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 for hydrograph modeling. Since a large number of their parameters have to be identified, there may be problems at the calibration stage. Hence, sensitivity analysis is used to limit the number of parameters. However, the current sensitivity analytical methods ignore the effect of the temporal distribution and intensity of precipitation in a rainfall event on the catchment outflow hydrograph. This article presents a methodology of constructing a simulator of catchment outflow hydrograph parameters (volume and maximum flow). For this purpose, uncertainty analytical results obtained with the use of the GLUE (generalized likelihood uncertainty estimation) method were used. A novel analysis of the sensitivity of the hydrodynamic catchment models was also developed, which can be used in the analysis of the operation of stormwater networks and underground infrastructure facilities. Using the logistic regression method, an innovative sensitivity coefficient was proposed to study the impact of the variability of the parameters of the hydrodynamic model depending on the distribution of rainfall, the origin of rainfall (on the Chomicz scale), and the uncertainty of the estimated simulator coefficients on the parameters of the outflow hydrograph. The developed model enables the analysis of the impact of the identified SWMM (Storm Water Management Model) parameters on the runoff hydrograph, taking into account local rainfall conditions, which have not been analyzed thus far. Compared with the currently developed methods, the analyses included the impact of the uncertainty of the identified coefficients in the logistic regression model on the results of the sensitivity coefficient calculation. This aspect has not been taken into account in the sensitivity analytical methods thus far, although this approach evaluates the reliability of the simulation results. The results indicated a considerable influence of rainfall distribution and intensity on the sensitivity factors. The greater the intensity and rainfall were, the lower the impact of the identified hydrodynamic model parameters on the hydrograph parameters. Additionally, the calculations confirmed the significant impact of the uncertainty of the estimated coefficient in the simulator on the sensitivity coefficients. In the context of the sensitivity analysis, the obtained results have a significant effect on the interpretation of the relationships obtained. The approach presented in this study can be widely applied at the model calibration stage and for appropriate selection of hydrographs for identification and validation of model parameters. The results of the calculations obtained in this study indicate the suitability of including the origin of rainfall in the sensitivity analysis and calibration of hydrodynamic models, which results from the different sensitivities of models for normal, heavy, and torrential rain types. In this context, it is necessary to first divide the rainfall data by origin, for which analyses will be performed, including sensitivity analysis and calibration. Considering the obtained results of the calculations, at the stage of identifying the parameters of hydrodynamic models and their validation, precipitation conditions should be included because, for the precipitation caused by heavy rainfall, the values of the sensitivity coefficients were much lower than for torrential ones. Taking into account the values of the sensitivity coefficients obtained, the calibration of the models should not only cover episodes with high rainfall intensity, since this may lead to calculation errors at the stage of applying the model in practice (assessment of the stormwater system operating conditions, design of reservoirs and flow control devices, green infrastructure, etc.).

Climate change and progressive urbanization result in an increase in the volume of stormwater outflow from catchments, which leads to flooding and deterioration of water quality in receivers (Crocetti et al., 2020; Fletcher et al., 2013). To reduce the incidence of these phenomena, runoff model generation is needed. This can be achieved using hydrodynamic models based on physical equations representing stormwater outflows. One of the common tools is the SWMM (Storm Water Management Model) program (Buahin and Horsburgh, 2015; Crocetti et al., 2020; Gironás et al., 2010). SWMM allows for simulation of sewage quantity (Guan et al., 2015), quality (Dotto et al., 2012), including objects located in sewer networks (separation and combined sewer networks. The program allows for simulation of surface runoff from a drainage basin including the flow in a network of pipes and analysis of interaction between hydraulic conditions in the system and sewage flooding on the ground (Fraga et al., 2016). The program's advantages also include the possibility to model green infrastructure facilities (McGarity et al., 2013). The source code of the program is available to users, which gives the possibility of its modification and adaptation to individual requirements. Due to the interactions between parameters identified in the models, they may be difficult to calibrate, and the results may be biased. Therefore, statistical models are used for the simulation of runoff, which has been shown in a number of studies (Gernaey et al., 2011; Yang and Chui, 2020). A serious drawback of many models (the so-called black-box techniques) is their inability to interpret structural parameters (Zoppou, 2001). Linear models, including multiple linear regression (MLR), as well as nonlinear models such as artificial neural networks (ANNs) and classification and regression trees (CRTs) with their modifications (Yang and Chui, 2020), are used for this purpose. Nonlinear models enable a more accurate description of hydrological processes in urban catchments, which results from the physics of the analyzed phenomena and is confirmed in the literature (Zoppou, 2001).

