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Tecnología y ciencias del agua
versão On-line ISSN 2007-2422
Tecnol. cienc. agua vol.11 no.6 Jiutepec Nov./Dez. 2020 Epub 15-Jun-2024
https://doi.org/10.24850/j-tyca-2020-06-10
Notes
Correction of design floods due to hydrological uncertainty
1Universidad Autónoma de San Luis Potosí, San Luis Potosí, México, campos_aranda@hotmail.com
The planning, design and/or hydrological inspection of hydraulic works is based
on what are called Design Floods (DF) or maximum river flows
associated with low probabilities of exceedance. DFs are estimated using
Frequency Analysis (FA), a statistical method that ignores the hydrological
uncertainties involved. When such indecisions are taken into account, DFs are
not unique magnitudes but a range of values, and therefore, the design and
inspection of hydraulic infrastructure becomes an indeterminate process. DFs can
be corrected, according to hydrological uncertainty, using a simple practical
procedure developed by Botto, Ganora, Claps and
Laio (2017). The method is based on a correction factor (
Keywords: frequency analysis; design floods; hydrological uncertainty correction; L moments; L moment ratios; LP3; GEV; LOG; LN3 and PT3 distributions; minimum absolute distance; standard error of fit
La planeación, diseño y/o revisión hidrológica de las obras hidráulicas se basa
en las llamadas crecientes de diseño (CD) o gastos máximos del río asociados con
bajas probabilidades de excedencia. Las CD se estiman por medio del análisis de
frecuencia (AF), método estadístico que ignora las incertidumbres hidrológicas
involucradas. Cuando tales indecisiones son tomadas en cuenta, las CD no son
magnitudes únicas, sino una gama de valores, y entonces, el diseño y la revisión
de la infraestructura hidráulica se vuelve un proceso indeterminado. La
corrección de las CD conforme a la incertidumbre hidrológica se puede realizar
de forma práctica y simple, con base en el procedimiento que han desarrollado
Botto, Ganora, Claps y Laio (2017),
fundamentado en un factor de corrección (
Palabras clave: análisis de frecuencias; crecientes de diseño; corrección de incertidumbre hidrológica; momentos L; cocientes de momentos L; distribuciones LP3; GVE; LOG; LN3 y PT3; distancia absoluta mínima; error estándar de ajuste
Introduction
Flood frequency analysis
Planning, design, construction, and inspection of hydraulic works, such as reservoirs, protective dikes, canals, bridges and urban drainage, are based on what are called Design Floods (DF), which are maximum flows of a river associated with low exceedance probabilities. The most reliable DF are estimated with the Flood Frequency Analysis (FFA).
Around the middle of the first decade of the 20th century probabilistic concepts were first applied to the problem of DF estimation. The objective was to replace techniques based on enveloping curves and empirical formulas with a more objective and precise method. By mid-20th century, the existence of broad registers of annual maximum flows and theoretical advances that developed new probabilistic models allowed arrival at a standard approach for FFA (Merz & Blöschl, 2008).
FFA consists basically of fitting a probability distribution function (PDF) to the ordered sequence of annual maximum flows to extrapolate its right tail and make predictions or estimations with low exceedance probabilities. This process comprises the following four steps: (1) verifying the randomness of the flood registers to be processed; (2) fitting several PDF using the moments, maximum likelihood and L moments methods; (3) selecting the best fit using quantitative measures, such as standard errors of fit and absolute mean; and (4) obtaining the relevant predictions (Kite, 1977; Stedinger, Vogel, & Foufoula-Georgiou, 1993; Rao & Hamed, 2000; Meylan, Favre, & Musy, 2012; Stedinger, 2017).
FFA involves several weaknesses or hydrological uncertainties that can be specified in three aspects. The first refers to the representativity of the available register of floods in the future, which can be altered by physical changes in the basin or by regional or global climate change. Also implied is that all the registers can contain errors in measurement or sampling. The second generator of uncertainty is associated with extrapolation that must be done to estimate return periods of 100, 500 and 1000 years because, generally, the available registers of floods do not exceed 80 years. Finally, the third source of uncertainty lies in the method itself because one PDF must be selected in order to make the predictions (Merz & Blöschl, 2008).
