TECTION IN SEA STATES TIME SERIES
In Chapter 4 it has been shown how the EVA of a time series of Hs can be consistently
performed via two different approaches: i) applying the work-flow explained in Section 2.3. to the whole series of Hsdata or ii) applying the same work-flow to subsets of the starting dataset,
and computing the overall distribution at a second time through a Monte Carlo simulation. The latter approach has shown to efficiently catch the long-term distribution of Hsfor datasets split
according either to WPs or waves incoming direction (if a well-defined difference in the peaks directionality is present, as in the example of Section 4.3.1.). The algorithm introduced could be refined by taking into account the time dependency of the peaks, without any loss of gener-ality. Indeed, the same bootstrap procedure can be applied also for time-varying parameters, for example those introduced in Equation (2.15). Hence, it is interesting to understand when does the need of accounting for trend in time series of data arise; in view of the above, this Chapter presents a research aimed at evaluating how to properly employ the most diffused trend analysis, through a practical application on wave height series in the Mediterranean Sea. This research was carried on with Prof. Giovanni Besio and Ms. Annalisa De Leo from the Department of Civil, Chemical and Environmental Engineering, University of Genoa (Italy); Prof. Riccardo Briganti, from the Faculty of Engineering, University of Nottingham.
5.1. Introduction & Outline of the research
Climate change is expected to significantly affects the main met-ocean parameters, at both large scale and the local scale (Weisse, 2010). Relevant changes are taking place in the upper-sea physics, and in particular in water temperature and salinity (Durack and Wijffels, 2010), large-scale circulation (Cai et al., 2005; Cai and Cowan, 2007), mean sea level (Nicholls and Cazenave, 2010), and wave heights and periods (Vanem, 2016; Morim et al., 2019). In view of these considerations, the evaluation and prediction of the upper-sea physics rate of change, i.e. trends, plays a crucial role in a plethora of geophysical studies and engineering applications, such as the erosion of the coasts (Stive, 2004), changing design from hard to soft engineering options (Hamm et al., 2002), flooding hazard and coastal vulnerability assessment and man-agement (Adger, 1999; Scavia et al., 2002; Dolan and Walker, 2006; Wdowinski et al., 2016), marine ecosystems (Harley et al., 2006; Richards et al., 2008; Doney et al., 2012). The study here introduced focuses on the variations in wave climate, in particular in trends of Hs; this is
an issue of primary concern, because these may affect the fluxes of energy between the ocean and the atmosphere and even storm surges (Young and Ribal, 2019). These variations would be especially important for the coastal areas, as they may in turn modify the equilibrium condi-tions of coastal beach profiles (Mori et al., 2010) and affect ports’ activity to a substantial extent (Koetse and Rietveld, 2009). It is therefore crucial to identify and quantify the trends in wave climate, to embed these information in engineering design.
Identification and quantification of trends are usually two separate stages in data analysis. The simplest approach to quantify a trend in sea state datasets is to perform a linear regression over the values of the time series. As an instance, Gulev and Grigorieva (2004) applied a lin-ear least square regression to annual mean Hs observed from ship routes over the last century
on a global scale. The same approach was used by Shanas and Kumar (2015) to analyse both the annual mean and 90th percentile of Hs in the Central Bay of Bengal, and by Musi´c and
Nickovi´c (2008) in the Eastern Mediterranean. In the context of wave climate in the Mediter-ranean Sea sub-basins, Piscopia et al. (2003) applied a linear regression over extremes Hs in
the Italian seas, selected with the Peak Over Threshold (POT) analysis. Similarly, Casas-Prat and Sierra Pedrico (2010) estimated the linear slopes of POT time series of Hs in front of
the Catalan coast, having previously selected the locations most-likely subjected to significant trends. Previous research made also use of a linear regression modified according to the model of Theil-Sen (Sen, 1968; Theil, 1992), resulting in a sounder slope estimate (hereinafter TS slope), because it is insensitive to possible outliers. TS slopes were used in trend analysis in Pomaro et al. (2017), evaluating monthly quantiles of Hs in the northern Adriatic sea, and in
Young and Ribal (2019), assessing Hsglobal trends for annual mean, mode and 90thpercentile.
In most of the applications that aim to evaluate changes in geophysical time series, the identification of trends is usually detected using the non-parametric Mann-Kendall statistical test (Mann, 1945; Kendall, 1955, hereinafter called MK), based on the samples rank correlation within a dataset. The MK, as well as many other statistical tests, allows to accept or reject the hypothesis it verifies (the so called null hypothesis, in this case the absence of a climate trend) on the basis of the variable pvalue, defined as the observed significance level for the test hypothesis.
The pvalueis compared with a significance level α, used as a threshold, to reject (if it is lower)
or accept (if it is higher) the null hypothesis. The common use of MK allows to state if a trend exists, but it provides no informations on its magnitude. In the context of trends of Hs, the
MK was employed, for example, by the aforementioned study by Casas-Prat and Sierra Pedrico (2010). A threshold of 0.1 was used to identify locations showing trends and linear regression was used to quantify these trends. Similarly, the TS slope was used by Pomaro et al. (2017) to quantify the trends in Hsin the Adriatic sea, identified by using the MK. Nevertheless, it should
be mentioned that there is no theoretical basis for the definition of the threshold value α, for that the binary use of pvaluehas been increasingly questioned over the last few years. According
to Wasserstein et al. (2016) and Greenland et al. (2016), the pvalue should be considered as
a continuous measure spanning the 0-1 range; 1 indicates that data behave consistently with the null hypothesis, while values tending to zero indicate that data behave progressively less consistently with the null hypothesis.
