NHESSNatural Hazards and Earth System SciencesNHESSNat. Hazards Earth Syst. Sci.1684-9981Copernicus PublicationsGöttingen, Germany10.5194/nhess-17-505-2017 Time clustering of wave storms in the Mediterranean SeaBesioGiovannigiovanni.besio@unige.ithttps://orcid.org/0000-0002-0522-9635BrigantiRiccardoRomanoAlessandroMentaschiLorenzoDe GirolamoPaoloDepartment of Civil, Chemical and Environmental Engineering, University of Genoa, Genoa, ItalyDepartment of Civil Engineering, University of Nottingham, Nottingham, UKDepartment of Civil, Architectural and Environmental Engineering, La Sapienza University, Rome, ItalyEuropean Commission, Joint Research Centre (JRC), Ispra, ItalyGiovanni Besio (giovanni.besio@unige.it)27March201717350551430September20163November20161March20173March2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://nhess.copernicus.org/articles/17/505/2017/nhess-17-505-2017.htmlThe full text article is available as a PDF file from https://nhess.copernicus.org/articles/17/505/2017/nhess-17-505-2017.pdf
In this contribution we identify storm time clustering in the Mediterranean
Sea through a comprehensive analysis of the Allan factor. This parameter is
evaluated from a long time series of wave height provided by oceanographic buoy
measurements and hindcast reanalysis of the whole basin, spanning the
period 1979–2014 and characterized by a horizontal resolution of about 0.1∘ in
longitude and latitude and a temporal sampling of 1 h
. The nature of the processes highlighted by the AF and
the spatial distribution of the parameter are both investigated. Results
reveal that the Allan factor follows different curves at two distinct timescales. The range of timescales between 12 h to 50 days is characterized
by a departure from the Poisson distribution. For timescales above 50 days, a
cyclic Poisson process is identified. The spatial distribution of the Allan factor reveals that the clustering at smaller timescales is present to the
north-west of the Mediterranean, while seasonality is observed across the whole
basin. This analysis is believed to be important for assessing the local
increased flood and coastal erosion risks due to storm clustering.
Introduction
In recent years the occurrence of different coastal storms in a short time
has been studied in the context of storm-driven erosion of beaches and dunes.
Indeed it has been shown by different authors
that storms occurring in quick succession may
result in greater beach erosion than the cumulated erosion induced by single
storms of far higher return periods.
In the events analysed in the aforementioned studies both the surge and the
wave components played an important role. While studies that identify
time clustering of storm surges are available (e.g. ;
), there is no study, to the best knowledge of the
authors, that analyses the clustering properties of wave storms alone. In
micro-tidal environments, such as the Mediterranean Sea, wave storms are the
principal driver of short term coastal erosion and flooding; hence it is
important to understand the occurrence of clustering.
The Mediterranean Sea wave climate has been extensively studied
(e.g. ) and it is known that throughout the basin winter is
richer in cyclones and, in turn, in wave storms. However, regional
differences are significant. linked the seasonality of
wave storms to local features of atmospheric pressure over the Mediterranean
basin, strongly suggesting that the local typical meteorological conditions
determine different temporal regimes of storm waves.
Value of significant wave height threshold in metres for the 98 %
percentile.
The present work addresses the gap in the knowledge of the occurrence of
time clustering of wave storms by carrying out an analysis of wave storms
sequences using the Allan factor hereinafter AF,, a well-established technique to study the time behaviour
of environmental processes. When the underlying process is characterized by
clustering, the AF of a specific sequence of events is larger than 1 and
shows a power-law behaviour at the timescales that exhibit departure from a
Poisson distribution. The simplicity of the AF analysis made it popular in
the study of time sequences of a number of physical processes such as
earthquakes , lightning
, rainfall or fires . However, the AF
can also be larger than 1 for non-homogeneous Poisson processes, as shown
in . Hence it is important to distinguish clustering
dynamics from cyclic Poisson processes. Methodologies that are suitable to
achieve this are presented in and .
