UBC Theses and Dissertations

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UBC Theses and Dissertations

A geostatical evaluation of the southern tail zone of Equity Silver Mines, Limited, Houston, B.C. Giroux, Gary Henri 1984

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A GEOSTATISTICAL EVALUATION OF THE SOUTHERN T A I L ZONE OF EQUITY SILVER MINES, LIMITED HOUSTON, B.C. BY GARY HENRI GIROUX B . A . S c , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF GEOLOGICAL SCIENCES) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e p e g u i r e d s t a n d a r d : The U n i v e r s i t y of May © Ga r y H e n r i B r i t i s h 1984 G i r o u x , C o l u m b i a 1 984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f G&-OCO G Y The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date §><^)j, / f f , I? 2 f i i ABSTRACT The Southern T a i l zone of E q u i t y S i l v e r Mines, Sam Go o s l y s i l v e r - c o p p e r d e p o s i t p r e s e n t s an i n t e r e s t i n g c h a l l e n g e not o n l y f o r g l o b a l and l o c a l ore r e s e r v e e s t i m a t i o n s , but a l s o f o r m a i n t a i n i n g q u a l i t y c o n t r o l on samp l i n g and a s s a y i n g . S i l v e r grades from both stockwork and massive g a l e n a - s p h a l e r i t e l e n s e s v a ry as much as, 3 o r d e r s of magnitude a c r o s s v e r y s h o r t d i s t a n c e s . G e o s t a t i s t i c a l methods have been d e v e l o p e d t o mini m i z e the e f f e c t s of e r r a t i c m i n e r a l i z a t i o n on r e s e r v e e s t i m a t i o n and q u a l i t y c o n t r o l . C o n d i t i o n a l p r o b a b i l i t y i s demonstrated as a g l o b a l ore r e s e r v e s e s t i m a t i o n method f o r the Southern T a i l zone u s i n g e x p l o r a t i o n diamond d r i l l d a t a . The e n t i r e p r o c e d u r e , i n c l u d i n g d e c l u s t e r i n g , s t r u c t u r a l a n a l y s i s and c o n d i t i o n i n g raw da t a d i s t r i b u t i o n s t o b l o c k grade d i s t r i b u t i o n s i s o u t l i n e d . R e s u l t s are p r e s e n t e d as a grade tonnage c u r v e and compared t o e s t i m a t e s c a l c u l a t e d by i n v e r s e d i s t a n c e . A s t a t i s t i c a l and g r a p h i c a l approach t o e v a l u a t i n g p a i r e d data i s p r e s e n t e d as a method of m o n i t o r i n g the q u a l i t y of sampling and a s s a y i n g . R e s u l t s from the Southern T a i l zone i n d i c a t e a b i a s i n s i l v e r a n a l y s e s between two l a b o r a t o r i e s , a l a r g e random e r r o r i n the s t a n d a r d tube sa m p l i n g t e c h n i q u e f o r b l a s t h o l e c u t t i n g s and an i n v e r s e r e l a t i o n s h i p between sampling e r r o r and sample volume. An i d e a l i z e d c a l c u l a t i o n based on p o s s i b l e improvement of sa m p l i n g method i n d i c a t e s t h a t a s i g n i f i c a n t i n c r e a s e i n metal r e c o v e r y c o u l d be a t t a i n e d through b e t t e r s a m p l i n g . For l o c a l b l o c k e s t i m a t i o n , a v a r i e t y of k r i g i n g t e c h n i q u e s i i i have been e v a l u a t e d and compared w i t h i n v e r s e d i s t a n c e and p o l y g o n a l e s t i m a t e s . Both c r o s s v a l i d a t i o n and comparison of k r i g i n g e r r o r s showed t h a t l o g k r i g i n g of s e l e c t i o n m i n i n g u n i t s u s i n g p a r t i t i o n e d sample data i s s u p e r i o r t o o t h e r methods, a l t h o u g h o n l y m a r g i n a l l y so i n some c a s e s . For a t e s t area of 100 b l o c k s on the 1310 N bench e s t i m a t e d by the v a r i o u s methods, the p o l y g o n a l method p r e d i c t e d 1.4 m i l l i o n grams of s i l v e r more than a l o g p a r t i t i o n e d method. W h i l e no c o n c l u s i o n s can be drawn as t o which method i s c o r r e c t , the n e c e s s i t y of m o n i t e r i n g the metal c o n t e n t from s p e c i f i c b l o c k s e a r l y i n the p r o d u c t i o n l i f e of a mine i s s t r e s s e d . TABLE OF CONTENTS ABSTRACT i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS x CHAPTER 1 : INTRODUCTION 1 CHAPTER 2 : GLOBAL RESERVES OF SOUTHERN TAIL ZONE BY CONDITIONAL PROBABILITY 4 A b s t r a c t 5 I n t r o d u c t i o n 6 Data 12 Data E v a l u a t i o n 14 S t r u c t u r a l (Semivariogram) I n t e r p r e t a t i o n 16 D e c l u s t e r i n g 21 C o n d i t i o n a l P r o b a b i l i t y 25 Grade and Tonnage E s t i m a t i o n . . . 28 C o n c l u s i o n s 33 Acknowledgements 35 Re f e r e n c e s 36 CHAPTER 3 : PRODUCTION QUALITY CONTROL EXPERIMENTS 37 A b s t r a c t 38 I n t r o d u c t i o n 39 Procedure f o r E v a l u a t i n g D u p l i c a t e ( P a i r e d ) D a t a . . . 41 Q u a l i t y C o n t r o l T e s t i n g a t E q u i t y S i l v e r Mine 47 D u p l i c a t e A n a l y t i c a l Data 48 D u p l i c a t e B l a s t h o l e Sampling 52 V B l a s t h o l e Sampling P r o c e d u r e s at E q u i t y S i l v e r Mine 55 E v a l u a t i o n of Sampling T e s t R e s u l t s 59 I d e a l i z e d Economic A n a l y s i s 65 C o n c l u s i o n s 70 Acknowledgements 73 R e f e r e n c e s 74 CHAPTER 4 : AN• • EVALUATION OF VARIOUS KRIGING TECHNIQUES AS APPLIED TO EQUITY SILVER MINES SOUTHERN TAIL ZONE 75 A b s t r a c t 76 I n t r o d u c t i o n 77 Data A c q u i s i t i o n 78 S t r u c t u r a l A n a l y s i s 79 T r a d i t i o n a l Model 79 Lognormal Model.. 83 P a r t i t i o n e d Model 86 Log P a r t i t i o n e d Model 89 M u l t i - G a u s s i a n Model 91 Back A n a l y s i s 92 B l o c k K r i g i n g 96 C o n c l u s i o n s 102 Acknowledgements.. 103 R e f e r e n c e s 104 CHAPTER 5 ,: CONCLUSIONS 105 REFERENCES 109 APPENDIX I - Comparison of E q u i t y Lab and a Check Lab on M i l l Sample Assays 111 APPENDIX I I - D u p l i c a t e B l a s t h o l e Sampling f o r Copper and S i l v e r 113 v i APPENDIX III - R e s u l t s f o r D i f f e r e n t Sampling Techniques f o r 42 Blocks 115 APPENDIX IV - C a l c u l a t i o n of Volumes f o r D i f f e r e n t Sampling Techniques 117 v i i L I S T OF TABLES CHAPTER 2 TABLE 1 - L i t h o l o g i c a l U n i t s 8 TABLE 2 - G e n e r a l P r o c e d u r e f o r G l o b a l R e s e r v e E s t i m a t i o n by C o n d i t i o n a l P r o b a b i l i t y . . . . 11 TABLE 3 - P a r a m e t e r s of 3 - D i m e n s i o n a l S p h e r i c a l Model S o u t h e r n T a i l Zone Diamond D r i l l D a t a 20 TABLE 4 - Grade and Tonnage C u r v e s ( S i l v e r ) CHAPTER 3 TABLE 1 - Summary of R e s u l t s f o r P a i r e d t - T e s t on T h r e e S e t s o f P a i r e d D u p l i c a t e A n a l y s e s f o r S i l v e r 51 TABLE 2 - R e s u l t s f o r P a i r e d t - T e s t f o r Copper and S i l v e r on 21 D u p l i c a t e B l a s t h o l e Samples 52 TABLE 3 - Sample T y p e s w i t h C o r r e s p o n d i n g Volumes and W e i g h t s 58 TABLE 4 - C o m p a r i s o n o f S a m p l i n g P r o c e d u r e s Comparing Volume, ' Amount of S c a t t e r (Sy) and R e g r e s s i o n P a r a m e t e r s ( y = A + Bx ) 63 TABLE 5 - T a b u l a t i o n o f B l a s t h o l e s M i s c l a s s i f i e d by Tube S a m p l i n g Assuming t h a t a B e s t ( C a l c u l a t e d ) B u l k E s t i m a t e i s C o r r e c t , and C u t - O f f G r a d e i s 30 g.AgE/tonne 66 CHAPTER 4 TABLE 1 - A Summary o f t h e Ranges i n t h e F o u r P r i n c i p a l D i r e c t i o n s 81 TABLE 2 - A Summary o f P a r a m e t e r s f o r t h e F o u r A n i s o t r o p i c S e m i v a r i o g r a m M o d e l s D e v e l o p e d 89 TABLE 3 - Mean S q u a r e d D i f f e r e n c e s o f E s t i m a t e d and T r u e B l a s t h o l e G r a d e s as a F u n c t i o n o f 3 Grade C a t e g o r i e s f o r 5 E s t i m a t i o n P r o c e d u r e s 94 LIST OF FIGURES CHAPTER 2 F i g u r e 1 : F i g u r e 2 : F i g u r e 3 : F i g u r e 4 : F i g u r e 5 : F i g u r e 6 : F i g u r e 7 : F i g u r e 8 : F i g u r e 9 : F i g u r e 10 F i g u r e 11 L o c a t i o n of E q u i t y S i l v e r Mines' Sam Goos l y D e p o s i t , P l a n view of the Southern T a i l zone showing c o l l a r l o c a t i o n s of 108 diamond d r i l l h o l e s C u m u l a t i v e l o g p r o b a b i l i t y p l o t of 2235 s i l v e r a s s a y s from e x p l o r a t i o n diamond d r i l l h o l e s , Southern T a i l zone, 1 3 1 5 Average v e r t i c a l r e l a t i v e semi-v a r i o g r a m f o r 3 m. composite s i l v e r grades 17 Average h o r i z o n t a l r e l a t i v e semi-v a r i o g r a m f o r s i l v e r grades i n the NW-SE d i r e c t i o n 18 Average h o r i z o n t a l r e l a t i v e semi-v a r i o g r a m f o r s i l v e r grades i n the NE-SW d i r e c t i o n 19 Naive mean s i l v e r v a l u e p l o t t e d f o r v a r i o u s c e l l s i z e s (s) i n or d e r t o determine the optimum c e l l s i z e f o r d e c l u s t e r i n g 23 Unbiased h i s t o g r a m f o r Southern T a i l zone s i l v e r from 3 m. composites w i t h h i s t o g r a m of raw d a t a super-imposed 24 Unbiased h i s t o g r a m f o r s i l v e r grades of 10 x 10 x 10 cu.m. b l o c k s , Southern T a i l zone, w i t h h i s t o g r a m from un-b i a s e d 3 m. composites superimposed.... 27 : Upper s l i c e ( i n c l u d i n g l e v e l s 1280 to 1235), Southern T a i l zone, showing d i s t r i b u t i o n of 45 m. b l o c k s . B l o c k s w i t h i n t he double l i n e s used t o c a l c u l a t e tonnage of 48.8 m i l l i o n t o n n es; b l o c k s c o n t a i n i n g d a t a a r e shaded 29 : Grade-tonnage c u r v e f o r Southern T a i l zone showing two tonnage c u r v e s : one based on a t o t a l m i n e r a l i z e d tonnage of 59 m i l l i o n tonnes and a second based on 48.8 m i l l i o n t o n n e s . . . . 32 i x CHAPTER 3 F i g u r e 1 : A s y s t e m f o r e v a l u a t i n g c h e c k a s s a y s . . . . 42 F i g u r e 2 : A s i m p l e l i n e a r model t o d e s c r i b e e r r o r s i n p a i r e d d a t a 44 F i g u r e 3 : I d e a l i z e d examples of p a t t e r n s e x h i b i t e d on X-Y p l o t s o f d u p l i c a t e a n a l y s i s 46 F i g u r e 4a : P l o t of c o m p a r i s o n between two l a b o r a t o r i e s showing 23 p o i n t s a b o u t an e q u a l v a l u e r e f e r e n c e l i n e (Y=X) f o r t a i l i n g s ( T ) , o r e (0) and c o n c e n t r a t e s (C) samples f o r c o p p e r 49 F i g u r e 4b : P l o t o f c o m p a r i s o n between two l a b o r a t o r i e s showing 23 p o i n t s about an e q u a l v a l u e r e f e r e n c e l i n e (Y=X) f o r t a i l i n g s ( T ) , o r e (0) and c o n c e n t r a t e s (C) samples f o r s i l v e r 49 F i g u r e 5a : P a i r e d r e s u l t s f o r c o p p e r from 21 d u p l i c a t e b l a s t h o l e s a m p l e s 53 F i g u r e 5b : P a i r e d r e s u l t s f o r s i l v e r from 21 d u p l i c a t e b l a s t h o l e s a m p l e s 53 F i g u r e 6 : P l a n v i e w of b l a s t h o l e c u t t i n g s s h o wing l o c a t i o n s o f t h e t u b e and c h a n n e l s a m p l e s (Cyr e t a l , 1980) 56 F i g u r e 7 : O r i g i n a l t u b e sample v e r s u s b e s t e s t i m a t e b u l k f o r 42 p a i r e d s i l v e r d a t a 60 F i g u r e 8 : A v e r a g e c h a n n e l sample v e r s u s b e s t e s t i m a t e b u l k f o r 42 p a i r e d s i l v e r d a t a 61 F i g u r e 9 : B u l k sample v e r s u s b e s t e s t i m a t e b u l k f o r 42 p a i r e d s i l v e r d a t a 62 F i g u r e 10 : P l o t s h o wing t h e s c a t t e r a b o u t t h e r e g r e s s i o n l i n e (Sy) v e r s u s sample volume w i t h a c u r v e f i t t e d e m p i r i c a l l y . . 64 F i g u r e 11 : O r i g i n a l t u b e sample v e r s u s b e s t e s t i m a t e b u l k f o r 42 p a i r e d s i l v e r e q u i v a l e n t d a t a 68 C H A P T E R 4 F i g u r e 1 : Average h o r i z o n t a l semivariograms f o r s e l e c t e d benches Southern T a i l zone, showing e x p e r i m e n t a l c u r v e s f o r d i r e c t i o n s 1 t o 4 80 F i g u r e 2 : S t r u c t u r a l e l l i p s e of ranges i n the f o u r p r i n c i p a l d i r e c t i o n s showing the d i r e c t i o n of minimum and maximum s t r u c t u r e 82 F i g u r e 3 Ar i thmet i c grams f o r b l a s t h o l e s and l o g t r a n s f o r m e d h i s t o -2575 s i l v e r grades from Southern T a i l zone 84 F i g u r e 4 : Average h o r i z o n t a l s e m i v a r i o g r a m i n the d i r e c t i o n of minimum and maximum s t r u c t u r e f o r l o g t r a n s f o r m e d s i l v e r g r a d e s , Southern T a i l zone , F i g u r e 5 : Lognormal c u m u l a t i v e d i s t r i b u t i o n c u r v e f o r 2773 s i l v e r grades from 1310 N, 1310 S and 1315 N benches, Southern T a i l zone 85 87 F i g u r e 6 : Average h o r i z o n t a l semivariograms i n the d i r e c t i o n of minimum and maximum s t r u c t u r e f o r p a r t i t i o n e d l o g n o r m a l s i l v e r g r a d e s , Southern T a i l zone F i g u r e 8 F i g u r e 9 F i g u r e 7 : S c a t t e r p l o t showing r e a l s i l v e r grade v e r s u s s i l v e r grade e s t i m a t e d by l o g p a r t i t i o n e d k r i g i n g f o r 1510 b l a s t h o l e s , Southern T a i l zone , Tes t a r e a of 1310 N bench showing s i l v e r a s s a y s f o r b l a s t h o l e s w i t h an a r b i t r a r y g r i d of 100 ( 5 x 5 x 5 c u . m.) b l o c k s superimposed , 90 93 97 Graph•showing r e l a t i v e k r i g i n g e r r o r ( u s i n g f o u r d i f f e r e n t k r i g i n g models) f o r 25 a r b i t r a r i l y s e l e c t e d b l o c k s from the 100 e s t i m a t e d 99 F i g u r e 10 : Test a r e a of 1310 N bench showing k r i g e d s i l v e r grade f o r 100 ( 5 x 5 x 5 cu.m.) b l o c k s . K r i g e d s i l v e r grade c a l c u l a t e d u s i n g l o g p a r t i t i o n e d model 100 x i ACKNOWLEDGEMENTS The a u t h o r would l i k e t o thank many p e o p l e b o t h d i r e c t l y and i n d i r e c t l y c o n c e r n e d w i t h t h i s s t u d y . F i r s t s p e c i a l t h a n k s a r e e x t e n d e d t o Dr. A . J . S i n c l a i r f o r s u p e r v i s i n g t h e r e s e a r c h , a s s i s t i n g w i t h p r o b l e m s and p r o v i d i n g c o n s t r u c t i v e c r i t i c i s m t h r o u g h o u t t h e s t u d y . The a u t h o r i s g r a t e f u l t o P l a c e r Development L i m i t e d f o r a l l o w i n g a c c e s s t o t h e d a t a and f o r f i n a n c i a l a s s i s t a n c e i n c o m p l e t i n g t h e work. Thanks a r e due t o Mr. E.T. L o n e r g a n , P r o j e c t S u p e r v i s o r a t P l a c e r f o r h i s c o - o p e r a t i o n and a s s i s t a n c e . Thanks a l s o t o t h e computer g r o u p o f P l a c e r and i n p a r t i c u l a r Mr. W. Gree n f o r h e l p i n p r e p a r i n g d a t a . T e c h n i c a l s u p p o r t and h e l p has been p r o v i d e d by t h e M i n i n g E n g i n e e r i n g D epartment a t E q u i t y S i l v e r M i n e s L t d . , most n o t a b l y J.H.L. M i l l e r , J . C y r , D. F r a s e r , P. B e a u d o i n and W. M y c k a t y n . S p e c i a l t h a n k s t o D. W a l k e r f o r d r a f t i n g and J.H. Montgomery and D.F. Symonds f o r h e l p and en c o u r a g e m e n t . L a s t l y t h a n k s t o Anne, C o l l e e n and K e v i n f o r t h e i r p a t i e n c e and p e r s o n a l s u p p o r t . 1 CHAPTER 1 INTRODUCTION 2 The Sam G o o s l y s i l v e r - c o p p e r d e p o s i t , owned by E q u i t y S i l v e r M i n e s L i m i t e d , i s i n c e n t r a l B r i t i s h C o l u m b i a 35 km. s o u t h e a s t of H o u s t o n . The d e p o s i t c o n s i s t s o f two p r i n c i p a l m i n e r a l i z e d b o d i e s , t h e M a i n zone and t h e S o u t h e r n T a i l z o n e . A s s a y d a t a c o l l e c t e d f r o m s e v e r a l s t a g e s i n t h e d e v e l o p m e n t -of t h e S o u t h e r n T a i l zone form t h e b a s i s of t h i s s t u d y . The d e p o s i t i s of s p e c i a l i n t e r e s t due t o t h e seeming e r r a t i c n a t u r e of t h e s i l v e r g r a d e s and t h e s u b s e q u e n t p r o b l e m s t h a t a r i s e i n a l l s t a g e s o f mine p l a n n i n g and d e v e l o p m e n t . The s t u d y i s o r g a n i z e d as a s e r i e s o f t h r e e i n d e p e n d e n t s e c t i o n s , e a c h f o r m i n g a c h a p t e r and e a c h d e a l i n g w i t h a p a r t i c u l a r s e t o f p r o b l e m s . C h a p t e r two d e a l s w i t h t h e e s t i m a t i o n of g l o b a l o r g e o l o g i c a l r e s e r v e s from e x p l o r a t i o n diamond d r i l l d a t a . A method of c o n d i t i o n a l p r o b a b i l i t y i s d e s c r i b e d w i t h r e s u l t s compared t o t h e method of i n v e r s e d i s t a n c e . The p r o b l e m s o f c l u s t e r e d d a t a i n h i g h g r a d e p o r t i o n s of t h e d e p o s i t and e s t i m a t i o n o f g e o l o g i c r e s e r v e s u s i n g t h e wrong s u p p o r t a r e c o v e r e d w i t h s o l u t i o n s p r o p o s e d . The p r o b l e m s of p r o d u c t i o n q u a l i t y c o n t r o l a r e d e a l t w i t h i n C h a p t e r t h r e e . A s i m p l e s t a t i s t i c a l and g r a p h i c a l p r o c e d u r e i s o u t l i n e d as a method o f e v a l u a t i n g p a i r e d d a t a . S e v e r a l s e t s of d u p l i c a t e d a t a a r e examined and r e s u l t s a r e p r e s e n t e d w i t h r e g a r d s t o : a c o m p a r i s o n of two l a b o r a t o r i e s , a c h e c k on t h e r e l i a b i l i t y of a s a m p l i n g method and f i n a l l y t h e c o m p a r i s o n o f s e v e r a l s a m p l i n g p r o c e d u r e s . The f i n a n c i a l r a m i f i c a t i o n s o f u s i n g a s a m p l i n g method w i t h a h i g h random e r r o r a r e d i s c u s s e d . In C h a p t e r f o u r l o c a l b l o c k e s t i m a t i o n i s examined a t t h e 3 S o u t h e r n T a i l z o n e. S e v e r a l d i f f e r e n t k r i g i n g p r o c e d u r e s a r e o u t l i n e d and compared w i t h e m p i r i c a l p r o c e d u r e s . The i m p o r t a n c e of good g e o l o g i c c o n t r o l on any o r e r e s e r v e e s t i m a t e i s s t r e s s e d w i t h s e v e r a l examples g i v e n . 4 CHAPTER 2 GLOBAL RESERVES OF THE SOUTHERN T A I L ZONE BY CONDITIONAL PROBABILITY 5 ABSTRACT The S o u t h e r n T a i l zone of t h e Sam G o o s l y s i l v e r - c o p p e r d e p o s i t o f E q u i t y S i l v e r M i n e s L i m i t e d i s an e r r a t i c a l l y m i n e r a l i z e d o r e zone which p r e s e n t s d i f f i c u l t i e s i n e s t a b l i s h m e n t of g l o b a l ( i n s i t u ) o r e r e s e r v e s . Here we o u t l i n e t h e p r o c e d u r e f o r e s t i m a t i n g g l o b a l o r e r e s e r v e s by t h e g e o s t a t i s t i c a l method o f c o n d i t i o n a l p r o b a b i l i t y b a s e d on (1) a d e c l u s t e r i n g p r o c e d u r e t o p r o d u c e an u n b i a s e d h i s t o g r a m (and mean and d i s p e r s i o n ) of a s s a y d a t a , (2) a 3 - d i m e n s i o n a l s e m i - v a r i o g r a m model t h a t p r o v i d e s t h e means of e s t i m a t i n g t h e d i s p e r s i o n v a r i a n c e f o r a v e r a g e g r a d e of b l o c k s o f any s i z e , and (3) an a f f i n e c o r r e c t i o n t h a t c o n d i t i o n s t h e h i s t o g r a m of b l o c k g r a d e s t o t h e h i s t o g r a m of raw d a t a . The h i s t o g r a m f o r 10 x 10 x 10 m 3 b l o c k g r a d e s f o r t h e S o u t h e r n T a i l zone was e s t i m a t e d i n t h e f o r g o i n g manner and i s p r e s e n t e d as a g r a d e - t o n n a g e c u r v e . Tonnage e s t i m a t e s above a . c u t - o f f g r a d e of 40 g.Ag/tonne a g r e e c l o s e l y w i t h c o m p a r a b l e e s t i m a t e s b a s e d on t h e method o f i n v e r s e d i s t a n c e . 