Introduction

Landslide, earthquake, flood, and volcano are among the prevalent natural disasters that annually impose huge financial and life losses to humans. By population growth and increasing construction in landslide-prone areas, global statistics of landslide-based casualties are continuously rising. According to the World Atlas of Natural Hazards, landslides are the most abundant and widespread natural disasters on Earth. In this regard, the susceptibility and damage maps of landslides can be considered as a basis for crisis management of those regions. The maps also are important tools for engineers, geologist, planners, and decision-makers to select suitable places for agriculture, construction, and other developmental activities (Ercanoglu and Gokceoglu 2002).

Rockfalls are the downward motion of rock fragments involving free falling, bouncing, and rolling, which, depending on the changes in the topographic profile, two or more forms of rockfalls may be seen. Rockfall is a very common event that occurs in mountainous slopes, coastal cliffs, volcanoes, riverside, and trenches. Although most of the rockfalls occur in the remote area, they can be a serious threat to residential area and transportation lines (Hungr et al. 1999; Chau et al. 2003; Corominas et al. 2005). The high hazard of rockfalls is mainly due to their dynamic behavior, the velocity of the block motion (30–100 km/h), and the difficulty of predicting the source of their occurrence. The high velocity and sudden occurrence of rockfalls make it different from other slope instabilities. The affected area by rockfalls can be identified through the existence of cracks and crashed blocks on the surface of the slope. It is worth pointing out that detection of rockfalls in the area is possible in the case of less vegetation. In areas susceptible to rockfall hazard, the occurrence of multiple rockfalls results in the formation of debris deposits in the heel of the slope (Corominas et al. 2017).

The unpredictable nature of most of the rockfalls has been a concern for authorities and decision-makers. Although the dimensions of rockfalls are limited, they are a very fast event with high kinetic energy that makes enormous damage. Turner and Jayaprakash (2012) presented a complete list of rockfalls, which showed that even rocks with less volume could be resulted in a significant destruction and blockage of roads, especially in railway lines. The recent research (Petley 2012) showed that the most landslide and rockfall-induced losses were observed in most of the less developed countries due to the lack of research and suitable resources.

In general, the trigger factor in rockfalls is related to the existence of previous shear or tensile fracture along the discontinuity, which led to the detachment and creation of potentially unstable blocks from the slope (Valagussa et al. 2014b). Gosar (2017) classified the rockfalls into five classes based on volume and frequency (Table 1). The blocks can be triggered by many factors including seismic movements of the earth, rainfall, increasing the pore pressure in the rock mass, melting–freezing cycle, chemical decomposition, advanced weathering of the rock mass, root wedging caused by winds, and progressive decomposition of rock mass (Varnes 1978). The rockfalls originate from cliffs with different size, lithology, and structure. Rockfalls and other kinds of landslides are secondary effects of large earthquakes. Therefore, rockfall hazard zonation is an important issue for planning the land use, especially in seismic regions with steep slopes.

Table 1 Distribution of rockfalls according to their size
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Given that no research has been conducted on zonation and mapping the susceptibility of seismic rockfalls using intelligent methods (such as ANFIS), the present research was carried out to implement the ANFIS and LR methods in the area stricken by Firooz Abad-Kojour earthquake of 2004, evaluate and select the optimum model with high accuracy in prediction, mapping the susceptibility of seismic rockfalls, and apply this method to predict probability of rockfall events induced by future earthquakes. In the present research, Geographic Information System (ArcGIS), PCI Gematica (Landsat Satellite Image Processing), SPSS 22 (to implement LR), and MATLAB 2014 (Neuro-fuzzy network implementation environment) were used.

Materials and methods

Study area description

Geographical setting of the study area

Topographically, the study area, which was stricken by Firooz Abad-Kojour earthquake of 2004, is one of the highlands of Alborz Mountain. Damavand Peak with a height of 5671 meters—the highest peak of Iran—is located in this area. The average height of the area is over than 2000 m (A.S.L). The existence of deep and steep valleys and long crests are the main characteristics of this area in Alborz. Based on the classification of plain and mountainous regions, more than 80% of the earthquake-stricken parts of the study area are located in a mountainous region. The plain and low-lying regions of the study area are limited to small parts of the valleys bottom and the southern margin of the Caspian Sea. The selected stricken areas are located in longitude of 51°00′–52°00′ N and latitude of 36°00′–36°30′E (Fig. 1). Firooz Abad-Kojour earthquake of 2004 resulted in casualties and financial losses from the south of Chalus, Firooz Abad, Kojour valley, Marzan Abad, Kelardasht, and Chalus Road to the western parts of Roudbar Alamut district of Qazvin province, Iran (Zare 2004).

