Hydrodynamic Modelling

Íñigo Aniel-Quiroga, Maria Ana Baptista, Jörn Behrens, Miquel Canals, Cléa Denamiel, Utku Kânoğlu, Galderic Lastras, Finn Løvholt, Jorge Macías, Shane Murphy, et al.

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Íñigo Aniel-Quiroga, Maria Ana Baptista, Jörn Behrens, Miquel Canals, Cléa Denamiel, et al. Hydrodynamic Modelling. Sørensen, M. Probabilistic Tsunami Hazard and Risk Analysis. A cookbook, 14, Springer, pp.223 - 263, 2026, Mathematics and Planet Earth Series, 10.1007/978-3-031-98115-9_5. hal-05530603

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Chapter 5
Hydrodynamic Modelling

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Íñigo Aniel-Quiroga Ⓔ, Maria Ana Baptista, Jörn Behrens Ⓔ,
Miquel Canals Ⓔ, Cléa Denamiel Ⓔ, Utku Kânoğlu, Galderic Lastras Ⓔ,
Finn Løvholt Ⓔ, Jorge Macías Ⓔ, Shane Murphy Ⓔ, and Raphaël Paris Ⓔ

**5.1 Introduction**

This chapter presents several methods to simulate tsunami propagation, inundation, and impact on land. The chapter is subdivided into five sections: Preparation of Data, Interfaces, Propagation, Inundation, and Testing. The order of the sections represents a typical sequence of tsunami modelling in a probabilistic tsunami hazard assessment (PTHA) workflow.

The first step is the preparation of the data for use in numerical models. The second step involves the interface between the source modelling described in Chap. 4 and the modelling of tsunami propagation and inundation. Finally, inundation modelling is described in the fourth step providing the output for exposure modelling, discussed in Chap. 6.

Tsunami models predict wave height, surface elevation, flow depth and inun-
dation areas. These parameters are crucial for either deterministic or probabilistic
tsunami hazard assessment methods. Moreover, they can compute real-time tsunami
travel time and offshore wave heights for early warning and evacuation purposes.
The propagation and inundation of tsunamis is generally governed by equations

Í. Aniel-Quiroga

Instituto de Hidráulica Ambiental (IH Cantabria), Universidad de Cantabria, Santander, Spain

e-mail: anieli@unican.es

M. A. Baptista

Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de Lisboa, Lisboa, Portugal

e-mail: mavbaptista@gmail.com

J. Behrens (✉)

Department of Mathematics, Universität Hamburg, Hamburg, Germany

e-mail: joern Behrens@uni-hamburg.de

M. Canals · G. Lastras

Department of Earth and Ocean Dynamics, CRG Marine Geosciences, Universitat de Barcelona,

Barcelona, Spain

e-mail: miquelcanals@ub.edu
of motion—the Navier-Stokes equations—and simplifications thereof. Due to the complexity of the equations and the geometry of the domain needed in practical simulations, no general analytical solutions exist. Therefore, numerical codes for tsunami simulation are used to solve the relevant equations. To do this, the general process includes simplification/elimination of terms of the Navier Stokes equations, generation of numerical grids incorporating boundary conditions and bathymetry/topography, application of numerical methods such as finite differences, finite elements or finite volumes to the equations of motion, solution of the inherent discrete equation sets—often by iterative solvers.

Tsunami inundation models predict tsunami propagation over land (onshore). These models rely heavily on topography and land use data and to date still comprise uncertainty, when considering detailed impact. For large-scale hazard assessments heuristic amplification factors or statistically based emulators, based on wave heights near to the shore and derived from tsunami propagation models, are frequently used to estimate inundation depths.

Code validation and verification is of paramount importance to produce reliable and reproducible results. As data from real time events are scarce, modellers recur to analytical solutions, laboratory experimental and observed data (Synolakis et al. 2008; NTHMP 2012, 2016; Lynett et al. 2017) to numerically benchmark the models.

G. Lastras

e-mail: glastras@ub.edu

C. Denamiel

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

e-mail: cdenami@irb.hr

U. Kânoğlu

Department of Aerospace Engineering, Dokuz Eylül University, Izmir, Turkey

e-mail: utku.kanoglu@deu.edu.tr

F. Løvholt

Offshore Energy, Norwegian Geotechnical Institute, Oslo, Norway

e-mail: finn.lovholt@ngi.no

J. Macías

Departamento de Análisis Mathemático, Estadística e Investigación Operativa y Mathemática

Aplicada, Universidad de Málaga, Málaga, Spain

e-mail: jmacias@uma.es

S. Murphy

Geo-Ocean, Ifremer, Plouzané, France

e-mail: shane.murphy@ifremer.fr

R. Paris

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

e-mail: raphael.paris@uca.fr
A summary of this chapter's contents and intention can be found in condensed form in Table 5.1.

## 5.2 Preparation of Data

Diverse types of data are important to tsunami modelling. Mathematically, tsunami modelling can be described as an initial and boundary value problem, thus results sensitively depend on the input data.

Most importantly, boundary data influencing the modelling results are topography and bathymetry data. A more in-depth description of their inclusion in tsunami modelling is given below. Initial data, more precisely an accurate representation of the tsunami source, is another crucial data type for accurate modelling results, mostly covered in the next section. Finally, topography, sometimes supplemented by land use data and corresponding tsunami modelling parameterizations are important for a realistic inundation behaviour of tsunami models. A condensed list of contents of this section is presented in Table 5.2.

### 5.2.1 Bathymetric and Topographic Data

Bathymetric and topographic data are key ingredients for the numerical modelling of tsunami propagation and its impact on target coastlines. A typical continental margin shallows shoreward from the trench up to the shelf. This configuration produces a shoaling effect in tsunami propagation, which is most noticeable over the continental shelf. Shoaling implies the decrease of tsunami wavelength and velocity and the increase of wave height when the wave propagates from deep to shallow water. The significance of shoaling depends on the shape and depth of each margin: the width and depth of its shelf, the gradient of its continental slope, the presence of submarine canyons dissecting the slope, the incision of these canyons into the continental shelf eventually with their heads close to the coastline, and the presence of seamounts, shallow banks or wide troughs, among many other features.

Such topographic features induce significant local changes to the wave characteristics. Tsunami arrival time to a coastline stretch located shoreward of a highly incised submarine canyon is short, and wave height is less, whereas enhanced wave setup occurs along the flanks of the canyon, with an opposite effect at both sides of the relevant coastal stretch (Iglesias et al. 2014). Extremely incised canyons favour the formation of trapped edge waves along the coast. In addition, seafloor and coastline features such as semi-enclosed bays, estuaries, and fjords can lead to resonance phenomena enhancing tsunami effects. Moreover, the complexity of bathymetric features, such as a sand spit, necessitates accurate topographic data for the correct estimation of runup. Thus, tsunami numerical modelling is sensitive to both the resolution and the accuracy of bathymetric and topographic data. The appropriate data
<table><caption>**Table 5.1** Hydrodynamic modelling</caption><tr><td>What does the ingredient describe?</td><td>Tsunami modelling is at the core of each PTHA/PTRA workflow. This ingredient describes the different steps of setting up a modelling workflow</td></tr><tr><td>Definitions of the ingredient</td><td>Input data: What are the inputs available or required<br>Sources: Different phenomena triggering tsunamis such as earthquake, volcano, landslide, meteorological sources<br>Interfaces: Sources trigger tsunamis via interfaces such as initial, boundary or forcing data<br>Equations: A tsunami model is represented by mathematical equations of the wave propagation and fluid dynamics<br>Discretization: The transformation of mathematical equations into a form consisting of elementary computational formulas, digestible by a computer<br>Coupling: Exchange of appropriate data between different models at their interfaces<br>Propagation: The phase in a tsunami model approach, in which tsunami waves travel through the ocean basin<br>Inundation: The phase, in which a tsunami penetrates land<br>Verification: Test that a model implementation is free of errors and behaves as it should according to its defined specification<br>Validation: Ensure that a model behaves in accordance with the physical world (up to its model assumptions/limitations)</td></tr><tr><td>What is/are the metric (s) used for measuring/ evaluating the ingredient?</td><td>On input (disturbances from a resting sea): These may be sea surface elevation, ocean bottom displacement, mass displacements at the boundaries, or forcing terms<br>On output: Hazard metrics such as sea surface elevation offshore and at the coast, inundation height, run-up, inundation area, momentum flux, flow velocity, etc</td></tr><tr><td>Possible uses/applications of the ingredient</td><td>Except for event-based hazard assessment (based on field data catalogues), modelling is one of the best-available ways to assess hazard from tsunami activity</td></tr><tr><td>Possible multi-hazard dimensions/ applications?</td><td>The multi-hazard aspect is represented through the source, or after the tsunami has hit in the risk analysis parts</td></tr></table>

(continued)
<table><caption>**Table 5.1 (continued)**</caption><tr><td>What does the ingredient describe?</td><td>Tsunami modelling is at the core of each PTHA/PTRA workflow. This ingredient describes the different steps of setting up a modelling workflow</td></tr><tr><td>Possible time-dependent/dynamic aspects</td><td>Sources may be instantaneous or dynamic. The modelling outputs may be maximum values or time-series</td></tr><tr><td>List of the available methods to be described below</td><td>Data Preparation<br>Interfacing<br>Propagation<br>Inundation<br>Validation</td></tr></table>

representation depends on the complexity of the shoreline. On a considerably plain and very long beach such as those of Southern California, a 100-m-grid resolution may be sufficient. Small offshore and onshore features likely to affect tsunami wave behaviour should be reflected in the numerical grids. If a location is fronted by a sand spit of width 100 m, a sufficient number of grid points are needed to provide accurate resolution of the flow over the spit, thus higher resolution is necessary. On the other hand, deep ocean propagation does not require such fine grid spacings. Therefore, adaptive or nested grids are often used for representing bathymetry and topography adequately.

