Introduction

Tsunamis are gravity-driven water waves, most of which generated by vertical displacement of the seabed that propagates through the water column to the surface. The resulting elevated surface wave collapses owing to gravity and then propagates outward from the source. Dispersion of the original wave generates a multiple wave train.

Tsunamis are mainly (∼ 80%) generated by earthquakes, but alternative mechanisms include subaerial and submarine landslides and volcanic collapse and eruption (Tappin 2017). Their main features are represented by both the long wavelength (Charvet et al. 2013) and the height that can vary significantly from the open sea toward the coastal areas (Ezersky et al. 2013), giving the known destructive effect of tsunamis (e.g., the dramatic events in Indian Ocean 2004 and Japan 2011). Where the water depth gradually decreases, the wave slows dramatically, becomes compressed, and grows steeper. Historically known events are referred to sensational episodes which registered significant values of run-up (Ru), accompanied by the highest risks with loss of life. The study of mechanisms generating tsunami is almost exclusively focused on seismic sources because of the presence of monitoring systems widespread on the ground that allow the phenomenon to be easily attributed to a seismic event, so it is a commonplace that tsunamis are generated mostly by earthquakes.. In recent decades, morpho-bathymetric and high-resolution seismic surveys of extensive marine areas made it possible to investigate in detail the presence of large mass transport deposits, and other morphological features attributed to tectonics, volcanism, and fluid seepage.

As a consequence, recent research has addressed tsunami sources other than earthquake, paying particular attention to landslides, which can generate large size tsunamis (Synolakis et al. 2002; Tinti et al. 2008; Liu et al. 2005; Glimsdal et al. 2013). Furthermore, medium-scale landslides (volumes lower than 1 km3) have a higher hazard potential as they occur at higher frequency (Casalbore et al. 2011). In order to produce an adequate assessment of the potential landslide-generated tsunami (LGT), there are several statistical, graphical, and analytical methods (Harbitz et al. 2006; Enet and Grilli 2007; Dao et al. 2013; Flouri et al. 2013). Hydrodynamic simulation of landslide-generated water waves is based on laboratory experiments (Wiegel 1955; Grilli and Watts 2005; Madsen and Furman 2008), numerical modeling (Grilli and Watts 1999; Ioulalalen et al. 2006), and analytical solutions. The latter are the most common methods for LGT assessment; they are founded generally on two characterizing parameters, length and amplitude of the wave near the point source, which can be determined with algorithms based on the morphometry of submarine landslides in the neighborhood of the source area (McAdoo and Watts 2004). The main parameter of tsunamis in the coastal area is the run-up, which is a measure of the maximum vertical raising of the wave, which is a function of both the height of the incident wave to the shore, and the morphology of the coastal sector where the water mass is impacting. In most of the coastal engineering applications, practical methods are necessary to evaluate conservative estimate of tsunami parameters to be used in project calculations by operators with no experience in running detailed numerical models (Kriebel et al. 2017). Despite of some inaccuracies, simplified methods can fruitfully be used for fast tsunami hazard assessments. For mapping coastal risk, a good estimate of the maximum height reached by the wave is essential because it is closely linked to the calculation of flooding. Overall, these data are used to define a buffer zone to protect the coastal infrastructure by tsunami events. The limitations of these methods are the simplification of the phenomenon, which consider a “shallow-water,” small amplitude, and long-wave model, without a detailed description of the wave shape (Didenkulova et al. 2009).

In order to verify simplified methods to assess the potential coastal and marine hazard, we analyzed the North Sicily continental margin (NSCM), because it is a tectonically active area with potential for tsunami generation, as in the cases of Messina 1908 tsunami event (Aversa et al. 2014; Billi et al. 2010) and Stromboli 2002 tsunami event (Chiocci et al. 2008; Tinti et al. 2005; Tinti et al. 2006). Morpho-bathymetric and seismic data, recently collected in the frame of MAGIC and CARG projects, revealed the occurrence of widespread submarine mass transport, genetically related to different processes affecting this margin (canyon, fluid seepage, volcanism, tectonics). In addition, the northern Sicily is characterized by coastal mountains and hills, which are part of the Sicily fold and thrust belt, which yields a great amount of sediments that contribute to the instability of the margin. We identified two different sectors of this margin, located in the western (Gulf of Palermo) and eastern (Patti offshore) coasts, chosen based on their morphological, stratigraphic, and structural features, which could act as sources of tsunamis. Considering that most of models describing the wave propagation and calculating the tsunami parameters were proposed for open oceanic areas and they are rarely suited to semi-confined and narrow basins, the main aims of this paper are to assess the potential hazard associated with LGT in the NSCM and to identify and verify a fast analytical methodology effective for a basin with the characteristics of the Mediterranean. We selected those methods based on simple empirical relationships or obtained as results of simulations, which therefore do not require complex computations, but overall which take into account morpho-bathymetric parameters. We compared the results obtained with different methods of run-up computation applied to the potential Sicilian LGT with those related to actual events of LGT. As a result, we obtained a simple empirical formula that can be used to quickly produce inundation maps and project monitoring systems in the coastal areas exposed to tsunami hazard.

