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UBC Theses and Dissertations

Linearized inverse theory applied to direct current resistivity measurements Plenderleith, Donald H. 1983

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LINEARIZED INVERSE THEORY APPLIED TO DIRECT CURRENT RESISTIVITY MEASUREMENTS by DONALD H . PLENDERLEITH B . S c . ( h o n o u r s G e o p h y s i c s ) , U n i v e r s i t y Of B . C . , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of G e o p h y s i c s And Astronomy We accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA May 1983 © Donald H . P l e n d e r l e i t h , 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of <JTe,op LySic r- ^iflS'f'ro/lvmy The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) A b s t r a c t Layered r e s i s t i v i t y models have been c o n s t r u c t e d from the p o t e n t i a l d i f f e r e n c e measurements made in d i r e c t c u r r e n t r e s i s t i v i t y soundings u s i n g Backus and G i l b e r t ' s l i n e a r i z e d i n v e r s e theory in a manner s i m i l a r to that of Oldenburg (1978). The e f f i c i e n c y of the o r i g i n a l method has been improved by a p p r o x i m a t i n g the o n e - d i m e n s i o n a l s t r u c t u r e wi th a f i n i t e number of l a y e r s , and by computing the Hankel t rans forms used in e v a l u a t i n g the F r e c h e t k e r n e l s , and s o l v i n g the forward e q u a t i o n , wi th a d i g i t a l f i l t e r . A l s o , the s u r f a c e r e s i s t i v i t y i s no longer r e q u i r e d thus making the method f u l l y a u t o m a t i c . The c o n s t r u c t e d model ' s r e p r e s e n t a t i o n of a h y p o t h e t i c a l r e s i s t i v i t y s t r u c t u r e was t e s t e d , v a r y i n g the s t a r t i n g model , adding up to 20 percent no i se to the s y n t h e t i c d a t a , and d e c r e a s i n g the data d e n s i t y . In a l l cases the h y p o t h e t i c a l s t r u c t u r e was r e c o g n i z a b l y r e p r o d u c e d . With the program's a b i l i t y to reproduce t h i s s t r u c t u r e v e r i f i e d , s i x Schlumberger soundings in the Anahim V o l c a n i c b e l t in B r i t i s h Columbia were i n v e r t e d . A l though the s e t t i n g was not i d e a l , models which f i t the data a c c o r d i n g to the c h i - s q u a r e d c r i t e r i o n were c o n s t r u c t e d . A s tandard d e v i a t i o n of 10 percent was a s s i g n e d to the data to account for measurement e r r o r s , and i n v a l i d i t y of the f l a t one-d i m e n s i o n a l e a r t h a p p r o x i m a t i o n made in s o l v i n g the p o t e n t i a l e q u a t i o n . i i i From the models c o n s t r u c t e d and the known n e a r - s u r f a c e g e o l o g y , t h r e e i n t e g r a t e d i n t e r p r e t a t i o n s were made. The f i r s t i s a p u r e l y g e o l o g i c a l i n t e r p r e t a t i o n of the low r e s i s t i v i t y zones . In the s e c o n d , they are a t t r i b u t e d to i n c r e a s i n g t empera tures w i t h d e p t h and to the n o r t h , and a heat source n o r t h of the survey r e g i o n i s h y p o t h e s i z e d from an observed n o r t h - s o u t h r e s i s t i v i t y g r a d i e n t . The t h i r d i s the most p r o b a b l e i n t e r p r e t a t i o n . In i t , the heat source to the n o r t h i s r e t a i n e d because i t i s i n f e r r e d from a t r e n d of d e c r e a s i n g average r e s i s t i v i t i e s i n d i c a t e d by the f o u r northernmost s o u n d i n g s , but one of the remain ing s o u t h e r n anomal i e s i s q u e s t i o n e d , and a g e o l o g i c a l e x p l a n a t i o n i s g i v e n for the o t h e r . There was not enough i n f o r m a t i o n i n t h i s da ta set to f a v o r e i t h e r the h o t - s p o t , or the edge e f f e c t h y p o t h e s i s f o r the g e n e s i s of the Anahim V o l c a n i c b e l t . The two soundings most i m p o r t a n t to the h e a t - s o u r c e -t o - t h e - n o r t h h y p o t h e s i s were l i n e a r l y a p p r a i s e d . Average models have been c o n s t r u c t e d w i t h a p r e d e t e r m i n e d s t a n d a r d d e v i a t i o n a n d , on the b a s i s the low r e s i s t i v i t y zones b e i n g p r e s e n t , i t can be s a i d that the f e a t u r e s which form the b a s i s of t h i s h y p o t h e s i s are r e s o l v a b l e . i v T a b l e of Contents A b s t r a c t i i L i s t of f i g u r e s v i i Acknowledgements ix AN INTRODUCTION 1 CHAPTER 1 The B a s i c E q u a t i o n s of R e s i s t i v i t y 5 1-1 The P o t e n t i a l E q u a t i o n 5 1- 2 O b t a i n i n g the F r e c h e t K e r n e l s 10 CHAPTER 2 C o n s t r u c t i o n of R e s i s t i v i t y Models 15 2- 1 Backus and G i l b e r t ' s L i n e a r i z e d C o n s t r u c t i o n 16 2-2 S m a l l e s t P e r t u r b a t i o n C o n s t r u c t i o n • 17 2-3 F l a t t e s t P e r t u r b a t i o n C o n s t r u c t i o n 20 2-4 E r r o r s and t h e i r Infuence 23 2-4.1 Sources of E r r o r 24 2-4.2 How E r r o r s are Accomodated 26 2-4.3 F i n a l C o n s i d e r a t i o n s Regard ing P e r t u r b a t i o n s 31 2-5 Computing Hankel Transforms 34 2-6 T e s t i n g the L i n e a r i z e d C o n s t r u c t i o n Program 40 2-6.1 O b j e c t i v e s 40 2-6.2 E f f e c t s of the S t a r t i n g Model 41 2-6.3 E f f e c t s of Noise and Data D e n s i t y 52 2-6.4 R e p r e s e n t a t i o n T e s t i n g 55 CHAPTER 3 A R e s i s t i v i t y Survey in the Anahim B e l t 59 3-1 The. "Anahiin R e s i s t i v i t y P r o j e c t 60 3-2 Causes of R e s i s t i v i t y Anomal ie s in Geothermal Areas . . 63 3-3 G e o l o g i c a l S e t t i n g s of the Schlumberger Soundings . . . . 6 5 3-4 C o n s t r u c t i o n s O b t a i n e d from the Schlumberger Data . . . . 6 9 3-5 R e s u l t s from the D i p o l e - D i p o l e L i n e 78 3-6 G e o l o g i c a l - G e o t h e r m a l I n t e r p r e t a t i o n s 79 3-6.1 S i m p l e s t Model w i th G e o l o g i c a l C o n s t r a i n t s 80 3-6.2 R e g i o n a l H e a t i n g Model 82 3-6 .3 Most Probab le Model 85 3- 7 C o r r e l a t i o n w i t h the L a r g e r S c a l e T e c t o n i c P i c t u r e . . . 8 6 3-7.1 G e o l o g i c a l H i s t o r y of the Anahim B e l t 87 3-7 .2 F i t to the Hot -Spot T r a c e H y p o t h e s i s 92 3- 7.3 F i t to the Edge E f f e c t H y p o t h e s i s 93 CHAPTER 4 A p p r a i s a l 95 4- 1 O b j e c t i v e s of A p p r a i s a l 95 4-2 L i n e a r i z e d A p p r a i s a l 96 4- 2.1 A v e r a g i n g F u n c t i o n s and Model Averages 97 4-2.2 D e l t a n e s s C r i t e r i o n 101 4-2 .3 The T r a d e o f f Between R e s o l u t i o n and A c c u r a c y . . . . 1 0 4 4-3 A p p r a i s a l of some Schlumberger C o n s t r u c t i o n s 107 4-4 Towards Comprehensive A p p r a i s a l 114 CONCLUSION BIBLIOGRAPHY 121 Appendix A E v a l u a t i o n of the Modal C o e f f i c i e n t A(X) 125 Appendix B S o l v i n g for H ( z , X ) in a Layered Medium 129 Appendix C F r e c h e t D i f f e r e n t i a b i l i t y 134 Appendix D E v a l u a t i n g I n d e f i n i t e I n t e g r a l s 141 Appendix E Winnowing E i g e n v a l u e s ' . . . . 1 4 4 Appendix F The Schlumberger Data Set 147 1.1 N o t a t i o n used f o r any e l e c t r o d e c o n f i g u r a t i o n 8 1.2 L a y e r i n g c o n v e n t i o n 9 2.1 P e r t u r b a t i o n p r o c e s s i n g 33 2.2 True s t r u c t u r e , apparent r e s . and s t a r t i n g models t e s t e d 42 2.3 S m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n from 7.8 ohm-meters ..44 2.4 S m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n from 13.4 ohm-meters .45 2.5 S m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n from 34 ohm-meters ...47 2.6 F l a t t e s t p e r t u r b a t i o n c o n s t r u c t i o n from 7.8 ohm-meters ..49 2.7 F l a t t e s t p e r t u r b a t i o n c o n s t r u c t i o n from 13.4 ohm-meters .50 2.8 C o n s t r u c t i o n u s i n g a u t o m a t i c s t a r t i n g model s e l e c t o r ....51 2.9 C o n s t r u c t i o n s from n o i s y d a t a 53 2.10 C o n s t r u c t i o n s o b t a i n e d v a r y i n g d ata d e n s i t y ....55 2.11 C o n s t r u c t i o n w i t h a h a l f - s p a c e as the t r u e s t r u c t u r e ...56 2.12 C o n s t r u c t i o n w i t h a box c a r as the. t r u e s t r u c t u r e 57 3.1 B.C. showing the Anahim V o l c a n i c b e l t 60 3.2 The Schlumberger survey r e g i o n .61 3.3 A p o t e n t i a l d i f f e r e n c e r e c o r d 62 3.4 E f f e c t s of temperature and s a l i n i t y on the r e s i s t i v i t y of water , 64 3.5 G e o l o g i c a l s e t t i n g of VES 1 66 3.6 G e o l o g i c a l s e t t i n g of VES 5 ..: 68 3.7 VES 1 c o n s t r u c t i o n s 71 3.8 VES 2 c o n s t r u c t i o n 72 3.9 VES 3 c o n s t r u c t i o n s 73 v i i i 3.10 VES 4 c o n s t r u c t i o n s 75 3.11 VES 5 c o n s t r u c t i o n s 76 3.12 VES 6 c o n s t r u c t i o n " 78 3.13 The r e s i s t i v i t y g r a d i e n t i n f e r r e d from VES 1,2,3,and 6 .85 3.14 P l a t e b o u n d a r i e s and v o l c a n i c c h a i n s 89 3.15 Mechanism f o r S t a c e y ' s edge e f f e c t h y p o t h e s i s 93 4.1 The r e s i s t i v i t y g r a d i e n t 108 4.2 T r a d e o f f c u r v e a t 150 m f o r VES 2 109 4.3 VES 2 Average model w i t h a=0.25 110 4.4 VES 1 Average model w i t h a=0.3 112 4.5 VES 5 c o n s t r u c t i o n s and average model 113 4.6 Schematic showing a p p r a i s a l methods and nonuniqueness ..116 i x Ac knowledqments I feel very fortunate at having had such a rewarding thesis topic, and I am grateful to Doug Oldenburg for suggesting i t . His intimate knowledge of inverse theory, and of the D.C. r e s i s t i v i t y problem in pa r t i c u l a r , were invaluable in thi s work — especially in getting the construction program operational. For their part in providing the data, and for several discussions regarding the Anahim r e s i s t i v i t y project, I wish to thank Greg . Shore and Michael Schlax of Premier Geophysics. I also appreciate Dr. J.G. Souther of the Geological Survey of Canada permitting us the use of this data before i t was released in the open f i l e report. Other faculty members I have enjoyed working with on f i e l d experiments and as a teaching assistant are Drs. R.M. Clowes, W.F. Slawson, and T. Watanabe. Through the M-T discussion group, I received exposure to the closely related magnetotelluric problem, and learned a few useful t r i c k s . Wang, Kerry, and Ken also gave me advice on various aspects of my problem. During my stay in the Department of Geophysics and Astronomy I have enjoyed the superb company of many of the other graduate students and their partners. Those whom I got to know well were: Brad Prager, Ian and Gemma Jones, Jim Horn, David Waldron, Max and A l l i s o n , Wang, Julian Cabrera, Ed Waddington, and Lynda Fisk. 1 AN INTRODUCTION Measur ing the e a r t h ' s r e s i s t a n c e to the flow of e l e c t r i c c u r r e n t i s the c e n t r a l purpose behind a l l of the e l e c t r i c and e l e c t r o m a g n e t i c methods used i n g e o p h y s i c s . When the r e s i s t i v i t y s t r u c t u r e i s known, i n f o r m a t i o n on rock t y p e s , p o r o s i t i e s , and temperatures of the e a r t h may then be i n f e r r e d . In the d i r e c t c u r r e n t r e s i s t i v i t y method, p o t e n t i a l d i f f e r e n c e s in the presence of an e l e c t r o - s t a t i c f i e l d are measured. These measurements c o n s t i t u t e a data set which can be i n v e r t e d to y i e l d r e s i s t i v i t y i n f o r m a t i o n as a f u n c t i o n of d e p t h . In t h i s t h e s i s l i n e a r i z e d i n v e r s e methods are used to c o n s t r u c t and a p p r a i s e r e s i s t i v i t y models . G e o l o g i c a l and geothermal i n t e r p r e t a t i o n s are made from data taken in the Anahim V o l c a n i c B e l t of B r i t i s h C o l u m b i a . The r e s i s t i v i t y method was f i r s t s u c c e s s f u l l y a p p l i e d i n 1912 by Conrad Sch lumberger . He was the f i r s t to i n t r o d u c e e l e c t r i c c u r r e n t s to the e a r t h u s i n g a c u r r e n t c i r c u i t and measure the p o t e n t i a l d i f f e r e n c e s wi th a s eparate c i r c u i t . T h i s i s s t i l l the essence of any r e s i s t i v i t y experiment done t o d a y . E a r l i e r workers had e i t h e r t r i e d to measure the r e s i s t a n c e between two e l e c t r o d e s wi th a c u r r e n t f l o w i n g between them, or used the r i g h t c i r c u i t s w i th f r e q u e n c i e s that were too h i g h . N e i t h e r of these methods a l l o w the e l e c t r i c f i e l d much depth of p e n e t r a t i o n . The u n d e r s t a n d i n g behind r e s i s t i v i t y measurements has advanced tremendously s i n c e 1912. By 1925 S t e f a n e s c o , K o s t i t z n , and M a i l l e t had a patent on measuring the depth to a h o r i z o n t a l l a y e r u s i n g r e s i s t i v i t y , and G i s h and Rooney had 2 p u b l i s h e d papers on d e t e r m i n i n g the t r u e r e s i s t i v i t y s t r u c t u r e from p o t e n t i a l measurements (Kunetz,1966). In 1933 R.E. Langer p u b l i s h e d "An i n v e r s e problem i n d i f f e r e n t i a l e q u a t i o n s " i n which he o b t a i n e d the t r u e v e r t i c a l r e s i s t i v i t y d i s t r i b u t i o n from apparent r e s i s t i v i t y c u r v e s . F u r t h e r , he showed how i t i s u n i q u e l y d e t e r m i n e d by the complete s e t of apparent r e s i s t i v i t i e s . T h i s s t i l l s tands as the uniqueness proof f o r r e s i s t i v i t y models. The f i r s t major p u b l i c a t i o n i n an American j o u r n a l appeared i n 1947 wherein M a i l l e t e x t o l l e d the v i r t u e s of the r e s i s t i v i t y method t o the American audience and i n t r o d u c e d Dar Zarrouk parameters as a means of c a l c u l a t i n g s u r f a c e p o t e n t i a l s . P r i o r to t h a t d a t a , i n t e r e s t i n the e l e c t r i c a l methods had e x i s t e d o n l y i n Europe, C h i n a , and the USSR. As the body of knowledge about r e s i s t i v i t y grew, i n t e r p r e t a t i o n methods became i n c r e a s i n g l y s o p h i s t i c a t e d . Conrad Schlumberger measured the p o t e n t i a l d i s t r i b u t i o n on a g r i d near a c u r r e n t e l e c t r o d e and c o n t o u r e d i t t o produce an e q u i p o t e n t i a l map. He compared t h i s t o the p o t e n t i a l d i s t r i b u t i o n of a h a l f - s p a c e and made i n t e r p r e t a t i o n s based upon the d i f f e r e n c e s between the two e q u i p o t e n t i a l maps. The number of measurements mapping the p o t e n t i a l i n v o l v e s i s i n the hundreds. A f t e r the p o t e n t i a l e q u a t i o n was s o l v e d , i t became p o s s i b l e to compute the apparent r e s i s t i v i t y c u r v e s f o r any l a y e r e d s t r u c t u r e , and c u r v e matching became the p r e m i e r i n t e r p r e t a t i o n method. S e v e r a l c a t a l o g u e s of apparent r e s i s t i v i t y c u r v e s e x i s t ; the o r i g i n a l , c o m p i l e d by O r e l l a and 3 Mooney (1966), had a hundred and f o r t y such c u r v e s . S h o r t l y t h e r e a f t e r , d i r e c t i n t e r p r e t a t i o n methods were s t a r t e d . Koefoed in 1966 used r a i s e d k e r n e l f u n c t i o n s to c a l c u l a t e r e s i s t i v i t i e s and l a y e r t h i c k n e s s e s from data over a one d i m e n s i o n a l e a r t h , and Zohdy deve loped a f a s t i t e r a t i v e scheme u s i n g m o d i f i e d Dar Zarrouk f u n c t i o n s (Zohdy, 1975). G e n e r a l i z e d i n v e r s e t h e o r y , and l a t e r r i d g e r e g r e s s i o n were employed by Inman to i n v e r t p o t e n t i a l d i f f e r e n c e s over a l a y e r e d e a r t h (Inman et a l , 1973; Inman, 1975). And in 1978, Oldenburg used Backus and G i l b e r t ' s l i n e a r i z e d i n v e r s e theory to c o n s t r u c t and a p p r a i s e c o n t i n u o u s one d i m e n s i o n a l models (Oldenburg , 1978). Models have a l s o been c o n s t r u c t e d u s i n g 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 method which , u n l i k e the l i n e a r i z e d method, s o l v e s the f u l l n o n l i n e a r e q u a t i o n (Cohen and Wu, 1981). At presen t though, the a b i l i t y to a p p r a i s e models l ags somewhat behind the a b i l i t y to c o n s t r u c t them, and the problem of nonuniqueness s t i l l looms. In t h i s t h e s i s l i n e a r i z e d i n v e r s e theory i s used to i n v e r t p o t e n t i a l measurements in a way very s i m i l a r to tha t f o r m u l a t e d by Oldenburg (1978) . Here though, l a y e r e d models are c o n s t r u c t e d , and knowledge of the s u r f a c e r e s i s t i v i t y i s not r e q u i r e d . A l s o , the f i l t e r method of Gosh (1971) i s used to c a l c u l a t e the Hankel t r a n s f o r m s used in the p o t e n t i a l e q u a t i o n and i n the F r e c h e t k e r n e l s . T h i s g r e a t l y reduces the computat ion t ime . The advantage of u s i n g Backus and G i l b e r t ' s l i n e a r i z e d c o n s t r u c t i o n i s tha t the k e r n e l f u n c t i o n s for l i n e a r i z e d a p p r a i s a l are a u t o m a t i c a l l y a v a i l a b l e . Those k e r n e l s are 4 combined to g i v e unique averages of e a r t h r e s i s t i v i t y , p r o v i d e d that the c o n s t r u c t e d models , and the e a r t h model , are l i n e a r l y c l o s e . Most o f t e n though, there i s not enough d a t a to guarantee l i n e a r c l o s e n e s s to the r e a l e a r t h , but l i n e a r i z e d a p p r a i s a l can be used to determine which f e a t u r e s are common to the set of models l i n e a r l y c l o s e to the c o n s t r u c t e d model . W i t h i n l i n e a r i z e d c o n s t r u c t i o n , v a r i o u s s e l e c t i o n c r i t e r i o n e x i s t for c h o o s i n g model p e r t u r b a t i o n s . H e r e , the s m a l l e s t and f l a t t e s t p e r t u r b a t i o n s are used to c o n s t r u c t models from r e a l and s y n t h e t i c d a t a , but there i s no l i m i t to the number of p e r t u r b a t i o n s t h a t c o u l d a l s o have been used . The models c o n s t r u c t e d from the Anahim data are i n t e r p r e t e d for temperatures and g e o l o g i c a l i n f o r m a t i o n , and t h r e e g e o l o g i c a l - g e o t h e r m a l e x p l a n a t i o n s for the models are g i v e n . L a s t l y , those models which were most important to the most p r o b a b l e i n t e r p r e t a t i o n are l i n e a r l y a p p r a i s e d . 5 Chapter 1  The B a s i c E q u a t i o n s of R e s i s t i v i t y The p o t e n t i a l d i s t r i b u t i o n over an e a r t h i n which the r e s i s t i v i t y v a r i e s as a f u n c t i o n of depth w i l l be d e v e l o p e d h e r e . A d i r e c t c u r r e n t source a t r=0 on the s u r f a c e produces the e l e c t o - s t a t i c f i e l d , and p o t e n t i a l d i f f e r e n c e s measured i n t h i s f i e l d a re i n v e r t e d t o o b t a i n a r e s i s t i v i t y model as a f u n c t i o n of depth. 1-1 The P o t e n t i a l E q u a t i o n J u s t as i n c i r c u i t r y , Ohm's Law r e l a t e s c u r r e n t and r e s i s t i v i t y to the e l e c t r o - s t a t i c p o t e n t i a l . E = p J ( 1 . 1 ) E x p r e s s i n g E as the g r a d i e n t of the p o t e n t i a l , V, and u s i n g the f a c t t h a t the d i v e r g e n c e of J i s z e r o when t h e r e are no c u r r e n t s o u r c e s i n the r e g i o n r e s u l t s ir: the c o n s e r v a t i o n of c u r r e n t c o n d i t i o n . y . j = V. -W = 0 ( 1 . 2 ) P When the r e s i s t i v i t y i s c o n s t a n t , e q u a t i o n ( 1 . 2 ) i s L a p l a c e ' s e q u a t i o n f o r V which i s e a s i l y s o l v e d ; however a depth dependent r e s i s t i v i t y w i l l g i v e r i s e to a second term. 6 1_ V 2 V + V 1 « W = 0 (1 .3) p ( z ) p ( z ) Since the p o t e n t i a l d i s t r i b u t i o n has c i r c u l a r symmetry around the c u r r e n t source and v a r i e s w i th d e p t h , c y l i n d r i c a l c o o r d i n a t e s are most amenable to s o l v i n g t h i s e q u a t i o n . Z i s p o s i t i v e downwards, and there i s no 9 dependence. Let V ( r , z ) = R(r ) H(z) . S e p a r a t i n g the v a r i a b l e s in e q u a t i o n (1 .3) y i e l d s a B e s s e l ' s equat ion f o r R, and an e q u a t i o n of the S t u r m - L i o u v i l i e type for H . R" + J _ R ' +X 2R = 0 (1 .4a) r H" - p*H' + X 2 H = 0 (1 .4b) Of the two k inds of z e r o t h o r d e r B e s s e l f u n c t i o n s which s o l v e ( 1 . 4 a ) , on ly the f i r s t i s r e t a i n e d because of the seconds c h a r a c t e r i s t i c s near r=0. At t h i s p o i n t the s o l u t i o n for the p o t e n t i a l i s an i n t e g r a l over the c o n t i n u o u s eigenmodes of these two s o l u t i o n s . A(X) i s a modal c o e f f i c i e n t . V ( r , z ) A(X) H ( X , z ) J 0 ( X r ) dX (1 .5) 7 There are many ways of e v a l u a t i n g A ( X ) . Langer (1933) , u s i n g the c o n d i t i o n that 3V = 0 everywhere on the e a r t h ' s s u r f a c e except at the c u r r e n t e l e c t r o d e o b t a i n e d : r V ( r , z ) = - I p n 2na H ( z , X ) s inXa J 0 ( X r ) dX H ' ( 0 , X ) where ' a ' i s the d iameter of the c u r r e n t e l e c t r o d e . S te fanesco(1930) s o l v e d for V in the near and f a r f i e l d and combined the s o l u t i o n s . V ( r , z ) = Ip< p o f J _ + 2 K(X, k , d ) J 0 ( X r ) dX " TTT |_ r J , where K ( X , k , d ) i s a k e r n e l dependent upon l a y e r t h i c k n e s s e s and r e s i s t i v i t i e s , and Oldenburg(1978) used the c o n d i t i o n tha t the s u r f a c e i n t e g r a l of c u r r e n t d e n s i t y over an i n f i n i t e l y long c y l i n d e r c e n t r e d around the c u r r e n t source must equa l the input c u r r e n t , I . T h i s c o n d i t i o n p r o v i d e s the most compact e x p r e s s i o n for the p o t e n t i a l and i s used ih appendix A to e v a l u a t e A ( X ) . A(X) IX 2TT H ' ( 0 , X ) T h e r e f o r e : V ( r , 0 ) = - I p 0 2TT X H(0 ,X) J 0 ( X r ) dX H ' ( 0 , X ) (1 .6) 8 F i e l d measurements, however, are p o t e n t i a l d i f f e r e n c e s between two p o i n t s , m and n , in the presence of two c u r r e n t e l e c t r o d e s at p o s i t i o n s a and b . The p o t e n t i a l d i f f e r e n c e i s then: AV = V ( r d ^ ) - V ( r a n ) + V(rfcn) " V ( r ^ m ) where r a m i s the d i s t a n c e between c u r r e n t e l e c t r o d e , m, and the p o t e n t i a l e l e c t r o d e , a . 'am • m 'an 4 ? F i g u r e 1.1 N o t a t i o n f o r any e l e c t r o d e c o n f i g u r a t i o n . Any type of r e s i s t i v i t y a r r a y can be s p e c i f i e d by the f o u r d i s t a n c e s ram, r a n , r b n . and rbm. R e f e r r i n g to the s e p a r a t i o n s i n t h i s way, g i v e s the method i t s u n i v e r s a l a p p l i c a b i l i t y . For the i ' t h e l e c t r o d e c o n f i g u r a t i o n , l e t C i ( X ) r e p r e s e n t Jo(Xram) " J o U r a n ) + J 0 (Xr/>/,) " J o U r ^ ) (1 .7) then , AVi = - I p ( 2TT r X H(0 , X ) C i ( X ) H ' ( 0 , X ) dX ( 1 .8) o 9 I f V and p were l i n e a r l y r e l a t e d , p would a p p e a r i n t h i s i n t e g r a l . T h e i r r e l a t i o n s h i p , however, i s n o n l i n e a r and p e n t e r s i n t o i t as a v a r i a b l e i n t h e S t u r m - L i o u v i l i e e q u a t i o n f o r H ( e q u a t i o n ( 1 . 4 b ) ) . C a l c u l a t i n g H(0,X) and H'(0,X) was a t i m e c o n s u m i n g s t e p i n t h e c o n t i n u o u s s t r u c t u r e i n v e r s i o n o f O l d e n b u r g ( 1 9 7 8 ) , but by a s s u m i n g a m u l t i - l a y e r e d e a r t h i n wh i c h p i s c o n s t a n t and i s o t r o p i c w i t h i n e a c h l a y e r , t h e d i f f i c u l t y and t i m e i n v o l v e d i n c o m p u t i n g H(0,X) and H'(0,X) can be g r e a t l y r e d u c e d . U s i n g t h i s a s s u m p t i o n , e q u a t i o n (1.4b) s i m p l i f i e s t o H j " ( z j ) - X 2 H j ( z j ) = 0 (1.9) i n t h e j ' t h l a y e r , where z j i s t h e d e p t h f r o m t h e t o p o f t h e j ' t h l a y e r . F i g u r e 1.2 L a y e r i n g c o n v e n t i o n . The l a y e r s a r e indexed by t h e i r lower boundary . The t h i c k n e s s o f the j 'th l a y e r i s hj , and z j i s the d i s t a n c e down from the top bounda r y . Each z j i s o n l y d e f i n e d from Oj to h j . The b o u n d a r y c o n d i t i o n s a r e c o n t i n u i t y o f p o t e n t i a l and c u r r e n t a c r o s s a l a y e r i n t e r f a c e . T h e s e , p l u s t h e c o n d i t i o n t h a t t h e c u r r e n t d e n s i t y must d e c a y t o z e r o a t g r e a t d e p t h , a r e u s e d i n a p p e n d i x B t o s o l v e f o r H ( 0 , X ) . The s o l u t i o n i s a n a l y t i c and t h e r e f o r e o b t a i n i n g H'(0,X) p r e s e n t s no added d i f f i c u l t y . 10 1-2 O b t a i n i n g the F r e c h e t K e r n e l s The approach taken here i n c o n s t r u c t i n g a r e s i s t i v i t y model which f i t s the data i s to f i r s t make an i n i t i a l guess , and then i t e r a t e to a b e t t e r - f i t t i n g mode l . S u c c e s s i v e i t e r a t i o n s are made, i f n e c c e s s a r y , u n t i l the model reproduces the observed data to w i t h i n the e x p e r i m e n t a l e r r o r bounds ( equat ion ( 1 . 8 ) ) . To i t e r a t e s u c c e s s f u l l y , the model must be p e r t u r b e d in such a way that the m i s f i t between measured and c a l c u l a t e d p o t e n t i a l d i f f e r e n c e s i s m i n i m i z e d . A c c o m p l i s h i n g t h i s r e q u i r e s r e l a t i n g changes in the model to changes in the d a t a , then model p e r t u r b a t i o n s i m p l i e d by the m i s f i t to the data can be made, r e s u l t i n g i n an improved model . T h i s procedure i s s i m p l y Newton's method wi th the e x t e n s i o n t h a t we are s o l v i n g for a v e c t o r , the model , i n p l a c e o f a s i n g l e v a r i a b l e , and i s repeated u n t i l the m i s f i t i s down to an a c c e p t a b l e l e v e l . N o r m a l i z i n g the p o t e n t i a l d i f f e r e n c e s to u n i t c u r r e n t , and a b s o r b i n g the geometr ic c o n s t a n t s r e s u l t s i n a more c o n v i e n i e n t q u a n t i t y to use as d a t a . The i ' t h datum i s : A l though the data are n o n l i n e a r in p, they are l i n e a r in the q u a n t i t y H ( 0 , X ) / H ' ( 0 , X ) . By d e f i n i n g H ( z , X ) / H ' ( z , X ) as S ( z , X ) , a l i n e a r r e l a t i o n s h i p i s e s t a b l i s h e d which f a c i l i t a t e s c a l c u l a t i n g the F r e c h e t k e r n e l . CO E i = AVi 2TT = - p 0 X H(0 ,X) C j ( X ) dX i TTToTxT (1.10) o 1 1 E i = - p 0 X S ( 0 , X ) C i ( X ) dX (1.11) The u s e f u l n e s s of S i s now c l e a r ; i t i s l i n e a r l y r e l a t e d to the datum, E i . A s m a l l p e r t u r b a t i o n i n S i s t h e r e f o r e l i n e a r l y r e l a t e d to a s m a l l p e r t u r b a t i o n in E i . 5Ei = p 0 X 6 S ( 0 , X ) C i ( X ) dX (1.12) With the d e f i n i t i o n of w(z) as p ' ( z ) / p ( z ) , and s u b s t i t u t i o n of both new v a r i a b l e s i n t o e q u a t i o n (1.4b) the R i c c a t t i e q u a t i o n in S i s o b t a i n e d (Oldenburg , 1978). S' + wS + X 2 -•1 = 0 (1.13) To f i n d 5S, S and w are p e r t u r b e d and s u b s t i t u t e d back i n t o (1.13). The p e r t u r b e d v e r s i o n s of S and w a r e : S , ( z , X ) = S ( z , X ) + 5 S ( z , X ) , (1 .14a) and w, (z ) = w(z) + 5w(z) (1.14b) Then the new f u n c t i o n s S, and w, are s u b s t i t u t e d i n t o e q u a t i o n ( 1 . 1 3 ) , and the u n p e r t u r b e d v e r s i o n i s s u b t r a c t e d o u t . A f t e r throwing away the second order terms an e q u a t i o n for 5 S ( z , X ) remains . 12 5S' + (w + 2SX 2 )5S + 6wS = 0 (1.15) Us ing s t a n d a r d t e c h n i q u e s , (Boyce+Diprima, 1977) the i n t e g r a t i n g f a c t o r H 1 2 / p i s found. (6S H' 2 ) ' +• 5wHH' = 0 P P I n t e g r a t i n g from 0 to °°, 5S H* 2 -6wHH' dz (1.16) as z — > ° = , H ' ( X , z ) — > 0 , ( f or X=0) t h e r e f o r e the f i r s t term has no c o n t r i b u t i o n at °°. so SS(0 ,X) H ' ( 0 ) 2 = p(0) 5S(0,X) = pp H' (0) 2 HH'6w(z) dz p(z) HH'6w(z) dz p(z) (1.17) Now through 5S, the change in data r e s u l t i n g from a p e r t u r b a t i o n in r e s i s t i v i t y of 6p(z) i s a v a i l a b l e . S u b s t i t u t i n g e q u a t i o n (1.17) i n t o ( 1 . 1 2 ) , and chang ing the order of i n t e g r a t i o n i d e n t i f i e s the k e r n e l for 5Ei in terms of known v a r i a b l e s . 13 where, SEi = J F i ( z ) 6 w ( z ) dz o F i ( z ) = - p n 2 /XH(z,X)H'(z.X) dX p(z) / H'(0,X) 2 ( 1 . 