Tsunamis: Bayesian Probabilistic Analysis
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Glossary
 Aleatory variability

In the present context, it is the assumed random variability of the parameters characterizing the future hazardous events or, in other words, the random variability in the model describing the physical system under investigation.
 Bayesian statistics

An approach to statistics which represents unknown quantities with probability distributions that in one interpretation represent the degree of belief that the unknown quantity takes any particular value. Data are considered fixed and the parameters of distributions representing the state of the world or hypotheses are updated as evidences are collected.
 Bias

The tendency of a measurement process or statistical estimate to over or underestimate the value of a population parameter on average.
 Conditional probability

The probability that an event will occur under the condition or given knowledge that another event occurs.
 Conjugacy

In Bayesian statistics, the property of parametric families of distributions for prior and likelihood that lead the posterior distribution to be of the same family as the prior distribution.
 Completeness

It is the extent to which all needed statistics are available. In geophysics, it is usually referred to catalogs of past events, and it refers to the spatiotemporal windows in which virtually no events in a given energetic range are missing.
 Epistemic uncertainty

The uncertainty deriving from limited knowledge of the physical process, usually treated with alternative models of the same process.
 Estimation

The process by which we make inferences about a population, based on information obtained from a sample.
 Exceedance probability

The probability that a given parameter will be larger than a threshold value over a time interval of interest.
 Frequentist statistics

An approach to statistical reasoning which considers the observed sample to be one realization of repeatable random experiment. The parameters to be estimated are considered to be constants, in contrast with Bayesian statistics where the parameters are treated as random variables.
 Inference

It is the act of generalizing from the data (“sample”) to a larger phenomenon (“population”).
 Joint probability distribution

Describes the simultaneous occurrence of two or more events treated as random variables.
 Likelihood function

A function of the unknown parameters conditioned on the given fixed observed data, which returns the likelihood that the parameters assume specific values.
 Probability density function (PDF)

Also known as the density of a continuous random variable, it is a function that describes the relative likelihood that a random variable takes a given value. The probability of the random variable falling within a particular range of values is given by the integral of this variable over the given range. The PDF is nonnegative everywhere, and its integral over the entire space is equal to one.
 Recurrence interval or average return period

It is the average time interval between events of a similar size or intensity.
 Runup

