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\begin{document}
\title{Identifying sources of rainfall derived infiltration and inflow using
impulse response functions}
\author[1]{Namjeong Choi}%
\author[2]{Arthur Schmidt}%
\affil[1]{US Geological Survey Southeast Region}%
\affil[2]{University of Illinois at Urbana-Champaign}%
\vspace{-1em}
\date{\today}
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\selectlanguage{english}
\begin{abstract}
Rainfall derived infiltration and inflow (RDII) are extraneous water in
a sanitary sewer system that are originated from rainfall in a surface
runoff form. Most RDII enters sanitary sewer systems through illegal
connections or mechanical faults especially in aged sewer systems. In
this study, the physical process of three major RDII sources: roof
downspout, sump pump, and leaky lateral, are investigated using
physics-based models. These three sources represent three different flow
paths: direct connection of impervious catchments, mixed flow through
course porous media followed by a direct connection, and percolated flow
through compacted soil, respectively. Due to the differences in medium
and the flow paths, flow responses of these three RDII sources differ in
time and magnitude and they can be distinctly identified from each
other. The typical flow response of each RDII source is represented as
an Impulse Response Function (IRF) that is a flow response to a
pre-specified representative rainfall computed using physics-based
models. The total RDII flow hydrograph is presented as a combination of
these three IRFs and the weighting factors of each IRF is calibrated
using a genetic algorithm (GA) technique in a test sewer catchment. The
results may shed light on identifying the contributions of different
RDII sources in a sewershed and help public water managers to better
understand the local RDII issues which in turn facilitates a more
effective management of a sewer system.%
\end{abstract}%
\sloppy
\section*{Acknowlegement}
{\label{acknowlegement}}
This research was supported by the Tunnel and Reservoir Plan (TARP)
project from the Metropolitan Water Reclamation District of Greater
Chicago (MWRDGC). We thank Illinois State Water Survey (ISWS) and U.S.
Geological Survey (USGS) for valuable data. We also thank Andrea Zimmer
and Sarah Berastegui-Vidalle for their scientific guidance related to
genetic algorithm technique. This paper is based on the model
methodology and result portion of the first author's Ph.D. dissertation
with the citation Choi, N. (2016) Understanding sewer infiltration and
inflow using impulse response functions derived from physics-based
models (Doctoral Dissertation), University of Illinois at
Urbana-Champaign, Retrieved from
\url{https://www.ideals.illinois.edu/bitstream/handle/2142/90540/CHOI-DISSERTATION-2016.pdf?sequence=1}.
\par\null
\section*{Introduction}
{\label{292263}}
Infiltration and inflow (I\&I) is an urban water resources term that
describes the unwanted water existing in sewer systems that are not
originated from the typical sewer sources e.g. domestic and industrial
discharge. Infiltration is water seeping into the sewer pipes preferably
through broken pipe cracks and joints (Figure 1). The origin of
infiltration can be surface water percolated down to the sewer pipes or
groundwater with the water table above the pipe invert. Inflow is
surface water entering the sewer system through direct connections from
runoff catchments or cross connections from storm sewer or combined
sewer. The term rainfall derived infiltration and inflow (RDII) is used
when I\&I origin is surface runoff which is caused by rainfall.
\par\null
I\&I is one of the major problems affecting sewer systems in terms of
flow overloading that causes sewer overflows, basement flooding, street
flooding, increase in pumping costs, water pollution, and decrease in
treatment efficiency in water treatment plants (Backmeyer, 1960; Field
\& Struzeski, 1972; Gottstein, 1976l; Lai, 2008). Based on the
estimation by Petroff (1996), roughly 50\% of the water entering
wastewater treatment plants in the U.S. is from I\&I. Depending on the
age and the condition of the sewer system, the relative volume of I\&I
to the dry weather flow (DWF) could be ranged from 0.4 to 9 (Bishop et
al., 1987; National Small Flows Clearinghouse, 1999; Ertl et al., 2002;
Weiss et al., 2002; Lucas, 2003; Pecher, 2003; Jardin, 2004; Kretschmer
et al., 2008; Bhaskar \& Welty, 2012). For example, I\&I for Baltimore
City was nine times greater than the DWF and it was also larger than the
gauged streamflow from the urban watershed (Bhaskar \& Welty, 2012).
This indicates that I\&I volume can affect the capacity of a sanitary
sewer system significantly.
\par\null
Various I\&I estimation modeling methods have been developed since the
1980s to quantify the amount of I\&I (De B\selectlanguage{ngerman}énédittis \&
Bertrand-Krajewski, 2005). Bishop et al. (1987) developed a simple
synthetic hydrograph method for 300 study basins to estimate I\&I and to
evaluate flow data. Gustafsson (2000) presented a leakage model that
takes account of the two way interaction between pipes and the aquifer
using MOUSE (Lindberg et al., 1989) and MIKE-SHE (DHI Software, 2007ab).
Karpf and Krebs (2004) also used the same leakage approach. The model
was calibrated using a leakage factor that is a function of groundwater
infiltration rate, groundwater level, water level in sewer pipe, and the
pipe surface to which the groundwater is exposed. Schulz et al. (2005)
used the same modeling approach to estimate potential benefits of sewer
pipe rehabilitation with different hypothetical infiltration rates. Qiao
et al. (2007) presented a groundwater infiltration model using a
two-reservoir approach: one reservoir for soil storage in unsaturated
zone and another for groundwater storage in saturated zone. The
elevations of the reservoir openings determine the trigger points that
initiate infiltration into sewer pipes.
