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Sensitivity based attribution of flood risk
Attribution du risque d'inondation bas? sur la sensibilit?
Jim Hall*, Richard Dawson*, Linda Speight**, Slobodan
Djordjevi?***, Dragan Savi?*** and Jorge Leandro***
*School of Civil Engineering and Geoscience, Newcastle University
Cassie Building, Newcastle upon Tyne, NE1 7RU, United Kingdom
**JBA Consulting  Engineers & Scientists
Magna House, South Street, Atherstone, CV9 1DF, United Kingdom
***Centre for Water Systems, University of Exeter
Harrison Building, North Park Road, Exeter, EX4 4QF, United Kingdom
RESUME
Cet article pr?sente l?exploration de nouvelles m?thodes d?attribution des risques
d?inondation ? l?aide d?un mod?le int?gr? synth?tique de r?seaux de drainage en
milieu urbain. La seule approche permettant de traiter le vaste nombre de variable
d?un syst?me urbain consiste en une simplification hi?rarchique du syst?me.
L?attribution de risques est analys?e ? plusieurs niveaux pour identifier les
composants responsables du risque d?inondation. L?attribution bas?e sur la sensibilit?
r?partit le risque entre les variables influen?ant le risque total. Cette approche utilise
des moyennes statistiques pour analyser les d?g?ts dus ? une s?rie d??v?nements,
d?g?ts bas?s sur la mod?lisation hydraulique d?terministe de l?inondation de rues.
Deux exemples d?attribution de risques bas?s sur les indices de sensibilit?s sont
pr?sent?s.
ABSTRACT
A synthetic integrated urban drainage system is used in this paper to explore
alternative methods for flood risk attribution. The only feasible approach to tackling
the problem of huge number of variables in urban systems is by hierarchical
simplification of the system, with the attribution analysis being applied at several
levels, to identify the system components responsible for flood risk. Sensitivitybased
attribution apportions risk between the variables that influence the total risk. In this
approach, statistical means are used to analyse damage from a series of events,
based on deterministic hydraulic modelling of street flooding. Two examples of risk
attribution based on sensitivity indices are shown.
KEYWORDS
Flood damage, integrated flood risk management, risk attribution, urban flooding
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124 NOVATECH 2007
1 INTRODUCTION
Integrated Flood Risk Management (IFRM) explicitly recognises the interrelationships
between all sources of flooding, risk management measures, their analysis, costs and
effectiveness, within changing social, economic and environmental contexts. The
main sources of flooding include pluvial runoff that leads to sewers backing up and
high surface flows, fluvial flooding caused by high river flows, coastal storm surges
and perhaps also groundwater floods. A given flood event could be caused by a
single source, or several sources acting in combination. The UK?s Department for
Environment Food and Rural Affairs (DEFRA) has identified IFRM as a key strategic
aim (DEFRA et al., 2005). Likewise, initiatives such as the Water Framework
Directive, Integrated Coastal Zone Management and proposed EU Floods Directive
are driving the need for ?joinedup? thinking across Europe.
In order to demonstrate the technical feasibility of IFRM a necessary methodological
advancement is the development of core concepts for a framework for unified
systemsbased flood risk analysis. After this introduction, we shall present these
concepts and present in greater detail a key aspect of these concepts: a methodology
for attributing risk between flood sources, management infrastructure and
stakeholders. Implementation of this approach on a synthetic system will be shown.
2 SYSTEMSBASED RISK ANALYSIS
The core principles of an integrated systemsbased flood risk analysis are now
defined as (Hall et al., 2006):
1) Risk is a ?common currency?. To enable interorganisational ?communication? of
flood risk information, the first step is that it is measured using a common metric.
Risk estimates provide the common currency which can be used to compare
risks from different sources on a common basis. In a situation where there are
several organisations responsible for risk management we wish to be able to
disaggregate the total risk and attribute it to different components in the system.
2) Risk is a multidimensional measure and should be a broad measure of all
losses (and gains) including social, environmental and economic.
3) Spatial and temporal profiles of this multidimensional measure of risk need to be
constructed to support long term planning.
