## 1. Introduction

A landslide is either geophysical or climate-related disaster that is described as a mass movement of earth surface material. This usually involves shear displacement of soil and/or rock masses along one or several slip surfaces [

1]. A landslide susceptibility map (LSM) is a promising solution for both understanding and predicting probable future landslides. It assists planners in decision-making phase aimed for further mitigation of landslide consequences. Accordingly, a LSM depicts areas likely to have landslides in the future by correlating some of the principal factors that contribute to landslides with the past distribution of slope failures [

2]. In this respect, production of LSM at the early stage of landslide assessments is of crucial importance for safe economic planning, such as urbanization activities and the engineering of structures. However, a standard procedure for the production of landslide susceptibility maps does not exist [

3].Thus, LSM can be accomplished by providing risk managers with easily accessible, continuous, and accurate information about landslide occurrence. The predictive capacity is poorly understood in LSM and is vague. In general, the spatial prediction of landslides is not easy due to the complex nature of landslides [

4]. LSM provide important information for predicting landslides hazards which include an indication of the time scale within which particular landslides are likely to occur [

5]. The associated vagueness can be dealt using fuzzy sets theory.

Introduced by Zadeh, fuzzy set theory handles indefiniteness arising from intrinsic ambiguity than from a statistical variation [

6]. A functional defined on the class of generalised characteristic functions (fuzzy sets), called “entropy”, is introduced using no probabilistic concepts in order to obtain a global measure of the indefiniteness connected with the situations described by fuzzy sets [

7]. The meaning of this quantity is quite different from the one of classical entropy because no probabilistic concept is needed in order to define it. This function gives a global measure of the “indefiniteness” of the situation of the problem at hand [

8]. Although there is a well-defined mathematical theory of probability, there is no universal agreement about the meaning of probability. Thus, for example, there is the view that probability is an objective property of a system and another view that it describes a subjective state of belief of a person. Then there is the frequentist view that the probability of an event is the relative frequency of its occurrence in a long or infinite sequence of trials. Thus, entropy is often used as a characterization of the information content of a data source, this information content is not absolute: it depends crucially on the probabilistic model [

9].

Effective LSMs could provide a proper understanding of “susceptible regions” [

10]. In order to better assist planners in understanding landslide hazard, a variety of GIS-based susceptibility mapping techniques are employed and developed [

11]. These approaches can be classified into three main groups: subjective, objective and hybrid methods. The subjective methods typically include inventory mapping and decision makers’ (DMs) evaluation in both standardisation and weighting of selected criteria [

12]. There are various GIS-based studies on LSM through the use of subjective approaches. Some of them used multi-criteria evaluation (MCE) techniques including: simple additive weighting [

13], ordered weighted average [

14], analytical hierarchy process [

15], analytical network process [

16], PROMETHEE [

17], etc. and some used different heuristic and knowledge driven techniques in order to assess landslide susceptibility mapping [

18,

19,

20]. Other studies, on the other hand, have shown a variety of objective methods in the assessment of the landslide susceptibility because of some limitations such as insufficient knowledge about the area of interest. The objective methods mostly rely on statistical [

21,

22,

23,

24,

25,

26,

27], soft computing [

4,

28,

29], deterministic analysis [

30], neuro-fuzzy [

4,

31], artificial neural network [

32,

33,

34], decision trees [

35,

36], and index of entropy [

37,

38,

39,

40,

41], which are more rigorous and mostly relying on objective assessments. On the other hand, there are various hybrid GIS-based LSM methods which are both subjective and objective. In other words, some hybrid GIS-based LSM methods used subjective standardisation and an objective weighing technique [

42,

43,

44], and vice versa.

