Forecasting dam height and stability of dams formed by rock slope

Based on an inventory of 69 dams formed by rock slope failures in southwestern Norway and published landslide 9 dam inventories from other parts of the World we developed semi-empirical relationships linking the maximum dam height 10 (HD.max in m) to dam volume (VD in 106 m3) and other relevant parameters such as valley width (WV in m) or dam area (AD 11 in km2). Power-laws are obtained for HD.max = f(VD) and HD.max = f(VD, WV), while a linear relationship links HD.max to the 12 ratio VD/AD. For dams in southwestern Norway, the linear relationship HD.max = 1.75×VD/AD has least uncertainties and 13 provides best results when comparing predicted dam heights with a validation dataset composed of existing dams in northern 14 Norway and numerically modelled dams for possible rock slope failures. To assess the stability of future dams we use the 15 predicted dam heights in the dimensionless blockage index DBI and relating this index to the probability of dam failure 16 derived from our dataset and other published databases on landslide dams. This study underlines the potential of semi17 empirical relationships for assessing dam height and stability that needs to be included in preliminary hazard and risk 18 assessment for unstable rock slopes, because damming of a river is an important secondary effect of landslides due to 19 upstream flooding and possible outburst floods in case of dam failure. 20


Introduction 21
Landslides, and more particularly large rockslides and rock avalanches, have formed natural dams in many mountainous 22 regions (Hewitt, 1982;Costa and Schuster, 1988;Korup, 2002;Casagli et al., 2003;Evans et al., 2011;Hermanns et al., 23 2011a;Weidinger, 2011;Dufresne et al., 2018). Even large dams with several millions m³ in volume may be unstable and 24 breach (Hewitt, 1998;Dai et al., 2005;Plaza et al., 2011). Many historic events of landslide dam failures are reported to 25 have occurred within a few days to years after the landslide event, causing catastrophic outburst floods in the valley 26 downstream of the dam (Groeber, 1916;Hewitt, 1982;Costa and Schuster, 1988;Evans, 2006) and leading to major 27 destruction and loss of live (Evans et al., 2011). 28 The National landslide database of Norway (NVE, 2020) includes at least 181 historical landslides that caused damming of 29 rivers. Most of them were earth and debris slides (153) and only 22 events were rockslides or rock avalanches. Many of 30 those events created only minor damming of rivers without significant consequences. Yet, there were several major events 31 with significant consequences in terms of loss of life or long-lasting landscape changes: the worst natural disaster in 32 Norway's history occurred on 21 September 1345 when the Gaula River was dammed by a massive debris slide that created 33 a 14 km long lake. After only 2-3 days the dam breached leading to a huge outburst flood in the Gaula Valley burying 48 34 farms and killing at least 500 persons (Furseth, 2006). In 1823, a rock avalanche dammed the Frondøla River and formed 35 the Lintuvatnet Lake (NVE, 2020). The lake is still existing today, even though the dam partially breached leading to an 36 https://doi.org/10.5194/nhess-2020-135 Preprint. Discussion started: 11 May 2020 c Author(s) 2020. CC BY 4.0 License. simulations of the landslide propagation (Hungr, 2011). Examples of such numerical models are the DAN3D code 48 (McDougall and Hungr, 2004) or the RAMMS software suite (Christen et al., 2012). However, these models require 49 numerous input parameters and extensive calibration in order to obtain reliable results, which precludes their cost-efficient 50 use for characterization of a large number of sites, as is required in regional studies. 51 Here we establish semi-empirical relationships for the rapid assessment of the maximum dam height, comparable to those 52 developed for landslide run-out (e.g. Scheidegger, 1973;Corominas, 1996) or landslide-generated displacement waves 53 (Oppikofer et al., 2019). We use an inventory of dams formed by rock slope failures (RSF dams) in southwestern Norway 54 ( Fig. 2a) along with other published databases on landslide dams (Ermini and Casagli, 2003;Hermanns et al., 2011a;Tacconi 55 Stefanelli et al., 2015) to evaluate the dam height as a function of landslide volume, valley width and dam area. This approach 56 addresses the need for a fast assessment of possible dam formation and stability for potential future RSF, as a part of the 57 systematic hazard and risk analysis of unstable rock slopes in Norway (Hermanns et al., 2012;Oppikofer et al., 2016aOppikofer et al., , 58 2016b. 59 2 Methodology 60