6 INTRODUCTION Sam G o o s l y d e p o s i t i s a b u l k s i l v e r d e p o s i t i n c e n t r a l B r i t i s h C o l u m b i a about 35 km. s o u t h e a s t of Ho u s t o n ( F i g u r e 1 ) . The d e p o s i t c o n s i s t s of two s e p a r a t e o r e z o n e s , t h e Main zone t o t h e n o r t h and t h e S o u t h e r n T a i l zone. The S o u t h e r n T a i l zone c o n t a i n s a b o u t 8.6 m i l l i o n t o n n e s of o r e b a s e d on a c u t - o f f g r a d e of 39 g / t o n n e s i l v e r e q u i v a l e n t . C o u n t r y r o c k i s a s e q u e n c e of M e s o z o i c p y r o c l a s t i c and s e d i m e n t a r y r o c k s s t r i k i n g a b o u t 15 d e g r e e s and d i p p i n g about 45 d e g r e e s w e s t e r l y ( W e t h e r e l l e t a l , 1979). T h e s e r o c k s a r e c u t by you n g e r i n t r u s i o n s and form an i n l i e r w i t h i n an a r e a u n d e r l a i n by g e n t l y d i p p i n g T e r t i a r y v o l c a n i c r o c k s . A s t r a t i g r a p h i c s e c t i o n i s p r e s e n t e d i n T a b l e 1 ( W e t h e r e l l , 1979). The two o r e zones a r e c h a r a c t e r i z e d by an a p p a r e n t l y e r r a t i c d i s t r i b u t i o n of s i l v e r g r a d e s and a l t h o u g h s e p a r a t e on th e b a s i s o f a r b i t r a r y economic u n i t s , a r e b o t h p a r t s of t h e same much l a r g e r but more i r r e g u l a r m i n e r a l i z e d volume ( W e t h e r e l l e t a l , 1979). P y r i t e i s by f a r t h e most abundant s u l p h i d e p r e s e n t and g e n e r a l l y g r e a t l y e x c e e d s t h e main o r e m i n e r a l s t e t r a h e d r i t e and c h a l c o p y r i t e . L e s s abundant o r e m i n e r a l s i n c l u d e g a l e n a , s p h a l e r i t e , p y r r h o t i t e and a v a r i e t y o f r a r e r s u l p h o s a l t s . L o c a l l y s i l v e r - r i c h z o n e s o f n e a r m a s s i v e g a l e n a - s p h a l e r i t e o c c u r and m i n i n g has shown t h e s e t o be somewhat more p r e v a l e n t t h a n a n t i c i p a t e d f r o m t h e r e s u l t s of e x p l o r a t o r y d r i l l i n g . The form o f m i n e r a l i z a t i o n i s somewhat d i f f e r e n t i n t h e two o r e z o n e s . B o t h open s p a c e f i l l i n g and r e p l a c e m e n t f e a t u r e s a r e e v i d e n t i n e a c h zone. The S o u t h e r n T a i l z o n e , however, i s c l e a r l y f r a c t u r e and b r e c c i a c o n t r o l l e d and 7 8 TABLE 1 - L i t h o l o g i c a l U n i t s ( f r o m W e t h e r e l l , 1979) PERIOD or EPOCH FORMATION or INTRUSION LITHOLOGY MIOCENE MIOCENE AND EOCENE EARLY OLIGOCENE AND LATE UPPER CRETACEOUS MIDDLE UPPER CRETACEOUS EARLY UPPER CRETACEOUS POPLAR BUTTES VOLCANIC ROCKS ENDAKO GROUP GOOSLY LAKE INTRUSIONS NANIKA INTRUSIONS OOTSA LAKE GROUP TI P TOP HILL VOLCANIC ROCKS BULKLEY INTRUSIONS KASALKA GROUP KASALKA INTRUSIONS LOWER CRETACEOUS SKEENA GROUP UPPER JURASSIC AND LATE MIDDLE JURASSIC BOWSER LAKE GROUP O l i v i n e b a s a l t s B a s a l t s and a n d e s i t e f l o w s and b r e c c i a s G a b b r o , m o n z o n i t e Q u a r t z m o n z o n i t e R h y o l i t e and d a c i t e f l o w s and p y r o c l a s t i c s p l u s minor a n d e s i t e s and b a s a l t s A n d e s i t e and d a c i t e b r e c c i a s and f l o w s , r h y o l i t e P o r p h y r i t e g r a n o d i o r i t e q u a r t z m o n z o n i t e Welded t u f f s , b r e c c i a s , l a t i t e - a n d e s i t e f l o w s , l a h a r s , r h y o l i t e , b a s a l c o n g l o b e r a t e P o r p h y r i t i c l a t i t e -a n d e s i t e , d a c i t e and d i o r i t e d i k e s , s i l l s and s t o c k s A m y g d a l o i d a l b a s a l t s and i n t e r m e d i a t e m a r i n e s a n d s t o n e s and s h a l e s M a r i n e and no n - m a r i n e s h a l e s , s a n d s t o n e s , and c o n g l o m e r a t e p l u s a n d e s i t e b r e c c i a s , t u f f s and f l o w s TABLE 1 ( C o n t . ) MIDDLE JURASSIC FRANCOIS LAKE INTRUSIONS EARLY MIDDLE HAZELTON GROUP JURASSIC AND LOWER JURASSIC LOWER JURASSSIC TOPLEY INTRUSIONS AND UPPER TRIASSIC LATE TRIASSIC TAKLA GROUP G r a n i t e , g r a n o d i o r i t e d i o r i t e and s y e n i t e b a t h o l i t h s A n d e s i t e and b a s a l t f l o w s , t u f f s , and b r e c c i a s p l u s i n t e r -c a l a t e d m a r i n e s h a l e and c o n g l o m e r a t e Q u a r t z m o n z o n i t e q u a r t z d i o r i t e A n d e s i t e and b a s a l t f l o w s , b r e c c i a s , and t u f f s w i t h i n t e r b e d -ded m a r i n e a r g i l l i t e , g r a ywacke, and c o n g l o m e r a t e 10 and commonly has t h e a p p e a r a n c e of a m i n e r a l i x e d s t o c k w o r k a l t h o u g h i n t e n s i v e r e p l a c e m e n t has p r o d u c e d l o c a l p y r i t e -- q u a r t z - s e r i c i t e z o n e s . The M a i n zone, i n c o n t r a s t , shows a dominance of m a s s i v e and d i s s e m i n a t e d o r e a l o n g p a r t i c u l a r l a y e r s of p y r o c l a s t i c r o c k . M i n e r a l i z e d f r a c u t r e s a r e p r e s e n t but much l e s s i n e v i d e n c e t h a n i n t h e S o u t h e r n T a i l z o n e . G l o b a l o r g e o l o g i c a l r e s e r v e s r e p r e s e n t a measure o f t h e t o t a l r e s o u r c e i n a d e p o s i t and a l t h o u g h commonly d e t e r m i n e d w i t h g e n e r a l c o n s i d e r a t i o n s of e conomic v i a b i l i t y i n mind a r e not b a s e d on d e t a i l e d c o n s i d e r a t i o n s of p r o f i t i b i l i t y . Thus, g l o b a l r e s e r v e s a r e not t o be c o n f u s e d w i t h r e c o v e r a b l e r e s e r v e s . A number of b o t h e m p i r i c a l and t h e o r e t i c a l p r o c e d u r e s have been u s e d f o r t h e e s t i m a t i o n of g e o l o g i c a l r e s e r v e s ( S i n c l a i r , 1978). We have e l e c t e d t o d e v e l o p g l o b a l r e s e r v e s f o r t h e S o u t h e r n T a i l zone o f E q u i t y S i l v e r M i n e s L i m i t e d u s i n g t h e g e o s t a t i s t i c a l a p p r o a c h of c o n d i t i o n a l p r o b a b i l i t y . Our m o t i v a t i o n has been i n p a r t t o t e s t t h e p r o c e d u r e a g a i n s t r e s u l t s of an e a r l i e r e m p i r i c a l s t u d y , and i n p a r t t o examine the p o t e n t i a l a d v a n t a g e of g r a d e - t o n n a g e c u r v e s t h a t a r e a n a t u r a l p r o d u c t of s u c h p r o c e d u r e s . The g e n e r a l o r g a n i z a t i o n o f our s t u d y i s p r o v i d e d i n T a b l e 2. 11 TABLE 2 - G e n e r a l P r o c e d u r e f o r G l o b a l R e s e r v e E s t i m a t i o n by C o n d i t i o n a l P r o b a b i l i t y 1. D a t a E v a l u a t i o n - e d i t i n g of d a t a , p r o d u c t i o n of p l a n o f d r i l l h o l e l o c a t i o n s , p r o d u c t i o n of c o m p o s i t e s o f a p p r o x i m a t e l y u n i f o r m s u p p o r t and d e t e r m i n a t i o n of s i m p l e s t a t i s t i c s . 2. S t r u c t u r a l a n a l y s i s - d e t e r m i n a t i o n of down h o l e e x p e r i m e n t a l s e m i - v a r i o g r a m s , h o r i z o n t a l e x p e r i m e n t a l s e m i - v a r i o g r a m s and a 3-d i m e n s i o n a l s e m i - v a r i o g r a m p o i n t model. 3. U n b i a s e d h i s t o g r a m - b a s e d on a g e n e r a l p r o c e d u r e f o r d e c l u s t e r i n g b i a s e d d a t a . 4. C o n d i t i o n a l P r o b a b i l i t y - a p p l i c a t i o n of v o l u m e - v a r i a n c e r e l a t i o n s h i p s b a s e d on d e c l u s t e r e d d a t a and t h e s e m i - v a r i o g r a m m o d e l . E s t i m a t i o n of g r a d e - t o n n a g e r e l a t i o n s h i p s f o r b l o c k s o f a s p e c i f i c s i z e . 1 2 DATA E x p l o r a t i o n a s s a y i n f o r m a t i o n f o r t h e S o u t h e r n T a i l zone was o b t a i n e d from 108 e x p l o r a t i o n diamond d r i l l h o l e s d r i l l e d i n t h e 1970's a l o n g a s e r i e s of c r o s s - s e c t i o n s a t i n t e r v a l s of 100 f e e t (30 m e t r e s ) ( F i g u r e 2 ) . N i n t y of t h e h o l e s were d r i l l e d a t -45 d e g r e e s , 89 of w h i c h had an a z i m u t h of 090 d e g r e e s , 17 were d r i l l e d v e r t i c a l l y and one p l u n g e d a t -74 d e g r e e s w i t h an a z i m u t h of 180. D r i l l c o r e was m o s t l y NA (1 7/8 i n c h d i a . o r 47.6 mm) and samples were a l l o b t a i n e d as s p l i t c o r e of about 10 f e e t (3 m) i n l e n g t h . H i g h g r a d e i n t e r s e c t i o n s of c o r e commonly were sampled o v e r much s h o r t e r d i s t a n c e s t h a n were l o w e r g r a d e s e c t i o n s . Samples were a s s a y e d f o r c o p p e r , z i n c , a n t i m o n y , s i l v e r , g o l d and a r s e n i c . Our g l o b a l s t u d y i s c o n c e r n e d o n l y w i t h s i l v e r , t h e p r i n c i p a l c o n t r i b u t o r t o o r e v a l u e . In o r d e r t o p r o d u c e m e a n i n g f u l s e m i - v a r i o g r a m s and u n b i a s e d h i s t o g r a m s t h e d a t a had t o be c o m p o s i t e d t o a common s u p p o r t (sample s i z e ) . To t h i s end, s m a l l h i g h g r a d e samples were combined w i t h a d j a c e n t samples i n s u c h a manner as t o p r o d u c e a p p r o x i m a t e l y 3 metre c o m p o s i t e s . g u r e 2 : P l a n v i e w o f t h e S o u t h e r n T a i l zone s h o w i n g c o l l a r l o c a t i o n s o f 108 diamond d r i l l h o l e s . 14 DATA EVALUATION Data were examined i n some d e t a i l t o p r o v i d e i n s i g h t i n t o l e v e l of e r r o r s i n s a m p l i n g and a s s a y i n g , s y s t e m a t i c v a r i a t i o n s i n s i l v e r g r a d e and t h e p o s s i b l e p r e s e n c e o f more t h a n a s i n g l e p o p u l a t i o n of a s s a y v a l u e s . F i g u r e 3 i s a p r o b a b i l i t y g r a p h of a v a i l a b l e 3 metre a s s a y s and 3 metre c o m p o s i t e s showing t h e p r o b a b l e e x i s t e n c e of two l o g n o r m a l p o p u l a t i o n s . The low g r a d e p o p u l a t i o n r e p r e s e n t i n g 67 % o f t h e a v a i l a b l e d a t a r e p r e s e n t s t h e b a c k g r o u n d v a l u e s i n m i n e r a l i z e d s e c t i o n s . The l e s s abundant h i g h g r a d e p o p u l a t i o n r e f l e c t s t h e common m i n e r a l o g i c a l o r e t y p e , p y r i t e - t e t r a h e d r i t e . I t i s w o r t h n o t i n g t h e s m a l l g a l e n a s i l v e r - r i c h pods i d e n t i f i e d a t t h e p r o d u c t i o n s t a g e were n o t r e p r e s e n t e d i n t h e e x p l o r a t i o n d a t a . The e x t r e m e l o c a l v a r i a b i l i t y o f g r a d e was r e c o g n i z e d e a r l y i n t h e e x p l o r a t i o n of t h e p r o p e r t y and l e d t o a s u b s t a n t i a l amount of c h e c k a s s a y i n g a t v a r i o u s s t a g e s i n t h e e x p l o r a t i o n and d e v e l o p m e n t of t h e p r o p e r t y . T h e s e p r o p r i e t a r y d a t a p r o v i d e two i m p o r t a n t c o n c l u s i o n s r e g a r d i n g d a t a q u a l i t y . F i r s t , a n a l y t i c a l e r r o r s a r e n e g l i g i b l e compared w i t h s a m p l i n g v a r i a b i l i t y ; and s e c o n d l y , s a m p l i n g v a r i a b i l i t y , a l t h o u g h l a r g e , a p p e a r s t o be random, t h a t i s , s a m p l i n g w i l l not i n t r o d u c e a b i a s i n t o o r e r e s e r v e e s t i m a t i o n s . An e x a m i n a t i o n of g r a d e c o n t o u r e d s e c t i o n s and p l a n s d i d not i n d i c a t e extreme t r e n d s i n m e t a l c o n t e n t . C o n s e q u e n t l y , i t a p p e a r s t h a t t h e b a s i c p r e m i s e o f our g e o s t a t i s t i c a l s t u d y , l o c a l s t a t i o n a r i t y , can be assumed w i t h c o n f i d e n c e . F i g u r e 3 : C u m u l a t i v e l o g p r o b a b i l i t y p l o t o f 2235 s i l v e r a s s a y s f r o m e x p l o r a t i o n d i a m o n d d r i l l h o l e s , S o u t h e r n T a i l z o n e . 16 STRUCTURAL (SEMI-VARIOGRAM) INTERPRETATION A s e m i - v a r i o g r a m i s ' h a l f t h e mean s q u a r e d d i f f e r e n c e ' f o r p a i r s of a v a r i a b l e ( g r a d e s ) as a f u n c t i o n of t h e d i s t a n c e ( l a g ) s e p a r a t i n g t h e p a i r s . The d e t e r m i n a t i o n of an a d e q u a t e m a t h e m a t i c a l r e p r e s e n t a t i o n of t h e s e m i - v a r i o g r a m f u n c t i o n i s r e f e r r e d t o commonly as a s t r u c t u r a l a n a l y s i s and p r o v i d e s f o r m a l q u a n t i f i c a t i o n of t h e 'zone of i n f l u e n c e ' c o n c e p t a p p l i e d t o a s s a y v a l u e s . The a p p l i c a t i o n of t h e s e m i - v a r i o g r a m model i n e s t i m a t i n g d i s p e r s i o n ( v a r i a n c e ) of b l o c k g r a d e s i s d i s c u s s e d i n a l a t e r s e c t i o n . We use a s t a n d a r d p r o c e d u r e t o d e v e l o p e a 3 - d i m e n s i o n a l s e m i - v a r i o g r a m model f o r t h e S o u t h e r n T a i l z o n e . S e p a r a t e e x p e r i m e n t a l s e m i - v a r i o g r a m s were examined f o r d i f f e r e n t down-h o l e o r i e n t a t i o n s ( v e r t i c a l and 45 d e g r e e s p l u n g e i n 090 d i r e c t i o n ) and f o r f o u r d i f f e r e n t h o r i z o n t a l d i r e c t i o n s . The down-hole e x p e r i m e n t a l s e m i - v a r i o g r a m and a f i t t e d m a t h e m a t i c a l ( s p h e r i c a l ) model b a s e d on 3m c o m p o s i t e s a r e shown i n F i g u r e 4. H o r i z o n t a l d a t a a r e much more e r r a t i c t h a n a r e down-hole d a t a . G e o l o g i c a l c o n s i d e r a t i o n s l e d us t o e x p e c t a p r o n o u n c e d a n i s o t r o p y t h a t became e v i d e n t i n t h e e x p e r i m e n t a l s e m i -v a r i o g r a m s p a r a l l e l i n g t h e two p r i n c i p a l d i r e c t i o n s o f t h e d e p o s i t . The two s e m i - v a r i o g r a m s a r e shown i n F i g u r e s 5 and 6 where t h e s i l l and nugget e f f e c t have been c o n t r o l l e d by t h e down-hole model ( F i g u r e 4) and t h e r a n g e s have been s e l e c t e d b a s e d on t h e p a t t e r n s i n F i g u r e 5 and F i g u r e 6 r e s p e c t i v e l y . The p r i n c i p a l weakness i n t h e model i s t h e s u b j e c t i v i t y i n d e t e r m i n i n g t h e range i n a NW-SE d i r e c t i o n ( F i g u r e 5 ) . 17 F i g u r e 4 : A v e r a g e v e r t i c a l r e l a t i v e s e m i -v a r i o g r a m f o r 3 m. c o m p o s i t e s i l v e r g r a d e s . 18 . a'=40m. 0 50 100 150 200 250 Lag h (m«tr«s) F i g u r e 5 : A v e r a g e h o r i z o n t a l r e l a t i v e s e m i -v a r i o g r a m f o r s i l v e r g r a d e s i n t h e NW-SE d i r e c t i o n . 19 5.0-4.0-7(h) 3.0^  2.0-I.OJ »(340) V / (370) | FITTED MODEL 7(h) = A/|*qC, A =8.29 ,1 = 13.4 A/j = .62 C, = 2.83 a=7l.6 1(404) — i — 50 —I— 100 — I — 150 l I 200 " 250 lag h (mttrts) F i g u r e 6 : A v e r a g e h o r i z o n t a l r e l a t i v e s e m i -v a r i o g r a m f o r s i l v e r g r a d e s i n t h e NE-SW d i r e c t i o n . 20 N e v e r - t h e - l e s s , t h e model a p p e a r s r e a s o n a b l e f o r p u r p o s e s of g l o b a l e s t i m a t i o n . P a r a m e t e r s of t h e 3 - d i m e n s i o n a l model a r e summarized i n T a b l e 3. TABLE 3 - P a r a m e t e r s of 3 - D i m e n s i o n a l S p h e r i c a l Model S o u t h T a i l Zone Diamond D r i l l D a t a Nugget E f f e c t (Co) 8.29/1 S i l l (C ) 2.83 Range V e r t i c a l (av) 1 1 m NE-SW (aNE) 72 m NW-SE (aNW) 27 m A n i s o t r o p y R a t i o N E / v e r t . 6 . 5 4 NE/NW 2.07 21 DECLUSTERING D e c l u s t e r i n g r e f e r s t o methods u s e d t o m i n i m i z e t h e e f f e c t s of b i a s e d s p a t i a l d i s t r i b u t i o n s o f sample d a t a . Commonly, i n m i n e r a l e x p l o r a t i o n and p r o p e r t y e v a l u a t i o n more d r i l l h o l e s a r e l o c a t e d i n h i g h g r a d e s e c t i o n s of a d e p o s i t r e l a t i v e t o low g r a d e p a r t s . C o n s e q u e n t l y , *a h i s t o g r a m of t h e d a t a c o n t a i n s t o o h i g h a p r o p o r t i o n of h i g h g r a d e s a m p l e s , l e a d i n g t o o v e r e s t i m a t e s of b o t h mean g r a d e and d i s p e r s i o n of t h e d a t a . To o f f s e t t h i s s o u r c e of b i a s i t i s n e c e s s a r y t o a p p l y l e s s w e i g h t t o t h o s e s a m p l e s t h a t o c c u r i n c l u s t e r s or a b n o r m a l c o n c e n t r a t i o n s of d a t a , h e n c e , t h e name d e c l u s t e r i n g . A t h e o r e t i c a l b a s i s f o r d e c l u s t e r i n g i s p r o v i d e d by J o u r n e l ( 1 9 8 3 ) . In b r i e f , t h e p r o c e d u r e i n v o l v e s s u p e r i m p o s i n g a 3-d i m e n s i o n a l g r i d on a d a t a s e t and a s s i g n i n g e q u a l w e i g h t f o r e a c h c e l l . F o r example, a c e l l c o n t a i n i n g 100 samples w i l l have t h e same t o t a l w e i g h t as a n o t h e r c o n t a i n i n g 30 s a m p l e s . I n t h e fo r m e r c a s e e a c h i n d i v i d u a l sample would have a r e l a t i v e w e i g h t of 1/100 , whereas, i n t h e l a t t e r c a s e t h e i n d i v i d u a l sample w e i g h t would be 1/30. The p r i n c i p a l p r o b l e m i n d e c l u s t e r i n g i s t o d e f i n e t h e optimum c e l l s i z e f o r a s s i g n i n g w e i g h t s t o a s s a y s . C o n s i d e r two e x t r e m e s : i f a l l t h e d a t a a r e i n one c e l l t h e n e v e r y sample has e q u a l w e i g h t and t h e b i a s i s not a c c o u n t e d f o r . S i m i l a r l y , i f the c e l l s i z e i s so s m a l l t h a t e a c h sample i s i n a s e p a r a t e c e l l , e a c h sample w i l l have t h e same w e i g h t and, a g a i n , t h e b i a s i s n ot a c c o u n t e d f o r . In b o t h c a s e s t h e w e i g h t e d mean o f t h e d a t a w i l l be t o o h i g h . F o r c e l l s i z e s between t h e s e two e x t r e m e s , v a r i a b l e w e i g h t s w i l l be a s s i g n e d s u c h t h a t t h e 22 w e i g h t e d mean of t h e d a t a w i l l be l e s s t h a n i n t h e two extreme c a s e s . The a p p r o x i m a t e c e l l s i z e f o r o b t a i n i n g u n b i a s e d mean e s t i m a t e s of a m i n e r a l i z e d f i e l d o c c u r s where t h e mean p a s s e s t h r o u g h a minimum between t h e two l i m i t i n g c a s e s c i t e d a b o v e . T h i s c e l l s i z e i s d e t e r m i n e d g r a p h i c a l l y by t r i a l and e r r o r a s i l l u s t r a t e d by 3 m c o m p o s i t e e x p l o r a t i o n d r i l l h o l e d a t a f o r t h e S o u t h e r n T a i l zone ( F i g u r e 7 ) . U s i n g a l l d r i l l h o l e d a t a f r o m th e S o u t h e r n T a i l zone t h e mean g r a d e of t h e d e p o s i t was c a l c u l a t e d f o r e a c h of a s e r i e s of c e l l s i z e s ( F i g u r e 7 ) . A c e l l d i m e n s i o n of 45 m e t r e s was c h o s e n as t h e optimum b a s e d on t h e i d e a l i z e d i n t e r p r e t a t i o n shown i n F i g u r e 7. The d e p a r t u r e o f t h e t r i a l and e r r o r r e s u l t s f r o m t h e i d e a l i z e d e x p e c t a t i o n beyond a 70 m c e l l s i z e r e s u l t s from g e o m e t r i c c o n s t r a i n t s as t h e c e l l s i z e a p p r o a c h e s one o f t h e d i m e n s i o n s of t h e d e p o s i t . The d a t a were t h e n d e c l u s t e r e d w i t h e a c h sample r e c e i v i n g a w e i g h t e q u a l t o 1 / L n i , where L i s t h e number o f 45 m etre b l o c k s c o n t a i n i n g a t l e a s t one sample and n i i s t h e number of samples i n b l o c k i . A h i s t o g r a m of t h e u n b i a s e d d e c l u s t e r e d d i s t r i b u t i o n i s shown as F i g u r e 8. The y - a x i s shows f r e q u e n c y as a p r o p o r t i o n w h i c h i s an a c c u m u l a t i o n of w e i g h t s i n e a c h c l a s s i n t e r v a l . The d i s t r i b u t i o n has a mean of 39.3 g.Ag/tonne w i t h a s t a n d a r d d e v i a t i o n of 164.2. As a r e f e r e n c e t h e h i s t o g r a m of raw d a t a i s a l s o shown on F i g u r e 8 as a s h a d e d a r e a . I t i s w o r t h n o t i n g t h a t t h e p r o p o r t i o n o f d a t a g r e a t e r t h a n a 40 g A g / t o n n e c u t - o f f f o r t h e raw d a t a i s 18.2 % whereas f o r t h e d e c l u s t e r e d d a t a i s 15.3 %. T h i s n o t o n l y p o i n t s o u t t h e need f o r d e c l u s t e r i n g but a l s o q u a n t i f i e s t h e m a g n i t u d e o f t h e o v e r e s t i m a t i o n e r r o r t h a t c o u l d 23 20 -I ' 1 I I | 25 50 75 100 125 S ( c t l l dimensions inmst rss ) F i g u r e 7 : Naive mean s i l v e r v a l u e p l o t t e d f o r v a r i o u s c e l l s i z e s (s) i n order to determine the optimum c e l l s i z e f o r d e c l u s t e r i n g . 