It is the maximum topographic height reached by inundation above a reference sea level, usually measured at the horizontal inundation limit during a tsunami event.
Definition of the Subject
Tsunamis are lowfrequency highconsequences major natural threats, rare events devastating vast coastal regions near and far from their generation areas. They may be caused by coseismic seafloor motions, subaerial and submarine mass movements, volcanic activities (like explosions, pyroclastic flows, and caldera collapses), meteorological phenomena, and meteorite ocean impacts. The probability of tsunami occurrence and/or impact on a given coast may be treated formally by combining calculations based on empirical observations and on models; this probability can be updated in light of new/independent information. This is the general concept of the Bayesian method applied to tsunami probabilistic hazard analysis, which also provides a direct quantification of forecast uncertainties. This entry presents a critical overview of Bayesian procedures with a primary focus on their appropriate and relevant applicability to tsunami hazard analyses.
Introduction
Bayesian inference is a process of learning from data and from experience (Gelman et al. 2013). We define prior information as the knowledge before observing new data and posterior information as the understanding after considering new data with the improvement of the prior knowledge due to the evidence in the data. Named after British mathematician the Reverend Thomas Bayes (1701–1761), Bayesian inference is based on statements of conditional probability. Bayes’ formula, at the roots of Bayesian methods, was introduced in special cases and was formalized in a posthumous paper presented at the Royal Society of London. Stigler (1983) attributes the principle of the Bayesian inference to Saunderson (1683–1739), a professor of optics who published a large number of mathematics papers. From 1774 to 1812, Laplace studied the general concept of the conditional probability independently, considering the inductive probability by reassessing the prior estimates if new relevant evidence has emerged. Due to the work of Laplace, Bayesian statistics was in common use for practical applications beginning around the late nineteenth to early twentieth centuries.
The awareness of the tsunami threat to coastal communities has been increased worldwide after catastrophic, globalscale tsunamis generated by great megathrust earthquakes in the Indian Ocean (26 December 2004), in Chile (27 February 2010), in Japan (11 March 2011) and by several other seismic events in the last decade (Lay 2015; Lorito et al. 2016). Hence, progressively more intense efforts have been spent on estimating longterm tsunami hazard and risk. Probabilistic tsunami hazard analysis (PTHA) provides a quantitative tool of the tsunami hazard assessments for tsunami risk mitigation plans and for the implementation of tsunami early warning systems (Geist and Parsons 2006; Geist and Lynett 2014; Grezio et al. 2017).
PTHA estimates the probability of exceeding specific tsunami intensities (wave heights, flowdepth, runup, velocity, etc.; see, e.g., TPSWG 2006) within a certain time period (exposure time) at given locations (key sites). The most common approaches to PTHA are either based on combining source probability with numerical modeling of the ensuing tsunamis (computationally based approach: Annaka et al. 2007; Burbidge et al. 2008; Davies et al. 2016; Gonzalez et al. 2009; Heidarzadeh and Kijko 2011; Hoechner et al. 2016; Horspool et al. 2014; Knighton and Bastidas 2015; Lane et al. 2013; Lorito et al. 2015; Mueller et al. 2015; Omira et al. 2015; Power et al. 2013; Sakai et al. 2006; Selva et al. 2016; Sørensen et al. 2012; Suppasri et al. 2012; Thio et al. 2010; Thio and Li 2015) or, less frequently, based on the observed tsunami frequency at a given coastal site (empirical approach: Geist et al. 2014; Orfanogiannaki and Papadopoulos 2007; Tinti et al. 2005; Yadav et al. 2013). Empirical methods are often inhibited by a paucity of information. For this reason, computationally based methods are often preferred for tsunamis; however, they can be tremendously computationally intensive, and the statistics and physics of the sources can be difficult to constrain and model (see discussion on the different approaches and their advantages/limitations in Geist and Lynett 2014).
Bayesian statistics allows merging these different approaches, homogeneously integrating the different kinds of information and thus may represent an effective method to assess tsunami hazard. In examples of computationally based versus empirical approaches, the computationally based analyses may constitute the core assessments for building a first attempt of a complete description of the probabilistic tsunami hazard. The computationally based approach covers all the tsunami sources that are considered possible and their (natural) aleatory variability. Direct numerical modeling of the tsunamis generated by each assumed source scenario is then performed. The exceedance probability at a given site is finally assessed by combining the simulation results with the source probability. Alternative (source and propagation) models span the epistemic uncertainty. Furthermore, this prior information may be updated considering the statistics of the observed tsunamis. Bayesian techniques enable the merging of models and observations into a coherent probabilistic framework, and they may be adopted at some stage of the computationally based PTHA, for example, to constrain the earthquake or tsunami magnitudefrequency relationships and associated uncertainties, or the likelihood of the earthquake focal mechanism at a given location (e.g., Shin et al. 2015; Selva et al. 2016; Yadav et al. 2013), or for the final results of the PTHA (e.g., Grezio et al. 2012; Parsons and Geist 2009).
Bayesian approaches have been applied not only for longterm PTHA but also for shortterm and/or timedependent tsunami hazard forecast, in the context of tsunami early warning (Blaser et al. 2011; Tatsumi et al. 2014).
Here we present the general Bayesian methodology, and two examples of Bayes’ theorem applied to the tsunami hazard analysis: (i) developing a tsunami forecast from numerical modeling and an empirical catalog and (ii) implementing weighting factors based on past tsunami data for statistical models estimating runup exceedance rates and runup forecast with subjective estimation of the variance. Finally, we discuss the advantages and the limitations of the Bayesian approach in tsunami probabilistic analysis, and we outline some future directions of the Bayesian approach in tsunami studies.
Methodology
In general terms, Bayesian statistics identifies a given number of unknown parameters Θ and tries to infer their values θ by accounting for some measurable data Y = {y_{1},…,y_{i}, …, y_{n}} and for the knowledge, independent from data, that we may have about their values. In all the steps of the analysis, the uncertainty is expressed through probability density functions (PDFs), and, in Bayesian analyses, all the parameters (including probability values) can be treated as unknowns (Draper 2009; Gelman et al. 2013).
