Analyses of extreme precipitation and runoff events including uncertainties and reliability in design and management of urban water infrastructure

Journal of Hydrology 544 (2017) 290–305

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Research papers

Analyses of extreme precipitation and runoff events including uncertainties and reliability in design and management of urban water infrastructure Teklu T. Hailegeorgis ⇑, Knut Alfredsen Department of Hydraulic and Environmental Engineering, Faculty of Engineering Science and Technology, NTNU, Norwegian University of Science and Technology, S.P. Andersens vei 5, N-7491 Trondheim, Norway

a r t i c l e

i n f o

Article history: Received 4 April 2016 Received in revised form 18 August 2016 Accepted 18 November 2016 Available online 23 November 2016 This manuscript was handled by Andras Bardossy, Editor-in-Chief, with the assistance of Vazken Andréassian, Associate Editor Keywords: Extreme precipitation-runoff events IDF curves Design flood Stormwater pipes Uncertainty Reliability

a b s t r a c t There is a need for assessment of uncertainties and hence effects on reliability of design and management of stormwater pipes due to the prevalence of urban floods trigged by modification of land cover and high precipitation intensities respectively due to increasing urbanization and changing climate. Observed annual maximum series (AMS) of extreme precipitation intensities of 17 durations (1-min to 1440-min) and runoff records of 27 years from a 21.255 ha (23% impervious, 35% built-up and 41% open areas) Risvollan catchment in Trondheim City were used. Using a balanced bootstrap resampling (BBRS) with frequency analysis, we quantified considerable uncertainty in precipitation and runoff quantiles due to the sampling variability of systematic observations (e.g.,
43% to +49% relative differences from the quantile estimates for the original sample). These differences are higher than suggested increase in design rainfall and floods by many countries for climate change adjustment. The uncertainties in IDF curves and derived design storm hyetographs are found to have large effects on the reliability of sizing of stormwater pipes. The study also indicated low validity of the assumptions on extreme precipitation and runoff relationships in the return period-based method for the partially paved urban catchment: (i) maximum of only 46% of the AMS of extreme precipitation and runoff events occurred concurrently and (ii) T-year return period extreme precipitation events do not necessarily result in T-year flood events. These indicate that there are effects of snowmelt seasonality, and probably catchment moisture states and interactions between the flows in subsurface media and pipes. The results substantiate the need for better understanding of relationships between precipitation and runoff extremes and urban runoff generation process, and importance of uncertainty assessment and application of reliability-based methods for design and management of water infrastructure. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction Prevalence of urban flooding triggered by modification of the built and natural environment that would alter the runoff response of urban catchments and the impacts of climate change is becoming one of the major challenges in cities. Pluvial flooding already challenges the capacity of drainage and sewerage system in urban areas in Scandinavia (Torgersen et al., 2014). In addition, flooding of drainage infrastructure for transportation facilities including culverts and bridges entail huge economic losses and impairs life quality both in urban and rural catchments. Due to the high degree of vulnerability of urban environments, proper design and man⇑ Corresponding author. E-mail addresses: [email protected] (T.T. Hailegeorgis), knut.alfredsen@ntnu. no (K. Alfredsen). http://dx.doi.org/10.1016/j.jhydrol.2016.11.037 0022-1694/Ó 2016 Elsevier B.V. All rights reserved.

agement of urban water infrastructure (e.g., stormwater sewer) under uncertainty are important to minimize the risks or increase the reliability of their performance during their lifetime. An ideal approach for estimation of design flood of a certain return period is statistical analysis of urban flood records. However, runoff data is not available for a sewer system that has not been built and transfer of information from other urban catchments may be challenging due to the heterogeneity among urban catchments attributed to their heavily modified nature. For stormwater network with available runoff data, flood frequency analysis can be performed for management of the existing stormwater system and to evaluate the accuracy of the approach for the design of stormwater system, which is based on deriving the design flood hydrographs from design storm hyetographs for return periods of interest.

