Advanced sensitivity analysis of the impact of the temporal distribution and intensity in a rainfall event on hydrograph parameters in urban catchments: a case study

Knowledge of the variability of the hydrograph of outflow from urban catchments is highly important for measurements and evaluation of the operation of sewer networks. Currently, hydrodynamic models are most frequently used 15 for hydrograph modeling. Since a large number of their parameters have to be identified, there may be problems at the calibration stage. Hence, the sensitivity analysis is used to limit the number of parameters. However, the current sensitivity analysis methods ignore the effect of the temporal distribution and intensity of precipitation in a rainfall event on the catchment outflow hydrograph. The article presents the methodology of construction of a simulator of catchment outflow hydrograph parameters (volume, maximum flow). For this purpose, uncertainty analysis results obtained with the use of the GLUE 20 (Generalized Likelihood Uncertainty Estimation) method were used. An innovative sensitivity coefficient has been proposed to study the impact of the variability of hydrodynamic model parameters depending on rainfall distribution, rainfall genesis (in the Chomicz scale), and uncertainty of estimated simulator coefficients on the parameters of the outflow hydrograph. The results indicated a considerable influence of rainfall distribution and intensity on the sensitivity factors. The greater the intensity and temporal distribution of rainfall, the lower the impact of the identified hydrodynamic model parameters on the 25 hydrograph parameters. Additionally, the calculations confirmed the significant impact of the uncertainty of the estimated coefficient in the simulator on the sensitivity coefficients, which has a significant effect on the interpretation of the relationships obtained. The approach presented in the study can be widely applied at the model calibration stage and for appropriate selection of hydrographs for identification and validation of model parameters. https://doi.org/10.5194/hess-2021-99 Preprint. Discussion started: 7 April 2021 c © Author(s) 2021. CC BY 4.0 License.

of this method, as it is possible to assess the model sensitivity to individual parameters at any point in the domain. An additional modification can be the use of a standardized local sensitivity analysis method based on logarithms of dependent and explanatory variables. This facilitates assessment of the effect of the percentage increase in the explanatory variable on the percentage increase in the dependent variable. 100

Study object
The analysis in this study was carried out in a catchment with a total area of 62 ha, located in the south-eastern part of the city of Kielce, central Poland ( Fig. 1). Six types of impervious surfaces were distinguished in the catchment: sidewalks, roads, parking lots, greenery, school playgrounds, and roofs (with 72.5% of their area directly connected to the stormwater sewer system). The main canal is 1.6 km long with a diameter in the range of Ø 0.60-1.25 m. Detailed information about the analyzed 105 catchment was provided by Kiczko et al. (2018). The analysis of measurement data (2010)(2011)(2012)(2013)(2014)(2015)(2016)   slopes of 0.1-2.7 %, which gives a retention capacity of 2032 m 3 . Maningʼs roughness coefficient for the conduit is in the 115 range of 0.010÷0.018 m -1/3 ·s. The average retention depth is 2.5 mm in the impervious areas and 6.0 mm in the pervious surfaces, which gives a weighted mean of 3.81 mm for the entire catchment. Stormwater is discharged from the catchment through the S1 channel to the diversion chamber (DC), and some part is discharged directly to the stormwater treatment plant (STP) to the filling level hm=0.42 m. After exceeding the hm value, the stormwater is discharged via the stormwater overflow (OV) into channel S2, which discharges the stormwater into the Silnica River. 120 As part of the continuous monitoring carried out in 2009-2011, the volume of stormwater outflow from the catchment was measured using a flow meter installed in the S1 channel at a distance of 3.0 m from the inlet to the DW chamber. In turn, in 2015, parallel MES1 and MES2 flow meters were installed in the inlet (S1) and discharge (S2) channels to measure the flow and stormwater level. A detailed description of the stormwater catchment and installed measuring equipment is provided in Szeląg (2016). 125 The catchment (Fig.1) had previously been analyzed to determine the variability of the quantity and quality of stormwater and the operation of the sewer system based on the catchment hydrodynamic model generated in the SWMM program. The model used in the study was subjected to deterministic (Szelag et al., 2016) and probabilistic (Kiczko et al., 2018) calibration and was used as the basis for the sensitivity analysis. It was also subjected to probabilistic calibration with the GLUE+GSA method (Szelag et al., 2016). The deterministic calibration is perceived in the present study as a computational case where the 130 uncertainty and interaction of calibrated parameters in the SWMM model is omitted. The parameters were determined with the method of successive substitutions to achieve a sufficiently high degree of agreement between the modeled and measured hydrographs.