## 5.2.2 Technical Requirements for and Availability of Topo-Bathymetry Data

Since most tsunami propagation tools employ low to medium order numerical methods (i.e. lower than 3rd order consistency), as a rule of thumb one may use that the relevant wavelength or feature size needs to be sampled by at least 10–20 grid points or cells for being adequately resolved, although this should be verified by sensitivity tests. To produce a good Digital Elevation Model (DEM) to simulate tsunami inundation, we need to merge high-resolution ocean depth data (bathymetry) and height data (topography). This merging process requires a common definition for the vertical reference surface. It is important to note that bathymetric and topographic maps use different vertical references. The bathymetric maps use a local reference to the main harbour in the area, whereas topographic maps use the mean sea level referred to a specific datum.

Tsunami numerical modelling requires computational grids, and for inundation modelling often also several nested grids, to model generation, propagation, and inundation phases. While the tsunami propagation requires accurate bathymetric data,
<table><caption>**Table 5.2** Summary table modelling data</caption><thead><tr><th colspan="2">Short description of method</th></tr></thead><tbody><tr><td>Theoretical/mechanical background for the method</td><td>Considered in this section are data that are not related to the tsunami source, such as topography and bathymetry data</td></tr><tr><td>Short outline of how the method works</td><td>Thorough consideration of accuracy requirements and compatibility with modelling assumptions is required. Guidelines are given in this section</td></tr><tr><td>How is/are the output (s) of the method represented?</td><td>The outputs of this method are input files to tsunami modelling tools, usually in standardized formats</td></tr><tr><td>In what way does the method take into account time-dependent/dynamic aspects?</td><td>Most of the data are time-independent relative to the time-scales of tsunami modelling</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>No</td></tr><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>N/A</td></tr><tr><td colspan="2"><em>Data and technical requirements for method</em></td></tr><tr><td>What input data does the method use?</td><td>Diverse sources of elevation data</td></tr><tr><td>What are the technical requirements?</td><td>Vertical accuracy is more important than horizontal accuracy, elevations around mean sea level are most critical</td></tr><tr><td>Are HPC computing resources necessary, would absence of these compromise the use?</td><td>No</td></tr><tr><td>Is the method/workflow available as a git/software/jupyter notebook?</td><td>Usually no unified workflows available, but sometimes individual/proprietary scripts, since the data processing depends on specific model requirements</td></tr><tr><td colspan="2"><em>Treatment of uncertainties for the method</em></td></tr><tr><td>Is the method itself deterministic or uncertainty based?</td><td>Deterministic</td></tr><tr><td>Which are potential sources of uncertainties? Which parameters are used to characterize them?</td><td>Data accuracy is a big issue. Definition of mean sea level may not be uniform, elevation data may include canopies and may not represent ground elevation</td></tr><tr><td>How are uncertainties quantified for the identified parameters? Are there significant correlations between different uncertain parameters?</td><td>Difficult to quantify, sensitivity is large, but little systematic assessment available</td></tr></tbody></table>

(continued)
<table><caption>**Table 5.2** (continued)</caption><thead><tr><td>Short description of method</td><td></td></tr></thead><tbody><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? If yes, how are uncertainties propagated?</td><td>This represents input uncertainty, there is no propagation</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>Relevant hazard metrics may be rendered incorrect</td></tr><tr><td>What are the consequences of ignoring or poorly treating uncertainties?</td><td>Results, in particular inundation maps, may be very incorrect</td></tr></tbody></table>

the inundation phase requires both bathymetric and topographic data in consistent combination. Several issues can be listed in setting up DEM for tsunami numerical modelling (Venturato 2007) that might contribute to cumulative error in the data. These can be listed as follows:

1. Most often, high-resolution bathymetric and topographic data are proprietary, or potentially restricted, specifically close to the shoreline where the highest accuracy is needed.

2. Bathymetric and topographic data can be obtained from different sources. Hence, there might be problems matching the data, especially for the shoreline.

3. Inherent error within the original data, conversion error due to several data sources (for example, converting from a map projection to geographic coordinates), interpolation error of spatially varying data sources, and error due to subjective interpretation.

4. A common source of error is the different datum (e.g. mean sea level definition) of sea maps versus topography maps, and this mismatch must be corrected for.

In terms of bathymetric data, global bathymetric grids such as ETOPO2022 (2022), GEBCO (2022), and others are derived from satellite altimetry, ship soundings, and multibeam bathymetry. The Nippon Foundation GEBCO Seabed 2030 project aims to bring together all available bathymetric data sets and produce a definitive, 100% complete seafloor map by 2030. These compilations are fed with seafloor bathymetry data, even from remote regions, and are useful to compute large-scale offshore tsunami propagation. Nevertheless, resolutions of such global maps may not be high enough to resolve tsunami propagation near the coastline, where shoaling effects take place. GEBCO (2022) has a 15-arc second grid size, which translates to approximately 450 m at the Equator or 300 m at 50° of latitude. As an example, continental shelves in the Mediterranean Sea, with a mean width of approximately 20 km, are described by half a hundred bathymetric points, and canyon heads incised in the shelf may be represented by half a dozen of soundings. This is not sufficiently fine for resolving the details needed for the near shore propagation.
EMODnet Bathymetry provides a service for viewing and downloading the best available harmonised Digital Terrain Model (DTM) for the European sea regions, together with a range of other bathymetric data, products, and services (https://emo.dnet.ec.europa.eu/en/bathymetry). Topographic data or Digital Elevation Models (DEMs) are available freely or commercially. DEMs are available in two forms: as Digital Terrain Models (DTM), which contain bare-earth elevation data, i.e., buildings, vegetation, and others removed; or as Digital Surface Models (DSM), which include all these features. Light Detection and Ranging (LiDAR), Airborne Interferometric Synthetic Aperture Radar (IFSR), Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER), Shuttle Radar Topography Mission (SRTM), and Multifunctional camera—High-Resolution Stereoscopic Camera (HRSC) are possible sources of topographic data (Griffin et al. 2015).

It is important to note that for both bathymetric as well as topographic data, the accuracy with respect to the vertical dimension is more important than the horizontal accuracy. To make this clear, it makes a big difference if the topographic height includes an uncertainty of the order of metre, because this can determine inundation or safe grounds. If a hill is relocated by one meter horizontally, this will hardly affect the result of an inundation model.

There are several data formats in use for bathymetric and topographic data sets. The use of self-describing binary data formats, such as NetCDF, HDF5 or GRIB data formats are helpful for replicability. However, ASCII formats as well as diverse geographical information system (GIS) file formats are also often in use. In many cases a rectangular grid format is used, but there are also triangulated bathymetry files or point data sets in use. In any case, it is common to interpolate from a given dataset to the computational mesh, this needs care.

### 5.2.3 Other Data Types

In many tsunami models the bottom roughness parameter is used to calibrate the model for realistic applications. This parameter is used to represent subgrid-scale processes, such as turbulent flows, wave breaking, interaction with structures in inundation modelling, etc. on the larger modelling scale. It is most common to use a Manning roughness between 0.25 and 0.35 but heterogeneous roughness parameterization covering a wider range of friction parameters for different types of land use are available in certain models.

Another parameter used in some tsunami models is an eddy diffusivity parameter, accounting for turbulent processes. In meteotsunami applications, a wind-stress tensor is used to parameterize the effect of wind shear as a forcing to the water wave and flow build-up. All these parameters are subject to uncertainty and usually some testing and calibration will need to be applied, based on field data. This indicates the importance of test cases, as described in the last section of this chapter.
## 5.3 Interface

This section is titled interface, since the simulation of tsunami propagation and inundation can be understood as a mapping of data to outcomes such as inundation height, fluxes, or wave amplitudes. So, tsunami simulation needs to be interfaced with models providing accurate input data. This interfacing needs to be consistent with the physical as well as numerical characteristics of the target model. Again, we condensed the contents of this chapter into a table for better overview and orientation, which can be found in Table 5.3.

### 5.3.1 Tsunami Modelling as an Initial Boundary Value Problem

In the input mapping there are several stages that need to be interfaced, these stages are described here. The core of the tsunami simulation is formed by an equation set representing the fluid motion, and these equations are further controlled by initial and boundary values as well as parameters and source terms. So, data as described in the previous section need to be interfaced appropriately with the modelling environment. It is paramount to adhere to the physical consistency of all stages and interface conditions to obtain accurate results.

In an abstract sense, the wave propagation and inundation of a tsunami is simulated by the equations of motion which in a very general way can be given in the following abstract form:

$$ \frac{\partial q}{\partial t} + \nabla \cdot F(q) = S(q) \qquad (5.1) $$

where $q$ contains the conserved quantities (density $\rho$, momentum $\rho u$ with $u$ the velocity, total energy $e$), $F(q)$ represents the flux terms, and $S(q)$ represents the source terms. Adding to this set of equations, initial conditions are required:

$$ q(t = 0) = I(q) $$

and possibly boundary conditions for the domain, in which the simulation is intended to be conducted.

