What’s the Difference: Fisher Distribution & Elliptical Fisher Distribution
The Fisher Distribution is the analog of the Normal (Gaussian) distribution on a sphere, and is theoretically justified when a sum of multiple, uncorrelated variations in rock properties and stress conditions cause the variation in fracture orientation. It is easy to fit and easy to explain. However, its major drawback is that it requires a symmetrical variation of orientation around the mean pole. The variability of strike is forced to be equal to the variability of dip, such that the variability of orientation plots as a circular cluster on an equal area stereonet. In practice, much of the fracture data we work with shows elliptically shaped clusters on stereonets, which indicates a difference in the variability of strike and dip. This variability can occur, for example, in a normal stress regime where Shmin and Shmax are similar; the local fracture normal is variable, even though the fractures are predominately vertical. The Elliptical Fisher (EF) Distribution implemented in FracMan 7.4 provides a powerful, flexible way to analyze and model fracture sets with different amounts of variability in fracture strike and dip that plot as an ellipse on an equal area stereonet.
The EF distribution has four parameters fit automatically by FracMan’s orientation module, Interactive Set Identification System (ISIS). The first two parameters are the same as for a Fisher distribution: the set mean pole (trend and plunge (Θ,Φ), and Fisher Concentration parameter (κ1). The direction in which orientation dispersion is greatest (trend and plunge [Θ,Φ]), and the ratio (κ2) of concentration in the directions of greatest-to-least orientation dispersion are two additional parameters used by EF Distribution.
To account for a distribution for fracture orientations where the variation of strike is greater than the variability of dip, FracMan includes several possibilities. Bivariate Normal Distribution can be used for very small (e.g., <±4º) variations in orientation, such that the Euclidean space bivariate normal distribution provides a tangent approximation. Bingham and Bivariate Fisher distributions can also be used. The Bingham Distribution is diagnostic of girdle distributions, such as those found by plotting data from multiple points in a fold structure. When data is found to fit a Bingham Distribution, users should consider using a local coordinate system to account for the variation across the fold.
Both types suffer from a lack of conveniently fitted parameters for representing nonsymmetrical orientation distributions.
This distribution works by transforming orientations—using the Eigenvalues and Eigenvectors of the orientation data—so they are symmetrical about the mean pole, with equal variation in all directions. When the orientations are transformed to be symmetrical about the mean pole, the Fisher distribution can be fit to obtain a value of the Fisher concentration parameter κ in the transformed space, and then fitting a Fisher distribution to the transformed data.
The probability density function of the Elliptical Fisher distribution is:
Where κ is the Fisher concentration parameter for the transformed data, θ is the mean pole trend (clockwise from north), φ is the mean pole plunge (down from horizontal), and R is the ellipse axial ratio.
When to use Elliptical Fisher distribution in FracMan
In general, use EF for fracture orientation data which plots as an ellipse on an equal area stereonet. For such data, also think about the geologic controls on the fracture system. A few important things to consider:
- What mechanisms formed the fractures and why would the data plot as an ellipse?
- Are the formations in question folded or faulted?
- Are there strong lithological or mechanical heterogeneities that need to be accounted for?
- Might fracture orientations be better modeled relative to a local coordinate system rather than using FracMan’s global coordinate system?
If you contour the data on a stereonet and observe a much larger variation in strike than dip—particularly when plotted in a local coordinate system like in the figure below—you will want to use the EF parameters from an ISIS analysis in FracMan to characterize the set.
 In FracMan 7.5, the Fisher “Dispersion” parameter κ, will be renamed as the Fisher “Concentration” parameter – since the higher the value of κ, the greater the concentration is.