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Article

Design Flood Estimation: Exploring the Potentials

and Limitations of Two Alternative Approaches

Kenechukwu Okoli 1,2 , Korbinian Breinl 2,3, Maurizio Mazzoleni 1,2 and

Giuliano Di Baldassarre 1, 2, *

1Department of Earth Sciences, Uppsala University, Villavägen 16, 752 36 Uppsala, Sweden;

kenechukwu.okoli@geo.uu.se (K.O.); maurizio.mazzoleni@geo.uu.se (M.M.)

2Centre of Natural Hazards and Disaster Science (CNDS), Villavägen 16, 752 36 Uppsala, Sweden;

breinl@hydro.tuwien.ac.at

3Institute of Hydraulic Engineering and Water Resources Management, TU Vienna, Karlsplatz 13/222,

1040 Vienna, Austria

*Correspondence: giuliano.dibaldassarre@geo.uu.se

Received: 22 February 2019; Accepted: 4 April 2019; Published: 9 April 2019

Abstract:

The design of ﬂood defence structures requires the estimation of ﬂood water levels

corresponding to a given probability of exceedance, or return period. In river ﬂood management,

this estimation is often done by statistically analysing the frequency of ﬂood discharge peaks.

This typically requires three main steps. First, direct measurements of annual maximum water

levels at a river cross-section are converted into annual maximum ﬂows by using a rating curve.

Second, a probability distribution function is ﬁtted to these annual maximum ﬂows to derive the

design peak ﬂow corresponding to a given return period. Third, the design peak ﬂow is used as

input to a hydraulic model to derive the corresponding design ﬂood level. Each of these three

steps is associated with signiﬁcant uncertainty that affects the accuracy of estimated design ﬂood

levels. Here, we propose a simulation framework to compare this common approach (based on the

frequency analysis of annual maximum ﬂows) with an alternative approach based on the frequency

analysis of annual maximum water levels. The rationale behind this study is that high water levels

are directly measured, and they often come along with less uncertainty than river ﬂows. While this

alternative approach is common for storm surge and coastal ﬂooding, the potential of this approach

in the context of river ﬂooding has not been sufﬁciently explored. Our framework is based on the

generation of synthetic data to perform a numerical experiment and compare the accuracy and

precision of estimated design ﬂood levels based on either annual maximum river ﬂows (common

approach) or annual maximum water levels (alternative approach).

Keywords: ﬂood defence; design ﬂoods; peak levels; peak ﬂows; uncertainty

1. Introduction

The design of river ﬂood defence requires the estimation of potential ﬂood levels [

1

,

2

]. In the past,

this was done by keeping track of the historical water levels, which were then used for the design of

ﬂood protection measures [

3

]. For example, ﬂood walls in Rome were raised at the end of the 19th

century to about one meter above the maximum water level that was reached during the catastrophic

1870 ﬂood event [4]. More recently, as hydrology has progressed, river ﬂows (instead of water levels)

have been used more and more by water engineers, as river discharge is more useful than water levels

for the study of hydrological processes and water management applications [

5

]. As a result, the design

of defence structures to prevent river ﬂooding is currently based on the analysis of ﬂood discharge

peaks. More speciﬁcally, the estimation of design ﬂood levels often consists of three main steps: (i)

Water 2019,11, 729; doi:10.3390/w11040729 www.mdpi.com/journal/water

Water 2019,11, 729 2 of 11

Time series of annual maximum ﬂows are derived from (directly measured) annual maximum water

levels using a rating curve; (ii) river discharges corresponding to the desired return periods (hereafter

“design peak ﬂows”) are derived from annual maximum ﬂows by means of one or more probability

distribution functions; and (iii) design ﬂood levels are estimated by using hydraulic models to simulate

design peak ﬂows [

6

]. As such, this common approach requires a cascade of three types of models:

A rating curve, a probability distribution function, and a hydraulic model. Numerous studies have

shown that signiﬁcant uncertainty affects each of these three steps.

First, direct measurements of river ﬂows are not error free [

7

]. Systematic and random errors

affect these measurements as a result of wrong calibration and precision limitations of measuring

instruments. Pelletier [

8

] provided a comprehensive discussion about sources of errors that affect

direct measurements of stream ﬂows and concluded that their combined effect is around 8% at the

95% conﬁdence level. Other studies report error estimates of about 5% as a result of an improvement

in measurement techniques [

9

]. In contrast, water levels are measured more accurately and errors

are considered to be around

±

3 mm at the 95% conﬁdence level [

10

]. Schmidt [

11

] reported the

uncertainty of a single water level gauge to vary from 3 to 14 mm. As such, the uncertainty in the direct

measurements of water levels is minimal, and essentially negligible for medium-to-large rivers [12].

