Introduction

Earthquakes and rainfall are the most common triggers of landslides. Thus, many papers have been published regarding their hazardous consequences on slope stability1,2,3,4 and their influences on landscape evolution5,6,7,8,9. Global predictive models separately exist for both earthquake- and rainfall-triggered landslides10,11. However, few predictive models include the compound effects of earthquakes and rainfall, and they are only available for specific areas, where both multi-temporal landslide inventories and proxies representing the triggering factors exist12.

The predictive capacity of models targeting landslides is limited primarily by the accuracy of model inputs and, in particular, the availability of landslide inventories and estimates of triggering factors13. Use of physical models decreases the strong data dependency with respect to landslide inventories, but characterizing shear strength at global and regional scales remains challenging14. But if the geotechnical characteristics of hillslopes, as well as the triggering factors of rainfall and seismic shaking, can be reasonably estimated, hazard from earthquake- and rainfall-triggered landslides can be modeled15,16,17,18. Nevertheless, identifying the compound effect of earthquakes and rainfall is still difficult19. For example, modeling hazard from rainfall-triggered landslides in an area recently struck by an earthquake requires a better understanding because post-seismic hillslope susceptibility varies mainly depending on landslide characteristics (e.g., type of landslide material, slope, and landslide geometry) and the mechanism controlling strain accumulation within the landslide body19,20.

Further complicating matters is that laboratory-scale analyses show that post-seismic hillslope strength can be unchanged, increased, or decreased compared to its pre-seismic counterpart19,20. Most commonly, however, strong ground shaking weakens hillslopes21,22. This weakening, which is also observed in regional-scale assessments using multi-temporal landslide inventories23,24,25,26, commonly is referred to as the earthquake legacy effect27,28,29. The magnitude and duration of the legacy effect can be estimated by examining reactivated or new landslides that occur in post-seismic periods29,30.

Coseismic hillslope weakening is caused by ground shaking that disturbs hillslopes whether or not landslides are triggered. This disturbance can cause a reduction in shear strength (RSS) of hillslope materials. To accurately characterize the resultant elevated landslide susceptibility in post-seismic periods using physical modeling, the amount of RSS needs to be estimated. Theoretically, this could be done by detailed geotechnical characterization of hillslope material in pre- and post-seismic periods. In this context, type of hillslope materials, hillslope geometry, and failure surface31,32 also need to be characterized. As a result of such investigations, the most appropriate failure criterion is selected for a given hillslope to estimate its strength. For instance, the stability of a hillslope governed by soil, intact rock, or discontinuities could be assessed by using Mohr-Coulomb33, Hoek-Brown34,35, and Barton-Bandis36,37 faliure criteria, respectively. However, in regional-scale analyses, such detailed geotechnical investigations on a regional scale are impractical because of the time and cost required.

To address these limitations, some methods coupling active geotechnical seismic surveys with outcrop-scale engineering desciptions/tests, which are used to classify rock masses based the geological strength index34, have been proposed38,39. However, even those methods fail to capture slope shear strength at a regional scale and thus do not address the RSS.

In light of these limitations of physical sampling and testing, we aim to capture the amount of RSS of hillslope materials following strong earthquakes by carrying out back analyses40,41,42. To do so, we analyze a subset of the area affected by the 2008 Mw 7.9 Wenchuan, China earthquake43. The earthquake triggered ~200,000 landslides around the Longmen mountains44. In addition to triggering widespread coseismic slope failures, the earthquake also increased the post-seismic landslide frequency21,25,45. Although it was argued that the legacy effect of the earthquake on hillslope stability could last several decades24, a study of the epicentral area of the where peak ground acceleration (PGA) ranged from 0.4 g–1.0 g suggested that the legacy effect lasted ~7 years46. In this study, we focus on the same area (Fig. 1)47.

Fig. 1: Maps showing the study area.
figure 1

a Location map. b Study area overlain by faults and earthquake epicenter. c Major geologic units in study area. d Peak ground acceleration map of the 2008 Wenchuan earthquake bounded by the study area. Surface rupture is digitized from Xu et al.47 and geologic map is taken from National Geological Data Library of China Geological Survey (http://www.ngac.org.cn).

