“Stationarity is dead” or “stationary is immortal” ? The statements reflect the fact that many environmental and hydrological processes exhibit time-varying changes of characteristic quantities, usually due to the presence of trends and shifts , but such changes are usually observed in the short-term period, whereas the overall process may exhibit long-term stationarity (Fig. ). In this regard, the term stationarity (also known as strict or strong stationarity) refers to the over-time steadiness of the probability distribution function (pdf) of the process . Mathematically speaking, we can write pX(X,t)=pX(X,t+c), with pX(X,t) the pdf of the process at time t, and c a constant. Accordingly, the steadiness condition can be written as ∂pX(X,t)/∂t=0. Therefore, for strictly stationary processes, all the moments (e.g., mean, variance, skewness, etc.) of the pX(X,t) function, as well as other statistical properties such as the above-threshold probability, are constant in time. When the time steadiness of pX(X,t) moments is satisfied up to a particular order K, the process is Kth-order stationary and generally referred to as weakly stationary . For instance, if only the mean and variance are constant, the process is 2nd-order stationary. On the contrary, when the pdf varies in time such that none of the moments show over-time stationary, the process is defined non-stationary (see Fig. ).

Regarding the specific case of hydrological and weather-related variables, several scientific studies have identified alterations of mean annual temperature, rainfall amount, and riverflow regime. The alterations in hydrological statistics are typically attributed to several factors, including natural variability of climate conditions, and systematic changes of the system's dynamics due to, for instance, anthropogenic activities (e.g., urbanization, land-use change, greenhouse gas emissions) which may impact the precipitation patterns, air temperature, and sea levels . Whether of climatic origin or not, significant changes also alter both the empirical and the theoretical probability distribution functions. For instance, reported the presence of trends in the maximum flow discharge of Swiss rivers, which can be significantly correlated to the alterations in heavy precipitation events and temperature . Similar results were obtained in the analysis of mean precipitation data , and sea level rise, both at the local scale e.g.,for an analysis of the Mediterranean area, and at the global scale , as well as for the rainfall amount , and the streamflow regime in the United States.

From an engineering point of view, water resources management and the design of hydraulic structures rely on quantifying hydrological extreme events, like floods and minimal flows, and their duration. The traditional/historical procedure to estimate the extreme-event magnitude and the average occurrence frequency is based on the return period concept, which is defined as the average intertime between extreme events greater (lower) than, or equal to, a specific magnitude (threshold). Such a definition, usually implied in the design of engineering applications, is inherently built on the assumption of independent and identically distributed (iid) events above a specific value, x (i.e., the threshold). Therefore, the return period, T‾(x), is usually taken as constant (in a stationary framework, as highlighted by the ⋅‾ notation) and associated with the time interval, Δτ, for which the observations are available (e.g., Δτ = 1 year, for annual observations). In terms of the occurrence probability of extreme events (with magnitude X≥x), the return period T‾(x) can be calculated as: 1T‾(x)=1P(X≥x)Δτ with P(X≥x) the exceedance probability related to extreme events (Fig. ).

The iid hypothesis implies the over-time (strict) process stationarity, which is not satisfied in the presence of trends in the signal . For instance, the return interval of a specific flood peak under non-stationary conditions can vary significantly over time, potentially ranging from thousands of years to less than a decade . Noteworthy, when local non-stationarities or alterations in the observed statistical properties of the process dynamics are present (e.g., Fig. ), the duration of the temporary changes must be much longer than the interval of observations (i.e., Δτ in Eq. ), and specifically in the order of magnitude (e.g., decades) of the typical time-scales for engineering applications or other practical considerations involving design risk.