Here we analyse the AF on long time series of wave height in the
Mediterranean Sea provided by hindcast reanalysis spanning the period 1979–2014
. This analysis is validated and compared
against the AF evaluated using the time series of wave measurements of the
Italian national Sea Wave Measurement Network (Rete Ondametrica Nazionale,
hereinafter RON). Subsequently we apply the methodology proposed in
to gain an insight into the type of process that is
described by the AF. The objective of this study is to identify the presence
of time clustering of wave storms in the whole of the Mediterranean basin and
examine the timescales at which events are correlated as well as the spatial
distribution of the clustering. To this end, after scaling properties of wave
storms are identified, they are mapped over the Mediterranean Sea.
The paper is organized as follows: after the Introduction, Sect.
explains the methodology used for the AF analysis, Sect.
describes the data sets used, Sect.
illustrates the results and Sect. discusses the results
and draws the conclusions of this work.
Hindcast control grid points (red circle) and RON buoys as reference
points (yellow circles).
Clustering analysis methodology
Sequences of natural events such as earthquakes, rainfall and wildfires, can be
seen as realizations of stochastic point processes. A process of this kind
describes events that occur randomly in time and is completely defined by
the times at which these events occur. Here time series of sea states are
considered. Each sea state is defined by a set of spectral parameters, such as
the significant wave height Hs, the peak period Tp, the mean
period Tm-1,0 and the mean direction of propagation θm. Waves are
always present on the sea surface; hence a sequence of storms needs to be
extracted from a time series of sea states by only considering events that
satisfy a certain criterion. A storm is commonly defined as a sequence of sea
states in which Hs exceeds a given threshold (e.g. ).
In this work, a threshold for each node is defined by considering the local
98% percentile of the Hs distribution, regardless of θm
(omnidirectional analysis; see Fig. for threshold values of
Hs obtained with the hindcast model used here). The time ti at which
the threshold is exceeded for the first time in each storm defines the event
as part of a point process. If the interval between two subsequent events is
below 12 h, the two are regarded as one event. This is common practice in
analysing storms and the value is deemed appropriate for the Mediterranean
Sea (e.g. ). Therefore, in each of the computational
nodes over the Mediterranean Sea (see Fig. for a map of
the domain and the location of few control grid points used in this study to
show the single-point behaviour of the AF), a point process is defined. An
example for the control point A and for the years 2004 and 2005, is given in
Fig. . In this figure it is evident that most of the
storms during the 2 years considered occur between November and May,
showing the pronounced seasonality that characterizes the basin.
Figure shows the number of events defined in each month over the
year in the hindcast record for the same reference point A during the
period 1979–2014 as a function of the percentile threshold (different wave heights).
The seasonal variability of the storms in the Mediterranean basin is again
recognizable. Note that the difference in the number of storms between the
different percentiles considered is maximum in the most active months and, if
the 99 % is chosen, the differences among seasons are small, although the
seasonal variability is still recognizable.
Storm occurrence for the northern Tyrrhenian reference point (A):
2004/2005 in the top panel, zoomed-in graph of winter 2004/2005 in the bottom
panel.
These point processes are studied by defining equally spaced time windows of
duration τ and counting the events in each window. The result is a
sequence of counts Nk (k= 1, …, M, where M is the number of time
windows). The clustering of the events is then studied with the Allan factor
, defined as the variance of successive
counts as
AF(τ)=<Nk+1(τ)-Nk(τ)2>2<Nk(τ)>.