In PTHA, the parameters under investigation are generally the exceedance probabilities of a given tsunami intensity at one site (Θ = P(Z > z; x, DT)), where the tsunami intensity is, for example, the runup. The data may be observed intensities at given sites due to past tsunamis or the observed frequency of exceedance in a past time interval DT (Grezio et al. 2010). The parameters in question may also be intermediate quantities required for building the PTHA, like earthquake magnitudes and focal mechanisms (Θ = {M,strike,dip,rake}) or landslide volume or shape (Grezio et al. 2012).
The current state of knowledge about the parameters Θ (i.e., the relative ignorance of the parameter values) and the relative uncertainty before considering new observations are expressed through the prior PDF, hereinafter indicated with P(Θ). This term allows to account for whatever source of information, from theoretical models to expert beliefs and for both quantitative assessments and qualitative information. If no independent information is available, it is possible to set noninformative prior distributions (Box and Tiao 1992).
In order to understand what the observables say about the parameters Θ, we have to set a parametric statistical model that links the observables Y to the parameters. Then, the likelihood of the observables, given a specific value of Θ, needs to be evaluated. This information is expressed through the likelihood PDF, hereinafter indicated as P(YΘ), which gives the probability density of the observed data for any choice of model parameters and follows from the assumed parametric statistical model.
The final goal is the quantification of the probability of the different values of the parameters Θ given the prior knowledge and the observations Y. This information is expressed through the posterior PDF, hereinafter indicated as P(ΘY). The posterior distribution is computed from the prior and the likelihood distributions through Bayes’ theorem, discussed in the following paragraph. In practice, the past data are utilized to compute the likelihood function and then update the prior beliefs. If new evidence is then gathered, such as new data, the described procedure may be applied iteratively, and the old posterior can play the role of the new prior, to be combined with the new likelihood to obtain a new posterior. The results of the Bayesian analysis are conditional on the assumptions on which the statistical models for the prior PDFs and likelihood functions are built. The results allow for multiple interpretations like hypothesis evaluation, inverse probability problem, prediction process, model evaluation, parameter ranges, and sensitivity analysis (Box and Tiao 1992; Congdon 2006; Gelman et al. 2013).
Bayes’ Theorem
the notation P(⋅) denotes the probability PDF and P(··) the conditional PDF, whose parameters are usually referred to as hyperparameters. Hence, conditional PDFs are central elements in the Bayesian framework.
P(Θ) is the prior probability distribution. The Θ parameter spans a range of possible values θ and defines the hypothesis space. The parameters are not estimated as a single point in the parameter space but are instead represented by a distribution and its statistics (e.g., prior mode or prior mean).
P(Y  Θ) is the likelihood function. It represents the information about Θ contained in the data Y.
P(Y) is a normalization constant ensuring posterior probability integrates to 1. For this reason, it is often omitted from the notation, reporting “proportional to” instead of “equal to” in Eq. 1 (Gelman et al. 2013).
The posterior distribution P(Θ  Y) quantifies the Bayesian inference about the parameters obtained through Bayes’ theorem from the prior and likelihood. As for the prior, the parameters are not estimated as a single point in the parameter space but are instead represented by a distribution. The posterior distribution may be seen as an update of the prior (a novel estimate of the same quantity) in light of new data.
Distribution Forms and Mathematical Techniques
Normal density (Θ ~ Nor (y  μ, σ) where μ is the mean and σ the standard deviation)
Poisson density (Θ ~ Pois (y  λ) where λ is a constant rate of occurrence in the considered time interval)
Gamma density (Θ ~ Gam (y  α, β) where the hyperparameters α and β, respectively, control the probability distribution immediately after each subevent and describe the longerterm rate)
Binomial density (Θ ~ Bin (y  n, θ) where n are the trials, given that the probability of success in one trial is θ)
Beta density (with Θ ~ Beta (y  α, β) where the two positive hyperparameters α and β are the exponents of the random variable)
Uniform density (with Θ ~ Unif (y  α, β) with all values between α and β equally probable)
Bayesian statistics may utilize the valuable mathematical properties of the conjugate analysis to find prior distributions for the likelihoods (denominated conjugate priors) in order to represent the results in an analytic form which simplifies the computations. When possible, this is obtained by picking a prior distribution of the same “family” of the likelihood function so that the resulting probability distribution is also in the family. In this way, a closedform expression for the posterior distribution is obtained, and, for example, numerical integration can be avoided. In tsunami problems, like estimating the frequency of events where the rate parameter is unknown but can be constrained somewhat with data, conjugate families are sometimes used. Geist and Parsons (2010) applied a PoissonGamma conjugate to model the probability of potentially tsunamigenic submarine landslides in the Santa Barbara Channel (Southern California), Port Valdez (Alaska), and Storegga Slide complex (Norwegian Sea). The landslide probability problem assumes that the landslides are independent and thus occur randomly in time according to a Poisson distribution (characterized by a rate parameter λ), and eventually earthquake probability is also considered. Landslide interevent time uncertainties associated with age dating of individual events and open time intervals were estimated. The seismically imaged landslides typically exhibited only the ages of the youngest and oldest underlying events. However, through the Bayesian approach, even not straightforward information are included. The most likely mean return time (1/λ) of the submarine landslides was estimated by this PoissonGamma model using the number of landslide occurrences and the observation period. Grezio et al. (2010) combined the prior Beta distribution with the Binomial likelihood function in the Messina Strait Area. The posterior distribution is a modified Beta distribution constrained by the past runup observations. In a similar formulation, Knighton and Bastidas (2015) employed the PoissonGamma conjugate model in the first step to estimate a likelihood function for the Poisson rate parameter of tsunamigenic events given in an historical catalog. Then, the likelihood function for the Poisson parameter was determined to find the probability of tsunamigenic events causing a hazard exceeding a critical value. The outcome probability was used into a BetaBinomial scheme like in Grezio et al. (2012). Selva et al. (2016) proposed an event tree procedure to quantify source uncertainties in a seismic PTHA. At one level of the event tree, the uncertainty on potential focal mechanisms of earthquakes is modeled. At this node, a Dirichlet distribution is used to represent the prior knowledge about the probability of occurrence of the different combinations of discrete intervals of strike, dip, and rake angles. This distribution is then updated by observations of such angles from two earthquake catalogs from the Ionian Sea region (central Mediterranean Sea), naturally distributed following a Multinomial distribution. The resulting posterior distribution for the probability of the different interval of angles is again a Dirichlet distribution.
