Introduction

Landslide is a major geomorphological process affecting landscape evolution in mountainous area (Roering et al. 2005); it may be triggered by rainfall or earthquake. During the 2008 Wenchuan earthquake, 197,481 slope failures were triggered in about 110,000 km2 (Xu et al. 2013), about 20,000 deaths were attributed to landslides (Huang and Li, 2008, 2009a). Numerous authors have analyzed the relations between landslide spatial distribution and influential factors so as to serve landslide hazard map (Huang and Li 2009a, b; Xu et al. 2009b, 2010; Chigira et al. 2010; Qi et al. 2010; Dai et al. 2011; Gorum et al. 2011; Guo and Hamada 2013). However, there are few studies having explored the effectiveness of various factors on landslide travel distance and its prediction in Wenchuan earthquake area, except that Qi et al. (2011) delineated six typical destructive long travel landslides and listed 66 valuable cases, but this research was limited to analyze the relationship between elevation loss and sliding area and travel distance. Yin et al. (2009) pointed out that at least 4,970 sites were highly potential to develop into geo-hazards. Therefore, if it was possible to preliminarily estimate landslide travel distance and endangered area, besides landslide spatial distribution, the quality of landslide hazard map would be highly improved and more applicable to disaster mitigation and relocation.

However, landslide travel distance prediction is a complicate issue because it was governed by the properties of materials (i.e., grain size distribution and water content), topographical factors, mobile mechanics of failed mass, the confinement attribute of travel path, and so on. In order to research this topic to mitigate geo-hazard caused by landslide, there are three techniques to evaluate landslide travel distance: empirical model (for example, Rickenmann 1999, 2005; Legros 2002), theoretical model (for example, Takahashi 1981, 1991), and numerical simulation model (for example, McDougall and Hungr 2003). Theoretical model requires two parameters which are difficult to be accurately estimated, that is, velocity of sliding mass and frictional parameter, whereas the predictive result of numerical simulation model highly depends on input data, which would be variable during sliding process. Simplifying assumptions are often made for theoretical model and numerical simulation model, especially where input parameters cannot be measured easily.

Since the parameters of a landslide may change during movement to avoid the usage of uncertainly and highly variable parameters to predict landslide travel distance, empirical model was widely applied to preliminary assessment of landslide travel distance, as a result of no requirement of the parameters of rheology or detail mechanics of movement, besides, it is a relatively simple tool to offer a practical means of prediction. Hence, there are lots of previous researches to use this approach, such as Scheidegger 1973; Corominas 1996; Fannin and Wise 2001; Hunter and Fell 2003; Okura et al. 2003; Berti and Simoni 2007; Prochaska et al. 2008; Hattanji and Moriwaki 2009, 2011. Among these studies, a common index for expressing landslide mobility was widely used and firstly designated by Heim (1932) as fahrböschung, which is the angle of the line connecting the head of sliding source and the toe of the deposited mass. Shreve (1968) and Scheidegger (1973) later called the tangent of this angle as equivalent coefficient of friction. More recently, some authors named this angle as angle of reach (Corominas 1996) or travel distance angle (Hunter and Fell 2003). Further, equivalent coefficient of friction has a significant correlation with landslide volume, as demonstrated by Scheidegger (1973), Corominas (1996), Fannin and Wise (2001), Rickenmann (1999, 2005) and also has a significant correlation with slope angle (Hunter and Fell 2003; Hattanji and Moriwaki 2009, 2011). However, these previous studies neither have revealed the comprehensive relationship between landslide travel distance and various influential factors nor have discussed the effectiveness of each influential factor on travel distance. Hence, a new model is needed to consider more influential factors on landslide travel distance simultaneously. Finlay et al. (1999) made use of multiple regression model to study over 1,100 man-modified slopes in Hong Kong; however, most of these landslide scales and travel distances are much smaller than those landslides from natural slopes. Meanwhile, most of previous researches were about non-seismically induced landslides; therefore, it is meaningful to explore the prediction of earthquake-induced landslide travel distance by simultaneously taking numerous influential factors into consideration.

This paper collected 54 well-documented landslides triggered by the 2008 Wenchuan earthquake to study the relations between landslide travel distance and six influential factors. It was divided into six sections and proceeded as follows: section 2 introduced the data source, section 3 reported the relations between landslide travel distance and landslide characteristic parameters, section 4 analyzed the effectiveness of influential factors on landslide travel distance and developed a new model to its prediction, and section 5 and section 6 presented the discussions and conclusions, respectively.