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Several methods are available for derivation of design storm, namely historical design storms (e.g. Reilly and Piechota, 2005) and synthetic design storms like the Intensity-DurationFrequency or IDF curves-based design storms (e.g. alternating blocks design based storm: Chow et al., 1988; Chicago design storm: Kiefer and Chu, 1957) and SCS design storms (USDA-SCS, 1956). The IDF curves, which are summary of quantiles of different return periods, are estimated from frequency analysis of systematic observed extreme precipitation events and are commonly used for estimation of design storm hyetographs. The design storm hyetographs or precipitation time series are used for estimation of design floods mainly using Precipitation-Runoff (P-R) models (e.g., Markus et al., 2007). Due to the uncertainties related to data and modelling for simulation of the hydro-climatic variables, improved methods are required for better understanding of relationships between extreme precipitation and runoff events in urban catchments, and for reliable quantile prediction of extreme hydro-climatic events. The results would help for improved decision-making for instance for reliability-based sizing of stormwater pipes and assessing their hydraulic performances and management (upgrading or retrofitting) of stormwater systems. Therefore, identification and quantification of major sources of uncertainties in estimation of IDF curves, design storm, design floods, and implications to the reliability of sizing of stormwater pipes would be necessary in the design and management of stormwater sewers. The following are the major sources of uncertainties: 1. Uncertainty due to data (methods of extraction of extremes, stationary assumption, length of data and sampling variability). 2. Uncertainty due to selection of statistical distribution and parameter estimation method. 3. Uncertainty in estimation of design storm hyetographs. 4. Uncertainty in estimation of design floods. Extraction of large samples of extreme events may reduce the sensitivity of predicted quantiles to sampling variations. In this case, the partial duration series (PDS) that extracts all peaks above a certain threshold value during the year is expected to be better than the annual maximum series (AMS) that extracts only a single peak event for a year (e.g. see Madsen et al., 1997). However, there are challenges related to the PDS mainly due to the autocorrelation among the series (e.g., Madsen et al., 1997) and the selection of the threshold values. There are several different guidelines in literature for precipitation (Bell, 1969; Van Montfort and Witter, 1986; Madsen et al., 2002) and for floods (Jarvis et al., 1936; Cunnane, 1973; Pikand, 1975; Koutsoyiannis, 2004; Trefry et al., 2005). However, the autocorrelation and the threshold values would affect the parameters and hence quantile estimations. The conventional frequency analysis of extreme events is based on the main assumption of stationarity of statistical properties of the extreme events. However, this assumption is challenged by the impacts of climate change as indicated by several studies for extreme precipitation events (Willems et al., 2012; ArnbjergNielsen et al., 2013; Caroletti and Barstad, 2010; Ntegeka and Willems, 2008; Räisänen and Alexandersson, 2003; ArnbjergNielsen, 2006; Madsen et al., 2009; Brunetti et al., 2001). Milly et al. (2008) further argued that stationarity is dead and we need to find ways to identify non-stationary probabilistic models of relevant environmental variables to optimize water systems. For better understanding of the impacts of climate change, analysis of statistical significance of trends or non-stationarity in extreme records are important. However, the results are affected by the periods of time series (Zhang et al., 2010; Hailegeorgis et al., 2013), the length of time series as identification of trends from the natural variability may require a time window of

