USE OF THE MULTINOMIAL FUNCTION TO MAP VARIATIONS IN THE DEGREE OF ORGANIZATION IN FRACTURE SYSTEMS

Polygonal fracture arrays are common (e.g. mudcracks, columnar jointing, polygonal fault systems, and some vein networks). Generated by internal processes such as dessication, cooling, and syneresis, the fracture strikes are expected to have a random distribution. This expectation can be tested using a Chi square test with a sufficient number of data points. In stratiform chalcedony vein arrays in the South Dakota Badlands and in published images of polygonal fault systems, however, fracture patterns that test as statistically random appear to be organized and non-random in portions. Thus, the fracture pattern consists of a mosaic of organized subareas that constitute a larger array without preferred directions. A low fracture n prevents the Chi-square test from being used for the subareas. However,the use of the multinomial probability function can provide a local measure of the degree of organization , allowing visualization of the mosaic pattern. The multinomial function is similar to the binomial function, but instead of looking at two possibilities or bins, the multinomial function can look at any number of bins. The approach to characterize the local degree of organization is to compute the probability that the observed strike distribution (in six, 30-degree bins) of the nearest n fractures would occur randomly. The assumption is that the smaller the probability the more organized the fractures are locally. The multinomial function is sensitive to changes in n, so n is held constant. This local measure of organization is applied over a grid, and the results contoured to makes maps that show the spatial variation in the relative degree of organization. This was applied to published polygonal fault maps by digitizing the faults into representative line segments. The resulting pattern has orders of difference in probability magnitude, so a log scale conversion was used.

The resulting maps identify some organized subareas readily evident to the eye, and some that are not, and help document the pattern where locally organized fractures contribute to a regionally random pattern. Comparing maps generated from the same data set, but with increasing n, provides additional useful information. Such maps could also explore why less organized subareas exist in ordered fracture sets.