Second, river discharge is almost never directly observed, especially during ﬂood conditions.

Measurements of water levels are typically converted into river ﬂows by using analytical functions

often referred to as rating curves [

13

]. Rating curves are often derived by ﬁtting a model to a limited

number of direct measurements of river ﬂow (Q) and corresponding water stage (h) [

14

]. The power

law model is commonly used in hydrology to deﬁne the relationship between water level and river

ﬂow, i.e., the so-called stage-discharge relation [

15

]. While the uncertainty in the direct measurements

of river ﬂows is limited, with expected errors of around 5%, river ﬂow values derived using the rating

curve (Q*) are affected by additional sources of uncertainty [

7

], which can lead to very high errors

(over 30%) during high ﬂow conditions [7,16].

Third, the signiﬁcant uncertainty present in estimated peak ﬂows derived from a rating curve

can be exacerbated when probability models are used to derive design events with a given probability

of occurrence, as discussed extensively in the literature [

17

–

22

]. Indeed, on top of the measurement

errors and rating curve uncertainty discussed above, additional sources of uncertainty affect the design

peak ﬂow due to: (i) The choice of a probability distribution function and a parameter estimation

method [

23

–

25

]; (ii) the assumptions about randomness, homogeneity, and independence; and (iii) the

type of sample, i.e., peak-over-threshold versus annual maximum ﬂows [26].

Fourth, the recent literature has shown that hydraulic models are also affected by signiﬁcant

uncertainty [

27

,

28

], especially when used for conditions different from the ones they were calibrated

for [

29

,

30

]. This is often the case as extreme ﬂood conditions are rare (by deﬁnition) and therefore

hydraulic models are typically calibrated by parameterizing the Manning’s roughness coefﬁcients to

reproduce more common ﬂow conditions.

Hence, the propagation of uncertainty from measurement errors to design ﬂood levels through

this cascade of three models (rating curve, statistical analysis, hydraulic modelling) can negatively

affect the precision and accuracy of design ﬂood levels, thereby limiting the reliability of ﬂood risk

assessment and defence measures based on these uncertain estimations.

An alternative approach to derive design ﬂood levels is to ﬁt probability distribution functions

directly to the time series of annual maximum water levels. As a matter of fact, high water levels

are the only type of data that is directly measured and they typically have high precision and

accuracy

[10–12]

. Such an approach is relatively common in modelling coastal ﬂoods caused by

storm surge events

[31–33]

. A similar approach has been developed to estimate extreme ground water

levels by Fürst et al. [

34

]. Apart from the technical note published by Dyhouse [

35

], there are no studies

exploring the potential of frequency analysis of water level peaks in the context of river ﬂooding,

despite direct analysis of high water levels being the traditional approach used by Egyptians, Romans,

and other past civilizations [3].

Water 2019,11, 729 3 of 11

Thus, we provide a new simulation framework to compare two approaches for the estimation of

design ﬂood levels (Figure 1): (i) The common three-step approach based on ﬂood discharge peaks;

and (ii) the alternative approach based on ﬂood level peaks. This paper also discusses opportunities

and limitations of these two alternative approaches.

Water 2019, 11, x FOR PEER REVIEW 3 of 11

Thus, we provide a new simulation framework to compare two approaches for the estimation

of design flood levels (Figure 1): (i) The common three-step approach based on flood discharge peaks;

and (ii) the alternative approach based on flood level peaks. This paper also discusses opportunities

and limitations of these two alternative approaches.

Figure 1. Virtual experiment proposed in this study. Graphical representation of the simulation

framework to compare the estimation of design flood levels based on two alternative approaches.

