Source Modelling

Finn Løvholt, Shane Murphy, Stefano Lorito, Fabrizio Romano, Carl B Harbitz, Jacopo Selva, Raphaël Paris, Anita Grezio, Cléa Denamiel

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Finn Løvholt, Shane Murphy, Stefano Lorito, Fabrizio Romano, Carl B Harbitz, et al.. Source Mod-
elling. Sørensen, M. Probabilistic Tsunami Hazard and Risk Analysis. A cookbook, 14, Springer,
pp.157 - 222, 2026, Mathematics and Planet Earth Series, 10.1007/978-3-031-98115-9_4. hal-
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Chapter 4
Source Modelling

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Finn Løvholt, Shane Murphy, Stefano Lorito, Fabrizio Romano,
Carl B. Harbitz, Jacopo Selva, Raphaël Paris, Anita Grezio,
and Cléa Denamiel

**4.1 Introduction**

This chapter presents modeling procedures for simulating all possible source
processes that are used in scenario based PTHA and PTRA (see Chap. 2 for a descrip-
tion of different frameworks). The methods for selecting the sources, their parameters,
and probabilities, are described in Chap. 3. The methods described herein capture
the mechanistic nature of each source process to different degree of complexity. For
most of the sources, the uncertainty can be treated by varying the source parameters
through Monte Carlo type analysis, a rigorous description of methods for analysing
the model sensitivity is not included here. We describe for each model the parameters
expected to influence uncertainty in tsunami generation the most. The source process
simulation is used as input to modeling tsunami generation (Chap. 5). The chapter is

F. Løvholt (✉)
Offshore Energy, Norwegian Geotechnical Institute, Oslo, Norway
e-mail: finn.lovholt@ngi.no

S. Murphy
Geo-Ocean, Ifremer, Plouzané, France
e-mail: shane.murphy@ifremer.fr

S. Lorito · F. Romano
Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy
e-mail: stefano.lorito@ingv.it

F. Romano
e-mail: fabrizio.romano@ingv.it

C. B. Harbitz
Natural Hazards, Norwegian Geotechnical Institute, Oslo, Norway
e-mail: carl.bonnevie.harbitz@ngi.no

J. Selva
Dipartimento di Scienze della Terra, dell' Ambiente e delle Risorse (DiSTAR), Università di
Napoli Federico II, Naples, Italy
e-mail: jacopo selva@unina.it
<table><caption>**Table 4.1** Source modeling</caption><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>What does the ingredient describe?</td><td>Modeling procedures for earthquake, landslide, volcanic, and atmospheric tsunami sources</td></tr><tr><td>Definitions of the ingredient</td><td>Source models that can be used to model tsunami generation</td></tr><tr><td>What is/are the metric(s) used for measuring/evaluating the ingredient?</td><td>Interface information used by the tsunami models, such as changes in the bathymetric depth as a function of time</td></tr><tr><td>Possible uses/applications of the ingredient</td><td>Modeling of tsunami generation. The models can also be relevant for modeling the source processes on their own right (independently of the tsunami generation)</td></tr><tr><td>Possible multi-hazard dimensions/applications? (if applicable)</td><td>Modeling of cascading / coupled geophysical phenomena</td></tr><tr><td>Possible time-dependent/ dynamic dimensions (if applicable)</td><td>Time dependency effects on tsunami due to finite generation time</td></tr><tr><td>List of the available methods to be described below</td><td>Various methods for modeling tsunami generation from earthquakes (kinematic and dynamic modeling), landslides (initial conditions, block sources, landslide dynamics models), volcanoes (pyroclastic flows, caldera collapse, underwater explosions), and atmospheric pressure fluctuations (synoptic index, synthetic catalogs, operational forecasts, climate modeling)</td></tr></tbody></table>

subdivided into four sections comprising the most frequent tsunami sources, namely earthquake sources, landslides sources, volcanic sources, and atmospheric sources. Volcanic sources may also be linked to landslides and atmospheric pressure, and in these cases the source description is given only in the landslide or atmospheric related subsections. Table 4.1 gives a brief description of the content of the ingredient.

This chapter describes the different sources in the separate Sections: Earthquakes in Sect. 4.2, including kinematic (slip distribution methods, and seafloor deformation models), and dynamic models; Landslides in Sect. 4.3, including descriptions of initial condition models, block models, and dynamic landslide models; Volcanoes in Sect. 4.4, which comprises pyroclastic flow models, caldera collapses, and underwater explosions; Atmospheric sources in Sect. 4.5, which includes methods based on

R. Paris

CNRS, IRD, OPGC, Laboratoire Magmas et Volcans, Université Clermont Auvergne, Aubière, France

e-mail: raphael.paris@uca.fr

A. Grezio

Istituto Nazionale di Geofisica e Vulcanologia, Bologna, Italy

e-mail: anita.grezio@ingv.it

C. Denamiel

Division of Marine and Environmental Research, Ruđer Bošković Institute, Zagreb, Croatia

e-mail: cdenami@irb.hr
synoptic indices, deterministic models, synthetic catalogs, and climate uncertainty. For each source type, a table that summarizes the method’s description, requirements, and uncertainty treatment is provided. Finally, Sect. 4.6 briefly outlines the pros and cons of the various methods in order to help users select the appropriate models.

## 4.2 Earthquakes

A description of the scientific state of the art and gaps associated with modelling earthquake sources in Seismic-Probabilistic Tsunami Hazard Analysis (SPTHA) has been provided by several authors (Geist and Oglesby 2014; Davies et al. 2015; Grezio et al. 2017; Behrens et al. 2021).

Section 4.2 deals with the two main approaches to the mechanical earthquake modelling: kinematic and dynamic. The kinematic source modelling starts with a prescribed slip distribution on the fault interface, from which the seafloor displacement can be calculated. The dynamic approach models the full mechanical process starting from pre-stress and frictional state on the fault interface, and the mechanical properties of the surrounding materials, to the seismic rupture evolution and the seafloor deformation.

The kinematic model is frequently used in PTHA or PTRA given its relative simplicity and smaller computational weight in comparison with dynamic modelling, which, conversely, provides a more physics-based representation of earthquakes.

### 4.2.1 Method: Kinematic Rupture Modeling

#### Short Description of the Method

Generally, in SPTHA, the earthquakes are modeled as individual ruptures by considering the kinematic parameters of the source in terms of fault dislocation (slip and rake). Often, the additional assumption that the rupture occurs quasi-instantaneously is introduced. This means that the speed of the rupture front and the dislocation rise-time on a given portion of the fault are much faster than the tsunami propagation. This assumption may not hold in all circumstances, for example, in the case of large earthquakes were rupture occurs over large areas (e.g., 2004 Sumatra-Andaman), or slow ruptures like tsunami earthquakes (e.g., 2006 Java, 2010 Mentawai).

Here, we describe the two main components of the kinematic rupture process, that is (i) the slip distribution modeling (Sect. 4.2.1.1) and (ii) the associated seafloor deformation which causes the tsunami initiation by moving the water column above it (Sect. 4.2.1.2).
### 4.2.1.1 Method-Slip Distribution Modeling

The slip distribution modeling can be broken down into three steps: (i) defining the fault plane, (ii) linking it to the global earthquake characteristics, (i.e. total area, average slip) (iii) assigning the slip distribution (local slip value, i.e. intensity, and direction, and possibly the time-history). The distribution of the slip vector in space (and time) then constitutes the input for the second component of the method, that is the seafloor deformation.

#### Fault Plane Representation

Within the variability ranges prescribed by the earthquake source model characterization as discussed in Chap. 3, modelling individual ruptures is the first step to simulate tsunamis for each considered scenario. The simplest way to model a seismic scenario, with relatively little (e.g., seismic and/or geological) information, and limited computational resources, is a rectangle with uniform slip (see Fig. 4.1a). However, with increasing computational resources and higher resolution imaging of fault interfaces available (e.g., Slab2.0, that maps the subduction zone at the global scale, Hayes 2018), fault geometry can be better approximated using structured or unstructured meshes (e.g., discretized with quadrangular or triangular cells as in Fig. 4.1b).

#### Earthquake Scaling Relations

As discussed in Chap. 3, there are several empirical scaling relations (see Fig. 4.2) expressing the earthquake area and average slip as a function of the magnitude (e.g., Wells and Coppersmith 1994; Strasser et al. 2010; Allen and Hayes 2017; Murotani et al. 2013; Blaser et al. 2010; Skarlatoudis et al. 2016; Goda et al. 2016).

<figure><img src="image_3.png" /><figcaption>**Fig. 4.1** Examples of different fault representations for a magnitude 8.5 event. Panel a) shows the simplest case: the earthquake rupture is approximated by a single planar fault with a uniform slip of 6 m. Panel b) shows a more complex modelling: the Scotia subduction plate interface as taken from Slab 2.0, and with a heterogeneous slip distribution (discussed towards the end of Sect. 4.2.1) featuring a maximum slip of 16.8 m. The same colorbar has been used for both representations</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Fig. 4.2** Example of different scaling relations between magnitude and rupture area. The magnitude range of the different scaling relations is based on the magnitude range used by the authors when deriving their best fit line to the data</figcaption></figure>

A comprehensive discussion on and a comparison between different scaling relations are provided by Stirling et al. (2013). An estimation on the uncertainty on the scaling is generally also provided as standard error and/or deviation which can be incorporated into PTHA.

Regardless of the scaling relation, the energy release from the simulated earthquake must match the prescribed magnitude that obeys the moment magnitude equation (Hanks and Kanamori 1979):

$$M_w = 2/3\log M_o - 6.03$$

where $M_o$ is the seismic moment (in Nm) defined as:

$$M_o = \mu A \delta$$

With $A$ being the slipping area (in m²), $\mu$ the average rigidity (in GPa) of the rock, and $\delta$ the average slip (in m). A typical rock rigidity value is 30 GPa (Kanamori and Brodsky 2004), yet in subduction zones the rigidity variation with depth is significant and at shallow depth smaller than 30 GPa (e.g., Geist and Bilek 2001). In order to preserve this relation, the slip distribution is multiplied by a constant to correct the mean slip so that the desired magnitude is produced.
Slip Distribution

After defining the characteristics of the earthquake, the final step is to assign a slip distribution onto the fault surface.

The simplest approach is to assume uniform slip values along the fault (Fig. 4.1a). A method better representig reality, although more complicated, is the use of heterogeneous slip distributions (Fig. 4.1b).

Andrews (1980) proposed the first static self-similar (e.g., fractal) slip distributions based on the observation that the self-similar nature of far-field displacement is consistent with a power law scaling of seismic rupture (Hanks 1979). Herrero and Bernard (1994) introduced the *k-squared model* to generate stochastic slip distributions in the frequency domain by imposing an inverse square scaling on the radial wavenumber. The description, in the frequency domain, is then converted to slip distribution using an inverse Fourier transform. As the phase is randomly assigned, each slip distribution is different.

With time, different methods for generating stochastic models have been proposed (e.g., Lavelee and Archuleta 2003; Song and Somerville (2010); Davies et al. 2015). For example, Mai and Beroza (2002), developed a slip model based on the fitting of a von Karman autocorrelation function with a database of finite fault slip inversions; the slip distribution is self-affine rather than self-similar (as in the k-squared model previously mentioned).

A limitation of many of these slip models is the use of discrete Fourier transform which requires a regularly spaced cartesian grid limiting their use to planar faults. However new slip models are emerging for use on non planar surfaces. For example Sepulveda (2017) used a combination of the Karhunen-Lòeve expansion method with the von Karman function to produce stochastic slip on a curved north Chilean subduction zone (Leveque et al. 2016). Another method is the combination of the composite source model (Zeng et al. 1994; Ruiz et al. 2007) which involves the placement of hierarchical sets of circular slip functions on a surface, with an accurate method of calculating the geodesic distance across the non-planar surfaces (Herrero and Murphy 2018) - an example of which is shown in Fig. 4.1b.

The advantage of this method is that it is fast. The disadvantage is that the physics of the earthquake rupture on a specific fault is not fully accounted for, as the stochastic slip retains just global statistical rupture properties. Moreover, the displacement is assumed instantaneous. However, while admitting the limitations in constraining the kinematic rupture parameters like the rupture nucleation point, the rupture propagation velocity, and the rise time in this kind of modeling, a simple kinematic stochastic rupture model including these parameters can be generated by following the approach proposed by Goda et al. (2018).

Regarding the slip direction in subduction zones (where most large tsunamis originate), it is generally assumed that the fault movement occurs dominantly with a thrust mechanism whereby the hanging wall (i.e., the medium above the fault) moves upwards relative to the footwall (rake = 90 degrees), or antiparallel to the direction of plate convergence. However, other faulting mechanisms, while much
less frequent, have generated tsunamis also, for example the 2018 Palu strike-slip earthquake (Ulrich et al. 2019b), splay faulting (Wendt et al. 2009) or outer rise earthquakes (Baba et al. 2020). The forecasting of the faulting mechanisms was briefly addressed in Chap. 3.

### 4.2.1.2 Method-Static Coseismic Seafloor Deformation Modeling

The seafloor deformation is computed based on the faulting mechanism (i.e., strike-slip, reverse, normal) and the amount of slip on the fault. This is known as the *dislocation model*, and in the simplest case it assumes that the Earth consists of one layer in which the fault is buried, with a planar surface. The advantage of this technique is that analytical solutions can be solved rapidly. Analytical equations have been developed for point and rectangular dislocations in half space media (Chinnery 1963; Mansinha and Smylie 1971; Iwasaki and Sato 1979), which Okada (1985) provided in a compact form, widely used nowadays with the code available in the public domain. Recently, similar analytical solutions with triangular surfaces have been developed (Meade 2007; Nikkhoo and Walter, 2015, the latter reference fixes some issues present in the former). These two have become popular as they can provide Green's functions from the triangular elements suited for describing non-planar fault surfaces, but also because the numerical code is available in the public domain. Care is required in all cases when cells touch the seafloor surface due to the creation of singularities along cell edges.

To model heterogeneous slip distributions, the fault plane is subdivided into smaller rectangles or triangles (the subfaults) with different slip placed in each cell based on a stochastic slip distribution (see Sect. 4.2.1.1). The total deformation is then computed by summing up the contribution from each of the individual subfaults (the Green's functions) using the principle of linear superposition which applies in the framework of linear elasticity. In the case of dislocation point sources, a Gaussian quadrature is used to numerically integrate the different sources (Segall 2010).

