The performance of urban drainage systems is typically examined using
hydrological and hydrodynamic models where rainfall input is uniformly
distributed, i.e., derived from a single or very few rain gauges. When models
are fed with a single uniformly distributed rainfall realization, the
response of the urban drainage system to the rainfall variability remains
unexplored. The goal of this study was to understand how climate variability
and spatial rainfall variability, jointly or individually considered, affect
the response of a calibrated hydrodynamic urban drainage model. A stochastic
spatially distributed rainfall generator (STREAP –
Urban drainage systems are designed to ensure safe wastewater disposal
(focus: dry weather) and adequate storm water handling (focus: wet weather).
Whereas the variability of dry weather flows is rather low and very
predictable, rain-induced flow dynamics scale over several orders of
magnitude and require stochastic analysis due to the high rainfall
variability. The latter is often addressed by summarizing the rainfall input
in the form of intensity–duration–frequency (IDF) curves
A common practice in evaluating the performance of urban drainage systems for
different forcing situations is by using a model with a hydrological
component to simulate the runoff at the urban catchment scale, and a
hydrodynamic component to simulate the flow in the drainage system itself.
Rainfall is defined as the most important input required by these models
Rainfall input may be given by observations of rain gauges and weather radar;
however, this constrains the analysis to storms observed in a limited time
period. On the other hand, stochastic modeling of space–time rainfall fields
allows a full exploration of the potential impacts of space–time variability
in rainfall on the urban drainage system. The
The use of stochastic rainfall generators that account for spatial rainfall
variability and/or climate variability in urban hydrology applications is
still rather new.
Location map of the case study catchment (bounded with black line).
The black mesh represents the 1.5 km
The main objective of this paper is to investigate the relative contribution of the spatial versus climatic rainfall variability for flow peaks at different locations in the drainage network and for different return periods. We apply a new and advanced stochastic rainfall generator to simulate rainfall inputs for a small urban catchment in Lucerne (Switzerland) and to simulate flow dynamics in the sewer system. This work demonstrates the potential of using stochastic rainfall generators for urban applications and the benefits gained compared to other methods, such as bootstrapping rainfall events from a long rainfall series.
The case study is an urban catchment located near the city center of Lucerne,
Switzerland (Fig.
The flow rate at the outlet of the combined sewer system (location B) was monitored for a period of 12 months from July 2014 to June 2015. In order to reduce measurement uncertainty, the water level and flow velocity were recorded using two different combi-sensors with different monitoring techniques (in situ Doppler-ultrasound technique, ex situ ultrasound-radar technique) in parallel. The recording interval was set to 1 and 15 min for the Doppler-ultrasound sensor and the ultrasound-radar sensor, respectively.
A stochastic space–time rainfall generator was used to simulate multiple
realizations of 2-D rain fields for a 30-year period. The rainfall was
generated for four distinct cases which were defined in order to explicitly
account for the climate variability, spatial rainfall variability, and total
variability of the flow. The generated rainfall was used as an input into a
hydrodynamic model. For each of the four cases, IDF curves were computed for
the annual maxima of the mean areal rainfall and
flow–duration–frequency (FDF) curves were computed for annual flow maxima
simulated at three different network elements, representing different aspects
in the assessing of the performance of urban drainage systems. The total flow
variability was partitioned into the part originating from climate
variability and the additional contribution due to spatial rainfall
variability. The methods are illustrated in Fig.
A schematic illustration of the methods used in this study: (i) the STREAP model was used to simulate multiple realizations of 2-D rain fields based on radar and gauged data (Sect. 3.2); (ii) rainfall was generated for four distinct cases which were defined in order to explicitly account for the climate variability, spatial rainfall variability, and total variability of the flow (3.3); (iii) the EPA SWMM model was used to calculate the flow over the catchment (3.4); (iv) IDF and FDF curves were computed for the annual maxima of the mean areal rainfall and flow, respectively, at three different locations (3.5); and (v) the total flow variability was partitioned (3.6).
Rainfall data originate from two sources: a rain gauge located about 2 km
west of the case study catchment (Fig.
The tipping bucket rain gauge records rainfall in 10 min intervals with a precision of 0.1 mm. A 34-year record was used in this study, covering the period 1981–2014. High-resolution 10 min rainfall intensities were benchmarked with hourly rainfall data (validated record provided by MeteoSwiss) and obvious deviations were corrected. The length of the observed record allows an adequate estimation of the statistical rainfall characteristics, especially regarding high rainfall intensities of short durations and with return periods of up to 30 years. Climatological stationarity has been assumed for the observed record.
