Flood Frequency Analysis Using the Bivariate Logistic Model with Non-Stationary Gumbel and GEV Marginals

Abstract

Flood frequency analysis is essential for designing resilient hydraulic infrastructure, but traditional stationary models fail to capture the influence of climate variability and land-use change. This study applies a bivariate logistic model with non-stationary marginals to eight gauging stations in Sinaloa, Mexico, each with over 30 years of maximum discharge records. We compared stationary and non-stationary Gumbel and Generalized Extreme Value (GEV) distributions, along with their bivariate combinations. Results show that the non-stationary bivariate GEV–Gumbel distribution provided the best overall performance according to AIC. Importantly, GEV and Gumbel marginals captured site-specific differences: GEV was most suitable for sites with highly variable extremes, while Gumbel offered a robust fit for more regular records. At station 10086, where a significant increasing trend was detected by the Mann–Kendall and Spearman tests, the stationary GEV estimated a 50-year return flow of 772.66 m³/s, while the non-stationary model projected 861.00 m³/s for 2075. Under stationary assumptions, this discharge would be underestimated, occurring every ~30 years by 2075. These findings demonstrate that ignoring non-stationarity leads to systematic underestimation of design floods, while non-stationary bivariate models provide more reliable, policy-relevant estimates for climate adaptation and infrastructure safety.

1. Introduction

Floods constitute one of the most critical natural hazards, producing extensive economic impacts, infrastructure deterioration, and considerable risks to human life. Reliable flood frequency analysis (FFA) is therefore essential, as it provides the basis for estimating the magnitude of extreme events given a return period (T) and for supporting the safe design and management of hydraulic structures and flood-risk mitigation strategies. In this context, Genest and Favre [1] provided a brief review of univariate and multivariate approaches to FFA.

Building on this perspective, research has increasingly turned to multivariate methods, which allow for a more complete representation of flood phenomena. Multivariate approaches to FFA have been applied globally. These studies often consider flood peaks, volumes, and durations [1,2,3]. While many analyses incorporate two or more hydrological variables to capture joint flood behavior, some studies have focused on a single variable across multiple stations within a multivariate framework [4,5]. These multivariate frameworks improve the realism of flood representation and support more reliable design criteria and risk management strategies in complex hydrological contexts.

A central limitation of conventional FFA, whether univariate or multivariate, is the assumption of stationarity. As defined by Milly et al. [6], stationarity implies that the probability distribution of a variable remains constant over time. However, growing evidence indicates that this assumption is often invalid due to climatic and non-climatic drivers, undermining the applicability of traditional statistical methods [7,8,9]. In response, a large body of research has focused on non-stationary FFA, with applications spanning diverse regions and methodological approaches [10,11].

Most non-stationary flood frequency studies rely on extreme value distributions such as GEV and Gumbel, combined with trend-detection techniques like the Mann–Kendall test. Time is typically used as the main covariate, consistently revealing linear trends, while large-scale climate indices such as ENSO, PDO, NAO, and global mean temperature have also been incorporated, highlighting their significant role in flood frequency dynamics [12].

While these non-stationary approaches provide a more flexible and realistic framework, their adoption also presents important challenges. First, they involve greater mathematical and computational complexity, as the number of parameters increases and convergence of iterative estimation can be difficult. Second, their performance depends critically on the appropriate selection of covariates (e.g., large-scale climatic indices or land-use metrics). When relevant covariates are unavailable or unstable, there is a risk of model misspecification and reduced predictive skill. Third, despite their flexibility, non-stationary models cannot fully eliminate the uncertainty in projecting future extremes, since climatic and land-use conditions evolve in ways that may not be captured by historical relationships [7]. Finally, the assumptions underlying the selected statistical distributions (such as GEV or Gumbel) may not always hold across diverse hydrological settings, requiring careful validation before generalization [13]. In addition, when copula-based approaches are used to model dependence, their validity depends strongly on the copula family selected, which may introduce further subjectivity and potential misspecification [1,2]. Recognizing these limitations is essential to ensure that non-stationary bivariate models are applied responsibly and interpreted with due caution in flood frequency analysis.

In the multivariate domain, copula-based methods have become the prevailing approach for modeling flood dependence structures. Copulas are flexible because they separate marginal behavior from the joint dependence function, and they have been applied extensively in hydrology [2,3,4]. However, they typically require the prior specification of copula families and can be computationally demanding when extended to non-stationary contexts. By contrast, logistic models for extremes [14,15] provide a direct parametric representation of dependence, allowing both marginal distributions and the dependence structure to evolve with covariates. This framework offers a simpler alternative to copulas while retaining the ability to capture changing joint dynamics.

