One of the key parameters constituting the basis for the
operational assessment of stormwater systems is the annual number of storm
overflows. Since uncontrolled overflows are a source of pollution washed
away from the surface of the catchment area, which leads to imbalanced
receiving waters, there is a need for their prognosis and potential
reduction. The paper presents a probabilistic model for simulating the
annual number of storm overflows. In this model, an innovative solution is
to use the logistic regression method to analyze the impact of rainfall
genesis on the functioning of a storm overflow (OV) in the example of a catchment
located in the city of Kielce (central Poland).
The developed model consists of two independent elements. The first element
of the model is a synthetic precipitation generator, in which the simulation
of rainfall takes into account its genesis resulting from various processes
and phenomena occurring in the troposphere. This approach makes it possible
to account for the stochastic nature of rainfall in relation to the annual
number of events. The second element is the model of logistic regression,
which can be used to model the storm overflow resulting from the occurrence
of a single rainfall event. The paper confirmed that storm overflow can be
modeled based on data on the total rainfall and its duration. An
alternative approach was also proposed, providing the possibility of
predicting storm overflow only based on the average rainfall intensity.
Substantial simplification in the simulation of the phenomenon under study
was achieved compared with the works published in this area to date. It is
worth noting that the coefficients determined in the logit models have a
physical interpretation, and the universal character of these models
facilitates their easy adaptation to other examined catchment areas.
The calculations made in the paper using the example of the examined
catchment allowed for an assessment of the influence of rainfall characteristics
(depth, intensity, and duration) of different genesis on the probability of
storm overflow. Based on the obtained results, the range of the variability
of the average rainfall intensity, which determines the storm overflow, and
the annual number of overflows resulting from the occurrence of rain of
different genesis were defined. The results are suited for the
implementation in the assessment of storm overflows only based on the
genetic type of rainfall. The results may be used to develop warning systems
in which information about the predicted rainfall genesis is an element of
the assessment of the rainwater system and its facilities. This approach is
an original solution that has not yet been considered by other researchers.
On the other hand, it represents an important simplification and an
opportunity to reduce the amount of data to be measured.
Introduction
One of the important criteria for assessing the operation of stormwater
systems is the annual number of storm overflows, which is confirmed by
foreign guidelines (US EPA, 1995; Zabel et al., 2001; ÖWAV, 2003). The
physics of the phenomenon is complex and depends on the dynamics of
rainfall, the changes in rainfall over time, and the characteristics of urban
catchment areas with storm overflows. Currently, the annual number of
overflows in the catchment areas can be assessed based on long-term
observations of their operation (Price, 2000; Andrés-Doménech et
al., 2010; Gamerith et al., 2011), but it is a costly solution due to the
need for the continuous monitoring of flows. An alternative approach is to
build a hydrodynamic model of the catchment, which requires detailed data
about the basin, precipitation from a long period (30 years according to the German Association for Water, Wastewater and Waste's (DWA) standard DWA-A 118E, 2006) and flows to calibrate the model from a period of at least 2 years
(Szeląg et al., 2016). Using the model of the catchment built based on
long-term rainfall measurements, it is possible to perform the so-called
continuous simulation, which will allow for an estimation of the annual number
of storm overflows. Such a solution may be a source of a reliable estimation
of the number of overflows, although its technical implementation is complex,
and the results obtained (numerical simulations of the catchment model) are
not always satisfactory (Romanowicz and Beven, 2006; Beven and Binley,
2014).
Considering the above, the article uses probabilistic models to predict the
annual number of overflows, which take into account the stochastic nature of
rainfall and the complex nature of runoff in urban catchment areas. This
problem was discussed by Thorndahl and Willems (2008), who used the first-order reliability model
(FORM) method to simulate the storm overflow (OV) event;
its application in engineering practice is limited due to the complexity of
its implementation. In subsequent works addressing this problem, rainfall
generators and episode the of storm overflow models (Grum and Aalderink, 1999)
were used to simulate the annual number of overflows. Multidimensional
scaling methods and fractal geometry (Rupp et al., 2009; Licznar et al.,
2015; Müller-Thomy and Haberlandt, 2015) are used to simulate rainfall
series. An alternative solution is an approach based on the multidimensional
distributions created based on theoretical distributions and copula
functions (Vandenberghe et al., 2010; Vernieuwe et al., 2015). Despite
numerous applications, these solutions are relatively complex and require
expert knowledge. For the storm overflow simulation, hydrodynamic models are
usually used, and less frequently, empirical models are used (Szeląg et
al., 2018). Nevertheless, this approach to the simulation of the annual
number of overflows is very local and, in many cases, requires the
construction of a catchment model. Szeląg et al. (2018) presented a
model of simulation of storm overflow (single rainfall event) determined by
the logistic regression method. A significant disadvantage of the above-mentioned solutions is the fact that in the probabilistic models, simplified
rainfall generators were used, and the variability of precipitation
characteristics was taken into account in relation to only one episode of
overflow. At the same time, the issue of precipitation genesis was not
addressed at all. Therefore, the question arises as to whether the information on
the nature of precipitation (e.g., season of the year or precipitation
genesis) could not find practical use in its modeling. It seems puzzling
that the time course and the dynamics of the rainfall as the result of air
masses advection (Vicente-Serrano et al., 2009; Alhammoud et al., 2014; Dayan et
al., 2015) were not taken into account when rainfall generators were used to
simulate storm overflows. The problem of the modeling of complex
atmospheric phenomena is the subject of many works (Madsen et al., 1995;
Paquet et al., 2006; Vicente-Serrano et al., 2009; Garavaglia et al., 2010;
Abushandi and Merkel, 2011). The models created consider simulations of
meteorological conditions changing in time and determining the distribution
of temperature, pressure, and humidity, which affects the dynamics of air
advection and, consequently, the patterns of precipitation phenomena.
According to the literature, the information concerning the genesis of
precipitation allows for a preliminary assessment (quantitative and
qualitative) of the course and estimation of the average rainfall intensity
(Suligowski, 2004). This information may be the basis for control of the
systems and the development of an early-warning system against the risks of
flash floods. This problem has not been considered so far and, at the same
time, it seems possible that modeling the functioning of the stormwater
system and the facilities located in it based only on forecasts and
the identification of the rainfall genesis could be accomplished.
Rainfall is universally classified into three types (Sumner, 1988; Smith,
1993): convective, cyclonic, and orographic. The main distinguishing feature
between convective precipitation in an air mass and frontal precipitation in
mid-latitudes is its spatial extent and duration. The range of convective
precipitation associated with local air circulation is much smaller than in
the case of traveling extratropical cyclones with weather fronts.
Convective precipitation induced by single thunderstorm cells, their
complexes, or squall lines is short-lived but is characterized by high
average intensity (Kane et al., 1987) and causes flash floods (Gaume et al.,
2009; Marchi et al., 2010; Bryndal, 2015). On the other hand, the lifespan
of the mechanisms of creating cyclonic precipitation is much longer than
that of convective precipitation – on the order of days rather than hours.
Hence, the effect of this is long-term rainfall with a high depth (Frame et
al., 2017), often causing regional floods (Barredo, 2007).
The presented classification of precipitation types distinguished by Sumner (1988) due to the origin, developed for the British Isles and Western
Europe, cannot be directly applied in practical hydrology in other regions
of the continent, especially in its eastern and central parts due to
the exceptional variability of meteorological conditions occurring in the
temperate zone of the warm transition climate – on the border of air masses
coming from the Atlantic and continental masses from the east (Twardosz and
Niedźwiedź, 2001; Niedźwiedź et al., 2009; Twardosz et al.,
2011; Łupikasza, 2016). The analysis of maximum rainfall of different
durations in Poland carried out at the end of the 1990s (Kupczyk and
Suligowski, 1997, 2011), supplemented by an analysis of the synoptic
situation (based on surface synoptic charts of Europe, published in the
Daily Meteorological Bulletin of the Institute of Meteorology and Water
Management in Warsaw; IMGW) and a calendar describing the types of
atmospheric circulation together with air masses and air fronts
(Niedźwiedź, 2019), led to the separation of three types of genetic
precipitation: convective in air mass, frontal, and generated in the convergence
zone.
