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Estimation of confidence bounds for mean trace length of discontinuities using scanline surveys
Int. J. Rock Mech. Min. Sci. & Geomech, Abstr. Vol. 27. No. 3. pp. 207-212. 1990 Printed in Great Britain. All rights reser',ed
0148-9062,90 $3.00 + 0.00 Copyright ~ 1990 Pergamon Press pie
Technical Note Estimation of Confidence Bounds for Mean Trace Length of Discontinuities Using Scanline Surveys ZHANG XINGt LIAO G U O H U A t
INTRODUCTION It is generally recognized that the behaviour of a rock mass is strongly influenced by the discontinuities that are present in the rock mass. Quantitative descriptions o f discontinuities have been proposed by the International Society for Rock Mechanics [1]. Discontinuity geometry is mainly characterized by location, orientation, spacing and persistence. On 2-D exposures, the trace length is related to persistence so it is important to estimate the mean trace length of discontinuities. Sampling plans used for trace length measurements can be divided into three types. They are: (I) sampling the traces that intersect a line drawn on the exposure, known as a scanline survey; (2) sampling all the traces on an outcrop; and (3) sampling the traces on a finite-sized exposure made on the outcrop. In sampling for trace length with any one of the above three methods, sampling errors can occur due to the following biases: (a) Size bias--large joints have a greater probability of being sampled than small joints. This bias affects the results in two ways [2]. The first one is the difference between the actual frequency and the frequency on an infinite outcrop. The second one is the difference between the frequency of a sample on the infinite outcrop and that of a sample on a finite area of the outcrop. (b) Censoring bias--joints whose ends can be seen to provide only lower bound estimations of their lengths. (c) Truncation bias--joint lengths below some known cutoff length are not recorded.
than the confidence bounds for mean trace length of joints on a finite-size exposure. Making a point estimation is usually not enough, because an engineer often wants to know the precision or possible error of an estimation for the trace length of the joints. In addition, the results of point estimations for mean trace length of the joints are related to the size of the exposure on which the joints are sampled. Generally, the estimated mean trace length increases as the exposure size increases. Thus, no criterion may be used to determine the exposure size and the size of samples with a point estimation method. This note presents a method that can produce not only point estimation but also confidence bounds for mean trace length of the joints recorded on a finite-sized exposure. The method described in this paper follows the same basic steps of sampling plans for trace length measurement as that used by Priest and Hudson, and the same result for point estimations is obtained, although this method is different from their method. Finally, some examples are given to illustrate the use of the method to be presented. POINT ESTIMATION FOR MEAN TRACE LENGTH OF JOINTS
Priest and Hudson have shown that mean semi-trace length is half the mean complete trace length sampled using a scanline, so a point estimation for mean trace length can be made by taking the sampling bias of censored semi-trace length into account. Figure I illustrates the case where the size of the exposure and orientation of joints limit the maximum observable semi-trace length to some value C,,. If there Effects of truncation bias on data can be made are n joints intersecting the scanline on an exposure, negligible by choosing a cutoff length which is small there should be r joints (r <~n) with a semi-trace length compared to average trace length [2]. The remainder of less than C and their semi-trace lengths are L), the biases had been studied by Priest and Hudson [3], L_,. . . . . L, (where L, ~< L_, ~< • .. ~< L, and L, = C). The Kulatilake and Wu [4]. and some corrections for the remainder of the joints on the exposure, (n-r) joints, biases have been proposed in their papers. However. should be censored at C, i.e. they have a semi-trace their methods can only make the point estimation rather length beyond C. For a random variable with a continuous density function, a likelihood can be used to make a point tDepartment of Mining Engineering. Beijing University of Iron & estimation of the random variable. In order to obtain the Steel Technology. Beijing. People's Republic of China. 207 RMMS
27~--E
208
ZHANG XING and LIAO GUOHUA: TECHNICAL NOTE
likelihood of the semi-trace length observed on a finite size exposure, it is necessary to know the probability of the above observations. The probability that the termination of a joint Lr with a semi-trace length less than C lies in the range L, to L , + d L , is f ( L , ) d L , (i = 1,2 . . . . . r). The probability that the remaining (n - r) censored joints are observed is: (n - r)
Thus the likelihood of the semi-trace length is: ) dL,.
.. " f ( L , ) d L r ( n
-r)
f(L)dL.
S ( L , ) = Lt + Lz + • " • + L , + (n - r ) L , = L,r + (n - r ) L ,
ESTIMATION
L ( L J~) = [p exp(-/~L~) dLt 1[/~ e x p ( - ~ L : ) dL,
"'" [I~ e x p ( - ~ L , ) d L ] [ e x p ( - l ~ L , ) ] " - "
Let (3)
We can obtain:
r
- - [L, + L: + . .
• + L , + (n - r)L~] = O.