The accuracy of LSM mostly depends on the amount and quality of available data, the working scale and the selection of the appropriate methodology for analysis and modelling [

17]. In methodology implementation and its assessment, landslide casual criteria play a key role. In this study, we decipher the optimality of predictive solutions for objective criteria weighting. In an attempt to find an optimal solution, we show how modified Shannon entropy algorithm in association with fuzzy set theory can be successfully applied to the numerical solution of the LSM while there is no sufficient knowledge about the area of interest. In other words, the main objective of the present study is to extend a hybrid GIS-based LSM method within which fuzzy membership functions (FMFs) have been applied for criteria standardisation using “global knowledge” about landslides, while no “local knowledge” is utilised for criteria weighting. In literature, although different GIS-based models have been used for landslide susceptibility mapping, however, LSM map extracted from modified Shannon entropy algorithm in association with fuzzy set theory has seldom been carried out. Therefore, this study aims to fill this identified gap in the relevant literature.

Since the LSM deals with a various sets of criteria it can be assumed that integration of fuzzy set theory with information theory, and in particular with Shannon entropy, will assist in performing accurate landslide susceptibility mapping. This accurate LSM is due to the flexibility of fuzzy membership functions and objective evaluation of criteria weights. Based on this assumption, the present research is an attempt to propose a novel hybrid method, which contributes to the objective decision making for regional landslide management. In other words, by using only the entropy values of previous landslide events for each criterion and regardless of experts’ opinions, we intend to facilitate criteria weighting process while improving or preserving LSM predictive accuracy compared with accurate subjective methods.

The paper is organized as follows: after a description of the study area in

Section 2, a detailed definition of the material and methods of the research is described in

Section 3.

Section 4 presents results while

Section 5 discusses the achieved results and contributions, respectively. At the end, we provide the conclusions of this research in

Section 6.

## 2. Description of the Study Region

Izeh is located in the eastern part of Khuzestan province, in south-western Iran (see

Figure 1), where the high susceptibility for a mass movement and in particular landslides is considered as a potential natural hazard for human society and their activities such as the hydropower plants in Izeh. According to the inventory of landslides compiled by the Ministry of Natural Resources [

45], there are 108 recorded landslide events in the region.

The climate is a temperate in north, while in south a warm climate prevails. Similarly, mean annual precipitation within the study area varies from 450 to 700 mm. The region is important in terms of the agricultural activities and in particular hydropower plants. The Karun River, the main and longest river in all of Iran, passes through this area. The suitable topography of Karun canyon provides the possibility of constructing hydropower plants and three main dams have been constructed so far on different branches of the Karun River [

44].

Geologically, there are several minor faults and one major thrust in the region along with the 13 types of geologic formations cropping out in the region. The Izeh fault zone is a transverse fault zone with right-lateral strike slip (and some reverse component) in the Zagros Mountains, south-western Iran. That is majority controlled by the subsidence and sedimentation of the embayment. In terms of 13 types of geologic formations, nearly all of them composed of sedimentary rocks including, marl, shale, limestone, gypsum, siltstone and other Quaternary deposits. It also should be mentioned that in the case of any triggering cause, there will be a significant chance of landslide occurrence within the south and south-east where the rough topography coincides with major thrust fault, Karun canyon and susceptible lithology. In other words, where there is susceptible lithology, proximity to faults contributes to slope instability, affecting not just surface structures but also terrain permeability. Eventually, the erosion associated with Karun River in nearby areas further leads to slope instability and generally increases the rate of subsequent slope failure. This is considered another prominent reason for the notable landslide recurrence in the region [

44].

## 5. Discussion

The accuracy of predictive models is considered a major concern in the majority of environmental modelling applications including LSM [

50]. The predictive accuracy of subjective LSM models can be affected by the inherent bias that emanates from DMs’ point of view during both data standardisation and criteria weighting. Moreover, the absence of expert DMs may be a serious hindrance in the LSM process when using a subjective method. Considering criteria standardization schemes, by applying a more computationally intensive approach we attempted to preserve the original quality of spatial data. In this respect using a variety of FMFs positively affected the validity and accuracy of input spatial criteria. Therefore, missing or generalised values can represent otherwise precise data. Further, the proposed methodology shows promising results to predict landslide susceptibility values regardless of experts’ opinion. According to the obtained results, the accuracy of the proposed hybrid model is improved significantly compared with the accuracy of accurate subjective approaches, which have been previously implemented in the study area using the same dataset [

44].