Full size image

Landslides that occurred in our study area were mapped by Fan et al.43,48. They generated one coseismic and five post-seismic landslide inventories obtained by comparing successive images acquired in 2011, 2013, 2015, 2017, and 2018 (Supplementary Fig. 1). They assessed the average uncertainty in landslide polygons as ± 19 % in terms of total landslide area based on comparison between five geomorphologists that mapped the same area. They mapped remobilized landslides and new landslides triggered in post-seismic periods of the 2008 Wenchuan earthquake. We exclude remobilized landslides from our analyses because we lack access to multi-temporal digital elevation models (DEMs) of hillslope geometry before and after the remobilization. Considering remobilized cases would introduce additional uncertainties in the back-analysis. Therefore, we focus exclusively on coseismic landslides and the newly activated ones that occurred post-seismically (Supplementary Table 1) to estimate the RSS responsible for their initiation.

To estimate the shear strength of slope materials, we limit our study area to locations where post-seismic landslides occurred. Specifically, we back-analyze unfailed hillslopes during the coseismic event and both unfailed and failed hillslopes in the post-seismic periods. By doing so, we aim to estimate shear strength at locations where hillslopes did not fail in the coseismic event but rather in one of the post-seismic periods. We then calculate the RSS, which represents the decrease between the shear strength in the coseismic and post-seismic periods. Ultimately, we analyze the spatial distribution of RSS in relation to topographic and seismic variables using a data-driven model (Gaussian Generalized Linear Model, GLM) and extrapolate estimated reductions over larger areas. Our findings provide a comprehensive insight into post-seismic hillslope reaction to strong ground shaking.

Results

Estimating strength parameters

The study area includes five lithologic units (National Geological Data Library of China Geological Survey, http://www.ngac.org.cn): andesite, phyllite and schist, lithic sandstone, granite, and diorite (Fig. 1). To estimate shear strengths of these lithologic units, we estimate the average landslide thickness in the study area. We use published landslide area-volume relationships for landslide thicknesses as great as 5 m (Fig. 2)49,50. We include a sensitivity test for a range of potential values (2 m, 3 m, 4 m, and 5 m) to be separately introduced in our back-analytical routine51,52.

Fig. 2: Estimated landslide thickness based on two landslide area-volume relationships proposed for the landslides triggered by the Wenchuan earthquake.
figure 2

a Landslide thickness distribution. b Curves representing the landslide area-volume relationship.

Full size image

For each post-seismic landslide location, we assign material and shear-strength properties (\(\gamma ,{\varphi }^{{\prime} },{c{{\hbox{'}}}}\)) to each grid cell (Table 1) based on published values corresponding to the five lithologic units in the area14,15,53,54,55,56. For the identified range of cohesion and friction-angle couples, we iteratively test shear-strength values from the literature that meet the failure and stability conditions for both coseismic and post-seismic periods. Figure 3 shows an example for granite hillslopes where landslide thickness is 3 m. We run the same analyses for each lithologic unit for the four landslide thicknesses (see Supplementary Fig. 2).

Table 1 Cohesion and friction-angle couples for the five lithologic units
Full size table
Fig. 3: An example for 3-m-thick landslides on granite slope showing the shear-strength range in each time window.
figure 3

The black curves are the shear strength calculated by random cohesion and friction-angle couples built based on the Gaussian distribution. The horizontal boxplots are selected shear-strength ranges fulfilling unfailed hillslopes in coseismic (blue boxplots) and failed hillslopes in post-seismic phases (red boxplots). The panels show the shear strength range for each time windows of (a) coseismic phase, (b) 2008–2011, (c) 2011–2013, (d) 2013–2015, (e) 2015–2017, (f) 2017–2018. Supplementary Fig. 2 shows similar plots for other lithologic units and landslide thicknesses.

Full size image

Estimating RSS

The approach presented above allows us to identify the mean shear strength of hillslope materials for post-seismic landslide locations in both coseismic and post-seismic periods. We then compare these post-seismic shear strengths with their coseismic counterparts and calculate the difference, which yields the RSS due to the earthquake.