High-resolution radar rainfall data (2 km and 5 min) for an 8-year
period (2003–2010) were derived from a third-generation weather radar system
of MeteoSwiss
Rainfall fields at a high spatial and temporal resolution were generated
using the STREAP model (
For this analysis, rainfall was generated with a spatial resolution of
100 m
The calibration process of STREAP using weather radar products was discussed
in detail by
Changes were also applied to the second module. Originally, mean areal
intensity and fraction of wet area during a storm were simulated as
bi-variate auto-correlated stochastic processes that also depend on storm
duration. Here, due to the small extent of the spatial domain, the wet area
ratio was assumed to be equal to zero during intra-storm periods and assumed
to be equal to one during storms; i.e., during storms all grid cells over the
catchment experience rainfall. The mean areal intensity is simulated using an
AR(1) model which simulates a normalized quantile time series that is later
inverted using a mixed-exponential function
No modifications were needed for the last module. However, some model
parameters (e.g., rainfall coefficient of variation) could not be directly
estimated from the weather radar data as the spatial resolution of the radar
product (2 km) is too coarse compared to the model resolution (100 m).
Therefore the required parameters were first estimated using the weather
radar data for a coarse spatial resolution and then downscaled to higher
resolution using power-law functions (Fig. S1 in the Supplement) as described
in
The evaluation of STREAP, its ability to reproduce the rainfall intensity over the domain (with emphasis on the high rainfall intensity), and its performance with regard to the natural climate variability of the annual maxima in rainfall intensity, are discussed below.
The ability of STREAP to reproduce the rainfall intensity over the domain is
shown using the inverse cumulative distribution function for the 10 min mean
areal rainfall intensity (Fig.
An inverse cumulative distribution function of the 10 min mean
areal rain intensity over the catchment (the 0.1–1 quantile range is
presented in
An example of STREAP's ability to spatially distribute the annual
maxima rainfall intensity over the catchment. The annual maxima recorded by
the Lucerne gauge for the year 1981 is 80.4 mm h
Four rainfall cases were defined in order to account for climate variability
and spatial rainfall variability and to allow the investigation of their
effect on the urban drainage.
Case 1: consists of one time series of rainfall derived from the Lucerne rain gauge
(observed data, 34 years long). For this case, rainfall was not spatially
distributed using STREAP but was uniformly distributed; i.e., the same
rainfall intensity was assigned to all sub-catchments for a given time step.
In this case the rain gauge time series also represents the mean areal
rainfall over the catchment. This is a common and critical assumption in
hydrological studies, where point rainfall is used to represent areal
rainfall Case 2: consists of 30 realizations of the same time series (rain gauge
observations) that was used in case 1, but spatially distributed using
STREAP. Cases 1 and 2 differ in the spatial configuration of the rainfall
(uniformly distributed vs. spatially distributed) which will later allow one
to explicitly analyze how the spatial rainfall variability affects the flow. Case 3: consists of 30 realizations of 30 years generated by STREAP. For this
case, STREAP was set to generate only the mean areal rainfall and to
uniformly distribute it over the sub-catchments (similar to case 1).
Comparing the urban drainage response to the rainfall given from cases 1 and 3
will allow us to account for the climate variability component directly, as
case 3 represents 30 alternative and equiprobable trajectories of the
rainfall series given in case 1. Case 4: consists of 900 realizations accounting for both the spatial rainfall
variability and the climate variability. Each of the 30 realizations
generated for case 3 were re-generated 30 times using STREAP. The forcing has
a different spatial distribution of the rainfall over the sub-catchments for
each re-generation. This allows computing of urban drainage dynamics
subjected to the total variability.
Flow simulations were conducted using the US EPA's Storm Water Management
Model (EPA SWMM), a dynamic 1-D model coupling rainfall–runoff processes
with hydrodynamic channel flow
EPA SWMM is composed of two modules: the surface runoff (hydrological) and the in-sewer flow (hydraulic) model. The hydrological model calculates the direct runoff under consideration of initial precipitation losses (i.e., evaporation and wetting losses) and soil infiltration (here using the Horton method). The resulting surface runoff is then used as input for the hydraulic model to simulate the pipe flow using the 1-D Saint-Venant equations. The diffusive wave approximation and a routing step of 10 s were applied for all simulations. Surface flooding is accounted for by allowing excess water to leave a manhole in case sewer capacity is exceeded. Due to the lack of detailed land use and surface topography data at meter scale it was found inadequate to further define a manhole-specific “ponding area” allowing the water to spread at the surface around a manhole. Hence excess water leaving the manhole is routed into a virtual sink and does not re-enter the system even though sewer capacity is available again.