Euler Equations of fluid motion often serve as the master equations from which simplifications are derived and are given by the following definitions for $q$, $F$, and $S$ (where $P$ is pressure)

$$ q = \begin{bmatrix} \rho \\ \rho u \\ \rho e \end{bmatrix}, F(q) = \begin{bmatrix} \rho u \\ \rho u \times u + P \\ (\rho e + P)u \end{bmatrix}, S(q) = \begin{bmatrix} f_C \\ -\rho \nabla \phi + f_M \\ -\rho u \nabla \phi + f_E \end{bmatrix} $$
<table><caption>**Table 5.3** Summary table interfaces</caption><thead><tr><th colspan="2">Short description of method</th></tr></thead><tbody><tr><td>Theoretical/mechanical background for the method</td><td>Source modelling is considered as an interface problem, which may either be an input as is, or needs additional treatment for consistent representation</td></tr><tr><td>Short outline of how the method works</td><td>Input from heuristic or physical models for diverse sources is taken and processed such that it is compatible with propagation modelling</td></tr><tr><td>How is/are the output (s) of the method represented?</td><td>The outputs of this method are usually directly linked to tsunami propagation models. They are often integrated into the models themselves</td></tr><tr><td>In what way does the method take into account time-dependent/dynamic aspects?</td><td>Dynamical modelling is often considered and poses one of the challenges to the compatibility requirements. So processing data, such that temporal scale differences can be bridged is one of the method's main purposes</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>Some approaches use ML for a sophisticated interpolation of data. Parallel processing may also be applied, in particular, if the method is implemented in the same software framework as the target model</td></tr><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>N/A</td></tr><tr><td colspan="2"><em>Data and technical requirements for method</em></td></tr><tr><td>What input data does the method use?</td><td>Diverse source models</td></tr><tr><td>What are the technical requirements?</td><td>Compatibility with modelling assumptions, as well as with simulation software infrastructure is required</td></tr><tr><td>Are HPC computing resources necessary, would absence of these compromise the use?</td><td>Not necessary, but often utilized</td></tr><tr><td>Is the method/workflow available as a git/software/jupyter notebook?</td><td>No</td></tr><tr><td colspan="2"><em>Treatment of uncertainties for the method</em></td></tr><tr><td>Is the method itself deterministic or uncertainty based?</td><td>Usually deterministic, but may consider uncertainties by varying the source parameters</td></tr><tr><td>Which are potential sources of uncertainties? Which parameters are used to characterize them?</td><td>Uncertainty may arise from improper treatment of sources</td></tr><tr><td>How are uncertainties quantified for the identified parameters? Are there significant correlations between different uncertain parameters?</td><td>Difficult to quantify, sensitivity may be large</td></tr></tbody></table>

(continued)
<table><caption>**Table 5.3 (continued)**</caption><thead><tr><th>Short description of method</th><th></th></tr></thead><tbody><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? If yes, how are uncertainties propagated?</td><td>This method may consider and propagate uncertainty by sampling the possible input parameter space</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>Relevant hazard metrics may be rendered incorrect</td></tr><tr><td>What are the consequences of ignoring or poorly treating uncertainties?</td><td>Results, in particular inundation maps, may be very incorrect</td></tr></tbody></table>

Here we do not discuss all the terms in these equations, because we do not want to focus on the mathematical representation. This only serves as an illustration. To summarise, there are several parts in the mathematical representation of a tsunami model that need to be specified:

* $q$ represents the set of variables that need consideration, usually mass (water height/density), momentum (including horizontal and possibly vertical velocities) and optionally energy.

* $F$ represents the inherent physical balances and is usually only determined by the choice of the model representation and needs no further specification

* $S$ contains forcing terms that may be externally given and needs various types of data, depending on the modelling case.

* $I$ contains the initial conditions, which can be resting states or predefined conditions. These conditions apply only instantaneously and will not carry a time dependency.

* Finally, boundary conditions will need specification of the domain and will usually also not be time-dependent, except for inundation boundary conditions or tidal boundary forcing.

Turning to the conditions, note that $S(q)$ represents source terms that partly are fed from data and partly from physical processes. $f_M$ and $f_E$ represent friction terms that either relate to internal friction, to bottom friction or (in case of meteorological forcing) to wind stress. $f_C$ is usually set to zero (and then corresponds to mass conservation), but is used as a forcing term in mass movement induced tsunamis. The gradient terms often include pressure forcing (from atmospheric disturbances) or time dependent changes in the bathymetry. However, an instantaneous uplift from e.g. an earthquake rupture can also be modelled by an initial condition and will then enter in the term $I(q)$ above.
### 5.3.2 Interfacing Sources of Tsunamis in the Modelling Process

While most modelling approaches separate the source modelling from the tsunami propagation and inundation modelling steps, there are still many examples where source and tsunami models are integrated. Examples with 3D finite difference schemes using staggered pressure and velocity grids have been used by Saito and Furumura (2009), and Maeda and Furumura (2013). A variation on this is presented by Lotto and Dunham (2015) who implemented a different numerical scheme using summation by parts (SBP) finite difference operators with simultaneous approximation terms (SAT) for implementing boundary conditions. Panza et al. (2000) provided an alternative technique using the Haskell method to solve the equations of motion for a planar multilayer ocean and elastic layered model. In Chap. 4, other examples of coupled landslide tsunami models are described.

Interfacing sources with tsunami models often involves transferring physical processes that may not be representable with respect to the modelling approach in one of the parts. For example, fast seismic Rayleigh waves are not straightforwardly representable in a shallow water approximation to the hydrodynamic tsunami wave propagation models partly because they would result in supercritical wave speeds. Another example is the dispersion of the uplift signal through the vertical water column, which is not representable in a one-layer model without correction terms for the hydrodynamic response. The simplest method for modelling the hydrodynamic response is to copy the seabed deformation directly to the sea surface. This may work well for long wavelength seafloor deformations (e.g. wavelengths of more than 20 water depths), but for shorter seafloor disturbance, it is necessary to model the hydrodynamic response.

A commonly used technique to handle this later problem is a parameterization of this process. By considering the velocity potential, $\varphi$, which is related to the fluid velocity by $v = \nabla\varphi$. For non-breaking waves, it is possible to model this process using for instance boundary integral methods (e.g. Grilli et al. 1989; Grilli and Watts 2005; Pedersen 2008) taking into account full potential theory (for fully non-linear, fully dispersive non-breaking waves), but this is often not feasible due to the computational cost.

A frequently used method is to use close form expressions assuming linearity and instantaneous seabed deformation. A celebrated method for this purpose is the analytical solution of Kajiura (1963), which assumes constant depth and instantaneous deformation for a point source using linear potential theory. A similar solution assuming a rectangular deformation and slowly varying topography was further proposed by Nosov and Kolesov (2011). By summing different unit sources for an earthquake source, it is possible to take into account the hydrodynamic response for compound sources. Glimsdal et al. (2013) updated this to also take into account time dependent sources such as landslides, but this method could in principle also take into account time dependent earthquakes or volcano processes as well. To reduce computational efforts further, it is possible to simply assume smoothing length on
the depth deformations and utilise any kind of analytical smoothing function (in a similar way as done for the box shaped slides, Harbitz et al. 1991; Harbitz 1992), e.g. through Gaussian unit sources for earthquake tsunamis (e.g. Molinari et al. 2016). This process is faster than Kajiura filtering or Nosov and Kolesov's method, but will not take into account the correct hydrodynamic response as the smoothing length and shape is fixed.

### 5.3.3 Approximations and Simplifying Assumptions

The most common simplification to the equations of motion in tsunami modelling is the shallow water assumption, i.e. that the relevant length scales in the horizontal dimensions are large compared to those in the vertical, that the wave height is small compared to the water depth and that the wavelength is large compared to the water depth. Under these conditions, the vertical motion within the water column can be neglected and the two-dimensional shallow water equations can be derived by averaging over the depth. The shallow water equations omit the third energy-related equation in (5.1) and have simplified terms in the first and second equations.

$$q = \begin{bmatrix} h \\ hu \end{bmatrix}, F(q) = \begin{bmatrix} hu \\ hu \times u + \frac{1}{2}h^2 \mathbb{I}_2 \end{bmatrix}, S(q) = \begin{bmatrix} 0 \\ -gh\nabla b + f_M \end{bmatrix}$$

Under this assumption frequency dispersion is not possible. Also, non-hydrostatic velocity amplification over bathymetry is not well represented. It is also important to note that in this regime, only one homogeneous water layer exists, and thus a horizontal displacement of the bottom uplift shifts the whole water column (which may be an unrealistically large transfer of momentum). Some of the above-mentioned deficiencies may be parameterized. For example, the frequency dispersion effect can be parameterized to some extent (assuming constant water depth) by an artificial diffusive approach through the numerical resolution. The missing smoothing of the sea surface response may be parameterized by the above-mentioned Kajiura filter.

Alternatively, it can be assumed that only vertical deformation is considered in the generation of the tsunami. However, in the presence of a bathymetric slope this is not the case. The horizontal movement of the slope will cause a vertical displacement of the water column which contributes to the tsunami source. Tanioka and Satake (1996) calculated the total vertical displacement of water, D, using:

$$D = u_x \frac{\partial H}{\partial x} + u_y \frac{\partial H}{\partial y} + u_z$$

where $u_x$ and $u_y$ are the horizontal deformation of the seafloor, $u_z$ is the vertical displacement and $H$ is water depth (i.e. positive downwards). The effect of the additional uplift caused by accounting for the bathymetric gradient can be significant for
tsunami generation, particularly in the case of shallow earthquakes near the trench (Tanioka and Seno 2001).

A common further simplification is to assume the fluid is incompressible ($\nabla \cdot \mathbf{v} = 0$), (e.g. Saito and Furumura 2009; Maeda and Furumura 2013; Kajiura 1963); however this is not universal (e.g. Lotto and Dunham 2015; Nosov and Kolesov 2007).

A more complex, but still simplifying approximation to the full equations of motion are Boussinesq-type models (see e.g. Fuhrman and Madsen 2009). These modelling approaches provide more realistic wave frequency dispersion relations and a better representation of wave-induced flow patterns, compared to (hydrostatic) shallow water theory, while maintaining a certain amount of efficiency, compared to the full 3-dimensional Navier-Stokes or Euler Equations.

Note that for the presented simplifications of equations there are no significant differences in the input requirements. However, the representable physical processes differ substantially. While a fully 3-dimensional Navier-Stokes equation model is capable of representing (and thus are accepted as input) fast Rayleigh waves, as well as horizontal dislocations, this is not the case for shallow water equation based models, as described above.

