Editorial Type:
Article
Article Type:
Research Article

Uncertainty Propagation Using Polynomial Chaos Expansions for Extreme Sea Level Hazard Assessment: The Case of the Eastern Adriatic Meteotsunamis

Cléa Denamiel Institute of Oceanography and Fisheries, Split, Croatia

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Xun Huan University of Michigan, Ann Arbor, Michigan

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Jadranka Šepić Institute of Oceanography and Fisheries, Split, Croatia

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Ivica Vilibić Institute of Oceanography and Fisheries, Split, Croatia

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Online Publication:
01 Mar 2020
Print Publication:
01 Apr 2020
DOI:
https://doi.org/10.1175/JPO-D-19-0147.1
Page(s):
1005–1021
Free access

Abstract

This study quantifies the hazard associated with extreme sea levels due to eastern Adriatic meteotsunamis—long waves generated by traveling atmospheric disturbances—and assesses the sensitivity of the ocean response to the disturbances responsible for those events. In this spirit, a surrogate model of meteotsunami maximum elevation based on generalized polynomial chaos expansion (gPCE) methods, is implemented. The approach relies on the definition of a synthetic pressure disturbance—depending on six different stochastic parameters known to be important for meteotsunami generation, which is used as forcing to produce series of meteotsunami simulations defined with sparse grid methods (up to 10 689 used in this study). The surrogate model and the sensitivity study are then obtained with a pseudo-spectral approximation (PSA) method based on the chosen meteotsunami simulations. This study mainly presents the developed methodology and discusses the feasibility of implementing such gPCE-based surrogate models to assess the hazard and to study the sensitivity of meteorologically driven extreme sea levels.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: C. Denamiel, cdenamie@izor.hr

Abstract

This study quantifies the hazard associated with extreme sea levels due to eastern Adriatic meteotsunamis—long waves generated by traveling atmospheric disturbances—and assesses the sensitivity of the ocean response to the disturbances responsible for those events. In this spirit, a surrogate model of meteotsunami maximum elevation based on generalized polynomial chaos expansion (gPCE) methods, is implemented. The approach relies on the definition of a synthetic pressure disturbance—depending on six different stochastic parameters known to be important for meteotsunami generation, which is used as forcing to produce series of meteotsunami simulations defined with sparse grid methods (up to 10 689 used in this study). The surrogate model and the sensitivity study are then obtained with a pseudo-spectral approximation (PSA) method based on the chosen meteotsunami simulations. This study mainly presents the developed methodology and discusses the feasibility of implementing such gPCE-based surrogate models to assess the hazard and to study the sensitivity of meteorologically driven extreme sea levels.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: C. Denamiel, cdenamie@izor.hr
Keywords: Atmosphere-ocean interaction; Probability forecasts/models/distribution; Ocean models; Stochastic models; Flood events; Risk assessment

1. Introduction

In the past decades, extensive research has been undertaken in the development of polynomial chaos expansion (PCE; Wiener 1938; Ghanem and Spanos 1991) and generalized PCE (gPCE) methods (Xiu and Karniadakis 2002a; Soize and Ghanem 2004), as well as their applications in uncertainty quantification (UQ) and sensitivity analysis. The main advantages of using gPCEs for performing these tasks compared to sampling approaches (e.g., Monte Carlo simulations) are twofold. First, gPCEs serve as surrogate models—based on expansions that decompose into deterministic coefficients and random orthogonal bases—that are highly efficient for propagating the uncertainties of model inputs to outputs [e.g., Knio and Le Maître (2006) and Najm et al. (2009) provide detailed discussions in the context of computational fluids applications]. Second, gPCEs allow for the analytical computation of Sobol’ indices often used in global sensitivity analysis (Sobol’ 2001; Saltelli et al. 2008; Sudret 2008), directly from a closed-form formula from the gPCE coefficients without needing any additional Monte Carlo sampling. Furthermore, gPCE methods have been demonstrated with success in a wide range of applications including solid mechanics (Ghanem 1999a,b; Anders and Hori 1999; Foo et al. 2007), fluid mechanics (Le Maître et al. 2001; Xiu and Karniadakis 2002b, 2003; Zhang and Lu 2004; Ghanem and Dham 1998; Hou et al. 2006; Rupert and Miller 2007), thermodynamics (Xiu and Karniadakis 2002c; Lin and Karniadakis 2006; Jardak et al. 2002), electrical engineering (Su and Strunz 2005; Agarwal and Aluru 2007; Hover and Triantafyllou 2006), chemical engineering (Nagy and Braatz 2007; Hauptmanns 2008), differential equations (Xiu and Karniadakis 2002a; Wan and Karniadakis 2005; Williams 2006; Wan and Karniadakis 2005), water resources (Rupert and Miller 2007), solid–fluid interactions (Witteveen et al. 2007), and so on. More specifically, in the field of geosciences, some applications also adopted the use of gPCE, for example, Formaggia et al. (2013) built a surrogate model of basin-scale geochemical compaction, Wang et al. (2016) studied the acoustic uncertainty predictions, Sraj et al. (2014) estimated the wind drag parameter forcing an ocean model, and Giraldi et al. (2017) documented propagation of the uncertainty in the earthquake ocean floor displacement model to the tsunami wave parameters at a location. In the ocean community, the use of the PCE methodology mainly focused on the improvement of the model physics by considering the uncertainties linked to either the boundary condition forcing (Thacker et al. 2012) or the parameterization of the wind stress and the mixing processes (Winokur et al. 2013). To the best of our knowledge, however, there has not been any work that leverage gPCE to perform UQ in atmospherically driven extreme sea level modeling, an area that is highly important for developing warning systems, and the focus application of this paper.

The forecast of meteorologically driven extreme sea level events highly depends on the quality of the modeled atmospheric forcing. To quantify the associated ocean hazard, it is thus primordial to account for the uncertainties linked to this forcing. In this sense, the atmospheric forcing can be seen as an uncertain input depending on various stochastic parameters defining its shape and temporal scale. When the atmospheric forcing can be described analytically with a reasonably small number of parameters (e.g., in hurricane modeling), the use of gPCE methods appears highly attractive for enabling efficient UQ computations in extreme sea level hazard assessment.

This study focuses on the hazard assessment of the eastern Adriatic meteotsunamis—long waves generated by traveling atmospheric disturbances, commonly identified as internal gravity waves (IGWs; Vilibić and Šepić 2009; Horvath et al. 2018). Even though the physics of ocean models theoretically allows for simulating meteotsunami events (Rabinovich et al. 1999; Liu et al. 2003; Vilibić et al. 2004; Dragani 2007), resolution and bathymetry of these models (Vilibić et al. 2008; Orlić et al. 2010) as well as meteorological forcing (Horvath and Vilibić 2014) are often not accurate enough to deterministically forecast the extreme sea levels of such phenomenon. Additionally, the evaluation of the Adriatic Sea and Coast (AdriSC) Meteotsunami Forecast system (Denamiel et al. 2019a,b) has shown that the capacity of the ocean models to deterministically reproduce or forecast the observed floods occurring during the meteotsunami events is extremely sensitive to the capacity of the atmospheric models to reproduce the meteotsunamigenic disturbances. Given the scarcity of the well-documented eastern Adriatic meteotsunami events (i.e., the lack of knowledge of the meteotsunami maximum elevation distributions at sensitive locations) and the difficulty of reproducing deterministically the atmospheric mesoscale processes with state-of-the-art numerical models, the assessment of meteotsunami extreme sea level hazard via a stochastic method was judged necessary. A gPCE-based meteotsunami surrogate model was thus developed in order to 1) describe the meteotsunami maximum elevation (i.e., a scalar quantity) distribution as its projection onto a polynomial basis in a similar way the empirical orthogonal function (EOF) describes the different modes of a random field (e.g., a spatially dependent quantity) as its projection onto its eigenfunctions; 2) account for and propagate the uncertainty in the atmospheric forcing; and 3) better understand the sensitivity of the ocean response to the atmospheric disturbance parameters (e.g., location, speed, amplitude). The meteotsunami early warning system developed in the eastern Adriatic Sea, which is fully described and evaluated in Denamiel et al. (2019b), was thus designed to 1) deterministically forecast and detect the atmospheric disturbances responsible for the meteotsunami events at least 24 h before any potential event; 2) measure these disturbances with pressure sensors (three along the Italian coastline and two in the middle of the Adriatic Sea) at least 2 h before they hit sensitive locations (e.g., harbors) along the Croatian coastline; 3) extract the values of the IGW parameters (e.g., location, amplitude, period) from both the forecasted and measured disturbances; and 4) run the surrogate model for input parameters varying around these extracted values in order to provide a hazard assessment based on the propagation of the atmospheric disturbance uncertainties to the resulting meteotsunami maximum elevation distributions at the sensitive locations.

In this paper, the methodology used to create the surrogate model is first described in detail, including 1) the parameterization of the atmospheric meteotsunamigenic disturbances; 2) the gPCE model construction for the meteotsunami maximum sea level application; and 3) the numerical framework of the deterministic ocean model. The performance of gPCE, the quantified uncertainty for the maximum sea level, and the sensitivity study with respect to the inputs are then presented as the main results of this paper. Finally, conclusions are drawn concerning the use of gPCE-based methods to assess meteotsunami hazard and more generally meteorologically driven extreme sea levels.

