The TS methodology consists of three steps: transforming a non-stationary time series y(t) into a stationary series x(t), performing a stationary EVA, and back-transforming the resulting extreme value distribution into a time-dependent one.

The transformation y(t) → x(t) we propose is x(t)=f(y,t)=y(t)-Ty(t)Cy(t), where Ty(t) is the trend of the series, i.e., a curve representing the long-term, slowly varying tendency of the series, and Cy(t) is the long-term, slowly varying amplitude of a confidence interval that represents the amplitude of the distribution of y(t). In particular, if Cy(t) equals the long-term varying standard deviation Sy(t) of the series y(t), Eq. (1) reduces to a simple time-varying renormalization of the signal: x(t)=f(y,t)=y(t)-Ty(t)Sy(t). For simplicity, in the remainder of this paper we will limit our analysis to Eq. (2), knowing that all the considerations can be easily extended to any time-varying confidence interval Cy(t).

Equation (2) guarantees that the average of x(t) and its standard deviation are uniform in time, which is a necessary condition for x(t) to be stationary. In particular, the transformed signal x(t) has a mean equal to 0 and a variance equal to 1. It is worth noting that the transformed series x(t) is not necessarily stationary: a series with a constant trend and a uniform standard deviation may still have a time-dependent auto-covariance that would invalidate the hypothesis of stationarity (i.e., the condition of a series with statistical moments constant in time). Before proceeding with the analysis, therefore, a stationarity test should be carried out to ensure that x(t) is stationary and that its annual maxima can be fitted by a stationary extreme value distribution. For example, a simple test can be performed to ensure that higher order statistics such as skewness and kurtosis are roughly constant along the series.

Once the hypothesis of stationarity of x(t) is verified, we can estimate the distribution GEVX(x) that best fits its extremes, for example through MLE. GEVX(x) is then given by GEVX(x)=Pr(X<x)=exp⁡-1+εxx-μxσx-1/εx, where the shape (εx), scale (σx), and location (μx) parameters do not depend on the time. To find the time-dependent distribution GEVY(y, t) that fits the non-stationary time series y(t) we can write that GEVY(y)=Pr[Y(t)<y]=Prf-1(X,t)<y=Pr[X<f(y,t)]=GEVX[f(y,t)], where f(y, t) is the transformation from y to x given by Eq. (1), and f-1(x, t) is its inverse: f-1(x,t)=y(t)=Sy(t)⋅x+Ty(t). It is always possible to compute GEVY(y, t) from GEVX(x) because f(y, t) is a monotonically increasing function of y for every time t, because the standard deviation Sy(t) is always positive.

Using Eqs. (3) and (5) in Eq. (4), we find GEVY(y,t)=GEVX[f(y,t)]=exp⁡-1+εxy-Ty(t)Sy(t)-μxσx-1/εx=exp⁡-1+εxy-Ty(t)-μx⋅Sy(t)σx⋅Sy(t)-1/εx. Therefore, if x(t) is fitted by the stationary distribution GEVX(x) then y(t) is fitted by the time-dependent distribution GEVY(y, t) with shape, scale, and location parameters given by εy=εx,σy(t)=Sy(t)⋅σx,μy(t)=Sy(t)⋅μx+Ty(t). It can be shown that the time-dependent GEV parameters given by Eqs. (7)–(9) are the same as the time-varying parameters εns, σns, and μns of a non-stationary distribution GEVns that would be obtained from a non-stationary MLE on the series y(t), and which are given by εns=const.,σns=Sy(t)⋅a,μns=Sy(t)⋅b+Ty(t), for varying parameters a and b. In fact, if pGX(x) is the probability density function (PDF) associated with the distribution GEVX(x), then the MLE for GEVX(x) is estimated so that ∑log⁡pGX(x)=max, which involves vanishing derivatives of Eq. (13) on the GEV parameters εx, σx, and μx. For example, considering the scale parameter σx, we see that ∑∂∂σxlog⁡pGXx,σx=0. The non-stationary MLE maximizes the log-likelihood of the non-stationary PDF pGns(y, t) associated with GEVns, in function of the parameters a and b. For example, considering the parameter a, we impose ∑∂∂alog⁡pGns(y,a,t)=0. Let us assume that pGns(y, t) coincides with the PDF pGY(y, t) associated with the distribution GEVY(y, t) given by Eq. (6) and that a = σx. Considering that pGY(y,t)=∂∂yGEVY(y,t)=pGX(x)∂∂yf(y,t)=pGX(x)Sy(t), we obtain ∑∂∂alog⁡pGns(y,a,t)=∑∂∂σxlog⁡pGYy,σx,t=∑∂∂σxlog⁡pGXx,σxSy(t)=∑∂∂σxlog⁡pGXx,σx-log⁡Sy(t)=∑∂∂σxlog⁡pGXx,σx=0, where the last step is possible because Sy(t) does not depend on σx.

