## Abstract

Displacement waves (or tsunamis) generated by sub-aerial landslides cause damage along shorelines over long distances, making run-up assessment a crucial component of landslide risk analysis. Although site-specific modelling provides important insight into the behaviour of potential waves, more general approaches using limited input parameters are necessary for preliminary assessments. We use a catalogue of landslide-generated displacement waves to develop semi-empirical relationships linking displacement wave run-up (*R* in metres) to distance from landslide impact (*x* in kilometres) and to landslide volume (*V* in millions of cubic metres). For individual events, run-up decreases with distance according to power laws. Consideration of ten events demonstrates that run-up increases with landslide volume, also according to a power law. Combining these relationships gives the SPLASH equation: *R* = *a**V ^{b}*

*x*, with best-fitted parameters

^{c}*a*= 18.093,

*b*= 0.57110 and

*c*= −0.74189. The 95% prediction interval between the calculated and measured run-up values is 2.58, meaning that 5% of the measured run-up heights exceed the predicted value by a factor of 2.58 or more. This relatively large error is explained by local amplifications of wave height and run-up. Comparisons with other displacement wave models show that the SPLASH equation is a valuable tool for the first-stage preliminary hazard and risk assessment for unstable rock slopes above water bodies.

The impact of sub-aerial landslides into lakes, fjords, the sea, or large rivers can generate displacement waves (Hermanns *et al.* 2013*a*), also called landslide-generated tsunamis (e.g. Bornhold & Thomson 2012). Such waves can be several tens of metres high and cause severe consequences along the shoreline over distances of several tens of kilometres, and they occur in many regions in the world (Fig. 1) (Roberts *et al.* 2014). Near-field wave run-up can be extremely large resulting from the temporary displacement of a large volume of water out of the reservoir (e.g. the 1905 Loen I, 1953 Vaiont and 1958 Lituya Bay events), as well as complex wave behaviour due to the influence of short-wavelength components. Long-wavelength motion predominates in far-field settings, resulting in lower but potentially very damaging run-up and wave behaviour that is more similar to seismically generated tsunamis. Due to this dual behaviour we prefer the term landslide-generated ‘displacement wave’ instead of ‘tsunami’, as discussed in Hermanns *et al.* (2013*a*).

In Norway alone, over 200 sub-aerial landslide events are reported to have generated displacement waves in lakes and fjords (Hermanns *et al.* 2014). These displacement waves were caused by rockfalls (92 out of 212 events), snow avalanches (59), quick clay slides (25), rock avalanches (17), debris flows (14) and unknown landslide types (5). Even though rock avalanches generated only 8% of the historic events in Norway, they account for 283 out of 339 fatalities related to landslide-generated displacement waves (Hermanns *et al.* 2014). Three disastrous events in the twentieth century caused 175 fatalities alone (i.e. rock avalanches at Loen Lake in 1905 and 1936, and in Tafjord in 1934; Fig. 1c, Table 1). The largest historic rock avalanche in Norway occurred in 1756 at Tjellefonna and generated up to 50 m high displacement waves that killed 32 people (Fig. 1c) (Sandøy *et al.* 2017).

These historic data underline the necessity to assess the run-up of landslide-generated displacement waves when analysing the risk of landslides near water bodies. This is particularly important for unstable rock slopes that might lead to massive rock slope failures and generate waves with heights of tens of metres (Table 1). The Geological Survey of Norway systematically maps, investigates and analyses fractured bedrock slopes that show signs of post-glacial deformation and that might fail catastrophically in the future (Hermanns *et al.* 2013*b*). Over 110 unstable rock slopes that will impact fjords or lakes in case of a catastrophic failure have so far been discovered in Norway (NGU 2017), setting the requirement for cost-effective displacement wave assessments for preliminary risk analyses.

