r e q u i r e d , but a p r e - o c c u p a t i o n w i t h m a t h e m a t i c a l p r e c i s i o n i s u n j u s t i f i e d i f the d a t a a r e known t o c o n t a i n e r r o r s , and can l e a d t o p h y s i c a l l y m e a n i n g l e s s r e s u l t s ( L a n c z o s , 1961, p l 3 1 ) . The p r a c t i c e of "winnowing" e i g e n v a l u e s s m a l l e r than a c e r t a i n " c u t - o f f " v a l u e i s a common f e a t u r e of i n v e r s i o n a l g o r i t h m s . There i s no u n e q u i v o c a l r u l e f o r s e l e c t i n g the " c u t o f f " , and Wiggins (1972, pp260-261) and S a b a t i e r (1977a, p l 2 5 ) r e f e r t o a number of d i f f e r e n t s t r a t e g i e s . In HLEM.INV a c h i - s q u a r e d c r i t e r i o n has been adopted, f o l l o w i n g P a r k e r 73 ( 1 9 7 7 a ) . E i g e n v a l u e s (and t h e i r a s s o c i a t e d e i g e n v e c t o r s ) a r e d i s c a r d e d i n i n c r e a s i n g o r d e r u n t i l t h e s q u a r e d m i s f i t , x 2 , e x c e e d s i t s maximum p e r m i s s i b l e v a l u e , X2MAX. In o t h e r words, t h e winnowing c o n t i n u e s u n t i l x£ > N + NSD^2N = X2MAX (2.46) where x£ i s t h e x 2 - m i s f i t when L e i g e n v a l u e s r e m a i n . A s i m p l e e x p r e s s i o n e x i s t s f o r x£ i n terms of oi. and t h e e i g e n v a l u e s of Jl. F o r t h e d e r i v a t i o n of t h e f o r m u l a , i t i s c o n v e n i e n t t o i n t r o d u c e t h e NxN m a t r i x U, wh i c h i s i d e n t i c a l t o U i n i t s f i r s t L c o l u m n s , b ut w h i c h c o n t a i n s z e r o s i n i t s l a s t N-L c o l u m n s . I f ">2L d e n o t e s t h e b e s t a p p r o x i m a t i o n t o -J^ i n terms of L e i g e n v e c t o r s , t h e n from (2.44) y} = u. AS? (2.47) Hence, t h e s q u a r e d m i s f i t when L e i g e n v e c t o r s r e m a i n i s g i v e n by *L - \n ~ \ I 2 = " « J A 5 ? | 2 where i t i s assumed w i t h o u t l o s s of g e n e r a l i t y t h a t A j > • • • • > A(^ • Even a f t e r a p p l i c a t i o n o f t h e c h i - s q u a r e d c r i t e r i o n , t h e a m p l i t u d e o f t h e "winnowed" p e r t u r b a t i o n i s sometimes t o o g r e a t f o r t h e l i n e a r i s a t i o n , ( 2 . 1 1 ) , t o be v a l i d . In t h i s e v e n t t h e m o d i f i e d c o n d u c t i v i t y , c r 1 , w i l l n ot c o n s t i t u t e an i m p r o v e d N = 2 1 A- oc? (2.48) 74 e s t i m a t e of an a c c e p t a b l e model u n l e s s the s i z e of the p e r t u r b a t i o n i s reduced. J u s t as f o r "winnowing", t h e r e a re a number of d i f f e r e n t s t r a t e g i e s f o r p e r t u r b a t i o n "containment" d e s c r i b e d i n the l i t e r a t u r e ; see, f o r example, Wiggins (1972, p259), or Glenn et a l . (1973, p l l 2 0 ) . There i s , m a n i f e s t l y , no u n i v e r s a l l y - a p p l i c a b l e c r i t e r i o n f o r r e c o n c i l i n g the d e s i r e f o r r a p i d convergence ( s a t i s f i e d by r e l a t i v e l y l a r g e p e r t u r b a t i o n s ) w i t h the confinement t o s m a l l a m p l i t u d e imposed by l i n e a r i s a t i o n . HLEM.INV employs the s i m p l e e x p e d i e n t of m u l t i p l y i n g &<r~ , the winnowed s o l u t i o n , by a c o n s t a n t k d e f i n e d by k = DLNMAX/| |max (2.49) where *T n = Scrn-/crn n = l,NL (2.50) Thus the s c a l i n g f a c t o r k i s chosen t o reduce |£7|max t o a p r e s c r i b e d v a l u e , DLNMAX. The c h o i c e of DLNMAX i s l e f t t o the u s e r ; i n p r a c t i c e a DLNMAX v a l u e of 0.5 has proved t o be s u i t a b l e a t the f r e q u e n c i e s used i n t h i s s t u d y , but a t h i g h e r f r e q u e n c i e s a somewhat s m a l l e r v a l u e would p r o b a b l y be more a p p r o p r i a t e . The a t t r a c t i o n of s c a l i n g w i t h r e s p e c t t o k i s t h a t the a c t u a l changes i n the responses c l o s e l y resemble the ex p e c t e d changes, k^ , i f k i s s u f f i c i e n t l y s m a l l , i . e . the a c t u a l changes w i l l tend t o be i n f i x e d p r o p o r t i o n t o the d e s i r e d changes. However, i f 6<r -/a" » DLNMAX i n some l a y e r s , a c c e p t i n g k<fcr~ as the d e s i r e d (approximate) s o l u t i o n c o u l d be 75 c o u n t e r - p r o d u c t i v e because the p e r t u r b a t i o n might be reduced t o an u n j u s t i f i a b l y s m a l l v a l u e i n some l a y e r s ; under t h e s e c i r c u m s t a n c e s | £ | can t r u n c a t e d a t a l e v e l DLNMAX i n s t e a d . To complete t h i s o u t l i n e of the c o n s t r u c t i o n p r o c e d u r e , i t remains o n l y t o c l a r i f y the meaning of the term " a c c e p t a b l e model", and hence t o d e f i n e the g l o b a l convergence c o n d i t i o n . In HLEM.INV convergence i s deemed t o have been a t t a i n e d when e i t h e r (2.32) or (2.33) i s s a t i s f i e d . When the e r r o r s a r e independent, the two c r i t e r i a a r e complementary. In p r a c t i c e , the c r i t e r i a d i f f e r t o the e x t e n t t h a t the x 1 - m i s f i t i s l e s s s t r o n g l y a f f e c t e d by the o c c u r r e n c e of one or two l a r g e r e s i d u a l s and i s more " r o b u s t " i n t h i s sense ( C l a e r b o u t and M u i r , 1973); thus x 2 - c o n v e r g e n c e i s a more s t r i n g e n t r e q u i r e m e n t than x 1 - c o n v e r g e n c e . 11-4 S p e c i a l F e a t u r e s of HLEM.INV 1 1 - 4 ( i ) Data O p t i o n s : At p r e s e n t HLEM.INV i s a b l e t o i n v e r t s i x t e e n d i f f e r e n t t ypes of d a t a . The magnetic components, Hr and Hz, may be i n v e r t e d i n d i v i d u a l l y or i n c o m b i n a t i o n . In a d d i t i o n , the program can i n v e r t w a v e - t i l t d a t a , where the w a v e - t i l t W i s d e f i n e d by Wj = W(r,0,Wj ) = Hr(r,0,wj )/Hz(r,0,Wj ) = Hrj /HZj (2.51) A s m a l l change, <5w, i n w a v e - t i l t i s r e l a t e d t o s m a l l changes i n 76 Hr and Hz a c c o r d i n g t o <$W = (JHr - W<$Hz)/Hz (2.52) Hence,, a t any f r e q u e n c y wj t h e w a v e - t i l t F r e c h e t k e r n e l D W J i s computed from t h e c o r r e s p o n d i n g Hr and Hz k e r n e l s , Drj and Dzj , as f o l l o w s : F o u r o p t i o n s a r e a v a i l a b l e f o r e a c h of t h e f o u r s e t s of m e a s u r a b l e s : 1. A m p l i t u d e o n l y 2. Phase o n l y 3. A m p l i t u d e and p h a s e 4. R e a l and i m a g i n a r y p a r t s To d e r i v e t h e F r e c h e t k e r n e l s f o r a m p l i t u d e and phase d a t a , o b s e r v e t h a t a s m a l l change i n an o b s e r v a b l e P may be e x p r e s s e d i n t e r m s o f s m a l l c h a n g e s i n t h e a m p l i t u d e , |P|, and p h a s e , t9p, Dw.( r , z ) = [ Dr.( r , z ) - Wj DZj( r , z ) ] /Hzj (2. 53) of P: Jp ~ 6 |P|exp(i0p) + iPcfflp (2.54) Hence, cf|P| = Re{<JP.exp(-i0p)} (2.55) cf0p = Im{c$P/P} 77 In d i r e c t analogy w i t h ( 2 . 5 5 ) , the F r e c h e t k e r n e l s f o r a m p l i t u d e and phase of the h y p o t h e t i c a l measurable P are g i v e n by: D f p J(r,z,w) = R e { D p ( r , z , w ) e x p [ - i 0 p ( r , 0 , w ) ]} (2.56) D 0 p(r,z,w) = Im{Dp(r,z,w)/P(r ,0,w) } (2.57) where Dp i s the F r e c h e t k e r n e l f o r P. O f t e n t h e r e i s no freedom as t o which of the d a t a o p t i o n s s h o u l d be used, e.g. o n l y |W| i s measured i n some contemporary EM systems ( S i n h a , 1980). I f , on the o t h e r hand, both Hr and Hz are c o m p l e t e l y known, a l l s i x t e e n o p t i o n s a r e a v a i l a b l e . A l t h o u g h i n some cases i n v e r s i o n of a p a r t i c u l a r measurable may prove t o be most e x p e d i t i o u s , i t i s u s u a l l y i m p o s s i b l e t o p r e d i c t b eforehand and i n p r a c t i c e the c h o i c e of o p t i o n i s l a r g e l y s u b j e c t i v e . T h i s l a t i t u d e i s not an embarrassment p r o v i d e d a l l d a t a o p t i o n s can be i n v e r t e d i n a r e l a t i v e l y s m a l l number of i t e r a t i o n s . The v i a b i l i t y of a l l d a t a o p t i o n s i n t h i s sense i s demonstrated i n I I - 5 below. The range of o p t i o n s a v a i l a b l e i n HLEM.INV i s not e x h a u s t i v e by any means. Some a d d i t i o n a l q u a n t i t i e s c o u l d be used as i n p u t a f t e r t r i v i a l m o d i f i c a t i o n s t o the code; f o r example, time d e r i v a t i v e s of magnetic f i e l d (measured d i r e c t l y by i n d u c t i o n c o i l s ) c o u l d be i n v e r t e d a f t e r i n c l u s i o n of a f a c t o r iw i n a l i m i t e d number of s t a t e m e n t s . I n v e r s i o n of more c o m p l i c a t e d q u a n t i t i e s such as e l l i p t i c i t y (Ryu e t a l . , 1970, p867) i s p o s s i b l e a l s o , though the m o d i f i c a t i o n s a r e a l i t t l e 78 more s u b s t a n t i a l . I n v e r s i o n of the magnetic components and w a v e - t i l t s has been f a v o u r e d i n t h i s study on the grounds of m a t h e m a t i c a l s i m p l i c i t y . I I - 4 ( i i ) M o d i f i c a t i o n of Depth P a r t i t i o n : One s h o r t c o m i n g of c o n v e n t i o n a l p a r a m e t r i c i n v e r s i o n i s the 'a p r i o r i ' i n i t i a l s p e c i f i c a t i o n of the number of l a y e r s . T h i s i s a s e r i o u s l i m i t a t i o n because the a l g o r i t h m may be unable t o converge i f t h e r e a re too few l a y e r s ( F i g . 2.1b). To c i r c u m v e n t t h i s problem, the number of l a y e r s may be i n c r e a s e d (up t o a p r e s c r i b e d maximum, NLMAX) i n HLEM.INV. More e x p l i c i t l y , the t h i c k n e s s of the k t h l a y e r i s h a l v e d i f l D i , k * r " D;k I > STEP | Djk | (2.58) f o r any sounding f r e q u e n c y w^  , where STEP i s an i n p u t parameter. On t h i s b a s i s , e x t r a l a y e r s a r e i n s e r t e d i n depth ranges where the F r e c h e t k e r n e l s undergo r a p i d v a r i a t i o n s . When STEP = 0, eve r y l a y e r i s h a l v e d a f t e r each i t e r a t i o n , whereas i f STEP i s l a r g e the performance of the r o u t i n e p a r a l l e l s t h a t of the c o n v e n t i o n a l programs. The p r e c i s e v a l u e of NLMAX i s not c r i t i c a l ( p r o v i d e d i t i s s u f f i c i e n t l y l a r g e f o r convergence) because the assessment of the p h y s i c a l s i g n i f i c a n c e of s t r u c t u r a l f e a t u r e s i s conducted i n d e p e n d e n t l y , as d i s c u s s e d i n Chapter I I I . M o d i f i c a t i o n of the p a r t i t i o n i s not n e c e s s a r y i f the number of l a y e r s i s chosen t o be l a r g e from the o u t s e t ; t h i s i s e f f e c t i v e l y the approach adopted by Backus and G i l b e r t . However, 79 s u c h a p o l i c y can r e s u l t i n n e e d l e s s e x p e n s e , e s p e c i a l l y when t h e s t a r t i n g model i s a p o o r e s t i m a t e of any a c c e p t a b l e m o d e l . M o r e o v e r , by e n a b l i n g s p u r i o u s s m a l l - s c a l e f e a t u r e s t o a p p e a r , e x c e s s l a y e r s can hamper c o n v e r g e n c e i f t h e c u r r e n t model i s f a r from an a c c e p t a b l e model; f o r t h i s r e a s o n t h e p a r t i t i o n i s n o t m o d i f i e d i n HLEM.INV u n l e s s x 2 < 10*X2MAX. An i n i t i a l c o a r s e p a r t i t i o n a c t s as an i n t r i n s i c " l o w - p a s s " f i l t e r , a l l o w i n g o n l y b r o a d s c a l e a l t e r a t i o n s of t h e s t a r t i n g m o d e l . The p a r t i t i o n becomes f i n e r as t h e i t e r a t i o n s p r o c e e d , a l l o w i n g s m a l l - s c a l e d e t a i l t o a p p e a r s u c h as i s n e c e s s a r y t o f i t t h e d a t a t o t h e d e s i r e d a c c u r a c y . F i g . 4.2 p r o v i d e s a good i l l u s t r a t i o n of t h e manner i n wh i c h f i n e r s t r u c t u r e i s p r o g r e s s i v e l y i n t r o d u c e d d u r i n g c o n s t r u c t i o n . 1 1 — 4 ( i i i ) E x p l i c i t T r e a t m e n t of E r r o r s : I t has been s t r e s s e d i n 11-3 t h a t t h e p e r t u r b a t i o n e q u a t i o n s (2.34) s h o u l d not be s o l v e d e x a c t l y when t h e d a t a a r e c o n t a m i n a t e d w i t h e r r o r s . Thus i n t h e s p e c t r a l e x p a n s i o n method an i n e x a c t s o l u t i o n i s d e v e l o p e d by d e g r a d i n g t h e e x a c t s o l u t i o n u n t i l t h e p r e d i c t e d m i s f i t e x c e e d s a c e r t a i n l e v e l , i n a c c o r d a n c e w i t h ( 2 . 4 6 ) . An a l t e r n a t i v e a p p r o a c h i s t o i n t r o d u c e t h e d i m e n s i o n l e s s r e s i d u a l s , f o r t h e p e r t u r b e d model, c?1 = cr- + &<r , i n t o t h e e q u a t i o n s e x p l i c i t l y as a d d i t i o n a l unknowns, and t o s o l v e s u b s e q u e n t l y f o r b o t h ^ 1 and Scr-. In t h i s c a s e t h e p e r t u r b a t i o n e q u a t i o n s become mScr + = (2.59) 80 ,where, i n analogy w i t h ( 2 . 2 4 ) , y^1 i s d e f i n e d as the n o r m a l i s e d d i f f e r e n c e between oj , the j t h o b s e r v a t i o n , and c.1 , the j t h response f o r the p e r t u r b e d model, c r 1 : T . 1 - (oj - c / ) / g j (2.60) E q u a t i o n (2.59) can be e x p r e s s e d more s i m p l y i n terms of an augmented k e r n e l m a t r i x , D +, an augmented " t h i c k n e s s m a t r i x " , H +, and an augmented v e c t o r of unknowns, 6~m: D + H+<$m = 7$ (2.61) where N L A D + = N N (2.62) N L N H + = H 0 I N L (2.63) N 81 j=l,NL (2.64) j =NL+1,NL+N The augmented system (2.61) may be s o l v e d by s p e c t r a l e x p a n s i o n as b e f o r e , and the s m a l l e s t s o l u t i o n d e t e r m i n e d . In p r a c t i c e the m i n i m i s a t i o n of |J'm||2 has a mod e r a t i n g e f f e c t on Sir i n the sense t h a t e x t r e m e l y l a r g e p e r t u r b a t i o n s (sometimes n e c e s s a r y t o s o l v e the da t a e x a c t l y i n the c o n v e n t i o n a l f o r m u l a t i o n ) become u n f a v o u r a b l e , and a c e r t a i n "minimum m i s f i t " , II^MI 2 ' i s t o l e r a t e d i n s t e a d . I f the s t a r t i n g model i s f a r from an a c c e p t a b l e model, \rf ||2 may be g r e a t e r than X2MAX, i . e . the minimum squared m i s f i t may exceed the l e v e l p r e s c r i b e d by the c h i - s q u a r e d c r i t e r i o n . There i s no c o n t r a d i c t i o n i f t h i s i s the case because i n the s p e c i f i c a t i o n of the l e v e l of a c c e p t a b l e m i s f i t , i t i s assumed t h a t the m i s f i t i s e n t i r e l y a t t r i b u t a b l e t o e r r o r s i n the d a t a ; i n f a c t the m i s f i t i s due t o e r r o r s a l o n e o n l y i f the p e r t u r b e d model, c^"1 , i s a c c e p t a b l e . Hence, the c h i - s q u a r e d c r i t e r i o n p r o v i d e s a lower bound on the exp e c t e d m i s f i t when the model i s f a r from an a c c e p t a b l e model because i t i s most u n l i k e l y t h a t convergence w i l l be a t t a i n e d i n a s i n g l e i t e r a t i o n under those c i r c u m s t a n c e s . The "minimum m i s f i t " a r i s i n g i n the m o d i f i e d f o r m u l a t i o n i s t h e r e f o r e a d e s i r a b l e f e a t u r e because i t reduces the l i k e l i h o o d t h a t the e q u a t i o n s w i l l be s a t i s f i e d too a c c u r a t e l y , and i n so d o i n g e x e r t s a m o d e r a t i n g i n f l u e n c e on the c o n d u c t i v i t y p e r t u r b a t i o n . As per ( 2 . 3 6 ) , the s m a l l e s t model i s a l i n e a r c o m b i n a t i o n of the augmented k e r n e l s : 82 dm = D + b + (2.65) where b + i s an unknown N-v e c t o r of c o e f f i c i e n t s . R e c a l l i n g (2.62) and ( 2 . 6 4 ) , i t f o l l o w s i m m e d i a t e l y from (2.65) t h a t b* = TJ1 (2.66) S u b s t i t u t i n g (2.65) and (2.66) i n t o ( 2 . 6 1 ) , where, on the b a s i s of (2.62) and ( 2 . 6 3 ) , the augmented i n n e r p r o d u c t m a t r i x i s g i v e n by SV = D + H + D + T= DHDT + 1 (2.68) Comparison of (2.68) and (2.38) r e v e a l s t h a t SL * d i f f e r s from the i n n e r p r o d u c t m a t r i x SL i n the c o n v e n t i o n a l f o r m u l a t i o n o n l y by the a d d i t i o n of the i d e n t i t y m a t r i x . S u b s t i t u t i n g from (2.39) i n t o (2.68) and i n v o k i n g the o r t h o n o r m a l i t y of the e i g e n v e c t o r s i n U, the s p e c t r a l decompositon for.J2. + f o l l o w s d i r e c t l y : J l + = U A 2 U T + I = U( A 2 + D U T (2.69) Thus the a d d i t i o n of the i d e n t i t y m a t r i x i n c r e a s e s a l l the e i g e n v a l u e s by one, w i t h the r e s u l t t h a t the s m a l l e i g e n v a l u e s of the o r i g i n a l i n n e r p r o d u c t m a t r i x can no l o n g e r a m p l i f y any 83 e r r o r s i n the d a t a i n the manner d e s c r i b e d by Lanczos (1961, p l 3 1 ) . Hence, the net e f f e c t i s q u i t e s i m i l a r t o t h a t r e s u l t i n g from the winnowing of s m a l l e i g e n v a l u e s from A -The a d d i t i o n of a c o n s t a n t t o the main d i a g o n a l i s a w e l l known d e v i c e f o r i m p r o v i n g the c o n d i t i o n number of a m a t r i x , f i n d i n g common useage i n g e o p h y s i c a l i n v e r s i o n a l g o r i t h m s (Inman, 1975). N o r m a l l y the unknown a d d i t i v e c o n s t a n t i s i n t r o d u c e d i n an 'a p r i o r i ' manner, p u r e l y as a c o m p u t a t i o n a l d e v i c e . The n o v e l f e a t u r e here i s t h a t the the augmentation of the main d i a g o n a l o c c u r s " n a t u r a l l y " as a d i r e c t consequence of the m o d i f i e d f o r m u l a t i o n , ( 2 . 6 1 ) . I t has been remarked a l r e a d y t h a t when a l l e i g e n v a l u e s a r e r e t a i n e d i n the . s o l u t i o n of the augmented system, the net x 2 - m i s f i t e x p e c t e d i s jl?^1!!2' I t : * s now of i n t e r e s t t o a s c e r t a i n the manner i n which the m i s f i t v a r i e s as e i g e n v a l u e s are d i s c a r d e d . To determine x£, the e x p e c t e d x 2 - m i s f i t when L e i g e n v a l u e s remain, note from (2.59) t h a t D H < ^ = ^ (2.70) where cfc^ , >a "*2 1 a r e t* l e b e s t a p p r o x i m a t i o n s t o which can be c o n s t r u c t e d from L e i g e n v e c t o r s . For any v e c t o r j$, PL = 2j£f (2.71) where U. i s i d e n t i c a l t o U i n the f i r s t L columns, and z e r o i n the l a s t N-L columns, as b e f o r e . In terms of -T^, , and T^ 1, x 2 i s g i v e n by 84 K - \n - ( \ - £ } i 2 - i « ? - £ > + ^ i 2 ( 2 . 7 2 ) A A The v e c t o r >j " " rlL i s n e c e s s a r i l y o r t h o g o n a l t o the subspace spanned by the e i g e n v e c t o r s i n U t, whereas ^ i s an element of t h a t subspace as per ( 2 . 7 1 ) . T h e r e f o r e , ^ ~VL ^ s o r t h o g o n a l t o 7^ 1 , so from (2.72) K = I f " 7 t 12 + l ^ l 2 < 2 - 7 3 ) The f i r s t term on the RHS of ( 2 . 7 3 ) c o n s t i t u t e s the t o t a l e x p e c t e d m i s f i t i n the c o n v e n t i o n a l f o r m u l a t i o n as per ( 2 . 4 8 ) ; t h i s term i n c r e a s e s m o n o t o n i c a l l y as more e i g e n v a l u e s a r e d i s c a r d e d . The c o n t r i b u t i o n from ip^ , on the o t h e r hand, d e c r e a s e s as more e i g e n v a l u e s a r e r e j e c t e d because fewer e i g e n v e c t o r s a re i n c l u d e d i n the o r t h o n o r m a l e x p a n s i o n f o r t^1 of the form ( 2 . 7 1 ) . N e v e r t h e l e s s , x£ i n c r e a s e s m o n o t o n i c a l l y as L d e c r e a s e s , j u s t as i n the c o n v e n t i o n a l f o r m u l a t i o n . To prove t h i s s t a t e m e n t , i t i s c o n v e n i e n t t o i n t r o d u c e the symbol 6L t o r e p r e s e n t the ex p e c t e d d i m e n s i o n l e s s r e s i d u a l when L e i g e n v a l u e s a r e r e t a i n e d , i . e . I - n - 1 + ( 2 . 7 4 ) ~ A L e t SL be the v e c t o r of c o e f f i c i e n t s of 6L w i t h r e s p e c t t o the o r t h o n o r m a l b a s i s i n U: 85 By the d e f i n i t i o n , ( 2 . 7 4 ) , 4; where 7^ 1 = U7"^1 7 = 2 7 (2.76) The e x p e c t e d squared m i s f i t can be computed r e a d i l y from Hence, u s i n g ( 2 . 7 5 ) , 2 _ y 2 (2.77) where y£ and i ^ 1 a r e the L t h components of ?7 and •ij1 r e s p e c t i v e l y . The t r a n s f o r m e d d i m e n s i o n l e s s r e s i d u a l f o r the p e r t u r b e d model, yj1 > c a n ^ e s i m p l y e x p r e s s e d i n terms of ^ , i t s c o u n t e r p a r t f o r the s t a r t i n g model. The r e l a t i o n between the two r e s i d u a l s may be d e r i v e d by s u b s t i t u t i o n from (2.69) i n t o (2.67) and m u l t i p l i c a t i o n from the l e f t by U T, whence 86 ( A2 + = *} (2.78) Thus rj.1 = ^/( X/ + 1) (2.79) T h i s r e s u l t i s of i n t e r e s t i n i t s e l f . I t s i g n i f i e s t h a t the f i t t o the d a t a i n o r t h o g o n a l " d i r e c t i o n s " c o r r e s p o n d i n g t o s m a l l e i g e n v a l u e s i s , almost i d e n t i c a l f o r cr" and c r 1 , whereas the exp e c t e d r e s i d u a l i s reduced t o almost z e r o i n the d i r e c t i o n s c o r r e s p o n d i n g t o v e r y l a r g e e i g e n v a l u e s . S u b s t i t u t i n g from (2.79) i n t o ( 2 . 7 7 ) , the m o n o t o n i c i t y of x£ can be e s t a b l i s h e d : xL-i " XL = 5f L 2 [1 - ( X J+D - 2 ] > 0 (2.80) Gi v e n ( 2 . 8 0 ) , winnowing i s conducted i n p r e c i s e l y the same manner as f o r the c o n v e n t i o n a l f o r m u l a t i o n when || f 1 ||2 < X2MAX, and i s t e r m i n a t e d when (2.46) i s v i o l a t e d . I f H^MI2 > X2MAX, a l l e i g e n v a l u e s a r e r e t a i n e d . The r a t e s of convergence u s i n g the c o n v e n t i o n a l and m o d i f i e d f o r m u l a t i o n s a r e compared f o r two sample c o n s t r u c t i o n s i n F i g . 2.1. S i x t e e n i t e r a t i o n s were r e q u i r e d by each f o r m u l a t i o n f o r an a c c e p t a b l e f i t f o r the case i n F i g . 2.1b. Convergence i s slow d u r i n g i n t e r m e d i a t e i t e r a t i o n s because too few l a y e r s were p e r m i t t e d a t t h a t s t a g e . The r a t e of dec r e a s e i n m i s f i t i s b e t t e r s u s t a i n e d f o r the example i n the upper frame ( F i g . 2.1a). In t h i s i n s t a n c e seven i t e r a t i o n s were r e q u i r e d 87 F i g u r e 2.1 Examples of Convergence f o r C o n v e n t i o n a l and Modi f i e d Formulat i o n s (a) x 1 - m i s f i t v e r s u s i t e r a t i o n number f o r c o n s t r u c t i o n of a model s a t i s f y i n g t h e o r e t i c a l w a v e - t i l t a m p l i t u d e and phase d a t a (Table 2.3). The c o n v e n t i o n a l f o r m u l a t i o n (open symbols) r e q u i r e d seven i t e r a t i o n s , w h i l e the m o d i f i e d f o r m u l a t i o n ( c r o s s e s ) converged i n f i v e i t e r a t i o n s . The m i s f i t d e c r e a s e s m o n o t o n i c a l l y f o r both f o r m u l a t i o n s . The m o d i f i e d f o r m u l a t i o n c u r v e c o n s t i t u t e s the convergence h i s t o r y f o r the a c c e p t a b l e model i n F i g . 2.6c. (b) x 2 - m i s f i t v e r s u s i t e r a t i o n number f o r c o n s t r u c t i o n of a model s a t i s f y i n g v e r t i c a l magnetic r e a l and ima g i n a r y p a r t s d a t a (Table 2.2). Both c o n v e n t i o n a l and m o d i f i e d f o r m u l a t i o n s (open symbols and c r o s s e s r e s p e c t i v e l y ) r e q u i r e d s i x t e e n i t e r a t i o n s i n t h i s example. The " p l a t e a u " i n the m i s f i t between i t e r a t i o n s 7 and 11 i s a consequence of an i n s u f f i c i e n t number of l a y e r s ; the f i n a l approach t o a c c e p t a b l e models, b e g i n n i n g a t i t e r a t i o n 12, was s t i m u l a t e d by the ( a u t o m a t i c ) i n t r o d u c t i o n of a d d i t i o n a l l a y e r s . The cu r v e f o r the m o d i f i e d f o r m u l a t i o n ( c r o s s e s ) d e l i n e a t e s the convergence h i s t o r y f o r the a c c e p t a b l e model d e p i c t e d i n F i g . 2.5d. 88 Convergence for | W | & 0 w data 40 G O X 0 I i I . I i I < I 0 2 4 6 8 F i g . 2.1a Convergence for Re iHz^&lmiHz? data iteration § Fig . 2.1b 89 u s i n g the c o n v e n t i o n a l f o r m u l a t i o n , and f i v e u s i n g the m o d i f i e d f o r m u l a t i o n . I I - 4 ( i v ) Compensation f o r C o i l M i s a l i g n m e n t : C o i l m i s a l i g n m e n t i s the term used t o d e s c r i b e n o n - p a r a l l e l i s m between the a x i s of the t r a n s m i t t e r c o i l and the " v e r t i c a l " of the r e c e i v e r . In p r a c t i c e i t i s d i f f i c u l t t o a l i g n the t r a n s m i t t e r and r e c e i v e r t o b e t t e r than a few d e g r e e s , even i n r e g i o n s of almost f l a t topography l i k e Grass V a l l e y . I f both components of the magnetic f i e l d were always s i m i l a r i n magnitude, a s l i g h t m i s a l i g n m e n t would not u s u a l l y be a s e r i o u s c o n c e r n . However, i n HLEM soundings i t i s not uncommon f o r one component t o be c o n s i d e r a b l y l a r g e r than the o t h e r ; the v e r t i c a l component i s o f t e n pre-dominant a t low f r e q u e n c i e s , and the r a d i a l component at h i g h f r e q u e n c i e s . When the two components are markedly d i f f e r e n t i n magnitude, a s m a l l m i s a l i g n m e n t can cause s e r i o u s d i s t o r t i o n of the s m a l l e r component. The e f f e c t on i n t e r p r e t a t i o n can be e s p e c i a l l y pronounced when r a t i o s (such as w a v e - t i l t s ) a r e i n v e r t e d ( S i n h a , 1980). I f the m i s a l i g n m e n t a n g l e were known, i t s e f f e c t c o u l d be n u l l i f i e d by s t r a i g h t f o r w a r d t r i g o n o m e t r y . In p r a c t i c e the m i s a l i g n m e n t a n g l e i s d i f f i c u l t t o measure, and i t i s n e c e s s a r y to compensate f o r i t s e f f e c t s by o t h e r ( l e s s p r e c i s e ) means. In HLEM.INV the m i s a l i g n m e n t a n g l e i s r e g a r d e d as the r e a l i s a t i o n of a Normal random v a r i a b l e , (£> , w i t h mean z e r o and a s m a l l s t a n d a r d d e v i a t i o n , y , where V < 8.5x10" 2 rad ( a p p r o x i m a t e l y 5°). C o i l m i s a l i g n m e n t then has the e f f e c t of augmenting the v a r i a n c e of each datum i n a manner d e t e r m i n e d by the type and 90 v a l u e s of the o b s e r v a t i o n s ; the d e r i v a t i o n of the a p p r o p r i a t e " c r o s s - c o n t a m i n a t i o n " c o e f f i c i e n t s i s d e t a i l e d i n Appendix D. I f ^Mj i s the c o e f f i c i e n t a p p r o p r i a t e f o r the j t h o b s e r v a t i o n , and i f Q; i s the random v a r i a b l e a s s o c i a t e d w i t h the e r r o r from a l l o t h e r s o u r c e s f o r t h a t o b s e r v a t i o n , then the net e r r o r may be rega r d e d as a r e a l i s a t i o n of the Normal random v a r i a b l e Q.+, d e f i n e d by Q.+ = y U j $ + Qj (2.81) I f Q; has mean z e r o and v a r i a n c e q ? as b e f o r e , the augmented e r r o r has mean z e r o and v a r i a n c e ( q t ) 2 , where ( q j ) 2 = y U . 2 V 2 + q / (2.82) I t i s t a c i t l y assumed here t h a t ^ i s independent of Qj f o r a l l j = l , N . C o n s i d e r a t i o n of (2.81) r e v e a l s t h a t the augmented random v a r i a b l e s , {Qj : j = l , N } , a r e not independent, each b e i n g d e f i n e d i n terms of <J> . P h y s i c a l l y , t h i s common dependence on i s r e s p o n s i b l e f o r c o r r e l a t e d e r r o r s : i f perchance the mi s a l i g n m e n t i s l a r g e , the c o n t a m i n a t i o n of a l l o b s e r v a t i o n s w i l l be r e l a t i v e l y s e v e r e . M a t h e m a t i c a l l y , the s t a t i s t i c a l dependence i s m a n i f e s t e d as n o n - d i a g o n a l elements i n the c o v a r i a n c e m a t r i x f o r { Q + } . E x p l i c i t l y , from ( 2 . 8 1 ) , Cov[Q.+ ,Qk+] = y U j ^ a / 2 + q}<$ik (2.83) 91 The convergence c r i t e r i a (2.32) and (2.33) were d e v i s e d assuming independent e r r o r s . I f t h e s e same c r i t e r i a were i n v o k e d to a s s e s s convergence f o r d a t a c o n t a m i n a t e d w i t h m i s a l i g n m e n t e r r o r , the f i t t o the d a t a might be u n n e c e s s a r i l y poor. As an i l l u s t r a t i o n , observe from (2.81) t h a t i t i s p o s s i b l e t o d e f i n e l i n e a r c o m b i n a t i o n s of the random v a r i a b l e s {Q +} which a r e independent of e.g. v"jQk VfcQj V * J Q * Thus i f V 2 >> q.2 f o r a l l j = l , N , the o v e r a l l " l e v e l of e r r o r " f o r a s e t of such l i n e a r c o m b i n a t i o n s of da t a c o u l d be d r a m a t i c a l l y l e s s than f o r the o r i g i n a l d a t a ; indeed the e f f e c t of <^ c o u l d be t o t a l l y removed i f i t were p e r m i s s i b l e t o d i s c a r d one datum. More g e n e r a l l y , f o l l o w i n g G i l b e r t (1971), i t i s p o s s i b l e t o d e f i n e a s e t of u n c o r r e l a t e d random v a r i a b l e s by e x p l o i t i n g the A s p e c t r a l d e c o m p o s i t i o n of £ + , the d i m e n s i o n l e s s c o v a r i a n c e m a t r i x a s s o c i a t e d w i t h the augmented random v a r i a b l e s , {Q +}. S + has elements 4k = t ^ i / ' f c ^ 2 + ^]64U ) / ( 3 j + ( 3 k = Cov[QJ ,Q* ] (2.84) where, i n ana l o g y w i t h ( 2 . 2 7 ) , the d i m e n s i o n l e s s augmented random v a r i a b l e s {Qt: j=l,N} a re d e f i n e d by Qj/qj (2.85) A j 92 A L i k e t h e i n n e r p r o d u c t m a t r i c e s e n c o u n t e r e d e a r l i e r , £ * i s p o s i t i v e d e f i n i t e and s y m m e t r i c , and t h e r e f o r e a d m i t s a d e c o m p o s i t i o n i n terms o f e i g e n v e c t o r s and e i g e n v a l u e s c o m p l e t e l y a n a l o g o u s t o t h a t i n ( 2 . 3 9 ) : £ + = F V * 2 F T (2.86) The c olumns of F c o n t a i n o r t h o n o r m a l e i g e n v e c t o r s , w h i l e 2£ 2 ^ s a d i a g o n a l m a t r i x of e i g e n v a l u e s . M u l t i p l y i n g (2.86) s u c c e s s i v e l y from t h e l e f t by FT and from t h e r i g h t by F has t h e A e f f e c t of d i a g o n a l i s i n g £ * : F T S + F = 2 (2.87) ~ ~ ~ rz? By i n s p e c t i o n of ( 2 . 