Regionalized Favorability Theory for Information

8/18/2019 Regionalized Favorability Theory for Information

1/29

Mathemat ical Geology , Vol . 25 , No. 5 , 1993

R e gion a l i z e d Favor ab i l i ty Th e or y for I n for mat ion

S yn th e s is in M in e r a l Ex p lor at ion 1

G u o c h e n g P a n 2

T hi s pap e r pre se n t s a r e g i ona l iz e d m e t hod f o r t he e s t im a t i on o f a f av orab i l i t y f unc t i on t h rough

general i zat ion o f a l l re levant var iables (explanatory an d target ) in to random func t io ns . The new

m e t hod a l l ows t he use o f cros s -c ov ari anc e f unc t i ons i n add it i on t o o rd i nary c ov ar i anc es f o r e x -

t rac t ing spat ia l jo in t in forma t ion, which i s v i r tual ly over looked in the convent ional analyses . The

op t i m a l we i gh t s f o r a f av orab i l i t y e qua t i on are de r i v e d f rom so l v ing a ge ne ra l i z e d e i ge n -sy s te m

e s t ab l ishe d by t he m ax i m i za t ion o f c ov ar i anc e s be t we e n a f av orab i l i t y f unc t i on and t he pr i nc i pa l

comp onents o f a se t o f pre -s e lec ted targe t variables . Var ious corre lat ion coe ff ic ients ma y be com-

pu t e d t o a s s is t i n i n t e rpre t a ti on o f the f av orab i l i t y e s t im a t e s . Bo t h f av ora b i l #y f unc t i ons and c or -

re la t ion coe ff ic ients ma y be es t imated fo r a po int or a b lock . The regional i zed fav ora bi l i ~ theory

c an be c om pa re d t o c ok r i g ing i n tha t bo t h use t he sam pl e - sam pl e c ov ar i anc es t o ac c oun t f o r t he

sam pl e - sam pl e r e l a ti ons and t he po i n t - sam pl e c ov ar i anc es t o ac c oun t f o r t he po i n t - sa m p l e c onf ig -

urations . The ne w technique i s dem onstra ted on a te s t case s tudy , w hich involves the in tegrat ion

o f ge oc he m i c a l , a i rborne -ge ophy s i c a l, and s t ruc tura l da t a s e t s f o r t he t a rge t s e l ec t ion o f hy dro -

t he rm al go l d - s i l v e r de pos i ts .

K E Y W O R D S regionalized favorability theory, cross-variogram, information synthesis, explo-

ration target, airborne geophysics, cokriging.

I N T R O D U C T I O N

I n f o r m a t i o n s y n t h e s is o f m u l t i p le g e o l o g i c a l , g e o c h e m i c a l , a n d g e o p h y s i c a l d a t a

i s a c e n t r a l t a s k i n t h e s e l e c t i o n o f m i n e r a l e x p l o r a t i o n t a r g e t s a n d t h e q u a n t i t a t iv e

e s t i m a t i o n o f m i n e r a l r e s o u r c e s . M u l t iv a r i a te s t a t is ti c al a n a l y s i s w a s i n t ro d u c e d

a s a m a j o r t o o l f o r c o m b i n i n g m u l t i p l e g e o - m a p p a t t e rn s o r l a y e r s ) in o r d e r t o

e s t im a t e e i t h er f a v o m b i l it y o r p r o b a b il it y o f o c c u r r e n c e o f k n o w n a n d u n k n o w n

m i n e r a l d e p o s i ts H a r r i s, 1 9 8 4 ) . A t y p i c a l f a v o r a b i l it y fu n c t i o n is a l in e a r c o m -

b i n a t i o n o f a s e t o f r e l e v a n t g e o s c i e n c e v a r i a b l e s , w h i c h c a n b e q u a n t i t a t i v e ,

n o m i n a l b i n a r y o r t e r n ar y ) , o r c a t e g o r i c a l . A c o m m o n p r a c t ic e in m i n e r a l e x -

~Received 9 A pril 199 2; accepted 30 June 1992.

2Independence M ining Company, 525 1 DTC Parkway, 3700 Englewood, Colorado 80111.

603

0882 8121/93/0700 0603507.00/I © 1993

Internat ional s socia t io n for Mathematica l

GeoJogy


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6 4 P a n

p lo r a t ion a nd r e sou r c e a s se s sm e n t is t o c o l l e c t a ll pos s ib l e r e l e va n t ge o log ic a l ,

ge oc he mic a l , ge ophys i c a l , a nd r e mote s e ns ing da t a . Ge o- da t a a r e o f t e n d ive r se

in f o r m; some a r e nume r i c a l , e . g . , so i l ge oc he mic a l s a mple s a nd ma gne t i c

in t e ns i ty me a su r e me n t s , wh i l e o the r s a r e nonnume r i c a l , e . g . , hyd r o the r ma l a l -

t e r a t ion , l i t ho logy , a nd f a u l t. I n t he p r a c t ic e o f mine r a l e xp lo r a t ion , t he se da t a

se ts a r e qua n t i f i e d , un i f ie d , f i l t er e d , e nha nc e d , a nd f ina l ly c omb ine d to p r oduc e

a f a vo r a b i l i t y o r p r oba b i l i t y m e a su r e f o r oc c u r r e nc e o f mine r a l de pos i t s o f c on -

c e r n .

A n u m b e r o f m e t h o d s h a v e b e e n p r o p o s e d i n t h e la s t d e c a d e f o r e s ti m a t in g

a f a vo r a b i l i t y f unc t ion ba se d on a s e t o f ge o log ic a l f e a tu r e s . C ha r a c t e ri s t ic a na l -

y s i s ( B o tbo l e t a l . , 1978 ; Mc C a m m on e t a l. , 1983 ; P a n a nd W a n g , 1987) i s a

we l l known e xa mple . A f e w o the r no ta b l e a lgo r i t hms inc lude c ons t r a ine d l e a s t

squa r e s ( Luo , 1990) , t he sub je c t ive we igh t ing ( R e d dy a nd Koc h , 1988a , b ), a nd

the we igh te d c r i te r ion (Ha r r i s a n d P a n , 1987 , 1990 ; P a n , 1989) . C a non ic a l

c o r r e l a t i on a na lys i s (S e be r , 1984 ; Johnson a nd W ic he r n , 1988 ; Da v i s , 1986) i s

a l so a u se fu l a pp r oa c h f o r t he e s t ima t ion o f a t yp i c a l c ombina t ion . Th e e s t im a te d

inde x usua l ly c ha r a c t e r i z e s f a vo r a b le de g r e e s o f a mine r a l oc c u r r e nc e a t e a c h

spa t i a l s a mple l oc a t ion . The inde x va lue s a r e c ommonly e s t ima te d in a p r e -

d e t e r m i n e d e q u a l - a r e a g r id a n d t h e y m a y b e c o n t o u r e d t o p r o d u c e a f a v o ra b i li ty

ma p . Th e r e g ions show ing the h ighe s t f a vo r a b i l i t y va lue s a r e i de n ti f ie d a s min -

e r a l e xp lo r a t ion ta r ge t s. Th e f a vo r a b i l i t y g r id ma y a l so be c onve r t e d to a n ima ge ,

wh ic h c a n be f i l t e r e d , c o lo r e d , a nd e nha nc e d in va r ious wa ys in o r de r t o p r e -

c i s e ly de l ine a t e t he bound a r i e s o f e xp lo r a t ion t a rge t s .

Mos t t e c hn ique s f o r e s t ima t ing a f a vo r a b i l i t y f unc t ion c a r r y c e r t a in a mbi -

gu i t ie s i n t e r ms o f t he m e a n ing o f t he e s t ima te s ( P a n a nd H a r r i s , 1992a ). I n

pa r t i c u l a r , t he e s t ima te d inde x va lue s ma y no t ne c e s sa r i l y imp ly the de g r e e o f

f a vo r a b i l i t y o f mine r a l oc c u r r e nc e whe n d i r e c t i n f o r ma t ion a bou t mine r a l de -

pos i t s i s no t e xp l i c i t l y i nc o r po r a t e d in to t he e s t ima t ion . M os t e xp la n a to r y va ri -

a b l e s i n t he f unc t ion u sua l ly p l a y a r o l e i n t he i nd i r e c t i n f e r e nc e o f t he pos s ib i li t y

o f mine r a l oc c u r r e nc e s . W he th e r a n e s t ima te o f the f a vo r a b i l i t y func t ion is

m e a n ing f u l o r no t t o mine r a l oc c u r r e nc e l a r ge ly de pe nd s upon the c r i t e r ion f o r

de r iv ing the e s t ima te a nd the i n f o r ma t ion c a r r i e d by e xp la na to r y va r i a b l e s . I n

o r de r t o mi t iga t e t he se d i f f i c u l t i e s , P a n a nd Ha r r i s ( 1992a ) p r opose d a me thod ,

r e f e r r e d to a s t he c a non ic a l f a vo r a b i l i t y me thod , wh ic h wa s de mons t r a t e d on

t a r g e t s e le c t io n f o r h y d r o t h e r m a l g o l d - s i l v e r d e p o s i ts i n th e W a l k e r L a k e q u a d -

r a n g le o f N e v a d a a n d C a l i fo r n i a .

Th e a m oun t o f i n f o r m a t ion e x t r a c t e d by a s t at is t ic a l mode l i s a ke y c r i t e rion

in j udg ing the m ode l f o r a pa r ti c u l a r da t a s e t a nd f o r a g ive n s tudy ob je c t ive .

W he n a da t a s e t c on ta in s on ly a s ing l e va r i a b l e o r a g r oup o f i nde pe nde n t

va r i a b l e s , a f u l l e x t r a c t ion o f t he ma r g ina l i n f o r ma t ion f r om e a c h ind iv idua l

va r i a b l e w ou ld su f f ic e t o a t t a in t he be s t e s t im a te o f the f a vo r a b i l i ty f unc t ion .

W he n the va r i a b l e s a r e i n t e r r e l a t e d , howe ve r , qua n t i f i c a t ion o f t he j o in t i n f o r -


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Regionalized Favorabili ty Theory 605

m a t i o n b e c o m e s a m o r e d e m a n d i n g t a s k . T h e a u t o - a n d c r o s s - c o r r e l a t i o n s i n

s p a c e c o m p r i s e s a n i m p o r t a n t c o m p o n e n t o f th e jo i n t i n f o r m a t i o n . G e o s t a t is t ic a t

t e c h n i q u e s a r e w e l l - k n o w n t o o l s f o r c a p t u r i n g t h is t y p e o f s p a ti a l i n f o r m a t i o n

( Dav i d , 1 9 7 7 ; J o u rn e l an d H u i jb reg t s , 1 9 7 8 ; I s aak s an d S r i v as t av a , 1 9 8 9 ) .

E a ch k r i g i n g t ech n i qu e i s d e s i g n ed i n o n e w ay o r an o t h e r fo r in te rp ola tio n ~

F o r e x a m p l e , c o k r i g i n g s i m u l t a n e o u s ly e s t im a t e s a s e t o f c o r e g i o n a l i z e d ra n d o m

v a r ia b l e s ( M y e r s , 1 9 8 2 , 1 9 8 3 , 1 9 8 4 ). T h e m e t h o d h a s b e e n d e m o n s t r a te d w i t h

a n u m b e r o f c a s e s tu d i es ( D a v i s a n d G r e e n e s , 1 9 8 3; C a r t a n d M c C a l l is t e r, t 9 8 5 ;

U n a l a n d H a y c o c k s , 1 9 8 6 ) . S e v e r a l c o k r i g i n g p r o g r a m s c a n b e f o u n d i n t h e

p u b l i s h e d l it e r a t u re ( C a r r e t a l . , 1 9 8 5 ; M a rco t t e , 1 9 9 1 ; Pan e t a1 . ,1 9 9 2 ) . A l -

t h o u g h c o k r i g i n g i s cap ab l e o f d ea l i n g w i t h m u l t i p l e v a r i ab l e s , i t i s n o t s u i t ab le

fo r e s t i m a t i o n o f a f av o rab i l i t y fu n c t i o n , s i n ce t h e f av o rab i l i t y fu n c t i o n i s n o t

d i r e c t ly o b s e r v a b l e .