The analysis of extreme events has usually been performed by considering the annual maxima and the empirical definition of the occurrence probability of events above thresholds characterizing their extreme magnitude . Some works additionally proposed the change of the return period definition, based on the expected waiting time until an event or the expected number of events over a given period, to better quantify the risk under non-stationarity . As a consequence, the traditional quantification of extreme events based on the return period concept, such as the Peak Over Threshold (POT) analysis, is increasingly questioned and a proper framework that can dynamically account for the time-evolution of probability distributions is currently missing . Noteworthy, the change in return period due to shifts and trends may affect not only the peak values, but also time-integrated quantities. From a hydrological point of view, this condition may reflect in changes of the flow volume associated with a specific hydrograph and the duration of specific trajectories (mean first passage times) above, below, or between characteristic thresholds . In the literature, some methods that involve a time-varying probability distribution (Fig. ) have been formulated to address this challenge . According to , the return period, T̃0(x) at the current time (noted by the subscript 0) under non-stationary conditions (highlighted by the ⋅̃ notation) can be computed as: 2T̃0(x)=Δτ∑i=0∞∏j=0i-11-P(X>x,jΔτ).

Due to non-stationary conditions, Eq. () defines the return period T̃0(x) as the first moment of the time-discrete non-homogeneous distribution of extreme events, whose occurrence at a specific timeframe iΔτ is calculated as the product of the non-occurrence probability, 1-P(X>x,jΔτ), in the previous timeframes (i.e., j up to i-1). The return period, T̃0(x), is then calculated by summing up all these product terms. As a matter of course, Eq. () simplifies to Eq. () under stationary conditions. However, from a practical point of view, Eq. () presents some drawbacks due to the infinite number of terms within the summation, which may prevent practitioners from using it due to the required computational time. Simplified approaches to Eq. () consider the sample mean of the probability distribution function, but the estimation error strongly depends on the sample length, and, as a matter of fact, the method can be satisfactorily applied to determine the non-stationary return period of past time-series e.g.,. Unfortunately, despite an approximate solution may be retrieved by computing fewer terms, the minimal number of terms required to obtain an acceptable error may still range from tens to thousands, depending on the process parameters (we will remark on this aspect in the Discussion). Furthermore, uncertainty in the estimation of potential trends and their associated changes in the probabilistic distribution, as well as the complexity of available tools, often prevents practitioners from including non-stationarity in their approaches .

In this work, we develop a framework to estimate the return period by extending to weakly non-stationary processes the simpler formulation provided by Eq. (). The non-stationary return period analysis is tackled by relying on the General Extreme Value distribution, with time-varying parameters due to trends in the signal process. The validity of the proposed approach is mathematically derived and discussed based on the hypothesis of iid events above specific thresholds. Accordingly, the definition weakly non-stationary is proposed for processes satisfying the derived conditions. As a result, the return period associated with specific magnitude values can be retrieved as a function of the timeframe and the time-varying parameters, thus allowing for improved definitions of design criteria and strategy management when dealing with weakly non-stationary processes.

We consider a generic stochastic process, X(t), modeling the continuous time-evolution of a random variable. Although the analysis is focused on hydro-climatic quantities (e.g., rainfall amount or flow discharge), the framework can be readily extended to other random variables. For the sake of simplicity, we consider the case of above-threshold extreme events only (e.g., floods, earthquakes), but the considerations can be readily applied to the case of below-threshold events (e.g., low flow analysis, meteorological droughts, etc.). In a non-stationary framework, it is straightforward to allow for potential time-variability if proper conditions linking the process observation time and the return period are imposed. In the following, the notation T̃(x,t) is adopted to indicate the conditional return period of an ensemble of non-stationary stochastic trajectories.

For the process X(t), the Fisher–Tippett–Gnedenko's theorem implies that the extreme values above specific, asymptotically high, thresholds are distributed following either the Gumbel, the Fréchet, or the reversed (i.e., upper bounded) Weibull functions . Hereafter, when referring to the Weibull type, the term “reversed” is omitted for brevity. Mathematically, the three functions can be summarized by the General Extreme Value (GEV) distribution, according to the value of a shape parameter ξ. In non-stationary processes, the most general formulation of the GEV distribution can be written as : 3GEV(X,t)=exp⁡-1+ξ(t)X-μ(t)σ(t)-1ξ(t) with X the modeled variable, μ(t) and σ(t) the time-dependent mean and variance of the process, respectively, and ξ(t) the time-dependent shape parameter. Equation () resembles a Weibull type for ξ(t)<0, and a Fréchet type for ξ(t)>0. In the limit case of ξ(t)→0, Eq. () simplifies to the classic double-exponential Gumbel distribution. For the sake of the analysis, we define G[x,t], the positive part of the exponential function in Eq. (), such that: 4GEV(X,t)=exp⁡[-G(X,t)]