24 .040 J . 0 3 3 . .030—1 . 0 2 5 J >-O z 3 . 0 2 0 . o .013 _l .010 _l .008 RAW DATA DISTRIBUTION n DECLUSTERED UNBIASED DISTRIBUTION 30 100 ISO SILVER ( g . / t o n n « ) 200 F i g u r e 8 : U n b i a s e d h i s t o g r a m f o r S o u t h e r n T a i l zone s i l v e r from 3 m. c o m p o s i t e s w i t h h i s t o g r a m of raw d a t a su p e r i m p o s e d . 25 a r i s e i f d e c l u s t e r i n g were not done. CONDITIONAL PROBABILITY We have a t t h i s p o i n t o b t a i n e d an e s t i m a t e of t h e u n b i a s e d or t r u e d i s t r i b u t i o n of s i l v e r -grades i n t h e S o u t h e r n T a i l Zone. However, m i n i n g s e l e c t i o n w i l l be made on t h e b a s i s of a b l o c k s u p p o r t and n o t a sample s u p p o r t . I t s t a n d s t o r e a s o n t h a t as b l o c k s i z e i n c r e a s e s t h e e s t i m a t i o n v a r i a n c e w i l l d e c r e a s e and t h e r e f o r e t h e b l o c k s w i l l p r o d u c e a ' t i g h t e r ' d i s t r i b u t i o n a b o u t t h e same mean. T h e r e a r e s e v e r a l methods a v a i l a b l e f o r e s t i m a t i n g t h e b l o c k d i s t r i b u t i o n ( h i s t o g r a m ) from t h e c o r e sample d i s t r i b u t i o n , a l l b a s e d on a s s u m p t i o n s as t o t h e f o r m of t h e b l o c k g r a d e d i s t r i b u t i o n . In t h e a f f i n e c o r r e c t i o n p r o c e d u r e , w h i c h we use h e r e , t h e a s s u m p t i o n i s t h a t t h e g e n e r a l f o r m of t h e b l o c k g r a d e d i s t r i b u t i o n m i r r o r s t h e sample g r a d e d i s t r i b u t i o n , b u t , of c o u r s e , w i t h s m a l l e r v a r i a n c e . The a f f i n e r e l a t i o n s h i p ( J o u r n e l and H u i j b r e g t s , 1981) between t h e d i s t r i b u t i o n o f c o r e g r a d e s (Z) and t h e d i s t r i b u t i o n o f b l o c k s (Y) i s as f o l l o w s : Y = a y / a z ( Z - m ) + m where a2 y i s t h e v a r i a n c e of d i s t r i b u t i o n Y a2 z i s t h e v a r i a n c e of d i s t r i b u t i o n Z m i s t h e d e c l u s t e r e d mean of t h e d a t a and z i s any v a l u e o f t h e v a r i a b l e i n d i s t r i b u t i o n Z. 26 The o n l y unknowns i n t h i s e q u a t i o n a r e t h e d i s t r i b u t i o n Y we a r e s e e k i n g and t h e v a r i a n c e of t h a t d i s t r i b u t i o n . The v a r i a n c e c a n be o b t a i n e d u s i n g R r i g e ' s a d d i t i v i t y r e l a t i o n s h i p ( J o u r n e l and H u i j b r e g t s , 1981) w h i c h s t a t e s : o2 y = D 2 ( v / G ) = D 2 ( c / G ) - D 2 ( c / v ) where D 2 ( v / G ) i s t h e d i s p e r s i o n v a r i a n c e o f g r a d e s of m i n i n g u n i t s v i n m i n i n g volume G. D 2 ( c / G ) i s t h e d i s p e r s i o n v a r i a n c e o f c o r e g r a d e s i n volume G ( o b t a i n e d f r o m t h e h i s t o g r a m ) . and D 2 ( c / v ) i s t h e d i s p e r s i o n v a r i a n c e o f c o r e g r a d e s i n m i n i n g u n i t v ( o b t a i n e d u s i n g s e m i -v a r i o g r a m m o d e l ) . The d i s t r i b u t i o n o f g r a d e s f o r 10 metre b l o c k s i s shown i n F i g u r e 9. The b l o c k s w i l l have t h e same mean g r a d e of 39.3 g A g / t o n n e and a s t a n d a r d d e v i a t i o n of 143.6. I t i s o b v i o u s t h a t i f t h e c o r e sample d i s t r i b u t i o n were u s e d i n a d v e r t e n t l y f o r i n s i t u g r a d e e s t i m a t e s a s e r i o u s o v e r e s t i m a t i o n o f h i g h g r a d e o r e w o u l d r e s u l t . 0 SO 100 130 ZOO ZOO SILVER (g./tonna) F i g u r e 9 : U n b i a s e d h i s t o g r a m f o r s i l v e r g r a d e s o f 10 x 10 x 10 cu.m. b l o c k s , S o u t h e r n T a i l z o n e , w i t h h i s t o g r a m f r o m un-b i a s e d 3 m. c o m p o s i t e s s u p e r i m p o s e d . 28 GRADE AND TONNAGE ESTIMATION U s i n g t h e d i s t r i b u t i o n f o r 10 metre b l o c k s shown i n F i g u r e 9 i t i s a s i m p l e p r o c e d u r e t o c a l c u l a t e t h e p r o p o r t i o n and g r a d e of a l l m a t e r i a l above any g i v e n c u t - o f f . The l a s t r e m a i n i n g p r o b l e m t o overcome i n p r o d u c i n g an i n s i t u r e s e r v e e s t i m a t i o n i s d e c i d i n g on t h e volume and s p e c i f i c g r a v i t y o f t h e m a t e r i a l t o be e s t i m a t e d . The upper l e v e l of t h e S o u t h e r n T a i l zone i s shown as F i g u r e 10 w i t h 45 metre b l o c k s o u t l i n e d . The b l o c k s w i t h i n t h e b o u n d a r i e s i n d i c a t e d , but w i t h no d a t a p r e s e n t , were i n c l u d e d s i n c e t h e i r r e m o v a l would be n e c e s s a r y t o r e a c h o r e below. A d e n s i t y of 2.89 t o n n e s / m 3 was o b t a i n e d from E q u i t y M i n e s L i m i t e d , a s t h e d e n s i t y u s e d i n p r e v i o u s g l o b a l e s t i m a t i o n s . The t o t a l t o n n a ge t o be c o n s i d e r e d from t h e t h r e e 45 metre l e v e l s was 59,000,000 t o n n e s . F o r a c u t - o f f o f 40 g.Ag/tonne t h e p r o p o r t i o n of tonnage above t h i s c u t - o f f was .15296 ( f r o m T a b l e 4 ) . The tonnage of m a t e r i a l above 40 g.Ag/tonne i s t h e r e f o r e 9,024,000 t o n n e s a v e r a g i n g 125.1 g.Ag/tonne. A more c o n s e r v a t i v e e s t i m a t e was a r r i v e d a t by e l i m i n a t i n g p e r i f e r a l b l o c k s w i t h d a t a below c u t - o f f . The t o t a l t o n n a ge o f t h e r e d u c e d volume was 48,800,000 t o n n e s , w i t h a tonnage above a 40 g.Ag/tonne c u t - o f f , of 7,460,000 t o n n e s a v e r a g i n g 125.1 g. A g / t o n n e . I t i s i m p o r t a n t t o n o t e t h a t t h i s method o f i n s i t u o r e r e s e r v e e s t i m a t i o n makes no a t t e m p t t o i d e n t i f y where t h e o r e o c c u r s w i t h i n a g i v e n volume. T h e s e r e s u l t s compare w i t h a g e o l o g i c a l o r e r e s e r v e c a l c u l a t e d p r i o r t o p r o d u c t i o n and b a s e d on a s i m i l a r 40 g.Ag/tonne c u t - o f f f o r 10 me t r e b l o c k s by t h e 29 Owl K) ro oo (0 * • • \ 8||l|| - -• • . ; 1 fllllj 'V'. • -» - IllSf mm §1111 • o t--<o CM 00 co 10 <M CM o o cn 1 8 o o I o CM o I 2 eg o o • 8 10 o o co o •o o o <0 (0 o o 10 I o o 10 ~ CM F i g u r e 10 : Upper s l i c e ( i n c l u d i n g l e v e l s 1280 t o 1235), S o u t h e r n T a i l z o n e , s h o w i n g d i s t r i b u t i o n o f 45 m. b l o c k s . B l o c k s w i t h i n t h e d o u b l e l i n e s u s e d t o c a l c u l a t e t o n n a g e o f 48.8 m i l l i o n t o n n e s ; b l o c k s c o n t a i n i n g d a t a a r e s h a d e d . 30 TABLE 4 - Grade and Tonnage C u r v e s ( S i l v e r ) CUT-OFF GRADE TONNAGE USING TONNAGE USING ( g / t o n n e ) ( g / t o n n e ) ( T o t a l of 59 ( T o t a l of 48.8 m i l l i o n t o n n e s ) m i l l i o n t o n n e s ) 20.0 89.9 14 056 000 11 626 000 30.0 114.1 10 293 000 8 513 000 40.0 125.1 9 025 000 7 465 000 50.0 1 35.8 7 983 000 6 603 000 60.0 1 49.0 6 877 000 5 688 000 70.0 1 56. 1 6 327 000 5 233 000 80.0 161.7 5 917 000 4 894 000 90.0 170.3 5 331 000 4 409 000 1 00.0 180.2 4 71 1 000 3 897 000 110.0 185.7 4 389 000 3 630 000 1 20.0 188.6 4 221 000 3 491 000 1 30.0 196.5 3 754 000 3 1 05 000 1 40.0 203.3 3 368 000 2 786 000 150.0 206. 1 3 212 000 3 657 000 160.0 210.0 2 988 000 2 471 000 170.0 212.7 2 823 000 2 335 000 180.0 218.3 2 469 000 2 042 000 1 90.0 218.7 2, 440 000 2 018 000 200.0 220. 1 2, 291 000 1 895 000 210.0 220.9 2, 1 62 000 1 789, 000 220.0 221 .4 2, 028 000 1 678, 000 31 i n v e r s e d i s t a n c e method w h i c h i n d i c a t e d 8,124,000 t o n n e s of o r e a v e r a g i n g 121.5 g . A g/tonne. A d v a n t a g e s of t h e c o n d i t i o n a l p r o b a b i l i t y method of e s t i m a t i n g g l o b a l r e s e r v e s l i e i n t h e f a c t t h a t a g r a d e - t o n n a g e c u r v e i s a n a t u r a l end r e s u l t of t h e p r o c e d u r e ( F i g u r e 1 1 ) . T h us, an i n d i c a t i o n of p o t e n t i a l r e c o v e r y i s p r o v i d e d f o r v a r i o u s c u t - o f f g r a d e s . I t i s an e a s y m a t t e r t o p r o d u c e s u c h g r a d e - t o n n a g e c u r v e s f o r s e l e c t i o n b a s e d on s e v e r a l b l o c k s i z e s . Of c o u r s e , we must r e - e m p h a s i z e t h a t s u c h g l o b a l e s t i m a t e s p r o v i d e no i n d i c a t i o n o f where i n a m i n e r a l i z e d volume, t h e b l o c k s above c u t - o f f a r e l o c a t e d . P r e c i s e b l o c k s e l e c t i o n d u r i n g m i n i n g e v e n t u a l l y w i l l be b a s e d on a r e l a t i v e l y t i g h t g r i d of p r o d u c t i o n b l a s t h o l e a s s a y s , and i t i s t h i s more d e t a i l e d d a t a t h a t w i l l l e a d t o b l o c k s e l e c t i o n . F i g u r e 11 : Grade-tonnage c u r v e f o r S o u t h e r n T a i l zone showing two tonnage c u r v e s : one based on a t o t a l m i n e r a l i z e d tonnage of 59 m i l l i o n tonnes and a second based on 48.8 m i l l i o n t o n n e s . 33 CONCLUSIONS G l o b a l ( i n s i t u ) o r e r e s e r v e e s t i m a t e s f o r t h e S o u t h e r n T a i l zone of E q u i t y S i l v e r M i n e s L i m i t e d have been g e n e r a t e d u s i n g t h e method of c o n d i t i o n a l p r o b a b i l i t y i n w h i c h an a f f i n e c o r r e c t i o n i s a p p l i e d t o an u n b i a s e d h i s t o g r a m of sample d a t a c o m p o s i t e d t o a common s u p p o r t . T h i s p r o c e d u r e assumes t h a t t h e h i s t o g r a m of b l o c k g r a d e s has t h e same g e n e r a l f o r m as t h e h i s t o g r a m o f sample, d a t a , t h a t i s , t h e b l o c k h i s t o g r a m i s c o n d i t i o n e d t o t h e sample d a t a . The l e s s e r d i s p e r s i o n of b l o c k g r a d e s r e l a t i v e t o sample g r a d e s i s e s t i m a t e d u s i n g K r i g e ' s a d d i t i v i t y r e l a t i o n s h i p , a f u n d a m e n t a l g e o s t a t i s t i c a l r e l a t i o n s h i p i n c o r p o r a t i n g t h e s e m i - v a r i o g r a m f u n c t i o n . In t h i s c a s e r e s u l t s compare w e l l w i t h g l o b a l r e s e r v e s b a s e d on 10 m b l o c k e s t i m a t e s by a method of i n v e r s e d i s t a n c e w e i g h t i n g . The p r i n c i p a l a d v a n t a g e s o f t h e c o n d i t i o n a l p r o b a b i l i t y a p p r o a c h u s i n g an a f f i n e c o r r e c t i o n a r e : (1) no r i g i d a s s u m p t i o n s must be made r e g a r d i n g t h e form of t h e h i s t o g r a m of e i t h e r sample o r b l o c k g r a d e s ( e . g . l o g n o r m a l i t y and c o n s e r v a t i o n o f l o g n o r m a l i t y ) . I n s t e a d , t h e b l o c k s a r e c o n d i t i o n e d t o a v a i l a b l e but u n b i a s e d d a t a . (2) e f f e c t s of a u t o c o r r e l a t i o n a r e i n d i c a t e d i t t h e r e s e r v e e s t i m a t e v i a t h e s e m i - v a r i o g r a m and K r i g e ' s a d d i t i v i t y r e l a t i o n s h i p . (3) A c o m p l e t e g r a d e - t o n n a g e c u r v e c a n be o b t a i n e d f o r many b l o c k s i z e s w i t h o u t t h e c o s t l y n e c e s s i t y o f r e g e n e r a t i n g new g r a d e e s t i m a t e s as w o u l d be r e q u i r e d i n t h e c a s e o f an i n v e r s e d i s t a n c e model or any o t h e r e x a c t i n t e r p o l a t i o n p r o c e d u r e . 35 ACKNOWLEDGEMENTS T h i s s t u d y was i n i t i a t e d a t t h e s u g g e s t i o n of Dr. N. H i l l h o u s e and Mr. E.T. L o n e r g a n of P l a c e r Development L t d . and W. Myck a t y n of E q u i t y S i l v e r M i n e s L i m i t e d . T e c h n i c a l s u p p o r t has been p r o v i d e d by P. B e a u d o i n , J . C yr and J.H.L. M i l l e r of th e E n g i n e e r i n g Department of E q u i t y S i l v e r M i n e s L t d . and Mr. W. Green of P l a c e r Development L t d . We thank a l l of t h e s e p e o p l e f o r t h e i r c o o p e r a t i o n and i n v o l v e m e n t i n t h e s t u d y . F i n a n c i a l s u p p o r t was p r o v i d e d by P l a c e r Development L t d . 36 REFERENCES J o u r n e l , A.G. and Ch. H u i j b r e g t s (1981) M i n i n g g e o s t a t i s t i c s ; A c a demic P r e s s ( I n c . ) London L t d . , 600 p. J o u r n e l , A.G. (1983) N o n - p a r a m e t r i c e s t i m a t i o n of s p a c i a l d i s t r i b u t i o n s ; J o u r . I n t . A s s o c . Math. G e o l . v.15, no. 3., p. 445-468 Ney, C.S., J.M. A n d e r s o n and A. P a n t e l e y e v (1972) D i s c o v e r y , g e o l o g i c s e t t i n g and s t y l e of m i n e r a l i z a t i o n , Sam G o o s l y d e p o s i t , B r i t i s h C o l u m b i a ; I n s t . M i n . M e t a l l . B u l l . v. 65, no. 723, p 53-64. S i n c l a i r , A . J . (1978) S a m p l i n g a m i n e r a l d e p o s i t f o r f e a s i b i l i t y s t u d i e s and m e t a l l u r g i c a l t e s t i n g ; i n M u l a r , A.L. and R.B. Bhappu, M i n e r a l P r o c e s s i n g P l a n t D e s i g n ; Am. I n s t . M i n . Eng. New Y o r k , p 115-134. W e t h e r e l l , D.G (1979) G e o l o g y and Ore G e n e s i s of t h e Sam G o o s l e y c o p p e r - s i l v e r - a n t i m o n y D e p o s i t , U n p u b l i s h e d MSc T h e s i s , D e p t . o f G e o l o g i c a l S c i e n c e s , U n i v e r s i t y of B.C., V a n c o u v e r B.C., 208 p. W e t h e r e l l , D.G., A . J . S i n c l a i r and T.G. S h r o e t e r , (1979) P r e l i m i n a r y r e p o r t on t h e Sam G o o s l y c o p p e r - s i l v e r d e p o s i t ; B.C. M i n i s t r y of E n e r g y , M i n e s and P e t r o l e u m R e s o u r c e s , Paper 1979-1, pp. 132-137. CHAPTER 3 PRODUCTION QUALITY CONTROL EXPERIMENTS 38 ABSTRACT Q u a l i t y c o n t r o l p r o c e d u r e s a r e e s s e n t i a l t o m a i n t a i n a d e q u a t e a c c u r a c y and p r e c i s i o n i n s a m p l i n g and a s s a y i n g , p a r t i c u l a r y i n e r r a t i c a l l y m i n e r a l i z e d d e p o s i t s such as t h e S o u t h e r n T a i l zone of E q u i t y S i l v e r M i n e s L t d . T h e r e , d u p l i c a t e a n a l y s i s and d u p l i c a t e s a m p l i n g t e s t s have d e m o n s t r a t e d 1. a s i g n i f i c a n t b i a s between two l a b o r a t o r i e s i n t h e i r s i l v e r a n a l y s e s , 2. a l a r g e random e r r o r i n t h e s t a n d a r d t u b e s a m p l i n g p r o c e d u r e u s e d t o p r o v i d e g r a d e e s t i m a t e s f o r p r o d u c t i o n b l a s t h o l e s , and 3. a s y s t e m a t i c r e l a t i o n s h i p between g r a d e r e p r o d u c i b i l i t y and volume of samples from b l a s t h o l e c u t t i n g s . T h e s e r e s u l t s i n d i c a t e t h a t an i m p r o v e d s a m p l i n g t e c h n i q u e f o r b l a s t h o l e c u t t i n g s w i l l p r o v i d e a marked i n c r e a s e i n m e t a l r e c o v e r y t h r o u g h i m p r o v e d b l o c k s e l e c t i o n , p e r h a p s w i t h a n e t b e n e f i t i n e x c e s s of $ 1 m i l l i o n p e r y e a r . 39 INTRODUCTION O p t i m a l b l o c k s e l e c t i o n i n m i n i n g depends on two c r i t i c a l p r o c e d u r e s , v i z . 1. o b t a i n i n g r e p r e s e n t a t i v e sample a s s a y s and 2. w e i g h t i n g a s e l e c t i o n of a s s a y s i n an a p p r o p r i a t e f a s h o n t o • b e s t e s t i m a t e a g i v e n b l o c k ( c f . Raymond, 1979). Here we a r e c o n c e r n e d w i t h t e s t s d e s i g n e d t o e v a l u a t e v a r i o u s s o u r c e s o f e r r o r i n t h e a s s i g n m e n t o f a s s a y v a l u e s t o p r o d u c t i o n b l a s t h o l e s . H i g h q u a l i t y s a m p l i n g and a n a l y t i c a l p r o c e d u r e s a r e e s s e n t i a l f o r o p t i m a l o r e and waste d i s c r i m i n a t i o n i n a l l e f f i c i e n t m i n i n g o p e r a t i o n s . S a m p l i n g i s p a r t i c u l a r l y c r i t i c a l i n t h e c a s e of b l o c k s e l e c t i o n f o r an e r r a t i c a l l y m i n e r a l i z e d d e p o s i t s u c h as t h e S o u t h e r n T a i l zone of E q u i t y S i l v e r M i n e s L t d . The S o u t h e r n T a i l zone o f t h e Sam G o o s l y s i l v e r d e p o s i t l i e s w i t h i n an i n l i e r o f C r e t a c e o u s s e d i m e n t a r y , p y r o c l a s t i c and v o l c a n i c r o c k s ( W e t h e r e l l 1979; W e t h e r e l l e t a l , 1978). The i n l i e r i s i n t r u d e d i n t h e west by a q u a r t z m o n z o n i t e s t o c k and i n t h e e a s t by a g a b b r o - m o n z o n i t e c o m p l e x . S i l v e r - c o p p e r -a n t i m o n y o r e s a r e t h o u g h t t o be e p i g e n e t i c i n o r i g i n and b o t h s t r u c t u r a l l y , and t o a l e s s e r d e g r e e , s t r a t i g r a p h i c a l l y c o n t r o l l e d . The S o u t h e r n T a i l zone i s c r u d e l y t a b u l a r i n form w i t h a s t r i k e of ab o u t 025 d e g r e e s and a w e s t e r l y d i p of 30 t o 50 d e g r e e s . S u l p h i d e s o c c u r m a i n l y i n a s t o c k w o r k o r as a cement between b r e c c i a f r a g m e n t s . S a m p l i n g t e s t s on p r o d u c t i o n b l a s t h o l e c u t t i n g s were begun d u r i n g t h e e a r l y s t a g e s o f mine d e v e l o p m e n t by E q u i t y ' s 40 E n g i n e e r i n g D e p a r t m e n t . A d d i t i o n a l s t u d i e s were c a r r i e d out i n r e s p o n s e t o t h e f i n d i n g s o f a g e o s t a t i s t i c a l s t u d y o f t h e d e p o s i t . T h i s r e p o r t d e a l s s e q u e n t i a l l y w i t h an e v a l u a t i o n of t h e r e s u l t s o f t h e s e s t u d i e s , a l l of w h i c h i n v o l v e t h e c o m p a r i s o n of d u p l i c a t e d a t a . 