### 5.3.4 Input

Modelling is generally performed in a two-step approach: in the first step the seafloor deformation or—in case of meteotsunamis—the sea-surface perturbation is simulated; in the second step the output from this simulation is used as an input for the hydrodynamic modelling. In this case there is no interaction between the two models. The advantage of this approach is that the complexity of the simulation is reduced, often decreasing the possibility of instabilities (e.g. at the fluid-solid interface). The disadvantage to this technique is the removal of the interaction between fluid and solid domains, consequently the full physics of the phenomena are not represented. See Chap. 4 for a description of source modelling. Coupled physics-based models which incorporate both source and hydrodynamic modelling do exist with a discussion on coupled source simulations provided in Chap. 4.

With this “separation of concerns” type of approach in mind, the interface may simply represent a layer between the models to interpolate between different mesh layouts or even just a driver to couple correct file formats. However, if different physical processes are involved, then the interfaces layer may also add some sort of filtering to provide the receiving model with appropriately modified information. This layer is also responsible for modifying outputs from one model to be suitable for the other. An example is the computation of wind stress values from wind velocities in meteotsunami modelling. A second example is the conversion of rupture velocity in complex earthquake modelling into bottom uplift. A third example can be a time dependent full potential filter conveying seabed deformation to sea surface perturbations due to various types of sources (i.e. earthquakes, landslides, volcanoes).
Note, that this filtering computation may be embedded directly into one of the major models. An analytical half space model based on a dislocation on a fault surface is usually included in tsunami hydrodynamic wave models (see Sect. 5.4.2), so that these can be initialised with simple text files containing the (possibly time-dependent) parameter sets.

### 5.3.5 Outputs

The output from the interface modelling is dependent on the source phenomena in question; it can be in the form of:

* *source terms* in the case of atmospheric tsunamis

* *initial conditions* such as sudden sea surface displacement for instantaneous earthquake or volcanic explosions

* *boundary conditions* like a time-dependent sea surface displacement for landslide, volcano (pyroclastic flow, eruptive column collapse, caldera collapse), and earthquake sources.

### 5.3.6 Data Requirements for the Interfaces

The input data varies depending on the source and the type of model. For the following discussion, it is assumed that source simulations are separate from the hydrodynamic models (i.e. uncoupled models).

* **Earthquake:** in most cases the static seafloor deformation is used (see Sect. 5.4.2). This assumes that the earthquake process is instantaneous and the rupture velocity (i.e. the velocity at which the slipping area spreads across the fault) is much faster than the tsunami wave propagation. Alternatively, the time series of the dynamic seafloor deformation can be used, which would account for the temporal evolution of the earthquake but will increase computational resources. It is worth mentioning that time-dependent source interfacing may be important in particular in large source area situations. Sensitivity of the exact source location and timing is large for near-field tsunami simulations. In case of an instantaneous seafloor deformation initial conditions may be defined. Otherwise, the seafloor deformation is given as a forcing term to the equations.

In cases where ground motions from dynamic simulations are being used, care needs to be taken when interfacing to simple (e.g. shallow water) approximations to the wave propagation simulation. In particular Rayleigh waves may create numerical artefacts, since their propagation speed is much faster than the shallow water gravity wave speed and can therefore be removed in such simulations. A viable approach may be to filter such waves (Madden et al. 2021). Bathymetric data
is required to accurately account for the contribution that horizontal deformation will make to the tsunami source.

* Landslide (submarine, subaerial): A variety of methods are used (most are time dependent, while only some assume an initial condition). For time dependent models it is necessary to employ an interface model at a given time step in the propagation. Then it is necessary to make sure that time stepping related to updating the (volumetric) source contribution to the wave model is sufficiently frequent (Zengaffinen et al. 2020; Zengaffinen-Morris et al. 2022).

For each individual update of the source model feeding into the tsunami model, it is clear that the choice of the interface model will have a strong flavour of the selected source model (see Chap. 4). For initial conditions, the shape should consider integral contribution of both the source mechanics and the water response, and no filtering or interface model should in principle be needed. For block models and depth averaged landslide models, smoothing filters are needed, either of Boundary Integral Methods (BIM) type or Kajiura type taking into account the physics, or simpler models that have as the main purpose to remove noise. For block models, smoothing the box shape may render interface models unnecessary if the sole purpose is to remove noise. For coupled models the tsunami modelling is already embedded, and interface models are in principle not necessary. On the other hand, it is stressed that the physics in some coupled models may involve simplified coupling terms (e.g. copying the landslide response to the wave model without filtering), and that the user must assess if the coupling mechanism in each model is sufficiently accurate.

For the models where the landslide modelling is carried out independently of the wave model, the source displacement is conveyed to the wave model at several output times. Ideally, this displacement pattern should be fed into the wave model at every time step. However, interface modelling may require significant computational resources. Therefore, it might be necessary to set a larger time interval for conveying the source information into the wave model. Convergence tests (e.g. Zengaffinen et al. 2020) should be undertaken to optimise this time interval, as the requirement may vary from case to case.

* Volcano: Tsunami sources in a volcanic context are diverse: underwater explosions, caldera collapses, pyroclastic flows, eruptive column collapse, landslides, etc. There is therefore no single solution for modelling the volcanic sources of tsunamis. A distinction will be made between: (1) static or instantaneous sources, such as underwater explosions, where a displacement of the surface of the water is imposed as the initial condition; (2) time-dependent sources, such as pyroclastic flows, eruptive column collapse, and caldera collapse (see Sects. 4.3.3 and 4.4.3 for more details on these source mechanisms and their modelling). Caldera collapses can be treated with a simple approach, using sea floor motion as a finite-time forcing term in a depth-averaged model, whereas pyroclastic flows and eruptive column collapses require the use of complex CFD full Navier-Stokes
type models. Indeed, the physics of tsunami generation by pyroclastic flows and eruptive column collapse is so complex that it justifies the use of advanced models considering several fluids and their interactions. However, this type of advanced modelling is computationally very demanding and currently not applicable for PTHA.

* Atmosphere: There are several terms that control the tsunami behaviour of a meteorological tsunami. Most importantly, the air pressure forcing at the sea surface, provided as a time-dependent perturbation and causing an uplift or downdraft of the surface elevation, and the wind stress pushing the sea surface by a friction term. To have a significant tsunamigenic effect, these terms need to be represented with sufficient spatial and temporal resolution. The air pressure as well as the wind stress are tsunamigenic only if they correspond to resonance frequencies related to the topo-bathymetric situation. Therefore, exact location, timing, extent and magnitude of the air pressure perturbation and wind speed anomaly need to be used to force the tsunami model.

Therefore, the continuous time-dependent mean sea level atmospheric pressure and wind stress forcing must be introduced within the model equations. Consequently, adding two supplementary terms in the equations, reading the atmospheric forcing at a given time-step (e.g., every 1-min to 5-min) and interpolating this forcing at every tsunami model time-step is required. Further, for weather related events (i.e., events driven by mesoscale atmospheric processes), meteotsunamis are generally an order of magnitude smaller than tsunamis driven by other sources. Tides and storm surges can thus play a critical role in the final extreme sea level hazard. It is also worth mentioning that up to 70% of the meteotsunami wave transformation and amplification occur nearshore where the resolution of the tsunami models should have an order of just tens of metres. Finally, for sonic boom related events (i.e., events driven by planetary acoustic-gravity waves, such as the Hunga Tonga-Hunga Ha'apai event, see Lynett et al. 2022), the tsunami model should be able to run globally (see propagation section below).

### 5.3.7 Technical Requirements

Technical requirements are dependent on the type of coupling between source and interface simulations. In the case of direct coupling, the output from the source simulation will be automatically included. Separate simulations for the source and hydrodynamics will involve file transfer in the form of the output from the source model used as an input of the hydrodynamic model. In this case either self-defining binary data formats such as NetCDF or HDF5 or standard ASCII formats such as Esri™ ASCII are employed (the binary formats will be preferable because of their storage requirements and standardised inclusion of metadata).

An alternative approach is to include a kinematic model of the source embedded directly in the tsunami propagation model. The last technique is commonly employed when modelling earthquake-generated tsunamis with the earthquake represented as
a slipping surface (see Sect. 4.1 for a discussion on the topic). HPC computing resources are generally not used for this method.

Interfacing to landslide or meteorological sources is usually more involved. The inputs used in the interface to a landslide forcing are typically binary or ASCII formatted files in the local source area. As the source domain is typically smaller and more finely resolved, interpolation in time and space is needed. The time steps of the inputs from the source model typically range from a few seconds (e.g. Zengaffinen et al. 2020) to about one minute (e.g. Kim et al. 2019). Typically, one needs to employ smoothing (see Sect. 4.3) either using box type sources or physically based full potential type filters to remove short wavelength components. The inputs used in the interface to meteorological forcing are 1-min to 5-min mean sea level pressure (and/or wind stress) time and space varying forcing that is interpolated on the tsunami model grid/mesh. This forcing is either generated by atmospheric models (realistic forcing: see Deterministic Atmospheric Models in Sect. 4.5.2) or consisting of parametric equations (synthetic forcing; see Synthetic Catalogue in Sect. 4.5.3). Due to the amount of data, binary formats (NetCDF or HDF5) should be used to store the multidimensional inputs when realistic forcings are used. Additionally, tidal information and appropriate forcing needs to be considered.

### 5.3.8 Treatment of Uncertainties for the Method

In this section first we discuss the propagation of uncertainty in the representation of the source, afterwards uncertainties within the method are analysed. Single realisations of the interface methods are deterministic. However, the method is able to propagate uncertainty in the input due to the small computational requirements in the case where the interface between seafloor and sea surface is represented by a Kajiura filter. This allows uncertainty representation from the source (e.g. heterogenous slip for earthquake) by including multiple source representations in probabilistic analysis, by assigning conditional probabilities to each model realisation.