4) Attribution of risk. The contribution towards risk from different flooding sources
and components of flooding pathways, including infrastructure components, is
critical information to support riskbased decisionmaking:
a) Risk ownership. There are several organisations with a role in flood risk
management. We wish to know, in broad terms, what proportion of the risk
each is responsible for.
b) Estimation of capacity to reduce risk. Ideally, risk should be owned by
organisations with the greatest capacity to manage it. Capacity to reduce
flood risk is related to the potential to change the characteristics of the
flooding system.
c) Asset management. An organisation with responsibility for management of
flood defence or drainage infrastructure should rationally invest resources
so that they maximise impact in terms of risk reduction. Within a specified
set of system components it is therefore necessary to identify those
components that contribute most to risk and compare potential measures to
reduce risk with the cost of implementing those measures in order to
develop an optimum intervention strategy. A secondary problem is to target
monitoring strategies so that resources are invested in data acquisition that
makes the greatest contribution to reducing uncertainty.
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2.1 Formulation of the risk problem
Consider a system which is described by a vector of loading variables S and a vector
of variables that describe the flood management infrastructure system R. We write all
of the basic variables as X = (S, R). The resistance variables R might include the
height or other dimensions of dikes, the properties that determine dike failure or the
dimensions of the sewer system. Their variation might be continuous (e.g. a height
variable) or discrete (e.g. a ?blocked? or ?not blocked? descriptor of a pipe.
The variability in the loading and resistance is described by a joint probability
distribution ?(X). We may often be able to assume that many of the variables in R are
statistically independent and we will often assume that S and R are independent.
There is a damage function D(X) where the units of D are ? (British Pound) or some
other suitable measure of impact. The risk r associated with the system is therefore
?
?
=
?
0
()()rXDXdx (1)
The risk integral can be further extended to address antecedent conditions either by
including antecedent variables in the loading vector S, or, alternatively, by extending
the analysis so that S is a function of time. At any point in time the damage is D(X);
the risk is the instantaneous expected value of this function. A further attraction of the
approach is that it can deal with other variations in the system state variables with
time, for example due to deterioration in the condition in the variables describing the
system state or changes in the loading due to climate change or other environmental
changes.
2.2 Standards based attribution
Consider an organisation with responsibility for urban drainage (hereafter a UDO),
providing a specified level of service to discharge rainfall events up to return period
T
s
, although it is likely that through degradation etc. the system only conforms to
T
s
? T
s
?, the sewer and drainage capacity
(even assuming no blockages) will certainly be exceeded.
A flood model can be used to estimate the damage D(T
s
) and D(T) (by definition
D(T
s
?) = 0). Damage attributable to the UDO is D(T
s
)D(T
s
?) and damage not
attributable to the UDO is D(T)D(T
s
). This can be extended to give the expected
attributed damage over the distribution of rainfall L:
?
?
()
0
Expected attributed damage for UDO = ( ) ( )
s
lT
LDLdl (2)
where l(T
s
) is the rainfall with return period T
s
.
This may be extended further to consider the situation in which due to blockage or
some other sewer failure the damage is not D(T) but D(TF) where F indicates some
failure event in the sewer system attributable to the UDO. The damage not
attributable to the water service provider is still D(T)D(T
s
), so the damage that is
attributable to them is now D(TF)D(T)+D(T
s
). The expected attributed damage
calculation now requires a probability distribution over the various possible blockage
states F
j
:
Expected attributed damage = ??
??
=
?
?
??
2
1
0()
()()( ) ()()
n
s
jj
j
lT
PF LDL F ds LDLds (3)
However, P(F) is notoriously difficult to estimate for sewer systems and so application
of Equation 3 is likely to be limited.
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2.3 Sensitivity based attribution
An intuitive measure of influence or sensitivity is the extent to which variation in a
factor of interest (or a set of factors) has on a system performance, in our case flood
risk r. This is the classical sensitivity analysis problem to which there are a number of
solutions. However, relating sensitivity analysis to risk attribution is, in general, not
straightforward.
If each of the loading variables (e.g. fluvial flows, rainfall) were the unequivocally
responsibility of a particular agent, then sensitivity analysis would provide a basis for
definition of risk ownership. Risk ownership could be disaggregated on the basis of
sensitivity to the relevant loading variable. However, rainfall runoff, for example, is
dealt with in sewer and highway drainage systems as well as urban water courses.
Hence it is necessary to consider the variables R that define system performance.
Evidently, this is also necessary to make asset management prioritisation decisions.