In general terms, a point process is called fractal when a number of the
relevant statistics shows scaling with related scaling exponents
. This implies that the AF depends on τ with a
power law, with exponent α, which indicates the presence of clusters
of points over a number of timescales τ. For a fractal process with
0 <α< 3 this power law reads as follows:
AF(τ)=1+ττ1α,
where τ1 is the fractal onset time that marks the lower limit for
significant scaling behaviour for the AF. For times smaller than τ1
there is no significant time correlation, while for times greater than τ1
a characteristic fractal trend can be derived from the value of the
exponent. If the storm's process is Poissonian, the arrival times are
uncorrelated; hence α is expected to be zero and the AF will be near
unity. If non-Poissonian processes are present over a significant range of
timescales it will be possible to identify α> 0 and AF > 1.
demonstrated that cyclic, hence non-homogenous,
Poisson processes show AF > 1 for timescales associated to cyclic
components. It is therefore necessary to identify and separate the timescales
at which clustering occurs from those at which the point process is
Poissonian. To this end it is necessary to compare the AF pattern found in
the wave time series with that of a process of known properties. A cyclic
Poisson process is generated here with the same integrate and fire (IF)
technique used in . The cyclic components are
selected by looking at the dominant harmonic components obtained with the
Fourier analysis.
Number of storms vs. threshold for the northern Tyrrhenian reference
point (A).
The exponent α is estimated for the timescales
at which the process is not Poissonian. Note that different ranges of τ can
reveal different time scaling (clustering) of the same process through
different slopes of Eq. () due to different kinds of forcing
.
The occurrence of subsequent wave storms can be interpreted as a
realization of stochastic temporal point process that could attain a
clustered character when a number of its underlying features exhibit
some scaling as a function of some scaling power law. The presence of
such characteristics reveals that the process follows some kind of
clustering in time (). There are
different statistical measures available in the literature to characterize
the counting process of a general physical phenomena. In the present
study we decided to employ the Allan factor thanks to the fact that it does not
saturate the exponent α at unity as other indicators
such as the Fano factor do. The Allan factor is defined as the
variance of successive counts for a specific counting time window T
divided by two times the mean number of counts in the same counting window
AF(τ)=<Nk+1(τ)-Nk(τ)2><Nk(τ)>.
For a fractal process the Allan factor recovers a power law of the type
AF(τ)=1+ττ1α
over an extended range of counting windows τ, where α is the so-called fractal exponent that for white noise time series attains values
close to zero (i.e. the signal is characterized by the absence of time
correlations, homogeneous Poissonian process), while for time-clustered
processes it shows values greater than zero. τ1 represents the fractal
onset time and marks the lower limit for significant scaling behaviour for the
Allan factor: for times smaller than τ1 there is no significant time
correlation, while for times greater than τ1 a characteristic trend can
be derived from the value of the exponent; furthermore different time windows
can reveal different time scalings of the same process through different
slopes of Eq. () due to different kind of forcing
. demonstrated that cyclic,
hence non-homogenous, Poisson processes show AF > 1 and power law behaviour
for timescales associated to cyclic components. It is therefore necessary to
compare the AF pattern found in the wave time series with that of a process
of known properties. The same technique used in has
been used here to simulate surrogate non-homogeneous point processes and
compare them with the reference ones using a Monte Carlo approach. The
processes have been generated using the same IF
technique in , to which the reader is referred for details.
This further clarifies the nature of the process described by the AF and the
role of the different cyclic components that contribute to generate
above-threshold events.
Wave dataWave hindcast
Wave hindcast in the Mediterranean Sea has been implemented on a time window
covering 36 years, from the first of January 1979 to 31 December 2014
(http://www.dicca.unige.it/meteocean/hindcast.html).
The wave model is forced by the 10 m wind fields obtained by means of the
non-hydrostatic model WRF-ARW (Weather Research and Forecasting – Advanced
Research WRF) version 3.3.1 . In the present study a
Lambert conformal grid covering the whole of the Mediterranean Sea with a resolution
of about 0.1 degree in longitude and latitude has been used. Initial and
boundary conditions for atmospheric simulations were provided by the CFSR
(Climate Forecast System Reanalysis) database . Use of CFSR
reanalysis data for wave modelling provides reliable results, even if
sometimes extreme wave conditions are not properly modelled
. For
further details of the set-up and validation of the meteorological model,
readers can refer to .
Generation and propagation of sea waves have been modelled using
WavewatchIII®, version 3.14 .