Fig. 1
figure 1

Geographical setting map of the region affected by Firooz Abad-Kojour earthquake of 2004

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Geological conditions of the study area

The study area is located in the central part of the Alborz Mountains. Alborz mountain range was created by severe Alpine orogenic movement. Today, these tectonic movements are observed in forms of folds, faults, and thrusts. Although the general trend of folds and their structure affected by Alborz Mountain and it is East–West, there are many irregularities in this trend. In Stratigraphic view, the main part of the study area is covered by Karaj Formation, which is often pyroclastic and includes a wide range of tuff, agglomerate, shale, sandstone, and volcanic and intrusive rocks. It followed by sandy-shale Shemshak Formation. In addition to Karaj and Shemshak formations, the rock units related to Precambrian until the end of the third era are outcropped in a different part of the area, and the multiple fractures and folds caused by tectonic movement are the common point of the rock units. In the eastern and central parts, the calcareous deposits and pyroclastic rocks are outcropped, respectively. From central part to the western part of the study area, the Neogene marl and conglomerates deposits are covered the dominant part of the area (Amini Hosseyni et al. 2004; Mahdavifar 2006). The geological map of study is shown in Fig. 2. Besides, lithological description of the geological terms used in Fig. 2 is presented in Table 2.

Fig. 2
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Geological map of the area stricken by the Firooz Abad-Kojour earthquake of 2004

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Table 2 Lithological description of the geological terms used in Fig. 2
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Characteristics of the Firooz Abad-Kojour earthquake of 2004 and its causative fault

In 28th May 2004, the Firooz Abad-Kojour earthquake of 2004 shook the northwest of central Alborz and part of western Alborz at 17:08:46 h (local time). The earthquake with magnitudes of 6.2 (Mw) and focal depth of 18 km was felt in south of Chalus, Kojour valley, and Marzan Abad area, which along with casualties and financial loss from Firooz Abad (Kojour), Marzan Abad, Kelardasht, and Chalus road (Mazandaran province) areas to western parts of Roudbar Alamut district of Qazvin province. According to the recorded strong motion data in the main shock and locating the earthquake focal point, the epicenter was located at 36.35°N, 51.35°E. Based on the report of Harvard University website, the earthquake focal mechanism was introduced as compressive with a small strike–slip component. Given the location of earthquake focus, seismic epicenter zone, and faults in the area, the north fault of Alborz was introduced as a causative fault (Fig. 3) (Zare 2004).

Fig. 3
figure 3

Iso-intensity map of the Firooz Abad-Kojour earthquake of 2004

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Based on the damages in the seismic epicenter, field observation of Baladeh to Kojour Valley, observing the residential area in Chalus valley, and due to the focus of the rockfalls in the range of Hezarcham to Siyahbishe on the Chalus Road, it seems that, in epicenter zone and in the area of rockfalls in Chalus road, the intensity of the earthquake was in the VII+ and VIII+ (on the EMS scale), respectively. On the other hand, the expansion of the iso-seismical line in the epicentral zone showed the earthquake intensity of north and North West of Tehran which was VI (Fig. 3) (Zare 2004).

Rockfalls induced by the Firooz Abad-Kojour earthquake of 2004 in the study area

Firooz Abad-Kojour earthquake of 2004 due to the topographic conduction and existence of Mountains and steep areas resulted in numerous slope instability in a large part of the Central Alborz. The instabilities were expanded in the area of 12,000 km2 (around the earthquake epicenter) in forms of rockfalls and small and large slides. In the study area, about 74 falls and falling zone were recorded (Fig. 4). The furthest distance from the falling event to the earthquake epicenter is about 101 km, which occurred within 27 km of the Qazvin road to Moallem Kalayeh. This distance was in agreement with the distance of seismic landslides presented by Keefer (1984). The rockfalls are concentrated in the central, western, southern, and southwest areas of the iso-intensity line (Chalus road, Baladeh, and adjacent areas of Sama village, Firooz Abad, Manjir, etc.). In geological point of view, these areas consist of Precambrian (sandstone, slits of Kahar formation) and Cambrian formations (sandstone and shale). The presence of weak rocks and rough topography has greatly increased the potential for landslides occurred in the region. The landslides commonly occur in the length of the steep valley and more dispersed in areas with a mild topography. The most abundant type of seismic landslides is shallow and severely disrupted falls, and rocks and debris slide that has spread throughout the region. Coherent slump and block slides occur in rocks with higher strength. The highest number of seismic rockfalls occurred in the range of Chalus road (between the HaftBaradar and Hezarcham) and the linking road of Sama village to Manjir in Sanj Mountain. In some parts, more than 40% of slopes were affected by the earthquake (Amini Hosseyni et al. 2004). The depth of landslides varies from several decimeters to several meters and affects the dry and unconsolidated materials. The volume of these instabilities varies from a fraction of cubic meters to several thousand cubic meters. Within the Chalus Road, the highest casualties and damage are caused by seismic rockfalls. Besides, the similar rockfalls were observed in the range of Sama Village to Manjir (close to the epicenter) and the roads around it. Seismic rockfalls resulted in many financial losses and casualties in villages, one can name Anarak, Firooz Abad, Sama, Manjir, Larak, etc.