1 8 ) T h i s form of the p e r t u r b a t i o n equation, although mathematically f i n e , r e q u i r e s s o l v i n g 6w(z) f o r 6p(z). Choices of p e r t u r b a t i o n s other than 5w(z) are p o s s i b l e p r o v i d e d that the k e r n e l s are mod i f i e d c o r r e s p o n d i n g l y . Recent work by Wang (pe r s o n a l communication, 1982) seems to i n d i c a t e that the r e s o l u t i o n width i n l i n e a r i z e d a p p r a i s a l i s dependent upon the ch o i c e of model, and ge n e r a l r e s i s t i v i t y l e v e l of the environment. In h i g h l y r e s i s t i v e environments c o n d u c t i v i t y models gave b e t t e r r e s o l u t i o n than r e s i s t i v i t y models. A p r i o r i knowledge c o u l d be an a s s e t here, as i t would be i n many asp e c t s of in v e r s e theory, however i t trades o f f a g a i n s t u n i v e r s a l a p p l i c a b i l i t y . Only one form of model w i l l be used here. Choosing In(p(z)/ohm-meters) as that dimensionless model and r e l a t i n g i t to 5w(z) us i n g the chain r u l e and the fundamental theorem of C a l c u l u s , m(z) = In(p(z)/ohm-meters) d m(z) = Sp'(z) = 5w(z) (1.19) dz 6p(z) 1 4 S u b s t i t u t i n g t h i s e x p r e s s i o n for 6w(z) i n t o e q u a t i o n ( 1 . 1 8 ) , and i n t e g r a t i n g by p a r t s y i e l d s the F r e c h e t k e r n e l s for these models , G i ( z - ) . 5Ei = Gi (z )6m(z ) d z , where G i ( z ) = - p 0 2 /XI X 2 H ( z , X ) 2 + H ' ( z , X ) 2 . H ' ( 0 , X ) 2 J 0 ( X z ) dX (1.20) i s the F r e c h e t k e r n e l for the i ' t h datum. Through i t , the change in the model r e q u i r e d to min imize the m i s f i t to the da ta can be found . Throughout t h i s d e r i v a t i o n i t has been assumed tha t the f u n c t i o n a l E i ( m ( z ) ) v a r i e s c o n t i n u o u s l y w i th m(z) ; t h i s i s the requirement for F r e c h e t d i f f e r e n t i a b i l i t y . See Appendix C for the d e f i n i t i o n of F r e c h e t d i f f e r e n t i a b i l i t y . I t must be remembered tha t the F r e c h e t d i f f e r e n t i a l (1.20) i s o n l y a l i n e a r a p p r o x i m a t i o n to the change 6Ei brought about by 6m. Thus , the a c c u r a c y of e q u a t i o n (1.20) depends upon the l i n e a r i t y of the f u n c t i o n a l in the r e g i o n of the model , or e q u i v a l e n t l y , upon the e x i s t a n c e of h i g h e r order F r e c h e t d e r i v a t i v e s . 1 5 Chapter 2  C o n s t r u c t i o n of R e s i s t i v i t y Models Given the set of p o t e n t i a l d i f f e r e n c e s measured in a r e s i s t i v i t y exper iment , the c o n s t r u c t i o n problem i s to f i n d a o n e - d i m e n s i o n a l r e s i s t i v i t y model which reproduces those data to w i t h i n the e x p e r i m e n t a l e r r o r bounds. By the n o n l i n e a r nature of the p h y s i c s i n v o l v e d , data are more e a s i l y c a l c u l a t e d g i v e n a c e r t a i n model , than a model i s , g i v e n a set of d a t a . T h i s i s put to our advantage in an i t e r a t i v e s o l u t i o n for a model . An a c c e p t a b l e model , one which f i t s the d a t a , i s c o n s t r u c t e d i n a number of s t e p s , the f i r s t of which i s j u d i c i o u s l y c h o o s i n g a s t a r t i n g model , m 0 . A c o n s t a n t r e s i s t i v i t y h a l f - s p a c e i s the most f r e q u e n t l y chosen s t a r t i n g model for two r e a s o n s . The data are e a s i l y c a l c u l a t e d from t h i s model s i n c e the e q u a t i o n i s a n a l y t i c , E i = p 0 ( J _ " _1_ + J _ - J _ ) (2.0) r<3ro F a n rbn rbm for the i ' t h c o n f i g u r a t i o n . (see f i g u r e 1.1) and s e c o n d l y , when c o n s t r u c t i o n i s s t a r t e d from a model wi th no s t r u c t u r e , t h e r e i s no a m b i g u i t y about which p a r t s of the f i n a l model were c o n s t r u c t e d , and which are s i m p l y remnants of the i n i t i a l model . I f a h a l f - s p a c e i s c h o s e n , i t shou ld l i e w i t h i n the upper and lower bounds of the apparent r e s i s t i v i t y c u r v e , and a l s o be the best f l a t l i n e through that set of 1 6 p o i n t s . The s t a r t i n g model ' s e f f e c t on the f i n a l model i s i n v e s t i g a t e d in s e c t i o n 2 . 6 . Once the data from a s t a r t i n g model have been computed v i a the forward e q u a t i o n , the d i f f e r e n c e between each of the measured da ta and the c o r r e s p o n d i n g computed datum i s c a l c u l a t e d . I f t h i s d i f f e r e n c e i s l e s s than the e x p e r i m e n t a l e r r o r , an a c c e p t a b l e model was chosen . O t h e r w i s e , the F r e c h e t k e r n e l s are c a l c u l a t e d at the s t a r t i n g model , and the m i s f i t s to the data are approximated by F r e c h e t d i f f e r e n t i a l s . E q u a t i o n (1 .20) i s then s o l v e d for the p e r t u r b a t i o n v e c t o r , 5m(z) . The p e r t u r b a t i o n i s then added to the i n i t i a l model , and data are recomputed for the updated model . M i s f i t s are a g a i n c a l c u l a t e d , and depending upon t h e i r s i z e , another i t e r a t i o n i s made • to improve the model f u r t h e r . T h i s i s a d i r e c t a p p l i c a t i o n of the l i n e a r i z e d c o n s t r u c t i o n method of Backus and G i l b e r t (1967) . 2-1 Backus and G i l b e r t ' s L i n e a r i z e d C o n s t r u c t i o n The essence of Backus and G i l b e r t c o n s t r u c t i o n i s in the s o l u t i o n of the p e r t u r b a t i o n e q u a t i o n . A v e c t o r 5m(z) i s found which s o l v e s the l i n e a r e q u a t i o n s . r 5Ei = Gi (z )6m(z ) dz for ( i=1 ,n) ( 2 . 1 ) J o Here 5Ei i s the d i f f e r e n c e between the i ' t h measured datum, and the i ' t h computed datum for a c e r t a i n model , m. The 1 7 F r e c h e t k e r n e l , G i ( z ) i s a l s o model dependent , and i s „ sometimes w r i t t e n as G i ( z , m ( z ) ) to emphasize t h i s dependency. As i t has been shown i n appendix C , the data f u n c t i o n a l s in the D . C . r e s i s t i v i t y problem are F r e c h e t d i f f e r e n t i a b l e for any model v e c t o r in L 2 , t h e r e f o r e the e x i s t e n c e of the F r e c h e t k e r n e l for any model in L 2 i s g u a r a n t e e d . The i n t e g a l (2 .1) a c t u a l l y expresses the d i f f e r e n t i a l dEi (m;6m) , the l i n e a r change in the f u n c t i o n a l e x t r a p o l a t e d from m where the F r e c h e t d e r i v a t i v e s were c a l c u l a t e d to m+6m. Thus the problem i s l i n e a r i z e d by a p p r o x i m a t i n g 6Ei w i th dEi (m;6m) . T h i s a p p r o x i m a t i o n becomes exact i n the l i m i t as ||6m||2—>0 s i n c e the f u n c t i o n a l s are c o n t i n u o u s . Were the f u n c t i o n a l s l i n e a r in m, the a d d i t i o n of 6m(z) to m 0 , the s t a r t i n g model , would e x a c t l y s o l v e the forward p r o b l e m . As they are n o t , t h i s a d d i t i o n b r i n g s the da ta s y n t h e s i z e d at t h i s new argument c l o s e r to that which was measured, h o p e f u l l y w i t h i n a l i n e a r reg ime. I f t h i s i s so , the a d d i t i o n of the next p e r t u r b a t i o n v e c t o r which s o l v e s e q u a t i o n (2.1) f or the r e s u l t i n g S E i ' s w i l l s o l v e the forward e q u a t i o n . G e n e r a l l y , three or four such l i n e a r i z a t i o n s are a l l t h a t are needed to c o n s t r u c t a model which f i t s a g i v e n set of d a t a . 2-2 S m a l l e s t P e r t u r b a t i o n C o n s t r u c t i o n Backus and G i l b e r t (1967) have shown that s o l u t i o n s to the p e r t u r b a t i o n e q u a t i o n are nonunique; i n f i n i t e l y many p e r t u r b a t i o n v e c t o r s Sm(z) may s o l v e e q u a t i o n ( 2 . 1 ) . High f r e q u e n c i e s , f or i n s t a n c e , not c o n t a i n e d i n the spectrum of G i ( z ) may be a r b i t r a r i l y added to 5m(z) wi thout a f f e c t i n g the 18 v a l u e of the i n t e g r a l . Other c l a s s e s of nonuniqueness are a l s o p r e s e n t , most n o t a b l y that i n t r o d u c e d by the a c c e p t a b l e m i s f i t to the d a t a . The problem now i s to d e f i n e c r i t e r i a for choos ing p h y s i c a l l y reasonab le s o l u t i o n s f o r 6m(z) out of the set of p o s s i b l e s o l u t i o n s . One of these i s the v e c t o r which p e r t u r b s the p r e v i o u s model the l e a s t at every p o i n t — the s m a l l e s t p e r t u r b a t i o n v e c t o r . We w i l l s o l v e equat ion (2 .1) w i th the v e c t o r which has the s m a l l e s t norm in an L 2 sense . Thus , e q u a t i o n (2 .1) w i l l be s a t i s f i e d and the two norm, II 6m(z) ||2 = 5m(z) 2 dz (2.2) w i l l be m i n i m i z e d . T h i s i s now a v a r i a t i o n a l problem most amenable to the method of Lagrange m u l t i p l i e r s (Backus and G i l b e r t , 1967). The o b j e c t i v e f u n c t i o n a l to be min imized i s f ormula ted as f o l l o w s : iH 6m) r . r Sm(z) 2 dz - 2La; { 5 E j - Gj(z)6m(z)dz} (2 .3) where the a j ; j = 1 ,n are the Lagrange m u l t i p l i e r s . Then the g l o b a l minimum of v H S m ) i s o b t a i n e d by f i n d i n g the 6m for which the d i f f e r e n t i a l di^(6m;s) i s zero f o r some n o n - z e r o i n c r e m e n t a l v e c t o r , s. The d i f f e r e n t i a l i s dvM5m;s) = 2Jom(z)s dz - 2Laj J G j ( z ) s 1 9 dz = 0, (2 .4) . o Combining the i n t e g r a l s we have, {6m(z) - Eaj Gj'(z) }s dz = 0. (2 .5) S o l v i n g t h i s w i t h a non-zero i n c r e m e n t a l v e c t o r s, because s i s a r b i t r a r y , r e v e a l s the 5m which min imizes i//(5m). 5m(z) = E a ; G j ( z ) (2 .6 ) The s m a l l e s t p e r t u r b a t i o n i s thus a l i n e a r cbmbinat ion of F r e c h e t k e r n e l s . Upon s u b s t i t u t i n g t h i s s o l u t i o n for 6m(z) i n t o the p e r t u r b a t i o n e q u a t i o n , a set of n l i n e a r e q u a t i o n s for the a c o e f f i c i e n t s i s o b t a i n e d . o O r 6Ei = La; j o G i ( z ) G j ( z ) dz -(2.7) The set of inner p r o d u c t s formed by i n t e g r a t i n g G i ( z ) G j ( z ) from zero to i n f i n i t y for i = 1,n and j = 1,n c o n s t i t u t e s an n x n p o s i t i v e symmetric m a t r i x , T. Some of the p r o p e r t i e s of V a r e : the e i g e n v a l u e s w i l l be r e a l ( t h i s i s known as the s p e c t r a l theorem. ( S h i e l d s , 1968, theorem 17) ) , and i t i s 20 p o s s i b l e to f i n d a s e m i - o r t h o g o n a l matr ix U such that • r = UAU T (2 .8) where A i s a d i a g o n a l m a t r i x of the e i g e n v a l u e s such t h a t Aij = bjj X/ (Lanczos , 1961). T h i s i s c a l l e d the s i n g u l a r v a l u e d e c o m p o s i t i o n of r . E q u a t i o n (2 .7) i s s o l v e d by forming the i n v e r s e of V which i s U A ~ 1 U T , and m u l t i p l y i n g i t to both s i d e s . 5E U A " 1 U T = a (2 .9) Once the a c o e f f i c i e n t s have been o b t a i n e d , the s m a l l e s t p e r t u r b a t i o n v e c t o r 5m(z) i s immediate ly a v a i l a b l e through the l i n e a r combinat ion ( 2 . 6 ) . 2-3 F l a t t e s t P e r t u r b a t i o n C o n s t r u c t i o n Another g e o p h y s i c a l l y reasonable s o l u t i o n to the p e r t u r b a t i o n equat ion i s that which has the minimum s l o p e or maximum f l a t n e s s . I t s s i g n i f i c a n c e i s i n t h a t i t produces models wi th g r a d a t i o n a l f e a t u r e s ; m i n i m i z i n g the s lope l e a d s to models w i t h g r a d u a l changes in the r e s i s t i v i t y s t r u c t u r e . The p e r t u r b a t i o n h a v i n g the minimum L 2 norm of i t s d e r i v a t i v e i s thus c a l l e d the f l a t t e s t p e r t u r b a t i o n . Other c r i t e r i a a l s o e x i s t for c h o o s i n g a c c e p t a b l e p e r t u r b a t i o n s . The second d e r i v a t i v e may be m i n i m i z e d , 21 thereby m i n i m i z i n g c u r v a t u r e , or p e r t u r b a t i o n s can be chosen u s i n g l i n e a r programming, and o b j e c t i v e f u n c t i o n s can be m i n i m i z e d wi th norms o ther than the L 2 . B u t , s i n c e the set of p e r t u r b a t i o n v e c t o r s which w i l l s a t i s f y e q u a t i o n (2.1) i s open, the c h o i c e of which c r i t e r i o n i s used must r e f l e c t the s t r u c t u r e expected in the f i n a l model . W i t h i n l i n e a r i z e d c o n s t r u c t i o n , the f l a t t e s t or smoothest p e r t u r b a t i o n s sh ou ld be chosen where a c o n t i n u o u s s t r u c t u r e i s thought to e x i s t , and where a l a y e r e d s t r u c t u r e i s expec ted , models s h o u l d be c o n s t r u c t e d wi th s m a l l e s t p e r t u r b a t i o n s . In t h i s way, some g e o l o g i c a l c o n s i d e r a t i o n s can be i n j e c t e d back i n t o model c o n s t r u c t i o n . An e q u a t i o n r e l a t i n g the d e r i v a t i v e of the p e r t u r b a t i o n to the m i s f i t i s o b t a i n e d by i n t e g r a t i n g the p e r t u r b a t i o n e q u a t i o n by p a r t s . SEi = 6m(z)' H i ( z ) o o r Sm(z)* H i ( z ) dz (2.10) o where r H i ( z ) = G i ( u ) d u The r e l a t i o n s h i p s i m p l i f i e s c o n s i d e r a b l y upon i n s p e c t i n g the f i r s t t erm. It expresses the product of 6m(z)' and the i n d e f i n i t e i n t e g r a l of the F r e c h e t k e r n e l , e v a l u a t e d at the e n d p o i n t s . At the zero '-endpoint there i s no c o n t r i b u t i o n , as the i n t e g r a l has no w i d t h , and at i n f i n i t y the p e r t u r b a t i o n must be t o t a l l y f l a t because the data c o n t a i n no i n f o r m a t i o n at tha t d e p t h . T h i s renders 6m(°°) , and thus the t erm, z e r o . The f l a t t e s t p e r t u r b a t i o n e q u a t i o n i s then: 22 - S E i = H i ( z ) 6 m ( z ) ' dz (2.11) o S i m i l a r o p e r a t i o n s to those done in f i n d i n g the s m a l l e s t a c c e p t a b l e p e r t u r b a t i o n are now performed on e q u a t i o n ( 2 . 1 1 ) . The two norm of 6m(z) ' , 5m(z) ' || 2 = ( 6m(z ) ' } 2 dz , (2.12) i s min imized w h i l s t e q u a t i o n (2.11) p r o v i d e s the c o n s t r a i n t s . T h i s o b j e c t i v e f u n c t i o n a l , \p(8m') i s r <//(om') = 6 m ( z ) ' 2 dz - 2 /^3; {SEJ+ 5m(z) 'Hj (z )dz} , (2.13) which , when min imized wi th re spec t to 6m', p r o v i d e s the means for c a l c u l a t i n g the f l a t t e s t p e r t u r b a t i o n . S e t t i n g the d i f f e r e n t i a l to zero y i e l d s Sm(z) ' = Z/3: Hj (z) j'X J (2.14) A matr ix equat ion for 0 s i m i l a r to (2 .9) r e s u l t s when (2.14) i s s u b s t i t u t e d i n t o e q u a t i o n ( 2 . 1 1 ) . I t i s s o l v e d for /3 a l s o by s i n g u l a r va lue d e c o m p o s i t i o n . Once 6m(z)' has been c a l c u l a t e d v i a ( 2 . 1 4 ) , i t must be i n t e g r a t e d to o b t a i n the p e r t u r b a t i o n 6m(z) which i s then added to the p r e v i o u s model . 23 Appendix D d e s c r i b e s how t h i s , and the H i ( z ) i n d e f i n i t e i n t e g r a l s are computed. Both of these methods f o r o b t a i n i n g p e r t u r b a t i o n s r e q u i r e the l i n e a r independence of k e r n e l f u n c t i o n s (Backus and G i l b e r t , 1967). I f t h i s i s not the c a s e , s i n g u l a r v a l u e decompos i t i on w i l l expose an e i g e n v a l u e of z e r o , and the set of b a s i s v e c t o r s for p r o d u c i n g the p e r t u r b a t i o n w i l l be r e d u c e d . The k e r n e l s are in f a c t n e a r l y l i n e a r l y dependent , w i th those for the f l a t t e s t p e r t u r b a t i o n s be ing even more so than those for the s m a l l e s t p e r t u r b a t i o n . T h i s c o n d i t i o n m a n i f e s t s i t s e l f by making the columns of the i n n e r product m a t r i c e s n e a r l y l i n e a r l y dependent too , g i v i n g r i s e to some very smal l e i g e n v a l u e s . In matr ix i n v e r s i o n , s m a l l e i g e n v a l u e s can c r e a t e * l a r g e d i f f i c u l t i e s . They are o f t e n at the l i m i t of machine r e s o l u t i o n , and are t h e r e f o r e n u m e r i c a l l y i n a c c u r a t e , but t h e i r i n f l u e n c e i s grea t because t h e i r r e c i p r o c a l s are i n v o l v e d i n o b t a i n i n g the a or 0 c o e f f i c i e n t s ( equat ion 2 . 9 ) . A g a i n , t h i s problem more s e v e r e l y e f f e c t s the f l a t t e s t p e r t u r b a t i o n s and the way i t i s d e a l t w i t h , which i s to d e l e t e the smal l e i g e n v a l u e s , g e n e r a l l y reduces the s i z e of the p e r t u r b a t i o n — sometimes to the p o i n t of p r e v e n t i n g convergence . The method f o r d e a l i n g w i t h s m a l l e i g e n v a l u e s , and t h e i r r e l a t i o n s h i p to da ta e r r o r s i s deve loped in the f o l l o w i n g s e c t i o n . 2-4 E r r o r s and t h e i r I n f l u e n c e P o t e n t i a l d i f f e r e n c e measurements c o n t a i n e r r o r s and i n f o r m a t i o n . E r r o r s p l a y an important r o l e in l i n e a r i z e d 24 c o n s t r u c t i o n . Not on ly do they d i c t a t e what degree of m i s f i t i s j u s t i f i a b l e between the measured and computed d a t a , but they are a l s o used to -winnow out the s m a l l e i g e n v a l u e s . 2-4.1 Sources of E r r o r The sources of e r r o r in the data are m a n i f o l d . A l though the method i s known as D . C . r e s i s t i v i t y , the c u r r e n t which produces the e l e c t r i c f i e l d i s a square wave wi th a p e r i o d s u f f i c i e n t l y long that i n d u c t i o n e f f e c t s have ceased w e l l be fore the sw i t ch in p o l a r i t y o c c u r s . I d e a l l y , the p o t e n t i a l r e c o r d i s a square wave t o o , h a v i n g i t s l e a d i n g edges rounded by i n d u c t i o n , and the a c t u a l p o t e n t i a l measurement i s the wave a m p l i t u d e . Records however are u s u a l l y t a i n t e d wi th no i se of both i n s t r u m e n t a l and g e o p h y s i c a l o r i g i n s . A l l ins truments whether d i g i t a l , or ana logue , have t h e i r own i n t e r n a l n o i s e which i s superimposed upon the incoming s i g n a l . A l though the n o i s e l e v e l i s s m a l l , i t can a f f e c t low l e v e l r e a d i n g s such as those taken at very l a r g e e l e c t r o d e s p a c i n g s . Of g r e a t e r i n f l u e n c e though, i s the g e o p h y s i c a l n o i s e from t e l l u r i c c u r r e n t s . S ince they flow c o n t i n u a l l y , the p o t e n t i a l e l e c t r o d e s w i l l always p i c k up some l e v e l of t e l l u r i c n o i s e , and removing i t can present some d i f f i c u l t y because of i t s broad bandwidth . P e r i o d s range from 0.1 second, to 1000 seconds . The h i g h frequency t e l l u r i c s s imply add h i g h frequency no i se to the r e c o r d but the lower f r e q u e n c i e s add time v a r y i n g b i a s e s , the e f f e c t s of which i n c r e a s e wi th p o t e n t i a l e l e c t r o d e s e p a r a t i o n and in severe cases can d e s t r o y the square wave response . T h i s i s not 2 5 u s u a l l y a problem in Schlumberger sound ings , but i t can be in p o l e - p o l e and l a r g e d i p o l e - d i p o l e exper iments where the e l e c t r o d e s e p a r a t i o n s may be tens or hundreds of k i l o m e t e r s . O n - l i n e computers have been used to c a l c u l a t e the s t a b l e D . C . potent i a l . I n v a l i d i t y of the p h y s i c a l s i m p l i f i c a t i o n s made to s o l v e the forward e q u a t i o n i s another source of e r r o r , and as such w i l l add to the degree of m i s f i t which must be expected of an a c c e p t a b l e model . These s i m p l i f i c a t i o n s w i l l be examined. They were: 1. . The e a r t h i s f l a t . 2. The r e s i s t i v i t y s t r u c t u r e v a r i e s w i th depth o n l y . 3. R e s i s t i v i t y s t r u c t u r e i s l a y e r e d . 4. L a y e r s are i s o t r o p i c . The c u r v a t u r e of the e a r t h i s u n l i k e l y to cause s i g n i f i c a n t e r r o r s in c a l c u l a t i n g p o t e n t i a l s f or any exper iments done to d a t e , but the topography e f f e c t can be s i g n i f i c a n t . Even i f e l e c t r o d e s are at the same e l e v a t i o n , e r r o r s w i l l be i n t r o d u c e d i f the e a r t h ' s s u r f a c e i s not f l a t in the v i c i n i t y . These e r r o r s are u s u a l l y not d e b i l i t a t i n g however, i f they are c o r r e c t l y e s t i m a t e d and i n c l u d e d i n the t o t a l e r r o r e s t imate for each datum. The data s u c c e s s f u l l y i n v e r t e d i n Chapter 3 a t t e s t to t h i s f a c t , as they were taken in ex tremely mountainous t e r r a i n . Assuming that the e a r t h ' s s t r u c t u r e v a r i e s wi th depth o n l y i s the most f r e q u e n t l y made a p p r o x i m a t i o n in g e o p h y s i c s , w i th i t s v a l i d i t y depending upon where i t i s made. Over sed imentary s t r u c t u r e s i t i s o f t en q u i t e c o r r e c t to assume 26 there i s no l a t e r a l v a r i a t i o n , but in igneous and metamorphic environments t h i s s i m p l i f i c a t i o n may be q u i t e e r r o n e o u s . F a u l t s and v e r t i c a l c o n t a c t s , both of which are p r e s e n t i n the r e g i o n where data in Chapter 3 were t a k e n , a l s o v i o l a t e the o n e - d i m e n s i o n a l assumption so the d a t a e r r o r e s t i m a t e s must be i n c r e a s e d in such r e g i o n s to a l l o w a o n e - d i m e n s i o n a l model to converge . The a p p r o x i m a t i o n that the o n e - d i m e n s i o n a l s t r u c t u r e i s l a y e r e d was made to f a c i l i t a t e c a l c u l a t i o n of the p o t e n t i a l equat ion k e r n e l and the F r e c h e t k e r n e l s . S i n c e the a c c u r a c y of t h i s depends upon the f i n e n e s s of the l a y e r i n g scheme, e r r o r s can be made n e g l i g i b l e by hav ing many l a y e r s . At p r e s e n t , the c o n s t r u c t i o n program has a c a p a c i t y for s e v e n t y -f i v e l a y e r s . ' Sedimentary l a y e r s , p a r t i c u l a r l y those which have undergone some metamorphism, o f t e n conduct e l e c t r i c i t y more r e a d i l y p a r a l l e l to the l a y e r i n g than p e r p e n d i c u l a r to i t . I t i s common for the p e r p e n d i c u l a r r e s i s t i v i t y to be four or f i v e t imes g r e a t e r than that p a r a l l e l to the l a y e r i n g ( M a i l l e t , 1947, p . 530) . Us ing the method of Dar Zarrouk parameters , apparent r e s i s t i v i t y c u r v e s may be c a l c u l a t e d over such s t r a t a p r o v i d e d the a n i s o t r o p y i s known. As our c o n s t r u c t i o n method does not d e a l wi th l a y e r a n i s o t r o p i c s , models c o n s t r u c t e d over such s t r u c t u r e s w i l l have l a y e r r e s i s t i v i t i e s somewhere i n between the two. The l a y e r b o u n d r i e s w i l l be c o r r e c t l y p l a c e d , however the a b s o l u t e r e s i s t i v i t i e s w i l l be i n c o r r e c t . A d d i t i o n a l m i s f i t t o l e r e n c e may be n e c c e s s a r y to account for a n i s o t r o p y when c o n s t r u c t i n g models over such a medium. 27 2-4 .2 How E r r o r s are Accomodated by the C o n s t r u c t i o n Method If the a p p r o x i m a t i o n s made in f o r m u l a t i n g t h i s problem are v a l i d , and the p o t e n t i a l measurements are c o m p l e t e l y a c c u r a t e , then the p e r t u r b a t i o n equat ions c o u l d be s o l v e d e x a c t l y , and an a c c e p t a b l e model would be c o n s t r u c t e d in a few i t e r a t i o n s . However, n e i t h e r of these c o n d i t i o n s are ever met. E r r o r s in the measurements and e r r o r s in our i d e a l i z a t i o n s must t h e r e f o r e be accomodated by the c o n s t r u c t i o n a l g o r i t h m . Sets of l i n e a r equat ions are e s p e c i a l l y s e n s i t i v e to e r r o r s . S m a l l e r r o r s in the data a f f e c t the inner product m a t r i x , and would cause l a r g e e r r o r s in the a or j3 c o e f f i c i e n t s , were the e q u a t i o n s s o l v e d e x a c t l y . The approach taken here i s to assume that e r r o r s a r i s e from random p r o c e s s e s w i th G a u s s i a n d - i s t r i b u t i o n s and zero means. The one s t a n d a r d d e v i a t i o n l e v e l f or each measurement i s a s c e r t a i n e d , and i s used in d e t e r m i n i n g the c h i - s q u a r e d m i s f i t to the data which measures g l o b a l convergence . Each datum i s d i v i d e d by i t s c o r r e s p o n d i n g s t a n d a r d d e v i a t i o n , thereby n o r m a l i z i n g the s tandard d e v i a t i o n s to u n i t y . I f a measured va lue i s w i t h i n one s t a n d a r d d e v i a t i o n of the t r u e v a l u e , Ei(meas) = E i ( t r u e ) ± a (2 .15) then Ei(meas) = E i ( t r u e ) ± 1 The n o r m a l i z e d i ' t h datum w i l l be w r i t t e n as E i . To m a i n t a i n e q u a l i t y , the p o t e n t i a l k e r n e l s and the F r e c h e t k e r n e l s are 28 a l s o d i v i d e d by the a p p r o p r i a t e s t a n d a r d d e v i a t i o n s . Thus the n o r m a l i z e d f o r w a r d , and p e r t u r b a t i o n e q u a t i o n s a r e : •a E i = /pp X S ( 0 , X ) C i ( X ) dX SEi = " G i ( z ] J °i )5m(z) d z . (2 .16a) (2.16b) N o r m a l i z i n g the F r e c h e t k e r n e l s a l s o changes the m a t r i x e q u a t i o n for c o e f f i c i e n t s . SEi = h £ a; a; G i ( z ) G j ( z ) dz (2 .16c) °J C o m p u t a t i o n a l l y , n o r m a l i z i n g the s t a n d a r d d e v i a t i o n s i s one of the f i r s t o p e r a t i o n s p e r f o r m e d . N o r m a l i z e d data are compared for the m i s f i t ; n o r m a l i z e d F r e c h e t k e r n e l s are i n t e g r a t e d to form the i n n e r p r o d u c t m a t r i x , and n o r m a l i z e d m i s f i t s are used to compute the a or /3 c o e f f i e n t s . In s h o r t , n o r m a l i z e d q u a n t i t i e s are used throughout the c o n s t r u c t i o n o p e r a t i o n . The c h i - s q u a r e d s t a t i s t i c for the m i s f i t to the data i s g i v e n by the f o l l o w i n g : rt L hi Ei(meas) - E i ( c a l c ) (2. 17a) which for n o r m a l i z e d d a t a / i s : . 2 _ = I |E i (meas ) - E i ( c a l c ) | 2 . (2 .17b) C h i - s q u a r e d w i l l equa l n , the number of d a t a , when the L 2 average of the n o r m a l i z e d m i s f i t s i s down to u n i t y . T h i s 29 c o n s t i t u t e s c h i - s q u a r e d g l o b a l convergence and i t e r a t i o n i s s topped when i t i s a t t a i n e d . The c h i - s q u a r e d - c r i t e r i o n i s a l s o used in s o l v i n g the m a t r i x e q u a t i o n s for a and (3. S ince the m i s f i t to the i ' t h datum a l s o has a s tandard d e v i a t i o n of aL , s o l v i n g the matr ix e q u a t i o n any more e x a c t l y than t h i s i s u n j u s t i f i a b l e . Thus the c h i - s q u a r e d m i s f i t between v e c t o r s 6E and Ta*a must a l s o be a p p r o x i m a t e l y n . L (6E; - r- ; a; a; ) 2 < n . (2.18) i'l A s o l u t i o n wi th t h i s degree of m i s f i t i s o b t a i n e d by winnowing e i g e n v a l u e s from s m a l l e s t to l a r g e s t u n t i l the d e s i r e d m i s f i t i s a t t a i n e d . Winnowing p r o g r e s s e s i n t h i s d i r e c t i o n because the s m a l l e s t e i g e n v a l u e s are o f t e n at or beyond the l i m i t of machine r e s o l u t i o n , and are t h e r e f o r e i n a c c u r a t e . T h e i r i n f l u e n c e however, can be very l a r g e because t h e i r r e c i p r o c a l s are used to o b t a i n the a c o e f f i c i e n t s . In the s i n g u l a r va lue d e c o m p o s i t i o n form of the m a t r i x , winnowing c o r r e s p o n d s to d e c r e a s i n g the rank of A , the d i a g o n a l m a t r i x of e i g e n v a l u e s , and d e c r e a s i n g the number of columns i n U . The f o l l o w i n g e q u a t i o n c o d i f i e s the n o r m a l i z e d s o l u t i o n wi th n minus m e i g e n v a l u e s winnowed. a-o = U A ' 1 U T 5E (2.19) ( n , l ) (n,m) (m,m) (m,n) ( n , l ) T h i s i s c a l l e d the t r u n c a t e d s o l u t i o n , and c h i - s q u a r e d 30 measures the d i f f e r e n c e between the t r u n c a t e d data m i s f i t s which i t determines and 5E*i. Z {5Ei - 6 E i + } 2 = x 2 (2.20) D e t e r m i n i n g the number of e i g e n v a l u e s to winnow i s most e f f i c i e n t l y done in the r o t a t e d m i s f i t space . The m i s f i t A T v e c t o r , 6E i s r o t a t e d by U , and i t s e lements are squared and summed backwards, from n to 1. The maximum number of r o t a t e d m i s f i t s that can be summed in t h i s d i r e c t i o n u n t i l the va lue of n i s reached i s / , the number of e i g e n v a l u e s which w i l l be winnowed. Then, the d i f f e r e n c e between n and X i s m, the number of e i g e n v a l u e s which w i l l be k e p t . (The a l g e b r e i c j u s t i f i c a t i o n for t h i s i s g i v e n in Appendix E) Z^SErot 2 - < n , (2.21 ) t--n 1 where 6Erot = U T 5 E . The f i r s t m r o t a t e d m i s f i t s are d i v i d e d by t h e i r a s s o c i a t e d e i g e n v a l u e s , A " 1 S E r o t (2.22) (m,m) (m,1)> then r o t a t e d back to the i n i t i a l frame by U and d i v i d e d by t h e i r s t a n d a r d d e v i a t i o n s to o b t a i n the c h i - s q u a r e d a c c e p t a b l e c o e f f i c i e n t s . These c o e f f i c i e n t s , which most c l o s e l y s a t i s f y 3 1 the e q u a l i t y c o n d i t i o n in equat ion (2.18) w h i l e e r r i n g towards the i n e q u a l i t y , are the ones used to c o n s t r u c t the p e r t u r b a t i o n . 