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Statistical Distributions Overview

If you are defining a joint network in RS2, parameters such as joint spacing, length, persistence and orientation, can be defined as random variables by assigning a statistical distribution to the variable(s). The following statistical distributions are available:

The type of statistical distribution, together with the distribution parameters (mean, standard deviation, minimum and maximum values), define a probability density function (PDF) for a random variable. A PDF describes the distribution of possible values that a random variable may assume, for a hypothetical, infinite set of observations of the variable.

In most cases, very limited data is available, on which to decide what statistical distribution and standard deviation to use. Therefore, the engineer must often rely on "best estimates", when defining the PDF for a random variable. Some suggestions are noted below.

Distribution Selection

A Normal Distribution is commonly used for statistical analysis in geotechnical engineering. When the true distribution of a variable is not known, a Normal Distribution is often assumed. By making a best estimate of the minimum and maximum values of the variable, a standard deviation can be estimated, as described in the Normal Distribution topic.

Other distributions may have the following applications.

  • Variables which can only have positive values (e.g. joint length, persistence), often have PDFs which are NOT well modeled by a Normal Distribution. For such variables it is often more appropriate to use a non-symmetric distribution such as Exponential or Lognormal, rather than a Normal distribution.
  • A Uniform Distribution can be useful if you wish to specify an equal probability of the variable taking on any value between the minimum and maximum values.
  • A Fisher distribution is applicable for joint orientation data, for the Baecher and Veneziano joint network models.

In general, the applicable statistical distributions will depend on the random variable and joint model you are considering.

Standard Deviation

The Standard Deviation of a random variable is a measure of the variance or scatter of the variable about the mean value. The larger the Standard Deviation, then the wider the range of values which the random variable may assume (within the limits of the Minimum and Maximum values). NOTE:

  • The Standard Deviation is applicable for Normal, Lognormal and Fisher distributions.
  • It is NOT APPLICABLE for Uniform or Exponential distributions. If you are using one of these distributions, then you will NOT be able to enter a Standard Deviation.
  • For tips on estimating values of Standard Deviation, see the Normal Distribution topic.

Minimum / Maximum Values

The Minimum and Maximum values define the allowable limits of a random variable. For the purposes of data input, the Minimum / Maximum values are specified as Relative quantities (i.e. as distances from the Mean), rather than as absolute values. This simplifies the data input and is much less prone to error.

During the analysis, the Relative Minimum and Maximum values are converted to the actual Minimum and Maximum values, when the statistical sampling is carried out for each random variable, i.e.

MINIMUM = MEAN – Relative MINIMUM

MAXIMUM = MEAN + Relative MAXIMUM

EXAMPLE: if the Mean Dip = 35, and the Relative Minimum = Relative Maximum = 10, then the actual Minimum = 25 degrees, and the actual Maximum = 45 degrees.

For each random variable, you must specify non-zero values for the Relative Minimum and the Relative Maximum.

If you are using a Normal distribution (or other distribution which is symmetric about the Mean), the Relative Minimum and Relative Maximum values will usually be equal. However, for non-symmetric distributions they do not necessarily have to be equal.

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