Taking into account the above considerations, an innovative probabilistic
model for calculating the number of storm overflows is proposed in the
paper. This model is composed of two independent elements: a synthetic
rainfall generator and a model for predicting storm overflows. The
identification of overflows takes place only based on information about
average rainfall intensity. In the rainfall generator, the genesis of
rainfall is taken into account, which allows for determining the curves that
show the influence of rainfall genesis on the occurrence of storm overflow
in a single rainfall event. Based on the performed analyses, the ranges of
variability of the average rainfall intensity assigned to rainfall of
different genesis, for which storm overflow may occur in the examined city
catchment, were determined. The calculation experiments carried out in the
study facilitated the determination of the influence of the distribution of
the annual number of rainfall events of different genesis on the variability
of the number of storm overflows.
Object of study
The object of analysis was a 62 ha urban catchment located in the
southeastern part of Kielce (Fig. 1). The city covers an area of 109 km2 and is located in Świętokrzyskie Voivodeship. The average
population density of the area in question is 21.4 people ha-1. Impervious areas, including pavements, roads,
parking lots, roofs, and school playgrounds, constitute 47.2 %. On the
other hand, pervious areas, i.e., lawns, including green areas, occupy 52.8 %. On this basis, it was established that the weighted average value of
the catchment retention is dav=3.81 mm (Szeląg et al., 2016). The
length of the main canal is 1.6 km, and its diameter changes between
0.60 and 1.25 m. The height difference in the catchment is 12.0 m, and the
average slope is 7.1 %. The analysis of the measurement data (2008–2017)
concerning the analyzed catchment area showed that the antecedent period
lasted 0.16–60 d, and storms occurred 27–47 times a year. The average
annual air temperature during the period under consideration varied from 8.1
to 9.6 ∘C. In addition, the analysis of the measurement data of
flows recorded with the MES1 flow meter showed that in the antecedent
periods, the temporary stream of the stormwater was from 0.001 to 0.009 m3 s-1, which indicates the occurrence of infiltration in the stormwater
system under study.
Diagram of the analyzed urban catchment.
Stormwater from the catchment is discharged to the Silnica River. Detailed
information about the catchment can be found in Dąbkowski et al. (2010).
The stormwater from the catchment is drained via channel S1 into the
diversion chamber (DC). If the chamber level is less than h=0.42 m, the
stormwater is discharged into the stormwater treatment plant (STP). If the
chamber level is filled above h, the stormwater is discharged by a storm
overflow into the S2 canal, from where it flows into the Silnica River.
In the years 2009–2011, the amount of stormwater outflowing the
catchment area was measured with the use of the MES1 flow meter located in
the canal (S1) at a distance of 3.0 m from the inlet to the DC chamber. On
the other hand, since 2015 in the inlet (S1) and discharge (S2) channels,
MES1 and MES2 flowmeters measuring the values of filling and flows have been
installed. A detailed description of the installed measuring equipment can
be found in Szeląg et al. (2016).
Rainfall data and analysis
The source material for the study presented was data from May–October
1961–2000 obtained from the records of a traditional float pluviograph
(precipitation depth, duration, and mean intensity) installed at the IMGW
meteorological station in Kielce. Only these data were taken into account in
the conducted analyses, as the launch of a new device, the SEBA electronic
rain gauge (tipping-bucket SEBA rain gauge), a few years later in the state
measuring network resulted in the recording of significantly lower
precipitation levels (by several percent). The data concern events with a
high intensity and short duration (Kotowski et al., 2011). In the period
1961–2000 there were 1312 precipitation events in Kielce with a depth above
3 mm, with an average of 32.8 events per year. The greatest number of
rainfall events (54) was observed in 1974.
Positional statistics of rainfall depth (a) and rainfall intensity (b) in genetic types.
In Kielce, precipitation classified as the first genetic type (convective in
an air mass) lasts up to 150 min. This precipitation is caused by single
convective cells with intensive ascending and descending currents or by
complexes of cells forming systems in the form of bands (squall line). The
average annual number of these precipitation events in Kielce is 14.3
(1961–2000). These precipitation events are characterized by a low depth
(Fig. 2a) but rapidly increase with the increase in duration (maximum of 40.5 mm
in 137 min). Due to the short duration of all rainfall events (from 5 min to
2.5 h), these events have a high intensity (median of 7.97 L s-1 ha-1; maximum of 97.8 L s-1 ha-1) (Fig. 2b).
The second type (frontal rainfall) forms a group of precipitation in Kielce
in which the duration is very variable and ranges from 2.5 to 10.5 h.
These are the most frequent precipitation events in Kielce (average of 16
events per year). These precipitation events are associated with the
movement of weather fronts, while the fast cold front together with dynamic
processes in its zone leads to a high intensity of precipitation lasting
2.5–5.5 h (maximum of 53.8 L s-1 ha-1). On the
other hand, the three processes in the zone affected by the warm front usually
generate higher precipitation levels but, due to their duration (5.5–10.5 h), also a twofold lower precipitation intensity (maximum up to 10.8 L s-1 ha-1). The transformation of the air masses over
the western part of the continent, the lower speeds of the air advection, and
the weakening of the dynamics of processes in the front zone cause
precipitation in Kielce to differ in intensity and duration in relation to
precipitation defined by Sumner (1988) as cyclonic.
On the other hand, the precipitation associated with convergence zone occurs
in Kielce on average 2.3 times a year. These events are the result of the
passage of deep low-pressure centers or a series of low-pressure centers with two
clearly marked frontal areas. The high dynamics and magnitude of processes
operating within them cause long-term continuous precipitation
(>10.5 h) that is recorded near the ground surface, with a
clearly increasing sum (Fig. 2a) and variability (weakening of intensity
after passing a warm front and an increase in intensity on a cold front), although
with a low mean intensity (Fig. 2b).
Methodology
An innovative probabilistic model is proposed for modeling the annual
number of storm overflows (Fig. 3). This model allows for the prediction
of the annual number of overflows and the simulation of the number of events
per year, taking into account the genesis of rainfall, which is typical for
countries located in central Europe and other regions of the world. Although
the paper focuses on the genesis of rainfall developed by Kupczyk and
Suligowski (1997, 2011), the proposed approach is universal. The
distribution of rainfall data may be based on local conditions determining
the advection of air masses, which plays a key role in the dynamics of
rainfall events. The time range of particular rainfall groups can then be
determined based on meteorological, synoptic, and statistical analysis in the
periods of high precipitation totals or precipitation intensity in a given
area (Llasat, 2001; Rigo and Llasat, 2004; Millán et al., 2005; Langer
and Reimer, 2007; Federico et al., 2008; Lazri et al., 2012; Berg and
Haerter, 2013). The classification of precipitation proposed by Sumner (1988) for Western Europe can also be used in such analyses. A literature
review (Vandenberghe et al., 2010) shows that the seasons of the year were
taken into account in models for predicting rainfall distribution based on
copula functions. The aspects related to the genesis of rainfall have so far
not been included in probabilistic models for the operational analysis of
stormwater systems operation. The model proposed in the present study
consists of three components. The first components are synthetic
precipitation generators, which are realized in two variants. In the first
variant it was assumed that the basis for the simulation of rainfall series
is their genesis. In the second variant, precipitation is predicted
regardless of its origin – in the annual cycle. Another component is a
logit model, which is used to simulate the occurrence of a storm overflow.
The third component of the model is a calculation block, in which the annual
number of overflows is simulated for the generated rainfall series. On this
basis, distribution functions (CDFs) are determined that describe the
probability of non-exceeding the number of storm overflows.
Calculation diagram of the algorithm for building a probabilistic
model for modeling the annual number of storm overflows.
