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Exploring magnetotelluric nonuniqueness using inverse scattering methods Whittall, Kenneth Patrick 1987

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EXPLORING MAGNETOTELLURIC NONUNIQUENESS USING INVERSE SCATTERING METHODS by KENNETH PATRICK WHITTALL M.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1982 B.Sc.(Honours), U n i v e r s i t y of B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1987 © Kenneth P a t r i c k W h i t t a l l , 1987 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t The U n i v e r s i t y of B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g 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 n o t 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 . 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 The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : A p r i l 1987 ABSTRACT I p r e s e n t two a l g o r i t h m s w h i c h s o l v e t h e o n e - d i m e n s i o n a l m a g n e t o t e l l u r i c (MT) p r o b l e m of f i n d i n g t h e e l e c t r i c a l c o n d u c t i v i t y a(z) as a f u n c t i o n o f d e p t h i n t h e e a r t h . T o g e t h e r , t h e s e a l g o r i t h m s r e s t r i c t and e x p l o r e t h e n o n u n i q u e n e s s o f t h e n o n l i n e a r MT i n v e r s e p r o b l e m . They a c c e p t c o n s t r a i n t s w h i c h l i m i t t h e s p a c e o f a c c e p t a b l e c o n d u c t i v i t y m o d e ls and t h e y c o n s t r u c t d i v e r s e c l a s s e s o f o(z) i n o r d e r t o e x p l o r e t h i s s p a c e . To a v o i d p i t f a l l s d u r i n g i n t e r p r e t a t i o n , i t i s e s s e n t i a l t o i n v e s t i g a t e t h e e x t e n t o f t h e n o n u n i q u e n e s s p e r m i t t e d by t h e MT d a t a . A l g o r i t h m 1 i s a t w o - s t a g e p r o c e s s b a s e d on t h e i n v e r s e s c a t t e r i n g t h e o r y o f W e i d e l t . The f i r s t s t a g e u s e s t h e MT f r e q u e n c y - d o m a i n d a t a t o c o n s t r u c t an i m p u l s e r e s p o n s e a n a l o g o u s t o a d e c o n v o l v e d s e i s m o g r a m . S i n c e t h i s i s a l i n e a r p r o b l e m (a L a p l a c e t r a n s f o r m ) , numerous i m p u l s e r e s p o n s e s may be g e n e r a t e d by l i n e a r i n v e r s e t e c h n i q u e s w h i c h h a n d l e d a t a e r r o r s r o b u s t l y . I m i n i m i z e f o u r norms of t h e i m p u l s e r e s p o n s e i n o r d e r t o c o n s t r u c t v a r i e d c l a s s e s o f l i m i t e d s t r u c t u r e m o d e l s . Two l e a s t - s q u a r e s norms m i n i m i z e t h e e n e r g y i n t h e i m p u l s e r e s p o n s e o r t h e e n e r g y i n i t s d e r i v a t i v e w i t h r e s p e c t t o d e p t h . Two l e a s t a b s o l u t e v a l u e norms m i n i m i z e t h e m a g n i t u d e s o f t h e r e s p o n s e o r i t s d e r i v a t i v e . I t i s p o s s i b l e t o use o t h e r norms. The d i f f e r e n t c l a s s e s sample t h e r a n g e o f a c c e p t a b l e models and t h e minimum s t r u c t u r e c r i t e r i o n i s i i u n l i k e l y t o a l l o w m o d e ls w i t h s p u r i o u s f e a t u r e s . The s e c o n d s t a g e of A l g o r i t h m 1 c o n s t r u c t s t h e c o n d u c t i v i t y model from t h e i m p u l s e r e s p o n s e u s i n g any o f f o u r F f e d h o l m i n t e g r a l e q u a t i o n s o f t h e s e c o n d k i n d . I e v a l u a t e t h e p e r f o r m a n c e o f e a c h of t h e f o u r mappings and recommend t h e B u r r i d g e and G o p i n a t h - S o n d h i f o r m u l a t i o n s . I a l s o e v a l u a t e t h r e e a p p r o x i m a t i o n s t o t h e s e c o n d - s t a g e e q u a t i o n s . One o f t h e s e i s e q u i v a l e n t t o t h e B o rn a p p r o x i m a t i o n w h i c h assumes t h e i m p u l s e r e s p o n s e has n e g l i g i b l e m u l t i p l e r e f l e c t i o n s . The a p p r o x i m a t i o n t h a t i n c l u d e s f i r s t - o r d e r m u l t i p l e r e f l e c t i o n s i s t h e most a c c u r a t e and g i v e s c o n d u c t i v i t y m o d e l s s i m i l a r t o t h o s e g i v e n by t h e i n t e g r a l e q u a t i o n s . A l g o r i t h m 2 s o l v e s an i n t e g r a l f o r m o f a n o n l i n e a r R i c c a t i e q u a t i o n r e l a t i n g t h e m e asured f r e q u e n c y - d o m a i n d a t a t o a f u n c t i o n of t h e c o n d u c t i v i t y . The i t e r a t i v e s o l u t i o n scheme s a c r i f i c e s t h e e f f i c i e n c y o f a d i r e c t i n v e r s i o n p r o c e s s s u c h as A l g o r i t h m 1 f o r t h e a d v a n t a g e s o f i n c o r p o r a t i n g l o c a l i z e d c o n d u c t i v i t y c o n s t r a i n t s . The l i n e a r programming f o r m u l a t i o n r e a d i l y a c c e p t s a wide v a r i e t y o f e q u a l i t y and i n e q u a l i t y c o n s t r a i n t s on a(z). I use t h e s e c o n s t r a i n t s i n two ways t o combat t h e n o n u n i q u e n e s s of t h i s n o n l i n e a r i n v e r s e p r o b l e m . F i r s t , I impose p h y s i c a l c o n s t r a i n t s d e r i v e d f r o m e x t e r n a l s o u r c e s t o r e s t r i c t t h e n o n u n i q u e n e s s and c o n s t r u c t o(z) models t h a t a r e c l o s e r t o r e a l i t y . S e c o n d , I impose c o n s t r a i n t s s p e c i f i c a l l y d e s i g n e d t o e s t i m a t e t h e e x t e n t o f t h e n o n u n i q u e n e s s and e x p l o r e t h e r a n g e o f a c c e p t a b l e o(z) p r o f i l e s . The f i r s t t e c h n i q u e e n h a n c e s t h e r e l i a b i l i t y o f an i n t e r p r e t a t i o n and t h e s e c o n d a s s e s s e s t h e p l a u s i b i l i t y o f p a r t i c u l a r c o n d u c t i v i t y f e a t u r e s . The c o n v e r g e n c e o f A l g o r i t h m 2 i s good b e c a u s e A l g o r i t h m 1 p r o v i d e s v a r i e d i n i t i a l a(z) w h i c h a l r e a d y f i t t h e d a t a w e l l . i v TABLE OF CONTENTS A b s t r a c t i i T a b l e o f C o n t e n t s v L i s t o f F i g u r e s v i i A cknowledgements i x C h a p t e r 1. I n t r o d u c t i o n 1 C h a p t e r 2. W e i d e l t I n v e r s e S c a t t e r i n g 14 2.1. Review of W e i d e l t ' s A p p r o a c h 15 2.2. Frequency-Wavenumber T r a n s f o r m a t i o n 24 2.3. Wavenumber-Domain R e s p o n s e s and E r r o r s 26 2.4. I mpulse R e s p o n s e I n v e r s i o n 30 2.4.1. B a c k u s - G i l b e r t C o n s t r u c t i o n 32 2.4.2. L i n e a r Programming C o n s t r u c t i o n 41 • 2.4.3. B a c k u s - G i l b e r t A p p r a i s a l 46 2.5. C o n s t r a i n e d I m p u l s e R e s p o n s e I n v e r s i o n 55 2.5.1. P h y s i c a l R e a l i z a b i l i t y C o n s t r a i n t s ..56 2.5.2. A P r i o r i I mpulse R e s p o n s e 66 2.5.3. W e i g h t e d Impulse R e s p o n s e Norms 69 2.6. C o n d u c t i v i t y M a p p i n g s 72 2.6.1. I n t e g r a l E q u a t i o n s 73 2.6.2. N u m e r i c a l S o l u t i o n s 78 2.6.3. A p p r o x i m a t e M a p p i n g s 84 C h a p t e r 3. R i c c a t i E q u a t i o n I n v e r s i o n s 92 3.1. R i c c a t i E q u a t i o n T h e o r y 93 3.2. N u m e r i c a l S o l u t i o n s 98 C h a p t e r 4. C o n c l u s i o n s R e f e r e n c e s LIST OF FIGURES 1. Conductivity Test Models 18 2. Impulse Responses R(x) and B(x) 20 3. Conductivity from True Impulse Response 22 4. Histograms of Noisy Data 27 5. Smooth Model Wavenumber-Domain Data Part 1 28 6. Smooth Model Wavenumber-Domain Data Part 2 29 7. BG R(x) Impulse Response 36 8. BG B(x) Impulse Response 37 9. Determining Surface Conductivity from BG Impulse Responses 40 10. LP R(x) Impulse Response 43 11. LP B(x) Impulse Response 44 12. Kernels and Tradeoff Contour Diagram 49 13. Impulse Response Averaging Functions 53 14. Physical R e a l i z a b i l i t y Constraint Functions 60 15. Contours of f(s) Bound Exceeded 62 16. The JDF Data Set 64 17. Physical R e a l i z a b i l i t y Constraints on B(x) 65 18. A P r i o r i Impulse Response Inversion 68 19. Weighted Impulse Response Inversion 70 20. Conductivity from BG Impulse Response Models 81 21. Conductivity from LP Impulse Response Models 83 22. Approximate Mappings of the True Impulse Response 88 23. Approximate Mappings of Impulse Response Models 89 24. Integrated R i c c a t i Equation Kernels 102 v i i 25. C o n s t r a i n e d C o n d u c t i v i t y I n v e r s i o n s ....104 v i i i ACKNOWLEDGEMENTS My s p e c i a l t h a n k s t o Doug O l d e n b u r g who has p r o v i d e d much needed s c i e n t i f i c , m o r a l , and f i n a n c i a l s u p p o r t t h r o u g h o u t t h i s p r o j e c t . I thank Bob E l l i s f o r h i s . w i l l i n g n e s s t o r e v i e w p r e l i m i n a r y m a n u s c r i p t s and h i s a d m i n i s t r a t i v e acumen. I thank a l s o M a t t Y e d l i n and R o b e r t M i u r a f o r k e e p i n g me h o n e s t . I th a n k Ron Cl o w e s f o r g i v i n g me t h e c h a n c e t o w r i t e TRACE. To t h e i n n u m e r a b l e g r a d u a t e s t u d e n t s I have had t h e p l e a s u r e o f w o r k i n g w i t h o v e r t h e y e a r s : My t h a n k s f o r s h a r i n g w i t h me y o u r u n i q u e p e r s p e c t i v e s on l i f e , g e o p h y s i c s , and d a r t s . T h i s r e s e a r c h was s u p p o r t e d by a N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l p o s t g r a d u a t e s c h o l a r s h i p , NSERC g r a n t 67-4270, and an H.R. M a c M i l l a n F a m i l y F e l l o w s h i p . i x CHAPTER 1. INTRODUCTION The m a g n e t o t e l l u r i c (MT) s o u n d i n g method u s e s s u r f a c e measurements of n a t u r a l l y o c c u r r i n g e l e c t r i c and m a g n e t i c f i e l d s t o d e t e r m i n e t h e e l e c t r i c a l c o n d u c t i v i t y w i t h i n t h e e a r t h . Many n a t u r a l s o u r c e s c o n t r i b u t e t o t h e o b s e r v e d s p e c t r u m o f e l e c t r o m a g n e t i c (EM) e n e r g y . L i g h t n i n g s t o r m s a r e a m a j o r g e n e r a t o r o f s h o r t - p e r i o d o s c i l l a t i o n s l e s s t h a n / s e c o n d . M i c r o p u l s a t i o n s i n t h e m a g n e t o s p h e r e c a u s e v a r i a t i o n s o v e r a wide band of p e r i o d s r a n g i n g from / t o 600 s e c o n d s . M a g n e t i c s t o r m s d o m i n a t e t h e s p e c t r u m a t l o n g e r p e r i o d s o f h o u r s t o d a y s . The f i e l d f l u c t u a t i o n s can be decomposed i n t o p l a n e EM waves p r o p a g a t i n g down t o t h e s u r f a c e f r o m v a r i o u s d i r e c t i o n s . A t t h e s u r f a c e , most o f t h e wave e n e r g y i s r e f l e c t e d away due t o t h e e x t r e m e c o n d u c t i v i t y c o n t r a s t between t h e a i r and t h e e a r t h . However, a s m a l l amount r e f r a c t s , and d i f f u s e s v e r t i c a l l y i n t o t h e e a r t h . S h o r t - p e r i o d o s c i l l a t i o n s d i f f u s e and a r e a t t e n u a t e d more r a p i d l y w i t h d e p t h t h a n l o n g - p e r i o d o n e s . Waves w i t h p e r i o d s of s e v e r a l h o u r s have s i g n i f i c a n t a m p l i t u d e s down t o h u n d r e d s of k i l o m e t e r s i n t h e e a r t h . The r a t e o f d e c a y a l s o depends upon t h e c o n d u c t i v i t y ; t h e more c o n d u c t i v e t h e e a r t h , t h e g r e a t e r t h e a t t e n u a t i o n . T h u s , a v e r a g e s of c o n d u c t i v i t y down t o p r o g r e s s i v e l y l a r g e r d e p t h s i n f l u e n c e t h e d i f f u s i o n o f waves of p r o g r e s s i v e l y l o n g e r p e r i o d s . The s u r f a c e e l e c t r o m a g n e t i c impedance i s t h e r a t i o of a 1 I n t r o d u c t i o n / 2 h o r i z o n t a l e l e c t r i c f i e l d component t o a h o r i z o n t a l m a g n e t i c component. A t e a c h f r e q u e n c y , t h e impedance q u a n t i f i e s t h e e f f e c t o f t h e c o n d u c t i v i t y s t r u c t u r e o f t h e e a r t h on t h e d i f f u s i n g EM p l a n e wave. T h e r e f o r e , t h e e s s e n c e of t h e MT i n v e r s e p r o b l e m i s t o measure t h e impedance o v e r a b r o a d band o f f r e q u e n c i e s i n o r d e r t o e s t i m a t e t h e c o n d u c t i v i t y o f t h e e a r t h . MT e x p e r i m e n t s d e s i g n e d f o r r e s o u r c e p r o s p e c t i n g y i e l d c o n d u c t i v i t y a n o m a l i e s w h i c h may be i n t e r p r e t e d a s m i n e r a l d e p o s i t s , g e o t h e r m a l a r e a s , o r p o t e n t i a l h y d r o c a r b o n r e s e r v o i r s . MT s u r v e y s d e s i g n e d f o r g e n e r a l g e o p h y s i c a l m o d e l l i n g use c o n d u c t i v i t y and i t s c o r r e l a t i o n s w i t h o t h e r m a t e r i a l p r o p e r t i e s t o c o n s t r a i n t h e r m a l , p e t r o l o g i c a l , and d y n a m i c a l m o d e l s o f t h e c r u s t and upper m a n t l e . The r a n g e o f d e p t h s e x p l o r e d by t h e MT method i s s e t by t h e band o f measured f r e q u e n c i e s . Strangway e t a l . (1973) r e c o r d e d e l e c t r i c and m a g n e t i c f i e l d s ( g e n e r a t e d by t h u n d e r s t o r m s ) i n t h e a u d i o - f r e q u e n c y r a n g e o f 10 Hz t o 10 kHz. T h i s band i s s e n s i t i v e t o m i n e r a l d e p o s i t s o c c u r r i n g a t d e p t h s down t o ab o u t one k i l o m e t e r . A u d i o - f r e q u e n c y MT u s i n g e i t h e r n a t u r a l o r c o n t r o l l e d s o u r c e s i s a l s o an e f f e c t i v e t o o l f o r g e o t h e r m a l p r o s p e c t i n g b e c a u s e o f t h e h i g h c o n d u c t i v i t y of t h e a s s o c i a t e d h o t b r i n e , r o c k s and a l t e r a t i o n m i n e r a l s (Hoover e t a l . , 1978; S a n d b e r g and Hohmann, 1982). K o z i a r and Strangway (1975) u s e d a u d i o - f r e q u e n c y MT t o map t h e i n t e r f a c e I n t r o d u c t i o n / 3 between r e s i s t i v e p e r m a f r o s t and t h e u n d e r l y i n g , more c o n d u c t i v e , u n f r o z e n e a r t h . V o z o f f (1972) and V o z o f f e t a l . (1975) u s e d f r e q u e n c i e s between 0.001 Hz and 10 Hz t o e x p l o r e s e d i m e n t a r y b a s i n s . The c o n d u c t i v i t y o f s e d i m e n t a r y r o c k s depends m a i n l y on wa t e r s a l i n i t y and w a t e r - s a t u r a t e d p o r o s i t y so t h a t t h e MT method c a n d e l i n e a t e p o s s i b l e p e t r o l e u m r e s e r v o i r s . MT o b s e r v a t i o n s a t p e r i o d s of h o u r s t o d a y s g i v e c o n d u c t i v i t y m o d e ls o f t h e l o w e r c r u s t and u p p e r m a n t l e . These m o d e ls c o n s t r a i n many l a r g e s c a l e p r o p e r t i e s of t h e e a r t h b e c a u s e of t h e c o n n e c t i o n s between c o n d u c t i v i t y and o t h e r g e o p h y s i c a l p a r a m e t e r s . Gough ( 1 9 7 4 ) , G a r l a n d ( 1 9 7 5 ) , and Adam (1980) r e v i e w e d t h e c o r r e l a t i o n s between c o n d u c t i v i t y and g e o p h y s i c a l f a c t o r s s u c h as h e a t f l o w , t e m p e r a t u r e , s e i s m i c v e l o c i t y and a t t e n u a t i o n , p e t r o l o g y , p a r t i a l m e l t i n g , and t e c t o n i c f e a t u r e s . F o r example, b e c a u s e c o n d u c t i v i t y i s a t h e r m a l l y a c t i v a t e d p r o c e s s , i t i s a s e n s i t i v e i n d i c a t o r o f r o c k t e m p e r a t u r e . O t h e r g e o p h y s i c a l methods a r e n o t as s t r o n g l y d e p e n d e n t on t e m p e r a t u r e . Hermance and G r i l l o t ( 1 970,1974), O l d e n b u r g ( 1 9 8 1 ) , and J o n e s (1982) u s e d c o n d u c t i v i t y p r o f i l e s f r o m MT i n t e r p r e t a t i o n s t o g e n e r a t e t e m p e r a t u r e m o d els o f t h e e a r t h . T h i s a p p r o a c h t o f i n d i n g a g e o t h e r m i s i n d e p e n d e n t o f t h e method w h i c h s o l v e s t h e h e a t f l o w e q u a t i o n g i v e n i n i t i a l c o n d i t i o n s and a t h e r m a l c o n d u c t i v i t y f o r t h e e a r t h . I n t r o d u c t i o n / 4 The bulk c o n d u c t i v i t y of upper mantle rocks i s a l s o s e n s i t i v e to the assumed p e t r o l o g y and degree of p a r t i a l m e l t i n g (Waff, 1974, 1980; Shankland and Waff, 1977; Haak, 1980). A small percentage of melt causes a dramatic i n c r e a s e i n c o n d u c t i v i t y . H i g h - c o n d u c t i v i t y zones near 100 km depth i n t e r p r e t e d from geomagnetic and MT o b s e r v a t i o n s c o r r e l a t e w e l l with the (perhaps p a r t i a l l y molten) asthenosphere (Oldenburg, 1981). Furthermore, F i l l o u x (1980) and Oldenburg (1981) noted that the depth to these zones i s p o s s i b l y age dependent. The zone depths are r e l a t e d to the l i t h o s p h e r e t h i c k n e s s p r e d i c t e d by thermal models such as that of Parker and Oldenburg (1973), and to s e i s m i c l o w - v e l o c i t y zones i n t e r p r e t e d , f o r example, from R a y l e i g h wave d i s p e r s i o n data (Weidner, 1974; F o r s y t h , 1975). The c o r r e l a t i o n s between c o n d u c t i v i t y and other m a t e r i a l p r o p e r t i e s enable a u n i f i e d i n t e r p r e t a t i o n based upon g e o l o g i c i n f o r m a t i o n and s e v e r a l g e o p h y s i c a l methods. The e x t r a c o n s t r a i n t s p r o v i d e d by j o i n t i n t e r p r e t a t i o n s enhance the p l a u s i b i l i t y of the f i n a l e a r t h model (Hermance and G r i l l o t , 1970; Vozoff and Jupp, 1975). T h e r e f o r e , c o n d u c t i v i t y models d e r i v e d from MT o b s e r v a t i o n s p r o v i d e v a l u a b l e i n f o r m a t i o n about the resource p o t e n t i a l , temperature, composition, s t r u c t u r e , and dynamics of the e a r t h . However, the MT i n v e r s e problem must be s o l v e d before any i n t e r p r e t a t i o n s i n v o l v i n g e l e c t r i c a l c o n d u c t i v i t y are I n t r o d u c t i o n / 5 p o s s i b l e . In p a r t i c u l a r , the one-dimensional MT i n v e r s e problem of f i n d i n g c o n d u c t i v i t y as a f u n c t i o n of depth z has been s t u d i e d e x t e n s i v e l y . D e s p i t e the mathematical s i m p l i c i t y of the governing equations, a s p e c t s remain which are not yet f u l l y s o l v e d . The main impediment to a complete s o l u t i o n i s the inherent nonuniqueness; 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 a(z) s a t i s f y the f i n i t e number of i n a c c u r a t e o b s e r v a t i o n s a c q u i r e d on the e a r t h ' s s u r f a c e , and the s t r u c t u r a l shapes of these f u n c t i o n s d i f f e r s u b s t a n t i a l l y . The extent of t h i s f u n c t i o n a l v a r i a b i l i t y was not f u l l y a p p r e c i a t e d u n t i l Parker (1980) showed that f o r a given set of incomplete, i n a c c u r a t e responses, the o(z) model which regenerates these data with the h i g h e s t f i d e l i t y c o n s i s t s of a number of d e l t a f u n c t i o n s i n c o n d u c t i v i t y . These models are mathematically optimal but p h y s i c a l l y u n r e a l i s t i c . They c o n s i s t of v a n i s h i n g l y t h i n l a y e r s of i n f i n i t e c o n d u c t i v i t y (but f i n i t e conductance) separated by broad zones of i n f i n i t e r e s i s t i v i t y . A d e l t a f u n c t i o n model i s the l i m i t i n g case of a l a y e r e d model. Layered models have t h i c k l a y e r s of f i n i t e c o n d u c t i v i t y and p r o v i d e p h y s i c a l l y reasonable i n t e r p r e t a t i o n s of d i s c o n t i n u o u s g e o l o g i c formations such as sedimentary b a s i n s . Parker and Whaler (1981) and Hooshyar and Razavy (1982) showed how to c o n s t r u c t l a y e r e d models whose l a y e r s have a constant p e n e t r a t i o n depth. For these models, each l a y e r a t t e n u a t e s the I n t r o d u c t i o n / 6 f i e l d s by an equal amount (Loewenthal, 1975). Wu (1968), Jupp and V o z o f f (1975), Shoham et a l . (1978), and Larsen (1981) l i n e a r i z e d the MT equations and s o l v e d f o r l a y e r e d o(z) using i t e r a t i o n and damped l e a s t - s q u a r e s . Hermance and G r i l l o t (1974), Jones and Hutton (1979), and Connerney et a l . (1980) used Monte-Carlo methods to randomly generate thousands of simple l a y e r e d models. The few that a c t u a l l y f i t the data d e f i n e d the range of a c c e p t a b l e p r o f i l e s . Smooth c o n d u c t i v i t y models a l s o s a t i s f y MT o b s e r v a t i o n s . The smooth c l a s s of a(z) i s a p p l i c a b l e to areas where g e o l o g i c formations vary c o n t i n u o u s l y with depth. However, any d i s c o n t i n u i t i e s i n the e a r t h w i l l be w e l l approximated by r a p i d changes in a smooth a(z). In p r a c t i c e , MT data never have adequate r e s o l u t i o n to r e s o l v e an a c t u a l d i s c o n t i n u i t y . Becher and Sharpe (1969) developed a method of c o n s t r u c t i n g a smooth a(z) based on the s y n t h e s i s of t r a n s m i s s i o n l i n e s . B a i l e y (1970,1973) d e r i v e d and s o l v e d a n o n l i n e a r equation f o r a(z) u s i n g the c a u s a l i t y of a response f u n c t i o n . Parker and Whaler (1981) gave an a l g o r i t h m based on the theory of G e l ' f a n d and L e v i t a n (1955) which c o n s t r u c t s i n f i n i t e l y d i f f e r e n t i a b l e c o n d u c t i v i t y p r o f i l e s . Weidelt (1972) and M a r c h i s i o (1985) presented i n v e r s i o n schemes a l s o based on G e l ' f a n d - L e v i t a n theory. The above f i v e approaches are a l l exact s o l u t i o n s of the n o n l i n e a r MT e q u a t i o n s . I t i s a l s o p o s s i b l e to l i n e a r i z e the MT equations and i t e r a t e towards an I n t r o d u c t i o n / 7 acc e p t a b l e smooth, continuous a(z). Hobbs (1977,1982) and Oldenburg (1979) gave a l g o r i t h m s based on l i n e a r i z a t i o n . No s i n g l e c o n s t r u c t i o n a l g o r i t h m produces the most g e o p h y s i c a l l y r e a l i s t i c model f o r every data s e t . In some areas o(z) might be a smooth continuous f u n c t i o n , while i n other areas o(z) might be d i s c o n t i n u o u s . Moreover, no s i n g l e a l g o r i t h m c o n s t r u c t s a l l p o s s i b l e models which s a t i s f y one p a r t i c u l a r data s e t . Oldenburg et a l . (1984) demonstrated the n e c e s s i t y of i n v e r t i n g MT data with s e v e r a l d i s t i n c t a l g o r i t h m s . Strong trends observed on models from one i n v e r s i o n technique are not as prominent on models from another technique. Each i n v e r s i o n a l g o r i t h m c o n s t r u c t s a p a r t i c u l a r type of c o n d u c t i v i t y p r o f i l e from a d i f f e r e n t small region of the i n f i n i t e - d i m e n s i o n a l space of a c c e p t a b l e models. V a r y i n g the parameters w i t h i n a s i n g l e a l g o r i t h m produces d i f f e r e n t p r o f i l e s but does not completely explore the vast space of s o l u t i o n s which i n c l u d e s d e l t a f u n c t i o n , l a y e r e d , and i n f i n i t e l y smooth a(z). The s t r u c t u r e d e r i v e d from one a l g o r i t h m may a c c u r a t e l y r e p r e s e n t the true e a r t h but, to av o i d p i t f a l l s d u r i n g i n t e r p r e t a t i o n , the nonuniqueness of the MT i n v e r s e problem must be i n v e s t i g a t e d . I t i s e s s e n t i a l to sample as many a(z) s t r u c t u r e s as p o s s i b l e i n order t o ex p l o r e the e n t i r e range of models allowed by the data. In Chapter 2, I present the Weidelt (1972) MT in v e r s e s c a t t e r i n g s o l u t i o n augmented by l i n e a r i n v e r s e techniques. I n t r o d u c t i o n / 8 The a l g o r i t h m c o n s i s t s o f two main s t a g e s . The f i r s t s t a g e c o n s t r u c t s an i m p u l s e r e s p o n s e f r o m t h e f r e q u e n c y - d o m a i n MT d a t a . The s e c o n d s t a g e c o n s t r u c t s t h e c o n d u c t i v i t y f r o m t h e i m p u l s e r e s p o n s e . T h i s s e c o n d r e l a t i o n s h i p i s a r e l a t i v e l y s t r a i g h t f o r w a r d , n o n l i n e a r mapping w h i c h c a u s e s few p r o b l e m s b u t w h i c h has l i t t l e f l e x i b i l i t y . I n c o n t r a s t , t h e f i r s t s t a g e i s l i n e a r and so i t i s . f l e x i b l e enough t o c o n s t r u c t d i v e r s e i m p u l s e r e s p o n s e m o d e l s and a c c e p t e x t r a c o n s t r a i n t s . Hence, t h e f i r s t - s t a g e l i n e a r p r o b l e m i s t h e key t o e x p l o r i n g t h e n o n u n i q u e n e s s i n h e r e n t i n t h e MT d a t a . The u t i l i t y of t h e l i n e a r i n v e r s i o n f o r t h e i m p u l s e r e s p o n s e has n o t been f u l l y a p p r e c i a t e d o r e x e r c i s e d u n t i l now. T h e r e a r e many r e a s o n s why t h e new i n v e r s e s c a t t e r i n g method augmented by l i n e a r i n v e r s e t e c h n i q u e s p e r f o r m s b e t t e r t h a n o t h e r e x i s t i n g i n v e r s i o n schemes. F i r s t , t h e i n v e r s