Fig. 4
figure 4

Distribution map of Rockfalls induced by the Firooz Abad-Kojour earthquake of 2004 within the study area

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The Firooz Abad-Kojour Earthquake of 2004 led to 41 life losses and huge financial loss such as destruction of buildings and vehicles. The highest rate of causalities was related to the falling of huge rocky blocks and hit cars passing through Chalus to Tehran road (Amini Hosseyni et al. 2004; Mahdavifar 2006). The samples related to Firooz Abad-Kojour seismic rockfalls are presented in Figs. 5a–d and 6a, b. Moreover, Table 3 presents geological and morphological properties of some dominant discontinuities effective in rockfall events induced by Firooz Abad-Kojour of 2004 recorded during the field survey.

Fig. 5
figure 5

Rockfalls triggered by the Firooz Abad-Kojour earthquake of 2004 within the study area: a falling rocks on the Baladeh road, 19 km from Haraz to Baladeh; b rockfall and blockage of road from Kojour to Baladeh; c rockfalls on the link road of Sama village to Manjir in Sanj Mountain; d farms stricken by rockfalls around of Sama village; consider the mud trace resulting from rockfalls on walls

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Fig. 6
figure 6

Rockfalls triggered by the Firooz Abad-Kojour earthquake of 2004 within the study area: a falling zone in Chalus road, 12 km from Baladeh three ways to Chalus; b blockage of some parts of Chalus road due to rockfalls in 14 Km of Baladeh three ways to Karaj

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Table 3 Geological and morphological properties of some dominant discontinuities effective in rockfall events induced by Firooz Abad-Kojour of 2004
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Background of the research

Estimating the rockfalls hazard for zonation requires the determination of the event probability (Cancelli and Crosta 1993; Corominas et al. 2005; Jaboyedoff et al. 2005; Straub and Schubert 2008; Agliardi et al. 2009). For seismic rockfalls, the occurrence probability is affected by event probability of earthquakes with different intensity. For other types of landslides, expanded studies were performed on the relationship between earthquake and landslide occurrence through experimental (Keefer 1984; Rodrıguez et al. 1999; Lee et al. 2008; Miles and Keefer 2009a, b), physical, or numerical methods (Jibson et al. 2000; Wasowski and Del Gaudio 2000; Capolongo et al. 2002; Del Gaudio et al. 2003; Del Gaudio and Wasowski 2004; Uchida et al. 2006; Peng et al. 2009; Rapolla et al. 2010; Motamedi and Liang 2013). In fact, less number of research (Marzorati et al. 2002; Valagussa et al. 2014a, b; Lari et al. 2014; Massey et al. 2015; Mahdavifar et al. 2016) was conducted on seismic rockfalls hazard zonation and susceptibility mapping. In these studies, the seismic rockfall probability was predicted by hierarchical methods and statistical methods such as information value and logistic regression (LR) generally. The above-mentioned methods, however, are not efficient in areas that the rockfall probability is affected by the earthquake.

Required information layers for LR and ANFIS methods

The region stricken by the Firooz Abad-Kojour earthquake of 2004 has a 4983 km2 area. In this research, various information layers with a pixel size of 10 m × 10 m were prepared. The determination of cell size is based on the minimum precision of all causative factors to guarantee the quality of output maps and avoid misleading results. Unit categorization of the study area was performed using the ArcGIS environment. Through this process, the spatial position of each pixel is fixed with respect to all information layers. Therefore, for a given pixel with fixed spatial coordinates, the predefined information and values are recorded. Finally, each pixel contains data from all information layers as well as the presence or lack of seismic rockfall.

There are much affecting and triggering factors used in the literature for landslide hazard zonation. The most important triggering factors are earthquake and rainfall, and most important affecting factors are slope angle and ground strength (Yin and Yan 1988; Budimir et al. 2015). Furthermore, the importance of affecting factors changes for different landslide types. For example, soil moisture is not an important factor in rockfall occurrence, while it is one of the most important factors in soil slump occurrence. In this research, seismic rockfalls’ susceptibility mapping and hazard zonation are investigated, and hence, important factors for rockfall occurrence are chosen for analysis.

The initial information layers used in this research are distribution map of the rockfalls induced by the Firooz Abad-Kojour earthquake of 2004, aspect, distance from the river, distance from the road, ground strength class, hypocentral distance, shake intensity (MMI), slope angle, and slope height in the study area. Using the distribution map of the rockfalls triggered by the Firooz Abad-Kojour earthquake of 2004, the other information layers were classified.

Ground strength class information layer is established based on the description and information provided by Keefer (2000) about the relationship between the qualitative characteristics of slope materials (lithology) and concentration of seismic rockfalls. The ground strength class is related to the lithology unit ranking based on their susceptibility to seismic rockfalls. To prepare this information layer, the 1:50,000 geological maps of the area stricken by the Firooz Abad-Kojour earthquake of 2004 were used, and then, these maps were elaborated using the field visit data and 1:20,000 aerial photos. For each lithology unit in the study area, a ground strength class value was assigned in the range of 1–5 and accuracy of 0.5 (Fatemi Aghda and Bagheri 2015). These values are proportional to the concentration of seismic rockfalls calculated for each lithology. This information layer is a representative of ground conditions, material properties such as shear strength, and rock mass joints. The lower ground strength class values suggest the weaker and higher susceptibility of the ground-to-seismic landslide occurrence (Miles and Keefer 2007).

