The analyzed catchment was divided into subcatchments (Szeląg et al., 2022a) that constituted the study areas for the identification of stormwater flooding. Due to the limited number of rainfall data, the obtained model for the simulation of stormwater overflow did not include all of the important factors, such as the dry-period duration between rainfall events and catchment retention, that impact flooding phenomenon; this meant that the model had a limited predictive capability. A detailed description of the subcatchments used for the construction of the flooding identification model and the justification of their selection were presented in Szeląg et al. (2022b). In reference to the approach proposed by Duncan et al. (2012), Jato-Espino et al. (2018), and Li and Willems (2020), the number of subcatchments used for the development of a logit model was increased to eight in the current analysis (Fig. 2). The subcatchments' boundaries and data on the spatial development and stormwater network (Table 1) were determined based on maps for design purposes, which were discussed in detail in Szeląg (2013).

Data were verified using an independent analysis performed by Wałek (2019), who used the QGIS program to construct a spatial development model and stormwater network for Kielce. The location of closing cross-sections of subcatchments (J, K, L, M, M, O, P, R, and S) along the main interceptor channel were additionally supported by the simulation results of outflow hydrographs developed by Wałek (2019), with use of the Hydrologic Engineering Center – Hydrologic Modeling System (HEC-HMS) model, as well as by Szeląg et al. (2016, 2022b), with use of the SWMM.

The value of the specific flood volume was defined as the stormwater flooding per unit paved area, which can be expressed using the following formula (Siekmann and Pinnekamp, 2011): 1κ=∑i=1KVt(i)Apav. Here, Vt is the volume of stormwater flooding from the ith sewer utility hole of the stormwater network, K is the number of sewer utility holes, and Apav is paved area. Siekmann and Pinnekamp (2011), based on continuous simulations with hydrodynamic models for three urban catchments, found that the specific flood volume ranged from 0 to >20 m3 ha-1.

On this basis, they established a limiting κ value that expressed the need to improve the operating conditions of the drainage system. Specifically, they showed that a κ>13 m3 ha-1 inferred that the drainage system requires adaptation. This was also confirmed by the calculations of Kotowski et al. (2013) for a catchment in Wroclaw and by Szeląg et al. (2021a) for a catchment in Kielce. This allows us to conclude that the κ value quoted above can be used as a decision-making criterion for urban catchments (e.g., in Poland and Germany) with respect to the necessity for corrective action on the drainage network.

The concept of the proposed tool – a simulator integrated with risk assessment and a sensitivity analysis – to evaluate the operation of a sewage system is presented in Fig. 2. Applying the MCM of an urban catchment with separate subcatchments (varying land use and topology), a specific flood volume simulator was developed as an alternative approach to the SWMM. A logistic regression model simulator based on rainfall data, catchment and stormwater network characteristics, and SWMM parameters (width of runoff path, retention depth of impervious areas, the Manning roughness coefficient of impervious areas, the correction coefficient of impervious areas, and the Manning roughness coefficient of channels) was proposed. The resulting tool enables fast analysis of sewer network performance, even with limited data access, and can be applied to other catchments. The proposed methodology is based on the extension of algorithms given by Szeląg et al. (2021a, 2022a). In contrast to previous studies (Szeląg et al., 2022b), the current approach considers the retention of the catchment and the sewer network, and the performance criterion of the sewer network was the volume of flooding, not just the fact that it occurred. Integration of the simulator with an analytical relationship for sensitivity coefficient calculations for logistic regression allows fast evaluation of the impact of MCM parameters on flooding for arbitrary catchment characteristics and topological data.

In order to provide more reliable simulation results, the proposed risk assessment considered the uncertainty in the SWMM parameters and enabled the optimization of the operation of the sewer network based on the maximum allowable values of the channel Manning roughness coefficients.

