487

Analytical probabilistic flood routing for urban

stormwater management purposes
Yiping Guo and Zhengji Zhuge

Abstract: The main purpose of flood routing calculations conducted in urban stormwater management studies is to deter-
mine the flood frequency distribution at a downstream location knowing the upstream catchments, channel reaches, and
detention ponds. As the current state of practice, the design storm approach is widely used for this purpose and for the
planning and design of related stormwater management facilities. Building upon previous work, methods and procedures
are proposed here so that flood routing for stormwater management purposes can be completed probabilistically using ana-
lytical equations. Results from the proposed methods and procedures are compared with design storm modeling results for
a number of routing scenarios. The comparison study not only establishes the credibility of the proposed methodology, but
also highlights potential problems associated with the application of the design storm approach.
Key words: design storms, detention pond, flood control, drainage design, master drainage plan.
Résumé : L’objectif principal des calculs de détournement des eaux de crue effectués lors d’études de gestion des égouts
pluviaux est de déterminer la distribution statistique des crues à un endroit en aval en connaissant les bassins récepteurs en
amont, les tronçons de canaux et les bassins de rétention. Dans la pratique actuelle, l’approche de l’averse type est large-
ment utilisée à cette fin et pour la planification et la conception des installations connexes de gestion des eaux pluviales.
Des méthodes et des procédures basées sur les travaux antérieurs sont proposées afin que le calcul du détournement des
eaux de crue à des fins de gestion des eaux pluviales puisse être réalisé de manière probabiliste en utilisant des équations
analytiques. Les résultats des méthodes et des procédures proposées sont comparés aux résultats de la modélisation des
averses types dans plusieurs scénarios de détournement. L’étude comparative a non seulement établit la crédibilité de la
méthode proposée, mais a souligné les problèmes potentiels reliés à l’utilisation de l’approche utilisant les averses types.
Mots-clés : averses types, bassin de rétention, contrôle des crues, débit type, plan principale de drainage.
[Traduit par la Rédaction]

1. Introduction logic model may be considered as a design model that is

used without calibration to convert a set of hydrologic inputs
One of the tasks in urban stormwater management is the to discharge hydrographs. Hydrologic models developed
estimation of the frequency distributions of runoff volumes specifically for stormwater management purposes are often
and peak discharge rates at locations of interest throughout referred to as stormwater models.
an urban or urbanizing watershed. Stormwater management The vast majority of existing stormwater models are de-
facilities (e.g., detention ponds, artificial wetlands, infiltra- terministic in nature because neither the input nor the output
tion trenches, etc.) may be constructed to satisfy site-specific of these models is described as a random variable. The prob-
stormwater management requirements. As long-term ob- ability of occurrence and annual or seasonal averages of
served flow data rarely exist in small urban watersheds, hy- some of the output variables of interest can only be deter-
drologic models have to be used as the basic tool for mined through either the use of design storms of various re-
estimating the frequency distributions of runoff volume and turn periods or the use of continuous simulation with long-
peak discharge and for evaluating the performance of storm- term observed rainfall data as input followed by frequency
water management facilities. For these purposes, a hydro- analysis on the output. Currently, the design storm approach
is used predominantly in stormwater management studies
Received 17 May 2007. Revision accepted 17 October 2007. (ASCE and WEF 1992).
Published on the NRC Research Press Web site at cjce.nrc.ca on The basic assumption made in the use of design storms is
13 May 2008. that the return period of the resulting runoff and peak flow
Y. Guo.1 Department of Civil Engineering, McMaster is the same as that of the input design storm. This assump-
University, 1280 Main St. West, Hamilton, ON L8S 4L8, tion may result in significant errors in some cases (Linsley
Canada. et al. 1982; Adams and Howard 1986). A series of research
Z. Zhuge. Water Resources Department, Marshall Macklin (e.g., Marsalek 1978; Marsalek and Watt 1984; Urbonas
Monaghan Ltd., 80 Commerce Valley Dr. East, Thornhill, ON 1979; Packman and Kidd 1980; Beaudoin et al. 1983;
L3T 7N4, Canada. Wenzel and Voorhees 1984; Voorhees and Wenzel 1984;
Written discussion of this article is welcomed and will be Nnadi et al. 1999; Levy and McCuen 1999; etc.) has been
received by the Editor until 30 September 2008. conducted to understand the design storm approach’s limita-
tions and ensure its proper use in engineering practice. Most
1Corresponding author (e-mail: guoy@mcmaster.ca). of these studies focused only on flood peak estimation be-

Can. J. Civ. Eng. 35: 487–499 (2008) doi:10.1139/L07-131 # 2008 NRC Canada
488 Can. J. Civ. Eng. Vol. 35, 2008