The paper follows a workflow that provides (a) the morpho-bathymetric analysis of the selected areas in the NSCM and the morphometric characterization of the main recognized landslides, (b) the calculation of the tsunami wave parameters in the source points and the computation of coastal run-ups comparing different methods, (c) the verification with historical tsunami events and selection of the best-fitting method, and (d) the formulation of an empirical law based on simple parameters, as amplitude and depth at the starting point.

Geological setting

The NSCM extends in the southern Tyrrhenian Sea, from the north Sicily coastal belt to the Marsili bathyal plain (Fig. 1). The margin is located in the transitional area between the Sicilian–Maghrebian chain to the south (Agate et al. 1993) and the Tyrrhenian back-arc basin to the north (Kastens et al. 1988). This region originated as a consequence of a complex interaction of compressional events, crustal thinning, and strike-slip faulting (Trincardi and Zitellini 1987; Pepe et al. 2005). Tectonic activity started in the Miocene with the thrusting of the Kabilian-Calabrian units followed by the deformation of the Sicilian–Maghrebian chain until the early Pliocene (Sulli 2000), while the opening of the Tyrrhenian Sea led to the subsidence of the margin since the Late Tortonian (Fabbri et al. 1981). E–W, NW–SE, and NE–SW trending both extensional and compressional faults with a local strike-slip component exerted control on the morphology of the present day shelf and coastal areas during the Pleistocene.

Fig. 1
figure 1

Structural map of the NSCM with the main tectonic features (data from Bigi et al. 1992; Sulli 2000). In the bottom left corner, the regional setting of the central Mediterranean (from Gasparo Morticelli et al. 2015)

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On the continental shelf, Pleistocene deposits are truncated by an erosional surface formed during the last glacio-eustatic oscillation. Prograding sedimentary wedges of coastal deposits, formed during the Last Glacial Maximum (LGM, about 20 ka), are present along the shelf margin (Caruso et al. 2011). In the Patti sector, the occurrence of fault segments and the alternation of rocks with different competence, namely, the Hercynian metamorphic basement, its Meso-Cenozoic carbonate, Oligocene-Miocene terrigenous cover, and Pleistocene deltaic sands and gravels (Carbone et al. 1998), produced a very uneven and steep submarine morphology.

Due to active tectonics, sedimentation, and sea-level fluctuations, both in the Gulf of Palermo (western NSCM) and the Patti offshore (eastern NSCM), swath-bathymetry shows a very active continental shelf-slope system, incised by several submarine canyons and related mass failure. The shaping of the canyons is due to both downslope evolution along tectonic lineaments, often in correspondence of the mouth of the torrential rivers, and concurrent upslope retrogressive mass failures (Lo Iacono et al. 2011; Lo Iacono et al. 2014).

The upper plate seismicity of the NSCM is defined by compressional focal mechanisms to the west and extensional to strike-slip mechanisms to the east. Shallow (< 25 km) seismic events of low to moderate magnitude (max M 5.6 in September 2002) occur along an ENE–WSW trending belt, coinciding with the Kabilian-Calabrian thrust, coupled with a NW–SE compressive offset direction (Agate et al. 2000; Giunta et al. 2009). The main seismicity of the Patti area (max M 6.1 in 1978) is linked to right-lateral NNW-SSE transcurrent systems (Neri et al. 1996).

