the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# An improved method of Newmark analysis for mapping hazards of coseismic landslides

### Mingdong Zang

### Shengwen Qi

### Yu Zou

### Zhuping Sheng

### Blanca S. Zamora

Coseismic landslides can destroy buildings, dislocate
roads, sever pipelines, and cause heavy casualties. It is thus important but
challenging to accurately map the hazards posed by coseismic landslides.
Newmark's method is widely applied to assess the permanent displacement
along a potential slide surface and model the coseismic response of slopes.
This paper proposes an improved Newmark analysis for mapping the hazards of
coseismic landslides by considering the roughness and effect of the size of the
potential slide surfaces. This method is verified by data from a case study
on the 2014 *M*_{w} 6.1 (the United States Geological Survey) Ludian
earthquake in Yunnan Province, China. Permanent displacements due to the
earthquake ranged from 0 to 122 cm. The predicted displacements were
compared with a comprehensive inventory of landslides triggered by the
Ludian earthquake to map the spatial variation in the hazards of coseismic
landslides using the certainty factor model. The confidence levels of
coseismic landslides indicated by the certainty factors ranged from −1 to
0.95. A hazard map of the coseismic landslide was generated based on the
spatial distribution of values of the certainty factor. A regression curve
relating the predicted displacement and the certainty factor was drawn, and
can be applied to predict the hazards of coseismic landslides for any
seismic scenario of interest. The area under the curve was used to compare
the improved and the conventional Newmark analyses, and revealed the
improved performance of the former. This mapping procedure can be used to
predict the hazards posed by coseismic landslides, and provide guidelines
for decisions regarding the development of infrastructure and
post-earthquake reconstruction.

Earthquakes are recognized as one of the major causes of landslides (Keefer, 1984). Hazards caused by coseismic landslides have drawn increasing attention in recent years (e.g., Jibson et al., 1998, 2000; Khazai and Sitar, 2003; Qi et al., 2010, 2011, 2012; Chen et al., 2012; Xu et al., 2013; Yuan et al., 2014). The damage caused by seismically triggered landslides is sometimes more severe than the direct damage caused by the earthquake (Keefer, 1984). Estimating where a specific shaking is likely to induce a slope failure plays an important role in the regional assessment of coseismic landslides.

Pseudo-static analysis formalized by Terzaghi (1950), and finite-element modeling applied by Clough and Chopra (1966) have been employed to assess the seismic stability of slopes in early efforts (Jibson, 2011). Newmark (1965) first introduced a relatively simple and practical method, which is still commonly used nowadays to estimate the coseismic permanent displacements of slopes (Jibson, 2011). Studies have shown that Newmark's method yields reasonable and practical results when modeling the dynamic performance of natural slopes (Wilson and Keefer, 1983; Wieczorek et al., 1985; Jibson et al., 1998, 2000; Pradel et al., 2005). Rathje and Antonakos (2011) recently presented a unified framework for predicting coseismic permanent sliding displacement based on Newmark's method. Chen et al. (2018) used Newmark's method to calculate the minimum accelerations required for coseismic landslides in the region affected by the 2014 Ludian earthquake. Chen et al. (2019) subsequently developed an easy operation mapping method to assess hazards posed by coseismic landslides in the zone struck by the 2014 Ludian earthquake using Newmark's method.

Such applications generally start from an analysis of the dynamic stability of slopes, which is quantified as the critical acceleration. Barton model (Barton, 1973) has been widely used in rock mechanics and engineering to predict the shear strength of rock joints, which plays a crucial role in the calculation of critical acceleration. However, researchers have not adequately attended to the shear strength of rock joints during the assessment of coseismic landslides. To better estimate the dynamic stability of slopes, in this paper, we introduce the Barton model (Barton, 1973) to Newmark analysis to develop an improved modeling method for mapping the hazards posed by coseismic landslides using data from the 2014 Ludian earthquake in Yunnan Province in southwestern China. As predictions of coseismic landslides are not based on exact results, i.e., the computed permanent displacements, but are also mingled with unformalized expertise, i.e., the interpreted landslides, we present a model of inexact reasoning, i.e., the certainty factor model (CFM), that defies analysis, as an application of sets of inference rules that are expressed in predicate logic (Shortliffe and Buchanan, 1975) to produce a map of the hazards posed by coseismic landslides.

