1490
Forecasting seasonal to interannual variability in extreme
sea levels
Melisa Menendez, Fernando J. Mendez, and Inigo J. Losada
Menendez, M., Mendez, F. J., and Losada, I. J. 2009. Forecasting seasonal to interannual variability in extreme sea levels. – ICES Journal of Marine
Science, 66: 1490 – 1496.
A statistical model to predict the probability of certain extreme sea levels occurring is presented. The model uses a time-dependent
generalized extreme-value (GEV) distribution to fit monthly maxima series, and it is applied for a particular time-series record for the
Atlantic Ocean (Newlyn, UK). The model permits the effects of seasonality, interannual variability, and secular trends to be identified
and estimated in the probability distribution of extreme sea levels. These factors are parameterized as temporal functions (linear, quadratic, exponential, and periodic functions) or covariates (for instance, the North Atlantic Oscillation index), which automatically yield
the best-fit model for the variability present in the data. A clear pattern of within-year variability and significant effects resulting from
astronomical modulations (the nodal cycle and perigean tides) are detected. Modelling different time-scales helps to gain a better
understanding of recent secular trends regarding extreme climate events, and it allows predictions to be made (for example, up to
2020) about the probability of the future occurrence of a particular sea level.
Keywords: climate variability, generalized extreme-value model, high-water levels, nodal cycle, perigean influence, seasonality, statistical model.
Received 15 August 2008; accepted 1 March 2009; advance access publication 16 April 2009.
M. Menendez, F. J. Mendez and I. J. Losada: Environmental Hydraulics Institute “IH Cantabria”, Escuela de Caminos, Canales y Puertos, Universidad
de Cantabria, Avda de los Castros s/n 39005, Santander, Spain. Correspondence to M. Menendez: tel: þ34 942 201810; fax: þ34 942 201860;
e-mail: [email protected].
Introduction
Sea level is a key variable in marine, climate, and coastal processes
studies. Sea level studies focus mainly on temporal changes in
mean sea level because of global warming (Church et al., 2001).
Few studies have concentrated on likely changes in extreme sea
level events.
Investigations of coastal flooding and design of marine facilities
require estimates of the highest water levels at a site. These estimates are usually obtained using the definition of the statistical
distribution of extreme sea levels. A good characterization of the
extreme sea level distribution consequently permits a more accurate evaluation of risks and hazards. Moreover, the prediction of
the probability of exceeding a given extreme sea level considering
certain climate variability is useful information for coastal planners to assist with the evaluation of risk variation over time
(Marbaix and Nicholls, 2007).
After averaging out wind surface waves, sea level is composed of
three components: mean sea level, tidal level, and zero mean surge
level (Pugh, 1987). Extreme sea levels usually correspond to a joint
event of storm surge and a high astronomical tide. Two statistical
techniques are available to estimate probabilities of extreme sea
levels, namely the direct approach, estimating the extreme value
long-term distribution from observed maximum values (Mendez
et al., 2007), and the indirect approach, using a joint tide-surge
probability estimation, which separates the deterministic origin
of the tidal record from the stochastic behaviour of the surge
residual (see details of the revised joint probability method,
RJPM, in Tawn and Vassie, 1989). In this study, we used the
direct approach. However, most of the methodology developed
could also be applied to the RJPM.
Two different data sources are available to obtain sea level
variability: tide gauge records and satellite-altimeter data. Several
studies have investigated sea level variability with altimeter data
that provide large-scale spatial information (Woolf et al., 2003).
However, these datasets are relatively short (only starting in
1987), making it difficult to analyse extreme values and lowfrequency climate variability in combination. Consequently, long
sequences from high-quality tide gauge records are better suited
for analyses of extreme sea level event, because the larger sample
of maxima facilitates a more effective estimation of the likelihood
of extreme sea level events.