cause runoff volume was not the main concern before storm- Their comparison studies showed that the APSWM results
water quality control was required. Urbonas (1979) con- are fairly close to SWMM continuous simulation results for
cluded that it is possible to develop design storms that, Fort Collins. However, for Santiago, the comparisons are not
when used together with a stormwater model, can reason- as good as those for Fort Collins. The authors pointed out
ably predict peak flows from small urban basins of various the causes of these differences and suggested methods to
recurrence intervals. Comparing design storm and continu- further improve APSWM. Quader and Guo (2006) applied
ous simulation results, Packman and Kidd (1980) pointed APSWM for the estimation of peak discharge rates in a
out that the proper selection of design storms and antecedent practical design case with multiple subcatchments in
catchment conditions is of paramount importance if the Kingston, Ontario, Canada, and compared results with the
probabilities of rainfall and runoff are to be considered design storm approach. Differences in meteorological data
equal. Studying the applications of the design storm ap- analysis and representation of rainfall input, subcatchment
proach for different locations, Beaudoin et al. (1983), aggregation, and the treatment of the catchment time of con-
Wenzel and Voorhees (1984), Voorhees and Wenzel (1984), centration between the two approaches were identified as the
and Nnadi et al. (1999) all concluded that significant param- three main causes contributing to the discrepancy in peak
eter sensitivity does exist, but an appropriate choice of de- discharge estimates. In spite of the differences, peak dis-
sign storm parameters can produce peak flows of desired charge estimates from the two approaches are generally
return periods. comparable for the actual design case.
Despite the fact that the basic assumption of the design Implemented in a computer program, APSWM is com-
storm approach is still not proven to be always acceptable, putationally efficient and easier to use than either the de-
the consensus of the engineering community seems to be sign storm or the continuous simulation approach.
that the design storm approach can produce peak discharges However, the analytical expressions forming the basis of
of desired return periods with acceptable levels of accuracy. APSWM do not explicitly consider the effect of flood
The acceptance of this approach is partly due to the lack of routing through a channel reach. This is due to the com-
other feasible design and analysis methods with a comparable plexities encountered in mathematical derivations. In this
level of accuracy. Thus, the state of practice in stormwater study, a simplified method is proposed for use with
management is that, unless runoff volume is of specific inter- APSWM to account for the flood peak attenuation effect
est or the project is of great importance, continuous simula- of individual channel reaches. In addition, multiple storm-
tion is not conducted even though it may provide more water detention ponds are sometimes designed to control
accurate estimates of peak discharge frequencies. stormwater from different subareas in a master drainage
The analytical probabilistic approach was recently devel- plan study for a large area. However, the analytical expres-
oped as an alternative or complementary approach to provid- sions developed so far can only determine flood frequen-
ing estimates of frequency distributions of flood peaks and cies immediately downstream of a detention pond and still
runoff volumes from urban catchments. Details on the analyt- cannot explicitly represent the impact of an upstream de-
ical probabilistic approach can be found in Guo and Adams tention pond on the performance of a downstream deten-
(1998a, 1998b, 1999a, 1999b). Closed-form analytical equa- tion pond or the flood frequencies at locations further
tions are used in the analytical probabilistic approach to esti- downstream. A practical procedure is provided here so
mate flood peaks and runoff volumes of various return that cases involving multiple detention ponds can also be
periods. These equations are collectively referred to as the modeled properly by APSWM.
analytical probabilistic stormwater models (APSWM). The new methods and procedures added to the existing
For single catchments of various imperviousness, slopes, APSWM framework constitute an analytical probabilistic
and soil characteristics, the appropriateness of simplifying approach for flood routing through channel reaches and
assumptions made in the development of APSWM was detention ponds. They are intended for use in master drain-
verified in Guo and Adams (1998a, 1998b, 1999a, 1999b) age plan studies and in the planning and design of small-
by comparing APSWM results with continuous simulation scale stormwater management facilities. To evaluate their
results. Rainfall data from Toronto, Ontario, Canada were appropriateness for engineering applications, results from
used in these verification studies. Guo (2001) provided the the new methods and procedures will be compared with de-
first comparison between the analytical probabilistic, design sign storm modelling results. Design storm modelling will
storm, and continuous simulation approaches for a test be conducted in accordance with established procedures
catchment in Chicago, Illinois, USA. It was found that the [e.g., those outlined in Packman and Kidd (1980)] in addi-
three approaches can generate similar results for the tion to local regulatory requirements. Furthermore, design
prediction of peak discharges of various return periods from storms of two durations and two different hyetographs will
urban catchments and for the design of flood control also be used to quantify the impact of the duration and de-
detention ponds servicing urban catchments. It was also tailed hyetograph of design storms on peak discharge esti-
demonstrated that appropriate design storm durations and mation.
hyetographs must be chosen in order for the design storm
approach to provide similar results as compared with contin- 2. Brief description of the analytical
uous simulation and APSWM. probabilistic approach
Rivera et al. (2005) applied APSWM for two locations:
Fort Collins, Colorado, USA, and Santiago, Chile. The 2.1. Probabilistic models of storm event characteristics
APSWM results were compared with continuous storm Instead of using individual storms as with the design
water management model (SWMM) simulation results. storm approach, the analytical probabilistic approach uses
# 2008 NRC Canada
Guo and Zhuge 489

probabilistic models of storm (or rainfall) event characteris- rainfall event. If the fraction of impervious areas of an urban
tics. These probabilistic models are obtained by first identi- catchment is h, combining the runoff volumes from imper-
fying individual rainfall events at a specific location as vious and pervious areas of the urban catchment, the event-
recorded in the continuous rainfall series of that location. based runoff generation is represented in APSWM as
After the identification of individual rainfall events, the rain- 8
fall volume (v) and rainfall duration (t) of each rainfall <0 for v  Sdi
event, as well as the interevent time (b) following each rain- ½1
vr ¼ hðv  Sdi Þ for Sdi < v  Sil þ fc t
:
fall event are determined. Treating the individual v, t, and b v  Sd  fc ð1  hÞt for v > Sil þ fc t
values as realizations of three random variables, frequency
analyses can be conducted with histograms prepared and In eq. [1], vr is the runoff event volume resulting from the
probability density functions fitted. An average annual num- input of a rainfall event with duration t and volume v;
ber of storm events can also be obtained from these statisti- Sd = hSdi + (1 – h)Sil is the area-weighted depression storage
cal calculations. It has been found that exponential of the impervious areas and the initial losses of the pervious
probability density functions often fit the v, t, and b histo- areas of the urban catchment. Using eq. [1] and the proba-
grams satisfactorily (Eagleson 1972, 1978; Howard 1976; bilistic models of storm event characteristics, the frequency
US Environmental Protection Agency 1979; Adams et al. distribution of vr was mathematically derived. This fre-
1986; Guo and Adams 1998a; Guo 2001). quency distribution is expressed in closed-form analytical
The development of APSWM employs the exponential equations. The average annual or seasonal runoff volume is
distributions for rainfall event characteristics as given in Ta- also expressed in a closed-form analytical equation. Detailed
ble 1. In Table 1,
, , and are distribution parameters; derivations and results can be found in Guo and Adams
and q is the average annual or seasonal (e.g., nonwinter sea- (1998a).
sons) total number of rainfall events. For a specific location, Routing of runoff over a catchment is modelled in
the four parameters
, , , and q need to be known to use APSWM by the incorporation of the catchment time of con-
the probabilistic models to describe local rainfall character- centration and the assumption that individual runoff hydro-
istics. The value of the three distribution parameters may be graphs are approximately triangular in shape. The duration
estimated from the average event volume, the average event of the runoff event is estimated as t + tc; i.e., the duration
duration, and the average interevent time determined from of the input rainfall event plus the catchment time of con-
the local rainfall record. For locations across Canada, these centration. The time of concentration is defined as the aver-
rainfall statistics may be obtained from Adams and Papa age time required for runoff to travel from the most remote
(2000). portion of the catchment to its outlet or design point. The
peak discharge rate Qp of a runoff event can therefore be es-
2.2. Catchment rainfall-runoff model timated as
In developing APSWM, the surface runoff generation of a 2vr
catchment is modelled in a very similar way as conventional ½2
Qp ¼
t þ tc
stormwater models. Runoff generation from the pervious
and impervious areas of an urban catchment are calculated Using eq. [2] to estimate peak discharge rates of runoff
separately. For impervious areas, the user specifies the value events, the typical center-peaked hyetographs of input rain-
of depression storage, Sdi; for pervious areas, the user speci- fall events are partly taken into consideration. The probabil-
fies the value of pervious area depression storage, Sdp, initial istic models of storm event characteristics together with
soil wetting infiltration depth, Siw, and the ultimate infiltra- eq. [2] were used to derive the frequency distribution of Qp.
tion capacity, fc. The initial soil-wetting infiltration as de- The resulting frequency distribution is expressed in closed-
fined in detail in Guo and Adams (1998a) takes into form analytical equations. Detailed derivations and results
account the recovery of the soil’s infiltration capacity during can be found in Guo and Adams (1998b).
interevent times. It was shown in Guo and Adams (1998a)
that the value of Siw can be estimated from the infiltration 2.3. Detention pond flood control and stormwater quality
parameters of the soil and parameters of the probabilistic control models
models of storm event characteristics. To simplify expres- Routing of runoff hydrographs through detention ponds is
sions, the sum of Sdp and Siw is denoted as Sil and is referred modelled for sizing individual ponds in APSWM by assum-
to as the pervious area initial losses. Similar to the effect of ing that outflow hydrographs are also approximately triangu-
Sdi on impervious areas, rainfall must first fill the need of Sil lar in shape. This is shown in Fig. 1. The storage–discharge
before runoff can generate from pervious areas. characteristics of a detention pond are represented the same
The APSWM considers the input storm event and the out- way as with the use of conventional stormwater models, i.e.,
put runoff event individually in their entireties. A runoff through the specification of pairs of storage–discharge val-
event is characterized by runoff event volume, runoff event ues. Details can be found in Guo and Adams (1999a). Math-
duration, and peak discharge rate. The overall runoff gener- ematical derivations enabled the expression in analytical
ation from an urban catchment is the area-weighted sum of form of the frequency distribution of the peak outflow from
runoff from the pervious and impervious portions of the a detention pond downstream of a catchment. Using these
catchment. In an urban catchment, Sdi is usually less than expressions, the required storage–discharge relationship of
Sdp. Therefore, if v is less than Sdi, then v must be less than the detention pond to achieve a specific level of flood con-
(Sil + fct), as fc and t are both positive. The sum (Sil + fct) is trol can be determined expediently. Furthermore, the water
the maximum total losses from the pervious area during a quality control aspects of detention ponds were investigated
# 2008 NRC Canada
490 Can. J. Civ. Eng. Vol. 35, 2008

