The effects of slope length and slope gradient on the size distributions of loess slides: Field observations and simulations

Geomorphology 300 (2018) 69–76

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The effects of slope length and slope gradient on the size distributions of loess slides: Field observations and simulations Haijun Qiu a,b,c,⁎, Peng Cui b, Amar Deep Regmi b, Sheng Hu a, Xingang Wang d, Yuzhu Zhang a a

College of Urban and Environmental Science, Northwest University, Xi'an 710127, China Institute of Mountain Hazard and Environment, Chinese Academy of Sciences, Chengdu, Sichuan 610041, China Institute of Earth Surface System and Hazards, Northwest University, Xi'an 710127, China d State Key Laboratory of Continental Dynamics, Department of Geology, Northwest University, Xi'an 710069, China b c

a r t i c l e

i n f o

Article history: Received 10 May 2016 Received in revised form 9 July 2017 Accepted 24 October 2017 Available online xxxx Keywords: Loess slides Size distribution Simulations Rollover

a b s t r a c t In this study, we characterize and consider the effects of slope length and slope gradient on the size distributions of loess slides. To carry out this study, we employ data on 275 loess slides within Zhidan County, Central Loess Plateau, China. These data were collected in the field and supplemented by the interpretation of remote sensing images. Both the field observations and slope stability analysis show that loess slide size increases with the slope length. Slide sizes is significantly correlated with slope length, showing a power law relationship in both cases. However, the simulation results show that slope gradient is not associated with loess slide size. The main part of the link between slope gradient and slide size seen in the observations is only apparent, as indicated by the strong connection between slope gradient and length. Statistical analysis of the field observations reveals that slope gradient decreases with increasing slope length, and this correlation interferes with the potential relationship between landslide sizes and slope gradient seen in the field observations. In addition, the probability densities of the areas of loess slides occurring on slopes of different slope lengths are determined using kernel density estimation. This analysis shows that slope length controls the rollover of the frequency-size distribution of loess slides. The scaling exponent increases with slope length. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Loess sediments cover approximately 10% of the earth's surface (Liu, 1985). Loess is extensively distributed in the arid regions of China, and it accounts for approximately 6.6% (631,000 km2) of the total area (Liu, 1985; Tu et al., 2009). Loess landslides, which are a major and common engineering problem, are a persistent threat to human activities in loess-covered terrain (Derbyshire, 2001; Xu et al., 2013). Increases in population, unplanned urbanization and the excavation of slopes to produce road cuts amplify the impact of loess landslides (Derbyshire, 2001; Zhang and Liu, 2010). Meanwhile, owing to the high costs of controlling these landslides through engineering measures and rational land-use planning, casualties and economic losses are becoming more severe in steep mountainous regions (Derbyshire, 2001). Proper characterization of the size distributions of landslides is extremely important in determining landslide hazard (Guzzetti et al., 2005), quantifying the integrated effects of erosion and sediment yield caused by landslips (Hovius et al., 2000; Stark and Hovius, 2001; Martin et al., 2002; Brardinoni and Church, 2004; Guthrie and Evans, ⁎ Corresponding author at: College of Urban and Environmental Science, Northwest University, Xi'an 710127, China. E-mail address: [email protected] (H. Qiu).

https://doi.org/10.1016/j.geomorph.2017.10.020 0169-555X/© 2017 Elsevier B.V. All rights reserved.

2004; Hungr et al., 2008; Chen, 2009) and evaluating the magnitude of landslide events (Hungr et al., 1999; Guzzetti et al., 2002; Malamud et al., 2004a; Van Den Eeckhaut et al., 2007; ten Brink et al., 2009). Topography affects slope hydrological processes and thus slope stability, as has been studied by several researchers (Sidle and Onda, 2004; Gao and Maro, 2010; Xu et al., 2013). Frattini and Crosta (2013) examined the effects of topography on the distribution of landslide size using the virtual tiling method and investigated the effects of material properties on landslide size through 2D limit equilibrium slope stability analyses. Similarly, Liu and Koyi (2013) noted that landslide size increases with increasing material strength. Katz et al. (2014) studied the controls on the sizes and geometries of individual landslides using a numerical two-dimensional discrete element model and found that landslide size increases with decreasing slope gradient for a given material strength. Chen et al. (2016) performed a study of the controls on landslide size using limit equilibrium simulations and stated that sliding volume increases with decreasing slope gradient. However, controversy regarding on the relationship between landslide size and slope gradient still exists. In particular, few studies have examined landslide size with respect to slope length. Thus, it is very necessary to ascertain the controls on landslide size using field observations and develop a proper simulation model to enable understanding of the field observations.

