Discontinuity spacing and RQD estimates from finite length scanlines

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.

Vol. 21, No. 4, pp. 203-212, 1984

Printed in Great Britain. All rights reserved

0148-9062/8453.00+0.00 Copyright ~ 1984 Pergamon Press Ltd

Discontinuity Spacing and RQD Estimates from Finite Length Scanlines Z. SEN*

A. KAZIt

Distributions of discontinuity spacings along finite length scanlines are derived on the basis of negative exponential and log-normal probability density functions. The error percentages are presented for mean discontinuity spacings estimated from short scanlines. Procedures are presented allowing unbiased mean discontinuity spacing values to be obtained for negative exponential as well as log-normal distributions of spacings between discontinuities. Analytical expressions for the expectation and variance of the Rock Quality Designation (RQD ) have been derived for short scanlines, which are assumed to be samples of spacings from negative exponential or log-normal probability density functions. The validity of modelling discontinuity spacings with log-normal distributions has been demonstrated using field measurements obtained in Saudi Arabia.

INTRODUCTION

Discontinuities occur in the forms of fissures, bedding planes, joints or faults within any rock mass. Their presence strongly effects the mechanical and hydrogeological properties of a rock mass, in terms of its strength, deformability, stability, porosity, water storage capacity and transmissivity. These properties play a maior role in the design of civil works, as well as in the occurrence and movement of groundwater. General guidelines for the description of rock mass discontinuities have been presented by the Geological Society Engineering Group Working Party [1] and International Society for Rock Mechanics [2]. These descriptions, among other factors, lay emphasis on assessment of spacing between adjacent discontinuities. The spacing or frequency of discontinuities can be determined from analysis of scanline measurements and from other techniques such as drill core description and terrestrial photogrammetry. In any study of discontinuity spacings along a scanline, variables such as the number of discontinuities along the scanline, the intact lengths and the frequencies have to be accurately and precisely estimated for the prediction of rock mass behaviour [3-6]. One common property in these variables is that they are all random variables and therefore their quantification justifies the application of stochastic methods, i.e. probability and statistics, rather than deterministic approaches. For

instance, the probabilistic description of discontinuities in rock masses is necessary in computations of slope failure possibilities in rocks [7]. For engineering purposes, some descriptive parameters have been defined in terms of discontinuity spacings and numbers, such as the Rock Quality Designation (RQD) [8] and the Rock Mass Rating (RMR) [9]. Due to its relative simplicity, the RQD has been extensively employed in the classification of rocks for engineering purposes. Since RQD is a function of the discontinuity spacings and numbers, it is also a random variable and therefore it has its own distribution function. The estimation of RQD is based on scanlines readings in the field, which requires an effective sampling procedure to yield unbiased and consistent estimates. However, most often, one is confronted with small areas of outcrops, excavations or short boreholes, as a result of which the scanline readings reflect only the finite length properties. These properties are biased and cannot be representative for large rock masses [10]. Prior to determining any engineering design variable, these biases must be corrected in the discontinuity spacings as well as RQD. The purpose of this paper is to model the discontinuity spacings in short scanline measurements and hence to develop an unbiased estimation of RQD. DISCONTINUITY SPACINGS

* Associate Professor of Civil Engineering, Technical University of Istanbul, Taskisla, Taksim, Istanbul, Turkey (on leave to the King Abdulaziz Univm~ity, Faculty of Earth Sciences, Jeddah). t Proft.sor of Engineering Cmology, King Abdulaziz University, Faculty of Earth Sciences, P.O. Box 1744, Jeddah, Saudi Arabia.