Data selection of landslide and its parameters

In this paper, the collected data excluded not only landslides obstructed by relatively large river, valley, and infrastructures but also travel path sharply confined by steep lateral slope and landslides with big deflection travel path, as a result of that, the attributes of travel path had significant effect on the mobilization of failed mass (Shaller 1991; Corominas, 1996). However, the characteristics of confinement are difficult to be quantitatively expressed. Hence, the travel path of selected landslides was relatively open or confined by gentle lateral slope and the travel path with relatively small deflection. Totally, 54 landslides were collected based on remote sensing interpretation, field investigation, and descriptions of published papers and books. The source volume range of these landslides is 4.5 × 104–2.75 × 107 m3; horizontal travel distance is within 347–4,170 m.

Figure 1 illustrates the basic terminologies and notations. The longitudinal profile of a slope was obtained by using topographic map. Based on the density of contour lines, slope was divided into several segments. Contour lines appear relatively sparser; the inclination of corresponding segment will be smaller and the contour lines become much denser; the corresponding segment has relatively larger slope angle. The interval changing segments of contour lines (marked by ellipse in Fig. 1) along longitudinal section correspond with the inclination changing segments along a slope. The boundary of contour line interval apparent changing in the topographic map along slope longitudinal section was regarded as the turning point of different segments of a slope; hence, a slope will be divided into several segments, and the average inclination and slope height of each segment will be obtained. Slope angle, θ, is the average inclination of the sectional slope with failed part. Slope height, h, is the elevation difference of the sectional slope with failed part. Travel distance, L max, is the horizontal distance between the crest of the sliding source and the distal of debris. Landslide height, H max, is the maximum elevation loss between sliding source and debris. α represents fahrböschung (Heim 1932) or travel distance angle (Hunter and Fell 2003). Slope transition angle, β, is the angle between the upper slope (failed section) and lower slope, indicating the change degree of slope inclination. During investigation, the area of sliding source, in situ outlined on the map, was calculated by ArcGIS software, and the volume of sliding source was obtained from multiplying the area of sliding source by average collapse depth. The average collapse depth was estimated from typical longitudinal profile of slope, as shown in Fig. 2. According to the rock strength and weathered degree, rock materials were assorted into two types, such as hard rock and soft rock, further divided into two subclasses, respectively, as listed in Table 1.

Fig. 1
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The sketch of landslide for the definitions of geometric parameters

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Fig. 2
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Typical longitudinal profile of landslide

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Table 1 Classification standard of rock type (Chang et al. 2006)
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Topographical factors play an important role in the mobilization of landslide (Okura et al. 2003). The topography of a slope can be described by slope angle (θ), slope transition angle (β), and slope height (h). Slope angle has positive correlation between internal friction coefficient (Okura et al. 2003), which decides the energy consumption by internal friction; meanwhile, the steeper the slope is, the bigger the component of gravity to accelerate the sliding mass is and higher consumption of kinetic energy due to the impact at the foot of upper slope. The changing degree of slope inclination between upper slope and lower slope can be expressed by slope transition angle (β), which has a strong effect on the impact of sliding mass; it will affect the postimpact mobile motion of sliding mass. For slope height (h), it implies the preimpact potential energy of sliding mass and governs the available space to accelerate the failed mass; the bigger slope height, the faster sliding velocity; the larger kinetic energy was consumed during impact at the slope transition section.

The influence of seismic wave on landslide travel distance was explored in terms of energy by Kokusho et al. (2007, 2009). Furthermore, Wang and Lin (2011) researched seismic effect on landslide initiation and displacement by shaking model test and proposed that “the final runout displacement is not proportional with the relationship of law of similitude but can be obtained from a normalized displacement relationship with maximum loading amplitude.” Since peak ground acceleration is one of the most important parameters to represent seismic energy, and horizontal peak ground acceleration was widely used to evaluate slope stability; hence, horizontal peak ground acceleration (PHA) was applied during the following study about influential factors on landslide travel distance, but it is impossible to install monitoring equipment at each investigated landslide before Wenchuan earthquake, so PHAs of landslides were estimated by Eq. (1), which was derived from 183 seismic stations and considered hanging-foot wall effect (Guo and Hamada 2013).

