Annual report: April 1, 1953 to March 31, 1954

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3

TABLE OF CONTENTS

ABSTRACT I. INTRODUCTION

II. DRILLING PATTERNS

III. THE PROBLEM OF ASSOCIATION IV. ANALYSIS OF POLAR MESA DATA

V. ANALYSIS OF DATA FROM MESA V, LUKACHUKAl MOUNTAINS

APPENDIX A. CONSTRUCTION OF DRILLING PATTERNS APPENDIX B. TABLE OF RANDOM NUMBERS

APPENDIX C. ON TESTING THE ASSOCIATION OF MINERAL OCCURENCE WITH A SET OF OBSERVABLE

CHARACTER-4

5 6 11 21 33

44

50 ISTICS 55

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ANNUAL REPORT- APRIL 1, 1953 -MARCH 31, 1954 By M. R. Mickey, Jr. and H. W. Jesperson, Jr.

ABSTRACT

The objective of work under this contract is the consideration of the application of statistical methods and concepts to problems in the exploration for uranium ore. This report discusses the problems con-sidered by the project during the past year and summarizes the work performed.

The material discussed falls into three parts. The first deals with pattern drilling as an exploration technique. A basis for the comparison and selection of drilling patterns is presented and the details involved in pattern selection are presented. Secondly, a basis for considering the question of whether a set of observable characteristics is associated with the presence of mineralization is presented. The ideas presented here are further elaborated by application to analysis of data from the Polar Mesa and the Lukachukai Mesa V drilling projects.

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5

I. INTRODUCTION

This report is submitted as the annual technical report for the period 1 April 1953 to 31 March 1954. The work reported on was performed under contract AT{30-l)-1377 between Iowa State College and the Atomic Energy Commission. The general objective of this work is the consideration of possible applications of statistics to problems of uranium exploration in the Colorado Plateau.

The work has been undertaken as a research project at the Iowa State College Statistical Laboratory. Dr. T. A. Bancroft, Director of the Statis-tical Laboratory, is the project leader. Dr. C. J. Roy, Head, Geology Department, Iowa State College, is geological advisor to the project.

Dr. M. R. Mickey, principal investigator, and Mr. H. W. Jespersen corn-prise the remainder of the project personnel.

The work reported on was carried out at Iowa State College. Mickey and Jespersen visited the plateau for an approximately seven week period during the months of July, August, and September 1953. The greater portion of this time was spent at the Dry Valley field camp near Moab, Utah. The objective of this trip was the observation of field practices, difficulties, and problems.

We would like at this point to express our thanks to the personnel of the Grand Junctions Operations Office for the many courtesies extended during our stay.

This report presents work performed on four interrelated problems. These may be described as:

1. The problem of drilling patterns.

z.

The problem of association.

3. Analysis of data from the Polar Mesa drilling project. 4. Analysis of data from Mesa V, Lukachukai Mountains.

Each of these problems is discussed in a separate section of this report. In addition, two appendices are provided. These contain elaborations of

certain aspects of the problems described by {1) and {2) above.

Since the previously submitted annual report1 covered the period 1 August 52 to 31 July 53, part of the work performed during the period of the present report has already been discussed. There does not appear to be any point in re-discussing this material.

The material of section V of this report was prepared by Jespersen and Mickey on the basis of work of Jespersen. The remainder of the report was prepared by Mickey, with computational assistance provided by Jespersen and the Statistical Laboratory computing service.

1.

Mickey, M. ·R., Annual Report from August 1, 1952 to July 31, 1953. RME-3056, 24 July 1953.

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II. DRILLING PATTERNS

During the course of our stay on the Plateau, the question of pattern drilling frequently came up for discussion. In particular, it was wondered

if there was a 11_best11 _{pattern form.} _In_{view of the interest expressed, the} problem was taken under consideration.

In considering such a problem it is first necessary to establish a con-ceptual basis for the formulation of the problem. Once this is done, it is a matter of detail to obtain "answers". There appear to be two more or less distinct approaches to the exploration problem. One of these may be charac-terized by the presence of well defined search targets. That is, if one takes an observation at a particular place he is able to say, on the basis of that observation alone, whether or not there is a target at that location. The mineralized bodies themselves may form such a target. A region of high gamma ray intensity may form such a target; a region of abundance of carbon-aceous material may form such a target. Such targets might be defined in an indefinite number of ways. The final objects of the search, of course, are the ore bodies. But these are ordinarily small and require much effort to locate. Consequently, the possibility of a more inclusive target offers possibilities for less expensive search programs. There then arise the problems of the selection of suitable search targets and the selection of appropriate search procedures. We are concerned here with the latter. We further limit the problem by confining the search region to regions of drilling project size. The use of drilling patterns as a search procedure then naturally suggests itself. It is particularly useful to study the properties of pattern drilling, since rather definite results can be obtained. One can then evaluate the capabilities of this search method.

By way of contrast, we consider the second general approach to the explora-tion problem. This approach has a more predictive nature than the first. In

general, one seeks features which control the deposition of the ore; recon-naissanse effort attempts to locate the controlling features and the final

effort is concentrated at the predicted locations. An example of this approach is given by the use of channels in localizing area of final search.

It is to be pointed out that these two approaches are not mutually ex-clusive. The degree to which they may be combined will depend upon the size of the predicted favorable area. This area will ordinarily be sufficiently large that it can be designated as a search area which can then be more fully

explored by the technique of pattern drilling.

In considering pattern drilling, it is necessary first to say what is

meant by the term pattern. The use of the term is restricted to mean a two dimensional array of points with the following property. There exist two families of evenly spaced parallel lines such that the set of pattern points coincides with the set of points of intersection of intersecting lines. Such patterns can be generated in a number of ways. For example the plane may be 11_filled11 _{with congruent parallelograms; pattern points lie at the vertices}

of the parallelograms. Hence we may speak of the parallelogram that

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•,

7

congruent triangles, since the parallelogrru:n may be decomposed into tri-angles. From another point of view, a pattern can be considered as a set of "fences"· For purposes of elaboration, we will speak mostly of the paral-lelogrru:n generation of patterns. It turns out that it suffices to confine

con-sideration to parallelogrru:ns of a diamond shape; that is, one in which the two diagonals are mutually perpendicular.