T h i s p a p e r p r e s e n ts a r e g i o n a l i z e d fa v o r a b i l i ty e s ti m a t i o n m e t h o d f o r d a t a

i n t e g ra t io n a n d m i n e r a l e x p l o r a t i o n . T h e n e w m e t h o d i s d e m o n s t r a t e d o n a t e st

c a s e s t u d y i n th e s e l e c t i o n o f m i n e r a l e x p l o r a t i o n t a rg e t s f o r h y d r o t h e r m a l g o l d -

s i l v e r d ep o s i t s b a s ed u p o n a s e t o f g eo l o g i ca l ( s t ru c t u ra l ) , g eo ch em i ca l ( s o i l

s a m p l e s ) , a n d a i r b o r n e g e o p h y s i c a l ( e le c t r o m a g n e t i c ) o b s e r v a t io n s . A f a v o r -

ab i l it y m ap i s p ro d u ce d t o s h o w t h e t a rg e t s fo r g o l d - s i l v e r p o ten t i a ls . A s u s e fu l

s u p p l em en t a ry i n fo rm a t i o n , s ev e ra l co r r e l a t i o n co e f f i c i en t s a r e a l so g en e ra t ed t o

a i d i n t h e i n t e rp re t a t i o n o f t h e f av o rab i l i t y e s t i m a t e s .

B A C K G R O U N D R E V I E W

L e t Z I , Z ~ . . . . Zm b e m g e o l o g ic a l , g e o c h e m i c a l , a n d g e o p h y s i c a l r a n d o m

v a r i a b le s . T h e o b j e c t i v e h e r e i s t o c o n s t r u c t a f a v o r a b il i ty i n d e x , F , c o m b i n e d

f r o m t h e s e t o f e x p l a n a t o r y v a r ia b l e s . F i s a w e i g h t e d l i n e a r c o m b i n a t io n :

F = a l w l Z l +

a2w2Z2 o. . q-

a m w m Zm ( 1 )

w h e r e w l , w 2 . . . . wm a r e a p r i o r i s e l e c t e d w e i g h t s f o r t h e m v a r i a b le s a n d

a ~ , a 2 . . . . . a m a re u n k n o w n co e f f i c i en t s a s s o c i a t ed w i th Z ~ , Z 2 . . . , Z m. L e t

W = D i a g ( w l , w 2 . . . . w in ) a n d a = ( a l , a 2 . . . . a ~ j r . T h e n , t h e f a v o r a b i l i ty

i n d e x m a y b e r e w r i t t e n i n t h e m a t r i x f o r m :

F = Z W a (2 )

w h e r e Z = ( Z1, Z 2 . . . . . Z m ), a r o w v e c t o r o f t h e m r a n d o m v a r i a b le s .

E x p l a n a t o r y v a r i a b l e s u s u a l ly p r o v i d e i n d i re c t s ig n a t u re s o f m i n e r al o c c u r -

r e n c e s . F u r t h e r m o r e , t h e i r m a g n i t u d e s a n d s i g n s m a y n o t u n i q u e l y c o r r e s p o n d

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

m a g n e t i c l o w s o r h i g h s r a r e l y c o r r e s p o n d t o t h e m i n e r a l iz a t i o n o f i n t e re s t in a

u n i q u e w a y . T h i s ty p e o f d a ta , h o w e v e r , h a s a n i m p o r t a n t a d v a n t a g e . T h e y c a n

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


4/29

606 Pan

d a t a a r e v a l u a b l e f o r th e p r e d i c t io n o f m i n e r a l o c c u r r e n c e i n u n k n o w n r e g io n s .

A n o t h e r t y p e o f d a t a , t e r m e d t a r g e t v a r i a b l es , p r o v i d e s f i rm a n d d i re c t i n fo r -

m a t i o n o n m i n e r a l o c c u r re n c e . A t y p i c a l e x a m p l e i s t h e c o n c e n t ra t io n o f m e t a l ,

w h i c h c a n b e u s e d f o r a c l e a r- c u t j u d g m e n t a s to t h e d e g r e e s o f m i n e r a li z a ti o n .

U n f o r t u n a t e l y , i n m a n y p r a c ti c a l c as e s , m o s t ta r g e t v a r ia b l e s a re o n l y a v a i l a b l e

i n t he be s t e xp l o r e d s uba r e a s a nd t he y c a nno t be r e a d i l y ob t a i ne d w i t hou t u s i ng

h i gh c os t t e c hn i que s , e . g . , d r i ll i ng . T hus , t a r ge t va r i a b l e s , c l o s e l y re l a t e d t o the

l e ve l o f e xp l o r a t i on , a r e no t e a s i l y i nc o r po r a t e d i n to t he f a vo r a b i l i t y e qua t i on t o

e s t i m a t e m i n e r a l o c c u r r e n c e i n u n e x p l o r e d r e g i o n s . N e v e r t h e l e s s , t a r g e t v a r i -

a b l e s s hou l d be u s e d a s c r i t e r i a i n de f i n i ng a nd i n t e r p r e t i ng a f a vo r a b i l i t y i nde x

F ) . T h i s o b j e c t i v e m a y b e a c h i e v e d b y m a x i m i z i n g t h e c o r r e l a t i o n s b e t w e e n

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

s e t , w h i c h c on t a i n s s u f fi c ie n t i n f o r m a t i on o n bo t h t yp e s o f va r i a b l e s .

L e t Y = Y i , Y 2 . . . . . Y p ) b e a r o w v e c t o r o f p t a r g e t v a r i a b le s . A b r i e f

r e v i e w o f t h e c a n o n i c a l f a v o r a b i l it y m e t h o d C F M ) P a n a n d H a r r i s , 1 9 9 2a )

w i l l he l p t o s how t he c onne c t i on be t w e e n t he r e g i ona l i z e d t he o r y de ve l ope d i n

t he ne x t s e c t i on a nd e x i s t i ng t e c hn i que s .

L e t E y y de no t e t he o r d i na r y non - s pa t i a l ) c ova r i a nc e m a t r i x o f t he p t a r ge t

v a r i a b l es a n d G j t h e j t h p r i n c ip l e c o m p o n e n t o f t h e c o v a r i a n c e m a t r ix Z y y , i . e . ,

G = YI~ 3)

w h e r e

G = G 1 , G 2 . . . . , G q )

w ith q _< p an d I~ = 131, 1~2 . . . . ha) is the p

× q m a t r i x c on t a i n i ng q s t a nda r d i z e d e i ge nve c t o r s a s s oc i a t e d w i t h t he q l a r ge s t

e i ge n va l ue s o f ~yy. C F M de t e r m i ne s t he c oe f f i c ie n t ve c t o r a i n E q . 2 ) by

m a x i m i z i n g t h e s u m o f t h e s q u a r e d c o v a r ia n c e s b e t w e e n F a n d e a c h G j . D e n o t e

q

~ b Z a ) = ~ ] C o y 2

G j , F ) = a r Q a 4 )

j l

w h e r e Q = W ~ 2 z y B B V E y z W . T h e qua n t i t y in 4 ) i s m a x i m i z e d s ub j e c t t o t he

c ond i t i on V a r F ) = a V D a = 1 w i t h D : W ~z z W . T h i s l e a ds t o t he ge ne r a l i z e d

e i g e n - s y s t e m :

Q a = / 3 D a , a V D a = 1 5 )

T h e s o l u t i on i s g i ve n by t he s t a nda r d i z e d e i g e nv e c t o r a s s oc i a t e d w i t h t he l a r ge s t

e i g e n v a l u e o f s y s t e m 5 ) .

R E G I O N A L I Z E D F A V O R A B I L I T Y T H E O R Y

T h e f o r m u l a t i o n a b o v e a s s u m e s t h a t f e a t u r e s Z : l = 1 , 2 . . . . . m ) a n d

Y k k = 1 , 2 . . . . . p ) a r e r a n d o m v a r i a b l e s . N o w l e t u s e x t e n d t h e s e f e a t u r e s

t o be r a nd om f unc t i ons . T h e s e t { Z i , Z 2, • • . ,

Z m }

i s c a l l e d t he r e g i ona l i z e d

e xp l a na t o r y s y s t e m . S i m i l a r l y , t he s e t { Y 1 , Y 2 . . . . . Y p } i s ca l l ed the reg ion-


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Regionalized Favorability Theory 6 7

a l i z e d t a r g e t s y s t e m . C l e a r l y , t h e f a v o r a b i l i t y in d e x , a s a l i n e a r c o m b i n a t i o n o f

t h e e x p l a n a t o r y v a r i a b l e s , s h o u l d t h e n b e a r a n d o m f u n c t i o n . I t is fu r t h e r a s -

s u m e d t h a t a l l r e g i o n a t i z e d s y s t e m s a r e s e c o n d - o r d e r s t a t i o n a r y . S u p p o s e t h a t

t h e e x p e r i m e n t a l d a t a o n t h e r e g i o n a l i z e d e x p l a n a t o r y s y s t e m t o b e u s e d i n t h e

e s t i m a t i o n o f a f a v o r a b i l i ty f u n c t i o n c o n s i s t o f a se t o f s a m p l e s : { Z j x i ) ; j = l ,

2 , . . . , m , i = 1 , 2 . . . . . n } , w h e r e x~ r e p r e s e n t s th e i t h s p a ti a l l o c a t i o n . T h e

r e g i o n a l i z e d f a v o r a b i l i ty f u n c t i o n , F , i s d e f i n e d a s a l i n e a r c o m b i n a t i o n o f t h e

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

v a r i a b l e s h a v e a n e q u a l a p r i o r i w e i g h t ) :

f x o ) = ~ , ,

) k i j Z j x i )

=

Z x,-))~/ 6)

t l j = l i = l

w h e r e ? ~ i = ~ , i l , ) k i 2

. . . . .

) k im ) T is a v e c t o r o f m u n k n o w n c o e f fi c ie n t s f o r t h e

m e x p l a n a t o r y f u n c t i o n v a l u e s a t lo c a t i o n x i .

B a s i c T h e o r y

S u p p o s e t h a t t h e e x p e r i m e n t a l d a t a o n t h e r e g i o n a l i z e d t a r g e t s y s t e m c o n -

s i st s o f a s e t o f s a m p l e s : { Y ~ x t); k = 1 , 2 , . . . , p , t = 1 , 2 . . . . . l } , w h e r e

x t is th e t th s a m p l e l o c a t i o n . A s i n C F M , t h e ta r g e t r a n d o m f u n c t io n s a r e

c o n v e r t e d t o a se t o f p ri n c ip a l c o m p o n e n t s :

G ( x t ) = Y ( x , ) ~

w h e r e G x t ) = [ G l x t ) ,

. . . , G q x t ) ]

a n d U = i ll . . . . . f i q ) q _< p ) . O b -

v i o u s l y , t h e p r i n c i p a l c o m p o n e n t s e t { G 1 , G 2 . . . . . G q } s h o u l d b e c o n s i d e r e d

a s a r e g i o n a l i z e d p r i n c i p a l c o m p o n e n t s y s t e m o f

{ Y 1 , Y 2 . . . . . Y p } .