#### 5.1. Obtained Results and Relevance to the Previous Studies

Considering the high frequency of landslides ocurring in several areas of southern Izeh, there was a demand to establish an accurate landslide susceptibility map. The expected accuracy of LSM depends not only on the presence of concise and comprehensive data, in terms of data scale and accuracy, but also on the selection of the appropriate methodology of data processing and modelling [

15]. Regardless of data scale and accuracy, the present study aimed to explore landslide susceptibility of southern Izeh by developing a hybrid GIS-based LSM that uses neither DM's evaluation nor sophisticated objective methods. This is an integrated strategic LSM framework with an emphasis on structuring the decision-making process problem. Within this approach, Shannon entropy was employed to determine the criteria weightings from an objective evaluation of spatial domain while different fuzzy membership functions were employed for criteria standardization.

Obtained results of ROC curve analysis (AUC = 0.934) (see

Figure 7) and simple overlay technique (see

Figure 8) signify that the proposed hybrid fuzzy Shannon entropy evaluation technique is a promising tool for integrating multiple raster-based criteria for LSM while there is not sufficient knowledge about the criteria weights with respect to landslide mechanism of the study region. The previous study using the same dataset through extended fuzzy multi-criteria evaluation which was built on the basis of DMs’ evaluation achieved AUC value of 0.894 [

44]. This further approves the capability of proposed hybrid model for prediction of landslide susceptibility values. In other words, achieved results of accuracy metrics comparison approves that the proposed LSM model can achieve superior prediction accuracy to what that can be achieved by using DMs’ points of view (

Table 3), with significant time saving.

#### 5.2. Spatial Information Extraction and Prediction

This study contributed in the area of the spatially structured dilemma of predicting landslide susceptibility values for specific geographic locations. This may be implemented through standardising and subsequent summing of landslide casual criteria. In this paper, we attempted to present an assessment of LSM, carried out by the implementation of hybrid fuzzy Shannon entropy evaluation within which fuzzy set theory has been used for criteria standardisation, and Shannon entropy algorithm was used for weighting of some factors that may affect the landslide susceptibility. Therefore, the prepared hybrid susceptibility map is the result of a pixel-based combination of nine standardised criteria affecting the degree of landslide susceptibility. The optimal criteria weights are obtained objectively by a precise mathematical solution through the proposed entropy-based model [

79]. In this respect, the lower the landslide entropy of a criterion, the higher the weight is. In other words, a lower landslide entropy within certain criteria (i.e., distance to faults and distance to river) indicates the presence of predictive spatial frequency and vice versa.

Further, as expected, the estimated data driven (objective) criteria weights using Shannon entropy algorithm do not conform to the subjective criteria weights estimated using an aggregation of DMs’ votes from our prior research (

Figure 9).

According to the obtained fuzzy Shannon entropy criteria weighting scheme results the distance to fault is the most important criterion, followed by distance to river and rainfall criteria, respectively. Therefore, considering the estimated criteria weights, the spatial distribution of landslide susceptibility values is mostly controlled by these mentioned criteria. This may be further proved by the high concentration of recorded landslide events along the Karun River (

Figure 6). Nonetheless, considering the DMs’ evaluation slope is referred to as the most significant criterion followed by lithology and distance to road layers. Considering these two weighting approach, fuzzy Shannon entropy seems more realistic for predictive modelling of spatial pattern of landslides compared to the latter method. Even though the slope criterion is of paramount importance in any shape of slope instability, it is not the only constituent of landslides. Accordingly, the spatial pattern of landslides (at least in the study region) is controlled by other important but less geographically available landslide casual criteria (distance to fault, distance to river and rainfall criteria). In other words, if similar high susceptible values of slope (or any other criteria) are prevailing all around a region while the landslide distribution pattern is represented by a different spatial order (

