A MATHMW
TICAL MODL OF GIEOPHYSICAL EXPLORATION
FOR ELONGATED ORE DPOXSITS
by
ES
ROGER C.GORE
Geophysal nSieer
Ceorado sool of Mine
SUBMITTED IN PARTIAL ULFIU LM
OF THE RtTWQUI(r.EIMTS FOR
DEGREE OF XASTER OF
mE
Solnfor
at th
MASSACHUSETTS INSTITUTE OF
T EIHIOLOO
5tedbo?1
Signature of Author.
Departent oft
Certified by.............
Aocept4 by.............
.p
1958
..
v.
r.,..r..
4
sand Geophystes, Ausget 25, 1958
04
,-.WV
.. W.,
r
C1Irman,
artuetal
#
....
..
dtee
an Graduate Stgnsxts
•
AB3TRACT
A MATHEMATICAL MODEL OF GF
ICA$ EXPLORATICS
FOR ELOGATED ORE DEPOSIS
by
Roger C. Gore
"ubmaitted
t
the Dear
A aot 2S, 198
fol the te~e
the
itpattil
ofMaster
itmnt otf
ol
fitlMuent
f Saeenes,
In an attempt to establish the batsi
lte
of pbnleft
elnaif
t
t
ai. Geophystes on
of the requtiresents
eeneeptv for
lAn
erived wh-h P
general MatheMatlal model 1to
eMaeted return fromaa ertan &MA of ez
a
et the
*aptta spent This goeral fl-el is
a ltd
to the
problem of sarbing for elongated oebtdint
with
sostant
stke
Q
n s rgiton in whi h Virtless deposits
give
tti l gnphyseal realtshe
ee
osed is
sea
to provide a perfe*t eat a
ot the gradle/
ength etf the rebodies ant it to asmsed that parallel
saro h lAeN pepeaular
to the strike are used. x in
**ats are seamed to be tinar ft etiet of value and
ill*
tas ects are aasued eenstant for p
t!VitrtIezlr type of
depeost.
of the
umrre shoving the expeoted return for various values
ime spacing and 1mnima aecepted gme/
/ength
are oeanted for the ease of an exploratioa Pgram
New Br~drwiek Canda. The ret its ttea~e
I
it
n noisy
areas the optiai line spsaing is equal to the maxiamu
posible iagth
n
of orebodies in New Brunaswik
in less
tqy reas any line spaoing will be poftta0 eand
an the
average but the eptima line spactgs al
Z)as close
tgether
As posetsbe, I abtn the a a
psible length
of orebodtes.
ThDesis 8 pe okiert .
Dr. Patrick P4.
uthrlq
Titlet Professor of Geology
TABLE Or OQT0
Mtj~mttili Ft' datta of fle RottrI flww0tom
Co lot.A' flpr0*elon Or a.o R.tnn= POIn
AMfltntMn Of the Oo~pte MAo4 to Mtb0*n
**
**.**#*
G~b7SI1
?WRS*t0
II
NW
'9..
**2
*estq*#*o..0**#.e*0.4t#***3
Afl~2t1
*e***4*
Z
*e*o~a~ge.*q*o*#*#qoeqeeqj
Ol
WI
A itgl?
a
se
bes w1e"OeO
?te,
t wee" "t Aupete
a grat 4Aat bt
of e
1 As new
i
he
oas nOa,
be!as dae
th
Uatoae
has beato
co'ci
treosent
aend
eaetp
i
to
e
ewapIuS nowmy
Ua
of
*roble
haO 4a4:tay bee d
U
Uapet
i
ent
Lphys
new
ert
of
to the
bti
taes8t 840
t Wt theats
thfnt to be on ep*Untory efftrt
p
es n eiUtd to 4eIapt$
nreeffiat
e4
a be ooeitere
theasutaen
ofa
e
ewloratio
etwevetr, i orer to kehp tthis thesis from evel-O
tag to
a hi
y abfte lt rehaa of om
seass, the
enerat probs
of eaptlratioan eoffinp
4 to tho proble
with fted strite.
of the dtitoultt
problem. but the
of seareohing for -3.1
has bee spetial
ated. deposits
this spetiw problem includes say
easeount oerd In solving nr* attrtst
446Ama
ties invelved to ecasiderably
Ins intriate: a destrable qauaity in a preliminary work.
Every attempt is madoe in this thesis to solve the parti.
otaur proposed probeal
this thesis has nt
the sin hope in writing
,oweer,
bee to develop spaifle solutions
to apete problems, but rather to deelop a broad en*qp l base ftr the solution of s* similar problems
en**toed in the pnag
of an exploration progra.
Far
te
reason,
htirever neosesary data vas not available,
2
the necessary ftcts were hypothesized as closely as possible
from nast experience in order thrt a total modtel miiht be
coxoleted to t t ost if the general a pVroa.h till
produce
a result conformnble with expcrience.