Rainfall and separation of independent rainfall events 135
The methodology described in the DWA-A 118E (2006) guidelines was adopted in the study to separate independent rainfall events. The interval between successive independent rainfall events was 4 hours. The minimum rainfall depth (3.0 mm) constituting a rainfall event was adopted as in the studies conducted by Fu et al. (2011) and Fu and Butler (2014). Independent rainfall events were distinguished based on series of rainfall (2010-2016) measured at the rainfall station located at a distance of 2 km from the Si9 collector catchment and the definition of a rainfall event specified above. The number of events in the 140 individual study years was estimated at 36 -58. The rainfall duration (tr) in the events was 20 -2366 min, and the length of the dry period was 0.16 ÷ 60 days. The rainfall depth (Ptot) in the rainfall events was in the range of 3.0 -45.2 mm. https://doi.org/10.5194/hess-2021-99 Preprint. Discussion started: 7 April 2021 c Author(s) 2021. CC BY 4.0 License.

Scheme of model analysis
In the present study, a method of model sensitivity analysis was proposed to predict the stormwater volume (maximum instantaneous flow, hydrograph volume) with the use of logistic regression (Fig. 2). The method presented here represents a 145 group of sensitivity analysis methods based on empirical models. It was assumed that the variable rainfall distribution may exert different effects on the sensitivity of the model and induce changes in the calibrated parameters. It was also assumed that the sensitivity of the model may change as a result of an increase in the maximum instantaneous stormwater flow and the volume of the outflow hydrograph. Due to the non-linearity between the modeled hydrograph parameters and the calibrated model coefficients, the use of the linear approach is limited (Chan et al., 2018); therefore, the classification model (logit) was 150 used in the study. Appropriate threshold values of hydrograph parameters constituting the basis for substitution of continuous values with classes were selected in the model.
On the one hand, this approach is based on the precipitation dynamics during rainfall events specified in the DWA-A 118E (2006) guidelines (distribution R1 -constant rainfall intensity during a rainfall event, distribution R2 -maximum rainfall intensity in the middle of the rainfall event, i.e. t/tr=0.50, distribution R3 -maximum rainfall intensity for dla t/tr=0.85-1.00, 155 and distribution R4 -maximum rainfall intensity the initial phase of rainfall). On the other hand, the modeled hydrograph parameter values were combined with the rainfall classification, which facilitated generalization of the analysis results.
Compared to the local and global analysis methods, detailed analysis of changes in the sensitivity to the effect of calibrated coefficients is possible the proposed approach, taking into account values of the modeled parameters of the catchment outflow hydrograph. This has been scarcely considered in this approach so far. The calculation algorithm presented in this study 160 consists of three elements (Fig. 2). The first one comprises a simulator of parameters of the catchment outflow hydrograph (statistical model generated with the logistic regression method), which includes rainfall characteristics and coefficients calibrated in the hydrodynamic catchment model (here: SWMM -Storm Water Management Model).  The simulator was constructed based on simulations performed with the use of calculations in the catchment model, which included the uncertainty of the identified coefficients subjected to calibration. The approach proposed here is applied in computational experiments at the stage of generation of mathematical models for urban catchments, as described by Thorndahl 170 et al. (2009). It is important that the distribution of coefficients (tab. 1) used for GLM identification should result from their https://doi.org/10.5194/hess-2021-99 Preprint. Discussion started: 7 April 2021 c Author(s) 2021. CC BY 4.0 License. actual variability. The distribution can be determined by probabilistic identification of calibrated coefficients. The GLUE methodology, in which the variability of calibrated coefficients is determined by selecting the so-called behavioral simulations, was employed in this study. Based on a posteriori distributions of calibrated coefficients in the catchment model determined by observation data, simulations of catchment outflow hydrographs were performed based on the separated rainfall events in 175 continuous rainfall time series (2010)(2011)(2012)(2013)(2014)(2015)(2016), for which typical rainfall distribution was assumed independently (R1, R2, R3, R4). This was the basis for determination of the outflow hydrograph parameters -maximum instantaneous flow (Qm) and hydrograph volume (V).
The second stage consisted in establishment of the so-called threshold values of maximum flow (Qm,g) and hydrograph volume (Vg), which served as the basis for the division into rainfall events with different intensity and their distribution in the time 180 series (ξ=R1, R2, R3, R4). Establishment of general rules for selection of threshold values may be very difficult, as they are the result of the response of the catchment to the rainfall, which is catchment-specific. These may be characteristic values of flows influenced by the presence of objects in the sewer network (stormwater overflows, etc.) at which they begin to operate.
An alternative approach is to apply rainfall classification measures (proposed by Chomicz (1951), Sumner (1988), etc.), which allow determination of the characteristic parameters of hydrographs. The rainfall classes in the Sumner scale determine the 185 extremely different hydraulic conditions prevailing in the sewer network, which may not always be used in practice for measurements and calibration. In the present study, the reference rainfall values determined in the regional classification scale proposed by Chomicz (1951) were the basis for the selection of threshold values (maximum instantaneous flow, hydrograph volume) in accordance with the following equation: where: trrainfall duration, Ptot -rainfall depth equal to its efficiency, α0rainfall efficiency coefficients taking into account the normal, heavy, and torrential rain types.
Based on the Chomicz (1951) classification of rainfall, outflow hydrographs were calculated, their parameters (Qm, V) were determined, and classification variables were defined. When the calculated values Q(Ptot, tr, ζ, θ) and V(Ptot, tr, ζ, θ) (where: ζ is a function describing the temporal intensity distribution; θ is a function taking into account the uncertainty of the calibrated 195 parameters in the catchment model) are smaller than the threshold values, they have the value of 0; otherwise, they are equal to 1.
In the third stage, logistic regression models were developed for the values of the explanatory variables (Ptot, tr, ζ, and xjvalues of calibrated coefficients in the catchment model; rainfall characteristics) and for the established dependent (zero-one) variables for the adopted threshold values (Qg,m and Vg) and temporal rainfall distribution (ζ). The subsequent stage of the 200 analyses consisted in determination of the values of the sensitivity coefficients (Sxj) in accordance with the methodology described later in this study.
Based on the calculation scheme described above, the paper presents the next stages of construction of a logit model. A catchment model generated in the SWMM program was used for this purpose. The threshold values were determined in accordance with the Chomicz (1951) classification, in which the following categories of rainfall were defined: normal rain (α0=1.00), heavy rain (α0=1.40), and torrential rain (α0=5.66), assuming a constant temporal rainfall distribution and rainfall duration tr=15 min. For these assumptions, the depth of rainfall was determined from Eq. (1), and catchment outflow hydrographs were simulated using the calibrated catchment model.