Intersections of discontinuities with the scanline in any direction appear as a sequence of irregularly located points along the line. Discontinuity spacing corresponds to the distance between an adjacent pair of points 203

204

SEN and KAZI: SHORT SCANLINE RQD ESTIMATES

measured along the scanline. The spacing largely controis the size of individual blocks of intact rock, secondary porosity, rock mass permeability and seepage characteristics, types of failures and deformations. The total number of discontinuities and lengths of spacings constitute the population of these variables within the rock mass concerned. However, their appearance and measurement on a finite length scanline is one set of equally likely samples, referred to as a realization. In practice, the problem is to infer the population characteristics from the finite scanline measurements. Depending on the degree of irregularity of the discontinuity spacings, the distribution of intact lengths may assume normal, log-normal, exponential or some other type of distribution. Priest and Hudson [3,6, 14] have shown through their studies that for randomly distributed discontinuities, in the absence of a large predominance of evenly spaced discontinuities, the exponential distribution represents satisfactorily the intact lengths frequency of occurrence. They based their arguments on long scanlines where the effect of scanline length is ignored. However, in practice, one is frequently restricted to short lengths of borehole core, small excavations and outcrops; and the short length of scanline introduces some bias into the discontinuity spacings estimations. In general, let the frequency density function of a given discontinuity spacing, x, be denoted by f(x). For the population of discontinuity spacing, x varies from zero to infinity and hence

o.

e "

5

x w

~'=~-9-~E1.0

2

o

i

0

1.0

20

Mean discontinuity spacing (~--), m, ( population )

Fig. 1. Effectof ~ l i n e length, L, on the mean discontinuityspacing (negative exponential distribution). censored distribution become

1 E(x) - (1 - e -~L) [1 - (1 + 2L)e ->'L]

(5)

and V(x) =

fo

~ f ( x ) dx = 1,

(0~
(1)

In the case of a finite scanline length, L, the discontinuity spacings longer than L cannot be observed and therefore the distribution becomes censored, including only intact lengths from zero to L. The censored distribution function, f'(x), can be expressed in terms of the parent distribution as f'(x) = f(x)

zo

f(x) dx,

(0 ~
(2)

Equations (1) and (2) can be applied to any given density function for the intact lengths. For instance, for negative exponentially distributed intact lengths, the frequency density function is given as f ( x ) = 2 e -z',

(0~
(3)

where 2 i~ the frequency of discontinuities in the population i.e. along a theoretically infinite length of scanline. The distribution of intact lengths in a finite scanline length can be obtained from equation (2), as 2 f ' ( x ) = l - e -~Le-z~'

(0~
(4)

The expected value, E(x), and variance, V(x), of this

,[12

- 22(1 _ e_a. ) 1 - (1 + 2L)e -~L

}

(6)

respectively. For a very large scanline, these two last expressions approach 1/2 and 1/~. 2, respectively, which are the mean intact length and its variance for the exponential distribution given in equation (3). Figure 1 shows the relation between the population and finite length scanline mean discontinuity spacings. It is clear that for scanlines longer than approximately 20/2, the population value and scanlino estimate become almost equal. Otherwise, for short scanlines, the population mean discontinuity spacing is always greater than the sample mean discontinuity spacing, i.e. scaniine measurements yield underestimated mean discontinuity spacings. Therefore, they must be corrected to allow for this sampling bias prior to any analytical treatment. Especially for small scanlines with large values of mean discontinuity spacing, the assumption that the sample mean discontinuity spacing is equal to the population counterpart will be erroneous. For instance, assuming that the discontinuity spacings are negative exponentially distributed and the moan discontinuity spating is found to be 0.6 m in a scanlin¢ of length 2 m, then the corresponding population value can be obtained as 0.76 m from Fig. 1. In this example, the relative error of estimation of the population value is 21%. In any design

SEN and KAZI: SHORT SCANLINE RQD ESTIMATES

for engineering structures within or on the rock mass concerned, the mean discontinuity spacing in this example should be considered as 0.76 m. There is also, of course, the addition factor that in repeated measurements there is a greater degree of imprecision associated with estimates from short scanlines. Figure 1 gives also the upper limit of unbiased estimation regions for each scanline length. For instance, for L = 1.0 m, a mean discontinuity spacing less than 0.2 m corresponds to the population value without any significant bias correction. For a scanline length of 2 m this upper limit is about 0.35 m. In this connection, Fig. 1 can be used for deciding on the length of a scanline that will yield an almost unbiased estimation of a desired mean discontinuity spacing. The necessary scanline length to obtain unbiased estimates of a population mean discontinuity spacing value of, say, 1.2 m is 10 m. Any scanline length shorter than this amount will lead to significantly biased estimations. However, scanlines larger than 10 m will not provide any further information. The relative error percentage in the mean discontinuity spacing estimation can be defined as 0t= 100 {(l)/21-~2E(x) } .

