Subjective Sampling Approaches to Resource Estimation

R M - 7 5 - 2 9

Second P r i n t i n g

S U B J E C T I V E S A M P L I N G APPROACHES T O RESOURCE E S T I M A T I O N G r e g o r y B . B a e c h e r

J u n e 1 9 7 5

R e s e a r c h M e m o r a n d a a r e i n f o r m a l p u b l i c a t i o n s r e l a t i n g t o o n g o i n g o r p r o j e c t e d a r e a s of re- s e a r c h a t I I A S A . T h e v i e w s e x p r e s s e d a r e t h o s e of t h e a u t h o r , a n d do n o t n e c e s s a r i l y r e f l e c t t h o s e of I I A S A .

T h i s p a p e r was o r i g i n a l l y p r e p a r e d u n d e r t h e t i t l e " M o d e l l i n g f o r Management" f o r p r e s e n t a t i o n a t a N a t e r R e s e a r c h C e n t r e

(U.K. ) Conference on " R i v e r P o l l u t i o n C o n t r o l " , Oxford, 9 - 1 1 A s r i l , 1979.

Subjective Sampling Approaches to Resource Estimation Gregory B. Baecher

Abstract

This paper suggests deficiencies in present sampling approaches to regional resource estimation, and ways in which these deficiencies might be remedied.

General approaches to resource estimation are dis- cussed, as are requirements which well conceived

approaches should satisfy. Using presently available theory, a comprehensive sampling approach to estimation can be formulated. The results of such an analysis are directly incorporable into decisions concerning exploration strategy optimization. However, further computational and experimental work are required before this approach is operational.

I. Introduction

Resource estimation techniques can be broadly grouped into two classes: macroanalytic approaches which model empirical relationships in aggregated discovery or pro- duction data, and microanalytic approaches which model structural relationships in the exploration process.

Perhaps the most well known examples of each of these are Hubbert [141 and Allais [l].

This paper addresses microanalytic approaches.

In particular, it sets about broadening present sampling

theory techniques to encompass more of what we know about

the exploration process, and to include prior geological opinion. This broadening is seen as necessary if estimates based on microanalytic methods are to be comprehensive and valid.

The main purpose of the paper is not to mathematically solve formulae associated with the broadening, but to

indicate directions toward which continuing work should be moving.

11.

Macroanalytic versus Microanalytic Approaches Macroanalytic models, which in essence are trend

extrapolation procedures, assume an unspecified "uniformity- of-nature." They assume that exploration and production operate within a fixed (or at most, gradually changing) environment which leads to aggregate behavior according to simple relationships among important variables. Taking this to be true, empirically fitted relationships may be extrapolated into the future, either in time or along some other dimension (e.g., cumulative drilling length).

Macroanalytic approaches do not use structural relation- ships among facets of exploration and production, and lump together economic, geological, and exploratory variables.

Arguments for and against macroanalytic approaches appear in the geologic literature (Ryan

[ 2 6 1 ,

Hubbert [14], Moore [181

)

as well as in the literature of other disciplines where similar tools are used for estimation or forecasting

(e.g., economics). Specifically, two properties limit their

u s e f u l n e s s f o r r e s o u r c e e s t i m a t i o ~ . F i r s t , t h e y l e a d t o d e t e r m i n i s t i c p r e d i c t i o n s , t h e u n c e r t a i n t y o f w h i c h i s d i f f i c u l t t o j u d g e ( e . g . , c h a n g i n g f r o m o n e f a m i l y o f c u r v e s t o a n o t h e r , o r f r o m o n e method o f f i t t i n g t o

a n o t h e r , d r a s t i c a l l y c h a n g e s e s t i m a t e s - - C f . , H u b b e r t [ 1 4 ] , Moore [ 1 8 1 ) . S e c o n d , t h e y d e p e n d o n a s u b s t a n t i a l

h i s t o r y o f d i s c o v e r y a n d p r o d u c t i o n . W h i l e t h i s h i s t o r y e x i s t s f o r a r e a s l i k e t h e U n i t e d S t a t e s , f o r s p a r s e l y e x p l o r e d a r e a s t h e a n a l y s i s o f t e n b e g i n s b y p r e d i c t i n g t o t a l r e s o u r c e s some o t h e r way, a n d t h e n c a l c u l a t e s t i m e s t r e a m s o f p r o d u c t i o n ( H u b b e r t [ 1 4 1 ) .

M i c r o a n a l y t i c a p p r o a c h e s a l s o s u f f e r d r a w b a c k s , w h i c h a g a i n a r e g e n e r i c t o t h e a p p r o a c h a n d n o t l i m i t e d t o

r e s o u r c e e s t i m a t i o n . F i r s t , t h e y r e q u i r e d e t a i l e d d a t a o n a r e g i o n b y r e g i o n b a s i s o f g e o l o g i c a n d g e o m e t r i c p r o p e r t i e s , n u m b e r s a n d s i z e s o f d i s c o v e r i e s , a n d a m o u n t s a n d p a t t e r n s o f e x p l o r a t i o n a l l o c a t i o n s . S e c o n d , t h e y r e q u i r e o r d e r s o f m a g n i t u d e mvre c o m p u t a t i o n e f f o r t t h a n m a c r o a n a l y t i c a p p r o a c h e s , a s g r o s s e s t i m a t e s a r e f o r m e d by f i r s t m a k i n g r e g i o n a l e s t i m a t e s a n d t h e n a g g r e g a t i n g .

T h e s e r e q u i r e m e n t s make m i c r o a n a l y t i c a p p r o a c h e s d i f f i c u l t a n d l a k o r i o u s t o a p p l y . T h i r d , a l t l ~ o u g h n o t n e c e s s a r i l y a s h o r t c o m i n g , m i c r o a n a l y t i c a p p r o a c h e s d o n o t a c c o u n t f o r e c o n o m i c o r p r o d u c t i o n f a c t o r s . They d e a l p u r e l y w i t h g e o l o g i c a l a n d s t a t i s t i c a l v a r i a b l e s . Economic v a r i a b l e s m u s t b e c o n s i d e r e d s e p a r a t e l y u s i n g t h e g e o l o g i c a l / s t a t i s t i c a l a n a l y s i s a s i n p u t ( e . g . , MacAvoy a n d P i n d y c k [ 1 7 1 ) .

On t h e o t h e r h a n d , m i c r o a n a l y t i c a p p r o a c h e s h a v e f o u r v e r y f a v o r a b l e p r o p e r t i e s w h i c h recommend them f r o m t h e p r e s e n t p e r s p e c t i v e :

1) t h e y a l l o w i n c l u s i o n o f g e o l o g i c a l i n p u t on a r e g i o n a l b a s i s ;

2 ) t h e y may b e a p p l i e d t o r e g i o n s w h i c h h a v e b e e n o n l y s p a r s e l y e x p l o r e d ;

3 ) t h e y o f t e n a l l o w q u a n t i f i c a t i o n o f u n c e r t a i n t y ; a n d 4 ) t h e i r o u t p u t c a n b e r e a d i l y i n c o r p o r a t e d i n s t r a t e g y

o p t i m i z a t i o n f o r l o c a l o r r e g i o n a l e x p l o r a t i o n . 111. M i c r o a n a l v t i c Models

M i c r o a n a l y t i c a p p r o a c h e s p r o c e e d by making r e s o u r c e e s t i m a t e s f o r s m a l l r e g i o n s w h i c h a r e assumed g e o l o g i c a l l y homogeneous, t h e n a g g r e g a t i n g o v e r a l l r e g i o n s .

