The spatio-temporal dynamics of landslide path dependency was recently quantified for the Collazzone study area (Samia et al., 2017a) and was implemented in landslide susceptibility modelling at the scale of slope units (Samia et al., 2018). Our previous quantification of landslide path dependency used simplified information about the spatial overlap among landslides in a polygon-based multi-temporal landslide inventory (Samia et al., 2017b). The novel aspect of the present paper is that now, at finer spatial resolutions, we quantify landslide path dependence simultaneously in space and time. For this quantification, we use Ripley's K function (Ripley, 1976; Diggle et al., 1995). Ripley's K function has been used mainly in spatial point pattern analysis and reflects the degree of spatial clustering of events (e.g., landslides – Tonini et al., 2014; forest fires – Gavin et al., 2006; crimes – Levine, 2006; and disease outbreaks – Hinman et al., 2006). The function determines whether events are clustered, dispersed or randomly distributed. A modified Ripley's K function was also used to quantify the degree of clustering of point events in space and time (Lynch and Moorcroft, 2008; Ye et al., 2015). In the landslide path-dependency context, we used Ripley's space–time K function to reflect the degree to which landslides occur near previous landslides and how this changes with increasing distance to the previous landslide in space and time. The starting point to derive Ripley's K is a point-based multi-temporal landslide inventory consisting of points in the geometric centre of polygons of landslides (Fig. 1).

Ripley's space–time K function tests whether the number of events that is observed in a space–time cylinder around an initial event is equal to what is expected given the average point density in space and time (Ripley, 1976, 1977; Diggle et al., 1995). The space–time cylinder I(h,Δ) (Fig. 4) is defined as 1I(h,Δ)dij,tij=1,dij≤handtij≤Δ0,otherwise, where h shows the spatial distance increment, Δ shows the time increment, i and j are two landslide centre points, and d and t reflect the distance and time between the two landslide centre points, respectively.

The expected Ripley K function for one space–time cylinder of size h and Δ is defined as 2K(h,Δ)=1λst∑j≠iEI(h,Δ)dij,tij, where E is the expected number of landslides in the cylinder, and λst reflects the average space–time intensity of the landslides, i.e., the expected number of landslides per unit of space–time volume, which is calculated as 3λst=na(R)×tmax-tmin, where n is the number of landslides in the entire inventory, t is time and a(R) reflects the size of the area. Therefore, the expected Ripley space–time K function for the space–time cylinders around each landslide point is defined as 4K(h,Δ)=1n⋅λst∑i=1n∑j≠iEI(h,Δ)dij,tij. Similarly, the observed Ripley space–time K function is calculated from the landslide inventory: 5K^(h,Δ)=1n⋅λ^st∑i=1n∑j≠iI(h,Δ)dij,tij. Finally, we defined the space–time clustering (STC) measure, which reflects how much more likely it is that a landslide will occur given a time and space distance from a previous landslide, as follows: 6EmpiricalSTC(h,Δ)=K^(h,Δ)K(h,Δ)-1. STC values > 0 indicate clustering, and values < 0 indicate dispersion. We calculated STC (h, Δ) for a wide range of h and Δ: values of h ranged from 0 to 500 m in 30 steps, and values of Δ ranged from 0 to 38 years in 30 steps. This yielded 900 empirical values of STC (h, Δ). We then fitted an exponential-decay function of h and Δ to the empirical STC values. This exponential-decay function was used to calculate STC values for each pixel, depending on when and where a landslide last occurred closely to that pixel.

Based on this, we calculated two landslide path-dependency variables (Fig. 5). The first variable reflects the maximum value of all STC values for all previous landslides near a pixel. This variable results in high values when one particular previous nearby landslide is expected to have a large impact on the susceptibility of landslides. The second variable is the sum of all STC values of all previous landslides near a pixel. This variable results in high values when all previous nearby landslides are expected to have a large impact on the susceptibility of landslides. This approach mirrors what we did in our slope-unit-based susceptibility model (Samia et al., 2018) in the sense that the variables separate the impact of the most impactful previous nearby landslide from the impacts of all previous nearby landslides.