BOUNDS OF
FOR
JOINTS
In order to obtain a pair of bounds for mean semitrace length at ( i - ~) confidence level, i.e. to obtain a pair of statistics such that: P(~ ~
(2)
=0.
OF CONFIDENCE
MEAN TRACE L E N G T H
= i d e x p { - i L [ L ~ + L , + . . . + L,
? L , L ( L I~L)
(7)
L , = (Lt + L,. + . • • + L,)/r.
(1)
In the case w h e r e f ( L ) is a negative exponential form, the likelihood is:
+ (n - r)L,]}.
(6)
= L,r + (n - r ) C ,
The result of equation (5) is the same as that obtained by Priest and Hudson. This confirms the idea that the point estimation for mean trace length can be made by measuring the semi-trace length of joints on a finite sized exposure using a scanline.
f(L) dL.
L ( L I1~) = f ( L ~ ) d L , f ( L ,
where
(8)
We must seek a function of samples, with parameter it, which are collected on an exposure, and the distribution of the function without parameter y must be known. S ( L , ) in equation (6) is the function of the samples with some censored semi-trace lengths and is related to the number of these censored joints r. It may be rewritten as:
(4)
S ( L , ) = n ( L l - Lo) + (n -
Thus, the point estimation of parameter I~ is:
I ) ( L : - Lj)
+ . . . + ( n - r + I ) ( L , - L,_~)
r
= ~ (n-i+l)(L,-L~_~),
l~ = S ( L , )
(9)
t=l
where
I
L 0 ~ 0.
Then, let:
or
wi=(n-i+l)(L,-L,_l), 1
t"
(5)
• ConceaLed ///~LJpper Limit of exposure
•
.............. ._j_; A c t u a l semi - t r a c •TLengt Ch
/Vf
/
/
,~,
(/~,
/
/
/
Concealed
// /
,~ ,r , 4 / "
Scanline
i = 1 , 2 . . . . . r,
(11)
att
Concealed .
(10)
According to the distribution of order statistics, the random variables wt, w: . . . . . w, can also be shown as negative exponential distributions with parameter/~ and they are independent of each other. A negative exponential distribution is one of the gamma-distribution family F ( p , ~). In other words, a gamma-distribution becomes a negative exponential distribution when its first parameter p equates to unity. Thus, the distribution of the random variables w, ( i = 1 , 2 . . . . . r) is: w,~F(l,~),
o
i = 1 , 2 . . . . . r.
I
Fig. I. Diagrammatic representation of joint traces intersecting a scanline on a planar face of limited extent.
where the first parameter p = 1, and the second parameter 1
fl=-
IL
ZHANG XING and LIAO GUOHUA:
If x~, x,. ..... x, are independent random variables such that xi have a gamma-distribution
TECHNICAL NOTE
209
According to equations (15) and (16), we have:
P~_,,2(2r) <~2#S(L,)
~<;(:~,:(2r)] = l - ' ! .
From the transformation of the inequality in equation (17), we can obtain:
i = 1 , 2 . . . . . r, then
/'Xf_,.:(2r) ~< ;(2'2(2r)~ = I - ~t. P\ ~ ~<# 2 S ( L , ) ]
i=1
(18)
Thus the confidence bounds for parameter # at confidence level ( I - =) are:
has a gamma distribution
_. [-xf_:.(2r) X2,2(2r)1 ("-'"J=L ~s(-~,) ~j" '
S(L,) is:
Thus, the distribution of the random variable
S(L,).,, F(r, ~).
(12)
After S(L,) times 2/a, the distribution of 2pS(L,) can be obtained from the distribution of the function of S(L,): 2#S(L,) ~
(17)
r(r, ~).
In statistics, the gamma-distribution , F p, with P = ~ 7 and
(13)
Also, the confidence bounds for the mean semi-trace length of joints on a finite exposure are:
F
(L, /':.) = Lz;,2(2r--~--~-~,~ ), ;(~r)J"
T.-L_
-=# 2
is referred to as the Z-'-squared distribution with r degrees of freedom and is denoted by z2(r). Clearly, the random variable 2/~S(L,) is a Z2-squared distribution with 2r degrees of freedom, i.e. (14)
2/zS(L,) ~ ;(-'(2r).
X2-squared density is an often used distribution and its cumulative distribution function is given in many texts. For an arbitrary (0 < ~t < !), we can obtain two cumulative distribution functions of ;(-'-squared density, ;(2,:2(2r ) and ;(~_,,2~(2r) with two quantities ct/2 and ( 1 - ct/2), respectively (Fig. 2), such that:
P[2#S(L,) > ;(~2(2r)] = ct/2, P[211S(L,) > ;(~_,.2(2r)] = 1 - =/2.
(15) (16)
l
An engineer usually wants to know the accuracy of the confidence bounds estimated so a relative quantity 6 is constructed to show the idea: 6 =--
,,
(19)
2L
x 100%.