F o r s i m -

p l ic i ty , w e a s s u m e t h a t t h e e i g e n v e c t o r m a t r i x I J d o e s n o t v a r y w i t h s p a t ia l

l o c a ti o n s . T h e p r i n c i p a l c o m p o n e n t s a t th e e s t i m a t i o n p o i n t Xo a r e g i v e n b y

G x o ) = G x o ) I J 7 )

D e f i n e

q

~ 2 ( ) k l , ) k 2 . . . . . ) kn ) = Z C O V 2 [ F ( x o ) , Gk x0) ]

k = l

C o m p u t e t h e c o v a r i a n c e

C o v [ F x o ) , G k x o )] = ~

C o v [ Z x i ) X i ,

Y xo)fik]

i = 1

= Z X / r C o v [ Z r ( x i ) , g ( X o ) 10 ~ = ~ X f I- I~ ° ~a ~

i = l i = l

w h e r e t t i ° ) = C o v [ Z T x i ) , Y ~ ) ] . T h u s ,

( 8 )


6/29

6 0 8 P a n

C o v 2 [F xo ) , Gk X0)] ~--- ~ ~

k T l g l ( i O ) ~ ~ T l l ( O j ) ' k

* i u t U k U k a . X t~ .j

i = l j = l

A c c o r d i n g l y , E q . 8 ) m a y b e re w r i tt e n a s

=

w he re H 0 = I - I i ° ) lJU r I t ° r ) .

n

i=

i = l j = l

n n

= Z Z

i = l j = l

i = l j = l

n

k r i [ k~ = H i° ) fik f i~ H ° J )X j

k ri [ H i ° ) k ~ = l i ~ f i r ) H ° J ) )

X f [ n i°) 0 3 T H ° J )] X j

xfn jxj

T h e f o l l o w i n g c o n s t r a i n t i s i m p o s e d :

x 2 X t , . . . , X n) = V a r [ F x o ) ] - - 1

E x p l i c i t l y ,

. . . . . A :

i = I j = l

w h e re C U = C o v [ f x D , Z x j) }.

D e n o t e ) T = ) , [ , ) , ~ . . . . . ~,nT~, a n d

I - Il l H 1 2 . . .

H = , C =

I H n 2 n \ C , I C n 2

T h e n , E q s . 1 0 ) a n d 1 1 ) m a y b e s i m p l if ie d t o :

f f2 X ) = X r H X

c ~2 X ) = X T C h = 1

. . . C 1 .

• . , Cnn

9 )

1 0 )

1 1 )

1 2 )

1 3 )


7/29

Regionalized Favorab il ity Theory 6 9

T h e o b j e c t iv e is t o m a x i m i z e 1 2 ) s u b j e ct t o c o n d i t io n 1 3 ). C o n s e q u e n t l y , f o rm

the La g r a nge f unc t ion :

V h , t o ) = ~ 2 ~ ) _ to o ~2 _ 1 )

= h r H h - c o h r C h - 1)

wh e r e t o is t he L a g r a ng e m u l t ip l i e r . T a k ing the pa r ti a l de r iva t ive s o f V w i th

r e spe c t to X a nd co a nd se t ti ng the m to z e r o , we o b ta in t he sys t e m

HX = toC S . , ~ k T c ~ k = 1 14)

Th e be s t so lu t ion f o r h i s the s t a nda r d i z e d e ige n ve c to r a s soc i a t e d w i th t he l a r ge s t

e i g e n v a l u e o f t h e s y s t e m .

Th e sys t e m in 1 .4 ) c a n be r e a d i ly mod i f i e d to i nc o r po r a t e

a p r i o r i

in for -

ma t ion . Assume tha t t he a p r i o r i we igh t s a r e no t spa ti a l ly r el a t e d . Le t W =

D i a g

w l , w 2 . . . . . w in )

be the

a p r i o r i

we igh t ma t r ix . Th e f a vo r a b i l i ty func t ion

F in 6) i s m odi f ied to be :

FW xo) = ~

Z x i ) W X i

15)

i = 1

The n , t he we igh te d f a vo r a b i l i t y f unc t ion c a n be e s t ima te d th r ough the sys t e m

14) , p r ov ide d tha t m a t r i c e s H i j a nd C U a r e r e p l a c e d b y

/¢ w

H ~j = W H o W , Cgj = W C / j W 1 6 )

W he n e xpe r t s a r e a va i l a b l e i n a mine r a l e xp lo r a t ion p r o j e c t ,

a p r i o r i

we igh t s

c a n be a pp r op r i a t e ly de t e r mine d to a c c oun t f o r t he i n f o r ma t ion pe r t i ne n t to e a c h

e xp la na to r y va r i a b l e .

I n o r de r to so lve 14 ) , C shou ld be pos i t i ve de f in i t e . I f th i s c ond i t i on is

sa t i s f ied , then C = C 1 / 2 C 1 / 2 . A c c o r d i n g l y , th e s y s t e m 1 4 ) c a n b e c o n v e r t e d t o

a n o r d ina r y e ige n - sys t e m:

f i x = t o ~ , X r X = 1 1 7 )

w he re I7I = C-1/2 H C- 5/2 and X = C l/2 ~° Le t ~x be the e ig en vec tor a ssoc ia ted

wi th t he l a r ge s t e ige n va lue o f m a t r ix 171. Th e n , t he be s t so lu t ion f o r X is g ive n

b y

F ina l ly , t he op t ima l e s t ima to r f o r t he f a vo r a b i l i t y f unc t ion i s ob t a ine d

n

P x o ) = Z Z x l ) ~ i 1 8 )

i = t

He r e we use d the f a c t t ha t

C - j / 2

i s s y m m e t r ic .


8/29

61 Pan

C o r r e l a t i o n C o e f f ic i e n t s

T h e i n t er p r e ta t io n o f th e f a v o r a b i l i ty e s t i m a t e s c a n b e a i d e d b y c o m p u t i n g

t h e c o r r e l a t i o n c o e f f i c i e n t s b e t w e e n t h e e s t i m a t e d f a v o r a b i l i t y f u n c t i o n a n d e x -

p l a n a t o r y , t a r g e t v a r ia b l e s a s w e l l a s th e p r i n c ip a l c o m p o n e n t s o f t h e t a rg e t

v a r i a b l e s . T h e c o v a r i a n c e v e c t o r b e t w e e n F a n d Z a t l o c a ti o n X o i s

C o v [ F x o ) , Z x o ) ]

= C o v l i = ~ Z ( x i ) X i , Z ( x o ) ]

= ~ X T C o y [ Z r x i ) , Z x o ) ] = X C - i° ) 1 9 )

i = l i = l

w h e r e C i °) = C o y [ Z r x i ) , Z x o ) ]. E q u a t i o n 1 9 ) s u g g e s t s t h a t t h e c o v a r i a n c e

- - Z Z

b e t w e e n t h e f a v o r a b i l i t y f u n c t i o n a n d e a c h e x p l a n a t o r y v a r i a b l e a t l o c a ti o n Xo

is a l i n e a r c o m b i n a t i o n o f th e c o v a r i a n c e s b e t w e e n e x p l a n a t o r y v a r i a b le a t l o -

c a t i o n s X o a n d a ll e x p l a n a t o r y v a r i a b l e s a t a ll s a m p l e l o c a t i o n s x l . . . . , x n .

G i v e n t h e c o n d i t i o n V a t F ) = 1 , t h e c o r r e l a t i o n c o e f f i c ie n t v e c t o r o f F a n d Z

a t l o c a t i o n x o is g i v e n b y

n

C o r r [ F x o ) , Z x o ) ] = ~ X C - i ° )E - I /2

- i - z ~ - 20 )

i = 1

w h e r e ~ = D i a g a ~ , o ~ . . . . . a ~ ) w i t h o J = V a r Z j ) . I n p r a c t ic e , t h e c o r r e -

l a t io n c o e f f i c i e n t v e c t o r i n 2 0 ) i s e s t i m a t e d b y r e p l a c i n g X b y i ts e s t i m a t e X .

N o t e t h a t h e r e w e u s e d t h e s e c o n d o r d e r s t a ti o n a ri ty a s s u m p t i o n f o r t h e e x p l a n -

a t o r y r a n d o m f u n c t i o n s . C l e a r l y , t h e c o r r e l a ti o n c o e f f i c ie n t s v a r y w i th s p a t ia l

c o o r d i n a t e s .

T h e c o v a r i a n c e v e c t o r o f F a n d Y a t l o c at io n x o is

C ° v [ F ( x ° ) Y ( x ° ) ] = C ° v [ ~ Z ( x i ) ~ k i

n

~ k T{ ~ ( i 0 ) 2 1 )

= ~ h /r C o v [ Z r x i ) , Y x o )] = - , ~ z y

i = 1 i = 1

w h e r e 17 i °) = C o v [ Z r ( x i ) , Y x o ) ]. U s i n g t h e c o n d i t io n t h a t V a r F ) = 1 a n d

v y

t h e a s s u m p t i o n o f t h e s e c o n d o r d e r s t a t io n a r i t y , t h e c o r r e l a t io n c o e f f i c ie n t o f F

a n d Y a t lo c a t i o n X o i s g i v e n b y

~ a a ~ h l [ ~ ( i O ) v 1 / 2 2 2 )

C o r r [ F x o ) , Y X o )]

= - i ~ z y - -

i = l

2 = V a r Y j ) . T h e c o r r e l a t io n c o e f -

h e r e V = D i a g v 2 , v 2 , . . . , @ ) w i t h v i

f ic i e n t 2 2 ) v a r i e s w i t h s p a t i a l l o c a t i o n . I n r e a li t y , th e c o r r e l a t i o n s a r e e s ti m a t e d

b y s u b s t i tu t i n g ~ i n t o E q . 2 2 ) .


9/29

Regionalized F avorabilityT h e o r y 6 1 1

S i mi l a r l y , th e co v a r i an ces v ec t o r b e t w een F an d t h e p r i n ci p a l co mp o n en t s

G of the t a rge t var i ab les a t loca t ion Xo are g iven b y

C o v [ F ( x o ) , G ( xo )] = C o v [ ii = ~ Z ( x i ) h ; , Y (x o)U 1

- - ~ Xf C ov [ZT(x i ) , Y(xo)] r~ ~ x r c ( i ° ) f l (23)

- - ~ ~ i v z y

i = 1 i = l

U s i n g th e c o n d i ti o n t h a t V a r ( F ) = I a n d t h e a s s u m p t i o n o f th e s e c o n d o r d e r

s ta t ionar i ty , the cor re la t ion coef f i c ien t vec to r o f F and G a t loca t ion Xo i s g iven

b y

n

C orr [F (x0) , G(xo)] = ~ X'FC-(i°)fl~-t/2

, - - zy v _ (24 )

i = 1

w h e r e O = D ia g(0 ~ , 0 2 . . . . . 0 2) w i t h 02 = V a r ( Q ) .

F u r t h e rm o re , t h e qu an t i t y ¢ i n (8 ) ma y b e r ew r i t ten a s

q

~bZ(X) = N C ov 2[F (xo ), Gk(x0)

k - - I

q

= ~ 02 Co rr2[F(x0 ), G~(x0)]

k = l

q

= Z 0 2 0 2 (25)

k = l

w here Ok = C orr [F (xo ) , Gk(xo)], w hich i s the cor re la t ion coef f i c ien t o f favor -

ab i l i t y fu n c t i o n an d t h e k t h p r i n c i p a l co mp o n en t o f t h e t a rg e t v a r i ab l e s a t l o -

ca t i o n x 0 . A cco rd i n g l y , max i mi za t i o n o f 1~ 2 s u b j ec t t o co n d i t i o n V ar (F ) = 1 i s

equ i v a l en t t o t h e max i mi za t i o n o f t h e w e i g h t ed s u m o f t h e s qu a red co r r e l a t i o n

coef f i c ien t s be tween the favorab i l i ty func t ion a t loca t ion x0 and a l l p r inc ipa l

co m p o n en t s o f t h e t a rg e t v a r iab l e s a t t h e s am e l o ca t i o n . T h e w e i g h t s 0 2 , 0 2 ,

. . . . 0 ~ d es c r i b e d i f f e r en t l ev e l s o f co n t r ib u t i o n s fro m d i f f e r en t p r i n c ip a l co m-

p o n en t s . Na t u ra l l y , t h e m ax i m i za t i o n g i v es mo re em p h as i s to t h o s e p ri n c ip a l

co mp o n en t s w h i ch ch a rac t e r i ze t h e d i r ec t i o n s w i t h l a rg e r v a r i ab i l i t i e s i n t h e

t a rg e t v a r iab l e s p ace an d s u p p ress e s o t h e r co m p o n en t s w i t h s m a l l e r v a r i ab i li ti e s .

