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A geostatistical analysis of the Venus Mine, Yukon Territory Stubens, Thomas Camillo 1988

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A GEOBTATISTICAL ANALYSIS OF THE VENUS MINE,'YUKON TERRITORY By Thomas Cami l l o Stubens B.A.Sc, U n i v e r s i t y of Toronto, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Mining and M i n e r a l Process Engineering) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard © THE UNIVERSITY OF BRITISH COLUMBIA DECEMBER 19 88 Copyri g h t Thomas Camillo Stubens, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of /Ht*/wC * / / V g / ? A < . ryilec&%s / ~ / ' The University of British Columbia Vancouver, Canada Date V/  DE-6 (2/88) ABSTRACT T h i s t h e s i s d e s c r i b e s a g e o s t a t i s t i c a l a n a l y s i s of the Venus d e p o s i t i n the Yukon T e r r i t o r y . The Venus Mine i s l o c a t e d on T a g i s h Lake, approximately 1.5 miles north of the Yukon/B.C. border. I t i s a g o l d , s i l v e r , l e a d , z i n c b e a r i n g quartz v e i n of l a t e Cretaceous to e a r l y T e r t i a r y age o c c u r r i n g i n p o r p h y r i t i c a n d e s i t e of the Mount Nansen Group v o l c a n i c s . Three companies have attempted to mine Venus s i n c e i t s d i s c o v e r y i n the l a t e 1890's. In a l l three cases the o p e r a t i n g company c o u l d not p r o f i t a b l y mine the d e p o s i t and was f o r c e d to shut down. Ove r e s t i m a t i o n of the ore r e s e r v e s and problems with grade c o n t r o l were the main problems encountered by the f a i l e d o p e r a t i o n s . The study r e p o r t e d here a p p l i e d a w e l l - e s t a b l i s h e d g e o s t a t i s t i c a l methodology to the Venus d e p o s i t . Over 3500 c h i p samples of the v e i n , c o l l e c t e d a t r e g u l a r i n t e r v a l s along a l l development d r i f t s and r a i s e s , were used i n t h i s a n a l y s i s . Seml-variograms were c a l c u l a t e d of 3 v a r i a b l e s : v e i n t h i c k n e s s , g o l d accumulation and s i l v e r accumulation. A f t e r f i t t i n g semi-variogram models to the c a l c u l a t e d curves the 3 v a r i a b l e s were k r i g e d and the r e s u l t s were p l o t t e d on maps showing the k r i g e d estimates i n each of the 2039 b l o c k s . The maps i l l u s t r a t e c l e a r l y the s i z e , l o c a t i o n and o r i e n t a t i o n of the ore zones. The ore r e s e r v e s of the Venus mine were c a l c u l a t e d u s i n g the k r i g e d block e s t i m a t e s . The proven and probable i i r e s e r v e s above a c u t - o f f of $130/ton (Cdn), assuming g o l d and s i l v e r p r i c e s of $520/oz (Cdn) and $7.80/oz (Cdn) are 108,577 tons gr a d i n g 0.3 oz Au/ton (+/- 1.18%) and 6.05 oz Ag/ton (+/- 1.29%). I n t e r p r e t a t i o n of the maps and the ore res e r v e estimates l e d the author to make s e v e r a l recommendations f o r f u t u r e mining of the Venus d e p o s i t . i i i Table of Contents A b s t r a c t i i Table of Contents i v L i s t of Tables v i L i s t of F i g u r e s v i i i Acknowledgements x i i I n t r o d u c t i o n 1 1. L o c a t i o n and H i s t o r y 4 2. Geology 10 3. Data 12 4. S t a t i s t i c s 1.4 4.1 Scattergrams and C o r r e l a t i o n C o e f f i c i e n t s 14 4.2 Frequency D i s t r i b u t i o n s 17 4.2.1 Gold Grades 18 4.2.2 S i l v e r Grades 23 4.2.3 Thickness 27 4.2.4 Gold Accumulation 3 4 4.2.5 S i l v e r Accumulation 38 4.3 Summary and Co n c l u s i o n s 42 5. G e o s t a t i s t i c s 43 5.1 Theory of R e g i o n a l i z e d V a r i a b l e s 43 5.2 Procedure 49 5.3 S t a t i o n a r i t y and the I n t r i n s i c Hypothesis 51 6. Semi-Variogram R e s u l t s : Thickness 54 6.1.1 Thickness i n the Upper North 54 6.1.2 S t a t i o n a r i t y of Thickness i n the Upper North 56 6.2.1 Thickness i n the Lower North 58 6.2.2 S t a t i o n a r i t y of Thickness i n the Lower North 60 6.2.3 A n i s o t r o p y of Thickness i n the Lower North 62 6.3.1 Thickness i n the South 66 6.3.2 S t a t i o n a r i t y of Thickness i n the South 68 6.3.3 A n i s o t r o p y of Thickness In the South Quadrants 71 6.4 Summary and C o n c l u s i o n s 77 i v 7. Semi-Variogram R e s u l t s : Gold Accumulation 79 7.1.1 Gold Accumulation i n the North 79 7.1.2 S t a t i o n a r i t y of Gold Accumulation i n the North 81 7.1.3 A n l s o t r o p y of Gold Accumulation i n the Northern Areas 85 7.2.1 Gold Accumulation i n the South 89 7.2.2 S t a t i o n a r i t y of Gold Accumulation i n the South 91 7.2.3 A n i s o t r o p y of Gold Accumulation i n the Southern Areas 95 7.3 Summary and Co n c l u s i o n s 98 8. Semi-Variogram R e s u l t s : S i l v e r Accumulation 100 8.1.1 S i l v e r Accumulation i n the North 100 8.1.2 S t a t i o n a r i t y of S i l v e r Accumulation i n the North 102 8.1.3 A n i s o t r o p y of S i l v e r Accumulation i n the North Areas 106 8.2.1 S i l v e r Accumulation i n the South 110 8.2.2 S t a t i o n a r i t y of S i l v e r Accumulation i n the South 112 8.2.3 A n i s o t r o p y of S i l v e r Accumulation i n the South Areas 116 8.3 Summary and Co n c l u s i o n s 121 9. K r i g i n g 123 9.1 Theory 123 9.2 Procedure 125 9.3 R e s u l t s of K r i g i n g 128 9.4 Ore Reserve E s t i m a t i o n 132 9.5 P r o d u c t i o n Estimates 139 Co n c l u s i o n s and Recommendations 141 B i b l i o g r a p h y 145 v L i s t of Tables 4.1.1 C o r r e l a t i o n C o e f f i c i e n t s 15 4.3.1 Summary of Means and V a r i a n c e s 42 6.1.1 Semi-variogram parameters of t h i c k n e s s i n the Upper North 54 6.1.2 Mean Thickness and Variance i n the Upper North 56 6.2.1 Semi-variogram parameters of t h i c k n e s s i n the Lower North 58 6.2.2 Mean Thickness and Variance i n the Lower North 60 6.2.3 Semi-variogram parameters of t h i c k n e s s i n the Lower North Areas 62 6.3.1 Semi-variogram parameters of t h i c k n e s s i n the South 66 6.3.2 Mean Thickness and Va r i a n c e i n the South Quadrants 68 6.3.3 Semi-variogram parameters of t h i c k n e s s i n the South Quadrants 72 6.4.1 Semi-variogram parameters of t h i c k n e s s 78 7.1.1 Semi-variogram parameters of l o g transformed g o l d accumulation i n the North 79 7.1.2 Mean Gold Accumulation and Va r i a n c e i n the North 81 7.1.3 Semi-variogram parameters of log-transformed g o l d accumulation i n the North Quadrants 82 7.1.4 Semi-variogram parameters of log-transformed g o l d accumulation i n areas N l and NA 86 7.2.1 Semi-variogram parameters of l o g transformed g o l d accumulation i n the South 89 7.2.2 Mean Gold Accumulation and Variance i n the South 91 7.2.3 Semi-variogram parameters of log-transformed g o l d accumulation i n the South Quadrants 92 v i 7.2.4 Seml-variogram parameters of log-transformed g o l d accumulation i n areas S3 and SA 95 7.3.1 Semi-variogram parameters of log-transformed gold accumulation 99 8.1.1 Semi-variogram parameters of l o g transformed s i l v e r accumulation i n the North 100 8.1.2 Mean S i l v e r Accumulation and Variance i n the North 102 8.1.3 Semi-variogram parameters of log-transformed s i l v e r accumulation i n the North Quadrants 103 8.1.4 Semi-variogram parameters of log-transformed s i l v e r accumulation i n areas NI and NA 107 8.2.1 Semi-variogram parameters of l o g transformed s i l v e r accumulation i n the South 110 8.2.2 Mean S i l v e r Accumulation and Variance i n the South 112 8.2.3 Semi-variogram parameters of log-transformed s i l v e r accumulation i n the South Quadrants 113 8.2.4 Semi-variogram parameters of log-transformed s i l v e r accumulation i n areas SI, S2 and SU 117 8.3.1 Semi-variogram parameters of log-transformed s i l v e r accumulation 122 9.2.1 Semi-variogram models of t h i c k n e s s , g o l d accumulation and s i l v e r accumulation 126 9.4.1 K r i g e d i n s i t u r e s e r v e estimate #1 135 9.4.2 K r i g e d i n s i t u r e s e r v e estimate #2 136 9 .4.3 K r i g e d ore r e s e r v e estimate #1 137 9 .4 .4 K r i g e d ore r e s e r v e estimate #2 138 9.5.1 T o t a l r e p o r t e d p r o d u c t i o n 139 v i i L i s t of F i g u r e s 1.1 L o c a t i o n Map 5 1.2 P l a n of the Venus Mine Workings 7 4.1.1 Scattergrams of Au and Ag vs. Nor t h i n g 16 4.2.1 R e l a t i v e Frequency D i s t r i b u t i o n s of Gold Assays 20 4.2.2 Cumulative Frequency D i s t r i b u t i o n of Gold Assays i n the North 21 4.2.3 Cumulative Frequency D i s t r i b u t i o n of Gold Assays i n the South 22 4.2.4 R e l a t i v e Frequency D i s t r i b u t i o n s of S i l v e r Assays 24 4.2.5 Cumulative Frequency D i s t r i b u t i o n of S i l v e r Assays i n the North 25 4.2.6 Cumulative Frequency D i s t r i b u t i o n of S i l v e r Assays i n the South 26 4.2.7 R e l a t i v e Frequency D i s t r i b u t i o n s of Thickness 28 4.2.8 R e l a t i v e Frequency D i s t r i b u t i o n s of Thickness i n the Upper and Lower North 30 4.2.9 Cumulative Frequency D i s t r i b u t i o n of Thickness i n the Upper North 31 4.2.10 R e l a t i v e Frequency D i s t r i b u t i o n s of Thickness i n the Upper and Lower South 33 4.2.11 R e l a t i v e Frequency D i s t r i b u t i o n s of Gold Accumulation 35 4.2.12 Cumulative Frequency D i s t r i b u t i o n of Gold Accumulation i n the North 36 4.2.13 Cumulative Frequency D i s t r i b u t i o n of Gold Accumuation i n the South 37 4.2.14 R e l a t i v e Frequency D i s t r i b u t i o n s of S i l v e r Accumulation 39 4.2.15 Cumulative Frequency D i s t r i b u t i o n of S i l v e r Accumulation i n the North 40 v i i i 4.2.16 Cumulative Frequency D i s t r i b u t i o n of S i l v e r Accumulation i n the South 41 5.1.1 Sample semi-variogram model. 45 5.1.2 S p h e r i c a l and E x p o n e n t i a l Semi-variogram Models. 47 5.1.3 A Nested S p h e r i c a l Semi-variogram Model. 48 5.3.1 Quadrant Boundaries O v e r l a y i n g a P l a n of the Mine Workings 53 6.1.1 Semi-variograms of Log-Transformed Thickness i n the Upper North 55 6.1.2 Semi-variograms of Log-Transformed Thickness i n Areas NU1 and NU2 57 6.2.1 Semi-variograms of Thickness i n the Lower North 59 6.2.2 Semi-variograms of Thickness i n Areas NL1 and NL2 61 6.2.3 Semi-variograms of Thickness i n NLl 64 6.2.4 Semi-variograms of Thickness i n NL2 65 6.3.1 Semi-variograms of Thickness i n the South 67 6.3.2 Semi-variograms of Thickness i n the South Quadrants a t 15 degrees 69 6.3.3 Semi-variograms of Thickness i n the South Quadrants a t 105 degrees 70 6.3.4 Semi-variograms of Thickness i n Area SI 73 6.3.5 Semi-variograms of Thickness i n Area S2 74 6.3.6 Semi-variograms of Thickness i n Area S3 75 6.3.7 Semi-variograms of Thickness i n Area S4 76 7.1.1 Semi-variograms of Gold Accumulation i n the North 80 7.1.2 Semi-variograms of Gold Accumulation i n the North Quadrants a t 25 degrees 83 7.1.3 Semi-variograms of Gold Accumulation i n the North Quadrants a t 115 degrees 84 i x 7.1.4 Semi-variograms of Gold Accumulation i n the Area Nl 87 7.1.5 Semi-variograms of Gold Accumulation i n the Area NA 88 7.2.1 Semi-variograms of Gold Accumulation i n the South 90 7.2.2 Semi-variograms of Gold Accumulation i n the South Quadrants a t 25 degrees 93 7.2.3 Semi-variograms of Gold Accumulation i n the South Quadrants a t 115 degrees 94 7.2.4 Semi-variograms of Gold Accumulation i n the Area SA 96 7.2.5 Semi-variograms of Gold Accumulation i n the Area S3 97 8.1.1 Semi-variograms of S i l v e r Accumulation i n the North 101 8.1.2 Semi-variograms of S i l v e r Accumulation i n the North Quadrants a t 0 degrees 104 8.1.3 Semi-variograms of S i l v e r Accumulation i n the North Quadrants a t 90 degrees 105 8.1.4 Semi-variograms of S i l v e r Accumulation i n the Area N l 108 8.1.5 Semi-variograms of S i l v e r Accumulation i n the Area NA 109 8.2.1 Semi-variograms of S i l v e r Accumulation i n the South 111 8.2.2 Semi-variograms of S i l v e r Accumulation i n the South Quadrants a t 60 degrees 114 8.2.3 Semi-variograms of S i l v e r Accumulation i n the South Quadrants a t 150 degrees 115 8.2.4 Semi-variograms of S i l v e r Accumulation i n the Area SI 118 8.2.5 Semi-variograms of S i l v e r Accumulation i n the Area S2 119 8.2.6 Semi-variograms of S i l v e r Accumulation i n the Area SU 120 x 9.3.1 K r i g e d v e i n Thickness Pocket 1- ( 0 jji 9.3.2 Kriged Gold Accumulation Eojckei—2 ^ C M 9.3.3 Kriged S i l v e r Accumulation Pocket -3 x i ACKNOWLEDGEMENTS I would l i k e t o thank my a d v i s o r Alan Reed f o r i n t r o d u c i n g me to g e o s t a t i s t i c s and f o r h i s advice and support. I thank Reg Faulkner f o r h i s f r i e n d s h i p , encouragement, advi c e and f o r many b r a i n s t o r m i n g s e s s i o n s . I thank the Cy and Emerald Keyes Foundation f o r t h e i r f i n a n c i a l support. U n i t e d Keno H i l l Mines s t a f f , Dennis P r i n c e , Ken Watson, Bob S t e r l i n g and L o r i Walton were a l l very h e l p f u l i n g i v i n g me f r e e access to r e p o r t s , data and t h e i r e x p e r t i s e . I acknowledge the a s s i s t a n c e of Paul McGuigan and Michael Pond of Tecucomp G e o l o g i c a l f o r the use of t h e i r computer d r a f t i n g equipment. The love and support of my wife Ruth have been a constant source of s t r e n g t h and encouragement. I thank her f o r s t a n d i n g by me. x i i INTRODUCTION Th i s t h e s i s d e s c r i b e s a g e o s t a t i s t i c a l a n a l y s i s of the Venus d e p o s i t i n the Yukon T e r r i t o r y . The Venus Mine i s l o c a t e d on T a g i s h Lake, approximately 1.5 m i l e s north of the Yukon/B.C. border. I t i s a g o l d , s i l v e r , l e a d , z i n c b e a r i n g quartz v e i n of l a t e Cretaceous to e a r l y T e r t i a r y age o c c u r r i n g i n p o r p h y r i t i c a n d e s i t e of the Mount Nansen Group v o l c a n i c s . The Venus v e i n has an average s t r i k e of 20 degrees and a d i p of 30 degrees W and has been t r a c e d along s t r i k e f o r over a m i l e . I t averages 2.7 f e e t i n t h i c k n e s s . Three companies have attempted to mine Venus s i n c e i t s d i s c o v e r y i n the l a t e 1890's: - C o l . Conrad from 1904 to 1912; - Venus Mines L t d . from 1966 to 1971; - United Keno H i l l Mines 1978 to p r e s e n t . In a l l three cases the o p e r a t i n g company co u l d not p r o f i t a b l y mine the d e p o s i t and was f o r c e d to shut down. In the l a s t two cases the reasons were: - o v e r e s t i m a t i o n of the ore r e s e r v e s ; - problems with grade c o n t r o l ; - i n a b i l i t y t o feed a 300 tpd m i l l (Venus mines); and - a sudden drop i n commodity p r i c e s (UKHM 1981). The study r e p o r t e d here a p p l i e d a w e l l - e s t a b l i s h e d g e o s t a t i s t i c a l methodology to the Venus d e p o s i t . The data f o r t h i s study c o n s i s t of over 3500 c h i p samples taken a c r o s s the f u l l width of the v e i n . These samples were -1-c o l l e c t e d a t 5 f o o t i n t e r v a l s a l o n g a l l development d r i f t s and r a i s e s . The mine i s developed on f i v e l e v e l s : 2600, 2650, 2700, 2800 and 2850 f e e t above sea l e v e l , and i n 35 con n e c t i n g r a i s e s and stopes. Both Venus Mines and UKHM sample data were u t i l i z e d . The g e o s t a t i s t i c a l a n a l y s i s c o n s i s t e d of the f o l l o w i n g s t e p s : 1) P r o d u c t i o n of s c a t t e r p l o t s and c a l c u l a t i o n of c o r r e l a t i o n c o e f f i c i e n t s t o look Cor gross trends w i t h i n the d a t a . 2) A n a l y s i s of frequency d i s t r i b u t i o n s of the f o l l o w i n g v a r i a b l e s : g o l d assays, s i l v e r assays, v e i n t h i c k n e s s , g o l d accumulation and s i l v e r accumulation. 3) C a l c u l a t i o n of semi-variograms of 3 v a r i a b l e s ( t h i c k n e s s , g o l d accumulation and s i l v e r accumulation) and f i t t i n g of semi-variogram models to the c a l c u l a t e d c u r v e s . 4) K r i g i n g of the 3 v a r i a b l e s u s i n g the semi-variogram parameters d e r i v e d above. 5) Generation of maps showing the k r i g e d estimates i n each of the 2039 b l o c k s . 