HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-3353-2017Technical note: Design flood under hydrological uncertaintyBottoAnnaGanoraDanieledaniele.ganora@polito.ithttps://orcid.org/0000-0003-0605-6200ClapsPierluigihttps://orcid.org/0000-0002-9624-7408LaioFrancescoDepartment of Environment, Land and Infrastructure Engineering,
Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129
Turin, Italynow at: the Department of Civil, Environmental and
Architectural engineering, Università di Padova, Via Marzolo,
9, Padua, ItalyDaniele Ganora (daniele.ganora@polito.it)6July20172173353335830November20162December20162June20177June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/3353/2017/hess-21-3353-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/3353/2017/hess-21-3353-2017.pdf
Planning and verification of hydraulic infrastructures require a
design estimate of hydrologic variables, usually provided by frequency
analysis, and neglecting hydrologic uncertainty. However, when hydrologic
uncertainty is accounted for, the design flood value for a specific return
period is no longer a unique value, but is represented by a distribution of
values. As a consequence, the design flood is no longer univocally defined,
making the design process undetermined.
The Uncertainty Compliant Design Flood Estimation (UNCODE) procedure is a
novel approach that, starting from a range of possible design flood estimates
obtained in uncertain conditions, converges to a single design value. This is
obtained through a cost–benefit criterion with additional constraints that
is numerically solved in a simulation framework. This paper contributes to
promoting a practical use of the UNCODE procedure without resorting to
numerical computation. A modified procedure is proposed by using a correction
coefficient that modifies the standard (i.e., uncertainty-free) design value
on the basis of sample length and return period only. The procedure is robust
and parsimonious, as it does not require additional parameters with respect
to the traditional uncertainty-free analysis.
Simple equations to compute the correction term are provided for a number of
probability distributions commonly used to represent the flood frequency
curve. The UNCODE procedure, when coupled with this simple correction factor,
provides a robust way to manage the hydrologic uncertainty and to go beyond
the use of traditional safety factors. With all the other parameters being
equal, an increase in the sample length reduces the correction factor, and
thus the construction costs, while still keeping the same safety level.
Illustrative example (without uncertainty) of the application of the
cost–benefit framework to compute the design flood. Two generic cost and
damage functions are reported in (a), while (b) shows the
linear functions adopted in the UNCODE framework.
Introduction
The flood frequency curve is commonly used to derive the design flood as the
quantile QT corresponding to a fixed return period T. For practical
reasons, QT is commonly expressed only as a single value; however, QT
can only be expressed in this way if its frequency distribution and its
parameters are known perfectly. In practice, one can only estimate the
frequency distribution and its parameters using a sample of observed data,
thereby inflating the uncertainty in the estimate of QT. However, the
design of a hydraulic infrastructure requires a single design value to be
selected. A gap therefore exists between theory and practice. Quantitative
methods to measure the uncertainty associated with the quantiles of the flood
frequency curve (e.g., through their variance or probability distribution)
have been proposed
e.g.,,
but very few suggestions are provided about how to extract a single design
value from the probability distribution of possible design values.
, with the development of the Uncertainty Compliant
Design Flood Estimation (UNCODE) procedure, have shown that it is possible to
select meaningful flood quantiles from their distribution by considering an
additional constraint based on a cost–benefit criterion. Hence, the output
is a unique design flood value QT*. Before illustrating the UNCODE
approach, it is worth recalling the working principles of the cost–benefit
analysis, which is a core element of the procedure. Cost–benefit analysis
can be used to estimate the design flood as the flow value which minimizes
the total expected cost function, defined as the sum of the actual cost to
build a flood protection infrastructure (cost function) and the expected
damages caused by a flood event. An illustrative example of this approach is
reported in Fig. a. The cost function is rather
easy to understand, being an increasing function of the design flood.
Instead, the expected damage function needs to be computed point-by-point:
for any single tentative design flood value (see the inset in
Fig. a) it equals the integral of the product
of the probability density function (pdf) of the flood flow values and a
specific damage function. The latter indicates the damage occurring when the
flood exceeds the flow value used to design the infrastructure. The damage
function depends on a number of parameters such as the exposure and
vulnerability of the flooded goods, the flooding dynamics and the topography,
to mention a few. For these reasons the damage function turns out to be very
site-specific and often unavailable, due to the lack of information needed to
compute it ; in these cases the cost–benefit method is inapplicable.