Therefore, the pvalueof MK (referred to as pM K) can be used as a measure of compatibility
between the data and the hypothesis that they are not characterized by a long term trend. A similar use of pvalueis found in Solari et al. (2017), where the pvalue of the Anderson-Darling
statistic was used as a Goodness-Of-Fit measure, to check whether their data were best repre-sented by a Generalized Pareto Distribution. In case of trend analysis, one further limitation of the traditional use of the MK is that a value of α is required to evaluate the sign of a trend. On the contrary, the slope of the best fitting line immediately reveals whether the data of a series are most likely to increase (positive slopes) or decrease (negative slopes) in time, however it refers to a linear rate of change, which may represent a too limiting assumption. As such, in this study
it was evaluated whether the TS slope can be efficiently employed to quantify the sign and the magnitude of a trend, even if the underlying trend is not linear. To this end, the hindcast data introduced in Section 3.2. were employed, retaining the annual maxima, the annual mean and the annual 98thpercentile of Hsover the whole Mediterranean Sea. First, it was evaluated the
relation between the TS slopes of the reference time series and their respective pM K; indeed,
MK does not postulate the linearity of the underlying trend. Subsequently, TS estimates were compared against the outcomes of another method that is not bounded by the hypothesis of linear trend (the so called Innovative Trend Analysis, hereinafter referred to as ITA, S¸en, 2011, 2013). Finally, once the suitability of the TS slope for detecting long-term trends is proved, its spatial distribution over the MS is evaluated to get a first insight of the historical trends in the basin.
The next Section introduces the data employed and the methodology developed; Results are then presented in Section 5.3.. Finally, Section 5.4. presents the discussion of the results and the final conclusions.
5.2. Data & Methods
In order to detect trends in extreme sea state time series, the events considered to be extremes were extracted from the whole time series of Hs under study. In the extreme value analysis
framework, the POT has become a well-settled methodology, often preferred to the Annual Maxima (AM) approach, above all for relatively short time series. However, the POT requires to define a Hs threshold, as shown in the application presented in the previous Chapter; if the
POT is best suited in the framework of EVT, the selection of the threshold may significantly affect the subsequent trend analysis, either in terms of magnitude and number of resultant peaks (Laface et al., 2016; Liang et al., 2019). The value of the threshold may also be affected by climatic trends, e.g. if the Hscorresponding to the 98th percentile is taken as threshold for the
POT, this value will vary in time in presence of a trend. Additionally, the number of events above a given threshold varies every year and different nodes in the grid considered might have different number of events for a given year, posing additional problems of homogenity of the reference population, with respect to AM values. On the basis of these considerations, for this study the AM Hs, annual 98th percentile of Hs and annual mean Hs of the hindcast locations
introduced in Section 3.2. were chosen, assuring that one sample per year is used across the grid in all cases. It is worth to recall that the WWIII hindcast was forced using the CFSR wind for the period 1979-2018, and CFSv2 for the period 2011-2018. To rely on different wind fields may introduce homogeneity issues that could influence in turn the apparent historical trends in the dataset. However, the wind fields used to drive WWIII were uniformly downscaled over finer resolutions for both periods. Moreover, no abrupt changes in the NCEP wind fields took place at the mid-latitudes (while a similar problem was appreciated starting from 1994 in the Southern Ocean, see Stopa and Cheung, 2014), thus the hindcast data can be reliably employed for the trend estimations with no need of additional pre-processing.
5.2.1. Trend detection and quantification
The TS slope
The simplest trend is a linear one, hence the slope of the linear fit of a series of data can be used to quantify a trend. The value of this slope can be computed following the TS method that is insensitive to outliers, and it is therefore preferred to other common tools, such as the least squares regression, for the problem under study.
Considering a series of values xi (i = 1...n, n being the number of samples) the estimate of
the TS slope (b) is computed as:
b = Median xj− xl j − l
∀l < j , l, j = 1...n (5.1)
where xj and xlare the jthand lthdata of the series, respectively.
The Mann-Kendall test
The MK is aimed at evaluating whether either an upward or downward monotonic trend is present within a dataset. The null hypothesis of the test is that there is no monotonic trend in the time series. The test statistic, considering a time series of n elements xi, i = 1...n, is computed
as:
ZM K =
num
pσ2(S) (5.2)
where num is equal to:
num = S − 1, if S > 0 0, if S = 0 S + 1, if S < 0 , (5.3)
and S and σ2(S) are computed as:
S = n−1 X k−1 n X j−k+1 δj−k (5.4) σ2(S) = 1 18 " n (n − 1) (2n + 5) − g X p−1 tp(tp− 1) (2tp+ 5) # , (5.5)
with δj−k being an indicator function that takes 1, 0 or -1 value according to the sign of
xj−xk(positive, null or negative, respectively); g is the number of tied groups in the time series,
with tp being the number of elements in each pth group (p = 1, 2, ..g). The value of ZM K is
then evaluated as a percentile of the standard normal distribution, leading to the corresponding pM K of the statistic. In most of the applications that take advantage of the MK, the pM K is
successively compared to α. In such a case, the use of α also allows to detect the sign of the trend (whether it is upward or downward oriented) using the following relationships:
ZM K > φ−1(1 − α/2) → positive trend ZM K < −φ−1(1 − α/2) → negative trend ZM K < |φ−1(1 − α/2) | → no trend , (5.6)
where φ stands for the transform in the normal space.
