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Note on a Weibull property for joint spacing analysis
Int. J. Rock Mech. ,Win. Sci. & Geomech. Abstr. Vol. 27, No, 2, pp. 133-134. 1990
0148-9062,90 $3.00 + 0.00 Copwight ~O 1990 Pergamon Press plc
Printed in Great Britain. All rights reserved
Technical Note Note on a Weibull Property for Joint Spacing Analysis W. E. B A R D S L E Y t T. J. MAJOR'{" M. J. SELBY#
parameter estimates into the relevant expressions. Maximum likelihood equations for the Weibull parameters are listed in [4]. This approach of using parameter estimates is more objective than, say, reading a modal value from a histogram. The Weibull modal value is given by:
INTRODUCTION
Probability distributions provide a convenient means of describing the relative frequency of discontinuity spacings recorded along a scanline. Distributions employed to date include the exponential [1], the normal [2], the Iognormal [3] and the Weibull [4]. Of the above distributions, only the exponential has theoretical support in that it arises under conditions of random location of discontinuities along a line. This random location implies a Poisson frequency of discontinuities per unit length, making subsequent analysis particularly simple. More flexible distributions are required to describe spacing sets which fail a test of exponentiality. Those more flexible alternatives should ideally include the exponential distribution as a special case. This arises from the fact that non-random discontinuity locations must represent the effect of some physical process. The exponential distribution must therefore arise as a limit case for situations where the effect of this process is only just evident. Of the standard "text book" distributions, only the gamma and Weibull include the exponential distribution as special cases. This paper is concerned with summarising some Weibull properties, with particular reference to making inferences concerning smallest distances between adjacent discontinuities.
GENERAL
Mo=O
M0 = ~t[(fl - l)/#l '/a for fl > 1.
(3)
E(X) = ~r(l + #-'),
var(X) = ~-'{r[l + ( 2 / # ) ] - r ~ ( l where E function. Unlike butions, inverted,
+ #-')},
(4)
denotes expectation and F is the gamma the normal, gamma and Iognormal distrithe Weibull distribution function can be giving: x = ~t[-ln(l -
F,)] j/~,
(5)
where Fx is some specified value of pr(X ~< x). The expression (5) is useful for simple calculation of the median and other quantile values. It is evident from the distribution function (1) that In(X) is a linear function of l n l n { [ l - p r ( X ~ < x ) ] - I } . This enables quick graphical tests of goodness of fit using a plotting position formula to obtain estimated probabilities of a smaller value from each of the data points ([5], p. 263). The average frequency of discontinuities per unit length is a necessary input for calculating RQD values. For non-exponential distributions of spacings, a problem sometimes arises in that the average frequency is not constant along the scanline. This difficulty really arises out of the "unit lengths" being perceived as consecutive. However, it will make little difference from an engineering viewpoint if the unit lengths are defined to be located randomly along the line. Given this definition, the mean frequency per unit length is simply the reciprocal of the mean of the spacing distribution. RQD values can thus be obtained directly from the Weibuli parameter values (or the parameter values of any other utilised spacing distribution).
The Weibull cumulative distribution function is given by:
(i)
where fl and ct are shape and scale parameters, respectively. The exponential distribution corresponds to # = I. A near-normal form results for fl near 3.6. As with most distributions, quantities such as the mean and mode can be obtained directly by substituting
t D e p a r t m e n t of Earth Sciences, University of Waikato, Hamilton, New Zealand. RMMS 27 2--E
(2)
The mean and variance are respectively:
PROPERTIES
pr(X ~< x) = I - exp[-(x/~t)P],
for # ~< i,
133
134
BARDSLEY et al.:
SMALLEST SPACINGS Closely-spaced discontinuities may lead to an increased risk of structural failure. It is thus of some practical value to make some inference concerning X'm,n. the smallest spacing along some length L. For example. L might be the length of a proposed tunnel through jointed rock. The nature of the Weibuli distribution means that it is particularly simple to obtain working expressions relating to the smallest probable spacing. This follows from the fact that the Weibull distribution is itself an extreme value distribution of smallest extremes. As with all extreme value distributions, the Weibull distribution reproduces itself with respect to sample extremes [5]. Explicitly, the smallest member of a Weibull sample of size N is itself a Weibull random variable. The shape parameter fl remains unchanged and the new scale parameter is simply 2' = a N - ~ ([5], p. 254). Substituting for ~ in (1) gives the equation: pr(X,,i, ~
The distribution functions (6) and (7) can both be inverted. In particular, the respective expressions for the median of the smallest spacings are given by: median X~,, = :~(0.693/N) I~ median Xm,, = --In
I
l
0.693] ~--:-:7:.,,/:c.