To face this problem made the assumption that costs
and damages can be represented by linear functions, with slope c and d,
respectively, as illustrated in Fig. b. Given
this assumption, the total cost, CTOT, can be computed as
CTOT=c⋅Q*+∫Q*∞d⋅Q-Q*⋅pQ|ΘdQ,
where Q* is the generic design flood value and pQ|Θ
is the probability density function of the flood flow with parameters
Θ. The optimal design flood of the (uncertainty-free) cost–benefit
framework can then be calculated as the value that minimizes
Eq. (). Examples of cost–benefit analysis in the
hydrologic/hydraulic context can be found in the literature
, with only a few
of them accounting for uncertainty
.
further demonstrated that the optimal design flood
obtained from the cost–benefit analysis with linear cost and damage
functions is equivalent to the design flood QT obtained from the standard
frequency analysis, provided that uncertainty is not accounted for and the
ratio between d and c equals the return period T. This result can be
shown by setting to 0 the derivative of CTOT with respect to Q*,
in order to find the minimum of Eq. (); this leads to the
equivalence
dc=11-PQ*|Θ=T,
where P(⋅) is the cumulative distribution function of the flood values
and T is the return period. This is valid provided that the probability
distribution used in the cost–benefit framework is the same used in the
standard frequency analysis.
The UNCODE approach is founded on the joint use of the cost–benefit approach
of Eq. () and the constraint derived in Eq. (). The
rationale behind this approach is that it is possible to apply the
cost–benefit framework with standard, but meaningful, cost and damage
functions. This is particularly convenient because the cost–benefit
framework can be easily extended to include the estimation uncertainty
inherent in the limited sample length of hydrological records. Consequently,
the UNCODE framework (which is a particular case of cost–benefit analysis)
can also be extended to account for this kind of uncertainty. In uncertain
conditions, the parameters of the flood frequency distribution,
Θ, become a random vector; hence, the uncertainty can be
included in the cost benefit analysis by compounding CTOT over all
the possible values of Θ. In mathematical terms, the
cost–benefit framework with uncertainty is summarized by the equation
QT*=argminQ*∫ΘCTOTQ*|c,d,pΘ⋅hΘdΘ,
where hΘ is the joint pdf of the parameters of
the flood frequency curve. Equation () represents the full
UNCODE model, which adopts linear cost and damage functions and accounts for
uncertainty in a cost–benefit framework.
It is worth noting that, as a consequence of the inherent equivalence of
Eq. (), there are no additional parameters in the cost–benefit
framework; in fact, c and d are related through the known value of the
return period T. The remaining free parameter can be shown to affect only
the magnitude of the integral in Eq. () but not the
position of its minimum, thus avoiding the need for further parameters in the
UNCODE framework with respect to the standard design flood procedure.
To simplify the UNCODE application, which requires the use of numerical
computation of QT*, we provide here an approximated yet reliable method
to estimate QT* starting from QT. Other than a useful practical tool
for design purposes, the analysis reported in this note also provides a
method to quantify the “value” of newly available hydrological information
or the effect of data scarcity on QT* due to uncertainty.
Practical estimation of the UNCODE design flood
The UNCODE design flood, QT*, results in a systematically larger value
than its corresponding standard value QT, as shown by
. Moreover, the relative difference between the two
values,
y=QT*-QTQT,
has been reported to increase with the return period (as the quantile
uncertainty increases) as well as, for fixed T, with the standard deviation
of the probability distribution of QT (i.e., with the uncertainty of
QT). We propose calculating the approximated estimate of the UNCODE design
flood, hereafter referred to as Q^T*, directly by inversion of
Eq. (), without resorting to the numerical solution of
Eq. (). This solution reads as
Q^T*=(1+y^)⋅QT,
where the correction factor y^ (i.e., the approximated estimator of
y) needs to be computed separately. Given this background, we propose
modeling y^ according to the equation
y^=10-2⋅expa0+a1n+a2lnT,
where T is the return period and n is the sample length which can be
considered as a proxy of the standard deviation of QT; n can be computed
from at-site records or as an equivalent sample length from the regional
estimate of QT.