A sea level –climate variability analysis can indicate a nonstationary behaviour at different time-scales (seasonal, interannual, decadal, and secular). Variation within a given year
(intra-annual) is manifested in three sea level components
(Pugh, 2004): monthly mean sea levels (Tsimplis and
Woodworth, 1994), surge levels (Zhang et al., 2000), and the
highest tide levels (Zetler and Flick, 1985; Dixon and Tawn,
1999). Interannual fluctuations and decadal variability can also
be discerned in extreme sea level behaviour. This variability can
result from the atmospheric-ocean circulation [expressed as teleconnection indices, such as the North Atlantic Oscillation
(NAO) index] and astronomical forcing, such as the nodal cycle
(Sobey, 2004), which emphasizes the important contribution of
high astronomical tides and storm surges to extreme sea-level
events. With respect to secular trends, an increase in mean sea
# 2009 International Council for the Exploration of the Sea. Published by Oxford Journals. All rights reserved.
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Forecasting seasonal to interannual variability in extreme sea level
level is one of the principal contributors to the occurrence of
extreme sea levels (Woodworth et al., 1999; Flick et al., 2003).
Moreover, there is global evidence of long-term changes in the
intensity and frequency of storms (Meehl et al., 2000;
Woodworth and Blackman, 2004; Méndez et al., 2007). This
leads to a legitimate cause for concern, because changes in such
extreme events can increase the risk of flooding in coastal areas.
This study focused on an analysis of extreme sea levels at different time-scales, using a non-stationary extreme-value model. We
first had to estimate seasonal to interannual variability and
trends in extreme sea levels. Results are presented based on an
analysis of a long-term hourly tidal-gauge record for Newlyn,
UK. In addition, we were able to predict the probability of exceeding a given sea level in the future, based on the methodology that
we applied.
Data and methods
Data
The Newlyn hourly sea level tide gauge was used as the basis for
this analysis. These data (#170/161) were obtained from the
British Oceanographic Data Centre (BODC) and they covered
the period 1915–2006 (92 years). This station is recognized as
the best available tide record in the UK (Woodworth et al.,
1999; Araujo and Pugh, 2008). The data for the period 1915–
2001 were used to fit the model, and the period 2002–2006 was
used to evaluate the ability to forecast future sea level events.
Monthly maxima and monthly mean water levels are illustrated
in Figure 1. Annual maxima are indicated with a circle. The timeseries reveals an important positive trend (increase in mean sea
level), as well as possible changes in the variance of the data,
although some information is likely hidden, because of visual
scale effects. Some descriptive statistics are also given to allow a
more detailed examination of the different time-scales involved.
All monthly maxima were plotted against the day of year (1 –
366) in Figure 2. This reveals a clear intra-annual seasonal variability. Two separate peaks are evident (March and September); these
correspond to the equinoctial spring tides. However, greater variability is evident during winter. The clear seasonal pattern (with
Figure 1. Monthly mean (solid line), monthly (dots) and annual
(circles) maxima series for the Newlyn tide gauge. Elevations are
relative to Chart Datum.
Figure 2. Monthly maximum sea level for Newlyn, UK.
likely annual and semi-annual cycles) suggested the need to fit a
seasonal model to the data.
With respect to interannual variability, extreme sea levels are
influenced by the NAO in the study area (Woodworth et al.,
2007; Araujo and Pugh, 2008), and the interannual modulations
of the astronomical component might have an influence on
extremes (Woodworth and Blackman, 2004). The highest percentile time-series reduced to median sea level height at Newlyn
(Figure 3) illustrate an important role of the 18.61-year nodal
cycle and the 4.4-year perigean tides in extreme sea level events.
Extreme-value model
The first step in the model was to apply a generalized extremevalue (GEV) distribution for block maxima (monthly maxima),
j
given by F(z;u) = expf2[1 + j (z2 m)/c]21/
g, where u = (m, c,
+
j) is the vector parameter and [x]+ = max (0,x). The location
parameter is m, which specifies where the distribution is
centred, c . 0 is a scale parameter, which represents the dispersion, and j is a shape parameter, which determines the shape
of the upper tail of the distribution. Negative j values indicate
bounded tails (or Weibull distributions), null values indicate a
Gumbel distribution (an unbounded tail that decreases exponentially), and positive j values indicate heavy-tailed (or Fréchet)
distributions.