In order to be able to compare patterns, it is necessary to be able to characterize both cost and performance considerations. In general the cost will be simply related to the number of wells that are required. It then

suffices to consider the required density of wells, that is,the number of wells per unit area. The characterization of performance offers greater difficulties. We submit that the performance of drilling patterns can be studied by means of probability ideas.

In order to elaborate the ideas involved, we may strip the problem of its complexity by considering first a special case. Suppose that there is a single target in the area to be explored. Suppose that the pattern under con-sideration can be generated by a rectangle, and that the target is such that it can be completely contained in a generating rectangle. The situation is illustrated in figure 1.

Fig. I-llustrative drilling target and pattern rectangle.

In such a case it might be argued that an appropriate measure of the probability of hitting the target with the pattern of wells is given by the ratio:

area of target

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-The justification of this formula is as follows. Whether or not the target is hit obviously depends upon where the pattern is "located". We ask, loosely; how many ways can the pattern be located, and of these what pro-portion will result in the target being hit? We suppose that the pattern is

to be oriented in a given direction. One and only one pattern point will fall inside the rectangle of figure 1 for any choice of the location of the pattern, except for the case in which the illustrated rectangle coincides with a rec-tangle of the chosen pattern. Thus the area of the recrec-tangle provides a measure of the 11_number11 _{of possible pattern locations.} _In_{the same way,} the area of the target provides a mea&ure of the "number" of pattern loca-tions that will result in a. hit on the target. Thus the above ratio can be taken as a. measure of the probability of hitting the target with a pattern generated by the illustrated rectangle.

As it stands, s:uch a. measure of probability is difficult to interpret in terms of experience. Fortunately, it is easy to resolve the matter. All that is necessary is to locate the pattern by means of a randomly chosen point. The technique for doing this is elaborated in appendix A of this report. The probability is then interpreted in terms of experience with various games of chance. Whether or not the target will be hit with a randomly located pat-tern is equivalent to whether or not certain chance events will be realized in appropriate chance experiments. For example, suppose that the hit prob-ability has the value one-half. Consider the chance experiment of shuffling a. deck of cards and picking a card from the deck. The probability that the chosen card will be red is also one half. Thus we may say that we have exact-ly the same degree of assurance that the target will be hit as we have that the drawn card will be 11_red_{11 •} _{By utilizing the device of random location} of the pattern, we ensure that all probability statements have an interpreta-tion in terms of experience with chance events. The random locainterpreta-tion of the pattern plays essentially the same role as the shuffling of the deck of cards. It guarantees the interpretation of the odds, and hence provides a. basis for evaluation.

The manner in which hit probabilities provide a basis for the evaluation of performance of drilling patterns in the simple case under consideration is now apparent. The hit probability characterizes our expectations. Thus given two patterns which have the same cost requirements, we will prefer the one which yields the greater hit probability. On this basis we are able to compare and select patterns.

It is appropriate to note that the above simple expression for the hit probability will hold provided that it is not possible for two or more wells to

simultaneously penetrate the target. If this condition is not fulfilled, the calculation becomes more complex. The calculation can be carried out by graphical means involving the use of overlays. If the target can be charac-terized mathematically, it will be possible to express the hit probability by means of formulas or by numerical computational procedures.

We have so far discussed the performance evaluation of drilling pa.1:terns in terms of a. single target. Conceptually, the same principles apply to the more complicated cases. For example, if there were several targets in an area., then given appropriate information, it would be possible to compute the probability that any given selection of the targets would be simultaneously

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c '

9

hit. The problem rapidly becomes very complex, and it is not possible to give general results that are of interest. Special cases need to be considered. For example, one can perform pencil and paper drilling exercises, in which a set of targets is laid out in a drilling area, and randomly located patterns are laid out to locate the targets. Such exercises are essential to the apprecia-tion of the performance of pattern drilling.

For practical purposes, it is necessary to provide a simpler means of comparing and selecting the drilling patterns to be used. We have considered the following idea. We assume that it is possible for the searcher to express information about the search area in terms of an anticipated target. We assume further that this anticipated target can be adequately described as an ellipse of given size, shape, and orientation. The selection problem can then be formulated in the following dual manner.

i. Am.ong those patterns that achieve a given hit probability for the anticipated target, select patterns that require minimum well density.

ii. Am.ong those patterns that require a given well density, select patterns that maximize the hit probability for the anticipated target.

The duality involved is that any set of solutions to the one formulation is also a set of solutions to the other.

The formalities of the solution of this problem will be presented in the final technical report. It turns out that there are an indefinitely large number of solutions to the problem. A basis for further limitation of the solutions lies in considering the sensitivity of the solutions to variations in the antici-pated target. For example, if the orientation of the anticipated target is a bit "off", how much does this affect the hit probability? This approach has not been explored theoretically. Two of the possible solutions have been

singled out on the basis of relative ease in construction. There is some in·-dication that these two solutions represent extremes in the sensitivity, parti-cularly with respect to orientation errors, at least for certain ranges of the parameters that describe the anticipated target.

The two solutions mentioned may be described as being generated by diamonds. Since the generating diamond is determined by the lengths and orientations of the two diagonals, it suffices to describe the manner in which these are to be chosen. The procedure is described in detail in appendix A

of this report. For the present we may say that one diagonal of the diamond is to be oriented in the direction of the longer axis of the ellipse that describes the anticipated target. Its length is obtained by multiplying the length of the longer axis of the ellipse by an appropriate factor which depends upon the desired hit probability. (See table A-I, appendix A). Similarly, the other diamond diagonal is to be oriented in the direction of the shorter axis of the ellipse; its length is obtained by multiplying the length of the shorter axis of the ellipse by an appropriate factor. The two solutions mentioned are obtained by interchanging the two multiplying factors.

The one solution is relatively insensitive to small errors in orientation of the target, but is relatively more sensitive to larger errors. For example,

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for a desired hit probability of 0. 75, an orientation error of 25°- 300 will

have negligable effect on the one solution, The range of insensitivity will increase as the desired hit probability decreases, Hence the one solution is to be recommended if it can be considered that the orientation is fairly well known, Otherwise, the second solution is to be recommended.

