We propose an integrated statistical procedure (containing stochastic elements) to compute the spatial probability of a given area (technically, a given GIS pixel) to be affected by a landslide either through its release or through its propagation. We first consider release and propagation separately and finally combine the two concepts. The entire work flow is illustrated in Fig. 1, its components are introduced in detail in Sects. 2.2–2.6.

Two newly developed raster modules of the open source software package GRASS GIS 7 (Neteler and Mitasova, 2007; GRASS Development Team, 2015) are combined:

r.landslides.statistics allows inventory subsetting, estimation of the spatial probability of landslide release, and the generation of a zonal probability function.

An issue of central importance consists in the strict separation of the data used for model development and the data used for model application and evaluation. In this sense, most operations are performed either for the model development area (MDA) or for the model evaluation area (MEA), but not for both. The only exception from this rule applies to the initial step of inventory subsetting.

All probabilities used in the context of the present work are summarized in Table 1.

Landslide inventories often suffer from a missing – or unsatisfactory – subsetting into release, transit and deposition areas. The reason for this problem, which applies also to our case study, is not necessarily related to deficient mapping efforts, but rather to the impossibility to identify each zone in the field or from remotely sensed data. Appropriate subsetting, however, is required before using the inventory for statistical analyses of landslide release or propagation. We therefore suggest a reproducible procedure to deal with this problem.

We analyze the geometric properties of all landslides in a given inventory in terms of inclination, minimum and maximum elevation, elevation range, central, maximum and average 2-D and 3-D length and width, 2-D and 3-D areas. Lengths and widths are defined as Euclidean distances (the central 2-D and 3-D lengths L2-D and L3-D as well as the elevation range H are shown in Fig. 2). On this basis we compute the height ratio rH for each observed landslide pixel: rH=Hp-Hmin⁡Hmax⁡-Hmin⁡, where Hp is the elevation at the considered pixel, Hmin⁡ is the minimum elevation of the landslide and Hmax⁡ is the maximum elevation of the landslide (see Fig. 2).

In the present work, we consider all observed landslide pixels with rH≥rR as release pixels and all observed landslide pixels with rH≤rD as deposition pixels. rR and rD are defined by the user. All other observed landslide pixels are considered as unknowns regarding release and deposition. Following these rules, we obtain three landslide inventory maps:

observed release areas (ORA), where all release pixels are considered observed positives (OP), the rest of the landslide areas are considered no data, and all non-landslide pixels are considered observed negatives (ON);

Statistical analyses of landslide spatial release probability (landslide susceptibility) have been treated exhaustively in previous studies (see Sect. 1 for references). In the context of the present work we are bound to a method yielding spatial probabilities in the range 0–1. In this sense, we employ a simple approach building on the spatial overlay of classified predictor maps. Considering separately each of the resulting combinations of predictor classes, we compute the fraction fR of observed landslide release pixels related to all pixels. For this step we consider only the MDA. Building on the assumption that possible future landslides in the MEA are spatially related to the predictors in the same way as the observed landslides in the MDA, the release probability PR (see Table 1) for each pixel in the MEA is set to the value of fR associated to the corresponding combination of predictor classes.

The true positive (TP), true negative (TN), false positive (FP) and false negative (FN) pixel counts are derived for selected levels of PR. An ROC Curve is produced by plotting the true positive rate TP/OP against the false positive rate FP/ON.

It is useful for many purposes to work with pixel-based spatial release probabilities (PR). They can be averaged in order to characterize the spatial probability of landslides for any type of zone (such as slope units, catchment basins, administrative entities or larger pixels). However, the average of PR over a certain zone does not tell us how likely it is that a landslide occurs in a zone at all. For this purpose we introduce the zonal release probability PRZ (see Table 1) which increases with study area size. When considering one single pixel, PRZ=PR. For large areas (mountainous catchments or entire countries) PRZ=1 as there will always be at least one landslide pixel. PRZ may be useful for assessing how likely it is that a certain object (such as a road) is affected by a landslide at all. It is further the appropriate parameter when validating landslide probability at the level of slope units or other entities larger than single pixels. In the present work it is needed primarily as a basis to compute the integrated spatial landslide probability PL (see Sect. 2.6). It is further used to aggregate the model results at the level of slope units.

PRZ cannot be computed in a fully analytic way. We suggest an empirical approach to approximate PRZ (Fig. 3):

a subset of the MDA with a randomized size and randomized centre coordinates is selected. PRO is the observed pixel-based spatial probability of landslide release in this subset (i.e. the fraction of ORA pixels out of all pixels);

This procedure results in a line cloud of PRZO plotted against Z (one line for each subset; Fig. 3b). A logistic regression is fitted to the average value of PRZO, μPRZO, for each tested value of Z: μPRZO(Z)=(1-μPRO)1+e-(a2+a3Z)+μPRO, where a2 and a3 are the regression coefficients and μPRO is the fraction of the observed landslide area within the considered zone. We will come back to the function introduced in Eq. (2) in Sect. 2.6.