2-4 .3 F i n a l C o n s i d e r a t i o n s Regard ing P e r t u r b a t i o n s E m p i r i c a l l y , i t has been found t h a t t h i s raw p e r t u r b a t i o n , a l t h o u g h i t s a t i s f i e s the p e r t u r b a t i o n e q u a t i o n , i s not n e c e s s a r i l y the best one to add to the p r e v i o u s mode l . Through s o l v i n g the M-T and D . C . r e s i s t i v i t y prob lems , a sequence of o p e r a t i o n s which are a p p l i e d to the raw p e r t u r b a t i o n be fore i t i s added to the model has e v o l v e d . T h i s procedure seems to produce the best p e r t u r b a t i o n s p o s s i b l e for g u i d i n g a model towards convergence ( O l d e n b u r g , 1979, 1978, and p e r s o n a l communicat ion) . The ' r e c i p e ' has no r e a l mathemat ica l • j u s t i f i c a t i o n — i t j u s t works . A p e r t u r b a t i o n i s opera ted on when i t shows l a r g e a m p l i t u d e s or h i g h frequency o s c i l l a t i o n s , both of which are u s u a l l y d e t r i m e n t a l to the c o n s t r u c t i o n of a p h y s i c a l l y r e a l i s t i c model . I t must be remembered that the a p p r o x i m a t i o n which makes the p e r t u r b a t i o n e q u a t i o n s o l v a b l e i s tha t 6Ei = d E i ( m , 5 m ) , and when 6m i s l a r g e t h i s i s l i k e l y i n v a l i d . I t i s t h e r e f o r e r e a s o n a b l e to l i m i t the maximum p e r t u r b a t i o n a l l o w e d in a s i n g l e i t e r a t i o n , at the c o s t of p o s s i b l y r e d u c i n g the c h i -squared m i s f i t more s l o w l y . A maximum a b s o l u t e v a l u e of 1.4 f o r 6m(z), which can i n c r e a s e or decrease the r e s i s t i v i t y by a f a c t o r of f o u r , has worked w e l l in both f l a t t e s t and smoothest c o n s t r u c t i o n s . T h i s i s impl imented by t r u n c a t i n g any l a r g e r 32 or s m a l l e r v a l u e s of 5m(z) at tha t l e v e l . Less f r e q u e n t l y , Sm(z) has some very l a r g e va lues which not on ly i n v a l i d a t e the l i n e a r i z a t i o n a p p r o x i m a t i o n , but would i n v a r i a b l y p r e c l u d e convergence i f they were a p p l i e d . When the p e r t u r b a t i o n i s anywhere g r e a t e r than 5 .0 , another e i g e n v a l u e i s winnowed and new c o e f f i c i e n t s are computed. As these c o e f f i c i e n t s s o l v e the m i s f i t equat ion l e s s c l o s e l y , the ensu ing p e r t u r b a t i o n s are most o f t e n s m a l l e r . T h i s summarizes the treatment of l a r g e ampl i tude f e a t u r e s , but the frequency c h a r a c t e r i s t i c s are important as w e l l . S ince c o n s t r u c t i o n u s u a l l y s t a r t s from a h a l f - s p a c e , i t must f i r s t shape the g e n e r a l s t r u c t u r e of the model , and then add f i n e r s c a l e d e t a i l s f o r a r e a l i s t i c e a r t h mode l . Whi le i t may be p o s s i b l e to f i t r e s i s t i v i t y data wi th a d e l t a - f u n c t i o n -l i k e model as Parker (1981) has done for M-T, t h i s ' i s u n d e s i r a b l e for comparison wi th the r e a l e a r t h . I n s t e a d , l a r g e s c a l e f e a t u r e s are d e s i r e d as they are l i k e l y more r e s o l v a b l e (see Chapter 4 ) , and most r e s i s t i v i t y i n t e r p r e t a t i o n s c e n t r e around t r e n d s r a t h e r than g l i t c h e s . For these r e a s o n s , the h i g h frequency content of a p e r t u r b a t i o n i s suppressed u n t i l the model i s f a i r l y c l o s e to c h i - s q u a r e d convergence . T h i s i s done by c o n v o l v i n g the p e r t u r b a t i o n wi th a t r i a n g u l a r smoothing window. In t h i s c o n s t r u c t i o n program, as in tha t of Oldenburg (1978) the ex t en t to which the h i g h f r e q u e n c i e s are suppressed i s p r o p o r t i o n a l to the maximum p e r t u r b a t i o n . The smoothing f u n c t i o n i s passed through the p e r t u r b a t i o n a number of t imes e q u a l to the i n t e g e r va lue of the l a r g e s t p e r t u r b a t i o n — up to 33 a l i m i t of t h r e e . Thus , a p e r t u r b a t i o n v e c t o r having no e lements l a r g e r than one w i l l remain unsmoothed. F i n e s c a l e f e a t u r e s of t h i s low. an a m p l i t u d e are n e c c e s s a r y in the l a s t s tages of c o n s t r u c t i o n to b r i n g the c h i - s q u a r e d down to s t a t i s t i c a l convergence . The f o l l o w i n g f i g u r e summarizes the treatment of a p e r t u r b a t i o n v e c t o r p r i o r to a p p l y i n g i t to a model . Perturbation Processing in QJ ^ 3 cn 9 cr. c-° 1 a CO 0 untrun c a r e a Truncated ar 1.4 2. 3. 4. rla.ximum Um(zJ! One more eigenvalue i s uinnaued. uinnoved F i g u r e 2.1 Depend ing upon the s i z e of the l a r g e s t e lement in a p e r t u r b a t i o n v e c t o r , i t may be smoothed, t r u n c a t e d , or b o t h . A v e c t o r hav i ng an e lement l a r g e r than 5 i s not a c c e p t a b l e , and i s recomputed w i t h one more e i g e n v a l u e winnowed. Any e lements h a v i n g an a b s o l u t e v a l u e g r e a t e r than 1.4 a re t r u n c a t e d at tha t l e v e l , then smoothed, w i t h the number of t imes a p e r t u r b a t i o n i s smoothed depend ing upon i t s l a r g e s t v a l u e . The number of t imes a smooth ing f i l t e r i s passed over the t r u n c a t e d v e c t o r i s equal to the i n t e g e r v a l u e of the l a r g e s t e lement p r i o r to t r u n c a t i o n . 34 2-5 Computing Hankel Transforms The s i n g l e most important improvement in t h i s l i n e a r i z e d c o n s t r u c t i o n program oyer the o r i g i n a l i s tha t the Hankel t r a n s f o r m s are now computed more e f f i c i e n t l y u s i n g the d i g i t a l f i l t e r t echn ique of Gosh (1971), r a t h e r than be ing i n t e g r a t e d n u m e r i c a l l y . Not o n l y were they d i f f i c u l t to compute as i n t e g r a l s , but s i n c e s o l v i n g the forward e q u a t i o n s r e q u i r e s p e r f o r m i n g four per datum as does e v a l u a t i n g each F r e c h e t k e r n e l , G i ( z ) at every d e p t h , the d i f f i c u l t y q u i c k l y m u l t i p l i e s . For twenty d a t a , and a good l a y e r i n g scheme, hundreds of Hankel t r a n s f o r m s must be performed per i t e r a t i o n . number of t r a n s f o r m s = 4 x (no. of data + 2(no . of l a y e r s ) ) eg . 480 t r a n s f o r m s = 4 x ( 20 data + 2( 50 l a y e r s )) In a d d i t i o n to there be ing hundreds to compute, they must be a c c u r a t e because the four tha t are added and s u b t r a c t e d per datum or F r e c h e t k e r n e l are n u m e r i c a l l y very c l o s e . F o r t u a t e l y , both speed and a c c u r a c y are a t t r i b u t e s of the f i l t e r t e c h n i q u e . U s i n g i t , Hankel t r a n s f o r m s are a c c u r a t e l y computed in l e s s time than a n u m e r i c a l i n t e g r a t i o n r o u t i n e would r e q u i r e . Moreover , they are computed o n l y once for the range of arguments , and s p l i n e - f i t wherever needed r e d u c i n g the CPU time for each i t e r a t i o n f u r t h e r . Now, that models can be c o n s t r u c t e d more e f f i c i e n t l y , l i n e a r i z e d c o n s t r u c t i o n i s a q u i c k and p r a c t i c a l way of i n t e r p r e t i n g r e s i s t i v i t y measurements. 35 Computing i n t e g r a l s of B e s s e l f u n c t i o n s by c o n v o l v i n g the k e r n e l w i t h a set of f i l t e r weights was f i r s t i n t r o d u c e d by Gosh i n 1971. S ince t h e n , o ther workers have improved the a c c u r a c y of the w e i g h t s , and d e s i g n e d f i l t e r s for s p e c i f i c problems i n r e s i s t i v i t y and E-M work. A n d e r s o n ' s 1979 paper summarizes the development to da te and i n c l u d e s a computer program c o n t a i n i n g the most a c c u r a t e set of weights in the l i t e r a t u r e (Anderson , 1979). H i s program performs Hankel t r a n s f o r m s of o r d e r s zero and one, and c o n t a i n s 283 p o i n t f i l t e r s f o r each t r a n s f o r m . The f i l t e r weights have seven f i g u r e a c c u r a c y , and h a l f - s p a c e p o t e n t i a l s computed w i t h them agreed w i t h the exact v a l u e s to f o u r or f i v e dec ima l p l a c e s . The f o r m u l a t i o n of the f i l t e r method and our exact t a i l o r i n g of i t w i l l be deve loped h e r e . The g e n e r a l form of a z e r o t h o r d e r Hankel t r a n s f o r m i s K ( r ) = k(X) J 0 ( X r ) dX (2.23) J o where k(X) i s the t r a n s f o r m k e r n e l , and r i s the t r a n s f o r m argument. At the d i s t a n c e r from a c u r r e n t source I, the e x p r e s s i o n f o r the p o t e n t i a l i s «o V ( r ) = - I p p ' 2ir XH(0,X) Jp ( X r ) dX (2.24a) H' ( 0 ,X) o and the F r e c h e t k e r n e l at the depth z for a s i n g l e source i s g i v e n by 36 r G ( z , r ) = pp "2~pTzl X f X 2 H ( z , X ) 2 + H'(z ,X) 2 1 J 0 ( X r ) dX (2.24b) L H » ( 0 , X ) 2 ' J Both of these are Hankel t r a n s f o r m s and can be t r a n s f o r m e d i n t o c o n v o l u t i o n i n t e g r a l s by a p p l y i n g the f o l l o w i n g change of v a r i a b l e s , (Gosh,1971) r = exp(x) x = l n ( r ) X = exp( -y ) y = l n ( l / X ) dX =-exp(-y)dy and m u l t i p l y i n g the Hankel t r a n s f o r m by e x p ( x ) . Now, x x e K(e ) = r - y x-y x-y - k ( e ) J 0 ( e )e dy (2 .25) and the l i m i t s become: X = 0 —> y = » X = oo — > y = - c F i n a l l y the d i r e c t i o n of the i n t e g r a l needs to be r e v e r s e d by chang ing the s i g n of the i n t e g r a l . T h i s i s now a c o n v o l u t i o n i n t e g r a l in l o g space and may be w r i t t e n as x x y x x e K(e ) = k(e ) * J 0 ( e ) e (2 .26) y x x x x where k(e ) i s the i n p u t , f J 0 ( e )e i s the f i l t e r , and e K(e ) i s the output ( Anderson , 1979 ) . The change of v a r i a b l e s i s done a u t o m a t i c a l l y in A n d e r s o n ' s program. Computing Hankel t r a n s f o r m s in t h i s way r e q u i r e s knowing the f i l t e r , and Anderson has computed the f i l t e r s by t a k i n g a n a l y t i c Hankel 37 t r a n s f o r m p a i r s and d e c o n v o l v i n g them. He found that the most a c c u t a t e f i l t e r s were o b t a i n e d from t r a n s f o r m s hav ing r a p i d l y d e c r e a s i n g k e r n e l s and output f u n c t i o n s . The z e r o t h order p a i r exp(-r 2/4) < > X e x p ( - X 2 ) 2 meets t h i s c o n d i t i o n , and was used f o r the z e r o t h order f i l t e r . The s teps i n v o l v e d i n computing the f i l t e r s are o u t l i n e d below. 1. Both f u n c t i o n s have the change of v a r i a b l e s a p p l i e d to them and are d i g i t i z e d a t a c o n s t a n t sampl ing i n t e r v a l i n n a t u r a l l o g space; A=.2 was u s e d . T h i s c o r r e s p o n d s to t a k i n g samples a t o r d e r s o f 1.221 on a l i n e a r s c a l e . 2 . The D i s c r e t e F o u r i e r t r a n s f o r m s of bo th f u n c t i o n s a r e computed, and that of the output f u n c t i o n i s d i v i d e d by tha t of the i n p u t . F o u r i e r t r a n s f o r m d e c o n v o l u t i o n i s the most h i g h l y r e s o l v e d d e c o n v o l u t i o n method when both s i g n a l s are known. 3. M u l t i p l y i n g the deconvo lved spectrum by the spectrum of the sampl ing s i n e f u n c t i o n removes the s p e c t r a repeated at 1 /Ay i n t e r v a l s w h i l s t p r e s e r v i n g the f i l t e r f r e q u e n c i e s from the n e g a t i v e to p o s i t i v e N y q u i s t . Had a c o n t i n u o u s F o u r i e r t r a n s f o r m been o b t a i n e d , t h i s s t ep would be u n n e c c e s s a r y . 4. The spectrum i s then i n v e r s e F o u r i e r t rans formed p r o v i d i n g the f i l t e r w e i g h t s . They are the s ine response 38 of the f i l t e r , r a t h e r than the impulse response because the f u n c t i o n s were sampled at f i n i t e i n t e r v a l s . Once the f i l t e r weights have been o b t a i n e d , the z e r o t h order Hankel t r a n s f o r m of k ( r ) may be e v a l u a t e d by c o n v o l v i n g i t w i th the f i l t e r w e i g h t s , both w i t h the change of v a r i a b l e s a p p l i e d . For the cont inuous case t h i s would be: K ( r ) = 1 . -y l n ( r ) - y M e ) W(e ) dy (2.27) r O - CO but s i n c e t h i s i s a d i g i t a l f i l t e r , i»3 -y; l n ( r ) - y . K ( r ) = 1 1 k(e ) W(e ) Ay (2.28) T~ i-i One can see that as the argument becomes l a r g e r , the f i l t e r i s d i s p l a c e d to the l e f t w i th r e s p e c t to the k e r n e l ; t h e r e f o r e , to keep f u l l u t i l i z a t i o n of the f i l t e r , the k e r n e l must extend as f a r l e f t as the f i l t e r w i l l be d i s p l a c e d . Every f a c t o r of 1.221 i n c r e a s e in r s h i f t s the f i l t e r l e f t by another increment , and s i m i l a r l y arguements s m a l l e r than 1.0 w i l l s h i f t the weights r i g h t . By chang ing the arguments so that the k e r n e l i s now s h i f t e d to the r i g h t over the w e i g h t s , and r e t u r n i n g to our o r i g i n a l v a r i a b l e s , the f i l t e r i n g o p e r a t i o n becomes a c r o s s c o r r e l a t i o n between the s h i f t e d k e r n e l and f i l t e r w e i g h t s . K ( r ) = 1 I M X , / r ) W(X, ) (2.29) T i -130 X i s g iven by .735 x 1.221 39 Thus the endpo in t s of the f i l t e r s t r e t c h from .735 x 1.221 to .735 x 1.221 , i n s t e a d of from 0 to » a l l o w i n g the f i l t e r to g ive an a c c u r a t e v a l u e for the t r a n s f o r m , p r o v i d e d the k e r n e l has decayed to zero w e l l i n s i d e the endpo in t s and i s not h i g h l y o s c i l l a t o r y . With t h i s e q u a t i o n and A n d e r s o n ' s we ight s , the z e r o t h order Hankel t r a n s f o r m of any such k e r n e l can be e a s i l y and a c c u r a t e l y computed. In our a p p l i c a t i o n , the Hankel t r a n s f o r m s of j u s t one k e r n e l f o r the computed d a t a , and two at each l a y e r boundary for the F r e c h e t k e r n e l s a r e r e q u i r e d per i t e r a t i o n , f or a l l the arguments r . They c o u l d be computed s e p a r a t e l y for each t r a n s f o r m argument by r e - e v a l u a t i n g the k e r n e l a t the 283 new p o i n t s X '^ / r i m p l i e d by every new r , and t h i s would c e r t a i n l y p r o v i d e the a c c u r a c y a f f o r d e d by the f i l t e r method, but l i t t l e or no s a v i n g s of time would be r e a l i z e d . Each t r a n s f o r m would r e q u i r e 283 k e r n e l f u n c t i o n e v a l u a t i o n s and the summation of 283 k e r n e l - f i l t e r p r o d u c t s . T h i s would have to be done f o r every k e r n e l at each argument. To make f u l l use of the time s a v i n g s p o s s i b l e , another i m p l i c a t i o n of the ana logy wi th c o n v o l u t i o n i s used . I f the f i l t e r and k e r n e l were a v a i l a b l e i n t h e i r c o n t i n u o u s forms, the most e f f i c i e n t way of o b t a i n i n g K ( r ) would be to do the c r o s s c o r r e l a t i o n by m u l t i p l y i n g the c o n j u g a t e s of the c o n t i n u o u s s p e c t r a , a n d f i n v e r s e F o u r i e r t r a n s f o r m i n g . Then K ( r ) would be a v a i l a b l e f o r any r . T h i s i s a p p r o x i m a t e l y what i s done. The k e r n e l i s e v a l u a t e d at i n t e r v a l s of 0.2 i n n a t u r a l l o g space over the e n t i r e range i n which i t w i l l be needed. That i s from two 40 i n t e r v a l s l e f t of X , / rmax to two i n t e r v a l s r i g h t of X ^ / r m i n . Then the c r o s s c o r r e l a t i o n i s c a r r i e d out in frequency domain and , upon i n v e r s e t r a n s f o r m i n g , the Hankel t r a n s f o r m s K(r^' ) are a v a i l a b l e ; rj = rmax (1 . 221 ) 1 "J . In the z e r o l a g p o s i t i o n i s the Hankel t r a n s f o r m w i t h the l a r g e s t argument, and at subsequent g r e a t e r l ags are Hankel t r a n s f o r m s wi th arguments s u c c e s s i v e l y s m a l l e r by f a c t o r s of 1.221. Hankel t r a n s f o r m s of a l l the arguments needed are then p i c k e d o f f the s p l i n e - f i t K ( r y ) f u n c t i o n , and combined to form the computed data or F r e c h e t k e r n e l s . With t h i s method, the number of f u l l c r o s s c o r r e l a t i o n s c a l c u l a t e d i s two per l a y e r p l u s one to compute the d a t a . T h u s , the computer time i s p r o p o r t i o n a l to the number of depths at which the F r e c h e t k e r n e l s are e v a l u a t e d , but independent of the number of d a t a . -2-6 T e s t i n g the L i n e a r i z e d C o n s t r u c t i o n Program Immediately upon programming a method for a n a l y z i n g d a t a , the performance and l i m i t a t i o n s must be found . One hopes that the program w i l l not l i m i t the method's c a p a b i l i t i e s , and f u r t h e r , t h a t any c o n d i t i o n s p l a c e d upon the d a t a , i n terms of q u a l i t y and sample d e n s i t y , w i l l not be so r e s t r i c t i v e tha t most of the p r e - e x i s t i n g data i s u n s u i t a b l e . A l s o , in t e s t i n g an i n v e r s i o n program, such as l i n e a r i z e d c o n s t r u c t i o n for r e s i s t i v i t y models , the degree to which nonuniqueness can a f f e c t models i s p a r t i a l l y q u a l i f i e d by comparing models c o n s t r u c t e d from s y n t h e t i c data to those from which t h e . d a t a was d e r i v e d . 41 2-6.1 Objec t i ve s There are three main o b j e c t i v e s in t e s t i n g t h i s program. 1. The l i m i t a t i o n s of the program must be found; the extent to which a s t a r t i n g model a f f e c t s the f i n a l model must be d e t e r m i n e d . More s p e c i f i c a l l y , can the program converge upon a model r e p r e s e n t a t i v e of the t r u e • s t r u c t u r e when i t e r a t i o n s t a r t s from a p o o r l y chosen h a l f - s p a c e ? 2. L i m i t a t i o n s on the data w i l l be i n v e s t i g a t e d by add ing n o i s e to a s y n t h e t i c da ta set to see how a model w i l l d e t e r i o r a t e wi th i n c r e a s i n g e r r o r s in the d a t a . The e f f e c t s data s p a c i n g has on r e s o l v i n g the r e s i s t i v i t y s t r u c t u r e w i l l a l s o be i n v e s t i g a t e d . 3. In both of the p r e c e e d i n g e x p e r i m e n t s , the d i f f e r e n c e s between s y n t h e t i c and c o n s t r u c t e d models w i l l be taken as i l l u s t r a t i o n s of nonuniqueness . I f there i s l i t t l e resemblence between the two, then there i s l i t t l e to be g a i n e d from c o n s t r u c t i n g models . Some wide ly d i f f e r e n t t h e o r e t i c a l models w i l l a l s o be used in an attempt to c a t a l o g u e the c o n s t r u c t e d models s i g n a t u r e s of p a r t i c u l a r s t r u c t u r e s . In t h i s way, some i n t u i t i o n . can be a c q u i r e d about what a c o n s t r u c t e d model i s s a y i n g about the u n d e r l y i n g s t r u c t u r e . Less i n t u i t i o n i s needed here than in o t h e r methods of i n t e r p r e t i n g r e s i s t i v i t y d a t a , . such as apparent r e s i s t i v i t y c u r v e s or p s e u d o - s e c t i o n s , but some u n d e r s t a n d i n g can c e r t a i n l y be ga ined from t h i s c o m p a r i s o n . 2-6 .2 E f f e c t s of the S t a r t i n g Model As mentioned e a r l i e r in t h i s c h a p t e r , the s t a r t i n g model 42 shou ld be c a r e f u l l y c h o s e n . I t i s e v i d e n t tha t a model w i l l be more e a s i l y c o n s t r u c t e d from a s t a r t i n g model that l i e s c o m f o r t a b l y w i t h i n the range of apparent r e s i s t i v i t i e s , than one o u t s i d e tha t range , but the s p e c i f i c e f f e c t s a poor c h o i c e has on the f i n a l model s h o u l d be known. Then , the s t a r t i n g model c o u l d be a d j u s t e d when the e f f e c t s are r e c o g n i z e d . Another aim of t h i s experiment i s to d e v e l o p some i n t u i t i o n on c h o o s i n g the best s t a r t i n g model from an apparent r e s i s t i v i t y c u r v e . A s i x t e e n l a y e r e d s t r u c t u r e was the h y p o t h e t i c a l model and twenty Schlumberger da ta wi th AB/2 r a n g i n g from 5 to 3620 meters were s y n t h e s i z e d from i t . A nominal s t a n d a r d d e v i a t i o n of 0.5 percent was a s s i g n e d to each datum for n o r m a l i z a t i o n p u r p o s e s , a l t h o u g h the da ta were a c c u r a t e . The expans ion f a c t o r for AB/2 was 1.41, the square root of 2, w i t h the p o t e n t i a l e l e c t r o d e s e p a r a t i o n was always one t e n t h tha t of the c u r r e n t e l e c t r o d e s . 4 True S t r u c t u r e 4 Rho' Apparent  t j 1 i 1 i 1 i h - 1 1 i i 1 f i i i H 10° IQ' IQ2 IQ3 IQ4 10° 10' 102 103 104 Depth (meters) RB/2 (meters) F i g u r e 2.2 On the l e f t , the r e s i s t i v i t y s t r u c t u r e used to s y n t h e s i z e the d a t a f o r t h i s e x p e r i m e n t i s shown. The mean v a l u e i s 7 .39 ohm-meters . . Twenty Sch lumberger d a t a were computed , and the r e s u l t i n g appa ren t r e s i s t i v i t y c u r v e i s shown on the r i g h t . The h o r -i z o n t a l dashed l i n e s r e p r e s e n t h a l f - s p a c e s o f c o n s t a n t r e s i s t i v i t y which were used as s t a r t i n g m o d e l s . The models c o n s t r u c t e d from these a re compared to i d e n t i f y the s t a r t i n g mode 1's e f f e c t . 43 The t r u e s t r u c t u r e and apparent r e s i s t i v i t y c u r v e are shown i n f i g u r e 2.2 wi th three d i f f e r e n t c h o i c e s of s t a r t i n g models . For the top 2.5 meters , the r e s i s t i v i t y i s 7.39 ohm-meters then i t drops to 2.7 at 10 meters , and i n c r e a s e s to 22 ohm-meters by 500 meters be fore d r o p p i n g back down to the o r i g i n a l va lue of 7.39 ohm-meters for the u n d e r l y i n g h a l f space . C o n s t r u c t i o n s were done u s i n g f i r s t the s m a l l e s t p e r t u r b a t i o n s , then the f l a t t e s t , wi th a l l the c o n s t r u c t e d models hav ing 73 l a y e r s . From the apparent r e s i s t i v i t y c u r v e , a h a l f space of r e s i s t i v i t y equa l to exp(2 .05) seems the best and most obv ious c h o i c e f o r a good s t a r t i n g model . T h i s i s 7.8 ohm-meters, and from i t a model was converged upon in three i t e r a t i o n s . Below a depth of 12 meters , the c o n s t r u c t e d model matches the t r u e s t r u c t u r e very w e l l , whi le above, i t has under e s t i m a t e d the low r e s i s t i v i t y l a y e r , and compensated by i n c r e a s i n g the r e s i s t i v i t y j u s t above and below that r e g i o n . O v e r l e a f : F i g u r e 2.3 In t h r e e i t e r a t i o n s , a model i s c o n s t r u c t e d from the twenty Sch lumberger d a t a wh ich r e p r o d u c e s the t r u e s t r u c t u r e v e r y w e l l . S t a r t i n g f rom a h a l f space o f 7.8 ohm-meters , s m a l l e s t p e r t u r b a t i o n s a r e used to improve the p r e c e d i n g model and reduce the m i s f i t to the computed d a t a . The f i n a l model s a t i s f i e s the c h i - s q u a r r e d c r i t e r i o n The m i s f i t a t each i t e r a t i o n i s g i v e n be low. RMS r e l a t i v e m i s f i t Chi squa red s t a r t i n g model 59 . 7% 2.84 x 105" f i r s t i t e r a t i o n 1 2 . 5% 1 . 24 x IO1* second i t e r a t i o n 1 . 1% 9.18 x 1 0 1 t h i r d i t e r a t i o n 0. 3% 9.27 44 S ta r t ing Model F i r s t I t e r a t i on 10° 10' 1Q2 103 104 Second I t e r a t i o n 1Qa 10' 102 103 104 Third I t e r a t i on 10fl 10' 1Q2 1Q3 Depth (meters) 1Q4 10° 10' 102 103 Depth (meters) 104 When the s t a r t i n g r e s i s t i v i t y i s i n c r e a s e d to the maximum on the apparent r e s i s t i v i t y c u r v e , 13.4 ohm-meters, one more i t e r a t i o n i s r e q u i r e d to produce an a c c e p t a b l e model . T h i s i n c r e a s e i s c h a r a c t e r i z e d by h i g h e r frequency models at every i t e r a t i o n . In t h i s c o n s t r u c t i o n , the d e c r e a s i n g a b i l i t y of the F r e c h e t k e r n e l s to a f f e c t p e r t u r b a t i o n s below a c e r t a i n depth i s seen . As z —> the k e r n e l s decay to z e r o , and so must the p e r t u r b a t i o n v e c t o r s i n c e they are i t s b a s i s f u n c t i o n s (see e q u a t i o n 2 . 6 ) . Hense, the i l l u s i o n of an i n c r e a s e in r e s i t i v i t y i s c r e a t e d in a r e g i o n where the c o n s t r u c t e d model c o n t a i n s no r e l i a b l e i n f o r m a t i o n . S ta r t i ng Model F i r s t I t e r a t i on Second I t e r a t i on . Third I t e r a t i on i — i — i — i — | — \ — i — i — u 10° 10' IO2 io3 io4 Depth (meters) F i g u r e 2.4 When c o n s t r u c t i o n s t a r t s from a h a l f - s p a c e of 13.4 ohm-meters, one more i t e r a t i o n i s needed to c o n v e r g e upon an a c c e p t a b l e model i n the c h i - s q u a r e d sense T h i s model i s s l i g h t l y more o s c i l l a t o r y than tha t s t a r t e d from 7.8 ohm-meters, but i s s t i l l a v e r y good r e p r e s i n t a t i o n of the t rue s t r u c t u r e . The appa ren t i n c r e a s e i n r e s i s t i v i t y below 1000 meters i s an a r t i f a c t of the s t a r t i n g mode l , and the the c o n s t r u c t i o n method. I t s i g n i f i e s l o s s of r e s o l u t i o n . 46 I n c r e a s i n g the s t a r t i n g r e s i s t i v i t y to 34 ohm-meters, f a r o u t s i d e the range of apparent r e s i s t i v i t i e s , r e s u l t s i n the f o l l o w i n g c o n s t r u c t i o n ( f i g u r e 2 . 5 ) . A f t e r f i v e i t e r a t i o n s the model f i t s the d a t a , and reproduces the t r u e s t r u c t u r e very w e l l . I t i s no more sp ikey in the r e g i o n from 1 to 10 meters than the c o n s t r u c t i o n which s t a r t e d from 7.8 ohm-meters , and from 10 to 500 meters i t o v e r l i e s the t r u e s t r u c t u r e e x a c t l y . I t f a l l s away from the t r u e s t r u c t u r e at 1000 meters d e p t h , however, and i s the l e a s t r e p r e s e n t a t i v e model in tha t r e g i o n . V a r y i n g the s t a r t i n g models over the range i n v e s t i g a t e d has had very l i t t l e e f f e c t on the models c o n s t r u c t e d . In f a c t , the c o n s t r u c t i o n which was s t a r t e d from the worst s t a r t i n g model f i t the t r u e s t r u c t u r e most c l o s e l y down to about 1000 metes , where the k e r n e l s beg in to l oose t h e i r r e s o l v i n g power. Below the depth where t h i s happens, the models s t a r t e d from 13.4 and 34 ohm-meters f a l l s away from the t r u e s t r u c t u r e more than the f i r s t c o n s t r u c t i o n ( f i g u r e 2 . 3 ) . However, t h i s i s dependent upon the d i f f e r e n c e between the s t a r t i n g model and the t r u e s t r u c t u r e at those d e p t h s , not j u s t upon the s t a r t i n g model . S i n c e the same p h y s i c a l i n t e r p r e t a t i o n s would l i k e l y be drawn from any of these three models , we can say that the model c o n s t r u c t e d i s f a i r l y independent of the s t a r t i n g model . More i t e r a t i o n s are r e q u i r e d though , when the s t a r t i n g model i s p o o r l y c h o s e n . 47 -t 1 r- -i 1 r-10Q 10' 102 IQ3 IO4 H (- -i 1 1 h 10° 10' IQ2 IQ3 104 Fourth I t e r a t i on (meters) _ J 1 1QC F i f t h I t e r a t i on !Q! !G" 1Q: Depth (meters) !Q4 F i g u r e 2.5 C o n s t r u c t i o n s t a r t e d from a h a l f - s p a c e of 34 ohm-meters. In t h i s t e s t a model r e p r e s e n t a t i v e of the t r ue s t r u c t u r e down to 1000 meters was c o n s t r u c t e d in f i v e i t e r a t i o n s . In the range from 1 to 1000 m e t e r s , i t i s as good or b e t t e r than the two p r e v i o u s mode l s . V a r y i n g the s t a r t i n g model d i d not c a u s e any n o t i c a b l e d e t e r i o r a t i o n of the f i n a l mode l s . 