e s c a t t e r i n g method p r o v i d e s an e x a c t s o l u t i o n t o t h e n o n l i n e a r MT e q u a t i o n s . No l i n e a r i z a t i o n o r i t e r a t i o n i s n e c e s s a r y . No a p p r o x i m a t i o n s a r e r e q u i r e d o t h e r t h a n d i s c r e t i z a t i o n f o r computer i m p l e m e n t a t i o n . S e c o n d , t h e i n v e r s e t h e o r y f o r m u l a t i o n makes t h e i n v e r s e s c a t t e r i n g method s t a b l e and r o b u s t . W e i d e l t (1972) and B a i l e y (1970,1973) b o t h s t a t e d t h a t t h e i r e x a c t i n v e r s i o n schemes do n o t p e r f o r m w e l l on t h e n o i s y , b a n d l i m i t e d d a t a t y p i c a l o f any MT s u r v e y . However, i n v e r s e t h e o r y a d m i t s from t h e o u t s e t t h a t t h e d a t a a r e i n c o m p l e t e and i n a c c u r a t e . The f o r m a l i s m I n t r o d u c t i o n / 9 g e n e r a t e s minimum s t r u c t u r e m o d e l s c o m p a t i b l e w i t h r e a l i s t i c d a t a . The minimum s t r u c t u r e s o l u t i o n s a r e i m p o r t a n t b e c a u s e t h e y a r e u n l i k e l y t o have s p u r i o u s f e a t u r e s t o m i s l e a d t h e i n t e r p r e t e r . T h i s a p p r o a c h i s i n c o n t r a s t t o many a l g o r i t h m s w h i c h a t t e m p t t o c o n s t r u c t m o d e ls w h i c h e x a c t l y r e p r o d u c e t h e measured r e s p o n s e s . Such a l g o r i t h m s a r e doomed t o p r o d u c e m o d els more r e p r e s e n t a t i v e o f t h e d a t a n o i s e t h a n o f t h e e a r t h . T h i r d , t h e augmented a l g o r i t h m c o n s t r u c t s l a y e r e d and smooth o(z) p r o f i l e s i n o r d e r t o e x p l o r e t h e s p a c e of a c c e p t a b l e m o d e l s . T h i s f l e x i b i l i t y a l s o p e r m i t s an i n t e r p r e t e r t o c h o o s e i n a d v a n c e t h e model t y p e i n a c c o r d a n c e w i t h t h e l o c a l g e o l o g y and so enhance t h e r e l i a b i l i t y o f t h e i n t e r p r e t a t i o n . F o u r t h , t h e i n v e r s e t h e o r y f o r m u l a t i o n a p p r a i s e s t h e n o n u n i q u e n e s s i n t h e f i r s t s t a g e of t h e t w o - s t a g e i n v e r s e s c a t t e r i n g p r o c e d u r e . U n f o r t u n a t e l y , q u a n t i f y i n g t h e a b i l i t y o f a p a r t i c u l a r MT d a t a s e t t o r e s o l v e t h e i m p u l s e r e s p o n s e r e s u l t s i n o n l y q u a l i t a t i v e e s t i m a t e s o f t h e r e s o l u t i o n o f a(z) . F i f t h , t h e augmented a l g o r i t h m a c c e p t s c o n s t r a i n t s on t h e c o n d u c t i v i t y . E x t r a g e o p h y s i c a l and g e o l o g i c c o n s t r a i n t s on o(z) r e s t r i c t t h e ra n g e o f a c c e p t a b l e p r o f i l e s and h e l p c o n s t r u c t models t h a t a r e more r e p r e s e n t a t i v e of t h e t r u e e a r t h . In a d d i t i o n , c o n s t r a i n t s s p e c i f i c a l l y d e v i s e d t o I n t r o d u c t i o n / 10 e x p l o r e t h e s p a c e o f a c c e p t a b l e m o d e l s t e s t whether o r n o t i n t e r e s t i n g c o n d u c t i v i t y f e a t u r e s a r e a l l o w e d o r r e q u i r e d by t h e d a t a . C o n s e q u e n t l y , c o n s t r a i n t s a r e v i t a l t o a m e a n i n g f u l i n t e r p r e t a t i o n . G l o b a l c o n s t r a i n t s on t h e t o t a l v a r i a t i o n of t h e i m p u l s e r e s p o n s e a r e a p p l i e d by t h e l i n e a r i n v e r s e f o r m u l a t i o n and r e s u l t i n minimum s t r u c t u r e a(z) m o d e l s . The augmented method c a n a l s o a c c e p t a g l o b a l c o n s t r a i n t t o c o n s t r u c t c o n d u c t i v i t y models c l o s e t o an a p r i o r i e s t i m a t e . M o r e o v e r , t h e a l g o r i t h m a l l o w s f o r w e i g h t i n g f u n c t i o n s i n t h e f i r s t - s t a g e i n v e r s i o n f o r t h e i m p u l s e r e s p o n s e . W e i g h t i n g f u n c t i o n s p r o v i d e an e x t r a d i m e n s i o n of f l e x i b i l i t y t o i n f l u e n c e t h e c o n d u c t i v i t y o v e r l o c a l i z e d d e p t h r a n g e s . S i x t h , t h e augmented method u s e s two new i n t e g r a l e q u a t i o n mappings o f t h e i m p u l s e r e s p o n s e t o t h e c o n d u c t i v i t y ( S o n d h i and G o p i n a t h , 1971; B u r r i d g e , 1980). The two mappings u s e d by W e i d e l t (1972) p e r m i t o(z) t o have d i s c o n t i n u i t i e s o n l y i n i t s d e r i v a t i v e s . In c o n t r a s t , t h e new mappings can e x a c t l y r e p r o d u c e d i s c o n t i n u o u s m o d e l s . I t i s a l s o p o s s i b l e t o use i n v e r s e s c a t t e r i n g t h e o r y t o d e r i v e t h r e e a p p r o x i m a t e mappings (Howard, 1983). The most a c c u r a t e of t h e a p p r o x i m a t e mappings g e n e r a t e s models v e r y s i m i l a r t o t h o s e from t h e e x a c t i n t e g r a l e q u a t i o n s . F i n a l l y , t h e augmented i n v e r s e s c a t t e r i n g a l g o r i t h m i s d i r e c t l y a p p l i c a b l e t o t h e o n e - d i m e n s i o n a l r e f l e c t i o n s e i s m i c and dc r e s i s t i v i t y i n v e r s e p r o b l e m s . Most o f t h e r e s u l t s I n t r o d u c t i o n / 11 p r e s e n t e d h e r e have a l r e a d y been a p p l i e d t o t h e s e i s m i c c a s e . However, f o r dc r e s i s t i v i t y , t h e i n v e r s e t h e o r y a p p r o a c h y i e l d s many new r e s u l t s w h i c h complement t h e work o f Coen and Yu (1981) and S z a r a n i e c ( 1 9 8 2 ) . In f a c t , any t w o - s t a g e , o n e - d i m e n s i o n a l i n v e r s e s c a t t e r i n g p r o b l e m may b e n e f i t f r o m t h e s o l u t i o n s g i v e n i n t h i s t h e s i s . F o r t h e s e r e a s o n s , t h e i n v e r s e t h e o r y f o r m u l a t i o n of MT i n v e r s e s c a t t e r i n g y i e l d s a v e r y f l e x i b l e and r o b u s t a l g o r i t h m . The method i s e s p e c i a l l y d e s i g n e d t o combat t h e n o n u n i q u e n e s s by c o n s t r u c t i n g d i v e r s e models and by a c c e p t i n g e x t r a c o n s t r a i n t s . C o n s t r a i n t s a r e v e r y i m p o r t a n t f o r e x p l o r i n g t h e range o f a(z) models p e r m i t t e d by t h e d a t a and i n c r e a s i n g t h e r e l i a b i l i t y of an i n t e r p r e t a t i o n . However, i t i s d i f f i c u l t t o i n c o r p o r a t e l o c a l i z e d c o n s t r a i n t s i n t o t h e i n v e r s e s c a t t e r i n g a p p r o a c h . The p r o c e d u r e a p p r o x i m a t e s l o c a l i z e d c o n s t r a i n t s on o(z) by w e i g h t i n g t h e i n v e r s i o n f o r t h e i m p u l s e r e s p o n s e . T h i s method i s n o t p r e c i s e b e c a u s e o f t h e n o n l i n e a r r e l a t i o n s h i p between t h e i m p u l s e r e s p o n s e and t h e c o n d u c t i v i t y . An awkward i t e r a t i o n between t h e t r u e d e p t h z and a p s e u d o d e p t h x f u r t h e r c o m p l i c a t e s t h e a p p r o a c h b a s e d on i m p u l s e r e s p o n s e w e i g h t i n g f u n c t i o n s . B e c a u s e o f t h e n o n l i n e a r i t y i n t h e MT p r o b l e m , i t may be t h a t i n c o r p o r a t i n g l o c a l i z e d c o n s t r a i n t s w i l l a l w a y s r e q u i r e i t e r a t i o n . T h e r e f o r e , r a t h e r t h a n use a d i r e c t c o n s t r u c t i o n method and i t e r a t e t o a l l o w f o r t h e e x t r a c o n s t r a i n t s , I I n t r o d u c t i o n / 12 p r o p o s e an i t e r a t i v e scheme w h i c h a c c e p t s l o c a l i z e d c o n s t r a i n t s d i r e c t l y . In C h a p t e r 3, I d e r i v e an i n t e g r a l form o f a R i c c a t i e q u a t i o n t h a t r e l a t e s a f r e q u e n c y - d o m a i n MT r e s p o n s e t o a s i m p l e f u n c t i o n o f t h e c o n d u c t i v i t y . A l t h o u g h t h e i n t e g r a l e q u a t i o n i s e x a c t , i t i s n o n l i n e a r and must be s o l v e d by i t e r a t i o n . No i t e r a t i v e scheme c a n s u p e r s e d e a more e f f i c i e n t , e x a c t i n v e r s i o n method. The d i s a d v a n t a g e s o f i t e r a t i o n a r e t o l e r a t e d b e c a u s e t h e y a r e o u t w e i g h e d by t h e a d v a n t a g e t h a t t h e a l g o r i t h m a c c e p t s l o c a l i z e d c o n s t r a i n t s d i r e c t l y on o(z). In f a c t , t h e s o l e p u r p o s e o f t h e i t e r a t i v e a l g o r i t h m i s t o use c o n s t r a i n t s t o a s s e s s t h e n o n u n i q u e n e s s and r e f i n e c o n c l u s i o n s drawn f r o m an e x a c t i n v e r s i o n method. U s i n g R i c c a t i e q u a t i o n s t o s o l v e t h e MT i n v e r s e p r o b l e m i s n o t new. E c k h a r d t (1968) d e r i v e d R i c c a t i e q u a t i o n s f o r t h e MT impedance. He u s e d s i m p l e g r a p h i c a l and n u m e r i c a l t e c h n i q u e s f o r f o r w a r d and i n v e r s e m o d e l l i n g . B a i l e y (1970) gave an e x a c t i n v e r s i o n scheme b a s e d on a R i c c a t i e q u a t i o n , b u t was p e s s i m i s t i c a b o u t t h e p r a c t i c a l u s e s o f h i s a l g o r i t h m . O l d e n b u r g (1979) i n t e g r a t e d a l i n e a r i z e d MT R i c c a t i e q u a t i o n and u s e d l i n e a r i n v e r s e t h e o r y t o c o n s t r u c t i t e r a t i v e i mprovements t o an i n i t i a l o(z). G j e v i k e t a l . (1976) and N i l s e n and G j e v i k (1978) u s e d i t e r a t i o n t o s o l v e a R i c c a t i e q u a t i o n f o r t h e s e i s m i c r e f l e c t i o n p r o b l e m . Of t h e s e a p p r o a c h e s , o n l y O l d e n b u r g u s e d r o b u s t , i n v e r s e t e c h n i q u e s t o c o n s t r u c t s o l u t i o n s from t h e i n a c c u r a t e , I n t r o d u c t i o n / 13 b a n d l i m i t e d d a t a . The a l g o r i t h m I p r o p o s e s h a r e s t h i s m a j o r a d v a n t a g e o v e r p a s t s o l u t i o n s u s i n g R i c c a t i e q u a t i o n s . M o r e o v e r , t h e a t t e n d a n t p r o b l e m s o f c o n v e r g e n c e and o f c h o o s i n g an i n i t i a l o(z) a r e m i t i g a t e d by s t a r t i n g t h e i t e r a t i o n s w i t h one o f t h e d i v e r s e a c c e p t a b l e models from t h e MT i n v e r s e s c a t t e r i n g a l g o r i t h m . F i n a l l y , a l t h o u g h c o n s t r a i n e d l e a s t - s q u a r e s s o l u t i o n s a r e p o s s i b l e , I use a f a s t e r l i n e a r programming f o r m u l a t i o n w h i c h r e a d i l y a c c e p t s a wide v a r i e t y o f e x t r a e q u a l i t y and i n e q u a l i t y c o n s t r a i n t s on o(z). T h e r e f o r e , t h e m o t i v a t i o n f o r t h e s e two new a l g o r i t h m s d e r i v e s m a i n l y from a need t o a p p r a i s e t h e n o n u n i q u e n e s s of t h e MT i n v e r s e p r o b l e m . I n t e r p r e t a t i o n s b a s e d on a s i n g l e c o n d u c t i v i t y model o r c l a s s o f model c a n be m i s l e a d i n g . Some e f f o r t must be made t o r e p r e s e n t t h e range o f f e a t u r e s a l l o w e d o r r e q u i r e d by t h e d a t a . The t e c h n i q u e s d e s c r i b e d h e r e i n r e d u c e t h e a m b i g u i t i e s due t o n o n u n i q u e n e s s by c o n s t r i c t i n g t h e s p a c e o f a c c e p t a b l e models and by e x p l o r i n g t h e r e m a i n i n g s p a c e w i t h d i v e r s e model c l a s s e s . Hence, t h i s u n i f i e d a p p r o a c h e f f i c i e n t l y d o e s t h e work o f s e v e r a l a l g o r i t h m s . U s i n g t h e i n v e r s e s c a t t e r i n g a p p r o a c h i n c o n j u c t i o n w i t h a new n o n l i n e a r a p p r a i s a l t e c h n i q u e by O l d e n b u r g (1983) g i v e s v e r y c o m p l e t e and r e s p o n s i b l e MT i n t e r p r e t a t i o n s . CHAPTER 2. WEIDELT INVERSE SCATTERING In t h i s chapter, I present a f l e x i b l e MT inversion algorithm based on the inverse scattering formulation of Weidelt (1972). The solution separates into two p r i n c i p a l stages. The f i r s t stage (Sections 2.4 and 2.5) concerns the solution of a linear inverse problem to construct an impulse response function from the frequency-domain MT data. The second stage (Section 2.6) constructs the conductivity from the impulse response via a stable, Fredholm integral equation of the second kind. This two stage formulation was also pursued by Khachay (1978), Parker (1980), and Marchisio (1985). However, they used the spectral function rather than the impulse response. The f l e x i b i l i t y of model construction l i e s in the f i r s t stage because t h i s stage i s open to attack by linear inverse theory (Backus and G i l b e r t , 1967, 1968, 1970; Parker, 1977; Oldenburg, 1984). Sections 2.4.1 and 2.4.2 show how to construct varied impulse responses which f i t the observations. Each such impulse response, when used as input in the second stage of the solution, y i e l d s a d i f f e r e n t conductivity model. Hence, th i s single algorithm samples d i f f e r e n t regions in the space of acceptable models. I use Backus-Gilbert appraisal in Section 2.4.3 to quantify the a b i l i t y of frequency-domain MT data to resolve the impulse response. This analysis requires the measurement 14 W e i d e l t I n v e r s e S c a t t e r i n g / 15 f r e q u e n c i e s but no a c t u a l d a t a v a l u e s . T h e r e f o r e , b e f o r e any f i e l d work i s u n d e r t a k e n , i t c a n be u s e d t o i n d i c a t e t h e number and b a n d w i d t h of f r e q u e n c i e s n e c e s s a r y t o a c h i e v e a g i v e n l e v e l o f r e s o l u t i o n . In S e c t i o n 2.5, I a p p l y t h r e e t y p e s o f c o n s t r a i n t s t o t h e i m p u l s e r e s p o n s e i n v e r s i o n and examine t h e e f f e c t on t h e c o n d u c t i v i t y . Such c o n s t r a i n t s a r e v i t a l b e c a u s e t h e y g r e a t l y r e d u c e t h e n o n u n i q u e n e s s i n h e r e n t i n t h e MT method. In S e c t i o n s 2.6.1 and 2.6.2, I i n v e s t i g a t e f o u r i n t e g r a l f o r m u l a t i o n s o f t h e s e c o n d s t a g e : t h e G e l ' f a n d - L e v i t a n and M a rchenko e q u a t i o n s d i s c u s s e d by W e i d e l t ( 1 9 7 2 ) , t h e G o p i n a t h - S o n d h i e q u a t i o n ( S o n d h i and G o p i n a t h , 1971) and t h e B u r r i d g e (1980) e q u a t i o n . As an a l t e r n a t i v e t o t h e f o u r i n t e g r a l e q u a t i o n s , t h e r e a r e a p p r o x i m a t e methods w h i c h f i n d a o(z) p r o f i l e f r o m t h e i m p u l s e r e s p o n s e . In S e c t i o n 2.6.3, I e v a l u a t e t h r e e a p p r o x i m a t i o n s a n a l o g o u s t o t h o s e p r e s e n t e d by Howard ( 1 9 8 3 ) . 2 . 1 . REVIEW OF WEIDELT'S APPROACH F i r s t , I o u t l i n e t h e i n v e r s i o n p r o c e d u r e by W e i d e l t ( 1 9 7 2 ) . C o n s i d e r an e l e c t r i c a l l y c o n d u c t i n g h a l f - s p a c e i n w h i c h t h e c o n d u c t i v i t y i s i s o t r o p i c and v a r i e s w i t h d e p t h z o n l y . A v e r t i c a l l y p r o p a g a t i n g , p l a n e e l e c t r o m a g n e t i c wave w i t h t i m e v a r i a t i o n e l u t i m p i n g e s on t h e s u r f a c e . The c o o r d i n a t e s y s t e m i s C a r t e s i a n w i t h t h e o r i g i n a t t h e s u r f a c e W e i d e l t I n v e r s e S c a t t e r i n g / 16 and z p o s i t i v e downwards. Assume t h e e l e c t r i c f i e l d i s p o l a r i z e d i n t h e y d i r e c t i o n a nd t h e m a g n e t i c f i e l d has o n l y an x component. The complex a m p l i t u d e s o f t h e s e f i e l d s a t f r e q u e n c y w a r e E(Z,<J) and H(z,a), r e s p e c t i v e l y . The d i f f e r e n t i a l e q u a t i o n s a t i s f i e d by t h e e l e c t r i c f i e l d i s ( T i k h o n o v , 1950; C a g n i a r d , 1953) E"(z,u) - i u0a)o(z)E(z,CJ) = 0, (1) where a p r i m e d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o z. The f o l l o w i n g t r a n s f o r m a t i o n maps t h i s MT e q u a t i o n i n t o a S c h r o d i n g e r e q u a t i o n . L e t k2 = iu0oto0, (2) z x = J [ o ( t ) / o 0 ] ' / 2 d t , (3) 0 u(x) = \ a ( z ) / a 0 V / \ (4) and f(x,k) = u(x)E(z,u)/E(0,u>) , (5) where u0 i s t h e p e r m e a b i l i t y o f f r e e s p a c e and o0=o(z=0) i s t h e s u r f a c e c o n d u c t i v i t y . S e c t i o n s 2.2, 2.4.1, and 2.5.1 a l l g i v e p r o c e d u r e s f o r e s t i m a t i n g a0. U s i n g t h e t r a n s f o r m a t i o n g i v e n by e q u a t i o n s (2) t h r o u g h ( 5 ) , e q u a t i o n (1) becomes f"(x,k) - [k2+V(x)]f(x,k) = 0, (6) where t h e p o t e n t i a l V(x) i s g i v e n by V(x) = u"(x)/u(x). (7) The v a r i a b l e k i s a complex wavenumber w i t h d i m e n s i o n s of i n v e r s e l e n g t h . k has e q u a l r e a l and i m a g i n a r y p a r t s . The W e i d e l t I n v e r s e S c a t t e r i n g / 17 v a r i a b l e x o f e q u a t i o n (3) i s a p s e u d o d e p t h d e r i v e d from a n o n l i n e a r s t r e t c h i n g of t h e z c o o r d i n a t e . In t h e s e i s m i c p r o b l e m , t h i s s t r e t c h i n g i s e q u i v a l e n t t o a t r a n s f o r m a t i o n from d e p t h t o t r a v e l t i m e u s i n g t h e s p e e d o f p r o p a g a t i o n . B e c a u s e t h e e l e c t r i c f i e l d d i f f u s e s a t a r a t e w h i c h v a r i e s a s 1/V'a, a h i g h - c o n d u c t i v i t y peak a p p e a r s t h i c k e r i n t h e x domain t h a n i n t h e z domain. A l t e r n a t i v e l y , a h i g h l y r e s i s t i v e r e g i o n i s n a r r o w e d . The e f f e c t o f t h e t r a n s f o r m a t i o n , e q u a t i o n ( 3 ) , i s seen by c o m p a r i n g t h e l e f t h a l f o f F i g u r e 1, w h i c h c o n t a i n s two a(z) t e s t m o d e l s , w i t h t h e r i g h t h a l f w h i c h c o n t a i n s t h e c o r r e s p o n d i n g o(x) m o d e l s . R e t u r n i n g t o t h e S c h r o d i n g e r e q u a t i o n ( 6 ) , n o t e t h a t t h e c o n d u c t i v i t y s t r u c t u r e i s e n c o d e d i n t h e p o t e n t i a l V(x). One f i n d s V(x) u s i n g s c a t t e r i n g t h e o r y and an e l e c t r o m a g n e t i c r e s p o n s e r e c o r d e d on t h e s u r f a c e . F o r a r e s p o n s e , W e i d e l t u s e d t h e complex r a t i o E(0,io) E ( 0 , u ) C(CJ) - - = - , E'(0,u)) i uou)H(0,u) w h i c h i s o b t a i n e d from s u r f a c e measurements. The r e s p o n s e s c(u.) j = l , 2 , . . . N , or where A: = [ //u0w . o 0 J 1 / 2 r p l u s t h e a s s o c i a t e d s t a n d a r d d e v i a t i o n e s t i m a t e s , c o n s t i t u t e t h e d a t a s e t t o be i n v e r t e d . However, i n s t e a d o f u s i n g c(k) d i r e c t l y i n t h e i n v e r s i o n , W e i d e l t u s e d two complex wavenumber-domain r e f l e c t i v i t y f u n c t i o n s Weidelt Inverse S c a t t e r i n g / 18 10-1 10 20 30 40 50 Depth z 20 40 60 80 100 Pseudodepth x FIGURE 1 C o n d u c t i v i t y Test Models Two a(z) t e s t models are shown on the l e f t , panels (a) and (b) . Both curves represent roughly the same c o n d u c t i v i t y f u n c t i o n except that one i s continuous and one i s l a y e r e d . The corresponding a(x) models given by equation (3) are p l o t t e d i n panels (c) and (d) on the r i g h t . The t r a n s f o r m a t i o n from z to x i s analogous to a change from depth to t r a v e l t i m e i n the seismic problem. l-kc(k) r(k) = (8) l + kc(k) and b(k) = [l-kc(k)]/2. (9) S e c t i o n 2.3 demonstrates how these t r a n s f o r m a t i o n s map the measurement e r r o r s in c(k) to e r r o r s i n r(k) and b(k). W e i d e l t I n v e r s e S c a t t e r i n g / 19 A L a p l a c e t r a n s f o r m r e l a t e s r(k) and b(k) t o t h e i r c o r r e s p o n d i n g p s e u d o d e p t h - d o m a i n i m p u l s e r e s p o n s e s R(x) and B(x). T h a t i s , oo r(k) = / R(x)e~kxdx (10) 0 and O O b(k) = / B(x)e~kxdx, (11) 0 where R(x) and B(x) a r e r e a l f u n c t i o n s o f x. N o t e t h a t , by a n a l o g y w i t h t h e s e i s m i c r e f l e c t i o n p r o b l e m , t h e i m p u l s e r e s p o n s e R(x) has i n t e r n a l b u t no s u r f a c e m u l t i p l e r e f l e c t i o n s . The r e s p o n s e B(x) i n c l u d e s t h e s u r f a c e m u l t i p l e s . F i g u r e 2 shows t h e i m p u l s e r e s p o n s e s R(x) and B(x) c o r r e s p o n d i n g t o t h e two a(x) m o d e l s of F i g u r e s 1c and 1d. The two-way d e p t h h o r i z o n t a l c o o r d i n a t e i s a n a l o g o u s t o t h e s e i s m i c two-way t r a v e l t i m e . The i m p u l s e r e s p o n s e down t o 2x i s n e c e s s a r y t o r e c o n s t r u c t t h e c o n d u c t i v i t y down t o l e v e l x. I c a l c u l a t e t h e s e r e s p o n s e s r e c u r s i v e l y i n a manner s i m i l a r t o Wuenschel ( 1 9 6 0 ) . I b r e a k t h e o(x) model i n t o homogeneous l a y e r s and a s l i g h t l y m o d i f i e d W uenschel a l g o r i t h m g i v e s t h e d e l t a f u n c t i o n i m p u l s e r e s p o n s e . F o r t h e MT p r o b l e m , t h e r e f l e c t i o n c o e f f i c i e n t a t i n t e r f a c e / i s d e f i n e d by W e i d e l t (1972) a s rt = — — 1 = 0 , 1 , 2 , . . . Here I b r o a d e n t h e d e l t a f u n c t i o n s i n t o b o x c a r s h a p e s w h i l e W e i d e l t I n v e r s e S c a t t e r i n g / 20 0 40 80 120 160 200 Two-Way Depth 2x 0 40 80 120 160 200 Two-Way Depth 2x A 0.0 oz • -0.2 ——1 r— b ; 1 1 , 1 . 