B l o c k E s t i m a t io n

In the pract ice of mineral exploration, a s tudy region is usual ly subdivided

into an equal-area (volume) grid and each e lementary area (volume) i s referred


10/29

6 2 Pan

t o as a c e l l b l o c k ) . T r a d i t i o n a l f a v o r a b i l i t y a n a l y s i s a s s i g n s a f a v o r a b i l i ty v a l u e

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

h o we v e r , su c h a n a s s i g n m e n t i s n o t su f f i c i e n t f o r a c c u r a t e l y d e sc r i b i n g t h e d e -

g r e e o f f a v o r a b i l i t y t o m i n e r a l o c c u r r e n c e i n t h e c e l l , s i n c e t h e c e l l i s a n a r e a

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

d e g r e e o f f a v o r a b i l it y f o r a c e l l is t o c o m p u t e a n a v e r a g e v a lu e :

1 F x ) d x = ~ Z x i ) 1 / A ) Xi u ) du 26)

F A ) = ~ t A i = | A

w h e r e A i s a s e c t i o n , a r e a , o r v o l u m e f o r o n e - , t w o - , o r t h r e e - d i m e n s i o n a l

f a v o r a b i l i ty a n a l y s i s . N o t e E q . 2 6 ) u se d t h e a s su m p t i o n o f t h e f in i te n e s s o f t h e

i n t e g r a l f u n c t i o n . R e l a t i o n 2 6 ) sh o w s t h a t e s t i m a t i o n o f t h e a v e r a g e fa v o r a b i l i t y

f u n c t i o n i n a b l o c k i s e q u i v a l e n t t o e s t i m a t i o n o f t h e a v e r a g e o f t h e we i g h t s i n

t h e b l o c k .

T h e a v e r a g e in 2 6 ) is u s u a l l y a p p r o x i m a t e d n u m e r i c a l l y b y t h e a v e r a g e o f

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

wi th in A, t ha t i s ,

s n s

F A )

= - = - 27)

S k l S i l k l

w h e r e F i s g i v e n i n 1 8 ) ; ~ k ) i s t h e e s t i m a t e o f t h e c o e ff i c ie n t v e c t o r a t p o i n t

x k f o r Z a t l o c a t i o n x i ; s i s t h e t o t a l n u m b e r o f d i s c r e t e p o i n t s e v e n l y d i s t r i b u t e d

i n A . T h e c h o i c e o f s d e p e n d s o n h a r d w a r e , s o f t w a r e , a s w e l l a s t h e p r e c i s i o n

r e q u ir e d f o r b l o c k e s t i m a t i o n . T h e a b o v e p r o c e d u r e r e q u ir e s e s t im a t i n g a s e t o f

we i g h t c o e f f ic i e n t s b y so l v i n g t h e e i g e n - sy s t e m 1 4 ) a t e a c h s e l e c t e d p o i n t .

S imi l a r ly , t he cor re l a t ion coe f fÉc ien t s be tween favorab i l i t y func t ion and

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

t h e a v e r a g e c o r r e l a t i o n c o e f fi c i e n ts i n 2 0 ) in a b l o c k m a y b e a p p r o x i m a t e d b y

--~1 c iA ) E 1/2

C o r r [ F A ) , Z x i ) ] = l / A ) X f f i)

d f i - z z -

28)

i A

w h e r e C~izz ) i s t h e c o v a r i a n c e m a t r i x o f e x p l a n a t o r y v a r i a b l e s b e t w e e n s a m p l e i

a n d t h e b l o c k A . Ag a i n , r e l a t i o n 2 8 ) u se d t h e a s su m p t i o n o f i n t e r c h a n g a b i l it y

b e t w e e n s u m a n d i n t e g r a l .

S O M E C O M P A R I S O N S

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

c a n o n i c a l fa v o r a b i l i ty m e t h o d p r o p o s e d b y P a n a n d H a r r i s 1 9 9 2 a ) o f fe r s a

s p e c i a l a d v a n t a g e i n t e r m s o f t h e u s e a n d q u a n t i f i c a t i o n o f i n f o r m a t i o n . T h e


11/29

Regionalized avorability Theory 6 3

C F M e x p l i ci tl y e m p l o y s a s e t o f t a rg e t v a r i a b l e s t o d e f in e t h e m e a n i n g o f t he

f a vo r a b i l i t y f unc t i on . E xp l a na t o r y va r i a b l e s , w h i c h a ppe a r i n t he f a vo r a b i l i t y

e qua t i on , a r e e nha nc e d i n t he i r c on t r i bu t i ons t o t he va r i a b i l i t i e s i n t he t a r ge t

v a r i a b le s . T h e C F M , a s w e l l a s o t h e r e s t im a t i o n m e t h o d s , i s e s s e n ti a ll y a n o n -

s pa t i a l a pp r oa c h , s i nc e i t doe s no t f u l l y qua n t i f y t he i n f o r m a t i on a bou t s pa t i a l

a u t o - a nd c r o s s - c o r r e l a t i ons o f the e xp l a na t o r y a nd t a r ge t va r i a b l e s .

T h e g e n e r a l i z e d e i g e n - s y s t e m 1 4 ) h a s t h e s a m e f o r m a s s y s t e m 6 ) f o r t h e

C F M . T h e i r g e o l o g i c a l im p l i ca t i o n s , h o w e v e r , a r e r a d i c a ll y d if fe r en t . T h e s y s -

t e m 14 ) c on t a i n s a l l i n f o r m a t i on c on t r i bu t e d f r om t he s pa t ia l a u t o - a nd c r o s s -

c o r r e l a t i ons be t w e e n t he f a vo r a b i l i t y f unc t i on a t t he e s t i m a t i on l oc a t i on xo a nd

a l l e xp l a na t o r y a nd t a r ge t va r i a b l e s a t a l l s a m p l e l oc a t i ons . T h i s ge ne r a l i z a t i on

c a n on l y be don e t h r ough t he ge n e r a l i z a t ion o f t r a d i ti ona l r a nd om v a r i a b l e s i n to

r a n d o m f u n c t io n s r e g i o n a li z e d r a n d o m v a r i a b le s ) . B e c a u s e t h e se r a n d o m f u n c -

t i ons a r e i n t e r r e l a t e d , t he r e g i ona l i z e d s y s t e m i s a n a de qua t e no t i on t o de s c r i be

t he c ha r a c t e r i s t i c s o f c o r e g i ona l i z a t i on .

T h e m a t r ix Q i n t h e s y s t e m 6 ) c o n t a in s o n l y t h e o r d i n a r y c o v a r ia n c e s

be t w e e n t a r ge t a nd e xp l a na t o r y va r i a b l e s r e ga r d l e s s o f t he i r s pa t i a l l oc a t i ons .

T he m a t r i x H i nc l ude s t he s pa t i a l c ova r i a nc e s be t w e e n t a r ge t a nd e xp l a na t o r y

r a nd om f unc t i ons a t a l l s a m p l e l oc a t ions . E a c h e n t r y o f Q i s a s c a la r , w h i l e

e a c h e n t r y o f H i s a n m × m s ub - m a t r i x . S i m i l a r l y , D i n t he s y s t e m 6 ) c on t a i n s

o n l y o r d i n a ry c o v a r i a n c e s b e t w e e n e x p l a n a t o r y v a r ia b l e s , w h i l e C in th e s y s t e m

14) i nc l ude s t he s pa t i a l c ova r i a nc e s be t w e e n e xp l a na t o r y r a ndom f unc t i ons a t

a l l s a m p l e l o c a t i o n s . T h i s c h a n g e d o e s n o t c r e a t e a n y m a t h e m a t i c a l p r o b l e m s

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

t i m e - i n t e ns i ve .

A n o t h e r u s e fu l c o m p a r i s o n c a n b e m a d e i n t e r m s o f e s t im a t i o n . O n c e a

f a vo r a b i l i t y e qua t i on i s e s t i m a t e d on t he ba s i s o f a s e t o f s a m p l e s , t he e s t i m a t o r

P i n C F M w i l l be a pp l i e d t o t he e s t i m a t i on o f t he f a vo r a b i l i t y va l ue s a t e a c h

u n k n o w n l o c a t i o n b a s e d u p o n o n l y t h e e x p l a n a t o r y v a r i a b l e o b s e r v e d a t t h a t

s a m e l oc a t i on . T h e f a vo r a b i l i ty e qua t i on i n the r e g i ona l i z e d ve r s i on , how e ve r ,

i s e s t i m a t e d a t e a c h s pa t i a l l oc a t i on ba s e d upon t he s e t o f obs e r va t i ons on t he

e x p l a n a t o r y v a r i a b l e s f r o m b o t h t h a t l o c a ti o n a n d o t h e r n e a r b y s a m p l e d p o i n t s.

M or e o ve r , t he w e i gh t s i n t he r e g i ona l i z e d f a vo r a b i l i t y e s t i m a t o r a r e r e - e s t i m a t e d

w h e n t h e e s t i m a t i o n m o v e s f r o m o n e l o c a t i o n t o a n o t h e r .

S o m e p h y s i c a l a n d c h e m i c a l g e o l o g i c a l f e a t u r e s a r e n o t b e s t u s e d b y o n l y

l ook i ng a t t he qua n t i t y a t a s a m p l e po i n t , s i nc e a s i ng l e po i n t va l ue m a y no t

c onve y s u f f i c i e n t i n f o r m a t i on a bou t m i ne r a l oc c u r r e nc e . F o r i n s t a nc e , a t o t a l

m a gne t i c i n t e ns i t y obs e r va t i on a t a po i n t doe s no t i nd i c a t e w he t he r i t i s a nom -

a l ous o r no t . H ow e ve r , w he n t he s e f e a t u r e s a r e t r e a t e d a s f i e l d s by l ook i ng a t

t he va l ue s a t t he e s t i m a t i on po i n t a s w e l l a s i t s ne a r by s a m p l e s , t he i r a nom a l ous

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

a b i l it y t he o r y i s w e l l c om pa r e d t o t he no t i on o f ge o - f i e l d s , w h i c h e n t a i l s s ta t is -

t i ca l spa t i a l s t ruc tures .


12/29

614 Pan

In ad d i t i o n , a co mp ar i s o n b e t w een t h e ap p ro ach d ev e l o p ed h e re w i t h co -

kr ig ing (My ers , 1982, 1983) i s in te res t ing . Sy s tem (14) invo lves bo th sam ple- -

s amp l e co v a r i an ce an d p o i n t - s amp l e co v a r i an ce ma t r i ce s w h i ch ap p ea r i n an

o r d i n a r y c o k r ig i n g s y s t e m . I n c o k r ig i n g , t h e s a m p l e - - s a m p l e c o v a r ia n c e m a t ri x

i s r equ i r ed t o b e p o s i t iv e d e f i n i te i n o rd e r t o g u a ran t ee p o s i t i v e e s t im a t i o n v a ri -

an ce ( s ee M y er s , 1 9 8 8 a ,b ). I n t h e r eg i o n a l ized f av o rab i l i ty t h eo ry , C i s a l s o

requ i r ed t o b e p o s i t i v e d e f i n i t e , n o t f o r a p o s i t i v e e s t i ma t i o n v a r i an ce , b u t t o

en s u re a u n i t v a r i an ce fo r t h e f av o rab i l i t y fu n c t i o n e s ti ma t e s . A s i n co k r i g in g ,

C t ak es acco u n t o f s amp l e co n f i g u ra t i o n s i n v o l v ed i n t h e e s t i ma t i o n . I n co k f i g -

i n g , t h e r e l a t i v e d eg rees o f co n t r i b u t io n s o f t h e i n d i v id u a l s amp l es a n d v a r i ab l e s

i n v o l v ed i n an e s t i ma t i o n a r e en t a i l ed b y t h e p o i n t - s amp l e co v a r i an ce ma t r i x .