\par\null
One of the most common practices of estimating I\&I contribution to
sewer flow is the RTK method that was developed by Camp Dresser and
McKee (CDM) Inc. et al. (1985). According to Lai (2008) ``the RTK method
is probably the most popular synthetic unit hydrograph (SUH) method'' in
the stormwater management field. This method uses unit hydrographs to
estimate the response times associated with the effect of fast,
moderate, and slow I\&I by a linear convolution. A user may calibrate
the model by comparing to an observed I\&I hydrograph. This method is
embedded in EPA SWMM5 (Rossman, 2010) and EPA SSOAP toolbox
(Vallabhaneni et al., 2008). Despite its popularity, the model does not
reflect the underlying physics of each I\&I response and it may leave a
user with a vast number of possible solutions. Also there is a little
guidance for calibrating these models and for I\&I modeling in general
(Allitt, 2002).
~
InfoWorks CS (Innovyze, 2011) is another popular stormwater modeling
tool that has an option for I\&I simulation. InfoWorks simulates I\&I
using two components: rainfall-induced infiltration, and groundwater
infiltration. In the InfoWorks CS infiltration module, the percolation
flow from the surface depression storage is assigned to the soil storage
reservoir after a runoff occurs. When the soil reaches the percolation
threshold, a proportion of this percolation flow goes to the sewer
network. This represents RDII. The remainder of the percolation flow
goes down to the groundwater storage reservoir. When the groundwater
level reaches the sewer system invert level, groundwater infiltration
occurs. The method enables engineers to model groundwater infiltration
into a sewer system but the full physical process is not taken into
account. For example, according to the model assumption, groundwater
infiltration occurs when the groundwater level is higher than the pipe
invert elevation not the water level in the sewer pipe. InfoWorks CS is
widely used because it provides easy to use representation of RDII and
it is useful for operational design. However, the empirical
approximations in this approach to model RDII and infiltration limit the
ability to use this model to provide understanding of process behind
I\&I for a given system.
~
Both SWMM and InfoWorks take simple I\&I estimation approaches that
represent I\&I with unit hydrographs or constant rates. Simplified
modeling methods are labor- and cost-effective and easy to apply but
such approaches do not provide understanding of processes and need much
more calibration data for parameter estimation. Various I\&I prediction
methods, including the above methods, are well documented by Crawford et
al. (1999), Wright et al. (2001), Vallabhaneni et al. (2007), and Lai
(2008).
~
In terms of modeling, often the sources and origins of the RDII are not
identified due to the complexity of the system or lack of data. Though
for a convenience the I\&I sources are often categorized as fast,
medium, and slow sources. The RTK method is a good example of this
practice where three triangular hydrographs are used to represent
short-term, intermediate-term, and long-term responses, respectively
(Rossman, 2010).
~
In physical world, the fast I\&I source indicates a direct connection of
impervious surface runoff catchments e.g. roof downspout, connected to a
sewer pipe. The slow I\&I is the infiltration component of I\&I that
indicates flow through porous media. The medium speed I\&I falls in
between of the fast and slow I\&I in terms of the time to peak. Walski
et al. (2007) defined medium response as ``more delayed and attenuated
response to rainfall'' and this is also referred to as ``rapid
infiltration.'' Hodgson and Schultz (1995) used footer drain as an
example of the medium response. Nogaj and Hollenback (1981) pointed out
that foundation drains and storm sumps are not expected to be highly
sensitive to changes in rainfall intensity, which makes these inflow
sources classified as medium sources.
~
The fast and medium sources are examples of illegal connections to
sanitary sewer systems that lead surface water into sewer pipes. The
standard practice of treating the runoff from impervious areas is to
``drain to light'' or drain to a gravity flow -- a ditch, a storm sewer,
or an overland flow surface, ideally with a permeable soil. In case the
storm sources are connected to sanitary sewer systems the extra water
becomes RDII. Contrast to the fast and medium sources, slow infiltration
occurs when the sewer system fails to keep groundwater out of the
system.
~
The objective of this paper is to identify three representative RDII
sources and understand the hydrologic characteristics of the flow using
the impulse response functions (IRFs). The model is calibrated using a
genetic algorithm (GA) technique in a study area and eventually used to
verify the relative predominance of each RDII source in the test
community.
\section*{Data and Method}
{\label{data-and-method}}
Three RDII sources were selected based on the type of flow paths: roof
downspout, sump pump, and leaky lateral. Each flow path was
characterized using physics-based models in a spatial domain of a
simplified residential lot. The three RDII sources represent: flow
through a direct connection from runoff catchments, flow through coarse
porous media, and flow through compacted soil. These three flow paths
can be simply refereed as fast, medium, and slow paths for convenience
though it is ideal to identify them based on flow patterns and the
medium that is involved in the processes.
The three IRFs are identified for the test sewershed that includes
Hickory Hills, Palos Hills, and Bridgeview, Illinois (IL) where sewer
system configurations and sewer flow monitoring data are available.
Hickory Hills is a city in Cook County, IL with the size of 7.33
km\textsuperscript{2} and the population of 14,049. The areal size of
Palos Hills and Bridgeview, IL is 11.12 km\textsuperscript{2} and 10.75
km\textsuperscript{2}, respectively and the population of the cities is
17,484 and 16,446, respectively (U.S. Census Bureau, 2010).
\par\null
\subsection*{2.1 Physics-based models}
{\label{physics-based-models}}
\subsubsection*{2.1.1 Roof connection model}
{\label{roof-connection-model}}
The roof connection model consists of a sloped roof area, flat gutter,
and vertical downspout. Roof area receives rainfall and conveys the flow
to the rain gutter by gravity. The rain gutter is connected to a
downspout(s) to transport flow to a drainage system. When the downspout
is connected to a sewer system it becomes RDII.