Seismic ground motion in the study area ranged from 0.4 g–1.0 g; thus, we calculate the RSS of hillslopes across this range (Fig. 4). Figure 4a shows that the RSS ranges from 40–80 kPa where the landslide thickness is 2 m. Results also show that the RSS is positively correlated with PGA. This indicates that hillslopes affected by stronger ground shaking have higher RSS in post-seismic periods. We also show that greater RSS is required for deeper landslides to fail. For example, for a landslide thickness of 5 m, the estimated RSS is 120–180 kPa (Fig. 4d), whereas its counterpart for a landslide thickness of 2 m is 40–80 kPa. This does not mean that the percentage of RSS is necessarily higher for deeper landslides because this amount changes depending on the initial shear strength in the coseismic phase. Shear strength along the failure surface increases as the thickness of a landslide increases because of increasing normal stress at depth. Therefore, a higher RSS is required to cause failure of a deeper landslide.

Fig. 4: Reduction in shear strength (RSS) in response to range of PGA values.
figure 4

RSS generated for landslide thicknesses of (a) 2 m, (b) 3 m, (c) 4 m, and (d) 5 m. Gray shaded area indicates the 99% confidence interval.

Full size image

To express RSS in terms of percentages, we compare them with their coseismic counterparts (Fig. 5). Results show that for all thicknesses, RSS generally varies between 60% and 75% (Fig. 5e).

Fig. 5: Reduction in shear strength (RSS) expressed as percentages for landslide thicknesses.
figure 5

RSS generated for landslide thicknesses of (a) 2 m, (b) 3 m, (c) 4 m, (d) 5 m. Panel (e) shows mean values estimated from all landslide thicknesses, and panel (f) shows best linear fits provided for all thicknesses. Gray shaded area indicates the 99% confidence interval.

Full size image

Extrapolating RSS through a data-driven model

This stage of the analysis estimates RSS only within mapped coseismic landslides because we lack information about unfailed slopes. Therefore, we include a final step aimed at regionalizing the RSS model. We use a Gaussian Generalized Linear Model (GLM) trained to explain the spatial variability in RSS as a function of PGA, local relief (LR), and topographic position index (TPI). Because the RSS model is based on slopes that failed, it reflects the maximum possible RSS, a worst-case scenario. By regionalizing this model, this scenario propagates everywhere and thus overestimates the instability of slopes that might have experienced some RSS but not enough to cause failure. This limitation cannot be overcome unless RSS is estimated in unfailed slopes, a capability beyond our approach. Nevertheless, extrapolating the maximum RSS constitutes a useful experiment. Figure 6 summarizes the numerical aspects of the model: the dominant covariate is LR (mean βLR = −2.62), followed by PGA (mean βPGA = 1.89) and TPI (mean βTPI = 0.19). All variables are significant (i.e., the distribution of regression coefficients does not intersect the zero line, Fig. 6a), which also is confirmed by p-values < 0.001 calculated for LR, PGA, and TPI.

Fig. 6: Outputs of the data-driven model (Gaussian GLM).
figure 6

a Regression coefficients. b Observed versus fitted values with residuals. c Pearson Correlation Coefficient (PCC). d Mean Absolute Error (MAE) of the validation results.

Full size image

The sign and the absolute value of the regression coefficients indicate the role of each covariate. Our findings show that LR is strongly negative, PGA is strongly positive, and TPI is slightly positive. We interpret these findings in the discussion section.

The model fits the data well, as shown by the clustering of observed and predicted RSS values along a 45° line (Fig. 6b). This is maintained from low to high quantiles, with the exception of very large observed RSS values, which are slightly underestimated (as shown via the probability density function (PDF) of the residuals, centered at zero and with few very positive cases). The deviation from the fitted model for RSS values of >70% might be due to the range of RSS values in the target dataset. Figure 5 shows that a large number of our landslides have RSS values ranging between ~65% and ~70%.

Applying a predictive equation spatially requires assessing both the goodness of fit and the predictive performance of the data-driven model. Therefore, we include random cross-validation with a 90/10 split, iterated 100 times. Results are shown in Fig. 6c, d, with very high values of the Pearson Correlation Coefficient (PCC, centered at 0.81) associated with very low values of Mean Absolute Error (MAE). Notably, the PCC distribution appears to be positively skewed, implying that in most of the 100 cross-validation replicates, the prediction is better than what is shown through the mean.