This paper briefly introduces the characteristics and spatial distribution of landslides triggered at the chosen site, describes the method of modeling used for the analysis of the stability of seismic slopes, presents the mapping procedure of the confidence level of seismic slope failure, and finally discusses the results of the assessment of seismic hazard, as well as a comparison with the conventional Newmark analysis.

The epicenter of the 2014 *M*_{w} 6.1 (the United States Geological Survey)
Ludian earthquake was located on the southeastern margin of the Tibetan
Plateau. A rectangular area lying immediately around the epicenter
containing dense concentrations of the induced landslides was chosen for
study (Fig. 1). The elevation of the area ranged from 785 to 3085 m above
sea level. Three rivers – the Niulanjiang River, Shaba River, and Longquan
River – pass through the study area (Fig. 1). The topography ranges from
flat in the river valleys to nearly vertical in the slopes on the banks of
the rivers. According to Chen et al. (2015), Niulanjiang River flows from
the southeast (SE) to the northwest (NW), and incises to a depth between
1200 and 3300 m, resulting in about 80 % of the slopes having angles
greater than 40^{∘} distributed along the banks. The predominant
geological units of the study area have an age that varies from the
Proterozoic to the Mesozoic, including dolomite, limestone, shale,
sandstone, basalt, and slate (Fig. 2).

An inventory of 1416 landslides triggered by the 2014 Ludian earthquake
(Fig. 1) was compiled by visual inspection through comparisons between
pre-earthquake satellite images obtained from Google Earth (30 January
2014) and 0.2 m high-resolution post-earthquake aerial images (7 August
2014; data provided by the Digital Mountain and Remote Sensing Applications
Center, Institute of Mountain Hazards and Environment, Chinese Academy of
Sciences, and Beijing Anxiang Power Technology Co., ltd.). A majority of
landslides triggered by the earthquake were shallow, flow-like landslides
(shallower than 3 m), developing in particularly dense concentrations along
steeply incised river valleys. The total area of these interpreted
landslides was 7.01 km^{2} within a study area of 705 km^{2}. A detailed
study showed that 846 of the mapped landslides were greater than 1000 m^{2} in area, occupying 6.74 km^{2} and accounting for 96.1 % of the
total area of landslides, of which 279 were greater in area than 5000 m^{2}, occupying 5.37 km^{2} and accounting for 76.6 % of the total
landslide area.

## 3.1 Modeling method

In the context of the analysis of the dynamic stability of a slope, Newmark (1965) proposed a permanent displacement analysis that bridges the gap between simplistic pseudo-static analysis and sophisticated but generally impractical finite-element modeling (Jibson, 1993). Newmark's method simulates a landslide as a rigid plastic friction block with a known critical acceleration on an inclined plane (Fig. 3) and calculates the cumulative permanent displacement of the block, as it is subjected to an acceleration time history of an earthquake. Newmark (1965) showed that the dynamic stability of a slope is related to the critical acceleration of a potential landslide block and can be expressed as a simple function of the static factor of safety and the geometry of the landslide (Jibson et al., 1998, 2000):

where *a*_{c} is the critical acceleration in terms of *g*, i.e., the acceleration
due to the Earth's gravity, *F*_{S} is the static factor of safety, and
*α* is the angle from the horizontal at which the center of the slide
block moves when displacement first occurs (Jibson et al., 1998, 2000). For
a planar slip surface parallel to the slope, this angle generally
approximates to the angle of the slope.

Natural slopes often develop a group of shallow unloading joints (Fig. 4) parallel to the surface due to valley incisions (Gu, 1979; Hoek and Bray, 1981). Studies have shown that rock slopes behave as collapsing and sliding failures of shallow unloading joints under strong earthquakes, and 90 % of coseismic landslides are shallow falls and slides (Harp and Jibson, 1996; Khazai and Sitar, 2003; Dai et al., 2011; Tang et al., 2015). According to Qi et al. (2012), two typical kinds of landslides are triggered by earthquakes: (a) shallow, flow-like landslides with a depth of less than 3 m in general and (b) rockfalls thrown by the shaking caused by the earthquake that usually occur at the crest of the slope. For both types, unstable blocks of rock are often cut and activated along the rock joints. Therefore, the static factor of safety in terms of the critical acceleration in these conditions is related to the peak shear strength of the rock joints. For the purpose of regional analysis, we use a limit equilibrium model of an infinite slope (Fig. 3) by referring to the simplification of Newmark's method by of Jibson et al. (1998, 2000). The value of the static factor of safety against sliding given by the ratio of resistance to the driving forces is determined by conventional analysis without considering accelerations, expressed as follows:

where *τ* is the peak shear strength of the rock joint, *L* is the
length of the rock joint, *m* is the mass of the failure rock block, *γ* is the unit weight of the rock mass, and *t* is the thickness of the
failure rock block.