If the parameter Θ is a mdimensional vector {θ_{1}, …, θ_{m}}, as in the last example of Selva et al. (2016), the probability distributions and the normalization constant P(Y) are mdimensional problems. If the conjugacy property of prior/likelihood functions is not used to directly obtain the posterior distribution in a closed form, advanced sampling techniques should be adopted. Markov chain Monte Carlo (MCMC) methods are intensive computational techniques used to approximate the highdimensional integrals associated with the posterior probability distribution in Bayes’ theorem. Markov chain samples from the posterior distribution for a time sufficient long to reach the equilibrium within the required approximation (Draper 2009). These techniques extend the range of the singleparameter sampling method to multivariate situations where each parameter or subset of parameters in the overall posterior density may have different density (Congdon 2006). Knighton and Bastidas (2015) evaluate the hazard to the hypothetical coastal facility within a 30year time period by the Monte Carlo analysis through sampling the likelihood distribution of interevent timing of the tsunami sources and the beta distribution which pertains to the binomial distribution of the hazard parameter.
Epistemic and Aleatory Uncertainties
All probabilistic analyses typically address the problem of epistemic and aleatory uncertainties. Many authors supported this division, based on either theoretical (Marzocchi and Jordan 2014) or practical (PatéCornell 1996) reasoning. In its general interpretation, aleatory uncertainty represents the unreducible natural variability of the studied phenomenon, while the epistemic uncertainty arises from the limited knowledge on the system that does not allow to perfectly quantifying the aleatory uncertainty. In this way, it is possible to distinguish the uncertainty that may be reduced by increasing the knowledge of the modeled system (the epistemic uncertainty) from the irreducible unpredictability of the system itself (the aleatory uncertainty). Also, the separation allows to report the effective variability of the results in a more robust format and to make any probabilistic analysis a testable experiment (Marzocchi and Jordan 2014; Marzocchi et al. 2015).
In probabilistic tsunami hazard analysis, epistemic uncertainty emerges from the substantial lack of understanding of the tsunamigenic processes (e.g., the longterm earthquake rates or the dynamics of “tsunami earthquakes”; see, e.g., Polet and Kanamori 2009) and of the tsunami evolution after generation, or even from approximations in the tsunami numerical modeling made for the sake of practical feasibility (e.g., the common shallowwater approximation), or from the lack of accurate enough digital elevation models.
Both the physicsbased and datadriven concepts should address the appropriate hypotheses on the statistical experiment settings. Tsunami events of largest intensity are rare, and the data are often not sufficient to constrain properly the variability of the controlling parameters. As a consequence, a large epistemic uncertainty arises, and many scientifically acceptable alternative models may be formulated.
In the Bayesian paradigm, both types of uncertainty are automatically quantified for the potential reduction of the possible epistemic uncertainty by accounting for all relevant available information. In a Bayesian analysis, the epistemic uncertainties are represented as uncertain parameters, whereas aleatory uncertainties are represented with the choice of probability density functions appearing in the selected parametric statistical model. Since in tsunami applications relatively few data are generally available, parameters are (usually) poorly constrained, and few additional data can feed the likelihood. Thus, the weight of the prior is larger compared to the weight of the likelihood functions.
Advanced approaches tend to extend the exploration of epistemic uncertainty by including alternative statistical models of the aleatory uncertainty, that is, developing the prior probability distributions by implementing different statistical models (Knighton and Bastidas 2015) or implicitly adopting an ensemble modeling approach (Marzocchi et al. 2015; Selva et al. 2016).
Bayesian PTHA
Forecasting tsunamis is typically an underinformed exercise because the mean return time of a given event is often longer than the period we have had to observe it. Thus, we can only rarely develop a complete empirical distribution that satisfactorily captures the aleatory variation. We then rely on numerical models, indirect paleoevidence, and/or incomplete historical observations to develop probability density function parameters. Many of these models have a stochastic component that attempts to cover the possible range of behaviors. Each of these datasets or model results may capture different aspects of the hazard process. Here we discuss two paradigmatic analyses in which the Bayesian method aggregates a variety of information sources, specifically using likelihood functions shaped from measurement uncertainties and/or stochastic distributions to integrate and weight results. If advanced models, additional and even sparse data, improved instrumental measurements, or new observations become available, an update of the posterior inferences is possible in the Bayesian statistical framework, keeping track of the assumptions on the prior knowledge and the introduced further information.
Example of Tsunami Forecast from Numerical Modeling and an Empirical Catalog
A 30year probability of tsunami runup in excess of 0.5 m in cells that contain population concentrations in 20 by 20 km cells for representative Caribbean countries and territories. Population given as a relative measure of risk throughout the region. Values were calculated as uniform over cell areas and are not intended to convey any detail at selected cities but are presented for comparison purposes. Dashes indicate negligible calculated probability
Country  Nearest coastal city  Latitude  Longitude  Population  30yr probability 