Having shown that our RSS prediction is reliable both for interpretation and performance, we produce a predictive equation for the entire study area (Fig. 7). RSS increases from northwest to southeast. As would be intuitively expected, it is thus greatest near the fault, where ground shaking was the most intense.

Fig. 7: Spatial distribution of estimated RSS based on the fitted model.
figure 7

The red star indicates the epicenter of the Wenchuan earthquake; black lines show the coseismic surface rupture. The equation shown above corresponds to the linear combination of each model component.

Full size image

Discussion

Geotechnical characterization of hillslope materials is by far the greatest challenge for physically based models. Most regions of the world lack available high-quality shear-strength data. Perhaps more challenging is the inherent spatial variability in the physical properties of mapped geologic units and consequent difficulty in characterizing rock-mass properties at regional scale7,34,35. For bedrock slopes, overall rock-mass strength is governed more by pre-existing discontinuities (faults, joints, bedding planes, etc.) than by the strength of intact material such as might be sampled and tested in the laboratory. The shape and dimensions of bedrock landslides commonly are defined by the presence, orientation, and spacing of such discontinuities35,57,58. And in both rock and soil, mapped geologic units generally include layers of widely differing strength properties, which makes assigning a single strength to a unit inherently uncertain14. In spite of these difficulties, laboratory-based cohesion and friction-angle couples are used widely at regional scale in physically based slope stability analyses15,59,60. Selection of cohesion and friction-angle couples is a source of uncertainty in these and other mechanistic approaches.

Our study aims to minimize the uncertainties detailed above by combining multiple approaches. First, we apply back-analyses for all multi-temporal landslide inventories. This means that we identify cohesion and friction-angle couples fulfilling stability and failure conditions in coseismic and post-seismic phases, respectively, though the initial shear-strength parameters are material properties taken from the literature. We then build theoretical Gaussian distributions around these mean values to simulate a wider range of values to model the instability conditions. To minimize or unfold various source of uncertainties regarding parameter selection, we replicate our analyses for a range of values for additional parameters. For example, we test a wide range of COV values to identify the most suitable Gaussian distribution that cohesion and friction-angle values could follow. We should stress that although shear strength parameters generally follow a normal distribution61,62,63, this assumption may not hold everywhere. Therefore, upon acquisition of site-specific geotechnical analyses that would yield skewed distributions of shear-strength parameters, these would need to be addressed accordingly. In other words, one would need to fit a suitable distribution to the data and simulate from there. However, in the context of the analyses presented here, such information is not available.

Also, we replicate our analyses for different landslide thicknesses to assess the associated sensitivity in the RSS estimates. We run this sensitivity test for landslide thicknesses ranging between 2 m and 5 m. Where landslide thickness is <2 m or >5 m, the given range of shear strength will not fulfill the failure condition in the post-seismic phase; thus, we do not expand the sensitivity test for those shallower and deeper landslides. Despite this limitation, the performed test still demonstrates the influence of landslide thickness in RSS calculations.

Second, we complement this approach by calculating the difference in shear strength corresponding to stable and unstable conditions, which we express as a percentage. In a regional analysis, strength differences between units are more important than highly accurate strength estimations for each unit15. This certainly is true in our analysis, and in a similar sense, the RSS is of greater interest than the absolute strength. Therefore, targeting not the actual shear strength but the required reduction of them by testing a range of possible cohesion and friction-angle couples, as well as landslide thicknesses, is a way to minimize uncertainty.

Other approaches to capture the uncertainty in model inputs have been suggested, the most common of which is logic trees, wherein estimates of the upper, lower, and most likely values of input parameters are estimated, and weights are assigned to each value for model iteration51,64. These approaches are simple to use and intuitive, but choosing weighting factors can be subjective and difficult to rationalize65.

Our approach does not completely remove the uncertainties described above because it is based on several simplifications. Hillslopes do not necessarily even experience a weakening effect after an earthquake; they can be strengthened or weakened in post-seismic periods depending on type of hillslope material, hillslope geometry, intensity of ground shaking, and accumulated shear strain along the failure surface19,20. Nevertheless, characterizing these parameters requires detailed geotechnical field and laboratory investigations, which is practical for regional-scale analyses.