The same principle can be applied differentiating ∑log⁡[pGY(x, μx, t)] = 0 on the location parameter μx to maximize the log likelihood, finding the condition ∑∂∂μxlog⁡pGYx,μx,t=∑∂∂μxlog⁡pGXx,μx=0. Therefore, if x is stationary, the condition of maximum likelihood for pGX(x) coincides with the condition of maximum likelihood for pGY(y, t), and applying MLE for fitting the stationary parameters (σx, μx) is equivalent to fitting the parameters (a, b) of by Eqs. (10)–(12) by non-stationary MLE. The equivalence between the two methodologies suggests that the TS approach is dual to the EM approach, meaning that any implementation of EM is equivalent to an implementation of the TS approach for some transformation f(y, t) : y(t) → x(t) (see Appendix A for a more detailed discussion). One can also prove that Eq. (1) allows a general TS formulation with a constant shape parameter, i.e., all the TS models with a constant εy can be connected to Eq. (1) (see Appendix A). This last result is remarkable, because it shows that Eq. (1) is exhaustive for all the TS models with a constant shape parameter.

The findings drawn above are general and can be applied also to peak over threshold (POT) methodologies, because the GPD is formally derived from the GEV as the conditional probability that an observation beyond a given threshold u is greater than x. In particular, the POT/GPD parameters are given by uy(t)=Sy(t)⋅ux+Ty(t),εy=εx=const.,σGPDy(t)=σy(t)+εyuy(t)-μy(t)=Sy(t)⋅σGPDx, where ux(t) and uy(t) are the thresholds of the x and y time series, εy = εx is the shape parameter, σGPDx and σGPDy(t) are the GPD scale parameters of x, and y, σy, and μy are the scale and location parameters of a GEV associated with the GPD, which have been included in Eq. (19) to make it clear how the parameter σGPDy(t) can be derived.

It is worth noting that the TS methodology is neutral for a stationary series, i.e., the application of this methodology to a stationary series leads to the same results as a stationary EVA with the same underlying statistical model. That is because in such case Ty and Sy are constant, and Eq. (2) reduces to a constant translation and scaling.

Often, we would like to model extreme events that show seasonality, for example with local winter extremes that differ in magnitude from summer extremes. A simple way to add the seasonal cycle to Eqs. (7)–(9) is by expressing the trend Ty(t) and the standard deviation Sy(t) as Ty(t)=T0y(t)+sT(t),Sy(t)=S0y(t)⋅sS(t), where T0y(t) and sT(t) are, respectively, the long-term varying and seasonal components of the trend, S0y(t) is the long-term varying standard deviation and sS(t) is the seasonality factor of the standard deviation. In the notation, the subscript 0 denotes the long-term varying components. Applying Eqs. (22)–(24) to Eq. (2), we obtain x(t)=y(t)-T0y(t)-sT(t)S0y(t)⋅sS(t). The time-varying GEV parameters can be expressed as εy=εx=const.,σy(t)=S0y(t)⋅sS(t)⋅σx,μy(t)=S0y(t)⋅sS(t)⋅μx+T0y(t)+sT(t), and the time-varying POT/GPD parameters can be expressed as uy(t)=S0y(t)⋅sS(t)⋅ux+T0y(t)+sT(t),εy=εx=const,σGPDy(t)=S0y(t)⋅sS(t)⋅σGPD.

The implementation of the TS methodology is illustrated in Fig. 1. The fundamental input is represented by the series itself, and the core of the implementation consists of a set of algorithms for the elaboration of the time-varying trend T0y(t), standard deviation S0y(t), and seasonality terms sT(t) and sS(t).