Hydrological regionalization studies traditionally test homogeneity using L-moment-based approaches [16] or related statistical tests [17]. In this study, we complement these classical approaches by applying Andrew’s curves [18], a visualization-based technique that clusters stations according to morphometric attributes and their relationship with return levels, thereby providing an alternative justification of regional homogeneity prior to multivariate modeling [19,20].

Extending FFA to bivariate non-stationary marginals therefore represents a promising avenue to better capture the joint dynamics of flood-related variables under changing conditions. In this study, we address this gap by applying a bivariate logistic model with non-stationary Gumbel and Generalized Extreme Value (GEV) marginals to flood frequency analysis.

2. Materials and Methods

According to Durocher et al. [20], at-site frequency analysis based on annual maxima has a long-standing tradition in hydrology and remains one of the most widely adopted methods for quantile estimation [21]. Theoretical principles support the use of the GEV distribution, which represents the limiting distribution of block maxima [22]. In addition to the GEV distribution, this study also applies the Gumbel distribution in its stationary form, as part of the modeling framework [15].

Gumbel and GEV distributions:

$$F\left(x\right)=e^{-e^{-\left({\displaystyle \frac{x-v}{\propto }}\right)}},$$

(1)

$$F\left(x\right)=e^{-{\left[1-\left({\displaystyle \frac{x-v}{\propto }}\right)\beta \right]}^{{\displaystyle \frac{1}{\beta }}}}.$$

(2)

2.1. Bivariate Logistic Model

Yue and Wang [14] emphasize that the bivariate logistic model (LM) is more adaptable than the mixed model, as it accommodates a broader range of correlation coefficients and dependency indices. This flexibility makes it a preferable choice for addressing frequency analysis problems in hydrology. The LM is defined as follows [15]:

$$F\left(x,y,m\right)=exp\left\{-{\left[{\left(-lnF\left(x\right)\right)}^m+{\left(-lnF\left(y\right)\right)}^m\right]}^{1/m}\right\},$$

(3)

where $F\left(x\right)$ and $F\left(y\right)$ represent the marginal distribution functions, and $m (m>1)$ is the bivariate association parameter. When $m=1$, the bivariate distribution simplifies to the case of independence: $F\left(x,y,m\right)=F\left(x\right)F\left(y\right)$.

Using notation 1 for the Gumbel distribution and notation 2 for the GEV distribution, the possible combinations of bivariate extreme value (BEV) distributions are denoted as BEV11, BEV12, BEV21, and BEV22 [4,5]. The probability distribution and density function for each BEV combination are shown in Appendix A along with the return level functions.

2.2. Estimation of Bivariate Parameters

The likelihood function for $n$ random variables is defined as the joint density of those variables and serves as a function of the parameters. Given a random sample $\left(x_1,y_1\right)$ from a bivariate density, the corresponding likelihood function is formulated accordingly.

$$L\left(x,y,{\theta }_i\right)={\prod }_{i=1}^{n}f\left(x_i,y_i,{\theta }_i\right).$$

(4)

Due to unequal record lengths in the analyzed samples (Figure 1 and Figure 2), a flexible formulation is required to accommodate all possible data arrangements. This formulation is based on the generalization proposed by Anderson [23].

The likelihood function corresponding to the maximum sample arrangement is derived accordingly.

$$L\left(x,y,{\theta }_i\right)={\left[{\prod }_{i=1}^{n_1}f\left(p_i,{\theta }_i^{\left(1\right)}\right)\right]}^{I_1}·{\left[{\prod }_{i=1}^{n_2}f\left(x_i,y_i,{\theta }_i^{\left(2\right)}\right)\right]}^{I_2}·{\left[{\prod }_{i=1}^{n_3}f\left(q_i,{\theta }_i^{\left(3\right)}\right)\right]}^{I_3},$$

(5)

where the following abbreviations are used:

• $n_1,n_3=$ univariate lengths of record before and after the common period $n_2$
• $n_2=$ common bivariate period
• $\mathrm{p}=$ variable with univariate record before the common period $n_2$
• $\mathrm{q}=$ variable with univariate record after the common period $n_2$
• $\left(x,y\right)=$ variable with bivariate record during the common period $n_2$
• $I_i=$ indicator number such that $I_i$ = 1 if $n_i>0$ or $I_i$ = 0 if $n_i=0$
• ${\theta }_i$ = parameter vector