Nevertheless, in this research the pM K of each annual statistic was evaluated in the whole
0-1 range. In this way, pM K was used to assess the intensity of trends over the wave height
series, as previously explained. Innovative Trend Analysis
The approach known as ITA is here briefly introduced. This method requires to split a series in two halves, each with elements sorted in ascending order, and plotted versus each other in a square plot. This allows to evaluate how the scatters diverge from the bisecting line, which represents the no-trend condition. Therefore, the ITA allows to quickly check for increasing or decreasing trends (whether the scatter lies above or below the bisector respectively). An example can be seen in Figure 5.1, in which three realizations of generic time series (x1 and x2)
are first ordered according to the ITA procedure and then plotted against each other.
x 1 x 2 i no trend positive trend negative trend
Figure 5.1: ITA plot for datasets characterized by positive (black crosses), negative (black x) and no trend (black circles)
δi is the distance between the ith (1...n/2) element of the series and the no-trend line (see
Figure 5.1). Consequently, for each wave height dataset treated using the ITA, a series of n/2 δi can be built (where n, in the present analysis, is the number of years over which the hindcast
data are defined). Note that the sign of δi indicates that the ith value lays below (δi < 0) or
above (δi > 0) the no-trend line. If δi > 0 the trend is positive, and it is negative for δi < 0,
while change in sign of δi indicates that not all data behave consistently with the presence of a
5.2.2. Analysis of the correlation between the variables employed
The proposed methodology makes it possible to combine b and pM K without restriction
for the rejection of the null hypothesis. Subsequently, b is shown to be correlated with the parameters of the population of δiin the ITA method.
The correlation between pM K and b was analysed for all the hindcast locations following
Genest and Favre (2007). Correlations were graphically evaluated in the unit-square space, spanning the 0-1 range and populated by the scaled ranks (SRi) of the investigated variables,
SRi =
mi
n + 1 (5.7)
where mi is the position of the ith data within the sorted series it belongs to, whereas n in
our case equals the number of years covered by the hindcast. The scatter plot of ranks of pM K
versus ranks of b is a visual tool that indicates the presence of correlation, anti-correlation, or no correlation at all. In the case of correlation, high (low) ranks of pM K occurr frequently together
with high (low) ranks of b. In the case of anti-correlation, high (low) ranks of either variables tend to occur with low (high) ranks of the other. No correlation is characterized by the absence of either of the previous patterns (Genest and Favre, 2007).
Correlation levels were then quantified through the Spearman grade correlation coefficient (ρs). Said Rp.iand Rb,ithe ithranks of pM K and b respectively, the following expression
gener-ally applies: ρs = 12 n (n + 1) (n − 1) n X i=1 Rp,iRb,i− 3 n + 1 n − 1. (5.8)
ρswas selected because it has the advantage of being always defined, unlike other commonly
employed correlation coefficients, such as the classical Pearson coefficient, which directly de-pends on the second-order moments of the variables of interest, that is not always guaranteed (De Michele et al., 2007). The values of ρs span from -1 (for series perfectly anti-correlated)
to 1 (series perfectly correlated); ρs equal to 0 indicates that no correlation exists between the
investigated series. Correlations were iteratively evaluated by varying the significance level α within the 0-1 range with a 0.01 incremental step. For every iteration, only the series with pM K < α were retained for the analysis, i.e. just the series allowing to reject the MK null
hypothesis according to the binary use of pM K. When α equals 1, no data are excluded and all
the hindcast locations are taken into account; indeed, the maximum value that pM K can attain
is exactly 1, therefore in the latter case no filtering on the series due to the value of α is applied. The second step of the developed methodology, requires to check whether the values of b are consistent with the δi obtained by the ITA, referred to the respective series (i.e. the Hs annual
statistics of the hindcast locations).
The reliability of the linear trend hypothesis was first evaluated by analysing the empirical cumulative distribution function (ecdf) of δi series for four hindcast locations characterized by
different values of b. This allows to check rapidly if the δi series increase or decrease according
to the values of b the ecdf is linked to, meaning that the larger is b, the larger are the δi. However,
number of available datasets. Therefore, the sum of the δifor each Hstime series was computed,
thus using this sum as a single parameter for the analysis instead of n/2 data. This allowed to perform a direct comparison between datasets of equal length (e.g. the number of hindcast locations): one containing the values of b and the other with the sum of δi (for each of the
annual statistics taken into account).
