Determination of karst area using Dempster–Shafer theory and Fuzzy method (Case Study: Northwest of Fars Province)

Karst lands provide water for 25% of the world's population. Karst's economic resources, the importance of karst lands in agriculture for identify geomorphic developments is determined. On the other hand, the study of karsts in arid areas is very important due to the suitable areas for water storage. Recently, numerous studies have been conducted on karst with different models (Bai et al., 2013; Febles-González et al., 2012; Wang et al., 2004; Xu et al., 2015). Considering the importance of the subject, the aim of this study was to determine the karstic regions using a new algorithm called Dempster–Shafer theory and comparing it with fuzzy method. In order to determine the susceptible areas of karst, geological data, distance from fault, rainfall, elevation, temperature, distance from the river, slope in GIS environment were used.
Dempster–Shafer theory
The Dempster–Shafer theory is based on two non-additive evidential measures: belief and plausibility, which can be estimated from basic probability assignment. The value of the basic probability assignment, also called mass, for a set Xi, represented as m(Xi), expresses the amount of evidence supporting the claim that an element of the universal set X belongs to the set Xi. The basic probability assignment is defined on a universal set X as a function of the power set (PX) in the interval [0,1] that satisfies two conditions: and =: Sets with nonzero mass are called focal elements. Measurement of belief for a set Xi, Bel (Xi), represents the minimum belief in the claim that an element of the universal set X belongs to the set Xi. It satisfies the following conditions: and Bel (x)=1
Fuzzy-AHP The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by Lotfi Zadeh. Zadeh, in his theory of fuzzy sets, proposed using a membership function (with a range covering the interval (0,1) operating on the domain of all possible values. For any set X a membership function on X is any function from X to the real unit interval [0,1].
For preparing the fuzzy map for each parameters should definite membership function. A membership function assigns to each object a grade ranging between 0 and 1. The value 0 means that x is not a member of the fuzzy set, while the value 1 means that x is a full member of the fuzzy set. (Mc Bratney and Odeh, 1997): A fuzzy set is an extension of a classical set. If X is the universe of discourse and its elements are denoted by x, then a fuzzy set A in X is defined as a set of ordered pairs:µA(x) is called the membership function (or MF) of x in A. The membership function maps each element of X to a membership value between 0 and 1.
The Analytic Hierarchy Process (AHP) is a theory of measurement by pairwise comparisons and relies on the judgments of experts to derive priority scales (Saaty, 2008). The first step in the AHP is the estimation of the pertinent data. That is, the estimation of the aij and Wj values of the decision matrix. This is described in the next sub-section.
The weights of importance of the criteria are also determined by using pairwise comparisons.

The results of the Dempsstrapher method showed that by increasing the level of reliability and reducing the riskiness of the karstic areas, it decreases. Also, the results of the Dempsstrapher model have better results and better dispersion than the fuzzy method. Finally the results showed that the likelihood of karst in the west of the study area was higher than the other parts. According to the results of DST method, it can be generated karst map with different levels of confidence, which according to the conditions of economic and importance of the subject in the study area can be from one of these maps for management and planning.