2. Institute of Geosciences, State University of Campinas (UniCamp), Campinas, São Paulo, Brazil
Landslides are a type of catastrophic events. As typical geological hazards, landslides can destroy roads and buildings, and lead to human injury or even loss of life. In the landslide literature, increasing attention is being paid to the spatial distribution characteristics of landslides (e.g., Gorum and Carranza, 2015; Ghosh and Carranza, 2010), temporal and magnitude probabilities of landslides (e.g., Ghosh et al., 2012a, b; Guzzetti et al., 2005) and predictive mapping of landslide susceptibility (Ghosh et al., 2011, 2010, 2009; Poli and Sterlacchini, 2007). These aspects of landslide analysis are essential to understand and recognize the key conditioning factors of landslide formation in order to mitigate and prevent the occurrence of landslides.
Landslides can be triggered either by earthquakes or rainfalls, or they can be induced by man's activities. The various conditioning factors of landslide formation typically include faults, rivers, roads, geological boundaries, elevation, slope angle, slope aspect, and many others, depending on the type of landslides. Landslide susceptibility mapping is a hot topic in the field of landslide risk assessment. In a given study area, there are several steps for mapping landslide susceptibility: (ⅰ) conducting an inventory of past/present landslides and constructing a landslide inventory map; (ⅱ) identifying the conditioning factors of landslide formation; (ⅲ) generating spatial maps of conditioning factors; (ⅳ) measuring the spatial relationship between the conditioning factors and past/present occurrences of landslides in order to assign weights to conditioning factors; (ⅴ) integrating multiple layers of conditioning factors using a suitable spatial decision model; (ⅵ) validating and evaluating the performance of landslide susceptibility map. The various methods that have been successfully applied to assign weights to conditioning factors and to integrate multiple layers of conditioning factors can be divided into qualitative and quantitative groups. The group of qualitative methods consists of field geomorphological analysis and use of index or parameter maps. The group of quantitative methods comprises statistical analysis, geotechnical engineering approaches, and spatial decision models. How to measure and assign the weights of conditioning factors is crucial for quantitative assessment of landslides risk. The various quantitative methods for mapping landslide susceptibility include evidential belief function (e.g., Lee et al., 2013; Althuwaynee et al., 2012), logistic regression (e.g., Tsangaratos and Ilia, 2016; Ghosh et al., 2011), artificial neural networks (e.g., Tsangaratos and Benardos, 2014; Lee et al., 2004), fuzzy logic (e.g., Alimohammadlou et al., 2014; Zhu et al., 2014), support vector machine (e.g., Hong et al., 2015; Pradhan, 2013), fuzzy based study (e.g., Barrile et al., 2016; Oh and Pradhan, 2011), and geographicallyweighted principal component analysis (e.g., Faraji Sabokbar et al., 2014).
The fractal/multifractal theory and model has been applied to investigate landslides' characteristics. Studies have been demonstrated that (1) the frequencyarea statistics of landslides exhibited a powerlaw distribution (Ghosh et al., 2012b; Trigila et al., 2010; Turcotte and Malamud, 2004; Guzzetti et al., 2002); (2) the frequencysize statistics of landslides satisfied powerlaw (fractal) distributions (Malamud et al., 2004; Iwahashi et al., 2003; Pelletier et al., 1997); (3) the cumulative frequency of the landslide occurrence satisfied a power law function (fractal) of the landslidetriggering rainfall (Li et al., 2011), and (4) the spatial distribution of landslides was not a homogeneous fractal structure but a heterogeneous one (Rouai and Jaaidi, 2003). However, the question of how to evaluate the weights of conditioning factors from a fractal perspective is not well explored. The purpose of this paper is to propose a different method for measuring the spatial associations between conditioning factors and landslides in order to estimate/assign weights to conditioning factors. Although there are various conditioning factors of landslide formation, only natural as well as manmade linear features are considered here namely faults, rivers, geological boundaries, and roads. Two case studies from South China are presented to demonstrate the usefulness of the method proposed in this study.
1 METHOD AND DATA 1.1 Fractal ModelThe fractal model is proposed in this study to estimate the relative importance of conditioning factors of the spatial distribution of landslides. Wang et al. (2015) and Zuo (2016) reported a nonlinear controlling function of geological features on magmatichydrothermal mineralization, and proposed an alternative method to measure the spatial relationships between geological features and mineral deposits from a fractal and multifractal theory. Mineral deposits and geohazards (e.g., landslides, earthquakes) are, depending on scale, geological point processes (Zuo et al., 2009) and are singular physical processes associated with anomalous amounts of energy release or material accumulation within a narrow spacialtemporal range (Cheng, 2007). Geological point processes can be and have been modelled by fractal/ multifractal modelling (e.g., Zuo, 2016; Wang et al., 2015; Blenkinsop, 2014; Agterberg, 2013; Zuo et al., 2009; Raines, 2008; Cheng and Agterberg, 1995; Carlson, 1991).
