We present a database of pre-calculated tsunami waveforms for the
entire Mediterranean Sea, obtained by numerical propagation of uniformly
spaced Gaussian-shaped elementary sources for the sea level elevation. Based
on any initial sea surface displacement, the database allows the fast
calculation of full waveforms at the 50 m isobath offshore of coastal sites
of interest by linear superposition. A computationally inexpensive procedure
is set to estimate the coefficients for the linear superposition based on the
potential energy of the initial elevation field. The elementary sources size
and spacing is fine enough to satisfactorily reproduce the effects of

After the 2004 Indian Ocean tsunami, particular attention has
been devoted to the improvement of tsunami warning systems (TWS) and
probabilistic tsunami hazard analysis (PTHA), which currently represent two
pillars in risk mitigation policies for the authorities of each country
exposed to tsunami threat

However, the computational cost of numerical simulations still limits the
feasibility for approaches which require (i) a very fast response and/or
(ii) a massive amount of simulations, thus encouraging the development of
efficient approximated solutions. Pre-calculated tsunami sources are commonly
adopted by TWS to rapidly forecast tsunami effects which follow strong
earthquakes. For example, stored scenarios are used in inversions of tsunami
observations at DART buoys and of seismic and geodetic data (e.g. NOAA/PMEL
for the Pacific Ocean and GI-INA-TEWS project for Indonesia) or interpolated
on the basis of real-time earthquake parameters (e.g. JMA for Japan and
CENALT for France, for the north-eastern Atlantic Ocean and Western
Mediterranean)

These limitations can be overcome by defining a database of ES for the sea level elevation that, properly queried and combined, is
able to reproduce any tsunami initial condition and the corresponding tsunami
impact, while significantly limiting the computational effort. No a priori
assumptions about the seismic source geometry and the kind of tsunamigenic
source are necessary, as long as the linear propagation of tsunami waves in
deep water and the superposition principle hold. This methodology has been
proposed in several studies, using Gaussian-shaped

We present here a database of tsunami waveforms stored at densely spaced
observation points (OPs) along the 50 m depth isobaths, obtained from a very
large number of Gaussian-shaped tsunami ES covering the
whole Mediterranean Sea. Given any static tsunami initial condition, the
proposed procedure provides a rapid approximation of the corresponding full
time history at any OP by LCs of the pre-calculated waveforms
associated to each selected ES. In addition to being independent of the source
mechanism, the unit source size and density is suitable to satisfactorily
reproduce not only the tsunamis generated by large earthquakes but also
those generated by events as small as M6 earthquakes. The performance of this
tool is analysed by quantifying its limits and errors in recovering an
initial water displacement field and by assessing its usability in several
different possible applications, such as probabilistic tsunami hazard
analysis, tsunami source inversions and tsunami warning systems: for example,
by propagating the estimated uncertainty in the probability distribution of
the tsunami forecast

In this section we illustrate the approach followed to calculate the approximate tsunami waveforms generated by any given seismic source. The method is based on LCs of the contributions of elementary sea level displacement recorded at the 50 m isobath contour.

Spatial distribution of the Gaussian-shaped elementary sources (black dots) covering the Mediterranean Sea and position of tsunami receivers on the 50 m isobaths (red dots) where the pre-computed tsunami waveforms are evaluated. Yellow (maximum considered magnitude up to 8.0) and magenta (maximum considered magnitude up to 8.5) stars mark the epicentres used in our performance analysis presented in Sect. 3 (Al is Algeria, Li is Liguria, Ca is Calabria and Gr is Greece).

The whole Mediterranean Sea is covered with a dense grid of

Numerical simulations have been performed using the Tsunami-HySEA code

We follow two main steps to reproduce the tsunami generated by a given
seismic source: (i) finding the coefficients for an approximated
representation of the initial (I) water vertical displacement

To find the coefficients, the ES whose centres fall in the area where

Focal mechanisms and depths of the top of the faults considered testing the performances of the database in the Mediterranean Sea.

In the next section, we test this method extensively to quantify the accompanying uncertainty, trace back the relative contributions of the different uncertainty sources and revise the method accordingly, in order to reduce the bias introduced by our approach.

To test the proposed approach, we first visually compared the original and
reconstructed initial conditions for several earthquake scenarios. Each
scenario is represented with a rectangular fault with length and width
assigned by the

The initial conditions are qualitatively well reproduced in most of the
cases; some examples are shown in Fig.

A more thorough quantitative analysis is deemed necessary to assess
limitations and uncertainties introduced by the method. Therefore, we test a
large number of realistic earthquake scenarios with epicentres located in
four areas where tsunamigenic earthquakes may occur (Fig.

First, we analyse the misfit (Sect. 3.1) between LC and NS waveforms and then we also perform a comparison between the LC and NS maximum wave amplitudes (Sect. 3.2) in order to quantify the uncertainty related to different quantities, possibly required by different specific applications.

We argue that the main uncertainty sources are (i) the misfit between

Validation results:

The overall agreement between the waveforms predicted by LC and the
corresponding ones obtained by NS is evaluated through the calculation of the
misfit for each scenario and each OP. The misfit is defined through a cost
function frequently used to compare tsunami signals in source inversions

Overall, the waveforms are reproduced quite well, as shown by the rather
narrowly peaked misfit distribution (Fig.