WIERAL MODFL
AW attept to enstret or 4devt
Ite at w
ot
prems
req ir
a.
rmstia sir a
*
*.
the peta, ayp
ep
l
S es
a s
ab
a psste s.4ma
th
n stt$nt.te
w eetgUt
aese
.
a
oe
*taq f 0,4Gwil b* as
t
a
or wlass at the a prtofl a
by th.
psn tAt th*: mAsl to basst.
$eat00
I
. suhh a
h
neG
l "
s aet
kae
n
aea
rp
bo-st.arr
btSy,
tmatrated
.xtte
as
t4trasrtatM athMastty
f
*flw t
Rothe"sanrolistedaaEi
avt4
bt Atah
a pstoaiarso
Ms
ail
tehe
ehsese e
be eplord
bay
a
ole oftboar
toun
PApeitflta ate ti a, ouras
ta*f abl
iso**
'non
area
It
Airaleo.
at
Nurther, ve asswn tht
the gle
hisaw eans
r 1m101otor
in
words.
ns.m that 41 tSypes e sati *f*
dpnite are equally
UZatooL to*ear evr-here In the obese
area. V also
postulate that another kIA of mterial ocars similarly
to the valuable material ea gires a geophystleat
respoase inseparable from the response of the valuable
deposits but this other material is worthles; to
simplit the terminology, deposits contaling valuable
material will be called ore deposits and those deposits
which contain no valuable material aatal
will be refered
to as noise deposits.
Geophysilt
Rypothesee
We shall assume that some sort of geophysieal
Instruaent whiah responds to ore depests a
deposits In a similar ma
r is used whicha
noise
n provide
an accurate estimate of the Srade or apparent grade
( if the deposit is a noise deposit ) of the anomalams
deposit.
Thi instrument is to be transported along equally
spaced parallel lines aoross the chosen area perpendioular
to the known strike, aed from the response of the
instrument certain small, as compared to the chosen area,
areas will be selected for further evalurition.
EVution Hypotheses
We ase me that all anomalies a oepted are evtluated
by diamond drilling. The eost of a sufficiently exact
evaluation of an anomaly is assumed to be a canstant for
all ore depsit anomalies and noise deposit anomalies,
howveer, the two constants any be different. By " sufficiently exact " is meant that enough drilling is done so
that it can be reliably determined whether an anomnly
is assoiated with a ore dposit or a noise deposit and,
if there is an ore deposit, whether this ore deposit can
be produoed at a profit.
C
A very itmprtant assuption that we make is that
sUftlient funds are available, perhaps from some
" benevolent banker , such that =l accepted anomalies
may be completely evaluated; however, these funds mIst
be repaid and so they are Oonsidered to be part of the
exploration eost sa.
Productioln lypothesea
In produoing an ere deposit we asam that first
an initial investment, I, must be made and then for every
dollar which is ultimately earned frm the sale of the
prodeed valuable material a eertain amout mast be paid
for production and extraction costs say , * Therefore
the total aost of developing, producing, sad extracting
an orebo4y of value v a c.v"
the minima
4 We *an then express
value of an rebody which it
sto profitable
to produce from
;,f
O
, or
.,,
,
or
for brevity letting -c, =-a , r,.,
"
In this model the only sourse of revenue eonsidered
is that revenue from the sale of the produced material.
No attempt s made to include revenue from sale of elaims,
stock, or information.
QUANTITATIVE WPR SSION OF TH
OTRAL MODEL
G
The general model, as detseribed in the prellminary
hypothese, oft an exploration program can be characterized
by a mathemateal model whih has three states each state
having a partioular assemblage of probabilit- functions.
The first state is what might be called the ' natural "
state in which the probability functions are these which
naturally occur in the chosen area of exploratlon. The
three most important probability funtions of this state
are: f')
, the probability density funation of the value
of any dhosen ore deposit; PCX) , the probability
ftnction of the total number of ore deposits which occur
in the chosen area P(~a) , the probability function of
the total rmber of noise derosits in the hoden area,
In proceeding from the first or " natural " state
to the second or " evaluation " state we must go through
the geophysical
stage of exploratlon.
Ator the gee.'
physteal survey certain deposits have been aooepted for
evaluation and others have been rejejected; we hope that
the more valuable dpposits have been aooepted and the
submarginal and noise deposits rejected. We therefore
have three different rbp;4ity functions: gv)
, the
density function of the accepted ore deposits;
PU()
,
the probability funation of the total number of ore
deposits which have bee aceepted;
yP.,) , the probability
funtion of the total number of noise deposits which have
been aecepted
The transition stage from the second state to the
third or " production " state is
purpose of the drilling stage is
the drilling stage.
iThe
the rejection of all
submarginal and noise deposits since the geophysical stage
will probably do an imperfect job of seleetion. The
V7-
" produotin " state will therefore be characterized by two
probability functions: hf)
, the density function of
the value of all ore deposits which wIll be brouaght into
prod tion; and P()
, the probability funetion of the
total nmber of ore deposits broa
t tinto production.