For a Newmark analysis, it is customary to describe the shear strength of
rocks instead of rock joints in terms of Coulomb's constants, i.e., friction
angle (*φ*) and cohesion (*c*). However, both are not only
stress dependent but also scale dependent (Barton and Choubey, 1977).
According to Barton (1973), a more satisfactory empirical relationship for
predicting the peak shear strength of a joint can be written as follows:

where *σ*_{n} is the effective normal stress, JRC is the joint
roughness coefficient, JCS is the joint wall compressive strength, and
*ϕ*_{b} is the basic friction angle, i.e., the angle of frictional sliding
resistance between rock joints, which can be obtained from residual shear
tests on natural joints (Barton, 1973).

The effective normal stress (*σ*_{n}) generated by gravity acting on
the rock block is as follows:

Considering the impact of size on JRC and JCS, the formulations developed by Barton and Bandis (1982) are shown as follows:

where the nomenclature adopted incorporates (0) and (*n*) for values of the
laboratory scale and the in situ scale, respectively.

Hence, the static factor of safety (*F*_{S}) of a slope can be written as follows:

After calculating the angle of the slope and static factor of safety, the
critical acceleration of the slope can be determined. Once the time history
of the earthquake' acceleration has been selected, portions of the record
lying above the critical acceleration *a*_{c} (Fig. 5a) are integrated once
to derive a velocity profile (Fig. 5b); the time history of velocity is then
integrated a second time to obtain the profile of cumulative displacement of
the block (Fig. 5c). Users finally determine the dynamic performance of the
slope based on the magnitude of the Newmark displacement (Jibson et al.,
1998, 2000; Jibson, 2011). The detailed procedure for conducting a Newmark
analysis with the Barton model is discussed in the following sections.

## 3.2 Static factor-of-safety map

Considering that the mapped landslides greater than 1000 m^{2} in area
occupied 96.1 % of the total landslide area, we selected a 30 m × 30 m digital elevation model (DEM) from the ASTER Global Digital Elevation
Model (https://doi.org/10.5067/ASTER/ASTGTM.002; NASA/METI/AIST/Japan Spacesystems and U.S./Japan ASTER Science Team, 2009), which facilitated the subsequent hazard analysis. A basic
slope algorithm was applied to the DEM to produce a slope map (Fig. 6),
where the slope was identified as the steepest downhill descent from a cell
to its neighbors (Burrough and McDonell, 1998). The slopes ranged from
greater than 60^{∘} along the banks of the Niulanjiang River, Shaba
River, and Longquan River to less than 20^{∘} in low mountains and
hills in the north and east.

According to Jibson et al. (1998, 2000), slopes steeper than 60^{∘}
remain unstable even at high strengths. We assume that Newmark's rigid
plastic block is unsuitable for such a steep sliding surface. In this case,
sliding occurs along a plane at an angle (*α*) of $\mathrm{45}{}^{\circ}+{\mathit{\varphi}}_{\mathrm{b}}/\mathrm{2}$ with the horizon (Fig. 7). Therefore, we assigned
an angle (*α*) of $\mathrm{45}{}^{\circ}+{\mathit{\varphi}}_{\mathrm{b}}/\mathrm{2}$ to slopes
steeper than 60^{∘} to avoid too small a value of *F*_{S} in the
Newmark analysis.

Friction angle (*φ*), cohesion (*c*), and unit weight
(*γ*) were derived from the Geological Engineering Handbook
(Geological Engineering Handbook Editorial Committee, 2018).