48 F l a t t e s t p e r t u r b a t i o n c o n s t r u c t i o n s are more s e n s i t i v e to the c h o i c e of s t a r t i n g models . The p e r t u r b a t i o n i s the i n d e f i n i t e i n t e g r a l of 6m'(z) and as the r e s i s t i v i t y i s more l i k e l y to be known at the s u r f a c e than at d e p t h , 6m'(z) i s i n t e g r a t e d from z=0 to the maximum depth wi th the i n t e g r a t i o n cons tant set to zero (see equat ion 2.14 and Appendix D ) . 6m(z ) = 5m'(u) du (2.30) S ince the i n t e g r a l i s z ero at the s u r f a c e , and remains s m a l l a t sha l low d e p t h s , a l a r g e change in the s t a r t i n g model i s i m p o s s i b l e t h e r e . U n l e s s the s t a r t i n g model i s very c l o s e to the t r u e s t r u c t u r e at sha l low d e p t h s , the p e r t u r b a t i o n s w i l l compensate for t h i s at g r e a t e r d e p t h s , and an u n r e p r e s e n t a t i v e model w i l l be c o n s t r u c t e d . The f i r s t c o n s t r u c t i o n wi th f l a t t e s t p e r t u r b a t i o n s was s t a r t e d from a h a l f space of 7.8 ohm-meters, the best c h o i c e from the apparent r e s i s t i v i t y c u r v e , and a model very r e p r e s e n t a t i v e of the t r u e s t r u c t u r e was c o n s t r u c t e d i n t h r e e i t e r a t i o n s . At sha l low depths , the c o n s t r u c t e d r e s i s t i v i t y matches the t rue s t r u c t u r e even more c l o s e l y than the s m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n s d i d , and at a l l depths from 1 to 1000 meters , i t o v e r l i e s the t r u e . s t r u c t u r e . I t must be remembered, that a l t h o u g h the c o n s t r u c t i o n i s l a y e r e d , i t i s an a p p r o x i m a t i o n to a con t inuous r e s i s t i v i t y d i s t r i b u t i o n , and not an attempt to p a r a m e t e r i z e the r e s i s t i v i t y s t r u c t u r e . T h e r e f o r e the c o n s t r u c t e d model i s not expected to reproduce the s tepped nature of the t r u e s t r u c t u r e . 49 Starting Ma del F i r s t I teration 10° 10' 102 10' !04 Depth (meters) Second I terat ion 10° 10' 10' 103 104 Depth (meters) Third I terat ion 10C 10' Depth 10 103 (meters) 104 10C IO1 Depth 102 103 (meters) 1Q4 F i g u r e 2.G P e r t u r b i n g the s t a r t i n g model of 7.8 ohm-meters w i th the f l a t t e s t p e r t u r -b a t i o n s y i e l d s the above mode l . At s h a l l o w depths i t o v e r l a y s the t r ue s t r u c t u r e more c l o s e l y than the c o r r e s p o n d i n g s m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n ( f i g u r e 2 . 3 ) . and a t a l l dep ths f rom 1 to 2000 meters the f i t to the t r ue s t r u c t u r e i s ve ry good . When the near s u r f a c e r e s i s t i v i t y i s known, and used as the s t a r t i n g mode l , the f l a t -t e s t p e r t u r b a t i o n s c o n s t r u c t a smooth ly v a r y i n g mode l . I n c r e a s i n g the s t a r t i n g r e s i s t i v i t y to 13.4 ohm-meters p r e v e n t s the c o n s t r u c t i o n of a r e p r e s e n t a t i v e model because of the i n h e r e n t l i m i t a t i o n in the f l a t t e s t p e r t u r b a t i o n s . The p e r t u r b a t i o n s are not l a r g e enough at sha l low depths to b r i n g 50 the model down to the t r u e s t r u c t u r e , and c h i - s q u a r e d convergence was not a t t a i n e d h e r e . T h i s c o n s t r u c t e d model can on ly t r a c e the s t r u c t u r e of the t h e o r e t i c a l model wi th a r e s i s t i v i t y b i a s equa l to the d i f f e r e n c e between the s t a r t i n g model and the sha l low r e s i s t i v i t y . T h e r e f o r e , when c o n s t r u c t i n g a model w i th f l a t t e s t p e r t u r b a t i o n s i t i s c r u c i a l that i t e r a t i o n i s s t a r t e d from the near s u r f a c e r e s i s t i v i t y v a l u e . If t h i s i s not done, an a c c e p t a b l e model w i l l not be found, and c o n s t r u c t i o n w i l l be to no a v a i l . 4 S t a r t ing Model 4 F i r s t I t e r a t i on t — 1 — i 1—1 1— 1 1 H I—\—1 i 1 j—1 1 H 10fl 10' 102 103 ]Q4 io* IQ< I 0 2 1 0 * 1 Q < 4 Second I t e r a t i on 4 Third I t e r a t i on 10° ]Q' 102 10"J IQ4 10° 10' 102 103 IQ4 Depth (meters) Depth (meters) F i g u r e 2.7 The l i m i t a t i o n s in c o n s t r u c t i n g models u s i n g the f l a t t e s t p e r t u r b a t i o n a r e c l e a r l y shown in t h i s f i g u r e . The s t a r t i n g model was 13.4 ohm-meters, and they a r e i n s u f f i c i e n t to b r i n g the model down to the t r ue s t r u c t u r e . I n s t e a d , i t s imp l y t r a c e s the t r u e s t r u c t u r e , k e e p i n g a mean v a l u e of 13.4 ohm-meters. The f i n a l up -swing i s a r e s u l t of the a l l o w a b l e m i s f i t in the p e r t u r b a t i o n e q u a t i o n . b 1 An automat ic s t a r t i n g model s e l e c t o r has been i n v e s t i g a t e d for the f l a t t e s t p e r t u r b a t i o n s . I t se ts the s t a r t i n g model to a h a l f - s p a c e of the s h a l l o w e s t apparent r e s i s t i v i t y v a l u e . I f the r e s i s t i v i t y s t r u c t u r e i s cons tant from the s u r f a c e down to a depth comparable to the AB/2 s p a c i n g of t h i s r e a d i n g , a very good s t a r t i n g model i s chosen, but when t h e r e i s a r e s i s t i v i t y v a r i a t i o n of more than a f a c t o r of two, the automat ic s e l e c t o r h i n d e r s c o n s t r u c t i o n . In environments where the r e s i s t i v i t y v a r i e s g r e a t l y wi th d e p t h , the f l a t t e s t p e r t u r b a t i o n s w i l l not c o n s t r u c t a very r e p r e s e n t a t i v e model w i t h any s t a r t i n g model . Smoothly v a r y i n g s t r u c t u r e s are c e r t a i n l y t h e i r f o r t e . In the data be ing used , the s h a l l o w e s t sounding has an AB/2 s p a c i n g of 5 meters , and an apparent r e s i s t i v i t y of 6.44 ohm m e t e r s . When t h i s i s s e l e c t e d a u t o m a t i c a l l y for the s t a r t i n g model , the f o l l o w i n g c o n s t r u c t i o n r e s u l t s . S t a r t i n g H o d e i F o u r t h I t e r a t i o n r i t 3. cr. 10° 10' 102 1Q'J Depth .(meters) 10' 10° 10' 102 103 10' Depth (meters) F i g u r e 2.8 U s i n g the s h a l l o w e s t apparen t r e s i s t i v i t y as a s t a r t i n g model i t was d i f f i c u l t to c o n s t r u c t a d a t a - f i t t i n g mode l . The l a s t two i t e r a t i o n s were made u s i n g the s m a l l e s t p e r t u r b a t i o n s . Had t h i s not been done, the model would not have c o n -v e r g e d . T h e i r ext reme s e n s i t i v i t y to s t a r t i n g models makes f l a t t e s t p e r t u r b a t i o n s i m p r a c t i c a l f o r g e n e r a l u s e . but used f o r the f i r s t . o r f i r s t two i t e r a t i o n s t h e i r i n f l u e n c e i s n o t i c e a b l e , and the model c o n v e r g e s . 52 2-6 .3 The E f f e c t s of Noise and Data D e n s i t y There are two aspec t s to data q u a l i t y : a c c u r a c y , and sample d e n s i t y . The degree to which each of these independent ly a f f e c t s the c o n s t r u c t i o n of a r e p r e s e n t a t i v e model shou ld be known. A c c u r a t e data i s that which c o n t a i n s no e r r o r s , from e i t h e r g e o p h y s i c a l or i n s t r u m e n t a l s o u r c e s . There are no e f f e c t s from l a t e r a l i n h o m o g e n e i t i e s , no human e r r o r s , and no n o i s e in a c c u r a t e d a t a . The o ther a s p e c t , data d e n s i t y , r e f e r s to how w e l l the e l e c t r i c f i e l d has been sampled w i t h p o t e n t i a l measurements. In a Schlumberger sound ing , data d e n s i t y i s q u a n t i f i e d by the expans ion f a c t o r . The c l o s e r i t i s to one, the h i g h e r the data d e n s i t y ; 1.41 i s o f t en used , and has been shown by Oldenburg (1979) to a f f o r d good r e s o l u t i o n . In the f i r s t exper iment , no i s e i s added to the data in the form of random numbers. T h e i r mean i s z ero and t h e i r s t a n d a r d d e v i a t i o n i s one. They are s c a l e d by the percent n o i s e l e v e l b e i n g t e s t e d , then m u l t i p l i e d by the data to g i v e the no i se e r r o r e;. £i = x ^ l e v e l E i 100% (2.31) X i i s the random number T h i s i s added to the i ' t h datum, and the s t a n d a r d d e v i a t i o n i s taken as a i = l e v e l E i 100% (2.32) No i se l e v e l s of 3 , 5 , 1 0 , and 20 p e r c e n t were t r i e d . The 53 minimum e r r o r in a p o t e n t i a l d i f f e r e n c e measurement i s r e a l i s t i c a l l y about 3 p e r c e n t . With these four n o i s e l e v e l s added to the d a t a , the f o l l o w i n g f i n a l models were c o n s t r u c t e d wi th the s m a l l e s t p e r t u r b a t i o n s . 3% Noise added 5% Noise added ' 1 1 1 H r- 1 1 M I 1 1 1 1 1 1 1 H 10° 10' 102 103 104 10° 10' 102 !Q3 104 . 10% Noise added 20% Noise added Depth (meters) Depth (meters) F i g u r e 2 .9 In each o f t hese f rames i s a c h i - s q u a r e d a c c e p t a b l e model c o n s t r u c t e d from d a t a to wh ich the d e s i g n a t e d amount of random n o i s e had been added . The models w i t h 3 and 5 p e r c e n t n o i s e i n the d a t a took two i t e r a t i o n s to c o n v e r g e , w h i l e t hose w i t h 10 and 20 p e r c e n t , r e q u i r e d one more i t e r a t i o n . The shape of the f i n a l model does not change v e r y much between 5 and 20 p e r c e n t . Knowing the n o i s e l e v e l s , the same g e n e r a l r e s i s t i v i t y s t r u c t u r e would p r o b a b l y be i n t e r p r e t e d from any of these mode l s . A l l of the above models s a t i s f y the c h i - s q u a r e d c r i t e r i o n f o r the n o i s e l e v e l s a p p l i e d . The l a r g e s t change i n f i n a l models o c c u r s between l e v e l s of 3 and 5 p e r c e n t . From 5 to 20 54 p e r c e n t , there i s l i t t l e change, and the r e s i s t i v i t y s t r u c t u r e i n f e r r e d from any of the four models would most l i k e l y be the same. However, i t i s easy to sea how e r r o r s of 5 percent might cause a m i s i n t e r p r e t a t i o n over more complex s t r u c t u r e . To f i n d the dependence upon data d e n s i t y , Schlumberger data were c a l c u l a t e d u s i n g expans ion f a c t o r s g r e a t e r than 1.41. The degree to which t h i s a f f e c t s the r e l i a b i l i t y of the c o n s t r u c t e d model a l s o dependents upon the s c a l e l e n g t h of the r e s i s t i v i t y s t r u c t u r e ; o b v i o u s l y , very wide ly spaced da ta would be s a t i s f a c t o r y over a h a l f space . T h e r e f o r e , t h i s i n v e s t i g a t i o n w i l l o n l y r e t u r n the maximum . expans ion f a c t o r p o s s i b l e for r e s o l v i n g t h i s p a r t i c u l a r s t r u c t u r e . Because there i s a uniqueness theorum for r e s i s t i v i t y measurements (Langer , 1933), i t i s t r u e that i n f i n i t e l y many a c c u r a t e da ta would r e s o l v e any o n e - d i m e n s i o n a l s t r u c t u r e . I t i s of p r a c t i c a l i n t e r e s t though, to determine how a few, w i d e l y spaced data can r e s o l v e a c e r t a i n s t r u c t u r e . I n c r e a s i n g the expans ion f a c t o r from 1.41 to 1.71 had l i t t l e e f f e c t on the f i n a l model o ther than to improve the f i t i n the r e g i o n from 1 to 10 meters d e p t h . The second i n c r e a s e to 2 .28 , l i k e w i s e had l i t t l e e f f e c t , except p o s s i p l y a l l o w i n g the p o s i t i o n of the minimum r e s i s t i v i t y to s h i f t lower . By any s t a n d a r d s , these are good f i t s to the t rue s t r u c t u r e . No d e t e r i o r a t i o n i s apparent w i t h these expans ion f a c t o r s . Had a s t r u c t u r e wi th a s m a l l e r s c a l e l e n g t h been used , r e s o l u t i o n wi th the s c a l e f a c t o r of 2.28 may have been inadequate s i n c e h i g h e r d e n s i t y data i s n e c c e s s a r y to r e s o l v e those f e a t u r e s . Ex Factor 2.28 Ex. Factor 1.71 I — i — i — i — i — i — | — i — | - J I — i — i — i — i — i — i — i — H 10° 10' 102 103 10< 10° 10' 102 103 104 Depth (meters) Depth (meters) F i g u r e 2 .10 D e c r e a s i n g the d a t a d e n s i t y in t h i s range d i d not a f f e c t the f i n a l models a p p r e c i a b l y . The o n l y n o t i c e a b l e d i f f e r e n c e between these two i s tha t the r e s i s t i v i t y low i s d i s p l a c e d s l i g h t l y s h a l l o w e r when the d a t a d e n s i t y i s at i t s lowest i n the frame on the l e f t . There a re 13 d a t a in the range from AB/2 o f 5 t o 3650 f o r an e x p a n s i o n f a c t o r o f 1.71, and 9 i n the same range w i t h the e x p a n s i o n f a c t o r o f 2 . 2 8 . The maximum and minimum a re c o r r e c t l y p l a c e d w i th e x p a n s i o n f a c t o r s o f 1.71, and 1.41 (see f i g u r e 2 . 3 ) . 2-6.4 R e p r e s e n t a t i o n T e s t i n g The f i n a l way in which the l i n e a r i z e d c o n s t r u c t i o n program s h o u l d be t e s t e d i s to c o n s t r u c t models from the two extremes of s t r u c t u r a l v a r i a t i o n s . I t s response to s t r u c t u r e s wi th no v a r i a t i o n and l a r g e v a r i a t i o n s would be u s e f u l for i n t e r p r e t i n g c o n s t r u c t e d models , and shou ld t h e r e f o r e be known. When there i s e s s e n t i a l l y no v a r i a t i o n in the t r u e s t r u c t u r e , and i t e r a t i o n i s s t a r t e d from a h a l f - s p a c e v a l u e o ther than that of the t rue s t r u c t u r e , i t w i l l be easy to see the v a r i a t i o n between the c o n s t r u c t e d and the t r u e mode l s . T h i s w i l l i l l u s t r a t e how a r e g i o n of c o n s t a n t r e s i s t i v i t y may be r e p r e s e n t e d on a c o n s t r u c t e d model . 56 Starting ModeJ Fourth Iteration 10° IQ' 102 103 104 1Q° 10' 102 103 104 Depth (meters) Depth (meters) F i g u r e 2.11 The t r u e s t r u c t u r e i s a h a l f - s p a c e o f c o n s t a n t r e s i s t i v i t y , and i t Is f i t a f t e r f o u r I t e r a t i o n s by t h i s o s c i l l a t o r y mode l . N o t i c e t ha t the c o n s t r u c t e d model matches the t r u e s t r u c t u r e most c l o s e l y i n the m i d d l e r e g i o n , f rom 10* *1 .5 t o 10* *2 .5 m e t e r s . There the model i s c o n s t r a i n e d by bo th s h a l l o w e r and deepe r s e e i n g d a t a . T h i s f i g u r e i s meant t o i l l u s t r a t e the d i f f e r e n c e s t h a t a r e p o s s i b l e between a c h i - s q u a r e d a c c e p t a b l e model and the t r u e s t r u c t u r e . With c o n s t r u c t i o n programs response to a h a l f - s p a c e known, i t remains to be seen how a l a r g e change in r e s i s t i v i t y over a s h o r t d i s t a n c e w i l l be r e p r e s e n t e d . To t e s t t h i s , the t r u e s t r u c t u r e used i s a box c a r f u n c t i o n w i t h the r e s i s t i v i t y h i g h an o r d e r of magnitude g r e a t e r than the s u r r o u n d i n g h a l f -space . From the p r e v i o u s e x p e r i m e n t s , i t has been l e a r n e d tha t r e p r e s e n t a t i v e models can be c o n s t r u c t e d from twenty da ta s t a r t i n g from AB/2=5 wi th an expans ion f a c t o r of 1.41, when the t r u e s t r u c t u r e has o n l ^ l a r g e s c a l e f e a r t u r e s . With t h i s n o t e d , we would expect the model c o n s t r u c t e d to be a smoothed v e r s i o n of the box c a r due to the l a c k of r e s o l u t i o n , and to f a l l away from the t r u e s t r u c t u r e at about 1000 meters d e p t h . T h i s does i n f a c t d e s c r i b e the c o n s t r u c t i o n . 57 In f i g u r e 2.12, the s t a r t i n g , f i r s t , and f i n a l models are shown w i t h the box c a r superimposed upon each.. The c o n s t r u c t e d model smooths the r e s i s t i v i t y i n c r e a s e and d e c r e a s e , and matches the box c a r best i n the midrange. At 700 meters, the d e c r e a s e i s somewhat b e t t e r f i t than the i n c r e a s e at 20 meters, but c l e a r l y the r e s o l u t i o n i s l o s t j u s t below t h e r e s i n c e the model d i p s below the t r u e s t r u c t u r e a t 1000 meters. Starting Model First Iteration 10° 10' 102 10: Depth (meters) 104 10° 10' 102 103 Depth (meters) 104 Fifth Iteration 10' 10' 10' 10: 104 Depth (meters) F i g u r e 2.12 T h i s t e s t was to i l l u s t r a t e the c o n s t r u c t e d m o d e l ' s r e p r e s e n t a t i o n of an o r d e r o f magn i tude change in r e s i s t i v i t y . The t r ue s t r u c t u r e , which i s a 4 .5 ohm-meter h a l f - s p a c e w i t h a 45 ohm-meter l a y e r from 20 to 700 meters dep th i s matched q u i t e c l o s e l y by the model c o n s t r u c t e d a f t e r f i v e i t e r a t i o n s . As e x p e c t e d , the c o n s t r u c t e d model is somewhat smoother . H igh f r e q u e n c y f e a t u r e s such as the c o r n e r s of the s t r u c t u r e cannot be r e p r o d u c e d because of the r e s o l u t i o n of the d a t a ; t h e r e f o r e monoton i c v a r i a t i o n s which take the c o n s t r u c t e d model th rough a l a r g e change in r e s i s t i v i t y s h o u l d be i n t e r p r e t e d as l a y e r b o u n d r i e s . 58 The g e n e r a l f i n d i n g s in t e s t i n g the l i n e a r i z e d c o n s t r u c t i o n program have been e n c o u r a g i n g . It i s a f a i r l y robust program, c o n s t r u c t i n g r e p r e s e n t a t i v e models from data wi th e r r o r s , and c o n s t r u c t i n g reasonable models in every one of the t e s t s per formed . A wide range of t h e o r e t i c a l s t r u c t u r e s have been reproduced from s y n t h e t i c d a t a . The degree of c l o s e n e s s between the c o n s t r u c t e d and t h e o r e t i c a l models does not seem very dependent upon the c h o i c e of s t a r t i n g models , but a model i s converged upon in fewer i t e r a t i o n s when c o n s t r u c t i o n i s s t a r t e d from the best f l a t l i n e through the apparent r e s i s t i v i t y c u r v e . F u r t h e r , the c o n s t r u c t i o n program was q u i t e i n s e n s i t i v e to d a t a d e n s i t y over the s t r u c t u r e used in the f i r s t two exper iment s , and nine data wi th an expansion f a c t o r of 2.28 would have a d e q u a t e l y reproduced i t . 59 CHAPTER 3 A R e s i s t i v i t y Survey in the Anahim B e l t In 1981 a l a r g e s c a l e r e s i s t i v i t y survey was c a r r i e d out in the Anahim v o l c a n i c b e l t in B r i t i s h Columbia by Premier Geophys i c s under c o n t r a c t from the G e o l o g i c a l Survey of Canada. The r e g i o n surveyed i s at the e a s t e r n end of the v o l c a n i c b e l t , where the recency of e r u p t i o n s may s i g n i f y the presence of economic geothermal energy . The Anahim b e l t i s a c h a i n of Miocene and Quaternary v o l c a n o e s running from west to east a p p r o x i m a t e l y a l o n g the 52nd p a r a l l e l . I t s western l i m i t i s at B e l l a C o o l a , and i t s e a s t e r n l i m i t i s at Blue R i v e r (see f i g u r e 3 . 1 ) . The l a s t e r u p t i o n at the western edge was 14.5 m.y. be fore p r e s e n t , w h i l e in the v i c i n i t y of Blue R i v e r 10 thousand year o l d l a v a s e x i s t . T h i s g r a d a t i o n in ages of l avas from eas t to west suggests that the e a s t e r n edge has the h i g h e s t p o t e n t i a l for geothermal energy , and has been used as ev idence for a mantle hot spot beneath the area ( B e v i e r et a l . 1979). R e s i s t i v i t y s u r v e y i n g was used here in an e f f o r t to l o c a t e the low r e s i s t i v i t i e s commonly a s s o c i a t e d wi th h i g h t e m p e r a t u r e s . Some such anomal ies were found, but t h e i r c o r r e l a t i o n wi th h i g h temperatures at depth i s s t i l l u n e s t a b l i s h e d as they have yet to be d r i l l e d . The r e s i s t i v i t y experiment done i n the Anahim b e l t w i l l be d e s c r i b e d , and r e s i s t i v i t y models c o n s t r u c t e d from the Schlumberger soundings w i l l be p r e s e n t e d . Geothermal i n t e r p r e t a t i o n s w i l l then be made from the models c o n s t r u c t e d and the e x i s t i n g g e o l o g i c a l i n f o r m a t i o n . 60 F i g u r e 3.1 l o c a t i o n s o f some r e c e n t v o l c a n o e s In the Anahim v o l c a n i c b e l t , and the d i p o l e - d i p o l e s u r v e y l i n e . The Sch lumberger s i t e s were a l o n g the d i p o i e - d l p l o e l i n e . F i g u r e 3.2 shows t h e i r exac t l o c a t i o n s , ( f rom Sch l ax and Shore 1981) 3-1 The Anahim R e s i s t i v i t y Exper iment Whi le the main purpose of the survey was to a s s e s s the a r e a ' s geothermal p o t e n t i a l , i t was a l s o used for r e c o n n a i s s a n c e purposes s i n c e the g e o p h y s i c a l da ta base i s so minute . N e i t h e r aeromagnet ic nor g r a v i t y data e x i s t for the a r e a . Thus the r e s i s t i v i t y survey c o n s i s t e d of two p a r t s : a long d i p o l e - d i p o l e l i n e from which the g e o e l e c t r i c c r o s s s e c t i o n of the b e l t was o b t a i n e d , and s i x Schlumberger soundings f o r the more d e t a i l e d p i c t u r e and for c o r r e l a t i o n w i t h the d i p o l e - d i p o l e l i n e . The d i p o l e - d i p o l e l i n e r a n . r o u g h l y n o r t h - s o u t h a c r o s s the suspec ted hot spot t r a c e , and the Schlumberger soundings were expanded a l o n g the l i n e and in a d j o i n i n g v a l l e y s to the west, 61 p e r p e n d i c u l a r to the l i n e (see f i g u r e 3 . 2 ) . D i p o l e s of l e n g t h s one and two k i l o m e t e r s were used , w i t h s e p a r a t i o n s from 1 to 8 k i l o m e t e r s for the 1 km d i p o l e s , and 1 to 6 k i l o m e t e r s for the l a r g e r ones . T h i s p r o v i d e d a d e e p - p r o b i n g p s e u d o - s e c t i o n a c r o s s the b e l t whi l e the Schlumberger a r r a y s , which had a maximum AB/2 s p a c i n g of 2 km, produced data r e f l e c t i n g sha l lower f e a t u r e s . T h e i r maximum p e n e t r a t i o n was between 1 and 2 k i l o m e t e r s . K AZ A S H U S W A P S tauno l i t e-Kyan i t<T "*- ^_ S i 1 imam te METAMORPHIC C O M P L E X Legend metamorph i c — — -b o u n d a r i e s •+• rock g roup . boundary d i po1e 1i ne • ••• f a u l t ~ ' w F i c u r e 3 2 N o r t h Thompson R i v e r v a l l e y showing the s i x Sch lumberger s i t e s and the major g e o l o g i c b o u n d r i e s . The t h i c k dashed l i n e s r e p r e s e n t the boundary between the Kaza Group and the Shuswap Complex , and the t h i n d a s h e d l i n e s a r e b o u n d a r i e s between metamorph ic z o n e s , ( f rom G . S . C . Map 1967-15. Canoe R i v e r ) b2 In both of the s u r v e y s , the p o t e n t i a l d i f f e r e n c e s were r e c o r d e d on a H e w l e t t - P a c k a r d c h a r t r e c o r d e r w i t h a maximum r e s o l u t i o n of 10 m i c r o v o l t s , w h i l e a Pheonix G e o p h y s i c a l t r a n s m i t t e r s u p p l i e d the c u r r e n t . A square wave c u r r e n t , w i t h an 8 second p e r i o d , and 100 p e r c e n t duty c y c l e was used. Data q u a l i t y f o r the Schlumberger soundings was g e n e r a l l y v e r y good, as can be seen i n f i g u r e 3.3. T e l l u r i c , and i n s t r u m e n t a l n o i s e l e v e l s were both low. F i g u r e 3.3 P o t e n t i a l d i f f e r e n c e r e c o r d from VES 1 a t AB/2 = 150m. The s c a l e i s 0 .02 v o l t s / c m . Bo th i n s t r u m e n t a l and t e l l u r i c n o i s e l e v e l s a r e ve r y low. The u n c e r t a i n t y l e v e l a s s o c i a t e d w i t h t h i s r e a d i n g which i s an AB/2 s p a c i n g of 150m on VES 1 i s o n l y about 2 p e r c e n t , but a 10 p e r c e n t s t a n d a r d d e v i a t i o n was a s s i g n e d t o accomodate f o r e r r o r s a r i s i n g from the o n e - d i m e n s i o n a l a p p r o x i m a t i o n and nonplanar topography. Where t e l l u r i c n o i s e was e x c e s s i v e , a me c h a n i c a l f i l t e r i n g t e c h n i q u e was employed f o r p i c k i n g out the square wave's a m p l i t u d e . (Premier G e o p h y s i c s , 1979) 63 3-2 Causes of R e s i s t i v i t y Anomalies in Geothermal Areas Here, as in most geophysical experiments, the quantity measured is not that which is ultimately desired. Rather, i t i s the physical property most correlatable with what we r e a l l y want to find — in this case a heat source. The fact that high temperatures near the surface correlate with low r e s i s t i v i t i e s makes the r e s i s t i v i t y survey a useful geophysical method in geothermal exploration. Geochemical, and geological data are also essential for developing the entire picture of a geothermal area. High temperatures near a .heat source are manifested in anomalously low r e s i s t i v i t y values for the following reasons. 1. High temperatures increase the permiability by fracturing rocks thereby allowing more ground water to enter. This in turn lowers the bulk r e s i s t i v i t y . 2. The r e s i s t i v i t y of water decreases with increasing temperature independent of s a l i n i t y . (see figure 3.4) 3. Soluble minerals present, such as s a l t , go into solution more readily as a result of higher water temperature and decrease the r e s i s t i v i t y further. Sulpher minerals, which are often present in geothermal water systems, have the same effect as s a l t . The less soluble minerals p e r c i p i t a t e out as the temperature decreases near the surface, and c r y s t a l i z e on the sides of the conduits. The influence of temperature and s a l i n i t y on the r e s i s t i v i t y of water is shown in figure 3.4. Data from the 64 Meager Creek geothermal area ( a l s o in B . C . ) i s i n c l u d e d as a r e f e r e n c e p o i n t for geothermal water . Any one, or a l l , of these e f f e c t s w i l l be- i n s t r u m e n t a l in p r o d u c i n g anomously low r e s i s t i v i t i e s near a heat source making i t s d e t e c t i o n wi th a r e s i s t i v i t y survey p o s s i b l e . ofrw HM\tr and FriictiknechT (1966) S A L I N I T Y . G R A M S P E R L I T R E 3 cr h-Z I z I o > > r— in c/> 001 M E A G E R C R E E K HOT S P R I N G S P E B B L E C R E E K HOT S P R I N G S ffrcTscMax a'nd Shore* f r e S i S i ^ ^ " " h temperature and s a H n U y . 65 3-3 G e o l o g i c a l S e t t i n g s of the Schlumberger Soundings U s i n g the Geology map e n t i t l e d Canoe R i v e r ( 1 5 - 1 9 6 7 ) , . t h e g e n e r a l g e o l o g i c a l environment of the survey area w i l l be d e s c r i b e d , and the l o c a l geology at each Schlumberger s i t e w i l l be d e t a i l e d in a n t i c i p a t i o n of the c o n s t r u c t i o n s . The l a r g e s c a l e geology i s dominated by the North Thompson f a u l t , and two Precambrian rock groups : the Kaza Group , and the Shushwap Metamorphic Complex. Both rock groups have undergone e x t e n s i v e thermal metamorphism i n the survey a r e a , and a l l but one of the Schlumberger soundings are in the S t a u r o l i t e - K y a n i t e or Garnet zones of these rock g r o u p s . The Nor th Thompson f a u l t i s a west s i d e up normal f a u l t h a v i n g at l e a s t four k i l o m e t e r s of v e r t i c a l throw ( P e l l and Simony, 1981 i n S c h l a x and Shore , 1981). I t extends from the township of Blue R i v e r , n o r t h through the survey a r e a , and ends at the B . C . - A l b e r t a border (see f i g u r e s 3.1 and 3 . 