6) C a l c u l a t i o n of i n s i t u r e s e r v e s and ore r e s e r v e s with 84% c o n f i d e n c e l i m i t s . The ore res e r v e c a l c u l a t i o n s employed the same c u t - o f f as those of UKHM. 7) A t e s t of the acc u r a c y of the k r i g e d ore re s e r v e estimate through a comparison of estimated mine p r o d u c t i o n w i t h r e p o r t e d p r o d u c t i o n f i g u r e s . T h i s study i s the f i r s t t o employ a r i g o r o u s , -2-s t a t i s t i c a l methodology to the c a l c u l a t i o n of ore re s e r v e s at the Venus Mine. The r e s u l t s are acc u r a t e and more r e l i a b l e than the e x i s t i n g e s t i m a t e s . The maps i l l u s t r a t e c l e a r l y the s i z e and l o c a t i o n of the ore zones and l e d the author to make s p e c i f i c recommendations f o r f u t u r e mining of the Venus d e p o s i t . - 3 -1. LOCATION AND HISTORY The Venus Deposit i s l o c a t e d on the n o r t h shore of the Windy Arm of T a g i s h Lake, approximately 1.5 m i l e s n o r t h of the Yukon/B.C. border. The mine i s 60 mi l e s south of Whitehorse, beside the Klondike Highway which connects Whitehorse t o Skagway, A l a s k a . The ne a r e s t community i s Car c r o s 3 , 14 m i l e s to the north ( f i g u r e 1.1). The Venus v e i n was f i r s t d i s c o v e r e d a t the t u r n of the c e n t u r y by p r o s p e c t o r s on t h e i r way to the Klondike g o l d f i e l d s . In 1900, Jack Stewart and Jack Pooley staked the Venus c l a i m group which they s o l d four years l a t e r to C o l . J.H. Conrad. From 1904 to 1912, Conrad developed the Venus v e i n i n three l o c a t i o n s : the V a u l t , Venus #1 and Venus #2 workings. The most e x t e n s i v e work was c a r r i e d out a t Venus #2 where two d r i f t s e x p l o i t e d a 1000 f o o t s t r i k e l e n g t h of the v e i n . A 1500 f o o t long a e r i a l t r a m l i n e was c o n s t r u c t e d t o c a r r y ore from the 3150 f o o t l e v e l t o a g r a v i t y c o n c e n t r a t o r on the lake shore ( e l e v a t i o n 2200 f e e t ) . S i x thousand tons of ore were mined from Venus 82 (Watson, 1979). In 1916, the Harper Syndicate a c q u i r e d the r i g h t s t o the Venus p r o p e r t y . In four years, 2650 tons of hand cobbed ore, g r a d i n g 0.64 oz Au/ton and 68.2 oz Ag/ton, were shipped t o the smelter (Watson, 1979). T h i s m a t e r i a l i s b e l i e v e d t o have come from the Venus #2 workings. -4-F i g u r e 1 .1 : L o c a t i o n Map. - 5 -The area then saw l i m i t e d a c t i v i t y u n t i l 1966 when Venus Mines L t d . (VML) a c q u i r e d the p r o p e r t y and began underground e x p l o r a t i o n of the Venus v e i n . A c t i v i t y was concentrated below the Venus #1 workings ( e l e v a t i o n 2900 f e e t ) . Two a d i t s were c o l l a r e d a t the 2600 and 2700 foot l e v e l s and c r o s s - c u t s were d r i v e n to the v e i n . The Venus v e i n was developed i n f i v e d r i f t s a t e l e v a t i o n s of 2600, 2650, 2700, 2800 and 2850 f e e t above sea l e v e l and i n 35 connecting r a i s e s ( f i g u r e 1.2). In 1970, ore r e s e r v e s of 134,845 tons of ore grading 0.4 oz Au/ton and 11.5 oz Ag/ton were c a l c u l a t e d u s i n g a minimum 4 f o o t mining width. The d e c i s i o n was made to put the mine i n t o p r o d u c t i o n and to b u i l d a m i l l capable of h a n d l i n g 300 tons of ore per day. In 1971, a f t e r p r o c e s s i n g 65,231 tons of ore g r a d i n g o n l y 0.19 oz Au/ton and 6.12 oz Ag/ton, the mine c l o s e d (Watson, 1979). The low head grades achieved d u r i n g t h i s phase of the mine's o p e r a t i o n were the r e s u l t of two main f a c t o r s . VML overestimated the grade and c o n t i n u i t y of the v e i n ' s m i n e r a l i z a t i o n and on the b a s i s of t h e i r ore reserve estimate, they c o n s t r u c t e d a m i l l which was too b i g f o r the mine to feed. Problems with grade c o n t r o l were encountered due to an i n a b i l i t y to v i s u a l l y d i f f e r e n t i a t e ore from waste (MacDonald, 1971 and M c F a u l l , personnal communication). The lower than expected and e r r a t i c mine grades f o r c e d the company i n t o bankruptcy. -6--7-The Tagish Lake Syndicate a c q u i r e d the Venus claims i n 1975 and optioned them to United Keno H i l l Mines L t d . (UKHM) i n 1978. In 1979, the author was p a r t of a f i v e man team t h a t mapped and sampled s e l e c t e d p o r t i o n s of the Venus mine workings to check the r e p r o d u c e a b i l i t y of previous sampling and to t e s t unsampled ar e a s . Using a l l a v a i l a b l e sample data and d i l u t i n g the assays to a minimum 5 f o o t mining width, UKHM estimated r e s e r v e s of 77,580 tons g r a d i n g 0.27 oz Au/ton and 7.2 oz Ag/ton. E i g h t diamond d r i l l holes d r i l l e d below the 2600 l e v e l i n d i c a t e d a f u r t h e r 42,000 tons a t 0.13 oz Au/ton and 5.5 oz Ag/ton. A d e c i s i o n was made to put the Venus mine back i n t o p r o d u c t i o n . In 1980, c o n s t r u c t i o n began on a 100 ton per day m i l l . Development d r i f t i n g began on the 2800 and 2850 l e v e l s . Low mine grades caused by ex c e s s i v e d i l u t i o n i n the stopes, environmental problems and f a l l i n g metal p r i c e s f o r c e d the mine's c l o s u r e i n October 1981. The m i l l was mothballed before i t had processed a s i n g l e ton of ore. The mine's p r o d u c t i o n , 4,641 tons g r a d i n g 0.18 oz Au/ton, 5.03 oz Ag/ton, 1.62% Pb and 1.09% Zn, were s t o c k p i l e d a t the m i l l s i t e ( S t e r l i n g , 1981). In 1984, UKHM undertook f u r t h e r development on the 2850 l e v e l . The d r i f t was extended 545 f e e t to the north and 518 f e e t to the south. Two r a i s e s were d r i v e n up from the l e v e l to f o l l o w the up d i p e x t e n s i o n of high grade i n t e r s e c t i o n s i n the d r i f t . A f u r t h e r 13,163 tons of -8-development muck grad i n g 0.17 oz Au/ton, 4.3 oz Ag/ton, 1.29% Pb and 0.72% Zn were added to the s t o c k p i l e a t the m i l l s i t e ( S t e r l i n g , 1984). The p u b l i s h e d r e s e r v e s a t Venus now stand a t 68,286 tons of proven and probable ore g r a d i n g 0.32 oz Au/ton, 8.85 oz Ag/ton, 2.45% Pb and 1.46% Zn ( P r i n c e , 1984). T h i s estimate i s based upon a minimum mining width of 5 f e e t and a c u t - o f f value of $130/ton g i v e n a g o l d p r i c e of $390/oz and a s i l v e r p r i c e of $7.80/oz. A l l p r i c e s i n t h i s study w i l l be g i v e n i n Canadian d o l l a r s . -9-2. GEOLOGY The Venus Deposit i s a Au/Ag/Pb/Zn be a r i n g quartz v e i n of l a t e Cretaceous to e a r l y T e r t i a r y age o c c u r r i n g i n intermediate v o l c a n i c rocks of what i s known as the Montana Mountain V o l c a n i c Complex' (Roots, 1980). The v o l c a n i c s themselves are p a r t of the Mount Nansen Group, a s u i t e of v o l c a n i c flows, b r e c c i a s and p y r o c l a s t i c s which have been mapped i n many l o c a t i o n s i n the Yukon and B r i t i s h Columbia (Walton, 1987). In the Montana Mountain area, the Mount Nansen V o l c a n i c s are green to grey-green i n t e r m e d i a t e flows and b r e c c i a s i n t r u d e d by f e l s i c d i k e s (Roots, 1980). The Venus v e i n i s found on the south-east slope of the roughly c i r c u l a r Montana Mountain complex. I t has been in t r u d e d along a zone of weakness which i s the co n t a c t between p o r p h y r i t i c a n d e s i t e to the south and a f e l s i c d i k e to the north (Walton, 1987). The Venus v e i n i s one of many found i n the Montana Mountain area but having been t r a c e d f o r over 2 k i l o m e t e r s along s t r i k e , i t i s by f a r the l a r g e s t . The Venus v e i n i s a simple, banded f i s s u r e - f i l l i n g c o n t a i n i n g q u a r t z , a r s e n o p y r i t e , p y r i t e , galena and s p h a l e r i t e . In a study of the geology and geochemistry of the v e i n , Walton (1987) concluded t h a t "the paragenesis can be d i v i d e d i n t o two m i n e r a l o g i c a l l y d i s t i n c t s t a g e s : Stage I - e a r l y q u a r t z , a r s e n o p y r i t e and p y r i t e , and Stage II -l a t e r q u a r t z , s p h a l e r i t e , galena and g o l d . Gold and galena occur together as f r a c t u r e f i l l i n g s or replacements i n -10-e a r l i e r d e p o s i t e d a r s e n o p y r l t e and p y r i t e " (p. 40). The s i l v e r occurs In the form of a r g e n t i f e r o u s galena and thus should a l s o be c l o s e l y a s s o c i a t e d with the g o l d . The Venus v e i n has an average s t r i k e of 20 degrees over the m a j o r i t y of i t s l e n g t h but i n the south end of the mine the v e i n bends s h a r p l y t o a s t r i k e of 60 degrees. T h i s bend appears t o predate the emplacement of the v e i n m a t e r i a l and does not a f f e c t the d i p . V e i n d i p s a t Venus va r y between 0 and 60 degrees NW and average 30 degrees MW. The t h i c k n e s s of the Venus v e i n averages 2.7 f e e t but has been observed t o var y from s e v e r a l inches t o over 10 f e e t . The ore shoots a t Venus appear to be long, narrow and f i n g e r - l i k e , o r i e n t e d w i t h t h e i r long a x i s p a r a l l e l t o the v e i n d i p ( P r i n c e , 1984). Surface o x i d a t i o n extends as much as 250 f e e t down d i p and has a l t e r e d the s u l p h i d e minerals to an assemblage c o n t a i n i n g a r s e n i c s u l p h i d e s : r e a l g a r and orpiment, as w e l l as l i m o n i t e , s c o r o d i t e and yu k o n i t e . Gold and s i l v e r grades are s i g n i f i c a n t l y lower i n the o x i d i z e d m a t e r i a l . Below t h i s oxide zone, there does not appear to be a zone of supergene enrichment ( C a r l y l e , 1984). -11-3. DATA The data f o r t h i s study c o n s i s t of 3673 c h i p samples taken a c r o s s the f u l l width of the v e i n . The sampling covers an area 2900 f e e t along the s t r i k e l e n g t h of the v e i n and 1050 f e e t up and down d i p . Four separate sampling regimes provide the sample data: a) Venus Mines L t d . , 1969/70 (2176 samples) - p r o d u c t i o n and development sampling of d r i f t s and r a i s e s ; b) United Keno H i l l Mines, 1979 (745 samples) -resampling of s e l e c t e d areas of mine workings; - f i l l - i n sampling of p r e v i o u s l y unsampled stopes and development headings; c) UKHM, 1981 (287 samples) -resampling and f i l l - i n as above; - p r o d u c t i o n and development headings; d) UKHM, 1984 (465 samples) -development headings. Sample l o c a t i o n s were d e r i v e d from mine plans using a Complot S e r i e s 7000 d i g i t i z i n g t a b l e t connected to a Hewlett Packard 9845 computer. The n o r t h i n g , e a s t i n g and e l e v a t i o n of each sample were then entered onto the UBC -12-main-frame computer a l o n g with the g o l d , s i l v e r , l e a d and z i n c a ssays, the v e i n t h i c k n e s s and v e i n d i p . A f t e r an i n s p e c t i o n of the data i t was decided to exclude a l l d u p l i c a t e samples. The areas of resampling caused a c l u s t e r i n g of the sampling, thus b i a s i n g the data towards t h i c k e r and higher grade s e c t i o n s of the d e p o s i t . Hence 598 samples were excluded. A f u r t h e r 94 samples were excluded when i t was observed t h a t they had been d i l u t e d to a minimum 4 f o o t mining width. Since v e i n t h i c k n e s s i s an important v a r i a b l e i n d e t e r m i n i n g the economic v i a b i l i t y of a v e i n d e p o s i t l i k e Venus, i t i s important t h a t o n l y the t r u e t h i c k n e s s e s be used. The Venus v e i n has been observed t o take a sharp t u r n i n the extreme southern p o r t i o n of the mine (see f i g u r e 1.2). In order to perform a v a l i d t e s t of a n i s o t r o p y i n the g e o s t a t i s t i c a l p a r t of t h i s study, the database was d i v i d e d a l o n g the hinge l i n e of t h i s bend. The hinge has the e q u a t i o n : N = -0.778 * E + 49946 where N = n o r t h i n g E = e a s t i n g The r e s u l t a n t North and South p o p u l a t i o n s have 2227 and 758 samples r e s p e c t i v e l y . -13-4. STATISTICS 4.1 S c a t t e r P l o t s and C o r r e l a t i o n C o e f f i c i e n t s The f i r s t s t a t i s t i c a l t e s t a p p l i e d to the Venus data was the p l o t t i n g of s c a t t e r p l o t s and the c a l c u l a t i o n of c o r r e l a t i o n c o e f f i c i e n t s between the v a r i a b l e s . In p a r t i c u l a r , the e f f e c t of sample l o c a t i o n on gold and s i l v e r assays was examined to look f o r gross z o n a t i o n s i n the plane of the v e i n . The gold and s i l v e r versus n o r t h i n g (approximate v e i n s t r i k e ) s c a t t e r p l o t s show a c o n c e n t r a t i o n of high assays i n f i v e areas which i n d i c a t e ore shoots (Figure 4.1.1). When the go l d and s i l v e r assays were p l o t t e d a g a i n s t e l e v a t i o n , g o l d showed a s l i g h t i n c r e a s e i n the upper p o r t i o n s of the mine while s i l v e r remained r e l a t i v e l y c o n s t a n t . These r e s u l t s confirmed the ob s e r v a t i o n s of the mine g e o l o g i s t s t h a t the ore shoots are long, up and down d i p , and narrow along s t r i k e . C o r r e l a t i o n c o e f f i c i e n t s are l i s t e d i n Table 4.1.1. As expected, there i s a s t r o n g c o r r e l a t i o n between lead and s i l v e r (.78), between lead and z i n c (.6) and between s i l v e r and z i n c (.5). But o n l y a weak c o r r e l a t i o n e x i s t s between gold and le a d (.2) and between gold and s i l v e r (.25). T h i s does not come as a s u r p r i s e i n s p i t e of Walton's (1987) c o n c l u s i o n t h a t g o l d i s g e n e t i c a l l y a s s o c i a t e d with galena (PbS). Walton a l s o found t h a t a r s e n o p y r i t e and p y r i t e must be present f o r gold to p r e c i p i t a t e and th a t t h i s requirement i s so s t r o n g t h a t i t masks the g o l d / l e a d -14-c o r r e l a t i o n . Gold versus a r s e n i c s c a t t e r p l o t s of s u r f a c e t r e n c h samples were generated by Dennis P r i n c e , E x p l o r a t i o n Manager a t UKHM. He r e p o r t e d a s t r o n g c o r r e l a t i o n between these elements ( p e r s o n a l communication, 1985). U n f o r t u n a t e l y , the mine samples were not assayed f o r a r s e n i c so i t i s not p o s s i b l e to examine the go l d versus a r s e n i c r e l a t i o n s h i p i n the underground sampling. No other s i g n i f i c a n t c o r r e l a t i o n s were found. Au A3. Pb Dip Thick E l e v .06 .025 - .020 -.063 Thick -.134 .091 - -.165 -North .051 -.039 - - -Dip .004 -.072 - - -Zn .096 .508 .597 - -Pb .197 .782 - - -Ag .249 - - - -Table 4.1.1: C o r r e l a t i o n c o e f f i c i e n t s . -15-SCATTER PLOT: Au vs. North 11500 11750 12000 Norlh (ft) SCATTER PLOT: Ag vs. North 130^ 140-110-120-110-100-">1 •0 - | 70-40 -30 -40-J 0 -20-10-0-•*J.yr 11300 11730 12000 Norlh III) F i g u r e 4.1.1: S c a t t e r p l o t s of gold grade (above) and s i l v e r grade (below) are shown p l o t t e d a g a i n s t n o r t h i n g . F i v e c o n c e n t r a t i o n s of high grade assays can be seen. These i n d i c a t e the presence of ore shoots. -16-4.2 Frequency D i s t r i b u t i o n s The t h e o r y of r e g i o n a l i z e d v a r i a b l e s assumes t h a t the v a r i a n c e i s symmetrical about the mean, i e . the data i s normally d i s t r i b u t e d ( J o u r n a l and H u i j b r e g t s , 1978). Before proceeding to the c a l c u l a t i o n of semi-variograms i t was necessary to determine whether the v a r i a b l e s were normally d i s t r i b u t e d . Where the frequency d i s t r i b u t i o n s were not normal a t r a n s f o r m a t i o n was made to normalize them. Frequency d i s t r i b u t i o n s were c a l c u l a t e d on the U n i v e r s i t y of B r i t i s h Columbia's Amdahl V8 main-frame computer and p l o t t e d on the author's A t a r i 130XE micro-computer u s i n g programs w r i t t e n by the author. Frequency d i s t r i b u t i o n s were c a l c u l a t e d f o r the f o l l o w i n g v a r i a b l e s : gold grade, s i l v e r grade, t h i c k n e s s and the accumulations (grade * t h i c k n e s s ) of go l d and s i l v e r . In many p r e c i o u s metal d e p o s i t s , assays are lo g - n o r m a l l y d i s t r i b u t e d . The cumulative frequency of a log-normal d i s t r i b u t i o n when p l o t t e d on l o g - p r o b a b i l i t y paper w i l l produce a s t r a i g h t l i n e . I f , as was the case a t Venus, the p l o t i s not s t r a i g h t , a t h i r d component may be added to normalize the d i s t r i b u t i o n (Rendu, 1978). The a d d i t i o n of a "beta a d d i t i v e c o n s t a n t " to the v a r i a b l e causes the 3-parameter log-normal d i s t r i b u t i o n to p l o t as a s t r a i g h t l i n e on l o g - p r o b a b i l i t y paper. The t r a n s f o r m of a 3-parameter log-normal d i s t r i b u t i o n i s : ln(x+B). -17-The beta a d d i t i v e constant B i s c a l c u l a t e d u s i n g the formula: 2 B = m - f l * f2 £1 + £2 - 2m where: m = median (50th p e r c e n t i l e ) f l = value a t p (as low as p o s s i b l e %) f2 = value a t 100%-p (Rendu, 1978) The mean of a 3-parameter log-normal d i s t r i b u t i o n i s not the simple a r i t h m e t i c mean of the p o p u l a t i o n . I t i s c a l c u l a t e d u sing the formula: mean = m(exp(V/2))-B where: V = the n a t u r a l l o g a r i t h m i c v a r i a n c e (Rendu, 1978) The v a r i a n c e of a 3-parameter log-normal d i s t r i b u t i o n i s c a l c u l a t e d u s i n g the formula: 2 Var = mean * (e x p ( V ) - l ) ( K r i g e , 1978) -18-4.2.1 Gold Grades The r e l a t i v e frequency d i s t r i b u t i o n s of gold i n the north and south p a r t s of the mine were p l o t t e d ( F igure 4.2.1) and found t o be p o s i t i v e l y skewed d i s t r i b u t i o n s . The cumulative frequency d i s t r i b u t i o n s were then p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper ( F i g u r e s 4.2.2 and 4.2.3), and beta a d d i t i v e c onstants were c a l c u l a t e d to a d j u s t the 3-parameter log-normal d i s t r i b u t i o n s . For the north d a t a s e t a beta a d d i t i v e constant of 0.036 oz/ton was c a l c u l a t e d . The mean of the p o p u l a t i o n i s 2. 