Let us consider the presence of trends in the stochastic dynamics whose effects on the process's statistics are sufficiently low in relation to the given observation interval, Δτ. We define such a process as weakly non-stationary. This allows assuming Eq. () is still valid conditional on the duration of the observation. Accordingly, we can rewrite Eq. () as: 5T^(x,t)=11-GEV(x,t)Δτ where Δτ is still the time interval of observations, and the notation T^ highlights the proposed simplified solution for weakly non-stationary processes, in comparison to the (fully) non-stationary value, T̃(x,t), given by Eq. (). The validity of Eqs. () and () is based on two hypotheses, which can be summarized as :

the process parameters (i.e., μ(t), σ(t), and ξ(t) in Eq. ) can be calculated based on the data available within the time interval Δτ at each time t;

For the sake of clarity, the GEV parameter estimation and their time-dependence (i.e., non-stationary trends in ξ(t), μ(t), and σ(t)) go beyond the scope of this work. Herein, we assume that they are known based on the plethora of models available in the literature, usually relying on a pre-assumed definition of the non-stationary trend . In this regard, one may consider the (Generalized) Maximum Likelihood Estimation, (G)MLE e.g.,, the Differential Evolution Markov Chain approach based on the Bayesian inference through MonteCarlo simulations, DE-MC e.g., , or the GAMLSS (Generalized Additive Models for Location, Scale and Shape) tool proposed by . Equation () can now be used to evaluate the change in return period, for events with the same threshold X, at a different time, t1. The change in return period can be easily calculated through the difference between the two values, T^(x,t1) and T^(x,t), in relation to the difference t1-t. As a result, we can write: 6T^(x,t1)-T^(x,t)t1-t=11-GEV(x,t1)-11-GEV(x,t)×Δτt1-t which, by straightforwardly imposing the limit for t1-t→0 and by 1st-order approximation, yields: 7∂T^(x,t)∂t=Δτ(1-GEV(x,t))2∂GEV(x,t)∂t

Following the definition of weakly non-stationary processes, the change in return period must be as low as possible (eventually zero under stationary conditions). Accordingly, we can write: 8∂T^(x,t)∂t≤a where a represents a small, positive dimensionless quantity and the absolute value accounts for positive or negative changes in the return period, T^(x,t). Combining Eqs. () and (), by accounting for Eq. (), leads to: 9∂G(x,t)∂t≤aΔτ(1-GEV(x,t))2GEV(x,t) which can be easily solved in terms of the function G(x,t). By using the boundary condition T^(x,0)=T‾0(x) (i.e., the current return period under stationarity), the solution of Eq. () reads: 10log⁡T‾0(x)+atT‾0(x)+at-Δτ≤G(x,t)≤log⁡T‾0(x)T‾0(x)-Δτ∂G(x,t)∂t≤0log⁡T‾0(x)T‾0(x)-Δτ≤G(x,t)≤log⁡T‾0(x)-atT‾0(x)-at-Δτ∂G(x,t)∂t≥0 with the two conditions corresponding to the decrease or increase of the return period T^(x,t), respectively. Equation () represents the most general solution for the function G(x,t) satisfying the definition of weakly non-stationary conditions. In terms of the process parameters, the general solution and a simplified version considering the shape parameter ξ(t) constant are given in the Appendix . For practical applications, it is interesting to analyze the limit solution of Eq. () (i.e., the equal sign at the boundary depending on the parameter a), which can be written as: 11G±(x,t)=log⁡T‾0(x)±atT‾0(x)±at-Δτ where the ± sign refers to the increasing (+) or decreasing (-) behavior of the G(x,t) function. Eventually, the quantity on the right-hand side of Eq. () can be evaluated in a=0 (stationary conditions). In this case, the notation G0(x) is adopted for brevity, and the function G±(x,t) reduces to 12G0(x)=log⁡T‾0(x)T‾0(x)-Δτ

To solve Eq. () in terms of the acceptable trends in the process parameters, the actual type of the GEV distribution must be known (e.g., Weibull, Fréchet, or Gumbel). Furthermore, we consider the simplified case ξ(t)=ξ.