Due to to
the effect of drilling P(sae
for no
, in
other words all noise deposits hrve been rejected,
The question of a criteria of succes
of suach a
complex operation is of course one open to a considerable
amount of variation in solutn
soluti
and it s obtful if any
one criteria could be developed which would satisfy all
exploration concepts, To me, the most natural criterion
seems to be the total ultimate returnr
where I define i,
returan as i
value * costs.
Since we have assumed that all revenue comes from
profaeed wealth, the ultimate rlue of all the deposits
discover-ed by exploration is
,S '; , the sum of the
individual values of the discovered deposite.,
However
we have three oe*t funtions: geophysles costs, drilling
costs, and production costs.
The geophysies costs I
shall represent by C ; the drilling costs are N,Od t
404
, where D/ and P
are the cost of a. complete
evaluation by drilling an ore deposit and a noise deposit,
respeattely; the production costs are, as already d&sOtsed,
(c0,") , The
Th total ultimate return is thust
R,-;- - ;cri
- N, -a..
or
rM
Rra
-
-n .
. ,
i.-c-
I,
Rowever,
at the beginning of an exploration survey
we are concerned with the future return and so we have
to consider a probability function for R . Therefore in
planning an exploration survey the quantity whieh we will
be trying to maximize is the expected retum,,
(R) .
From the theorem which states that the expeated value of a
e expected values ( Wadsworth,
sum is equal to the sum of
- D eur,)
sj -E(N
f.
1956 ) , we have: t R) oEl
. Unfortuxntely it is imposaeble to let
- 0, E(A,)- C.
is
since N
-r £(m 4 )
- -E )H
,
itself a random variable. There also is no probability
function for C since C is one of the parameters whose
choice should be indicated by this model. Howvevr, it
can be shown that if N Anad v are stochastically independent
( See Appendix 1 ).
)
)
E(
then EC, 3P9
D E(NM) - Do E C.) - C.
) -'AThereftre; E(It) GI)[a
Etrl
vE
XATREMTI0M FORtAlO N OP THF IRVURN FUpOTNC
As outlined in the previous secttion, a complete
exploration progrn Ean be divided into thre states:
natura, ealuation, and
uprodetion. With each one of
these states we en assooiate a roup of probabilites
which deeribe the 4dstribution of the total madber
of ore deposits, the total aber of noise aposits,
and the ~Uae of an erbody. ( See fig
) If the
exploration program is eonsidered in an abstrat sense
as a systea AiEh passes throuh these varias states,
then the transition st4ags, geophysi
ant Ar ling, sne
be eemf$Aered as treastorming rahanisms wbh trans
form the probability d1etributions of one State into
the probablty distfibutions of the next state. There-
fare if the " natursl " state probabiities are known
or tasS
an ftthe transforming mehanis oan be
derivtd or approxtmt,"
it will be possible to Aerive
the probability funetions of the esucceeding states.
After the neeessary probability funotions are derived,
the return f3ttio E()
man then be ftonlated in terms
of these probability f~antions. Speoifically frEa
(nR)- E( N,) aE r) -A
_
- 06,
E() - , E(.,) - C
()
we realize that we need to derive: P), the probability
that a ore deposits will sacur in the produation state;
P(n) , the probability that an ore deposit which oc
rs
in the production state will have a
kalue v; P(N,)
the probability that there will be N ore deposits in the
evaluation state; and P(k,), the probability that there will
be nnise deposits in the evaluation stte.,
A
Natural State
f 6.) , P
.) , PC*)
Geophysics
Stage
Evaluation State
Drilling
Stage
Production State
M(v), Pi N), P
OG.)
PHASES OF PT EXPLORATION PROGRAM
fig. 1
Natural State Probabilities
Since this thesis i soncerned with discovering
elongated, needle-haped, orebodies and sine a basie
assatimt
Is that geophystles an provde a measure
of the mrdheof the deposit, It wifl be more eonvenient
to consider the probabilities of the length
, at
the naue per unit I
U, A, at then eombine them
thaurgh the equation v: A/ to obtain the proability
distribution of s wh
aeoeslary, than to as sue a
probability distribution of Initiafly. The probles n
aM A the
~
arises as to what distributions to use for
solution of this problem 1t extremely 4UiOUIt
tue to
the s
tty at data.
of Minig
Seeaaue of the natUw4 retioence
Oapanies to release info mtten oonoerning
any eamine deposits eept
those I prttion,
most
available infobration oncernet only profitable orebodies
art even this infteration is so
e restricted in
nature. Therefore aw deeision made entirely on avaiable
data vwl zwaeessarfly have to ignore the proportaiohlly
neth higher nubbr of submarginal and nearly worthless
deposits.