Inventory and characteristics of landslide dams 61
Systematic mapping of RSF dams in southwestern Norway (approximately 120 000 km² in surface) was carried out by 62 Jakobsen (2015) using the online orthophoto map service "Norge i bilder" (Norwegian Mapping Authority, 2020b) and its 63 associated web map service (WMS) in a geographical information system (GIS) (Fig. 1b). This aerial photo analysis focused 64 on present-day lakes as an indicator for possible dams, with the aim of identifying lakes that were impounded by RSF. The 65 analysis investigated therefore the immediate downstream surroundings of lakes, looking for deposits, debris and scars of 66 RSF, but also debris from a possible downstream flooding due to a dam breach. It must be noted that dams without remaining 67 lake are therefore not included in present inventory. 68 The detected dams were mapped and registered in a geospatial database, and their geomorphologic characteristics determined 69 based on orthophotos and the national 10-m digital elevation model (DEM) (Norwegian Mapping Authority, 2020a). These 70 dam characteristics include: 71 the type of landslide that formed the dam, chiefly rock avalanches (massive RSF with several hundred thousand to 72 millions of cubic meter in volume and high mobility) and rockslides/rockfalls (RSF with several thousands to hundred 73 thousands of cubic meter in volume, but without high mobility) or other landslide types; 74 the morphologic dam classifications in plan view and in across-valley and along-valley profiles according to Hermanns 75 et al. (2011b) (Fig. 3); 76 https://doi.org/10.5194/nhess-2020-135 Preprint. A total of 69 landslide dams are mapped in southwestern Norway (Fig. 2a). Thirty-eight dams were formed by rock 118 avalanches, 29 by rockslides/rockfalls and 2 by debris-flows. We discarded those generated by debris-flows from further 119 analyses because the aim of these empirical relationships is to determine the maximum dam height of future RSF. 120 The frequency of rock avalanches in Norway was highest shortly after the last deglaciation, i.e. between 14 000 and 10 000 121 years BP depending on the location (e.g. Böhme et al., 2015;Hermanns et al., 2017). We therefore assume that also most of 122 the RSF dams in southwestern Norway were formed shortly after the retreat of the Scandinavian ice sheet. However, three 123 dams are most likely influenced by glaciers, notably by depositing on decaying glaciers or on dead-ice bodies in the valley. 124 For 10 other dams such a glacial influence is possible. We excluded these 13 dams from further analyses because their 125 dimensions may have been altered by glaciers and are thus not representative for the present-day situation. 126 According to the landform classification by Etzelmüller et al. (2007), most of the 54 remaining dams are in regions with 127 "extreme Alpine relief with over-deepened glacial valleys" or in "high paleic mountain regions with glacial incisions" (Fig.  128 2a). In Rogaland County in southern Norway several clusters of RSF dams are observed in the landform types "glacially 129 scoured low mountains and valleys" and "mountain plateaus" (Fig. 2c). These clusters are closely related to WSW-ENE-130 trending faults (Gabrielsen et al., 2002) forming escarpments that are prone to RSF. Twenty-one dams are intact with a 131 dammed lake and 10 other dams are filled by sediments except a small residual lake. On the side of unstable dams, 16 dams 132 are classified as eroded because no deposits of an outburst flood are visible, and 7 dams have failed and likely led to an 133 outburst flood as suggested by related deposits downriver. 134 The morphologic dam classification in plan view according to Hermanns et al. (2011b) reveals that most dams are formed 135 by a RSF completely crossing the valley (Type IIa, n=36) (Fig. 3a). Partial damming of the valley by a RSF occurred in 5 136 cases (Type IIc), and 5 dams have multiple lakes (Type IIIa). The across-valley profiles can be classified as symmetrical 137 deposits in a symmetrical valley in 24 cases (Type i), and as asymmetrical with thickest deposits in the distal part in 19 cases 138 (Type ii) (Fig. 3b). The classification of the along-valley profiles reveals 21 dams with low thickness and gentle slopes (Type 139 1) due to the absence of constraints in the valley morphology (Hermanns et al., 2011b), and 29 dams with high thickness 140 and steep slope (Type 2) in a confined valley setting (Fig. 3c). 