The digital geological map from the China Geological Survey (CGS) was
rasterized at a 30 m grid spacing to assign material properties throughout
the study area. According to the literature, JRC_{0} and JCS_{0}
depend strongly on lithology (Coulson, 1972; Barton and Choubey, 1977;
Bandis et al., 1983; Bilgin and Pasamehmetoglu, 1990; Priest, 1993; Singh et
al., 2012; Alejano et al., 2012, 2014; Giusepone and da Silva, 2014; Yong et al., 2018).
Representative values of *γ*, JRC_{0}, JCS_{0}, and *ϕ*_{b} assigned to each rock type exposed in the study area were estimated
using the test data listed in Table 1. The selected values were near the
middle of the ranges represented in the references. These JRC_{0} and
JCS_{0} values were considered in a laboratory scale for a length of 100
mm as *L*_{0}. For each grid cell in the regional analysis, the length of
the engineering dimension, *L*_{n}, can generally be set as a 10-fold range
of *L*_{0}. This is because the value of JRC_{n}∕JRC_{0}
(JCS_{n}∕JCS_{0}) is nearly constant when the value of
*L*_{n}∕*L*_{0} is greater than 10 (Bandis et al., 1981). The values of
JRC_{n} and JCS_{n} were then calculated by inserting the values
of JRC_{0} and JCS_{0}, and *L*_{0}, and *L*_{n} into Eqs. (5) and
(6), respectively. Figures 8 and 9 show the spatial distributions of
JRC_{n} and JCS_{n}, respectively. The basic friction angle (*ϕ*_{b}) map and unit weight (*γ*) map are shown in Figs. 10 and 11,
respectively.

For the sake of simplicity, the thickness of the modeled block *t* was taken
to be 3 m, which reflected the typical slope failures of the Ludian
earthquake. The static factor-of-safety map was produced by combining these
data layers (*α*, JRC_{n}, JCS_{n}, *ϕ*_{b}, and *γ*) in Eq. (7). In the initial iteration of the calculation, grid cells in
steep areas with static factors of safety smaller than 1 indicated that
the slopes were statically unstable, but this did not necessarily mean that they
were moving under shaking induced by the earthquake. In this condition, to
avoid conservative results, we neither increased the strengths of the rock
types with statically unstable cells nor adjusted the strengths of other
rock types to preserve the differences in relative strength between them
(as in Jibson et al., 1998, 2000). Instead, we assigned a minimal static
factor of safety of 1.01, merely above limit equilibrium (Jibson et al.,
1998, 2000), to these slopes to avoid a negative value of the critical
acceleration *a*_{c}. According to Keefer (1984), most landslides triggered
by earthquakes occur with a slope of at least 5^{∘}. The static
factors of safety resulting from slopes of angles smaller than 5^{∘}
were very high. These slopes were unlikely to fail under the Ludian
earthquake and did not produce a statistically significant sample in the
analysis. Therefore, slopes less steep than 5^{∘} were not analyzed
in the second iteration. After the adjustment, the static factors of safety
ranged from 1 to 17.4, as shown in Fig. 12.

## 3.3 Critical acceleration map

According to Newmark (1965), a pseudo-static analysis in terms of the static
factor of safety and the slope angle was employed to calculate the critical
acceleration of a potential landslide. The map of critical acceleration
(Fig. 13) was generated by combining the static factor of safety and the
slope angle in Eq. (1). The critical accelerations were derived from the
intrinsic properties of the slope (topography and lithology), regardless of
the given shaking. Therefore, the map of critical acceleration indicated the
susceptibility of coseismic landslides (Jibson et al., 1998, 2000). The
calculated critical accelerations ranged from nearly zero in areas that were
more susceptible to coseismic landslides to greater than 1 *g* in areas that
were less susceptible.

## 3.4 Shake map

There were 23 strong motion stations within 100 km of the epicenter of the Ludian earthquake (Fig. 14). Each station's record contained the three components of the peak ground acceleration (PGA), south–north direction, east–west direction, and up–down direction, as listed in Table 2 (the dataset was provided by the China Earthquake Data Center, http://data.earthquake.cn, last access: 16 June 2016). We calculated the average PGA of the two horizontal components of each strong motion recording and plotted a contour map (Fig. 15) using an inverse distance-weighted (IDW) interpolation algorithm. It determined the cell values using a linearly weighted combination of a set of sample stations with weights inversely proportional to distance (Watson and Philip, 1985). In addition, given that input stations far from the epicenter, where the prediction was made, might have had poor or no spatial correlation, we eliminated the input stations beyond 100 km from the epicenter from the calculation.