The flow from the roof is calculated using the one-dimensional kinematic
wave model for rainfall-runoff. Two governing equations describe the
rainfall-runoff process when using kinematic wave theory:
one-dimensional continuity equation for unit width of sheet flow, and
Manning's equation as a momentum equation for one-dimensional steady
uniform flow per unit width. The one-dimensional continuity equation is
as follows:
\(\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=I\)(1)
where~\emph{h} = water depth {[}L{]},~\emph{t} = time {[}T{]},~\emph{q}=
flow rate per unit width
{[}L\textsuperscript{2}T\textsuperscript{-1}{]},~\emph{x} = distance in
down slope (measured from upstream end of plane) {[}L{]},~\emph{I} =
rainfall intensity {[}LT\textsuperscript{-1}{]}.
Manning's equation can be used as a momentum equation for
one-dimensional steady uniform flow per unit width as following.
\(q=\frac{1.49}{n}S_{0}^{\frac{1}{2}}h^{\frac{5}{3}}\) (2)
where \emph{n} = Manning's roughness
coefficient,~\emph{S}\textsubscript{0} = bottom slope.
The equation (1) and (2) can be expressed as one equation
\(\frac{\partial q}{\partial x}+\alpha\beta q^{\beta-1}\frac{\partial q}{\partial t}=I\)(3)
where\(\alpha=\left(\frac{1.49}{n}S_{0}^{\frac{1}{2}}\right)^{-\beta}\)and \selectlanguage{greek}\emph{β} \selectlanguage{english}= 3/5, which is the governing
equation of kinematic wave model with \emph{q} as only dependent
variable.
The gutter is treated as a simple bucket and the outlet of downspout is
treated as a weir or orifice depending on the flow condition. The gutter
is modeled using the standard level-pool routing method (Chow et al.,
1988). Level-pool routing is a lumped flow routing method that is
suitable for a case with horizontal water surface in storage unit. The
storage is a function of its water surface elevation. By using the
stage-storage relation of the rain gutter and the stage-discharge
relation of the downspout this equation can be solved. Stage-discharge
relations of the rain gutter-outlet are derived using an orifice and a
weir equation.
\par\null
\subsubsection*{2.1.2 Sump pump connection
model}
{\label{sump-pump-connection-model}}
To derive the IRF from a sump pump, the commercial software MIKE-SHE
(DHI Software, 2007ab) is used to model flow to the sump in the single
residential lot. MIKE-SHE is a spatially distributed hydrologic model
that simulates surface water flow and groundwater flow in
three-dimensional gridded form. The one-dimensional gravity flow
equation in MIKE-SHE is selected as the unsaturated zone equation. The
gravity flow equation is a simplified version of the Richards equation,
which ignores the pressure head term. The vertical driving force is
entirely due to gravity. By selecting the gravity flow module, the
dynamics in the unsaturated zone caused by capillarity are ignored. This
is typically a valid assumption for coarse soils and drainage trench
around a house is usually filled with coarse materials. This is suitable
to calculate the recharge rate of groundwater and faster and more stable
than Richards equation (Graham \& Butts, 2005). The governing equation
for Richards equation is presented as following.
\emph{h} = \emph{z} + \selectlanguage{greek}\emph{ψ} \selectlanguage{english}(4)
Then the gravity equation drops the pressure term
\emph{h} = \emph{z} (5)
where \emph{h} is hydraulic head {[}L{]}, \emph{z} is gravitational head
{[}L{]}, and \selectlanguage{greek}\emph{ψ} \selectlanguage{english}is pressure head {[}L{]}.
The vertical gradient of the hydraulic head is the driving force to
transport water. Thus, for the Richards equation,
\(h=\frac{\partial h}{\partial z}\) (6)
and for the gravity equation,
\(h=\frac{\partial h}{\partial z}=1\) (7)
The volumetric flux that is obtained from Darcy's law for the gravity
equation is
\(q=-K\left(\theta\right)\frac{\partial h}{\partial z}=-K\left(\theta\right)\)(8)
where \emph{K} (\selectlanguage{greek}\emph{θ} \selectlanguage{english}) is unsaturated hydraulic conductivity
{[}L\textsuperscript{3}T\textsuperscript{-1}{]}.
For incompressible soil matrix and soil water with constant density, the
continuity equation is:
\(\frac{\partial\theta}{\partial t}=-\frac{\partial q}{\partial z}-S\left(z\right)\)(9)
where \selectlanguage{greek}\emph{θ} \selectlanguage{english}is volumetric soil moisture {[}L\textsuperscript{2}{]}
and \emph{S} is root extraction sink term
{[}L\textsuperscript{2}T\textsuperscript{-1}{]}. The sum of root
extraction over the entire root zone depth is equal to the total actual
evapotranspiration. Direct soil evaporation is computed only in the
first node below the surface.
Substituting equation (18) onto equation (19), the following expression
is derived.
\(\frac{\partial\theta}{\partial t}=-\frac{\partial K\left(\theta\right)}{\partial z}-S\left(z\right)\)(10)
This can be also expressed using the soil water
capacity,\(C=\frac{\partial\theta}{\partial\psi}\)
\(C\frac{\partial\psi}{\partial t}=\frac{\partial K\left(\theta\right)}{\partial
z}-S\left(z\right)\)(11)
This is called the gravity equation. This equation is used to calculate
the unsaturated zone flow into a sump pump, which is used to derive the
sump pump IRF.
The drainage trench around the house enables surface water to percolate
down to the bottom of the building then feeds into the sump pump. In
MIKE-SHE, sink cells are placed under the building to mimic sump pump
behavior and extract the water from the foundation. Unsaturated zone
flow at the foundation level of the drainage trench area is interpreted
as the total sump pump flow from the house. When the outlet of this sump
pump is connected to a sewer system this becomes I\&I.