A 336 × 180 regular grid covers the whole of the
Mediterranean Sea with a resolution of 0.1273∘× 0.09∘,
corresponding to about 10 km at the latitude of 45∘ N. Spectral
resolution is characterized by 24 bins in direction and 25 frequencies
ranging from 0.06 to 0.7 Hz with a step factor of 1.1. The output has been
recorded hourly in all points of the computation grid for integrated
quantities (i.e. significant wave height Hs, mean period Tm-1,0, peak
period Tp, mean direction θm, peak direction θp,
directional spreading Δθ). The validation of the wave hindcast
has been carried out through extensive comparison of simulated quantities and
wave buoy data see
and has already been employed for different applications such as wave energy
resource assessment and extreme and wave climate analysis
.
Buoy data
The Italian Sea Wave Measurement Network (Rete Ondametrica Nazionale RON)
started operating in July 1989 .
The locations of the buoys are indicated in Fig. . Until
1998 the network was made of eight pitch-roll directional buoys located
offshore, in deep water conditions, of several sea areas equally spaced along
the Italian peninsula. These original eight stations were La Spezia,
Alghero, Ortona, Ponza, Monopoli, Crotone, Catania and Mazara del Vallo. The
statistical wave parameters (i.e. significant wave height Hs, mean
period Tm, peak period Tp, mean direction θm) were originally
retrieved every 3 h, below a station-dependent threshold for Hs,
and every half an hour above this threshold. The wave data time series,
measured by the RON buoys, that have been analysed in the present study,
cover a time window of 20 years, from the summer of 1989 to the spring of 2008
for the original eight buoys. For the cluster analysis performed using
the RON records, data were considered every 3 h for all the stations.
ResultsComparison between hindcast and buoy measurements
In order to assess the reliability of the hindcast time series related to
storm cluster analysis, the results of AF for the RON buoys are analysed and
compared to the corresponding grid points of the hindcast model. These
results are shown in Figs. –. Results
obtained on the basis of the RON data and hindcast series show a good
qualitative and quantitative agreement, especially for lower threshold
conditions (98 % percentile), while for higher thresholds (99.5 % percentile)
they tend to present stronger differences, e.g. in Alghero (see
Fig. ). These findings can be explained by the fact that
increasing the threshold limit would select just the most energetic wave
conditions that are the most difficult to be reproduced by numerical models
a.o. and sometimes to be recorded by wave buoys
(breakdown, damages or even loss of the instrumentation). Also, differences
are usually larger for smaller timescales, i.e. 0.5 <τ< 50 days and for
the 99.5 % percentile (e.g. Alghero and Mazara in Fig. ).
These results confirm that the hindcast data and the wave buoys show very
similar scaling properties.
Comparison of Allan factor between RON and hindcast data series for
different threshold percentiles (98 and 99.5 %).
Comparison of Allan factor between RON and hindcast data series for
different threshold percentiles (98 and 99.5 %).
Comparison of Allan factor between hindcast data series for 98 %
percentile (black line) and 1000 simulated cyclic Poisson processes (grey
lines). The AF corresponding to the 95 % percentile of the AF distribution
is also plotted (dashed line). Top left shows point A (northern Tyrrhenian). Top
right shows point G (southern Tyrrhenian). Bottom shows point O (south-eastern
Mediterranean).
Allan factor (AF) as a function of counting window τ and of the
wave height threshold (different percentiles as in the legend) for different
locations in the Mediterranean Sea (cf. Fig. ).
Allan factor (AF) as a function of counting window τ and of the
wave height threshold (different percentiles as in the legend) for different
locations in the Mediterranean Sea (cf. Fig. ).
Allan factor (AF) as a function of counting window τ and of the
wave height threshold (different percentiles as in the legend) for different
locations in the Mediterranean Sea (cf. Fig. ).
