an informational approach - J-Stage

Geoinformatics, vol.14, no.3, pp.233-240, 2003

ARTICLE

FORMALIZATION AND ESTIMATION OF INTEGRATED GEOLOGICAL INVESTIGATIONS: AN INFORMATIONAL APPROACH Lev Eppelbaum1 ＊, Vilen Eppelbaum2 and Zvi Ben-Avraham1 1

Dept. of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel 2 Geological Consultant, 5/68 Burla St., Be’er-Sheva, Israel ＊ Corresponding author: E-mail: [email protected], Fax: +972 3 6409282, Tel: +972 3 6405086 (Received: 28 November 2002; Accepted: 8 March 2003)

Abstract : All geological investigations are used in the definite succession in the time and space. For this different geological means are employed (mining works, drilling, geophysical and geochemical investigations, etc). The final expected aim of geological prospecting is the best recognition of studied area by given limitations. Effectiveness of geological mean application is based on three factors: cost, time and informational criterions. Cost and time criterions maybe estimated by simple calculation, but determination of last criterion is a complicated problem. Informational criterion model is composing from three factors: (1) quantitative estimation of information, (2) estimation of informational reliability and (3) estimation of informational value by degree of aim achievement according to the pragmatic criterion. The main aim of the paper is a problem of determination of set of means composing the notion “geological prospecting” (relative to some fixed feature) by assumed reliabilities of the means. The reliability of geological prospecting means is considered at the level of local determination. Reliabilities of information obtaining by separate mean and set of means are analyzed in detail. Suggested procedure of determining reliability for means and sets of means relative to feature is based on improved methodology of conditional probability utilization. The ways providing the increment of reliability of geological means are proposed. The applicability of proposed methods is shown on simplified examples. Utilization of the methods will allow to finding the most optimal combinations of geological means in different physical-geological conditions. Introduction Key words : Ceological investigation, Geological prospecting, Informational criterion model, Conditional probability utilization, Geological means, Physical-geological conditions

INTRODUCTION All types of geological investigations are applied in the definite succession in the time and space. For this different geological

Criterions C and T can be easy determined by a direct calculation, but estimation of criterion Π is a complex investigation problem.

means are employed (geological mapping, mining works, drilling,

All available geological/environmental information can rep-

geophysical and geochemical investigations, etc). The final aim of

resent in the classic three-level variant: (a) syntactical - number of

geological prospecting is the best recognition (the most precise

information; (b) semantic - substance of information; (c) prag-

identification of targets) of studied area by some given limita-

matic - value of information.

tions (Figure 1). The problem of geological means rational integration can be solved using following criterions: (1) Necessary expenditures for realization of the integration (cost criterion C); (2) Necessary time for realization of the integration (time criterion T); (3) Informativeness of the integration (informational criterion Π).

A principal logical-heuristic model of the geological information can describe in the following form: Π=Q∪R∪V,

(1)

where Q is the quantitative estimation of information, R is the estimation of informational reliability corresponding to the semantic criterion, V is the estimation of informational value by degree of aim achievement according to the pragmatic criterion, ∪ is the symbol of unification.

Lev Eppelbaum, et al.

234

Figure 1. Simplified scheme of the notion “Geological prospecting information”. This algorithm is based on the fundamental terms of infor-

Eppelbaum, 1988; Eppelbaum and Eppelbaum, 1995; Eppelbaum

mation theory and combined with the structural (hierarchical)

et al., 2001) allows us to suggesting that application of expert

approach. This approach allows to constructing each geological

methods in many situations (set of logical and math-statistical

indicator as a structure reflected a set of typical situations. After

procedures) will be most effective.

this the depth of searching recognition is estimated and calculated

We will consider the reliability of geological prospecting at

using developed informational approach. Realization of proposed

the level of local determination, and examine reliabilities of infor-

strategy provides quantitative calculation and effective control of

mation obtaining by a separate mean or set of means. The main

geological/environmental studies.