As this model is
an attempt to look at the entire
asspeet of exploration for erebedtes, an attempt m st
be made to inlude ndat adjust for the*e low value deposits.
Sinee data is
laeking, the next best approach is to
asease a distribution which coneords as elosely as
possible with the faets which are ayailable and with
what we know to be likely from past e erience.
i
One
tnotion which seems to agree quite closely with what
we ight expect the actual distribution to be in nature
-9 o
S
*
o.
-<
oexAX
is
however the total
area under the ourve to finite ant
equal to
~
I"
t a ensity fntion is then tfonr
as
, we
n have a probability density fctton
itich is a mdified fore
f the Beta Distribution.
The eumlativeletribution
F(s) :
L'* ;n ?
n
tion
JA s
has a verieal elope at the ori4n for
c 0 and ftr
deoreaszg di has an inereastas probability in the
smaller a.es
00*X
of
( See fEtotia
i . 2)
,
The
, - I < ortes
&0
toble ideajy
suited for the escript n of the
f
oeornto
of -eettain
propertes in nature sUce by propet etioe oft v e *an
mfke as lag
oar
1as
in
a proprtion of the total astritaton
small a range near zee
atIll have a tntiizst
the varPiate
as we AI*
ft
probability ftistribtin
er
entire r'wge of variation.
Ntth I MaA seem to tall int
the
lass of
natural variablee whtia might be described by ths
Of the eta Distribation and we we shafl let
Pg)r
A
faj
*Olt L
foAs
A
,-I<ers
,-
r A0
tfo
L
(S)
iooording
to one of the basic hypotheses of this
model the entlre area of interest is geoleogicaly
honogeneousj this is the same as saying that any given
part of the Chosen area is equally likely to contain a
deposit,
If the further stipulation it mde that the
Oocurrenee of one deposit has no effect on the probability of oeQarrenoe of any other deposit, then itSan
be said that deposits ocuer individually =nd collectivly
at random in the area of interest.
If the mean n
fber
of deposits per unit area Is some constant ,
then it
#an be rigorously shown ( Fry,1928; Operations Research
Notes, 1953 ) that the total anuber of deposits
,T
t
T
I,
1.0
"'.'
If:!:
.
.
..
...
1?
.....
.. .....
....
F::
14L-
m_
r:L
Z ,_
f,
.
L :
:
.....
.....
. ...
per wait area
s thed Potsson Distribut on, i.e.,
A0or dtiny wo
an l
thes probability istributton of
rtorke deposits
th. total
PCi vc
e
as
6,2
'-wo,2i aje
athe
astural state.
toe &adDrtling Probabilities
Geop
lnphytes an di
at
theb
from
e
abitties
r
t
t in test
te
th
na
bot
drilg
tna W tie one*Atdia state tate. they
bot alter the distribtion of
by
-sps
o agt
tszAe Plstrs ensthe pre'tw *tteto
~et r4.eS
thapr
aw
t-
to
ii the sueeflng state,
U
*b*eet
Mas a
apobsilitty Kf at the oerreose
s
t property
the obojet givn
the probability of aceptig
* at
x sto lI)whoMe A referss to the -evwt of
p ety
te
SAoptsaoe, tht
p (Ala) pr) z P A,)
the probability
Pfr etar
at aeopting an objest with the property x.
or
tears of course, P(AIx) represents the *7phys*s
drtflin
probabilities eat Pt) represents the preceding
state probabiliti ess
ith regar4
to the gaophysies probabiities, in
lines must
order to aeept a deOPost one of the tsy
first
ooss it; the probability of orem"sing a deposit
v
italnspacI
P(Al
ej.
e10)
1
I
A
)
For the drilling transition probabilities we have
already stipulated that the drilling will reeet all
unprofitable deposits, therefore
PCA / v, .,) : 0
o
C v.,
()
=I
v,;, .rL V
Noise Probabilities
Little information is released oonoernin low
value deposits, but even lese informaton is avalable
oncerning the
ILse,
shape, and ==ber of worthless
deposits which have been discovered. With so little
informtion, asy taomtions we may assin to noise
deposit occurrene lie tn the realm of pure conjecture,
but, nevOfthless if a complete strumtue is to be
onsidered, some attempt mast be made to arrive at
realistle fntions. One way to get aroamd this
difficlty is to assme that nature oonstruated noise
deposits with the same probability finotions as ore
deposits but possibly with different Valune of the
parameters &eoorinftaly we haves
,O I,
P'n.) .-
I
le
f(Aa,
J
at
Cr)
- -
a.