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P30 years (e.g., Longobardi and Villani, 2009; Sonali and Kumar, 2013; Herold et al., 2016) and methods for detection of trends (e.g., Sonali and Kumar, 2013; Gado and Nguyen, 2016a, 2016b). However, shorter record length with varying periods can also be used to detect any patterns of non-stationarity to check the validity of the stationarity assumption in the frequency analysis of extreme precipitation and flood events (e.g., Zhang et al., 2010; Hailegeorgis et al., 2013) and relationships between them. Different approaches are available in literature to address the impacts of climate change on extreme precipitation events. One approach is using scenarios of climate change projections using climate models (e.g., Ashley et al., 2005; Semadeni-Davies et al., 2008; Mailhot et al., 2012). However, there is large uncertainty in projected changes in extreme precipitation events (e.g., Frei et al., 2006; Fowler et al., 2007) especially when applied to fine temporal and spatial resolutions that are relevant for the size of urban catchments (e.g., Olsson et al., 2009). Precipitation-runoff models are used for projections of extreme runoffs using projected extreme precipitation outputs from climate models (e.g., see a review by Madsen et al., 2014). However, there are several sources of uncertainties associated with the precipitation-runoff models, which operate on the climate projection outputs (e.g., see Refsgaard et al., 2014). Khaliq et al. (2006) presented a review of non-stationary frequency analysis of extreme events geared towards addressing the issues of climate change and nonstationarity. Methods that considers time-dependent location and scale parameters of distributions are commonly used for nonstationarity frequency analysis (Strupczewski et al., 2001a, 2001b; Strupczewski and Kaczmarek, 2001; Cox et al., 2002; Cunderlik and Burn, 2003; Zhang et al., 2004; Mailhot and Duchesne, 2010; Cheng and AghaKouchak, 2014; Panagoulia et al., 2014; Gado and Nguyen, 2016a, 2016b). Some of these studies indicated marked differences in quantile estimates between the non-stationary and stationary approaches. However, quantile estimation using this method is subject to uncertainties due to various assumptions, for instance, models for time dependency (trend) of location and scale parameters, time-independent shape parameter and continuation of the trend in the future. For instance, Gado and Nguyen (2016b) obtained marked differences in estimated flood quantiles among various non-stationary models, and in some years the non-stationary models resulted in quantiles smaller than those computed from the stationary models. A simpler approach is suggested in many countries for practical applications to take care of the impacts of climate change by allowing a relative increase of design rainfall or design floods under a future climate (e.g., 10– 30% in Niemczynowicz, 1989; 10–40% in Semadeni-Davies, 2004; 10–30% in Defra, 2006; 30% in Willems, 2011; 5–30% in SWWA, 2011; 0–40% in Lawrence and Hisdal, 2011). The guidelines vary from country to country and are subjective. The magnitudes of uncertainty in quantile estimates from other sources may outweigh the uncertainty related to the issues of climate change or non-stationarity and the suggested increase in design magnitudes. The uncertainty in quantile estimation from the existing systematic records would be a useful indicator of the uncertainty in quantile estimation due to climate change or nonstationarity. Therefore, frequency analysis of extreme precipitation events that is augmented by uncertainty quantification, for instance, by confidence intervals and evaluations of implications of the uncertainties on the reliability and lifetime reliability of drainage infrastructure are important. Short length of data and sampling variability are major sources of uncertainty in prediction of extreme quantiles and hence IDF curves. The Monte Carlo sampling approach (e.g. McKay et al., 1979) is commonly used for uncertainty analysis and hence the method can be used to illustrate the uncertainty due to short samples. The bootstrapping approach (Efron, 1979) has also become a