High uplift rates during the last 125 ky were found along the eastern NSCM (0.8–1.63 mm/year) (Ferranti et al. 2010). The vertical rates show a decrease from E to W and highlight coseismic activity between adjacent sectors, while comparison between onshore and offshore sectors suggests the activity of fault systems parallel to the coastline, causing differential vertical movements (subsidence vs. uplift) (Sulli et al. 2013). The western NSCM shows a present-day stability, except for local vertical movements in the Castellammare area, where uplift rates reach 0.1–0.2 mm/year (Mauz et al. 1997; Antonioli et al. 2006).

Methods

To identify the landslides that could generate tsunamis in the NSCM, we used morpho-bathymetric data, acquired from 2001 to 2010 by the Marine GeoGroup of the University of Palermo, in the frame of CARG (national official geological cartography) and MAGIC (Marine Geohazards along the Italian Coasts) projects (oceanographic cruises DFP04, R/V Universitatis; EGUS2001, R/V Thetis; PUMA2009, R/V Universitatis; MACS2010, R/V Universitatis; Patti2003, R/V Thetis).

In the EGUS2001 and Patti2003 cruises, a Reson SeaBat 8111 MultiBeam EchoSounder (MBES), generating 105 beams with a frequency of 100 KHz and operational range of 20–900 m bsl, was used; in the cruises DFP04 and PUMA2009, a Reson SeaBat 8160 MBES, with a lower frequency (50 kHz), but able to investigate at higher depth (3000 m), was used. We post-processed the collected data with the software PDS2000, applying graphic removal of erroneous beams, noise filtering, processing of navigation data, and correction for sound velocity. To obtain the digital terrain model (DTM), we chose different footprint resolutions depending on the depth, with a cell size of 10 m in the shelf and 20 m in the continental slope (mean resolution of 15 m).

The workflow was developed through the following steps (Fig. 2):

  1. 1)

    Analysis of the morpho-bathymetric data. The first step was to identify and to map a large amount of submarine landslides, potentially generating tsunamis, in the Gulf of Palermo and in the Patti offshore. Golden Software Surfer and Global Mapper were used to obtain 3D maps and bathymetric profiles DTM and to analyze the morpho-bathymetric features.

  2. 2)

    Morphometric characterization. We measured the main morphometric parameters useful to calculate the potential landslide-generated anomalous wave (Fig. 3): length (b) and width (w) of landslide; its depth (ds) measured in the center point; slope angle at the source area (θ); thickness in the central point (T).

  3. 3)

    Calculation of parameters of the landslide-generated wave near the source point. Assuming each interpreted submarine landslide as tsunamigenic sources, we calculated the characterizing parameters of the landslide-generated wave near the source point, wave amplitude (As) [Eq. 1], and length (Ls) [Eq. 2] (Fig. 3), by using McAdoo and Watts (2004) equations.

Fig. 2
figure 2

The workflow used in this analysis

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Fig. 3
figure 3

Morphometric characterization of the landslide and related landslide-generated wave (for symbols, see the text)

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$$ As=0.224\ T\ \left(\frac{w}{w+ Ls}\right)\left[\sin {\theta}^{1.29}-0.746\sin {\theta}^{2.29}+0.170\sin {\theta}^{3.29}\right]{\left(\frac{b}{ds}\right)}^{1.25} $$
(1)
$$ Ls=3.8\left[{\left(\frac{b\ ds}{\sin \theta}\right)}^{0.5}\right] $$
(2)

Tsunamis are characterized by large wavelengths, so comparing their half-wavelengths to the ocean depths, they can be considered shallow-water waves; their amplitudes change during propagation, in agreement with the principle of energy conservation, considering that the wave speed, which is function of water depth, decreases coastward due to the fast depth decrease. The wave speed near the coast (c) can be calculated using Zhao et al. (2010) equations [Eq. 3].

$$ c=\sqrt{g\ \left( Ac+ dc\right)} $$
(3)

where dc is the depth considered conventionally as the interference point of wave with sea bottom (10 m), Ac is the wave height near the coast, and g is the gravity acceleration. This parameter represents the shoaling process, that is, how the wave amplitude changes with the depth variation. To calculate Ac, we used the Green law [Eq. 4] that relates the wave amplitude at a generic point (Ac) to the amplitude at the source point (As).

$$ Ac= As\ {\left(\frac{ds}{dc}\right)}^{0.25} $$
(4)

The obtained value of Ac is a rough estimate, good in correspondence of the interference point, and with the Green Law approximation, which can be accepted if we consider that wavelengths are much longer than the water depth, reducing the dispersive effects (Grilli and Watts 2005).

