Implementation of Fuzzy Random Variable in Imprecise Modelling of Contaminant Migration through a Soil Layer

Fuzzy random variables possess several interpretations. Historically, they were proposed either as a tool for handling linguistic label information in statistics or to represent uncertainty about classical random variables. Accordingly, there are two different approaches to the definition of the variance of a fuzzy random variable. In the first one, the variance of the fuzzy random variable is defined as a crisp number that makes it easier to handle in further processing. In the second case, the variance is defined as a fuzzy interval, offering a gradual description of our incomplete knowledge about the variance of an underlying, imprecisely observed, classical random variable. In this work, we first discuss another view of fuzzy random variables that comes down to a set of random variables induced by a fuzzy relation describing an ill-known conditional probability. This view leads to yet another definition of the variance of a fuzzy random variable, in the context of the theory of imprecise probabilities. The new variance is a real interval, which achieves a compromise between both previous definitions in terms of representation simplicity. Our main objective is to demonstrate, with the help of simple examples, the practical significance of these definitions of variance induced by various existing views of fuzzy random variables. Finally, fuzzy random variable concept is implemented in modeling of contaminant migration through a soil layer. The transport of contaminant through a saturated soil layer is modeled by advection, dispersion, sorption and first order degradation. The parameters of the contaminant migration model such as seepage velocity, porosity of soil, dispersion coefficient and distribution coefficient are taken into consideration as fuzzy random variable due to their dual nature of fuzziness and randomness.