0.232 oz/ton and the v a r i a n c e i s 0.105 (oz/ton) . The south d a t a s e t , transformed u s i n g a beta a d d i t i v e c onstant of 0.054 oz/ton, has a mean of 0.116 oz/ton and a 2. v a r i a n c e of 0.017 (oz/ton) . -19-FREQUENCY DISTRIBUTION AU_N oz/ton MEAN= 8.379 UARIANCE= 0.332 FREQUENCY DISTRIBUTION AU_5 oz/ton MEAN= 8.28 UARIANCE= 8.175 F i g u r e 4.2.1: The r e l a t i v e frequency d i s t r i b u t i o n s of go l d i n the north (above) and south (below) are shown. Both are p o s i t i v e l y skewed d i s t r i b u t i o n s . -20-F i g u r e 4.2.2: Au: North The cumulative frequency d i s t r i b u t i o n of gold i n the north i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=18%, a Beta a d d i t i v e constant of 0.036 oz/ton was c a l c u l a t e d . The a d j u s t e d p l o t i s shown by a dashed l i n e . -21-PERCENTAGE 2 5 10 15 20 30 40 50 60 70 80 90 95 98 30 15 i.O i.5 5.0 5.5 6.0 6.5 7.0 PROBITS F i g u r e 4.2.3: Au: South The cumulative frequency d i s t r i b u t i o n of gold i n the south i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=26%, a Beta a d d i t i v e constant of 0.054 oz/ton was c a l c u l a t e d . The a d j u s t e d p l o t i s shown by a dashed l i n e . -22-4.2.2 S l i v e r Grades S i l v e r i s the second element of value i n the Venus mine. The r e l a t i v e frequency d i s t r i b u t i o n s of s i l v e r i n the north and south p a r t s of the mine were p l o t t e d and found t o be p o s i t i v e l y skewed ( F i g u r e 4.2.4). The cumulative frequency d i s t r i b u t i o n s were then p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper, and beta a d d i t i v e c o n s t a n t s were c a l c u l a t e d to a d j u s t the 3-parameter log-normal d i s t r i b u t i o n s . The constants were 0.624 oz/ton i n the nor t h and 0.573 oz/ton i n the south ( F i g u r e s 4.2.5 and 4.2.6) The n o r t h d a t a s e t has a mean of 4.69 oz/ton and a v a r i a n c e of 42.76 (oz/ton) . In the south the mean i s 6.67 oz/ton and the v a r i a n c e i s 74.05 (oz/ton) . I t should be noted t h a t the south p o r t i o n of the Venus d e p o s i t has a higher average s i l v e r grade than the n o r t h . T h i s i s the i n v e r s e of the t r e n d i n the g o l d a s s a y s . -23-FREQUENCY DISTRIBUTION AG-S MEAN = 8.78 UARIANCEr 117.75 Fig u r e 4.2.4: The r e l a t i v e frequency d i s t r i b u t i o n s of s i l v e r i n the north (above) and south (below) are shown. Both are p o s i t i v e l y skewed d i s t r i b u t i o n s . -24-PERCENTAGE 40 50 60 F i g u r e 4.2.5: Ag: North The cumulative frequency d i s t r i b u t i o n of s i l v e r i n the north i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=18%, a Beta a d d i t i v e c o nstant of 0.624 oz/ton was c a l c u l a t e d . The a d j u s t e d p l o t i s shown by a dashed l i n e . -25-F i g u r e 4.2.6: Ag: South The cumulative frequency d i s t r i b u t i o n of s i l v e r i n the south i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=10%, a Beta a d d i t i v e constant of 0.573 oz/ton was c a l c u l a t e d . The a d j u s t e d p l o t i s shown by a dashed l i n e . -26-4.2.3 Thickness In p r e v i o u s ore r e s e r v e c a l c u l a t i o n s of the Venus d e p o s i t , the t h i c k n e s s of the v e i n was e l i m i n a t e d as a v a r i a b l e by a v e r a g i n g a l l samples to a minimum 4 or 5 f o o t mining width. At Venus the v e i n t h i c k n e s s v a r i e s from s e v e r a l inches to over ten f e e t . Because i t i s so v a r i a b l e , t h i c k n e s s can s i g n i f i c a n t l y a f f e c t the economic v i a b i l i t y of the mine. In t h i s study, v e i n t h i c k n e s s i s t r e a t e d as a v a r i a b l e . P l o t s of the frequency d i s t r i b u t i o n s of t h i c k n e s s In the north and south p a r t s of the mine showed p e c u l i a r yet s i m i l a r bimodal d i s t r i b u t i o n s ( F i g u r e 4.2.7). There i s a p o s i t i v e l y skewed d i s t r i b u t i o n i n the n o r t h and a n e g a t i v e l y skewed d i s t r i b u t i o n i n the south. Both d i s t r i b u t i o n s have a s p i k e a t a v e i n t h i c k n e s s of 4 f e e t . These s p i k e s are thought t o be due to s p o r a d i c averaging of the v e i n to a minimum 4 f o o t mining width i n s e v e r a l of the workings. -27-FREQUENCY DISTRIBUTION TH_N MEAN= 2.47 VARIANCE:: 1.88 FREQUENCY DISTRIBUTION TH-S Feet MEAN= 3.29 UARIANCE= 2.182 Fig u r e 4.2.7: The r e l a t i v e frequency d i s t r i b u t i o n s of th i c k n e s s i n the no r t h (above) and south (below) are shown. I t can be seen t h a t both d i s t r i b u t i o n s are bi a s e d towards 4 f e e t . -28-F u r t h e r a n a l y s i s of the data was c a r r i e d out by d i v i d i n g each of the north and south d a t a s e t s i n t o upper and lower h a l v e s . The d i v i d i n g l i n e chosen f o r the north p a r t of the d e p o s i t was the 2700 l e v e l , an e l e v a t i o n of 2750 f e e t . The r e s u l t i s two very d i f f e r e n t d i s t r i b u t i o n s ( F igure 4.2.8). The lower h a l f has a normal d i s t r i b u t i o n with a sp i k e a t four f e e t while the upper h a l f has a p o s i t i v e l y skewed d i s t r i b u t i o n . I t i s the i n t e r s e c t i o n of these two d i s s i m i l a r d i s t r i b u t i o n s which r e s u l t e d i n the bi-modal curve d e s c r i b e d above. An examination of the mine plans showed t h a t of the 107 assays o b v i o u s l y averaged to 4 f e e t , 94 are i n the lower p o r t i o n of the north end of the d e p o s i t . The e x c l u s i o n of these samples r e s u l t s i n a more normal d i s t r i b u t i o n with a mean of 2.9 f e e t and a v a r i a n c e of 1.8 f t 7 " . F i g u r e 4.2.9 shows a p l o t of the cumulative frequency d i s t r i b u t i o n of the upper north a r e a . There i s 3-parameter log-normal d i s t r i b u t i o n with a beta a d d i t i v e constant of 2. 0.37 f e e t , a mean of 1.2 f e e t , and a v a r i a n c e of 0.38 f t . -29-FREQUENCY DISTRIBUTION TH_.NL MEAN- 2.89 VARIANCE- 1.79 Fi g u r e 4.2.8: The r e l a t i v e frequency d i s t r i b u t i o n s of th i c k n e s s i n the upper north (above) and the lower north (below) are shown. The upper curve i s p o s i t i v e l y skewed. The lower curve i s the d i s t r i b u t i o n a f t e r the e x c l u s i o n of the averaged samples. Both curves show a s l i g h t spike at 4 f e e t . - 3 0 -Figure 4.2.9: Th: Upper North The cumulative frequency d i s t r i b u t i o n of t h i c k n e s s i n the upper no r t h i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=8%, a Beta a d d i t i v e constant of 0.37 f e e t was c a l c u l a t e d . The ad j u s t e d p l o t i s shown by a dashed l i n e . -31-In the south, the 2800 l e v e l , an e l e v a t i o n o£ 2795 f e e t , was chosen as the d i v i d i n g l i n e . The r e s u l t a n t upper and lower d i s t r i b u t i o n s are s i m i l a r , e r r a t i c d i s t r i b u t i o n s dominated by a s p i k e a t a t h i c k n e s s of 4 f e e t (Figure 4.2.10). The mean t h i c k n e s s i n the south i s 3.29 f e e t and the v a r i a n c e i s 2.1. Thickness at Venus has been bi a s e d towards 4 f e e t . When VML d i d the o r i g i n a l ore r e s e r v e estimate, a minimum mining width of 4 f e e t was used and many of the assays were d i l u t e d to t h a t t h i c k n e s s . Whether c o n s c i o u s l y or not, i t i s l i k e l y t h a t t h i s a f f e c t e d the l a t e r sampling of the development d r i f t s and r a i s e s . In the north p a r t of the mine, the b i a s of the d i s t r i b u t i o n s has been l a r g e l y e l i m i n a t e d by d i v i d i n g the p o p u l a t i o n i n h a l f about the 2750 f o o t e l e v a t i o n and e l i m i n a t i n g the samples o b v i o u s l y averaged to 4 f e e t . In the south, the problem i s more complex. No advantage c o u l d be gained by d i v i d i n g the south i n t o two p o p u l a t i o n s s i n c e the b i a s appears to be u n i v e r s a l and the upper and lower d i s t r i b u t i o n s are s i m i l a r . The south p a r t of the Venus mine has seen the m a j o r i t y of the s t o p i n g and r a i s i n g . The sampling of the v e i n i n these workings, mined 4 f e e t high, has b i a s e d the southern d i s t r i b u t i o n s more than those of the n o r t h . -32-R 25 1 E L A 20 T I E i 5 * i o H 5 -0 FREQUENCY DISTRIBUTION TH_SU 1 1 1 j 1 1 1 1 1 1.5 3 4.5 6 7.5 F e e t MEAN- 2.99 VARIANCE- 2.09 F i g u r e 4.2.10: The r e l a t i v e frequency d i s t r i b u t i o n s of t h i c k n e s s i n the upper south (above) and the lower south (below) are shown. The upper curve i s biased towards 4 f e e t . The lower curve i s dominated by a spi k e a t 4 f e e t . -33-4.2.4 Gold Accumulation Accumulation i s the product of grade and t h i c k n e s s . At Venus, v e i n t h i c k n e s s i s v a r i a b l e and every grade i s measured over a d i f f e r e n t t h i c k n e s s . Accumulation i s used to i n c o r p o r a t e the v a r i a b i l i t y of grade and t h i c k n e s s i n t o a s i n g l e v a r i a b l e . The frequency d i s t r i b u t i o n s of the accumulation of gold grade were c a l c u l a t e d f o r the north and south p a r t s of the Venus mine (Figure 4.2.11). As was the case with the gold grades, the frequency d i s t r i b u t i o n s of gold accumulation are p o s i t i v e l y skewed d i s t r i b u t i o n s i n both p a r t s of the mine. Cumulative frequency d i s t r i b u t i o n s of gold accumulation were p l o t t e d f o r the north and south and both were found to be 3-parameter log-normal d i s t r i b u t i o n s ( F i g u r e s 4.2.12 and 4.2.13). Beta a d d i t i v e c o n s t a n t s , 0.135 f t oz/ton i n the nor t h and 0.1425 f t oz/ton i n the south, were used to normalize the two d a t a s e t s . The mean and v a r i a n c e of gold accumulation i n the north were 2. c a l c u l a t e d to be 0.398 f t oz/ton and 0.208 ( f t oz/ton) r e s p e c t i v e l y . In the south, the mean i s 0.484 f t oz/ton and the v a r i a n c e i s 0.456 ( f t oz/ton) . A f t e r n o t i n g a b i a s i n the d i s t r i b u t i o n s of t h i c k n e s s , i t was expected t h a t a s i m i l a r b i a s would appear i n the accumulation of g o l d . T h i s was not the case. The v a r i a n c e of g old grade i s so high, r e l a t i v e to the mean, that i t overshadows any b i a s t h a t may e x i s t i n t h i c k n e s s . -34-FREQUENCY DISTRIBUTION AU—ACC_N 2 3 oz/ton * f t MEAN- 0.787 VARIANCE= 1.351 FREQUENCY DISTRIBUTION R 25 -1 AU_ACC_S L " A 28 " oz/ton * f t MEAN- 0.993 VARIANCE^ 2.614 F i g u r e 4.2.11: The r e l a t i v e frequency d i s t r i b u t i o n s of gold accumulation i n the north (above) and south (below) are shown. Both are p o s i t i v e l y skewed d i s t r i b u t i o n s . - 3 5 -PERCENTAGE 40 50 60 4 3 2 3 2 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 PROBITS F i g u r e 4.2.12: Au-Acc: North The cumulative frequency d i s t r i b u t i o n of gold accumulation in the north i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=19%, a Beta a d d i t i v e constant of 0.135 f t oz/ton was c a l c u l a t e d . The adju s t e d p l o t i s shown by a dashed l i n e . -36-F i g u r e 4.2.13: Au-Acc: South The cumulative frequency d i s t r i b u t i o n of gold accumulation i n the nor t h i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=24%, a Beta a d d i t i v e c o n s t a n t of 0.1425 f t oz/ton was c a l c u l a t e d . The a d j u s t e d p l o t i s shown by a dashed l i n e . -37-4.2.5 S l i v e r Accumulation The frequency d i s t r i b u t i o n s of the s i l v e r accumulation were c a l c u l a t e d f o r the north and south p a r t s of the Venus mine (Figure 4.2.14). As was the case with the s i l v e r grades, the frequency d i s t r i b u t i o n s of s i l v e r accumulation are p o s i t i v e l y skewed d i s t r i b u t i o n s i n both p a r t s of the mine. Cumulative frequency d i s t r i b u t i o n s of s i l v e r accumulation were p l o t t e d f o r the north and south and both were found to be 3-parameter log-normal d i s t r i b u t i o n s ( F i g u r e s 4.2.15 and 4.2.16). Beta a d d i t i v e c onstants of 2.405 f t oz/ton i n the n o r t h and 2.453 f t oz/ton i n the south, were used to normalize the two d a t a s e t s . The mean and v a r i a n c e of gold accumulation i n the north were 2. c a l c u l a t e d to be 8.57 f t oz/ton and 149.4 ( f t oz/ton) r e s p e c t i v e l y . In the south, the mean i s 20.4 f t oz/ton and 2. the v a r i a n c e i s 834.1 ( f t oz/ton) . A f t e r n o t i n g a b i a s i n the d i s t r i b u t i o n s of t h i c k n e s s , i t was expected t h a t a s i m i l a r type of b i a s would appear i n the accumulation of s i l v e r . As i s the case with g o l d , the v a r i a n c e of s i l v e r grades i s so high, r e l a t i v e to the mean, th a t i t overshadows the b i a s i n t h i c k n e s s . -38-R E L A T I v E y. 30 i 24 " 18 " 12 6 -0 FREQUENCY DISTRIBUTION AG-ACC_N i — i — r 24 48 oz/ton * f t 128 MEAN- 19.78 VARIANCE- 1143.7 R E L A T I V E 30 1 24 " 18 " FREQUENCY DISTRIBUTION AG-ACC-S y. i 1 — r 48 72 oz/ton * f t r 96 120 MEAN- 30.3 VARIANCE- 1857 F i g u r e 4 .2.14: The r e l a t i v e frequency d i s t r i b u t i o n s of s i l v e r accumulation i n the north (above) and south (below) are shown. Both are p o s i t i v e l y skewed d i s t r i b u t i o n s . - 3 9 -F i g u r e 4.2.15: Ag-Acc: North The cumulative frequency d i s t r i b u t i o n of s i l v e r accumulation i n the nor t h i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=28%, a Beta a d d i t i v e constant of 2.405 f t oz/ton was c a l c u l a t e d . The ad j u s t e d p l o t i s shown dashed. - 4 0 -F i g u r e 4.2.16: Ag-Acc: South The cumulative frequency d i s t r i b u t i o n of s i l v e r accumulation i n the north i s shown as a s o l i d l i n e p l o t t e d on 3-cycle l o g - p r o b a b i l i t y paper. Using p=13%, a Beta a d d i t i v e constant of 2.453 f t oz/ton was c a l c u l a t e d . The adjust e d p l o t i s shown by a dashed l i n e . -41-4.3 Summary and Conclusions The mean and v a r i a n c e of each v a r i a b l e are summarized i n Table 4.3.1. I t can be seen t h a t gold grade i s higher n o r t h of the hinge l i n e than i t i s to the south. The in v e r s e i s true of s i l v e r which i s higher i n the south. The v e i n i s t h i c k e s t i n the south and i n the lower n o r t h . T h i s r e s u l t s i n the average accumulations of g o l d and s i l v e r being higher i n the south. I t should a l s o be noted that the v a r i a n c e s of both the gold and s i l v e r assays are c o n s i d e r a b l y higher r e l a t i v e to t h e i r means, than those of t h i c k n e s s . For t h i s reason the b i a s i n the t h i c k n e s s d i s t r i b u t i o n s does not a f f e c t the d i s t r i b u t i o n s of the accumulations. Element Mean Var iance Beta Au-N 0.232 0 .105 0 .036 Au-S 0.116 0.017 0 .054 Ag-N 4.69 42.76 0 . 624 Ag-S 6.67 74.05 0.573 Th-NU 1.2 0 . 38 0 . 37 Th-NL 2.9 1.8 -Th-S 3.29 2.1 -Au:Acc-N 0 . 398 0.208 0.135 Au:Acc-S 0. 484 0. 456 0.1425 Ag:Acc-N 8.57 149 . 4 2 .405 Ag:Acc-S 20. 40 834 .1 2 .453 Table 4.3.1: Summary of the mean and v a r i a n c e of a l l the elements and accumulations. -42-5. GEOSTATISTICS 5.1 Theory ot R e g i o n a l i z e d V a r i a b l e s G e o s t a t i s t i c s i s the a p p l i c a t i o n of the theory of r e g i o n a l i z e d v a r i a b l e s to the s c i e n c e of ore reserve e s t i m a t i o n . "A random f u n c t i o n Z(x) can be seen as a s e t of random v a r i a b l e s Z ( x ( i ) ) d e f i n e d a t each p o i n t x ( i ) of the d e p o s i t D.... The random v a r i a b l e s Z ( x ( i ) ) are c o r r e l a t e d and t h i s c o r r e l a t i o n depends on both the ve c t o r h (modulus and d i r e c t i o n ) s e p a r a t i n g two p o i n t s x ( i ) and x(i+h) and the nature of the v a r i a b l e c o n s i d e r e d . " ( J o u r n e l and H u i j b r e g t s 1978 p.11) In t h i s way a r e g i o n a l i z e d v a r i a b l e "has p r o p e r t i e s i n t e rmediate between a t r u l y random v a r i a b l e and one completely d e t e r m i n i s t i c . " (Davis, 1986 p.239) The main t o o l f o r mo d e l l i n g the c o r r e l a t i o n of a r e g i o n a l i z e d v a r i a b l e i s the semi-variogram. The semi-variance, the mean squared d i f f e r e n c e between a l l p o i n t s a " l a g l e n g t h " or s e p a r a t i o n h u n i t s a p a r t , i s c a l c u l a t e d using the formula: n 2 Gamma(h) = 1 y ( Z ( x ( i ) ) - Z ( x ( i + h ) ) ) 2n i = l where n = the number of p a i r s Z ( x ( l ) ) = the value of the v a r i a b l e a t p o i n t x ( i ) h = l a g le n g t h By c a l c u l a t i n g Gamma(h) f o r many m u l t i p l e s of h i t i s p o s s i b l e to graph the experimental semi-variogram curve. -43-The f i t t i n g of a model to the experimental curve enables one to q u a n t i f y s e v e r a l c h a r a c t e r i s t i c s of the p o p u l a t i o n under examination. (Figure 5.1.1.) The p o i n t a t which the model i n t e r s e c t s the gamma(h) a x i s i s c a l l e d the "nugget e f f e c t " (Co). At t h i s p o i n t h=0. The nugget e f f e c t i s t h e r e f o r e an i n d i c a t i o n of the randomness of the v a r i a b l e or the r e p r o d u c e a b i l i t y of assays. The value of gamma(h) a t which the semi-variogram model l e v e l s o f f i s c a l l e d the s i l l (C). In theory, the sum of the s i l l and the nugget e f f e c t i s equal to the po p u l a t i o n v a r i a n c e . The d i s t a n c e a t which the model curve reaches the s i l l i s c a l l e d the "range of i n f l u e n c e " ( a ) . T h i s i s the d i s t a n c e up to which the v a r i a b l e i s s p a t i a l l y dependant and beyond which i t i s independant. I f a semi-variogram's s i l l equals the nugget e f f e c t then the p o p u l a t i o n i s random and the best estimate of such a v a r i a b l e i s simply the a r i t h m e t i c mean of the p o p u l a t i o n . -44-1*5_ EXAMPLE A = 55 C = 8.7 | Co= 8.3 88 — I — r 128 —1 1 1 168 288 H Cfeet) F i g u r e 5.1.1: A sample semi-variogram model i s shown above. The nugget e f f e c t (Co) i s denoted by the lower dashed l i n e , the s i l l (C) by the s o l i d h o r i z o n t a l l i n e and the range (A) by the dashed v e r t i c a l l i n e . The upper dashed l i n e i s the p o p u l a t i o n v a r i a n c e (1.0). In t h i s example the sum of the s i l l and nugget e f f e c t i s equal to the v a r i a n c e . The u n i t s of gamma(h) i s the square of the u n i t s of the v a r i a b l e ( f e e t , ounces, l n ( o z ) , e t c .) -45-S e v e r a l models can be f i t t e d to the experimental semi-variogram to mathematically r e p r e s e n t the c o r r e l a t i o n of a r e g i o n a l i z e d v a r i a b l e . The most common models are "e x p o n e n t i a l " and " s p h e r i c a l " . The e x p o n e n t i a l model (Figure 5.1 . 2 ) has the equation: Gamma(h) = C(1-exp(-h/a)) + Co The s p h e r i c a l model (Figure 5.1 . 2 ) has the equation: Gamma(h) = C^3h _ 1 ^ ^+ Co f o r h<=a Gamma(h) = C + Co f o r h>=a In some d e p o s i t s i t has been found t h a t two nested s p h e r i c a l models are i n e f f e c t . One i s a l o c a l , s h o r t range model with a range of a l and a s i l l of CI while the other i s a more r e g i o n a l , longer range model with range a2 and s i l l C2 (Figure 5.1 . 3 ) . The equation of such a model i s : Gamma (h) = C 2 / 3h _ l / h \ \ + C l / l h _ l / h \ \+ Co f o r h<=al I 2a2 2J^a2J J i . 2 a l 2(al) j Gamma(h) = C2/^3h _ l / h X \+ CI + Co f o r al<=h<=a2 \2a2 2^2J / Gamma(h) = C2 + CI + Co f o r h>=a2 ( S i n c l a i r and Deraisme, 1974) -46-1.5, EXAMPLE SPHERICAL MODEL CO-0.3 C=8. 7 A=5 5 40 80 — i — r 120 — I — 168 I 200 H (feet) i.5_ 0 EXAMPLE EXPONENTIAL MODEL CO-0.3 C1=0.7 A=55 ~T~ 40 80 T —I 1 1— 120 160 T 1 200 H Cfeet) Figure 5.1.2: Two semi-variogram models are shown: a s p h e r i c a l model above and an ex p o n e n t i a l model below. Both models have the same nugget e f f e c t (Co), s i l l (C) and range (A). The p o p u l a t i o n v a r i a n c e , shown by a dashed l i n e , i s 1.0 i n each case. -47-EXAMPLE 28 108 1 1 1 1 1 1 1 1 1 48 88 128 168 288 H Cfeet) 1.5.. G 1.2-H -M 8.9-A _ H 8.6. 8.3-EXAMPLE JPHEHICAL M6PEL UL co=e ci=e.36 oi=2e ~ee ' 128 H tfeet) 1.5-G A 1.2. H M -H 0.9-A H 8.6. 