  1. 4)

    Computation of run-ups with different simplified, approximate methods. The run-up (Ru) is one of the main parameters that identifies the magnitude of a tsunami event. In order to estimate the tsunamigenic potential associated with submarine landslides, we calculated the Ru of waves at the coast, through different algorithms (Synolakis 1987 [Eq. 5]; Federici et al. 2006 [Eq. 6]; Didenkulova et al. 2009 [Eq. 7]; Zhao et al. 2010 [Eq. 8]; Bryant 2014 [Eq. 9]), obtained from simple empirical relationships, fluid dynamic laboratory simulations, or numerical modeling (Fig. 4), but we did not consider dispersive wave models, not consistent with a rapid computation.

Fig. 4
figure 4

Schematic models of run-up computation proposed by different authors: a Zhao et al. (2010), b Federici et al. (2006), c Synolakis (1987), d Bryant (2014), and e Didenkulova et al. (2009). See text for explanation of abbreviations

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In detail, Synolakis (1987) proposed an approximate theory supported by laboratory experiments and considered the run-up of solitary waves on plane beaches, describing different regimes for non-breaking or breaking waves. Federici et al. (2006) developed a GRASS-based analytical method, therefore giving emphasis to the morphological factors. Their hydrodynamic model takes into account some basic assumptions dealing with wave shape, morphology, and energy propagation. Didenkulova et al. (2009) developed an analytical method based on the solution of the non-linear shallow-water theory, but it does not take into account the wave shapes. Zhao et al. (2010) used a numerical model based on the high order Boussinesq equations, describing analytical solution of shallow-water equations for different wave shapes. Bryant (2014) introduced a numerical model based on shallow-water long-wave equations with different morphological scenarios and different wave shapes.

Other methods (Gjevik and Pedersen 1983; Grilli et al. 1997; Gedik et al. 2005; Madsen and Fuhrman 2008) are based on similar assumptions, but are referred to peculiar conditions (type of wave, morphology, source, and so on). It is noteworthy to mention that the equations proposed by Synolakis (1987), Zhao et al. (2010), and Bryant (2014) were formulated for events generated by earthquakes.

In order to simplify the computation, the equations refer to a theoretical coastal sector, where the height of the waves does not change with the morphology of the coast and during the propagation of the wave, the energy does not decrease.

$$ Ru= dc\ 2.831\left(\sqrt{\cot \alpha}\right){\left(\frac{Ac}{dc}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} $$
(5)
$$ Ru= Ac+\sqrt{\frac{c^2}{2g}\tan \alpha \left( Lc+\frac{Ac}{2\tan \alpha}\right)} $$
(6)
$$ Ru=3.5\ Ac\sqrt{\frac{p}{\ Lc}} $$
(7)
$$ Ru=3.043\ Ac\ \sqrt{c\sqrt{\frac{3 Ac}{4\ {dc}^3}}\left(\frac{dc\cot \alpha }{\sqrt{g\ dc}}\right)} $$
(8)
$$ Ru=2.83\ \sqrt{\cot \alpha }\ {Ac}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} $$
(9)

To apply these algorithms, we had to calculate other parameters:

  • α—slope of the coastal strip (we assumed it equal to 1°, considering the investigated regions);

  • p—Breaking point–coastline interval (considering a breaking point depth of 10 m and a constant slope of 1°, it geometrically corresponds to a constant value of 588.23 m).

  • k—wave number (according to Zhao et al. 2010) [Eq. 10]

$$ k=\sqrt{\frac{3 Ac}{4\ {dc}^3}} $$
(10)

  • Lc—wavelength near the coastal sector (according to Federici et al. 2006) [Eq. 11]

$$ Lc=\frac{4\ q\ dc}{\sqrt{3\frac{Ac}{dc}}} $$
(11)
  1. 4.1)

    Computation of run-ups with different equations: run-ups of historical cases. The considered equations were applied to actual historical cases. We considered historical cases of LGT, extending worldwide because Mediterranean tsunamis are almost completely attributed to earthquakes. We selected LGT with more detailed information, to apply the described equations and to compare computed theoretical results with real values, to identify the equation that provides the most reliable results. Morphometric parameters were extrapolated from available data or integrated with low-resolution morpho-bathymetric data available online (http://www.gebco.net).