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convenient method for estimating uncertainties (Davison and Hinkley, 1997) due to sampling variability since a single sample set of extreme precipitation or runoff events would only provide a point estimate. The approach has been applied in hydrology to estimate confidence intervals of quantiles of extreme values (e.g., Zucchini and Adamson, 1989; Burn, 2003; Hall et al., 2004; Cannon, 2010). Obeysekera and Salas (2014) further applied the approach for non-stationary flood frequency analysis using timevariant parameters of the extreme value distributions and noted that the method provided more realistic confidence intervals than the other methods. Hailegeorgis et al. (2013) applied the approach for quantification of uncertainty in IDF curves due to sampling variability and found large uncertainty bounds. Willems (2013) used the approach to detect trends and multidecadal variations in extreme quantiles. Garavaglia et al. (2011) used the approach to compute a score that provides a quantitative value of confidence interval overlap of extreme quantiles of sub-samples of a long data series as a measure of robustness of uncertainty estimates of models. The estimated uncertainty ranges due to sampling variability may cover the other sources of uncertainty in frequency analysis of extreme events and hence needs to be quantified. However, Hall et al. (2006) argued that uncertainty estimation without validation is worthless and the uncertainty estimates need to be scrutinized. Renard et al. (2013) assessed the performance of various local and regional frequency analysis implementations in terms of reliability and stability indices, and reliability of their uncertainty estimation in terms of predictive distribution. The authors split the data in to calibration and validation sets, and evaluated the reliability indices in a predictive mode, i.e. using the data that are not used for calibration. The authors noted that the splitsample (Klemes, 1986) procedures are challenging to apply since they require a large amount of data. Furthermore, the main objective of the regional frequency analysis was also to augment short at-sites (local) records by combining records from several sites in a homogeneous region (‘trading-space-for-time’). The splitsample validation procedures are widely used for rainfall-runoff models in which continuous records of few years may allow reliable evaluation of performance measures (e.g. Seibert, 2003; Coron et al., 2012) and predictive uncertainty (Thyer et al., 2009). Since statistical frequency analysis is highly influenced by sample length and sampling variability, using full length of available records is expected to provide more reliable estimation of quantiles that is required for practical applications of design and management of water infrastructure than the sub-samples or split-samples (e.g., see Garavaglia et al., 2011; Willems, 2013). Choice of distributions based on their goodness-of-fit and robustness is important. Moreover, distributions that has shape parameters would provide better predictions of the tails of distributions and hence the extreme quantiles. In literature, there are tendencies of fitting the Generalized Pareto or GP distribution to the PDS data (Pikand, 1975) and fitting the Generalized Extreme Value or GEV distribution to the AMS data (Smith, 1984). However, selection of ‘‘best-fit” distributions for extreme precipitation events of different durations and floods based on goodness-of-fit tests is important to reduce the uncertainty in quantile estimation. Similarly, selection of best parameter estimation methods is important. For at-site frequency analysis with small or moderate sample size, Hosking et al. (1985) and Hosking and Wallis (1987) found that the method of L-moments is often efficient than the method of maximum likelihood. The L-moments have several advantages including unbiasedness, robustness and consistency with respect to conventional moments (Asquith, 2007). The uncertainty in estimation of IDF curves results in uncertainty in design storm hyetographs and hence design floods. The main uncertainty in estimation of design floods occurs in transfor-

mation of the precipitation time series or hyetographs to runoff time series or design flood using the P-R models due to lack of complete knowledge on relationships between occurrence of extreme precipitation and runoff, and complex nature of runoff generation processes in urban catchments. Extreme floods mainly occur due to extreme precipitation events in the form of rainfall or snowmelt or rainfall on snowmelt. Some samples of extreme precipitation intensities that are used in the frequency analysis may occur during winter seasons when precipitation is in the form of snowfall. However, extreme snowmelt rates during melt seasons may cause extreme runoff impacts. Typically, the return period based design approach is based on the assumptions that a T-year return period design storm is assumed to generate a T-year return period flood and the peak floods occur at the so-called the time of concentration. These assumptions disregards the effects of antecedent moisture conditions of the catchment on runoff response of the catchment, contributions of dry weather flows (DWF) to the flows in the stormwater pipes in terms of infiltration from soil and ground water and exfiltration out of the pipes to the subsurface flows on flood events. Therefore, studying relationships between extreme precipitation and runoff events using data from catchments that have sufficient records of both extreme precipitation and runoff events are useful in reducing uncertainties and hence for reliable decision making in transforming precipitation time series to runoff time series or the design storm hyetographs to design flood hydrographs by using the P-R models for design and management of stormwater infrastructure. Seasonality analysis of observed extreme precipitation and runoff records would shed lights on the relationships between occurrences of extreme precipitation and runoff events and hence the extent of the validity of the assumptions in the return period approach for design and management of water infrastructure. The uncertainty in estimation of design flood affects the reliability of sizing the stormwater pipes. Therefore, for decisionmaking under uncertainty, reliability-based design approach is suitable. Tung (2002) proposed a risk-based design for flood systems based on tradeoff between investment cost and expected economic costs due to failures and a return period as a design variable instead of being a pre-selected design parameter. However, the author noted that there is difficulty in practical applications of the method due to lack of reliable flood damage data. Assessment of the reliability of sizing the stormwater pipes based on the quantified uncertainty ranges of quantiles of extreme events would be useful for improved decision-making without requiring data on costs of damage. In addition, application of the time-dependent reliability analysis (e.g., Ellingwood and Mori, 1993) to assess the lifetime reliability of the capacity of stormwater pipes is important for management cases. The objectives of the present study are (a) to assess and quantify uncertainties in the flood quantiles and the IDF curves; (b) to quantify the uncertainty in synthetic design storm hyetographs that is derived from the IDF curves and (c) to show the implications of uncertainty in estimation of flood quantiles on the reliability of sizing the capacity of stormwater pipes and importance of timedependent reliability for management cases.

