STATISTICAL DESCRIPTION OF ROCK PROPERTIES AND SAMPLING
by
NICHOLAS ANTHONY JANNEY
BS, Tufts Univesity
(1972)
MS, University of Michigan
(1974)
Submitted in partial fulfillment
of the requirements for the degree of
Master of Science
at the
Massachusetts Institute of Technology
January, 1978
Signature of Author..
Department of Civil Engineering, anuary, 1978
Certified by........ .
S ........
.
Thesis
Supervisor........
Thesis Supervisor
and......
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. . . . ....
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.
.
.
........ e..o............-Supervisor.
Thesis Co-Supervisor
^
Accepted by.........
....
A. .
...................................
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-
Chairman, Depaftmertal Committee on Graduate Students of the
Department of Civil Engineering
Archives
/9
78
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37
2
ABSTRACT
STATISTICAL DESCRIPTION OF ROCK PROPERTIES AND SAMPLING
by
NICHOLAS ANTHONY LANNEY
Submitted to the Department of Civil Engineering on January 24, 1978 in
partial fulfillment of the requirements for the degree of Master of Science.
Statistical description of rock mass properties is essential for two
reasons: 1- Analyses in rock engineering require statistical description
to take the distributive nature of properties into account and 2- Field
sampling requires statistical descriptions to develop sampling plans and
to draw inferences from the data. For both purposes, it is essential to
know the appropriate distributions of rock mass properties.
Based on the evaluations of a large number of joint data from five
geologically distinct sites and taking previous works into account, it was
determined that the best fitting distribution for joint length is lognormal
and for joint spacing exponential. Based on these conclusions a model was
developed for inferring joint parameters and estimating the intensity of
jointing (joint surface area per volume). Analytical relationships (inference equations) were developed from the model. These relationships rigorously account for sampling biases, such as truncation, censoring, and those
associated with various sampling plans (line, circle and area survey).
Herbert H. Einstein
Associate Professor of Civil Engineering
Gregory B. Baecher
Assistant Professor of Civil Engineering
3
ACKNOWLEDGEMENTS
I am indebted to Professors Herbert H. Einstein and Gregory B.
Baecher, my thesis advisors, for their advice, encouragement and insight.
I am also indebted to Mr. Leo Martin and Tim Bruno of Stone and Webster
for supplying much of the joint data.
In addition, I thank Mr. Bak Kong
Low for his assistance in processing the raw data and Mr. Bill Roberds
for reviewing the manuscript. Finally, I thank my wife, Barbara, for
her support during my stay at M.I.T.
TABLE OF CONTENTS
Page
Title Page
Abstract
Acknowledgements
Table of Contents
List of Figures
List of Tables
Chapter I -
Introduction
Chapter II - Method of Analysis
Approaches
Determination of Probability Models
Development of Sampling Inferences
9
11
11
13
21
Chapter III - Orientation
23
Chapter IV - Persistence and Joint Length
33
Literature Survey
Present Investigation
Chapter V - Joint Spacing
35
41
53
Literature Survey
Present Investigation
Practical Results
53
57
67
Chapter VI - Joint Surveys and Sampling Inferences
70
Principles of Sampling
Sampling Theory Applied to Joint Surveys
Sampling Inferences
70
76
98
Chapter VII -Conclusions and Recommendations for Future Study
115
References
117
5
TABLE OF CONTENTS
(continued)
Page
Appendix A - Geology of Sampled Sites
122
Appendix B - Probability Models
134
Appendix C - Contoured Pole Diagrams of Orientation Data
147
Appendix D - Joint Length Distributions
158
Appendix E - Joint Spacing Distributions (Simulation)
193
LIST OF FIGURES
Figure
Title
Page
II-la
PDF of Normal Model and Histogram of Observed Data
16
II-lb
PDF of Exponential Model and Histogram of Observed
Data
16
II-2a
Plot of Exponential, Lognormal and Normal Distributions on Exponential Probability Paper
17
II-2b
Plot of Exponential, Lognormal and Normal Distributions on Normal Probability Paper
18
II-2c
Plot of Exponential, Lognormal and Normal Distributions on Lognormal Probability Paper
19
III-1
Example of Contoured Pole Diagram
24
111-2
Deviation of Unit Vector (Pole) Vi from an Axis Z
26
IV-1
Illustration of Joint Persistence
34
IV-2
Illustration of Joint Intensity
34
IV-3
Examples of Truncation and Censoring
36
IV-4
Lognormal Density Sorted Into Different Sized
Intervals
38
IV-5
Histogram of Length Data from Site B
42
IV-6
Histogram of Length Data from Pine Hills
43
IV-7
Listing of Program Used to Analyze Joint Length
44
IV-8
Results from USPC Subroutine
46
IV-9
Randomly Oriented Ellipses
52
V-la
Relation Between Mean Pole and Sampling Line
54
V-lb
Computation of Average Spacing from Two Joint
Sets
54
V-2
Theoretical Discontinuity Spacing Distributions
56
V-3
Input Parameters for Simulation Program
59
LIST OF FIGURES
(continued)
Figure
Title
Page
V-4a
Equal Angle Plot of Simulation Program ResultsSite A, Top of Rock, Joint Set 1
61
V-4b
Equal Angle Plot of Simulation Program Results-
62
Site A, Bottom of Excavation, Joint Set 1
V-5
Plot of Mean Vs. Standard Deviation of Spacing
63
V-6
Illustration of "Three Hole" Problem
69
VI-la
Idealized Relationship Between Target and Sampled
Populations
72
VI-lb
More Typical Relationship Between Target and
Sampled Populations
72
VI-2
Systematic Relationship Between Outcrops and
Joints
78
VI-3
Relationship Between Joint Planes and Outcrop
Surface
80
VI-4
Examples of Joint Termination
83
VI-5
Empirically Observed Measurement Error in Strike
and Dip
85
VI-6
Error in Measured Dip Due to Misalignment of
Brunton Compass by io
80
VI-7
Alignment of Axes with Study Area
88
VI-8
Example of Upper Level Sampling
89
VI-9
Example of Line Survey
91
VI-10
Example of Area Survey
95
VI-11
Example of Circle Survey
96
VI-12
Conceptual Model of Joints as Two-Dimensional
Discs
99
VI-13
Size Bias in Circle Survey
111
LIST OF TABLES
Table No.
Title
Page
III-i
Summary of Orientation Data and Goodness-of-Fit
Testing for Fisher Distribution
30
IV-1
Example Illustrating How a Lognormal Can Pass
37
for an Exponential Distribution
IV-2
Summary of Parameter Values for Joint Length
47
IV-3
Results of Goodness-of-Fit Tests for Joint
Length
48
V-I
Summary of Results from Spacing Simulation
Program
64
V-2
Summary of Results of Actual Spacing Data
65
V-3
Results of Goodness-of-Fit Testing for
66
Actual Spacing Data
VI-1
Evaluation of Equation A.9
103
CHAPTER I
INTRODUCTION
Stability analyses of rock slopes necessarily involve high levels of
uncertainty in the critical joint parameters such as joint size and density
(number per volume) controlling the mechanical stability of the rock mass.
This uncertainty arises because the parameters describing these properties
are not unique, but distributed (random variables).
Traditionally, this
uncertainty is incorporated in the stability analysis through a factor-ofsafety.
However, factors-of-safety do not relate parameter distribution
to the probability of failure or excessive deformations.
A more rational approach is to determine the distribution of these
parameters.
This requires us to make direct measurements of the joint
properties which these parameters describe.
to make these measurements.
We cannot enter a rock mass
Therefore, we must rely on measurements made
on those joint properties observable in outcrop, i.e., orientation, length
and spacing, and use them to make inferences about joint shape, size and
density.
In order to do this we need to:
(1) Collect data on and determine the distributional forms of joint
orientation, length and spacing, by bringing to bear statistical (goodnessof-fit) tests on the data.
In addition, empirical analyses will be compared
with theoretical considerations on the processes governing joint formation
to ensure that the distributional forms obtained for orientation, length
and spacing are consistent with theoretical understanding.
(2) Develop a conceptual model of a jointed rock mass which is consistent with our findings about the distributional form of orientation,
length and spacing, and which will serve as a basis for developing sampling
inferences.
(3) Develop analytical relationships for inferring the distributions
of parameters such as joint size and density and which rigorously account
for sampling biases.
In Chapter II we discuss Classical and Bayesian inferences and how we
use them to meet our objectives.
In Chapters III, IV,and V the appropriate
distribution form for joint orientation, length and spacing, respectively,
are sought. In Chapter VI we discuss the sampling of joints and develop
analytic relationships.
In Chapter VII a summary of the thesis is presented
and topics for future study are suggested.
CHAPTER II
METHODS OF ANALYSIS
The primary objectives of this thesis are to:
(1) Determine the probability model or distributional form for
joint orientation, length and spacing, and
(2) Develop analytical relationships for inferring distributions of
parameters such as joint area from sample data.
These equations must
also rigorously account for sampling bias.
APPROACHES
In achieving these objectives one can use either of two statistical
approaches:
Classical or Bayesian.
tinguishing features.
The Classical approach has two dis-
It regards sample data as the sole quantifiable
form of relevant information, and defines probability as the long term
frequency under essentially similar conditions (Barnett, 1973).
trate the latter point, consider the following example.
To illus-
Under constant
conditions, a "large" number, M, of trials are made, where a trial may be
the flip of a coin or the toss of a die.
An outcome, E, occurs r times.
Then the relative frequency, or probability, of E is r/M.
The Classical approach does not allow probabilities on the state of
nature, rather probabilities or confidence limits are placed on an estimating
statistic or testing procedure upon which inferences are made (Baecher, 1972).
The testing procedure is established so that the chance of rejecting a true
hypothesis is kept to a minimum, usually 5.0%. Rejection of an hypothesis
provides no additional information as to what hypothesis should replace it.
Classical theory gives answers in the form of the probability of different
sets of data occurring, given some hypothesis.
The practical problem is
just the reverse. The problem is what is the chance of some hypothesis
being true, given the data.
Despite these misgivings, the Classical theories of inference are well
established and they require only moderate amounts of computation. Also,
using subjective judgement and past information, an experimenter can choose
reasonable hypotheses to test.
On the other hand,
Bayesian inference assumes that quantified a priori
information may be available, and that probability represents a degree-ofbelief in some state of nature. The state of nature is considered a random
variable in the sense that different values are possible with different
probabilities, degrees-of-belief or weights (Barnett, 1973).
A priori beliefs are modified or "updated" in view of sample data by
means of the sample likelihood function.
The sample likelihood function is
the probability of obtaining the sample data conditioned on some state of
nature. The updating is accomplished by means of Bayes Theorem, and the
updated degree-of-belief is called the posterior probability. The posterior
is in terms of the probability of an hypothesis being true given the data.
(Classical inference does not allow probabilities on an hypothesis being
The hypothesis is either true or not true.
true.
Classical inference
seeks to minimize the probability of rejecting a true hypothesis.)
To illustrate the preceding points, consider the following example:
The census bureau wants to know the mean income of the people living in
Boston.
A random sample is taken.
The Classical statistician would take
this sample and compute its statistics, i.e., mean, variance, etc.
He
would hypothesize a mean income for the entire city and test whether or
13
not the sample mean could come from a population with this hypothesized
mean.
There will be any number of hypothesized population means which
could give rise to the sample mean. However, since probabilities cannot
be placed on the truth of the hypothesis, the census bureau has no insight
into which value to use.
A Bayesian would seek prior information on the distribution of incomes
in the Boston area. (Inthis case, prior information will probably be very
informative.
However, prior information need not be informative, in which
case the prior is called diffuse.
are equally probable.)
update his beliefs.
By diffuse, we mean all states of nature
He would develop the likelihood function and then
Hopefully, the posterior distribution on income will
reinforce his previous ideas.
DETERMINATION OF PROBABILITY MODELS
In determining the probability model underlying some physical processes,
one commonly follows four steps:
(1) Collection of data
(2) Selection of plausible models
(3) Estimation of parameters
(4) Verification of the model
The determination of the model consisting of these four steps was
accomplished using a Classical approach.
Since mathematical difficulties
were encountered when evaluating likelihoods, the originally attempted
Bayesian approach was abandoned.
Data Collection
Data collection methods (sampling plans) are many.
Regardless of
which is chosen, the plan must be random, or if not, the exact nature of
the sampling biases must be known (Kendall and Stuart, 1967).
There are
several biases, both natural and artificial, present while joint data is
being collected.
They are discussed in Chapter VI.
Selection of Model
A literature survey should be conducted in order to determine what
models might fit the data. This can be supplemented by plotting the data
as histograms, and comparing them to the shape of probability distributions.
Parameter Estimation
To obtain a point estimate of the parameter o based on data, we consider an estimate 6
eo(X).
The quantity 6
eo(x) is a realized value of the
point estimator 6(X), where X is the random variable of which the data x
constitute an observation (Barnett, 1973).
from which to choose.
There are many point estimators
Common criteria for the selection include the
following (Barnett, 1973; Benjaming and Cornell, 1970):
(1) Unbiasedness.
(2) Consistency.
e is an unbiased estimator of 6 if E(e)=e.
§ is a consistent estimator of e if as the sample
sizes increase, 6+e.
(3) Efficiency.
e is efficient if it has the minimum expected
squared error among all possible unbiased estimators.
(4) Sufficiency. e is sufficient if the extent to which the data
gives information about 0 is obtained from 6 alone.
A review of the various point estimators reveals that the method of
maximum likelihood is the most popular. This popularity is based partly
on the ease with which maximum likelihood estimators (MLE) can be obtained,
15
but more importantly, on the desirable properties they possess (Hoel, 1971).
They meet the four criteria mentioned above, either for all sample sizes
or asymptotically as the sample size increases (Benjamin and Cornell, 1970;
Freeman, 1963).
(The MLE used in this thesis are presented in Appendix B.)
The sample sizes in the current work were generally greater than 100.
Model Verification
In Classical inference, model verification is commonly referred to as
goodness-of-fit testing.
titatively.
It can be performed either qualitatively or quan-
Qualitatively, it is done by comparing the form of the model
to that of the data and subjectively assessing the goodness-of-fit.
Quantitatively, it is done by determining if the hypothesized model deviates
from the data by a statistically significant amount (Benjamin and Cornell,
1970).
The latter method is a more powerful method of measuring goodness-
of-fit because it objectively defines an allowable amount of deviation.
There are two qualitative tests.
The first test consists of plotting
the data in the form of a histogram against the probability density function
(PDF) of the model (Figure II-1). By simple visual inspection those models
which obviously do not fit can be eliminated from consideration. The
second method consists of plotting the cumulative mass function (CMF)
of the data on probability paper.
Probability paper provides properly
scaled ordinates such that the cumulative distribution function of the
probability model plots as a straight line (Benjamin and Cornell, 1970).
With such paper, comparison between the model and the data is reduced to
a comparison between the CMF of the data and a straight line.
Commonly
used types of probability paper are normal, lognormal and exponential
(Figures II-2a, b and c).
A more detailed discussion of the uses of
No.
obs
inter
8225
0
Fig.
8325
8425
8525
II-la
8625
8725 Ib
PDF of normal model and histogram of
observed data.
(From Benjamin and. Cnrnell, 1970)
30
t )= 0.130 e-0.1
'
31No. of observation in interval
< - Proposed PDF
13 _-
1-
Observed hisTogram
PDF,
0
Fig.
II-lb
6
9
1
15
!8
21
24
27
30
33 sec
t
P1iF (if e(x1,oilent i:lmodel avd histogram
,t ob)scr\ cd datita.
(From Benjamin and Corn-1.l,
1970)
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3-.
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Figure II-2a
i: .1..11--~.
'--I
Plot of exponential,
lognormal and normal
distributions on exponential probability
paper
10
30o
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8
99.9 998
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20
probability paper is found in Benjamin and Cornell (1970).
The most commonly used quantitative goodness-of-fit tests are the
Kolmogorov-Smirnov and x2 goodness-of-fit tests.
Procedures for these
tests are well known (Hoel, 1971; Benjamin and Cornell, 1970), so only
the relative advantages and disadvantages will be mentioned.
In comparison
to the X2 test, the Kolmogorov-Smirnov has the advantage that it does not
lump the data and compare discrete categories, but uses data in an unaltered
form. The statistic is easier to estimate, being a function of only the
sample size, and the test is exact for all sample sizes.
On the other
hand, the Kolmogorov-Smirnov test is valid only for continuous distributions
and for models hypothesized independently of the data.
When the data are
used both to estimate the parameters and verify the model, it is not known
how to quantitatively adjust the Kolmogorov-Smirnov statistic in a manner
similar to subtracting degrees of freedom in a x2 test.
One can only say
that the statistic should be reduced in magnitude (Benjamin and Cornell,
1970).
Aside from the theoretical aspect of these tests, there is a practical problem to consider.
Using the Kolmogorov-Smirnov test for samples
greater than 100 can be time consuming, since each data point must be
plotted and then it must be determined whether or not each data point falls
within the Kolmogorov-Smirnov bounds.
Computerizing the Kolmogorov-Smirnov
procedure or using the x2 test eliminates this problem. Consequently,
for this study, when analyzing the goodness-of-fit for large samples, a
computerized approach using the Kolmogorov-Smirnov test was implemented.
For samples of less than 50, hand calculations using the x2 test were
performed.
Finally, neither test has a clear advantage over the other and
21
each is valid for testing goodness-of-fit, i.e., if the hypothesis fails
either or both of the tests it is rejected.
DEVELOPMENT OF SAMPLING INFERENCES
Bayesian Approach
The second objective was to develop equations for inferring the distribution of parameters from sample data, equations which rigorously
account for sampling bias.
An exact Bayesian analysis of sampling infer-
ence would use the sample likelihood function, and update distributions on
the population parameters a using Bayes theorem. The likelihood function
is the mathematical expression of the probability of observing specific
samples. Sampling bias is rigorously accounted for by appropriately modifyTo illustrate this point, let us recall the
ing the likelihood function.
example of mean income in the Boston area.
It is known that income is
normally distributed with a variance of 1.0. The likelihood function
would be:
L(xle)
=
1
J
e
ag
exp 1[
dxi
2
S= mean income
However, the sample was taken during the week between Christmas and New
Years.
Many people were away on vacation and not available to be sampled.
Thus, the likelihood is the probability of observing the sample, i.e.,
incomes of people in Boston during Christmas.
This sample is "biased"
since some members, especially the more affluent ones, would appear in
the sample with a different (lesser) probability than they do in the actual
population. This difference must be accounted for because the desired
result is the mean income for Boston residents, not the mean income of
people remaining in Boston during the week between Christmas and New
Years.
Thus, not accounting for this bias would render any inferences
regarding the mean income meaningless.
On the other hand, adding the
bias term into the likelihood function might give us an integral which is
very difficult to solve close form and must be evaluated numerically or
by approximate methods.
Approximate Methods
Either due to these mathematical difficulties or by preference, one
may choose to find the expected value of the likelihood function or use
an approximate second moment analysis.
(1963),
The former is treated by Freeman
Hoel (1971), Benjamin and Cornell (1970),
among others.
The approximate method allows us to estimate the moments of a dependent variable as a function of the moments of some related independent
variable.