$$ \ln \mathrm{PHA}={a}_1 \ln \left({D}_{\mathrm{rup}}+{a}_2\right)+{a}_3{D}_{\mathrm{rup}}+{a}_4 $$
(1)

Where PHA is the horizontal peak ground acceleration (centimeters per square second); D rup represents nearest distance from site to Yingxiu-Beichuan surface fault rupture (kilometer) located by using USGS result (Ji and Hayes, 2008); and a 1, a 2, a 3, and a 4 are the regression coefficients, listed in Table 2, in which R 2 stands for coefficient of determination.

Table 2 Regression parameters of seismic acceleration attenuation (Guo and Hamada 2013)
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The distribution of 46 landslides was illustrated in Fig. 3, and their parameters were listed in Table 3, which were used to obtain statistical model for landslide travel distance. Other eight landslides were applied to verify the validity of proposed statistical model in following section 4.

Fig. 3
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The distribution of 46 landslides (based on Qi. et al. 2011)

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Table 3 The parameters of 46 landslides induced by the 2008 Wenchuan earthquake
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Relationships between landslide travel distance and landslide parameters

Figures 4 and 5 illustrate the relations between landslide travel distance and sliding source volume and area, respectively; nevertheless, the statistical significance levels are relatively weak; these two results both suggest that landslide travel distance had exponential correlation with landslide scale (sliding source volume and area); it means landslide travel distance would generally and rapidly increase with landslide scale. Figure 6 illustrates that horizontal travel distance (L max) had a relatively strong linear correlation with landslide height (H max, total elevation loss). The relation between landslide horizontal travel distance and landslide height were further explored based on nonvolcanic and volcanic landslides (Hayashi and Self 1992; Legros 2002), as shown in Fig. 7, which also suggests that landslide travel distance increased with the elevation loss of failed mass. The reason was attributed to that failed mass was forced to travel down by gravity, and the potential energy was transformed to kinetic energy.

Fig. 4
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Landslide travel distance related with sliding source volume

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Fig. 5
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Landslide travel distance related with sliding source area

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Fig. 6
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Landslide travel distance related with landslide height

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Fig. 7
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The relations between travel distance and total elevation loss of nonvolcanic and volcanic landslides (data from Hayashi and Self 1992; Legros 2002)

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Statistical model of landslide travel distance

Although landslide travel distance (L max) had a relatively significant correlation with landslide height (H max) in Fig. 6, it cannot be used to predict landslide travel distance because the stop place of failed mass and the top of sliding source were both unaware before landslide occurrence. Since landslide movement was affected by numerous factors, hence, the predictive model of landslide travel distance had better consider as many influential factors as possible simultaneously. Under the linear assumption, multiple linear regression and backward elimination approach were applied to obtain optimization model of landslide travel distance in the aid of F test and t test. The regression procedures and results were listed in Table 4.

Table 4 Multiple analysis of landslide travel distance by backward elimination approach
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In Table 4, L max refers to landslide horizontal travel distance (meters); RT refers to rock type, it was qualitatively considered and 4, 3, 2, and 1 were, respectively, assigned to RT1, RT2, RT3, and RT4; V represents the volume of sliding source (cubic meter); β represents slope transition angle (°); h represents slope height of the sectional slope with failed part (meters); PHA refers to horizontal peak ground acceleration (meters per square second) estimated from Eq. (1); and θ denotes slope angle of the sectional slope with failed part (°). The basis of logarithm is 10. Multiple R means multiple correlation coefficient, which represents the correlated level between dependent variable and independent variables. F-stat denotes regressive F value, F0.05 denotes F test threshold value with 95 % significance level. If F-stat is bigger than F0.05, it suggests overall regression of the model satisfies statistical significance level. t-stat denotes regressive t value of each regression coefficient, t0.05 denotes t test threshold value with 95 % significance level; if t-stat is larger than t0.05, it suggests corresponding regression coefficient, n i , which satisfies significance level of regression. However, not all regression coefficients have statistical significance during regression procedures; therefore, the variable with smallest absolute t-stat value (bold digit in Table 4) was eliminated and then re-regressed step by step until overall regression of the model and all regressive coefficients both satisfy statistical significance level. These procedures are named as backward elimination regression. Because the units of independent variables affect regression coefficient, n i , therefore, standardized regression coefficient, n ′ i , was applied to exclude the effect of unit dimension, so as to have insight into the effectiveness of each independent variable (influential factors) on dependent variable (landslide travel distance).