The proposed algorithm includes the following steps:
separation of precipitation events in rainfall measurement series,
identification of independent variables (xi) in a logit model at the
accepted confidence level and estimation of empirical coefficients,
determination of empirical distributions and theoretical rainfall
characteristics of the different types of genetic precipitation (convective
in air mass, frontal, and in the convergence zone),
simulation of the number of precipitation events by means of the Monte
Carlo method or based on a fixed average number of rain events,
Monte Carlo simulation of rainfall characteristics for the number of rain
events generated,
determination of the number of storm overflows for the generated rainfall
series:
per year,
in different genetic types.
Simulation of the annual number of storm overflow events using a
hydrodynamic model
One of the possible solutions allowing for the verification of empirical
dependencies describing the operation of stormwater systems is the
simulation of this operation with the use of a calibrated hydrodynamic
model. It is an approach applied in engineering practice, which is confirmed
by a number of works in this field (Bacchi et al., 2008; Thordahl, 2009;
Andrés-Doménech et al., 2010). The simulations performed with a
hydrodynamic model based on rainfall data allow for the verification of the
prediction capabilities of the probabilistic models designed to simulate the
quantity and quality of stormwater and the operation of separate objects
located in the stormwater system (tanks and overflows).
Within the framework of the conducted analyses, a calibrated model of the
catchment basin made in the stormwater management model (SWMM) program was
used to verify the annual number of storm overflows (Fig. 1). The total
area of the analyzed catchment is 62 ha, while the area of partial
catchments ranges from 0.12 to 2.10 ha. The number of stormwater junctions
in the catchment area is 200, and the number of stormwater pipes is 72. The
retention depth of the imperviousness surfaces of the catchment area is 2.5 mm, and the retention depth of the pervious surfaces is 6.0 mm. The roughness
coefficient of impervious areas is equal to 0.025 m-1/3 s,
and that of pervious areas is 0.250 m-1/3 s. The roughness
coefficient of the stormwater channels is equal to 0.018 m-1/3 s.
The results of the simulation with the hydrodynamic model were compared with
the results of calculations made with the use of logit models for the data
from the examined catchment from the period 2008-2016, which allowed for the
verification of the determined relation p=f(x1, x2, x3, …,
xi). Simultaneously, using the measurement data from the period of
1961–2000, a simulation of the annual number of storm overflow events was
performed, taking into account the precipitation genesis, which enabled the
verification of the developed probabilistic model.
Logistics regression
The logistic regression model is also called the binomial logit model and is
usually used to simulate binary data. Therefore, this model is commonly used
for probability modeling. The logit model is often used to simulate
phenomena and processes in medicine, social sciences, and psychology (Bagley
et al., 2001). This model is also successfully used to model processes in
ecology, water engineering, geotechnics (Heyer and Stamm, 2013; Ingelmo et al.,
2011), and wastewater treatment (Bayo et al., 2006; Szeląg et al., 2018).
This model is also used to simulate the influence of constructions on flow
processes (Szeląg et al., 2018). The logit model takes the following
form:
p=exp(X)1+exp(X)=exp∑i=1jαi⋅xi+α01+exp∑i=1jαi⋅xi+α0,
where p (–) is the probability of occurrence of
storm overflow in a single rainfall event, αi is the empirical
coefficients estimated with the method of maximum likelihood estimation
(MLE), xi is the independent variables, which in this paper include the
rainfall depth (Ptot; mm), rainfall duration (tr; min), and
average intensity of rainfall event (i; L s-1 ha-1).
The calculations assume that a storm overflow occurs when p is not less than
0.50, which corresponds to the following condition:
∑i=1jαi⋅xi+α0=0.
To evaluate the predictive capacity of the logit model, the following
measures were used to match the calculation results to the measurements:
sensitivity (SENS; determines the correctness of data classification in
the set of data, including events when a storm overflow occurred), specificity (SPEC; determines the correctness
of data classification in the set of data constituting cases when no storm
overflow occurred), and counting error (Rz2; determines the
correctness of identification of the simulation of events, i.e., whether storm overflow
occurred or did not occur). These measures are discussed in detail in
McFadden (1963).
Two variants of the logit model were considered. In the first variant, the
rainfall depth and its duration were assumed to be independent variables, as
described in Szeląg et al. (2018). The second variant is a
simplification. This variant considers only a single independent variable,
i.e., average rainfall intensity. In the urban catchment area, continuous
flow measurements were carried out in the period 2008–2011 (69 storm
overflows during 188 rainfall events were separated at that time), whereas
in the years 2012–2014, only fillings in the diversion chamber were measured
(42 overflows during 93 rainfall events were separated). The reason for this
was the construction works carried out in the analyzed catchment and a large
amount of suspended solids limiting the operation of the measuring devices.
Since 2015, MES1 and MES2 flow meters have been installed, which also allow for
the measurement of the volume of stormwater discharged by overflow. Thus,
based on data from the period 2008–2014, a logit model was developed, while
data from the period 2015–2017 were used to verify it.
Separation of the rain event and synthetic rainfall simulator
One of the basic conditions allowing for the completion of a synthetic
precipitation generator is the separation of single independent rain events
in rainfall time series. For this purpose, the guidelines of DWA-A 118E (2006)
were used, in which the basis for precipitation separation is a minimum
antecedent period of 4 h. As a precipitation event, such a precipitation
episode has been assumed for which the rainfall depth is not less than 3.0 mm (Fu and Kapelan, 2013; Fu et al., 2014).
Based on the precipitation observed in the period 1961–2000, independent
precipitation events were separated for which statistical distributions
were determined. To obtain the best possible fit of the theoretical data to
the empirical precipitation data (including rainfall depth – Ptot,
rainfall duration – tr, and average rainfall intensity – i, for
precipitation of appropriate genesis), the following statistical
distributions were considered (Adams and Papa, 2000; Bacchi et al., 2008;
Andrés-Doménech et al., 2010): Weibull, chi-squared (Chi), exponential, generalized extreme value (GEV), Gumbel,
gamma, Johnson, log-normal, Pareto, and beta. Kolmogorov–Smirnov (KS) and
chi-squared tests were used to assess the conformity of the empirical and
theoretical distributions. The empirical distributions were also determined,
and the theoretical distributions were adjusted to the data describing the
number of precipitation events in the year and the number of events
resulting from convective precipitation, frontal rainfall (warm and cold
fronts), and rainfall in the convergence zone. Within the theoretical
distributions, Poisson, geometric, Bernoulli, and binomial distributions were
considered. Kolmogorov–Smirnov tests were used to assess the conformity of
empirical and theoretical distributions.
Taking into account the computational algorithm described at the beginning
of Sect. 4 (Methodology), based on the determined distributions of the
theoretical rainfall characteristics describing the operation of a storm
overflow, a model for the simulation of a synthetic rainfall series was
adopted for further analysis. The simulations carried out for this purpose
included the Monte Carlo method with modifications from Iman and Conover (1982).
This model provides the possibility of simulating the independent variables
based on the determined theoretical distributions. In this method, the
variability of the considered variables is described by boundary
(theoretical) distributions, and the basis for the evaluation of their
correlation is the Spearman correlation coefficient. The conditions that
must be met in order for the results obtained to be considered correct are
as follows:
In the data obtained from simulation and measurements, the mean values
(μ1(x1), μ2(x2), …, μi(xi))s and the standard deviations (σ1(x1), σ2(x2), …, σi(xi))s of the variables (xi) considered in j samples do not
differ by more than 5 %.
The theoretical distributions of xi variables obtained from
simulation should be consistent with those obtained from measurements; in
order to meet this condition it is recommended to use the Kolmogorov–Smirnov test.
The value of the correlation coefficient (R) between the individual
dependent variables (xi) obtained for the data from Monte Carlo (MC)
simulation does not differ by more than 5 % from the value of R obtained
for empirical data.
If the above-mentioned conditions are met, the results of the simulation
performed with the Iman–Conover (IC) method can be considered correct. If
these conditions are not met, the sample size of the MC needs to be
increased (Wu and Tsang, 2004). To limit the sample size and improve the
efficiency of the Iman–Conover algorithm, a modification has been
developed by using the Latin hypercube (LH) algorithm, which is part of the
layered sampling methods aimed at improving the “uniformity” of the numbers
generated from the boundary distributions.