The tool r.randomwalk (Mergili et al., 2015) is employed for routing mass points representing hypothetic landslides through the DEM. The specific impact probability PIR describes the probability of an arbitrary impact pixel to be hit by a mass point routed from a defined release pixel through the DEM. The impact probability PI∗ or PI results from the spatial overlay of all relevant values of PIR at a certain pixel (see Table 1). We define PIR on the basis of the angle of the path ω between the release pixel and a possible impact pixel. This approach follows the concept of the angle of reach (Heim, 1932; Fig. 4). PI is computed in three steps:

The CDF describing the probability that a moving mass point starting from an arbitrary release pixel leaves the OIA of the same landslide at or below a certain threshold of ω is derived for the MDA. This is done by back-calculating the observed angles of reach ωOT for all observed landslides (see Fig. 4a) and analyzing the resulting probability density (see Fig. 4b).

The integrated spatial landslide probability PL approximates the spatial probability that a landslide coincides spatially with an arbitrary pixel of the MEA, either through its release or through its impact (see Table 1). In principle, PL is computed by multiplying a release probability and an impact probability. Obviously, a simple overlay of PR and PI would be useless. Instead, we have to consider for each impact pixel with PI>0 the zonal release probability PRZ of the possible release zone (Fig. 5) relevant for this pixel. Z and the associated value of μPR (see Sect. 2.4) refer to the entire set of release pixels which may propagate all the way to the impact pixel. I.e. PRZ has to be computed separately for each impact pixel.

For this purpose, we come back to the function introduced in Eq. (2). Thereby we assume that the shape of the logistic regression function is insensitive to the zonal average of the computed values of PR, μPR, of any arbitrary subset of the study area with zone size Z (see Fig. 3c): 1-PRZ(Z)1-μPRZO(Z)∼1-μPR1-μPRO. Reformulating Eq. (3), PRZ(Z) is computed as PRZ(Z)∼1-(1-μPRZO(Z))1-μPR1-μPRO. For those pixels where PRZ⋅PI<PR, PL is set to PR. For all other pixels, PL is set to the product of PRZ and PI: PL=max⁡(PR,PRZ⋅PI). The resulting raster map of PL is evaluated against the OIA by means of an ROC Plot (see Sect. 2.3).

The expected error of PRZ is explored by comparing the empirical values of PRZO obtained for each subset and each zone size with the results of Eq. (2) (see Fig. 3d). It is expressed as a third-order polynomial regression function of the standard deviation of PRZ: σPRZ=b1+b2log⁡10Z+b3(log⁡10Z)2+b4(log⁡10Z)3, where σPRZ is the standard deviation of PRZ and b1–b4 are the regression coefficients. The standard deviation of PL, σPL, is derived as σPL=σPRZ⋅PI. Equation (7) only applies to those pixels where PRZ⋅PI≥PR.

We note that the described procedure is supposed to yield smoothed results due to averaging effects: (i) Eq. (5) builds on the simplification of a uniformly distributed release probability over the possible release zone. (ii) As highlighted in Sect. 2.5, PI represents the average of PIR of all mass points impacting a pixel. This type of averaging is necessary to ensure a consistent combination of PRZ and PI.

In the period from 7 to 9 August 2009, Typhoon Morakot triggered a high number of landslides in Taiwan. According to Lin et al. (2011), more than 22 000 landslides were recorded in Southern Taiwan. One of the hot spots was the Kao Ping Watershed (Wu et al., 2011), where extremely heavy rainfall (more than 2000 mm in a period of 90 h) caused an enormous amount of mass wasting and triggered a catastrophic landslide in the Hsiaolin Village (Kuo et al., 2013).

We consider a 637 km2 subset of the Kao Ping Watershed for computing the integrated spatial landslide probability PL (Fig. 6). 1399 landslides triggered by the Typhoon Morakot are mapped in the shale, sandstone and colluvium slopes of the area. A stereo-photogrammetrically generated 10 m DEM is used along with a landslide inventory derived from FORMOSAT-2 scenes recorded before and after the event. The landslide inventory delineates the OIA without differentiating between ORA and ODA, and without providing direct information on landslide volumes. Overlapping landslide polygons are aggregated to one polygon for the purpose of the statistical analyses.

The model tests are summarized in Table 2. The Kao Ping study area is divided into four subsets (A–D in Fig. 6) to separate between MDA and MEA. In each of the tests, three subsets are used as MDA and one subset is used as MEA. The division lines between the subsets follow catchment boundaries in order to ensure that all landslides are clearly assigned to one of the four subsets and no landslide may impact more than one subset. All tests are run at a pixel size of 20 m.