8 7 ) , i t i s a p p a r e n t t h a t t h e c o r r e l a t e d random v a r i a b l e s 1Q +} a s s o c i a t e d w i t h £ * g i v e r i s e t o a s e t of N u n c o r r e l a t e d random v a r i a b l e s when combined l i n e a r l y i n p r o p o r t i o n s d e t e r m i n e d by t h e e i g e n v e c t o r s i n F. D e n o t i n g t h e u n c o r r e l a t e d random v a r i a b l e s by {Sj : j = l , N } , i t f o l l o w s t h a t N R A S;= J E Fft' Ql (2.88) i-i E a c h c o m p o s i t e random v a r i a b l e , Sj , i s N o r m a l l y d i s t r i b u t e d , w i t h mean z e r o and v a r i a n c e s. 2, where sf = S V . (2.89) Any o p e r a t i o n p e r f o r m e d on t h e random v a r i a b l e s {Q +} must 93 a l s o be performed on the c o r r e s p o n d i n g e q u a t i o n s . Hence, the r e l e v a n t t r a n s f o r m e d system of e q u a t i o n s i s g i v e n by p*HS~r = 7 * (2.90) where, i n accordance w i t h (2.85) and (2 . 8 8 ) , (2.91) (2.92) The t r a n s f o r m e d datum, y j * , a s s o c i a t e d w i t h the s m a l l e s t e i g e n v a l u e , s N , i s the most a c c u r a t e l y d e t e r m i n e d l i n e a r c o m b i n a t i o n of the d a t a ; c o n v e r s e l y , the t r a n s f o r m e d datum, rj*, a s s o c i a t e d w i t h the l a r g e s t e i g e n v a l u e , st , i s the most p o o r l y known l i n e a r c o m b i n a t i o n . Normal random v a r i a b l e s w i t h z e r o c o v a r i a n c e a r e n e c e s s a r i l y independent ( J e n k i n s and Watts, 1969, p 7 4 ) , so {S} i s a s e t of independent random v a r i a b l e s . C o n s e q u e n t l y , the convergence c r i t e r i a (2.32) and (2.33) may be i n v o k e d t o a s s e s s a c c e p t a b i l i t y of models i n r e l a t i o n t o the t r a n s f o r m e d d a t a . In view of the d e f i n i t i o n s (2.22) and (2.23) f o r x 1 - and x 2 - m i s f i t , i t i s c o n v e n i e n t t o n o r m a l i s e each e q u a t i o n w i t h r e s p e c t t o the s t a n d a r d d e v i a t i o n of the a s s o c i a t e d e r r o r . Hence, the system a c t u a l l y s o l v e d i s D*H<$& = V* (2.93) 94 where, i n a n a l o g y w i t h (2.35) and ( 2 . 2 4 ) , (2.94) 7/ = 7 / / s i A l t h o u g h p r e s e n t e d i n c o n n e c t i o n w i t h the tr e a t m e n t of c o i l m i s a l i g n m e n t e r r o r , t h i s p r o c e d u r e i s a v a r i a t i o n of t h a t proposed by G i l b e r t (1971) and i s a p p l i c a b l e t o any l i n e a r or l i n e a r i s e d i n v e r s e problem, p r o v i d e d o n l y t h a t the c o v a r i a n c e m a t r i x i s known. 11-5' T h e o r e t i c a l Examples In o r d e r t o t h o r o u g h l y t e s t the performance of the HLEM.INV c o n s t r u c t i o n a l g o r i t h m , d a t a from one model have been i n v e r t e d u s i n g a l l s i x t e e n d a t a o p t i o n s d e s c r i b e d i n 1 1 — 4 ( i ) . The t r u e c o n d u c t i v i t y s t r u c t u r e i s shown i n F i g . 2.2; a c o n d u c t i v i t y of 0.1 S/m has been a s s i g n e d t o depths g r e a t e r than 4 km. I t was remarked i n 1-5 t h a t i n d u c t i o n sounding i s most e f f e c t i v e i n d e l i n e a t i n g e l e c t r i c a l s t r u c t u r e when the c o n d u c t i v i t y i n c r e a s e s m o n o t o n i c a l l y w i t h depth; the t r u e model s e l e c t e d f o r t h i s d e m o n s t r a t i o n , however, e x h i b i t s a decrease i n c o n d u c t i v i t y from 0.2 S/m t o 0.01 S/m i n the f i r s t 1.2 km, and t h e r e f o r e c o n s t i t u t e s a c h a l l e n g e t o the e f f e c t i v e n e s s of the a l g o r i t h m . The t r u e responses were computed a t f o u r f r e q u e n c i e s , and a s m a l l n o i s e component was i n t r o d u c e d ; the a c c u r a t e and c o n t a m i n a t e d d a t a a re r e c o r d e d i n T a b l e s 2.1-3. The l e v e l of 95 F i g u r e 2.2 "True" Model f o r T h e o r e t i c a l C o n s t r u c t i o n Examples. The " a c c u r a t e " responses i n T a b l e s 2.1-3 a p p l y t o t h i s conduct i v i t y . F i g u r e 2.3 S t a r t i n q Model f o r T h e o r e t i c a l C o n s t r u c t i o n Examples The depth range of i n t e r e s t was i n i t i a l l y d i v i d e d i n t o f o u r l a y e r s , w i t h i n t e r f a c e s a t depths i n d i c a t e d by the t i c k marks. The number of l a y e r s was p e r m i t t e d t o i n c r e a s e t o a maximum of ten i n the c o u r s e of c o n s t r u c t i o n . 96 "True" model 0.3 ^ 0.2 b 0.1 0.0 T r T 1 1 r 0 100 200 300 400 (X 10 1) Fig. 2.2 Starting model 0.3 ^ 0 . 2 b 0.1 0.0 0 100 200 300 400 (X 10 1) depth (m) Fig. 2.3 97 T a b l e 2.1 D a t a f o r Hr C o n s t r u c t i o n T h e o r e t i c a l r a d i a l m a g n e t i c f i e l d d a t a f o r model i n F i g . 2.2, u s e d f o r c o n s t r u c t i o n examples i n F i g s . 2.4 and 2.7. A c c u r a t e | I n a c c u r a t e | s.d. | w ( r a d / s e c ) 0.4759 | 0.467 | 9x10" 3 | 1 |Hr| | 3.5385 | 3.55 | 7x10' 2 1 1 0 (/^amp/m) | 2.3914 | 2.42 | 4x10" 2 | 100 0.7399 | 0.732 | 1.4x10" 2 | 1000 1.4457 | 1.448 | 2.5x10" 2 | 1 Br | 0.7837 | 0.768 | 2.5x10" 2 1 1 0 ( r a d ) | 5.3968 | 5.37 | 2.5x10" 2 | 100 5.5055 1 5 - 5 1 2.5x10" 2 | 1000 0.0594 | 0.06 | 6.4x10" 3 | 1 Re{Hr} | 2.5063 1 2.47 1 5x10" 2 1 1 0 (yW-amp/m) | 1.5118 1 1.5 | 3x10- 2 | 100 0.5272 | 0.53 | ,1x10" 2 | 1000 0.4722 | 0.48 | 6.4x10- 3 | 1 Im{Hr} | 2.4979 1 2-5 | 5x10- 2 1 1 0 (yXxamp/m) | -1.8529 | -1.83 | 3x10" 2 | 100 -0.5191 | -0.526 | 1x10" 2 | 1000 98 Table 2.2 Data f o r Hz C o n s t r u c t i o n T h e o r e t i c a l v e r t i c a l magnetic f i e l d d a t a f o r model i n F i g . 2.2, used f o r c o n s t r u c t i o n examples i n F i g s . 2.5 and 2.7. A c c u r a t e | I n a c c u r a t e | s .d. | w ( r a d / s e c ) 4.0124 | 4.04 | 3x10- 2 | 1 |Hz| | 5.0893 | 5.07 | 5x10" 2 1 1 0 (/<• amp/m) | 0.5932 | 0.594 | 1.2x10" 2 | 100 0.0578 | 0.0574 | 1x10" 2 | 1000 3.1774 | 3.185 | 9x10" 3 | 1 0z | 2.9189 1 2 - 9 1 1 9x10" 3 1 1 0 (rad) | 0.9408 | 0.925 | 1.7x10" 2 | 100 1.5684 1 1.59 | 3.5x10" 2 | 1000 -4.0098 | -4.03 | 2x10" 2 | 1 Re {Hz} | -4.9637 | -4.94 | 4x10" 2 1 1 0 (/*• amp/m) | 0.3495 | 0.345 | 8x10- 3 | 100 0.0001 | 0.0005 | 7x10' 3 | 1000 -0.1434 1 "0.12 | 2x10" 2 | 1 Im{Hz} | 1.1237 1 1-11 1 4x10" 2 1 i o (/iamp/m) | 0.4793 | 0.481 | 8x10- 3 | 100 0.0578 | 0.054 | 7x10" 3 | 1000 99 Table 2.3 Data f o r W C o n s t r u c t i o n T h e o r e t i c a l w a v e - t i l t data f o r model i n F i g . 2.2, used f o r c o n s t r u c t i o n examples i n F i g . 2.6. | A c c u r a t e | I n a c c u r a t e | s .d. | w ( r a d / s e c ) | 0.1186 | 0.115 | 3.3x10- 3 | 1 |W| | 0.6953 | 0.72 | 2x10" 2 1 1 0 | 4.0314 1 4- 1 1 0.16 | 100 | 12.8062 1 i i . o I 2.2 | 1000 | 4.5516 | 4.546 | 2.7x10" 2 1 1 fc?w | 4.1479 | 4.141 | 2.7x10" 2 1 1° (rad) | 4.4560 | 4.445 | 3x10" 2 | 100 | 3.9371 1 3.91 | 4.3x10- 2 | 1000 | -0.0190 | -0.02 | 2.3x10" 3 1 1 Re{W} | -0.3719 | -0.38 | 1.4x10" 2 1 i o | -1.0224 1 - i - o 1 0.11 | 100 | -8.9629 1 - 9 - 0 1 1.6 | 1000 | -0.1171 | -0.115 | 2.3x10" 3 1 1 Im{W} | -0.5874 | -0.57 | 1.4x10" 2 1 1 0 | -3.8996 1 -4-0 | 0.11 | 100 | -9.1468 1 _ 8 - 4 1 1.6 | 1000 100 n o i s e i s r a t h e r lower than t h a t which might be c o n s i d e r e d t y p i c a l of r e a l g e o p h y s i c a l d a t a . However, t h i s i s of no consequence because the p r e s e n t o b j e c t i v e i s t o conduct a thorough t e s t of the program; the i n v e r s i o n of r e a l d ata i s d e f e r r e d u n t i l Chapter IV. The s t a r t i n g model f o r each c o n s t r u c t i o n was the homogeneous h a l f - s p a c e w i t h c o n d u c t i v i t y 0.1 S/m, as shown i n F i g . 2.3. The h a l f - s p a c e was composed of f o u r l a y e r s w i t h i n t e r f a c e s a t depths 0.5, 1.0, 2.5, and 4.0 km, i n d i c a t e d by t i c k s i n the f i g u r e . W ith t h i s c h o i c e of i n i t i a l p a r t i t i o n , i t i s i m p o s s i b l e f o r HLEM.INV t o r e c o v e r the t r u e model e x a c t l y . T h i s i s because the number of l a y e r s i s i n c r e a s e d d u r i n g c o n s t r u c t i o n by h a l v i n g the t h i c k n e s s e s of e x i s t i n g l a y e r s ; t h u s an i n t e r f a c e a t 400m (as r e q u i r e d f o r the t r u e model) cannot be i n t r o d u c e d by e q u i - p a r t i t i o n of the f i r s t 500m l a y e r of the s t a r t i n g model. In o r d e r t o conduct a t r u e t e s t of the program, o n l y the dat a and the parameters i d e n t i f y i n g the p a r t i c u l a r d a t a o p t i o n were a l t e r e d between s u c c e s s i v e r u n s . Thus DLNMAX, (the maximum p e r m i s s i b l e change i n Scr/er per i t e r a t i o n ) was 0.5 t h r o u g h o u t , w h i l e the maximum number of l a y e r s a l l o w e d was t e n , i . e . NLMAX = 10. A complete parameter l i s t i s p r e s e n t e d i n Ta b l e 2.4. The s i x t e e n c o n s t r u c t e d models c o r r e s p o n d i n g t o t h i s c h o i c e of parameters a re d e p i c t e d i n F i g s . 2.4-7. A l l the a c c e p t a b l e models d e p i c t e d were c o n s t r u c t e d u s i n g the m o d i f i e d f o r m u l a t i o n d e s c r i b e d i n I I - 4 ( i i i ) ; i t i s t o be ex p e c t e d t h a t s i m i l a r r e s u l t s c o u l d have been o b t a i n e d u s i n g the c o n v e n t i o n a l f o r m u l a t i o n . The r e l e v a n t i n i t i a l and f i n a l m i s f i t s , the number, 101 Table 2.4 Parameter L i s t f o r C o n s t r u c t i o n Examples Parameter A Val u e 50 m F u n c t i o n r a d i u s of t r a n s m i t t e r c o i l R l HC T-R s e p a r a t i o n d i f f e r e n c e i n e l e v a t i o n between r e c e i v e r and t r a n s m i t t e r 10 ZM SIG1 50 A 4 km 0.1 S/m net t r a n s m i t t e r c u r r e n t maximum depth of i n t e r e s t DLNMAX c o n d u c t i v i t y s c a l e and c o n d u c t i v i t y of h a l f - s p a c e beneath depth ZM 0.5 1 maximum p e r m i s s i b l e 6v/<r i n a s i n g l e i t e r a t i o n ; see 11-3. NSD number of s t a n d a r d d e v i a t i o n s between e x p e c t e d v a l u e and maximum a c c e p t a b l e v a l u e of x 1 - or x 2 - m i s f i t STEP 0.2 c o n t r o l s the i n t r o d u c t i o n of new l a y e r s ; see (2.58) NLMAX | 10 | maximum number of l a y e r s 102 F i g u r e 2.4 C o n s t r u c t i o n w i t h Hr Data Models s a t i s f y i n g t h e o r e t i c a l r a d i a l magnetic f i e l d d a t a (Table 2.1) have been c o n s t r u c t e d from the s t a r t i n g model shown i n F i g . - 2 . 3 . The t r u e model i s d e p i c t e d i n F i g . 2.2. The c o n s t r u c t e d models are e i t h e r x 1 - or x 2 - a c c e p t a b l e , as per (2.32) or ( 2 . 3 3 ) . The symbols x£ (x.2) and x^ (x^ 2) denote r e s p e c t i v e l y the i n i t i a l and f i n a l v a l u e s of x 1 - ( x 2 - ) m i s f i t . ITER i s the t o t a l number of i t e r a t i o n s r e q u i r e d . T i s the CPU time f o r the complete c o n s t r u c t i o n on the U.B.C. Amdahl 470 V/6 computer. Other d a t a p e r t i n e n t t o the c o n s t r u c t i o n s appears i n Table 2.4. (a) A c c e p t a b l e model f o r r a d i a l magnetic a m p l i t u d e d a t a . (b) A c c e p t a b l e model f o r r a d i a l magnetic phase d a t a . (c) A c c e p a t a l e model f o r r a d i a l magnetic a m p l i t u d e and phase d a t a . (d) A c c e p t a b l e model f o r r a d i a l magnetic r e a l and i m a g i n a r y p a r t s d a t a . 103 |Hr| construct ion 0.3 0.2 CO b 0.1 0.0 r L i r-—i r x7 = 69.00 l x* = 3.85 x =4.40 max T 1 r ITER = 7 T = 5.29s 1 0 100 200 300 400 (X 10 1) Fig. 2.4a 6r construct ion 0.3 ^ 0.2 b 0.1 0.0 T 1 1 1 1 1 r x. = 19.05 ITER = 3 l |__ x j = 3.77 T = 3.30s • x 1 -U | m a x = 4.40 J 1 1 1 0 100 200 300 400 (X 10 1) depth (m) Fig. 2.4b |Hr| & 9r construction 0.3 0.2 co b 0.1 0.0 T 1 1 r xT = 88.04 I xf = 7.23 x = 8.09 max T 1 r ITER = 7 T - 6.54s ^ J 1 I I I I L 0 100 200 300 400 (X F i g . 2.4c ReiHr* & ImJHr! construction 0.3 ^ 0.2 CO b 0.1 0.0 1 -1 1 1 1 1 1 - xT = 133.3 l ITER = 6 x* = 7.99 T = 5.01s - x1 = 8.09 max -— --1 . . . . 1 J 1 1 0 100 200 300 400 (X depth (m) F i g . 2.4d 105 F i g u r e 2.5 C o n s t r u c t i o n w i t h Hz Data Models s a t i s f y i n g t h e o r e t i c a l v e r t i c a l magnetic f i e l d d a t a (Table 2.2) have been c o n s t r u c t e d from the s t a r t i n g model shown i n F i g . 2.3. The t r u e model i s d e p i c t e d i n F i g . 2.2. The c o n s t r u c t e d models are e i t h e r x 1 - or x 2 - a c c e p t a b l e , as per (2.32) or ( 2 . 3 3 ) . The symbols x£ (x?) and x j (x^) denote r e s p e c t i v e l y the i n i t i a l and f i n a l v a l u e s of x 1 - ( x 2 - ) m i s f i t . ITER i s the t o t a l number of i t e r a t i o n s r e q u i r e d . T i s the CPU time f o r the complete c o n s t r u c t i o n on the U.B.C. Amdahl 470 V/6 computer. Other data p e r t i n e n t t o the c o n s t r u c t i o n s appears i n Ta b l e 2.4. (a) A c c e p t a b l e model f o r v e r t i c a l magnetic a m p l i t u d e d a t a . (b) A c c e p t a b l e model f o r v e r t i c a l magnetic phase'data. The r e l a t i v e l y l a r g e number of i t e r a t i o n s r e q u i r e d i n t h i s case i s a r e s u l t of the f a c t t h a t too few l a y e r s were a v a i l a b l e d u r i n g the i n t e r m e d i a t e s t a g e s of c o n s t r u c t i o n ; see t e x t . (c) A c c e p t a b l e model f o r v e r t i c a l magnetic a m p l i t u d e and phase d a t a . (d) A c c e p t a b l e model f o r v e r t i c a l magnetic r e a l and ima g i n a r y p a r t s d a t a . 106 |Hz| construction 0.3 0.0 0 100 200 300 400 (X 10 1) F i g . 2.5a 0z construction 0.3 ^ 0 . 2 CO b 0.1 h 0.0 1 1 1 T xT = 54.91 l = 4.19 x = 4.40 max 1 1 1" ITER = 15 T = 13.61s \ 0 100 200 300 400 (X 10 1) depth (m) F i g . 2.5b |Hz| Sc 6z c o n s t r u c t i o n 0 . 3 ^ 0 . 2 b 0 .1 0 . 0 T r xT = 177.8 I = 7.91 ~T 1 r ITER =13 T = 13.57s x = 8.09 max 1 0 100 200 300 400 (X Fig. 2.5c R e i H z ? Sc \m\Hzl c o n s t r u c t i o n 0 . 3 ^ 0 . 2 ui b 0.1 0 .0 • i r 1 1 x * = 269.07 x * = 7.73 x 1 - 8.09 max ITER = 16 1 "1 1 T = 15.05s . 1 1 . 0 100 200 300 400 (X d e p t h ( m ) Fig. 2.5d 108 F i g u r e 2.6 C o n s t r u c t i o n w i t h W Data Models s a t i s f y i n g t h e o r e t i c a l w a v e - t i l t d a t a (Table 2.3) have been c o n s t r u c t e d from the s t a r t i n g model shown i n F i g . 2.3. The t r u e model i s d e p i c t e d i n F i g . 2.2. The c o n s t r u c t e d models are e i t h e r x 1 - or x 2 - a c c e p t a b l e , as per (2.32) or ( 2 . 3 3 ) . The symbols x£ (x£) and x£ (x^) denote r e s p e c t i v e l y the i n i t i a l and f i n a l v a l u e s of x 1 - ( x 2 - ) m i s f i t . ITER i s the t o t a l number of i t e r a t i o n s r e q u i r e d . T i s the CPU time f o r the complete c o n s t r u c t i o n on the U.B.C. Amdahl 470 V/6 computer. Other d a t a p e r t i n e n t t o the c o n s t r u c t i o n s appears i n T a b l e 2.4. I t may be observed t h a t the c o s t per i t e r a t i o n i s a p p r o x i m a t e l y t w i c e as h i g h f o r w a v e - t i l t data as f o r magnetic component d a t a ; t h i s i s because both Hr and Hz must be computed d u r i n g w a v e - t i l t c o n s t r u c t i o n , as per ( 2 . 5 1 ) . (a) A c c e p t a b l e model f o r w a v e - t i l t a m p l i t u d e d a t a . (b) A c c e p t a b l e model f o r w a v e - t i l t phase d a t a . (c) A c c e p t a b l e model f o r w a v e - t i l t a m p l i t u d e and phase d a t a . (d) A c c e p t a b l e model f o r w a v e - t i l t r e a l and i m a g i n a r y p a r t s d a t a . 109 |W| construction 0 . 3 £ ^ 0 . 2 ui b 0 .1 h 0 . 0 i 1 1 r x* = 23.03 x* = 4.33 x = 4.40 max -i 1 r ITER = 7 T = 14.92s 0 100 200 300 400 (X 10 1 ) Fig. 2.6a 0w construction 0 . 3 ^ 0 . 2 co b 0 .1 0 .0 T 1 1 r xt = 117.5 l • x^ = 5.9 x = 6.8 max i 1 r ITER = 4 T = 7.74s 0 100 200 300 400 (X 10 1 ) depth (m) Fig. 2.6b |W| Sc 0w construction 0 . 3 ^ 0 . 2 b 0 .1 0 .0 hi T r xT = 38.96 I x* = 8.06 T 1 1 r ITER = 5 T = 10.16s x = 8.09 max 1 1 1 0 100 200 300 400 (X Fig. 2.6c ReiW? (fc-lmJWS construction 0 . 3 0 . 2 Jl b 0.1 h 0 .0 i r 1 r xT = 41.63 x x* = 7.64 x1 - 8.09 max ITER = 12 T = 23.61 0 100 200 300 400 (X depth (m) Fig. 2.6d I l l F i g u r e 2.7 C o n s t r u c t i o n w i t h Hr and Hz Data Models s a t i s f y i n g t h e o r e t i c a l Hr and Hz d a t a ( T a b l e s 2.1 and 2.2) have been c o n s t r u c t e d from the s t a r t i n g model shown i n F i g . 2.3. The t r u e model i s d e p i c t e d i n F i g . 2.2. The c o n s t r u c t e d models are e i t h e r x 1 - or x 2 - a c c e p t a b l e , as per (2.32) or ( 2 . 3 3 ) . The symbols x{ (x t 2) and x j (x|) denote r e s p e c t i v e l y the i n i t i a l and f i n a l v a l u e s of x 1 - ( x 2 - ) m i s f i t . ITER i s the t o t a l number of i t e r a t i o n s r e q u i r e d . T i s the CPU time f o r the complete c o n s t r u c t i o n on the U.B.C. Amdahl 470 V/6 computer. (a) A c c e p t a b l e model f o r r a d i a l and v e r t i c a l magnetic a m p l i t u d e d a t a . (b) A c c e p t a b l e model f o r r a d i a l and v e r t i c a l magnetic phase d a t a . (c) A c c e p t a b l e model f o r r a d i a l and v e r t i c a l magnetic a m p l i t u d e and phase d a t a . (d) A c c e p t a b l e model f o r r a d i a l and v e r t i c a l magnetic r e a l and i m a g i n a r y p a r t s d a t a . 112 |Hr| & |Hz| construct ion 0.3 ^ 0.2 co b 0.1 0.0 T 1 1 r x. = 191.85 i -| 1 r ITER = 7 xf = 7.47 x = 8.09 max T = 10.21s 1 J 1 u 0 100 200 300 400 (X 101) F i g . 2.7a 9r & 6z construct ion 0.3 ^ 0.2 b 0.1 0.0 i 1 1 r xT = 73.95 I = 6.94 x 1 = 8.09 max -i 1 r ITER = 6 T = 10.19 0 100 200 300 400 (X 101) depth (m) F i g . 2.7b 113 |Hr|, 0r, |Hz|, & 9z construction 0 . 3 ^ 0 . 2 CO, b 0 .1 0 . 0 i 1 1 r xT = 265.81 l x* = 14.63 x 1 = 15.18 max i 1 r -ITER = 12 T = 23.84s 0 100 200 300 400 (X 1 0 1 ) F i g . 2.7c ReiHr,Hz* & lmiHr,Hz* construction 0 . 3 ^ 0 . 2 CO b 0 .1 0 . 0 T 1 1 r xT = 402.37 I x j = 12.84 x 1 = 15.18 max T 1 1— ITER = 13 T = 22.80s J L i 1 0 100 200 300 depth (m) 400 (X 10 1 ) F i g . 2.7d 114 ITER, of i t e r a t i o n s r e q u i r e d , and the CPU time on the U n i v e r s i t y of B r i t i s h Columbia Amdahl 470 V/6 computer a r e i n d i c a t e d on each of the diagrams. I t i s apparent from the f i g u r e s t h a t a l l d a t a o p t i o n s y i e l d e d a c c e p t a b l e models a f t e r a moderate number of i t e r a t i o n s . In g e n e r a l , the v a r i a b i l i t y i n the number of i t e r a t i o n s r e q u i r e d i s r e l a t e d d i r e c t l y t o the magnitude of the i n i t i a l m i s f i t ; i t i s not s u r p r i s i n g , f o r example, t h a t i n v e r s i o n of the Re{Hz}&Im{Hz} data ( F i g . 2.5d) r e q u i r e d more i t e r a t i o n s than i n v e r s i o n of the 9r d a t a ( F i g . 2.4b), g i v e n the c o n s i d e r a b l y l a r g e r i n i t i a l x ^ m i s f i t i n the former c a s e . One e x c e p t i o n t o t h i s r u l e i s p r o v i d e d by the i n v e r s i o n of the 9z d a t a . As f o r the c o n s t r u c t i o n summarised i n F i g . 2.1b, convergence was r e t a r d e d i n t h i s i n s t a n c e by an inadequate number of l a y e r s : s e v e r a l i n t e r m e d i a t e i t e r a t i o n s were performed t o l i t t l e e f f e c t , b e f o r e an i n t e r n a l "non-convergence f l a g " was a c t i v a t e d t o a l l o w the number of l a y e r s t o i n c r e a s e . As s t a t e d i n 1 1 - 4 ( i i ) , the p a r t i t i o n n o r m a l l y remains u n a l t e r e d i f x 2 > 10.X2MAX; t h i s r e q u i r e m e n t i s r e l a x e d i f the m i s f i t x * * x 2 does not d e c rease m o n o t o n i c a l l y . A f t e r the i n t r o d u c t i o n of a d d i t i o n a l l a y e r s , the program converged r a p i d l y t o a model s a t i s f y i n g the 9z d a t a . ' The f a c t t h a t a l l c o n s t r u c t i o n s f o r v e r t i c a l component da t a r e q u i r e d more i t e r a t i o n s than the c o r r e s p o n d i n g c o n s t r u c t i o n s f o r r a d i a l magnetic d a t a i s p r o b a b l y a p e c u l i a r i t y of t h i s example r a t h e r than a g e n e r a l c h a r a c t e r i s t i c . L i k e w i s e , the f a c t t h a t F i g . 2.4b e x h i b i t s a n o t a b l y p o o r e r resemblance t o the t r u e model than F i g . 2.4a i s presumably an a r t i f a c t of the p a r t i c u l a r c h o i c e of e r r o r s i n the d a t a , r a t h e r than a m a n i f e s t a t i o n of a 115 g e n e r a l " i n f e r i o r i t y " of 6r d a t a i n comparison w i t h |Hr| d a t a . By i n s p e c t i o n of F i g s . 2.4-2.7 i t i s c l e a r t h a t a l l the models a r e c h a r a c t e r i s e d by a c o n d u c t i v e s u r f a c e l a y e r of a p p r o x i m a t e l y 500m i n t h i c k n e s s , o v e r l y i n g a more r e s i s t i v e zone 1 km or more i n w i d t h . Thus a l l s i x t e e n a c c e p t a b l e models reproduce the p r i n c i p a l f e a t u r e s of the t r u e model i n F i g . 2.2, n o t w i t h s t a n i d n g the e r r o r s i n the da t a and the i n c o m p a t i b i l i t y of the p a r t i t i o n w i t h the depths of the t r u e i n t e r f a c e s . The q u a l i t a t i v e s i m i l a r i t y of the a c c e p t a b l e models, even though o n l y f o u r f r e q u e n c i e s were used, s u g g e s t s t h a t the importance of the uniqueness theorem f o r HLEM may not be r e s t r i c t e d t o p h y s i c a l l y u n r e a l i s a b l e c o n d i t i o n s . N e v e r t h e l e s s , the v a r i o u s a c c e p t a b l e models do e x h i b i t d i f f e r e n c e s ; a d i s c u s s i o n of the s i g n i f i c a n c e of these w i l l be d e f e r r e d u n t i l the next c h a p t e r . 116 Chapter I I I APPRAISAL A model c o n s t r u c t e d t o s a t i s f y c e r t a i n o b s e r v a t i o n s i s j u s t one of an i n f i n i t e s e t of a c c e p t a b l e models. C o n s e q u e n t l y , i n v e r s i o n i s not complete u n t i l the f e a t u r e s common t o a l l a c c e p t a b l e models have been i d e n t i f i e d , i . e . u n t i l the non-uniqueness a d m i t t e d by the o b s e r v a t i o n s has been a p p r a i s e d . I t i s on the b a s i s of t h i s a p p r a i s a l t h a t the p h y s i c a l i n t e r p r e t a t i o n s h o u l d be founded. The m a i n s t a y of the HLEM.INV scheme i s B a c k u s - G i l b e r t a v e r a g i n g , which i s e s s e n t i a l l y a l i n e a r a p p r a i s a l t e c h n i q u e . Backus and G i l b e r t (1968, p l 7 2 ) d e f i n e averages as i n n e r p r o d u c t s of the model w i t h c e r t a i n " a v e r a g i n g f u n c t i o n s " c o n s t r u c t e d from l i n e a r c o m b i n a t i o n s of the k e r n e l s . For l i n e a r i n v e r s e problems the B a c k u s - G i l b e r t averages a re common t o a l l models which f i t the d a t a , but i n n o n - l i n e a r problems t h e i r u n i v e r s a l i t y i s not a s s u r e d . In HLEM.INV the u n i v e r s a l i t y of the averages i s i n v e s t i g a t e d by d i r e c t comparison of averages c o r r e s p o n d i n g t o d i f f e r e n t a c c e p t a b l e models, i n complete analogy w i t h the comparison of models s a t i s f y i n g MT d a t a performed by Oldenburg (1979). In p r a c t i c e , the averages f o r the d i f f e r e n t models a r e o f t e n found t o be e q u a l ( w i t h i n e r r o r ) a t some depths and unequal a t o t h e r s . C o n s i s t e n c y of the averages at a c e r t a i n depth i m p l i e s t h a t the c o n d u c t i v i t y near t h a t depth i s w e l l - d e t e r m i n e d by the data f o r f e a t u r e s on the s c a l e of the 1 1 7 w i d t h of the a v e r a g i n g f u n c t i o n . On the o t h e r hand, i n zones where the averages v a r y s i g n i f i c a n t l y from one a c c e p t a b l e model to a n o ther i t i s not p o s s i b l e t o rea c h d e f i n i t i v e c o n c l u s i o n s about the l o c a l c o n d u c t i v i t y . D e s p i t e i t s l i m i t a t i o n s , B a c k u s - G i l b e r t a v e r a g i n g i s the best a v a i l a b l e a p p r a i s a l t e c h n i q u e f o r n o n - l i n e a r as w e l l as l i n e a r i n v e r s e problems. At the v e r y minimum, the B a c k u s - G i l b e r t averages f o r a . p a r t i c u l a r a c c e p t a b l e model are c h a r a c t e r i s t i c of an i n f i n i t e number of a c c e p t a b l e models which c l o s e l y resemble the g i v e n model. In the broader c o n t e x t of n o n - l i n e a r a p p r a i s a l t h e r e a r e grounds f o r i n v o k i n g B a c k u s - G i l b e r t a v e r a g i n g whenever t h e r e i s some l i k e l i h o o d t h a t a l l a c c e p t a b l e models w i l l bear some q u a l i t a t i v e resemblance t o one a n o t h e r ; f o r the HLEM problem the use of B a c k u s - G i l b e r t averages can t h e r e f o r e be j u s t i f i e d by a p p e a l t o the uniqueness theorem i n Appendix C. In the f i r s t two s e c t i o n s of t h i s c h a p t e r c u r r e n t a p p r a i s a l methods f o r l i n e a r i n v e r s e problems a r e o u t l i n e d t o p r o v i d e background f o r the subsequent d i s c u s s i o n of n o n - l i n e a r a p p r a i s a l i n 111-3. The uniqueness of i n v e r s i o n of HLEM frequency soundings i s c o n s i d e r e d i n g e n e r a l terms i n the f o u r t h s e c t i o n , p r i o r t o a complete d e s c r i p t i o n of the HLEM.INV a p p r a i s a l scheme i n I I I - 5 . A p p r a i s a l i n HLEM.INV i s conducted i n two s t a g e s . In the f i r s t stage a s e t of d i v e r s e a c c e p t a b l e models i s c o n s t r u c t e d from a g i v e n a c c e p t a b l e model u s i n g l i n e a r programming. Such a s u i t e of models i s a p r e - r e q u i s i t e f o r the second stage of HLEM.INV a p p r a i s a l i n which the u n i v e r s a l i t y of B a c k u s - G i l b e r t averages i s a s s e s s e d by comparing the averages f o r the d i f f e r e n t a c c e p t a b l e models. The performance of the 118 scheme i s i l l u s t r a t e d by means of t h e o r e t i c a l examples. 111-1 B a c k u s - G i l b e r t A p p r a i s a l f o r L i n e a r I n v e r s e Problems A p p r a i s a l i n l i n e a r i n v e r s e problems has been put on a v e r y s o l i d f o o t i n g by Backus and G i l b e r t (1968, 1970). T h e i r a v e r a g i n g t e c h n i q u e can be a p p l i e d t o g e n e r a l models which are v e c t o r f u n c t i o n s of more than one v a r i a b l e ; , the o u t l i n e below a p p l i e s t o the s i m p l e s t c a s e , namely when the model i s a s c a l a r f u n c t i o n of a s i n g l e v a r i a b l e . Backus and G i l b e r t e x p l o i t the f a c t t h a t N d a t a , {OJ : j=l,N}, can be r e p r e s e n t e d as i n n e r p r o d u c t s of an a c c e p t a b l e model w i t h known k e r n e l f u n c t i o n s , {Gj(z) : j = l ,N} , as per ( 2 . 1 ) . Thus a l l a c c e p t a b l e models, and i n p a r t i c u l a r the " t r u e " model, have the same i n n e r p r o d u c t w i t h any l i n e a r c o m b i n a t i o n of those k e r n e l s , i . e . (3.1) where m(z) i s any model s a t i s f y i n g the d a t a , and where b i s an a r b i t r a r y v e c t o r of c o e f f i c i e n t s . The t r u e model i s n o r m a l l y a c o n t i n u o u s f u n c t i o n of z and t h e r e f o r e has an i n f i n i t e number of degrees of freedom. C o n s e q u e n t l y , a f i n i t e number of d a t a cannot c o m p l e t e l y d e t e r m i n e the t r u e model. R a t h e r , the d a t a can o n l y p r o v i d e i n f o r m a t i o n about average p r o p e r t i e s of m(z) over c e r t a i n f i n i t e ranges of z. Backus and G i l b e r t (1968, p l 7 2 ) e s t a b l i s h e d the c o n n e c t i o n between the k e r n e l s and t h i s l i m i t e d " r e s o l v i n g power" of the d a t a by d e s i g n a t i n g c e r t a i n l i n e a r c o m b i n a t i o n s of 119 k e r n e l s as " a v e r a g i n g f u n c t i o n s " . An a v e r a g i n g f u n c t i o n c e n t r e d at depth z0 i s denoted by A ( z , z t f ) , and d e f i n e d by A ( z , z , ) = 21 W z , )G.(z) (3.2) f o r some s u i t a b l e c h o i c e of c o e f f i c i e n t s . The i n f o r m a t i o n p e r t a i n i n g t o depths near z0 i s c o n t a i n e d i n the av e r a g e , <m(z 0)>, t h i s b e i n g the i n n e r p r o d u c t of A ( z , z „ ) w i t h any model which f i t s the d a t a , i . e . Combining ( 3 . 1 ) , ( 3 . 2 ) , and ( 3 . 3 ) , i t f o l l o w s t h a t <m(z < >)> i s a-p a r t i c u l a r l i n e a r c o m b i n a t i o n of the da t a f o r any g i v e n b ( z ^ ) , i . e . ' Thus <m(z^)> i s common t o a l l a c c e p t a b l e models i n c l u d i n g , of c o u r s e , the t r u e model. The c h o i c e of b i n (3.2) cannot be a r b i t r a r y i f the a s s o c i a t e d average i s t o a c q u i r e p h y s i c a l s i g n i f i c a n c e as a l o c a l e s t i m a t e of the model. I f , however, i t i s p o s s i b l e t o f i n d a v e c t o r b ( z 0 ) f o r which the composite f u n c t i o n , A(z,zc), i s r e l a t i v e l y l a r g e and peaked near depth zQ but s m a l l everywhere e l s e , then <m(z 0)> i s a p h y s i c a l l y m e a n i n g f u l c h a r a c t e r i s a t i o n of the model near z0 . The n u m e r i c a l v a l u e of the average w i l l c o n s t i t u t e an e s t i m a t e of the magnitude of any a c c e p t a b l e model (3.3) (3.4) 120 near z0 i f the c h o i c e of b i s f u r t h e r r e s t r i c t e d t o a l l o w o n l y those f u n c t i o n s A which a r e u n i m o d u l a r , i . e . have u n i t a r e a . I f i t i s not p o s s i b l e t o c o n s t r u c t a f u n c t i o n of the form (3.2) which i s c o n c e n t r a t e d near z0 , i t may be c o n c l u d e d t h a t the d a t a do not f u r n i s h l o c a l i s e d i n f o r m a t i o n a t t h a t d e p t h . When the o b s e r v a t i o n s are c o n t a m i n a t e d w i t h n o i s e , so too i s any average v i a ( 3 . 