Risks arise because of phenomena whose future state is not known with certainty. If
the magnitude of a given load on a system was known with certainty then decision
making would be easy. We would take measures to reduce the predicted damage if it
was economical to do so and otherwise we would not. In other words we would know
future losses precisely and the notion of risk, which is associated with phenomena
that are only predicable in probabilistic terms, would be redundant. Because, in fact,
the future is uncertain we construct the concept of risk and design measures to
reduce risk i.e. to reduce the expected damage due to some uncertain hazards.
Variancebased methods seek to attribute risk to system variables on the basis of the
amount that those variables contribute to uncertainty and hence to risk.
Consider a model of the form Y = g(X
1
,?, X
k
). The sensitivity index I
i
represents the
fractional contribution of a given factor X
i
to the variance in a given output Y. In order
to calculate the sensitivity indices the total variance V in the model output Y is
apportioned to all the input factors X
i
as (Sobol, 1993):
<<<
=++ ++
??? 12...
...
iij ijl k
iijijl
VVV V V (4)
where
()
??
==
??
*

iii
VVEYX x (5)
===?
**
,
ij i i j j i j
VVEYX xX x VV (6)
()
??
=
??
*

ii
VEY X x is referred to as the Variance of the Conditional Expectation (VCE)
and is the variance over all values of
*
i
x in the expectation of Y given that X
i
has a
fixed value
*
i
x . This is an intuitive measure of the sensitivity of Y to a factor X
i
, as it
measures the amount by which
( )
*

ii
EY X x= varies with the value of
*
i
x , while all
the effects of the X
j
?s, j?i, are averaged. The first order (or ?main effect?) sensitivity
index S
i
for factor X
i
is therefore defined as:
=
i
i
V
I
V
(7)
Also of interest is the influence of factor X
i
when acting in combination with other
factors. There are 2
k
1 of such interactions, so it is usually impractical to estimate the
effect of all of them. A more practical approach is to estimate the k total sensitivity
indices, I
Ti
, where (Homma and Saltelli 1996):
??=
??
=?
*
~~
(
1
()
ii
Ti
VEY X x
I
VY
(8)
SESSION 1.2
NOVATECH 2007 127
where X
~i
denotes all of the factors other than X
i
. The total sensitivity index therefore
represents the average variance that would remain as long as X
i
stays unknown. The
total sensitivity indices provide an indicator of interactions within the model. For
example, factors with small first order indices but high total sensitivity indices affect
the model output Y mainly through interactions ? the presence of such factors is
indicative of redundancy in the model parameterisation.
In the case of flood risk analysis, the output quantity of interest is the damage D. The
probability density function of the annual damage estimate, f
D
(d):
?=
?
() (( ) ( )( )
Dd
fd IDXDX Xdx (9)
where I
d
(?) is the indicator function. Recall that risk r is the mean of D i.e.
?
?
=
?
0
()()rXDXdx (10)
The variance is
?
?
=?
?
2
0
() ( )( )Var d X D X r dx (11)
The variancebased sensitivity analysis described above is applied to this function.
3 IMPLEMENTATION
The risk attribution methodology is implemented in the first instance on a realistic (but
not real) system shown in Figure . Upstream of the urban area is a rural catchment of
50km
2
. Runoff from rural catchment is discharged into the river that is the recipient for
runoff from urban area. The area of the urban catchment is 1.5km
2
, with 4.6km of
storm sewer pipes (minor drainage system, with three outlets to the river) and 3.3km
of streets/roads with assumed wide trapezoidal crosssection (major drainage system,
with one outlet ? link 185163). Interaction between minor and major system can take
place through virual weirs that link manholes to surface network nodes.
Figure 1 Urban flood system (catchment boundary covers area of 1.2km by 1.25km)
SESSION 1.2
128 NOVATECH 2007
Pipes are ?designed? so that, at low river flows (i.e. at free outflow from all outlets),
sewer system can handle surface runoff from 1 in 10 year storms with surcharging but
without the hydraulic head reaching the terrain level. At more intense storms, minor
system capacity becomes insufficient and pluvial flooding takes place. On the other
hand, assuming zero runoff from urban area, fluvial flooding occurs at river flows
exceeding the 1 in 100 year flow rate. The urban area is susceptible to combined
pluvial/fluvial flooding when backwater influence from the river may reduce the
capacity of the sewer system. Properties (or damage points) are assumed to be
spaced at 20m intervals along the roads.
Figure 2 shows steps required to generate estimates of flood risk, described below:
1) Rainfall boundary conditions are defined as a series of 50% summer profile
storms (Butler and Davies, 2004) for return periods of 1 to 1000 years and
durations between 15min and 24h.