The model requires that block maxima of successive months
must be independent and identically distributed random variables.
The independence between consecutive months is achieved by
imposing a constraint that monthly maxima must be separated
at least by a duration t (following Mendez et al., 2007, we use t
= 48 h). However, an assumption of homogeneity of the random
variable is clearly not met. A visual inspection of the data
(Figures 1 –3) permitted us to identify the different time-scales
and the temporal dependence between consecutive monthly
maxima, which confirms that data did not follow identical
distributions.
To address the temporal variability in extreme values, we used
an extension of the standard models of extreme-value theory for
non-stationary variables (see Chapter 6 of Coles, 2001). Monthly
maxima of successive months were assumed independent
random variables, but the assumption of homogeneity throughout
consecutive months is not required (because they are not presumed homogeneously distributed). We assumed that monthly
maximum sea levels Zt observed in month t followed a GEV distribution over time, with a time-dependent location parameter m(t),
scale parameter c(t). 0, and shape parameter j (t). The
1492
M. Menendez et al.
Figure 3. Percentile time-series for Newlyn, UK (with each percentile time-series reduced to the median).
cumulative distribution function of Zi was then given by
h
8
i1=jðtÞ >
< exp 1 þ jðtÞ zcmðtÞðtÞ
jðtÞ = 0
þ
Ft ðz; uÞ ¼
:
n
h
i
o
>
:
exp exp zcmðtÞðtÞ
jðtÞ ¼ 0
ð1Þ
The probability density function (PDF) of Zi was derived by differentiating Equation (1) with respect to z, so that
ft ðz; uÞ ¼ dFt ðz; uÞ=dz. In the next section, we represent m(t),
c(t), and j (t) as functions of time (linear, quadratic, exponential,
and cosine functions) or covariates (e.g. the NAO index). For
illustrative purposes, Figure 4 shows the GEV PDF for the stationary case (Figure 4a) and a time-dependent GEV PDF considering
variation within a year (Figure 4b), or for longer time-scales
(Figure 4c). Note how the variability affects the shape of the
PDF and, consequently, the probability of occurrence of a
certain sea level value.
For non-stationary or time-dependent GEV parameters, the
time-dependent return level quantiles zR(t, u) associated with
the return period R (in years) were calculated using
zR ðt; uÞ ¼ zR ðmðtÞ; cðtÞ; jðtÞÞ
ðtÞ mðtÞ cjðtÞ
1 f logð1 1=RÞgjðtÞ jðtÞ = 0
¼
:
mðtÞ cðtÞ logf logð1 1=RÞg jðtÞ ¼ 0
ð2Þ
The estimate of the return sea level value z̄R(ti) for a certain year ti
was obtained by solving
ð ti þT
1
jðtÞðz R mðtÞÞ 1=jðtÞ
1 ¼ exp ½1 þ
dt ;
þ
R
cðtÞ
ti
ð3Þ
where T is equal to 1 year. Confidence intervals were obtained with
the delta method (Rice, 1994), assuming approximate normality
for the maximum likelihood estimators.
Incorporation of the variability into the model
Different factors were investigated to explain the non-stationary
behaviour in the distribution of sea level maxima: the mean sea
level rise, the annual and semi-annual cycles, the influence of
NAO, the perigean tides, the nodal cycle, and a possible secular
trend. All these factors could be parameterized to interpret
the variability in the data. In the following, we used subscripts
Figure 4. (a) GEV PDF for the stationary case, and (b)
time-dependent intra-annual and (c) interannual variations
(perigean and nodal modulations).
SLR for sea level rise, S for seasonal cycles, NAO for sensitivity
of maximum sea levels to NAO, P for perigean influence, N for
nodal cycle, and LT for long-term trends.