To prepare shake intensity data layer, first, moment magnitude of earthquake (MW), the distance of each unit from epicenter (R), and Eq. 1 (Mahdavifar et al. 2007) were used to provide Arias Intensity (Arias 1970) map (with pixel size of 10 m × 10 m). Then, using Eq. 2 (Wilson 1993), shake intensity map was based on the MMI (Fatemi Aghda and Bagheri 2015):

$$\log {I_a}= - 3.880+0.810M - \log R - 0.002R,$$
(1)

where Ia is Arias Intensity in m/s, M is the moment magnitude, and R is the source-to-site distance in km.

Source-to-site distance (R), which is defined as the shortest distance between accelerometer stations and the fault rupture plan at the earthquake focal depth (h) is obtained by \(R=\sqrt {{r^r}+{h^2}}\):

$$\log ({I_{\text{a}}})={\text{ }}0.527{\text{ }}{I_{{\text{MMI}}}} - 3.816,$$
(2)

where IMMI is the shake intensity in MMI.

Seismic rockfalls’ susceptibility mapping using LR method

A bivariate LR is a multivariate statistical method, which is generalized by linear models and introduced by Cox (1958). Besides, it forms a mathematical model to predicts the probability of occurrence of an event in the determined location. This model evaluates the absence or existence of a dependent variable in connection with the independent variables (Garsia et al. 2008). When the dependent variable is dichotomous (1 and 0), bivariate LR is the best choice for statistical analysis. In comparison with linear regression method, LR method has three advantages. First, in contrast to linear regression, variables in LR do not require to have a normal distribution. Second, independent variables in the LR may be continuously or categorized, or their combination. Third, LR method needs fewer assumptions (Chauhan et al. 2010). LR method is a quantitative method, which determines the effect of each independent variable on dependent variable quantitatively through coefficient (Lee and Sambath 2006; Garsia et al. 2008; Nefeslioglu et al. 2008; Dashti Marvily 2008; Yilmaz 2009). In bivariate LR model, the coefficient of independent variables (effective and trigger parameters) is estimated using maximum-likelihood method. The formula of LR is presented in the following equation:

$$Y={\text{Logit}}(p)=\ln \left( {\frac{p}{{1 - p}}} \right)=~{\beta _0}+{\beta _1}{X_1}+{\beta _2}{X_2}+ \cdots +{\beta _n}{X_n},$$
(3)

where Y is dependent variable, X1, X2 ,…, Xn is independent variables, Logit is probability logarithm or the natural amount of probability logarithm, p is probability of dependent variable (Y), \(\left( {\frac{p}{{1 - p}}} \right)\) is likelihood or disagreement ratio, β0 is constant coefficient, and β1, β2, …, βn are coefficients of independent variables.

The results indicate the relationship between the seismic rockfalls and their causative factors. In the LR method, the dependent input variable should be dichotomous, i.e., it only contains data coded as 1 (rockfalls) or 0 (non-rockfalls). The LR method does not directly indicate the susceptibility and, however, provide an inference to show the susceptibility rate using probability (Ayalew and Yamagishi 2005; Lee and Sambath 2006; Garsia-Rodríguez et al. 2008; Nefeslioglu et al. 2008).

Since the application of LR method is to identify the best model for creating a relationship between the trigger and effective parameters of seismic rockfalls with dichotomous dependent variable (coded with 1 and 0 for occurrence of rockfall and non-occurrence of rockfall), in the present research, to develop the LR method, the random sampling was performed in non-falling parts of the study area, provided that the stable and unstable points did not coincide (74 falling and 74 non-falling points; code 1 and 0, respectively). Table 4 not only provides a summary of the role of each variable in the model but also indicates that which variables remain in the model after performing LR. Table 4 is the most important tool in interpretation related to significance (Wald) and effective rate of each independent variable on dependent one (Exp (β)). It should be noted that Wald statistic was employed to find which independent variables have a significant effect on the dependent variable. Besides, to determine the effective rate of each independent variable on dependent one, odds ratio statistics were used. As shown in Eq. 4, the value of Wald test is calculated by the comparison of maximum-likelihood estimation of each coefficient (β) with standard estimated error (SE) of each coefficient. Interpretation of the Wald statistics values is performed by evaluating the significance level (Sig.) of each variable. Accordingly, if the significance level of each variable is less than 0.05, the existence of this variable in the model is beneficial and has a significant effect. The sign of independent variable coefficient (β) reveals the direct effect or reverse effect of the variable on the dependent variable (Chauhan et al. 2010). The coefficients obtained by LR method, the Wald values, and related Exp (β) are presented in Table 4.