The discussion of appendix A of this report is largely based upon the assumption that the desired hit probability is stated, If, on the other hand it is the density of wells that is given, it will be a simple matter to convert this to a desired hit probability, so that the one set of tables suffices for all practical purposes,

Part of the flexibility involved in the use of pattern drilling stems from the fact that nothing has so far been said about the size and nature of the area to be explored. The attitude here is that it is up to the exploration man to localize the search area. Once this has been done, however, there should be real advantage in completing the exploration by means of pattern drilling, Consequently, we recommend that serious consideration be given to the use of pattern drilling as an exploration technique, Since exploitation of the

possibilities of pattern drilling depends upon an appreciation of its properties, it is recommended that those concerned actually conduct a few pencil and paper "drilling projects "•

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11

III. THE PROBLEM OF ASSOCIATION

A large portion of the effort in an empirical approach to exploration

problems lies in the discovery and exploitation of associations. For example, it is considered that the presence of mineralization is associated with an abundance of carbonaceous material, and this leads to guiding rules for prospecting. The importance of the problem is clear and needs no further discussion.

There is a connec"f:i.on between the problem of association and the prob-lem of drilling patterns. One aspect of this connection lies in the consider-ation that characteristics that are associated with the presence of mineral-ization provide means of defining drilling targets. Hence the prtblem of association, as we view it, falls into the pursuit of the first point of view to-wards exploration that was presented in section II.

An important concleptual problem lies in deciding what is meant by the presence of an association. This leads to the methodological question of

how a suspected association is to be established. These questions are clearly of fundamental importance. The purpose of this section is to contribute to the discussion by exhibiting an approach to the problems.

The approach taken here is largely empirical in nature. We are not concerned with the basic problems of causal relations and explanations for the existence of certain associations. In a sense, the establishing of the existence of the associations comes first; the point of view is that it is necessary to establish the existence; theory must then account for the

associations. Clearly, an attempt at theorization is on un£irm ground unless the existence question has been satisfactorily answered. On the other hand, a purely empirical approach is likely to bog down due to the great number of possible associations.

These evident points are mentioned in order to give some perspective to the present discussion. Our primary consideration is a discussion of the concepts and methodology involved. The job of turning these ideas into use-ful field results lies in the province of the geologist

The essence of the problem has been expressed by Stokes1: "Stated in simplest terms solution of the problem of ore genesis consists of sorting out from a great number of factors those which are common in mineralized rock and those which are uncommon or absent in unmineralized rock11 • There is in this statement also the implication that those factors which are common in mineralized rock are not also common in unmineralized rock, and that those factors which are uncommon in unmineralized rock are not also un-common in mineralized rock. Thus to establish an association, it is neces-sary to establish that the mineralization is relatively more abundant where the factor is present than where the factor is absent.

I.

Stokes,

William

Lee, Primary Sedimentary Trend Indicators as Applied to Ore Finding in the Carrizo Mountains, Arizong and New Mexico, Part I. RME-3043, August, 1953. page 42.

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For purposes of discussion, we may suppose that the interest is in a single factor which is either present or absent. We further assume that the mineralization has this same characteristic of being either present or absent; that is, we are not concerned with assay values, etc. The area of interest may then be subdivided into four disjoint parts:

1. The factor and mineralization are both absent. 2. The factor is present but mineralization is absent. 3. The factor is absent but mineralization is present. 4. The factor and mineralization are both present. Let A₁, A

2, A3, A4 be the areas of the above parts. Then the fraction

A

+

i

gives a measure of the abundance of the mineralization where the

2 4 A

3

factor is present, and the fraction

A

+A gives a measure of the abundance 1 3

of mineralization where the factor is absent. Hence we may say that there is no association if the

two

fractions are equal, that is if

holds.

Since we can say what we mean by an absence of association, we can also say what we mean by the presence of an associatic;m: Namely, an a13so·ciation is present if the above equality does not hold. The stumbling block, when it comes to application of this definition, is that the values of the areas are never known. This is where statistics enters the picture. It is not necessary

to know the values of the areas; the entire area can be sampled in such a wa,y

that if there is an association, this fact will very likely become known. As an illustration of the idea involved, an example has been prepared in which both the presence and absence of association are involved. In figure 2 is illustrated a hypothetical drilling project in which the regions of the

presence of mineralization and the presence of the factor are indicated. The areas of these regions are given in Table I.

Table I. Areas, in square inches, of the regions indicated in figure 2

Factor Minerahzatlon TOtal

present absent

present

0.70 5. 24

5.94

lab sent

0.54

17.03

17.57

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0 0 0 0 0 0 ₀ 0 0 0 0 0 0 ₀ 0 ₀ 0

0

0 0 0 0 0 0 0 0 ₀ 0 0 0 0 0 0 0 0

L£G£NO:

o WELL LOCATION

v)1

BOUNDARY ENCLOSING REGION ON WHICH

,f'

FACTOR IS PRESENT

~ORE

BODY

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0 ₀ _c

0

0 0 ₀

eJo

'

0 0 ₀ / 0 ₀ ₀ 0 ₀ 0 0

!],

0 0 ₀ 0

d:?

0 0

~

0 0 0 0 ₀

!J'

0$

0 0 0 0

...

0 0 ₀ _II» 0 ₀ 0 0 0 0

LeGEND:

0 _{WELL LOCATION}

/

WHICH

BOUNDARY ENCLOSING REGION FACTOR IS PRESENT

IN

~ORE BODY

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15

From the values given in table I it is seen that the region in which the factor is present comprises Z5. 3% of the total drilling area, and that the region in which the factor is present cont<~.ins 56.5% of the area of the mineralized region. Consequently according to the concepts presented,

there is definitely present an association between the presence of the factor and the presence of mineralization.

By the way of contrast, figure 3 illustrates the case of no association. Here, the exploration region and the region of mineralization are the same as in figure Z, but the region of presence of the factor has been altered. The areas of the regions of figure 3 are presented in table II.

Table II. Areas, in square inches, of the regions indicated in figure 3

.l!"ac'tor M1nera.J.1zat1on Total present a'6sent

present 0.33 5.99 6. 3Z

aosent U.':/1 lb. GIS U.l':l

ITOta! l . G4

.:.:. u

.:::s.!>l

The region in which the factor is present now comprises Z6. 9% of the exploration region; it contains Z6. 6% of the area of the mineralized region • .Acc·ording to the above definition, there is a very slight association present, but the association is so slight as to be negligible, and figure 3 may be

taken as representing the case of the absence of an association.