1493
Forecasting seasonal to interannual variability in extreme sea level
We propose the following aggregations of factors for the
location, scale, and shape parameters, respectively:
best model following the principle of parsimony (see details in
Menendez et al., 2008).
mðtÞ ¼ bSLR t þ mS ðtÞ exp½bLT t þ mN ðtÞ þ mP ðtÞ þ bNAO NAOðtÞ;
cðtÞ ¼ cS ðtÞ exp½aLT t þ cN ðtÞ þ cP ðtÞ þ aNAO NAOðtÞ;
jðtÞ ¼ jS ðtÞ exp½gLT t:
ð4Þ
Results
Extreme sea level variability
After application of the automatic selection algorithm, the best
model for extreme sea levels, as measured by the Newlyn tide
gauge for the period 1915–2001, was defined with these terms:
Relative mean sea level rise was introduced as a fixed value, because
the rate of increase in the sea level at the Newlyn tidal gauge has
been well-documented (Woodworth et al., 1999), and it was introduced linearly into the location parameter as mSLR(t) = bSLRt,
where bSLR = 1.69 mm year21. The intra-annual variability
(annual and semi-annual cycles) was modelled with harmonic
functions (Katz et al., 2002; Menendez et al., 2009):
mðtÞ ¼ b0 þ bSLR t þ mS ðtÞ þ mN ðtÞ þ mP ðtÞ;
cðtÞ ¼ a0 þ cS ðtÞ þ cN ðtÞ;
j ¼ g0 :
mS ðtÞ ¼ b0 þ b1 cosðvtÞ þ b2 sinðvtÞ þ b3 cosð2vtÞ þ b4 sinð2vtÞ;
cS ðtÞ ¼ a0 þ a1 cosðvtÞ þ a2 sinðvtÞ þ a3 cosð2vtÞ þ a4 sinð2vtÞ;
jS ðtÞ ¼ g0 þ g1 cosðvtÞ þ g2 sinðvtÞ þ g3 cosð2vtÞ þ g4 sinð2vtÞ;
ð5Þ
where t is given in years, v = 2p/T, and T = 1 year. Long-term
trends over the mean sea level were computed using exponential
functions (4) with the regression parameters bLT, aLT, and gLT
for each parameter of the PDF. A possible acceleration (Church
and White, 2006) in the magnitude of extreme sea levels was
also tested using a quadratic function in the location parameter,
mLT = exp(bLT t + bLT2 t2). The modulations of the perigean influence and the nodal cycle were introduced as follows into the
location and scale parameters:
mN ðtÞ ¼ bN1 cosðvN tÞ þ bN2 sinðvN tÞ; and
cN ðtÞ ¼ aN1 cosðvN tÞ þ aN2 sinðvN tÞ;
mP ðtÞ ¼ bP1 cosðvP tÞ þ bP2 sinðvP tÞ; and
cP ðtÞ ¼ aP1 cosðvP tÞ þ aP2 sinðvP tÞ;
ð6Þ
where vN = 2 p/TN, vP = 2 p/TP, TN = 18.61 year, and TP = 4.4
year. Finally, the coefficients bNAO and aNAO represent the influence of the covariate NAO index in the magnitude and variability
of extreme sea levels, respectively.
The complete vector of p regression parameters (bi, ai, gi) was
denoted with u. We used standard likelihood theory to obtain the
model parameter estimates, u^ , and the confidence intervals
(Mendez et al., 2007). An automatic selection algorithm, based
on the Akaike Information Criterion, was used to achieve the
ð7Þ
A summary of values and standard errors of each estimated parameter for the final model is given in Table 1. The shape parameter
did not exhibit any kind of time-variation pattern, providing evidence of a bounded upper tail in the extreme distribution of
Newlyn tide levels (g0 =20.23). Consequently, intra-annual and
interannual variations dominated the variability in the magnitude
and variance of extreme sea level events. No significant long-term
trend in extremes for the mean sea level increase could be discerned, nor did the NAO appear to have exerted any significant
influence on sea levels (the model provides an 86% significance
to NAO index). This apparent weak influence agrees with the
outcome of previous studies (Woolf et al., 2003; Woodworth
et al., 2007; Araujo and Pugh, 2008), which demonstrated a low
sensitivity of sea levels to the NAO. Note that, depending on the
location of the tidal gauge, different combinations of parameters
were obtained. For example, in an analysis of the San Francisco
tidal gauge data (Mendez et al., 2007), we obtained a significant
contribution by the nodal cycle, a significant influence by SOI,
as well as an extra long-term trend associated with both the
location and scale parameters.
