Estimation of fracture trace length distributions using probability weighted moments and L-moments

The characterization of fracture trace length distributions is an initial and essential step in estimating three-dimensional fracture size distributions. Present challenge mainly lies in the accurate depiction for the distributional nature of trace lengths from various sizes of trace data, especially for small samples. The present paper is an attempt to solve this problem by using probability weighted moments (PWMs) and L-moments. To quantify the statistical property of trace lengths, the PWMs and L-moments of true trace lengths on an infinite surface from the measured trace lengths by an irregular convex window are estimated. A distribution-free method is then developed using the maximum entropy principle with PWMs for estimating the quantile functions of true trace lengths. Since there is no assumption regarding the type of trace length population distribution, the estimation obtained is distribution-free. For practicing engineers, a method using L-moments for estimating the common trace length distributions is also suggested. Examples that are tested showing the present method provides good approximations of the quantile functions and probability density functions of true trace lengths. The method is effective even for problems with outliers or highly skewed trace data, and can be used as a reliable tool for inferences from various sample sizes with good accuracies.