5.3. Results
First, the correlations between b and pM K for the AM Hs are shown in Figure 5.2. For
the sake of clarity, only results related to four levels of α are here reported (0.01, 0.05, 0.90, and 0.95). The panels show as well the values of ρs computed for the AM Hs series showing
pM K ≤ α. 0 0.2 0.4 0.6 0.8 1 R p 0 0.2 0.4 0.6 0.8 1 Rb : -0.45 a) 0 0.2 0.4 0.6 0.8 1 R p 0 0.2 0.4 0.6 0.8 1 Rb : -0.53 b) 0 0.2 0.4 0.6 0.8 1 R p 0 0.2 0.4 0.6 0.8 1 Rb : -0.92 c) 0 0.2 0.4 0.6 0.8 1 R p 0 0.2 0.4 0.6 0.8 1 Rb : -0.93 d)
Figure 5.2: Correlations between b and pM K due to different values of α. Panel a): α=0.05; panel b): α=0.1; panel c): α=0.9; panel d): α=0.95
If only the series characterized by pM K ≤ α , α = 0.05|0.1 are analysed (Figure 5.2, panels
a) and b), no appreciable correlation between pM K and b can be detected. In fact, the scatters
of Rb and Rp are almost randomly distributed over the unit-square space, and the values of the
respective ρs are far from -1, which indicates a perfect anti-correlation. On the other hand,
analyzing the correlation for pM K retained considering higher values of α (0.9|0.95, Figure 5.2
panels c) and d)), there is a striking anti-correlation between the two investigated parameters. As shown by the distributions of Rpand Rbin panels c) and d) of Figure 5.2, low values of pM K
are most likely to occur when b attains high values and vice-versa: the scatters of the scaled ranks are homogeneously distributed along the -1 bisector, and ρs reaches values close to -1.
As previously mentioned, the full range of α was explored; Figure 5.3 shows the results of ρs
as a function of α. For the sake of clarity, hereinafter values of b related to the AM, annual 98th percentile and annual mean Hsare referred to as bAM, b98and bM EAN, respectively. In panel a)
of Figure 5.3 it can be noticed how the anti-correlation between pM K and bAMbecomes stronger
(i.e. ρs tends to -1) proportionally to the level of α taken into account. Similar outcomes were
found for b98and bM EAN; in these cases, results of ρsseries are shown in Figure 5.3, panels b)
and c). 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -1 -0.5 0 0.5 1 s b AM a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -1 -0.5 0 0.5 1 s b 98 b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -1 -0.5 0 0.5 1 s b MEAN c)
Figure 5.3: ρs evaluated between b and pM K for different values of α. Panel a): AM data; panel b): annual 98thpercentile of Hs; panel c): annual mean Hs
Then, the series of b were further compared with the ITA results. To this end, four AM Hs
se-ries characterized by either very intense trends were considered (Point 001337 and Point 005995, upward and downward trend respectively), and by almost flat trends (Point 013330 and Point 021272), according to the values of the respective bAM, analyzing at a second time the ecdf of the
respec-tive δi.
The locations of the hindcast points taken into account are shown in Figure 5.4, while the AM Hsseries of the selected locations are shown in Figure 5.5.
Lon Lat 001337 013330 005995 021272 36oE 0o 9oE 18oE 27oE 28oN 32oN 36oN 40oN 44oN 48oN
Figure 5.4: Locations of the hindcast points employed for the graphical comparison between b and δi
1980 1985 1990 1995 2000 2005 2010 2015 time (years) 0 2 4 6 8 10 12 Hs [m] a) Point_001337 b 3cm/yr 1980 1985 1990 1995 2000 2005 2010 2015 time (years) 0 2 4 6 8 10 12 Hs [m] b) Point_005995 b -3cm/yr 1980 1985 1990 1995 2000 2005 2010 2015 time (years) 0 2 4 6 8 10 12 Hs [m] c) Point_013330 b 1mm/yr 1980 1985 1990 1995 2000 2005 2010 2015 time (years) 0 2 4 6 8 10 12 Hs [m] d) Point_021272 b -1mm/yr
Figure 5.5: Panels a) and c): AM Hsseries with respective TS slopes for upward trends. Panels b) and d): downward trends. Red markers: AM Hs characterized by positive trends; blue markers: AM Hs characterized by negative trends (original time series are not shown for the sake of clarity)
As Figure 5.6 shows, the series with almost flat trends (e.g. b close to zero, panels c)-d) of Figure 5.5) show δiwith an approximately vertical profile in the ecdf space; on the other hand,
series related to steeper trends (panels a)-b) of Figure 5.5) are characterized by δi more shifted
with respect to the 0 line. This analysis reveals how the higher (lower) values of δi are in turn
linked to the higher (lower) values of b. This applies both for upward (positive slopes, right half of Figure 5.6) and downward (negative slopes, left half of Figure 5.6) trends.
-1.5 -1 -0.5 0 0.5 1 1.5 i 0 0.2 0.4 0.6 0.8 1 ecdf 001337 013330 005995 021272
Figure 5.6: δiecdf for the locations characterized by different trend intensities for AM Hsseries shown in Figure 5.5
At a second time, ρs was computed using the sum of δi and b for all the populations
con-sidered (the subscripts AM , 98 and M EAN are used for δi with the same meaning as for b).