(7)

The substitution of equation (5) into this expression leads to Gt=100 { 1

( 1 - 1 -"~ L ) [ 1 - - ( I + ~ . L ) e - ~ L ] t .

(8)

The graphical representation of equation (8) has been given in Fig. 2 which facilitates the decision on the scanline length for a specified error percentage. It is obvious that, in general, small mean discontinuity spacings only require short scanlines for adequate estimation.

205

In an actual field survey of discontinuities at any site, the following steps are recommended. (i) In order to have a preliminary estimate of the mean discontinuity spacings, measure discontinuities on five to six scanlines of a few metres length each. Find the mean discontinuity spacings and calculate the mean values. (ii) Estimate the population mean discontinuity spacings from Fig. 1 for the adopted scanline length. (iii) Assume an error percentage and find the necessary scanline length from Fig. 2 for adequate measurement and estimation of discontinuity properties. Recommended error percentages in practice are 5-7%. For the scanline length determined in this manner, even the variance in equation (6) will become close to its population value. The mean discontinuity spacing estimate and its error given by Priest and Hudson [6] do not take the scanline length into consideration but rather concentrate on the number of discontinuities. In their approach they have assumed the population mean and variance of discontinuity for defining the error bands. It has been indicated by Attewell and Farmer [1 l] and Bridges [12] on the basis of field evidence that the discontinuity spacings for stiff clay, clay shale and chalk sites tend to conform with log-normal distributions. If the discontinuity lengths, y, were log-normally distributed then x = In (y) would be normally distributed. The general form of a log-normal distribution is given by Benjamin and Cornell [13] as fly)

=

1

exp

y(2r0½%

{

-

in(y/~hy)]}, (O~
where flay and ay are the two parameters of the distribution, namely, the median and standard deviation, re-

100

80

u

g

A-'Q1

60 Or

~

4o

o

ne 2O

2

4

6

8 $canline

10 length

12

14

16

18

20

(rn)

Fig. 2. Relative error percentage in the estimated mean discontinuity spacing with scanline length (negative exponential distribution).

206

SEN and KAZI:

SHORT SCANLINE RQD ESTIMATES

spacings estimation from the log-normally distributed intact lengths in a rock mass can be obtained in a similar way to equation (7) as

spectively. The relations between these and other parameters are given as

Thy=

m,. e - ~

(10)

a~ = In (V~ + 1)

(11)

~

100

=

and the coefficient of skewness, y, is 7

1 , exp Py(2rO~O'y

ln(y/gny)

-2

,

(0~
(12)

where Py is the probability of the intact lengths being less than the scanline length, L. The expectation of intact lengths along a finite scanline can be evaluated as

E(y) = fay e-~'~ {F [In ( ~ rh')

O'y]}

×

a,

.

(14)

The scanline measurements, including the number of discontinuities and the intact lengths, can be combined through the definition of RQD so as to provide an objective quantity for the classification of rock mass. The scanline estimate, RQD, of the population RQD value has been proposed by Deere [8] as

(13)

RQD = 100

x*

(15)

i=l

where x* is the ith intact length equal to or greater than a threshold value, t, and n is the number of such intact lengths along the scanline of length, L metres. A theoretical RQD expression, i.e. the expectation of the sample RQD, has been developed by Priest and Hudson [14], particularly for a negative exponential distribution of spacing values. An entirely different methodology will be proposed herein as follows. An inspection of equation (15) reveals that RQD is a function of two random variables, namely, discontinuity number and spacings. In fact, RQD is expressed as a