Since the maximum of a function and its logarithm occur at the same point, and the logarithmic form of the maximum likelihood Equation (5) is easier to manipulate, the logarithmic likelihood function is employed.

$$lnL\left(x,y,{\theta }_i\right)=I_1\left[{\sum }_{i=1}^{n}lnf\left(p_i,{\theta }_i^{\left(1\right)}\right)\right]+I_2\left[{\sum }_{i=1}^{n}lnf\left(x_i,y_i,{\theta }_i^{\left(2\right)}\right)\right]+I_3\left[{\sum }_{i=1}^{n}lnf\left(q_i,{\theta }_i^{\left(3\right)}\right)\right].$$

(6)

Maximum likelihood estimators for the parameters of bivariate extreme value distributions are obtained by maximizing Equation (6). Due to the complexity of the functions associated with bivariate probability densities, analytical solutions via differential calculus are not feasible. Instead, a direct search optimization process is required. Given the characteristics of the function to be optimized, Rosenbrock [24] recommended a restricted multivariate nonlinear optimization algorithm.

2.3. Non-Stationary GEV Models and Covariates

Coles [22] gives an example of an annual maximum series $X_t$ to which the GEV distribution can be applied, even though its record has changed linearly over the observation time, but in other respects, the distribution does not change. Using the notation $GEV\left(\nu ,\alpha ,\beta \right)$ to denote the GEV distribution with parameters $\nu ,\alpha$ and $\beta$, it follows that a suitable model for $X_t$, the annual maximum value in a year t, can be

$$X_t~GEV\left(\nu \left(t\right),\alpha ,\beta \right), \mathrm{where} \nu \left(t\right)={\nu }_0+{\nu }_{1}t,$$

(7)

for parameters ν₀ and ν₁. In this way, variations over time in the observed process are modeled as a linear trend in the location parameter of the appropriate extreme value model, which in this case is the GEV distribution. The parameter ν₁ corresponds to the annual rate of change in the annual maximum.

According to Coles [22], a different situation that may arise is that the extreme behavior of one series is related to that of another variable, called a covariate. For example, the values of a series of maximums may appear larger in years when the Southern Oscillation Index (SOI) is higher. This suggests the following model for $X_t$:

$$X_t~GEV\left(\nu \left(t\right),\alpha ,\beta \right), \mathrm{where} \nu \left(t\right)={\nu }_0+{\nu }_{1}SOI\left(t\right),$$

(8)

and SOI(t) denotes the Southern Oscillation Index in year t.

There is a structural unit in all these examples. In each case, the extreme value parameters can be written in the form

$$\theta \left(t\right)=h\left(X^{T}B\right),$$

(9)

where $\theta$ denotes $\nu ,\alpha$, or $\beta$, $h$ is a specific function, $B$ is a vector of parameters, and $X$ is a model vector.

One advantage of maximum likelihood over other parameter estimation techniques is its adaptability to changes in model structure. Take, for example, a non-stationary GEV model to describe the distribution $X_t$ for $t=1,\dots ,n$:

$$X_t~GVE\left(\nu \left(t\right),\alpha \left(t\right),\beta \left(t\right)\right),$$

(10)

where each parameter has an expression in terms of a vector of parameters and covariates of the type (9). Denoting the complete parameter vector as B, the likelihood is simply:

$$L\left(B\right)={\prod }_{t=1}^{n}g\left(x_t;\nu \left(t\right),\alpha \left(t\right),\beta \left(t\right)\right),$$

(11)

where $g\left(x_{\mathrm{t}};\nu \left(t\right),\alpha \left(t\right),\beta \left(t\right)\right)$ denotes the GEV density function with parameters $\nu \left(t\right),\alpha \left(t\right),\beta \left(t\right)$ evaluated at $x_{\mathrm{t}}$.

2.4. Covariate Choice and Functional Form

The covariates used to model non-stationarity in the location parameter were time, representing long-term gradual trends, and the large-scale climate indices PDO and SOI, representing interannual variability. The selection of these covariates was informed by preliminary correlation analyses and trend detection tests (Spearman and Mann–Kendall), which consistently indicated significant temporal trends and climatic teleconnections at the regional scale [15,22]. We adopted a linear form in the location parameter as a parsimonious first-order representation of these relationships, consistent with previous non-stationary extreme value studies [7,8,13]. While nonlinear terms or alternative covariates (e.g., precipitation anomalies, land-use metrics) may capture additional variability, our model comparison results showed limited improvements relative to linear formulations, suggesting that linear terms provide a robust and interpretable baseline [9].