Alternatives to the dislocation model also exist. The *crack model* where slip is tapered towards the edge of the slipping surface (Geist and Oglesby 2014b). This method produces a constant stress drop across the cell, with no stress singularities produced at the edge of the cells. Like the dislocation model, analytical solutions for the crack model exist for sources in a half space media. The advantage of using the analytical models is the speed of computation, and only a description of the fault plane is required; however, variations in the mechanical properties of the subsurface media are not represented. To account for more complex media with multiple layers and accretionary wedges, Finite Element Models (FEM) are used (Zhao et al. 2004; Romano et al. 2014). Such models require detailed information on the media surrounding the fault, generally acquired from seismic imaging techniques as well as the meshing of the model (fault and surrounding medium). Therefore, more time is taken to build the model and calculate the deformation, but the deformation will include more information.
The output is the static seafloor displacement, assuming that the displacement occurs within a timescale much shorter than the tsunami wave propagation. In some cases, the slip on the sub-faults is assumed to occur at different instants to mimic a time-dependent rupture evolution (e.g., Satake et al. 2013). In most cases, only the vertical seafloor displacement is considered, as the horizontal one is considered to have a negligible tsunamigenic capability, except when it occurs on steep oceanic slopes and it is larger or comparable to the vertical one as for low-angle subduction interfaces. This parameter is treated as an input in Chap. 5. To consider this contribution, a largely used technique is to project the horizontal displacement onto the bathymetric gradient (Tanioka and Satake 1996).

## Data and Technical Requirements for Method

The input for the slip distribution modeling is the parametrization of the fault plane. This can vary from a basic description of the width, length, position, and orientation in space of a planar fault, to a point cloud description of a non-planar surface. This parameterization does not require High-Performance Computing (HPC) resources.

The input for computing the coseismic seafloor deformation is the slip distribution defined on the fault surface. The general assumption is that the surrounding medium is elastic and homogeneous, but the level of complexity of the modeling can increase depending on the amount of information available and the type of numerical solver used; for example, using Finite Element Modeling (FEM) allows for using 3D crustal velocity models, topography, etc.).

For individual slip distributions, the seafloor deformation can be computed on standard computers without the requirement of additional computational resources. However, in PTHA, where large (thousands to millions) numbers of seafloor deformation calculations are sometimes used to model the natural (aleatory) variability of the seismic parameters and to quantify epistemic uncertainties with alternative model formulations, HPC usage may become almost mandatory. To save computational effort, Green's functions can be pre-computed by calculating the surface deformation caused by unitary slip (e.g. 1 m) on each individual cell, to be linearly combined to obtain virtually any desired scenario. This process is particularly useful when computationally demanding software (e.g. FEM) is adopted and/or the number of cells describing the fault is much less than the number of scenarios being simulated.

## Treatment of Uncertainties for the Method

The uncertainty in the location and description of the fault plane is highly dependent on the amount of data available. High-resolution seismic profiles and/or well located seismic catalogs can be used when available to reduce the uncertainty. Indeed, simplification of the fault into a planar surface has been shown (e.g., Tonini et al. 2020) to affect the generated tsunami height. In the absence of a good characterization of individual faults, the geometrical fault parameters should be treated as aleatory variables like the other earthquake parameters.

For empirical scaling relations, estimates of the uncertainty related to the fit of the equations to the data are provided in the literature. This uncertainty can be incorporated into PTHA by treating the corresponding parameters as aleatory. Additionally,
logic trees or other approaches can be used to account for the existence of different (epistemic, alternative) scaling relations, as discussed in Chapter 3.

Given all the other earthquake parameters, a certain number of stochastic slip distributions can be computed to account for a range of potential future scenarios, treating them as aleatoric uncertainty. Accounting for heterogeneous slip is particularly important for capturing the large variability in tsunami wave height in cases where the coast is in the near (Geist 2002) and far field (Davies and Griffin 2020). Davies (2019) demonstrated statistically that uniform slip models have a tendency to produce small tsunami's too often, on the contrary, variable rupture areas or heterogeneous models do not give rise to statistical underestimation.

An additional complexity that is only sometimes considered is the near-surface slip amplification as observed in Tohoku 2011 or tsunami earthquakes, which can lead to higher tsunami wave heights and hence hazard (Murphy et al. 2016). A simplified way of handling this phenomenon is scaling the slip with the inverse of shear modulus, which increases with depth (Bilek and Lay 1999; Geist and Bilek 2001). Scala et al. (2020) incorporated this depth dependent scaling as well as variable fault coupling into a PTHA for the Mediterranean and found that the mean hazard curves exhibited a higher probability for larger inundation when depth dependent scaling is accounted for.

The main advantage of the kinematic method is that it is fast, allowing for the uncertainty in the spatial distribution of slip to be propagated in PTHA. Analytical solutions which can be rapidly applied on standard computers assume simplified media. The displacement can be computed with different degrees of complexity as far as the elastic media parameterization is concerned (layers, 3D variations; e.g., Zhao et al. 2004). However, highly compliant material with inelastic behaviour, such as an accretionary wedge, has been shown to alter the orientation of surface deformation (Ma 2012).

## 4.2.2 Method: Dynamic Rupture Simulations

### Governing Equations

Dynamic rupture simulations can be used to provide a tsunami source that takes into account the properties of the medium surrounding the fault, the lithospheric plate, as well as stress conditions, while kinematic modelling at least in principle can produce unphysical scenarios. The elastic wave equations are solved numerically. The relationship between stress and displacement of an isotropic elastic material is described by Hooke's law. On the fault surface, an empirical relationship between the shear stress acting on the fault surface and the normal stress (Byerlee 1978) is applied through a coefficient of friction which controls the fault strength. The evolution of the friction coefficient during an earthquake is described by friction laws based on experiments in rock physics laboratories and is generally expressed as a function of slip, slip-rate, or state variables (e.g., Ida 1972; Ohnaka 1973; Di Toro et al. 2011;
Ruina 1983; Dieterich 1979). Defining an all-encompassing friction law remains an active field of research in laboratory rock-physics.

There are two approaches to couple tsunami simulations and dynamic rupture models. The first uses the ground motion resulting from the dynamic simulation as input into a tsunami simulator (e.g., Ryan et al. 2015; Madden et al. 2021). In the second approach tsunami and dynamic rupture simulation are fully coupled using equations of motion in both the elastic and fluid domains (e.g., Lotto and Dunham 2015; Abrahams et al. 2023). Modelling the “interface” between the source and the ensuing tsunami is dealt with in Chapter 5.

There are several numerical methods that have been used to solve the elastodynamic equations incorporating the earthquake rupture (a detailed overview in Igel 2017). However, it is beyond the scope of this section to provide an in-depth description of all the numerical techniques. Rather, a discussion on the use of the different numerical methods in relation to dynamic rupture, suitable to model non-planar faults and heterogeneous media as for example in subduction zones, is provided here with references to the relevant literature.

Numerical techniques used for dynamic rupture simulations in subduction zone environments are based on the spatial discretization of the media by geometrical elements that can describe complex fault geometries and velocity structures. Rupture codes using Finite Element (FEM; Barall 2009; Ma 2009); and Finite Volume Methods (FVM; Benjemaa 2009) have been developed and successfully applied to dynamic simulations in subduction zones (Ryan et al. 2015; Ma 2012). Spectral Element Method (SEM; Festaand Villote 2005; Galvez et al. 2014) and the Discontinuous Galerkin (DG) Method (de la Puente et al. 2009; Pelties et al. 2012) are higher order versions of the FEM and FVM techniques, provide higher accuracy while still leveraging the advantages of using adaptable meshing. The application of Finite Difference (FD) methods on non-planar or non-vertical faults is more complicated and requires the use of particular techniques such as (a) support operators (Ely et al. 2009) or (b) high-order summation by parts combined with simultaneous approximation term method (Lotto and Dunham 2015). Additionally, for FD, accuracy is a function of the spatial extent of the stencil size, and consequently care is required at boundaries, particularly for high order schemes.

Initial conditions for dynamic simulations include: the geological structure/media, fault geometry, frictional parameters, shear and normal stress on the fault (which can be heterogeneous). Outputs from the modeling are temporal and spatial evolution of (a) the seismic rupture on the fault plane, (b) the seismic waves including the resulting ground motion and (c) tsunami surface elevation.

The advantage of this method is that it can provide physical insight into the observed phenomena. Dynamic modelling can of course be used also for tsunami-genic earthquakes occurring on crustal faults. Examples of this include the role of regional stress in 2016 M 7.8 Kaikoura earthquake where a number of individual faults ruptured in cascade (Ulrich et al. 2019a); the effect of rupture dynamic and fault geometry in near surface amplification (Murphy et al. 2016); the role of the frictional variation with depth in earthquake type (Murphy et al. 2018); and the role of supershear rupture in tsunami generation during the 2018 Palu earthquake
(Ulrich et al. 2019b). One of the main disadvantages with dynamic modelling is that it is computationally demanding, making it challenging to investigate uncertainty by performing numerous simulations. This obstacle will likely decrease in the near future as computational resources continue to increase allowing the possibility to incorporate a variety of different geological conditions. Additionally, a full description of the physical environment around the fault must be constructed and meshed which can be labor intensive and is subject to data availability. Care is required in construction of the numerical model in terms of nodal spacing / element size so that the simulation accurately models wave propagation and rupture process (Day et al. 2005). Finally, care is also required in coupling the solid earth and water domains, which is described in Chapter 5. In non depth-averaged fully coupled models this is accounted for with seismic waves being converted into acoustic waves at the seafloor and propagating within the water column. In order to accommodate the acoustic tsunami generation, inclusion of compressible effects are customary. In traditional depth averaged tsunami models (e.g. shallow water or Boussinesq), this phenomenon is rarely considered.

## Data and Technical Requirements for the Method

Dynamic simulations require a complete description of the geophysical setting that the simulation occurs in. This includes the density as well as the P- and S-wave velocities throughout the model. The fault surface must be defined within the model, as well as the friction law on the fault plane and the stress distribution on it, which is often constrained by the regional stress field. Once the layers and fault plane are defined, meshing software (e.g., Gmsh, CUBIT) is needed to discretise the model domain into discrete elements (i.e., triangles, quadrangles, tetrahedrons, polyhedrons). Community based codes such as Seissol (https://seissol.org/) are optimized for running on HPC with inbuilt parallelism (MPI, and/or OpenMP) and/or written for GPU processors. 2D dynamic simulations are less computationally onerous than 3D simulations and can be run on standard computers but they can still take a large amount of time to run.

## Treatment of Uncertainties for the Method

This method is deterministic, particularly in 3D simulations due to large computational resources required to run one simulation. Consequently, at present, extensive sensitivity analysis is not possible; however, limited analysis running a small number of scenarios is feasible (Madden et al. 2021).

It is often assumed that the rock around the fault acts elastically. Some, more computationally expensive, numerical models account for plastic deformation of the material close to the fault (Andrews 2005; Ma 2012, Ulrich et al. 2019ab), which is a phenomenon that can be important for shallow earthquakes, as deformation in a compliant material, such as the accretionary wedge, slows down the rupture on the fault.

2D simulations are less computationally intensive and may be used on HPCs to run a large number of simulations to assess the uncertainty of certain phenomena, for example the dependence of slip with depth, which can be then used to inform
stochastic models (Murphy et al. 2016). However, due to the highly non-linear nature of rupture, small changes in the stress distribution or location of the nucleation can have a large effect on the subsequent slip distribution and ground motion. This nonlinearity makes it very challenging to account for uncertainty in dynamic simulations, especially in 3D environments.

### 4.2.3 Summary Table

See (Table 4.2).

## 4.3 Landslides

A description of the scientific state of the art and gaps associated with modeling landslide sources can be found in Løvholt et al. (2015), Yavari-Ramshe and Ataie-Ashtiani (2016), while adaptation to PTHA and challenges related to this are found in Grezio et al. (2017) and Behrens et al. (2021).

The landslide volume or frontal area (in the case of subaerial landslides) are the main factors influencing tsunami-genesis. In addition, the landslide dynamics and the water depth of generation also influence the tsunami generation strongly. Landslide parameters such as the local slope, external friction forces, rheological parameters, and fluid resistance parameters (e.g. viscous drag, skin friction, and added mass) influence the sensitivity to the tsunami-genesis (e.g. Grilli and Watts 2005; Watts et al. 2005; Heller and Hager 2010; Kim et al. 2019; Snelling et al. 2020).

Their sensitivity can be modeled by Monte Carlo type models systematically varying the parameters. The application of methods as described in Løvholt et al. (2020) provides examples embedding source parameter uncertainty into PTHA (see also the R8 Lyngen recipe on LPTHA). The influence of the water depth implies that landslide kinematic parameters such as the initial acceleration and Froude number correlate strongly with the tsunami-genesis (e.g. Ward 2001; Grilli and Watts 2005; Løvholt et al. 2015; Yavari-Ramshe and Ataie-Ashtiani 2016; Zengaffinen et al. 2020).