The methodological question that arises is: How can the two cases be distinguished on the basis o£ the results of a drilling program? It is

assumed that the only information that results from the drilling of a well is whether or not the factor is present and whether or not mineralization is present at the point at which the well is drilled. Since the amount of drilling effort is limited, one must rely upon a relatively small sampling of the

exploration region to provide an answer, and the sampling fluctuations of the results become of great importance.

In the present example, it is assumed that the exploration is to be done by means of a randomly located pattern. .A pattern was selected according to- the technique discussed in appendix A; the length of the longer diamond diagonal is Z. 0 inches and the length of the shorter diamond diagonal is 0. 35 inches. The pattern is indicated in figures Z and 3. The results of the drilling program are summarized in tables ill and IV.

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Table lli. Results of hypothetical drilling indicated in figure 2 [Factor Minerallzation Total

present absent

present 3 15 18

absent l 4b 47

Total 4 b! btl

Table IV. Results of hypothetical drilling indicated in figure 3. Factor Miner a lizatlon Total

present absent

present 1 !b

u

absent 3 45 48

Total 4 bl b5

On the basis of Table Ill we would estimate that 75% o£ the miner-alization is contained in 27. 7"/o o£ the exploration region, for the situation represented in figure 2. On the basis of Table IV we would estimate that 25"/o of the mineralization is contained in 26. 2% of the exploration region for the situation represented in figure 3. Clearly, the results of Table Ill are consistent with the assumption that an association exists, and the re-sults of Table IV ar.e consistent with the assumption that an association does not exist.

The argument must go beyond this observation, however. The degree to which confidence is to be placed in the assertion that the results of

Table ill support the supposition of association will depend upon a judgement of the likelihood of obtaining evidence at leas t this strong if no association is present. For if the probability of obtaining results as consistent with the assumption of association as those of Table Ill were fairly large for san1pling the exploration region of figure 3, then the fact that these results were ob-tained could not be considered as providing very strong evidence of associa-tion. Consequently, it is of fundamental importance to form same idea of the san1pling distribution of the entries of tables similar to Tables Ill and IV when san1pling from the non-association case.

To further illustrate these ideas a. sampling experiment was rep:l!lated ten times. The results are presented in Table V. The experiment

con-sisted of the random location of the pattern and the observation of the numbers of wells that fell into the four catagories of Tables Ill and IV. The san1e

locations were used for both figures .2 and 3.

The results of the san1pling experinlents indicate in general that the results of Table ill are relatively unlikely relative to the assumption of a lack of association. Consequently, there is a basis for regarding such re-sults as evidence of association.

Examination of Table V also indicates the desirability of reducing the data to more manageable proportions. A set of four numbers is too difficult to comprehend without further manipulation. One way of achieving this re-duction is to associate with each two-way table a single number which is de-signated as the value of the chi square statistic corresponding to th"' tabl'i!.

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Table V. Results of "pencil and paper" drilling experi-ments for examples representing both existence and absence of association.

Trial no. I

Factor Mineralization Total Factor Mineralization Total

present absent present absent

present 1 16 17 present 0 19 19

absent 1 49 50 absent 2 46 48

Total ~ 65 67 Total 2 65 67

chi square = 0. 66 chi square = 0. 82 Trial no. 2

Facto-r Mineralization Total Factor Mineralization Total

Total 4 60 64 Total 4 60 64

present I 16 17 present 1 18 19

Total 2 66 68 Total 2 66 68

;present absent present absent

present 3 16 19 present I I7 18

absent I 46 47 absent 3 45 48

Total 4 62 66 Total 4 62

66

Factor Mineralization Total Facto-r Mineralization Total

Total 5 62 67 Total 5 62 67

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Table Y concluded. Trial no. 6

- - -

--

-- ---

---

---Factor Mineralization Total Factor Mineralization Total

present 2 15 17 present l 17 18

Total 4 62 66 Total 4 62 66

chi square

=

1. 31 chi square

=

0. 01 Trial no. 7

Total 3 63 66 Total 3 63 66

Total 6 62 68 Total 6 62 68

Factor Mineraization Total Factor Mineralization Total

Total 5 58 63 Total 5 58 63

chi square

=

2. 63 chi square =

o.

35 Trial no. 10

Total 4 66 70 Total 4 66 70

chi square

=

0. 95 chi square

=

o.

02

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19 This is done as follows.

On the basis o£ the marginal totals of the table, an auxiliary table of "expected values 11 _{may be computed; the entries are in a sense the} values that would be expected if there were no association present. For example consider Table ilL From the marginal totals, one may say that 4/65 of the wells are expected to be mineralized, and that o£ these, the fraction 18/65 should have the factor present. Thus the value

65 :lC 4/65 "" 18/65 = 1. 107

is obtained as the expected number of mineralized-factor present wells. Other table entries are computed similarly, and the results are presented in Table V1.

Table VI. "Expected values 11 _{corresponding} to entries of Table ill

.

Factor Mineralization Total present absent

!Present 1.108 16.892 18. 000 absent z.892 44. 108 47.000

Total 4.000 61.000 65.000

The value of the chi square statistic is then computed according to the formula:

hi f {observed value- 'a£iected value)

3

c square

=

sum o _{expec e v ue}t d

_ (3-1.108)3

+

{1-Z. 89Z)3

+

(15-16. 892)3

+

{46-44.108)3

- 1.108 z. 892 16.892 44.1o8

= 4. 76

One reason for considering the chi square statistic is that its sampling distribution, relative to the assumption of a lack of association, can be approximated by the computed chi square distribution!. According to this approximation, it is about

1. An even bet that the value obtained will be less than 0. 455; z. A 4 to 1 bet that the value obtained will be less than 1. 64Z; 3. A 9 to 1 bet that the value obtained will be less than z. 706; 4. A 19 to 1 bet that the value obtained will be less than 3. 841. The results of Table V indicate that the approximation will not be good in all cases; this is to be expected due to the nature of the systematic sampling provided by pattern drilling; it is to be expected that in sampling from the

1.

Snedecor, G. W., Statistical Methods. Iowa State College Press. Ames, Iowa. Fourth addition, 1946, Page 190.

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no association case fewer large values of chi square will result than are indicated by the tabulated distribution. fu any event, the approximation should be sufficient for practical purposes.

The principal reason for considering the chi square statistic, however, is that it is to be expected that when an association is present, the value obtained will be relatively large, and that when no association is present the value obtained will be relatively small. fu the example considered {figure 2 and Table III) the chi square value was computed to be 4. 76. On the basis of comparison with the tabulated distribution, it would be of the order of a 19 to 1 bet against obtaining a value as large or larger if no association were present. Hence, one may be more inclined to regard the large value obtained as evidence of an association, than as the happening of an unlikely event.