2. Methodology

a. Synthetic atmospheric forcing

Due to the spatial and temporal scarcity of available 1-min pressure and sea level measurements (Monserrat et al. 2006), producing hazard maps and sensitivity studies of meteotsunami events is more easily achieved with synthetic atmospheric gravity waves forcing a regional ocean model than with in situ data (Ličer et al. 2017; Denamiel et al. 2018, Williams et al. 2019). In this study, the meteotsunami hazard is thus defined as the maximum elevation ξmax generated at certain sensitive locations (e.g., harbors) by a synthetic atmospheric pressure forcing Ps alone, as effects of winds and other forcing are documented to be of secondary importance for meteotsunamis in the eastern Adriatic (Vilibić et al. 2005; Orlić et al. 2010). This forcing, previously described in Denamiel et al. (2018), varies in space (x, y ∈ Dx × Dy) and time (t ∈ T) and is expressed as a stochastic process
Ps:{Dx×Dy×T×Ω→R⁡(x,y,t,ω)→Ps⁡(x,y,t,ω)
depending on a set of elementary events (ω ∈ Ω), such as
Ps⁡(x, y, t, ω)=P0+PGW⁡(x, y, t, ω).
The synthetic atmospheric pressure forcing is split into 1) a mean atmospheric pressure component P0 assumed constant over the entire Adriatic Sea and 2) a stochastic gravity wave component PGW.
In the eastern Adriatic Sea, atmospheric gravity waves responsible for meteotsunami events generally propagate from the eastern side of the Italian peninsula (Vilibić and Šepić 2009; Šepić et al. 2012, 2016). To limit the number of stochastic parameters used in the expression of the synthetic pressure wave, the points of origin of the stochastic gravity waves are taken along a virtual line (Fig. 1), mathematically defined by the function f depending only on the latitude y0(ω), such that x0 = f(y0) is the corresponding longitude of the chosen point of origin. Additionally, the stochastic variable θ(ω) represents the direction of propagation of the gravity waves and the rotated coordinate system (xx, yy) is expressed as
{xx=[x−f⁡(y0)] cos⁡(θ)+(y−y0) sin⁡(θ)yy=−[x−f⁡(y0)] sin⁡(θ)+(y−y0) cos⁡(θ).
Fig. 1.
Fig. 1.

Rotated coordinate system (xx, yy) depending on the point of origin [f(y0), y0] and the direction of propagation θ of each synthetic atmospheric gravity wave PGW. The red box represents the central eastern Adriatic, which is the focus area of the study.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0147.1

The formulation of the synthetic atmospheric gravity wave PGW in the rotated coordinate system is then given by
PGW⁡(xx,yy,t,ω)=D⁡(t)PA⁡(ω)e−5yy2d⁡(ω)2⁡ sin[2πT⁡(ω)ϕ⁡(t,xx)].
In Eq. (2), the stochastic parameters defining the shape of the atmospheric gravity wave (PGW) are the amplitude PA(ω), the period T(ω), the phase ϕ(t, xx) = {t − [xx/c(ω)]}, which depends on the velocity of propagation c(ω), and the attenuation in yy direction d(ω). Examples of synthetic gravity wave spatial and temporal properties are visualized in Fig. 3 of Denamiel et al. (2018). In addition, the exponential attenuation in time of the atmospheric wave D(t), assumed to be deterministic (i.e., not a stochastic variable), drops the wave amplitude to 0.5% of its original value after N = 73 number of wave periods.
Based on a set of pressure measurements recorded during meteotsunami events taking place in the eastern Adriatic, Šepić and Vilibić (2011) defined several warning metrics including parameters such as velocity, direction, amplitude, and period of the atmospheric gravity waves at different locations along the Croatian coastline. Through merging of these metrics with recently gained knowledge on the Adriatic meteotsunamis (Denamiel et al. 2018, 2019a), the range of variation of the six stochastic input parameters (y0, θ, PA, T, c, d) of the synthetic atmospheric gravity wave has been defined. However, given the sparsity of the available pressure measurements and rarity of strong meteotsunami events, the prior distribution of the stochastic parameters could not be assessed accurately. It is thus assumed that all the stochastic variables follow uniform distributions (i.e., within the range of the uniform distributions, the generation of any pressure wave is equally probable) given by
θ: {Ω→Rω→θ⁡(ω),θ⁡(ω)~U⁡([−π3,π2])~U⁡([0°N,150°N])c: {Ω→Rω→c⁡(ω), c⁡(ω)~U⁡([15 m s−1,40 m s−1])T: {Ω→Rω→T⁡(ω), T⁡(ω)~U⁡([300 s,1800 s])PA:{Ω→Rω→PA⁡(ω), PA⁡(ω)~U⁡([50 Pa,400 Pa])y0:{Ω→Rω→y0⁡(ω), y0⁡(ω)~U⁡([41.25°,43.65°])d: {Ω→Rω→d⁡(ω), d⁡(ω)~U⁡([30 km,150 km]).

The six stochastic variables (y0, θ, PA, T, c, d) used to define the synthetic atmospheric gravity waves [Eq. (2)] are also assumed to be mutually independent. Both uniform distribution and mutual independence assumptions follow the principle of maximum entropy. This principle states that when little is known about a distribution, then the distribution with the largest entropy should be chosen as the least-informative default since it involves the fewest assumptions (Jaynes 1957). In other words, as the range of definition of each uniform distribution has been chosen to cover all the conditions known to generate meteotsunamis along the Croatian coastline, all the synthetic tsunamigenic disturbances are taken into account in this study. However, with this choice, many random atmospheric gravity waves chosen from the defined distributions presumably have no potential to generate strong meteotsunami waves in the area of interest. This is in fact strong motivation for reducing the uncertainty distributions from additional data, which can be achieved through Bayesian statistical inference (e.g., von Toussaint 2011) or more broadly, model calibration or data assimilation; however, this is beyond the scope of the current paper, and will be investigated as future work.

b. Stochastic surrogate model for meteotsunami maximum elevation

In this study, both quantifying the uncertainties and analyzing the sensitivity of the meteotsunami hazard were identified as key objectives. As only a few meteotsunami events were properly monitored in this region of the world, these tasks could only be achieved via a numerical modeling approach. However, running ocean models can be expensive, and a computationally inexpensive stochastic surrogate model strategy—relying on the gPCE (Xiu and Karniadakis 2002a,b, 2003; Soize and Ghanem 2004)—was chosen instead to capture the maximum sea level elevation ξmax due to meteotsunami events generated by synthetic atmospheric gravity waves (described in section 2a). The main advantage of this approach is that it offers a convenient form for uncertainty propagation, where moments and sensitivity indices can be directly extracted from the coefficients of the gPCE decomposition. The appropriateness of the surrogate model choice is based on the assumption (Ernst et al. 2012) that a full gPCE for the maximum elevation due to a meteotsunami ξmaxgPCE [Eq. (4)] converges (i.e., the mean-square error between ξmaxgPCE and ξmax reduces to zero in the infinite expansion), which is valid if ξmax has finite variance—a requirement that is quite reasonable from a physics point of view. A truncated gPCE then can be shown to be optimal (lowest error) expansion for a given finite number of terms:
ξmaxgPCE=∑|α|≤pξαΨα⁡(R1,R2,…,Rr).
Let us define {Zi}i=1r, the “basis” random variables carrying a standard distribution form. In this application, since our stochastic input parameters defined in section 2a have uniform distributions, it is convenient to select Zi~U⁡([−1, 1]). The independent random input variables, denoted Ri, i ∈ 1, 2, …, r and used in Eq. (4), relate to Zi by
{Zi:{Ω→Rω→Zi⁡(ω), Zi~U⁡([−1,1])Ri=(ai+bi2)+(bi−ai2)Zi, Ri~U⁡([ai,bi]), i∈1,2,…,r.
Here, α = [α1, α2, …, αr] is a multi-index with αi indicating the polynomial degree associated with the ith random input variable Ri. In this study we include all polynomials of total order up to degree p: |α|=∑i=1rαi≤p, but other choices of the index set are also possible.
The multivariate orthogonal polynomial basis, Ψα⁡(R1,R2,…,Rr)=∏i=1r Leαi[⁡(2Ri−ai−bi)/⁡(bi−ai)], is expressed as the product of the Legendre polynomials Leαi, i∈1,2,…,r associated with each uniformly distributed input variable Ri. The Legendre polynomials are orthogonal with respect to uniform distributions such that E[Ψα, Ψβ]=δα,β‖Ψα‖2, where
δα,β={0, if α≠β 1, if α=β 
is the Kronecker symbol and the expectation E[Ψα, Ψβ] and norm ‖Ψα‖ are derived with respect to the uniform distributions.
One approach to compute the coefficients of the gPCE (ξα) is through a projection of the target response ξmax onto each polynomial basis of the gPCE:
ξα=1‖Ψα‖2E[ξmaxΨα⁡(R1,R2,…,Rr)].
However, since ξmax is not available analytically, the expectation E[ξmaxΨα⁡(R1,R2,…,Rr)] in Eq. (6) cannot be computed in closed form. Instead, numerical integration methods need to be used to approximate the expectation. In this study, the nonintrusive pseudo-spectral approximation (PSA) method (Gerstner and Griebel 1998; Constantine et al. 2012), a quadrature-based approach, is used to create the stochastic surrogate model. The pseudospectral approximation ξmaxPSA of the generalized polynomial chaos expansion ξmaxgPCE is based on a Smolyak sparse-grid construction, and can be shown to be the following (Smolyak 1963):
ξmaxPSA=∑max⁡(0, p−r+1)≤|α|≤p⁡(−1)p−|α|⁡(r−1p−|α|)⁡(Aα1⁡(1)⊗Aα2⁡(2)⊗⋯⊗Aαr⁡(r)),
where the full tensor product Aα⁡(1:r)=Aα1⁡(1)⊗Aα2⁡(2)⊗⋯⊗Aαr⁡(r) encompasses univariate operator Aαi⁡(i) i∈1, 2, ⋯, r given by
Aαi⁡(i)⁡(ξmaxi)=∑j=0αiQαi⁡(i)⁡(ξmaxiLej)‖Lej‖2Lej⁡(Ri).
Let us define a theoretical function ξmaxi that can be evaluated requiring only one random input variable Ri. The univariate quadratic rule Qαi⁡(i)⁡(ξmaxiLej)=∑k=1Liξmaxi⁡(sk⁡(i))Lej⁡(sk⁡(i)) wk⁡(i) is used to approximate the expectation E[ξmaxiLej⁡(Ri)] with Li evaluation abscissa points (i.e., chosen samples) of the random variable Ri: sαi⁡(i)={sk⁡(i)}k=1Li associated with Li weights: wαi⁡(i)={wk⁡(i)}k=1Li. Some of the most known quadrature rules for uniform distributions are the Gauss–Legendre (Abramowitz and Stegun 1964), the Clenshaw–Curtis (Clenshaw and Curtis 1960), and the Gauss–Patterson (Patterson 1968) rules. The choice of the quadrature rule used in the surrogate model is discussed in detail in section 2c(1).
For each α from max(0, p − r + 1) ≤ |α| ≤ p, the combined total set of all abscissa and weights needed to calculate the tensor product defined in Eq. (7), is given by sα⁡(1:r)=sα1⁡(1)⊗sα2⁡(2)⊗⋯⊗sαr⁡(r) and wα⁡(1:r)=wα1⁡(1)⊗wα2⁡(2)⊗⋯⊗wαr⁡(r), respectively. The ultimate multivariate quadratic rule that is needed to evaluate the real function ξmaxPSA in Eq. (7) is then expressed as
Qα⁡(1:r)⁡(ξmaxΨβ)=∑αξmax⁡(sα⁡(1:r))Ψβ⁡(sα⁡(1:r))wα⁡(1:r).
Finally, for any given α, the multi-index β = [β1, β2, …, βr] stores all the polynomial combinations needed to calculate the tensor product: Aα⁡(1:r)=∑β[Qα⁡(1:r)⁡(ξmaxΨβ)/‖Ψβ‖2]Ψβ⁡(R1,…,Rr).
Practically, in this study, for any given order p such as max(0, p − r + 1) ≤ |α| ≤ p, the PSA method follows three steps:

  1. carrying out the deterministic ocean simulations forced by all the needed atmospheric gravity wave conditions (defined by the unique combinations of the abscissa sα⁡(1:r)) and extracting the model results ξmax⁡(sα⁡(1:r));

  2. deriving [from Eqs. (8) and (9)] and storing, for each value of |α|, all the combinations of the multi-index β and all the associated coefficients C|α|,β=Qα⁡(1:r)⁡(ξmaxΨβ)/‖Ψβ‖2;

  3. creating the stochastic surrogate model estimating the maximum elevation due to meteotsunami events generated by any synthetic atmospheric gravity waves depending on the predefined stochastic variables (y0, θ, PA, T, c, d):

ξmaxPSA⁡(y0,θ,PA,T,c,d)=∑max⁡(0,p−r+1)≤|α|≤p⁡(−1)p−|α|⁡(r−1p−|α|) ∑βC|α|,βΨβ⁡(y0,θ,PA,T,c,d).

c. Deterministic meteotsunami simulations

1) Choice of the simulations

In the PSA method, the number and values of the synthetic atmospheric pressure conditions forcing the deterministic ocean model only depend on the total order p of the polynomial decomposition and the chosen quadrature rule Qα⁡(1:r)⁡(ξmaxΨβ).

For a given random input variable
Z:{Ω→Rω→Z⁡(ω) ,Z~U⁡([−1,1]),
the univariate quadrature problem consists in finding an approximation of the integral of a function g over the interval (−1, 1) such as ∫−11g⁡(Z) dZ≈∑i=1ng⁡(Si)Wi with {Si}i=1n the abscissas (or samples) of the rule and {Wi}i=1n their associated weights. The Gauss–Legendre quadrature formula (Abramowitz and Stegun 1964), which has a polynomial precision of 2n − 1, is the natural n point rule for uniform distributions. This quadrature thus produces the exact value of the integral of any function g with a polynomial degree 2n − 1 or less. However, the Gauss–Legendre rule requires a free choice of all the abscissas: none of the abscissas defined for the n + 1 point rule intersects with the abscissas defined for the n point rule. On the contrary, for nested quadrature rules, all abscissas of one rule are included in the next rule. The Clenshaw–Curtis (Clenshaw and Curtis 1960) and the Gauss–Patterson (Patterson 1968) rules are the most common nested rules defined for random input variables with uniform distributions. The major difference between these two nested rules is that the precision and growth rate of the Gauss–Patterson rules are nearly double the precision and growth rate of the Clenshaw–Curtis rules. For these nested rules the notion of level, which represents the chosen quadrature rules where their abscissas are nested, is introduced. For a given level l, the number of abscissas required for the Clenshaw–Curtis and Gauss–Patterson rules is 2l + 1 and 2l+1 − 1, respectively (see Table 1). However, in order to construct more efficiently the nested Gauss–Patterson rules (i.e., to limit the growth rate), the introduction of the next Gauss–Patterson rule can be delayed as presented in Table 1. This means that, for an identical level, the delayed Gauss–Patterson rule (Holtz 2011) uses a smaller sample of the abscissas defined for the Gauss–Patterson rule. The advantage of the delayed Gauss–Patterson nested rules is that while the precision of the rule is still high (at least 2l − 1), the number of abscissas needed for the quadrature is dramatically reduced. In addition, Holtz (2011) stated that “in many cases, the performance of the classical sparse grid method can be enhanced by the use of delayed sequences of one-dimensional quadrature rules.”
Table 1.

Number of abscissas required for the Clenshaw–Curtis, Gauss–Patterson, and delayed Gauss–Patterson nested rules, depending on the level n of the quadratic rule.

Table 1.

In this study, as the polynomial degree at which the PSA method will converge is unknown, it is important to use nested quadratic rules in order to limit the number of deterministic numerical ocean simulations needed. Because of their higher precision, the Gauss–Patterson (GP) and the delayed Gauss–Patterson (DGP) rules are preferred to the Clenshaw–Curtis (CC) rules. However, both GP and CC rules have shown to achieve similar performance in practice (Trefethen 2008).

Within the framework of the PSA method, for any random input variable Ri=[⁡(ai+bi)/2]+[⁡(bi−ai)/2]Zi,Ri~U⁡([ai,bi]),Zi~U⁡([−1,1]),i∈1,2,…,r, associated with the polynomial order αi, the level of the univariate quadrature is αi and the number of required abscissas, which depends on the chosen rule (GP or DGP), is Li (see Table 1). The abscissas and weights of Zi derived by Patterson (1989) are given by Sαi⁡(i)={Sk⁡(i)}k=1Li and Wαi⁡(i)={Wk⁡(i)}k=1Li, respectively, and are rescaled for the input variable Ri such as sαi⁡(i)=[⁡(ai+bi)/2]+[⁡(bi−ai)/2]Sαi⁡(i) and wαi⁡(i)=[⁡(bi−ai)/2]Wαi⁡(i). Finally, for the r = 6 random input variables (y0, θ, PA, T, c, d), depending on the level (previously referred as order) p of the PSA method, the set of unique combinations of samples sα⁡(1:r)=sα1⁡(1)⊗sα2⁡(2)⊗⋯⊗sαr⁡(r) of the multivariate GP or DGP quadrature rules—also called multidimensional (6D) sparse grid, defines the final number and values of the unique synthetic atmospheric pressure conditions (and thus the final number of ocean simulations) needed to calculate the PSA (Fig. 2).

Fig. 2.
Fig. 2.

Example of 2D sparse grids for both the Gauss–Patterson and delayed Gauss–Patterson rules, combining the period T and the speed c only and used in the definition of the synthetic atmospheric pressure. Depending on the level p of the PSA, the total numbers of unique synthetic atmospheric pressure conditions (6D sparse grid) needed to calculate the Gauss–Patterson and delayed Gauss–Patterson quadrature rules are given in the two tables.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0147.1

2) Numerical model setup

In this study, the barotropic version (2DDI) of the unstructured model ADCIRC (Luettich et al. 1991) is used to carry out the ocean simulations required by the PSA method. With the only forcing of the simulations being the atmospheric pressure PS [Eq. (1)] ADCIRC solves the following simplified equations of motion for a moving fluid on a rotating earth:
{∂H∂t+∂∂x⁡(UH)+∂∂y⁡(VH)=0∂U∂t+U∂U∂x+V∂U∂y−fcV=−g∂∂x⁡(ξ+PSgρ0)−τbxHρ0+RxH∂V∂t+U∂V∂x+V∂V∂y+fcU=−g∂∂y⁡(ξ+PSgρ0)−τbyHρ0+RyH.
The barotropic velocities of the model are given by U,V=(1/H)∫−hξu,υ dz with u, υ the horizontal baroclinic velocities, h the depth of the model (i.e., the bathymetry), ξ the free surface elevation, and H = ξ + h the total water column thickness. A nonuniform Coriolis parameter fc depending on the latitude is used. The terms Rx, Ry represent the sum of the vertically integrated lateral stress gradients and the horizontal momentum dispersion. The bottom stress is given by τb = (τbx, τby).