Different approaches exist to assess displacement waves generated by sub-aerial landslides: generally applicable equations from laboratory model tests (e.g. Fritz *et al.* 2009; Heller *et al.* 2009); numerical simulations of displacement waves (e.g. Pastor *et al.* 2009; Harbitz *et al.* 2014; Franz *et al.* 2016; Wang *et al.* 2015; Si *et al.* 2017); analytical solutions (Noda 1970); or empirical relationships (Slingerland & Voight 1979; Ataie-Ashtiani & Malek-Mohammadi 2006). Bornhold & Thomson (2012) provide an overview of these and other models for landslide-generated displacement waves. Due to complex wave behaviour and interaction close to the location of landslide, reliably representing near-field run-up is a major challenge for each of these. Most of the approaches require many input parameters related to wave generation, propagation and shoreline run-up, and therefore require detailed site-specific studies in order to provide reliable results. For instance, the Laboratory of Hydraulics, Hydrology and Glaciology, ETH Zurich (VAW) model based on laboratory model tests (Heller *et al.* 2009) requires landslide volume, density, porosity, width, thickness and fall height as well as slope angle, velocity and still-water depth at the impact location. Numerical simulations such as the GloBouss model (e.g. Harbitz *et al.* 2014) similarly require detailed input parameters characterizing landslide metrics and dynamics. Noda's (1970) analytical solution for predicting maximum-displacement wave amplitude is unique in its requirement for just two parameters: impacted water depth and landslide velocity. However, landslide velocity is often unknown. Similarly, the empirical relationships by Slingerland & Voight (1979) require data on a wide range of landslide and water-body parameters. The complexity of displacement-wave generation, propagation and run-up justifies their investigation using elaborated models with many input parameters. However, the data and computational requirements of these models preclude their cost-efficient use for characterization of a large number of sites, as is required in regional studies.

Here we establish a semi-empirical relationship enabling rapid assessment of run-up from landslide-generated displacement waves, analogous to those developed for landslide run-out (e.g. Scheidegger 1973; Corominas 1996). We use a global catalogue of landslide-generated displacement waves (Roberts *et al.* 2014) to estimate displacement wave run-up as a function of the landslide volume and distance from landslide impact. This approach addresses the need for a fast-assessment tool of possible displacement wave run-up areas for potential future rock slope failures, as a part of the systematic hazard and risk analysis of unstable rock slopes in Norway (Hermanns *et al.* 2012; Oppikofer *et al.* 2016*a*, *b*).

## Global event catalogue

We previously compiled a global catalogue of landslide-generated displacement waves based on published case studies and existing databases on displacement waves (Roberts *et al.* 2014). The catalogue contains 254 events, of which 153 occurred in Norway (excluding the 59 snow avalanches that generated displacement waves) (Hermanns *et al.* 2014). The catalogue contains general information on the landslide event (e.g. location, date, volume, fall height, event description and information sources), as well as details of the consequent wave (e.g. open-water wave height, location and height of run-up measurements).

Some landslide and displacement wave parameters are unavailable for many of the registered events, and cannot be retrieved based on the scarce available historic accounts. For instance, in many cases it is known only that a landslide occurred and that a displacement wave was generated, but no data about volume and run-up are reported. Because the focus of this study is large rock slope failures, we did not consider soil failures or volcano flank collapses in the following analyses. Twenty-eight rock slope failures from the preliminary global database include adequate details on landslide volume as well as the location and height of wave run-up (Table 1). Only ten rock slope failures have run-up registrations for more than two locations.

## Methodology

### Landslide and displacement wave data

Details of events identified from Roberts *et al.*’s (2014) catalogue of landslide-generated waves are from a range of primary sources: scientific literature and technical reports; Norwegian church records (Furseth 2006); the Norwegian national landslide inventory database (Jaedicke *et al.* 2009; NVE 2017); and tsunami catalogues, particularly the National Centers for Environmental Information (NCEI 2017) online global database of historic tsunamis. Landslide volumes and run-up heights are taken directly from the most detailed publications for each event (Table 1). Distances from the landslide impact location to each run-up measurement are measured in GoogleEarth using georeferenced maps from those publications. The expected paths of wave propagation are based on geomorphic recorders of wave impact direction in the run-up zone, suggesting nearshore wave motion (e.g. Roberts *et al.* 2013 for Chehalis Lake), and on wave fronts, suggested by numerical modelling studies (e.g. Mazzanti & Bozzano 2011 for Scilla; Huang *et al.* 2012 for Wu Gorge), where these details are available.