The proposed computation algorithm consists of eight modules. Modules 1, 2, 3, and 4 include identical steps to those in the work of Szeląg et al. (2021a, 2022a). In the present study, the scope of the analyses was extended: in addition to precipitation data and SWMM parameters (Szeląg et al., 2022a), the characteristics of the catchment and the stormwater network of the separated subcatchments were also included (module 1), which were used to determine the computational model. On the basis of spatial data (module 1a and 1b), a mechanistic model of the catchment was built (module 2), which allowed one to perform an uncertainty analysis using the generalized likelihood uncertainty estimation (GLUE) method (module 3). On this basis, simulations were performed in separated subcatchments for rainfall events (module 1e) under uncertainty (module 4). Based on the simulation results, a logistic regression model was developed (module 5) to calculate the local sensitivity coefficients for calibrated SWMM parameters, with respect to rainfall intensity and catchment characteristics (module 6). Modules 1, 2, 3, and 4 included analyses to determine a specific flood volume simulator that could be applied to any catchment. Thus, future algorithm implementation for the new catchment will ultimately only include modules 6, 7, and 8. Using adopted rainfall data, the sensitivity coefficients of the SWMM model parameters for subcatchments are computed, and maps showing sensitivity changes at the catchment scale are drawn (module 6). While the model is applied to identify stormwater flooding, the possible methods to improve stormwater network operation are analyzed inside modules 7 and 8. Computations using the developed algorithm consist of the following steps and substeps: 1.

Input data are collected (catchment characteristics – module 1a; stormwater network characteristics – module 1b; rainfall–runoff episodes – module 1c), independent rainfall episodes are separated (module 1d), and the characteristics of subcatchments are divided and determined (module 1e).

Based on SWMM simulation results including the uncertainty in the calibrated parameters (Table S1), the likelihood functions were determined (Kiczko et al., 2018). For the observed events (30 May 2010 and 8 July 2011) used to identify the SWMM parameters, it was found that 96 % of the measurement points included the calculated confidence interval. For the validation sets, 90 % of the observation points fall within the bands for the 15 September 2010 event and 70 % fall within the bands for the 30 July 2010 event (Fig. S1). The results of the likelihood function calculations for the calibrated SWMM model parameters are given in Figs. S2 and S3 in the Supplement.

The results of variation in the specific flood volume for the separate subcatchments are presented in Fig. 3. Based on the obtained curves, it was stated that the uncertainty in the SWMM parameters influenced the simulation results, which was confirmed by the great variability in the 1st and 99th percentile values for each subcatchment. The median values enabled one to identify that the largest specific flood volume was for subcatchment J (14.90 m3 ha-1), followed by subcatchment S (8.29 m3 ha-1) (Fig. 3). The simulation results for the 1st percentiles showed that, for the adopted rainfall events (Pt>5.0 mm and tr<150 min), stormwater flooding occurred in all subcatchments.

It was demonstrated that problems with the operation of the stormwater network are present in each subcatchment, as the calculated values of the (75th and 99th) percentiles are higher than 13 m3 ha-1. This indicates that the stormwater network requires modernization.

An LRM was built based on the operational simulation of the stormwater network. The model can be used to identify the specific flood volume and for decision-making regarding corrective action on the stormwater system. The relationship from Eq. (2) was described by the following linear combination: 7Xrain=4.05⋅Ptot-0.18⋅tr-54.15,8XSWMM=0.23⋅α-79.40⋅nimp+6.23⋅β+0.33⋅γ+234.12⋅nsew,9XCatchm=76.72⋅Imp+40.77⋅Impd-0.01⋅Vk-1967.04⋅Gk-1169.00⋅Gkd-20.33⋅Jkp. For other independent variables (Table S2), the determined coefficients were statistically insignificant in prediction confidence band 0.05. Standard deviations in the coefficients estimated from the LRM and the test probabilities are presented in Table S2. The best fit of the computed results to the measurement data was obtained for pm,cr=0.75. For the test data set (20 %), the following values were obtained: SPEC = 95.24 %, SENS = 84.62 %, and Acc = 87.87 % (where SPEC, SENS, and Acc denote specificity, sensitivity, and accuracy, respectively).

For the determined independent variables (Eqs. 7, 8), calculations were performed with the LRM and SWMM models (for 35 rainfall events, Pt≥5 mm and tr≤120 min), assuming values of catchment characteristics and topological data within ±0.2 in the separated subcatchments. The results of the validation of the developed model for the identification of the specific flood volume are given in Tables S5–S11 in the Supplement. The results obtained confirm that the determined LRM model can be applied to a wider range than that shown in Table 1. In the range of NF(SWMM)= (0–6), the relative difference in the number of episodes when κ≥13 m3 ha-1 did not exceed 20 %; for NF(SWMM)=〈6,19〉, the corresponding value was 15 %–33 % (Fig. 4).