The approximation is valid if the relationship between the de-
pendent and independent variable is sufficiently "well behaved" and the
coefficient of variation of the independent variable is not large. The
justification for these approximations lies in the observation that if the
coefficient of variation of the independent variable, Vx , is small, X is
very likely to lie close to its mean and hence a Taylor-series expansion
of the dependent variable about mx is suggested (Benjamin and Cornell,
1970).
In the present study, likelihood functions were developed, but because
of difficulties in evaluatinq them, a second moment analysis was also used.
CHAPTER III
ORIENTATION
The most widely used and generally accepted technique for analyzing
orientation data is the contoured pole diagram. This is due to several
factors.
First, pole diagrams are easy to construct. Second, joint
clusters or sets are easily identified and the "average" or modal orientation can be determined for each set (Figure III-1 and Appendix C).
Third, the relative size of each set is readily ascertained.
Despite these advantages, objections have been raised.
Distortion
is inherent in the equal area projection of the 1% counting circle used
to determine point concentrations (Mahtab et al., 1972; Scheidegger, 1977).
This distortion is attributable to the fact that in an equal area projection, a circle on the sphere projects as a circle on the plane only if
the center of the circle is a vertical pole.
all other circles are elliptical.
Equal area projections of
Mahtab et al. (1972) suggest that
densities obtained by counting points in a specified circular area are
inaccurate, and consequently, any attempt to find the mean or average
orientation of joint sets results in quantities which cannot be used
with "confidence".
We do not necessarily agree with this view since our
work indicates that average orientation obtained from a pole diagram is
within a few degrees of that obtained using the spherical mean (Table
III-1) (discussed below).
Scheidegger (1977) corrects for distortion
by using a series of curves which accurately represent equal area projection of circles.
JOINT ORIENTMTI
OATAT FRO
01RXINMUM *
*
INIMUM
-
e8CTrM CF tXCqVRTIch qT
GEOLOGtJOINTS PLOT
.320
ahE INCH 0.001
.2
CONTO.R INTERVAL -
2.000
N
N
!E
S
LOWER NHEMIHEPE PLOT
Figure III-1 Example of Contoured Pole Diagram
More importantly, the presentation of data as contours of equal
point density is not in a form amenable to a rigorous statistical analysis.
Consequently, sample statistics, such as mean and variance and their precision cannot be determined and inferences about the probability distribution underlying orientation cannot be made.
Nevertheless, the amount
of information that pole diagrams readily yield will insure their continued
use in rock mechanics and structural geology.
Two statistically more sophisticated approaches have been presented
by Kiraly (1969) and by Mardia (1972) and Watson (1966), respectively. Kiraly
(1969) developed a set of equations to calculate the "best axis" ofa joint
set and the variance about that axis (Figure III-2).
He defines the "best
axis" as that orientation which maximizes the expression
N
C
Icos
pi2
i=l
where cos 4i is the angle between the best axis X and a joint pole Vi
The variance of Vi about the best axis is
(Figure 111-2).
N
N'1
(1- Icos ýil) 2
i=l
The best axis can be determined by finding the three eigenvalues and eigenvectors of the symmetric square matrix of the sum of the direction cosines
of the V.'s and the sum of their products.
The symmetric square matrix is
SCOS22
cos
Z COS ý cOS ~a
cos a
cos y cosa
CcosS2
cos B cosy
CO
cos o cosy
coS
cos 2 y
cosy
x(BEST AX IS)
fi
Figure II'I-2
: Deviation of a unit vector V i from an axis Z,
(from Kiraly, 1969)
It has three real eigenvalues and three orthogonal eigenvectors.
vector associated with the largest eigenvalue is the best axis.
The eigenThe var-
iance is easily computed once the best axis is known.
Mardia (1972) and Watson (1966) propose an alternative method for
computing mean and variance.
The spherical normal mean is the direction
of the resultant of 1i , mi , and ni. (i=1,2,....N), where 1, m, and n are
the direction cosines of the data.
Let (T , mO , nO ) be the direction cosines of the resultant.
N
To =• ./R
i
1
N
m
i=l
N
mi/R
=
We have
n =
i=l
ni/R
i=l
where R is the length of the resultant given by
R=
SN
N
(I 1 )2 + ( mi)
=1i=l
2
N
+ (
ni)2
i=
R will approach N if the observations are clustered around a single
direction, whereas if the observations are dispersed, R will be small.
Hence, R is a measure of the concentration about a mean direction.
Con-
sequently, spherical variance, S, may be defined as
S = (N- R)/N
Barton (1976) and Savely (1971) used this technique to calculate the mean
orientation of joint orientation data.
The approaches of Kiraly (1969), Mardia (1972) and Watson (1966) permit
one to calculate the precision of the estimate. In addition, mean and variance
can be used in a second moment analysis.
However, inferences about the
distribution form cannot be made and their approaches provide no way to
separate joints into sets.
28
While these methods yield information that may be used in a preliminary analysis, a more exact and rigorous study of rock slope stability
would require knowing the distributional form of orientation. Presently,
the most promising distributional forms are those which describe the configuration of points on a sphere.
Five such distributions which have been
widely suggested (Mardia, 1972) are the uniform, Dimroth- Watson (or girdle),
Bingham, Bridges and Fisher (or spherical normal) distributions (Appendix
B).
From an analysis of the pole diagrams in Appendix C and based on the
author's experience, it is apparent that joint 0oles are not distributed
either as uniform or a Dimroth-Watson distribution.
Shanley and
Mahtab (1975) fit the Bingham distribution to fractures in porphyry
coppers, but the results are inconclusive.
Bridges (1976) developed an
elliptical normal version of the Fisher distribution and fit it to joint
orientation data.
However, his presentation of results is very sketchy,
and thus cannot be evaluated.
Efforts to *obtain more detailed in-
formation on the Bridges model have been fruitless to this time.
The Fisher distribution has been used extensively in the geologic
sciences, especially in the field of paleomagnetism (Watson, 1966). Mahtab
et al. (1972) tried fitting the Fisher distribution to orientation data
from coal and porphyry coppers.
(McMahon, 1971;
Their results were inconclusive. Others
Zabuk, 1977; Piteau and Martin, 1977) have suggested
that orientation may be normally distributed on a sphere based on qualitative evaluation rather than rigorous statistical testing.
Since the evidence suggests that the Fisher distribution may fit
joint orientation data, it was the first to be tested. Goodness-of-fit testing
was facilitated through the use of a computer program, PATCH, written by
Mahtab et al. (-1972).
This program identifies concentrations or clusters,
determines the spherical orientation mean of each cluster and compares the
data to the Fisher distribution (x2 test).
Results of the PATCH program
for the present data are summarized in Table III-1.
Data from the Blue
Hills, PineHill,Greene County and SiteAwere used. The results indicate
that the Fisher distribution provides a poor fit to any of our data. Research into other distributional forms, both spherical and bivariate, is
presently being conducted.
Pole diagrams of the data were also constructed (Appendix C).
Com-
parison of results from the PATCH program and the pole diagrams indicates
two interesting points.
First, joint sets identified by PATCH do not al-
ways correspond to those one may identify visually on a pole diagram.
It
appears that in some instances the program is too insensitive in that it
combines joint sets which should be separate.
Second, the mean joint set
orientation as computed by PATCH is within several degrees of the "average"
or modal orientation obtained from the pole diagrams (Table III-1).
This
result suggests that the distributional form for orientation may be symmetrical since distributional forms which are symmetric about an axis
have their mean and modal values equal.
Also, the contours of the pole
diagrams suggest a symmetrical distribution of joint poles (Appendix C).
Thus, we recommend using the average orientation from the pole diagram as
an approximation for the mean orientation.
In summary, we conclude the following:
(1) Pole diagrams are a valuable tool for dealing with orientation
data, and their use should be continued.
(2) The Fisher distribution does not fit orientation data, although
TABLE III-I
SUMMARY OF ORIENTATION DATA AND
GOODNESS-OF-FIT TESTING FOR FISHER DISTRIBUTION
SET
SAMPLE SIZE 1
SPHERICAL
NORMAL MEAN
x2
TEST
MODAL ORIENTATION
FROM POLE DIAGRAM
SITE A
Top of Rock
N83E, 70E
N11E, 70E
N07E, 75E
Bottom of Excavations
160
993
N83E, 75E
NO7E, 73E
Fail
Fail
N84E, 74E
NIOE, 70E
N12E, 70W
N76E,
N12E,
NO8E,
N42W,
Pass
Fail
Pass
Fail
N78E,
N1 OE,
NIOE,
N47W,
NO8E, 64E
NlOE, 78W
N43W, 6W
Pass
Pass
Pass
N04E, 65E
NlOE, 75W
0
0
88E
76E
55E
39E
45W
Fail
Fail
Fail
Pass
Pass
NO9W,
N74E,
N60W,
N38E,
Sides of
Containment
194
14
194
Sides of Other
Excavations
207
18
203
74E
64E
78W
8W
74E
64E
74W
9W
BLUE HILLS
NO6W,
N75E,
N72W,
N35E,
N67W,
1
2
3
4
5
90
70E
66E
37E
N75E, 58W
GREENE COUNTY 2
Trench A
547
N17E, 67W
Fail
N14E, 58W
N22W, 74W
N42E, 70W
TABLE III-1
SUMMARY OF ORIENTATION DATA AND
GOODNESS-OF-FIT TESTING FOR FISHER DISTRIBUTION
(CONTINUED)
SET
SAMPLE SIZE 1
SPHERICAL
NORMAL MEAN
x 2 TEST
MODAL ORIENTATION
FROM POLE DIAGRAM
GREENE COUNTY (Cont.)
Trench B
Fail
N45W, 90
N45E, 80W
N18E, 74E
88W
85W
69W
65E
Fail
Pass
Pass
Pass
N32E, 90
N87W, 84W
132
N14W, 80E
Fail
N40W, 90
N04W, 79E
N39E, 59E
110
N16E, 86W
Fail
N44W, 64W
Fail
N22E,
N05W,
N18W,
N42W,
N22E,
N74W, 34E
N18E, 45E
Fail
Fail
Trench C
N22E,
N82W,
N40W,
N54W,
Trench T
PINE HILLS
1
80W
82W
82W
62W
72E
Refers to sample size used to compute sphertcal mean and in
goodness-of-fit testing.
2 PATCH
was unable to correctly separate joints in their respective
sets for the Greene Co. data. Therefore, the computed spherical means
represent the mean orientation of 2 or more sets,
32
the distribution of orientation is probably symmetric.
(3) The modal or "average" orientation obtained from the pole
diagram can be used to estimate mean orientation.
33
CHAPTER IV
PERSISTENCE AND JOINT LENGTH
Persistence, the percentage of the area of a plane through a rock
mass which contains joints coincident with that plane (Barton, 1976)
(Figure IV-I), must be evaluated since the shear resistance on a potential
failure plane is directly related to it. The shear resistance, T, can be
expressed by
T = c.
=
i
1
1
oin
tot a l
+ Cn tan4
n
where c. = cohesion of the intact rock
Ajoint
Atotal
=
persistence ratio
Ajoint = surface area of the jointed rock section in the joint "plane"
Atotal = total surface area of the joint "plane" including open
sections and sections of intact rock
On = normal stress on the joint plane
tan 4 = friction angle (assumed to be equal for the joint and intact
rock)
Due to the usually high value of ci , the persistence in many cases will
have a predominant effect on the shearing resistance.
However, a failure surface and the joint planes it contains are
usually not confined to a single plane.
In fact, as the failure surface
propagates through the rock mass, it may go through portions of intact
and jointed rock as shown in Figure IV-2.
Clearly, persistence ratio
does not model the actual situation during failure.
A more rational
approach would be to determine joint intensity, the joint area per volume
FAILURE
SURFACE
14
m
N
Persistence ratio
=
Aj
i-totl
Atotal
Figure IV-1
Illustration of Joint -Persistence
....... FAiLURE THRU INTACT
ROCK
N
Joint Intensity =A
Volume
Figure IV-2 Illustration of Joint Intensity
of rock, of the failure zone of finite width.
JOint intensity could then
be used in a manner similar to the persistence ratio.
In order to evaluate intensity, a mathematical relationship between
the spacing and length of joints observed in outcrops and the density
(number of joints/unit volume of rock) and the area of joints in a rock
mass must be developed.
In order to construct such a relationship we need
to:
(1) Determine the density function which best describes the distribution of joint length.
(2) Infer a reasonable shape for joints based on joint length
observations.
(3) Determine the density function which best describes the distribution of joint spacing.
(4) Develop inference equations which can be used to compute the
joint intensity from joint length and spacing.
Distribution of joint length
and inferences on joint shape are the
topics of this chapter. Joint spacing is discussed in the next chapter.
Inference equations for estimating intensity are developed in Chapter VI.
LITERATURE SURVEY
Robertson (1970), based on an analysis of 9000 joint lengths collected from the deBeer's mine in South Africa, drew three conclusions.
First-
ly, bivariate plots of strike length against dip length indicated that they
were equal, from which he inferred that joints are circular discs.
Second-
ly, he concluded that joint length data is censored (i.e., the entire length
of some joints in a sample cannot be measured since one or both ends are
Censored
edIe-iso-cd a,
at both
one end
limit of tie outcrop
Figure IV-3
Examples of Censoring and Truncation
TABLE IV-1
EXAMPLE ILLUSTRATING HOW A LOGNORMAL
CAN PASS FOR AN EXPONENTIAL DISTRIBUTION
Sample Size = 100
INTERVAL (FEET)
0-3
3- 6
6- 9
9- 12
12- 15
Lognormal
Frequency (fi)
44
26
12
6
4
(1= 1.25, a=1.0)
The mean value for this set of observations is found by solving:
- ý fi xi
X =
Yfi
where xi is the mid-point of the ith class.
Solving;
x = 4.24.
Computing the exponential frequency and the statistic
(f - fe) 2 /fe
Exponential
50.7
25
12.3
6.1
3.0
.89
.25
.01
.01
.33
(f - f)2
e
Comparing
(fi- f-)
f
to X 2 (4) suggests no reason to believe that the
data set cannot be described by an exponential distribution.
Thus, by
grouping the data into an interval close in magnitude to the sample mean,
we conclude that data from a known lognormal density can pass for an exponential distribution.
0.2
0.1
C.'
Figure
IV-4 Lognormal
Joint Lennth
Densi y Sorted into Different Sized fntervals
not visible, either because the outcrop stops or is covered by overburden
(Figure IV-3)), but does not correct for it. Thirdly, he found that joint
length wasexponentially distributed. However, we disagree with this
last conclusion on two points:
goodness-of-fit testing was by qualitative
methods only and the data were collected and grouped into 5-foot intervals,
i.e., 0-5 feet, 5-10 feet, etc.
These are relatively large intervals,
approaching, or in some cases exceeding, the sample mean.
Potential peaks
in the distribution, as would occur in a negatively skewed density function (gamma or lognormal, for example) might not be seen and the distribution would appear similar in shape to the exponential (Table IV-1 and
Figure IV-4).
Steffen et al. (1975) further discussed the problem of censoring
and developed inference equations conditioned on length following an exponential distribution, and there being a single point of censoring.
discussion of censoring in Appendix B.)
(See
However, this assumption regarding
the point of censoring is questionable since the point is in reality not
constant, i.e., there are joints of many different lengths whose true
length cannot be measured (see Figure IV-3).
Call et al. (1976) briefly discuss the problem of censoring and the
need to correct for it. They recommend against using only those joints
whose both ends are visible since this would not only bias the sample
but also severely limit its size.
Another problem discussed by Call et al.
(1976) is truncation. A truncated sample is one in which data below (or
above) a predefined value are excluded from the sample,
a discussion of truncated distributions.)
(See Appendix B for
They recommend a truncation point
of 15 to 30 cm since measurement errors usually increase with decreasing
length.
Finally, they simply state that length is exponentially distri-
buted and offer one graph in its support,
Barton (1976), McMahon (1974) and Bridges (1976) concluded that
length follows a lognormal distribution, but base this on qualitative
goodness-of-fit tests only (see Chapter II).
Bridges further stated
that fractures have rectangular shapes and that both strike and dip
lengths are lognormally distributed.
Barton concluded that joints are
equidimensional and that the mean area A is
4T
where T is the average trace length.
The source of this equation is not
apparent.
Cruden (1977) was the first to recognize that for joint lengths the
point of censorship is a random variable. He recommends that joints be
classified according to whether none, or one or two ends are visible in outcrop.
Finally, he presents evidence based on the X2 test which suggests
that length follows a censored exponential distribution. However, we disagree with this last result.
Cruden uses the standard, not the censored
form of the exponential distribution to calculate theoretical frequencies
for the X2 test.
Secondly, he ignored the fact that the point of censor-
ing is not a constant.
Finally, he grouped length data into intervals
approaching the sample mean in magnitude.
Thus, the criticism applied to
Robertson's (1970) work applies here too.
Based on these past studies, we conclude that:
(1) Either the exponential or the lognormal distribution may fit
length data.
(2) Joints are roughly equidimensional.
(3) Sampling errors such as truncation and censoring occur while
gathering length data.
Further examination of the first two conclusions is presented below.
The emphasis will be placed on ascertaining which probability density
function provides the best fit to length data.
The third conclusion
will be discussed in Chapter VI, in the context of sampling errors and
the methods available for their correction.
PRESENT INVESTIGATION
Distribution of Length
The length data gathered for this study came from the five sites
described in Appendix B. 5500 lengths were measured.
Histograms, con-
structed from the data, were studied and it was determined that the gamma
as well as the exponential and lognormal distributions may fit length
data (Figures IV-5 and IV-6).
Due to the large size of the sample, a
computer program was developed to evaluate maximum likelihood estimators,
compute the cumulative density functions of the sample and the model and
compute the Kolmogorov-Smirnov bounds around the sample data with a significance level of 0.05. The USPC subroutine of the International Mathematical and Statistical Library was used for the latter two tasks. The
program used for the analyses is included as Figure IV-7 and a sample output as Figure IV-8.
Values of the maximum likelihood estimators are
summarized in Table IV-2, and the results of the goodness of fit tests are
summarized in Table IV-3 and Appendix D. Inspection of this table indicates that the lognormal distribution best models joint length.
In 10 of
0
II
a
5 = 7.55 FT.
N=1279
C)
D
0CL
LUJ
w
LL
10
15
20
LENGTH-FEET
TRUNCATf
I
TRUNCATION
Figure IV-5 ,Histogram of Length Data from Site 8
25
0
16
`12-
z
cr
LL
8>-
X=-1.47 FEET
N = 266
U..
4-,
__ _
~1
=F-
LENGTH-FEET
Figure IV-6
Histogram of Length Data from Pine Hills
I
r[ n F
I
.
.
.
.