In this study, we propose algorithms based on running means and running statistics (see Sect. 2.2.1). Hence, an important aspect is the definition of a time window W for the estimation of the long-term statistics T0y(t) and S0y(t) and of a time window Wsn for the estimation of the seasonality. The computation of T0y(t) and S0y(t) acts as a low-pass filter removing the variability within W. Therefore, W should be chosen short enough to incorporate in the analysis the variability above the desired timescale but long enough to exclude noise, short-term variability, and sharp variations in the statistical properties of the transformed series. For example, in studies of long-term climate changes a reasonable choice is to impose W = 30 years, because this is the generally accepted time horizon for observing significant variations in climate (e.g., Arguez and Vose, 2011; Hirabayashi et al., 2013). It is worth stressing that the chosen value of W should be verified a posteriori to ensure that the transformed series is stationary. The time window Wsn is used to estimate the intra-annual variability of the standard deviation (see Sect. 2.2.1). In Fig. 1, the input corresponding to the seasonal time window Wsn is drawn in a dashed box because its value is easier to choose than the value of W. For the examined case studies, a value of 2 months for Wsn always resulted in a satisfactory estimation of the seasonal cycle.

In this implementation of the TS methodology, the estimation of the long-term statistics is separated from the estimation of the seasonality. This allows to study the long-term variability of the extreme values as is typically done when studying extremes on an annual basis, as well as the combination of long-term and seasonal variability to evaluate extremes on a monthly basis.

After the estimation of T0y(t), S0y(t), sT(t), and sS(t) we can apply Eq. (2) and perform a stationary EVA on the transformed series. It is important to stress that the stationary EVA is performed on the whole time horizon. The stationarity of the transformed signal allows us to apply different techniques for the EVA. In this study, we illustrate the GEV and GPD approaches, but an interesting development would be the elaboration of non-stationary techniques for other approaches such as those described by Goda (1988) or Boccotti (2000), based on the TS methodology.

The final step of the implementation is the back-transformation of the fitted extreme value distribution into a non-stationary one as given by Eqs. (10)–(12) and (25)–(27) for GEV and by Eqs. (19)–(21) and (28)–(30) for GPD.

To assess the generality of the approach, the TS methodology has been validated on time series of different variables, from different sources and with different statistical properties.

The analysis of annual and monthly maxima has been carried out on time series of significant wave height at two locations: the first located in the Atlantic Ocean, west of Ireland (-10.533∘ E, 55.366∘ N), and the second close to Cape Horn (60.237∘ E, -57.397∘ N). The data have been obtained by means of wave simulations performed with the spectral model Wavewatch III^® (Tolman, 2014) forced by the wind data projections of the RCP8.5 scenario (van Vuuren et al., 2011) of the CMIP5 model GFDL-ESM2M (Dunne et al., 2012) on a time horizon spanning from 1970 to 2100. This data set is referred to from now on as GWWIII. Here, the TS methodology is used in order to examine its applicability to climate change studies. The annual and monthly analyses have been repeated on a series of water level residuals offshore of the Hebrides Islands (Scotland, -7.9∘ E, 57.3∘ N) obtained from a 35-year hindcast of storm surges at European scale (Vousdoukas et al., 2016a, b) forced by the ERA-Interim reanalysis data (Dee et al., 2011). This data set is further referred to as JRCSURGES.

For the annual maxima of the considered series, we further compare the TS methodology with the SS technique as implemented by Alfieri et al. (2015) and Vousdoukas et al. (2016a). For this purpose, we extracted time series from projections of streamflow in the Rhine and Po rivers covering a time horizon from 1970 to 2100 (Alfieri et al., 2015), from now on referred to as JRCRIVER. Also, the two series of significant wave height of west Ireland and Cape Horn extracted from the GWWIII data set have been used in this comparison.

Finally, we compare the TS methodology and the EM for monthly maxima using time series of significant wave height extracted from a 35-year wave hindcast database (Mentaschi et al., 2015) near the locations of La Spezia and Ortona. The analysis of this data set, further referred to as WWIII_MED, focuses on a comparison between seasonal cycles modeled by the two approaches.

The validation of the TS methodology was performed first on the time series of significant wave height of west Ireland and Cape Horn from the GWWIII data set. We verified first the non-seasonal transformation given by Eq. (2) and the time-dependent GEV and GPD given by Eqs. (7)–(9) and (19)–(21), respectively. By ignoring the seasonality, this formulation is suitable for finding extremes and peaks on an annual basis. For technical reasons the two series do not have data in two time intervals, from 2005 to 2010 and from 2092 to 2095. The impact of the missing data on the analysis is small, however, especially if we choose a time window W large enough for the estimation of the trend and standard deviation using Eqs. (31) and (33). In particular, for this analysis we chose a time window of 20 years, which is long enough to ensure the accuracy of the results and short enough to include the multi-decadal variability of a 130-year time series.