(8) (9)
CONCLUSION The expressions given here for smallest spacings should be of practical value in helping to anticipate rock-strength problems in engineering projects. Of course, these equations are only applicable to those situations where discontinuity spacings can be approximated as Weibull random variables. This implies carrying out a prior test of goodness of fit. Also, a check should be made that the spacings are behaving as independent random variables--that is, the magnitudes of consecutive spacings should be independent.
(6) Accepted for publication 31 October 1989.
In the present context, (6) must be applied as an approximation since N is unknown and must be estimated prior to the application of the formula. The accuracy of the equation will be better for larger values of L/E(X). An exact equation can be written for the special case of exponential spacings: pr(X,,i. ~
TECHNICAL NOTE
(7)
where E(N) is the expected number of discontinuities in the length L. The exact expression differs from (6) in that N is recognised as a (Poisson) random variable rather than a constant.
REFERENCES I. Priest S. D. and Hudson J. A., Discontinuity spacings in rock. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 16, 339-362 (1976). 2. Priest S. D. and Hudson J. A., Estimation of discontinuity spacing and trace length using scanline surveys. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 18, 183-187 (1981). 3. Sen Z. and Kazi A., Discontinuity spacing and RQD estimates from finite length scanlines. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 21, 203-212 (1984). 4. Rouleau A. and Gale J. E., Statistical characterization of the fracture system in the Stripa Granite, Sweden. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 22, 353-367 (1985). 5. Johnson N. L. and Kotz, S., Continuous Unit'eriate Distributions-1. Wiley, New York (1970).
0148-9062,90 $3.00 + 0.00 Copwight ~O 1990 Pergamon Press plc
Printed in Great Britain. All rights reserved
Technical Note Note on a Weibull Property for Joint Spacing Analysis W. E. B A R D S L E Y t T. J. MAJOR'{" M. J. SELBY#
parameter estimates into the relevant expressions. Maximum likelihood equations for the Weibull parameters are listed in [4]. This approach of using parameter estimates is more objective than, say, reading a modal value from a histogram. The Weibull modal value is given by:
INTRODUCTION
Probability distributions provide a convenient means of describing the relative frequency of discontinuity spacings recorded along a scanline. Distributions employed to date include the exponential [1], the normal [2], the Iognormal [3] and the Weibull [4]. Of the above distributions, only the exponential has theoretical support in that it arises under conditions of random location of discontinuities along a line. This random location implies a Poisson frequency of discontinuities per unit length, making subsequent analysis particularly simple. More flexible distributions are required to describe spacing sets which fail a test of exponentiality. Those more flexible alternatives should ideally include the exponential distribution as a special case. This arises from the fact that non-random discontinuity locations must represent the effect of some physical process. The exponential distribution must therefore arise as a limit case for situations where the effect of this process is only just evident. Of the standard "text book" distributions, only the gamma and Weibull include the exponential distribution as special cases. This paper is concerned with summarising some Weibull properties, with particular reference to making inferences concerning smallest distances between adjacent discontinuities.
GENERAL
Mo=O
M0 = ~t[(fl - l)/#l '/a for fl > 1.