The size of the computational domain of the sump pump model is 50 meter
(m) lengthwise and 26 m widthwise. The cell size is 0.33 m x 0.33 m thus
total 150 \selectlanguage{ngerman}× 78 or 11,700 cells in total were created. The vertical cell
height is 0.2 m. The vegetation was assumed as uniform grass with Leaf
Area Index 5 and Root Depth 100 mm. The horizontal width of the drainage
trench is assumed as 0.33 m and the total number of cells in the
horizontal domain is 149 which corresponds to total 50 m length of
trench. The drainage trench goes down to the base level of the house, 4
m below the surface where the sump is located.
Three soil types are employed in the sump pump model: ambient soil,
impermeable soil, and extremely permeable soil. The hydraulic
conductivity of the ambient soil is calculated as the average hydraulic
conductivity of soil in Hickory Hills,
IL,~\emph{K}\textsubscript{ambient} = 2.19 · 10\textsuperscript{-7}
meter per second (m/s; Natural Resources Conservation Service
{[}NRCS{]}, 2019). Hydraulic conductivity of impermeable soil is assumed
as 1·10\textsuperscript{-12} m/s and that of extremely permeable soil is
assumed as 1·10\textsuperscript{0} m/s. The hydraulic conductivity value
of the extremely permeable soil, which represents backfill in the
drainage trench is within the range of the hydraulic conductivity for
gravels based on Chow et al. (1988). The Averjanov model (Vogel et al.,
2000) is used to simulate a hydraulic conductivity curve that shows the
relationship between soil moisture and hydraulic conductivity.
\(K\left(\theta\right)=K_{S}\left(\frac{\theta-\theta_{r}}{\theta_{S}-\theta_{r}}\right)^{m}\)(12)~
where~\emph{Ks} is saturated hydraulic conductivity
{[}LT\textsuperscript{-1}{]},~\selectlanguage{greek}\emph{θ\textsubscript{S}}\selectlanguage{english} is saturated
water content
{[}L\textsuperscript{3}L\textsuperscript{-3}{]},~\selectlanguage{greek}\emph{θ\textsubscript{r}}\selectlanguage{english}
is residual water content
{[}L\textsuperscript{3}L\textsuperscript{-3}{]}, and~\emph{m} is an
empirical constant. Following values are used for the sump pump
connection model: saturated moisture content~\selectlanguage{greek}\emph{θ\textsubscript{S}}\selectlanguage{english} =
0.38, residual moisture content~\selectlanguage{greek}\emph{θ\textsubscript{r}}\selectlanguage{english} = 0.01, and
empirical constant~\emph{m} = 13.
For the MIKE-SHE model setting, the Van Genuchten model (Van Genuchten,
1980) is used to estimate the retention curve, which is a relationship
between moisture content and pressure.
\(\theta\left(\psi\right)=\theta_{r}+\frac{(\theta_{S}-\theta_{r})}{\left[1+{(\alpha\psi)}^{n}\right]^{1-1/n}}\)(13)
where \selectlanguage{greek}\emph{θ} \selectlanguage{english}(\selectlanguage{greek}\emph{ψ} \selectlanguage{english}) is the water retention curve
{[}L\textsuperscript{3}L\textsuperscript{-3}{]}, \selectlanguage{greek}\emph{ψ} \selectlanguage{english}is suction
pressure {[}L{]}, \selectlanguage{greek}\emph{α} \selectlanguage{english}is an empirical constant as the inverse of
the air entry suction (\selectlanguage{greek}\emph{α} \selectlanguage{english}\textgreater{} 0)
{[}L\textsuperscript{-1}{]}, and \emph{n} is a measure of the pore-size
distribution (\emph{n} \textgreater{} 1). Following values are used for
the sump pump connection model: inverse of air entry suction \selectlanguage{greek}\emph{α} \selectlanguage{english}=
0.067, and pore-size distribution \emph{n} = 1.446.
Bulk density of ambient soil and extremely permeable soil is assumed as
1,700 kilograms per cubic meter (kg/m\textsuperscript{3}) and that of
impermeable soil is assumed as 1,600 kg/m\textsuperscript{3}.
Manning's~\emph{n} values for overland flow computation for each surface
type are estimated as 0.013, 0.025, and 0.030 for concrete side walk,
asphalt shingle rooftop, and grassed yard, respectively (Chow, 1959).
Evapotranspiration rate is set as 2.76 millimeters per day (mm/d) which
is a suggested value in the Chicago area according to Grimmond and Oke
(1999).
\subsubsection*{2.1.3 Leaky sewer lateral
model}
{\label{leaky-sewer-lateral-model}}
Similar to the sump pump model, the leaky sewer lateral model is
developed using MIKE-SHE. The equivalent medium approach from Carlier et
al. (2007) is adopted to simulate the actual flow into the sewer. This
simplified approach provided reasonable results in the flow simulation
in agricultural drainage trenches (Carlier et al., 2007). According to
the equivalent medium approach, the leaky sewer pipe and surrounding
drainage trench area can be modeled as a single soil layer with an
equivalent uniform hydraulic conductivity. Hydraulic conductivity of
this layer is assumed as 1\selectlanguage{ngerman}·10\textsuperscript{0} m/s. Infiltrated flow
is estimated as the sum, over the length of the pipe line, of
unsaturated flow into the top of the equivalent porous medium
representing the leaky lateral. The same values of soil type, soil
property, Manning's \emph{n} values, and evapotranspiration rate were
used from the sump pump model.
\par\null
\subsection*{2.2 Input data}
{\label{input-data}}
Rainfall data were obtained from the Illinois State Water Survey (ISWS)
by averaging rainfall data from four nearby ISWS rain gages: G11, G12,
G16, and G17 (Illinois State Water Survey, 2019). The sewer flow data
were collected by U.S. Geological Survey (USGS) at 17 monitoring
locations in spring and summer of 2009. Based on the data quality and
the length, the site located on 104th Street and east of Terry Drive in
a manhole was selected for this study. This location receives sanitary
sewer flow from Hickory Hills, Palos Hills, and Bridgeview, IL.