2 ) . The l a t e r a l d i s p l a c e m e n t i s s l i g h t , and the o n l y p l a c e where K a z a , and Shushwap rocks oppose each other a c r o s s the f a u l t i s at VES 5. Everywhere n o r t h of there the rock i s of the Kaza G r o u p , and everywhere s o u t h , i t i s of the Shushwap Complex. In the survey r e g i o n , the metamorphic grade i n c r e a s e s to the n o r t h h i n t i n g that a r e g i o n a l heat s o u r c e , whether past or p r e s e n t , l i e s in tha t d i r e c t i o n . Extreme t o p o g r a p h i c v a r i a t i o n s c h a r a c t e r i z e t h i s p a r t of the p r o v i n c e . That i s why the N o r t h Thompson R i v e r v a l l e y was chosen f o r the survey s i t e d e s p i t e the presence of such a l a r g e f a u l t . In a d d i t i o n to the two major rock g r o u p s , there are g l a c i a l a l l u v i u m d e p o s i t s on the v a l l e y f l o o r , and in most 66 of the drainages l e a d i n g i n t o the v a l l e y . These are presumed t h i n . VES 1 i s s i t u a t e d on a small f l o o d p l a i n on the e a s t e r n shore of the North Thompson R i v e r , approximately i n the middle of the survey area. I t s purpose was to determine the r e s i s t i v i t y of the a l l u v i u m which covers the ground at t h i s and some of the other Schlumberger s i t e s . Thus, i t was expanded as f a r as the a l l u v i u m d e p o s i t allowed. The maximum AB/2 spacing p o s s i b l e was 500 meters, then marshy ground was encountered to the n o r t h - e a s t . Kaza rock i n the Garnet zone u n d e r l i e s the a l l u v i u m here, and the f a u l t i s to the west. F i g u r e 3.5 An en l a rgement of the VES 1 s i t e . The e l e c t r o d e s were expanded out to the l i m i t s o f the a l l u v i u m d e p o s i t which o v e r l a y s Kaza Me tased imen t s . K A Z A A L L U V I U M G R O U P The v a l l e y i s very steep at VES 2, which prevented the e l e c t r o d e s from being expanded in a s t r a i g h t l i n e . Instead, they were expanded in a elongate W p a t t e r n . Being i n the 67 v a l l e y g o r g e , as i t i s , t h e s u r f a c e e x p r e s s i o n i s a l l u v i u m — sand a.nd g r a v e l . T h e r e i s no b e d r o c k e x p o s u r e , but t h e u n d e r l y i n g r o c k i s Kaza i n t h e S t a u r o l i t e — K a y a n i t e zone w i t h G a r n e t zone metamorphism a c r o s s t h e r i v e r . T h i s s o u n d i n g i s s u f f i c i e n t l y west o f t h e f a u l t t h a t o n l y t h e l a r g e s t e x p a n s i o n c o u l d r e g i s t e r i t s e f f e c t . VES 3 r u n s p e r p e n d i c u l a r t o t h e f a u l t and t h e d i p o l e -d i p o l e l i n e i n a n a r r o w d r a i n a g e t o t h e west o f t h e N o r t h Thompson R i v e r ( M i l e d g e C r e e k ) . A l l u v i u m c o v e r s t h e g r o u n d h e r e t o o , but Kaza G r o u p r o c k s i n t h e G a r n e t zone o u t c r o p on e i t h e r s i d e o f M i l e d g e C r e e k , and l i k e l y u n d e r l a y t h e a l l u v i u m a t a s h a l l o w d e p t h . The f o u r t h S c h l u m b e r g e r s o u n d i n g , VES 4, i s o r i e n t e d i n a s i m i l a r manner t o VES 3, but i n t h e B l u e R i v e r d r a i n a g e , t h i r t y k i l o m e t e r s s o u t h o f M i l e d g e C r e e k . T h i s d r a i n a g e i s b r o a d e r and l e s s s t e e p , so t h e a l l u v i u m w h i c h c o v e r s t h e f l o o r h e r e i s p r o b a b l y t h i c k e r t h a n a t VES 3. S i n c e t h e s u r r o u n d i n g t o p o g r a p h y i s q u i t e subdued and t h e f a u l t f a i r l y d i s t a n t , t h e f l a t e a r t h a p p r o x i m a t i o n i s p r o b a b l y q u i t e v a l i d . The one-d i m e n s i o n a l a p p r o x i m a t i o n may not be, however, as f o l i a t i o n or g n e i s s o s i t y has been measured as d i p p i n g a t 50° t o t h e s o u t h e a s t n e a r b y . T h i s i s t h e s o u t h e r n m o s t s o u n d i n g , and i s i n t h e S i l l i m a n i t e zone o f t h e Shushwap M e t a m o r p h i c Complex w h i c h i s t h e l o w e s t m e t a m o r p h i c g r a d e . VES 5 i s i n t h e Thompson R i v e r v a l l e y v e r y c l o s e t o t h e f a u l t t r a c e . I t i s a l s o c l o s e t o an o i l p i p e l i n e and r a i l r o a d t r a c k , b ut n e i t h e r o f t h e s e was j u d g e d t o have had any e f f e c t 68 the r e a d i n g s (Schlax and Shore , 1981). The spread was e n t i r e l y on a l l u v i u m , and j u s t west of the f a u l t t r a c e where rock of the Shushwap Metamorphic Complex in the S i l l i a n i t e zone l i e s o p p o s i t e tha t of the Kaza Group in the Garnet zone to the eas t (see f i g u r e 3 . 6 ) . T h i s was one of the l o n g e s t spreads and the c l o s e l y spaced da ta show no w i l d f l u c t u a t i o n s , however i t o v e r l i e s . some complex geology which makes the i n t e r p r e t a t i o n c o r r e s p o n d i n g l y d i f f i c u l t . S i n c e c u r r e n t s flow s ideways , i n a d d i t i o n to s t r a i g h t down, these measurements p r o b a b l y averaged the r e s i s t i v i t i e s of the two rock groups a f t e r . a c e r t a i n expans ion was r e a c h e d . T h i s must be taken i n t o c o n s i d e r a t i o n when VES 5 i s i n t e r p r e t e d . < t (D IB < SHUSWAP C O M P L E X K A Z A G R O U P F i g u r e 3.6 l o c a l g e o l o g y a t VES 5. I t is on the west s i d e of the f a u l t t r a c e , i n rock of the Shuswap Metamorphic Complex, but the p r o x i m i t y of the Kaza Group a c r o s s the f a u l t w i l l l i k e l y a f f e c t the measurements. VES 6 was expanded halfway between VES 5 and VES 2. I t was a l s o on a l l u v i u m tha t o v e r l i e s rock of the Kaza G r o u p . 69 The spread i t s e l f was j u s t west of the f a u l t which put i t in the Garnet zone, but the grade to the east i s S t a u r o l i t e -K y a n i t e . (see f i g u r e -3.2) Most of these measurements w i l l have been a f f e c t e d by the f a u l t zone , and those taken at l a r g e r AB/2 s p a c i n g s may show an e f f e c t r e l a t e d to the change i n metamorphic zone a c r o s s the f a u l t . Any r e s i s t i v i t y c o n t r a s t between the two metamorphic grades shou ld be n o t i c a b l e on the apparent r e s i s t i v i t y c u r v e . 3-4 C o n s t r u c t i o n s O b t a i n e d from the Schlumberger Data R e s i s t i v i t y models have been c o n s t r u c t e d from each of the s i x Schlumberger data s e t s . In most c a s e s , on ly the s m a l l e s t p e r t u r b a t i o n s were used because of t h e i r more robust convergence , but for VES 5 a model was e a s i l y c o n s t r u c t e d w i t h the f l a t t e s t p e r t u r b a t i o n s t o o . The g e n e r a l p r o b l e m ' w i t h these c o n s t r u c t i o n s i s t h a t most of the data s e t s seem a f f e c t e d by l a t e r a l inhomogene i t i e s because they v i o l a t e the s l o p e r u l e f o r o n e - d i m e n s i o n a l s t r u c t u r e somewhere ( K e l l e r and F r i s c h k n e c h t , 1977) The o n l y way to compensate for t h i s i s to i n c r e a s e the s t a n d a r d d e v i a t i o n a s s i g n e d to the p o i n t s which s i m p l y a l l o w s a l o o s e r f i t to these d a t a . The mode l ' s f i t to the data can be seen i n the f i r s t frame of each of the f o l l o w i n g f i g u r e s . Apparent r e s i s t i v i t y c u r v e s were p r e d i c t e d from each of the f i n a l models , and superimposed upon the apparent r e s i s t i v i t y v a l u e s d e r i v e d from the d a t a . The one s t a n d a r d d e v i a t i o n e r r o r bars are shown for those apparent r e s i s t i v i t i e s . From the degree of i n t e r s e c t i o n , one can see which data cannot be f i t by the program and h y p o t h e s i z e about 70 the i n f l u e n c e s — such as l a t e r a l l y v a r y i n g s t r u c t u r e . Data p o i n t s which d i d not break the s lope r u l e were n o r m a l l y a s s i g n e d a s t a n d a r d d e v i a t i o n of 10 p e r c e n t , whi l e those which d i d were e i t h e r a s s i g n e d a g r e a t e r s t a n d a r d d e v i a t i o n or o m i t t e d , depending upon the s e v e r i t y . VES 1 Models were c o n s t r u c t e d from two s t a r t i n g models for t h i s data s e t : a h a l f - s p a c e , and a t w o - l a y e r e d s t r u c t u r e . The l a t t e r i s s t r o n g l y suggested by the apparent r e s i s t i v i t y c u r v e s , and i s shown i n f i g u r e 3 .7b to f i t the da ta w i t h l i t t l e change . S i n c e the minimum AB/2 s p a c i n g i s 25 m, the r e s i s t i v i t y of the top l a y e r cannot be c o n s t r a i n e d very w e l l . With t h i s n o t e d , the model c o n s t r u c t e d from a h a l f - s p a c e matches tha t s t a r t e d from the two l a y e r e d model over the r e s t of i t s range . Both models show a d i s t i n c t change in r e s i s t i v i t y between 20 and 30 meters , c o r r e s p o n d i n g to the t h i c k n e s s of the f i r s t l a y e r . Below t h a t , the r e s i s t i v i t y of both models drops to around 150 ohm-meters at 100 meters depth and i n c r e a s e s s l i g h t l y w i t h g r e a t e r d e p t h . The c o n s t r u c t e d models are r e l i a b l e on ly down to two hundred meters ( i n d i c a t e d w i t h an a r r o w ) . R e s o l u t i o n d e c r e a s e s r a p i d l y below t h a t , as the model r e t u r n s to i t s i n i t i a l v a l u e . The upswing between 100 and 200 meters i s i n d i c a t e d by the d a t a . Only the s m a l l e s t p e r t u r b a t i o n s were used for t h i s c o n s t r u c t i o n s . 71 Rho Apparent Second I t e r a t i o n AB/2 (meters) Depth (meters) Rho Apparent Second I t e r a t i o n 10° 10' 102" 103 104 10° 10' 102 TO3 104 AB/2 (meters) Depth (meters) F i g u r e 3.7 VES 1 c o n s t r u c t i o n s from a) a h a l f - s p a c e o f 700 ohm-meters, and b) 1500 ohm-meter l a y e r 22 meters t h i c k ove r a 300 ohm-meter h a l f s p a c e . The models d i f f e r o n l y i n the top 20 meters i l l u s t r a t i n g the l ack o f r e s o l u t i o n t h e r e . The dep th to which the model can be c o n s i d e r e d r e l i a b l e i s i n d i c a t e d w i t h an a r row. The i n t e r p r e t a t i o n i s s i m p l y a v e r i f i c a t i o n of the known n e a r - s u r f a c e geo logy . H i g h l y r e s i s t i v e a l l u v i u m o v e r l i e s the l e s s r e s i s t i v e rock of the Kaza Group . An approximate depth to the Kaza i n t e r f a c e i s 22 meters and , a l t h o u g h i t cannot be e x a c t l y p l a c e d , i t l i k e l y i s a s h a r p , g l a c i e r - s c o u r e d boundary . 72 VES 2 At VES 2 the s i t u a t i o n i s more complex. The apparent r e s i s t i v i t y curve i s w e l l sampled by 15 c l o s e l y spaced d a t a . I t shows one major low zone, and an i n f l e c t i o n i n the l a s t two d a t a . The c o n s t r u c t e d model however, shows three low r e s i s t i v i t y zones wi th minima at 20, 160, and 1200 m e t e r s . Only the main low at 160 m should be i n t e r p r e t e d though. The f i r s t low i s based upon a s i n g l e p o i n t , and i s a c c e n t u a t e d by the s h o u l d e r on the main anomaly (see f i g u r e 2 . 1 2 ) . At 1200 m, the decrease i s a l s o p r o b a b l y a shou lder anomaly. N o t i c e that the p r e d i c t e d apparent r e s i s t i v i t y curve shows o n l y one broad low. A g a i n , o n l y the s m a l l e s t p e r t u r b a t i o n s were u s e d . Rho Apparent Third I teration 10' 10' io-RB/2 (meters) 104 10' W 10' Depth (meters) 104 F i g u r e 3.8 VES 2 c o n s t r u c t i o n . The s t a r t i n g model was a 155 ohm-meter h a l f - s p a c e , and the f i n a l model was a c h i e v e d a f t e r 3 i t e r a t i o n s . From the l e f t f r ame , i t appea rs tha t the 10 p e r c e n t s t a n d a r d d e v i a t i o n s were used to overcome a l a t e r a l e f f e c t e n c o u n t e r e d a t l a r g e r s p a c i n g s . Only r e s i s t i v i t i e s r e p r e s e n t a t i v e of the Kaza Group are seen, wi th an i n t e r e s t i n g decrease between 50 and 300 m e t e r s . No h i g h l y r e s i s t i v e top l a y e r i s i n d i c a t e d here as i t was in VES 1. The l a c k of c o l i n e a r i t y among the e l e c t r o d e s was 73 compensated for and was found to have a n e g l i g a b l e e f f e c t . VES 3 VES 3 i s a p r o b l e m a t i c d a t a s e t . The d a t a wi th AB/2 spac ings g r e a t e r than 200 meters show a grea t d e a l of s c a t t e r ; the apparent r e s i s t i v i t y changes from 180 ohm-metrs at AB/2 of 400 meters to 1300 ohm-meters a t 500 m e t e r s . T h i s v i o l a t e s the maximum s lope r u l e for apparent r e s i s t i v i t y c u r v e s . E i t h e r l a t e r a l inhomogenity i s a f f e c t i n g the measurements or there i s some o ther problem, but these data cannot be f i t by the program. The apparent r e s i s t i v i t y c u r v e for the remain ing s i x p o i n t s l o o k s somewhat s i m i l a r to that of VES 1. A c o n s t a n t decrease from a h i g h near s u r f a c e v a l u e s which p r o b a b l y * r e p r e s e n t a l l u v i u m i s shown. C o n s t r u c t i o n s were s t a r t e d from both a 650 ohm-meter h a l f - s p a c e , and a model i d e n t i c a l to tha t used f o r VES 1, except tha t the top l a y e r was 6 meters t h i n n e r . B Rho A p p a r e n t a S e c o n d I t e r a t i o n 10" IQ1 10 2 IQ3 IQ4 10" 10' 10 2 IO 3 10 4 A 3 / 2 (metersj fosters) 74 (b) p, Rho A p p a r e n t F i r s t I t e r a t i on tn ^ 6 i 5. 4. -> • 1 1— .• ... i. . i 1 I i W 1Q! IO2 ]03 RS/2 (meters) IO4 10' 102 103 Depth (meters) F i g u r e 3.9 VES 3 c o n s t r u c t i o n s . The f i r s t s i x d a t a (up to AB/2 = 200 m e t e r s ; »<=. f i t i n one i t e r a t i o n from a s t a r t i n g from a s t a r t i n g model s i m i l a r to tha t used in VES 1 and i n two i t e r a t i o n s from a 650 ohm-meter h a l f - s p a c e . The s t a r t i n g models seem to have had l i t t l e e f f e c t on the f i n a l models s i n c e they a re so s i m i l a r between 10 and 200 m e t e r s . The o n l y i n f o r m a t i o n that can p o s s i b l y be g l eaned from the l a s t four data p o i n t s i s that they show a g e n e r a l i n c r e a s e in r e s i s t i v i t y . VES 4 The VES 4 data set a l s o shows e f f e c t s which are p r o b a b l y due to l a t e r a l i n h o m o g e n e i t i e s , but t h i s i s not s u r p r i s i n g as 5 0 ° f o l d i n g was i n d i c a t e d on the geology map. Apparent r e s i s t i v i t y c u r v e s as s t eep as those which appear here are i m p o s s i b l e over a o n e - d i m e n s i o n a l s t r u c t u r e ( K e l l e r and F r i s c h k n e c h t , 1977). N e v e r t h e l e s s a very low r e s i s t i v i t y zone i s shown in the data and reproduced in the c o n s t r u c t e d model . G r e a t e r e r r o r than 10 percent was needed for the data p o i n t at the AB/2 s p a c i n g of 500 meters , and the v a l i d i t y of p o i n t s at 750 and 1000 meter AB/2 spac ings must a l s o be q u e s t i o n e d . A rock type change was c r o s s e d on the s u r f a c e when the c u r r e n t e l e c t r o d e s were expanded to an AB/2 of 500 meters , so perhaps a l l of the data p o i n t s from t h e r e out w i l l i n t r o d u c e e r r o r s i n t o o n e - d i m e n s i o n a l c o n s t r u c t i o n s . For t h i s 75 reason another model was c o n s t r u c t e d u s i n g o n l y the data up to an AB/2 s p a c i n g of 400 meters (see f i g u r e 3 . 1 0 b ) . I t was e s s e n t i a l l y a h a l f space of 1200 ohm-meters. T h i s r e s i s t i v i t y va lue i s very d i f f e r e n t from those observed in the VES 1-3 c o n s t r u c t i o n s and i s i n d i c a t i v e of a d i f f e r e n t rock group , the Shushwap Metamorphics . The downturn present i n f i g u r e 3.10b now a r i s e s from o n l y one p o i n t and i s t h e r e f o r e open to q u e s t i o n . If i t i s c o n s i d e r e d r e l i a b l e , an i n t e r e s t i n g decrease e x i s t s and i f no t , then the r e s i s t i v i t y of the f i r s t 200 meters i s a p p r o x i m a t e l y 1200 ohm-meters. Rho .Apparent 1 a, F i r s t I t e r a t i o n 10° IQ1 IQ' 10: R B / 2 (meters) 10' 10' 10r MO 2 10'J Depth- (meters.) 10' F i g u r e 3 .10 )a Model c o n s t r u c t e d from the f i r s t n i n e d a t a in VES 4. a l l w i th 10 p e r c e n t s t a n d a r d d e v i a t i o n . The 1100 ohm-meter s t a r t i n g model f i t s the d a t a a f t e r one i t e r a t i o n . )b shows the c o n s t r u c t i o n o b t a i n e d from a l l twe lve d a t a . The l a s t two d a t a were a s s i g n e d 20 p e r c e n t s t a n d a r d d e v i a t i o n s , and 40 p e r c e n t was a s s i g n e d to the da ta p o i n t at AB/2=500m. 7 6 VES 5 The VES 5 model has s i m i l a r r e s i s t i v i t y v a l u e s to VES 4 down to 200 meters ; i t i s q u i t e r e s i s t i v e . These two soundings are i n marked c o n t r a s t w i t h those i n the Kaza G r o u p . From VES 4 and VES 5, the Shushwap Complex i s seen to be s i g n i f i c a n t l y more r e s i s t i v e than the Kaza Group at shal low d e p t h s . 5. 1.0° 17 Rho A p p a r e n t Th i rd I t e r a t i on in' 102 103 Rho A p p a r e n t 104 1(1 7 10'. 102 103 104 T h i r d I t e r a t i o n 1.0° 101 102 10/ A B / 2 (meters) 104 10° 10' 102 10: Depth (meters) 104 F i g u r e 3.11 S m a l l e s t and f l a t t e s t c o n s t r u c t i o n s f o r VES 5. A l l of these da ta were a s s i g n e d 10 p e r c e n t s t a n d a r d d e v i a t i o n s . )a shows the f i n a l model c o n s t r u c t e d w i t h s m a l l e s t p e r t u r b a t i o n s . )b shows the model c o n s t r u t e d u s i n g the f l a t t e s t p e r t u r b a t i o n s . The f l a t t e s t p e r t u r b a t i o n model shows tha t one low r e s i s t i v i t y zone can f i t the d a t a . 77 With the e x c e p t i o n of the data p o i n t at AB/2=200 meters , t h i s data set obeys the s lope c r i t e r i o n for one-d i m e n s i o n a l i t y . The downward t r e n d in the apparent r e s i s t i v i t y curve i s s u b s t a n t i a t e d by e i g h t data p o i n t s . An upturn i s i n d i c a t e d by the l a s t d a t a p o i n t ; however i t s r e l i a b i l i t y i s q u e s t i o n a b l e as i t i s o n l y one p o i n t . Models were c o n s t r u c t e d u s i n g both the s m a l l e s t and f l a t t e s t p e r t u r b a t i o n s ( f i g u r e s 3 .11a and 3.11b) and both show the same b a s i c f e a t u r e s . The r e s i s t i v i t y decreases from 900 to 150 ohm-meters over the range from 100 to 1000 meters . C l e a r t rends and r e l i a b l e depth i n f o r m a t i o n make these two c o n s t r u c t i o n s very v a l u a b l e . A g e o l o g i c a l e x p l a n a t i o n for the drop in r e s i s t i v i t y i s c u r r e n t leakage a c r o s s the f a u l t i n t o the l e s s r e s i s t i v e rock of the Kaza Group , but i t c o u l d a l s o be d e s c r i b e d g e o t h e r m a l l y by a very h i g h temperature g r a d i e n t at d e p t h . T h i s i s c e r t a i n l y an i n t e r e s t i n g and important s o u n d i n g . VES 6 A h a l f - s p a c e of 220 ohm-meters comes very c l o s e to f i t t i n g the data f o r VES 6. A f t e r one i t e r a t i o n a zone of lower r e s i s t i v i t y i s p l a c e d from 50 to 250 meters . The s h o u l d e r s on t h i s low r e s i s t i v i t y zone are p r o b a b l y i n s i g n i f i c a n t . They have been observed be fore when the t r u e s t r u c t u r e i s a h a l f - s p a c e wi th an i n t e r n a l l a y e r of d i f f e r e n t r e s i s t i v i t y (see f i g u r e 2 . 1 2 ) . T h e r e f o r e , the s t r u c t u r e i n d i c a t e d i s a 220 ohm-meter h a l f - s p a c e wi th a zone of 150 ohm-meters e x t e n d i n g from 50 to 250 meters i n d e p t h . 78 R e s i s t i v i t i e s in t h i s range (150-250 ohm-meters) now appear to be r e p r e s e n t a t i v e of the Kaza Group. Rho. Apparent F i r s t • I t e r a t i o n A B / 2 (meters) Depth, [meters) F i g u r e 3.12 C o n s t r u c t i o n o b t a i n e d f rom VES 6 d a t a , e s s e n t i a l l y a h a l f - s p a c e . 3-5 R e s u l t s from the D i p o l e - D i p l o e L i n e The p s e u d o - s e c t i o n o b t a i n e d from the d i p o l e - d i p l o e survey (Schlax .and Shore , 1981) a i d e d i n the i n t e r p r e t a t i o n of Schlumberger soundings VES 5, and VES 2. I t was a l s o c o n s u l t e d when f o r m u l a t i n g the i n t e g r a t e d i n t e r p r e t a t i o n s because i t extends on e i t h e r s i d e of the area where Schlumberger measurements were made. As mentioned e a r l i e r though, i t s i n f o r m a t i o n r e f l e c t s l a r g e r s c a l e and deeper f e a t u r e s . The most s t r i k i n g f e a t u r e on the p s e u d o - s e c t i o n i s a t r a n s i t i o n from apparent r e s i s t i v i t i e s a v e r a g i n g 400 to va lues of 2000-3000 ohm-meters which occurs about 10 km. south of B lue R i v e r t o w n s i t e . A c c o r d i n g to the Canoe R i v e r map, t h i s o c c u r s e n t i r e l y w i t h i n the Shuswap Complex. North of t h e r e , i n the Schlumberger r e g i o n , they g e n e r a l l y range from 150 to 450 ohm-meters. At VES 5, they decrease from 350 ohm-meters 7 9 at the s m a l l e s t s e p a r a t i o n of the 1 km d i p o l e s to 100 ohm-meters at 6 km s e p a r a t i o n . A g e n e r a l decrease in r e s i s t i v i t y wi th d i p o l e s e p a r a t i o n i s i n d i c a t e d . At the other l a r g e Schlumberger sound ing , VES 2, lower apparent r e s i s t i v i t i e s were measured. They average 200 ohm-meters at 1 km s e p a r a t i o n of the 1 km d i p o l e s and 100 ohm-meters at 2 km s e p a r a t i o n . At deeper p l o t t i n g p o i n t s , apparent r e s i s t i v i t i e s of 27, 28, and 31 ohm-meters were measured. The low r e s i s t i v i t y zone i n t h i s v i c i n i t y appears w e l l s u b s t a n t i a t e d . The four other Schlumberger soundings were e i t h e r too f a r from the d i p o l e - d i p l o e l i n e f o r c o r r e l a t i o n , or were not expanded to the p o i n t where they would sample depths comparable to those sampled by the d i p l o e - d i p o l e s u r v e y . 3-6 G e o l o g i c a l - G e o t h e r m a l I n t e r p r e t a t i o n s J u s t as the r e s i s t i v i t y models c o n s t r u c t e d from a set of data are nonunique, so are the g e o l o g i c a l models which f i t or e x p l a i n tha t set of c o n s t r u c t i o n s . However, t h e r e i s no automat ic i n v e r s i o n r o u t i n e which takes r e s i s t i v i t y models as data and produces a g e o l o g i c model , a l t h o u g h the same p r i n c i p l e s a p p l y . T h i s must be done wi th g e o l o g i c a l and g e o p h y s i c a l i n s i g h t . An analogue to the s m a l l e s t model or p e r t u r b a t i o n i s the s i m p l e s t g e o l o g i c a l model . And g e o l o g i c a l models which i n c o r p o r a t e such f e a t u r e s as the Nor th Thompson f a u l t , or a heat source may be produced by i n t r o d u c i n g them as c o n s t r a i n t s . The p r i n c i p l e of nonuniqueness a l s o a p p l i e s . No unique g e o l o g i c a l model can be c o n s t r u c t e d from the f i n i t e 80 number of r e s i s t i v i t y models . At b e s t , we w i l l be a b l e to d i s c e r n the .range of g e o l o g i c a l models which f i t our r e s i s t i v i t y c o n s t r u c t i o n s . 3-6.1 S i m p l e s t Model w i th G e o l o g i c a l C o n s t r a i n t s The s i m p l e s t model i s that in which there i s no heat s o u r c e , and the f a u l t has no e f f e c t o ther than to separate the two rock types i n one a r e a . T h i s model i s o n l y c o n s t r a i n e d to f i t the known geo logy . A l l of the r e s i s t i v i t y v a r i a t i o n s can be e x p l a i n e d by the rock types and the grades of metamorphism which they d i s p l a y . Both of the rock groups are t h e r m a l l y metamorphosed everywhere in the survey a r e a , w i th the metamorphism i n c r e a s i n g from south to n o r t h . Schlumberger soundings VES 1, VES 2, VES 3, and VES 6 were on g l a c i a l d e p o s i t s o v e r l y i n g the Kaza group of metasediments which i s a h i g h l y v a r i a b l e group of r o c k s . Soundings VES 4, and VES 5 are south of the o t h e r s in the Shushwap Metamorphic Complex w i t h VES 5 l y i n g very c l o s e to the f a u l t i n the zone where i t s e p a r a t e s the Shushwap Complex from the Kaza Group . In the soundings over the Kaza G r o u p , VES 1 i s a two l a y e r case s u g g e s t i v e of t h i c k a l l u v i u m over the Kaza rock and VES 2 has an average r e s i s t i v i t y c l o s e to tha t of the second l a y e r measured in VES 1. VES 3 i s a l s o l i k e l y a l l u v i u m over K a z a , w i th some l a t e r a l v a r i a t i o n s below 100 meters , and VES 6 i s very c l o s e to a h a l f space of the same average r e s i s t i v i t y seen in the second l a y e r in VES 1. The r e s i s t i v i t y l e v e l common to a l l of these i s a p p r o x i m a t e l y 200 ohm-meters which 81 t h e r e f o r e seems r e p r e s e n t a t i v e of the Kaza G r o u p . In the remain ing soundings , a more r e s i s t i v e rock wi th a low r e s i s t i v i t y zone a t 500 meters depth i s i n d i c a t e d . VES 4 however c r o s s e d an apparent v e r t i c a l c o n t a c t , and VES 5 i s c l o s e to Kaza rock a c r o s s the f a u l t . At l a r g e e l e c t r o d e s e p a r a t i o n s the c u r r e n t s c o u l d leak a c r o s s the f a u l t and d i f f u s e more e a s i l y through the l e s s r e s i s t i v i t y Kaza rock on the o ther s i d e . T h i s would produce the lower v o l t a g e r e a d i n g s which caused the low r e s i s t i v i t y zone to be c o n s t r u c t e d . The VES 5 models do approach the r e s i s t i v i t y of the Kaza Group at the deepest r e l i a b l e depths and t h e r e f o r e support t h i s i n t e r p r e t a t i o n . (see f i g u r e s 3.11) In g e n e r a l , the soundings i n the Shushwap showed much h i g h e r r e s i s t i v i t y l e v e l s than those in the Kaza Group . The average r e s i s t i v i t i e s are 900 -ohm-meters for the Sushwap and 200 ohm-meters for the K a z a . There i s n o t h i n g in these c o n s t r u c t i o n s which needs to be e x p l a i n e d by a heat source because i t cannot be by the g e o l o g y . The e f f e c t of the f a u l t i s hard to e v a l u a t e , but there are three soundings on the f a u l t t r a c e and one (VES 6) shows e s s e n t i a l l y a h a l f - s p a c e response down to a depth of 1 k i l o m e t e r , so i t appears to have very e f f e c t t h e r e . The s i m p l e s t i n t e r p r e t a t i o n of the soundings can be summarized as f o l l o w s : VES 1 20-25 meters of r e s i s t i v e a l l u v i u m over the Kaza Group VES 2 t y p i c a l range of Kaza r e s i s t i v i t i e s 82 Kaza Group wi th a measurable a l l u v i u m i n f l u e n c e t y p i c a l Shushwap r e s i s t i v i t i e s w i t h a v e r t i c a l c o n t a c t at AB/2=500 meters Shushwap r e s i s t i v i t i e s w i th c u r r e n t leakage a c r o s s the f a u l t i n t o the Kaza Group a h a l f - s p a c e of t y p i c a l Kaza r e s i s t i v i t i e s . 3-6 .2 R e g i o n a l H e a t i n g Model Another g e o l o g i c a l - g e o t h e r m a l model can be c o n s t r u c t e d u s i n g a r e g i o n a l heat f l u x from the n o r t h and h i g h temperatures a t depth under the Sushwap Complex to e x p l a i n some of the low r e s i s t i v i t y zones p r e v i o u s l y a t t r i b u t e d to bad data or c u r r e n t l e a k a g e . None of the c o n s t r u c t i o n s p r e c l u d e such a model; i n f a c t h i g h temperatures at depth p r o v i d e q u i t e a p l a u s a b l e e x p l a n a t i o n f o r the low r e s i s t i v i t y i n f l e c t i o n s i n VES 4 and VES 5. Deeper i n f o r m a t i o n from the d i p o l e - d i p o l e l i n e s u p p o r t s t h i s h y p o t h e s i s by showing d e c r e a s i n g r e s i s t i v i t i e s w i t h d e p t h . T h i s i n f o r m a t i o n i s o n l y p e r t i n e n t to the deepest Schlumberger soundings s i n c e the minimum d i p o l e s e p a r a t i o n was 1 k i l o m e t e r . The Schlumberger soundings show a t r e n d of d e c r e a s i n g r e s i s t i v i t y nor thward . The apparent r e s i s t i v i t y c u r v e s s h i f t lower at each s i t e when c o n s i d e r e d from south to n o r t h . On the d i p o l e - d i p o l e l i n e , the average apparent r e s i s t i v i t y drops a b r u p t l y at a l l depths about 10 k i l o m e t e r s south of Blue R i v e r , and remains at t h a t lower l e v e l t o the n o r t h e r n end of the l i n e . T h i s drop c o u l d s i g n i f y the southern ex ten t of heat VES 3 VES 4 VES 5 VES 6 83 f l o w i n g from a source to the n o r t h and the Schlumberger soundings may be showing a n o r t h - s o u t h temperature g r a d i e n t . Sch lax and Shore .(1 98 1 ) , h y p o t h e s i z e d tha t the p a t t e r n of a l t e r n a t i n g h i g h e r and lower r e s i s t i v i t y zones on the pseudo-s e c t i o n i n t h i s r e g i o n r e p r e s e n t s u p w e l l i n g and downgoing b r i n e c i r c u l a t i n g in c o n v e c t i o n c e l l s . The u p w e l l i n g b r i n e would be h o t t e r and t h e r e f o r e l e s s r e s i s t i v e than the downgoing. High temperatures at depth are a p o s s i b l e e x p l a n a t i o n for the two soundings which show i n d i s p u t a b l e low r e s i s t i v i t y zones at l a r g e AB/2 s p a c i n g s , VES 4 and VES 5. Both are i n the Shuswap Complex and are t h e r e f o r e more r e s i s t i v e than the o t h e r s down to about 200 meters f o r VES 4 and 300 meters f o r VES 5. At these depths the r e s i s t i v i t i e s beg in to d r o p . In VES 4, t h i s was e x p l a i n e d by a v e r t i c a l c o n t a c t , however, the geology map shows no rock boundary or f a c i e s change t h e r e (Canoe R i v e r 15-1967). A n d , i n VES 5, the e x p l a n a t i o n was c u r r e n t leakage i n t o the l e s s r e s i s t i v e Kaza Group . N o t i n g tha t the r e s i s t i v i t y s t a r t s d r o p p i n g when AB/2=300 m, where i t would be u n i n f l u e n c e d by the f a u l t , the temperature g r a d i e n t e x p l a i n s t h i s f e a t u r e b e t t e r . In a d d i t i o n , the northernmost of the two sound ings , VES 5, i s a p p r o x i m a t e l y 400 ohm meters l e s s r e s i s t i v e than VES 4 i n the top 200 meters which s u p p o r t s the heat from the n o r t h i n t e r p r e t a t i o n . The Shuswap Complex i s known to be l e s s a l t e r e d t h e r m a l l y and s t r u c t u r a l l y than the Kaza Group; i t d i s p l a y s o n l y S i l i m a n i t e zone metamorphism. S i n c e i t i s not so f r a c t u r e d , h i g h temperatures at depth would not be expected to produce a 84 low r e s i s t i v i t y zone which extends up to the s u r f a c e . The d i p o l e - d i p o l e data support the e x i s t e n c e of a low r e s i s t i v i t y zone below 1000 meters , and t h e r e f o r e support the h i g h temperatures at depth h y p o t h e s i s . Over the 30 k i l o m e t e r n o r t h - s o u t h span of the sound ings , two t h i n g s are for c e r t a i n : The f i r s t i s that the Shuswap Complex i s more r e s i s t i v e , at l e a s t at sha l low d e p t h s , than the Kaza G r o u p , and the second i s t h a t the Kaza Group becomes g e n e r a l l y l e s s r e s i s t i v e to the n o r t h (see f i g u r e 2 . 