4 ) . The v a r i a n c e , £ 2 , of an average can be computed from the c o v a r i a n c e s of the e r r o r s i n the i n d i v i d u a l o b s e r v a t i o n s (Backus & G i l b e r t , 1970, p l 3 3 ) ; s p e c i f i c a l l y , where the e r r o r s i n the d a t a a r e r e g a r d e d as r e a l i s a t i o n s of the random v a r i a b l e s { Q } : j = l , N } , and where £ i s the c o v a r i a n c e m a t r i x of t h o s e v a r i a b l e s . Backus and G i l b e r t (1970) show t h a t " r e s o l u t i o n " and " a c c u r a c y " a r e m u t u a l l y a n t a g o n i s t i c a t t r i b u t e s of an average, i . e . t h a t r e l i a b i l i t y i s enhanced o n l y a t the expense of a wider a v e r a g i n g "window". Thus, the most d e l t a - l i k e a v e r a g i n g f u n c t i o n s g i v e r i s e t o the most i n a c c u r a t e a v e r a g e s , w h i l e the a v e r a g i n g f u n c t i o n s f o r more r e l i a b l e averages a t the same depth are n e c e s s a r i l y more d i s t r i b u t e d . The "most s u i t a b l e " a v e r a g i n g f u n c t i o n s cannot be s p e c i f i e d 'a p r i o r i ' , but r a t h e r w i l l depend on the p a r t i c u l a r a p p l i c a t i o n ; r e s o l u t i o n might be the o v e r r i d i n g c o n s i d e r a t i o n i n some c a s e s , w h i l e a c c u r a c y c o u l d be of paramount importance i n o t h e r s . To guide s e l e c t i o n of a v e r a g i n g f u n c t i o n s , Backus and G i l b e r t (1970) i n t r o d u c e the n o t i o n of a " t r a d e - o f f c u r v e " f o r any depth of i n t e r e s t ; t h i s N (3.5) 121 c u r v e r e l a t e s the v a r i a n c e of any average t o the w i d t h of the c o r r e s p o n d i n g a v e r a g i n g peak f o r d i f f e r e n t v a l u e s of a " t r a d e - o f f parameter", 9. A c o n t i n u o u s d i s t r i b u t i o n of a v e r a g i n g f u n c t i o n s may be computed f o r any depth z0 by f i n d i n g the c o e f f i c i e n t s , b ( z e , 0 ) , which m i n i m i s e the f u n c t i o n a l , " ) ^ 2 , d e f i n e d by N y>2 = Y2cosc9 + £2sin0 + 2 A [ 1 - bj ^ Gj dz ] (3.6) In t h i s e q u a t i o n Y 2 i s a measure of the " d e l t a n e s s " of the a v e r a g i n g f u n c t i o n , £ 2 i s the v a r i a n c e of the av e r a g e , as per ( 3 . 5 ) , and A i s a Lagrange m u l t i p l i e r f o r the u n i m o d u l a r i t y c o n s t r a i n t . When Y 2 i s q u a d r a t i c i n b, m i n i m i s a t i o n of (3.6) w i t h r e s p e c t t o the components of b g i v e s r i s e t o a l i n e a r system of e q u a t i o n s , from which b(ze ,9) may be de t e r m i n e d . Backus and G i l b e r t (1970, p l 2 8 ) d e v i s e one p o s s i b l e measure of " d e l t a n e s s " , known as the "s p r e a d " ; t h i s c r i t e r i o n i s employed i n HLEM.INV a p p r a i s a l , as d e s c r i b e d i n I I I - 5 . Other measures of " d e l t a n e s s " i n common use a r e the 1 s t and 2nd D i r i c h l e t c r i t e r i a ( Oldenburg, 1976). C o n s i d e r the m i n i m i s a t i o n of "fi2 i n ( 3 . 6 ) . I t i s c l e a r t h a t when 9 = 0, o n l y Y 2 i s m i n i m i s e d , the o n l y c o n s t r a i n t on v a r i a n c e i n t h i s case b e i n g t h a t imposed by the u n i m o d u l a r i t y r e q u i r e m e n t ; thus 9=0 i s the s i g n a t u r e of h i g h l y r e s o l v e d , y e t p o s s i b l y i n a c c u r a t e a v e r a g e s . C o n v e r s e l y , when 9 = TT/2 the " d e l t a n e s s " of the a v e r a g i n g f u n c t i o n does not i n f l u e n c e the m i n i m i s a t i o n , and b i s non-zero o n l y by v i r t u e of the u n i m o d u l a r i t y c o n d i t i o n ; a t t h i s extreme, the averages a re 122 u s u a l l y r e l i a b l e , but p o o r l y r e s o l v e d . 111 -2 The R e s o l u t i o n M a t r i x In the c o n v e n t i o n a l p a r a m e t r i c i n v e r s i o n of Wiggins (1972), a p p r a i s a l i s performed by a more or l e s s s u b j e c t i v e e x a m i n a t i o n of a m a t r i x , known as the r e s o l u t i o n m a t r i x . As the name i m p l i e s , the r e s o l u t i o n m a t r i x i s c o n s i d e r e d t o c h a r a c t e r i s e the r e s o l v i n g power of the d a t a . In t h i s s e c t i o n the m a t h e m a t i c a l s i g n i f i c a n c e of the r e s o l u t i o n m a t r i x w i l l be examined, and the c o n n e c t i o n between the r e s o l u t i o n m a t r i x and the most h i g h l y r e s o l v e d B a c k u s - G i l b e r t a v e r a g i n g f u n c t i o n s w i l l be e s t a b l i s h e d . E s s e n t i a l t o an u n d e r s t a n d i n g of the r e s o l u t i o n m a t r i x i s an a p p r e c i a t i o n of the fundamental d e c o m p o s i t i o n a d m i t t e d by any m a t r i x . As per Lanczos (1961, p l 2 1 ) , any NxP m a t r i x A can be e x p r e s s e d as a product of t h r e e m a t r i c e s , i . e . A = U A V T (3.7) <V J*V 0* where the columns of U (V) c o n s i s t of o r t h o n o r m a l e i g e n v e c t o r s a s s o c i a t e d w i t h the rows (columns) of A, and where A i s a d i a g o n a l m a t r i x c o n t a i n i n g the e i g e n v a l u e s of A. I f J i s the l e a s t of N and P, t h e r e a r e a t most J non-zero e i g e n v a l u e s . The o r t h o n o r m a l i t y of the e i g e n v e c t o r s may be e x p r e s s e d by U TU = I = V TV (3.8) where I i s the i d e n t i t y m a t r i x . 123 In o r d e r to f a c i l i t a t e comparison between the work of W i g g i n s (1972) and the f o r m u l a t i o n d e s c r i b e d i n Chapter I I , W i g g i n s ' n o t a t i o n w i l l be f o l l o w e d as c l o s e l y as p o s s i b l e i n t h i s s e c t i o n . In p a r t i c u l a r , the l i n e a r e q u a t i o n s w i l l be w r i t t e n as r e l a t i o n s between changes i n responses and changes i n model p a r a m e t e r s ; however, t h i s s h o u l d not obscure the f a c t t h a t i n a t r u l y l i n e a r i n v e r s e problem the v a l i d i t y of the l i n e a r system i s not r e s t r i c t e d t o s m a l l p e r t u r b a t i o n s . F o l l o w i n g Wiggins (1972, p255), the p e r t u r b a t i o n e q u a t i o n s (2.34) r e l a t i n g r e s i d u a l s , 7], between responses and o b s e r v a t i o n s t o changes, cfc^ , i n c o n d u c t i v i t y are w r i t t e n i n the form ^ = b&V (3.9) A 'A where A = DH (3.10) St = *X6Tr As i n Chapter I I , H i s a d i a g o n a l m a t r i x of l a y e r t h i c k n e s s e s , and the c i r c u m f l e x i s used t o i n d i c a t e s c a l i n g w i t h r e s p e c t t o the s t a n d a r d d e v i a t i o n s . In (3.9) the m a t r i x A can be v i s u a l i s e d as a l i n e a r t r a n s f o r m a t i o n from a P - d i m e n s i o n a l "model space" t o an N - d i m e n s i o n a l "data space". U s i n g (3.7) and ( 3 . 8 ) , t h i s i n t e r p r e t a t i o n i s summarised m a t h e m a t i c a l l y by AV = UA (3.11) 124 Thus A maps an o r t h o n o r m a l b a s i s of t h e p a r a m e t e r s p a c e o n t o an o r t h o n o r m a l b a s i s -of d a t a s p a c e . C o n v e r s e l y , A T maps v e c t o r s from d a t a s p a c e i n t o model s p a c e : A T U = VA (3.12) I f r i s t h e rank of A, t h e r e a r e o n l y r n o n - z e r o e i g e n v a l u e s i n A . T h e r e f o r e A i s " a c t i v a t e d " o n l y i n r - d i m e n s i o n a l s u b s p a c e s of d a t a s p a c e and model s p a c e ( L a n c z o s , 1961, p l 2 3 ) . In an u n d e r d e t e r m i n e d p r o b l e m ( r ^  N < P) t h e r e a r e P-r e i g e n v e c t o r s i n V w i t h image z e r o under t h e o p e r a t i o n of A. The o t h e r r e i g e n v e c t o r s i n V ( c o r r e s p o n d i n g t o n o n - z e r o e i g e n v a l u e s ) c a n n o t span t h e P - d i m e n s i o n a l model s p a c e ; c o n s e q u e n t l y , t h e r e a r e s t r u c t u r e s ( c o m b i n a t i o n s o f p a r a m e t e r s ) w h i c h p r o d u c e no m e a s u r a b l e e f f e c t . The e x i s t e n c e o f s u c h n u l l v e c t o r s o r " a n n i h i l a t o r s " i s t h e c a u s e o f t h e i n f i n i t e n o n - u n i q u e n e s s of l i n e a r i n v e r s e p r o b l e m s . D i s c a r d i n g e i g e n v a l u e s a g g r a v a t e s t h i s c o n d i t i o n b e c a u s e t h e d i m e n s i o n of the p a r a m e t e r - s u b s p a c e w h i c h c an be e x p l o r e d i s e q u a l t o t h e number o f e i g e n v a l u e s r e t a i n e d . I f t h e p r o b l e m i s o v e r d e t e r m i n e d (r ^ P < N), A may have P n o n - z e r o e i g e n v a l u e s , i n w h i c h e v e n t i t i s " a c t i v a t e d " i n a l l d i m e n s i o n s of model s p a c e . However, t h e r e w i l l i n g e n e r a l be no e x a c t s o l u t i o n i n t h i s c a s e b e c a u s e r e i g e n v e c t o r s c a n n o t span t h e d a t a s p a c e ; i f t h e d a t a c a n n o t be r e p r e s e n t e d as a l i n e a r c o m b i n a t i o n o f t h e r a v a i l a b l e e i g e n v e c t o r s , t h e y a r e s a i d t o be " i n c o m p a t i b l e " . In l i n e a r i n v e r s e p r o b l e m s i n c o m p a t i b i l i t y i s due t o t h e p r e s e n c e o f e r r o r s i n t h e d a t a ; i n n o n - l i n e a r p r o b l e m s , when l i n e a r 125 e q u a t i o n s a r e n e c e s s a r i l y a p p r o x i m a t e , i n c o m p a t i b i l i t y may a r i s e from the i n e x a c t m a t h e m a t i c a l f o r m u l a t i o n as w e l l as from e r r o r s . Only when N = r = P i s i t p o s s i b l e f o r A t o op e r a t e i n a l l d i m e n s i o n s of i t s domain and range s i m u l t a n e o u s l y . In p r a c t i c a l problems i t i s u s u a l l y e s s e n t i a l t o d i s c a r d some s m a l l e i g e n v a l u e s i n d e f e r e n c e t o e r r o r s i n the d a t a , i t be i n g u n d e s i r a b l e t o s a t i s f y e r r o n e o u s d a t a e x a c t l y ( L a n c z o s , 1961, p l 3 1 ) . C o n s e q u e n t l y , r e g a r d l e s s of the r e l a t i v e .magnitudes of N and P, t h e r e i s a s e t of e i g e n v e c t o r s , m=l,M}, f o r which Av* ~ 0 m=l,M (3.13) In t h i s c o n t e x t the " ~ " s h o u l d be i n t e r p r e t e d t o mean t h a t when elements of {v*} are added t o a model, e i t h e r i n d i v i d u a l l y or i n l i n e a r c o m b i n a t i o n , the r e s u l t i n g changes i n the o b s e r v a t i o n s a r e t o t a l l y n e g l i g i b l e i n comparison t o the e m p i r i c a l u n c e r t a i n t i e s . Thus t h e r e a re c e r t a i n p h y s i c a l s t r u c t u r e s about which the o b s e r v a t i o n s do not e n l i g h t e n us a t a l l , so any " a p p r a i s a l " of a c c e p t a b l e models i s n e c e s s a r i l y r e s t r i c t e d t o the subspace spanned by the e i g e n v e c t o r s i n V w i t h " s u f f i c i e n t l y l a r g e " e i g e n v a l u e s . C o n s i d e r now the r e s o l u t i o n m a t r i x , R, d e f i n e d by £ ' 2L£ ( 3 - 1 4 ) Here V L i s the PxL m a t r i x w i t h columns c o r r e s p o n d i n g t o the e i g e n v e c t o r s a s s o c i a t e d w i t h " l a r g e " e i g e n v a l u e s , these b e i n g L 126 i n number. In the l i g h t of the p r e c e d i n g d i s c u s s i o n , R may be r e c o g n i s e d as a p r o j e c t i o n on parameter space (Hoffman and Runze, 1971, p211). Thus R p r o j e c t s any v e c t o r i n V i n t o the L - d i m e n s i o n a l subspace spanned by the e i g e n v e c t o r s i n VL. In o r d e r to g a i n some u n d e r s t a n d i n g of the s i g n i f i c a n c e of R, suppose t h a t cfp+ i s the t r u e "model", and t h a t Sp i s i t s p r o j e c t i o n on the subspace spanned by the e i g e n v e c t o r s i n V u. Then where ^ i s a v e c t o r of c o e f f i c i e n t s . I f e r r o r s a r e i g n o r e d , f - f j p -whence, Sp = R<Jp+ (3.15) A l t e r n a t i v e l y , <5p may be regar d e d as a v e c t o r of "averages" of S~p* , and the rows of R may be v i s u a l i s e d as f i l t e r s or " a v e r a g i n g f u n c t i o n s " . To e s t a b l i s h the c o n n e c t i o n w i t h B a c k u s - G i l b e r t a p p r a i s a l , l e t us seek the f i l t e r 6 which m i n i m i s e s <f>* = iis - skr lk where o e q u a l s one i n the k t h l a y e r , and i s z e r o everywhere 127 e l s e . M i n i m i s a t i o n of <fi2 r e p r e s e n t s an a p p l i c a t i o n of the 1st D i r i c h l e t " d e l t a n e s s " c r i t e r i o n . The d e s i r e d f i l t e r i s a l i n e a r c o m b i n a t i o n of the b a s i s v e c t o r s i n V, : 4= XLY (3.16) where y i s a v e c t o r of c o e f f i c i e n t s t o be d e t e r m i n e d . M i n i m i s i n g 0 2 w i t h r e s p e c t t o the components of y, i t t r a n s p i r e s t h a t _ » Y "ck r = J (3.17) S u b s t i t u t i n g f o r y i n ( 3 . 1 6 ) , the d e s i r e d a v e r a g i n g f u n c t i o n i s g i v e n by ^ - V L V ^ f e = R d f k (3.18) Thus, the most " d e l t a - l i k e " a v e r a g i n g f u n c t i o n c e n t r e d on the k t h l a y e r c o r r e s p o n d s t o the k t h row or column of the r e s o l u t i o n m a t r i x . From (3.18) i t f o l l o w s ' t h a t J p i s the best a v a i l a b l e l e a s t squares e s t i m a t e of Sp* which can be c o n s t r u c t e d from L e i g e n v e c t o r s (Hoffman and Kunze, 1971, p284). In t h i s sense, the 1st D i r i c h l e t a v e r a g i n g f u n c t i o n i s the most h i g h l y r e s o l v e d e s t i m a t e a t t a i n a b l e ; i n e v i t a b l y , the a s s o c i a t e d parameter e s t i m a t e s (averages) a r e the most u n r e l i a b l e (Backus and G i l b e r t , 1970). Jackson (1972, p l 0 4 ) has observed t h a t parameter e s t i m a t e s w i t h lower v a r i a n c e c o u l d be o b t a i n e d by winnowing e i g e n v a l u e s , i . e . by d e c r e a s i n g L. However, o n l y a f i n i t e 128 number of p a r t i c u l a r v a r i a n c e s a r e a t t a i n a b l e i f e i g e n v a l u e s a r e winnowed, whereas averages c o r r e s p o n d i n g t o any v a r i a n c e (over a c e r t a i n range) are a v a i l a b l e i n B a c k u s - G i l b e r t a p p r a i s a l . Thus B a c k u s - G i l b e r t a v e r a g i n g i s more g e n e r a l . 111-3 A p p r a i s a l Techniques f o r Non-Linear I n v e r s e Problems The e x i s t i n g t e c h n i q u e s f o r a p p r a i s a l of l i n e a r i n v e r s e problems can be c o n s i d e r e d adequate f o r most a p p l i c a t i o n s , w i t h the B a c k u s - G i l b e r t averages p r o v i d i n g a f i r m f o u n d a t i o n f o r any i n t e r p r e t a t i o n . However, f o r n o n - l i n e a r problems the development of a p p r a i s a l t e c h n i q u e s i s f a r from complete and, moreover, may never be complete. In view of the pre-eminence of B a c k u s - G i l b e r t a v e r a g i n g i n l i n e a r a p p r a i s a l , the d i f f i c u l t i e s e n c o u n t e r e d i n n o n - l i n e a r a p p r a i s a l w i l l be d e s c r i b e d w i t h i n the framework of a v e r a g i n g . The same d i f f i c u l t i e s , i n somewhat d i f f e r e n t g u i s e , p l a g u e the o t h e r a p p r a i s a l methods f o r n o n - l i n e a r problems. The fundamental c o m p l i c a t i o n i n a n o n - l i n e a r i n v e r s e problem i s the dependence of the k e r n e l s on the model. J u s t as f o r a l i n e a r problem, the responses can be r e g a r d e d as i n n e r p r o d u c t s of the model w i t h k e r n e l f u n c t i o n s , but each d i f f e r e n t model i s a s s o c i a t e d w i t h i t s own s e t of k e r n e l s i n the n o n - l i n e a r c a s e , as per ( 2 . 2 ) . I f the k e r n e l f u n c t i o n s change " c o n t i n u o u s l y " w i t h changes i n the model, i . e . i f the problem i s F r e c h e t d i f f e r e n t i a b l e , then " s i m i l a r " models w i l l be a s s o c i a t e d w i t h " s i m i l a r " k e r n e l s , and the i n v e r s e problem w i l l be a p p r o x i m a t e l y l i n e a r f o r some r e s t r i c t e d c l a s s of models s u f f i c i e n t l y " c l o s e " t o a g i v e n a c c e p t a b l e model. For any 1 2 9 F r e c h e t d i f f e r e n t i a b l e problem, t h e r e f o r e , i t i s p o s s i b l e t o a p p l y l i n e a r a p p r a i s a l t e c h n i q u e s t o an a c c e p t a b l e model, but w i t h the i m p o r t a n t q u a l i f i c a t i o n t h a t the scope of such an a p p r a i s a l i s r e s t r i c t e d t o a somewhat nebulous s e t of models which c l o s e l y resemble the o r i g i n a l a c c e p t a b l e model. At the v e r y minimum, t h e r e f o r e , B a c k u s - G i l b e r t averages a r e c h a r a c t e r i s t i c of an i n f i n i t e subset of a c c e p t a b l e models; however, the c h a r a c t e r i s a t i o n of an i n f i n i t e s e t of a c c e p t a b l e models i s of no p r a c t i c a l s i g n i f i c a n c e i f the s e t does not i n c l u d e the t r u e model. In o r d e r to c l e a r l y d e l i n e a t e the l i m i t a t i o n s of B a c k u s - G i l b e r t a v e r a g i n g i n n o n - l i n e a r a p p r a i s a l , suppose t h a t two a c c e p t a b l e models, m, and m2, have been found. The two models s a t i s f y the same d a t a , so the d i f f e r e n c e s between t h e i r t h e o r e t i c a l responses a r e i d e n t i c a l l y z e r o ( w i t h i n e r r o r ) . Hence, r e c a l l i n g (2.6) e q u a t i o n i s e q u a l l y v a l i d i f D.(z;m( ) i s r e p l a c e d by Blz;mz) , the c o r r e s p o n d i n g F r e c h e t k e r n e l f o r m2. I f the n o n - l i n e a r term i s n e g l i g i b l e , then 0 j = l , N (3.19) where {Dj( z ;m, ) : j = l ,N} a r e the F r e c h e t k e r n e l s f o r m, ; the D.(z;m, )m,dz = \D.(z;m z)m zdz = t j , say Thus d i f f e r e n t a c c e p t a b l e models g i v e r i s e t o the same i n n e r p r o d u c t s , i n c l o s e a n a l o g y w i t h the d e f i n i n g r e l a t i o n f o r a 130 l i n e a r i n v e r s e problem, ( 2 . 1 ) . C o n s e q u e n t l y , a v e r a g i n g f u n c t i o n s can be d e f i n e d as l i n e a r c o m b i n a t i o n s of k e r n e l s , as i n ( 3 . 2 ) , and the c o r r e s p o n d i n g averages may be e x p r e s s e d as l i n e a r c o m b i n a t i o n s of the d e r i v e d q u a n t i t i e s { t j : j = l , N } , i n analogy w i t h ( 3 . 4 ) . As b e f o r e , peaked a v e r a g i n g f u n c t i o n s can be sought at any d e p t h , and the a c c u r a c y and r e s o l u t i o n w i d t h of the r e s u l t i n g average can be a d j u s t e d t o s u i t the r e q u i r e m e n t s of the p a r t i c u l a r s u r v e y . I t must always be borne i n mind, however, t h a t the averages s o - o b t a i n e d a r e not n e c e s s a r i l y common t o a l l a c c e p t a b l e models because the the n o n - l i n e a r term i n (3.19) may not always be n e g l i g i b l y s m a l l . T h e r e f o r e , B a c k u s - G i l b e r t averages a l o n e do not c o n s t i t u t e a complete a p p r a i s a l of a n o n - l i n e a r i n v e r s e problem s i n c e they need not p e r t a i n t o the t r u e model. T h i s l i m i t a t i o n of B a c k u s - G i l b e r t a v e r a g i n g has been an i m p o r t a n t c o n s i d e r a t i o n i n the development of the a p p r a i s a l scheme i n HLEM.INV. I t has been e s t a b l i s h e d t h a t l i n e a r i s e d a p p r a i s a l of one a c c e p t a b l e model does not convey any i n f o r m a t i o n about a c c e p t a b l e models which a r e v e r y d i f f e r e n t from the g i v e n model. C o n s e q u e n t l y , as s t r e s s e d by Anderssen (1975), the f i r s t s t e p towards a comprehensive a p p r a i s a l f o r a n o n - l i n e a r i n v e r s e problem s h o u l d i d e a l l y be the i d e n t i f i c a t i o n of a l l p o s s i b l e s o l u t i o n s . C o n c e p t u a l l y , the most s t r a i g h t f o r w a r d approach t o the problem of l o c a t i n g a l l a c c e p t a b l e models i s t o embark on a random s e a r c h ; i n essence, t h i s i s the p h i l o s o p h y of the Monte C a r l o method ( P r e s s , 1970). Depending on one's p o i n t of view, the method can be h a i l e d f o r i t s o b j e c t i v i t y or b e r a t e d because i t s r e l i a n c e on a random s e a r c h does not a l l o w i t t o e x p l o i t any 131 s p e c i a l f e a t u r e s of the problem i n hand. Be t h a t as i t may, the method i s t h w a r t e d i n the f i n a l a n a l y s i s by p r a c t i c a l i t i e s because, except i n the s i m p l e s t of problems, i t i s not p o s s i b l e t o conduct an e x h a u s t i v e s e a r c h . (See the f o o t n o t e on page 126 of Backus and G i l b e r t , 1970, f o r example.) An a l t e r n a t i v e approach i s t o endeavour t o prove t h a t t h e r e i s o n l y one d i s t i n c t s o l u t i o n . E s s e n t i a l to t h i s approach i s a uniqueness theorem, such as t h a t p r e s e n t e d i n Appendix C. However, uniqueness p r o o f s i n v a r i a b l y r e l a t e t o i d e a l i s e d c o n d i t i o n s which a r e u n a t t a i n a b l e i n p r a c t i c e , and so do not c o n s t i t u t e a guarantee t h a t a l l models s a t i s f y i n g a s e t of r e a l o b s e r v a t i o n s w i l l i n f a c t resemble one a n o t h e r . N e v e r t h e l e s s , a uniqueness theorem does ensure t h a t the a c q u i s i t i o n of s u f f i c i e n t q u a n t i t i e s of ( a c c u r a t e ) d a t a w i l l u l t i m a t e l y remove any a m b i g u i t y . .111-4 The Uniqueness of HLEM I n v e r s i o n An i m p o r t a n t a t t r i b u t e of any i n v e r s i o n i s i t s " u n i q u e n e s s " , by which i s meant the a b i l i t y t o d e f i n e a unique s t r u c t u r e i n the h y p o t h e t i c a l case when an u n l i m i t e d amount of p e r f e c t l y a c c u r a t e d a t a i s a v a i l a b l e . On t h i s b a s i s g r a v i t y i n v e r s i o n i s "non-unique" (Grant and West, 1965, pp214-216), whereas i n v e r s i o n of r e s i s t i v i t y s oundings over a s t r a t i f i e d e a r t h i s "unique" (Langer, 1933). I f uniqueness c o u l d be e s t a b l i s h e d , i t might be p o s s i b l e t o g a i n i n s i g h t i n t o the e x t e n t t o which a f i n i t e number of HLEM d a t a c o n s t r a i n any a c c e p t a b l e model t o be " c l o s e " t o the t r u e model. For example, 132 i f a s m a l l number of d a t a s u f f i c e d t o d e f i n e s t r u c t u r e s w h i c h d i f f e r e d from t h e t r u e s t r u c t u r e i n s m a l l s c a l e d e t a i l o n l y a t c e r t a i n d e p t h s , t h e n a l l a c c e p t a b l e models would be s i m i l a r a t t h o s e d e p t h s . I f , on t h e o t h e r hand, v e r y many d a t a were needed t o e n s u r e t h a t a l l a c c e p t a b l e models r e s e m b l e d t h e " t r u e " model, a u n i q u e n e s s t h e o r e m would p r o v i d e v e r y l i m i t e d r e a s s u r a n c e . I t i s t a c i t l y assumed h e r e t h a t i f t h e i n v e r s e p r o b l e m i s u n i q u e , a l l a c c e p t a b l e models w i l l a p p r o a c h t h e t r u e model i n a " c o n t i n u o u s " way as t h e number o f d a t a i n c r e a s e s ; f o r t h e HLEM p r o b l e m , s u c h b e h a v i o u r i s g u a r a n t e e d by F r e c h e t d i f f e r e n t i a b i l i t y . The most d i r e c t way t o a s s e s s t h e u n i q u e n e s s of i n v e r s i o n i s by c o n s t r u c t i o n o f a se q u e n c e o f a c c e p t a b l e m o d e l s . O l d e n b u r g (1979, 1980) has employed t h i s s t r a t e g y i n an a t t e m p t t o e x p l o r e t h e s u b s p a c e of models s a t i s f y i n g c e r t a i n MT d a t a ; u s i n g d i v e r s e s t a r t i n g m o d e l s , he o b t a i n e d s o l u t i o n s w h i c h c l o s e l y r e s e m b l e d one a n o t h e r o v e r c o n s i d e r a b l e r a n g e s of d e p t h . G i v e n t h e c l o s e a n a l o g y between MT and HLEM s o u n d i n g , h i s f i n d i n g s augur w e l l f o r t h e d e f i n i t i v e n e s s of HLEM i n v e r s i o n . When a p p l i e d t o HLEM i n v e r s i o n , t h e same i d e a has p r o d u c e d c o r r e s p o n d i n g r e s u l t s , as shown i n F i g . 3.1. The s t a r t i n g m o d e l s f o r t h e two i n v e r s i o n s i m p l i c i t i n t h e f i g u r e d i f f e r e d by two o r d e r s of m a g n i t u d e , y e t t h e r e i s n o n e t h e l e s s a good q u a l i t a t i v e agreement between t h e two c o n s t r u c t e d m o d e l s , and between t h o s e models and t h e t r u e model ( F i g . 2 . 2 ) . The number of i t e r a t i o n s r e q u i r e d f o r t h e c o n s t r u c t i o n of t h e model i n Fig'. 3.1b i s not f l a t t e r i n g . I t i s t o be e x p e c t e d t h a t s i g n i f i c a n t l y fewer i t e r a t i o n s w ould be n e c e s s a r y i f l a r g e r p e r t u r b a t i o n s were a l l o w e d , i . e . I f a l a r g e r 133 F i g u r e 3.1 C o n s t r u c t i o n from Remote S t a r t i n g Models The t h i n l i n e i n each frame r e p r e s e n t s the " t r u e " model ( c f . F i g . 2.2) . (a) C o n s t r u c t e d model f o r t h e o r e t i c a l |Hr|,&r,|Hz|,&0z d a t a u s i n g h a l f - s p a c e w i t h c o n d u c t i v i t y 1 S/m as the s t a r t i n g model. (b) C o n s t r u c t e d model f o r t h e o r e t i c a l |Hr|,0r,|Hz|,&0z d a t a u s i n g h a l f - s p a c e w i t h c o n d u c t i v i t y 0.01 S/m as the s t a r t i n g model. 134 r|,0r,|Hz|,&0z construction 10° F " — I 1 1 1 1 1 1 a £ cn b 10-1 "ti 10 -2 x. = 852.1 I x* = 13.9 x = 15.2 max ITER = 19 I I I I 0 1 0 0 2 0 0 3 0 0 4 0 0 ( X 101) Fig . 3.1a |Hr|,0r,|Hz|,&0z construction 10° E b 10"1 U 10 -2 0 1 0 0 2 0 0 3 0 0 4 0 0 ( X 101) depth (m) F i g . 3.1b 135 v a l u e of DLNMAX ( d e f i n e d i n 11-3) had been chosen. However, the number of i t e r a t i o n s i s i n c i d e n t a l t o the d e m o n s t r a t i o n t h a t , a t l e a s t f o r the data s e t i n q u e s t i o n , t h e r e a re o b v i o u s s i m i l a r i t i e s between a l l a c c e p t a b l e models. In p r a c t i c e i t would never be n e c e s s a r y t o use such a poor s t a r t i n g model because the program c o u l d be i n v e s t e d w i t h the f a c i l i t y t o s e l e c t an a p p r o p r i a t e i n i t i a l guess; f o r example, the c o n d u c t i v i t y of the h a l f space p r o v i d i n g the be s t f i t t o the da t a c o u l d be det e r m i n e d . I n v e r s i o n of HLEM geometric soundings over a s t r a t i f i e d e a r t h i s known t o be unique ( S l i c h t e r , 1933), as i s i n v e r s i o n of o n e - d i m e n s i o n a l geomagnetic and MT p a r a m e t r i c soundings ( S i e b e r t , 1964; Chetaev, 1966; B a i l e y , 1970; Johnson and S m y l i e , 1970; W e i d e l t , 1972). However, f o r reasons expounded i n some d e t a i l i n Appendix C, none of the s e e a r l i e r r e s u l t s bears d i r e c t l y on the i n v e r s i o n of HLEM fr e q u e n c y s o u n d i n g s . C o n s e q u e n t l y , the uniqueness of HLEM p a r a m e t r i c i n v e r s i o n became an i m p o r t a n t t o p i c f o r i n v e s t i g a t i o n i n the c o u r s e of t h i s p r o j e c t , and the theorem i n Appendix C was d e r i v e d as a r e s u l t . I t has been proved t h a t i f p e r f e c t l y a c c u r a t e HLEM d a t a a r e a v a i l a b l e over some f i n i t e range of h i g h f r e q u e n c i e s , the M a c l a u r i n S e r i e s c o e f f i c i e n t s of c ( z ) (and hence the c o n d u c t i v i t y i t s e l f ) can be d e t e r m i n e d u n i q u e l y . Uniqueness f o l l o w s from c o n s i d e r a t i o n of the cr"-dependence of the c o e f f i c i e n t s i n an a s y m p t o t i c e x p a n s i o n f o r secondary magnetic f i e l d , A H , i n the l i m i t of h i g h f r e q u e n c i e s . E x p l i c i t l y , an exp a n s i o n of the f o l l o w i n g form i s d e r i v e d f o r A H : 136 oo A H ~*2L £^(<7" a s /3 — > ^ ( 3 . 2 0 ) where t h e c o e f f i c i e n t s { £T} a r e d e f i n e d by (B.61) and ( B . 6 5 ) , and where y3 = R^yu^d^i 0) , a s p e r ( 1 . 8 ) . On t h e b a s i s o f ( 3 . 2 0 ) , i t i s p o s s i b l e t o g a i n a q u a l i t a t i v e i n s i g h t i n t o t h e e x t e n t t o w h i c h t h e c o n d u c t i v i t y i s c o n s t r a i n e d by a f i n i t e number o f f r e q u e n c y s o u n d i n g s . To t h i s e n d , s u p p o s e t h a t e x a c t d a t a a r e a v a i l a b l e a t N d i f f e r e n t h i g h f r e q u e n c i e s . I f A H j d e n o t e s t h e t r u e v a l u e o f t h e s e c o n d a r y m a g n e t i c f i e l d a t f r e q u e n c y Wj , t h e n f r o m ( 3 . 2 0 ) H-I ^ H J = H * n f o n + TJ 3 = 1 ' N ( 3 ' 2 1 ) where Tj d e n o t e s t h e t r u n c a t i o n e r r o r ( a t f r e q u e n c y WJ ) a f t e r N t e r m s . By d e f i n i t i o n o f an a s y m p t o t i c e x p a n s i o n , | T \p~t — > 0 a s j3 —> 0 0 Thus t h e t r u n c a t i o n e r r o r a t a f i x e d v a l u e o f N d e c r e a s e s a s i n c r e a s e s . S u p p o s e t h a t t h e s o u n d i n g f r e q u e n c i e s a r e s u f f i c i e n t l y h i g h t h a t "Cj i s a l w a y s a s m a l l f r a c t i o n o f t h e c o r r e s p o n d i n g datum A H j . Under t h e s e c o n d i t i o n s t h e e q u a t i o n s , ( 3 . 2 1 ) , d e f i n e a l i n e a r i n v e r s e p r o b l e m w i t h i n a c c u r a t e " d a t a " , { ^ H J ' - T J I : j = l , N } , w h i c h c a n be s o l v e d t o y i e l d e s t i m a t e s ( " a v e r a g e s " ) o f t h e c o e f f i c i e n t s , { £ ^ ^ = 0,1^-1}. I t i s shown i n A p p e n d i x C t h a t k n o w l e d g e o f t h e f i r s t N c o e f f i c i e n t s i n t h e e x p a n s i o n ( 3 . 