Flood Frequency Analysis (FFA) Overview

Flood Frequency Analysis (FFA) is one of the basic estimates of Surface Hydrology, since it allows the planning, design and review of all the waterworks required by society, being these for use like reservoirs, or for protection like the embankments, channelings, control dams, bridges, urban drainage works, etc. FFA tries to define the relationship between the maximum flows of annual floods and their probability of being exceeded. Selected such probability or risk for Design Flood, its value should be estimated as accurately as possible, since a default error leads to an increase in the adopted risk, and an excess error leads to an increase in the cost of the project (^{Botto, Ganora, Laio, & Claps, 2014}).

The most accurate procedure for performing an FFA is to represent the available record of annual maximum flows by a probability distribution function (PDF) and based on such a probabilistic model, make the inferences sought or predictions. This technique can lead to inaccurate estimates mainly due to the following four factors: (1) measurement errors in data; (2) finite amplitude of the available record; (3) presence of different mechanisms for generating the floods and (4) choice of PDF (^{Merz & Blöschl, 2008}).

^{Francés-García (1995)} states that historically the evolution of the FFAs has gone through three stages: (I) search for and fit of a better PDF to the annual maximum flows until the 1960s; (II) improved methods for estimating PDF fitting parameters during the 1970s and (III) increase in estimate accuracy, since the 1980s, by increasing the maximum flow records, through regional analysis and/or incorporation of flood historical information.

A fourth evolutionary stage of FFAs, beginning in this 21st century, refers to the processing of non-stationary records, because they show trends and/or increase or decrease in variability, as a result of physical changes in basins or climate change (^{Khaliq, Ouarda, Ondo, Gachon, & Bobée, 2006}; ^{Katz, 2013}; ^{López-de-la-Cruz & Francés, 2014}; ^{Prosdocimi, Kjeldsen & Svensson, 2014}; ^{Campos-Aranda, 2018}).

FFA Based on Mixed PDFs

Since the middle of the last century, the interest for the presence of two statistical properties observed in the records of maximum annual flows raised; the first of them was called “dog leg effect”, named so by ^{Potter (1958)}, and it is observed at the moment of drawing data on the probability paper and finding a sudden slope change, due to the presence of outliers. The second property was named “phenomenon of separation”; it was studied and documented by ^{Matalas, Slack and Wallis (1975)}, who drew on the x-axis the average values of the asymmetry or bias coefficients, and on the y-axis their respective standard deviations from the synthetic records generated with the most common PDFs. By plotting the points in the available maximum flow historical records in each of the 14 hydrologically homogeneous regions of the USA; PDFs were observed to be below actual points, thereby indicating that such PDFs could not represent flood sample variability.

The above-mentioned PDF deficiencies identified by hydrologists used in late 1970s, their solution was proposed through the mixed probabilistic models, which consider the different origins in the floods of a record. Mixed models are applied under two different approaches: (a) independent and separable floods and (b) independent floods which are mixed. The classic example of the first approach is the model suggested by ^{Waylen and Woo (1982)}, applicable when snow melt and rainfall floods are fully identifiable, by their date of occurrence, in the available record of annual maximum flows. Snow melting floods lead to low flows and occur in spring-summer, and rainfall floods lead to higher flows and occur in winter from October to February. For the second approach, PDF mixtures have been proposed, for example: (1) the so-called Double Gumbel suggested by ^{González-Villarreal (1970)}; (2) the Wakeby distribution proposed by ^{Houghton (1978)}, which attempts to model the left and right portions separately; (3) the TCEV model developed by ^{Rossi, Fiorentino, and Versace (1984)}, and (4) the mixed Gumbel function recently applied by ^{Molina-Aguilar, Gutiérrez-López and Aparicio-Mijares (2018)}. ^{Rulfova, Buishand, Roth and Kysely (2016)} proposed the TCGEV (two-component generalized extreme value) distribution of two components with generalized extreme values, such as a more flexible model for the analysis of maximum annual precipitation of six hours duration.

TCEV distribution is described in detail in this article, because it has not been applied systematically in Mexico; neither under precept, nor as an option of the probabilistic models suggested in records of floods with mixed populations.