0.3-EXAMPLE I — 48 T T " — 1 — 168 288 SPHERICAL MODEL »2 "C0=8 C2=B.S 02=188 128 H C fee t ) 48 —r— 88 8 T 280 F i g u r e 5.1.3: A nested s p h e r i c a l semi-variogram model i s shown above. The two s p h e r i c a l models which nest together to form the nested model are shown below, l e f t and r i g h t . The p o p u l a t i o n v a r i a n c e , shown by a dashed l i n e , i s 1.0. - 4 8 -5.2 Procedure Programs w r i t t e n by the author were used t o c a l c u l a t e semi-varIograms on the UBC main-frame computer. The h and gamma(h) data were then entered i n t o the author's A t a r i 130XE micro computer where the experimental semi-variograms were modelled. The v a r i a b l e s used i n the c a l c u l a t i o n s were t h i c k n e s s and the log-transformed accumulations (grade * t h i c k n e s s ) of g o l d and s i l v e r . In c a l c u l a t i n g semi-variograms of t h i c k n e s s , the raw data was used i n the lower n o r t h and i n the south, while log-transformed data was used i n the upper north a r e a . Semi-variograms were c a l c u l a t e d i n four p r e f e r r e d d i r e c t i o n s to t e s t f o r a n i s o t r o p y of the m i n e r a l i z a t i o n . The four d i r e c t i o n s chosen were along the s t r i k e of the v e i n , up and down d i p and a t 45 degrees to each. T h e r e f o r e i n the n o r t h p a r t of the v e i n , the chosen d i r e c t i o n s were 0, 45, 90 and 135 degrees with a t o l e r a n c e angle of +/-22.5 degrees. In the south, semi-variograms were c a l c u l a t e d a t 15, 60, 105 and 150 +/- 22.5 degrees. The ranges of the semi-variogram models f i t t e d t o the experimental curves of a p a r t i c u l a r v a r i a b l e i n a data s e t were then p l o t t e d as r a d i i and an e l i p s e was drawn through the ends of the r a d i i . The major and minor axes of t h i s e l i p s e became the major and minor axes of the geometric a n i s o t r o p y of t h a t v a r i a b l e . In t heory, the sum of the s i l l and nugget e f f e c t of a -49-semi-variogram model equals the p o p u l a t i o n v a r i a n c e and i s the same i n every d i r e c t i o n . In some cases, the s i l l of the semi-variogram v a r i e s with the d i r e c t i o n being c o n s i d e r e d , r e s u l t i n g i n a "zonal a n i s o t r o p y " . A lower s i l l i s observed i n the d i r e c t i o n of the "ore shoots" and a higher s i l l normal to the ore shoots. - 5 0 -5.3 S t a t i o n a r i t y and the I n t r i n s i c Hypothesis In order to make v a l i d s t a t i s t i c a l i n f e r e n c e s from a s e t of data i t must f i r s t be proven t h a t the data s a t i s f y the laws of s t a t i o n a r i t y . Second order s t a t i o n a r i t y r e q u i r e s t h a t the mean e x i s t s and i s independent of any p o i n t i n space and t h a t the c o v a r i a n c e e x i s t s and depends o n l y on the s e p a r a t i o n d i s t a n c e h. T h i s i m p l i e s the s t a t i o n a r i t y of the v a r i a n c e and the semi-variogram ( J o u r n a l and H u i j b r e g t s , 1978). In g e o s t a t i s t i c a l a n a l y s e s i t i s necessary to s a t i s f y the i n t r i n s i c h y p o thesis o n l y . That i s , t h a t the semi-variogram (gamma(h)) e x i s t s and depends o n l y on the s e p a r a t i o n d i s t a n c e h. To prove the i n t r i n s i c h y p o thesis a t Venus the nor t h and south p a r t s of the d e p o s i t were each d i v i d e d i n t o q u a r t e r s . Experimental semi-variograms were c a l c u l a t e d f o r each v a r i a b l e i n each of the quadrants, then p l o t t e d and compared t o each o t h e r . I f a v a r i a b l e ' s semi-variogram was s i m i l a r i n two or more quadrants, the i n t r i n s i c h y p o thesis was assumed i n those quadrants. For the i n v e s t i g a t i o n of s t a t i o n a r i t y , the nor t h and south p a r t s of the d e p o s i t were d i v i d e d i n t o quadrants. In the n o r t h , the g r i d l i n e s 11,500 N and 50,150 E were used. The quadrants were l a b e l l e d N l t o N4, begin n i n g In the south-west and proceeding c l o c k w i s e from t h e r e . In the south, l i n e s p a r a l l e l t o the s t r i k e (60 degrees) and d i p of the v e i n were used. The equations of -51-these l i n e s a r e : N = 88,666 - 1.561 * E and N = 0.644 * E - 21,200 The southern quadrants were l a b e l l e d SI to S4, agai n beginning i n the west and proceeding c l o c k w i s e from t h e r e . See F i g u r e 5.3.1. -52-6. SEMI-VARIPGRAM RESULTS: THICKNESS Experimental semi-variograms of t h i c k n e s s were c a l c u l a t e d i n three a r e a s . The north p a r t of the Venus v e i n was separated from the south p a r t a t the hinge l i n e . The north was f u r t h e r d i v i d e d i n t o a l o g - n o r m a l l y d i s t r i b u t e d upper h a l f and a normally d i s t r i b u t e d lower h a l f above and below the 2750 f o o t e l e v a t i o n l i n e . 6.1.1 Thickness i n the Upper North In the upper north area, a beta a d d i t i v e constant of 0.37 f e e t was used to normalize the p o p u l a t i o n . Experimental semi-variograms were c a l c u l a t e d a t 22, 67, 112 and 157 (+/- 22.5) degrees as a t e s t f o r a n i s o t r o p y (Figure 6.1.1). The t h i c k n e s s semi-variogram i n the upper north was found to be i s o t r o p i c when very s i m i l a r semi-variogram models c o u l d be f i t to a l l of the d i r e c t i o n a l c urves. A nested s p h e r i c a l model with a range of 150 f e e t provided the best f i t to the g e n e r a l curve (Table 6.1.1). Area/Az Co CI a l C2 aJ2 NU 0.075 0.09 20 0.04 150 Table 6.1.1: Semi-variogram parameters of t h i c k n e s s i n the upper no r t h r e g i o n of the Venus mine. -54-0.2-0.1. L -TH-NU.22 NESTED SPHERICAL MODEL CO=0.075 Cl=0 .11 01=25 C2=0.035 02=150 40 80 T I 120 —I— 160 —r 200 H CfeetJ M M 0 H 2-0 . 1 . L - T H - N U . 6 7 NESTED SPHERICAL MODEL Co=0.075 Cl=0 .085 01=20 C2=8.045 02=150 I 40 — r -80 T i — r 120 —I— 160 200 H (feet! I TH-NU Co=0.075 C l = 0 . 0 9 01=20 C2=0.04 02=150 —I 1 1 1 1 1 1 1 1 40 80 120 160 200 F i g u r e 6.1.1: Semi-variograms of log-transformed t h i c k n e s s i n the upper north were c a l c u l a t e d a t 22, 67, 112 and 157 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. Since the model curves are ve r y s i m i l a r i n each d i r e c t i o n , i t was concluded that the semi-variogram i s i s o t r o p i c . The gen e r a l curve, o v e r l a i n by the best f i t t i n g nested s p h e r i c a l model, i s shown below. -55-6.1.2 S t a t i o n a r i t y of Thickness i n the Upper North To t e s t f o r s t a t i o n a r i t y , the upper north area was d i v i d e d i n t o halves (NU1 and NU2), nor t h and south of the l i n e 11,500 N. The subsets of the upper area s a t i s f y the i n t r i n s i c h ypothesis and the requirements of s t a t i o n a r i t y . The v a r i a n c e s of the subsets vary l i t t l e from those of the whole (Table 6.1.2) and the semi-variograms are s i m i l a r to the g e n e r a l model ( F i g u r e 6.1.2). Area Mean Variance Pts NU 1.203 0.234 1007 NU1 1.184 0.209 458 NU2 1.213 0.245 549 Table 6.1.2: Mean t h i c k n e s s , v a r i a n c e and number of p o i n t s i n each of the halves of the upper n o r t h . The i s o t r o p i c g e n e r a l model d e s c r i b e d above was used to r e p r e s e n t the upper north a r e a . -56-F i g u r e 6.1.2: Semi-variograms of log-transformed t h i c k n e s s i n areas NU1 and NU2 are shown o v e r l a i n by the g e n e r a l model (above) and the best f i t t i n g semi-variogram models (below). I t can be seen that the curves are very s i m i l a r and s a t i s f y the i n t r i n s i c h y p o t h e s i s . -57-6.2.1 Thickness i n the Lower North Semi-variograms of t h i c k n e s s i n the lower north were c a l c u l a t e d a t 22, 67, 112 and 157 (+/- 22.5) degrees as a t e s t f o r a n i s o t r o p y (Figure 6.2.1). The t h i c k n e s s semi-variogram models f i t to the curves were very s i m i l a r i n each d i r e c t i o n . I t was t h e r e f o r e concluded t h a t the t h i c k n e s s semi-variogram i n the lower north i s i s o t r o p i c and best f i t by a nested s p h e r i c a l model with a range of 150 f e e t . The model i s summarized i n Table 6.2.1. Area/Az Co CI a l C2 a2. NL 0.5 0.8 25 0.5 150 Table 6.2.1: Semi-variogram parameters of t h i c k n e s s i n the lower north r e g i o n of the Venus mine. -58-TH—NL H Cfeet) F i g u r e 6.2.1: Semi-variograms of t h i c k n e s s i n the lower north were c a l c u l a t e d a t 22, 67, 112 and 157 degrees to t e s t for a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen t h a t , s i n c e a l l of the models are v e r y s i m i l a r , the semi-variogram i s i s o t r o p i c . The i s o t r o p i c g e n e r a l curve with the best f i t t i n g nested s p h e r i c a l model i s a l s o shown. -59-6.2.2 S t a t i o n a r i t y of Thickness i n the Lower North To t e s t f o r s t a t i o n a r i t y , the lower north area was d i v i d e d i n t o halves (NL1 and NL2), north and south of the l i n e 11,500 N. Although the v a r i a n c e s of the subsets vary l i t t l e from that of the whole (Table 6.2.2), the semi-variograms are q u i t e d i f f e r e n t ( F igure 6.2.2). The r e l a t i v e s i z e of the C l and C2 parameters i n NL1 i s the i n v e r s e of those i n NL2. Area Mean Var iance Pts NL 2.89 1.792 1224 NL1 2.95 1.572 380 NL2 2.86 1.89 844 Table 6.2.2: Mean t h i c k n e s s , v a r i a n c e and number of p o i n t s i n the halves of the lower n o r t h . Since the subsets of the lower area do not s a t i s f y the i n t r i n s i c h y p o t h e s i s , they were t r e a t e d as separate p o p u l a t i o n s and each was t e s t e d f o r a n i s o t r o p y . -60-TH-HL1 H ( f e e t ) TH-NL2 G G A A M - M M 1.2- M A A H 0 .8 . H 8.4-' NESTED SPHERICAL MODEL C0=8.5 C l = 8 . 8 A l = 25 - C2=8.5 62=158 8 1 1 1 48 1 1 1 88 128 1 1 1 1 168 288 8 .8 . NESTED SPHERICAL MODEL C0=0.5 C l = 0 . 8 Al=25 C2=8.5 A2=158 48 i 88 T I 128 H C f e e t l —I— 168 T 1 288 2 A M TH—NLl 8.8. NESTED SPHERICAL MODEL Co=8.4 C l = 8 . 7 Al=20 C2=8.47 A2=188 48 88 128 H ( f e e t ) 168 288 TH_NL2 8.4-NESTED SPHERICAL MODEL C0=8.S C l = 9 . 5 A l = 1 7 . 5 C2=8.9 02=80 -1— 48 —J— 88 T I 120 T 168 i — r 288 H ( f e e t ) F i g u r e 6.2.2: Semi-variograms of th i c k n e s s i n areas NL1 and NL2 are shown o v e r l a i n by: the general model (above) and the best f i t t i n g semi-variogram models (below). I t can be seen that the curves are d i f f e r e n t and do not s a t i s f y the i n t r i n s i c h y p o t h e s i s . -61-6.2.3 A n i s o t r o p y of Thickness i n the Lower North Areas Semi-variograms of t h i c k n e s s i n the lower north areas NL1 and NL2, were c a l c u l a t e d a t 0, 45, 90 and 135 (+/-22.5) degrees to t e s t f o r a n i s o t r o p y . Thickness i n area NLl was found to have axes of a n i s o t r o p y which b i s e c t the above c a l c u l a t e d d i r e c t i o n s . The major a x i s trends 120 degrees and the minor a x i s i s 30 degrees. In NL2, the major and minor axes trend 90 and 0 degrees r e s p e c t i v e l y (Table 6.2.3) . Area/Az Co CI a l C2 a2 NLl 30 0.0 1.57 22.5 NLl 120 0.0 1 . 0 15 0.57 135 NL2 0 0.4 1.25 30 NL2 90 0.65 0.6 70 0.65 150 Table 6.2.3: Semi-variogram parameters of t h i c k n e s s i n the lower n o r t h areas NLl and NL2. The t h i c k n e s s semi-variograms i n NLl are best f i t by nested s p h e r i c a l models with axes of geometric a n i s o t r o p y t r e n d i n g 30 and 120 degrees (Figure 6.2.3). The range of i n f l u e n c e i n the d i r e c t i o n of the major a x i s (120 degrees) i s 135 f e e t . Along the minor a x i s (30 degrees) the range i s 22.5 f e e t . The minor and major axes of a n i s o t r o p y are very c l o s e to the s t r i k e and d i p d i r e c t i o n s r e s p e c t i v e l y . R e c a l l t h a t north of the hinge l i n e the Venus v e i n s t r i k e s 25 degrees. -62-In area NL2, the axes of a n i s o t r o p y are 0 and 90 degrees ( F i g u r e 6.2.4). The a n i s o t r o p y i s zo n a l with the nugget e f f e c t c o n t r o l l i n g the s i l l h e i g h t . The semi-variogram curve i n the d i r e c t i o n of the minor a x i s (0 degrees) i s best f i t by a simple s p h e r i c a l model with a range of 30 f e e t and a s i l l (Co + CI) of 1.65. In the d i r e c t i o n of the major a x i s (90 deg r e e s ) , the t h i c k n e s s semi-variogram i s best f i t by a nested s p h e r i c a l model with a range of 150 f e e t and a s i l l of 1.9. The he i g h t of the s i l l i n the d i f f e r e n t d i r e c t i o n s i s c o n t r o l l e d by the nugget e f f e c t . In the 0 degree model Co=0.4, while a t 90 degrees Co=0.65. -63-TH_NLl.e NESTED SPHERICAL MODEL CO=8.5 C l=8 .35 01=12.5 C2=0.72 02=45 ~i—r 40 T —I— 80 120 H C fee t ) T 160 T 200 2 . 5 , T H - N L 1 . 4 5 H C fee t ) 2.S. TH-NL1 .135 G A M M A 1.5-NESTED SPHERICAL MODEL Co=0 C l = l 01=15 C2=0.57 02=110 40 80 120 H Cfee t ) 160 200 2.5. 1.5-1 0.5-T H - N L 1 . 9 0 NESTED SPHERICAL MODEL C0=0.25 C l = 0 . 4 5 01=12.S C2=0.87 02=95 7e T 80 H C fee t ) ~T r 120 - 1 — 160 —I 200 2.5_ T H - N L 1 . 3 0 NESTED SPHERICAL MODEL CO=B C l = 1 . 5 7 01=22.5 C2=0 02=22.S T— 40 T -r— 80 - i — r 120 —I— 160 280 H C fee t ) 2.S. 5 2 ; T H - N L 1 . 1 2 0 M 1.5- 7l~ H * • NESTED SPHERICAL MODEL C0=8 C l = l 01=15 C2S0 .57 02=135 "To" T - T 88 T —i—r 120 —I— 160 —I 200 H C fee t ) Figure 6.2.3: Semi-variograms of t h i c k n e s s i n area N Ll, c a l c u l a t e d a t 0, 45, 90 and 135 degrees, are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen that the major and minor axes of a n i s o t r o p y b i s e c t these trends a t 120 and 30 degrees r e s p e c t i v e l y . Semi-variograms c a l c u l a t e d i n the d i r e c t i o n of the major and minor axes are a l s o shown. -64-TH-HL2 .135 2 g TH-ML2.90 H ( f e e t ) H ( f e e t ) Figure 6.2.4: Semi-variograms of t h i c k n e s s i n area NL2, c a l c u l a t e d a t 0, 45, 90 and 135 degrees, are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen that the major and minor axes of a n i s o t r o p y trend 90 and 0 degrees r e s p e c t i v e l y . The e f f e c t of Co on the s i l l height i s a l s o v i s i b l e . -65-6.3.1 Thickness i n the South Thickness semi-variograms of the Venus v e i n south of the hinge l i n e were c a l c u l a t e d a t 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y (Figure 6.3.1). The semi-variogram was best f i t by a nested s p h e r i c a l model with major and minor axes of the geometric a n i s o t r o p y t r e n d i n g 15 and 105 degrees r e s p e c t i v e l y (Table 6.3.1). The s t r i k e and d i p d i r e c t i o n s of the v e i n i n t h i s p a r t of the mine are 60 and 150 degrees r e s p e c t i v e l y . Area/Az Co CI a l C2 a2 S 15 0.0 1.0 10 1.1 100 S 105 0.25 1.0 20 0.85 37.5 Table 6.3.1: Semi-variogram parameters of t h i c k n e s s i n the d i r e c t i o n s of the a n i s o t r o p y axes. -66-H t f e e t J H CfeetJ F i g u r e 6.3.1: Semi-variograms of t h i c k n e s s i n the south were c a l c u l a t e d a t 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen t h a t the semi-variogram i s a n i s o t r o p i c with the major and minor axes of a n i s o t r o p y t r e n d i n g 15 and 105 degrees r e s p e c t i v e l y . -67-6.3.2 S t a t i o n a r i t y of Thickness i n the South South of the hinge l i n e , the d a t a s e t was d i v i d e d i n t o q u a r t e r s u sing the l i n e s d e s c r i b e d i n Chapter 5. The b i a s e d d i s t r i b u t i o n d e s c r i b e d i n Chapter 4 had an adverse e f f e c t on the t h i c k n e s s semi-variogram curves of the southern quadrants. Thickness i n the quadrants has d i f f e r i n g v a r i a n c e s (Table 6.3.2) and l i t t l e s i m i l a r i t y i s seen between the semi-variograms ( F i g u r e s 6.3.2 and 6.3.3). Area Mean Var iance Pts S 3.29 2.10 758 51 3.44 2.54 178 52 3.66 1.61 144 53 2.93 1.71 253 54 3.36 2.31 183 Table 6.3.2: Mean t h i c k n e s s , v a r i a n c e and number of p o i n t s i n each of the south quadrants. I t must be concluded t h a t t h i c k n e s s i n the southern p o r t i o n of the mine f a i l s to s a t i s f y the i n t r i n s i c h y p o t h e s i s . Each quadrant was t h e r e f o r e modelled and k r i g e d s e p a r a t e l y . -68-Figure 6.3.2: Semi-variograms of t h i c k n e s s i n the southern quadrants c a l c u l a t e d a t 15 degrees, the d i r e c t i o n of the major a x i s of a n i s o t r o p y , are shown o v e r l a i n by the general 15 degree model. I t can be seen t h a t the curves are d i f f e r e n t and do not s a t i s f y the requirements of s t a t i o n a r i t y . -69-F i g u r e 6.3.3: Semi-variograms of t h i c k n e s s i n the southern quadrants c a l c u l a t e d at 105 degrees, the d i r e c t i o n of the minor a x i s of a n i s o t r o p y , are shown o v e r l a i n by the g e n e r a l 105 degree model. I t can be seen that the curves are d i f f e r e n t and do not s a t i s f y the requirements of s t a t i o n a r i t y . - 7 0 -6.3.3 A n i s o t r o p y of Thickness i n the Southern Quadrants Thickness was t e s t e d f o r a n i s o t r o p y i n each of the southern quadrants. Semi-variograms were c a l c u l a t e d a t 15, 60, 105 and 150 degrees. In area SI, the semi-variogram of t h i c k n e s s i s best f i t by a simple s p h e r i c a l model with axes of geometric a n i s o t r o p y t r e n d i n g 15 and 105 degrees (Figure 6.3.4). The semi-variogram c a l c u l a t e d i n the d i r e c t i o n of the major a x i s , 15 degrees, has a range of 70 f e e t while a t 105 degrees the range i s 25 f e e t . The t h i c k n e s s semi-variogram of area S2 shows a geometric a n i s o t r o p y with major and minor axes t r e n d i n g 15 and 105 degrees. A s p h e r i c a l model, with a range of 40 f e e t , f i t the 15 degree t r e n d , while one with a range of 10 f e e t f i t the 105 degree trend (Figure 6.3.5). In area S3, the t h i c k n e s s semi-variogram d i s p l a y s zonal a n i s o t r o p y . The axes of a n i s o t r o p y were found to l i e between the d i r e c t i o n s c a l c u l a t e d above. The semi-variogram a t 35 degrees (the minor a x i s ) has a range of 20 f e e t and a s i l l of 1.55, while a t 125 degrees the range i s 50 f e e t and the s i l l i s 1.85 (Figure 6.3.6). Both semi-variograms are simple s p h e r i c a l models. The t h i c k n e s s semi-variogram of S4 i s g e o m e t r i c l y a n i s o t r o p i c . The 60 degree curve i s best f i t by a s p h e r i c a l model with a s i l l of 2.3 and a range of 100 f e e t . At 150 degrees, the semi-variogram model i s a l s o s p h e r i c a l with a range of 25 f e e t ( Figure 6.3.7). -71-The semi-variogram models are summarized i n Table 6.3.3. Area/Az Co CI a l C2 SI 15 0.4 2.14 70 SI 105 0.0 2.54 25 -S2 15 0.0 1.6 40 _ S2 105 0.0 1.6 10 -S3 35 0 . 