B y t h e s am e t o k en , t h e m a i n s o u rce o f i n fo rm a t i o n d e t e rm i n i n g th e e s t ima t e s

o f f a v o r a b il it y f u n c t io n a t a p o i n t o r b l o c k is m a i n l y f r o m t h e m a t ri x H , w h i c h

co n t a i n s t h e p o i n t - s amp l e co v a r i an ces .

V A R I O G R A M M O D E L I N G

A n i mp o r t an t s tep i n i mp l em en t i n g t h e r eg i o n a l i zed f av o rab i li t y an a l y s i s is

t h e co mp u t a t i o n o f co v a r i an ces an d c ro s s - co v a r i an ces b e t w een ex p l an a t o ry an d

t a rg e t v a r iab l e s a t e ach e s t i m a t i o n lo ca t i o n . U n d e r t h e co n d i t i o n s o f s eco n d o rd e r

s t a t io n a r i t y , co v a r i an ce an d v a r i o g ram a re r e l a ted a s fo l l o w s :

¥(h) = c (0 ) - c (h )

w h e r e c ( 0 ) is th e v a r ia n c e . T h e c o v a r i a n c e c a n b e c o m p u t e d f r o m t h e v a r io g m m :

c( h) = Si l l - 3 ,(h) (29)

T h e c ro s s -v a r i o g ram an d c ro s s - co v a r i an ce can b e ca l cu l a t ed each o t h e r t h ro u g h

the equ at ion (H ohn , 1988 , p . 141):

2~,i~(h = 2cjk(0) - cik(h ) - cky(h

A s s u m i n g t h a t c j k (h ) = c~j(h) , the abov e equat ion i s s imp l i f ied to

~ / k ( h ) = c jk ( 0 ) - % ( h )

w h ere c j k (0 ) i s equ a l t o t h e s i l l o f t h e c ro s s -v a r i o g mm (Cs i l l ) . T h u s , c ro s s -

c o v a r i a n c e c a n b e c o m p u t e d b y

c j k h )

= C si l l - 7 ik(h) (30)

In o rd i n a ry v a r i o g ram mo d e l i n g , i t i s a co mmo n p rac t i ce t o f i t o n e o f t h e

s t an d a rd mo d e l s ( sp h e r i ca l, ex p o n en t i a l , e t c . ) , i n c l u d i n g n es t ed mo d e l s to m ak e

a s u b j ec t i v e e s t i ma t e o f t h e mo d e l p a rame t e r s f ro m t h e p l o t s o f ex p e r i men t a l

v a r i o g rams . F o r c ro s s -v a r i o g ram m o d e l i n g , th e fo l l o w i n g p ro ced u re h as b een


13/29

R e g i o n a l i z e d F a v o r a b il it y T h e o r y 6 5

p r o p o s e d b y M y e r s ( 1 9 8 3 , 1 9 8 4 , 1 9 8 8 a ). F o r e a c h p a i r o f v a r i a b l es f o r m a n e w

v a r i a b le b y t h e s u m ; m o d e l t h e v a r i o g r a m f o r t h is n e w v a r i a b le ( 3 '~ ) . T h e n t h e

c r o s s - v a r i o g r a m c a n b e m o d e l e d b y :

3 , jk ( h ) = 0 . 5 [ ~ / ~ ( h ) - - y j (h ) - 3 ' ~ ( h ) ] ( 3 1 )

A l t e r n a t i v e l y , r e l a ti o n ( 3 1 ) m a y b e w r i t t en i n t e r m s o f c o v a r i a n c e s :

c j k ( h ) = 0 . 5 [ C j k ( h ) - - c y ( h ) - C k ( h ) ] ( 3 1 a )

A c r o s s - v a r i o g r a m a n d o r d i n a r y v a r i o g r a m s a r e r e q u i re d t o s a t is f y t h e C a u c h y -

S c h w a r t z ( C - S ) i n e q u a l i t y a s a n e c e s s a r y c o n d i t io n f o r p o s i t i v e d e f in i t e n e s s o f

t h e s a m p l e - s a m p l e c o v a r i a n c e m a t r i x . K i r s c h a n d P a w l o w s k y ( 1 9 8 5 ) d e v e l o p e d

a n u m e r i c a l a l g o r i th m t o e v a l u a t e th e C - S c o n d i t io n :

v - -

(V 3 'y - f~ k) 2 < 3~jk -< (4~% - + x /~Tk) (32 )

T h i s a l g o r i th m e v a l u a t e s t h e a b o v e r e l a t i o n a t a n u m b e r o f d i s c re t e d i s t an c e s

b e t w e e n s a m p l e s . A c r o s s - v a r i o g r a m m o d e l i s s a id to v i o l a te t h e C - S r u le , w h e n

i t s v a l u e s a t s e v e r a l p o i n t s d o n o t s a t i s f y t h e a b o v e i n e q u a l i t y .

T h e m e t h o d d e s c r i b e d a b o v e r e q u i r e s t h a t d a t a b e a v a i l a b l e o n b o t h v a r i -

a b l e s a t a ll d a t a l o c a t i o n s u s e d i n th e p r o c e s s . C l a r k e t a l . ( 1 9 8 9 ) ( a l so s e e

M y e r s , 1 9 9 1 ) p r o p o s e d t h e u s e o f a p s e u d o c r o s s - v a r i o g r a m :

~ jk (h ) = 0 . 5 E { [ Z j ( x i ) - Z k ( x , ) l 2 } ( 3 3 )

w h e r e h = Xe - x / . T h i s m e a s u r e m a y b e c o n s i d e r e d a s a n a t u r a l e x t e n s i o n o f

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

t re a t ed i n a c o m m o n E u c l i d e a n s p a c e . T h i s m e a s u r e i s i m p o r t a n t , s o to s p e a k ,

i n p r a c t i c a l m i n e r a l e x p l o r a t i o n , s i n c e i t d o e s n o t r e q u i r e t h a t al l v a r i a b l e s m u s t

b e s a m p l e d a t c o m m o n l o ca t i o n s .

I N T E R P R E T A T I O N O F F A V O R A B I L I T Y E S T I M A T E S

T h e f a v o r a b i l i ty f u n c t i o n i s v i e w e d a s a g e n e r a l iz a t io n o f th e m e a s u r e f o r

t h e p o s s i b i l i t y o f m i n e r a l o c c u r r e n c e . P r o b a b i l i t y a p p r o a c h e s s u c h a s l o g i s ti c

r e g r e s s io n m o d e l ( C h u n g a n d A g t e r b e r g , 1 9 8 0 ) a n d th e w e i g h t s o f e v i d e n c e

m e t h o d ( A g t e r b e r g , 1 9 8 9 , 1 9 9 2 ; A g t e r b e r g e t a l . , 1 9 9 0 ) , p r o d u c e a p r o b a b i l is t i c

m e a s u r e f o r t h e l ik e l i h o o d o f m i n e r a l o c c u r r e n c e t h r o u g h t h e c o m b i n a t i o n o f a

s e t o f i n d e p e n d e n t g e o - p a t t e rn s . T h e s e q u a n t i ta t iv e m e t h o d s a r e u s u a l ly i n-

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

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

E v e n s o , r e s t r ic t i n g t h e m e a s u r e t o a r e g u l a r i z e d r a n g e , e . g . , [ 0 , 1 ] , i s s t il l n i c e

f o r i n t e r p r e t a t i o n .

S o m e c o n v e n t i o n a l f a v o r a b i l i t y m o d e l s d o p r o d u c e a n i n d e x t h a t i s l im i t e d


14/29

6 6 P a n

t o a r e gu l a r i z e d ra nge . F o r i n s t a nc e , c ha r a c t e r i s ti c a na l y s i s M c C a m m on e t a l . ,

1983 ) ge ne r a t e s a f a vo r a b i l i t y i nde x w hos e v a l ue li e s i n t he r a nge [ - 1 , 1 ], w he n

e xp l a na t o r y va r i a b l e s a r e t e r na r y c ha r a c t e r i s t i c s . W he n da t a a r e b i na r y , c ha r -

a c t e r is t ic a na l y s i s p r od uc e s a n i nde x i n t he r a nge [ 0 , 1 ] P a n a nd H a r r i s , 1992b ) .

W h e n d a t a a r e c o n t i n u o u s , s o m e d i s cr e t e p r o b a b i l it y m e t h o d s b e c o m e d if fi cu l t

t o u s e a nd m os t f a vo r a b i l i t y m e t ho ds w i l l no t gua r a n t e e t he e s ti m a t e s i n a r e gu l a r

r a nge un l e s s a l l va r i a b l e s a r e a pp r op r i a t e l y p r e - t r a ns f o r m e d . S i nc e t he r e g i on -

a l i z e d f a vo r a b i l i t y t he o r y p r opos e d i n t h i s pa pe r a r e de s i gne d w i t hou t r e qu i r i ng

a pa r t i c u l a r da t a f o r m , t he f a vo r a b i l i t y e s t i m a t e s c a n be i n a ny r a nge de pe nd i ng

u p o n t h e l i m i ts o f e x p l a n a t o r y v a r i a b l e s.

T h e r e a r e , a t l e a s t , t w o w a ys t o c on ve r t t he f a vo r a b i l i t y e s t i m a t e s i n t o a

r e gu l a r r a nge . T he f i r s t a pp r oa c h i s t o a pp l y a n a pp r op r i a t e t r a ns f o r m a t i on t o

o r i g i na l va r i a b l e s , s o t ha t a l l o r i g i na l va r i a b l e s a r e r e gu l a r l y va l ue d . F o r e x -

a m p l e , a l l qua n t i t a t i ve va r i a b l e s m a y be d i s c r e t i z e d i n t o b i na r y o r t e r na r y f o r m

by p r e de t e r m i ne d t h r e s ho l d s . T he o t he r m e t hod i s t o d i r e c t l y a pp l y a r e gu l a r -

i z a t i on t o t he e s t i m a t e s o f t he f a vo r a b i l i t y f unc t i on . T he f a vo r a b i l i t y e s t i m a t e s

c a n be c onve r t e d t o t he r a nge [ 0 , 1 ] t h r ough t he t r a ns f o r m a t i on :

p t ) = P - m in F ) 34)

m a x F ) - m i n P )

w he r e / ~ t ) i s t he t r a ns f o r m e d f a vo r a b i l i t y va l ue . T he f a vo r a b i l i t y e s t i m a t e s a n

b e t r a n s f o r m e d i n to t h e r a n g e [ - 1 , 1] b y

/~ ,) = P - m in F ) + P - m ax F) 35)

m a x F ) - m i n F ) m a x F ) - m i n F )

N o t e t ha t the r e gu l a r i z a t i on m a y d i s t o r t t he a nom a l i e s o f a f a vo r a b i l i t y m a p .

T h e r e f o r e , i n t e r p r e t a t i on o f t he r e gu l a r i z e d f a vo r a b i l i ty va l ue s s hou l d be done

w i t h c a r e i n o r d e r t o a v o i d d r a w i n g m i s l e a d i n g c o n c l u s io n s .