Following the selection of the best method to calculate the LGT run-ups, we put together actual LGT and potential Palermo Gulf and Patti offshore LGT and found the relative best-fitting curve to extrapolate a fast empiric equation that allows calculating quickly run-ups.

Results

Throughout the study areas, we identified and parameterized, but not classified, 21 significant landslides, 18 in the Gulf of Palermo (Table 1) and 3 in the Patti offshore (Table 2) respectively, on the basis of bathymetric and morphological features, such as size, depth, and distance from the coast (Figs. 5 and 7).

Table 1 Main morphometric features of the landslides in the Gulf of Palermo
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Table 2 Main morphometric features of the landslides in the Patti offshore
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Fig. 5
figure 5

Morpho-bathymetric model of the Gulf of Palermo and localization of the 18 mapped landslides. A detail of the Priola landslide (PMO-10) is shown in the top right. Canyons, gullies, and pockmarks lie in the continental slope as well as the related pockmark scars and landslides

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The submarine mass failures in the NSCM. The Gulf of Palermo

The continental shelf in the Gulf of Palermo occupies an area of approximately 250 km2 and is 8 km wide on average. The Gulf of Palermo shows a very active continental shelf-slope system, incised by several submarine canyons, which locally indent the shelf-edge and flow into the Palermo intraslope basin, at a depth of around 1300 m. Most of the mass failures of the area are related to canyon shaping processes, and only few of them are not confined to the upper slope (Fig. 5). The continental shelf is characterized by mounds and pockmarks. The average height of the mounds is approximately 80 m, and their maximum area is about 50,000 m2, while the average depth of pockmarks is about 20 m and the maximum area is 85,000 m2 (Fig. 5). The canyons evolved through concurrent top-down turbiditic processes and bottom-up retrogressive mass failures, whose average length is about 5000 m, while the mean width is about 1000 m The main geological feature that controls the evolution of the canyons and induces sediment instability is the steep (1–10°) slope gradient. Faults and antiforms contributed to the regulation of mass failure processes, while the alignment of pockmarks and authigenic carbonates suggests a relationship between structural control, fluid escape processes, and mass failures (Lo Iacono et al. 2011).

The 18 main submarine landslides of the Palermo Gulf (Fig. 5), observed along the upper slope, are presented from the east to the west in Table 1. The main mass failure is the well-preserved Priola landslide (PMO-10), affecting the uppermost slope, at a depth of 150 m. The scar (Fig. 6), which is about 900 m wide and 100 m high, displays a semicircular shape and a failure plan flattening toward the detachment area, indicating a rotational component in a general translational mass movement. A detailed description of the morphometric characters of the landslides is presented in Table 1.

Fig. 6
figure 6

Line drawing from the DTM (above) and longitudinal profile (below) of the Priola landslide (PMO-10)

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The submarine mass failures in the NSCM. The Patti offshore

In the Patti offshore, the shelf margin is strongly uneven and mainly located at a depth of 140–145 m (Fig. 7). A dense network of canyons, gullies, channel-levee systems, structural steps and highs, submarine terraces and landslides, and sedimentary creep features shows that the margin is very young and continuously reactivated by tectonic processes.

Fig. 7
figure 7

Morpho-bathymetric model of the Patti offshore and localization of the three mapped landslides. A detail of the Orlando landslide (PTT-01) is shown in the inset

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The shaping of the canyons is due to both downslope evolution along tectonic lineaments, often in correspondence of the mouth of torrential rivers, and concurrent upslope retrogressive mass failures. The landslides analyzed in the Patti offshore are presented in Table 2.

The well-preserved Orlando landslide (PTT-01 in Table 2) is located in the upper slope, at a minimum depth of 153 m. The scar (Fig. 8), which is about 586 m wide and has a detachment surface of 0.48 km2 and a perimeter of 2.96 km, shows a semicircular shape and a plane flattening to the detachment zone, which indicates a rotational component in a translation mass movement. It is only 3.3 km far from the coast.