### Creation of semi-empirical relationships

We use Microsoft Excel and MatLab to establish semi-empirical relationships through analysis, plotting and fitting of selected parameters for the individual landslide events and for combinations of events. First, we plot run-up *R* against distance from landslide impact *x* for each of the ten events with more than two documented run-up values to find event-specific relationships for decreasing run-up with distance from the impact location. We then compare the parameters from each of the site-specific semi-empirical relationships to assess the influence of landslide volume *V* on those parameters. Finally, we combine these findings into a semi-empirical relationship for *R* as a function of *V* and *x*, called the SPLASH equation (semi-empirical prediction of landslide-generated displacement wave run-up heights).

To assess the uncertainties of the SPLASH equation, we compute the ratio *ρ* between the measured and predicted run-up. We then fit the cumulative frequency distribution of this ratio using a log-normal function to determine the 95th percentile *ρ*_{95}. The ratio *ρ*_{95} corresponds to the 95% prediction interval and means that 5% of the measured run-up values exceed the predicted value by a factor of *ρ*_{95}. We have also tried to include other relevant parameters in the semi-empirical relationships, such as water depth at slide impact as well as water depth and slope angle at the run-up location (Rem 2016) because of their important role in both initial wave generation and subsequent shoreline interactions (Heller *et al.* 2009). Those tests are not presented here; results were inconclusive since the uncertainties were not reduced by including these additional parameters. Other parameters, such as landslide impact velocity, width, thickness, frontal shape and density, influence wave generation (Sælevik *et al.* 2009; Harbitz *et al.* 2014; Si *et al.* 2017). However, they are not available for most landslide events in the catalogue and therefore could not be included in the semi-empirical relationship. In forecasting displacement waves from future sub-aerial landslides, such parameters can only be generally constrained through detailed modelling studies; they would therefore not be available during the preliminary regional studies that SPLASH is intended to support.

### Correction factors for wave propagation and run-up

Most of the landslides-generated displacement wave events used to establish the SPLASH equation occurred in water bodies with relatively simple plan-view geometries. In contrast, fjords (particularly in western Norway) commonly comprise multiple branches with complex geometries. Numerical simulations of landslide-generated displacement waves in these settings (e.g. Harbitz & Glimsdal 2011; Glimsdal 2013; Harbitz *et al.* 2014) show a reduction in displacement wave height by 30% or more for a 90° change in wave propagation direction. We account for this effect in the empirical assessment by multiplying the initially computed run-up by a factor *f*_{D} for each major direction change by an angle α. Factor *f*_{D} is calculated using Equation (1), based in part on empirical relations derived from model tests by Heller *et al.* (2009, fig. A-4, equation 3-13):

Decreases in basin depth and width, such as at the heads of lakes and fjords, greatly increase the displacement wave height and therefore the run-up. We use a correction factor *f*_{R} to account for the increase in run-up in shallow waters, bays and end of fjords or lakes, but also for direct impacts on valley sides directly opposite the landslide impact location. For each of these different settings, we computed the 95% prediction interval based on the log-normal distribution function of the ratio between measured and predicted run-up. We then rounded these 95% prediction intervals up to the nearest 0.5 to obtain *f*_{R} for different settings: *f*_{R} = 1.5 for shallow waters; *f*_{R} = 2 for bays; *f*_{R} = 3 for heads of water bodies; and *f*_{R} = 1.5 for direct impacts (values rounded). Numerical simulations of landslide-generated displacement waves (e.g. Harbitz & Glimsdal 2011; Glimsdal 2013; Harbitz *et al.* 2014) yield similar correction factors. Other effects that influence wave height, particularly wave interference and resonance, are not accounted for in the semi-empirical relationship. They are to a certain level included however, because such effects may have occurred in the events used in this study.