.A,,-
< Wes
40
I.SO
The geophysies is stipulated as being nable to
disoriminate between ore and noise; however, if the
maxim
length of the noise deposit is
the maXate
larger than
length of the ore d*posits sad if
J < 1.
then it
would be possible to re ject all noise diposits
with I, >L ; further, itf s>ltb a sa.then it would still
be possible to reject any noise deposit which interset
two
urvey lines ( assung
of course,
that the correlation
cani b a
*. froii tIn. to Una .)*
Theroft
*.twe hve it
0> /. t
(q)
S
LA
o if,
&-3~
3)4>
SJ
L< 3 1-4
-b
3~ 4 Lq
Sa * I- 1 1
3 >
If a wniso ' poslt is -aoceptode4hr"* to the
drIflIAS sage It t,
of otose, roejoete
tZs z'efte:
(i'O)
Pj(/I.II*we1) 1o
Evatuation State Probabilities
.quations' (9) aM (5), ad Using the relation
?ro
we Mn obtain by inspection the
probability of intereepting a deposit of length I,
P(it),
tame:
sL
J
P(41,1
tes
f(0,Pc=
PCA IAj
(s")
Sa
lot
I,
5
() satd (6), the
epting a depaosit of grade
P(AA)
Ner theenoe
s
o>
eqatIons
"
rly
probability of a
P(AA,)
I
.
=0
=~
jPeA
A
A
A,
L i/
-
e, 0 dA 1 .44
probabilities from equation's (8)
sad (9# we han
pCAeglq)
ot
t
,7
.4
As as
re
gejntL
s< .
Io,
s L t
S
l3
LU:*1
'd r
=
*~
t
Sd
ar A
"
s-
-
) - o
4+1
"" °
ndistin-
A.<
an
Aw
'd< Am
A LI
loJ
AM also since noise and ore deposits are
shablet
( '3)
E< .
t,'
(1
F)
e &A* A i
I AV$
the drilling stage is
AA*
governed by the total value of b deposit,
The aWseptanee meChanism in
thereO
re it
w~ald fatlitate
matters if
Probability density fnCtion of r
in
j&)
, the
the evaluation state,
= j is substituted into
j:./ and
If
is now derived.
) , the probability density
we obtain F (/
P('l)
of function of V given A :
I (YlZ ) ; :
V - AS
-
r
2&
s>
.
It will now be useful if the following substitutions
and since the various zones
are made sp L , . =hu,
of definition are beginning to be confusing figure 3
shows the various zones of definition.
iv
V
v.-
..---
...
v
PIV
--
p>I
fI .3
- (
Now
>o1)
=
f(r la)f A
A) andf
Substituting for f() we have f(r) ,
ard completing the substitution for
,,v
" I A 06k
O
f +4
V
f ( -/ A)
we have:
'
r
, o
ovip
pV~r
{p
3AN
V *.v*o* V, Pu'f+
v,
A)
-.
:
,
'
~B'-A
,,,
I-q-I
v
J~
I)
A.
V
06)+
V
A
s
a
,
V
*,E
Ds
w Y &-V *p PI, OR -/ 1 )
-A
n rp rl
3 5 (v)+:Lr
V & v
Ef tIX~4 !
.46
V-0
V
6 V,
p'"'!
myVsv
I.
The inte ration then gives
,4
14-
W,
+ (*O)(. i
0- -
j)
. 11
f
A
V'-'
c - we*
pO
-r*I
f
PC
*
-"
"
II(Is
I-
P6'
4MIf
.
Y4
9--v-
,,1
20
-P
To avoid confusion it
will be useful to s w
shematioally
V
limits Of detinition of the v&rious
along the
the
axis:
MV
0
New sisee the natural state total nubser of deposits
Is Poiteso~ disttibuted all that is needed sto the preba,
bility of aoePting a depslit, represtsed bv p, to show
that the probablity funtion in the evaluation state of
aU
bre of depo@ts is Poisson distributed with
the total
a (p
if we 1et
4
X
/
#A.
N.-N,
A~
N.!
N=
'
Now
at**~el
Sf~~~~
~~,~~~t
AN-~VN .!
C-,)
sinee the summtion terms
an~~~8
a~S& x9b
are jast the sers
P(,L) ei Jf.'9
expnsion t
of e; Ad so
t,-t.