44
DIVENSICN A(1500),D0(15CC),L(1500),W(7500)
REAL*4 LSUN,FAN,L,LS IC-A, LCEAN
COPMON THFTA,
4LPHA,BETA, LCMEAN,LSIGMA
EXTERNAL PCF1,PCF2,PDF4
WRITE(6,999)
999 FORMAT(IH ,'THREE PARAMETER CISTRIBUTICNS FOR JCINT LENGTH.
1'THE TI-RIC PARAVETER IS THE CUTCFF LENGTH.',
1'JCINT CATA FRCM THE BLLE HILLS')
READ(5,1CC) N,X
100 FCGRAT(15,F5.2)
READ(5,2CO) (L(J),J=1,h )
200 FCRVAT(16F5.1)
K=1
DO 350 J=1,N
D(K)=L(J)
K=K+1
350 CCKTINLE
ALPHA=. 8028S
BETA=1.33119
SUM=0.C
LSUM=O.C
DO 4CC J=1,N
SUM=SUM4L(J)
LSUP=LSLMVALCGIL(J))
400 CONTINLE
THETA=N/SUU
VEAK=SLIC/
LGMEAN=LSUP/h
C=C.0
6=0.0
DC 5CC J=1,N
C=C+(L(J)-MEN )**2
B=n+(ALCG(L(J))-LGMEAN)**2
500 CONTINLE
SIGMA=SCRT(C/N)
LSIGMA=SCRT( EI/N)
I=1
DO 6C00 J=1,N
A(I)=ALCG(L(J))
I=I+1
600 CONTINLE
WRITE(6,2CCO)
2C00 FCRMAT(IHC)
WRITE(6, CCO ) TFETA
WRITE(6, CC1 ) IEANSIGVA
WRITE (6,CC3 ) ALPHA,BETA
WRITE6, 1CC4 ) LGVEANLSIGMA
WRITE(6, 1CC5 ) N,X
1CO0 FCRVAT (1H , 'TEETA=',F7.5)
1001 FORMAT (1H , 'VEAN=',F6.3,1CX,' STAN.CEVIATICN=',F6.3)
1003 FORMAT(1H ,' ALPDA=',F1C.5,1CX, '8FTA=',F10.5)
1004 FCRVAT(1H ,' LSVEAN=',F7.5,1CX, 'LCG STAN. DEVIATICN=',F7.5)
1CC5 FORMAT(1H ,' SAMPLE SIZE IS',15 ,'CUTCFF LENGTH IS',F5.2)
CALL LSFC
CALL LSFC
CALL USPC
STCP
END
Figure IV-7
(PCFI,L,N,2,95, 1,1,W)
(PCF2,vD, t2,95, l,1,W)
(PCF4,A,N,2,95, 1,1,W)
Listing of Program Used to Analyze Length
45
PDF1
EXI'Oh LET IAL
SUBROUTINE PCEFI(XP)
CC'PCN THETA, ALPH4,BETA, L
REAL*4 LSUP IEAN,LLSI CA,L
P=1-(EXP(-THETA*X))
RETURN
FAN, L SIGMA
GAIVAN
END
PUF2
GAMIVMA
SUPRCUTINE PEF2(X,P)
EXTERNAL FUN2
REAL*4 LSU,PFAN,L,vLSIGPA,LC.EAN
CCPMCN
THETA, ALPHA,BET9
, LCEFAN,LSIGMA
P=SCUANK(C,X,1F-3,FIFTrFRRCRMFLN2)
RETLRN
END
FUN2
FUNCTICI
FUN2
tX)
COFMCN THETAt ALPHA,BETA,
LCGEAN,LSIGMA
REAL*4 LSU,tEA, L ,LSICPA, LGEAN
FUN2=X**(ALP-HA-1.0)*EXP(-X/EETA)/(GAMMA(ALPFA)*BETA**ALPHA)
RFTURN
ENC
PCF4
LOGLORMAL
SU3ROGUTINE PCF4(X,P)
REAL*4 LSUP,vFANv,L,LSIGA,L CMFAN
CEVMCN THFTA, ALPHA,BETA, L CVFA ,LSIGMA
AU=(X-LC~GAN)/(SCRT(2.C)*LS ICMA)
IF(AU.CE.C) F=.5*(1+ERF(AU)
AB=X+2* (LGVEAN-X)
AY=(AP-LGVEAN)/(SCRT(2.C) * L SIGVA)
P=1-(.5* (1+ERF(AY)))
RETURN
END
i'it-ure IV-7
(continued)
-ample
1,ii F,
FN(X)
-0.230258E+01
-O.189712E+01
-0.160944E+01
-0.138629E+01
-0.120397E+01
-O. 1049a2E+01
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0.0
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0.834581E+00
0.849621E+00
0.857140E+00
i44odel
Ff K)
0.376987E-02
95 PER CENT BAND
0.138011E-01
0.0
0.0
, 0.907892E-01)
* 0.109586E+00)
0.307393E-01
0.0
, 0.132143E+00)
0.534973E-01
0.0
, 0.150940E+001
0.31955BE-02. 0.169736E+00)
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0.877505E*00
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0.909544E#00
0.107144E-01, 0.177255E+00)
0.520676E-01,
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0.984959E+00
0.988718E+00
0.992478E+00
0.996237E+00
0.100000E+01
Figure IV-8
Results from U,5jC 6ubrou ine
S0116315E+01
0.915210E400
0.920449E*00
0.925298E+00
0.929790E400
0.944771E+00
0.e953503E+00
0.956035E£00
0.958400E+00
0.968154E+00
0.969757E+00
0.976453E+00
0.977568E£00
0.980552E+00
0.981863E+00
0.987572E*00
0.991761E+00
0.995298E+00
0.996050E*00
0.218608£E00)
0.233646E+00)
0.323871E+00)
0.33890'E+00)
0.387781E+00)
0.395299E*00)
0.399059E+00)
0.444171E+00)
0.470487E+00)
0.515600E+00)
0.519359E*00)
0.560712E+001
0.564472E+00)
0.628381E+00)
0.669735E+00)
0.699810E+001
0.714841E+OO)
0.737403E+00)
0.741163E+00)
0.759960E+00)
0.763719E+001
0.790035E+001
0.805072E+00)
0.812591E+00)
0.935147E+00)
0.838937E+00)
0.046426E+00)
0.853944E+00)
0.857704E+00)
0.865221E+00)
0.880260E+00)
0.895298E•00)
0.899057T+00)
0.902816E+00)
0.91t854E+00)
0.932681E+00)
0.940410E*00)
0.944170E+00)
0.951688E+00)
0.955448E+00)
0.966726E+00)
0.970485E+00)
0.974245E+00)
0.978004E+00)
0.993042E+00)
0.10000CE+01)
0.OOOO0000O01)
0.100000E+01)
0.100000E+01)
0.100000E+01)
0.OOO0000E+01)
0.100000E*01)
0.100000E+01)
0.1000030E01)
0.100000E+01)
0.100000E+01)
0.100000E+01O
0.100000E+01)
O.100000E+01)
0.100000E+01)
0.100000E+01)
0.100000E+01O
47
TABLE IV- 2
SUMMARY OF PARAMETER VALUES FOR JOINT LENGTH
GAMMA
NORMAL
LOGNORMAL
ALPHA
BETA
MEAN
a
Site A - Top of
Rock
3.23
4.24
13.74
Site A - Bottom
3.77
3.10
Site A - Sides
2.95
Site A - Sides
SITE
Ill nx
alnx
9.52
2.46
0.54
11.73
7.44
2.32
0.50
8.06
23.79
17.61
2.98
0.56
3.43
5.87
20.19
13.37
2.85
0.52
Trench A
2.25
3.31
7.48
5.87
1.77
0.67
Trench B
2.16
2.84
6.13
4.96
1.56
0.69
Trench C
2.60
2.31
6.02
4.59
1.59
0.60
Trench T
2.85
3.19
9.10
6.57
2.02
0.58
Blue Hills
1.80
1.33
2.40
2.35
0.57
0.75
Pine Hills
1.46
1.00
1.47
1.50
0.0068
0.86
Site B
2.16
3.38
7.55
6.08
1.77
0.70
Dip length, Set 1
16.06
21.10
7.96
13.88
2.70
2.89
0.44
0.53
Strike length, Set 2
28.27
25.24
21.78
17.06
3.15
3.05
0.58
0.56
Greene County
Site A
Strike length - use top and bottom of rock
NOTE:
Maximum likelihood estimator for the parameter of the exponential
distribution is 1/MEAN.
TABLE IV- 3
RESULTS OF GOODNESS-OF-FIT TESTS
FOR JOINT LENGTH
SITE
EXPONENTIAL
GAMMA
LOGNORMAL
Site A Top
fail
fail
fail
Site A Bottom
fail
fail
fail
Site A Sides
fail
fail
pass*
Site A Sides
fail
fail
pass*
Greene Co.
Trench A
fail
fail
pass
Greene Co.
Trench B
fail
fail
pass
Greene Co.
Trench C
fail
fail
pass
pass
pass
Greene Co.
Trench T
Site B
fail
fail
pass
Blue Hills
fail
fail
pass
Pine Hills
fail
fail
pass
NOTE:
The Kolmogorov-Smirnov test with a significance level of 0.05 was
used to verify the models. Where noted by* a significance level
of 0.01 was used.
the 12 cases the lognormal distribution passed the goodness-of-fit test
and in the two cases where it failed, it still provided the best fit of
the models tested (see Figures D-1 and D-2).1
Additional studies were conducted to verify Bridges' (1976) contention that both strike length and dip length are lognormal.
At Site A
there are two major joint sets; one dips steeply to the southeast,
Set i, and another is nearly horizontal, Set 2 (Figures C-1 to C-4).
On a horizontal surface such as found at the top of rock and the bottom of
the excavations at Site A, only the strike lengths of Set 1 could be recorded.
On the vertical excavations, one can measure the strike lengths
of Set 2 and the dip lengths of Set 1. It was found that both the strike
and dip lengths are lognormally distributed (Figures D-12 and D-13).
At Site B, the joints are also steeply dipping and the surface on
which length measurements were made was horizontal.
Thus lengths repre-
sent strike lengths, and they too follow a lognormal distribution.
Simi-
lar studies were not possible at the other sites because we were dealing
with natural rather than artificially created outcrops.
Finally, we found that length is lognormal regardless whether the
data came from one or many sets.
However, Atchinson and Brown (1957)
have shown that the superposition of independent lognormal distributions
is not itself lognormal.
But, if the means and variances of the super-
With a 95% confidence level (0.05 significance level) one would expect 5
samples in 100 to fail the test even if all samples were lognormal. By
failure we mean rejecting the hypothesis that the sample comes from a
lognormal distribution. In our case, even if all 12 samples are lognormal, there is still a 10% chance of exactly 2 of the 12 not passing and
only a 54% chance of all 12 passing the goodness-of-fit test.
imposed distributions are close in value, then it is quite possible that
the resulting distribution can be approximated by a lognormal density.
For example, the sides of the excavations at Site A contain two major
joint sets.
The mean and variance of one set are 2.8 and 0.5 respectively,
and of the other, 3.1 and 0.57.
Yet alone or taken together, the length
data from these sets pass the Kolmogorov-Smirnov test for a lognormal
distribution at the 5% level (Table IV-3).
We feel that the lognormal distribution is compatible with the
physical process of joint formation.
A random variable (inthis case
joint length) will be lognormally distributed if the combined effect of a
large number of mutually independent causes act in an ordered sequence
(i.e., multiplicatively) during its formation (Atchinson and Brown, 1957).
For joints, such causes may be lithology, initial state of stress and
changes in the state of stress, water pressure, etc.
The lognormal dis-
tribution provides the best description for related physical processes,
e.g.,the distribution ofsizes due to the crushing and breaking of particles
by mechanical means.
Finally, a number of other geologic attributes such
as the distribution of bedding thicknesses and the value and size of ore
bodies are lognormally distributed (Krumbein and Graybill, 1967).
Joint Shape
Inferences about joint shape can be made from a study of lengths in
both the strike and dip direction.
For example, if the average strike
length of a joint set is about equal to its average dip length, the joints
are either equidimensional or non-equidimensional (for example, elliptical or
rectangular) with their long axis randomly oriented (see Figure IV-9).
On the
other hand, if the average strike length differs greatly from the average dip
length, the joints are non-equidimensional.
For example, they may be ellipse
with their major axis parallel to the strike of the joint.
Strike lengths and dip lengths were obtained from the southeasterly
dipping joint set at Site A. Along strike the average length is 12 feet
and along dip it is 18 feet.
equidimensional.
This indicates that these joints are not
This conclusion lends support to Bridges' (1976) hy-
pothesis that joints are rectangular, but is in conflict with that of
Robertson (1970).
Obviously, more work is needed before any definitive
statement on this subject can be made,
In summary, the conclusions drawn (based on our data) are the following:
(1) Both strike and dip length are lognormally distributed.
(2) Joints may be non-equidimensional.
Picure PV-9
Randomly Oriented Ellipses
CHAPTER V
JOINT SPACING
As outlined in the last chapter, the third task needing completion
prior to evaluating the joint intensity is the determination of the distributional form of joint spacing.
Spacing is usually the distance bet-
ween adjacent joints of one set measured along a line parallel to its mean
pole (Call, 1971; Terzaghi, 1970; Bridges, 1976; Barton, 1976).
Joint
spacing measured in any other direction can be easily corrected to the
spacing along the mean pole by using the method of Terzaghi (1970) (Figure
V-1A).
Conversely, knowing the mean pole and the average spacing along
it,the spacing in any arbitrary direction can be determined.
the spacing from a number of sets can be combined.
In addition.
For example, given
two joint sets, one striking due north with a vertical dip and a spacing
of 1.0 feet (set 1), and another set striking due east with a vertical
dip and a spacing of 2.0 feet (set 2), then the average spacing along a
horizontal line trending at N45E is 0.95 feet (Figure V-lB).
A second, less restrictive definition of spacing is the distance
between adjacent joints, regardless of which set they belong to (Figure V-2a,
b), measured along an arbitrary line.
Priest and Hudson (1976) in their
study and the author in the present study adopted this defintion.
LITERATURE SURVEY
The most detailed statistical analysis on spacing to date is presented
by Priest and Hudson (1976).
They maintain that the intersections of joints
with a sampling line can be evenly spaced, clustered, random or some combination of these.
PROBE LINE
~~~~~-l
ILE
Si= St/cos Gi
1/8
1i
-
(Intensity)
•1 +
liav e
2
+...n
FIGURE V-lA
For Set I
S t = 1.0
For -et 2
t
0
=
dl= cos 9/t
FIGURE V-1B
= 0.70 /1. 0 = ýO.U
2= 0.
0 /2.0
0.70
ave
=0.}5
+ 0.35
= 1.0 /foot.
ave
cave
=
0. 5 ft
1/105
If joints are fairly evenly spaced, as in an evenly bedded sandstone
or columnar basalt, spacing will be normally distributed with the standard
deviation reflecting the uniformity of bedding or jointing (see Figure
V-2c).
Clustered spacing results when a high frequency of low spacing
values occurs within clusters and a low frequency of high spacing values
occurs between clusters.
in Figure V-2d.
ing effects.
The resulting frequency distribution is shown
Clustering can develop as a result of stress or weather-
In addition, cyclic variation in lithology such as alter-
nating layers of sandstone and highly fractured siltstone could produce
this distribution.
The intersection points between the joints and the sampling line
are random if the presence of one intersection point does not affect the
chance of another occurring in its neighborhood.
From statistical theory
(Appendix B), if each small segment of a sampling line has an equal but
small chance of containing a joint intersection point, the points form
a Poisson process and the spacings follow an exponential distribution.
In a geologically complex rock mass with a varied mechanical history,
it is likely that a combination of evenly spaced, clustered and random distributions, as shown in Figure V-2f, will occur. This results in a distribution similar to the exponential distribution (Figure V-2f).
If,
however, the mean spacing of the random distribution is large compared to
the evenly spaced distribution, the latter will dominate, and spacing will
appear to be normally distributed.
In all other combinations, the clusters
are largely unaffected while the even spacings are broken up by the random
joint pattern, thus exponential distribution of spacing results.
Dst-nce fromA to the ith
disconrlhnty Ad,
d, d,\
Id.
d
d.
Spoong values .x)gqven
os x d, -d.. for i: -I-n
a C,sconhnuy intersechon points, Olnq 0
strToghtMe
therOckmaos
(A) tirough
b S•r,
ine(measurn
rock face
SB
A
Drscontinuiy socng values,
,,
8
spacing values,x
Discontinuity
x
d
C Fairly eveny spaced dstrlbution
Clustered distribuhon
isteed
C N
C
CY
tope exposed
•n
dom I
Clustered aonJ
rardom
d-stributions mutually
reinforce at low spoc7gs I
Evernlyspacedand
random distributions
mutually interfere at
c
high spocings
C scntinuity
spacing
e. Random
values, x
distribution
Discontinuit spacing volues
f
x
Combination of distrib6uions
FIGURE V- 2 Theoretical discontinuity spacing distributions.
(from Priest and Hudson,1976)
Using this as background, Priest and Hudson analyzed spacing data
from three English tunnels.
They concluded that spacing was exponentially
distributed based on:
(1) data plotted close to a straight line on semi-logarithmic
(exponential probability) paper
(2) means and standard deviations of spacing data were approximately equal, a theoretical characteristic of the exponential
distribution (Rodgers, 1974)
Work of other investigators is less comprehensive. Call et al. (1976)
conclude,as did Priest and Hudson, that spacing is exponential, while Barton
(1976), Bridges (1976) and Steffen et al. (1975) maintain that spacing is
lognormally distributed.
None of these investigations used quantitative
goodness-of-fit testing.
As discussed in Chapter II, this is a less than
satisfactory means of model verification.
Consequently, our efforts were
directed toward ascertaining which density function provided the best fit
to spacing data as measured by quantitative goodness-of-fit testing.
PRESENT INVESTIGATION
In the present study, data were gathered from Site A, Blue Hills and
Pine Hill.
At Site A, data were obtained by simulating a line through a
jointed rock mass (described below) or by measuring spacings from maps of
the various excavations.
At the Blue Hills and Pine Hill, spacing was
measured in the field.
A computer program was used to simulate arbitrarily directed lines
through a jointed rock mass and calculate spacings between joints which intersect the line. Each joint is located in space by its pole and a point on
the plane.
The distance of the plane from the origin along a line p is
found by solving the equation:
Distance
w • r
w p
where w is the pole to the plane, r is the radius vector from the origin to
the point on the plane, and p is the sampling line (Figure V-3).
Due to
the limitations of the simulation program the joints are assumed to have
infinite extent. Spacing is calculated by subtracting consecutive intersection distances along the sampling line.
Next, exponential and lognormal
distributions are fitted to the data using the USPC subroutine (as described
in the last chapter).
Goodness-of-fit is measured by the Kolmogorov-Smirnov
test at a significance level of 0.05.
The simulation program was used to analyze joint data collected from
maps of the top of rock and the various excavations at Site A. Spacings
were computed for the following situations:
(1) Using data from the top of rock, spacing was computed between the
joints belonging to Set 1 (average strike of NIOE and average dip of 75 SE,
Figure C-1) along 7 lines.
The results are summarized in Figures V-4a, V-5
and Table V-l.
(2) Using data from the bottom of the excavations, spacing was computed
between the joints belonging to Set 1 along 7 lines.
The results are sum-
marized in Figures V-4b, V-5 and Table V-l.
(3) Using data from the sides of the excavations, spacing was computed
between the joints of Set 1 along 6 lines, the joints of Set 2 (horizontal
set, Figures C-3 and C-4) along 5 lines and between the joints of all sets
along 5 lines,
The results are summarized in Figure V-5 and Table V-l.