A s t h e a g g r e g a t i o n i s s t r a i g h t f o r w a r d , a t t e n t i o n i s drawn t o making e s t i m a t e s f o r e a c h r e g i o n . I n a t r a d i t i o n a l , j u d g m e n t a l way t h i s h a s a l w a y s b e e n d o n e by e x p l o r a t i o n y e a l o g i s t s . B a s e d o n e x p e r i e n c e , g e o l o g i s t s s u b j e c t i v e l y j u d g e t h e s i m i l a r i t y o f t h e r e g i o n t o b e t t e r known r e g i o n s , a n d i n c o m b i n a t i o n w i t h g e o l o g i c a l t h e o r y make p r e d i c t i o n s o f r e s o u r c e s ( e . g . , u p p e r and l o w e r b o u n d s ) . T h i s i s a v e r y b a s i c m i c r o a n a l y t i c a p p r o a c h , and i s t h e a p p r o a c h w h i c h H a r r i s [ I l l a t t e m p t s t o q u a n t i f y .

A s e c o n d a p p r o a c h i s t o c o r r e l a t e g e o l o g i c a l v a r i a b l e s w i t h r e s o u r c e s e i t h e r by r e g r e s s i o n o r f a c t o r a n a l y s i s

( H a r r i s [ 1 2 1 , D e G e o f f r o y a n d W i n g a l l [ 4 ] , D e G e o f f r o y a n d Wu [ 5 1 )

.

T h i s i s a s t r a i g h t f o r w a r d a p p r o a c h w i t h w h i c h

t h e r e i s e x p e r i e n c e i n many a p p l i c a t i o n s . However, i t s u f f e r s well-known l i m i t a t i o n s i n t h a t it i s a c o r r e l a t i o n a n d n o t a c a u s a l m o d e l . F a c t o r s w h i c h a r e h i g h l y c o r r e l a t e d w i t h m i n e r a l i z a t i o n o r d e p o s i t i o n i n o n e c o n t e x t a r e n o t n e c e s s a r i l y t h o s e w h i c h would b e c o r r e l a t e d w i t h it i n o t h e r s . A s t h e s e m e t h o d s a r e n o r m a l l y a p p l i e d t o known d e p o s i t s r a t h e r t h a n r e s o u r c e s , r e g r e s s i o n a n d f a c t o r a n a l y s i s c o n f o u n d g e o l o g i c a l a n d n o n - g e o l o g i c a l v a r i a b l e s . T h i s l e a d s t o t h e n o t too s u r p r i s i n g r e s u l t o f G r i f f i t h s a n d S i n g e r

[ l o ]

t h a t " m i n e r a l p o t e n t i a l " i s m o s t h i g h l y c o r r e l a t e d w i t h d . e g r e e o f d e v e l o p m e n t .

A t h i r d a p p r o a c h t r e a t s e s t i m a t i o n a s a p r o b l e m o f i n f e r e n c e f r o m s a m p l i n g . The s i z e d i s t r i b u t i o n a n d

s p a t i a l d i s p e r s i o n o f d e p o s i t s a r e m o d e l e d by f a m i l i e s o f p r o b a b i l i t y f u n c t i o n s , a n d p a r a m e t e r s f o r t h e s e d i s t r i - b u t i o n s a r e e s t i m a t e d a s s u m i n g known d e p o s i t s t o b e a

p r o b a b i l i t y s a m p l e o f t h e t o t a l i n s i t u p o p u l a t i o n ( A l l a i s [ 1 ]

,

U h l e r a n d B r a d l e y [ 3 2 ]

,

Kaufman [ I 5 1

,

S l i c h t e r [ 3 0 ]

,

G r i f f i t h s [ 9 ] ) . T o t a l r e s o u r c e e s t i m a t e s a r e made by e v a l u a t i n g t h e random sum

i n w h i c h a i i s a random v a r i a b l e drawn f r o m t h e d i s t r i - b u t i o n o f d e p o s i t s i z e s , a n d

N

i s a r . v . r e p r e s e n t i n g t h e number o f d e p o s i t s w i t h i n t h e r e g i o n ( U h l e r a n d B r a d l e y [ 3 2 1 ) .

Criticism of the sampling approach has primarily been based on the observation that known deposits are not a simple random sample of in situ deposits, the necessity of choosing families of distributions to model the size

distribution and spatial dispersion of deposits, and lack of geological input in the model.

The observation that known deposits are not a simple random sample of in situ deposits is not so much an argument against a sampling approach as an argument against uncriti- cal application of that approach. For example, Kaufman [15]

has presented a more rigorous analysis of the sampl- ing approach in which in situ deposits are treated as a finite random sample from some "super-population" (see also Ericson

[ 7 1 ) .

Then, the parameters of that super-popula- tion are estimated by assuming known deposits to be

a

sample of the in situ population selected with probability propor-

tional to size and without replacement (Figure 1). Very different estimates of super-population parameters are

obtained using this assumption from those using the simple random assumption.

A

point we will return to in Section 3 is that

similar special considerations must be made in estimating

parameters of the spatial dispersion function, in particular,

that the probability of finding n deposits within a subregion

is nonlinearly related to the amount of exploration effort

allocated to that subregion and to the distribution of

deposit sizes.

The p r o b l e m o f s e l e c t i n g a f a m i l y o f d i s t r i b u t i o n s t o model t h e s u p e r - p o p u l a t i o n a n d s p a t i a l d i s p e r s i o n o f d e p o s i t s i s common t o a l l a n a l y s e s ( i n t h a t m a t h e m a t i c a l l y s i m p l e f u n c t i o n s m u s t a l w a y s b e c h o s e n somehow). I n t h e p r e s e n t c o n t e x t , h o w e v e r , t h e r e i s c o n s i d e r a b l e e m p i r i c a l e v i d e n c e t o s u g g e s t t h a t l o g n o r m a l s u p e r - p o p u l a t i o n s do a c c u r a t e l y model g e o m e t r i c p r o p e r t i e s o f many g e o l o g i c a l p o p u l a t i o n s ( S l i c h t e r [ 3 0 ] ) , a n d t h a t t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n may a d e q u a t e l y model s p a t i a l d i s p e r s i o n , t h o u g h t h e l a t t e r p o i n t i s n o t a s c l e a r a s t h e f o r m e r . W e w i l l r e t u r n t o t h i s s p a t i a l model i n S e c t i o n 3. F u r t h e r m o r e , o n e s u s p e c t s

( s e e , e . g . , U h l e r a n d B r a d l e y [ 3 2 ] ) t h a t t h e t o t a l r e s o u r c e

e s t i m a t e s a r e f a i r l y r o b u s t t o c h a n g e s i n t h e f o r m o f t h e s u p e r - p o p u l a t i o n , a n d e v e n more s o t o t h e f o r m o f s p a t i a l d i s p e r s i o n .

An i n t e r e s t i n g d i r e c t i o n o f f u t u r e work would b e t o

q u a n t i t a t i v e l y e v a l u a t e t h e s e n s i t i v i t y of r e s o u r c e e s t i m a t e s t o t h e f o r m of t h e s e d i s t r i b u t i o n s . A r e f i n e m e n t f o l l o w i n g s u c h a n a l y s i s may b e t o f o r m s o - c a l l e d c o m p o s i t e B a y e s i a n d i s t r i b u t i o n s a s s u g g e s t e d by Wood [ 3 4 ] a n d Box a n d T i a o [ 3 ] , i n w h i c h d i s t r i b u t i o n m o d e l s a r e t h e m s e l v e s random

v a r i a b l e s .