S

(a)

o,

RQD ESTIMATION

where F [ . . . ] is the area under the standardized normal distribution from - o o to the value in the brackets. Figure 3a-c show the relation between the population and sample values of the mean discontinuity spacings for given scanline lengths. These are similar to Fig. 1 and are helpful for correcting the bias effects in the case of log-normally distributed intact lengths. It is clear that the bias amount increases with increasing standard deviation which implies that longer scanline lengths will be required to obtain unbiased estimates for larger standard deviations of intact lengths. The log-normal distribution gives the flexibility of allowing a different mean and standard deviation of the intact lengths. However, the negative exponential distribution does not give this opportunity because the mean and standard deviation have to be equal for its application. The percentage of relative error, ~ , in the mean discontinuity

I°

1 -- F L

Graphical representations of this equation are given in Fig. 4a--c. They provide useful tools in assessing the relative error percentages involved in the estimation of average discontinuity spacings for a given pair of finite scanline length and standard deviation of these spacings. Increase in standard deviation results in greater errors in estimations. It is therefore suggested to adopt rather longer scanlines, but restrictions in the field do not always allow such a choice.

= 3Vy + V~.

where my and Vy are the mean and coefficient of variation. The distribution of intact lengths in a finite length of scanline can be found from equation (2) as f(y)=

{ F'n"'m' 1t

,0,

//,o/

14

50

12

2O

7. 10 g m 18 c 16

2

g

"o

g 12

o)

0

I 02

I 04

I 06

I 0.8

I 1.0

A 12

I 14

I 1.6

I 18

2.0

[

I

Q2

Q4

Mean

I

I

06

08

discontinuity

I

1.0

[

[

I

12

14

1.6

spocinO

[

18

2.0 0

Q2

0.4

06

08

1.0

12

(--~-), m , ( p o p u l a t i o n )

Fig. 3. Effect of scanline length on the estimated mean discontinuity spacing (log-normal distribution).

1.4

16

1.B

20

~N and ~ I :

SHORTSCANLINERQD ESTIMATES

207

!00

80

60

40

20

0

g eo @

40

0

80

¢0

40

20

0

2

4

S

s $canli~

t0

s2

~4

!e;

~e

20

;enCth { m )

Fig~ 4~ Relative error ~¢¢ntag~ in the ~stimat~t mean discominaity spaeia8 with scanliae lcagth (~og-normatdiaribmion)~

summation of random variables, The expectation of RQD can be achieved after two stages. Firstly, the conditional expectation of RQD given the number of discontinuities is

tinuiti¢s per metre~ and whatever the distribution function of the spacings is, it is equal to the reciprocal of the mean discontinuity spacing, E(x). Therefore, equation (16) can be rewritten as

E(RQi3/n) = Ioo I n E(x*)

E(RQD) = I00

and subsequently the expectation of RQD becomes E(RQD) = ~ 100 t E(n)E(x*)

E(x*)

(I7)

By similar reasoning, one can find the second order moment of RQD as

(16)

where E(n)/L is equivalent to the number of discon-

E(R~)

=~

(I00)2~ [V(x*)E(n) + E2(x*)E(n~)]

208

SEN and KAZI:

SHORT SCANLINE R Q D ESTIMATES

and hence the variance of RQD can be obtained as V(RQD = E(RQD 2) - E: (RQD) -

(100) 2 [V(x*)E(n) + E2(x*)V(n)]. L

(18)

An implicit assumption in these derivations is that the occurrence of successive intact lengths are independent from each other. It is clear from equations (16) and (18) that the expectation and variance of RQD can be found provided that the same statistical properties are evaluated for the discontinuity number and intact lengths. So far, these equations are general and do not depend on any specific distribution. There exist various alternatives for the implementation of equation (17) such as (i) E(x) in the denominator can be assumed to be equal to the population value, and E(x*) to the expectation of an incomplete distribution of the intact lengths. This is tantamount to considering long scanlines as adopted by Priest and Hudson [3]. However, for short scanlines this procedure will yield erroneous results. (ii) E(x) can be taken as the short scanline sample mean of the intact lengths with the same E(x*) as in the previous alternative. (iii) E(x) can be taken as the population average and E(x*) as the expectation of a complete distribution function with the upper boundary being equal to the scanline length. Such a censored distribution can be obtained from the parent distribution through equation (2).