41 A PROCEDURE FOR EVALUATION OF DUPLICATE (PAIRED) DATA Q u a l i t y c o n t r o l of s a m p l i n g and a s s a y i n g r e s u l t s commonly i n v o l v e s t h e c o m p a r i s o n of p a i r e d d a t a . F o r example, i n e v a l u a t i n g t h e r e p r o d u c i b i l i t y o f an a n a l y t i c a l method i t i s s t a n d a r d p r o c e d u r e t o a n a l y z e e v e r y 10th o r 2 0 t h sample i n d u p l i c a t e . P a i r e d d a t a may d e r i v e from a p r o d u c t i o n medium ( e g . b l a s t h o l e c u t t i n g s ) or a n a l y t i c a l s t a n d a r d s , b o t h of w h i c h m i g h t be a n a l y z e d by t h e same or d i f f e r e n t methods. Such q u a l i t y c o n t r o l p r o c e d u r e s l e a d t o p a i r e d o r d u p l i c a t e d a t a t h a t a r e of i n t e r e s t b e c a u s e of t h e i r s i m i l a r i t i e s o r d i f f e r e n c e s . The d i f f e r e n c e s ( o r e r r o r s ) c a n be c l a s s e d as random o r s y s t e m a t i c e r r o r s ( b i a s ) and an e v a l u a t i o n of b o t h t y p e s i s f u n d a m e n t a l t o an a p p r e c i a t i o n o f d a t a q u a l i t y . A g e n e r a l and r o u t i n e a p p r o a c h t o t h e use of d u p l i c a t e s a m p l e . ( o r a n a l y t i c a l ) d a t a f o r e x a m i n a t i o n o f a s s a y q u a l i t y d u r i n g p r o p e r t y e v a l u a t i o n or p r o d u c t i o n i s i l l u s t r a t e d s c h e m a t i c a l l y i n F i g u r e 1. S e v e r a l s i m p l e g r a p h i c a l and s t a t i s t i c a l methods a r e u t i l i z e d i n o r d e r t o i d e n t i f y and q u a n t i f y v a r i o u s c l a s s e s o r p a t t e r n s o f e r r o r s t h a t m i g h t be r e p r e s e n t e d i n d i f f e r e n t t y p e s o f p a i r e d a s s a y d a t a . A c e r t a i n amount o f s u b j e c t i v e d e c i s i o n - m a k i n g i s r e q u i r e d i n a p p l y i n g t h e p r o c e d u r e a s w i l l become a p p a r e n t . In b r i e f , t h e e s s e n t i a l e l e m e n t s o f t h e e v a l u a t i o n s y s t e m a r e as f o l l o w s : 1. P a r a l l e l e v a l u a t i o n of b o t h raw and l o g t r a n s f o r m e d d a t a . 2. P r o b a b i l i t y g r a p h s as a v i s u a l c h e c k on t h e p o s s i b i l i t y o f m u l t i - m o d a l p r o b a b i l i t y d e n s i t y f u n c t i o n s . 42 I -PAIRED DATA li^Sw-^i fmmmmmJ • RAW DATA : LOG TRANSFORM PROBABILITY GRAPHS 2 -UNI MODAL CHI SQUARE TEST 3 -NON-PARAMETRIC TESTS PAIRED T-TEST 4 -X Y PLOTS SUBJECTIVE EVALUATION 5 - CURVE FITTING | Generally straight line ! F i g u r e 1 : A s y s t e m f o r e v a l u a t i n g c h e c k a s s a y s . 43 3. P a i r e d t - t e s t of raw o r l o g t r a n s f o r m e d d a t a as a p p r o p r i a t e , and n o n - p a r a m e t r i c s t a t i s t i c a l t e s t s ( e . g . W i l c o x o n p a i r e d rank t e s t ) where i n d i c a t e d by t h e form of t h e p r o b a b i l i t y d e n s i t y f u n c t i o n ( h i s t o g r a m ) . 4. E x a m i n a t i o n of x-y p l o t s o f raw and l o g t r a n s f o r m e d d a t a f o r s u b j e c t i v e e v a l u a t i o n of v a r i a t i o n i n n a t u r e of e r r o r as a f u n c t i o n of c o n c e n t r a t i o n l e v e l . 5. C u r v e f i t t i n g , e s p e c i a l l y a s i m p l e l i n e a r model, t o q u a n t i f y e r r o r s . In p a r t i c u l a r , we a r e c o n c e r n e d w i t h r e c o g n i z i n g random e r r o r , f i x e d b i a s and p r o p o r t i o n a l b i a s as i l l u s t r a t e d by an i d e a l , s i m p l e l i n e a r model i n F i g u r e 2. More complex models may be n e c e s s a r y i n i n d i v i d u a l c a s e s but e x p e r i e n c e i n d i c a t e s t h a t t h e s i m p l e l i n e a r model p r o v i d e s an a d e q u a t e r e p r e s e n t a t i o n f o r most e r r o r p a t t e r n s . Y = a + b X + e where Y and X a r e p a i r e d v a l u e s , a i s t h e y - i n t e r c e p t ( f i x e d b i a s ) , b i s t h e s l o p e o f t h e l e a s t s q u a r e s l i n e ( p r o p o r t i o n a l b i a s ) and e i s t h e e r r o r i n y a b o u t t h e l i n e (random e r r o r ) . Where l a r g e numbers o f sample p a i r s a r e a v a i l a b l e t h e y a r e d i v i d e d i n t o s u b g r o u p s a c c o r d i n g t o i d e n t i f i a b l e t i m e p e r i o d s . E a c h s u b g r o u p i s t r e a t e d s e p a r a t e l y t h r o u g h o u t t h e e v a l u a t i o n p r o c e d u r e and as a r u l e , s h o u l d c o n t a i n a t l e a s t 30 t o 40 p a i r s . 44 ( c ) x = a p p r o x i m a t i o n t o t r u e v a l u e . y = sample r e s u l t FIGURE 2 A SIMPLE LINEAR MODEL TO DESCRIBE ERRORS IN PAIRED DATA. (a) Simple random e r r o r (b) Simple random e r r o r p l u s p r o p o r t i o n a l b i a s (c) Simple random e r r o r p l u s f i x e d b i a s (d) Simple random e r r o r p l u s p r o p o r t i o n a l and f i x e d b i a s e s ( d ) 45 In t h i s f a s h o n , t h e e v a l u a t i o n p r o c e d u r e a t t e m p t s t o d e a l w i t h v a r i a t i o n i n l a b o r a t o r y a n a l y s i s methods i n c o r p o r a t e d i n d u p l i c a t e d a t a a s a f u n c t i o n of t i m e . F o r s h o r t e r p e r i o d e r r o r f l u c t u a t i o n s i t i s u s e f u l t o examine r e p l i c a t e a n a l y s i s d a t a f o r s t a n d a r d s a n a l y z e d by t h e l a b o r a t o r i e s i n s u c e s s i v e t - t e s t s a s e a c h new a n a l y t i c a l p a i r i s added t o a d a t a s e t , b u t t h i s p r o c e d u r e does not a p p e a r t o be e s s e n t i a l i n most c a s e s . A c o n s i d e r a b l e emphasis i s p l a c e d on a s u b j e c t i v e e v a l u a t i o n of p r o b a b i l i t y g r a p h s and s c a t t e r d i a g r a m s . The r e a s o n f o r s u c h an a p p r o a c h i s t o r e c o g n i z e t h e v e r y common s i t u a t i o n s i n e a r t h s c i e n c e d a t a of : 1. m u l t i m o d a l h i s t o g r a m s and 2. v a r i a t i o n s i n t h e t y p e o f e r r o r a s a f u n c t i o n of c o n c e n t r a t i o n . S e v e r a l h y p o t h e t i c a l but common examples d e m o n s t r a t i n g v a r i a t i o n s i n e r r o r as a f u n c t i o n o f c o n c e n t r a t i o n a r e p r o v i d e d i n F i g u r e 3. 46 (e) (f) FIGURE 3 IDEALIZED EXAMPLES OF PATTERNS EXHIBITED ON X-Y PLOTS OF DUPLICATE ANALYSES (a) Random e r r o r p l u s o u t l i e r (b) Two l e v e l s of random e r r o r as a f u n c t i o n of c o n c e n t r a t i o n ( a n a l y t i c a l e f f e c t at low values) (c) P r o p o r t i o n a l b i a s at low valu e s , random e r r o r at high values (d) Two l e v e l s of random e r r o r as a f u n c t i o n of c o n c e n t r a t i o n (nugget e f f e c t at high values) (e) E r r o r r e s u l t i n g from s.g. e f f e c t in one component ( f ) p r o p o r t i o n a l b i a s a r i s i n g from d i l u t i o n s of an i n c o r r e c t standard 47 QUALITY CONTROL TESTING AT EQUITY SILVER MINES S i n c e t h e e a r l y s t a g e s of e x p l o r a t i o n , t h e S o u t h e r n T a i l zone has been r e c o g n i z e d a s b e i n g m i n e r a l i z e d e r r a t i c a l l y , and t h u s , p o t e n t i a l l y s u b j e c t t o s a m p l i n g and a n a l y t i c a l p r o b l e m s . P r o d u c t i o n began i n 1980 and from t h e e a r l i e s t s t a g e s e f f o r t s were d i r e c t e d t o w a r d s an e v a l u t i o n of t h e q u a l i t y o f a n a l y t i c a l and s a m p l i n g p r o c e d u r e s . The i m p o r t a n c e of t h i s work can be a p p r e c i a t e d i f one i s aware t h a t b l o c k s e l e c t i o n ( 5 x 5 x 5 m 3) a t E q u i t y i s b a s e d on a c o m p o s i t e t u b e sample a s s a y composed o f f o u r s c o o p s of c u t t i n g s from a s i n g l e , c e n t r a l l y l o c a t e d b l a s t h o l e . T h i s s a m p l i n g t e c h n i q u e , common t o s e v e r a l open p i t o p e r a t i o n s i n B r i t i s h C o l u m b i a ( e . g . Ewanchuck, 1968), u s e s a r e l a t i v e l y s m a l l p r o p o r t i o n of t h e t o t a l b l a s t h o l e c u t t i n g s a s a sample : a d e s c r i p t i o n of t h e p r o c e d u r e a s a p p l i e d t o t h e S o u t h e r n T a i l zone i s g i v e n i n a s u b s e q u e n t s e c t i o n . A g e n e r a l p r o g r a m t o c h e c k f o r a n a l y t i c a l and s a m p l i n g e r r o r s a t E q u i t y S i l v e r M i n e s L t d . c o n t a i n e d t h e f o l l o w i n g e l e m e n t s : 1. D u p l i c a t e a n a l y s i s by E q u i t y and a s e c o n d l a b o r a t o r y . 2. D u p l i c a t e b l a s t h o l e s a m p l i n g . 3. T e s t o f s e v e r a l b l a s t h o l e s a m p l i n g p r o c e d u r e s . 4. E v a l u a t i o n o f s a m p l i n g r e s u l t s . Some of t h e d e t a i l e d r e s u l t s o f t h i s p r o g r am w i l l be p r e s e n t e d i n s u b s e q u e n t s e c t i o n s . 48 DUPLICATE ANALYTICAL DATA The f i r s t and p e r h a p s e a s i e s t c h e c k on v a l i d i t y of d a t a can be made by c o m p a r i n g r o u t i n e a n a l y t i c a l r e s u l t s f o r ( i n t e r n a l ) s t a n d a r d s a m p l e s w i t h d u p l i c a t e r e s u l t s f r o m a s e c o n d o r even t h i r d l a b o r a t o r y . The p r i n c i p a l p u r p o s e of t h i s a p p r o a c h i s t o d e t e c t a b i a s . As an example, a t o t a l o f 24 m i l l s a m p l e s from E q u i t y ' s S o u t h e r n T a i l zone were a n a l y z e d f o r c o p p e r and s i l v e r by b o t h E q u i t y M i n e s ' L a b o r a t o r y and a V a n c o u v e r b a s e d L a b o r a t o r y . The r e s u l t s f r o m b o t h l a b o r a t o r i e s a r e i n c l u d e d as A p p e n d i x I . U n f o r t u n a t e l y , any c o m p a r i s o n s can o n l y i n d i c a t e i f t h e l a b s d i f f e r , and no i n d i c a t i o n i s g i v e n as t o w h i c h i s c o r r e c t . The e x e r c i s e d o e s , however, s e r v e n o t i c e of a p o t e n t i a l a n a l y t i c a l p r o b l e m w h i c h when r e c o g n i z e d c a n be a l l e v i a t e d . The a s s a y r e s u l t s f o r c o p p e r from t h e two l a b s show r e a s o n a b l e agreement ( F i g u r e 4 a ) , and p a i r e d t - t e s t s f o r t h e h i g h g r a d e and low g r a d e g r o u p s v e r i f y t h e a b s e n c e of any b i a s . In c o n t r a s t , a c o m p a r i s o n of s i l v e r r e s u l t s shows a s i g n i f i c a n t d i f f e r e n c e between t h e two l a b s ( F i g u r e 4 b ) . A b i a s i s i n d i c a t e d f o r s i l v e r a n a l y s e s b e c a u s e a l m o s t a l l p o i n t s p l o t above t h e e q u a l v a l u e r e f e r e n c e l i n e . The b i a s can be e x p l a i n e d e i t h e r as an E q u i t y o v e r e s t i m a t e o r as an u n d e r e s t i m a t e by t h e c h e c k l a b o r a t o r y . Here we assume t h a t d i f f e r e n t l e v e l s o f e r r o r w i l l o c c u r f o r ( i ) o r e and t a i l i n g and ( i i ) c o n c e n t r a t e s , and t r e a t e a c h o f t h e s e g r o u p s s e p a r a t e l y by p a i r e d t - t e s t i n o r d e r t o q u a n t i f y t h e f i x e d b i a s . R e s u l t s a r e t a b u l a t e d i n T a b l e 1 and c l e a r l y d e m o n s t r a t e an a v e r a g e b i a s o f 336 g.Ag./tonne f o r c o n c e n t r a t e s n = 2 3 • I -2 3 -4 5 14 16 IB 20 22 24 CHECK LABORATORY % Cu. FIGURE 4a PLOT OF COMPARISON BETWEEN TWO . LABORATORIES SHOWING 23 POINTS ABOUT AN EQUAL VALUE REFERENCE LINE (Y=X) FOR TAILINGS ( T ) , ORE (0) AND CONCENTRATE (C) SAMPLES FOR COPPER CHECK LABORATORY g. AgVlonne FIGURE 4b PLOT OF COMPARISON BETWEEN TWO LABORATORIES SHOWING 23 POINTS ABOUT AN EQUAL VALUE REFERENCE LINE (Y=X) FOR TAILINGS ( T ) , ORE (O) AND CONCENTRATE (C) SAMPLES FOR SILVER ^ 50 w i t h E q u i t y o v e r e s t i m a t i n g r e l a t i v e t o t h e s e c o n d l a b . F o r o r e and t a i l i n g d a t a 14 of 15 v a l u e s p l o t above t h e e q u a l v a l u e l i n e , a s i t u a t i o n w i t h a p r o b a b i l i t y much l e s s t h a n .01, a s s u m i n g no b i a s e x i s t s . C o n s e q u e n t l y , a b i a s i s i n d i c a t e d and t h e s i n g l e v a l u e p l o t t i n g below t h e l i n e c a n be i n t e r p r e t e d as an o u t l i e r . A p a i r e d t - t e s t e x c l u d i n g t h e o u t l i e r d e m o n s t r a t e s a p r o n o u n c e d b i a s w i t h E q u i t y a n a l y z i n g on a v e r a g e a b o u t 5.5 g.Ag./tonne h i g h r e l a t i v e t o t h e s e c o n d l a b . T h i s p r e l i m i n a r y s t u d y i n d i c a t e s t h e need f o r c o n s i d e r a b l e e f f o r t a t q u a l i t y c o n t r o l of a n a l y t i c a l r e s u l t s . At a c u t - o f f l e v e l of 30 g Ag / t o n n e f o r b l o c k s e l e c t i o n t h e i n d i c a t e d a v e r a g e d i s c r e p a n c y between l a b o r a t o r i e s i s 5.5 g Ag, i e . an e i g h t e e n p e r c e n t e r r o r . I t i s c l e a r t h a t t h e s o u r c e o f d i s c r e p e n c i e s between t h e two l a b o r a t o r i e s s h o u l d be i n v e s t i g a t e d t h r o u g h c o n t i n u e d r e p l i c a t e a n a l y s i s o f sam p l e s a s w e l l a s use o f l a b o r a t o r y s t a n d a r d s f o r r e p e a t e d a n a l y s e s by b o t h l a b o r a t o r i e s . R e p l i c a t e a n a l y s e s s h o u l d i n c l u d e a l l c a t e g o r i e s o f sample n o t o n l y t a i l s , c o n c e n t r a t e s , and o r e but a l s o c o r e and c h i p s a m p l e s . T h e s e r e s u l t s d e m o n s t r a t e t h e n e c e s s i t y t h a t a l a b o r a t o r y c o n s t a n t l y c o n d u c t i n t e r n a l c h e c k s a g a i n s t s t a n d a r d s of known c o n c e n t r a t i o n s , so t h a t l a r g e s y s t e m a t i c e r r o r s a r e n o t i n c o r p o r a t e d i n r e p o r t e d a s s a y v a l u e s . 51 TABLE 1 - Summary of R e s u l t s f o r P a i r e d t - T e s t on T h r e e S e t s of P a i r e d D u p l i c a t e A n a l y s e s f o r S i l v e r . GROUP n AVERAGE STANDARD STANDARD PAIRED DEVIATION ERROR DIFFERENCE ( g / t o n n e ) Ore and t a i l i n g 15 3.76 8.24 2.13 Ore and t a i l i n g minus o u t l i e r 1 4 5.53 4.75 1 .27 C o n c e n t r a t e s 8 335.70 444.90 148.30 52 DUPLICATE BLASTHOLE SAMPLING Twenty-one d u p l i c a t e b l a s t h o l e samples were t a k e n t o e v a l u a t e t h e l e v e l of s a m p l i n g v a r i a b i l i t y by t h e t u b e - s a m p l i n g t e c h n i q u e u t i l i z e d by E q u i t y ( A p p e n d i x I I ) . T h e s e p a i r e d r e s u l t s f o r c o p p e r and s i l v e r a r e d i f f i c u l t t o e v a l u a t e s o l e l y a s a t a b u l a t i o n but c a n be v i e w e d u s e f u l l y on an x-y p l o t ( F i g u r e 5) f o r q u a l i t a t i v e e v a l u a t i o n . F i g u r e 5 i n d i c a t e s t h a t b i a s i s u n l i k e l y i n t h e s a m p l i n g p r o c e d u r e and t h i s i s v e r i f i e d by a p a i r e d t - t e s t f o r b o t h c o p p e r and s i l v e r ( T a b l e 2 ) . However, t h e r e s u l t s a l s o i n d i c a t e a l a r g e random e r r o r , an e x p e c t e d p r o b l e m a t E q u i t y S i l v e r M i n e s and a p a r t i c u l a r l y c r u c i a l one b e c a u s e b l o c k s e l e c t i o n a t p r e s e n t i s b a s e d on s u c h i n d i v i d u a l b l a s t h o l e r e s u l t s . TABLE 2 - R e s u l t s f o r P a i r e d t - T e s t f o r Copper and S i l v e r on 21 D u p l i c a t e B l a s t h o l e S a m p l e s . GROUP n AVERAGE PAIRED DIFFERENCE STANDARD DEVIATION STANDARD ERROR Copper 21 .001 .0473 .01 03 S i l v e r 21 7.14 31.01 6.77 Ag g r a d e s i n g / t o n n e and Cu g r a d e s i n % 0 I 2 3 -4 5 -6 ORIGINAL ASSAY % Cu. FIGURE 5a PAIRED RESULTS FOR COPPER FROM 21 DUPLICATE BLASTHOLE SAMPLES 10 30 50 * 70 9 0 110 130 150 ORIGINAL ASSAY (g. Ag./tonne) FIGURE 5b PAIRED RESULTS FOR SILVER FROM ^ 21 DUPLICATE BLASTHOLE SAMPLES 54 Even w i t h t h e l i m i t e d d a t a r e p r e s e n t e d i n F i g u r e 5 t h e r e i s an e v i d e n t d i f f e r e n c e i n t h e n a t u r e of s i l v e r v a r i a b i l i t y a s a f u n c t i o n of c o n c e n t r a t i o n . Where b o t h v a l u e s of a p a i r of d u p l i c a t e s a r e below a b o u t 70 g.Ag/tonne t h e d a t a p l o t f a i r l y t i g h t l y a b o u t t h e e q u a l v a l u e s l i n e . Where one v a l u e of a p a i r i s g r e a t e r t h a n 70 g.Ag/tonne t h e s c a t t e r of d u p l i c a t e s a m p l e s i s i n c r e a s e d d r a m a t i c a l l y . Below 70 g Ag/tonne t h e d a t a of F i g u r e 5 c a n be f i t t e d t o a l i n e w i t h t h e e q u a t i o n y = .87x - 0.068. The e r r o r i s t h e mean s q u a r e d d e v i a t i o n about t h e l i n e and g i v e s a g e n e r a l i d e a of t h e random s a m p l i n g p l u s a n a l y t i c a l e r r o r . At t h e 30 g.Ag/tonne c u t -o f f g r a d e one s t a n d a r d ( r e l a t i v e ) e r r o r i s 30 p e r c e n t . These d a t a p o i n t t o t h e n e c e s s i t y of e v a l u a t i n g t u b e s a m p l i n g r e s u l t s r e l a t i v e t o o t h e r p o t e n t i a l s a m p l i n g p r o c e d u r e s i n an e f f o r t t o r e d u c e t h e l a r g e random e r r o r i n h e r e n t i n t h e t u b e - s a m p l i n g p r o c e d u r e . 55 TESTING BLASTHOLE SAMPLING PROCEDURES (AT EQUITY SILVER MINES) E q u i t y S i l v e r M i n e s L i m i t e d a t t h e i r S o u t h e r n T a i l zone d e p o s i t employ a t u b e s a m p l i n g p r o c e d u r e f o r o b t a i n i n g samples from 'cones' of p r o d u c t i o n b l a s t h o l e c u t t i n g s . A somewhat c o m p a r a b l e p r o c e d u r e has been d e s c r i b e d , w i t h r e g a r d t o i t s a p p l i c a t i o n by B e t h l e h e m Copper M i n e s L t d . (Ewanchuk 1968). Upon c o m p l e t i o n of a b l a s t h o l e t h e d r i l l e r s c r a p e s away a p r e d e t e r m i n e d t h i c k n e s s o f c u t t i n g s from t h e t o p o f t h e p i l e w h i c h e x p e r i m e n t a t i o n has shown r e p r e s e n t s ' o v e r d r i l l ' m a t e r i a l from t h e t o p p o r t i o n o f t h e n e x t u n d e r l y i n g b e n c h . A 3 i n c h d i a m e t e r t u b e i s t h e n p u s h e d t h r o u g h t h e c u t t i n g s s u c c e s s i v e l y a t f o u r s y m m e t r i c a l l y l o c a t e d sample s i t e s ( F i g u r e 6 ) . The i n d i v i d u a l samples s c o o p e d from t h e 4 s i t e s a r e t h e n combined i n one bag t o form a c o m p o s i t e ' t u b e ' sample. In view of t h e l a r g e e r r o r s i n d u p l i c a t e tube sample a n a l y s e s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n t h e d e c i s i o n was made t o t e s t t h e tube s a m p l i n g p r o c e d u r e a g a i n s t two a l t e r n a t i v e s : c h a n n e l and b u l k s a m p l i n g . As a t e s t , 42 b l a s t h o l e s were sampled i n t h e f o l l o w i n g ways : a) f o u r 3 - i n c h d i a m e t e r t u b e samples were t a k e n i n t h e manner d e s c r i b e d p r e v i o u s l y ( F i g u r e 6) and combined t o form one t u b e sample ( c o m p o s i t e ) . b) f o u r c h a n n e l s a m p l e s were t a k e n w i t h a 5 i n c h wide t r a c k s h o v e l from l o c a t i o n s shown i n F i g u r e 6. The f o u r c h a n n e l samples were each r i f f l e d once w i t h one s p l i t k e pt a s t h e sample 56 CHANNEL 3 FIGURE 6 PLAN VIEW OF BLASTHOLE CUTTINGS SHOWING LOCATIONS OF THE TUBE AND CHANNEL SAMPLES ( C y r e t a l ( 1 9 8 0 ) . 1 57 f o r t h a t c h a n n e l and t h e r e m a i n i n g s p l i t s f r o m a l l f o u r c h a n n e l s c o m b i n e d t o form a c o m p o s i t e c h a n n e l sample. T h i s c o m p o s i t e sample was r i f f l e d u n t i l one sample bag of m a t e r i a l r e m a i n e d . c) t h e r e m a i n i n g m a t e r i a l i n t h e p i l e was t h e n r i f f l e d u n t i l one sample bag of c u t t i n g s 1 r e m a i n e d . T h i s sample r e p r e s e n t e d a r e s i d u a l b u l k a n a l y s i s . The t u b e sample, t h e f o u r c h a n n e l s a m p l e s , t h e c o m p o s i t e c h a n n e l sample and t h e r e s i d u a l b u l k sample were t h e n a s s a y e d w i t h t h e r e s u l t s f o r e a c h b l a s t h o l e l i s t e d i n A p p e n d i x I I I . In any c o m p a r a t i v e s t u d y i t i s c o n v e n i e n t but r a r e l y p o s s i b l e t o have t h e c o r r e c t answer w i t h w h i c h t o compare a s i n g l e t y p e o f r e s u l t . In t h i s p a r t i c u l a r c a s e we formed a ' b e s t e s t i m a t e ' g r a d e f o r ' e a c h b l a s t h o l e by c o m b i n i n g a l l r e s u l t s and w e i g h t i n g e a c h sample by i t s c o r r e s p o n d i n g volume. The volumes were c a l c u l a t e d , a s s u m i n g a u n i f o r m i d e a l shape f o r t h e cone of b l a s t h o l e c u t t i n g s , ( s e e A p p e n d i x IV f o r c a l c u l a t i o n s ) and a r e l i s t e d i n T a b l e 3. 