The m o s t i m p o r t a n t c r i t i c i s m of s a m p l i n g a p p r o a c h e s i s t h a t t h e y g e n e r a l l y n e g l e c t p r i o r g e o l o g i c a l i n f o r m a t i o n . T h i s i s c e r t a i n l y t r u e of t h e " f r e q u e n t i s t " a p p r o a c h e s t o

i n f e r e n c e , a n d e v e n t h e B a y e s i a n a n a l y s e s h a v e r e m a i n e d t i e d t o

" u n i n t o r m e d p r i o r s . " I n making e s t i m a t e s o f n a t u r a l r e s o u r c e s w e h a v e c o n s i d e r a b l y more i n f o r m a t i o n a v a i l a b l e t h a n m e r e l y

t h e number and s i z e s o f a l r e a d y d i s c o v e r e d d e p o s i t s . T h i s p r i o r i n f o r m a t i o n comes f r o m r e g i o n a l g e o l o g y , e x p e r i e n c e i n s i m i l a r r e g i o n s , and b a s i c c o n c e p t s o f g e o l o g i c a l p r o - cesses. C o m p r e h e n s i v e e s t i m a t e s m u s t a c c o u n t f o r t h i s i n f o r m a t i o n . I t i s o n l y when t h e a v a i l a b l e d a t a s e t i s s o l a r g e t h a t i n f e r e n c e s become i n s e n s i t i v e t o p r i o r i n - f o r m a t i o n t h a t t h e l a t t e r c a n b e n e g l e c t e d . G i v e n t h e s m a l l amount o f i n f o r m a t i o n w h i c h comes f r o m f i n d i n g o r n o t f i n d i n g d e p o s i t s ( r e l a t i v e Lo t h e i n f e r e n c e s a b o u t r e g i o n a l g e o l o g y a n d s t r u c t u r e w h i c h a r e m a d e ) , t h i s i s s e l d o m t h e c a s e . G i v e n human b i a s e s t o w a r d n e g l e c t i n g p r i o r i n f o r m a t i o n i n t h e f a c e of new, " h a r d " d a t a ( T v e r s k y

[ 3 1 1 ) i n c l u s i o n of g e o l o g i c a l i n f o r m a t i o n m u s t b e e x p l i c i t . I V . R e q u i r e m e n t s f o r a C o m p r e h e n s i v e S a m p l i n g A p p r o a c h

To t h i s p o i n t w e h a v e d i s c u s s e d macro- and m i c r o a n a l y t i c a p p r o a c h e s t o r e s o u r c e e s t i m a t e s , a n d i n d i c a t e d a d v a n t a g e s a n d d i s a d v a n t a g e s of e a c h . From t h i s d i s c u s s i o n i t seems a p p a r e n t t h a t s a m p l i n g a p p r o a c h e s o f f e r a m e t h o d o l o g i c a l framework w i t h i n w h i c h a c o m p r e h e n s i v e a n d r e a l i s t i c model o f e x p l o r a t i o n a n d e s t i m a t i o n m i g h t b e d e v e l o p e d . W e now t u r n t o w a r d n e c e s s a r y m o d i f i c a t i o n s o f t h e s a m p l i n g a p - p r o a c h .

Two r e q u i r e m e n t s w h i c h p r e s e n t s a m p l i n g a p p r o a c h e s d o n o t e n t i r e l y s a t i s f y , b u t w h i c h t h e y m u s t t o b e comprehen- s i v e and r e a l i s t i c a r e t h a t :

a ) p r i o r g e o l o g i c a l i n f o r m a t i o n a n d o p i n i o n b e a c c o u n t e d f o r ;

b ) t h e r e a l l i k e l i h o o d o f d e p o s i t s b e i n g d i s c o v e r e d b e r e f l e c t e d .

L o g i c a l l y , t h e s e f a c e t s o f i n f e r e n c e a r e s e p a r a b l e and may b e combined by B a y e s ' Theorem,

H e r e , - 8 and - R a r e t a k e n t o b e g e o l o g i c a l p a r a m e t e r s d e s c r i - b i n g t h e s i z e a n d s p a t i a l d i s t r i b u t i o n o f d e p o s i t s ;

f' ( 8 , Q ) - - a n d f

'

( 0 , ^- - bt/data) a r e t h e p r i o r a n d p o s t e r i o r p r o b - a b i l i t y d i s t r i b u t i o n of t h e p a r a m e t e r s , r e s p e c t i v e l y ; a n d L ( d a t a / B , R ) - - i s t h e l i k e l i h o o d o f o b s e r v i n g t h e d a t a w e r e 8 a n d R t h e t r u e p a r a m e t r i c v a l u e . P r i o r g e o l o g i c a l i n f o r - - -

m a t i o n i s c o n t a i n e d i n f 0 ( 8 , R ) ; - - c h a r a c t e r i s t i c s o f t h e e x p l o r a t i o n p r o c e s s a r e c o n t a i n e d i n L ( d a t a / B , R ) . - -

4 . 1 P r i o r I n f o r m a t i o n a n d S u b i e c t i v i t v

One e n t e r s n e a r l y a l l i n f e r e n t i a l s i t u a t i o n s w i t h some p r i o r i n f o r m a t i o n o r s u s p i c i o n s . A r e g i o n seems f a v o r a b l e f o r e x p l o r a t i o n b e c a u s e it i s s i m i l a r t o known a r e a s o f d e p o s i t i o n o r b e c a u s e it h a s g e o l o g i c a l p r o p e r t i e s a s s o c i - a t e d w i t h d e p o s i t i o n . However, e a c h i n d i v i d u a l h a s d i f - f e r e n t e x p e r i e n c e s a n d c o n c e p t s o f g e o l o g y a n d t h u s a s s e s s e s f a v o r a b i l i t y d i f f e r e n t l y . T h i s i s t h e t r a d i t i o n a l r o l e of t h e e x p l o r a t i o n g e o l o g i s t . G e o l o g i c a l s t r u c t u r e s a r e h i g h l y c o m p l e x , a n d c o m p a r a t i v e l y f e w o b s e r v a t i o n s a r e made i n

e x p l o r a t i o n . T h e r e f o r e , e x p e r i e n c e a n d judgment a r e i m - p o r t a n t . T h i s i s why g e o l o g i s t s a r e c a l l e d upon t o make

r e s o u r c e e s t i m a t e s r a t h e r t h a n o t h e r p e o p l e ( s e e R o b i n s o n [ 2 4 1 , f o r a n i l l u s t r a t i o n o f t h e i m p o r t a n c e o f s u b j e c t i v e c o n c e p t s i n i n t e r p r e t i n g e x p l o r a t i o n d a t a ) .

A g e o l o g i s t c o n s i d e r s t h e r e s u l t s o f e x p l o r a t i o n i n t h e c o n t e x t o f h i s p r i o r f e e l i n g s . To t h e e x t e n t t h e two a r e c o n s i s t e n t h e g i v e s more o r l e s s c r e d i b i l i t y t o h i s f e e l i n g s . However, t h i s i n f e r e n t i a l p r o c e s s , a n d t h u s e x p l o r a t i o n a s a w h o l e , i s p u r e l y s u b j e c t i v e . T h u s e x p l o r a t i o n c a n n o t b e a d e q u a t e l y m o d e l e d w i t h o u t i n t r o - d u c i n g t h e c o n c e p t o f s u b j e c t i v e p r o b a b i l i t y ( B a e c h e r [ 2 ] ) . The u s e a b l e r e s u l t s o f e x p l o r a t i o n a r e h y p o t h e s e s .

T h e s e h y p o t h e s e s a r i s e s u b j e c t i v e l y a n d a r e g i v e n c r e d e n c e s u b j e c t i v e l y ; " h a r d " d a t a o n l y e n t e r i n m o d i f y i n g t h e d e g r e e - o f - c r e d i b i l i t y g i v e n t o h y p o t h e s e s . U n c e r t a i n t i e s a s s o c i a t e d w i t h e x p l o r a t i o n a r e t h o s e a s s o c i a t e d w i t h h y p o t h e s e s , s o u n c e r t a i n t i e s , t o o , a r e n e c e s s a r i l y s u b - j e c t i v e . Only when t h e amount o f d a t a becomes s o l a r g e t h a t i n f e r e n c e s c e a s e t o b e a f f e c t e d . by p r i o r f e e l i n g s c a n e x p l o r a t i o n b e s p o k e n of a s " o b j e c t i v e . " T h i s o c c u r s f o r o n l y t h e m o s t i n t e n s i v e l y e x p l o r e d r e g i o n s .