E(RQD) = 100(1 + 0.1).)e .... ~

(22)

V(RQD) = [100(1 + 0.12)e-°1'] 2

(23)

and

respectively. In the case of the second derivation, the expected numbers of points occurring along the portion of the scanline forming the basis for the RQD definition can be written as (L - 02. Hence, from equations (16) and (19) one can find E(RQD) = 1 0 0 ( 1 - L ) ( I + :~t)e-~-'

(24)

which for long scanlines converges to equation (20). Hence an explicit effect of the scanline length on the RQD estimation is given by this equation. For instance, in a scanline of length l m and with t = 0 . 1 m, the E(RQD) values will be reduced by 10~o compared to the long scanline value. In the third derivation the probability density function for the intact lengths longer than t can be found from equation (2). For a negative exponential distribution both truncated and censored, one at the t and the other at the L level, 2 f(x*)= e_~,_e_~Le-~-e,

(t~
(25)

The expectation and variance can be evaluated as

Negative exponential distribution If the rock mass is homogeneous, in the sense that the averages of the discontinuity number and spacings do not change with location within the rock domain, and if clustered as well as evenly spaced discontinuities are absent, then the discontinuities occur according to a Poisson generating mechanism, which gives rise to negative exponentially distributed spacings. The aforementioned first alternative expectations will be E ( x ) = 1/)` and E(x*)=)` ft ~ e- ,~td t = ~1( l + 2 t ) e

scanline. Since in the conventional RQD definitions, the intact lengths greater than 0.1 m are considered, equations (20) and (21) become

-at.

(19)

1

E(x*) =

)`(e-;.t _ e-aL) x [(1 + 2t)e -at -- (1 + )`L)e -aL] (26)

and V(x*)=(e_a ' -e_aL )

_

L2+

), +

t2 + 2 ~ +

e -;.g

e -A'

_ 22(e_:. t -

[(1 + 2t)e -at - (1 + )`L)e-aL] 2

e-,~L)2

(27)

The substitution of this value into equation (17) leads to E(RQD) = 100(1 + )`t)e -at.

(20)

This expression has been originally derived by Priest and Hudson [14]. Also, the variance of RQD can be found as

respectively. It is clear that for t--)0 and L - ) o o these become equal to 1/)` and 1/), 2, respectively. By taking the average number of discontinuities, as in the first derivation, and employing equation (19), we obtain

(21)

100 (RQD) = e_~t - e_,~L [(1 + 2t)e -at - (1 + )`L)]e -~L. (28)

Surprisingly, the variance appears as the square of the expectation of RQD. Equations (20) and (21) offer a means of estimating rock quality parameters simply by counting the number of discontinuities along a known

For very long scanlines, this equation yields a relatively linear relation. Figure 5a-d show the graphical representation of equation (28) for finite scanline lengths of

V(RQD) = [100(1 + )`t)e-at] 2.

SEN and KAZI: SHORT SCANLINE RQD ESTIMATES

,oo

T

209

o.'o......

80

60

40

20

A r.~ O n,IM

0 100 0 ~~

o

I

L=l"Sm

I

80

60

4'0•,.

40

20

0

I

I

0.2

0.4

I

0.6

I

0.8

I

I

1.0

1.2

Mean

I 0.4

I

1.4

discontinuity

0 spacing

0.2

I

[ --~ ][m

I 0.6

I 0.8

I 1.0

I 1.2

1.4

]

Fig. 5. Variation of RQD expectationwith mean discontinuityspacing for a given scanline length (negativeexponential distribution). 1.0, 1.5, 2.0 and 3.0 metres, respectively. It is clear that, as the scanline length increases, the RQD expectation also increases, especially for large mean discontinuity spacings. Eventually, as L tends to infinity (very long scanlines) the curves becomes asymptotical to and RQD value of 100% as obtained by Priest and Hudson [3]. Decreases in RQD in Fig. 5, after the peak value, are due to the bias effects already mentioned in the previous section.

or simply E(x*) = ri%.e½"~{ 1 -

E(x*) =

exp

--~

dy (31)

(32)

The substitution of equation (32) into equation (17) by considering from equation (10) that rhy = E(x) = my e½°.,' leads to

r'n' m' L ~>.