Ideally, we would segregate the analysis based on different types of landslides. Also, focusing on landslide source areas rather than entire landslide areas would yield more accurate results. Unfortunately, the landslide inventories analyzed make no such distinctions.

Another simplification is expressing ground shaking using only PGA provided by empirical equations. Although this ignores site effects, frequency content, and duration of shaking66, it is a widely used approach in regional-scale slope-stability assessments14,41 because numerical earthquake simulations of these parameters over broad regions are both uncommon and time consuming67,68.

Also, we use m, a value representing the proportion of a landslide block that is saturated in response to the extreme rainfall events, because identifying the absolute groundwater level at regional scale is impossible. However, m could also be assigned, for instance, considering average or cumulative precipitation for a given period69. Therefore, in the Method section, we provide sensitivity analysis on RSS depending on the m value.

Our findings show that RSS has a strong positive correlation with landslide thickness and that variation in RSS can be expected at different landslide depths (see Fig. 4). To further explore this variation, we examine the converted RSS values in percentages and observe consistent results, though depth differences are far less pronounced than for the absolute values and converge at higher ground motions (see Fig. 5). One possible reason for the depth-dependence of RSS is that coseismic slope damage occurs at several scales, from weakening interparticle cohesion to fracturing slope material at centimeter to meter scale. Thus, deeper slides characterized by larger volumes would accumulate slope damage across all scales, and shallow slides are likely to have proportionally less damage.

Overall, our analyses show that near-surface rock strength can be reduced as much as 75% if a hillslope experiences ground shaking >0.9 g. Although this is the maximum theoretical strength reduction, it still is a dramatic hillslope weakening process.

We extrapolate our approach to parts of the study area unaffected by mapped landslides by building a data-driven model. We apply the calculated RSS values to the rest of the study area and then interpret the role of our predictive variables based on regression coefficients. LR is anti-correlated with RSS. Greater values of LR correspond to large local variations in elevation, and thus this it is generally correlated with stronger rocks, which are more resistant to erosion by frost cracking or landsliding70. As for PGA, higher shaking levels could damage both unconsolidated and consolidated materials, though perhaps not to the same extent. Most unconsolidated materials have some cohesion, which would likely be reduced or eliminated by strong shaking. More consolidated materials would likewise undergo reduction in cohesion and possible fracture formation. However, we lack sufficient observations in each lithologic unit in post-seismic landslide locations to derive a statistically sound conclusion. Therefore, further analyses are required to test the transferability of our approach to other landscapes. Also, the RSS of different lithologic units requires further examination. In this context, multi-temporal landslide inventories generated for other earthquake-affected areas12 and, in particular, inventories representing the temporal variations in spatial distributions of landslides71 are primary targets to test the transferability of the proposed approach. RSS would be different for reactivated versus new landslides. Also, landslide polygons separately generated for source and runout areas would provide better insight into RSS because the source is the main area needs to be taken into account while checking slope stability. However, landslide inventories rarely differentiate source areas from runouts72,73.

The marked increase in RSS with proximity to the fault zone (see Fig. 7) shows the dominant role of seismic ground motion in weakening slope materials. In the near-fault area, static strain induced by fault movement also can weaken near-surface material74. The combined effects of fault-driven near-surface deformation and extreme ground shaking make this zone near the fault particularly susceptible to RSS and consequent slope failure. However, these aspects also require further investigations and seismic surveys, which have been used to assess hillslope strength39,75.