Objective

In this paper, the theoretical origin of TCEV (two-component extreme value) distribution is detailed, with four fitting parameters, which leads to a probabilistic model appropriate to annual flood records that come from two physically different mechanisms, but that are mixed. The TCEV distribution fitting method through maximum likelihood is described in its two versions: (1) Successive Substitution Method and (2) Optimization Method. With both methods, six records of floods taken from the specialized literature are processed, which vary from 31 to 72 data and have between three and six outliers. Then, six return period predictions for 10, 25, 50, 100, 500 and 1 000 years are contrasted, each with six values coming from the fit of TCEV, Log-Pearson type III, General Extreme Values, Generalized Logistics and Wakeby distributions. Finally, the study conclusions are formulated.

Other Fitting Methods and Applications of TCEV Distribution

^{Francés (1998)} expanded the maximum likelihood method for the joint use of systematic and historical information of floods. ^{Beran, Hosking and Arnell (1986)} put forward three equations to quantify the probability weighted moments (PWM) and indicate that such expressions must be resolved in an iterative manner, which makes them more complicated than the maximum likelihood method. The procedure of the PWM method, has been exposed by ^{Singh (1998)}, as well as the one based on the maximum entropy principle, originally suggested by ^{Fiorentino, Arora and Singh (1987)}.

Moreover, ^{Fiorentino, Versace and Rossi (1985)} were the first ones to state that TCEV distribution showed a good fit in maximum annual daily precipitation records in southern Italy, which were processed as series of partial duration or magnitudes greater than a threshold value. Later, ^{Cannarozzo, D'Asaro, and Ferro (1995)} applied the TCEV distribution regionally in Sicily, Italy, using rainfall and flood records. ^{Ferro and Porto (2006)} use the TCEV model with the regional hierarchical approach in an FFA for Sicily, Italy. Such a regional approach was originally proposed by ^{Fiorentino, Gabriele, Rossi and Versace (1987)}. ^{Boni, Parodi and Rudari (2006)}, as well as ^{Aronica and Candela (2007)} use the TCEV distribution in the regional rainfall analyses they have described. ^{Escalante-Sandoval and Reyes-Chávez (2004)} perform a flood bivariate analysis using TCEV as marginal distributions.

Mixed Models PDF

^{Waylen and Woo (1982)} call R the random variable representing the annual floods caused by precipitation, which are the largest ones, and S to those generated by the melting of snow, which are the basic or minor ones; X_i is a new random variable that is the maximum of both, i is the year and n the amplitude of the available record. As the hypothesis is accepted that R and S are independent, its PDF is equal to the product of both, and each one is estimated separately with the corresponding n data taken from the registry. So, the non-exceedance probability function of X_i is (^{Francés-García, 1995}):

In the simplest case, F_R (x) and F_S (x) are Gumbel PDFs with two fitting parameters, i.e., location and scale, each. The application of five fitting parameter Double Gumbel distribution has been solved in a practical way through not restricted optimization (^{Gómez, Aparicio, & Patiño, 2010}); its PDF is (^{González-Villarreal, 1970}):

In the above equation, p is the probability of having ordinary events. In the Wakeby distribution (^{Houghton, 1978}), the new random variable is defined by the addition, since the annual maximum flood is produced by the combination in time of the two generating mechanisms and then its PDF is explicitly expressed as:

The estimation of its five fitting parameters has been solved by means of the L-moment method (^{Hosking & Wallis, 1997}) and by restricted optimization (^{Campos-Aranda, 2001}). Finally, the mixed Gumbel distribution has been fitted based on modern search algorithms (^{Molina-Aguilar et al., 2018}), its PDF is similar to Equation (3), this is:

TCEV Distribution Genesis

Fitting by Maximum Likelihood

Standard Error of Fit

For purposes of quantitative comparison of the fitting achieved with the TCEV distribution and the method of maximum likelihood, as opposed to other probabilistic models, it was proposed to estimate the standard error of fit (SEF), defined as (^{Kite, 1977}):

being n total number of data, Qo_i the maximum annual flow observed measured from lowest to highest , and Qc_i the maximum flow calculated with the TCEV distribution (Equation (15) or Equation (18)), for the same non-exceedance probability assigned to the observed flow by the Weibull formula (Equation (26)). Finally, np is the number of fitting parameters, in this case four (λ₁ or ε₁, θ₁, λ₂ or ε₂, θ₂).