r ≥ 0.5 m (%)  
Antigua and Barbuda  St. John’s  17.1167°  −61.8500°  24,226  5.74 
Belize  Belize City  17.4847°  −88.1833°  70,800  – 
Cayman Islands  George Town  19.3034°  −81.3863°  20,626  10.79 
Columbia  Cartagena  10.4000°  −75.5000°  895,400  0.08 
Costa Rica  Puerto Limon  10.000°  −83.0300°  78,909  8.32 
Cuba  Santiago de Cuba  20.0198°  −75.8139°  494,337  2.31 
Dominica  Roseau  15.3000°  −61.3833°  14,847  11.94 
Dominican Republic  Santo Domingo  18.5000°  −69.9833°  913,540  17.56 
France, Guadeloupe  BasseTerre  16.2480°  −61.5430°  44,864  11.79 
France, Martinique  FortdeFrance  14.5833°  −61.0667°  94,049  5.33 
Grenada  St. George’s  12.0500°  −61.7500°  7500  2.48 
Guatemala  Puerto Barrios  15.7308°  −88.5833°  40,900  – 
Haiti  PortauPrince  18.5333°  −72.3333°  1,277,000  0.01 
Honduras  La Ceiba  15.7667°  −86.8333°  250,000  – 
Jamaica  Kingston  17.9833°  −76.8000°  660,000  21.95 
Netherlands Antilles  Willemstad  12.1167°  −68.9333°  125,000  7.04 
Nicaragua  Bluefields  12.0000°  −83.7500°  45,547  – 
Panama  Colon  9.3333°  −79.9000°  204,000  17.56 
St. Kitts and Nevis  Basseterre  17.3000°  −62.7333°  15,500  6.95 
St. Lucia  Castries  14.0167°  −60.9833°  10,634  5.52 
St. Vincent and the Grenadines  Kingstown  13.1667°  −61.2333°  25,307  11.32 
Trinidad and Tobago  Port of Spain  10.6667°  −61.5167°  49,031  – 
Turks and Caicos  Cockburn Town  21.4590°  −71.1390°  5567  3.57 
UK, Virgin Islands  Road Town  18.4333°  −64.5000°  9400  13.85 
USA, Puerto Rico  San Juan  18.4500°  −66.0667°  434,374  22.24 
USA, Virgin Islands  Charlotte Amalie  18.3500°  −64.9500°  18,914  17.56 
Venezuela  Cunana  10.4564°  −64.1675°  305,000  6.27 
For each earthquake in the synthetic catalogs, vertical and horizontal coseismic seafloor displacements are the initial conditions for tsunami modeling (Tanioka and Satake 1996). Displacements are calculated using Okada’s (1985) analytic functions. A finite rise time of 20 s was applied uniformly, with no preferred rupture propagation direction. The propagation of the tsunami wavefield is modeled using a finitedifference approximation to the linear longwave equations (Aida 1969; Satake 2002). A 2 arc minute bathymetric grid (Smith and Sandwell 1997) was used with an 8 s time step that satisfied the CourantFriedrichsLewy stability criterion for the Caribbean region. A reflection boundary condition was imposed at the 250 m isobath, whereas a radiation boundary condition was imposed along the openocean boundaries of the model (Reid and Bodine 1968). Runup (R_{0}) was approximated from the coarsegrid model by finding the nearest model grid point to the coastline and then multiplying the peak offshore tsunami amplitude by a factor of 3 that roughly accounts for shoaling amplification and the runup process itself (Satake 1995, 2002; Shuto 1991).
We conducted two experiments with a single 4442year synthetic runup catalog and another with 50,500year catalog. We found that the 50,500year catalogues captured more variability in spatial runup distribution than did the 4442year catalogues. This resulted from the multiple catalogs having more varieties of earthquake locations since a few very large events can dominate the distribution of moment, and consequently regional tsunami runup distribution, due to the GutenbergRichter constraint. We thus used the set of 50,500year catalogues to determine mean rates and uncertainties in the probability calculations.
The primary sources of epistemic uncertainty include (1) tsunami sources not explicitly known or included in the model, (2) seismic coupling coefficient of the Caribbean plate boundary zones, and (3) the degree of completeness in the empirical tsunami catalog. To encompass these uncertainties into probability estimates, a Bayesian framework is created to build tsunami runup rate estimates within 20 by 20 km cells that contain coastlines throughout the Caribbean region. The key advantage of this approach is that the empirical and model results end up being combined and weighted by their attendant uncertainties.
Having independent empirical and modelderived rate estimates in each spatial cell enables some of the runuprate uncertainty to be addressed. Monte Carlo fitting of empirical intervals as shown in Fig. 3 along with results from 50 numerical model runs (e.g., Fig. 2) provides arrays of possible runup rate values at each cell. Unknown/unaccountedfor tsunami sources can be partly accounted for because some of the empirical rates result from sources not accounted for in the numerical model (the most affected areas can be seen by comparing the panels of Fig. 4); the forecast may suffer from incomplete knowledge if events not covered by numerical models have also not occurred in the empirical catalog over the past 500 years. Seismic coupling is a difficult parameter to estimate with certainty; a broad range is captured because the historic earthquake catalog implies a low coupling value of 0.32 (found by comparing seismic moment release to expected slip on Caribbean faults), whereas the numerical models have coupling coefficients of 1.0. Completeness is addressed because lowrate plateboundary events potentially not seen in the empirical catalog are accounted for with the 50 numerical model runs.
Likelihood functions are used to weight rate models over a range from 0 to 10 events in the 500year observation period. Rates between 0 and 10 events in 500 years are considered for all cells, assuming no further prior information. Final rates are found by weighted means of the posterior rates. To summarize the process, where model and empirical values are both absent for a given rate, the posterior distribution was zeroed. When one model provides rates, its likelihood function was used to update the priors, and when both empirical and numerical rate estimates are available, likelihood is developed through combination and renormalization using Eq. 4, which is then used to update the priors. Combining empirical and modeled rates makes up for some of the deficiencies in each approach; the empirical catalog is likely not a complete record of all possible interplate tsunami sources, whereas the numerical model did not account for accommodating intraplate faults and/or landslide sources that appear likely causes of tsunamis in the empirical record.
Example of Past Data Weighting Factors for Statistical Models and Subjective Estimation of the Variance
The SSSs are localized on active faults, and the relative epicenters are extracted from the instrumental Catalogue of the Italian Seismicity with a completeness magnitude of 2.5 (Castello et al. 2007) at depths smaller than 15 km within the shallow part of the crust. Instrumental magnitudes are recorded since 1981, and no tsunami occurred in this short time. In order to consider a large set of potentially tsunamigenic SSSs, magnitudes in the range 5.5–7.5 M_{w} were introduced consistently with the regional seismotectonic studies and weighted using the GutenbergRichter distributions (Gutenberg and Richter 1944). Finally the magnitudes are associated with the catalog epicenters, and the seafloor deformations are calculated via the analytical formulas by Okada (1992) in order to compute the initial tsunami sea surface waves. The relative fault parameters (width, length, and slip) and focal mechanisms (strike, dip, and rake) are provided, respectively, by the empirical relationships in Wells and Coppersmith (1994) and the Earthquake Mechanisms of the Mediterranean Area database (Vannucci and Gasperini 2004). The SSS spatial distribution is considered uniform.
The SMFs are spatially identified using marine geology background knowledge. Their propensity to fail is evaluated on the basis of the mean slope and mean depth, and it is associated to bathymetry cells. In each cell, potentially tsunamigenic SMFs are simulated with volumes spanning from 5 × 10^{5} to 5 × 10^{10} m^{3} as indicated by the historical SMF sizes identified in the Tyrrhenian and Ionian basins. Additionally, spatial conditional probabilities are introduced considering that the past SMF scars represent instability areas. The other geometric parameters and the initial tsunami waves are estimated, respectively, by the rigid body approximation and the empirical formulas in Grilli and Watts (2005) and Watts et al. (2005). In analogy with the subaerial mass failures, the SMF frequencysize relationship is assumed to be a power law.
The runups Z caused by the SSS and SMF tsunamigenic sources were calculated trough empirical formulas (Synolakis 1987). Uncertainties related to the Z parameter would be reduced through the modeling of source directivity (tsunami energy is not spread isotropically around the source), wave propagation effects (refraction, diffraction, etc.), and other, also nonlinear processes during shoaling and coastal inundation (wave breaking, bores, friction, etc.).
By setting the parameter Λ to specific values, we assign both the subjective reliability to the prior model and the relative confidence interval. The parameter Λ weights the prior model and represents an estimate of the epistemic uncertainties due to the limited knowledge of the process. In general, a large Λ value corresponds to a large reliability of the prior model, so that the prior distribution needs a great number of past data or observations in the likelihood to be modified significantly. On the contrary, Λ must be small if the prior model is only a firstorder approximation of the process, so that even a limited number of observations in the likelihood can heavily modify the prior distribution. The minimum possible value of Λ is 1, representing the maximum possible epistemic uncertainty or maximum of level of ignorance. As Λ increases, the Beta function becomes more and more spiked around the given mean. The endmember is a Dirac’s function judging the epistemic uncertainty negligible when a large amount of data is available (Marzocchi and Lombardi 2008). Here, Λ is assumed equal to 10 on the basis of practical and expert judgement; it means that more than 10 real data can change drastically the prior probability distribution (Grezio et al. 2012). After computing the expected value E and the variance V, the α and β hyperparameters of the Beta distribution are finally constrained in Eq. 6, and the prior PDF is determined.
The analysis shows that SSS and SMF posterior probability generally increases by one or more order of magnitude, and both types of tsunamigenic sources present the same order of magnitude in the Messina Strait Area. Therefore, both sources must be considered and combined in order to produce a reliable PTHA in this area. Conversely, the posterior variances are reduced by one order of magnitude in the SSS case and by two orders of magnitude in the SMF case. The epistemic uncertainty decreases when the number of past data and/or historical information increase and the Beta distribution results more spiked because of Λ_{post} = Λ + y^{i}.
Final means and variance of the posterior probability distribution that a tsunami runup overcoming 0.5 m occurs in the time interval of 1 year due to the SSSs and SMFs
Key sites  Mean  Variance 