Based on the determined boundary distributions of rainfall characteristics,
the simulations of the synthetic rainfall series were performed with the use
of the Monte Carlo method with Iman–Conover modifications and
taking into account the Latin hypercube algorithm.
Simulator of annual synthetic rainfall series
The paper presents two approaches to the simulation of rainfall series. In
the first approach to the simulation, the average annual number of
precipitation events (convective rainfall in air mass, frontal rainfall, and
rainfall in the convergence zone) was assumed (Fig. 4). In the second approach,
it was assumed that the number of rainfall events of the appropriate genetic
type is stochastic.
Algorithm of the simulation of annual rainfall series, taking into
account the generation of rainfall. N – number of modeled samples for M
rainfall events (N=500); K – number of samples of rainfall
characteristics modeled with the IC + LH method (K=1000); explanations of the other symbols are in the Appendix.
Within the framework of the conducted analyses, an attempt was made to
establish simplifications related to the omission of the impact of rainfall
genesis on the annual number of storm overflows. Based on the calculation
algorithm (Figs. 3, 4) and the proposed rainfall generator, several
simulation variants were considered. In the simplest case, all rainfall
events were considered regardless of their genesis. Thus, the average number
of rain events per year – M=const(xi) – was assumed. In the next case
rainfall of different genesis (convective, frontal, and in the convergence zone)
was taken into account. This approach reflects the different dynamics of
rainfall and the changing number of rainfall events in the year – M=var(xi).
In addition, an attempt was made to analyze the intermediate variants. The
differentiated number of rainfall events in the year was taken into account,
but the genesis – M=f(xi,M=var) – was omitted.
Results
Following the above-mentioned methodology concerning the structure of the
individual elements of the probabilistic model (Fig. 3), calculations were
made. The calculations consisted of the determination of the logit model,
the identification of empirical distributions and theoretical rainfall
characteristics, the simulation of synthetic rainfall series with the
inclusion of rainfall genesis, and the modeling of the annual number of
overflows.
Logit model and verification
Based on the results of the measurements of storm overflow and rainfall
described in detail in the sections above and in the works of Szeląg et al. (2013, 2018), independent rain events were separated, on the basis of which
the logit model was determined. In the case of independent variables such as
Ptot and tr, the logit model takes the form (Szeląg et al.,
2018)
p=exp0.566⋅Ptot-0.004⋅tr-2.1521+exp0.566⋅Ptot-0.004⋅tr-2.152.
The logit model is characterized by satisfactory predictive abilities
due to the values of SPEC =96.90 % (out of 106 overflows, in 103 events
the model correctly identified the event), SENS =98.20 % (out of 165
events, 162 overflows were correctly classified), and Rz2=97.78 % (out of 271 observed events, in 265 events the calculation results were
consistent with the measurements).
If only the average rainfall intensity (i) was included in the analyses, a
logit model of the following form was obtained:
p=exp0.312⋅i-1.2571+exp0.312⋅i-1.257.
An interesting result of the research is the fact that it is possible to
simulate storm overflows with satisfactory accuracy only on the basis of
rainfall intensity. This fact is important from the perspective of
constructing models for modeling rainfall. The result obtained in the study
indicates the possibility of significant simplification of the construction
of the probabilistic model for the simulation of the annual number of
overflows. This model is also characterized by satisfactory predictive
abilities due to the values of SPEC =84.90 % (out of 106 overflows, in
90 events the model correctly identified the event), SENS =90.20 % (out
of 165 events, 148 overflows were correctly classified using the model) and
Rz2=87.82 % (out of 271 observed events, in 229 events, the
calculation results were consistent with the measurements).
The values of the SENS and SPEC coefficients are
usually calculated to assess the predictive capability of the models.
However, SENS =f(SPEC) may also be used for this purpose. In this case, the
greater the area value (the maximum value is AUC =1; area under curve) between the SENS = SPEC and SENS =f(SPEC) curves
is, the more accurate the model will be (Fig. 5). To verify the predictive
capabilities of the logit models and their dependencies, a calibrated
hydrodynamic model was used. The results of the simulation obtained with the
hydrodynamic model for data from the period 2008–2017 were compared with the
results of the calculations made with the use of the logit models p=f(i) and p=f(Ptot,tr) (see
Table 1).
Relation SENS =f(SPEC) for the
logit curves p=f(i) and
p=f(Ptot,tr).
Comparison of the measurements of the annual number of
overflow events with the calculation results obtained with hydrodynamic and
logit models.
∗ Storm overflow events in the period 2012–2014 determined on the basis
of the maximum filling of the diversion chamber. Explanations of
symbols are in the Appendix.
The above relations (3) and (4) are of local character and can be applied in
principle only in relation to the analyzed catchment area. With this in
mind, an attempt was made to build a universal model by modifying these
relations. Based on theoretical considerations of Thorndahl and Willems (2008), who investigated the relationships between the characteristics of
catchment areas, including the limit retention that determines the volume of
overflow, the following equations were determined:
50.566Ptot-0.004tr-2.152=0/⋅10.566=Ptot-0.007tr-3.802=0,6Ptot-0.007tr-3.802=Ptot-0.007tr-dav=0,70.312i-1.257=0/⋅10.312=i-4.03=0,8i-4.03=i-dav=0.
On the basis of these relationships, it can be concluded that the values of
the intercept obtained in them are similar to the weighted average value of
the catchment retention (dav). The relative difference between the
values of intercept and retention of the catchment area does not exceed 5 %. The simulation calculations of the annual number of overflow events in
the tested catchment area confirm the above statement (Table 1). In
addition, this fact is supported by the results obtained in Fig. 5, which
indicate that the results of the simulation of the annual number of overflow
events using the hydrodynamic model are within the scope of a probabilistic
solution. However, because the presented analyses were performed only for a
single catchment, it is necessary to verify the obtained results using
examples of other catchments with different physical and geographical
characteristics.
Identification of the empirical and theoretical distributions of the
selected rainfall characteristics
Table 2 presents the results of the Kolmogorov–Smirnov and chi-squared tests of fitting the empirical distributions to theoretical
distributions for the rainfall characteristics (Ptot, tr, i, and the number of rainfall events – M)
depending on the genesis of precipitation and shows the determined
parameters in equations describing theoretical distributions.
Results of KS and Chi tests and parameters of theoretical models
for the considered random variables (Ptot, tr, i, and M) depending on rainfall
genesis.
VariableDistributionModel parametersp (KS)p (Chi)Ptot (all events)Weibullβ=0.772; γ=5.158; μ=3.000.1210.096tr (all events)GEVζ=0.466; σ=129.355; μ=1080.0960.071i (all events)Log-normalσ=1.932; μ=0.8550.1120.096M (all events)Poissonλ=32.800.6240.053Ptot (convective)Weibullβ=0.821; γ=3.102; μ=3.000.4770.412tr (convective)Betaα=1.391; β=1.173; c=5.5; d=1500.2680.173i (convective)Log-normalσ=2.557; μ=0.6940.2380.211M (convective)Poissonλ=14.330.8710.756Ptot (frontal)Weibullβ=0.968; γ=6.054; μ=3.000.3530.314tr (frontal)Weibullβ=1.201; γ=164.99; μ=1500.6390.589i (frontal)Log-normalσ=1.485; μ=0.6440.9060.878M (frontal)Poissonλ=15.950.3720.831Ptot (frontal, type I)Weibullβ=0.862; γ=4.535; μ=3.000.6310.425tr (frontal, type I)Betaα=1.221; β=1.372; c=150; d=2700.2000.145i(frontal, type I)Log-normalσ=1.701; μ=0.6120.1040.085Ptot (frontal, type II)Weibullβ=1.065; γ=7.222; μ=3.000.3970.342tr (frontal, type II)Betaα=0.829; β=1.562; c=266; d=6500.2700.226i (frontal, type II)Log-normalσ=1.289; μ=0.6110.0590.056Ptot (in the convergence zone)log-normalσ=0.603; μ=3.000.9690.856tr (in the convergence zone)Weibullβ=0.802; γ=276.138; μ=6500.9470.879i (in the convergence zone)Log-normalσ=1.296; μ=0.4970.9420.923M (in the convergence zone)Poissonλ=2.550.0670.652
Type I – cold front; type II – warm front.