1 , 1 , 0 40 80 120 160 200 Two-Way Depth 2x o.o -0.2 -— i — i — i , .1 i i i l_ llhi..fli. ..lit : 1 |ll . i . i . i •• 1 0 40 80 120 160 200 Two-Way Depth 2x FIGURE 2 Impulse Responses R(x) and B(x) P a n e l s (a) and (b) on t h e l e f t g i v e t h e i m p u l s e r e s p o n s e R(x) [a(x<0)=oo] c o r r e s p o n d i n g t o t h e smooth and m u l t i l a y e r a(x) models o f F i g u r e s 1c and 1d. S i m i l a r l y , p a n e l s ( c ) and (d) on th e r i g h t g i v e t h e i m p u l s e r e s p o n s e B(x) [o(x<0)=0]. The two-way d e p t h 2x i s a n a l o g o u s t o t h e s e i s m i c two-way t r a v e l t i m e . m a i n t a i n i n g t h e i r a r e a a s r^ . The w i d t h i s e q u a l t o "the d i s c r e t i z a t i o n i n t e r v a l w i d t h . T h i s b l o c k y n a t u r e of R(x) and B(x) i s e v i d e n t f o r t h e m u l t i l a y e r model i n F i g u r e s 2b and 2d, but i t was a r b i t r a r i l y s u p p r e s s e d when p l o t t i n g t h e smooth model r e s p o n s e s i n F i g u r e s 2a and 2c. F i g u r e s 2c and 2d i l l u s t r a t e how t h e r e s p o n s e B(x) i s c l u t t e r e d by what a r e a n a l o g o u s t o s e i s m i c m u l t i p l e r e f l e c t i o n s f r o m t h e s t r e s s - f r e e W e i d e l t I n v e r s e S c a t t e r i n g / 21 s u r f a c e . The r e s p o n s e R(x) i n F i g u r e s 2a and 2b i s f r e e o f t h i s c o n t a m i n a t i o n , and l a r g e e v e n t s t h e r e a r e more l i k e l y t o c o r r e s p o n d t o a c t u a l e a r t h i n t e r f a c e s . The i m p u l s e r e s p o n s e s a r e u s e d i n t h e Marchenko e q u a t i o n x A(x,y) = R(x+y) + J A(x, t )R(y+t) dt \y\±x, (12) -y or t h e G e l ' f a n d - L e v i t a n e q u a t i o n x A(x,y) = B(x+y) + J A(x, t) [ B (y+t) +B (y-1) ] dt \y\<x, (13) -x t o f i n d t h e r e a l f u n c t i o n o f y, A(x,y), f o r e a c h v a l u e of t h e p a r a m e t e r x. The p s e u d o d e p t h - d o m a i n c o n d u c t i v i t y i s t h e n x a(x)/a0 = uk(x) = [1 + j A(x,t)dt]\ (14) - x w i t h t h e f i n a l map from o(x) t o a(z) g i v e n by r e a r r a n g i n g e q u a t i o n (3) as z = / [ o 0 / a ( t ) ] ' / 2 d t . (15) 0 E q u a t i o n (14) i s a s h o r t c u t w h i c h b y p a s s e s c a l c u l a t i n g V(x) from A(x,y) and f i n d i n g u(x) from e q u a t i o n (7) . F i g u r e 3 shows t h e a(x) models ( d a s h e d l i n e s ) c a l c u l a t e d f r o m d i s c r e t e v e r s i o n s of e q u a t i o n s (12) and (13) o p e r a t i n g on t h e i m p u l s e r e s p o n s e s o f F i g u r e 2. The t r u e c o n d u c t i v i t y model ( s o l i d l i n e ) i s a l s o p l o t t e d i n e a c h f r a m e . The Marchenko and G e l ' f a n d - L e v i t a n r e c o n s t r u c t i o n s o f t h e smooth t e s t model shown i n F i g u r e s 3a and 3c a r e a l m o s t i d e n t i c a l t o t h e t r u e W e i d e l t I n v e r s e S c a t t e r i n g / 22 _J I i I i I 0 20 40 60 80 100 Pseudodepth x IO"3 20 40 60 80 100 P s e u d o d e p t h x 10- 1 F 20 40 60 80 100 Pseudodepth x 20 40 60 80 100 P s e u d o d e p t h x FIGURE 3 C o n d u c t i v i t y from True Impulse Response P l o t t e d i n e a c h p a n e l i s t h e t r u e a(x) t e s t model ( s o l i d l i n e ) a l o n g w i t h an i n t e g r a l e q u a t i o n r e c o n s t r u c t i o n d e r i v e d u s i n g an e x a c t i m p u l s e r e s p o n s e f rom F i g u r e 2. P a n e l s (a) a n d (b) on t h e l e f t e a c h g i v e a Marchenko r e c o n s t r u c t i o n u s i n g R(x) ( d a s h e d l i n e ) . P a n e l s ( c ) and (d) on t h e r i g h t e a c h g i v e a G e l ' f a n d - L e v i t a n r e c o n s t r u c t i o n u s i n g B(x) ( d a s h e d l i n e ) . T h e s e e q u a t i o n s p r o d u c e a good match t o t h e smooth t e s t m o d e l . However, f o r l a y e r e d m o d e ls t h e r e a r e s m a l l i n a c c u r a c i e s . The B u r r i d g e and G o p i n a t h - S o n d h i e q u a t i o n s d i s c u s s e d i n S e c t i o n 2.6.1 g i v e r e c o n s t r u c t i o n s i d e n t i c a l t o t h e t r u e smooth o r m u l t i l a y e r m o d e l . a(x). However, t h e l a y e r e d model r e c o n s t r u c t i o n s ( F i g u r e s 3b and 3d, d a s h e d l i n e s ) show s m a l l i n a c c u r a c i e s . T h i s i s b e c a u s e t h e d e r i v a t i o n o f e q u a t i o n s (12) and (13) p e r m i t t e d a(z) t o have d i s c o n t i n u i t i e s o n l y i n i t s d e r i v a t i v e s ( W e i d e l t , 1972). W e i d e l t Inverse S c a t t e r i n g / 23 T h i s c o r r e s p o n d s to a d e l t a - f u n c t i o n p o t e n t i a l (Ware and A k i , 1969). L a y e r e d models v i o l a t e t h i s r e s t r i c t i o n but the r e c o n s t r u c t i o n s are s t i l l very c l o s e to b e i n g e x a c t . For n u m e r i c a l c o m p u t a t i o n s , both the smooth and m u l t i l a y e r models are r e p r e s e n t e d by homogeneous l a y e r s . However, the smooth model r e f l e c t i o n c o e f f i c i e n t s have magnitudes l e s s than 0.04. In c o n t r a s t , the m u l t i l a y e r model has s e v e r a l w i t h magnitudes near 0.40. The s m a l l e r the d i s c o n t i n u i t i e s in o(z), the s m a l l e r the r e f l e c t i o n c o e f f i c i e n t s and the b e t t e r the r e c o n s t r u c t i o n . When o p e r a t i n g on the t r u e impulse re sponse , the B u r r i d g e and G o p i n a t h - S o n d h i e q u a t i o n s d i s c u s s e d in S e c t i o n 2 .6 .1 reproduce a l l l a y e r e d models e x a c t l y . I summarize W e i d e l t ' s i n v e r s e s c a t t e r i n g f o r m u l a t i o n as f o l l o w s . Conver t from frequency u> to wavenumber k. T r a n s f o r m the observed complex response c(k) p l u s i t s measurement e r r o r s to a wavenumber-domain r e f l e c t i v i t y r(k) or b(k) w i t h e r r o r s . T r a n s f o r m these data to an impulse response R(x) or B(x). Then for each f i x e d x, s o l v e the Marchenko or G e l ' f a n d - L e v i t a n e q u a t i o n f o r A(x,y), \y\<x. I n t e g r a t e A(x,y) t o f i n d the c o n d u c t i v i t y a at the pseudodepth x. F i n a l l y , use a l l the p r e v i o u s a v a l u e s to f i n d the depth z c o r r e s p o n d i n g to x. In a more c o n c i s e form, these s teps are Step 1: c(u>) => c(k), e q u a t i o n (2) ; Step 2: c(k) => r (k) or b(k), e q u a t i o n (8) or (9 ) ; Step 3: r(k) or b (k) => R(x) or B(x) , e q u a t i o n (10) or (11); Weidelt Inverse S c a t t e r i n g / 24 Step 4: R(x) or B(x) => A(x,y), equation (12) or (13); Step 5: A(x,y) => a(x), equation (14); Step 6: a(x) => a(z), equation (15). In S e c t i o n s 2.2, 2.4.1, and 2.5.1, I d i s c u s s procedures for e s t i m a t i n g the parameter o 0 r e q u i r e d by step 1. The only concern f o r step 2 i s how the standard d e v i a t i o n s of c(k) transform to e r r o r s i n r(k) and b(k). S e c t i o n 2.3 shows that Gaussian e r r o r s i n c(k) l e a d to Gaussian e r r o r s i n r(k) and b(k). Weidelt s t a t e d that step 3 i s the most d i f f i c u l t . He presented two a n a l y t i c a l g o r i t h m s but they are not completely s a t i s f a c t o r y because they r e q u i r e p r e c i s e data over a broad frequency range. In S e c t i o n s 2.4 and 2.5, I present the in v e r s e theory approach which i s more robust s i n c e i t admits from the ou t s e t that the data are incomplete and i n a c c u r a t e . In S e c t i o n 2.6.1, I e v a l u a t e the performance of the Marchenko and G e l ' f a n d - L e v i t a n equations as w e l l as two new i n t e g r a l f o r m u l a t i o n s of step 4. Steps 5 and 6 present no d i f f i c u l t i e s . The numerical d e t a i l s are given i n S e c t i o n 2.6.2. I d i s c u s s three approximate f o r m u l a t i o n s combining steps 4 and 5 i n Se c t i o n 2.6.3. 2.2. FREQUENCY-WAVENUMBER TRANSFORMATION The t r a n s f o r m a t i o n i n equation (2) from angular frequency u> to wavenumber k r e q u i r e s a s p e c i f i c a t i o n of the su r f a c e c o n d u c t i v i t y o0. Geologic i n t e r p r e t a t i o n s or w e l l l o g s near W e i d e l t I n v e r s e S c a t t e r i n g / 25 t h e MT s u r v e y s i t e may p r o v i d e one e s t i m a t e o f aQ. A n o t h e r e s t i m a t e f o l l o w s f r o m t h e h i g h - f r e q u e n c y b e h a v i o u r o f t h e c(u) s o u n d i n g c u r v e . I f t h e phase o f c(co) a t t h e h i g h e s t measured f r e q u e n c y i s near -45 d e g r e e s ( t h e v a l u e c o r r e s p o n d i n g t o a c o n s t a n t c o n d u c t i v i t y h a l f - s p a c e ) t h e n t h i s f r e q u e n c y i s r e s p o n d i n g o n l y t o t h e s u r f a c e l a y e r o f c o n d u c t i v i t y a 0 . W e i d e l t (1972) showed t h a t a s a>->», c(cj)->l/k. Hence, a t t h e h i g h e s t measured f r e q u e n c y o)y , c(co,) 1/ky - l/[ i (f 0w, o0 ] 1 / 2 . S o l v i n g f o r o0 g i v e s o Q « 7 / [ / x 0 " i I c(<*>0 | 2 ] = oa(uO , where o ^ f w j i s t h e a p p a r e n t c o n d u c t i v i t y ( i n v e r s e of a p p a r e n t r e s i s t i v i t y ) a t f r e q u e n c y CJ, . oa(u>i) i s t h e c o n d u c t i v i t y o f t h e h a l f - s p a c e w h i c h b e s t f i t s t h e datum c(u>^) . I t i s a l s o p o s s i b l e t o e s t i m a t e o0 as t h a t h a l f - s p a c e c o n d u c t i v i t y w h i c h b e s t f i t s t h e two o r t h r e e h i g h e s t f r e q u e n c y c(u>) d a t a . T h i s g e n e r a l i z a t i o n t o f i t t i n g s e v e r a l d a t a d e c r e a s e s t h e s u s c e p t i b i l i t y of t h e o0 e s t i m a t e t o a b a d l y b i a s e d datum. I use two o t h e r t e c h n i q u e s t o e s t i m a t e oQ. The f i r s t method u s e s t h e s u r f a c e v a l u e of t h e i m p u l s e r e s p o n s e ( S e c t i o n 2 . 4 . 1 ) , and t h e s e c o n d u s e s t h e p h y s i c a l r e a l i z a b i l i t y c o n s t r a i n t s ( S e c t i o n 2 . 5 . 1 ) . W e i d e l t I n v e r s e S c a t t e r i n g / 26 2.3. WAVENUMBER-DOMAIN RESPONSES AND ERRORS E q u a t i o n s (8) and (9) show how t o t r a n s f o r m c(k) t o r(k) and b(k), r e s p e c t i v e l y . F o r p r a c t i c a l d a t a s e t s , t h e s t a n d a r d d e v i a t i o n s o f c(k) must a l s o be t r a n s f o r m e d t o t h e c o r r e s p o n d i n g e r r o r s i n r(k) and b(k). I a c c o m p l i s h t h i s u s i n g a s i m p l e n u m e r i c a l s i m u l a t i o n . I assume t h a t t h e e r r o r s i n t h e r e a l and i m a g i n a r y p a r t s o f c(k) a r e i n d e p e n d e n t and G a u s s i a n w i t h z e r o mean. I t h e n g e n e r a t e a G a u s s i a n d i s t r i b u t i o n o f n o i s y c(k) v a l u e s c e n t e r e d on t h e measured datum and h a v i n g i t s s t a n d a r d d e v i a t i o n . F i g u r e 4a g i v e s h i s t o g r a m s o f one r e a l i z a t i o n of a c(k) d i s t r i b u t i o n a t a s i n g l e wavenumber. The s o l i d l i n e c o r r e s p o n d s t o t h e r e a l p a r t o f c(k) and t h e d a s h e d l i n e c o r r e s p o n d s t o t h e i m a g i n a r y p a r t . F o r a G a u s s i a n d i s t r i b u t i o n , 68 p e r c e n t o f t h e v a l u e s f a l l w i t h i n ±1 s t a n d a r d d e v i a t i o n and 95 p e r c e n t f a l l w i t h i n ±2 s t a n d a r d d e v i a t i o n s . F o r t h i s r e a l i z a t i o n , t h e c o r r e s p o n d i n g p e r c e n t a g e s a r e 70 and 87, i n d i c a t i n g t h a t t h e t a i l s a r e s l i g h t l y t o o l o n g . I f e a c h of t h e n o i s y c(k) v a l u e s i n F i g u r e 4a i s t r a n s f o r m e d by e q u a t i o n (8) t o an r(k) v a l u e t h e n t h e d i s t r i b u t i o n s i n F i g u r e 4b r e s u l t . The s o l i d a nd d a s h e d l i n e s c o r r e s p o n d t o t h e r e a l and i m a g i n a r y p a r t s of r(k), r e s p e c t i v e l y . T h e s e h i s t o g r a m s of r(k) v a l u e s a r e n o t b i a s e d , t h a t i s , t h e y a r e c e n t e r e d on t h e r(k) v a l u e d e r i v e d f r o m e q u a t i o n (8) u s i n g t h e o r i g i n a l m e asured c(k) datum. M o r e o v e r , t h e h i s t o g r a m s c l o s e l y r e s e m b l e G a u s s i a n d i s t r i b u t i o n s b e c a u s e 68 and 86 p e r c e n t of t h e v a l u e s W e i d e l t I n v e r s e S c a t t e r i n g / 27 100 - 4 - 2 0 2 Standard Deviation u 100 80 60 40 20 v r 1 ' d: ii - 4 - 2 0 2 Standard Deviation FIGURE 4 Histograms of Noisy Data P a n e l (a) c o n t a i n s h i s t o g r a m s o f n o i s y c(k) v a l u e s d e r i v e d by a d d i n g pseudorandom, zero-mean G a u s s i a n n o i s e t o an a c c u r a t e c(k) datum a t one wavenumber. The s o l i d l i n e c o r r e s p o n d s t o t h e r e a l p a r t o f c(k) and t h e d a s h e d l i n e c o r r e s p o n d s t o t h e i m a g i n a r y p a r t . P a n e l (b) shows r(k) h i s t o g r a m s g e n e r a t e d by e q u a t i o n (8) a n d t h e n o i s y c(k) v a l u e s i n ( a ) . P a n e l ( c ) shows s i m i l a r b(k) h i s t o g r a m s . P a n e l (d) shows a p p a r e n t c o n d u c t i v i t y ( s o l i d l i n e ) and phase ( d a s h e d l i n e ) h i s t o g r a m s d e r i v e d from t h e n o i s y c(k) v a l u e s i n ( a ) . Hence, G a u s s i a n n o i s e on c(k) maps t o G a u s s i a n n o i s e on r(k), b(k), and aa(k) and 4>(k) d a t a . l i e between ±7 and ±2 s t a n d a r d d e v i a t i o n s , r e s p e c t i v e l y . The t a i l s f o r t h e s e d i s t r i b u t i o n s a r e a g a i n t o o l o n g b e c a u s e t h e t a i l s o f t h e o r i g i n a l c(k) d i s t r i b u t i o n s were t o o l o n g . F i g u r e 4c shows t h e ( u n b i a s e d ) d i s t r i b u t i o n s of b(k) v a l u e s d e r i v e d f r o m F i g u r e 4a and e q u a t i o n (9). F i g u r e 4d shows t h e W e i d e l t I n v e r s e S c a t t e r i n g / 28 70 1? 60 50 -40 t E j i d i i i 1 1 1 i n t i i i i IO" 2 I O r 1 Wavenumber tc (m - 1 ) IO" 2 IO" 1 Wavenumber K (m - 1 ) FIGURE 5 Smooth Model Wavenumber-Domain Data P a r t 1 P a n e l s (a) and (b) g i v e t h e r e a l and n e g a t e d i m a g i n a r y p a r t s o f t h e complex r e s p o n s e c(k) f o r t h e smooth a(z) t e s t m o d e l . The h o r i z o n t a l a x i s i s t h e r e a l ( o r i m a g i n a r y ) p a r t o f t h e complex wavenumber k. The smooth l i n e s a r e t h e t r u e r e s p o n s e s and t h e symbols a r e n o i s y v a l u e s . The e r r o r b a r s on t h e s ymbols c o r r e s p o n d t o one s t a n d a r d d e v i a t i o n e q u a l t o 5 p e r c e n t o f \c(k)\. P a n e l s ( c ) and (d) p l o t t h e e q u i v a l e n t a p p a r e n t c o n d u c t i v i t y and p h a s e d a t a . The e r r o r b a r s h e r e a r e d e r i v e d n u m e r i c a l l y from t h e n o i s y c(k) d a t a . d i s t r i b u t i o n s o f a p p a r e n t c o n d u c t i v i t y °a(k) ( s o l i d l i n e ) and p h a s e <}>(k) ( d a s h e d l i n e ) d e r i v e d from F i g u r e 4a and ajk) = o0/\kc(k)\2 <t>(k) = arct an[lm{c(k)}/Re{c(k)}] * 180/n + 90. The d i s t r i b u t i o n s have 70 and 86 p e r c e n t o f t h e i r v a l u e s W e i d e l t I n v e r s e S c a t t e r i n g / 29 IO"2 10"1 Wavenumber K (m - 1) 8 o.io IO"2 1 Q - 1 Wavenumber JC (m - 1) IO"2 1 0 - i 1 0 - 2 1 Q - i Wavenumber tc (m - 1) Wavenumber K (m _ 1) FIGURE 6 Smooth Model Wavenumber-Domain Data Part 2 P a n e l s (a) and (b) g i v e t h e r e a l and n e g a t e d i m a g i n a r y p a r t s o f t h e complex r e s p o n s e r(k) f o r t h e smooth a(z) t e s t m o d e l . The h o r i z o n t a l a x i s i s t h e r e a l ( o r i m a g i n a r y ) p a r t o f t h e complex wavenumber k. The smooth l i n e s a r e t h e t r u e r e s p o n s e s and t h e symbols a r e n o i s y v a l u e s . The e r r o r b a r s on t h e symbols c o r r e s p o n d t o one s t a n d a r d d e v i a t i o n and a r e d e r i v e d n u m e r i c a l l y f r o m t h e n o i s y c(k) d a t a i n F i g u r e s 5a and 5b. P a n e l s ( c ) and (d) p l o t t h e e q u i v a l e n t b(k) d a t a f o r t h e smooth o(z) t e s t m o d e l . w i t h i n ±1 and ±2 s t a n d a r d d e v i a t i o n s . F i g u r e 4 d e m o n s t r a t e s t h a t zero-mean G a u s s i a n e r r o r s i n c(k) d a t a t r a n s f o r m t o zero-mean G a u s s i a n e r r o r s i n r(k), b(k), and o~Q(k) and <j>(k) d a t a . F i g u r e s 5 and 6 g i v e t h e d a t a c o r r e s p o n d i n g t o t h e smooth W e i d e l t I n v e r s e S c a t t e r i n g / 30 o(z) t e s t m o d e l . The d a t a f o r t h e m u l t i l a y e r model a r e q u i t e s i m i l a r . I n t h e s e two f i g u r e s , t h e c o n t i n u o u s l i n e s a r e t h e e x a c t r e s p o n s e s p l o t t e d a s f u n c t i o n s o f K, w h i c h i s t h e r e a l o r i m a g i n a r y p a r t o f t h e complex wavenumber k. The sym b o l s w i t h one s t a n d a r d d e v i a t i o n e r r o r b a r s a r e t h e e x a c t r e s p o n s e s a t 75 wavenumbers c o n t a m i n a t e d by one r e a l i z a t i o n o f zero-mean G a u s s i a n n o i s e . I a r b i t r a r i l y s e t t h e s t a n d a r d d e v i a t i o n s of t h e s e i n a c c u r a t e c(k) d a t a e q u a l t o 5 p e r c e n t o f |cf<U| ( F i g u r e s 5a and 5 b ) . A l l o t h e r n o i s y d a t a and s t a n d a r d d e v i a t i o n s a r e d e r i v e d from t h e s e n o i s y c(k). F i g u r e s 5c and 5d g i v e t h e a c c u r a t e and i n a c c u r a t e °Q(k) and <t>(k) d a t a . The s t a n d a r d d e v i a t i o n s d e r i v e d from t h e c(k) e r r o r s a r e a b o u t 10 p e r c e n t f o r oQ(k) and 6 t o 10 p e r c e n t f o r <t>(k). F i g u r e s 6a and 6b show t h e complex r e f l e c t i v i t y r(k) and F i g u r e s 6c and 6d show b(k). In s u b s e q u e n t s e c t i o n s , I use t h e d a t a f r o m F i g u r e 6 t o t e s t t h e p e r f o r m a n c e of t h e i n v e r s i o n a l g o r i t h m s . 2 . 4 . IMPULSE RESPONSE INVERSION The l i n e a r i n v e r s e p r o b l e m o f e q u a t i o n s (10) and (11) c a n be s t a t e d as f o l l o w s . G i v e n a f i n i t e number N o f i n a c c u r a t e , complex d a t a c(k.) j = l,2,...N, f o r m r(kj) and b(kj) u s i n g e q u a t i o n s (8) and ( 9 ) . The r e l a t i o n s h i p s between t h e s e d a t a and t h e i m p u l s e r e s p o n s e s a r e t h e n 0 0 - I r r(k.) = f R(x)e jXdx, (16) J 0 and W e i d e l t I n v e r s e S c a t t e r i n g / 31 b(k ) = J* B(x)e~kjxdx, (17) J 0 where j = l , 2 , . . . N . I s o l v e t h e s e e q u a t i o n s f o r R(x) and B(x) u s i n g s t a n d a r d t e c h n i q u e s of l i n e a r i n v e r s e t h e o r y (Backus and G i l b e r t , 1967; P a r k e r , 1977; O l d e n b u r g , 1984). U n i q u e R(x) and B(x) c a n n o t be f o u n d f r o m o n l y N complex and i n a c c u r a t e d a t a . F u r t h e r , e a c h d i f f e r e n t R(x) or B(x) c o m p a t i b l e w i t h t h e measured d a t a p r o d u c e s a d i f f e r e n t a(z) v i a t h e i n t e g r a l e q u a t i o n s (12) or ( 1 3 ) . T h e r e f o r e , i n o r d e r t o e x p l o r e t h e s p a c e o f a c c e p t a b l e o(z), I m i n i m i z e s e v e r a l norms o f R(x) and B(x). T h e s e a r e t h e L, and L2 norms o f t h e i m p u l s e r e s p o n s e o r t h e i m p u l s e r e s p o n s e g r a d i e n t w i t h r e s p e c t t o p s e u d o d e p t h . I f i n d , f o r example, an R(x) w h i c h s a t i s f i e s e q u a t i o n (16) and a t t h e same t i m e m i n i m i z e s one o f t h e f o l l o w i n g norms: 00 CO 01 = ; w(x) \R(x) \ 2dx, <p3 = S w(x)\R(x)\dx, 0 0 (18) oo oo ~" 0 2 = ; w(x)\R' (x) | 2dx, 0« = / w(x)\R' (x)\dx, 0 0 where w(x) i s a p o s i t i v e w e i g h t i n g f u n c t i o n . The L2 norms 0, and 02 g e n e r a t e " s m a l l " and " f l a t " m o d e l s w i t h minimum e n e r g y and g r a d i e n t e n e r g y , r e s p e c t i v e l y . <f>3 and 0ft a l s o g e n e r a t e s m a l l and f l a t models b u t i n t h e L% norm s e n s e . Minimum norm models a r e i m p o r t a n t b e c a u s e t h e y do n o t o v e r e s t i m a t e t h e f e a t u r e s r e q u i r e d by t h e d a t a . I d e m o n s t r a t e i n S e c t i o n 2.6.3 how s u c h i m p u l s e r e s p o n s e s l e a d t o , i n some s e n s e , minimum s t r u c t u r e c o n d u c t i v i t y p r o f i l e s . I n t e r e s t i n g r e g i o n s of t h e s e Weidelt Inverse S c a t t e r i n g / 32 models are more l i k e l y to be r e p r e s e n t a t i v e of the true e a r t h . Note that a l t e r i n g the weighting f u n c t i o n w(x) i s an e f f e c t i v e means of producing v a r i e d impulse response models ( S e c t i o n 2.5.3). 2.4.1. B a c k u s - G i l b e r t C o n s t r u c t i o n F i r s t , I use the methods of Backus and G i l b e r t to c o n s t r u c t small models which minimize #i s u b j e c t to the data - k x c o n s t r a i n t s . The data r(kj) a n <3 the k e r n e l s e j i n equation (16) are complex, whereas the impulse response R(x) i s r e a l . If there are N complex data, the r e a l and imaginary p a r t s give 2N equations of the form CO Re{r(k.)} = / R( x) e~ K jx c os ( K . x) dx J 0 J and (19) CO -lm{r(k.)} = J R(x)e~KjXsin(K.x)dx, J 0 J where KJ = [U0OJ O0/2] 1 / 2 and j = l,2,...N. Represent the two d i f f e r e n t k e r n e l s of equations (19) by g.(x) j=1, 2,... 2N, and the corresponding data by e. j=l,2,...2N. Equations (19) become e. = J R(x) g .(x) dx j = l,2,...2N. (20) The impulse response model which minimizes <t>, i s 2N R(x) = L a . g . (x) . j = l J J W e i d e l t I n v e r s e S c a t t e r i n g / 33 The c o e f f i c i e n t s a. a r e f o u n d by s o l v i n g t h e m a t r i x s y s t e m e =Ta, (21 ) where T i s t h e s y m m e t r i c , p o s i t i v e d e f i n i t e , i n n e r p r o d u c t m a t r i x o f t h e k e r n e l s g.(x). T h a t i s , oo r . , = J g (x)g (x)dx. 1 0 J N u m e r i c a l i n t e g r a t i o n i s n o t r e q u i r e d t o f i n d t h e e l e m e n t s o f th e i n n e r p r o d u c t m a t r i x b e c a u s e a n a l y t i c s o l u t i o n s e x i s t . The sy s t e m o f e q u a t i o n s (21) i s s o l v e d u s i n g s t a n d a r d 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 s ( P a r k e r , 1977). When t h e d a t a e r r o r s a r e i n d e p e n d e n t and G a u s s i a n w i t h z e r o mean and s t a n d a r d d e v i a t i o n £j t h e n t h e c h i - s q u a r e d s t a t i s t i c d e t e r m i n e s t h e a c c u r a c y w i t h w h i c h t h e e q u a t i o n s a r e s o l v e d . The x 2 m i s f i t i s 2N X 2 = 2 (e.-eP)2/e* (22) J=l J J J where ep a r e t h e d a t a p r e d i c t e d by t h e c o n s t r u c t e d model. The p r e f e r r e d model w i l l have x2-2N, w h i c h i s t h e e x p e c t e d v a l u e of x 2 . I f x2«2N, t h e d a t a a r e f i t t o o a c c u r a t e l y a nd s t r u c t u r e t h a t i s m e r e l y an a r t i f a c t o f t h e n o i s e w i l l be e v i d e n t on R(x). A l t e r n a t i v e l y , i f x2»2N, t h e d a t a a r e n o t f i t c l o s e l y enough and i n f o r m a t i o n a b o u t t h e model c o n t a i n e d i n t h e d a t a w i l l be l o s t . The x 2 v a l u e o b t a i n e d h e r e by c o m p a r i n g e. w i t h ep. from a c o n s t r u c t e d i m p u l s e r e s p o n s e i s v e r y c l o s e t o t h a t f o u n d when t h e f i n a l c o n d u c t i v i t y p r o f i l e i s u s e d t o g e n e r a t e ep. T h u s , W e i d e l t I n v e r s e S c a t t e r i n g / 34 t h e s e c o n d - s t a g e i n t e g r a l mappings do n o t s i g n i f i c a n t l y p e r t u r b t h e x 2 m i s f i t o f t h e f i r s t s t a g e . P a r k e r (1980) showed t h a t f o r t h e s p a c e o f 1-D p r o f i l e s t h e r e i s a minimum v a l u e o f X 2 g i v e n by h i s D+ d e l t a f u n c t i o n m o d e l s . The 7 5 wavenumber, n o i s y d a t a s e t shown i n F i g u r e s 5 and 6 c o r r e s p o n d i n g t o t h e smooth a(z) m o d e l , has a D+ minimum x2 = 20.6. The m u l t i l a y e r model d a t a has a minimum x2=23. 4. By s o l v i n g e q u a t i o n ( 2 0 ) , one may c o n s t r u c t an i m p u l s e r e s p o n s e w i t h a s m a l l e r x 2 , but t h e n t h i s i m p u l s e r e s p o n s e no l o n g e r c o r r e s p o n d s t o any a(z) and t h e s e c o n d - s t a g e mappings g i v e u n p h y s i c a l r e s u l t s ; t h a t i s , t h e y g i v e i n t e r f a c e r e f l e c t i o n c o e f f i c i e n t s w i t h m a g n i t u d e s g r e a t e r t h a n u n i t y . K u n e t z (1972) and W e i d e l t (1972) d i s c u s s e d e x t r a c o n s t r a i n t s a p p l i c a b l e t o B(x) ( S e c t i o n 2.5 .1 ) w h i c h g u a r a n t e e t h a t i t i s one s i d e of an a u t o c o r r e l a t i o n f u n c t i o n and hence c o r r e s p o n d s t o a r e a l i z a b l e a(z) . B a c k u s - G i l b e r t (BG) t h e o r y c a n a l s o c o n s t r u c t f l a t m o d e l s w h i c h m i n i m i z e t h e i m p u l s e r e s p o n s e g r a d i e n t e n e r g y <t>2 s u b j e c t t o t h e d a t a c o n s t r a i n t s . I n t e g r a t i n g e q u a t i o n (16) by p a r t s g i v e s k.r(k.) - R(0) = J Ry(x)e K j x d x . (23) J J Q A s u r f a c e l a y e r of c o n d u c t i v i t y a0 i m p l i e s t h a t R(0)=0. B r e a k i n g (23) i n t o r e a l and i m a g i n a r y p a r t s , and w r i t i n g Weidelt Inverse S c a t t e r i n g / 35 2N R'(x) = Z 0 g (x), (24) j = l J J y i e l d s the same system (21) except that a. r e p l a c e s 0. and the data e. are now r e a l and imaginary p a r t s of k r(k )-R(0). S o l v i n g f o r the c o e f f i c i e n t s /3. u s i n g s p e c t r a l expansion and the x 2 c r i t e r i o n (22), and i n t e g r a t i n g the summation of k e r n e l s (24) g i v e s the BG f l a t model f o r R(x). F i g u r e s 7 and 8 give examples of BG R(x) and B(x) models c o n s t r u c t e d u s i n g the L2 small and f l a t norms, 0, and <f>2, from equations (18). The l e f t h a l v e s of these two f i g u r e s show BG small ( s h o r t dashes) and BG f l a t (long dashes) models c o n s t r u c t e d from 15 a c c u r a t e data. The data f o r the smooth t e s t model i n F i g u r e s 7a and 8a are taken from the smooth curves i n F i g u r e 6 at the same 15 l o g a r i t h m i c a l l y spaced wavenumbers i n d i c a t e d by the symbols f o r the i n a c c u r a t e data. The m u l t i l a y e r data used f o r F i g u r e s 7b and 8b are s i m i l a r to those i n F i g u r e 6 and are not p l o t t e d . The true impulse responses of F i g u r e 2 are r e p l o t t e d i n F i g u r e s 7 and 8 as s o l i d l i n e s . The true responses corresponding to the m u l t i l a y e r model are s c a l e d to f i t w i t h i n the frames. An i n f i n i t e number of accurate data would e x a c t l y reproduce the true impulse responses. However, the sparse sampling of data used here has two d e l e t e r i o u s e f f e c t s . F i r s t , m i s s i n g high and low f r e q u e n c i e s o u t s i d e the f i n i t e bandwidth decrease the r e s o l u t i o n of shallow and deep s t r u c t u r e s , r e s p e c t i v e l y . Second, m i s s i n g f r e q u e n c i e s w i t h i n the bandwidth cause a W e i d e l t I n v e r s e S c a t t e r i n g / 36 120 160 200 Two-Way Depth 2x 120 160 200 Two-Way Depth 2x 0.06 0 40 80 120 160 200 Two-Way Depth 2x 05 0.02 0.01 0.00 -0.01 --0.02 L. - | 1 1 1 1 1 1 1 r 7 d T II.. I . I ™|L^^^!g•^*ll^l»=^|J*^ —i I i I i L 0 40 80 120 160 200 Two-Way Depth 2x FIGURE 7 BG R(x) Impulse Response P a n e l s (a) and (b) c o n t a i n BG s m a l l ( s h o r t d a s h e s ) and BG f l a t ( l o n g d a s h e s ) R(x) models f o r t h e smooth and m u l t i l a y e r t e s t m o d e l s , r e s p e c t i v e l y . T h e s e R(x) were c o n s t r u c t e d u s i n g 15 a c c u r a t e complex r(k). The t r u e i m p u l s e r e s p o n s e s o f F i g u r e s 2a a n d 2b a r e r e p l o t t e d h e r e as s o l i d l i n e s . The c o n s t r u c t i o n s f o l l o w t h e t r u e smooth r e s p o n s e i n F i g u r e 7a w e l l . The d i s c r e p a n c i e s i n F i g u r e 7b a r e due t o t h e i n h e r e n t ' l i m i t a t i o n s o f r e p r e s e n t i n g d e l t a f u n c t i o n s h apes w i t h smooth k e r n e l f u n c t i o n s . P a n e l s ( c ) a n d (d) show BG s m a l l ( s h o r t d a s h e s ) and BG f l a t ( l o n g d a s h e s ) R(x) c o n s t r u c t e d u s i n g 15 i n a c c u r a t e complex r(k). F i t t i n g t h e n o i s y d a t a t o w i t h i n an a c c u r a c y c o r r e s p o n d i n g t o t h e e r r o r s a l l o w s v e r y s i m p l e R(x) m o d e l s . g e n e r a l b r o a d e n i n g of t h e t r u e f e a t u r e s . The s m a l l and f l a t BG models a r e s i m i l a r b e c a u s e t h e k e r n e l s o f t h e i n v e r s e p r o b l e m i n e q u a t i o n (20) a r e e x p o n e n t i a l f u n