Both rainfall records and the sewer monitoring records are presented in
Figure 2 in the period of April 17, 2009--August 3, 2009. The base flow
shows daily fluctuation of dry weather flow except when storm event
occurs high flow peaks are observed which tend to sync in time with the
arrivals of rainfall peaks.
In order to only focus on the RDII portion of the sewer record, dry
weather flow (DWF) needs to be estimated and separated from the sewer
record. The average DWF was estimated using the DWF estimation component
in Special Contributing Area Loading Program (SCALP), which is developed
by Hydrocomp, Inc. (Hydrocomp 1979). SCALP is a flow routing model
mainly developed for use in the Chicago area. DWF is determined on a per
capita basis and distributed in time by coefficients: average DWF
loading, monthly pattern, daily pattern, and hourly pattern using
following equation (Espey et al., 2009; Miller \& Schmidt, 2010).
DWF = average DWF loading x monthly pattern x daily pattern x hourly
pattern (14)
These DWF coefficients are estimated using data from a 14-day dry period
from July 17, 2009 to July 31, 2009. The 14 days of DWF is averaged and
the set of best DWF coefficients is derived by adjusting each value
until the best fit to the average DWF was achieved. Nash-Sutcliffe model
efficiency coefficient is used to find the best fit (Nash \& Sutcliffe,
1970).
The monthly pattern is the pattern describing the variability among
months within a year. The monthly pattern values are all set to one
throughout the year due to insufficient data to define them. The daily
pattern describes the variability among days within a week and the
hourly pattern describes the variability among the hours of the day. The
average DWF loading is calculated as 0.12 m\textsuperscript{3}/s (4.40
ft\textsuperscript{3}/s). The daily pattern shows that DWF is greater
during weekends than weekdays. The hourly pattern shows two peaks during
a day: in mornings and evenings, and minimum DWF at 4 am.
\par\null
\subsection*{2.3. Impulse Response Function
derivation}
{\label{impulse-response-function-derivation}}
To derive the three IRFs from the three models, a representative
rainfall was introduced as a model input. Based on the rainfall record
in Hickory Hills, IL, a total of 702 mm of rainfall was recorded in the
period of January 1--July 31, 2009. Seventeen distinct storm events were
identified manually during this period; hence the average rainfall
volume for a single event was assumed as 41 mm (as 702 mm divided by
17). The maximum rainfall intensity during the same period is 14 mm/hr.
Three hours of 14 mm/hr of rainfall produces a total of 42 mm of
rainfall volume. Therefore, 3-hour 14-mm/hr uniform precipitation is
selected as a representative rainfall. The representative uniform
rainfall was used as an input of the three physics-based models to
derive the IRF of each RDII process described in the models.
The representative rainfall can be used directly for the roof connection
model however, it cannot be used directly for the gravity flow models
that are used to derive the sump pump IRF and leaky lateral IRF.
Infiltration and runoff processes are affected by the ground conditions
e.g. land cover, land use, soil type, vegetation, seasonality,
antecedent moisture condition, etc. To eliminate the variability of
ground conditions, the representative rainfall was added to the actual
rainfall hyetograph at random times and the resulted RDII hydrograph was
subtracted by the RDII hydrograph resulted from the unaltered rainfall
record. The representative rainfall was added to the actual rainfall
hyetograph at 10 randomly selected times during the period of June
1--January 31, 2009 and the IRF was calculated by averaging the
individual IRF which is the difference between the hydrographs resulted
from the altered and unaltered rainfall hyetographs.
In case of the roof connection model, antecedent moisture condition has
a minimal effect on the flow response of the roof runoff thus rainfall
manipulation was not necessary.
Three IRFs derived from the roof downspout, sump pump, and leaky lateral
models using the representative rainfall are presented in Figure 3. The
flow discharge units are normalized using the contributing areas of each
model so that effective flowrates can be compared among the models. The
peak values of each IRF are 0.0942, 0.0427, and 0.00902
m\textsuperscript{3}/day/m\textsuperscript{2} for the roof downspout,
sump pump, and leaky lateral models, respectively. By integrating the
flow over time, resulting RDII volume per unit contributing area are
0.0118, 0.0319, and 0.0842 m
(m\textsuperscript{3}/m\textsuperscript{2}). The result indicates that
the roof IRF sports the shortest response time although the total RDII
volume per unit area is the smallest. At the same time the leaky lateral
IRF shows the longest response time with the largest volume per unit
area. Total volume of each IRF is 2.89, 1.54, and 1.63
m\textsuperscript{3}, however the values are not good indicators of
showing the impact of each RDII source as the total volume is dependent
on the size and the condition of each model domain. The order of total
response time for each IRF was hours, days, and weeks for the roof
downspout, sump pump, and leaky lateral, respectively.
To understand the long term behavior of the three IRFs, each IRF is
weighted based on the actual rainfall intensity record in the period of
April 17--July 16, 2009. The total rainfall depth in this period was 372
mm and the peak precipitation rate was 13 mm/hr. Based on the assumption
that resulting RDII hydrographs from each source are proportional to the
rainfall, three independent hydrographs were created for the same time
period. Then each hydrograph was divided by the effective contributing
area to compare the net RDII volume. Figure 4 indicates the flow
duration curves of the three RDII responses for the time period. The
roof connection response presented in the black solid line shows a steep
curve, which indicates a greater amount of RDII for a short period of
time. This displays a strong evidence that the flow is rain-caused. The
leaky lateral response, which is presented in a grey solid line, shows a
flatter curve. This indicates that the leaky lateral IRF displays a
longer flow duration than the roof IRF due to the delayed percolation
through porous media. The sump pump IRF in the black dashed line falls
between the roof IRF and the leaky lateral IRF. The sump pump flow path
also involves a flow through a porous media but it is ``faster'' than
the leaky lateral flow path as the travel distance of surface water in
the sump pump model is shorter than that of the leaky lateral model and
the medium has a larger hydraulic conductivity. The shapes of the three
IRFs are easily distinguishable from one another which in turn makes
them suitable as building blocks of an RDII hydrograph.