Based on the data in Table 2, it appears that the empirical distributions can
be expressed by means of the following theoretical distributions:
Weibull:
fx=βγ⋅x-μγβ-1⋅e-x-μγβ,
Log-normal:
fx=1x⋅σ⋅2⋅π⋅elnx-μ22⋅σ2,
Generalized maximum value (GEV):
fx=1σ⋅(1+ξ⋅(x-μ)/σ)-1ξ-1⋅e-1⋅1+ξ⋅(x-μ)/σ-1ξ,
Beta:
fx=Γ(α+β)Γ(α)⋅Γβ⋅x-cα-1⋅x-dβ-1(d-c)α+β-1,
Poisson:
fx=e-λ⋅λxx!,
where β, γ, λ, μ, σ, and ξ are the parameters of distributions determined by the maximum likelihood
estimation (MLE) method.
Comparison of empirical and theoretical quantiles concerning the
number of rainfall events and distinguishing rainfall types for (a) all events, (b) convective, (c) frontal, and (d) in the convergence zone.
The calculations performed (Table 2) showed that in most cases the Weibull
distribution in the form (Eq. 9) is the best suited to the empirical data
describing the variability of the total depth of rainfall (Ptot) in a
rainfall event. In addition, the variation in rainfall duration of the
events resulting from rainfall of different genesis is described, in most
cases, by the Weibull distribution and only in the case of the data measured
over an annual cycle is it expressed better by the GEV distribution (Eq. 11). The values of the rainfall intensities of different genesis are, in
most cases, described by log-normal distributions (Eq. 10), whereas in the
case of the rain intensity it is caused by frontal precipitation, and its dynamics are
described by a generalized distribution of extreme values. Satisfactory
adjustment of the calculation results to the measurements of the tested
variables of a continuous nature is confirmed by the curves shown in Figs. 6–7.
Comparison of empirical and theoretical quantiles concerning
Ptot and tr values for (a) all events, (b) convective, (c) frontal,
and (d) rainfall in the convergence zone.
The curves in Figs. 6 and 7, illustrating the fitting of the
quantile values obtained from the measurements and based on the specific
theoretical distributions in most of the cases considered, show a high
agreement of the data in relation to the number of rainfall events per year
(Mc,f,cz), precipitation altitude, and duration (Ptot,tr). For the
distribution describing the annual number of events, the values of the
minimal quantiles are overstated by 5, while the maximal values are
understated by 7. As a result, this distribution may affect the results of
the determined number of storm overflows.
Impact of rainfall genesis on the probability of storm overflow
occurrence
Based on the models determined for the modeling of storm overflow and the
determined theoretical rainfall distributions, the impact of rainfall
genesis on the occurrence of a single storm overflow was calculated in the
first place. Taking into account the different predictive abilities of the
logit models obtained, the analyses were limited to calculations with a
model that best represented the existing state. The results of the
simulations are shown in Fig. 8.
Impact of rainfall genesis on (a) the probability of storm
overflow and (b) rainfall intensity distribution determining storm overflow.
Within the framework of the conducted analyses, the division of frontal
rainfall events of the duration not longer than 4.5 h (related to the cold
front – frontal type I) and exceeding the given value (due to the
displacement of the warm front – frontal type II) was additionally
distinguished. Based on the curves shown in Fig. 8a, it can be concluded
that the greater probability of storm overflow was obtained for events
caused by convective rainfall events. For precipitation of this genesis, the
minimum probability of storm overflow is not less than 0.40, and the
percentile value of 0.50 is as high as 0.90. In order to verify the obtained
relationship using a probabilistic model, empirical curves are plotted based on
the measurement data (Fig. 8b). Curves obtained on the basis of
measurements and simulations show high compliance, which confirms the
usefulness of the determined model.
Impact of rainfall genesis on the average rainfall intensity
occurrence of a storm overflow
Based on the results of the calculations obtained above, the ranges of the
variability of the average rainfall intensity were determined depending
on the genesis conditioning the occurrence of a storm overflow (Fig. 8).
Based on the determined curves, it can be stated that among the analyzed
rainfall, the highest range of variability of values i=4.49–75.00 L s-1 ha-1 was obtained for convective
rainfall (Fig. 8b). From the point of view of modeling and forecasting,
this result seems to be interesting because the information about the
genesis of precipitation (it is now generally available information) enables
assessing, to some extent, the conditions of the functioning stormwater
systems. Based on the analysis of the variability of curves from Fig. 8a
and b, it can be stated that the lowest probability of storm overflow is
obtained in the case of rainfall in the convergence zone i=4.49–10.94 L s-1 ha-1 and rainfall in the warm-front
zone (with a duration longer than 4.5 h), for which i=4.49–28.50 L s-1 ha-1. In the analyzed case, it was
determined that for p=0–0.57 in the examined catchment the lowest
probability value is obtained for events caused by precipitation in the warm
front zone (tr≥4.5 h), while for higher p values, the lowest
probability of overflow occurrence is obtained as a result of rainfall in
the convergence zone (Fig. 8a). This fact can be justified by the physics
of the studied phenomenon. In the case of rainfall in the convergence zone,
lasting a relatively long time, the average rainfall intensity is lower than
in the case of frontal rainfall with a rainfall duration longer than 4.5 h.
Dependencies requiring commentary were also obtained for frontal
precipitation (tr≥2.5 h), for which i=4.49–35.50 L s-1 ha-1. The conducted analyses showed
that with the increase in p values, the difference in absolute values of the
probability of storm overflow caused by precipitation being the effect of
the cold-front (tr=2.5–4.5 h) and warm-front (tr≥4.5 h)
increases (Fig. 8a, b). The difference between individual values of p
(pI–pII) between individual curves increases to 0.20 for p=0.60
and then decreases. It is worth noting that this difference (its maximum
value) includes events in the case of frontal precipitation caused by a cold
and warm front (it corresponds to i=4.49–35.50 L s-1 ha-1) when storm overflows take place, i.e., when
p>0.50. Attention is drawn to the fact that the variability
of the curve describing the probability of storm overflow by the effect of
frontal rainfall, for which tr≤4.5 h, shows that in the range of p=0–0.55 a similar variability to the curve obtained for rainfall data
regardless of their genesis. For p>0.55, a greater probability
of storm overflow due to all rainfall events than precipitation lasting
tr=2.5–4.5 h was found. The curves determined in Fig. 8b may
indicate lower values of i for rainfall events tr=2.5–4.5 h than
when i>4.86 L s-1 ha-1 which
corresponds to all rainfall.
Impact of rainfall genesis on the annual number of overflow
events
The simulation calculations performed with the use of the logit model
described by Eq. (3) showed that the annual number of overflows resulting
from convective rainfall, frontal rainfall, and rainfall in the convergence zone
is lower in the case of using the simplified model taking into account only
the i value (Fig. 9). These results are confirmed by the value of average
retention in logit models, i.e., when d=3.80 mm (in the model that takes
into account the genesis of rainfall) and d=4.32 mm (in the simplified
model with lower predictive capabilities).
Distribution function (CDF) showing the annual number of storm
overflow events (Z) caused by (a) convective rainfall, (b) frontal rainfall, and (c) rainfall in the convergence zone. (d) Curve showing the probability of
non-exceeding the annual number of overflows (Z). Explanations of the other
symbols are in the Appendix.
Based on the calculations, it was found that the inclusion in the simulation
of the annual number of storm overflows with a stochastic character of rain
events has a significant impact on the results of the analyses. The analyses
showed that in the case of precipitation caused by convection in air mass
and frontal rainfall, the annual number of overflows (percentile value of 0.50)
is lower when the number of events is described by Poisson's distribution.
