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Fisher Distribution

A Fisher Distribution is commonly used for modelling the distribution of 3-dimensional orientation vectors (Fisher, 1953), such as the distribution of joint orientations (pole vectors) on a sphere. A Fisher Distribution describes the angular distribution of orientations about a mean orientation vector and is symmetric about the mean. The probability density function can be expressed as:

probability density function
Eqn. 1


  • θ = angular deviation from the mean vector, in degrees
  • K = "Fisher constant" or dispersion factor

In RocSlope3, a Fisher Distribution can be used to define statistical joint orientation.

Fisher K

The Fisher K value describes the tightness or dispersion of an orientation cluster. A larger K value (e.g. 50) implies a tighter cluster, and a smaller K value (e.g., 20) implies a more dispersed cluster, as shown in the following figure.

effect of fisher K on samples
Effect of Fisher K on a Randomly-Generated Joint Set of 500 Samples

The Fisher K value can be estimated from Eqn. 2, for data sets with greater than approximately 30 vectors (poles) (Fisher, 1953).

fisher k equation
Eqn. 2


  • N = number of poles
  • R = magnitude of the resultant vector (i.e., the magnitude of the vector sum of all pole vectors in the set)

Standard Deviation

By analogy with a Normal Distribution, it can be shown that the standard deviation of a Fisher Distribution can be estimated from Eqn.3 (Butler, 1992).

fisher k standard deviation
Eqn. 3


  • θ = "angular standard deviation" or "angular dispersion" of the Fisher Distribution
  • K = Fisher constant
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