This study presents a new method to analyse the properties of the sea-level
signal recorded by coastal tide gauges in the long wave range that is in
a window between wind/storm waves and tides and is typical of several
phenomena like local seiches, coastal shelf resonances and tsunamis. The
method consists of computing four specific functions based on the time
gradient (slope) of the recorded sea level oscillations, namely the
instantaneous slope (IS) as well as three more functions based on IS, namely the
reconstructed sea level (RSL), the background slope (BS) and the control function
(CF). These functions are examined through a traditional spectral fast Fourier transform (FFT) analysis
and also through a statistical analysis, showing that they can be
characterised by probability distribution functions PDFs such as the
Student's

This paper is focused on the analysis of coastal long waves, for which we mean the period window above the wind and storm waves and below the tides, which is typical of phenomena like harbour resonant oscillations, coastal-basin seiches, meteo-tsunamis, tsunamis etc. The study of such long waves is an important component of sea-level studies for coastal areas because it is known that they can be very damaging. Large amplitude waves, like tsunamis or extreme seiches, may cause severe flooding of coastal areas, with concomitant danger and damage to people, facilities and properties. And even waves of smaller amplitudes, especially in harbours, are capable of inducing large oscillations and strong currents that might limit operability and stress/break moorings destabilising ships (Wiegel, 1964; Miles, 1974; Rabinovich, 2009; Kwak et al., 2012).

One of the main topics of the literature on coastal long waves regards resonant properties of harbour basins, which is understandable due to the great economic and strategic relevance of port installations and facilities in all marine countries of the world. Nowadays free-oscillation and longwave analysis are performed mostly through numerical simulations and hydraulic laboratory modelling (Beltrami et al., 2003; Bellotti, 2007; Luick and Hinwood, 2008; Bellotti et al., 2012; Hinwood and Luick, 2012; Lopez et al., 2012; Guerrini et al., 2014). The main goal in harbor and port engineering studies is to design breakwaters, wharves and quays that are able to withstand the meteo-marine-induced wave conditions as well as current conditions and to design berths and quays such that the moored vessel movements of various types would enable sufficient yearly operability of the vessels at berth during loading or unloading operations.

Despite the recognised importance of longwave studies and coastal resonances,
direct measurements of long waves are not common, and they have been carried
out especially for research purposes (Okihiro et al., 1993; Okihiro and Guza,
1996; Lara et al., 2004; Bellotti and Franco, 2011; Guerrini et al., 2014).
In recent years, the upgrade of the traditional coastal tide gauge networks,
that usually recorded data every 10 or 15

In this paper, a study of sea level in the longwave range has been carried out for the harbour of Targia, located a few km north of the town of Siracusa, in Italy, as an example of application of a new method of analysis of coastal tide-gauge records. Since the toponym of Siracusa is known much better than Targia, we will hereafter refer to this station as the Siracusa tide-gauge station. From sea-level traces the method computes four functions that are analysed in two ways: (i) considering them as functions of time, one computes and examines their fast Fourier transform (FFT) spectra and (ii) interpreting them as time series of random wave fields, one computes their statistical properties by searching for the probability density functions (PDFs) that best fit the experimental occurrences. The functions we introduce here were first defined in a different context, i.e. within a tsunami early detection algorithm (TEDA) with the purpose of detecting tsunamis and hazardous long waves (see Bressan and Tinti, 2011, 2012 and Bressan et al., 2013), and were proven to be suitable to characterise the sea-level signal in the tsunami frequency band. Before using TEDA for a specific site, a calibration process is needed to determine the parameters of TEDA computational procedures and mathematical expressions. It was during the TEDA calibration for the Siracusa tide gauge that the new analysis method presented in this paper was first conceived. This is the reason why (i) the method is applied here to Siracusa station data, (ii) the longwave functions are calculated with the TEDA setting resulting from calibration and (iii) the calibration procedure is synthetically illustrated in the Appendix, with the main body of the paper dedicated to the new method.