The calculations show that in the case of storm overflows resulting from
convective precipitation, the stochastic nature of the precipitation has a
significant impact on the values of the lower and upper percentiles (Fig. 9a). For example, for the percentile value of 0.05, the difference in the
annual number of overflows obtained with the assumption of M=var is 5 times greater than the solution when the average number of precipitation
events per year was assumed. For the percentile value of 0.95, the difference
in the annual number of overflows between the considered solutions is 4. In
the case of storm overflows resulting from frontal rainfall (Fig. 9b), the
difference in the annual number of overflows obtained for the variants when
M=const and M=var is much smaller than for convective rainfall (Fig. 9a). This difference may be due to a significant variation in the number of
convective and frontal rainfall events over an annual cycle and the
variation in rainfall intensity in both cases. By analyzing the results of
the simulation, it can be concluded that when the number of precipitation
events (convective, frontal, and in the convergence zone) is determined by
Poisson's distribution, the calculated annual number of overflows for p<0.50 is smaller than when M=const (Fig. 9). On the other hand,
for p>0.50, the inverse relation is maintained. The influence of
the theoretical distribution of the number of rainfall events per year on
the values of percentiles 0.50–0.99 is confirmed by Szeląg et al. (2018). Based on the implemented simulations and designed curves, it can be
established that the average annual number of overflows resulting from
convective rainfall is 15 (Fig. 9a), and in the case of frontal rainfall
and rainfall in the convergence zone, it is much smaller and equals 7 and 1,
assuming that M=const (Fig. 9b, c). In the case of storm overflows
caused by frontal overflow, it was found based on the determined curves that
out of seven overflows, as many as five are caused by rainfall connected with a
cold front (type I), for which the duration of rainfall does not exceed
tr=4.5 h (Fig. 9b). Very interesting results were obtained in the
calculation variant, in which the number of rain events in a year was taken
into account, regardless of their genesis. The average annual number of
storm overflows for p>0.5 is then greater than the number of overflows taking
into account the rainfall genesis (Fig. 9d). In conclusion, the omission
of rainfall genesis in the calculations may result in the overestimation of
the average annual number of storm overflows. This overestimation affects
the designed storm overflow height and the functioning of facilities located
in the sewage network, e.g., stormwater treatment plants.
The innovative synthetic precipitation generator proposed in the paper
enables the quantification of the impact of rainfall genesis on the annual
number of storm overflows (Fig. 9d), which until now, has not been
included in the models developed by other authors. This approach can be
transferred to other facilities located in stormwater systems and used to
assess the effectiveness of stormwater drainage systems. Ultimately, the
results obtained with this approach may be the basis for the construction of
an expert early-warning system for torrential phenomena caused by
heavy rainfall. Another advantage of the developed probabilistic model is
the possibility it provides to perform simulations of a long-term nature
(multiannual), and thus, to assess the impact of the distribution of
individual precipitation types on the annual number of storm overflows and
its variability.
The simulation calculations performed with the calibrated hydrodynamic model
of the catchment area show that the results of the simulation of the annual
number of storm overflow events (Fig. 9d) caused by precipitation of
different genesis are within the scope of the solution obtained with the
probabilistic model. This finding confirms that the probabilistic model
developed in the paper is an alternative solution to the hydrodynamic model.
This fact is also confirmed by the annual number of overflow events caused
by rainfall of different genesis obtained on the basis of measurements
(Fig. 9d). The conformity of the simulation results obtained with the
probabilistic and hydrodynamic model may indicate that the values of the
intercept determined in the relevant equations may correspond to the depth
of the weighted average retention of the catchment area.
Conclusions
The calculations performed showed that the measurements of average rainfall
intensity can be used to simulate (using a logit model) the storm overflow
occurrence in a single rain event. The simulation results obtained do not
differ significantly from the calculations made based on the rainfall depth
and duration. In both models, it was shown that the numerical value of the
intercept in the model equations does not differ by more than 10 % from
the depth of the weighted average retention of the catchment area. This fact
has a significant practical meaning because it provides the possibility of
using the results obtained in other urban catchments. However, in order to
confirm this, further analyses on catchment areas with different physical
and geographical characteristics are advisable.
The computational experiments carried out in the study allowed for an assessment
of the influence of the rainfall genesis (convective in air mass, frontal,
and in the convergence zone) on the occurrence of storm overflow. Moreover,
the ranges of the variability of the average rainfall intensity were
determined, for which storm overflows were found. The information obtained
may be used in engineering practice because, on this basis, it is possible to
determine whether a storm overflow will take place.
The identification of the operational state of the stormwater system (in
this case, storm overflow) based on forecasting the weather front may be of
practical significance. This identification of the operational state
provides an opportunity to develop an emergency early-warning system (for the spilling of stormwater to the surface, hydraulic
overload of pipes, and overfilling of tanks) in stormwater systems.
List of symbols and abbreviations
AUCarea under curveCDFcumulative distribution function of probability densityChichi-squared testDCdiversion chamberdavweighted mean of the retention depth of the catchment areaf(x)probability density functionF(x)ctheoretical distribution to simulate rainfall characteristics due to convective rainfallF(x)ftheoretical distribution to simulate rainfall characteristics due to frontal rainfallF(x)cztheoretical distribution to simulate rainfall characteristics due to rainfall in the convergence zoneF(ζ)ctheoretical distribution to simulate the annual number of convective rainfall eventsF(ζ)ftheoretical distribution to simulate the annual number of frontal rainfall eventsF(ζ)cztheoretical distribution to simulate the annual number of rainfall events in the convergence zoneFORMfirst-order reliability modeliaverage rainfall intensity; 166.7Ptot/tr (L s-1 ha-1)ICIman–Conover methodKnumber of samples modeled using the Monte Carlo method of annual rainfall seriesKSKolmogorov–Smirnov testLHLatin hypercube methodMannual number of rainfall eventsM=const(i)calculation variant (annual number of overflow events), in which for the simulation, the constant average annual number of rainfall events is used; for identifying the overflow in a rainfall event, the following logit model was applied: logit model p=f(i)M=const(Ptot,tr)calculation variant (annual number of overflow events), in which for the simulation, the constant average annual number of rainfall events is used, and for identifying the overflow in a rainfall event, the following logit model was applied: logit model p=f(Ptot,tr)M=var(i)calculation variant (annual number of overflow events), in which the annual number of rainfall events caused by precipitation (convective, frontal, and in the convergence zone) is modeled; for identification of overflow in a rainfall event, the following logit model was applied: logit model p=f(i)M=var(Ptot,tr)calculation variant (annual number of overflow events), in which the annual number of rainfall events caused by precipitation (convective, frontal, and in the convergence zone) is modeled; for identifying overflow in a rainfall event, the following logit model was applied: logit model p=f(Ptot,tr)Mcannual number of convective rainfall eventsMfannual number of frontal rainfall eventsMczannual number of rainfall events in the convergence zoneMCMonte Carlo simulationMES1, MES2flowmeter measuresMLEmaximum likelihood estimationNnumber of samples in the Monte Carlo simulationOVstorm overflowpprobability of a storm overflow eventPtottotal rainfall (mm)trrainfall duration (min)RSpearman's correlation coefficientRz2counting errorSENSsensitivitySPECspecificitySTPstormwater treatment plantSWMMstormwater management modelZannual number of storm overflow eventsZcannual number of storm overflow events due to convective rainfallZfannual number of storm overflow events due to frontal rainfallZczannual number of storm overflow events due to rainfall in the convergence zone
xiindependent variables included in the logit modelαivalues of estimated coefficients in the logit modelα, β, γ, λ, μ, σ, ζparameters of theoretical distributions(μ1(x1), …, μi(xi))smean value of variable xi in the data set obtained from simulation using the Iman–Conover method(σ1(x1), …, σi(xi))svalue of standard deviation of variable xi in the data set obtained from simulation using the Iman–Conover method
Data availability
In order to access the data, we kindly ask researchers to contact the corresponding author.
Author contributions
BS designed the experiment. BS and RS carried it out. BS and RS performed all analyses. BS and RS prepared the paper with contributions from JS and FDP.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors would like to thank the Polish Institute of Meteorology and Water Management (IMGW) of the National Research Institute in Warsaw for access to precipitation data. We are grateful to the reviewers for their helpful comments and suggestions.