Logistic regression
The logistic regression model, also known as the binomial logit model, is usually employed for analysis of binary data and can 210 be used to determine the probability and identify the occurrence of events (Jato-Espino et al., 2018;Li and Willems, 2019;Szeląg et al., 2020). The maximum amount of stormwater outflow from the catchment and the hydrograph parameters of any rainfall event can be calculated using hydrodynamic models, e.g. SWMM. An alternative solution are statistical models (hydrograph simulators are considerably easier to implement than physical models), for instance the generalized linear model (GLM), which comprises the variability of rainfall characteristics and the uncertainty of calibrated coefficients, shown in the 215 following equation: ( ) = 0 + 1 · + 2 · + 3 · 1 + 4 · 2 + ⋯ + +2 · (2) where: α0 -intercept, α1, α2, …, αj+2empirical coefficients determined with the maximum likelihood method, Ptotrainfall depth, trrainfall duration, x1, 2, j=ncalibrated coefficients in the SWMM model, Qmlink function determining the relationship of the mean value of the dependent variable μ with the linear combination of predictors. 220 Assuming that μ=p and introducing the link function referred to as logit, it is possible to transform the modeled values of dependent variables included in Eq. (2) into a new (zero-one) system describing the probability values: This approach may prove especially useful when the results of calculations in the multiple linear regression model exhibit unsatisfactory convergence (R 2 <0.70) and it is therefore advisable to introduce classification variables, which is a simplifying 225 solution. Moreover, this approach makes it possible to emphasize and include relationships that might be omitted in the calculations of multiple linear regression, as demonstrated in many reports (Hosmer and Lemeshow, 2000;Kleinbaum and Klein, 2010;Myers et al., 2010). Since the continuous values Q(Ptot, tr, xj)m are transformed into the probability space p by the logit function in this case, it is reasonable to equate them with the determined p(Ptot, tr, xj) values for a given threshold Qg,m Qg,m for the given explanatory variables (Ptot, tr, x1, x2, x3, …,xn) equal to p=0.50 (most of the considerations in the present 235 analyses related to the value p=0.50, as this value corresponds to that of Qg,m in the probability scale p). As indicated in Fig. 3, an increase in xj,g by Δxj results in a decrease in the Qg,m value by ΔQm and yields a flow value of Qξ, 240 which facilitates determination of the numerical value of the sensitivity coefficient described by Eq. (4). In the transformed space (see Fig. 2b), the increase in the xj,g value (corresponding to p=0.50 and the threshold value Qg,m) to the value of xj,g + Δxj is accompanied by a decline in the p value by Δp to the value p * . In these analyses, the determined p * value corresponds to Qm * , which can be defined as Qg,mf(p, p * ), and the relationship can be expressed as follows: where: εempirical coefficient for conversion of the Qm * value into p * . The p * value can be related to Qm * <Qm,g; hence, the effect of changes in the xj value on the calculation results can be inferred and the sensitivity coefficient can be determined from Eq. (5). Assuming a p-value of 0.50 for the analyses was driven by the fact that the logit models determined should be universal, which is important from the point of view of being able to generalise the results obtained and apply them also to other urban catchments (Jato -Espino et al., 2018;Li and Willems 2019;Szeląg et. al., 2020). 250 The following parameters were included in the assessment of the predictive abilities of logit models: sensitivity-SENS (reflects the correctness of classification of data in a dataset p > p(Qg,m)), specificity -SPEC (reflects the correctness of classification of data in a dataset p<p(Qg,m)), and calculation error Rz 2 (reflects the correctness of classification of events at p<p(Qg,m) and p<p(Qg,m)), as described in detail by Hosmer and Lemeshow (2000) and Szeląg et al. (2020).