S o , t h e s a m p l i n g a p p r o a c h w e would l i k e t o d e v e l o p

m u s t b e b a s e d o n s u b j e c t i v e p r o b a b i l i t y . T h i s i s n o t u n i q u e t o t h e e x p l o r a t i o n l i t e r a t u r e , a l t h o u g h a t h o r o u g h a t t e m p t a t a r i g o r o u s f u n d a m e n t a l l y s u b j e c t i v e a p p r o a c h may b e . ¹

rayso son's

^{[ 8 1} well-known a p p l i c a t i o n o f s t a t i s t i c a l d e c i s i o n t h e o r y t o o i l a n d g a s d r i l l i n g d e c i s i o n s i s , o f c o u r s e , a r i g o r o u s a n d e a r l y a p p l i c a t i o n o f s u b j e c t i v e p r o b a b i l i t y t o g e o l o g i c a l e x p l o r a t i o n . However, f o r w h a t - e v e r r e a s o n s , s u b j e c t i v i s m h a s n e v e r b e e n a d o p t e d by

" g e o s t a t i s t i c i a n s " a n d t h u s t h e r e s o u r c e --- e s t i m a t i o n l i t e r - a t u r e r e m a i n s n o n - s u b j e c t i v s t a n d ( w i t h t h e e x c e p t i o n o f Kaufman) n o n - B a y e s i a n .

Kaufman [15] bases his analysis on a Bayesian approach, but does not use geological information to assess

priors (adopting "uninformed" priors instead). Harris [ll]

and Harris, et al. [13] use subjective probability in a one-step procedure for making resource estimates without exploration data (i.e., using only geological naps).

However, this is a degenerate case of resource estimation, and they seem to use subjective probability merely as a pragmatic tool when other data are not available.

Assessing Subjective Probabilities

Applicability of subjectivist theory rests ultimately on our ability to adequately assess probability distri- butions. Adequacy here means the ability to quantify an

individual's true personal feelings in a probability measure.

There is not room here to review the literature on behavioral decision theory and quantification of subjective probabili- ties. However, this work is extensive and rather consistent.

Feelings can be reliably quantified if a careful, rigorously based technique is employed. People do exhibit bias in

quantifying their feelings, (Tversky [31]) but these biases may not be great. Individuals may exhibit consistent con-

servative biases in updating their prior feelings by sample data ( ~ d w a r d s

16

I

)

, but in highly complicated, real

problems this conservatism seems to diminish or even dis-

appear (Winkler and Murphy [33]). In some meteorological

experiments, experts' measured, subjective probabilities

have been shown to be better forecasters of natural

o c c u r r e n c e s t h a n s t r u c t u r a l m o d e l s (Murphy a n d W i n k l e r [ 221)

.

I n s h o r t , w e c a n a d e q u a t e l y a s s e s s s u b j e c t i v e - p r o b a b i l i t i e s , b u t t h e s e a s s e s s m e n t s s h o u l d b e c a r e f u l l y made i n t h e c o n t e x t o f p a s t r e s e a r c h . A s w i t h a n y t e c h - n o l o g y , h a p h a z a r d a p p l i c a t i o n l e a d s t o u n r e l i a b l e r e s u l t s .

C o a l e s c i n s c ~ o l u a j c a l O p i n i o n

A d o p t i n g a s u b j e c t i v i s t p h i l o s o p h y o f c o u r s e l e a d s t o t h e p r o b l e m o f d i f f e r i n g e x p e r t o p i n i o n , a n d s p e a k i n g o f

"good" a n d " b a d " a s s e s s o r s c e a s e s t o make s e n s e . P r o b a b i l - i t i e s r e f l e c t o n l y i n d i v i d u a l f e e l i n g s , w h i c h i n t u r n may n o t r e f l e c t r e a l i t y . T h e s e d i f f e r e n c e s a r e n o s u r p r i s e , h o w e v e r , a s t h e l i t e r a t u r e c o n t a i n s w i l d l y d i f f e r i n g

r e s o u r c e e s t i m a t e s a l r e a d y , a n d p o l i c y m a k e r s h a v e a l w a y s h a d t o d e a l w i t h d i f f e r i n g e x p e r t o p i n i o n s .

The t r a d i t i o n a l way t o c o a l e s c e d i f f e r i n g s u b j e c t i v e p r o b a b i l i t i e s h a s b e e n t h e D e l p h i m e t h o d , w h i c h i s a

d i s c u s s i o n a n d a v e r a g i n g p r o c e s s . T h i s p r o c e d u r e h a s r e c e i v e d c o n s i d e r a b l e c r i t i c i s m , b u t i s w i d e l y u s e d ( P i l l

[ 2 3 1 ) . A c t u a l l y , it i s more c o n s e n s u s s e e k i n g t h a n a t r u e c o a l e s c e n c e . H a r r i s [ I l l u s e s t h i s a p p r o a c h i n h i s m i n e r a l

p o t e n t i a l e s t i m a t e s o f S o n o r a .

A more r i g o r o u s method b a s e d e n t i r e l y on B a y e s i a n p h i l o s o p h y h a s b e e n r e c e n t l y p r o p o s e d by M o r r i s [ 1 9 , 2 0 1 , a n d t h i s a p p r o a c h c o u l d b e a d o p t e d f o r c o a l e s c i . n g g e o l o g i c a l o p i n i o n . Assume t h a t t h e u l t i m a t e p o l i c y a n a l y s t , c a n him- s e l f a s s i g n some p r i o r s u b j e c t i v e d i s t r i b u t i o n t o t h e e x t e n t o f d e p o s i t i o n o r m i n e r a l i z a t i o n . L e t t h e s e e s t i m a t e s b e

expressed in terms of two sets of parameters which cor- respond respectively to the size distribution of deposits

( 8 ) , and to spatial dispersion

( 1 2 ) -

. These prior probabili- ties could be taken as uniform. Opinion is taken individ- ually from several geologists in the same terms, that is, in the form of probability distributions on the parameters

8

and R. To the analyst or policy maker these probability

-

^-

distributions (representing expert opinion) are information and he may coalesce them by the normal Ra-yesian argument, using his own feelings as a prior,

f

'

(8,

- - R

1 ^experts ' ^opinion)

This formulation offers a rigorous relationship for coalese- ing expert opinion. The difficulties of evaluating

"credibility" of experts are concentrated (some might say transferred) to developing a likelihood function for their opinion conditioned on what the actual parametric values

- 8

and

-

R might be. While this is straightforward, it becomes untidy when the likelihoods of individual experts' opinions are not independent.

But how can the likelihood function be estimated? As

Morris argues, no matter how one proceeds with a statistical

analysis, likelihood functions are always established sub-

jectively. For convenience, we may assign families of dis-

tributions to those as we do to other things (e.g., a normal

likelihood) but always this is judgmentally done. Just as we assess subjective probability, so also we may assess likelihood functions based on the policy-analysts' or de- cision-maker's feelings relative to his expertst credibility.

This reflects the central argument in tavor ot all

quantitative decision analysis: quantitative analysis does not make decisions for the decision maker, rather it allows him to decompose a decision (or estimation), treat each part in isolation, then reaggregate in a logically consis- tent manner to draw deductive conclusions. Always, the conclusion drawn rests on the judgment of the person who draws it. To deny this is misleading.

A strength of this approach is that it allows the analyst also to establish the expected value of expert opinion (or the marginal expected value of an additional opinion). This process is established exactly as the

"expected value of sample information" is evaluated in any Bayesian Decision Theoretic application.

A Proposal for Including Geological Opinion in Resource Estimates

Entering a new estimation task there are four types of prior information to be included: individual experience, documented experience, geological theory, and local

characteristics. Were there only documented experience

and local conditions, priors could be generated by regression

or related techniques. However, individual experience and

theory serve to modify direct correlations with the "hard"

d a t a o f p r e v i o u s l y e x p l o r e d a r e a s by d e g r e e s t o w h i c h t h e r e g i o n u n d e r c o n s i d e r a t i o n i s o r i s n o t s i m i l a r t o p r e v i o u s a r e a s , and t h e ways i n w h i c h it seems a n o m a l o u s i n t e r m s of b a s i c g e o l o g i c a l p r o c e s s e s . IT c o m b i n i n g t h e s e s o u r c e s of i n f o r m a t i o n t h e g e o l o g i s t f u n c t i o n s somewhat a s a s u b j e c t i v e i n f o r m a t i o n p r o c e s s o r ( F i g u r e 2 ) .