c t i o n s . The e x t r a i n t e g r a t i o n s r e q u i r e d t o f i n d Weidelt Inverse Scattering / 37 0 40 80 120 160 200 Two -Way Depth 2x 0 40 80 120 160 200 Two -Way Depth 2x 0.06 -0.03 0 40 80 120 160 200 Two -Way Depth 2x 0.02 -0.01 0.00 -0.02 —1 7/ —i i r ^ i . . i I . I . i . i . " 0 40 80 120 160 200 Two -Way Depth 2x FIGURE 8 BG B(x) Impulse Response Shown a r e BG s m a l l ( s h o r t d a s h e s ) and BG f l a t ( l o n g d a s h e s ) B(x) models c o n s t r u c t e d i n t h e same way as f o r F i g u r e 7. f l a t m odels do not r a d i c a l l y change t h e s e k e r n e l s . Hence, l i n e a r c o m b i n a t i o n s of s m a l l o r f l a t k e r n e l s c a n g i v e s i m i l a r i m p u l s e r e s p o n s e m o d e ls. A n o t h e r c o n s e q u e n c e o f t h e e x p o n e n t i a l k e r n e l s i s t h a t t h e r e s u l t i n g i m p u l s e r e s p o n s e i s i n f i n i t e l y d i f f e r e n t i a b l e . S u c h a smooth f u n c t i o n i s a v e r y good r e c o n s t r u c t i o n i f t h e t r u e i m p u l s e r e s p o n s e i s a l s o smooth ( F i g u r e s 7a and 8 a ) . However, t h e BG method c a n n o t p r e c i s e l y r e p r e s e n t d e l t a f u n c t i o n i m p u l s e r e s p o n s e s d e r i v e d from l a y e r e d o(z) ( F i g u r e s Weidelt Inverse S c a t t e r i n g / 38 7b and 8b). N e v e r t h e l e s s , r e c o n s t r u c t i o n s from accurate data are good r e p r e s e n t a t i o n s of the true impulse responses. In a l l p r a c t i c a l cases, the data are not only f i n i t e i n number but i n a c c u r a t e as w e l l . The r i g h t h a l v e s of F i g u r e s 7 and 8 g i v e R(x) and B(x) c o n s t r u c t i o n s from 15 n o i s y r(k) and b(k) data. The smooth o(z) model data are p l o t t e d as the symbols with e r r o r bars i n F i g u r e 6. The f i n i t e bandwidth of the data and t h e i r e r r o r s c o n s p i r e to allow models with l i t t l e s t r u c t u r e to pass the x 2 c r i t e r i o n of equation (22). The m i s f i t f o r a l l models i n F i g u r e s 7c, 7d, 8c, and 8d i s x2=30, which i s the expected value of x 2 . In the absence of a d d i t i o n a l i n f o r m a t i o n these models might be p r e f e r r e d because they have a minimum of s t r u c t u r e c o n s i s t e n t with the i n a c c u r a t e d a t a . Hence, these models imply that the true impulse response i s most l i k e l y p o s i t i v e f o r 0£2x<60, and somewhat negative t h e r e a f t e r . In S e c t i o n 2.4.3, I d i s c u s s how BG a p p r a i s a l q u a n t i f i e s the a b i l i t y of wavenumber-domain MT data to r e s o l v e f e a t u r e s i n the impulse response. The t r a n s f o r m a t i o n i n equation (2) from frequency CJ to wavenumber k r e q u i r e s an estimate of the s u r f a c e c o n d u c t i v i t y o0. It i s p o s s i b l e to estimate o0 from the behaviour of the BG impulse response model near the s u r f a c e . The f i r s t value of the impulse response i s a c t u a l l y the r e f l e c t i o n c o e f f i c i e n t r0 f o r the s u r f a c e i n t e r f a c e . T h i s r e f l e c t i o n c o e f f i c i e n t must equal zero to give c o n d u c t i v i t y models c o n s i s t e n t with the Weidelt Inverse S c a t t e r i n g / 39 assumption of a su r f a c e l a y e r of c o n d u c t i v i t y a^=o0. Hence, R(0)=0. I imposed these c o n d i t i o n s f o r BG f l a t model c o n s t r u c t i o n s but they must a l s o h o l d f o r BG small models. Using the true o0 to transform from frequency-domain data to wavenumber-domain data r e s u l t s i n impulse responses which are zero at the s u r f a c e . Using an i n c o r r e c t a 0 r e s u l t s i n l a r g e i n i t i a l v a l u e s of the recovered impulse response which correspond to r a p i d c o n d u c t i v i t y changes from the erroneous a0 to the proper v a l u e . However, the c o r r e c t i o n i n t r o d u c e s l a r g e o s c i l l a t i o n s i n t o the impulse response which p e r s i s t to great depths. T h e r e f o r e , wavenumber-domain data d e r i v e d using the c o r r e c t o0 g i v e impulse response models which begin at zero and have minimal o s c i l l a t i o n s . F i g u r e 9a gi v e s three BG small R(x) models, each c o n s t r u c t e d from a d i f f e r e n t set of 15 i n a c c u r a t e r(k). I generated each r(k) data set using the same i n a c c u r a t e c(W c o r r e s p o n d i n g to the smooth o(z) t e s t model but us i n g oo=0.001 (dashed l i n e ) , the true o0=0.002 ( s o l i d l i n e ) , and ao=0.004 ( d o t t e d l i n e ) . The impulse responses based on an i n c o r r e c t aQ have very l a r g e values near the s u r f a c e as w e l l as l a r g e o s c i l l a t i o n s below. Note the change of s c a l e s between F i g u r e s 9a and 2a. F i g u r e 9b shows three BG f l a t R(x) models c o n s t r u c t e d from s i m i l a r data. In t h i s case, the imposed c o n d i t i o n that R(0)=0 i n t r o d u c e s l a r g e o s c i l l a t i o n s throughout the models based on an i n c o r r e c t a0. A l l impulse responses i n W e i d e l t I n v e r s e S c a t t e r i n g / 40 0 5 10 15 20 0 5 10 15 20 Two-Way Depth 2x Two-Way Depth 2x Chi—squared FIGURE 9 Determining Surface C o n d u c t i v i t y from BG Impulse Responses P a n e l s (a) and (b) show t h r e e BG s m a l l R(x) and t h r e e BG f l a t R(x) m o d e l s , r e s p e c t i v e l y . A l l c o n s t r u c t i o n s u se d a t a c o r r e s p o n d i n g t o t h e smooth a(z) t e s t model but u s i n g o0=0.001 ( d a s h e d l i n e s ) , t h e t r u e ao=0.002 ( s o l i d l i n e s ) , a n d oQ=0.004 ( d o t t e d l i n e s ) . The i m p u l s e r e s p o n s e s b a s e d on an i n c o r r e c t o0 have l a r g e o s c i l l a t i o n s and t h e s m a l l models do n o t b e g i n a t z e r o . P a n e l ( c ) i s a c o n t o u r p l o t o f R(0) f r o m BG s m a l l R(x) c o n s t r u c t i o n s u s i n g t h e smooth a(z) t e s t d a t a . C o n t o u r s numbered 1 t h r o u g h 7 c o r r e s p o n d t o R(0)=-0.5, -0.2, -0.05, 0.0, 0.05, 0.1, and 0.2, r e s p e c t i v e l y . The z e r o c o n t o u r (number 4) f o l l o w s t h e c o r r e c t o o=0.002 f o r a l l x 2 v a l u e s g r e a t e r t h a n t h e D+ minimum o f 20. 6. Weidelt Inverse S c a t t e r i n g / 41 F i g u r e s 9a and 9b have a x 2 m i s f i t of 36. In Fi g u r e 9c, I contoured the x=0 value of BG small R(x) models f o r the smooth o(z) t e s t data as a f u n c t i o n of x 2 m i s f i t and s u r f a c e c o n d u c t i v i t y a0 . The R(0)=0 contour (number 4) i s very c l o s e to the c o r r e c t value of ao=0. 002 f o r a l l v a l u e s of x 2 g r e a t e r than the minimum x 2 of 20. 6 d e f i n e d by the D+ a l g o r i t h m of Parker and Whaler (1981). I t i s a l s o p o s s i b l e to estimate o0 by f i n d i n g t h a t value which minimizes a g l o b a l norm of the impulse response. S i m i l a r l y , M a r c h i s i o (1985) contoured the norm of the s p e c t r a l f u n c t i o n versus x 2 and o 0 to d e f i n e a range of a0 values which produce smooth a(z) models. 2.4.2. L i n e a r Programming C o n s t r u c t i o n The second impulse response c o n s t r u c t i o n technique I d i s c u s s uses l i n e a r programming (LP) to minimize 0 3 or 0„ of equations (18) s u b j e c t to the c o n s t r a i n t s p r o v i d e d by the data. Assume X is a l a r g e value of x f o r which the ke r n e l s i n equations (19) have decayed to near zero. P a r t i t i o n the i n t e r v a l [0,X] i n t o L s u b i n t e r v a l s with x,=0 and x^+1=X. The i n t e g r a l s i n equations (19) become the summations L Re{r(k .)} = L a . ,R, , J { = J Jt ' and (25) L -Im{r(k.)} = L b .,R, , J i = 1 J 1 1 where j = l , 2 , . . . N . The value s of the c o e f f i c i e n t s W e i d e l t I n v e r s e S c a t t e r i n g / 42 and b.^ depend on t h e p a r a m e t e r i z a t i o n o f t h e i m p u l s e r e s p o n s e . D i f f e r e n t c l a s s e s o f i m p u l s e r e s p o n s e y i e l d d i f f e r e n t o(z) models c o m p a t i b l e w i t h t h e d a t a . I f R(x) i s composed o f a s e r i e s o f L d e l t a f u n c t i o n s e a c h w i t h a r e a R^ a t t h e b e g i n n i n g o f t h e / t h s u b i n t e r v a l , t h e n a . , = e KjXI cos (K .x,) j I J J I and K . x, b . j l = e j I s i n (Kj x {) , f o r j = J , 2 , . . . N . I t i s e q u a l l y p o s s i b l e t o s p e c i f y R(x) as a c o n s t a n t R^ o v e r e a c h s u b i n t e r v a l . In t h i s c a s e , x.l+l -K.X j I = S e j c o s (Kj x) dx xl and , x.l +1 - K . x . , b . , = j e J si n(K.xj dx. J 1 v J xl An a c c e p t a b l e LP s m a l l model i s c o n s t r u c t e d by m i n i m i z i n g o r m a x i m i z i n g t h e o b j e c t i v e f u n c t i o n L 03 = 2 w \R | 1=1 1 1 s u b j e c t t o t h e c o n s t r a i n t s o f e q u a t i o n s ( 2 5 ) . The c o e f f i c i e n t s a r e d i s c r e t i z a t i o n s of t h e w e i g h t i n g f u n c t i o n w(x). In a s i m i l a r manner t h e LP f l a t model f o l l o w s from a d i s c r e t i z e d v e r s i o n o f e q u a t i o n ( 2 3 ) . In t h i s c a s e t h e o b j e c t i v e f u n c t i o n i s L <t>n = L w.\R' , \ , 1=1 1 1 W e i d e l t I n v e r s e S c a t t e r i n g / 43 OS 0.02 -0.01 0.00 -0.01 h -0.02 -i 1 1 r a x x X X 40 80 120 160 200 Two-Way Depth 2x -i 1 1 r X X c: X 40 80 120 160 200 Two-Way Depth 2x 0.06 0.00 -0.03 'J — i — i 1 1 T 1 1 1 T T l b • i I . I . I . 0 40 80 120 160 200 Two-Way Depth 2x 0 40 80 120 160 200 Two-Way Depth 2x FIGURE 10 LP R(x) Impulse Response P a n e l (a) c o n t a i n s a ( b r o a d e n e d ) d e l t a f u n c t i o n LP s m a l l R(x) and a p i e c e w i s e c o n s t a n t LP f l a t R(x) from t h e 15 a c c u r a t e smooth model d a t a p l o t t e d i n F i g u r e s 6a and 6b. P a n e l (b) c o n t a i n s s i m i l a r LP s m a l l and f l a t R(x) models from t h e 15 a c c u r a t e m u l t i l a y e r d a t a . The t r u e R(x) models a r e p l o t t e d i n F i g u r e s 2a and 2b. P a n e l ( c ) g i v e s an LP s m a l l and an LP f l a t R(x) model c o n s t r u c t e d u s i n g i n a c c u r a t e smooth model d a t a . P a n e l (d) g i v e s LP s m a l l and LP f l a t models f o r t h e m u l t i l a y e r d a t a . A l l s m a l l models a r e n o r m a l i z e d by a f a c t o r o f 10 t o f i t on t h e same p l o t w i t h t h e f l a t m o d e l s . where R*{ i s t h e a r e a of t h e / t h d e l t a f u n c t i o n c o r r e s p o n d i n g t o t h e d e r i v a t i v e of a p i e c e w i s e c o n s t a n t R(x). The i m p u l s e r e s p o n s e may a l s o be p a r a m e t e r i z e d as h a v i n g a c o n s t a n t d e r i v a t i v e o v e r e ach s u b i n t e r v a l i n w h i c h c a s e R(x) i s p i e c e w i s e l i n e a r . N o t e t h a t , f o r a p a r t i c u l a r R(x) W e i d e l t I n v e r s e S c a t t e r i n g / 44 x m 0.01 0.00 -0.01 -0.02 — 1 1— —i—i—i—i—i—i—i— a: . 1 . 1 . 1 . ' 0 40 80 120 160 200 T w o - W a y Depth 2x 0 40 80 120 160 200 T w o - W a y Depth 2x 0.06 0.03 -0.00 -0.03 T 1 1 1 1 r 4-r-_L 0 40 80 120 160 200 T w o - W a y Depth 2x 0 40 80 120 160 200 T w o - W a y Depth 2x FIGURE 11 LP B ( x ) Impulse Response Shown a r e LP s m a l l and LP f l a t B(x) models c o n s t r u c t e d i n t h e same way as f o r F i g u r e 10. p a r a m e t e r i z a t i o n , t h e c o e f f i c i e n t s a.^ and b.^ a r e t h e same f o r t h e s m a l l and f l a t m o d e l s , o n l y t h e d a t a change by a m u l t i p l i c a t i v e f a c t o r o f t h e wavenumber k. F i g u r e s 10 and 11 g i v e LP R(x) and LP B(x) c o n s t r u c t i o n s , r e s p e c t i v e l y . E a c h frame c o n t a i n s a s m a l l model p a r a m e t e r i z e d a s a s e r i e s o f ( b r o a d e n e d ) d e l t a f u n c t i o n s and a p i e c e w i s e c o n s t a n t f l a t model. A m p l i t u d e s o f t h e s m a l l models a r e r e d u c e d by an a r b i t r a r y f a c t o r o f 10 t o f i t on t h e same p l o t w i t h t h e f l a t m o d e l s . The l e f t h a l v e s of F i g u r e s 10 and 11 W e i d e l t I n v e r s e S c a t t e r i n g / 45 show t h a t c o n s t r u c t i o n s f r o m a c c u r a t e d a t a a r e good a p p r o x i m a t i o n s t o t h e t r u e i m p u l s e r e s p o n s e s g i v e n i n F i g u r e 2. T h i s i s e s p e c i a l l y t r u e f o r t h e m u l t i l a y e r r e s p o n s e s i n F i g u r e s 10b and 11b. The LP s m a l l norm d e l t a - f u n c t i o n m o d e ls a r e n a t u r a l r e p r e s e n t a t i o n s o f i m p u l s e r e s p o n s e s from l a y e r e d a(z). However, d e l t a f u n c t i o n s c a n n o t a c c u r a t e l y r e p r e s e n t t h e smooth R(x) and B(x) i n F i g u r e s 10a and 11a. LP f l a t norm, p i e c e w i s e c o n s t a n t f u n c t i o n s a r e somewhat more s u c c e s s f u l . In t h i s s e n s e t h e n , t h e BG and LP methods a r e c o m p l e m e n t a r y : t h e BG a p p r o a c h has a b e t t e r c h a n c e o f r e p r o d u c i n g a smooth i m p u l s e r e s p o n s e and t h e LP a p p r o a c h has a b e t t e r c h a n c e o f r e p r o d u c i n g a d i s c o n t i n u o u s i m p u l s e r e s p o n s e . F i g u r e s 1Oc, l O d , 11c, and 11d show R(x) and B(x) models c o n s t r u c t e d f r o m i n a c c u r a t e d a t a . The x 2 m i s f i t s a r e a l l n e a r t h e e x p e c t e d v a l u e o f 30 f o r t h e 15 complex d a t a . T h e s e m o d e l s a l o n g w i t h t h o s e f r o m t h e BG c o n s t r u c t i o n s i n F i g u r e s 7 and 8 i l l u s t r a t e t h e n o n u n i q u e n e s s of t h e i m p u l s e r e s p o n s e i n v e r s e p r o b l e m . D e l t a f u n c t i o n , p i e c e w i s e c o n s t a n t , and i n f i n i t e l y d i f f e r e n t i a b l e models a l l f i t t h e e r r o n e o u s d a t a e q u a l l y w e l l . The d i f f e r e n t c l a s s e s map t o d i f f e r e n t o(z) c l a s s e s w h i c h sample t h e s p a c e o f a c c e p t a b l e m o d e l s and g i v e an i n t e r p r e t e r a c h o i c e as t o w h i c h i s more r e p r e s e n t a t i v e o f t h e l o c a l g e o l o g y . The BG o r LP method g i v e s R(x) and B(x) models c o n s t r u c t e d f r o m i n a c c u r a t e d a t a w h i c h a r e v e r y s i m i l a r t o Weidelt Inverse S c a t t e r i n g / 46 each o t h e r . T h i s i s because the t r u e R(x) and B(x) are q u i t e s i m i l a r down to about 2x=100. At g r e a t e r depths where the true impulse responses d i f f e r , the r e s o l u t i o n ( S e c t i o n 2.4.3) i s so poor that the models take on v a l u e s which h e l p minimize the a p p l i e d norm. Small models tend t o zero and f l a t models tend to a constant not n e c e s s a r i l y equal to zero. The l a r g e r the e r r o r s on the data, the g r e a t e r t h i s tendancy. The p r e d i s p o s i t i o n of f l a t models towards nonzero co n s t a n t s causes problems f o r the c o n d u c t i v i t y mappings d i s c u s s e d i n S e c t i o n 2.6.2. Next, I d i s c u s s BG a p p r a i s a l which generates unique i n f o r m a t i o n about the d i v e r s e impulse responses f i t t i n g the data. Moreover, a p p r a i s a l determines how w e l l the wavenumber-domain MT data r e s o l v e f e a t u r e s of the impulse response. 2 . 4.3. B a c k u s - G i l b e r t A p p r a i s a l Many techniques c o n s t r u c t impulse responses which f i t the l i m i t e d number of i n a c c u r a t e MT data. The BG and LP fo r m u l a t i o n s are j u s t two. However, i t i s not p o s s i b l e to overcome nonuniqueness using any combination of c o n s t r u c t i o n methods. There i s no guarantee t h a t impulse response f e a t u r e s which p e r s i s t on BG and LP c o n s t r u c t i o n s of small and f l a t models must represent r e a l i t y . BG a p p r a i s a l (Backus and G i l b e r t , 1968,1970; Parker, Weidelt Inverse S c a t t e r i n g / 47 1977; Oldenburg, 1984) i s the a l t e r n a t i v e to c o n s t r u c t i o n . A p p r a i s a l a s s e s s e s the nonuniqueness p e r m i t t e d by the data. For l i n e a r i n v e r s e problems, a p p r a i s a l forms averaging f u n c t i o n s which q u a n t i f y the data r e s o l v i n g power. These f u n c t i o n s i n turn form unique averages common to a l l models reproducing the data. Consider a l i n e a r combination of both s i d e s of . equation (20) : 2N 2N Z a (x0)e = Z a.(x0) / R(x) g .(x) dx j=l J J j=i J o J °° 2N = j R(x) Z a (x0) g . (x) dx, 0 j=l J J where a^(x0) are c o e f f i c i e n t s f o r a p a r t i c u l a r x=x0. The averaging f u n c t i o n or r e s o l u t i o n k e r n e l A(x,x0) i s a l i n e a r combination of the k e r n e l s . That i s , 2N A(x,x0) = Z a (x0) g (x) . (26) j = l J J Hence, CO <R(x0)> = J R(x)A(x, x0)dx, 0 where 2N <R(x0)> = Z a (xQ) e . j = l J 1 i s an average of the model R(x) near x=x0. For accurate data, these averages are unique because every model reproducing the data e. has the same average when i n t e g r a t e d with A(x,x0). For W e i d e l t I n v e r s e S c a t t e r i n g / 48 i n a c c u r a t e d a t a , t h e a v e r a g e s have a v a r i a n c e d e r i v e d from t h e d a t a c o v a r i a n c e m a t r i x . The w i d t h o f t h e a v e r a g i n g f u n c t i o n q u a n t i f i e s t h e d a t a r e s o l v i n g power. I f A(x,xQ) i s s i m i l a r t o a d e l t a f u n c t i o n c e n t e r e d on x0, t h e n t h e c o r r e s p o n d i n g a v e r a g e i s c l o s e t o t h e t r u e i m p u l s e r e s p o n s e v a l u e . I f t h e n a r r o w e s t p o s s i b l e A(x,x0) has a b r o a d peak d i s p l a c e d from x0t t h e n t h e d a t a have l i t t l e r e s o l v i n g power a t x0. R e s o l u t i o n and t h e v a r i a n c e of t h e a v e r a g e s a r e a n t a g o n i s t i c . H i g h l y r e s o l v e d e s t i m a t e s have a l a r g e v a r i a n c e , whereas p o o r l y r e s o l v e d e s t i m a t e s have a s m a l l e r v a r i a n c e . BG a p p r a i s a l a c h i e v e s t h e o p t i m a l t r a d e o f f between r e s o l u t i o n a nd v a r i a n c e by f i n d i n g t h o s e a.(xQ) w h i c h m i n i m i z e 4>(x0) = cosd W(x0) + sine e2(x0) 0<6<n/2, (27) where W(x0) i s a measure o f t h e a v e r a g i n g f u n c t i o n w i d t h , e2(x0) i s t h e v a r i a n c e of t h e a v e r a g e e s t i m a t e and 6 i s t h e t r a d e o f f p a r a m e t e r . Two p o s s i b l e d e f i n i t i o n s f o r W(xQ) a r e t h e s p r e a d c r i t e r i o n 0 0 W(x0) = 12 S (x-x0)2[A(x,x0)]2dx, 0 and t h e D i r i c h l e t c r i t e r i o n CO W(x0) = J [A(x, x0)-8(x-x0)]2dx. 0 The s p r e a d c r i t e r i o n r e q u i r e s an e x t r a c o n s t r a i n t t h a t t h e a v e r a g i n g f u n c t i o n h as u n i t a r e a t o p r e v e n t i t from b e i n g W e i d e l t I n v e r s e S c a t t e r i n g / 49 0 40 80 120 160 200 0 40 80 120 160 200 Two-Way Depth 2x Two-Way Depth 2x Two-Way Depth 2x FIGURE 12 K e r n e l s and T r a d e o f f C o n t o u r D i a g r a m P a n e l s (a) and (b) show k e r n e l s c o r r e s p o n d i n g t o t h e r e a l and n e g a t e d i m a g i n a r y p a r t s o f 5 of t h e 15 complex smooth model d a t a . P a n e l ( c ) i s t h e t r a d e o f f c o n t o u r d i a g r a m f o r t h e 15 wavenumber d a t a s e t . The c o n t o u r s g i v e t h e w i d t h o f a v e r a g i n g f u n c t i o n s . R e s o l u t i o n a t l a r g e d e p t h s or f o r a c c u r a t e a v e r a g e e s t i m a t e s i s p o o r . W e i d e l t I n v e r s e S c a t t e r i n g / 50 i d e n t i c a l l y z e r o . F o r t h e D i r i c h l e t c r i t e r i o n , t h e n u m e r i c a l v a l u e o f W(x0) c a n n o t be c a l c u l a t e d b e c a u s e of t h e s q u a r e o f t h e d e l t a f u n c t i o n . N e v e r t h e l e s s , t h e p r e c i s e v a l u e i s n o t e s s e n t i a l t o t h e m i n i m i z a t i o n o f <f>(x0). F i g u r e s 12a and 12b show t h e k e r n e l s of e q u a t i o n s (19) -k x w h i c h a r e t h e r e a l and n e g a t e d i m a g i n a r y p a r t s o f e j . F o r c l a r i t y , I p l o t t e d k e r n e l s c o r r e s p o n d i n g t o o n l y 5 o f t h e 75 wavenumbers u s e d f o r t e s t i n v e r s i o n s ( F i g u r e 6 ) . The 5 c o n s i s t of e v e r y t h i r d wavenumber b e g i n n i n g w i t h t h e s m a l l e s t o f t h e 15. T h e s e k e r n e l s a r e 10 o f t h e t o t a l o f 30 a v a i l a b l e t o c o n s t r u c t a v e r a g i n g f u n c t i o n s u s i n g e q u a t i o n ( 2 6 ) . The p l o t s i n d i c a t e q u a l i t a t i v e l y t h a t t h e i m p u l s e r e s p o n s e r e s o l u t i o n i s good f o r s m a l l d e p t h s b e c a u s e t h e k e r n e l s change r a p i d l y and have s i g n i f i c a n t a m p l i t u d e t h e r e . C o n s e q u e n t l y , some l i n e a r c o m b i n a t i o n o f k e r n e l s i s l i k e l y t o p r o d u c e a narrow a v e r a g i n g f u n c t i o n a t s h a l l o w d e p t h s . At l a r g e d e p t h s , t h e r e s o l u t i o n i s p o o r b e c a u s e high-wavenumber k e r n e l s have d e c a y e d t o n e a r z e r o and low-wavenumber k e r n e l s a r e c h a n g i n g v e r y s l o w l y . A t t h e s e d e p t h s , i t i s d i f f i c u l t t o c o n s t r u c t a l o c a l i z e d a v e r a g i n g f u n c t i o n . A r o u g h e s t i m a t e o f t h e b e s t r e s o l u t i o n p o s s i b l e f o r a d a t a s e t i s g i v e n by t h e n o r m a l i z e d s e p a r a t i o n o f k e r n e l p e a k s f o r a d j a c e n t wavenumbers. The d e p t h s of k e r n e l e x t r e m a a r e p r o p o r t i o n a l t o 7r/(4/c^.) where K. = [ Mo wy o0/2] 1 / 2 . The s e p a r a t i o n of t h e f i r s t maxima of two i m a g i n a r y k e r n e l s n o r m a l i z e d by t h e W e i d e l t I n v e r s e S c a t t e r i n g / 51 a r i t h m e t i c mean o f t h e i r d e p t h s g i v e s t h e r e s o l u t i o n e s t i m a t e I K . - K . I „ , - 2 . L L L ! d. The same s e p a r a t i o n n o r m a l i z e d by t h e g e o m e t r i c mean g i v e s _ i y 7 - Kj\ P2 - y • F o r t h e l o g a r i t h m i c a l l y s p a c e d , 15 wavenumber t e s t d a t a s e t , b o t h t h e s e e s t i m a t e s a r e c o n s t a n t f o r a l l a d j a c e n t wavenumbers and t h e i r v a l u e i s a p p r o x i m a t e l y 0. 23. F o r t h i s s a m p l i n g of d a t a t h e n , t h e b e s t p o s s i b l e r e s o l u t i o n a t a l l d e p t h s x i s r o u g h l y x/4. Hence, l o g a r i t h m i c a l l y s p a c e d wavenumbers i m p l y c o n s t a n t r e s o l u t i o n when measured by t h e l o g a r i t h m i c Lx/x r a t i o . O l d e n b u r g (1979) d e r i v e d a n o t h e r measure of r e s o l u t i o n g i v e n by a r a t i o of s p r e a d c r i t e r i o n w i d t h t o d e p t h . F o r t h i s d a t a s e t , h i s f o r m u l a g i v e s a s i m i l a r r e s o l u t i o n e s t i m a t e of a b o u t 0.21. A s e l e c t i o n o f wavenumbers w h i c h c o n c e n t r a t e s p e a k s o f t h e k e r n e l f u n c t i o n s n e a r a p a r t i c u l a r d e p t h w i l l have e n h a n c e d r e s o l u t i o n t h e r e . The MT u n i q u e n e s s t h e o r e m ( T i k h o n o v , 1965; B a i l e y , 1970) q u a r a n t e e s t h a t r e s o l u t i o n becomes p e r f e c t " as t h e number of wavenumbers a p p r o a c h e s i n f i n i t y . F i g u r e 12c i s a t r a d e o f f c o n t o u r d i a g r a m f o r t h e 15 wavenumber d a t a s e t . T h i s d i a g r a m q u a n t i f i e s t h e D i r i c h l e t c r i t e r i o n r e s o l u t i o n by c o n t o u r i n g t h e a v e r a g i n g f u n c t i o n w i d t h f o r a ra n g e of d e p t h s and s t a n d a r d d e v i a t i o n s o f t h e W e i d e l t I n v e r s e S c a t t e r i n g / 52 a v e r a g e s . The w i d t h i s d e f i n e d a s t h e i n v e r s e o f t h e maximum of t h e a v e r a g i n g f u n c t i o n . T h i s w i d t h i s v e r y c l o s e t o t h e f u l l w i d t h a t h a l f t h e maximum v a l u e . The c o n t o u r e d w i d t h s a r e , f r o m l e f t t o r i g h t , 20, 40, 60, 80, 100, 150, and 200. The s t a n d a r d d e v i a t i o n s of t h e a v e r a g e s p l o t t e d on t h e v e r t i c a l a x i s a r e d e r i v e d from t h e s t a n d a r d d e v i a t i o n s o f t h e smooth model i n a c c u r a t e r(k) d a t a . The c o r r e s p o n d i n g p l o t f o r t h e m u l t i l a y e r model i s v e r y s i m i l a r . A t a p a r t i c u l a r x Q , 6 i s v a r i e d between 9=0 and 0=7r/2 t o g e n e r a t e a t r a d e o f f c u r v e w h i c h i s a v e r t i c a l s l i c e o f t h i s p l o t . A s e r i e s o f t r a d e o f f c u r v e s c o m p l e t e s t h e d i a g r a m . F o r a s p e c i f i e d s t a n d a r d d e v i a t i o n , t h e r e s o l u t i o n d e c r e a s e s w i t h i n c r e a s i n g d e p t h b e c a u s e i t becomes more d i f f i c u l t t o c o n s t r u c t l o c a l i z e d a v e r a g i n g f u n c t i o n s from t h e k e r n e l s s u c h as t h o s e p l o t t e d i n F i g u r e s 12a and 12b. R e s o l u t i o n a l s o d e c r e a s e s when fewer k e r n e l s a r e a v a i l a b l e t o combine i n t o a v e r a g i n g f u n c t i o n s . The h i g h e s t r e s o l u t i o n o b t a i n s when 6=0 b e c a u s e e q u a t i o n (27) t h e n m i n i m i z e s o n l y W(x0) w i t h o u t r e g a r d t o e 2 ( x 0 ) . In t h i s c a s e , t h e s t a n d a r d d e v i a t i o n of t h e a v e r a g e e s t i m a t e i s a maximum. At 8=0, a p l o t o f t h e a v e r a g i n g f u n c t i o n w i d t h v e r s u s two-way d e p t h c o n f i r m s t h e b e s t r e s o l u t i o n e s t i m a t e s py and p 2 made e a r l i e r . F o r t h e 15 wavenumber d a t a s e t , t h e r a t i o o f w i d t h t o d e p t h f o r 0=0 i s a b o u t 0. 