The sewer model application is based on infrastructure data from the
municipality's cadaster database. The model has carefully been calibrated and
validated (split-sample approach) using the above-mentioned 1-year flow data
record. Flow dynamics can be adequately reproduced throughout the year
despite the rather coarse 10 min rainfall input data resolution. More
details on the catchment, particularly on the urban land use characteristics,
the monitoring setup, and the model calibration procedure, are given in
The runoff-generating surfaces are distributed over
the entire catchment. This is represented by 158 individual sub-catchment
entities with an area ranging between 0.02 and 0.84 ha. The rainfall fields
generated by STREAP were intersected with the sub-catchment areas and
rainfall intensity was assigned for each sub-catchment based on the weighted
sum of the intersect area
The Generalized Extreme Value (GEV) distribution
IDF curves were calculated for two datasets: observed data derived from the Lucerne rain gauge and simulated data that were generated using STREAP. For the observed dataset, one IDF curve was computed for the 34 years of records. For the simulated dataset, 30 IDF curves were calculated for the 30 stochastic realizations (of 30 years each). The curves were calculated for a 10 min duration.
FDF curves were calculated for the simulated flow at three locations which
were chosen according to their function within the drainage network (see
Fig.
Note that no condition was imposed on the time concurrency of annual maxima of mean areal rainfall intensity and conduit flow; i.e., annual peak flow can precede, overlap, or follow the annual maxima of mean areal rainfall intensity.
The partition method used in this study follows the guidelines suggested by
The climate variability, CLM, is defined as the 5–95 quantile range of
the flow that is calculated using the 30 spatial uniform climate realizations
simulated for case 3 (i.e., the outcome is one flow range for a given return
period). For each of the 30 climate realizations, the spatial flow
variability, SPT, is defined as the 5–95 quantile difference of the flow
calculated using the spatially variable rainfall simulated for case 4. The
outcome is 30 different flow ranges, SPT
An example of the partition method (illustrative) for the 2-year
return period (zoomed panel). Three climate trajectories are plotted (red
lines) for which the 5–95 quantile range is calculated
(
In the following, we present computed IDF (rain) and FDF (flow) curves and
discuss the contributions of individual rainfall variabilities to the
modeled sewage flow variability at three
different locations: A – inner network node (Figs.
Rainfall and flow results for cases 1 and 2. In
Same as Fig.
Same as Fig.
Same as Fig.
The ratio between the climate variability and the total flow variability for a given return period and for different locations within the urban drainage system is represented in dark blue. The remaining contribution is due to the addition of spatial rainfall variability (light blue).
The effect of spatial rainfall variability on the flow can be directly
estimated by examining the flow variability from case 2
(Figs.
The effect of the climate variability over the catchment is calculated from
the 30 rainfall realizations stochastically simulated for cases 3 and 4 (left
panels in Figs.
The individual effect of the climate variability on the flow is estimated
from case 3 (Figs.
The total flow variability is calculated using the data of case 4
(Figs.
The rainfall generator was used to simulate rainfall for the weather radar
subpixel scale, i.e., at a finer spatial resolution than can be estimated
using the MeteoSwiss radar. The rainfall data required for a complete
validation of the rainfall generator for this resolution can be obtained from
a dense rain gauge network
No automatic calibration process exists for STREAP. The model requires not only high-resolution rainfall data, but also an expert user for the calibration process, as modifications to the calibration procedure (e.g., scaling at higher spatial resolution) are needed in order to tailor STREAP to a given application.