### Displacement wave inundation

Details on the extent of displacement wave inundation (i.e. wave run-up area) are necessary to improve assessment of displacement wave hazard and risk. We produce inundation maps for potential future events by interpolating between empirical run-up estimates for selected points along the shoreline. The density of these points depends on the complexity of the water body. For each event they include, at least, all settlements and locations of sensitive infrastructure as well as intermediate tie points, for example where the water body changes direction. The area affected by the displacement wave is then obtained by interpolating the run-up between these points and intersecting the run-up with a digital elevation model (10 m cell size in this study).

## Results

### Run-up as a function of distance for single historic events

All ten of the studied landslide events exhibit a general trend of decreasing run-up with increasing distance from landslide impact. Wave amplification in shoals, bays and the heads of fjords and lakes leads to local-scale variability in run-up and therefore deviations from the general trend (Fig. 2). The decrease of run-up *R* (in metres) with distance from landslide impact *x* (in kilometres) is approximated by the power law (Equation (2)):

Given the complexity of the phenomenon, we consider fits with *r*^{2} values of 0.45 or higher, as obtained for eight out of ten landslide events (Table 2), to be relatively good. For the 1783 Scilla landslide in Italy (*r*^{2} = 0.39) and the 2008 Aratozawa landslide in Japan (*r*^{2} = 0.12), available run-up measurements do not show the expected decrease with distance. These two events are therefore discarded from further analyses.

Factor *a*_{S} is equivalent to the run-up at *x* = 1 km. Landslide events that generated several tens to hundreds of metres high run-up also yielded high values of *a*_{S}. Calculated *a*_{S} values for the eight events with good fits range from 4.812 (Wu Gorge) to 211.054 (Lituya Bay).

Exponent *c* in the power law describes the rate of decrease of *R* as a function of *x*. This exponent is fairly uniform between the eight landslide events with good fits. It varies over the range −0.63037 (Isla Mentirosa) to –0.95123 (Lituya Bay) with an average of −0.74385 ± 0.10186 (1*σ*). Heller *et al.* (2009) use an exponent of −2/3 for the decrease in wave height in the initial phase of wave propagation, which is within the range and only slightly less than the mean calculated here. For the eight landslide events that showed good power-law fits, the 95% prediction interval, that is, the 95th percentile of the cumulative frequency distribution of the ratio between measured and predicted run-up *ρ*_{95}, ranges from 1.65 (Wu Gorge) to 3.26 (Lituya Bay) with an average of 2.46 (Table 2).

### Run-up as a function of landslide volume and distance from impact

#### Comparison of parameters for multiple events

To assess the influence of landslide volume (*V* in Mm^{3}) on the factor *a*_{S} and exponent *c* in the power law in Equation (2), we plot these power-law parameters against *V* (Fig. 3a). While exponent *c* shows no clear dependency on landslide volume, the factor *a*_{S} generally increases with *V*. The relationship between *a*_{S} and *V* is again best-fitted by a power-law (Equation (3), Fig. 3a):
*a*_{S} values (19.391 and 20.169, respectively) with respect to their volumes (8 and 14 Mm^{3}, respectively). This is tentatively explained by their failure types and coeval nature. Isla Mentirosa likely failed as three or more separate rock slides or small rock avalanches, not as a single 8 Mm^{3} failure (Sepúlveda & Serey 2009). Punta Cola began as a rock avalanche that remobilized some of the sediments in the valley to form a debris avalanche (Oppikofer *et al.* 2012). Its volume and velocity were likely greatly reduced as a result of debris deposition over its >1.5 km travel path to Aysén Fjord. Furthermore, since these landslides were coeval – both having been triggered by the 21 April 2007 Aysén earthquake (*M*_{W} 6.2) – run-up at many locations within the fjord cannot be confidently attributed to one landslide or the other, and was likely influenced by wave interference. Such factors appear to have reduced the expected run-up in both cases, resulting in smaller *a*_{S} values and justifying their removal from the power law in Equation (3).

#### Normalized run-up as a function of distance from impact

In order to verify the independency of exponent *c* from landslide volume, we normalized the measured run-up *R* (in m) by dividing by the volume-dependent factor *a*_{S} for the eight landslide events that showed good power-law fits (using Equation (2)). Jointly assessing all values of normalized run-up *R*_{n} for these eight events shows a decrease with *x*, which again follows a power law (Fig. 3b, Equation (4)):

The exponent in this power-law is very close to the average of exponents determined in Equation (2), suggesting that exponent *c* is independent of the landslide volume. The factor in the power law is very close to 1, as it should be for a normalized dataset.