× 10^{−3}  × 10^{−5}  
Messina  5.9  1.1 
Reggio Calabria  7.9  1.5 
Pellaro  5.9  1.1 
Catania  5.9  1.2 
Augusta  5.9  1.2 
Siracusa  3.9  0.8 
Milazzo  2.0  0.4 
Capo d’Orlando  2.0  0.4 
Cefalù  2.0  0.4 
Stromboli  11.9  2.3 
Capo Vaticano  5.9  1.2 
Roccella  7.9  1.6 
Discussion and Conclusions
A mathematical consequence of the Bayesian procedure is that the results are always within the range of hypotheses, similar to most of the statistical techniques. Thus for the Bayesian methods to be considered objective, the results must depend on the assumed prior statistical model and observed data. In fact, different prior parameter determinations (with their own probability distributions considered as the random variables) may reach different conclusions, in particular when few past data are available.
Information at the base of prior PDFs and likelihood functions sometimes cannot be completely independent in practical applications, leading to potential double counting. For example, in longterm applications where all the data of rare events should be considered, this issue can seriously affect Bayesian inferences if not properly accounted for (e.g., discussion in Selva and Sandri 2013). To overcome this issue, in practice, two assertions can be considered: (i) the tsunami data used to define the prior probability models should be extracted by different catalogs (e.g., generic earthquakes, not necessarily tsunamigenic earthquakes) and/or used only in aggregated forms (e.g., the prior information may be derived by tsunami events that occurred globally and support general knowledge about tsunami processes), whereas (ii) the data used to create the likelihood functions should be mainly local, in order to reduce as much as possible the potential effect of possible double counting. Therefore all assumptions made in formulating the prior probabilities and in selecting the parameters should be stated explicitly so that the results can be properly assessed.
It enables merging of several kinds of available information in a homogeneous framework. Different statistical methods, theoretical deductions, background knowledge, physical beliefs, empirical laws, numerical models, analytical results, historical data, and instrumental measurements are combined and integrated. Bayesian techniques make use of all information, even sparse data, while keeping track of the assumptions about the prior knowledge or the level of ignorance. For a given probability model, an update of the final inferences is possible as soon as new models and/or additional data become available.
It enables accounting for different sources of uncertainty, i.e., aleatory and epistemic uncertainty. The uncertainties are specified and synthesized in the statistical distributions, and the Bayesian procedure, considering potential all sources of information, enables a quantification and, in principle, a controlled reduction of the inherent epistemic uncertainties.
It allows for propagating all the uncertainties from all the levels of the assessment. The most relevant sources of uncertainty from the tsunami source generation process to wave propagation and impact on the coasts may be reported and incorporated in the tsunami hazard computation (Marzocchi et al. 2004, 2008; Grezio et al. 2010, 2012; Gelman et al. 2013; Knighton and Bastidas 2015; Selva et al. 2016). Additionally, different types of potentially tsunamigenic sources may be included in the analysis in order to reduce biases (Grezio et al. 2015).
Future Directions
A key role in the future is PTHA testability against real and independent data. In its Bayesian interpretation, the probability represents a state of knowledge, and it is intrinsically subjective because all probabilities are degrees of belief that cannot be measured (Lindley 2000) and/or eventually rejected (Jaynes 2003). In PTHA, this means that the probabilistic quantification strictly refers to the next time window, and its results cannot be tested. The frequentist interpretation instead intrinsically connects the probability definition to a measurable quantity (the past frequency) that can be theoretically known by analyzing an “infinite” sequence of outcomes for repeatable event (Popper 1983). This makes such frequencies formally testable against real data. An unificationist approach (Marzocchi and Jordan 2014) may then be adopted for PTHA, in which the expert opinion is regarded as a model distribution describing the longrun frequencies determined by the datagenerating process. These frequencies, which characterize the aleatory variability, have epistemic uncertainty described by the experts’ distributions. As far as the knowledge of the system increases, our capability of assessing the true value of such frequencies is refined, that is, the epistemic uncertainty is reduced. Therefore, following this definition for the PTHA and related uncertainty, if “infinite” dataset is made available, Bayesian and classical PTHA will lead to equivalent results, since any subjective choice regarding priors is completely overcome by the infinite dataset perfectly constraining the longrun frequencies. However, we are unfortunately far from this case, being tsunamis relatively rare events as compared to our observation window.
Acknowledgments
We wish to thank Gareth Davies and Eric Geist for the constructive comments during the review process.
Bibliography
 Aida I (1969) Numerical experiments for the tsunami propagation – the 1964 Niigata tsunami and the 1968 TokachiOki tsunami. Bull Earthquake Res Inst 47:673–700Google Scholar
 Annaka T, Satake K, Sakakiyama T, Yanagisawa K, Shuto N (2007) Logictree approach for probabilistic tsunami hazard analysis and its applications to the Japanese coasts. Pure Appl Geophys 164:577–592ADSCrossRefGoogle Scholar
 Blaser L, Ohrnberger M, Riggelsen C, Babeyko A, Scherbaum F (2011) Bayesian network for tsunami early warning. Geophys J Int 185(3):1431–1443. https://doi.org/10.1111/j.1365246X.2011.05020.xADSCrossRefGoogle Scholar
 Box GEP, Tiao GC (1992) Bayesian inference in statistical analysis. Wiley Classics Library, New York, p 588CrossRefGoogle Scholar
 Burbidge D, Cummins PR, Mleczko R, Thio HK (2008) A probabilistic tsunami hazard assessment for Western Australia. Pure Appl Geophys. https://doi.org/10.1007/S000240080421XADSCrossRefGoogle Scholar
 Castello B, Olivieri M, Selvaggi G (2007) Local and duration magnitude determination for the Italian earthquake catalogue (1981–2002). Bull Seismol Soc Am 97:128–139CrossRefGoogle Scholar
 Congdon P (2006) Bayesian statistical modelling, Wiley series in probability and statistics. Wiley, Chichester, p 529CrossRefGoogle Scholar
 Davies G, Griffin J, Løvholt F, Glymsdal S, Harbitz C, Thio HK, Lorito S, Basili R, Selva J, Geist E, Baptista MA (2016) A global probabilistic tsunami hazard assessment from earthquake sources, Accepted Manuscript, “Tsunamis: geology, hazards and risks”. GSL Special Publications, LondonGoogle Scholar
 Draper D (2009) Bayesian statistics. Enc Complexity Earth Syst Sci 1:445–476Google Scholar
 Favalli M, Boschi E, Mazzarini F, Pareschi MT (2009) Seismic and landslide source of the 1908 Straits of Messina tsunami (Sicily, Italy). Geophys Res Lett 36:L16304. https://doi.org/10.1029/2009GL039135ADSCrossRefGoogle Scholar