This section outlines the main classes of methods presently used for modeling landslide tsunamis: We describe first the simplest one that is the static initial condition, where an analytical shape is provided as input to the tsunami model. The second method we describe is the block model, using a fixed shape moving block as a tsunami generator. Finally, we describe landslide dynamics methods, which may inherit different degrees of sophistication (depth averaged or vertically resolved) or different rheologies (e.g. granular or viscoplastic).
<table><caption>**Table 4.2** Summary table: earthquake sources</caption><thead><tr><th></th><th>Kinematic models</th><th>Dynamic models</th></tr></thead><tbody><tr><td>Short description of method(s)</td><td></td><td></td></tr><tr><td>Theoretical / mechanical background for the method(s)</td><td>Phenomenologically-based computationally lightweight way of simulating the earthquake kinematics by prescribing a stochastic slip distribution on a particular fault plane as input for the calculation of Green's functions of the static seafloor deformation</td><td>Various models which account for the time-dependent dynamics of earthquake rupture based on elastic/plastic theory, using friction laws and pre-stress on the fault surface</td></tr><tr><td>Short outline of how the method(s) works</td><td>Spatial description of an earthquake as one or a series of cells which move instantaneously as input to the elastic static seafloor deformation. The slip distribution is stochastic with a spatial probability on the fault plane which is usually uniform, yet different scaling relations for the rupture dimension and average slip, and different stochastic or physics-based slip models can be used (e.g. self-similar, self-affine, shallow slip amplification)</td><td>Numerical methods are used to solve elastodynamic equations with rupture physics represented by boundary conditions. Models differ widely in complexity with some modeling solely the solid earth while others incorporate hydrodynamics</td></tr><tr><td>How is/are the output(s) of the method(s) represented?</td><td>Residual static or time-history of the vertical seafloor displacement. A more accurate output includes contributions from the horizontal displacement times the gradient of the bathymetry</td><td>Dependent on the model complexity. All involve time dependent variation of slip evolution on fault, seismic wave field, ground motion, and in coupled models also the tsunami generation. Sometimes only the residual static displacement (as described in the kinematic case) is used as a tsunami initial condition</td></tr><tr><td>In what way do(es) the method(s) take into account time-dependent/dynamic aspects?</td><td>Generally the earthquake and the ensuing seafloor displacement are assumed to occur instantaneously, but a kinematic representation is possible</td><td>The evolution of the stress, slip-rate and slip are calculated across the fault plane as well as ground motion at the sea floor and in some cases tsunami wave generation</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.2 (continued)**</caption><thead><tr><th></th><th>Kinematic models</th><th>Dynamic models</th></tr></thead><tbody><tr><td>Do(es) the method(s) use parallelisation, machine learning, surrogate models, advanced simulation schemes, other computational reduction strategies (e.g. cluster analysis, etc.)?</td><td>Parallelism is not generally required. A common strategy used to reduce computational resources is to precalculate the displacement Green's functions caused by unit slip on subfaults (i.e. smaller spatial elements that describe the fault surface) which can then be rapidly used in subsequent simulations of different heterogeneous slip distributions</td><td>Generally, parallelisation is used due to large computational resources required for each simulation</td></tr><tr><td>What are the pros and cons and peculiarities of this specific method(s)? Are there known shortcomings / benefits of the method(s)? In what cases is this method(s) a good/poor choice?</td><td>Pros: Stochastic slip generation techniques and analytical solutions are fast; pre-computed Green's functions are a computationally efficient way to simulate a large number of seafloor displacements using the same fault after upfront cost<br>Cons: Physical consistency of random nature of models is not verified; analytical solutions greatly simplify mechanical properties of the subsurface. Time-dependent ruptures are often simplified to be instantaneous</td><td>Pros: Models the full physics of an earthquake<br>Cons: Construction of model can be labor intensive; simulations are often computationally demanding; parametric uncertainties regarding friction, pre-stress and structure can also be problematic due to lack of specific observational constraints</td></tr><tr><td colspan="3"><strong>Data and technical requirements for method</strong></td></tr><tr><td>What input data does the method(s) use?</td><td>Description of fault surface and faulting mechanism, mechanical properties of subsurface, and bathymetry</td><td>Complete physical description of the fault geometry, frictional parameters and surrounding material: density, Lame's parameters, bathymetry</td></tr><tr><td>What are the technical requirements?</td><td>Software/scripts for fault discretization, stochastic slip generation, seafloor displacement calculation</td><td>Software/scripts for fault discretization, softwares for dynamic modelling</td></tr><tr><td>Are HPC computing resources necessary, would absence of these compromise the use?</td><td>Can be used in batch mode for uncertainty quantification. Also, complex FEM models may require more computational resources</td><td>In most cases large HPC facilities required, although simple models (e.g. 2D) can be run on less powerful computers</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.2 (continued)**</caption><thead><tr><th></th><th>Kinematic models</th><th>Dynamic models</th></tr></thead><tbody><tr><td>Is the method(s)/ workflow(s) available as a git/ software/jupyter notebook?</td><td>Various softwares are freely available online</td><td>Some softwares are freely available online</td></tr><tr><td>Treatment of uncertainties for the method</td><td></td><td></td></tr><tr><td>Is the method(s) itself deterministic or uncertainty based?</td><td>Deterministic</td><td>Deterministic</td></tr><tr><td>Which are potential sources of uncertainties? Which parameters are used to characterize them?</td><td>Lack of data for fault and surrounding medium characterization, slip randomness and depth-independence, uncertainty on earthquake mechanism and scaling relations, simplified treatment of the elastostatic problem (e.g. homogeneous medium)</td><td>Initial conditions on fault plane (initial stress, nucleation), fault geometry, material parameters, friction law and its spatial variation</td></tr><tr><td>How are uncertainties quantified for the identified parameters? Are there significant correlations between different uncertain parameters?</td><td>Aleatory uncertainty in slip location and earthquake scaling is accounted for by generated large ensembles; the procedure can be repeated using different assumptions (e.g. depth dependent rigidity, coupling conditions), or with different material properties and bathymetries, to quantify the epistemic uncertainty. The earthquake scaling relations also define the correlation among the different earthquake parameters</td><td>Uncertainties are rarely accounted for because of practical and computational limitations. Sensitivity tests are sometimes performed especially with 2D models</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.2** (continued)</caption><thead><tr><th></th><th>Kinematic models</th><th>Dynamic models</th></tr></thead><tbody><tr><td>Is the method(s) able to digest and propagate uncertainty in input (previous ingredient) and propagate it to the output (next ingredients)? Does the assessment of this ingredient require uncertainty propagation? If yes, how are uncertainties propagated?</td><td>Yes, through parameter variation and multiple simulations slip distributions and of the ensuing seafloor displacement. Uncertainty is naturally propagated to the next step which will require simulating a tsunami for each initial condition in each ensemble</td><td>Complex models rarely account for uncertainty due to computational cost for running each simulation, simpler 2D models can be used to test some initial conditions in the model by running large batches of simulations with varying conditions, in a similar fashion to the kinematic models</td></tr><tr><td>What are the consequences of lack of data in input or for boundary conditions, or for characterizing certain parameters?</td><td>Method highly dependent on definition of fault geometry and material properties. For example, using a wrong fault dip or neglecting features such as compliant accretionary wedges may cause large differences</td><td>Large epistemic uncertainty in describing conditions on the fault plane which produces large model bias and uncertainty in results</td></tr><tr><td>What are the consequences of ignoring or poorly treating uncertainties?</td><td>First order biases or underestimation of the uncertainty may occur, which are comparable to those regarding the earthquake occurrence characterization (Chap. 3)</td><td>The nonlinear nature of earthquake rupture makes it challenging to account for uncertainty in dynamic simulations</td></tr></tbody></table>
### 4.3.1 Method: Static Initial Conditions

#### Short Description of the Method

Static initial conditions represent the simplest method for modeling landslide tsunamis, and are often used in the absence of landslide dynamics models or possibilities of including general time-dependent sources in tsunami models. An example of a static initial condition model was first presented by Watts (1998), combining theoretical (constant slope) block motion with dimensional scale analysis. Later model derivations have followed a more rigorous mathematical development to yield analytical solutions for dispersive waves (Lo and Liu 2017, 2021).

Static initial conditions cannot take into account the local geometry where the slide occurs, and may therefore represent local generation effects and effect of the complex bathymetry poorly, which is a shortcoming compared to the more sophisticated generation models described below. In other words, the dynamic evolution of the landslide over time and complex topography is only represented by a single instance, which is a simplified representation. The advantage of using static initial models is that they are simple, on the other hand they may have significant biases related to the tsunami generation, both in terms of tsunami heights and the wave characteristics as the model equations are generally not adapted to the local bathymetry, nor take time-dependent landslide dynamics with deformation into account.

#### Data and Technical Requirements for the Method

The model of Watts (1998) yields as output both the surface elevation and an estimate of the wavelength along a transect. Necessary inputs include the dimensions of the landslide block (length, width and thickness), landslide density, slope angle, water depth, as well as values for the skin friction, pressure drag, and added mass coefficients. It is necessary to make further assumptions about the azimuthal distribution of the wave field using ad hoc assumptions, such as analytical shape functions (e.g. Synolakis et al. 2002). This provides a 2D output of the surface elevation. The 2D wave field was included in the later development of Lo and Liu (2017) who took into account the spreading of the wave. These models involve analytical expressions that only require small computational power, thus HPC is not needed.

#### Treatment of Uncertainties for the Method

The main parameters influencing the uncertainty of the tsunami generation are described in the intro paragraph of Sect. 4.3, and they can be modeled using Monte Carlo type methods varying the parameters. The lack of direct inclusion of physics and time-dependency in the generation (see above) results in the possibility of significant biases in the simulations. They are also less tested than the two other model classes when it comes to forecasting ability. Hence, there are also more considerable uncertainties and biases associated with these methods than with the more sophisticated methods described below.
## 4.3.2 Method: Block Sources

### Short Description of the Method

The method uses moving blocks as sources in the tsunami model. Block sources are fixed shaped 3D objects that move as flexible blankets over the seafloor, and the time-dependent motion is used for displacing the water mass in tsunami models (see Fig. 4.3). Harbitz (1992) described the first applications of such sources, and later analysis is given in e.g. Løvholt et al. (2005), Løvholt et al. (2015).

The block has a given shape determined by the user and can either have prescribed motion or be modeled according to simple ordinary differential equations of a sliding block on an inclined plane (e.g. Gylfadottir et al. 2017). The moving block is then given as a time dependent thickness field as input to the wave model, either as direct input in the volume displacement terms in the tsunami model (e.g. Løvholt et al. 2005) or subject to time-dependent source filtering of the seabed displacement before being used as input to the tsunami model (see Løvholt et al. 2015, and further discussion in Chap. 5). As described in Chap. 5, low-pass filtering is necessary to remove unphysical wave components during generation. This can also be partly mitigated by rounding the shape of the block. The frequency of feeding time-dependent mass into the wave model must be tested for convergence in a similar fashion as for grid refinement tests for the time step (e.g. Zengaffinen-Morris et al. 2022), by updating the seafloor deformation sufficiently frequent in the tsunami model. The block sources have the advantage over static initial conditions to more rigorously capture the bathymetric effect and the time dependent tsunami generation, while still being relatively simple. It is further more broadly tested with respect to modelling real events (e.g. Løvholt et al. 2005; Harbitz et al. 2014). But they do require the tsunami model to take into account time dependent forcing, which represents increased complexity for the user compared to the simpler static initial conditions. It can also generate large datasets since the model input may need to be fed to the tsunami model with short time steps. When combined with source filtering, the computational time can increase significantly, but otherwise this method is computationally relatively cheap. In most cases, block models do not take into account the landslide deformation (an

<figure><img src="image_5.png" /><figcaption>**Fig. 4.3** Examples of block sources: Left panel, transect of the front of a smoothed block representing a subaerial block sliding down a slope. Right panel, shape of block used in early estimates of the Storegga slide tsunami</figcaption></figure>
exception is the retrogressive block model by Løvholt et al. 2016, 2017), which can
be important, in particular for submarine landslides.

**Data and Technical Requirements for the Method**

This method uses a grid file with topo-bathymetry as input, and gives time-dependent
fields of the water depth deformation as a function of time in the same reference
system as the tsunami model as output. As the governing equations are simple, they
do not require much computational power or HPC use themselves (although the
source filtering may require significant computational resources).

**Treatment of Uncertainties for the Method**

The main parameters influencing the uncertainty of the tsunami generation are
described in the intro paragraph of Sect. 4.3, and they can be modeled using Monte
Carlo type methods varying the parameters. The landslide motion (e.g. landslide
kinematic or dynamic parameters) cannot so far be measured in the field, and there is
a limited number past events to calibrate the model against. Hence, for testing models
to establish plausible parameter ranges, it may be necessary to rely on the experience
from modeling representative events where the field evidences are comprehensive
(such examples could for instance comprise Lake Askja 2014: Gylfadottir et al. 2017,
Rauter et al. 2022; Tyndall Glacier 2015: George and Iverson 2014; Anak Krakatoa
2018: Grilli et al. 2021), to generate plausible parameter ranges as input to the uncer-
tainty analysis. It should be stressed that such evidence represents global data that
do not necessarily have local source information. Tailoring inputs to benchmark
tests (e.g. Kirby et al. 2022) can further help the calibration process. On the other
hand, most of the community benchmark tests proposed by Kirby et al. (2022) are
composed only of laboratory tests. Forecasting full-scale hazard events then necessi-
tates an assessment of the upscaling from laboratory to field conditions, and landslide
dynamics is not scale invariant (Lucas et al. 2014), hence testing against field events
is imperative (e.g. Rauter et al. 2022).

**4.3.3 Method: Landslide Dynamics Models**

**Short Description of the Method**

Numerical landslide models represent a large family of tools that can be used to model the time-dependent landslide dynamics as source input in tsunami modeling. Due to the large number of such numerical models that are available, we will not describe them in detail, but rather go through the main principles of how they work (some more details are given in the reviews by Yavari-Ramshe and Ataie-Ashtiani 2016; Løvholt et al. 2022). The main difference compared to the abovementioned block models is that these landslide models may take into account the landslide material behavior (considering the rheology) and deformation. They comprise many
different categories, but here we separate only between depth averaged and vertically resolved (generalised) CFD type models:

1. **Depth averaged landslide models (model type 1):** These are based on the shallow water approximation (as also most tsunami models are) typically including non-Newtonian fluid rheologies. The models will in most cases assume that the landslide is completely fluidised after release, although some models also consider the failure process by heuristically taking material remolding into account (e.g. BingClaw, Løvholt et al. 2017; Kim et al. 2019). Fluid resistance can be included by one-way forcing via viscous drag terms and added mass in the model (e.g. Kim et al. 2019). Examples of output and induced tsunami due to the landslide volume displacement is shown in Fig. 4.4. Depth-averaged models can further include hydrostatic (e.g. George and Iverson 2014; Macias et al. 2015) or non-hydrostatic (e.g. Zhang et al. 2021) layers of both the landslide and the ambient fluid including tsunami modeling in coupled layers. It is stressed that due to the depth-averaging and hydrostatic assumptions not all the physics in the wave propagation is necessarily included, this will depend on the model formulation. Some of the listed models may capture dispersion and wave-breaking as well as depth-averaged landslide rheology to different degrees of sophistication depending on the model complexity, but they cannot capture more complex 3D features involving vorticity, such as landslide-water mixing, wave plunging, or crater formations during wave generation.

2. **Vertically resolved Computational Fluid Dynamics (CFD) type models (model type 2):** A family of numerical models that resolve the vertical structure of the flow, includes the ambient fluid in addition to the landslide itself, and to different extent allows for interaction and mixing between the phases. These models capture more of the physics of the tsunami generation for subaerial landslides, and in particular to account for the 3D effects that require coupling with the water column such as impact cratering, vertical jet formations etc. They inherit few simplifications (and are hence often coined primitive models, based on primitive equation sets). On the other hand they will most often need extensive HPC resources and parallelisation in order to solve real world problems, are generally too computationally intensive to be used in PTHA and hence not discussed in further detail here. Examples of primitive models from the literature include e.g. Abadie et al. (2012), Ma et al. (2013), Horillo et al. (2013), Chen et al. (2020), Si et al. (2018), Rauter et al. (2022), and they include complexities related to the representation of the landslide rheology to various degrees. Various numerical implementations exist (e.g. continuum methods, Rauter et al. (2022), or particle methods such as SPH, Panizzo et al. (2007)) Just a few of these models incorporate complex time dependent evolution of the rheology such as of $\mu(I)$ type (see below), e.g. Si et al. (2018) and Rauter et al. (2022).