In the next sections of the paper, we first introduce the four functions of the method and the basic data set we use. Then we illustrate the results of the FFT analysis of these functions, which is the first part of the method, and the results of the statistical analysis, which is the second part. Discussion of the outcomes and of their potential will conclude the paper.

The method for longwave analysis we propose is based on the definition of four functions, the first of which is the instantaneous sea-level slope (IS). From it, further longwave functions are derived: the reconstructed sea-level (RSL), the background slope (BS) and the control function (CF). These functions will be defined here below, and an example of them is shown in Fig. 1.

The function IS is computed by least-squares fitting the de-tided time
series of the sea-level data within the time interval

Sea-level of the Siracusa tide gauge (top panel), the de-tided reconstructed sea-level RSL (second panel), the sea-level slope IS and background slope BS (third panel) and the control function CF for a sample of 4 h (forth panel).

The longwave reconstructed sea level RSL represents a filtered
sea-level signal, and it is computed by integrating the function IS
over the time interval

The background slope BS is defined as the maximum sea-level slope (in
absolute value) computed over the time interval

The control function CF is defined as the positive ratio of the sea level and
background slope, i.e. as follows:

The de-tiding method we adopted works by removing the tide trend directly from
the sea-level slope IS, rather than by removing a fitting synthesis of
tidal harmonics from the sea-level record. The de-tiding procedure consists of
the following steps. First, one computes a temporary raw sea-level slope

In general, for small tidal heights (in the range of 0.2–0.3

In the frame of the TSUNET project, a local sea-level monitoring network has been installed by the University of Bologna for the coasts of eastern Sicily, Italy, including three coastal tide gauges in Tremestrieri (south of Messina), Catania and Targia (north of Siracusa), that will be named as the Siracusa tide gauge in this paper. This tide gauge is installed in the inner wall of the main breakwater forming the little harbour of Targia in the bay of Augusta, that is divided in two sectors (north and south) by the natural peninsula of Magnisi (see Fig. 2). The bay has a great strategic value for military, commercial and industrial reasons also related to the industrial area of Priolo (with petrochemical, refinery, electric-power-station installations) located in the northern sector.

The Siracusa tide-gauge station started recording on 4 May 2011 with
a 5

Average power spectra of the entire 25 month long interval of analysis from May 2011 to June 2013 (Total) and of 2 years, the first from May 2011 to April 2012 and the second from May 2012 to April 2013. The stability of the spectra is evident, which allows the identification of typical spectral peaks.

Average power spectral densities (PSDs) for the year-long interval
June 2012–May 2013 (tot), and for the corresponding calendar months from
1

In this section we compute traditional FFT spectra of the sea-level signal
including tides for a subset of the available data spanning the time interval
from May 2011 to June 2013. We calculate power spectral densities (PSDs), in a
way similar to Welch (1967), over consecutive 12

The yearly stability and seasonal variability of sea-level spectra in the intermediate wave regime has been found for many coastal stations (see e.g. Rabinovich, 2009; Bressan and Tinti, 2011, 2013), with typical spectra changing from station to station. It has to be noticed that, while offshore it is possible to refer to a universal shape for the shortwave amplitude spectrum, as for example the the JONSWAP spectrum (resulting from the Joint North Sea Wave Observation Project, Hasselman et al., 1973), that can be adapted to local conditions (such as fetch), on the contrary spectral characteristics at the coast differ substantially from place to place. Indeed, spectral peaks may change in number, position and width, depending on the morphology and bathymetry of the coastal area. In general, the association between spectral peaks and coastal basins or sub-basins is not trivial and has not been carried out in this study. We simply suggest that for Siracusa tide-gauge station, the semi-closed Targia harbour basin, the southern part of the bay of Augusta bounded by the Magnisi peninsula, and the whole Augusta bay (see Fig. 2) might act as resonant basins and might be responsible for some (if not all) of the observed peaks.