A potential source of uncertainty in this method is the effect of simplifying the physics of hydrodynamics. As mentioned above, a common assumption used in many studies is that sea water is an incompressible fluid. An additional uncertainty may be triggered in the way different physics interact (e.g. between the elastic and acoustic domains). Several full potential based filter models assume constant depth and deep-water tsunami generation (linearity). Uncertainty in parameters such as the density of the water or the kinematic viscosity are generally not accounted for, but methodologies based on multiple simulations using Monte-Carlo type variations or a probability density function for expected ranges of parameters could conceivably be applied to account for such uncertainty.

The consequences of a lack or poor resolution of input data depend on the sensitivity of each case. For example, deep bathymetric data may be safely interpolated while the resolution of disturbance data in atmospheric tsunami sources are critical.
In the case of seismic sources, the sensitivity of the tsunami source to the bathymetric gradient is dependent on geometry and the depth of the slip, i.e. large slip at shallow depths and fault dip can be highly sensitive to gradient. The representation of the physics and the treatment of uncertainty between the source and subsequent models need to be consistent with particular importance on the interface between models where sensitivity may be high (e.g. bathymetric gradient). Poor treatment of source uncertainty will propagate through PTHA, this has been demonstrated for seismic sources where simplification of earthquakes to fixed size rectangles with uniform slip produces lower hazard appraisals compared to more complicated models (Davies 2019; Davies and Griffin 2020).

## 5.4 Propagation

### 5.4.1 Introduction

As discussed above, the modelling and simulation of tsunami propagation is based on mostly non-linear partial differential equations. Three equation sets are commonly used to model the tsunami propagation and inundation, namely the (linear or non-linear) shallow water models, Boussinesq type long wave models that consider non-linearity and frequency dispersion, and complete fluid dynamic (i.e. Navier-Stokes) type models.

Numerical tsunami propagation models describe the propagation of tsunamis from the source area towards coastal waters. Forecasts of tsunami sea surface elevation and first wave arrival along a coastline and approximations for inundation maps use the outputs of those simulations to serve warning purposes and coastal management and planning.

The equations are solved using numerical methods, such as finite difference, finite volume, or Galerkin-type methods, to obtain a solution to the tsunami wave propagation problem. Hydrodynamic models solve empirical and governing equations at different complexities. The limitations of tsunami modelling result from the limitations of the input data, which may contain errors, interpolations, or estimates. An overview of the contents of this section can be found in Table 5.4.

### 5.4.2 Physical Approximations

#### Linear and non-linear Shallow Water Models

The shallow water equations (SWE) describe the evolution of the water surface elevation with time and the depth-averaged water particle velocity, or alternatively, the momentum evolution. They are derived from the Euler equations by depth averaging and neglecting vertical components of acceleration and velocity. To be valid, such
<table><caption>**Table 5.4** Summary table propagation</caption><thead><tr><th colspan="2">Short description of method</th></tr></thead><tbody><tr><td>Theoretical/mechanical background for the method</td><td>This method solves hydrodynamic equations to propagate the tsunami wave, initiated by a source to the coast</td></tr><tr><td>Short outline of how the method works</td><td>The method works as a partial differential equation solver and may utilize diverse discretization schemes, such as finite differences, finite volumes, finite elements, etc.</td></tr><tr><td>How is/are the output (s) of the method represented?</td><td>The outputs of this method may be direct lists of hazard metrics (such as wave height at the coast) or maps</td></tr><tr><td>In what way does the method take into account time-dependent/dynamic aspects?</td><td>This method usually is fully dynamical and processes large numbers of timesteps</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>Most propagation models use parallel processing and HPC infrastructure. Some models use GPU acceleration. ML is rarely used for this method, however surrogate models have been proposed</td></tr><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>There is no alternative method</td></tr><tr><th colspan="2">Data and technical requirements for method</th></tr><tr><td>What input data does the method use?</td><td>Input from interface method and data method</td></tr><tr><td>What are the technical requirements?</td><td>Well-tested modelling software needs to be used</td></tr><tr><td>Are HPC computing resources necessary, would absence of these compromise the use?</td><td>Not strictly necessary, but almost always utilized. In particular if probabilistic approaches are intended then large numbers of simulations require the use of HPC infrastructure</td></tr><tr><td>Is the method/workflow available as a git/ software/jupyter notebook?</td><td>There are many different models available, and listed in the chapter</td></tr><tr><th colspan="2">Treatment of uncertainties for the method</th></tr><tr><td>Is the method itself deterministic or uncertainty based?</td><td>Fully deterministic, but used for probabilistic assessment by Monte-Carlo approaches</td></tr><tr><td>Which are potential sources of uncertainties? Which parameters are used to characterize them?</td><td>Uncertainty may arise from improper choice of model for sources</td></tr><tr><td>How are uncertainties quantified for the identified parameters? Are there significant correlations between different uncertain parameters?</td><td>Sensitivity analysis should be applied to quantify uncertainty</td></tr></tbody></table>

(continued)
<table><caption>**Table 5.4** (continued)</caption><thead><tr><th>Short description of method</th><th></th></tr></thead><tbody><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? If yes, how are uncertainties propagated?</td><td>This method may consider and propagate uncertainty only by ensemble modelling</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>Relevant hazard metrics may be rendered incorrect</td></tr><tr><td>What are the consequences of ignoring or poorly treating uncertainties?</td><td>Results may be very incorrect</td></tr></tbody></table>

simplifications require the surface elevation to be small and the wavelength to be large compared to the total water depth, which is true for tsunami offshore propagation.

These equations constitute a good approximation for tsunamis generated by subduction zone earthquakes since the seafloor deformation over a large region generally leads to tsunamis with a wavelength of several tens of kilometres or more, compared to the average ocean depth of only few kilometres. A gravity wave behaviour can represent tsunami propagation in the deep ocean, with negligible particle velocities (Behrens and Dias 2015). The shallow water equations describe the conservation of mass and momentum and are solved numerically.

The negligible particle velocity allows for a further simplification: by omitting the (velocity driven) advection term linear shallow water models (LSW) are obtained. Linearity leads to better numerical stability properties enabling faster computing times and simpler discretization schemes. In any case the two prognostic variables of such models are the wave height and the momentum or (particle) velocity.

As these models are limited to a single layer of water, they can only account for depth-averaged motions (which might be too simplistic for complex tsunami problems). They are only valid for relatively small amplitude waves.

LSW models can adequately predict tsunami waves in deep water, up to a water depth corresponding to near-shore regions (say 50 m), while non-linear models including moving boundary treatment may be used to simulate inundation as well (Behrens and Dias 2015). When the simulation involves a large region and propagation time (transoceanic propagation), the Coriolis' effect should not be neglected (see e.g. Omira et al. 2022). Also, earth's curvature needs to be taken into account in this modelling regime.

The computational stability depends on the numerical method used, and in case of explicit methods on the relationship between the time step and grid cell size. Temporal and spatial accuracy should be maintained to ensure the accuracy and stability of the computations which in turn can minimise unrealistic effects on the solution. The numerical stability can be assessed with the CFL (Courant-Friedrichs-Lewy) stability criterion. This criterion states that wave velocity, in other words the information
propagation velocity, does not exceed the discretely representable velocity. Other stability criteria may also be applicable, but the discussion of numerical stability goes beyond the scope of this chapter.

It is worth noting that explicit time stepping methods are mandatory for tsunami modelling, since implicit methods are damping the gravity wave propagation too severely (an early analysis of the effect of explicit vs. implicit numerical methods on shallow water gravity wave propagation can be found in Harding 1982). While in other disciplines, where the flow regime is the important feature, this smoothing effect is desirable for improved stability, in tsunami wave propagation, it would render the simulation results unrealistically smoothed.

The inputs to the model include the source model and bathymetric grid. The results of the simulations depend heavily on the quality of bathymetric data (see Sect. 5.2) and on the numerical scheme used to solve the equations. Non-linear shallow water models (NLSW) including near-shore and on-shore (inundation) computations often necessitate additional input parameters related to treating the variable shoreline and bed friction, in particular for the inundation modelling. Usually, the complex small-scale behaviour of a tsunami inundating land is represented by a roughness parameterization, given by (a generalised) Manning roughness. Computational results are sensitive to this parameter (see Sect. 5.5).

Most SWE codes are computationally fast and inexpensive and run on a simple PC. High-speed processors (e.g. GPUs, de la Asunción et al. 2013) or compute clusters may be used to shorten the computation time. Non-linear and linear shallow water models are commonly used for early warning purposes. They provide—within the range of uncertainties given by the limited forecast time (see Greenslade et al. 2014)—accurate forecasts for expected wave heights, arrival times and possibly inundation areas.

For commonly used and validated codes refer to Lynett et al. (2017). Another set of models specific for landslide induced tsunamis can be found in Kirby et al. (2022).

## Business Models

There are situations in which the shallow-water equations are inadequate, in particular for short wavelength tsunamis and when non-linear wave-to-current interactions are of importance (e.g. in ports, see Glimsdal et al. 2013; Ma et al. 2012; Løvholt et al. 2015; Lynett et al. 2012).

Boussinesq type models are depth-averaged models accounting for nonlinear and frequency dispersion. These models are based on the Boussinesq equations, and can to different extent consider friction, Coriolis force, and consider wave breaking and energy dissipation. Tsunamis travelling long distances can be significantly modified by effects of frequency dispersion (Glimsdal et al. 2013). Dispersion is also relevant for certain non-seismic sources such as landslides and volcano flank collapses (Løvholt et al. 2015). The inputs of the code are the source model and bathymetric grid, similar to shallow water models. In general, these models provide accurate predictions of wave height, arrival time, and momentum flux.