The hydrodynamic model must be calibrated to reflect the conditions prevailing in the real system. Calibration of the catchment model is a complex task aimed at determining the optimal values of parameters with a satisfactory goodness of fit of calculation outcomes and measurement results (Bárdossy, 2007; Dotto et al., 2012; Guan et al., 2015). Parameter values are determined for an appropriate form of the objective function in which one or more criteria (maximum instantaneous flow, hydrograph volume, or mean relative or absolute error of flow prediction) can be included. Since the description of the stormwater outflow from the catchment is complicated, modeling the phenomenon requires knowledge of many parameters (physical and geographic characteristics of the catchment and sewer network). A number of these parameters can be determined using detailed spatial data (GIS, geographic information system), as has been indicated in numerous studies (Fraga et al., 2016; Leandro and Martins, 2016). This helps to reduce the number of variables included in the calibration. However, since a large number of parameters must be included in the models, there may be problems with the identification of their values. Therefore, the aim is usually to simplify the calibration process by eliminating factors that have a negligible impact on simulation results. Hence, model sensitivity analysis is employed. To understand the modeled processes in urban catchments and to determine the influence of interactions between the identified parameters on the simulation results, an uncertainty analysis (GLUE – generalized likelihood uncertainty estimation) is performed. This method is widely used in the analysis of the quantity and quality of stormwater for models of urban and agricultural catchments (Dotto et al., 2012; Mirzaei et al., 2015), retention reservoirs (Kiczko et al., 2018), stormwater flooding (Fraga et al., 2016), etc., which is reflected in a large number of publications in this field. In this approach, the empirical distributions of parameters identified in hydrodynamic models are determined (catchment retention, roughness coefficients of pervious and impervious areas, roughness of channels, etc.) and a confidence interval is determined (e.g., 95 %), containing the data obtained from the measurement results.

As shown by the literature (Chisari et al., 2018; Tolley et al., 2019; Xu et al., 2019), the analysis is often applied at the stage of calibration of the mathematical models. In practice, local and global sensitivity analytical methods, which can be implemented for statistical and physical relationships, are used (Link et al., 2018; Morio, 2011; Cristiano et al., 2019). In the case of the local sensitivity analysis, the calculations consist of determination of the derivative value at a given point, which is the basis for assessment of the effect of the variance of the variables on the modeled value (Razavi and Gupta, 2015). One of the drawbacks of the local sensitivity analysis is the fact that the variability of the analyzed phenomenon and the effect of variables are considered in the narrow domain of the modeled variable (Pianosi et al., 2016). This approach ignores the fact that the sensitivity of the model in the domain of the output values may change, which may be important for calibration of the model at the validation stage and its course. In the case of nonlinear models, the local sensitivity analysis does not take into account the character of the relationships between the explanatory variables and dependent variable. Then, the sensitivity coefficient is calculated only for the mean level of the explanatory variable. Nevertheless, this method is widely used in the analysis of the sensitivity of models describing runoff in urban catchments, which has been confirmed by numerous studies (Ballinas-González et al., 2020; Liu et al., 2020; Yang et al., 2019). Another shortcoming of sensitivity analysis based on partial derivatives is the fact that the effects of individual variables on the output variable are estimated while the other variables are kept constant. This is rarely observed in the case of complex relationships, as the explanatory variables are then correlated to some extent. The ceteris paribus analysis does not take this fact into account. Consequently, the effects of individual variables may be overestimated.