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Considerable research has addressed landslide size distributions, and a rollover effect has been identified below a certain threshold within landslide frequency-size distributions (Guzzetti et al., 2002; Malamud et al., 2004a). Many empirical observations have shown that landslide size distributions obey a power-law (fractal) correlation for large landslides (Brunetti et al., 2009; Frattini and Crosta, 2013). Some authors have attributed the rollover effect to undersampling and influence of soil moisture, topography and material properties (Pelletier et al., 1997; Stark and Hovius, 2001; Malamud et al., 2004a; Frattini and Crosta, 2013). Though the frequency distribution of landslides shows a power-law scaling for landslides with sizes greater than the threshold, the causes of the rollover effect are not clearly understood (Guzzetti et al., 2002; Frattini and Crosta, 2013). In the present study, the effects of slope length and slope gradient on the size distributions of loess slides are evaluated. We quantify the response of loess slide size to slope length and slope gradient using a power-law form and a containing field observations of 275 loess slides. In addition, a 2D slope stability analysis using the general limit equilibrium method is performed to reproduce the field observations. Finally, the rollover effect of the loess slide probability distribution is analysed using kernel density estimation, and the contributing factors to the rollover effect are also considered. 2. Materials and methods 2.1. Study area Zhidan County is bounded by the latitudes 36°21′23″–37°11′47″ N and the longitudes 108°11′56″–109°3′48″ E, and it covers an area of approximately 3781 km2 in the central part of the Loess Plateau of northern China (Fig. 1). This area contains rugged and steep mountainous terrain, which arises from the combined effects of intermittent tectonic uplift, valley incision and soil erosion on hillslopes (Gao et al., 2016). The elevations within the area range from 1054 to 1714 m above sea level; the mean and standard deviation of these elevations are 1427.40 m and 103.64 m, respectively. The area experiences a

typical temperate continental monsoon climate, and the mean annual rainfall and air temperature are 524.5 mm and 7.8 °C, respectively. The amount of rainfall that occurs in different seasons varies significantly. Inspection of the historical precipitation record indicates that approximately 56% of the annual precipitation falls in the rainy season (June to August), and mass movements and severe water erosion also occur during this season. The river network has a dendritic pattern. A dense drainage network dissects the basin, which has a drainage density of 3.3 km−1. Loess is a homogeneous, porous, friable and predominantly silt-sized sediment type that forms through the accumulation of wind-blown silt (Liu, 1985; Frechen, 2011). Loess particles primarily range from coarse to medium silt (0.01–0.06 mm) (Derbyshire, 2001). Vertical joints and large pores are well developed in loess (Derbyshire, 2001; Qiu et al., 2017). The study area is underlain by a thick Quaternary loess deposit. The late Pliocene Red Clay overlies a pre-Tertiary base and ranges in thickness from 20 to 50 m. The middle Pleistocene Lishi Loess overlies the Red Clay and is approximately 60–100 m in thickness. The lower part of the Lishi Loess contains paleosols and calcareous nodules. The late Pleistocene Malan Loess, which lies atop the Lishi Loess, is widely distributed in Zhidan, and it has thicknesses of approximately 10–30 m (Liu, 1985; Derbyshire et al., 1991; Derbyshire, 2001). The main exposed outcrops are of loess deposits. The pre-Tertiary units are exposed only at the toes of slopes in the bottoms of deep river valleys. Area with relief of b200 m covers N85% of the study area, and most of the landslides are observed within these low-relief areas. According to the field observations, all of the loess slides actually occurred in the loess substrate. None of the slides are cut in other pre-Tertiary basement rocks. Only a few of the loess slides move along a bedrock surface. 2.2. Landslide data The Loess Plateau can be considered to be relatively homogeneous from a geological point of view (Liu, 1985). The area's highly fragmented topography, which is very prone to sliding, was caused by the intermittent tectonic uplift that started during the middle

Fig. 1. Location of the study area in Zhidan County, Shaanxi province, China. Red polygon in the figure denotes the study area. The sizes of the white dots represent the log-transformed areas of the loess slides.