Like shallow water models, Boussinesq type models are affected by the method used to discretize the model equations. Due to the existence of higher order and
mixed derivatives in the equation set, stability is much more difficult to achieve in Boussinesq-type models (e.g. Løvholt et al. 2009; Løvholt et al. 2013) compared to shallow water models. While higher order Boussinesq equations comprise mixed higher derivatives and are often difficult to discretize in a stable way, an alternative (and for some equation sets equivalent, see Jeschke et al. 2017) approach is given by a non-hydrostatic correction method (Casuli and Stelling 1998; Yamazaki et al. 2009). The correction then empowers shallow water models to represent wave frequency dispersion and non-linear velocity amplification. To ensure the stability of the model it is important to use suitable numerical techniques and appropriate high resolution meshes and time stepping schemes. Combining finite volume techniques, use of limiters, and turn off dispersion terms in shallow water have also improved stability in the near shore propagation (e.g. Shi et al. 2012; Kim et al. 2009, 2017).

A related class of models are the layered models, that assume depth averaging in each layer (e.g. Ma et al. 2012; Macias et al. 2021). They may extend the validity range compared to Boussinesq models, but will generally require more computational resources.

These models are adequate to simulate tsunami propagation in coastal regions, to properly describe transoceanic tsunami propagation, and to some extent even to represent tsunami structure interaction.

## Full Navier Stokes models

The three-dimensional or Full Navier-Stokes equations are the most complete description of fluid movement. This description is needed to describe complex wave interactions and general treatment of vorticity. Full Navier-Stokes simulation is a complex sophisticated process. It involves solving the incompressible Navier-Stokes equations in a two- or three-dimensional domain to model the flow of water. The use of these models requires sophisticated numerical methods on parallel computers to calculate the models accurately. In most cases this is computationally much too costly, and for most tsunami applications the vertical structure of the flow does not need to be modelled directly (Grezio et al. 2017).

The physical parameters like the water density, gravity, water surface elevation, and boundary conditions must be properly specified in the model. After that, the model can be used to simulate the evolution of a tsunami from the source over a certain duration of time. The simulated results can be used to study the effects of tsunamis on coastal regions and assess the potential local hazards posed by tsunamis.

It should be noted that Navier-Stokes equations are usually not adequate for simulating large scale geophysical phenomena due to the geometry and relatively thin layer characteristic of such processes. Therefore, certain approximations (e.g. hydrostatic layering, spherical geometry, etc.) can be made to adjust the equation set to such applications. Currently, the authors are not aware of any successful application of a three-dimensional Navier-Stokes-based model to global or large-scale tsunami modelling.

However, local phenomena, such as fluid structure interaction, debris transport, inundation processes, or fluvial landslide tsunamis, often require a sophisticated equation set given by Navier-Stokes equations (see Sect. 5.5).
### 5.4.3 Discretization Methods

Numerical methods used in tsunami simulations must consider the physical approximations used to model wave behaviour. The most used numerical methods for solving the shallow water wave equations are the finite-difference (FD) and finite-volume (FV) methods. Besides these, also Galerkin methods—finite element (FE) or discontinuous Galerkin (DG) methods—and smoothed particle hydrodynamics (SPH) methods are in use. Except for SPH, most of these methods rely on discretizing the domain into a grid of small cells. In the finite-difference method, the equations are approximated using discrete differences between cell centres or cell vertices, while the finite-volume method involves integrating the differential equations over each cell volume.

#### Finite Differences

The finite difference method uses approximations to the derivatives by finite differences to approximate the analytical equations of motion and the associated exact solutions. This means that the domain is discretized into grid points (nodes) and the differential terms of the equation are converted into an algebraic equation, the so-called finite-difference equation. The dependent variable values are stored at the nodes only. Early tsunami simulation codes used the FD method (see Imamura 1996). It should be noted that FD discretization methods are suitable for modelling large domains, but they require fine meshes to capture the wave behaviour accurately.

Several tsunami simulation codes use FD discretization schemes, including for instance: TUNAMI-N2 (Imamura et al. 2006) MOST - Method of Splitting Tsunamis (Titov and Gonzalez 1997), COMCOT (Liu et al. 1998), NSWING Nonlinear Shallow Water (Miranda et al. 2014).

#### Finite Volumes

The finite volume method is based on the Eulerian perspective of fluid flows. The domain of interest is partitioned into (small) discrete volumes (cells) and the conserved variables in cells are evolved in time using the fluxes with neighbouring cells. By this approach FV methods achieve exact discrete conservation of the main conservation properties of the governing equations, such as mass and momentum conservation for the shallow water equations.

Since FV methods assume averaged quantities within cells, they are usually of low convergence order. On the other hand, because no rigorous assumptions on the shape and size of the cells are necessary, FV methods are well suited to represent complicated geometries and are also suitable for locally refined meshes. Nevertheless, rectangular meshes are mostly used and nested grids implemented for high-resolution inundation.

Examples of numerical tsunami codes using Finite Volumes: Tsunami-HySEA (Macías et al. 2017), Geoclaw (LeVeque et al. 2010), or ANUGA from Australian National University and Australia Geosciences (Nielsen et al. 2005).
## Galerkin Methods

Galerkin methods, such as FE and DG, are based on the idea of representing the solution functions by polynomial approximations. So, instead of looking for the true continuous solution function, a simpler approximation function—a polynomial, which has nice properties with respect to derivatives and integrals—is derived. To be efficient, the domain is again partitioned into small cells (the elements) in order to provide local support for the approximating polynomials and eventually leading to an algebraic equation with a sparse matrix. In FE methods a piecewise continuous polynomial representation of the solution is achieved. DG methods on the other hand use the same polynomial approximation within cells, allow for discontinuities between elements and represent their coupling by numerical flux functions. Therefore, DG methods can be viewed as high-order FV type methods.

Finite element methods are conservative in an integral sense, so mass and momentum are conserved, when these quantities are integrated over the whole domain. DG methods—as well as FV—are locally conservative, meaning that balances between cells are exactly represented discretely. Galerkin type methods are well suited for local mesh refinement, even dynamic adaptive mesh refinement implementations have been presented.

Examples of Galerkin type codes are TsunAWI (Behrens 2008; Rakowsky et al. 2013), TsunaFlash (Pranowo et al. 2008), StormFlash (Vater et al. 2019).

## Smoothed Particle Hydrodynamics

Smoothed-Particle Hydrodynamics (SPH) is a computational method for simulating fluid flow, based on the Lagrangian representation of fluids by particles. The equations of motion and properties of these particles are determined from the continuum equations of fluid dynamics taking interactions of particles into account by interpolation from neighbouring particles (Monagan 1992, 2005, 2012; Rogers and Dalrymple 2008). SPH is meshless, it does not require the use of a grid to calculate spatial derivatives obtained by analytical differentiation of interpolation formula.

The SPH algorithm has several advantages over classical computational methods. Firstly, it can efficiently simulate complex flows involving moving and deforming objects. Secondly, the SPH method naturally conserves mass, momentum, and energy by satisfying the continuity and momentum equations. It has the standard features of a robust numerical solver and is highly accurate with no numerical dissipation along Lagrangian paths (see Sarfaraz and Pak 2017; Pringgana et al. 2021; Reis et al. 2022). SPH can handle complex geometries and simulate the fluid flow around obstacles being particularly useful in problems of fluid structure interaction or in our case in tsunami structure interaction. The method can be computationally demanding but offers high accuracy and versatility in modelling complex fluid flows, such as those involving turbulence, shocks, or free surfaces. SPH methods are mostly used for near field simulations, since in far field propagation the particle density tends to decrease due to spreading, leading to more and more diffusive results.

Well-established SPH software for tsunami simulation are not yet available to the knowledge of the authors.
## 5.5 Inundation

### 5.5.1 Introduction

The aim of this method is to present methods for near shore propagation and inundation modelling, i.e. interaction between the tsunami and the coast includes near shore propagation and inundation.

The numerical simulation of inundation aims at determining the areas that are inundated, as well as describing the variables that characterize such inundation and the process in which the inundation occurs.

The inundated area is determined by the interaction between waves and elements near-shore and onshore. This interaction can be assessed following several approaches, mainly depending on the way in which the numerical model solves the frontier from offshore to onshore, from bathymetry to topography. An overview of this section's contents is included in Table 5.5.

### 5.5.2 Calculation Approaches

A first approach is the application of models that solve the inundation as a continuation of, or coupled to, the propagation, using the same equations or codes applied for the propagation. This is the case of the models mentioned in the previous propagation section: in particular the Non-Linear-Shallow-Water-Equations (NSWE) models, such as Tsunami-HySEA (Macías et al. 2017), GeoClaw (Berger et al. 2011) COMCOT (Wang and Liu 2011), MOST (Titov et al. 2016), StormFlash (Vater et al. 2019), and many others.

Other approaches separate the processes that occur before and after the shoreline, by coupling two different models. The reasons for this separation can be several, such as using the models that are best suited for each process, or avoiding the high computational cost that is usually linked to the simulations of this type of process. There are two ways in which this coupling is addressed. First, by sharing files between models, meaning that the output of the propagation model is processed to be used as input for an inundation model. And second, an actual coupling in which the whole process occurs in a continuous calculation.