2) The rainfall is propagated through a hydrological model ARNO (Todini, 1996) to
give the upstream river flow rates for the hydrodynamic model.
3) The rainfall and river flow are used as inputs to the coupled surface/subsurface
hydrodynamic model, SIPSON (Djordjevi? et al., 2005).
4) Maximum flood depths obtained by the hydrodynamic model are integrated over
functions describing depthdamage relationships for properties (PenningRowsell
et al. 2003) to calculated the flood damage, D, for a given event.
5) The flood risk, expressed in terms of expected annual damage, is calculated
using Equation (1).
6) The sensitivitybased analysis is subsequently applied using Equation (10).
Figure 2 Model linkages for integrated flood risk assessment
In addition to varying loading parameters (rainfall duration, peak intensity and river
flows), the following infrastructure parameters were varied:
1) Pipe size (uniform distribution, range 40% to +50% of the ?designed? diameters).
2) Percentage of impermeable area (normal distribution, range 30% to 90%).
3) River bottom width (beta distribution, range 1.5m to 11m).
Calculate
rainfall inputs
Hydrological
model
Hydrodynamic
surface/ sub
surface model
Flood
depths
Depth
damage
curves
Flood
damages
River flows
Urban
rainfall
Catchment
rainfall
Risk analysis
and attribution
SESSION 1.2
NOVATECH 2007 129
Based on sensitivity indices, the pie chart in Figure 3 shows the total contribution of
each variable to risk. As is evident, duration, peak rainfall and pipe size are the most
important loading variables. River width has no effect on flood risk and the influence
of peak flow and impermeable area is generally insignificant. It should be noted that
the obtained figures are very much casespecific and therefore should not be
considered to have any general relevance. Instead, they should be taken as an
illustration of the proposed methodology.
Figure 3 Total sensitivity indices
The influence of sewer blockages was analysed. Sensitivity indices were calculated
for individual pipes (assuming their blockage) to identify which ones contribute most
to the flood risk. As expected (see Figure 4), blockage of the lowest of the three
outlets (pipe 165163) contributes most to flood risk. Somewhat surprisingly, blockage
of the middle outlet (pipe 172171) would contribute insignificantly to flood risk. Other
investigations made in this study included combinations of blocked pipes and analysis
of the effect of climate change assuming different scenarios (Speight, 2006).
Figure 4 Total sensitivity to blockages in the sewer system
duration
41%
peakflow
7%
peak rain
31%
change in pipe size
19%
river width
0%
Impermeable area
2%
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130 NOVATECH 2007
4 DISCUSSION AND CONCLUSIONS
Integrated flood risk analysis requires that risk is measured using a common metric.
We have identified core principles and identified two approaches to disaggregating
the contribution to risk from different loadings, system components and stakeholders.
The standardsbased attribution methodology does not require significant
computational resource, but because of the difficulties associated with estimating
sewer failure probabilities is limited in practise to risk attribution of loadings only (i.e.
the contribution towards the total risk from urban rainfall and river flow).
The sensitivitybased attribution methodology can be readily used to explore the
contribution from specific infrastructure components (eg. flood defences, sewer
network). However, drainage systems involve thousands of variables. The only
feasible approach to tackling this problem is therefore by hierarchical simplification of
the system, with the attribution analysis being applied at several levels, from a very
broad scale to identify the main influences on flood risk, to a detailed scale for small
well defined problems, to identify the components that are responsible for flood risk.
Approach to integrated flood risk management presented in this paper uses statistical
methods to analyse results of series of simulations made by deterministic fulldynamic
flood model. Consequent flood damage is interpreted using spatial integration of
maximum flood depths linked to corresponding depthdamage curves. Asset
management decisions based on sensitivitybased attribution of flood risk are clearly
much sounder than those made upon standardsbased analysis, which are based on
a single event (or a limited number of events).
Future research will look at possibilities for implementing risk attribution methodology
at a broader scale on real systems. In these studies, the importance of other groups
of elements such as pumps, storage or SUDS will be analysed. Description of
damage will be enhanced to include spatially variable housing density and value
(using GIS), traffic disruption, health impacts and other damages.
5 ACKNOWLEDGEMENTS
The research upon which this paper is based is funded through the EPSRC Flood
Risk Management Research Consortium: (Work Packages 4.5 and 6.1): Grant
GR/S76304/01.
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