2.5. Choice of Non-Stationary Model

Coles [22] presents an alternative method for quantifying uncertainty in maximum likelihood estimators using the deviation function, defined as

$$D=2\left\{{\mathcal{l}}_1\left(M_1\right)-{\mathcal{l}}_0\left(M_0\right)\right\},$$

(12)

Here, ${\mathcal{l}}_0\left(M_0\right)$ and ${\mathcal{l}}_1\left(M_1\right)$ represent the maximized log-likelihoods under models M₀ and M₁, respectively. A high value of $D$ indicates that model M₁ explains significantly more variation in the data than M₀, while a low value suggests that increasing model complexity does not enhance explanatory power.

Model M₀ is rejected at a significance level α if $D>c_{\alpha }$, where $c_{\alpha }$ is the (1−α) quantile of the ${\mathcal{X}}_k^2$ distribution, and k is the difference in dimensionality between models M₁ and M₀. This provides a formal criterion for preferring model M₁ based on the magnitude of D.

To reduce statistical uncertainty and design safer hydraulic structures, this study proposes analyzing annual maximum discharges using a bivariate logistic model with non-stationary marginals. Figure 3 presents a schematic representation of the methodology followed, highlighting the main steps from data collection to flood frequency modeling.

2.6. Regional Homogeneity

The hydrological homogeneity of the study region was assessed using Andrew’s curves [18], which is a multivariate visualization technique that maps each observation into a continuous function of the form

$$f\left(t\right)={\displaystyle \frac{X_1}{\sqrt{2}}}+X_{2}sin\left(t\right)+X_{3}cos\left(t\right)+X_{4}sin\left(2t\right)+X_{5}cos\left(2t\right)+\cdots ,$$

(13)

where $X_1$, $X_2,$ … $X_n$ are standardized attributes of each station, and the function is plotted over the range −π to π.

Attributes correspond to morphometric and climatological properties and their empirical relation to return levels. Candidate attributes were screened via multiple regression following Nathan and McMahon [19] and standard hydrologic practice [13,16,17], retaining drainage area (A), mean annual precipitation (MAP), and drainage density (DD) as most relevant for regionalization.

3. Results

3.1. Data

Eight stations with at least 30 years of records, located in a homogeneous region of Sinaloa in northern Mexico, were selected to apply the bivariate logistic model with non-stationary marginals for flood frequency analysis. Data were obtained from the official database of the National Commission of Water (CONAGUA) [25].

Table 1. Some characteristics of the stations analyzed.

Station	Period	Length of Record (Years)	Mean	Standard Deviation	Kurtosis	Skewness	Variation Coefficient
10027	1938–1995	58	285.79	260.83	13.68	2.74	0.91
10065	1953–1999	47	1218.73	1081.76	13.59	2.84	0.89
10066	1955–2005	51	311.89	278.67	16.36	3.18	0.89
10079	1959–1999	41	1016.91	1642.41	19.86	3.74	1.62
10083	1960–1992	33	458.64	425.13	5.85	1.58	0.93
10086	1960–1992	33	236.33	156.80	4.48	1.19	0.66
10111	1958–2009	52	1315.93	1558.22	14.23	3.03	1.18
10137	1958–2008	51	1058.15	992.91	7.59	2.02	0.94

lists record lengths and basic statistics. Outliers were screened (box plot, Grubbs test, three-sigma), independence verified (Anderson–Mantilla–Amigó), homogeneity tested (Helmert, Student’s t, Cramer, Pettit, Standard Normal, Buishand, Von Neumann), and trends assessed (Spearman, Mann–Kendall). Results are shown in

Table 2. Independence, Homogeneity, and Trend Test Results.

Station	Independent?	Homogeneous?	Trend?	Increasing?
10027	Yes	Yes	No	Yes
10065	Yes	Yes	No	No
10066	Yes	No	No	No
10079	Yes	Yes	Yes	Yes
10083	Yes	Yes	No	Yes
10086	Yes	No	Yes	Yes
10111	Yes	Yes	No	No
10137	Yes	Yes	No	No

.