The above example has been elaborated at some length as an illustration of the ideas involved in the study of associations. The simplifications of the example of the case of a single factor does not represent any real re-striction. Likewise, the restriction of the factor to be either present or absent is not essential. The essential ingredients are that the data .should arise from a uniform sampling of the exploration region, and that whether or not the results are to be regarded as evidence of the existence of

associations is to be decided according to the likelihood of the results relative to the assumption that no association exists. These ideas are further elaborated in the discussions of the analyses of data from the drilling projects at Polar Mesa and at Mesa V, Lukachukai Mountains.

A somewhat broader discussion of the problem of association testing was presented at a symposium at the December 1953 meeting of the

American Statistical Association. This paper is reproduced as appendix C of this report.

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I

Zl

IV. ANALYSIS OF POLAR MESSA DATA

Early in 1953, certain results from the drilling project at Polar Mesa were forwarded to the Statistical Laboratory for our examination, The information supplied consisted of the following:

l, Footage yellow-brown to brown sandstone,

z.

Footage blue to green mudstone,

3. Footage gray sandstone, 4. Footage siltstone

5, Total footage sandstone

6, Footage red to brown mudstone, 7. Footage red sandstone,

8. Total footage mudstone.

These data were given, for a designated unit, at each of a number of wells. Data from both exploratory and offset wells were provided, In addition data from gamma ray logs were given for certain of the wells.

Two problems appeared to be of interest, It was decided to consider the data from the standpoint of the prediction of "favorable areas 11_, _{and to examine} the data to see whether the factors could be considered to be associated with the presence of mineralization, We consider first the £avorability problem,

The principle underlying the present approach to the problem is the following:

1, Points for which the values of corresponding characteristics are the same are to be regarded as being equally favorable,

z.

A coefficient of the favorability at any given point can be obtained from a coefficient at the degree of likeness of the characteristics of the given point to the characteristics of the most similar point where there is mineralization.

As an illustration of these ideas, suppose that only two characteristics were of interest, the footage of sandstone and the footage of mudstone in the ore unit. Then to each well there corresponds a pair of numbers, Thus each well may be represented as a point in a plane. Two wells which have the same values of the two characteristics are represented by the same point. It is clear that a me measure of the similarity between the characteristics of any two wells is given by the distance, in this plane representation, between the points representing the two wells, To obtain a coefficient of favorability for any given well, that well may be compared with each of the mineralized wells on the above basis. This comparison gives a set of numbers, the distances between the point re-presenting the given well and the points rere-presenting the mineralized wells. The favorability coefficient for the given well is the smallest of these distances; i.e. it is the distance of the closest point representing a mineralized well, In this way a favorability coefficient may be assigned to each. well; a contour map may then be prepared. The value of the coefficient at any mineralized hole must

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22

always be zero; relatively src.all values of the coefficient are

to

be con-sidered favorable.

The above idea is readily formalized and extended to the case of more than two factors. Let d .. be the value of the coefficient of likeness between the characteristics of v.vMl.s i and j, and let S denote the set of indices

(well numbers) corres·ponding to mineralized wells. Then the favorability coefficient at well i is ,given as

f. =min d ..

1 1J

jeS

The set of number~> d .. may be thought of as a two way table, in which the subscript i den-otes

tlfJ

row and the subscript j denotes the column of the entry. By confining j to the well identification numbers of mineralized wells, we obtain a table in which there are as many rows as there are wells and as many columns as there are mineralized wells. The value of the favo:'ii\.bility coefficieL\t at well number i is then the smallest entry value in the i row of the table.

Suppose that there· .are k characteristics under consideration. Then any given well, say well no. i, can be represented by the ordered set of the values of the characteristics,

(vil' viZ' • • • • v ik)

where vi is the value of the 11th characteristic -of well no. i. There are many wa{/s in which a coefficient of likeness may be constructed. Perhaps the simplest of these is given by the computational formula,

k

d ..

=

I: )\

{(v. - v. )z]

lJ n=l

<{

111 Jll

Here we use the symboln as the index of summation over the k character-istics. The constant coefficients,

1\ ,

are introduced to allow different weighting I> of the factors. If it is des\red to give the factors equal weights, the values for these coefficients may be chosen according to the formula

1 n - ·z

-.... I: _n~~_{._} (v. - v ) 1 111 11

1-where n is the number of wells and

v

is the average value of the a. th characteristic, i. e. vn

=

u n 1 I: v. n i=₁ 1u

A computationally rp.ore suitable expression for )\ is given as a.

.A :

1

fn

I; V.z - - ( I ; 1

n

v. )Z

:I

11 _{n-1 i= 1 111} _{n i= 1 111}

(23)

23

The type of calculation required in the above is well suited to punched card machine methOds. For a moderate number of wells, the table of dij values can be computed in a few hours. The remaining effort then consists of sorting out the smallest value in each row of the table, and constructing a contour map.

This type of computation was carried out for the data from Polar Mesa. Only data from exploration wells were used. The factors previously listed c<mtain certain redundancies, and it is appropriate to eliminate these. Further it was considered desirable to utilize dimensionless quantities as far as

possible. The factors utilized in this analysis were as follows. 1. Total footage sandstone.

2. Total footage mudstone.

3. Per cent sandstone, yellow to yellow brown. 4. Per cent sandstone, gray

5. Per cent mudstone, blue to green.

The resulting contour map is illustrated in figure 4. This figure represents considerable "artistic effort", since the "control" for contouring leaves much to be desired. Since many arbitrary choices are involved in drawing the map, the values of the favorability coefficients are tabulated in Table VII.

The use of a procedure such as this requires either that mineralized wells are encountered fairly early in the exploration program, or that results from mineralized wells from a "similar" drilling area are available. It would be possible to incorporate a procedure such as the above into a randomly located pattern drilling progr.am, in which several phases of drilling were contemplated.