The results of the analysis for the two time series are illustrated in Figs. 2 and 3. Panel (a) of each figure shows the original time series and its slowly varying trend and standard deviation. Panel (b) illustrates the normalized series obtained through the transformation given by Eq. (1), allowing an evaluation at a glance of the stationarity of the normalized series. The mean and the standard deviation of the normalized series plotted in panel (b) are 0 and 1, respectively. Higher order statistics such as skewness and kurtosis are included in the graphics to support the assumption of stationarity of the normalized series. From the normalized time series we extracted the annual maxima and estimated the corresponding non-stationary GEV as given by Eqs. (7)–(9) (see Figs. 2c and 3c). Moreover, we performed a POT selection of the extreme events on the normalized series. The threshold was defined in order to have on average five events per year, following Ruggiero et al. (2010), corresponding for both of the series to the 97th percentile. From the resultant POT sample we estimated the corresponding non-stationary GPD as given by Eqs. (19)–(21) (see Figs. 2d and 3d). In Fig. 2c and d and Fig. 3c and d, the shape parameters ε estimated by the MLE for the GEV and the GPD are also reported. Inter-decadal oscillations in the annual maxima are modeled for both of the series, though they are more pronounced for the west Ireland time series. Moreover, for both series there is a tendency for the annual maxima to increase. This is more pronounced for the Cape Horn series, where the increase in the annual maxima of significant wave height estimated by GWWIII is about 2 m.

It is worth noting that for both of the considered series, the statistical mode of GEV and GPD grows faster in time than the slowly varying trend Ty(t). This is due to the fact that the growth of the location parameter μy(t) of the non-stationary GEV (Eq. 7) and of the threshold uy(t) of the non-stationary GPD (Eq. 19) are related not only to the growth of Ty(t) but also to the growth of Sy(t). The upper tail of the distributions grows even faster because the scale parameter is also proportional to Sy(t).

The impact of the statistical error in the slowly varying trend and the standard deviation on the uncertainty of the distribution parameters have been examined using Eqs. (48)–(50) and (51)–(53), which, for the non-seasonal analysis, reduce to Errεy=Errεx,Errσy=Sy⋅Errσx2+Errsy⋅σx2,Errμy=Sy⋅Errμx2+ErrTy⋅μx2+Err2Ty, for the GEV, and to Erruy=Sy⋅Errux2+ErrSy⋅ux2+Err2Ty,Errεy=Errεx,ErrσGPDy=Sy⋅ErrσGPDx2+ErrSy⋅σGPDx2, for the GPD. The result is that for the non-seasonal analysis the error due to the estimation of the trend and standard deviation is negligible with respect to the error associated with the stationary MLE. In Table 1, the values of the different components of the compared error in Eqs. (55)–(57) and (58)–(60) are reported together with the total error estimated for each parameter of the non-stationary GEV and GPD. Since the threshold ux of the stationary GPD was selected to have on average five events per year, the error has been computed as the uncertainty related to this definition. The percentage contribution to the squared error is also reported in Table 1 in a single column because the percentages estimated for the two series are roughly equal. The error for both GEV and GPD and for the two series is clearly dominated by the error associated with the estimation of the parameters of the stationary distributions ([Sy ⋅ Err[σx]] and [Sy ⋅ Err[μx]] for the GEV and [Sy ⋅ Err[σGPDx]] and [Sy ⋅ Err[ux]] for the GPD).

The seasonal formulation of the approach is suitable to estimate extreme value distributions on a monthly basis. Hence, we applied Eq. (24) to estimate the normalized series, then fitted a stationary GEV of monthly maxima by means of a MLE that was back-transformed into a non-stationary GEV through Eqs. (25)–(27). It is worth stressing that for the stationary MLE, the entire normalized series was used, covering a time horizon of 130 years. For the GPD, we selected the threshold in order to have on average 12 events per year, corresponding to the 93rd percentile for both series. Results are displayed in Fig. 4 for the location of west Ireland and in Fig. 5 for Cape Horn. To make the seasonal cycle distinguishable in these figures, we plotted only a slice of 5 years from 2085 to 2090. The meaning of the four panels in Figs. 4 and 5 is the same as in Figs. 2 and 3. The non-stationary extreme value distribution estimated for the location of west Ireland presents a strong seasonal cycle with higher and more broad-banded extremes during winter. For Cape Horn, the seasonal cycle is weaker, with the extremes of significant wave height slightly lower during the local summer. The estimated PDF for the seasonal GEV and GPD are significantly lower than those estimated for the non-seasonal analysis because in the seasonal analysis we consider monthly extremes, while in the non-seasonal one we consider annual extremes.