Results are shown in Table 5.1, from which it appears that bAM and bM EAN have similar ρs,
while the correlation is slightly lower between b98and the respective δi.
P δiAM − bAM P δi98− b98 P δiM EAN − bM EAN
ρs 0.79 0.67 0.85
Table 5.1: Values of ρscomputed between the series of sums of δi and b for the annual statistics taken into account
5.3.1. Practical implications of trends in wave height series
In the previous Section, it has been demonstrated how series characterized by different val-ues of b result in intense trends also according to the other tests adopted. Even thought account-ing for a linear trend is the simplest way to explain the time variability of a series of data, this assumption has shown to be reasonable. Therefore, it is interesting to evaluate how the EVA for extreme Hs may change under a linear trend, taken as granted the reliability of such
assump-tion. In fact, the hypothesis of linear trend is often adopted in the framework of Non-stationary Extreme Value Analysis (NEVA): a pragmatic approach to embed the intra-period trend in the
NEVA, is that of modeling the distribution parameters as functions of time, and in particular by linearly varying the location parameter in the Generalized Extreme Value (GEV) and/or the Generalized Pareto distributions (Coles et al., 2001). This approach was applied to several en-vironmental data, such as ocean waves (M´endez et al., 2006; Vanem, 2015), air temperature (Wang et al., 2013), residual water level and river discharge (Mentaschi et al., 2016), droughts (Burke et al., 2010) etc.
Here, the series used as a reference for the ITA analysis were employed to compute the long-term distribution of Hsthrough a stationary model (see Equation (2.9)) and a non-stationary one
(Equation (2.15)). In this case, the parameters of the distributions were computed through the MLE method. The results of non-stationary analysis are shown in Table 5.2.
Point b [m/yr] µ1 [m/yr] σµ1 µL[cm/yr] µU [cm/yr] Hst100[m] Hnst100[m]
005995 -0.03 -0.04 0.017 -0.057 -0.023 9.36 7.95
021272 -0.001 -0.007 0.012 -0.019 0.005 8.06 7.94
013330 0.001 0.002 0.01 -0.008 0.012 8.22 8.26
001337 0.03 0.03 0.01 0.02 0.04 10.78 11.19
Table 5.2: Parameters of non-stationary EVA for the locations of Figure 5.4
where µ1is the fourth parameter in the time-dependent GEV (cfr. Equation (2.15)) and σµ1
is the standard error resulting from fitting the parameters to the data, which leads to the lower (µL) and upper (µU) values of µ1. Hst100 and Hnst100 are the 100 year wave heights computed
according to the stationary and non-stationary GEV, respectively.
It can be noticed how the series characterized by high values of b show in turn high values of µ1, both in case of upward (Point 001337) and downward (Point 005995) trend. On the
contrary, series characterized by mild b slopes, show negligible µ1 as well (Point 021272 and
Point 013330). Furthermore, in the latter case, the uncertainty on the µ1 estimate (σµ1), is one
order of magnitude higher than the parameter itself, resulting in lower and upper values of µ1
that indicate opposite trends. Consequently, in this case it is not useful to consider a time-varying location, and this is reflecting on long term Hs. Indeed, the 100 years Hs does not
significantly vary among the two approaches, i.e. Points 021272 and 013330 show divergences of 12 cm and 4 cm, respectively, as shown in Figure 5.7.
Figure 5.7: Comparison between Hst100 and Hnst100 for AM Hsseries. Left panel: Point 021272; right
panel: Point 013330. The black dashed line indicates the µ1slope
On the other hand, as far as Points 005995 and 001337 are concerned, much more relevant differences can be pointed out. In the former case, the non-stationary EVA leads to Hst100 which
is 1.4 m lower than Hnst100; in the latter case, i.e. when a positive trend is expected, the 100
years Hsis increased instead of approximately 40 cm (cfr. Figure 5.8).
Figure 5.8: Comparison between Hst100 and Hnst100 for AM Hsseries. Left panel: Point 005995; right
panel: Point 001337. The black dashed line indicates the µ1slope
It is worth to mention that, in the examples above the choice of referring to a GEV distri-bution is due to the selection of the data, which indeed are AM. Nonetheless, the same analysis could be performed on POT data by employing different time-dependent model without any loss of generality, under the hypothesis that a long term trend exists. However, when refer-ence is made to the peak series introduced in the previous Chapter (B4 and B7), no evidrefer-ence of significant trends can be appreciated.
In this case, since POT data are analyzed, trend analysis can be performed both over the Hs
peaks and the yearly number of events detected (referred to as λi). Results are reported in Table
5.3 and 5.4. WP#1 location variable pM K b B4 Hs 0.14 -0.004 λi 0.50 0 B7 Hs 0.31 -0.001 λi 0.13 0
Table 5.3: Results of trend analysis for the peaks series belonging to WP#1
WP#2 location variable pM K b B4 Hs 0.09 0.0006 λi 0.80 0 B7 Hs 0.24 -0.0003 λi 0.17 0.048
Table 5.4: Results of trend analysis for the peaks series belonging to WP#2
From the trend tests, it can be appreciated how for both the locations and the WPs consid-ered, pM K attains high values for Hs and λi, meaning that the null hypothesis can be accepted
with a high level of confidence. The only exception is represented by the Hs series of WP#2
selected in B4, though they still would pass the test at the 5% level of confidence. Accordingly, b show negligible values for both the variables investigated; in this case, it has to be mentioned that for POT data the TS slope for Hs explains the rate of change between two successive
storms, while for λi it refers to the variations in the yearly number of storms.