#### Combined empirical relationship for multiple events

Combining the findings of Equations (3) and (4) leads to a power law that links run-up *R* to both distance from impact *x* and landslide volume *V* (Equation (5)):
*a*, *b* and *c* in Equation (5), and then perform a least-squares fitting of these parameters to the measured run-up values. For this least-squares fitting we considered three different sub-groups of the landslide events listed in Table 2: (1) the eight events exhibiting good fits in Equation (2) (i.e. excluding the 1783 Scilla and 2008 Aratozawa landslides); (2) the six events exhibiting good fits in Equation (2) and without high uncertainties in landslide volume and failure complexity (i.e. also excluding the 2007 Isla Mentirosa and Punta Cola landslides); and (3) subsampling of the number of run-up measurements of the 1936 Loen III, 1958 Lituya Bay and 2007 Chehalis Lake events to 26 measurements each (i.e. the number of measurements for the 1934 Tafjord event), to avoid overrepresentation of single events in the total dataset.

Sub-group (3) yields the best results based both on least-squares error between the measured and predicted run-up and on expected behaviour of the function, that is, a decrease in run-up with distance from impact and an increase in run-up with landslide volume. Equation (6) provides the best-fitted parameters using sub-group (3):

This forms our semi-empirical SPLASH equation with *R* in metres, *x* in kilometres and *V* in millions of cubic metres. The exponent *c* = −0.74189 closely matches the average of values for this parameter determined for individual events (Equation (2), Table 2), as well as the value for volume-normalized run-up (Equation (4)). Exponent *b* = 0.57110 is significantly lower than values for this parameter determined in Equation (3), but is counterbalanced by a higher factor *a* = 18.093 (*cf*. 14.700 in Equation (3)). Consequently, run-up values predicted using Equation (6) are higher for small volumes (less than *c.* 3 Mm^{3}) than if the original values from Equation (3) were used. We explain this discrepancy by the 1958 Lituya Bay event. Its maximum run-up of 524.3 m – by far the highest of any historic landslide-generated wave – leads to a very high *a*_{S} value in Equation (2), which thereafter greatly influences exponent *b* in Equation (3). When combining multiple landslide events, the influence of extreme run-up from the Lituya Bay event is reduced. Consequently, run-up heights of this magnitude are not characterized using this empirical relationship (Equation (6)). In contrast, most of the measured run-up values generated at Lituya Bay in 1958 are well predicted by Equation (6) (Fig. 2f). The extreme run-up heights could also not be matched when fitting the single event (Equation (2)).

The uncertainties of the SPLASH model are again expressed as the 95% prediction interval computed from the ratio between the measured and predicted run-up. For the empirical relationship in Equation (6), the 95% prediction interval *ρ*_{95} equals 2.58. This means that 5% of the measured run-up values exceed the predicted value by a factor of 2.58 or more.

## Discussion

### Comparison with other models

To test the validity of our SPLASH equation, we compare predicted run-up with two other models: (1) numerical simulations made by the Norwegian Geotechnical Institute (NGI) (Harbitz & Glimsdal 2011; Glimsdal 2013); and (2) a model using generally applicable equations from laboratory model tests at the VAW (Heller *et al.* 2009). We apply the VAW model for this study for the unstable rock slope of Opstadhornet, Norway, based on principles outlined in Oppikofer *et al.* (2016*a*, *b*).