## 3.5 Newmark displacement map

In case of a landslide, in practice it is impossible to conduct a rigorous
Newmark analysis when accelerometer records are unavailable. It is also
impractical and time consuming to produce a displacement in each cell during
the regional analysis. Therefore, empirical regressions (Ambraseys and Menu,
1988; Bray and Travasarou, 2007; Jibson, 2007; Saygili and Rathje, 2008;
Rathje and Saygili, 2009; Hsieh and Lee, 2011) have been proposed to
estimate Newmark displacement as a function of the critical acceleration and
peak ground acceleration or the Arias intensity. Rathje and Saygili (2009)
developed a vector model for displacement in terms of the critical
acceleration (*a*_{c}), peak ground acceleration (PGA), and moment
magnitude (*M*_{w}) based on an analysis of over 2000 strong motion
recordings:

where *D* is the predicted displacement in centimeters, and *a*_{c} and
PGA are in units of *g*.

This model is a preferred displacement model at a site where acceleration time recordings are not available. Incorporating multiple parameters of ground motion into the analysis typically results in less variation in the prediction of displacement (Rathje and Saygili, 2009).

The Newmark displacement of each cell was calculated by combining the corresponding values of the critical acceleration, peak ground acceleration, and moment magnitude in Eq. (8). The predicted displacements ranged from 0 to 122 cm, as shown in Fig. 16.

## 3.6 Coseismic landslide hazard map

According to Jibson et al. (1998, 2000), predicted displacements provide an
index of the seismic performance of slopes, where larger predicted
displacements relate to a greater incidence of slope failures. But the
displacements do not correspond directly to measurable slope movements in
the field. To produce a coseismic landslide hazard map, we chose a model of
inexact reasoning, the certainty factor model (CFM) created by Shortliffe
and Buchanan (1975) and improved by Heckerman (1986), to explore the
relationship between the occurrences of landslides and their predicted
displacements. The CFM was created as a numerical method, initially used in
MAYCIN, a backward-chaining expert system used in medical contexts (Shortliffe and
Buchanan, 1975), for managing uncertainty in a rule-based system. In this
model, the certainty factor (CF) represents the net confidence in a
hypothesis *H* based on the evidence *E* (Heckerman, 1986). Certainty
factors range between −1 and 1. A CF with a value of −1 means a total lack
of confidence, whereas a CF with a value of 1 means total confidence.
Values greater than zero favor the hypothesis while those less than zero
favor its negation. According to Heckerman (1986), the probabilistic
interpretation of CF is as follows:

where CF is the certainty factor, *p*(*H*|*E*) denotes the conditional probability for a posterior hypothesis
that relies on evidence, the posterior probability, and *p*(H)
is the prior probability before any evidence is known. In the displacement
analysis, *p*(*H*|*E*) was defined as the
proportion of the area of the landslide within a specific displacement area
and *p*(H) was defined as the proportion of the landslide area
within the entire study area, excluding slopes less steep than 5^{∘}. In this way, the values of CF represented the confidence level for
coseismic landslides. Positive values corresponded to an increase in the
confidence level in slope failure while negative quantities corresponded to
a decrease in this confidence. Higher positive values indicated higher
confidence levels for coseismic landslides.

Given the above definition, we produced a coseismic landslide hazard map in terms of the certainty factors. First, displacement cells every 1 cm were grouped into bins such that all cells with displacements between 0 and 1 cm were grouped into the first bin, those with displacements between 1 and 2 cm were grouped into the second bin, and so on. The displacements were grouped into 123 bins, from 0 to 122 cm. We then calculated the proportion of cells occupied by areas of landslides in each bin. This proportion was considered the posterior probability of each bin as previously defined. The prior probability calculated by dividing the entire landslide area by the entire study area was the same in each bin. Finally, the values of CF were computed in each bin by using Eq. (9) to combine the corresponding values of the posterior and prior probabilities. The certainty factors ranged from −1 to 0.95. The values of CF indicated the confidence level of the occurrence of a landslide for each bin in the study area and provided the basis for producing the coseismic landslide hazard map.

As shown in the hazard map (Fig. 17), 73.2 % of landslides triggered by the Ludian earthquake were in areas with higher confidence levels with CF values greater than 0.6. The interpreted landslides were covered on the map to demonstrate their goodness of fit for the predicted confidence levels for coseismic landslides (Fig. 17).

