"Science, Faculty of"@en . "Earth, Ocean and Atmospheric Sciences, Department of"@en . "DSpace"@en . "UBCV"@en . "Fullagar, Peter Kelsham"@en . "2010-03-26T23:06:07Z"@en . "1981"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "A detailed study of electromagnetic induction in a sequence of conductive layers has been completed for the case when the inducing fields are generated by an alternating current in a horizontal loop. The study was undertaken with a view to the development of a computer program to perform automatic inversion of horizontal loop electromagnetic (HLEM) frequency soundings taken over horizontally stratified ground. The program constitutes a new implementation of the general approach of Backus and Gilbert (1967, 1968, 1970). By means of a linearised iterative scheme, it constructs layered conductivities which satisfy a given set of observations to an accuracy consistent with the observational uncertainties. Subsequently, the non-uniqueness admitted by the limited amount of data can be appraised by computing averages of the original constructed model and comparing them with averages corresponding to other dissimilar models which also satisfy the data. In examples the Backus-Gilbert averages faithfully reflect the character of the \"true\" conductivity in regions of high conductivity, but they are of limited value in delineating resistive zones. The program has been applied successfully to the inversion of real data from Grass Valley, Nevada.\r\nA uniqueness theorem is presented for inversion of HLEM frequency soundings. It has been proved that an unlimited quantity of perfectly accurate HLEM frequency soundings (at a fixed receiver location) suffices to completely determine the\r\n\r\nconductivity as a function of depth. This result, which is believed to be new, enhances the credibility of conclusions based on inversion of HLEM soundings."@en . "https://circle.library.ubc.ca/rest/handle/2429/22712?expand=metadata"@en . "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 \u00C2\u00A9 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 \u00E2\u0080\u00A2 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 \u00E2\u0080\u0094 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 \u00E2\u0080\u0094 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, -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) 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\u00E2\u0080\u009Ee\"' 2 + B\u00E2\u0080\u009E 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\u00E2\u0080\u009E2E = -2u,B e0 as A\u00E2\u0080\u0094> 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 \u00C2\u00A3, 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) \u00C2\u00A3 ( 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\u00E2\u0080\u009E (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 \u00C2\u00A3 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\u00C2\u00A3j + 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= \u00E2\u0080\u0094 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 \u00E2\u0080\u0094> , 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 \u00E2\u0080\u0094 > and hence t h a t t a n h ( u j h y ) \u00E2\u0080\u0094 > 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 \u00E2\u0080\u0094 > 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, \u00E2\u0080\u0094 > - i a J ( A a ) exp( -uah)/( u, + uc ) as w >\u00E2\u0080\u00A2 \u00C2\u00AB* (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 \u00E2\u0080\u0094> Z0 \u00E2\u0080\u0094>\u00E2\u0080\u00A2 -i/A as per (1.42) and (1.36),- so from (1.52) - i a . . E. \u00E2\u0080\u0094 > J.(Aa)e as A \u00E2\u0080\u0094 > \u00C2\u00B0\u00C2\u00B0 (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 } > \u00E2\u0080\u00A2* (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 \u00C2\u00A3j ) 2 + /3 4 2 (1.57) $j = O.Stan- 1 { ji2oj/{ X2 - y 2\u00C2\u00A3j )} (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 \u00C2\u00BB 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 (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\u00C2\u00B0 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 \u00C2\u00BB 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 ! \u00E2\u0080\u0094 [ 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; \u00C2\u00A3 , ' - EJV, = 2 H \u00C2\u00BB 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, - 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 ) * \u00E2\u0080\u0094 ^ 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 \u00C2\u00A3 ) 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\u00C2\u00A3 \u00C2\u00BBw 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 \u00C2\u00B10.36\u00C2\u00B0. 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, /32z)E( A , z , w )cr( z ) dz/2u\u00E2\u0080\u009E (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: ; E + cfE = E + when crcr + Scr ; then from (2.6) where \&m\ i s the L 2-norm of <** 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 , \u00C2\u00A3[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 \u00E2\u0080\u00A2CC 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 \u00C2\u00A3 (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-\u00C2\u00A3A *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\u00C2\u00A3 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[ N + NSD^2N = X2MAX (2.46) where x\u00C2\u00A3 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\u00C2\u00A3 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 = \" \u00C2\u00AB 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 > \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 > A(^ \u00E2\u0080\u00A2 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 & 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 \u00E2\u0080\u0094 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- + &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\u00C2\u00A3f (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 - ( \ - \u00C2\u00A3 } i 2 - i \u00C2\u00AB ? - \u00C2\u00A3 > + ^ 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\u00C2\u00A3 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\u00C2\u00A3 and i ^ 1 a r e the L t h components of ?7 and \u00E2\u0080\u00A2ij1 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\u00C2\u00A3 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 \u00C2\u00A7 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 , (\u00C2\u00A3> , 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\u00C2\u00B0). 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 . 