59
P
Fig•ure
V-j
Input Parameters for
5rimul1tion Program
Based on these results we conclude that;
(1) Spacings from Set 1 are exponentially distributed.
(2) Spacings from Set 2 are lognormally distributed (This set might
be sheet jointing,)
(3) Spacing from all sets follow neither,
This result is not sur-
prising since the superpositioning of a lognormal and an exponential distribution should be neither exponential nor lognormal.
(4) For a given joint set, the mean spacing along a line is a function of the cosine of the angle beteen the mean pole and the sampling line
(Figures V-4a and V-4b).
Using the same joint data, actual spacings were measured from the maps
and the data tested against the exponential and lognormal distributions
using the X2 test.
Spacing from the top of rock and the bottom of the
excavation is exponentially distributed (Tables V-2 and V-3).
Neither
distribution consistently fits the data from the sides of the excavation
(Tables V-2 and V-3).
Mostly notable is the fact that spacings measured
along vertical lines are not clearly lognormal.
These spacings represent
for the most part spacing the horizontal joints of Set 2. This is surprising because spacing between Set 2 joints from the simulation program
was consistently lognormal (Table V-1).
Finally, the exponential and lognormal distributions were fit to
spacing data from Blue Hills and Pine Hill.
The results suggest that
spacing is exponential (Tables V-2 and V-3).
Considering the results of Priest and Hudson (1976) and the present
study, we conclude:
(1) Spacing appears to be exponential regardless of the direction of
N
5C
MCAhIU
0^1
TVLL
LOWER HEMISPHERE
Figure V-4A Equal Angle Plot of Simulation Program
Results - Site A, Top of Rock, Joint Set 1
N
D
IALI
I~L~I~
DflI
C Dq•!
lUll'AM
TVLL;
LOWER HEMISPHERE
iiiure
V-4B
Equal Angle Plot of ~imulation Program
Results- ite A, Bottom of Excavation,
Joint Set 1
8.0
S6.0
r-I
-P
4.0
u 2.0
2.0
4.0
6.0
8.0
Mean
A
Site A, Top of Rock, Joint Set 1, Simulation
y
Site A, Bottom of Excavation, Joint Set 1, Simulation
*
Site A, Sides of Excavation, Joint Set 2, Simulation
A
Blue Hills
o
Site A, Sides of Excavations, All Sets, Actull Data
Site A, Bottom of Excavations, All Sets, Actual Data
V7
Note: For data obtained by simulation, the points plotted
in the above graph must be reduced an order of
magnitude in order to correspond to those in Table V-1.
Figure V-5 Plot of Mean vs. Standard Deviation
TABLE V-1
SUMMARY OF RESULTS FROM SPACING SIMULATION PROGRAM
LINE DIRECTION
N
MEAN
EXPONENTIAL
LOGNORMAL
A) Site A, Top of Rock - Joint Set 1 Only
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Fail
Pass
Pass
Fail
Fail
Fail
Fail
Site B, Bottom of Excavation - Joint Set 1 Only
0.360
321
0.379
100
N45E
Pass
Pass
250
N45E
259
0.511
0.493
100
0.285
0.304
Pass
N90E
641
0.335
Pass
250
630
0.317
N90E
200
550
0.352
0.462
Pass
N80W
Pass
100
358
0.344
0.341
S45E
0.421
Pass
250
S45E
335
0.401
Pass
Pass
Fail
Fail
Fail
Fail
Fail
10
250
100
250
200
100
250
B)
N45E
N45E
N90E
N90E
N80W
S45E
S45E
139
113
400
367
363
193
160
0.871
1.166
0.456
0.535
0.531
0.634
0.831
0.961
1.337
0.509
0.530
0.563
0.699
0.943
Site A, Sides of Containment - Joint Set 1
100 @ N45E
311
1.502
2.258
250 @ N45E
249
1.847
2.842
100 @ N90E
250 @ N90E
290
290
0.682
0.959
0.720
1.072
100 @ S45E
278
1.372
3.290
273
1.689
2.885
250 @ S45E
Site A, Sides of Containment - Joint Set 2
4.274
1.213
197
700 @ NOOE
700
700
450
450
E)
@ N60E
@ S60E
@ S60W
@ N60W
Site A, Sides
450 @ NOOE
450 @ N60E
450 @ S60E
450 @ S60W
450 @ N60W
197
197
195
191
0.912
0.973
1.286
1.313
2.507
4.045
4.651
4.064
of Containment - All Joints
353
0.405
0.473
318
0.519
0.650
313
0,515
0.775
377
0.734
1.777
400
0.709
1.287
Fail
Pass
Pass
Pass
Fail
Fail
Pass
Pass
Pass
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Pass
Pass
Pass
Pass
Pass
Pass
Pass
Fail
Fail
Fail
Fail
Fail
Pass
Pass
Pass
N is the number of intersections of joints with the sampling line.
Mean is the mean spacinq in feet.
S.D. is the standard deviation of spacing in feet.
TABLE V-2
SUMMARY OF RESULTS OF ACTUAL SPACING DATA
LINE DIRECTION
N
L
MEAN
S.D.
11.9
6.8
16.8
10.0
6.0
5.5
0.398
0.342
0.579
0.452
0.352
0.294
0.65
0.30
0.40
0.341
0.32
0.200
24.0
5.0
0.78
0.39
0.94
0.26
225
332
284
204
216
170
318
217
7.46
9.76
8.61
8.87
8.31
18.89
14.45
14.47
10.46
9.62
8.91
9.41
9.65
12.00
19.00
14.60
150
167
155
164
150
3.97
5.76
3.78
7.13
5.17
4.28
6.01
3.34
8.91
4.18
59
46
304
218
89
2.15
2.93
2.95
2.25
1.78
1.43
2.17
2.38
2.14
1.43
A) Blue Hills
00
120
20
260
00
00
@ NOOE
@ N78W
@ N65W
@ NO5W
@ N55W
@ N60W
B) Pine Hill
00 @ NOOE
50 @ S30W
C) Site A - Top of Rock
00
00
00
00
00
00
00
00
D)
@
@
@
@
@
@
@
@
N90E
N90E
N90E
N90E
N45E
N45E
S45E
S45E
Site A - Bottom of Excavation
00
00
00
00
00
@ N90E
@ N90E
@ S45E
@ S45E
@ N45E
E) Site A - Sides of Excavation
00 @ NOOE
26
16
450 @ SOOE
Vertical
Vertical
Vertical
103
97
50
N is the number of intersections with the sample line
L is the length of the sample line in feet
Mean is the mean spacing in feet
S.D. is the standard deviation of spacing in feet
TABLE V-3
RESULT OF GOODNESS-OF-FIT TESTING FOR ACTUAL SPACING DATA
EXPONENTIAL
(f - f )2
LINE DIRECTION
fee
LOGNORMAL
(f
x 2 (df)
- f )2
fe
x2 (df)
A) Blue Hills
NOOE
N78W
N65W
N05W
N55W
N60W
00
120
20
260
00
00
6.17
6.38
14.23
2.56
3.45
3.65
11.07(5)
9.49(4)
12.59(6)
14.07(7)
11.07(5)
7.78(3)
15.46
2.56
15.51(8)
7.8 (3)
23.09
1.49
14.07(7)
5.99(2)
23.20
16.05
16.9
19.2
15.47
2.33
25(15)
25(15)
25(15)
25(15)
25(15)
25(15)
18.2
29.08
28.9
32.7
23.68
9.00
23.68(14)
23.68(14)
23.68(14)
23.68(14)
23.68(14)
23.68(14)
14.07(7)
11.07(5)
14.07(6)
22.36(13)
22.36(13)
9.62
4.72
5.80
21.22
10.34
12.05(6)
9.49(4)
11.07(5)
21.03(12)
21.03(12)
3.8
8.68
9.28
8.15
6.28
6.724
9.49(4)
7.81(3)
11.07(5)
12.5 (6)
9.49(4)
5.99(2)
B) Pine Hill
00 @ NOOE
50 @ S30W
C) Site A - Top of Rock
N90E
N90E
N90E
N90E
S45E
S45E
D)
Site A - Sides of
Vertical
00 @ NOOE
450 @ SOOE
Vertical
Vertical
NOTE:
Excavation
22.13
5.41
5.48
19.64
9.78
f = expected frequency
f
=
observed frequency
df = degrees of freedom
the sampling line. The fact that the spacing from joint set 2 at Site A
is lognormally distributed is not sufficient to reject the conclusion that
spacing is generally exponential.
We should expect observations from time
to time contrary to this general property.
(2) From the exponential distribution of spacing, we conclude that
the intersections of joints with a sampling line are a Poisson process,
and infer that the center point of joints are randomly and independently
distributed in space, forming a Poisson field.
(3) Spacing along any line can be estimated if the mean spacing along
the mean pole is known.
PRACTICAL RESULTS
Practical results of our conclusions are:
(1) Given the distribution of spacing, the RQD of an unweathered,
tightly jointed rock mass can readily be estimated.
(2) Given the orientation and average spacing of a single joint set
along three non-parallel boreholes, the orientation of that set can be
estimated.
This is the "3-hole" problem.
The analysis is approximate
because the joints are assumed to be parallel lines.
RQD Estimation
The first result was arrived at by Priest and Hudson (1976) who showed
L
that for an exponential distribution RQD= J 100 xe- x dx, For a long
0.1 m
sampling line, terms containing &L
e
can be ignored and thus RQD
100 e- 0.1
(0.l'+
), where A is the mean joint spacing in meters.
Orientation Estimation
The second result follows from the fact that the spacing along a
68
borehole is a function of the angular relationship of the joints with respect to the orientation of the borehole.
Defining oip as the angle between
the mean Dole of the joint set under consideration and the borehole B., S.
1
1
as the spacing along Bi and Strue as the spacing along a line parallel to
the mean pole, then Strue= Si x cos
ip. Unfortunately, the only value
known from the borehole data is Si. However, if we have three non-parallel
boreholes, we can write the following set of equations which will yield
the desired answer.
Sj cos/ (p -ebj)2+(
p-Ybj)2= Sj+ 1 cos/
Sj+ 1cos/ (ep-eb (j+l)2
(ep-Ob(j+l ))2+ (Yp-Yb(j+ l ))2
b(j+l) )2= Sj+2co s p (6 epb
(p-b(j+2))2
(j +2 ))2-
where the Obj's and the Ybj's are the trend and plunge of the boreholes, the
Sj's are the spacings along the boreholes and ep and yp are the trend and
plunge of the mean pole.
In order to use this method, the spacings must be measured between
joints of a single set.
sets in a borehole.
It is often difficult to distinguish between joint
If previous experience indicates that only one joint set
exists or that there are two sets with very different orientations, this
method may be used.
However, if a complex jointing pattern is indicated,
it will be very difficult to distinguish between joint sets.
this method cannot be used.
In this case,
In conclusion, we recommend that this method
of determining orientation be used cautiously because;
(1) It is approximate because it assumes parallel joint planes.
(2) It requires being able to distinguish between joints of individual
sets.
\0B1
B3
B3
82
yY-,
x
PLAN VIEW OF BORINGS
B1
B2
1/:
S //j
ISt
LMEAN POLE
IL$S2
IL-
CROSS-SECTION X-X
B2
B3
CROSS-SECTION Y-Y
Figure V-6
Illustration of "Three Hole" Problem
CHAPTER VI
JOINT SURVEYS AND SAMPLING INFERENCES
This sixth and final chapter is devoted to the topics of how to
collect joint data and what to do with it. Ideally, we seek to obtain
information on jointing within reasonable limits of time and money and
then to use this information to obtain an accurate, quantitative description of a jointed rock mass.
In Part A, the general requirements which sampling plans must meet
are presented. In addition, the common types of errors which occur during
sampling are discussed.
In Part B, we apply the general requirements of
sampling to the sampling of joint populations.
Errors which occur during
joint sampling are discussed and several specific joint sampling plans
are presented in detail.
In Part C, a conceptual model of jointing is
developed based on the conclusions regarding joint length, shape and
spacing.
This model is used to develop inference equations on joint
size and density. Finally, some of the sampling biases presented in
Part B are rigorously accounted for.
PART A - PRINCIPLES OF SAMPLING
Statistical or Probability Sampling
Measuring every member or element of a large population is usually
infeasible, and sometimes impossible. Consequently, we desire to secure
a sample from this population from which we can infer population characteristics as closely as possible.
This is accomplished by requiring
that the procedures used to gather the sample be such that the sample
will yield estimates of the large population with determinable errors
of estimates (Freeman, 1963).
These procedures, known as "statistical"
or "probability" sampling have the following characteristics
(Cochran
et al., 1954):
(1) Each element of the population has a non-zero chance of entering
the sample.
(2) For each pair of elements in the sampled population, the relative chance of each entering the sample is known.
(3) In analyzing the sample data each element is given weight
inversely proportional to its chance of being sampled such that
(relative weight) x (relative chance) -1 = constant
(4) For any two possible samples, the sum of the reciprocals of
the relative weights of all individuals in the sample are the same.
In sampling, concern is often expressed about the representativeness
of the sample.
Note that it is the sampling plan and not the sample that
must be representative. This is accomplished not by giving each element
the same chance of being sampled, but by compensating for the differences
in the probabilities of being sampled through weighing.
Those members
more likely to enter the sample are given less weight; those less likely
are given more weight (Baecher, 1972).
The net effect is to give each
element an equal chance to affect the weighted sample mean.
Finally, the large population which is the object of our interest and
about which inferences are to be made from sample observations is called
the "target" population.
The "sampled" population is that part of the
target population from which actual samples are drawn (Figure IV-I).
Each
element in the sampled population has some chance of entering the actual
A. Ideal Situation
Qtatistical
Judgement
B. More typical situation
Figure VI-1 Relation Between dample, Sampled
Population and Target Population
sample. When all elements of the target population are directly available
for sampling, the target and sampled populations are identical (Cochran,
Mosteller and Tukey, 1954).
In many rock mechanics problems, the sampled population is more
restricted than the target population.
For example, the target population
may be the joints 500 feet below the surface and the sampled population
may be the joints exposed in outcrop.
In such cases, inferences drawn
from the sample apply only to the sampled population. The extent to
which these inferences can extend to the target population is a geologic
question.
Sources of Sampling Errors
Discrepancies between the sample characteristics and the sampled
population characteristics can be attributed to the following sources:
(1) Sampling error
a. Bias error (Yates, 1960) or non-probability sampling
(Cochran, 1963)
b. Random error (Yates, 1960)
(2) Measurement errors (Baecher, 1972)
a. systematic
b. random
(3) Estimation errors (Baecher, 1972)
Sampling Errors
Bias is the faulty selection of members of a sample such that the
members are not selected in proportion to their occurrence in the sampled
population.
It forms a constant component of error which does not de-
crease, in a large population, as the sample size increases (Yates, 1960).
There are many ways by which faulty selection can arise.
The main causes
are (Yates, 1960):
(1) Deliberate selection of a "representative" sample.
For example,
one may select the tallest and shortest individuals in the Boston area
and then claim that the average of their heights is the average height
for people in the Boston area.
(2) Conscious or unconscious bias in the selection of a "random"
sample.
If a proper random process is not strictly adhered to, the inves-
tigator, although claiming that his sample is random, may allow his desire
to obtain a certain result to influence his selection.
This bias may be
particularly serious since its existence may not be immediately apparent.
(3) Substitution.
Investigators often substitute another convenient
member of the population when difficulties are encountered in obtaining information. Thus, in a house-to-house survey the next house may be taken
when there is no reply at the first.
This may lead to a preponderance
of houses that are occupied all day, e.g.,
houses with families.
(4) Failure to cover the whole of the chosen sample. If a return
visit is not made to houses from which no reply was received, there will
still be a bias though no substitution is attempted.
Returns of question-
naires sent through the mail are likely to be received from individuals
who are specially interested in the objects of the survey, or possess other
characteristics which make them unrepresentative of the whole population.
If bias exists, no fully objective conclusions can be drawn.
Thus,
it is essential for any sampling procedure that all important sources of
bias be eliminated or accounted for.
The simplest, and only universally
certain way of removing bias is for the sample to be drawn completely at
random. Certain biases, such as orientation and size bias (which will be
discussed in Part B), cannot be removed by random sampling and must be
accounted for by the weighing process described above.
Random sampling errors are due to chance error differences between
the members of the population included in the sample and those not included.
Since random sampling errors are approximately inversely propor-
tional to the square root of the sample size, they can be reduced by increasing the sample size.
They also depend on the variability of the
sampled population.
Measurement Errors
Measurement errors are caused by inaccuracies in either the instrument or measurement of the reading of the instrument (Baecher, 1972), and
are of two types, random and systematic.
Random errors are unpredictable in both magnitude and algebraic sign.
They are accidental and cannot be avoided.
The treatment of random errors
usually assumes that the magnitude of the error is normally distributed
with zero mean. This allows us to place confidence limits on the measurements themselves.
Finally, random errors can be reduced by increasing
the sample size.
Systematic errors are errors with non-zero means.
will, therefore, be consistently high or low.
Measured values
Systematic errors, like
random errors, are inevitable; however, unlike random errors, they cannot
be reduced by large samples.
The only strategy that can be used against
them is to hold them to a "reasonable level".
We never know whether the sys-
tematic errors have been sufficiently reduced. Examples of systematic errors
are rounding off in the same direction, a defective instrument which always
reads 5 units too low, etc.
Estimation Errors
Estimation errors are caused by differences in the sample statistics
from sample to sample caused by the random nature of samples (Baecher,
1972).
For example, since samples of concrete cylinders will not yield
the same average strength, they will lead to different estimates of the
intact strength of the concrete pour from which they came.
Since representativeness is not inherent in the individual sample,
we cannot evaluate the uncertainty in sample results without evaluating
the similarity (or lack of similarity) between the obtained sample and
samples which a sampling plan might have provided (Cochran et al., 1954).
The degree of similarity between individual samples is called "stability".
A simple empirical procedure for evaluating stability is to use a sampling
plan involving interpenetrating replicate samples. A large sample of
N members is randomly divided into M smaller samples called replicates.
We can evaluate each of the replicates as if they were chosen individually
at random from the sampled population, and thus calculate the sample
statistics for each replicate. Then we can empirically assess the
similarity of the statistics.
This, in turn, can be used as a yardstick
for the stability of the lumped sample (Cochran et al., 1954).
PART B - SAMPLING THEORY APPLIED TO JOINT SURVEYS
The application of statistics to joint data must be accomplished
judiciously.
Frequently, the sampled and target populations are not coin-
cident. The extent to which inferences regarding the sampled population
apply to the target is a geologic question.
It is usually related to the
geologic conditions in the region of the physical separation between the
sampled and target populations.
For example, joint information from near-
by outcrops and borings in a geologically homogeneous area can be used to
design a shallow rock cut (FigureVI-1A). At the other extreme, if the
target population is jointing at the elevation of a proposed powerhouse
1000 feet below the surface, then the similarity between the two populations is a geologic question (Figure VI-lB).
Another difficulty is to determine the correspondence between the
sample and the sampled population.
Outcrops, which are the source of the
sampled population, are actually presampled clusters of joints.
(By pre-
sampling, we mean that only certain parts of the rock surface are exposed
and thus available to be sampled. The amount of exposure is related to
the rock mass and its geologic history.)