## Numerical Treatment of the Depth Averaged Models

The depth-averaged types of landslide dynamics models (model type 1) share some similarities with the block models as they both provide time-dependent displacement
<figure><img src="image_6.png" /><figcaption>**Fig. 4.4** Simulated landslide dynamics for the translational part of the 1929 Grand Banks landslide (thickness as a function of time) using the depth-averaged code bingclaw (left panels), and the induced wave field (right panels). Snapshots (from top to bottom) are shown after 5 and 10 min, respectively. Figure modified from Løvholt et al. (2019)</figcaption></figure>

fields as input to the tsunami models, but they are computationally more intensive. However, HPC resources are usually not needed for single model runs. The equations are nonlinear, and Finite Volume schemes or similar are needed for robust treatment of shock-front formation and improved stability. The time resolution for feeding the depth-dependent information frequently enough into the tsunami models needs convergence testing, similar to the application of the block models (above), and discussed by Zengaffinen-Morris et al. (2022).

Model type 1 can accurately model the displacement pattern and the landslide dynamics needed for the tsunami analysis, as they consider both the material behavior and the complex terrain. They are suitable to be combined with PTHA, but when combined with source filtering (described in Chapter 5), the computational time and data need can potentially increase significantly.

## Rheology of the Depth Averaged Models

There are two main classes of landslide rheologies. The largest group consists of the frictional-collisional (granular type) models, where the internal stresses are mainly
described by the collisions of particles, sometimes also incorporating pore-fluid pres-
sure and internal drag forces in addition to the granular-induced stresses. These
models can describe landslides with relatively high concentrations and large parti-
cles, e.g. sands or rock avalanches. The models differ with respect to complexity,
depending on the mathematical model description, e.g. relatively simple Savage-
Hutter or Voellmy models (e.g. VolcFlow Kelfoun and Druitt 2005; VoellmyClaw
Kim2014; Landslide-HySEA Macias et al. 2015), to more advanced Coulomb-type
materials (e.g. Iverson and Denlinger 2001; DClaw, George and Iverson 2014; Pitman
and Le 2005; Pudaisini 2012). Another class is the cohesive (visco-plastic) type of
landslides that primarily represent more clayey material, such as e.g. the two-layer
BingClaw described by a Herschel-Bulkley fluid (Løvholt et al. 2017; Kim et al.
2019).

A development within granular rheology, μ(I) (Jop 2006), show that the granular
stresses can be described as a viscosity parameterized through the inertia number
(labeled I) in dry granular media. This and related rheologies unify the granular and
cohesive material description. More recent developments also included the effects
of pore-fluid into the μ(I) description (Guazzelli and Poliquen 2018). μ(I) rheologies
may be combined consider time-dependent material effects due to contraction and
dilatancy in the granular material and may thus include time-dependent frictional
properties of the material. While μ(I) of shallow-water type has not been used much
for tsunami modeling so far, there are still depth-averaged models available using
this type of rheology (Gray and Edwards 2014), and we expect that they will be used
more frequently in future implementations for tsunami hazard analysis.

Data and Technical Requirements for the Method

The depth-averaged landslide models need the material parameters for the land-
slide, and topo-bathymetric grids, as input. They also need several model-dependent
settings, such as the selection of the numerical scheme, this will depend on the model
(some libraries such as GeoClaw, Leveque 2002, allow for adapting the numerical
scheme as input to the model). The main output to the tsunami model is the time
dependent seafloor displacement. The landslide simulations typically require similar
or less computational resources compared to typical tsunami propagation codes.
They often operate on a much smaller domain than the computational domain for the
tsunami model and need shorter simulation times. On the other hand, they often
require finer grids to resolve the undulations in the slide material and the slide
front. When used as input to tsunami models, the necessary data output may be
considerable.

The input to primitive models is similar to the depth-averaged models, but setting
up the numerical models and gridding the input are generally much more time-
consuming. For feasibility, the codes need to be efficient and parallelized and the
codes must run on large clusters (typically, at least hundreds of CPUs are required).
## Treatment of Uncertainties for the Method

The treatment of uncertainties shares the same challenges described for the block models, and would necessitate the calibration process in the same way. Hence, using past tsunami events to calibrate the uncertainty range is also necessary here. For more details, see Sect. 4.3.2 on block sources.

### 4.3.4 Summary Table

See (Table 4.3).

## 4.4 Volcanoes

Numerical modeling of volcanic tsunamis is still challenging because these sources are often complex and partly not well understood, for reasons that we discussed in Chap. 3.4. There is currently no up-to-date review on the modeling of volcanic tsunamis. In the following section we provide some guidelines for three types of volcanic tsunami sources: pyroclastic flows, caldera collapse, and underwater explosion. The proposed modeling approach will be different from one type of source to another, because of the different physical processes implied, and because the mechanisms of wave initialization are intrinsically different.

### 4.4.1 Methods for Tsunamis from Pyroclastic Flows

#### Short Description

The phenomenon of tsunami generation by a pyroclastic flow is complex, and different from landslide tsunamis for several reasons: (1) Pyroclastic flows are highly-mobile, (2) they are hot (typically 200–800 °C) and might produce steam explosion when entering the water, (3) they are internally structured in a lower dense component and an upper dilute component, (4) and they are often heterometric (from block to ash-size clasts). Watts and Waythomas (2003) demonstrated that the most energetic and coherent water waves are produced by the dense, basal debris flow component of the pyroclastic flow. Preliminary reviews on tsunami generation by pyroclastic flows were presented by deLange (2001), Watts and Waythomas (2003), and Maeno and Imamura (2007), Maeno and Imamura (2011). Direct observations of tsunamis generated by pyroclastic flows are rare, and the dense part of the flow (i.e., the one that is likely to generate a tsunami) is masked by the dilute cloud. Hence, the tsunami-genesis from pyroclastic flows is less well understood than for (dense flow)
<table><caption>**Table 4.3** Summary table: Summary of basis for the different landslide models</caption><thead><tr><th></th><th>Static initial conditions</th><th>Block sources</th><th>Landslide dynamics models</th></tr></thead><tbody><tr><td colspan="4"><strong>Short description of methods</strong></td></tr><tr><td>Theoretical / mechanical background for the method</td><td>Analytical models for landslide shape</td><td>Time dependent motion / kinematics of a box with a fixed shape</td><td>Depth averaged or fully resolved numerical models describing the equations of motion, with or without water layers embedded</td></tr><tr><td>Short outline of how the methods work</td><td>Analytical shape of the wave as an initial condition based on landslide parameter inputs</td><td>Analytical or numerical description of a moving flexible blanket that displaces water as a function of time</td><td>Numerical models solving Partial Differential Equations for the time-dependent dynamics of landslides, with or without taking into account the hydrodynamic resistance or coupling to the fluid layer (air and / or water)</td></tr><tr><td>How is/are the output(s) of the method represented?</td><td>Shape function of initial tsunami surface elevation</td><td>Time-dependent perturbations of the slide thickness as a function of time and space</td><td>Time-dependent perturbations of the slide thickness as a function of time and space. For some models, the coupling to wave dynamics is also incorporated</td></tr><tr><td>In what way do the methods take into account time-dependent/ dynamic aspects?</td><td>Static output, no time dependency in the generation model</td><td>Time-dependent effects of the landslide motion and the wave generation is included. Landslide dynamics models also take into account time dependent slide deformation</td><td>Time-dependent effects of the landslide motion and the wave generation is included. Landslide dynamics models also take into account time dependent slide deformation</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.3 (continued)**</caption><thead><tr><th></th><th>Static initial conditions</th><th>Block sources</th><th>Landslide dynamics models</th></tr></thead><tbody><tr><td>Do the methods use parallelisation, machine learning, surrogate models, advanced simulation schemes, other computational reduction strategies (e.g. cluster analysis, etc.)?</td><td>No</td><td>No</td><td>Parallelisation is most often crucial for the vertically resolved models</td></tr><tr><td>What are the pros and cons and peculiarities of these methods? Are there known shortcomings / benefits of the methods? In what cases are these methods good/poor choices?</td><td>Pros: Simple to use. Robust with respect to instabilities<br>Cons: Severely simplify the generation process, the time-dependent landslide motion and local bathymetry is not taken into account</td><td>Pros: Robust method with respect to noise or instabilities<br>Cons: Does not take into account landslide deformation. Spanning the uncertainty space for multiple parameters can be difficult to convey into PTHA for the time dependent models. Can lead to large data outputs when coupled to tsunami generation models</td><td>Pros: More accurate, they take into account complex landslide deformation, rheology, and bathymetry effects<br>Cons: Stability can be an issue. Need for sophisticated filtering. Spanning the uncertainty space for multiple parameters can be difficult to convey into PTHA for the time-dependent models. Can lead to large data outputs when coupled to tsunami generation models. Vertically resolved models are too resource intensive to be used in PTHA</td></tr><tr><td>Data and technical requirements for method</td><td></td><td></td><td></td></tr><tr><td>What input data does the method(s) use?</td><td>Water depth, slope, density, friction, hydrodynamic resistance parameters (pressure drag, skin friction, added mass). Initial volume parameters (length, width, thickness)</td><td>Topo-bathymetric data (water depth, slope), rheological information (e.g. density, friction angle, yield strengths, etc.), and hydrodynamic resistance parameters (pressure drag, skin friction, added mass). Initial volume parameters (length, width, thickness). Sometimes also prescribed motion</td><td>Topo-bathymetric data (water depth, slope), rheological information (e.g. density, friction angle, yield strengths, etc.), and hydrodynamic resistance parameters (pressure drag, skin friction, added mass). Initial volume parameters (length, width, thickness).</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.3** (continued)</caption><thead><tr><th></th><th>Static initial conditions</th><th>Block sources</th><th>Landslide dynamics models</th></tr></thead><tbody><tr><td>What are the technical requirements?</td><td>None</td><td>Topo-bathymetric data</td><td>Topo-bathymetric data</td></tr><tr><td>Are HPC computing resources necessary, would absence of these compromise the use?</td><td>No</td><td>No</td><td>HPC is obligatory only for the vertically resolved models, but on the other hand, they are too computationally intensive for use in PTHA / PTRA</td></tr><tr><td>Is the method(s)/workflow(s) available as a git/software/jupyter notebook?</td><td>No</td><td></td><td></td></tr><tr><td colspan="4">Treatment of uncertainties for the method</td></tr><tr><td>Are the method(s) itself deterministic or uncertainty based?</td><td>Deterministic</td><td>Deterministic</td><td>Deterministic</td></tr><tr><td>Which are potential sources of uncertainties? Which parameters are used to characterize them?</td><td>Potential bias due to model simplifications. Scale-effects if empirical data from laboratory experiments are used as the basis for the model</td><td>Lack of material parameters, or knowledge of the landslide material / frictional behavior and scaling. Model parameters are often set based on the experience of retrofitting past events as several parameters from real landslides in the field cannot be measured. In many settings a relatively large epistemic uncertainty related to this is expected</td><td>Lack of material parameters, or knowledge of the landslide material / frictional behavior and scaling. Model parameters are often set based on the experience of retrofitting past events as several parameters from real landslides in the field cannot be measured. In many settings a relatively large epistemic uncertainty related to this is expected</td></tr><tr><td>How are uncertainties quantified for the identified parameters? Are there significant correlations between different uncertain parameters?</td><td colspan="3">For certain parameters, such as the hydrodynamic resistance factors, there might be correlations between them (e.g. for coefficients for pressure drag, skin friction, added mass)</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.3 (continued)**</caption><thead><tr><th></th><th>Static initial conditions</th><th>Block sources</th><th>Landslide dynamics models</th></tr></thead><tbody><tr><td>Are the method(s) able to digest and propagate uncertainty in input (previous ingredient) and propagate it to the output (next ingredients)?<br>Does the assessment of this ingredient require uncertainty propagation? If yes, how are uncertainties propagated?</td><td>No, only through parameter variation in the probabilistic framework, i.e. representing different model realizations (and parameters) with different probabilities</td><td></td><td></td></tr><tr><td>What are the consequences of lack of data in input or for boundary conditions, or for characterizing certain parameters?</td><td></td><td>There is rarely local data available describing the parameters. The lack of data implies that epistemic uncertainties should be modeled through parameter variation. With lack of data, use of benchmark tests and calibration towards other events may be necessary to get a grip of the uncertainty, even if they do not represent local conditions</td><td></td></tr><tr><td>What are the consequences of ignoring or poorly treating uncertainties?</td><td></td><td>Model bias. Vice versa, such models would often necessitate broad parameter variations to cover epistemic uncertainties</td><td></td></tr></tbody></table>
landslides, and an understanding of the respective roles of the different parameters implied in the wave generation is yet to be firmly established.

The important parameters controlling the interactions between pyroclastic flows and water bodies are the bulk density of the flow and its preservation offshore, the mass flux, the momentum angle, and the angle of incidence (Cas and Wright 1991; Carey et al. 2000; Paris et al. 2014). Experimental granular flows with a bulk density similar to water (e.g., pumice-rich flows) generate waves, regardless of their temperature (Freundt 2003). Experiments on fluidised granular flows by Bougouin et al. (2020) demonstrated that the wave features are mostly controlled by mass flux and the volume of the flow, unlike the water depth (Fig. 4.5). This is confirmed by the simulations presented in Fig. 4.6, where a 10⁶ m³/s flow generates waves considerably larger than a 10⁵ m³/s flow. The grain size may also affect the wave amplitude, as it controls the ability of water to penetrate the granular material (Bougouin et al. 2020). Interestingly, experimental fine-grained granular flows (e.g., ash-rich flows) behave as single-phase water flows in terms of wave generation.