An example of the behavior of the limit solutions for the Frechét and Weibull distributions (Eq. ) is shown in Fig.  for both the distributions, by varying the initial return period, T‾0(x). For the sake of simplicity, we have considered a=1. Initial values of μ0=302, σ0=170, and ξ = ± 0.354 are taken from the application example of .

The global solution (Eq. ) at a specific time, t^, is represented by the area contained between the limit curve corresponding to the time t^ and the initial curve at t=0y (orange line in Fig. ). Therefore, any possible trend that keeps the pair of parameter values inside the aforementioned area for t≤t^ satisfies the weak non-stationary hypothesis (Eq. ). For instance, the dotted line in Fig. a highlights a possible general trend of the parameter μ(t) and σ(t). Considering the three limit solutions provided in the plot (at t = 10 y, at t = 20 y, and at t = 30 y), the trend satisfies the weakly non-stationary condition if the corresponding points Pt1 (black crosses in Fig. a) are reached at a time larger than, or equal to t1. In other words, weakly non-stationary processes are represented by those pairs of parameters [σ(t),μ(t)] that change along the curvilinear axis depicting the trend slower than (or equal to) the limit solution.

The domain of allowed pairs of the process parameters strongly depends on the initial return period, T‾0(x), and decreases with increasing T‾0(x). As shown in the panels of Fig. , the distance between the limit curves is narrower in the case of greater initial return period (T‾0(x) = 25 y in Fig. a and c, and T‾0(x) = 50 y in Fig. b and d). Similar considerations can be done when comparing the limit curves for the decreasing or increasing return periods (solid and dashed, respectively, lines in Fig. ): in the latter case, the limit conditions on the parameters appear more restrictive. Furthermore, for the same combination of initial return period, T‾0(x), and timeframe, t, Fig.  shows that the allowed range of validity is larger for the Fréchet distribution than for the Weibull (e.g., compare panel a and c); The behavior of the limit solutions for the Gumbel distribution is hereafter analyzed in terms of the allowed trends for the mean frequency, λM(x,t), and mean magnitude, γM(x,t), of stochastic events (Fig. ). Figure  shows the behavior of the limit solution described by Eq. () for two different values of the initial return period, T‾0(x), and for various future timeframes, t.

For the tested initial return periods (T‾0(x) = 25 y in panel a; T‾0(x) = 50 y in panel b), a reasonable variation of the parameter λM(t) spans two orders of magnitude, whereas the variation of the parameter γM(t) is limited by approximately -20 % and +40 %. The validity of the solution at a specific time horizon, t, follows the same considerations as in the case of the Fréchet and Weibull distribution (Fig. ). Furthermore, Figs. a, c, and a show the absence of the curve at t = 30 y for the case of decreasing return period (solid line). Indeed, the proposed framework loses validity for t≥T‾0(x)-Δτa in the case of decreasing return period, due to the argument of the logarithm within the function G±(x,t) in Eq. (). The limitation does not apply in the case of increasing return period (dashed lines in Figs.  and ). In the case of the Gumbel distribution, this is highlighted in Fig. , where the simplified solutions (i.e., the trend affects only one parameter at a time, Eqs.  and ) show the presence of an asymptote at t=T‾0(x)-1 (Δτ = 1 year and a=1 in the shown examples), for the case of decreasing return period (solid lines). The asymptote applies to the limit curves of λM(x,t) (solid red line) and γM(x,t) (solid blue line).

The return period is a concept that exists theoretically for strictly stationary and ergodic time series and has been extensively used in several engineering disciplines for design and risk analysis purposes. Despite the return period concept possessing a rigorous theoretical foundation, its practical use has ever since encountered the challenging issue of dealing with observations of limited duration. This required loosening the strict assumption of stationarity and ergodicity when statistical moments and parameters are empirically calculated. Several procedures to estimate the process parameters, as well as to forecast their values in the future (i.e., trends and shifts), have been proposed in the literature e.g., and were therefore not the scope of this work. Herein, we assumed that the parameters of the non-stationary process and their variation in time are known based on some formulated assumptions and data fitting models e.g.,. Based on this assumption, the proposed framework has been developed to define under which statistical conditions such a time series can be assumed as weakly non-stationary. Accordingly, it allows for performing a return period analysis strictly valid within Δτ (equal to 1 year for the shown results, and in general for annual maxima analysis), by determining a condition linking the maximum change of return period in relation to the timeframe, t. We've shown that constraining the maximum change of the return period over a given timeframe, t, and quasi-stationary properties within Δτ, leads to some limits in the range of variability of the process parameters.