Opstadhornet is a 11.2 Mm^{3} rockslide on the southern side of the island Otrøya (Fig. 1) moving at approximately 2 mm a^{−1}. Its hazard is qualitatively assessed to medium (Hermanns *et al.* 2016) and the annual likelihood of failure is estimated as <1/5000 (Blikra *et al.* 2016). Catastrophic failure of the Opstadhornet rockslide would form a rock avalanche, which would generate a large displacement wave in Romsdalsfjord, affecting settlements along the shoreline including the city of Molde with >20 000 inhabitants (17.8 km away) and a planned hospital facility at Hjelset (36.1 km away). As part of the hospital planning process, displacement wave run-up was simulated by Glimsdal (2013) for three rockslide scenarios of 6, 10 and 20 Mm^{3}. Maximum predicted run-up heights are 0.8 and 1.8 m for volumes of 10 and 20 Mm^{3}, respectively. The run-up heights are doubled according to Glimsdal (2013) to account for ‘special wave effects such as diffraction, reflection and interference, which are not included in the simulations’. Additionally, 0.7 m is added to the run-up to compensate an assumed future sea-level rise (Glimsdal 2013, p. 18). A run-up of 2.3 m at Hjelset is therefore predicted by the NGI simulations for a 10 Mm^{3} rockslide from Opstadhornet. By comparison, the SPLASH equation (Equation (6)) predicts a run-up of 5.0 m (*V* = 11.2 Mm^{3}, *x* = 36.1 km) at Hjelset. Including the correction factor *f*_{D} for a 90° change in propagation direction (see ‘Methodology’ section) yields a corrected run-up of 3.5 m. Using the same volume as in the numerical simulations (*V* = 10 Mm^{3}), the run-up predicted by the SPLASH equation would be 4.7 m before and 3.3 m after correction.

The VAW model yields a much higher run-up estimate of 7.5 m at Hjelset. Several factors could explain this ≥50% increase over the other prediction tools: (1) several of the constraints in the VAW model are not satisfied, notably the still water depth at impact is too small compared to the width and mass of the rock avalanche and the run-up angle is smaller than values used in the model tests; (2) the exponent for the decrease in wave height with distance is reduced from −2/3 in the initial propagation step to −4/15 in the following propagation steps, leading to comparatively greater run-up at large distances from the landslide; and (3) exclusion of the role of wave breaking in shallow waters, as might occur in the narrow fjord section close to Molde. For the majority (58 of 72) assessed run-up locations, however, values predicted by the VAW model are lower than those predicted by the SPLASH equation (including correction factors *f*_{D} and *f*_{R}) (Fig. 4).

Unfortunately, Glimsdal (2013) did not perform displacement wave run-up models for other areas in the affected fjord system, necessitating estimation of run-up from the map of maximum wave heights (Glimsdal 2013, fig. 8 for 10 Mm^{3}) and using wave height to run-up conversion factors from Harbitz & Glimsdal (2011). Such assessment is only approximate, but provides an indication of possible run-up. At the small island of Tautra (*x* = 2.2 km, Fig. 5), run-up estimates differ between the numerical simulation (40 m), the VAW model (37 m) and the SPLASH equation (60 m) (Fig. 4). The higher run-up predicted by the SPLASH equation results from the correction factor for a direct impact (*f*_{R} = 1.5), without which the predicted run-up would be identical to that of the numerical simulation (40 m). In Tomra (*x* = 12.0 km, Fig. 5), predicted run-up values for the numerical simulation, the VAW model and the SPLASH equation are 14–19, 40 and 34 m, respectively (Fig. 4). At Molde (*x* = 17.8 km, Fig. 5) the predicted run-up values are *c.* 4, 9 and 6 m, respectively. These comparisons show that the SPLASH equation is a valuable tool for the first-stage preliminary hazard and risk assessment for unstable rock slopes above water bodies when correction factors for direction changes (*f*_{D}) and shoaling (*f*_{R}) are applied.

### Application in preliminary risk analyses

Our SPLASH equation lacks the precision required for detailed quantitative risk analyses. Uncertainty in run-up prediction is greatest along near-field shorelines, likely reflecting greater variably in wave-motion components near the landslide. However, it is a suitable tool for delimiting areas that could be affected by displacement waves from specific landslide scenarios. With few exceptions the semi-empirical relationship is conservative, predicting moderately higher run-up than the VAW model (Heller *et al.* 2009) and numerical simulations made by Glimsdal (2013). We therefore consider SPLASH to be appropriate for susceptibility maps, which are intended to indicate the maximum possible extent of hazardous phenomena.