The a p p r o a c h p r o p o s e d i s t h a t e a c h g e o l o g i s t b e g i v e n i n f o r m a t i o n i n t h e form of g e o l o g i c a l p r o p e r t i e s a n d e s - t i m a t e s o f ^- 8 a n d ^- 52 f o r g r o s s l y s i m i l a r r e g i o n s i n which more e x t e n s i v e e x p l o r a t i o n h a s b e e n c o n d u c t e d , and l o c a l c h a r a c t e r - i s t i c s o f t h e r e g i o n i n q u e s t i o n . Then t h r o u g h a p r o c e s s o f c a r e f u l q u e s t i o n i n g and gaming d i r e c t l y a s s e s s h i s f e e l i n g s a b o u t p o s s i b l e v a l u e s of

%

a n d Q,, t h e l o c a l p a r a m e t e r s , i n t e r m s o f p r o b a b i l i t y m e a s u r e s . T h i s p r o c e s s m i g h t b e e x - t e n d e d by p r e c o n d i t i o n i n g d a t a f r o m o t h e r r e g i o n s i n t e r m s of l o c a l c h a r a c t e r i s t i c s ( i . e . , r e g r e s s i o n o r f a c t o r a n a l y s e s a p p l i e d t o t h e new r e g i o n ) . I n t h i s way, e a c h e x p e r t b a s e s h i s judgment p r i m a r i l y upon t h e same h a r d d a t a s e t , and i n - c o r p o r a t e s h i s p a s t i n d i v i d u a l e x p e r i e n c e a n d c o n c e p t s o f g e o l o g i c a l p r o c e s s e s p u r e l y s u b j e c t i v e l y .

A s M o r r i s p o i n t s o u t , it i s n o t a s i m p l e t a s k t o a s - c e r t a i n t h e i n d e p e n d e n c e o f e x p e r t o p i n i o n . I f t h e o p i n i o n

i s i n d e p e n d e n t , t h e l i k e l i h o o d f u n c t i o n of e q u a t i o n ( 3 ) re- d u c e s t o t h e s i m p l e m u l t i p l i c a t i v e f o r m of t h e m a r g i n a l

l i k e l i h o o d s ; b u t i f i t d o e s n o t , i n t e r d e p e n d e n c i e s must b e m o d e l l e d , a n d t h e s e may h a v e complex a n d n o n - o b v i o u s f o r m s .

I n p a r t i c u l a r , i f e x p e r t s b a s e t h e i r j u d g m e n t s p a r t i a l l y upon t h e same d a t a , t h e n t h e i r o p i n i o n s a r e n o t i n d e p e n d e n t . P r o - c e e d i n g a s o u t l i n e d a b o v e , h o w e v e r , m i t i g a t e s t h i s d e p e n d e n c y by f o r m i n g o p i n i o n s w h i c h a r e z o n d i t i o n e d o n t h e d a t a s e t , a n d t h u s may b e c o n d i t i o n a l l y i n d e p e n d e n t w h i c h would a l l o w a s i m p l e m u l t i p l i c a t i v e f o r m .

M i t i g a t i n g t h e p r o b l e m o f d e p e n d e n c e c a u s e d by s i m i l a r g e o l o g i c a l t h e o r y i s n o t s o e a s i l y a c h i e v e d , a n d i n d e e d w i l l r e q u i r e f u r t h e r a t t e n t i o n t o t h e d e s i g n o f a s s e s s m e n t s c h e m e s .

The s e c o n d s t e p o f t h e p r o c e s s i s c o a l e s c i n g o p i n i o n . How c a n l i k e l i h o o d s o f g e o l o g i s t s ' o p i n i o n b e g e n e r a t e d ? C u r r e n t l y t h i s p r o b l e m i s d i f f i c u l t t o t r e a t e x c e p t i n s i m p l i s t i c w a y s , b u t t h e t h e o r e t i c a l b a s e o f t h i s a p p r o a c h i s o n l y now e x p a n d i n g ( e . g . , M o r r i s [ 2 0 ] ) . A s a f i r s t a p p r o x i m a t i o n o n e c a n a s s u m e t h a t t h e l i k e l i h o o d of a p r e d i c t i o n i s r e l a t e d o n l y t o t h e a b s o l u t e v a l u e o f t h e d i s c r e p a n c y f r o m t h e t r u e p a r a m e t r i c v a l u e . T h a t i s , t h a t e x p e r t s ' o p i n i o n s a r e u n b i a s e d a n d t h a t e r r o r i s s y m m e t r i c a b o u t t r u e v a l u e s . I f o n e a s s u m e s a s i m p l e a n a l y t i c a l f u n c t i o n , e . g . , a n o r m a l d i s t r i b u t i o n , t o r e p r e s e n t t h i s e r r o r , t h e n 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 i s a s u f - f i c i e n t d e s c r i p t i o n of e x p e r t c r e d i b i l i t y . I t would f a l l t o t h e a n a l y s t o r p o l i c y maker t o s u b j e c t i v e l y d e c i d e upon v a l u e s o f t h i s v a r i a n c e ( i . e . , " c r e d i b i l i t y " ) f o r e a c h e x p e r t h e c o n s u l t s - - b u t t h i s i s a l w a y s t h e t a s k o f t h e a n a l y s t

w h e t h e r h e a c h i e v e s it q u a n t i t a t i v e l y o r q u a l i t a t i v e l y .

Symbolically, this analysis is of the form

Where fO(-) is the probability distribution used as a prior in subsequent resource estimates, fa

( )

is the analysts prediction of the parameters (which might be uniform),

fi

( a )

is the ith geologist Is prediction, and fn(- l~o,fio,z)

is the normal distribution (in this case the likelihood) with mean ~ o , ~ o (i.e., the assumed true values) and variance matrix -C.

^{- -}

As a first approximation, it seems reasonable

to assume that errors in the estimate of

_- 8,

the parameters of the size distribution, and R ,

_-

the parameters of spatial dispersion are independent. So,

in which oie is the credibility assigned to geologist i's estimate of

- C ,

and oiR is the credibility assigned to his estimate of

-

R.

The approach just described, clearly, is very rough.

Considerable effort, and in particular attempts to apply such methodologies, would need to be invested before a

workable and practical procedure could be developed. Never-

theless, an approach somewhat of the type outlined is needed

to analytically include geological opinion within the context

o f r e g i o n a l r e s o u r c e e s t i m a t i o n . I g n o r i n g t h i s p r i o r

i n f o r m a t i o n l e a d s t o e s t i m a t e s w h i c h a r e n o t c o m p r e h e n s i v e , o v e r l y d . i f f u s e , a n d p o s s i b l e e r r o n e o u s .

4 . 2 L i k e l i h o o d F u n c t i o n

I n t h e B a y e s i a n s c h e m e , e q u a t i o n ( 2 ) , c h a r a c t e r i s t i c s o f t h e s a m p l i n g p l a n a r e e n t i r e l y c o r l t a i n e d w i t h i n t h e l i k e l i h o o d

f u n c t i o n . T h i s i s t h e p r o b a b i l i t y o f o b t a i n i n g t h e s a m p l e a c t u a l l y o b s e r v e d - - t h a t i s , t h e d e p o s i t s a c t u a l l y d i s c o v e r e d - - c o n d i t i o n e d o n v a l u e s of t h e p a r a m e t e r s - 8 and -

R.

T h i s p r o b a b i l i t y may o r may n o t d e p e n d o n t h e o r d e r o f d i s c o v e r y .

F o r s i m p l e random s a m p l i n g e a c h o b s e r v a t i o n i s assumed i n d e p e n d e n t , a n d t h e i r o r d e r i n g u n i m p o r t a n t . I f d e p o s i t s o f s i z e x _i a r e d i s c o v e r e d i n t h i s way, t h e i r l i k e l i h o o d i s

w h e r e f ( a 1 8 ) - i s t h e d i s t r i b u t i o n o f d e p o s i t s i z e s f r o m w h i c h d i s c o v e r i e s a r e made.