In addition, there are other approaches to inundation calculation. Analytical solutions remain as an alternative, especially in cases where not all the necessary data for numerical modelling is available (Park 2015; Madsen 2010; Aniel-Quiroga et al. 2018). The method of amplification factors is another simplified method that uses look up tables for extracting inundation heights globally or regionally (Løvholt et al. 2012; Glimsdal et al. 2019). These methods incorporate the uncertainty treatment of the maximum inundation height. A related but more elaborate method is the Energy Grade line approach (Kriebel et al. 2017). Machine Learning and emulators trained
<table><caption>**Table 5.5** Summary Table Inundation</caption><thead><tr><th>Short description of method</th><th></th></tr></thead><tbody><tr><td>Theoretical/mechanical background for the method</td><td>This method computes the inundation by tsunami waves on land. It is either based on hydrodynamic modelling or on heuristically relating wave height at the coast, terrain characteristics to inundation height</td></tr><tr><td>Short outline of how the method works</td><td>The method works locally or regionally and computes some of the most important and relevant hazard metrics, such as flow depth, flux, or inundation height</td></tr><tr><td>How is/are the output(s) of the method represented?</td><td>Either as metric values at points of interest or as maps</td></tr><tr><td>In what way does the method take into account time-dependent/dynamic aspects?</td><td>This method may output time dependent values, but is mostly used to compute maximum or statistical (fixed) values</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>Most hydrodynamic inundation models use parallel processing and HPC infrastructure. Heuristic inundation schemes are often implemented as postprocessing schemes and may use parallelism in probabilistic (Monte-Carlo) settings</td></tr><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>This method has to be used with care, since inundation results are very sensitive with respect to accuracy of topography data as well as land use structure</td></tr><tr><td colspan="2"><em>Data and technical requirements for method</em></td></tr><tr><td>What input data does the method use?</td><td>Input from hydrodynamic propagation models, and topographic data</td></tr><tr><td>What are the technical requirements?</td><td>Very accurate and validated topographic data sets are required, as well as accurate and validated parameterizations (usually roughness parameterization) to account for land use characteristics</td></tr><tr><td>Are HPC computing resources necessary, would absence of these compromise the use?</td><td>For hydrodynamic modelling HPC resources are required</td></tr><tr><td>Is the method/workflow available as a git/ software/jupyter notebook?</td><td>There are many different models available and listed in the chapter</td></tr><tr><td colspan="2"><em>Treatment of uncertainties for the method</em></td></tr><tr><td>Is the method itself deterministic or uncertainty based?</td><td>Hydrodynamic modelling is fully deterministic, but heuristic methods are often based on probabilistic approaches</td></tr><tr><td>Which are potential sources of uncertainties? Which parameters are used to characterize them?</td><td>Uncertainty may arise from incorrect data or improper calibration of roughness parameterization</td></tr></tbody></table>

(continued)
<table><caption>**Table 5.5** (continued)</caption><thead><tr><th>Short description of method</th><th></th></tr></thead><tbody><tr><td>How are uncertainties quantified for the identified parameters? Are there significant correlations between different uncertain parameters?</td><td>Sensitivity analysis should be applied to quantify uncertainty. Land use and topography data may contain interdependent errors, since data may be spoiled by canopy effects</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? If yes, how are uncertainties propagated?</td><td>This method may consider and propagate uncertainty, e.g. by ensemble modelling</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>Relevant hazard metrics may be rendered incorrect</td></tr><tr><td>What are the consequences of ignoring or poorly treating uncertainties?</td><td>Results may be very incorrect</td></tr></tbody></table>

using simulations are becoming increasingly popular for rapid inundation height estimation (e.g. Mulia et al. 2022). A typical simplification that has been traditionally applied consists of the so-called Bathtub approach, in which the maximum run-up (maximum elevation gained in the flooding process) of the inundation is directly assumed as the tsunami height on the coast. This simplification provides a conservative approach. The bathtub approach is often combined with simplified inundation models for extrapolating the inundation height over topography.

### 5.5.3 Calculation Scales

The tsunami inundation analysis can be conducted at regional or local scale, which is determined by the aim of the assessment, the resolution of the analysis and the quality of the available data. Due to the scale differences between the wider ocean basin and local structures at the coast, variable resolution techniques play a crucial role for inundation simulations. Many tools employ a nested grid technique, whereas others use non-uniform unstructured meshes or adaptive mesh refinement (Harig et al. 2008; Pranowo et al. 2008; LeVeque et al. 2010). Thus, the numerical simulation of the tsunami inundation starts with the preparation of topo-bathymetric data into appropriate high-resolution grids required by the numerical model. The grid-spacing must be high enough to resolve coastal features and to guarantee accurate results. Some previous sensitivity analysis and calibration with test cases (see the following Sect. 5.6) may be necessary. To build these grids, the topographic and bathymetric data is interpolated. Special care needs to be taken in the definition of the coastline since it is common to observe a gap between the last topographic value and the first bathymetric one (see Sect. 5.2 on data preparation, above).
### 5.5.4 Interaction Between the Tsunami and the Coastal Elements

Inundation models, or those parts of the model that evaluate the inundation, involve solving the interaction between the flooding wave and the land. In this interaction the roughness of the type of land onshore is included by means of a friction coefficient (usually de Chezy or Manning roughness is considered). These friction coefficients incorporate the way in which the interaction retards the wave in the inundation process (e.g. Gibbons et al. 2022). Different kinds of land surfaces can be represented by different values of the roughness coefficient to mimic differences in surface resistance (see e.g. Kaiser et al. 2011 for a discussion).

In addition, if the topographic data, given by a Digital Terrain Model (DTM) or Digital Elevation model (DEM), are accurate enough, buildings or other small-scale obstacles existing in the potentially flooded area can be included in the grid, working as a barrier for the inundation.

Similarly, coastal protection structures can be represented in the small-scale grids. These structures are the first barrier that the coast presents to the tsunami, and their performance is crucial for diminishing tsunami hazard (Rosetto 2011; Aniel-Quiroga 2019). As they are part of the coastline, coastal structures are included directly into the topographic grid. In addition, some models calculate this interaction in a vertically two-dimensional approach, such as volume-of-fluid models. Finally, both coastal buildings and structures can be modelled by means of full three-dimensional Navier-Stokes based models (e.g. OpenFOAM, Guler et al. 2018), that may represent the interaction of structure and fluid flow, including debris, more accurately, although the computational cost is huge, and normally not feasible for implementation in PTHA.

### 5.5.5 Outputs

The final result of the inundation is the characterization of the inundated area. This is commonly represented by a tsunami inundation map containing the calculated variables.

These variables are normally related to the following definitions (see also IOC Tsunami Glossary 2019):

* Inundation depth ($I_{Depth}$), more often referred to as *flow depth*: Depth, or height of the tsunami above the ground, at a specific grid point of the output map.

* Inundation length or inundation distance ($I_{Length}$): The horizontal distance inland that a tsunami penetrates, generally measured perpendicularly to the shoreline.

* Maximum run-up (Ru): Difference between the elevation of maximum tsunami penetration (inundation line) and the sea level at the time of the tsunami.

* Maximum inundation height (MIH): The maximum elevation reach above sea level along a transect between the coastline and the trim line.
* Tsunami travel time: Time required for the first tsunami wave to propagate from its source to a given point on a coastline.

* Trim line: the iso-line of maximum inundation lengths.

* Flow velocity (u): the particle velocity of the fluid, mostly relevant only in inundation simulations.

* Momentum flux (d_f): velocity squared multiplied with flow depth, relevant for loads on structures in inundation simulation.

* Inundation Area: Area flooded with water by the tsunami.

* Forces on structures: Buildings, coastal structures

Commonly, these variables are directly obtained from the numerical simulations and represented in tsunami inundation maps that have several applications. One of the most important ones is its use as input for evacuation models towards the proposal and design of evacuation routes and plans. While the evaluation of the tsunami travel time in deep water can be computed accurately, tsunami behaviour onshore is a quite more complex problem, involving friction coefficients, land use, presence of coastal structures and buildings, etc. For determining it accurately, an accurate near shore bathymetry is necessary. Tsunami traversal, offshore and onshore, provides important information used to design potential evacuation routes in a tsunami-prone area.

Finally, the calculation of the inundated area by a tsunami opens the door to setting the limits of the exposed assets and helps to properly assess their vulnerability. Their combination allows addressing the analysis towards tsunami risk assessment.

## 5.6 Testing, Validation, and Verification

### 5.6.1 Short Description of Method

The aim of the present method is to characterise the models: their behaviour to uncertain input data and parameterizations, the accuracy/precision of models compared to mathematically or numerically derived benchmark solutions, in reproducing laboratory data and, finally, compared to observed data. This is necessary to ensure the suitability of the model for physically robust and sound simulations in parameter regimes that lack previous measured data. Or in other words, a validated model is reliable for forecasting purposes.

Corresponding methods for testing and validation are:

1. Sensitivity to input data (in particular to uncertainties in the generating source, to bathymetry/topography uncertainties and errors, to the numerical resolution, or bottom friction parameterizations, friction coefficient, etc.).

2. Verification, i.e. testing the model specifications. In other words, does the program work without errors and does it produce the expected results for specified inputs.
3. Validation, or testing the model against the real examples, making sure that it represents the modelled phenomenon in a useful way, which can be tested using experimental data or field data.

4. Testing of a complete workflow in particular applications as, for example, early warning systems.

## Sensitivity analysis to model parameters

Sensitivity analysis consists in testing the behaviour and variation in model results when the different model parameters are varied within a range of reasonable values. When model parameters are too many this leads to a large number of combinations to be simulated in order to cover the full set of parameter combinations. This also includes sensitivity to bathymetry as it is a source of uncertainty (possible errors and gaps, coarse resolution, inhomogeneities, etc.). The role of numerical resolution is also relevant here and must be addressed as simulations in general need to have a compromise between precision and computational cost. We also mention parameterisation of friction parameters, in particular when inundation is addressed. Nevertheless, when only propagation is computed this is not a key issue. The use of constant friction vs variable friction coefficients is also an important issue for high-resolution inundation studies.

## Verification

Verification is the first step in model benchmarking, this is defined as producing model results that conform expected behaviour in a defined and known context (within model specifications). This is achieved by comparison of model results against analytical solutions, quasi-analytical approximate solutions and model properties, such as mass and momentum conservation or other metrics. This means verification can to some extent be automatized. It is good practice to use automatic code checking and analysis as well as reproducibility standards.