5.3.2. Wave climate trends in the Mediterranean Sea
First, it is interesting to analyse the spatial distribution of Hstrends when the most common
usage of the MK, relying on the threshold α=0.05, is applied. Figure 5.9 shows only the loca-tions characterized by trends according to the aforementioned method for the AM data, annual 98thpercentile, and annual mean Hs(panels a), b) and c) of Figure 5.9). The sign of the trends
is computed using Eq. (5.6). The AM Hs results show large areas characterized by negative
trends in the south Tyrrhenian Sea and in the Ionian Sea, while smaller areas and isolated spots characterized by positive trends are present, for instance, in south-east of the Aegean Sea and in the northernmost areas of the MS. The results for the annual 98thpercentile of H
sshow
pos-itive trends limited to the south of the MS between Sicily and Libya, while negative trends are limited to very few locations. Results for the annual mean Hsshow negative trends limited to
the south-east basin of the MS and positive trends limited to small spots in front of the Libyan coast and along the coastlines of Italy and Greece.
Lon Lat AM H s a) 36oE 0o 9oE 18oE 27oE 28oN 32oN 36oN 40oN 44oN 48oN Lon Lat 98th H s b) 36oE 0o 9oE 18oE 27oE 28oN 32oN 36oN 40oN 44oN 48oN Lon Lat mean H s c) 36oE 0o 9oE 18oE 27oE 28oN 32oN 36oN 40oN 44oN 48oN
Figure 5.9: Locations characterized by MK trends for α equal 0.05. Panel a): AM Hs; panel b): annual 98thpercentile of Hs; panel c): annual mean Hs. Red dots indicate positive trends, blue dots indicate negative trends
On the other hand, Figure 5.10 shows the values of b computed for the annual statistics of Hs
over the MS, for both downward and upward trends. It can be can seen that the most significant b for the AM Hsare between -5cm/year and 3cm/year. The areas subjected to the most intense
negative trends are the south of the Tyrrhenian Sea (in front of the northern coasts of Calabria and Sicily) and the Ionian Sea, opposite the Greek coasts. On the other hand, the Aegean Sea and the Tyrrhenian Sea (on east Corsica and Sardinia), together with areas spread within the Balearic Sea, show wide areas subject to positive trends of the AM Hs. Results for the mean
and the 98th percentile of H
s change dramatically with respect to those of AM Hs, with more
softened intensities (for the mean values these are in the order of mm/year) and with a different spatial trends distribution. For both the latter statistics, negative trend areas can be appreciated in the south-east of the MS and in the north Tyrrhenian Sea, while positive trends are found in the west of Sardinia and in the area between Libya and the Ionian Sea.
To the best of the Author’s knowledge, it is the first time that an analysis of wave climate trends is performed on the whole MS with such a resolution. Therefore, comparison with pre-vious results can be carried out only considering local analysis in the literature. Casas-Prat and Sierra Pedrico (2010) and Casas Prat and Sierra Pedrico (2010) evaluated trends for different di-rectional sectors along the Catalan coast; Piscopia et al. (2003) carried out a study on the Italian seas. However, these works computed trends on sea storms extracted by de-clustering threshold exceedance within the POT approach, thus a direct comparison with the present analysis would be not significant. Pomaro et al. (2017) evaluated trends on several monthly percentiles of Hs
in the northern Adriatic Sea, showing a reduction in extremes and an increase in storminess that seems not fully consistent with the results of Figure 5.10, where b98 is positive but no trend
in AM Hsis found. Statistics based on annual intervals in eastern Mediterranean were carried
out by Musi´c and Nickovi´c (2008), further employing a simple linear regression for computing trends; their analysis of annual mean Hs returned negative trends of order of magnitude
con-sistent with the present work, however, in their research, no positive trend is identified. As far their AM Hs analysis, local analysis within the Aegean Sea agrees well with the outcomes of
the present work. On the other hand, Musi´c and Nickovi´c (2008) showed a slightly negative trend in front of the coast of Lebanon, while the present analysis suggests an area subject to homogeneous positive trends. On the contrary, the trend for 90thpercentile in the same area is positive and apparently more consistent with the outcomes of the present work for the annual 98thpercentile of H
Lon Lat
a)
36oE 48oN 0o 9oE 18oE 27oE 28oN 32oN 36oN 40oN 44oN -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 b AM [m/yr] Lon Latb)
36oE 48oN 0o 9oE 18oE 27oE 28oN 32oN 36oN 40oN 44oN -0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 b 98 [m/yr] Lon Latc)
36oE 48oN 0o 9oE 18oE 27oE 28oN 32oN 36oN 40oN 44oN -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 b MEAN [m/yr] 10-3Figure 5.10: Spatial distribution of b in the MS. From top to bottom: Panel a): bAM; panel b): b98; panel c): bM EAN
5.4. Discussion & Conclusions
In most of the studies that use the MK, pM K is evaluated against a threshold value of α to
check for the presence of a trend. For the typical values of α used, the values of b seem not to be significantly related to the pM K they refer to. Therefore, no useful considerations can be
inferred for b, regardless the assumptions made about the use of pM K. On the contrary, when
pM K is considered in its whole range, a clear anti-correlation with b can be appreciated. In this
case, it follows that the magnitude of b can be retained to evaluate how strong is the increas-ing/decreasing trend of the dataset under study. Indeed, the MK null hypothesis is the absence of a trend in a dataset. Close-to-0 values of pM K mean that the data behave consistently with
