Estimating Mean Fracture Trace Length and Density from Observations in Convex Windows

Abstract

Although window samples of fracture traces are widely recognized to be length-biased and censored, they are often the best source of data for inferring statistical parameters of fracture sets. This paper presents new estimators for mean fracture trace length and density that correct for the effects of bias and censoring. A stereological estimator of mean trace length is derived for parallel traces in a rectangular sampling window, an end-point estimator of mean trace length is derived for windows with arbitrary convex boundaries and for arbitrary trace length distributions, and an end-point estimator of trace density is derived for windows with arbitrary convex boundaries and for arbitrary trace length and trace orientation distributions. Results for rectangular and circular windows are obtained as special cases of the general solutions for arbitrary convex windows. When applied to circular windows, the end-point estimator of mean trace length is, in addition, independent of the trace orientation distribution. The estimators are easily determined from field data. The end-point estimator of trace density requires knowing only the area of the window and the number of end-points inside the window. The end-point estimator of mean trace length, when applied to circular windows, requires, in addition, the number of end-points outside the window (of those traces that intersect the window) and the stereological estimator of mean trace length requires only the apparent mean trace length and the height of the window.