*
e p') probability doenity fMtotinsi
ve in their entirety, j( )ir,
P ( interseotia, a~ept=an , ,
&
v *-rv ),
- P ( interseotion, aooeptae e ) '
and so 4 (0) d
Thereforsi
,
rV.
em
I"(qrV)dr
-At*
S ",o,,
41 1
Nr)
,,
--
The probability f=unetions of the properties of
noise depo tstea be obtained in an i4ent- l fashion
to the derivation of the ore dept it protbablttie
tA.the the
providiag that L,9 A however, if
hiter
oektion rate of the longer nose deposits
of the
A great tal
ould be taken into aecount.
labor tnvolved in the or deposit dterivtias an be
avie4
in derlvins the nolte deposit funttons since
ve oh
need
abilityp'
PC),
enad therefore only need the probthat a noie deposit will be aeoepted, this
., a)
p'c ff (,i,
*ffy
(qj
I
,A)
fa
A
ef2.) d)a
) F(ft dA/
where a,i refer to the event of aeeeptance and interA more speeifie formulation of
seetion respeatively.
p' will mneeesatate making a specifle assumption of
the maximu
length, L. , of a noise deposit,
Pmrofdtuetton State Prob bilities
The production state probabilities are obviously
just truneated forms of the evaluation state probabilities;
(-
Cru
h (v)(V)
o,
I.v
1 vA4
11.14
6
(xo)
v
-IM
CL- I)
P(j
whre
-
vc I
voP,,..,
COXPLETE EXPRFSSION OF TH:
RMTURN FUNCTION
The expected value of a Poisson variate is
just
its mean ( see Wadaworth ,1956 for a complete disoussion
of the properties of the Poisson Distribution ).
aecount of this,equation (1) becomes
V, qjv) J,, (, E(r)-e]-
EA)
,
,J,
On
-
-
after substituting from equations (18) and (21).
)
From
equation 20, it follws that E5v)is just the submean
of the distribution p) taken over the interval
v.;.
V
V
V
and so
VV
vgu
vdi
Q-h)
Sbstituting (23) into (22) then gives:
e(r)
. Vf(v)iv
f4 , (tol)c - D4
Ar-.%
(lJ.
where j(e) is defined by equatotWs (17) and fig. 4.
-
-4
1$4
APPLICATION OF TRE COPLTE MODEL TO
AIRBORNE PROSPCTING IN NEW BRUNSWIK
En omposting this
anthematieal
s
0del quite a few
gm as ptions had to be mde in oer
simpinty
to keep
lox as to be
frma be
so *
reslt
the ultimte
eif
1 which
oapletely ~
rehensible; even so the
has been derIved is seomlex enoug to defy 4itreet interin which
orr this reason an aetual ase* stuy
pretation
the, moel
. appied to a speettie problem beooaes a
necessity i oxer to ob tain some idea whAt this model
predlt,,
of Pareameters for Case Study
Estizatle
As I bhae alr
seare
exeeiny
frq
f
"
stated data relevant t
a
sareity
~this
th"
neessitatea :aking
" asf
suptions which are attempts
to eFroly arrive at estimates whiah are of the eorreet
u te. Fr instance one reasonable estimate
order of a
of,a t .rs thi is the same as sayin 95% of the value of
ore Mi
iek
a1sIs used to pay prouing
tn lNew
A reasma be esttate t of
costs.
is $ 1t00,000 I therefore
1.;
ties uI'
t*
.Nov
we do have a reference
( St~~h, 195 )wh'ith states that New larder
ine has
retmntly dlscontinued operatione; aooordtng to data
obtaned from
Nts
presentO naae of ore in
.Mrrate( June 14, 19956 ) the
the New Latrder U Mine is
approx-
ltely
4 20#000#000 and so our estimates of a, seem to
be reasoable.
A rea
estimate oftwas obtained by placing a piece
of paper with a hole in it on a geologo map
Brunsewk by randoly looating the paper on
aomtisg the total naber of mineralizations
oeourred within the hole, whose area was the
of New
the map and
which
equivalent of
100 square miles, a maxima likelehaod estimation of Ar
ould be obtained ( Appendi
11 )4
turned out to be approximtely*:
This estimation
2 deposits per 100
square tI.e4
lrom the
1956
we obtainerd an estimate of value oft the 4
i
the
Brunswick No. 12 Min. whish is lose to $ ?00,000000.
No
oe
of reserves of other sites comes
ther estates
tosthis fig oe and so it seems fair to say
Drieftig osts are agaitn onjeeo turd Wt if 30 holes
at $ 5 000 ap eoe are noessary to evtaUate a deposit
geophyss, roads,
and$ 100,000 is alleed for g~
t,. then 4 250,000. Zf only 3 Gill holes
ela4
are sattotent to deteraine if a deposit is noise and
$
5,i000 is allowed for
roads the C6$
mi
eoxphyobstes, .lai-ts
anM
500o04
This
.
O#
I arbitrartly let r:m-. rY r ,
?Per anoats to sayng that on the average 9 out of 10 deposits
have a 1engt less than j their maxim= possible length
an
19 out of 20 deposits have a grade/length less than
I believe these are stringent
theL maximna grade/lengthW
enoub eistimations to keep the values within reason.