Log-normal distribution

The negative exponential distribution has been extensively employed in discontinuity spacing descriptions and the associated analytical expressions have been derived [3-6]. Although these would help to speed up the RQD estimation and eliminate the tedium of measurement, the value of the operation is dependent upon the discontinuity spacings following a distribution of negativ~ exponential form. The log-normal distribution function provides a potential alternative and has been claimed to account for some discontinuity spacings other than those effectively modelled as having random locations along a scanline [11]. The first of the proposed alternatives for the RQD evaluation requires the expectation of intact length as

F[! n(t/--rn')

°1t

This relation is plotted in Fig. 6a-c for different standard deviation values. These figures show that with the lognormal distribution, the conventional RQD is sensitive for mean discontinuity spacings between zero and almost 0.3 m for small standard deviations. However, large standard deviations, as in Fig. 6c, cause a reduction in this region, for instance, down to 0.15 m with ay = 2. In general, the sensitivity region increases by increasing the threshold value. The relation between RQD and the number of discontinuities per metre can be found from equation (33) by considering equation (10) and 2 = 1~my as, E(RQ[)) =

lO0 II -- F Fln(t e~'~)

210

SEN and KAZI: SHORT SCANLINE RQD ESTIMATES 100

so -'~ 8o

o~

~_

o

o

" ~ -

40 20

(o)

i i I I I i -.----------'--

100~ 80 "°~o~ o ~ so 40

w

(b)

20

O

I

100 "

F

6o 480 0

I

I

~

I

I

I

l

~

.,o ~

/

/

I

''"

~

~

-

~

I

I

(c) 20

0

0.2

04

06

O8

Mean

1.0

1.2

I

I

I

1.4

1.6

18

discontinuity

I

2.0

I

I

1

I

2.2

2.4

Z6

2.8

30

spacing (-1~x ) , rn

Fig. 6. Variation of RQD expectation with mean discontinuity spacing (log-normal distribution).

The relation given in equation (34) is also plotted in Fig. 7a-c. For the specific case of 10 discontinuities per metre in the scanline and with standard deviations ranging from 0.5 to 2 m, the conventional RQD increases from 50 to 94~ which, according to the classification of Deere [8], the rock quality changes from poor to excellent. It is, therefore, interesting to note that, on the average, increase in the standard deviation of discontinuity spacing indicates better rock quality. Two scanline measurements can be compared on the basis of the mean and standard deviation of discontinuity spacings. If they have the same mean discontinuity value, then the one with greater standard deviation will indicate better quality rock.

FIELD MEASUREMENTS Field studies were carried out in an area (20°35'N, 4l°lS'E), l l 0 k m southeast of the city of Taif, Saudi Arabia. The area mainly consists of a succession of different kinds of schists. These rocks, forming fairly

steep slopes, arc nicely exposed along the course of Wadi Shauqab. The stepped nature of outcrops, resulting from the presence of orthogonal sets of joints contained in the rock, provided an excellent opportunity for making discontinuity spacing measurements on some selected scanlines. For the sake of uniformity, all measurements were made on one rock type, namely, the quartzfeldspathic schist. The rock is cut by a number of joints. A contour diagram of the poles of these joints is shown in Fig. 8. It can be sccn that there are three sets of joints. Sets II and III are steep, striking in NE-SW and NW-SE directions respectively, whereas set I is generally characterized by low amounts of southwesterly and northeasterly dips. Separate scanlines were used for different sets of joints and the orientation of the scanlinc in each case was determined such that it was normal to the strike of the selected set of subparallet joints. It was seldom possible to establish long scanlines on natural outcrops. In many instances, the length of

100 O"y • 20

80

~ 6o

~ 4o w

Cry =1.0

ZO

4

e

12

16

20

24

28

32

I 36

J 40

i 44

J 48

J 5;)

I 516 CoO

Average number of discontinuities

Fig. 7. Variation of RQD expectation with average number of discontinuities (log-normal distribution).