In the context of literature studies (Xu et al., 2019), the results of LSAs (local sensitivity analyses) may lead to simplifications in the interpretation of hydrological processes in catchments. From the point of view of the appropriate selection of the identified parameters of urban catchment models, the local sensitivity analytical method has limited application and may lead to problems with calibration (Morio, 2011). Global sensitivity analysis does not have many of the aforementioned disadvantages. One of the simplest methods used in many cases is based on multiple linear regression (Ashley and Parmeter, 2020; Touil et al., 2016). However, the results of the sensitivity analysis can be considered reliable when the coefficient of determination reflecting the relationship between the dependent variable and explanatory variables is not lower than 0.70. When this requirement is not met, other methods for global sensitivity analysis should be applied (Saltelli et al., 2007). Variance methods, which facilitate estimation of the contribution of the individual parameters to the model output variance using the Monte Carlo method, are more precise and more computationally complex. The global sensitivity analysis (GSA) method is one of the commonly used approaches. It has been subjected to modifications, as described in Iooss and Lemaître (2015). Variance methods are currently gaining increasing interest, which is confirmed by the number of publications in this field. However, since implementation is complicated, simplified methods are used in many cases despite the major advantage of variance approaches over local analytical methods. The implementation of global sensitivity analytical methods is not a simple task, as it requires complex mathematical tools, which limits their application. Despite the limitations of the local sensitivity analytical method and the complex implementation of the global sensitivity analysis, in both cases, the aspects related to local precipitation conditions are treated to a limited extent. Recent studies of urban catchments indicate that the temporal and spatial distributions of rainfall are very important factors that strongly influence the catchment response (Schilling, 1991; Berne et al., 2004; Ochoa-Rodriguez et al., 2015; Cristiano et al., 2017). However, a number of issues have not been fully clarified. In the currently used methods, the influence of rainfall origin on the results of the sensitivity analysis is neglected. It is not clear how the sensitivity of the model (maximum flow rate and hydrograph volume) changes for rainfall events resulting from high (convective) or low intensity (convergence zone) rainfall. The LSA and GSA methods ignore the influence of rainfall temporal distribution on the sensitivity coefficients, which is contrary to the information from the literature (Schilling et al., 1991) describing the analyses conducted for different urban catchments. It is important to select outflow hydrographs from the catchment area for the identification of parameters and their validation in the context of rainfall parameters (rainfall origin, rainfall intensity, and temporal distribution). It is also of great methodological importance in the context of modifying the currently used methods of sensitivity analysis of hydrodynamic catchment models. In the sensitivity analytical methods based on statistical models, the influence of the uncertainty of the estimated coefficients on the sensitivity coefficients is neglected. From the point of view of the reliability of the obtained results, this is important when deciding on the selection of the method of parameter identification in hydrodynamic models (GIS, maps, etc.) to reduce the uncertainty of the simulation results.

Given the information specified above, this paper presents an original application of the logistic regression method for sensitivity analysis. This is one of the first studies to analyze the sensitivity of the model in terms of the temporal variability of rainfall. The advantage of the model is the fact that it has the form of a statistical relationship; hence, without the need for complex analyses, it can be used to determine the effects of parameters included in the calibration of the catchment model, precipitation characteristics, and absolute values of the modeled dependent variables on the parameters of outflow hydrographs (maximum instantaneous flow and hydrograph volume). The approach proposed in the present study also facilitates analysis of the sensitivity of selected explanatory variables, depending on the numerical values of the modeled hydrograph parameters of catchment runoff. At the stage of sensitivity analysis, the effect of the uncertainty of coefficients estimated in the statistical model (logistic regression) on the calculated results is included, which is reflected in the determined sensitivity coefficients. Since the model is constructed based on simulation results provided by the Monte Carlo method, which is typical for global sensitivity analytical methods, this approach can complement and extend the results of GSA calculations. In summary, (Saltelli et al., 2007; Razavi and Gupta, 2015), the sensitivity analysis used in the present study represents a fusion of local and global sensitivity analysis through a combination of logistic regression in phenomenon modeling with partial derivatives. Since logistic regression is not an example of a black-box method, as it has an explicit form of dependence between the modeled probability of success and explanatory variables, the use of partial derivatives for assessment of the sensitivity of the model to individual parameters seems reasonable. Especially in the case of an implicit, complex, and nonlinear dependence, it is recommended that variance-based techniques such as the Sobol method be employed. Partial derivatives used in the logistic regression model increase the flexibility 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 analytical 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.