2 0 ) i s t a n t a m o u n t t o k n o w l e d g e o f t h e f i r s t N M a c l a u r i n S e r i e s c o e f f i c i e n t s f o r c r ' ( z ) ; t h u s , r o u g h l y s p e a k i n g , 137 N f r e q u e n c y soundings support an Nth o r d e r p o l y n o m i a l a p p r o x i m a t i o n f o r the c o n d u c t i v i t y . The i n s u f f i c i e n c y of the N secondary magnetic f i e l d d a t a i s communicated v i a the t r u n c a t i o n e r r o r s , and i s t r a n s f o r m e d i n t o u n c e r t a i n t i e s i n the M a c l a u r i n S e r i e s c o e f f i c i e n t s . I t i s i n t u i t i v e l y more p r o b a b l e t h a t these u n c e r t a i n t i e s w i l l f i n d e x p r e s s i o n i n a s i n g l e " c o r r i d o r " of p o s s i b l e models, r a t h e r than i n d i s t i n c t model regimes which somehow c o a l e s c e i n t o the unique c o n d u c t i v i t y as N i n c r e a s e s . The o b s e r v a t i o n s { A H j : j = l,N} are complex, and are t h e r e f o r e e q u i v a l e n t t o 2N r e a l d a t a . In the s c e n a r i o d e c r i b e d above o n l y N ( r e a l ) M a c l a u r i n S e r i e s c o e f f i c i e n t s were det e r m i n e d from the N complex d a t a , whereas (assuming some v e s t i g e of l i n e a r i t y ) 2N r e a l parameters c o u l d have been s p e c i f i e d . The i m p l i c a t i o n i s t h a t N redundant c o n d i t i o n s are imposed on the N r e a l M a c l a u r i n S e r i e s c o e f f i c i e n t s by the N complex d a t a . There i s no q u e s t i o n of redundancy i f the unknown p h y s i c a l p r o p e r t y i s complex, however; t h i s would be the case i f d i s p l a c e m e n t c u r r e n t s were taken i n t o a c c o u n t , i n which case the M a c l a u r i n S e r i e s c o e f f i c i e n t s of [ y 2£(z) + iy3 2 0"(z) ] would be sought. A l t h o u g h the uniqueness of i n v e r s i o n of HLEM fr e q u e n c y soundings has not been prove d f o r cases when d i e l e c t r i c p r o p e r t i e s p l a y an a p p r e c i a b l e r o l e i n d e t e r m i n i n g the e l e c t r o m a g n e t i c response of the ground, the apparent redundancy r e v e a l e d here and the f a c t t h a t S l i c h t e r (1933) proved uniqueness of both £(z) and <7"(z) f o r geometric HLEM soundings s u g g e s t s t h a t i t may be p o s s i b l e t o g e n e r a l i s e the theorem i n Appendix C t o i n c l u d e d i e l e c t r i c m a t e r i a l s . 138 111-5 A p p r a i s a l _in HLEM. INV The a p p r a i s a l scheme i n HLEM.INV c o u l d be d e s c r i b e d as an extended B a c k u s - G i l b e r t approach. The " e x t e n s i o n " t a k e s the form of a d i r e c t comparison of averages f o r a s u i t e of d i f f e r e n t a c c e p t a b l e models i n an attempt t o a s s e s s the u n i v e r s a l i t y of the a v e r a g e s . T h i s b r u t e - f o r c e comparison i s performed because, as d i s c u s s e d i n 111 —4, B a c k u s - G i l b e r t averages a re not n e c e s s a r i l y common t o a l l a c c e p t a b l e models i n a n o n - l i n e a r i n v e r s e problem. I f the r e s u l t s of such a comparison were always t o t a l l y n e g a t i v e , i . e . i f no u n i v e r s a l averages c o u l d be found, the u t i l i t y of B a c k u s - G i l b e r t a v e r a g i n g i n HLEM a p p r a i s a l would be i n s e r i o u s doubt. However, the uniqueness theorem i n Appendix C p r o v i d e s some grounds f o r optimism t h a t c e r t a i n f e a t u r e s w i l l be common t o a l l a c c e p t a b l e models, and hence f o r the e x p e c t a t i o n t h a t the averages f o r d i f f e r e n t a c c e p t a b l e models w i l l be i d e n t i c a l w i t h i n e r r o r a t some d e p t h s . G i v e n the F r e c h e t d i f f e r e n t i a b i l i t y of the problem, i t i s always p o s s i b l e t o f i n d an a r b i t r a r i l y l a r g e number of a c c e p t a b l e models which d i f f e r o n l y m a r g i n a l l y from a g i v e n a c c e p t a b l e model. However, i n o r d e r t o t e s t the u n i v e r s a l i t y of averages i t i s d e s i r a b l e t o f i n d a s e t of a c c e p t a b l e models which a r e as d i f f e r e n t from each o t h e r as p o s s i b l e , and such a set i s sought i n the f i r s t s t age of HLEM.INV a p p r a i s a l . A h i g h degree of c o n s i s t e n c y between averages f o r any l i m i t e d s e t of a c c e p t a b l e models does not. c o n s t i t u t e p r o o f t h a t those averages a r e c h a r a c t e r i s t i c of the t r u e s t r u c t u r e ; n e v e r t h e l e s s , i n view of the uniqueness theorem i t i s not un r e a s o n a b l e t o p o s t u l a t e 139 t h a t c e r t a i n averages a r e u n i v e r s a l "beyond r e a s o n a b l e doubt" i f they are common t o a number of d i v e r s e a c c e p t a b l e models. I I I - 5 ( i ) C o n s t r u c t i o n of a S u i t e of A c c e p t a b l e Models As e x p l a i n e d above, a p r e - r e q u i s i t e f o r HLEM.INV a p p r a i s a l i s a s u i t e of d i s s i m i l a r a c c e p t a b l e models. I t i s d e s i r a b l e t h a t t hese models be i n d i v i d u a l l y d i s t i n c t i v e i n o r d e r t o l a t e r pose a severe t e s t f o r the u n i v e r s a l i t y of a v e r a g e s . In HLEM.INV a s e t of e x t r e m a l models i s used f o r the comparison of a v e r a g e s , namely the s e t of a c c e p t a b l e models f o r which the c o n d u c t i v i t y of a p a r t i c u l a r l a y e r i s maximum or minimum. These e x t r e m a l models are c o n s t r u c t e d u s i n g l i n e a r programming, as w i l l now be d e s c r i b e d . The s t a r t i n g p o i n t f o r a p p r a i s a l i n any n o n - l i n e a r i n v e r s e problem i s an a c c e p t a b l e model, c o n s t r u c t e d i n the manner d e s c r i b e d i n Chapter I I or o t h e r w i s e . A l t h o u g h c o n s t r u c t i o n of e x t r e m a l models c o u l d be i n i t i a t e d i m m e d i a t e l y , i t i s u s u a l l y d e s i r a b l e t o f i r s t f i n d a n o t h e r a c c e p t a b l e model, the " b a s e l i n e model", on a c o a r s e r depth p a r t i t i o n . T h i s p r e l i m i n a r y s t e p i s performed p r i m a r i l y f o r reasons of economy: the s m a l l e r the number of l a y e r s , the lower the c o s t . In a d d i t i o n , the bounds f o r t h i c k l a y e r s a re u s u a l l y more p h y s i c a l l y e n l i g h t e n i n g than those f o r t h i n l a y e r s because the c o n d u c t i v i t y of a v e r y t h i n l a y e r can v a r y enormously w i t h o u t a f f e c t i n g the responses a p p r e c i a b l y . There are good grounds, t h e r e f o r e , f o r d e f i n i n g a set of t h i c k e r l a y e r s . Such a s e t c o u l d be s e l e c t e d a c c o r d i n g t o the r e s o l v i n g w i d t h s of B a c k u s - G i l b e r t a v e r a g i n g f u n c t i o n s , f o r example, or by i n v o k i n g an 'a p r i o r i ' c r i t e r i o n of some k i n d . In 140 t h i s study the l a t t e r c o u r s e has been f o l l o w e d ; o f t e n the number of l a y e r s i s s i m p l y h a l v e d by removing every second i n t e r f a c e . Once a s u i t a b l e s e t of l a y e r s has been d e f i n e d , the immediate o b j e c t i v e i s t o f i n d an a c c e p t a b l e model on the new p a r t i t i o n . T h i s b a s e l i n e model s e r v e s s u b s e q u e n t l y as the s t a r t i n g model f o r the c o n s t r u c t i o n of the e x t r e m a l models. The b a s e l i n e model i s a l s o of i n t e r e s t i n i t s e l f , because i t i s a s i m p l i f i e d v e r s i o n of the o r i g i n a l c o n s t r u c t e d model i n t h a t i t n e c e s s a r i l y i n c l u d e s l e s s s m a l l - s c a l e s t r u c t u r e than the o r i g i n a l model. In a d d i t i o n , by v i r t u e of i t s e x i s t e n c e , the b a s e l i n e model guarantees t h a t a c c e p t a b l e models can i n f a c t be found on the m o d i f i e d p a r t i t i o n . I f , on the c o n t r a r y , i t proves i m p o s s i b l e t o f i n d a b a s e l i n e model a t f i r s t , then e i t h e r the p a r t i t i o n must be a l t e r e d (perhaps by the i n t r o d u c t i o n of a d d i t i o n a l l a y e r s ) , or the " a c c e p t a b i l i t y " c r i t e r i o n r e l a x e d by i n c r e a s i n g NSD i n (2 . 3 2 ) . The b a s e l i n e model i s c o n s t r u c t e d by i t e r a t i v e improvement of an i n i t i a l e s t i m a t e or s t a r t i n g model, i n c l o s e analogy w i t h the c o n s t r u c t i o n procedure d e s c r i b e d i n Chapter I I . The i n i t i a l e s t i m a t e of the b a s e l i n e model i s chosen by i n v o k i n g the f a m i l i a r n o t i o n t h a t the c o n d u c t i v i t y - t h i c k n e s s p r o d u c t , or c o n d u c t a n c e , i s the c r u c i a l f a c t o r i n d e t e r m i n i n g the EM response of a l a y e r e d s t r u c t u r e . A c c o r d i n g l y , the i n i t i a l b a s e l i n e model e s t i m a t e i s the model on the m o d i f i e d p a r t i t i o n w i t h conductance i d e n t i c a l , as f a r as i s p o s s i b l e , t o t h a t of the o r i g i n a l a c c e p t a b l e model. More e x p l i c i t l y , i f * denotes the c o n d u c t i v i t y of the j t h l a y e r of the b a s e l i n e model e s t i m a t e , and i f t h i s l a y e r extends from depth z.* t o depth z.* , 141 t h e n C T * = J [ * d z / ( z * + , " zf) (3.22) where £ d e n o t e s t h e c o n d u c t i v i t y of t h e o r i g i n a l a c c e p t a b l e m o d e l . The b a s e l i n e model e s t i m a t e i n F i g . 3.2a was computed i n t h i s way f r o m t h e o r i g i n a l a c c e p t a b l e model f o r Re{Hr,Hz} & Im{Hr,Hz} d a t a ( F i g . 2 . 7 d ) . A c c o r d i n g t o t h e d e f i n i t i o n ( 3 . 2 2 ) , t h e i n t e g r a t e d c o n d u c t i v i t y of C * ( z ) i s i d e n t i c a l t o t h a t f o r 9 ( z ) a t e a c h of t h e i n t e r f a c e s o f t h e new p a r t i t i o n , i . e . where NL* i s t h e number of l a y e r s i n t h e new p a r t i t i o n . D u r i n g c o n s t r u c t i o n of t h e b a s e l i n e model t h e p e r t u r b a t i o n e q u a t i o n s , ( 2 . 3 4 ) , a r e s o l v e d u s i n g l i n e a r programming. L i n e a r programming a l l o w s c o n s i d e r a b l e f l e x i b i l i t y i n t h e c h o i c e of c o n s t r a i n t s ; f o r example, t h e m a g n i t u d e s o f t h e c o n d u c t i v i t y p e r t u r b a t i o n s c an be r e s t r a i n e d d i r e c t l y i n t h e l i n e a r programming " f o r m u l a t i o n , whereas p e r t u r b a t i o n " c o n t a i n m e n t " must be p e r f o r m e d as a s e p a r a t e s t e p i n t h e c o n v e n t i o n a l l e a s t s q u a r e s a p p r o a c h . T h i s and o t h e r p o i n t s w i l l be e l u c i d a t e d i n t h e o u t l i n e of t h e l i n e a r programming f o r m u l a t i o n w h i c h now f o l l o w s . L e t D* d e n o t e t h e NxNL* m a t r i x of d i s c r e t e F r e c h e t k e r n e l s f o r t h e b a s e l i n e model e s t i m a t e C * , and l e t "c* r e p r e s e n t t h e c o r r e s p o n d i n g t h e o r e t i c a l r e s p o n s e s . L e t c r 1 d e n o t e t h e e s t i m a t e of t h e b a s e l i n e model a f t e r one i t e r a t i o n , and l e t "c 1 be t h e k=l,NL* o o 142 F i g u r e 3.2 C o n s t r u c t i o n of B a s e l i n e Model (a) B a s e l i n e model e s t i m a t e f o r Re{Hr,Hz}&Im{Hr,Hz} d a t a . The i n t e g r a t e d conductance a t each i n t e r f a c e i s i d e n t i c a l t o t h a t f o r the a c c e p t a b l e model i n F i g . 2.7d (see t e x t ) . In t h i s case the b a s e l i n e model e s t i m a t e i s a c c e p t a b l e ; computation of the b a s e l i n e model below was e f f e c t e d p u r e l y f o r purposes of i l l u s t r a t i o n . (b) B a s e l i n e model f o r t h e o r e t i c a l Re{Hr,Hz}&Im{Hr,Hz} d a t a computed i n one i t e r a t i o n from the e s t i m a t e i n F i g . 3.2a u s i n g l i n e a r programming. 143 ReiHr,Hz* & lmiHr,Hz* baseline model estimate 0 . 3 ^ 0 . 2 b 0 .1 0 . 0 1 r — r i r I " T — i -l X = 12.64 x 1 = 15.18 max — --• 1 1 1 1 1 0 100 200 300 400 (X 10 1 ) F i g . 3.2a RelHr,Hz? & ImiHr.Hz* baseline model 0 . 3 ^ 0 . 2 b 0 .1 0 . 0 T 1 1 1 r x = 9.03 x = 15.18 max ITER = 1 1 0 100 200 300 400 (X 10 1 ) depth (m) F i g . 3.2b 144 v e c t o r of t h e o r e t i c a l responses f o r t h a t model. By d e f i n i t i o n of F r e c h e t k e r n e l s , the f o l l o w i n g e q u a t i o n c o n n e c t s t h e s e q u a n t i t i e s : D j* h * c£r = c; - c* (3.23) fc»l where n o n - l i n e a r terms have been n e g l e c t e d , c . f . ( 2 . 6 ) , and where h* denotes the t h i c k n e s s of the k t h l a y e r on the m o d i f i e d p a r t i t i o n , i . e . h* = z* - z* k=l,NL* k kH k In k e e p i n g w i t h the n o t a t i o n i n t r o d u c e d i n Chapter I I , l e t oj be the measured datum f o r the j t h f r e q u e n c y , and l e t 7£* and Ty1 be the r e s i d u a l s f o r the s t a r t i n g model and p e r t u r b e d model r e s p e c t i v e l y , i . e . — * k •>-;*=: O - C* (3.24) yj1 = "o - C 1 Thus ly* i s known and vg1 i s unknown. Adding Oj t o both s i d e s of (3 . 2 3 ) , r e - a r r a n g i n g and i n v o k i n g ( 3 . 2 4 ) , the p e r t u r b a t i o n e q u a t i o n s may be e x p r e s s e d i n the form N L * £ D * h*Scr + 7J.1 = ty* (3.25) k s , Jk k k tj <j T h i s statement of the e q u a t i o n s has been c o n s i d e r e d p r e v i o u s l y 145 i n I I - 4 ( i i i ) . Other c o n s t r a i n t s a re imposed on the p e r t u r b a t i o n , i n a d d i t i o n t o the da t a c o n s t r a i n t s ( 3 . 2 5 ) . F i r s t l y , e l e c t r i c a l c o n d u c t i v i t y i s n o n - n e g a t i v e , so P o s i t i v i t y c o n s t r a i n t s have been a p p l i e d p r e v i o u s l y u s i n g l i n e a r programming i n both l i n e a r and n o n - l i n e a r i n v e r s e problems ( S a b a t i e r , 1977; Kennett and N o l e t , 1978, p420). In k e e p i n g w i t h the r e s t r i c t i o n t o s m a l l a m p l i t u d e s imposed by the l i n e a r i s a t i o n ( 2 . 1 1 ) , the change i n c o n d u c t i v i t y i n a s i n g l e i t e r a t i o n i s r e q u i r e d t o be l e s s than o n e - t e n t h the maximum c o n d u c t i v i t y , i . e . By the i n c l u s i o n of these i n e q u a l i t i e s , p e r t u r b a t i o n "containment" i s a p p l i e d d i r e c t l y i n the l i n e a r programming f o r m u l a t i o n , as noted e a r l i e r . The f r a c t i o n , o n e - t e n t h , i s an 'a p r i o r i ' c h o i c e ; any o t h e r s m a l l f r a c t i o n c o u l d be s u b s t i t u t e d i n i t s p l a c e . In o r d e r t o enhance convergence when s e a r c h i n g f o r a b a s e l i n e model, m i n i m i s a t i o n of the ex p e c t e d x 1 - m i s f i t i s an e x p e d i t i o u s s t r a t e g y where, r e c a l l i n g (2.22) and (3 . 2 4 ) , cr* + Scr > o k=l,NL* (3.26) 1 ^ 1 < ° ' l c C a x k=l,NL* (3.27) N (3.28) 146 As d i s c u s s e d i n 11-3, m i n i m i s a t i o n of m i s f i t i s an u n f a v o u r a b l e o b j e c t i v e i n a l i n e a r i n v e r s e problem w i t h i n a c c u r a t e d a t a i f the data a r e s a t i s f i e d e x a c t l y as a r e s u l t . However, i n the c o n s t r u c t i o n of the b a s e l i n e model, the c o n s t r a i n t s (3.26) and (3.27) p r e c l u d e the w i l d changes i n the c o n d u c t i v i t y which a r e the s i g n a t u r e of e x a c t s o l u t i o n s . C o n s e q u e n t l y , m i n i m i s a t i o n of e x p e c t e d m i s f i t i s a s u c c e s s f u l s t r a t e g y i n t h i s c o n n e c t i o n . I t e r a t i o n s c o n t i n u e u n t i l the x 1 - m i s f i t s a t i s f i e s the g l o b a l convergence c o n d i t i o n , i . e . u n t i l A b a s e l i n e model f o r the t h e o r e t i c a l Re{Hr,Hz} & Im{Hr,Hz} d a t a i s d e p i c t e d i n F i g . 3.2b; t h i s model was c o n s t r u c t e d i n one i t e r a t i o n from the e s t i m a t e i n F i g . 3.2a. Given a s u i t a b l e b a s e l i n e model, c o n s t r u c t i o n of the e x t r e m a l models may commence. The b a s e l i n e model s e r v e s as the s t a r t i n g model f o r the m a x i m i s a t i o n and m i n i m i s a t i o n of the i n d i v i d u a l l a y e r c o n d u c t i v i t i e s and c o n s t r u c t i o n proceeds by l i n e a r i s e d i t e r a t i v e improvement, as b e f o r e . The unknown cf<r i n (3.25) i s r e q u i r e d t o s a t i s f y the c o n s t r a i n t s (3.26-27), as f o r the c o n s t r u c t i o n of the b a s e l i n e model, and e s s e n t i a l l y the o n l y m o d i f i c a t i o n t o the l i n e a r programming problem l i e s i n the c h o i c e of o b j e c t i v e f u n c t i o n . For c o n s t r u c t i o n of the e x t r e m a l x 1 < XIMAX (3.29) where, r e c a l l i n g ( 2 . 3 2 ) , XI MAX (3.30) 147 models, the o b j e c t i v e i s of c o u r s e t o maximise or m i n i m i s e the (change i n ) c o n d u c t i v i t y of a p a r t i c u l a r l a y e r , w i t h the c o n d u c t i v i t y of the o t h e r l a y e r s f r e e t o v a r y w i t h i n the c o n f i n e s of (3.26) and ( 3 . 2 7 ) . In t h i s c o n t e x t , " m i n i m i s e " s h o u l d be i n t e r p r e t e d t o mean t h a t the "most n e g a t i v e " , not the " s m a l l e s t " , change i s sought. S u p e r f i c i a l l y the approach i n HLEM.INV p a r a l l e l s t h a t of Kennett and N o l e t (1978, pp419-422), but t h e r e a r e a number of d i f f e r e n c e s . One im p o r t a n t d i s t i n c t i o n a r i s e s from the f a c t t h a t Kennett and N o l e t r e g a r d the bounds as s i g n i f i c a n t i n t h e m s e l v e s , whereas i n t h i s s tudy the " e x t r e m a l " models are c o n s t r u e d o n l y as the means t o an end. There i s c o n s i d e r a b l e doubt about the s i g n i f i c a n c e of the bounds because the e x t r e m a l c o n d u c t i v i t i e s a re a f f e c t e d by the depth p a r t i t i o n as w e l l as by the d a t a . T h i s p o i n t i s i l l u s t r a t e d i n F i g . 3.3, where a change i n p a r t i t i o n i s seen t o i n f l u e n c e the bounds q u i t e s t r o n g l y . Another d i s t i n g u i s h i n g f e a t u r e of the HLEM.INV scheme i s the f a c t t h a t the c o n s t r u c t i o n of the " e x t r e m a l models" i s i t e r a t i v e ; Kennett and N o l e t r e l y h e a v i l y on l o c a l l i n e a r i t y and e n v i s a g e c o m p u t a t i o n of the bounds i n a s i n g l e s t e p . However, the bounds a r e not always a t t a i n e d i n HLEM.INV, even a f t e r s e v e r a l i t e r a t i o n s , as w i t n e s s e d by F i g . 3.3b; the arrowheads i n the f i g u r e i n d i c a t e t h a t some upper bounds were not a c h i e v e d . T h i s i s of no consequence i n HLEM.INV a p p r a i s a l , because the a c c e p t a b i l i t y of the models i s the paramount c o n c e r n . In Kennett and N o l e t a p p r a i s a l , however, f a i l u r e t o a t t a i n a bound c o u l d l e a d t o e r r o n e o u s i n t e r p r e t a t i o n s . The t r e a t m e n t of e r r o r s i n HLEM.INV r e p r e s e n t s a t h i r d 148 F i g u r e 3.3 E f f e c t of Depth P a r t i t i o n on C o n d u c t i v i t y Bounds (a) C o n d u c t i v i t y bounds f o r t h e o r e t i c a l 9r d a t a . The shading i n d i c a t e s the range of r e a l i s a b l e c o n d u c t i v i t i e s f o r each l a y e r . (b) Approximate c o n d u c t i v i t y bounds f o r the same da t a as i n F i g . 3.3a. The d i f f e r e n c e s between the two s e t s of bounds may be a t t r i b u t e d t o the d i f f e r e n c e s i n the depth p a r t i t i o n . The arrowheads i n d i c a t e t h a t the upper bounds f o r the deepest t h r e e l a y e r s were not a t t a i n e d i n a l l o w e d number of i t e r a t i o n s . 149 0r bounds 0 . 3 s £ 0 . 2 co b 0 . 1 0 . 0 i 1 1 1 1 1 r J * i » i 0 1 0 0 2 0 0 3 0 0 4 0 0 (X 1 0 1 ) F i g . 3.3a Or bounds 0 . 3 0 1 0 0 2 0 0 3 0 0 4 0 0 (X 1 0 1 ) depth (m) F i g . 3.3b 150 s i g n i f i c a n t d i g r e s s i o n from t h e a p p r o a c h a d o p t e d by S a b a t i e r (1977) and K e n n e t t and N o l e t (-1978). In t h e HLEM.INV f o r m u l a t i o n t h e m i s f i t s , T p 1 , e n t e r e x p l i c i t l y i n t o t h e p r o b l e m v i a (3.25) as a d d i t i o n a l unknowns, and a r e c o n s t r a i n e d c o l l e c t i v e l y by t h e x ^ a c c e p t a b i l i t y c o n d i t i o n , ( 2 . 3 2 ) , w h i c h i s r e p e a t e d h e r e f o r c o n v e n i e n c e : In t h e f o r m u l a t i o n s o f S a b a t i e r and of K e n n e t t and N o l e t t h e model p a r a m e t e r s ( c o n d u c t i v i t i e s ) a r e r e g a r d e d as t h e o n l y unknowns, and t h e s o l u t i o n i s r e q u i r e d t o s a t i s f y e a c h o b s e r v a t i o n t o w i t h i n one s t a n d a r d d e v i a t i o n . S t a t i s t i c a l l y , t h e HLEM.INV f o r m u l a t i o n i s s u p e r i o r : i f N i s " l a r g e " and t h e e r r o r s a r e i n d e p e n d e n t , t h e p r o b a b i l i t y t h a t x 1 w i l l assume a v a l u e w i t h i n t h e bounds p r e s c r i b e d i n (3.31) i s a p p r o x i m a t e l y 0.95 when NSD=2, whereas t h e p r o b a b i l i t y t h a t N i n d e p e n d e n t Normal random v a r i a b l e s w i l l a l l l i e w i t h i n two s t a n d a r d d e v i a t i o n s of t h e i r mean s i m u l t a n e o u s l y (as demanded by t h e p r e v i o u s f o r m u l a t i o n s ) i s 0.95 . Beyond s a t i s f y i n g t h e c o n d i t i o n ( 3 . 3 1 ) , t h e m i s f i t s i n t h e HLEM.INV f o r m u l a t i o n a r e f r e e t o a d j u s t t h e m s e l v e s i n t h e manner most f a v o u r a b l e f o r t h e p a r t i c u l a r m a x i m i s a t i o n o r m i n i m i s a t i o n . The p e r f o r m a n c e o f t h e HLEM.INV scheme i s i l l u s t r a t e d i n F i g . 3.4 where " e x t r e m a l m o d e l s " s a t i s f y i n g t h e t h e o r e t i c a l |Hr j &|Hz| d a t a from C h a p t e r II a r e d e p i c t e d . I t i s c l e a r by e x a m i n a t i o n of t h e d i a g r a m s t h a t t h e d a t a a r e s a t i s f i e d by some d i v e r s e c o n d u c t i v i t y d i s t r i b u t i o n s ; t h i s i s most e v i d e n t f r o m a XIMAX (3.31) 151 F i g u r e 3.4 Examples of E x t r e m a l Models (a) Model f o r which the c o n d u c t i v i t y i n the f i r s t l a y e r has been maximised, s u b j e c t t o the requirement t h a t i t s a t i s f i e s the t h e o r e t i c a l |Hr[&|Hz| d a t a . (b) Model 'for which the c o n d u c t i v i t y i n the f i r s t l a y e r has been m i n i m i s e d , s u b j e c t t o the requirement t h a t i t s a t i s f i e s the t h e o r e t i c a l |Hr|&|Hz| d a t a . (c) Model f o r which the c o n d u c t i v i t y i n the second l a y e r has been maximised, s u b j e c t t o the requirement t h a t i t s a t i s f i e s the t h e o r e t i c a l |Hr|&|Hz| d a t a . (d) Model f o r which the c o n d u c t i v i t y i n the second l a y e r has been m i n i m i s e d , s u b j e c t t o the requirement t h a t i t s a t i s f i e s the t h e o r e t i c a l |Hr|&|Hz| d a t a . (e) Model f o r which the c o n d u c t i v i t y i n the t h i r d l a y e r has been maximised, s u b j e c t t o the requirement t h a t i t s a t i s f i e s the t h e o r e t i c a l j Hr j &|Hz| d a t a . ( f ) Model f o r which the c o n d u c t i v i t y i n the f o u r t h l a y e r has been maximised, s u b j e c t t o the requirement t h a t i t s a t i s f i e s the t h e o r e t i c a l |Hr|&|Hz| d a t a . (g) Model f o r which the c o n d u c t i v i t y i n the f i f t h l a y e r has been maximised, s u b j e c t t o the requ i r e m e n t t h a t i t s a t i s f i e s the t h e o r e t i c a l |Hr|&[Hz| d a t a . (h) Model f o r which the c o n d u c t i v i t y i n the f i f t h l a y e r has been m i n i m i s e d , s u b j e c t t o the requ i r e m e n t t h a t i t s a t i s f i e s the t h e o r e t i c a l |Hr|&|Hz J d a t a . I t i s apparent t h a t a h i g h l y c o n d u c t i v e zone a t de p t h , such as t h a t i n F i g . 2.7a, i s not n e c e s s a r y i n o r d e r t o f i t the d a t a . 152 M a x i m i s e d l a y e r #1 c o n d u c t i v i t y 0 . 3 £ 0 . 2 b 0.1 0 .0 0 100 200 300 400 (X 10 1 ) F i g . 3.4a M i n i m i s e d l a y e r # 1 c o n d u c t i v i t y 0 . 3 ^ 0 . 2 b 0.1 i ~r — i 1 1 — I 1 - 1 X = 8.05 1 X = 8.09 max - ITER = 7 — -1 1 1 . i 1 0 100 200 300 400 (X 10 1 ) d e p t h ( m ) F i g . 3 .4b 1 1 7 " T 1 1 - 1 X = 8.06 -1 X = 8.09 max ITER = 5 — -1 i . 1 , 153 M a x i m i s e d l a y e r # 2 c o n d u c t i v i t y 0 . 3 ^ 0 . 2 b 0.1 0 .0 ~l 1 1 1 1 T x = 7 .95 x 1 = 8 .09 max ITER = 5 0 100 200 300 400 (X 10 1 ) F i g . 3 . 4 c M i n i m i s e d l a y e r # 2 c o n d u c t i v i t y 0 . 3 ^ 0 . 2 b 0 .1 0 .0 "i 1 1 1 1 1 r x 1 = 8 .03 x 1 = 8 .09 max ITER = 5 0 100 200 300 400 (X 10 1 ) d e p t h ( m ) F i g . 3 . 4 d I 154 M a x i m i s e d l a y e r # 3 c o n d u c t i v i t y 0 . 3 ^ 0 . 2 b 0 .1 0 .0 i r x = 8.01 x 1 = 8.09 max ITER = 8 0 100 200 300 400 (X 10 1 ) F i g . 3.4e M a x i m i s e d l a y e r # 4 c o n d u c t i v i t y 0 . 3 0 . 2 GO b 0.1 0 .0 T 1 1 1 1 1 I x = 8.01 x 1 = 8.09 max ITER = 5 0 100 200 300 400 (X 10 1 ) d e p t h ( m ) F i g . 3.4f 155 M a x i m i s e d l a y e r #5 c o n d u c t i v i t y 0 . 3 ^ 0 . 2 oo 0 .0 1 1 1 1—1 1 1 x = 8.02 x 1 = 8.09 max — ITER = 5 — -1 1 1 1 0 100 200 300 400 (X 10 1 ) F i g . 3.4g M i n i m i s e d l a y e r #5 c o n d u c t i v i t y 0 . 3 ^ 0 . 2 b 0 .1 0 .0 T 1 1 r 1 x = 7.80 x 1 = 8.09 max ITER = 7 1 1 1 0 100 200 300 400 (X 10 1 ) d e p t h ( m ) F i g . 3.4h 156 comparison of F i g s . 3.4g and 3.4h. In p a r t i c u l a r , i t i s apparent from F i g s . 3.4 h and 3.4e t h a t t h e r e i s no n e c e s s i t y f o r a h i g h l y c o n d u c t i v e "basement" a t depths i n excess of 2 km. T h i s o b s e r v a t i o n i s c o m p l e t e l y c o n s i s t e n t w i t h the n a t u r e of the t r u e model ( F i g . 2.2), and amply demonstrates the p i t f a l l s i n h e r e n t i n any pr o c e d u r e which p l a c e s heavy r e l i a n c e on a s i n g l e a c c e p t a b l e model. A l t h o u g h t h e r e a re q u i t e pronounced d i f f e r e n c e s between some of the models i s F i g . 3.4, they a l l have a c o n d u c t i v e zone near the s u r f a c e , and e x h i b i t a g e n e r a l d e c r e a s e i n c o n d u c t i v i t y down t o a depth of a p p r o x i m a t e l y 1 km. Other c o n s i s t e n c i e s can be r e c o g n i s e d a f t e r c l o s e r i n s p e c t i o n of the i n d i v i d u a l models. In p a r t i c u l a r , i t may be observ e d i n F i g . 3.3a t h a t the m a x i m i s a t i o n of of i s accompanied by a de c r e a s e i n c£, whereas i n F i g . 3.4b i t i s p l a i n t h a t <7£ has i n c r e a s e d d u r i n g the m i n i m i s a t i o n of cr . F u r t h e r t o t h i s , i f F i g s . 3.4c-3.4h are examined i n t u r n , i t i s apparent t h a t m a x i m i s a t i o n or m i n i m i s a t i o n of a p a r t i c u l a r l a y e r i s always "compensated" by changes of o p p o s i t e p o l a r i t y i n the c o n d u c t i v i t y of the a d j a c e n t l a y e r s . These o b s e r v a t i o n s a r e q u a l i t a t i v e l y c o m p a t i b l e w i t h the n o t i o n t h a t conductance governs the EM response of a l a y e r e d ground. A more q u a n t i t a t i v e c o r r o b o r a t i o n of t h i s t e n e t i s f u r n i s h e d by F i g . 3.5, i n which the conductances f o r a l l the |Hr|&[Hz| e x t r e m a l models a r e superimposed. There i s a v e r y good corres p o n d e n c e between these conductances over the f i r s t k i l o m e t r e , and i t might be c o n j e c t u r e d t h a t t h i s depth marks the e f f e c t i v e l i m i t of p e n e t r a t i o n of the i n d u c i n g f i e l d s . These g e n e r a l remarks a p p l y e q u a l l y w e l l t o e x t r e m a l models 157 C o n d u c t a n c e f o r | H r | & | H z | e x t r e m a l m o d e l s (X 1 0 1 ) 0 100 200 300 400 (X 10 1 ) d e p t h ( m ) F i g . 3.5 V. 158 f o r t h e o r e t i c a l |Hr|&0r, and Re{Hr,Hz}&Im{Hr,Hz} d a t a ; the models themselves a r e not i n c l u d e d as f i g u r e s f o r reasons of b r e v i t y . I l l — 5 ( i i ) B a c k u s - G i l b e r t A v e r a g i n g The aim of a p p r a i s a l i s t o d e l i n e a t e f e a t u r e s which a re common t o ' a l l a c c e p t a b l e models, and which a re t h e r e f o r e c h a r a c t e r i s t i c of the t r u e s t r u c t u r e . In t h i s quest B a c k u s - G i l b e r t a v e r a g i n g f u n c t i o n s assume the r o l e of "windows" through which the data e n a b l e the ground s t r u c t u r e t o be viewed. In a l i n e a r i n v e r s e problem the r e s u l t i n g B a c k u s - G i l b e r t averages a r e r e p l i c a t e d by a l l a c c e p t a b l e models, but t h i s u n i v e r s a l i t y i s not a s s u r e d i n n o n - l i n e a r problems. In a n o n - l i n e a r problem, a s e t of B a c k u s - G i l b e r t averages computed f o r a p a r t i c u l a r a c c e p t a b l e model c o u l d a p p l y o n l y t o those a c c e p t a b l e models which c l o s e l y resemble the g i v e n model. N e v e r t h e l e s s , i t i s p o s s i b l e t h a t a t c e r t a i n depths the averages are i n f a c t common t o a l l a c c e p t a b l e models. The c r u x of n o n - l i n e a r a p p r a i s a l i s t h e r e f o r e t o e s t a b l i s h whether or not t h e r e a r e ranges of depth f o r which the averages a r e common t o a l l a c c e p t a b l e models. I n t u i t i v e l y , i t i s t o be e x p e c t e d t h a t the averages w i l l t end t o be c o n s i s t e n t from one a c c e p t a b l e model t o another i n r e g i o n s where the r e s o l u t i o n i s good, and i n c o n s i s t e n t i n r e g i o n s where the da t a p r o v i d e l i t t l e l o c a l i s e d i n f o r m a t i o n . These p r e d i c t i o n s have been borne out i n p r a c t i c e , both i n t h i s study and i n the work of Oldenburg (1979). F u r t h e r m o r e , zones of good r e s o l u t i o n have been found t o c o i n c i d e w i t h c o n d u c t i v e b e l t s , c o n s i s t e n t w i t h the s e n s i t i v i t y 159 of MT and EM soundings t o the l o c a t i o n of induced c u r r e n t s . In o r d e r t o s e v e r e l y t e s t the u n i v e r s a l i t y of a v e r a g e s , i t i s d e s i r a b l e t o compare the averages f o r a s e t of d i v e r s e a c c e p t a b l e models; thus the e x t r e m a l models c o n s t r u c t e d i n the f i r s t s t a g e of HLEM.INV a p p r a i s a l c o n s t i t u t e the raw m a t e r i a l f o r the second s t a g e . The b a s i c p r o c e d u r e i s t o compute averages over a range of depths f o r a l l known a c c e p t a b l e models, and t o a s s e s s t h e i r u n i v e r s a l i t y on the b a s i s of ( 3 . 