26. F i g u r e 13 g i v e s examples o f a v e r a g i n g f u n c t i o n s d e r i v e d from t h e 15 wavenumber d a t a s e t a n d t h e D i r i c h l e t r e s o l u t i o n W e i d e l t I n v e r s e S c a t t e r i n g / 53 C 0.04 60 Cj 0.02 QO £ 0.00 > ^ -0.02 1 1 1 1 1 1 ~" "~ A • 1 T ^ b : . 1 , 1 , I . I . 40 80 120 160 200 Two-Way Depth 2x 40 80 120 160 200 Two-Way Depth 2x FIGURE 13 Impulse R e s p o n s e A v e r a g i n g F u n c t i o n s P a n e l s (a) t o (d) c o n t a i n D i r i c h l e t c r i t e r i o n a v e r a g i n g f u n c t i o n s f o r d e p t h s o f 40, 80, 120, and 160, r e s p e c t i v e l y . The s o l i d l i n e s a r e t h e h i g h e s t r e s o l u t i o n a v e r a g i n g f u n c t i o n s f o r t h e 15 wavenumber d a t a s e t . These a v e r a g i n g f u n c t i o n s r e s u l t i n a v e r a g e s w i t h s t a n d a r d d e v i a t i o n s much l a r g e r t h a n t h e a v e r a g e s t h e m s e l v e s . The d a s h e d l i n e s a r e a v e r a g i n g f u n c t i o n s c o r r e s p o n d i n g t o a v e r a g e s w i t h a r e a s o n a b l e s t a n d a r d , d e v i a t i o n o f 0.001. R e s o l u t i o n i s s a c r i f i c e d t o a c h i e v e more a c c u r a t e e s t i m a t e s o f t h e a v e r a g e s . c r i t e r i o n . The two-way d e p t h l o c a t i o n s 2x0 a r e a t 40, 80, 120, and 160, f o r F i g u r e s 13a t h r o u g h 13d, r e s p e c t i v e l y . The s o l i d l i n e s a r e t h e h i g h e s t r e s o l u t i o n (0=0) a v e r a g i n g f u n c t i o n s . T h e s e c o r r e s p o n d t o a v e r a g e s w i t h u n a c c e p t a b l y l a r g e s t a n d a r d d e v i a t i o n s (much g r e a t e r t h a n t h e a v e r a g e v a l u e s t h e m s e l v e s ) W e i d e l t I n v e r s e S c a t t e r i n g / 54 b e c a u s e o n l y W(x0) i n e q u a t i o n (27) i s m i n i m i z e d . T h e s e a v e r a g i n g f u n c t i o n s depend o n l y on t h e k e r n e l s w h i c h a r e d e f i n e d by a p a r t i c u l a r c h o i c e o f measured f r e q u e n c i e s . T h e r e f o r e , q u a n t i t a t i v e e s t i m a t e s o f t h e b e s t p o s s i b l e r e s o l u t i o n o f th e i m p u l s e r e s p o n s e c a n be made b e f o r e any f i e l d work i s done. To o b t a i n a v e r a g e s w i t h a c c e p t a b l e s t a n d a r d d e v i a t i o n s , r e s o l u t i o n must be s a c r i f i c e d . One c h o i c e f o r an a c c e p t a b l e e r r o r i s 0.001. T h i s i s a b o u t 10 p e r c e n t o f t h e a m p l i t u d e o f t h e m a j o r f e a t u r e s i n t h e smooth model i m p u l s e r e s p o n s e i n F i g u r e 2a. F i g u r e 12c shows t h a t t h i s s t a n d a r d d e v i a t i o n c o r r e s p o n d s t o a r e s o l u t i o n r a t i o o f Lx/x-1. The d a s h e d l i n e s i n F i g u r e 13 a r e t h e 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 . T h e s e c u r v e s d e m o n s t r a t e t h a t i t i s n o t p o s s i b l e t o c a l c u l a t e a c c u r a t e as w e l l as l o c a l i z e d a v e r a g e s of t h e i m p u l s e r e s p o n s e below d e p t h s o f about 2x=80. At l a r g e d e p t h s ( F i g u r e s 13c and 13d, d a s h e d l i n e s ) , t h e r e s o l u t i o n i s p o o r . The a v e r a g e s and a v e r a g i n g f u n c t i o n s a r e q u a n t i t a t i v e m e a s u r e s of t h e a b i l i t y o f wavenumber-domain MT d a t a t o r e s o l v e t h e i m p u l s e r e s p o n s e . However, t h e mappings from R(x) t o o(x) and from o(x) t o a(z) c o m p l i c a t e t h e e x t r a p o l a t i o n o f t h e s e r e s u l t s t o q u a n t i t a t i v e measures o f c o n d u c t i v i t y r e s o l u t i o n . I f a f i n a l a(z) i n t e r p r e t a t i o n i s a v a i l a b l e , t h e n t h e x t o 2 mapping i s c o m p l e t e l y known. E q u a t i o n (3) g i v e s t h e c o r r e s p o n d i n g o(x) model so t h a t z-domain f e a t u r e s c a n be Weidelt Inverse S c a t t e r i n g / 55 l o c a t e d i n the x-domain. The mapping from R(x) to o(x) i s a more d i f f i c u l t problem. I g i v e a l i n e a r approximation i n S e c t i o n 2.6.3 which s t a t e s that the c o n d u c t i v i t y at l o c a t i o n x depends on the i n t e g r a l of the impulse response down to two-way depth 2x. Hence, the r e s o l u t i o n of a at x depends on the a b i l i t y to c o n s t r u c t an averaging f u n c t i o n s i m i l a r to a step f u n c t i o n which i s u n i t y from 0 to 2x and zero t h e r e a f t e r . That i s , 2x <R(x)> = / R(t)dt. 0 However, a l i n e a r combination of a f i n i t e number of k e r n e l s can never reproduce a true s t e p - f u n c t i o n . Moreover, the l i n e a r i z e d r e l a t i o n s h i p between R(x) and a(x) i s only an approximation. T h e r e f o r e , t h i s approach does not g i v e more q u a n t i t a t i v e i n f o r m a t i o n than the impulse response r e s o l u t i o n . In general then, i f the impulse response r e s o l u t i o n i s good down to two-way depth 2x, then the c o n d u c t i v i t y i s w e l l - r e s o l v e d down to depth x. 2 . 5 . CONSTRAINED IMPULSE RESPONSE INVERSION Any c o l l e c t i o n of c o n s t r u c t i o n methods can not overcome the nonuniqueness p l a g u i n g the l i n e a r i n v e r s e s o l u t i o n f o r the impulse response. However, i n c o r p o r a t i n g e x t r a g e o l o g i c and g e o p h y s i c a l c o n s t r a i n t s i n t o the i n v e r s i o n enhances the c r e d i b i l i t y of the c o n s t r u c t e d models. Here I impose three types of c o n s t r a i n t s on the impulse response c o n s t r u c t i o n and W e i d e l t I n v e r s e S c a t t e r i n g / 56 examine t h e e f f e c t on t h e c o n d u c t i v i t y . S u b s e q u e n t l y , f o r t h e n o n l i n e a r R i c c a t i e q u a t i o n o f C h a p t e r 3, I a p p l y c o n s t r a i n t s d i r e c t l y t o t h e c o n d u c t i v i t y o(z). 2.5.1. P h y s i c a l R e a l i z a b i l i t y C o n s t r a i n t s E v e r y p h y s i c a l l y r e a l i z a b l e c o n d u c t i v i t y model must have r e f l e c t i o n c o e f f i c i e n t s w i t h m a g n i t u d e l e s s t h a n u n i t y . A r e f l e c t i o n c o e f f i c i e n t g r e a t e r t h a n u n i t y c o r r e s p o n d s t o an u n p h y s i c a l n e g a t i v e c o n d u c t i v i t y . I f an a r b i t r a r y s e r i e s o f numbers i s l a b e l l e d a s an i m p u l s e r e s p o n s e t h e n t h e c o r r e s p o n d i n g r e f l e c t i o n c o e f f i c i e n t s c a n e a s i l y have m a g n i t u d e s g r e a t e r t h a n one. An i m p u l s e r e s p o n s e d e r i v e d f r o m MT d a t a and e q u a t i o n (16) o r (17) a l m o s t a l w a y s y i e l d s p h y s i c a l l y r e a l i z a b l e r e f l e c t i o n c o e f f i c i e n t s i f x 2 o r a n o t h e r m i s f i t c r i t e r i o n i s a d o p t e d . However, i n some c a s e s e x t r a c o n s t r a i n t s a r e r e q u i r e d a s , f o r i n s t a n c e , when u s i n g w e i g h t f u n c t i o n s ( S e c t i o n 2 . 5 . 3 ) . K u n e t z ( 1 9 7 2 ) , W e i d e l t ( 1 9 7 2 ) , and Coen and Yu (1981) u s e d t h e c o n d i t i o n C O f(s) = f B(x) cos (sx)dx < 1/2 0<s<°° (28) 0 t o g u a r a n t e e t h a t o(z) has i n t e r f a c e r e f l e c t i o n c o e f f i c i e n t s l e s s t h a n u n i t m a g n i t u d e . E q u a t i o n (28) f o l l o w s from t h e p r o p e r t i e s o f t h e MT p r o p a g a t o r m a t r i x . T h i s m a t r i x i s a n a l o g o u s t o t h e s e i s m i c p r o p a g a t o r m a t r i x d i s c u s s e d , f o r example, by R o b i n s o n (1982) Weidelt Inverse S c a t t e r i n g / 57 and Aki and Richards (1980, p.664). U r s i n (1983) reviewed i n d e t a i l the propagation of e l a s t i c and EM waves i n l a y e r e d -2kh media. For the MT case, the Z-transform v a r i a b l e i s Z=e where k i s the complex s u r f a c e wavenumber, h i s the x-domain t h i c k n e s s of every l a y e r , and the f a c t o r of two i m p l i e s a two-way depth. The x spacing i s constant, and i s analogous to the constant l a y e r t r a v e l t i m e assumed f o r the s e i s m i c problem. The MT propagator matrix i s u l + J ( z ) z-1/2 1 r tZ Ul (Z) D[+1(Z) / ' Z . D> (Z) where i s the r e f l e c t i o n c o e f f i c i e n t given i n S e c t i o n 2.1, t^=[1-ri 2 ] 1 / 2 i s the t r a n s m i s s i o n c o e f f i c i e n t , Z i s the Z-transform v a r i a b l e , and U[(Z) and D^(Z) are the Z-transforms of the upgoing and downgoing waves, r e s p e c t i v e l y , at the top of l a y e r /. The s u r f a c e boundary c o n d i t i o n s are that the upgoing wave i s the impulse response and that the downgoing wave i s the u n i t impulse source p l u s the p e r f e c t r e f l e c t i o n of the upgoing wave. That i s , U, (Z) = B(Z) Dy(Z) = 1 - B(Z) where B(Z) i s the Z-transform of the impulse response B(x). The boundary c o n d i t i o n i n the u n d e r l y i n g h a l f - s p a c e i s that there i s no upgoing wave. That i s , Weidelt Inverse S c a t t e r i n g / 58 u L + 1 ( z ) = 0. Using the MT propagator matrix, i t i s easy to show that the q u a n t i t y D(Z)D(1/Z) - U(Z)U(1/Z) i s the same i n every l a y e r . In the seismic case, t h i s corresponds to the c o n s e r v a t i o n of the net downgoing energy i n every l a y e r . A p p l y i n g the boundary c o n d i t i o n s , the conserved q u a n t i t y becomes 1-B(Z)-B(1/Z) = DL+1(Z)DL+1(1/Z) . The i n v e r s e Z-transform of the ri g h t - h a n d s i d e of t h i s equation corresponds to an a u t o c o r r e l a t i o n . The F o u r i e r transform of t h i s a u t o c o r r e l a t i o n must be grea t e r than or equal to zero. The same must be tr u e f o r the l e f t - h a n d s i d e of the equation. Hence, / [h(x)-B(x)-B(-x) ]e~' SXdx > 0. — CO The q u a n t i t y w i t h i n square b r a c k e t s i s the impulse response completed by the i n i t i a l p u lse and by symmetry. The above F o u r i e r transform reduces to equation (28) on us i n g the c a u s a l i t y of B(x). The LP fo r m u l a t i o n r e a d i l y i n c o r p o r a t e s the i n e q u a l i t y c o n s t r a i n t s i n equation (28) as e x t r a rows i n the LP t a b l e a u matrix. The BG fo r m u l a t i o n a l s o a c cepts i n e q u a l i t y c o n s t r a i n t s but they are more d i f f i c u l t to implement and r e s u l t i n a l a r g e inner product matrix. However, i t i s p o s s i b l e to check on the Weidelt Inverse S c a t t e r i n g / 59 p h y s i c a l r e a l i z a b i l i t y of a BG B(x) model c o n s t r u c t e d without the e x t r a c o n s t r a i n t s of equation (28). A BG impulse response i s the summation of k e r n e l s 2N B(x) = Z a g (x), j = ] J J where 8j(x) r e p r e s e n t s the r e a l and imaginary p a r t s of small model k e r n e l s . S u b s t i t u t i n g t h i s summation i n t o equation (28) giv e s 2N c o f(s) = Z a. / g . (x) cos (sx)dx < 1/2. j=l 1 0 J The i n t e g r a t i o n has an a n a l y t i c s o l u t i o n so that the c o n s t r a i n t reduces to a simple summation f o r each value of the wavenumber s. I n t e g r a t i n g equation (28) by p a r t s g i v e s a c o n s t r a i n t on f l a t B(x) models. That i s , CO f(s) = - - / B' (x) si n(sx) dx < 1/2 0<s<*>. S 0 T h i s i n t e g r a l i s used to c o n s t r a i n LP f l a t B(x) models and to t e s t BG f l a t B(x) models f o r p h y s i c a l r e a l i z a b i l i t y . F i g u r e 14 shows the behaviour of f(s) f o r BG B(x) models c o n s t r u c t e d without the e x t r a c o n s t r a i n t s of equation (28). Because the BG i n v e r s i o n s are un c o n s t r a i n e d there are B(x) models which v i o l a t e f(s)<l/2 f o r some value s of the x 2 m i s f i t to the b(k) data. F i g u r e s 14a and 14b p l o t the percentage of s v a l u e s f o r which f(s)>l/2 as a f u n c t i o n of x 2 f o r the smooth Weidelt Inverse S c a t t e r i n g / 60 20 30 C h i - s q u a r e d 200 100 0 -100 h -200 nm 1 i 11 inn i i ium—i i ium—i 11mm |MA/Vm ; iiui t i • iiuJ fl/VV I IIIIIJ i i IIIIIJ • t m i n i IO"3 IO"2 IO"1 10° IO1 Wavenumber s ( l / m ) 60 w -r-l 40 A 01 «M 20 fr? 1 1 — i ' ' ' b ; \ — 1 10 20 30 40 C h i - s q u a r e d 0.6 0.4 2 0.2 0.0 -0.2 Tnnr i i i i u m i I I lllll i M i n i i i IJJIIII -7 * d . - -i m i • ' i n " ' & • i 111 II J 1 1 I I I I U IO"3 IO"2 IO - 1 10° IO1 Wavenumber s ( l / m ) FIGURE 14 P h y s i c a l R e a l i z a b i l i t y C o n s t r a i n t F u n c t i o n s P a n e l (a) p l o t s , a s a f u n c t i o n of x 2 m i s f i t , t h e p e r c e n t a g e o f s v a l u e s f o r w h i c h f(s)>l/2 f o r BG s m a l l ( s o l i d l i n e ) and BG f l a t ( d a s h e d l i n e ) models d e r i v e d u s i n g t h e smooth model d a t a . The v e r t i c a l l i n e i n d i c a t e s t h e minimum x 2 g i v e n by t h e D+ a l g o r i t h m . P a n e l (b) i s t h e same p l o t u s i n g t h e m u l t i l a y e r d a t a . P a n e l ( c ) shows an u n p h y s i c a l f(s) f u n c t i o n c o r r e s p o n d i n g t o a BG s m a l l B(x) w i t h \2=I5 w h i c h i s l e s s t h a n t h e D+ minimum. P a n e l (d) shows f(s) f u n c t i o n s c o r r e s p o n d i n g t o BG s m a l l B(x) w i t h x 2 = 24 ( s o l i d l i n e ) a nd x2=30 ( d a s h e d l i n e ) . and m u l t i l a y e r d a t a , r e s p e c t i v e l y . In t h e s e two f i g u r e s , t h e v e r t i c a l l i n e i s t h e minimum x 2 g i v e n by t h e D+ a l g o r i t h m o f P a r k e r and Whaler ( 1 9 8 1 ) . F o r BG s m a l l B(x) models ( s o l i d l i n e s ) , t h e p e r c e n t a g e o f f(s) v a l u e s e x c e e d i n g t h e bound d r o p s r a p i d l y a s t h e x 2 m i s f i t i n c r e a s e s above t h e D+ minimum. W e i d e l t I n v e r s e S c a t t e r i n g / 61 The bound i s e x c e e d e d j u s t a few t i m e s o r not a t a l l f o r x 2 g r e a t e r t h a n t h e e x p e c t e d v a l u e o f x 2 w h i c h i s 30. F o r BG f l a t B(x) models ( d a s h e d l i n e s ) , t h e bound i s o f t e n e x c e e d e d even f o r l a r g e x 2 m i s f i t s . T h i s i s b e c a u s e BG f l a t norm i m p u l s e r e s p o n s e s t e n d t o a n o n z e r o c o n s t a n t f o r l a r g e x. W h i l e t h i s i s an e f f e c t i v e s t r a t e g y f o r m i n i m i z i n g t h e f l a t norm, i t p r o d u c e s u n p h y s i c a l r e f l e c t i o n c o e f f i c i e n t s , u s u a l l y a t l a r g e d e p t h s . The c o r r e s p o n d i n g c o n d u c t i v i t y i s a c c e p t a b l e down t o th e d e p t h a t w h i c h t h e f i r s t u n p h y s i c a l r e f l e c t i o n c o e f f i c i e n t o c c u r s . As s t a t e d i n S e c t i o n 2.4.1, i t i s p o s s i b l e t o c o n s t r u c t an i m p u l s e r e s p o n s e w i t h x 2 l e s s t h a n t h e D+ minimum but t h i s r e s u l t s i n u n p h y s i c a l r e f l e c t i o n c o e f f i c i e n t s . F i g u r e 14c shows t h e f(s) f u n c t i o n c o r r e s p o n d i n g t o a BG s m a l l B(x) model w i t h x 2=75. T h i s f u n c t i o n e x c e e d s t h e bound a b o u t 50 p e r c e n t of t h e t i m e . F i g u r e 14d shows two f(s) f u n c t i o n s c o r r e s p o n d i n g t o BG s m a l l B(x) models w i t h x2 = 24 ( s o l i d l i n e ) and x2=30 ( d a s h e d l i n e ) . In p r a c t i c e , s m a l l e x c u r s i o n s o f f(s) c u r v e s beyond t h e bound r a r e l y c a u s e u n p h y s i c a l r e f l e c t i o n c o e f f i c i e n t s when t h e i m p u l s e r e s p o n s e i s d i s c r e t i z e d and n u m e r i c a l l y mapped t o a(z). The f u n c t i o n f(s) c a n be u s e d t o . d e t e r m i n e a v a l u e f o r the s u r f a c e c o n d u c t i v i t y a0. The c o r r e c t o0 p r o d u c e s b(k) d a t a w h i c h i n t u r n p r o d u c e BG B(x) m o d e l s f o r w h i c h t h e c o n s t r a i n t f(s)<l/2 i s r a r e l y v i o l a t e d . F i g u r e 15 shows t h e r e s u l t s W e i d e l t I n v e r s e S c a t t e r i n g / 62 O o O cd CO 10 -2 10 -3 _ 20 30 40 Chi—squared FIGURE 15 C o n t o u r s o f f ( s ) Bound E x c e e d e d T h i s f i g u r e c o n t o u r s , a s a f u n c t i o n o f x 2 m i s f i t and s u r f a c e c o n d u c t i v i t y a0, t h e p e r c e n t a g e of s f o r w h i c h f(s)>l/2 f o r BG s m a l l B(x) models d e r i v e d u s i n g t h e smooth model d a t a . F o r m i s f i t s g r e a t e r t h a n t h e D+ minimum of x2 = 20.6, t h e c o n t o u r s i n d i c a t e t h e c o r r e c t c h o i c e o f o0=0. 002. d e r i v e d u s i n g t h e smooth model d a t a . F o r e a c h v a l u e o f a0 w i t h i n a s p e c i f i e d r a n g e , I g e n e r a t e d a b(k) d a t a s e t and c o n s t r u c t e d a s e r i e s o f BG s m a l l B(x) models w i t h d i f f e r e n t x 2 m i s f i t s . F o r e a c h of t h e s e models I c a l c u l a t e d t h e p e r c e n t a g e o f f(s)>l/2 and c o n t o u r e d t h e r e s u l t s v e r s u s o0 and x 2 • F o r x 2 l e s s t h a n t h e D+ minimum o f 20.6, no v a l u e o f o0 g i v e s an i m p u l s e r e s p o n s e w i t h an a c c e p t a b l e f(s) f u n c t i o n . F o r x 2 Weidelt Inverse S c a t t e r i n g / 63 between the minimum and the expected value of 30 the percentage of times the bound i s exceeded f a l l s s t e e p l y . The contours i n F i g u r e 15 f o r x2>30 c l e a r l y i n d i c a t e the c o r r e c t value of aQ=0. 002. As the x 2 m i s f i t i n c r e a s e s t o very l a r g e v a l u e s , the B(x) models approach zero and so s a t i s f y equation (28) f o r any choice of o0 . The contour p l o t f o r the m u l t i l a y e r model data i s very s i m i l a r . To demonstrate how the c o n s t r a i n t s of equation (28) prevent u n p h y s i c a l a(z), I i n v e r t e d a set of ocean bottom MT data recorded on the Juan de Fuca r i d g e by Law and Greenhouse (1981). The JDF data are p l o t t e d i n Fi g u r e s 16a and 16b. These data were a l s o analyzed by Oldenburg (1981), Oldenburg et a l . (1984), and M a r c h i s i o (1985). F i g u r e 16c i s a contour map of the s u r f a c e value of the impulse response, R(0), p l o t t e d a g a i n s t x 2 m i s f i t and s u r f a c e c o n d u c t i v i t y c h o i c e a0. The contour v a l u e s range from -3x10'5 (number 1) , through 0 (number 4), to 4x10'6 (number 7 ) . The expected value of x 2 f o r the 11 complex JDF data i s 22. For values of x 2 near the expected value, the R(0)=0 contour (number 4) i n d i c a t e s that the proper c h o i c e f o r s u r f a c e c o n d u c t i v i t y i s approximately oo=0.02 S/m. However, a l l contour l e v e l s below t h i s are of the order of 10'6 implying that the lower contoured area i s f l a t and t hat oQ i s not narrowly c o n s t r a i n e d . F i g u r e 16d i s a contour diagram of the percent of f(s)>l/2 f o r the JDF data. The broad region w i t h i n the zero contour i n d i c a t e s t h at f o r BG W e i d e l t I n v e r s e S c a t t e r i n g / 64 ^ 6 0 I 40 a 20 a > i i 111II IO"1 1 0 3 10 4 Per iod T (s) C h i - s q u a r e d 40 & 6 0 a> — 40 20 11 II ' * • ' T £ 10 3 10 4 Per iod T (s) C h i - s q u a r e d FIGURE 16 The JDF D a t a S e t P a n e l s (a) a n d (b) g i v e t h e a p p a r e n t r e s i s t i v i t y and pha s e d a t a , r e s p e c t i v e l y , f o r t h e JDF o c e a n - b o t t o m l o c a t i o n . The e r r o r b a r s r e p r e s e n t one s t a n d a r d d e v i a t i o n . P a n e l ( c ) i s a c o n t o u r p l o t o f R(0) f o r BG s m a l l R(x) models d e r i v e d from t h e JDF d a t a . The m i d d l e c o n t o u r number 4 i s t h e R(0)=0 c o n t o u r . However, c o n t o u r number 7 has a v a l u e o f o n l y 4xl0~6 i n d i c a t i n g t h a t t h e lo w e r h a l f of t h e frame i s v e r y f l a t a nd c l o s e t o z e r o . P a n e l (d) c o n t o u r s t h e p e r c e n t a g e o f s f o r w h i c h f(s)>l/2 f o r JDF BG s m a l l B(x) models. m o dels a good c h o i c e f o r a 0 i s between a b o u t 0. 006 and 0. 02 S/m. S m a l l e r oQ c h o i c e s a r e p o s s i b l e b ecause m o dels from t h e 5 and 8 p e r c e n t c o n t o u r a r e a do not a l w a y s r e s u l t i n u n p h y s i c a l r e f l e c t i o n c o e f f i c i e n t s a t d e p t h s w i t h i n t h e r e g i o n o f i n t e r e s t . W e i d e l t I n v e r s e S c a t t e r i n g / 65 Pseudodepth x (km) P s e u d o d e p t h x (km) FIGURE 17 P h y s i c a l R e a l i z a b i l i t y C o n s t r a i n t s on B(x) P a n e l (a) shows t h e u n p h y s i c a l i n t e r f a c e r e f l e c t i o n c o e f f i c i e n t s r e s u l t i n g f r o m an LP s m a l l B(x) c o n s t r u c t i o n u s i n g t h e JDF d a t a and a h e a v i l y w e i g h t e d o b j e c t i v e f u n c t i o n . P a n e l (b) g i v e s t h e s u b s e q u e n t n o n s e n s i c a l a(x) p r o f i l e . P a n e l (c) shows t h e more r e a s o n a b l e i n t e r f a c e r e f l e c t i o n c o e f f i c i e n t s f r o m a s i m i l a r LP s m a l l B(x) c o n s t r u c t i o n w h i c h i n c o r p o r a t e d t h e c o n s t r a i n t s of e q u a t i o n ( 2 8 ) . P a n e l (d) g i v e s t h e c o r r e s p o n d i n g a(x). G i v e n t h e weak c o n s t r a i n t s on a0 p r o v i d e d by t h e BG method, I c h o s e oo=0. 003 S/m t o c o r r e s p o n d r o u g h l y t o p r e v i o u s i n v e r s i o n s . In any c a s e , I u s e d t h e LP method t o e n f o r c e t h e p h y s i c a l r e a l i z a b i l i t y c o n s t r a i n t s and B(0)=0. W i t h t h i s c h o i c e of o0, I g e n e r a t e d a b(k) d a t a s e t from t h e JDF r e s p o n s e s . To c r e a t e a c o n d u c t i v i t y model w h i c h was r e s i s t i v e W e i d e l t I n v e r s e S c a t t e r i n g / 66 i n t h e u p p e r l i t h o s p h e r e , I u s e d a h e a v i l y w e i g h t e d LP o b j e c t i v e f u n c t i o n ( S e c t i o n 2.5.3) t o c o n s t r u c t an LP s m a l l B(x) w i t h s h a r p l y r e d u c e d v a l u e s f o r 2x<180 km. F i g u r e s 17a and 17b show t h a t t h e w e i g h t i n g s u c c e e d s i n m i n i m i z i n g t h e c o r r e s p o n d i n g r e f l e c t i o n c o e f f i c i e n t s and a(x) f o r x<90 km. However, f o r x>170 km t h e r e a r e r e f l e c t i o n c o e f f i c i e n t s w i t h m a g n i t u d e g r e a t e r t h a n one w h i c h g e n e r a t e a n o n s e n s i c a l a(x). F i g u r e s 17c and 17d show t h e r e s u l t s of t h e same i n v e r s i o n i n c o r p o r a t i n g t h e e x t r a c o n s t r a i n t s of e q u a t i o n ( 2 8 ) . A l l o f th e r e f l e c t i o n c o e f f i c i e n t s a r e now l e s s t h a n u n i t m a g n i t u d e . The c o r r e s p o n d i n g a(x) p r o f i l e i s more r e a l i s t i c . W i t h o r w i t h o u t t h e e x t r a c o n s t r a i n t s , t h e B(x) i n v e r s i o n s have a x 2 m i s f i t o f 26. However, t h e x 2 m i s f i t s f o r t h e s u b s e q u e n t o(x) models a r e 200 f o r F i g u r e 17b and 26 f o r F i g u r e 17d. 2.5.2. A P r i o r i Impulse R e s p o n s e In t h i s s e c t i o n , I c o n s t r a i n t h e i m p u l s e r e s p o n s e c o n s t r u c t i o n t o y i e l d m odels c l o s e t o an a p r i o r i R(x) o r B(x). The r e s u l t i n g c o n d u c t i v i t y i s c l o s e , i n some s e n s e , t o the c o r r e s p o n d i n g a p r i o r i a(z). The p u r p o s e o f a p r i o r i c o n s t r a i n t s i s t o d e t e r m i n e how c o m p a t i b l e a p a r t i c u l a r d a t a s e t i s w i t h c o n d u c t i v i t y p r o f i l e s d e r i v e d f r o m o t h e r g e o p h y s i c a l s u r v e y s , g e o l o g i c i n f o r m a t i o n , o r n e a r b y w e l l l o g s . T h i s i s t h e s i m p l e s t c o n d u c t i v i t y c o n s t r a i n t t o a p p l y . F o r a g l o b a l c o n s t r a i n t s u c h a s t h i s , t h e x t o z mapping i s Weidelt Inverse S c a t t e r i n g / 67 known p r e c i s e l y because the a p r i o r i a(z) completely s p e c i f i e s a(x) over the i n t e r v a l of i n t e r e s t . I d i s c u s s the more d i f f i c u l t case of l o c a l i z e d c o n d u c t i v i t y c o n s t r a i n t s i n Se c t i o n s 2.5.3 and 3.2. The a p r i o r i data equations f o r R(x) a r e : -k x r (k.) = / R (x)e KjXdx, (29) a J Q a where ra(kj) a n d R a ( % ) a r e known and correspond to an a p r i o r i a(z). S u b t r a c t i n g equations (16) and (29) g i v e s 0 0 -k r(k ) - r (k ) = / [R(x)-R (x)]e 'j*'dx, J J Q or 00 -k br(k.) = J SR(x)e jXdx. (30) J 0 BG or LP i n v e r s i o n s of equation (30) use the r e s i d u a l data br(kj) to c o n s t r u c t a r e s i d u a l impulse response 8R(x) with minimum L2 or Ly norm. The f i n a l impulse response which d e v i a t e s l e a s t from R (x) i s a R(x) = RQ(x) + bR(x). M a r c h i s i o (1985) c o n s t r u c t e d s p e c t r a l f u n c t i o n s d e v i a t i n g l e a s t i n the L2 sense from an a p r i o r i e s timate. Here, as an example, I use an L, f o r m u l a t i o n on the impulse response. Note that an a p r i o r i a(z) equal to a constant i m p l i e s that ra ( k j ) = 0 a n d Ra(x)=0. T h e r e f o r e , equations (16) and (18) co n s t r u c t an impulse response corresponding to a o(z) c l o s e to a uniform h a l f - s p a c e . The same i s true f o r B(x) and equation Weidelt Inverse Scattering / 68 S \ CO £> 10-1 > • r H | 10-8 O O 1 0 - 3 6 \ co £ 10 • r H 1 1 i — i — T 1 1 1 a. I r H : l • • t . i . 