Due to the extensive nature of the conducted analyses, the article has been divided into several sections, including characteristics of the research object and methodology, which presents an innovative algorithm for the development of a logistic regression model and subsequent calculation steps, i.e., determination of a hydrodynamic model of a catchment, identification of the threshold values of the outflow hydrograph parameters from a catchment using a hydrodynamic model, uncertainty analysis using the GLUE method, development of a logit model and its verification, analysis of the influence of rainfall origin and temporal distribution of rainfall on the calculated sensitivity coefficients, and assessment of the impact of uncertainty of the identified coefficients in the logit model on the values of the sensitivity coefficients.

Scheme of the hydrodynamic model of the catchment generated in the SWMM program.

The analysis in this study was carried out in a catchment with a total area of 62

The analyzed sewer system consists of 200 manholes and 100 conduit sections with

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

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

The developed methodology of the sensitivity analysis of hydrodynamic models included several independent stages: preparation of data for the construction of the model and its implementation, conducting the uncertainty analysis using the GLUE method, development of a logit model for specific threshold values of the hydrograph parameters and model verification, and calculation of sensitivity coefficients, taking into account the rainfall origin, the temporal distribution of rainfall, and the evaluation of the impact of the uncertainty of the identified coefficients in the logit model on the results of the sensitivity analysis.

The methodology described in the DWA-A 118E (2006) guidelines was applied in the study to separate independent rainfall events. The interval between
successive independent rainfall events was 4 h (Dunkerley, 2008; Joo et al., 2014; Szeląg et al., 2021). The minimum rainfall depth (3.0

Calculation algorithm scheme in a logit model.

In the present study, a method of model sensitivity analysis was proposed to predict the stormwater volume (maximum instantaneous flow and hydrograph volume) with the use of logistic regression (Fig. 2). The method presented here represents a group of sensitivity analytical 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 nonlinearity between the modeled hydrograph parameters and the calibrated model coefficients, use of the linear approach is limited (Chan et al., 2018); therefore, the classification model (logit) was 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.,

Ranges of SWMM coefficients (Kiczko et al., 2018).

On the other hand, the modeled hydrograph parameter values were combined with the rainfall classification, which facilitated generalization of the
analytical results. Compared with the local and global analytical methods, detailed analysis of changes in the sensitivity to the effect of calibrated
coefficients was possible with the proposed approach, taking into account values of the modeled parameters of the catchment outflow hydrograph. This
has been scarcely considered in this approach thus far. The calculation algorithm presented in this study consists of three elements (Fig. 2). The
first 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. 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 et al. (2009). It is important that the distribution of coefficients (Table 1) used for GLM (generalized
likelihood model) identification should result from their 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 continuous rainfall time
series (2010–2016), for which typical temporal rainfall distribution was assumed independently (R1, R2, R3, and R4). This was the basis for
determination of the outflow hydrograph parameters – maximum instantaneous flow (

The second stage consisted of establishing the so-called threshold values of maximum flow (

In the present study, the reference rainfall values determined at the regional classification scale proposed by Chomicz (1951) were the basis for the
selection of threshold values (maximum instantaneous flow and hydrograph volume) in accordance with the following equation:

Based on the Chomicz (1951) classification of rainfall, outflow hydrographs were calculated, their parameters (

In the third stage, logistic regression models were developed for the values of the explanatory variables (

Determination of the hydrodynamic model of the catchment;