(21)
The basic steps of estimating confidence bounds for mean semi-trace length of joints on a finite-sized exposure using a scanline survey are as follows: 1. Recording the joints intersecting the scanline, the semi-trace length of r joints which are completely measured within censoring level C, then calculate the sum of all the semi-trace lengths of joints S(L,) using equation (6). 2. Choosing confidence level (i - ~ ) , obtaining the Z2,:2(2r) and Z~- =;2(2r) from any table of cumulative distribution functions of ;(2-squared density. 3. Calculating the mean L and the confidence bound L_, /~ for semi-trace lengths of joints using equations (5) and (20). 4. Calculating the relative accuracy of the confidence bonds estimated 6, using equation (21).
f(x) SOME CASE STUDIES
:_x
Xf-az212r) f(x)
o/2 O
~
)~Z/z( 2r )
.--I
Fig.2. Densityfunctionof Z:-squarcddistribution.
Some case studies are presented to provide a simple, practical demonstration of the theoretical ideas in the above sections. Detailed measurements of joint trace lengths were obtained by scanline sampling of two outcrops on the slope face in E-Kou Open Pit Mine, Shanxi Province, China. Locations l and 2 were placed in the outcrop of quartz-mica-schist. The observable semi-trace length was censored at 8 and 6.8 m, respectively. A total of 93 joints was intersected with the scanline at the first location and 122 at the second location. A total of 75 semi-trace lengths less than C were obtained at the first location and 115 at the second location. Histograms showing the distributions of semi-trace length, presented in Fig. 3, are in these particular cases
210
ZHANG XING and LIAO GUOHUA: 30
.••_•
20
#= 0.38 rn -1
10
8m
0
~ 3o
Location 2
~1
n = 127 r=115 ~.= 0.534 m -1
I
20
" - ~ \ ~
10
~
. I
I
. I
.
I
I
I
. I
~ i
[ --'n'---P--i-- --I
6.8m
0
Semi- trace length
Fig. 3. Histograms of sampled semi-trace length for locations I and 2.
of general negative exponential form. In the first case, the negative exponential form satisfies goodness-of-fit criteria at the 5% confidence level a n d at the ! % confidence level in the second case.
TECHNICAL NOTE
For each location, values of the mean trace length L and the confidence b o u n d s for the mean L_. E at three different confidence levels (1 - zc) were determined from the individual trace length m e a s u r e m e n t s for four different values of C. The resulting values are listed in Table 1 and plotted in Fig. 4 against C for both cases. The data of the other two cases were from [3], o b t a i n e d by Priest a n d H u d s o n from two locations in an O r d o v i c a n m u d s t o n e o u t c r o p a n d a C a m b r i a n sandstone outcrop. The results of estimating the mean a n d the confidence b o u n d s for trace length are listed in Table 2 a n d plotted in Fig. 5 against C for both cases. Moreover, in order to show the relation between the relative accuracy estimated a n d the censored level C, the relative accuracy 6 is, respectively, plotted in Figs 6 a n d 7 against C for all cases. Clearly, the accuracy of confidence b o u n d s estimated at the same confidence level (1 - ~) increases as the censored level C increases for all cases.
CONCLUSIONS M e a n trace length o f joints a n d its confidence b o u n d s can be estimated by measuring the semi-trace lengths of
Table I. Computation of mean and confidence bounds for locations I and 2 Location 1 n =93 C = L, (m) r/n r
2 0.43 40
L (m) L (m)
E (m)
(f (%)
= 0.4 = 0.1 = 0.01 = 0.4 = 0. I = 0.01 = 0.4 = 0.1 = 0.01
Location 2 n = 127
3.5 0.613 57
6 0.731 68
8 0.806 75
1.5 0.535 68
3 0.740 94
4.5 0.850 108
6 0.906 I 15
3.82
3.87
4.39
4.50
2.240
2.266
2.450
2.497
3.381 3.010 2.652 4.416 5.088 6.075
3.484 3.156 2.833 4.356 4.893 5.649
3.993 3.645 3.298 4.898 5.469 6.198
4.108 3.766 3.421 4.989 5.516 6.235
2.035 1.858 1.681 2.496 2.775 3.159
2.198 2.033 1.864 2.614 2.857 3.183
2.268 2.108 1.944 2.667 2.896 3.200
2.318 2.159 1.995 2.711 2.937 3.234
13.55 27.20 44.80
11.27 22.44 36.38
10.31 20.77 33.03
9.79 19.44 31.27
10.29 20.47 32.99
9.18 18.18 29.10
8.14 16.08 25.56
7.87 15.58 24.81
Quartz - m i c a - schist n=127
Location 2
Location I
8--
n=93
• Point estimation tx = = 0 . 0 1
~ ==0.10 o o =0.40
......
x
x
--~
x
x
.J
i 3
I
I 5
2
I
4
I
6
I
8
C(m)
Fig. 4. Variation of estimated confidence bounds at three different confidence levels (1 -:c) and mean for trace length with censoring level C for locations I and 2.