The ADCIRC unstructured mesh used in this study is identical to the one used in the AdriSC Meteotsunami Forecast Component (Denamiel et al. 2019a). Its resolution ranges from about 5 km in the deepest part of the Adriatic Sea to 10 m in the locations where meteotsunamis are known to occur (i.e., Vela Luka, Stari Grad, Vrboska, Rijeka dubrovačka, Ston, Mali Ston, Viganj). The mesh is composed of 286 336 nodes forming 513 340 triangular elements and includes 477 islands and islets. The bathymetry is interpolated from a digital terrain model (DTM) incorporating offshore bathymetry from ETOPO1 (Amante and Eakins 2009), nearshore bathymetry from navigation charts CM93 2011, topography from GEBCO 30-arc-s grid 2014 (Weatherall et al. 2015), and coastline data generated by the Institute of Oceanography and Fisheries (Split, Croatia). The structure and bathymetry of the mesh for the entire Adriatic Sea as well as zooms at six locations of interest are presented in Fig. 3.

Fig. 3.
Fig. 3.

Structure and bathymetry of the ADCIRC mesh for the entire Adriatic Sea and seven locations of interest: Vela Luka, Stari Grad, Vrboska, Rijeka dubrovačka, Ston, Mali Ston, and Viganj.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0147.1

3. Results

a. Performance of the PSA method

Two different steps are needed to prove the feasibility of using a surrogate model, based on the PSA method, to estimate the meteotsunami hazard in the eastern Adriatic Sea. First, the accuracy and the convergence of the PSA should be assessed and second, the maximum elevation distribution obtained with the surrogate model should be compared to the one obtained with the ADCIRC model. To investigate the reliability of the PSA method depending on the quadrature rule choice, the coefficients of the PSA were derived at the seven locations of interest (Vela Luka, Rijeka dubrovačka, Stari Grad, Vrboska, Viganj, Mali Ston, and Ston) with the Gauss–Patterson rules up to a level 5 (PSA GP 5) and with the delayed Gauss–Patterson rules up to a level 6 (PSA DGP 6). In practice, a total of N = 10 689 unique ADCIRC simulations were performed: 10 625 for the PSA GP 5 and 64 additional for the PSA DGP 6, which also uses 4097 simulations already derived for the PSA GP 5 (4161 simulations in total as described in Fig. 2). For each location of interest l ∈ 1, …, 7, the ensemble of ADCIRC maximum elevations {ξmaxl,n}n=1N is thus used to test the accuracy of the surrogate model results, {ξGP,pl,n}n=1N and {ξDGP,pl,n}n=1N, depending on the level p of the PSA and the chosen quadrature rule (GP or DGP). In addition, as the PSA can produce unphysical negative maximum elevations for polynomials of low order, all negative surrogate model results are assumed to be equal to zero.

In theory, in order to assess the accuracy of the PSA, the ensemble of simulations used to calculate the error should be 1) sampled with a Monte Carlo method and 2) large enough (104 or 105) to provide viable results. Such an estimator is, however, not affordable, since the very principle of constructing the PSA is to limit the number of ADCIRC runs. But, for the highest orders, reusing the simulations implemented to derive the PSA is not an ideal option due to overfitting. Consequently, the error for the PSA GP 5 method is most probably underestimated, as it is calculated with only 64 additional simulations (only used to derive the PSA DGP 6). For the DGP 6 method, the error is calculated with 6528 additional simulations (only used to derive the PSA GP 5), which is close to the theoretical setup of 104 Monte Carlo samples and thus overfitting is avoided. In this study, at each level p and for each location l, the normalized root-mean-square errors are thus given by eGP,pl=∑n=1N⁡(ξmaxl,n−ξGP,pl,n)2/∑n=1N⁡(ξmaxl,n−ξGP,0l,n)2 and eDGP,pl=∑n=1N⁡(ξmaxl,n−ξDGP,pl,n)2/∑n=1N⁡(ξmaxl,n−ξDGP,0l,n)2, respectively, for the GP and DGP PSA methods. Figure 4 shows, for the two methods, the evolution of the normalized errors ⁡(eGP,pl, eDGP,pl) depending on the locations and the PSA level. The first interesting feature revealed by Fig. 4, which is a known behavior of the PSA methods, is that for a given level and a given method, the normalized error is highly dependent on the studied location. For example, at level 5, the normalized error varies between 0.17 and 0.41 for the GP method and between 0.32 and 0.83 for the DGP method. For both methods, Ston is the location which has the highest errors for PSA levels above 2. Second, and most importantly, Fig. 4 shows that the normalized error generally decreases when the PSA level increases and thus the PSA methods (GP and DGP) seem to converge for all locations. However, for some locations like Rijeka dubrovačka or Ston, the error does not necessarily decrease with the increase of the PSA level even if the error is minimized respectively at levels 5 and 6 for the GP and DGP methods. This behavior can be explained by two factors. First, the choice of zeroing all negative results obtained with the surrogate model slightly affects the calculation of the error for lower-level PSA (the number of negative results tends to decrease when the PSA level increases). Second, given their latitude and their orientation, Rijeka dubrovačka and Ston are more likely to be affected by atmospheric disturbances coming from the edge of the start location y0 and direction θ intervals of definition (Šepić et al. 2016). Finally, although the PSA DGP method converges more slowly than the PSA GP method, the normalized errors obtained with the PSA GP 5 and the PSA DGP 6 methods are quite similar for all the locations except Viganj and Mali Ston. The mean and variance of the absolute difference between the errors is 0.048 and 0.002, respectively. In addition, normalized errors vary between 0.17 and 0.41 for the PSA GP 5 and between 0.18 and 0.42 for the PSA DGP 6. Keeping in mind that the error of the PSA GP 5 is probably underestimated, the PSA DGP 6 is thus performing well.

Fig. 4.
Fig. 4.

Evolution of the normalized errors derived for the seven locations of interest (Vela Luka, Rijeka dubrovačka, Stari Grad, Vrboska, Viganj, Mali Ston, and Ston) depending of the level p of the PSA and the chosen quadrature rule: (top) Gauss–Patterson and (bottom) delayed Gauss–Patterson.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0147.1

As the convergence of the PSA methods has been demonstrated, smaller errors would be achieved by increasing the PSA level. However, this would require additional ADCIRC simulations: nearly 30 000 more for the PSA GP level 6 and more than 4000 for the PSA DGP level 7. Given the exponential growth of the number of simulations required to achieve higher PSA levels, the PSA GP 5 and PSA DGP 6 were assumed to be close enough to the ADCIRC simulations, although their normalized errors are above 20%.

To qualitatively compare the distributions of maximum elevations obtained with the two PSA methods (GP and DGP) and the ADCIRC simulations, violin plots (Hintze and Nelson 1998)—based on box plots but including the full distribution of the data via a kernel density estimation—are shown in Fig. 5. They highlight that the distributions obtained with the PSA DGP 6 are generally better at each location than the ones obtained with the PSA DG 5. This is particularly clear for locations like Vela Luka, Stari Grad, Vrboska, and Mali Ston, where both extrema and median values from the PSA DGP 6 are closer to the ADCIRC simulations than the PSA DG 5 values. For the other locations (Rijeka dubrovačka, Viganj, and Ston) the median values of both methods are similar, but the extrema of the ADCIRC simulations are better represented with the PSA DGP 6. None of the two methods seems to perfectly catch the extreme values of the ADCIRC simulations, indicating that higher PSA level is probably required to better represent these values. In addition, Fig. 5 shows that the meteotsunami maximum elevations modeled with ADCIRC forced by synthetic atmospheric conditions is always under 0.2 m in Mali Ston. This location is thus not considered to be affected by extreme sea levels due to meteotsunami waves. This result is in accordance with a previous study from Vilibić et al. (2004) showing that, during meteotsunami events, Mali Ston is more affected by strong currents in constrictions, due to the complex bathymetry of the area, than by extreme elevations. As extreme currents are out of scope in this study, Mali Ston is excluded from further analysis.

Fig. 5.
Fig. 5.

Violin plots (with median only) of ADCIRC model results and PSA of level 5 for the Gauss–Patterson rule (GP 5) and PSA of level 6 for the delayed Gauss–Patterson rules (DGP 6) depending on the location of interest. “R. dub.” stands for Rijeka dubrovačka.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0147.1

Finally, the quantile–quantile and correlation analysis are presented in Fig. 6. In terms of quantile distribution (Fig. 6a), the PSA DGP 6 (all locations included) mostly matches the distribution of the ADCIRC simulation for value higher than 0.5 m while the PSA GP 5 clearly underestimates all maximum elevations above 0.5 m. However, in terms of correlation (Fig. 6b), the PSA GP 5 reaches 0.6 while the PSA DGP 6 reaches 0.65. The scatterplot also shows that the PSA DGP 6 largely overestimates some of the maximum elevations below 0.5 m, while the PSA GP 5 underestimates more than 80% of the ADCIRC simulation results.