The three locations analyzed in this study were deliberately chosen according
to their functional hierarchy within the combined drainage system
(i.e., inner network node, carry-on flow, and overflow). By doing so, we can
clearly differentiate the effect of spatial and climatological rainfall
variability on elements depending on their function within the network. On
the other hand, previous studies showed a tendency that conduits located
upstream, not affected by hydraulically constraining structures, are more
sensitive to rainfall spatial variability in comparison to conduits located
downstream
Rainfall records were obtained from a rain gauge that is located about 2 km
west of the case study catchment. It was chosen for three main reasons:
(i) its proximity to the catchment; (ii) it has a sufficiently long record
(34-year) that is adequate for statistical climatology analysis; and
(iii) records have been verified by MeteoSwiss, ensuring sufficient
consistency. In contrast to these advantages, the 10 min temporal resolution
of the rain data requires critical consideration when simulating the dynamics
of the flow response
The computational cost of running a rainfall generator combined with an urban drainage model may constrain the use of the proposed approach for practical applications. But given the advances in the availability of computing capacity, also for non-scientific institutions, such an application will become feasible in the near future. We have used a powerful 20-core desktop machine (Intel Xeon CPU E5-2687W) to run the 961 stochastic rainfall realizations with STREAP in approximately 4 days. We estimated that the time needed to run SWMM using the same stand-alone machine would have been about 4 months, which is an impractically long duration. Therefore, we have used a high-performance computing (HPC) cluster with hundreds of computing nodes allowing SWMM simulations in less than 48 h.
Output from a stochastic rainfall generator was used as input into an urban drainage model to investigate the effect of spatial rainfall variability and climate variability on peak flows in an urban drainage system located in central Switzerland. We found that the climate variability is the main contributor (74 % on average) to the total flow variability, but that the relative contribution of the addition of spatial rainfall variability increases with return period. This implies that the use of spatially distributed rainfall data can supply valuable information for sewer network design (based on return periods of 5 to 15 years), but it will become even more relevant when assessing the risk of urban flooding as a consequence of intense rain events of larger return periods.
The analysis presented in this study focused on three different locations in the urban drainage system which reflect different system functions. Deviations in flow quantities and dynamics were expected and are, in fact, observed within the catchment, depending on the corresponding location (i.e., upstream or downstream of the overflow structure, or the overflow itself). Despite this, in agreement for all three locations, we found that the climate variability is the dominant contributor to the flow variability for all return periods.
We present a single case study, a relatively small, but typical urban catchment located in the foothills of the Swiss Alps. We argue that the variability partitioning is likely to be similar for most small- to medium-sized urban catchments. That is to say, the climate variability will constitute the largest contribution to the overall flow variability also in other urban catchments, and spatial variability will gain more importance as longer return periods are being considered. Further investigations are needed to examine the contributions of the variability components in larger catchments (potentially more prone to spatial rainfall variability) with a more complex drainage network (potentially with more flow attenuation) and for different climates.
Stochastic spatially distributed rainfall generators should become an
integral part of the urban hydrologist toolbox, particularly when estimating
hazards of urban flooding. However, these models are still not commonly used
by planning engineers for designing and evaluating urban drainage systems. We
identify four main aspects that contribute to the reluctant acceptance in the
field of urban drainage.
High-resolution rainfall data are required (from a weather radar system or
from a dense rain gauge network) as well as an expert user for the
calibration process. Setting up an automatic calibration process is an
unrealistic option due to the spatio-temporal differences between weather
radar systems and the need to tailor the rainfall generator to specific
locations. The high computational cost of running a rainfall generator combined with an
urban drainage model may be prohibitive for common applications. Today the
resources required for an efficient computation (e.g., HPC cluster) are often
not available. The struggle to overcome old engineering paradigms towards accepting
variability ranges as useful information for design and performance assessment. The difficulty for rainfall generator modelers in transparently conveying the
modeling chain, its results, and uncertainties.
These aspects should be addressed in future applications of stochastic
rainfall generators in order to make them more accessible to the urban drainage community.
Rainfall data (from a rain gauge and a C-band weather radar composite) was provided by MeteoSwiss, the Swiss Federal Office of Meteorology and Climatology. Data associated with the drainage flow (infrastructure and land use data) were provided by the municipality for exclusive use in this study. Measured flow data were collected by the authors and are published in a previous study (Tokarczyk et al., 2015).
The authors declare that they have no conflict of interest.
This project is partly funded by the Swiss Competence Center for Energy Research – Supply of Electricity. We are grateful to MeteoSwiss, the Swiss Federal Office of Meteorology and Climatology, and the city of Lucerne for providing us with precipitation and infrastructure data. We furthermore would like to thank the Engineering Consultants from HOLINGER AG, Bern, for assisting us with details on the hydraulic model and extracting operation data from the central database. We thank the reviewers (Susana Ochoa-Rodriguez, Li-Pen Wang, and an anonymous reviewer) and Marie-Claire ten Veldhuis, the editor, for their contributions leading to a significantly increased quality of the paper. Edited by: M.-C. ten Veldhuis Reviewed by: S. Ochoa Rodriguez and one anonymous referee