In the first step of our method, the longwave functions introduced above are
seen as a function of time, since they can be computed at every sampling time

Average power spectra of the sea-level slope function IS
computed with

Power spectra of the long-wave reconstructed sea-level function
RSL, computed with

Regarding the seasonal cycle, this also can be identified through the power
spectra of the longwave functions. Variations within the year going from
June 2012 to May 2013 for these functions are shown in Fig. 7 in a way that
is an alternative to the graph type proposed in Fig. 4. For all functions we
compute the maximum envelope spectrum, say

Figure 7c–d allows one also to see that the power spectra of functions
BS and CF cannot be used for peak characterisation since they
lose either substantially or the total information about resonances.
Moreover, it is worth observing that the PSD of BS is almost linear
with trend of

In the second step of the analysis method, one regards the longwave functions
IS, RSL, BS and CF as the result of a random
process and the empirical value at each sampling time

Normalised empirical frequency distributions for the variable
IS corresponding to different time intervals: 20

The most important result is that the normalised EFDs of the longwave
variables are stable, that is they exhibit a characteristic shape over
monthly and multiyear periods. These typical shapes are illustrated in Fig. 9 and are
bell-like, unimodal, symmetric and centred around the origin, for IS and
RSL, whereas they are half-bell, unimodal, positive and decreasing for
CF, and positive, asymmetric, long right-tailed for BS. These
results do not change even if one considers different time constants such as
e.g.

The following step of the statistical analysis is to find the parent PDF that
might be associated to the obtained EFDs. We have tried with a number of
classical PDFs and have measured the misfit or error

Examples of normalised empirical distributions for the four longwave
functions of the method (IS upper left, RSL upper right,
CF lower left, BS lower right) and the corresponding fitting
PDFs (solid green curve). Units are

Normal Student's

The second relevant result is that we have been able to find that IS
and RSL can be satisfactorily fitted by means of a Student's

The variable IS has been found to obey a Student's

Normalised empirical frequency distributions of September 2013 for
the variable IS (left) and RSL (right) with four different
fitting Student's

Statistical analysis of IS. Monthly EFDs mean (in

For the variable IS we have estimated the parameters

The analysis of the

The variable RSL can be studied in the same way as the variable
IS, and remarkably, the finding is the same. First the Student's

Statistical analysis of RSL. See captions of Fig. 12 for
details. The measure units are cm for the mean and for the scale parameter

Statistical analysis of the variable CF. Since CF is
dimensionless, also dimensionless are mean, variance and scale. Misfits are
computed by keeping all the three parameters

The EFDs of the variable CF are unimodal, positive, monotonically
decreasing, right-tailed, with one inflection point. We found that they can
be satisfactorily approximated by a three-parameter beta distribution, as
given here below:

A study of recorded sea level in the longwave range has been carried out for the harbour of Siracusa, Italy, as an example of application of a new method of analysis for coastal oscillations that is based on functions originally introduced in the early detection algorithm called TEDA, devised for real-time identification of tsunamis and high-amplitude long waves on coastal recordings.

The classical tool for studying sea level is spectral analysis by means of which one can (1) characterise high-amplitude peaks and resonant properties of the basin or basins where the tide gauge is installed and (2) observe seasonal (inter-year) amplitude variations with respect to a multiyear typical spectrum. On carrying out the first step of our method, consisting in computing the FFT spectra of functions IS, RSL, BS and CF, we have proven that both issues, i.e. spectral peak sequence and seasonal cycle, can be recognised also in spectra of IS and RSL, that can therefore be considered as carrying the same information content as the spectra of the original sea level in the longwave window. Spectra of BS and CF have been found to show totally different features. In particular, we have seen that BS spectra show no peaks, but only a linear trend, and that CF spectra are stable and show no appreciable seasonal variations.