aim of the paper is a problem of determination of set of means

In real conditions many random factors disturb the results

composing the notion “geological prospecting” (relative to some

obtained by a set of geological means. One of the essential prob-

fixed feature) by assumed reliabilities of the means (in contrast to

lems consists of impossibility to obtaining satisfactory formal-

Khesin and Eppelbaum, 1997, where mainly statistical criterions

ized description of factors influencing to results of local determi-

were analyzed). Solving this problem will allow to finding the

nations. The similar situations are known in the theory of “deci-

most optimal combinations of geological means in different phys-

sion making” where a full math formalization of the investigated

ical-geological conditions.

problem is complicated. Our experience (Borisovich and

Determination of reliabilities of separate means, broadly

FORMALIZATION AND ESTIMATION OF INTEGRATED GEOLOGICAL INVESTIGATIONS: AN INFORMATIONAL APPROACH

235

speaking, maybe obtained by results of control observations or on

case we cannot speak about one value of feature R generating by

the basis of expert methods.

a set S. Then the following question is arising: having a set of obtained values R - (rτ 1, rτ 2 …, rτ n ), which alternative rτ needs to be

Necessary Math Background

recognized as value of feature R. By other words, which a priori

Let’s assume the following designations:

hypotheses from set k (the real value of R equals to rτ , 1 ≤ t ≤ k)

Feature R − any independent characteristics of geological

maybe assumed as the most suitable. Selection of the best (in a

target: thickness, density, color, etc.

sense) hypothesis should be, obviously, realized by use of some

Mean S − geological prospecting procedure providing

algorithm (rule). Formally such a rule maybe considered as map-

information about the direct or circumstantial geological feature

ping ℵ of a set of possible indications of means to the set of val-

(features): drilling, geophysical prospecting, geological mapping,

ues of features :

geochemical analyses, etc. Definition I. Beneath reliability of mean S with respect to feature R we will comprehend a probability of truth for hypothesis Γ: value of feature R equals to value obtaining by mean S. __ Symbol S={S i}i=1,n will designate an arbitrary integration of the geological means. If values of feature are measured on some numerical scale,

__ ℵ:{(rτ 1, rτ 2 …, rτ n )}(τ 1, τ 2 …, τ n ) → {r t}t = 1 , k

(2)

If we will associate some fixed rule ℵ = ℵ(S, R) with each __ feature R and set of means S={S i}i=1,n , then we will consider that S uniquely determines R. Really, series of local observations of feature R defines a set of alternatives rτ 1, rτ 2 …, rτ n and rule ℵ gives to the set one single value ℵ(rτ 1, rτ 2 …, rτ n ) of feature R.

we will consider this feature as quantitative, otherwise will

Taking into account that formalizability of problems associ-

believe that the feature as qualitative one. The main difference

ated with finding of reliability is mathematically complicated, we

between the quantitative and qualitative means are those that

have no ways for an identical definition of the rule ℵ. In a com-

numerical scales of the quantitative features are ordered, at that

mon case, if ℵ1 and ℵ2 are two different rules, we could not find

time in a common case scales of qualitative parameters have no

a simple method for their comparison.

order.

Proposed here probabilistic approach to definition of relia-

Let’s S and R are the fixed mean and feature, respectively

bility allows to formulate criterion for comparison of rules and to

and {r1, r2, …, rk } are set of values, which may include feature R.

solve simultaneously a problem of selection (in a sense of the

We will consider that result of local determination of feature R by

criterion) rule.

mean S always includes some alternative rτ , 1 ≤ r ≤ k which generally speaking, may differ from the real value of feature R.

Let’s suppose for definiteness that real value of feature R is r1. Results of determination R by mean S may differ from r1

If continuous scale of feature R has been divided to intervals,

because of the determination inaccuracy. Obviously, set of possi-

then determination of R reduces to finding the concrete interval to

ble indications of mean S maybe described using some probabili-

which pertains this feature. In this case we can consider that r1, r2,

ty distribution:

…, r k are the points belonging to intervals of dividing (for instance, middles of these intervals). Obviously that among the values r1, r2, …, rk is always such rt , which belongs to the same

