The Acheron rock avalanche in Canterbury, New Zealand (Fig. 4), was triggered approx. 1100 years BP (Smith et al., 2006). Within the present study, the release volume, V=6.4 million m3, is approximated from the reconstruction of the pre-failure topography and is lower than the value of V=7.5 million m3 estimated by Smith et al. (2006). We use a 10 m resolution DEM derived by stereo-matching of aerial photographs. Impact, release and deposition areas are derived from field and imagery interpretation as well as from data published by Smith et al. (2006). All random walks start from the highest pixel of the release area.

We use this case study for demonstrating how to compute the impact indicator index III from an elevation map, the release area, and the release volume. Before doing so, we have to analyse the influence of the pixel size and the parameters nwalks, Rmax⁡, Lctrl, Lseg, fβ, and fd on the model result. Preliminary tests have shown that r.randomwalk yields plausible results with the number of random walks: nwalks=104, Rmax⁡= 10 m, Lctrl= 1000 m, Lseg= 100 m, fβ= 5, fd=2, and a pixel size of 20 m. These values are taken as a basis to explore the sensitivity of the model results to the variation of each parameter and the best fit of the parameters in terms of the travel distance, AUCROC, and the size of the predicted impact area (Table 4). ωT=11.62∘ , the angle of reach observed for the Acheron rock avalanche, is applied as the break criterion for all tests. Some of the tests are run in the back-calculation mode (flag b; see Tables 2 and 4).

III is computed by executing r.randomwalk 100 times, with the parameter values optimized according to Table 4. We explore an empirical–statistical relationship for ωT derived from a compilation of 127 case studies (Fig. 5). The offset of the equation (b in Eq. 4 and Fig. 5) is randomly sampled between the lower and upper envelopes of the regression. The quality of the prediction is evaluated using the ROC plot (see Figs. 1 and 3). Note that the Acheron rock avalanche (not included in the relationship developed in Fig. 5) is found close to the lower envelope, meaning that it was very mobile compared to most of the other events.

Figure 6 summarizes the findings of the test s 1–3 (see Table 4). Test 1 leads to the expected result that the predicted impact area increases with the number of random walks. However, the predicted impact area is also a function of the pixel size: with larger pixels, less random walks are needed to cover an area of similar size than with smaller pixels. Figure 6a further indicates that the possible impact area is not fully covered even at 105 random walks: no substantial flattening of the curves is observed. We conclude that (i) a very high value of nwalks would be necessary to fully cover the possible impact area, and (ii) this would lead to a substantial overestimation of the observed impact area.

On the other hand, the quality of the prediction in terms of AUCROC reaches a maximum at nwalks≈102 (pixel size 40 m) or nwalks≈103 (pixel size 20 m), decreasing with higher values of nwalks. At a pixel size of 10 m, AUCROC reaches a constant level at nwalks≈104 (see Fig. 6b). We may conclude that excessive numbers of random walks lead to an overestimation of the impact area rather than to a better prediction quality. Coarser pixel sizes allow one to achieve the same level of coverage and the same prediction quality at lower values of nwalks. However, the pixel size has to be fine enough to account for the main geometric characteristics of the process under investigation (see Sect. 4). All further tests are performed with nwalks=104.

Figure 6c illustrates that, at Lctrl=1000 m, the travel distance computed within the observed impact area decreases with increasing values of Lseg (tests 2 and 3 in Table 4). This pattern is well explained by Fig. 2b. At short segment lengths, the effects of flow paths frequently changing their direction are particularly evident for pixel sizes of 10 m and 20 m. Lmax⁡ drops below the observed value of 3550 m (see Fig. 4b) at 75 ≤Lseg≤ 100 m. With Lseg≥3050 m, corresponding to the Euclidean distance between the release point and the terminal point of the Acheron rock avalanche, Lmax⁡ would also take a value of 3050 m. At Lctrl=50 m (only shown for a pixel size of 20 m), r.randomwalk tends to predict too long travel distances, compared to the observation. This phenomenon occurs as flow directions are not well defined in the relatively plane deposition zone of the Acheron rock avalanche; therefore, flow paths may frequently change their direction or even go backwards or in a circular way if such a behaviour is not impeded by sufficiently high values of Lctrl (see Fig. 2a). Figure 6d indicates that this undesired behaviour (visible in the area marked by the X in the gray circle) disappears at Lctrl > 200 m.