Logistic regression is considered a reference model in statistically based landslide susceptibility modelling (Reichenbach et al., 2018). Relations between presence and absence of landslides as a binary target variable are explained by a set of independent variables such as slope steepness and slope position in logistic regression. In this paper, DEM derivatives (Sect. 2 and Fig. 2) as well as the two landslide path-dependency variables computed using Ripley's space–time K function (see Sect. 3.1) were used as independent variables. Landslide presence or absence was the binary target variable.

When using a multi-temporal landslide inventory in landslide susceptibility modelling, the selection of time slices for the training and testing is crucial. In Rossi et al. (2010) and Samia et al. (2018), a sequential splitting sampling strategy was used in such a way that landslides in older time slices were used to train the model and landslides in newer time slices were used to test the model. However, such a sequential sampling strategy does not provide an equal range of landslide histories between training and testing datasets, and this could bias the role of time in path-dependent landslide susceptibility modelling. To avoid such a timing inequality, Samia et al. (2018) also introduced a non-sequential sampling strategy in which the span of timing segregation among time slices in the training and the testing datasets is comparable. In this study, we used this sampling strategy to split the multi-temporal landslide inventory into training and testing datasets. To achieve this, all landslides in the time slices of 1947, 1954, 1981, 1985, 1999, May 2004, and March and May 2010 were used for training, and all landslides in the time slices of 1965, 1977, 1991, 1997, December 2004 and 2005, and April 2013 and 2014 were used for testing (Fig. 1). It is important to note that the time slice in 1939 was used only for quantification of landslide history of the other time slices and not for training or testing. Thus, the first time slice in the training dataset is 1947 (Fig. 1).

The number of pixels with landslides was smaller than the number of pixels without landslides in both training and testing datasets. Therefore, we randomly selected 5000 pixels with landslides and 5000 pixels without landslides from both training and testing datasets in order to create equal datasets both for training and testing of the models. This random selection of pixels was repeated 10 times both in the training and testing datasets. Therefore, we trained the conventional, conventional plus path-dependent and purely path-dependent landslide susceptibility 10 times and later tested 10 times. After preparation of the 10 training datasets, logistic regression was applied to the 10 training datasets with an entry probability of 0.05 and removal probability of 0.06 for independent variables to diminish the chance of overfitting in the model. We only allowed inter-variable correlations of less than 0.8 to avoid multicollinearity. Conventional landslide susceptibility was modelled using DEM derivatives only once for the defined training dataset and was tested using the independent testing dataset. The conventional plus path-dependent landslide susceptibility model was constructed using DEM derivatives plus the two landslide path-dependency variables. The purely path-dependent landslide susceptibility was modelled only by using the two landslide path-dependency variables. All three models were constructed only once. Model performance was assessed using AUC and AIC values. The AUC values for testing were assessed using 10 training models and 10 independent testing datasets. The models with highest performance in terms of AUC values were used to map susceptibility to landslides. Finally, we compared landslide susceptibility maps resulting from conventional, conventional plus path-dependent and purely path-dependent susceptibility.

Ripley's space–time K function confirmed the existence of landslide path dependency at small spatial and small temporal distances from a previous landslide (Fig. 6). The STC measure (Eq. 6) is high in the space–time vicinity of an earlier landslide, and it then decreases rapidly. Apparently, landslide susceptibility is relatively high immediately after occurrence of an earlier, nearby landslide.

The exponential-decay function that was fitted to the empirical STC values is 7SmoothedSTC(td)=0.44⋅e(-t/16.7)⋅e(-d/58.8). This function shows that the STC measure decays exponentially over a characteristic timescale of 16.7 years and characteristic spatial scale of 58.8 m. The residual standard error of the exponential function is 0.01, in units of STC (–), which compares favourably with the actual values that range up to 0.44.