The results of the application of this procedure to the Polar Mesa project provide no evaluation of the possibilities of the technique. Due to the method of construction of the favorability map, the known mineralization is certain to lie in a favorable area. Due to the poor control over a large part of the drilling area, there is not much basis for prediction of the location of mineral-ization that was not discovered in the drilling program. It is the authorls understanding that further drilling is not contemplated in the area, so that the above results can be considered only as illustrative of a method of construction of 1_{'favorability" maps.}

We turn now to questions of associations. The ideas underlying the analysis are essentially those presented in appendix C of this report. An example of the type of question considered is: does the evidence support the assertion that mineralization is more likely to occur where the thickness of the sand lies in a ''middle range"?

In approaching such questions, the method of well location has considerable bearing. For example, if sampling is confined to locations where the sand thickness lies in the ''middle range", it is clear that the data cannot be used either to support or refute the contention. In the following, the analyses will precede on the assumption that the method of well location provides a uniform sampling of the drilling area. Since this assumption cannot be regarded as

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79 177 0 0 182 180 0 0 142 136 0 ,,

19~

189 0 -q, / \ / 195 0 63 13 0 0 0

)

159 0 1 -150- ----2

_:)153'

-~,

172 0 0 ·-..., __ 0 --- -2 173 0 174 0 166 0 167 0

Fig. 4-Contours of favorability index for Polar Mesa Drilling Project. Favorability index is based upon: sandstone thickness, mudstone thickness, per cent sandstone thickness yellow to yellow brown, per cent sandstone thickness gray, and per cent mudstone thickness blue green.

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25

Table VII. Values of favorability index for Polar Mesa wells Well No. Index Value Well No. Index Value

131 4,40

162 5,46

132

10.38

163 2,00

133 1. 44

164 3, 46

134 1. 35

165 3,15

135 o.oo

166 1,68

136 o.

42

167 1,84

137 3. 22

170 0,69

138 o.

73

171 0,76

139 o.

32

172 0,87

140 0,64

173 2, 29

141 4,85

174 2. 00

142 o.oo

175 o.

40

143 o.

00

176 1. 47

144 o.

00

177 2. 29

145 3. 82

178 o.oo

146

2.55

179 1. 02

147 o.

84

180 o.oo

148 3. 42

181 o.oo

149 o.

23

182 0,74

150 4. 89

183 1. 01

151 o.

35

184 4,82

152 o.

25

185

0.06

153 4, 12

186 o.oo

154 2. 75

187 1. 82

155 3. 35

188 1. 18

156 1. 23

195 o.

00

157 3, 66

196

6.48

158 o.

22

198 0,42

159 3. 05

199 o.oo

160 o.

62

203 1. 09

161 1. 58

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fulfilled, the results must be regarded as suggestive only; they do not have a firm foundation.

Consider the question relating to the sand thickness. Suppose that most of the discovered mineralil?:ation was found where the sand thickness lies in the "m.iddle range". One possible explanation lies in the assumption that an association exists. Another possible explanation is ·that for most of the area explored, the sand thickness lies in the middle range, so that it is to be expect-ed that the sand thickness at most of the points of mineralization will also fall in the middle range. It is clear that in this case, the data cannot be used to support the contention of an association.

What is required is a comparison of the distribution of sand thickness among the mineralized wells with the distribution of sand thickness among

the complete set of wells. In the present case, there are 61 exploration wells, and of those 10 encountered mineralization at least as intense as that designated as 11_{weak mineralization". Now suppose that a random selection of 10 of the}

61 wells were made and suppose that these selected wells were regarded as having penetrated mineralization. Examine the sand thickness at the selected wells to see to what extent these data would support the hypothesis of associa-tion. Clearly, if it is a reasonable bet that one can obtain evidence of

association in this

way,

that is, just as "strong" or "stronger" than that provided by the set of wells that actually did penetrate mineralization,

there is little justification for regarding the obtained data as supporting the hypothesis of association.

It is asking rather too much of human capacities for comprehension to ask that judgments be made merely by looking at the data. It is necessary to reduce the data to more manageable proportions. We do this by computing a few functions of the observations, or statistics, that are more readily intelligible. For example, a set of data is often rendered more intelligible by considering an average value and a measure of the dispersion of the observations about the average value. It is convenient to introduce some notation at this point in the discussion. Let the set of data be represented by the numbers x₁, x

2, ••. , x • Then a useful average value is_ that given by the arithmetic mean, whicWwe will designate by the symbol x. The computa-tional formula is:

(xl

+ • • •

x= m

+x )

m = m 1 !: x. m i=l 1

A convenient measure of the dispersion about the mean is given by the mean square, or the root mean square, of the deviations from the mean. We may designate these values by the notation s2 and s. The computational formulae for these statistics are:

(xl - :;qz

+ • • • +

(xm - :;qz m-1 m 1

=

m !: (x. - :;qz

=

- 1 i=l 1 m-

r;

X~_

mXZ7.

F=l

1

J

(27)

27 and

s

=

We need now to consider how these statistics may be useful in the matter of interpretation. Suppose, for example, that the sandstone is predominantly thin in the exploration area. Then the average sand thickness for all of the exploration wells would be relatively small. If the occurrence of mineralization were confined to the middle range of sand thickness values, then the average sand thickness for the mineralized wells would be rather larger than the overall average. But how much larger? We can get a line on this question by considering the average values that would result from the experiment of making random

selections of the wells to be regarded as mineralized. In case of the thin sand area, the average value actually obtained from the mineralized well.s would be but rll.rely exceeded by the average values resulting from the sampling experiment. Conversely, if the obtained average value is rarely exceeded in the sampling

experiment, then this fact can be placed in evidence of the existence of an associa-tion. It indicates that the obtained value is larger than would ordinarily be ex-pected on the basis of sampling fluctuations.

A simpler interpretation would obtain if the sand in the area were pre-dominantly thick. In this case the average sand thickness over all of the wells would be large, and, in the case of association, the average value of the thick-ness for the mineralized wells would be rather smaller than the overall average.

The above argument is not relevant if the sand is neither predominantly thick nor predominantly thin, and the overall average thickness itself lie~> in the middle range. Here the measure of dispersion enters. For if the mineral-ization were confined to the middle range, then the range of variation of the aand thickness among the mineralized wells should be rather smaller than the range of the variation of sand thickness among the complete set of wells. This should be reflected in a smaller value of the dispersion statistic, computed for the mineralized wells, than would ordinarily result from the sampling experi-ment. The argument carries on from there.