On the other hand, the value of Lctrl should not be chosen too high as this may negatively impact the model performance. In the case of the Acheron rock avalanche, a drop in AUCROC is observed between Lctrl≈2000 and Lctrl≈2500 m (Fig. 7a). This drop is explained by an increasing number of false negative pixels in those areas, which cannot be reached by the random walks due to the strict constraint of flow direction.

Within the tested ranges of parameter values, the quality of the prediction is highest at values of Rmax⁡≈5–10 m (see Fig. 7b) and fβ≥5 (see Fig. 7c), whilst it reaches it maximum at fd≈2–3 (see Fig. 7d). The predicted impact area increases with increasing Rmax⁡ and fd whilst it decreases with increasing fβ.

Figures 6 and 7 indicate that the initial values of nwalks, Lctrl, LsegRmax⁡, fβ, fd, and the pixel size suggested in Sect. 3.1.1 and Table 4 are within the optimum range of values (see Sect. 4). Therefore, they are used for computing the impact indicator index for the Acheron rock avalanche (Fig. 8a). Concerning the break criteria, this can be classified as a forward analysis. As expected from Fig. 5, where the Acheron rock avalanche falls in between the envelopes of the relationship employed, the upper part of the observed impact area displays a value of III = 1, whilst the remaining part of the observed impact area displays values of 1 > III > 0, decreasing towards the terminus. As the event was comparatively mobile within the context of the relationship used (see Sect. 3.1.1 and Fig. 5), the values of III are close to zero in the terminal area, and the area with III > 0 does not reach far beyond the observed terminus. Note that the maximum value of III is 0.8, meaning that 20 % of all model runs did not even start due to very high values of ωT yielded with the randomized values of b (see Fig. 5). Evaluation against the observed deposit yields a value of AUCROC=0.94 (see Fig. 8b). All values of AUCROC shown in Figs. 6 and 7 and the ROC plot of Fig. 8b build on normalized ON areas (see Sect. 2.5).

III was generated within a computational time of 188 s.

Between 7 and 9 August 2009, Typhoon Morakot struck Taiwan and triggered enormous landslides, causing significant land cover change (Fig. 9). More than 22 000 landslides were recorded in southern Taiwan (Lin et al., 2011). One of the hot spots of mass wasting was the Kao Ping Watershed (Wu et al., 2011), where the extremely heavy rainfall (in total, more than 2000 mm depth and 90 h duration) triggered a catastrophic landslide in the Hsiaolin Village (Kuo et al., 2013).

We consider a 61.5 km2 subset of the Kao Ping Watershed for computing the landslide impact probability PI, based on the observed landslide release areas. In all, 207 landslides are mapped in the shale, sandstone, and colluvium slopes (see Fig. 9). A 10 m DEM is used along with an inventory of the landslide impact areas. Release and deposition areas are extracted from the inventory. We employ the values of nwalks, Rmax⁡, fβ, fd, Lctrl, Lseg resulting from the optimization procedure for the Acheron rock avalanche (see Sect. 3.1.1), and a pixel size of 20 m. PI is computed as follows:

A set of random walks (nwalks=104) is started from each release point (i.e., the highest pixel of each landslide). Each random walk stops as soon as it would leave the impact area of the same landslide (back calculation, flag b).