Analysis of the uncertainty of estimated coefficients in the logit model
The study comprised the analysis of the effect of the parametric uncertainty of the logit models on the results of calculations of probability p as propagation of the uncertainty of the model coefficients. Moreover, the values of the sensitivity coefficients of individual factors Sxi were determined. The calculation of uncertainty in the scheme presented in Fig. 1   determination of probability curves for exceeding the Qg,m value, i.e. p * =f (Ptot, tr, ζ, xn, N(μα, σα)) and sensitivity coefficients Sxi * =F(Ptot, tr, ζ, xn, N(μα, σα)) from Eq. (4) as well as the relevant percentiles. 265 On the basis of the determined logit models for the assumed cut-off thresholds Qg,m depending on the temporal rainfall distribution (ζ), probability curves described by Eq. (3) were plotted and the values of sensitivity coefficients Sxi were determined from Eq. (4) for individual explanatory variables.

GLUE (Generalized Likelihood Uncertainty Estimation)
The model uncertainty was estimated using Generalized Likelihood Uncertainty Estimation (Beven and Binley, 1992). It was 270 assumed that model uncertainty can be described by the random variability of its calibrated coefficients. The coefficients variability ranges for the SWMM of the Kielce basin were investigated in previous studies (Kiczko et al., 2018;Szelag et al., 2016). They are shown in Table 1. In the previous studies conducted by Kiczko et al. (2018) and Szeląg et al. (2016), the parameter identification was performed along with the Bayesian approach, using likelihood functions. The parameters were identified on the basis of Bayesian estimation (Beven and Binley, 1992): 275 where ( ) stands for a priori (Table 1) calibrated coefficients distribution (uniform distribution was applied in the present study), ( / ) a likelihood function used to calculate weights for the Monte Carlo sample, depending on the model fit to the observed basin flows Q and ( / ) resulting in a posteriori distribution of model coefficients . The following formula was used as the likelihood function (Romanowicz and Beven 2006): 280 where and ̂ -i-th value from the times series of observed and computed flows; is the scaling factor for the variance 2 of model residuals, used to adjust the width of the confidence intervals. In the study conducted by Kiczko et al. (2018), the value of was determined, ensuring that 95% of observed flow points are enclosed by 95% confidence intervals of the model output.
The coefficients in the ranges given in Table 1 were uniformly sampled 5000 times, and the model was evaluated for each set.
The simulation goodness-of-fit was determined as the standard deviation of computed and observed outflow hydrographs. The behavioral simulations were selected using a threshold value of deviation, i.e. simulations with poorer fit were rejected. The threshold value was determined iteratively to ensure that confidence intervals explained the model uncertainty in respect of the observation. The goal was to enclose 95% observation points within 95% confidence intervals. Confidence intervals were 290 calculated on the basis of empirical cumulative distribution functions of an ensemble of modeled hydrographs. The value of the threshold was iteratively increased to reach the above assumption. Coefficients were identified and the threshold was adjusted for two rainfall events of 24 July 2011 and 15 September 2010.
The size of the behavioral set was as 5000. It should be noted that it is assumed in the above approach that the simulations from the behavioral set are equally probable. In this study, analyses were limited to four parameters in the SWMM model. This is because the calculations performed by Szeląg et al. (2016) for the study catchment showed that the coefficients in the Horton's model as well as the Manning's roughness coefficient and the retention depth of pervious area have an insignificant 300 influence on the results of the catchment outflow hydrograph calculations. With precise spatial data about the catchment, it was shown that the uncertainty in the identification of impervious areas also has a insignificant influence on the modeled outflow hydrogram (Szeląg 2013(Szeląg , 2016