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Pleistocene, coupled with severe soil erosion (Zhang et al., 2014). Loess landslides are abundant and frequent in the study area due to its lithological, morphological and climatic properties. According to the landslide classification of Cruden and Varnes (1996), most of the observed landslides are loess slides. Although remote sensing provides a convenient and efficient means of identifying landslides, its application is limited by the presence of forest canopies (Brardinoni and Church, 2004). Thus, a series of intensive field investigations were conducted to detect loess slides and measure their volumes within each traversed polygon. These investigations were aided by remote sensing, which was used to identify the locations of loess slides before the field surveys were conducted, and recent advances in mobile technologies and unmanned aerial vehicles (UAVs) were applied to improve the collection of field-data. As a result, 275 loess slides were mapped and stored in a GIS (ArcGIS by Esri) landslide database, which permits efficient data analysis (Fig. 1). The information in this database includes the geographical locations of the loess slides and their associated slope lengths and slope gradients, as well as the geometrical properties of the loess slides, including their lengths, widths, depths, areas and volumes. We use the rupture surface, which does not include the displacement component, to represent the lengths of most of the loess slides. It is difficult to separate the rupture surface from the displacement length in some cases, and we included both in those cases. However, such landslides are insignificant in number. Although the determination of landslide length is sometimes dependent on the experience and skill of the geomorphologist, the measurement of landslide length in the field is sufficiently accurate for our purposes. Slope length is the distance between slope crest and toe. The slope gradient represents the average value over the whole slope length. The depths of loess slides were estimated in the field. However, not all the information in the database is available for all the loess slides. Some geometrical properties are known accurately, whereas others are inferred from maps and the highly accurate DEMs obtained using UAVs. Area is estimated as the product of length and width (Brardinoni and Church, 2004). The volumes of a loess slide is estimated as the product of the area and the estimated mean depth (Brardinoni and Church, 2004; Guzzetti et al., 2009). However, geomorphological considerations suggest that the measurement of volume is sufficiently accurate (Guzzetti et al., 2009). The overwhelming majority of loess slides in the study area are small- and medium-scale slides, and these features account for approximately 88% of the total number of loess slides. Approximately 75% of the loess slides were triggered by intense or prolonged rainfall. Some were induced by human interventions on slopes, such as excavation. Most of the loess slides are shallow, and the mean depth is approximately 8 m. The slides with depths b 10 m make up approximately 85% of the total number of loess slides. About 80% of loess slides are retrogressive landslides. The loess slides examined in this study do not include a flow component. In addition, we did not distinguish translational and rotational slides in the field; instead, these loess slide types were mixed within the data set. Fig. 2a, c shows photographs of two typical loess slides within the study area. Fig. 2b, d represents the corresponding DEMs of these loess slides, which were obtained using a UAV. Fig. 2a, b shows typical shallow loess slides, which belong to the dominant type. The slides shown in Fig. 2c, d were induced by road cuts. 2.3. Methods 2.3.1. Frequency–size relations Magnitude–frequency statistics have been widely used in seismology; a power-law relationship between cumulative frequency and earthquake magnitude describes the Gutenberg-Richter relation in a robust manner (Gutenberg and Richter, 1954). By analogy, accumulating evidence suggests that the relationship describing landslide size versus frequency is scale invariant and exhibits power-law (fractal) statistics

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under a wide variety of circumstances (Guzzetti et al., 2002; Turcotte et al., 2002; Corominas and Moya, 2008). The linear portion of the segment of this non-cumulative power-law relation can be expressed as: NL ¼ CAL −α