<figure><img src="image_7.png" /><figcaption>**Fig. 4.5** Snapshots of an experimental fluidized granular flow (i.e. pyroclastic flow) entering water at different times after gate opening (t = 0; modified from Bougouin et al. 2020). ©2020. American geophysical union</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 4.6** Simulated water surface elevations in Santorini for different types of tsunamis generated by an eruption at Kolumbo volcano, Aegean Sea. **a** pyroclastic flow (plume collapse) with a volume $V = 0.1 \text{ km}^3$ and a discharge rate $q = [10^5: 10^6] \text{ m}^3/\text{s}$ entering the sea; **b** caldera collapse with different geometries (model 1 is a full-volume single-block collapse, model 2 is a half-volume single-block collapse, and model 3 corresponds to the deepening of a pre-existing caldera); **c** underwater explosions with different energies and initial water surface elevations $\eta_0 = [50: 250] \text{ m}$ (modified from Ulvrova et al. 2016)</figcaption></figure>

## Data and Technical Requirement

For such a complex tsunami source, well-controlled laboratory experiments are necessary to validate the numerical simulations. Facing the diversity and complexity of the physical processes acting during the impact of a pyroclastic flow with a water body, there is no single type of model that could be recommended. Different numerical models can be developed depending on the scale of interest and objectives of the simulations. Coupled CFD-DEM models (Computational Fluid Dynamics - Discrete Element Method) can simulate the discrete grain motion and its interaction with water, but their high computational cost restricts their use to very small scale (and not to natural cases). Multi-fluid CFD models based on the Navier-Stokes equations are probably the best option to simulate the generation of a tsunami by a pyroclastic flow. Laboratory experiments can also provide empirical correlations or scaling laws. As an example, Bougouin et al. (2020) proposed a dimensionless number that predicts
the initial wave amplitude from experimental data. As stressed in Sect. 4.6, CFD type analysis tools are well suited for characterizing the phenomenon, but cannot presently be used in PTHA due their computational intensity.

## Treatment of Uncertainty

The level of epistemic uncertainty remains high until we better understand the process of tsunami generation by pyroclastic flows. The main uncertainties are both related to the physics of the phenomenon itself, and to the parameterization of the flow in the numerical models. In terms of understanding the physics, open questions are: (1) What are the consequences of the impact with water on the characteristics of the flow and most particularly on flow segregation (dense vs. dilute components)? (2) How to address the role of temperature (200–800 °C before the impact with water)? Ongoing experimental studies will help better understand these complex flow-water interactions, and define a reasonable level of approximation in the numerical models, which in any case will struggle to reproduce the complexity of such a phenomenon.

## 4.4.2 Methods for Tsunamis from Caldera Collapse

### Short Description

The diameter of a caldera typically ranges from hundreds of meters to several kilo-meters. The duration of a caldera collapse is poorly constrained (from minutes to hours, keeping in mind that large collapses typically last hours) and the geometry varies from a single block to multi-stage subsidence of sub-blocks (Fig. 4.6). The collapse first generates a subsidence of the water surface, which is followed by the formation of a dome of water (e.g., Nomanbhoy and Satake 1995; Maeno et al. 2006; Ulvrova et al. 2016). Consequently, the wave train is characterized by a leading trough, followed by much larger positive crests. The initial subsidence of the water surface depends on the geometry (diameter, depth, single collapse or multi-blocks collapse) and duration of the collapse. Using a piston-like plunger model, Maeno and Imamura (2011) found that the water elevation above the caldera was the largest for a dimensionless collapse speed $VC^*$ of 0.01 ($VC^* = VC/\sqrt{gh}$). With a similar approach, Ulvrova et al. (2016) tested different collapse geometries and durations (Fig. 4.7 - right), and they found that only fast (and probably unrealistic) caldera collapses (<5 min) are efficient in terms of tsunami generation. Real-case caldera collapses usually last from ~ 30 min (e.g., the Pinatubo 1991 event, Philippines) to 12 days (e.g., the Fernandina 1968 Galapagos event).

### Data and Technical Requirement

As explained above, a caldera collapse is not an instantaneous process. Consequently, it is required to adopt a finite time initialization for the tsunami generation. The motion of the sea floor (i.e. the caldera collapse) is used as a forcing term. There is no specific technical requirement to treat this kind of initialization, except that the grid should
<figure><img src="image_9.png" /><figcaption>**Fig. 4.7** Example of a simulated tsunami generated by a caldera collapse at kolumbo volcano, Aegean Sea. (Left) map of the initial water subsidence following the caldera collapse, the colorbar depict the simulated tsunami water levels. The red line shows the distance from the volcano to Santorini Island where the maximum amplitudes are recorded; (Right) The graph represents the maximum simulated tsunami wave amplitudes extracted in different sites of the eastern and southern coasts of Santorini versus duration of the caldera collapse (modified from Ulvrova et al. 2016). © 2016 Elsevier B.V</figcaption></figure>

have a resolution high enough to accurately reproduce the contours of the caldera. For the tsunami propagation, a dispersive model might be necessary, depending on the water depth and diameter of the collapse area, and the duration of the sea floor motion.

### Treatment of Uncertainty

The main source of uncertainty is the duration of the caldera collapse, although it is also difficult to predict the geometry of the collapse. There are few well-documented examples of submarine caldera collapse. It is thus recommended to define a range of different collapse volumes to investigate their impact. Examples of caldera collapses (mostly subaerial), their volumes and kinematics, can be found in the literature. The duration of the collapse should exceed 5 min, and different values of duration should be tested until a threshold is reached; threshold for which the collapse does not generate waves higher than a given value (determined by the operator).

### 4.4.3 Methods for Tsunami from Underwater Explosions

#### Short Description

Compared to other sources of underwater explosions (e.g. industrial, military explosions), the dynamics of phreatomagmatic eruptions is complex. The physics of magma-water interactions is controlled by many parameters: water depth, geometry of the vent and magma-water interface, transfer of thermal energy, processes of intermingling and mixing between magma and water, metastability of superheated
water, and quantity of gas in the ascending magma (Kokelaar 1986; Wohletz 1986). Field observations of underwater eruptions and laboratory experiments show two different types of fountains at the water surface, dome-regime and finger-regime fountains, depending on explosion intensity and water depth (Shen et al. 2021a). There is a critical water depth at which an explosion with a given intensity generates the largest waves (Shen et al. 2021b). Underwater explosions in the ocean typically generate waves of shorter period and greater dispersion compared to earthquakes, thus reducing their far-field impact (e.g., the Myojin-Sho event 1952). On the contrary, violent explosions in shallow waters, and more particularly in lakes, have the potential to produce high-runup local tsunamis (e.g., the Karymskoye Lake event 1996).

Tsunami generated by underwater explosions typically consists of two main positive waves followed by smaller undulations propagating radially from the source, as demonstrated by experiments (Le Méhauté and Wang 1996; Kedrinskii 2005) and numerical simulations (Torsvik et al. 2010; Ulvrova et al. 2016; Paris and Ulvrova 2019). The water is initially pushed upward, forming a crater with a cylindrical bore that expands radially to form the leading wave, followed by a wave trough. Initial surface displacement (i.e., maximum height of the bore) can be estimated directly as a function of explosion energy at a given water depth (Le Méhauté 1971, Fig. 4.8 - right panel). The water crater collapses as in a dam break model, generating a steep dome of water that turns to a second cylindrical bore (Le Méhauté and Wang 1996).

For simulating the tsunami source, we recommend a semi-analytical approach that neglects complex interactions between dispersed fragments of lava, gas bubbles and water (which would require extremely complex multi-phase models). The underwater explosion is approximated by imposing a specific initial water surface displacement ($\eta_0$) that depends on the explosion depth and energy released during the explosion (e.g., Torsvik et al. 2010; Ulvrova et al. 2014).

<figure><img src="image_10.png" /><figcaption>**Fig. 4.8** (Left) Profile of initial water surface displacement for an underwater explosion. (Right) relationship between the explosion energy and the initial water surface displacement (calculated from the empirical equation of LeMéhauté 1971). Two cases of explosions should be distinguished: deep explosions with a high-scaled depth of explosion (compared to a water depth of 200 m in this case), and shallow explosions with a low-scaled depth (LeMéhauté and Wang 1996)</figcaption></figure>
**Data and Technical Requirement**

The source of the tsunami is considered as instantaneous here (i.e. instantaneous formation of a crater in the water, as an initial water surface deformation). Therefore, the modeling of tsunami generation is an independent step and does not require HPC. In order to have a homogenous circular source at the apron of the explosion, it is recommended to use fine grid resolution (typically < 50 m) or to refine the available grid. A poor resolution of the source (few grid points) might generate artificial directivity of propagation (Paris and Ulvrova 2019). Underwater explosions typically generate short-period waves.

**Treatment of Uncertainty**

In the case of a tsunami generated by a volcanic subaqueous explosion, the source of uncertainty comes from the estimation of the size of the explosion (i.e., its energy and diameter), or the definition of a range of possible explosions with a given energy and probability (Fig. 4.6). The explosion energy can be estimated indirectly using the size of the volcanic craters formed by previous explosions (Sato and Taniguishi 1997; Goto et al. 2001), assuming that the diameter of the crater corresponds to the size of the largest explosion. In fact, the morphology of the crater often results from multiple vertically and horizontally-migrating explosions, and post-eruptive erosional processes.

**4.4.4 Summary Table**

See (Table 4.4).

**4.5 Atmospheric Sources**

To this date, modeling of atmospheric sources has mostly been implemented for either the hindcast or the forecast of weather related events. However, the methods developed within this context, and previously introduced in Sect. 3.5, form the basis needed for Atmospheric Probabilistic Tsunami Hazard Analysis (APTHA).

Section 4.5 deals with three main approaches to the atmospheric sources modeling mostly developed for weather related events: “Synoptic Index” which allows to select meteotsunamigenic conditions from global atmospheric models at resolution of the order of 100 km, “Deterministic Atmospheric Model” which is used to downscale the global atmospheric fields (about 100 km resolution) to the kilometer-scale needed to represent the mesoscale conditions driving the meteotsunami events and “Synthetic Catalog” that can be used to account for the uncertainty associated with the deterministic models. Finally, the last method, “Climate Uncertainty”, presents how
<table><caption>**Table 4.4** Summary table: Volcanic sources: pyroclastic flows, caldera collapses and underwater explosions</caption><thead><tr><th></th><th>Pyroclastic flows</th><th>Caldera collapses</th><th>Underwater explosions</th></tr></thead><tbody><tr><td colspan="4"><strong>Short description of method</strong></td></tr><tr><td>Theoretical / mechanical background for the method</td><td>CFD model (full Navier-Stokes)</td><td>Finite-time sea floor motion to simulate the collapse of a submarine caldera</td><td>Instantaneous formation of a water crater as initial water surface deformation</td></tr><tr><td>Short outline of how the method(s) works and references to existing case studies implementing this method</td><td colspan="3">The described methods were applied to several case studies mentioned in the main text</td></tr><tr><td>How is/are the output(s) of the method represented?</td><td>Full or layered velocity field and displacement of the water surface as a function of time</td><td>Velocity field and displacement of the water surface as a function of time</td><td>Velocity field and displacement of the water surface as a function of time</td></tr><tr><td>In what way does the method take into account time-dependent/dynamic aspects?</td><td>Time-dependent motion of the coupled flow, water body, and air</td><td>Time-dependent motion of the caldera floor</td><td>Static (instantaneous initialization)</td></tr><tr><td>Does the method use parallelisation, machine learning, surrogate models, advanced simulation schemes, other computational reduction strategies (e.g. cluster analysis, etc.)?</td><td>Parallelisation and access to cluster resources is recommended but not mandatory</td><td>No</td><td>No</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.4 (continued)**</caption><tbody><tr><td>What are the pros and cons and peculiarities of this specific method? Are there known shortcomings / benefits of the method? In what cases is this method a good/poor choice?</td><td>Pyroclastic flows<br>Pros: few approximations, reproduce interactions between the different fluids, reliable if benchmarked by laboratory experiments<br>Cons: high computational cost, difficult to set up PTHA. It is recommended to limit the use of the Full Navier-Stokes model to the near-field (tsunami initiation/ generation)</td><td>Caldera collapses<br>Pros: low computational cost<br>Cons: other processes involved in a caldera collapse are not taken into account (e.g. rise and collapse of the eruptive column during the on-going caldera collapse itself)</td><td>Underwater explosions<br>Pros: low computational cost<br>Cons: other processes involved in a volcanic subaqueous explosion are not taken into account (e.g. upward flux of fragmented magma), the geometry of the initial water crater is simplified (e.g. there exist different fountain regimes)</td></tr><tr><td colspan="4">Data and technical requirements for method</td></tr><tr><td>What input data does the method use?</td><td>Bathymetry and coastal topography data, flow parameters</td><td>Bathymetry and coastal topography data, collapse geometry and timing</td><td>Bathymetry and coastal topography data, explosion parameters</td></tr><tr><td>What are the technical requirements?</td><td>CFD model (full Navier-Stokes) for tsunami generation, and Depth-averaged weakly dispersive model for tsunami propagation</td><td>Depth-averaged weakly dispersive model</td><td>Depth-averaged weakly dispersive model</td></tr><tr><td>Are HPC computing resources necessary, would absence of these compromise the use?</td><td>Recommended but not mandatory</td><td>No</td><td>No</td></tr><tr><td>Is the method/workflow available as a git/software/jupyter notebook?</td><td>No</td><td>No</td><td>No</td></tr><tr><td colspan="4">Treatment of uncertainties for the method</td></tr><tr><td>Is the method itself deterministic or uncertainty based?</td><td>Deterministic</td><td>Deterministic</td><td>Deterministic</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.4 (continued)**</caption><thead><tr><th></th><th>Pyroclastic flows</th><th>Caldera collapses</th><th>Underwater explosions</th></tr></thead><tbody><tr><td>Which are potential sources of uncertainties? Which parameters are used to characterize them?</td><td>Uncertainties on flow parameters (density, mass flux, structure, segregation dense-dilute, temperature, etc.) and its interactions with air and water</td><td>Uncertainties on the geometry (single block vs multiple blocks) and timing (duration, incremental vs massive) of the caldera collapse</td><td>Uncertainties on the explosion energy (it is recommended to test different values of energy, taken from the literature), and the explosion depth (keeping in mind that there is a critical water depth at which an eruption with a given source intensity will generate the largest waves)</td></tr><tr><td>How are uncertainties quantified for the identified parameters? Are there significant correlations between different uncertain parameters?</td><td colspan="3">Uncertainties on the physical processes are not quantified</td></tr><tr><td>Is the method able to digest and propagate uncertainty in input (previous ingredient) and propagate it to the output (next ingredients)? Does the assessment of this ingredient require uncertainty propagation?</td><td>No</td><td>No</td><td>No</td></tr><tr><td>What are the consequences of lack of data in input or for boundary conditions, or for characterizing certain parameters?</td><td>Uncertainty may result in a low reliability of the results. This can be compensated by an intermediate stage of numerical simulation of laboratory experiments, which helps estimating the influence of parameters</td><td>Uncertainty may result in a low reliability of the results. An overestimation of the level of hazard if a short (&lt; 5 min) and unrealistic collapse duration is selected</td><td>Uncertainty may result in a low reliability of the results. The reliability of results decrease if the parameters are arbitrarily defined (e.g. overestimation of the possible range of energies)</td></tr><tr><td>What are the consequences of ignoring or poorly treating uncertainties?</td><td colspan="3">Poor reliability of the results if the source parameters are not well-defined</td></tr></tbody></table>
these three approaches can be combined within an atmospheric source modeling
framework needed for APTHA.