50 100 150 Depth z (km) 200 i _ § 10 o « 10 - 8 _ -3 1 r 1-J" r — ' , b, i i i linn I : • 1 • I . I . 50 100 150 200 Depth z (km) 50 100 150 Depth z (km) 200 10-3 50 100 150 Depth z (km) 200 FIGURE 18 A P r i o r i Impulse Response Inversion The d a s h e d l i n e s a r e t w o - l a y e r and t h r e e - l a y e r a p r i o r i o(z) models f o r t h e JDF d a t a . The s o l i d l i n e s a r e a(z) models d e r i v e d f r o m LP s m a l l 8R(x) i n v e r s i o n s ( l e f t - h a n d s i d e ) and LP f l a t bR(x) i n v e r s i o n s ( r i g h t - h a n d s i d e ) . The two a p r i o r i a(z) have u n a c c e p t a b l y h i g h x 2 m i s f i t s , but t h e f o u r o(z) models a l l have x 2 m i s f i t s n e a r t h e e x p e c t e d v a l u e o f x 2 f o r t h e JDF d a t a . ( 1 7 ) . F i g u r e 18 shows JDF o(z) models ( s o l i d l i n e s ) d e r i v e d f r o m LP s m a l l 8R(x) i n v e r s i o n s ( l e f t - h a n d s i d e ) and LP f l a t 8R(x) i n v e r s i o n s ( r i g h t - h a n d s i d e ) . The a p r i o r i a(z) models ( d a s h e d l i n e s ) have u n a c c e p t a b l y h i g h x 2 v a l u e s of 70 and 49 f o r t h e t w o - l a y e r and t h r e e - l a y e r models, r e s p e c t i v e l y . T h e r e Weidelt Inverse Scattering / 69 is l i t t l e difference between the f i n a l o(z) derived using the small or f l a t norm because the norm applies only to the perturbation. The x 2 m i s f i t s are 21, 16, 24, and 25 for Figures 18a through I8d, respectively. These m i s f i t s are a l l near the expected value of 22 for the JDF data. These inversions show that the JDF data (with a choice of oo=0. 003 S/m) are more compatible with a lithosphere and asthenosphere conductivity model which increases to peak near z=75 km and then decreases again. 2 . 5 . 3 . W e i g h t e d I m p u l s e R e s p o n s e Norms A nonuniform weight function w(x) in equations (18) influences the constructed impulse response. Minimizing these norms implies that where w(x) i s large, R(x) i s small and where w(x) i s small, R(x) may be large. In Section 2.6.3, I show that, to f i r s t order, the logarithm of the conductivity varies as the integral of R(x). Also note that to f i r s t order, B(x) i s the same as R(x). Hence, the effe c t s on a(x) of perturbing the impulse response over small regions in x are known approximately. However, the nonlinear mapping from x to z makes i t impossible to know the precise depth z where the constraints on R(x) a f f e c t the conductivity. In Chapter 3, the R i c c a t i equation inversions allow constraints d i r e c t l y on the conductivity in the z domain. The problem with t h i s approach, however, i s that the equation i s nonlinear and the solution Weidelt Inverse S c a t t e r i n g / 70 (X IO"3) X m 0.02 -0.00 -0.02 1 r Tl. . — i — i — r — - i — i — . I . I . I . 0 200 400 600 800 Two-Way Depth 2x (km) 50 100 150 Depth z (km) 200 -0.02 0 200 400 600 800 Two-Way Depth 2x (km) a 10° t io-1 -4-> o 5 io ~2 o <-> 10-3 — _ — , — , — , — 1 ^ A d : [r *—\ z . i i . i 50 100 150 Depth z (km) 200 FIGURE 19 Weighted Impulse Response Inversion Panel (a) shows an LP small B(x) c o n s t r u c t e d from the JDF data. Panel (b) g i v e s the corresponding a(z) p r o f i l e . Panel (c) shows an LP small B(x) c o n s t r u c t e d with an o b j e c t i v e f u n c t i o n weighting a p p l i e d to reduce the impulse response f o r 2x<180 km. Panel (d) shows hov o(z) i s reduced f o r z<60 km. must be found by i t e r a t i o n . The mathematical f o r m u l a t i o n f o r BG i n v e r s i o n f o r a weighted impulse response i s s i m i l a r to the i n v e r s i o n when w(x)=l. The model i s a l i n e a r combination of the weighted k e r n e l s gj(x)/w(x) and T becomes a weighted inner product matrix. Weighted LP i n v e r s i o n simply r e q u i r e s a change to the o b j e c t i v e f u n c t i o n c o e f f i c i e n t s . W e i d e l t I n v e r s e S c a t t e r i n g / 71 As a t e s t o f t h e w e i g h t e d i m p u l s e r e s p o n s e norms, I u s e d t h e JDF d a t a t o c o n s t r u c t t h e LP s m a l l B(x) shown i n F i g u r e 19a. An i n t e g r a l e q u a t i o n mapping gave t h e a(z) p r o f i l e shown i n F i g u r e 19b. T h i s model i n c r e a s e s m o n o t o n i c a l l y o v e r t h e f i r s t 50 km and t h e n s t a y s c o n s t a n t . The x 2 m i s f i t i s 18. N e x t , I i n t r o d u c e d a w e i g h t i n g f u n c t i o n t o r e d u c e t h e i m p u l s e r e s p o n s e v a l u e s f o r 2x<180 km ( F i g u r e 1 9 c ) . F i g u r e 19d shows t h a t t h e c o n d u c t i v i t y i s s u b s t a n t i a l l y r e d u c e d f o r d e p t h s z<60 km. A t g r e a t e r d e p t h s t h e c o n d u c t i v i t y i n c r e a s e s r a p i d l y t o peak n e a r 80 km. The x 2 v a l u e i s 26. T h e r e f o r e , u s i n g i m p u l s e r e s p o n s e w e i g h t s , I have q u i c k l y and e a s i l y c o n f i r m e d two maj o r p o i n t s made by O l d e n b u r g e t a l . ( 1 9 8 4 ) : (1) t h e r e need n o t be a l o w - c o n d u c t i v i t y zone f o r z>100 km ( F i g u r e 19b); and (2) t h e o n s e t o f t h e h i g h - c o n d u c t i v i t y zone need not o c c u r a t d e p t h s l e s s t h a n z=60 km ( F i g u r e I 9 d ) . The l i n e a r i n v e r s e p r o b l e m o f f i n d i n g t h e MT i m p u l s e r e s p o n s e i s t h e key t o e x p l o r i n g t h e ra n g e o f a c c e p t a b l e c o n d u c t i v i t y p r o f i l e s a(z). In S e c t i o n s 2.2 t h r o u g h 2.5 I p r e s e n t e d e f f i c i e n t , p r a c t i c a l methods f o r c a l c u l a t i n g t h e i m p u l s e r e s p o n s e f r o m t h e g i v e n i n c o m p l e t e and i n a c c u r a t e d a t a c(ix>). I a p p l i e d f o u r norms t o c o n s t r u c t v a r i e d , minimum s t r u c t u r e i m p u l s e r e s p o n s e s w h i c h a l l f i t t h e d a t a e q u a l l y w e l l . I a p p l i e d g l o b a l and l o c a l c o n s t r a i n t s t o e x p l o r e f u r t h e r t h e s p a c e of a c c e p t a b l e i m p u l s e r e s p o n s e m o d e l s . The c o n d u c t i v i t y mappings i n S e c t i o n 2.6 g i v e m o d els w h i c h e x p l o r e W e i d e l t I n v e r s e S c a t t e r i n g / 72 t h e c o r r e s p o n d i n g s p a c e o f o(z) and h e l p a v o i d m i s i n t e r p r e t a t i o n s . 2 . 6 . CONDUCTIVITY MAPPINGS G i v e n a p a r t i c u l a r e s t i m a t e o f t h e i m p u l s e r e s p o n s e R(x) o r B(x), i t i s r e l a t i v e l y e a s y t o c o n s t r u c t t h e c o r r e s p o n d i n g c o n d u c t i v i t y p r o f i l e o(z). I c o n s i d e r f o u r methods w h i c h a c c o m p l i s h t h i s s e c o n d s t a g e o f t h e i n v e r s e s c a t t e r i n g f o r m u l a t i o n . E a c h i s an inhomogeneous F r e d h o l m i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d . The Marchenko and B u r r i d g e e q u a t i o n s use R(x), and t h e G e l ' f a n d - L e v i t a n and G o p i n a t h - S o n d h i e q u a t i o n s use B(x). T h e s e e q u a t i o n s p e r f o r m t h e same t a s k as t h e dynamic d e c o n v o l u t i o n of R o b i n s o n ( 1 9 7 5 ) , t h e r e c u r s i o n r e l a t i o n s of K u n e t z ( 1 9 7 2 ) , and t h e a l g o r i t h m s g i v e n by Bube and B u r r i d g e ( 1 9 8 3 ) . R o b i n s o n (1982) p r e s e n t e d an e x c e l l e n t d i s c u s s i o n o f s e v e r a l of t h e s e a p p r o a c h e s a l o n g w i t h t h e Marchenko method ( w h i c h he c a l l e d t h e G e l ' f a n d - L e v i t a n e q u a t i o n ) . He showed t h a t t h e K u n e t z and L e v i n s o n r e c u r s i o n s a r e i d e n t i c a l . B a l a n i s (1982) and G r a y (1983) d i s c u s s e d an e q u a t i o n w h i c h i s e q u i v a l e n t t o t h e B u r r i d g e e q u a t i o n . T h i s i s a l s o v e r y s i m i l a r t o one g i v e n by S a n t o s a (1982, e q u a t i o n 2 . 1 9 ) . I a l s o e v a l u a t e t h r e e a p p r o x i m a t e c o n d u c t i v i t y mappings i n S e c t i o n 2.6.3. W e i d e l t I n v e r s e S c a t t e r i n g / 73 2.6.1. I n t e g r a l E q u a t i o n s M a rchenko F o r c o m p l e t e n e s s I r e p e a t t h e March e n k o e q u a t i o n i n v o l v i n g , t h e i m p u l s e r e s p o n s e R(x). I t i s x A(x,y) = R(x+y) + / A(x, t) R(y+t) dl \y\*x. -y G e l ' f a n d - L e v i t a n The G e l ' f a n d - L e v i t a n i n t e g r a l e q u a t i o n e m p l o y i n g B(x) i s x A(x,y) = B(x+y) + J Mx, t ) [ B (y+t ) +B (y- I ) ] dt \y\*x. -x B u r r i d q e B u r r i d g e (1980) s o l v e d a s y s t e m o f two, c o u p l e d f i r s t - o r d e r d i f f e r e n t i a l e q u a t i o n s . The s y s t e m a r i s e s f r o m t r a n s f o r m i n g t h e MT e q u a t i o n (1) t o t h e p s e u d o d e p t h x domain u s i n g e q u a t i o n s (2) and ( 3 ) . Hence, E"(x,k) + 2>j(x)E' (x, k) - k2E(x,k) = 0 (31) where a p r i m e d e n o t e s 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 x, t h e f r e q u e n c y u> dependence i s r e p l a c e d by a dependence on t h e wavenumber k= [ i IM0CJO0 ] 1 / 2 , and t h e r e a l f u n c t i o n a* (x) y(x) = 4a(x) i s t h e c o n t i n u o u s a n a l o g u e o f t h e i n t e r f a c e r e f l e c t i o n c o e f f i c i e n t . D e f i n e p(x, k) = kE(x, k) , and Weidelt Inverse S c a t t e r i n g / 74 q(x,k) = u2(x)EUx,k), where equation (4) d e f i n e s u2(x)=[a(x)/o0]1/2. S u b s t i t u t i n g these i n t o the above second-order equation g i v e s u2 (x)kp(x,k) - q'(x,k) = 0 u2(x)p'(x,k) - kq(x,k) = 0. An i n v e r s e Laplace t r a n s f o r m with k and two-way depth y as conjugate v a r i a b l e s c a s t s these equations i n t o a form i d e n t i c a l to that c o n s i d e r e d by Burridge (1980). I t i s a l s o the same form as that used by Sondhi and Gopinath (1971). The i n t e g r a l equation s o l u t i o n to the system given by Burridge (1980) i s x J(x,y) = 1 + / J(x,t)R(y+t)dt \y\^x, (32) -y where I changed the s i g n of the impulse response R(x) to conform to Weidelt's d e f i n i t i o n . The c o n d u c t i v i t y i s a(x)/a0 = J*(x,x). Here, the c o n d u c t i v i t y at x depends only on the boundary value of the s o l u t i o n J(x,y) i n c o n t r a s t to the e n t i r e f u n c t i o n A(x,y) as i l l u s t r a t e d by equation ( 1 4 ) . T h i s r e s u l t s i n a small but s i g n i f i c a n t decrease i n computation. For impulse responses d e r i v e d from d i s c o n t i n u o u s models one must use (Burridge, 1980) a(x)/a0 = {^[ / J(x,t)dt] - I}2. (33) For the l a y e r e d model r e c o n s t r u c t i o n s i n Fi g u r e 3b, t h i s Weidelt Inverse S c a t t e r i n g / 75 formula e x a c t l y reproduces the t r u e c o n d u c t i v i t y ( s o l i d l i n e ) . I n v e r s i o n s based on the Marchenko and G e l ' f a n d - L e v i t a n equations (dashed l i n e s ) do not reproduce the t r u e l a y e r e d a(z). These l a t t e r two methods s o l v e a Schrodinger equation (6) d e r i v e d from equation (31) with the a d d i t i o n a l change of dependent v a r i a b l e given by equations (4) and ( 5 ) . In terms of y(x) , the Schrodinger p o t e n t i a l V(x) i s V(x) = 7 ' f x ; + y2(x). T h e r e f o r e , a c o n d u c t i v i t y p r o f i l e d e r i v e d from V(x) using the Marchenko or G e l ' f a n d - L e v i t a n methods r e q u i r e s second-order d i f f e r e n t i a b i l i t y of a(x). T h i s c o n d i t i o n may be r e l a x e d somewhat s i n c e Ware and Aki (1969) showed t h a t a d e l t a f u n c t i o n p o t e n t i a l i s p e r m i s s i b l e . T h i s i s e q u i v a l e n t to the statement by Weidelt (1972) that a(x) have d i s c o n t i n u i t i e s o n l y i n i t s d e r i v a t i v e s . However, the Burridge method combined with equation (33), and the Gopinath-Sondhi method d i s c u s s e d l a t e r , are s u p e r i o r because they can r e c o n s t r u c t d i s c o n t i n u o u s c o n d u c t i v i t y models. The Burridge equation (32) i s s i m i l a r to one d e r i v e d by B a l a n i s (1982) and d i s c u s s e d by Gray (1983). The B a l a n i s equation i s x+y . x J R(t)dt + K(x,y) + f K(x,t)R(y+t)dt = 0, 0 -y where I have changed the s i g n of R(x) as be f o r e . Rearranging t h i s equation and changing the l i m i t s of i n t e g r a t i o n g i v e s W e i d e l t I n v e r s e S c a t t e r i n g / 76 X X K(x,y) = - J R(y+t)dt - / K(x, t)R(y+t)dt . -y -y C o m b i n i n g t h e i n t e g r a l s and a d d i n g 1 t o b o t h s i d e s g i v e s x [1+K(x,y)] = / - f [1+K(x, t) ]R(y+t)dt . -y T h i s e q u a t i o n h as t h e same form a s t h e B u r r i d g e e q u a t i o n ( 3 2 ) . C h a n g i n g t h e s i g n o f R(x) makes them e q u i v a l e n t w i t h J(x, y) = 1+K(x,y). The o n l y e f f e c t o f n e g a t i n g R(x) i s t o i n v e r t t h e f i n a l c o n d u c t i v i t y p r o f i l e . T hus, t h e r i g h t - h a n d s i d e o f e q u a t i o n (33) w i t h / r e p l a c e d by 1+K g i v e s a0/a(x) m o d e l s . G o p i n a t h - S o n d h i S o n d h i and G o p i n a t h (1971) d e r i v e d an i n t e g r a l e q u a t i o n t o f i n d t h e a r e a o f t h e v o c a l t r a c t a s a f u n c t i o n o f d i s t a n c e from t h e g l o t t i s . T h e i r e q u a t i o n i s ; x F(xsy) + j J F(x,t)H(\y-t \)dt = 1 \y\±x. (34) - x They d e r i v e d i t from t h e same f i r s t - o r d e r s y s t e m a s g i v e n i n t h e p r e v i o u s d i s c u s s i o n o f t h e B u r r i d g e e q u a t i o n . The d i f f e r e n c e i s t h a t t h e i r r e s p o n s e H(x) i s d i r e c t l y p r o p o r t i o n a l t o B(x). I show t h i s u s i n g t h e f o l l o w i n g e q u a t i o n from S o n d h i and R e s n i c k ( 1 9 8 3 ) : x H(x) = -2R(x) - / H(t )R(x-t) dt . (35) 0 I c h a n g e d t h e s i g n of R(x) t o c o n f o r m t o W e i d e l t ' s d e f i n i t i o n . T a k i n g t h e L a p l a c e t r a n s f o r m o f e q u a t i o n (35) and u s i n g t h e Weidelt Inverse S c a t t e r i n g / 77 c o n v o l u t i o n theorem g i v e s h(k) = -2r(k) - h(k)r(k), (36) where the wavenumber-domain impulse response i s CO - kx h(k) = / H(x)e 0 dx. (37) Rearranging equation (36) giv e s -2r (k) h(k) = = -2b(k), (38) l + r(k) where the second e q u a l i t y f o l l o w s from equations (8) and ( 9 ) . Equations (38), (37) and (11) imply that H(x) = -2B(x). S u b s t i t u t i n g t h i s i n equation (34) g i v e s the f i n a l form, x F(x,y) = J + S F(x,t)B(\y-t \)dt \y\<x. (39) -x By analogy with the r e s u l t s of Gopinath and Sondhi f o r the v o c a l t r a c t and Weidelt f o r the MT case the c o n d u c t i v i t y i s a(x)/o0 = FUx,x) = F*(x,-x). For d i s c o n t i n u o u s c o n d u c t i v i t y p r o f i l e s one must use (Burri d g e , 1980) In F i g u r e 3d, the Gopinath-Sondhi equation used with equation (40) g i v e s a l a y e r e d a(x) ( s o l i d l i n e ) i d e n t i c a l to the tr u e c o n d u c t i v i t y . T h i s method i s s u p e r i o r to the G e l ' f a n d - L e v i t a n method (dashed l i n e ) which can not e x a c t l y reproduce l a y e r e d o(x)/o0 = / F(x, t)dt ]}2. (40) 0 W e i d e l t I n v e r s e S c a t t e r i n g / 78 m o d e l s . E q u a t i o n s (33) and (40) a r e o n l y r e q u i r e d f o r l a y e r e d m o d e l s w i t h l a r g e r e f l e c t i o n c o e f f i c i e n t s . As t h e r e f l e c t i o n c o e f f i c i e n t s o f d i s c o n t i n u o u s a(z) d e c r e a s e , t h e e r r o r s due t o s e t t i n g t h e c o n d u c t i v i t y p r o p o r t i o n a l t o t h e f o u r t h power o f a bo u n d a r y e l e m e n t of J or F d e c r e a s e . 2.6.2. N u m e r i c a l S o l u t i o n s The n u m e r i c a l s o l u t i o n s of t h e f o u r i n t e g r a l e q u a t i o n s a r e s i m i l a r . I f A i s t h e d i g i t i z a t i o n i n t e r v a l o f t h e i m p u l s e r e s p o n s e , t h e n x—>nA and y—>mA w i t h -n<m<n. The numbers n and m may be e i t h e r b o t h i n t e g e r s o r b o t h h a l f - o d d i n t e g e r s t o keep t h e argument of t h e i m p u l s e r e s p o n s e (n±m)L an i n t e g e r m u l t i p l e o f A. The i n d e x o f a summation i n v o l v i n g n and m a l w a y s i n c r e a s e s by u n i t y . U s i n g t h e t r a p e z o i d a l r u l e f o r i n t e g r a t i o n , t h e Marchenko e q u a t i o n (12) becomes n A(n,m) = R(n+m) + A L A(n, i )R(m+i ) - LA(n, n) R( n+m) / 2 (41) / =-m where A(n,m) i s s h o r t h a n d f o r A(nA,mA) and so on. The assumed s u r f a c e l a y e r of c o n d u c t i v i t y a0 means t h a t R(0)=0. T h i s c o n d i t i o n i m p l i e s t h a t t h e summation need o n l y b e g i n w i t h i=-m+l, and t h a t t h e m=-n e q u a t i o n r e d u c e s t o A(n,-n)=0 and so i t c a n be d r o p p e d . T h e r e f o r e , e a c h v a l u e o f t h e p a r a m e t e r n=l/2,1, 3/2, 2,. . . w i t h l-n<m<n g e n e r a t e s a new 2nx2n s y s t e m of e q u a t i o n s . W e i d e l t I n v e r s e S c a t t e r i n g / 79 A compact f o r m of e q u a t i o n (41) w h i c h y i e l d s a sym m e t r i c m a t r i x f o l l o w s f r o m d e f i n i n g a(n,m) as a(n,m) = A(n, m)/[ l-AA(n, n)/2]. (42) S u b s t i t u t i n g e q u a t i o n (42) i n t o e q u a t i o n (41) g i v e s n a(n,m) = R( n+m) + A Z a(n, i) R(m+i ) . (43) / =-m The c o n d u c t i v i t y i s d e r i v e d from a t r a p e z o i d a l d i s c r e t i z a t i o n o f e q u a t i o n ( 1 4 ) . Hence, i n terms o f a(n,m), n u(n) = [1 + A Z a(n,i)]/[l + ba(n,n)/2] (44) / =-n and a(nA)/o0 = u V « ; (45) where n=1/2, I, 3/2, 2,... Berryman and G r e e n e (1980) m a n i p u l a t e d e q u a t i o n s ( 4 3 ) , ( 4 4 ) , and (45) f u r t h e r t o d e r i v e a f a s t s o l u t i o n a l g o r i t h m a n a l o g o u s t o L e v i n s o n r e c u r s i o n . T h i s a l g o r i t h m r e d u c e s t h e number o f o p e r a t i o n s r e q u i r e d t o compute a a(x) p r o f i l e o f N p o i n t s from o r d e r Na t o o r d e r N2. The d i s c r e t e G e l ' f a n d - L e v i t a n e q u a t i o n (13) becomes t h e 2nx2n s y s t e m n a(n,m) = B(n+m) + A Z a (n, i) [B(m+i) + B(m-i)] i = - n where n=1/2,1, 3/2, 2,. . . and l-n<m<n, on u s i n g t h e t r a p e z o i d a l r u l e , B(0)=0, and e q u a t i o n ( 4 2 ) . E q u a t i o n s (44) and (45) g i v e t h e c o n d u c t i v i t y j u s t as f o r t h e Marchenko e q u a t i o n . The d i s c r e t e form of t h e B u r r i d g e e q u a t i o n (32) r e q u i r e s o n l y a r e c t a n g u l a r r u l e a p p r o x i m a t i o n a l o n g w i t h a d i s c r e t e Weidelt Inverse S c a t t e r i n g / 80 form of equation (33) to reproduce e x a c t l y the d i f f i c u l t l a y e r e d models. The form i s a symmetric 2nx2n system (the m=-n equation, J(n,-n)=l, i s not used) n J(n,m) = 1 + A I J(n, i)R(m+i) i =-m where n=1/2,1, 3/2, 2,. . . and l-n<m<n. The c o n d u c t i v i t y i s a(nA)/a0 = [S(n) - S(n-l/2) - l]2 where n S(n) = 2 2 J(n, i) , S(0) = 0. i=l-n S i m i l a r l y , the Gopinath-Sondhi equation (39) r e q u i r e s only a r e c t a n g u l a r r u l e approximation p l u s a d i s c r e t e form of equation (40). The r e s u l t i s a (2n+I)x(2n+1) T o e p l i t z system n F(n,m) = 1 + A Z F(n, i)B(\m-i \) i =-n where n=1/2,1, 3/2, 2,. . . and -n<m<n. The c o n d u c t i v i t y i s o(nA)/o0 = [S(n) - S(n-l/2)}2 where n S(n) = L F(n, i) , S(0) = 1. i =- n Sondhi and Resnick (1983) presented a r e c u r s i v e s o l u t i o n a l g o r i t h m analogous to Levinson r e c u r s i o n e x p l o i t i n g the T o e p l i t z s t r u c t u r e of the s u c c e s s i v e l y l a r g e r m a t r i c e s . Bube and Burridge (1983) compared v a r i o u s numerical schemes f o r s o l v i n g i n v e r s e s c a t t e r i n g equations. They found that downward c o n t i n u a t i o n a l g o r i t h m s , Levinson r e c u r s i o n , and W e i d e l t I n v e r s e S c a t t e r i n g / 81 10 20 30 40 50 Depth z 10 20 30 40 50 Depth z 10 -1 E—" 1 r > •l-t •S 10-aL--o o ° 10-3 i i — i — i _ i fr bi \ \ X X x 10 20 30 40 50 Depth z 10 20 30 40 50 Depth z FIGURE 20 Conductivity from BG Impulse Response Models P a n e l s (a) and (b) g i v e B u r r i d g e e q u a t i o n o(z) r e c o n s t r u c t i o n s u s i n g t h e BG s m a l l ( s h o r t d a s h e s ) and BG f l a t ( l o n g d a s h e s ) R(x) m o d e l s from F i g u r e s 7c and 7d. The t r u e a(z) i s t h e s o l i d l i n e . P a n e l s ( c ) and (d) g i v e t h e e q u i v a l e n t G o p i n a t h - S o n d h i r e c o n s t r u c t i o n s u s i n g t h e BG s m a l l ( s h o r t d a s h e s ) and BG f l a t ( l o n g d a s h e s ) B(x) models from F i g u r e s 8c and 8d. G o p i n a t h - S o n d h i r e c u r s i o n s a l l r e q u i r e o n l y o r d e r N2 o p e r a t i o n s . The l e f t h a l f of F i g u r e 20 shows B u r r i d g e e q u a t i o n o(z) r e c o n s t r u c t i o n s u s i n g t h e BG s m a l l ( s h o r t d a s h e s ) and BG f l a t ( l o n g d a s h e s ) R(x) c a l c u l a t e d from i n a c c u r a t e d a t a . The r i g h t h a l f shows G o p i n a t h - S o n d h i r e c o n s t r u c t i o n s u s i n g t h e e q u i v a l e n t BG B(x) m o d e l s . The t r u e c o n d u c t i v i t y i s p l o t t e d a s W e i d e l t I n v e r s e S c a t t e r i n g / 82 a s o l i d l i n e . In g e n e r a l , BG s m a l l i m p u l s e r e s p o n s e i n v e r s i o n s y i e l d o(z) models c l o s e r t h a n BG f l a t m odels t o t h e t r u e o(z). The r e a s o n i s t h a t BG f l a t i m p u l s e r e s p o n s e models f r o m F i g u r e s 7c, 7d, 8c, and 8d do n o t d e c a y t o z e r o a s r a p i d l y a s BG s m a l l m o d e l s . An a p p r o x i m a t e l y c o n s t a n t i m p u l s e r e s p o n s e has a v e r y s m a l l f l a t norm so t h a t t h e m i n i m i z a t i o n o f <p2 c a n p r o d u c e s u c h r e g i o n s . However, t h e c o r r e s p o n d i n g a(z) i s l i k e a ramp c r e x p o n e n t i a l f u n c t i o n . The l a r g e r t h e i m p u l s e r e s p o n s e o f f s e t f r o m z e r o , t h e f a s t e r t h e c o n d u c t i v i t y c h a n g e s . Hence, i n F i g u r e 20 t h e c o r r e s p o n d i n g o(z) m o d e l s ( l o n g d a s h e s ) do n o t a p p r o a c h a c o n s t a n t v a l u e but p l u n g e s t e e p l y . T h e r e f o r e , BG s m a l l i m p u l s e r e s p o n s e models a r e p r e f e r r e d o v e r BG f l a t m o d e l s b e c a u s e t h e y a r e n o t as l i k e l y t o p r o d u c e s u c h r a p i d l y c h a n g i n g c o n d u c t i v i t y f e a t u r e s . One way t o improve t h e BG f l a t i n v e r s i o n i s t o a p p l y a w e i g h t i n g f u n c t i o n a t l a r g e d e p t h s w h i c h d i s c r i m i n a t e s a g a i n s t l a r g e i m p u l s e r e s p o n s e v a l u e s and, h e n c e , l a r g e r e f l e c t i o n c o e f f i c i e n t s t h e r e . F i g u r e 21 i s t h e same as F i g u r e 20 e x c e p t t h a t t h e a(z) r e c o n s t r u c t i o n s a r e b a s e d on LP R(x) and LP B(x) models f r o m F i g u r e s 10c, I0d, 11c, and 11d. The c o n d u c t i v i t y p r o f i l e s f r o m LP s m a l l m o d els ( s h o r t d a s h e s ) a r e , as w i t h F i g u r e 20, l e s s l i k e l y t h a n t h e LP f l a t m odels ( l o n g d a s h e s ) t o p l u n g e t o low v a l u e s . The x 2 m i s f i t s f o r a l l a(z) models i n F i g u r e s 20 and 21 a r e n e a r t h e e x p e c t e d v a l u e o f 30 w h i c h i s t h e m i s f i t u s e d Weidelt Inverse S c a t t e r i n g / 83 10-1 10 20 30 40 50 Depth z 10 -2 o 10 ti o ° i o - 3 - i 1 1 1 1 1 r=~i Ck d ! l _ 10 20 30 40 50 Depth z FIGURE 21 C o n d u c t i v i t y from LP Impulse Response Models P a n e l s (a) and (b) g i v e B u r r i d g e e q u a t i o n a(z) r e c o n s t r u c t i o n s u s i n g t h e LP s m a l l ( s h o r t d a s h e s ) and LP f l a t ( l o n g d a s h e s ) R(x) models from F i g u r e s 10c and I 0 d . The t r u e o(z) i s t h e s o l i d l i n e . P a n e l s ( c ) and (d) g i v e t h e e q u i v a l e n t G o p i n a t h - S o n d h i r e c o n s t r u c t i o n s u s i n g t h e LP s m a l l ( s h o r t d a s h e s ) and LP f l a t ( l o n g d a s h e s ) B(x) models f r o m F i g u r e s 11c and 11d. f o r t h e f i r s t - s t a g e i m p u l s e r e s p o n s e c o n s t r u c t i o n s . F i g u r e s 20 and 21 i l l u s t r a t e a c o n n e c t i o n between t h e norm o f t h e i m p u l s e r e s p o n s e and t h e norm o f o(x). In g e n e r a l , t h e s e c o n d s t a g e i n t e g r a l e q u a t i o n s t a k e an i m p u l s e r e s p o n s e and g e n e r a t e a smoother c o n d u c t i v i t y p r o f i l e . F o r example, a d e l t a - f u n c t i o n R(x) g i v e s a p i e c e w i s e c o n s t a n t o(x) ( F i g u r e W e i d e l t I n v e r s e S c a t t e r i n g / 84 21a, s h o r t d a s h e s ) , and a p i e c e w i s e c o n s t a n t R(x) g i v e s a more c o n t i n u o u s a(x) ( F i g u r e 21a, l o n g d a s h e s ) . A f i r s t - o r d e r a n a l y s i s i n t h e n e x t s e c t i o n c o n f i r m s t h i s c o n n e c t i o n between norms. T h e r e f o r e , minimum s t r u c t u r e i m p u l s e r e s p o n s e s y i e l d ( i n some s e n s e ) minimum s t r u c t u r e c o n d u c t i v i t y m o d e l s . 