ð1Þ

where NL is the non-cumulative number of landslides in a region with sizes (e.g., areas) greater than or equal to the landslide size AL and C and α are constants. Moreover, landslide frequency-size distributions are also characterized by a rollover towards small slide sizes. 2.3.2. Loess slide simulations and experimental setup A 2D slope stability analysis was carried out using the general limit equilibrium (GLE) method using the SLOPE/W (GeoStudio 2007) software package. The GLE formulation was developed by Fredlund in the 1970s and satisfies two factors of safety (Fs) equations (Fredlund and Krahn, 1977; Chen et al., 2016). One equation gives the Fs regarding moment equilibrium, while the other equation gives the Fs regarding horizontal force equilibrium (Fredlund, 1974; Fredlund and Krahn, 1977). The SLOPE/W routine offers some options that enable searching for the critical slip surface within a slope body. In this work, we did not specify a slip surface; instead, we used the Auto-locate option in this software package to identify the critical slip surface, since it can lead to a more reasonable result (GEO-SLOPE International Ltd., 2010). SLOPE/W generates 1000 trial slip surfaces to search for the most likely minimum slip surface and applies an optimization procedure to determine the minimum Fs (GEO-SLOPE International Ltd., 2010). Finally, the trial slip surface with the lowest value of Fs is considered to be the critical slip surface of the slope (Chen et al., 2016). The input slope material strength parameters included the cohesion, C; the internal friction angle, φ; and the material unit weight, γ. Based on the average values of the material properties of the loess in the study area (Lei, 2014), the values of 45 kPa, 30° and 19 kN m− 3 were used for these parameters. According to the field observations, the loess slides within the study area can be categorized into three classes, based on their potential shear surfaces. These classes include landslides in loess, red clay contact landslides and bedrock contact landslides. This area has thick loess deposits and a deep groundwater table. In addition, it is difficult to obtain information on the position of groundwater level for all of the landslides. Most importantly, the purpose of this work is to determine how landslide size changes with slope length and slope gradient. Hence, reasonable results can be obtained only when the values of other parameters, such as pore pressure, are fixed. Based on this consideration, variations in the water table geometry and shear surfaces are not considered in our simulations. We change only the slope length and slope gradient and use fixed values for the others parameters. 3. Results 3.1. Statistics of loess slides sizes There exists a distinct linear relationship in log-log coordinates between areas and volumes of loess slides that spans multiple orders of magnitude and has a scaling exponent of 1.1845 (Fig. 3). The volumes of individual landslides cover 4 orders of magnitude and vary from 450 m3 to 3,618,000 m3. The mean volume of all loess slides is approximately 452,390 m3, and the corresponding coefficient of variation is 11.66. The 10 largest loess slides, which represent 3.64% of the total number, account for 22.81% of the total volume of the loess slides (Fig. 3). 3.2. Probability densities of the areas of loess slides The probability densities of the areas of loess slides were examined using kernel density estimation for all the studied loess slides in Zhidan

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Fig. 2. Photographs and corresponding DEMs of typical loess slides in the study area. (a) (c) Typical photographs of loess slides; (b) (d) Highly accurate DEMs obtained from a UAV.

County. Fig. 4 shows the non-cumulative distributions of the areas of loess slides. This relation is not strictly linear in a log-log coordinates; instead, the so-called rollover effect is present. The loess slide frequencysize curves depart from the linear fractal relationship when the size of loess slides is smaller than 24,800 m2.The abundance of loess slides increases with area up to maximum value, followed by a decrease

along a power-law. The thick red line in Fig. 4 is the power-law obtained by least squares regression of the areas and probability densities of the largest 130 largest loess slides in the database, and the scaling exponent of this power-law is 1.988.

Fig. 3. Empirical relationship between the areas and volumes of loess slides measured in the field. The solid line indicates the regression line. The box plot shows descriptive statistics of the areas and volumes of the loess slides.

Fig. 4. Probability densities of the areas of loess slides within Zhidan County area displayed using kernel density estimation on logarithmic axes. The right tail of the distribution scales as a power law (thick red line) with a scaling exponent of α = 1.988.

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3.3. Variations in loess slide size with slope length in loess terrain 3.3.1. Variations in loess slide size with slope length in the field observations Fig. 5 shows the variations in the area of loess slides, which is a proxy for landslide size, as slope length changes. Although there is little theoretical basis for choosing a function to describe the relationship between the sizes of loess slides and their slope lengths, the significant increase in loess slide size with slope length is well described (R2 = 0.7058; P ≪ 0.01) by a simple power-law. AL ¼ 0:31125LL 1:8320 where AL is the area of a loess slide, and LL is the slope length. 3.3.2. Variations in loess slide size with slope length in the simulations To study the effects of slope length on the sizes of loess slides, we conducted 30 numerical experiments to determine the cross-sectional areas of sliding masses, given different slope lengths and slope gradients appropriate for natural slopes. This proposed proxy reflects landslide size only approximately. We calculated the cross-sectional areas of the sliding masses using the two-dimensional GLE method incorporated in the software SLOPE/W of GeoStudio 2007 using slope lengths of 50, 100, 150, 200, 250, 300, 350, 400, 450 and 500 m and slope gradients of 20°, 30° and 40°. Fig. 6 shows the cross-sectional areas of the sliding masses as a proxy for landslide size in relation to the slope length for different slope gradients. The sizes of loess slides are clearly significantly correlated with slope length (Fig. 6). The sizes of loess slides increase with slope length, and this behaviour is well described by the simple power-law relation. 3.4. Variations in loess slide size with slope gradient in loess terrain 3.4.1. Variations in loess slide size with slope gradient in the field observations Fig. 7 shows the sizes (areas) of loess slides size in relation to the slope gradient. The sizes of loess slides decrease rapidly with increasing slope gradient. The larger loess slides occur mainly on gentle slopes. Loess slides larger than 105 m2 occur only at slope gradients below 35°. The areas of loess slides area fitted empirically with a power-law regression.

