Generalities

Design rainfalls are maximum precipitations of a certain duration associated to low exceedance probabilities, based on which, by means of hydrological methods, the Design Floods are estimated, when hydrometric records are not available. Details of the above can be found in ^{Teegavarapu (2012)} and ^{Mujumdar and Nagesh Kumar (2012)}. Design floods allow the sizing of hydraulic works, during their planning or when they are checked for safety. The most reliable estimate of design rainfalls is based on the probabilistic analysis (AP) of the annual maximum data, which assumes that the random process that generates the observed precipitations is stationary and therefore, its statistical properties do not change over time. In the AP, a known probability distribution function (FDP) is adjusted to the precipitation record and based on it, the predictions sought or design rainfalls are made. Due to the scarcity of rainfall recorder stations and the relative abundance of rain-gauge stations, the most commonly processed extreme rainfall records are those of the annual maximum daily precipitation (PMD).

At present, all rain-gauge stations are subject to the effects of global climate change, or their geographical environment suffers physical changes generated by human activities, among the most striking are urbanization, deforestation, dewatering of lagoons and the construction of reservoirs, which alter local atmospheric processes, giving rise to series or records of annual non-stationary PMD, since they present trends (^{Strupczewski & Kaczmarek, 2001}; ^{Khaliq, Ouarda, Ondo, Gachon, & Bobée, 2006}; ^{El Adlouni, Ouarda, Zhang, Roy, & Bobée, 2007}; ^{El Adlouni & Ouarda, 2008}).

To perform the AP of non-stationary records, the statistical theory of extreme values has been extended to adjust the classical model which follows asymptotically the maximum hydrological data series, that is, the FDP General of Extreme Values (GVE₀) stationary with three fit parameters (u,α,k). The above based on the introduction of so-called covariables, one of the most used is the time (t) in years, through which the trend observed in the data series can be taken into account, adopting variable the location parameter (u _t ). When the trend is linear (u _t = μ ₀ + μ ₁·t) the GVE₁ model of four fit parameters (μ ₀, μ ₁, α, k) is fitted. If the trend is curved (u _t = μ ₀ + μ ₁·t +μ ₂·t ²) the GVE₂ model of five fit parameters (μ ₀, μ ₁, μ ₂, α, k) is applied; in this model two covariables (u _t = μ ₀ + μ ₁·t +μ ₂·h) can be used (^{Khaliq et al., 2006}; ^{Prosdocimi, Kjeldsen, & Miller, 2015}).

Regarding to other non-stationary distributions, there are several, for example the Log-Normal (^{Vogel, Yaindl, & Walter, 2011}; ^{Aissaoui-Fqayeh, El Adlouni, Ouarda, & St. Hilaire, 2009}) and the Generalized Logistics models (^{Kim, Nam, Ahn, Kim, & Heo, 2015}) and Generalized Pareto (^{Rao & Hamed, 2000}), which are susceptible to an identical treatment to the one that will be exposed for the GVE distribution. Other covariates can also be used instead of time (t), for example some global or regional climate indexes (^{Prosdocimi, Kjeldsen, & Svensson, 2014}; ^{López-de-la-Cruz & Francés, 2014}; ^{Álvarez-Olguín & Escalante-Sandoval, 2016}; ^{Campos-Aranda, 2018}).

Objective

The objective of this study was to present the fitting theory of the non-stationary distributions GVE₁ and GVE₂, by means of the generalization of the method of L moments, initially proposed by ^{El Adlouni and Ouarda (2008)} and subsequently applied by ^{Gado and Nguyen (2016)}. In addition, four series of annual PMD are exposed and processed, all with trend and the results of the fitting of the model GVE₁ or GVE₂ are described, highlighting the simplicity and usefulness of the extension of the L-moments method to obtain the predictions within the historical record and to the future.