The sampling experiment was carried out twenty times for the sand thick-ness data from Polar Mesa. That is, values of

x

and s were computed from the sand thickness data provided by the ten mineralized wells, Then, ten wells were selected at random with the aid of a table of random numbers from the set of 61 wells. Values of

x

and s were computed from the sand thickness data provided by the selected wells. This random selection expe.dment was repeated 20 times. The results are tabulated in table VIII.

The observed value of x, 26. 8ft, was exceeded. in ouly two of the ~>ampJ.ing

experiments; we may estimate that the odds are about 9 to 1 that this observed value will fail to be exceeded in single performance of a sampling experiment. This estimate may be placed in evidence that the sand thickness is greater than average at points of mineralization, and hence that sand thickness and mineralization are associated. This evidence islnot very strong, bU'I: it cannot be iguored. Likewise, it may be noted that only one of the

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Table

vm.

Results of sampling experiments on Polar Mesa sand thickness data

Trial No. X s t v 00** 24. z 10.01 0* Z6.8 7. oz 1. 27 0.50 1' 25.7 6.83 o. 75 0.47 z Z3.9 7. 17 -0.15 o. 51 3 zz. 0 11.18 -0.68 1. 25 4 18.7 9. Z5 -Z. 06 o. 85 5 Z3. Z 9.45 -0.37 o. 89 6 z6. 3 12.04 o. 60 1. 45 7 22.6 9.08 -0.61 O. 8Z 8 23.3 10.80 -0. Z9 1. 16 9 Z6.4 9. Z8 o. 82 0.8& 10 Z4.7 11.94 0.14 1. 4Z 11 z£..9 8.4Z 1. 10 o. 71 lZ 22. z 11.1& -0.6Z 1. Z4 13 24.5 9.72 0.10 o. 94 14 zo. 6 10.~6 -1.24 1. 01 15 24.9 8.33 o. Z8 o. 69 16 Z3. 9 9.69

-o.

11 0.94 17 Z8.9 11.03 1. 47 1. Z1 18 24.4 7.50 0.09 o. 56 19 21.3 1. 75 -1. 3() o. 60

zo

23.1 7. 87 -0.49 0. 6Z

*Based on observed mineralized wells. **Based on complete set of 61 wells.

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29

sampling experiments yielded a value of the statistic s less than the observed value, 7. 02. The interpretation of this is that the values of sandstone thick-ness are more concentrated for the observed mineralized wells than would ordinarily result if mineralization were not associated with sand thickness. Thus consideration of the statistic s provides another piece of evidence that sand thickness and mineralization are associated.

It should be clear by now that the basis for interpretation of the observed values of the

x

and s statistics lies in the comparison of the obtained values with values that would ordinarily result from the performance of sampling experiments. The question arises as to whether it is always necessary to actually perform the sampling experiments in order to be able to form an idea of the sampling distributions of the desired statistics. The sampling distributions have been worked out and tabulated for a number of the more important statistics, but these tabulations are relative to certain assump-tions regarding the population from which the samples are taken. It turns out, however, that some of these distributions are not very sensitive to the assumptions that enter the mathematical derivations, and hence that

the theoretical results may be applied with a reasonable degree of confidence. To illustrate this, two additional statistics have been tabulated in

table VTII. These are designated as "t" and "v". The formula for the 1_'t11 statistic is given as

t

=r;;.N

'./~

x-X

s

In this expression,

x

denotes the arithmetic mean of the sample observations, as before; also as before s denotes the root mean square of deviations about the sample mean; X denotes the arithmetic mean of the values for the entire set of wells; n denotes the number of observations in the sample, i.e., the number of mineralized wells, and N denotes the number of observations in the population, i. e. , the total number of wells under consideration. The sampling distribution of the "t" statistic can in many yases be satisfactorily approximated by the tabulated "student t" distribution • In the cases consider-ed the tabulatconsider-ed results indicate that it is about an even bet that a sampling experiment will result in a value of "t" lying in the range -0. 703 to 0. 703.

In the experiment performed 13 of the twenty values fall in this range. Con-sidering sampling fluctuations, this is quite a satisfactory agreement. Sim-ilarly, it is about a 4 to 1 bet that the observed value will lie in the range -1.383 to 1. 383; that is, on an average basis, 16 of the 20 values can be expected to fall in this range. In the performed sampling experiments 18 of the 20 values fall in this range. Again the agreement is satisfactory. It is about a 19 to 1 bet that the observed value will fall in the range -2.262 to

2. 262; all 20 observed values fall in this range.

The 11_t11 _{statistic plays essentially the same role in the interpretation} of the results as the

x

statistic. The difference is that one can make use 1. Siiedecor, G. W. loc. cit.

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of the theoretical distribution oft in interpreting the results without resort-ing to the sampling experiment. On the basis of the sampling theory, the t value computed from the data of the mineralized wells, 1. 27, would be exceeded with probability approximately 0. 125. In the twenty trials, only a single larger value was obtained. While this result does not represent good agreement with the theory, it would not materially affect the inter-pretation of the results.

In a similar way, the 11_{v" statistic can be used to replace the sampling} experiment in the interpretation of the 11_{s" statistic. The computational} formula for the v statistic is

v=

In this expression, s2 is as previously defined and S2 is defined by the expression

sz

=

1

~

(X. -

X

)2

=

1 [ : X. 2 - N(X)2

l

N=1

i=l 1 N-l i=l 1

Here

x

1,

x

2, ••• , XN denote the values for the complete set of wells. The sampling distribution of the statistic

A

2

=

(n-l)v will in many cases closely follow the tabulated distribution of the chance variable ordin-arily designated as chi square. It should be about an even bet that the performance of a sampling experiment will yield a value of v greater than 0. 927; a 4 to 1 bet that the value will exceed 0. 598; a 9 to 1 bet that the value will exceed 0. 463; and a 19 to 1 bet that the value will exceed 0. 438. The results of the sampling experiment tabulated in table Vill show fair agreement with the above odds. It should be pointed out that the above odds for the t statistic as well as for v are dependent upon the number of miner-alized wells; that is, the sampling distribution of these statistics depends upon the size of the sample.