The ideal validation of our modeling approach would require a field survey to measure hillslope material strengths in both pre-seismic and post-seismic periods. Such data collection generally is unfeasible, and we know of no cases where it has been done. In a few instances, pre-existing geotechnical data have been compiled after a major earthquake to facilitate modeling slope conditions at the time of the seismic shaking. For example, after the Northridge, California (USA) earthquake in 199476, Jibson et al.15 compiled pre-existing geotechnical data across the region to model pre-seismic slope conditions. A program to sample and test a statistically significant number of samples in the dozens of geologic units in the area after the earthquake would have been cost-prohibitive and nearly impossible in practice. Jibson and Michael59 compiled geotechnical data to construct a seismic landslide hazard map of Anchorage, Alaska (USA), which provided needed pre-seismic data when the 2018 Anchorage earthquake occurred77. Again, though, collecting and testing large numbers of soil and rock samples following the earthquake could not reasonably be undertaken. Pre-seismic geotechnical data generally come from local government repositories that contain reports from private consultants who perform laboratory shear-strength testing for building projects. Such repositories can contain hundreds or thousands of geotechnical reports that accumulate over years to decades and thus have a statistically robust amount of data in different geologic units in many locations. Any attempt to replicate such a dataset after an earthquake would be impossible. Also, the coseismic damage to slope materials occurs at landscape scale; collecting samples small enough for geotechnical testing would miss the larger extent of the coseismic disturbance. This is the main justification for our experimental design. At present the only approach we know of to estimate coseismic RSS is back-analysis, and we offer one approach to this. This may prove to be useful in other seismically active areas, albeit more experiments are required to explore its portability, reliability, and consistency, especially for different earthquake magnitudes, geologic settings, and terrain complexities.

Methods

Inputs

The multi-temporal landslide inventories associated with the 2008 Wenchuan earthquake were mapped mainly based on high-resolution (SPOT-5 and 6, Worldview-2, Pleiades) satellite imagery43. The dataset provides polygons of remobilized and new landslides.

Source and runout areas of landslides were not differentiated in the dataset; thus, to estimate representative topographic, seismic, and meteorological variables for each landslide, we determine the highest elevation point in each polygon. For those points, we extract the corresponding factors for both back-analysis and developing the data-driven model. Specifically, for the back-analyses we identify the main environmental and triggering conditions controlling the slope stability, including lithology, slope steepness, ground shaking, and rainfall. We use NASA’s (earthdata.nasa.gov) DEM (12.5 m spatial resolution) for topography, the U.S. Geological Survey (USGS) ShakeMap of the 2008 Wenchuan earthquake78 for PGA (~1 km spatial resolution), and rainfall data from the Climate Hazards Group InfraRed Precipitation with Station (CHIRPS, ~5 km spatial resolution). Lithologic descriptions are taken from the National Geological Data Library of China Geological Survey (http://www.ngac.org.cn). As for the data-driven model, we use PGA, LR, and TPI.

Estimating reduction in shear strength

We follow a four-step methodology: (1) pre-processing, (2) back-analysis, (3) estimating the RSS, and (4) extrapolating the estimated RSS via a data-driven model (Fig. 8).

Fig. 8: Flowchart for identifying the reduction in effective shear strength (RSS).
figure 8

\({c{\prime} }_{{co}}\) and \({\varphi {\prime} }_{{co}}\) are cohesion and friction angle of post-seismic landslide locations during the earthquake where they were still stable. \({c{\prime} }_{{post}}\) and \({\varphi {\prime} }_{{post}}\) refer to their post-seismic counterparts representing the failure condition.

Full size image

Step-1

Step-1 involves pre-processing the data. We gather cohesion and friction angles (Table 1) proposed in the literature for the lithologic units we have in our study area. We use these couples later on to estimate shear strength of hillslope materials. We assume that effective shear-strength (τ′) parameters follow the Mohr-Coulomb failure criterion:

$$\tau {\prime} ={\sigma }^{{\prime} }\tan \varphi {\prime} +c{\prime}$$
(1)

where φ′ is the effective internal friction angle, c′ is the effective cohesion, and σ′ is effective normal stress, which can be expressed as:

$${\sigma }^{{\prime} }=(\gamma t-{\gamma }_{{{{{{\rm{w}}}}}}}{tm})\cos \alpha$$
(2)

where γ is the unit weight of the soil/rock material, γw is the unit weight of water, α is the slope angle (°), t is the landslide thickness, and m is the proportion of the landslide block that is saturated. Thus, m = 0 refers to dry conditions, whereas m = 1 indicates fully saturated conditions.

Step-2

In this step, we back-analyze the slopes to identify cohesion and friction-angle couples fulfilling the stability or failure conditions based on a set of assumptions for coseismic and post-seismic periods, defined below.