In estimatin C ve ca arrive at a somewhat more exact
value than ftr the other parameters since most airborne
surveys ru in the neighborhood of $ 20/11ne mile. If
we let
",o b
the length and width in ailes of the chosen
area then c:r=omn'V
i'vg.
the data available,
the two largest deposits in the Bathurst-weweastle dis-
triot have a length of 1200 feet, therefore 1500 feet is
probably the maximum length of any possible ore deposits
in the district. If we decide to consider a 100 square
Oci'. "
mile area in this district, then C=7,M
Computational Form of the Return Function
For the purposes of computing, equation (24) can be
put into a more convenient form:
F (4 :
where
.,.
t )-, (.
,, Y)
a r
. ,o
Or
-W-s ,
-,r ')-0
.
P*)
(0 ,,,
,
f t))
- c
From figure 4 now we can see that there are ten possible
return functions depending upon which interval l'.falls
Fortunately,for this case !-w,,sand so unlesswe use
into.
very small p or MpmV&r/..,>.We now have only three diff-.
I.
rI ,pr
erent return functions: p,>I s~,
,:
Forp
For m<
-
V)v
I:
t t
V
PL4~ k
g01"V~
fn
Y.aS
v
,I,,,',
-"
(,Y
, J
217
Foro wL .9vp
j-
O
-IA
+(.':
*1
After inea
)7
(IlulaI
mn
and.45
#416
~UO,~.,-,,~w
C 4I)
(
all~ th
ia+
-.
O
.,
onsans
as
aOny faantr raiB
The
ottb1*
it
P .)m
vhih up to now has bee
s ely avod*
i New rEeW
via ft of the Prineipal
*f 3t4s0 * staebn
eephys ties are l ua g gr p h ite ba a m
s l~
ve id enti al
reats
an Mber"b on
ate
syst
0t
ar
th r
it seem llkely that A.otad v ea se aat."ions (13),
(14.) (19), if v anlt
P (q .
C ,),)
5"//"(
Lot us now prope
:A
zA.
to
'
(t~')
bap'
(S
L
)I
It t
I- A )Ifpg
that 1.,iadt t
can this is the $am
as saying tUat the notse deposits ftollw the Sae &Mtl
buttan but oly ver a wider range ad arresponds with
the ru that Stphite
Im
**our*s In very Lmg beats.
Sine sMe fttle
is nmoe about th distributims of noise
eposits, let us peostu38e two sodeti
z. f'4,
1.
a=,:O
,d7t.'
o
CO
We may nw Oarry et the ealstuations for the two
diffnre
noite .sd -keeping the remainder of the parameter, at the values we have already assigned. For overall
rs~rm factions" for ea lora M4m in New Brtnviek we then
P sa): x , =To ,)->*
., e-.<,P,') -i.o
e:.)- 0 (-is)
E(k)
Due to the
tin
to Fr) ~A ?
waplezity of the
J.o P) - rIP(.-.
o9lets
(26) the boavier of this f
.
P () 'C
easpression o
atson
* 14
equa
ly unmdder-
stood by setually
ptzg values t the return ttnotion.
Piares 5 and 6 show the result obtaned by varying p,
---i ---
.
..
S..
. ... .. ..
...
.
I.
4-
--
-
..
..
000
.
o
i
i.
i
.
..
*
:
r
*
I
-
I
.
..
f
i
.
.o
4
.
L
S-
,
---
... i .. ... . .
i
'
.... ..... . .
"
_L __.:
..... --.--i - --
'
I.
.. .
-.
---....
'-
.
. ..
. ..
.... .. . . ......
------.
----
-
~--i
.........
:
-
-- .....
.
.
- ----.-. ..
--------
------
..
--------" '1- .€
..
--------
0a
i
....
....
j -.: :::::
_-
. . ...
...
._ fI---~-1
- --
... ,----o
0 Itoo',
00NCLUSIONS
I s
thesis has been essentially a pret
inrestigatton ofa partiater eplttrn
primary putoee no to establlsh a.sof
irtn.
of expVta&t a p
probl
poblem,
ItS
0 the
eopewat
t14t
e seeaMial
purpose wa to .ei i
at more apecialsed
Ifor
the oben problmt.
The raen 4 a
1
epres 00idn
t
this thes, Lees p nnie
a ramrk
r the estation of
othe prbles beetJe the oe svnttetl1o44 wth
via
tis thess
an tar as the asel
mel has a more
wrest~le sppteabiltty this theEs Bas
As. ftr
el* ated
i
e
te~d ant
draft tntaah
GMastia
St
partiewlar
s
proble
th
fit
ta
ti pnbablry mare (eta ne
the ntl
betwe any teante.
it
,l
fr
t o, be
mya
n be
a stMy with an tepee aof earaaty,
the objetIe of
Inazriolat deta, i we agree
.e
.ante.