55 1.0 20 _ S3 125 0.55 1.3 50 -S4 60 0 . 4 1.9 100 — S4 150 0.2 2.1 25 -Table 6.3.3: Semi-variogram parameters of t h i c k n e s s i n southern quadrants. -72-F i g u r e 6.3.4: Semi-variograms of t h i c k n e s s i n area SI were c a l c u l a t e d a t 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen t h a t the semi-variogram i s a n i s o t r o p i c with the major and minor axes of a n i s o t r o p y t r e n d i n g 15 and 105 degrees r e s p e c t i v e l y . -73-H ( f e e t ) H ( f e e t ] F i g u r e 6.3.5: Semi-variograms of t h i c k n e s s i n area S2 were c a l c u l a t e d at 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen t h a t the semi-variogram i s a n i s o t r o p i c with the major and minor axes of a n i s o t r o p y t r e n d i n g 15 and 105 degrees r e s p e c t i v e l y . -74-T K . S 3 . 1 5 0 NESTED SPHERICAL MODEL CO=8.78 C l = 8 . 2 7 61=5 C2=0.66 02=58 40 T 80 — i — r 120 160 ~l 1 200 H ( f e e t ) 2.5. TH-S3 .105 H C fee t ) F i g u r e 6.3.6: Semi-variograms of t h i c k n e s s i n area S3 were c a l c u l a t e d a t 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. The axes of a n i s o t r o p y can be seen to trend midway between these d i r e c t i o n s , at 35 and 125 degrees. Semi-variograms c a l c u l a t e d i n the d i r e c t i o n s of the major and minor axes of a n i s o t r o p y are a l s o shown. -75-F i g u r e 6.3.7: Semi-variograms of t h i c k n e s s i n area S4 were c a l c u l a t e d a t 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. A geometric a n i s o t r o p y can be seen with a long range model t r e n d i n g 60 degrees i n the d i r e c t i o n of the major a x i s and the s h o r t range model t r e n d i n g 150 degrees. -76-6.4 Summary and Co n c l u s i o n s Thickness i n the north p a r t of the Venus d e p o s i t i s composed of two p o p u l a t i o n s : one, l o g normally d i s t r i b u t e d above the e l e v a t i o n of 2750 f e e t and the o t h e r , normally d i s t r i b u t e d , below. The upper n o r t h p o p u l a t i o n i s s t a t i o n a r y and the semi-variogram i s best f i t by an i s o t r o p i c , nested s p h e r i c a l model. The lower n o r t h p o p u l a t i o n was separated i n t o two p a r t s to s a t i s f y the i n t r i n i c h y p o t h e s i s . The t h i c k n e s s semi-variogram i n NL1 shows a geometric a n i s o t r o p y with the major and minor axes of a n i s o t r o p y t r e n d i n g 120 and 30 degrees r e s p e c t i v e l y . In area NL2, the a n i s o t r o p y i s zo n a l with the s i z e of the nugget e f f e c t governing the height of the s i l l . The axes of a n i s o t r o p y l i e a t 0 and 90 degrees. The o r i e n t a t i o n of the major axes of a n i s o t r o p y i n the lower n o r t h p a r a l l e l s a pproximately the d i p d i r e c t i o n i n t h i s p a r t of the mine. T h i s c o r r e l a t e s with the r e p o r t e d o r i e n t a t i o n of the "ore zones" whose long axes are f e l t t o run up and down d i p . In the upper no r t h , where the mean v e i n t h i c k n e s s i s narrower, t h i s t r e n d does not appear. As demonstrated i n Chapter 4, the t h i c k n e s s d i s t r i b u t i o n south of the hinge l i n e i s b i a s e d . Consequently, when d i v i d e d i n t o quadrants, i t f a i l e d t o s a t i s f y the requirements of s t a t i o n a r i t y and the i n t r i n s i c h y p o t h e s i s . Each quadrant was t h e r e f o r e t r e a t e d as a separate p o p u l a t i o n . -77-Thickness semi-variograms i n areas SI and S2 have the same axes of geometric a n i s o t r o p y (15 and 105 degrees). The models d i f f e r , however, i n the he i g h t of t h e i r s i l l s and i n t h e i r ranges of i n f l u e n c e . In area S3, the semi-variograms of t h i c k n e s s show zo n a l a n i s o t r o p y with the high s i l l , long range a x i s o r i e n t e d a t 125 degrees and the low s i l l , s h o r t range a x i s o r i e n t e d a t 35 degrees. The geometric a n i s o t r o p y model i n S4 has major and minor axes t r e n d i n g 60 and 150 degrees r e s p e c t i v e l y . The t h i c k n e s s semi-variogram models are summarized i n Table 6.4.1. Area/Az Co C l a l C2 a2 NU 0 .075 0 . 09 20 0.04 150 NLl 30 NLl 120 0.0 0.0 1.57 1.0 22.5 15 0.57 135 NL2 0 NL2 90 0.4 0.65 1.25 0.6 30 70 0.65 150 SI 15 SI 105 0.4 0.0 2.14 2 .54 70 25 -S2 15 S2 105 0.0 0.0 1.6 1.6 40 10 -S3 35 S3 125 0 .55 0 . 55 1.0 1. 3 20 50 -S4 60 S4 150 0.4 0.2 1.9 2.1 100 25 -: 6.4.1: Semi- variogram parameters of t h i c k n e s s i n each of the p o p u l a t i o n s . -78-7. SEMI-VARIPGRAM RESULTS; GOLD ACCUMULATION 7.1.1 Gold Accumulation i n the North Semi-variograms of the log-transformed g o l d accumulation were c a l c u l a t e d i n the nor t h p a r t of the Venus mine. A beta a d d i t i v e c onstant of 0.135 f t oz/ton was used t o normalize the p o p u l a t i o n . Experimental semi-variograms were c a l c u l a t e d i n four d i r e c t i o n s , 0, 45, 90 and 135 +/-22.5 degrees, t o t e s t f o r a n i s o t r o p y . In every case the best f i t was accomplished w i t h nested s p h e r i c a l models of the form d e s c r i b e d i n Chapter 5. The major a x i s of a n i s o t r o p y was found t o t r e n d halfway between 90 and 135 degrees ( F i g u r e 7.1.1). Semi-variograms were then c a l c u l a t e d i n the d i r e c t i o n of the major and minor axes of the geometric a n i s o t r o p y , 115 and 25 degrees r e s p e c t i v e l y . Since the Venus v e i n s t r i k e s 25 degrees n o r t h of the hinge l i n e , the o r i e n t a t i o n of the a n i s o t r o p y confirms the ob s e r v a t i o n s of mine g e o l o g i s t s t h a t ore shoots a t Venus are long i n the d i p d i r e c t i o n and narrow a l o n g s t r i k e . The parameters of the g e n e r a l semi-variogram models are shown i n Table 7.1.1. Area/Az Co CI a l C2 a2 N 25 0.3 0.17 20 0.37 95 N 115 0.3 0.25 25 0.29 200 Table 7.1.1: Semi-variogram parameters of log-transformed g o l d accumulation i n the d i r e c t i o n s of the a n i s o t r o p y axes. -79-Figure 7.1.1: Semi-variograms of the log-transformed gold accumulation i n the north were c a l c u l a t e d a t 0, 45, 90 and 135 degrees. The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. The major a x i s of a n i s o t r o p y can be seen to f a l l halfway between the 90 and 135 degree semi-variograms which have s i m i l a r models. Semi-variograms c a l c u l a t e d i n the d i r e c t i o n s of the minor and major axes of a n i s o t r o p y , 25 and 115 degrees r e s p e c t i v e l y , are a l s o shown. -80-7.1.2 S t a t i o n a r i t y of Gold Accumulation i n the North The c a l c u l a t i o n of semi-variograms i n the d i r e c t i o n of the axes of a n i s o t r o p y i n the north quadrants showed t h a t one quadrant d i d not s a t i s f y s t a t i o n a r i t y . When a comparison of the v a r i a n c e s (Table 7.1.2) and semi-variograms ( F i g u r e s 7.1.2 & 7.1.3) of the quadrants was made, i t was found t h a t N2, N3 and N4 c o u l d be grouped to form a s i n g l e p o p u l a t i o n "NA". In other words, gold accumulation i s s t a t i o n a r y w i t h i n area NA. Because the v a r i a n c e and semi-variogram of Nl d i f f e r from those of the other quadrants, N l s a t i s f i e s n e i t h e r s t a t i o n a r i t y nor the i n t r i n s i c h ypothesis and was t r e a t e d s e p a r a t e l y . Area Mean Variance Pts N 0.3975 0.839 2312 Nl 0.3605 0.695 480 N2 0.4026 0.858 865 N3 0.3975 0.839 565 N4 0.3977 0.840 402 Table 7.1.2: Mean gold accumulation, v a r i a n c e and number of p o i n t s i n each of the north quadrants. The semi-variogram parameters of the models which best f i t the curves i n each quadrant are l i s t e d i n Table 7.1.3. They show t h a t the i n t r i n s i c hypothesis i s s a t i s f i e d i n areas N2, N3 and N4. -81-Area/Az Co CI a l C2 a2 N 25 0.3 0.17 20 0.375 95 Nl 25 0.25 0.1 20 0.475 150 N2 25 0.25 0.25 25 0 . 35 100 N3 25 0 . 3 0.3 25 0.24 130 N4 25 0.3 0 .17 20 0.37 130 N 115 0 . 3 0.25 25 0 .29 200 Nl 115 0.3 0.2 20 0.2 120 N2 115 0 .35 0.2 40 0.29 160 N3 115 0.2 0.35 20 0.29 150 N4 115 0.2 0.35 20 0.29 130 Table 7.1.3: Semi-variogram parameters of log-transformed gold accumulation i n the northern quadrants. -82-LA—AU—NI•25 L A - A U - N 2 . 2 5 H t f e e t ) H Cfeet ) 1 T L A - A U - N 4 . 2 5 NESTED SPHERICAL MODEL Co=0.3 C l = 0 . 1 7 A l = 20 C2=0.37 62=95 I 40 T —r~ 80 T - 1 — 120 —I— 160 1 200 H C fee t ) L A - A U - N 3 . 2 5 NESTED SPHERICAL MODEL C0=0.3 C l=0 .17 Al=20 C2=0.37 A2=9S I 40 T 80 T" I 120 160 200 H Cfeet) F i g u r e 7.1.2: The semi-variograms of the log-transformed gold accumulation i n the northern quadrants a t 25 degrees, the d i r e c t i o n of the minor a x i s of a n i s o t r o p y are shown o v e r l a i n by the gen e r a l 25 degree model. I t can be seen that s t a t i o n a r i t y e x i s t s i n NA (N2 f N3 + N4) and that Nl must be t r e a t e d s e p a r a t e l y . -83-LA_AU_N1.115 NESTED SPHERICAL MODEL Co=0.3 C l=0 .25 01 = 25 C2=0.29 A2=280 -1— 48 T" I 80 T" 128 —I 160 200 H Cfeet) LA_AU_N2.115 NESTED SPHERICAL MODEL Co=0.3 Cl=8 .25 01=25 C2=8.29 02=200 -1— 40 T 88 - — r 120 T 160 T —I 200 H Cfeet) L A _ A U _ N 4 . i l ! NESTED SPHERICAL MODEL Co=0.3 C l=0 .25 01=25 C2=8.29 02=200 T 1 1 1 1 1 1 1 80 120 160 280 I— 48 L A - A U - N 3 . 1 1 5 8.2-NESTED SPHERICAL MODEL CO=0.3 Cl=0 .25 01=25 C2=0.29 02=200 -1— 40 88 T —i r 128 —I— 168 280 H Cfeet) H Cfeet) F i g u r e 7.1.3: The semi-variograms of the log-transformed gold accumulation i n the northern quadrants a t 115 degrees, the d i r e c t i o n of the major a x i s of a n i s o t r o p y , are shown o v e r l a i n by the gene r a l 115 degree model. These curves a l s o show t h a t s t a t i o n a r i t y e x i s t s i n NA (N2 + N3 + N4). - 8 4 -7.1.3 A n i s o t r o p y of Gold Accumulation In the Northern  Areas A t e s t f o r a n i s o t r o p y of gold accumulation was c a r r i e d out i n the s t a t i o n a r y p o p u l a t i o n s Nl and NA. In N l , the gold accumulation semi-variogram i s z o n a l l y a n i s o t r o p i c and i s best f i t by a nested s p h e r i c a l semi-variogram model (Figure 7.1.4). The d i r e c t i o n s chosen for the t e s t were those of the axes of the g e n e r a l model, 25 and 115 degrees, and the intermediate d i r e c t i o n s , 70 and 160. A l l four d i r e c t i o n a l semi-variograms had the same range of i n f l u e n c e , but the s i l l s of the 25 and 160 degree semi-variograms were s l i g h t l y higher than the p o p u l a t i o n v a r i a n c e (Figure 7.1.4). I t was concluded t h a t one a x i s of a n i s o t r o p y f a l l s a t 0 degrees and the other a t 90 degrees. The c a l c u l a t i o n of the semi-variogram a t 0 degrees produced a s p h e r i c a l model with a range of 150 f e e t and a s i l l of 0.825. At 90 degrees, the model was a nested s p h e r i c a l semi-variogram with a range of 125 f e e t and a s i l l of 0.69. In NA, the semi-variogram i s best f i t by a nested s p h e r i c a l model with the major and minor axes of zonal a n i s o t r o p y t r e n d i n g 25 and 115 degrees r e s p e c t i v e l y ( Figure 7.1.5). The model f i t to the major a x i s semi-variogram (25 degrees) has a s i l l of 0.93 and a range of 125 f e e t . The semi-variogram a t 115 degrees has a lower s i l l (0.72) and a s h o r t e r range (60 f e e t ) . The semi-variogram models are summarized i n Table -85-7.1.4. Area/Az Co CI a l C2 a2 Nl 0 0.325 0 . 5 150 _ — Nl 90 0.0 0.325 12.5 0.365 125 NA 25 0.25 0.225 15 0.455 125 NA 115 0.2 0.25 10 0.27 60 Table 7.1.4: Semi-variogram parameters of log-transformed gold accumulation i n areas N l and NA. -86-Lo_AU-N1.16e , L f t _ J . U _ H i . i l S H Cfeet ) H Cfeet ) F i g u r e 7.1.4: Semi-variograms of log-transformed gold accumulation i n " N l " were c a l c u l a t e d a t 25, 70, 115 and 160 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. The semi-variogram's major a x i s of a n i s o t r o p y passes between the high s i l l models f i t a t 25 and 160 degrees. Semi-variograms c a l c u l a t e d at 0 and 90 degrees are shown below the four d i r e c t i o n a l c u rves. -87-NESTED SPHERICAL MODEL CO=0.2 Cl=0.23 01=15 C2=0.42 02=95 40 T 60 H Cf»*t) "T 120 — I — 160 200 5 0.8. A M M 0.6. A H »*J 0.2 0 LA-AU-NA.115 NESTED SPHERICAL MODEL COS0.2 Cl=0.25 01=18 C2=0.27 02=68 40 T —r— 80 120 "TJi T — I 200 H Cf»*t) F i g u r e 7.1.5: Semi-variograms of log-transformed gold accumulation i n "NA" were c a l c u l a t e d a t 25, 70, 115 and 160 degrees to t e s t for a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen that the major and minor axes of a n i s o t r o p y trend 25 and 115 degrees r e s p e c t i v e l y . -88-7.2.1 Gold Accumulation i n the South In the south p a r t of the Venus mine, a beta a d d i t i v e constant of 0.1405 f t oz/ton was used to normalize the p o p u l a t i o n . Semi-variograms were c a l c u l a t e d a t 15, 60, 105 and 150 degrees. The major a x i s of a n i s o t r o p y was found to f a l l between 105 and 150 degrees (Figure 7.2.1). A d d i t i o n a l semi-variograms were c a l c u l a t e d i n the d i r e c t i o n of the major and minor axes of a n i s o t r o p y , 115 and 25 degrees r e s p e c t i v e l y . The experimental curves were best f i t by the nested s p h e r i c a l models which are summarized i n Table 7.2.1. Area/Az Co CI a l C2 a2 S 25 0.35 0.20 25 0.53 55 S 115 0 . 4 0.3 45 0 . 38 150 Table 7.2.1: Semi-variogram parameters of log-transformed g o l d accumulation i n the d i r e c t i o n s of the a n i s o t r o p y axes. -89-1.2- L A - A U - S . I S NESTED SPHERICAL MODEL 8 .21 CO=0.45 C l = 8 . 1 5 Al=48 C2=8.48 A2=88 I 40 T 88 T" I 120 —I 160 T 200 H Cfee t ) 1.2., 0 .2 8 L A - A U - S . 60 SPHERICAL MODEL CO=0.45 C=8.63 I— 40 I 80 120 A=60 —I 160 —I 200 H Cfeet ) 1 > 2 „ G 1 A M M 0, A m 0. L A - A U - S . 1 5 0 ~ * CP-NESTED SPHERICAL MODEL 0.21 C0=0.4 C l=0 .3 Al=58 C2=8.38 02 = 95 I— 48 T 88 T — i — r 128 —I 168 208 H Cfee t ) 0 .2 0 L A - A U - S . 1 0 5 NESTED SPHERICAL MODEL CO=0.4 C l - 8 . 3 01=45 C2=8.38 02=158 -1— 40 T 80 120 T" —I 160 200 H Cfeet ) LO-AU-S . 25 ^X7 NESTED SPHERICAL MODEL CO=8.35 C l = 8 . 2 01=25 C2=8.53 02=55 40 — r ~ 80 —I 120 —I— 160 1 200 H Cfee t ) 1 .2 , G i : A M M 0.7-0 H ° - 5 8.2-L0_0U_5.115 NESTED SPHERICOL MODEL Co=0.4 C1=0.3 01=45 C2=0.38 02=150 —1— 40 T 80 T" 120 T - —I 160 —I 200 H Cfeet ) F i g u r e 7.2.1: Semi-variograms of log-transformed gold accumulation i n the south a t 15, 60, 105 and 150 degrees are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t i s ev i d e n t t h a t the major a x i s of a n i s o t r o p y f a l l s between 105 and 150 degrees. The semi-variograms c a l c u l a t e d i n the d i r e c t i o n of the major and minor axes of an i s o t r o p y , 115 and 25 degrees r e s p e c t i v e l y are a l s o shown. 7.2.2 S t a t i o n a r i t y of Gold Accumulation i n the South In the south, semi-variograms of gold accumulation were c a l c u l a t e d a t 25 and 115 degrees i n the southern quadrants. I t was found t h a t i n quadrants SI, S2 and S4, the v a r i a n c e s (Table 7.2.2) and the semi-variograms ( F i g u r e s 7.2.2 & 7.2.3) are s i m i l a r , thus s a t i s f y i n g the requirements of s t a t i o n a r i t y . Since the v a r i a n c e and semi-variogram i n S3 d i f f e r s i g n i f i c a n t l y from those i n the other quadrants, area S3 was t r e a t e d as a separate p o p u l a t i o n . Area Mean Variance Pts S 0.484 1.081 758 51 0.499 1.127 178 52 0.503 1.141 144 53 0.420 0.866 253 54 0.484 1.080 183 Table 7.2.2: Mean gold accumulation, v a r i a n c e and number of p o i n t s i n each of the south quadrants. The semi-variogram models which best f i t the experimental curves c a l c u l a t e d i n each of the quadrants are summarized i n Table 7.2.3. They c o n f i r m that gold accumulation i s s t a t i o n a r y i n "SA" (SI + S2 + S4). -91-Area/Az Co CI a l C2 a2 S 25 0.35 0.2 25 0.53 55 SI 25 0.1 0.45 20 0.58 50 S2 25 0 . 3 0 .35 15 0.49 50 S3 25 0.2 0.40 20 0.40 50 S4 25 0.3 0.25 20 0.55 50 S 115 0.4 0.3 45 0.38 150 SI 115 0.55 0.25 30 0.33 75 S2 115 0.45 0.05 20 0.64 75 S3 115 0.2 0.20 30 0.35 75 S4 115 0.3 0.4 50 0.4 75 Table 7.2.3: Semi-variogram parameters of log-transformed g o l d accumulation i n the southern quadrants. -92-LA_AI_.S1.2S 208 2 _ LA_AU_S2.25 G A M M 1. A 1.6. .2-H 0.8. 8. NESTED SPHERICAL MODEL CO=8.35 Cl=8.2 01=23 C2=8.53 A2=55 30 T 68 I T T — I 1 1 128 138 H Cfeet) LA-AU-S4.25 LA_AU-S3.2S H <fe«t) H ffeet) F i g u r e 7.2.2: The semi-variograms of the log-transformed gold accumulation i n the southern quadrants c a l c u l a t e d a t 25 degrees, the d i r e c t i o n of the minor a x i s of a n i s o t r o p y , are shown o v e r l a i n by the g e n e r a l 25 degree model. I t can be seen that t h i s model f i t s the semi-variogams of quadrants SI, S2 and S4 and not S3. -93-LA_AU-S1 .115 H ( f e e t ) g 1.6. M M 1.2-A H 8 .8 . 8 . 4 - ^ L A - A U - S 2 . 1 1 5 NESTED SPHERICAL MODEL CO=0.4 C l = 8 . 3 Al=45 C2=8.38 A2=158 —I— 48 T I 88 T - l — r 128 I— 168 — f 288 H ( f e e t ) L A - A U _ S 4 . 1 1 5 L A - A U - S 3 . 