**4.5.1 Method: Synoptic Index**

**Short Description**

For weather related events, the derivation of meteotsunami synoptic indices is based
on the identification of atmospheric conditions favorable to meteotsunami events
using both synoptic conditions (e.g., vertical profiles and spatial distributions of
wind, temperature, etc.) extracted from numerical models, or echo sounding data,
and long-term records of Non-Seismic Sea-Level Oscillations at Tsunami Timescales
(NSLOTT) at coastal stations. In practice, the synoptic index approach is imple-
mented at a given coastal station depending only on a combination of several wisely
chosen atmospheric variables extracted from numerical models. The main advantage
of the method is its low computational cost. However, its main drawbacks are that
(1) it produces a large number of false positive events (detected events that are not
leading to meteotsunamis) as the local and mesoscale effects are ignored and (2)
it cannot be derived for acoustically driven events. Consequently, the primary use
of the synoptic index method for hazard assessments is to extract potential weather
related meteotsunami events from existing ensembles of global climate models (e.g.,
Coupled Model Intercomparison Project Phase 6 or CMIP6, Eyring et al. 2016). At
this time, two different approaches have been successfully implemented: (1) Linear
Combination (LC) and (2) Neural Network (NN).

**Linear Combination (LC):** In this framework, the synoptic index ($\mu_n$), as defined in Zeminik et al. (2022) for stations worldwide (Fig. 4.9), is given by: $\mu_n(t) = \sum_{i=1}^{n} a_i x_i(t)$, with *n* the number of synoptic variables, $x_i$ the synoptic variables, $a_i$ the coefficient obtained by LC, as described below, and *t* the time. The derivation of the synoptic index is as follows:

1. *Derivation of the NSLOTT ranges*—hereafter **h(t)**. Resampling NSLOTT to the temporal sampling of the synoptic variables;

2. *Choice of the first synoptic variable.* Solving the equation $h(t) = \mu_1(t)$ with least square method (Barrett et al. 1994) to obtain $a_1$ for $x_1(t)$ with the highest correlation with $h(t)$;

3. *Choice of the second synoptic variable.* Finding $x_2(t)$ with the highest cost function for the remaining $n-1$ synoptic variables and calculating $a_2$;

4. *Choice of third synoptic variable.* If synoptic index not optimal, finding the highest cost function for the remaining $n-2$ variables and calculating $a_3$;

5. *Next synoptic variables.* Carrying out iteratively the procedure for the remaining $n-3$ synoptic variables till the optimal synoptic index is derived.

The LC approach can easily be implemented without parallelization using soft-
wares like Python, Matlab or R. However, depending on the temporal and spatial
<figure><img src="image_11.png" /><figcaption>**4.9** Derivation of the meteotsunami synoptic index using a linear combination (LC) approach.
Adapted from Zemunik et al. (2022), licensed under CC-BY 4.0. See text for explanations of
variables</figcaption></figure>

resolution of the reanalysis products used to extract the synoptic variables, it might
require substantial storage.

**Neural Network (NN):** In this framework, the synoptic index is defined following the procedure presented in Vich and Romero (2021). A Neural Network is a computational system capable of reproducing reality without a rulebook (Bishop et al. 1995; Haykin 2004). It is formed by at least three layers: (1) input, (2) output and (3) hidden. For the derivation of the meteotsunami synoptic index, the input consists of atmospheric variables on several vertical layers while the output layer is the daily maximum NSLOTT range. In order to create and test the NN for the synoptic index, two associated datasets are needed: (1) observed daily maximum NSLOTT ranges in the harbor where meteotsunamis occur and (2) observations (or reanalysis product) of the atmospheric vertical structure in the vicinity of this harbor. The maximum daily
NSLOTT range dataset (and the associated atmospheric dataset) is then randomly split into two sets: a training database (70% of the cases) and a testing database (the remaining 30%). The best NN configuration is based on three criteria: (1) limited spread (i.e., NN prediction as close as possible to the NSLOTT range observations), (2) symmetry (i.e., as few over-predictions as under-predictions) and (3) low degree of overfitting (i.e., discarding configurations that fit the training subset too closely). The NN approach can easily be implemented using softwares like Python, Matlab or R including artificial intelligence packages.

## Data and Technical Requirements

NSLOTT data can be extracted from tide gauge observations provided with at least a 5-min temporal resolution. The tide gauge observations are filtered with a 2-h Kaiser-Bessel filter to produce high-frequency NSLOTT signal which has then to be quality-checked. To this date, the MInute SEa-Level Analysis (MISELA) dataset (Zemunik et al. 2021a, 2021b) is the only publicly available quality-checked NSLOTT world-wide dataset. It includes 331 tide gauges mostly extracted from the Intergovernmental Oceanographic Commission Sea Level Station Monitoring Facility (IOC SLSMF, http://www.ioc-sealevelmonitoring.org).

Synoptic variables can be extracted from global or regional reanalysis products or observational data. However, the forecast of weather related events with synoptic indices requires extracting the synoptic variables from global or regional weather forecasts (e.g., produced by the European Centre for Medium-Range Weather Fore-casts - ECMWF or by the National Centers for Environmental Prediction-NCEP). The extraction of the synoptic variables does not require the use of High Performance Computing (HPC) facilities in operational mode.

The LC approach described in Zemunik et al. (2022) derives synoptic indices for 307 worldwide stations of the MISELA dataset. The seven synoptic variables are extracted from the hourly ECMWF/ERA5 reanalysis dataset (Hersbach et al. 2020) and consist in: (1) vertical profiles of wind speed, temperature and relative humidity from the surface up to a pressure level of 400 hPa at the ERA5 grid point closest to the corresponding MISELA station, and (2) horizontal fields of mean sea-level pressure, temperature at 850 hPa and geopotential at 550 hPa in a rectangular area of [-10°, 10°]x[-10°, 10°] in longitude and latitude centered over the point closest to the MISELA station.

The NN approach described in Vich and Romero (2021) derives a synoptic index for meteotsunamis occurring in Ciutadella (Balearic Islands, Spain). The NN is trained with two datasets obtained between July 1981 and July 2018: (1) a maximum daily NSLOTT range dataset available in Ciutadella and (2) an associated echo sounding dataset (i.e., temperature, zonal and meridional wind components as well as dew point interpolated on 21 fixed vertical levels) measured at 00 and 12 UTC in Palma station (100 km west of Ciutadella). The NN is created and tested with 126 meteotsunami days (i.e., NSLOTT ranges above 70 cm) and 549 days without meteotsunamis extracted from these datasets. In its current setting, the forecast of the
meteotsunami events in Ciutadella is envisioned to be derived, at nearly no numerical cost, with the operational outputs of the NCEP/Global Forecasting System (GFS; Wang et al. 2019).

## Treatment of Uncertainties

The synoptic index method, whether obtained with a LC or a NN approach, is by nature deterministic. However, for hazard assessments, it can be used with an ensemble of global or regional climate models and can thus produce a time-varying distribution of potential meteotsunami events at a given coastal location. Switching from a deterministic to a probabilistic hazard assessment at the climate scale is crucial as, even at the synoptic scale, the climate models are different due to their resolutions, their numerical schemes, etc. Consequently, using a single climate model increases the probability to miss potential meteotsunami events and, hence, decreases the reliability of the hazard assessments.

## 4.5.2 Method: Deterministic Atmospheric Models

### Short Description

**Weather related events:** Deterministic atmospheric models allow the downscaling of global models to resolve the mesoscale conditions during meteotsunami events at hotspot locations. The equations of the atmospheric models used in state-of-the-art weather/climate models are usually based on the fully compressible Euler nonhydrostatic equations, the scalar-conserving flux form for prognostic variables and the complete Coriolis and curvature terms. They often include the possibility to use multiple nests and to set up mass-based terrain following coordinates with a vertical grid-spacing varying with height. The set of equations used in atmospheric models are: (1) the mass continuity equation depending on the wind velocity and the density; (2) the momentum equations depending on the wind velocity, the potential temperature, the water vapor, the atmospheric pressure and the forcing terms arising from model physics, turbulent mixing and the Earth's rotation; (3) the ideal gas law and (4) the moisture equation depending on the transport of the water vapor. The atmospheric model physics can be represented by many numerical schemes and parametrizations and include (1) microphysics, (2) cumulus parameterizations (for coarser grids), (3) surface physics, (4) planetary boundary layer physics and (5) atmospheric radiation physics.

In the framework of meteotsunami hazard assessments, atmospheric models should be able to capture meteotsunamigenic processes such as mesoscale gravity waves (MGWs), convective clouds, squall lines, etc. This requires the use of (1) telescopic nested grids downscaling the global climate models to high horizontal spatial resolution – at least 2 km (Horvath and Vilibić 2014), as MGWs typically range between 10 and 1000 km (Plougonven and Zhang 2014), (2) terrain following coordinates with an increased resolution in the first 2 km above the Earth surface, and (3)
substantially reduced model time-steps (below 1-min), in order to properly represent the generation and the propagation of the tsunamigenic processes (characterized by atmospheric variations of the order of a few minutes) and to preserve the numerical stability of the model. In terms of numerical resources, kilometer-scale atmospheric forecasts require fully parallelized models that can run in a timely manner to produce 3 to 5 day-long simulations.

**Acoustically driven events:** In the barotropic framework (Kasahara 1977), a horizon-
tally propagating surface gravity wave has a phase speed $c = \sqrt{gD}$, where $g$ is the
gravity and $D$ is the mean fluid depth. This approach extends into the baroclinic atmo-
sphere, where $D$ becomes the equivalent depth (Cohn and Dee 1989). Consequently
Lamb waves can be simulated following the rotating shallow-water equations on the
sphere (Vasylkevych and Žagar 2021). To properly represent the deflection of Lamb
waves by major mountain ranges, such models should include a realistic orography
on a global computational grid. For example, the T170 grid (about 70 km horizontal
resolution) which corresponds to the regular Gaussian grid of 680 and 320 grid points
in the longitudinal and latitudinal directions has proven to have an adequate reso-
lution (Denamiel et al. 2023a). Further, acoustic internal gravity waves can also be
generated by volcanic explosions or asteroid impacts but would require a different
modeling framework (e.g., baroclinic high-resolution models). This framework has
not been described in detail, to this date. It remains to be properly developed and
implemented for practical applications to acoustically driven events. Due to their
relatively low resolution and their barotropic nature, global Lamb wave models are
far less demanding in terms of numerical resources and storage and much faster to
run than the local kilometer-scale models used for weather related events.

**Data and Technical Requirements**

**Weather related events:** In the meteotsunami community, the Weather Research and Forecasting model (WRF; Skamarock et al. 2005) is the atmospheric model of choice for the forecast of weather related events. For example, it is used in the Mediterranean within both Balearic Islands (Mourre et al. 2021) and Adriatic Sea (Denamiel et al. 2019b) early warning systems (see Fig. 4.10) while it is also tested in Korea (Kim et al. 2022). Additionally, the atmospheric model needs (1) pre-defined nested grids adapted to the region of interest, (2) initial conditions and lateral boundary conditions provided within a relaxation zone, (3) top boundary conditions at constant pressure level, and (4) bottom boundary conditions like sea-surface temperature, soil moisture and temperature, etc. In climate studies, these boundary conditions are extracted from the three dimensional fields of global or regional climate models. Finally, the use of kilometer-scale atmospheric models requires access to High Performance Computing (HPC) facilities. For example, in the Adriatic Sea, the WRF model at 3-km resolution runs on 64 CPUs to produce 5 day-long simulations on the ECMWF ATOS HPC (https://www.ecmwf.int/en/about/media-centre/news/2022/taking-ecmwfs-new-high-performance-computing-facility-operation; see Adriatic APTH Recipe, R10).
<figure><img src="image_12.png" /><figcaption>**Fig. 4.10** Existing meteotsunami modeling systems in the Mediterranean Sea: BRIFS (in red) in the Balearic Islands and AdriSC (in black) in the Adriatic Sea. Adapted from Vilibić et al. (2021)</figcaption></figure>

**Acoustically driven events:** Modeling of acoustic-gravity waves following for instance volcanic explosions is still at its very beginning. However, after the Hunga Tonga-Hunga Ha'apai volcano eruption on 15 January 2022, several models were used to reproduce the generated Lamb wave (e.g., Amores et al. 2022; Themens et al. 2022). For example, a few days after the event, the Lamb wave propagation was reproduced with the Transient Inertia-Gravity And Rossby wave dynamics model (TIGAR; Vasylkevych and Žagar 2021) solving the presented primitive equations with realistic orography using terrain-following coordinates and the Hough harmonics as the basis functions. A large amplitude Gaussian geopotential height perturbation centered on the volcano location with an amplitude of $\Delta p = 35.1$ hPa and a scale parameter of $\sigma = 2.7^\circ$ was specified as the initial condition of the simulation. The parameters of the initial condition were optimized to match the observed wavelength of the wave response, while the dissipation time scale $\tau = 3.4$ days was obtained by a best fit of the wave amplitude to the observed pressure fluctuations at evaluation sites (Themens et al. 2022). It should be noted that, to this date, none of the Lamb wave models have been used operationally which would require some significant developments in terms of coupling with the hydrodynamic models and quantification of the uncertainty related to the initial condition parameters.