The maximum difference between the LRM and SWMM simulations (NF(SWMM)-NF(LRM)=4) was obtained for Imp = 0.49, Impd = 0.66, Gk = 0.011 m ha-1, and Vk=1500 m3, which correspond to the extreme values of the catchment characteristics and the topology of the sewer network. Verification results showed that the maximum difference in the number of events when κ>13 m3 ha-1 using the ML model and SWMM for Imp = 0.26–0.50, Impd = 0.32–0.66, Gk = 0.0068–0.011 m3 ha-1, and Gkd = 0.0009–0.0013 m3 ha-1 did not exceed four episodes (Fig. 4). The calculations performed confirm the good fit of the calculations with measurements of the number of episodes when the specific flood volume exceeds 13 m3 ha-1.

For a rainfall depth of Ptot=10 mm and rainfall duration of tt=30–90 min, the sensitivity coefficients for the SWMM were determined based on Eq. (4). For the calculation of Sxi, the values established during calibration were adopted (Kiczko et al., 2018). The computation results for two parameters of the SWMM (β and nimp) are presented in Fig. 5.

These two parameters appeared to have the most significant impact on the specific flood volume and, at the same time, they present a vastly different impact on the dynamics of changes regarding Sxi=f(tr, Imp, Impd, Vk, and Jkp); the calculation results for the other SWMM model parameters are given in Figs. S4–S8 in the Supplement. Figures 5 and S4–S8 indicated that, for the adopted values of tr, Imp, Impd, Vk, and Jkp, the highest values of Sxi were obtained for correction coefficient percentage of impervious areas (β), the Manning roughness coefficient for sewer channels (nsew), and the Manning roughness coefficient for impervious areas (nimp). The retention depth of impervious areas (dimp) had the lowest impact on the results of the specific flood volume. An increase in the rainfall duration results in higher values of Sβ and Snimp (Fig. 5). The lowest sensitivity coefficients were obtained for tr=30 min, whereas the highest values were obtained for tr=90 min. An increase in the Imp and Impd results in a decrease in the Sβ and Snimp sensitivity coefficients. For instance, an increase in Imp from 0.34 to 0.36 results in a decrease in Sβ from 1.23 to 0.28; identical values were obtained for Impd (Fig. 5). Moreover, an increase in Vk, Jkp, and Gk leads to an increase in the Sβ and Snimp sensitivity coefficients. Among the analyzed catchment characteristics, the density of the stormwater network (Gk) had the highest impact on the sensitivity coefficients, whereas the longitudinal slope of the canal (Jkp) was of the lowest significance; these results are confirmed by the variability in the obtained curves for the subsequent SWMM parameters (Fig. 5). For example, when Vk increased from 400 to 500 m3, Sβ increased from 0.29 to 0.82. Additionally, a 10 % growth in Sβ was observed due to a change from Jkp = 0.004 to Jkp = 0.010. Finally, when Gk increased from 0.0075 to 0.009, Sβ also increased from 0.29 to 3.03 (Fig. 5).

Due to the fact that an exceedance of specific flood volume was observed in the analyzed stormwater network, possible improvements to the network were considered in terms of correcting catchment imperviousness (Imp) and enhancing terrain retention and channel capacity. The results of pm computations are presented in Fig. 6, while Fig. 7 shows Sβ for variants I, II, and III for subcatchments.

Simulation results for the sensitivity coefficients of other SWMM model parameters (Table S1) and the probability of specific flood volumes are presented in Figs. S9–S17 in the Supplement. A 10 % decrease in Imp in subcatchment J has a negligible impact on the pm value, whereas it results in a 10 % decrease in the specific flood volume probability in subcatchment S (Fig. 6a, b). It was found that a decrease in catchment imperviousness (variant I) leads to improvement in stormwater system operation (Fig. 6).

The greatest reduction in flooding volume was obtained for variant III: pm values decreased by 2 % and 36 % for subcatchments J and S (Fig. 6d). Based on the pm values for catchments J, M, N, and S for corrective action variant III, it was found that, despite the increase in retention depth and channel capacity and the reduction in the imperviousness of the catchments, there was hydraulic overloading (κ>13 m3 ha-1) in the subcatchments. This indicates the need for further changes in both the catchment and the stormwater network than were assumed. For variants I and III, the Imp values for the subcatchment were below the applicability range of the LRM; therefore, MCM simulations were performed to verify the results (Table S4). The results of the model calculations confirm their high agreement: out of 72 cases, identical results were obtained in 68 cases. The calculations performed (variants I, II, and III) for the subcatchments showed a greater influence of changes in terrain retention and channel capacity on the sensitivity coefficients than on the influence of the probability of a specific flood volume (Fig. 7). For catchments J and S, a 10 % decrease in Imp (variant I) increased Sβ by 7.55 times and 17.50 times (Fig. 7a, d). For variant II (increasing catchment retention), sensitivity coefficients were found to be higher than 51 % (catchment S) and 59 % (catchment J) compared with variant I, and the highest Sβ was obtained for variant III. The Sβ values for subcatchment S are 20.7 times, 19.3 times, and 14.7 times higher than in catchment J for variants I, II, and III, respectively. These results provide relevant information for planning retentive infrastructure that reduces outflow.