Coseismic conditions

For coseismic conditions, we estimate the shear strength (\({\tau {\prime} }_{{{{{{\rm{co}}}}}}}\)) of hillslope materials, where post-seismic landslides are stable during the earthquake, using two assumptions: (1) hillslope materials are dry and (2) the hillslope remains stable if the critical acceleration (ac) is greater than the estimated PGA at any given location79. We check the first assumption by calculating the standard precipitation index (SPI)80, a widely used index expressing the droughtiness of an examined period for a given location81,82. We calculate SPI for our study area for the month of the earthquake (May 2008) and for the two prior months using the 10-year CHRPS daily precipitation dataset between 2008 and 2017. SPI values for those periods are −1.09 (moderately dry) and −1.58 (severely dry), respectively. Based on these findings, we assume that the contribution of precipitation to landsliding was negligible during the coseismic event.

As for the second assumption, we characterize seismic slope stability based on the Newmark approach79, which models the sliding mass as a rigid block on an inclined plane. The block has a known critical acceleration, the seismic acceleration required to overcome basal shear resistance and initiate permanent downslope displacement. The critical acceleration can be approximated accurately as follows:

$${a}_{{{{{{\rm{c}}}}}}}=(FS-1)g\,\sin \alpha$$
(3)

where FS is the static factor of safety, g is the acceleration of gravity (9.81 m/s2), and α is the slope angle79. A further simplification generally applied is that landslides are infinite-slope failures, which allows calculation of FS as follows:

$${FS}=\frac{{c}^{{\prime} }}{\gamma {tsin}\,\alpha \,}+\frac{{\tan \varphi }^{{\prime} }}{\tan \alpha \,}-\frac{m{\gamma }_{{{{{{\rm{w}}}}}}}{\tan \varphi }^{{\prime} }}{\gamma \tan \alpha \,}$$
(4)

in which the first term accounts for the cohesive component of the shear strength, the second term for the frictional strength, and the third term for the reduction in frictional strength due to pore-water pressure15.

Post-seismic periods

We quantify the post-seismic shear strength (\({\tau {\prime} }_{{{{{{\rm{post}}}}}}}\)) analogously to the coseismic situation by assuming that the post-seismic landslides were triggered by extreme rainfall events. We use a range of m values representing the groundwater response to the extreme rainfall events recorded during the examined time windows69. We calculate the mean of extreme daily precipitation (above 95th percentile) in each grid for each post-seismic period (Supplementary Fig. 3); then, we assign m values to each grid by rescaling the calculated extreme precipitation values between 0 and 1. Failure is assumed to have occurred in cells where FS < 1. This approach is consistent with other studies where precipitation is used to diminish the influence of meteorologic effects from post-seismic landsliding and to isolate the role of earthquake legacy effect28.

Based on this approach, we also quantify the influence of m in the calculation of RSS for the coseismic phase, which we assume it is dry. Despite this assumption, supported by SPI, we also run our analyses for a hypothetical fully saturated condition, where m = 1. The result shows a maximum uncertainty of 3.36% in RSS caused by groundwater level (Supplementary Fig. 4).

Back-analyses

We first test the role of two variables capable of influencing the slope-stability model: landslide thickness and the coefficient of variation (COV) of distributions of cohesion and friction-angle pairs.

For each landslide thickness, we randomly create shear-strength parameters using a Monte Carlo simulation, a commonly used approach for addressing the uncertainty of soil and rock properties in engineering geological problems63,83,84. To run the simulations, we estimate the distribution of shear-strength parameters. Ranges of cohesion and friction angle for a given rock or soil unit commonly are represented using a Gaussian distribution61,62,63. However, the shape of the distribution needs to be identified because different variances around the mean can lead to large simulation differences. Therefore, we test different COV values, calculated as the standard deviation divided by the mean, to control the width of the Gaussian distribution. Because we assume both cohesion and friction angle follow the Gaussian distribution, while testing COV values for one of these shear-strength parameters, we keep the other constant. Following Phoon and Kulhawy85, we set the COV of friction angle at 0.15. For cohesion, we assign different COVs for different landslide thicknesses. This step does not involve back-analysis, but rather the direct solution of the Newmark method for the coseismic landslide inventory. We use the cohesion and friction-angle couples suggested in the literature as the beginning means (Table 1) and then build theoretical Gaussian distributions around these mean values for each rock unit (Fig. 1). This allows estimation of the COVs that best match the unfailed hillslopes.