UNCODE Procedure
Botto, Ganora, Laio and Claps (2014) proposed the procedure called Uncertainty Compliant Design Flood Estimation, or UNCODE. This is a novel approach that parts from the interval, or range, of possible values for each DF and converges in a single design. Using UNCODE, DFs can be estimated in accord with hydrological uncertainty since it permits selecting significant flood estimations from its PDF that exist in each return period value (Tr); when additional restrictions are posed based on the cost-benefit criterion and are resolved numerically by means of a simulation scheme.
The criterion minimum cost-benefit can be consulted in Campos-Aranda (2015). Botto, Ganora, Claps and Laio (2017) present a summary of the UNCODE operative process, and in Kjeldsen, Lamb and Blazkova (2014) general aspects relative to uncertainty in the FFA can be consulted, and more specific aspects in Burn (2003), and Cheng, Chang and Hsu (2007).
Objective
Botto et al. (2017)
were able to reach concrete results of UNCODE using a quite simple procedure
that corrects DF estimated with FFA using a correction factor, register size
function (n), and the return period (Tr).
The result is an estimation of the DF considering hydrological uncertainty (
Methods and materials
Estimation of the correction factor
Botto et al. (2014)
have demonstrated that the DF obtained by applying UNCODE (
According to Botto et al.
(2017), y increases with Tr due
to the increase in QTr
estimation uncertainty and, for a fixed Tr, it also grows
with variability in the probability distribution that allows estimating
QTr , which can be accepted
to be in function of size (n) of the available register of
annual maximum flows used in the FFA. From equation 1, we can obtain the
estimation of
The best
The coefficients a 0, a 1 and a 2, cited in Table 1, depend on the probability distribution used in the FFA and were evaluated by applying UNCODE extensively. Botto et al. (2017) used five parent distributions, eight sizes (n) from 30 to 100 in intervals of ten, and 100 synthetic sequences for each combination of source model and size n. The parent distributions were: Log-Pearson type III (LP3), General of Extreme Values (GEV), Generalized Logistic (LOG), Log-Normal (LN3) and Pearson type III (PT3).
Parent distribution | a0 | a0 | a0 | R2 | MAE | MSE |
---|---|---|---|---|---|---|
LP3 | 0.78 | -0.26 | 0.687 | 0.89 | 0.0235 | 0.0363 |
GVE | -2.27 | -0.30 | 1.110 | 0.85 | 0.0190 | 0.0321 |
LOG | -2.36 | -0.25 | 0.994 | 0.85 | 0.0096 | 0.0145 |
LN3 | -0.82 | -0.25 | 0.809 | 0.94 | 0.0107 | 0.0160 |
PT3 | 0.59 | -0.24 | 0.567 | 0.96 | 0.0080 | 0.0115 |
The parent distributions of Table 1 are cited in chronological sequence, the order in which they were accepted as PDF of the application under precept. For this reason, the LP3 distribution was presented first, suggested in the late 1960s in the USA, then the GEV proposed in England in 1975, and finally, LOG, which substituted GEV as of 1999. LN3, of generalized application, was then cited in the FFA in the 1970s, and finally, the little used PT3 in FFA.
Botto et al. (2017)
indicated that in 90% of the synthetic registers, their statistical
properties oscillated in the following intervals for L moment ratios:
variation 0.28 ≤
In general terms, the obtained regressions and those proposed by Botto et al. (2017) are reliable, their coefficient of determination varying 0.85 to 0.96. Moreover, its analysis of residues by MAE y MSE in Table 1, define an overall mean value of 0.0181, that is, in the order of a 2% variation in y in estimation of a DF, for practical purposes is negligible.