1 1 5 288 H ( f e e t ) H ( f e e t ) F i g u r e 7.2.3: The semi-variograms of the log-transformed gold accumulation i n the southern quadrants c a l c u l a t e d at 115 degrees, the d i r e c t i o n of the major a x i s of a n i s o t r o p y , are shown o v e r l a i n by the general 115 degree model. In co n j u n c t i o n with F i g u r e 7.2.2, t h i s f i g u r e shows that s t a t i o n a r i t y e x i s t s i n SA (SI + S2 + S4). -94-7.2.3 A n i s o t r o p y of Gold Accumulation In the southern Areas Semi-variograms of g o l d accumulation were c a l c u l a t e d i n area SA (the union of areas SI, S2 and S4) and i n S3 to t e s t f o r a n i s o t r o p y . In area SA, the experimental semi-variogram i s best f i t by a model with a z o n a l a n i s o t r o p y ( F i g u r e 7.2.4). The major and minor axes of the a n i s o t r o p y , 115 and 25 degrees r e s p e c t i v e l y , are the same as those of the g e n e r a l model. The semi-variogram model i n the d i r e c t i o n of the minor a x i s i s s p h e r i c a l with a range of 50 f e e t . In the d i r e c t i o n of the major a x i s , a nested s p h e r i c a l model was f i t with a range of 120 f e e t . The semi-variogram of g o l d accumulation i n S3 i s g e o m e t r i c a l l y a n i s o t r o p i c with the major and minor axes of a n i s o t r o p y t r e n d i n g 135 and 45 degrees r e s p e c t i v e l y ( F i g u r e 7.2.5). Both semi-variograms were best f i t by a simple s p h e r i c a l model with ranges of 110 and 25 f e e t r e s p e c t i v e l y . The r e s u l t s are summarized i n Table 7.2.4. Area/Az Co CI a l C2 a2 SA 25 0.45 0.66 50 _ SA 115 0.4 0.35 50 0.36 120 S3 45 0 0.87 25 — _ S3 135 0.25 0.62 110 - -Table 7.2.4: Semi-variogram parameters of log-transformed g o l d accumulation i n areas SA and S3. -95-LA-AU-SA.25 LA-AU-SA.78 NESTED SPHERICAL MODEL CO=Q.2 C1=B.4 01=10 C2=0.5i 02=75 — I — r 120 — 1 — 160 — I 200 H (feet) H (feet) 1.5^ G £ 1.2-M M 8.9-A H 8.6. 0.3-LA_AU_SA . 168 SPHERICAL MODEL Co=8.5 C=0.61 i — T : — i — . 1 I I 48 — r 88 128 H (feet) — I 168 0=75 T I 288 1.5-G 1.2-A M M 8.9-A H 8.6. 8.3-LA_AU-.SA.115 NESTED SPHERICAL MODEL CO=0.4 Cl=8.35 01=50 C2=8.36 02=128 40 —I 88 T ( — r 128 — T — 168 1 288 H (feet) F i g u r e 7.2.4: Semi-variograms of log-transformed gold accumulation i n SA (SI + S2 + S4) are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen that a n i s o t r o p y e x i s t s with the major, longest range, a x i s at 115 degrees and the minor a t 25 degrees. - 9 6 -LA-AU-S3.25 LA—AU—S3 • 70 i—i—r 128 T 1 1 160 200 H (feet) H (feet) 1.5. LA-AU-S3.160 1.5-G 1.2-A M • M 0 . 9-A H 0.6. 0.3-LA-AU-S3.115 'NESTED SPHERICAL MODEL CO=0.2 Cl=0.15 Al=25 C2=0.52 02=188 40 T — i — r 80 H (feet) —i— 120 160 200 LA-AU-S3.45 LA-AU-S3.135 SPHERICAL MODEL C0=0 C=0.87 -1— 40 —r 80 1 1 "1 . 120 160 T 1 200 SPHERICAL MODEL CO=8.25 C=0.62 0=110 1— 40 H (feet) T — i — r 80 H (feet) -l—r 120 —I—I—I 160 200 F i g u r e 7.2.5: Semi-variograms of log-transformed gold accumulation i n quadrant S3 are shown o v e r l a i n by the best f i t t i n g semi-variogram models. A n i s o t r o p y e x i s t s with the major a x i s o r i e n t e d midway between 115 and 160 degrees. Semi-variograms i n the d i r e c t i o n s of the minor and major axes of a n i s o t r o p y , 45 and 135, are shown below. -97-7.3 Summary and Conclusions Gold accumulation a t Venus has been d i v i d e d i n t o four p o p u l a t i o n s . In the north, Nl was separated from the other quadrants because i t s v a r i a n c e and semi-variogram d i f f e r e d s i g n i f i c a n t l y from those of the other quadrants. The semi-variogram of g o l d accumulation i n Nl i s best f i t by an a n i s o t r o p i c , nested s p h e r i c a l model. The major and minor axes of the zonal a n i s o t r o p y t r e n d 0 and 90 degrees r e s p e c t i v e l y . NA i s the union of quadrants N2, N3 and N4. Gold accumulation i n t h i s area i s z o n a l l y a n i s o t r o p i c with major and minor axes t r e n d i n g 25 and 115 degrees r e s p e c t i v e l y . The o r i e n t a t i o n of the axes of a n i s o t r o p y i n Nl i s 25 degrees d i f f e r e n t t o those i n the remainder of the n o r t h . T h i s can be a t t r i b u t e d to the v e i n ' s e r r a t i c s t r i k e and f l a t t e r d i p i n t h i s quadrant, as shown i n F i g u r e 5.3.1. In NA, the high s i l l , long range a x i s runs p a r a l l e l to the s t r i k e of the v e i n (25 degrees), while the low s i l l , s h o r t range model p a r a l l e l s the v e i n d i p . T h i s agrees with the o b s e r v a t i o n s of UKHM's mine g e o l o g i s t s , s i n c e the low s i l l of a zonal a n i s o t r o p y should be observed i n the d i r e c t i o n of the ore shoots. South of the hinge l i n e , g o ld accumulation was d i v i d e d i n t o two p o p u l a t i o n s . S3 was separated from the other quadrants to s a t i s f y s t a t i o n a r i t y . SI, S2 and S4 were combined to form SA. The semi-variograms of both -98-p o p u l a t i o n s are g e o m e t r i c a l l y a n i s o t r o p i c with major and minor axes i n SA t r e n d i n g 115 and 25 degrees r e s p e c t i v e l y . The axes i n S3 t r e n d 135 and 45 degrees r e s p e c t i v e l y . In each case the major a x i s f o l l o w s n e i t h e r the v e i n ' s d i p d i r e c t i o n nor i t s s t r i k e . I t must be concluded, t h e r e f o r e , t h a t south of the hinge l i n e the go l d c o n c e n t r a t i o n s do not f o l l o w the d i p of the v e i n . The semi-variograms models of g o l d accumulation i n each s t a t i o n a r y p o p u l a t i o n are summarized i n Table 7.3.1. Area/Az Co CI a l C2 a2 HI 0 0.325 0.5 150 _ _ N l 90 0.0 0.325 12.5 0.365 125 NA 25 0.25 0.225 15 0.455 125 NA 115 0.2 0.25 10 0.27 60 SA 25 0.45 0.66 50 _ — SA 115 0.4 0.35 50 0.36 120 S3 45 0 0.87 25 — _ S3 135 0.25 0.62 110 - -Table 7.3.1: Semi-variogram parameters of log-transformed g o l d accumulation i n each s t a t i o n a r y p o p u l a t i o n . -99-8. SEMI-VARIOGRAM RESULTS: SILVER ACCUMULATION 8.1.1 S i l v e r Accumulation i n the North Semi-variograms of the log-transformed s i l v e r accumulation were c a l c u l a t e d i n the north p a r t of the Venus d e p o s i t . A beta a d d i t i v e constant of 2.405 f t oz/ton was used to normalize the 3-parameter log-normal d i s t r i b u t i o n . To t e s t f o r a n i s o t r o p y , semi-variograms were c a l c u l a t e d i n four p r e f e r r e d d i r e c t i o n s : 0, 45, 90 and 135 +/- 22.5 degrees (Figure 8.1.1). The nested s p h e r i c a l models which best f i t the experimental semi-variogram curves show zonal a n i s o t r o p y i n t h i s a r e a . The four d i r e c t i o n a l semi-variograms have the same s h o r t range s t r u c t u r e with the d i f f e r e n c e l y i n g i n the second, long range model. The major a x i s , t r e n d i n g 0 degrees, has a s i l l of 1.25 and a range of 130 f e e t . In the d i r e c t i o n of the minor a x i s , 90 degrees, the s i l l i s 1.05 and the range i s 80 f e e t . These axes are approximately 20 degrees o b l i q u e to the v e i n ' s s t r i k e and d i p d i r e c t i o n s . The g e n e r a l semi-variogram models are summarized i n Table 8.1.1: Area/Az Co CI a l C2 a2 N 0 0.2 0.4 22.5 0.65 130 N 90 0.2 0.35 20 0.5 80 Table 8.1.1: Semi-variogram parameters of log-transformed s i l v e r accumulation i n the d i r e c t i o n s of the a n i s o t r o p y axes. -100-LA_AG_N.8 NESTED SPHERICAL MODEL CO=0.2 Cl=0.4 Al=22.5 C2=8.6S A2=130 48 80 H (feet) 128 — I — 160 T I 280 1.5-G 1.2. A M • M 0.9-A H 0.6. 8.5-LA-AG-N.45 NESTED SPHERICAL MODEL C0=0.2 Clr8.42 A1=22.S C2=8.58 A2=110 -1— 40 T -T— 80 T —I 1 I 1 1 120 160 288 H Cfeet) 1.5. LA_AG_N.135 NESTED SPHERICAL MODEL C0=8.25 Cl=0.35 Al=22.5 C2=8.S1 A2=12S 1— 48 T — i — r 80 H (feet) - i—r 120 - 1 — 160 T 1 200 l.S. LA—AG—N.98 NESTED SPHERICAL MODEL C0=8.2 C1=8.3S Al=20 C2=0.5 02=88 T — 48 T 88 T — I 120 160 T 1 200 H (feet) F i g u r e 8.1.1: Semi-variograms of log-transformed s i l v e r accumulation c a l c u l a t e d at 0, 45, 90 and 135 degrees are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen that the major and minor axes of the zonal a n i s o t r o p y t r e n d 0 and 90 degrees r e s p e c t i v e l y . -101-8.1.2 S t a t i o n a r i t y of S i l v e r Accumulation i n the North As a t e s t of the s t a t i o n a r i t y of the s i l v e r accumulation a t Venus, semi-variograms were c a l c u l a t e d a l o n g the a n i s o t r o p y axes i n each of the northern quadrants. S i l v e r accumulation i n quadrants N2, N3 and N4 was found to have s i m i l a r v a r i a n c e s (Table 8.1.2) and s i m i l a r semi-variograms ( F i g u r e s 8.1.2 & 8.1.3), thus s a t i s f y i n g the i n t r i n s i c h y p o t h e s i s . N2, N3 and N4 were combined to form "NA". In quadrant Nl the v a r i a n c e i s c o n s i d e r a b l y higher than i t i s i n the combined north d a t a s e t . Since the semi-variograms of t h i s quadrant d i f f e r g r e a t l y from the g e n e r a l model and from those of the other quadrants, n e i t h e r s t a t i o n a r i t y nor the i n t r i n s i c hypothesis are s a t i s f i e d . Nl was t r e a t e d as a separate p o p u l a t i o n . Area Mean Var iance Pts N 8.57 1.11 2225 Nl 9.64 1.296 431 N2 8.61 1.118 826 N3 8.41 1.081 566 N4 7.72 0.959 402 Table 8.1.2: Mean s i l v e r accumulation, v a r i a n c e and number of p o i n t s i n each of the northern quadrants. The semi-variogram models t h a t best f i t the curves c a l c u l a t e d i n each of the quadrants are shown i n Table 8.1.3. T h i s i n f o r m a t i o n f u r t h e r supports the c o n c l u s i o n -102-t h a t s i l v e r accumulation In area NA s a t i s f i e s the i n t r i n s i c h ypothesis and t h a t a r e a N l must be t r e a t e d s e p a r a t e l y . Area/Az Co CI a l C2 a2 N 0 0.2 0.4 22.5 0.65 130 Nl 0 0 0.5 20 1.55 170 N2 0 0.35 0.35 27 0.55 130 N3 0 0.15 0.35 20 0.75 130 N4 0 0.1 0.5 20 0.35 110 N 90 0.2 0.35 20 0.5 80 Nl 90 0.35 0.95 85 „ , N2 90 0.2 0.35 20 0.55 90 N3 90 0.2 0.4 20 0.48 125 N4 90 0.1 0.35 20 0.5 90 Table 8.1.3: Semi-variogram parameters of log-transformed s i l v e r accumulation i n the no r t h e r n quadrants. -103-F i g u r e 8.1.2: Semi-variograms of the log-transformed s i l v e r accumulation i n the northern quadrants a t 0 degrees, the d i r e c t i o n of the major a x i s of a n i s o t r o p y , are shown o v e r l a i n by the general 0 degree model. The f i t of t h i s model to the curves of N2, N3 and N4 s a t i s f i e s the i n t r i n s i c hypothesis i n area NA. Nl must be t r e a t e d as a separate p o p u l a t i o n . -104-LA—AG—Nl>90 LA-AG-N2.90 H Cf*«t) LA-AG-N4.90 SPHERICAL MODEL 2 C1=0.3S 01=20 C2=0.5 02=80 1 — 40 — r — 8 0 T - T 128 160 T — I 2 0 0 H (f»*t) LO-AG-N3.90 H <fO«t> F i g u r e 8.1.3: Semi-variograms of the log-transformed s i l v e r accumulation i n the northern quadrants at 90 degrees, the d i r e c t i o n of the major a x i s of a n i s o t r o p y , are shown o v e r l a i n by the gen e r a l 90 degree model. This i s f u r t h e r evidence that NA s a t i s f i e s the i n t r i n s i c h y p o t h e s i s . -105-8.1.3 A n i s o t r o p y of S i l v e r Accumulation i n the Northern Areas The two subsets of the north p o r t i o n of the d a t a s e t were t e s t e d f o r a n i s o t r o p y by c a l c u l a t i n g semi-variograms at 0, 45, 90 and 135 degrees. The semi-variogram models of the areas are summarized i n Table 8.1.4. S i l v e r accumulation i n area Nl d i s p l a y s zonal a n i s o t r o p y . In an attempt to c l a r i f y the a n i s o t r o p y , a d d i t i o n a l semi-variograms were c a l c u l a t e d at 25, 68, 115 and 158 degrees ( F i g u r e 8.1.4). The major a x i s of a n i s o t r o p y was found to trend 0 degrees. The model i s a nested s p h e r i c a l one which has a f i n a l s i l l of 2.05 and a range of 170 f e e t i n t h i s d i r e c t i o n . In the d i r e c t i o n of the minor a x i s , 90 degrees, the model i s s p h e r i c a l with a range of 85 f e e t and a s i l l equal to the p o p u l a t i o n v a r i a n c e of 1.3. In area NA (N2 + N3 + N4), the nested s p h e r i c a l semi-variogram model i s a n i s o t r o p i c with the major a x i s o r i e n t e d midway between 90 and 135 degrees (Figure 8.1.5). A d d i t i o n a l semi-variograms were c a l c u l a t e d i n the d i r e c t i o n of the major and minor axes, 113 and 23 degrees r e s p e c t i v e l y , bearings which p a r a l l e l the d i p and s t r i k e d i r e c t i o n s of the v e i n north of the hinge l i n e . The model of the semi-variogram of s i l v e r accumulation i n NA i s z o n a l l y a n i s o t r o p i c . -106-Area/Az Co CI a l C2 a2. Nl 0 0 0.5 20 1.55 170 Nl 90 0.35 0.95 85 NA 23 0.2 0.4 20 0.47 70 NA 113 0.3 0.4 25 0.275 150 Table 8.1.4: Semi-variogram parameters of log-transformed s i l v e r accumulation i n the 2 northern d a t a s e t s . -107-L0_AG_N1.11S , LA_0G_N1.98 H (feet) H (feet) F i g u r e 8.1.4: Semi-variograms of the log-transformed s i l v e r accumulation i n area N l were c a l c u l a t e d a t 0, 25, 45, 68, 90, 115, 135 and 158 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. A zonal a n i s o t r o p y was d i s c o v e r e d with the major and minor axes t r e n d i n g 0 and 90 degrees r e s p e c t i v e l y . -108-1.6-G 1.2-A M -M e.9. A H 8.6. 8.3-LA—AG—MA•133 SPHERICAL MODEL CO=8.4 C=8.6 7e~ —r~ 88 H Cfe»tJ 128 ' 161 A=128 —r —i 288 LA—AG—NA.98 NESTED SPHERICAL MODEL C0=8.2 Cl=0.4 Al=28 C2=8.47 A2=135 I 48 88 —1 128 168 — I 288 H Cf««t) F i g u r e 8.1.5: Semi-variograms of the log-transformed s i l v e r accumulation i n area NA were c a l c u l a t e d a t 0, 45, 90 and 135 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen t h a t the major a x i s of a n i s o t r o p y passes midway between 90 and 135 degrees. The semi-variograms c a l c u l a t e d i n the d i r e c t i o n s of the minor and major axes of a n i s o t r o p y , 23 and 113 degrees r e s p e c t i v e l y , are shown above. -109-8.2.1 S i l v e r Accumulation In the South South of the hinge l i n e , a beta a d d i t i v e constant of 2.453 f t oz/ton was used to normalize the 3-parameter log-normal d i s t r i b u t i o n of s i l v e r accumulation. To t e s t f o r a n i s o t r o p y , semi-variograms were c a l c u l a t e d a t 15, 60, 105 and 150 degrees ( F i g u r e 8.2.2). Nested s p h e r i c a l semi-variogram models best f i t the experimental curves and r e v e a l e d a geometric a n i s o t r o p y with major and minor axes t r e n d i n g 150 and 60 degrees r e s p e c t i v e l y . These bearings p a r a l l e l those of the v e i n ' s d i p and s t r i k e . The semi-variogram parameters are summarized i n Table 8.2.1 Area/Az Co CI a l C2 a2 S 60 0.15 0.45 35 0.5 65 s 150 0.2 0.4 35 0.5 95 Table 8.2.1: Semi-variogram parameters of log-transformed s i l v e r accumulation i n the d i r e c t i o n s of the a n i s o t r o p y axes. -110-F i g u r e 8.2.1: Semi-variograms of log-transformed s i l v e r accumulation c a l c u l a t e d a t 15, 60, 105 and 150 degrees are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen t h a t the major and minor axes of the geometric a n i s o t r o p y trend 150 and 60 degrees r e s p e c t i v e l y . - I l l -8.2.2 S t a t i o n a r i t y of S i l v e r Accumulation i n the South In the south, the semi-variograms were c a l c u l a t e d along the axes of the a n i s o t r o p y , 60 and 150 degrees, to t e s t f o r s t a t i o n a r i t y of s i l v e r accumulation. The upper quadrants, S3 and S4, have s i m i l a r v a r i a n c e s (Table 8.2.2) and semi-variogram curves (F i g u r e s 8.2.2 and 8.2.3) thus s a t i s f y i n g the requirements of s t a t i o n a r i t y . They were combined to form area "SU". The other two quadrants, SI and S2, d i f f e r from each other and from the upper quadrants. They were t r e a t e d s e p a r a t e l y . Area Mean Variance Pts S 20.40 1.097 758 51 16.96 0.771 178 52 23.82 1.376 144 53 19.32 1.001 253 54 20.30 1.089 183 Table 8.2.2: Mean s i l v e r accumulation, v a r i a n c e and number of p o i n t s i n each of the south quadrants. The semi-variogram models which best f i t the curves of each quadrant are summarized i n Table 8.2.3. They show th a t s i l v e r accumulation s a t i s f i e s the i n t r i n s i c hypothesis i n quadrants S3 and S4, but not i n SI or S2. A l l three areas were t e s t e d f o r a n i s o t r o p y . -112-Area/Az Co CI a l C2 a2 s 60 0.15 0.45 35 0.5 65 SI 60 0.25 0.35 30 _ S2 60 0 1.38 45 - -S3 60 0.4 0.2 35 0.5 65 34 60 0.1 0.5 35 0.5 60 S 150 0.2 0.4 35 0.5 95 SI 150 0.0 0.2 20 0.4 35 S2 150 0.2 1.18 110 - -S3 150 0.2 0.4 35 0.6 90 S4 150 0.1 0.5 35 0.5 90 Table 8.2.3: Semi-variogram parameters of log-transformed s i l v e r accumulation i n the southern quadrants. -113-2-5-. i2 M M 1.5-1 A H i e.5. e LA_.AG_S1.60 NESTED SPHERICAL MODEL CO=8.15 Cl-8.45 61=35 C2=B.5 62=65 I 40 I 80 — i — r 120 - 1 — 160 T 1 200 H (feet) LA_AG_S2.68 NESTED SPHERICAL'MODEL Co=8.15 Cl-8.45 61=35 C2=8.5 62 = 65 T -48 I 80 - 1 — r 120 I— 160 — I 200 H (feet) F i g u r e 8.2.2: Semi-variograms of log-transformed s i l v e r accumulation i n the southern quadrants c a l c u l a t e d a t 60 degrees, the d i r e c t i o n of the major a x i s of a n i s o t r o p y , are shown o v e r l a i n by the gen e r a l 60 degree model. The f i t of the g e n e r a l model to the curves of quadrants S3 and S4 s a t i s f i e s the i n t r i n s i c h ypothesis i n area SU, but not i n SI and S2. -114-F i g u r e 8.2.3: Semi-variograms of the log-transformed s i l v e r accumulation i n the southern quadrants c a l c u l a t e d a t 150 degrees, the d i r e c t i o n of the major a x i s of a n i s o t r o p y , are shown o v e r l a i n by the general 150 degree model. Further evidence of the s t a t i o n a r i t y i n SU. -115-8.2.3 A n i s o t r o p y or S i l v e r Accumulation i n the Southern Areas S i l v e r accumulation i n the 3 subsets of the south p a r t of the Venus d e p o s i t was t e s t e d f o r a n i s o t r o p y by c a l c u l a t i n g semi-variograms a t 15, 60, 105 and 150 degrees. In area SI, the s i l v e r accumulation semi-variogram proved to be i s o t r o p i c . I t was best f i t by a nested s p h e r i c a l model with a range of 60 f e e t . The semi-variogram of s i l v e r accumulation i n area S2 i s a n i s o t r o p i c with major and minor axes t r e n d i n g 150 and 60 degrees r e s p e c t i v e l y . These bearings c o i n c i d e with those of the d i p and s t r i k e d i r e c t i o n s south of the hinge l i n e . In area SU, the s i l v e r accumulation semi-variogram i s a n i s o t r o p i c with the major a x i s o r i e n t e d midway between 105 and 150 degrees. A d d i t i o n a l semi-variograms were c a l c u l a t e d i n the d i r e c t i o n s of the major and minor axes of a n i s o t r o p y , 120 and 30 degrees r e s p e c t i v e l y . The SU semi-variograms were best f i t by simple s p h e r i c a l models. The semi-variogram models of s i l v e r accumulation i n the three areas are summarized i n Table 8.2.4. -116-Area/Az Co CI a l C2 a2 SI 0.0 0.45 