The susceptibility map of displacement wave run-up developed for the Opstadhornet rockslide (Fig. 5) shows where people and infrastructure might be at risk. According to the Norwegian Planning and Building Act (Byggteknisk forskrift TEK 17 §7.3 and §7.4: Lovdata 2017) new constructions may only be sited in areas where the nominative yearly probability of a landslide is smaller than 1/1000 for individual houses and 1/5000 for housing blocks, schools, hotels and other large buildings. Hospitals, emergency services and other critical infrastructure must be located outside of any hazardous areas. Applied to landslide-generated displacement waves, the Norwegian Planning and Building Act implies that susceptibility maps of displacement wave run-up are currently sufficient for all unstable rock slopes that have an annual likelihood of failure smaller than 1/5000, except for critical infrastructure. The latter case is illustrated by numerical simulations made for the planned hospital facility at Hjelset, which was assumed to be exposed to a displacement wave generated by the 11.2 Mm^{3} rockslide from Opstadhornet (Glimsdal 2013).

The current practice complying with Norwegian building codes is to perform detailed models of landslide-generated displacement waves for all unstable rock slopes with a likelihood of failure greater than 1/5000 years and that might reach water bodies. The run-up areas depicted by these models become hazard maps to be implemented in municipal land-use planning. Exceptions to this general procedure are remote and scarcely populated areas, where a displacement wave would cause none to only a few potential fatalities. Another exception is for small landslide events (few tens of thousands of cubic metres) if the empirical displacement wave assessment shows that no infrastructure is situated in the run-up area.

## Conclusions

The SPLASH equation presented here provides an alternative approach for predicting the run-up of landslide-generated displacement waves from the limited input parameters of landslide volume and distance from impact. The semi-empirical relationship is formulated from the eight best-documented landslide-generated displacement waves from a worldwide catalogue of 254 historic events. All eight events are sub-aerial rock slope failures of *c.*0.3–30 Mm^{3} entering into longitudinal water bodies. Consequently, the SPLASH equation may be less applicable for other landslide types (e.g. very large volcanic flank collapses into seas, predominantly soil failures). Five percent of the actual run-up magnitudes exceed those predicted by the empirical relationship by a factor 2.58 or more. Underestimation of run-up is mitigated using correction factors for direction changes in the wave propagation path and for shoaling in the run-up area.

Forecasting potential displacement waves from the 11.2 Mm^{3} Opstadhornet rockslide, western Norway, provides a preliminary test of the SPLASH equation. Comparison with numerical simulations performed by NGI and generally applicable equations based on model tests by VAW show that run-up predicted by the SPLASH equation is generally in good agreement with established displacement wave models. The semi-empirical relationship developed here generally predicts slightly higher run-up compared to the VAW model (in more than 80% of the assessed locations). This tendency for modest overestimation of run-up makes the SPLASH equation appropriate for first-order assessment of landslide-generated displacement waves, such as susceptibility mapping of the inundation area.

The quality of this predictive tool will likely be improved through inclusion of additional datasets. We therefore plan to revise the SPLASH equation using further events as their details become available. These additions require thorough investigations of future events and, where historic records exist, analysis of past events. Landslide events with volumes smaller than 0.3 Mm^{3} and larger than 3 Mm^{3} and those impacting large basins such as seas are particularly limited in the presently used database. Furthermore, the SPLASH equation is only valid for rock slope failures. Similar semi-empirical relationships can be envisaged for other landslide types, but multiple events which have been thoroughly documented are required first.

## Acknowledgements

We are grateful to Gro Sandøy (Geological Survey of Norway), Øyvind Rem (Norwegian University of Science and Technology) and Juditha Schmidt (Technical University of Munich) for help in preparing datasets, and to Pierrick Nicolet (Geological Survey of Norway) for assistance with the uncertainty assessment.

## Funding

This work was financed by the Norwegian Water Resources and Energy Directorate through the national mapping program for unstable rock slopes in Norway.

- © 2018 The Author(s). Published by The Geological Society of London. All rights reserved