The coefficients a0, a1 and a2 depend on the probability
distribution adopted in the frequency analysis. They have been evaluated from
an extensive simulation study in which the full UNCODE procedure has been
systematically applied to many simulated records, created by combining the
following criteria.
The parent distribution P is selected among the most
common distributions used in flood frequency: log-normal (LN3), generalized
extreme value (GEV), generalized logistic (GLO), Pearson type III (PE3) and
log Pearson type III (LP3). For details on the probability distribution
equation and on the relationship between parameters and
L-moments, the reader is referred to
. The LP3 corresponds to the PE3 with
log-transformed values.
The sample length n of annual maxima is selected from the list 30, 40, 50, 60, 70 80, 90, 100.
We generated 100 records for each combination of P and n.
Looking at the properties of the L-moments, 90 % of the synthetic records
fall within the ranges 0.28 ≤ L-CV ≤ 0.40,
0.14 ≤ L-skewness ≤ 0.40 and
0.07 ≤ L-kurtosis ≤ 0.32, which correspond well to values
typically encountered in real-world applications. The standard design flood
QT as well as the (exact) UNCODE estimator QT* have been computed for
each record of the simulated dataset. This step has been performed by
adopting a suitable fitting distribution F to the whole synthetic
dataset. To make the results more general, F has been selected
from the list LN3, GEV, GLO, PE3, LP3. Note that any F is used to
fit records from any parent P, as in real cases the exact parent
distribution is not known a priori. In this way, the error due to the
misspecification of the fitting distribution is included in the results. The
correction factor y (Eq. ) has been computed for all the
available records in the simulated dataset and for different return periods
T (respectively, equal to 50, 100, 200, 500 and 1000 years). It depends on
the fitting distribution F adopted in the frequency analysis.
Finally, the exact y values have been regressed against n and T to
obtain their estimate y^ (using an ordinary least squares linear
regression on the log-transformed terms of Eq. ). Different forms
of Eq. () have also been tested, but are not reported as they
provide less accurate results.
Coefficients to be used to estimate y^ based on the sample
length n and the return period T (Eq. ) and corresponding
regression diagnostics, for different three-parameter fitting distributions
(LN3: log-normal; GEV: generalized extreme value; GLO: generalized logistic;
PE3: Pearson type III; LP3: log Pearson type III). The LP3 corresponds to the
PE3 with log-transformed variate.
Coefficients a0, a1 and a2 are reported in Table
for different fitting distributions commonly used in hydrological practice to
compute the design flood (in fact, the fitting distribution is always known,
while the parent is not). It can be noticed that, when increasing the sample
length n, the difference between QT* and QT is reduced, due to the
negative value of the coefficient a1. Table also reports
some diagnostics of the regressions used to estimate the coefficients. The
global performance of the regressions has been evaluated using the
coefficient of determination and residuals analysis (through the mean
absolute error, MAE, and root mean squared error, RMSE) for each fitting
distribution. The value of the coefficient of determination ranges from 0.96
in case of the PE3, and 0.94 for the LN3, to 0.85 for the GEV and GLO. The
MAE and the RMSE take values around 0.02, corresponding to a 2 %
variation in the design flood estimation, which is negligible in many
situations. In general, the PE3 probability distribution results in the best
performance in terms of residuals analysis and Radj2, as can be
appreciated by looking at the results reported in Table .
Comparison between the exact, QT*, and approximated,
Q^T*, UNCODE estimators of the design flood for a pool of six flood
records considered in Table 1 with at least
30 years of data. Different return periods are listed in the legend. The
reference distribution used for this flood frequency analysis is the
three-parameter log-normal (LN3) in (a) and the generalized extreme
value (GEV) in (b).
Values of the correction factor y^ from Eq. ()
for some values of the sample length n and return period T and for
different three-parameter fitting distributions (LN3: log-normal; GEV:
generalized extreme value; GLO: generalized logistic; PE3: Pearson type III;
LP3: log Pearson type III). The LP3 corresponds to the PE3 with
log-transformed variate.