the presence of a marked trend (i.e. the null hypothesis is rejected), and this is more likely to occur for high values of b, as shown in Figure 5.2 for the AM Hs (similar considerations
hold for annual 98th percentile of Hs and annual mean Hs). Furthermore, b was proved to be
correlated with the ITA outcomes. In fact, both the graphical analysis of the δi ecdf for the
selected locations, and the correlation analysis of the sum of δi for all the hindcast locations,
reveal a strong correlation of δi itself and b, in particular for the annual mean and maxima Hs
(as shown in Table 5.1). It follows that the δi identified by the ITA graphical method is in turn
correlated to the pM K. Therefore, this paper allows to use the information provided by b to
quantify trends, because of the correlation with pM K, and it provides theoretical support to the
ITA. The application here considered shows that the general use of pM K, as recommended in
Greenland et al. (2016), expands, with respect to the more usual usage of the MK, the knowl-edge of possible trends attaching to each value of b a measure of the consistency with the null hypothesis, without any a priori selection based on a threshold.
The estimates of b have also been proved to be consistent with the outcomes of non-stationary EVA. Long term Hs of series characterized by high values of the TS slope considerably vary
depending on the parent distribution, i.e. if a non-stationary rather than a stationary model is taken into account. Therefore, information provided by b and pM K may also be employed to
assess whether stationary or non-stationary EVA have to be employed. As for the POT data, the peak series analyzed for the WPs analysis presented in Chapter 4 did not show evidences of long term trends. On the contrary, the examples reported in Section 5.3. are worthy of attention. In this case, the result of Point 001337 showed lower deviations between the two approaches with respect to results of Point 005995. As a matter of fact, the computation of µ1 follows different
schemes than that of b, and equal values of the latter parameter do not necessarily correspond to equal values of the latter one.
Hence, the values of b were used to gain an insight into the spatial distribution of wave cli-mate trends over the MS. The analysis revealed similar patterns among the spatial distributions of trends for the annual 98th percentile of Hs and annual mean Hs, while trends of AM Hs
are differently spread over the MS, moreover they are characterized by more intense values of b (order of cm/year). These outcomes were then compared with previous researches aimed at detecting and computing trends over isolated spots in the MS. The order of magnitude of the annual rate of changes show good consistency with the values of b computed in this work, while there are slight deviations in the sign of trends for some locations.
Finally, it is worth mentioning that, although the paper focuses on sea states, the analysis here introduced can be extended to other parameters without loss of generality, and the applica-tion of this methodology to different geophysical time series is therefore straightforward.
This thesis introduces new methodologies for the analysis of extreme sea states, showing practical examples focused on the Mediterranean Sea. The data employed come from the hind-cast service of the Department of Civil, Chemical and Environmental Engineering of the Uni-versity of Genoa. Each hindcast points contain forty years of hourly-defined wave parameters; therefore, being continuously defined over a long period, these data represent appropriate ref-erences for reducing the uncertainty of the methodologies proposed. Nevertheless, implemen-tation and running of numerical models are not always feasible: third-generation models for the generation and propagation of sea waves require massive efforts for their setting calibration, and the availability of relevant computational power. On the other hand, direct measurements may provide more reliable wave data, but they are characterized by a trade-off: installation costs of wave-sampling devices are often not negligible, moreover the maintenance of instru-ments placed into the sea never comes with ease. As such, in addition to the devices commonly employed, in the past years the use of HF-radars has started to be investigated. Indeed, these instruments have the advantage of being installed on the ground, and they allow to measures velocity currents and wave parameters at the same time. However, the signal sampled by radars is known to efficiently detect the current velocity, while it needs non trivial post-processing to carry out the wave parameters. In view of the above, the use of HF-radars for wave measure-ments need to be further investigated. In this framework, Chapter 3 shows a study comparing HF-radar wave measurements and numerical simulations in the Gulf of Naples (Southeastern Tyrrhenian Sea). First, numerical simulations were validated against experimental data acquired by a buoy installed offshore the gulf; such buoy is not co-located with respect to the HF-radars, therefore it could not be employed for a direct comparison with the latter. Subsequently, the agreement between HF-radar measurements and model hindcasts was evaluated through the estimate of statistical error indexes for the main wave characteristics (significant wave height, mean period and mean direction). Results showed reasonable consistency between wave param-eters retrieved by HF-radars and hindcasted by the models, opening the way to future integration of the two systems as well, as to the potential utilization of HF-radar wave parameters that could be envisaged for data assimilation in wave models.