It is worth stressing that in the study of the monthly maxima, the long-term trend is also estimated even if it cannot be appreciated in Figs. 4 and 5 due to the short time horizon represented.

Table 2 reports the components of the statistical error due to the uncertainty in the estimation of the seasonality, together with the components of the stationary MLE. The error components relating to the uncertainty in the estimation of T0y and S0y were omitted as they are negligible compared with the error associated with the fitting of the stationary extreme value distribution (see Sect. 3.1). In Table 2, we can see that, as for the non-seasonal analysis, the error for both GEV and GPD and for the two series is clearly dominated by the uncertainty associated with the estimation of the parameters of the stationary distributions, though in this case the error related to the stationary MLE is significantly smaller than that found for the non-seasonal analysis due to the larger sample of data.

To verify the performance of the TS methodology on a series from a different source, of a different size, and with different statistical characteristics, we tested it on a series of water level residuals extracted from the JRCSURGES data set for an off-shore location of the Hebrides Islands, Scotland (-7.9∘ E, 57.3∘ N). This series is characterized by a flat trend Ty(t) because the model results are approximately constant averaged. Therefore, almost all the variability is modeled by the TS methodology in the standard deviation Sy(t). Since the time horizon of this series is shorter than that of the GWWIII projections, a time window of 6 years was adopted for the computation of the trend to better identify its inter-annual variability. The results of the TS analysis of the yearly maxima are shown in Fig. 6. The series displays also a strong seasonal behavior with annual maxima usually occurring during the local winter (for brevity, the seasonal analysis is not illustrated).

An interesting aspect is that the estimated standard deviation Sy(t) presents a strong correlation (ρ = 0.79) with the annual means of the North Atlantic Oscillation (NAO) index. This is illustrated in Fig. 7, where the scatter plot of Sy(t) vs. the annual means of the NAO index (Fig. 7a) and the two time series (Fig. 7b) are represented. As a consequence, the estimated annual maxima are also correlated with the NAO index.

A comparison was carried out between the TS methodology and the SS technique, consisting of a stationary analysis on quasi-stationary slices of data. This analysis was carried out on river discharge projections for the Po and the Rhine extracted from the JRCRIVER data set and on the projections of significant wave height extracted from the GWWIII data set for the locations of west Ireland and Cape Horn. The TS methodology was applied with a time window of 30 years to estimate a non-stationary GPD of annual maxima. The SS technique was carried out using a GPD approach on time slices of 30 years from 1970 to 2000, 2020 to 2050, and 2070 to 2100. For both methodologies, the threshold was selected to have on average five peaks per year.

Results are illustrated in Fig. 8, where the return levels of the projected discharge of the Rhine are shown for three time slices. In Fig. 8 the continuous black line and the green band represent the return levels and the 95 % confidence interval estimated by the TS methodology, where the dashed black line represents the return levels estimated by the stationary EVA on the considered slice (labeled in the legend as SS). The return levels estimated for short return periods by the two methodologies are close, while they tend to spread for high return periods. This fact is also evident from Fig. 9, where the return levels estimated by the two methodologies are plotted against each other for the river discharge of the Rhine and the Po and for the significant wave height of west Ireland and Cape Horn. We can see that for the analyzed time series the two methodologies are in good agreement for return periods below 30 years while they spread for larger return periods. Some quantitative data about this fact are shown in Table 3, which reports the normalized bias NBI of the return levels of the two methodologies, defined as NBI=RLTS-RLcmp‾/RLcmp‾, where RLTS and RLcmp are the return levels obtained by the TS and the SS methodology, respectively. Table 3 also includes the maximum deviation between the return levels estimated by the TS and by the SS methodology, as well as the 95 % confidence interval amplitude expressed as a percentage of the return level. The NBI and the maximum deviations were obtained by comparing results of the two techniques on the three 30-year time windows. From Table 3 we can see that the maximum deviation for return periods up to 30 years is always below 6 %, while for higher return periods it increases up to 13 % for the discharge of the Po. Moreover, the confidence intervals estimated for SS are always larger than those for TS, especially for large return periods. This is mainly due to the fact that for the stationary analysis on the quasi-stationary time slices we consider a sample of only 30 years, which leads to wider uncertainty ranges especially in the estimation of large return periods such as 100 and 300 years. This also explains the sharp variations of high return levels that we find between the three time windows using the SS approach. These variations are likely more related to the uncertainty in estimating the levels associated with long return periods rather than to climatic changes. The TS methodology allows a more accurate estimation of high return levels because it uses the whole sample of 130 years, and this represents one of the strengths of the TS methodology vs. SS. It is finally worth noting that the relative confidence interval estimated by both methodologies for the series of river discharge is larger than that estimated for the series of significant wave height. This is because for wave height data the minimum distance between two peaks has been set to at least 3 days, while for river discharge it has been set to 7 days.