A s e t o f c o r r e l a ti o n c o e f fi c ie n t s m a y b e c o m p u t e d i n e a c h b l o c k , p r o v i d in g

a u s e f u l gu i de f o r t he i n t e r p r e ta t i on o f f a vo r a b i l i t y e s t i m a t e s . A n a no m a l y o f

t he f a vo r a b i l i t y e s t i m a t e s c a n be be t t e r unde r s t ood w he n t he c o r r e l a t i on c oe f f i -

c i en t s a re j o i n t l y e x a m i n e d . F o r e x a m p l e , w h e n a n a n o m a l o u s f a v o r a b i l it y a r e a

c o r r e s ponds t o h i gh c o r r e l a t i on c oe f f i c i e n t s be t w e e n t he f a vo r a b i l i t y e s t i m a t e s

a n d a t a r g e t v a r i a b le , t h e a n o m a l y s h o u l d b e u n d e r s t o o d i n t e r m s o f th e g e o l o g -

i c a l i m p l i c a t i ons o f t ha t t a r ge t va r i a b l e . A n a nom a l ous r e g i on , w h i c h i s c o i n -

c i de n t t o s e ve r a l c o r r e l a t i on a no m a l i e s , s ho u l d be i n t e r p r e t e d i n t e r m s o f s e ve r a l

t a r ge t a nd / o r e xp l a na t o r y va r i a b l e s . G l o ba l a ve r a g e s o f t he c o r r e l a t ion c oe f f i-

c i e n t s i n t he e n t i r e s t udy r e g i on p r ov i de a s i m p l e w a y t o e xp r e s s t he r e l a t i ve

l e v e l s o f i m p o r t a n c e o f d i ff e re n t t a rg e t a n d e x p l a n a t o r y v a r i ab l e s .


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M A J O R S T E P S IN A P P L I C A T I O N

The r e g iona l i z e d f a vo r a b i l i t y a na lys i s i s pe r f o r me d in two ma jo r s t a ge s .

The f i r s t s t a ge invo lve s t he va r iog r a m mode l ing o f e xp la na to r y a nd t a r ge t

va r i a b l e s . The se c ond s t a ge e s t ima te s t he f a vo r a b i l i t y va lue s a nd va r ious c o r -

r e l a ti on c oe f f ic i e n t s a t e a c h spa ti a l l oc a t ion . Va r iog r a m m ode l ing r e qu i r es a

da t a ba se c ons t r uc t e d f r om a p r e - se l e c t e d c on t r o l a r e a , whe r e bo th t a r ge t a nd

e xp la na to r y va r i a b l e s a r e s a mple d .

S e ve r a l ne c e s sa r y s t e ps i n u s ing the r e g iona l i z e d f a vo r a b i l i t y t he o r y f o r

se l e c t ion o f mine r a l t a r ge t s a r e summa r i z e d be low:

a ) I de n t i f y a s e t o f the m os t r e l e va n t e xp la n a to r y va r ia b l e s wh ic h w i ll

a ppe a r i n t he f a vo r a b i l i ty e qua t ion , a nd a s e t o f a p r i o r i se lec ted ta rge t va r iab les

wh ic h w i l l be u se d to de f ine t he me a n ing o f t he f a vo r a b i l i t y e s t ima te s .

b ) S e l e c t t he c on t r o l a r e a s ) w h ic h c on ta in s t he be s t kn ow le dge o f the

t a r ge t va r i a b l e s a nd mine r a l oc c u r r e nc e s . The se a r e a s a r e u sua l ly t he be s t e x -

p lo r e d r e g ions . M or e o ve r , t he c on t r o l ar e a a ) is r e qu i r e d to c on ta in a su f f ic i e n tly

l a r ge num be r o f sa mple s f o r bo th t a r ge t a nd e xp la na to r y va r i a b le s .

c ) C o m p u t e t h e m a t ri x I ] w i t h c o l u m n s b e i n g t h e e i g e n v e c t o rs o f th e

c o v a r i a n c e m a t r i x

yy

of t he t a r ge t va r i a b l e s ba se d on the s a mple s c o l l e c t e d

f r om the c on t r o l a r e a s ) .

d ) Es t ima te t he c ova r i a nc e s a nd c r os s c ova r i a nc e s o f t he t a r ge t a nd e x -

p l a na to r y va r i a b l e s ba se d on the s a mple s i n t he c on t r o l a r e a a ) . As in k r ig ing ,

the u se o f va r iog r a m s a nd c r os s - va r iog r a m s in p l a c e o f c ova r i a nc e s a nd c r os s -

c ova r i a nc e s u sua l ly s imp l i f i e s t he mode l ing p r ob le ms .

e ) Es t a b l ish a s e a r c h s t r a t e gy f o r t he ne a r by sa m ple s a r ound a n e s t ima t ion

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

v a r i o g r a m m o d e l s . O f c o u r s e , t h e m i n i m u m a n d m a x i m u m n u m b e r s o f sa m p l e s

in e a c h se a r c h e l l i p so id a r e u sua l ly c ons t r a ine d .

f ) C o m p u t e m a t ri c e s H a n d C b a s e d u p o n t h e n e a rb y s a m p l e s o f t h e

e s t ima t ion po in t a nd the n so lve the e ige n - sys t e m 14 ) o r 17 ) . An e ige n - sys t e m

is e s ta b l i she d a nd so lve d to p r od uc e a s e t o f op t ima l w e igh t s ~ a t e a c h e s t ima t ion

loca t ion .

g ) C o m pu te a ve r a ge b loc k f a vo r a b i l i t y e s t ima te ba se d on a s et o f e st ima te s

a t a p r e - se l e c t e d num be r o f po in t s i n e a c h b loc k c e l l) . Th e d i sc r e te po in t s shou ld

be e ve n ly d i s t r i bu t e d in e a c h b loc k .

h ) The f a vo r a b i l i t y e s t ima te s ma y be no r ma l i z e d th r ough a t r a ns f o r m to

[ 0 , 1 ] o r [ - 1 , 1 ] t o e a se i n t e rp r e t a t ion .

i ) C o m pu te t he c o r r e l a t i on coe f f ic i e n t s be tw e e n the f a vo r a b i l i ty e s t ima te s

a nd e x p la na to r y , t a r ge t va r i a b l e s , a s we l l a s p r inc ipa l c om pon e n t s o f t he ta r ge t

va r i a b l e s . Ano m a lous c o r r e l a t i on va lue s a r e ide n t i f ie d a s supp le me n ta l e v ide nc e

in t he i n t e r p r e t a t i on o f f a vo r a b i l i t y a nom a l i e s .

j ) I de n t i f y mine r a l e xp lo r a t ion ta r ge t s in t hose r e g ions ha v ing the h ighe s t


16/29

6 8 P a n

b loc k f a vo r a b i l i t y va lue s . The t a r ge t s ma y be p r e c i se ly de l ine a t e d th r ough e n -

ha nc e m e n t o r op t im um d i sc r e ti z a tion o f t he c on tou r e d f a vo r a b i l it y ma ps o r c o lo r -

c ompos i t e d f a vo r a b i l i t y ima ge s .

k ) E v a l u a t e e a c h o f th e t a rg e t s b a s e d u p o n t h e m a g n i t u d e o f th e b l o c k

f a vo r a b i l i t y e s tima te s a nd the c o r r e l a t i on c oe f fi c i e n ts c om pu te d a t d i f f er e n t l o -

c a t ions . Th e t a r ge t s ma y be r e f ine d on the ba s i s o f o rig ina l e xp la na to r y o r ta r ge t

va r i a b le s a nd r a nke d a c c o r d ing to t he a ve r a g e o f the f a vo r a b i l i t y e s t ima te s a nd

va r ious c o r r e l a t i on c oe f f ic i e n t s w i th in e a c h t a r ge t.

T h e a b o v e p r o c e d u r e o n l y c o n s is t s o f t h e n e c e s s a r y s t e ps f o r a s u c c e s s fu l

a pp l i c a t ion to mine r a l e xp lo r a t ion . S e ve r a l o the r s t e ps , suc h a s da t a qua n t i f i -

c a t ion a nd un i f i c a t ion , nons t a t i ona r y t r e nd r e mova l , i n f o r ma t ion e nha nc e me n t ,

e t c . , wou ld be invo lve d in t he e n t i r e a na lys i s .

T E S T C A S E D E M O N S T R A T I O N

T h e r e g i o n a l i z e d f a v o r a b i l i t y t h e o r y d e v e l o p e d a b o v e i s d e m o n s t r a t e d o n

a t e s t c a se s tudy wh ic h invo lve s t he s e l e c t ion o f e xp lo r a t ion t a r ge t s f o r hyd r o -

the r ma l go ld - s i l ve r de pos i t s . T he s tudy r e g ion c on ta in s a s ign if ic a n t go ld - s i l v e r

mine r a l oc c u r r e nc e , wh ic h wa s d i s c ove r e d in a Te r t i a r y vo lc a n ic un i t . Ma jo r

type s o f l i tho log y pe r t i ne n t t o t he m ine r a l i z a tion inc lude po r phyr i t i c r hyo l i t e a nd

la t i te . S t rong hydro the rmal a l te ra t ions , inc luding a rg i l l iza t ion and s i l ic i f ica t ion ,

w e r e r e c o r d e d in s o m e w e l l - e x p o s e d a re a s. N o r t h - s o u t h a n d n o r t h - w e s t s t ri k in g

s t r uc tu r e s a r e p r e dom ina n t i n t he de ve lo pm e n t o f t e c ton ic s e tt i ngs i n the r e g ion .

F a u l t s a r e b e l i e ve d to b e a n im por t a n t c on t r o l l i ng f a c to r f o r spa t ia l d i s t ri bu t ion

o f t h e g o l d - s i l v e r m i n e ra l o c c u r r e n c e .

D a t a

Da ta a va i l a b l e i n t h i s r e g ion inc lude so i l ge oc he mic a l s a mple s , h igh r e s -

o lu t ion a i r bo r ne ge ophys i c a l su r ve ys , a s we l l a s r e g iona l a nd loc a l s t r uc tu r e s .

S t r uc t ur a l a t a

Th e s t r uc tu r a l da ta w e r e ob ta ine d b y d ig i t i z ing f a u lt s . B o th r e g iona l fa u l ts

go ing th r ough the s tudy r e g ion a nd loc a l f r a c tu r e s d i r e c t ly r e l a t e d to t he go ld -

s i l ve r mine r a l oc c u r r e nc e we r e d ig i t i z e d a nd the n c onve r t e d to a s e t o f d ig i t a l

da ta the spa t ia l coo rd ina te s ) . The fau l t s w ere c la ss i f ied in to two ca tegor ies :

r e g iona l f a u l t s wh ic h pa s s t h r ough the s tudy r e g ion a nd loc a l f a u l t s wh ic h a r e

e n t i r e ly c on f ine d w i th in t he s tudy r e g ion .

I n t he n e x t s t e p , s t r uc tu r a l va r i a b l es w e r e qua n t i f i e d in a r e gu la r g r id f r om

t h e d i g i t a l d a t a b y u s i n g a m o v i n g w i n d o w t e c h n i q u e b a s e d u p o n t h e w i n d o w

s iz e o f 500 by 500 f e e t . N ine s t r uc tu r a l va r i a b l e s ge ne r a t e d in t h i s s tudy a r e


17/29


1: The number of regional faults in a window.

2: The number of local faults in a window.

3: The total length of regional faults in a window.

4: The total length of local faults in a window.

5: The number of north-east local faults in a window.

6: The number of north-west local faults in a window.

7: The number of north-south local faults in a window.

8: The number of east-west local faults in a window.

9: The number of intersections between faults in a window.

These fault features were then combined to a structural score at the center

of each window by means of principal component analysis. The structural score,

S, is a linear combination of the nine structural variables:

S = -.60S~ + .31S2 + .29S3 - .03S4 + .01S5

+ .01S6 q- .28S7 - .01S8 q- .16S9 (36)

where the coefficients were obtained from the standardized eigenvector associ-

ated with the largest eigenvalue of the covariance matrix among the nine fault

features. Consequently, a set of structural scores were obtained by applying Eq.