The bias in this presampling
is, of course, unknown. At best, the outcrops are randomly located and
independent of jointing.
At worst, outcrops may be functionally related
to jointing. As an illustration of the latter, consider the example in
Figure VI-2, an interbedded sequence of sandstones and shale with different
jointing patterns in each. The worst case would occur if jointing information from the sandstone was used to make inferences about the shale.
Errors in All Joint Surveys
Sampling Errors
Bias errors in joint surveys can arise from a number of sources.
For
example, long or open joints are measured because they "strike" the eye,
outcrops which are difficult to reach are excluded from the sample, short
joints are intentionally not sampled.
This latter bias is common (Call
et al., 1976; Savely, 1972) and is employed when measuring joints over a
78
I
I
VI-2
(joii
o intJi
if
rnt in sands or eM
sh 1A
(frtom _baecher, 197L)
large area. The remaining observations make up a truncated sample.
The
mathematics for truncated distributions have been developed for the common
distributions (Appendix B), and thus this bias, if recognized, can be
accounted for.
To be sampled a joint must be a member of the sampled population.
The joint must intersect an outcrop or excavation, otherwise it cannot
be sampled (Baecher, 1972).
Therefore, the probability of a joint appear-
ing in a sampled population is proportional to its probability of intersecting an outcrop or excavation, which is in turn related to:
(1) the angle between the joint and the outcrop, boring or excavation, or "orientation" bias,
(2) the size of the joint, or "size" bias,
(3) the dimensions of the outcrop or excavations from which the
sample is taken.
For orientation, consider the two-dimensional case in Figure VI-3.
The probability of a joint of given orientation intersecting the ground
surface in a unit interval is
P(intersection) = (sin a)/d
Therefore, for a given spacing, d, the probability of being sampled is
proportional to sin a (Terzaghi, 1965).
In other words, joints which are
flatter with respect to the surface will appear less frequently in the
sample than joints which are steeper.
In order to be a probability sample
(as defined in Part A), these differences must be accounted for by weighing,
The procedures for weighing are presented by Terzaghi (1965).
She recom-
mends replacing the number of joints, N , which intersect the outcrop at
an angle a by N9 0 , the number of joints with the same orientation which
1_
-1
Figure VI-3 Relationship Between Joint Planes and
Outcrop Surface (from Baecher, 1972)
strike
dip
20 i---
@-0
O/
10
~=~
~
U
'-I
(,,0
10
20
30
40
50
60
70
Dip
Figure VI-6 Error in measured dip due to misalignment cf Brunton coimpass by ,
(from Baecher,
1972)
80
90o
would have been observed on an outcrop of the same dimension intersecting
the joints at an angle of 90'.
N90
This value is
N/sin a
No adequate correction can be made for low values of a because, in real
rock, the number of intersections is significantly affected by local variations in spacing and continuity of joints if a is small (Terzaghi, 1965).
If a joint survey is carried out in an area characterized by a great
variety of outcrop orientations the resulting data are likely to represent
a reasonably fair sample of the joints present in the area surveyed, and
weighing is not necessary.
However, if the observations come from drill
holes or bedrock exposures of nearly uniform orientation, joint data are
unlikely to provide even approximately correct information concerning the
relative occurrence of all sets.
In order to provide a basis for correcting errors associated with
uniform distribution of outcrop orientation, records must be kept of the
orientation of the outcrops on which the orientation of joints are measured.
Such records can be easily summarized by plotting the poles of the outcrop
orientation on a spherical projection.
Size bias or size effects influence the sample in three ways.
First,
the probability of a joint intersecting an outcrop is proportional to its
size.
Consider a set of joint planes having their center points located
at the same elevation and having the same orientation.
A set of random
planes through these joints will intersect each joint in proportion to its
size relative to the other joints.
Thus a joint 5 feet wide will be inter-
sected twice as many times as a joint 2.5 feet wide. Conversely, for a
single plane through a rock mass, the chance of that plane intersecting
the 5 foot wide joint is twice that of intersecting the 2.5 foot wide joint.
Second, joint size influences whether or not a joint will be part of
a sample.
veys.
Joints are usually sampled by means of area, circle or line sur-
Only those joints entering into an area or crossing the circum-
ference of the circle or the line are sampled.
The probability of doing
either is proportional to the joint's size.
Third, the entire intersection length of a joint with a rock surface
may not be visible because:
(1) The joints are covered with overburden.
(2) The joints continue into the sidewall of an excavation (Figure
VI-4b).
(3) The joint is presumed to extend beyond the edge of the outcrop
(Figure VI-4c).
As a result, a sample of joint lengths will consist of some members
whose true length is greater than the recorded length.
We would expect
and the work of Piteau (1970) confirms the fact that it is usually longer
joints whose true length cannot be measured.
Since length data is used
to make inferences about joint size and large joints produce long lengths,
we would expect that inferences made about joint size would be biased toward lower estimates of joint size.
(Statistically, the inability to
measure the true value of some members of a population comes under the
topic of censoring.)
A rigorous mathematical treatment of these three biases is put off
until the end of this chapter. This is necessary because they cannot be
rigorously accounted until a conceptual model of jointing is developed.
This will be done in the next section (Part C) of this chapter.
83
JOINT PLANE C: CIPL ENG H 3,4/
ib
ae)FPLA%•
JO;NYr ENSALON0 STRIKE
'OT V't CE, JO'N1 PRESUMED
CCNTNUE ALONJ SIRtKE
T')
\
v--LL
T ,P·N
FtO CSFUItVA'iC
fw
.JCINT PLANE
'A'
JINT END LIKELY
fS'~!KE bFYOND
FAC
CUT
ROFI
cf
JONY PLANE
'
OVEPVEO LENGT
.ON•,•STrKE-,IOm
JOI
T
EN
OBSERVED
LENGTH
ALONG DIP.le7.,
ExTE N-S
.OFI10 CUT
FACE ANG Of
DP
BOTHENOS ATJOINTREtJTO EXTLNDOBEYOND BENCH
TOP AND BOTTOM RESP
JOINT END
SEEN•/
TERIdNATE ON DIP
LENGlWN RECORDED
JOINT PLANE At STRIKE LENGTH -O0,7/2
D0P LENGTH
II,7/A
: DIP LENGTH
-83,5/2
JOINTYPIANE
Fivfure VI- 4 Examples of Joint Termiration
(from otef en et al., 1975)
Measurement Errors
Although random errors in the measurement of orientation arise from
a host of sources, several general comments can be made about them.
Baecher
(1972) and Steffen et al. (1975) have shown that random errors in the strike
direction are greater for "flatly" dipping joints than for "steeply" dipping
joints.
This results because it is more difficult to align the Brunton
compass along the line of strike when the joint plane is nearly horizontal,
and thus the measured strike is more prone to error (Figure VI-5).
Random errors in dip are greater for steeper joints.
These errors
come primarily from inaccurate alignment of the Brunton compass parallel
to the dip direction (Figure VI-6).
Other sources of error are inaccuracy
in leveling the inclinometer and roughness on the joint surface.
Steffen
et al. (1975) simulated joint surfaces of various roughness in the lab
and found that the variance doubled in going from a "smooth" to a "rough"
joint, but stayed about the same in going from "rough" to "very rough".
Systematic sources of error can be found by analyzing possible sources.
Some examples are:
(1) Incorrect setting of declination.
(2) Rounding off consistently in one direction.
(3) Mijadjusted leveling bubbles in the brunton.
(4) Holding a geology hammer always in the same hand while making
measurements with the compass.
Estimation Errors
Estimation errors are caused by differences in the sample statistics
from sample to sample. For example, the mean orientation of a joint set at
one outcrop may be different from the mean orientation of the same set at
x = 47.90
a = 0.890
N60
61
62
63
Strike
650 E
64
47
48
49
Dip
50
51 0 S
"steep" plane
i = 15.00
S= 68.350
-
N66
67
68
69
strike
70
-
71
n
a = 0.950
72 E
,, FTI
14
15
16
17
Dip
"flatter" plane
Figure VI-5 Empirically Observed lIeasurement Errors
in Strike and Dip (from 3aecber, 1972)
18S
another outcrop.
If this difference is due to the random nature of samples
and not to regional trends or structural controls, the stability of the
samples can be measured by using interpenetrating replicates.
Sampling Plans for Joint Surveys
Sampling plans for joint surveys must meet two criteria:
(1) They must allow valid statistical inferences to be drawn,
(2) They must be economical and easy to implement.
Of the many sampling plans available, two-stage cluster sampling
plans have long been favored because they require the least amount of time
to conduct (Baecher, 1972):
during the first stage, or "upper sampling
level", sampling sites are selected by some random process, and in the
second stage or "lower sampling level", samples are taken at each site by
a random process.
Upper Sampling Level
Prior to the selection of sampling sites, a geologic investigation of
the proposed study area needs to be conducted.
A literature survey as well
as preliminary field studies should be performed,
gations have a dual purpose.
The results of investi-
First, the geologic formations in the region
must be delineated since different formations may have different jointing
patterns.
Since it is not known beforehand whether the joint populations
are homogeneous from formation to formation, data sets for each formation
must be kept separately.
Second, the distribution of outcrop size and lo-
cation must be determined,
area no problems exist,
If the rock outcrops nearly continuously over the
However, if the outcrops are scattered, as will
normally be the case, it must be determined if the distribution of their
size and location is random.
The former can be determined by testing the
87
distribution of outcrop area against the Poisson distribution.
The latter
can be determined by the nearest neighbor technique (Baecher, 1972), or
by placing square cells of equal area over the study region, counting the
number of outcrops per cell and comparing this to the Poisson distribution
(Rodgers, 1974).
If the outcrops are nearly continuous or random, the sampling site
can be located randomly.
(Ifthe outcrop size and location is found to
be non-random for certain geologic formations in the study area, this procedure can still be used.
By keeping separate records of each formation,
those with random outcrops can be treated as probability samples and those
with non-random outcrops can be treated as "lower semi-probability" samples.)
We suggest the following procedure.
scale is placed over the study area.
A grid system of convenient
It should be located such that the
entire area lies to the right of the y-axis and above the x-axis (Figure
VI-7).
Next, pairs of random numbers are chosen from a random number table,
and treating them as X and Y coordinates, sampling points are located in
the area.
If the random number pairs exceed the maximum dimension of the
study area, they can be disregarded or modified by a constant (Yates, 1960).
Some sampling points will fall on rock outcrops and others will fall where
there are no outcrops (Figure VI-8).
In the latter case we recommend moving
the sampling location to the nearest outcropl.
(Ifwe disregard those
points not falling on an outcrop, there will be a tendency toward sampling
lOutcrops surrounded by more free space (areas where there are no outcrops)
have a greater probability of being sampled than those with less free space.
This arises because the more free space surrounding an outcrop, the higher
the probability of a random point falling into the free spacing, and the
higher the probability of the outcrop associated with this free space being
sampled.
-
-
-,
STUDY AREA
I
I -
1
-
----
-
- --
---
--
-
-
Improper Alignment of study Area with Axes
/
I----------------------------
-t
Proper Alignment of Study Area with Axes
Figure VI-7 Alignment of Study Area with Axes
1000
r------------------
Q
/,62
/o
-7
S
0
62
ia
C3,
0
29I
500
/
6
6
v
/
0
250
r G;
500o
1I
I
~62
I
m
Y
750
L
w
1000
Random lumber
Pairs
X
Y
1
712
240
2
908
757
3
875
88
4
762
5
596
335
6
426
290
7
664
686
8
525
528
9
305
217
Indicates sampling
point moved to nearest outcrop
266
Figure VI-8
Example of Upper Level zampling
90
only larger outcrops.)
At each sampling site, a sample would be collected
by some random process.
If it is determined that the outcrops are not random, first stage
random sampling becomes statistically meaningless.
However, all is not
lost, for once an outcrop is located the geologist is free to sample
the exposed rock layer.
Thus, in terms of Cochran, Tukey and Mosteller (1954),
the geologist may collect a 'lower semi-probability sample' in the sense
that although he has no control of the upper level of sampling, he does
have control of the lower level (Krumbein, 1960).
Results from such
sampling must be viewed cautiously. For example, if the outcrop size
distribution is related to jointing, then one might expect larger outcrops
to have fewer joints than small outcrops.
be biased.
Sampling in this situation will
Inferences from the sample to the sample population will be
correct, but they can be useless or misleading if there are great differences between the target and the sampled populations (Cochran et al., 1954).
However, if geologic investigations show that the sampled and target populations are similar, inferences are valid.
Lower Sampling Level
The lower level of sampling is performed by means of "spot sampling" in
the form of line, area or circle surveys.
The survey should be performed
on that portion of the outcrop which appears to have a "representative"
population of joints.
An experienced geologist can determine the represen-
tative joint sets during a preliminary field investigation of the study area.
Line Survey
The detailed line survey is the most common spot sampling technique
(Piteau, 1970; Robertson, 1970; Call, 1972).
At each selected sample site,
I
Mapping
0-2
Figure VI-9
\t
Example of Line Survey
Zone
a randomly oriented reference line, usually a tape measure, is placed on
the outcrop. For each joint crossing the line or falling within a specified distance on either side of the line (Call (1972) used 3 feet and Savely
(1972) used 2 feet as the specified distance), the following is recorded:
(1) Distance along the tape where the joint or its projection intersects the line.
(2) Orientation.
(3) Length and type of termination 2.
The major advantage of the line survey is that it directly yields information on spacing, orientation and length.
Spacing can be measured bet-
ween joints of a single set or between adjacent joints.
two disadvantages.
However, there are
First, orientation must be corrected for bias.
Joints
making a smaller angle with the sampling line have a smaller chance of
intersecting it than joints making a greater angle (Figure VI-9).
Robert-
son (1970) and Terzaghi (1965) have developed an equation for correcting
for this bias.
Second, line surveys usually require large outcrops.
For
example, in the present study, lines up to 25 feet long were needed to collect samples of 30 to 35 joints.
Call (1972) needed lines up to 50 feet
long to collect data on 100 joints.
Line surveys have been extensively used
in mapping open pit mines where there is almost continuous outcrop exposure.
Area Survey
An area survey is performed by placing a circular, square, rectangular,
etc. area on an outcrop and sampling every joint which is partially or wholly
contained within the area (Figure VI-10).
2Type
The orientation and length of
of termination refers to whether both ends of the joint are visible
in outcrop, one end or no ends. See Figure VI-6 for examples.
each joint is measured.
93
In order to measure spacing, a line must be placed
within the area and spacings measured along the line. Since the orientation
of these joints has been measured, spacings can be computed between joints
of a single set as well as between adjacent joints.
Area surveys require
only those corrections applicable to all sampling plans and do not introduce outcrop bias within the orientation plane.
Circle Survey
A circle survey is conducted by placing a circle on an outcrop and
sampling every joint which crosses the circumference of the circle.
Joints
which cross the circumference twice are sampled only once (Figure VI-11).
The orientation, length and type of termination need to be measured.
In
order to measure spacing, a line or lines are placed within the circle and
spacings are measured along them.
Since the orientation of all joints
crossing the line(s) is not known, only the spacing between adjacent joints
is recorded.
Circle surveys require only those corrections applicable to
all joint sampling plans.
Orientation bias is not present within the outcrop
plane since all orientations are equally likely on a circle.
Circle surveys, as well as area surveys, are well suited to regions
where outcrops are of limited areal extent.
For example, in the Blue Hills
where the average joint spacing is 0.4 feet, up to 60 joints were sampled
using a 9 foot diameter circle. At Pine Hill, where the average spacing
is 0.6 feet, a sample of 50 joints was obtained by using a circle 6 feet
in diameter. This compares with a sample of 30 to 35 joints being obtained
from a line 25 feet long at the Blue Hills and 28 joints being obtained from
a line 24 feet long at Pine Hill.
Field sampling for this thesis was performed by means of circle and
94
line surveys.
Circle surveys were used the majority of the time.
Their
use has not been discussed in the literature and so efforts were made to
assess their suitability as a device to sample joints.
In our opinion,
circle surveys were much easier to use as a sampling device than line surveys.
Their only disadvantage is that they do not yield information direct-
ly on spacing.
If a relationship between the number of joints intersecting
the circle's circumference and joint density can be established, this disadvantage can be removed.
e mesured
along this line
Notes:
1) All joints partially or wholly
within circle are measured.
2) Dashed joints are not measured.
3) Area can be other than circular.
Figure VI-10 Example of Area 6urvey
opaci
be me
alont
NIotes:
1) Only joint intersecting the
circumference of the circle
are measured.
Figure VI-11
Example of Circle Survey
97
This page is intentionally left blank,
PART C -SAMPLING INFERENCES
To integrate data commonly collected on joint length and spacing
and to make inferences about joint size
of joint geometry is needed.
and density, a conceptual model
Conclusions regarding the distribution
of
joint length and spacing would imply a model of jointing as two-dimensional convex discs randomly located in space with a lognormal distribution
of sizes.
The distribution of lengths one sees in outcrop is the set of
intersections of these random discs with a plane through space (Figure
VI-12).
In order to make inferences from sample data some assumptions must
be made. They are:
(1) Joints are two-dimensional circular discs.
At pre-
sent there is insufficient evidence to either support or reject this
assumption.
While stronger evidence is desired, this assumption offers a
starting point and it simplifies the analysis.
(2)The center points of joints are randomly and independently distributed in space, forming a Poisson field (Chapter V).
3
(3) Radii of joints are lognormally distributed. Baecher
at al. (1977) have shown that if joint radii are lognormal, the outcrop
length is also lognormal.
(4) Joint radius and dip are uncorrelated (statistically
independent.
(5)Joint radius and spatial location are uncorrelated
(statistically independent).
3
This is the conditional probability, given that the joint intersects the
the outcrop. It does not account for size bias.
PLAN
PROFILE
/
/
Figure VI-12 Conceptual Model of Joints as Two Dimensional
Discs
100
One might argue with these assumptions, but they offer a starting
point.
Assumptions 1, 2 and 3 are substantiated for the most part by the
data. Assumptions 4 and 5 are more open to debate.
To make them more
acceptable, one can restrict attention to each joint set, individually,
or divide the site into regions in which these assumptions are approximately
correct.
Joint Size
The probability of observing an intersection of length L conditioned
on lognormal radius is the product of the probability that a joint intersect an outcrop and the conditional probability of length L given that
it intersects the outcrop.
Pr(L,Ilr) = Pr(LIl,r) Pr(IJr)
A.1
where I is the event that a joint intersects the outcrop.
Transforming
the conditionality on r to the conditionality on the parameters on the
distribution of r,
f(L,IJr,p,a) =
f
f(LII,r,p,a)
f(Ijr,1 ,c) f(rli,a)
dr
A.2
r=½ L
where P = E(ln r) = mean of the logarithm of r
a = V(ln r) = variance of the logarithm of r
Assuming f(rlp,a) is lognormal
f(rjio)
=
1
exp -
ln r-
A.3
The conditional distribution of L given I and r depends on the randomness
101
in the distance of the line of intersection from the center of the joint.
If the joint and the outcrop plane are random and independent, this distance is uniformly distributed.