Based on the SWMM model parameters determined via the MCM method (Table S1), the probability of failure (pF) was computed for convectional rainfall in Kielce with a duration of tr=30 min and Ptot=9.61 mm. The following threshold values of nsew(m) were adopted for calculations: nsew(m)=0.015–0.045 m-1/3 s. This threshold was coupled with three variants of catchment characteristics: Imp = 0.36 and Impd = 0.40, Imp = 0.35 and Impd = 0.40, and Imp = 0.35 and Impd = 0.42. The impact of canal retention (Vk=750, 850, and 950 m3) and the density of the stormwater network (Gk = 0.0075, 0.0080, and 0.0085 m ha-1; Gkd = 0.005, 0.006, and 0.007 m ha-1) in upper and lower part of the catchment on the probability of failure (pF) were also analyzed. The Manning roughness coefficients of the channels (nsew) for the analyzed variants were presented as an empirical distribution (CDF). In Figs. 8a and 9a, the results for Imp = 0.36, Impd = 0.40, and Vk=750, 850, 950 m3 are presented; other variants are shown in Figs. S18 and S19.

Figure 8b presents the impact of nsew=f(nsew(m)) for percentiles 0.25 and 0.50 (based on the curves in Figs. 8b, 9b, 9c, 9d, S25, and S26 – the values of the respective percentiles for the analyzed nsew(m)) on the probability of failure (pF). Assuming that the Manning roughness coefficients (nsew(un)) determined by Monte Carlo (MC) simulation exceed the threshold and trigger corrective action on sewer pipes, resulting in a reduction in roughness below nsew(m), following the condition under which the stormwater network functions, pm=f(Xrain,XSWMMXCtchm)>0.75 for an independent rainfall event; thus, it was found that an appropriate decrease in the percentiles (0.25 and 0.50 – median) leads to improved network operation and to a lower failure probability (Fig. 8a, b). It was observed that the change in percentile 0.50 for nsew for a sample from MC simulation leads to a decrease from 0.028 to 0.021 m-1/3 s (as a result of correction nsew(un)<nsew(m)) and to improved stormwater network operation, understood as a lower probability of failure (decrease in pF from 0.68 to 0.42 for Imp = 0.36 and Impd = 0.40). These results confirm the significance of catchment characteristics (Imp and Impd) for the operability of a stormwater network. For Impd = 0.40, the reduction in catchment impervious area (Imp) from 0.36 to 0.35 at percentile nsew=0.019 m-1/3 s results in a decrease in the failure probability from pF=0.42 to pF=0.33 (Fig. 8b).

A great impact of channel retention (Vk) and the density of stormwater network in the upper and lower part of a catchment (Gkd and Gk, respectively) on the probability of failure pF were indicated (Fig. 9). For nsew<0.0215 m-1/3 s, pF reached higher values (max 0.41) than for Vk=850 m3 and Vk=950 m3.

The highest failure probability (pF=0.80) was obtained for Vk=750 m3 (nsew=0.031 m-1/3 s), whereas the lowest failure probability (pF=0.65) was obtained for Vk=950 m3 (Fig. 9b). Furthermore, the highest probability of failure (pF=0.79) was obtained for Gk = 0.0075 m ha-1 (nsew=0.031 m-1/3 s), whereas the lowest probability of failure was observed for Gk = 0.0085 m ha-1 (nsew=0.0276 m-1/3 s) (Fig. 9c). For nsew<0.023 m-1/3 s, it was established that computed values of pF for Gk = 0.0075 m ha-1 and Gk = 0.0080 m ha-1 were higher than 0.41. Moreover, the highest failure probability (pF) for nsew=0.035 m-1/3 s was equal to 0.82 for Gkd = 0.005 m ha-1, while it was 0.73 for Gkd = 0.007 m ha-1 (Fig. 9d).