A s Kaufman [ I 5 1 p o i n t s o u t , h o w e v e r , d i s c o v e r i e s o f m i n e r a l d e p o s i t s d o n o t f o l l o w a s i m p l e - r a n d o m p r o c e s s . F i r s t , t h e t o t a l p o p u l a t i o n o f i n s i t u d e p o s i t s i s f i n i t e ; a n d s e c o n d , l a r g e r d e p o s i t s h a v e a g r e a t e r p r o b a b i l i t v o f b e i n g f o u n d t h a n s m a l l e r o n e s . Once a d e p o s i t i s f o u n d i t i s "removed" f r o m t h o s e w h i c h m i g h t s t i l l b e f o u n d , a n d

thus s m p l i n g is "without replacement." This means that the order of discovery

-

is important.

IZaufman assumes that deposits appear in the sample with probability proportional to the ratio of their size to the cumulative size of still undiscovered deposits. This is the probability relative to other deposits appearing, or the probability conditioned on a discovery. He also postulates that in situ deposits he considered a simple-random sample from some infinite super-population, fs(x/e), then inters values of the parameters of that distribution. Considering equation (1) once again, this approach allows inferences on the distribution of the random variables x . in the resource

1

estimate, and also inferences about the sum of undiscovered sizes. It does not allow direct inferences of the in situ

number, N.

I

I

While this procedure offers an approach to estimating total ~

resources, it does not make use of all available information,

and does not yield spatial characteristics which might be

used in optimizing future exploration strategies. However,

it may be expanded to include the likelihood of numbers of

deposits being discovered and the non-uniform geographic

distribution of exploration, and thus to overcome these

objections.

Spatial Dispersion Function

The spatial dispersion of mineral deposits is most often treated as a point process in two dimensions.

²

Parameters of the spatial dispersion model are then esti- mated by dividing the geographic region into uuadrats and fitting curves to the distril,lltion of numbers of deposits per quadrat.

Empirical data displays more clustering than the Poisson model would predict, thus other mod.els have heen considered and at present there seems widespread satis- faction with the negative binomial model (DeGeoffroy and Wu [5] , ^Griffiths

^{[ 9 ]}

, Uhler and Bradley [32] --other models are discussed in 3ogers [251). Anlong the few criticisms of the negative binomial is that it tends to underestimate the frequency of quadrats with high numbers of deposits (Kaufman and Bradley [161) (Figure 3).

Previous work typically assumes the numher of known deposits per quadrat to be mutually independent samples from the spatial dispersion process; thus,

where q is the number of quadrats, ni the number of known deposits in quadrat i, and

- R

the parameters of the spatial

2 Kaufman and Bradley's [16] random-walk simulation

is one of the few exceptions.

dispersion process (whose values are to be inferred). This procedure leads to results which are ditficult to interpret for the following reasons:

1) Known numbers of deposits are not samples from the spatial process p(9lil) - but are lower bounds on the actual numb€.- in a quadrat.

2)

If the analysis is restricted to intensively

explored quadrats, which would yield truer samples of p(Nl

_{- S 2 ) ,}

the sample of quadrats is biased toward greater density (i.e., the most intensively

explored quadrats are also the ones with the most extensive mineralization or deposition).

3 )

If very sparsely explored quadrats are included,

the sample is biased toward low numbers per quadrat; the probability of discovering in situ deposits in these quadrats is small.

This approach clearly leads to incorrect estimates.

Search Effort

The number of deposits found in exploration obviously depends on the amount and spatial allocation of search effort. If this effort is non-uniformly distributed

geographically, then the probability of d-iscovery is non- uniform also. Although this principle is intuitively clear, it may explain certain anomalies in resource modelling,

and may lead to mitigation of the three objections just

mentioned.

Assume temporarily that deposits were actually dispersed according to a negative binomial process. Then let one

deposit be found in some quadrat, c, as shown in Figure

4.

As deposit locations are positively correlated, this increases the favorability of quadrat c for containing additional deposits. That is, the probability of c con- taining at least one more deposit is increased from

0.19 to 0.52 (using DeGeoffroy and Wu's parameters).

Therefore, an optimal exploration strategy would be to allocate more effort to exploring quadrat c than other quadrats. Since this process feeds back upon itself as more discoveries are made, high

_n

quadrats appear in

observations with probability disproportionately higher than their frequency in situ. Thus the objection of

Kaufman and Bradley may only reflect non-uniform exploration.

Returning to equation

( 7 ) ,

one sees that the likelihood is not merely the spatial dispersion model, hut must be

modified by the probability of finding in situ deposits.

Me will call this latter relation the detection function.

The detection function has the property that when there is no exploration effort

( + = 0 )

the probability of a discovery is zero (p(n

=

0)

=

1.0), and as +

⁺

^p(n

⁼

^N)

⁺

^1.0.

Here n and N are the number of discovered deposits and the

total number of in situ deposits, respectively. So, as is

intuitively clear, the probability of discovering deposits

within a quadrat depends on the number of deposits present

and the effort exerted to find them.

Form of the Detection Function

While the detection function begins at zero and reaches an asymptote of 1.0, its exact form depends on the strategy of allocating search effort and the distribution of deposit sizes.

Consider a quadrat of a r ? a A which contains a single deposit of area a. If

$

units of search effort are randomly allocated to points within the quadrat, the probability of finding the deposit is (Figure

5 ) ,

If a systematic allocation is used (i.e., a grid), tken p(find) depends both on the target and grid geometries, as illustrated in Figures 6a to

6 3 .

Similar curves can be

generated for other systematic allocations (e.g., geophysical methods) or for "optimal search" when prior locations

probabilities can be specified (Morse [21]) .

Without detailed information on the way exploration has been conducted, there is no way to precisely reconstruct the detection function. Therefore, in making resource

estimates we must make assumptions on its shape. On one

hand, exploration may be viewed as the uncoordinated. effort

of many separate decision makers. If this is so, then a

random model seems appropriate. On the other hand,

e x p l o r a t i o n may b e c a r r i e d o u t by o n e d e c i s i o n maker a s i n t h e c a s e o f a g o v e r n m e n t m i n i s t r y o r l a r g e c o r p o r a t i o n . Were t h i s t h e c a s e , t h e n a p u r e l y s y s t e m a t i c model m i g h t b e a p p r o p r i a t e . B o t h a r e c r u d e a p p r o x i m a t i o n s , b u t p e r h a p s s a t i s f a c t o r y f i r s t a t t e m p t s .

B o t h random a n d g r i d s e a r c h c a n b e a p p r o x i m a t e d by a n e x p o n e n t i a l d e t e c t i o n f u n c t i o n o f t h e form 3

i n w h i c h k i s a c o n s t a n t . T h i s c a n b e m o d i f i e d f o r u n c e r - t a i n t y i n d e p o s i t s i z e i n t h e n o r m a l way,

3 ~ y a n ' s 1 2 6 , 2 7 1 d e t e r m i n i s t i c model o f d i s c o v e r y

w i t h i n a p l a y i s o f t h i s f o r m , t h o u g h h e d o e s n o t d i r e c t l y t r e a t it a s a d e t e c t i o n f u n c t i o n . I n h i s model c u m u l a t i v e - n e w f i e l d - w i l d c a t s i s u s e d a s a m e a s u r e o f

+,

a n d h e i n t r o - d u c e s a c o n s t a n t m u l t i p l i e d by

+

t o a c c o u n t f o r " g e o l o g i c a l k n o w l e d g e . " To f i n d t h e r e g i o n a l r e s o u r c e h e e q u a t e s r a t e o f new d i s c o v e r i e s t o t h e p r o d u c t o f r e s o u r c e a n d d e t e c t i o n f u n c t i o n .

w h e r e

R = r a t e o f d i s c o v e r y , U = t o t a l r e s o u r c e ,

m

B,k = c o n s t a n t s ,

w = c u m u l a t i v e new w i l d c a t s .