Integrating out the uncertainty in the
distance to the center point (Kendall and Moran, 1963) we get
f(LII,r) =
L
A.4
2r(4r 2 - L2) 2
Assuming the probability of a joint intersecting an outcrop is proportional to its radius (this accounts for the first of the three size biases
discussed in Part B), we get
p(I r) = r/Co
A.5
where Co is a constant. Substituting A.3, A.4 and A.5 into A.2, we get
f(Lli,a)
=
L
r=½L 2C (4r2 - L 2)
2avi
1
exp
n r-
-
dr
A.6
22
Attempts to integrate this directly were unsuccessful.
However, by
substituting r = (L/2) cosh(x), we get
f(L11,a)
=
K
-exp
K
-
-
lnL/2 - cosh(x)
1n(L/4 +x
dx +
dx
The second part of this expression can be integrated directly and
A.7
is equal
to
L
erf (z)
erf = error function
A.8
102
The likelihood function reduces to
3
L
f(Llj,a) = Kf L exp
L
4 C
1
2
In[L/2 cosh(x)] o
x
22+
+
erfi1n(L/2)+x-x •
2a
A.9
3
Evaluating both parts of the likelihood, we find that the second
term does not contribute significantly to the magnitude of the likelihood
(Table VI-1), and therefore, can be neglected for all practical purposes.
An exact analysis of sampling inference would use equations A.7 or
A.9 and update the distribution of the parameters p and a using Bayes theorum.
Ideally, we would separate the likelihood function into two parts:
one which is a function of the parameters and the other a function of the
data.
The former can be evaluated for reasonable values of the parameters
and the results tabulated. Then it would only be necessary to determine
the sample statistics in order to evaluate the likelihood.
Thus far, we
have not been able to do this. Instead, the entire likelihood must be
evaluated numerically and the procedure is time consuming.
A simpler
approach is to use an approximate second moment analysis and obtain point
estimators of the mean and variance of r.
Considering the distribution of joint radii intersecting an outcrop,
f(r
II), the mean and variance of f(rl I) can be estiamted by taking a
Taylor's expansion about the mean intersection length (Benjamin and Cornell, 1970).
Thus
E(r) = g(E(L), E(k)) +
1
()
EL2
E L)
A
62
V(L) + k
62
2Ek
V(k)
E k)
A.10
103
TABLE VI-1
EVALUATION OF EQUATION A.9
L. = 2, 4, 6, 8, 10
1
"ln r
0Inr
= 0.5, 1.0
= 0.5, 1.0
3
kf
exp
1 1n[z/2 cosh(k)] -]2
dx
O
L
i1
a
0.5
0.5
0.5
1.0
6.22
7.30
4.38
2.14
1.00
4.56
6.64
7.04
6.78
6.28
In(/4) + x -- erf Ierf
1.0
0.5
1.0
1.0
5.73
11.99
12.83
10.23
7.10
4.73
8.63
10.61
11.43
11.60
.0045
0
0
0
0
.082
.045
.045
0
0
L
2
4
6
.0001
.0001
0
3
.0351
.0124
.004
0
0
104
where k is U( ¼,'/,).
Next, we must correct for the bias of larger joints
appearing disproportionately in the sample.
f(r,I) = f(r I)
f(I) =
f(II r) f(r)
f(II r) f(r)
f(I)
A.11
f(II r) = r
f(I) =
A.12
f f(II r) f(r) dr
r
=
fr
f(r) dr = E(r)
r
A.13
Substituting,
f(rl I)
r f(r)
E(r)
A.14
Rearranging terms, we get
f f(r) dr =
E(r)
0
f 1/r
f(r I)
dr
A.15
0
E(r) = E- 1 (1/r
I)
Similarly,
E(r k ) =
E(rk- II)
E(I/r II)
A.16
Now we can estimate the mean and variance of the actual distribution
of joint radii.
Using this approximate analysis (A.10 and A.16) we get
E(r) = [ 1.86 E-1(L) + 1.73 E-3(L) V(L)] -1
E(r 2 ) = 0.64 E(L) E(r)
105
V(r2 ) = [0.362 E3(L) + 0.58 E(L) V(L)] E(r) [0.64 E(L) V(L)] 2
where E(L) is the average intersection length and V(L) is the variance.
From the top of rock data at Site A,
E(r) = 4.72
E(L) = 13.1
E(r2 ) = 39.57
V(L) = 90.63
V(r2 ) = 5525
E(rl I) = 8.38
Since the expected radius of those joints appearing in outcrop is 8.38 ft.,
correction for sampling bias substantially lowers the estimate of joint
size.
Joint Density
The density of jointing, that is the number of joint centers per
unit volume of rock, can be inferred from either spacing data or the number
of joints appearing in outcrop.
dered here.
Inferences from spacing data are consi-
For a joint of radius r to intersect a line through a rock
mass parallel to its pole(this can be a boring or a line inscribed on an
outcrop), the joint center must lie within a cylinder having the line as
its axis.
Thus, given the number of intersections n per length of line t
one can make inferences about the density of joint centers, x , according
to the familiar Poisson process equations
f(n
, )=
e
A V
n!
A.17
where V is the volume of the cylinder.
For a joint whose pole makes an angle
0
with the sampling line, any
106
joint whose center is within the elliptical cylinder of major axis r and
minor axis r sin e will intersect the line. The orientation of the major
diameter of this cylinder is parallel to the strike of the joint.
For all joints of the same orientation and radius, the number of
intersections along a line at e would be distributed as:
f(nlr,e)
- (wr2sin20)
=
e(r 2sin 2n!
n
A.18
A.18
rr2in2
As the radius and orientation of joints are not constant, but distributed, the expression of equation A.18 must be compounded to;
7T/2
f(n) =
f f f(nfr,e) f(e) f(r) de dr
O
A.19
O
where f(O) is a transformation of the distribution of joint poles and f(r)
is the distribution of radii,
In this thesis, we do not consider pole
distributions since their distributional form is not yet known.
There-
fore, 0 is taken as a constant. The distribution on r may be transformed
to a distribution on r2 and the equation takes on the form of a compound
Poisson distribution in which the parameter is lognormal.
e
f(n) =
o
2
r 2 sin 2 n (Xrrr 2sin 8)
n "2
1
I
exp
- - exo
a 22r
-
22L
nr-l
01
dr
dr
A.20
The expectation of this distribution is (Moran, 1968) is
E(n) = r x
sin o E(r 2 )
where a is the length of the sampling line.
Thus, the estimator
A.21
107
=
T
N/I
sin e E(r 2 )
A.21
where N is the number of joints intersected.
Another approach would be to take the likelihood of joint density
(A.20) and transform it into a distribution on the parameters of r. Thus,
f(n) = e- aE(r 2 l,
)sin 2e (A_T sin 2 0 E(r21,
n!
A.22)n
A.22
Joint Intensity
From the distribution of joint area and spatial density, the distribution of joint intensity can be exactly calculated.
A1
. . . . . . AN
For N joints with areas
the area of jointing is
N
A. =
A.
A.23
and the area of jointing per volume is A./V where V is the volume of rock
in which the N joints occur.
(Baecher et al. (1977) have shown that if r
is lognormal, then the area of the joint is also lognormal.)
As we assume joints to be random in number and size, both N and Aj
are random variables.
Equation A.23 is the sum of a random number of
random area, where N is a Poisson variable with parameter
-XV
f(NIX) = e -V
N
V)
A.24
and the Ai's are independent identically distributed (IID) variables with
parameters ua and aa'
Barouch and Kaufman(1976) show that the distribution of the sum of
IID lognormal variables has a complicated form, but can be approximated
108
asymptotically for large N. For a Bayesian predictive analysis such an
approximation to the full distribution of A. would be necessary.
However,
given the second-moment approach of the present analysis, the mean and
variance of Aj constitute a satisfactory description.
The mean and variance of A. can be calculated from the mean and
variance of random sums(Papoulis, 1965).
E(Aj) = E(N) E(Ai)
=(V) exp(p++q2/2)
V(A)
A.25
= E2(Ai) V (N2 ) + V(Ai ) E(N)
= exp( 2p +C2 ) (XV) + AV
in which V is the volume of rock.
exp(2 + ) (exp(G 2 )-
I)
A.26
We can transform these equations into
parameters on the mean and variance of the radius by substituting the
following into the above equations.
Ia = In7+ 21r
a = 4 or
Corrections for Sampling Errors
The likelihood function (A.7) is the probability that the entire length
of all joints intersecting an infinitely large outcrop are measured. (The
likelihood function does account for smaller joints intersecting the outcrop a disproportionately fewer number of times.)
In reality, only a por-
tion of the joints intersecting an outcrop are sampled with a line, circle
or area survey.
In addition the entire intersection length of some joints
cannot be measured, and so lengths are censored.
Finally, shorter joints
109
may be purposely excluded from the sample. We can account for these factors by appropriately modifying
the likelihood function so that it
describes the actual situation.
Line Survey
In a line survey only those joints intersecting the line are sampled.
The probability of intersecting the line is
co
Pr(L,It j I,r,p,c ) =
f
Pr(I
L,Is,r,
r=O
Pr(LI Is
Pr(Isl r,
r ,
}i,
,o )
X
,1 ) X
a) X Pr(rlp,o)dr
Pr(I. ) = sin a L a x
A.27
Ik = intersection of the joint with the line
where
a = angle between the joint and the line
X = joint density
a = length of the sampling line
L = joint length
The remainder of the equation is the same as A.2 and A.6. The likelihood
becomes
c0
Pr(I1
L ,Is,r,},y) = K1
f
L2 (4r
2
-L2)
-½
(r,,a)dr
r =½ L
1
o sin ax- A
K = C
1
Using a second moment approach and obtaining point estimators of the
mean and variance of r, we have
Pr(I ,Is r) f(r)
Pr(rIlI,S)
f(r) f(I0,IIr)
-f
J
110
_ r2 f(r)
A.28
E(r2)
1
f - f(rlIs5,
)dr = E (r2) j f(r)dr
or
o
E[2-Is,
5I
= E'[r2]
Similarly,
E=E(rk)
A.29
E(/r21iz ,I
s )
Circle Survey
The probability of a joint intersecting the circumference of a circle
is proportional to its length and the radius of the circle.
line segment G.
Consider the
It intersects a circle of radius P. The chord formed
by this intersection is of length g (Figure VI-13).
If the joint
length, L, is longer than g, then in order to intersect the circumference,
midpoint of the joint must lie L/2 either side of the circle or along g
(Figure VI-13).
Pr(Ic L)
f(g) =
Therefore,
2(L/2) + h(P) f(g)
9
2P
V4P2
A.30
-
g2
where g is uniformly distributed from 0 to 2r.
For joints less than g,
the midpoint of the joint must lie L/2 either side of the circumference
of the circle (Figure VI-13).
Pr(IclL)o- 4(L/2)
Therefore,
A.31
111
L/2
L/2
L/2
L g
Figure VI-13 Size Bias In Circle Survey
112
Taking Equation A.7 and multiplying it by A.30 and A.31 gives us the
likelihood of seeing a sample of joint lengths which intersect the circumference of a circle, given the parameters of joint radius.
Area Survey
The probability of a joint intersecting an area is
Pr(Ia IL) = 7A + Lp
A.32
where A = area of the shape
L
=
joint length
p = perimeter of the shape
Truncation
The intentional exclusion of shorter joints from the sample is called
truncation.
than 1.0.
Truncation results in the area under the likelihood to be less
In order to account for this, the likelihood function must be
renormalized (Appendix B).
Cf
or= L /4r
o
Pr(Lly,o) =
l-C 0
=0
L
2 - L2
A. (rlv,a) dr
L > L
4r2 -L
A.33
A rj1,o dr
L<L
0
where L = truncation point
Censoring
In a censored length sample of size t, there are n joints whose entire
intersection length can be seen, and its true length measured.
Also,
there are m joints whose entire intersection length cannot be seen, and
whose measured length is therefore less than its true length.
113
For the simple case of censoring, n joints will lie below the point
of censoring, Lc , and the remaining m joints will be greater than Lc.
This case is treated in Appendix B and by Hald (1949).
However, in dealing
with joint length, the point of censoring is not constant, and this can
complicate the situation.
Using Equation A.7, we would have to evaluate
it twice, once for the true (uncensored) lengths and again for the censored lengths.
The two would be then multiplied together to give us the
desired results.
Pr(LJI,a)
=
..... A (r ~,c)
r=½L
X
f ......•(rJl,o)
dr
r=½Lc
We conclude this chapter with an outline of the procedures for sampling
joints.
Upper Sampling Level
1) Define limits of study area and collect relevant geologic information.
2) Determine the extent of the various geologic formation and if the
distribution of outcrop size and area is random.
If their distribution is
random, the sample is treated as a probability sample.
If not, the sample is
treated as a lower semi-probability sample.
3) Place the line, area or circle on the outcrop and sample the joints.
If using line surveys, one should use randomly oriented lines.
a) For a line survey, every joint falling within a specified distance either side of the line is sampled.
b) For an area survey, every joint partially or wholly contained
within the area is sampled.
c) For a circle survey every joint crossing the circumference is
sampled.
114
Sampling Inferences
1) Joint size
2) Corrections for size effects or bias
a) truncation
b) censoring
c) sampling plan corrections
3) Joint density
4) Joint intensity
5) Joint orientation
a) correction for orientation bias, if necessary
115
CHAPTER VII
Conclusions and Recommendations for Future Study
This thesis represents the initial phase of research into the
development of a comprehensive set of analytical relationships between the
joint properties observed in outcrop and their actual properties in the rock
mass.
We have tried to answer a number of questions, and have raised just
as many. Specifically, we have found the following:
(1)The Fisher distribution does not fit orientation data.
However, for each joint set or cluster, it appears that the data are symmetrically arranged about a mean orientation and this mean can be approximated by the modal orientation obtained from a pole diagram.
(2)Joint length is lognormally distributed, in both
strike and dip direction.
It appears that joints are equidimensional or
slightly elongate in shape.
(3)Spacing appears to be exponentially distributed.
Spacing data can be used to estimate RQD and joint orientation.
(4) Two-stage cluster sampling is the best sampling plan
for collecting joint data.
During the upper sampling stage, sites to be
sampled are located randomly.
During the lower level data are best col-
lected by the circle survey, if not from a theoretical viewpoint, at least
a practical one.
(5)Using our conclusions regarding joint length and
spacing , we propose that joints are two-dimensional convex discs randomly
distributed
in space with a lognormal distribution of sizes.
Using this
model likelihood equations and point estimators of mean and variance of
116
joint size and density were formulated.
In addition point estimators for
the mean and variance joint intensity were developed.
In most cases sam-
pling biases were rigorously accounted for.
Among the questions we still seek to answer are:
(1)What is the distribution of joint orientation ?
(2)Can we find a stronger indication of the true shape
of joints?
(3)Can we breakdown the various likelihood functions into
two parts: one which is a function of the parameters and the other which is
a function of the data?
The latter could be evaluated for reasonable values
of the parameters and the results tabulated.
Then it would only be neces-
sary to evaluate the sample statistics in order to determine the likelihood.
117
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"Analysis of Fracture Orientations for Input to Structural Models
of Discontinuous Rock," USBM, R.I. 7669.
Mardia, K.V. (1972), Statistics of Directional Data, Academic Press, New
York, 357 pp.
McMahon, B. (1974), "Design of Rock Slopes Against Sliding on Pre-existing
Surface," 3rd Intl. Symp. on Rock Mechanics, Vol. II-B, pp. 803-808.
McMahon, D. (1971), "A Statistical Method for the Design of Rock Slopes,"
Proc. Ist Australia-New Zealand Conf. on Geomechanics, pp. 314-321.
Moran, P.A.P. (1968). Introduction to Probability Theory, Oxford Press,
London.
119
REFERENCES
(continued)
Papoulis, A. (1965), Probability, Random Variables and Stochastic Processes,
McGraw-Hill, New York, 580 pp.
Piteau, D.R. (1970), "Geological Factors Significant to the Stability of
Slopes Cut in Rock," Proc. Symp. on Theoretical Background to the
Planning of Open Pit Mines with Special Reference to Slope Stability,
pp. 33-53.
Piteau, D.R. and Martin, D.E. (1977), "Slope Stability Analysis and Design
Based on Probability Techniques at Cassiar Mine," Canadian Mining
and Metallurgical Journal, March.
Priest, S.D. and Hudson, J. (1976), "Discontinuity Spacings in Rock,"
Int. J. Rock Mech. and Mining Science, Vol. 13, pp. 135-148.
Robertson, A. (1970), "The Interpretation of Geological Factors for Use
in Slope Stability," Proc. Symp. on the Theo. Background to the
Planning of Open Pit Mines with Special Reference to Slope Stability,
pp. 55-71.
Rodgers, A. (1974), Statistical Analysis of Spatial Dispersion, Pion Limited,
London, 164 pp.
Savely, J.P. (1972), "Orientation and Engineering Properties of Jointing
in Sierrita Pit, Arizona," M.S. Thesis, Univ. of Arizona.
Scheidegger, A. (1977), "On the Theory of the Evaluation of Joint Orientation Measurements," Rock Mechanics, Vol. 9, pp. 9-25.
Shanley, R.J. and Mahtab, M. (1975), "FRACTAN: A Computer Code for Analysis
of Clusters Defined on a Unit Hemisphere," USBM, IC 8671.
Siddiqui, M.M. and Weiss, G.M. (1963), "Families of Distributions of Hourly
Median Power and Instantaneous Power of Received Radio Signals,"
Journ. Research of Nat. Bureau of Standards, Vol. 67D, pp. 753-762.
Steffen, 0., Kerrich, 0. and Jennings, J.E. (1975), "Recent Developments
in the Interpretation of Data from Joint Surveys in Rock Masses,"
6th Reg. Conf. for Africa on Soil Mech. and Found., Vol. II, pp. 17-26.
Terzaghi, R. (1965), "Sources of Error in Joint Surveys," Geotechnique,
V. 15(3), pp. 287-304.
Watson, G. (1966), "The Statistics of Orientation Data," J. of Geology,
V. 74, pp. 786-794.
Yates, F. (1960), Sampling Methods for Censuses and Surveys, 3rd edition,
Hafner Publishing Company. New York.
120
REFERENCES
(continued)
Zabak, Caner (1977), "Statistical Interpretation of Discontinuity Contour
Diagrams," Int.
of Rock Mech. and Mining Sci., Vol. 14, pp. 111120.
121
APPENDICES
122
APPENDIX A
GEOLOGY OF SAMPLED SITES
SITE A
Site A is located in southern Connecticut, about 50 miles east of
Bridgeport.
The joint data was gathered from excavations made prior to
the construction of a power plant at the site.
The country rock at the site is the Monson Gneiss, a thinly banded
rock with feldspathic and biotitic layers.
northwest, iswell developed.
This banding, which trends
The gneiss is intruded by a series of peg-
matitic and granitic bodies which run parallel to the banding.
The Monson
Gneiss, originally deposited as an Early Paleozoic felsic volcanic series,
was extensively deformed and metamorphosed during the Middle Paleozoic.
The intrusion of the granitic and pegmatitic bodies occurred during the
Permian.
The initial excavation consisted of removing overburden from an area
roughly 375 feet on a side (see Figure A-1).
The uncovered bedrock sur-
face displays features typical of glaciated areas.
It is rounded, striated
and generally fresh; however, in highly jointed areas, the rock is slightly to moderately weathered and iron oxide coated.