 Geist EL (2002) Complex earthquake rupture and local tsunamis. J Geophys Res 107:ESE2–1–ESE 2–16CrossRefGoogle Scholar
 Geist EL, Lynett PJ (2014) Source processes for the probabilistic assessment of tsunami hazards. Oceanography 27(2):86–93. https://doi.org/10.5670/oceanog.2014.43CrossRefGoogle Scholar
 Geist EL, Oglesby DD (2014) Tsunamis: stochastic models of occurrence and generation mechanisms. In: Meyers RA (ed) Encyclopedia of complexity and systems science. Springer, New York. https://doi.org/10.1007/9783642277375_5951CrossRefGoogle Scholar
 Geist EL, Parsons T (2006) Probabilistic analysis of tsunami hazards. Nat Hazards 37:277–314. https://doi.org/10.1007/s110690054646zCrossRefGoogle Scholar
 Geist EL, Parsons T (2010) Estimating the empirical probability of submarine landslide occurrence. In: Mosher DC et al (eds) Submarine mass movements and their consequences. Advances in natural and technological hazards research, vol 28. Springer, Dordrecht, p 377CrossRefGoogle Scholar
 Geist EL, Ten Brink US, Gove M (2014) A framework for the probabilistic analysis of meteotsunamis. Nat Hazards 74:123–142. https://doi.org/10.1007/S1106901412941CrossRefGoogle Scholar
 Gelman A, Carlin JB, Stern HS, Rubin DB (2013) Bayesian data analysis. Chapman & Hall/CRC Press, Boca Raton, p 667Google Scholar
 Gonzalez FI, Geist EL, Jaffe B, Kaˆnoglu U, Mofjeld H, Synolakis CE, Titov VV, Arcas D, Bellomo D, Carlton D, Horning T, Johnson J, Newman J, Parsons T, Peters R, Peterson C, Priest G, Venturato A, Weber J, Wong F, Yalciner A (2009) Probabilistic tsunami hazard assessment at Seaside, Oregon, for near and farfield seismic sources. J Geophys Res 114:C11023. https://doi.org/10.1029/2008JC005132ADSCrossRefGoogle Scholar
 Gregory P (2005) Bayesian logical data analysis for the physical sciences. Cambridge University Press, Cambridge, p 468CrossRefGoogle Scholar
 Grezio A, Marzocchi W, Sandri L, Gasparini P (2010) A Bayesian procedure for Probabilistic Tsunami Hazard Assessment. Nat Hazards 53:159–174. https://doi.org/10.1007/s1106900994188CrossRefGoogle Scholar
 Grezio A, Marzocchi W, Sandri L, Argnani A, Gasparini P (2012) Probabilistic tsunami hazard assessment for messina strait area (Sicily – Italy). Nat Hazards. https://doi.org/10.1007/s110690120246xCrossRefGoogle Scholar
 Grezio A, Tonini R, Sandri L, Pierdominici S, Selva J (2015) A methodology for a comprehensive probabilistic tsunami hazard assessment: multiple sources and shortterm interactions. J Mar Sci Eng 3:23–51. https://doi.org/10.3390/jmse3010023CrossRefGoogle Scholar
 Grezio A, Babeyko A, Baptista MA, Behrens J, Costa A, Davies G, Geist EL, Glimsdal S, González FI, Griffin J, Harbitz CB, LeVeque RJ, Lorito S, Løvholt F, Omira R, Mueller C, Paris R, Parsons T, Polet J, Power W, Selva J, Sørensen MB, Thio HK (2017) Probabilistic tsunami hazard analysis: multiple sources and global applications. Rev Geophys 55. https://doi.org/10.1002/2017RG000579ADSCrossRefGoogle Scholar
 Grilli ST, Watts P (2005) Tsunami generation by submarine mass failure, I: modeling, experimental validation, and sensitivity analyses. J Waterway Port Coast Ocean Eng 131:283–297CrossRefGoogle Scholar
 Gutenberg B, Richter C (1944) Frequency of earthquakes in California. Bull Seism Soc Am 34:185–188Google Scholar
 Heidarzadeh M, Kijko A (2011) A probabilistic tsunami hazard assessment for the makran subduction zone at the Northwestern Indian Ocean. Nat Haz 56:577–593CrossRefGoogle Scholar
 Hoechner A, Babeyko AY, Zamora N (2016) Probabilistic tsunami hazard assessment for the Makran region with focus on maximum magnitude assumption. Nat Hazards Earth Syst Sci 16:1339–1350. https://doi.org/10.5194/nhess1613392016ADSCrossRefGoogle Scholar
 Horspool N, Pranantyo I, Griffin J, Latief H, Natawidjaja DH, Kongko W, Cipta A, Bustaman B, Anugrah SD, Thio HK (2014) A probabilistic tsunami hazard assessment for Indonesia. Nat Hazards Earth Syst Sci 14:3105–3122. https://doi.org/10.5194/nhess1431052014ADSCrossRefGoogle Scholar
 Jaynes ET (2003) In: Bretthorst GL (ed) Probability theory the logic of science. Cambridge University Press, Cambridge, p 727CrossRefGoogle Scholar
 Kagan YY (2002a) Seismic moment distribution revisited: I, statistical results. Geophys J Int 148:520–541ADSCrossRefGoogle Scholar
 Kagan YY (2002b) Seismic moment distribution revisited: II, moment conservation principle. Geophys J Int 149:731–754ADSCrossRefGoogle Scholar
 Kagan YY, Jackson DD (2000) Probabilistic forecasting of earthquakes. Geophys J Int 143:438–453ADSCrossRefGoogle Scholar
 Knighton J, Bastidas LA (2015) A proposed probabilistic seismic tsunami hazard analysis methodology. Nat Hazards. https://doi.org/10.1007/s1106901517417CrossRefGoogle Scholar
 Lane EM, Gillibrand PA, Wang X, Power W (2013) A probabilistic tsunami hazard study of the auckland region, part II: inundation modelling and hazard assessment. Pure Appl Geophys 170:1635–1646. https://doi.org/10.1007/s0002401205389ADSCrossRefGoogle Scholar
 Lay T (2015) The surge of great earthquakes from 2004 to 2014. Earth Planet Sci Lett Invited Front Pap 409:133–146. https://doi.org/10.1016/j.epsl.2014.10.047ADSCrossRefGoogle Scholar
 Lindley DV (2000) The philosophy of statistics. Statistician 49:293–337Google Scholar
 Lorito S, Selva J, Basili R, Romano F, Tiberti MM, Piatanesi A (2015) Probabilistic hazard for seismically induced tsunamis: accuracy and feasibility of inundation maps. Geophys J Int 200(1):574–588. https://doi.org/10.1093/gji/ggu408ADSCrossRefGoogle Scholar
 Lorito S, Romano F, Lay T (2016) Tsunamigenic earthquakes (2004–2013): source processes from data inversion. In: Meyers RA (ed) Encyclopedia of complexity and systems science, vol 2015. Springer Science+Business Media, New York. https://doi.org/10.1007/9783642277375_6411CrossRefGoogle Scholar
 Maramai A, Graziani L, Alessio G, Burrato P, Colini L, Cucci L, Nappi R, Nardi A, Vilardo G (2005a) Near and far field survey report of the 30 December 2002 Stromboli (Southern Italy) tsunami. Mar Geol 215(93):106Google Scholar
 Maramai A, Graziani L, Tinti S (2005b) Tsunami in the Aeolian Islands (southern Italy): a review. Mar Geol 215(11):21ADSGoogle Scholar
 Marzocchi W, Jordan TH (2014) Testing for ontological errors in probabilistic forecasting models of natural systems. PNAS 111(33):11973–11978. https://doi.org/10.1073/pnas.1410183111//DCSupplementalADSCrossRefGoogle Scholar
 Marzocchi W, Lombardi AM (2008) A double branching model for earthquake occurrence. J Geophys Res 113:B08317. https://doi.org/10.1029/2007JB005472ADSCrossRefGoogle Scholar
 Marzocchi W, Sandri L, Gasparini P, Newhall C, Boschi E (2004) Quantifying probabilities of volcanic events: the example of volcanic hazard at Mount Vesuvius. J Geophys Res 109:B11201. https://doi.org/10.1029/2004JB003155ADSCrossRefGoogle Scholar
 Marzocchi W, Sandri L, Selva J (2008) BET_EF: a probabilistic tool for long and shortterm eruption forecasting. Bull Volcanol 70:623–632. https://doi.org/10.1007/s004450070157yADSCrossRefGoogle Scholar