141 Table 1 summarizes the dimensions of the RSF dams in the inventory. The dam length LD ranges from 45 to 1600 m with a 142 median length of 200 m, whereas the dam width WD tends to be larger by a factor of 1.7 (median of ratio WD/LD) and ranges 143 from 45 to 2800 m with a median width of 330 m. The dam area covers three orders of magnitude with values between 144 5000 m² to 2.7 km² with a median of 53 000 m². The maximum dam heights HD.max vary between 5 and 210 m, whereas the 145 mean dam heights HD.mean vary between 2 and 113 m. The median dam heights are 21 m and 12 m for HD.max and HD.mean, 146 respectively. The dam volume VD computed as the product of AD and HD.mean, spans five orders of magnitude (12 000 m³ to 147 135 × 10⁶ m³). The median dam volume is approximately 1.0 × 10⁶ m³. The cumulative distributions of these dam 148 dimensions can all be fitted by lognormal distributions with very high correlation coefficients (r² > 0.95 except for WD) 149 (Table 1). 150 https://doi.org/10.5194/nhess-2020-135 Preprint. Discussion started: 11 May 2020 c Author(s) 2020. CC BY 4.0 License.

Semi-empirical relationships 151
We created semi-empirical relationships for the 54 RSF dams in southwestern Norway that were not influenced by glaciers. 152 First, we linked the maximum dam height HD.max (in m) to the dam volume VD (in 10⁶ m³) ( (2) 155 The exponent of ⅓ is given by dimensional analysis, whereas the scale factor of 24.5 was fitted with a high correlation 156 coefficient r² of 0.73. The ratio ρ between the measured and predicted maximum dam heights ranges from 0.46 to 1.94, and 157 its cumulative frequency distribution can be fitted by a lognormal distribution. The 95 th percentile of this distribution 158 (ρ95 = 1.81) yields the upper bound of the 90% prediction interval of Eq. (2). This implies that approximately 5% of RSF 159 dams in southwestern Norway have a maximum height exceeding the predicted value by 81% or more. 160 Similar power-law functions can be derived from datasets from other studies (Ermini and Casagli, 2003;Hermanns et al., 161 2011a;Tacconi Stefanelli et al., 2015), with different scale factors, however ( Table 2). The scale factor of landslide dams in 162 the Andes (Hermanns et al., 2011a) is much lower than those from other studies (10.1 vs. 21.5 to 24.5). Compared to our 163 inventory, other databases have a larger spread of the data indicated by higher ρ95-values (Table 2). 164 Power-law functions are commonly used in landslide studies to relate the landslide volume to landslide frequency (e.g. 165 Dussauge et al., 2003;Guzzetti et al., 2003), but also other landslide characteristics, such as landslide area (e.g. Hovius, 166 1997). Similarly, the relationship between landslide volume and Fahrböschung, i.e. the ratio between the landslide fall height 167 and travel distance, can be fitted by power-law functions (e.g. Scheidegger, 1973;Nicoletti and Sorriso-Valvo, 1991;168 Erismann and Abele, 2001;De Blasio, 2011). Furthermore, Oppikofer et al. (2019) found a power-law function linking the 169 run-up height of landslide-generated displacement waves to the landslide volume and distance from impact. 170 Regarding the influence of the morphologic dam classification on the dam height (Table 2), dams classified as asymmetrical 171 with thickest deposits in the distal part (Type ii in across-valley profile) are higher than dams with symmetrical deposits in 172 a symmetrical valley (Type i), but smaller than those partially blocking a valley (Type iv). In along-valley profiles, Type 2 173 dams with high thickness and steep slope are higher than Type 1 dams with low thickness and gentle slopes. Too few data 174 are available for the other dam types in along-or across-valley profiles and in plan view. 175 In narrow valleys the RSF deposits are more confined leading to thicker deposits and thus to a higher dam compared to wide 176 valleys where the deposits are unconfined and spread out over a larger surface. We calculated therefore ratio VD (in 10⁶ m³) 177 over valley width WV (in m) and fitted following power-law with the exponent given by dimensional analysis (Fig. 6, Eq. 178 (3)): 179 Ratio ρ between the measured and predicted maximum dam heights ranges from 0.52 to 2.36. The 95 th percentile of the 181 lognormal distribution fitted to the cumulative frequency distribution of ρ equals 1.76 (ρ95). This value is slightly smaller 182 than for Eq.  