P11=P(r1o |r1r ), P12=(r2o |r1r ), … , P1k =P(rko |r1r ),

(3)

where P1τ=P(rτo |r1r) is the conditional probability of that results of

interval that and real value of feature R. Then difference between

determination is rτ if the real value of feature R is r1 (indexes “o”

the real value of feature and value rt does not exceed a length of

and “r” designate the “observed” and “real” values, respectively).

respective interval, and value rt maybe considered as real since we

Broadly speaking, value r2 of feature R corresponds to another

propose that dividing of scale for feature R is being with the nec-

set of probabilities: P21, P22, …, P2k . In the common case probabili-

essary accuracy. Further under real value of feature R we will

ties Pt1, Pt2, …, Ptk depend on t (i.e. on the real value of feature R).

imply the mentioned value rt .

We will consider a matrix of conditional probabilities __ {Ptr= P(rτo |rtr )}t, τ =1, k for each pair (S,R) by introducing expert

Problem Statement Introduced notion of reliability of mean S relative to feature R is a quantitative measure of frequency of coinciding feature R (obtained by use of mean S) with its real value.

methods since to obtain it by a logical way is practically impossible:

 P11 … P1k    ℜ(S, R)＝ … … …   Pk1 … Pkk 

(4)

A case when feature R is determining not by one mean S but __ by a set of means S={S i}i=1,n has the following peculiarity. Any

but also on the concrete physical-geological conditions of geo-

series of observations (investigations) realized by a set S is

logical prospecting. However, relationship of ℜ(S, R) from dif-

defined, obviously, not one alternative rτ , but a set of alternatives

ferent geological (physical, chemical, etc.) factors is not discussed

rτ 1, rτ 2 …, rτ n , which, broadly speaking, are different ones. In this

here since we believe that the factors in real conditions are fixed.

We must note that the matrix depends not only on S and R

Lev Eppelbaum, et al.

236

It may appear that obtaining matrix ℜ(S, R) containing k2 numbers is practically insolvable problem. However, in real con-

(
;

) 

Σ τ

1

ditions number of independent elements in this matrix is greatly reduced.

=

It was mentioned above that probabilities Ptτ are conditional

1



Σ

1   

 τ 1

Σ    τ

1

 (ττ )

 (τ  )

1  τ =

 (  )



Σ

Σ  τ 1



1

   (τ  ) 

ττ .

ones. Ptr= P(rτo |rtr ) is the conditional probability of observed

A few trivial examples have been given below:

value rτ of feature R by condition that rτ is the real value of the

Example1.1. Exampl

feature. However, on the practice an inverse problem may have a vital importance: when using observed indication of mean is necessary to give a probabilistic estimation of real value. Mathematically this offers to the problem of determination of ~ __ probabilities { Pτt =P(rτr |rt0 )}t, τ=1, k . In contradiction to Ptτ , indexes “o” and “r” here are interchanged. ~ Probabilities Pτt , generally speaking, do not associated with Ptτ by a hard analytical relationship. However, if a priori proba~ bilities P(rt ) of alternative values of R are known, then values Ptr maybe expressed through Ptr using well-known Bayes’s expres-

~

P (rτo|rtr )P(rt) . Σ kt=1 P (rτo|rtr )P(rt)

(5)

If we have no any initial information about the values of this case formula (5) will be simplified: (6)

 0.8  0.3

1

)=

 0.9  0.4

2

)=

0.2 0.7

; 

d (S ; R

1

) = 0.5(0.8 + 0.7) = 0.75.

; 

d (S ; R

1

) = 0.5(0.9 + 0.6) = 0.75.

1

Example2.2. Exampl  ℜ(S , R 2

Example3.3. Exampl  ℜ(S , R 3

d (S ; R 3

2

  

0.1 0.6

0.8

0.1

0.1

0.1

0.8

0.1

0.1

0.1

0.8

2

  

;

) = 1/3(0.8 + 0.8 + 0.8) = 0.8.

themselves.