Comparison with a simulated non-homogeneous point process
The AF patterns of both the model and data show a consistent behaviour
across the Mediterranean basin. The AF is greater than one for τ greater
than 12–24 h (0.5–1 days) and a distinct slope is recognizable, generally
between 0.5 to 20–50 days at many of the stations. For larger values of τ,
the AF increases to reach a maximum at 180 days. It is necessary to
clarify the nature of the processes described by the AF patterns seen and, in
particular, it is necessary to identify whether deviation from a cyclic
Poisson process is present. To this end, the AF pattern found from hindcast
time series is compared with that of a simulated non-homogeneous Poisson
process. This is generated using the IF technique employed in
. The rate function of the simulated non-homogeneous
Poisson process is generated as a sum of sinusoidal components with
amplitudes, periods and phases obtained from the Fourier analysis of the
reference signal. A Monte Carlo simulation of 1000 time series is then
carried out and the simulated population of AF is compared with the reference
one. Hindcast points A, G and O (see Fig. ) are chosen for
this analysis because they show different AF patterns in the timescales
τ< 50 days.
This analysis reveals that, as expected, the dominant
cyclic component for all the considered time series is the one with a 1-year
period. This was also noted for the RON data in ,
where the amplitude of the annual cycle component was estimated to be around
0.25 m in Alghero, which is consistent with what was found in the present work.
Together with the annual cycle the components with periods of 6,
3 and 1 months and 1 week have also been considered to simulate the
non-homogeneous Poisson processes. The results of the comparison are shown in
Fig. . For all three points it is clear that the
simulated cyclic Poisson process explains the pattern of the AF at
τ> 50 days well in all cases. As expected, this is the signature of the annual
cycle, which strongly influences the occurrence of above-threshold events.
The AF departs from the Poisson distribution at τ< 50 days, above all in
points A and G. The departure from Poissonian behaviour at these timescales occurs even at very low values of α, for example in point O.
However, data often show oscillations, above all for α< 0.1, and
it is not possible to make conclusions about the existence of a clustering regime.
AF results over the Mediterranean Sea
Results from the control points located over the basin (see Fig. )
are shown in Figs. –. The analysis of the AF curves
reveal that these can be divided in two groups:
The first group clearly shows the slope corresponding to the departure from
the Poisson regimes. The change in regimes occurs at around τ=50 days
in most cases. α varies significantly from point to point. A
well-defined slope is very evident at points A (northern Tyrrhenian Sea),
B (Gulf of Lion), D (Alboran Sea), and E (Algerian Sea). In all these cases a
uniform value of α can be defined and the exponent value is in the
interval 0.15–0.3. In other cases the slope is not so well defined or it is
significantly smaller than 0.2. Points that show either or both
characteristics are point R (Adriatic Sea), C (western Sardinia), F (Tunisian
coast), G (southern Tyrrhenian Sea), M (Ionian Sea) and Q (Aegean Sea). At point
Q (Aegean), α is virtually naught.
In the second group only the cyclic Poissonian regime is clearly
recognizable, generally for τ> 20 days. At smaller scales the slope that
is associated with the departure from the Poisson distribution is not
present. This is the case for the southern Mediterranean points H (Egypt),
I (western Libya), L (north-eastern Libya), O (south-eastern Mediterranean Sea) and
P (southern Turkey).
Allan factor (AF) as a function of counting window τ and of the
wave height threshold (different percentiles as in the legend) for different
locations in the Mediterranean Sea (cf. Fig. ).
The spatial distribution of the slope for small timescales is shown in
Fig. . This figure has been obtained by determining the best
fit value of α at different timescales. In order to take into account
the local differences in determining the transition between slopes and the
different regimes seen in the representative points, the slope has been
estimated using four different ranges of τ. Clustering in the range
12 <τ< 72 h (3 days) is presented in Fig. a, for
12 <τ< 120 h (5 days) results are shown in Fig. b and finally Fig. c shows the
results for 12 <τ< 240 h (10 days). Within this range the
small-scale slope is higher in the north-western Mediterranean Sea and, in
particular in the northern Tyrrhenian Sea and in the Balearic Sea. Here
α reaches values up to 0.3. Areas with α around 0.2 are present in
the Adriatic Sea, on the Syrian and Lebanese coast and along the Tunisian
coast. The effect of widening the range of τ is to decrease the best fit
value of α. This effect reduces the regions that show α
significantly higher than zero, in particular in the Adriatic Sea and on the
eastern coast of Tunisia. When the interval 12 <τ< 240 h (0.5–10 days) is
used (Fig. c), the best fit of α is
significantly higher than zero only in the north-western Mediterranean Sea with
the average α around 0.2 and zones with α> 0 are present in
the eastern part of the Adriatic Sea and on the Syrian coast.