Hydrodynamic model
The SWMM 5.1 model was used to simulate the outflow from the catchment. The hydrodynamic model considered in this study consists of 92 partial catchments, 200 manholes, and 72 conduit sections. The proportion of impervious areas in the individual sub-catchments ranges from 5% to 90%, and the average slope of the area is 0.5-6%. The surface area of the partial 315 catchments varies from 0.12 ha to 2.10 ha. After calibration, the Manning roughness coefficient for the sewer channels had a value nsew=0.018 m -1/3 ·s, the roughness coefficient and retention depth for the impervious areas were nimp=0.020 m -1/3 ·s and dimp=1.65 mm, respectively, and the flow path width expressed as W=αS·A 0.50 was αS=2.00 (Kiczko et al., 2018). The analyzed catchment model was calibrated and used in the analysis of the quantity and quality of stormwater outflow from the catchment, the operation of stormwater treatment plant, and the function of the stormwater system, which was reported in 320 detail by Szeląg et al. (2016) and Kiczko et al. (2018). The sensitivity analysis and calibration of the catchment model were performed with the GLUE+GSA method as well (Szeląg et al., 2016).

Verification of generated logit models for analysis of hydrograph parameters
The suitability of the generated logit models for simulation tasks in the case of the stormwater catchment analyzed in the study was verified vs. measurement data. Since the temporal rainfall distributions in the rainfall events derived from measurements 325 varied, they were assessed and adjusted to the theoretical distributions presented in this study (see Fig. B1 -Appendix B) based on the value of the correlation coefficient (R) expressing the goodness-of-fit of empirical distributions = ( ) to the theoretical distributions (R1, R2, R3, R4).

Establishment of threshold values 330
The values of calibrated parameters shown in Table 1 served for the SWMM model calculations. Assuming rainfall intensity values corresponding to normal (Ptot,u=3.7 mm), heavy (Ptot,m=5.8 mm), and torrential (Ptot,g=21.9 mm) rain, outflow hydrographs were determined for tr=15 min; the Q(t) values were determined with at 10-s resolution. The simulations revealed the following values of maximum instantaneous flow and hydrograph volumes: Q(qu)m=0.275 m 3 /s and V(qu)=450 m 3 , Q(qs)m=0.735 m 3 /s and V(qs)=812 m 3 , and Q(qg)m=2.95 m 3 /s and V(qg)=3500 m 3 . It is worth noting that the values of the 335 catchment outflow hydrographs were identical with the rainfall intensity distributions R1, R2, R3, and R4, as demonstrated by Szeląg et al. (2016).

GLUE (Generalized Likelihood Uncertainty Estimation)
Parameters were identified using outflow time series for two rainfall events of 24 July 2011 and 15 September 2010 (Kiczko et al., 2018). The threshold value of the correlation coefficient ensuring that 95% of the observations were enclosed within 95% confidence intervals was 0.920. The size of the behavioral obtained set was 3375. The confidence intervals were verified for two rainfall events of 30 May 2010 and 30 July 2010 (see Fig. B2 -Appendix B). The percentage values of the enclosed observation points were as follows: 30 May 2010: 91% and 30 July 2010: 47% (Kiczko et al., 2018). The poorer performance for 30 July 2010 results from the bias of the model output, whereas the maximum stormwater flows were predicted correctly.