Table 2). 185 Equation (3) has the expected behaviour with an increase in HD.max for higher volumes and a decrease for wider valleys. The 186 lateral spreading of the landslide deposits in the valley is, however, not accounted for. This could be achieved by including 187 the dam width WD as additional parameter in a semi-empirical relationship. However, WD is not independent from VD and is 188 not easily predictable when using the semi-empirical equations to forecast the dam height for future landslides, except if the 189 run-out area is known. In that case, the dam area AD (in km²) can be assessed, and the average dam height HD.mean (in m) can 190 be computed as the ratio VD/AD as an alternative proxy. For the RSF dams in southwestern Norway, HD.max (in m) increases 191 linearly with HD.mean (Fig. 7, Eq. (4)): The ratio ρ ranges from 0.57 to 1.72 with a value of ρ95 of 1.48 (lognormal distribution). This implies that approximately 5% 194 of landslide dams in southwestern Norway have a maximum height exceeding the predicted value by 48% or more. Both the 195 range of ρ and its 95 th percentile are significantly smaller than for the other semi-empirical relationships. Again, only the 196 database by Tacconi Stefanelli et al. (2015) contains AD for few dams. However, we calculated AD from the published dam 197 width WD and dam length LD assuming an elliptic shape of the landslide dam. Using those calculated dam areas in Eq. (4) 198 provides a scale factor of 1.35 (Fig. 7, Table 2). Lower r² and higher ρ95-values (0.65 and 1.84, respectively) indicate again 199 a larger spread of the data compared to the inventory of RSF dams in southwestern Norway. (1) to assess the likelihood of a dam failure pf. 216 6 Application to predict dam height and stability 217

Prediction of maximum dam height 218
We use the semi-empirical relationships (Eq. (2), (3) and (4)) to predict the maximum dam height generated by a future rock 219 slope failure damming a valley. We thereby use following assumptions and methods: 220 -The dam volume VD is equal to the slide volume VS times a bulking factor of 1.25 (25% volume increase due to fracturing 221 of the rock mass and porosity of the deposits) (Hungr and Evans, 2004). This implies that the entire volume reaches the 222 valley and forms the dam. This is obviously the worst-case scenario as shown by Ermini and Casagli (2003) with an 223 average ratio VD/VS of 40% for rainfall-triggered landslides and 57% for earthquake-triggered landslides. In Norway, 224 however, numerical run-out modelling for the six unstable rock slopes used for the validation of the semi-empirical 225 relationships (see Table 3 plus HD.max) to obtain a new approximation of AD, which in turn is used in Eq. (4) for a new estimation of HD.max; (c) this 234 procedure is repeated until the difference between successive estimations of HD.max is smaller than a threshold of 1 m. 235 The area of the impounded lake corresponds to the contour line of the estimated dam elevation (elevation of the valley floor 236 plus HD.max) (Fig. 9a). 237

Prediction of dam stability 238
The maximum dam height HD.max predicted by the semi-empirical relationships can then be used to assess the dam stability 239 using the DBI (Ermini and Casagli, 2003) (Eq. (5)). The catchment area AC upstream of the dam can be easily assessed with 240 a "flow accumulation" GIS-function provided that the DEM covers the entire upstream catchment area. The resulting  values are in turn used in Eq. (1) to assess the probability of failure pf. 242

Validation of semi-empirical relationships 243
To test the semi-empirical relationships for RSF dams in southwestern Norway, we analyzed four RSF dams in northern 244 Norway as validation dataset. Those dams are presently stable or infilled (Fig. 2b). In addition, the relationships were 245 validated by comparing predicted dam heights with results from detailed numerical run-out modelling for six unstable rock 246 slopes (see Böhme et al., 2016 for an example; NGU, 2020). 