Z H A N G X I N G and LIAO G U O H U A :
T E C H N I C A L NOTE
211
Table 2. Computation of mean and confidence bounds for Ordovican mudstone and Cambrian sandstone Ordovican mudstone C = L, (m) r~ n r L (m)
. = 122
Cambrian sandstone
n = 113
I 0.648 79
2 0.803 98
3 0.869 106
4 0.926 113
I 0.735 83
2 0.876 99
3 0.929 105
4 0.956 108
0.950
1.078
1.189
1.221
0.683
0.781
0.833
0.867
t_ ( m )
~, = 0.4 :t = 0.1 = 0.01
0.869 0.798 0.727
0.993 0.920 0.845
1.100 1.022 0.941
1.132 1.054 0.973
0.626 0.579 0.528
0.721 0.668 0.613
0.770 0.716 0.659
0.802 0.745 0.687
E (m)
z = 0.4 = 0.1 = 0.01
1.050 1.158 1.306
1.177 1.284 1.427
1.295 1.408 1.557
1.326 1.437 1.584
0.753 0.828 0.930
0.854 0.930 1.033
0.908 0.988 1.092
0.943 1.025 1.131
(%)
= 0.4 :t = 0.1 :t = 0.01
9.53 18.95 30.47
8.53 16.88 26.99
8.20 16.23 25.90
7.94 15.68 25.02
9.30 18.23 29.43
Cambrian
Ordovican mudstone n=122
8.51 16.77 26.89
8.28 16.33 25.99
8.13 16.15 25.61
sandstone n=113
1.1 • Point estimation a == 0.01 x a = 0.10 oa = 0.40
~o x~
~
~ O ~ o
1.o
/
_1
..... : 0.6
o
: /.1
"~-
I
I
I
I
I
t
I
I
1
2
3
4
1
2
3
4
C(m)
Fig. 5. Variation of estimated confidence bounds at three different confidence levels (I - " , ) and mean for trace length with censoring level C. These data for mean are from [3].
Quartz-mica Location 1
n=93
-schist Location 2
40--
n= 127
60
"~x.......
a=O01
40
t~....~
a
= 0.01
ao
20 x
""
"--x
x~
""-...
a= 0.I0 x
x
a=0.10
20 o-
..o.-
= ='
0.40 o
I
I
I
I
2
4
6
8
0
I
3
I
5
C (m)
Fig. 6. Variation of estimated accuracy ~ at different confidence level (I - ' , ) with censoring level C for the results in Fig. 4.
212
Z H A N G XING and LIAO G U O H U A :
TECHNICAL NOTE
Ordovican m u d s t o n e
Cambrian sandstone n:122
~z~. ~
ao
n=113
40--
40
•
=
0.01
~
20
%
20 x~_~__
x
a =0,I0
o-
.o
a
~
=
a=O, lO
x__
~ x
o
a = 0.01
I
I
I
I
2
3
4
x
x
a = 0.40
0.40 --~
1
......
o--
o
-o--
-o,_
.o
I
I
I
I
1
2
3
4
CtmI Fig. 7. Variation of estimated accuracy 6 at different confidence levels (I - ~) with censoring level C for the results in Fig. 5.
the joints on a finite-sized exposure using scanline surveys. Through four case studies in both countries, China and Britain, it is shown that the accuracy of confidence bounds estimated at the same confidence level (I - ~ ) increases as the censored level C used in measuring these joints increases or the confidence level ( I - :¢) with the same accuracy 6 increases as the censored level C. Importantly, the method proposed in this paper provides a simple, practical means of choosing the censored level C according to the confidence level (I - :~) and the relative accuracy 6 when engineers makes the estimation for trace length of joints. Moreover. this method also confirms the idea, proposed by Priest and Hudson, that the point estimation for mean trace length of joints can be made by measuring the semi-trace lengths of the joints on a finite-sized exposure using a scanline.
Acknowledgements--The data in the work described in this paper were collected by the authors and Mr Yong Qing. The authors are grateful to Mr Yong Qing for his help in the work
Received for publication I0 October 1989.
REFERENCES I. International Society for Rock Mechanics Commission on Standardization of Laboratory and Field Tests, Suggested methods for the quantitative description of discontinuities in rock masses. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 15, 319-368 (1978). 2. Baecher G. B. and Lanney N. A., Trace length biases in joint surveys. Proc. 19th U.S. Syrup. on Rock Mechanics, Nevada, Vol. I, pp. 56-65 (1978). 3. Priest S. D. and Hudson J. A., Estimation of discontinuity spacing and trace length using scanline surveys. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 18, 183-197 (1981). 4. Kulatilake P. H. S. W. and Wu T. H., Estimation of mean trace length of discontinuities. Rock Mech. Rock Engng 17, 215-232 (1984). 5. Bickel P. J. and Doksum K. A., Mathematical Statistics, Basic Ideas and Selected Topics. Holden-Day, San Francisco (1977).
