The Analysis of Two-Dimensional Orientation Data

Abstract

It is often useful to measure and analyze the orientations of various geological features-fabric elements, structural features, etc. Such measurements are more easily interpreted if they are summarized by statistics representing: (1) a preferred orientation direction, (2) the degree of preferred orientation, and (3) the probability that the preferred orientation is real and not merely due to chance. Application of conventional linear statistical methods, such as are used for grain-size analyses, presents some difficult problems, because orientation data are in the form of circular frequency distributions. A method of analysis which avoids these difficulties is discussed in this paper. A logical approach to the problem is to treat the distribution directly in its circular form rather than to divide it into a linear frequency distribution. This avoids the difficulty that the mean and standard deviation of such a distribution varies with the choice of origin or dividing point. A resultant vector is obtained for the circular distribution by treating each observation as a vector and summing components. The direction of this vector is the preferred orientation direction, and the vector magnitude is a measure of the dispersion, or degree of orientation. These descriptive statistics have the advantage that they are independent of any a priori reference direction or origin. An excellent correlation exists between vector mganitude and the standard deviation calculated around the vector direction. A test of the significance of the difference between an empirical orientation distribution and randomness is presented. Comparison shows that statistical significance is in many cases more easily attained with this test than with the chi-square and the F tests. Two empirical distributions with rather low vector magnitudes (high standard deviations), taken from sand-grain orientation studies, are tested for goodness of fit against several model distributions. The best fits appear to be the circular normal distribution and a distribution obtained by wrapping an unlimited linear normal curve around the center point of a polar co-ordinate plot.