Table 1. Probabilistic models of local rainfall characteristics and rainfall statistics for Toronto,
Ontario, Canada [based on rainfall data from Lester B. Pearson International Airport (Adams
and Papa (2000)].

Distribution Statistics for

Rainfall characteristic Exponential PDF parameter Toronto
Volume, v (mm) fV ðvÞ ¼
expð
vÞ
¼ 1= v
= 0.1060
Duration, t (h) fT ðtÞ ¼ expðtÞ  ¼ 1= t  = 0.1294
Interevent time, b (h) fB ðbÞ ¼ expð bÞ ¼ 1= b = 0.0097
Average annual or seasonal total Not applicable q q = 57.2
number of rainfall events
Note: PDF, probability density function.

Fig. 1. Routing of runoff hydrograph through detention ponds and channel reaches.

in Guo and Adams (1999b). Analytical equations were de- input rainfall hyetograph in a similar way as detention ponds
rived for the estimation of the long-term average detention transform inflow hydrographs: attenuating and delaying the
time and runoff volume capture efficiency provided by a peak input. Using APSWM, catchment rainfall-runoff rout-
stormwater quality control pond. ing is accomplished through the adoption of the triangular
The advantages provided by APSWM’s analytical expres- hydrograph assumption and the use of time of concentration.
sions are that the probability distribution of input rainfall is The longer the time of concentration, the greater the degree
transformed correctly using the derived distribution theory of peak attenuation. If outflow hydrographs from a detention
(Benjamin and Cornell 1970) to that of the output runoff pond can still be approximated as a triangle, the flood peak
characteristics. The equality assumption about the probabil- reduction effect of a pond may be accounted for by increas-
ity of occurrence between an input design storm and its re- ing the time of concentration of the upstream catchment that
sulting output runoff characteristics, as inevitably adopted in discharges into the pond. The key is to estimate the addi-
the design storm approach, is avoided completely. However, tional time of concentration that the pond contributes.
the existing APSWM can only be used for lumped catch- In the original derivation of APSWM equations for flood
ments. To further expand APSWM’s capability, the effects control analysis (Guo and Adams 1999a), it was assumed
of channel reaches and multiple detention ponds need to be that the detention pond is empty at the beginning of a rain-
explicitly represented and modeled. fall event, and the interevent time following the rainfall
event is long enough for the pond to drain completely. This
3. Probabilistic detention pond flood routing is valid if the focus is on heavy rainfall events that occur
If there is a detention pond upstream of a detention pond infrequently. By incorporating these assumptions, the rout-
under study, as shown schematically in Fig. 2, the effect of ing of the hydrograph through a detention pond for each
the upstream detention pond may be taken into account by runoff event is depicted in Fig. 1. In Fig. 1, t is the duration
modifying the parameters of the catchment draining into the of the causal rainfall event, tc is the time of concentration of
downstream detention pond. However, this needs to be done the upstream catchment and te is the time required for the
properly to ensure that the flood routing effect of the up- pond to empty itself after one runoff event.
stream pond is recognized and represented as accurately and Let SQq and Qq be the maximum storage volume utilized
explicitly as possible. In essence, catchments transform the and the maximum outflow rate reached, respectively, during
# 2008 NRC Canada
Guo and Zhuge 491

Fig. 2. Routing diagram of a typical case where an internal deten- Fig. 3. Routing diagram of a typical case where a channel reach
tion pond needs to be explicitly modeled. needs to be explicitly modeled.