Reliability of set of means Si relative to feature R Let’s consider definition of feature R by set of means __ S={S i}i=1,n The proposed methodology is based on realizing the __ Axiom. A sequence of indications of means S={S i}i =1,n

(replacing feature R) is independent one. This means that

Thus, if for all τ is fulfilled the following equality: k t=1

Σ P (rτo|rtr ) =1, r 1

)=

following axiom:

P (rτo|rtr ) Pτ t = P (r |r = k . Σ t=1 P (rτo|rtr ) r o t τ)

o τ

1

(S2, R2) are different, but their probabilities are equal between

feature, we will consider that P(r1) = P(r2) = … = P(rk) = 1/k. In

~

1

In first two examples matrixes corresponding to pairs (S1, R1) and

sion (Daston, 1995; Ulrych et al., 2001):

Pτt = P (rtr |rτo) =

 ℜ(S , R

(9)

P ( rτ1 , rτ 2, o

o

r r r r o o o o … , rτn | rt ) = P ( rτ1 | rt )・P ( rτ 2 | rt )・…・P ( rτ n | rt )

(7)

r o t τ

then P(r |r ) = P(r |r ). Equality (7) indicates that a sum of ele-

(10) As it was mentioned above, we are needed to agree, which value of R we will select as the most plausible hypothesis about

ments of any column of the matrix ℜ (S, R) = 1. Further we will suppose that P(r1) = P(r2) = … = P(rk) = 1/k,

the real value of R (by each fixed set of indications of means). A

i.e. alternatives r1, r2, …, rk have a priori the equal probabilities.

variant of the possible analysis of geological information is pre-

Fulfilling of equality (7) is not obligatory.

sented in Figure 2. After series of observations of feature R (using a set of __ means S={S i}i=1,n ), we will receive a set of alternatives rτo1, rτo2 …,

Reliability of mean S relative to feature R From the definition I and description of the total probability

rτon . Which of the following k hypotheses Γt (rtr=rto ) we should

(Daston, 1995) follows that reliability of mean S relative to fea-

adopt as the most plausible? Obviously, this must be hypothesis,

ture R maybe calculated using the following formula

for which the respective probability

 (, )  Σ τ  (
τ)・ (
τ

 ) .

(8)

1

Taking into account that P (rτo) =

Στ

k

=1

P ( rt | rτ1 , rτ2 , r

o

o

… , rτn ) o

(11)

will admit the maximum value.

P (rtr ) P (rτo|rtr ) =

1

k Σ

k t=1

P (rτo|rtr )

and

Let’s designate that ℵ* is mapping (rule) placing in requirement to each sequence of means such alternative value rt* of feature R, on which is reaching the maximal value of (11).

P (rτo|rτr ) P (rτr |rτo) = k Σ t= P (rτo|rtr )

,

1

expression (8) has been transformed to the following form:

Definition II. Reliability of set of means S relative to feature R is the probability of coinciding feature R (determined by rule ℵ*) with the real value of R. Taking into account definition II and expression of total

FORMALIZATION AND ESTIMATION OF INTEGRATED GEOLOGICAL INVESTIGATIONS: AN INFORMATIONAL APPROACH

237

Figure 2. Flow-chart of geological investigation process using informational comparative probability we have

tionship (14) for each fixed set (τ1 , τ2 , …, τn).

d (S , S , …, Sn ; R) = Σ ( τ 1

2

1,

τ2 , … , τn )

o τ1 ,

o τ2 ,

o , τn

P (r r

… r )・

Let’s explain expression (13) by use of the following trivial example.

P ( r tr*| rτo , … , rτon ). 1

(12)

Example 4. Let’s feature R is determined by means S1 and S2.

Transforming (12) analogously to conversion of (8) to (9),

It is necessary by given matrixes ℜ(S1, R) and ℜ(S2, R) to calcu-

we will receive the following expression for calculation of relia-

late a reliability of means S1 and S2 relative to R (R may take val-

bility of set S relative to feature R :

ues r1 and r2 ):

d (S1, S2, …, Sn ; R) =

1 k

Σ ( τ , τ , … , τ ) P *τ ・P *τ ・…・P *τ 1

2

n

t

1

t

2

t

n

(13) *

The rule ℵ setting up a correspondence between the set of possible indications of means {rτo1, rτo2 …, rτon} and set of values of feature {rt}, sets up simultaneously correspondence between the indexes: ℵ* : ( τ1, τ2, … , τn ) → t * .