From a fractal/multifractal theory point, the density of landslides (ρ) versus ε can be expressed as (cf., Agterberg, 2012; Cheng, 2012, 2008, 2007)
$ \rho \left(\varepsilon \right) = C{\varepsilon ^{  d}} $  (1) 
where ε denotes distance to conditioning factors (e.g., faults, rivers, geological boundaries, or roads), d is the fractal dimension and C is a constant.
Based on Eq. (1), the fractal dimension (d) can be estimated from the following linear relation
$ \log \rho = \log C  d\log \varepsilon $  (2) 
With the support of GIS, buffer analysis with i number of round buffer zones can be implemented, the number of landslides (N_{i}) occurring within each buffer zone (ε_{i}) was counted. The density of landslides ρ_{i} was then calculated as N_{i}/ε_{i}. A straight line can be fitted to a paired dataset of [ρ_{i}, ε_{i}] using the leastsquares method. The d can be estimated from the slope of the fitted straight line based on Eq. (2). The fractal dimension d characterizes how the density of landslides in areas, with buffer width ɛ, arounds a linear conditioning factor changes when the value of ɛ increases. A value of d > 0 indicates that the density of landslides is a decreasing function of the buffer width ɛ, which means that the closer the distance to conditioning factors, the higher the density of landslides is. Therefore, a value of d > 0 represents a significant positive spatial association between a specific conditioning factor and landslides, and increasing positive d ( > 0) values indicate increasing significant positive spatial correlation. Conversely, a value of d < 0 indicates a negative spatial association. A value of d=0 suggests spatial independence.
1.2 DataLinear natural or manmade features as conditioning factors of landslides, which include faults, rivers, geological boundaries, and roads, are selected for demonstration of the proposed method in the two case studies. Faults are an important factor of landslides because they represent zones of weaknesses where landslides can form/occur. Rivers influence the occurrence of landslides because they represent areas of high erosion activity and areas close to rivers are more susceptible to slope instability (Lee and Chi, 2011). Geological boundaries are a significant factor of landslides because areas proximal to geological boundaries have geomechanical values that are lower than those that are distal to geological boundaries, and hence are more susceptible to landslides (Tsangaratos and llia, 2016; Kawabata and Bandibas, 2009). The construction of roads could decrease the stability of rocks and soils, and therefore areas proximal to roads are usually more susceptible to landslides than areas distal to roads (Pradhan and Lee, 2010).
Two separate datasets consisting of a geological map, faults, rivers, roads, and locations of landslides were digitized for two case study areas in South China (Fig. 1). The first dataset from Yongjia County in Zhejiang Province of China (Zhang et al., 2005) consists of locations of landslide occurrences, a geological map, rivers, and roads. Four conditioning factors including faults, rivers, geological boundaries, and roads are shown in Fig. 2. The small landslides are dominant in this dataset. The second dataset from Lianhua County situated at the middle Luoxiao Mountains, Jiangxi Province of China includes 163 locations of landslides, and 14 conditioning factor maps (Hong et al., 2006). Maps of three conditioning factors of faults, rivers, and roads were selected for analysis in this study (Fig. 3). The landslide inventory in the Lianhua Country was identified through on aerial photographs and extensive field surveys. The extent of the smallest and largest landslides in this area is 15 m^{2} and 48 000 m^{2}, respectively. The detailed geology, hydrogeology and geomorphology, and the datasets used in this study for each of the two case study areas can be found in Zhang et al. (2005) and Hong et al. (2016), respectively.
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Maps of distances to faults, rivers, geological boundaries, and roads are created using a GIS. A 10ring round buffer was constructed at intervals of 200 m around faults, rivers, geological boundaries, and roads (Figs. 2 and 3). The width of buffer of 200 m used in this study is randomly selected in performing buffer analysis. From a spatial statistic point, the selection of buffer width do not influence the slope of the loglog plot of landslide density versus distance to linear conditioning factors, and therefore, the resulting fractal dimension could not change when different buffer widths were used. However, in a particular case, the area of influence of linear conditioning factors such as crushed material and transitional boundaries, and other factors should be considered when choosing the buffer width for performing such analysis.
It was observed in both Figs. 4 and 5 that the greater the proximity to conditioning factors, the greater the number of landslides, and the loglog plots of landslide density versus distance to conditioning factors can be fitted with straight lines by the leastsquares method. This phenomenon suggest a nonlinear spatial relationship between landslide density and distance to conditioning factors, meaning that landslide density ρ can be expressed as a powerlaw function of distance to conditioning factors (ε).