2. Methodology

Case studies cannot allow insights about the potential and limitations of estimating design

floods based on flood level peaks instead of flood discharge peaks, as the real design flood is

obviously unknown. Thus, we propose a virtual experiment based on the use of synthetic data. A

Comparison &

Errors

Annual Maximum

Water Levels

Common approach

(based on flood discharge peaks)

Alternative approach

(based on flood level peaks)

Rating

Curve

Probability

Model

Annual Maximum

Flows

Design Peak Flow

Hydraulic

Model

Design Flood Level

Design Flood Level

Probability

Model

Design Peak Flow Design Flood Level

(Reference Value)

Parent

Distribution

Hydraulic

Model

(B)(A)

(D)

(E) (F)

(G)

Synthetic

Estimated

Annual Maximum

Flows

Hydraulic

Model

(C)

Parent

Distribution

Figure 1.

Virtual experiment proposed in this study. Graphical representation of the simulation

framework to compare the estimation of design ﬂood levels based on two alternative approaches.

2. Methodology

Case studies cannot allow insights about the potential and limitations of estimating design

ﬂoods based on ﬂood level peaks instead of ﬂood discharge peaks, as the real design ﬂood is

obviously unknown. Thus, we propose a virtual experiment based on the use of synthetic data.

A fundamental assumption of the proposed experiment is that a well-established parent distribution

and a well-established hydraulic model of a river reach are used to generate synthetic data, which

enables a comparison of the estimates provided by the two approaches.

Water 2019,11, 729 4 of 11

Our Monte-Carlo simulation framework consists of the following steps (Figure 1):

We start by assuming a well-established probability distribution function as the Parent Distribution

(Figure 1), and use it to generate the design peak ﬂow with a return period of Tyears (step A).

We then apply a well-established Hydraulic Model to propagate the design peak ﬂow generated in

step A, to derive a synthetic T-year design ﬂood level (step B). This value is used as a reference (i.e.,

the “truth”) for comparing the two different approaches.

The Parent Distribution is likewise used to generate a sample of synthetic annual maximum ﬂows

with a length of Myears (step C). This sample from step C is then converted into the synthetic annual

maximum water levels with the Hydraulic Model (step D). These water levels are assumed to be a

typically observed sample, as they are often the only direct measurements available in the real world.

The water levels are the key input for the two approaches:

Common approach (Figure 1, E): The synthetic annual maximum water levels are converted into

estimated annual maximum water ﬂows by using a rating curve. Then, a Probability Model is used to ﬁt

these annual maximum ﬂows and estimate the design peak ﬂows for a T-year return period. Lastly,

these design peak ﬂows are propagated by the Hydraulic Model to estimate a T-year design ﬂood level.

Alternative approach (Figure 1, F): The sample of annual maximum levels obtained in step D is

directly fed into a Probability Model (Figure 1) to estimate a T-year design ﬂood level.

The design ﬂood levels estimated in step E (common approach) and step F (alternative approach)

are compared (step G). Errors are deﬁned as differences between these two estimates and the synthetic

design ﬂood level estimated in step B is used as a reference value.

Steps (C–G) are repeated Ntimes to explore the role of sampling.

3. Example Application

A 98 km reach of the Po River between Cremona and Borgoforte (Italy), and a ﬂoodplain conﬁned

by two continuous levee systems was investigated. Its width varies from 400 m to 4 km [

6

]. A high

quality 2 m digital terrain model (DTM) of the middle-lower portion of the Po River (covering an

extension of about 350 km) was made available by the Po River Basin Authority. Information regarding

the river geometry essential for hydraulic modelling was extracted from the DTM. A 90 year record of

annual maximum ﬂows (1920–2009) at the gauge station of Pontelagoscuro was available and used in

this study.

3.1. Hydraulic Modelling

The 1D hydrodynamic Hydrologic Engineering Center’s River Analysis System (HEC-RAS)

model [

36

] was applied for simulating the hydraulic behaviour of the 98 km reach of the Po River

between Cremona and Borgoforte under steady ﬂow conditions. A calibrated 1D hydrodynamic

model by Brandimarte and Di Baldassarre [

6

] was used in this study as the Hydraulic Model. Many

studies have used HEC-RAS for hydraulic modelling in simulating ﬂows in natural rivers [

37

–

41

].

The river geometry is described by 88 cross sections extracted from the DTM. HEC-RAS solves the

governing equations for gradually varied ﬂow, and water proﬁles are computed using the standard

step procedure. Details about the HEC-RAS model code can be found in the Hydraulic Reference

Manual [

36

]. The modelling exercise neglects the effects of unsteady ﬂow and sediment transport that

can inﬂuence the stage–discharge relationship at a given location.