T h i s d e t e r m i n i s t i c model c l o s e l y f i t s e m p i r i c a l r a t e s o f d i s c o v e r y w i t h i n i n d i v i d u a l p l a y s , a n d t h u s a d d s c r e d i b i l i t y t o t h e random e x p l o r a t i o n model. However, h i s e q u a t i o n h a s t h r e e a d j u s t a b l e p a r a m e t e r s a n d t h u s i s f l e x i b l e .

To form the likelihood function for inferring values of the spatial parameters,

-

R, the number of discoveries must be related to the number of in situ deposits by an equation of the form

Here, p(nlN,q) is a modification of the detection function to account for multiple deposits, and ~ ( N J R )

-

is the spatial dispersion process. Unfortunately, p(nl~,$) is not a simple relationship.

Let n deposits be found in a particular quadrat in the order

alI a2, . - . ^an

^I

with

$2,

^--• I

qn

increments of exploration effort, respectively. Given that the first j-1 of these have been found, the probability of finding the jth with one additional quantum of effort is

n N

I ^.ai - I ^ai

i=j i=n

p(findlalI. ..,aj-l,~=l)

=

pj

=

j-1

I

A - 1 ^ai

i=l (12)

and the probability of having discovered the jth deposit

with the increment of effort, qjI is

x Pr(a find) x f(ajlg) j

As N is a random variable with parameters

-

R, this becomes

in which the term S=

1

ai, the sum of undiscovered deposits.

i=n

is an uncertain quantity depending both on

- O and

-

R.

The likelihood of discoveries then is

1

expt-k$.p.lpj 3 3 J f ( a . 10)

N ^{3 -}

L

^ai

+ s

i-j

Clearly, this equation is difficult to deal with, although as Kaufman has done, this might be approached by Monte Carlo simulation. It does account for exploration effort, however, and conceptually at leastallows inferences to be drawn about the spatial dispersion of deposits.

The point of this short discussion is that inferences about the number of deposits in a quadrat or region (and thus about their spatial dispersion and the total amount of resources) must account for how and how hard they were looked for. Further, inferences about spatial dispersion

are not independent of inferences about size distribution; the simple-random sampling model is not satisfactory for this

purpose.

Conclusions

This paper has discussed the place of sampling approaches to resource estimation in a broad context, and it has attempted 1

to indicate that sampling approaches could lead to a more com- prehensive analysis than is currently employed. Specifically, discussion has concentrated on three points about exploration I 1

and inferences drawn from it: I

a) Geological exploration is fundamentally and

necessarily a subjective undertaking; prior

judgment ot geologists based on tindings in

other regions and on concepts of geological

processes must be included.

b) The analytical methods for including geological opinion from multiple experts must be theoret- ically rigorous and reflect current knowledge of probability assessment, judgmental biases, and subjective info-rmation processing.

C)

The procedure used for modifying prior opinion by the local results of exploration should

include consideration of exploration effort and

its allocation through some detection function.

SUPER POPULAT

F I G U R E 1

DOCUMENTED EXPERIENCE

1 ^GEOLOGIST I

GEOLOGICAL SCIENCE

I ^REGIONAL I

I ^GEOLOGY

t EXPERIENCE

SPERSION

I ^INITIAL

I PREDICTIONS I

F I G U R E 2

DEGEOFFROY &WU 1.0 (1970)

SLICHTER (1960)

UHLER & BRADLEY (1970)

FIGURE 3: FIT OF 'THE NEGATIVE

BIN OMlNAL TO EMPIRICAL DATA

OBSERVED

0 ^PREDICTED

NUMBER OF DEPOSITS

PER QUADRAT

QUADRAT

FlGLlRE4:PROBABILITIES OF31 AD- DITIONAL DEPOSIT CONDITIONEG ON DISCOVERIES- NEGATIVE B I - ~ NOMIAL DISTRI ., DATA AFTER ~

DEGEOFFROY & WU ,1970 ~

NUMBER OF FINDS, n, I N QUADRAT

0.4 0.6 0.8 1 .O DIAMETER / GRID SPACING

C.

-5-u ( FROM BAECHER ,1972) ,85 0o *'' - - / PN

-

BLACKED - IN SYMBOLS IEJDICATE NUMERICALLY EVALUATED DATA - I 1 I I I FIGURE 6a : PROBABILITY OF FINDING CIRCULAR AND SQUARE BODIES WITH SQUARE POINT GRID ( EFFORT PROPORTIONAL TO ( GRID SPACING)-2

h= alb 1 .O 2 .O 3.0 GRID SPACING/SQUARE ROOT OF BODY AREA Fi GURE 6c : PROBABILITY OF Fl NDlNG ELLIPTICAL BODIES WITH SQUARE POINT GRIDS.

A= alb 1 .o 2.0 3 .o GRID SPACING / SQUARE ROOT OF BODY AREA FIGURE 6d:PROBABILITY OF FINDING RECTANGULAR BODY WITH SQUARE POINT GRID

R e f e r e n c e s

[ l ] A l l a i s , M . "Method o f A p p r a i s i n g E c o n o m i c P r o s p e c t s o f M i n i n g E x p l o r a t i o ? - , o v e r L a r g e T e r r i t o r i e s . "

Management S c i e n c e , 3 ( 1 9 5 7 ) , 2 8 5 - 3 4 7 . -

[ 2 ] B a e c h e r , G.B. " S i t e E x p l , . ~ r a t i o n : A p r o b a b i l i s t i c a p p r o a c h . " Ph.D. d i s s e r t a t i o n , M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , 1 9 7 2 .

[ 3 ] Box, G.E. a n d T i a o , G.C. B a y e s i a n I n f e r e n c e i n --- S t a t i s t i c a l A n a l y s i s . A d d i s o n - W e s l e y , 1 9 7 3 . [ 4 ] D e G e o f f r o y , J . a n d W i g n a l l , T . K . " A P r o b a b i l i s t i c

A p p r a i s a l o f M i n e r a l R e s o u r c e s i n a P o r t i o n o f t h e G r e n v i l l e P r o v i n c e o f t h e C a n a d i a n S h i e l d . "

E c o n o m i c G e o l o g y , - 66 ( 1 9 7 1 ) , 4 6 6 - 4 7 9 .

[51 D e G e o f f r o y , J . a n d Wu, S.M. "A S t a t i s t i c a l S t u d y o f O r e O c c u r r e n c e s i n t h e G r e e n s t o n e B e l t s o f t h e C a n a d i a n S h i e l d . " E c o n o m i c G e o l o g y , - 6 5 ( 1 9 7 0 ) , 496-504.

161 E d w a r d s , W . " C o n s e r v a t i s m i n Human I n f o r m a t i o n P r o c e s s i n g " i n F o r m a l R e p r e s e n t a t i o n o f Human J u d g m e n t . E d . , B . K l e i n m o n t z . W i l e y , 1 9 6 8 , p p . 1 7 - 5 2 .

[ 7 ] E r i c s o n , W.A. " S u b j e c t i v e B a y e s i a n M o d e l s i n S a m p l i n g 3 1 F i n i t e ~ o ~ u l a t i o n s . " J o u r . Roy. S t a t . S o c . ,

B,

-

( 1 9 6 9 )

,

1 9 5 - 2 3 3 .

[ 8 ] G r a y s o n , C . J . D e c i s i o n s U n d e r U n c e r t a i n t y : D r i l l i n g D e c i s i o n s b y O i l a n d Gas O p e r a t o r s . H a r v a r d U n i v e r s i t y B u s i n e s s S c h o o l , 1 9 6 0 .

[ 9 ] G r i f f i t h s , J . C . " E x p l o r a t i o n f o r N a t u r a l R e s o u r c e s . "

O p e r . R e s . ,

-

1 4 ( 1 9 6 6 )

,

1 8 9 - 2 0 9 .