247 Table 3 shows the measured or modelled dam characteristics (VD, WV, AD, AC, HD.max) and the predicted maximum dam 248 heights HD.max using the semi-empirical relationships in Eq. (2), (3) and (4). This comparison shows that Eq. (4) provides the 249 best match with measured/modelled dam heights in 8/10 cases, whereof all six potential future rock slope failures. For Eq. 250 (4) the average relative error is ±13%, which is very small considering the relatively large uncertainties on the semi-empirical 251 relationship itself with a ρ95 of 1.48 (see above). For Eq. (2) and (3) the average relative errors are also acceptable when 252 considering only the four existing RSF dams in northern Norway (±29% and ±20%, respectively). Regarding the six future 253 RSF dams however, the average relative errors become inacceptable (±267% and ±202%, respectively). Possible reasons for 254 this huge discrepancy are discussed below. Based on this validation dataset we consider Eq. (4) as best possible semi-255 empirical relationship to predict the maximum dam height HD.max. 256 7 Discussion 257

Differences between landslide dam inventories 258
The inventory of landslide dams in southwestern Norway and other inventories used in this study (Ermini and Casagli, 2003;259 Hermanns et al., 2011a;Tacconi Stefanelli et al., 2015) contain significant differences, notably the landslide processes 260 considered, the geological settings and the volume estimations. The relationship between the maximum dam height and dam volume (Fig. 5) shows a wide spread in values, i.e. a RSF dam 269 with a volume of 1 × 10⁶ m³ can lead to a dam height ranging from 4 to 55 m. However, there is no significant difference 270 between our inventory and the datasets by Ermini andCasagli (2003) andTacconi Stefanelli et al. (2015), which is reflected 271 in the power-law distributions fitted to the different inventories ( Table 2). The Andean inventory (Hermanns et al., 2011a) 272 shows, however, significantly lower dam heights for a given volume compared to the other datasets (Fig. 5, Table 2). This 273 is related to the different geomorphic/tectonic settings of the Andean inventory with often tens of kilometer wide valleys, 274 compared to more Alpine settings used in our and other inventories. 275 Finally, the assessment of the dam volume is a crucial parameter for all semi-empirical relationships established in this study. The DBI-values for landslide dams in southwestern Norway cover a similar range than those from other inventories (Fig. 8). 294 It is however surprising to have several stable landslide dams with DBI-values significantly higher than the "unstable limits" 295 defined in other studies, i.e. 3.08 in Ermini and Casagli (2003)  of unstable dams in the bin with highest DBI-values is indeed significantly lower (4 unstable dams out of 10 dams) than in 298 the bin with second-highest DBI-values (9 out of 11) (Fig. 8b). Possible reasons for this difference with other inventories 299 are: 300 -In the creation of our inventory, we focused on existing lakes impounded by landslide deposits as identification criteria. 301 Landslide dams without remaining lake are thus not included, yet many of those dams were likely unstable. Extending 302 the inventory to all RSF dams might thus increase the overall proportion of unstable dams (23 out of 54), especially also 303 for higher  -Most dams in our inventory formed in prehistoric times and the stability assessment of these paleo dams is solely based 305 on geomorphologic observations. In other datasets (Ermini andCasagli, 2003, Tacconi Stefanelli et al., 2015) most 306 landslide events occurred in historic times and available historical records help distinguishing between intact, eroded and 307 breached dams. 308 -The RSF deposits impounding the lakes in southwestern Norway often have a large grain size (Fig. 1c, e, f). Grain size 309 analysis of RSF dams shows a median diameter of 0.6 to 0.9 m, and boulders of more than 2 m in diameter form up to Norway could explain the relatively higher stability compared to (possibly) finer grained deposits in other parts of the 313 World. Deposits with larger grain size are more resistant to erosion and favour drainage through the rock avalanche 314 deposits (Casagli et al., 2003;Dunning, 2006;Weidinger, 2011) (Fig. 1c). 