Identification of a posteriori distributions of the calibrated parameters in the model of the catchment;

Monte Carlo sampling of the identified parameters of the SWMM and calculation of the parameters of the outflow hydrograph from the catchment for the separated rainfall events (described by the temporal distribution of rainfall and duration of rainfall, as well as rainfall depth);

Identification of the threshold values of the outflow hydrograph parameters from the catchment, taking into account the origin of rainfall (on the Chomicz scale);

Determination of logit models and their validation using a hydrodynamic model;

Calculation of sensitivity coefficients, taking into account the origins of rainfall and the temporal distribution of rainfall;

Determination of the influence of uncertainty of estimated coefficients in logistic regression models on the sensitivity coefficients.

Based on the calculation scheme described above, this 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 (

The logistic regression model, also known as the binomial logit model, is usually employed for the analysis of binary data and can 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 is statistical models (hydrograph simulators are considerably easier to implement than physical models); for instance,
the generalized linear model (GLM – generalized likelihood model), which comprises the variability of rainfall characteristics and the uncertainty of
calibrated coefficients, is shown in the following equation:

Assuming that

This approach may prove especially useful when the results of calculations in the multiple linear regression model exhibit unsatisfactory convergence
(

Calculation diagram showing the effect of changes in the

As indicated in Fig. 3, an increase in

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

In the deterministic solution, the values of the sensitivity coefficients (

The study comprised the analysis of the effect of the parametric uncertainty of the logit models on the results of calculations of probability

Determination of mean coefficient values (

T-fold sampling of the

Determination of probability curves for exceeding the

On the basis of the determined logit models for the assumed cutoff thresholds

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 subcatchments ranges from 5 % to 90 %,
and the average slope of the area is 0.5 %–6 %. The surface area of the partial catchments varies from 0.12 to 2.10

The model uncertainty was estimated using generalized likelihood uncertainty estimation (Beven and Binley, 1992). It was assumed that model
uncertainty can be described by the random variability of its calibrated coefficients. The coefficient 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), 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):

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 a poorer fit were rejected. The threshold value was determined iteratively to ensure that
confidence intervals explained the model uncertainty with respect to the observation. The goal was to enclose 95 % observation points within
95 % confidence intervals. Confidence intervals were 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 on 24 July 2011 and 15 September 2010. The size of the behavioral set was 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. Computer simulations (Szeląg et al., 2016) using the presented catchment model (SWMM) integrated with the MATLAB
algorithms, in which the GLUE

With precise spatial data about the catchment, it was shown that the uncertainty in the identification of impervious areas also had an insignificant
influence on the modeled outflow hydrograph (Szeląg, 2014, 2016). Based on the continuous rainfall series from the 2010–2016 period and the determined a posteriori distributions of calibrated
coefficients in the SWMM, simulations of the combinations of numerical values [

The suitability of the generated logit models for simulation tasks in the case of the stormwater catchment analyzed in this study was verified
vs. measurement data. Since the temporal rainfall distributions in the rainfall events derived from measurements 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 (

Following the developed computational algorithm, the “Results and discussion” section includes the following steps: determining the threshold values of the outflow hydrograph parameters using the hydrodynamic model of the catchment, as well as uncertainty analysis; developing the logistic regression model and its verification; sensitivity analysis, in which the influence of rainfall origin and temporal rainfall distribution on the parameters of the hydrograph is analyzed (volume and maximum flow rate); and analysis of the impact of uncertainty of the estimated coefficients in the logit model on the determined sensitivity coefficients.

Independent rainfall events were distinguished based on a series of rainfall events (2010–2016) measured at the rainfall station located at a
distance of 2

The values of calibrated parameters shown in Table 1 served for the SWMM calculations. Assuming rainfall intensity values corresponding to normal
(

Parameters were identified using outflow time series for two rainfall events on 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 on 30 May 2010 and 30 July 2010 (see Fig. B2 – Appendix B). The percentage values of the enclosed observation points were as follows: 91 % for 30 May 2010 and 47 % for 30 July 2010 (Kiczko et al., 2018).