We compared performance of the conventional, conventional plus path-dependent and purely path-dependent landslide susceptibility models, using AUC (greater is better) and AIC (lower is better) values as measure of performance (Table 1 and Fig. 7). The best-performing landslide susceptibility model was the conventional plus path-dependent model, both when expressed as AUC values and as AIC values (Table 1). The purely path-dependent landslide susceptibility model, constructed with only the two landslide path-dependency variables, performed better than the conventional landslide susceptibility model with its 16 DEM-derived variables.

For conventional susceptibility models, six DEM derivatives were selected in all 10 models (Table 2). Adding two landslide path-dependency variables into DEM derivatives variables affected the inclusion and exclusion of DEM-derivative variables only slightly. For example, the variables of TPI and distance to river were selected four and seven times, respectively, in the conventional susceptibility models, whereas after adding the two landslide path-dependency variables, these variables were selected five and four times, respectively. The variable eastness, which was selected twice in the conventional susceptibility models, was never selected in the conventional plus path-dependent susceptibility models.

In all the training and the testing datasets, the contingency tables (Table 3) showed that conventional landslide susceptibility models differed substantially from the conventional plus path-dependent and path-dependent landslide susceptibility models. In particular, the percentage of false positives (the percentage of pixels without landslides predicted with landslides) for the conventional susceptibility models is higher than for the two other susceptibility models. However, there are also fewer true negatives (the percentage of pixels without landslides predicted without landslides) in the conventional than in the conventional plus path-dependent and path-dependent susceptibility models. The variation in the differences is larger in the training datasets than the testing datasets, suggesting that all fitted models are robust.

The landslide susceptibility maps derived from the three models illustrate different patterns of landslide susceptibility (Fig. 8). For the models that include path dependency, the presented maps give the average values of all simulated time slices. Differences between the maps correspond with the considerable differences in the performance of their landslide susceptibility models in terms of AUC and AIC values (Table 1). The path-dependent landslide susceptibility map is visually different from both other landslide susceptibility maps, with the pattern dominated by regions of high susceptibility around locations where landslides previously occurred.

The 16 conventional plus path-dependent landslide susceptibility maps are dynamic and change over time (Fig. 9). These changes reflect the exponential decay with increasing time, since previous nearby landslides (Fig. 6) and the sudden increase in susceptibility in areas close to recent landslides. The gradual decrease in susceptibility levels is clearest when comparing the 1981 and 2004 susceptibility maps, whereas the sudden increase is clearest when comparing the 2004 and 2014 maps. The 2014 susceptibility map has higher susceptibility levels because of the impact of recent landslides in the year 2013.

Similar dynamics are visible when comparing landslide susceptibility maps constructed with the purely path-dependent model for different years (Fig. 10). These maps show only the pure influence of earlier landslides on susceptibility to future landslides (Fig. 6). Again, the susceptibility of landslides decreases where distance from earlier landslides in space and time increases but jumps back up when more recent landslides become part of the landslide history. The pure influence of each individual landslide on the susceptibility to the future landslide is strong when a landslide is fresh, which is reflected in the high percentage of susceptibility levels of 0.6–0.8 and 0.8–1.0 in 1947 and 2014. As time passes since the previous landslide has occurred, the susceptibility decreases, with an exponential-decay response which is reflected in the low percentage of susceptibility levels of 0.6–0.8 and 0.8–1.0 in 1981 and 2004.

The quantification of landslide path dependency using Ripley's space–time K function (Ripley, 1976; Diggle et al., 1995) indicates, in our study area, an exponential-decay response in the STC values (Fig. 6). This means that there is a positive influence of earlier nearby landslides on susceptibility that decays exponentially in time and space with a characteristic timescale of about 17 years and a characteristic spatial scale of about 60 m. This is in accordance with our previously quantified landslide path dependency using a follow-up landslide fraction in which the decay period of landslide path dependency was found to be about 2 decades (Samia et al., 2017b). Landslide clustering manifests in the form of spatial association among landslides, where follow-up landslides occur immediately after and close to a previous landslide (Samia et al., 2017a). Samia et al. (2017b) discussed the possible mechanism in the formation of clusters of landslides in which the size of the initial landslide and changes in hydrology of slope destabilized by a landslide could facilitate the occurrence of follow-up landslides and hence clusters of landslides.