ship is nonlinear, runoff events that fill the pond to different

maximum levels would have different te values. An average
can still be estimated as the average of the te values
calculated from each pair of ðSQq ; Qq Þ on the pond storage–
discharge curve. This average te, denoted as tp, can then be
added to the upstream subcatchment’s time of concentration
to estimate the peak detention pond outflow resulting from
any rainfall event. That is
the passage of a runoff event. From Fig. 1, it can be seen 2vr
that ½4
Qp ¼
t þ tc þ tp
1
½3
SQq ¼ Qq te where vr is the same as expressed in eq. [1] from the up-
2 stream subcatchment. As the storage–discharge relationship
where ðSQq ; Qq Þ is also a point on the pond storage– of a pond provides the complete information about the
discharge curve. Equation [3] illustrates that if the incom- pond that affects its flood routing function, tp captures and
ing runoff event is such that the maximum storage volume condenses this complete information. To emphasize its im-
utilized is SQq , then the time required for the pond to portance in flood routing, tp is defined as the time of disper-
empty itself (te) is 2SQq =Qq . Guo and Adams (1999b) sion caused by a detention pond. This definition is
showed that SQq =Qq is equal to the average detention time introduced recognizing that a detention pond temporally dis-
that the runoff receives. Therefore, te is two times the perses the spatially concentrated inflow, whereas a catch-
average detention time. For the purpose of accounting for ment concentrates spatially distributed overland flow
the peak reduction effect of a pond, this te can be consid- resulting in temporal dispersion of the input hyetograph.
ered as the dispersion in time to incoming flood hydro- The temporal dispersion effect to flood hydrographs from a
graphs caused by the pond. The impact of te to an detention pond and a catchment is therefore similar.
individual flood hydrograph is the same as that of tc, i.e., Equation [4] would be the same as eq. [2] for a single
elongating the time base of the flood hydrograph. subcatchment if the subcatchment has (tc + tp) as its time of
Values of te for runoff events that fill the pond to the concentration. Thus, the APSWM equations for a single
same maximum level can be considered the same. If the catchment can be used with minor modifications for sub-
pond’s storage–discharge relationship is linear, 2SQq =Qq catchment and detention pond combinations. With the
would be a constant, te for runoff events of any magnitude known value of tp for a detention pond downstream of a
would be the same. If the pond’s storage–discharge relation- subcatchment with parameters of Sdi, Sil, fc, h, tc, and area,
# 2008 NRC Canada
492 Can. J. Civ. Eng. Vol. 35, 2008

the modified analytical equations [the original analytical bution of the peak outflow from a detention pond are as fol-
equations are derived and presented in Guo and Adams lows:
(1998b)] that can be used to determine the probability distri-

(1) For subcatchment and pond combinations with fc < Sdd =ðtc þ tp Þ,
½5
P½Qp > qp

8

>
> 2h
ðtc þ tp Þ
>
> exp 
Sdi  qp for qp < 2fc h
>
> 2h þ
qp 2h  
>
>
>
> 2
ð1  hÞðqp  2fc hÞ ½
Sil  ðtc þ tp Þ  fc
ðtc þ tp Þ
qp  2fc
hSdi þ 2hSdd
>
< exp 
ð2h þ
qp Þð2 þ
qp þ 2fc
 2hfc
Þ  qp  2fc h
¼
>
> 2h
ðtc þ tp Þ 2hSdd
>
> þ exp 
Sdi  qp for 2fc h  qp <
>
> 2h þ
qp 2h tc þ tp
>
>  
>
> 
ðt c þ tp Þ 2hS dd
>
: exp 
Sd  qp for qp 
 þ
½ðqp =2Þ þ fc  fc h
2 tc þ tp
(2) For subcatchment and pond combinations with fc  Sdd =ðtc þ tp Þ,
½6
P½Qp > qp

8  
>
> 2h
ðtc þ tp Þ 2hSdd
>
> exp 
Sdi  qp for qp <
>
> 2h þ
qp 2h  tc þ tp 
>
>
>
> 2
ð1  hÞð2fc h  qp Þ ½
Sil  ðtc þ tp Þ  fc
ðtc þ tp Þ
qp  2fc
hSdi þ 2hSdd
>
< exp 
ð2h þ
qp Þð2 þ
qp þ 2fc
 2hfc
Þ  qp  2fc h
¼
>
> 2
ðt c þ tp Þ 2hSdd
>
> þ exp 
Sd  qp for  qp < 2fc h
>
> 2 þ
ðq þ 2f  2f
c  c hÞ 2 t c þ tp
>
>
p 
>
> 
ðtc þ tp Þ
>
: exp 
Sd  qp for qp  2fc h
 þ
½ðqp =2Þ þ fc  fc h
2

In the above equations, Qp is the peak outflow from the ilarly, to determine the flood frequency at the outlet of the
pond regarded as a random variable, qp is a specific peak downstream detention pond, the areas upstream of the down-
discharge value, and P[Qp > qp] is the exceedance probabil- stream detention pond are aggregated into one equivalent
ity per rainfall event that Qp is greater than qp. The conver- subcatchment first; this equivalent subcatchment is then ag-
sion from exceedance probability per rainfall event to return gregated with the downstream detention pond — the flood
period is as follows: routing effect of each pond is explicitly represented by
their respective tp values.
1
½7
TR ¼
 P½Qp > qp

4. Probabilistic channel flood routing
where TR is the return period (in years) of the given peak
discharge rate qp; and q is the average number of rainfall The objective of traditional deterministic channel flood
events per year. Flood routing through detention ponds can routing is to determine the effect of a channel reach on an
therefore be analytically performed, the effect of a pond is inflow hydrograph as it propagates downstream. A river
passed downstream explicitly through the incorporation of reach usually makes two distinct modifications to an inflow
its time of dispersion. hydrograph: attenuation of the peak and translation of the
For the example catchment shown in Fig. 2, conventional hydrograph. For stormwater management planning and de-
stormwater models may model it as comprised of two sub- sign purposes, we are usually only interested in flood peaks
catchments and two detention ponds. There may be a chan- corresponding to various return periods rather than flood
nel reach with intermittent flows downstream of the upstream peaks resulting from individual incoming flood events.
detention pond. Just for the purpose of setting up an exam- Thus, in stormwater management analyses using either the
ple, the effect of this channel reach may be neglected due design storm or the continuous simulation approach, channel
to its shortness. The assumption of equality in the proba- flood routing is conducted not to predict in real time the
bility of exceedance between input design storms and out- flood wave movement, but to determine the frequency distri-
put runoff characteristics would have to be made for all butions of flood peaks at locations of interest.
locations of interest. Using APSWM, to determine the flood When a channel reach is downstream of a subcatchment
frequency at the outlet of the upstream detention pond, the as shown schematically in Fig. 3, the peak attenuation effect
pond and its upstream subcatchment is aggregated into one exerted by the channel reach is mainly a result of the valley
equivalent subcatchment with the addition of tp to tc. Sim- storage that the channel reach provides. This peak-reducing
# 2008 NRC Canada
Guo and Zhuge 493