2. Study site and data The study site is a Risvollan urban catchment, which is located about 4 km southeast of the center of the city of Trondheim in midNorway (Fig. 1). The ground elevation of the catchment ranges from 85 m asl to 134 m asl. The catchment has separate stormwater networks of about 21.255 hectare (ha) residential area and has been an active urban research catchment since 1987. The

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Fig. 1. Location map, land use and stormwater networks of the Risvollan catchment.

catchment is equipped with instruments for measuring precipitation, temperature, solar radiation, wind speed, relative humidity and stormwater runoff. The AMS data of extreme precipitation events for Risvollan station from 1987 to 2014 but missing for the year 1998 (27 years), which was obtained from the Norwegian Meteorological Institute, were used. Extreme precipitation intensities of seventeen different durations (aggregation times) ranging from 1-min to 1440-min that are denoted as P:1min, P:2min, P:3min, P:5min, P:10min, P:15min, P:20min, P:30min, P:45min, P:60min, P:90min, P:120min, P:180min, P:360min, P:720min, P:1080min and P:1440min were used. The mean annual precipitation for the catchment is about 985 mm and the catchment experiences higher mean monthly rainfall in September and higher monthly snowfall in December. Owing to the small catchment size, the precipitation records at the Risvollan site is assumed to represent the whole catchment by neglecting any spatial variability of precipitation in the catchment. The AMS data of extreme runoff events from 1988 to 2014 (27 years) were extracted from the stormwater runoff records for the catchment that was obtained from the Norwegian water and energy directorate (NVE). The maximum values of runoff records at a temporal resolution of 1-min were used and hereafter denoted as Q:1min. There are missing runoff records immediately after AMS runoff magnitudes of 355.67 l/s, 416.27 l/s and 385.5 l/s for 2001, 2007 and 2014 that respectively occurred on 13.07.2001 11:14, 29.07.2007 00:46 and 10.07.2014 17:36. In addition, the extreme historical rainfall intensities recorded at the Risvollan catchment for P:1min (198.00 mm/h), P:2min (174.00 mm/h), P:3min

(164 mm/h), P:5min (117.60 mm/h), P:10min (67.80 mm/h), P:15min (46.80 mm/h) and P:20min (35.40 mm/h) were occurred on 13/08/2007 from 18:26 to 18:42. On this date, the runoff record for the Risvollan catchment at 18:28 was 288.93 l/s, but runoff data were missing starting from 18:29. However, the flood occurred in the City of Trondheim on 13/08/2007 caused several damages (Thorolfsson et al., 2008). These missing data bring speculation that the weir at Risvollan was flooded and hence the actual runoff were missing. However, these runoff magnitudes are far lower than the maximum of the AMS records of 645.28 l/s that was recorded on 12.08.2013 16:16. Hence, we do not have a strong evidence that the missing data are much larger than the recorded runoff or they are greater than the weir capacity. Therefore, we did not consider right censoring (i.e., 355.67+, 416.27+ and 385.5+) of these records. The land cover of the catchment is 23% impervious (14% roofs or buildings and 9% paved roads), 35% built-up areas (e.g., lawns, walkways, parking spots and playgrounds), 2% vegetation (trees) and 41% open areas (e.g., grasses). 3. Methods 3.1. Stationarity test The short record lengths of less than 30 years in the present study is not suitable for analysis of trends of the extreme events related to the impacts of climate change. Trend analysis directly on the quantiles, for instance, by performing quantile estimations for extreme precipitation intensities for data set with the same