3.2. Rating Curve

To replicate the typical estimation of design ﬂood levels using frequency analysis of the ﬂood

discharge peaks (as shown in Figure 1), it is necessary to estimate the rating curve information at any

given cross-section along the river. We applied the hydraulic model to derive a number of synthetic

pairs (h,Q), which were then used to parameterize the estimated rating curve equation. The synthetic

rating curve was derived by simulating in HEC-RAS ﬂows ranging from 100 m

3

/s to 1000 m

3

/s with

steps of 100 m

3

/s, followed by simulations of discharge ranging from 1000 m

3

/s to 13,000 m

3

/s with

Water 2019,11, 729 5 of 11

steps of 1000 m

3

/s. After that, the estimated rating curve was assessed by ﬁtting the synthetic data

pairs

(Q,h)

using a power law relationship between the water level and river ﬂow. Data pairs

(Q,h)

for the statistical ﬁt are discharge data ranging from 1000 m

3

/s to 6000 m

3

/s (with steps of 1000 m

3

/s)

and their corresponding stages derived from the hydraulic simulations. The power law relationship is

expressed as:

Q∗=(h−h0)b(1)

where Q* is the estimated river ﬂow,

h0

is the cease-to-ﬂow stage,

h

is the measured water stage, and

C

and

b

are calibration parameters that can be determined by the method of non-linear least squares.

Once the power law relationship is established, one can extrapolate to higher ﬂow values.

The two rating curves are shown in Figure 2and the full circles represent the data pairs

(Q,h)

used in deriving the estimated rating curve. Figure 2refers to two cross-sections at 58 and 68 km

downstream of Cremona. Extrapolation beyond the range of measurements leads to a systematic

underestimation of discharges for this particular case.

Water 2019, 11, x FOR PEER REVIEW 5 of 11

rating curve was derived by simulating in HEC-RAS flows ranging from 100 m3/s to 1000 m3/s with

steps of 100 m3/s, followed by simulations of discharge ranging from 1000 m3/s to 13,000 m3/s with

steps of 1000 m3/s. After that, the estimated rating curve was assessed by fitting the synthetic data

pairs (𝑄,ℎ) using a power law relationship between the water level and river flow. Data pairs (𝑄,ℎ)

for the statistical fit are discharge data ranging from 1000 m3/s to 6000 m3/s (with steps of 1000 m3/s)

and their corresponding stages derived from the hydraulic simulations. The power law relationship

is expressed as:

𝑄

∗

=

(

ℎ

−

ℎ

)

(1)

where Q* is the estimated river flow, ℎ is the cease-to-flow stage, ℎ is the measured water stage,

and 𝐶 and 𝑏 are calibration parameters that can be determined by the method of non-linear least

squares. Once the power law relationship is established, one can extrapolate to higher flow values.

The two rating curves are shown in Figure 2 and the full circles represent the data pairs (𝑄,ℎ)

used in deriving the estimated rating curve. Figure 2 refers to two cross-sections at 58 and 68 km

downstream of Cremona. Extrapolation beyond the range of measurements leads to a systematic

underestimation of discharges for this particular case.

3.3. Parent Distribution

The GEV distribution, as defined by Jenkinson [42] and parameterised using a 90-year record of

annual maximum flows available from a gauging station along the Po River, was assumed to be the

Parent Distribution. Our justification for selecting the GEV as a parent lies with the fact that the focus

is on modelling the block maxima of water levels, and the theoretical arguments from extreme value

theory are considered sufficient for this task, assuming that the discharge, 𝑄, follows a GEV

distribution, 𝑄 ~GEV(𝜉,𝛼,𝑘), ξ, α, and 𝑘 are the location, scale, and shape parameters, respectively.

The tail behaviour of the GEV distribution is influenced by the value of the shape parameter and

generally falls in three classes: The Gumbel type (𝑘 = 0) is characterised by a light upper tail, the

Frechet type (𝑘 > 0) has a heavy upper tail, and the Weibull type has (𝑘 < 0) and is bounded

above. Hence, GEV encompasses three asymptotic distributions that can be selected to model the

right tail of a distribution and as such is a grand-parent.

(A)

Water 2019, 11, x FOR PEER REVIEW 6 of 11

(B)

Figure 2. Synthetic and estimated rating curves derived using the Hydrologic Engineering Center’s

River Analysis System (HEC-RAS) model and the power law models for two cross-sections located

58 km (A) and 68 km (B) downstream of Cremona (Po River, Italy).

