Probabilistic Modelling of Soil Shear Strength by Maximum Entropy Quantile Functions

Abstract

The 1990 Nipigon River landslide that occurred north of the township of Nipigon, northwestern Ontario, is listed among the catastrophic landslides in Canada. Since 1990, many additional landslides of various scale and consequence have occurred in the Nipigon River area. To investigate the mechanism behind initiation of these landslides, a series of soil sampling and laboratory soil direct shear testing were conducted to measure the shear strength of watershed soils (cohesion and the angle of friction). Given that the shear strengths exhibit high variability (and hence uncertainty), they are amenable to a comprehensive probabilistic treatment. Conventionally, random variables are characterized using a probability density function or a cumulative distribution function; the type of function is usually determined from histograms and from the classical distributions such as normal and lognormal distributions, and the distribution parameters are estimated using the method of moments or the method of maximum likelihood.

This paper proposes a novel probabilistic method to model soil properties using quantile functions, based on fractional probability-weighted moments, principle of maximum entropy, and Akaike information criterion. The quantile function is a counterpart to distribution functions of a random variable since the quantile function is mathematically the inverse cumulative distribution function. The maximum entropy method is presented to generate unbiased quantile functions for measured soil properties. The use of the fractional probability-weighted moments facilitates more accurate quantification of soil uncertainties by the entropy-based quantile functions than the probability density or cumulative distribution functions. Akaike information criterion is then used to locate the optimal order of maximum entropy quantile functions. Maximum entropy quantile distributions are compared to the traditional quantile distributions to evaluate their performance. The analytical entropy quantile distribution obtained can be used in probabilistic reliability analysis.