Fig. 6. Cross-sectional areas of sliding masses versus slope length for different slope gradients. The solid line is the best-fit regression line, which has a power-law form.

3.4.2. Relationship between loess slide size and slope gradient in the simulations and further explanation We further analysed the cross-sectional areas of sliding masses that occur with slope gradients of 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50° and 55° for slope lengths of 150 m and 300 m by using the SLOPE/W software to investigate the effects of slope gradient on loess slide size. As illustrated in Fig. 8, there is clearly no correlation between loess slide size and slope gradient. However, further analysis shows that a relationship between slope gradient and slope length exists in the field observations. This relationship can explain the controversial conclusion that we derive from our field observations and simulations. As shown in Fig. 9, the slope gradient decreases with increasing slope length for the collected loess slides. There is an obvious powerlaw relationship between slope gradient and slope length that can be (R2 = 0.6022, P b 0.01). represented by the function: S = 5.200 L−0.421 L 3.5. Probability distributions of loess slide size for different slope length

where S is the tangent of the slope gradient.

To assess the influence of slope length on the frequency-size distributions of loess slides, we calculated the probability densities of the areas of loess slides using kernel density estimation for slope lengths of b 150 m, 150–200 m and N 200 m (Fig. 10). Power-law distributions with exponents of − 1.693, − 2.019 and − 2.301 are observed for large areas.

Fig. 5. Relationship between slope length and loess slide area. The solid line shows the regression line.

Fig. 7. Relationship between slope gradient and loess slide area. The solid line shows the regression line.

AL ¼ 9 966:4S−2:44

  R2 ¼ 0:3702; P b 0:05

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Fig. 8. Cross-sectional areas of sliding masses versus slope gradient for different slope lengths.

4. Discussion Landslides result from geomorphic processes (Stark and Hovius, 2001). Slope geometry influences the sizes of landslides (Schmidt and Montgomery, 1995; Brardinoni and Church, 2004; Guthrie et al., 2008). In this study, two methods, i.e., statistical analyses of field data and numerical simulations, were used to study the roles of slope length and slope gradient in determining the size distributions of loess slides. Thus, the results of the numerical simulation are consistent with those obtained from field observations. 4.1. Relationship between landslide volume and area In previous studies, many authors have investigated the empirical relationships between landslide volume and area in global data sets (Guzzetti et al., 2009; Larsen et al., 2010; Klar et al., 2011). Larsen et al. (2010) found that the scaling exponents derived from landslide data sets for bedrock landslides ranges from 1.3 to 1.6, whereas the values for soil slides ranged from 1.1 to 1.3. Loess slides are typical soil slides. Guzzetti et al. (2009) collected existing relationships that connect landslide area to volume from the literature and showed that the scaling exponents of power-law relationships range from 0.88 to 1.95, whereas the values of the intercept fall between 0.00004 and 12.273 (Guzzetti et al., 2009; Larsen et al., 2010; Klar et al., 2011). The relationship identified in the present study is a power-law with a scaling

Fig. 9. Relationship between slope gradient and slope length. The solid line shows the regression line.

Fig. 10. Probability density of loess slides areas for different slope lengths (A) and slope gradients (B). Dots show kernel density estimation of the areas of loess slides; Green lines show that the probability densities of loess slides that are larger than a certain area can be approximated by a power-law relationship.