315 Using the proportion of unstable dams in bins of DBI-values for the combined inventory (Ermini and Casagli, 2003, Tacconi 316 Stefanelli et al., 2015 and our dataset) yields a much broader range for the transition zone between the "stable domain" and 317 "unstable domain" than in previous studies (Fig. 8). This reanalysis of the joint dataset is robust as it considers possible 318 outliers and it is less dependent on single values. One could for example argue to set the upper limit DBIupper to the highest 319 DBI-value of all stable dams (4.37 instead of 5.0). This would imply that DBIupper is solely depending on a single landslide 320 dam, which is not appropriate given the complexity of the phenomena and the uncertainties in the inventories. We did not 321 include the dataset by Hermanns et al. (2011a) into the combined inventory due to the significantly higher proportion of 322 unstable dams for given DBI-classes (Fig. 8a, b). A possible reason is the relatively lower dam heights in the Andes compared 323 to other datasets (see discussion above), which leads to lower DBI-values. Other causes for this difference could be the grain 324 size of deposits, climatic conditions and the age of the Andean dams, which are up to 60 ka old (Hermanns et al., 2004, 325 2011a, Costa and González Díaz, 2007 It would be interesting to perform this stability assessment for different geological, geomorphological and climatic 327 environments, in order to obtain lower and upper DBI-limits for different conditions. This requires however more complete 328 inventories, as at least 100 or 150 landslide dams are required to obtain a sufficient number of bins (10 to 15 bins) containing 329 each a sufficient number of dams (≥10). 330

Prediction of dam height using semi-empirical relations or numerical modelling 331
Two of the proposed semi-empirical relationships rely only on the dam volume VD (Eq. (2)), or on the ratio VD over valley 332 width WV (Eq. (3)). These equations are thus a quick tool to assess the dam height, yet comparison with numerical modelling 333 shows that these relationships overestimate the maximum dam height (Table 3). The third proposed semi-empirical 334 relationship using the ratio VD over dam area AD (Eq. (4)) provides a better match with numerical modelling results, requires 335 however a simple run-out analysis to assess the run-out area and estimate AD (Fig. 9). A first assessment of the landslide run-336 out area can be achieved by calculating the landslide run-out length L as a function of the landslide fall height H and the 337 volume-dependent angle of reach α (e.g. Scheidegger, 1973;Nicoletti and Sorriso-Valvo, 1991;Erismann and Abele, 2001;338 De Blasio, 2011) (Fig. 4b). The angle of reach α is also used in more advanced computer programs, such as CONEFALL 339 (Jaboyedoff andLabiouse, 2011) or Flow-R (Horton et al., 2013), which require little to no calibration and can thus be 340 quickly applied to assess the run-out area. Yet, these tools do not provide the thickness of deposits and thus the dam height. 341 The third semi-empirical relationship HD.max = f(VD/AD) (Eq. (4)) yields the maximum dam height based on the landslide run-342 out area and dam area. 343 Using detailed numerical simulations of the landslide propagation and run-out, such the DAN3D code (McDougall and 344 Hungr, 2004) or the RAMMS software suite (Christen et al., 2012), directly provide the thickness of landslide deposits and 345 allows to find the lowest elevation of the post-slide topography up to which a lake can form (see Oppikofer et al., 2016a, 346 Fig. 9). However, these simulations require many input parameters and extensive calibration in order to obtain reliable 347 results. These requirements impede their cost-efficient use in regional studies, where a large number of potential landslide 348 dams need to be assessed. 349 The proposed semi-empirical relationships are a conservative method because they assess the maximum dam height and thus 350 not the lowest elevation where dam overtopping may occur. Numerical simulations on the other hand provide the dam height 351 and elevation where overtopping would occur. This difference partly explains the discrepancy between numerically modelled and empirically predicted dam heights (Table 3). Another possible reason for this discrepancy is the difference between 353 observed and modelled run-out areas. The effective run-out area of a landslide can be significantly smaller than numerically 354 simulated ones: the latter generally cover the entire area potentially affected by a landslide, while the real run-out area of a 355 landslide event may only cover parts of the total area. As the landslide volume in reality may spread over a smaller area than 356 simulated, the average and maximum dam heights obtained by numerical simulations or by Eq. (4) may be too small. Yet, 357 the possible overestimation of AD is counterbalanced by conservative estimate of VD being the entire landslide volume VS 358 times a bulking factor of 1.25. More back-analyses of landslide-generated dams are required to ascertain these possible 359 differences between modelled and real run-out areas. In turn, this could lead to an improved workflow for assessing the dam 360 height and reducing uncertainties. 