valley storage bears similarity to the storage provided by de- terest in stormwater management studies are in the order of
tention ponds. Therefore, a time of dispersion to flood hy- minutes to a maximum of about 2 h, whereas average rain-
drographs caused by the channel reach (referred to as the fall event durations are in the order of 4 to 10 h. Therefore,
channel reach time of dispersion tr) may be added to the up- the triangular hydrograph assumption should still be reason-
stream subcatchment’s tc to account for the peak attenuation able for most cases. Besides, if a channel reach is too long
effect of the channel reach. The key is to find a suitable to be modeled as a single unit, it can be divided into sub-
method to calculate tr so that the physical characteristics of reaches and modeled separately with lateral inflows from
the channel reach that affect its peak reduction capability the additional subcatchments added as the calculation moves
most are captured and properly represented. gradually downstream.
We know from derivations for detention ponds that a It is observed that for an extremely short reach, inflow
pond’s time of dispersion is two times the average detention into the reach may spread instantly throughout the reach
time that the pond provides. Average detention time is also and the total potential peak reduction capability of the reach
the average travel through time experienced by water par- would be utilized, therefore the peak reduction behavior of
ticles. On one hand, the total potential peak reduction capa- the reach is the same as that of a pond and tr for this reach
bility of a channel reach may be assumed to be the same as should be 2tt rather than tt. To obtain this result, the base
that of a pond with a detention time equaling the average time of the outflow hydrograph from the channel reach
time of travel, tt, through the reach. This simplifying as- should be (t + tc + 2tt), still in agreement with the basic as-
sumption is reflected by the base time of (t + tc + 2tt) of the sumption adopted earlier, but with the low rising limb barely
outflow hydrograph from a channel reach resulting from a existing. The disappearance of the low rising limb is be-
rainfall event with duration t falling onto an upstream sub- cause inflows can instantly spread throughout an extremely
catchment with time of concentration tc. Equivalently, we short reach. Inflow into a regular reach cannot spread in-
are assuming that it would also take a channel reach two stantly throughout the reach — the tr values for these
times the detention time that it provides to finish discharg- reaches should therefore be less than 2tt. The revelation of
ing the flood volumes after a runoff event. This assumption the behavioural connection between a pond and an ex-
seems to be reasonable given the similarity of the storage tremely short reach seems to further justify the approxima-
effect provided by a channel reach and an on-line detention tion of tr with tt for most channel reaches.
pond. On the other hand, the difference between a channel The modified closed-form analytical equations that ac-
reach and a detention pond is that inflow into the reach count for the flood peak reduction effect of a channel reach
does not spread instantly throughout the reach, whereas in- are therefore the same as those for detention ponds (i.e.,
flow into a pond would much more likely spread instantly eqs. [5] and [6]) with tp replaced by tr. The time of travel,
throughout the pond. This difference must be considered. tt, through a river reach can be calculated as the length of
As shown in Fig. 1, the outflow hydrograph from a channel the reach divided by the average flow velocity. The average
reach is no longer triangular, but approximated as having a flow velocity can be estimated based on the physical charac-
low rising limb followed by a triangle representing the dis- teristics of the reach and a representative discharge. Proba-
persed flood hydrograph. Therefore, unlike a detention pond, bilistic flood routing through a channel reach can, therefore,
the total potential peak reduction capability of a channel be conducted in a similar way as through a detention pond.
reach can only be partially utilized during the passage of For the example case shown in Fig. 3, to add the effect of
the majority of flood hydrographs. the downstream subcatchment that flows laterally into the
The channel reaches considered in this study do not have channel reach, this downstream subcatchment would be con-
base flows. The low rising limb in Fig. 1 results from rain- sidered as parallel to the combination of the upstream sub-
fall that has fallen directly on the river. The length of this catchment and its downstream channel reach. The APSWM
low rising limb is equal to tt, the average time of travel would aggregate the upstream subcatchment with the chan-
through the reach. This is because it takes about time tt nel reach into an equivalent subcatchment first, and then ag-
from the start of the rainfall event for the first part of the gregate this equivalent subcatchment with the downstream
incoming flood hydrograph to appear at the downstream subcatchment before applying the analytical expressions for
end of the reach. For the estimation of peak outflow, this single lumped catchments. As discussed in Quader and Guo
low rising limb for the majority of flood events can be ne- (2006), aggregation of parallel subcatchments follows the
glected; that is, the outflow flood hydrograph from a reach principle of area average.
can still be approximated as a triangle starting at time tt and It can be seen from the above that the analytical probabil-
ending at time (t + tc + 2tt). Thus, the base time of this tri- istic flood routing methods and procedures proposed in this
angular hydrograph is (t + tc + tt), indicating that the channel paper focus on the peak reduction effect of subcatchments,
reach time of dispersion (tr) is equal to tt rather than 2tt. The reaches, and detention ponds, and aggregate these different
fact that tr is equal to the time of travel through a reach units so that their respective peak reduction effect is repre-
whereas tp is two times the time of travel through a pond is sented properly and explicitly through the use of time of
a result of the hydrologic differences between the two units concentration and time of dispersion. Aggregation is neces-
of modeling. sary so that probability transformations from input rainfall
The above-described simplification is acceptable as long to output peak flow rates can be traced directly and analyti-
as the channel reach is short (as measured by tt) so that the cally. Aggregation of subcatchments, reaches, and detention
outflow hydrograph from a channel reach resulting from an ponds following the proposed procedures does not necessa-
input rainfall event is approximately triangular in shape. The rily result in lower accuracy, given that the probability trans-
travel times through typical channel reaches that are of in- formation is properly traced and that the overall study area
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is small. Moreover, the lack of measured flow data as usu- upstream–downstream orders (e.g., runoff from A discharges
ally encountered in stormwater management studies makes to the channel reach and then combines with runoff from B),
it infeasible and unnecessary to calibrate more distributed as many as six subcatchment and channel reach configura-
hydrologic models. tions were modeled. This was done to examine the possibil-
ity that APSWM results may agree with design storm
modeling results for some configurations but not for others.
5. Comparison study cases and results
The results of the comparison of peak discharge at the
The proposed methodology for probabilistic flood routing downstream end of the channel reach for one of the config-
through channels and detention ponds was examined by urations is shown in Fig. 4. The comparisons for the other
comparing the expanded APSWM and design storm model- five configurations are similar.
ing results using micro-interactive design of urban storm- To obtain the APSWM results as illustrated in Fig. 4, the
water systems (MIDUSS) (Smith 2004), which was subcatchment tc values were calculated using the effective
developed in 1978 and continuously improved and main- rainfall intensities corresponding to each of the return peri-
tained ever since by A.A. Smith Inc. in Ontario, Canada. It ods of interest (i.e., 2, 5, 10, 25, 50, and 100 years). The ef-
is a stormwater modeling software used for the detailed de- fective rainfall intensity of a specific return period was
sign of a range of devices for centralized or on-site storm- determined by subtracting the average loss rate of the catch-
water management facilities. The MIDUSS is chosen in this ment from the rainfall intensity of the same return period.
study because it is accepted and widely used in Canada. Rainfall intensity of a specific return period was determined
Comparisons are made for hypothetical test catchments using the exponential distribution fitted to the average rain-
located in the vicinity of Toronto, Ontario. The rainfall input fall event intensities (Adams and Papa 2000). With known
data for MIDUSS are provided by the Atmospheric Environ- effective rainfall intensity, catchment length, slope, and
ment Services (AES) of Environment Canada. In this study, roughness, the corresponding time of concentration was esti-
to evaluate the variability resulting from input design mated using the kinematic wave formula (Wanielista et al.
storms, two different design storm types are used: AES 1 h 1997). For each return period of interest, the corresponding
design storms (Watt et al. 1986) and 3 h Chicago-type de- peak discharge was then estimated using APSWM with the
sign storms developed from rainfall intensity–duration– input of tc estimated from the effective rainfall intensity of
frequency (IDF) relationships. Both design storm types are the same return period. It was found that if a constant tc
recommended for use in Canada. Detailed input rainfall value corresponding to a return period of 5 years is used,
data for both models are listed in Tables 1, 2, and 3. To eval- the results are not significantly different from what is re-
uate the differences caused by the use of different catchment ported in Fig. 4. This indicates that a constant tc value for
surface runoff routing algorithms of the design storm ap- each subcatchment may be used in APSWM for practical
proach, three surface runoff routing methods were used in design purposes.
the comparison study: triangular Soil Conservation Service Deducing from the above observation, a constant tt value
(SCS) [now Natural Resources Conservation Service (i.e., not dependent on discharge rates) for each channel
(NRCS)] unit hydrograph, rectangular unit hydrograph, and reach may also be used in APSWM for practical design pur-
SWMM runoff routing algorithm (Huber and Dickinson poses. Therefore, constant tc values estimated using an ef-
1988). All three surface runoff routing algorithms are suit- fective rainfall intensity of a return period of 5 years for
able for small urban catchments and are available in MID- subcatchments and constant tt values estimated using an
USS as options; comparing results from using each one of average channel flow velocity corresponding to the dis-
them against those from APSWM also assists in identifying charge of a return period of 5 years for channel reaches
the one conventional surface runoff routing method that were used in comparison group 2. Knowing the catchment
may be considered approximately equivalent to the simpli- upstream of a channel reach, the discharge of a return period
fied surface runoff routing approach used in APSWM. of 5 years was determined using the APSWM equations.
Two groups of comparisons were conducted to examine Given this discharge value and knowing the channel reach’s
the suitability of the proposed methods for various subcatch- slope and cross-sectional characteristics, the average channel
ment, channel reach, and detention pond combinations: flow velocity was estimated using Manning’s equation for
group 1 consists of combinations where subcatchments dis- steady uniform flows (Wanielista et al. 1997).
charge to a downstream channel reach and group 2 consists The importance of tc in affecting a catchment’s peak dis-
of catchments with both internal and downstream detention charges has long been recognized (Henderson 1963). Taking
ponds. The group 1 comparison studies were performed to advantage of the calculation efficiency of APSWM, the im-
verify the proposed analytical probabilistic channel flood pact of tc on peak discharge is further quantified by making
routing method. Figure 3 illustrates a typical case where additional APSWM computations using tc values 25%
runoff from an upstream subcatchment is routed through a higher and lower than the estimated tc value. Figure 4 indi-
channel reach and then combined with runoff from a down- cates that a 25% variation associated with tc estimation
stream subcatchment. Three test subcatchments and one test caused 18%, 20%, and 22% differences in peak flow estima-
channel reach were set up to model this general case. The tion for return periods of 2, 10, and 100 years, respectively.
input parameters of the three test subcatchments, denoted as In terms of variation in peak flow caused by variation in tc,
A, B, and C, as well as those of the hypothetical channel similar results were obtained from the other comparison
reach are listed in Tables 4 and 5, respectively. Soils of the group and are shown in Fig. 5.
three subcatchments are representative of clay, silt, and The MIDUSS results as illustrated in Fig. 4 indicate that
sand. By arranging subcatchments A, B, and C in different if the AES design storms are used, use of the SWMM rout-
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Guo and Zhuge 495