0148-9062,90 $3.00 + 0.00 Copyright ~ 1990 Pergamon Press pie
Technical Note Estimation of Confidence Bounds for Mean Trace Length of Discontinuities Using Scanline Surveys ZHANG XINGt LIAO G U O H U A t
INTRODUCTION It is generally recognized that the behaviour of a rock mass is strongly influenced by the discontinuities that are present in the rock mass. Quantitative descriptions o f discontinuities have been proposed by the International Society for Rock Mechanics [1]. Discontinuity geometry is mainly characterized by location, orientation, spacing and persistence. On 2-D exposures, the trace length is related to persistence so it is important to estimate the mean trace length of discontinuities. Sampling plans used for trace length measurements can be divided into three types. They are: (I) sampling the traces that intersect a line drawn on the exposure, known as a scanline survey; (2) sampling all the traces on an outcrop; and (3) sampling the traces on a finite-sized exposure made on the outcrop. In sampling for trace length with any one of the above three methods, sampling errors can occur due to the following biases: (a) Size bias--large joints have a greater probability of being sampled than small joints. This bias affects the results in two ways [2]. The first one is the difference between the actual frequency and the frequency on an infinite outcrop. The second one is the difference between the frequency of a sample on the infinite outcrop and that of a sample on a finite area of the outcrop. (b) Censoring bias--joints whose ends can be seen to provide only lower bound estimations of their lengths. (c) Truncation bias--joint lengths below some known cutoff length are not recorded.
than the confidence bounds for mean trace length of joints on a finite-size exposure. Making a point estimation is usually not enough, because an engineer often wants to know the precision or possible error of an estimation for the trace length of the joints. In addition, the results of point estimations for mean trace length of the joints are related to the size of the exposure on which the joints are sampled. Generally, the estimated mean trace length increases as the exposure size increases. Thus, no criterion may be used to determine the exposure size and the size of samples with a point estimation method. This note presents a method that can produce not only point estimation but also confidence bounds for mean trace length of the joints recorded on a finite-sized exposure. The method described in this paper follows the same basic steps of sampling plans for trace length measurement as that used by Priest and Hudson, and the same result for point estimations is obtained, although this method is different from their method. Finally, some examples are given to illustrate the use of the method to be presented. POINT ESTIMATION FOR MEAN TRACE LENGTH OF JOINTS
Priest and Hudson have shown that mean semi-trace length is half the mean complete trace length sampled using a scanline, so a point estimation for mean trace length can be made by taking the sampling bias of censored semi-trace length into account. Figure I illustrates the case where the size of the exposure and orientation of joints limit the maximum observable semi-trace length to some value C,,. If there Effects of truncation bias on data can be made are n joints intersecting the scanline on an exposure, negligible by choosing a cutoff length which is small there should be r joints (r <~n) with a semi-trace length compared to average trace length [2]. The remainder of less than C and their semi-trace lengths are L), the biases had been studied by Priest and Hudson [3], L_,. . . . . L, (where L, ~< L_, ~< • .. ~< L, and L, = C). The Kulatilake and Wu [4]. and some corrections for the remainder of the joints on the exposure, (n-r) joints, biases have been proposed in their papers. However. should be censored at C, i.e. they have a semi-trace their methods can only make the point estimation rather length beyond C. For a random variable with a continuous density function, a likelihood can be used to make a point tDepartment of Mining Engineering. Beijing University of Iron & estimation of the random variable. In order to obtain the Steel Technology. Beijing. People's Republic of China. 207 RMMS
27~--E
208
ZHANG XING and LIAO GUOHUA: TECHNICAL NOTE
likelihood of the semi-trace length observed on a finite size exposure, it is necessary to know the probability of the above observations. The probability that the termination of a joint Lr with a semi-trace length less than C lies in the range L, to L , + d L , is f ( L , ) d L , (i = 1,2 . . . . . r). The probability that the remaining (n - r) censored joints are observed is: (n - r)
Thus the likelihood of the semi-trace length is: ) dL,.
.. " f ( L , ) d L r ( n
-r)
f(L)dL.
S ( L , ) = Lt + Lz + • " • + L , + (n - r ) L , = L,r + (n - r ) L ,
ESTIMATION
L ( L J~) = [p exp(-/~L~) dLt 1[/~ e x p ( - ~ L : ) dL,
"'" [I~ e x p ( - ~ L , ) d L ] [ e x p ( - l ~ L , ) ] " - "
Let (3)
We can obtain:
r
- - [L, + L: + . .
• + L , + (n - r)L~] = O.
BOUNDS OF
FOR
JOINTS
In order to obtain a pair of bounds for mean semitrace length at ( i - ~) confidence level, i.e. to obtain a pair of statistics such that: P(~ ~
(2)
=0.