(36) to all cells in the study region. The score values characterize the relative

extensiveness of the structural development and possibly their associations to

the occurrence of gold-silver mineralization.

i rborne Geophys ical Data

The geophysical dataset was obtained from a high resolution airborne elec-

tromagnetic survey conducted through a multi-coil, multi-frequency electro-

magnetic system hooked on a helicopter. The flight line was east-west with the

line spacing about a quarter mile. The flight altitude was approximately a hundred

feet above ground. The data include total magnetic and electromagnetic at multi-

frequency bands. Total magnetic fields and apparent resistivities at 900 and

7200 Hz were employed in this analysis. The data were processed by various

filtering and enhancing techniques for noise removal and enhancement of local

anomalies.

The earth magnetic component (IGRF) was first removed from total mag-

netic fields. The data were then filtered to remove high frequency noises. Since

subsurface pictures can be visualized more readily in the vertical direction,


18/29

62 Pan

ve r t i c a l ma gne t i c c ompone n t s a r e u sua l ly p r e f e r r e d in i n t e r p r e t a t i on ( P a msn i s ,

1986) . The ve r t i c a l ma gne t i c a noma l i e s c ompu te d f r om the no i se - f r e e f i e ld s

we r e sub je c t to va r ious e nh a nc e m e n t s . F i r s t, t he d a t a we r e f i l te r e d f o r supp r e s s-

i n g h ig h f r e q u e n c y co m p o n e n t s . S e c o n d , a l i n e a r t r en d w a s r e m o v e d f r o m t h e

f i l t e r e d da t a t o e nha nc e loc a l a noma l i e s . The r e s idua l s we r e t he n h igh - pa s se d

to p r oduc e the ma gne t i c a noma l i e s ma in ly a s soc i a t e d w i th sha l low ge o log ic a l

bod ie s .

R e s i s t i v i t y , t he i nve r se o f c onduc t iv i ty , i s a n impor t a n t e l e c t r oma gne t i c

c ha r a c t e ri s t ic o f ma ny h yd r o th e r m a l - r e l a t e d ge o log ic a l bod ie s . R e s i s t i v i ty me th -

ods a r e pa r t i c u l a r ly u se f u l f o r de t e c t ing o r e bod ie s hos t e d by s t r ong a r g i l l i z e d

( o r s il ic i fi e d ) r oc ks . F r a se r ( 1990) be l i e ve s t ha t r e s i s t iv i t y ma pp ing c a n a ugm e n t

e l e c t ro m a g n e t i c a n o m a l y m a p p i n g f o r m i n i n g e x p l o r a t io n s u r v e y s . T h e r es is ti v -

i ty d a ta e m p l o y e d h e r e w e r e c o m p u t e d f r o m i n p h a s e a n d q u a d r a tu r e c o m p o n e n t s

o f e l e c t r oma gne t i c f ie ld s ba se d u pon a p se ud o la y e r ha l f - spa c e mo de l ( F r a se r ,

1 9 7 8 ). T h e u s e o f tw o o r m o r e f r e q u e n c i e s p r o v i d e s i n f o r m a t io n o n t h e v a r i at io n

o f r e s i s ti v i t y w i th de p th . An a ppa r e n t r e s i s t i v it y ma p c a n b e p r odu c e d f o r e a c h

f r e que nc y to a l l ow a qua l i t a t i ve i n te r p r e t a ti on . R e s i s t iv i t y da t a o f bo th 90 0 H z

a nd 72 00 H z ba nds we r e e m plo ye d in th i s a na lys i s . R e s i s t iv i t y f ie ld s a t 900 H z

c a n pe ne t r a t e up to 500 f e e t f r om the g r ou nd , wh e n su r f i ci a l c o ve r is h igh ly

r e s i st i ve . R e s i s t i v i t y a noma l i e s a t 7200 Hz a r e u sua l ly c a pa b le o f de t e c ting

ge o log ic a l c ond uc to r s a bo ve 200 f e e t a t de p th , a ga in de pe nd ing u pon r e s is t i v it y

c ha r a c t e r i s t i c s o f su r f i c i a l c ove r . The r e s i s t i v i t y da t a i n t h i s r e g ion we r e p r o -

c e s s e d f o r th e r e m o v a l o f n o i se a n d e n h a n c e m e n t o f l o c a l a n o m a l ie s .

Geochemical ata

The ge oc he mic a l da t a a va i l a b l e i n t he s tudy r e g ion we r e c o l l e c t e d f r om a

so i l g r id su r ve y . T w o e l e m e n t s , A u a nd Ag , we r e a na lyz e d a t e a c h so i l s a mple .

The so i l s a mpl ing wa s c onduc te d on a 50 - f e e t r e gu la r g r id c ove r ing the suba r e a

w i t h a m a j o r k n o w n g o l d - s i l v e r m i n e ra l o c c u r r e n c e . G e o c h e m i c a l d a t a p ro v i d e

d i r e c t i n f o r ma t ion f o r i n t e ns i ty a nd oc c u r r e nc e o f mine r a l i z a t ion .

T a r g e t a n d E x p l a n a t o r y V a r i a b l e s

Ta r g e t va r i a b l es s e r ve a s a ve h ic l e t o t r a ns po r t t he r e l e va n t i n f o r ma t ion

c a r r i e d by e xp la na to r y va r i a b l e s t o t he f a vo r a b le de g r e e s o f mine r a l oc c u r r e nc e

o f c onc e r n . Ex p la na to r y va r i a b l e s a r e u sua l ly t he phys i c a l a nd c he m ic a l a tt ri -

bu t e s pe r t i ne n t t o t he ge o log ic a l ob j e c t s o f i n t e r e s t a nd the y ge ne r a l ly p l a y a

r o l e a s i nd i r e c t i n t e rp r e t e r s o f mine r a l oc c u r r e nc e .

The so i l ge oc he mic a l a noma l i e s i n t h i s r e g ion a r e c ons i s t e n t w i th t he oc -

c u r r e nc e o f go ld - s i l ve r mine r a l i z a t ion . The r e f o r e , t he f o l lowing a r e i de n t i f i e d

as ta rge t va r iab les :

I11: G old con cen t ra t ion in so i l samp les ( in ppb) .

112: S i lve r conce nt ra t ion in so i l sam ples ( in ppm ) .


19/29

Regional i zed Favorabi li ty Theory 621

+ +

+

+ + 4- + + +

+ + +

i r e c t i on a l V ar i ogr am ~ for

S t r u c % u r a l S c o r e

0 4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0

D i s t a n c e i n

e e t

~ l + 4 . ~ + + + + + + +

~,( ÷ + + + + + + + + 4-

D i r e c t i on a l V ar l ogr am ~ f or

. . . . .

. . . .

c ~ , -+ - i ; . I i i l ~ + - - - 4 i

0 4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0

Distance in F e e t

. . . . . . . . . . . . . . . . - - X - / ~ . . . . . - 1 i

t + + + +

/ i / . i

4- / / /

d

Vertical gagnetic

Gradien+~

0 4 0 0 1 3 0 0 l ~ O 0 1 0 0 0 2 0 0 0

Distance in

F e e t

o + + 4- + + + + + + + ]

o + + + ÷ , +

+ + + + + + + + +

+ + + + + + 4- .~ +

irectional V~rio gra~s for

Resis[ivity

at 7200 HZ

4 0 0 8 0 0 1 2 0 0 I £ i0 0 2 0 0 0

D i s L a n c e i n F e e t

d

d

5 o

d-

~ L .....

+ + + + +

+ + + + +

+ + + + + + + +

i r e c t i o n a l V a r i o g r a m ~ f o r

~ + G o l doncentrate

4 - . - - , - { ] I t - - ~ ] J '

4 0 0 8 0 0 1 2 0 1 6 0 0 2 0 ' 0 0

Distance in

F e e t

o

+ + + ÷ +

+ + + + +

+ + + + f + + +

D i r e c t i on a l V ar i ogr am s f or

Silver ortcentrate

4 0 0 8 0 0 ] 2 0 0 1 6 0 0 2 0 0 0

D i s t a n c e

in

e e t

Fig . 1. Directional variograms for four explanatory variables.

Structural and geo phy s ica l data are used as exp lanatory var iab les:

Z l : S tructura l s core cons tructed from fau l t features .

Z~: F i l tered ver t i ca l magnet ic grad ient ( in gamma) .

Z3 : F i l tered res i s t iv i ty f i e ld a t 900 Hz ( in decade) .

Z4 : F i l tered res i s t iv i ty f i e ld a t 7200 Hz ( in decade) .

S ince the targe t var iab le data are on ly ava i lab le in a l imi ted subreg ion , a

contro l area was ident i f i ed to serve as a mode l in e s t imat ion of var iograms and


20/29

6 2 2 P a n

i J ~ I [ [ [ I [ ]

C ~ q _

c ~

~.~ -I-

~cA

•

c o f

c~ ~ L M an g e tl ~ G r a d l e n t

I I I I I I I

0 0 0

D i s t a n c e i n F e e t

I I I I i I I L I [

, ~ ~

h -

r r l

/ ¢ J

o f

A u a n d R e s i s ~ r i t y a t 9 0 0 H z

~ / ~

I 4 ; 0 . 1 8 ; 0 I 1 2 0 0 1 1 6 1 0 0

[ 2 0 1 0 0

D i s t a n c e i n F e e t

F i g . 2 . D i r e c t i o n a l v a r i o g r a m s f o r t h e s u m v a r i a b l e s b e

t w e e n g o l d c o n c e n t r a t e a n d v e r t i c a l m a g n e t i c g r a d i e n t a n d

r e s i s t i v i t y a t 9 0 0 H z .

0

c

c r o s s - v a r i o g r a m s . T h e s u b r e g i o n , w h i c h c o n t a i n s a m a j o r m i n e r a l o c c u r r e n c e

a n d is c o v e r e d b y t h e s o i l g r id , w a s s e l e c t e d a s a c o n t r o l a r e a, f r o m w h i c h a

c o m p l e t e d a t a s e t in c l u d i n g a l l t a rg e t a n d e x p l a n a t o r y v a r i a b le s w a s c o n s t r u c t e d .

V a r i o g r a m s

S i n c e e s t i m a t i o n o f t h e c r o s s - v a r i o g r a m s b y E q . 3 1 ) i n v o l v e s t h e o r d in a r y

v a r i o g r a m s f o r s u m s o f p a i r s o f v a r i a b l e s , a l l v a r i a b l e s w e r e t r a n s f o r m e d i n t o

n e w o n e s h a v i n g s i m i la r m a g n i t u d e s i n or d e r t o m i n i m i z e p o s s i b le d i st o rt io n s

o f v a r i o g r a m e s t i m a t i o n . A c c o r d i n g l y , a l l t a rg e t a n d e x p l a n a t o r y v a r i a b le s w e r e

s t a n d a r d i z e d s u c h t h a t e a c h h a s a u n i t s t a n d a r d d e v i a t i o n . T o m a k e u s e o f E q .

3 1 ) , a s e t o f s u m v a r i a b l e s w e r e c r e a t e d . T h e t o ta l o f 2 0 v a r i o g r a m m o d e l s a r e

n e e d e d , i n c l u d i n g s i x f o r i n d i v id u a l t a r ge t a n d e x p la n a t o r y v a r i a b l e s a n d 1 4 f o r

t h e s u m s .