## Validation

Validation is a second step in benchmarking where experimental and observed field data are used to check model results. The formal definition of validation is to ensure that model results compare reasonably with the physical world the model is designed for. This part is crucial as we want to reproduce real world events and at the same time represents a great challenge as high-quality observed data are scarce and they will always include larger measurement errors. Expert knowledge is necessary for model validation, since often uncertainty in representing the real world needs to be considered in setting up quality metrics.

## Testing

This part consists in checking the correct functioning of a workflow including simulations in the process. In a general framework, a complete workflow could include acquisition and preprocessing of input data, filtering, model simulations,
post-processing of output data and visualisation. While each component of the work-
flow needs to be verified and validated, this alone does not guarantee the functional
efficiency and operational reliability of the whole system, because the interplay of
the components needs to be tested as well.

For all the above steps it is necessary to formulate useful testing protocols with
appropriately defined metrics for quality control. This involves how much deviation
from predefined expected output values are allowed in the verification process, and
involves decision of the importance of testing metrics, e.g. whether the inundation
area of the model should correspond to the inundation area of the real event at
specified points or in an integral sense. It should also be ensured that the model
works for different tests without changing the parameter set for individual test cases.

**5.6.2 Examples and Case Studies**

Sensitivity analysis studies in seismic tsunami propagation and inundation modelling
may usually try to investigate the effects of different fault plane parameters on
different hazard measures (for example, maximum wave amplitude or inundated
area).

We first make an example for earthquake tsunamis: As the Okada model for the
bottom deformation (see Sect. 4.2) is mostly used, the parameters studied are usually
the location of the epicentre, focal depth, rake, dip and strike angles, fault plane
dimensions, and slip displacement (Gica et al. 2007). It is also a way to deal with
uncertainties in tsunami modelling. In González et al. (2022) this is done to estimate
the economic cost associated with tsunamis in Spain for the insurance sector.

For landslide generated tsunamis, the level of variation due to different values for
the parameters involved is larger and associated uncertainties in model results, sensi-
tivity studies for this kind of tsunamis are many (Grilli and Watts 2005; González-
Vida et al. 2019; Zengaffinen et al. 2020; Zengaffinen-Morris et al. 2022, to name
few).

Synolakis et al. (2008) propose a suite of increasingly complex benchmarks for
verifying and validating tsunami propagation and inundation models. All such models
involved in hazard assessment and used as tools for tsunami mitigation plans need
to be tested and benchmarked. This process must include both verification and vali-
dation. This must be a continuous process as new knowledge and data are acquired
or are made available.

A pioneering effort on model benchmarking was proposed in the early 1990s
and manifested itself in a first joint National Science Foundation (NSF) and National
Oceanographic and Atmospheric Administration (NOAA) National Tsunami Hazard
Mitigation Program (NTHMP) workshop, documented in Yeh et al. (1997). A number
of workshops followed, including one in 2011 where participating models were
approved for tsunami inundation modelling and results were documented in the
“Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop”
(NTHMP 2012). Besides NTHMP (2012) and references therein, for NTHMP-benchmarked tsunami models, other authors have performed similar benchmarking efforts, as is the case of Nicolsky et al. (2011), Apotsos et al. (2011), Tolkova (2014), Macías et al. (2017) or Vater et al. (2019). Similar benchmarking initiatives exist for hazard modelling of tsunami induced currents in harbors (Lynett et al. 2017) or landslide induced tsunamis (Kirby et al. 2022).

### 5.6.3 Further Testing Considerations

The usual output in sensitivity analysis studies is the range of model values for the variable or variables for which the sensitivity study is performed (for example, maximum wave height and/or run-up at several locations). They are usually represented as bar plots.

In validation/verification studies comparison of model results with analytical, measured or observed values is the final aim. Model goodness is assessed using maximal differences or NRMSD (Normalised Root Mean Square Deviation).

Sensitivity analysis studies and model verification mainly use maximum values of some variables (usually run-up or wave height) at several locations. For validation and in some cases for verification, when data exists, time series are also used. In any case, sensitivity studies can use any variable being maximum values or time series as they only rely on computed data.

Any method requiring numerical simulations is susceptible to use parallel computational resources. Many scenarios used in a Monte-Carlo sensitivity approach as well as operationally in scenario-based probabilistic hazard assessment were computed on parallel clusters in (Rakowsky et al. 2013; Gibbons et al. 2020).

We see these methods as mandatory for reliable, reproducible and robust numerical modelling. However, verification and validation test cases can face shortcomings regarding the quality or quantity of observed data, often not avoidable. Care must be taken to collect suitable data and make best use of available data to compare and verify numerical model results.

### 5.6.4 Treatment of Uncertainties for the Method

Tsunami simulations are often subject to various sources of uncertainty, including uncertainties in the initial conditions, boundary conditions, topography, bathymetry, and model parameters.

Aleatory variability refers to the unpredictability (or randomness) of a physical process. This randomness is expressed through the objective probability of the PTHA while epistemic uncertainty expresses limitations in our understanding of natural processes and different opinions regarding these processes, resulting in uncertainty on
quantifying these objective probabilities (Grezio et al. 2017). Sources of uncertainties
include:

* Uncertainties in the initial condition. The computation of the initial condition includes uncertainties in the bottom displacement, and unknowns on the earthquake mechanism in case of earthquake generated tsunamis, and uncertainties in the bottom movement in case of landslide generated tsunamis. For all sources of tsunamis (earthquake, volcanic, meteorological, and atmospheric tsunamis) uncertainties on source should be accounted for (Fukutani et al. 2015; Cienfuegos et al. 2018; Davies et al. 2018; Basili et al. 2021; Gibbons et al. 2020).

* Uncertainties in bathymetric data constitute another source of perturbations in tsunami propagation models (Behrens et al. 2009; Griffin et al. 2015). The quality of the data is crucial to compute the bathymetric domain to run the simulation. GEBCO and EMODNET data sets are examples of free sources of bathymetry data suited to run deep water simulations but not suitable to assess the inundation.

* Uncertainties in model physics. The shallow water equations are derived with the assumptions that the wavelength is larger than the water depth and that the vertical velocity component is constant in the water column. These assumptions are no longer valid for example for water flowing through the structures of a harbour or through streets (see Sect. 5.5), for impact tsunamis, or dispersive waves.

* Uncertainties in parameters such as the bottom friction. This issue is of particular importance when dealing with high-resolution inundation simulations.

* Uncertainties introduced by limitations in mesh resolution.

To account for these uncertainties, various methods and techniques can be
employed in tsunami simulations, some of which are:

1. Ensemble simulations: This involves running multiple simulations with different sets of initial conditions, model parameters, or boundary conditions. The results of these simulations can be used to estimate the uncertainties and provide probabilistic forecasts of the tsunami wave height, inundation, and arrival time.

2. Sensitivity analysis: This involves varying one or more input parameters of the numerical model and observing its impact on the simulation results. This can help identifying the most critical parameters affecting the simulation results and hence uncertainties and model bias.

3. Data assimilation: This involves integrating observational data, such as tsunami wave height, sea level, or GPS data, into the numerical model to reduce the uncertainties in the input data and improve the accuracy of the simulation results.

4. Uncertainty quantification: This involves estimating the uncertainty in the simulation output using statistical methods such as Monte Carlo simulations and Bayesian inference.
## 5.7 Choosing What Method to Use

Modelling the tsunami obviously is one of the key components of a PTHA/PTRA workflow. It is therefore paramount to select an appropriate modelling approach. The following considerations may help to decide on the appropriate model.

1. What is the aim of the analysis? Is the analysis directed towards a global, regional, or local area of coverage? For a global analysis it may suffice to neglect inundation computation, while for a local analysis it may be necessary to compile detailed inundation data.

2. What data is available? Are accurate topography and bathymetry data available? If so, then a detailed inundation model may be viable, otherwise the uncertainty in the data will be much larger than the suggested model accuracy.

3. What computational resources are available? Some models require high-performance computing devices or graphics processing units, while others may run on laptops. Detailed non-linear modelling usually requires much larger computational effort than linear shallow water theory.

4. How large is the uncertainty in the sources? The larger the source uncertainty (i.e. the epistemic uncertainty in the possible triggering events) the less severe is a model error. While model error should in general be omitted, some inaccuracy in discretizing the equations, or solving the discrete model can usually be accepted. More accurate models are usually also more demanding with respect to computational resources.

5. How detailed is the model approach? Are the physical processes accurately represented? More precisely, are the approximations to the real world in each of the modelling steps balanced? If one step in the analysis workflow represents physical processes that the others cannot capture, it may not be adequate to include such processes.

6. Which software is available? Some tools may be very accurate and efficient; however, they may not be easily available. Others are easily downloadable from open repositories but may not be fit for the purpose. It is by any means necessary to use well validated tsunami modelling tools. Community standards for validation are described in Synolakis et al. (2008).

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Electronic supplements (notebooks, codes, references to web services, etc.)
## Codes

Tsunami-HySEA. https://zenodo.org/record/6400815#.ZB8f8OzMJQJ

Landslide-HySEA. https://zenodo.org/record/6400825#.ZB8gf-zAJQI

TsunAWI. https://tsunami.awi.de/

GeoClaw. https://depts.washington.edu/clawpack/geoclaw/

## Benchmarks

EDANYA benchmarks. https://edanya.uma.es/hysea/index.php/benchmarks

NOAA PMEL benchmark suite. https://nctr.pmel.noaa.gov/benchmark/

## Open access topography data sets

GEBCO. https://www.gebco.net

ETOPO. https://www.ncei.noaa.gov/products/etopo-global-relief-model

EMODNET. http://www.emodnet-hydrography.eu

## Useful Resources for modelling the interface are

Gareth Davie's Code for PTHA. https://github.com/GeoscienceAustralia/ptha

RuptGen (Babeyko et al. 2010)

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