Section 3 shows that the TS methodology is mathematically equivalent to a particular implementation of the EM methodology as described for example by Coles (2001), Izaguirre et al. (2011), Menéndez et al. (2009), and Sartini et al. (2015). For the sake of completeness, we show here the results of a comparison between the performances of TS and of a different formulation of the EM methodology. In its formulation, the parameters of the non-stationary GEV of the monthly maxima are expressed as μ(t)=β0+∑i=1Nμβ2i-1cos⁡(iωt)+β2i-1sin⁡(iωt),σ(t)=α0+∑i=1Nσα2i-1cos⁡(iωt)+α2i-1sin⁡(iωt),ε(t)=γ0+∑i=1Nεγ2i-1cos⁡(iωt)+γ2i-1sin⁡(iωt), where β0, α0, and γ0 are the stationary components; βi, αi, and γi are the harmonics' amplitudes; ω = 2πT-1 is the angular frequency, with T corresponding to 1 year; Nμ, Nσ, and Nε are the number of harmonics; and t is expressed in years. Therefore, the parameters βi, αi, and γi have been optimized through a non-stationary MLE in order to fit the monthly maxima of the non-stationary series. Different combinations of Nμ, Nψ, and Nε have been tested and the best model was chosen as the one presenting the lowest value of the Akaike criterion (Akaike, 1973) given by AIC=2k-2log⁡(L), where k is the number of degree of freedoms of the model and L is the likelihood. In particular, the maximum value tested for Nμ and Nψ is 3, while the maximum considered value of Nε is 2. In general, this model can be extended to incorporate long-term trends, but the two series examined in this test display flat trends. Hence, Eqs. (62)–(64) are adequate to model them.

In the comparison, the EM and the seasonal TS methodology (GEV only) were applied to the same series of significant wave heights relative to the WWIII_MED data set described in Sect. 2.3. For the transformed-stationary approach, a 10-year time window was used for the computation of the long-term trend. The results of the two methodologies are similar, with a roughly flat trend and strong seasonal pattern. The comparison of the seasonal cycles estimated by the two techniques is represented in Fig. 10 for the two series. Here, the continuous red and green lines are the location and scale parameters (μ and σ, respectively) as estimated by the TS approach. The dashed red and green lines are the location and scale parameters estimated through the EM. The blue dots represent the monthly maxima, while the color scale represents the time-varying probability density estimated by the transformed-stationary methodology. Since for both of the series the models selected based on the Akaike criterion have a constant shape parameter ε, these are reported together with those estimated by the TS methodology.

The GEV parameters estimated by the two approaches are in good agreement. The small differences have relatively small impact on the return levels as one can see in Fig. 11, where the return levels estimated by the two methodologies for the month of January are plotted. For both series, the return levels estimated by EM lie within the 95 % confidence interval estimated by TS. Table 4 reports the values of normalized bias (NBI) between the return levels estimated by TS and EM, defined as in Eq. (61), and the mean 95 % confidence interval amplitude expressed as a percentage of the return level. In Table 4 the values of NBI are reported for the four seasons for return periods of 5, 10, 30, 50, and 100 years, for both La Spezia and Ortona. In the definition of seasons that is used, winter starts on 1 December, spring on 1 March, summer on 1 June, and autumn on 1 September. We did not report return levels of periods greater than 100 years because the extension of the data covers only 35 years, hence the estimates for such periods are inaccurate for both methodologies. The average deviation between RLTS and RLcmp for the considered time series is rather small and remains below 7 % for all seasons. The confidence intervals estimated for TS are smaller than those estimated for EM, because the stationary MLE of TS has fewer degrees of freedom than the non-stationary one of EM, and is therefore affected by smaller uncertainty.