58 TABLE 3 - Sample Typ e s w i t h C o r r e s p o n d i n g Volumes and W e i g h t s SAMPLE VOLUME WEIGHTS * Tubes 113 i n . 3 .027 C h a n n e l 1 C h a n n e l 2 C h a n n e l 3 C h a n n e l 4 720 i n . 3 300 i n . 3 360 i n . 3 300 i n . 3 .396 R e s i d u a l B u l k 2447 i n . 3 .577 * p r o p o r t i o n s of t o t a l v o l u m e s . 59 EVALUATION OF SAMPLING TEST RESULTS F o r p u r p o s e s of d e m o n s t r a t i o n we w i l l o n l y c o n s i d e r t h e a n a l y t i c a l r e s u l t s f o r s i l v e r b e c a u s e of t h e r e l a t i v e l y minor economic i m p o r t a n c e of c o p p e r g r a d e s i n b l o c k s e l e c t i o n . F u r t h e r m o r e , s i l v e r g r a d e s a r e much more v a r i a b l e t h a n a r e c o p p e r g r a d e s and so p r o d u c e more s i g n i f i c a n t e r r o r s i n b l o c k s e l e c t i o n . Some c o n c e r n e x i s t e d r e g a r d i n g t h e p o s s i b l e d e n s i t y s e p a r a t i o n o f m a t e r i a l i n a cone of b l a s t h o l e c u t t i n g s . C o n s e q u e n t l y t h e f o u r i n d i v i d u a l c h a n n e l s a m p l e s were examined t o see i f one d i r e c t i o n r e l a t i v e t o t h e ' t a i l ' o f t h e c u t t i n g s was b i a s e d . No b i a s was f o u n d . The f o u r i n d i v i d u a l c h a n n e l samples were combined i n a w e i g h t e d a v e r a g e and compared w i t h t h e c o r r e s p o n d i n g c h a n n e l c o m p o s i t e s . No b i a s was f o u n d t o e x i s t a l t h o u g h t h e d i s p e r s i o n of v a l u e s was r e l a t i v e l y h i g h . We a t t r i b u t e t h i s l a r g e l y t o t h e a c c u m u l a t e d s a m p l i n g e r r o r s of t h e f o u r i n d i v i d u a l samples and use t h e c o m p o s i t e c h a n n e l sample r e s u l t s f o r c o m p a r i s o n of s a m p l i n g methods. Tube sample d a t a show c o n s i d e r a b l y more d i s p e r s i o n t h a n do e i t h e r c o m p o s i t e c h a n n e l r e s u l t s o r r e s i d u a l b u l k r e s u l t s r e l a t i v e t o t h e ' b e s t e s t i m a t e ' g r a d e ( F i g u r e s 7,8 and 9 ) . C o n s e q u e n t l y , c o m p o s i t e c h a n n e l s a m p l e s o r b u l k s a m p l e s of b l a s t h o l e s p r o v i d e a b e t t e r e s t i m a t e of t h e t r u e s i l v e r g r a d e t h a n do t h e t u b e sample r e s u l t s . One measure of t h i s r e l a t i v e 'goodness' i s p r o v i d e d by t h e s c a t t e r of d a t a a b o u t t h e r e g r e s s i o n l i n e s f i t t e d t o t h e p l o t s of F i g u r e s 7, 8, and 9 and summarized i n T a b l e 4. The a n a l y t i c a l r e s u l t s show c l e a r l y t h a t t h e q u a l i t y of 60 • CI 5 ? . , , a •—, to . or ~ o GO a a Y = X area A Y = 7 5 U 8 9 4 X N= 42 j .0 a BO. a :BO.D 240.0 320.0 BEST ESTIMATE BULK 400 .0 a a . 03 !Z) I— _ l GC CD or o a . area 'W detail Y=X • • • / A g _ ' • • // / / / / A Y =75! + 894 • / / / m • • • ^ / N= 42 / / • • 0 . 0 20 0 40 .0 6 0 . 0 8 0 . 0 BEST ESTIMATE BULK 100.0 F I G U R E 7 O R I G I N A L T U B E SAMPLE VERSUS B E S T E S T I M A T E - BULK FOR 42 P A I R E D S I L V E R DATA 61 m ud° (_) UJ (_3 a X o o r ^ • U J -> X a -I CD Y=X area A see below Y=374 + 974X N= 42 J .0 80.0 160.0 240.0 BEST ESTIMATE 3ULK 320.0 400.0 C3 a . co x" 3 X (_) UJ x ° M „ 11 area A detai Y=X . Y = 3.74+974 X M= 42 0.0 20.0 40.0 60.0 80.0 BEST ESTIMATE BULK 100.0 FIGURE 8 AVERAGE CHANNEL SAMPLE VERSUS BEST ESTIMATE BULK FOR 4 2 PAIRED SILVER DATA 62 o m Y=X see below A g _ Y = -2.83+1.02 X N= 42 0.0 , I , , 80.0 160.0 240.0 320 0 . B E S T E S T I M A T E B U L K 400.0 Area "A" detai 00 Y=.:X I a -ac CO 5 ? -a a -t"NJ C3 a * Ag_ •Z83+I.02X N= 42 o'.o — I 1 1 - , — 20.0 40.0 60.0 80.0 B E S T E S T I M A T E B U L K !00.0 F I G U R E 9 B U L K S A M P L E V E R S U S B E S T E S T I M A T E B U L K F O R 4 2 P A I R E D S I L V E R D A T A 63 a s a m p l i n g p r o c e d u r e i s r e l a t e d t o sample volume, as e v i d e n c e d i n T a b l e 4 and F i g u r e 10. TABLE 4 - C o m p a r i s o n of S a m p l i n g P r o c e d u r e s C o m p a r i n g Volume, Amount o f S c a t t e r (Sy) and R e g r e s s i o n P a r a m e t e r s ( Y= A + BX ). SAMPLING PROCEDURE SAMPLE VOLUME ( a s a p r o p o r t i o n of t o t a l ) sy A B Tube s a m p l i n g vs b e s t .027 43.37 7.510 0.894 C h a n n e l s a m p l i n g v s b e s t .396 23.37 2. 129 1 .022 B u l k vs b e s t .577 12.36 -2.834 1 .022 64 60. CO co c c o 'to CO <D k_ CT) or 4 0 J 3 o o o o CO 20 J AxTUBE SAMPLES A CHANNEL COMPOSITES "^BULK Y =50-46^X~ I .2 .4 .6 .8 VOLUME (as proportion of total sample) ~i i.o FIGURE 10 PLOT SHOWING SCATTER ABOUT THE REGRESSION LINE VERSUS SAMPLE VOLUME WITH A CURVE F I T T E D 65 IDEALIZED ECONOMIC ANALYSIS To e m p h a s i z e t h e i m p o r t a n c e of a c c u r a t e b l a s t h o l e g r a d e e s t i m a t e s economic i m p l i c a t i o n s must be e x a m i n e d . A s s a y d a t a f o r t h e 42 b l a s t h o l e s t h a t were t e s t - s a m p l e d c an be c o n s i d e r e d a s r e p r e s e n t i n g a h y p o t h e t i c a l 2 - d i m e n s i o n a l a r r a y of 5 x 5 x 5 m 3 b l o c k s s i m u l a t i n g a p a r t o f a p r o d u c t i o n b e n c h . E a c h b l o c k i s assumed t o c o n t a i n a s i n g l e b l a s t h o l e , and t h e b l o c k s i z e i s assumed t o d e f i n e t h e s e l e c t i v e m i n i n g u n i t . N e i t h e r of t h e s e c o n d i t i o n s i s e x a c t l y t r u e a t E q u i t y b u t t h e assumed geometry and s e l e c t i o n u n i t a l l o w us t o c o n d u c t a s i m p l e e conomic a n a l y s i s w h i c h , even t h o u g h h y p o t h e t i c a l , d r a m a t i c a l l y e m p h a s i z e s t h e i m p o r t a n c e o f h i g h q u a l i t y s a m p l i n g i n b l o c k s e l e c t i o n . The f o l l o w i n g q u a n t i t a t i v e p a r a m e t e r s a r e s p e c i f i e d f o r t h e economic a n a l y s i s : (1) a s e l e c t i v e m i n i n g u n i t ( b l o c k ) i s 341 t o n n e s . (2) s e l e c t i o n i s b a s e d on s i l v e r - e q u i v a l e n t a s s a y s (AgE) where AgE ( g / t ) = Ag ( g / t ) + Cu (%) / 0.04. (3) c u t - o f f g r a d e s o f 30, 70 and 90 g.AgE/tonne w i l l be compared. A c u t - o f f o f 30 g.AgE/tonne p r e v a i l e d i n 1980 when t h e e x p e r i m e n t s were c o n d u c t e d whereas t h e p r e s e n t (1984) c u t - o f f g r a d e i s 90 g. A g E / t o n n e . F i g u r e 11 shows t h e s c a t t e r p l o t w i t h a 30 g.AgE/tonne c u t - o f f imposed. The two h a t c h e d a r e a s c o r r e s p o n d t o o r e c l a s s i f i e d a s waste, and waste c l a s s i f i e d a s o r e . B o t h 66 t y p e s of m i s c l a s s i f i c a t i o n s c o s t mines d e a r l y i n t h a t e c o n o m i c a l l y r e c o v e r a b l e m e t a l i s s h i p p e d t o t h e dump and waste m a t e r i a l i s i n a d v e r t e n t l y c o n t a m i n a t i n g t h e m i l l c i r c u i t . I t s h o u l d be r e - e m p h a s i z e d a t t h i s p o i n t t h a t b l o c k c l a s s i f i c a t i o n assumes t h a t t h e g r a d e of a b l o c k i s g i v e n by t h e b e s t e s t i m a t e of t h e g r a d e o f a c e n t r a l l y l o c a t e d b l a s t h o l e . G r a d e s of - t h e s e i n d i v i d u a l b l a s t h o l e s (and by e x t e n s i o n , t h e i r c o r r e s p o n d i n g b l o c k s ) w h i c h a r e m i s c l a s s i f i e d by t h e o r i g i n a l c o m p o s i t e t u b e s a m p l i n g method a r e l i s t e d i n T a b l e 5. TABLE 5 - T a b u l a t i o n of B l a s t h o l e s M i s c l a s s i f i e d by Tube S a m p l i n g Assuming t h a t a B e s t ( C a l c u l a t e d ) B u l k E s t i m a t e i s C o r r e c t , and C u t - o f f G r a d e i s 30 g.AgE/tonne. ORIGINAL TUBE SAMPLE MISCLASSIFIED AS WASTE ORIGINAL TUBE SAMPLE MISCLASSIFIED AS ORE TUBE BEST ESTIMATE TUBE BEST ESTIMATE Cu Ag AgE Cu Ag AgE Cu Ag AgE Cu Ag AgE .02 .05 .03 .02 .02 16 28 28 27 10 16.5 29.3 28.8 27.5 10.5 .09 .10 .04 .04 .06 82 68 45 34 30 84.3 70.5 46.0 35.0 31 .5 10 06 07 55 51 31 57.5 52.5 32.8 .06 .03 .06 28 21 27 29.5 21.8 28.5 Cu g r a d e s i n %, Ag g r a d e s i n g / t , AgE g r a d e s i n g / t . 67 The d a t a of T a b l e 5 can be u s e d f o r an i n t e r e s t i n g mass b a l a n c e c a l c u l a t i o n . From T a b l e 5 and F i g u r e 11 i t i s e v i d e n t t h a t o r i g i n a l tube samples r e s u l t i n t h e m i s c l a s s i f i c a t i o n o f 5 b l o c k s of o r e as waste and 3 b l o c k s of waste as o r e . The 3 waste b l o c k s i d e n t i f i e d as o r e c o n t a i n a t o t a l of 27,211 grams of s i l v e r (1023 t o n n e s a v e r a g i n g 26.6 g . A g E / t o n n e ) . The 5 o r e b l o c k s i n c o r r e c t l y a s s i g n e d t o waste by t h e o r i g i n a l t u b e samples c o n t a i n 91,217 grams o f s i l v e r (1705 t o n n e s a v e r a g i n g 53.5 g . A g E / t o n n e ) . Thus, c o r r e c t b l o c k s e l e c t i o n w ould r e s u l t i n s e n d i n g t h r o u g h t h e m i l l an a d d i t i o n a l 64,006 grams of s i l v e r a t a c o s t of m i n i n g and m i l l i n g 2 a d d i t i o n a l b l o c k s (682 t o n n e s ) o f o r e . A p r a c t i c a l q u e s t i o n t h e n a r i s e s as t o whether t h e g a i n i n r e c o v e r a b l e m e t a l i s s u f f i c i e n t t o o f f s e t t h e added c o s t s of improved s a m p l i n g p r o c e d u r e . In t h i s p a r t i c u l a r c a s e t h e c o s t o f m i n i n g and m i l l i n g t h e a d d i t i o n a l 682 t o n n e s i s e q u i v a l e n t t o 30 g A g E / t o n n e ( t h e assumed c u t - o f f g r a d e ) , f o r a t o t a l c o s t of 20,460 grams of s i l v e r . C o n s e q u e n t l y , t h e n e t b e n e f i t r e s u l t i n g from i m p r o v e d b l o c k s e l e c t i o n i s (64,006-20,460) 43,546 grams of s i l v e r f o r 42 b l o c k s (14,322 t o n n e s ) . I f t h e s e same p r o p o r t i o n s were e x t e n d e d o v e r one y e a r s p r o d u c t i o n ( s a y 5000 t p d f o r 300 d a y s ) t h e n e t b e n e f i t w ould be more t h a n 4,000,000 grams of s i l v e r h a v i n g a v a l u e o f more t h a n 1.0 m i l l i o n d o l l a r s ( a s s u m i n g $10.00 C a n a d i a n p e r ounce o f s i l v e r ) . Of c o u r s e , t h e s p e c i f i c f i g u r e s i n t h i s c a l c u l a t i o n a r e h i g h l y i d e a l i z e d and a r e a p p r o x i m a t e a t b e s t . N e v e r - t h e - l e s s , t h e m a g n i t u d e o f p o t e n t i a l i n c r e a s e d r e v e n u e from i m p r o v e d b l o c k e s t i m a t e s o b v i o u s l y demands h i g h q u a l i t y s a m p l i n g and may w e l l w a r r a n t r e l a t i v e l y h i g h l e v e l s o f PI CDS i . — ' C N , , Cl (-3d -,co_| o CO 68 area A Y=X equiv. <' 7=765+ .903X see below N= 42 .3.0 80.0 160.0 240 0 BEST ESTIMATE BULK 320.0 400.0 c? LTJ • —1° area A detai WASTE CLASSIFIED AS ORE Y=X Ag. equiv Y=7S5+9C3X T ORE CLASSIFIED AS N = ^XP WASTE • '^ C — 0 T 2 0 . 0 <K>.0 G O . O BO 0 BEST EST IMATE SULK 1 0 0 . 0 F I G U R E 1 1 O R I G I N A L T U B E S A M P L E V E R S U S B E S T E S T I M A T E B U L K F O R 4 2 P A I R E D S I L V E R E Q U I V A L E N T D A T A 69 e x p e n d i t u r e on sample c o n t r o l . S i m i l a r mass b a l a n c e c a l c u l a t i o n s b a s e d on a c u t - o f f g r a d e of 70 g.AgE/tonne i n d i c a t e a n e t s a v i n g of 3,500,000 g.AgE per y e a r ; w i t h a 90 g.AgE/tonne c u t - o f f t h e c a l c u l a t e d a n n u a l s a v i n g i s 39,000 g.AgE. R e g a r d l e s s of c u t - o f f g r a d e t h e g e n e r a l m a g n i t u d e of l o s s by a s a m p l i n g t e c h n i q u e w i t h l a r g e random e r r o r i n o u r h y p o t h e t i c a l b l o c k s e l e c t i o n p r o c e d u r e i s e v i d e n t . T h e s e c a l c u l a t i o n s e m p h a s i z e t h e need f o r r e p r e s e n t a t i v e s a m p l e s . F i g u r e 10 d e m o n s t r a t e s t h a t t h e e r r o r i n g r a d e e s t i m a t i o n o f a b l a s t h o l e i s m i n i m i z e d i f t h e e n t i r e volume o f c u t t i n g s i s t h o r o u g h l y mixed and s u b s a m p l e d as a b a s i s f o r a s s a y s . T h i s p r o c e d u r e would be done b e s t w h i l e d r i l l i n g was i n p r o g r e s s . Sample c o l l e c t i o n and s p l i t t i n g c o u l d be s t o p p e d b e f o r e th e d r i l l e n t e r e d t h e u n d e r l y i n g b e n ch so t h a t s a m p l i n g v a r i a b i l i t y r e s u l t i n g f r o m d r i l l i n g i n t o t h e u n d e r l y i n g bench would be m i n i m i z e d . 70 CONCLUSIONS The i m p o r t a n c e of a c c u r a c y and p r e c i s i o n i n s a m p l i n g a m i n e r a l d e p o s i t c a n n o t be o v e r e m p h a s i z e d . C a r e i n b o t h s a m p l i n g p r o c e d u r e and a n a l y t i c a l methods s h o u l d be e x e r c i s e d a t e v e r y s t a g e of p r o p e r t y d e v e l o p m e n t . D u p l i c a t e s a m p l i n g and a n a l y s e s • r e p r e s e n t an e f f e c t i v e q u a l i t y c o n t r o l p r o c e d u r e f o r i s o l a t i n g p o t e n t i a l p r o b l e m s i n s a m p l i n g p r o g r a m s , as w e l l a s m o n i t o r i n g l a b o r a t o r y p r e c i s i o n . The i m p l e m e n t a t i o n of s u c h a c h e c k program, however, i s not an end i n i t s e l f ; t h e s a m p l i n g and a s s a y i n g must be m o n i t o r e d c o n t i n u a l l y . The s i m p l e s t a t i s t i c a l and g r a p h i c a l a p p r o a c h a p p l i e d h e r e t o d a t a from t h e S o u t h e r n T a i l zone o f E q u i t y S i l v e r M i n e s L t d . i s one example o f how t o e f f e c t i v e l y e v a l u a t e q u a l i t y c o n t r o l r e s u l t s by b o t h i d e n t i f y i n g and q u a n t i f y i n g s a m p l i n g e r r o r s t h a t m i g h t o t h e r w i s e go unnot i c e d . At E q u i t y c o m p a r i s o n of a n a l y t i c a l r e s u l t s between two l a b o r a t o r i e s showed a s m a l l b u t s i g n i f i c a n t b i a s i n a n a l y t i c a l r e s u l t s f o r s i l v e r by one of t h e two l a b o r a t o r i e s . F o l l o w - u p i n v e s t i g a t i o n s a r e o b v i o u s l y n e c e s s a r y . D u p l i c a t e b l a s t h o l e s a m p l e s , f o r t h e S o u t h e r n T a i l zone, c o l l e c t e d u s i n g a t u b e - s a m p l i n g t e c h n i q u e , showed a l a r g e random e r r o r f o r s i l v e r a s s a y s , t h e r e b y i n d i c a t i n g t h e need f o r t e s t i n g a l t e r n a t e s a m p l i n g t e c h n i q u e s . A p r o g r a m of c o m p a r a t i v e s a m p l i n g , u s i n g t u b e s a m p l i n g , c h a n n e l s a m p l i n g and b u l k s a m p l i n g methods was u n d e r t a k e n and r e s u l t s were compared w i t h a ' b e s t e s t i m a t e ' sample c a l c u l a t e d by w e i g h t i n g a l l sample t y p e s ( t u b e , c h a n n e l , r e s i d u a l b u l k ) by t h e i r r e s p e c t i v e v o l u m e s . As t h e volume of sample i n c r e a s e s t h e amount o f random s a m p l i n g 71 e r r o r d e c r e a s e s and a l l r e s u l t s i n d i c a t e . a t o t a l b u l k sample t o be t h e most a c c u r a t e p r o c e d u r e . The c o l l e c t i o n and c u t t i n g o f t o t a l b u l k s a m p l e s of b l a s t h o l e c u t t i n g s , however, r e q u i r e s more p e r s o n n e l and s o p h i s t i c a t e d equipment t h a n needed f o r t u b e s a m p l i n g . Hence, t h e c o s t p e r sample w i l l i n c r e a s e . T h e r e f o r e , i t must be shown t h a t t h e a d d e d p r e c i s i o n g a i n e d by b u l k s a m p l i n g w i l l o f f s e t t h e h i g h e r s a m p l i n g c o s t s . An i d e a l i z e d c o m p a r i s o n of t u b e s a m p l i n g r e s u l t s of c o n e s of b l a s t h o l e c u t t i n g s , w i t h ' b e s t e s t i m a t e s ' o b t a i n e d from t h e e n t i r e p i l e of c u t t i n g s shows t h a t 8 o f 42 b l a s t h o l e s a r e m i s c l a s s i f i e d by t h e t u b e s a m p l i n g method. Assuming t h a t e a c h of t h e 8 b l a s t h o l e s i s c e n t r a l l y l o c a t e d w i t h i n a b l o c k ( s e l e c t i o n m i n i n g u n i t ) and t h a t e a c h i s r e p r e s e n t i t i v e o f i t s r e s p e c t i v e b l o c k , c o r r e c t c l a s s i f i c a t i o n of t h e 8 b l o c k s would r e s u l t i n 64,000 a d d i t i o n a l grams of s i l v e r b e i n g s e n t t h r o u g h t h e m i l l but h a v i n g a n e t b e n e f i t of a b o u t 43,000 g a t t h e added c o s t of m i n i n g and m i l l i n g an a d d i t i o n a l 682 t o n n e s (2 b l o c k s ) o f o r e . In terms of a n n u a l p r o d u c t i o n t h i s same p r o p o r t i o n of i m p r o v e d r e c o v e r y c o u l d r e s u l t i n a n e t b e n e f i t w e l l i n e x c e s s o f 1 m i l l i o n d o l l a r s . I t i s w o r t h r e - e m p h a s i z i n g t h a t a l l r e s u l t s p r e s e n t e d a r e b a s e d on t h e e s t i m a t i o n o f a b l o c k g r a d e by a s i n g l e c e n t r a l l y l o c a t e d s a m p l e . No e f f o r t has been made t o c o n s i d e r improvements i n r e c o v e r y t h a t m i g h t r e s u l t from t h e use of moving a v e r a g e ( w e i g h t i n g ) e s t i m a t i o n p r o c e d u r e s , as Raymond ( 1 9 7 9 ) , among o t h e r s , has shown t o be of c r u c i a l i m p o r t a n c e i n e r r a t i c a l l y m i n e r a l i z e d d e p o s i t s . 72 The q u a l i t y c o n t r o l e x p e r i m e n t s r e p o r t e d h e r e were c o n d u c t e d i n 1980 s h o r t l y a f t e r p r o d u c t i o n was i n i t i a t e d from t h e S o u t h e r n T a i l z o n e . R e s u l t s have s e r v e d as a b a s i s f o r t h e c o n t i n u e d e x a m i n a t i o n and d e v e l o p m e n t of g e o s t a t i s t i c a l p r o c e d u r e s f o r i m p r o v i n g t h e q u a l i t y of b l o c k e s t i m a t i o n s f o r m i n i n g s e l e c t i o n ( e . g . S i n c l a i r and G i r o u x , 1983). 73 ACKNOWLEDGEMENTS T h i s s t u d y was i n i t i a t e d i n d i s c u s s i o n s w i t h W.H. Myck a t y n f o r m e r Mine E n g i n e e r a t E q u i t y S i l v e r M i n e s L t d . and has b e n e f i t e d f r o m t h e c o n t r i b u t i o n s by D. F r a s e r and J . Cyr E q u i t y S i l v e r M i n e s L t d . , and Mr. E.T. L o n e r g a n , P l a c e r Development L t d . F i n a n c i a l s u p p o r t was p r o v i d e d by P l a c e r Development L t d . 74 REFERENCES C y r , J . , J . M i l l e r (1980) B l a s t h o l e S a m p l i n g P r o c e d u r e S t u d y , P r i v a t e r e p o r t , E q u i t y S i l v e r M i n e s L t d . , November, 11 p. Ewanchuck, H. (1968) Grade C o n t r o l a t B e t h l e h e m C opper, Can. I n s t . M i n . M e t a l l . S p e c . V o l . 