Since Equation (15), Equation (16), Equation (17) and Equation (18) have no inverse solution, the approach was to use the bisection method with error tolerance equal to 0.0001, between the calculated probability (Equation (26)) and that obtained with Equation (15) or Equation (18), using lower limit values of 0.001 of Qo_i and upper of 3 times Qo_i . The evaluation of the SEF was attempted at the end of the calculation of the initial values of the fitting parameters (Equation (27), Equation (28), Equation (29), Equation (30), Equation (31) and Equation (32)) and sometimes the method did not work; for example, when the TCEV distribution does not match the data. For such cases, the use of the SEF subroutine is suppressed. The other SEF calculation was performed upon completion of the numerical substitution method or Rosenbrock algorithm and generally concluded satisfactorily.

Flood Records to be processed

Six series of maximum annual flows (m³/s) were processed, taken from the specialized literature, which are presented in progressive order of amplitude (n), in Table 1. Its origin is as follows: (1) taken from ^{Haan, Barfield, and Hayes (1994)} for the place called Beargrass Creek with 31 values; (2) the record of 37 values of Santa Cruz station has been exposed by ^{Molina-Aguilar et al. (2018)}; (3) ^{Francés-García (1995)} presented the Turia river one at station E-25 with 41 data; (4) the record of 53 data at Huites station comes from ^{Campos-Aranda (1999)}; (5) ^{Gómez et al. (2010)} showed the record of 58 data in La Cuña station; and finally, (6) ^{Kite (1991)} presented the data of St. Mary’s River, with 72 data.

For the three gauging stations in Mexico, their keys, locations and registration periods are placed immediately. (1) Santa Cruz: 10040, Rio San Lorenzo, of the Hydrological Region No. 10 (Sinaloa), 1944-1980; (2) Huites: 10037, Río Fuerte, of the Hydrological Region No. 10 (Sinaloa), 1941-1993 and (3) La Cuña: 12054, Río Verde, of the Hydrological Region No. 12-3 (Río Santiago), 1947-2004.

Record Homogeneity Verification

For the results of the FFA to be reliable, the data to be used must come from a stationary random process, which implies that it has not changed over time. Then, flood records must be composed of independent data, which are free of deterministic components, for such a record to be homogeneous.

To verify the aforementioned, seven statistical tests were applied, i.e., one general, the Von Neumann test, and six specific: two persistence tests (Anderson and Sneyers), two trend tests (Kendall and Spearman), one for the change in the mean (Cramer) and the last one to look for inconsistencies in dispersion (Bartlett). These tests are available at ^{WMO (1971)}, and ^{Machiwal and Jha (2008)}. All the tests mentioned were applied with a level of significance (α) of 5 % and six of them show that the selected records are homogeneous. Bartlett test detects loss of homogeneity for excess variability due to the presence of outliers.

In addition, the Wald-Wolfowitz test was applied, which is a non-parametric test that has been used by ^{Bobée and Ashkar (1991)}, and by ^{Rao and Hamed (2000)} to test independence and stationarity in annual maximum flow records (x_i ). Based on this test, the statistical quality of the records to be processed was also approved.

Fittings According to Numerical Substitution Method

Table 2 and Table 3 show the results of the TCEV distribution fitting to the six processed records, based on Equation 24 and Equation (25). It is noted that the number of iterations varied from 1 to 15 and that only in the Turia river record 3 could the initial SEF not be assessed. The evaluation of the quality of the achieved fitting with this method will be estimated during the prediction contrast.

In Figure 1, the Gumbel-Powell probability paper shows data recorded at the Huites station in five-value intervals from 1 to 40, and then one at a time. Dotted lines show the straight lines representing the two populations, and continuous line shows the TCEV distribution calculated with the successive substitution method. It is observed that the TCEV model follows approximately the basic floods up to number 42, but it deviates from the extraordinary ones from number 43 on, and coincides with the last one, which is data 53.