1 9 ) . A c c o r d i n g l y , i f the d i f f e r e n c e s between the averages a t each depth a re not z e r o w i t h i n e r r o r , they can be a t t r i b u t e d t o the presence of an a p p r e c i a b l e n o n - l i n e a r term. The b a s e l i n e model s e r v e s as a " r e f e r e n c e model" f o r the comparison of the a v e r a g e s ; i n p a r t i c u l a r , the a v e r a g i n g f u n c t i o n s a r e c o n s t r u c t e d from i t s k e r n e l s . The a c t u a l computation of the averages i n HLEM.INV i s performed u s i n g m o d i f i e d v e r s i o n s of s u b r o u t i n e s w r i t t e n by Oldenburg (Oldenburg and R u s s e l l , 1980). As e x p l a i n e d i n I I I - l , a continuum of a v e r a g i n g f u n c t i o n s can be c o n s t r u c t e d f o r any depth of i n t e r e s t by m i n i m i s i n g "J^2 i n (3.6) f o r a l l v a l u e s of & between 0 and 7Z/2. The f u n c t i o n a l "j^ 2 i s a weighted sum of two terms, the f i r s t c o n s t i t u t i n g a measure of the " d e l t a n e s s " of the a v e r a g i n g f u n c t i o n and the second r e p r e s e n t i n g the v a r i a n c e of the c o r r e s p o n d i n g a v e r a g e . In HLEM.INV the spre a d c r i t e r i o n i s used t o c h a r a c t e r i s e " d e l t a n e s s " (Backus and G i l b e r t , 1970, p l 2 8 ) . Spread, s, i s d e f i n e d by s ( z e ; A ) = 1 2 \ ( x - z c ) 2 A 2 ( x , z , ) d x (3.32) 160 where K{z,z0) i s a l i n e a r c o m b i n a t i o n of k e r n e l s , c . f . ( 3 . 2 ) . The f a c t o r t w e l v e i s a n o r m a l i s i n g c o n s t a n t i n t r o d u c e d t o ensure t h a t the spread of a box-car f u n c t i o n c o r r e s p o n d s t o i t s t r u e w i d t h . When u s i n g the s p r e a d c r i t e r i o n , the term "Y 2" i n (3.6) becomes Y 2 = b TSb (3.33) where Sy = j"(z-z, ) aD.(z;a*)Dj(z;*-)dz i = l , N ; j = l , N (3.34) and where D.(z;cr) i s the F r e c h e t k e r n e l a t f r e q u e n c y w: f o r the a c c e p t a b l e model, V . S u b s t i t u t i n g (3.33) i n t o (3.6) and m i n i m i s i n g w i t h r e s p e c t t o the components of b y i e l d s the f o l l o w i n g system of l i n e a r e q u a t i o n s : cos© Sb + K s i n f l £ b = - A u (3.35) where U j = j l ) ^ z ;<r)dz (3.36) The w e i g h t i n g f a c t o r , K, i s an a r b i t r a r y p o s i t i v e number, i t s i n t e n d e d f u n c t i o n b e i n g t o compensate f o r n u m e r i c a l mismatch between the c o v a r i a n c e and spread m a t r i c e s ; one i n f o r m e d c h o i c e of K i s t h a t proposed by Backus and G i l b e r t (1970, p l 4 5 ) . E q u a t i o n (3.35) i s s o l v e d f o r b u s i n g a t r a n s f o r m a t i o n which s i m u l t a n e o u s l y d i a g o n a l i s e s both S, and £ ( G i l b e r t , 1971); t h i s t r a n s f o r m a t i o n m a t r i x need o n l y be computed once at each d e p t h , i . e . i s independent of 9. I f the a v e r a g i n g f u n c t i o n i s 161 r e q u i r e d a t o n l y one v a l u e of 0, i t i s more e f f i c i e n t t o s o l v e f o r b by s t a n d a r d mehtods. The d e r i v a t i o n of the a p p r o p r i a t e t r a n s f o r m a t i o n m a t r i x has been d e s c r i b e d by G i l b e r t (1971). For p r e s e n t purposes i t s u f f i c e s t o p o s t u l a t e the e x i s t e n c e of a m a t r i x T w i t h the p r o p e r t i e s T TST = I (3.37) TTST = A (3.38) where A i s a d i a g o n a l m a t r i x w i t h p o s i t i v e e n t r i e s . The r e l a t i o n s (3.37) and (3.38) can be brought t o bear on (3.35) v i a the i n t r o d u c t i o n of t r a n s f o r m e d v e c t o r s g and v d e f i n e d by b = Tg (3.39) v = T T u (3.40) S u b s t i t u t i n g from (3.39) f o r b i n ( 3 . 3 5 ) , m u l t i p l y i n g from the l e f t by T T, and i n v o k i n g ( 3 . 4 0 ) , i t f o l l o w s t h a t cos© T TSTg + K s i n 0 TT£ Tg = - Av R e c a l l i n g (3.37) and ( 3 . 3 8 ) , t h i s e q u a t i o n s i m p l i f i e s t o Wg = -Av (3.41) 162 where W i s a d i a g o n a l m a t r i x , d e f i n e d by W = cos0 I + Ksin$A (3.42) By s i m p l e i n v e r s i o n of ( 3 . 4 1 ) , the ( t r a n s f o r m e d ) s o l u t i o n i s g = -Aw^v (3.43) A p p l y i n g the u n i m o d u l a r i t y c o n s t r a i n t , 1 = u T b = u TTT- 1b = v T g (3.44) whence from (3.43) A" 1 = -vrVi-lv (3.45) The spr e a d and v a r i a n c e admit p a r t i c u l a r l y s i m p l e r e p r e s e n t a t i o n s i n terms of the t r a n s f o r m e d s o l u t i o n v e c t o r , q: s = b TSb = g T T T S T g = g T g (3.46) S2 = b T C b = g T T r £ Tg = g T A g (3.47) *" " — Once b has been d e t e r m i n e d , the d e s i r e d a v e r a g i n g f u n c t i o n f o l l o w s i m m e d i a t e l y , as per ( 3 . 2 ) : N A ( z , z , ) =21 blz0 ,0)D.(z;<r) (3.48) j s l J J 163 As mentioned i n I I I - l , a p l o t of v a r i a n c e v e r s u s spread a t any depth,, known as a t r a d e - o f f c u r v e , p r o v i d e s a c o n v e n i e n t summary of the range of averages which are a v a i l a b l e a t t h a t d epth. The p r o p e r t i e s of t r a d e - o f f c u r v e s a r e d e s c r i b e d i n e x h a u s t i v e d e t a i l by Backus and G i l b e r t (1970). The t r a n s f o r m a t i o n m a t r i x , T, g r e a t l y e x p e d i t e s the c o mputation of the t r a d e - o f f c u r v e a t any depth because i t i s independent of 0. G i v e n T,, the d e s i r e d v e c t o r *g, and hence b, can be d e t e r m i n e d r e a d i l y f o r any v a l u e of 6 u s i n g ( 3 . 4 2 ) , ( 3 . 4 3 ) , and ( 3 . 4 5 ) . Four s e t s of t r a d e - o f f c u r v e s a r e p r e s e n t e d i n F i g . 3.6. In each frame, t r a d e - o f f c u r v e s are p r o v i d e d a t f o u r d e p t h s , c o r r e s p o n d i n g t o the m i d p o i n t s of the l a y e r s of the t r u e model ( F i g . 2.2). I t i s apparent t h a t v a r i a n c e i s a m o n o t o n i c a l l y d e c r e a s i n g f u n c t i o n of s p r e a d , as indeed i t must be (Backus and G i l b e r t , 1970, p l 4 1 ) . T h i s m o n o t o n i c i t y i s a m a n i f e s t a t i o n of the mutual antagonism of r e s o l u t i o n and a c c u r a c y . The arrangement of the c u r v e s i n each frame i s such t h a t the spread a s s o c i a t e d w i t h a p a r t i c u l a r v a r i a n c e i n c r e a s e s as the depth i n c r e a s e s , c o n s i s t e n t w i t h the e x p e c t a t i o n t h a t r e s o l u t i o n w i l l d e t e r i o r a t e w i t h depth i n g e n e r a l . The t r a d e - o f f c u r v e s i n F i g s . 3.6a and 3.6b were a l l computed u s i n g the k e r n e l s f o r the model c o n s t r u c t e d f o r the |Hr|&c9r d a t a ( F i g . 2.4c); comparison of t h e s e f i g u r e s t h e r e f o r e p r o v i d e s a d i r e c t i l l u s t r a t i o n of the e f f e c t of i n c l u s i o n of a d d i t i o n a l d a t a . I t i s c l e a r t h a t the a t t a i n a b l e ranges of v a r i a n c e and s p r e a d a r e f a r broader i n 3.6b than i n 3.6a, c o n s i s t e n t w i t h the f a c t t h a t more k e r n e l s (and 164 F i g u r e 3.6 Examples of Trade-Qf f Curves (a) T r a d e - o f f c u r v e s f o r t h e o r e t i c a l 9r d a t a at depths 200m, 800m, 1900m, and 3300m. (b) T r a d e - o f f c u r v e s f o r t h e o r e t i c a l |Hr|&0r data at depths 200m, 800m, 1900m, and 3300m. (c) T r a d e - o f f c u r v e s f o r t h e o r e t i c a l |Hr|&|Hz| d a t a a t depths 200m, 800m, 1900m, and 3300m. (d) T r a d e - o f f c u r v e s f o r t h e o r e t i c a l Re{Hr,Hz}&Im{Hr,Hz} da t a a t depths 200m, 800m, 1900m, and 3300m. 6r t r a d e - o f f c u r v e s 1 1 1 " i i 1 1 i i 1 — r 1 1 1 II jb — 1900 y3300 s -.800 ^200 I A Nil 1 t 1 i i i i i l i i .1 1 1 1 1 1 103 104 105 F i g . 3 . 6 a | H r | & 0 r t r a d e - o f f c u r v e s — i I I I i m i i T i 11 m i i i i i n n - .3300 -— 800 \ -— 1900 \ -— i200 \ -l i i i mil i i i i mil i i i i m i 102 103 104 105 s ( m ) F i g . 3 .6b 166 Hr|&|Hz| t rade-o f f curves E \ GO CM CO 10-1 l O ' 2 10~3 10"5 10"6 10"7 Z 1 1 1 HUH 1 1 TTTTTTl— 1 1 1 llllll -\3300 -I .1900 \ Mil 1 .800^^ ^ -=- 2^00 ^ ^ ^ N ^ ^ 1 II I 1 II 1 " " . I I I m i l l i i i mill i i 11 mil 102 103 104 105 Fig. 3.6c (NI Re{Hr,HzS&lmSHr,Hz* t rade-o f f curves 1 0"~* t ' i kniin i i i 11 l O " 2 i i i ium—=i 102 103 104 105 s (m) Fig. 3.6d 167 hence more d e g r e e s of freedom) a r e a v a i l a b l e d u r i n g c o n s t r u c t i o n of a v e r a g i n g f u n c t i o n s f o r t h e |Hr|&0r d a t a . In t h e same v e i n , t h e r e s o l u t i o n ' a t any p a r t i c u l a r v a r i a n c e i s seen t o be enh a n c e d by t h e i n t r o d u c t i o n of a d d i t i o n a l d a t a ; f o r example, t h e s p r e a d a t d e p t h 200m i s a l w a y s g r e a t e r t h a n 1000m i n F i g . 3.6a, and a l w a y s l e s s t h a n 1000m i n F i g . 3.6b. In t h e a s s e s s m e n t of t h e u n i v e r s a l i t y of a v e r a g e s i n HLEM.INV, a v e r a g e s w i t h t h e same v a r i a n c e a r e s o u g h t a t a ra n g e of d e p t h s . When p l o t t e d a g a i n s t d e p t h , s u c h e q u a l l y a c c u r a t e a v e r a g e s v a r y i n a f a i r l y smooth f a s h i o n , whereas a v e r a g e s w i t h d i s p a r a t e v a r i a n c e s c o u l d f l u c t u a t e f r o m one d e p t h t o t h e n e x t and n e e d l e s s l y c o n f u s e t h e i s s u e . Thus i t i s a d v a n t a g e o u s t o be a b l e t o s e l e c t an a v e r a g i n g f u n c t i o n c o r r e s p o n d i n g t o a p r e s c r i b e d v a r i a n c e ; t h i s i s ta n t a m o u n t t o a f a c i l i t y f o r f i n d i n g t h e v a l u e of f o r t h a t v a r i a n c e . A c c o r d i n g l y , HLEM.INV has been i n v e s t e d w i t h t h e c a p a b i l i t y t o f i n d t h e v a l u e of 6> a s s o c i a t e d w i t h a s p e c i f i c v a r i a n c e by t h e s i m p l e s t a p p l i c a t i o n of Newton's method. The d e r i v a t i v e d£2/d9 may be e v a l u a t e d by d i r e c t d i f f e r e n t i a t i o n of ( 3 . 4 7 ) , ( 3 . 4 3 ) , ( 3 . 4 2 ) , a n d - ( 3 . 4 5 ) . The p e r t i n e n t f o r m u l a e a p p e a r below: W = 2 g de de W J J V J wjf !J2to = - s i n * IJJ + KcoseAjj (3.49) 168 The p r o c e d u r e f o r assessment of the u n i v e r s a l i t y of the averages w i l l now be o u t l i n e d . F i r s t t he a v e r a g i n g f u n c t i o n s w i t h the d e s i r e d v a r i a n c e , VARDES, a r e de t e r m i n e d a t a number of dept h s ; t h e s e a v e r a g i n g f u n c t i o n s a r e l i n e a r c o m b i n a t i o n s of the F r e c h e t k e r n e l s f o r a " r e f e r e n c e model" ( u s u a l l y the b a s e l i n e model). At each d e p t h , the average f o r every e x t r e m a l model i s then computed as an i n n e r p r o d u c t w i t h the a p p r o p r i a t e a v e r a g i n g f u n c t i o n , as per ( 3 . 3 ) . F i n a l l y , a t each depth the r e f e r e n c e model average i s s u b t r a c t e d from e v e r y e x t r e m a l model average. A c c o r d i n g t o ( 3 . 1 9 ) , the d i f f e r e n c e s computed i n t h i s way s h o u l d be z e r o ( w i t h i n e r r o r ) i f t h e n o n - l i n e a r term i s n e g l i g i b l e . I f the averages a re i d e n t i c a l w i t h i n e r r o r , the d i s c r e p a n c i e s a t each depth s h o u l d be d i s t r i b u t e d about z e r o i n a p p r o x i m a t e l y G a u s s i a n f a s h i o n . Thus a t each depth about 65% of the d i s c r e p a n c i e s s h o u l d l i e w i t h i n the u n i t s t a n d a r d d e v i a t i o n e n v e l o p e , w i t h 95% i n s i d e a band t w i c e t h i s w i d t h . A p p l i c a t i o n of t h i s p r o c e d u r e t o the t h e o r e t i c a l |Hr|&|Hz| da t a y i e l d e d the r e s u l t s p l o t t e d i n F i g . 3.7. The averages themselves a r e d e p i c t e d i n F i g . 3.7a, and the d i f f e r e n c e s between the e x t r e m a l and r e f e r e n c e a v e r a g e s appear i n F i g . 3.7b. By i n s p e c t i o n of the lower frame, i t may be obser v e d t h a t the averages a r e i d e n t i c a l ( w i t h i n e r r o r ) from the s u r f a c e t o a depth of a p p r o x i m a t e l y 800m. At g r e a t e r depths the averages a re i n c o n s i s t e n t . C o n s i d e r a t i o n of the c u r v e s i n the upper frame r e v e a l s a pronounced d e c r e a s e i n average c o n d u c t i v i t y over the f i r s t k i l o m e t r e . A l t h o u g h c o n c l u s i o n s c o n c e r n i n g the n a t u r e of the t r u e c o n d u c t i v i t y w i l l u l t i m a t e l y be based on the appearance 169 F i g u r e 3.7 IHr|& 1 Hz 1 Averages, and Assessment of t h e i r U n i v e r s a l i t y (a) Averages w i t h s t a n d a r d d e v i a t i o n 0.01 S/m f o r the e x t r e m a l models d e p i c t e d i n F i g . 3.4. (b) D i f f e r e n c e s (x) between the averages i n F i g . 3.7a and those f o r the b a s e l i n e model ( F i g . 3.2b). I f the averages f o r the v a r i o u s models are e q u a l w i t h i n e r r o r , the d i f f e r e n c e s a t each depth s h o u l d be N o r m a l l y d i s t r i b u t e d about z e r o . The s o l i d l i n e s d e l i n e a t e the u n i t s t a n d a r d d e v i a t i o n e n v e l o p e . 170 | H r | & | H z | a v e r a g e s 0 . 2 co^  £ 0 .1 V 0 . 0 0 50 100 150 200 (X 10 1 ) Fig. 3.7a U n i t s . d . e n v e l o p e 0 . 04 \ 0 .02 GO A 0 .00 b V - 0 .02 - 0 . 0 4 T 1 1 1 1 1 r * L * x * x x x x x * x x x x i m 1 1 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) Fig. 3.7b 171 c u r v e s such as t h e s e , i t s h o u l d be borne i n mind t h a t they do not r e p r e s e n t models which f i t the d a t a . In a c o m p l e t e l y analogous way, the r e l i a b i l i t y of the averages f o r |Hr|&0r d a t a and f o r Re{Hr,Hz}&Im{Hr,Hz} data can be a s s e s s e d by e x a m i n a t i o n of F i g s . 3.8 and 3.9 r e s p e c t i v e l y . As b e f o r e , the averages f o r the v a r i o u s e x t r e m a l models e x h i b i t a s i g n i f i c a n t degree of ( q u a l i t a t i v e ) s i m i l a r i t y t o one another when p l o t t e d a g a i n s t d e p t h ; thus the e x i s t e n c e of a h i g h s u r f i c i a l c o n d u c t i v i t y and the g e n e r a l d e c r e a s e over the f i r s t k i l o m e t r e can be c o n s i d e r e d t o have been e s t a b l i s h e d . However, the |Hr|&0r averages a r e not s t r i c t l y c o n s i s t e n t w i t h i n e r r o r a t depths between about 300m and 400m. T h e r e f o r e , the averages of the t r u e model i n t h a t depth range cannot be det e r m i n e d from t h i s d a t a s e t . In c o n t r a s t , the averages f o r the Re{Hr,Hz}&Im{Hr,Hz} models appear t o be c o n s i s t e n t t o a depth of a p p r o x i m a t e l y 1.5 km. The g r e a t e r degree of correspondence between a c c e p t a b l e models i n t h i s case i s a r e f l e c t i o n of the t w o f o l d i n c r e a s e i n the number of d a t a . In a s i m i l a r v e i n , comparison of F i g s . 3.7b and 3.8b suggest t h a t i n t h i s example the |Hr|&#r data c o n s t r a i n the model l e s s e f f e c t i v e l y than the |Hr[&|Hz| d a t a ; t h i s o b s e r v a t i o n may be a t t r i b u t e d t o the s e v e r i t y of the e r r o r s i n the t h e o r e t i c a l phase d a t a r e l a t i v e t o those i n the a m p l i t u d e d a t a , as r e v e a l e d d u r i n g c o n s t r u c t i o n i n I I - 5 . I t has been demonstrated t h a t the |Hr|&|Hz| averages a r e common t o a l l a v a i l a b l e a c c e p t a b l e models a t depths l e s s than 800m, and t h a t the c o r r e s p o n d i n g range of commonality f o r Re{Hr,Hz}&Im{Hr,Hz} avera g e s extends about 1.5 km beneath the 172 F i g u r e 3.8 1Hr|&flr Averages, and Assessment of t h e i r U n i v e r s a l i t y (a) Averages w i t h s t a n d a r d d e v i a t i o n 0.01 S/m f o r |Hr|&0r e x t r e m a l models. (b) D i f f e r e n c e s (x) between the averages i n F i g . 3.8a and those f o r the |Hr|&0r b a s e l i n e model. I f the averages f o r the v a r i o u s models are e q u a l w i t h i n e r r o r , the d i f f e r e n c e s a t each depth s h o u l d be N o r m a l l y d i s t r i b u t e d about z e r o . The s o l i d l i n e s d e l i n e a t e the u n i t s t a n d a r d d e v i a t i o n e n v e l o p e . I t may be i n f e r r e d t h a t the averages a r e m u t u a l l y c o n s i s t e n t over the f i r s t 800m. 173 |H r |&#r a v e r a g e s 0 . 0 0 50 100 150 200 (X 10 1 ) Fig. 3.8a 0 .04 If 0 .02 CO A 0 .00 b V - 0 .02 - 0 . 04 r x x U n i t s . d . e n v e l o p e i 1 1 r 2 x x x s , x ' x x x x x ' x x * x g k u . x S x ^ X X y x ^ Y y X ) ( D ( X ) C r ^ x j j x x x x x x x x x x x x ) c X v x x " x x * x x x x x x x x>r-  I— x. X 1 1 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) Fig. 3.8b 174 F i g u r e 3.9 Re{Hr,Hz}&Im{Hr, Hz] Averages, and Assessment of t h e i r U n i v e r s a l i t y (a) Averages w i t h s t a n d a r d d e v i a t i o n 0.01 S/m f o r Re{Hr,Hz}&Im{Hr,Hz} e x t r e m a l models. (b) D i f f e r e n c e s (x) between the averages i n F i g . 3.8a and those f o r the Re{Hr,Hz}&Im{Hr,Hz| b a s e l i n e model. I f the averages f o r the v a r i o u s models a r e e q u a l w i t h i n e r r o r , the d i f f e r e n c e s a t each depth s h o u l d be N o r m a l l y d i s t r i b u t e d about z e r o . The s o l i d l i n e s d e l i n e a t e the u n i t s t a n d a r d d e v i a t i o n e n v elope. I t may be i n f e r r e d t h a t the averages are m u t u a l l y c o n s i s t e n t over the f i r s t 1500m. 175 RefHr.Hzi&lmfHr.HzS averages 0 .0 0 50 100 150 200 (X 10 1 ) F i g . 3.9a Unit s.d. envelope E 0 . 04 \ CO 0 .02 A b 0 .00 V - 0 .02 T 1 1 1 1 1 T - X ¥ X X x x x x x * x x I 0 50 100 150 200 (X 10 1 ) depth (m) F i g . 3.9b 1 7 6 s u r f a c e . On the b a s i s of these r e s u l t s , i t w i l l be i n f e r r e d t h a t the averages i n the d e s i g n a t e d depth ranges a re common t o a l l a c c e p t a b l e models. In the f i n a l a n a l y s i s t h i s e x t r a p o l a t i o n i s not f u l l y j u s t i f i e d . I n a t h e o r e t i c a l example such as t h i s , knowledge of the t r u e model f u r n i s h e s a c o n s i s t e n c y check f o r the i n f e r e n c e s r e g a r d i n g u n i v e r s a l i t y of averages because the t r u e model averages s h o u l d be i d e n t i c a l ' w i t h i n e r r o r t o the b a s e l i n e model a v e r a g e s . As seen i n F i g . 3.10, the t r u e averages do i n f a c t comply w i t h t h i s r e q u i r e m e n t . T h i s concordance does not prove t h a t the b a s e l i n e averages a re common t o a l l a c c e p t a b l e models, however. Assuming t h a t the averages a r e u n i v e r s a l over the s p e c i f i e d depth i n t e r v a l s , i t remains o n l y t o examine the c o r r e s p o n d i n g a v e r a g i n g f u n c t i o n s i n o r d e r t o a s s e s s the p h y s i c a l s i g n i f i c a n c e of the a v e r a g e s . The spre a d of the a v e r a g i n g f u n c t i o n s can be read d i r e c t l y from the a p p r o p r i a t e t r a d e - o f f c u r v e s . For example, a t 200m the a v e r a g i n g f u n c t i o n f o r Hr|&|Hz| data has a sprea d of a p p r o x i m a t e l y 200m, as per F i g . 3.6c; the r e l e v a n t a v e r a g i n g f u n c t i o n i s d e p i c t e d i n F i g . 3.11a. (By h y p o t h e s i s , i t i s u n i m p o r t a n t t h a t the t r a d e - o f f c u r v e s and a v e r a g i n g f u n c t i o n s are a s s o c i a t e d w i t h k e r n e l s c o r r e s p o n d i n g t o d i f f e r e n t a c c e p t a b l e models; the t r a d e - o f f c u r v e s were computed w i t h r e s p e c t t o the k e r n e l s f o r the o r i g i n a l a c c e p t a b l e model, whereas the a v e r a g i n g f u n c t i o n s a r e c o n s t r u c t e d from the b a s e l i n e model k e r n e l s . ) S i m i l a r l y , the spread of the |Hr|&|Hz| a v e r a g i n g f u n c t i o n a t 800m f o r a v a r i a n c e of 1 0 - 4 S2/m2 i s a p p r o x i m a t e l y 800m a c c o r d i n g t o the a p p r o p r i a t e t r a d e - o f f c u r v e 177 F i g u r e 3.10 Comparison of Averages of the "True" Model w i t h B a s e l i n e Model Averages (a) D i f f e r e n c e s (x) between averages of the " t r u e " model ( F i g . 2.2) and those of the |Hr|& J Hz| b a s e l i n e model ( F i g . 3.2b) u s i n g the same a v e r a g i n g f u n c t i o n s as f o r the r e s u l t s i n F i g . 3.7. The f a c t t h a t the " t r u e " and b a s e l i n e averages a r e e q u a l w i t h i n e r r o r over the f i r s t 800m i s c o n s i s t e n t w i t h the u n i v e r s a l i t y of the averages i n f e r r e d from F i g . 3.7b. (b) D i f f e r e n c e s (x) between averages of the " t r u e " model ( F i g . 2.2) and t h o s e . o f the Re{Hr,Hz}&Im{Hr,Hz} b a s e l i n e model u s i n g the same a v e r a g i n g f u n c t i o n s as f o r the r e s u l t s i n F i g . 3.9. The f a c t t h a t the " t r u e " and b a s e l i n e averages a r e eq u a l w i t h i n e r r o r over the f i r s t 1500m i s c o n s i s t e n t w i t h the u n i v e r s a l i t y of the averages i n f e r r e d from F i g . 3.9b. 178 |Hr|<8e|Hz| "true" differences E \ co A b V 0.02 0.01 0.00 •0.01 T 1 1 1 1 1 r X w X X X|XX t h X - » — * — I — « -x x x xi: 0 50 100 150 200 (X 101) F i g . 3.10a ReiHr,Hz?&lmfHr,Hz* "true" differences E cn A b V 0.02 0.01 0.00 -0.01 i 1 1 1 1 1 r ~ X X X V x X x x : * ' ' » • * T 7 - + X X, XX XxJc 0 50 100 150 200 (X 101) depth (m) F i g . 3.10b 179 F i g u r e 3.11 |Hr1 & 1 Hz| A v e r a g i n g F u n c t i o n s (a) A v e r a g i n g f u n c t i o n f o r depth 200m w i t h v a r i a n c e 10" 4 S 2/m 2. The r e l a t i v e l y l a r g e spread i s due t o the f a c t t h a t the maximum o c c u r s at about 125m. (b) A v e r a g i n g 800m. f u n c t i o n w i t h v a r i a n c e 1 0" 4 S 2/m 2 f o r depth 180 |Hr|&|Hz| averaging function 1 . 0 o o ^ 0 . 5 N • 0 . 0 0 50 100 150 200 (X 10 1 ) F i g . 3.11a |Hr|&|Hz| averaging function 1.0 o o ° ° 0 . 5 N 0 . 0 0 50 100 150 200 (X 10 1 ) depth (m) F i g . 3.11b i 1 1 1 r C = 0.01 S/m s = 262m 181 i n F i g . 3.6c; the v i a b i l i t y of the spre a d as an e s t i m a t e of peak w i d t h i s v e r i f i e d by i n s p e c t i o n of the a v e r a g i n g f u n c t i o n i t s e l f i n F i g . 3.11b. The a v e r a g i n g f u n c t i o n s a re h i g h l y c o n c e n t r a t e d near the a p p r o p r i a t e d e p t h s , and so the a s s o c i a t e d averages c o n s t i t u t e a m e a n i n g f u l e s t i m a t e of the l o c a l c o n d u c t i v i t y . The p h y s i c a l s i g n i f i c a n c e of an average i s u s u a l l y , but not n e c e s s a r i l y , a s s u r e d by a s m a l l s p r e a d ; i t i s sound p r a c t i c e t o examine the a c t u a l a v e r a g i n g f u n c t i o n s because the peak may not be p r o p e r l y c e n t r e d on the depth of i n t e r e s t (Backus and G i l b e r t , 1970, p l 2 8 ) . The a v e r a g i n g f u n c t i o n s a t 200m and 800m f o r the Re{Hr,Hz}&Im{Hr,Hz} d a t a a r e shown i n F i g . 3.12. These are even more l o c a l i s e d than those i n F i g . 3.11, i n keeping w i t h the g r e a t e r number of da t a and, hence, k e r n e l s . A g a i n , t h e r e i s good agreement w i t h the a c t u a l peak w i d t h s and the spreads c o r r e s p o n d i n g t o v a r i a n c e 1 0 " 4 S 2/m 2 i n F i g . 3.6d. Given the i n f e r r e d u n i v e r s a l i t y of the a v e r a g e s , and the h i g h l y l o c a l i s e d a v e r a g i n g f u n c t i o n s , i t may be c o n c l u d e d once more t h a t the n u m e r i c a l v a l u e s of the averages a r e i n d i c a t i v e of the magnitude of the t r u e l o c a l c o n d u c t i v i t y . T h i s c o n c l u s i o n i s s u b s t a n t i a t e d by comparison of F i g s . 3.9a and 2.2. 182 F i g u r e 3.12 Re{Hr,Hz}&Im{Hr,Hz} A v e r a g i n g F u n c t i o n s (a) A v e r a g i n g f u n c t i o n w i t h v a r i a n c e 1 0" 4 S 2/m 2 f o r depth 200m. The maximum i s much c l o s e r t o 200m than t h a t f o r the c o r r e s p o n d i n g |Hr|&|Hz| a v e r a g i n g f u n c t i o n i n F i g . 3.11a; hence the c o n s i d e r a b l y lower s p r e a d . (b) A v e r a g i n g f u n c t i o n w i t h v a r i a n c e 1 0 " 4 S 2/m 2 f o r depth 800m. The main peak i s n o t i c e a b l y narrower than t h a t f o r the c o r r e s p o n d i n g |Hr|&|Hz| a v e r a g i n g f u n c t i o n i n F i g . 3.11b. 183 RefHr,Hzi&lmfHr,Hz* averaging function F i g . 3.12a ReiHr,Hz$&lm*Hr,Hz* averaging function 1.0 o o °° 0.5 N 0.0 0 50 100 150 200 (X 10 1) depth (m) F i g . 3.12b 184 Chapter IV INVERSION OF DATA FROM GRASS VALLEY, NEVADA The purpose of t h i s c h a p t e r i s t o i l l u s t r a t e the performance of HLEM.INV when a p p l i e d t o r e a l d a t a . The HLEM soundings i n v e r t e d below were o b t a i n e d by a team from U. C. B e r k e l e y i n Grass V a l l e y , Nevada, d u r i n g the summer of 1978. The g e o p h y s i c a l i n t e r e s t i n the a r e a a r i s e s from the anomalously h i g h observed heat f l o w ( F i g . 4.1). The anomalous heat f l o w i s m a n i f e s t e d a t the s u r f a c e i n the form of hot s p r i n g s , and the data c o n s i d e r e d here were o b t a i n e d near Leach's Hot S p r i n g which i s marked i n the f i g u r e . The HLEM survey was one component i n a broader i n v e s t i g a t i o n of the geothermal p o t e n t i a l of the a r e a . The c o n d u c t i v e t a r g e t was the groundwater, and i t was hoped t h a t HLEM soundings would h e l p t o d e l i n e a t e the r e s e r v o i r s u p p l y i n g the hot s p r i n g s . S i n c e the geology i n Grass V a l l e y i s c h a r a c t e r i s e d by g e n t l y - d i p p i n g s t r a t a , and s i n c e the d i s t r i b u t i o n of hot water w i l l t end t o be governed by s t r a t i g r a p h i c a l l y - r e l a t e d v a r i a b l e s such as p e r m e a b i l i t y , i t i s r e a s o n a b l e t o assume t h a t the c o n d u c t i v i t y v a r i e s w i t h depth o n l y ( a t l e a s t t o a f i r s t a p p r o x i m a t i o n ) . On the s t r e n g t h of t h i s a s s u m p t i o n , and i n view of the almost p e r f e c t l y f l a t topography i n Grass V a l l e y , i t i s a p p r o p r i a t e t o a p p l y HLEM.INV t o the d a t a . The r o l e of HLEM.INV, and hence the scope of t h i s c h a p t e r , i s r e s t r i c t e d t o the d e t e r m i n a t i o n of a c o n d u c t i v i t y s t r u c t u r e which s a t i s f i e s the o b s e r v a t i o n s ; the 4 . 1 : Hot Springs in Northwestern Nevada ( a f t e r S a s s e t a l . , 1971) 186 g e o p h y s i c a l s i g n i f i c a n c e of t h a t c o n d u c t i v i t y , and i t s importance i n an o v e r a l l assessment of the geothermal p o t e n t i a l of the a r e a , are much wider q u e s t i o n s which w i l l not be add r e s s e d h e r e . The two s e c t i o n s of t h i s c h a p t e r a r e devoted r e s p e c t i v e l y t o c o n s t r u c t i o n and a p p r a i s a l . In the cour s e of a p p r a i s a l the e x i s t e n c e of a s t e e p c o n d u c t i v i t y g r a d i e n t over the f i r s t 800m depth i s e s t a b l i s h e d ; the average c o n d u c t i v i t y a t 800m i s a p p r o x i m a t e l y 0.4 S/m, f o u r t i m e s the average v a l u e near the s u r f a c e . A n o v e l a s p e c t of the i n v e r s i o n of the Grass V a l l e y d a t a , which d i d not a r i s e i n the t h e o r e t i c a l i n v e r s i o n s i n Chapter I I , i s the compensation f o r c o i l m i s a l i g n m e n t e r r o r ; t h i s i s a c c o m p l i s h e d i n the manner d e s c r i b e d i n I I - 4 ( i v ) , I I I — 5 ( i i ) , and Appendix D. IV-1 C o n s t r u c t i o n The w a v e - t i l t a m p l i t u d e and phase d a t a from B e r k e l e y -s t a t i o n T3-R2 i n Grass V a l l e y a r e p r e s e n t e d i n Ta b l e 4.1. The o b s e r v a t i o n s were made a t t w e l v e f r e q u e n c i e s s p r e a d over t h r e e decades, from 0.1 Hz t o 200 Hz. The r e c e i v e r was s i t u a t e d 1003.9m from a source c o i l 50m i n r a d i u s . Two s e t s of s t a n d a r d d e v i a t i o n s a r e i n c l u d e d i n the t a b l e . The f i r s t c o r r e s p o n d s t o {q}, d e f i n e d i n 11-3, and r e p r e s e n t s the e r r o r due t o a l l s o u r c e s o t h e r than c o i l m i s a l i g n m e n t . A s t a n d a r d d e v i a t i o n of 5% has been a s s i g n e d t o the a m p l i t u d e v a l u e s , w h i l e the c o r r e s p o n d i n g e r r o r f o r the phase d a t a i s assumed t o have s t a n d a r d d e v i a t i o n 5x10' 2 r a d ( a p p r o x i m a t e l y 187 T a b l e 4.1 W a v e - T i I t D a t a f r o m G r a s s V a l l e y , N evada O b s e r v a t i o n s . d . (q) w| | 4. 06x1 0 " 2 1 2 x l 0 ' 3 9.1x10" 3 6.283x10" 1 0w ( r a d ) | -1.32 5 x 1 0 - 2 8.4x10" 1 w| | 6. 64x10" 2 1 3.3x10" 3 3.5x10" 3 1.257 0W ( r a d ) | -1.61 5 x 1 0 " 2 5.3x10" 1 W| | 9. 90x10- 2 1 5 x 1 0 " 3 9.2x10" 3 1.885 0W ( r a d ) | -1.79 5 x 1 0 - 2 3.5x10- 1 W| | 1. 60x 1 0 " 1 1 8 x 1 0 " 3 1.6x10" 2 3.770 0W ( r a d ) | -1.98 5 x 1 0 " 2 2 x 1 0 " 1 W| | 2. 24x10" 1 1 1 . 1 x 1 0 " 2 2.2x10" 2 6.283 0W ( r a d ) | -2.12 5 x 1 0 " 2 1.4x10" 1 W| | 3. 27x1 0 " 1 1 1 . 5 x 1 0 " 2 2.8x10" 2 1.257X10 1 0w ( r a d ) | -2.22 5 x 1 0 - 2 9.2x10" 2 W| | 4. 05x10" 1 1 2 x 1 0 " 2 [ 3.2x10" 2 1.885X10 1 0W ( r a d ) | -2.24 5 x 1 0 - 2 j 7.6x10" 2 W| | 5. 79x1 0 " 1 1 2 . 9 x 1 0 " 2 | 4.2x10" 2 3.770X10 1 0W ( r a d ) | -2.28 5 x 1 0 " 2 | 5.9x10' 2 |w| I 7. 50x10" 1 1 3 . 