Table 4 Influence coefficients (β), Wald values, and Exp (β) of trigger and effective parameters of Firooz Abad-Kojour seismic rockfalls (2004) using LR based on backward conditional rule in SPSS software
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$${\text{Wald}}={\left( {\frac{\beta }{{SE}}} \right)^2}.$$
(4)

Considering the results of fourth step in Table 4, the independent variables, including shake intensity, slope angle, ground strength class, distance from the road, and distance from the river, are able to predict the changes in the dependent variables of occurrence or non-occurrence of seismic rockfall. Besides, their prediction ability is significant at < 0.05. On the other hand, among the remaining significant independent variables, shake intensity and ground strength class, respectively, have the highest and lowest abilities to susceptibility mapping of Firooz Abad-Kojour seismic rockfalls (2004). Based on the fourth step of Table 4, the independent variables of shake intensity, slope angle, ground strength class, distance from the road, and distance from the river, have a positive and direct relationship with the dependent variable of occurrence or non-occurrence of rockfall triggered by Firooz Abad-Kojour earthquake of 2004.

In the present research, to weight the parameters and related classes to each parameter, the LR analysis of SPSS software and fuzzy logic of ArcGIS software were, respectively, employed. After performing four steps of LR analysis based on “backward condition” rule in the SPPS software (Table 4) and determining the suitable weights for selected parameters (shake intensity, slope angle, ground strength class, distance from the road, and distance from the river), first, the information layers were fuzzified in the ArcGIS software to weight the sub-criteria of those parameter. Then, the weights obtained from Step 4 were multiplied and overlapped with corresponding parameters, such that to prepare the map of seismic rockfall susceptibility (Fig. 7). In ArcGIS software, the seismic rockfall susceptibility zonation map is classified into five classes of “very low hazard”, “low hazard”, “moderate hazard”, “high hazard”, and “very high hazard” due to the natural breaks (Swets 1988; Bednarik et al. 2010; Constantin et al. 2011; Pourghasemi et al. 2012a, 2013). By this way, the boundary between the susceptibility levels of seismic rockfalls was determined by the abrupt change in the corresponding histogram curve.

Fig. 7
figure 7

Hazard zonation map of the rockfalls induced by the Firooz Abad-Kojour earthquake of 2004 using logistic regression (LR) method

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As shown in Fig. 7, the maximum area is related to “very low hazard” and “low hazard” classes (shown in dark green and green regions, respectively). In general, the percentage of susceptible area for seismic rockfalls was reduced by increasing the hazard in related zones. As can be seen, the majority of rockfalls triggered by Firooz Abad-Kojour earthquake of 2004 is placed in “high hazard” to “very high hazard” zones.

Besides, all parts close to linking roads, rivers, the parts with a steep slope, the areas around earthquake epicenter, and parts with consisting of low-strength geological formation show a “high”-to-“very high” rockfall susceptibility level. Since rockfalls mostly occur in scraps and steep areas and considering the rough morphology of streams and roadside features, the high concentration of rockfalls within the area stricken by Firooz Abad-Kojour earthquake of 2004 along these points is logical (Fig. 7). Therefore, it can be concluded the desirable accuracy of LR analysis in preparation of hazard zonation map for the rockfalls induced by the Firooz Abad-Kojour earthquake of 2004.

Seismic rockfalls’ susceptibility mapping using ANFIS method

In most problems of the real world, by increasing the number of variables, determination of linguistic rule is being difficult. Therefore, due to the impossibility of complete defining the mentioned rules, learning algorithms are used to synthesize the fuzzy model, which is called neuro-fuzzy modeling (Jang 1993). Although the fuzzy models are able to show the complicated process using concepts and rules of “if-then”, they are not able to learn. Besides, in the case of a high number of variables in the description of a problem, the choice of membership function and suitable “if-then” rule is difficult in the fuzzy model and the adjustment step of the fuzzy model will be unlimited. This problem can be solved using artificial neural learning algorithm such as optimization methods (Backpropagation algorithm) (Yesilnacar 2005; Polat and Güneş 2006). In this method, learning algorithm automatically chooses the suitable parameter for membership function in the fuzzy model. Although the neural networks are able to learn, they cannot explain the complexity of the mentioned system. Therefore, adaptive neuro-fuzzy inference system (ANFIS) method is used, which is a developed fuzzy model and trained by learning algorithm of neural network theory (Polat and Güneş 2006). The other capability of ANFIS, which almost overcomes the disadvantages of the fuzzy model, is its self-learning feature. ANFIS creates a fuzzy inference system (FIS) using input and output data sets. The parameters of the membership function of ANFIS are adjusted through back propagation algorithm or hybrid learning principle (combination of back propagation algorithm with least squares method) (Ying and Pan 2008; Oh and Pradhan 2011). The adjustment operations allow the fuzzy system to learn own structure from the data set.

ANFIS structure with two inputs is shown in Fig. 8. Each input has two membership functions; consequently, this structure has four principles. Besides, ANFIS model is a five-layered structure and each of them is described here.

Fig. 8
figure 8

ANFIS with two inputs and one output (Sugeno 1985)

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The first layer: in this layer, the membership degree of each input to corresponding fuzzy set is calculated (Eqs. 5 and 6):

$$O_{i}^{1}=\mu {A_i}(x),$$
(5)
$$O_{i}^{1}=\mu {B_i}(y),$$
(6)

where x and y are input variables of i node, A and B are a linguistic label, and µAi (x) and µBi (x) are fuzzy membership function associated with this node. Parameters are referred to as “premise parameters”.