1 3 ) . These two o b s e r v a t i o n s suggest a r e g i o n a l heat source to the n o r t h . The h i g h temperatures i n the Shuswap Complex c o u l d be p l u n g i n g i sotherms from the n o r t h e r n heat s o u r c e , or c o u l d be r e l a t e d to something e n t i r e l y d i f f e r e n t . The r e g i o n a l h e a t i n g model can be summarized as f o l l o w s : (The soundings are a r r a n g e d from south to n o r t h . ) VES 4 c o l d Shuswap rock down to a p p r o x i m a t e l y 300 meters where the temperature beg ins to i n c r e a s e . VES 5 low r e s i s t i v i t y zone at 500 meters s i g n i f y i n g h i g h temp-e r a t u r e s at that d e p t h . VES 1 very r e s i s t i v e a l l u v i u m over a warm Kaza basement. VES 6 h a l f - s p a c e of warm Kaza rock VES 3 an a l l u v i u m l a y e r over warmer Kaza rock VES 2 northernmost sounding and the lowest r e s i s t i v i t y v a l u e s , c l o s e s t to the heat s o u r c e . 85 3 - 6 . 3 The Most Probable Model I t i s u n l i k e l y that e i t h e r the s i m p l e s t model or the r e g i o n a l h e a t i n g model i s e n t i r e l y c o r r e c t . The most p r o b a b l y c o r r e c t model i s one which combines the most p l a u s a b l e e x p l a n a t i o n s from each and uses them, a l o n g w i t h the geo logy of the a r e a , to complete the g e o l o g i c a l - g e o t h e r m a l p i c t u r e . T h i s d e f i n e s the most p r o b a b l e model . G i v e n the Schlumberger data and the v o l c a n i c h i s t o r y of the a r e a , we have s i g n i f i c a n t ev idence f o r r e g i o n a l h e a t i n g from the n o r t h . We a l s o see a l a r g e r e l a t i v e d i f f e r e n c e i n r e s i s t i v i t y between the two major rock groups i n the area wi th the Kaza be ing the l e s s r e s i s t i v e of the two. Lowest R e s i s t i v i t y Values 3. 6. 9. 12. North-South Project ion (km) is. F i g u r e 3.13 T h i s graph shows the lowest r e s i s t i v i t y v a l u e s c o n s t r u c t e d from the sound-ings m the Kaza Metasedlments. They a r e p l o t t e d a g a i n s t d i s t a n c e s o uth of V E S 2 . which was the northernmost sounding. The g r a d i e n t seen here forms the b a s i s of the heat-from-t h e - n o r t h model. The l e a s t s q u a r e s f i t t o the p o i n t s has the f o l l o w i n g f o r m u l a : = exp(3.9 + 0.11x) ohm-meters where x i s the d i s t a n c e s o uth of V E S 2 i n km. The N o r t h Thompson f a u l t must p l a y a r o l e i n p r o d u c i n g the measurements because of the enormous v e r t i c a l d i s p l a c e m e n t that has o c c u r e d a long i t . F a u l t s i g n a t u r e s u s u a l l y man i f e s t themselves w i th low r e s i s t i v i t i e s r e s u l t i n g from i n c r e a s e d p o r o s i t y . T h u s , VES 5 may have been a f f e c t e d by the f a u l t 86 zone i t s e l f , r a t h e r than the rock on the o ther s i d e , or h i g h t e m p e r a r u r e s . I t i s indeed u n f o r t u n a t e tha t the v a l l e y bottom, which p r o v i d e s the deepest vantage p o i n t a l s o c o n t a i n s such a l a r g e f a u l t . The low r e s i s t i v i t y zone i n d i c a t e d on VES 4 i s a l s o of q u e s t i o n a b l e r e l i a b i l i t y . When the da ta p o i n t s which v i o l a t e the s l o p e r u l e are o m i t t e d , the e x i s t e n c e of the low r e s i s t i v i t y zone h inges upon o n l y one da ta p o i n t . U n f o r t u n a t e l y the d i p o l e - d i p o l e l i n e i s too d i s t a n t for c o r r e l a t i o n , so t h i s low r e s i s t i v i t y zone cannot be substant i a t e d . Without the low r e s i s t i v i t y zone on VES 4, and e x p l a i n i n g the low zone on VES 5 by the f a u l t i t s e l f , t h e r e i s on ly ev idence f o r r e g i o n a l h e a t i n g from the n o r t h . S ince every sounding c o n t r i b u t e s to t h i s h y p o t h e s i s , and i t i s supported by t h e i r average r e s i s t i v i t y l e v e l s r a t h e r than i n f l e c t i o n s , heat f l o w i n g from a source to the n o r t h i s the most p r o b a b l e model . 3.7 C o r r e l a t i o n wi th the L a r g e r S c a l e T e c t o n i c P i c t u r e A g e o l o g i c a l overview of the Anahim v o l c a n i c b e l t and i t s r e l a t i o n s h i p to the west coas t t e c t o n i c p i c t u r e w i l l be p r e s e n t e d w i t h the i n t e n t i o n of p u t t i n g our most p r o b a b l e model i n t o i t s proper p e r s p e c t i v e . I t s " f i t " i n t o the l a r g e s c a l e h y p o t h e s e s , or models , c o n c e r n i n g the g e n e s i s of the Anahim v o l c a n i c b e l t w i l l then be examined. 87 3-7.1 G e o l o g i c a l H i s t o r y of the Anahim V o l c a n i c B e l t The Anahim v o l c a n i c b e l t i s composed of a wide v a r i e t y of e x t r u s i v e and p l u t o n i c . f e a t u r e s . E x t r u s i v e f e a t u r e s range from s m a l l b a s a l t i c c i n d e r cones to huge p e r a l k a l i c 1 s h i e l d v o l c a n o e s , and the p l u t o n s i n c l u d e d i k e swarms and s u b v o l c a n i c p l u t o n s (Baer , 1973). From the r e l a t i v e p o s i t i o n s of l a v a s at many v o l c a n i c c e n t r e s , Souther (1977) c o n c l u d e d tha t most of the b a s a l t i c magma rose to the s u r f a c e q u i c k l y through s m a l l c o n d u i t s forming the many s m a l l s i n g l e - e r u p t i o n v o l c a n o e s . Only in p l a c e s where the magma p o o l e d and mel ted out a chamber d i d any d i f f e r e n t i a t i o n o c c u r . These gave r i s e to l a r g e s t r a t o - v o l c a n o e s one of which , the Rainbow Range s h i e l d v o l c a n o , has been i n v e s t i g a t e d i s o t o p i c a l l y for age i n f o r m a t i o n ( B e v i e r et a l . 1979). The l a s t e r u p t i o n there was 8.7 m i l l i o n years be fore presen t ( K - A r d e t e m i n a t i o n ) . T h i s i s a p p r o x i m a t e l y in the middle of the range of ages found in the Anahim b e l t which i s from 14.5 m i l l i o n y e a r s to 10 thousand years b e f o r e p r e s e n t . In the l a t e Miocene , at the time of the e a r l i e s t e r u p t i o n s in the Anahim, the p l a t e o r i e n t a t i o n s o f f the west coas t were s i m i l a r to today a l t h o u g h the E x p l o r e r P l a t e may not have moved i n d e p e n d e n t l y at tha t t ime . The Juan de F u c a , and P a c i f i c P l a t e s were s e p a r a t e d by a s p r e a d i n g r i d g e , as they are now, and the Juan de Fuca P l a t e was b e i n g subducted under the N o r t h American P l a t e w h i l e the P a c i f i c P l a t e moved 1 P e r a l k a l i c igneous rocks are those i n which the m o l e c u l a r p r o p o r t i o n of A l i s l e s s than that of Na and K combined. 88 n o r t h . A r i g h t l a t e r a l t r a n s f o r m f a u l t s epara te s i t from the Nor th American P l a t e (see f i g u r e 3 . 1 4 ) . The t r i p l e j u n c t i o n formed where t h i s s p r e a d i n g r i d g e meets the North American P l a t e i s known to have jumped around d u r i n g the l a s t 10 m i l l i o n y e a r s (Ridd ihough and Hyndman, 1976), making r e c o n s t r u c t i o n of the p l a t e p o s i t i o n s back to 10 m i l l i o n y e a r s be fore presen t very d i f f i c u l t . But R idd ihough (1977) has p l a c e d the t r i p l e j u n c t i o n o f f Brooks P e n i n s u l a on Vancouver I s l a n d d u r i n g the p e r i o d from 5 to 10 m i l l i o n years ago. I t was d u r i n g t h i s time that the A l e r t Bay vo lcanoes on the n o r t h e r n end of Vancouver I s l a n d became a c t i v e . (see f i g u r e 3.14) Subduct ion of the Juan de Fuca P l a t e under the N o r t h American P l a t e r e s u l t e d in a r c and back a r c v o l c a n i s m , and the Anahim b e l t , a c c o r d i n g to Stacey (1974) . The Pemberton and G a r a b a l d i v o l c a n i c c h a i n s are examples of a r c v o l c a n i s m . (see f i g u r e 3.14) These were not a c t i v e s i m u l t a n e o u s l y , however, and a c t i v i t y s h i f t e d from the Pemberton c h a i n , which i s f u r t h e r i n l a n d , to the G a r a b a l d i c h a i n about 2-3 m i l l i o n y e a r s ago ( B e v i e r et a l . 1979). The A l e r t Bay vo l canoes a l s o became e x t i n c t a t t h i s t i m e . R i d d i h o u g h (1977) proposed a t e c t o n i c cause for these changes: the r e o r i e n t a t i o n of the E x p l o r e r P l a t e , and subsequent changes i n the s u b d u c t i o n of the Juan de Fuca P l a t e . The t r i p l e j u n c t i o n was moving n o r t h a t t h i s t i m e . 89 F i g u r e 3.14 P l a t e b o u n d r i e s and v o l c a n i c b e l t s . T h i s shows the o r i e n t a t i o n o f the Juan de Fuca P l a t e i n r e l a t i o n to the Anahim v o l c a n i c b e l t . The p r e s e n t boundry between the Juan de F u c a , and E x p l o r e r P l a t e s i s ' s h o w n w i t h a dashed l i n e , and a r rows i n d i c a t e r e l a t i v e p l a t e m o t i o n s . ( f rom S t a c e y , 1974) The Anahim v o l c a n i c b e l t l i e s j u s t n o r t h of the s u b d u c t i n g l imb of the Juan de Fuca P l a t e . S tacey (1974) used t h i s r e l a t i o n s h i p and p h y s i o g r a p h i c e v i d e n c e to a r r i v e at h i s edge e f f e c t h y p o t h e s i s f or the o r i g i n of the b e l t . In t h i s h y p o t h e s i s , the l i t h o s p h e r e i s f o r c e d up by the u n d e r t h r u s t i n g Juan de Fuca P l a t e and f l e x e s where i t o v e r l a y s the n o r t h e r n end of the s u b d u c t i o n zone (see f i g u r e 3 . 1 5 ) . T e n s i o n c r e a t e d by the f l e x u r e would cause c r a c k i n g at the base of the l i t h o s p h e r e , and hot mantle m a t e r i a l c o u l d melt i t s way to the s u r f a c e as the c r a c k s opened. Over the f l e x u r e zone, c h a i n of v o l c a n o e s would ensue. For t h i s mechanism to have a c t i v a t e d the Anahim v o l c a n i c b e l t , the Juan de Fuca P l a t e would have to 90 u n d e r t h r u s t the Nor th American P l a t e f a r i n l a n d be fore s i n k i n g . At the e s t i m a t e d convergence r a t e of 2.5 cm/yr in the d i r e c t i o n N 3 5 ° E , (Atwater , 1970) i t would take 24 m.y. f o r a zone of t e n s i o n to propagate from the c o a s t to the presen t e a s t e r n edge of the Anahim v o l c a n i c b e l t . As the age d i f f e r e n c e between the v o l c a n o e s in the eas t and those i n the west i s 14 m . y . , the two t ime s c a l e s do not f i t v e r y w e l l . F u r t h e r m o r e , the Anahim b e l t l i e s e a s t - w e s t , not N 3 5 ° E . (see f i g 13.4) Another h y p o t h e s i s a t t r i b u t e s the Anahim v o l c a n i c b e l t to n o r t h - s o u t h r i f t i n g of the N o r t h American P l a t e ( T i p p e r , 1957, 1969). T h i s theory a r i s e s from the common a s s o c i a t i o n of A l k a l i c , and P e r a l k a l i c v o l c a n i c r o c k s w i t h t e n s i o n a l e n v i r o n m e n t s . However, the l a c k of e x t e n s i v e eas t -wes t normal f a u l t i n g makes t h i s theory the l e a s t c r e d i b l e . The Nor th Thompson f a u l t , because of i t s n o r t h - s o u t h o r i e n t a t i o n makes n o r t h - s o u t h r i f t i n g in the e a s t e r n end of the 'Anahim q u i t e u n l i k e l y , and t h i s theory l a c k s an i n t r i n s i c e x p l a i n a t i o n for the t r e n d of d e c r e a s i n g ages seen i n the Anahim v o l c a n i c b e l t . The most favored h y p o t h e s i s i s one which e x p l a i n s the Anahim b e l t as the t r a c e of a mantle h o t - s p o t over which the N o r t h American P l a t e has been d r i f t i n g . S t r o n t i u m i s o t o p e a n a l y s i s of the l a v a s from the l a r g e Rainbow Range S h i e l d V o l c a n o i n d i c a t e s that they o r i g i n a t e d i n the mantle ( B e v i e r et a l . 1979). The p e r a l k a l i c l a v a s which are found in v o l c a n i c c e n t r e s and r o o t s of s h i e l d v o l c a n o e s are b e l i e v e d to be a r e s u l t of f r a c t i o n a l c r y s t a l i z a t i o n of b a s a l t i c magma beneath the l a r g e v o l c a n o e s . 9 1 F u r t h e r support for the h o t - s p o t theory i s c o n t a i n e d i n M i n s t e r ' s 1974 r e s u l t s for the i n s t a n t a n e o u s v e l o c i t i e s of the major p l a t e s . In t h i s i n v e s t i g a t i o n , 106 earthquake motion v e c t o r s , 68 s p r e a d i n g r a t e s , and 62 f r a c t u r e zone t r e n d s were i n v e r t e d to o b t a i n the i n s t a n t a n e o u s v e l o c i t i e s . T h e i r s o l u t i o n s were s e l f - c o n s i s t e n t ; i n t e g r a t i n g the v e l o c i t y v e c t o r s over any c l o s e d path on the s u r f a c e of the e a r t h r e t u r n e d zero net mot ion , as i t s h o u l d . The d r i f t r a t e M i n s t e r et a l . o b t a i n e d for the Nor th America P l a t e was 2.7 cm/year r e l a t i v e to the mantle h o t - s p o t s beneath Y e l l o w s t o n e Park and Raton , New M e x i c o . U s i n g the g r a d a t i o n in ages and assuming a s t a t i o n a r y h o t - s p o t under the Anahim b e l t , B e v i e r (1979) c a l c u l a t e d a mean d r i f t v e l o c i t y of 2-3 .3 c m / y e a r . M i n s t e r ' s work a l s o uses the g l o b a l hot spot d i s t r i b u t i o n as a s t a t i o n a r y r e f e r e n c e . S t i l l more c o n v i n c i n g l y , the e a s t - n o r t h e a s t t r e n d of the Anahim v o l c a n i c b e l t at l a t t i t u d e 5 2 . 5 ° l i e s a l o n g a s m a l l c i r c l e of M i n s t e r ' s hot spot - Nor th America p o l e at 4 8 . 1 ° n o r t h and 9 7 . 9 ° e a s t . The v e l o c i t y p r e d i c t e d a t the c e n t r e of the Anahim v o l c a n i c b e l t i s 2.5 c m / y e a r , which i s w i t h i n the range of v e l o c i t i e s i n d i c a t e d by the p r o g r e s s i o n of ages . Thus the mantle hot spot h y p o t h e s i s seems to have the most ev idence s u p p o r t i n g i t , but t h e r e i s i n s u f f i c i e n t ev idence to r u l e out the o ther hypotheses . We now look to our most p r o b a b l e model to see whether one theory i s suppor ted over the o t h e r by the r e s i s t i v i t y d a t a , as u n f o l d i n g some of the g e o l o g i c h i s t o r y was one of the aims of the Anahim r e s i s t i v i t y s u r v e y . 92 3-7 .2 F i t to the H o t - S p o t T r a c e H y p o t h e s i s A heat source to the n o r t h of the survey a r e a was c a l l e d f o r in the most p r o b a b l e g e o l o g i c a l - g e o t h e r m a l model . I t would c r e a t e a s o u t h - n o r t h temperature g r a d i e n t i n the survey area which c o u l d account for the northward decrease i n r e s i s t i v i t y . A c c o r d i n g to the h o t - s p o t h y p o t h e s i s , the Anahim b e l t i s the t r a c e of a mantle plume which the North American P l a t e has been d r i f t i n g o v e r . I t i s h y p o t h e s i z e d to l i e under the e a s t e r n s e c t i o n of the b e l t at p r e s e n t . I f t h i s plume i s taken to be the heat source c a l l e d f o r , then the most p r o b a b l e model e a s i l y f i t s i n t o the h o t - s p o t h y p o t h e s i s . D e s c r i p t i v e l y , they support each o t h e r . Q u a n t i t a t i v e l y though, there i s not enough r e s i s t i v i t y or temperature data to v e r i f y the f i t . ' I f more measurements were a v a i l a b l e , then the support c o u l d be e v a l u a t e d by c h e c k i n g t h e i r f i t to the temperature d i s t r i b u t i o n f u n c t i o n expec ted from the h o t - s p o t s geometry. Heat f l o w i n g from a long column of magma in the l i t h o s p h e r e would have c i r c u l a r symmetry, f a l l i n g o f f as a f u n c t i o n of the d i s t a n c e from the c e n t r e of the co lumn. The t h e o r e t i c a l temperature d i s t r i b u t i o n c o u l d be mode l l ed by s o l v i n g the heat e q u a t i o n on the top of a s emi -i n f i n i t e s o l i d wi th a c y l i n d r i c a l heat s o u r c e . Then , the h o t -spot h y p o t h e s i s c o u l d be t e s t e d by comparing the p r e d i c t e d temperatures to those measured, or d e r i v e d from r e s i s t i v i t y measurements. 93 3-7 .3 F i t to the Edge E f f e c t H y p o t h e s i s S t a c e y ' s (1974) proposed mechanism a t t r i b u t e s the Anahim v o l c a n i c b e l t to c r a c k i n g at the base of the l i t h o s p h e r e over the n o r t h e r n end of the s u b d u c t i o n zone . The f o l l o w i n g diagram i l l u s t r a t e s t h i s h y p o t h e s i s . F i g u r e 3.15 the mechanism f o r S t a c e y ' s edge e f f e c t h y p o t h e s i s , a : s p r e a d i n g c e n t r e . b : J uan de Fuca P l a t e , c : P a c i f 1 c P l a t e , d : N o r t h Amer i can P l a t e , e :Queen C h a r l o t t e t r a n s f o r m f a u l t , f : z o n e o f t e n s i o n , g :Anah im v o l c a n i c b e l t , h r c o a s t a l v o l c a n i c a r c . The heat flow from such a mechanism would be s u b s t a n t i a l l y g r e a t e r than from a s i n g l e hot s p o t , and the v o l c a n i s m more a c t i v e . S u r f a c e temperatures would decrease wi th l a t e r a l d i s t a n c e from the edge of the s l a b , and near s u r f a c e r e s i s t i v i t i e s would i n c r e a s e . Our r e s i s t i v i t y c o n s t r u c t i o n s do show an i n c r e a s e in the 94 average r e s i s t i v i t i e s go ing from n o r t h to south which c o u l d be the r e s u l t of d e c r e a s i n g temperature (see f i g u r e 3 . 1 3 ) . However, w i th the soundings l y i n g rough ly a l o n g one l i n e , i t i s i m p o s s i b l e to gather support f o r an edge e f f e c t source from them. A g a i n , a q u a n t i t a t i v e approach c o u l d be taken i f the data warranted i t . By model ing the temperature d i s t r i b u t i o n that r e s u l t s from the edge e f f e c t heat geometry, and c h e c k i n g the f i t to a c t u a l measurements, some e v i d e n c e , e i t h e r f o r or a g a i n s t the theory c o u l d be o b t a i n e d . Before such an a n a l y s i s i s c a r r i e d o u t , a r e l i a b l e r e s i s t i v i t y to temperature c o n v e r s i o n for rock of the Kaza Group s h o u l d be d e v e l o p e d . In summary, the r e s i s t i v i t y models c o n s t r u c t e d in t h i s c h a p t e r support e i t h e r h y p o t h e s i s f o r the o r i g i n of the Anahim v o l c a n i c belt- . There, i s not enough i n f o r m a t i o n i n the s i x r e s i s t i v i t y models to support one h y p o t h e s i s over the o t h e r . That c o u l d o n l y come from a more d e t a i l e d exper iment . yb CHAPTER 4  APPRAISAL Once a c h i - s q u a r e d a c c e p t a b l e model has been c o n s t r u c t e d , r e g a r d l e s s of the method, i t s h o u l d be a p p r a i s e d . We s h o u l d know which f e a t u r e s of the model l i k e l y e x i s t in the r e a l e a r t h , and which are merely a r t i f a c t s of the c o n s t r u c t i o n method. T h i s i s e s p e c i a l l y important for those f e a t u r e s which a r e of h i g h i n t e r p r e t i v e v a l u e i n a l a r g e s c a l e g e o l o g i c a l i n t e r p r e t a t i o n . We want to be as c e r t a i n as p o s s i b l e that the e a r t h a c t u a l l y has the s t r u c t u r e s which form the b a s i s of any such i n t e r p r e t a t i o n . 4-1 OBJECTIVES OF APPRAISAL Models which f i t a set of r e s i s t i v i t y d a t a , or most o ther types of g e o p h y s i c a l data f o r tha t m a t t e r , can u s u a l l y be c o n s t r u c t e d by a number of d i f f e r e n t methods. H e r e , r e s i s t i v i t y models have been c o n s t r u c t e d wi th the s m a l l e s t and f l a t t e s t p e r t u r b a t i o n s in l i n e a r i z e d c o n s t r u c t i o n , but o ther methods a l s o e x i s t . G e l ' f a n d - L e v i t a n p r o c e d u r e s and m o d i f i e d D a r - Z a r r o u k f u n c t i o n s have a l s o been used to c o n s t r u c t r e s i s t i v i t y models a n d , there are d i f f e r e n t norms to min imize both g l o b a l l y and l o c a l l y w i t h i n the rea lm of l i n e a r i z e d c o n s t r u c t i o n . In each case d i f f e r e n t models w i l l be p r o d u c e d . T h i s p r e s e n t s the f o l l o w i n g d i l e m a : wi th such a range of models p o s s i b l e , what c e r t a i n t y can we have that a s i n g l e a c c e p t a b l e model bears any resemblence to the r e a l e a r t h ? V e r y l i t t l e , i t would seem from the number of models tha t can be c o n s t r u c t e d , but t h i s should be q u a l i f i e d . 96 Q u a l i f y i n g the degree to which the c o n s t r u c t e d models are s i m i l a r to the r e a l e a r t h model i s the o b j e c t of a p p r a i s a l . In e f f e c t , a p p r a i s a l i s the e r r o r a n a l y i s of c o n s t r u c t e d models . 4-2 L i n e a r i z e d A p p r a i s a l U n t i l r e c e n t l y , Backus and G i l b e r t ' s l i n e a r i z e d approach to a p p r a i s a l was c o n s i d e r e d adequate f o r the n o n l i n e a r problems . I t i s now known, however, t h a t i t s a b i l i t y to s u c c e s s f u l l y a p p r a i s e a model depends upon the v a l i d i t y of the assumption tha t a l l the models which f i t a set of data are l i n e a r l y c l o s e 1 . By i m p l i c a t i o n of t h i s a s s u m p t i o n , the t r u e e a r t h model i s l i n e a r l y c l o s e to any d a t a - f i t t i n g model . The problem w i t h t h i s approach has r e c e n t l y been i l l u s t r a t e d i n the case of m a g n e t o t e l l u r i c s , where a c c e p t a b l e models have been c o n s t r u c t e d from d e l t a f u n c t i o n conductances ( P a r k e r , 1981) which are not l i n e a r l y c l o s e to e i t h e r the r e a l e a r t h , or p r e v i o u s l y c o n s t r u c t e d mode l s . L i n e a r i z e d a p p r a i s a l of c o n t i n u o u s models had s a i d n o t h i n g about the e x i s t e n c e of these ; they are not l i n e a r l y c l o s e to c o n t i n u o u s models . In f a c t , l i n e a r i z e d a p p r a i s a l does not extend to any models v e r y d i f f e r e n t i n t h e i r gross f e a t u r e s from the one b e i n g a p p r a i s e d . One c o n c l u s i o n tha t may be drawn from t h i s , i s tha t the f i r s t s t e p towards a comprehensive a p p r a i s a l shou ld be the 1 The term l i n e a r l y c l o s e w i l l be d e f i n e d i n s e c t i o n 4-2.1 97 c o l l e c t i o n of as many varied models as i s possible. Then, i f a l l the models are l i n e a r l y close, one may proceed with l i n e a r i z e d appraisal. If not, the appraisal must be done q u a l i t a t i v e l y . But even i f a l l the col l e c t e d models were l i n e a r l y close, there would s t i l l be no guarantee that a l l acceptable models were. Some solace may be found in the fact that over a one-dimensional earth, there i s a uniqueness theorum for r e s i s t i v i t y measurements (Langer 1933). This very fortunately implies that as the number of data increases, the structure w i l l become better determined to the l i m i t where i n f i n i t e accurate data w i l l uniquely determine the real earth's r e s i s t i v i t y structure. 4-2.1 Averaging Functions and Model Averages Backus and Gilb e r t ' s l i n e a r i z e d method for appraising models in the non-linear problems such as D.C. r e s i s t i v i t y and M-T w i l l now be presented. The inherent assumptions, and the conditions under which they are true w i l l also be discussed. It w i l l be assumed that a chi-squared acceptable model m 0(z) has already been found. Now, by Frechet d i f f e r e n t i a b i l i t y of the data, there exists the function Gi(m 0(z),z) such that i f m,(z) i s any other earth model, then Ei(m,)-Ei(m 0) = Gi (z) [m, (z)-m 0 (z)]dz + oQm^mo) 2] for i = 1,n (4.1) 98 where E i ( m , ) and E i ( m 0 ) are the i ' t h data p r e d i c t e d by the r e s p e c t i v e models , and 0 [ ( m , - m 0 ) 2 J i s a q u a d r a t i c remainder term. When m, -m 0 i s s u f f i c i e n t l y s m a l l , as measured by a L 2 norm, the q u a d r a t i c term goes to zero and the d i f f e r e n c e i n data depends almost l i n e a r l y upon ra,-ni0 (Backus and G i l b e r t , 1968). The r e g i o n i n model space where t h i s i s t r u e i s c a l l e d the l i n e a r r e q i o n around m 0 , and the models in t h i s r e g i o n are s a i d to be l i n e a r l y c l o s e to m 0 . I t i s i n t u i t i v e l y obv ious tha t the s i z e of the l i n e a r r e g i o n i s dependent upon the r e l a t i o n s h i p between E i and m. I f the r e l a t i o n s h i p i s l i n e a r over a l l of model space , then we are back to a l i n e a r p r o b l e m . The k e r n e l , G i ( m ( z ) , z ) embodies t h i s r e l a t i o n s h i p through i t s m(z) dependence i n the f o l l o w i n g way. A k e r n e l which has no dependence upon m(z) i s r e p r e s e n t a t i v e of data which are l i n e a r i n m ( z ) , and in n o n - l i n e a r prob lems , the r e g i o n around m 0 ( z ) where G i ( m ( z ) , z ) i s c o n s t a n t w i t h r e s p e c t to m(z) i s the l i n e a r r e g i o n . Thus , the d i f f e r e n c e between the k e r n e l s of two models which f i t the data c o u l d p o s s i b l y be used as a measure of l i n e a r c l o s e n e s s . Assuming that both models f i t the d a t a , and are l i n e a r l y c l o s e so tha t the remainder term i s much s m a l l e r than the d i f f e r e n c e i n d a t a , we have then to the f i r s t a p p r o x i m a t i o n r G i ( z ) [ m , ( z ) - m 0 ( z ) ] dz = 0 i=1,n (4 .2) which can be s e p a r a t e d i n t o two e q u i v a l e n t i n t e g r a l s . 