Station 10086 exhibited an increasing trend, confirmed by graphical analysis (Figure 4) and statistical tests.

3.2. Delineation of Homogeneous Region

Attributes such as drainage area (A), mean annual precipitation (MAP), and drainage density (DD) were compiled and standardized (

Table 3. Raw values of drainage area (A), mean annual precipitation (MAP), and drainage density (DD) for study stations.

Station	A (km2)	MAP (mm)	DD (km/km2)
10027	388.259	912.9	0.2135
10065	6102.181	871.5	0.2474
10066	1373.576	777.45	0.2471
10079	1010.891	1031.19	0.2746
10083	827.432	754.8	0.2750
10086	226.588	831.2	0.2356
10111	5277.521	993.5	0.2471
10137	3300.188	982.9	0.2396

and

Table 4. Standardized morphometric and climatic attributes for study stations.

Station	Area (km2)	MAP (mm)	DD (km/km2)
10027	0.169	8.855	10.579
10065	2.656	8.454	12.259
10066	0.598	7.542	12.244
10079	0.440	10.003	13.607
10083	0.360	7.322	13.626
10086	0.099	8.063	11.674
10111	2.297	9.637	12.244
10137	1.436	9.535	11.872

). Stations from a homogeneous region tend to produce Andrews curves that cluster, indicating similar hydrological behavior and flood-generating mechanisms. The clustering observed in Figure 5 confirmed that the eight stations form a hydrologically homogeneous region, justifying joint bivariate non-stationary analysis.

This validated homogeneity provides a solid basis for applying the bivariate non-stationary framework in the subsequent frequency analysis.

3.3. Frequency Analysis

After verifying independence and randomness, the Gumbel (1) and GEV (2) distributions were fitted, followed by four bivariate marginal combinations (6). Models with (D > 3.84) were retained, and the one with highest log-likelihood and lowest AIC was selected at each site.

The Akaike Information Criterion (AIC) is defined as

$$AIC=-2l+2\theta$$

(14)

where l is the maximum log-likelihood and θ represents the number of estimated parameters within the model.

Table 5. Parameter estimates for Station 10086 under stationary and non-stationary univariate and bivariate models.

Distribution	Covariate	M	${\mathit{v}}_{01}$	${\mathit{v}}_{11}$	${\propto }_1$	${\mathit{\beta }}_1$	${\mathit{v}}_{02}$	${\mathit{v}}_{22}$	${\propto }_2$	$\mathcal{l}$	AIC	D
Gumbel	-	-	168.54		108.89					−203.88	411.75
GEV	-	-	155.56		96.66	−0.235				−204.12	414.23
BEV 21-10137	-	1.11	154.55		92.42	−0.094	681.09		574.59	−200.14	406.28
BEV 21-10137	Time	1.10	105.82	2.77	86.74	−0.123	745.27	−2.45	573.55	−198.55	405.11	11.13

presents detailed results for Station 10086 (stationary and non-stationary, univariate and bivariate).

Table 6. Estimated return levels (m³/s) for Station 10086 under stationary and non-stationary models across different return periods.

Distribution	Scenario	t	T (years)
			2	5	10	20	50	100
Gumbel	-	-	208.45	331.86	413.58	491.96	593.42	669.44
GEV	-	-	192.55	329.32	442.08	570.59	772.66	955.82
BEV 21-10137	-	-	189.01	303.43	386.20	471.27	590.31	686.62
BEV 21-10137 non-stationary in time	2025	66	320.99	431.32	513.29	599.33	722.62	824.71
	2050	91	390.17	500.51	582.47	668.52	791.81	893.90
	2075	116	459.36	569.69	651.66	737.71	861.00	963.08
	2125	166	597.74	708.07	790.03	876.08	999.37	1101.46

summarizes return levels across return periods under stationary and non-stationary assumptions. To complement tabular results, Figure 6 visualizes the differences in return levels between stationary GEV and non-stationary BEV21, making the divergence across time explicit. Non-stationary estimates reveal increasing return levels over time, underscoring the importance of incorporating trends into flood risk analysis.

Finally,

Table 7. Best-fitting distributions for stations in the homogeneous region.