Figure 5a), a secondary criterion (such as distance to river) with less availability may be the determinant factor of landslides’ spatial distribution (

Figure 5c,e,f). This indicates the insight of the proposed objective weighting scheme in local evaluation of the landslide casual criteria. In other words, in the current study area, the slope angle is usually sufficient to influence landsliding. Nonetheless, considering the spatial distribution of landslides, there is limited evidence which proves that the slope criterion plays an important role in landsliding. In the present study area, susceptible slope values are distributed almost evenly over the study region; however, actual landslide events are more or less concentrated along the Karun Canyon. This may be due to the fact that the required water for slope failure, as a triggering factor, is controlling landslide events in southern Izeh. Water is not always directly involved as the transporting medium in mass movement processes while it does play an important role. This is not only proved by the obtained criteria weights of the Shannon entropy method but also it can be recognized by visual inspection of landslide spatial patterns and frequency along the Karun River (

Figure 6).

Further, considering the results of our proposed objective weighting approach lithology criterion is the least important among all selected criteria, while the expert opinion refers to the rainfall layer as the least important. The achieved accuracy value of fuzzy Shannon entropy, however, is still remarkably superior.

#### 5.3. Decision Aiding and Planning

Many researchers, [

13,

80,

81] have pointed out that the traditional subjective weighting schemes usually suffer from sensitivity in decision-making and they are susceptible to intrinsic experts’ knowledge errors. Looking into the contribution to decision aiding, this study presents an integrated strategic weighting procedure using an objective method which determines the criteria weights by solving mathematical models. This is executed without any consideration of the decision maker’s preferences as it is a convention in subjective methods, such as the AHP method, OWA method, Delphi method, etc. In other words, this article introduces an objective approach that integrates fuzzy set theory and information theory algorithm (i.e., Shannon entropy), which could be a useful geospatial tool for integrating multiple features/attributes that affect the LSM process. This can largely compensate for the absence of expert DMs or the lack of local knowledge about study area when it comes to producing quality LSMs.

#### 5.4. Limitation of the Proposed Methodology in LSM

While information theory-based methods such as the one proposed in the present research have shown considerable potential in different predictive spatial modelling scenarios, they do have their own limitations. Even though the application of the proposed methodology as an objective weighting scheme is not dependent on decision maker’s expertise and judgment, it relies on quantification of defined attributes of landslide data points using step by step mathematical computations. This is conditional on the existence of a concise and representative database. In terms of the present research, the availability of a comprehensive and readily accessible landslide inventory database was quite beneficial in achieving the desired outcome.

Another limitation of the implemented methodology is observable in particular in the NE part of the study region where false alarms exist in the form of low slope areas indicated as a high susceptibility class (very few pixels as a very high susceptible class). This is mainly due to the fact that the slope angle is not characterised as a primary criterion shaping the landslide occurrence spatial pattern. Most LSM approaches end up with extremely high false positive rates in terms of high or very high susceptible areas compared to the total landslide areas. This problem is not only limited to our study, therefore, we would like to call the attention of the physical geography community, in particular methodological development researchers, to exploring ways to reduce the problem of over-estimated susceptibility in future studies.

Further, after fitting the desired membership function, the proposed fuzzy Shannon entropy technique considers the dataset as a collection of distributions, which may not be suitable to extract specific spatial structures embedded in the underlying features/attributes [

82]. Even though datasets with the same histogram certainly have the same entropy (i.e., distance to river and distance to fault in the present study), the distributions of their data values in space could be totally different. In addition, the result can be sensitive to the level of discretization caused by different membership functions (i.e., crisp or fuzzy) when using the histogram. We believe that further interest from researchers with access to larger data sample sizes is vital for developing more robust entropy-based LSM methods that can incorporate generalizable results.