The contribution by each factor in the location and scale parameters is represented in Figure 5. The plots reveal the annual
and semi-annual cycles required in the location parameter, mS
(with a total amplitude of 33.8 cm), and in the scale parameter,
cS (with an amplitude of 7.3 cm). The seasonal cycles found in
the location parameter of the sea level GEV in the Newlyn
dataset defined two maxima in early March and September,
caused by equinoctial spring tides. The seasonal cycle in the
scale parameter is explained by the increase in storms during
winter. The sea level GEV distribution in the Newlyn dataset exhibits a 18.61-year nodal variation, with a strong influence not only
on the magnitude of extreme sea levels (amplitude of 6.7 cm in the
Table 1. Summary of the final results for the estimated parameters of the best time-dependent extreme-value model at the Newlyn tidal
gauge.
Variable
Seasonality
mS(t)
Nodal cycle
mN(t)
Perigean influence
mP(t)
Location parameter [m(t)]
b^ 0 ðs:e:Þ
5.62 (0.005)
b^ 1 ðs:e:Þ
0.060 (0.007)
b^ 2 ðs:e:Þ
20.052 (0.007)
b^ 3 ðs:e:Þ
20.053 (0.007)
b^ 4 ðs:e:Þ
0.107 (0.007)
b^ N1 ðs:e:Þ
20.033 (0.007)
b^ N2 ðs:e:Þ
0.007 (0.007)
b^ P2 ðs:e:Þ
0.002 (0.008)
b^ P1 ðs:e:Þ
20.031 (0.008)
cS(t)
cN(t)
Scale parameter [c(t)]
a^ 0 ðs:e:Þ
0.15 (0.004)
a^ 1 ðs:e:Þ
0.030 (0.005)
a^ 2 ðs:e:Þ
0.004 (0.004)
a^ 3 ðs:e:Þ
0.010 (0.005)
a^ 4 ðs:e:Þ
20.008 (0.006)
a^ N1 ðs:e:Þ
0.0 (0.005)
a^ N2 ðs:e:Þ
20.010 (0.004)
Shape parameter
g^ 0 ðs:e:Þ
20.23 (0.018)
The standard errors are presented in parentheses. All the regression parameters are in metres for the dimensionless shape parameter g0.
1494
M. Menendez et al.
Figure 5. Contribution of the different time-scales in the location and scale parameters for the best extreme-value model (SLR, mean sea level
rise; S, seasonality; N, nodal component; P, perigean influence).
location parameter, mN), but also in the variance (amplitude of
2 cm in the scale parameter, cN). The notable influence of the
nodal cycle is consistent with other studies of variations in high
seawater levels around the UK coast (Woodworth, 1999). The
influence of the perigean tides at Newlyn is also detected in the
magnitude of the extreme sea levels values, with amplitude of
6.1 cm in the location parameter, mP.
The accuracy of the fitted model was assessed with diagnostic
plots (probability and quantile plots, see details in Mendez et al.,
2007). The non-stationary model required modification to apply
such diagnostics. Hence, the maximum Zt was standardized using
"
Zt m^ ðtÞ
log 1 þ j^ ðtÞ
Z~ t ¼
c^ ðtÞ
j^ ðtÞ
1
!#
;
ð7Þ
to compare empirical and fitted distributions. Therefore, data
closer to the unit diagonal revealed an adequate goodness-of-fit.