22 0.175 60 S2 60 0.0 1.38 45 — — S2 150 0.2 1.18 110 - — SU 30 0.2 0.85 35 — -SU 120 0.35 0.7 125 - -Table 8.2.4: Semi-variogram parameters of log-transformed s i l v e r accumulation i n the 3 southern d a t a s e t s . -117-H (feet) H (feet) F i g u r e 8.2.4: Semi-variograms of the log-transformed s i l v e r accumulation i n area SI were c a l c u l a t e d a t 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. A l l of the curves reach a s i l l below the p o p u l a t i o n v a r i a n c e . The i s o t r o p i c nested s p h e r i c a l model i s a l s o shown. -118-Figure 8.2.5: Semi-variograms of the log-transformed s i l v e r accumulation i n area S2 were c a l c u l a t e d at 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. It can be seen that the axes of a n i s o t r o p y trend 150 and 60 degrees. -119-F i g u r e 8.2.6: Semi-variograms of the log-transformed s i l v e r accumulation i n area SU were c a l c u l a t e d at 15, 60, 105 and 150 degrees to t e s t f o r a n i s o t r o p y . The curves are shown o v e r l a i n by the best f i t t i n g semi-variogram models. I t can be seen that the major a x i s of a n i s o t r o p y passes between 105 and 150 degrees. The semi-variograms c a l c u l a t e d along the minor and major axes of a n i s o t r o p y , 30 and 120 degrees, are shown. -120-8.3 Summary and C o n c l u s i o n s S i l v e r accumulation a t Venus was d i v i d e d i n t o 5 p o p u l a t i o n s t o s a t i s f y the requirements of s t a t i o n a r i t y and the i n t r i n s i c h y p o t h e s i s . The semi-variogram models are summarized i n Table 8.3.1. North of the hinge l i n e , N l was separated from the other quadrants because of i t s higher v a r i a n c e and d i f f e r e n t semi-variogram c u r v e s . The model of the s i l v e r accumulation semi-variogram i s z o n a l l y a n i s o t r o p i c with the major and minor axes t r e n d i n g 0 and 90 degrees r e s p e c t i v e l y Areas N2, N3 and N4 were merged to form area NA. S i l v e r accumulation i n NA i s z o n a l l y a n i s o t r o p i c with the major and minor axes of a n i s o t r o p y t r e n d i n g 113 and 23 degrees. The low s i l l models of the z o n a l a n i s o t r o p i c s i n the n o r t h e r n p o p u l a t i o n s f i t the semi-variograms t r e n d i n g 90 degrees i n N l and 113 degrees i n NA. In both cases the semi-variograms are i n the d i r e c t i o n of the v e i n d i p . One must conclude t h a t the s i l v e r c o n c e n t r a t i o n s run p a r a l l e l to the v e i n d i p . South of the hinge l i n e , s i l v e r accumulation was d i v i d e d i n t o 3 p o p u l a t i o n s : SI, S2 and SU (the union of S3 and S4). In area S i , the s i l v e r accumulation semi-variogram proved to be i s o t r o p i c and best f i t by a nested s p h e r i c a l model. The semi-variogram of s i l v e r accumulation i n area S2 -121-Is g e o m e t r i c l y a n i s o t r o p i c with major and minor axes t r e n d i n g 150 and 60 degrees r e s p e c t i v e l y . In area SU, the s i l v e r accumulation semi-variogram i s g e o m e t r i c a l l y a n i s o t r o p i c with the major and minor axes t r e n d i n g 120 and 30 degrees r e s p e c t i v e l y . South of the hinge l i n e one can conclude t h a t the s i l v e r c o n c e n t r a t i o n s f o l l o w the v e i n d i p i n a l l but quadrant SI. In S2 and SU the major axes of the geometric a n i s o t r o p i c s p a r a l l e l the v e i n ' s d i p d i r e c t i o n . Area/Az Co CI a l C2 a2 Ml 0 0 0.5 20 1.55 170 Nl 90 0.35 0.95 85 - — NA 23 0.2 0.4 20 0.47 70 NA 113 0.3 0.4 25 0.275 150 SI 0.0 0.45 22 0.175 60 S2 60 0.0 1.38 45 — — S2 150 0.2 1.18 110 - -SU 30 0.2 0.85 35 — — SU 120 0.35 0.7 125 - -Table 8.3.1: Semi-variogram parameters of log-transformed s i l v e r accumulation i n the s t a t i o n a r y d a t a s e t s . -122-9. KRIGING 9.1 Theory K r i g i n g i s a method of weighted-distance a v e r a g i n g which uses the parameters of the semi-variogram model to d e r i v e the weights. Samples i n the v i c i n i t y of the block being estimated are weighted i n p r o p o r t i o n t o : a) t h e i r d i s t a n c e from the b l o c k ; b) t h e i r l o c a t i o n r e l a t i v e to the block and c) t h e i r l o c a t i o n r e l a t i v e t o the other samples i n the v i c i n i t y . The k r i g e d estimate a t p o i n t k i s c a l c u l a t e d u s i n g the f o l l o w i n g e q u a t i o n : i = l where: Z(k) = the k r i g e d estimate a t p o i n t k Z ( i ) = the sample value a t p o i n t 1 W(i) = the k r i g i n g weight a s s i g n e d to Z ( i ) K r i g i n g weights are c a l c u l a t e d by s o l v i n g a system of simultaneous l i n e a r e q u a t i o n s . The use of a Lagrangian m u l t i p l i e r and summing the weights to 1 i n s u r e s t h a t the r e s u l t i n g estimate i s unbiased and the e s t i m a t i o n v a r i a n c e i s minimized, thus producing the Best L i n e a r Unbiased E s t i m a t o r (BLUE). The simultaneous l i n e a r equations f o r a s i t u a t i o n where three samples are used to estimate the grade a t a p o i n t pk are shown below: n Z(k) W(i) * Z ( i ) -123-w l * C ( p l , p l ) + w2*C(pl,p2) + w3*C(pl,p3) - L = C(pl,pk) wl*C(p2,pl) + w2*C(p2,p2) + w3*C(p2,p3) - L = C(p2,pk) wl*C(p3,pl) + w2*C(p3,p2) + w3*C(p3,p3) - L = C(p3,pk) wl + w2 + w3 = 1 where: wl = the k r i g i n g weights a s s i g n e d t o p i C(pl,p3) = the covariogram between p t s . p i and p3 (the covariogram = Co+Cl+C2 - the semi-variogram) L = the Lagrangian m u l t i p l i e r pk = the p o i n t being e s timated The k r i g i n g v a r i a n c e i s a f u n c t i o n of the block v a r i a n c e , the Lagrangian m u l t i p l i e r and the weighted sum of the c o v a r l a n c e s between the k r i g e d block and the sur r o u n d i n g p o i n t s . I t i s c a l c u l a t e d u s i n g the f o l l o w i n g e q u a t i o n : m m n V(k) = 5> C ( p ( h ) , p ( i ) ) - L - ^ W( j ) * C ( p ( j ) , p ( k ) ) m h=l i = l j = l where: V(k) = the k r i g i n g v a r i a n c e C(,) = the co v a r i a n c e W() = the weighting f a c t o r L = the Lagrangian m u l t i p l i e r p(h) = a p o i n t h w i t h i n k r i g e d block p ( i ) = a p o i n t i w i t h i n k r i g e d block p ( j ) = a p o i n t j p(k) = the ce n t r e p o i n t of the k r i g e d block m & n = counters -124-9.2 Procedure To c a r r y out the k r i g i n g of the Venus mine data, the author adapted a program w r i t t e n by Kim and Knudsen (1977). T h i s program was o r i g i n a l l y w r i t t e n to handle stratabound, r o l l - f r o n t uranium d e p o s i t s and i s i d e a l l y s u i t e d to the a n a l y s i s of v e i n d e p o s i t s . The o p e r a t i o n i s reduced to a two dimensional problem with the d e p o s i t ' s t h i c k n e s s (the t h i r d dimension) t r e a t e d as a v a r i a b l e . The k r i g i n g was c a r r i e d out on the UBC main-frame computer. Before commencing the k r i g i n g o p e r a t i o n , the f i r s t s t e p was to c o n v e r t the semi-variogram parameters of the l o g - n o r m a l l y d i s t r i b u t e d p o p u l a t i o n s . Since the nugget e f f e c t and s i l l v a l u e s are v a r i a n c e s , the equation used i n chapter 4 c o u l d a g a i n be used. The e q u a t i o n i s : 2 Var = mean * ( e x p ( V ) - l ) ( K r i g e , 1978) As the semi-variogram parameters were converted, the models were s i m p l i f i e d and reduced to 7 parameters: Co, CI, a l , C2, a2, a h o r i z o n t a l a n i s o t r o p y f a c t o r (HAF) and the azimuth of the major a x i s of a n i s o t r o p y . The h o r i z o n t a l a n i s o t r o p y f a c t o r i s the r a t i o of the ranges (a2) i n the d i r e c t i o n s of the major and minor axes of a n i s o t r o p y . In s i m p l i f y i n g the models, an average s i l l (C2) was assumed i n cases where a z o n a l a n i s o t r o p y model had p r e v i o u s l y been f i t t e d . T h i s o p e r a t i o n g r e a t l y s i m p l i f i e d the k r i g i n g e q u a t i o n s , lowered computing c o s t s and d i d not a d v e r s e l y -125-a f f e c t the accurac y of the e s t i m a t e . The semi-variogram models used to k r i g e the d i f f e r e n t areas of the Venus d e p o s i t are shown i n Table 9.2.1: Area Co CI a l C2 a2 HAF Az Th-NU 0.113 0.136 20 0.059 150 1.0 0 Th-NLl 0.0 1.0 15 0.57 135 6.0 120 Th-NL2 0.60 1.05 80 0.0 80 2.67 90 Th - S l 0.4 2.14 70 0.0 70 2.8 15 Th-S2 0.0 1.6 40 0.0 40 4.0 15 Th-S3 0.55 1.15 50 0.0 50 2.5 125 Th-S4 0.4 1.9 100 0.0 100 4.0 60 Au-Nl 0.0 0.049 12.5 0.078 150 1.0 0 Au-NA 0.041 0.046 15 0.068 90 1.0 0 Au-SA 0.121 0.103 50 0.106 120 2.4 115 Au-S3 0.05 0.152 110 0.0 110 4.4 135 Ag-Nl 38.94 147.36 85 0.0 85 1.0 0 Ag-NA 19.85 34.37 25 36.38 150 2.14 113 Ag-Sl 0.0 163.47 22 55.01 60 1.0 0 Ag-S2 125.62 1279.11 110 0.0 110 2.44 150 Ag-SU 165.12 399 .44 125 0.0 125 3.57 120 Table 9.2.1: Semi-variogram models of t h i c k n e s s , g o l d accumulation and s i l v e r accumulation i n each of the s t a t i o n a r y p o p u l a t i o n s are shown. Thi c k n e s s , g o l d accumulation and s i l v e r accumulation were k r i g e d i n t o b l o c k s 25 f e e t square over the e n t i r e Venus d e p o s i t . A s e a r c h r a d i u s of 100 f e e t was used f o r each v a r i a b l e . The s e a r c h was a d j u s t e d by the h o r i z o n t a l a n i s o t r o p y f a c t o r to f i t the a n i s o t r o p y of the semi-variogram. The 20 assays nearest t o the k r i g e d p o i n t and w i t h i n the s e a r c h r a d i u s were used to estimate the v a r i a b l e . I f fewer than 2 samples were found w i t h i n the -126-search r a d i u s , the block was not k r i g e d . Where a quadrant boundary was found to cut a k r i g e d b l o c k , the block was assumed to f a l l i n t o the quadrant i n which the c e n t r e p o i n t of the block was l o c a t e d . -127-9.3 R e s u l t s of K r i g i n g The k r i g e d t h i c k n e s s , gold accumulation and s i l v e r accumulation were p l o t t e d as o v e r l a y s on a plan of the mine workings. These maps, f i g u r e s 9.3.1, 9.3.2 and 9.3.3, are found i n pockets numbered 1 to 3. The on l y blocks p l o t t e d were those f o r which a l l 3 v a r i a b l e s had been k r i g e d . A t o t a l of 2039 blocks were p l o t t e d . The maps were d i g i t i z e d and generated by the author u s i n g a computer a s s i s t e d d r a f t i n g program a t the o f f i c e of Tecucomp G e o l o g i c a l I n c o r p o r a t e d . Each block i n f i g u r e 9.3.1 shows the k r i g e d t h i c k n e s s of t h a t block and the conf i d e n c e i n t e r v a l of +/- 1 standard d e v i a t i o n . F i g u r e s 9.3.2 and 9.3.3 show k r i g e d estimates of gold and s i l v e r accumulation r e s p e c t i v e l y , with t h e i r c o nfidence i n t e r v a l s . A l l confidence i n t e r v a l s are expressed as a percentage of the k r i g e d v a l u e s . A t h i r d number, p l o t t e d i n the c e n t r e of each block, corresponds to the contour i n t e r v a l of the block as f o l l o w s : 0 = l e s s than the mean 1 = mean to mean + 1 standard d e v i a t i o n (Sd) 2 = mean + 1 Sd to mean + 2 Sd 3 = g r e a t e r than mean + 2 Sd S e v e r a l unmarked blocks appear on the maps. They correspond to areas where the sample r e s u l t s had been averaged t o a minimum 4 f o o t mining width. As e x p l a i n e d i n -128-Chapters 3 and 4, these samples were e l i m i n a t e d from the d a t a s e t . Examination of the k r i g e d t h i c k n e s s map ( f i g u r e 9.3.1) confirms the o b s e r v a t i o n s made i n Chapter 4; namely, the v e i n i s t h i c k e s t south of the hinge l i n e and below the 2700 l e v e l i n the n o r t h . In the upper no r t h a r e a , the v e i n t h i c k n e s s r a r e l y exceeds 2.4 f e e t . In a d d i t i o n , the map shows t h a t the m a j o r i t y of the stoped areas are found i n the t h i c k e r p a r t s of the v e i n . T h i s may be due to the problems pr e v i o u s o p e r a t o r s had i n v i s u a l l y d i f f e r e n t i a t i n g ore from waste. With nothing e l s e t o go by, they mined the t h i c k e r p a r t s of the v e i n . However, these d i d not n e c e s s a r i l y c o n t a i n the h i g h e s t grade m a t e r i a l , as can be seen by the weak c o r r e l a t i o n s between t h i c k n e s s and g o l d accumulation (0.220), and between t h i c k n e s s and s i l v e r accumulation (0.398). In f i g u r e 9.3.2, we see t h a t the higher g o l d accumulations, as shown by the two upper contour i n t e r v a l s , are concentrated above the 2700 l e v e l . R e c a l l t h a t the v e i n t h i c k n e s s e s t i m a t e s are lowest above t h i s l e v e l . We can t h e r e f o r e conclude t h a t the g o l d grades above the 2700 l e v e l are e s p e c i a l l y h i g h . F u r t h e r examination of f i g u r e 9.3.2 r e v e a l s 5 zones of high g o l d accumulation. Two of these are found i n the 2850 no r t h d r i f t , a t and beyond the northernmost r a i s e . One of these has a l r e a d y been t r a c e d up from the 2700 l e v e l . The other has o n l y been i n t e r s e c t e d by the 2850 l e v e l . -129-Two a d d i t i o n a l zones c o n t a i n i n g h i g h g o l d v a l u e s are found i n the c e n t r a l p o r t i o n of the mine. One has been f o l l o w e d up from the 2600 l e v e l by r a i s e s but does not extend v e r y f a r beyond the 2850 l e v e l . I t appears to be cut o f f by s u r f a c e o x i d a t i o n a t t h i s p o i n t . T h i s i s the o n l y g o l d - b e a r i n g shoot to extend below the 2700 l e v e l . The other zone of high g o l d accumulation i n the c e n t r a l p o r t i o n of the mine i s more d i f f u s e and extends from the 2700 l e v e l as f a r as the 2850 l e v e l . L a s t l y , t here i s an area of high g o l d accumulation i n the south p a r t of the mine which extends from the 2700 l e v e l to above the 2800 l e v e l . T h i s area has seen much s t o p i n g but there i s a l a r g e amount of ore grade m a t e r i a l s t i l l to be mined here. With the e x c e p t i o n of the most n o r t h e r l y zone, a l l of the areas of high g o l d accumulation are longer i n the d i p d i r e c t i o n than they are along s t r i k e . They can be t r a c e d from one l e v e l to another with r e l a t i v e ease. The n o r t h zone has o n l y been i n t e r s e c t e d by the 2850 l e v e l but the author i s c o n f i d e n t t h a t a r a i s e from the 2700 l e v e l would soon encounter h i g h g o l d v a l u e s i n the bottom of t h i s zone. High s i l v e r accumulations are more e v e n l y s p l i t between the upper and lower p a r t s of the mine ( f i g u r e 9.3.3). In the upper areas of the mine there i s a good c o r r e l a t i o n between the areas of h i g h g o l d accumulation d e s c r i b e d above and areas of high s i l v e r accumulation. The areas h i g h i n s i l v e r accumulation however, are more -130-l o c a l i z e d and l e s s wide spread than those of gold accumulation. Below the 2700 l e v e l o n l y one shoot i s high i n both gold and s i l v e r . I t i s found on the 2600 l e v e l and has been t r a c e d as f a r as the 2850 l e v e l i n r a i s e s and stopes. In the c e n t r a l p o r t i o n of the mine between the 2650 and 2700 l e v e l s , two zones of high s i l v e r accumulations are found which do not have c o r r e s p o n d i n g l y high gold v a l u e s . Although a l a r g e amount of t h i s m a t e r i a l has been mined out, over h a l f s t i l l remains. The c o r r e l a t i o n between s i l v e r accumulation and t h i c k n e s s (0.398) i s s t r o n g e r than i t i s between gold accumulation and t h i c k n e s s (0.220). T h i s i s l a r g e l y a f u n c t i o n of s i l v e r ' s d i s t r i b u t i o n throughout the mine versus gold's c o n c e n t r a t i o n i n the narrower p a r t s of the v e i n above 2700 l e v e l . L i k e the areas of high gold accumulation d e s c r i b e d above, the areas of high s i l v e r accumulation a l s o have long axes p a r a l l e l to the d i p and are narrow along s t r i k e . -131-9.4 Ore Reserve E s t i m a t i o n ore r e s e r v e s were estimated from the k r i g e d t h i c k n e s s and accumulation d a t a . In c a l c u l a t i n g the ore r e s e r v e s , d i s t i n c t i o n s were made between proven, probable and p o s s i b l e r e s e r v e s on the b a s i s of the confidence i n t e r v a l of the estimates of the gold accumulations. Gold was chosen because i t accounts f o r approximately 75% of the t o t a l value of the ore. Given an 84% confidence l i m i t on a l l estimates of gold accumulation, the re s e r v e s i n a block were co n s i d e r e d proven r e s e r v e s when the confidence i n t e r v a l was l e s s than 20%. Probable r e s e r v e s are those with a confidence i n t e r v a l of between 20% and 50%. P o s s i b l e r e s e r v e s have a confidence i n t e r v a l of gr e a t e r than 50% For blocks which had been p a r t i a l l y mined, the s u r f a c e area was a d j u s t e d to r e f l e c t the amount of v e i n m a t e r i a l remaining. A tonnage c o n v e r s i o n f a c t o r of 10 c u b i c f e e t per ton was used. Since the t h i c k n e s s and accumulations were estimated by k r i g i n g , the average grade i s simply: grade = accumulation t h i c k n e s s C a l c u l a t i n g the v a r i a n c e of the grade i s accomplished with the f o l l o w i n g e q u a t i o n : -132-2 V(g) = g * V(a) + V ( t ) - 2*Cc(a,t) * Sd(a) * Sd(t) 2 2 a t a t where: V() = the v a r i a n c e Sd() = the standard d e v i a t i o n g = the grade a = the accumulation t = the t h i c k n e s s C c ( a , t ) = c o r r e l a t i o n c o e f f i c i e n t between a and t The i n s i t u r e s e r v e s of v e i n m a t e r i a l a t Venus are shown i n Table 9.4.1. The proven and probable r e s e r v e s t o t a l 138,642 tons (+/- 0.71%) gr a d i n g 0.404 oz Au/ton (+/-0.92%) and 7.67 oz Ag/ton (+/- 1.14%). The average v e i n t h i c k n e s s i s 2.4 f e e t (+/- 0.71%). When d i l u t e d to a minimum 5 f o o t mining width, the i n s i t u proven and probable r e s e r v e s t o t a l 284,928 tons g r a d i n g 0.197 oz Au/ton (+/- 0.9 2%) and 3.73 oz Ag/ton (+/- 1.14%). See t a b l e 9.4.2. The r e s e r v e s were r e c a l c u l a t e d u s i n g November, 1988 g o l d and