 Marzocchi W, Taroni M, Selva J (2015) Accounting for epistemic uncertainty in PSHA: logic tree and ensemble modeling. Bull Seismol Soc Am 105:2151–2159. https://doi.org/10.1785/0120140131CrossRefGoogle Scholar
 Mueller C, Power W, Fraser S, Wang X (2015) Effects of rupture complexity on local tsunami inundation: implications for probabilistic tsunami hazard assessment by example. J Geophys Res Solid Earth 120:488–502. https://doi.org/10.1002/2014JB011301ADSCrossRefGoogle Scholar
 O’Loughlin KF, Lander JF (2003) Caribbean tsunamis: a 500year history from 1498–1998. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
 Okada Y (1985) Surface deformation due to shear and tensile faults in a halfspace. Bull Seismol Soc Am 75:1135–1154Google Scholar
 Okada Y (1992) Internal deformation due to shear and tensile faults in a halfspace. Bull Seismol Soc Am 82:1018–1040Google Scholar
 Omira R, Baptista MA, Matias L (2015) Probabilistic tsunami hazard in The Northeast Atlantic from near and farfield tectonic sources. Pure Appl Geophys 172:901–920. 2014 Springer, Basel. https://doi.org/10.1007/S000240140949XADSCrossRefGoogle Scholar
 Orfanogiannaki K, Papadopoulos G (2007) Conditional probability approach of the assessment of tsunami potential: application in three tsunamigenic regions of the Pacific Ocean. Pure Appl Geophys 164:593–603ADSCrossRefGoogle Scholar
 Parsons T (2008) Monte Carlo method for determining earthquake recurrence parameters from short paleoseismic catalogs: example calculations for California. J Geophys Res 113. https://doi.org/10.1029/2007JB004998
 Parsons T, Geist EL (2009) Tsunami probability in the caribbean region. Pure Appl Geophys 165(2008):2089–2116. https://doi.org/10.1007/s0002400804167ADSCrossRefGoogle Scholar
 PatéCornell M (1996) Uncertainties in risk analysis: six levels of treatment. Reliab Eng Syst Saf 54:95–111CrossRefGoogle Scholar
 Polet J, Kanamori H (2009) Tsunami earthquakes. In: Meyers A (ed) Encyclopedia of complexity and systems science. Springer, New York. https://doi.org/10.1007/9780387304403_567CrossRefGoogle Scholar
 Popper KR (1983) Realism and the aim of science. Hutchinson, LondonGoogle Scholar
 Power W, Wang X, Lane EM, Gillibrand PA (2013) A probabilistic tsunami hazard study of the auckland region, part I: propagation modelling and tsunami hazard assessment at the shoreline. Pure Appl Geophys 170:1621. https://doi.org/10.1007/s000240120543zADSCrossRefGoogle Scholar
 Reid RO, Bodine BR (1968) Numerical model for storm surges in Galveston Bay. J Waterways and Harbors Div ACE 94:33–57Google Scholar
 Sakai T, Takeda T, Soraoka H, Yanagisawa K, Annaka T (2006) development of a probabilistic tsunami hazard analysis in Japan. In: Proceedings of ICONE14 international conference on nuclear engineering, Miami, 17–20 July, ICONE14–89183Google Scholar
 Satake K (1995) Linear and nonlinear computations of the 1992 Nicaragua earthquake tsunami. Pure App Geophys 144:455–470ADSCrossRefGoogle Scholar
 Satake K (2002) Tsunamis. In: Lee WHK, Kanimori H, Jennings PC, Kisslinger C (eds) International handbook of earthquake and engineering seismology, vol 81A. Academic Press, Amsterdam, pp 437–451CrossRefGoogle Scholar
 Selva J, Sandri L (2013) Probabilistic seismic hazard assessment: combining cornelllike approaches and data at sites through Bayesian inference. Bull Seismol Soc Am 103(3):1709–1722. https://doi.org/10.1785/0120120091CrossRefGoogle Scholar
 Selva J, Tonini R, Molinari I, Tiberti MM, Romano F, Grezio A, Melini D, Piatanesi A, Basili R, Lorito S (2016) Quantification of source uncertainties in seismic probabilistic tsunami hazard analysis (SPTHA). Geophys J Int 2016. https://doi.org/10.1093/gji/ggw107ADSCrossRefGoogle Scholar
 Shin JY, Chen S, Kim TW (2015) Application of Bayesian Markov Chain Monte Carlo Method with mixed gumbel distribution to estimate extreme magnitude of tsunamigenic earthquake. KSCE J Civ Eng 19(2):366–375. https://doi.org/10.1007/s1220501504300CrossRefGoogle Scholar
 Shuto N (1991) Numerical simulation of tsunamis – its present and near future. Nat Hazards 4:171–191CrossRefGoogle Scholar
 Smith WHF, Sandwell DT (1997) Global seafloor topography from satellite altimetry and ship depth soundings. Science 277:1957–1962Google Scholar
 Sørensen MB, Spada M, Babeyko A, Wiemer S, Grünthal G (2012) Probabilistic tsunami hazard in the Mediterranean Sea. J Geophys Res 117:B01305. https://doi.org/10.1029/2010JB008169ADSCrossRefGoogle Scholar
 Stigler SM (1983) Who discovered Bayes Theorem? Am Stat 37:290–296zbMATHGoogle Scholar
 Suppasri A, Imamura F, Koshimura S (2012) Probabilistic tsunami hazard analysis and risk to coastal populations in Thailand. J Earthquake and Tsunami 06:1250011. [27 Pages]. https://doi.org/10.1142/S179343111250011xCrossRefGoogle Scholar
 Synolakis CE (1987) The runup of solitary waves. J Fluid Mech 185:523–545ADSMathSciNetCrossRefGoogle Scholar
 Tanioka Y, Satake K (1996) Tsunami generation by horizontal displacement of ocean bottom. Geophys Res Lett 23:861–865ADSCrossRefGoogle Scholar
 Tatsumi D, Calder CA, Tomita T (2014) Bayesian nearfield tsunami forecasting with uncertainty estimates. J Geophys Res Oceans 119:2201–2211. https://doi.org/10.1002/2013JC009334ADSCrossRefGoogle Scholar
 Thio HK, Li W (2015) Probabilistic tsunami hazard analysis of the cascadia subduction zone and the role of epistemic uncertainties and aleatory variability. In: 11th Canadian conference on earthquake engineering, Victoria, pp 21–24Google Scholar
 Thio HK, Somerville P, Polet J (2010) Probabilistic tsunami hazard in California, PEER Report 2010/108 Pacific Earthquake Engineering Research CenterGoogle Scholar
 Tinti S, Maramai A, Graziani L (2004) The new catalogue of Italian Tsunamis. Nat Haz 33(439):465Google Scholar
 Tinti S, Armigliato A, Tonini R, Maramai A, Graziani L (2005) Assessing the hazard related to tsunamis of tectonic origin: a hybrid statisticaldeterministic method applied to Southern Italy coasts. ISET J Earthquake Tech 42:189–201Google Scholar
 Tinti S, Argnani A, Zaniboni F, Pagnoni G, Armigliato A (2007) Tsunamigenic potential of recently mapped submarine mass movements offshore eastern Sicily (Italy): numerical simulations and implications for the 1693 tsunami. IASPEI—JSS002—abstract n. 8235 IUGG XXIV General Assembly, Perugia, 2–13 July 2007Google Scholar
 TPSWG Tsunami Pilot Study Working Group (2006) Seaside, Oregon tsunami pilot study— modernization of FEMA flood hazard maps. NOAA OAR Special Report, NOAA/OAR/PMEL, Seattle, p 94 + 7 appendicesGoogle Scholar
 Vannucci G, Gasperini P (2004) The new release of the database of earthquake mechanisms of the mediterranean area (EMMA2). Ann Geophys Suppl 47:307–334Google Scholar
 Watts P, Grilli ST, Tappin D, Fryer GJ (2005) Tsunami generation by submarine mass failure, II: predictive equations and case studies. J Waterway Port Coast Ocean Eng 131:298–310CrossRefGoogle Scholar
 Wells DL, Coppersmith KJ (1994) New empirical relationships among magnitude, rupture length, rupture width, rupture area and surface displacement. Bull Seismol Soc Am 84:974–1002Google Scholar
 Yadav RBS, Tsapanos TM, Tripathi JN, Chopra S (2013) An evaluation of tsunami hazard using Bayesian approach in the Indian Ocean. Tectonophysics 593:172–182ADSCrossRefGoogle Scholar