The predicted displacements represent the cumulative sliding displacements for a given time history of acceleration. Based on the statistically significant sizes of the areas, displacements less than 60 cm, which was around the middle of the range of displacement, occupied about 80 % of the study area, while displacements greater than 80 cm occupied a very small area. Jibson et al. (1998, 2000) assumed that shallow falls and slides in brittle, weakly cemented materials fail following a relatively small displacement, whereas slumps and block slides in more compliant materials likely fail following a larger displacement. That is to say, the study area was more susceptible to rockfalls and shallow, disrupted slides that fail following a relatively small displacement. By contrast, it was subjected at a lower probability to coherent, deep-seated slides that would fail following a larger displacement. Indeed, the majority of landslides triggered by the Ludian earthquake were shallow, disrupted slides and rockfalls (Zhou et al., 2016). Although a few catastrophic rock avalanches, such as the Hongshiyan landslide (Chang et al., 2017), occurred in the field, they did not produce statistically significant samples that could meaningfully contribute to the model, which is consistent with the statistical results as discussed above. Therefore, the model should relate well to typical kinds of earthquake-induced landslides in the study area, thus demonstrating its usefulness in predicting the probability of other types of landslides.

According to Jibson et al. (1998, 2000), a function of CF and Newmark displacement would make it possible to predict the spatial variation in coseismic landslides in any scenario of interest involving the ground shaking. As mentioned above, 80 % of the study area featured predicted displacements of less than 60 cm. The numbers of the Newmark displacement cells were uneven. There were more cells in 1 cm bins for smaller displacements and fewer cells in 1 cm bins for larger ones. This might have affected the statistical significance of the function of CF and Newmark displacement. Therefore, the predicted displacement cells were grouped into bins based on quantile statistics. The breakpoints were 0, 10, 30, 39, 46, 51, 55, 59, 63, and 122. In this way, the number of cells in each bin was equal. Figure 18 shows, for each bin, the CF value of the Newmark displacement as plotted as a dot. As CF values ranged from −1 to 1, and not from 0 to 1, the Weibull (1939) curve developed by Jaeger and Cook (1969) is unsuitable here. Therefore, we modified the functional form as follows:

where CF is the certainty factor, *k* is the maximum CF value
represented by the data, *D* is the predicted displacement, and *a* and *b* are
the regression constants. The regression curve based on data from the Ludian
earthquake is

From the curve shown in Fig. 18, when the predicted displacement increased, the value of CF increased monotonically, meaning that the confidence level for slope failure grew and landslide would probably occur. Such a procedure is consistent with the interpretation of certainty factor theory. Therefore, we were able to obtain estimates of the hazard different from the one used in this study using the same procedure described here.

When fitting the results of shear tests using Coulomb's linear relation, the shear strengths varied widely from high normal stress in the laboratory to low normal stress in the field (Barton, 1973). We introduced the Barton model to the Newmark analysis to reduce the variation in shear strength in terms of Coulomb's constants. We also considered the impact of scale effects by using Eqs. (5) and (6) to prevent Newmark's method from underestimating the shear strength of geological units in regional analysis. In addition, for the Barton model, the joint roughness coefficient (JRC) was estimated from tilt tests, or by matching Barton's joint standard roughness profiles as defined by the International Society for Rock Mechanics (ISRM, 1978). The joint wall compressive strength (JCS) was estimated by Schmidt hammer index tests. These tests helped make a quick estimate of the shear strength in situ, which can facilitate the use of Newmark's method in an emergency hazard and risk assessment after an earthquake.

It is difficult for a statically stable slope to fail under an earthquake.
Earthquakes usually cause slopes to fail in the state of limit equilibrium.
For this reason, it is important to characterize the shear strength of the
slope accurately. The shear strengths were assigned to the geological units
using the results of hundreds of shear tests reported in the references
provided in Table 1. We assigned the original shear strengths to the
geological units, instead of increasing them to render the cells statically
stable, as was done by Jibson et al. (1998, 2000). This would have changed the
statically stable level of the entire study area, especially the slopes in
the state of limit equilibrium. In addition, we considered the size effect
of the potential slide surface, which could yield a lower *F*_{S} and, in
turn, a higher displacement. However, the inventory of landslides was used
to calibrate the predicted displacements, and the confidence levels
indicated by the certainty factors fitted well with the spatial distribution
of coseismic landslides, as shown in the hazard map (Fig. 17).