9 9 E q u a l i t y i s m a i n t a i n e d when the i n t e g r a l s are m u l t i p l i e d by the c o e f f i c i e n t s a; which minimize the d i f f e r e n c e between L a i G i ( z ) and 6 ( z - z 0 ) , and we o b t a i n o O r A ( z , z 0 ) m , ( z ) dz = A ( z , z 0 ) m 0 ( z ) dz (4.3) where Z a t G i ( z ) = A ( z , z 0 ) , ft and A ( z , z 0 ) i s the a v e r a g i n g f u n c t i o n c e n t r e d at z 0 . The models as seen through the a v e r a g i n g f u n c t i o n s are e q u a l , and the o n l y c o n d i t i o n s on m,(z) were that i t f i t s the d a t a , and tha t i t i s l i n e a r l y c l o s e to m 0 ( z ) . Models s a t i s f y i n g these c o n d i t i o n s w i l l have i d e n t i c a l averages a t any p o i n t z 0 i n t h e i r range . <m,(z 0 )> = <m 0 (z 0 )> (4.4) Average models can be c o n s t r u c t e d p o i n t by p o i n t to determine what f e a t u r e s a l l of the l i n e a r l y c l o s e models have in common. O r , the a v e r a g i n g f u n c t i o n i t s e l f can be c o n s t r u c t e d at depths of i n t e r e s t to determine over what range of depths the model i s a v e r a g e d . T h i s p r o v i d e s a way of v i s u a l i z i n g the depth r e s o l u t i o n of a mode l . To o b t a i n i n f o r m a t i o n about the r e a l e a r t h , though, from l i n e a r i z e d a p p r a i s a l we must know u n e q u i v i c a l l y that the model m 0 ( z ) i s l i n e a r l y c l o s e to the t r u e e a r t h model me(z ) . Then , and o n l y t h e n , w i l l averages of the c o n s t r u c t e d model equa l t r u e e a r t h model a v e r a g e s . I t i s here t h a t the problem wi th 1 00 l i n e a r i z e d a p p r a i s a l l i e s . Where i t was p r e v i o u s l y thought tha t a l l a c c e p t a b l e models were l i n e a r l y c l o s e to each o ther and the t r u e model , i t has been shown to be f a l s e . Recent ev idence has shown that in the M-T prob lem, a l l a c c e p t a b l e models are not l i n e a r l y c l o s e to each o t h e r . Parker (1981) has c o n s t r u c t e d models for M-T data compr i sed of a set of D i r a c - d e l t a f u n c t i o n c o n d u c t a n c e s , where a c o n t i n u o u s model had a l s o been found a c c e p t a b l e . The i m p l i c a t i o n t h a t t h i s c a r r i e s over to D . C . r e s i s t i v i t y i s tha t the r e a l e a r t h cannot be guaranteed l i n e a r l y c l o s e to a c e r t a i n c o n s t r u c t e d model . As a r e s u l t , model averages may not convey any i n f o r m a t i o n about the r e a l e a r t h s t r u c t u r e . I f the c o n s t r u c t e d models m,(z) and m 0 ( z ) f i t the d a t a , but are not l i n e a r l y c l o s e , the q u a d r a t i c term o £ ( m , - m 0 ) 2 ] w i l l not be n e g l i g i b l e compared to E i ( m , ) - E i ( m 0 ) , and e q u a t i o n 4.2 w i l l be i n v a l i d . T h e r e f o r e , d i f f e r e n t model averages w i l l g e n e r a l l y r e s u l t when the models are seen through a v e r a g i n g f u n c t i o n s formed from e i t h e r set of k e r n e l s . In g e n e r a l , ^ A ( z , z 0 ) m , ( z ) dz / Jkiz,z0)m0{z) dz (4 .5) where e i ther" ° £ a £ G i ( m 0 , z ) = A ( z , z 0 ) or tai G i ( m , , z ) = A ( z , z 0 ) In these c a s e s , l i n e a r i z e d a p p r a i s a l may not r e v e a l what f e a t u r e s a c o n s t r u c t e d model has in common wi th the r e a l , e a r t h . Used w i t h a d i f f e r e n t i n t e n t though, l i n e a r i z e d a p p r a i s a l 101 can f u r n i s h some u s e f u l i n f o r m a t i o n . I t can t e l l us what f e a t u r e s on a c o n s t r u c t e d model are not guaranteed to e x i s t in the r e a l e a r t h . Thus , by l i n e a r l y a p p r a i s i n g s p e c i f i c f e a t u r e s on a c o n s t r u c t e d model , we can determine whether or not they are common to a l l l i n e a r l y c l o s e models . I f they are n o t , then they are two s tages removed from be ing f e a t u r e s in the r e a l e a r t h , and shou ld not be p i v o t a l in any p h y s i c a l i n t e r p r e t a t i o n of the model . F e a t u r e s which do pass t h i s t e s t can be c o n s i d e r e d more r e l i a b l e , a l t h o u g h there i s s t i l l no guarantee that they e x i s t in the r e a l e a r t h . T h i s approach w i l l be taken h e r e . 4-2 .2 D e l t a n e s s C r i t e r i o n For a p e r f e c t l y r e s o l v e d model , the a v e r a g i n g f u n c t i o n s would be d e l t a f u n c t i o n s c e n t r e d over the depths of i n t e r e s t , and c o n s t r u c t e d model v a l u e s would be r e t u r n e d as the average model . Owing to the l i m i t e d number of d a t a , however, and the r e s o l v i n g power of the F r e c h e t k e r n e l s , o n l y a p p r o x i m a t i o n s to d e l t a f u n c t i o n s can be a c h i e v e d . Of ten t o o , the a v e r a g i n g f u n c t i o n s cannot be c e n t r e d over the r e g i o n of i n t e r e s t . So i n p r a c t i c e , c o n s t r u c t i n g an a v e r a g i n g f u n c t i o n amounts to f i n d i n g the set of c o e f f i c i e n t s w h i c h , when m u l t i p l i e d by the k e r n e l s and.summed t o g e t h e r , g i v e s the best a v e r a g i n g f u n c t i o n p o s s i b l e under the c i r c u m s t a n c e s — s u b j e c t to the c o n d i t i o n t h a t the area under i t i s u n i t y . S e v e r a l c r i t e r i a e x i s t f or comparing an a v e r a g i n g f u n c t i o n to a t r u e d e l t a f u n c t i o n , and the a l p h a c o e f f i c i e n t s are chosen so as to min imize the d i f f e r e n c e between the two 1 02 a c c o r d i n g to one of these c r i t e r i a . In the Spread c r i t e r i o n of Backus and G i l b e r t , an a v e r a g i n g f u n c t i o n ' s narrowness , s u b j e c t to the u n i t area c o n d i t i o n , measures i t s s i m i l a r i t y to a d e l t a f u n c t i o n . To 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 w i th t h i s c r i t e r i o n , the w i d t h i s min imized wh i l e the area i s c o n s t r a i n e d to be one. F u n c t i o n s o b t a i n e d in t h i s way g e n e r a l l y have no n e g a t i v e s i d e l o b e s , u n l i k e those o b t a i n e d u s i n g other c r i t e r i a . The spread of an a v e r a g i n g f u n c t i o n i s d e f i n e d a s : S(A) = 12 r ( z - z 0 ) 2 A ( z , z 0 ) 2 d z (4 .7) o I f A ( z , z 0 ) i s a b o x - c a r f u n c t i o n , of h e i g h t 1/w and width w, the spread i s the'n a l s o w. 2.. r by symmetry, w = 24 ( z - z n ) 2 1 dz (4 .8) w2 The ( z - z 0 ) 2 term h e a v i l y weights any area under the a v e r a g i n g f u n c t i o n away from i t s in tended c e n t r e . Thus , a sharp a v e r a g i n g f u n c t i o n d i s p l a c e d from i t s in tended p o s i t i o n w i l l have a l a r g e s p r e a d , j u s t as a wide one w i l l . In the F i r s t D i r i c h l e t c r i t e r i o n , the squared d i f f e r e n c e between the a v e r a g i n g f u n c t i o n and a t r u e d e l t a f u n c t i o n c e n t r e d at z 0 i s taken as the measure of d e l t a n e s s . The a l p h a c o e f f i c i e n t s which minimize the two-norm of t h i s d i f f e r e n c e w i l l c o n t r u c t the o p t i m a l a v e r a g i n g f u n c t i o n a c c o r d i n g to the F i r s t D i r i c h l e t c r i t e r i o n . 1 03 [ l a ; G i ( z ) - 6 ( z - z 0 ) ] 2 dz (4 .9) M i n i m i z i n g t h i s d i f f e r e n c e would seem to be a more d i r e c t way of c o n s t r u c t i n g an a v e r a g i n g f u n c t i o n than u s i n g the Spread c r i t e r i o n , but i t o f t en produces f u n c t i o n s w i th n e g a t i v e s i d e l o b e s around the main peak (see Backus and G i l b e r t , 1968, Appendix B, f i g u r e 4 . 1 ) . An a v e r a g i n g f u n c t i o n wi thout n e g a t i v e s i d e l o b e s i s more e a s i l y i n t e r p r e t e d , as i t graphs the r e l a t i v e importance of the model at every depth to the average mode l . F u r t h e r m o r e , n e g a t i v e s i d e l o b e s c o u l d c o n c e i v a b l y produce a n e g a t i v e average model , which in the case of r e s i s t i v i t y would convey no meaning o t h e r than poor r e s o l u t i o n . O f f s e t t i n g t h i s d i sadvantage though, i s the e m p i r i c a l f i n d i n g that F i r s t 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 are o f t en narrower than Spread 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 . So, the c h o i c e of d e l t a n e s s c r i t e r i o n used i s t r u l y a s u b j e c t i v e one. Another d e l t a n e s s c r i t e r i o n seeks to min imize the d i f f e r e n c e between a Heavys ide f u n c t i o n set at z 0 , and the i n d e f i n i t e i n t e g r a l of A ( z , z 0 ) . T h i s i s the Second D i r i c h l e t c r i t e r i o n . T h e o r e t i c a l l y , t h e r e are as many c r i t e r i a as there are ways of a p p r o x i m a t i n g d e l t a f u n c t i o n s , and they c o u l d a l l be employed to 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 . Only the Spread c r i t e r i o n w i l l be used h e r e , however, in our a p p r a i s a l of the Schlumberger models . 1 04 4-2 .3 The T r a d e o f f Between R e s o l u t i o n and A c c u r a c y The presence of e r r o r s in the data makes a p p r a i s a l more c o m p l i c a t e d than has been d e s c r i b e d thus f a r . Sources of e r r o r , which are d i s c u s s e d i n s e c t i o n 2 . 3 , are e v e r p r e s e n t i n g e o p h y s i c a l exper iments so a method f o r accomodat ing the e r r o r s which a r i s e must be b u i l t i n t o l i n e a r i z e d a p p r a i s a l . In t h e i r 1970 paper , Backus and G i l b e r t a l l o w for the presence of e r r o r s i n the d a t a and deve lop the idea of the t r a d e o f f c u r v e . They show that when a model i s b e i n g a p p r a i s e d , some s a c r i f i c e in the r e s o l u t i o n of a model average w i l l o f t en be d e s i r a b l e to b r i n g down the v a r i a n c e a s s o c i a t e d wi th i t . I f e r r o r s A E i , i=1 ,n were made in measur ing the d a t a , and a model which f i t those erroneous d a t a , w h i l e be,ing l i n e a r l y c l o s e to the r e a l model was c o n s t r u c t e d , then the e r r o r i n t r o d u c e d to the model average i s : A<m e ( z 0 )> = Z d ; G i [m 0 ( z ) - m e ( z ) J dz , (4.10) The e r r o r in a model average i s thus the a combinat ion of the measurement e r r o r s . A<m e ( z 0 )> = Z a ^ A E i s i n c e A E i = G i [m0 ( z ) - m e ( z ) ] d z . (4.11) Of course the data e r r o r s themselves are never known, a l t h o u g h 105 through repeated measurements, t h e i r s t a t i s t i c s may be l e a r n e d . I t w i l l be assumed that they are random, and f o l l o w a Gauss ian d i s t r i b u t i o n with zero mean. I t w i l l be f u r t h e r assumed that the e r r o r s in the i 1 t h datum are independent of those in the j ' t h datum. We are assuming: 1. The e r r o r s have zero mean; A E i = 0, f or i = 1 , n . 2. The c o v a r i a n c e matr ix of A E i and AEj i s zero except on the t r a c e . So, E i j = A E i AEj = 6,j a\ where a\ i s the v a r i a n c e of the i ' t h measurement. Knowing the c o v a r i a n c e m a t r i x , the v a r i a n c e in the model average can now be e x p r e s s e d . v a r < m c ( z 0 ) > = Z a; a; E;; (4.12) Thus , both v a r i a n c e in the d a t a , and the a l p h a c o e f f i c i e n t s de termine the v a r i a n c e in the model a v e r a g e . I f the c o e f f i c i e n t s which maximize d e l t a n e s s are chosen wi thout r e g a r d to the v a r i a n c e which they d i c t a t e , the v a r i a n c e may be so l a r g e tha t the model average i s h i g h l y u n c e r t a i n . In order to reduce the v a r i a n c e , we must be p r e p a r e d to accept a l e s s d e l t a - l i k e a v e r a g i n g f u n c t i o n . R e s o l u t i o n , as c o n t r o l l e d by the a v e r a g i n g f u n c t i o n , and a c c u r a c y , as expressed by the v a r i a n c e , are m u t u a l l y a n t a g o n i s t i c q u a l i t i e s in the a p p r a i s a l of mode l s . 106 I f hav ing a smal l v a r i a n c e i s p r e f e r e d to hav ing good r e s o l u t i o n , the c o e f f i c i e n t s which minimize v a r i a n c e should be found , and i f good r e s o l u t i o n i s d e s i r e d , then the c o e f f i c i e n t s which maximize d e l t a n e s s shou ld be found. However, one q u a l i t y w i l l a lways be s a c r i f i c e d for the sake of the o t h e r . There are p a r a l l e l s between t h i s phenomenon and the He i senburg u n c e r t a i n t y p r i n c i p l e . Backus and G i l b e r t ' s t r a d e o f f curve r e l a t e s the a v e r a g i n g f u n c t i o n ' s width to the v a r i a n c e in a model average at a p a r t i c u l a r depth (Backus and G i l b e r t , 1970). The curve i s p l o t t e d on axes of model v a r i a n c e , and s p r e a d . To c o n s t r u c t the t r a d e o f f c u r v e at the depth z 0 , spread and v a r i a n c e are weighted a c c o r d i n g to whichever we are more d e s i r o u s of m i n i m i z i n g and both m i n i m i z e d . The area under the a v e r a g i n g f u n c t i o n must be c o n s t r a i n e d to be one to exc lude the ai=0 s o l u t i o n for i = 1 , n , which g i v e s zero w i d t h , and zero v a r i a n c e . T h i s c o n d i t i o n i s added through a Lagrange m u l t i p l i e r X. The r e s u l t i n g o b j e c t i v e f u n c t i o n i s then min imized and the r e s u l t i n g spread and v a r i a n c e v a l u e s are c o o r d i n a t e s of a p o i n t on the c u r v e . By chang ing the w e i g h t i n g c o n t i n u o u s l y the whole curve i s swept o u t . T h i s i s done by m u l t i p l y i n g the spread term by c o s 0 , and the v a r i a n c e term by s i n 0 , and v a r y i n g 9 from 0 to i r / 2 . The o b j e c t i v e f u n c t i o n i s then: \p = spread (cost?) + var<m(z 0 )>(sin#) + X u n i m o d u l a r i t y c o n s t r a i n t (4.13) Spread or v a r i a n c e can be m i n i m i z e d i n d e p e n d e n t l y by s e t t i n g 6 to 0, or IT/2 r e s p e c t i v e l y . Most o f t en though, some t r a d e o f f 1 07 of width i s made i n order to b r i n g the v a r i a n c e down to an a c c e p t a b l e l e v e l when c o n s t r u c t i n g an average model . For a v e r a g i n g f u n c t i o n s c o n s t r u c t e d u s i n g the spread c r i t e r i o n , the o b j e c t i v e f u n c t i o n i s : \p = cose Zaia-T;] + W sintf La(a;E;; + X ( 1 - Ia; /Gi (z )dz) t,'r-i J i.j'i J J in J where _ » (4.14) Vi} = 12 / ( z - z 0 ) 2 G i ( z ) G j ( z ) d z , and X i s the Lagrange m u l t i p l i e r , and W i s a c o n s t a n t which h e l p s t h e t a to vary u n i f o r m l y a l o n g the t r a d e o f f c u r v e . The a c t u a l m i n i m i z a t i o n i s done by d i f f e r e n t i a t i n g \p w i th r e s p e c t to al and X to f i n d the minumum wi th r e s p e c t to the a l p h a c o e f f i c i e n t s and X. Now the f e a t u r e s in the Schlumberger c o n s t r u c t i o n s which our h e a t - s o u r c e - t o - t h e - n o r t h h y p o t h e s i s was based upon w i l l be a p p r a i s e d . 4-3 L i n e a r i z e d A p p r a i s a l of some Schlumberger C o n s t r u c t i o n s In Chapter 3, r e s i s t i v i t y models were c o n s t r u c t e d from data taken i n the Anahim V o l c a n i c B e l t and temperature i n f o r m a t i o n was i n t e r p r e t e d from them. Three g e o l o g i c a l -geothermal models for e x p l a i n i n g the temperature d i s t r i b u t i o n were i n f e r r e d from the models and d e s c r i b e d . The most p r o b a b l e model c a l l e d f o r a heat source to the n o r t h of the area where the soundings were t a k e n . E v i d e n c e f o r the heat source to the n o r t h h y p o t h e s i s comes from a g e n e r a l t r e n d of 108 d e c r e a s i n g r e s i s t i v i t y to the n o r t h . T h i s shows up on a l l s i x models , but on ly in the northernmost f o u r , a l l in the Kaza Group , i s i t i n t e r p r e t e d as be ing very s i g n i f i c a n t . ( f i g u r e 4.1) The low r e s i s t i v i t y zones which determine t h i s t r e n d w i l l be l i n e a r l y a p p r a i s e d . Lowest R e s i s t i v i t y Values 6 -cr 4 ) ( VES 2 3 . — < - - NorTh South —> | . 1 j ; ( H 0. 3. 6. 9. 12. 15. N o r t h - S o u t h P r o j e c t i o n (km) F i g u r e 4.1 N o r t h - s o u t h r e s i s t i v i t y g r a d i e n t o f the c o n s t r u c t e d models i n the Kaza G r o u p . The e x i s t a n c e o f t h i s g r a d i e n t depends h e a v i l y upon the r e s i s t i v i t y low on the VES 2 m o d e l . It t h e r e f o r e must be a p p r a i s e d . The model most important to the h y p o t h e s i z e d r e s i s t i v i t y g r a d i e n t i s VES 2. Without i t s low r e s i s t i v i t y zone, there would be scant e v i d e n c e for a t r e n d at a l l . I t i s t h e r e f o r e important to the h y p o t h e s i s that t h i s zone shou ld pass l i n e a r i z e d a p p r a i s a l by be ing p r e s e n t on average models . F i r s t , a t r a d e o f f curve w i l l be c o n s t r u c t e d in t h i s low r e s i s t i v i t y zone to q u a l i f y the i n t e r r e l a t i o n of spread and v a r i a n c e t h e r e . Then , the maximum a c c e p t a b l e v a r i a n c e for r e s o l v i n g t h i s low zone w i l l be de termined from the c o n s t r u c t e d model ( f i g u r e 4 . 3 a ) . An average model w i l l be VES 6 X VES 3 X X VES I i u y c o n s t r u c t e d by p l o t t i n g model a v e r a g e s , each hav ing that v a r i a n c e , at many d e p t h s . Each p o i n t on t h i s model w i l l have a known s t a n d a r d d e v i a t i o n , and i f the low r e s i s t i v i t y zone cannot be n u l l i f i e d by two 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 , i t w i l l be c a l l e d r e s o l v a b l e . The t r a d e o f f curve was c o n s t r u c t e d at the depth of 150 meters ( f i g u r e 4 . 2 ) . I t decreases m o n o t o n i c a l l y from a v a r i a n c e of 63 w i t h a spread of 81 meters a t 6=0.0, to a v a r i a n c e of 0 .0007, and a spread of 377 meters a t ir/2. The model average i n c r e a s e s m o n o t o n i c a l l y as 6 —> n/2 from l n ( p ) = 4 . 0 to l n ( p ) = 4 . 8 . As the a v e r a g i n g f u n c t i o n ge t s w i d e r , i t samples the f l a n k s of the low r e s i s t i v i t y zone as w e l l . F i g u r e 4 .2 T r a d e o f f c u r v e and ave rage model f o r VES 2 a t 1SOm. The t r a d e o f f cu r ve does not i n c l u d e the p o i n t s at 9= ff/2 because the s p r e a d i s so l a r g e . I n s t ead i t shows the c u r v e f o r 0.0< 8 <1.55 . 1 1 0 The a m p l i t u d e of the r e s i s t i v i t y low on the VES 2 model i s at l e a s t one l o g a r i t h m i c u n i t . T h e r e f o r e to r i g i r o u s l y a p p r a i s e i t , i t shou ld be u n e q u i v o c a l l y present on the average model hav ing a two s tandard d e v i a t i o n l e v e l of 0.5. T h i s means a v a r i a n c e of 0.0625 for every p o i n t . When the model i s computed, we see that t h i s i s borne o u t . Average Model: c r -0 .25 !0 a 10' 102 10' 104 Depth (meters] 10£ Spread vs. Depth 10' 102 IO3 Depth (meters) IO4 10' T h i r d I t e r a t i o n 10: 10' 10" Depth (meters) 104 Average Model; 9=0.0 1Q( 10' 10' 10" Depth (meters) F i g u r e 4 .3 a) Average models f o r VES2. Each p o i n t has a s t a n d a r d d e v i a t i o n of 0 .25 a s s o c i a t e d w i t h i t . Even at the two s t a n d a r d d e v i a t i o n l e v e l , the low zone i s r e s o l v a b l e . )b Sp read as a f u n c t i o n o f dep th f o r each of the p o i n t s on the ave rage mode l . N o t i c e how i t l e aves the s = z l i n e at a round 1000 m. )c S m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n s f o r VES 2. )d Ave rage model c o n s t r u c t e d a t 9 = O.O. Each p o i n t has an unknown v a r i a n c e . 104 111 The low r e s i s t i v i t y zone i s present on the average model , even i f the e r r o r s were at the two s t a n d a r d d e v i a t i o n l e v e l ( f i g u r e 4 . 3 a ) . The width of the Spread 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 at t h i s v a r i a n c e i s a l s o shown as a f u n c t i o n of d e p t h . T h i s p l o t can be used to d e c i d e below what depth the model has l o s t i t s r e s o l v a b i l i t y and c o n t a i n s no more u s e f u l i n f o r m a t i o n . C e r t a i n l y when the spread i s much g r e a t e r than the depth at which i t i s c a l c u l a t e d , the a v e r a g i n g f u n c t i o n i s not l o c a l i z i n g i n f o r m a t i o n , and the model average looses i t s meaning. The l i n e where the spread equa l s the depth has been p l o t t e d for c o m p a r i s o n . Comparing the a=.25 model to the 0=0 model , l i t t l e r e s o l u t i o n appears to have been l o s t i n s l i d i n g down the t r a d e o f f curve to t h i s p o i n t . More i m p o r a n t l y , the s t a n d a r d d e v i a t i o n of each p o i n t on the cr=.25 curve i s known, and the low r e s i s t i v i t y zone i s the major f e a t u r e on both models . I t i s i n t e r e s t i n g to note tha t the t h i r d d i p i n r e s i s t i v i t y shown on the s m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n for VES 2 ( f i g u r e 4.3a) i s not i n c l u d e d in the a=0.25 average model , and i s o n l y h i n t e d at at i n the 9=0.0 a v e r a g e . I t i s t h e r e f o r e not r e s o l v a b l e and i s not common to the l i n e a r l y c l o s e models . At the southern end of the Kaza Group t h e r e were t h r e e c l o s e l y spaced Schlumberger sound ings , VES 1, VES 3, and VES 6. H e r e , a p p r a i s a l i s l e s s important to the heat h y p o t h e s i s for two r e a s o n s . F i r s t l y , the common r e s i s t i v i t y v a l u e below 50 m i s s u b s t a n t i a t e d by three models , and s e c o n d l y , the VES 6 model i s e s s e n t i a l l y a h a l f - s p a c e of c o n s t a n t r e s i s t i v i t y , so 1 1 2 that data p o i n t on f i g u r e 4.1 does not r e l y upon r e s o l u t i o n for i t s r e s i s t i v i t y v a l u e . N e v e r t h e l e s s , the VES 1 model c o n s t r u c t e d from a h a l f - s p a c e w i l l be a p p r a i s e d to determine the r e s o l v a b i l i t y of i t s major low. A two s t a n d a r d d e v i a t i o n l e v e l of 0.6 at every p o i n t would s u f f i c e for r e s o l v i n g t h i s major f e a t u r e . T h u s , model averages hav ing a v a r i a n c e of 0.09 were computed and p l o t t e d in f i g u r e 4 . 4 . The average model o b t a i n e d v e r i f i e s the r e s o l v a b i l i t y of the t h i s low r e s i s t i v i t y zone . B Average rlode.l: <r=0.3 fl Spread vs. Depth 10° 10' 10' 10* 10* 10° 10' 10' 103 10* Depth (meters.) Depth (metersJ Depth (meters) F i g u r e 4 .4 a) Average model f o r VES 1. The low r e s i s t i v i t y zone has an a m p l i -ude g r e a t e r than the 2 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 . The two s t a n d a r d d e v i a t i o n 1evel i s 0 . 6 . b) Sp read o f the a v e r a g i n g f u n c t i o n s . T h i s shows tha t good a v e r a g i n g f u n c t i o n s can be c o n s t r u c t e d down to 500 meters d e p t h . c ) S m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n f o r VES 1 from a h a l f - s p a c e . 1 1 3 The s t r u c t u r e s which determine the r e s i s t i v i t y g r a d i e n t have been v e r i f i e d as r e s o l v a b l e . The low v a l v e s on VES 1. are the major f e a t u r e on the average models , and i n VES 2, the low i s a l s o s t r o n g l y p r e s e n t , so we have not l o s t any e v i d e n c e for the r e s i s t i v i t y g r a d i e n t , or the h e a t - s o u r c e - t o - t h e - n o r t h e x p l a n a t i o n for i t . A l t h o u g h the most p r o b a b l e model i n t e r p r e t a t i o n does not r e l y on VES 5, i t was a p p r a i s e d t o o . The i n t e r e s t i n g outcome was that the average models showed a s imple s t r u c t u r e h a v i n g f a i r l y h i g h r e s i s t i v i t y down to about 200 meters . T h i s i s c l e a r l y an average of the c o n s t r u c t i o n s o b t a i n e d u s i n g the s m a l l e s t and f l a t t e s t p e r t u r b a t i o n s . The s m a l l e s t p e r t u r b a t i o n model had the decrease at about 200 meters , but i t showed much o s c i l l a t i o n , and the f l a t t e s t p e r t u r b a t i o n model had on ly one d e c r e a s e , but i t was somewhat deeper . T h i r d I t e r a t i o n Thi rd I t e r a t i o n io° io' io 2 io 3 io- ioD io' io 2 io 3 io 4 Depth (meters] Depth (meters] F i g u r e 4 .5 a) S m a l l e s t p e r t u r b a t i o n c o n s t r u c t i o n f o r VES 5. b) F l a t t e s t p e r t u r b a t i o n c o n s t r u c t i o n f o r VES 5. c ) Ave rage model w i t h cr = 0 .25 f o r VES 5. 1 1 4 Average riddel: cr—0.25 1.0° 10' 102 IO3 Depth (meters) 104 L i n e a r i z e d a p p r a i s a l does a good job of what i t p u r p o r t s to do , and i t i s a worthwhi le e x e r c i z e a f t e r an i n t e r p r e t a t i o n has been made. In g e n e r a l l i n e a r l y a p p r a i s i n g models w i l l t each an i n t e r p r e t e r not to p l a c e too much importance in whatever s m a l l s c a l e f e a t u r e s a model may have . I n s t e a d , on ly the gros s s c a l e f e a t u r e s shou ld be c o n s i d e r e d r e a l i s t i c , and i n t e r p r e t e d . One would q u i c k l y l e a r n what i s r e s o l v a b l e , and what i s n o t , and l i n e a r i z e d a p p r a i s a l would become a f o r m a l i t y in i n t e r p r e t a t i o n . 4.4 Towards Comprehensive A p p r a i s a l In the n o n l i n e a r problems such as D . C . r e s i s t i v i t y and M - T , the crux of the problem wi th a p p r a i s a l i s tha t models c o n s t r c t e d from a few data are h i g h l y nonunique . The degree of nonuniqueness i s so great tha t a c c e p t a b l e models very d i f f e r e n t from those converged upon by l i n e a r i z e d c o n s t r u c t i o n may f i t the d a t a . Such models would not be l i n e a r l y c l o s e to a l i n e a r i z e d c o n s t r u c t i o n model and would t h e r e f o r e be beyond the reach of l i n e a r i z e d a p p r a i s a l . 1 1 5 T h i s i s w e l l known to be the problem, and some pseudo-s o l u t i o n s to i t have been c o n c e i v e d , such as Monte C a r l o m o d e l l i n g , and s imply comparing models o b t a i n e d by d i f f e r e n t c o n s t r u c t i o n methods. N e i t h e r of these are r e a l l y the s o l u t i o n because they cannot guarantee that they have swept out the e n t i r e space of a c c e p t a b l e models in r e t u r n i n g those few. A more s y s t e m a t i c approach has been deve loped by Oldenburg (1983) where s m a l l s e c t i o n s of the model are maximized and min imized to p r o v i d e i n f o r m a t i o n about the average va lue of the model . T h i s i s expres sed by the f u n n e l f u n c t i o n c u r v e s . He a l s o maximizes and min imizes the i n d e f i n i t e i n t e g r a l of the model thus c r e a t i n g the g l o b a l bounds of the i n d e f i n i t e i n t e g r a l of the model . I n t u i t i v e l y these methods seem to reach much f a r t h e r i n t o model space than l i n e a r i z e d a p p r a i s a l , but problems w i t h e v a s i v e r e g i o n s p o s s i b l y h a r b o u r i n g an a c c e p t a b l e model s t i l l r emain . One way to e n v i s i o n the r e l a t i o n s h i p between data and models and nonuniqueness i s on a two d i m e n s i o n a l c o o r d i n a t e system wi th the axes of E i ( m ( z ) ) , and m(z ) , where every p o i n t on the m(z) a x i s r e p r e s e n t s a model , and wi th p o i n t s p r o x i m i t y to each o ther r e p r e s e n t a t i v e of t h e i r degree of s i m i l a r i t y . 1 1 6 Ei (m(zj ) E i (meas) <—m (z ) — • F i g u r e 4 .6 A s c e m a t i c r e p r e s e n t a t i o n o f nonun iqueness and a p p r a i s a l . L i n e a r i z e d a p p r a i s a l (a) w i l l o n l y a p p r a i s e models in the l i n e a r r e g i o n a round the c o n s t r u c t e d mode l . More o f model space i s s canned in c o n s t r u c t i n g g l o b a l bound c u r v e s ( b ) , but one c anno t be c e r t a i n t ha t i n a c c e s s i b l e models such as *2 do not e x i s t . The f u n c t i o n a l E i ( m ( z ) ) i s p l o t t e d out by p r e d i c t i n g the i ' t h datum from every model in model space . Then at every p o i n t a l o n g the m(z) a x i s where E i (m(z ) )=Ei (meas ) for i=1,n there e x i s t s an a c c e p t a b l e model . Now, the a p p r a i s a l problem i s to f i n d and compare a l l the models m(z) which p r e d i c t the measured data . E i ( m e a s ) , f o r i=1,n. L i n e a r i z e d a p p r a i s a l scans o n l y the r e g i o n around m 0 ( z ) where the r e l a t i o n s h i p between E i and m 0 ( z ) i s l i n e a r but cannot guarantee a g a i n s t the e x i s t e n c e of the models *1 or *2. The methods of Oldenburg may scan much f a r t h e r and f i n d *1 but s t i l l may miss the model at *2. Perhaps they o f f e r the most comprehensive a p p r a i s a l that w i l l be p o s s i b l e . 1 1 7 C o n c l u s i o n In the p r e c e e d i n g c h a p t e r s an e f f i c i e n t and robust method for i n v e r t i n g d i r e c t c u r r e n t r e s i s t i v i t y measurements has been p r e s e n t e d . Backus and G i l b e r t ' s l i n e a r i z e d i n v e r s e theory has been a p p l i e d in e s s e n t i a l l y the same way as Oldenburg (1978) wi th the o n e - d i m e n s i o n a l s t r u c t u r e now approximated by a f i n i t e number of l a y e r s , and the Hankel t rans forms computed u s i n g the c o n v o l u t i o n a l f i l t e r method of Gosh (1971). One-d i m e n s i o n a l models can g e n e r a l l y be c o n s t r u c t e d in three or fewer i t e r a t i o n s which take 35 seconds on an Amdhal 470 V / 8 . The c o n s t r u c t i o n program has been shown to be f a i r l y i n s e n s i t i v e to the s t a r t i n g model chosen and to the presence of moderate e r r o r s in the d a t a . W i t h i n the ranges t e s t e d , n e i t h e r f a c t o r prevented convergence of the program, and v a r y i n g the s t a r t i n g model seemed not to a f f e c t the c o n s t r u c t e d models r e p r e s e n t a t i o n of the h y p o t h e t i c a l s t r u c t u r e . On the b a s i s of these r e s u l t s , an e f f i c i e n t , r e l i a b l e , and robust method for i n v e r t i n g D . C . r e s i s t i v i t y measurements has been p r e s e n t e d . A l t h o u g h models may be c o n s t r u c t e d more q u i c k l y u s i n g the d i r e c t method of Cohen and Yu (1981), the advantage of l i n e a r i z e d c o n s t r u c t i o n remains in the a spec t of a p p r a i s a l . With l i n e a r i z e d c o n s t r u c t i o n , the k e r n e l s f or l i n e a r i z e d a p p r a i s a l are a u t o m a t i c a l l y a v a i l a b l e once the model has c o n v e r g e d . In cases where the c o n s t r u c t e d model i s l i n e a r l y c l o s e to the t r u e e a r t h model , unique model averages can be o b t a i n e d by combining the k e r n e l s i n t o an a v e r a g i n g f u n c t i o n 1 18 and forming i t s i n n e r product wi th the model . L i n e a r c l o s e n e s s to the t r u e s t r u c t u r e cannot be guaranteed wi th s m a l l data s e t s , but because there i s an uniqueness theorem, the l i k e l i h o o d of t h i s i n c r e a s e s wi th the number of a c c u r a t e d a t a . Not s u r p r i s i n g l y , the i m p l i c a t i o n that t h i s c a r r i e s over to f i e l d exper iments i s tha t a l a r g e q u a n t i t y of good g u a l i t y data i s d e s i r a b l e f o r r e s o l v i n g the u n d e r l y i n g r e s i s t i v i t y s t r u c t u r e . There i s no o ther way to conquer the nonuniqueness p r o b l e m . These two f a c t o r s were taken i n t o c o n s i d e r a t i o n i n the Anahim r e s i s t i v i t y exper iment . The data d e n s i t y was h i g h ; an expans ion f a c t o r of 1.24 was used i n s t e a d of the u s u a l 1.41. So for the a r r a y s i z e s used , a r e l a t i v e l y l a r g e q u a n t i t y of data was t a k e n , and the measurement e r r o r s on most soundings were l e s s than 3 percent (see f i g u r e 3 . 