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start year (1987) but varying end years (2005–2014) is also not a suitable approach due to the fact that this procedure entails change in the length of data and hence probably change in estimated parameters and ‘‘best-fit” distribution. Any trend in extreme events may not necessarily result in similar trend in estimated quantiles due to the effects of change in parameters and ‘‘bestfit” distribution on the estimated quantiles. In addition, sensitivity of low and high return period quantiles to the length of data, values of parameters and types of ‘‘best-fit” distributions is different. Hence, conducting trend analysis directly on estimated quantiles may provide misleading results. Therefore, we used the MannKendall non-parametric test (Mann, 1945; Kendall, 1975) to observe any patterns of non-stationarity in the extreme events only for checking the validity of the stationarity assumption in the frequency analysis. Due to the dependency of the test on a data used, different start years (from 1987 to 2005) were used. The end years for all cases are 2014 until which recent data were available. If absolute values of observed Z-values or |Zobs| is greater than critical values of Z or Zcrit, we have an evidence to reject a null hypothesis of stationarity and hence conclude that there is significant patterns of non-stationarity in the data. A significance level (a) = 5% or a confidence level of 95% was used in the present study. 3.2. Frequency analysis of extreme events The L-moments method of parameter estimation (Hosking and Wallis, 1997) was used in the present study. The method is based on order statistics. Let X be a real-valued random variable for a set of ordered data by x1:n 6 x2:n 6,    , 6xn:n, certain linear combinations of the elements of an ordered sample contain information about the location, scale and shape of the distribution from which the sample is drawn (Hosking and Wallis, 1997). We computed the L-moments and L-moment ratios based on the equations given in Hosking (1990), Serfling and Xiao (2007) and Hailegeorgis et al. (2013). The choice of ‘‘best-fit” frequency distributions is usually performed based on goodness-of-fit measures, which indicate how much the candidate distributions fit the sample data. For selection of ‘‘best-fit” distributions, the L-moment ratio diagrams or LMRD (Vogel and Fennessey, 1993; Hosking and Wallis, 1997; Peel et al., 2001) that are based on comparing plots of L-skewness versus L-kurtosis of the samples with the theoretical L-moment ratio curves of the candidate distributions are used in the present study. A distribution to which the sample L-moment ratios are closer to the theoretical curve is selected as a ‘‘best-fit”. Five 3-parameter candidate distributions that are Generalised Extreme Value (GEV), Generalised Logistic (GLO), Lognormal (LN), Generalized Pareto (GP) and Pearson Type III (P3) were tested. When several distributions fit the data adequately, Hosking and Wallis (1997) suggested selecting a robust distribution, which is capable of giving good quantile estimates even though future values may come from a distribution somewhat different from the fitted one. Parameter estimation for the frequency distributions was carried out by their relationship with the L-moments and L-moment ratios as given by Hosking and Wallis (1997). In the present study, we limited our analysis mainly up to a return period of 25y because the available records of only 27 years may not provide reliable quantiles for higher return periods. Return periods of 20y and 30y of design rains are used in Norway for separate sewer systems respectively for full pipe capacity and surcharge to critical level (see Nie et al., 2011). 3.3. Uncertainty in quantile estimation In the present study, we focused on the two main sources of uncertainty related to the data used namely short length of records

and sampling variability. We illustrated the uncertainty due to short length of records of the AMS of extreme precipitation events of some durations. In this case, we used a parametric Monte Carlo simulation by drawing random samples of N = 10,000 from the ‘‘best-fit” distribution and estimated parameters of the original samples (n = 27) using the method of inverse. The uncertainty was presented in terms of the percentage of times the ranked original samples of extreme precipitation intensities were exceeded (Perexc). The Monte Carlo simulation for the GEV distributions was performed as:

xi ¼ e þ

a f1
exp ½j lnð
lnðui ÞÞg j

ð1Þ

The Monte Carlo simulation for the GLO distributions was performed as:

xi ¼ e þ








a 1 1
exp j ln
1 ui j



ð2Þ

where ui denotes random numbers simulated from a uniform distribution U[0, 1] and e, a and j are the location, scale and shape parameters estimated from the original samples of extreme precipitation events. To estimate the uncertainty due to sampling variability, we applied a non-parametric balanced bootstrap resampling (Davison et al., 1986), which is a random sampling from the original observations of extreme events with replacement. The uncertainty was summarized in terms of 95% confidence intervals for quantiles of extreme precipitation intensities (IDF curves) and floods. We followed the procedures by Faulkner and Jones (1999) and Carpenter (1999) for estimation of the confidence intervals. In the balanced bootstrap resampling (BBRS), the total number of occurrences of each sample point in the whole resamples is the same and is equal to the number of bootstrap resamples (Nr). We used Nr = 10,000 and hence the 95% upper and lower confidence intervals respectively are given by the 250th and the 9750th bootstrap residuals. The bootstrap residuals are differences of estimated quantiles from the resampling and the quantiles estimated from the original samples ranked in ascending order. Some of the previous applications of the BBRS method include Burn (2003) and Hailegeorgis et al. (2013). The non-parametric bootstrap resampling (Efron, 1979) does not assume any statistical distribution for the quantile estimates to compute the confidence intervals unlike the parametric normally distributed approximation. However, the ‘‘best-fit” distribution for the extreme events may be different for different resamples, which substantiate the need for selection of ‘‘best-fit” distribution for each resamples of extreme events in order to carry out parameters and quantile estimation accordingly. In the present study, we estimated the quantiles for all resamples from the ‘‘bestfit” frequency distribution obtained from the original samples like in the previous applications of the BBRS method (e.g., Burn, 2003; Hailegeorgis et al., 2013). However, in the present study, we illustrated the change in the ‘‘best-fit” distributions for the extreme events among the resamples by presenting the LMRD for the resamples. 3.4. Relationships between design storm and design floods We used the IDF curves to derive design storm hyetographs based on the alternating blocks design (Chow et al., 1988). We propagated the estimated uncertainty in the IDF curves to the design storm hyetographs and presented the 95% confidence intervals for the design storm hyetographs. To evaluate the assumptions in the conventional method of transforming design storm to design floods, we studied the relationships between the occurrences of extreme precipitation and runoff events using seasonality

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measures based on the directional statistics method (Mardia, 1972; Bayliss and Jones, 1993) to estimate the Julian day of occurrences within a year of the AMS precipitation and runoff events. In the method, a length r of the mean vector is computed from the mean date of occurrences in Cartesian co-ordinates (see Burn, 1997; Cunderlik and Burn, 2006). This length is a dimensionless measure of the degree of variability or regularity of occurrence of the AMS events. The magnitude of r is bounded between zero and one where r = 0 indicates low regularity or low seasonality and r = 1 indicates high regularity or high seasonality. 3.5. Reliability and time-dependent reliability of the capacity of stormwater pipes The stormwater network is a complex system of pipes connected in series or parallel, which require system reliability analysis based on the load (floods) and the resistance (capacity) of each system components. In the present study, data on extreme runoff events are available only for the pipe at the outlet of the network. Therefore, we illustrated a reliability-based approach of sizing the capacity of the stormwater pipes under uncertainty in flood quantile estimation by assessing the reliability of sizes (capacity) of only the outlet stormwater pipe. Denoting the load variable S and the resistance variable C, the limit state function or the performance function is given by:

gðC; SÞ ¼ C
S

ð3Þ

The S in this case is flood quantiles corresponding to different return periods, which were estimated from the frequency analysis. For C and S that are random variables, takes only positive values and independent, the probability of failure or Pf ¼ PrðgðC; SÞ 6 0Þ is given by:

Z Pf ¼

1

F C ðsÞf S ðsÞds

ð4Þ

0

where FC denotes the CDF of the capacity and fS(s) is the PDF of S. The above equation is a convolution integral. However, simulation methods are usually used to calculate the probability of failure. We considered the uncertainty in S or flood quantiles due to the sampling variability that was estimated from the BBRS. We did not consider the uncertainty in the capacity since we do not have sufficient information regarding the capacity of the existing outlet pipe. We rather illustrated the reliability-based approach for different hypothetical values of fixed (deterministic) capacity of the outlet pipe. We computed the probability of failure as Pf ¼ nðgðC; SÞ 6 0Þ=Nr and the reliability R ¼ 1
P f , where n(g(C, S) 6 0) is the number of BBRS which provided the limit state function less than or equal to zero, Nr is the total numbers of BBRS (=10,000) and the Pr denotes a probability. In addition, for prediction of reliability of performance during a service life or any future time, we applied a time-dependent reliability analysis of the capacity of stormwater pipes. The limit state function for the time-dependent reliability is given by:

gðCðtÞ; SðtÞÞ ¼ CðtÞ
SðtÞ

ð5Þ

Probability of failure P f ¼ PrðgðCðtÞ; SðtÞÞ 6 0Þ and time-dependent reliability LðtÞ ¼ 1
P f . A resistance can be considered as timedependent e.g., by accounting for possible degradation of the capacity of pipes due to aging and as a random variable by considering the uncertainty (randomness) in hydraulic parameters like roughness coefficient of pipes. However, better knowledge on timedependent degradation function and a probability distribution are required. Therefore, we illustrated a simple case of timedependent reliability by considering different hypothetical fixed or deterministic capacity of pipes or resistance. We used the AMS