.
pm wbta the .aelas A.eleped
eato
reaenbly well With experlensee, thm eartain
rather lateot
ee n tas
ea be dramva from the
wrk aoplts
0t
, In reoe-ons whia 4 not have
extensvet
ane
Spots, as eharaserte
iet
perhaps by
dael 1, prfi
slly
exptentfot
Wamy
proSar
voui have. a
psitive
a
peoted retwn; however:
there do appear to be
two elati
a~a
I) lpa e the Ite
as widely as the
legth of ore depslts a
ce*ept praetisallyf
all
despoeits
d
the geohstes sta e, 2) spaw* the lines very
olasel tSkthe"
r sA aeopt fhr triling all 4epoits
iatrne
et,
Ina notq areas, as ehareterised perhaps by
Mol
.X1 the prblsm Of exploration pladmag beonos
w4"acm
oritesl sta
if the Itns are tneeorestly
spaeed no mount ts4t1ortmiatiea ill pevet a negative
expeoted profit.
In point of raet, the only method by
which a positive expected return as be assured is by
spacing the lines as wide as the maximm length of ore
deposits.
?etse deductions aght peraps explain the disillusioned attitude of many people who have dealt with geophystes in the past,sor it the exploration pronsa has
not ben planned earefully with reliable geolgical data,
the -line spaetg might be too wide or nar
e and in a
noisy area this mght well mean a negative expected retour
This poitts up sather
conxluston them an exploration
propm sho"ld not be plemed without extentive knowledsge
of the area invOlved especially with respet
to the costs
of mini:ng
the
alogy, and most espeially the dtsti
bution of noise deposits, Again such a eonalusion was
arrived at taig before this work vas done but now we an
perhaps Munderstnl just wtat types of information are
necessary and why this Itformaton is se important,
I fee1 that my own experienoe in working with this
problem has providod some inftrmation whiah might be
usefal to any other researchers in this field. I did,
of course, attempt to fforalate other models besides the
one presented; all the other models wwere vastly more
eomplitated althouh they were also more realiste.
All
these other models were
arried through the analytical
p*ortln of their formElation, bt
all these models were
so intrie1te that it was
Vset le to derive a coherent
result from thp and It was
timpossible to test if
they afforded realstic
answers,
Judgins fro my expertenoe, I feel that future work dealing with more eoox1qt
models should be done using simulation technques if
specifice results are desired.
APP~DIX 1
Proof that E [
Let
; J - E(N,) E(v)
4a,
A(s) be the generating function of
the generating funttop of
funotion of the sum . 3j
PIC)
.
fv),
Bls
be
, and C(s) the generating
Feller (1950) has shown
pt
that
P4)
C (s)
(I
A s)
and further that if v , and Na are stoohastically independent then the riht
side of (19) is just the Taylors
series expansion of
fore C() Z 8 e4
B(s) with s replaced by A(s),
. Now if a variate X has a
There-
:
.(X) /,
, and so
ation c0,
khsn E
senewtins
Elr
il
T*O
is.
For any generating
funetion of a random variable, 6r) = 1 ; therefore,
I
ECM,)
EY. ad
z+4/
6()j
and E
;
-0
N)
NO-
34
Maxi
APPENDIX IX
Likelihood Estimator of the Mean
of the Poisson Distribution
If a probability density f notion -fr() has a para-
meter
,a.
, and
a if we have a group of observations of
,n
. then if we.4 fine L= ., f(e),
L
T
-
t
)
t
)
defines the maximum likelihood estimation of in terms of
S(Wadeworth.1t56). For the Poisson Distribution
KN) = k
and thus
) A',
-
J. +-1,
which simplifies to -m
2wN,)
N
- ---
NO
o or
IV
1S
3,
BIBLIOGRAPHY
A os, w. WB
"The Effect of Iine Spaoingr
Illustrated
by Marmora, Ontario Airborne Magnetometer Control
hra.
,Spaiag,"
G
sa the Deteruemation of Opti.
vwO.
Altas
. 4,
20,
tober, 195, p. 871.
pr~peets of
Exploration over Large Territories,-4
X. "Meths et Appratsting EaeMno
S4g
Feller, Vill aa.
Pry, Thornton,
New Yorkt
nAu
Y
Is
.
. Van W* tran ,
.
no
"Mineal Revew of New ranvsiok '
Eol. 77, no. 1, February, 194.
MNt, A, N*.a
L
la0rthe men 87
: 3
19%
so4
ts 0
atlat o6
e
abridge, Nasaehusetts.
SmIth, J.
0. '"ew Brunsik Reviewi."
l~aal, VtL 79, no. 2, February,
adiaworth,
G. P,
Pt,
,
,...s an
Copyrighted, January, 1
Note.
.