Fig. 6.
Fig. 6.

(a) Q–Q plot and (b) scatterplot: all locations of interest included, of the PSA results (GP 5 and DGP 6) depending on the ADCIRC model results.

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0147.1

In summary, the results clearly demonstrate that, in order to obtain conservative results and to catch the more extreme conditions, the PSA DGP 6 should be used to define the surrogate model of meteotsunami maximum elevation. However, because the comparison of the results obtained with the GP and DGP grids was not performed for the same PCE levels, no conclusion can be drawn concerning the overall accuracy of the PSA GP approach, which may have provided better results for a PCE of level 6 than the one presented for the PSA DGP 6 but is far less efficient (i.e., require nearly 10 times more simulations to reach a PCE of level 6).

b. Sensitivity analysis

An important question is how to quantify which atmospheric disturbance parameters (y0, θ, PA, T, c, d) or combinations of these parameters best explain the variability of the maximum elevation (and thus the meteotsunami hazard) at a location of interest, that is, to which parameters is sensitive the hazard assessment done by surrogate modeling. This can be achieved with variance decomposition techniques (ANOVA) using the Sobol’ indices (Sobol’ 2001; Saltelli 2002), which can be directly derived from the gPCE coefficients (Crestaux et al. 2009). Following the notations of section 2b, the input parameters are given by Ri i ∈ 1, 2, …, 6, the multi-index β = [β1, β2, …, βr], β ∈ A stores all the polynomial combinations needed to calculate the PSA DGP 6 and the PSA coefficients are expressed as
ξβ′^=⁡(−1)6−|α|⁡(56−|α|)C|α|,β′,
where 1 ≤ |α| ≤ 6 and β′ ∈ A′, A′ ⊂ A. The variability of the meteotsunami maximum elevation at each location is then given by D=Var[ξDGP,6]=∑β∈Aβ≠0ξβ2^, and the total, first-order, and second-order Sobol’ indices for each parameter or combination of two parameters—SRiT, SRi, and SRi,Rj, respectively—are given by
{SRiT=DiTD,DiT=∑β∈AiT⁡(ξ^β)2,AiT={β∈A:βi> 0}SRi=DiD,Di=∑β∈Ai⁡(ξ^β)2,Ai={β∈A:βi> 0,∀j≠i,βj= 0}SRi,Rj=DijD,Dij=∑β∈Aij⁡(ξ^β)2,Aij={β∈A:βi> 0,j≠i,βj> 0,∀k≠j,βk= 0}. .
In this study, the total and main sensitivities, given by the Sobol’ indices [Eq. (12)], quantify the relative impact (between 0 and 1) of each atmospheric disturbance parameter (or combination of parameters) to the meteotsunami maximum elevation predictions obtained at the six locations of interest (as Mali Ston was excluded) with the surrogate model. The total sensitivities given by the total Sobol’ indices ⁡(Sy0T, SθT, SPAT, STT, ScT, SdT) give the total impact of each parameter. The sum of the total sensitivities can be greater than 1 as for each parameter its impact alone and in combination with all the other parameter(s) is taken into account (the combinations of parameters are thus counted more than once). Consequently, if a total sensitivity is negligible (in practice below 0.01) the associated parameter can be set to a deterministic value without affecting much the meteotsunami maximum elevation distributions. In this study, as the impact of higher-order Sobol’ indices is often negligible, the main sensitivities are only given for the first- and second-order Sobol’ indices. The first-order main sensitivities ⁡(Sy0, Sθ, SPA, ST, Sc, Sd) show how much each parameter alone influences the distribution of the meteotsunami maximum elevation and allow the determination of which parameter(s) shall be investigated in priority. The second-order main sensitivities indicate the influence of combination (interaction) of two different parameters.

In Fig. 7, the total sensitivities at the six locations of interest are given for each of the parameters (y0, θ, PA, T, c, d) of the atmospheric disturbance. It can clearly be seen that, all locations included, the width of the atmospheric disturbance is the parameter which has the least influence on the meteotsunami hazard (total sensitivities below 0.1). For the other parameters, their relative impact highly depends on the studied locations. For example, the period of the atmospheric disturbance is not so influential in Vela Luka and Viganj (total sensitivities of 0.12 and 0.11, respectively), but is the most important meteotsunamigenic parameter in Vrboska and Stari Grad (0.33). The amplitude is generally the second less important parameter after the width, except for Vela Luka (0.27) and Viganj (0.27). Overall, the two most influential parameters are the speed (up to 0.55 in Viganj) and/or the period (up to 0.33 in Stari Grad). The start location is quite important for Ston meteotsunamis, while being of least importance for other stations, particularly for Viganj, Vrboska, and Stari Grad. The total sensitivity analysis is thus showing that, for all the considered locations, the speed and period of the atmospheric disturbances, closely followed by the start location and the direction, are going to be the most influential parameters in terms of meteotsunami impact in the eastern Adriatic.

Fig. 7.
Fig. 7.

Total sensitivities calculated for each of the six parameters of the atmospheric disturbance at each of the seven locations of interest [Vela Luka, Rijeka dubrovačka (R. dubro.), Stari Grad, Vrboska, Viganj, and Ston].

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0147.1

However, the total sensitivity of a given parameter also includes its higher-order interactions with the other parameters and, to better understand the influence of each parameter separately, the first- and second-order results of the main sensitivity analysis should be jointly analyzed (Fig. 8). Together, they are responsible for almost all the meteotsunami impacts: 0.95 in Vela Luka, 0.94 in Rijeka dubrovačka, 0.95 in Stari Grad, 0.97 in Vrboska, 0.95 in Viganj, and 0.96 in Ston. The first-order main sensitivities represented by the matrix diagonals in Fig. 8 are influencing the meteotsunami maximum elevation distributions up to 0.64 in Vela Luka, 0.59 in Rijeka dubrovačka, 0.70 in Stari Grad, 0.71 in Vrboska, 0.65 in Viganj, and 0.69 in Ston. Individually, each first-order sensitivity does not surpass 0.24 and the highest first-order sensitivities are generally found for the speed, period or start location parameters. The second-order main sensitivities also play an important role (more than 0.25 in total at each location) on the meteotsunami maximum elevation distributions. In fact, even if the individual second-order sensitivities are small (<0.05), their sum can be large due to their large number (15 in total). For example, in Vela Luka, Rijeka dubrovačka, Stari Grad, and Viganj, the sum of the interactions between the speed and the other parameters reaches more than 0.10 (up to 0.20 in Viganj). Both total and first-order analysis show that speed is a really important parameter concerning meteotsunami elevations. Unfortunately, it is also a difficult parameter to extract precisely from the numerical simulations. In addition, the main sensitivity analysis confirms that the width is a parameter that can be assumed constant in the stochastic atmospheric disturbance definition, as both its first- and second-order sensitivities are below 0.03.

Fig. 8.
Fig. 8.

First- and second-order main sensitivities for each of the six parameters of an atmospheric disturbance at six locations of interest [Vela Luka, Rijeka dubrovačka (R. dubro.), Stari Grad, Vrboska, Viganj, and Ston].

Citation: Journal of Physical Oceanography 50, 4; 10.1175/JPO-D-19-0147.1

In summary, the sensitivity analysis shows that, excluding the width, all the remaining parameters (speed, period, direction, start location, and for a smaller part amplitude) and their second-order interactions can play an important role on the variability of the meteotsunami maximum elevations, depending on the location of interest.

4. Discussion and conclusions

Atmospherically driven extreme sea level events are one of the major threats to people and assets in the coastal regions. Assessing the hazard associated with such events together with uncertainty quantification in a precise and timely manner is thus of primary importance in modern societies. As running ensemble of deterministic models can be computationally expensive and slow, this study has investigated the feasibility of developing a computationally inexpensive gPCE-based surrogate model of meteotsunami maximum sea level elevation in the eastern Adriatic Sea. The main advantage of such a method is that, once the surrogate model is created, only few minutes of computation are needed to assess, with no additional deterministic simulation, the hazard of any given typical event. In other words, for any atmospheric disturbance with parameters within the ranges of definition of the surrogate model, less than 3 min are needed to produce the resulting meteotsunami maximum elevation distribution based on more than 10 000 samples. However, the main disadvantage of the gPCE-based methods is that they only perform well for processes with relatively smooth responses. As meteotsunamis are rare events and only few measurements were available, the smoothness of the maximum elevation distributions at the various locations of interest along the Croatian coastline were not known beforehand in this study. The main aim of this work was thus to develop a gPCE-based surrogate model for meteotsunami maximum elevations and to demonstrate its performances for UQ and sensitive analysis.