Financial support
This research has been supported by the Minister of Science and Higher Education's program “Regional Initiative of Excellence” for 2019–2022 (grant no. 025/RID/2018/19; financing amount of PLN 12 000 000).
Review statement
This paper was edited by Markus Weiler and reviewed by two anonymous referees.
ReferencesAbushandi, E. and Merkel, B.: Rainfall estimation over the Wadi Dhuliel arid catchment, Jordan from GSMaP_MVK+, Hydrol. Earth Syst. Sci. Discuss., 8, 1665–1704, 10.5194/hessd-8-1665-2011, 2011.
Adams, B. J. and Papa, F.: Urban Stormwater Management Planning with
Analytical Probabilistic Models, John Wiley & Sons, Chichester, UK, 2000.Alhammoud, B., Claud, C., Funatsu, B. M., Beranger, K., and Chaboureau, J.
P.: Patterns of Precipitation and Convection Occurrence over the
Mediterranean Basin Derived from a Decade of Microwave Satellite
Observations, Atmosphere, 5, 370–398, 10.3390/atmos5020370,
2014.Andrés-Doménech, I., Múnera, J. C., Francés, F., and Marco, J. B.: Coupling urban event-based and catchment continuous modelling for combined sewer overflow river impact assessment, Hydrol. Earth Syst. Sci., 14, 2057–2072, 10.5194/hess-14-2057-2010, 2010.Bacchi, B., Balistrocchi, M., and Grossi, G.: Proposal of a semiprobabilistic
approach for storage facility design, Urban Water J., 5, 195–208,
10.1080/15730620801980723, 2008.Bagley, S. C., White, H., and Golomb, B. A.: Logistic regression in the
medical literature: standards for use and reporting, with particular
attention to one medical domain, J. Clin. Epidemiol., 54, 979–985,
10.1016/S0895-4356(01)00372-9, 2001.Barredo, J. I.: Major flood disasters in Europe: 1950–2005, Nat. Hazards,
42, 125–148, 10.1007/s11069-006-9065-2, 2007.Bayo, J., Angosto, J. M., Serrano-Aniorte, J., Cascales-Pujalte, J. A.,
Fernández-López, C., and López-Castellanos, J.: Evaluation of
physicochemical parameters influencing bulking episodes in a municipal
wastewater treatment plant, WIT Trans. Ecol. Envir., 95, 531–541,
10.2495/WP060521, 2006.Berg, P. and Haerter, J.: Unexpected increase in precipitation intensity
with temperature – a result of mixing of precipitation types?, Atmos. Res.,
119, 56–61, 10.1016/j.atmosres.2011.05.012, 2013.Beven, K. and Binley, A.: GLUE: 20 years on, Hydrol. Process., 28,
5897–5918, 10.1002/hyp.10082, 2014.Bryndal, T.: Local flash floods in Central Europe: a case study of Poland,
Norsk Geogr. Tidsskr., 69, 288–298,
10.1080/00291951.2015.1072242, 2015.
Dąbkowski, S. L., Górska, K., Górski, J., and Szeląg, B.:
Introductory results of examining precipitation sewage in one Kielce
channels, Gaz Woda i Technika Sanitarna, 34, 20–24, 2010.Dayan, U., Nissen, K., and Ulbrich, U.: Review Article: Atmospheric conditions inducing extreme precipitation over the eastern and western Mediterranean, Nat. Hazards Earth Syst. Sci., 15, 2525–2544, 10.5194/nhess-15-2525-2015, 2015.
DWA-A 118E: Hydraulic Dimensioning and Verification of Drain and Sewer
Systems, DWA German Association for Water, Wastewater and Waste, Hennef,
Germany, 2006.Federico, S., Avolio, E., Pasqualoni, L., and Bellecci, C.: Atmospheric patterns for heavy rain events in Calabria, Nat. Hazards Earth Syst. Sci., 8, 1173–1186, 10.5194/nhess-8-1173-2008, 2008.Frame, T., Harrison, G., Hewson, T., and Roberts, N.: Meteorological risk:
extra-tropical cyclones, tropical cyclones and convective storms, in:
Science for disaster risk management 2017: Knowing better and losing less,
Publications Office of the European Union, Luxembourg, 246–256,
10.2788/688605, 2017.Fu, G. and Kapelan, Z.: Flood analysis of urban drainage systems:
probabilistic dependence structure of rainfall characteristics and fuzzy
model parameters, J. Hydroinform., 15, 687–699,
10.2166/hydro.2012.160, 2013.Fu, G., Butler, D., Khu, S. T., and Sun, S.: Imprecise probabilistic
evaluation of sewer flooding in urban drainage systems using random set
theory, Water Resour. Res., 47, 1–13,
10.1029/2009WR008944, 2014.
Gamerith, V., Bertrand-Krajewski, J. L., Mourad, M., and Rauch, W.:
Implications of long-term stormwater quality modelling for design of
combined sewer infrastructure, Urban Water J., 8, 155–166, 2011.Garavaglia, F., Gailhard, J., Paquet, E., Lang, M., Garçon, R., and Bernardara, P.: Introducing a rainfall compound distribution model based on weather patterns sub-sampling, Hydrol. Earth Syst. Sci., 14, 951–964, 10.5194/hess-14-951-2010, 2010.Gaume, E., Bain, V., Bernardara, P., Newinger, O., Barbuc, M., Bateman, A., Blaškovičova, L., Blöschl, G., Borga, M., Dumitrescu, A., Daliakopoulos, I., Garcia, J., Irimescu, A., Kohnowa, S., Koutroulis, A., Marchi, L,, Matreata, S., Medina, V., Preciso, E., Sempere-Tores, D., Stancalie, G., Szolgay, J., Tsanis, I., Velasco, D., and Viglione, A.: A compilation of
data on European flash floods, J. Hydrol., 367, 70–78,
10.1016/j.jhydrol.2008.12.028, 2009.Grum, M. and Aalderink, R. H.: Uncertainty in return period analysis of
combined sewer overflow effects using embedded Monte Carlo simulations,
Water Sci. Technol. 39, 233–240,
10.1016/S0273-1223(99)00063-3, 1999.Heyer, T. and Stamm, J.: Levee reliability analysis using logistic regression models – abilities, limitations and practical considerations, Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 7, 77–87, 10.1080/17499518.2013.790734, 2013.Iman, R. L. and Conover, W. J.: A distribution-free approach to inducing
rank correlation among input variables, Commun. Stat. Simulat., 11,
311–334, 10.1080/03610918208812265, 1982.Ingelmo, F., Molina, M. J., de Paz, J. M., and Visconti, F.: Soil saturated
hydraulic conductivity assessment from expert evaluation of field
characteristics using an ordered logistic regression model, Soil Till. Res.,
115–116, 27–38, 10.1016/j.still.2011.06.004, 2011.Kane Jr., R. J., Chelius, C. R., and Fritsch, J. M.: Precipitation
Characteristics of Mesoscale Convective Weather Systems, J. Clim. Appl.
Meteorol., 26, 1345–1357, available at: https://www.jstor.org/stable/26183088?seq=1 (last access: 10 February 2020), 1987.
Kotowski, A., Dancewicz, A., and Kaźmierczak, B.: Accuracy of
measurements of precipitation amount using standard and tipping bucket
pluviographs in comparison to Hellmann rain gauges, Environ. Prot.
Eng., 37, 23–34, 2011.
Kupczyk, E. and Suligowski, R.: Statistical description of the rainfall
structure as the input to hydrological models, in: Prediction of the design
storms and floods, edited by: Soczyńska, U., University of Warsaw
Publisher, Warsaw, Poland, 191–212, 1997.