SEN

KAZI:

and

211

SHORT SCANLINE RQD ESTIMATES N

N

1I

rrr

Fig. 10. Orientation of composite scanlines.

< 2%

2-4%

4-6%

6-8%

BB

m 10-12%

8-10%

12-14%

14-16%

> 16%

Fig. 8. Contour diagram of joint poles (lower hemisphere equal area projection).

scanline was generally limited by the size of outcrop and more often the greater portion of the rock face was covered with fallen rock debris. However, meaningful results could be obtained by slightly shifting the position of scanline up or clown the face along the traces of a selected set, without affecting its orientation. For every set of joints, five scanlines, each containing at least 20 discontinuities, were established. The total number of discontinuities obtained from all the lines was counted and the distances between adjacent pairs

1::F

& O

Se) I Set "n-

o

Set 'n'r

CONCLUSIONS

1.0 >, 5 ._= 0.5

8 o

0.1

recorded. The latter were then distributed in classes and the number of intact lengths corresponding to each class calculated. Finally, the probability of occurrence of a given length equal to or greater than a certain value was estimated. Figure 9 corresponding to sets I, II and III shows the probability distribution of intact lengths (log scale). In this figure, the degree to which the plotted field data pertaining to a given set lies on a straight line illustrates its closeness of fit to a log-normal distribution. Additional measurements were carried out on two more scanlines to determine if the distribution of intact lengths between pairs of discontinuities, irrespective of the sets they represented, also followed a log-normal form. The two scanlines (A and B) were oriented in the NE-SW and NW-SE directions. In order to differentiate these scanlines from those described earlier, they will be referred to as composite scanlines. Figure 10 shows the dips and strikes of rock faces on which the composite scanlines were established. Each of these scanlines ineluded 40-50 discontinuities. After conducting the necessary measurements, the intact lengths were grouped in classes and their probabilities determined. Figure 11 shows that once again the distribution of discontinuity spacing values, along composite scanlines, can also be approximated by a log-normal form.

99 9 5 9 O

5O Probability

10

5

%

Fig. 9. Probability distribution functions of intact lengths.

The distributions of discontinuity spacings along straight lines have been modelled by negative exponential and log-normal probability density functions. It has been shown analytically that finite length scanlines yield biased estimations of the population value of mean discontinuity spacing as well as the Rock Quality Designation (RQD). The amount of bias, however, decreases as the length of scanline increases. Furthermore, the length of scanline required for essentially unbiased estimates depends on the value of mean discontinuity

lo / x

212

SEN and KAZI:

SHORT SCANLINE RQD ESTIMATES

been verified for some field measurements obtained in the Kingdom of Saudi Arabia.

05

Received 8 July t983: revised 30 November 1983.

O1 o

/x

/

x/ REFERENCES

005

t~

mD O01

I I

I

l

99 9 5 9 0

I

I

I

50

10

5

Probability %

Fig. 11. Probability distribution function of intact lengths on composite scanlines.

spacing. Large discontinuity spacings require longer scanlines for unbiased estimates. The log-normal probability density function provides flexibility in the sense that, in addition to the average discontinuity spacing value, the variance of the discontinuity spacings is also taken into consideration in any scanline evaluations. Increase in the standard deviation gives rise to greater errors in estimations. The mean and variance of the RQD have been explicitly derived for the two distributions. It has been observed that in the case of negative exponentially distributed values, the mean and standard deviation of the RQD are the same, which sugg¢sts that the RQD distribution may also Ix of negative exponential type. The applicability of the log-normal distribution has

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