Table 2. Rainfall input data of Atmospheric Environment Services 1 h design storms

(Watt et al. 1986).

Return period (years)

Input data 2 5 10 25 50 100
Average rainfall intensity, i (mm) 23.5 31.2 36.2 42.7 47.4 52.1
Rainfall duration (min) 60 60 60 60 60 60
Time to peak* (min) 21 21 21 21 21 21
Decay factor* 7 7 7 7 7 7
*For Toronto, Ontario.
Table 3. Rainfall input data of 3 h Chicago-type design storms.

Return period (years)

Input data 2 5 10 25 50 100
Storm parameter, a 403.040 526.753 610.344 714.935 790.246 869.583
Storm parameter, bs 0 0 0 0 0 0
Storm parameter, c 0.717 0.709 0.706 0.703 0.701 0.700
Peak fraction, r 0.400 0.400 0.400 0.400 0.400 0.400
Rainfall duration, td (min) 180 180 180 180 180 180
Note: The average intensity of a design storm, i, used for the construction of Chicago-type design storms is
based on the following rainfall intensity–duration–frequency (IDF) relationship: i = a/(td + bs)c, where td is in
minutes and i is in mm/h.

Table 4. Physical characteristics of test subcatchments.

Subcatchment
Physical characteristics A C B
Area, A (ha) 30 4 40
Imperviousness, h 1 0.20 0.70

Impervious portion
Overland flow length (m) 750 10 800
Manning’s n 0.014 0.013 0.02
Depression storage, Sdi (mm) 1.5 1.5 1.5
Pervious portion
Overland flow length (m) 0 40 343
Manning’s n 0.25 0.25 0.25
Ultimate (minimum) infiltration capacity, fc (mm/h) 3.6 0.36 36
Initial infiltration capacity, f0 (mm/h) 76.2 25.4 127
The exponential decay time constant, Kh (h) 0.25 0.25 0.25
Depression storage, Sdp (mm) 4.5 4.5 4.5
Initial soil wetting infiltration volume, Siw (mm) 13.2 4 17.3
Initial losses, Sil = Siw+ Sdp (mm) 17.7 8.5 21.8

Slope (pervious and impervious), S 0.01 0.005 0.005

Table 5. Physical characteristics of the test channel reach.