Calculated coefficients (

Based on the determined values of the dependent variables and the corresponding explanatory variables (

Results of validation of logit models shown in Table 2.

As shown in Table 2, no less than 95.79 % of the cases were correctly identified at the calculated values of

The results of calculations of the goodness of fit measures of the logit models for the temporal rainfall distributions R1, R2, R3, and R4 associated with the normal, heavy, and torrential rains confirm the high goodness of fit of the calculated and measured results. This confirms the suitability of the models for further analyses.

Comparison of measurement and calculated results in the analyzed period.

The analyses showed that in 237 of the 248 events for which the empirical and theoretical rainfall distribution exhibited high convergence
(

The table shows the agreement of the calculated results for the hydrograph parameters obtained via simulation with the SWMM and logistic regression with regard to the classification of maximum flows and hydrograph volumes. The data presented in Table 4 indicate agreement of the logit model-based calculated results with the measurement results.

In Table 4, the format

The calculated results confirm that the proposed logit models include the key determinants of the variability of hydrograph parameters, which has been
confirmed in theoretical studies and the 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 characteristics by the logit model and the calibrated values of the SWMM was six events, which was noted for
2015. In this case, and in the other years, this was associated with problems with agreement between empirical and theoretical distributions specified
in DWA-A 118E (2006). This was 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 in studies on the use of typical DWA-A 118E (2006) rainfall distributions to model 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 of
determining sensitivity (

The analyses performed in this study (Table 2) indicate a strong effect of the flow path width (

The plotted curves indicated that the smaller the volume of the calibrated catchment outflow hydrograph, the greater the 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 (

The analysis of the results of calculations of the probability of exceeding the threshold values

Comparison of the calculated results (deterministic and probabilistic solutions) of sensitivity coefficients (

Comparison of the calculated results (deterministic and probabilistic solutions) of sensitivity coefficients (

The curves in Fig. 4e–h show that apart from the rainfall origin (average rainfall intensity as a result of normal, heavy, and torrential rainfall), the temporal distribution of rainfall has an impact on the values of the determined sensitivity coefficients. This result is the effect of the fact that the temporal distribution of rainfall and the intensity of rainfall have a significant impact on the values of the modeled maximum flow rates, which was confirmed by the analysis by Schilling (1991). The obtained curves (Fig. 5) prove that the volume of the outflow hydrograph depends on the origin of rainfall and hence the variability of the determined values of the sensitivity coefficients for normal, heavy, and torrential rainfall.

In turn, the impact of rainfall distribution was not found, which was confirmed by studies in the field of modeling the volume of runoff from urban catchments (Grum and Aalderink, 1999), including computer simulations by Szeląg et al. (2014) for the urban catchment under consideration.

Based on the plotted curves (probabilistic solution), it can be concluded that when the

The results of the present analyses may be highly important in engineering practice, as they confirm that, with the

The plotted curves (probabilistic solution) with the deterministic solutions showed that the greater the rainfall intensity (rising

The relationships presented in this study have been scarcely analyzed by other researchers (Barco et al., 2008; Krebs et al., 2014; Li et al., 2014) in terms of catchment outflow modeling. These relationships, which confirm the significant effect of rainfall intensity distribution on hydraulic phenomena occurring in a sewer network, were described by Jato-Espino et al. (2018) in their study of stormwater overflow. The authors showed a statistically significant effect of the rainfall intensity distribution on the relationship between stormwater overflow onto the land surface and catchment characteristics. A certain analogy with the calculated results described in the present study may be suggested. This is related to the fact that, along with the increase in rainfall intensity, Jato-Espino et al. (2018) reported a decline in the sensitivity of the model to the values of selected catchment characteristics; this is equivalent to a decrease in the sensitivity of the model to the calibrated parameters.