## Treatment of Uncertainties

Deterministic modeling of meteotsunami events often fails due to the uncertainty linked to the modeling of the atmospheric conditions. Consequently, to derive meteotsunami hazards, using “Ensemble Modeling” (for weather related events only) or “Uncertainty Quantification (UQ) techniques” (presented under the “Synthetic Catalog” method) to account for the atmospheric condition uncertainties is necessary.
In the Balearic Islands, following Mourre et al. (2021), the “Ensemble Modeling” approach, using more than one atmospheric model configuration, improves the forecast of meteotsunami events. Mourre et al. (2021) thus implemented an ensemble of nine simulations varying the schemes and parametrizations within the WRF model. The ensemble was used to reproduce the nine largest past meteotsunamis in the Balearic Islands between 2014 and 2019. All simulations were found to realistically represent the meteotsunamigenic disturbances but their mesoscale characteristics (e.g., location, speed) were significantly modified with the change of parameterization. It should also be noted that no set of parameterizations overall led to a better forecast of the weather related events. In fact, by using this ensemble of nine simulations, a large range of meteotsunami surges – including the observed surges in 70% of the cases – was obtained. Consequently, in the probabilistic sense, ensemble modeling was found to improve the accuracy of meteotsunami predictions compared to single deterministic forecasts. However, ensembles of kilometer-scale atmospheric models have enormous computational cost. For meteotsunami hazard assessments, the “Ensemble Modelling” approach can be used to downscale meteotsunami events from an ensemble of global or regional climate models in order to account for the climate uncertainty.

### 4.5.3 Method: Synthetic Catalog

#### Short Description

Meteotsunamis are rare events and reliable observations of such events are consequently scarce. This particularly applies to air pressure (or wind) high-resolution measurements as 1-min records are not the standard in meteorological observations. Consequently, synthetic catalogs should be used to produce any kind of statistics, synthetic scenarios as input to hazard analysis, or sensitivity studies. Hereafter, the following steps of the synthetic catalog methodology are presented for meteotsunamis driven by pressure disturbances:

1. **Definition of the pressure field as a stochastic process ($P_s$).** The pressure field should first be defined as a stochastic process $P_s(x, y, t, \omega) = P_0 + P_A(x, y, t, \omega)$ with the horizontal coordinates following longitude and latitude $(x, y)$, the time coordinate $t$, the stochastic variable $\omega$, the mean atmospheric pressure $P_0$ and the stochastic process representing the analytical formulation of the atmospheric disturbance $P_A(x, y, t, \omega)$.

2. **Definition of the idealized atmospheric disturbance as a function of stochastic variables.** Several analytical formulations can be used for the different types of atmospheric disturbances driving meteotsunami events worldwide. Generally the analytical formulation of the pressure disturbance includes several stochastic variables such as the direction of propagation $\theta(\omega)$, the amplitude $P_A(\omega)$, the period $T(\omega)$, and the propagation speed $c(\omega)$. The number of
stochastic variables is highly variable depending on the choice of the analyt-
ical formulation and can be further reduced when a parameter is proven to have
negligible impact on the final results (Denamiel et al. 2018, 2020).

3. **Definition of the stochastic variable distributions.** The stochastic variables used to define the meteotsunamigenic disturbance should be mutually independent and associated with a specific distribution. These distributions can be based on pressure measurements recorded during meteotsunami events but, if these measurements are too scarce, uniform distributions can be used for all variables. Both uniform distribution and mutual independence assumptions follow the principle of maximum entropy. This principle states that when little is known about a distribution, then the distribution with the largest entropy should be chosen as the least-informative default since it involves the fewest assumptions (Jaynes 1957).

4. **Sampling of the stochastic variable distributions.** Different strategies can be adopted concerning the sampling of the distributions (e.g., Monte Carlo, Markov Chain Monte Carlo, sparse grids, adaptive sparse grids, etc.). The choice of the sampling methodology highly depends on the final usage of the catalog and the numerical resources available. Indeed, every single Mesoscale Gravity Wave (MGW) defined with the combination of parameters sampled from the stochastic variable distribution has to be used as a forcing of a numerical hydrodynamic model to calculate the meteotsunami surges.

Overall, the generation of synthetic catalogs is numerically expensive, particularly when the number of stochastic variables is high (e.g., like in Denamiel et al. 2020). On the other hand, such catalogs can bring valuable insights concerning the potential hazards caused by rare events like meteotsunamis. Further, this methodology can also be applied for acoustically driven events whose intensity and location (in the case of asteroid impacts) can be highly uncertain.

**Data and Technical Requirements**

Synthetic catalogs are generally implemented when observations are scarce and are
designed to work even with little knowledge of the prior distributions of the stochastic
variables. However, the use of 1-min atmospheric records can help better constrain
these prior distributions and hence, build more targeted synthetic catalogs – i.e.,
catalogs mostly formed of atmospheric disturbances that will lead to meteotsunami
events at hotspot locations.

In terms of numerical resources, the generation of the atmospheric disturbances
does not require the use of HPC facilities. However, to produce the final meteot-
sunami surges, sub-kilometer-scale hydrodynamic models are run for thousands of
simulations forced by the sampled atmospheric disturbances. Consequently, when
creating synthetic catalogs, HPC facilities are needed for both parallelization and
storage.

**Treatment of Uncertainties**

Synthetic catalogs are essential to the implementation of Uncertainty Quantification - (UQ) - methods. For APTHA, UQ methods are useful as simulations of weather- and
acoustically- driven meteotsunami waves at hotspot locations are highly influenced by the exact reproduction of the atmospheric disturbance parameters (e.g., start location, speed, period, etc.) which might not be captured deterministically. To this date two different UQ methods have been developed for weather related events: (1) Probabilistic Analysis and (2) Surrogate Model. However, these methods are based on a single application for a single geographical location.

The “Probabilistic Analysis” approach was implemented in the northeast U.S. by Geist et al. (2014) for isolated atmospheric pressure disturbances from squall lines and derechos. It is based on multiple synthetic catalogs and a Monte Carlo (MC) scheme. First, the atmospheric disturbances related to the squall lines in the northeast U.S. are parameterized and the observations from Automated Surface Observing System stations between 2000 and 2013 are used to establish a continuous analytic distribution of each parameter as well as the overall Poisson rate of occurrence. Then, the probabilistic equations are implemented using a MC scheme and a synthetic catalog of squall lines is compiled by sampling the parameter distributions. Finally, multiple synthetic catalogs are generated by resampling from the parent parameter distributions.

The “Surrogate Model” approach was implemented in the Adriatic Sea by Denamiel et al. (2019a, 2020, 2021). As the “Probabilistic Analysis”, it is based on the generation of synthetic catalogs but for atmospheric disturbances assumed to be MGWs. The main difference between the two methods is that the sampling of the uniform distributions follows a sparse grid rule in order to build the surrogate model following the generalized Polynomial Chaos Expansion (gPCE) method that can be found at: https://github.com/cdenamiel/Adriatic_Sea_meteotsunami_surrogate_model.

Further, ANalysis Of VAriance (ANOVA), based on the gPCE method and providing the sensitivity to the atmospheric stochastic variables, was also performed at hotspot locations (e.g., Vela Luka and Stari grad) along the eastern Adriatic coast (e.g., Total and Main Sensitivities presented in Fig. 4.11). It reveals that the most important parameters for the generation of meteotsunamis are the speed and the period of the atmospheric disturbances.

To this date, UQ has not yet been developed for, nor applied to, acoustically driven events but the above presented approaches could also be used to derive the APTHA of planetary meteotsunami waves driven by volcanic eruptions or asteroid impacts.

### 4.5.4 Method: Climate Uncertainty

#### Short Description

Contrarily to the other tsunami sources, the long-term frequency and intensity of the atmospherically-driven tsunami sources are impacted by climate change (Vilibić et al. 2018; Denamiel et al. 2022). However, accounting for the climate uncertainties in meteotsunami modeling present two main challenges. On one hand, weather
<figure><img src="image_13.png" /><figcaption>**Fig. 4.11** Sensitivity analysis at Vela Luka and Stari Grad, two meteotsunami hotspot locations of the Adriatic Sea. Total (left panel) and 1st and 2nd order main sensitivities which represent 95% (0.95) of the meteotsunami impacts (right panels) calculated for the six parameters of the atmospheric disturbance used in the “Surrogate Model” approach. Adapted from Denamiel et al. (2020)</figcaption></figure>

related meteotsunami events can not be dynamically reproduced by the available atmospheric Global or Regional Climate Models (GCMs or RCMs) with resolutions of the order of 100 km and 10 km respectively but can be extracted from these models at the synoptic scale. On the other hand, atmospheric kilometer-scale models, which downscale the results of the GCMs or RCMs, can represent the meteotsunami-genic disturbances but their numerical cost and slowness is an obstacle for long-term climate studies. For example, the coupled atmosphere-ocean AdriSC model running in the Adriatic Sea (Denamiel et al. 2019a) would take 7 years continuously using 256 CPUs to downscale one single climate scenario from one single GCM or RCM for the 1950–2100 period. Consequently, the methodology proposed by Denamiel et al. (2023b) to derive the frequency and intensity of the meteotsunamigenic disturbances in the future, as well as their associated uncertainties, at an affordable numerical cost is based on (1) synoptic indices that preselect a substantial number of short-term meteotsunami episodes from the GCMs and (2) a suite of atmospheric models that downscale these episodes from an ensemble of GCMs to the sub-kilometre-scale.

To this date, the methodology has not yet been fully tested but could practically be applied anywhere in the world by following 6 steps (Fig. 4.12):
<figure><img src="image_14.png" /><figcaption>**Fig. 4.12** Workflow of the methodology for projecting meteotsunami hazards using synoptic indices designed with global reanalysis products (ERA5 Design) and GCM historical simulations (CMIP6 Design) to extract meteotsunamigenic atmospheric disturbances including the climate uncertainty by downscaling (km downscaling) historical (CMIP6 Eval.) and future GCM simulations and using surrogate models to account for the atmospheric model uncertainties. Adapted from Denamiel et al. (2023b), licensed under CC-BY 4.0</figcaption></figure>

1. **Synoptic Index Design:** Implementation of a synoptic index based on atmospheric reanalysis fields (e.g., ERA5) and available multi-year observations including high-frequency sea level measurements

2. **Synoptic Index Evaluation:** First, the synoptic index derived in step 1 is used with the historical simulations of the selected GCMs (e.g., CMIP6). Second, the sea-level distributions derived with the synoptic index for the reanalysis product
and the GCM historical simulations are compared. Then, if the comparison is successful – e.g., sea-level distributions significantly similar following the Z-test—the extracted meteotsunami events from the reanalysis product and the GCM historical simulations are downscaled to the kilometer-scale. Finally, the atmospheric and sea-level distributions obtained at hotspot locations by downscaling reanalysis products and GCM historical simulations are compared.

3. **Synoptic Index Redesign:** if the previous step fails – e.g., distributions significantly different following the Z-test, the synoptic index is redesigned by considering both GCM historical simulations (e.g., CMIP6) and reanalysis product (e.g., ERA5) until success of the Synoptic Index Evaluation step.

4. **Kilometer-scale Downscaling:** Downscaling of the GCM future climate scenario ensembles for the meteotsunami events extracted with the synoptic index.

5. **Selection of meteotsunamigenic conditions:** False positive events are excluded from the ensemble of kilometer-scale simulations for both historical and future simulations. The selection of the events is based on the automatic detection of meteotsunamigenic disturbances from the 1 km resolution atmospheric results – e.g., as presented in Denamiel et al. (2019b) for the Adriatic Sea.

6. **Uncertainty Quantification:** It is extremely likely that meteotsunamigenic disturbances do not generate meteotsunamis at hotspot locations due to a shift in location, speed, etc. In this case, surrogate models can be used to account for the uncertainty.

In this methodology, the derivation of the synoptic indices for meteotsunami events has been well tested and validated (Ramis and Jansà 1983; Vich and Romero 2021; Zemunik et al. 2022). Further, kilometer-downscaling – where telescopically nested domains at a decreasing horizontal grid spacing with boundary conditions provided by GCMs or reanalysis until kilometer-scales are reached – can also be considered state-of-the-art (Prein et al. 2015). However, the combination between climate and modeling uncertainties has never really been envisioned in the climate modeling community. Despite these untested aspects, the presented methodology remains, to this date, the best avenue to derive and project meteotsunami hazards (i.e., proper APTHA) under the on-going climate warming at an affordable numerical cost.

## Data and Technical Requirements

Three main sources of data are needed to implement the method: (1) an ensemble of GCMs, (2) a reanalysis product and (3) sea-level observations. The best available ensemble of GCMs is, to this date, provided by the Coupled Model Intercomparison Project Phase 6 (CMIP6, Eyring et al. 2016) which includes between 14 and 16 members providing 6-hourly pressure level results at 100 km resolution for both Shared Socioeconomic Pathways (SSP) 2-4.5 and 5-8.5 scenarios. For the synoptic index derivation, the 6-hourly ERA-Interim atmospheric reanalysis (Dee et al. 2011) or the hourly ERA5 reanalysis at 25 km resolution (Hersbach et al. 2020) can be used. Indeed, ERA-Interim has a horizontal resolution, and thus a representation of atmospheric physics, closer to the above mentioned CMIP6 GCMs while ERA5 is the latest and probably the best reanalysis product from ECMWF. Finally, for
the long-term quality-checked high-frequency sea-level observations, the MISELA dataset (Zemunik et al. 2021a, 2021b) can be used.

In terms of the numerical resources needed to implement the methodology, kilometer-scale models should be fully parallelized in order to efficiently down-scale the GCMs. In addition, surrogate models used to account for the atmospheric model uncertainties require the generation of synthetic catalogs which are numerically expensive to implement. Finally, the use of GCM simulations as well as the generation of ensembles of kilometer-scale simulations require access (even temporarily) to nearly unlimited storage resources (e.g., tape system). Consequently, for the “Climate Uncertainty” method, even within the presented framework, access to HPC facilities is mandatory. For example, in the Adriatic Sea, the kilometer-scale modeling suite runs on 256 CPUs on the new ECMWF ATOS HPC (https://www.ecmwf.int/en/about/media-centre/news/2022/taking-ecmwf-new-high-performance-computing-facility-operation).

## Treatment of Uncertainties

The presented method deals with two sources of uncertainties: (1) the climate uncertainty linked to the downscaling of an ensemble of GCMs and (2) the uncertainty linked to the atmospheric disturbance modeling.