L moments and quotients
L moments (λi) are linear combinations of the probability weighted moments (βi), developed by Greenwood, Landwehr, Matalas and Wallis (1979); they are statistical parameters associated with ordered data. L moments are an efficient and robust system for fitting PDF currently in use or established under precept. The equations for calculating them are (Stedinger et al., 1993; Hosking & Wallis, 1997; Rao & Hamed, 2000; Stedinger, 2017):
Moreover, L moment ratios (
In a sample size n, with its xi elements arranged
in ascending order (
L quotient diagram
Has
Generalized logistic (LOG):
Generalized Pareto (PAG):
og-Normal (LN3):
Pearson type III (PT3):
and General of Extreme Values (GEV):
being:
Using the logarithms of the data in Equation (4), Equation (5), Equation (6), Equation (7), Equation (8), Equation (9), Equation (10) and Equation (11), we obtain the L logarithmic quotients and so we can use expression 15 to evaluate PDF Log-Pearson type III (LP3). Figure 1 shows the diagram of L moment quotients from Hosking and Wallis (1997)).
Selection of the best PDF
One of the recent approaches for selecting the best PDF, to be used in FFA,
consists of taking the sample values (
where
Fitting source PDF
Exclusively, the LP3 distribution was applied with the moments method in the logarithmic dominion (WRC, 1977; Campos-Aranda, 2015); while Botto et al. (2017) fit it as a PT3 based on the logarithmic L quotients. The rest of the distributions from Table 1 were fit with the method of L moments, following the procedures described by Hosking and Wallis (1997). In all cases the standard error of fit (SEF) was evaluated.
Standard error of fit
The standard error of fit is the most common indicator for contrasting PDF against real data (Chai & Draxler, 2014). It was established in the mid-1970s (Kite, 1977), and has been applied in Mexico using the empirical formula of Weibull (Benson, 1962). Now, its application is recommended for use with the formula of Cunnane (Equation (18)), according to Stedinger (2017), leading to non-exceedance probabilities (p) approximately unbiased for many PDF; that is:
The expression for the standard error of fit is:
In which Qi are annual maximum flows ordered from
least to highest and whose number is n, and
Flood registers to be analyzed
To develop the numerical applications, we chose to process the seven largest registers (43 ≤ n ≤ 56) of hydrometric information processed by Campos-Aranda (2014) of Hydrological Region Num. 10 (Sinaloa), Mexico, corresponding to the Huites, Santa Cruz, Jaina, Naranjo, Acatitán, Zopilote and El Bledal stations. Table 2 cites two general characteristics and the L moment ratio values of the seven selected stations.
Station | A(km2) | n |
|
|
|
|
---|---|---|---|---|---|---|
1. Huites | 26 057 | 51 | 0.49086 | 0.29757 | 0.14918 | 0.14510 |
2. Santa Cruz | 8 919 | 52 | 0.42451 | 0.35919 | -0.08131 | 0.17382 |
3. Jaina | 8 179 | 56 | 0.47970 | 0.34935 | 0.04559 | 0.18304 |
4. Naranjo | 2 064 | 45 | 0.40967 | 0.20800 | -0.03623 | 0.11706 |
5. Acatitán | 1 884 | 43 | 0.35069 | 0.22836 | -0.27398 | 0.20318 |
6. Zopilote | 666 | 56 | 0.20325 | 0.06008 | -0.20501 | 0.09740 |
7. El Bledal | 371 | 56 | 0.38055 | 0.27257 | -0.02134 | 0.15315 |
Figure 1 shows the graphic position of each register, according to its asymmetry and kurtosis L quotient. We deduced that the convenient PDF (because of their closeness) are the following: in Huites PAG, in Santa Cruz LOG, in Jaina GEV, in Naranjo PT3, in Acatitán LN3, in Zopilote PAG and in El Bledal GVE.
Results and analysis
Verification of register randomness
For the results of the FFA to be exact, the register of annual maximum flows to be processed must have been generated by a stationary random process, which implies that it has not changed over time. Therefore, the flood register should integrate independent data, exempt of deterministic components.