If the sampling experiment had not been performed, it would have been judged that the probability of obtaining a value of v smaller than the observed 0. 50 is about 0. 13. The performance of the set of 20 sampling experiments resulted in a single value less than the observed one. In either case the interpretation of the result that may be suggested is that the sandstone thick-ness values are more "grouped" than would ordinarily be expected if there were no association present, i.e., if the actually observed mineralized wells were a random selection of the total set of wells. .Although the observed v value may be placed in evidence of an association, in neither case would the "grouping" be termed 11_striking11_, _{or the evidence strong.}

The same method of analysis was applied to the other characteristics for which data were reported. The results are presented in table IX. Perhaps the outstanding result of this analysis is the large value of the 11_t11

statistic for the characteristic: per cent of total sandstone thickness that is

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31

Table IX. Individual analyses for Polar Mesa data characteristic

Characteristic X s t v

x

s

Total footage sandstone 26.8 7.02 l. 27 0.50 24.2 10.01 Footage yellow-brown to

brown sandstone 19.8 11.44 2.16 0.98 12.6 ll. 58 Per cent of total sandstone

yellow brown to brown 72.1 30.11 2. 66

o.

75 49.0 34.85 Footage gray sandstone 6. l 7. 23 -2.04

o.

59 10.4 9.43 Per cent total sandstone

gray 24.9 28.44 -2.42

o.

74 44.9 33.11

Footage red sandstone 0.8 l. 40 -0.82 0.38 1.13 2. 27 Total footage mudstone 15.3 7.02 -1. 17

o.

49 17.7 9.69 Footage blue to green

mudstone 5.7 4.16 (1.02 0.87 5.67 4.47

Per cent total mudstone

blue to greer. 35.9 16.69 0.73 0.63 32.4 21.11

Footage red to brown

mudstone 9.5 4.50 -1.92 0.35 12.0 7.66

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yellow brown to brown. Since the corresponding value of the 11_v11 _statistic, 0. 75, is not small, the indication is that the values of the characteristic c'lrresponding to mineralized wells are not particularly well grouped among the values for the complete set of wells. With respect to uniform sampling of the exploration area, these results may be considered to provide fairly strong evidence of association. It would appear that the characteristic: per cent of total sandstone thickness that is yellow brown to brown, is the most dis·crlmjnating single characteristic that was reported.

Consideration of the values of the

"v"

statistic would suggest that the characteristics: footage red sandstone and footage red to brown mudstone can be considered to be associated with mineralization. Examination of the

data,

however, reveals that the small

''v" values are the result of

single observations that are excessively large, and hence that the usual distribution considerations do not apply.

The results of this analysis may be summarized as follows. Relative to the assl,UUption of uniform sampling of the exploration area, the results of the analysis may be submitted in evidence of the existence of association of the several characteristics with the presence of mineralization. The results suggest that the characteristics may be ordered as follows on the basis of degree of association.

1. Per cent of total sandstone, yellow brown to brown Z. Per cent tot.U sandstone, gray

3. Total footage sandstone 4. Total footage mudstone

These results may be interpreted physically. Such interpretations are not appropriate here, however, and are left to others.

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33

V. ANALYSIS OF DATA FROM MESA V, LUKACHUKAI MOUNTAINS

In the fall of 1953, Grand Junction Operations Office forwarded to the Statistical Laboratory strip logs that had been constructed from. the cores obtained during the exploration of Mesa V, Lukachukai Mountains, Arizona. Various data were taken from the logs and analyzed from the point of view of the questions of association. Presented here are discussionsof the results and the methodology used in obtaining them.

The work to be discussed was performed without the benefit of advice from exploration geologists. Consequently, it is to be expected that the results in themselves will present a somewhat superficial appearance. It should be further noted that the items of analysis are the recorded obser-vations. For example, we are not able to say what is meant by the term: abundant CaC0

3 in the ore sand. It is hoped, however, that the results will be sufficiently suggestive that the questions of methodology of analysi·s will be viewed from a more appropriate perspective.

Altogether, 79 strip logs were available for study. Of the wells repre-sented, 21 were listed as containing mineralization. All of these were

included in the present study. A selection of 29 of the non-mineralized wells was made. The basis for this selection was the aim of achieving a fairly uniform coverage of the exploration area.

The items selected for study consisted of the following: 1. Thickness of the ore sand

2. Thickness of mudstone in the ore sand 3. Colors of the ore sand

4. Presence and colors of the mudstones above and below the ore sand

5. The presence or absence of carbonaceous material in the ore sand

6. The presence or absence of limonite staining in the ore sand 7. The presence or absence of inter bedding above and below the

ore sand

8. The relative abundance of CaC0

3 in the ore sand.

Data for these items were transferred from the strip logs to tabulation sheets. The only exceptions were a few cases in which it was not possible to obtain the information with respect to item 7.

Items 1 and 2, thickness of ore sand and thickness of mudstone within the ore sand were treated by the methods discussed in section IV of this report. The analysis is given in table X.

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Table X. Summary statistics for thickness of ore sand and thickness of mudstone in ore sand data

Characteristic X s t v

X

s

Ore sand thickness

38.24

6.58 -1.78 0.74

40.18

7.66

Mudstone thickness

I. 59

1.

99 -0.37 0.99

1. 72

1.99

These results indicate an association between the thickness of the ore sand unit and the presence of mineralization. li there were no such associa-tion, the set of ore sand thicknesses observed at the mineralized wells should represent essentially a random selection of

21

of the

50

values. Under such random selections, the odds are about

19

to 1 against obtaining a value of the "t" statistic as small or smaller than the value that was in :fact obtained. Consequently, it appears more reasonable to regard the results as evidence of association. One may say that the ore sand tends to be somewhat thinner where the mineralization 1s present than where it is

absent. The relatively large value of the 11_V11 _{statistic indicates, however,} that the sand thickness values for the mineralized wells are not particularly well concentrated relative to the values for the entire set of wells, so that

this association may be of relatively little importance to the exploration program.

The results of table X do not indicate the presence of an association be-tween values of mudstone thickness and the presence of mineralization. It is to be pointed out that the reproducibility of these observations is very

much open to question, so that not much weight is to be attached to the results. The remaining items,

3-8,

possess a discrete nature, and may be analyzed by the methods of contingency table analysis. This is merely an extension of a method presented in section ill of this report. For example, in case of the characteristic which has been termed the relative abundance of CaC0₃, the following table was obtained. .

Table XI. Number of wells classified according to presence of mineralization and abundance of CaC0₃

CaCO

Mineralization Abundant Some None

Present

15

1

5

Absent

10

5

14

25

6

19

21

29

50

Upon inspection, Table XI certainly appears to support the contention that one is more likely to find mineralization where CaC0₃is abundant than where