OF CONFIDENCE
MEAN TRACE L E N G T H
= i d e x p { - i L [ L ~ + L , + . . . + L,
? L , L ( L I~L)
(7)
L , = (Lt + L,. + . • • + L,)/r.
(1)
In the case w h e r e f ( L ) is a negative exponential form, the likelihood is:
+ (n - r)L,]}.
(6)
= L,r + (n - r ) C ,
The result of equation (5) is the same as that obtained by Priest and Hudson. This confirms the idea that the point estimation for mean trace length can be made by measuring the semi-trace length of joints on a finite sized exposure using a scanline.
f(L) dL.
L ( L I1~) = f ( L ~ ) d L , f ( L ,
where
(8)
We must seek a function of samples, with parameter it, which are collected on an exposure, and the distribution of the function without parameter y must be known. S ( L , ) in equation (6) is the function of the samples with some censored semi-trace lengths and is related to the number of these censored joints r. It may be rewritten as:
(4)
S ( L , ) = n ( L l - Lo) + (n -
Thus, the point estimation of parameter I~ is:
I ) ( L : - Lj)
+ . . . + ( n - r + I ) ( L , - L,_~)
r
= ~ (n-i+l)(L,-L~_~),
l~ = S ( L , )
(9)
t=l
where
I
L 0 ~ 0.
Then, let:
or
wi=(n-i+l)(L,-L,_l), 1
t"
(5)
• ConceaLed ///~LJpper Limit of exposure
•
.............. ._j_; A c t u a l semi - t r a c •TLengt Ch
/Vf
/
/
,~,
(/~,
/
/
/
Concealed
// /
,~ ,r , 4 / "
Scanline
i = 1 , 2 . . . . . r,
(11)
att
Concealed .
(10)
According to the distribution of order statistics, the random variables wt, w: . . . . . w, can also be shown as negative exponential distributions with parameter/~ and they are independent of each other. A negative exponential distribution is one of the gamma-distribution family F ( p , ~). In other words, a gamma-distribution becomes a negative exponential distribution when its first parameter p equates to unity. Thus, the distribution of the random variables w, ( i = 1 , 2 . . . . . r) is: w,~F(l,~),
o
i = 1 , 2 . . . . . r.
I
Fig. I. Diagrammatic representation of joint traces intersecting a scanline on a planar face of limited extent.
where the first parameter p = 1, and the second parameter 1
fl=-
IL
ZHANG XING and LIAO GUOHUA:
If x~, x,. ..... x, are independent random variables such that xi have a gamma-distribution
TECHNICAL NOTE
209
According to equations (15) and (16), we have:
P~_,,2(2r) <~2#S(L,)
~<;(:~,:(2r)] = l - ' ! .
From the transformation of the inequality in equation (17), we can obtain:
i = 1 , 2 . . . . . r, then
/'Xf_,.:(2r) ~< ;(2'2(2r)~ = I - ~t. P\ ~ ~<# 2 S ( L , ) ]
i=1
(18)
Thus the confidence bounds for parameter # at confidence level ( I - =) are:
has a gamma distribution
_. [-xf_:.(2r) X2,2(2r)1 ("-'"J=L ~s(-~,) ~j" '
S(L,) is:
Thus, the distribution of the random variable
S(L,).,, F(r, ~).
(12)
After S(L,) times 2/a, the distribution of 2pS(L,) can be obtained from the distribution of the function of S(L,): 2#S(L,) ~
(17)
r(r, ~).
In statistics, the gamma-distribution , F p, with P = ~ 7 and
(13)
Also, the confidence bounds for the mean semi-trace length of joints on a finite exposure are:
F
(L, /':.) = Lz;,2(2r--~--~-~,~ ), ;(~r)J"
T.-L_
-=# 2
is referred to as the Z-'-squared distribution with r degrees of freedom and is denoted by z2(r). Clearly, the random variable 2/~S(L,) is a Z2-squared distribution with 2r degrees of freedom, i.e. (14)
2/zS(L,) ~ ;(-'(2r).
X2-squared density is an often used distribution and its cumulative distribution function is given in many texts. For an arbitrary (0 < ~t < !), we can obtain two cumulative distribution functions of ;(-'-squared density, ;(2,:2(2r ) and ;(~_,,2~(2r) with two quantities ct/2 and ( 1 - ct/2), respectively (Fig. 2), such that:
P[2#S(L,) > ;(~2(2r)] = ct/2, P[211S(L,) > ;(~_,.2(2r)] = 1 - =/2.
(15) (16)
l
An engineer usually wants to know the accuracy of the confidence bounds estimated so a relative quantity 6 is constructed to show the idea: 6 =--
,,
(19)
2L
x 100%.