We also ran a conventional Newmark analysis using the assigned strengths,
such as friction angle (*φ*) and cohesion (*c*), as shown in
Table 1. The predicted displacements calculated by the conventional Newmark
analysis ranged from 0 to 121 cm, compared with 0 to 122 cm as
obtained by the proposed method. Figure 19 shows the hazard map produced
using conventional Newmark analysis. The CFs ranged from −1 to 0.94,
indicating a very similar result to that of the proposed method above.
However, there were large differences along the Shaba River and upstream of
the Niulanjiang River between the methods. By comparing Fig. 17 with Fig. 19, we see that the confidence levels of the proposed method fitted the data
better than those of the conventional method, especially near upstream portions of
the Niulanjiang River. The area under the curve (AUC) was employed to
compare the performance of the methods. To create an AUC plot, the
cumulative area of CFs within each interval of the calculated values, from
the maximum to the minimum, was determined as a proportion of the total
study area (*x* axis) and plotted against the proportion of cumulative
landslides falling within those CFs (*y* axis) (Miles and Keefer, 2009). A
value of 0.5 of the AUC indicates that performance is not better than a
random guess and that of 1 indicates perfect performance (Miles and Keefer,
2009). Figure 20 shows the results of the AUC analysis of both methods. The
calculated value for the proposed method was 0.58 while that for the
conventional Newmark's method was 0.53. That is to say, the method
introduced here yielded better results and is an improvement over the
conventional Newmark analysis.

Newmark's method is a useful physical model to estimate the seismic stability of natural slopes. The mapping procedure for data on the 2014 Ludian earthquake shows the feasibility of a Newmark analysis combined with Barton's shear strength criterion. Such a method has practical applications in the assessment of regional seismic hazard. We also considered the size effect of parameters of shear strength, such as the joint roughness coefficient (JRC) and the joint wall compressive strength (JCS), in regional analyses. Moreover, linking the Newmark displacements to the certainty factor model improved the utility of Newmark's method to predict the hazard posed by coseismic landslides. Finally, the results of an AUC analysis indicate that the proposed method is more reliable than the conventional Newmark method.

The digital geological map hosted by the China Geological Survey (CGS) can be made available upon request. The pre-earthquake satellite images are available from Google earth, a public desktop software (https://www.google.com/earth/, last access: 30 January 2014). The 0.2 m high-resolution post-earthquake aerial images from the Digital Mountain and Remote Sensing Applications Center, Institute of Mountain Hazards and Environment, Chinese Academy of Sciences, and Beijing Anxiang Power Technology Co., ltd. are restricted and cannot be accessed publicly but may be requested from the corresponding author. The 30 m × 30 m ASTER Global Digital Elevation Model is distributed by NASA EOSDIS Land Processes DAAC (https://doi.org/10.5067/ASTER/ASTGTM.002; NASA/METI/AIST/Japan Spacesystems and U.S./Japan ASTER Science Team, 2009). The dataset of the strong motion stations is provided by the China Earthquake Data Center (http://data.earthquake.cn; China Earthquake Data Center, 2016).

SQ initiated and led this research. MZ designed the analytical framework of this study, produced maps and figures, performed the data analysis and interpretation, and wrote the paper. YZ helped interpret landslides and collect the records of the strong motion stations. SQ, ZS, and BSZ reviewed and edited the paper.

The authors declare that they have no conflict of interest.

This research was supported by the National Natural Science Foundation of China (grant nos. 41825018, 41672307, and 41807273), the Science and Technology Service Network Initiative of the Chinese Academy of Sciences (grant no. KFJ-EW-STS-094), and the China Scholarship Council (grant no. 201704910537).

This research has been supported by the National Natural Science Foundation of China (grant nos. 41825018, 41672307, and 41807273), the Science and Technology Service Network Initiative of the Chinese Academy of Sciences (grant no. KFJ-EW-STS-094), and the China Scholarship Council (grant no. 201704910537).

This paper was edited by Mario Parise and reviewed by two anonymous referees.

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*M*

_{w}6.1 Ludian earthquake in 2014 and compared with a conventional Newmark analysis using area under the curve analysis.