3.4. Probability Model

The Gumbel distribution was selected as the Probability Model to fit samples drawn from the

parent distribution. The Gumbel distribution is quite popular among engineering practitioners for

most parts of the world. For instance, it is the distribution of choice to describe flood probabilities at

the Tiber River in Rome, Italy [43].

As such, we aim to mimic the common situation in the real world, in which simple models (the

Gumbel distribution in our example application) are used to fit reality (the GEV distribution in our

example application). Previous studies in statistical hydrology used similar research methods, by

using other simple distribution functions, such as the Lognormal, as a probability model and more

complex distribution functions, such as the Kappa and Wakeby, as parent distributions [22,24].

The GEV and Gumbel quantile function is given as

𝑋

(

𝐹

)

𝜉

+

𝛼

{

1

−

(

−

log

𝐹

)

}

/

𝑘

𝑘

≠

0

𝜉

−

𝛼

log

(

−

log

𝐹

)

𝑘

=

0

(2)

where 𝑋 represents a quantile estimate (for either discharge or water stage) corresponding to a

certain return period, in this case 100 years, while 𝐹 is the non-exceedance probability. The

parameters of the GEV were estimated by the method of maximum likelihood using the available 90

year record of annual maximum flows. These parameters were fixed and used for generating

synthetic annual maximum flows by random sampling from its inverse cumulative distribution

function. Equation (2) for the condition of 𝑘 ≠ 0, which is the quantile function for GEV, was used

to estimate the 100-year design peak flow. The 100-year peak flow was simulated using HEC-RAS to

get the estimated design flood level, ℎ.

3.5. Application of the Simulation Framework

Based on the hydraulic model (Section 3.1), rating curve (Section 3.2), parent distribution

(Section 3.3), and probability model (Section 3.5), we applied our simulation framework (Section 2)

to the 98-km reach of the Po River and by using a return period of 100 years (T = 100), repeating steps

C–G for 1000 times (N = 1000), and exploring the role of sample size by using M equal to 30, 50, and

100. A generated sample size of 30 and 50 years reflects the typical length of historical observations,

while samples of a length equal to 100 years represent an optimistic case in hydrology.

Figure 2.

Synthetic and estimated rating curves derived using the Hydrologic Engineering Center’s

River Analysis System (HEC-RAS) model and the power law models for two cross-sections located

58 km (A) and 68 km (B) downstream of Cremona (Po River, Italy).

Water 2019,11, 729 6 of 11

3.3. Parent Distribution

The GEV distribution, as deﬁned by Jenkinson [

42

] and parameterised using a 90-year record

of annual maximum ﬂows available from a gauging station along the Po River, was assumed to be

the Parent Distribution. Our justiﬁcation for selecting the GEV as a parent lies with the fact that the

focus is on modelling the block maxima of water levels, and the theoretical arguments from extreme

value theory are considered sufﬁcient for this task, assuming that the discharge,

Q

, follows a GEV

distribution,

Q

~GEV

(ξ,α,k)

,

ξ

,

α

, and

k

are the location, scale, and shape parameters, respectively.

The tail behaviour of the GEV distribution is inﬂuenced by the value of the shape parameter and

generally falls in three classes: The Gumbel type (

k=

0) is characterised by a light upper tail, the Frechet

type (

k>

0) has a heavy upper tail, and the Weibull type has (

k<

0) and is bounded above. Hence,

GEV encompasses three asymptotic distributions that can be selected to model the right tail of a

distribution and as such is a grand-parent.

3.4. Probability Model

The Gumbel distribution was selected as the Probability Model to ﬁt samples drawn from the

parent distribution. The Gumbel distribution is quite popular among engineering practitioners for

most parts of the world. For instance, it is the distribution of choice to describe ﬂood probabilities at

the Tiber River in Rome, Italy [43].

As such, we aim to mimic the common situation in the real world, in which simple models (the

Gumbel distribution in our example application) are used to ﬁt reality (the GEV distribution in our

example application). Previous studies in statistical hydrology used similar research methods, by using

other simple distribution functions, such as the Lognormal, as a probability model and more complex

distribution functions, such as the Kappa and Wakeby, as parent distributions [22,24].