21/29

Region alized Favor ability Th eory 6 3

Table I. Spherical Variogram Models for All Target, Explanatory Variables, and Their Sums

Variable CO Ci Rt R~ Angle

Y~ + z~ 0.07 0.63 1500.0 1200.0 45.0

Y2 + Z~ 0.05 0.71 1500.0 1200.0 0.0

Y~+ Z2 0.10 0.85 1500.0 1300.0 135.0

Y2 + Z2 0.08 0.81 1300.0 1200.0 90.0

Y, + z3 0.06 1.35 1500.0 1200.0 45.0

Y2 + z3 0.02 1.30 1200.0 1200.0 0.0

Y~+ Z4 0.03 1.38 1350.0 1300.0 45.0

Y2 + z4 0.02 t.40 1200.0 1200.0 0.0

Y, 0.12 0.38 1350.0 1100.0 0.0

} 2 0.05 0.43 1000.0 1000.0 0.0

Zi 0.00 0.30 1300.0 1300.0 0.0

Z~ + Zz 0.00 0.70 1400.0 1300.0 45.0

Z~ + Z3 0.00 0.85 1400.0 1300.0 45.0

Z, + Z4 0.00 0.91 1600.0 1600.0 0.0

Z2 0.00 0.22 1500.0 1500.0 0.0

Z~_+ Z3 0.00 0.95 1600.0 1600.0 0.0

Z2 + Z4 0.00 0.90 1300.0 1300.0 0.0

Z3 0.00 0.62 1300.0 t350.0 135.0

Z3 + Z4 0.00 0.80 1300.0 1300.0 0.0

Z4 0.00 0.52 1300.0 1300.0 0.0

Four directional experimental variograms were comput ed for each variable,

starting from zero degree in azimuth, for easting, to 135 degrees in increments

of 45 degrees. The directional variograms for the two target variables and four

explanatory variables are shown in Fig. 1, suggesting that the spatial structures

of all variables are slightly zonal-anisotropic. All variograms were fitted by a

single spherical model. For comparis on, the directional variograms for the sum

of gold conce ntration of soil geochemica l samples and the filtered vertical mag-

netic gradient and the sum of gold and the resistivity at 900 Hz are given in

Fig. 2, which depict a blend of the spatial structures of the two component

variables.

An anisotropic variogram model was then constructed for each variable or

sum. The variogram parameters are summarized in Table 1. The cross-vario-

grams were indirectly computed from these ordinary variograms based on the

relation (31).

E s t i m a t i o n a n d I n t e r p r e t a t i o n

The entire study region was subdivided into an equal-area grid with cell

size of 200 by 200 feet. On the basis of the variograms derived above, a block

favorability value was estimated in each cell by solving a generalized eigen-


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6 2 4 P a n

Z - - ~ L , ~ , ~ , ~ ~ J ~ - ~ I , . j . ~ , ~ . , j . ~ I - L - ~ , , L ~ I _ _

o

o

r--

Z

o

o

~3

Z

z

oo-

.e

Z

o _

~ q

Z

oo -

Z

o ~

2

Z

c 10 0 0 E ~ 0 0 0 C 3 0 0 0 4 0 0 0 E 5 0 0 0 E 6 0 0 0

Fig. 3. Contoured map of the favorability estimates, showing a known gold

rnineml occurrence, so il g old geochem ical anom alies, and potential targets for

gold-silver deposits.

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

s e a r c h s t r a t e g y f o r e a c h e s t i m a t i o n w a s b a s e d o n t h e s p a t ia l r a n g e s o f th e o r -

d i n a ry a n d c r o s s - v a r io g r a m s . I n o rd e r t o r e d u c e t h e e f fe c t o f o v e r - s m o o t h i n g ,

t h e m a x i m u m n u m b e r o f s a m p l e s f o r th e e s t i m a t io n i n e a c h c e ll is f ix e d to

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

t h a t e a c h e s t i m a t e i s r e a s o n a b l y r e li a b le .

T h e f a v o r a b i l it y e s t i m a t e s w e r e n o m a a l i z e d t o th e r a n g e [ 0, 1] b y m e a n s

o f t h e t r a n s f o r m ( 3 4) t o e a s e t h e i n te r p r e ta t io n . T h e n o r m a l i z e d f a v o r a b i l it y

v a l u e s w e r e t h e n c o n t o u re d a s s h o w n in F i g . 3 . T h e m a j o r g o l d a n o m a l y f r o m

s o i l s a m p l e s i s p e r f e c t l y c o n s i s t e n t w i t h t h e m i n e r a l o c c u r r e n c e . M o r e i m p o r -

t a n t ly , t h e a n o m a l o u s f a v o r a b i l i ty r e g io n s i n t h e e a s t p a r t o f t h e m a p a r e e x a c t l y

t h o s e c o n t a i n i n g t h e b e s t g e o c h e m i c a l a n o m a l i e s a n d t h e m i n e r a l o c c u r r e n c e .


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regionalized Favorab ility The ory 625

z

r~

z

z

z

k . _ _ j -

0 E lO 0 0 ~ 2 0 0 0 E 3 0 0 0 E 4 0 0 0 E 5 0 0 0 E 6 0 0 0 E

Fig. 4. Contoured map of the filtered structural score synthesized from nine

quantified fault features by principal componentmethod.

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

p o t e n t ia l a r e a s f o r g o l d a n d s i l v e r m i n e r a l i z a t io n . N o t e t h a t t h e t a r g e t b o u n d a r i e s

m a y b e d e t e r m i n e d m o r e p r e c i s e l y b y u s i n g o p t i m u m d i s c r e t i z a t i o n m e t h o d s

( s ee Pan an d H ar r i s , 1 9 9 0 ) .

T o en h a n ce t h e i n t e rp re t a t i o n , t h e co n t o u r m a p s fo r t h e s t ru c tu ra l s co re ,

v e r t i c a l m a g n e t i c g r a d i e n t , r e s i st iv i ty a n o m a l i e s a t 9 0 0 a n d 7 2 0 0 H z a r e p r o v i d e d

i n F i g s . 4 - 7 . A l t h o u g h t h e g e n e r a l p a t t e r n s a m o n g t h e m a p s a r e d i v e r s e , t h e y

a r e r e l a te d i n o n e w a y o r a n o t h e r t o t h e g o l d - s i l v e r a n d f a v o r a b i l it y a n o m a l ie s .

T w o g o l d - s i l v e r a n o m a l ie s a r e d i r e c t ly a ss o c i a te d w i th m a g n e t i c h ig h s , w h e r e a s

a l l m a jo r g o l d an o m a l i e s a r e h i g h l y co r re l a t ed t o t h e l o ca l r e s i s t i v i t y h i g h s

( s il i c if i ed a rea s ) a t b o t h 9 0 0 a n d 7 2 0 0 H z . T h e f av o rab i l i ty m ap ( F i g . 3 ) cap t u re s

b o t h t y p es o f ch a rac t e r i s t i c s , a l t h o u g h t h e o v e ra l l ch a rac t e r i s t i c s o f F i g . 3 s u g -

g es t t h a t t h e r e s i s t i v i t y an o m a l i e s p l ay a m o re i m p o r t an t ro l e i n t h e e s t i m a t i o n ,


24/29

6 6 P a n

z

z

z

o o

.S

z

o o

z

o

cu

z

o

I L

I I J I I I I I

1 E 2 E 3 E 4 E 5 E 6

Fig. 5. Contouredmap o f the standardizedverticalmagnetic gradient computed

from filteredand high-pass magnetic data.

Th i s obse r v a t ion i s e v ide n t ly supp or t e d by the f a c t t ha t t he r e s i s ti v i t y fi e ld s a r e

h i g h l y c o r r el a t e d t o t h e g o l d - s i l v e r g e o c h e m i c a l an o m a l i e s a n d t h e k n o w n m i n -

e r a l oc c u r r e nc e .

Th e a bo ve in t e r p r e ta t i on is f u r the r e nha n c e d by c om pu t ing the c o r r e l a ti on

c oe f f i c ie n t s be tw e e n the f a vo r a b i l i t y func t ion , e xp la na to r y , a nd t a r ge t va ri a b l e s .

Us ing Eqs . 20 ) , 22 ) , a nd 24 ) , a s e t o f c o r r e l a t i on c oe f f i c ie n t s we r e e s t ima te d

in e a c h c e l l . F o r s im p l i c i t y , t he g loba l a ve r a ge s o f t he se c o r r e l a t ion c oe f fi c i e n ts

i n th e e n t i re r e g i o n w e r e c o m p u t e d a n d l is t ed i n T a b l e I I. T h e s e n u m b e r s s h o w

tha t t he r e s i s ti v i t y fi e ld a t 900 H z i s t he m os t im por t a n t e xp la na to r y va r i a b l e in

e s t ima t ion o f go ld - s i l ve r po t e n t i a l s . Ac c o r d ing to t he c o r r e l a t i on c oe f f i c i e n t s ,

t he go ld c onc e n t r a t e p l a ys a ma jo r r o l e i n t he i n t e r p r e t a t i on o f t he f a vo r a b i l i t y

e s t ima te s . F u r th e r m or e , t he f a vo r a b i l i ty f unc t ion i s s ign i fi c a n tly c o r r e l a te d to

the f i rs t p r inc ipa l c om pon e n t o f t he t a r ge t va r ia b l e s . T h i s is c ons i s t e n t w i th the


25/29

Regionalized Favorability Theo ry

J _

i l i I I

z

3

z

z

o o

z

z

o

z

E l O OO F 2 0 0 0 Z 3 0 0 0 E ~ O O O E 5 O J O O E 8 0 0 0 E

Fig . 6 . Con toured map of the standardized resistivity values at 900 H z.

6 7

f a c t t ha t t he fi rs t p r i nc i pa l c o m po ne n t c on t a in s ove r 95 o f t he t o t al va r i a t i on

in the t a rge t va r i ab le da ta se t .

C O N C L U D I N G R E M A R K S

F a vo r a b i l i t y a na l y s i s i s a n i m p or t a n t m e a ns o f ge o - d a t a i n t e g r a ti on a nd

t a r ge t s e l e c t i on i n m i ne r a l e xp l o r a t i on . A l l t r a d i t i ona l e s t i m a t i on m e t hods , how -

e ve r , a r e e s s e n t i a ll y nons pa t i a l . T he y do no t a l l ow t he e xp l i c i t u s e o f va r i og r a m s

a nd c r o s s va r i og r a m s in the f a vo r a b i l i t y e s t i m a t i on . A n e s t i m a t e o f t he f a vo r -

a b i l i t y f unc t i on a t a l oc a t i on i s ob t a i ne d by u s i ng t he on l y da t a a va i l a b l e a t t ha t

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

l oc a t ions . T hu s , t he f a vo r a b i l i ty va l ue s a t the s e unk now n l oc a t i ons a r e i nd ir e c t ly

de r i ve d f r om i n t e r po l a t i on o r e x t r a po l a t i on .

T h e r e g i ona l i z e d f a vo r a b i l i t y t he o r y p r opo s e d i n t h i s pa p e r is de s i gna t e d


26/29

6 2 8 P a n

z

8

o

~t3

z

o o -

o

z

8 -

g

z

z

o _

2

g 1 0 0 0 Z 2 0 0 0 Z 3 0 0 0 E 4 0 0 0 E 5 0 0 0 Z 6 0 0 0

z

g

8

Fig . 7. Contoured map o f the standardized resistivity values at 7200 Hz.

Ta ble 2 Global Averages of Correlation Coefficients

z~ Z2 z~ z4

F -0.12 0.41 0.52 0.49

YI Y2 Gi G2

F 0.78 0.54 0.59 0.09

t o e x t r a ct m o r e i n f o r m a t i o n r e l e v a n t t o s p a t ia l s t ru c t ur e s . T h e n e w m e t h o d , a n

e x t e n s i o n o f t h e c a n o n i c a l f a v o r a b i l i t y m e t h o d d e v e l o p e d b y P a n a n d H a r r i s

( 1 9 9 2 a) , i s d e r i v e d t h ro u g h g e n e r a l i z a t io n o f e x p l a n a t o r y a n d t a r g e t v a r i a b le s

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

Regionalized Favorability Theory for Information

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