A comparison of the quantile plot of the stationary model and
the best-fitted model is presented in Figure 6. A notable improvement, especially for large values of z, is evident for the best nonstationary model.
Logically, the ability of the model to extrapolate the results
should depend on the stochastic and deterministic behaviour of
the time-dependent terms. Future changes in astronomical variations (e.g. the nodal influence) are negligible on the human timescale of interest. Currently, one of the major uncertainties is the
secular changes in the mean sea level rise and a likely extra longterm trend in extreme values. Special caution must be taken
when the fitted model is used for projections. We consider the
model’s forecast accuracy not to extend beyond 2020 (say, one
nodal cycle order).
To evaluate the ability to forecast accurately, a validation of the
statistical model is required. We fitted the model for the period
1915–2001 and used the 2001–2006 period for validation.
Figure 7 demonstrates that the instantaneous 100-year quantile
followed a seasonal variation according to the monthly maxima
values. Note that for the validation period (2001–2006), the
100-year aggregated quantile predicted a decrease in the magnitude of maxima. The detected downward trend during the
period 2001–2006 was attributed to the nodal cycle contribution.
Moreover, the joint influence of an increase in the mean sea level
and the nodal and perigean modulations would produce an
increased likelihood of high sea levels in 2019.
Forecast validation
Once the significant time-dependent terms had been included in
the extreme-value model for a particular site, an estimation of
the probability of a high extreme sea level value outside the
fitted interval (extrapolation) could be obtained. This approach
was not only able to assess a certain return value, but also the
changes over time of these values with low occurrence rates,
including their inherent variability.
Conclusions
An extreme-value model was developed to analyse the seasonal,
interannual, decadal, and long-term variations of extreme sea
levels using mathematical expressions in the parameters of the
statistical distribution. Modelling the different time-scales contributed to a better understanding of recent secular trends in extreme
climate events, which are currently one of the main concerns
1495
Forecasting seasonal to interannual variability in extreme sea level
Figure 6. Quantile plots for the stationary model (left) and for the best non-stationary model estimated (right).
Figure 7. Observed values of monthly maxima sea level (dots), the annual aggregated sea level (bold line), and instantaneous 100-year return
sea level in the 1990 – 2020 period. Results are for the best model applied to the Newlyn data.
relating to climate change. The model provides a better characterization of extreme values with time-dependent quantiles and confidence intervals. The different results obtained with the
San Francisco (Mendez et al., 2007) and Newlyn tide records
demonstrate the flexibility of the statistical model, which allows
separate analyses of sea level datasets covering different time-scales
of interest.
The model was applied to a well-known large dataset, the
Newlyn tide-gauge dataset. The results revealed that the most significant contributing factors for sea level extremes can be separated
into intra-annual variability (annual and semi-annual cycles), a
perigean influence, the nodal cycle, and the increase in mean sea
level. The modelled seasonal variation caused fluctuations of
16.9 cm over the averaged extreme sea level value and a variability
of 30% in the variance. The nodal cycle exerted a strong influence
on the magnitude of extreme sea level events, causing an increased
probability of a given extreme sea level event every 19 years.
Perigean tides caused an extra modulation of the nodal cycle of
high seawater levels. Consequently, the estimated 100-year return
period sea level in year 2009 was estimated as 6.35 m, whereas
for 2020 this quantile reached 6.5 m. This increase was not only
associated with an increase in sea level, but also with the joint contribution of the nodal cycle and the perigean tides.
Finally, we proposed a methodology to produce forecasts of the
probability of exceeding a given extreme sea level by considering
the natural variability in high seawater levels, therefore allowing
the determination of flood risk.
Acknowledgements
The first author is indebted to the Spanish Ministry of Science and
Innovation (MCI) for the funding provided. The study was partly
funded by the project “GRACCIE” (CSD2007-00067, Programa
Consolider-Ingenio 2010) from MCI and by the project C3E
(200800050084091) from the Spanish Ministry of Environment.
We thank two anonymous reviewers for useful comments, which
notably improved the manuscript.
1496
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