(21)
The basic steps of estimating confidence bounds for mean semi-trace length of joints on a finite-sized exposure using a scanline survey are as follows: 1. Recording the joints intersecting the scanline, the semi-trace length of r joints which are completely measured within censoring level C, then calculate the sum of all the semi-trace lengths of joints S(L,) using equation (6). 2. Choosing confidence level (i - ~ ) , obtaining the Z2,:2(2r) and Z~- =;2(2r) from any table of cumulative distribution functions of ;(2-squared density. 3. Calculating the mean L and the confidence bound L_, /~ for semi-trace lengths of joints using equations (5) and (20). 4. Calculating the relative accuracy of the confidence bonds estimated 6, using equation (21).
f(x) SOME CASE STUDIES
:_x
Xf-az212r) f(x)
o/2 O
~
)~Z/z( 2r )
.--I
Fig.2. Densityfunctionof Z:-squarcddistribution.
Some case studies are presented to provide a simple, practical demonstration of the theoretical ideas in the above sections. Detailed measurements of joint trace lengths were obtained by scanline sampling of two outcrops on the slope face in E-Kou Open Pit Mine, Shanxi Province, China. Locations l and 2 were placed in the outcrop of quartz-mica-schist. The observable semi-trace length was censored at 8 and 6.8 m, respectively. A total of 93 joints was intersected with the scanline at the first location and 122 at the second location. A total of 75 semi-trace lengths less than C were obtained at the first location and 115 at the second location. Histograms showing the distributions of semi-trace length, presented in Fig. 3, are in these particular cases
210
ZHANG XING and LIAO GUOHUA: 30
.••_•
20
#= 0.38 rn -1
10
8m
0
~ 3o
Location 2
~1
n = 127 r=115 ~.= 0.534 m -1
I
20
" - ~ \ ~
10
~
. I
I
. I
.
I
I
I
. I
~ i
[ --'n'---P--i-- --I
6.8m
0
Semi- trace length
Fig. 3. Histograms of sampled semi-trace length for locations I and 2.
of general negative exponential form. In the first case, the negative exponential form satisfies goodness-of-fit criteria at the 5% confidence level a n d at the ! % confidence level in the second case.
TECHNICAL NOTE
For each location, values of the mean trace length L and the confidence b o u n d s for the mean L_. E at three different confidence levels (1 - zc) were determined from the individual trace length m e a s u r e m e n t s for four different values of C. The resulting values are listed in Table 1 and plotted in Fig. 4 against C for both cases. The data of the other two cases were from [3], o b t a i n e d by Priest a n d H u d s o n from two locations in an O r d o v i c a n m u d s t o n e o u t c r o p a n d a C a m b r i a n sandstone outcrop. The results of estimating the mean a n d the confidence b o u n d s for trace length are listed in Table 2 a n d plotted in Fig. 5 against C for both cases. Moreover, in order to show the relation between the relative accuracy estimated a n d the censored level C, the relative accuracy 6 is, respectively, plotted in Figs 6 a n d 7 against C for all cases. Clearly, the accuracy of confidence b o u n d s estimated at the same confidence level (1 - ~) increases as the censored level C increases for all cases.
CONCLUSIONS M e a n trace length o f joints a n d its confidence b o u n d s can be estimated by measuring the semi-trace lengths of
Table I. Computation of mean and confidence bounds for locations I and 2 Location 1 n =93 C = L, (m) r/n r
2 0.43 40
L (m) L (m)
E (m)
(f (%)
= 0.4 = 0.1 = 0.01 = 0.4 = 0. I = 0.01 = 0.4 = 0.1 = 0.01
Location 2 n = 127
3.5 0.613 57
6 0.731 68
8 0.806 75
1.5 0.535 68
3 0.740 94
4.5 0.850 108
6 0.906 I 15
3.82
3.87
4.39
4.50
2.240
2.266
2.450
2.497
3.381 3.010 2.652 4.416 5.088 6.075
3.484 3.156 2.833 4.356 4.893 5.649
3.993 3.645 3.298 4.898 5.469 6.198
4.108 3.766 3.421 4.989 5.516 6.235
2.035 1.858 1.681 2.496 2.775 3.159
2.198 2.033 1.864 2.614 2.857 3.183
2.268 2.108 1.944 2.667 2.896 3.200
2.318 2.159 1.995 2.711 2.937 3.234
13.55 27.20 44.80
11.27 22.44 36.38
10.31 20.77 33.03
9.79 19.44 31.27
10.29 20.47 32.99
9.18 18.18 29.10
8.14 16.08 25.56
7.87 15.58 24.81
Quartz - m i c a - schist n=127
Location 2
Location I
8--
n=93
• Point estimation tx = = 0 . 0 1
~ ==0.10 o o =0.40
......
x
x
--~
x
x
.J
i 3
I
I 5
2
I
4
I
6
I
8
C(m)
Fig. 4. Variation of estimated confidence bounds at three different confidence levels (1 -:c) and mean for trace length with censoring level C for locations I and 2.