General of Extreme Values Distribution

The maximum annual values are events that occur in the right tail of the FDP, which defines the behavior of the hydrological random variable. Therefore, the design rainfalls can be predicted based on the FDP, as the maximum value corresponding to a certain average interval of recurrence or return period (Tr), whose probability of exceedance is p = 1/Tr. The theory of extreme values established that the extreme data follow asymptotically to the FDP General of Extreme Values (GVE), whose application has been recommended (^{Stedinger, Vogel, & Foufoula-Georgiou, 1993}; ^{Hosking & Wallis, 1997}; ^{Coles, 2001}; ^{Khaliq et al., 2006}; ^{Papalexiou & Koutsoyiannis, 2013}; ^{Gado & Nguyen, 2016}) to model extreme hydrological data (dh) with a probability of non-exceedance [F(dh)] that is:

In the previous expression, u, α and k are the location, scale and shape parameters of the GVE₀. When k = 0 the Gumbel distribution is obtained, which is a straight line in the Gumbel-Powell probability paper, whereby the interval of the variable is: -∞< dh <∞. When k > 0 the distribution is Weibull which is a curve with concavity downwards and upper limit, whereby: -∞< dh ≤ u + α/k. Finally, if k < 0 the distribution is Fréchet which is also a curve but with a concavity upwards and lower border, so: u+α/k ≤ dh <∞. The predictions sought (DH _Tr ) are obtained with the inverse solution of Equation 1:

L-Moments of the data sample

The L-moments method is perhaps the simplest of the reliable procedures for estimating the fitting parameters of the PDFs used in hydrology. This is due to the fact that the L-moments, which are linear combinations (^{Hosking & Wallis, 1997}) of the weighted probability moments (βr), are not affected significantly by the scattered values (outliers) of the sample. The first three L-moments of a sample (l ₁, l ₂, l ₃) and the asymmetry L-ratio (t ₃), are estimated through the unbiased estimator (b _r ) of the β _r , as follows:

The unbiased estimator of the β _r is (^{Hosking & Wallis, 1997}):

where r = 0, 1, 2,…. and dh _i are the data of the available sample or series of hydrological data of size n, ordered from low to dh1≤dh2≤⋯≤dhn.

Fit parameters of the stationary distribution GVE

For the GVE₀ model with the L-moments method, its three fit parameters are calculated with the equations (^{Stedinger et al., 1993}; ^{Hosking & Wallis, 1997}; ^{Rao & Hamed, 2000}; ^{Campos-Aranda, 2018}):

being:

For the estimation of the Gamma Γ(ω) function the Stirling formula (^{Davis, 1972}) was used, shown in the following equation (12):

AP non-stationary with GVE₁ and GVE₂

According to ^{Strupczewski and Kaczmarek (2001)} the fit of the non-stationary distributions GVE₁ and GVE₂ implies two assumptions: (1) it is accepted that the non-stationarity of the annual PMD series is caused by gradual changes in the geographical environment and/or global climate change, generating a slight alteration of its statistical parameters and (2) it is accepted that the FDP is independent of time, so the distribution GVE, with fit parameters that are variable over time, is acceptable for modeling extreme non-stationary data. Therefore the mean (μ _t ) of the distribution GVE must be considered variable over time whose expression is (^{Rao & Hamed, 2000}):

Predictions in the Abritas station, S.L.P.

First, equations of Appendix 1 were applied and μ ₀ = 167.6315, μ ₁ = -1.6057 y r _xy = -0.4619 were obtained. Then the linear trend was tested with equations of the Appendix 2, a DS = -3.7558 and DS _c = 2.0066 were obtained, which was highly significant. The stationary series (PMDeμ, Equation 15) accepts a GVE₀ model with scale and shape parameters of: α = 41.4364 and k = 0.0691; however, Equation 2 using the variable location parameter (Equation 14) leads to a 34.8 mm standard error of fit (EEA). Table 2 shows part of the predictions within the historical record and an extrapolation to a decade (year 2025), since at future they are smaller due to the downward slope. Predictions with the stationary model GVE₀ of return periods 2 and 100 years are: 114.9 and 290.8 mm, with an EEA = 6.0 mm. Figure 1 shows the chronological series of the data and the lines of the predictions, which are plotted, based on the predictions in Table 2, within the historical record with the values of the covariate t that vary from 1 to n.

Figure 1 Chronological series of annual PMD and estimated prediction lines with the distribution GVE₁ in the Abritas station, S.L.P., Mexico. 