Second layer: every node in this layer is a fixed-node labeled Π. The output of each node is a multiplication of all the input signals to the node, which equivalents to “if” part of the fuzzy rule (Eq. 7):

$$O_{i}^{2}={w_i}=\mu {A_i}(x) \times \mu {B_i}(y),\quad i=1,2,$$
(7)

where wi is weight or firing strength of each fuzzy principle of “if–then” rule.

Third layer: every node in this layer is a fixed-node labeled N. The nodes of this layer calculate the normalized output of second layer (Eq. 8):

$$O_{i}^{3}={\bar {w}_i}=\frac{{{w_i}}}{{{w_1}+{w_2}}},\quad i=1,2,$$
(8)

where \(O_{i}^{3}\) is the weight of normalized firing strength of the third layer.

The fourth layer: the output of this layer is the multiplication of previous layer output by a first-order polynomial (first-order Sugeno model) (Eq. 9):

$$O_{i}^{4}={\bar {w}_i}{f_i}={\bar {w}_i}({p_i}x+{q_i}y+{r_i}),$$
(9)

where \({\bar {w}_i}\) is normalized firing strength of third layer, and pi, qi, and ri are the parameter set of i nodes. These are referred to as consequent parameters, which can be adjusted during the training process.

Fifth layer: the single node in this layer is labeled ∑, which computes the overall output as the summation of all incoming signals (Eq. 10):

$$O_{i}^{5}={\text{overalloutput}}=\sum {\bar {w}_i}{f_i}=\frac{{\sum {w_i}{f_i}}}{{\sum {w_i}}},i=1,2,$$
(10)

where \(O_{i}^{5}\) is the ith node output of the fifth layer (Wang and Elhag 2008; Bui et al. 2012).

The final output of the ANFIS is the mean weighted outputs of all rules (Eq. 11):

$$y(k)=\frac{{\sum\nolimits_{{i=1}}^{l} {\left\{ {{\text{rul}}{{\text{e}}_i}(k)\left[ {{r_{1,i}}(k)+{p_{1,i}}(k)x+{q_{1,i}}(k)y} \right]} \right\}} }}{{\sum\nolimits_{{i=1}}^{l} {{\text{rul}}{{\text{e}}_i}(k)} }}.$$
(11)

To implement ANFIS for Firooz Abad-Kojour seismic rockfalls’ susceptibility mapping, MATLAB software was employed. It should be noted that, to determine the efficiency of ANFIS and modeling, data were normalized before entering to the software. Besides, to design ANFIS, five selected input parameters (shake intensity, distance from the road, slope angle, distance from the river, and ground strength class) were used based on the LR method (the fourth step of Table 4). After normalization and preparing data, the data of study area were divided into two groups of “train” and “test” data. In the present research, 70% and 30% of data from each hazard zone, respectively, were used as train and test data. The next step was ANFIS design. There are several methods to design ANFIS such as “Grid Partition” and “Subtractive Clustering”. In the present research, “Subtractive Clustering” method was used. In this method, not only there is no need to determine the type and number of the membership function, but also the designed algorithm by network determines the number of the membership function. It means that the number of the membership function is determined based on the frequency and distribution of information. In addition, in this method, the type of input and output membership functions is considered “Gaussian” and “Linear” by default, respectively. The advantage of “Subtractive Clustering” method is that the data mining of data obtain from study area and determination of whether the designed network needs adding complexity or not. Besides, it is possible to assign every data to different membership function. Therefore, there is no need to determine the number of membership function in modeling. On the other hand, if the number of the membership function is high, the number of rules will be increased, and there will be the possibility of Meta-learning; consequently, the system will lose its efficiency (Pradhan et al. 2010).

The designed ANFIS model for Firooz Abad-Kojour seismic rockfalls susceptibility mapping with 200 training epochs is shown in Fig. 9. It is worth pointing out that the number of training steps continues where the error-epoch curve becomes horizontal and error reaches a minimum value. In other words, the error reaches a constant value; therefore, increasing the number of training epochs has no effect on reducing the error value. As shown in Fig. 9, error-epoch curve became horizontal after 200 training epochs and the minimum error was 0.024992. Besides, based on the “Subtractive Clustering” method, 12 membership functions were assigned to each input parameter (Fig. 9).

Fig. 9
figure 9

Error-epoch curve of designed ANFIS model for Firooz Abad-Kojour seismic rockfalls’ susceptibility mapping

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The Firooz Abad-Kojour seismic rockfalls’ susceptibility zonation map using ANFIS is shown in Fig. 10. In the ArcGIS software, the seismic rockfalls’ susceptibility zonation map is classified into five classes of “very low hazard”, “low hazard”, “moderate hazard”, “high hazard”, and “very high hazard” due to the natural breaks (Swets 1988; Bednarik et al. 2010; Constantin et al. 2011; Pourghasemi et al. 2012a, 2013).