Figure 1 Fitting by the successive substitution method of the TCEV distribution to the Huites station floods, in the Gumbel-Powell probability paper. 

Fittings According to the Optimization Method

In the records of Santa Cruz and La Cuña, the initial values defined by the points F1, X1 to F3, X3 did not allow the start of the Rosenbrock algorithm and were therefore slightly modified, as shown in Table 4 and Table 5, from the results of the TCEV function fitting to the six processed records. It is observed that sometimes this method improved fit (lower L_fmv and SEF), as in the first and last records processed, and on other occasions, it did not exceed the results of the numerical substitution method, as was the case with the records of the Turia and Huites rivers. Therefore, it is recommended that both maximum likelihood fitting methods be applied normally.

In Figure 2, the Gumbel-Powell probability paper shows data recorded at La Cuña station in five-value intervals from 1 to 50 and then one at a time. Dotted lines show the straight lines representing the two populations and continuous line shows the TCEV distribution calculated with the maximization of the objective function method. It is observed that TCEV model represents in an excellent way all the floods.

Figure 2 Fitting with the maximization of the objetive function method of the TCEV distribution to La Cuña station floods, in the Gumbel-Powell probability paper. 

Contrasts of TCEV Distribution Predictions

A first contrast was made between the predictions obtained with each of the two TCEV distribution fitting methods applied. It follows that both methods lead to similar predictions in all the records; the latter outstanding due to its similarity, and Huites station one due to its differences. This confirms the suggestion to fit the TCEV model with both exposed methods.

The second prediction contrast was carried out using the three distributions whose application has been suggested under precept. The Log-Pearson Type III (LP3) in USA; the Generalized Extreme Value (GEV) in England, from 1975 to 2000, and the Generalized Logistics (LOG) currently applied in England (^{Shaw, Beven, Chappell, & Lamb, 2011}). In addition, Wakeby distribution (^{Houghton, 1978}) was applied, which has shown great versatility and descriptive ability, being a mixed function.

LP3 distribution was fitted with the moment method, in the logarithmic domain (^{WRC, 1977}) and in the real domain (^{Bobée, 1975}), adopting the lower SEF; in contrast, GEV, LOG, and Wakeby functions were applied with the L-moment method (^{Stedinger, Vogel & Foufoula-Georgiou, 1993}; ^{Hosking & Wallis, 1997}; ^{Campos-Aranda, 2018}). Table 6 shows the estimated predictions for each of the four cited probabilistic models.

In general terms, LP3, GEV, LOG and WAK distributions lead to higher predictions in all processed records, in return periods (Tr) of 500 and 1 000 years. Adopting the criterion of selection of the smallest SEF, in the first three records, function LP3 would be the option to take, but this leads to high predictions in the 500- and 1 000-year Tr. The above is remarkable in the Turia River and in Santa Cruz, and it could be acceptable in record 1. The opposite occurs in the Huites and La Cuña records, in which the best option is the TCEV distribution, in the first one, through the numerical substitution method and in the second one, with the optimization method (Figure 1 and Figure 2). Finally, in the St. Mary’s River records, the distribution with the lowest SEF is Generalized Logistics, whose predictions are the largest, but of the same order of magnitude as those obtained with the other probabilistic models.

It should be noted that the predictions of the TCEV distribution in the Huites station (Figure 1), practically coincide with those obtained with the global approach by ^{Campos-Aranda (1999)}.

Another appreciation detected in Table 5 is the following: when the percentage of outliers (q) is close to or exceeds 10 %, distributions that are applied under precept and Wakeby can lead to very high predictions in the Tr over 100 years, especially if such outliers are very large compared to the with the ordinary floods, case of the records of the river Turia, Huites and Santa Cruz. In such records, TCEV model is an option that must always be used, since its theoretical origin justifies its application in samples of floods that come from two different physical mechanisms of formation, but that are mixed and therefore show outliers.