9, pp. 302-307. M i l l e r , J . (1981) B l a s t h o l e S a m p l i n g P r o c e d u r e , P r i v a t e r e p o r t E q u i t y S i l v e r M i n e s L t d . , May, 5 p. Raymond, G., (1979) Ore e s t i m a t i o n p r o b l e m s 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 o r e body, Can. I n s t . M i n . M e t a l l . B u l l e t i n , J u n e , pp. 90-97. S i n c l a i r , A . J . , and G.H. G i r o u x , (1983) G e o l o g i c a l c o n t r o l s o f s e m i - v a r i o g r a m s i n p r e c i o u s m e t a l d e p o s i t s ; G e o s t a t i s t i c s f o r N a t u r a l R e s o u r c e s C h a r a c t e r i z a t i o n , NATO ASI S e r i e s , D . R e i d e l P u b l i s h i n g Company, D o r d r e c h t , H o l l a n d , pp. 965-978. W e t h e r e l l , D. (1979) G e o l o g y and Ore G e n e s i s of t h e Sam G o o s l e y c o p p e r - s i l v e r - a n t i m o n y D e p o s i t , U n p u b l i s h e d MSc T h e s i s 1979, D e p t . o f G e o l o g i c S c i e n c e s , U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , 208 p. W e t h e r e l l , D., A . J . S i n c l a i r and T . S c h r o e t e r , (1979) P r e l i m i n a r y r e p o r t on t h e Sam G o o s l y c o p p e r - s i l v e r d e p o s i t ; i n Paper 1979-1, B.C. M i n i s t r y of E n e r g y , M i n e s and P e t r o l e u m R e s o u r c e s , pp. 132-137. CHAPTER 4 AN EVALUATION OF VARIOUS KRIGING TECHNIQUES AS APPLIED TO EQUITY SILVER MINES' SOUTHERN TAIL ZONE 7 6 ABSTRACT A v a r i e t y of a p p r o a c h e s t o b l o c k k r i g i n g have been a t t e m p t e d f o r t h e S o u t h e r n T a i l zone o f E q u i t y S i l v e r M i n e s L t d . and compared w i t h two e m p i r i c a l p r o c e d u r e s v i z . i n v e r s e d i s t a n c e s q u a r e d and p o l y g o n a l ( c e n t r a l b l a s t h o l e a s s a y a s s i g n e d t o e n c o m p a s s i n g b l o c k ) . C r o s s v a l i d a t i o n methods do not d i s t i n g u i s h c l e a r l y among a l l t h e e s t i m a t i o n methods a l t h o u g h l o g k r i g i n g of p a r t i t i o n e d d a t a i s m a r g i n a l l y t h e b e s t p r o c e d u r e . F o r b l o c k e s t i m a t i o n p u r p o s e s k r i g i n g p r o c e d u r e s a r e b e t t e r t h a n e m p i r i c a l p o i n t methods by an amount t h a t must a w a i t t h e a v a i l a b i l i t y of known b l o c k m e t a l c o n t e n t s w i t h w h i c h t o make c o m p a r i s o n s . B l o c k k r i g i n g e r r o r s a l s o s u g g e s t t h a t l o g p a r t i t i o n k r i g i n g i s t h e b e s t of t h e p r o c e d u r e s t e s t e d . A t r i a l s e t o f 100 b l o c k s f r o m 1310N bench, e s t i m a t e d by s e v e r a l t e c h n i q u e s show d i f f e r e n c e s i n t o t a l e s t i m a t e d m e t a l c o n t e n t s of as much as 1,400,000 grams o f s i l v e r by t h e two most extreme r e s u l t s , p o l y g o n a l ( h i g h e s t ) and a m o d i f i e d l o g k r i g i n g ( l o w e s t ) . Such extreme d i f f e r e n c e s e m p h a s i z e t h e need t o know m e t a l c o n t e n t s of some b l o c k s e a r l y i n t h e p r o d u c t i o n h i s t o r y of a mine so t h a t e s t i m a t e s c a n be compared w i t h r e a l i t y and an optimum b l o c k e s t i m a t i o n p r o c e d u r e a t t a i n e d . 77 INTRODUCTION K r i g i n g i s a moving a v e r a g e t e c h n i q u e commonly r e f e r r e d t o as t h e " b e s t l i n e a r u n b i a s e d e s t i m a t o r " b e c a u s e i t p r o v i d e s w e i g h t s f o r an a r r a y of d a t a p o i n t s s u c h t h a t t h e mean s q u a r e d random e r r o r o f e s t i m a t i n g a p o i n t ( o r a b l o c k ) i s m i n i m i z e d . A v a r i e t y of k r i g i n g p r o c e d u r e s have e v o l v e d and i t i s not a l w a y s c l e a r why one a p p r o a c h s h o u l d be p r e f e r r e d t o a n o t h e r . D i f f e r e n t k r i g i n g p r o c e d u r e s c a n r e s u l t i n s u b s t a n t i a l d i f f e r e n c e s i n e s t i m a t i o n s f o r p a r t i c u l a r p o i n t s or b l o c k s and, what i s more, s u b s t a n t i a l l y d i f f e r e n t e r r o r e s t i m a t e s . C l e a r l y , a l l k r i g i n g methods c a n n o t be " b e s t " . S o u t h e r n T a i l zone o f E q u i t y M i n e s L t d . p r o v i d e s a p r a c t i c a l o p p o r t u n i t y t o compare s e v e r a l k r i g i n g a p p r o a c h e s . T h i s p r e l i m i n a r y s t u d y i s of i n t e r e s t b e c a u s e o f t h e s e e m i n g l y e r r a t i c n a t u r e of g r a d e v a r i a b i l i t y i n t h e d e p o s i t and t h e a t t e n d a n t d i f f i c u l t y of h a v i n g c o n f i d e n c e i n b l o c k e s t i m a t e s f o r s e l e c t i o n p u r p o s e s ( G i r o u x and S i n c l a i r , 1984b) The S o u t h e r n T a i l zone of t h e Sam G o o s l y b u l k s i l v e r d e p o s i t l i e s w i t h i n an i n l i e r of C r e t a c e o u s s e d i m e n t a r y , p y r o c l a s t i c and v o l c a n i c r o c k s ( W e t h e r e l l , 1979). The i n l i e r i s i n t r u d e d i n t h e west by a q u a r t z m o n z o n i t e s t o c k and i n t h e e a s t by a g a b b r o - m o n z o n i t e complex. S i l v e r - c o p p e r - a n t i m o n y o r e s a r e t h o u g h t t o be e p i g e n e t i c i n o r i g i n and b o t h s t r u c t u r a l l y , and t o a l e s s e r d e g r e e , s t r a t i g r a p h i c a l l y c o n t r o l l e d . The S o u t h e r n T a i l zone i s c r u d e l y t a b u l a r i n f orm w i t h a s t r i k e o f a b o u t 025 d e g r e e s and a w e s t e r l y d i p o f from 30 t o 50 d e g r e e s . S u l p h i d e s o c c u r m a i n l y i n a s t o c k w o r k or as a cement between b r e c c i a f r a g m e n t s . 78 DATA ACQUISITION D u r i n g a v i s i t t o E q u i t y M i n e s L i m i t e d i n September 1980 raw d a t a were c o l l e c t e d f r o m 3 benches of t h e S o u t h e r n T a i l z o n e . The d a t a base c o n s i s t e d o f bench p l a n s f o r t h e 1310, 1315 and 1320 l e v e l s showing sample numbers and l o c a t i o n s on a s c a l e of 1:500, and a s s a y s h e e t s showing t h e a s s a y r e s u l t s from - b l a s t h o l e s a m p l e s a n a l y z e d f o r c o p p e r , s i l v e r , a n t i m o n y and a r s e n i c a t e a c h sample l o c a t i o n . The bench p l a n s were d i g i t i z e d by t h e w r i t e r a t t h e V a n c o u v e r o f f i c e of P l a c e r Development L t d . S i x f i l e s were c r e a t e d , one f o r each o f a n o r t h e r n and s o u t h e r n p a r t f o r e a c h of t h e t h r e e l e v e l s . The sample number and a s s a y s f o r e a c h b l a s t h o l e were th e n k e y p u n c h e d and b o t h s e t s of d a t a e n t e r e d i n t o a c o m p u t e r . Check p l o t s of t h e d i g i t i z e d sample l o c a t i o n s were p r o d u c e d and l o c a t i o n e r r o r s c o r r e c t e d . L i s t i n g s of t h e a s s a y d a t a a l s o were e d i t e d . The c o r r e c t e d f i l e s were t h e n merged t o p r o d u c e a d a t a base of sample number, x and y l o c a t i o n s and a s s a y i n f o r m a t i o n f o r e a c h b l a s t h o l e . 79 STRUCTURAL ANALYSIS The S o u t h e r n T a i l zone i s c h a r a c t e r i z e d by l o c a l l y e r r a t i c a s s a y d a t a ( G i r o u x and S i n c l a i r , 1984b). B l a s t h o l e s c a n v a r y f r o m 2 g Ag / t o n n e t o 2000 g Ag/tonne a c r o s s d i s t a n c e s a s s m a l l as t h e e s t i m a t i o n b l o c k s i z e ( 5 x 5 x 5 m 3 ) . T h i s h i g h d e g r e e of v a r i a b i l i t y makes any form of l o c a l e s t i m a t i o n v e r y d i f f i c u l t and i n g e o s t a t i s t i c a l t e r m s l e a d s t o l a r g e nugget e f f e c t s , h i g h s i l l v a l u e s and hence l a r g e e s t i m a t i o n e r r o r s . A v a r i e t y of t e c h n i q u e s were a t t e m p t e d t o t r y and m i n i m i z e t h e e s t i m a t i o n e r r o r and t h u s make t h e b e s t p o s s i b l e e s t i m a t i o n of 5 x 5 x 5 m 3 s e l e c t i v e m i n i n g u n i t s . A l t h o u g h a s s a y s were a v a i l a b l e f o r s e v e r a l v a r i a b l e s , o n l y s i l v e r r e s u l t s a r e shown , b e c a u s e , o f a l l m e t a l s r e c o v e r e d , s i l v e r i s t h e most i m p o r t a n t e c o n o m i c a l l y , and a l s o i s d i s t r i b u t e d by f a r t h e most e r r a t i c a l l y . T r a d i t i o n a l M odel To d e t e r m i n e t h e s t r u c t u r e a l o n g t h e p l a n e of t h e b e n c h e s , f o u r d i r e c t i o n a l s e m i v a r i o g r a m s were p r o d u c e d f o r e a c h o f t h e s i x d a t a f i l e s and compared. F o u r e x amples a r e shown as F i g u r e 1 fr o m b e n c h e s 1310N, 1310S, 1315S and 1320S. Two p o i n t s a r e a p p a r e n t f r o m a c o m p a r i s o n of t h e s e s e m i v a r i o g r a m s . F i r s t , a l l benches w i t h t h e e x c e p t i o n o f 1315S, e x h i b i t t h e same g e o m e t r i c a l a n i s o t r o p i c f o r m w i t h r a n g e s i n d i r e c t i o n s N-S ( l ) and SW-NE (4) l a r g e r t h a n i n d i r e c t i o n s E-W (2) and NW-SE ( 3 ) , and w i t h a common s i l l f o r a l l 4 d i r e c t i o n s . S e c o n d , t h e l e v e l of v a r i a b i l i t y ( s i l l ) i s p r o p o r t i o n a l t o t h e mean g r a d e o f t h e 80 TO 20" 30~~ 40 ° !0 20 30 40 Lag (m) L a g (m) F i g u r e 1 : A v e r a g e h o r i z o n t a l s e m i v a r i o g r a m s f o r s e l e c t e d b e n c h e s S o u t h e r n T a i l z o n e , s h o w i n g e x p e r i m e n t a l c u r v e s f o r d i r e c t i o n s 1 t o 4. 81 s a m p l e s . The d i s c r e p a n c i e s o b s e r v e d i n 1315S were d i s c u s s e d w i t h t h e mine g e o l o g i s t s from E q u i t y . I t was d i s c o v e r e d t h a t f i l l f rom o u t s i d e t h e m i n e r a l i z e d a r e a was b u l l d o z e d i n t o p r e p a r e d r i l l s i t e s on t h i s b e n c h . As a r e s u l t t h e d a t a from 1315S were not i n c l u d e d i n t h e s t u d y . The r e s u l t s from t h e r e m a i n i n g 5 bench segments were a v e r a g e d t o p r o d u c e a s i n g l e s e t o f f o u r - d i r e c t i o n a l s e m i v a r i o g r a m s . The r a n g e s o f t h e s e f o u r d i r e c t i o n s a r e summarized i n T a b l e 1. TABLE 1 - A Summary o f t h e Ranges i n t h e F o u r P r i n c i p a l D i r e c t i o n s . DIRECTION RANGE (a) N o r t h - S o u t h (1) 20 m. E a s t - W e s t (2) 10 m. S o u t h e a s t - N o r t h w e s t (3) 10 m. S o u t h w e s t - N o r t h e a s t (4) 18 m. The r a n g e s i n t h e f o u r d i r e c t i o n s were p l o t t e d and an e l l i p s e f i t t o d e t e r m i n e t h e d i r e c t i o n s o f maximum (22.5 d e g r e e s a z i m u t h ) and minimum (112.5 d e g r e e s ) r a n g e s ( F i g u r e 2 ) . Semi-v a r i o g r a m s were t h e n p r o d u c e d f o r e a c h of t h e 5 b e n c h e s i n t h e s e F i g u r e 2 : S t r u c t u r a l e l l i p s e o f r a n g e s i n t h e f o u r p r i n c i p a l d i r e c t i o n s s h o w i n g t h e d i r e c t i o n o f minimum and maximum s t r u c t u r e . 83 two d i r e c t i o n s of minimum and maximum a v e r a g e s t r u c t u r e . The s e c o n d n o t e w o r t h y c h a r a c t e r i s t i c o f t h e s e m i v a r i o g r a m s o f raw d a t a ( F i g u r e 1) was t h e f l u c t u a t i o n of s i l l l e v e l w i t h t h e mean g r a d e o f t h e s a m p l e s . T h i s phenomenon i n d i c a t e s a p r o p o r t i o n a l e f f e c t i s p r e s e n t w h i c h must be t a k e n i n t o a c c o u n t i n any d e v i s e d model. P l o t s of t h e f i v e nugget e f f e c t s (CO) and t h e c o r r e s p o n d i n g s t r u c t u r a l components (C1) v e r s u s t h e mean g r a d e s q u a r e d were p r o d u c e d and a l i n e a r model f i t t e d t o p o i n t s . By f o r c i n g t h e b e s t - f i t l i n e t h r o u g h t h e a v e r a g e of t h e f i v e d a t a s e t s a and (3 i n t h e e q u a t i o n s CO = a m 2 , and C1 = 0 m 2 were d e t e r m i n e d . The r e s u l t i n g s e m i v a r i o g r a m model f o r s i l v e r i n t h e S o u t h T a i l Zone i s summarized i n T a b l e 2. L o g n o r m a l M o d e l A h i s t o g r a m of s i l v e r v a l u e s shows t h e d a t a t o be s t r o n g l y p o s i t i v e l y skewed ( F i g u r e 3 ) . A t r a n s f o r m a t i o n was made w i t h t h e d a t a c o n v e r t e d t o l o g v a l u e s . The l o g h i s t o g r a m of F i g u r e 3 i s a l s o p o s i t i v e l y skewed and shows t h a t s i l v e r g r a d e s a r e n o t e x a c t l y l o g n o r m a l . N e v e r - t h e - l e s s , f o r c o m p a r i t i v e p u r p o s e s we p r o c e e d e d t o d e v e l o p e a s e m i v a r i o g r a m model f o r l o g t r a n s f o r m e d d a t a . E x p e r i m e n t a l s e m i v a r i o g r a m s f o r t h e l o g . v a l u e s were p r o d u c e d i n t h e maximum and minimum d i r e c t i o n s , e s t a b l i s h e d p r e v i o u s l y , f o r e a c h o f t h e f i v e b e n c h e s and a v e r a g e d . ( F i g u r e 4 ) . The p a r a m e t e r s of t h e l o g n o r m a l model a r e summarized i n T a b l e 2. T h i s a p p r o a c h t a k e s t h e p r o p o r t i o n a l e f f e c t i n t o a c c o u n t by t h e l o g o r i t h m i c t r a n s f o r m a t i o n . The c u r v e s of F i g u r e 4 i n d i c a t e some non s t a t i o n a r i t y i n a d d i t i o n t o a s i m p l e p r o p o r t i o n a l e f f e c t . The model f i t t e d i s f o r p u r p o s e s o f l o c a l 84 80.0 _ 70.0 _ 60.0 _ t>5 >-L J LU 50.0 _ o UJ 30.0 _ 20.0 -]0.0 >-UJ 12.5 Ll_ 7 . S ! . LOG (bosel0) HISTOGRAM N = 2576 HERN = 1.037 GM/T S.C. = .718 GM/T C.I. = .200 GM/T L a . - . 4 0 0 0 I . 4 E ' » ! 1.2000 1.2E 2) 2 . 0 0 0 0 4 .4000 I . 6 E 31 I . 3 E SI I 1 6 .0000 7 .6000 ( . I E 7) I . 4 E 61 SILVER .ARITHMETIC HISTOGRAM L W N = 2576 MEAN = 49.200 GM/T 5.D. = 145.821 GM/T C. I. = 30.000 GM/T T .000 2*0.000 '460. 000 720.000 960.000 1200.000 SILVER F i g u r e 3 : A r i t h m e t i c and l o g t r a n s f o r m e d h i s t o -grams f o r 2575 s i l v e r g r a d e s f r o m b l a s t h o l e s , S o u t h e r n T a i l z o n e . 85 0.5-"I 1 1 1 1 1 1— 0 8.0 16.0 24.0 32.0 40.0 48.0 L A G h (metres) F i g u r e 4 : A v e r a g e h o r i z o n t a l s e m i v a r i o g r a m i n t h e d i r e c t i o n o f minimum and maximum s t r u c t u r e f o r l o g t r a n s f o r m e d s i l v e r g r a d e s , S o u t h e r n T a i l z o n e . 86 e s t i m a t i o n w i t h i n d a t a f i e l d s whose d i m e n s i o n s a r e l e s s t h a n 15 x 15 m2 and w i t h i n w h i c h l o c a l s t a t i o n a r i t y i s assumed. P a r t i t i o n e d M odel The h i g h nugget e f f e c t and s i l l i n b o t h of t h e f o r g o i n g m o d e l s , c a u s e d by t h e t r e m e n d o u s l y e r r a t i c n a t u r e of s i l v e r g r a d e s , l e d t o a s e a r c h f o r a l t e r n a t i v e a p p r o a c h e s t o s t r u c t u r a l a n a l y s i s . A l o g n o r m a l c u m u l a t i v e d i s t r i b u t i o n p l o t ( F i g u r e 5) i n d i c a t e s t h e p r e s e n c e of t h r e e p o p u l a t i o n s w i t h i n t h e s i l v e r d a t a . The l o w e r two (B and C) a r e v e r y s i m i l a r t o t h e b i m o d a l d i s t r i b u t i o n of s i l v e r e s t a b l i s h e d from t h e e x p l o r a t i o n diamond d r i l l i n g ( G i r o u x and S i n c l a i r , 1984a) and r e p r e s e n t s t o c k w o r k s u l p h i d e s (mean = 55 g A g / t o n n e ) and t h e a l t e r a t i o n b a c k g r o u n d s i l v e r (mean = 3.5 g A g / t o n n e ) r e s p e c t i v e l y . The upper o r A p o p u l a t i o n ( 2 % of t h e d a t a ) r e p r e s e n t s l e n s e s of s i l v e r - b e a r i n g g a l e n a (mean = 630 g A g / t o n n e ) v i s i b l e i n t h e S o u t h e r n T a i l p i t but not i n t e r s e c t e d i n e x p l o r a t i o n d r i l l i n g . A t h r e s h o l d o f 340 g Ag/tonne e f f e c t i v e l y s e p a r a t e s t h e A and B p o p u l a t i o n s . Of paramount i m p o r t a n c e a t t h e p r o d u c t i o n s t a g e , i s t h e l o c a l e s t i m a t i o n o f s e l e c t i v e m i n i n g u n i t s , b a s e d on b l a s t h o l e d a t a . The c r i t i c a l a r e a of e s t i m a t i o n i s n o t f o r v e r y h i g h or v e r y low g r a d e s , but r a t h e r f o r t h o s e s e l e c t i v e m i n i n g u n i t s w i t h g r a d e s c l o s e t o t h e c u t - o f f g r a d e . B l o c k s c o n t a i n i n g a sample f r o m t h e u p p e r A p o p u l a t i o n i . e . a sample g r e a t e r t h a n 340 g A g / t o n n e , w i l l be s e l e c t e d as o r e by any l o c a l e s t i m a t i o n t e c h n i q u e . Of m a j o r c o n c e r n a r e t h e b l o c k s c o n t a i n i n g s a m p l e s i n t h e range 20 t o 100 g Ag/tonne where i n c o r r e c t c l a s s i f i c a t i o n w i l l s end o r e t o t h e dump and waste t h r o u g h t h e m i l l . W i t h t h i s 87 SILVER FROM I3ION.I3IOSJ3I5N C U M U L A T I V E % F i g u r e 5 : .Lognormal c u m u l a t i v e d i s t r i b u t i o n c u r v e f o r 2773 s i l v e r g r a d e s from 1310 N, 1310 S and 1315 N b e n c h e s , S o u t h e r n T a i l z o n e . 88 c r i t e r i o n i n mind, t h e h i g h p o p u l a t i o n was removed from th e d a t a s e t f o r t h e p u r p o s e of s t r u c t u r a l m o d e l l i n g . To a c c o u n t f o r t h e known p r o p o r t i o n a l e f f e c t , r e l a t i v e s e m i v a r i o g r a m s were u s e d f o r t h i s model ( 7 ( h ) / m 2 ) . The model p a r a m e t e r s i n t h e p r e v i o u s l y e s t a b l i s h e d minimum and maximum d i r e c t i o n s f o r t h e a v e r a g e of t h e f i v e b e n c h e s a r e shown i n T a b l e 2. A s t r u c t u r a l e l l i p s e f o r t h e f o u r p r i n c i p a l d i r e c t i o n s was c h e c k e d t o c o n f i r m t h a t t h e same d i r e c t i o n s f o r a n i s o t r o p y e x i s t e d . By e l i m i n a t i n g t h e e r r a t i c h i g h g r a d e p o p u l a t i o n t h e r a t i o of nugget e f f e c t t o s i l l was r e d u c e d from 0.5 i n t h e t r a d i t i o n a l model t o 0.25 i n t h e p a r t i t i o n e d m o d e l , whereas t h e r a n g e of i n f l u e n c e was i n c r e a s e d t o 60 m f r o m 22 m i n t h e d i r e c t i o n o f maximum r a n g e . To r e i t e r a t e , t h i s a p p r o a c h assumes t h a t our m o d el, when u s e d f o r l o c a l e s t i m a t i o n n e a r th e c u t - o f f g r a d e w i l l use l o c a l d a t a o n l y of p o p u l a t i o n s B and C of F i g u r e 5. D i s c u s s i o n s w i t h E q u i t y mine g e o l o g i s t s i n d i c a t e d t h e m o d e l s d e s c r i b e d t o d a t e , d i d not f i t t h e s t r u c t u r e s o b s e r v e d a t t h e e x t r e m e n o r t h end o f t h e d e p o s i t . The m i n e r a l i z e d f r a c t u r e s n o r t h of t h e E a s t Dyke had a d i f f e r e n t o r i e n t a t i o n f r o m t h e b u l k o f t h e d a t a t o t h e s o u t h . A s e p a r a t e s e m i v a r i o g r a m s t u d y on o n l y d a t a n o r t h of t h e E a s t Dyke p r o d u c e d an a n i s o t r o p i c model w i t h a z i m u t h o f maximum ran g e 68 d e g r e e s ( S i n c l a i r and G i r o u x , 1983). T h i s example and o t h e r s , d i s c u s s e d i n t h i s p a p e r , p o i n t t o t h e a b s o l u t e n e c e s s i t y o f i n c o r p o r a t i n g as much g e o l o g i c i n f o r m a t i o n a s i s p o s s i b l e i n any g e o s t a t i s t i c a l s t u d y . 