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 \u00C2\u00A3 + , 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 , \u00C2\u00A3 * 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 ) : \u00C2\u00A3 + = 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\u00C2\u00A3 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 \u00C2\u00A3 * : 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 \u00C2\u00A3 * 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 \u00E2\u0080\u0094 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 (/<\u00E2\u0080\u00A2 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 (/*\u00E2\u0080\u00A2 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\u00C2\u00B0 (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/ 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 , , 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 \u00E2\u0080\u009E ) 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 )> 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 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 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 , \u00C2\u00A3 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 \u00C2\u00A3 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 + \u00C2\u00A32sin0 + 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 , \u00C2\u00A3 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) * = 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 ^ ( 3 . 2 0 ) where t h e c o e f f i c i e n t s { \u00C2\u00A3T} 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 \u00E2\u0080\u0094 > 0 a s j3 \u00E2\u0080\u0094> 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 , { \u00C2\u00A3 ^ ^ = 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\u00C2\u00A3(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 \u00C2\u00A3(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 \u00E2\u0080\u00944, 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 \u00C2\u00A3 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 \u00E2\u0080\u0094 r i r I \" T \u00E2\u0080\u0094 i -l X = 12.64 x 1 = 15.18 max \u00E2\u0080\u0094 --\u00E2\u0080\u00A2 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\u00C2\u00A3r = c; - c* (3.23) fc\u00C2\u00BBl 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\u00C2\u00A3* 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 . \u00E2\u0080\u0094 * k \u00E2\u0080\u00A2>-;*=: 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 * \u00C2\u00A3 D * h*Scr + 7J.1 = ty* (3.25) k s , Jk k k tj o k=l,NL* (3.26) 1 ^ 1 < \u00C2\u00B0 ' 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 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\u00C2\u00A32/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^ \u00C2\u00A3 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\u00E2\u0080\u0094 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 \u00C2\u00A5 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 \u00E2\u0080\u00A20.01 T 1 1 1 1 1 r X w X X X|XX t h X - \u00C2\u00BB \u00E2\u0080\u0094 * \u00E2\u0080\u0094 I \u00E2\u0080\u0094 \u00C2\u00AB -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 : * ' ' \u00C2\u00BB \u00E2\u0080\u00A2 * 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 \u00E2\u0080\u00A2 0 . 0 0 50 100 150 200 (X 10 1 ) F i g . 3.11a |Hr|&|Hz| averaging function 1.0 o o \u00C2\u00B0 \u00C2\u00B0 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 \u00C2\u00B0\u00C2\u00B0 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 \u00E2\u0080\u0094 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\u00C2\u00B0). 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\u00C2\u00B0); 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\u00C2\u00B0 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 ( A B* -iaJ ((A a ) e \" X h / 2 A as A \u00E2\u0080\u0094 \u00C2\u00BB ~> ( 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 \u00E2\u0080\u0094 > 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 \u00E2\u0080\u0094> 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 \u00E2\u0080\u0094> 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 \u00C2\u00BB 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 \u00C2\u00B0\u00C2\u00B0 E ( A,0,w) = \u00E2\u0080\u0094 ^ 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 \u00E2\u0080\u009E E ( A,0,w) - 2u\u00E2\u0080\u009EB*] (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; ~ (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 { \u00C2\u00A3*} 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 \u00C2\u00AB = - - r t - | z = \u00E2\u0080\u009E ( 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> 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 \u00C2\u00BB 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\" = 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 \u00E2\u0080\u0094 , 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\u00C2\u00AB Elz=* \" E ' \ X m 0 = 2u,B* (C.24) where u0 and B\u00C2\u00A3 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'\u00C2\u00A5(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 * \u00C2\u00A3 Yn ocn as oc\u00E2\u0080\u0094> <*> (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, = - \u00C2\u00A3 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 \u00E2\u0080\u0094 , E, \u00E2\u0080\u0094 > 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 \u00E2\u0080\u0094 * 1 , i s the v a l u e r e q u i r e d . As fi \u00E2\u0080\u0094 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, \u00E2\u0080\u0094> 2 A B*// ifi2/\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 \u00E2\u0080\u0094 . 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 Aoa ( 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 >\u00E2\u0080\u009E-i fi A\"\"'\" A 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 { \u00C2\u00A3\"} 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\u00E2\u0080\u0094>\u00C2\u00B0\u00C2\u00B0 : ft <*> ft $ K B * # \u00E2\u0080\u009E 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 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 \u00E2\u0080\u0094> 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- \u00E2\u0080\u0094 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 \u00E2\u0080\u0094>\u00C2\u00B0* 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\u00E2\u0080\u0094*\u00C2\u00B0\u00C2\u00B0 (C.58) n = 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 "Thesis/Dissertation"@en . "10.14288/1.0052908"@en . "eng"@en . "Geophysics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Inversion of horizontal loop electromagnetic soundings over a stratified earth"@en . "Text"@en . "http://hdl.handle.net/2429/22712"@en .