First, the atmospheric disturbances responsible for the eastern Adriatic meteotsunamis were modeled with a simple wave equation [Eq. (2)] depending on six stochastic parameters (amplitude, speed, period, direction, start location, and width) having uniform prior distributions [Eq. (3)]. The main assumption of this approach is that the atmospheric disturbances can be identified as internal gravity waves propagating, without any kind of transformation, from the Italian coast to the eastern Adriatic. In theory, this is achievable through the wave-duct mechanism (Lindzen and Tung 1976), which is found active during most of the Adriatic meteotsunamis (Vilibić and Šepić 2009; Horvath et al. 2018). In reality, due to the highly nonlinear processes involved in the spatial and temporal evolution of the IGWs during the crossing of the Adriatic Sea, the characteristics (amplitude, speed, period, etc.) of the atmospheric disturbances measured along the Italian and Croatian coastlines are changing in time and space (Šepić et al. 2016). Also, there might be meteotsunami events generated by disturbances propagating along the Adriatic, from the northwest, and maintained by other atmospheric mechanisms, like on 27 June 2003 (Belušić et al. 2007). However, as little is presently known about the complex evolution of these atmospheric disturbances, the ranges of definition of the six stochastic parameters were extracted from observations along the Croatian coastline and assumed to be representative of the meteotsunamigenic conditions. Then, the surrogate model was developed with a nonintrusive pseudospectral approximation (PSA) methodology [Eq. (10)] used with Gauss–Patterson sparse quadrature (Fig. 2). Finally, the convergence of the method as well as its capability to describe the distributions of meteotsunami maximum elevations at specific locations were investigated. Given the relatively slow convergence of the method (Fig. 4), higher order of gPCE and thus higher levels of sparse grids (GP 5 and DGP 6) were found to be able to describe the distributions well (i.e., to minimize the errors). In addition, the use of delayed Gauss–Patterson grids reaching higher polynomial orders with a lower number of simulations was found to represent the meteotsunami maximum elevation distributions with the most accuracy (Figs. 5 and 6) but also to systematically underestimate the extreme values of these distributions (Fig. 5). The meteotsunami hazard assessments based on the presented surrogate model would in consequence provide a systematic bias toward lower estimates which could be corrected by a Bayesian calibration based on available measurements or the use of higher polynomial orders in the gPCE method. Given the presented results, the authors believe that the feasibility of creating a gPCE-based surrogate model of meteotsunami maximum elevations in the eastern Adriatic is shown. However, to obtain higher accuracy for the surrogate model, the use of adaptive sparse grids (Conrad and Marzouk 2013)—not investigated in this study—would be a more efficient way to reach even higher polynomial orders with limited computational resources.

The other important goal of this study was to take advantage of the gPCE decomposition to easily derive the ANOVA sensitivities and thus to better understand to which parameter(s) of the atmospheric forcing the meteotsunami maximum elevation is the most sensitive. In particular, the width of the atmospheric disturbance has been proven to be an unimportant parameter. This means that, in the eastern Adriatic, the method can be used with only five stochastic parameters and that, in consequence, if the method is extended to higher polynomial orders, fewer deterministic simulations will be needed. The sensitivity study has also revealed that the speed of the atmospheric disturbance is probably the most critical parameter while, surprisingly, the amplitude of the air-pressure disturbance has little influence. To draw better conclusions about the general sensitivity of meteotsunamis, the methodology should be applied in different places in the world (e.g., Menorca in Spain, the Great Lakes in the United States). However, to our knowledge, it is the first time that meteotsunami sensitivity to atmospheric disturbances has been systematically undertaken for six different parameters (start location, direction, speed, period, amplitude, and width) and gPCE-based methods seem worth investigating even with the sole purpose of doing sensitivity studies.

A first attempt to build a gPCE-based surrogate model of atmospherically driven extreme maximum elevations was presented in this study, but more experience has to be gained using this approach (i.e., different atmospheric forcing in different locations) to fully understand its limitations. For example, a surrogate model could be built to forecast the storm surge hazard in the Venice lagoon caused by the sirocco winds that regularly blow in the Adriatic Sea (Zampato et al. 2016). For this purpose, the sirocco wind field can be assumed to be homogeneous over the entire Adriatic region and described by the following stochastic parameters: direction, wind intensity and duration of the event. As the tidal forcing cannot be neglected in the Venice lagoon, the mean sea level can also be considered as an additional stochastic parameter which can be decreased or increased in order to emulate the tidal effects in a simplistic way. However, in this example, the ADCIRC model has also to be used in its coupled version with the SWAN model in order to properly generate the storm surge linked to the nonlinear wind–wave interactions. In consequence, the convergence of the method should be carefully retested to prove the feasibility of using the gPCE-based surrogate model in this context. In addition, as for any other method, the PCE approach is not universal and can present some shortcomings in some specific applications as discussed, for example, in 1) Lu et al. (2015), which found that the cost of using polynomial chaos expansions to accurately derive Bayesian solutions of inverse problems was often too high, or in 2) the theoretical study of Branicki and Majda (2013), which highlighted that truncated PCE methods struggle with representing correctly intermittent dynamic systems (i.e., turbulent signals in the energy transfer or energy dissipation regimes). In this study, however, the meteotsunami surrogate model is based on a PCE approach which solves a forward problem for a static quantity of interest (i.e., it describes the meteotsunami maximum elevation distribution at sensitive locations along the Croatian coastline) and thus currently not exposed to these challenges.

An important final point about the presented method is that it fully relies on the quality of the ocean model used to generate the deterministic simulations. As discussed in Denamiel et al. (2018), the ADCIRC model implemented for this study has numerous limitations, including the resolution and the bathymetry used to generate its mesh, the oversimplifications made by neglecting both the tidal and the wind forcing and its physics which cannot fully represent the atmosphere–ocean chaotic system. The successful assessment of the meteotsunami hazard may thus combine deterministic models, measurement networks and stochastic approaches as described in Denamiel et al. (2019b).

To conclude, even if the feasibility and the accuracy of using gPCE-based surrogate models of maximum elevations to provide reliable hazard assessments can only be proven by improving, adapting and applying the presented method to different locations and different atmospheric forcing (e.g., hurricanes in the Gulf of Mexico), this study opens the door to an alternative way to assess sea level hazard during atmospherically driven extreme events.

Acknowledgments

Acknowledgement is made for the support of the ECMWF staff, in particular Xavier Abellan, as well as for ECMWF’s computing and archive facilities used in this research, which has been supported by projects MESSI (UKF Grant 25/15), ADIOS (Croatian Science Foundation Grant IP-2016-06-1955), and ECMWF Special Project (The Adriatic decadal and inter-annual oscillations: modelling component). The authors would also like to thank the anonymous reviewers for their valuable contributions. The data and MATLAB programs used to create the gPCE-based meteotsunami surrogate model, as well as the sensitivity study, can be obtained under the Open Science Framework (OSF) FAIR data repository at https://osf.io/jysqu/ (doi:10.17605/OSF.IO/JYSQU).

REFERENCES

  • Abramowitz, M., and I. A. Stegun, Eds., 1964: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, Vol. 55, Dover Publications, 1046 pp.

    • Crossref
    • Export Citation

  • Agarwal, N., and N. Aluru, 2007: A stochastic Lagrangian approach for geometrical uncertainties in electrostatics. J. Comput. Phys., 226, 156–179, https://doi.org/10.1016/j.jcp.2007.03.026.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Amante, C., and B. W. Eakins, 2009: ETOPO1 1 arc-minute global relief model: Procedures, data sources, and analysis. NOAA Tech. Memo. NESDIS NGDC-24, 25 pp., https://www.ngdc.noaa.gov/mgg/global/relief/ETOPO1/docs/ETOPO1.pdf.