8 x 1 0 " 2 | 5.3x10" 2 6.283x10^ 0w ( r a d ) | -2.32 5 x l 0 " 2 | 5.3x10-2 |w| | 1.26 6 x 1 0 " 2 | 8.2x10" 2 1 . 8 8 5 x l 0 2 0W ( r a d ) | -2.23 5x10- 2 | 5.3x10" 2 |w| I 2.47 1.2x10" 1 | 1.6x10" 1 6 . 2 8 3 x l 0 2 0W ( r a d ) | -2.02 5 x 1 0 " 2 | 8.3x10" 2 |W| | 4.1 4x10" 1 | 4.2x10" 1 1 . 2 5 7 x l 0 3 0w ( r a d ) | -1.78 8 . 5 x l 0 " 2 | 1.6x10" 1 a u g . s . d . ( q + ) | w ( r a d / s e c ) 188 3°). The a c t u a l raw f i e l d d a t a were d e t e r m i n e d t o an a c c u r a c y much s u p e r i o r t o t h i s u s i n g a c r y o g e n i c magnetometer and d i g i t a l s i g n a l a v e r a g e r . Thus the a s s i g n e d e r r o r s do not r e f l e c t i n s t r u m e n t a l l i m i t a t i o n s , but a r e chosen r a t h e r t o account f o r the e f f e c t s of l a t e r a l inhomogeneity and a n i s o t r o p y , i . e . the d i s c r e p a n c i e s a r i s i n g from the l i m i t a t i o n s of the m a t h e m a t i c a l model. I t i s h i g h l y u n l i k e l y t h a t t h e r e i s a h o r i z o n t a l l y - l a y e r e d , i s o t r o p i c c o n d u c t i v i t y which s a t i s f i e s the d a t a t o an a c c u r a c y c o n s i s t e n t w i t h the s m a l l i n s t r u m e n t a l u n c e r t a i n t i e s . The second s e t of s t a n d a r d d e v i a t i o n s c o r r e s p o n d s t o ( q + l , d e f i n e d by ( 2 . 8 2 ) ; these a r e the s t a n d a r d d e v i a t i o n s which have been augmented t o a l l o w f o r c o i l m i s a l i g n m e n t e r r o r . In t h i s i n s t a n c e the s t a n d a r d d e v i a t i o n , v , of the random v a r i a b l e $ r e p r e s e n t i n g the m i s a l i g n m e n t a n g l e has been a s s i g n e d a v a l u e 3.5x10" 2 r a d ( a p p r o x i m a t e l y 2°); t h u s t h e r e i s assumed t o be a 65% p r o b a b i l i t y t h a t the d e v i a t i o n i s l e s s than 2° i n magnitude. By i n s p e c t i o n of the t a b l e i t i s apparent t h a t f o r t h i s d ata s e t the e f f e c t of m i s a l i g n m e n t i s most pronounced a t low f r e q u e n c i e s ; f o r example, the augmented e r r o r f o r 0w a t a n g u l a r f r e q u e n c y 0.6283 rad/sec i s almost seventeen t i m e s the o r i g i n a l s t a n d a r d d e v i a t i o n . The c o n s t r u c t i o n of an a c c e p t a b l e model f o r the Grass V a l l e y d a t a i s summarised i n F i g . 4.2. The s t a r t i n g model was a u n i f o r m h a l f - s p a c e (<?f = 0.1 S/m) c o m p r i s e d of f i v e l a y e r s w i t h i n t e r f a c e s a t depths 0.5, 1.0, 2.0, 3.5, and 5.0 km. The number of l a y e r s was i n c r e a s e d t o a maximum (NLMAX) of t w e n t y - f o u r as the c o n s t r u c t i o n p r o g r e s s e d . Only the uppermost 2 km was 189 F i g u r e 4.2 C o n s t r u c t i o n of Model s a t i s f y i n g Grass V a l l e y W a v e - T i l t Data from B e r k e l e y S t a t i o n T3-R2. Three i n t e r m e d i a t e models and the f i n a l a c c e p t a b l e model are d e p i c t e d . The s t a r t i n g model was a u n i f o r m h a l f space ( = 0.1 S/m) d i v i d e d i n t o f i v e l a y e r s . The i n i t i a l x 2 - m i s f i t f o r t h i s model was 501.0. The t o t a l CPU time r e q u i r e d on the U.B.C. Amdahl 470 V/6 computer was 148.36sec. (a) E s t i m a t e d G r a s s V a l l e y s t r u c t u r e a f t e r one i t e r a t i o n . The main f e a t u r e s of the a c c e p t a b l e model a r e a l r e a d y e v i d e n t , though g r e a t l y muted. (b) E s t i m a t e d Grass V a l l e y s t r u c t u r e a f t e r f o u r i t e r a t i o n s . The upper 5 km i s now d i v i d e d i n t o t e n l a y e r s . (c) E s t i m a t e d Grass V a l l e y s t r u c t u r e a f t e r e i g h t i t e r a t i o n s . The number of l a y e r s i n the f i r s t 5 km i s now t w e n t y - f o u r ; t h i s i s the maximum a l l o w e d (NLMAX). (d) A c c e p t a b l e model f o r the Grass V a l l e y d a t a . 190 T3-R2 construction: ITER=1 0 . 6 ^ 0 . 4 co b 0 . 2 0 .0 T 1 1 r i r x = 352.5 0 50 100 150 200 ( X 10 1 ) Fig. A.2a T3-R2 construction: ITER=4 o . o 0 50 100 150 200 ( X 10 1 ) depth (m) Fig. A.2b 191 T3-R2 construction: ITER=8 0 . 6 £ ^ 0 . 4 co b 0 . 2 h 0 .0 0 50 100 150 200 (X 10 1 ) F i g . 4.2c T3-R2 construction: ITER=13 0 . 6 ^ 0 . 4 co b 0 . 2 h 0 .0 0 50 100 150 200 (X 10 3 ) depth (m) F i g . 4.2d 192 m o d i f i e d a p p r e c i a b l y d u r i n g c o n s t r u c t i o n , so o n l y t h i s range of depths i s d e p i c t e d i n F i g . 4.2. Due t o the presence of m i s a l i g n m e n t e r r o r , the p e r t u r b a t i o n e q u a t i o n s u n d e r l y i n g each i t e r a t i o n were t r a n s f o r m e d i n accordance w i t h (2.90-92); the t r a n s f o r m e d e q u a t i o n s were s o l v e d u s i n g the m o d i f i e d f o r m u l a t i o n ( w i t h e x p l i c i t t r e atment of e r r o r s ) d e s c r i b e d i n I I — 4 ( i i i ) . The c o n s t r u c t e d model i s x 2 - a c c e p t a b l e , as per ( 2 . 3 3 ) , w i t h NSD=0. The p r i n c i p a l f e a t u r e of the a c c e p t a b l e model i n F i g . 4.2d i s the broad c o n d u c t o r c e n t r e d a t a depth of a p p r o x i m a t e l y 750m. Comparing F i g s . 4.2a and 4.2d, i t i s apparent t h a t the a l g o r i t h m i n t r o d u c e s such a c o n d u c t o r ( i n embryonic form) i n the f i r s t i t e r a t i o n , even though the s t a r t i n g model i s v e r y " f a r " from any a c c e p t a b l e model; b e h a v i o u r of t h i s s o r t augurs w e l l f o r the v i a b i l i t y of l i n e a r i s e d c o n s t r u c t i o n i n HLEM i n v e r s i o n . The f a c t t h a t f i v e i t e r a t i o n s were r e q u i r e d t o complete the t r a n s i t i o n from the model i n F i g . 4.2c t o the ve r y s i m i l a r ( a c c e p t a b l e ) model i n F i g . 4.2d i s perhaps a l i t t l e d i s a p p o i n t i n g . T h i s slow f i n a l convergence may be a t t r i b u t e d t o s h o r t c o m i n g s of the t e c h n i q u e s f o r c h o o s i n g the most a p p r o p r i a t e approximate s o l u t i o n t o the p e r t u r b a t i o n e q u a t i o n s a t each i t e r a t i o n . Even so, the o v e r a l l number of i t e r a t i o n s r e q u i r e d i s not e x c e s s i v e and, moreover, the c o n s t r u c t i o n was c o m p l e t e l y s t a b l e i n the sense t h a t the program converged m o n o t o n i c a l l y w i t h o u t o p e r a t o r i n t e r v e n t i o n . 193 IV-2 A p p r a i s a l I t i s of some p r a c t i c a l importance t o det e r m i n e whether or not a l l a c c e p t a b l e models have h i g h c o n d u c t i v i t y near depth 750m and, i f so, t o what e x t e n t the s t r u c t u r e resembles t h a t i n F i g . 4.2d. The aim of a p p r a i s a l i s t o r e s o l v e t h e s e q u e s t i o n s . T h i s aim f a l l s f a r s h o r t of an u n d e r t a k i n g t o produce a comprehensive e x p l a n a t i o n f o r the cause of any c o n d u c t i v e anomaly because an i n t e r p r e t a t i o n based on an i n v e r s i o n of HLEM soundings a l o n e would be h i g h l y s p e c u l a t i v e . For example, a zone of h i g h c o n d u c t i v i t y beneath Grass V a l l e y c o u l d be i n d i c a t i v e of an e l e v a t e d t e m p e r a t u r e , or i t c o u l d be due t o i n c r e a s e s i n e i t h e r the s a l i n i t y of the groundwater or the p e r m e a b i l i t y of the h o s t r o c k . As d i s c u s s e d i n I I I — 5 ( i ) , the f i r s t s t age i n HLEM.INV a p p r a i s a l i s the c o n s t r u c t i o n of a b a s e l i n e model, l a t e r t o serve as the s t a r t i n g model f o r the d e t e r m i n a t i o n of e x t r e m a l a c c e p t a b l e models. The b a s e l i n e model i s n o r m a l l y d e f i n e d on a c o a r s e r p a r t i t i o n than the o r i g i n a l c o n s t r u c t e d model, and f o r the t h e o r e t i c a l examples i n I I - 5 i t proved e x p e d i t i o u s t o s i m p l y h a l v e the number of l a y e r s by e l i m i n a t i n g e v e r y second i n t e r f a c e . When t h i s approach was f o l l o w e d i n the p r e s e n t example, no a c c e p t a b l e model c o u l d be found on the new p a r t i t i o n . C o n s e q u e n t l y , the b a s e l i n e model p a r t i t i o n f i n a l l y adopted d i f f e r s l i t t l e from the p a r t i t i o n f o r the o r i g i n a l c o n s t r u c t e d model over the f i r s t two k i l o m e t r e s : o n l y the i n t e r f a c e s a t 1250m and 1750m have been removed. Depths below 2 km were a s s i g n e d a c o n d u c t i v i t y 0.1 S/m, and were not c o n s i d e r e d i n a p p r a i s a l . The b a s e l i n e model on the ( s l i g h t l y ) 194 m o d i f i e d p a r t i t i o n was c o n s t r u c t e d u s i n g l i n e a r programming, and i s d e p i c t e d i n F i g . 4.3. In view of the s i m i l a r i t y of t h e i r depth p a r t i t i o n s , i t i s not s u r p r i s i n g t h a t the b a s e l i n e model c l o s e l y resembles the o r i g i n a l c o n s t r u c t e d model i n F i g . 4.2d. U s i n g the b a s e l i n e model as a s t a r t i n g model, a set of e x t r e m a l models has been computed by m a x i m i s i n g and m i n i m i s i n g the c o n d u c t i v i t y of i n d i v i d u a l l a y e r s . As e x p l a i n e d i n 1 1 1 - 6 ( i ) , t h i s s e t of e x t r e m a l models w i l l l a t e r c o n s t i t u t e the raw m a t e r i a l f o r an assessment of the u n i v e r s a l i t y of B a c k u s - G i l b e r t a v e r a g e s . A s e l e c t i o n of the e x t r e m a l models i s p r e s e n t e d i n F i g . 4.4. On the b a s i s of F i g s . 4.2d, 4.3, and 4.4 i t may be i n f e r r e d t h a t the presence of a co n d u c t o r a t a depth of a p p r o x i m a t e l y 750m i s e s s e n t i a l f o r a s a t i s f a c t o r y f i t t o the d a t a . However, t h e r e i s c o n s i d e r a b l e v a r i a b i l i t y i n the d e t a i l e d appearance of the c o n d u c t i v e zone; t h i s i s most e v i d e n t from comparison of F i g s . 4.4g and 4.4h, i t b e i n g apparent t h a t two narrow c o n d u c t o r s between depths 500m and 1000m can s a t i s f y the d a t a as w e l l as a s i n g l e broad c o n d u c t o r i n t h a t zone. The p r i n c i p a l o u t s t a n d i n g q u e s t i o n now con c e r n s whether the s t r u c t u r a l v a r i a t i o n s a r e r e s o l v a b l e , or whether a l l the known a c c e p t a b l e models a r e i d e n t i c a l w i t h i n the r e s o l u t i o n of the da t a s e t . The f i r s t i n d i c a t i o n of the r e s o l v i n g power of the da t a i s p r o v i d e d by t r a d e - o f f c u r v e s . As d e s c r i b e d i n 11 1-6(i i ) , a t r a d e - o f f c u r v e i s a p l o t of v a r i a n c e v e r s u s s p r e a d a t any dep t h , and as such summarises the c h a r a c t e r i s t i c s of a l l p o s s i b l e B a c k u s - G i l b e r t averages which can be o b t a i n e d a t t h a t d e p t h . The t r a d e - o f f c u r v e s f o r depths 200m, 500m, 900m, and 195 T3-R2 baseline model 0 . 6 ^ 0 . 4 cn, b 0 . 2 0 .0 T r Jl r. T 1 r x = 20 .A x = 24 max JJ 0 50 100 150 200 (X 10 1 ) depth (m) F i g . 4 . 3 196 F i g u r e 4.4 E x t r e m a l Models f o r the Grass V a l l e y Wave-TiIt Data A l l the e x t r e m a l models are e i t h e r x 1 - or x 2 - a c c e p t a b l e , as per (2.32) or ( 2 . 3 3 ) , w i t h NSD=0. (a) U s i n g the b a s e l i n e model as a s t a r t i n g model, the c o n d u c t i v i t y of the f i r s t l a y e r has been m i n i m i s e d , s u b j e c t t o the c o n s t r a i n t s imposed by the d a t a . (b) U s i n g the b a s e l i n e model as a s t a r t i n g model, the c o n d u c t i v i t y of the second l a y e r has been maximised, s u b j e c t t o the c o n s t r a i n t s imposed by the d a t a . (c) U s i n g the b a s e l i n e model as a s t a r t i n g model, the c o n d u c t i v i t y of the f o u r t h l a y e r has been maximised, s u b j e c t t o the c o n s t r a i n t s imposed by the d a t a . (d) U s i n g the b a s e l i n e model as a s t a r t i n g model, the c o n d u c t i v i t y of the f o u r t h l a y e r has been m i n i m i s e d , s u b j e c t t o the c o n s t r a i n t s imposed by the d a t a . (e) U s i n g the b a s e l i n e model as a s t a r t i n g model, the c o n d u c t i v i t y of the s i x t h l a y e r has been maximised, s u b j e c t t o the c o n s t r a i n t s imposed by the d a t a . ( f ) U s i n g the b a s e l i n e model as a s t a r t i n g model, the c o n d u c t i v i t y of the s i x t h l a y e r has been m i n i m i s e d , s u b j e c t t o the c o n s t r a i n t s imposed by the d a t a . (g) U s i n g the b a s e l i n e model as a s t a r t i n g model, the c o n d u c t i v i t y of the t e n t h l a y e r has been m i n i m i s e d , s u b j e c t t o the c o n s t r a i n t s imposed by the d a t a . (h) U s i n g the b a s e l i n e model as a s t a r t i n g model, the c o n d u c t i v i t y of the t w e l v t h l a y e r has been m i n i m i s e d , s u b j e c t t o the c o n s t r a i n t s imposed by the d a t a . 197 M i n i m i s e d l a y e r #1 c o n d u c t i v i t y 0 . 8 0 . 6 0 . 4 0 . 2 h d.o T 1 1 1 1 T r 1 Lf x = 19.01 x 1 = 19.15 max 0 50 100 150 200 (X 1 0 1 ) F i g . 4.4a M a x i m i s e d l a y e r # 2 c o n d u c t i v i t y 0 . 8 0 . 6 0 . 4 0 . 2 0 . 0 T r x = 18.94 x 1 = 19.15 max i 0 50 100 150 200 (X 1 0 1 ) d e p t h ( m ) F i g . 4.4b 198 M a x i m i s e d l a y e r # 4 c o n d u c t i v i t y 0 50 100 150 200 (X 10 1 ) F i g . 4.4c M i n i m i s e d l a y e r # 4 c o n d u c t i v i t y 0 . B ^ 0 . 6 \ ^ 0 . 4 0 . 2 0 .0 T 1 1 1 r n x = 18.76 x 1 = 19.15 max 1 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) F i g . 4.4d 199 M a x i m i s e d l a y e r # 6 c o n d u c t i v i t y 0 . 8 0 . 6 0 . 4 -0 . 2 In 0 .0 T 1 1 1 1 1 r x = 18.79 x 1 = 19.15 max fu I "1 0 50 100 150 200 (X 10 1 ) Fig. 4.4e M i n i m i s e d l a y e r # 6 c o n d u c t i v i t y 0 . B 0 . 6 0 . 4 0 . 2 k J 0 . 0 — i 1 r x 1 = 14.93 x 1 = 19.15 max 1 1 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) Fig. 4.4f 200 M i n i m i s e d l a y e r # 1 0 c o n d u c t i v i t y 0 . 8 0 . 6 0 . 4 0 . 2 t r 0 .0 T r f l —i 1 r x 1 = 14.94 x 1 = 19.15 max 0 50 100 150 200 (X 10 1 ) Fig. 4.4g M i n i m i s e d l a y e r # 1 2 c o n d u c t i v i t y 0 . 8 0 . 6 0 . 4 0 . 2 0 .0 ~i 1 1 1 — — r x 1 = 17.95 x 1 = 19.15^ max 1 1 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) Fig. 4.4h 201 1400m a r e d e p i c t e d i n F i g . 4.5. By i n s p e c t i o n of t h e s e c u r v e s i t i s p o s s i b l e t o p r e d i c t t h a t the d i f f e r e n c e s between the a c c e p t a b l e models i n F i g . 4.4 a r e not r e s o l v a b l e by the d a t a , w i t h the p o s s i b l e e x c e p t i o n of the secondary s t r u c t u r e s at s h a l l o w d e p t h s . In p a r t i c u l a r , the s m a l l - s c a l e v a r i a t i o n s i n c o n d u c t i v i t y a t depths near 750m cannot be r e s o l v e d because t h e i r w i d t h i s a s m a l l f r a c t i o n of the spread of the a v e r a g i n g f u n c t i o n s a t t h a t depth f o r the a p p r o p r i a t e range of v a r i a n c e s : by i n t e r p o l a t i o n between the t r a d e - o f f c u r v e s f o r 500m and 900m, i t f o l l o w s t h a t the a v e r a g i n g f u n c t i o n a t 750m c o r r e s p o n d i n g t o s t a n d a r d d e v i a t i o n 5x10' 2 S/m has a spread of a p p r o x i m a t e l y 400m, whereas t h e r e a r e s i g n i f i c a n t v a r i a t i o n s on a s c a l e of 125m i n F i g . 4.4. These p r e d i c t i o n s based on the t r a d e - o f f c u r v e s a re borne out i n F i g . 4.6 where the u n i v e r s a l i t y of the averages w i t h s t a n d a r d d e v i a t i o n 5 x l 0 ~ 2 S/m i s demonstrated. In F i g . 4.6b i t i s c l e a r t h a t a t almost a l l depths the avera g e s f o r the d i f f e r e n t e x t r e m a l models l i e w i t h i n one s t a n d a r d d e v i a t i o n of the b a s e l i n e model a v e r a g e s ; thus a l l the known a c c e p t a b l e models g i v e r i s e t o averages which a r e i d e n t i c a l t o an a c c u r a c y c o n s i s t e n t w i t h the l e v e l of e r r o r . I f i t i s i n f e r r e d on t h i s b a s i s t h a t the t r u e c o n d u c t i v i t y i s c h a r a c t e r i s e d by these same a v e r a g e s , then i t may be c o n c l u d e d from e x a m i n a t i o n of F i g . 4.6a t h a t t h e r e i s a s u b s t a n t i a l ( a p p r o x i m a t e l y f o u r f o l d ) i n c r e a s e i n c o n d u c t i v i t y i n the f i r s t k i l o m e t r e beneath G r a s s V a l l e y . The appearance of the a v e r a g i n g f u n c t i o n s i n F i g s . 4.7a and 4.7b l e n d s s u p p o r t t o t h i s c o n c l u s i o n : the a v e r a g i n g f u n c t i o n s a t depths 200m and 500m a r e h i g h l y c o n c e n t r a t e d a t the a p p r o p r i a t e T 3 - R 2 t r a d e - o f f c u r v e s s ( m ) F i g . 4.5 T r a d e - o f f c u r v e s f o r G r a s s V a l l e y w a v e - t i l t d a t a a t d e p t h s 200, 500, 900, and 1400m. 203 F i g u r e 4.6 Averages of the C o n d u c t i v i t y f o r A c c e p t a b l e Grass V a l l e y  S t r u c t u r e s , and Assessment of t h e i r U n i v e r s a l i t y (a) Averages w i t h s t a n d a r d d e v i a t i o n 5x10" 2 S/m p l o t t e d as a f u n c t i o n of depth f o r Grass V a l l e y b a s e l i n e and e x t r e m a l models. The c o r r e s p o n d i n g a v e r a g i n g f u n c t i o n s were c o n s t r u c t e d from l i n e a r c o m b i n a t i o n s of the b a s e l i n e model F r e c h e t k e r n e l s . The averages f o r a l l models e x h i b i t a marked i n c r e a s e w i t h depth over the f i r s t 800m. (b) D i s c r e p a n c i e s (x) between the b a s e l i n e model averages and the averages f o r the d i f f e r e n t e x t r e m a l models. The s o l i d l i n e s d e l i n e a t e the u n i t s t a n d a r d d e v i a t i o n e n v e l o p e ; i f the averages are i d e n t i c a l t o an a c c u r a c y c o n s i s t e n t w i t h the l e v e l of e r r o r , a t l e a s t 65% of the d i s c r e p a n c i e s s h o u l d l i e w i t h i n t h i s e n v e l o p e . In t h i s i n s t a n c e i t i s apparent t h a t the averages are i d e n t i c a l w i t h i n e r r o r a t a l l d e p t h s . 204 T 3 - R 2 a v e r a g e s ^ 0 . 4 S 0 . 2 V 0 . 0 £ = 0.05 S/m 0 50 100 150 200 (X 10 1 ) F i g . 4.6a Un i t s . d . e n v e l o p e E ui A b V 0 . 1 0 0 . 0 5 0 . 0 0 •0 .05 •0 .10 -x-l X*H*XXXXXXX$ x x x S 1 1 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) F i g . 4.6b 205 F i g u r e 4.7 A v e r a g i n g F u n c t i o n s f o r Grass V a l l e y A p p r a i s a l The a v e r a g i n g f u n c t i o n s i n t h i s f i g u r e were a l l c o n s t r u c t e d from the F r e c h e t k e r n e l s f o r the b a s e l i n e model ( F i g . 4.3). In each case the s t a n d a r d d e v i a t i o n of the average a s s o c i a t e d w i t h the a v e r a g i n g f u n c t i o n i s 5x10" 2 S/m. (a) A v e r a g i n g f u n c t i o n f o r depth 200m. (b) A v e r a g i n g f u n c t i o n f o r depth 500m. The spre a d of t h i s window, and t h a t i n F i g . 4.7c, i s c o n s i d e r a b l y g r e a t e r than the l e n g t h s c a l e of the c o n d u c t i v i t y e x c u r s i o n s e x h i b i t e d by the models i n F i g . 4.4 between depths 500m and 1 km. C o n s e q u e n t l y , t h e s e v a r i a t i o n s a r e not r e s o l v a b l e . (c) A v e r a g i n g f u n c t i o n f o r depth 900m. U n l i k e the a v e r a g i n g f u n c t i o n s i n (a) and ( b ) , t h i s a v e r a g i n g f u n c t i o n i s not p e r f e c t l y c e n t r e d , i . e . i t s peak i s a t a p p r o x i m a t e l y 750m, not 900m. (d) A v e r a g i n g f u n c t i o n f o r depth 1400m. T h i s window i s v e r y p o o r l y c e n t r e d , and the r e s u l t i n g average cannot be i n t e r p r e t e d as an e s t i m a t e of the c o n d u c t i v i t y i n the v i c i n i t y of 1400m. 206 T 3 - R 2 a v e r a g i n g f u n c t i o n 1.0 ^ 0 . 7 o o ^ 0 . 5 IN < 0 . 2 0 . 0 — T T r — i 1 1 1 1 C= 0.05 S/m — J I s = 102m L i r i -. . I I I . 0 50 100 150 200 (X 10 1 ) F i g . 4.7a T 3 - R 2 a v e r a g i n g f u n c t i o n 1.0 ^ 0 . 7 o o ^ 0 . 5 N ^ 0 . 2 0 . 0 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) F i g . 4.7b T 3 - R 2 a v e r a g i n g f u n c t i o n 0 50 100 150 200 (X 10 1 ) F i g . 4.7c T 3 - R 2 a v e r a g i n g f u n c t i o n 0 o 1 1 1 — 1 — 1 — 1 — 1 — t — 1 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) F i g . 4.7d 208 d e p t h s , so the averages at those depths a re i n d i c a t i v e of the l o c a l c o n d u c t i v i t y . However, the a v e r a g i n g f u n c t i o n s f o r 900m and, more e s p e c i a l l y , 1400m ( F i g s . 4.7c and 4.7d) are c e n t r e d a t a p p r o x i m a t e l y 750m and 800m r e s p e c t i v e l y ; t h e r e f o r e the r e s u l t i n g averages do not c h a r a c t e r i s e the l o c a l c o n d u c t i v i t y a t the depths w i t h which they are n o m i n a l l y a s s o c i a t e d . Thus the dat a do not p r o v i d e l o c a l i s e d i n f o r m a t i o n a t depths g r e a t e r than 900m; p h y s i c a l l y , t h i s i s c o m p a t i b l e w i t h the f a c t t h a t such depths a re s h i e l d e d by the c o n d u c t i v e m a t e r i a l above. A f u r t h e r p o i n t of c o n s i s t e n c y between a l l known a c c e p t a b l e models i s the conductance, as w i t n e s s e d by F i g . 4.8. There i s a near c o i n c i d e n c e of the c u r v e s down t o a depth of a p p r o x i m a t e l y lkm; t h i s depth might be c o n s i d e r e d t o mark the approximate l i m i t of p e n e t r a t i o n of the source f i e l d s . 2 0 9 T 3 - R 2 c o n d u c t a n c e s (X 10 1 ) 0 50 100 150 200 (X 10 1 ) d e p t h ( m ) F i g . 4 . 8 Integrated conductivity for Grass Valley baseline and extremal models. 210 CONCLUSION The p r e c e d i n g c h a p t e r s document the c a p a b i l i t i e s and demonstrate the performance of a new a l g o r i t h m , HLEM.INV, which e f f e c t s a u t o m a t i c i n v e r s i o n of HLEM frequency soundings taken over a s t r a t i f i e d e a r t h . The program was w r i t t e n i n response t o a p e r c e i v e d need f o r improved i n v e r s i o n t e c h n i q u e s f o r g e o p h y s i c a l EM methods, and t h e r e a re grounds f o r c l a i m i n g t h a t HLEM.INV supersedes a l l p r e - e x i s t i n g i n t e r p r e t a t i o n a l a i d s f o r HLEM so u n d i n g s . The a p p l i c a b i l i t y of the program i s not r e s t r i c t e d t o p a r a m e t r i c s o u n d i n g s , i t b e i n g r e a d i l y a d a p t a b l e t o the i n v e r s i o n of time-domain HLEM d a t a ( v i a a f a s t F o u r i e r t r a n s f o r m ) and, w i t h somewhat more s u b s t a n t i a l m o d i f i c a t i o n , t o i n v e r s i o n of geometric soundings ( i n c l u d i n g a i r b o r n e EM), s u b j e c t always t o the r e s t r i c t i o n t h a t the ground may be c o n s i d e r e d h o r i z o n t a l l y s t r a t i f i e d . B r o a d l y s p e a k i n g , HLEM.INV may be regard e d as the l a t e s t i m p l e m e n t a t i o n of the g e n e r a l approach t o i n v e r s i o n proposed by Backus and G i l b e r t (1967, 1968, 1970). As such i t i s a c l o s e r e l a t i v e of p r e v i o u s B a c k u s - G i l b e r t i n v e r s i o n r o u t i n e s , e.g. those w r i t t e n by Oldenburg (1978, 1979). The mark of the B a c k u s - G i l b e r t approach i s the c l e a r d i s t i n c t i o n they draw between c o n s t r u c t i o n of a model which s a t i s f i e s a g i v e n s e t of d a t a , and a p p r a i s a l of the e x t e n t t o which any a c c e p t a b l e model i s r e p r e s e n t a t i v e of a l l models which reproduce the d a t a . A c c o r d i n g l y , HLEM.INV i n c l u d e s s u b r o u t i n e s t o p e r f o r m both 211 c o n s t r u c t i o n and a p p r a i s a l , whereas the u t i l i t y of the a l t e r n a t i v e a i d s f o r HLEM i n v e r s i o n , namely c u r v e - m a t c h i n g and the Glenn et a l . (1973) a l g o r i t h m , i s r e s t r i c t e d t o c o n s t r u c t i o n . In the realm of a p p r a i s a l , t h e r e f o r e , HLEM.INV has no r i v a l s . W i t h i n the sphere of c o n s t r u c t i o n , HLEM.INV c o n s t i t u t e s a c o n s i d e r a b l e improvement over c u r v e - m a t c h i n g because i t i s a u t o m a t i c and because i t i s not r e s t r i c t e d t o s t r u c t u r e s w i t h a s m a l l number of l a y e r s . The HLEM.INV c o n s t r u c t i o n r o u t i n e s a re a l s o s u p e r i o r t o the Glenn et a l . a l g o r i t h m i n some r e s p e c t s ; most n o t a b l y , HLEM.INV has been d e s i g n e d t o cope w i t h u n c e r t a i n t i e s i n the da t a (independent or c o r r e l a t e d ) , whereas Glenn et a l . d e f e r a comprehensive tre a t m e n t of e r r o r s . In the development of the HLEM.INV c o n s t r u c t i o n s u b r o u t i n e s a c o n s i d e r a b l e e f f o r t was expended i n o r d e r t o m i n i m i s e the c o m p u t a t i o n a l l a b o u r . Most i m p o r t a n t l y , the e v a l u a t i o n of the Hankel t r a n s f o r m i n t e g r a l s r e q u i r e d i n the s o l u t i o n of the fo r w a r d problem has been s t r e a m l i n e d v i a the use of a n a l y t i c a l f ormulae ( b e l i e v e d t o be new) f o r the p r i m a r y f i e l d components and by a p p l i c a t i o n of the d i g i t a l l i n e a r f i l t e r i n g of Koefoed et a l . (1972). C o n s t r u c t i o n proceeds by i t e r a t i v e improvement of a s t a r t i n g model v i a the s o l u t i o n of a s u c c e s s i o n of l i n e a r p e r t u r b a t i o n problems. The method of s o l u t i o n of the system of l i n e a r e q u a t i o n s a r i s i n g i n each i t e r a t i o n i s e s s e n t i a l l y t h a t of P a r k e r (1977a), though c e r t a i n i n n o v a t i o n s have been i n t r o d u c e d i n an attempt t o enhance both the g e n e r a l i t y and n u m e r i c a l s t a b i l i t y of the program (see 11 -4) . The most n o v e l of these a r e a s i m p l i f i e d scheme t o compensate 212 f o r the e r r o r a r i s i n g from a m i s a l i g n m e n t of t r a n s m i t t e r and r e c e i v e r , and a r e f o r m u l a t i o n of the p e r t u r b a t i o n e q u a t i o n s i n which e r r o r s appear e x p l i c i t l y : the m i s a l i g n m e n t a n g l e i s m o d e l l e d as the r e a l i s a t i o n of a Normal random v a r i a b l e , and the r e s u l t i n g i n c r e a s e i n v a r i a n c e f o r each datum can be c a l c u l a t e d u s i n g formulae d e r i v e d i n Appendix D; the r e f o r m u l a t e d e q u a t i o n s enjoy the same " s t a b i l i t y " a t t a i n a b l e by r i d g e r e g r e s s i o n , but w i t h o u t the i n t r o d u c t i o n of unknown parameters. HLEM.INV c o n s t r u c t i o n r o u t i n e s have performed v e r y s a t i s f a c t o r i l y i n a number of examples, both w i t h a r t i f i c i a l l y g e n e r a t e d data and w i t h a c t u a l o b s e r v a t i o n s from Grass V a l l e y , Nevada, as documented i n I I - 5 and Chapter IV r e s p e c t i v e l y . In every case the program converged a u t o m a t i c a l l y t o an a c c e p t a b l e model i n a moderate number of i t e r a t i o n s , w i t h o u t o p e r a t o r i n t e r v e n t i o n . A u n i f o r m h a l f - s p a c e was always chosen as the s t a r t i n g model, t h i s b e i n g a r g u a b l y the most " o b j e c t i v e " c h o i c e p o s s i b l e ; the a v a i l a b i l i t y of a b e t t e r s t a r t i n g model would n e c e s s a r i l y reduce the number of i t e r a t i o n s r e q u i r e d . I t i s i n the c o n t e x t of a p p r a i s a l t h a t HLEM.INV most o b v i o u s l y r e p r e s e n t s an advance, s i n c e no p r e - e x i s t i n g i n t e r p r e t a t i o n a l a i d f o r HLEM soundings has any c a p a c i t y t o d i s t i l l from the d a t a those f e a t u r e s of the c o n d u c t i v i t y which are common t o a l l a c c e p t a b l e models. The m a i n s t a y of HLEM.INV a p p r a i s a l i s the e l e g a n t a v e r a g i n g of Backus and G i l b e r t (1968, 1970). The averages a r e d e f i n e d as i n n e r p r o d u c t s of c e r t a i n " r e s o l u t i o n windows" ( a v e r a g i n g f u n c t i o n s ) w i t h any a c c e p t a b l e model; t h e r e b e i n g d i f f e r e n t windows f o r d i f f e r e n t d e p t h s . In a l i n e a r i n v e r s e problem the averages a r e common t o a l l a c c e p t a b l e 213 models, but i n a n o n - l i n e a r problem (such as t h a t posed by HLEM data) the u n i v e r s a l i t y of the averages i s c o n d i t i o n a l on the degree of s i m i l a r i t y of a l l a c c e p t a b l e models. Thus, b e f o r e b a s i n g c o n c l u s i o n s on B a c k u s - G i l b e r t a v e r a g e s , i t i s n e c e s s a r y to e s t a b l i s h (as f a r as t h i s i s p o s s i b l e ) t h a t a l l a c c e p t a b l e models do. i n f a c t g i v e r i s e t o the same a v e r a g e s , a t l e a s t over c e r t a i n r e s t r i c t e d ranges of depth. In p a r t i c u l a r , the c r e d i b i l i t y of averages as a r e f l e c t i o n of the t r u e s t r u c t u r e i s enhanced i f i t can be shown t h a t a l l a c c e p t a b l e models become more and more a l i k e i n the l i m i t as the number of ( a c c u r a t e ) o b s e r v a t i o n s becomes i n f i n i t e . H e r e i n l i e s the p r a c t i c a l s i g n i f i c a n c e of the uniqueness theorem p r e s e n t e d i n Appendix C. I t i s proved t h e r e t h a t an u n l i m i t e d q u a n t i t y of p e r f e c t l y a c c u r a t e HLEM frequency soundings s u f f i c e s t o u n i q u e l y determine the c o n d u c t i v i t y as a f u n c t i o n of dep t h . T h i s p r o o f (thought t o be a new r e s u l t ) i n c r e a s e s the l i k e l i h o o d t h a t a l l a c c e p t a b l e models w i l l resemble one a n o t h e r . A l s o , by c o r o l l a r y , the e x i s t e n c e of d i s s i m i l a r a c c e p t a b l e models may be c o n s i d e r e d d i a g n o s t i c of a d e f i c i e n c y of the d a t a , e i t h e r i n q u a n t i t y or q u a l i t y . A l t h o u g h the theorem i n s t i l l s a c e r t a i n c o n f i d e n c e t h a t o a v e r a g i n g w i l l succeed i n d e l i n e a t i n g the c h a r a c t e r i s t i c s common to a l l a c c e p t a b l e models, the o n l y way t o a s s e s s u n i v e r s a l i t y of averages i n p r a c t i c e i s by d i r e c t comparison of the averages f o r a number of a c c e p t a b l e models, u s i n g the same a v e r a g i n g f u n c t i o n s f o r each ( c f . Oldenburg, 1979). Such a comparison ( f o r a l i m i t e d number of models) can be performed by HLEM.INV a p p r a i s a l s u b r o u t i n e s as a matter of c o u r s e . In an e f f o r t t o 214 f i n d a c c e p t a b l e models which a r e d i s s i m i l a r t o a g i v e n a c c e p t a b l e model and hence to t e s t u n i v e r s a l i t y as s e v e r e l y as p o s s i b l e , a s e r i e s of a c c e p t a b l e models i s c o n s t r u c t e d by ma x i m i s i n g and m i n i m i s i n g the c o n d u c i t v i t y of each l a y e r i n t u r n u s i n g l i n e a r programming. T h i s i s b e l i e v e d t o be the f i r s t such a p p l i c a t i o n of l i n e a r programming t o a n o n - l i n e a r i n v e r s e problem. The v a r i a b i l i t y of a c c e p t a b l e models i s most pronounced when the l a y e r s a r e " r e s o l v a b l y t h i c k " , whereas the o r i g i n a l c o n s t r u c t e d model i s u s u a l l y c o m p r i s e d of r e l a t i v e l y t h i n l a y e r s . T h e r e f o r e i t i s advantageous t o d e f i n e the e x t r e m a l models on a c o a r s e r depth p a r t i t i o n ; a c c o r d i n g l y , they a r e c o n s t r u c t e d by i t e r a t i v e improvement of a " b a s e l i n e model", t h i s b e i n g an a c c e p t a b l e model on the m o d i f i e d p a r t i t i o n . A f u l l HLEM.INV a p p r a i s a l has been performed i n a number of examples w i t h both t h e o r e t i c a l and r e a l d a t a , i . e . averages f o r the a p p r o p r i a t e e x t r e m a l models were computed a t a range of depths and were compared u s i n g the b a s e l i n e model a v e r a g e s as a r e f e r e n c e . In every case the averages were u n i v e r s a l i n c o n d u c t i v e zones, i m p l y i n g t h a t a l l the c o n s t r u c t e d models e x h i b i t l o c a l s i m i l a r i t y near good c o n d u c t o r s . P h y s i c a l l y t h i s i s c o n s i s t e n t w i t h the f a c t t h a t o b s e r v a t i o n s of magnetic f i e l d a r e most s e n s i t i v e t o zones of h i g h c u r r e n t d e n s i t y ; g e o p h y s i c a l l y , the d e l i n e a t i o n of c o n d u c t i v e zones i s the p r i n c i p a l aim of HLEM s u r v e y s , and i t has been demonstrated i n the examples t h a t averages a r e a b l e t o p r o v i d e i n f o r m a t i o n c o m p a t i b l e w i t h t h i s aim. 