The reliability of the approximated correction factor y^ estimated
with the regression model has also been evaluated by comparing the
QT*^ value obtained through Eqs. () and ()
with its exact counterpart calculated with the full UNCODE procedure
(Eq. ). As a reference, time series listed in
Table 1 with at least 30 years of record length
have been analyzed, assuming the LN3 and the GEV as possible fitting
distributions and different return periods. Results show a very good
agreement between the exact (QT*) and approximated (Q^T*)
UNCODE design flood values, as reported in Fig. ,
where each panel shows the estimates for all series and all the return
periods.
A synthesis of the obtained results is shown in Fig. , where the
values of y^ have been reported for the studied distributions, based
on a set of typical sample length and return period values. As mentioned, a
direct comparison of the results between different distributions is not
possible, but it is relevant to observe that for all the distributions
y^ evolves in the same way for varying n and T values. In general,
the correction factor does not exceed 10 % of the standard value QT
for intermediate return periods (e.g., T=200 years) even for small samples,
although a significative variability is associated with the distribution
type. It is around 10 % for T=500 years with sample length values
(n=50) commonly available at many gauged stations. On the other hand, the
sample length plays an important role: for example, considering
T=500 years, the GEV distribution and, varying the sample size, the
reduction of the y value is about 0.075 between n=30 and n=50, and to
0.040 between n=50 and n=70.
Discussion of the application conditions
The UNCODE approach to flood frequency analysis provides a solution to
quantify the design flood estimate when considering the uncertainty of the
distribution quantile; however, application of the full UNCODE procedure may
be cumbersome and computationally demanding for the practitioner. An
approximate but reliable framework has been proposed here to allow easy
computation of the UNCODE design flood value from the standard value using a
correction factor, y^.
The extensive simulation analysis at the base of this study shows that the
coefficients relating the UNCODE value QT*^ to the traditionally
computed value QT are distribution-dependent. For the most used
distributions in flood frequency analysis, they have been computed and
provided. The choice of the distribution and the quantification of its
associated uncertainty is a problem of model selection; hence, it cannot be
solved by the UNCODE procedure, but depends on the methods of standard flood
frequency analysis.
The obtained results demonstrate that an increase in the length of relatively
short samples has a noticeable impact in terms of reduction of y^ that
results in a reduction of the UNCODE estimate QT*^. This implies
that, while the infrastructure keeps the same safety level (or, equivalently,
is designed with the same return period), and with all other parameters being
equal, additional data reduce uncertainty and consequently the construction
costs. The UNCODE design value is indeed reduced with respect to the UNCODE
estimate computed with less data. Consequently, the coefficient y^ can
be considered a measure of the value of data. The mentioned results agree
with findings recently obtained by in a
study on the relative role of regional and at-site flood frequency modeling
approaches, where the value of at-site data has been highlighted and regarded
as a reliable way to improve regional predictions, even with short records.
Under this perspective, the correction factor can be used as a metric for
uncertainty comparison and quantification, thus providing a further tool to
combine different modeling approaches, similarly to the applications of
and , who, with different
methodologies, have exploited measures of hydrologic uncertainty to merge
regional and at-site information. Finally, the correction factor is a new and
easy-to-implement design tool which provides a quantitative way to determine
the design flood value accounting for hydrologic uncertainty while keeping
the same design hazard level considered in standard uncertainty-free
analyses. This is a novel approach when compared to common engineering
practice, which accounts for hydrologic uncertainty by considering, for
instance, the hydraulic freeboard. The use of the freeboard is equivalent to
increasing the design flood value, but without accounting for the size of the
system (e.g., the basin area), or for the hydrologic information available at
the section (i.e., observed of the equivalent record length used to compute
the standard design flood); therefore, this approach is not tailored to the
specific case study. The correction factor represents an advance with respect
to the use of “all-encompassing” safety factors and towards a clearer way
to manage the different sources of uncertainty in hydrological and hydraulic
design.
The work is
based on simulated data. The results can be reproduced by randomly
generating datasets as described in the text of this paper.
The authors declare that they have no conflict of interest.
Acknowledgements
Funding from ERC Consolidator Grant 2014 no. 647473 “CWASI – Coping with
water scarcity in a globalized world” is acknowledged. Daniele Ganora also
acknowledges the RTD Starting Grant from Politecnico di Torino. Edited by:
Giuliano Di Baldassarre Reviewed by: Alessio Pugliese and one
anonymous referee
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