Estimation of coefficients in the logit model and assessment of goodness-of-fit 345
Based on the determined values of the dependent variables and the corresponding explanatory variables (Ptot, tr, α, dimp, nimp, nsew) for the assumed rainfall distributions (R1, R2, R3, R4), logit models were generated for calculation of the probability of exceeding the threshold values: maximum instantaneous flows (Qg,m) and outflow hydrographs (Vg). Table 2  Rz 2 ) revealed that the proposed logit models were characterized by satisfactory classification abilities. As shown in Table 2, not less than 95.79% of the cases were correctly identified at the calculated value of p<p(Qg,m; Vg) and p ≥ p(Qg,m; Vg). The model was validated on 40000 independent rainfall events, for R1, R2, R3, R4 rainfall distribution (Table   3). 360 The results of calculations of the goodness-of-fit measures of the logit models for the temporal rainfall distributions R1, R2, R3, R4 associated with the normal, heavy, and torrential rains confirm the high goodness-of-fit of the calculated and measured 365 results. This confirms the suitability of the models for further analyses.

Verification of the generated logit models vs. measurement data
The analyses showed that, in 237 of the 248 events for which the empirical and theoretical rainfall distribution exhibited high convergence (R ≥ 0.96), the calculation results from the logit models were consistent with the simulation data provided by the  Table 4 indicate agreement of the logit model-based calculation results with the measurement results. where: x1/x2number of rainfall events in a year with an exceeded x1=Qg,m/x2=Vg threshold value; calibrated values α, ninp, dimp, nsew specified in section "Hydrodynamic model" were used for verification calculations in the logit models shown in Table 4.
The calculation results confirm that the proposed logit models include the key determinants of the variability of hydrograph 390 parameters, which has been confirmed in theoretical studies and results of field studies conducted by many authors (Gironás et al., 2010;Guan et al., 2015;Thorndahl, 2009). The maximum difference between the number of rainfall events where the parameters of the catchment outflow hydrograph were identified correctly based on rainfall distribution and rainfall https://doi.org/10.5194/hess-2021-99 Preprint. Discussion started: 7 April 2021 c Author(s) 2021. CC BY 4.0 License.
characteristics by the logit model and the calibrated values of the SWMM model is 6 events, which was noted for 2015. In this case and in the other years, this is associated with problems with agreement between empirical and theoretical distributions 395 specified in DWA-A 118E (2006). This is confirmed by the local nature of the dynamics of rainfall events in some urban catchments in Europe, as reported by various authors (De Paola and Ranucci, 2012;Todeschini et al., 2012) investigating the variability of temporal rainfall distribution in a rainfall event. Hence, there is a need to construct regional rainfall models that take into account the variability of measured rainfall distribution in an event rather than that assumed for another region (Wartalska et al., 2020). However, this may be the only solution in the absence of measurement data, which has been confirmed 400 in studies on the use of typical DWA-A 118E (2006) of rainfall distributions to model the sewer network operation (Siekmann and Pinnekamp, 2011). Analysis of the data compiled in Table 3 demonstrates that, in addition to their theoretical value and the possibility to determine sensitivity (Qm, Vg), the proposed models can be used for identification of an event with a probability of exceeding the Qm,g or Vg values in the analyzed catchment.
The analyses performed in the study (Table 2) (Rabori and Ghazavi, 2018), where the local sensitivity analysis was carried out. The analysis of the values of coefficients αj in the logit models indicates that only an increase in the flow path width (α) leads to an increase in the 415 probability of exceeding Qg,m as well as Vg, which is confirmed by the analyses performed by Barco et al. (2008). An inverse correlation was found for the other parameters in the SWMM model (nimp, dimp, nsew). The results of the nsew simulations relative to Qg,m and Vg are confirmed by the calculations reported by Barco et al. (2008) and Li et al. (2014). The catchment analyzed by Li et al. (2014) was situated in China (Changsha city, area catchment of 11.7 ha). The impervious area accounted for 56% of the catchment. The increase in the nsew value reported by many authors (Barco et al., 2008;Fraga et al., 2016;Li et al., 2014) 420 indicated an opposite relationship to that observed in this study. This shows that an increase in the nsew value results in a shorter stormwater flow time and accumulation of flow from channels, which leads to a rise in the stormwater level and reduction of the instantaneous flow stream in the cross-section closing the catchment (Leandro and Martins, 2016). The calculations performed by Li et al. (2014) confirmed the Qm=f(nimp) relationship obtained in the study; however, these analyses did not include the rainfall distribution and genesis. The nimp and dimp simulation results obtained in the study are relevant in the 425 nonlinear reservoir SWMM model for simulation of the catchment outflow (Gironás et al., 2010;Rossman, 2015). An increase https://doi.org/10.5194/hess-2021-99 Preprint. Discussion started: 7 April 2021 c Author(s) 2021. CC BY 4.0 License.
in the catchment retention leads to a reduction in the amount of stormwater flowing into the sewer channels, which has an impact on the simulation results of the outflow in the cross-section closing the catchment.

