Summary
Trace lengths of discontinuities observed on finite exposures are biased due to sampling errors. These errors should be corrected in estimating mean trace length. A technique, which takes into account the sampling errors, is proposed for estimating the mean trace length on infinite, vertical sections from the observations made on finite, rectangular, vertical exposures. The method is applicable to discontinuities whose orientation is described by a probability distribution function. The method requires that the numbers of discontinuities with both ends observed, one end observed, and both ends censored be known. The lengths of observed traces and the density function of trace length are not required. The derivation assumes that midpoints of traces are uniformly distributed in the vertical plane. Also independence between trace length and orientation is assumed. Data on a Pennsylvania shale in Ohio, U. S. A., were used as an example.
Access this article
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Similar content being viewed by others
Abbreviations
- α:
-
dip direction
- γ:
-
direction of sampling plane
- δ:
-
acute angle between dip direction and sampling plane
- θ:
-
dip angle
- θ A :
-
apparent dip angle
- λ:
-
mean density of trace mid-points per unit area
- μ:
-
mean trace length
- D :
-
diameter of discontinuity
- f (.),g (.):
-
probability density function
- h :
-
height of rectangular window
- \(\bar L\) :
-
estimator of mean trace length
- m :
-
sample size, number of discontinuities intersecting window
- m 0 :
-
number of discontinuities intersecting window with both ends censored
- m 2 :
-
number of discontinuities intersecting window with both ends observed
- n, N :
-
expected number of discontinuities intersecting the window
- n 0,N 0 :
-
expected number of discontinuities intersecting the window with both ends censored
- n 2,N 2 :
-
expected number of discontinuities intersecting the window with both ends observed
- Pr (.):
-
probability\(R_0 = \frac{{N_0 }}{N}, R_2 = \frac{{N_2 }}{N}\)
- w :
-
width of rectangular window
- x :
-
trace length
References
Baecher, G. B., Lanney, N. A., Einstein, H. H. (1977): Statistical Description of Rock Properties and Sampling. Proc. 18th U. S. Symp. on Rock Mech., Colorado, 5C1.1–5C1.8.
Baecher, G. B., Lanney, N. A. (1978): Trace Lengths Biases in Joint Surveys. Proc. 19th U. S. Symp. on Rock Mech., Nevada, 1, 56–65.
Bridges, M. C. (1976): Presentation of Fracture Data for Rock Mechanics. Proc. 2nd Australian—New Zealand Conf. on Geomech., 144–148.
Cruden, D. M. (1977): Describing the Size of Discontinuities. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.14, 133–137.
International Society for Rock Mechanics Commission on Standardization of Laboratory and Field Tests (1978): Suggested Methods for the Quantitative Description of Discontinuities in Rock Masses. Int. J. Rock Mech. Min. Sic. & Geomech. Abstr.15, 319–368.
Kendall, M. G., Moran, P. A. P. (1963): Geometrical Probability, Griffins Monograph Series. London: C. Griffin & Co., Ltd.
Kulatilake, P. H. S. W., Wu, T. H. (1984): Sampling Bias on Orientation of Discontinuities. Rock Mech. and Rock Engg.17, 243–254.
Mardia, K. W. (1975): Statistics of Directional Data. J. Royal Stat. Soc., Series B37 (no. 3), 349–371.
Pahl, P. H. (1981): Estimating the Mean Length of Discontinuity Traces. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.18, 221–228.
Priest, S. D., Hudson, J. A. (1981): Estimation of Discontinuity Spacing and Trace Length Using Scanline Surveys. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr.18, 183–197.
Robertson, A. M. (1970): The Interpretation of Geological Factors for Use in Slope Theory. In Planning Open Pit Mines. Van Rensburg, P. W. J. (Ed.). Cape Town, South Africa: Balkema.
Williams, R. L. (1982): Progressive Failure of Red Conemaugh Shale. Ph. D. Dissertation, Dept. of Civil Engg., Ohio State University, Columbus, Ohio.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kulatilake, P.H.S.W., Wu, T.H. Estimation of mean trace length of discontinuities. Rock Mech Rock Engng 17, 215–232 (1984). https://doi.org/10.1007/BF01032335
Issue Date:
DOI: https://doi.org/10.1007/BF01032335