The GEV and Gumbel quantile function is given as

XT(F)(ξ+α{1−(−log F)k}/k k 6=0

ξ−αlog(−log F)k=0(2)

where

XT

represents a quantile estimate (for either discharge or water stage) corresponding to a certain

return period, in this case 100 years, while

F

is the non-exceedance probability. The parameters of

the GEV were estimated by the method of maximum likelihood using the available 90 year record

of annual maximum ﬂows. These parameters were ﬁxed and used for generating synthetic annual

maximum ﬂows by random sampling from its inverse cumulative distribution function. Equation (2)

for the condition of

k6=

0, which is the quantile function for GEV, was used to estimate the 100-year

design peak ﬂow. The 100-year peak ﬂow was simulated using HEC-RAS to get the estimated design

ﬂood level, h100.

3.5. Application of the Simulation Framework

Based on the hydraulic model (Section 3.1), rating curve (Section 3.2), parent distribution

(Section 3.3), and probability model (Section 3.5), we applied our simulation framework (Section 2) to

the 98-km reach of the Po River and by using a return period of 100 years (T= 100), repeating steps

C–G for 1000 times (N= 1000), and exploring the role of sample size by using Mequal to 30, 50, and

100. A generated sample size of 30 and 50 years reﬂects the typical length of historical observations,

while samples of a length equal to 100 years represent an optimistic case in hydrology.

4. Results

The application of the simulation framework described in Section 2and depicted in Figure 1, to a

98-km reach of the Po River (Section 3), allows us to compare the accuracy and precision of the two

approaches for the estimation of design ﬂood levels.

Water 2019,11, 729 7 of 11

Figure 3shows boxplots of errors when the estimation of design ﬂood levels is based on the two

methods that are considered in this study.

Water 2019, 11, x FOR PEER REVIEW 7 of 11

4. Results

The application of the simulation framework described in Section 2 and depicted in Figure 1, to

a 98-km reach of the Po River (Section 3), allows us to compare the accuracy and precision of the two

approaches for the estimation of design flood levels.

Figure 3 shows boxplots of errors when the estimation of design flood levels is based on the two

methods that are considered in this study.

(A)

(B)

Figure 3. Boxplots of error estimates for the two approaches with Gumbel distribution used for design

flood level estimation at cross-sections A and B. The red lines represent the median (50th percentile),

the lower and upper ends of the blue box represent the 25th and 75th percentile, respectively. Outliers

are represented by red dots.

The left panel of Figure 3 refers to the common approach (when the estimation is based on

annual maximum flows), and it shows a substantial underestimation, which is larger in section A,

where the rating curve has larger errors (Figure 2). Moreover, and there is no substantial

improvement in accuracy and precision after increasing the sample size from 30 to 100 years.

The right panel of Figure 3 refers to the alternative approach (when the estimation is based on

annual maximum levels), and it shows an underestimation of the design flood by 0.8 m with a more

Figure 3.

Boxplots of error estimates for the two approaches with Gumbel distribution used for design

ﬂood level estimation at cross-sections (

A

,

B

). The red lines represent the median (50th percentile),

the lower and upper ends of the blue box represent the 25th and 75th percentile, respectively. Outliers

are represented by red dots.

The left panel of Figure 3refers to the common approach (when the estimation is based on annual

maximum ﬂows), and it shows a substantial underestimation, which is larger in section A, where

the rating curve has larger errors (Figure 2). Moreover, and there is no substantial improvement in

accuracy and precision after increasing the sample size from 30 to 100 years.

The right panel of Figure 3refers to the alternative approach (when the estimation is based

on annual maximum levels), and it shows an underestimation of the design ﬂood by 0.8 m with a

more signiﬁcant reduction of uncertainty after increasing the sample size from 30 to 100 years. These

differences are substantial when the estimation of design ﬂood levels is used for ﬂood defence design

or risk assessment.

Water 2019,11, 729 8 of 11

5. Discussion

Our results demonstrate that the alternative approach can provide lower errors in assessing the

design ﬂood level. The main advantage of the alternative approach is that less sources of uncertainty

come into play, such as the ones related to rating curve estimation and hydraulic modelling structure

and calibration [

44

,

45

]. In addition, the alternative approach can be easily implemented due to the

low data requirement, no dependency on hydraulic modelling, and computational power. Moreover,

one advantage in using high water levels is that information regarding the water stage is in abundance,

or at least it can be, given the relative ease of measuring it. Also, remote sensing using satellite altimetry

provides the opportunity to derive water stage data. Some studies have explored the direct use of

water levels (instead of river ﬂows) in the study of ungauged catchments [

46

]. Studies focusing on

low ﬂows, especially in large rivers, cannot be investigated using the alternative approach because of

hydraulic sensitivity at the section control [47].