Spatial distribution of the exponent α for the whole
of the Mediterranean basin.
Discussion and conclusions
The results presented highlighted the presence of a departure from the
Poisson distribution for timescales shorter than τ< 1200 h
(50 days). This regime is characterized by α= 0.15–0.3 and is more evident
in the north-west of the Mediterranean Sea. In the rest of the basin
α is closer to zero and the AF pattern is characterized by oscillations,
without a well defined regime.
For τ> 50 days the arrival of above-threshold storms is dominated by the
effect of seasonal and interseasonal oscillations and can be described as a
cyclic Poisson process. Similar scaling regimes have been observed in other
phenomena with seasonal behaviour, e.g. fires . These
results match with the findings by , who found that the
northern basin RON buoys (e.g. Ponza and La Spezia buoys in the Tyrrhenian
Sea) showed lower seasonality than the buoys in the southern basin
(e.g. Crotone, in the Ionian Sea). La Spezia buoy, for example, is located in the
Ligurian Sea, a region where departure from the Poisson distribution is
higher. Although in the region the cyclogenesis in the Gulf of Genoa shows
marked seasonality, cyclones are present throughout the year
. This persistence of
cyclonic events helps to explain the behaviour at smaller scales
(i.e. τ< 1200 h, 50 days). The clustering at scales of days indicates that
meteorological conditions favour the occurrence of multiple events over a few
days. It is not the case that this behaviour is seen in the most active
cyclonic region of the Mediterranean Sea, e.g. the north-west according to
. Similar considerations apply to the northern Adriatic
Sea. In other parts of the basin, where these persistent conditions do not
occur, the arrival of storms is well described as a cyclic-Poisson process.
The values of α found in the present study do not allow us to draw
conclusions on whether this deviation from a Poisson distribution is large or
small for the phenomenon at hand, as there is no comparison with other
basins. Because of this, it is important to analyse other basins.
The clustering at the timescales found has the potential to exacerbate local
beach erosion generated by individual storms, as shown in
; hence it will be important to understand the
implication of these time regimes on the dynamics of the Mediterranean
coastal regions.
Wave hindcast data are available for research purposes on
request. Please contact Prof. Giovanni Besio at giovanni.besio@unige.it.
Giovanni Besio and Lorenzo Mentaschi developed the wave
hindcast and the Allan factor analysis for the Mediterranean Sea;
Riccardo Briganti coordinated the work and gave the theoretical ideas to
develop the analysis; Alessandro Romano and Paolo De Girolamo developed the
analysis for the RON buoy data set and carried out the comparison with the
simulated non-homogeneous Poisson point process. All the authors participated
actively in the preparation and writing of the manuscript.
The authors declare that they have no conflict of interest.
Acknowledgements
The work described in this publication was supported by the European
Community's Horizon 2020 Research and Innovation Programme through the grant
to HYDRALAB-PLUS, Contract no. 654110. Riccardo Briganti expresses his gratitude to
the Engineering and Physical Sciences Research Council (EPSRC) for providing
the funding through the FloodMEMORY project (grant number: EP/K013513/1). The
authors would like to thank Thomas Wahl and an anonymous reviewer for having
contributed to the improvement of the manuscript. The authors are grateful to
Francesco Serinaldi for the proficuous discussion during the revision of
the manuscript and for having made the routines for the simulation
of cyclic Poisson processes available. Edited by: I. Didenkulova
Reviewed by: T. Wahl and one anonymous referee
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