89 Log P a r t i t i o n e d Model U s i n g t h e a s s u m p t i o n s o f l o g n o r m a l i t y (model 2) and m u l t i -p o p u l a t i o n s (model 3) a f o u r t h model was d e v e l o p e d . As i n model 3 o f T a b l e 2 o n l y t h e two l o w e r p o p u l a t i o n s (< 340 g A g / t o n n e ) were u s e d , b u t i n t h i s c a s e t r a n s f o r m e d t o l o g v a l u e s . Semi-v a r i o g r a m s were p r o d u c e d i n e a c h of t h e 5 a c c e p t a b l e d a t a f i l e s a l o n g t h e d i r e c t i o n s 22.5 and 112.5 d e g r e e s and t h e n a v e r a g e d t o p r o d u c e t h e e x p e r i m e n t a l c u r v e s shown i n F i g u r e 6. A g a i n a g e o m e t r i c a l a n i s o t r o p i c model was f i t t e d t o t h e d a t a , as summarized i n T a b l e 2. TABLE 2 - A Summary of P a r a m e t e r s f o r t h e F o u r A n i s o t r o p i c s e m i v a r i o g r a m M o d e l s D e v e l o p e d . MODEL NUGGET S I L L RANGE RANGE TYPE 1 CO C 22.5 112.5 m e t r e s m e t r e s T r a d i t i o n a l 2 4.13 m 2 8.26 m 2 22 1 0 Log 0.75 2.25 30 1 2 P a r t i t i o n e d 0.80 3.10 60 1 2 Log P a r t i t i o n e d 0.45 1 .55 33 1 5 1 See t e x t f o r d e t a i l s o f e a c h model t y p e 2 m i s t h e mean v a l u e o f d a t a t o w h i c h model i s a p p l i e d 90 AVERAGE SEMI-VARIOGRAM (5 BENCHES) PARTITIONED LOGNORMAL SILVER 0 f1 1 1 1 1 ' 11 i i i 11 i 1111 i i i 1 1 111111i 11111 i 1 1 i 111 i 111 11 0 10 20 30 40 50 LAG h (metres) F i g u r e 6 : A v e r a g e h o r i z o n t a l s e m i v a r i o g r a m s i n t h e d i r e c t i o n o f minimum a n d maximum s t r u c t u r e f o r p a r t i t i o n e d l o g n o r m a l s i l v e r g r a d e s , S o u t h e r n T a i l z o n e . 91 M u l t i - G a u s s i a n Model S t i l l a n o t h e r a l t e r n a t i v e , i s t o t r a n s f o r m t h e raw d a t a t o a G a u s s i a n ( n o r m a l ) d i s t r i b u t i o n by a g r a p h i c a l t r a n s f o r m a t i o n ( V e r l y , 1983). T h i s method would a c c o u n t f o r t h e h i g h g r a d e t a i l o f t h e d i s t r i b u t i o n by b r i n g i n g i t i n t o w a r d t h e mean and m i n i m i z e the- e f f e c t s of e r r a t i c m i n e r a l i z a t i o n on s t r u c t u r a l m o d e l l i n g and l o c a l e s t i m a t i o n . The p r o c e d u r e , however, r e q u i r e s a s t r i c t s t a t i o n a r i t y of t h e v a r i a b l e s i n q u e s t i o n ( V e r l e y 1983), a r e s t r a i n t t h a t d o e s not e x i s t i n E q u i t y d a t a . A model was d e v e l o p e d , t e s t e d and f o u n d t o be u n a c c e p t a b l e . An a d d e d c o n s e q u e n c e of a p p l y i n g a G a u s s i a n t r a n s f o r m a t i o n t o a m u l t i n o r m a l d i s t r i b u t i o n , a s o c c u r s i n t h e S o u t h e r n T a i l z o n e , was o b s e r v e d . When t h e t a i l of t h e d i s t r i b u t i o n i s b r o u g h t i n t o w a r d s t h e mean, an upper p o p u l a t i o n can be t o t a l l y l o s t i n t h e t r a n s f o r m a t i o n . 92 BACK ANALYSIS To t r y and e v a l u a t e t h e r e l a t i v e q u a l i t y o f t h e d i f f e r e n t m o d e l s summarized i n T a b l e 2, a method of b a c k - a n a l y s i s on 1310N b e n ch was u s e d . The method c o n s i s t s of r e m o v i n g a s i n g l e v a l u e and t h e n e s t i m a t i n g t h e removed p o i n t by k r i g i n g u s i n g one a u r e o l e o f t h e s u r r o u n d i n g d a t a . E a c h of 1723 b l a s t h o l e samples was c a l c u l a t e d i n t h i s manner. The upper p o p u l a t i o n ( > 340 g A g/tonne ) was removed from t h e d a t a f o r t h i s t e s t t o a l l o w a f a i r c o m p a r i s o n of t h e d i f f e r e n t methods, b e c a u s e two of t h e models were not a p p l i c a b l e t o t h e upper p o p u l a t i o n . I n v e r s e d i s t a n c e s q u a r e d was a l s o u s e d t o e s t i m a t e e a c h p o i n t . The r e s u l t s f o r e a c h of t h e f i v e methods showed a s i g n i f i c a n t , b i a s i n e s t i m a t i n g p o i n t s a m p l e s , w i t h a l l t e c h n i q u e s u n d e r e s t i m a t i n g h i g h g r a d e and o v e r e s t i m a t i n g low g r a d e . T h i s i s not u n e x p e c t e d a s , i n a r e a s w i t h s c a t t e r e d i s o l a t e d h i g h v a l u e s , i t i s n o t p o s s i b l e t o e s t i m a t e t h e h i g h sample from t h e s u r r o u n d i n g low v a l u e s . C o n v e r s e l y , f o r e a c h e s t i m a t e of a low v a l u e i n t h e n e i g h b o r h o o d of a h i g h v a l u e , any form of w e i g h t i n g w i l l c a u s e an o v e r e s t i m a t e . R e s u l t s f o r t h e l o g p a r t i t i o n e d model a r e shown t o i l l u s t r a t e t h i s b i a s ( F i g u r e 7 ) . To e v a l u a t e t h e d i f f e r e n t methods, i t was t h e r e f o r e d e c i d e d t o use mean s q u a r e d d i f f e r e n c e a s a measure o f t h e r e l a t i v e d i s p e r s i o n o f " e s t i m a t e d " f r o m " t r u e " v a l u e s . In an e f f o r t t o a c c o u n t f o r t h e known b i a s , e a c h method was e v a l u a t e d s e p a r a t e l y i n t h r e e a s s a y range c a t e g o r i e s : l e s s t h a n 20 g A g / t o n n e , 20 t o 100 g A g / t o n n e and g r e a t e r t h a n 100 g A g / t o n n e . The r e s u l t s shown i n T a b l e 3 a r e f a r f r o m c o n c l u s i v e , but t h e l o g 93 •260.00 LU CT O 210.00 _ 140.00 _ 70.00 N = I5I0 .00 .00 70.00 140.00 210.00 280.00 350.00 R E A L RG V A L U E E Q U I T Y P O I N T K R I G I N G F i g u r e 7 : S c a t t e r p l o t s howing r e a l s i l v e r g rade v e r s u s s i l v e r g r ade e s t i m a t e d by l o g p a r t i t i o n e d k r i g i n g f o r 1510 b l a s t h o l e s . S o u t h e r n T a i l zone. 94 p a r t i t i o n e d model g i v e s t h e l e a s t s c a t t e r i n t h e range of 20 t o 100 g Ag/ t o n n e where b l o c k s e l e c t i o n i s most c r i t i c a l b e c a u s e t h e range i n c l u d e s a l l c u t - o f f g r a d e s f o r t h e f i r s t y e a r s of p r o d u c t i o n . TABLE 3 - Mean S q u a r e d D i f f e r e n c e s of E s t i m a t e d and T r u e B l a s t h o l e G r a d e s as a F u n c t i o n of 3 G r a d e C a t e g o r i e s f o r 5 E s t i m a t i o n P r o c e d u r e s . POINT ESTIMATION MEAN SQUARED DIFFERENCE METHOD VALUE<20 20<VALUE<100 VALUE>100 I n v e r s e D i s t . S q . 22.00 37.09 122.76 T r a d i t i o n a l K r . 24.47 36.73 125.87 Log K r i g i n g 1 5.00 35.47 146.79 P a r t i t i o n e d K r . 27. 1 7 39.37 126.54 Log P a r t . K r . 1 4.22 34.77 146.03 The back e s t i m a t i o n method o f e v a l u a t i n g a s e m i v a r i o g r a m model i s o f l i m i t e d use i n t h i s s i t u a t i o n , where f o r e a c h p o i n t e s t i m a t e d , t h e a v a i l a b l e d a t a w i t h which t o make t h e e s t i m a t e a r e f a i r l y r e g u l a r l y d i s t r i b u t e d . C o n s e q u e n t l y , w e i g h t i n g f a c t o r s a s s i g n e d t o t h e s a m p l e s c a n n o t be v e r y d i f f e r e n t , r e g a r d l e s s o f w h i c h k r i g i n g p r o c e d u r e i s u s e d . I t i s of i n t e r e s t t o n o t e t h a t i n v e r s e d i s t a n c e s q u a r e d a s a p o i n t e s t i m a t o r i s c o m p a r a b l e i n q u a l i t y t o v a r i o u s k r i g i n g p r o c e d u r e s , d e s p i t e t h e 95 f a c t t h a t i n v e r s e d i s t a n c e i g n o r e s t h e p r e s e n c e of a n i s o t r o p y . T h i s . u n i f o r m i t y o f q u a l i t y of e s t i m a t i o n c e r t a i n l y does not e x t e n d t o b l o c k e s t i m a t i o n . Any improvement b r o u g h t a b o u t by k r i g i n g c an o n l y be s u r m i s e d q u a l i t a t i v e l y u n t i l s p e c i f i c b l o c k s of o r e a r e f o l l o w e d t h r o u g h t h e m i l l and e s t i m a t e s can be compared w i t h p r o d u c t i o n . We a n t i c i p a t e d t h a t t h e p a r t i t i o n i n g a p p r o a c h on l o g t r a n s f o r m e d d a t a would p r o v i d e a c l e a r improvement i n t h e e s t i m a t i o n o f b l o c k s n e a r t h e c u t - o f f . T h i s was not t h e c a s e a l t h o u g h t h e method seems t o be m a r g i n a l l y b e t t e r t h a n e i t h e r u s i n g a t r a d i t i o n a l k r i g i n g w i t h a p r o p o r t i o n a l e f f e c t model o r l o g k r i g i n g . 9 6 BLOCK KRIGING As a f u r t h e r t e s t of t h e q u a l i t y of v a r i o u s s t r u c t u r a l m o d e l s , a s e c t i o n of 1310N b e n c h was s e l e c t e d f o r b l o c k e s t i m a t i o n . One h u n d r e d 5 x 5 x 5 m 3 b l o c k s were e s t i m a t e d by k r i g i n g u s i n g e a c h of t h e f o u r models of T a b l e 2. The g r i d and a l l p e r t i n e n t b l a s t h o l e i n f o r m a t i o n a r e shown on F i g u r e 8. The e r r a t i c n a t u r e of t h e d e p o s i t i s w e l l d e m o n s t r a t e d i n t h i s d a t a s e t w i t h b l a s t h o l e v a l u e s r a n g i n g from 0.5 t o 1820 g A g / t o n n e . The t r a d i t i o n a l and l o g m o d els u s e d a l l a v a i l a b l e d a t a i n a s i n g l e s u r r o u n d i n g a u r e o l e t o e s t i m a t e i n d i v i d u a l b l o c k g r a d e s . The two p a r t i t i o n e d m o d e l s , on t h e o t h e r hand, i g n o r e v a l u e s from th e upper (A) p o p u l a t i o n , i . e . b l a s t h o l e s a s s a y i n g g r e a t e r t h a n 340 g A g / t o n n e . I f a b l o c k c o n t a i n e d a v a l u e g r e a t e r t h a n 340 g Ag/tonne i t was not e s t i m a t e d by k r i g i n g . I n s t e a d , t h e a v e r a g e g r a d e of t h e A p o p u l a t i o n (630 g A g / t o n n e ) was a s s i g n e d t o t h e b l o c k , t h e r a t i o n a l b e i n g t h a t s u c h b l o c k e r r o r s t h o u g h c o n s i d e r a b l e , would compensate o v e r a l a r g e number of b l o c k s . The s e c o n d c a s e , i n v o l v i n g h i g h g r a d e s a m p l e s , o c c u r s where one o r more samples from A p o p u l a t i o n were s e l e c t e d w i t h i n t h e s e a r c h r a d i u s s u r r o u n d i n g a b l o c k t o be e s t i m a t e d . In t h i s s i t u a t i o n , r a t h e r t h a n c o m p l e t e l y i g n o r e t h e h i g h s a m p l e s , t h e l o c a l a v e r a g e of t h e r e m a i n i n g p o i n t s w i t h i n t h e s e a r c h r a d i u s was a s s i g n e d t o t h e h i g h g r a d e b l o c k . U n d o u b t e d l y , t h o s e b l o c k s would s t i l l c o n t a i n B and C p o p u l a t i o n m a t e r i a l i n a d d i t i o n t o A p o p u l a t i o n s i n c e t h e m a s s i v e g a l e n a - s p h a l e r i t e l e n s e s a r e v e r y t h i n r e l a t i v e t o b l o c k d i m e n s i o n s . The v a r i o u s r e s u l t s f r o m b l o c k k r i g i n g a r e d i f f i c u l t t o e v a l u a t e w i t h o u t t h e b e n e f i t o f a c o r r e c t v a l u e w i t h which t o 97 o ID i O o OJ 2 3 •3 8360 E ;838QE 840QE 5 9 •85 '17 > I 6 .14 - 5 ' i s * 3 *l 5 -4 .134 ' 1 0 0 - 9 . 2 5 *I8 *883 -II *2I •I • 2 .5 4*9 23*1 3 5 3 2 6 6 18 4 0 3 4 15 'A V •|49 *88 2*41 5 7 15*8 118 14 106 • 3 *2 ' 5 3 9 7 8 5 54 8 6 •9 58 •fe- *3 11 8 72 31 9 52 2*0 1*5 • -V *I0 73 11 81 •-4.6E • 18 20 3 5 "3 : " '3 : j *I0 .250 :6l;3 : 5 81 1677 27 •2 •3 II *3 4 9 4 2 8 7*1 17 • 210 10 92 "is 1 1 I 2 9 5 2 8*2 4 5 II. 137 97 8 I 13 4 9 *I2 12 13 *2 33 13b •4 4 9 •5 18 87 *6 II 4*0 55 8 4 '15 -4 236 ' 2 0 7 119 "74 *4 'I i |55 4 4 ' 7 ' 3 -193 •20*102 F i g u r e 8 : T e s t a r e a o f 1310 N b e n c h s h o w i n g s i l v e r a s s a y s f o r b l a s t h o l e s w i t h an a r b i t r a r y g r i d o f 100 ( 5 x 5 x 5 c u . m.) b l o c k s s u p e r i m p o s e d . 98 compare b l o c k e s t i m a t e s . The one common f a c t o r t h a t can be compared, however, i s t h e k r i g i n g e r r o r . E r r o r s f o r an a r b i t r a r y s e l e c t i o n of 25 b l o c k s of t h e 100 e s t i m a t e d , r e p r e s e n t i n g t h e e n t i r e r a n g e of s i l v e r v a l u e s p r e s e n t , a r e shown as F i g u r e 9. The g r a p h shows r e l a t i v e k r i g i n g e r r o r s as p r o p o r t i o n s of t h e c o r r e s p o n d i n g k r i g e d e s t i m a t e s f o r t h e f u l l r a n g e of s i l v e r v a l u e s e n c o u n t e r e d . The r e s u l t s i n d i c a t e t r a d i t i o n a l k r i g i n g t o have t h e h i g h e s t r e l a t i v e e r r o r s , r a n g i n g f r o m 50 t o 175 p e r c e n t . L og t r a n s f o r m a t i o n of t h e d a t a o r p a r t i t i o n i n g and re m o v i n g t h e h i g h p o p u l a t i o n b o t h improve r e l a t i v e k r i g i n g e r r o r s t o ab o u t t h e same e x t e n t (50 t o 60 p e r c e n t ) . The l o g p a r t i t i o n e d model g i v e s t h e l o w e s t r e l a t i v e e s t i m a t i o n e r r o r s , r a n g i n g f r o m 38 t o 46 p e r c e n t . The t e s t a r e a showing t h e 100 b l o c k s . e s t i m a t e d by l o g p a r t i t i o n e d k r i g i n g i s shown a s F i g u r e 10. A p p l y i n g a 30 g Ag/tonne c u t - o f f t o t h e s e r e s u l t s w o u l d s e l e c t 43 b l o c k s o r 14,700 t o n n e s o f m a t e r i a l a v e r a g i n g 145 g A g / t o n n e . I f c e n t r a l l y l o c a t e d b l a s t h o l e s were u s e d f o r m i n i n g s e l e c t i o n , 46 b l o c k s o r 15,700 t o n n e s of m a t e r i a l , a p p a r e n t l y a v e r a g i n g 225 g Ag / t o n n e , would be p r e d i c t e d ( F i g u r e 8 ) . I f one compares t h e d i f f e r e n c e s i n m e t a l c o n t e n t between t h e s e two methods, c e n t r a l d r i l l h o l e e s t i m a t i o n p r e d i c t s an a d d i t i o n a l 1,403,215 grams of s i l v e r w o r t h a b o u t $ 415,000.00 ( b a s e d on 1 o z . Ag =U.S.$ 9.00 ). S i n c e i t i s i m p o s s i b l e t o p r o v e w h i c h e s t i m a t i o n p r o c e d u r e i s , on a v e r a g e , c l o s e s t t o r e a l i t y , w i t h o u t t r a c i n g t h e e s t i m a t e d m a t e r i a l t h r o u g h t h e m i l l , t h e f i n a l c o n c l u s i o n f o r t h e b e s t e s t i m a t i o n t e c h n i q u e r e m a i n s u n r e s o l v e d . I f however, 99 F i g u r e 9 : G r a p h s h o w i n g r e l a t i v e k r i g i n g e r r o r ( u s i n g f o u r d i f f e r e n t k r i g i n g m o d e l s ) f o r 25 a r b i t r a r i l y s e l e c t e d b l o c k s f r o m t h e 100 e s t i m a t e d . 100 BLOCKS WITH >30g./tonne Ag. AVERAGE GRADE =14 5 g./tonne Ag. F i g u r e 10 : T e s t a r e a o f 1310 N b e n c h s h o w i n g k r i g e d s i l v e r g r a d e f o r 100 ( 5 x 5 x 5 cu.m.) b l o c k s . K r i g e d s i l v e r g r a d e c a l c u l a t e d u s i n g l o g p a r t i t i o n e d m o d e l . 101 one c o n s u l t s s t u d i e s on s i m u l a t e d d e p o s i t s , where t h e c o r r e c t answer i s known, ( J o u r n e l and H u i j b r e g t s , 1981) and u t i l i z e methods a s s i g n i n g t h e g r a d e of c e n t r a l l y l o c a t e d d a t a t o a volume of m a t e r i a l ( p o l y g o n a l e s t i m a t e s ) , t h e so c a l l e d r e g r e s s i o n e f f e c t p r e v a i l s and c o n s i s t a n t o v e r e s t i m a t i o n o f h i g h g r a d e b l o c k s r e s u l t s . Numerous o t h e r s t u d i e s e x i s t i n w h i c h k r i g i n g p r o c e d u r e s a r e shown t o g i v e u n b i a s e d b l o c k e s t i m a t o r s r e l a t i v e t o p r o d u c t i o n ( e . g . R o y l e and K h o s r o w s h a h i , 1983). C o n s e q u e n t l y , one must c o n c l u d e t h a t t h e use of p o l y g o n a l e s t i m a t i o n p r o c e d u r e s , i n w h i c h c e n t r a l l y l o c a t e d d a t a a r e a s s i g n e d t o an e n c o m p a s s i n g b l o c k , r e s u l t i n s e r i o u s e s t i m a t i o n p r o b l e m s 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 body s u c h as t h e S o u t h e r n T a i l z o n e . P r a c t i c e e l s e w h e r e and t h e o r y s u g g e s t t h a t k r i g i n g , an u n b i a s e d e s t i m a t i o n p r o c e d u r e , r e p r e s e n t s a s i g n i f i c a n t improvement i n q u a l i t y of e s t i m a t i o n o v e r p o l y g o n a l methods. However, w i t h o u t knowledge o f t h e t r u e g r a d e of some b l o c k s , t h e e x t e n t o f improvement r e m a i n s u n c e r t a i n . In t h e c a s e o f t h e S o u t h e r n T a i l zone a l a r g e g r o u p of b l o c k s must be f o l l o w e d t h r o u g h t h e m i l l so t h a t p r o d u c t i o n can be compared w i t h e s t i m a t e s . Such t e s t s a r e d i f f i c u l t i n p r a c t i c e , b e c a u s e even o v e r s h o r t t i m e i n t e r v a l s o r e f o r m i l l i n g i s d e r i v e d f r o m s e v e r a l p i t s i t e s , as w e l l a s f r o m s t o c k p i l e d m a t e r i a l . 102 CONCLUSIONS The S o u t h e r n T a i l zone p r o v i d e s a p r a c t i c a l c a s e h i s t o r y t o compare s e v e r a l d i f f e r e n t b l o c k k r i g i n g t e c h n i q u e s . M o d e l s b a s e d on a t r a d i t i o n a l p r o p o r t i o n a l e f f e c t , a l o g t r a n s f o r m , a p a r t i t i o n e d d a t a s e t and a l o g t r a n s f o r m e d p a r t i t i o n e d d a t a s e t were d e v i s e d and t e s t e d . C r o s s v a l i d a t i o n p r o c e d u r e s of e s t i m a t i n g i n d i v i d u a l b l a s t h o l e a s s a y s were i n c o n c l u s i v e due t o a c o m b i n a t i o n of h i g h l y e r r a t i c sample v a l u e s and t h e f a i r l y r e g u l a r s p a c i n g o f d a t a . L o g n o r m a l p a r t i t i o n e d k r i g i n g p r o d u c e d m a r g i n a l l y b e t t e r r e s u l t s i n t h e c r i t i c a l r a n g e of v a l u e s n e a r c u t - o f f . C o m p a r i s o n s of k r i g i n g r e s u l t s f o r t h e d i f f e r e n t methods showed l o g n o r m a l p a r t i t i o n e d k r i g i n g t o p r o d u c e t h e l o w e s t e s t i m a t i o n e r r o r s . F o r any v a l i d c o m p a r i s o n , however, i t would be n e c e s s a r y t o m o n i t o r a c t u a l b l o c k g r a d e s t h r o u g h t h e m i l l . A t e s t s e t o f 100 b l o c k s were e s t i m a t e d by s e v e r a l t e c h n i q u e s w i t h p o l y g o n a l e s t i m a t e s p r e d i c t i n g 1,400,000 grams of s i l v e r more than l o g p a r t i t i o n e d k r i g i n g . The m a g n i t u d e of t h e d i f f e r e n c e i n d i c a t e s , a g a i n , t h e need t o know m e t a l c o n t e n t s of a p a r t o f a d e p o s i t e a r l y i n t h e l i f e of a mine so t h e b e s t l o c a l e s t i m a t o r can be s e l e c t e d . 1 03 ACKNOWLEDGEMENTS T h i s s t u d y was i n i t i a t e d a t t h e s u g g e s t i o n of Dr. N. H i l l h o u s e and Mr. E.T. L o n e r g a n of P l a c e r Development L t d . Computer s u p p o r t has been p r o v i d e d by t h e computer g r o u p a t P l a c e r and p a r t i c u l a r t h a n k s a r e e x t e n d e d t o B i l l G r e e n . T e c h n i c a l s u p p o r t and a d v i c e was p r o v i d e d by t h e E n g i n e e r i n g D epartment of E q u i t y M i n e s L i m i t e d and i n p a r t i c u l a r J.H.L. M i l l e r and J.B. C y r . F i n a n c i a l s u p p o r t was p r o v i d e d by P l a c e r D evelopment L t d . 104 REFERENCES G i r o u x , G.H. and A . J . S i n c l a i r , (1984a) G l o b a l R e s e r v e s o f S o u t h e r n T a i l zone by C o n d i t i o n a l P r o b a b i l i t y ; W e s t e r n M i n e r ( i n p r e p a r a t i o n ) . G i r o u x , G.H. and A . J . S i n c l a i r , (1984b) P r o d u c t i o n q u a l i t y c o n t r o l e x p e r i m e n t s ; W e s t e r n M i n e r ( i n p r e p a r a t i o n ) J o u r n e l , A.G. and Ch. H u i j b r e g t s (1981) M i n i n g g e o s t a t i s t i c s ; A cademic P r e s s ( I n c . ) London L t d . , 600 p. R o y l e , A.G. and S. K h o s r o w s h a k i , (1983) V a l u a t i o n o f a l l u v i a l t i n and g o l d d e p o s i t s ; T r a n s . I n s t . M i n . and M e t a l l . ( S e c . A, M i n e r a l I n d u s t r y ) , v o l . 