Compared to four models in Table 4, F tests suggest that all of these hypothetical models have statistical significance and satisfy linear assumption. Based on backward elimination procedures and those absolute values of each standardized regression coefficient in six variables model, they both suggest that rock type, sliding source volume, and slope transition angle were more influential than slope height, seismic acceleration, and slope angle on landslide travel distance. Furthermore, the optimization model with three variables satisfies not only overall statistical significance (F-test) but also the significance of each regression coefficient (t test), listed in Table 4, the empirical model for predicting landslide travel distance based on presented dataset in Wenchuan earthquake area was shown as Eq. (2):

$$ \begin{array}{cc}\hfill \log {L}_{\max }=0.136 RT+0.159 \log V+0.529 \sin \beta +1.497\hfill & \hfill \left({R}^2=0.51\right)\hfill \end{array} $$
(2)

However, coefficient of determination (R 2) indicates the regressive correlation of optimization empirical model is moderate; its validity need to be confirmed. By using Eq. (2) to predict the travel distance of surveyed 46 landslides, the comparison between predictions and observations was shown in Fig. 8; the estimation error of each landslide, |L predicted − L observed|/L observed × 100%, was listed in the column of “Error” in Table 3. This self-verification suggests that proposed statistical model for travel distance prediction is valid for most landslides. In order to further verify its applicability, eight landslides in Wenchuan earthquake area were used to check the validity of Eq. (2); the parameters of each landslides and the predictive results were listed in Table 5, and the corresponding eight landslides were compared; their observed and predicted travel distances in Fig. 8, which further suggest that the proposed model may be valid. Therefore, it might be useful to preliminarily assess the travel distance of potential landslide in Wenchuan earthquake area and other similar geological and geomorphological regions. Nevertheless, the reliability of this empirical model needs to be further verified by some techniques, such as ROC curves and success rate curves (Frattini et al. 2010).

Fig. 8
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Observed versus predicted horizontal travel distance

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Table 5 The verification of proposed statistical model for landslide travel distance by case studies
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Discussions

Based on the empirical model, it suggests that landslide travel distance had positive correlation with rock type (from soft rock to hard rock), landslide volume, and the sine of slope transition angle. It was inferred that landslide consisting of much softer rock (RT smaller), more kinetic energy would be consumed by friction, then failed mass would have shorter travel distance. The relation between landslide volume and travel distance was shown in Fig. 9 besides the data from Wenchuan earthquake, which reveals that landslide travel distance increased with landslide volume. Regarding the correlation between travel distance and the sine of slope transition angle, when a slope had several sections with different inclination, the failed mass would impact at the portion of slope transition. The fragment effect due to impact would enable failed mass to become high mobility, then making landslide travel longer. The range of slope transition angle of surveyed landslides was within 140° ∼ 176°, with the decrease of slope transition angle (namely, the increment of sinβ); the fragment effect became more significant, causing failed mass to have higher mobility and travel longer. Therefore, landslide travel distance had positive relation with the sine of slope transition angle.

Fig. 9
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The relation between landslide horizontal travel distance and volume [the data of nonvolcanic and volcanic landslide are from Hayashi and Self (1992) and Legros (2002)]