For the simple Poisson process with marked magnitudes being considered, the limits of applicability of our analysis for the case of trends affecting either the mean frequency or the mean magnitude are shown in Fig. . For example, consider a process with a threshold corresponding to a present-day T‾0(x) = 25 y (Fig. a) or T‾0(x) = 100 y (Fig. b), and the case of decreasing return period (continuous lines in Fig. ). Then, our analysis may be applied over a time horizon t if any trend affecting the process, either in the frequency or the magnitude, produces a change in such parameters less than the value of the corresponding curves. As said, an increase in the frequency or magnitude of events yields a reduction of T^(x,t) over that timeframe. Such a change in the mean frequency or magnitude increases with the timeframe length, ultimately reaching an infinite value at the asymptote t=T‾0(x)-Δτa, in the case of decreasing return period (see Fig. ).

However, the increasing value of the limit condition is counteracted by the shorter timeframe for which the value is allowed. Specifically, if we consider the limit condition on the mean frequency parameter at a specific timeframe t^>0, λM(x,t^), its value is valid for t^≤t≤T‾0(x)-Δτa, and such a range decreases as t^ increases. Similar considerations can be assessed on the parameter γM(x,t). Nevertheless, Fig.  shows that the graphs of γM(x,t) and λM(x,t) are strictly monotonic throughout the whole range of timeframe validity. Consequently, it suggests that a range of linear trends satisfying the limit condition exists for the case of a single time-varying parameter. The maximum slope of such a range is given by the derivative of the limit condition at t=0 in Fig. . Without losing generality of the following considerations, hereafter we focus on the case of the Gumbel distribution (ξ→0 in Eq. ).

In this case, by considering Eqs. () and (), and a linear trend in either one of the parameters in the form A(t)=A0(1+kAt) with A=γ or A=λ and kA the coefficient of the linear trend, the maximum slope can be retrieved as: 22kγ(x)=1γ0∂γM(x,t)∂t|t=0=aT‾0(x)ΔτT‾0(x)-Δτ2xγ0γ02x+γ02×log⁡T‾0(x)T‾0(x)-Δτ-1 for a linear trend in the parameter γ(t) (horizontal dashed-dotted line in Fig. a), and 23kλ(x)=1λ0∂λM(x,t)∂t|t=0=aT‾0(x)ΔτT‾0(x)-Δτlog⁡T‾0(x)T‾0(x)-Δτ-1 for a linear trend in the parameter λ(t) (vertical dashed-dotted line in Fig. a). For a given initial return period, T‾0(x), both Eqs. () and () show that the maximum allowed linear trend mainly depends on the value of the parameter a. For such linear trends, under the hypothesis of the weakly non-stationary framework, the time variation (Eq. ) of the initial return period T‾0(x) = 100 y is shown in Fig.  for different values of the parameter a. For the explored range of a-values, minor differences are shown between the case with a trend in γ(t) only (Fig. a) and the case with a trend in λ(t) only (Fig. b). Particularly, the trend on the mean magnitude, γ(t), seems more restricting, as the allowed change in time is lower, compared to the case with maximum linear trend in the average frequency, λ(t).