92, pp. A13-A18. S i n c l a i r , A . J . and G.H. G i r o u x , (1983) G e o l o g i c a l c o n t r o l s of s e m i - v a r i o g r a m s i n p r e c i o u s m e t a l d e p o s i t s ; G e o s t a t i s t i c s f o r N a t u r a l R e s o u r c e s C h a r a c t e r i z a t i o n , NATO ASI S e r i e s , D . R e i d e l P u b l i s h i n g Company, D o r d r e c h t , H o l l a n d , pp. 965-978. V e r l y , G. (1983) The m u l t i g a u s s i a n a p p r o a c h and i t s a p p l i c a t i o n s t o t h e e s t i m a t i o n s of l o c a l r e s e r v e s ; J o u r n a l o f M a t h e m a t i c a l G e o l o g y , V o l . 15, No.2, pp. 263-290. W e t h e r e l l , D. (1979) G e o l o g y and Ore G e n e s i s o f t h e Sam G o o s l e y c o p p e r - s i l v e r - a n t i m o n y D e p o s i t , U n p u b l i s h e d MSc T h e s i s 1979, D e p t . o f G e o l o g i c S c i e n c e s , U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , 208 p. CHAPTER 5 CONCLUSIONS 1 06 G e o s t a t i s t i c a l p r o c e d u r e s have been a p p l i e d a t v a r i o u s s t a g e s i n t h e d e v e l o p m e n t of E q u i t y S i l v e r M i n e s ' S o u t h e r n T a i l zone as an a i d t o h a n d l i n g p r o b l e m s c a u s e d by t h e e r r a t i c n a t u r e of t h e s i l v e r g r a d e s . The s t u d y i s d i v i d e d i n t o t h r e e p a r t s : an e s t i m a t i o n o f g l o b a l r e s e r v e s ; an e x a m i n a t i o n of q u a l i t y c o n t r o l i n b l a s t h o l e s a m p l i n g and a t e s t of d i f f e r e n t g e o s t a t i s t i c a l p r o c e d u r e s t o d e t e r m i n e l o c a l b l o c k e s t i m a t i o n s . C o n d i t i o n a l p r o b a b i l i t y p r o v i d e s a g e o l o g i c o r e r e s e r v e e s t i m a t e f o r t h e S o u t h e r n T a i l zone w i t h t h e f o l l o w i n g b e n e f i t s : 1) t h e g e o l o g i s t most f a m i l i a r w i t h t h e d e p o s i t c h o o s e s t h e m i n e r a l i z e d volume t o be e s t i m a t e d 2) i r r e g u l a r d r i l l h o l e p a t t e r n s t h a t c o u l d l e a d t o a b i a s i n r e s e r v e s a r e a c c o u n t e d f o r . 3) no a s s u m p t i o n s on t h e form o f t h e g r a d e d i s t r i b u t i o n s a r e made 4) t h e g r a d e d i s t r i b u t i o n i s c o n d i t i o n e d t o t h e m i n i n g s u p p o r t and t h e r e f o r e o v e r e s t i m a t i o n s of g r a d e and t o n n a g e s a r e a v o i d e d 5) a g r a d e t o n n a g e c u r v e f o r t h e m i n e r a l i z e d volume w h i c h i s e x t r e m e l y u s e f u l i n r e s o u r c e p l a n n i n g i s p r o d u c e d . Two e s t i m a t e s f o r i n s i t u s i l v e r r e s e r v e s f o r t h e S o u t h e r n T a i l zone were made. The r e s u l t s s t r a d d l e d t h e to n n a g e i n d i c a t e d by i n v e r s e d i s t a n c e and an a v e r a g e g r a d e o f 125 g Ag/tonne a t a 40 g A g / t o n n e c u t - o f f was p r e d i c t e d . The i m p o r t a n c e of p r e c i s i o n and a c c u r a c y i n s a m p l i n g a r e d e a l t w i t h i n C h a p t e r 3. A s i m p l e s t a t i s t i c a l and g r a p h i c a l a p p r o a c h f o r h a n d l i n g d u p l i c a t e a n a l y s i s i s p r e s e n t e d w i t h 107 e m p h a s i s p l a c e d on b o t h i d e n t i f y i n g and q u a n t i f y i n g s a m p l i n g e r r o r s . A p p l y i n g t h e s e p r o c e d u r e s t o s e v e r a l s e t s of d u p l i c a t e d a t a from t h e S o u t h e r n T a i l zone r e s u l t e d i n : 1) f i n d i n g a s m a l l but s i g n i f i c a n t b i a s i n a n a l y t i c a l r e s u l t s f o r s i l v e r by one of t h e two l a b o r a t o r i e s u s e d 2) showing a l a r g e random e r r o r f o r s i l v e r a s s a y s i n d u p l i c a t e b l a s t h o l e samples c o l l e c t e d by t h e method o f t u b e s a m p l i n g 3) p r o v i n g t h a t t h e random s a m p l i n g e r r o r c o u l d be r e d u c e d by i n c r e a s i n g t h e s i z e o f t h e sample 4) showing t h a t w h i l e c o l l e c t i n g l a r g e r s amples would i n c r e a s e t h e c o s t of s a m p l i n g , t h e n e t f i n a n c i a l b e n e f i t s a t t a i n e d by l o w e r i n g t h e s a m p l i n g v a r i a b i l i t y c o u l d be i n e x c e s s of 1 m i l l i o n d o l l a r s p e r y e a r . F i n a l l y a v a r i e t y of g e o s t a t i s t i c a l a p p r o a c h e s f o r l o c a l b l o c k e s t i m a t i o n have been t e s t e d on b l a s t h o l e d a t a f o r t h e S o u t h e r n T a i l z o n e . M o d e l s b a s e d on a t r a d i t i o n a l p r o p o r t i o n a l e f f e c t , a l o g t r a n s f o r m , a p a r t i t i o n e d d a t a s e t and a l o g t r a n s f o r m e d p a r t i t i o n e d d a t a s e t were d e v i s e d and t e s t e d a g a i n s t e m p i r i c a l methods. A d e t a i l e d s t r u c t u r a l a n a l y s i s of t h e 1310 bench e m p h a s i z e d t h e need f o r good g e o l o g i c c o n t r o l . When t h e b e n c h was d i v i d e d i n t o two domains by g e o l o g i c mapping two d i s t i n c t l y d i f f e r e n t s e m i v a r i o g r a m m o d els were o b t a i n e d . C r o s s v a l i d a t i o n p r o d e c u r e s f o r e s t i m a t i n g i n d i v i d u a l a s s a y s f r o m s u r r o u n d i n g d a t a were i n c o n c l u s i v e , p r o b a b l y due t o 108 a c o m b i n a t i o n of e r r a t i c sample a s s a y s and t h e f a i r l y r e g u l a r s p a c i n g of b l a s t h o l e d a t a . A c o m p a r i s o n of k r i g i n g r e s u l t s f o r t h e v a r i o u s models showed l o g n o r m a l p a r t i t i o n e d k r i g i n g t o p r o d u c e t h e l o w e s t e s t i m a t i o n e r r o r s . The r e s u l t s of e s t i m a t i n g 100 b l o c k s , u s i n g t h e d i f f e r e n t t e c h n i q u e s d e v e l o p e d , showed a wide d e s c r e p a n c y w i t h p o l y g o n a l e s t i m a t e s i n d i c a t i n g 1.4 m i l l i o n grams of s i l v e r more t h a n l o g n o r m a l p a r t i t i o n e d k r i g i n g . T h i s p o i n t s t o t h e i m p o r t a n c e o f m o n i t e r i n g grade c o n t e n t of m i n i n g b l o c k s e a r l y i n t h e l i f e o f a mine so t h a t t h e t r u e m e t a l c o n t e n t i s known and t h e b e s t , most c o s t e f f e c t i v e , e s t i m a t i o n t e c h n i q u e can be s e l e c t e d . 109 REFERENCES C y r , J . , J . M i l l e r (1980) B l a s t h o l e S a m p l i n g P r o c e d u r e S t u d y , P r i v a t e r e p o r t , E q u i t y S i l v e r M i n e s L t d . , November, 11 p. Ewanchuck, H. (1968) Grade C o n t r o l a t B e t h l e h e m C o p p e r , Can. I n s t . M i n . M e t a l l . S p e c . V o l . 9, pp. 302-307. G i r o u x , G.H. and A . J . S i n c l a i r , (1984a) G l o b a l R e s e r v e s of S o u t h e r n T a i l zone by C o n d i t i o n a l P r o b a b i l i t y ; W e s t e r n M i n e r ( i n p r e s s ) . G i r o u x , G.H. and A . J . S i n c l a i r , (1984b) P r o d u c t i o n q u a l i t y c o n t r o l e x p e r i m e n t s ; W e s t e r n M i n e r ( i n p r e p a r a t i o n ) . J o u r n e l , A.G. and Ch. H u i j b r e g t s (1981) M i n i n g g e o s t a t i s t i c s ; A c a demic P r e s s ( I n c . ) London L t d . , 600 p. J o u r n e l , A.G. (1983) N o n - p a r a m e t r i c e s t i m a t i o n of s p a c i a l d i s t r i b u t i o n s ; J o u r . I n t . A s s o c . Math. G e o l . v.15, no. 3., p. 445-468 M i l l e r , J . (1981) B l a s t h o l e S a m p l i n g P r o c e d u r e , P r i v a t e r e p o r t E q u i t y S i l v e r M i n e s L t d . , May, 5 p. Ney, C.S., J.M. A n d e r s o n and A. P a n t e l e y e v (1972) D i s c o v e r y , g e o l o g i c s e t t i n g and s t y l e o f m i n e r a l i z a t i o n , Sam G o o s l y d e p o s i t , B r i t i s h C o l u m b i a ; I n s t . M i n . M e t a l l . B u l l . v. 65, no. 723, p 53-64. Raymond, G., (1979) Ore e s t i m a t i o n p r o b l e m s 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 o r e body, Can. I n s t . M i n . M e t a l l . B u l l e t i n , J u n e , pp. 90-97. R o y l e , A.G. and S. K h o s r o w s h a h i , ( 1 9 8 3 ) , V a l u a t i o n of a l l u v i a l t i n and g o l d d e p o s i t s ; T r a n s . I n s t . M i n . and M e t a l l . ( S e c . A, M i n e r a l I n d u s t r y ) , v o l 92, pp. A13-A18. S i n c l a i r , A . J . (1978) S a m p l i n g a m i n e r a l d e p o s i t f o r f e a s i b i l i t y s t u d i e s and m e t a l l u r g i c a l t e s t i n g ; i n M u l a r , A.L. and R.B. Bhappu, M i n e r a l P r o c e s s i n g P l a n t D e s i g n ; Am. I n s t . M i n . Eng. New Y o r k , p 115-134. S i n c l a i r , A . J . and G.H. G r i o u x , (1983) G e o l o g i c a l c o n t r o l s of s e m i - v a r i o g r a m s i n p r e c i o u s m e t a l d e p o s i t s ; G e o s t a t i s t i c s f o r N a t u r a l R e s o u r c e s C h a r a c t e r i z a t i o n , NATO ASI S e r i e s , D. R e i d e l P u b l i s h i n g Company, D o r d r e c h t , H o l l a n d , pp. 965-978. V e r l y , G. (1983) The m u l t i g a u s s i a n a p p r o a c h and i t s a p p l i c a t i o n s t o t h e e s t i m a t i o n s of l o c a l r e s e r v e s ; J o u r n a l of M a t h e m a t i c a l G e o l o g y , V o l . 15, No.2, pp. 263-290. n o W e t h e r e l l , D.G (1979) G e o l o g y and Ore G e n e s i s of t h e Sam G o o s l e y c o p p e r - s i l v e r - a n t i m o n y D e p o s i t , U n p u b l i s h e d MSc T h e s i s , D e p t . o f G e o l o g i c a l S c i e n c e s , U n i v e r s i t y of B.C., V a n c o u v e r B.C., 208 p. W e t h e r e l l , D.G., A . J . S i n c l a i r and T.G. S h r o e t e r , (1979) P r e l i m i n a r y r e p o r t on t h e Sam G o o s l y c o p p e r - s i l v e r d e p o s i t ; B.C. M i n i s t r y of E n e r g y , M i n e s and P e t r o l e u m R e s o u r c e s , Paper 1979-1, pp. 132-137. 111 APPENDIX I COMPARISON OF EQUITY LAB AND A CHECK LAB ON MILL SAMPLE ASSAYS 1 1 2 SAMPLE NUMBER CU CU AG AG CHECK EQUITY CHECK EQUITY T a i l 6 g . 1 2 . 1 2 44.0 54.0 T a i l 6d .09 .10 36.0 43.7 T a i l 8g . 12 .12 38.0 44.0 T a i l 8a .09 .08 31.0 35.9 Ta i 1 1 1 g .09 .10 52.0 56.9 T a i l 14g .07 .08 28.0 29.0 T a i l 14d . 1 2 . 1 2 53.0 57.5 CoF 6g .24 .25 86.0 97.0 CoF 6d .26 .26 82.0 90.9 CoF 8g .29 .29 82.0 94.0 CoF 8a .33 .34 98.0 1 07.0 CoF 11g .19 .19 98.0 100.0 CoF 14g .21 .22 88.0 67.0 CoF 14d .23 .23 93.0 98. 1 CoF 14a .29 .24 1 04.0 104.0 Cone 6g 14.8 15.5 4840 571 5 Cone 8g 17.8 18.8 4800 5440 Cone 8a 19.4 19.4 5446 631 0 Cone 1 1 g 20. 1 18.9 9040 851 5 Cone 14g 18.2 19.0 7480 7735 Cone 14d 18.1 18.2 7200 7285 R/Gcon a 3.36 3.30 1 340 1 361 Cone 2d 25.2 26.7 7360 7800 Cone 15n 18.2 18.4 5920 6286 APPENDIX II DUPLICATE BLASTHOLE SAMPLING FOR COPPER AND SILVER 1 1 4 ORIGINAL CHECK DI F F . ORIGINAL CHECK DI F F . CU CU CU AG AG AG .17 . 1 5 .02 1 67 1 1 7 50 .07 .07 .00 85 73 1 2 .09 . 1 6 .07 61 1 24 63 .04 .06 .02 43 35 8 .06 .06 .00 53 54 1 .31 .34 .03 53 62 9 .22 .17 .05 77 48 29 .30 .23 .07 62 49 13 .51 .54 .03 40 22 18 .33 .48 .15 87 1 29 42 .03 .03 .00 7 8 1 .05 .02 .03 5 7 2 .02 .02 .00 17 10 7 .03 .02 .01 38 50 1 2 .08 .07 .01 1 1 8 3 .05 .05 .00 7 7 0 .04 .03 .01 24 10 14 .02 .02 .00 8 9 1 .07 .02 .05 1 30 32 98 .12 .06 .06 58 40 18 .03 .02 .01 54 43 1 1 M e a n . d i f f e r e n c e Cu = .0295 % S = .0364 % Mean d i f f e r e n c e Ag = 19.62 g/tonne S = 24.73 g/tonne APPENDIX I I I RESULTS FOR DIFFERENT SAMPLING TECHNIQUES FOR 42 BLOCKS EQUITY SOUTH T A I L ZONE D U P L I C A T E A N A L Y S I S O R I G I N A L T U B E C H A N N E L 4 CHANNEL 1 CHANNEL 2 • H CU AG H CU AG H CU AG # CU AG 4 6 8 4 0 . 1 4 5 . 3 0 5 0 . 2 1 1 2 . 3 0 2 0 . 2 1 1 5 . 3 0 3 0 . 3 0 21 . 1 1 3 0 7 0 . 0 6 2 0 . 3 1 7 0 . 0 4 1 3 . 3 1 4 0 . 0 4 1 5 . 3 1 5 0 . 0 3 1 3 . 4 9 7 6 0 . 0 2 11 . 31 1 0 . 0 1 3 . 3 0 8 0 . 0 2 1 0 . 3 0 9 0 . 0 2 1 3 . 1 1 2 9 2 0 . 0 8 1 2 3 . 3 2 3 0 . 0 5 5 4 . 3 2 0 0 . 13 2 0 6 . 3 2 1 0 . 0 7 9 0 . 4 9 8 2 0 . 0 3 3 6 . 3 3 5 0 . 0 7 8 5 . 3 3 2 0 . 13 1 4 9 . 3 3 3 0 . 0 5 5 4 . 5 0 7 1 0 . 6 9 2 7 4 . 3 4 1 0 . 8 9 3 3 5 . 3 3 8 0 . 7 3 3 9 5 . 3 3 9 0 . 7 8 3 8 0 . 5 1 1 5 0 . 6 9 1 4 8 . 3 4 7 0 ! 5 8 161 . 3 4 4 0 . 7 6 1 6 8 . 3 4 5 0 . 4 3 1 0 2 . 5 2 5 4 0 . 12 1 0 . 3 5 4 0 . 11 9 . 3 5 1 0 . 11 8 . 3 5 2 0 . 13 21 . 1 1 6 3 8 0 . 0 6 7 6 . 3 6 0 0 . 0 5 4 2 . 3 5 7 0 . 0 7 6 9 . 3 5 8 0 . 13 4 6 . 5 5 7 1 0 . 1 1 8 5 . 3 6 6 0 . 1 1 9 5 . 3 6 3 0 . 16 1 4 2 . 3 6 4 0 . 15 1 4 3 . 1 1 8 0 0 0 . 0 3 2 8 . 3 8 4 0 . 0 3 3 0 . 3 8 1 0 . 0 7 7 7 . 3 8 2 0 . 0 6 6 5 . 1 1 7 3 8 0 . 0 2 2 7 . 3 7 8 0 . 0 3 2 8 . 3 7 5 0 . 0 3 3 4 . 3 7 6 0 . 0 3 3 5 . 1 1 7 3 2 0 . 2 8 1 0 8 . 3 7 2 0 . 0 5 5 5 . 3 6 9 0 . 0 - 0 . 3 7 0 0 . 0 3 4 2 . 1 1 7 8 2 0 . 0 7 7 6 . 3 9 6 0 . 0 8 9 9 . 3 9 3 0 . 14 1 0 5 . 3 9 4 0 . 0 9 9 5 . 1 1 7 8 1 0 . 0 9 1 2 6 . 4 0 2 0 . 13 1 5 7 . 3 9 9 0 . 0 8 1 0 7 . 4 0 0 0 . 0 7 8 0 . 1 1 7 8 3 0 . 0 6 4 2 . 3 9 0 0 . 0 4 3 2 . 3 8 7 0 . 0 7 5 5 . 3 8 8 0 . 0 6 4 0 . 8 5 2 6 0 . 4 7 1 1 . 4 5 4 0 . 7 9 1 4 . 4 5 1 0 . 6 5 1 9 . 4 5 2 0 . 4 0 9 . 8 5 4 8 0 . 5 6 3 2 5 . 4 6 0 0 . 16 4 4 . 4 5 7 0 . 4 6 2 3 4 . 4 5 8 0 . 5 1 11 . 8 5 6 7 0 . 0 0 . 4 6 6 0 . 0 2 4 . 4 6 3 0 . 0 2 7 . 4 6 4 0 . 0 1 6 . 8 5 9 1 0 . 14 7 8 . 4 7 2 0 . 0 4 1 6 . 4 6 9 0 . 0 5 1 5 . 4 7 0 0 . 0 5 1 9 . 8 6 1 1 0 . 3 3 1 9 9 . 4 7 8 0 . 2 2 1 0 5 . 4 7 5 0 . 2 5 1 1 8 . 4 7 6 0 . 4 2 2 2 3 . 1 1 7 6 7 0 . 0 7 31 . 4 0 8 0 . 0 7 3 6 . 4 0 5 0 . 0 6 3 2 . 4 0 6 0 . 0 5 3 2 . 7 3 2 6 0 . 0 4 1 9 . 4 2 4 0 . 0 4 1 7 . 4 2 1 0 . 0 3 2 0 . 4 2 2 0 . 2 5 5 0 . 1 3 6 5 7 0 . 0 4 11 . 4 3 0 0 . 0 3 7 . 4 2 7 0 . 0 4 1 5 . 4 2 8 0 . 0 3 1 2 . 8 3 8 7 0 . 0 4 2 9 . 4 3 6 0 . 0 6 4 4 . 4 3 3 0 . 0 2 1 3 . 4 3 4 0 . 0 1 8 . 8 4 7 8 0 . 4 2 2 2 0 . 4 4 2 0 . 4 4 2 6 8 . 4 3 9 0 . 3 8 231 . 4 4 0 0 . 4 0 2 2 2 . 8 4 9 5 0 . 18 5 2 . 4 4 8 0 . 13 2 9 . 4 4 5 0 . 15 2 4 . 4 4 6 0 . 0 6 1 3 . 8 7 4 5 1 . 0 2 3 0 1 .• 4 9 1 0 . 8 6 2 0 4 . 4 8 8 0 . 7 0 1 5 5 . 4 8 9 0 . 2 3 9 6 . 9 8 6 8 0 . 2 4 4 3 . 5 2 2 0 . 2 5 8 8 . 5 1 9 0 . 3 1 9 0 . 5 2 0 0 . 16 9 3 . 1 4 6 3 5 0 . 0 2 1 0 . 5 1 0 0 -. 16 9 2 . 5 0 7 0 . 1 5 31 . 5 0 8 0 . 0 7 5 6 . 1 4 6 5 8 0 . 19 1 5 . 5 0 4 0 . 1 1 27 . 5 0 1 0 . 17 2 8 . 5 0 2 0 . 14 3 6 . 1 4 6 7 5 0 . 2 4 4 . 5 1 6 0 . 2 0 4 . 5 1 3 0 . 2 1 4 . 5 1 4 0 . 2 0 5 . 1 5 3 8 4 0 . 15 141 . 5 4 0 0 . 2 6 2 6 0 . 5 3 7 0 . 2 3 2 4 5 . 5 3 8 0 . 2 9 3 1 0 . 1 5 4 0 7 0 . 0 6 51 . 5 3 4 0 ; 0 2 14 . 5 3 1 0 . 0 2 9 . 5 3 2 0 . 0 2 1 6 . 1 5 3 3 4 0 . 0 2 1 6 . 5 4 6 0 . 2 2 2 1 8 . 5 4 3 0 . 0 3 2 3 . 5 4 4 0 . 0 5 4 4 . 1 5 3 3 5 0 . 0 3 2 2 . 5 5 2 0 . 0 2 14 . 5 4 9 0 . 0 2 1 4 . 5 5 0 0 . 0 3 1 9 . 1 5 3 5 6 0 . 0 7 6 . 5 2 8 0 . 0 1 5 . 5 2 5 0 . 0 1 5 . 5 2 6 0 . 0 1 1 3 . 1 5 5 0 7 0 . 0 5 2 8 . 7 2 4 0 . 12 8 3 . 7 2 2 0 . 1 0 4 9 . 7 2 5 0 . 0 6 3 4 . 2 1 0 4 1 0 . 1 0 5 5 . 7 3 0 0 . 0 5 2 5 . 7 2 8 0 . 0 7 3 6 . 7 3 1 0 . 0 7 1 4 . 2 0 5 8 4 0 . 2 0 6 8 . 7 3 6 0 . 12 4 5 . 7 3 4 0 . 1 2 4 0 . 7 3 7 0 . 3 3 1 0 8 . 1 5 6 4 5 0 . 0 4 9 . 5 5 7 0 . 0 4 7 . 5 5 5 0 . 0 5 9 . 5 5 8 0 . 0 6 9 . 21 1 8 5 0 . 0 2 . 7 4 2 0 . 0 1 4 . 7 4 0 0 . 0 1 3 . 7 4 3 0 . 0 1 2 . CHANNEL 3 C H A N N E L C O M P O S I T E B U L K B E S T E S T I M A T E - H CU AG » CU AG H CU AG cu AG 3 0 4 0 . 3 0 3 5 . 3 0 6 0 . 2 5 1 8 . 3 0 7 0 . 2 6 2 2 . - 0 . 2 5 2 0 3 1 6 0 . 0 3 1 3 . 3 1 8 0 . 0 5 2 2 . 3 1 9 0 . 0 4 1 5 . 0 . 0 4 14 3 1 0 0 . 0 4 1 8 . 3 1 2 0 . 0 2 6 . 3 1 3 0 . 0 1 4 . 0 . 0 1 6 3 2 2 0 . 0 9 1 3 2 . 3 2 4 0 . 0 7 1 0 3 . 3 2 5 0 . 0 7 1 0 1 . 0 . 0 7 102 3 3 4 0 . 0 3 4 0 . 3 3 6 0 . 0 6 7 0 . 3 3 7 0 . 0 4 5 4 . 0 . 0 5 6 5 3 4 0 0 . 8 6 3 9 4 . 3 4 2 0 . 9 7 3 9 3 . 3 4 3 0 . 8 7 3 8 9 . 0 . 8 5 3 7 7 3 4 6 0 . 9 7 3 8 7 . 3 4 8 0 . 6 0 1 6 3 . 3 4 9 0 . 5 2 151 . 0 . 5 8 166 3 5 3 0 . 13 1 8 . 3 5 5 0 . 1 0 8 . 3 5 6 0 . 16 2 0 . 0 . 14 17 3 5 9 0 . 0 6 6 2 . 3 6 1 0 . 0 6 6 2 . 3 6 2 0 . 0 5 4 6 . 0 . 0 6 4 9 3 6 5 0 . 15 1 3 6 . 3 6 7 0 . 15 1 4 2 . 3 6 8 0 . 14 1 1 7 . 0 . 14 1 18 3 8 3 0 . 0 3 2 2 . 3 8 5 0 . 0 3 3 2 . 3 8 6 0 . 0 4 4 7 . 0 . 0 4 4 5 3 7 7 0 . 0 3 2 8 . 3 7 9 0 . 0 3 2 7 . 3 8 0 0 . 0 4 3 7 . 0 . 0 4 3 4 3 7 1 0 . 0 3 4 4 . 3 7 3 0 . 0 7 5 5 . 3 7 4 0 . 0 3 5 0 . 0 . 0 4 4 8 3 9 5 0 . 0 5 5 9 . 3 9 7 0 . 0 9 9 8 . 3 9 8 0 . 0 9 8 6 . 0 . 0 9 8 8 4 0 1 0 . 12 2 3 3 . 4 0 3 0 . 0 9 1 6 0 . 4 0 4 0 . 12 1 7 8 . 0 . 1 1 164 3 8 9 0 . 0 6 2 0 . 3 9 1 0 . 0 7 3 5 . 3 9 2 0 . 0 6 4 8 0 . 0 6 4 3 4 5 3 0 . 3 2 1 0 . 4 5 5 0 . 4 4 131 . 4 5 6 0 . 3 8 11 0 . 4 7 12 4 5 9 0 . 4 1 2 0 3 . 4 6 1 0 . 4 3 2 1 7 . 4 6 2 0 . 4 5 2 2 6 . 0 . 4 1 179 4 6 5 0 . 0 5 1 6 . 4 6 7 0 . 0 2 5 . 4 6 8 0 . 0 3 7 . 0 . 0 3 7 4 7 1 0 . 10 4 8 . 4 7 3 0 . 0 9 4 5 . 4 7 4 0 . 0 8 4 0 . 0 . 0 7 3 4 4 7 7 0 . 3 9 1 8 0 . 4 7 9 0 . 2 4 1 2 3 . 4 8 0 0 . 3 0 1 4 7 . 0 . 3 0 148 4 0 7 0 . 0 7 3 2 . 4 0 9 0 . 0 7 2 8 . 4 1 0 0 . 0 5 2 3 . 0 . 0 6 2 7 4 2 3 0 . 0 4 1 6 . 4 2 5 0 . 0 4 18 . 4 2 6 0 . 0 3 1 4 . 0 . 0 5 18 4 2 9 0 . 0 2 1 1 . 4 3 1 0 . 0 3 1 0 . 4 3 2 0 . 0 3 1 0 . 0 . 0 3 1 0 4 3 5 0 . 11 9 5 . 4 3 7 0 . 0 2 1 8 . 4 3 8 0 . 0 3 21 . 0 . 0 4 2 9 4 4 1 0 . 4 7 2 9 7 . 4 4 3 0 . 3 9 2 3 3 . 4 4 4 0 . 4 3 2 6 8 . 0 . 4 3 2 6 2 4 4 7 0 . 5 3 2 6 9 . 4 4 9 0 . 1 1 21 . 4 5 0 0 . 0 9 16 0 . 13 3 7 4 9 0 0 . 4 5 1 3 3 . 4 9 2 0 . 7 6 1 7 5 . 4 9 3 0 . 7 9 181 . 0 . 7 3 176 5 2 1 0 . 3 5 1 6 3 . 5 2 3 0 . 2 7 8 9 . 5 2 4 0 . 2 1 7 4 0 . 2 3 8 5 5 0 9 0 . 0 2 1 0 . 51 10 . 0 2 9 . 5 1 2 0 . 0 2 1 1 . 0 . 0 6 3 0 5 0 3 0 . 2 0 2 0 . 5 0 5 0 . 15 2 6 . 5 0 6 0 . 16 2 4 . 0 . 15 2 5 5 1 5 0 . 19 4 . 5 1 7 0 . 1 9 4 . 5 1 8 0 . 2 3 3 . 0 . 2 2 4 5 3 9 0 . 3 1 3 1 5 . 5 4 1 0 . 2 5 2 4 0 . 5 4 2 0 . 17 1 6 7 . 0 . 21 2 1 0 5 3 3 0 . 0 5 5 0 . 5 3 5 0 . 0 3 2 2 . 5 3 6 0 . 0 3 21 . 0 . 0 3 21 5 4 5 0 . 0 2 1 9 . 5 4 7 0 . 0 6 5 7 . 5 4 B 0 . 0 7 6 6 . 0 . 0 9 8 2 5 5 1 0 . 0 1 1 0 . 5 5 3 0 . 0 1 1 0 . 5 5 4 0 . 0 2 1 3 . 0 . 0 2 14 5 2 7 0 . 0 4 1 8 . 5 2 9 0 . 0 3 1 2 . 5 3 0 0 . 0 1 7 . 0 . 0 1 8 7 2 3 0 . 0 6 2 7 . 7 2 6 0 . 0 8 4 5 . 7 2 7 0 . 1 1 7 8 . 0 . 1 0 6 8 7 2 9 0 . 0 8 4 5 . 7 3 2 0 . 0 3 17 . 7 3 3 0 . 0 5 2 6 . 0 . 0 6 2 8 7 3 5 0 . 2 0 6 7 . 7 3 8 0 . 19 6 8 . 7 3 9 0 . 16 5 4 . 0 . 17 5 7 5 5 6 0 . 0 9 2 5 . 5 5 9 0 . 0 6 1 8 . 7 2 1 0 . 0 4 8 . 0 . 0 5 9 7 4 1 0 . 0 1 2 . 7 4 4 0 . 0 1 2 . 7 4 5 0 . 0 1 2 , 0 . 0 1 2 APPENDIX IV CALCULATION OF VOLUMES FOR DIFFERENT SAMPLING TECHNIQUES 118 Volume of Tube Samples - 3 i n c h d i a m e t e r - sample l e n g t h assumed t o be 4 i n c h e s - volume o f sample i s 28.3 c u . i n . - sample c o m p o s i t e i s composed of 4 t u b e s a m p l e s - t h e r e f o r e t o t a l t u b e sample volume = 113 c u . i n . Volume of C h a n n e l Samples - 4 s e p a r a t e c h a n n e l s u s i n g a 5 i n c h wide s h o v e l - h e i g h t o f sample cone assumed t o be 8 i n c h e s - r e f e r t o F i g u r e 6 C h a p t e r 3 f o r d i r e c t i o n s - D i r e c t i o n 1 - Volume = 3 6 x 5 x 8 x . 5 = 720 c u . i n . - D i r e c t i o n 2 - Volume = = 15 X 5 X 8 X .5 = 300 c u . in.. - D i r e c t i o n 3 - Volume = = 18 X 5 X 8 X .5 = 360 c u . i n . - D i r e c t i o n 4 - Volume = = 15 X 5 X 8 X .5 = 300 c u . i n . Volume of B u l k Sample - assume a u n i f o r m c y l i n d r i c a l p i l e o f c h i p s - h e i g h t o f cone assumed t o be 8 i n c h e s a s above - volume o f t o t a l sample w i l l e q u a l volume o f t h e s h aded a r e a i n d i a g r a m 119 1 1. y= .67 x + 13.3 x= 1.49 y - 19.9 x= C-^y : + d^ 2. y=-2.x - 8.0 x= -5.y - 4.0 x^ C 2 y + d 2 (-8,8) (-20,0) ^ Volume to be estimated (0,13.3) V Integrate between these limits. 1 . x •BLASTHOLE ^(0,-8) v 2 2 V = J ( T T x - Tlx ) d y u 1 2 2 2 {(C y + d ) - (C y + d ) }dy 1 2 2 u 1 Y- 2 2 2 2 2 = IT J) { ( C - C )y + 2(C d -C d ) y + (d - d )}dy u 1 2 1 1 2 2 1 1 v 2 2 3 2 2 2 = TT { (C - C ) y + (C d - C d ) y + (d -d )y} u 1 2 1 1 2 2 1 2 2 2 3 2 2 2 = { ( 1 , 4 9 - ( - . 5 ) * ( 8 ) + ( ( 1 . 4 9 * - l 9 . 9 ) - ( - . 5 * - 4 ) * ( 8 ) + ( l 9 . 9 - 4 ) * 8 3 4243.19 s a y 4220 c u . i n . 120 - Volume of b u l k sample = t o t a l volume - volume of t u b e s and c h a n n e l s 4240 - 1793 = 2447 c u . i n . 

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