STC values and their exponential decay to some extent depend on the method that we have chosen to determine the centre point of landslides when converting polygons of landslides to points of landslides. Our approach was to take the geometric centre, but other options exist (Haines, 1994) and their impact should be explored. Also, in the computation of STC values with Ripley's space–time K function, distance between landslides was calculated using the Pythagorean theorem without distinguishing between distances in the x and y direction. Also we did not include differences in the elevation of centre points in our distance calculations. For future work, it could be interesting to define one dimension as the distance along the slope in the downslope direction and another dimension as the distance in the slope parallel direction, keeping these two spatial dimensions separate in addition to the temporal dimension.

Our results demonstrated that including the landslide path-dependency effect in a pixel-based landslide susceptibility model constructed by DEM derivatives improves model performance substantially. This is in line with high AUC and low AIC values for the conventional plus path-dependent landslide susceptibility model (Table 1 and Fig. 7). This confirms our main hypothesis that adding the effect of landslide path dependency boosts the performance of landslide susceptibility models and is in accordance with our previous expectations regarding the stronger effect of landslide path dependency in a pixel-based landslide susceptibility model than in a slope-unit-based landslide susceptibility model (Samia et al., 2018). Landslide path dependency is a local effect (apparently with characteristic spatial scale of about 60 m) in which an earlier landslide increases the likelihood of follow-up landslide occurrence. Such a local effect is obviously more visible at pixel resolution of 10 m rather than at slope-unit resolution (with a median size of 51 486 m2 in our study area).

Strikingly, the purely path-dependent landslide susceptibility model constructed with only the two landslide path-dependency variables performs better than the conventional landslide susceptibility model made by DEM-derivative variables (Table 1 and Fig. 7). This is potentially interesting, since it implies that the landslide inventory itself can be used to map susceptibility to landslides without using DEM derivatives which have been conventionally used in landslide susceptibility modelling (Varnes, 1984; Guzzetti et al., 2005). The performance of this path-dependency-only model thus highlights that proximity to previous landslides can adequately capture susceptible locations. It also suggests that the path-dependent models' success in our experiments may be partly due to the fact that they capture static spatial effects that have not been resolved with our explanatory factors. It is attractive to imagine follow-up work that attempts to disentangle this static spatial effect that is unrelated to landslide history from dynamic spatial effects that are related to landslide history. The key to such disentangling should be that the former does not decay over time, whereas the latter does. More advanced statistical approaches that simultaneously estimate purely spatial and spatio-temporal effects may be needed. More complex explanatory variables such as geology, soil and land use can also be used along with DEM derivatives to improve landslide susceptibility models and maps. However, these are not always available. In fact, considering the landslide path-dependency effect in such complete explanatory factors improves their performance as well. We confirmed this in an additional exploration where we constructed a conventional landslide susceptibility model used in this paper, with the same DEM derivatives but also with land use and geology as explanatory factors. The results demonstrated that adding our two landslide path-dependency variables to such an improved conventional landslide susceptibility increased its performance (from an AUC value of 0.771 to an AUC value of 0.801).

Another important aspect of considering the landslide path-dependency effect in landslide susceptibility modelling is providing dynamic landslide susceptibility maps. Landslide susceptibility maps are usually classified into five levels of probability of landslide occurrence, ranging from 0 to 1. In the conventional landslide susceptibility map (Fig. 8a), the five probability levels of susceptibility by definition remain constant over time, since the DEM derivatives in the model are constant (although DEM derivatives also change when a landslide occurs, but DEMs are not updated frequently enough to reflect this). The usage of conventional static landslide susceptibility maps and dynamic landslide susceptibility maps taking landslide path dependency depends on the goal and task of audience. In reality, static susceptibility maps created (either with a conventional susceptibility model or as the static portion of a conventional plus path-dependent model) can be used in sustainable planning, whereas dynamic susceptibility maps can be considered in short-term land use planning.