361 These considerations highlight the necessity to assess uncertainties on dam height and stability by using various approaches, 362 including different semi-empirical relationships, but also numerical simulations for critical areas. To assess uncertainties, 363 we calculate for example the DBI and pf using HD.max for the potential RSF dams of the validation dataset (Table 3). 364 Compared to the results from numerical simulations, the DBI increases in average by 0.64 and 0.56 for Eq. (2) and (3),  365 respectively. This leads in turn to an average increase of pf of +16% and +14%, respectively. This comparison highlights 366 that despite large uncertainties, the influence on dam stability and thus on the consequences assessment is relatively 367 moderate. 368

Conclusions & perspectives 369
The semi-empirical relations presented here provide a rapid approach for predicting the maximum dam height of dams that 370 might result from the future failure of an unstable rock slope. All relations require only limited input parameters, chiefly the 371 slide volume, the valley width and the dam area based on simple run-out assessments. These semi-empirical relationships 372 are established from an inventory of 54 RSF dams in southwestern Norway with dam volumes ranging from 12 000 m³ to 373 135 × 10⁶ m³. Only dams generated by catastrophic rockslides or rock avalanches and without any glacial influence were 374 included in the analyses. Consequently, the semi-empirical relations presented here may be less or not applicable for other 375 landslide types (e.g. debris-flows, shallow landslides) and other volume classes. The upper bounds of the 90% prediction 376 intervals of these semi-empirical relationships range from 1.48 to 1.81, meaning that approximately 5% of the actual 377 maximum dam heights exceed the predicted value by 48% to 81% or more. 378 Validation of the semi-empirical relationships was performed using four RSF dams in northern Norway, but also results 379 from detailed numerical run-out simulations for six unstable rock slopes. The maximum dam heights predicted by the semi-380 empirical relations are generally in good agreement with the measured/modelled dam heights from the validation dataset. 381 Best validation results are obtained for the relationship linking maximum dam height to landslide volume and dam area with 382 only a modest overestimation of the maximum dam heights (average relative error of 18%). This semi-empirical relationship 383 provides thus an appropriate tool for the first-order assessment of dams generated by rock slope failures at a local to regional 384 scale. Using limited input parameters, this relationship allows the prediction of the maximum dam height and thus the 385 upstream inundation area, but also to quickly forecast the dam stability using the dimensionless blockage index. 386 Possible improvements of these semi-empirical relationships are the inclusion of additional datasets, notably existing 387 landslide dams from other regions in Norway. Similar datasets could be collected for other mountainous regions in the 388 World, possibly leading to semi-empirical relationships with different parameters than those presented here for dams from 389 rock slope failures in southwestern Norway. Another possible major improvement consists in the addition of those dams that 390 do not possess a lake or residual lake at present. This requires however very time-intensive screening over large regions to 391 detect the landslide deposits that might have blocked a river in the past. Furthermore, the presented semi-empirical 392 relationships are only valid for rockslides and rock avalanches. Similar semi-empirical relationships can be imagined for 393 https://doi.org/10.5194/nhess-2020-135 Preprint. Discussion started: 11 May 2020 c Author(s) 2020. CC BY 4.0 License. other landslide types, but more complete datasets on those landslide dams are required first. We strongly suggest using the 394 predictive tools developed here to assess landslide dam formation and stability, which should be an integral part of risk 395 assessment for future landslide events. 396          (64) 28 (44) 8 (11) 3.37 (3.54) 57% (62%)