The availability of continuous dataset is crucial for developing analysis of extremes. As a matter of fact, datasets defined over short periods and/or characterized by lack of data might significantly increase the uncertainty of statistical analysis, undermining the reliability of the following outcomes. To this end, this thesis refer to the aforementioned hindcast data, in order to characterize in detail extreme significant wave heights occurring in several location along the Italian coastline. Chapter 4 reports the application of a new methodology to perform Extreme Value Analysis (EVA), by first grouping a series of extreme Hsdue to the climatic forcing most
likely generating them. Several physical quantities may concur in driving intense waves, e.g. variation in the mean sea level pressure, tidal excursions, soil vibrations, long fetches allowing swells to travel for long distances without dissipating energy. Nonetheless, this study referred to the spatial distribution of wind fields. In fact, the Mediterranean Sea is an enclosed basin with lots of island affecting the waves propagation, thus the wind is reasonably expected to be
the main forcing driving the local wave climates. The study took advantage of eight hindcast points, differently located along the coasts of Italy in order to cover different wave climates. For each study location, the extreme significant wave heights were carried out starting form the whole time series of data through a Peak Over Threshold approach. Then, they were grouped in different clusters through a k-means algorithm, fed with the wind velocities occurring some hours before the wave peaks over the whole Mediterranean Sea. The time and space domains for wind fields were selected analyzing the correlations maps between local wave heights and wind velocities of all the hindcast locations for several time lags. Subsequently, the mean sea level pressure fields for the event corresponding to each cluster were averaged, allowing to char-acterize the weather patterns corresponding to each cluster identified at a synoptic scale. Two well-known cyclonic systems were detected as those possibly driving the extreme wave con-ditions for the investigated locations: one characterized by a low moving NW-SE toward the Balkan area, and the other forming in the lee of the Atlas mountain and traveling north-easterly. Such systems were analyzed in the context of the Mediterranean sea atmospheric climatology, and were shown to be in line with those highlighted by previous studies developed on the same topic. Lastly, as for the EVA, the methodology implies to refer to a Monte Carlo simulation, in order to compute the overall distribution for the significant wave height-high return period quantiles, starting from the distributions fitted to each subset. Indeed, different families of extremes were identified as belonging to different weather patterns, each being characterized by a particular frequency of occurrence and waves incoming direction. However, the advan-tage of the proposed methodology is that it allows to differentiate also peaks resulting in the same direction of propagation, thus overcoming the limitations possibly arising when a direc-tional classification is taken into account. Hence, EVA were performed independently on single subsets through a Generalized Pareto Distribution; the set of parameters related to the overall distribution were built at a second time due to a bootstrap of the single sets. It was shown how the high return period quantiles of significant wave height match those resulting from the usual EVA. In conclusion, the proposed methodology was proved to be capable of identifying clearly differentiated subsets driven by homogeneous atmospheric processes, and ultimately allowing the proper computation of the significant wave height long-term curves.
The methodology introduced in Chapter 4 allows to perform independent EVA over the sub-sets of a series of extreme data, computing at a second time the overall long term distribution for the variable under investigation. The Monte Carlo simulation scheme holds regardless the particular model applied to each subset (i.e. data selection and chosen distribution), with the only requirement this is consistent among the subsets. The application shown relied on the stationary Generalized Pareto Distribution, implying that the referring extremes were expected not to significantly change in the long term. Literally, the parameters of the distribution used do not depend on time. Therefore, in this case variations of the data are just due to the natural variability of the parent distribution (i.e. population), and not to climatic trends inducing in-creases or decrements in the data. Such assumption should always be properly tested: to this end, Chapter 5 analyzes the use of linear regressions for detecting and quantifying long-term trends in datasets. In particular, it was evaluated the reliability of a linear trend slope estimate, modified in order to filter possible outliers. This slope was compared against the outcomes of two methods used for trend detection: the Mann-Kendall test and the Innovative Trend Analysis. The former is a non-parametric test that refer to the samples rank correlation within a dataset;
the latter implies a graphical analysis of the datataset, split in two halves according to the time frame over which they were observed. It is important to point out that both these methods do not imply the hypothesis of linear trends. An application to significant wave height time series over the Mediterranean Sea was presented. Time series of the hindcast data were analyzed, and the methodology presented was applied to the respective annual maxima, the annual 98th
percentile and the annual mean significant wave height. The outcomes of the three methods were compared for all the resulting series, and results proved that the use of the investigated linear slope is meaningful and sound. Therefore, such parameter could be also employed to evaluate for the need of non-stationary EVA, which result in high return period significant wave heights considerably different to those related to the stationary model if intense trends in the data are present. Finally, the linear slopes of the annual statistics were used to assess the spatial distribution of trends in the Mediterranean Sea. In general, trends characterizing extreme waves are more intense with respect to those shown by the milder ones, and the spatial distribution of historical trends varies among the annual statistics investigated. The order of magnitude of the rate of changes for both the extremes and the mean waves were shown to be consistent with pre-vious works conducted for punctual locations, while some divergences were pointed out on the trends’ sign, in particular in the Adriatic Sea and in the south-east basin of the Mediterranean Sea. However, interesting analogies were highlighted, and leave room for further investiga-tions, e.g. understanding what is driving the trends in wave height, and evaluating how these may affect the design and costs of coastal and off-shore structures.
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