The analysis of three-dimensional orientation data

Abstract

For many observed orientation patterns of three-dimensional geological data no statistical tests are available that define the probability with which a measured sample represents the parent population. Consequently, there is no accepted definition of an appropriate sample size and patterns can only be compared qualitatively by visual inspection. Empirical solutions to these problems are proposed and applied to quartz c-axis orientation patterns. An estimate of the distribution of orientations in a parent population is provided by an orientation diagram prepared by contouring a measured sample represented on the surface of a sphere using a counting circle 100/n% of the area of the projected hemisphere, where n is the sample size. Contouring the sample as it is accumulated allows identification of the sample size beyond which there is no significant change in the measured pattern. Data from similar-sized samples of computer-simulated random patterns provide outside estimates of the likely differences between the measured sample and the parent population. Pairs of patterns are compared using the mean absolute difference between the matrices used to prepare the contoured diagrams. A measure of their similarity is provided by the magnitude of this difference as a function of the departure of the two patterns from randomness, as indicated by the amount of empty space present. Characterization of the traditionally recognized types of quartz c-axis orientation patterns, using the parameters of the orientation patterns or their normalized eigenvectors, is successful only with point maximum, partial girdle, girdle, and random orientation patterns.