Stereological relationship between trace length and size distribution of elliptical discontinuities

L. Zhang, H. H. Einstein, W. S. Dershowitz

Research output: Contribution to journalArticlepeer-review

79 Scopus citations


This paper presents an approach for estimating the size distributions of elliptical discontinuities. This is an extension of approaches for estimating size distributions of circular discontinuities. By assuming an elliptical shape and changing the aspect ratio k of the ellipses (i.e. the ratio of the major to minor axes), many real discontinuities can be reasonably represented. For elliptical discontinuities, a general stereological relationship between distributions of trace length and distributions of discontinuity size (expressed by the major axis length, a, of the ellipse) is derived for area (or window) sampling. From this relationship, expressions are derived between the mean (μa) and standard deviation (σa) of the discontinuity size, a, and the mean (μl) and standard deviation (σl) of the trace length, l, respectively for log-normal, negative exponential and gamma distributions of discontinuity sizes. Using the derived expressions, the effect of sampling plane orientation on trace lengths is investigated. The results show that the sampling plane orientation has an important effect on both the mean and the standard deviation of trace lengths. Finally, the derived stereological relationship between the trace length distribution and the discontinuity size distribution was used in a simulation with the FracMan discrete fracture code. The results show that the approach for inferring the major axis orientation, aspect ratio and size distribution of discontinuities from observations on outcrops is satisfactory.

Original languageEnglish (US)
Pages (from-to)419-433
Number of pages15
Issue number6
StatePublished - Aug 2002


  • Geology
  • Rocks/rock mechanics
  • Sampling
  • Site investigation
  • Statistical analysis

ASJC Scopus subject areas

  • Geotechnical Engineering and Engineering Geology
  • Earth and Planetary Sciences (miscellaneous)


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