[ l o ] G r i f f i t h s , J . C . a n d S i n g e r , D . A . " U n i t R e g i o n a l V a l u e o f Non-Renewable N a t u r a l R e s o u r c e s a s a M e a s u r e o f P o t e n t i a l f o r D e v e l o p m e n t o f L a r g e R e g i o n s . "

S p e c . P u b l s . G e o l . S o c . A u s t . , - 3 ( 1 9 7 1 ) , 2 2 7 - 2 3 8 . [ l l ] H a r r i s , D.P. "A S u b j e c t i v e P r o b a b i l i t y A p p r a i s a l o f

Metal Endowment o f N o r t h e r n S o n o r a , M e x i c o . "

E c o n . G e o l . , 6 8 ( 1 9 7 3 )

- ,

222-242.

H a r r i s , D.P. "An A p p l i c a t i o n o f M u l t i v a r i a t e S t a t i s t i c a l A n a l y s i s t o M i n e r a l E x p l o r a t i o n . " Ph.D. d i s s e r t a t i o n , P e n n . S t a t e U n i v e r s i t y , 1 9 6 5 .

H a r r i s , D . P . , F r e y m a n , A . J . , a n d B a r r y , G.S. "The

M e t h o d o l o g y Employed t o E s t i m a t e P o t e n t i a l M i n e r a l S u p p l y o f t h e C a n a d i a n N o r t h w e s t - - A n A n a l y s i s B a s e d u p o n G e o l o g i c O p i n i o n a n d S y s t e m s S i m u l a t i o n . "

9 t h I n t l . Symp. T e c l - n i q u e s f o r D e c i s i o n - M a k i n g i n t h e Min. I n d . . 1 9 7 0 .

H u b b e r t , M . K . " E n e r g y R e s o u r c e s " i n R e s o u r c e a n d Man,

N a t i o n a l Academy o f S c i e n c e s . F r e e m a n a n d Company, 1 9 6 9 . Kaufman, G . M . " S t a t i s t i c a l M e t h o d s f o r P r e d i c t i n g t h e

Number a n d S i z e o f U n d i s c o v e r e d H y d r o c a r b o n D e p o s i t s "

i n M e t h o d s o f E s t i m a t i n g t h e Volume o f U n d i s c o v e r e d O i l a n d G a s R e s o u r c e s , K . H . C r a n d a l l a n d J . W . H a r b a u g h . S t a n f o r d , 1 9 7 4 .

Kaufman, G . M . a n d B r a d l e y , P.G. "Two S t o c h a s t i c M o d e l s U s e f u l i n P e t r o l e u m E x p l o r a t i o n . " Am. A s s n . P e t . G e o l . , A r c t i c G e o l . M e m . , - 1 9 ( 1 9 7 3 )

,

6 3 3 - 6 3 7 .

MacAvoy, P.W. a n d P i n d y c k , R.S. " A l t e r n a t i v e R e g u l a t o r y P o l i c i e s f o r D e a l i n g w i t h t h e N a t u r a l G a s S h o r t a g e . "

B e l l J o u r . M g t . S c i . , - 5 ( 1 9 7 4 ) , 454-498.

M o o r e , C.L. " L i m i t a t i o n s o f S t a t i s t i c a l M e t h o d s f o r P r e d i c t i n g P e t r o l e u m a n d N a t u r a l G a s R e s e r v e s a n d A v a i l a b i l i t y - - A R e p l y . " J o u r . P e t . T e c h n o l o g y ,

-

1 8 , 3

( 1 9 6 6 )

,

2 8 6 .

Morris, P.A. B a y e s i a n E x p e r t R e s o l u t i o n . Ph.D. d i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y , 1 9 7 1 .

Morris, P.A.

S c i e n c e ,

" D e c i s i o n A n a l y s i s E x p e r t 20 ( 1 9 7 4 )

,

1 2 3 3 - 1 2 4 1 .

U s e .

"

Management M o r s e , P.M. S e a r c h T h e o r y . O p e r . R e s . C e n t e r , M I T .

W o r k i n g P a p e r 0 1 - 7 4 , 1 9 7 4 .

Murphy, A . H . a n d W i n k l e r , R.L. " C r e d i b l e I n t e r v a l T e m p e r a t u r e F o r e c a s t i n g : Some E x p e r i m e n t a l R e s u l t s . " M o n t h l y

W e a t h e r R e v i e w , 1 0 2 ( 1 9 7 4 )

,

7 8 4 - 7 9 4 .

P i l l , J . " T h e D e l p h i Method: S u b s t a n c e , C o n t e x t ,

A C r i t i q u e , a n d An A n n o t a t e d B i b l i o g r a p h y . " S o c i o . - E c o n . P l a n . S c i . , 5 ( 1 9 7 1 ) , 57-71.

R o b i . n s o n , J . M . " N a t u r e a n d S i g n i f i c a n c e o f G e o l o g i c a l Maps" i n F a b r i c o f G e o l o g y . E d . , C.C. A l b r i t t o n . F r e e m a n a n d Company, 1 9 6 3 .

R o g e r s , A . S p a t i a l D i s p e l s i o n . P i o n M o n o g r a p h s i n

~ n v i r o n m e n t a l a n d S p a t i a l S y s t e m s . P i o n , 1 9 7 4 . R y a n , J . T . "An A n a l y s i s 2 f C r u d e - O i l D i s c o v e r y R a t e

i n A l b e r t a . " ~ u l l . C a n . P e t . G e o l . ,

-

2 1 ( i 9 7 3 )

,

2 1 9 - 2 3 5 .

R y a n , J . T . "An E s t i m a t e o f t h e C o n v e n t i o n a l C r u d e - O i l P o t e n t i a l i n A l b e r t a . " B u l l . C a n . P e t . G e o l . , 2 1 _-

-

( 1 9 7 3 )

,

2 3 6 - 2 4 6 .

S a v i n s k i i , J . P r o b a b i l i t y T a b l e s f o r L o c a t i n g E l l i p t i c a l U n d e r g r o u n d M a s s e s w i t h a R e c t a n g u l a r G r i d . N e w Y o r k , C o n s u l t a n t s B u r e a u , 1 9 6 5 .

S i n g e r , D . A . a n d Wickman, F . E . " P r o b a b i l i t y T a b l e s f o r L o c a t i n g E l l i p t i c a l T a r g e t s w i t h S q u a r e , R e c t a n g u l a r a n d H e x a g o n a l P o i n t - N e t s

. ^"

M i n e r a l S c i

.

E x p . S t a .

,

P e n n . S t a t e U n i v e r s i t y . S p e c . P u b . 1 - 6 9 , 1 9 6 9 .

S l i c h t e r , L.B. " T h e Need o f a New P h i l o s o p h y o f P r o s p e c t i n g . "

Min. E n g r . ,

-

1 2 ( 1 9 6 0 ) , 5 7 0 - 5 7 6 .

T v e r s k y , A . " A s s e s s i n g U n c e r t a i n t y . " J o u r . Roy. S t a t . S o c . ,

36 ( 1 9 7 4 ) , 1 4 8 - 1 6 0 . i

U h l e r , R.S. a n d B r a d l e y , P.G. "A S t o c h a s t i c Model f o r D e t e r m i n i n g t h e E c o n o m i c P r o s p e c t s o f P e t r o l e u m

E x p l o r a t i o n o v e r L a r g e R e g i o n s . " J o u r . Am. S t a t . A s s n . , 6 5 ( 1 9 7 0 ) , 6 2 3 - 6 3 0 .

-

W i n k l e r , R.L. a n d Murphy, A . H . "On t h e G e n e r a l i z a b i l i t y o f E x p e r i m e n t a l R e s u l t s " i n T h e C o n c e p t o f P r o b a b i l i t y i n P s y c h o l o g i c a l E x p e r i m e n t s . E d . , C. -A.S

.

S t a E l v o n H o l s t e i n . D o r d r e c h t - I J o l l a n d , R e i d e l P u b l i s h i n g Company, 1 9 7 4 , p p . 1 0 3 - 1 2 6 .

Wood, E . F . "A B a y e s i a n A p p r o a c h t o A n a l y z i n g U n c e r t a i n t y Among S t o c h a s t i c M o d e l s . " I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s , L a x e n b u r g , A u s t r i a , RR-74-16,

1 9 7 4 .