We refer to this work flow as test 1 and repeat the analysis with starting random walks not only from the release points but also from all the pixels within the observed release areas (test 2). This means that the CDF is derived from a much larger sample of data than when considering only one point per landslide for starting random walks. We exclude all sets of random walks yielding Lmax⁡ < 100 m, use a log-normal CDF and start a set of only 103 random walks from each release pixel for computing PI.

Starting sets of 104 random walks from the highest points of all landslides (test 1) results in a range of values of 16.0 ≤ωT≤43.5∘, an average of 30.4∘, and a standard deviation of 5.2∘ (derived from n=132 landslides, excluding those with Lmax⁡ < 100 m). Repeating the analysis with 104 random walks started from each pixel within the landslide release areas (test 2), we observe a range of values 16.4 ≤ωT≤44.1∘, an average of 26.9∘, and a standard deviation of 4.8∘ (n=1563). Figure 10 illustrates the histograms, probability density, and cumulative density functions derived from both analyses. Even though the ranges of values are similar in both tests, test 1 yields (i) a higher average of ωT and (ii) a broader range of values than test 2; (i) is explained by the fact that those random walks starting from lower parts of the release areas are expected to leave the observed impact area at lower values of ωT; (ii) is most likely the consequence of a number of rather small landslides with high or low values of ωT strongly reflected in the statistics. Such outliers are less prominent in the statistics of test 2 due to the much higher number of cases, most of them related to the larger landslides.

Each of the impact probabilities shown in Fig. 11 represents the overlay of 100 analyses where random sets of 80 % of the landslides are used for deriving the CDF and the remaining 20 % are used for computing the impact probabilities. The maps illustrate the maximum values of PI out of the overlay of the 100 results. Each of the results is derived using a slightly different CDF. Both tests yield largely similar patterns of PI. We note that (i) test 2 predicts larger impact areas and higher values of PI than test 1, and (ii) some random walks take the wrong direction in test 2 (indicated by “1” in the yellow circle in Fig. 11b), a phenomenon not observed for test 1; (i) is explained by the higher number – and the broader distribution – of release pixels in test 2, compared to test 1. The reason for (ii) is that random walks starting from the highest point of an observed landslide are forced to flow into the observed landslide area (test 1), a constraint not applicable when starting random walks from each release pixel (test 2). In this case it happens that pixels located at or near a crest produce random walks in both directions. In test 1, the computational time amounted to 63 s for deriving the CDF and 8613 s for calculating PI. In test 2, these times increased to 1719 and 9752 s, respectively. The relatively slight increase with regard to PI results from the reduced value of nwalks in test 2.

The prediction quality is tested for each of the 100 model runs for the two tests, producing sets of 100 ROC curves (Fig. 12). AUCROC=0.917±0.038 for test 1 and 0.920 ± 0.029 for test 2, both computed with the original number of TN pixels (see Sect. 2.5).

In contrast, the procedures demonstrated in the two tests vary strongly in their scope of applicability. We have demonstrated the methodologies by back calculating observed landslides. As soon as this is done, one may go one step further:

The methodology shown in test 1 can be employed to make forward predictions for defined expected future landslides, given that a sufficient set of observed landslides of similar behaviour is available to derive the CDF.

In either case the statistics (see Fig. 10) have to be derived with the same type of approach later used for producing the PI map.

As most mountain areas worldwide, the Pamir of Tajikistan experiences a significant retreat of the glaciers. One of the consequences thereof consists in the formation and growth of lakes, some of which are subject to glacial lake outburst floods (GLOFs), which may evolve into destructive debris flows (Mergili and Schneider, 2011; Mergili et al., 2013; Gruber and Mergili, 2013). No records of historic GLOFs in the test area are known to the authors. However, in August 2002 a GLOF in the nearby Shakhdara Valley evolved into a debris flow, which destroyed the village of Dasht, claiming dozens of lives (Mergili et al., 2011).

The frequency of such events is low and historical data are sparse. Consequently, possible travel distances of GLOFs may not be derived in a purely statistical way. Instead, we have to use published empirical–statistical relationships and simple rules to produce an impact indicator score (IIS) map.