\section*{Results}
{\label{results}}
A genetic algorithm (GA) is used to optimize the three scaling factors
for the RDII impulse response functions (IRFs). The same method was used
to calibrate the total sewer flow simulated by the SWMM RTK method for a
comparison. The efficiency of both RDII estimation methods is compared
using the modified Nash-Sutcliffe coefficient.
\(E_{j}=1-\frac{\sum_{t=1}^{T}{W_{t,j}{(Q_{0}^{t}-Q_{m}^{t})}^{2}}}{\sum_{t=1}^{T}{W_{t,j}{(Q_{0}^{t}-{Q_{0}})}^{2}}}\) (15)
where \(Q_{0}^{t}\) is observed discharge at time \emph{t}
{[}T{]},\(Q_{m}^{t}\) is modeled discharge at time
\emph{t}{[}L\textsuperscript{3}T\textsuperscript{-1}{]}, and
\({Q_{0}}\) is the average of observed discharge
{[}L\textsuperscript{3}T\textsuperscript{-1}{]}. The coefficient ranges
from -\selectlanguage{english}[?] to 1 and \emph{E} = 1 corresponds to a perfect match between the
observed discharge and the modeled discharge.\emph{j} is a weighting
factor (\emph{j} = 1, 2, and 3).\emph{W\textsubscript{j}} is a weighting
factor with the index \emph{j~}= 1 is applied to low flows, \emph{j} = 2
is applied to medium flows, and \emph{j} = 3 is applied to peak flow
values. In the conventional Nash-Sutcliffe method, all three weighting
factors are identical (\emph{W}\textsubscript{1} =
\emph{W}\textsubscript{2} =~\emph{W}\textsubscript{3}). By using the
modified Nash-Sutcliffe method, smaller runoff values are
under-emphasized and larger peaks are over-emphasized.
The calibration period was from May 9, 2009 to June 7, 2009 and the
validation period was from June 9, 2009 to July 8, 2009. The IRF method
has three parameters to calibrate: roof connection scaling factor (R),
sump pump connection scaling factor (S), and leaky lateral scaling
factor (L). The RTK method has nine parameters to calibrate: R1, R2, R3,
T1, T2, T3, K1, K2, and K3. R is a ratio of I\&I discharge volume to the
rainfall volume: R1 is for a fast inflow element, while R2 and R3
represent slower infiltration elements. T is the time to peak in each
hydrograph (typically expressed in hours), and K is the ratio of time of
recession to the time to peak.
For the GA optimization conditions, size of the population was set as
100 and the maximum number of generations was set as 300 for both models
approaches. Value 0.95 is selected as the probability of crossover for
both IRF and RTK calibration. The probability of mutation is set as
0.06.
The calibrated parameter solutions for the IRF and RTK methods are
presented in Table 1. The Nash-Sutcliffe model efficiency coefficient of
the IRF solution is 0.534 in the calibration period and 0.560 in the
validation period. The modified Nash-Sutcliffe coefficients for the IRF
solution were 0.892 for the calibration period and 0.866 for the
validation period when the Nash-Sutcliffe weighting factors were set
as~\emph{W}\textsubscript{1} = 3 for \emph{Q} \textgreater{} 90-th
percentile, \emph{W}\textsubscript{2} = 2 for 80- \textless{}
\emph{Q~}\textless{} 90-th percentile, \emph{W}\textsubscript{3} = 1 for
\emph{Q~}\textless{} 80-th percentile. Assigning larger weighting
factors for high flows improved the model fit significantly. The
Nash-Sutcliffe coefficient of the best RTK solution was 0.848 in the
calibration period and 0.795 in the validation period.
Though the model fitness was improved by using the modified
Nash-Sutcliffe method, model efficiency based on the RTK method was
higher since the RTK method has three times more parameters to adjust,
nine instead of three parameters. However, in the validation period,
model efficiency was increased for the IRF solution while it was
decreased for the RTK solution. This may imply the pitfall of the RTK
method that the method is not consistent and may not be very robust.
The optimal solution of the IRF scaling factors using the GA is: R =
3,359 for roof, S = 22,653 for sump pump, and L = 19,985 for lateral.
These values can be interpreted as RDII volume contribution of each RDII
source (Table 1). Contributing flow volume of each RDII source is
derived by multiplying the per-unit-area flow volume of IRFs and the IRF
weighting coefficients (Table 2). Then the contributing RDII volume from
the roof, sump pump, and lateral become 9,710 m\textsuperscript{3},
22,653 m\textsuperscript{3}, and 32,543 m\textsuperscript{3},
respectively, and they are 15\%, 35\%, and 50\% of total estimated RDII
flow volume. This simple calculation shows that IRF result can be
interpreted as RDII volume contribution of different RDII sources, which
shows the most problematic RDII contributor in the system volume-wise.
These values need to be interpreted with a caution as the IRF model
application in this study is only one realization of a real system and
each sewershed is unique in terms of factors that contribute to RDII.
However, this result still can provide insights to RDII behavior of the
system by providing physical meaning of the solutions.
