An elliptically symmetric fisher distribution and its application in statistics of discontinuities in rock mass

The Fisher distribution is recognized as the primary and fundamental probability model for analyzing the discontinuity orientation data in rock mass, which suggests that the orientation data displays a circular symmetry around their mean vector. However, the discontinuity orientation exhibits elliptical symmetry frequently, which cannot be characterized by Fisher distribution accurately. This paper thus proposes the elliptical Fisher distribution as a substitutive probability model specifically designed to handle orientation data with elliptical symmetry effectively. By transforming the Fisher distribution using generalized polar coordinates, the elliptical Fisher distribution facilitates a smooth transition between circular and elliptical symmetry. Serving as a general form of the Fisher distribution, the model retains the simplicity, interpretability, and ease of calculation of the Fisher distribution while expanding its applicability. Parameter estimation and data simulation algorithms for the model are presented, demonstrating its efficiency and versatility. Overall, the proposed elliptical Fisher distribution offers a promising framework for analyzing orientations with elliptical symmetry, providing a practical solution to the limitations of traditional elliptical models.