exponent 1.18 and an intercept value 1.0439 and is in general agreement with similar relationships published in previous studies. The values of the scaling exponent and intercept differ somewhat among different geological and geomorphological settings. 4.2. The role of slope length on loess slide size Several authors have investigated the relationship between mean local relief and landslide size and have found that the contribution of large landslides should increase as local relief increase (Korup et al., 2007). Montgomery and Brandon (2002) proposed that approximately 75% of giant landslides occur in regions with mean local relief N 1000 m. However, slope height and relief vary with both slope gradient and length. In the Loess Plateau, where the local relief does not vary substantially compared to that in mountainous settings, slope height or relief is intuitively more closely connected to the gradient, which has already been addressed, whereas length refers primarily to the potential lengths of the landslides that can occur on particular slopes. To better characterize the main control on landslide size, we used slope length instead of slope height in this study. To address how slope length and the sizes of individual landslides interact, we studied the effects of slope length on the sizes of loess slides using field observations and limit equilibrium simulations. Our results show that loess slide size correlates positively with slope length with strong coefficients of determination. Similar increases in the areas of sliding masses as slope length increases are observed in loess areas in both field observations and simulations. Furthermore, 2D slope stability analysis using the GLE method shows that this trend is not influenced by the slope gradient. Both approaches indicates that slope length is a major controlling factor of loess slide size on the loess hillslopes. This result provides new insight into the controls on the sizes of natural landslides. We note that landslide area (which is a proxy for size) differs in the observations and in the simulations. The surface area of a landslide observed in the field is given by the product of its length and width. On the other hand, the reference area of a simulated landslide is equal to its cross-sectional area, which is the product of its length and depth. However, it is often assumed that depth and width are related. We note that the slide width increases with increasing slope height. Despite these differences in the proxies for size (planform area in the data and cross-sectional area in the simulations), we obtained similar exponents for the slide area-slope length relationship in both cases.

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4.3. The role of slope gradient on loess slide size As is widely recognized, slope gradient is the factor that influences landslides most strongly (Dai and Lee, 2002; Trigila et al., 2015). Several authors have studied the controls on the size and geometry of an individual landslides using numerical simulations (Frattini and Crosta, 2013; Katz et al., 2014; Chen et al., 2016). In particular, Frattini and Crosta (2013) reported that small landslides are usually associated with high slope angles. Katz et al. (2014) found that landslide size increases with increasing slope gradient for a given material strength using the numerical two-dimensional discrete element method. On the other hand, Chen et al. (2016) showed that the sliding volume decreases with increasing slope gradient using limit equilibrium simulations. However, these previous studies include several limitations (Chen et al., 2016). For a fixed slope height, increasing gradient implies a decreasing slope length and thus a decreasing maximum potential slide length. Our simulation results differ strongly from those of previous studies (Frattini and Crosta, 2013; Katz et al., 2014 and Chen et al., 2016), who observed that the landslide size increases or decreases with the increase in slope gradient in the simulations. When slope length is held constant in the simulations, there appears to be no correlation between loess slide size and slope gradient (Fig. 8). The apparent effect of slope gradient on slide size seen in the field observations is only indirect; it is simply a result of the dependence of slope gradient on slope length (Fig. 9). This finding is confirmed by the statistical results derived from the field survey. As shown in Fig. 9, the slope gradient decreases with increasing slope length. Data and modeling actually agree in showing that the main control on landslide size is slope length. Summarizing these observations and simulations, we suggest that larger loess slides occur preferentially on longer slopes. Owing to the moderate local relief, such slopes generally also have smaller slope gradients. 4.4. Rollover of the size distribution of loess slides and its influencing factors The probability densities of landslide sizes is one of the most important components of landslide risk assessment (Corominas and Moya, 2008). However, the size distribution typically displays a rollover effect at smaller sizes, and this effect has been observed and discussed by many authors (Hungr et al., 1999; Stark and Hovius, 2001; Martin et al., 2002). One possible explanation for the rollover suggests that small landslides have been artificially undercounted (Stark and Hovius, 2001; Malamud et al., 2004a). Pelletier et al. (1997) proposed that the rollover seen in the cumulative frequency-size distributions of landslides depends on soil moisture and slope angle. This study focuses on the frequency distributions of loess slide size in loess terrain. We determine the probability densities of the areas of loess slides on different slope lengths using kernel density estimation. We observe that loess slide distributions typically exhibit heavy-tailed behaviour for loess slide areas exceeding certain threshold, consistent with the previous studies (Stark and Hovius, 2001; Guzzetti et al., 2002; Malamud et al., 2004a; Frattini and Crosta, 2013). Van Den Eeckhaut et al. (2007) compiled many reports on landslide frequency distributions in mountainous areas and found that most published values for the exponent of the power-law relation lie within the range [2.1, 2.6], and the average value of the exponent is 2.3. In this study, the overall power-law exponent is −1.988, which is close to the lower values reported by previous studies (Guzzetti et al., 2002; Van Den Eeckhaut et al., 2007). The difference in scaling exponents may be a consequence of the underlying geology and the type of failure event (Brunetti et al., 2009; Hurst et al., 2013). However, further analysis shows that the slope length constrains the power-law scaling. We obtained an exponent of only ~ 1.8 only for short slopes, and slope lengths exerts a control over the exponent within the usual range of values for slope lengths between 150 m and N200 m. The power-law exponents obtained in this work are within