(3)
E(X) = ~r(l + #-'),
var(X) = ~-'{r[l + ( 2 / # ) ] - r ~ ( l where E function. Unlike butions, inverted,
+ #-')},
(4)
denotes expectation and F is the gamma the normal, gamma and Iognormal distrithe Weibull distribution function can be giving: x = ~t[-ln(l -
F,)] j/~,
(5)
where Fx is some specified value of pr(X ~< x). The expression (5) is useful for simple calculation of the median and other quantile values. It is evident from the distribution function (1) that In(X) is a linear function of l n l n { [ l - p r ( X ~ < x ) ] - I } . This enables quick graphical tests of goodness of fit using a plotting position formula to obtain estimated probabilities of a smaller value from each of the data points ([5], p. 263). The average frequency of discontinuities per unit length is a necessary input for calculating RQD values. For non-exponential distributions of spacings, a problem sometimes arises in that the average frequency is not constant along the scanline. This difficulty really arises out of the "unit lengths" being perceived as consecutive. However, it will make little difference from an engineering viewpoint if the unit lengths are defined to be located randomly along the line. Given this definition, the mean frequency per unit length is simply the reciprocal of the mean of the spacing distribution. RQD values can thus be obtained directly from the Weibuli parameter values (or the parameter values of any other utilised spacing distribution).
The Weibull cumulative distribution function is given by:
(i)
where fl and ct are shape and scale parameters, respectively. The exponential distribution corresponds to # = I. A near-normal form results for fl near 3.6. As with most distributions, quantities such as the mean and mode can be obtained directly by substituting
t D e p a r t m e n t of Earth Sciences, University of Waikato, Hamilton, New Zealand. RMMS 27 2--E
(2)
The mean and variance are respectively:
PROPERTIES
pr(X ~< x) = I - exp[-(x/~t)P],
for # ~< i,
133
134
BARDSLEY et al.:
SMALLEST SPACINGS Closely-spaced discontinuities may lead to an increased risk of structural failure. It is thus of some practical value to make some inference concerning X'm,n. the smallest spacing along some length L. For example. L might be the length of a proposed tunnel through jointed rock. The nature of the Weibuli distribution means that it is particularly simple to obtain working expressions relating to the smallest probable spacing. This follows from the fact that the Weibull distribution is itself an extreme value distribution of smallest extremes. As with all extreme value distributions, the Weibull distribution reproduces itself with respect to sample extremes [5]. Explicitly, the smallest member of a Weibull sample of size N is itself a Weibull random variable. The shape parameter fl remains unchanged and the new scale parameter is simply 2' = a N - ~ ([5], p. 254). Substituting for ~ in (1) gives the equation: pr(X,,i, ~
The distribution functions (6) and (7) can both be inverted. In particular, the respective expressions for the median of the smallest spacings are given by: median X~,, = :~(0.693/N) I~ median Xm,, = --In
I
l
0.693] ~--:-:7:.,,/:c.
(8) (9)
CONCLUSION The expressions given here for smallest spacings should be of practical value in helping to anticipate rock-strength problems in engineering projects. Of course, these equations are only applicable to those situations where discontinuity spacings can be approximated as Weibull random variables. This implies carrying out a prior test of goodness of fit. Also, a check should be made that the spacings are behaving as independent random variables--that is, the magnitudes of consecutive spacings should be independent.
(6) Accepted for publication 31 October 1989.
In the present context, (6) must be applied as an approximation since N is unknown and must be estimated prior to the application of the formula. The accuracy of the equation will be better for larger values of L/E(X). An exact equation can be written for the special case of exponential spacings: pr(X,,i. ~
TECHNICAL NOTE
(7)
where E(N) is the expected number of discontinuities in the length L. The exact expression differs from (6) in that N is recognised as a (Poisson) random variable rather than a constant.
REFERENCES I. Priest S. D. and Hudson J. A., Discontinuity spacings in rock. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 16, 339-362 (1976). 2. Priest S. D. and Hudson J. A., Estimation of discontinuity spacing and trace length using scanline surveys. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 18, 183-187 (1981). 3. Sen Z. and Kazi A., Discontinuity spacing and RQD estimates from finite length scanlines. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 21, 203-212 (1984). 4. Rouleau A. and Gale J. E., Statistical characterization of the fracture system in the Stripa Granite, Sweden. Int. J. Rock Mech. Min. Sci. d Geomech. Abstr. 22, 353-367 (1985). 5. Johnson N. L. and Kotz, S., Continuous Unit'eriate Distributions-1. Wiley, New York (1970).