Z H A N G X I N G and LIAO G U O H U A :
T E C H N I C A L NOTE
211
Table 2. Computation of mean and confidence bounds for Ordovican mudstone and Cambrian sandstone Ordovican mudstone C = L, (m) r~ n r L (m)
. = 122
Cambrian sandstone
n = 113
I 0.648 79
2 0.803 98
3 0.869 106
4 0.926 113
I 0.735 83
2 0.876 99
3 0.929 105
4 0.956 108
0.950
1.078
1.189
1.221
0.683
0.781
0.833
0.867
t_ ( m )
~, = 0.4 :t = 0.1 = 0.01
0.869 0.798 0.727
0.993 0.920 0.845
1.100 1.022 0.941
1.132 1.054 0.973
0.626 0.579 0.528
0.721 0.668 0.613
0.770 0.716 0.659
0.802 0.745 0.687
E (m)
z = 0.4 = 0.1 = 0.01
1.050 1.158 1.306
1.177 1.284 1.427
1.295 1.408 1.557
1.326 1.437 1.584
0.753 0.828 0.930
0.854 0.930 1.033
0.908 0.988 1.092
0.943 1.025 1.131
(%)
= 0.4 :t = 0.1 :t = 0.01
9.53 18.95 30.47
8.53 16.88 26.99
8.20 16.23 25.90
7.94 15.68 25.02
9.30 18.23 29.43
Cambrian
Ordovican mudstone n=122
8.51 16.77 26.89
8.28 16.33 25.99
8.13 16.15 25.61
sandstone n=113
1.1 • Point estimation a == 0.01 x a = 0.10 oa = 0.40
~o x~
~
~ O ~ o
1.o
/
_1
..... : 0.6
o
: /.1
"~-
I
I
I
I
I
t
I
I
1
2
3
4
1
2
3
4
C(m)
Fig. 5. Variation of estimated confidence bounds at three different confidence levels (I - " , ) and mean for trace length with censoring level C. These data for mean are from [3].
Quartz-mica Location 1
n=93
-schist Location 2
40--
n= 127
60
"~x.......
a=O01
40
t~....~
a
= 0.01
ao
20 x
""
"--x
x~
""-...
a= 0.I0 x
x
a=0.10
20 o-
..o.-
= ='
0.40 o
I
I
I
I
2
4
6
8
0
I
3
I
5
C (m)
Fig. 6. Variation of estimated accuracy ~ at different confidence level (I - ' , ) with censoring level C for the results in Fig. 4.
212
Z H A N G XING and LIAO G U O H U A :
TECHNICAL NOTE
Ordovican m u d s t o n e
Cambrian sandstone n:122
~z~. ~
ao
n=113
40--
40
•
=
0.01
~
20
%
20 x~_~__
x
a =0,I0
o-
.o
a
~
=
a=O, lO
x__
~ x
o
a = 0.01
I
I
I
I
2
3
4
x
x
a = 0.40
0.40 --~
1
......
o--
o
-o--
-o,_
.o
I
I
I
I
1
2
3
4
CtmI Fig. 7. Variation of estimated accuracy 6 at different confidence levels (I - ~) with censoring level C for the results in Fig. 5.
the joints on a finite-sized exposure using scanline surveys. Through four case studies in both countries, China and Britain, it is shown that the accuracy of confidence bounds estimated at the same confidence level (I - ~ ) increases as the censored level C used in measuring these joints increases or the confidence level ( I - :¢) with the same accuracy 6 increases as the censored level C. Importantly, the method proposed in this paper provides a simple, practical means of choosing the censored level C according to the confidence level (I - :~) and the relative accuracy 6 when engineers makes the estimation for trace length of joints. Moreover. this method also confirms the idea, proposed by Priest and Hudson, that the point estimation for mean trace length of joints can be made by measuring the semi-trace lengths of the joints on a finite-sized exposure using a scanline.
Acknowledgements--The data in the work described in this paper were collected by the authors and Mr Yong Qing. The authors are grateful to Mr Yong Qing for his help in the work
Received for publication I0 October 1989.
REFERENCES I. International Society for Rock Mechanics Commission on Standardization of Laboratory and Field Tests, Suggested methods for the quantitative description of discontinuities in rock masses. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 15, 319-368 (1978). 2. Baecher G. B. and Lanney N. A., Trace length biases in joint surveys. Proc. 19th U.S. Syrup. on Rock Mechanics, Nevada, Vol. I, pp. 56-65 (1978). 3. Priest S. D. and Hudson J. A., Estimation of discontinuity spacing and trace length using scanline surveys. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 18, 183-197 (1981). 4. Kulatilake P. H. S. W. and Wu T. H., Estimation of mean trace length of discontinuities. Rock Mech. Rock Engng 17, 215-232 (1984). 5. Bickel P. J. and Doksum K. A., Mathematical Statistics, Basic Ideas and Selected Topics. Holden-Day, San Francisco (1977).