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Estimation of mean trace length of discontinuities

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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.

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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

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Authors and Affiliations

  1. Department of Mining and Geological Engineering, University of Arizona, Tucson, Arizona, USA

    P. H. S. W. Kulatilake

  2. Department of Civil Engineering, Ohio State University, Columbus, Ohio, USA

    T. H. Wu

Authors
  1. P. H. S. W. Kulatilake
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  2. T. H. Wu
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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

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