The IRF approach tends to be more robust because three parameters adjust
three IRF that represent processes based on physics. Each IRF shape is
defined independently using physics-based models and the weighting
parameters reflect the contribution from each of the three IRF. The IRF
solutions are a unique solution no matter how randomly the initial
population was selected. In contrast, RTK method gives different
solutions every time the model runs. As an example, 30 sets of three RTK
hydrograph solutions display widely variable results as presented in
Figure 5. Within the user specified range for each hydrograph, the
solution can be vastly different for each run. The Nash-Sutcliffe
coefficient of the best case was 0.848 and that of the worst case was
0.681. Depending on the user-specified ranges of each parameter, the
results can vastly differ and the performance is not guaranteed.
RTK method has many local optimal solutions, which indicates that nine
coefficients are not independent. Thus the starting points or
constraints of the parameters cause other parameters to adjust to obtain
a local optimum that behaves similarly good for calibration data. Box
plots of the nine RTK parameters from the 30 model runs are presented in
Figure 6. Greater variability is observed in RTK parameters for the
second and third triangular hydrographs, especially the third one. This
is because the model tries to adjust these parameters according to the
given constraints of earlier parameters. Technically, different RTK
local solutions can result in the same model fitness. Change in one
hydrograph affects other two hydrographs to simply achieve the best
fitness. This indicates the problem of the RTK method that physical
processes are not reflected in the modeling.
Figure 7 shows the prediction of the monitored flow hydrograph using the
IRF solution and the best case of the RTK solutions during the
calibration period (Figure 7(a)) and the validation period (Figure
7(b)). In June 24, both methods predict flow peaks but the peak is not
observed in the monitored flow record. The flow peak might have happened
in such a short time period and the flow monitor might have failed to
capture the peak. Overall, RTK method tends to follow the monitored
hydrograph well especially at the falling limbs of peaks while IRF tends
to underestimate the flow at the falling limbs.
The volume and the peak flow values for the estimated DWF, observed
sewer flow, IRF model result, and RTK model result are summarized in
Table 3. Flowrate 0.3 m\textsuperscript{3}/s is selected to define the
beginning and the end of each storm. The observed sewer flow, IRF
results, and RTK results are compared to the estimated DWF using the
following equation.
\(\text{Compare\ to\ DWF}=\frac{\text{Observed\ sewer}}{\text{Estimated\ DWF}}\times 100\)(16)
The observed sewer flow is three to four times of DWF in volume and
three to six times in peaks during the storms. Considering the
monitoring location is sanitary only, a great deal of RDII exists in the
area.
The IRF result and RTK result are compared to the observed sewer flow
using the following equation.
\(\text{Compare to observed RDII}=\frac{\text{Predicted RDII}- \text{Observed RDII}}{\text{Observed RDII}}\times100\)(17)
Both models underestimated the flow volume; IRF method underestimates
flow volume by 9\% to 28\% and RTK method underestimates flow volume by
4\% to 26\% compare to monitoring volume. In terms of flow peaks, IRF
method overestimated peak flowrate for May 13, May 27, and June 11
storms by 19\%, 25\%, and 9\%, respectively. At the same time IRF method
underestimated peak flowrate for May 15, and June 16 by 15\% and 8\%,
respectively. RTK method overestimated peak flowrate consistently from
1\% to 16\%.
Residual plots of the IRF and the best RTK solutions for the calibration
period and the validation period are presented in Figure 8. Residuals
are the difference between the observed value of the dependent variable
and the predicted value. Each data point has one residual and is defined
with the following equation.
Residual = Observed value -- Predicted value (18)
Residuals are plotted against the observed value in the \emph{x} axis.
There are clusters of points at low flowrate, which represent tails in
the hydrographs. In Figure 8(a), IRF underestimates the peaks as most of
the residuals are in the positive side. These points are from the storms
in May 15, 2009 and May 27, 2009. This trend is also observed in the
validation period and the outliers are from the storms in June 11, 2009
and June 16, 2009 (Figure 8(b)). In validation period, RTK also
underestimated peaks as most of high flow points are in the positive
side. This means the best RTK solution for the calibration period loses
the efficiency in the validation period. This explains the decrease of
Nash-Sutcliffe coefficient of RTK method in the validation period as
presented in Table 1 and supports that RTK method is more of a curve
fitting method with limited physical meaning.
\section*{Conclusion}
{\label{conclusion}}
In this study, the physical process of three major rainfall derived
infiltration and inflow (RDII) sources: roof downspout, sump pump, and
leaky lateral, were investigated using physics-based models. These three
sources represent three different flow patterns through: a direct
connection of runoff catchments, course porous media, and compacted
soil, respectively. The typical flow response of each RDII source was
expressed as impulse response functions (IRFs) that indicate the flow
responses to a representative rainfall. Three IRFs were superposed to
produce the total RDII flows and the weighting factors of each IRF were
calculated using the actual sewer flow data in a few neighborhoods in
Illinois based on a genetic algorithm (GA) technique. The results can
shed light on identifying the contributions of different RDII sources in
a sewershed that enables decision makers to better understand the local
RDII issues which in turn facilitates a more effective management of the
sewer system.