We use values of rR=0.75 and rD=0.25 (see Sect. 2.2). Preliminary tests have shown that the following two parameters are suitable as predictors for computing PR: (i) local slope (five classes); and (ii) aspect (2 classes). For reasons of the regional geology, NE–E–SE–S–SW exposed slopes are more affected by landslides than W–NW–N exposed slopes. Both predictors are derived from a modified version of the DEM: noise reduction is applied to the DEM through a low pass filter building on the mean of all values within in a radius of 50 m.

For back-calculating ωOT and for evaluating PI∗ we start a set of 103 random walks from each pixel in the ORA of the MDA and the MEA, respectively. For computing PL we start a set of 102 random walks from each pixel in the MEA. We use Gaussian distributions to generate the CDFs. The input parameters governing the routing procedure in r.randomwalk are chosen in accordance with the suggestions provided by Mergili et al. (2015).

Preliminary tests have further indicated that the largest, deep-seated landslides in the test area are poorly predicted by the statistical model applied. We hypothesize that landsides of this type are governed by other factors than those which can be derived directly from the DEM or other surface data. The analyses are therefore repeated excluding all landslides with a total size of the OIA ≥1 km2. All pixels within the OIA of those landslides are set to no data (Tests 2A–D in Table 2).

We further run the model with a spatially constant value of PR (identical to the observed density of ORA in the MDA) in order to quantify the component of PL (and of the model performance) associated to the zone size used for computing PRZ (see Sect. 2.6; Tests 3A–D in Table 2).

The model results are evaluated against the observed landslides at two spatial levels using ROC Plots:

The pixel level. PR is evaluated against the ORA, PI∗ is evaluated against the ODA, and PL is evaluated against the OIA.

Figure 7 illustrates the result maps for test 1C. For reasons of clarity, we show only a subset of the test area (see Fig. 6). However, the general patterns of the results are well represented in this area and are also valid for the other tests. Figure 7a shows the result of the inventory subsetting, the spatial variation of PR is displayed in Fig. 7b. Whilst the patterns of PI∗ related to the observed landslide release pixels (see Fig. 7c) clearly reflect the decreasing probabilities in downslope direction, the values of PI related to all possible release pixels (see Fig. 7c) are high where large contiguous steep slopes are present i.e. where the average slope angles are high. The probability density function and the CDF of ωOT computed for the relevant MDA (including the zones A, B and D; see Fig. 6) are shown in Fig. 8a. According to the Figs. 4 and 8a, PI∗=1 for those areas where ω≥ the maximum of ωOT. For PI, this is only true where PIR=1 for all mass points possibly impacting the considered pixel as PI represents the average of all relevant values of PIR (see Fig. 5)

The largest values of Z are displayed in those areas with large catchments i.e. in the valleys (see Fig. 7e). Whilst the maxima exceed 10 km2 in zone C, the median of Z for all pixels in zone C is 0.043 km2. The zonal release probability (see Fig. 7f) strongly reflects the patterns of Z, clearly dominating over the influence of PR (see Figs. 3 and 7b). This phenomenon is explained by the limited spatial variation of PR (see Fig. 7b) and the resulting dominance of the zone size reflected in PRZ. Figure 9a illustrates the dependency of the observed zonal release probability PRZO from the zone size (see Fig. 3).

Note that high values of PRZ are not associated to those areas with high release probabilities, but to the source areas of the random walks determining PI of the corresponding pixel (see Fig. 5). However, PI is usually low in those areas with very high values of PRZ as they are located in the valleys at some distance from the steep slopes. Therefore, the integrated spatial landslide probability PL reaches its maxima on the lower slopes and in narrow gorges, where both PRZ and PI are relatively high (see Fig. 7g). The standard deviation shown in Fig. 7h is derived from the standard deviation function of Fig. 9b (see Eqs. 6 and 7). σPRZ remains at a moderate level and is highest in those areas where also PRZ is high.

Figure 10 shows the distribution of PL for the entire test area. The maps for the tests 1A–1D – each of them covering the corresponding MEA – are combined into one map.

Considering all observed landslides (tests 1A–D), 7.5 % of the entire test area are classified as OIA (i.e. the observed integrated spatial landslide probability). The average value of PL=9.3 %, meaning that we arrive at a reasonable estimate of the integrated spatial landslide probability, even though we overestimate PL. The same is true for the landslide release areas, where 1.4 % of the test area are classified as ORA, with a similar average value of PR. Whilst the excellent correspondence of observed and modelled release probabilities is forced by the type of statistical approach employed, the still reasonable correspondence with regard to PR indicates a certain validity of the suggested workflow. The key parameters characterizing the outcomes of each test (see Table 2) are summarized in Table 3. Observed and computed percentages are lower for the tests 2A–D as some landslide areas are removed from the analysis.