To numerically identify the best match, we target unfailed hillslopes during the co-seismic phase. For those stable hillslopes, we test different COV values and accordinly generate cohesion and friction-angle couples. Based on those couples, we calculate critical acceleration using Eq. 5. If the given COV value of cohesion does not ensure the stability of a given hillslope (ac < PGA), then we label the hillslope as False Negative (FN); otherwise (ac < PGA), we label it as True Negative (TN). As a result, we calculate the True Negative Rate (TNR) for all tested COVs of cohesion and define the best match as giving highest value (Fig. 9):

$${{{{{\rm{TNR}}}}}}=[{{{{{\rm{TN}}}}}}/({{{{{\rm{TN}}}}}}+{{{{{\rm{FN}}}}}})]\times 100$$
(5)
Fig. 9: Variation in COV values with respect to True Negative Rate (TNR).
figure 9

Panels showing (a) map of estimated TP, FN, and TN, (b) cohesion distribution of granite using different COVs, and (c) variation between cohesion COVs and TNR for different landslide thicknesses.

Full size image

These analyses show that COVs of 0.39, 0.78, 1.2, and 1.55 give the highest TNRs for thicknesses of 2 m, 3 m, 4 m, and 5 m, respectively. As mentioned above, these values are estimated for a fixed COV of friction angle (COV = 0.15). However, to assess the uncertainty of using such a fixed value, we also test COVs ranging from 0.12 to 0.22. Specifically, for different COVs of friction angle, we keep its counterpart for the cohesion and calculate its influence on RSS. This test show that the variations in RSS due to the choice of COV is <2.1% (Supplementary Fig. 5).

Step-3

We run 1,000 Monte-Carlo simulations to repeat all steps summarized above to minimize the error due to a fixed parameter choice and to determine a final result as the mean of 1,000 simulations. In each simulation, we randomly select 5,000 cohesion and friction-angle couples to calculate FS and ac. Then, we calculate the RSS between co-seismic (\({\tau {\prime} }_{{{{{{\rm{co}}}}}}}\)) and post-seismic (\({\tau {\prime} }_{{{{{{\rm{post}}}}}}}\)) phases, represented by actual values (RSSv) and percentages (RSSp):

$${{RSS}}_{{{{{{\rm{v}}}}}}}={{\tau {\prime} }_{{{{{{\rm{co}}}}}}}-\tau {\prime} }_{{{{{{\rm{post}}}}}}}$$
(6)
$${{RSS}}_{{{{{{\rm{p}}}}}}}=\left[\frac{\left({{{\tau }^{{\prime} }}_{{{{{{\rm{co}}}}}}}-{\tau }^{{\prime} }}_{{{{{{\rm{post}}}}}}}\right)}{{{\tau }^{{\prime} }}_{{{{{{\rm{co}}}}}}}}\times 100\right]$$
(7)

Up to this point, the estimated RSSs are generated only for post-seismic landslide locations. However, RSS likely occurred across the entire region. We therefore introduce a final step to regionalize the greatest expected RSS across the study area. To do so, we use a Generalized Linear Model (GLM) where the RSS is the response (dependent) variable, assumed to have a Gaussian distribution. We use three input (independent) variables: (1) Local Relief (LR), the difference in elevation between each grid cell and the mean of all grid-cells within a 100 m radius; (2) Topographic Position Index (TPI86), a terrain classification method indicating the relevant position of a given location with respect to its neighborhood; and (3) PGA.

The resulting GLM can be denoted as

$$E(Y)={\beta }_{0}+{\beta }_{{{{{{\rm{LR}}}}}}}{LR}+{\beta }_{{{{{{\rm{TPI}}}}}}}{TPI}+{\beta }_{{{{{{\rm{PGA}}}}}}}{PGA}$$
(8)

where E(Y) is the expected shear strength reduction (RSS), β0 is the global intercept, and βLR, βTPI, βPGA are the regression coefficients estimated for each corresponding covariate.

To verify the functionality of the model, we cross-validate by sampling, without replacement, 100 times the data, and extracting a subset made of 90% of the data for calibration and 10% for validation. At each iteration, we store the Pearson Correlation Coefficient (PCC) and the Mean Absolute Error (MAE).