ing method versus rectangular unit hydrograph method Physical characteristic Value
caused 30%, 26%, and 22% average differences in peak dis- Manning’s n for channel 0.04
charge estimates for the 2, 10, and 100 year return periods, Channel base width (m) 0.60
respectively. The average difference here (and hereafter) is Channel left bank slope (H:1V) 3.00
calculated as half of the differences in the two peak esti- Channel right bank slope (H:1V) 3.00
mates divided by the mean of the two peak estimates and Channel gradient 0.005
expressed in percentages. Examination of Fig. 4 also reveals
that if the same surface runoff routing method is used for
both types of design storms, the use of AES design storms comparisons, Fig. 4 shows that APSWM results are closer
and Chicago-type design storms did not cause significant to the results from MIDUSS using the SWMM surface run-
differences in peak discharge estimates. For this group of off routing method. In comparison group 2, the SWMM sur-
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496 Can. J. Civ. Eng. Vol. 35, 2008

Fig. 4. Peak discharge rates versus return periods for comparison group 1 with the configuration where runoff from subcatchment A dis-
charges to the channel reach and then combines with runoff from subcatchment B. AES, Atmospheric Environment Services; APSWM,
analytical probabilistic stormwater models; MIDUSS, micro-interactive design of urban stormwater systems; SCS, Soil Conservation Ser-
vice; SWMM, storm water management model.

face runoff routing method was still used, whereas the rec- is different from group 1, where use of the two sets of de-
tangular unit hydrograph method was not used. sign storms resulted in only minor differences. Figure 5 in-
A sketch of the configurations studied in comparison dicates that APSWM results are in between MIDUSS results
group 2 is presented in Fig. 2. The same test subcatchments obtained with AES 1 h design storms and the SWMM sur-
as used in group 1 were used in this comparison group. The face runoff routing algorithm and those obtained with 3 h
orders of upstream and downstream test subcatchments were Chicago design storms and the SWMM surface runoff rout-
varied to examine the possible impact on results. It was ing algorithm.
found that only minor variations in the comparison resulted
when different combinations were used. The case reported
6. Conclusions and recommendations
here is the one with B as the upstream subcatchment and C
as the downstream subcatchment. The storage–discharge re- Throughout this study, single-event design storm model-
lationships of the two detention ponds are presented in Ta- ing was carried out following established procedures and
ble 6. The comparison of peak outflows from the downstream regulations regarding total storm depths, durations, detailed
detention pond is shown in Fig. 5. hyetographs, and antecedent soil moisture conditions. Differ-
The MIDUSS results as illustrated in Fig. 5 indicate that ent surface runoff routing methods have been used for the
use of the SWMM surface runoff routing method versus tri- purpose of illustrating the possible level of variability even
angular unit hydrograph method caused 16%, 20%, and 23% if these established procedures are followed. It was shown
average differences in peak discharge estimates for the 2, that selection of different surface runoff routing methods re-
10, and 100 year return periods, respectively. In this case, sulted in significant differences in peak discharges for both
even if the same surface runoff routing method is used for comparison groups. The magnitude of this difference was
both types of design storms, the use of AES 1 h and Chi- shown to be about 20% (expressed in average differences as
cago 3 h design storms still resulted in significant differen- defined earlier) between results obtained from the SWMM
ces in peak discharge estimates. Figure 5 shows that, when surface runoff routing method and the triangular unit hydro-
using the same SWMM surface runoff routing method, input graph method. This may cause problems in enforcing storm-
of AES 1 h storms and Chicago 3 h storms resulted in a water management regulations. To ensure uniformity and
10%, 14%, and 17% average differences in peak discharges consistency, it is recommended that regulatory agencies
for the 2, 10, and 100 year return periods, respectively. This specify not only the set of design storms, but also the appli-
# 2008 NRC Canada
Guo and Zhuge 497

Fig. 5. Peak discharge rates versus return periods for comparison Group 2 with the case where the upstream subcatcment B discharges to a
detention pond, runoff from the detention pond is combined with runoff from the downstream subcatchment C and then routed through the
downstream detention pond. AES, Atmospheric Environment Services; APSWM, analytical probabilistic stormwater models; MIDUSS, mi-
cro-interactive design of urban stormwater systems; SCS, Soil Conservation Service; SWMM, storm water management model.

capacity of the catchment (including depression storages, in-

Table 6. Storage–discharge relationships of detention
filtration during rainfall events, and storage provided by
ponds.
channel reaches and detention ponds) is small as compared
Discharge rate (m3/s) with the total volumes of the design storms. However, when
multiple detention ponds are used (as in group 2), use of
Storage Downstream Upstream pond AES 1 h and Chicago 3 h design storms resulted in signifi-
volume (m3) pond in group 2 in group 2
cant differences in peak discharge values at the outlet of the
0 0.000 0.000 catchment. This may have resulted from the fact that the to-
4 200 0.412 0.389 tal storage capacity of the catchment is comparable to the
6 100 0.468 0.565 total volumes of the design storms, and consequently, the
7 000 0.555 0.648
volumes of the design storms affected routing of runoff
8 000 0.707 0.741
through detention ponds such that significant differences in
9 000 0.823 0.833
peak outflows resulted. For a specific return period,
10 050 0.946 0.931
although the peak intensities contained in the two sets of de-
11 000 1.201 1.019
sign storms are similar, the total volumes contained in the
13 000 1.539 1.204
3 h Chicago storms were much larger than those contained
in the AES 1 h storms. To avoid potential problems, it is
recommended that the duration of design storms be specified
taking into consideration not only catchment time of concen-
cable and (or) adopted synthetic unit hydrograph or other trations, but also storage capacities that will be provided by
surface runoff routing method. stormwater management facilities. The time-of-dispersion
It was also shown that when there is no internal detention concept introduced in this paper may assist in the selection
pond (i.e., group 1), the AES 1 h and Chicago 3 h design of suitable durations of design storms.
storms generated similar peak discharge results at the outlet Given the problems associated with the design storm ap-
of the catchment when the same surface runoff routing proach, other feasible design approaches and tools are al-
method is used. This is probably due to the fact that both ways desirable. The proposed methods of representing the
design storms contain similar peak intensity and the storage flood peak attenuation effect of channel reaches and deten-
# 2008 NRC Canada
498 Can. J. Civ. Eng. Vol. 35, 2008