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\section*{Tables}
{\label{tables}}
Table 1. IRF and RTK results with Nash-Sutcliffe coefficients for the
calibration period (May 9, 2009 to June 7, 2009) and validation period
(June 9, 2009 to July 8, 2009)\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Method\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Estimated parameters\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Nash-Sutcliffe coefficient\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Nash-Sutcliffe coefficient\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Calibration\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
Validation\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
IRF\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
R+ = 3,359, S = 14,663, L = 19,985\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
0.534\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
0.560\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
IRF with weighted Nash-Sutcliffe++\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
R = 3,359, S = 14,663, L = 19,985\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
0.899\strut
\end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright\strut
0.866\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
RTK\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
R1 = 0.02, T1 = 0.338, K1 = 2 R2 = 0.0478, T2 = 1, K2 = 10 R3 = 0.123,
T3 = 8.5493, K3 = 14.6686\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
0.848\strut
\end{minipage} & \begin{minipage}[t]{0.24\columnwidth}\raggedright\strut
0.795\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}
(+R = Roof connection scaling factor, S = Sump pump connection scaling
factor, L = Leakey sewer lateral scaling factor,
++Weighting factors used for the weighted Nah-Sutcliffe model
are~\emph{W}\textsubscript{1} = 3 for \emph{Q} \textgreater{} 90-th
percentile,~\emph{W}\textsubscript{2} = 2 for 80- \textless{} Q
\textless{} 90-th percentile, and \emph{W}\textsubscript{3} = 1
for~\emph{Q} \textless{} 80-th percentile.)
Table 2. Contributing flow volume of three RDII sources using the IRF
volume and the weighting coefficients\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
& Roof & Sump & Lateral\tabularnewline
Effective contributing area (m\textsuperscript{2}) & 246 & 48 &
19\tabularnewline
Flow volume under IRF (m\textsuperscript{3}) & 2.89 & 1.54 &
1.63\tabularnewline
Flow volume per unit area (m) & 0.012 & 0.032 & 0.084\tabularnewline
IRF weighting coefficients & 3,359 & 14,663 & 19,985\tabularnewline
Total contributing volume (m\textsuperscript{3}) & 9,701 & 22,653 &
32,543\tabularnewline
Contributing volume / total RDII volume (\%) & 15 & 35 &
50\tabularnewline
\bottomrule
\end{longtable}
Table 3. Volume and peak of the DWF, estimated and observed RDII using
IRF and RTK for five storm events\selectlanguage{english}
\begin{longtable}[]{@{}lllllllll@{}}
\toprule
\begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Storm event in 2009\strut
\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Estimated DWF by flow separation\strut
\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Estimated DWF by flow separation\strut
\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Observed RDII\strut
\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Observed RDII\strut
\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Predicted RDII using IRF\strut
\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Predicted RDII using IRF\strut
\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Predicted RDII using RTK\strut
\end{minipage} & \begin{minipage}[t]{0.11\columnwidth}\raggedright\strut
Predicted RDII using RTK\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Volume+\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Peak++\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Volume+\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Peak++\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Volume+\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Peak++\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Volume+\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Peak++\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Compare to DWF (by multiplication; observed/DWF)\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Compare to DWF (by multiplication; observed/DWF)\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Compare to observed RDII (in percent; {[}observed --
predicted{]}/observed x 100)\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Compare to observed RDII (in percent; {[}observed --
predicted{]}/observed x 100)\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Compare to observed RDII (in percent; {[}observed --
predicted{]}/observed x 100)\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
Compare to observed RDII (in percent; {[}observed --
predicted{]}/observed x 100)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
May 13\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
12.23\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.15\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
40.49\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.53\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
34.74\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.63\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
35.49\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.58\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
3.31\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
3.53\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-14\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
19\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-12\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
9\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
May 15\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
15.16\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.17\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
51.03\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.74\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
40.8\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.63\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
48.8\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.84\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
3.37\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
4.35\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-20\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-15\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-4\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
14\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
May 27\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
10.77\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.15\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
43.36\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.83\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
39.47\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
1.04\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
40.77\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.96\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
4.03\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
5.53\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-9\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
25\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-6\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
16\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
June 11\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
13.75\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.15\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
60.35\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.88\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
43.28\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.96\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
44.94\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.89\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
4.39\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
5.87\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-28\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
9\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-26\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
1\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
June 16\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
19.08\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.15\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
77.63\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.85\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
56.56\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.78\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
59.69\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
0.93\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
4.07\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
5.67\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-27\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-8\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
-23\strut
\end{minipage} & \begin{minipage}[t]{0.08\columnwidth}\raggedright\strut
9\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}
\section*{Figure legends}
{\label{figure-legends}}
Figure 1. Root intrusion through cracks and joints of sewer pipes as
examples of sewer infiltration passage (Urbana Champaign Sanitary
District, 2012)
Figure 2. Rainfall and sewer flow data (a) rainfall record from ISWS (b)
sewer flow data from USGS sewage monitoring site in the period of April
17--August 3, 2009
Figure 3. Impulse response functions from the roof connection, sump
pump, and leaky lateral models with the representative rainfall
expressed as flow discharge per unit contribution area
Figure 4. Long term RDII responses of roof connection, sump pump, and
leaky lateral using exceedance probability of three RDII responses per
unit area in log scale in the period of April 17--July 16, 2009
Figure 5. Three RTK triangular hydrographs from 30 different model runs
Figure 6. Box plots of RTK solutions
Figure 7. Calibrated IRF and the best RTK results in the (a) calibration
period (May 9, 2009 to June 7, 2009) and the (b) validation period (June
9, 2009 to July 8, 2009)
Figure 8. Residual plots of IRF and RTK methods for (a) calibration
period and (b) validation period
\section*{Data Availability Statement}
{\label{data-availability-statement}}
The data that support the findings of this study are available from the
corresponding author, Namjeong Choi, upon reasonable request.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig1-1/fig1-1}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig1-2/fig1-2}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig2(a)/fig2(a)}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig2(b)/fig2(b)}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig3/fig3}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig4/fig4}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig5-1/fig5-1}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig5-2/fig5-2}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig5-3/fig5-3}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig6/fig6}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig7(a)/fig7(a)}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig7(b)/fig7(b)}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig8(a)/fig8(a)}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig8(b)/fig8(b)}
\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
\end{document}