Fig. 10
figure 10

Hazard zonation map of the rockfalls induced by the Firooz Abad-Kojour earthquake of 2004 using adaptive neuro-fuzzy inference system (ANFIS)

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As shown in Fig. 10, the share of area susceptible to seismic rockfalls is reduced by increasing the hazard level in the corresponding zones and the majority of rockfalls triggered by Firooz Abad-Kojour earthquake of 2004 are placed at “high hazard”-to-“very high hazard” zones. Besides, the parts close to linking roads, rivers, the parts with a steep slope, the areas around earthquake epicenter, and parts with consisting of low-strength geological formation show a “high”-to-“very high” rockfall susceptibility level. Therefore, it can be concluded the higher and very desirable accuracy of ANFIS model in hazard zonation of rockfalls triggered by the Firooz Abad-Kojour earthquake of 2004.

Since rockfall is a prevalent event in mountainous slopes, coastal cliffs, volcanoes, river margins, and trenches, the rockfalls occur along the rivers and road’s transverse (Wyllie 2014). Since rockfalls mostly occur in cliffs and steep areas, and due to the steep morphology of waterway path and the road’s transverse in mountainous regions, the high concentrations of rockfalls in the area stricken by Firooz Abad-Kojour earthquake of 2004 seem logical (Fig. 10).

Results’ assessment and analysis

After providing Firooz Abad-Kojour seismic rockfalls’ susceptibility map using the LR and ANFIS methods, and classifying them into five classes based on natural breaks (very low, low, medium, high, and very high susceptibility), their accuracy was evaluated using two introduced indexes by Gee (1992), including density ratio (Dr) and quality sum (Qs), and also, receiver-operating characteristic (ROC) curve.

The dimensionless Dr index was independently used to compare the hazard classes of each map (Eq. 12) (Shariat Jafari 2009):

$${\text{Dr}}=\frac{{\frac{{{S_i}}}{{{A_i}}}}}{{\frac{{\sum\nolimits_{{i=1}}^{n} {{S_{\text{t}}}} }}{{\sum\nolimits_{{i=1}}^{n} {{A_{\text{t}}}} }}}},$$
(12)

where Dr is falling Dr of each susceptibility class, Ai is area ith susceptibility class in a zonation map, Si is the sum of seismic rockfalls in each susceptibility class, At is total area of the study region, St is the total number of seismic rockfalls in the study area, and n is the number of susceptibility classes. The better separation is performed between susceptibility classes by Dr results in a more useful map. In a given rockfall hazard zonation map, a class with Dr of 1 indicates the moderate hazard probability. Besides, the Dr less and higher than 1 shows the low and high hazard probability, respectively (Shariat Jafari 2009). The values of Dr index for both LR and ANFIS methods are presented in Tables 5 and 6, respectively. As shown in Tables 5 and 6, the Dr index of rockfalls has ascending trend with steep slope from very low to very high hazard class, which indicated the accuracy and good quality of zonation map classification by the LR and ANFIS methods.

Table 5 Hazard zonation map information generated using the LR model
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Table 6 Hazard zonation map information generated using ANFIS model
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To compare the outputs of Firooz Abad-Kojour seismic rockfalls susceptibility models by LR and ANFIS methods, Qs index was used. The Qs was calculated for each map by the following equation:

$${\text{Qs}}=\mathop \sum \limits_{{i=1}}^{n} \left( {{{\left( {{\text{Dr}} - 1} \right)}^2} \times \% ~{\text{area}}} \right),$$
(13)

where i is hazard class number in a rockfall hazard zonation map, and n is the total number of hazards classes. The Qs values indicate the performance quality of the model in predicting the hazard of rockfalls in the study area. Typically, there is no theoretical limit to the value of Qs index (Shariat Jafari 2009). In other words, the higher Qs values indicate the optimum model superior in rockfall hazard zonation.

The values of Qs for Firooz Abad-Kojour seismic rockfall hazard zonation map using both LR and ANFIS methods are presented in Tables 5 and 6, respectively.

The information of Firooz Abad-Kojour seismic rockfalls susceptibility map using the LR is presented in Table 5. The table also shows the area percentage of each rockfall susceptible zone and percentage of seismic rockfalls occurred in each zone. Then, by obtaining those values and applying Eqs. 12 and 13, the Dr of each hazard zone and Qs of the hazard zonation map prepared using the LR were calculated (Table 5). According to the calculated Dr for each rockfall hazard zone in Table 5, Dr increases ascendingly with an increase in the hazard level. Moreover, the high value of Qs index (21.04) indicates the proper accuracy of LR method in Firooz Abad-Kojour seismic rockfalls susceptibility mapping.

Table 6 presents the information of Firooz Abad-Kojour seismic rockfalls susceptibility map using the ANFIS. The table also shows the area percentage of each rockfall susceptible zone and percentage of seismic rockfalls occurred in each zone. Then, by obtaining those values and applying Eqs. 12 and 13, the Dr of each hazard zone and Qs of the hazard zonation map prepared using the ANFIS were calculated (Table 6). According to the calculated Dr for each rockfall hazard zone in Table 6, Dr increases ascendingly with an increase in hazard level. Moreover, the higher value of Qs (26.75592) indicates the very desirable and acceptable accuracy of ANFIS method in Firooz Abad-Kojour seismic rockfall susceptibility mapping.