ℜ 5  4  












 

ℜ 5  4  












 

In this case a set of possible indications of means S1 and S2 consists of four elements: r1o , r1o ; r1o , r2o ; r2o , r1o ; r2o , r2o . The relationships between probabilities of means and real

(14)

It is supposed that t * in equation (13) is defined from rela-

values of feature R are compiled in Table 1.

Lev Eppelbaum, et al.

238

Table 1. Values of conditional probabilities P(rτo1, rτo2 | rtr ).

Table 2. Values of conditional probabilities for Example 5.

    

    

o

r r

o

o

r r


o

o

r
r

o

o

r
r


    

    

o

r

r

・

・


・


・

r


r

・

・

 ・

 ・

It easy to see that for data (R, S1, S2) mapping (rule) ℵ*

o

r r

o

o

r r


o

o

r
r

o

o

r
r


r

r

・

・


・


・

r


r


・


・

・

・

The rule ℵ* in this case maybe written as

maybe described by the following manner:

r1o , r1o → r1r , since 0.8・0.7 > 0.2 ・0.3;

r , r → r , since 0.8・0.9 > 0.3 ・0.4;

r1o , r2o → r2r , since 0.8・0.3 < 0.2 ・0.7;

r1o , r2o → r2r , since 0.8・0.1 < 0.3 ・0.6;

r2o , r1o → r2r , since 0.2・0.7 < 0.8 ・0.3;

r2o , r1o → r2r , since 0.2・0.9 < 0.7 ・0.4;

r2o , r2o → r2r , since 0.2・0.3 < 0.8 ・0.7

o 1

o 1

o

r 1

r2o , r2o → r2r , since 0.2・0.1 < 0.7 ・0.6.

.

Correspondingly, d(S1, S2; R) = 0.5(0.56 + 0.24 + 0.56 + 0.24) = 0.8.

Then reliability d(S1,S2; R) maybe determined as d(S1, S2; R)=0.5(0.8・0.9 + 0.3・0.6 + 0.7・0.4 + 0.7・0.6)=0.8.

Inasmuch as d (S 1 ; R) = 0.5(0.8 + 0.8) = 0.8 and d (S 1 ;

We must note that

R) = 0.5(0.7 + 0.7) = 0.7 then for the considered example in (16)

@ (5 ; 4) = 0.5 ( 0.8 + 0.7) = 0.75  . @ (5 ; 4) = 0.5 ( 0.9 + 0.6) = 0.75 

we have a sign of equality.

1

It is important also that values d(Si; R) , 1 ≤ i ≤ n in the com-

1

In the inspected case

mon case do not allow to determining identically the value d(S1,

@ (5 , 5 ; 4 ) > @ (5 ; 4 ) = @ (5 ; 4 ) . 1

2

1

(15)

2

It can be shown that non-strict inequalities analogical to (15) __ will be fulfilled always for any set of means S ={S i}i=1,n and for

S2 , …, Sn ; R). For instance, if in Example 4 we will replace matrix ℜ (S2, R) by matrix ℜ(5 , 4) = 

0.6

 0.1

3

any matrixes ℜ (S1, R). This condition will be realized in the case

0.4 0.9

, 

if probability Pττ , 1 ≤ τ ≤ k in the line with number τ in any

then for a new pair of matrixes ℜ (S1, R) and ℜ (S3, R) we will

matrix will have the most value. The last requirement has the fol-

have: d (S1; R) = d(S3; R) = 0.75 (as in the previous example), but

lowing significance: the most probable variant for each indication

d(S1, S3; R) = 0.5(0.48 + 0.32 + 0.12 + 0.63) = 0.775.

of mean is an output corresponding to the real value of feature R.

d (S , S , …, Sn ; R) ≥ max{d(Si ; R)} 1

Thus, reliabilities of means S1, S2 and S1 relative to feature R are equal, because d(S1, R) = d(S2, R) = d(S3, R) = 0.75. However,

Thus, inequality 2

(16)

i =1, n

__ is realizing for any R and {S i}i=1,n .

reliability of pair of means (S1, S2) relative to R is higher that reliability of pair (S1, S3 ): d (S1, S2; R) = 0.8 > d(S1, S3; R) = 0.775

At the same time in (16) left part may equal right part (by __ definite R and {S i}i=1,n ). Such a variant will have place when each

Let’s introduce a new notion “ increment of reliability ” __ I ∆d (S1, S2, …, Sn ; R) for each R and set of means S={S i}i=1,n .

of matrixes ℜ (S1, R) and ℜ (S2, R) has the following form

This value maybe defined using the following expression: ∆ d I (S , S , …, Sn ; R) = d (S , S , …, Sn ; R) − max{d(Si ; R)}.