The joints on the sur-
face are tight and mostly linear. A majority are coated with chlorite,
while many exhibit iron oxide coatings.
The jointing is independent of
rock type, cutting across both the gneiss and the pegmatites.
Orienta-
tion data from the top of rock is summarized in Figure C-1, length data
is summarized in Table IV-2 and spacing in Tables V-l and V-3.
The
length data does not include joints less than 2.0 feet long.
Additional joint information was collected from the sides and bottom
123
Approximate limit of overburcen excavation
0
50
100
FEET
Fi 'ure A-1
1larn View of
ixctevcatioLns
'ite
A
t
0
0
0
0
0
A
L
/'
-U
U
0
-0
-
-nl
m
S-10 -
-10 m
-4
,--20
-20-
SECTION A-A'
B
B
-A1 0
0
_-+10
0
-n
LU
L-10
m
--
10
-20
--20
SECTION B-B'
0
Fi gqure A-2
50
FEET
100
Cross-Sections of the Txcavations at Site A.
-0
0
125
of the excavations shown in Figures A-I and A-2. Joint orientations from
the bottom of the excavations (Figure C-2) are similar to those encountered at the top of rock.
However, mapping of the sides reveals the pre-
sence of another set which is nearly horizontal (Figures C-3 and C-4).
This set was not found elsewhere because it has a very small chance of
intersecting the almost horizontal surfaces of the top of rock and the
bottom of the excavations.
Length data is summarized in Table IV-2,
and spacing in Tables V-I and V-2. The average joint length from the
top of rock is 13.7 feet, from the bottom of the excavations 11.4 feet
and from the sides of the excavations 21 feet.
The lengths measured
from the top of rock and bottom of the excavation are from one major
joint set (Figures C-I and C-2).
Lengths measured from the sides come
from the two major joint sets (Figures C-3 and C-4).
The average spacing
from the top of rock is about twice that from the bottom of the excavations (see Tables V-i and V-3).
All orientation data was gathered with a brunton compass using standard geologic techniques (Compton, 1962).
The end points of each joint
were surveyed in,and the position and length of each joint determined
from the coordinates of their end points.
This method of determining
joint length does not allow the consideration of curved joints since
only the end points are used to calculate length. However, since most
joints are linear, this has little effect.
GREENE COUNTY SITE
The Greene County site is located on the west bank of the Hudson
River, 30 miles south of Albany, New York.
Joint data was gathered from
4 trenches which were excavated to the top of rock.
The trenches are
126
N
NCH A
10
Dio of
Bed d in
-4
TRENCH C
I
Sync !ire,
Arrow I'oirnts
in Dir ction
Of" -'u
TREN(
ENCH
RENCH T
100
0
FEE*
1iue
NCH B
.lan liew of
1
Ir'ee-ne '3ouit.1'
,
127
15 to 20 feet wide and have a total length of 1570 feet (see Figure A-3).
The bedrock at the site is the Ordovician Austin Glen Formation.
It is a sequence of medium to thickly bedded, marine sandstones, siltstones and shales which was deformed into a series of northerly plunging
folds during the Taconic orogeny (late Ordovician).
The site itself is
underlain by an asymmetric syncline. The spatial relationship of the
trenches with respect to the syncline is shown in Figure A-3.
The bedrock surface is rounded by glacial action, but striae are
not abundant.
The majority of the joints are straight and tight to open.
Many are filled in with secondary calcite and quartz or washed in soil.
Heavily weathered zones are rare. The joints in the sandstone and siltstone are not continuous into adjacent shale beds.
Joints in the shale
are limited to bedding plane joints and these were not measured.
The joint orientation data is summarized in Figures C-5 to C-8.
As can be seen in these figures there are 3 major sets.
The nearly
north-south striking set is missing in Trench A. This is the case because the strike of this set is parallel to that of Trench A. Length
data is summarized in Table IV-2.
The average lengths are 7.5, 6.2, 6.0
and 9.1 for Trenches A, B, C and T, respectively. The data from Trenches A
and B does not include lengths less than 1.0 foot and data from Trenches
C and T does not include lengths less than 2.0 feet.
Joint orientation was measured with a brunton compass, utilizing
standard techniques (Compton, 1962).
ruler.
Lengths were measured with a tape
Curvature of the joints was taken into account.
SITE B
Site B is located 5 miles east of Oswego, New York along the eastern
128
L
0
I
F
FEET
Figure A-4
80
Plan View of Excavations at Site B
129
shore of Lake Ontario. The joint data was obtained from the bottom of
excavations made at the site as part of the construction for a power
plant (see Figure A-4).
The bedrock at the site is the Ordovician-Silurian Oswego Formation,
a fossiliferous interbedded sequence of marine sandstones, siltstones and
The rock is essentially flat lying.
shales.
and confined to the sandstone beds.
rare.
The joints are tight, planar
Calcite or any other fillings are
Only joint location and length were measured.
The presence of
construction equipment made it impossible to accurately measure orientation.
Length data is summarized in Table IV-2.
The average length is
7.5 feet.
BLUE HILL AND PINE HILL SITES
Two areas in the vicinity of M.I.T. were designated as sites where
we would perform our own joint surveys.
The first site is located about
10 miles south of M.I.T. in an area known as the Blue Hills Reservation
(Figure A-5) and the other site is 7 miles north of M.I.T. in the Pine
Hill/Middlesex Fells Reservation (Figure A-7).
Both sites are open to
the public and operated by the Metropolitan District Commission.
The study area in the Blue Hills covers an area of roughly 2.5
square miles.
Within the study area there are two rock units, the Blue
Hills Porphyry and the Mattapan volcanics (Chute, 1969 and Billings, 1976)
(Figure A-6).
Ordovician age.
The Blue Hills is a dark gray granite porphyry of probable
The volcanics consist of red, pink, brown and gray ash
flows, tuff, breccia and lava flows.
bedding dip steeply southward.
comagmatic.
The flow structure and occasional
The porphyry and the volcanics are probably
130
The many rock outcrops in the Blue Hills are fresh and usually rounded by glacial action, although some outcrops show a preferential orientation due to jointing. The joints range from tight to slightly open, and
are rarely filled with any material, except for some that are coated with
iron oxide.
Sampling of joints was accomplished by means of 14 circle and 4 line
surveys.
The locations are shown in Figure A-6. The procedures for con-
ducting these surveys are outlined in Chapter VI.
Orientation data is
summarized in Figure C-9, length in Table IV-2 and spacing in Table V-3.
The average length is 2.4 feet and the average spacing is 0.40 feet.
The Pine Hill area covers an area of roughly 1.2 square miles (Figure
A-7).
No geologic map exists for the area, thus the geologic description
follows from the author's limited experience in the area.
The oldest rock
is the Dedham Granodiorite. It has been intruded by volcanic rock similar
in lithology and probably similar in age to those in the Blue Hills. Both
the granite and volcanics have been intruded by a series of basalt and
gabbrodiorite dikes of probable Mesozic age.
Joints in the Pine Hills
are similar in appearance to those in the Blue Hills.
Six circle and one line survey were conducted.
shown in Figure A-7.
Their locations are
The orientation data is summarized in Figure C-10,
and length and spacing in Table IV-2 and V-3, respectively. The average
spacing is 0.6 feet and average length is 1.5 feet.
131
0
L
2000
A
0 Sampling Locations
/QU
ASSAN OCON
ION
QUADRANGLE LOCAT
Figure A-5
Topographic Pfap of
Blue Hills 6ite
132
0
I
2000
I
FEET
I
]
Blue Hills Porphyry
1,
•iattaýpan Volcanics
*
Circle ,urvey
i
Circle and Liine ourvey
S
only
COriginal 3ample Kite i.oved
Because ]o Outcrop
Figure A-6
Sampling Locations
in the Blue Hills
N
\.~· ·
1·.K
i
-''.4,
2000
FEET
*
Circle Survey
*
Circle and Line Surveys
V
Volcani cs
Only
MASS
/
QUA
DG
8
N
F LOCAON
QUAfA•NGI E LOCATION
Dedham Granodiorite
Basalt Dikes
Figure A-7
Topographic-Sampling
Location Map Pine
Hills Site
134
APPENDIX B
PROBABILITY MODELS
Appendix B presents a brief description of the probabilistic models
used in this thesis.
The discussion concentrates on the functional form
of the distribution, the underlying physical process and the maximum
likelihood estimators.
POISSON DISTRIBUTION
A random point pattern is defined as a disposition of points on a
plane, generated by a process that satisfies 2 conditions:
(1) Any point has an equal probability of occurring at any position
on the plane.
Therefore, any subregion of the plane has the same proba-
bility of containing a point as that of any other subregion of equal area.
(2) The position of a point on the plane is independent of the position of any other point.
Imposing a third condition that the number of subregions goes to infinity,
Rodgers (1974) showed that the mathematical formulation for the random
disposition of points is the Poisson distribution (and the process is
called Poisson).
Benjamin and Cornell (1970) extend the process to in-
clude the distribution of events in time and Baecher et al. (1977) to
the distribution of points in three dimensions.
The probability mass function (pmf) for the Poisson distribution is
n
-X
f(nlX) =
where X is the density of points.
n= 1,2,3,......,n
Examples of the Poisson distribution
for different values of X are shown in Figure B-1.
135
71'
,-0.(n
0.5
K
C.2
P(0.7)
_L
__'
0o
0
2
4
e-
25
6
8
10
(2.5)x
0.2
P(2.5)
0
2
4
6
8
10
x
10
x
e-5.0i5.0) x
0.4
P(5.0)
0.2
0
2
4
U
8
(from Benjamin and Cornell,
1970)
Figure B-1
ixamples of Poisson
Distribution for
Different Values of
136
EXPONENTIAL DISTRIBUTION
If events in time or space are generated by a Poisson process,
then the distribution of the distances between events is described by
the exponential density function.
The probability density function (pdf) for the exponential distribution is
f(xjl)
A
e- x
=
=0
x> 0
x< 0
Figure B-2 illustrates the influence of varying A on the form of the distribution.
n
The maximum likelihood estimator (MLE) for A is n/Yx
i
(Benjamin
i=l
and Cornell, 1970).
GAMMA DISTRIBUTION
If events in time or space are generated by a Poisson process, then
the time or distance to the kth event is distributed as the gamma distribution.
The pdf for the gamma distribution is
ax-x
e-X/6
f(xa,)
a r(ca)
=0
;
xa
x> 0 where r(r)
e - x dx
0
;
x< 0
The gamma distribution with different values of a is shown in Figure B-3.
The reader is referred to the paper by Siddequi and Weiss (1963)
for the mathematics necessary to find the MLE of a,ý.
Briefly, the geo-
metric and arithmetic means are computed from the data and tables are
used to estimate a,6.
1 17
=
1.0
0.5
.\-
Figure B-2
Examples of Exponential Distribution for Various
Values of X
fI
.0= I
-
49= 2
I
1
2
3
4
5
6
7
8x
Figure B-3
Examples of Gamma Distribution for
Various Values of o:( = 1. (from Benjamin
and Cornell, 1970)
138
LOGNORMAL DISTRIBUTION
The pdf of the lognormal distribution is
1log
f(xjlP,o)
x.i
- 11
exp
=
>-
Examples of the influence of mean and variance on the form of the lognormal
density function are presented in Figure B-4.
The lognormal distribution arises as a result of the joint effect of
a large number of mutually independent causes acting in an ordered sequence
during the formation or destruction of an object (Atchinson and Brown, 1957).
An example of such a mechanism occurs in breakage processes, such as
the crushing of aggregate. The process is such that the size of the Xj
particle is equal to the size of the Xj_ l particle times a reduction factor W (Benjamin and Cornell, 1970).
n
The MLE for the mean is 1-Y log
Sn
ni=
-- (log x -ix)
xi,
and for the variance is
(Atchinson and Brown, 1957).
1=1
The reader is referred to Atchinson and Brown (1957) for a detailed
discussion of the lognormal distribution, its properties and various forms.
MODIFIED DISTRIBUTIONS
Truncated Distribution
A density function may appear to be a particular function, except
that, that part of the distribution for which X< X is removed because
such values of X either cannot or are not observed.
This distribution
is said to be incomplete, or more commonly, "truncated", and X is termed
the point of truncation.
Figure B-5 illustrates the difference between
139
I,
J
2
.,1
= 1.0
Figure B-4
Examples of Lognormal Densities for Various
Values of Mean and Variance (from Atchinson
and Brown, 1957)
(,(-)
fz(x)=O0x,'(x) x>146
/
I
I
UV XO= 140
2Ou
300
- Figure B-5
The Truncated Normal Distribution
(from Benjamin and Cornell, 1970)
400 x
140
a complete and a truncated normal distribution.
If the original population has a probability density function, f(x),
and a cumulative density function F(x), and if the random variable X is
truncated below xo, then the pdf of X is zero up to xo and k* f(x) for X
greater than x o ,
where k= 1/(
-F(x )) (Benjamin and Cornell, 1970).
The maximum likelihood estimator for the parameter of truncated
distributions described by only one parameter is relatively easy to
derive, as shown below. For example, for the exponential distribution:
f(x)
=
Xe-Xx/e
n
L = Xn e
-X xo
i/(e-xO)n
n
LL = n log X -
dLL
n
+ nx
ý xi + nXxo
i=l
n0
-
xi = 0
i=l
X.- nx
The maximum likelihood estimators for two parameter truncated distributions
are more difficult to derive.
Hald (1949) presents a method for evaluating
the MLE for the mean and variance of a truncated normal distribution, and
Atchinson and Brown (1957) for the lognormal.
Censored Distributions
Censored distributions differ slightly from the truncated in that
the total population is present but the exact frequencies are only known
up to (or beyond) a certain value xo. Stated in a slightly different
way, a density function is considered censored if given a sample of size n,
141
there are n1 observations greater than the point of censorship whose
exact frequencies are known, and n2 observations less than xo whose
exact values and frequencies are not known.
The density function for
such a distribution is
f(x) = k1 * fl(x)
where k
and fl(x) is the form of the density function had
o
ffl(x)dx
it not been censored.
An example of censored distribution is presented
in Figure B-5A.
Hald (1949) and Atchinson and Brown (1957) present methods for
estimating the parameters for the normal and lognormal distributions,
respectively.
DISTRIBUTIONS ON A SPHERE
Spherical distributions describe the configuration of points on a
sphere.
With respect to joints, the points represent the spherical
coordinates of the joint poles.
There are several spherical distributions
which have been extensively studied (Mardia, 1972). However, only the
"spherical normal" or Fisher distribution will be presented in detail,
while the others will be treated only briefly.
This is done since the
Fisher distribution is used more extensively in this thesis than the
others.
Fisher Distribution
The Fisher distribution is defined by the density
f(vIK) = (47sin h(k))-1 exp(Kcos ')
where K is a non-negative constant controlling the scatter and T is the
1-42
Y,E
00
X,N
Xi = Ii = sin Oi cos 98i
Z i = n i = cos
Yi
=
mi
=
sin 4i sin &i
i
M = Intersection of mean vector (colatitude
,
longitude
')
with the hemisphere.
i = Angle between the mean and observation i
Figure B-7
Rectangular Coordinates (or Direction Cosines) of a Poirt i (With Colatitude 1,iro I, LI-ahtab et al., 1972)
and Longitude =- i) on a Unit Sphere.
i
143
angle between the mean or preferred direction and an observation (Watson,
1966) (see Figure B-6 and B-7, respectively).
Defining the positive
z-axis as the mean direction, p as the colatitude and 0 as the longitude
of an observation (Figure B-7), Mardia (1972) shows that the above equation is transformed to
g(p,e) = (4rrsin h(k))
0<
-1
exp(Kcos ) sinq dýdeo
00<< O< 27
v
The density of points does not depend on 6 since e is uniformly distributed
as (2a)
-
1.
Rather it depends only on 4 which has a density function:
f(e) = (4r sin h(K))
- 1
exp (Kcos ) sin
O< 0 <2r,
<
K> 0
Figure B-8, which shows the value of the above density for selected values
of K, indicates that the probability mass concentrates about O=0 as K
increases (Mardia, 1972).
The shape of the above density can be visualized by imagining the
mass to be distributed on a unit sphere. The distribution is rotationally
symmetric about the z-axis.
The mass can be roughly imagined as distri-
buted on the sphere in the shape of a top with the spinning axis as the
z-axis and the spinning head towards the positive z-axis.
The preceding discussion was based on the mean pole coinciding with
the positive z-axis.
Frequently, it does not, and prior to goodness-of-
fit testing of any data against the Fisher distribution, the mean pole
(and its associated observations) must be rotated so that it coincides
with the z-axis.
a sphere.
Mardia (1972) presents equations for rotating points on
144
!-
10
-.
0
L~
I--L
20
.. W
40
~.I,I'
-
tO
80
10
that 50, 75, a•l 95 per c•nt of the '
giQin the angle ,,, as a funcLin of K,~Uc
:.
Tlus 1' is, the ,m anigk
tions fall N\ithin ' L, r., of the m", ,iro•'i 'Wi Se di' ria!t:2 n
of concentration about the mean dit•.
:..
(_ 'roI.
, Gt SO
s 0n I I
b )
Fi-Au Ire B-6.--Curvc,
,,,
K =30
K•=10
K=5
150
;35
i: -UI'
.-- C Density of
for the Fisher distribution for
(fr•o
K
-= 1, 5, 10, 30.
a::di , 1
)
145
The MLE for the mean direction expressed in terms of direction
cosines of the data is
i=l
1
o
1.iM
-
R
R
m
o
N
N
i=l
1
i=l
n
n
=
o
=
R
N
R= /I (1li)Z
+ (= mi)2++ (ni()2
th
where 1i , mi , and ni are the direction cosines of the it h observation and
1 , m-0 and no are the direction cosines of the mean direction.
The MLE
for K is
N - 1/N- R
where N is the number of observations.
Other Spherical Distributions
Uniform
As the name implies, the points are uniformally distributed on the
sphere (Mardia, 1972).
Dimroth-Watson Distribution (Mardia, 1972)
This distribution is characterized by a concentration of points
around a great circle and is rotationally symmetric about the mean direction.
It is also called a girdle distribution. Taking the z-axis as
the mean direction, the density of this distribution is
f(ý,O) = b(K)/27r exp(-Kcos 24) sinq
Bingham Distribution (Bingham, 1974; Mardia, 1972; Shanley and
Mahtab, 1975)
The Bingham distribution allows elliptical symmetry about the mean
direction of a joint set. The Bingham pdf is:
146
exp
(v
1
' x)
X[l2 +
( (2
'
X) 2 +
3 ( 3
'
x) 2 ]ds
47 I F1 (1/2; 3/2; Z)
Here Ci, C2, C3 are dispersion parameters and, using Kiraly's
terminology for p 3
' , and 2 , (Chapter III, this thesis)
is the best axis, or the mean of the distribution,
and is perpendicular to both pi and p2'
p is the axis of best zone,
3,
½2 is the axis of a plane containing V1 and
Z =
0
0
C2
P3'
0
0 C
and 1F, (1/2; 3/2; Z) is a hypergeometric function of matrix
argument. Bingham imposes the constraint r = 0 to render the
maximum likelihood estimate of Z unique.