To evaluate and compare the accuracy of both LR and ANFIS models, ROC curve was used (Yesilnacar 2005; Nefeslioglu et al. 2008; Mathew et al. 2009; Mohammady et al. 2012; Pourghasemi et al. 2012b, c; Pradhan 2013; Regmi et al. 2014). Finally, a map with maximum accuracy was suggested as final Firooz Abad-Kojour seismic rockfalls’ susceptibility map.

ROC curve is one of the most important and efficient methods in providing a characteristic, probabilistic identification, and prediction of a system that quantitatively measures the accuracy of the model (Yesilnacar 2005). In ROC curve, the true positive rate (TPR) was drawn against False-Positive Rate (FPR). In the study area, TPR or “sensitivity” and FPR or “specificity”, respectively, indicate the probability degree of pixels with and without rockfalls, which were determined correctly by model. The surface under ROC curve, which is known as area under the curve (AUC), shows the model accuracy in predicting the hazard of seismic rockfalls. The ideal model has the highest AUC, which ranges from 0.5 to 1. The qualitative–quantitative correlation of AUC and the estimation evaluation are as follows: 0.9–1 excellent, 0.8–0.9 very good, 0.7–0.8 good, 0.6–0.7 medium, and 0.5–0.6 weak (Yesilnacar 2005). The ROC curve associated with both LR and ANFIS methods, as well as AUC values are presented in Fig. 11.

Fig. 11
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ROC curve and AUC values associated with Firooz Abad-Kojour seismic rockfall hazard zonation map using the LR and ANFIS methods

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Based on ROC curve and AUC values (Fig. 11), the AUC for LR and ANFIS methods were calculated 0.972 and 0.984, respectively. Considering the AUC values for LR and ANFIS methods, the ANFIS-based hazard zonation map had higher accuracy compared to LR-based hazard zonation map.

Conclusions and discussion

Given that no research has been conducted on zonation and mapping the susceptibility of seismic rockfalls using intelligent methods (such as ANFIS), the present research was carried out to implement the ANFIS and LR methods in the area stricken by Firooz Abad-Kojour earthquake of 2004, evaluate and select the optimum model with high accuracy in prediction, mapping the susceptibility of seismic rockfalls, and apply this method to predict probability of rockfall events induced by future earthquakes. In LR method, after four steps of analysis, the suitable weights for selected parameters (shake intensity, distance from the road, slope angle, distance from the river, and ground strength class) were obtained based on “backward conditional” rule in SPSS software. Considering the LR-based hazard zonation map, the majority of Firooz Abad-Kojour seismic rockfalls falls in the “high”-to-“very high” hazard zones. Moreover, the areas close to linking roads, rivers, and low-strength geological formations had “high”-to-“very high” rockfall susceptibility levels. The minimum error of designed ANFIS model for Firooz Abad-Kojour seismic rockfalls susceptibility mapping with 200 training epochs was 0.024992. Considering the ANFIS-based hazard zonation map, the majority of Firooz Abad-Kojour seismic rockfalls was matched on “high”-to-“very high” zones. In addition, the areas close to linking roads, rivers, parts with a steep slope, the area around earthquake epicenter, and low-strength geological formations had “high”-to-“very high” rockfall susceptibility levels.

Rockfalls are very common events that occur in mountainous slopes, coastal cliffs, volcanoes, riverside, and trenches. Since rockfalls mostly occur in cliffs and steep area, and due to the steep morphology of waterway path and road’s transverse, we focused on rockfalls in the area stricken by Firooz Abad-Kojour earthquake of 2004.

After preparing the Firooz Abad-Kojour seismic rockfalls susceptibility map using LR and ANFIS methods, criteria including Dr, Qs, and ROC were used to evaluate the considered map and determine its accuracy. The Dr of rockfalls had an ascending trend with high slope from “very low” to “very high” hazard class, suggesting the good quality and accuracy of LR and ANFIS-based hazard zonation maps. By analyzing the susceptibility maps and considering the Qs index obtained by LR (21.04184) and ANFIS (26.75592), it could be observed that the Qs value of ANFIS is higher than that of LR. Moreover, based on the obtained value of AUC from LR (0.972) and ANFIS (0.984) methods, ANFIS provided a higher accuracy in zonation and susceptibility mapping of rockfalls triggered by Firooz Abad-Kojour earthquake of 2004 compared to LR method. Therefore, based on the verification done using the criteria such as “Dr” and “Qs”, and ROC curve, ANFIS-based Firooz Abad-Kojour seismic rockfalls susceptibility mapping was in agreement with the rockfall distribution map of the area and showed a higher accuracy compared to LR method.

Performing an analysis on the effect of geomechanical properties of geological material on probability of rockfall event, the generalization of this approach for synergetic seismic and hydrological activities (such as snow melting and rainfall), and evaluating the accuracy of ANFIS model in comparison with artificial neural network (such as MLP and RBF) are recommended for future research.