To test this, we applied the Wald-Wolfowitz Test, which is a non-parametric test used by Bobée and Ashkar (1991), Rao and Hamed (2000), and Meylan, Favre, and Musy (2012) to verify independence and stationarity in registers of annual maximum flows. Its application in the seven selected registers showed that they are integrated by independent random data.
The best PDF, according to minimum DA
Based on L moment ratios from Table 2, we used Equation (17) to find the two minimum DA and, in this way, define the two most suitable PDF in Table 3. The second option will be used in the Huites and Zopilote stations since Botto et al. (2017) did not establish the Generalized Pareto as the parent distribution in Table 1.
Design predictions and their correction
Table 4 presents the estimated
predictions with each of the most convenient PDF for the five return periods
that Botto et al. (2017) analyzed.
Moreover, their correction factors (
Station PDF SEF (m3/s) | Calculation (n) | Return period in years | ||||
---|---|---|---|---|---|---|
50 | 100 | 200 | 500 | 1000 | ||
Huites | QTr | 15 613 | 21 776 | 29 993 | 45 130 | 60 942 |
LP3 |
|
0.050 | 0.081 | 0.130 | 0.244 | 0.392 |
957.1 |
|
16 394 | 23 540 | 33 892 | 56 142 | 84 831 |
Santa Cruz | QTr | 4 335 | 5 891 | 7 971 | 11 843 | 15 948 |
LOG |
|
0.008 | 0.015 | 0.030 | 0.075 | 0.149 |
277.5 |
|
4 370 | 5 979 | 8 210 | 12 731 | 18 324 |
Jaina | QTr | 4 419 | 6 101 | 8 367 | 12 614 | 17 148 |
GVE |
|
0.008 | 0.018 | 0.039 | 0.108 | 0.234 |
360.3 |
|
4 454 | 6 210 | 8 693 | 13 976 | 21 161 |
Naranjo | QTr | 3 090 | 3 973 | 4 976 | 6 499 | 7808 |
LP3 |
|
0.056 | 0.090 | 0.145 | 0.273 | 0.439 |
136.2 |
|
3 263 | 4 331 | 5 698 | 8 273 | 11 236 |
Acatitán | QTr | 3 440 | 4 263 | 5 178 | 6 537 | 7 689 |
LN3 |
|
0.020 | 0.035 | 0.062 | 0.130 | 0.229 |
150.5 |
|
3 509 | 4 412 | 5 499 | 7 387 | 9 450 |
Zopilote | QTr | 1 269 | 1 432 | 1 576 | 1 739 | 1 844 |
LP3 |
|
0.046 | 0.074 | 0.119 | 0.223 | 0.359 |
67.1 |
|
1 327 | 1 538 | 1 764 | 2 127 | 2 506 |
El Bledal | QTr | 1 090 | 1 404 | 1 792 | 2 446 | 3 077 |
GVE |
|
0.008 | 0.018 | 0.039 | 0.108 | 0.234 |
64.0 |
|
1 099 | 1 429 | 1 862 | 2 710 | 3 797 |
We can observe in Table 4 that the
smallest corrections (
Conclusions
Because of its great simplicity, the correction method for Design Floods because of hydrological uncertainty, developed by Botto et al. (2017) and based on Equation (2) and Equation (3), the merit of this Note is exclusively that of dissemination.
In the seven numerical applications described, we use the largest registers of annual maximum flows of Hydrological Region Num. 10 (Sinaloa), Mexico, making use of the L quotient diagram and of minimum absolute distance to select objectively the best PDF for each available register.
The correction factors (
When n and Tr values are equal, in Table 4
we observed that the
Acknowledgements
We are grateful to the three anonymous referees B, D and E for their comments and corrections that allowed us to improve the writing, highlight results and broaden the theoretical concepts of hydrological uncertainty that occurs in flood frequency analysis.
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Received: February 04, 2020; Accepted: April 08, 2020