 2 1 −2  ,  1 −2 2 

1



 

2

i ≤1≤n

(17) Introduced value ∆d (S1, S2, …, Sn ; R) has the following __ significance. It shows how much the reliability of S ={S i}i =1,n

Example 5. Let’s




1

I

i.e. they are symmetrical.

ℜ 5  4  

2

   

ℜ 5  4   



 

exceeds the reliability of the most precise mean from the set Si .

   

We could also consider a new parameter ∆d I I : ∆ d II(S ,S , …, Sn ; R) 1

o τ1

o τ2

r t

In this case table of conditional probabilities P(r , r | r ) will have the following form:

=

2

d (S ,S , …, Sn ; R) − max {d(Si , Si ; R)}, 1≤ i , i ≤ n 1

2

1

2

1

2

(18) __ which defines how much the reliability of set S={S i}i=1,n exceeds the reliability of the most precise pair of means (in terms of d(Si1, Si2, …, Sn ; R), involving to the set S. Analogically we can define ∆d III (S1, S2, …, Sn ; R) , ∆d IV (S1,

FORMALIZATION AND ESTIMATION OF INTEGRATED GEOLOGICAL INVESTIGATIONS: AN INFORMATIONAL APPROACH

S2, …, Sn ; R) and so on.

239

value d (S1, S2, S3; R) = 0.5 ( 2・0.8・0.8・0.8 + 6・0.2・0.8・0.8)

For the above-mentioned Example 4 we have: I

I

∆d (S1, S2 ; R) = 0.8−0.75 = 0.05 ;

= 0.896.

∆d (S1, S3 ; R) = 0.025.

For Example 5 d(S1, S2 ; R) = d (S1; R) , consequently d (S1, S2 ;

whence

R)=d(S1; R) and d (S1, S2 ; R)= max{d (Si; R)}. Therefore, ∆d I (S1,

∆d I (S1, S2, S3 ; R) = 0.896− 0.8 = 0.096.

S2 ; R) = 0 .

If we will introduce to the above considered example

i=1,2

The last result does not contradict to logics, as it may appear

ℜ(5 , 4) = ℜ(5 , 4) = ℜ(5 , 4) = 

for a first view. If we will again refer to the rule ℵ* constructed

1

2

3

for Example 5, we could see that the values of feature R determined by the rule ℵ* precisely correspond to indications of first

0.7

. 

d(S1, S2, S3; R) = 0.5 ( 2・0.7・0.7・0.7 + 6・0.3・0.7・0.7)

feature R will be equal to reliability of mean S1. Therefore, here is realized an equality d(S1, S2 ; R) = d(S1; R) , whence ∆d (S1, S2 ; R)

0.3

we receive

mean Si. Then and reliability of pair of means (S1, S2) relative to I

0.7

 0.3

= 0.784. and ∆d I (S1, S2, S3 ; R) = 0.784− 0.7 = 0.084.

=0. In Example 4 the described above situation has no place,

Conclusion

therefore corresponding value ∆d I (S1, S2 ; R) is positive one. Let’s examine additional example where number of means n = 3.