Although GCM ensembles do not necessarily represent the full uncertainty in climate projections (Knutti 2010), they allow investigating uncertainty in future projections and particularly examining the maximum–minimum climate change scenarios. To avoid erroneous interpretations of the obtained climate impact assessments, several strategies have been derived to select subsets of GCMs (e.g., smaller biases in the historical climate simulations, coverage of the widest possible uncertainty range of future projections; Reichler and Kim 2008; Cannon 2015; Mendlik and Gobiet 2016). Within the proposed method, all selected meteotsunami events from all appropriate GCMs (i.e., all GCMs with 100 km resolution) are downscaled. Consequently, the subsampling procedure is not applied to select GCMs but to detect meteotsunami events. Provided that the synoptic index is efficient (i.e., that enough meteotsunami events are extracted), the range of the climate uncertainty derived with the proposed approach will thus be similar to the one derived with the GCM ensemble. However, a bias can be introduced due to the specific setup of the kilometer-scale modeling suite implemented to downscale the GCMs.

The treatment of the atmospheric model uncertainty is already described within the “Synthetic Catalog” method.

### 4.5.5 Summary Table

See (Table 4.5).
<table><caption>**Table 4.5** Summary table: atmospheric sources</caption><thead><tr><th></th><th>Synoptic Index</th><th>Deterministic Model</th><th>Synthetic Catalog</th><th>Climate Uncertainty</th></tr></thead><tbody><tr><td colspan="5"><strong>Short description of methods</strong></td></tr><tr><td>Theoretical/mechanical background for the method</td><td>Used to identify meteotsunami events with synoptic conditions extracted from global atmospheric models</td><td>Used to downscale the fields from global atmospheric models for weather sources or to generate the acoustic-gravity waves for acoustic sources</td><td>Used to generate all potential scenarios of a given atmospheric forcing (at a given location)</td><td>Used to quantify the long-term variability and frequency of meteotsunamigenic atmospheric disturbances under climate change</td></tr><tr><td>Short outline of how the methods work</td><td>Based on local sea level observations and atmospheric reanalyses/observations</td><td>Based on complex atmospheric models</td><td>Based on synthetic atmospheric forcing and catalogs of events</td><td>Based on downscaling global climate models only during meteotsunami events</td></tr><tr><td>How is/are the output(s) of the method represented?</td><td>Provides the dates of potential meteotsunami events at hotspot locations</td><td>Provides 2D/3D fields like pressure and winds needed to force meteotsunamis but also temperature, humidity, etc</td><td>Provides a catalog of events that can be used to derive/ implement statistics, hazard curves, surrogate models and sensitivity studies</td><td>Provides long-term projections of the intensity and frequency of the meteotsunamigenic atmospheric disturbances</td></tr><tr><td>In what way do the methods take into account time-dependent/dynamic aspects?</td><td>Included in the atmospheric global models/observations</td><td>Included in the complex kilometer-scale atmospheric models</td><td>Included in the time and space varying synthetic forcing/ long-term observations,</td><td>Included in the complex kilometer-scale atmospheric models</td></tr><tr><td>Do the methods use parallelisation, machine learning, surrogate models, advanced simulation schemes, other computational reduction strategies (e.g. cluster analysis, etc.)?</td><td>Uses machine learning tools</td><td>Uses parallelisation and optimization of the numerical schemes as well as large storage (e.g., tapes)</td><td>Used to implement surrogate models</td><td>Uses parallelisation and optimization of the numerical schemes as well as large storage (e.g., tapes)</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.5 (continued)**</caption><thead><tr><th></th><th>Synoptic Index</th><th>Deterministic Model</th><th>Synthetic Catalog</th><th>Climate Uncertainty</th></tr></thead><tbody><tr><td>What are the pros and cons and peculiarities of these methods?<br>Are there known shortcomings / benefits of the methods? In what cases are these methods good/poor choices?</td><td>Pros: cheap to build and simple to use<br>Cons: qualitative method often missing strongest events and providing a great number of false positives</td><td>Pros: dynamical solution based on model physics<br>Cons: numerically expensive and not highly reliable</td><td>Pros: used for hazard assessments and uncertainty quantification<br>Cons: synthetic forcing not as complex as real atmospheric forcing</td><td>Pros: full climate uncertainty by only downscaling meteotsunamigenic disturbances<br>Cons: cumulates all problems of the other methods</td></tr><tr><td colspan="5">Data and technical requirements for method</td></tr><tr><td>What input data does the method(s) use?</td><td>Needs 1-min tide gauge observations and atmospheric reanalyses/observations</td><td>Needs initial/boundary conditions and/or topography/ surface layer characterization from global models</td><td>Needs prior distributions of the atmospheric synthetic forcing stochastic parameters</td><td>Needs all of the inputs of the other methods</td></tr><tr><td>What are the technical requirements?</td><td>Uses machine learning</td><td>Set-up of kilometer-scale and/ or global atmospheric models</td><td>Sampling techniques and surrogate model methods</td><td>All of the technical requirements from the other methods</td></tr><tr><td>Are HPC computing resources necessary, would absence of these compromise the use?</td><td>No</td><td>Yes. Without HPC the method is impossible to implement</td><td>Yes. Without HPC, oversimplification of the synthetic forcing or decrease of the ocean model resolution are required</td><td>Yes. Without HPC the method is impossible to implement</td></tr><tr><td>Is the method(s)/workflow(s) available as a git/software/ jupyter notebook?</td><td>No</td><td>No</td><td>Yes</td><td>No</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.5** (continued)</caption><thead><tr><th></th><th>Synoptic Index</th><th>Deterministic Model</th><th>Synthetic Catalog</th><th>Climate Uncertainty</th></tr></thead><tbody><tr><td colspan="5">Treatment of uncertainties for the method</td></tr><tr><td>Are the method(s) itself deterministic or uncertainty based?</td><td>Mostly deterministic</td><td>Mostly deterministic</td><td>Fully uncertainty based</td><td>Fully uncertainty based</td></tr><tr><td>Which are potential sources of uncertainties? Which parameters are used to characterize them?</td><td>Sources: local effects not accounted for and knowledge of future climate changes Parameters: atmospheric temperature, wind, pressure at the synoptic scale</td><td>Sources: inaccurate topography and surface layer characterizations or initial and boundary conditions as well as model parameterizations Parameters: pressure and wind at the mesoscale for weather sources</td><td>Sources: sparse observations and/or poor knowledge of the atmospheric process shape Parameters: direction, start location, amplitude, speed, period, attenuation in time, Earth's surface temperatures</td><td>Sources: cumulates all of the previous method sources Parameters: cumulates all of the previous method parameters</td></tr><tr><td>How are uncertainties quantified for the identified parameters? Are there significant correlations between different uncertain parameters?</td><td>Based on ensemble modeling</td><td>Based on ensemble modeling</td><td>Based on prior distributions of mutually independent synthetic forcing stochastic parameters</td><td>Based on ensembles of climate models and future climate change scenarios</td></tr><tr><td>Are the method(s) able to digest and propagate uncertainty in input (previous ingredient) and propagate it to the output (next ingredients)? Does the assessment of this ingredient require uncertainty propagation? If yes, how are uncertainties propagated?</td><td>Uncertainty propagation uses different indices as an ensemble of conditions</td><td>Uncertainty propagation uses ensembles of kilometer-scale atmospheric models</td><td>Uncertainty propagation uses probabilistic and surrogate model approaches</td><td>Uncertainty propagation uses all the previous method approaches.</td></tr></tbody></table>

(continued)
<table><caption>**Table 4.5 (continued)**</caption><thead><tr><th></th><th>Synoptic Index</th><th>Deterministic Model</th><th>Synthetic Catalog</th><th>Climate Uncertainty</th></tr></thead><tbody><tr><td>What are the consequences of lack of data in input or for boundary conditions, or for characterizing certain parameters?</td><td>Consequences: poorly built synoptic index</td><td>Consequences: incapacity of reproducing the atmospheric disturbances</td><td>Consequences: inaccurate prior distributions and synthetic atmospheric forcing</td><td>Consequences: all of the previous method consequences as well as wrong climate uncertainties</td></tr><tr><td>What are the consequences of ignoring or poorly treating uncertainties?</td><td>Inaccurate: selection of the meteotsunamigenic atmospheric disturbances</td><td>Inaccurate: modeling of the atmospheric mesoscale conditions driving meteotsunami events</td><td>Inaccurate: meteotsunami hazard quantification</td><td>Inaccurate: climate hazard assessments</td></tr></tbody></table>
## 4.6 Choosing What Method to Use

### 4.6.1 Earthquakes

In PTHA modeling the earthquake scenario parameterization plays a key role. The choice of which method to use (rectangular or 3D faults, homogeneous or heterogeneous slip) depends on the amount of available information and the size of the earthquake. As a rule of thumb, the larger the earthquake, the larger the rupture area and thus 3D faults and heterogenous slip distributions are more suitable to describe these scenarios. Rectangular geometries can be used for crustal earthquakes, whereas the modeling of the ruptures along subduction faults requires a complex fault parameterization that honors as much as possible the slab geometry. For the same reasons, particular attention should be paid on using appropriate earthquake scaling relations depending on the magnitude range (see e.g. Figure 4.2) and the type of tectonic regime/faulting type.

The selection of the method to model the seafloor deformation also depends on the available information and the computational resources. The use of analytical methods is valid for both planar (rectangular) or 3D complex fault geometries; the latter requires a tessellation of the fault surface. If information such as 3D seismic tomography is available, the medium surrounding the fault can be characterized by the elastic parameters (e.g., shear modulus, Young modulus) and the deformation computed by means of FEM. Inclusion of the bathymetry allows computation of contribution of horizontal displacement to the tsunami source which can be significant in cases of shallow rupture.

Dynamic rupture models require the use of well-known faults and additional data related to the mechanical properties of the fault embedding rock volume that are often unavailable, or available just as average estimates. Assumptions are also required to the type of friction law, the spatial variability of its parameters as well as the distribution of initial stress across the fault plane.

### 4.6.2 Landslides

For landslides, the volume or frontal area, as well as rheology and landslide deformation with time control the tsunami generation. The simplest model is the static initial condition, while the block models and finally the depth averaged landslide models introduce better physical source representation but also more complexity. To embed the tsunami generation most correctly, it is preferential to use the block and landslide dynamics rather than the static initial conditions if resources allow it as these models better represent the source physics, whereas static initial conditions introduce large simplifications. In all cases, the models are associated with significant uncertainties related to the tsunami generation that require calibration towards past events before using them in PTHA.
Static initial condition methods for tsunami elevation are by far the least resource intensive. They may be easily embedded into PTHA models as they are instantaneous much like Okada-type sources for earthquakes. Benefits of these models are that they are easy to use and require almost no compute power. On the other hand they cannot integrate the effect of local bathymetry and the time dependence of the landslide into the tsunami model, and this limits the accuracy compared to all other models reviewed here. They may involve significant bias due to the many simplifying assumptions when describing a more complex type of source factors.

Block models are the simplest alternative when it comes to time-dependent source models that incorporate bathymetric effects. They are cheaper, more stable, and somewhat simpler to deploy than landslide dynamics models that require solving partial differential equations. Yet, projecting them into tsunami models needs careful effort, and this may be a challenge in mass-production simulations such as PTHA.

Numerical depth averaged landslide models include the deformation pattern of the landslide and can include rheology as well, and incorporate more physics than static initial conditions and block models. They may be feasible to include in a PTHA although they require more computational resources than static initial conditions and block models. Full three dimensional models are presently computationally too expensive to be embedded into PTHA, but are important for testing the representation of accurate physics, in particular for subaerial landslide tsunamis.

### 4.6.3 Volcanoes

There exist different approaches to model tsunamis induced by volcanic activity, depending on the type of source. The choice of the method depends on the targeted tsunami source (explosion, pyroclastic flow, etc.), the quality of available grids used for calculation (keeping in mind that volcanic tsunami sources often require high-resolution grids to be correctly defined, unlike seismic sources), and the objectives of the simulations. The strategy might be different between, for instance, (a) a deterministic approach aiming at accurately reproducing a past event at a local scale, and (b) a probabilistic approach aiming at simulating a database of scenarios for a hazard study.

Instantaneous initialization (static source model) is suitable if tsunami generation happens rapidly compared to the shallow water wave speed. This works for underwater explosions only (note that an instantaneous caldera collapse initiation is not realistic because a caldera collapse typically lasts minutes to hours). If the forcing happens over a specified time at the start of the modeling, then a time dependent initialization is required. In the case of a caldera collapse, a common approach is to apply the time dependent sea floor motion as a forcing term. Our understanding of tsunamis generated by pyroclastic flows is still in its infancy, but in the current state of knowledge, it appears strongly recommended to use multi-fluid CFD models (full Navier-Stokes models). These models are computationally too expensive for PTHA, therefore, it is often necessary to limit their use to the modeling of the source, and then
use less advanced models for the propagation and integration into PTHA. Another case of finite time generation is an ongoing forcing by a pressure anomaly, such as Lamb waves produced by large explosions in the atmosphere (see also atmospheric sources).

### 4.6.4 Atmospheric Sources

In the context of PTHA, contrary to other tsunami sources, weather related meteotsunami events are dependent on the climatic changes and are thought to be highly influenced by the on-going climate warming. Hence classical time dependent PTHA methods are likely to have shortcomings. Instead, a climate change approach attempting to extract the past and future events from climate models in order to derive statistics on their behavior and characteristics should be used. However, to this date, only few generic methods are implemented for operational forecast or research purposes within the meteotsunami community, but on the other hand not yet put to use for Atmospheric PTHA.

Most of the generic methods (e.g., synoptic index, deterministic model) so far only deal with short term forecast meteotsunami events with the aim to issue public warnings at hotspot locations. Synoptic indices can predict the occurrence of meteotsunami events up to five days in advance but not their intensity and they are known to generate many false alarms. In terms of providing input to APTHA, synoptic indices can be used to extract meteotsunami events from global climate models in order to derive the past and future frequency of weather related events depending on the chosen climate scenarios. Concerning the deterministic atmospheric numerical models, they have been proven to be able to successfully simulate the meteotsunami-genic disturbances. In terms of providing input to APTHA, they can thus be used to downscale the global climate models to the kilometer-scale needed to represent the intensity of the atmospheric disturbances driving the meteotsunami events. However, deterministic models have difficulties to properly reproduce the exact location, speed, period, etc., of these disturbances. Consequently, the need to account for the uncertainty in the atmospheric simulations (e.g., ensemble modeling, surrogate models) has become increasingly important in recent years. APTHA should thus also include the representation of the atmospheric model uncertainties.

Practically, the only method that can be used for APTHA of weather related events is the climate uncertainty method combining synoptic indices, deterministic models and uncertainty quantification linked to both climate projections and atmospheric modeling. However, the climate uncertainty method has not yet been successfully implemented due to both the lack of reliability of the synoptic indices and the extreme numerical costs of kilometer-scale atmosphere-ocean modeling.
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