s i l v e r p r i c e s of $520/oz and $7.90/oz. The c a l c u l a t i o n a l s o assumed a c u t - o f f of $130/ton and a 5 f o o t minimum mining width. By t h i s r e c a l c u l a t i o n , the proven and probable r e s e r v e s t o t a l 108,577 tons g r a d i n g 0.3 oz Au/ton (+/- 1.18%) and 6.05 oz Ag/ton (+/- 1.29%). See Table 9.4.3. A comparison of the t o t a l i n s i t u r e s e r v e s with the t o t a l ore r e s e r v e s shows t h a t o n l y 20% of the t o t a l v e i n m a t e r i a l i s ore. T h i s f i g u r e underscores the f a c t t h a t a c o n s i d e r a b l e amount of e x p l o r a t i o n and development d r i f t i n g i s r e q u i r e d to l o c a t e new ore r e s e r v e s a t Venus. -133-A f u r t h e r comparison was made with the UKHM 1984 ore rese r v e estimate quoted i n Chapter 1. T h i s estimate a l s o assumed a c u t - o f f of $130/ton and a minimum 5 f o o t mining width. Commodity p r i c e s of the day were $390/oz f o r g o l d and $7.80/oz f o r s i l v e r . They a r r i v e d a t 68,286 tons g r a d i n g 0.32 oz Au/ton and 8.85 oz Ag/ton ( P r i n c e , 1984). When the k r i g e d data were used t o estimate the ore re s e r v e s under the same c o n d i t i o n s , the proven and probable r e s e r v e s t o t a l l e d 69,392 tons g r a d i n g 0.34 oz Ag/ton (+/-1.35%) and 7.55 oz Ag/ton (+/- 1.41%). See t a b l e 9.4.4. As the next s e c t i o n w i l l demonstrate, the accuracy of the p r o d u c t i o n e s t i m a t e s confirms the acc u r a c y of the ore re s e r v e e s t i m a t e . That UKHM's ore re s e r v e estimate i s not s i g n i f i c a n t l y d i f f e r e n t from the k r i g e d e s t i m a t e , i s a t r i b u t e t o t h e i r g e o l o g i s t s and t h e i r rule-of-thumb methods. However, as we have seen, t h i s g e o s t a t i s t i c a l a n a l y s i s p r o v i d e s more than an acc u r a t e ore res e r v e e s t i m a t e . -134-VENUS IN SITU RESERVE ESTIMATE ttl Gold P r i c e ($Cdn/oz): . — S i l v e r P r i c e : C u t - o f f ($Cdn/ton): 0.00 Minimum Mining Width: n/a Proven Reserves Tonnage: Vein T h i c k n e s s : Gold Grade: S i l v e r Grade: 47,432 tons 2.6 f t 0.592 oz/ton 10.79 oz/ton +/• +/• +/-+/• 0.93% 0.93% 1.05% 1.17% Probable Reserves Tonnage: Vei n T h i c k n e s s : Gold Grade: S i l v e r Grade: 91,210 tons 2.4 f t 0.307 oz/ton 6.05 oz/ton +/-+/-+/-+/-0.97% 0.97% 1.46% 1.92% P o s s i b l e Reserves Tonnage: Vei n T h i c k n e s s : Gold Grade: S i l v e r Grade: 113,892 tons 2.2 f t 0.124 oz/ton 3.94 oz/ton +/-+/-+/-+/-1.12% 1.12% 3.46% 3.38% Proven + Probable Reserves Tonnage: V e i n Thickness Gold Grade: S i l v e r Grade: 138,642 tons 2.4 f t 0.404 oz/ton 7.67 oz/ton +/• +/• +/-+/• 0.71% 0.71% 0.92% 1.14% T o t a l Reserves Tonnage: Vei n Thickness Gold Grade: S i l v e r Grade: 252,535 tons 2.3 f t 0.278 oz/ton 5.99 oz/ton +/-+/-+/• +/• 0.64% 0.64% 1.01% 1.28% Table 9.4.1: The k r i g e d i n s i t u r e s e r v e s a t Venus. The a c t u a l v e i n width i s used t o c a l c u l a t e the tonnage. -135-VENUS IN SITU RESERVE ESTIMATE #2 Gold P r i c e ($Cdn/oz): . — s i l v e r P r i c e : C u t - o f f ($Cdn/ton): 0.00 Minimum Mining Width: 5.0 f t Proven Reserves Tonnage: 91,358 tons V e i n T h i c k n e s s : 2.6 f t +/- 0.93% Gold Grade: 0.307 oz/ton +/- 1.05% S i l v e r Grade: 5.60 oz/ton +/- 1.17% Probable Reserves Tonnage: 193,570 tons V e i n T h i c k n e s s : 2.4 f t +/- 0.97% Gold Grade: 0.144 oz/ton +/- 1.46% S i l v e r Grade: 2.85 oz/ton +/- 1.92% P o s s i b l e Reserves Tonnage: 254,725 tons V e i n T h i c k n e s s : 2.2 f t +/- 1.12% Gold Grade: 0.055 oz/ton +/- 3.46% S i l v e r Grade: 1.76 oz/ton +/- 3.38% Proven + Probable Reserves Tonnage: 284,928 tons V e i n T h i c k n e s s : 2.4 f t +/- 0.71% Gold Grade: 0.197 oz/ton +/- 0.92% S i l v e r Grade: 3.73 oz/ton +/- 1.14% T o t a l Reserves Tonnage: 539,652 tons Vein T h i c k n e s s : 2.3 f t +/- 0.64% Gold Grade: 0.130 oz/ton +/- 1.01% S i l v e r Grade: 2.80 oz/ton +/- 1.28% Table 9.4.2: The k r i g e d i n s i t u r e s e r v e s a t Venus. The v e i n m a t e r i a l has been d i l u t e d t o a minimum 5 f o o t mining width. -136-VENUS ORE RESERVE ESTIMATE #1 Gold P r i c e ($Cdn/oz): 520.00 S i l v e r P r i c e : 7.90 C u t - o f f ($Cdn/ton): 130.00 Minimum Mining Width: 5.0 f t Proven Reserves Tonnage: V e i n T h i c k n e s s : Gold Grade: S i l v e r Grade: 73,942 tons 2.7 f t 0.339 oz/ton '6.34 oz/ton +/-+/-+/• 1.06% 1.17% 1.26% Probable Reserves Tonnage: V e i n Thickness Gold Grade: S i l v e r Grade: 34,635 tons 2.4 f t 0.218 oz/ton 5.44 oz/ton +/-+/-+/-2.42% 3.06% 3.21% P o s s i b l e Reserves Tonnage: Vei n T h i c k n e s s : Gold Grade: S i l v e r Grade: 1,157 tons 3.1 f t 0.074 oz/ton 14.29 oz/ton +/-+/-+/-12.66% 30.86% 11.75% Proven + Probable Reserves Tonnage: V e i n T h i c k n e s s : Gold Grade: S i l v e r Grade: 108,577 tons 2.6 f t 0.300 oz/ton 6.05 oz/ton +/-+/-+/-1.04% 1.18% 1.29% T o t a l Reserves Tonnage: Vei n T h i c k n e s s : Gold Grade: S i l v e r Grade: 109,734 tons 2.6 f t 0.298 oz/ton 6.14 oz/ton +/-+/-+/-1.04% 1.18% 1.27% Table 9.4.3: The k r i g e d ore r e s e r v e s u s i n g November, 1988 commodity p r i c e s . A c u t - o f f of $130/ton i s used and the v e i n m a t e r i a l i s d i l u t e d t o a minimum 5 f o o t mining width. -137-VENUS ORE RESERVE ESTIMATE #2 Gold P r i c e ($Cdn/oz): 390.00 S i l v e r P r i c e : 7.80 C u t - o f f ($Cdn/ton): 130.00 Minimum Mining Width: 5.0 f t Proven Reserves Tonnage: Vei n T h i c k n e s s : Gold Grade: S i l v e r Grade: 57,493 tons 2.8 f t 0.369 oz/ton 7.28 oz/ton +/-+/-+/• 1.20% 1.32% 1.36% Probable Reserves Tonnage: V e i n T h i c k n e s s : Gold Grade: S i l v e r Grade: 11,899 tons 2.9 f t 0.223 oz/ton 8.87 oz/ton +/-+/-+/-4.11% 5.22% 4.58% P o s s i b l e Reserves Tonnage: Vei n T h i c k n e s s : Gold Grade: S i l v e r Grade: 606 tons 2.8 f t 0.056 oz/ton 17.44 oz/ton +/-+/-+/-15.44% 42.17% 14.17% Proven + Probable Reserves Tonnage: Vei n Thickness; Gold Grade: S i l v e r Grade: 69,392 tons 2.8 f t 0.344 oz/ton 7.55 oz/ton +/-+/-+/-1.23% 1.35% 1.41% T o t a l Reserves Tonnage: Vei n T h i c k n e s s : Gold Grade: S i l v e r Grade: 69,997 tons 2.8 f t 0.341 oz/ton 7.64 oz/ton +/-+/• +/• 1.23% 1.35% 1.39% Table 9.4.4: The k r i g e d ore r e s e r v e s u s i n g the same commodity p r i c e s as those used by UKHM i n 1984. A cut o f f of $130/ton i s used and the v e i n m a t e r i a l i s d i l u t e d t o a minimum 5 f o o t mining width. -138-9. 5 P r o d u c t i o n Estimates The r e p o r t e d p r o d u c t i o n from the Venus mine workings t o t a l s 83,035 tons of m a t e r i a l a v e r a g i n g 0.186 oz Au/ton and 5.77 oz Ag/ton. T h i s t o t a l i n c l u d e s p r o d u c t i o n data from VML and UKHM as shown below: Au Ag Source Tons o/t (oz) o/t (oz)  VML 70/71 Stoping 34,966 0.24 (8,391.8) 8.48 (296,511.7) D r i f t i n g 30,265 0.132 (4.003.5) 3.39 (102,650.2) T o t a l 65,231 0.19 (12,393.9) 6.12 (399,213.7) UKHM 1981 Stoping 4,050 D r i f t i n g 591 T o t a l 4,641 0.18 (835.4) 5.03 (23,344.2) UKHM 1984 R a i s i n g 5,065 D r i f t i n g 8.098  T o t a l 13,163 0.17 (2,237.4) 4.3 (56,600.9) Grand T o t a l 83,035 0.186 (15,467.0) 5.77 (479,158.8) Table 9.5.1: T o t a l r e p o r t e d p r o d u c t i o n from the Venus mine. The d r i f t s a t Venus were d r i v e n 8 f e e t square, but experienced a 20% d i l u t i o n ( S t e r l i n g , personnal communication). T h e r e f o r e , the 8900 f e e t of d r i f t i n g on v e i n measured from the mine p l a n s , t o t a l s 68,352 tons. The amount of d r i f t muck accounted f o r above i s 38,954 tons, l e a v i n g 29,398 tons which were presumably dumped as waste. The t o t a l amount of m a t e r i a l mined a t Venus i s 112,433 tons. -139-I t i s estimated from k r i g i n g , t h a t 53,060 tons (+/-0.7%) of v e i n m a t e r i a l g r a d i n g 0.314 oz Au/ton (+/- 0.91%) and 8.74 oz Ag/ton (+/- 0.91%) were mined by VML and UKHM. These f i g u r e s when converted t o an estimate of t o t a l ounces removed, y i e l d 16,696 (+/- 152) ounces of g o l d and 463,669 (+/- 4219) ounces of s i l v e r . A comparison with the r e p o r t e d ounces mined shows a d i f f e r e n c e of 1229 ounces of g o l d or 7.4%, and -15,469 ounces of s i l v e r or -3.3%. T h i s a v e r y good estimate, p a r t i c u l a r l y when one c o n s i d e r s the d i s c r e p a n c y between VML's mine and m i l l r e p o r t s . Over the mine's ten month l i f e , the mine r e p o r t s underestimated the tonnage shipped to the m i l l by 1750 tons, and overestimated the c o n t a i n e d ounces of g o l d and s i l v e r by 1752 and 144,818 ounces r e s p e c t i v e l y (MacDonald, 1971). The a c c u r a c y of the estimate of the mined v e i n m a t e r i a l has a d i r e c t c o r r e l a t i o n with the accuracy of the ore r e s e r v e e s t i m a t e s . Having been a b l e t o estimate the p r o d u c t i o n f i g u r e s with such p r e c i s i o n , an e x t r a measure of conf i d e n c e can now be p l a c e d on the k r i g e d ore r e s e r v e s of the Venus mine. -140-CONCLUSIONS and RECOMMENDATIONS On the b a s i s of the g e o s t a t i s t i c a l a n a l y s i s of the Venus d e p o s i t presented i n t h i s t h e s i s , the f o l l o w i n g c o n c l u s i o n s can be drawn: 1. There i s s t r o n g evidence f o r the e x i s t e n c e of two po p u l a t i o n s north and south of the hinge l i n e . As w e l l as a change i n s t r i k e , there i s a t h i c k e n i n g of the v e i n i n the south. The average v e i n width i s 3.29 f e e t i n the south as compared to 2.9 f e e t i n the lower north and 1.2 f e e t i n the upper n o r t h . S i l v e r grades are higher i n the south: 6.67 oz/ton versus 4.69 oz/ton i n the no r t h . But the opposite i s t r u e of gold which has an average grade of 0.116 oz/ton i n the south and 0.232 oz/ton i n the no r t h . Both the mean g o l d and s i l v e r accumulations are higher i n the south than i n the n o r t h . 2. The d i s t r i b u t i o n s of g o l d , s i l v e r , g o ld accumulation, s i l v e r accumulation and t h i c k n e s s i n the upper north are 3-parameter log-normal d i s t r i b u t i o n s . A t r a n s f o r m was r e q u i r e d to normalize them before c a l c u l a t i n g semi-variograms. 3. The ore shoots a t Venus are long, narrow zones o r i e n t e d with t h e i r axes up and down the d i p of the v e i n . T h i s i s p a r t i c u l a r l y c l e a r north of the hinge l i n e . The s c a t t e r p l o t s of g o l d grade versus n o r t h i n g and of s i l v e r grade versus n o r t h i n g show four d i s t i n c t c o n c e n t r a t i o n s of higher grade assays i n the n o r t h p a r t of the mine. The south appears to be one l a r g e zone. The m a j o r i t y of the geometric a n i s o t r o p i e s of the semi-variograms were o r i e n t e d with t h e i r major axes p a r a l l e l t o the d i p of the v e i n . Where a zonal a n i s o t o p y was modelled, most of the curves with the low s i l l s were i n the d i p d i r e c t i o n . The maps of the k r i g e d g o l d and s i l v e r accumulations add f u r t h e r support to t h i s c o n c l u s i o n . With o n l y one e x c e p t i o n , the zones of high g o l d and s i l v e r v a l u e s can be e a s i l y t r a c e d from one l e v e l to the next s i n c e they are longer i n the d i p d i r e c t i o n than a l o n g s t r i k e . 4. The m a j o r i t y of the high g o l d accumulations i n the k r i g e d b l o c k s are c o n c e n t r a t e d above the 2700 l e v e l . Only one zone with high g o l d values extends down to the 2600 l e v e l . I t i s t h e r e f o r e recommended t h a t f u r t h e r e x p l o r a t i o n f o r g o l d a t Venus con c e n t r a t e on the upper p o r t i o n of the mine above 2700 l e v e l . The e x p l o r a t i o n should extend the present workings to the n o r t h . 5. High s i l v e r accumulations correspond w e l l with the h i g h g o l d values above 2700 l e v e l . The zones are , however, s m a l l e r and more l o c a l i z e d than t h e i r g o l d c o u n t e r p a r t s . Below 2700 l e v e l , o n l y one zone of high s i l v e r accumulation -142-i s a s s o c i a t e d with c o r r e s p o n d i n g l y high gold accumulation. The other two zones below 2700 have o n l y low gold accumulations. 6. The k r i g e d i n s i t u ore r e s e r v e s d i l u t e d to a minimum 5 f o o t mining width are 284,928 tons g r a d i n g 0.197 oz Au/ton (+/- 0.92%) and 3.73 oz Ag/ton (+/- 1.14%). These c o n s t i t u t e proven and probable r e s e r v e s . 7. The proven and probable k r i g e d ore r e s e r v e s above a c u t - o f f of $130/ton, assuming gold and s i l v e r p r i c e s of $520/oz and $7.80/oz are 108,577 tons grading 0.3 oz Au/ton (+/- 1.18%) and 6.05 oz Ag/ton (+/- 1.29%). T h i s would a l l o w UKHM to operate f o r 3 years, g i v e n t h e i r c o n c e n t r a t o r ' s c a p a c i t y of 100 tons per day. I t i s t h e r e f o r e recommended t h a t United Keno H i l l Mines reopen the Venus mine. 8. A comparison of the t o t a l i n s i t u r e s e r v e s ( a l l c l a s s e s ) to the t o t a l ore r e s e r v e s shows t h a t o n l y 20% of the v e i n m a t e r i a l i s ore. T h i s underscores the f a c t t h a t a c o n s i d e r a b l e amount of e x p l o r a t i o n i s r e q u i r e d to expand the ore r e s e r v e s a t Venus. 9. K r i g i n g a c c u r a t e l y estimated the t o t a l amount of g o l d and s i l v e r mined by Venus Mines L i m i t e d and United Keno H i l l Mines L i m i t e d . The d i f f e r e n c e between the estimated -143-and the r e p o r t e d ounces mined was +7.4% i n the case of g o l d and -3.3% i n the case of s i l v e r . The accuracy of t h i s estimate i s a good i n d i c a t i o n of the r e l i a b i l i t y of the k r i g e d ore r e s e r v e s . 10. Thi3 g e o s t a t i s t i c a l a n a l y s i s i s the f i r s t t o a p p l y r i g o u r o u s s t a t i s t i c a l methodology to the e s t i m a t i o n of ore r e s e r v e s a t the Venus mine. The estimate of the produced ounces of g o l d and s i l v e r i s evidence of i t s accuracy. K r i g i n g by d e f i n i t i o n produces the best l i n e a r unbiased estimate so the low c o n f i d e n c e i n t e r v a l s on a l l of the e s t i m a t e s show i t s p r e c i s i o n . 11. The maps of the k r i g e d estimates i n 2039 blocks show g r a p h i c a l l y the areas of ore and waste. In a mine where i t i s v ery d i f f i c u l t t o v i s u a l l y d i f f e r e n t i a t e ore from waste, t h i s i s c r i t i c a l . I t i s recommended t h a t k r i g i n g be used on an ongoing b a s i s to estimate the ore r e s e r v e s and as a guide to s t o p i n g . To achieve t h i s , the data base should be updated with a l l new sample data as they become a v a i l a b l e . The area a n a l y s e d i n t h i s study c o u l d be s i g n i f i c a n t l y i n c r e a s e d by resampling o l d Venus workings. In t h i s way U n i t e d Keno H i l l Mines would have the i n f o r m a t i o n on which to base sound d e c i s i o n s about f u t u r e p r o d u c t i o n and e x p l o r a t i o n a t the Venus mine. -144-BIBLIOGRAPHY C a r l y l e , L.W., 1984, P r i v a t e company r e p o r t . C l a r k , I . , 1979, P r a c t i c a l G e o s t a t i s t i c s , A p p l i e d Science P u b l i s h e r s L t d . , London. David, M., 1977, G e o s t a t i s t i c a l Ore Reserve E s t i m a t i o n , E l s e v i e r S c i e n t i f i c P u b l i s h i n g Co., Amsterdam. Davis, J.C. 1986, S t a t i s t i c s and Data A n a l y s i s i n Geology, Second E d i t i o n , John Wiley & Sons, Inc.. J o u r n a l , A.G. and H u i j b r e g t s , Ch.J., 1978, Mining  G e o s t a t i s t i c s . Academic Pr e s s , London. Kim, Y.C. and Knudsen, H.P., 1977, G e o s t a t i s t i c a l Ore  Reserve E s t i m a t i o n f o r a R o l l - F r o n t Type Uranium  Deposit, United S t a t e s Energy Research and Development A d m i n i s t r a t i o n , Grand J u n c t i o n , Colorado, GJBX-3(77). K r i g e , D.G., 1978, Lognormal-de W i j s i a n G e o s t a t i s t i c s f o r  Ore E v a l u a t i o n . South A f r i c a n I n s t i t u t e of Mining and Metalurgy, Johannesburg. Matheron, G., 1963, " P r i n c i p l e s of G e o s t a t i s t i c s " , Economic  Geology. V o l . 58, No. 8, p. 1246 - 1266. Matheron, G., 1971, The Theory of R e g i o n a l i z e d V a r i a b l e s  and I t s A p p l i c a t i o n . Les C a h i e r s du Centre de Morphologie Mathematique de F o n t i a n e b l e a u , No. 5. MacDonald, C.H., 1971, P r i v a t e company r e p o r t . M c F a u l l , J . , 19 82, P r i v a t e company r e p o r t . Montgomery, J.H., S i n c l a i r , A.J., Symonds, D.F. and Giroux, G.H., 1977, " G e o s t a t i s t i c a l study of the Ladner Creek g o l d d e p o s i t of C a r o l i n Mines L t d . " . , Western Miner. March 1977, p.11 - 20. P r i n c e , D.R., 1984a, P r i v a t e company r e p o r t . P r i n c e , D.R., 1984b, P r i v a t e company r e p o r t . R a l f s , K., 1975, A Study of the Mineralogy. Paragenesis,  Zoning and Wall Rock A l t e r a t i o n . Venus V e i n .  Montana Mountain, S.W. Yukon, Unpublished B.Sc. t h e s i s , U n i v e r s i t y of B r i t i s h Columbia. Raymond, G., 1979, "Ore r e s e r v e e s t i m a t i o n problems i n an e r r a t i c a l l y m i n e r a l i z e d orebody", CIM B u l l e t i n , June 1979, p. 90 - 98 -145-Rendu, J.M., 1978, An I n t r o d u c t i o n t o G e o s t a t l s t l c a l  Methods of Ore Reserve E v a l u a t i o n , South A f r i c a n I n s t i t u t e of Mining and Metalurgy, Johannesburg. Roots, C.F., 1980, General Geology of the Montana Mountain  Area, South C e n t r a l Yukon, Unpublished F i e l d Report. S i n c l a i r , A.J. and Deraisme, J . , 1974, "A G e o s t a t i s t i c a l Study of the Eagle V e i n , Northern B r i t i s h Columbia", CIM B u l l e t i n , June 1974, p. 131 - 142 S t e r l i n g , R.W., 1981, P r i v a t e company r e p o r t . S t e r l i n g , R.W., 1984, P r i v a t e company r e p o r t . Walton, L.A., 1987, Geology and Geochemistry of the Venus Au-Ag-Pb-Zn Vei n Deposit, Yukon T e r r i t o r y , Unpublished M.Sc. t h e s i s , U n i v e r s i t y of A l b e r t a . Watson, K.W., 1979, P r i v a t e company r e p o r t . -146-

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