Sensitivity coefficients (hydrograph volume vs. maximum instantaneous flow)
The plotted curves indicated that the smaller the volume of the calibrated catchment outflow hydrograph, the greater the 430 sensitivity of the model to changes in the calibrated coefficients identified in the catchment model ( Fig. 5a-d). As part of the present calculations, the effect of the rainfall intensity distribution (ξ) and the threshold value (Qg,m and Vg) on sensitivity coefficients Sxj was assessed. The analyses focused on the temporal R2 distribution, i.e. Euler type II, as this distribution is used for assessment of the effectiveness of the operation of sewer networks (Siekmann and Pinnekamp, 2011) and is thus highly important in engineering considerations. The analyses of the subsequent rainfall distributions (R1, R2, R3, R4) were 435 based on the maximum flow caused by normal rainfall (Q=0.3 m 3 /s), which is determined by the occurrence of stormwater overflow in the case of the above-mentioned value. The results of these analyses are presented in Fig. 4-5.
The analysis of the results of calculations of the probability of exceeding the threshold values Vg revealed that the rainfall intensity distribution did not influence the model sensitivity, which was confirmed by simulation experiments in the analyzed urban catchment (Szelag et al., 2016). The plotted curves (Fig. 5) indicated that the calibrated volume in the domain of the Vg 440 value exhibits the greatest sensitivity (deterministic solution) to changes in dimp and α. This relationship was confirmed by Skotnicki and Sowiński (2015), who simulated outflows from a 6.7 km 2 catchment in Poznań and employed local sensitivity analysis. Similar results were also obtained by Rabori and Ghazavi (2018) in their analyses of a catchment outflow in Iran.
These correlations were also confirmed by the calculations reported by Mrowiec (2009), who modeled hydrographs in the urban catchment in Częstochowa (120 ha). The present analysis results were also are confirmed by Ballinas-Gonzáles et al. 445 (2020), who demonstrated a major impact of the characteristics of impervious areas on the variability of the catchment outflow hydrograph. Different sensitivity analysis results were reported by Li et al. (2014), who demonstrated a crucial effect of nsew on the outflow hydrograph volume. Among the explanatory variables considered in this study (for any p in Eq. (4)), nimp was found to exert the lowest effect on the probability of exceeding Vg at any p value. The course of the curves and their variability   In terms of the selection of hydrographs for calibration followed by validation (SWMM model), the present results have an 460 engineering aspect. This is associated with the fact that different relations V(Q)=f(xi) can be obtained by validation of the model coefficients at the calibration stage, which is crucial for minimization of the difference between measurement and simulation values.

Sensitivity coefficient (maximum instantaneous flow vs. rainfall distribution)
Based on the plotted curves (probabilistic solution), it can be concluded that, when the Qm value is calibrated in the region of 465 Qg,m=0.30 m 3 ·s -1 (uniform rainfall distribution R1, normal rain), the model shows the greatest sensitivity (percentile 0.50) to changes in nimp (deterministic solution), as confirmed by the value Snimp= -2.47 (Fig. 4g) Fig. 4h), flow path width (Sα=1.25; Fig. 4e), and retention depth of impervious areas (Sdimp= -1.03; Fig. 4f) have a lower impact. The plotted curves and the deterministic solution indicate that the absolute Snimp and Snsew values for the R2 and R3 temporal rainfall distributions (deterministic solution) are lower than for the R1 distribution 470 ( Fig. 4e-f). In turn, in the case of Sα (Fig. 4e) and Sdimp (Fig. 4f), it was found that the absolute values of the sensitivity coefficients calculated for the R1 distribution have lower values than for R2 and R3. When the model is calibrated based on hydrographs reflecting the reaction of the analyzed catchment to normal rain (constant temporal rainfall distribution in an event -R1), the greatest effect on the Qm in the Qg,m domain is exerted by nimp and the lowest impact is shown by dimp (in terms of absolute values); this is indicated by the curves in Fig. 4f-g. In turn, a different relationship, i.e. the greatest effect of dimp, α 475 and the lowest effect of nimp, was found for the R2 distribution ( Fig. 4e-f). These relationships indicate a significant effect of temporal rainfall intensity distributions on the model sensitivity to changes in the coefficients calibrated in the domain of Qg,m values.
The results of the present analyses may be highly important in engineering practice, as they confirm that, with the Qm values assumed as the basis for calibration, the hydrograph should be selected in a way facilitating identification of the coefficients 480 (α, dimp, nimp, nsew) and validation, so that the values will be a result of rainfalls with similar intensity dynamics. Therefore, it should be underlined that, in the hydrograph intended for identification of model coefficients and validation, the relationship between the dependent variables and the calibrated coefficients must have a similar form.
the increase in rainfall intensity, Jato-Espino et al. (2018) reported a decline in the sensitivity of the model to the values of 500 selected catchment characteristics; this is equivalent to a decrease in the sensitivity of the model to the calibrated parameters.