2.6.3. A p p r o x i m a t e M a p p i n g s The a l g o r i t h m s d i s c u s s e d i n t h e p r e v i o u s two s e c t i o n s a r e e f f i c i e n t , e x a c t methods f o r f i n d i n g t h e c o n d u c t i v i t y from t h e i m p u l s e r e s p o n s e . In t h i s s e c t i o n , I examine t h r e e a p p r o x i m a t e mappings of R(x) t o a(x). T h e s e t e c h n i q u e s o f t e n c o n s t r u c t a(x) p r o f i l e s w i t h a c c e p t a b l e v a l u e s o f x 2 . However, t h e r e i s no g u a r a n t e e t h a t t h i s i s a l w a y s s o . One a p p r o x i m a t i o n g i v e s a u s e f u l l i n e a r r e l a t i o n s h i p between t h e i m p u l s e r e s p o n s e and t h e l o g a r i t h m o f t h e c o n d u c t i v i t y . A n o t h e r g i v e s some p h y s i c a l i n s i g h t i n t o t h e f u n c t i o n A(x,y). Coen e t a l . (1983) p r e s e n t e d an MT i n v e r s i o n method b a s e d on t h e B o r n a p p r o x i m a t i o n of an i n t e g r a l e q u a t i o n . The wavenumber-domain r e f l e c t i v i t y f u n c t i o n r(k) and t h e i m p u l s e r e s p o n s e R(x) f o l l o w d i r e c t l y f r o m t h i s a p p r o x i m a t i o n . A n o n l i n e a r t r a n s f o r m a t i o n of t h e i n t e g r a l o f R(x) g i v e s t h e c o n d u c t i v i t y : a(x)/o0 ~ 2 1 + S(2x) 1 - S(2x) (46) Weidelt Inverse S c a t t e r i n g / 85 where 2x S(2x) = / R(t)dt . 0 Howard (1983) simultaneously developed the e q u i v a l e n t e x p r e s s i o n f o r the 1-D seismic i n v e r s e problem to f i n d the a c o u s t i c impedance. These exp r e s s i o n s are exact f o r a s i n g l e uniform l a y e r o v e r l y i n g a uniform h a l f - s p a c e . The approximation e f f e c t i v e l y assumes a constant c o n d u c t i v i t y below each depth x, implying that there are no l a r g e primary r e f l e c t i o n s i n the impulse response R(t) f o r t>2x. Moreover, the method assumes that the i n t e g r a l of the m u l t i p l e r e f l e c t i o n s c o n t r i b u t i n g t o R(t) f o r t>2x i s s m a l l . Howard (1983) d e r i v e d two a d d i t i o n a l approximate methods based on a Neumann s e r i e s s o l u t i o n of a matrix Marchenko equ a t i o n . Given the s c a l a r Marchenko equation (12) I w r i t e x A,(x,y) = R(x+y) - J A2 (x, t ) R( y+t ) dt (47) -y and x A2(x,y) = - J A,(x,t)R(y+t)dt , (48) -y where A(x, y)=A,(x, y)-A2(x, y). E l i m i n a t i n g A2 from equations (47) and (48) gi v e s x x A,(x,y) = R(x+y) + / / A, (x, T ) R( t+T ) R( y+t ) dr dt . (49) -y -I By analogy with equations (41) and (44) of Howard (1983) and equation (14) here, the c o n d u c t i v i t y i s Weidelt Inverse S c a t t e r i n g / 86 x a(x)/o0 = exp{8 J A^(t,t)dt} 0 2x = exp{4 / A,(t/2, t/2)dt] . (50) 0 Note that the f u n c t i o n Ay(x,x) i s p r o p o r t i o n a l to the continuous r e f l e c t i o n c o e f f i c i e n t y(x) presented i n S e c t i o n 2.6.1 and used throughout Chapter 3. The f i r s t two approximations to A^ from the f u l l Neumann s e r i e s s o l u t i o n of equation (49) are A, (x, y) - R(x+y) (51 ) and x x Ay(x,y) * R(x+y) + J / R(X+T)R(t + T)R(y+t)drdt . (52) -y -t As more terms are added to the Neumann s e r i e s , the m u l t i p l e r e f l e c t i o n s i n R(2x) are g r a d u a l l y c a n c e l l e d so that Ay approaches y(x). From equations (50) and (51) the c o n d u c t i v i t y i s 2x a(x)/a0 ~ exp{4 J R(t)dt}. (53) 0 T h i s approximation t r e a t s every nonzero p o i n t of the impulse response as a primary r e f l e c t i o n . Thus, m u l t i p l e r e f l e c t i o n s are i n c o r r e c t l y i n t e r p r e t e d as changes i n c o n d u c t i v i t y . Equation (53) i s a l i n e a r i z a t i o n of equation (50) and so i t i s r e f e r r e d to as a Born approximation. Note that equation (53) g i v e s an approximate r e l a t i o n s h i p between the norm a p p l i e d to R(x) and the r e s u l t i n g norm on Weidelt Inverse S c a t t e r i n g / 87 a(x). R e w r i t i n g equation (53) as 2x 4 S R(t)dt =* I og[o(x)/o0], 0 and t a k i n g the d e r i v a t i v e with r e s p e c t to x g i v e s 8R(2x) » {/og[o(x)/o0]}' . T h i s equation i m p l i e s that m i n i m i z i n g R(x) over a range i n x minimizes the d e r i v a t i v e of the l o g a r i t h m of the normalized c o n d u c t i v i t y . Hence, to f i r s t order I have j u s t i f i e d the c l a i m made p r e v i o u s l y that a p a r t i c u l a r norm on the impulse response leads t o a smoother norm on the c o n d u c t i v i t y . M a r c h i s i o (1985) gave a r i g o r o u s d i s c u s s i o n of the correspondance between norms on the s p e c t r a l f u n c t i o n and the c o n d u c t i v i t y . A b e t t e r approximation which i n c o r p o r a t e s f i r s t - o r d e r m u l t i p l e s f o l l o w s from equations (50) and (52). Employing s e v e r a l changes of v a r i a b l e s and s e t t i n g y=x g i v e s 2x x! A%(x,x) « R(2x) + J R(x^) j R(x2)R(2x+x2-xJ dx2dx: . (54) 0 0 Note that the arguments of R are a l l p o s i t i v e i n keeping with the c a u s a l i t y of the impulse response. S u b s t i t u t i n g equation (54) i n t o equation (50) g i v e s 2x a(x)/a0 - exp{4 J R(t)dt + 0 2x t x, * S J* J* R(xjR(x2)R(t+x2-xJdx2dx%dt}. (55) 0 0 0 The f i r s t term i n equation (55) i s i d e n t i c a l to equation (53). W e i d e l t I n v e r s e S c a t t e r i n g / 88 0 10 20 30 40 50 0 10 20 30 40 50 D e p t h z D e p t h z FIGURE 22 A p p r o x i m a t e Mappings o f t h e T r u e Impulse Response P a n e l s (a) and (b) g i v e t h e t r u e a(z) ( s o l i d l i n e ) p l u s an a p p r o x i m a t e i n v e r s i o n ( d a s h e d l i n e ) u s i n g e q u a t i o n (46) on t h e t r u e i m p u l s e r e s p o n s e s o f F i g u r e s 2a and 2b. P a n e l s ( c ) and (d) g i v e t h e t r u e a(z) ( s o l i d l i n e ) p l u s two a p p r o x i m a t e i n v e r s i o n s o f t h e t r u e i m p u l s e r e s p o n s e u s i n g e q u a t i o n s (53) ( d a s h e d l i n e ) and (55) ( d o t t e d l i n e ) . The s e c o n d t e r m i s a c o r r e c t i o n f o r f i r s t - o r d e r m u l t i p l e r e f l e c t i o n s . R e s n i c k (1982) p r e s e n t e d s u c c e s s i v e a p p r o x i m a t i o n s t o t h e B u r r i d g e e q u a t i o n which a r e s i m i l a r i n form t o e q u a t i o n s (46) and (53) expanded t o s e c o n d o r d e r i n S(2x). R e s n i c k a l s o p r e s e n t e d e r r o r bounds f o r t h e s e and h i g h e r - o r d e r a p p r o x i m a t i o n s . W e i d e l t I n v e r s e S c a t t e r i n g / 89 0 10 20 30 40 50 0 10 20 30 40 50 D e p t h z D e p t h z FIGURE 23 A p p r o x i m a t e M a p p i n g s of Impulse Response M o d e l s P a n e l s (a) and (b) g i v e t h e t r u e a(z) ( s o l i d l i n e ) p l u s two a p p r o x i m a t e i n v e r s i o n s d e r i v e d u s i n g e q u a t i o n (55) on t h e BG s m a l l ( s h o r t d a s h e s ) and BG f l a t ( l o n g d a s h e s ) R(x) models i n F i g u r e s 7c and 7d. P a n e l s ( c ) and (d) g i v e s i m i l a r o(z) d e r i v e d f r o m e q u a t i o n (55) and t h e LP s m a l l ( s h o r t d a s h e s ) and LP f l a t ( l o n g d a s h e s ) R(x) models i n F i g u r e s 10c and !0d. F i g u r e 22 shows t h e r e s u l t s of t h e t h r e e a p p r o x i m a t e methods o p e r a t i n g on t h e t r u e i m p u l s e r e s p o n s e s o f F i g u r e s 2a and 2b. The t r u e o(z) p r o f i l e i s t h e s o l i d l i n e . F i g u r e s 22a and 22b g i v e t h e r e s u l t s o f e q u a t i o n (46) ( d a s h e d l i n e ) . T h i s a p p r o x i m a t i o n o v e r s h o o t s t h e t r u e c o n d u c t i v i t y h i g h and m i s s e s t h e low n e a r z=30. F i g u r e s 22c and 22d g i v e t h e t r u e o(z) ( s o l i d l i n e ) p l u s t h e Born a p p r o x i m a t i o n from e q u a t i o n (53) Weidelt Inverse S c a t t e r i n g / 90 (dashed l i n e ) and the approximation (55) (dotted l i n e ) . F i g u r e 22d shows how m u l t i p l e s i n R(x) are t r e a t e d by the Born approximation as p r i m a r i e s , r e s u l t i n g i n spurious s t r u c t u r e w i t h i n the broad c o n d u c t i v i t y low near z=30. Equation (55) generates c o n d u c t i v i t y p r o f i l e s c l o s e r to the true o(z) f o r these t e s t c a ses. F i g u r e s 23a and 23b show a(z) r e c o n s t r u c t i o n s using equation (55) on BG s m a l l (short dashes) and BG f l a t (long dashes) R(x) models from F i g u r e s 7c and 7d. These o(z) p r o f i l e s are very s i m i l a r to the exact B u r r i d g e equation r e c o n s t r u c t i o n s i n F i g u r e s 20a and 20b. F i g u r e s 23c and 23d give s i m i l a r a(z) r e c o n s t r u c t i o n s from LP small (short dashes) and LP f l a t (long dashes) R(x) models from F i g u r e s 10c and I0d. Again, these curves are very c l o s e to the exact Burridge equation r e s u l t s i n F i g u r e s 21a and 21b. In t h i s chapter, I d i s c u s s e d the i n v e r s e theory f o r m u l a t i o n of Weidelt's MT i n v e r s e s c a t t e r i n g . The great f l e x i b i l i t y of the method l i e s i n the l i n e a r f i r s t - s t a g e i n v e r s i o n f o r the impulse response. D i f f e r e n t norms a p p l i e d at t h i s stage y i e l d d i f f e r e n t c l a s s e s of c o n d u c t i v i t y models: piecewise c o n s t a n t , piecewise l i n e a r , and smooth. A l s o , l o c a l i z e d c o n s t r a i n t s on the impulse response i n v e r s i o n have p r e d i c t a b l e , although approximate, e f f e c t s on the c o n d u c t i v i t y . The power of t h i s augmented i n v e r s e s c a t t e r i n g a l g o r i t h m i s i n i t s a b i l i t y to r e s t r i c t and explore the space W e i d e l t I n v e r s e S c a t t e r i n g / 91 of a c c e p t a b l e c o n d u c t i v i t y p r o f i l e s . Thus, t h e p i t f a l l s o f b a s i n g an i n t e r p r e t a t i o n on a s i n g l e a(z) model o r on models f r o m a s i n g l e a l g o r i t h m a r e a v o i d e d . The R i c c a t i i n v e r s i o n s p r e s e n t e d n e x t use e f f e c t i v e l i n e a r programming t e c h n i q u e s t o f u r t h e r c o n s t r a i n and e x p l o r e t h e s p a c e o f c o n d u c t i v i t y m o d e l s . CHAPTER 3. RICCATI EQUATION INVERSIONS C o n s t r a i n t s a r e v i t a l t o a m e a n i n g f u l MT i n t e r p r e t a t i o n . P h y s i c a l c o n s t r a i n t s f r o m e x t e r n a l s o u r c e s r e s t r i c t t h e n o n u n i q u e n e s s and h e l p c o n s t r u c t c o n d u c t i v i t y models t h a t a r e c l o s e r t o t h e t r u e e a r t h . C o n s t r a i n t s can a l s o be d e s i g n e d t o a s s e s s t h e e x t e n t of t h e n o n u n i q u e n e s s and e x p l o r e t h e r a n g e of a c c e p t a b l e m o d e l s . The t w o - s t a g e MT i n v e r s i o n p r o c e d u r e d i s c u s s e d i n C h a p t e r 2 a c c e p t s c o n s t r a i n t s on t h e f i r s t - s t a g e i m p u l s e r e s p o n s e i n v e r s i o n . The p h y s i c a l r e a l i z a b i l i t y and a p r i o r i c o n s t r a i n t s b o t h have a w e l l - d e f i n e d e f f e c t on t h e c o n d u c t i v i t y p r o d u c e d by a s e c o n d - s t a g e mapping o f t h e c o n s t r a i n e d i m p u l s e r e s p o n s e . However, f o r t h e i m p o r t a n t c a s e of l o c a l i z e d c o n s t r a i n t s , t h e e f f e c t i s n o t as r i g o r o u s l y d e f i n e d b e c a u s e o f t h e c o m p l e x i t y o f t h e c o n d u c t i v i t y mappings and t h e n o n l i n e a r t r a n s f o r m a t i o n f r o m d e p t h z t o p s e u d o d e p t h x. To c o n s t r a i n a(z) d i r e c t l y i t i s n e c e s s a r y t o r e f o r m u l a t e t h e MT i n v e r s e p r o b l e m . In S e c t i o n 3.1, I d e r i v e a R i c c a t i e q u a t i o n w h i c h r e l a t e s t h e wavenumber-domain r e f l e c t i v i t y r(k) t o a s i m p l e f u n c t i o n of o(z). T h i s e q u a t i o n f o l l o w s f r o m r e w r i t i n g t h e MT e q u a t i o n (1) as a s y s t e m of two, c o u p l e d , f i r s t - o r d e r e q u a t i o n s . R e c a s t i n g t h i s s y s t e m i n t o u p g o i n g and downgoing waves g i v e s a n a t u r a l b a s i s f o r t h e R i c c a t i e q u a t i o n i n v o l v i n g r(k) b e c a u s e t h e r e f l e c t i v i t y i s s i m p l y t h e r a t i o o f u p g o i n g t o downgoing e n e r g y . The R i c c a t i e q u a t i o n a l s o f o l l o w s f r o m g e n e r a l i z i n g 92 R i c c a t i E q u a t i o n I n v e r s i o n s / 93 c(k) and r(k) t o n o n z e r o d e p t h s and s u b s t i t u t i n g t h e s e e x p r e s s i o n s i n t o t h e o r i g i n a l MT e q u a t i o n ( 1 ) . I i n t e g r a t e t h e R i c c a t i e q u a t i o n so t h a t i t i s i n a form amenable t o s o l u t i o n by l i n e a r i n v e r s e t e c h n i q u e s . The i n t e g r a l e q u a t i o n i s e x a c t b u t i t i s n o n l i n e a r and so must be s o l v e d by i t e r a t i o n . The R i c c a t i f o r m u l a t i o n and i t e r a t i v e s o l u t i o n i s n o t i n t e n d e d t o r e p l a c e t h e more e f f i c i e n t , e x a c t i n v e r s i o n scheme of C h a p t e r 2. I t s p u r p o s e i s t o complement t h e i n v e r s e s c a t t e r i n g a p p r o a c h by u s i n g c o n s t r a i n t s t o a s s e s s t h e n o n u n i q u e n e s s and r e f i n e i n t e r p r e t a t i o n s b a s e d on t h e e x a c t method. In S e c t i o n 3.2, I p r e s e n t t h e n u m e r i c a l d e t a i l s o f t h e s o l u t i o n o f t h e i n t e g r a t e d R i c c a t i e q u a t i o n . The a t t e n d e n t p r o b l e m s o f c o n v e r g e n c e and of c h o o s i n g an i n i t i a l - o(z) a r e m i t i g a t e d by s t a r t i n g t h e i t e r a t i o n s w i t h one of t h e d i v e r s e a c c e p t a b l e m o d e l s from t h e MT i n v e r s e s c a t t e r i n g a l g o r i t h m . A l t h o u g h c o n s t r a i n e d l e a s t - s q u a r e s s o l u t i o n s a r e p o s s i b l e , I use a f a s t e r l i n e a r programming f o r m u l a t i o n w h i c h r e a d i l y a c c e p t s a wide v a r i e t y of e x t r a e q u a l i t y and i n e q u a l i t y c o n s t r a i n t s on a(z). 3.1. RICCATI EQUATION THEORY In t e r m s o f t h e s u r f a c e wavenumber k=[ i u0coo0 ] 1 / 2 and t h e d e p t h - d e p e n d e n t wavenumber v(z) = [i u0u>o(z) ] 1 / 2 , t h e o r i g i n a l MT e q u a t i o n (1) i s E"(z,k) - v2(z)E(z,k) = 0. (56) R i c c a t i Equation I n v e r s i o n s / 94 T h i s equation may be w r i t t e n as the f o l l o w i n g matrix system: Y' = AY (57) where E(z, k) Y = £' (z, k) and A = 0 1 v2(z) 0 F o l l o w i n g Claerbout (1976, p.169), I decompose the system (57) i n t o the upgoing and downgoing waves U(z,k) and D(z,k). Note that the waves f o r the MT problem are e x p o n e n t i a l l y attenuated because of the complex wavenumber v(z). The matrix C of column e i g e n v e c t o r s of t r a n s f o r m a t i o n . Let Y = CV where g i v e s the r e q u i r e d (58) C = }/v(z) 1 1 v(z) -v(z) and the ve c t o r V of upgoing and downgoing waves i s C" 1Y or V = Vv(z) E(z,k) + £' (z, k)/v(z) E(z,k) - EUz,k)/v(z) (59) S u b s t i t u t i n g equation (58) i n t o equation (57) and R i c c a t i Equation I n v e r s i o n s / 95 p r e m u l t i p l y i n g the r e s u l t by C" 1 g i v e s V = [C"'AClV - [ C " ' C ] V . The expanded v e r s i o n of the pr e c e d i n g equation i s (60) 9 U v(z) 0 U 0 1 u — = + n(z) 3z D 0 -v(z) D 1 0 D where the r e a l f u n c t i o n v'(z) o'(z) y(z) = = (61 ) 2v( z) 4a( z) i s the continuous analogue of an i n t e r f a c e r e f l e c t i o n c o e f f i c i e n t . y(z) i s a c o u p l i n g c o e f f i c i e n t which, depending on the c o n d u c t i v i t y g r a d i e n t , transforms downgoing energy i n t o upgoing energy and v i c e v e r s a . Ignoring y(z) and the second matrix on the right-hand s i d e of equation (60) r e s u l t s i n a decoupli n g of the waves U(z,k) and D(z,k). Equation (60) i s the continuous e q u i v a l e n t of the d i s c r e t e MT propagator matrix d i s c u s s e d i n S e c t i o n 2.5.1. The complex r e f l e c t i v i t y at depth z and wavenumber k i s simply the r a t i o r(z,k) = -U(z,k)/D(z,k) . (62) The minus s i g n i s r e q u i r e d to s a t i s f y W e i delt's p o l a r i t y c o n v e n t i o n . Equations (60) and (62) y i e l d a R i c c a t i equation f o r r (z, k) , namely rUz.k) = 2v(z)r(z,k) + y(z)[r2(z,k) - / ] . (63) There e x i s t other n o r m a l i z a t i o n s of the matrix of e i g e n v e c t o r s R i c c a t i Equation I n v e r s i o n s / 96 C. These give s l i g h t l y d i f f e r e n t e x p r e s s i o n s f o r U(z,k) and D(z,k) but the same R i c c a t i equation f o r t h e i r r a t i o r(z,k). The i n v e r s e Laplace transform of the x-domain form of equation (63) g i v e s an equation i n terms of the impulse response R(x,y) that would be recorded at pseudodepth x f o r two-way depths y. T h i s equation was d i s c u s s e d by Corones et a l . (1983) and Bregman et a l . (1985), and i t i s another mapping of the impulse response to a c o n d u c t i v i t y p r o f i l e . Every second-order, homogeneous, l i n e a r equation may be mapped to a n o n l i n e a r R i c c a t i e q uation. For the R i c c a t i equation (63) the mapping from the MT equation (56) i s 1- v ( z) c ( z, k) r(z,k) = , (64) l + v(z)c(z, k) where c(z, k) = -E(z, k)/E' (z,k) . These are g e n e r a l i z a t i o n s of c(k) and r(k) to nonzero depths. S o l v i n g equation (64) f o r E'(z,k), d i f f e r e n t i a t i n g and s u b s t i t u t i n g f o r E"(z,k) from equation (56) g i v e s the R i c c a t i equation (63). An i n t e g r a t i n g f a c t o r f o r equation (63) i s exp{-2 } v(t)dt] = e~2kx 0 where the pseudodepth x i s the f u n c t i o n of z d e f i n e d by equation ( 3 ) . Using t h i s i n t e g r a t i n g f a c t o r , equation (63) becomes R i c c a t i Equation I n v e r s i o n s / 97 [r(z, k)e~2KXV = y(z)[r2(z,k)-'\]e~2kx. I f i t i s assumed that below a l a r g e depth Z the c o n d u c t i v i t y i s c onstant, then the upgoing r e f l e c t e d energy U(Z,k)=0 and the r e f l e c t i v i t y r(Z,k) = 0. I n t e g r a t i n g the pr e c e d i n g equation from z=0 to z=Z and using t h i s boundary c o n d i t i o n g i v e s r(k) = J y(z)[l-r2(z,k)]e~2kxdz (65) 0 where r (k)=r (0, k) . Equation (65) i s a n o n l i n e a r i n t e g r a l equation r e l a t i n g the measured data r(k) to a f u n c t i o n of the c o n d u c t i v i t y y(z). The n o n l i n e a r i t y a r i s e s because the kern e l s of equation (65), — 2 kx [1-r2(z, k)]e , i n v o l v e r(z, k) and x which depend on the as yet unknown model y(z). To s o l v e equation (65) I i t e r a t e as f o l l o w s . I make an i n i t i a l estimate of the c o n d u c t i v i t y p r o f i l e and c a l c u l a t e the corresponding r2(z,k) and x using equations (56), (64), and (3). I then use the r e s u l t i n g approximate k e r n e l s and l i n e a r i n v e r s e techniques to sol v e f o r y(z). The corresponding c o n d u c t i v i t y i s , a f t e r i n t e g r a t i n g equation (61), z o(z)/a0 = exp{4 J y(t)dt}. (66) 0 T h i s new o(z) g i v e s a new estimate f o r the f u n c t i o n s r2(z,k) and x. I repeat the process u n t i l convergence i s a t t a i n e d . The convergence c r i t e r i o n i s that the x 2 m i s f i t i n equation (22) must be near i t s expected value of 2N but l e s s than the 95 R i c c a t i Equation I n v e r s i o n s / 98 percent c o n f i d e n c e l i m i t of approximately 2N+2(4N)1/2. Note that equation (66) guarantees that the c o n d u c t i v i t y i s p o s i t i v e . 3.2. NUMERICAL SOLUTIONS The numerical s o l u t i o n of the i n t e g r a t e d R i c c a t i equation i n v o l v i n g r (k) i s s i m i l a r to BG and LP s o l u t i o n s of equations (16) and (17) f o r the impulse response. Let e. j=l,2,...2N r e p r e s e n t the r e a l and imaginary p a r t s of the complex data r(k.) i n equation (65). Let g.(z) j=1, 2,...2N represent the — 2 kx c o r r e s p o n d i n g p a r t s of the complex k e r n e l s [l-r2(z,k)]e Equation (65) becomes Z e, = J y(z)g (z)dz j = l , 2 , . . . 2 N . (67) 1 0 J At each i t e r a t i o n I c o u l d s o l v e equation (67) f o r y(z) using standard i n v e r s e techniques. If an i n i t i a l or i t e r a t e d estimate of a(z) i s known then the c u r r e n t estimate of the ke r n e l s Sj(z) m a Y D e c a l c u l a t e d by s o l v i n g the MT equation (56) and us i n g equation (64) f o r the r e f l e c t i v i t y . At d i s c o n t i n u i t i e s i n a(z) the boundary c o n d i t i o n s are that E(z,k) and E'(z,k) are continuous across the i n t e r f a c e . Together, these imply that c(z,k) i s continuous. The value s of r(z,k) i n the ke r n e l s are c a l c u l a t e d r e c u r s i v e l y using equation (64) and matching the boundary c o n d i t i o n s . Note that from one i t e r a t i o n to the next, the maximum R i c c a t i Equation I n v e r s i o n s / 99 a l l o w a b l e m i s f i t to the data i s u s u a l l y reduced from an i n i t i a l l y h igh l e v e l . T h i s i s because the s t a r t i n g model may be f a r from any s o l u t i o n with an a c c e p t a b l e x 2 v a l u e . As the i t e r a t i o n s p r o g r e s s , the a l l o w a b l e m i s f i t i s reduced and the s o l u t i o n i s g r a d u a l l y molded i n t o one with an a c c e p t a b l e x 2 m i s f i t . The BG method generates a convergent s e r i e s of smooth c o n d u c t i v i t y models but i t i s very time consuming to r e c a l c u l a t e the inner product matrix f o r every i t e r a t i o n . Moreover, the BG method does not r e a d i l y accept e x t r a c o n s t r a i n t s on a(z). The d e s i r e to impose d i r e c t c o n d u c t i v i t y c o n s t r a i n t s was the reason f o r not using an exact, n o n i t e r a t i v e a l g o r i t h m such as i n v e r s e s c a t t e r i n g . An LP s o l u t i o n of equation (67) i s much f a s t e r than a BG s o l u t i o n and LP e a s i l y i n c o r p o r a t e s the e x t r a c o n d u c t i v i t y c o n s t r a i n t s necessary to r e s t r i c t and explore the space of a c c e p t a b l e models. The LP approach to s o l v i n g equation (67) p a r t i t i o n s the i n t e r v a l [0,Z] i n t o L l a y e r s with z,=0 and z^+j=Z. The model y(z) i s assumed to be composed of d e l t a f u n c t i o n s at the top of each l a y e r . Equation (66) then r e s t r i c t s a(z) to the c l a s s of models composed of homogeneous l a y e r s . Equation (67) becomes L e. = L 7,*,, j = 2 , . . . 