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the range of values published in previous studies, which range from 1.4 to 3.5 (Van Den Eeckhaut et al., 2007). We attribute the rollover effect of the probability distribution partly to the incompleteness of the loess slides inventory. However, the rollover effect also exists for welldocumented landslide records (Malamud et al., 2004b). Furthermore, the exponent depends on local geomorphic and geological conditions (Van Den Eeckhaut et al., 2007). Moreover, in terms of absolute values, the exponent increases with increasing slope length (Fig. 10). The graphs also suggest that the slope length controls the rollover seen in the frequency-size distributions of loess slides. The effects of slide size-controlling factors (such as slope length) on frequency-magnitude distributions can be quantified to some extent. The above analyses suggest that slope length plays an important role in the sizes and size distributions of loess slides. This result suggests that we can use this quantitative relationship to improve the models used in landslide hazard assessment in the future. Further studies are needed to enable a better understanding of the causes of this rollover. 5. Conclusions This study developed a loess slide inventory in Zhidan County using a combination of remote sensing images and field investigations. This study has shown that local topographic conditions play an important role in influencing landslide size. The results of both observational and simulation data sets show that slope length is the fundamentally control on landslide size. Loess slide size increases with increasing slope length. Furthermore, observational results indicate that loess slide size decreases with increasing slope gradient, whereas the simulation results indicate no relationship between landslide sizes and slope gradient. Further analysis reveals that the relationship between loess slide size and slope gradient is only indirect and is determined by the strong link between slope gradient and length. Slope gradient decreases with increasing slope length. In addition, the size distributions display a power-law scaling for large loess slides and a rollover effect. We observed that the power-law exponents depend on the slope length. The exponent increases as slope length increases. We anticipate that this work will be of use for loess landslide-related hazard assessment. Acknowledgements This work was funded by the National Natural Science Foundation of China (grant no. 41771539) and International Partnership Program of Chinese Academy of Sciences (grant no. 131551KYSB20160002), We warmly thank the anonymous reviewer and the editor, whose constructive and helpful comments substantially improved this manuscript. References Brardinoni, F., Church, M., 2004. Representing the landslide magnitude–frequency relation: Capilano River Basin, British Columbia. Earth Surf. Process. Landf. 29 (1), 115–124. ten Brink, U.S., Barkan, E.L., Andrews, B.D., Chaytora, J.D., 2009. Size distributions and failure initiation of submarine and subaerial landslides. Earth Planet. Sci. Lett. 287 (1–2), 31–42. Brunetti, M.T., Guzzetti, F., Rossi, M., 2009. Probability distributions of landslide volumes. Nonlinear Process. Geophys. 16 (2), 179–188. Chen, C.Y., 2009. Sedimentary impacts from landslides in the Tachia River Basin, Taiwan. Geomorphology 105, 355–365. Chen, X.L., Liu, C.G., Chang, Z.F., Zhou, Q., 2016. The relationship between the slope angle and the landslide size derived from limit equilibrium simulations. Geomorphology 253, 547–550. Corominas, J., Moya, J., 2008. A review of assessing landslide frequency for hazard zoning purposes. Eng. Geol. 102 (3), 193–213. Cruden, D.M., Varnes, D.J., 1996. Landslide types and processes. In: Turner, A.K., Schuster, R.L. (Eds.), Landslides: Investigation and Mitigation. Special-Report 247, Transportation Research Board, National Research Council. National Academy Press, Washington, DC, pp. 36–75. Dai, F.C., Lee, C.F., 2002. Landslide characteristics and slope instability modeling using GIS, Lantau Island, Hong Kong. Geomorphology 42, 213–228. Derbyshire, E., 2001. Geological hazards in loess terrain, with particular reference to the loess regions of China. Earth Sci. Rev. 54 (1), 231–260.

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