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In order to directly estimate landslide travel distance, Rickenmann (1999) and Legros (2002) developed empirical models for this issue, being listed in Table 6. Making use of these two previous models to estimate landslide travel distance of these 54 landslides in Wenchuan earthquake area, the results were shown in Fig. 10. It suggests that the model proposed by Rickenmann (1999) and Legros (2002) cannot obtain satisfactory assessment; the average errors of these two models are 248.26 and 116.77 %, respectively. From the above plot of Fig. 10, most of predicted travel distances by the model of Rickenmann (1999) are significantly longer than the observations (above the dash line with 2:1 gradient); this comparison suggests that the model proposed by Rickenmann is not suitable for the prediction of landslides triggered by the 2008 Wenchuan earthquake because Rickenmann model is based on debris flow dataset. However, dataset in this paper mainly included rock (soil) slide and debris (rock) avalanche; therefore, it may draw a conclusion that the mechanism of debris flow is very different from rock (soil) slide and debris (rock) avalanche and results in the failed application of Rickemann model in Wenchuan earthquake area; on the other hand, most of predictions by Rickemann model are significantly bigger than the observations (above the dash line with 2:1 gradient), and there is no prediction smaller than observation; it means that under the same conditions (volume, rock type, and slope transition angle, etc.), debris flow usually will travel longer than rock (soil) slide and debris (rock) avalanche. The reason was inferred that the lubrication of water content in the failed mass of debris flow increased travel distance. This comparison reconfirmed and implied that landslide travel distance was heavily affected by mobile mechanism of sliding mass, especially failed mass with relatively abundant water content; it would cause the mobilization of debris flow that was quite different from other types of landslides. For the model of Legros, its invalidity of usage in Wenchuan earthquake area might be due to the difference of geological and hydrogeological conditions.

Table 6 Landslide travel distance prediction models and their comparisons
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Fig. 10
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Predicted landslide travel distance by two previous models versus the observed

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The mobile motion of slope failure was affected by numerous factors, such as topography, rock (soil) properties, mobile mechanisms, ground water, and so on. As many influential factors as the authors could collect, 46 well-documented landslides with nine parameters were used to develop empirical model of landslide travel distance, and eight landslides were applied to verify the validity of presented model. Our data exclude landslides obstructed by relatively large river, valley, and infrastructures that neither included travel path sharply confined by steep side slope nor big deflection of travel path. However, it was difficult to judge the confinement degree during the investigation; the boundary between partly confined and confined not only relied on the experience of investigator but also affected by the relative magnitude between landslide volume and obstacles or topographical changes; hence, the estimation error of several predictions was too large to be valid, such as landslide no. 11 and no. 17 in Table 3. Moreover, travel distance of 46 landslides in Table 3 is 91.3 % (42 out of 46) shorter than 2000 m; it may cause the predicted results of landslides with over 2,000 m observed travel distance that are more scattering than others in Fig. 8. However, compared with observations, the overall outcome by presented empirical model mainly yielded satisfactory results, which implies that the judgment about the degree of confinement during data collection and linear hypothesis for statistical model were both reasonable. One more reason for the scattering of comparison plot might be attributed to the discretization of data because some data are from field investigation and some data are from the published papers and books.

The limitation of this study is that the effects of ground water, geological structure, and landslide type on travel distance were not discussed. Previous studies have been found that pore-water pressure caused shear strength significant reduction (Takarada et al. 1999; Major and Iverson 1999; Wang and Sassa 2003; Sassa et al. 2004, 2005), and then resulting in failed mass might perform different mobile motions during sliding. The effect of geological structure on slope failure mode was demonstrated by Aydan and Hamada (2006, 2009), and Corominas (1996) proposed that different landslide types appeared different mobile abilities based on empirical relations of different landslide types. Therefore, discussion on landslide travel distance should consider the influence of geological structure and landslide type. However, in this paper, our data are limited to 54 landslides, and the types of slope failure mainly included rock (or soil) slide, rock avalanche, and debris avalanche. If these data were further classified into several groups, the regression model of each type of landslide would be uncertainty. Hence, a more generalized model of landslide was derived to estimated landslide travel distance. In spite of ignoring the influences of ground water, geological structure, and landslide type on landslide travel distance in proposed model, most of the predicted results are reasonable compared with observations, as listed in Table 3 and Fig. 8. Notwithstanding, the applicability of proposed model needs to be further verified, and more influential factors on landslide travel distance were recommended to take into consideration during improvement.

Conclusions

In this study, the authors collected 54 well-documented landslides triggered by the 2008 Wenchuan earthquake, and landslides were delineated by following parameters, such as slope angle (θ), slope transition angle (β), slope height (h), horizontal travel distance (L max), landslide height (H max), the volume of sliding source (V) and sliding source area (A), rock type (RT), and horizontal peak ground acceleration (PHA). The results of multivariable analysis revealed that rock type, sliding source volume, and slope transition angle had predominant effect on landslide travel distance during Wenchuan earthquake and followed by slope height, seismic acceleration, and slope angle. An empirical model, as Eq. (2), was developed to predict landslide travel distance in similar geomorphological and geological region as Wenchuan earthquake affected area.