2N, (68) J l = 1 i Ji R i c c a t i Equation I n v e r s i o n s / 100 where 7 / o g ( a ^ / a ^ _ j ) ] / 4 , o^=a0, and g^ i s the average of the k e r n e l v a l u e s on e i t h e r s i d e of the i n t e r f a c e at z^ . The average i s necessary because the d e l t a f u n c t i o n l o c a t i o n s are p r e c i s e l y where the k e r n e l s are d i s c o n t i n u o u s . D i f f e r e n t assumptions about y(z) are p o s s i b l e . For i n s t a n c e , i t may be convenient to r e q u i r e i t to be c o n s t a n t , or of constant g r a d i e n t , w i t h i n each l a y e r . The c o r r e s p o n d i n g form of . a(z) f o l l o w s from equation (66). These l a s t two p a r a m e t e r i z a t i o n s are not c o n s i d e r e d here but may be used to generate d i f f e r e n t c l a s s e s of o(z) p r o f i l e s . An LP model i s c o n s t r u c t e d by m i nimizing or maximizing the Lf norm o b j e c t i v e f u n c t i o n L <t> = Z w, | 7, I f (69) / = ; ' ' s u b j e c t to the c o n s t r a i n t s of equation (68). The f i n a l c o n d u c t i v i t y i s given by a d i s c r e t e form of equation (66). That i s , / o,/o0 = exp{4 Z 7 }. (70) Many accept a b l e LP models can be generated by changing the p a r t i t i o n of the i n t e r v a l [0,Z] and the weighting f u n c t i o n w^ . However, the f u l l power of the LP f o r m u l a t i o n i s r e a l i z e d only when used to impose l o c a l i z e d c o n s t r a i n t s on y(z) and hence o(z). Within the LP method i t i s easy to impose ex t r a i n e q u a l i t y , e q u a l i t y , upper bound or lower bound c o n s t r a i n t s R i c c a t i Equation I n v e r s i o n s / 101 on any l i n e a r combination of the 7^. Moreover, equation (70) i n d i c a t e s t h at monotonically i n c r e a s i n g or d e c r e a s i n g a(z) f o l l o w from r e q u i r i n g 7^  to be nonnegative or n o n p o s i t i v e , r e s p e c t i v e l y . Any i t e r a t i v e s o l u t i o n a l g o r i t h m f o r a n o n l i n e a r equation must address the problems of the ch o i c e of a s t a r t i n g model and convergence. An unbiased s t a r t i n g model f o r the i t e r a t i o n s of equation (67) i s a homogeneous h a l f - s p a c e of c o n d u c t i v i t y a 0 . On the f i r s t i t e r a t i o n , t h i s i m p l i e s that r2(z,k) = 0 because there i s no upgoing r e f l e c t e d energy. T h i s s t a r t i n g model l i n e a r i z e s equation (67) so that the r e s u l t i n g y(z) i s a Born approximation. Coen et a l . (1983) d i s c u s s e d t h i s model which i s c l o s e ( i n some sense) to the homogeneous h a l f - s p a c e that generated the k e r n e l s . I t e r a t i v e improvements to the Born approximation are u s u a l l y necessary to f i n d a model which f i t s the data. I t i s e q u a l l y p o s s i b l e to begin the i t e r a t i o n s with a nonuniform a(z) p r o f i l e . A good i n i t i a l o(z) may e l i m i n a t e the need f o r i t e r a t i o n ; a poor estimate may cause many i t e r a t i o n s and suggest that f e a t u r e s on the s t a r t i n g a(z) are not allowed by the data. Uniform s t a r t i n g models g e n e r a l l y do not f i t the data w e l l and can r e s u l t i n a to r t u o u s path to convergence. F i g u r e 24 shows, however, that the convergence i s good even f o r the d i f f i c u l t case of uniform s t a r t i n g models. F i g u r e s 24a and 24b show r e a l and imaginary p a r t k e r n e l s , r e s p e c t i v e l y , R i c c a t i E q u a t i o n I n v e r s i o n s / 102 0 10 20 30 40 50 0 10 20 30 40 50 Depth z Depth z FIGURE 24 Integrated R i c c a t i Equation K e r n e l s P a n e l s (a) and (b) show r e a l and i m a g i n a r y p a r t k e r n e l s c o r r e s p o n d i n g t o t h e s m a l l e s t wavenumber datum from t h e smooth a(z) i n a c c u r a t e d a t a s e t . The t r u e k e r n e l s a r e t h e s o l i d l i n e s , t h e k e r n e l s on t h e f i r s t i t e r a t i o n a r e d o t t e d l i n e s , and t h e k e r n e l s on t h e f i f t h i t e r a t i o n a r e d a s h e d l i n e s . P a n e l s ( c ) and (d) show t h e same k e r n e l s f o r t h e s m a l l e s t wavenumber datum from t h e m u l t i l a y e r a(z) i n a c c u r a t e d a t a s e t . c o r r e s p o n d i n g t o t h e s m a l l e s t wavenumber datum from t h e smooth o(z) i n a c c u r a t e d a t a s e t . These k e r n e l s have t h e l a r g e s t p e n e t r a t i o n d e p t h and so show t h e e f f e c t s o f i t e r a t i o n most c l e a r l y . The s o l i d l i n e s a r e t h e c o r r e c t k e r n e l s , t h e d o t t e d l i n e s a r e t h e k e r n e l s on t h e f i r s t i t e r a t i o n when r2(z,k)=0, and t h e d a s h e d l i n e s a r e i t e r a t i o n 5. The c o n v e r g e n c e i s good. R i c c a t i E q u a t i o n I n v e r s i o n s / 103 The r i g h t h a l f of F i g u r e 24 shows t h e same k e r n e l s f o r t h e s m a l l e s t wavenumber f r o m t h e m u l t i l a y e r o(z) i n a c c u r a t e d a t a s e t . The g r e a t e s t m i s f i t i s a t l a r g e d e p t h s , e s p e c i a l l y f o r t h e i m a g i n a r y p a r t k e r n e l . T h i s m i s f i t i s due t o t h e p o o r r e s o l u t i o n a t t h e s e d e p t h s and a l s o t o t h e f a c t t h a t t h e i m a g i n a r y p a r t datum has a l a r g e s t a n d a r d d e v i a t i o n . In p r a c t i c e , t h e p o t e n t i a l l y d i f f i c u l t c o n v e r g e n c e f r o m a homogeneous s t a r t i n g model i s c o m p l e t e l y a v o i d e d by s t a r t i n g w i t h an a c c e p t a b l e o(z) c o n s t r u c t e d by t h e i n v e r s e s c a t t e r i n g a p p r o a c h . In t h i s c a s e , a R i c c a t i i n v e r s i o n i n c o r p o r a t i n g e x t r a c o n s t r a i n t s has a smoother p a t h t o c o n v e r g e n c e . To i l l u s t r a t e how c o n s t r a i n t s improve i n t e r p r e t a t i o n s , I a p p l y t h e i n v e r s i o n a l g o r i t h m t o two d a t a s e t s . The f i r s t s e t i s t h e m u l t i l a y e r model i n a c c u r a t e d a t a w h i c h a r e s i m i l a r t o t h e smooth model d a t a shown i n F i g u r e s 6a and 6b. The s e c o n d i s t h e JDF d a t a s e t shown i n F i g u r e s 16a and 16b. F i g u r e 25a shows t h e t r u e m u l t i l a y e r model ( s o l i d l i n e ) , t h e r e c o n s t r u c t i o n a f t e r one i t e r a t i o n ( d o t t e d l i n e ) , and a f t e r c o n v e r g e n c e on t h e f i f t h i t e r a t i o n ( d a s h e d l i n e ) . The x 2 m i s f i t s f o r t h e s e two i t e r a t i o n s a r e 75 and 37, r e s p e c t i v e l y . The e x p e c t e d v a l u e of x 2 i s 30 f o r t h e s e d a t a . The s t a r t i n g model f o r t h e i t e r a t i o n s was a homogeneous h a l f - s p a c e o f c o n d u c t i v i t y a=0.002. The d a s h e d l i n e f o l l o w s t h e t r u e a(z) w e l l f o r z<20. However, t h e major c o n d u c t i v i t y low n e a r z=30 i s n o t p r e s e n t on t h e r e c o n s t r u c t e d m o d e l . A l l o f t h e l a y e r s 10 -1 o 10 ~* d o u 10 -a : 1 1 r —r~ 1 — I 1 i 1 I 1 1 IL — r h ~r.r.: i j u i • i . I . I R i c c a t i E q u a t i o n I n v e r s i o n s / 104 B m >> 10 20 30 Depth z 40 50 50 100 150 Depth z (km) 200 io-» 10 20 30 40 50 Depth z CO io-3 IO" 2 a o O 1 0 - 3 1 ' r - ' i ' ' 1 1 1 1 i >—: ! i - U 1 1 I 1 , 1 , 50 100 150 Depth z (km) 200 FIGURE 25 C o n s t r a i n e d C o n d u c t i v i t y I n v e r s i o n s P a n e l (a) shows the t r u e o(z) ( s o l i d l i n e ) a l o n g w i t h t h e r e c o n s t r u c t i o n s a f t e r one i t e r a t i o n ( d o t t e d l i n e ) and a f t e r f i v e i t e r a t i o n s ( d a s h e d l i n e ) . No e x t r a c o n s t r a i n t s were imposed. P a n e l (b) shows the t r u e a(z) ( s o l i d l i n e ) and t h e f o u r t h i t e r a t i o n w i t h c o n s t r a i n t s on th e c o n d u c t i v i t y h i g h and low n e a r z=17 and z=30, r e s p e c t i v e l y ( d a s h e d l i n e ) . P a n e l ( c ) shows an u n c o n s t r a i n e d c o n s t r u c t i o n u s i n g t h e JDF d a t a ( s o l i d l i n e ) and a m o n o t o n i c a l l y i n c r e a s i n g JDF model ( d a s h e d l i n e ) . P a n e l (d) shows a JDF model c o n s t r a i n e d t o have o(z)<0.05 S/m from z=30 t o z=84 km ( s o l i d l i n e ) and a model c o n s t r a i n e d t o have a(z)>0.3 S/m from z=65 t o z=75 km (dash e d l i n e ) . a r e not f u l l y r e s o l v e d b e c a u s e t h e r e a r e measurements a t o n l y IS f r e q u e n c i e s . M o r e o v e r , l a r g e jumps i n c o n d u c t i v i t y w i l l g e n e r a l l y n o t appear on r e c o n s t r u c t e d m o d e ls b e c a u s e m i n i m i z i n g <j> i n equation (69) minimizes, v i a equation (70), R i c c a t i Equation I n v e r s i o n s / 105 changes i n a(z). Hence, i n the absence of e x t r a c o n s t r a i n t s , the LP method generates models with a minimum of s t r u c t u r e . E x t e r n a l g e o p h y s i c a l or g e o l o g i c c o n s t r a i n t s improve the agreement between the t r u e and r e c o n s t r u c t e d p r o f i l e s . Suppose i t i s known that o(z)=0. 07±0. 005 from z=16 to z=19 and that o(z) = 0. 004±0. 0004 from z = 22 to z = 36. Imposing these e x t r a c o n s t r a i n t s and s t a r t i n g the a l g o r i t h m with the dashed l i n e i n F i g u r e 25a g i v e s the i n v e r s i o n shown i n F i g u r e 25b (dashed l i n e ) a f t e r four i t e r a t i o n s . The x 2 m i s f i t i s 38. Note that each i t e r a t i o n produces a model which s a t i s f i e s the c o n s t r a i n t s but which may have l a r g e v a r i a t i o n s i n the u n c o n s t r a i n e d r e g i o n s . In t h i s case, four i t e r a t i o n s were r e q u i r e d to converge to a minimum s t r u c t u r e model with a good X 2 v a l u e . The c o n s t r a i n t s enhance the r e c o n s t r u c t i o n of the major f e a t u r e s of the m u l t i l a y e r model. While these c o n s t r a i n t s are too p r e c i s e f o r a c t u a l MT i n t e r p r e t a t i o n s , t h i s example shows how e x t r a c o n s t r a i n t s s i g n i f i c a n t l y improve the agreement between the t r u e and r e c o n s t r u c t e d o(z). There remain, however, c o n s i d e r a b l e v a r i a t i o n s i n the c o n d u c t i v i t y v a l u e s allowed by the data. Next, I d i s c u s s c o n s t r a i n t s s p e c i f i c a l l y designed to e x p l o r e t h i s nonuniqueness. Many d i f f e r e n t c o n d u c t i v i t y p r o f i l e s f i t the JDF data e q u a l l y w e l l . T h i s ambiguity makes i t p a r t i c u l a r l y d i f f i c u l t to g i v e a r e l i a b l e i n t e r p r e t a t i o n . F i g u r e s 25c and 25d g i v e 4 d i v e r s e JDF models c o n s t r u c t e d by the i t e r a t i v e a l g o r i t h m to R i c c a t i E q u a t i o n I n v e r s i o n s / 106 d e m o n s t r a t e i t s f l e x i b i l i t y . A f t e r 11 i t e r a t i o n s t h e a l g o r i t h m c o n v e r g e d t o t h e JDF model shown i n F i g u r e 25c ( s o l i d l i n e ) . No e x t r a c o n s t r a i n t s were a p p l i e d . The s t a r t i n g model was a h a l f - s p a c e o f c o n d u c t i v i t y a=0. 003 S/m. The x 2 m i s f i t i s 18 w h i c h i s n e a r t h e e x p e c t e d v a l u e o f 22 f o r t h e 11 complex JDF d a t a . The a l g o r i t h m r e q u i r e d 11 i t e r a t i o n s f o r t h i s u n c o n s t r a i n e d i n v e r s i o n b e c a u s e t h e h a l f - s p a c e s t a r t i n g model was f a r f r o m a c c e p t a b l e . The u s u a l mode o f o p e r a t i o n i s t o b e g i n t h e i t e r a t i o n s w i t h an a c c e p t a b l e model c o n s t r u c t e d by th e i n v e r s e s c a t t e r i n g method, and add e x t r a c o n s t r a i n t s t o t e s t v a r i o u s h y p o t h e s e s a b o u t t h e c o n d u c t i v i t y s t r u c t u r e . The d a s h e d l i n e i n F i g u r e 25c i s an example o f a more u s u a l use o f t h e a l g o r i t h m . I o b t a i n e d t h i s r e c o n s t r u c t i o n by-r e q u i r i n g a l l 7^ t o be n o n n e g a t i v e a nd m a x i m i z i n g t h e o b j e c t i v e f u n c t i o n <l>. The x 2 m i s f i t i s 30 and t h e s t a r t i n g model u s e d i s p l o t t e d i n F i g u r e 19b. U n f o r t u n a t e l y , s u c h a ma x i m i z e d m o n o t o n i c o(z) i s n o t a g l o b a l u p p e r bound on t h e c o n d u c t i v i t y , b e c a u s e i t depends on t h e s t a r t i n g m o d e l . N e v e r t h e l e s s , t h e s e e x t r a c o n s t r a i n t s r e s u l t i n a model w h i c h c l e a r l y shows t h a t t h e d a t a a l l o w h i g h l y c o n d u c t i v e r e g i o n s a t d e p t h . F u r t h e r m o r e , t h e d a t a do n o t r e q u i r e a c o n d u c t i v e zone n e a r z=70 km embedded between two r e s i s t i v e l a y e r s a s t h e s o l i d l i n e s i n F i g u r e s 25c and 19d s u g g e s t . F i g u r e 25d i l l u s t r a t e s t h e r a n g e of c o n d u c t i v i t y v a l u e s n e a r z=70 km p e r m i t t e d by t h e JDF d a t a . F o r t h e s o l i d l i n e , I R i c c a t i Equation I n v e r s i o n s / 107 imposed the c o n s t r a i n t s that o(z)<0. 05 S/m from z=30 to 84 km. These c o n s t r a i n t s e l i m i n a t e the c o n d u c t i v e zone completely and give a model s i m i l a r to a l a y e r over a h a l f - s p a c e . The x 2 m i s f i t i s 25 and the s t a r t i n g model i s p l o t t e d i n F i g u r e 19b. Only one i t e r a t i o n was r e q u i r e d . For the dashed l i n e i n F i g u r e 25d, I set a(z)>0.3 S/m f o r z=65 to 75 km. The x 2 m i s f i t i s 21. These p r . o f i l e s sample the range of c o n d u c t i v i t i e s allowed near z=70 km and warn a g a i n s t i n t e r p r e t a t i o n s based on a s i n g l e model with a p a r t i c u l a r c o n d u c t i v i t y v a l u e . In t h i s chapter, I presented an i t e r a t i v e i n v e r s i o n a l g o r i t h m which uses l i n e a r programming to s o l v e a n o n l i n e a r i n t e g r a l equation fo r o(z). The scheme i s not guaranteed to converge, and the sequence of i t e r a t e d models depends upon the i n i t i a l a(z) estimate. These disadvantages are outweighed by the advantage that the a l g o r i t h m accepts l o c a l i z e d c o n d u c t i v i t y c o n s t r a i n t s . I used such c o n s t r a i n t s to r e s t r i c t the nonuniqueness ( F i g u r e s 25a and 25b) and to explore the range of c o n d u c t i v i t y f e a t u r e s allowed or r e q u i r e d by the data ( F i g u r e s 25c and 25d). CHAPTER 4. CONCLUSIONS In t h i s t h e s i s , I p r e s e n t e d two a l g o r i t h m s f o r s o l v i n g t h e MT i n v e r s e p r o b l e m f o r a o n e - d i m e n s i o n a l c o n d u c t i v i t y p r o f i l e o(z). T o g e t h e r , t h e s e a l g o r i t h m s use c o n s t r a i n t s and d i f f e r e n t norms t o r e s t r i c t and e x p l o r e t h e s p a c e o f a c c e p t a b l e a(z). The n o n u n i q u e n e s s o f t h i s n o n l i n e a r i n v e r s e p r o b l e m must be a s s e s s e d i n t h i s way i n o r d e r t o a v o i d p i t f a l l s d u r i n g i n t e r p r e t a t i o n . The f i r s t a l g o r i t h m d i s c u s s e d i n C h a p t e r 2 i s an e x p a n s i o n o f t h e work o f W e i d e l t ( 1 9 7 2 ) . U s i n g i n v e r s e t h e o r y t e c h n i q u e s , I made W e i d e l t ' s e x a c t , t w o - s t a g e i n v e r s e s c a t t e r i n g method p r a c t i c a l and r o b u s t . The a l g o r i t h m now e a s i l y i n v e r t s t h e n o i s y , b a n d l i m i t e d d a t a t y p i c a l o f any MT f i e l d e x p e r i m e n t . I c a p i t a l i z e d on t h e f i r s t - s t a g e l i n e a r i n v e r s i o n f o r t h e MT i m p u l s e r e s p o n s e . By a p p l y i n g d i f f e r e n t minimum s t r u c t u r e norms, I g e n e r a t e d d i f f e r e n t c l a s s e s o f i m p u l s e r e s p o n s e : d e l t a f u n c t i o n , p i e c e w i s e c o n s t a n t , and smooth. I showed e m p i r i c a l l y t h a t t h e c o r r e s p o n d i n g c l a s s e s o f a(z) a r e one o r d e r smoother i n e a c h c a s e . F o r example, a d e l t a - f u n c t i o n i m p u l s e r e s p o n s e g i v e s a p i e c e w i s e c o n s t a n t c o n d u c t i v i t y . I a l s o gave a f i r s t - o r d e r f o r m u l a w h i c h c o n f i r m s t h e e m p i r i c a l r e s u l t s by r e l a t i n g t h e l o g a r i t h m o f t h e c o n d u c t i v i t y t o t h e i n t e g r a l o f t h e i m p u l s e r e s p o n s e . Hence, t h e d i f f e r e n t c l a s s e s of i m p u l s e r e s p o n s e s , c o n s t r u c t e d u s i n g l i n e a r i n v e r s e t e c h n i q u e s , r e s u l t i n v a r i e d , minimum s t r u c t u r e 108 C o n c l u s i o n s / 109 c o n d u c t i v i t y m o d e l s . T h e s e minimum s t r u c t u r e o(z) a r e u n l i k e l y t o h a v e s p u r i o u s f e a t u r e s t o m i s l e a d t h e i n t e r p r e t e r . At t h e same t i m e , t h e i n t e r p r e t e r c a n c h o o s e , f r o m t h e d i f f e r e n t c l a s s e s of a(z), t h a t c l a s s w h i c h b e s t r e p r e s e n t s t h e l o c a l g e o l o g y and so e nhance t h e r e l i a b i l i t y of t h e i n t e r p r e t a t i o n . I a p p l i e d B a c k u s - G i l b e r t a p p r a i s a l t o t h e f i r s t - s t a g e i n v e r s i o n t o q u a n t i f y t h e a b i l i t y o f f r e q u e n c y - d o m a i n MT d a t a t o r e s o l v e f e a t u r e s o f t h e i m p u l s e r e s p o n s e . I f i m p u l s e r e s p o n s e r e s o l u t i o n i s good down t o a p a r t i c u l a r d e p t h , t h e n t h e c o n d u c t i v i t y i s w e l l - r e s o l v e d down t o t h a t d e p t h . Such an a n a l y s i s d e t e r m i n e s t h e number and s p a c i n g of f r e q u e n c y - d o m a i n o b s e r v a t i o n s n e c e s s a r y t o a t t a i n a c e r t a i n r e s o l u t i o n . I u s e d t h r e e k i n d s of c o n s t r a i n t s on t h e i m p u l s e r e s p o n s e i n v e r s i o n and e x amined t h e e f f e c t s on t h e c o n d u c t i v i t y . F i r s t , p h y s i c a l r e a l i z a b i l i t y c o n s t r a i n t s e n s u r e t h a t t h e i m p u l s e r e s p o n s e c o r r e s p o n d s t o a o(z) w h i c h i s a l w a y s p o s i t i v e . S e c o n d , u s i n g an i n i t i a l e s t i m a t e o f a o(z) p r o f i l e as an a p r i o r i c o n s t r a i n t g e n e r a t e s c o n d u c t i v i t y models w h i c h d e v i a t e l e a s t ( i n some s e n s e ) from t h e i n i t i a l g u e s s . T e s t i n g t h e c o m p a t i b i l i t y of t h e d a t a w i t h a p r i o r i m o d e l s i s a n o t h e r way o f e x p l o r i n g t h e s p a c e of a c c e p t a b l e o(z). The t h i r d c o n s t r a i n t t y p e i n v o l v e s a n o n u n i f o r m w e i g h t i n g o f t h e i m p u l s e r e s p o n s e norm. In a r e g i o n where t h e w e i g h t i n g i s l a r g e , m i n i m i z i n g t h e norm i m p l i e s t h a t t h e i m p u l s e r e s p o n s e w i l l t e n d t o be s m a l l . I showed t h a t t h e c o r r e s p o n d i n g c o n d u c t i v i t y C o n c l u s i o n s / 110 c h a n g e s o v e r t h i s r e g i o n a r e s m a l l . Hence, t h e w e i g h t i n g f u n c t i o n s p r o v i d e an e x t r a d i m e n s i o n o f f l e x i b i l i t y t o i n f l u e n c e a(z) o v e r l o c a l i z e d d e p t h r a n g e s . I gave t h r e e methods f o r e s t i m a t i n g t h e s u r f a c e c o n d u c t i v i t y o0 r e q u i r e d by t h e i n v e r s e s c a t t e r i n g a l g o r i t h m s . One method u s e s t h e h i g h - f r e q u e n c y d a t a and i s n o t new. I t s i m p l y f i n d s t h e b e s t - f i t t i n g a0 t o t h e f i r s t few h i g h - f r e q u e n c y o b s e r v a t i o n s . The o t h e r two methods a r e new. F i r s t , t h e c o r r e c t c h o i c e o f a0 l e a d s t o 3G B(x) models w h i c h s a t i s f y t h e p h y s i c a l r e a l i z a b i l i t y c o n s t r a i n t s , even t h o u g h BG c o n s t r u c t i o n s do not i n c o r p o r a t e t h e s e i n e q u a l i t y c o n s t r a i n t s . A p l o t o f a0 v e r s u s t h e p e r c e n t a g e o f c o n s t r a i n t s v i o l a t e d c l e a r l y shows t h a t t h e c o r r e c t c h o i c e r e s u l t s i n t h e f e w e s t t r a n s g r e s s i o n s . S e cond, t h e c o r r e c t a0 g i v e s BG i m p u l s e r e s p o n s e m o d e l s w h i c h a r e i n i t i a l l y z e r o and have m i n i m a l o s c i l l a t i o n s . U s i n g a d i f f e r e n t o0 r e s u l t s i n l a r g e i n i t i a l v a l u e s c o r r e s p o n d i n g t o a r a p i d c o r r e c t i o n f r o m t h e e r r o n e o u s a0 t o t h e p r o p e r v a l u e . The n o n z e r o i n i t i a l v a l u e and l a r g e o s c i l l a t i o n s i n t h e i m p u l s e r e s p o n s e a t g r e a t e r d e p t h s a r e no d o u b t r e s p o n s i b l e f o r t h e v i o l a t i o n o f t h e p h y s i c a l r e a l i z a b i l i t y c o n s t r a i n t s . T h e r e f o r e , t h e two t e c h n i q u e s may be a s p e c t s o f t h e same i m p u l s e r e s p o n s e s t r u c t u r e . The a b i l i t y of t h e f i r s t - s t a g e l i n e a r i n v e r s i o n t o c o n s t r u c t d i v e r s e , minimum s t r u c t u r e i m p u l s e r e s p o n s e m o d e l s , t o a d m i t a r e s o l u t i o n a n a l y s i s , t o a c c e p t c o n s t r a i n t s , and t o C o n c l u s i o n s / 111 e s t i m a t e t h e s u r f a c e c o n d u c t i v i t y makes i t an i n d i s p e n s a b l e t o o l f o r e x p l o r i n g t h e s p a c e o f a c c e p t a b l e c o n d u c t i v i t y p r o f i l e s . T h i s u t i l i t y h a s n o t been f u l l y r e c o g n i z e d o r d e v e l o p e d u n t i l now. I added two new c o n d u c t i v i t y mappings t o t h e s e c o n d - s t a g e of t h e i n v e r s e s c a t t e r i n g a p p r o a c h : t h e B u r r i d g e e q u a t i o n w h i c h u s e s R(x), and t h e G o p i n a t h - S o n d h i e q u a t i o n w h i c h u s e s B(x). The s e two i n t e g r a l e q u a t i o n s c a n e x a c t l y r e p r o d u c e d i s c o n t i n u o u s c o n d u c t i v i t y m o d e l s t h a t t h e two e q u a t i o n s g i v e n by W e i d e l t (1972) c an n o t . A p p l y i n g t h e e f f i c i e n t a l g o r i t h m s d e v e l o p e d f o r t h e s e i s m i c i n v e r s e p r o b l e m makes t h e n u m e r i c a l s o l u t i o n o f t h e s e i n t e g r a l e q u a t i o n s v e r y f a s t . I e v a l u a t e d t h r e e a p p r o x i m a t e c o n d u c t i v i t y m a p p i n g s . One i s t h e l i n e a r B o r n a p p r o x i m a t i o n w h i c h e q u a t e s t h e l o g a r i t h m of a(x) t o t h e i n t e g r a l o f R(x). T h i s e q u a t i o n p r o v i d e s a f i r s t - o r d e r r e l a t i o n s h i p between norms on t h e c o n d u c t i v i t y and th e i m p u l s e r e s p o n s e . A n o t h e r a p p r o x i m a t i o n g i v e s some p h y s i c a l i n s i g h t i n t o t h e f u n c t i o n A(x,y). I t shows t h a t p a r t of A(x,y) i s b u i l t up o f t h e c o n t i n u o u s r e f l e c t i o n c o e f f i c i e n t j(x). The most a c c u r a t e a p p r o x i m a t e mapping g i v e s c o n d u c t i v i t y m o d e ls s i m i l a r t o t h o s e f r o m t h e e x a c t i n t e g r a l e q u a t i o n s . A l t h o u g h I d i d not i n v e s t i g a t e t h e a p p l i c a t i o n , t h e t e c h n i q u e s d e v e l o p e d h e r e f o r MT a r e e a s i l y a d a p t e d t o r e s t r i c t and e x p l o r e t h e s p a c e o f a c c e p t a b l e dc r e s i s t i v i t y m o d e l s . Coen and Yu (1981) u s e d G e l ' f a n d - L e v i t a n t h e o r y and C o n c l u s i o n s / 112 t h e r e s p o n s e B(x) t o i n v e r t r e s i s t i v i t y d a t a . D i f f e r e n t norms and c o n s t r a i n t s c o u l d be a p p l i e d t o y i e l d v a r i e d , minimum s t r u c t u r e i m p u l s e r e s p o n s e s . S z a r a n i e c (1982) u s e d t h e e q u i v a l e n t o f dynamic d e c o n v o l u t i o n f o r t h e s e c o n d - s t a g e o f t h e dc r e s i s t i v i t y i n v e r s i o n . The o t h e r i n t e g r a l e q u a t i o n s and a p p r o x i m a t i o n s u s e d h e r e a r e e q u a l l y a p p l i c a b l e . I n C h a p t e r 3 o f t h i s t h e s i s , I d e v e l o p e d a n o t h e r MT i n v e r s i o n scheme. I t i s b a s e d on an i n t e g r a t e d R i c c a t i e q u a t i o n w h i c h r e l a t e s r (k) t o t h e c o n t i n u o u s r e f l e c t i o n c o e f f i c i e n t y(z). The e q u a t i o n i s n o n l i n e a r and must be s o l v e d by i t e r a t i o n . However, t h e a l g o r i t h m i s n o t i n t e n d e d t o r e p l a c e t h e more e f f i c i e n t , e x a c t i n v e r s e s c a t t e r i n g a p p r o a c h . I t s v a l u e i s t h a t i s a c c e p t s d i r e c t , l o c a l i z e d c o n s t r a i n t s on a(z) and i t u s e s a f a s t and f l e x i b l e l i n e a r programming f o r m u l a t i o n f o r n u m e r i c a l c o m p u t a t i o n . The p u r p o s e o f t h e a l g o r i t h m i s t o use c o n s t r a i n t s t o e x p l o r e t h e n o n u n i q u e n e s s and r e f i n e c o n c l u s i o n s drawn from t h e e x a c t i n v e r s i o n method. The i t e r a t i o n s c o n v e r g e w e l l b e c a u s e t h e i n v e r s e s c a t t e r i n g a l g o r i t h m i s a s o u r c e o f d i v e r s e y e t a c c e p t a b l e s t a r t i n g m o d e l s . U s i n g t h i s a l g o r i t h m , I showed how t o improve i n t e r p r e t a t i o n s by a p p l y i n g p h y s i c a l c o n d u c t i v i t y c o n s t r a i n t s t h a t c o u l d be d e r i v e d f r o m w e l l - l o g s o r g e o l o g i c i n f o r m a t i o n . In a d d i t i o n , I a p p l i e d l o c a l i z e d upper bound, l o w e r bound, and m o n o t o n i c c o n s t r a i n t s s p e c i f i c a l l y d e s i g n e d t o e x p l o r e t h e C o n c l u s i o n s / 113 s p a c e o f a c c e p t a b l e a(z) and a s s e s s t h e n o n u n i q u e n e s s . The m o t i v a t i n g p h i l o s o p h y f o r t h e s e two a l g o r i t h m s i s not s i m p l y t o c o n s t r u c t a model w h i c h s a t i s f i e s t h e o b s e r v a t i o n s , b u t a l s o t o d e f i n e t h e n o n u n i q u e n e s s i n h e r e n t i n t h i s n o n l i n e a r i n v e r s e p r o b l e m . The a p p r o a c h e s d e s c r i b e d h e r e i n r e s t r i c t and e x p l o r e t h e n o n u n i q u e n e s s and so t h e y g i v e more c o m p l e t e and r e l i a b l e m a g n e t o t e l l u r i c i n t e r p r e t a t i o n s . 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