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In Yongjia County, the fractal dimensions of faults, rivers, geological boundaries, and roads were 0.54, 0.59, 0.57, and 0.59, respectively (Fig. 4). The fractal dimensions for rivers and roads are greater than those for faults and geological boundaries, indicating that the former have stronger spatial relationships with landslide occurrences in Yongjia County. This suggests that rivers and roads played a much more important role in landslide occurrence in Yongjia County.
The Lianhua data were compiled from Hong et al. (2016), in which only three linear conditioning factors are considered. Therefore, in this study, three linear conditioning factors including faults, rivers, and roads were analyzed. The fractal dimensions of faults, rivers, and roads were 0.14, 0.45, and 0.48 (Fig. 5). Roads have the largest fractal dimension among the three studied conditioning factors, suggesting that roads have a critical role in landslide occurrence in Lianhua County. In other words, in Lianhua County, much more attention should be given to areas near to roads for the prevention and/or mitigation of landslide hazards.
The fractal dimension (d) provides an objective measure of spatial associations between conditioning factors and landslide occurrence. Based on the value of d, the order of relative importance of conditioning factors on landslide occurrence can be ranked. In Yongjia Country, the order of relative importance is roads = rivers > faults > geological boundaries; whereas in Lianhua Country, the order is roads > rivers > faults. These orders suggest that the local geology, hydrogeology and geomorphology influence the locations of roads and, in turn, the locations of roads have a critical role in landslide occurrence, and thus provide critical information for road planning for landslides prevention. The goodness of fit of the loglog plot of landslide density versus distance to faults (Fig. 5a) is significantly less (0.66) as compared to all others because of a gentle slope of this plot linked to a lower value of fractal dimension, suggesting that in Lianhua Country the faults play a less important role in the spatial distribution of landslides than other conditioning factors.
The following linear equation can be used to integrate multiple conditioning factors for mapping landslide susceptibility
$ Y = {b_1}{x_1} + {b_2}{x_2} + ... + {b_n}{x_n} = \sum\nolimits_{i = 1}^n {{b_i}{x_i}} $  (3) 
where Y denotes landslide susceptibility, increasing values of which mean increasing landslide susceptibility, b_{i} (i=1, 2, …, n) is the weight of x_{i} conditioning factor, and n is the total number of conditioning factors. The fractal dimension can be assigned as the values of b_{i} for the conditioning factors x_{i}. The value of b_{i} can be directly estimated from the slope of the straight line fitted to a loglog plot of landslide density versus distance to conditioning factor x_{i}. Therefore, in Yongjia Country, landslide susceptibility can be estimated as Y=0.54x_{faults}+0.59x_{rivers}+0.57x_{geological boundaries}+ 0.59x_{roads}; whereas in Lianhua Country, landslide susceptibility can be estimated as =0.14x_{faults}+0.45x_{rivers}+0.48x_{roads}.
The fractal (or powerlaw) relation between landslide density and distance to conditioning factors can be used as factor (or evidence) value per cell in raster layer of conditioning factor. In mapping of landslide susceptibility, each conditioning factor map (x_{i}) is often converted into a raster map with multiple classes and each class is assigned as a factor (or evidence) value in the [0, 1] range based on either statistical analysis or expert opinion. Based on the powerlaw relation of Eq. (2), let where w is a constant to derive values of f(ε) in the [0, 1] range. Then, using Eq. (4), a raster map with values of f(ε) can be obtained. This raster map can be further used for mapping landslide susceptibility.
$ f(\varepsilon) = \frac{\rho }{w} = \frac{C}{w}{\varepsilon ^{  d}} $  (4) 
In this study, a fractal model is proposed for measuring the spatial relationships between conditioning factors and landslide occurrence, and the following conclusions are obtained: (1) landslide density follows a fractal relation with distance to linear features such as faults, rivers, geological boundaries, and roads; (2) the fractal dimension is a robust parameter for measuring the relative importance of conditioning factors of landslide occurrence, and provides critical information for the prevention and/or mitigation of landslide hanzard; and (3) the fractal relation proposed in this study also can be used as factor (or evidence) value per cell in raster layer of conditioning factor.
ACKNOWLEDGMENTS
The authors thank two anonymous reviewers for their valuable comments that helped us improve the manuscript, and Ziye Wang and Yihui Xiong from China University of Geosciences (Wuhan) for preparing parts of the datasets. Renguang Zuo thanks Wang Yang from China University of Geosciences (Wuhan) for providing a part of dataset used in this study. This research benefited from the joint financial support from the National Natural Science Foundation of China (No. 41522206), and the MOST Special Fund from the State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences (No. MSFGPMR033). The final publication is available at Springer via http://dx.doi.org/10.1007/s1258301707722.
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