3 ) . I f the topography was more subdued, and there were fewer i n d i c a t i o n s of l a t e r a l i n h o m o g e n i t i e s , both g e o l o g i c a l l y and in the d a t a , then a g r e a t d e a l of c o n f i d e n c e c o u l d be p l a c e d i n the mode l s . The u n d e r l y i n g rock groups however, are metamorphic , and v a r y in c o m p o s i t i o n thus d e p r e c i a t i n g the v a l i d i t y of the one-d i m e n s i o n a l a p p r o x i m a t i o n . F a u l t s , v e r t i c a l c o n t a c t s , and nonplanar topography a l s o v i o l a t e t h i s a p p r o x i m a t i o n . N e v e r t h e l e s s , i t i s f e l t that some u s e f u l i n f o r m a t i o n has been o b t a i n e d from the c o n s t r u c t e d models — e s p e c i a l l y when they are i n t e r p r e t e d as a group . I t i s immediate ly apparent tha t the models d i v i d e the soundings i n t o two g r o u p s . The two southern soundings produce models w i th r e s t i v i t i e s of 800-900 ohm-meters in the f i r s t 1 19 hundred meters d e p t h , whi l e models w i th n e a r - s u r f a c e r e s i s t i v i t i e s a v e r a g i n g 200 ohm-meters are c o n s t r u c t e d from the four n o r t h e r n s o u n d i n g s . T h i s i s i n d i c a t i v e of the change in rock types which a l s o d i v i d e s the soundings i n t o the same two g r o u p s . When the four n o r t h e r n soundings in the Kaza Metasediments are examined as a group , a t r e n d of d e c r e a s i n g average r e s i s t i v i t i e s i s seen. There are two p o s s i b l e e x p l a n a t i o n s for t h i s . I t c o u l d be the r e s u l t of some nonrandom c o m p o s i t i o n a l v a r i a t i o n s w i t h i n the Kaza group o r , of a temperature g r a d i e n t i n c r e a s i n g to the n o r t h . The temperature g r a d i e n t was chosen as the more p r o b a b l e because i t i s a l s o i n d i c a t e d by the metamorphic g r a d a t i o n , f u r t h e r m o r e , the two models in the Shuswap group to the south show the same t r e n d . A heat source to the n o r t h was i n f e r r e d from t h i s g r a d i e n t , and i s c o m p a t i b l e wi th both of the major t h e o r i e s for the g e n e s i s of the Anahim V o l c a n i c b e l t : the mantle h o t -spot t h e o r y , and the e d g e - e f f e c t h y p o t h e s i s . I t i s f e l t that ev idence which would support one theory over the o t h e r c o u l d be o b t a i n e d through e i t h e r a more e x t e n s i v e r e s i s t i v i t y exper iment , or from a net of heat flow measurements. A h o t -spot heat source would produce a temperature d i s t r i b u t i o n wi th r a d i a l symmetry, wh i l e a l i n e source such as the e d g e - e f f e c t h y p o t h e s i s c a l l s f o r , would have l a t e r a l l y symmetric i s o t h e r m s . I f such an experiment s t r o n g l y i n d i c a t e d e i t h e r heat source geometry, o t h e r parameters such as d e p t h , and power output c o u l d be e s t i m a t e d through mathemat ica l 1 20 m o d e l l i n g . The thermal c o n d u c t i v i t y of the Kaza Metasediments would have to be known, and i f r e s i s t i v i t y measurements were be ing used , the r e s i s t i v i t y of the Kaza rock would have to be c a l i b r a t e d in terms of t e m p e r a t u r e . In any event , a l a r g e amount of good q u a l i t y da ta would be r e q u i r e d . 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Stacey , T i p p e r , Watson, Y o r a t h , 1 25 Appendix A E v a l u a t i o n of the Modal C o e f f i c i e n t A(A) S e p a r a t i n g the v a r i a b l e s in the c o n s e r v a t i o n of c u r r e n t equat ion r e s u l t e d i n the f o l l o w i n g e x p r e s s i o n for the p o t e n t i a l . V ( r , z ) i s a c o n t i n u o u s B e s s e l s e r i e s e x p a n s i o n , or Hankel T r a n s f o r m . The modal c o e f f i c i e n t s , A(X) must be e v a l u a t e d . U s i n g a c o n d i t i o n i m p l i e d by the c o n s e r v a t i o n o f c u r r e n t , Oldenburg (1978) c a l c u l a t e d a c o m p u t a t i o n a l l y economica l e x p r e s s i o n for A ( X ) . S p e c i f i c a l l y , the c o n d i t i o n used i s t h a t the' s u r f a c e i n t e g r a l of c u r r e n t f l u x j , through an i n f i n i t e l y long c y l i n d e r c e n t e r e d on the c u r r e n t source must equa l I , the source c u r r e n t . r V ( r , z ) = A ( X ) H ( z , X ) J 0 ( X r ) dX (A. 1 ) e I j • r 0 d 9 « d z The c u r r e n t , I , equa l s r XT r j ( r 0 , z ) • r0dd dz (A.2) dd, and dz are u n i t v e c t o r s in the 8, and z d i r e c t i o n s , and r 0 i s the r a d i u s of the c y l i n d e r . r I = 2irr. j r ( r 0 , z ) dz (A.3) And from Ohm's Law, j ( r 0 , z ) = - W ( r 0 , z ) ~pTz1 , so the r component of j ( r 0 , z ) i s j / _ ( r 0 , z ) = -1 3 V ( r 0 , z ) p (z ) T r (A.4) U s i n g the e x p r e s s i o n f o r V ( r , z ) from ( A . 1 ) , j r ( r 0 , z ) = L _ A ( X ) J 0 ' ( r 0 X ) H ( z , X ) dX p(z) (A.5) I t can be proven (Berg+McGregor,1966) that J 0 ' ( r X ) = X J , ( r X ) r j r ( r 0 , z ) = 1 p(z ) X A ( X ) J , ( r 0 X ) H ( z , X ) dX (A.6) 1 27 T h e r e f o r e , I = 27rr, X A ( X ) J , ( r 0 X ) U Q r H ( z , X ) p (z j dzdX (A.7) I t i s easy to see that i f A ( X ) = I I p{z) dz 2TTX H (Z,X) the r e s u l t a n t would be s i m p l y : oO I = I r , J , ( r 0 X ) dX, which c o n t a i n s one of the a n a l y t i c Hankie Trans forms (Gradshteyn + R y z h i k , no . 6 . 5 1 1 . 1 , p . 6 6 5 , 1965). 1 • = J , ( r 0 X ) dX (A.8) A ( X ) can be f u r t h e r s i m p l i f i e d by n o t i n g from e q u a t i o n ( 1 . 4 b ) , H ( z , X ) p(z) 1 /£r X 2 n ( z , X ) p(z) , and i n t e g r a t i n g both s i d e s from 0 to °°. r H ( z , X ) dz = 1 H ' ( z , X ) p(z) X 2 p(z) H ( z , X ) i s a m o n o t o n i c a l l y d e c r e a s i n g funct ion (Langer ,1933) , and c o n s e q u e n t l y goes to zero as z goes to i n f i n i t y . Thus r H ( z , X ) dz = -1 H ' ( 0 , X ) p(z) U ~pT0) (A.9) o a 128 s i m p l i f y i n g A(X) t o A(X) = - l P o X 2TT H' (0,X) , (A.10) where p0 i s the s u r f a c e r e s i s t i v i t y . The f i n a l form of the p o t e n t i a l e q u a t i o n i s : V ( r , z ) = - I p 0 2TT X H(z,X) J n ( r X ) dX H'(0,X) (A.1 1 ) J o 1 29 Appendix B S o l v i n g for H (Z,A ) in a L a y e r e d Medium In a c o n t i n u o u s r e s i s t i v i t y s t r u c t u r e p ( z ) , the Sturm-L i o u v i l l e e q u a t i o n , must be s o l v e d b e f o r e the p o t e n t i a l d i s t r i b u t i o n can be c a l c l a t e d . Both H(0 ,X) and H ' ( 0 , X ) are needed from X=0 to X=°° to per form the i n t e g r a l in the p o t e n t i a l d i f f e r e n c e e q u a t i o n . O b t a i n i n g H(0 ,X) and H ' ( 0 , X ) for t h i s k i n d of s t r u c t u r e i s an onerous t a s k . I f , however, p(z) can be approx imated by a f i n i t e number of l a y e r s , each i s o t r o p i c , hav ing a c o n s t a n t r e s i s t i v i t y , then e q u a t i o n (B.1) i s v a s t l y s i m p l i f i e d . The p' term i s z ero w i t h i n each l a y e r so the e q u a t i o n H" - p ' H ' + X 2 H = 0 P (B.1) X H(0 ,X) C j ( X ) dX H ' ( 0 , X ) (B.2) o H" + X 2 H = 0 (B.3) d e s c r i b e s H w i t h i n any s i n g l e l a y e r . The s o l u t i o n of t h i s i n the j ' t h l a y e r i s of c o u r s e e x p o n e n t i a l f u n c t i o n s . 1 30 -Xzj Xzj Hj ( z j , X) = A j e + ' Bj< (B.4) where z j i s t h e d e p t h from the t o p o f t h e j ' t h l a y e r . T h i s g e n e r a l s o l u t i o n a p p l i e s i n e v e r y l a y e r . The bou n d r y c o n d i t i o n s f o r the c u r r e n t and p o t e n t i a l a t l a y e r i n t e r f a c e s d e t e r m i n e t h e A and B c o e f f i c i e n t s f o r e a c h l a y e r . The l a y e r i n g scheme and v a r i a b l e s u s e d a r e i l l u s t r a t e d below. //jyj/ / / ;// H(0,X) = A, + B, V ) J / J ! 1 J I ) J / / I I 1 / / / / / / H ( z 1 ( A ) = A i e"~V B i e * 2 1 1 H(z A ) = A 2e " A Z , 4 B a e A Z ; H(ZJ , A) = Aj e~AZ''+ Bj e A ZJ r h n - l 1 .Zn-1 1 - X z n H(z n,7\) = Ane 131 In the deepest l a y e r , Bn must be zero because the c u r r e n t d e n s i t y decays to zero at great d e p t h . A c r o s s every l a y e r boundary, the p o t e n t i a l d i s t r i b u t i o n , and the v e r t i c a l component of c u r r e n t d e n s i t y v a r y c o n t i n u o u s l y ( K e l l e r and F r i s c h k n e c h t , 1977). At the bottom of the j ' t h l a y e r , the p o t e n t i a l i s c o n t i n u o u s . Vj ( r , hj ) = VJVJ (r , 0) ~ I P o / -Xhj Xhj Aje + Bje | j 0 ( X r ) dX * B 7 Vj + i ( r , 0 ) = - Ipp •a / A j « , + BJ+, ) J 0  \-A-j T BTj / (Xr) dX (B.5) E q u a t i n g the k e r n e l s , -Xhi Aje B j e Xhj A j + i + B j + 1 (B.6) The v e r t i c a l component of c u r r e n t d e n s i t y i s a l s o c o n t i n u o u s , J * ( r , h ; ) = J r ( r , 0 ) ]_ _3_V j (hj , r) J 3 V j * , (0 , r ) (B.7) - A j e -Xhj B j e Xhj = PJ ("Aj*,+Bj+,) (B.8) Combining equat ions (B.6) and (B.8) r e s u l t s in the r e l a t i o n s for Aj and Bj in terms of A j ^ . , and Bj-»-, , . 1 32 Xhj Aj = e ["(1+_pj_)/2 A j + , ( X ) + (1-pj_)/2 Bj+,(X)"| (B.9a) L Pj*l Pj*l J - X h j Bj=e R l - p j _ ) / 2 A j + 1 ( X ) + ( 1+p_j_)/2. B j + 1 (X)"| L PJ P3 J (B.9b) U s i n g the boundary c o n d i t i o n that B i n the deepest l a y e r i s z e r o , and s e t t i n g A i n tha t l a y e r to u n i t y , r enders A , and B in every o ther l a y e r up to the s u r f a c e immediate ly a v a i l a b l e through the r e c u r r s i o n r e l a t i o n s (B.9a) and ( B . 9 b ) . The c o n s t a n t to which A i n the deepest l a y e r i s equated i s of no p a r t i c u l a r s i g n i f i c a n c e (a l though every A and B i s s c a l e d by i t ) , because i t -is n o r m a l i z e d out of the f i n a l e x p r e s s i o n s for the p o t e n t i a l d i f f e r e n c e and the F r e c h e t k e r n e l . In a l a y e r e d medium, these e q u a t i o n s a r e : AVi = I p 0 2TT f A , (X) + B 1 ( X )~| L-A,(X) + B i(X)J C i ( X ) dX (B.10) o and , G i ( z j ) = po f x f Aj(X)e** J'+ B j ( X ) e * r J ~ | C i ( X ) dX TT I L A i ( X ) *"2A, (X)B , (X)+B, (X) ( B . 1 1 ) I d e a l l y , Aj and Bj are very easy to c a l c u l a t e w i t h the r e c u r r s i o n r e l a t i o n s ( B . 9 ) , but c o m p l i c a t i o n can a r i s e however, when c a l c u l a t i n g A ' s and B' s n u m e r i c a l l y . The A ' s may at some p o i n t become too l a r g e for machine s torage when 1 3 3 c a l c u l a t i n g s u c c e s s i v e l y s h a l l o w e r A ' s . L i k e w i s e Bj may become too s m a l l . T h i s problem i s s k i r t e d by s t a r t i n g the r e c u r r s i o n r e l a t i o n s at a s h a l l o w e r depth wi th the f o l l o w i n g boundary c o n d i t i o n s . A + B = 0 (B.12a) A - B = 1 (B.12b) i Appendix C F r e c h e t D i f f e r e n t i a b i l i t y of the Data F u n c t i o n a l s For the s u c c e s s f u l convergence of an i t e r a t i v e scheme, such as the one used here to c o n s t r u c t r e s i s t i v i t y models , c o n t i n u i t y i s e s s e n t i a l . I t s convergence depends upon the d e r i v a t i v e s of the f u n c t i o n a l s w i th r e s p e c t to the model v e c t o r i n d i c a t i n g the d i r e c t i o n in which to p e r t u r b a model towards f i t t i n g the d a t a . And i f the data f u n c t i o n a l s are not c o n t i n u o u s wi th r e s p e c t to t h e i r model argument, they w i l l not be everywhere d i f f e r e n t i a b l e and areas w i l l e x i s t in data space for which no a c c e p t a b l e model can be converged upon u s i n g Newton's method. T h e r e f o r e , the data f u n c t i o n a l s must be c o n t i n u o u s l y d i f f e r e n t i a b l e f o r any model v e c t o r in model space . T h i s i s the F r e c h e t d i f f e r e n t i a b i l i t y c o n d i t i o n . C-1 Development of the I d e a . I w i l l now show that the mere e x i s t e n c e of the d e r i v a t i v e of a f u n c t i o n a l (or a f u n c t i o n a l of many v a r i a b l e s ) does not guarantee c o n t i n u i t y , but tha t c o n t i n u i t y can be e s t a b l i s h e d u s i n g the d i f f e r e n t i a l f u n c t i o n . R e g r e s s i n g momentar i ly to f u n c t i o n s of a s i n g l e v a r i a b l e , the d e r i v a t i v e , f ' ( x ) , i s d e f i n e d as the l i m i t l i m f(x+X) - f (x ) ( C . I ) X->0 X where i t e x i s t s ( M i l n e ,1 9 8 0 ) . Such a f u n c t i o n i s s a i d to be 135 d i f f e r e n t i a b l e where t h i s l i m i t e x i s t s , moreover, the c o n t i n u i t y of f (x ) at x i s c e r t i f i e d by the e x i s t e n c e of the l i m i t ( C l a r k , 1972, p83 ) . For f u n c t i o n s of many v a r i a b l e s , however, c o n t i n u i t y a t (x , , x 2 , x 3 , . . . . x n ) i s not i m p l i e d by the e x i s t e n c e of the p a r t i a l d e r i v a t i v e s . An example of t h i s i s g i v e n in M i l n e (1980, p285) . i f x, or x 2 = 0 otherw i se At ( 0 , 0 ) , both p a r t i a l s e x i s t , a l t h o u g h the f u n c t i o n i s d i s c o n t i n u o u s . And as s t a t e d e a r l i e r , c o n t i n u i t y i s c r u c i a l f or the s u c c e s s f u l convergence of an i t e r a t i v e scheme such as we use . T h i s ' somewhat s u p r i s i n g r e s u l t i s a l s o t r u e for f u n c t i o n a l s ; d e r i v a t i v e s may e x i s t wi thout the f u n c t i o n a l be ing c o n t i n u o u s . We cannot then take the e x i s t e n c e of d e r i v a t i v e s of our data to mean that they are F r e c h e t d e r i v a t i v e s or that the f u n c t i o n a l s are c o n t i n u o u s . T h i s d i f f i c u l t y in a n a l y s i n g c o n t i n u i t y i s overcome through the use of the d i f f e r e n t i a l . For f u n c t i o n s of a s i n g l e v a r i a b l e , i t i s d e f i n e d wherever f ' ( x ) e x i s t s as f o l l o w s : df (x;s ) = f ' (x)s ( C 2 ) where s i s an increment a l on g the x a x i s . G r a p h i c a l l y , d f ( x ; s ) i s a l i n e a r a p p r o x i m a t i o n to the change i n f ( x ) over the i n t e r v a l from x to x+s. When X i s r e p l a c e d by s i n the 1 36 d e f i n i t i o n of the d e r i v a t i v e , we have: df (x;s ) 1 im s - > 0 f(x+s) - f ( x ) , ( C . 3 ) and the f o l l o w i n g i n e q u a l i t y , f(x+s) - f (x ) - f ' ( x ) ( C . 4 ) p r o v i d e d | s | i s s u f f i c i e n t l y s m a l l . A l t e r n a t i v e l y , for every e > 0 there e x i s t s an i n t e r v a l c o n t a i n i n g x and s such that | f(x+s) - f (x ) - f ' ( x ) s | < e | s | . ( C . 5 ) By e x t e n d i n g the concept of the d i f f e r e n t i a l to f u n c t i o n s of n r e a l v a r i a b l e s , d f ( x ; s ) = I 3 f ( x ) s el 3x t-where s = ( s , , s 2 , s 3 , s n ) and x= (x , , x 2 , x 3 , x h ) , the c o n n e c t i o n between the d i f f e r e n t i a l and c o n t i n u i t y can now be e l u c i d a t e d . I f for any e > 0 t h e r e e x i s t s an s in the neighbourhood of x, such that f(x+s) - f (x ) - d f ( x ; s ) | < e| |s | | 2 , ( C . 6 ) 1 37 f (x ) i s d i f f e r e n t i a b l e at x. T h i s c o n d i t i o n i s s u f f i c i e n t for the c o n t i n u i t y of f (x ) at x. In the example from M i l n e , for e<1, there i s no s m a l l s f or which the i n e q u a l i t y w i l l h o l d , so the f u n c t i o n i s not d i f f e r e n t i a b l e at ( 0 , 0 ) . We now g e n e r a l i z e f u r t h e r to o p e r a t o r s which take a v e c t o r from R h to R m . Le t E be an o p e r a t o r which takes a v e c t o r , x, from a v e c t o r space V to a normed v e c t o r space U. I f the l i m i t d E ( x ; s ) = l i m E(x+\s ) - E(x) (C.7) X->0 X e x i s t s , i t i s the Gateaux D i f f e r e n t i a l of E a t x. I f , i n a d d i t i o n , d E ( x ; s ) e x i s t s for a l l s in V, then E i s Gateaux d i f f e r e n t i a b l e at x. I t s h o u l d be noted that tthe e x i s t e n c e of a Gateaux d i f f e r e n t i a l does not guarantee Gateaux d i f f e r e n t a i b i l i t y at that x; i t may be the r e s u l t of some f o r t u n a t e c h o i c e of s. C o n t i n u i t y of E i s t h e r e f o r e not proven by the e x i s t e n c e of d E ( x ; s ) ( M i l n e , 1980, p289) . In a d d i t i o n , V must be normed before c o n t i n u i t y of E can be d e f i n e d . I f V and U are both normed v e c t o r spaces , and E takes a v e c t o r i n V to U, i t i s F r e c h e t d i f f e r e n t i a b l e a t x c v i f the c o n t i n u o u s l i n e a r o p e r a t o r E ' ( x ) e x i s t s such that 1 38 E(x+s) - E(x) = E ' ( x ) s + e (x ; s ) ( C . 8 ) for a l l v e c t o r s s c v w i th the l i m i t l i m e (x;s ) v = 0. s||v->0 | | s | / E*(x) i s the F r e c h e t d e r i v a t i v e at x and i t i s u n i q u e . In the l i m i t || s|| — >0, the s t a n d a r d e x p r e s s i o n f o r the d e r i v a t i v e , w i th the c o r r e c t norms, a p p l i e s . l i m E(x+s) ~ E(x) = E ' ( x ) (C.9) C-2 A p p l i c a t i o n to the D . C . R e s i s t i v i t y Problem L e t us now move to the s p e c i f i c case . o f the data f u n c t i o n a l s i n the D . C . r e s i s t i v i t y p r o b l e m . The n o n l i n e a r f u n c t i o n a l s ( E i ; i = 1 , n ) are a subset of v e c t o r o p e r a t o r s which take the model v e c t o r s from L 2 to p o i n t s i n R. Both the spaces L 2 and R are normed, w i th the f o l l o w i n g i n t e g r a l d e f i n i n g the f u n c t i o n a l s . OO E i = Jx S ( 0 , X ) C i ( X ) dX (C.10a) o where S' + wS + X 2 S 2 - 1 = 0 (C.10b) A l i n e a r a p p r o x i m a t i o n to the d i f f e r e n t i a l may be o b t a i n e d by l e t t i n g 6w be the i n c r e m e n t a l v e c t o r s i n the p r e v i o u s formal i sm. 1 39 dEi(w;6w) * Ei(w+5w) - Ei(w) (C.11) Now the c o n t i n u i t y of Ei (w) must be checked . E i i s c o n t i n u o u s i f for any e>0, there e x i s t s a 6>0 such that |Ei(w+5w) - E i ( w ) | < e whenever ||6w|| 2<5 (C.12) for any w in L 2 ( M i l n e , 1980, p129) . We know tha t Ei(w+6w) and 5S i s l i n e a r in 6w ( S ( 0 , X ) + 5 S ( 0 , X ) ) C i ( X ) dX (C.13) 5S(0 ,X) = pp HH'6 w(z) dz (C.14) H' ( 0) TTzl which makes the d i f f e r e n c e 6Ei(6w) a l s o l i n e a r i n 5w. To f i n d 6, the i n e q u a l i t y i s expressed f o r the n data f u n c t i o n a l s , and the r e s u l t i n g set of l i n e a r e q u a t i o n s (5Ei(6w) = e; i=1,n) i s s o l v e d for the s m a l l e s t 6w, the L 2 norm of which i s our 6. C l e a r l y s i n c e 6Ei i s the i n n e r product of the F r e c h e t d e r i v a t i v e and 5w ( equat ion 1 .18) , 6Ei w i l l be l e s s than e f or any 6w hav ing a two norm l e s s than 6. The i n e q u a l i t y h o l d s ; E i j j ; c o n t i n u o u s wi th r e s p e c t to w. T h e r e f o r e , the data f u n c t i o n a l s a r e F r e c h e t d i f f e r e n t i a b l e . T h i s i s an i n t u i t i v e l y c o r r e c t r e s u l t , as we would expect a s m a l l change in the r e s i s t i v i t y to produce a c o r r e s p o n d i n g l y s m a l l change in the d a t a . Thus for smal l | |6w| | 2 , the d i f f e r e n c e 1 40 SEi = Ei(w+6w) - E i (w) ( C I S ) approximates the F r e c h e t d i f f e r e n t i a l , dEi (w;6w). In the i t e r a t i v e c o n s t r u c t i o n method, we make t h i s a p p r o x i m a t i o n u s i n g the m i s f i t to the i ' t h datum as 6 E i , then s o l v e for the p e r t u r b a t i o n . 141 Appendix D  E v a l u a t i n g I n d e f i n i t e I n t e g r a l s Computing the f l a t t e s t p e r t u r b a t i o n i n v o l v e s m a n i p u l a t i n g i n d e f i n i t e i n t e g r a l s of F r e c h e t k e r n e l s and , as the l a s t s t e p , i n t e g r a t i n g 6m'(z) to o b t a i n the p e r t u r b a t i o n , 6m(z). We want to use e v e r y t h i n g we know about these f u n c t i o n s to i n t e g r a t e them as a c c u r a t e l y and as q u i c k l y as p o s s i b l e . The two i n d e f i n i t e i n t e g r a l s needed a r e : and Hi (z) 5m(z) = G i ( u ) du f o r i=1 , n o 5m'(u) du (D. 1 ) (D.2) D-1 I n d e f i n i t e I n t e g r a l s of F r e c h e t K e r n e l s The F r e c h e t k e r n e l s are o b t a i n e d through a Hankel t r a n s f o r m (equat ion 1 .28) , and are e v a l u a t e d above and below each l a y e r boundary. They are assumed to v a r y l i n e a r l y w i th z i n s i d e tha t l a y e r . Thus , w i t h i n the l ' t h l a y e r G i ( z ^ ) i s r e p r e s e n t e d by GHOi) + {Gi (hj ) - G i (Oy ) }zJt 1 (D.3) Gi fh , 1 42 The i n d e f i n i t e i n t e g r a l over t h i s l a y e r has both a l i n e a r term and a squared term in z^. P G i ( u ) du = G i ( 0 ^ ) z / + {Gi ( h z ) - G i (0/) }_zj^  (D . 4 ) 2h, T h i s must be added to the i n t e g r a l s of G i ( z ) from 0 to h „ , f or the 1-1 s h a l l o w e r l a y e r s . The f u l l i n d e f i n i t e i n t e g r a l , H i ( z ^ ) i s t h e r e f o r e , I {Gi ( OK ) + G i ( h « )} h« + Gi(0^)z^+ {Gi (h / ) - G i ( 0^) } z 2z (D . 5 ) U n l i k e G i ( z ) , H i ( z ) needs to be e v a l u a t e d o n l y once at every l a y e r boundary because the i n d e f i n i t e i n t e g r a l i s c o n t i n u o u s . Thus H i ( z ) at the 1 1 t h boundary i s : I {Gi (0„ )+Gi (h K ) } JL« «.I 2 (D . 6 ) The i n n e r product of H i ( z ) and H j ( z ) i s c a l c u l a t e d by m u l t i p l y i n g e q u a t i o n D . 5 by i t s j ' t h c o u n t e r p a r t and i n t e g r a t i n g the p r o d u c t , l a y e r by l a y e r , from z=0 to z=Zmax. D-2 O b t a i n i n g 6m(z) from Sm'(z) Once the inner product m a t r i x has been i n v e r t e d , and the c h i - s q u a r e d a c c e p t a b l e /3 c o e f f i c i e n t s have been c a l c u l a t e d , the p r e t u r b a t i o n ' s d e r i v a t i v e i s a v a i l a b l e . 6m' (z) = Zjii Hi (z) ( D . 7 ) 1 43 Since t h e r e i s a g r e a t e r p r o b a b i l i t y of knowing the s u r f a c e r e s i s t i v i t y than that at d e p t h , the i n t e g r a t i o n c o n s t a n t w i l l be set to z e r o at the s u r f a c e . So, 7. 6m(z) = J*5m' (u) du (D.8) o Now, i n s t e a d of l i n e a r l y i n t e r p o l a t i n g 6m'(z) between l a y e r b o u n d a r i e s , i t i s more a c c u r a t e to i n t e g r a t e e q u a t i o n D.7 d i r e c t l y . T h i s i s p o s s i b l e because the c h a r a c t e r of H i ( z ) between the l a y e r s i s known. Thus , 6m(z) = Z/3- { H i ( u ) d u , (D.9) and the i n d e f i n i t e i n t e g r a l of H-i(z) i s the second i n d e f i n i t e i n t e g r a l of G i ( z ) . z. r u. H i ( u ) du J G i ( v ) dv du (D.10) Assumptions r e g a r d i n g the nature of 6m'(z) were a v o i d e d . E q u a t i o n D.10 r e s u l t s i n an e x p r e s s i o n s i m i l a r to D .5 for the i n d e f i n i t e i n t e g r a l of H i ( z ) in the l ' t h l a y e r . The c o n s t a n t s are unchanged w i t h the f i r s t term now m u l t i p l i e d by z / f the second term by zJL2/2, and the t h i r d by z / 3 / 6 . 1 44 Appendix E  Winnowing E i g e n v a l u e s Owing to the presence of e r r o r s in the d a t a , the s m a l l e s t and f l a t t e s t p e r t u r b a t i o n e q u a t i o n s s hou ld not be s o l v e d e x a c t l y . I n s t e a d , c o e f f i c i e n t s which s o l v e the n e q u a t i o n s a c c o r d i n g to the c h i - s q u a r e d c r i t e r i o n s hou ld be found. For the s m a l l e s t p e r t u r b a t i o n s t h i s means f i n d i n g the a c o e f f i c i e n t s which s o l v e 5Ei = r,j aj f or i = 1 , n ( E . 1 ) such tha t I r S E i - F , i aj 1 2 = n (E.2) i m l L 51 J In p r a c t i c e , n o r m a l i z e d p e r t u r b a t i o n e q u a t i o n s are s o l v e d for n o r m a l i z e d a l p h a c o e f f i c i e n t s . L [SEi-f j : a: ] 2 = n (E.3) When e i g e n v a l u e s have been winnowed to o b t a i n a we w i l l c a l l i t a T , and c a l l the p r o d u c t s fjj a-j t r u n c a t e d data m i s f i t s , A 1. 6 E i T . Now e q u a t i o n E.3 can be w r i t t e n as L [ 5 E i - 6 E i f ] 2 = n ( E . 4 ) 1 45 5E = UAU T a (E .5a) 5E + = UAU T af (E .5b) T r u n c a t e d a l p h a c o e f f i c i e n t s are o b t a i n e d by t r u n c a t i n g the e i g e n v a l u e matr ix and the or thonormal matr ix U (see s e c t i o n 2-4 .2 , e q u a t i o n 2 . 1 9 ) . a* = U + A " l f U T + 5 E (E .6 ) where and I t f o l l o w s from e q u a t i o n s E . 5 a , and E . 5 b , t h a t 6E = U A U T U A" 1 U T 6 E _ = U U T 5 E 5 E T = U A U T U f A " L F UTHE = U I* U T * 5 E (E .7 ) (E .8 ) The n x m matr ix I T i s a zero m a t r i x w i th an mxm i d e n t i t y m a t r i x in the top m rows, where n-m i s the number of e i g e n v a l u e s that have been winnowed. I t i s the product AU TU FA- 1 * m r v. 0 0 1 0 1 0 1 o o o o o m and e q u a t i o n E . 4 can now be w r i t t e n as l e n g t h of the 1 46 d i f f e r e n c e v e c t o r , s q u a r e d . | |U{U T 5E - I t U T f 8 E } | | 2 = n ( E . 9 ) I t i s apparent tha t I ^ U ^ S E i s a v e c t o r in which the f i r s t m v a l u e s are the n o r m a l i z e d data m i s f i t s r o t a t e d by U T , and the l a s t n-m v a l u e s are z e r o s . I d e n t i f y i n g U T 5 E as 6 E r o t , e q u a t i o n E . 9 i s the squared sum from 1 to n of the f o l l o w i n g v e c t o r [ 0 , , 0 2 , . . . , 0 m , 6 E r o t w t 1 , , 5 E r o t h ] (E .10 ) a f t e r i t has been r o t a t e d by U . S i n c e U i s o r t h o n o r m a l , t h i s -l a s t r o t a t i o n w i l l not change the l e n g t h , and t h e r e f o r e w i l l make no d i f f e r e n c e to n . Thus , the m i s f i t c o n d i t i o n i s : [ 0,+ 0 2 + . . . + 0m+ 6Erot^ ,+ + S E r o t 2 , ] = n , ( E . 1 1 ) and the v a l u e of m for which makes t h i s t r u e i s most e a s i l y found by summing the v e c t o r SErot in the d i r e c t i o n from n to 1. Then the minimum m f o r which m*i A L 5Erot x 2 < n (E .12) i s the number of e i g e n v a l u e s which are k e p t . T h i s i s the way in which the number of e i g e n v a l u e s to be winnowed for a c h i -squared a c c e p t a b l e set of a l p h a c o e f f i c i e n t s i s d e t e r m i n e d . 1 47 Appendix F  The Schlumberger Data L i s t e d below are the apparent r e s i s t i v i t i e s and AB/2 d i s t a n c e s which formed the da ta set used i n c h a p t e r 3. The t h i r d column in each case i s the s t a n d a r d d e v i a t i o n l e v e l in percent tha t was a s s i g n e d to the datum. C u r r e n t and v o l t a g e i n f o r m a t i o n were not d i r e c t l y a v a i l a b l e , but were r e c o v e r e d u s i n g the Schlumberger apparent r e s i s t i v i t y f o r m u l a . pa x MN TT(AB/2) 2 f or MN < . 1 x AB/2 S c h l u m b e r g e r C o n f i g u r a t i o n 1 48 VES 1 pa AB/2 (meters) s t d . 1600 25 10 810 50 10 460 75 10 21 0 100 10 1 40 1 50 10 1 50 200 10 1 60 300 10 180 400 1 0 230 500 10 pa 210 190 195 180 180 1 60 1 20 82 82 1 10 1 20 1 40 1 45 1 20 1 10 VES 2 AB/2 (meters) 25 50 75 100 1 50 200 300 400 500 737' 975 1 225 1 495 1725 2000 ( c o r r e c t e d for n o n c o l i n e a r i t y ) s t d . dev. 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 VES 3 pa AB/2 (meters) s t d . dev . 1200 25 . 10 800 50 10 490 75 10 480 100 10 330 150 10 260 200 10 data not used 360 300 180 400 1250 500 550 750 VES 4 pa AB/2 (meters) s t d . dev 1350 15 10 1350 25 10 1200 50 10 1150 75 10 1200 100 10 1175 150 ' 10 1100 200 10 1100 300 10 800 400 15 450 500 40 750 750 20 1200 1000 20 VES 5 pa AB/2 (meters) s t d . dev 860 25 10 760 50 10 660 75 10 640 100 10 710 150 10 900 200 10 650 300 10 550 400 10 530 500 10 460 750 10 390 1000 10 310 1250 10 250 1500 10 230 1750 10 240 2000 10 VES 6 pa AB/2 (meters) s t d . dev . 230 25 10 250 50 10 240 75 10 240 100 10 220 150 10 205 200 10 190 300 10 210 400 10 220 500 10 290 750 10 240 1000 10 

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