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
(

Different relationships were observed in the analysis of the variability of

Modeling of outflows and calibration of hydrodynamic models with the 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 reality, as well as what is possible, on the one hand. On the other hand, with their acceptable time and cost efficiency in the retrieval and analysis of data, the methods should have the potential to be used in practice by a wide group of engineers. The currently used methods of analyzing the sensitivity of hydrodynamic models neglect the origin of rainfall and the temporal distribution of rainfall. Moreover, in the methods based on statistical models, the influence of the uncertainty of the estimated coefficients in the logit model on the values of the calculated sensitivity coefficients is not taken into account. Neglecting the abovementioned conditions may result in problems with the calibration of models and simplification in the interpretation of the physics of hydrological processes in catchments, which makes them difficult to understand. This study showed that the logistic regression model can be used for analyses of the sensitivity of the maximum flow in a hydrograph and hydrograph 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. 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 analyses performed in this work showed that the origin of rainfall and the temporal distribution of rainfall in the event have a large impact on the sensitivity of the model. However, this aspect has been neglected until now in sensitivity analytical methods. The results of the calculations showed that the lowest values of the sensitivity coefficients were obtained for the outflow hydrographs resulting from heavy rainfall, while the highest values of the sensitivity coefficients were obtained for normal rain. In the context of the currently used methods of sensitivity analysis and calibration, it seems advisable to modify them by introducing an additional calculation step consisting of the classification of the measured rainfall data in terms of the origin of rainfall (accounting for average rainfall intensity) and the temporal distribution of rainfall. For this purpose, it is possible to use unsupervised machine learning methods (hierarchical cluster analysis, Kohonen neural networks, etc.). In the context of the obtained calculated results, it is advisable to select the rainfall–runoff events for calibration and validation in such a way that the determined sensitivity coefficients do not show significant variability. It is important for the appropriate selection of the values of calibrated parameters and their potential correction at the stage of model validation. The sensitivity coefficient proposed in this study facilitates the determination of the impact of selected parameters of the SWMM on the outflow hydrograph parameters with consideration of rainfall genesis and variability of temporal rainfall distribution in a rainfall event. Furthermore, it was demonstrated that 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 parameters to the maximum outflow and hydrograph volume. The calculations 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 the identified model parameters.

The computational methodology proposed in this paper is universal in nature and can be applied to any urban catchment area. The simulation results presented in this paper refer to a single catchment area. Therefore, further analyses are required to verify the developed model for catchments with different physical and geographic characteristics. Thus, it is advisable to determine the applicability range of the developed computational model. Considering the usefulness of the obtained dependencies, as well as the large influence of rainfall origin and rainfall temporal distribution on the sensitivity coefficients, further studies are needed. The purpose of these analyses should be to expand the developed methodology of the sensitivity analysis aimed at additionally taking into account the shape and area of the catchment, land use, the path of the stormwater network, and the retention of the network. The analysis of the effect of the temporal distribution of rainfall, together with the spatial distribution, seems to be a particularly interesting issue, especially because both distributions strongly depend on rainfall genesis. However, the design of an appropriate experiment seems challenging.

Dimensionless rainfall curves,

Comparison of measurement results of hydrographs of outflow from the catchment area with GLUE calculations.

Calculated likelihood function – scatter plots of

The authors announce that there is no problem sharing the used model and codes upon request to the corresponding author.

The authors confirm that data supporting the findings of this study are available from the corresponding author upon request.

Conceptualization of work was completed by BS. The methodology was prepared by BS, FF, AK, and DM. Investigation and formal analysis was conducted by BS, FF AK, DM, GŁ, and MM. The original draft paper was prepared by BS, AK, DM, and GŁ. The review and editing was performed by FF, BS, DM, GŁ, and JD. The supervision of work was conducted by FF, BS, and GŁ.

The authors declare that they have no conflict of interest.

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This article was prepared within the framework of project “Miniature 3” (2019/03/X/ST8/01452) entitled “Simulation of the impact of climate change and land use dynamics using statistical models on the performance of storm overflows for small urban catchments in the short and long term” funded by NCN (National Science Center).

This paper was edited by Roberto Greco and reviewed by two anonymous referees.