The second step of the method is the statistical analysis of IS,
RSL, BS and CF where these quantities are seen as random
variable occurrences. The main result is that the slope IS and the sea
level RSL have been found to be distributed as a two-parameter
Student's

In conclusion, if spectral analysis of the longwave functions (step 1 of the
method) cannot be considered to be an added value to the classical spectral
analysis of sea level data, the statistical analysis (step 2) provides an
innovative tool since it characterises the tide-gauge site by means of well-known theoretical distributions. This has the advantages that their
parameters are easy to determine (e.g. through the maximum likelihood method
adopted here) and that they are easy to handle in order to make any kind of
probability assessments: for example, it would be easily possible to estimate
the exceedance probability of a given threshold for the reconstructed sea
level or the sea level time gradient. Also noticeable is the separation
between the value content of the parameters for IS and RSL
statistics with

We believe that our new analysis method provides a general and synthetic characterisation of longwaves for coastal locations like Siracusa that could be routinely used to complement engineering studies in coastal areas, especially in harbours, where free oscillations and resonances might limit port operativity, even when their amplitudes do not imply flooding.

TEDA is an early detection algorithm designed to work in real-time that is
composed by two parallel algorithms to detect long waves: the tsunami
detection method (TDM) to detect long waves arriving with an impulsive front,
and the secure detection method (SDM) to detect long waves that pass
a specified threshold amplitude. The functions used by TEDA are the same
defined and analysed in this paper. The sea-level slope IS (called
instantaneous slope in TEDA) is computed in a short time interval including
the most recent datum. The background slope BS represents the present
sea-state in absence of hazardous long waves, and it is computed over
a longer time interval. The main difference between TEDA and other detection
algorithms, such as the ones based on the ratio

Results of TEDA detections with the optimal configuration setting
for the Siracusa tide gauge. Each panel shows the tsunami wave computed for
the Siracusa tide gauge from a given source. Vertical lines mark the time of
TEDA detection triggered either by the TDM (red) or by the SDM (blue). From
top to bottom the lines refer to different sea conditions of increasing
amplitude: calm, calm

The SDM is based on the filtered and de-tided reconstructed sea-level function
RSL and is triggered every time RSL passes a set amplitude
threshold, i.e. when

The detailed description of TEDA can be found in Bressan and Tinti (2011) and in Bressan et al. (2013).

Sea-level signals in the longwave range at the coast depend strongly on the
recording site since they are influenced by local conditions. Because of this
site-specific characterisation, TEDA, as well as other tsunami detection
algorithms, should be calibrated, i.e. optimised for the local sea-level
characteristics and for the tsunamis and longwave events typical of the site.
The calibration procedure can be summarised with the following steps (for
details see Bressan et al., 2013):

build a sea level background database;

build a tsunami database;

determine the parameters to test;

select thresholds by requiring no false detections on the background database;

test the tsunami database with all different combinations and thresholds;

select the parameters combination that detects the most events in the shortest time.

Four different background conditions have been selected for the calibration, including calm sea, rough sea, a seiche event and anomalous signal disturbances, that are probably due to the passage of boats close to the tide-gauge installation.

As for tsunamis, considering that there are no historical tsunami records for Siracusa, building a tsunami database means computing tsunami signals from potential sources. In our case we used sources identified by Tonini et al. (2011) to assess the tsunami hazard for the Catania coasts, since they are also considered relevant for Siracusa. Tsunami scenarios consider three remote seismic faults responsible for the 365 earthquake and tsunami, located in the Hellenic Arc west of Crete, two local sources (a seismic fault and a landslide) that could be responsible for the 1693 tsunami and two potential sources (seismic fault and a combination of a fault and a submarine landslide) that could be the cause of the 1908 Messina tsunami (see Tonini et al., 2011; Bressan et al., 2013). In total we consider 7 tsunami cases as schematised in Table A1. Tsunami simulations have been carried out by means of the model UBO-TSUFD developed and maintained by the Tsunami Research Team of the University of Bologna (see Tinti and Tonini, 2013).

Tsunami sources used for the tsunami hazard studies for the
Catania coasts (Tonini et al., 2011) Fx

The signals on which TEDA is tested are formed by combining (adding) the 4 selected recorded background conditions with the 7 computed tsunami time histories, giving a total of 28 cases.

From the calibration process the best performing parameter configuration
turns out to be

This work has been carried out in the framework of the EU project ASTARTE, grant agreement no. 603839, within FP7 ENV2013 6.4-3 and of the Italian project RITMARE.Edited by: I. Didenkulova Reviewed by: S. D. Rosen and one anonymous referee