Sensitivity coefficients (uncertainty of estimated coefficients in the logit model)
The calculations showed that the uncertainty of parameter estimation in logit models exerts a strong effect on the values of the sensitivity coefficients calculated for the analyzed cases. This is confirmed by the determined range of variability of the sensitivity coefficient values (Sα, Snimp, Sdimp, Snsew) depending on the size of the respective percentiles ( Fig. 4 -5). In most of 505 the calculation variants (with the exception of α; Fig. 4a, 4e, 5a), the difference between the determined values of the sensitivity coefficients (for the different temporal rainfall distributions R1, R2, R3 and rainfall genesis -normal, heavy, and torrential rains) was shown to decrease with the increase in the percentile values.
Different relationships were observed in the analysis of the variability of Sxi values shown in Fig. 5a. In this case, for percentiles below 0.36, the highest and the lowest Sα values were obtained for V(Qm,g=2.50 m 3 /s) and V(Qm,g=0.30 m 3 /s), respectively. 510 The analysis of the effect of rainfall distribution (R1, R2, R3) on the model sensitivity (calibrated Qm value) revealed an increase in the difference in the sensitivity coefficient Sα values with the increase in the percentiles. As shown by the analysis of the values of sensitivity coefficients Sα and Sdimp (Fig. 4a, 4b), the relationship Sα(dimp)(Qm=0.75 m 3 /s) > Sα(dimp)(Qm=2.50 m 3 /s) was obtained for percentile values above 0.42, whereas an inverse relationship was found for lower percentile values.

Summary and conclusions 515
Modeling of outflows and calibration of hydrodynamic models with design of tools supporting this task represent a relevant current research topic. It is necessary to search for methods that will yield reliable results reflecting the reality as well as possible on the one hand. On the other hand, with their acceptable time-and cost-efficiency in retrieval and analysis of data, the methods should have the potential to be used in practice by a wide group of engineers. This study has shown that the logistic regression model can be used for analyses of the sensitivity of the maximum flow in a hydrograph and hydrograph 520 volume in a rainfall event. The hydrograph parameters depended on the temporal rainfall intensity distribution in the rainfall event and parameters identified in the SWMM model. In addition to their scientific aspects, the proposed logit models may be a useful tool for forecasting the variability of the parameters of catchment outflow hydrographs, which confirms the usefulness of the developed tool.
The sensitivity coefficient proposed in the study facilitates determination of the impact of selected parameters of the SWMM 525 model on the outflow hydrograph parameters with consideration of rainfall genesis and variability of temporal rainfall distribution in a rainfall event. Furthermore, it has been demonstrated that the rainfall genesis and the temporal variability of rainfall intensity in a rainfall event should be included in the selection of hydrographs for calibration and validation of the model. It was found that the higher the rainfall intensity determining the modeled outflow hydrograph, the lower the sensitivity of the identified SWMM model parameters to the maximum outflow and hydrograph volume. The calculations have indicated that the uncertainty of the coefficients identified in the logit model has a significant impact on the determined sensitivity coefficients. The aspects discussed above are highly important for the procedure of hydrodynamic model calibration, which ultimately has a significant effect on the accuracy of identified model parameters.
Given the usefulness of the presented calculation results, further investigations are recommended to verify the logit models and relationships presented in this study. There is also a need for analyses of other urban catchments with different physical 535 and geographical characteristics, which may contribute to development of a universal model.