Predictions in the Zacatecas, Zac. Station

When applying the equations of Appendix 1: μ ₀ = 40.1247, μ ₁ = 0.2246 and r _xy = 0.2638 was obtained. The linear trend was then tested with equations of Appendix 2, obtaining a DS = 2.0463 and a DS _c = 2.0032, whereby it was scarcely significant. The stationary series (PMDeμ Equation 15) accepts a GVE₀ model with scale and shape parameters of: α =11.8358 and k = 0.0709: however, Equation 2 using the variable location parameter (Equation 14) leads to a 3.0 mm standard error of fit. Table 3 shows a part of the predictions within the historical and to future record in the years 2025, 2050 and 2100. As this record of 58 data ends in the year 2010, then, the value of time t in 2025 is 73, in 2050 is 98 and 2100 is 148. The predictions with the stationary model GVE₀ of return periods 2 and 100 years are: 45.2 and 86.7 mm, with an EEA = 1.5 mm. Figure 2 shows the chronological series of data and lines of the predictions, plotted according to the results of Table 3.

Figure 2 Chronological series of annual PMD and estimated prediction lines with the distribution GVE₁ in Zacatecas station, Zac., Mexico.  

Predictions in the Las Adjuntas station, S.L.P.

When applying equations of Appendix 1, μ ₀ = 74.9213, μ ₁ = 0.8813 and r _xy = 0.2981 was obtained. The linear trend was then tested with the equations in Appendix 2, a DS = 2.2520 and a DS _c = 2.0066 were obtained, therefore it was moderately significant. The stationary series

(PMDeμ Equation 15) accepts a GVE₀ model with scale and shape parameters of α = 26.5840 and k = -0.1617; however, Equation 2 using the variable location parameter (Equation 14) leads to a 15.6 mm standard error of fit. Table 4 shows a part of the predictions within the historical record and to future in the years 2025, 2050 and 2100. As this record of 54 data ends in the year 2015, then, the value of time t in 2025 is 64, in 2050 is 89 and 2100 is 139. The predictions with the stationary model GVE₀ of return periods 2 and 100 years are: 87.4 and 280.9 mm, with an EEA = 11.3 mm. Figure 3 shows the chronological series of data and lines of the predictions, plotted according to the results of Table 4.

Figure 3 Chronological series of annual PMD and estimated prediction lines with the GVE1 distribution in Las Adjuntas station, S.L.P., Mexico. 

Predictions in Los Filtros station, S.L.P.

Equations in Appendix 3 lead to the following values: μ ₀ = 43.7097, μ ₁ = -0.2339 and μ ₂ = 0.0049 as coefficients of the polynomial regression (equation A3.1). The stationary series (PMDeμ Equation 17) accepts a GVE₀ model with scale and shape parameters of: α =12.6389 and k = 0.05071: however, Equation 2 using the variable location parameter (Equation 16) leads to a 4.3 mm standard error of fit. Table 5 shows a part of the predictions within the historical record and to future in the years 2025 and 2050. As this record of 66 data ends in the year 2016, then, the value of time t in 2025 is 77 and in 2050 is 102. The predictions with the stationary model GVE₀ of return periods 2 and 100 years are: 41.5 and 87.5 mm, with an EEA = 3.7 mm. Figure 4 shows the chronological series of data and curves of predictions, plotted based on the results of Table 5.

Figure 4 Chronological series of annual PMD and estimated prediction curves with the distribution GVE₂ in Los Filtros station, S.L.P., Mexico. 

Predictions in the future

The four numerical applications described cover the most common cases that are known of non-stationary PMD _i records, which correspond to series with downward (severe) and upward (mild and moderate) linear trend and with upward curve trend (mild in an appreciative context) towards the future. Upward trends usually occur in rain-gauge stations located in cities, or where a large nearby reservoir has been built. In the first case there is an increase in temperature due to the effects of the heat island and in the second there may be an increase in relative humidity due to evaporation. Downward trends may be associated to regional climate change.

Naturally, the predictions associated with low probabilities of exceedance are less important in records with a downward trend, because they are lower in the future. The opposite occurs in series with upward trend, in which it is necessary to explain or justify the probable physical origin of such trend, to accept extrapolations of the predictions to future and to try to discern about the real scope of them, since it is extremely risky to adopt the behavior of trend in increase, when, for example, it is not known if urban development will continue.