Fig. 8
figure 8

Line drawing of the 3D model (above) and longitudinal profile (below) of the Orlando landslide (PTT-01)

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Landslide morphometry and run-up computation

With the aim to compute the expected run-up for theoretical LGT, we measured the morphometric parameters of each landslide (Table 3), considering them as potential source of tsunami waves. The morphometric analysis confirms that the Priola (PMO-10) landslide is the main feature in the Palermo Gulf, while the Orlando (PTT-01) landslide is the most prominent feature in the Patti offshore.

Table 3 Morphometric parameters calculated for submarine mass failures in the NSCM
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The As and Ls values were calculated to model the hypothetical wave generated near the source point from the identified submarine landslides, by using the McAdoo and Watts (2004) equations (Table 4).

Table 4 Wavelength and amplitude values at the source point
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Starting from these values, we calculated the basic parameters to compute the expected run-ups for each landslide (Table 5). In Table 5, we can notice that the maximum wave amplitude near the coast is 1.04 m (PMO-10), and the minimum value is 0.04 m (PMO-15); the maximum wavelength near the coast is related to PMO-15, and the minimum value is for PMO-10.

Table 5 Basic parameters useful for the calculation of the run-up. The calculations were carried out considering a theoretical coastal sector characterized by a slope of 1° (average value of inner continental shelf) and by a depth of interference with the seafloor of 10 m
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Subsequently, we computed the values of run-ups referred to the interpreted submarine landslides (Table 6). By applying the different equations, the run-up values change strongly for the analyzed landslides. In particular, from the comparison of the different methods used in both sectors (Gulf of Palermo and Patti offshore), characterized by different geological and morphological parameters, we found that the obtained Ru values are very different. In detail, the higher values of Ru are those obtained with the Bryant (2014) method (with maximum values of 21.9 and 17.83 m related to PMO-10 and PTT-O1 for Palermo and Patti respectively), while the lower values are those calculated with Didenkulova et al. (2009). From our results, it is also evident that there are no appreciable differences among the run-up values obtained with the methods proposed by Zhao et al. (2010) and Synolakis (1987), while the Federici et al. (2006) method provides Ru values included in a short interval between 5.34 and 9.09 m (Table 6).

Table 6 Computed run-ups, obtained by using five methods proposed by different authors
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Discussion

In general, the results show that most of the submarine landslides are confined on canyon walls. As a consequence, they could propagate without significant loss of energy, representing therefore a significant hazard and a threat of tsunamis along the coast. Moreover, these landslides could be element of potential risk also for underwater installations or cables.

But the comparison among swift methods used to assess run-ups related to both landslide- and earthquake-generated tsunamis showed that the values calculated with different equations can differ very much. In order to identify the best method to obtain the most realistic value of LGT run-ups, we tested all the algorithms with historical events, in order to verify the values obtained with different equations with actual values. To do this, we selected historical events of LGT where we found thorough data sufficient to get the parameters that we have to enter into the equations. In particular, we analyzed four cases of LGT:

  • Mona Passage (between Hispaniola and Puerto Rico), occurred on October 11, 1918 (López-Venegas et al. 2008)

  • Valdez (Alaska), where on March 27, 1964, a strong earthquake triggered a large submarine landslide that generated a tsunami, with run-ups of 6 m (Parsons et al. 2014; Nicolsky et al. 2010)

  • Nice (France), occurred on October 16, 1979 (Labbé et al. 2012)

  • Papua (New Guinea) occurred on July 17, 1998 (Tappin et al. 2008; Heinrich et al. 2000)

For these events, we calculated the morphometric parameters that are reported in Table 7.

Table 7 Morphometric parameters of the submarine mass failures responsible for historical events
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Subsequently, we calculated the parameters relative to the height and length of the tsunami wave near the source point (Table 8) and near the coast (Table 9).