1. Suggested methodology of determining reliability for means and sets of means relative to feature is based on that for each

Example 6. Let’s

pair of mean-feature (S, R) is assigned a matrix of conditional

ℜ(5 , 4) = ℜ(5 , 4) = ℜ(5 , 4) =  1

2

0.8

 0.2

3

probabilities. The reliability of set of means is determined

. 0.8  0.2

using (13) and increment of reliability ﾐ using (17) and (18). 2. Analysis of presented examples indicates that increment of reliability ∆d (・) (S1, S2,…,Sn ; R) especially depends on both

It is easy to show that @ (5 , 5 ; 4) = @ (5 , 5 ; 4) = @ (5 , 5 ; 4) = 0.8 1

2

1

3

2

3

and

∆ @ 1 (5 , 5 ; 4) = ∆ @ 1 (5 , 5 ; 4) = ∆ @ 1 (5 , 5 ; 4) = 0. 1

2

1

3

2

3

number of means involving to prospecting set and reliabilities

    

of these means and on method of the means integrating. For each feature R and any set of means S maybe established combinations of means and degree of their influence to the incre-

The necessary data for calculation of value d I (S1, S2, S3 ; R) are

ment of reliability. The obtained conclusion is significant for

compiled in Table 3.

optimization of geological prospecting set by a calculation of

Table 3. Values of conditional probabilities P(rτo1, rτo2 , rτo3| rtr ) . 

  

      

o

o

r r r

o

o

o

r r r


o

o

o

r r
r

o

o

r r
r


described in the presented paper. Matrixes ℜ (S1, R) and ℜ (S2,

o

r

・・ ・・
・
・ ・
・


r


r


・
・

・
・ 
・・

・・ o

o

r

r

r


r

o

o

o

r
r r


o

o

o

r
r
r

o

3. Besides the known procedures described in the theory of expert estimations, maybe used and analytical approach

o

r

r
r r

necessary expenditures.

o

o

r
r
r


R) developed by experts maybe adjusted by the way of comparing value ∆d I(S1, S2, ; R) with the expert estimation of the same value.

o


・・ 
・・

・
・ 
・
・


Acknowledgement

・
・
・
・ ・・
・・

The authors are grateful to unknown referees for their useful comments allowing to improving the manuscript.

The rule ℵ* in this case maybe presented as: r1o , r1o , r1o r1o , r1o , r2o r1o , r2o , r1o r1o , r2o , r2o r1o , r1o , r1o r1o , r1o , r2o r1o , r2o , r1o r1o , r2o , r2o

→ r1r , → r1r , → r1r , → r2r , → r1r , → r2r , → r2r , → r2r ,

since 0.8・0.8・0.8 > 0.2・0.2・0.2; since 0.8・0.8・0.2 > 0.2・0.2・0.8; since 0.8・0.2・0.8 > 0.2・0.8・0.2; since 0.8・0.2・0.2 < 0.2・0.8・0.8; since 0.2・0.8・0.8 > 0.8・0.2・0.2; since 0.2・0.8・0.2 < 0.8・0.2・0.8; since 0.2・0.2・0.8 < 0.8・0.8・0.2; since 0.2・0.2・0.2 < 0.8・0.8・0.8

References Borisovich, V.T. and Eppelbaum, V.M. (1988) Optimization of prospecting for economic ore minerals in mountainous regions. .

Nedra, Moscow, (in Russian), 239 pp. Daston L. (1995) Classical probability in the Enlightenment. Princeton University Press, Princeton, 451 pp. Eppelbaum, V.M. and Eppelbaum, L.V. (1995) Advanced approach to formalization and controlling of geological

Taking into account the data presented in Table 3 and rule ℵ*,

prospecting. Trans. of the Conference of the Geological Society

240

Lev Eppelbaum, et al.

of America, Rocky Mountain, USA, Vol. 27, no.4, pp. 10-11. Eppelbaum, L.V., Vaksman, V.L., Klepatch, C.A., Eppelbaum, V.M. and Itkis, S.E. (2001) Optimization of detailed searching intrusive objects in Israel using informational approach. Trans. of the Annual Meeting of the Europ. Geoph. Soc., Nice, France, GRA3, p. 1454. Khesin, B.E. and Eppelbaum, L.V. (1997) The number of geophysical methods required for target classification: quantitative estimation. Geoinformatics, vol. 8, no.1, pp. 31-39. Ulrych, T.J., Sacchi, M.D. and Woodbury, A. (2001) A Bayes tour of inversion: A tutorial. Geophysics, vol. 66, pp. 55-69.