We compute IIS with regard to GLOFs for a 2106 km2 study area in the Gunt Valley (Fig. 13). The analysis builds on the ASTER GDEM (Advanced Spaceborne Thermal Emission and Reflection Radiometer – Global Digital Elevation Model) V2 and the coordinates and characteristics (estimates of V and Qp) of 113 lakes in the area (Gruber and Mergili, 2013).

A set of random walks (nwalks=104) is routed from the outlet of each lake through the DEM. Six break criteria are combined to compute IIS, partly following Gruber and Mergili (2013). The relationships and rules employed as break criteria are summarized in Table 5. Rule 1 is applied with ωT=11∘ (test 1 – according to Haeberli, 1983 and Huggel et al., 2003, 2004a, b for debris flows from glacier- or moraine-dammed lakes, and Zimmermann et al., 1997 for coarse- and medium-grained debris flows) and with ωT=7∘ (test 2 – Zimmermann et al., 1997 for fine-grained debris flows). All other rules and relationships are used for both tests. For each pixel, IIS consists in the number of relationships or rules predicting an impact (i.e., IIS takes values in the range 0–6).

Rmax⁡, Lctrl, Lseg, fβ, and fd are set to the optimum values found for the Acheron rock avalanche, the pixel size is set to 60 m.

Figure 14 illustrates the possible impact areas of GLOFs in the Gunt Valley study area according to the relationships listed in Table 5.

Figure 14a shows the impact indicator score IIS i.e., the number of relationships predicting an impact, resulting from test 1 (rule 1 applied with ωT=11∘). Except for one prominent exception, IIS > 3 (possible debris flow impact) only for the largely uninhabited upper portions of the tributaries to the Gunt Valley. In contrast, a possible flood impact (1 ≤ IIS ≤3) is predicted for much of the main valley. test 2 (rule 1 applied with ωT=7∘) predicts a possible debris flow impact also for part of the main valleys (see Fig. 14b). The IF (per cent of random walks impacting each pixel) for test 1 is shown in Fig. 14c for a subsection of the test area, classified by quantiles. IF is strongly governed by the width of the movement, i.e. by the local topography, and may serve as a surrogate for the expected depth rather than as for the probability of an impact.

Note that Fig. 14 only indicates the tendency of an already released GLOF to impact certain pixels. It does not provide any information on the susceptibility of a certain lake to produce a GLOF at all. Earlier, Mergili and Schneider (2011) and Gruber and Mergili (2013) have attempted to combine GLOF release indicators with impact indicators and land cover maps to generate hazard and risk indicator maps. However, the results of their studies may underestimate the possible impact areas as the travel distance was computed on a pixel-to-pixel basis, possibly yielding too low values of ωT (see Figs. 2 and 6).

The robustness and appropriateness of the rules and relationships for low-frequency events, such as GLOFs (see Table 5), is questionable. The rules building on a unique value of ωT overpredict the possible impact areas for those lakes where not enough water is available to produce a flood in downstream valleys. Applying the rules and relationships for debris flows implies a blind assumption that enough entrainable sediment is available to produce a debris flow. Whilst ωT≥11∘ is considered the worst case for debris flows of GLOFs from glacier- or moraine-dammed lakes in the European Alps according to Haeberli (1983) and Huggel et al. (2002), ωT=9.3∘ was measured for the 2002 Dasht Event, the only well-documented GLOF near the test area (Mergili et al., 2011). Also the relationship proposed by Rickenmann (1999) severely underestimates the travel distance of this event, even when massive bulking is assumed. Applying ωT=7∘ as given by Zimmermann et al. (1997) for fine-grained debris flows might be more suitable as worst-case assumption for debris flows from GLOFs in the Pamir, even though this threshold leads to very conservative predictions.

We have measured computational times of 1520 s for test 1 and 1556 s for test 2.