Table 8 Wave parameters near the source point for historical events
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Table 9 Wave parameters near the coast for historical events. We considered a theoretical slope of 1° in the coastal sector
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Based on these parameters, we computed with the different equations the theoretical values of run-up (Table 10) and compared the results with the actual values. The comparison showed that with the methods proposed by Zhao et al. (2010), Synolakis (1987), and Bryant (2014), for events generated by earthquakes, as well as those coming from the Didenkulova et al. (2009) algorithm, we obtained theoretical values diverging very much from the actual values of run-ups, even more than 290 m higher (Papua Ru calculated with the Zhao et al. (2010) method) and more than 5.80 m lower (Valdez Ru calculated with the Didenkulova et al. (2009) equation). Differently, the method proposed by Federici et al. (2006) returns theoretical values comparable with the actual run-ups for all the considered events (Fig. 9). We motivate this result considering that this method takes into account the wave parameters (wavelength and speed) in the coastal sector, as well as morphological features (slope), so their model describes better the environmental conditions near the shore. While Zhao et al. (2010) do not consider slope and wavelength, Synolakis (1987) and Bryant (2014) equations do not include wavelength and speed, and Didenkulova et al. (2009) ignore slope and wave speed. Really, we have to consider however that Federici et al. (2006) developed their model in semi-confined and narrow basins (Mediterranean and Caribbean seas), similar to those of the considered historical events (lagoons, embayments, or restricted basins). Besides, this method does not consider increase in wave height in restricted areas and loss of energy during propagation and assumes that the wavefront is parallel to the coast.

Table 10 Theoretical values of run-ups for historical events obtained with the different methodologies. The last column shows their actual run-up values
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Fig. 9
figure 9

Comparison between Ru values (in meters) obtained with the different methods (with error bars) and the observed values of historical events (error bars not assessable)

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The results coming from the equation of Federici et al. (2006) are always higher than the measured ones. This is a very important aspect because it represents a precautionary element for the assessment of the geological hazard of LGT.

For this reason, we took into account the values returned by this algorithm to draw a graph in order to relate the expected run-up values (relative to both the NSCM and actual LGT events), according to Federici et al. (2006), to the wave amplitude near the source point (Fig. 10).

Fig. 10
figure 10

Relationship between the wave amplitude near the source point and computed run-ups of NSCM (blue dots) and actual (red dots) LGT. The occurrence of a critical amplitude value is outlined at 0.4 m. The R2 is higher than 0.7, pointing out a good fitting between values and curve

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Analyzing the resulting graph, it is clear that the distribution is ordered according to a curve where the highest values of run-ups are in direct proportionality, as expected, with the wave amplitude.

Nevertheless, a small section of the curve shows an opposite trend: for very low wave amplitude (< 0.4 m) values, an increase in run-up values is observed with decreasing amplitudes. It means that for amplitudes higher than 0.4 m, the greater contribution is given by the value of the amplitude, while for values less than 0.4 m, the major contribution to the run-up does not derive from the value of the wave amplitude at the source point. If we consider the second term of Eq. 7, which represents the maximum raising reached by the water respect to the wave amplitude (Zmax in Fig. 4b), the major contribution could be related to c (wave speed near the coast) and/or Lc (wavelength near the coast).

Based on the drawn curve, we deduced an empirical equation [Eq. 12] linking the wave amplitude to the expected run-ups and fitting the plotted values:

$$ Ru=\frac{3{As}^2+5 As+0.075}{As} $$
(12)

This equation has been applied both to the hypothetical LGT events in the NSCM studied areas and to the historical actual events. The results (Table 11), compared to the run-up values of the actual events, are surprisingly more fitting than those obtained from Federici et al. (2006), even if the differences are minimal. In this way, we propose an empirical method of tsunami computation that, notwithstanding the lack of theoretical backing, allows obtaining very promptly run-up values of LGT, which can be used in rapid assessment or to draw flooding maps in large regions.

Table 11 Spread (Δ) between the results obtained with the empirical equation compared to those obtained from Federici et al. (2006), both in relation to the values of actual LGT
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But end-users of this simplified approach must consider that it is an approximate method that takes into account only parameters of the source point, and not morphological features of the coastal areas, and overall that we tested it for LGT and not for earthquake-generated tsunamis. Furthermore, the values of run-up obtained from the curve are overestimated, representing precautionary conditions for the assessment of the geological hazard.

Conclusions

Analyzing the morpho-bathymetry data of the northern Sicily continental margin, a tectonically active sector of the central Mediterranean characterized by widespread instability, several submarine landslides were identified.

Assuming that each of them could be a potential tsunamigenic source, we calculated the associated theoretical run-ups, comparing approximation methods proposed in the literature.

In order to identify the methodology that better suits LGT values, we compared the known run-up values of actual cases with those calculated through the different methodologies. Once verified that the Federici et al. (2006) method better approximates the values of actual LGT events, we formulated an empirical law that allows calculating promptly the run-up associated with a generic LGT having as starting data the amplitude of the wave near the source point.