Station	Best-Fitting Distribution	Covariate	M	${\mathit{v}}_{01}$	${\mathit{v}}_{11}$	${\propto }_1$	${\mathit{\beta }}_1$	${\mathit{v}}_{02}$	${\mathit{v}}_{22}$	${\propto }_2$	$\mathcal{l}$	AIC
10027	BEV 21-10065	SOI	1.18	165.48	−4.68	112.91	−0.219	825.22	−75.56	584.65	−369.18	746.36
10065	BEV 21-10027	Time	1.15	804.27	−1.31	436.67	−0.191	190.17	−0.06	148.03	−361.49	730.98
10066	BEV 21-10079	SOI	1.07	192.23	4.25	103.89	−0.236	527.80	26.00	637.16	−320.53	649.06
10079	BEV 21-10066	Time	1.10	337.40	1.80	270.49	−0.478	257.48	−1.57	142.75	−301.79	611.58
10083	BEV 21-10065	Time	1.34	186.50	0.59	155.50	−0.455	909.26	−2.88	584.78	−223.72	455.45
10086	BEV 21-10137	Time	1.10	105.82	2.77	86.74	−0.123	745.27	−2.45	573.55	−198.55	405.11
10111	BEV 21-10065	Time	2.05	800.38	−6.16	338.99	−0.485	943.79	−2.84	626.07	−395.48	798.95
10137	BEV 21-10066	SOI	1.13	512.21	−44.16	344.23	−0.524	215.82	−4.84	146.25	−389.67	787.35

provides a regional overview of the best-fitting distributions for all stations in the homogeneous region, listing the selected covariates and model characteristics.

Although the analysis presented here focuses on point estimates of return levels, confidence intervals were not computed. We acknowledge that this represents a limitation, as measures of uncertainty are essential for decision-making in hydraulic design and flood risk management. The inclusion of confidence bounds would provide stakeholders with not only the expected magnitude of extreme events but also the range of plausible variability. In this study, the emphasis was on illustrating the methodological contribution of applying non-stationary bivariate models, and therefore the computation of uncertainty bands—through bootstrap resampling or Bayesian credible intervals—was beyond the present scope. We note, however, that these approaches constitute a natural extension of the framework and will be explored in future work to provide a more comprehensive quantification of flood frequency uncertainty [7].

4. Discussion

Traditional stationary univariate models assume that the statistical properties of extreme events remain constant over time. While simple to apply, they neglect temporal changes driven by climate variability, land-use, or anthropogenic impacts. As a result, return levels derived from stationary univariate approaches often misrepresent current and future risks [6].

Stationary bivariate models improve on this by accounting for dependence between two variables (for example, peak flow in two gauged stations), leading to more accurate joint return period estimates than univariate analyses [2]. However, they still impose the limitation of time-invariant marginals, which can bias estimates if the underlying processes evolve.

Traditional assumptions of stationarity are increasingly invalidated by anthropogenic influences such as urbanization, deforestation, and climate change, which significantly alter discharge records. Non-stationary univariate models allow distribution parameters to vary with time or covariates like PDO and SOI, capturing observed or expected trends [7,9,12]. Yet, they continue to overlook the dependence between variables, which is critical for compound flood risk.

The most comprehensive approach is the non-stationary bivariate model, which jointly models dependence while allowing marginals to evolve with time or climatic covariates [2,8,22,26,27]. This reduces misspecification bias in joint-tail estimation and improves return-level accuracy, while providing a platform for scenario analysis under future climate/land-use pathways. In practice, non-stationary bivariate models deliver policy-relevant results that reflect evolving hazards and their interactions-advantages confirmed here by the superior log-likelihood and AIC performance.

4.1. Performance of GEV and Gumbel Marginals

The best results were obtained when using GEV and Gumbel marginals to model annual maxima. The GEV distribution, with its shape parameter, offers flexibility for heavy, light, or bounded tails, while the Gumbel distribution, as a special case of the GEV, provides a parsimonious yet robust fit [21,22]. At both gauges, the fitted marginals captured the statistical properties of annual maxima.

The GEV was more suitable at the site with higher variability and occasional extreme peaks, while the Gumbel distribution fit better at the more regular site. This differentiation reflects catchment differences and translates to practice: return period estimates become more reliable when tailored to local discharge behavior.

4.2. Case Study: Station 10086

The analysis of station 10086 revealed a statistically significant increasing trend, detected by both the Spearman rank correlation test and the Mann–Kendall test [28,29]. This trend is reflected in the positive value of the location parameter ($v_{11}$) of the non-stationary model. Figure 5 illustrates the importance of explicitly incorporating non-stationarity when modeling extremes.