  • Anders, M., and M. Hori, 1999: Stochastic finite element method for elasto-plastic body. Int. J. Numer. Methods Eng., 46, 1897–1916, https://doi.org/10.1002/(SICI)1097-0207(19991220)46:11<1897::AID-NME758>3.0.CO;2-3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Belušić, D., B. Grisogono, and Z. B. Klaić, 2007: Atmospheric origin of the devastating coupled air-sea event in the east Adriatic. J. Geophys. Res., 112, D17111, https://doi.org/10.1029/2006JD008204.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Branicki, M., and A. J. Majda, 2013: Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities. Commun. Math. Sci., 11, 55–103, https://doi.org/10.4310/CMS.2013.v11.n1.a3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Clenshaw, C. W., and A. R. Curtis, 1960: A method for numerical integration on an automatic computer. Numer. Math., 2, 197–205, https://doi.org/10.1007/BF01386223.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Conrad, P. R., and Y. M. Marzouk, 2013: Adaptive Smolyak pseudospectral approximations. SIAM J. Sci. Comput., 35, A2643–A2670, https://doi.org/10.1137/120890715.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Constantine, P. G., M. S. Eldred, and E. T. Phipps, 2012: Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng., 229–232, 1–12, https://doi.org/10.1016/j.cma.2012.03.019.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Crestaux, T., O. P. Le Maître, and J.-M. Martinez, 2009: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf., 94, 1161–1172, https://doi.org/10.1016/j.ress.2008.10.008.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Denamiel, C., J. Šepić, and I. Vilibić, 2018: Impact of geomorphological changes to harbor resonance during meteotsunamis: The Vela Luka bay test case. Pure Appl. Geophys., 175, 3839–3859, https://doi.org/10.1007/S00024-018-1862-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Denamiel, C., J. Šepić, D. Ivanković, and I. Vilibić, 2019a: The Adriatic Sea and coast modelling suite: Evaluation of the meteotsunami forecast component. Ocean Modell., 135, 71–93, https://doi.org/10.1016/j.ocemod.2019.02.003.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Denamiel, C., J. Šepić, X. Huan, C. Bolzer, and I. Vilibić, 2019b: Stochastic surrogate model for meteotsunami early warning system in the eastern Adriatic Sea. J. Geophys. Res. Oceans, 124, 8485–8499, https://doi.org/10.1029/2019JC015574.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Dragani, W. C., 2007: Numerical experiments on the generation of long ocean waves in coastal waters of the Buenos Aires province, Argentina. Cont. Shelf Res., 27, 699–712, https://doi.org/10.1016/j.csr.2006.11.009.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ernst, O. G., A. Mugler, H.-J. Starkloff, and E. Ullmann, 2012: On the convergence of generalized polynomial chaos expansions. ESAIM:M2AN, 46, 317–339, https://doi.org/10.1051/m2an/2011045.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Foo, J., Z. Yosibash, and G. E. Karniadakis, 2007: Stochastic simulation of riser-sections with uncertain measured pressure loads and/or uncertain material properties. Comput. Methods Appl. Mech. Eng., 196, 4250–4271, https://doi.org/10.1016/j.cma.2007.04.005.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Formaggia, L., A. Guadagnini, I. Imperiali, V. Lever, G. Porta, M. Riva, A. Scotti, and L. Tamellini, 2013: Global sensitivity analysis through polynomial chaos expansion of a basin-scale geochemical compaction model. Comput. Geosci., 17, 25–42, https://doi.org/10.1007/s10596-012-9311-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gerstner, T., and M. Griebel, 1998: Numerical integration using sparse grids. Numer. Algorithms, 18, 209–232, https://doi.org/10.1023/A:1019129717644.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ghanem, R. G., 1999a: Ingredients for a general purpose stochastic finite elements implementation. Comput. Methods Appl. Mech. Eng., 168, 19–34, https://doi.org/10.1016/S0045-7825(98)00106-6.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ghanem, R. G., 1999b: Stochastic finite elements with multiple random non-Gaussian properties. ACSE J. Eng. Mech., 125, 26–40, https://doi.org/10.1061/(ASCE)0733-9399(1999)125:1(26).

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ghanem, R. G., and P. D. Spanos, 1991: Stochastic Finite Elements: A Spectral Approach. Springer, 214 pp.

    • Crossref
    • Export Citation

  • Ghanem, R. G., and S. Dham, 1998: Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp. Porous Media, 32, 239–262, https://doi.org/10.1023/A:1006514109327.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Giraldi, L., O. P. Le Maître, K. T. Mandli, C. N. Dawson, I. Hoteit, and O. M. Knio, 2017: Bayesian inference of earthquake parameters from buoy data using a polynomial chaos-based surrogate. Comput. Geosci., 21, 683–699, https://doi.org/10.1007/s10596-017-9646-z.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hauptmanns, U., 2008: Comparative assessment of the dynamic behaviour of an exothermal chemical reaction including data uncertainties. Chem. Eng. J., 140, 278–286, https://doi.org/10.1016/j.cej.2007.09.031.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hintze, H. L., and R. D. Nelson, 1998: Violin plots: A box plot-density trace synergism. Amer. Stat., 52, 181–184, https://doi.org/10.2307/2685478.

    • Search Google Scholar
    • Export Citation

  • Holtz, M., 2011: Sparse grid quadrature. Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance, 1st ed. Lecture Notes in Computational Science and Engineering, Vol. 77, Springer, 51–76.

    • Crossref
    • Export Citation

  • Horvath, K., and I. Vilibić, 2014: Atmospheric mesoscale conditions during the Boothbay meteotsunami: A numerical sensitivity study using a high-resolution mesoscale model. Nat. Hazards, 74, 55–74, https://doi.org/10.1007/s11069-014-1055-1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Horvath, K., J. Šepić, and M. Telišman Prtenjak, 2018: Atmospheric forcing conducive for the Adriatic: 25 June 2014 meteotsunami event. Pure Appl. Geophys., 175, 3817–3837, https://doi.org/10.1007/s00024-018-1902-1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hou, T. Y., W. Luo, B. Rozovskii, and H.-M. Zhou, 2006: Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. J. Comput. Phys., 216, 687–706, https://doi.org/10.1016/j.jcp.2006.01.008.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hover, F. S., and M. S. Triantafyllou, 2006: Application of polynomial chaos in stability and control. Automatica, 42, 789–795, https://doi.org/10.1016/j.automatica.2006.01.010.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jardak, M., C. Su, and G. E. Karniadakis, 2002: Spectral polynomial chaos solutions of the stochastic advection equation. J. Sci. Comput., 17, 319–338, https://doi.org/10.1023/A:1015125304044.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jaynes, E. T., 1957: Information theory and statistical mechanics. Phys. Rev., 106, 620–630, https://doi.org/10.1103/PhysRev.106.620.

  • Knio, O. M., and O. P. Le Maître, 2006: Uncertainty propagation in CFD using polynomial chaos decomposition. Fluid Dyn. Res., 38, 616–640, https://doi.org/10.1016/j.fluiddyn.2005.12.003.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Le Maître, O. P., O. M. Knio, H. Najm, and R. G. Ghanem, 2001: A stochastic projection method for fluid flow: Basic formulation. J. Comput. Phys., 173, 481–511, https://doi.org/10.1006/jcph.2001.6889.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ličer, M., B. Mourre, C. Troupin, A. Krietemeyer, A. Jansá, and J. Tintoré, 2017: Numerical study of Balearic meteotsunami generation and propagation under synthetic gravity wave forcing. Ocean Modell., 111, 38–45, https://doi.org/10.1016/j.ocemod.2017.02.001.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lin, G., and G. E. Karniadakis, 2006: A discontinuous Galerkin method for two-temperature plasmas. Comput. Methods Appl. Mech. Eng., 195, 3504–3527, https://doi.org/10.1016/j.cma.2005.06.024.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lindzen, R. S., and K. Tung, 1976: Banded convective activity and ducted gravity waves. Mon. Wea. Rev., 104, 1602–1617, https://doi.org/10.1175/1520-0493(1976)104<1602:BCAADG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Liu, P. L.-F., S. Monserrat, M. Marcos, and A. B. Rabinovich, 2003: Coupling between two inlets: Observation and modeling. J. Geophys. Res., 108, 3069, https://doi.org/10.1029/2002JC001478.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lu, F., M. Morzfeld, X. Tu, and A. J. Chorin, 2015: Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems. J. Comput. Phys., 282, 138–147, https://doi.org/10.1016/j.jcp.2014.11.010.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Luettich, R. A., R. H. Birkhahn and J. J. Westerink, 1991: Application of ADCIRC-2DDI to Masonboro Inlet, North Carolina: A brief numerical modeling study. Contractors Rep. to the US Army Engineer Waterways Experiment Station-DACW39-86-D-0004-0002, 19 pp., https://adcirc.org/files/2018/11/1991_Luettich02.pdf.

  • Monserrat, S., I. Vilibić, and A. B. Rabinovich, 2006: Meteotsunamis: Atmospherically induced destructive ocean waves in the tsunami frequency band. Nat. Hazards Earth Syst. Sci., 6, 1035–1051, https://doi.org/10.5194/nhess-6-1035-2006.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nagy, Z., and R. Braatz, 2007: Distributional uncertainty analysis using power series and polynomial chaos expansions. J. Process Contr., 17, 229–240, https://doi.org/10.1016/j.jprocont.2006.10.008.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Najm, H. N., B. J. Debusschere, Y. M. Marzouk, S. Widmer, and O. P. Le Maître, 2009: Uncertainty quantification in chemical systems. Int. J. Numer. Methods Eng., 80, 789–814, https://doi.org/10.1002/nme.2551.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Orlić, M., D. Belušić, I. Janeković, and M. Pasarić, 2010: Fresh evidence relating the great Adriatic surge of 21 June 1978 to mesoscale atmospheric forcing. J. Geophys. Res., 115, C06011, https://doi.org/10.1029/2009JC005777.

    • Search Google Scholar
    • Export Citation

  • Patterson, T. N. L., 1968: The optimum addition of points to quadrature formulae. Math. Comput., 22, 847–856, https://doi.org/10.1090/S0025-5718-68-99866-9.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Patterson, T. N. L., 1989: Algorithm 672: Generation of interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions. ACM Trans. Math. Software, 15, 137–143, https://doi.org/10.1145/63522.69649.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Rabinovich, A. B., S. Monserrat, and I. Fine, 1999: Numerical modeling of extreme seiche oscillations in the region of the Balearic Islands. Oceanology, 39, 16–24.

    • Search Google Scholar
    • Export Citation

  • Rupert, C., and C. Miller, 2007: An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems. J. Comput. Phys., 226, 2175–2205, https://doi.org/10.1016/j.jcp.2007.07.001.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Saltelli, A., 2002: Sensitivity analysis for importance assessment. Risk Anal., 22, 579–590, https://doi.org/10.1111/0272-4332.00040.

  • Saltelli, A., M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola, 2008: Global Sensitivity Analysis: The Primer. John Wiley & Sons, 304 pp.