However, adding landslide path dependency in landslide susceptibility models provides dynamic landslide susceptibility maps (Figs. 9 and 10) in which the levels of susceptibility change over time, reflecting the exponential-decay response of landslide path dependency (Fig. 6). The changes are in the places where landslides have already occurred, mainly in probability levels of susceptibility, ranging from 0.6 to 1.0. This suggests that the part of the area located in the high-probability level of susceptibility could switch to the low-probability level of susceptibility (0 to 0.6) after a decade. This is exemplified between the 1947 and 1954 landslide susceptibility maps, in which about 9 km2 of the study area drops by more than 0.1 in its probability of landslide occurrence. After adding the two path-dependency variables in the conventional landslide susceptibility modelled with DEM derivatives, it turns out that the coefficients of all DEM-derivative variables become lower (e.g., the LS factor becomes less important).

In landslide-prone areas where landslides are documented and mapped in the form of polygon-based multi-temporal inventories, the landslide path dependency can be quantified based on geographical overlap among landslides and hence used in landslide susceptibility modelling (Samia et al., 2017b, 2018). However, polygon-based multi-temporal landslide inventories are rare to the best of our knowledge, and hence in many areas geographical overlap among landslides cannot be computed. In this paper, we proposed using Ripley's space–time K function to compute landslide path dependency where point-based multi-temporal landslide inventories are used. Using such inventories, our STC measure (Eq. 6) can be used to quantify path dependency among landslides.

It is attractive to think that the STC measure (Eq. 6) and its parameters (Eq. 7) can be directly exported to landslide-prone areas with substantial geological and topographical similarities. However, to gain confidence in this approach, multi-temporal landslide inventories from such places (e.g., Schlögel et al., 2011) need to be questioned to find out whether path dependency occurs, whether it occurs over a similar space and timescales, and whether it adds value to susceptibility modelling. This would also allow us to start exploring what determines the characteristic space and timescales.

We have already modified the definition of conventional landslide susceptibility modelling (Varnes, 1984; Guzzetti et al., 2005) using spatial temporal dynamics of landslide path dependency (Samia et al., 2017a, b) as follows: 8Landslidesusceptibilitys,t=fconditioningattributess,landslidepathdependencys,t. In this study, both conventional plus path-dependent and path-dependent landslide susceptibility models turned out to perform better than a conventional landslide susceptibility model (Table 1 and Fig. 7). In both models, availability of a space–time component – reflecting the exponential decay of landslide path dependency – indicates that landslide susceptibility is dynamic. This challenges the way in which landslide hazard is assessed, as landslide susceptibility is an important element of landslide hazard.

In landslide hazard assessment, landslide susceptibility as a proxy of “where landslides occur” is combined with the temporal probability of landslide triggers (mainly rainfall) to determine “when landslides occur” (Guzzetti et al., 2006a). In this context, dynamic landslide susceptibility (Eq. 8) needs to be considered in combination with the temporal information of landslide triggers in the assessment of landslide hazard. When substantial landslides happen during a rainfall event, susceptibility in and around such landslides can be raised for a few decades in which moderate rainfall events may already cause substantial landslides, which raises susceptibility levels again (Fig. 11). Such dynamics have been observed in a site near Seattle, Washington, where several new landslides occurred in a slope that had recently experienced landslide activity, whereas a nearby hillslope with the same characteristics but without recently landslide activity did not experience new landslides (Mirus et al., 2017). If no substantial triggering event happens over the characteristic timescale of roughly 17 years, the increased susceptibility will be substantially reduced, and a later rainfall event may have less influence on landslides; the probability of experiencing a follow-up landslide will have decreased.