From a practical point of view, the limit solutions for the values of the GEV parameters (Eqs.  and  for the weibull-Fréchet type, and Eqs.  and  for the Gumbel distribution) have to be compared to forecasted trends from data analysis or modelled scenarios. As an example, Fig.  shows this comparison for the parameter λ and a sample modelled scenario of trend with decreasing return period. Due to the uncertainty in the estimation, at each future timeframe, t, the value of the forecasted parameter, λF(t), is provided with a probability distribution function (light blue curves in Fig. ) and the mean trend is represented by the continuous blue line. The shaded area of the pdf below the red curve (i.e., the limit solution) quantifies the probability that the forecasted trend satisfies the weakly non-stationary conditions in the proposed framework. In this regard, the deterministic timeframe limit of validity is defined by the point at which the forecasted trend (blue line) intersects the limit solution (red curve), i.e., t≃ 13 y in the shown example (Fig. ). Conversely, from a probabilistic point of view, even when the average trend is above the limit solution (e.g., at t = 15 y in Fig. ), a finite probability, although small, exists for a valid weakly non-stationary approach to the return period analysis.

The approximation error of the proposed weakly non-stationary framework is evaluated through Eq. () for different combinations of initial stationary return period T‾0(x), a-parameter, and the corresponding maximum linear trend in either γ(t) or λ(t). As expected from Eq. (), Fig.  shows that the greater the allowed change in time of the return period (i.e., the a parameter), the greater the approximation error ΔT^(x,t). Following the considerations on the maximum allowed value of the linear trend (kγ and kλ in Fig. ), ΔT^(x,t) is greater when the trend affects the mean frequency λ(t) (Fig. b) than the mean magnitude γ(t) (Fig. a), for the same combination of a and T‾0(x). Furthermore, the approximation error decreases according to the timeframe t, and maximizes at the current time (t = 0 y). This is because Eq. () accounts for the future time-varying values of the GEV distribution, whereas the proposed framework considers the local (in time) values. Therefore, the weakly non-stationary return period at t=0, T^(x,0) is equal to the initial return period, T‾0(x), under stationary conditions. By considering the best tested combination (corresponding to a=0.05), Fig.  shows that the approximation error is roughly in the order of 5 % for all the tested initial return periods, and this is achieved by one single calculation (Eq. ) for all the future timeframes, t. Conversely, Eq. () requires the computation of infinite terms, and the operation should be repeated for all the timeframes. An approximate solution of the fully non-linear return period, T̃(x,t) can be assessed by considering only the first N terms in the summation of Eq. (). In this case, the degree of approximation of the calculated value, T̃0,N(x), depends on the number of terms, N, on the threshold X (i.e., the value of the associated stationary return period T‾(x), and the effective trend in the process signal.

To give an example, Fig.  shows the degree of approximation of the solution with N terms in the summation, and the actual return period under non-stationary conditions using Eq. (), for a Poisson Process (Fig. a) with initial mean frequency λ0 = 73 y-1, and a linear trend in the mean frequency depending on the coefficient kλ, in the form λ(t)=λ0(1+kλt). Fig.  suggests that an approximated solution with few terms (N≤100) of Eq. () is reliable only in the case of a low initial return period and high trends in the process signal (e.g., T‾0(x) = 20 y and kλ = 0.01 y-1, dashed blue line in Fig. ). Conversely, for higher return periods (e.g., T‾0(x)≥ 50 y) and milder trends (kλ≤ 0.01 y-1), the required number of terms to get a good approximation of the return period in non-stationary conditions increases by an order of magnitude, in the shown example. For instance, in the case of T‾0(x) = 200 y and kλ = 10⁻⁴ y-1 500 terms lead to get an approximated value equal to 73 % of the actual one, whereas 1000 terms are necessary to obtain an approximation of 97 % (dotted yellow line in Fig. ).

A novel framework for the return period analysis of non-stationary time series is proposed. The model is based on the hypothesis of weak non-stationarity, by assuming that the changes in the governing parameters (presence of trends) occur on a timescale longer than that of the change of statistical characteristics of the process (e.g., return period). Closed-form solutions are derived for the maximum allowed trends for the General Extreme Value distribution, and specifically discussed in the case of a linear trend for the Gumbel distribution. As a result, once the GEV parameters are known from fitting models or forecasted scenarios, the maximum timeframe for the validity of the weakly non-stationarity analysis can be retrieved in the case of decreasing return period for specific values of the parameter a, which in turn drives the estimation error (e.g., Fig. ). The results are readily applicable, but not limited, to the design of hydraulic structures, for which the return period and its time-varying value may affect the failure statistics.