Kupczyk, E. and Suligowski R.: Rainfall types for hydrological purposes,
Prz. Geofiz., 3–4, 235–246, 2011.Langer, I. and Reimer, E.: Separation of convective and stratiform precipitation for a precipitation analysis of the local model of the German Weather Service, Adv. Geosci., 10, 159–165, 10.5194/adgeo-10-159-2007, 2007.Lazri, M., Hameg, S., Ameur, S., Brucker, J. M., Ouallouche, F., and Mohia, Y.: Behavior analysis of convective and stratiform rain using Markovian approach over Mediterranean region from meteorological radar data, Hydrol. Earth Syst. Sci. Discuss., 9, 6225–6250, 10.5194/hessd-9-6225-2012, 2012.Licznar, P., De Michele, C., and Adamowski, W.: Precipitation variability within an urban monitoring network via microcanonical cascade generators, Hydrol. Earth Syst. Sci., 19, 485–506, 10.5194/hess-19-485-2015, 2015.Llasat, M. C.: An objective classification of rainfall events on the basis
of their convective features, Application to rainfall intensity in the
North-East of Spain, Int. J. Climatol., 21, 1385–1400,
10.1002/joc.692, 2001.Łupikasza, E.: The climatology of air-mass and frontal extreme
precipitation. Study of meteorological data in Europe, Springer Atmospheric
Series, Springer International Publishing AG Switzerland,
10.1007/978-3-319-31478-5, 2016.
Madsen, H., Rosbjerg, D., and Harremoöes, P.: Application of the Bayesian
approach in regional analysis of extreme rainfalls, Stoch. Env. Res. Risk
A., 9, 77–88, 1995.Marchi, L., Borga, M., Preciso, E. and Gaume E.: Characterisation of
selected extreme flash floods in E,urope and implications for flood risk
management, J. Hydrol., 394, 118–133,
10.1016/j.jhydrol.2010.07.017, 2010.
McFadden, D.: Conditional logit analysis of qualitative choice behavior, in:
Frontiers in Econometrics, edited by: Zarembka, P., Academic Press, New
York, USA, 105–142, 1973.Millán, M. M., Estrela, M. J., and Miró, J.: Rainfall components:
Variability and spatial distribution in a mediterranean area (Valencia
Region), J. Climatol., 18, 2682–2705,
10.1175/JCLI3426.1, 2005.Müller-Thomy, H. and Haberlandt, U.: Temporal Rainfall Disaggregation
with a Cascade Model: From Single-Station Disaggregation to Spatial
Rainfall, J. Hydrol. Eng., 20, 04015026,
10.1061/(ASCE)HE.1943-5584.0001195, 2015.Niedźwiedź, T.: Catalogue of synoptic situations in the upper
Vistula river basin (1873–2018), Computer file available at: Department of
Climatology, Faculty of Earth Sciences, University of Silesia,
Będzińska 60, 41-200 Sosnowiec, Poland, available at:
http://klimat.wnoz.us.edu.pl (last access: 10 February 2020), 2019.Niedźwiedź, T., Twardosz, R., and Walanus, A.: Long-therm variability
of precipitation series in east central Europe in relation to circulation
patterns, Theor. Appl. Climatol., 98, 337–350,
10.1007/s00704-009-0122-0, 2009.
ÖWAV: ÖWAV-Regelblatt 19: Richtlinien für die Bemessung and
Gestaltung von Regenentlastungen in Mischwasserkanaelen, Oesterreichischer
Wasser- und Abfallwirtschaftsverband, Vienna, Austria, 2003.Paquet, E., Gailhard, J., and Garcon, R.: Evolution of GRADEX method:
improvement by atmospheric circulation classification and hydrological
modelling, Houille Blanche, 5, 80–90,
10.1051/lhb:2006091, 2006.Price, R.: Hydroinformatics and urban drainage: an agenda for the beginning
of the 21st century, J. Hydroinform., 2, 133–147,
10.2166/hydro.2000.0011, 2000.Rigo, T. and Llasat, M. C.: A methodology for the classification of convective structures using meteorological radar: Application to heavy rainfall events on the Mediterranean coast of the Iberian Peninsula, Nat. Hazards Earth Syst. Sci., 4, 59–68, 10.5194/nhess-4-59-2004, 2004.Romanowicz, R. J. and Beven, K. J.: Comments on generalized likelihood
uncertainty estimation, Reliab. Eng. Syst. Safe., 91, 1315–1321,
10.1016/j.ress.2005.11.030, 2006.Rupp, D. E., Keim, R. F., Ossiander, M., Brugnach, M., and Selker, J.: Time
scale and intensity dependency in multiplicative cascades for temporal
rainfall disaggregation, Water Resour. Res., 45, W07409,
10.1029/2008WR007321, 2009.
Smith, J. A.: Precipitation, in: Handbook of Hydrology, edited by: Maidment,
D. R., McGraw-Hill Education, New York, USA, 1993.
Suligowski, R.: Temporal and spatial structure of precipitation in Poland.
An attempt to regionalization, Prace Instytutu Geografii, Kielce, 12,
1–112, 2004.
Sumner, G.: Precipitation: process and analysis, John Wiley & Sons,
Chichester, UK, 1988.Szeląg, B., Górski, J., Bąk, Ł., and Górska, K.: Modelling of stormwater quantity and quality on the example of urbanized catchment in Kielce, Ecol. Chem. Eng. A, 20, 1305–1316, 10.2428/ecea.2013.20(11)118, 2013.
Szeląg, B., Kiczko, A., and Dąbek, L.: Sensitivity and uncertainty
analysis of hydrodynamic model (SWMM) for storm water runoff forecating in
an urban basin – A case study, Ochr. Sr., 38, 15–22, 2016.Szeląg, B., Kiczko, A., Studziński, J., and Dąbek, L.:
Hydrodynamic and probabilistic modelling of storm overflow discharges, J.
Hydroinform., 20, 1100–1110, 10.2166/hydro.2018.005,
2018.Thorndahl, S.: Stochastic long term modelling of a drainage system with
estimation of return period uncertainty, Water Sci. Technol., 59,
2331–2339, 10.2166/wst.2009.305, 2009.Thorndahl, S. and Willems, P.: Probabilistic modelling of overflow,
surcharge and flooding in urban drainage using the first-order reliability
method and parameterization of local rain series, Water Res., 42,
455–466, 10.1016/j.watres.2007.07.038, 2008.Twardosz, R. and Niedźwiedź, T.: Influence of synoptic situations on
the precipitation in Kraków (Poland), Int. J. Climatol., 21, 467–481,
10.1002/joc.620, 2001.Twardosz, R., Niedźwiedź, T., and Łupikasza, E.: The influence of
atmospheric circulation on the type of precipitation (Kraków, southern
Poland), Theor. Appl. Climatol., 104, 233–250,
10.1007/s00704-010-0340-5, 2011.US EPA: Combined Sewer Overflows. Guidance for Nine Minimum Controls, U.S.
Environmental Protection Agency 832-B-95-003, Office of Wastewater
Management, Washington, D.C., USA, 1995.
Vandenberghe, S., Verhoest, N. E. C., Buyse, E., and De Baets, B.: A stochastic design rainfall generator based on copulas and mass curves, Hydrol. Earth Syst. Sci., 14, 2429–2442, 10.5194/hess-14-2429-2010, 2010.Vernieuwe, H., Vandenberghe, S., De Baets, B., and Verhoest, N. E. C.: A continuous rainfall model based on vine copulas, Hydrol. Earth Syst. Sci., 19, 2685–2699, 10.5194/hess-19-2685-2015, 2015.Vicente-Serrano, S. M., Beguería, S., López-Moreno, J. I., El
Kenawy, A. M., and Angulo-Martínez, M.: Daily atmospheric circulation
events and extreme precipitation risk in Northeast Spain: the role of the
North Atlantic Oscillation, Western Mediterranean Oscillation, and
Mediterranean Oscillation, J. Geophys. Res.-Atmos., 114, D08106,
10.1029/2008JD011492, 2009.Wu, F. C. and Tsang, Y. P.: Second-order Monte Carlo uncertainty/variability
analysis using correlated model parameters: application to salmonid embryo
survival risk assessment, Ecol. Model., 177, 393–414,
10.1016/j.ecolmodel.2004.02.016, 2004.Zabel, T., Milne, I., and Mckay, G.: Approaches adopted by the European Union
and selected Member States for the control of urban pollution, Urban Water,
3, 23–32, 10.1016/S1462-0758(01)00019-X, 2001.