The ROC Plots for model evaluation are compiled in Fig. 11. PR is evaluated against the ORA, PI is evaluated against the ODA and PL is evaluated against the OIA. Only the MEA is taken into account. Considering the tests 1A–D, the predictors slope and aspect only explain part of the spatial variation of PR, indicated by moderate levels of AUCROC (0.569–0.661). The prediction level of test 1D even indicates model failure (see Fig. 11a). In contrast, the spatial variation of the observed deposition areas is comparatively well predicted by the modelled values of PI∗ (0.724≤AUCROC≤0.913; see Fig. 11b). This observation is not surprising as the possible path of movement is usually reasonably well constrained, and most mass points necessarily touch the observed impact areas whilst those pixels on slopes without observed landslides yield a large amount of “cheap” TN pixels (see Fig. 7c). Whilst PI derived by the routing of all possible release pixels (see Fig. 7d) is of theoretical nature and would be less useful to evaluate, PL again displays a moderate prediction level (0.605≤AUCROC≤0.685; see Fig. 11b) which is, however, better than PR. Considering the ROC Plots for PR and PI∗ indicates that the false predictions are a consequence of the uncertain release probability rather than of deficiencies in the routing procedure.

Removing the largest landslides (OIA ≥1 km2) from the data (Tests 2A–D) does not significantly change the general prediction quality with regard to PR (see Fig. 11d). However, in the test 2D AUCROC increases from 0.569 to a (still very low) value of 0.598, indicating that the large Hsiaolin Landslide located in zone D (see Fig. 6) is very poorly explained by the predictors used. The influence of removing large landslides (all of which are located in the zones C and D) on the model performance in terms of PI∗ is more obvious than in the case of PR (see Fig. 11e). The tests 2C and 2D display a significantly enhanced performance, compared to the tests 1C and 1D (0.784 to 0.863 and 0.724 to 0.900, respectively). This phenomenon is again a consequence of the particular settings associated to the large landslides (especially the Hsiaolin Landslide, the deposition area of which is very poorly predicted) yielding a large number of false negative pixels in the observed deposit. Coming back to Fig. 8b, the tests 1A–D yield lower peaks of the probability density and a shift of the curves towards lower values of ωOT, compared to the tests 2A–D (see Table 3). Those lower values of ωOT are associated to the large landslides excluded in the tests 2A–D. Consequently, ωT is underestimated – and therefore, the impact area is overestimated – for the majority of the observed landslides in the tests 1A–D. However, the shift in the model performance is related to the poor prediction of the large deposit of the Hsiaolin Landslide rather than to the changes in the CDF.

In accordance with the patterns observed with regard to PI, AUCROC increases for the tests 2C and D, compared to 1C and D (see Fig. 11f). In contrast, AUCROC for PL derived with the tests 2A and B decreases slightly, compared to the values obtained with the results for 1A and B. Figure 11g illustrates the ROC Curves yielded for PL, assuming a constant spatial pattern of PR (i.e. the fraction of observed landslide pixels in the MEA for each test 3A–D). The values of AUCROC are almost similar to those yielded with the tests 1A–D (see Fig. 11c). This observation indicates that the spatial differentiation of PR is almost completely covered by the patterns of PI and Z (see Fig. 7).

The ROC Plots shown in the Fig. 11h–l relate the modelled distribution of PR and PL to the distribution of OP and ON slope units of the entire test area (in each case, the combination of the results of the tests A–D). All slope units with at least one OP pixel are considered OP, the ROC Curves are weighted for the slope unit size. The AUCROC values derived for the average values of PR for each slope unit evaluated against the aggregated ORA are significantly higher than the AUCROC values derived at the pixel level (0.695 for the tests 1A–D and 0.723 for the tests 2A–D; see Fig. 11h and j). AUCROC further increases to 0.787 and 0.766, respectively, when the zonal values of PR for the slope units are considered. This would be the correct way. However, these zonal probabilities are extremely strongly correlated to the size of the associated slope unit (this phenomenon is already indicated by Fig. 9), so that validating the zone size against the ORA results in ROC Curves almost identical with those derived for the zonal probabilities. This means that, despite the high values of AUCROC, the zonal values of PR for the slope units have no predictive power in terms of differentiating between areas of varying environmental or topographic conditions. The high prediction quality just relies on the fact that larger slope units are more likely to contain OP pixels (see Sect. 2.4). This phenomenon was already indirectly shown by the comparison of the Fig. 11c and g.

Slope units are not the suitable level to spatially aggregate PL (see Fig. 11i, k and l). The average of PL for each slope unit evaluated against the aggregated OIA indicates random predictions for all the sets of tests (AUCROC=0.494–0.502). As for PR, the strong correlation between slope unit size and zonal values of PL results in high AUCROC values (0.771–0.779) in all tests. This implies limitations analogous to those described for PR.