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S o c , 46, p p l l - 2 2 . 221 Appendix A INDUCED CURRENT DISTRIBUTION FOR A PARTICULAR CONDUCTIVITY In o r d e r t o g a i n i n s i g h t i n t o the manner i n which an o s c i l l a t i n g l o o p c u r r e n t e x c i t e s the ground, d i s t r i b u t i o n s of i nduced c u r r e n t w i t h i n a p a r t i c u l a r l a y e r e d c o n d u c t i v i t y a r e d e p i c t e d i n F i g . A . l . The i n t e n s i t y of induced c u r r e n t ( s t r i c t l y l n | j | ) a t each of f o u r f r e q u e n c i e s (0.1, 1, 10, and 100 Hz) was c o l o u r coded and t r a n s f o r m e d i n t o a COMTAL image which was s u b s e q u e n t l y photographed (Condal et a l . , 1979). The images r e p r e s e n t v e r t i c a l s e c t i o n s of the ground. The l e f t hand edge of each image c o i n c i d e s w i t h the a x i s of the c o i l which, f o r a l a y e r e d e a r t h , i s an a x i s of symmetry f o r the problem. The w h i t e dot i n the upper l e f t - h a n d (LH) c o r n e r i n d i c a t e s where the c o i l ( r a d i u s 50m) c u t s the s e c t i o n . In t o t o , an a r e a 2.32 km square i s r e p r e s e n t e d . The a n n o t a t i o n "GV.STD" i s an a b b r e v i a t i o n f o r "Grass V a l l e y s t a n d a r d " because the c o n d u c t i v i t y s t r u c t u r e used f o r the i l l u s t r a t i o n ( F i g . A.2) bears a q u a l i t a t i v e resemblance t o the s t r u c t u r e thought t o u n d e r l y Grass V a l l e y , Nevada (e.g. F i g . 4.2d). Hence, the d i s t r i b u t i o n of c u r r e n t s d e p i c t e d i s p r o b a b l y not u n l i k e t h a t induced beneath Grass V a l l e y by the group from B e r k e l e y when g e n e r a t i n g the d a t a i n v e r t e d i n Chapter IV. E x a m i n a t i o n of the images i n o r d e r of i n c r e a s i n g f r e q u e n c y r e v e a l s t h a t the c u r r e n t s become p r o g r e s s i v e l y more l o c a l i s e d 222 F i g u r e A . l COMTAL Images of Induced C u r r e n t D i s t r i b u t i o n s C o l o u r - c o d e d induced c u r r e n t i n t e n s i t y f o r the c o n d u c t i v i t y d e p i c t e d i n F i g . A.2. The c o l o u r grades from a d u l l r e d , t h rough shades of mauve t o n e a r - w h i t e as the magnitude of l n | J ( w ) | i n c r e a s e s . (a) C u r r e n t d i s t r i b u t i o n a t 0.1 Hz. I n t e n s i t i e s are low, but p e n e t r a t i o n i s good. (b) C u r r e n t d i s t r i b u t i o n a t 1 Hz. (c) C u r r e n t d i s t r i b u t i o n a t 10 Hz. (d) C u r r e n t d i s t r i b u t i o n a t 100 Hz. I n t e n s i t i e s are h i g h near the c o i l , but p e n e t r a t i o n i s poor. 2 2 3 F i g . A . l a 224 F i g . A . l b 225 F i g . A . l c 226 F i g . A . I d 227 GV.STD conductivity 0 . 6 ^ 0 . 4 in b 0 . 2 h 0 . 0 100 200 depth (m) (x 101) Fig. A.2 228 near the c o i l a t h i g h e r f r e q u e n c i e s . T h i s i s c o n s i s t e n t w i t h the reduced depth of p e n e t r a t i o n e x p e c t e d a t those f r e q u e n c i e s . In a d d i t i o n i t may be observed t h a t the maximum i n t e n s i t y of induced c u r r e n t i s g r e a t e s t i n F i g . A . I d , as w i t n e s s e d by the n e a r - w h i t e tones near the c o i l . T h i s i s not unexpected i n view of the dependence of induced v o l t a g e on the time d e r i v a t i v e of magnetic i n d u c t i o n i n Faraday's Law. The b a s i c f e a t u r e s of the c o n d u c t i v i t y can be deduced by c o n s i d e r a t i o n of any one of the images because both i n t e n s i t y and l a t e r a l e x t e n t of induced c u r r e n t are enhanced by l a r g e c o n d u c t i v i t i e s . For example, the narrow f i n g e r of r e l a t i v e l y h i g h c u r r e n t d e n s i t y v i s i b l e a t the upper edge of each image i s i n d i c a t i v e of a t h i n c o n d u c t i v e l a y e r a t the s u r f a c e ; t h i s i n t e r p r e t a t i o n i s borne out i n F i g . A.2. 229 Appendix B EVALUATION OF INTEGRALS FOR THE DETERMINATION OF Ej_ HR^ HZ^ AND THE DISCRETE FRECHET KERNELS B - l P r i m a r y E l e c t r i c F i e l d I n t e g r a l As per ( 1 . 2 8 ) , d e t e r m i n a t i o n of the p r i m a r y e l e c t r i c f i e l d n e c e s s i t a t e s e v a l u a t i o n of the Legendre f u n c t i o n of the second k i n d , o r d e r 1/2. From Gradshteyn and Ry z h i k (1965, p l 0 1 6 ) , R n + n n ,/z)s~n'( Q(s) = —: : F[ (n + 2 ) / 2 , (n + l ) / 2 ; (2n + 3 ) / 2 , s " 2 ] ( B . l ) 2 T ( n + | ) Hence, s u b s t i t u t i n g from ( B . l ) i n t o ( 1 . 2 8 ) , the d i m e n s i o n l e s s p r i m a r y e l e c t r i c f i e l d i s g i v e n by Eo(r,z) = - i / a F ( 5/4 , 3/4 ; 2 ,1/s 2 ) /^2 7 r s 3 (B.2) where s = [ ( z + h ) 2 + a 2 + r 2 ] / 2 a r The h y p e r g e o m e t r i c f u n c t i o n , F ( u , v ; x , y ) , i s amenable t o r a p i d n u m e r i c a l e v a l u a t i o n as a s e r i e s (Gradshteyn and R y z h i k , pl039) . 230 B-2 Pr imary R a d i a l Magnet i c F i e l d I n t e g r a l The r a d i a l magnetic f i e l d may be r e g a r d e d as the sum of p r i m a r y and secondary f i e l d s : The i n t e g r a l f o r the secondary f i e l d , Hr, can be computed u s i n g the d i g i t a l l i n e a r f i l t e r i n g of Koefoed et a l . (1972) because i t s i n t e g r a n d decays t o z e r o f o r both s m a l l and l a r g e \. The p r i m a r y f i e l d , Hr^ , may be e v a l u a t e d a n a l y t i c a l l y i f d i s p l a c e m e n t c u r r e n t s i n a i r are d i s r e g a r d e d . S p e c i f i c a l l y , u s i n g (1.11) and ( 1 . 2 8 ) , where the prime denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o s. Note t h a t Hr t f(r,-h) i s z e r o , c o n s i s t e n t w i t h the p h y s i c a l r equirement t h a t t h e r e be no r a d i a l magnetic f i e l d i n the p l a n e of a c u r r e n t l o o p i n a i r . A c c o r d i n g to Gradshteyn & R y z h i k (1965, p l 0 1 9 ) , Hr = Hr„ + AHr (B.3) (B.4) Q^(s) = ( n + l ) [ Q n + , ( s ) - s Q n ( s ) ] / ( s 2 - l ) (B.5) Hence, r e c a l l i n g ( B . l ) , 37t [ 3 s - 2 F ( 7 / 4 , 5 / 4 ; 3 , s " 2 ) / 8 - F ( 5 / 4 , 3 / 4 ; 2 , s " 2 ) ] 8 v / 2 s ( s 2 - l ) (B.6) 231 The p r i m a r y r a d i a l magnetic f i e l d may be e v a l u a t e d by s u b s t i t u t i n g (B.6) i n ( B . 4 ) . B-3 P r i m a r y V e r t i c a l Magnet i c F i e l d I n t e g r a l P r o c e e d i n g as f o r the r a d i a l f i e l d , the net v e r t i c a l magnetic f i e l d i s decomposed i n t o p r i m a r y and secondary components: Hz = Hz0 + AHz The i n t e g r a l f o r the secondary component, A H z , i s s u i t a b l e f o r e v a l u a t i o n by d i g i t a l l i n e a r f i l t e r i n g . I f d i s p l a c e m e n t c u r r e n t s i n a i r are n e g l e c t e d , the p r i m a r y v e r t i c a l magnetic f i e l d , Hz 0 , may be d e t e r m i n e d a n a l y t i c a l l y by d i f f e r e n t i a t i o n of (1.28) i n accordance w i t h ( 1 . 1 2 ) . Hence, i d [aQ ( /(s) + ( r 2 - ( z + h ) 2 - a 2 ) Q M s ) / r ] Hz a(r,z) = ( r E ^ = - - (B.7) r d r QnJ a r 3 T h i s e x p r e s s i o n may be e v a l u a t e d u s i n g ( B . l ) and ( B . 6 ) . 232 B-4 I n t e g r a l s f o r D i s c r e t e F r e c h e t K e r n e l s i n the F i r s t L ayer From ( 2 . 1 6 ) , ( 2 . 1 3 ) , and (2.20) i t may be o b s e r v e d t h a t computation of the d i s c r e t e F r e c h e t k e r n e l s i n the f i r s t l a y e r i n v o l v e s d e t e r m i n a t i o n of the f o l l o w i n g i n t e g r a l s : oo I r = j*E( A ,0,w)E' (A ,0,w) A J,( A r)dA/(u, 2B*) (B.8) o oo Iz = | E (A , 0 , W ) E ' ( A , 0 , W ) A 2 A r ) d A / ( u 0 u 2 B * ) (B.9) o As above, these i n t e g r a l s a r e e v a l u a t e d by d e c o m p o s i t i o n i n t o two i n t e g r a l s , one s u i t a b l e f o r d i g i t a l f i l t e r i n g and the o t h e r a d m i t t i n g an a n a l y t i c a l f o r m u l a . In p a r t i c u l a r , an i n t e g r a l I i s w r i t t e n i n the form I = ( I - I , ) + I, where I, i s chosen so t h a t the i n t e g r a n d of I - I , decays t o z e r o f o r both s m a l l and l a r g e A• The " l a r g e - A i n t e g r a l " , I r, , c o r r e s p o n d i n g t o (B.8) i s o b t a i n e d by r e p l a c i n g E( A,0,w) and E'(A,0,w) w i t h Ejt \ ,0 ,w) and E^( A,0,w) u s i n g (1.73) and ( 1 . 7 4 ) . Hence, oo I r , = i a j" Jf( Aa) J(( A r ) d A / 2 A (B.10) o where i t s h o u l d be r e c a l l e d from ( 1 . 1 9 ) , ( 1 . 3 6 ) , and (1.70) t h a t 233 u, — » u0 —> A B* -iaJ ((A a ) e " X h / 2 A as A — » ~> ( B . l l ) I f h=0, Ir f f may be e v a l u a t e d u s i n g a f o r m u l a p r o v i d e d by Watson (1945, p405): I r = i a 2 / ( 4 r ) (B.12) To ensure t h a t the i n t e g r a n d of I z - I z ^ approaches z e r o as A — > 0 0 , the i n t e g r a n d of Iz0 may be chosen as the l a r g e - A l i m i t of the i n t e g r a n d of I z ; hence, from ( B . 9 ) , Iz0 = i a j" J,( A a) J0( A r)dA /2\ (B.13) o T h i s i s a s p e c i a l case of f o r m u l a #6.574 on page 692 of Gradshteyn and R y z h i k , whence lz0 = i a 2 F ( l / 2 , l / 2 ; 2 ; a 2 / r 2 ) / ( 4 r ) (B.14) The i n t e g r a n d i n (B.13) approaches aJ < >(Ar)/2 as A —> 0, whereas the i n t e g r a n d i n (B.9) v a n i s h e s a t A=0. The i n t e g r a n d of I z - I z ^ t h e r e f o r e does not decay t o z e r o as A —> 0. From a c o m p u t a t i o n a l v i e w p o i n t , t h i s i s an u n d e s i r a b l e s t a t e of a f f a i r s because the d i g i t a l f i l t e r i n g i s not w e l l s u i t e d t o the e v a l u a t i o n of i n t e g r a l s w i t h i n t e g r a n d s which do not d e c r e a s e t o z e r o f o r both l a r g e and s m a l l A. To e l i m i n a t e t h i s d i f f i c u l t y , a m o d i f i e d l i m i t i n g i n t e g r a l , I z ^ , i s employed, where 234 oo Izl = i a f [ J( A a) J( A r ) / A - aJ.( A r ) e " X a ]d A /2 (B.15) o The i n t e g r a n d of I z - I z , i s v a n i s h i n g l y s m a l l both when A << 1 and when A » 1, as r e q u i r e d f o r a c c u r a t e n u m e r i c a l i n t e g r a t i o n . An a n a l y t i c e x p r e s s i o n f o r I z ^ may be c o n s t r u c t e d from (B.14) and the f o l l o w i n g f o r m u l a (Gradshteyn and R y z h i k , 1965, p707): oo j e ~ A a Jo( A r )dA = 1/Ya 2 + r 2 (B.16) 235 Appendix C UNIQUENESS OF INVERSION OF HLEM FREQUENCY SOUNDINGS A fundamental c o n s i d e r a t i o n i n i n v e r s i o n i s the uniqueness of any a c c e p t a b l e model. In p r a c t i c e , many sources of non-uniqueness i n the p h y s i c a l experiment a r e d i s r e g a r d e d 'a p r i o r i ' when s i m p l i f y i n g assumptions a re made i n the f o r m u l a t i o n of the m a t h e m a t i c a l i n v e r s e problem; i n the HLEM problem, f o r example, the assumptions t h a t the c o n d u c t i v i t y i s i s o t r o p i c and a f u n c t i o n of depth o n l y remove from c o n s i d e r a t i o n an i n f i n i t e number of more c o m p l i c a t e d s t r u c t u r e s which c o u l d s a t i s f y the o b s e r v a t i o n s . Given the t h e o r e t i c a l r e s t r i c t i o n s on the form of the c o n d u c t i v i t y , the o n l y s o u r c e s of non-uniqueness c o n s i d e r e d i n HLEM.INV a p p r a i s a l a re the i n a c c u r a c y and i n s u f f i c i e n c y of the d a t a . In t h i s appendix the u l t i m a t e " d i s c r i m i n a t o r y p o t e n t i a l " of HLEM frequency soundings i s examined when the s e p r i n c i p a l causes of non-uniqueness a r e removed, i . e . when an u n l i m i t e d q u a n t i t y of p e r f e c t l y a c c u r a t e d a t a i s a v a i l a b l e . The problem t o be a d d r e s s e d may be s t a t e d e x p l i c i t l y as f o l l o w s : i f an i s o t r o p i c c o n d u c t i v i t y cr(z) has been found s a t i s f y i n g HLEM soundings a t a l l f r e q u e n c i e s , i s i t the o n l y such c o n d u c t i v i t y which r e p r o d u c e s the o b s e r v a t i o n s ? In the t h e o r e t i c a l d i s c u s s i o n s which f o l l o w , the c o n d u c t i v i t y i s assumed t o be f i n i t e everywhere. The problem would be t r i v i a l o t h e r w i s e because a zone of i n f i n i t e c o n d u c t i v i t y c o m p l e t e l y masks the s t r u c t u r e beneath i t ; the 236 c o n d u c t i v i t y beneath such a zone may be chosen a r b i t r a r i l y w i t h o u t a f f e c t i n g the responses a t the s u r f a c e , and the same o b s e r v a t i o n s would t h e r e f o r e admit an i n f i n i t e number of c o n d u c t i v i t y s t r u c t u r e s . Very n e a r l y t h i s s i t u a t i o n can a r i s e i n the f i e l d i n the presence of a g r a p h i t e h o r i z o n , f o r example; EM i n d u c t i o n methods are c o m p l e t e l y impotent i f such a zone o c c u r s c l o s e t o the s u r f a c e . The c o n d u c t i v i t y i s a l s o assumed t o be a n a l y t i c ( i n f i n i t e l y d i f f e r e n t i a b l e ) . T h i s may not be a s e r i o u s r e s t r i c t i o n i n p r a c t i c e because i t i s p o s s i b l e t o approximate any p i e c e - w i s e c o n t i n u o u s f u n c t i o n a r b i t r a r i l y c l o s e l y w i t h an a n a l y t i c f u n c t i o n , e.g. F o u r i e r S e r i e s . F r e c h e t d i f f e r e n t i a b i l t y of the HLEM problem e n s u r e s t h a t i t i s always p o s s i b l e t o f i n d a s u f f i c i e n t l y " s m a l l " change i n c o n d u c t i v i t y so t h a t the r e s u l t i n g changes i n the responses a r e s m a l l e r i n magnitude than a chosen l e v e l , however s m a l l . Uniqueness theorems have been e s t a b l i s h e d f o r c l o s e l y r e l a t e d problems, but these p r e v i o u s r e s u l t s do not bear d i r e c t l y on the uniqueness of a s t r u c t u r e s a t i s f y i n g HLEM d a t a , f o r reasons which w i l l emerge below. With the e x c e p t i o n of t h a t due t o S l i c h t e r (1933), a l l p r e - e x i s t i n g uniqueness theorems of r e l e v a n c e p e r t a i n t o m a g n e t o t e l l u r i c (MT) and geomagnetic soundings. The p r o o f of uniqueness i s i n v a r i a b l y a " c o n s t r u c t i o n " p r o c e d u r e , i t b e i n g shown e i t h e r t h a t the c o e f f i c i e n t s of the M a c l a u r i n S e r i e s f o r c r ( z ) can be unambiguously d e t e r m i n e d from the d a t a , or t h a t the c o n d u c t i v i t y can be deduced from the unique s o l u t i o n of an i n t e g r a l e q u a t i o n w i t h a data-dependent k e r n e l . The p r o o f s of S l i c h t e r (1933), 237 S i e b e r t (1964), Chetaev (1966), and B a i l e y (1970) belong t o the f i r s t group, w h i l e the r e s u l t s of Johnson and S m y l i e (1970), and W e i d e l t (1972) f a l l i n t o the l a t t e r c a t a g o r y . C - l S l i c h t e r ' s Theorem: The f i r s t r e l a t e d uniqueness theorem i s due t o S l i c h t e r (1933). He has proved t h a t a knowledge of e i t h e r the v e r t i c a l or r a d i a l component of magnetic f i e l d a t a l l t r a n s m i t t e r - r e c e i v e r (T-R) s e p a r a t i o n s f o r a s i n g l e source f r e q u e n c y s u f f i c e s t o s p e c i f y the c o n d u c t i v i t y c o m p l e t e l y (assumed a f u n c t i o n of depth o n l y ) . G i v e n Hz(r,0,w) or Hr(r,0,w) a t a l l d i s t a n c e s from the s o u r c e , E ( A,0,w) can be computed by a s t r a i g h t f o r w a r d Hankel t r a n s f o r m a t i o n ; s p e c i f i c a l l y , from ( 1 . 1 4 ) , ( 1 . 6 6 ) , ( 1 . 6 7 ) , ( 1 . 6 4 ) , and (1.75) i °° E ( A,0,w) = — ^ A H r ( r ,0,w)rJ,( A r ) d r + E j A,0,w) ( C . l ) OO E ( A,0,w) = J*AHz(r ,0,w)rJ t f( A r ) d r + E j A,0 , w ) (C.2) o where A H r and AHz are the secondary magnetic f i e l d components. I f E ( A,0,w) i s known, Hr( A>0,w) can be de t e r m i n e d u s i n g (1.11) and ( 1 . 7 1 ) ; hence, Hr ( A,0,w) = -i[u „ E ( A,0,w) - 2u„B*] (C.3) where B* i s the e f f e c t i v e source spectrum, d e f i n e d i n ( 1 . 7 0 ) . I t 238 f o l l o w s from ( C . l - 3 ) t h a t a complete knowledge of e i t h e r Hr(r,0,w) or Hz(r,0,w) p e r m i t s s p e c i f i c a t i o n of the s u r f i c i a l i n p u t a d m i t t a n c e , Y 1, a t a l l A , where Hr( A,0,w) Yl(\,w;cr) = (C.4) E( A,0,w) The r e l e v a n c e of the a d m i t t a n c e i n t h i s c o n t e x t d e r i v e s from the f a c t t h a t a t l a r g e A i t may be expanded as an a s y m p t o t i c s e r i e s : oo Y'( A,w;<r) ~ Za„(w;<r) X " " as A — ( C . 5 ) n-o The c o e f f i c i e n t s {a} can be de t e r m i n e d from the known v a l u e s of Y l ( A r W ) . A d a p t i n g the method used by Langer (1933) t o prove uniqueness f o r the DC r e s i t i v i t y problem, S l i c h t e r shows t h a t i t i s p o s s i b l e t o determine the M a c l a u r i n S e r i e s c o e f f i c i e n t s of the c o n d u c t i v i t y from the s e t {a}. I t may be sur m i s e d from (1.56-58) t h a t a t t e n u a t i o n becomes more pronounced f o r any cr(z) as A i n c r e a s e s . On t h i s b a s i s , s u c c e s s i v e terms i n (C.5) r e p r e s e n t the c o n t r i b u t i o n s from g r e a t e r and g r e a t e r depths because fewer and fewer terms a r e r e q u i r e d f o r a g i v e n a c c u r a c y as A i n c r e a s e s (Bender and Or s z a g , 1978, p l l 9 ) . T h i s i s r e f l e c t e d i n the dependence of {a} on the d e r i v a t i v e s of o~ a t the s u r f a c e : an i s dependent o n l y on d e r i v a t i v e s up t o o r d e r n - l ( S l i c h t e r , 1933, p418). A f a m i l i a r maxim i n EM sounding i s t h a t depth of p e n e t r a t i o n i n c r e a s e s w i t h i n c r e a s i n g T-R s e p a r a t i o n or d e c r e a s i n g i n p u t f r e q u e n c y . I t i s te m p t i n g t o t r y t o e x p r e s s 239 t h i s c o r r e s p o n d e n c e between the i n f l u e n c e s of R and w more e x a c t l y i n an attempt t o e x t e n d S l i c h t e r ' s p r o o f t o frequency s o u n d i n g s ; i t would be a s i m p l e matter t o prove uniqueness f o r the p a r a m e t r i c case i f soundings at f i x e d T-R s e p a r a t i o n and d i f f e r e n t f r e q u e n c i e s were e x a c t l y e q u i v a l e n t t o soundings at d i f f e r e n t T-R s e p a r a t i o n s f o r a f i x e d f r e q u e n c y . Of p o s s i b l e r e l e v a n c e i n t h i s quest i s the o b s e r v a t i o n t h a t the g o v e r n i n g DE can be w r i t t e n w i t h no e x p l i c i t w-dependence i f d i s p l a c e m n e n t c u r r e n t s are n e g l e c t e d . T h i s may be demonstrated by u s i n g 1// w/<.cr( 0) as the l e n g t h s c a l e i n s t e a d of the T-R s e p a r a t i o n , R, when d e f i n i n g the d i m e n s i o n l e s s form of the PDE ( 1 . 4 ) . In t h i s case the r e s u l t i n g d i m e n s i o n l e s s form of the PDE f o r E(r,z,w) i n the ground i s g i v e n by V 2 E - E / r 2 - i c ( z ) E = 0 (C.6) where (C.7) E = E/w/*l T a k i n g the Hankel t r a n s f o r m of ( C . 6 ) , as per (1.14), the z-dependence of E ( A , z , w ) i n the ground i s d e s c r i b e d by the ODE E" = ( \ 2 + icr )E (C.8) where primes denote d i f f e r e n t i a t i o n w i t h r e s p e c t t o z. 240 At f i r s t g l a n c e i t would appear t h a t (C.8) need o n l y be s o l v e d once, a s i n g l e s o l u t i o n a p p l y i n g a t a l l f r e q u e n c i e s . By the same t o k e n , the d i m e n s i o n l e s s T-R s e p a r a t i o n , fi = Rv/w/c«,^( 0) , a p p a r e n t l y p r o v i d e s a l i n k between ge o m e t r i c and p a r a m e t r i c s o u n d i n g , a l l p o s s i b l e s e p a r a t i o n s b e i n g sampled as w v a r i e s from z e r o t o i n f i n i t y . However, because the l e n g t h s c a l e i s f r e q u e n c y dependent, the d i m e n s i o n l e s s c o n d u c t i v i t y , c ( z ) , i s "compressed" or " s t r e t c h e d " w i t h r e s p e c t t o d i m e n s i o n l e s s depth as the f r e q u e n c y i s v a r i e d , w i t h the r e s u l t t h a t the s o l u t i o n s a t d i f f e r e n t f r e q u e n c i e s c o r r e s p o n d t o d i f f e r e n t s t r u c t u r e s . Thus the v a l u e of C T " ( 1 ) , f o r example, w i l l v a r y w i t h f r e q u e n c y . The o n l y s t r u c t u r e i n v a r i a n t w i t h r e s p e c t t o t h i s c h o i c e of s c a l i n g i s the u n i f o r m h a l f - s p a c e . For t h i s case a l o n e i s the e q u i v a l e n c e of p a r a m e t r i c and g e o m e t r i c soundings e s t a b l i s h e d . I t f o l l o w s i m m e d i a t e l y from S l i c h t e r ' s theorem t h e r e f o r e t h a t the c o n d u c t i v i t y of a u n i f o r m l y c o n d u c t i n g h a l f - s p a c e can be d e t e r m i n e d u n i q u e l y from f r e q u e n c y s o u n d i n g s . The l a c k of a d i r e c t m a t h e m a t i c a l c orrespondence between p a r a m e t r i c and g e o m e t r i c sounding i s u n d e r l a i n by a p h y s i c a l asymmetry between the r o l e s of the two p a r a m e t e r s , R and w. The T-R s e p a r a t i o n i s p u r e l y p a s s i v e , i n the sense t h a t a change i n R (w f i x e d ) does not a f f e c t the f i e l d s i n the ground; t o t a k e measurements at a l l s e p a r a t i o n s f o r one f r e q u e n c y i s t h e r e f o r e t o r e c o r d the s u r f a c e e x p r e s s i o n of a s i n g l e c u r r e n t d i s t r i b u t i o n . M a t h e m a t i c a l l y , a change i n R s i m p l y a l t e r s the argument of the Hankel t r a n s f o r m k e r n e l i n ( 1 . 6 8 ) , the response f u n c t i o n K b e i n g u n a f f e c t e d . V a r i a t i o n i n w, on the o t h e r hand, m o d i f i e s the d i s t r i b u t i o n of i n d u c e d c u r r e n t s , and t h e r e f o r e 241 a l t e r s the e a r t h response f u n c t i o n . P a r a m e t r i c sounding i s p h y s i c a l l y more c o m p l i c a t e d i n t h i s sense. C-2 B a i l e y ' s Theorem The second p r e - e x i s t i n g uniqueness theorem i s due t o B a i l e y (1970). E x p l o i t i n g the f a c t t h a t the magnetic f i e l d s can be decomposed i n t o i n t e r n a l and e x t e r n a l components a c c o r d i n g t o the l o c a t i o n of the c u r r e n t s c a u s i n g them, B a i l e y e x p r e s s e s the i n t e r n a l f i e l d s , $ , as the c o n v o l u t i o n of an impulse response f u n c t i o n , X , w i t h the e x t e r n a l f i e l d s , S : oo 4 ( A , z , t ) = f"X( A ,z,T ) £ ( A , z , t - r )dT (C.9) — oo In the f r e q u e n c y domain t h i s c o n v o l u t i o n may be e x p r e s s e d as a p r o d u c t : ff ( A,z,w) = X ( A ,z,w) £ ( A ,z,w) The F o u r i e r spectrum of the impulse response f u n c t i o n can t h e r e f o r e be d e t e r m i n e d from the e x p r e s s i o n s (1.33) and (1.34) f o r ^ ( A,z,w) and £ ( A , z , w ) . S i n c e the i n t e r n a l f i e l d s a r e induced by the e x t e r n a l f i e l d s , % ( T ) must be z e r o f o r n e g a t i v e X . T h i s c a u s a l i t y c o n d i t i o n on the response f u n c t i o n f u r n i s h e s a s u f f i c i e n t l y s t r o n g c o n s t r a i n t f o r B a i l e y t o be a b l e t o show t h a t soundings of one p a r t i c u l a r mode at a l l f r e q u e n c i e s c o m p l e t e l y determine the c o n d u c t i v i t y as a f u n c t i o n of d e p t h . The uniqueness of a 242 M a c l a u r i n S e r i e s e x p a n s i o n i s in v o k e d t o complete the p r o o f , the ex p a n s i o n c o e f f i c i e n t s b e i n g known f u n c t i o n s of the d a t a . P e r t a i n i n g as i t does t o frequency s o u n d i n g s , B a i l e y ' s theorem would appear t o be of more immediate r e l e v a n c e than S l i c h t e r ' s theorem t o the HLEM problem. However, an i m p l i c i t p r e - c o n d i t i o n i n B a i l e y ' s theorem i s a knowledge of the a m p l i t u d e s of p a r t i c u l a r modes. T h i s c o n d i t i o n can be met i n geomagneic depth sounding because the a m p l i t u d e s of i n d i v i d u a l s p h e r i c a l harmonic modes a r e e s t i m a t e d by a n a l y s i s of the o b s e r v a t i o n s from d i f f e r e n t s t a t i o n s . By a n a l o g y , t h e r e f o r e , i f i t were i n t e n d e d t o a p p l y B a i l e y ' s theorem t o HLEM soun d i n g s , a " h y b r i d " d a t a s e t would be r e q u i r e d c o n s i s t i n g of a r e c o r d of the magnetic f i e l d s over a l l f r e q u e n c i e s a t many ( s t r i c t l y a l l ) r e c e i v e r l o c a t i o n s . For ex a c t d e t e r m i n a t i o n of the a m p l i t u d e s of i n d i v i d u a l modes, i t would be n e c e s s a r y t o p e r f o r m Hankel t r a n s f o r m a t i o n of the o b s e r v a t i o n s , as per ( C . l ) and ( C . 2 ) . In p a r a m e t r i c sounding, when the magnetic f i e l d s a r e known a t o n l y one T-R s e p a r a t i o n f o r many f r e q u e n c i e s , the Hankel t r a n s f o r m cannot be performed u n l e s s i t i s p o s s i b l e t o r e l a t e changes i n fr e q u e n c y t o changes i n s e p a r a t i o n i n a p r e c i s e way. However, as d e s c r i b e d above, an e x a c t c o r r e s p o n d e n c e between g e o m e t r i c and p a r a m e t r i c sounding can be e s t a b l i s h e d o n l y f o r the t r i v i a l case of a u n i f o r m h a l f - s p a c e . From a d i f f e r e n t p e r s p e c t i v e , the f a c t t h a t an i n f i n i t e range of modes i s e x c i t e d i n HLEM s u r v e y s (as per (1.23)) can be a t t r i b u t e d t o the l o c a l i s e d s o u r c e geometry; a more r e s t r i c t e d s u i t e of modes would be e x c i t e d by a more e x t e n s i v e s o u r c e , e.g. s e v e r a l c o n c e n t r i c c u r r e n t l o o p s a c t i v a t e d s i m u l t a n e o u s l y . 243 However, by r e c i p r o c i t y , the soundings o b t a i n e d w i t h N c o n c e n t r i c c u r r e n t l o o p s can be repro d u c e d by a s u i t a b l e l i n e a r c o m b i n a t i o n of s i n g l e - l o o p soundings taken a t N d i f f e r e n t T-R s e p a r a t i o n s . The i n t e r c h a n g e a b i l i t y of source and r e c e i v e r i s e v i d e n t from the form of the i n t e g r a n d i n (1 . 