tion ponds together with the recommended procedures of ag- design storm concept in evaluating runoff peak flow. Journal of
gregating different hydrologic elements for use in the exist- the American Water Resources Association, 19(3): 483–487.
ing framework of APSWM resulted in an expanded doi:10.1111/j.1752-1688.1983.tb04607.x.
APSWM that is as versatile as conventional stormwater Benjamin, J.R., and Cornell, C.A. 1970. Probability, statistics and
models for the representation of the surface water hydrology decision for civil engineers. McGraw-Hill, New York.
of a watershed and its stormwater management systems. Eagleson, P.S. 1972. Dynamics of flood frequency. Water Re-
Considering that without the proper tracing of probability sources Research, 8(4): 878–898.
transformations, the design storm approach can only provide Eagleson, P.S. 1978. Climate, soil, and vegetation 2. The distribu-
inaccurate estimates for some cases, even those applied in tion of annual precipitation derived from observed storm se-
quences. Water Resources Research, 14(5): 713–721.
accordance with established procedures — the expanded
Guo, Y. 2001. Hydrologic design of urban flood control detention
APSWM is probably a viable alternative to the design storm
ponds. Journal of Hydrologic Engineering, 6(6): 472–479.
approach.
doi:10.1061/(ASCE)1084-0699(2001)6:6(472).
Flood routing in the expanded APSWM is performed in a Guo, Y., and Adams, B.J. 1998a. Hydrologic analysis of urban
probabilistic sense where the interest is not in the determina- catchments with event-based probabilistic models 1. Runoff vo-
tion of the modification that a channel reach or detention lume. Water Resources Research, 34(12): 3421–3431. doi:10.
pond makes to individual inflow hydrographs but in the de- 1029/98WR02449.
termination of the modification that the channel reach or de- Guo, Y., and Adams, B.J. 1998b. Hydrologic analysis of urban
tention pond makes to the probability distribution of peak catchments with event-based probabilistic models 2. Peak dis-
flows. The expression of the transformation in frequency charge rate. Water Resources Research, 34(12): 3433–3443.
distributions in analytical forms is made possible by aggre- doi:10.1029/98WR02448.
gation of different hydrologic elements and by simplifying Guo, Y., and Adams, B.J. 1999a. An analytical probabilistic ap-
assumptions related to hydrologic routing through individual proach to sizing flood control detention facilities. Water Re-
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of urban stormwater management studies, the accuracy 1999WR900125.
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may outweigh the loss of accuracy from those simplifying storm water quality control. Water Resources Research, 35(8):
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from APSWM and design storm modelling using some types Henderson, F.M. 1963. Some properties of the unit hydrograph.
of design storms and catchment surface runoff routing Journal of Geophysical Research, 68(16): 4785–4793.
Howard, C.D.D. 1976. Theory of storage and treatment plant over-
methods suggest that the expanded APSWM may be able to
flows. Journal of Environmental Engineering, 102(EE4): 709–
provide the degree of accuracy usually attainable in storm-
722.
water management calculations. Further verification of the
Huber, W.C., and Dickinson, R.E. 1988. Stormwater management
expanded APSWM using continuous simulation results or model, version 4: user’s manual. Environmental Research La-
observed long-term streamflow data is still desirable. boratory, Office of Research and Development, U. S. Environ-
mental Protection Agency, Athens, Ga.
Acknowledgments Levy, B., and McCuen, R. 1999. Assessment of storm duration for
This paper is a direct result of the 3 year project titled hydrologic design. Journal of Hydrologic Engineering, 4(3):
‘‘Transferring analytical probabilistic stormwater models to 209–213. doi:10.1061/(ASCE)1084-0699(1999)4:3(209).
practicing water resources engineers’’ supported by the Linsley, R.K., Kohler, M.A., and Paulhus, J.L.H. 1982. Hydrology
Centre for Earth and Environmental Technologies (E-Tech) for engineers. McGraw-Hill Book Company, New York.
of the Government of Ontario, Canada. Financial support Marsalek, J. 1978. Research on the design storm concept. Urban
from E-Tech, the two industrial partners of the project, i.e., Water Resources Research Program, American Society of Civil
Alan A. Smith Inc. and Weslake Inc. of Ontario, Canada, Engineers, Reston, Va. Technical Memorandum No. 33.
Marsalek, J., and Watt, W.E. 1984. Design storms for urban drai-
and the Natural Sciences and Engineering Research Council
nage design. Canadian Journal of Civil Engineering, 11: 574–
of Canada is gratefully acknowledged.
584. doi:10.1138/184-075.
Nnadi, F.N., Kline, F.X., Wray, H.L., Jr., and Wanielista, M.P.
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ROM]. Technical University of Denmark, Proceedings of Tech- h degree of imperviousness expressed as a fraction
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2.00 Rev2.00. Alan A. Smith Inc., Dundas, Ontario, Canada.
n Manning’s roughness coefficient
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P exceedance probability
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off, University of Kentucky, Lexington, Ky., 23–26 July 1979. Qp peak discharge rate of a runoff event regarded as a
pp. 27–35. random variable
U.S. Environmental Protection Agency. 1979. A statistical method Qq maximum outflow rate reached during the passage
for the assessment of urban stormwater. Report prepared by Hy- of a runoff event through a detention pond (m3/s)
droscience, Inc. for USEPA Nonpoint Sources Branch, EPA r peak fraction
440/3–79–023, Washington, D.C. S impervious and pervious area slope
Voorhees, M.L., and Wenzel, H.G., Jr. 1984. Urban design-storm Sdd difference between Sil and Sdi (mm)
sensitivity and reliability. Journal of Hydrology, 68(1–4): 39– Sdi depression storage of the impervious area (mm)
60. doi:10.1016/0022-1694(84)90203-8. Sdp depression storage of the pervious area (mm)
Wanielista, M., Kersten, R., and Eaglin, R. 1997. Hydrology: water Sil initial loss of the pervious area (mm)
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Watt, W.E., Chow, K.C.A., Hogg, W.D., and Lathem, K.W. 1986. passage of a runoff event through a detention pond
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219–236. te time for a pond to empty itself after a runoff event
(h)
List of symbols tp time of dispersion of a detention pond (h)
tr time of dispersion of a channel reach (h)
a storm parameter Tr return period (in years) of the given peak discharge
A area (ha) rate qp
b rainfall interevent time (h) tt time of travel through a channel reach (h)
b average rainfall interevent time (hours) v rainfall event volume (mm)
bs storm parameter v average rainfall event volume (mm)
c storm parameter vr runoff volume per rainfall event (mm)
f0 initial infiltration capacity of the soil (mm/h)
distribution parameter of rainfall event volume
fB(b) probability density function of interevent time (1/mm)
fc ultimate infiltration capacity of the soil (mm/h)  average number of rainfall events per year
fT(t) probability density function of rainfall event  distribution parameter of rainfall event duration
duration (1/h)
fV(v) probability density function of rainfall event distribution parameter of interevent time (1/h)
volume

# 2008 NRC Canada