When the data are concentrated about a preferred orientation, both cl and C2 will be negative. If ~ý 2 , then
C < C.2 and the contours of the distribution will1 be elliptical in shape. In this case, the axis ,P would be parallel
to the "minor axis" of the "ellipse," and the axis P2 would
be parallel to the "major axis" of the "ellipse." The axis
P3, that is, the best axis or the mean of the distribution,
would be perpendicular to both li and ½2. The parameters
C1 and C2 can be obtained by interpolating in tables presented by Bingham (1964) in which the C, are functions of
the w',where w, is the eigenvalue associated with pi.
R.S. Shanley and M. Mahtab. "FRACTAN: A Computer Code for Analysis of
Clusters Defined on a Unit Hemisphere," U.S. Bureau of Mines Information
Circular 8671, 1975.
147
APPENDIX C
Contoured Pole Diagrams of Orientation Data
148
GRID
LOWER HEMISPHERE PLOT
Poles from Top of Rock ,Site A
Figure C-I Contour Plot of 1009
149
CATA r7mm er5Ccm zF [ICqVRIk VT
MG:ECr JýiINTS PLOT
JINT ~RI[NTT:~I
OLXIN'UM ,
OY.3:00
C
*01IN!Mu"
0.001
COCMC.R
INCM
=
0.R,
INTEMVrL
-
N
LOdIEt
MwMtNEPE
PLOT
Figure C-2 - Contour Plot of 1620 Poles from Site A
150
iES CF CT4ER EXCAVATIONS
JaINT CRIENlqTah OATA Fr-C
MANIMUNM
SMINIMUM
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•11.q25
0.001
JOiIi
--LOT
CNE INCH -
0.25
CONTCUR INTERVAL -
2.000
LOJER NMEISPHERE PLOT
Figure C-3 - Contour Plot of 605 Poles from Site A
151
JUINT •lIENIATICe
IARXIMUM •
SMININUM
-
DATA FClM
S ID0ES
CNhTAINMENT EXCAVATIRlN
GEOLOLT JOIN1S PLCT
2.313
ONE INCHM
O.ZS
CONTe;.R INTERVAL -
0.001
2.000
N
LO.WER MtISPHE4E PLOT
Figure C-4 -' Contour Plot of 666 Poles from Site A
152
JOINT ORIENTATION DATR FRCO CREENE COUNkl TRENCH A
CEVLOCT JOINTS PLOT
MRAKIUMU
= '.i57
ONE INCH 0.25
6
IN!"UN -
0.001
CONTOUR INTERVAL -
2.OCC
LO0ER HEMISPHEqE PLOT
Figure C-5 - Contour Plot of 796 Poles
153
JOINT •M:ENTRTION
RXMU
04INIMUM *
RATRFR•M GREENE CJUNTY TRENCH 0
GEOLCr JOINTS PLCT
7.qZs
OhE INCH - 0.25
C.O00
CONTOUR INTERVAL -
2.0CC
LOAER HEMISPMERE PLOT
Figure C-6 - Contour Plot of 377 Poles
154
JOINT ORIENTATION ORTA FROM GREENE COUNIT TRENCH C
GEOLOCG JOINTS PLOT
RAXIMUN*
O.59.
ONE INCH 0.25
6H4ININUN -*
O.C
CONTOUR INTERVAL -
2.0CC
LOAER nEMISPHERE PLOT
Figure C-7 - Contour Plot of 103 Poles
155
JOINT ORIENTATION DATA FRCM GREENE CL~NTY TRENCH T
GEOLOGY JOINTS OLCT
MAXI|Um *
10.910
CNE INCH *
O.Z5
04INIMUM -
O.OC:
CONTOUR INTERVAL -
2.000
LOAER MEMISPHERE PLOT
Figure C-8 - Contour Plot of 165 Poles
156
•
JOINT ORIENT7TION ORTR FROM THE BLUE MILLS CIRCLES ONLY
GEOLOG JOINTS PLOT
.5258
3
ONE INCH *
0.25
• ARXIllU *
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*
0.001
CONTOUR INTERVAL -
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22N
H
E
2
P
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LO•EA HEMISPHMEAPLOT
Figure C-9 - Contour Plot of 435 Poles
157
JOINT
ARIENTATIEC
*
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Fiqure C-10 - Contour Plot of 266 Poles
158
APPENDIX D
Joint Length Distributions
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Fiqure D-2A - Plot of Length CFD Vs. Exponential Distribution for Site A, Bottom of Rock
CUMIILATIVE SAMPLE AID THEOIETICAL PDFS
******************e
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Figure D-2B - Plot of Length CDF Vs. Gamma Distribution for Site A, Bottom of Excavation
CUSULATIV
SANPLE AND THEOPETICAL PDFS
*
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Figure D-2C - Plot of Length CFD Vs. Lognormal Distribution for Site A, Bottom of Excavation
CUPULATIVF SAMPLE ANC THFORETICAL PCFS
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Finure D-3A - Plot of Length CFD Vs. Exponential Distribution for Site A, Sides of Excavation
CUPLLATIVE SAPPLE ANC THCCRETICAL PFUS
**
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Figure D-3B - Plot of Length CDF Vs. Gamma Distribution for Site A, Sides of Excavation
CLPLLAlIVE SAPPLE AND THEIRETICAL
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Figure D-3C - Plot of Lenqth CDF Vs. Lognormal Distribution for Site A, Sides of Excavation
*
0
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CUMULATIVE SAMPLE AND THIIEORTICAL PDFS
0.1000E*01 ************
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Figure D-4A - Plot of Lenqth CDF Vs.
Exponential Distribution for Site A, Sides of Excavation
•
•
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Figure D-4B - Plot of Length CDF Vs. Gamma Distribution for Site A, Sides of Excavation
CONULATIVE SANPLE AND THEORETICAL PDPS
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Figure D-4C - Plot of Lenqth CDF Vs. Lognormal Distribution for Site A, Sides of Containment
*
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9
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CUMULATIVF SAMPLE ASD THEOPETICAL PDFS
0.10031.01
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Figure D-5B - Plot of Length CDF Vs. Exponential Distribution for Greene Co., Trench A
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Figure D-7A - Plot of Length CDF Vs. Exponential Distribution for Greene Co., Trench C
SARPLE AND THEORETICAL PDFS
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Figure D-7B - Plot of Length CDF Vs. Gamma Distribution for Greene Co., Trench C
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Lognormal Distribution for Greene Co., Trench C
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Figure D-8A - Plot of Length CDF Vs. Gamma Distribution for Greene Co., Trench T
CUUOLATIVE SANPLE AND TFOIlTICAL PDFS
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Figure D-8B - Plot of Length CDF Vs. Lonnormal Distribution for Greene Co., Trench T
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Figure D-9A - Plot of Lenath CDF Vs. Exponential Distribution for Blue Hills
CLPULATIVE SAPPLF ANO THFORETICAL PCF5
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Figure D-9B - Plot of Lenqth CDF Vs. Gamma Distribution for Blue Hills
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Figure D-10OC - Plot of Length CDF Vs. Lognormal Distribution for Pine Hill
CUMULATIVE SAMPLE AND THEORTICAL PDIPS
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CUMULATIVE SANPLE AND THEORETICAL PDFS
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193
APPENDIX C
Joint Spacing Distribution- Simulation
CUMULATIVE SARPLE AND THEOHETICAL PDFS
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CUMULATIVE SARPLE AND THEORMTICAL PDFS
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Finure E-6B - Plot of Snacina CDF Vs.
4
Lonnormal Distribution for Site A - Too of Rock, Joint Set 1
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Figure E-7A - Plot of Spacing CDF Vs. Exponential Distribution for Site A
CUIULATIVE
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Finure E-7B - Plot of Soacinq CDF Vs. Lognormal Distribution for Site A - Top of Rock, Joint Set 1
COUULATIVE SAMPLE AND THEORETICAL PDFS
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CUMULATIVE SAMPLE AND TEOlITICAL PDFS
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Figure E-11A Plot of Spacing CDF vs Exponential Distribution for Site A-Bottom of Excavation, Joint Set 1
CUNOLATITV
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Figure E-22A - Plot of SDacina CDF Vs.
0.1221E*02
*
***
0.162E*02
.*****
0.2035E*02
VALUES
THEORETICAL PDF * 2
3 AND 4
Exponential
Distribution for Site A-Sides of Excavation, Joint Set 2
w
w
w
0
a
0
CUMULATIVE SAMPLE AND THECRETICAL PDFS
0.1000f+01
*
***********
**********
*444444444*4**44**
I 44
MM
1444
11
M
44
11 422
44
.
1 1 2
4 I
lit 22
44 I 1122
3
3 3
4
11122
333
4
1 22
333
44 1112
3 3
444
122
33
4
1M21 33
44
IM 333
44
IM 33
4
MM 331
444 112
3 I
44
12
33
44
M2
33 I*
*
*
0.900E400 *
*
*
0.O000E+000
*
*
0.7000E*00 *
*
*
Line Direction
P
R
0
B
A
8
*
*
*
*
0.5000EO00 +
*
L
*
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4
T 0.400oE.00 +
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0.3000[*00 +
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0.2000E*00 *
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0.1000E+00 44
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33
33
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4
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11
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211
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21
33
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33
44
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33
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33
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333
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33
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33
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13
SAMPLE
SAMPLE POF * I
*
*
*
1
I
I
1
I
*
*
*
1
I
I
I
*
*
*
I
I
I
*
*
***.*****************************,**S*******.
-0.2229E*00
-0.1841E*01
-0.3459E*01
*
I
I
I
VMMM3**3***3333**3333** 3******.********9**********.***
-0.50771E01
3
I
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I
44,
44
444
3
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I
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4 4 21
4
211
44 21
*
*
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33
2
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3
4 MM
3
4 MM
33
4 2M
3
44 211
33
44 211
33
44 211
33
44
21 33
44 2211
3
44
M
0.1395E*01
0.3013E*01
VALUES
THEORETICAL PDF * 2
CONFIDENCE BANDS * 3 AND 4
Finure E-22B - Plot of Spacing CDF Vs. Loqnormal
Distribution for Site.A-Sides of Excavation, Joint Set 2
0
•
CUMULATfIVE SAMPLE AND TIHEORETICAL PDFS
0.100I E*01
**44444·
M**
S4
1 II
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* 4 IM
N
* 4L22
0.9000o00 + 11
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3
3
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I
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********
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**
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3
3
3
33
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0
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0.8000E+CO 44M2
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*I2
Line Direction
70 @ S60E
0.7000E*00 #.2
*12
*M
*M2
0.6000E*00 *M2
4M2
4M
41M
0.5001E*00 4*
I
12
12
0.4000V#00 12
12
O.AOOOE0O0 M2
12
1M2
M2
0.3000EF00 M2
M2
M2
,
O. O00E*F0 m
0.?000E+00 M
0.
M
000E
-2+08f00
0.-002164E02
S
AMPL
SAMPLE
11*
SAMPLE POF * I
0.3246E+02
0•4327E*02
0.5409E#02
VALUES
VALUES
THEORETICAL POF * 2
CONFIOENCE BANDS * 3 AND 4
Figure E-23A - Plot of Spacing CDF Vs.
Exponential Distribution for Site A-Sides of Excavation, Joint Set 2
CUWULAIVE SAMPLE AND THEORETICAL POFS
0.1300E+31
****
***
*
***
***•• •*
I 44
*
144
4
11
4*
11
4441
1 2
4
I IM2
4
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44
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44
12 33
4
1122 3
44 112 1 3
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4
12 333
44 IM 31
44 I42 331
4 M 33 1
0.9000E+00
0
0.8000E400 0
*
0.7000E*00 *
:
Line Direction
70 @ S60E
O.6000E*CO
S44
*
*
*
0.5000E*00 .
*
*
*
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0.4000E00 *
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0.3000E*00
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0.2000E*CO *
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44
0.1000E400 44
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-0.36931E01
-01772E*01
0.1488E*00
SAMPLE
SAMPLE POF a I
1M
**
*
"
*
*
*
*
0
*
*
*
*******S*s*0*.
0.2070E+01
0.3991E*01
VALUE S
THEORETICAL POF * 2
CONFIDENCE BANDS * 3 AND 4
Figure E-23B - Plot of Spacing CDF Vs. Lognormal Distribution for Site-A-Sides of Excavation
Joint Set 2
CUMULATIVE SAMPLE AND THEORETICAL POFS
0.1000E.
01
•***4
* 4
*
4
M******M***
22Ml
MI
******
**********
*
*
4
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0.9000E.CO * 4112
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3133
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I
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0.O000E+00 *413M
*
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3
3
3
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0.6000E+CO
R
A
0
a
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5
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45 @ S60W
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4M
I
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L
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T
V
Line Direction
4M
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*
0.4000E+00 4M
I2
12
12
12
0.3000E00 M2
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a
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0.20000E00 M2
M
M
*
*
M
N
0.1000E+00 N
*
*
M
*
N
*
M
*
0.1831E-02
0.1155E+02
0.2309E*02
SAMPLE
SAMPLE POF *
I
0.3463E*02
0.4618E+02
0.5?772E*02
VALUES
THEORETICAL POF * 2
CONFIDENCE BANDS * 3 AND 4
Figure E-24A - Plot of Spacing CDF Vs. Exponential Distribution for Site A-Sides of Excavation, Joint Set 2
CUMULATIVE SAMPLE AND THEORETICAL PDfS
0.1000.01
I
I
*
*
I 44
*
I
144
*
44
0.9000E400 *
*
*
*
*
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0.7000E.O0
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0.6000E#00
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p
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0.4000E*CO *
*
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0
0
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0.2000E*00 *
*
9
*
4
4
0.1000ECO 4
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*
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M
N
•
Mn*.****+*9*N* *****
-0.6303E+01
2
*
1122
3
•
30333333
-0.4Z31+*O1
******
*
3•*
-O.I259E*01
S AMPLE
SAMPLE POF * I
VALUE
,
-::
1
1
I
I
I
I
I
I
I
1I
I
1
I
1
I
I
I
I
I
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13
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9
33
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44 211 33
44 211 33
3
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44211
3
3
44 21
33
44 211
4 211
33
221
33
442211 33
3
444 211
44
211L 33
33
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44
221
3
44
2 1
3
44
2111 3
4444
211
33
33
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4444
444
2MI
3
211
3
44
333
IMI
33
333
1 IMM2
0
0.0
44
45 @ S60W
0:
*
1
L
I
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y
Line Direction
0
0.5o0EO
M
11
1122
4
1122
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41 11122
3 333
41 11 27
33
411
2
333
3
4 1 2
4
1 2 33
4 112 333
44 12 33
4 1223
4 1223
44 M233
4 21 31
4 I 31
4 M133 I
4421 3 I
1
44211 3
44271
3 I
*
0.000E*00C
IlA
lMM
44
44
I
** ***************49
**
**
O.I•64E*01
-0.87?8E-01
*
***
0.4056E*01
S
THEORETICAL POF a 2
CONFIDENCE BANDS * 3 AND 4
Fiqure E-24B - Plot of SDacinq CDF Vs.
Loqnormal Distribution for Site A-Sides of Excavation, Joint Set 2
CUMULATIVE SAMPLE AND THEORETICAL PDFS
0.1303e.a* **e444*+*
**
4*
22
* 44
2 21 I1
*
4
MI I
*
4 11 2
0.9000E*CO * 411
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3
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O.OO00OEOO * 4132
* 1132
*
P
ne
11
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1
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********,****
4
1
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3
33
3
1
3
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132
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0.T000E*00 .413
*4432
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0.6000E*CO #132
0132
0
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0
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a
z*2
A
O.50OOE400 *M2
Line Direction
0
8
L
45 @ N60W
*1M2
,
,
*M2
*
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*
4M42
0.3300E*00
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0,
12
12
12
12
4
*
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0.2000E*00 M2
M2
M2
M
M
*
0.10000E*CO M
M
M
*
0.1984E-03
0.8451E+01
O.1690E*02
SAMPLE
SAMPLE POF *
CONFIDENCE
1
O.2535E*02
0.330E#02
0.4225?E*02
VALUES
THEORETICAL POF * 2
BANDS * 3 AND 4
Figure E-25A - Plot of Spacing CDF Vs. Exponential Distribution for Site A-Sides of Excavation, Joint Set 2
CUMULATIVE SAMPLI AND THIIEOTICAL PDFS
0.10001301
*
I
%
1
44
I
4
11
1
4
1
1 4
11 2
1 4
112
1*
44 1 22
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3
4
12
3
44
1N
33
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3
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4 21 33
*
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0.9000E.00
*
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0.8000O*00
*
*
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0.700oo0.oo
Line Direction
211
*
P
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0
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A
0.5000E*00 *
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0.4000E*00
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1
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-0.25691*01
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2
4422 113
0.60009100
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0.20001*00 .
*
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A
0.10009*00 *
*
*
*
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1 1
2
2
2 11 33
45 @ N60W
0.3000EO00
1
111
1 2
222
-0.10601*01
I AL
0.45013*00
0.1960E.01
S
TOIl02TICAL PDP * 2
3 AID
4
Figure E-25B - Plot of Spacinq CDF Vs. Lognormal Distribution for Site-A-Sides of Excavation, Joint Set 2
CUMULATIVE SAMPLE AND THEORETICAL PDFS
44
*
444
*
444
4o
*
*
0.9000E00 *
*
444
4
44
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44
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4
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3
3
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33
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0
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0
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41M3
+
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1
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43
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193
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4
0
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0
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0.7088E100
0.5188E-03
SIMPLE
SAMPLE PD
= I
0.2125E101
0.14178101
THEORETICAL PD? -
V ALU
0.2833E.01
0.3542E*01
S
2
CONPIDENCE BANDS * 3 AND 4
Figure E-26A Plot of Spacing CDF vs Exponential Distribution for Site A- Sides of Excavation, All Sets
CURULATIVR SAMPLE AND THEORIETICAL PDPS
*
0.10003.01 ******
*
*
*
*
****q*H
*
44
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4I
44
44
0.9000E*00 *
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Line Direction
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51
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S.**0***O* * ,*e***4l*q•9*Ol*l*M**33*33S*3333999*9*9S*9
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PLE
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1
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0
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***9****,9***O*0**
44
-0.5011E*00
0.1265i*01
VALUES
2
CONFIDENCE BANDS = 3 AND 4
Figure E-26B Plot of Spacing CDF vs Lognormal Distribution for Site'A-Sides of Excavation, All Sets
(.71
0
*
CUIMULATIVE SAMPLE AND THMEOIETICAL PDFS
0.1000Pn*0
.*,*,***********•****
*
*
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4
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AQA
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11
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4193
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AM
Line Direction
45 @ N60E
*
* 44M3
*
0.30001E0f
4
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0.2C000400
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493
0.10009100 4M3
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0.10582+01
0.27319-02
SAMPLE PDF
1I
0.21143201
0.31703*01
0.4226F#01
0.5281E*00
THEORETICAL PDF * 2
CONFIDEnCE BANDS a 3 AND A
Figure E-27A Plot of Spacing CDF vs Exponential Distribution for Site A-Sides of Excavation, All Sets
CUMULATIVE SAMPLE AND THEOILTICAL PDFS
O.1000oo01
rlo
*rO**e*e4*r4
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*
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-0.2876E*01
*
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33
1
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1
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1
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33*3333********************
-0.43qo9*01
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