    • Crossref
    • Export Citation

  • Šepić, J., and I. Vilibić, 2011: The development and implementation of a real-time meteotsunami warning network for the Adriatic Sea. Nat. Hazards Earth Syst. Sci., 11, 83–91, https://doi.org/10.5194/nhess-11-83-2011.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Šepić, J., I. Vilibić, and N. Strelec Mahović, 2012: Northern Adriatic meteorological tsunamis: Observations, link to the atmosphere, and predictability. J. Geophys. Res., 117, C02002, https://doi.org/10.1029/2011JC007608.

    • Search Google Scholar
    • Export Citation

  • Šepić, J., I. Međugorac, I. Janeković, N. Dunić, and I. Vilibić, 2016: Multi-meteotsunami event in the Adriatic Sea generated by atmospheric disturbances of 25–26 June 2014. Pure Appl. Geophys., 173, 4117–4138, https://doi.org/10.1007/s00024-016-1249-4.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Smolyak, S. A., 1963: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR, 148, 1042–1045.

    • Search Google Scholar
    • Export Citation

  • Sobol’, I. M., 2001: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul., 55, 271–280, https://doi.org/10.1016/S0378-4754(00)00270-6.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Soize, C., and R. G. Ghanem, 2004: Physical systems with random uncertainties: Chaos representations with arbitrary probability measure. SIAM J. Sci. Comput., 26, 395–410, https://doi.org/10.1137/S1064827503424505.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sraj, I., K. Mandli, O. Knio, C. N. Dawson, and I. Hoteit, 2014: Uncertainty quantification and inference of Manning’s friction coefficients using DART buoy during the Tohoku tsunami. Ocean Modell., 83, 82–97, https://doi.org/10.1016/j.ocemod.2014.09.001.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Su, Q., and K. Strunz, 2005: Stochastic circuit modelling with Hermite polynomial chaos. Electron. Lett., 41, 1163, https://doi.org/10.1049/el:20052415.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sudret, B., 2008: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf., 93, 964–979, https://doi.org/10.1016/j.ress.2007.04.002.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Thacker, W. C., A. Srinivasan, M. Iskandarani, O. M. Knio, and M. Le Hénaff, 2012: Propagating boundary uncertainties using polynomial expansions. Ocean Modell., 43–44, 52–63, https://doi.org/10.1016/j.ocemod.2011.11.011.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Trefethen, L. N., 2008: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev., 50, 67–87, https://doi.org/10.1137/060659831.

  • Vilibić, I., and J. Šepić, 2009: Destructive meteotsunamis along the eastern Adriatic coast: Overview. Phys. Chem. Earth, 34, 904–917, https://doi.org/10.1016/j.pce.2009.08.004.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Vilibić, I., N. Domijan, M. Orlić, N. Leder, and M. Pasarić, 2004: Resonant coupling of a traveling air-pressure disturbance with the east Adriatic coastal waters. J. Geophys. Res., 109, C10001, https://doi.org/10.1029/2004JC002279.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Vilibić, I., N. Domijan, and S. Čupić, 2005: Wind versus air pressure seiche triggering in the middle Adriatic coastal waters. J. Mar. Syst., 57, 189–200, https://doi.org/10.1016/j.jmarsys.2005.04.007.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Vilibić, I., S. Monserrat, A. B. Rabinovich, and H. Mihanović, 2008: Numerical modelling of the destructive meteotsunami of 15 June, 2006 on the coast of the Balearic Islands. Pure Appl. Geophys., 165, 2169–2195, https://doi.org/10.1007/s00024-008-0426-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • von Toussaint, U., 2011: Bayesian inference in physics. Rev. Mod. Phys., 83, 943–999, https://doi.org/10.1103/RevModPhys.83.943.

  • Wan, X., and G. E. Karniadakis, 2005: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys., 209, 617–642, https://doi.org/10.1016/j.jcp.2005.03.023.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wang, M., Z. Wan, and Q. Huang, 2016: A new uncertain analysis method for the prediction of acoustic field with random and interval parameters. Shock Vib., 2016, 3693262, https://doi.org/10.1155/2016/3693262.

    • Search Google Scholar
    • Export Citation

  • Weatherall, P., and Coauthors, 2015: A new digital bathymetric model of the world’s oceans. Earth Space Sci., 2, 331–345, https://doi.org/10.1002/2015EA000107.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wiener, N., 1938: The homogeneous chaos. Amer. J. Math., 60, 897–936, https://doi.org/10.2307/2371268.

  • Williams, D. A., K. J. Horsburgh, D. M. Schultz, and C. W. Hughes, 2019: Examination of generation mechanisms for an english channel meteotsunami: Combining observations and modeling. J. Phys. Oceanogr., 49, 103–120, https://doi.org/10.1175/JPO-D-18-0161.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Williams, M., 2006: Polynomial chaos functions and stochastic differential equations. Ann. Nucl. Energy, 33, 774–785, https://doi.org/10.1016/j.anucene.2006.04.005.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Winokur, J., P. Conrad, I. Sraj, O. Knio, A. Srinivasan, W. C. Thacker, Y. Marzouk, and M. Iskandarani, 2013: A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database. Comput. Geosci., 17, 899–911, https://doi.org/10.1007/s10596-013-9361-3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Witteveen, J. A. S., S. Sarkara, and H. Bijla, 2007: Modeling physical uncertainties in dynamic stall induced fluid-structure interaction of turbine blades using arbitrary polynomial chaos. Comput. Struc., 85, 866–878, https://doi.org/10.1016/j.compstruc.2007.01.004.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Xiu, D., and G. E. Karniadakis, 2002a: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput., 24, 619–644, https://doi.org/10.1137/S1064827501387826.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Xiu, D., and G. E. Karniadakis, 2002b: Stochastic modeling of flow-structure interactions using generalized polynomial chaos. J. Fluids Eng., 124, 51–59, https://doi.org/10.1115/1.1436089.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Xiu, D., and G. E. Karniadakis, 2002c: Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Eng., 191, 4927–4948, https://doi.org/10.1016/S0045-7825(02)00421-8.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Xiu, D., and G. E. Karniadakis, 2003: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys., 187, 137–167, https://doi.org/10.1016/S0021-9991(03)00092-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zampato, L., M. Bajo, P. Canestrelli, and G. Umgiesser, 2016: Storm surge modelling in Venice: Two years of operational results. J. Oper. Oceanogr., 9, S46–S57, https://doi.org/10.1080/1755876X.2015.1118804.

    • Search Google Scholar
    • Export Citation

  • Zhang, D., and Z. Lu, 2004: An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loève and polynomial expansions. J. Comput. Phys., 194, 773–794, https://doi.org/10.1016/j.jcp.2003.09.015.

    • Crossref
    • Search Google Scholar
    • Export Citation
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  • Abramowitz, M., and I. A. Stegun, Eds., 1964: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, Vol. 55, Dover Publications, 1046 pp.

    • Crossref
    • Export Citation

  • Agarwal, N., and N. Aluru, 2007: A stochastic Lagrangian approach for geometrical uncertainties in electrostatics. J. Comput. Phys., 226, 156–179, https://doi.org/10.1016/j.jcp.2007.03.026.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Amante, C., and B. W. Eakins, 2009: ETOPO1 1 arc-minute global relief model: Procedures, data sources, and analysis. NOAA Tech. Memo. NESDIS NGDC-24, 25 pp., https://www.ngdc.noaa.gov/mgg/global/relief/ETOPO1/docs/ETOPO1.pdf.

  • Anders, M., and M. Hori, 1999: Stochastic finite element method for elasto-plastic body. Int. J. Numer. Methods Eng., 46, 1897–1916, https://doi.org/10.1002/(SICI)1097-0207(19991220)46:11<1897::AID-NME758>3.0.CO;2-3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Belušić, D., B. Grisogono, and Z. B. Klaić, 2007: Atmospheric origin of the devastating coupled air-sea event in the east Adriatic. J. Geophys. Res., 112, D17111, https://doi.org/10.1029/2006JD008204.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Branicki, M., and A. J. Majda, 2013: Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities. Commun. Math. Sci., 11, 55–103, https://doi.org/10.4310/CMS.2013.v11.n1.a3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Clenshaw, C. W., and A. R. Curtis, 1960: A method for numerical integration on an automatic computer. Numer. Math., 2, 197–205, https://doi.org/10.1007/BF01386223.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Conrad, P. R., and Y. M. Marzouk, 2013: Adaptive Smolyak pseudospectral approximations. SIAM J. Sci. Comput., 35, A2643–A2670, https://doi.org/10.1137/120890715.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Constantine, P. G., M. S. Eldred, and E. T. Phipps, 2012: Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng., 229–232, 1–12, https://doi.org/10.1016/j.cma.2012.03.019.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Crestaux, T., O. P. Le Maître, and J.-M. Martinez, 2009: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf., 94, 1161–1172, https://doi.org/10.1016/j.ress.2008.10.008.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Denamiel, C., J. Šepić, and I. Vilibić, 2018: Impact of geomorphological changes to harbor resonance during meteotsunamis: The Vela Luka bay test case. Pure Appl. Geophys., 175, 3839–3859, https://doi.org/10.1007/S00024-018-1862-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Denamiel, C., J. Šepić, D. Ivanković, and I. Vilibić, 2019a: The Adriatic Sea and coast modelling suite: Evaluation of the meteotsunami forecast component. Ocean Modell., 135, 71–93, https://doi.org/10.1016/j.ocemod.2019.02.003.

    • Crossref