For example, under stationary assumptions, the GEV distribution estimates a 50-year return flow of 772.66 m³/s. In contrast, the non-stationary BEV21 model projects a 50-year return flow of 861.00 m³/s for the year 2075 scenario. This difference implies that a discharge of 772.66 m³/s, considered a 50-year event under stationary assumptions, would be expected approximately every 30 years by 2075 when non-stationarity is accounted for. Such a misrepresentation has direct implications for hydraulic design, as ignoring trends may lead to underestimation of design floods and compromise the safety and resilience of hydraulic infrastructure [30].

4.3. Practical and Policy Implications

The adoption of non-stationary bivariate models has important implications for both engineering design and water policy. For engineers, it means that hydraulic structures—such as dams, levees, and urban drainage systems—can be designed with greater confidence and safety margins, reflecting realistic compound risks under changing conditions. For policymakers, non-stationary bivariate approaches support adaptive management strategies, allowing the testing of “what-if” scenarios under different climate or socio-economic pathways. This provides a sound basis for long-term investment decisions, disaster risk reduction strategies, and compliance with national and international climate adaptation frameworks [31].

Ultimately, the progression from stationary to non-stationary, and from univariate to bivariate analysis, represents a critical methodological advance in flood frequency analysis. It ensures that design standards and water policies evolve in step with changing climatic and hydrological realities, providing a more resilient and cost-effective basis for flood risk management.

While these findings underscore the value of non-stationary bivariate models, it is also essential to recognize their inherent limitations, which qualify the interpretation of results and indicate directions for future research.

4.4. Limitations of the Proposed Models

Despite the advantages of non-stationary bivariate models, several limitations must be acknowledged. First, the assumption of linear tendencies in the marginal parameters may not be universally valid. Sudden shifts in climate conditions or nonlinear changes in land-use may lead to dynamics that cannot be fully captured by linear formulations, potentially biasing return level estimates. Second, the bivariate logistic model employed here assumes a symmetric dependence structure, which may not adequately represent more complex or asymmetric relationships between flood variables. This limitation could affect joint-tail estimation in catchments where dependence varies across different ranges of extremes. Finally, as with all statistical models, results are constrained by data availability and the inherent uncertainties of projecting future hydrological extremes. Recognizing these limitations is essential for balanced interpretation and highlights the need for future research exploring nonlinear covariate effects and alternative dependence structures (e.g., asymmetric copulas [1]) to further enhance the robustness of flood frequency analysis.

5. Conclusions

This study confirms that non-stationary bivariate models provide the most reliable estimates of return levels, since they capture both the dependence between variables and the temporal evolution of extremes. Compared with stationary or univariate approaches, this framework reduces bias, improves accuracy, and supports scenario-based risk assessment under changing climatic and land-use conditions.

The use of GEV and Gumbel marginals yielded the best overall performance for the maximum discharges at the two gauging sites. The GEV distribution was particularly suited to the site with greater variability and occasional extremes, while the Gumbel distribution offered a parsimonious yet robust fit at the more stable site.

Importantly, the case of station 10086 illustrates how trend detection alters return level estimates. A discharge of 772.66 m³/s, considered a 50-year event under stationary assumptions, is projected by the non-stationary model to occur every ~30 years by 2075. Ignoring such non-stationarity could lead to systematic underestimation of design floods and compromise the safety of hydraulic infrastructure.

The scientific contribution of this work lies in demonstrating the added value of combining a bivariate logistic model with non-stationary marginals for flood frequency analysis. By explicitly incorporating both dependence structures and time- or climate-driven variability, this framework advances beyond traditional copula-based or stationary approaches and provides a transparent, data-driven methodology applicable to regional flood risk management.

Nevertheless, uncertainties remain. The assumption of linear covariate effects and symmetric dependence may not fully represent nonlinear climatic shifts or asymmetric joint behaviors, and future research should address these issues by exploring alternative functional forms, additional covariates, and flexible dependence structures such as asymmetric copulas. Moreover, the lack of confidence intervals around return level estimates is a limitation that will be addressed in future extensions through bootstrap or Bayesian approaches, to provide decision-makers with explicit measures of uncertainty.

In sum, this study underscores the importance of adopting flexible non-stationary approaches for both engineering design and water policy, while also recognizing the methodological challenges that remain. By tailoring flood frequency analysis to evolving hydrological regimes, decision-makers can avoid under- or over-design, improve safety margins, and align planning with climate adaptation frameworks.