Yet, conducting ﬂood frequency analysis using water stages is not devoid of uncertainty. For rivers

with a gentle slope, water stages are heavily inﬂuenced by the downstream control (i.e., hydraulic

works, roughness, channel outfall), which might lead to backwater effects and in turn affect the accuracy

of the observed annual maximum water stage data. Water levels are also affected by backwater effects

and hydraulic jumps forming at the gauging station [

47

]. Seasonal variation in channel hydraulic

roughness affects the accuracy of observed water stages for a given discharge. Hence, the estimation

of design ﬂood levels based on records of water levels seems to have more potential for rivers with a

subcritical ﬂow regime and with a stable channel geometry. Di Baldassarre and Claps [

12

], for example,

showed that ﬂood water levels in the study area considered here (i.e., Po River) are not signiﬁcantly

inﬂuenced by changes in river geometry.

Moreover, while the design of levees only requires the estimation of a water level with a given

return period, the design of ﬂood control reservoirs requires an estimation of the volume of water

corresponding to a given return period. In such a case, the analysis of ﬂood discharges is needed.

Furthermore, using ﬂood discharges has other advantages, including the possibility to follow regional

methods for design ﬂood estimation, i.e., trading space for time and using multiple sites within a

homogeneous region [

48

]. On the other hand, however, hydrogeomorphic approaches can help identify

ﬂood water levels in ungauged areas [

49

]. Table 1summarizes the main advantages and disadvantage

of the two approaches.

Table 1. Summary of pros (+) and cons (−) of the two approaches.

Various Aspects and Applications Common Approach

(Flood Discharges)

Alternative Approach

(Flood Levels)

Sources of uncertainty −+

Computational time −+

Data requirement −+

Changes in the river channel (e.g., erosion) + −

Hydraulic effects on the channel (e.g., backwater) + −

Design of levees −+

Design of ﬂood-control reservoirs + −

Regionalization methods + −

Hydrogeomorphic methods −+

6. Conclusions

This paper presents a simulation framework (Figure 1) to compare two approaches for the

estimation of design ﬂood levels: (i) A common one based on the frequency analysis of annual

maximum ﬂows, and (ii) an alternative one based on the frequency analysis of annual maximum water

levels. We proposed a virtual experiment based on the use of synthetic data to gain insights about

potentials and limitations of these approaches.

Water 2019,11, 729 9 of 11

The example application demonstrates that the alternative approach can work better than the

common approach in terms of both accuracy and precision (Figure 3). These results were unavoidable

as they were associated with the speciﬁc test site, as well as the arbitrary assumptions about the

parent distribution and the hydraulic model. Still, they show that there is potential in this alternative

approach, which is essentially based on very old methods of estimating potential ﬂood levels along

rivers in the past. This result is only partly surprising. While the use of annual maximum ﬂows

remains theoretically more appropriate in the context of river ﬂooding, this approach is affected by

numerous sources of uncertainty due to the use of a cascade of three models (Figure 1). As such,

we suggest complementing the common approach with an alternative estimation of design ﬂood levels

directly based on annual maximum water levels, which are often the only type of data derived from

direct (and typically accurate) measurements.

Our work is an initial effort to compare common and alternative approaches for the estimation

of design ﬂood levels, which we see as complementary given their corresponding pros and cons

(Table 1). The simulation framework presented here can be applied elsewhere considering more

parent/probability distribution functions, as well as data from various hydrogeomorphic conditions.

As such, it offers new opportunities to gain more understanding about the potentials and limitations

of these two alternative approaches.

Author Contributions:

K.O., writing—original draft preparation; G.D.B., M.M. and K.B., writing—review and

editing; K.O., Investigation; K.O., data curation; K.B., M.M. and G.D.B., supervision.

Funding: This research received no external funding.

Acknowledgments:

We thank Alberto Montanari and Richard Vogel for nice discussions on previous versions of

our work. Two Anonymous Reviewers are also acknowledged for constructive comments to this paper.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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