Maximum offshore tsunami amplitude is a widely used metric for both TWS and
PTHA. For example, it is been used by

The differences between the maximum wave amplitudes predicted by the NS
(

The results have also been analysed by separating the scenarios according to
earthquake faulting mechanisms (Fig.

When grouping the events according to fault depth (Fig.

Since the LC procedure presented here results in an average overestimation
with respect the NS waveform maxima of

We then aim to reduce the bias introduced by the inaccurate reconstruction of
the tsunami initial condition

Validation results. Misfit between the considered LC and NS
waveforms using (top) the latitude and coastal crop (lat/crop) correction,
(bottom) lat/crop correction and potential energy preservation, for all
magnitudes and mechanisms.

The first refinement serves to correct the uneven sampling of the Gaussian ES
on a grid that has constant spacing of 30 arcsec in both north–south and
east–west directions. The ES constant width (

As a further step towards an improved representation of the initial water
displacement

The performances of the above corrections are tested against the direct
fully non-linear simulation (NS) of the tsunamis generated by the target
initial field; i.e. Fig.

We present here a source mechanism-free tool to rapidly reconstruct the full waveform and the maximum wave heights predicted by any static tsunami initial water displacement, independently from any a priori assumptions on fault geometry. The reconstruction is obtained through a linear combination of a pre-computed database of tsunami waveforms generated using tsunami elementary sources.

For the first time, the validity of the method has been systematically tested
against a wide range of realistic scenarios (

We point out that we have populated the ES database using non-linear shallow
water equation simulations since performing NS
with Tsunami-HySEA code increases the computational time by only
< 10 % with respect to the linear scheme version; moreover, we
were expecting only weak non-linearity, as the results confirmed, being most
of the propagation in deep enough waters. Hence, we have not judged necessary
to put efforts in switching off the non-linear terms in the code. Moreover,
one future project we envisage is to investigate other reconstruction
methods, such as the reduced base methods

We consider that the results provided by our method are satisfactory for
most of the practical applications such as probabilistic tsunami hazard
analysis, tsunami source inversion and tsunami warning systems. The present
tool, in fact, has been already successfully used to develop an event-tree-based PTHA methodology which accounts for both aleatory and epistemic
uncertainty

Moreover, since they basically contain no bias, the uncertainty introduced by
the approximations used can be propagated in a straightforward manner into
the uncertainty associated to the final results, for example when defining
the parameters of a log-normal distribution of the hazard impact metric, to
be convolved with the probability density function (PDF) of representing different sources of aleatory
uncertainty such as the natural variability of the earthquake source or the
contribution of the tidal stage (see

An important advantage in TWS applications is that our database will allow managing the regime of large epistemic uncertainty concerning the faulting mechanism, when either fast moment tensors or direct tsunami measurements are not immediately available after a potentially tsunamigenic earthquake. This is almost always the case in the Mediterranean or Caribbean seas, where, due to tectonic complexity combined to short tsunami arrival times at the coast, the faulting mechanism is highly unpredictable and its rapid estimation very challenging. The situation, however, applies to any potential source zone of large enough crustal earthquakes in the near-field of the coast.

A wide range of faulting mechanisms can be in fact readily explored using
this database; the search can be guided by prior knowledge of the regional
past seismicity and tectonic setting. For example, if an earthquake of a
certain magnitude happens with a certain hypocentre, the PTHA event tree

In the presence of fast moment tensor solutions, the forecast uncertainty can
be promptly reduced, while still incorporating errors in the real-time
seismic solutions, by combining the latter with a priori assumptions on the
source mechanism probability. These aspects will be better addressed in a
future study that deals with the implementation of this tool for the Italian
NEAMTWS Tsunami Service Provider

The underlying data, tsunami waveforms database and results are not available to the public. For scientific collaboration and data usage, interested researchers are invited to get in contact with the authors.

Irene Molinari, Roberto Tonini, Alessio Piatanesi and Stefano Lorito conceived the method and analysed the results; Irene Molinari, Roberto Tonini and Stefano Lorito wrote the manuscript; Irene Molinari and Roberto Tonini prepared the figures; Irene Molinari, Roberto Tonini and Andrea Hoechner performed numerical simulations and the performance analysis; Fabrizio Romano provided input and support to numerical simulations; Daniele Melini provided HPC support for numerical simulations; Jose M. González Vida, Jorge Macías, Manuel J. Castro and Marc de la Asunción provided the GPU code for numerical simulations. All authors reviewed the manuscript.

This work has been funded by: (i) the flagship project RITMARE funded by the
Italian Ministry of Research and Education; (ii) the INGV-DPC Agreement Annex
B2; (iii) the EU-project ASTARTE – Assessment, Strategy And Risk Reduction for
Tsunamis in Europe – FP7-ENV2013 6.4-3, Grant 603839; (iv) the TSUMAPS-NEAM
project, Agreement Number: ECHO/SUB/2015/718568/PREV26, co-financed by the
European-Union Civil Protection Mechanism; and (v) the Spanish Government
Research projects MTM2012-38383-C02-01 and MTM2015-70490-C2-1-R. We thank A. Armigliato and E. Pelinovsky
for constructive comments, that helped to
improve the manuscript. Figures have been prepared using the Generic Mapping
Tools