7 8 ) : the o b s e r v a b l e , E ( r , 0 , w ) , i s u n a f f e c t e d by a r e v e r s a l of the r o l e s of "a" and " r " . C o n s e q u e n t l y , t o measure the combined response of N c o n c e n t r i c c u r r e n t l o o p s w i t h r a d i i a, , a 2 , , a N a t a s i n g l e s t a t i o n , d i s t a n c e R from the t r a n s m i t t e r , i s e n t i r e l y e q u i v a l e n t t o a r i t h m e t i c a l l y c o m b ining s e p a r a t e r e a d i n g s a t s e p a r a t i o n s a, , a 2 , , a N f o r a source l o o p w i t h r a d i u s R. T h e r e f o r e , use of an extended source i n o r d e r t o l i m i t the range of modes i s p h y s i c a l l y e q u i v a l e n t t o p e r f o r m i n g geometric s o u n d i n g s . Hence, t h e r e i s a immutable i n t e r - r e l a t i o n s h i p between i d e n t i f i c a t i o n of i n d i v i d u a l modes and a v a i l a b i l i t y of dat a a t d i f f e r e n t p o s i t i o n s . C-3 G e l ' f a n d - L e v i t a n Methods: A d a p t i n g a g e n e r a l method f o r i n v e r s i o n of S t u r m - L i o u v i l i e problems ( G e l ' f a n d and L e v i t a n , 1955), Johnson and S m y l i e (1970) and W e i d e l t (1972) have d e v i s e d methods f o r the d i r e c t i n v e r s i o n of geomagnetic and MT d a t a . I f p e r f e c t l y a c c u r a t e d a t a were a v a i l a b l e a t a l l f r e q u e n c i e s , the c o n d u c t i v i t y c o n s t r u c t e d u s i n g one of the s e methods would be unique; i n t h i s i n s t a n c e the uniqueness f o l l o w s from the f a c t t h a t c e r t a i n V o l t e r r a i n t e g r a l e q u a t i o n s admit one and o n l y one s o l u t i o n ( W e i d e l t , 1972, p277; Johnson and S m y l i e , 1970, p45) I t has not been p o s s i b l e t o 244 modify the c i t e d work i n any s i m p l e way so t h a t i t bears d i r e c t l y on the uniqueness of HLEM i n v e r s i o n . At r o o t , the d i f f i c u l t y i s the same as t h a t d i s c u s s e d above i n the c o n t e x t of B a i l e y ' s theorem, namely t h a t the a n a l y s i s i s t a i l o r e d f o r i n d i v i d u a l modes and i t s r e l e v a n c e t o HLEM sounding i s s e v e r e l y l i m i t e d by the n e c e s s i t y f o r Hankel t r a n s f o r m a t i o n . That i s not t o say t h a t i t i s i m p o s s i b l e t o e s t a b l i s h uniqueness of HLEM i n v e r s i o n by a new a p p l i c a t i o n of G e l ' f a n d - L e v i t a n t h e o r y ; such an u n d e r t a k i n g w i l l not be attempted h e r e , however. C-4 A Uniqueness Theorem f o r HLEM Frequency Soundings In the t r a d i t i o n of S i e b e r t (1964) and Chetaev (1966), the theorem p r e s e n t e d here i s based on a WKBJ e x p a n s i o n f o r secondary magnetic f i e l d i n the l i m i t of h i g h f r e q u e n c y , i . e . uniqueness i s e s t a b l i s h e d by c o n s i d e r a t i o n of an a s y m p t o t i c e x p a n s i o n of the form OP A H m ( R , f l ) ~ ^ m r t / 3 n as /3 —> ~ (C.IO) where fi = PVW/*.^ 0 ) ( C . l l ) as per ( 1 . 8 ) , and where AH t f = A H z and AH, = A H r a r e the secondary v e r t i c a l and h o r i z o n t a l magnetic f i e l d s , d e f i n e d i n 1-6. Formulae w i l l be d e r i v e d below e x p r e s s i n g the c o e f f i c i e n t s { £*} i n terms of s u r f i c i a l d e r i v a t i v e s {s} of criz) , where s « = - - r t - | z = „ ( C ' 1 2 ) 245 I t i s found t h a t the d e r i v e d formulae are such as t o p e r m i t u n e q u i v o c a l d e t e r m i n a t i o n of t h e s e d e r i v a t i v e s when the n u m e r i c a l v a l u e s of { cj} are known e x a c t l y . The c o e f f i c i e n t s , { ^ } , can be d e termined from (C.10) i f p e r f e c t l y a c c u r a t e d a t a i s a v a i l a b l e over a c o n t i n u o u s range of h i g h f r e q u e n c i e s , i . e . i f Aum(R,p) i s known p r e c i s e l y f o r 1 << ft < < f*z < <*> . I f the c o n d u c t i v i t y i s an a n a l y t i c f u n c t i o n of d e p t h , a knowledge of the d e r i v a t i v e s i s tantamount t o a complete s p e c i f i c a t i o n of the c o n d u c t i v i t y v i a i t s M a c l a u r i n S e r i e s r e p r e s e n t a t i o n . The s t a r t i n g p o i n t f o r the p r o o f of uniqueness i s a s i m p l i f i e d form of ( 1 . 1 5 ) . The s i m p l i f i c a t i o n a r i s e s from the n e g l e c t of d i s p l a c e m e n t c u r r e n t s , t h e s e b e i n g n e g l i g i b l e i n most g e o l o g i c a l m a t e r i a l s a t t y p i c a l HLEM f r e q u e n c i e s . I n c l u s i o n of d i s p l a c e m e n t c u r r e n t s would, of c o u r s e , c o m p l i c a t e the mathematics somewhat, but i t i s a n t i c i p a t e d t h a t by f o l l o w i n g a p r o c e d u r e e n t i r e l y analogous t o t h a t adopted here i t would be p o s s i b l e t o prove t h a t both Biz) and c ( z ) are u n i q u e l y c h a r a c t e r i s e d by an u n l i m i t e d amount of p e r f e c t l y a c c u r a t e HLEM d a t a . T h i s e x p e c t a t i o n i s r e i n f o r c e d i n d i r e c t l y by the "redundancy" argument p r e s e n t e d i n 111-4, and by the f a c t t h a t S l i c h t e r ' s theorem a p p l i e s t o b oth c o n d u c t i v i t y and d i e l e c t r i c c o n s t a n t . For the p r e s e n t , however, c o n s i d e r the f o l l o w i n g s i m p l i f i e d DE f o r E (A,z,w) i n the ground ( s o u r c e - f r e e r e g i o n ) : E" = [A2 + iA2<r)E (C.13) where a l l q u a n t i t i e s ar 246 An a s y m p t o t i c expansion f o r E(A,0,w) i s sought, v a l i d i n the h i g h f r e q u e n c y l i m i t {fi» 1 ) . The r e s t r i c t i o n t o h i g h f r e q u e n c y soundings i s imposed f o r reasons of m a t h e m a t i c a l c o n v e n i e n c e , and s h o u l d not be c o n s t r u e d t o s i g n i f y t h a t low fre q u e n c y soundings a re i n h e r e n t l y i n f e r i o r i n s o f a r as they reduce the a m b i g u i t y of i n t e r p r e t a t i o n . Indeed i t may be p o s s i b l e t o prove the uniqueness of i n v e r s i o n of low frequency HLEM soundings u s i n g a l a r g e - A a s y m p t o t i c e x p a n s i o n l i k e t h a t d e v e l o p e d by O l v e r (1954, 1959). The work of O l v e r i s r e l e v a n t because i f A 2 >> fi2 i t i s of l i t t l e consequence whether A i s l a r g e or fi i s s m a l l i n s o f a r as i t a f f e c t s the form of a s e r i e s e x p a n s i o n f o r E i n (C.13). The p o s s i b i l i t y of d e v e l o p i n g a lo w - f r e q u e n c y e x p a n s i o n w i l l not be e x p l o r e d h e r e , however. For the l a r g e fi case i t i s c o n v e n i e n t i n i t i a l l y t o e x p r e s s E i n powers of a new parameter o(, d e f i n e d by oi = */X + ~P (C.14) Gi v e n the l a r g e magnitude of jZ , i t f o l l o w s t h a t oC » 1 f o r a l l v a l u e s of A, and hence t h a t the WKBJ e x p a n s i o n f o r E i n n e g a t i v e powers of oC w i l l be u n i f o r m l y v a l i d f o r a l l modes; t h i s i s i m p o r t a n t because an i n v e r s e Hankel t r a n s f o r m , c . f . (1 . 1 6 ) , must e v e n t u a l l y be a p p l i e d t o the ex p a n s i o n i n o r d e r t o r e l a t e i t t o measurable q u a n t i t i e s . D e f i n i n g a new f u n c t i o n q(A,z,w) by q = [X2 + i / 3 2 <7"(z) ]/<* 2 (C.15), 247 the g o v e r n i n g DE (C.13) may be r e - w r i t t e n i n the form: E" = <X2qE (C.16) P r o c e e d i n g as per S i e b e r t (1964), a s o l u t i o n i s sought i n the form z E = E e x p [ - 0C^^6v] (C.17) o where E(  A ,w) = E( \ ,0,w) must be det e r m i n e d from the boundary c o n d i t i o n a t the s u r f a c e . S u b s t i t u t i n g from (C.17) i n t o (C.16), " ocY2 + 0 ( q = 0 (C.18) A s e r i e s s o l u t i o n i s de v e l o p e d f o r "^f i n powers of 1/ex' , ' i . e . ao " ¥ " ( A,z,w) = A ,z,w )c^" n (C.19) where the f u n c t i o n s {"ft] a r e t o be determined from the R i c c a t i DE, (C.18). S u b s t i t u t i n g from (C.19) i n t o (C.18) and e q u a t i n g the c o e f f i c i e n t s of l i k e powers of OC, i t f o l l o w s t h a t (as per S i e b e r t , 1964, p29) = / q (c.20) V f ' # / 2 # (C.21) 248 and i n g e n e r a l t h a t n ")£•, = - ^ i ^ n ^ - r , ) / 2 i i f o r n > 2 (C.23) One f e a t u r e of these r e c u r r e n c e r e l a t i o n s which w i l l prove t o be c r u c i a l l a t e r i s the l i n e a r dependence of on . The boundary c o n d i t i o n E| X ( g a > = 0 i s i m p l i c i t i n the form of (C.17) p r o v i d e d Rel -}^} > 0 as z — , so i t remains o n l y t o s a t i s f y the boundary c o n d i t i o n a t the s u r f a c e . From ( 1 . 7 4 ) , i t f o l l o w s t h a t one form of the s u r f a c e c o n d i t i o n i s u« Elz=* " E ' \ X m 0 = 2u,B* (C.24) where u0 and B£ are d e f i n e d i n (1.19) and (1.70) r e s p e c t i v e l y . S u b s t i t u t i n g i n t o ( C . 2 4 ) f o r E'\ x s f f from (C.17), and r e p l a c i n g ua by A ( i n ke e p i n g w i t h the n e g l e c t of d i s p l a c e m e n t c u r r e n t s ) , the boundary c o n d i t i o n becomes AE, + oCE,^(0) = 2 A B * (C.25) Hence, E( = 2 A B*/[ A + oc'¥(Q) ] = 2 VB*/[ V + 2^(0) ] (C.26) where V = \/ot (C.27) 249 L e t the a s y m p t o t i c e x p a n sion f o r E, be w r i t t e n as E, ~ 2 V B * £ Yn ocn as oc—> <*> (C.28) where the c o e f f i c i e n t s {jf} a re t o be det e r m i n e d . S u b s t i t u t i n g (C.19) and (C.28) i n t o (C.26) and r e a r r a n g i n g , the boundary c o n d i t i o n becomes The c o e f f i c i e n t s , { ^ 1 , can now be found by e q u a t i n g c o e f f i c i e n t s of l i k e powers of oC i n (C.29); hence, ie = [ V ' + ' ^ O ) ] - 1 ( C . 3 0 ) Y, = - £ 2 ' V i ( 0 ) ( c . 3 1 ) K = K2[K ^ 2 { 0 ) ' ^ ( 0 ) ] ( C 3 2 ) and so on. I t i s p o s s i b l e t o check the p h y s i c a l c o n s i s t e n c y of ( C . 3 0 ) . As A — , E, — > Eo(0) = B*, as per ( 1 . 5 4 ) , ( 1 . 7 0 ) , and ( 1 . 7 3 ) ; from ( C . 3 0 ) , ( C . 2 0 ) , and ( C . 1 5 ) the l a r g e - A l i m i t of ^ i s 1 /2 which, r e c a l l i n g ( C . 2 8 ) and n o t i n g t h a t V — * 1 , i s the v a l u e r e q u i r e d . As fi — o n the o t h e r hand, i t f o l l o w s from ( C . 1 5 ) , ( C . 2 0 ) , and ( C . 2 8 ) t h a t E, —> 2 A B*// ifi2<r{ 0) , i n accordance w i t h ( 1 . 5 3 ) . The formulae d e r i v e d above c o m p l e t e l y d e f i n e the a s y m p t o t i c 250 e x p a n s i o n f o r E(A,0,w) v a l i d i n the l i m i t of l a r g e v a l u e s of oC . In o r d e r t o r e l a t e t h i s s o l u t i o n t o o b s e r v a t i o n s an i n v e r s e Hankel t r a n s f o r m a t i o n must be performed. To t h i s end, i t i s advantageous t o e x p r e s s the Hankel t r a n s f o r m i n the schematic form ( 1 . 6 8 ) , r e - s t a t e d below f o r c o n v e n i e n c e : OO A H m ( R , / 5 ) = ^K( A ,w;<r)B*( A ,a,h)#„ ( A ,R)dX (C.33) o where ( i ) A H m r e p r e s e n t s the observed secondary magnetic f i e l d , d e f i n e d i n (1.66) and (1 . 6 7 ) ; m=0 f o r the v e r t i c a l magnetic f i e l d , and m=l f o r the r a d i a l f i e l d . ( i i ) K i s the i n d u c t i o n response f u n c t i o n f o r the p a r t i c u l a r c o n d u c t i v i t y s t r u c t u r e ; a c c o r d i n g t o ( 1 . 6 9 ) , ( 1 . 6 4 ) , and ( 1 . 7 3 ) , K may be e x p r e s s e d as f o l l o w s : K(A,w;<T) = [E/B* - l ] l „ (C.34) ( i i i ) B£ i s the e f f e c t i v e source spectrum, d e f i n e d i n (1 . 7 0 ) ; i f u, i s r e p l a c e d by \ , B* i s g i v e n by -iaJ,(A a)exp(-Ah) B*(A,a,h) = (C.35) 2A ( i v ) £Cm i s the a p p r o p r i a t e Hankel t r a n s f o r m k e r n e l ; i f u, i s r e p l a c e d by A? (1.76) and (1.77) may be condensed i n t o a s i n g l e e q u a t i o n : K m ( A , r ) = ( - 1 U A 2 J m ( A r) m=0,l (C.36) 251 A l l t h e d e pendence on c W z ) and w on t h e RHS of (C.33) r e s i d e s i n t h e r e s p o n s e f u n c t i o n , K. R e c a l l i n g t h a t t h e o b j e c t i v e i s t o d e r i v e f o r m u l a e i n terms of {s} f o r t h e c o e f f i c i e n t s {*T} i n ( C . 1 0 ) , i t i s a p p r o p r i a t e t o seek t o d e l i n e a t e t h e c r - d e p e n d e n c e of t h e c o e f f i c i e n t s , {T}, i n t h e f o l l o w i n g a s y m p t o t i c e x p a n s i o n \ t o r K: K(A,w;<r) ~ 51 T n ( A ; <r)/?n as A ( c . 3 7 ) Once e x p r e s s i o n s have been d e r i v e d f o r {T} i n terms of t h e d e r i v a t i v e ' s {s} ( d e f i n e d i n ( C . 1 2 ) ) i t w i l l be p o s s i b l e t o f i n d t h e d e s i r e d f o r m u l a e f o r { } by i n v e r s e H a n k e l t r a n s f o r m a t i o n , i n a c c o r d a n c e w i t h ( C . 3 3 ) . D e r i v a t i o n o f f o r m u l a e f o r {T} r e q u i r e s " d e - c o u p l i n g " of t h e A - and ^ - d e p e n d e n c e i n t h e e x p a n s i o n ( C . 2 8 ) . The f i r s t s t e p i n t h i s e n d e a v o u r would be t o w r i t e t h e c o e f f i c i e n t s {y} e x p l i c i t l y i n terms of A and (S. In f a c t i t i s more e x p e d i t i o u s t o d e r i v e e x p r e s s i o n s f o r t h e s e t {k} o f e x p a n s i o n c o e f f i c i e n t s f o r K, where oo K / v - i + 2 ! k n o C ~ n (C.38) n-o The c o e f f i c i e n t s , {k}, a r e c l o s e l y r e l a t e d t o {%}, as may be v e r i f i e d by s u b s t i t u t i n g (C.28) i n t o ( C . 3 4 ) ; h e n c e , MA,w;s) = 2 i r y ; ( A ,w;s) (C.39) On t h e b a s i s of t h e r e s u l t s d e r i v e d a bove, i t i s a s t r a i g h t f o r w a r d , a l b e i t t e d i o u s , m a t t e r t o o b t a i n f o r m u l a e f o r 252 kolA ,w;s). The following expressions for k, , k, , and kz were derived using (C.30-32) and (C.20-22): k, = A / [ A + >/\2+ i/32 ] (C.40) -i / 3 2s , A < * k, = z = (C.41) 2( A + v7 X 2 + i / 3 2 ) 2 ( A 2 + i / 3 2 ) - A oc k {/34s,2 [ (5p+3A )/2p( A +p) ] + 2i / J 2s 2p} (C.42) 8( A + p ) 2 p 4 ' where p = y[~\2 + I In order to determine the c o e f f i c i e n t s {T} in (C.37), the expression for each c o e f f i c i e n t k n i s expanded in powers of A / y 3 using the Binomial Theorem. The expansions so obtained w i l l be v a l i d only when A < fi , whereas a l l values of A are included in the transform integral (C.33). It w i l l be demonstrated below that the contribution to the integral for A > fi become negli g i b l e as / 3 — . Consequently, " s m a l l -A " expansions for the c o e f f i c i e n t s {k} prove to be adequate for the purposes.of the proof. The f i r s t few terms in the series expansions for k,, k, , and k 2 are presented below: x -in \x x3 -ts x* k, = 2^-e T + 2 i A - i-p e 4 + 0 ( ^ r ) (C.43) 253 kzoC-> = ^ [e~4* (2s2 - 5 s , V 2 ) ^ - + O(-^r)] (C.45) I t may be i n f e r r e d f r o m t h e s e p a r t i a l e x p a n s i o n s t h a t i n g e n e r a l 1 k n o c n = ^ ^ n X ( t ) A - / f n A</3 (c.46) where ^"„i(s) i s some f u n c t i o n o f t h e d e r i v a t i v e s , { s } . S u b s t i t u t i n g (C.46) i n t o ( C . 3 8 ) , oo oo I K ~ -1 + ^  ^" r ti<s) 4 a s P~>oa ( C 4 7 ) C o m p a r i s o n o f (C.37) a n d (C.47) y i e l d s t h e f o l l o w i n g g e n e r a l e x p r e s s i o n f o r t h e c o e f f i c i e n t s { T }: Tn = IE >„-i fi<s> A""'" A</3 (C.48) The way i s now c l e a r f o r a d e t a i l e d e x a m i n a t i o n o f t h e of t h e d e p e n d e n c e o f K on { s } . U s i n g (C.43-45) a s a g u i d e , i t may be i n f e r r e d t h a t ^" n J0(s) i s a f u n c t i o n o f t h e f i r s t n d e r i v a t i v e s o n l y , i . e . 5;/(s) = J2„/(s, ,s, , , s B ) i=l,2,3, (C.49) F u r t h e r m o r e , b e a r i n g i n m i n d t h e g e n e r a l s t r u c t u r e e x h i b i t e d i n ( C . 2 3 ) , i t i s c o n c l u d e d t h a t B"n|Cs) is a l w a y s l i n e a r w i t h r e s p e c t t o s n , i . e . 254 ^ v i ( s ) = + ^ n i s 0 , s 1 , , s a . , ) f o r n > 1 (C.50) where {T } i s a s e t of computable c o n s t a n t s . These g e n e r a l p r o p e r t i e s are r e s p o n s i b l e f o r an o r d e r l y appearance of h i g h e r o r d e r d e r i v a t i v e s , one by one, i n s u c c e s s i v e terms of (C.37). More e x p l i c i t l y , c o n s i d e r a t i o n of (C.48) i n the l i g h t of (C.49) r e v e a l s t h a t the n t h o r d e r d e r i v a t i v e , s n , does not e n t e r i n t o the formulae f o r the f i r s t n + 1 c o e f f i c i e n t s , T0 , T, , ,Tn. Moreover, i n view of (C.50), i t e n t e r s l i n e a r l y i n t o the e x p r e s s i o n f o r T^, . These g e n e r a l c h a r a c t e r i s t i c s a r e p r e s e r v e d by Hankel t r a n s f o r m a t i o n , so h i g h e r o r d e r d e r i v a t i v e s e n t e r i n t o the formulae f o r s u c c e s s i v e c o e f f i c i e n t s { £"} i n (C.10) i n p r e c i s e l y the same o r d e r l y f a s h i o n . To c l a r i f y t h e s e p o i n t s , c o n s i d e r the f o l l o w i n g formulae f o r T, , Tz , and T3 which were o b t a i n e d by combining the c o e f f i c i e n t s of l i k e powers of l//h i n (C.43-45): T, = 2 Ae * (C.51) T 2 = i (2 A2+ As, /2) (C.52) -jTT T3 = - i e 4 [\3 + - A ( 2 s a - 5s, 2/2)/8] (C.53) As e x p e c t e d , T, i s ( i m p l i c i t l y ) a f u n c t i o n of s^ o n l y , w h i l e T z depends l i n e a r l y on s, (and o n l y on s0 o t h e r w i s e ) , whereas T 3 i s a f u n c t i o n of s 0 , s, , and s a w i t h l i n e a r dependence on sz . To o b t a i n the d e s i r e d formulae f o r (V) i n terms of { s } , i t 255 remains now o n l y t o p e r f o r m ( i n v e r s e ) Hankel t r a n s f o r m a t i o n i n accordance w i t h (C.33). To t h i s end, i t i s c o n v e n i e n t t o decompose (C.33) i n t o two i n t e g r a l s : (i oo A H m = [ K B * # m d A + J K B * f f M d A (C.54) S u b s t i t u t i n g f o r K from (C.36), the f i r s t i n t e g r a l i n (C.54) admits the f o l l o w i n g a s y m p t o t i c e x p a n s i o n asy3—>°° : ft <*> ft $ K B * # „ d A - 2:/f" K d A o nso o (C.55) oo oo oo j > [ fl"n\TnB*^mdA - /[_ / 3 n \ T n B * ^ m d A Z3 w i t h the c o e f f i c i e n t s {T} g i v e n by (C.48). A l t h o u g h (C.48) i s a v a l i d r e p r e s e n t a t i o n f o r T o n l y when A <y3 , the d e c o m p o s i t i o n i n (C.55) i s a m a t h e m a t i c a l t r u i s m p r o v i d e d the i n t e g r a l s from fi t o oo e x i s t . In f a c t , i f h > 0, t h e s e i n t e g r a l s become t r a n s c e n d e n t a l l y s m a l l as y3 —>  06 because 7 n Xh e _ / > h ^ / 3 J ( n - j ) ! ) A e" J(( A a) J m ( A r )d A | < 0.6 2- — i ^ j (C.56) A h jso h where (C.35) and (C.36) have been i n v o k e d . The f a c t o r 0.6 a r i s e s because s u p { | J ( x ) | } < 0.6 and s u p { | J ( x ) | } = 1. In view of (C.56), the i n t e g r a l s from j$ t o oo i n (C.55) a r e subdominant, and may be n e g l e c t e d . S i m i l a r l y , the second i n t e g r a l i n (C.54) decays e x p o n e n t i a l l y as /3 —>°* s i n c e , r e c a l l i n g (C.35-36), 256 7 7 tfPh ^ KB * dA | < 0.6|K|max jAe"* hdA = 0 . 6 | K | max^j- ( fi + 1/h) (C.57) P I* C o m b i n i n g ( C . 5 4 - 5 7 ) , and d i s r e g a r d i n g t h e s u b d o m i n a n t c o n t r i b u t i o n s , oo ~ A H m ~ /* JTrvBo a s fi—*°° (C.58) n =<? o C o m p a r i n g (C.58) a nd ( C . 1 0 ) , a n d r e c a l l i n g ( C . 3 7 ) , S n * = " [B*tf mdA ( C' 5 9 ) Oo = ^ T n B * p f m d A n = l , 2 , ( C.60) o S u b s t i t u t i n g (C.35) a nd (C.36) i n t o ( C . 5 9 ) , oo ^ m 0 ( R , a , h ) = < - l f * ' |- [ \ e " A h J(( A a ) J m ( A r ) d A (C.61) o T h e s e i n t e g r a l s may be e v a l u a t e d e x a c t l y u s i n g t h e f o r m u l a e (A.7) a n d (A.4) f o r t h e d i m e n s i o n l e s s p r i m a r y m a g n e t i c f i e l d c o m p o n e n t s ; h e n c e , 5"co = -Hz t f(R,0) (C.62) = - f a Q , ( v ) + ( R 2 - h 2 - a 2 ) Q ' ( v ) / R ] / ( 4 7 T / a R ^ ) *1 'z Xi0 = Hr t f(R,0) = - h Q ^ ( v ) / ( 2 7 t V a R 3 ) (C.63) 257 where v = (R 2+h 2+a 2)/2aR S u b s t i t u t i n g (C.48), (C.35), and (C.36) i n t o (C.60), the f o l l o w i n g g e n e r a l e x p r e s s i o n i s o b t a i n e d f o r n 0 0 £^ n(R,a,h;s) = a(-l)m2L ^ _,tn-«fc.|(s ) J A Ae" X h J,( A a ) J m ( A r )dA (C.64 Comparing (C.61) and (C.64), i t may be o b s e r v e d t h a t - 2 l < - l ) ^ , - 3 / n=l,2,3,... (C.65) Thus a l l the " c o e f f i c i e n t s i n (C.IO) may be e v a l u a t e d u s i n g a n a l y t i c formulae o b t a i n e d by d i f f e r e n t i a t i n g (C.62) and (C.63) w i t h r e s p e c t t o h. The r e s u l t , (C.65), i s q u i t e s i m i l a r t o a g e n e r a l a s y m p t o t i c f o r m u l a f o r i n t e g r a l s w i t h o s c i l l a t o r y k e r n e l s due t o W i l l i s (1948). I t i s now p o s s i b l e t o examine the r e l a t i o n s h i p between the d a t a and the c o n d u c t i v i t y u s i n g the c o e f f i c i e n t s { £ } as i n t e r m e d i a r i e s . The n u m e r i c a l v a l u e s of t h e s e c o e f f i c i e n t s a r e t o be c o n s i d e r e d known, h a v i n g been d e t e r m i n e d from the d a t a u s i n g (C.IO). As noted e a r l i e r , , the n a t u r e of the dependence of on the d e r i v a t i v e s i s governed by (C.49) and (C.50). For example, r e c a l l i n g (C.52-53), s, can be d e t e r m i n e d unambiguously from the known v a l u e of e i t h e r or £ l 2 , whereupon i t becomes p o s s i b l e t o e v a l u a t e sz from £* e 3 or "£ l 3 by s u b s t i t u t i o n of Sj . By e x t e n s i o n , a l l the d e r i v a t i v e s can be d e t e r m i n e d from p e r f e c t l y a c c u r a t e Hz or Hr d a t a by s u c c e s s i v e 258 s u b s t i t u t i o n . A knowledge of a l l the d e r i v a t i v e s a t the s u r f a c e i s tantamount t o a complete s p e c i f i c a t i o n of the M a c l a u r i n S e r i e s c o e f f i c i e n t s f o r c ~ ( z ) . S i n c e any f u n c t i o n a d m i t s one and o n l y one M a c l a u r i n S e r i e s e x p a n s i o n ( K r e y s z i g , 1967, p648), the uniqueness of HLEM i n v e r s i o n i s e s t a b l i s h e d . A complete set of M a c l a u r i n S e r i e s c o e f f i c i e n t s i s a m e a n i n g f u l c h a r a c t e r i s a t i o n of a f u n c t i o n o n l y i f i t i s a n a l y t i c , i . e . o n l y i f a l l i t s d e r i v a t i v e s a r e d e f i n e d and f i n i t e everywhere. A c o n d u c t i v i t y v a r y i n g as a n o n - i n t e g r a l power of z, f o r example, would be beyond the scope of t h i s r e p r e s e n t a t i o n . T h i s l i m i t a t i o n on the a l l o w a b l e c o n d u c t i v i t y v a r i a t i o n s i s common t o a l l the theorems c i t e d above, except those based on G e l ' f a n d - L e v i t a n t h e o r y which r e q u i r e o n l y t h a t cr be c o n t i n u o u s . S t r i c t l y s p e a k i n g , uniqueness has been e s t a b l i s h e d f o r h > 0, i . e . when t h e r e i s a d i f f e r e n c e i n e l e v a t i o n between t r a n s m i t t e r and r e c e i v e r . However, the e f f e c t of an a r b i t r a r i l y s m a l l v e r t i c a l s e p a r a t i o n between t r a n s m i t t e r and r e c e i v e r l e v e l s i s c o m p l e t e l y n e g l i g i b l e , and o f t e n o c c u r s i n p r a c t i c e due t o the f i n i t e d i m e n s i o n s of the r e c e i v e r . On t h i s b a s i s , i t c o u l d be argued t h a t the case h > 0 i s more g e o p h y s i c a l l y r e l e v a n t than h = 0. Moreover, i t i s i n t u i t i v e l y u n t e n a b l e t h a t measurements taken j u s t above the s u r f a c e can be i n v e r t e d u n i q u e l y , w h i l e those t a k e n on the s u r f a c e cannot. The c r u c i a l m a t h e m a t i c a l f e a t u r e e n s u r i n g uniqueness i s the l i n e a r dependence of " V ^ + ion i n (C.23), because i t i s t h i s r e l a t i o n s h i p which u l t i m a t e l y g u a r a n t e e s t h a t the h i g h e s t o r d e r d e r i v a t i v e i n f l u e n c i n g *cT* e n t e r s l i n e a r l y . Were the h i g h e s t ^ Try tx 259 o r d e r d e r i v a t i v e t o e n t e r q u a d r a t i c a l l y , f o r example, i t would not be p o s s i b l e t o choose between i t s two r o o t s and so a t l e a s t , two c o n d u c t i v i t i e s would s a t i s f y the d a t a e q u a l l y w e l l . A c u b i c dependence would a l l o w unique d e t e r m i n a t i o n of r e a l o " ( z ) ; however, c u b i c (or h i g h e r odd power) dependence would thwart unique r e c o n s t r u c t i o n of a complex p h y s i c a l p r o p e r t y , i . e . would p r e c l u d e any e x t e n s i o n of t h i s method t o the case where d i s p l a c e m e n t c u r r e n t s a r e s i g n i f i c a n t . The appearance of h i g h e r o r d e r d e r i v a t i v e s i n s u c c e s s i v e terms of the e x p a n s i o n f o r A H W has an i n t u i t i v e p h y s i c a l i n t e r p r e t a t i o n . In the l i m i t of e x t r e m e l y h i g h f r e q u e n c i e s , o n l y the s u r f a c e c o n d u c t i v i t y a f f e c t s the o b s e r v a t i o n s , so i t i s not unexpected t h a t the l e a d i n g terms i n (C.10) depend on cr (0) a l o n e . As f r e q u e n c y d e c r e a s e s the depth of p e n e t r a t i o n s t e a d i l y i n c r e a s e s , n e c e s s i t a t i n g the i n c l u s i o n of more terms, i . e . h i g h e r o r d e r d e r i v a t i v e s , i n the e x p a n s i o n t o m a i n t a i n the same a c c u r a c y . The p r o o f p r e s e n t e d above i s p o o r l y s u i t e d t o r o u t i n e p r a c t i c a l i m p l e m e n t a t i o n because of i t s heavy r e l i a n c e on s u b s t i t u t i o n s . Any e r r o r s i n the lower o r d e r d e r i v a t i v e s of cf would be a m p l i f i e d r e m o r s e l e s s l y , r e n d e r i n g e s t i m a t e s of the h i g h e r o r d e r d e r i v a t i v e s i n c r e a s i n g l y u n r e l i a b l e . T h i s i s c o n s i s t e n t w i t h p h y s i c a l i n t u i t i o n , i t b e i n g i n t r i n s i c a l l y more d i f f i c u l t t o determine s t r u c t u r e a t g r e a t e r d e p t h s . 260 Appendix D A SIMPLIFIED ANALYSIS OF COIL MISALIGNMENT ERROR Si n h a (1980) has drawn a t t e n t i o n t o the f a c t t h a t a s l i g h t m i s a l i g n m e n t of the t r a n s m i t t e r and r e c e i v e r " c o i l s " can produce s e r i o u s e r r o r s i n i n t e r p r e t a t i o n of HLEM so u n d i n g s . The s i m p l e s t p o s s i b l e case of i n t e r e s t c o r r e s p o n d s t o a r o t a t i o n of the r e c e i v e r a x i s , the o r i e n t a t i o n of the t r a n s m i t t e r assumed t o be p e r f e c t l y v e r t i c a l . I f i t i s f u r t h e r assumed t h a t the r o t a t i o n i s e n t i r e l y towards or away from the t r a n s m i t t e r , i . e . t h a t the r e c e i v e r and t r a n s m i t t e r axes l i e i n the same v e r t i c a l p l a n e , then the e r r o r may be e x p r e s s e d i n terms of a s i n g l e a n g l e , $ . S p e c i f i c a l l y , i f Hr' and Hz denote the t r u e magnetic components, and i f Hr and Hz a r e the c o r r e s p o n d i n g observed v a l u e s , .then H z T = Hz.cosjrf - Hr.sinfZ* ( D . l ) H r T = H z . s i n f * + H r . c o s ^ (D.2) These s i m p l e r e l a t i o n s a re an inadequate c h a r a c t e r i s a t i o n of c o i l m i s a l i g n m e n t i n the g e n e r a l case when both t r a n s m i t t e r and r e c e i v e r a re r o t a t e d i n a r b i t r a r y d i r e c t i o n s . The main purpose here i s t o i l l u s t r a t e an approach r a t h e r than t o p r e s e n t a f u l l t r e a t m e n t of m i s a l i g n m e n t e r r o r . In t h i s v e i n , e x p r e s s i o n s w i l l be d e r i v e d from ( D . l ) and (D.2) t o s e r v e as 2 6 1 e s t i m a t e s o f t h e m i s a l i g n m e n t e r r o r f o r e a c h o f t h e m e a s u r a b l e q u a n t i t i e s i n t r o d u c e d i n 1 1 - 4 ( i ) . I n t h e d e r i v a t i o n o f t h e s e f o r m u l a e i t i s a s s u m e d t h a t <fi i s s m a l l , a n d t h a t i t c a n b e r e g a r d e d a s a r e a l i s a t i o n o f a N o r m a l r a n d o m v a r i a b l e , w i t h m e a n z e r o a n d ( s m a l l ) v a r i a n c e , V 2 . D - l M i s a l i g n m e n t E r r o r s f o r V e r t i c a l M a g n e t i c F i e l d  M e a s u r e m e n t s F r o m ( D . l ) i t f o l l o w s t h a t , t o f i r s t o r d e r i n (fi, t h e e r r o r i n t r o d u c e d i n t o H z a s a r e s u l t o f m i s a l i g n m e n t i s g i v e n b y <$Hz = H z T - H z a: -HrJZ^ ( D . 3 ) H e n c e , V a r [ < j R e { H z ] ] ce. ( R e { H r } ) 2 V 2 ( D . 4 ) V a r [ d l m { H z ] ] a ( I m { H r } ) 2 V 2 ( D . . 5 ) I n o r d e r t o a s s e s s t h e e f f e c t o n | H z | a n d 9z m e a s u r e m e n t s , 6}iz i s w r i t t e n i n t e r m s o f cf|Hz| a n d 6$z ( b o t h a s s u m e d s m a l l ) : J H Z ~ c51 H z I e'' + i o f l z . H z ( D . 6 ) H e n c e , s u b s t i t u t i n g f r o m ( D . 3 ) f o r H z a n d r e - a r r a n g i n g , < J | H z | s R e { - ^ H r . e ~ * *} = - ^ | H r | c o s # w ( D . 7 ) 262 where 0w i s the phase of the w a v e - t i l t , W, d e f i n e d i n (2 . 5 1 ) : W = Hr/Hz = |W|e*** = Re{W} + i.Im{W} (D.8) T h e r e f o r e the v a r i a n c e of the m i s a l i g n m e n t e r r o r i n <5|Hz| i s , from (D.7), Var [<$ | Hz | ] 2r y 2 | Hr | 2 cos 20w (D.9) S i m i l a r l y , e s t i m a t e s f o r <$9z and Var[d#z] can be dev e l o p e d from (D.6) a n d ( D . 3 ) : SOz 2r Im{-Hr/Hz}^ = -Im{W}^ (D.10) Var[<f0z] ~ v 2 | W | 2sin 20w ( D . l l ) D-2 M i s a l i g n m e n t E r r o r s f o r R a d i a l Magnet i c F i e l d  Measurements By an e n t i r e l y analagous development t o t h a t p r e s e n t e d f o r Hz i n the p r e c e d i n g s e c t i o n , but based on (D.2) r a t h e r than ( D . l ) , i t may be shown t h a t 6lir d Hz.0 (D.12) Var [c$Re{Hr] ] Cz (Re {Hz } ) 2 V2 (D.13) 263 Var[<flm{Hr} ] C£ ( I m { H z } ) 2 V 2 (D.14) V a r | H r | ] d v 2|Hz| 2cos 20w (D.15) Var[cf0r] a y 2 s i n 2 £ w / | w | (D.16) D-3 Mi s a l i g n m e n t E r r o r f o r Wave-TiIt Measurements From (D.8), ( D . l ) , and (D.2) i t f o l l o w s t h a t the t r u e T w a v e - t i l t W may be e x p r e s s e d i n the form _ W + t a n ^ W = (D.17) 1 - Wtanjzf To f i r s t o r d e r i n tan , (D.17) s i m p l i f i e s t o WT = W + t a n ^ + W 2 t a n ^ (D.18) whence Jw a (1 + W 2 ) t a n ^ ~ ^|W|ettfw + i |W\66v.e**w (D.19) -Ldw M u l t i p l y i n g b oth s i d e s of (D.19) by e and s e p a r a t i n g the r e a l and im a g i n a r y p a r t s : <J|W| C£ R e C e ' ^ d + W 2)tan^} (D.20) 66v <a I m { ( l + W 2)tan#/W} (D.21) 264 Hence, Var[<5|W|] ~ |cos0w(l+Wr 2-Wi 2) + 2WrWi.sin0w| 2y 2 (D.22) V a r [ c f # w ] or |sin0w(l+Wr 2-Wi 2) - 2WrWi .cos0w | 2 v 2/| W| 2 (D.23) where Wi = Im{W} Wr = Re{W} The c o r r e s p o n d i n g e x p r e s s i o n s f o r t h e m i s a l i g n m e n t e r r o r i n Re{W} and Im{W} f o l l o w i m m e d i a t e l y from ( D . 1 8 ) : cfRe{W} <x R e { l + W 2}tanjzf chm{W} d Im{l + W2}tan05 T h e r e f o r e , Var[c^Re{W}] 11 + Wr 2 - W i 2 | 2 v 2 (D.24) Var[cflm{W}] 2|WrWi| 2 V 2 (D.25) 

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