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flood events as fluctuating loads following a Poisson process. By modelling the probability that the load events occur within the time interval (0, t] by a Poisson model, a time-dependent reliability for a time independent and fixed resistance can be obtained from (Hwang et al., 1987; Ellingwood and Mori, 1993):

LðtÞ ¼ expf
kt½1
F S ðCÞg

ð6Þ

where k is the load occurrence rate per unit time and kt is the mean number of load events within the time interval (0, t] and Fs is the CDF for the magnitudes of the loads or floods in this case, which was obtained by fitting a distribution to the AMS of the original sample of extreme runoff events. Hence, the uncertainty due to the sampling variability was not considered in this case. The value of k is equal to 1.0 for the AMS flood events. 4. Results and discussion 4.1. Test for stationarity The results of test for stationarity that are performed for different start years (1987–2005) and end year of 2014 for the extreme precipitation and runoff events are given in Fig. 2. The results of the Mann-Kendal test were found to be dependent on the data used. Statistically significant increasing patterns or non-stationarity at 95% confidence level were observed for only extreme precipitation events of shorter durations (2-min to 30-min) when the start years of analysis are within the ranges of 1993–1996. There is no statistically significant patterns of non-stationarity at 95% confidence level for longer duration extreme precipitation events (i.e. 45min to 1440-min). The P:15min exhibited significant patterns of non-stationarity for the whole start years between 1993 and 1996. Based on the Mann-Kendal test, there is no significant nonstationarity for the cases of complete data set of extreme precipitation events (1987–2014) and extreme runoff events (1988–2014), which were used for frequency analysis in the present study. Therefore, in the present study, the stationarity assumption is valid for the data set used for the frequency analysis. With minor violation of the stationarity assumption, the impacts of the non-stationarity on the ‘‘best-fit” distribution and quantile estimates for 2y to 15y were assessed for extreme precipitation events of 2-min to 30-min using only recent extreme records after the start year that showed the first significant pattern of nonstationarity. These analyses provided different ‘‘best-fit” distributions from that of the full records for some durations, for instance, GEV for P:2min, P:3min and P:5min, and GP for P:20min. The relative percentage change in quantile estimates from the full records range from
14.64% for P:30min 2y to +12.84% for P:2min 15y. Even if there is no change in the ‘‘best-fit” distribution for P:30min, excluding the records before the start year for nonstationarity pattern reduced the 2y quantile relatively by 14.46%. The change in the record length is expected to have a negligible effect on a 2y quantile. Therefore, it shows that the excluded older extreme events that do not exhibit significant increasing patterns but contains some larger extremes e.g., extremes greater than the mean or the index storm could affect the quantile estimates more than the increasing patterns of recent extremes. However, this magnitude of uncertainty is lower than the uncertainty due to the sampling variability quantified in the present study. Despite significant increasing patterns of non-stationarity observed for some periods of extreme precipitation events of shorter durations (2-min to 30-min), which are relevant to the catchment size based on the concept of peak runoff response of catchments at the time of concentration, no statistically significant patterns of non-stationarity were observed for the extreme runoff events. Increase in runoff is expected for growing urbanization and

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Fig. 2. The Z-values of the Mann-Kendall test for non-stationarity for different start years and end year of 2014.

hence increased effects on runoff response. For the small urban catchment of the present study, there was no significant increase in the extent of impervious and built-up areas during the period of analysis. However, the results invoke questions related to (i) some magnitude of change in extreme precipitation intensities may not directly result in change in extreme runoff due to the effects of dynamic catchment runoff response and/or interaction between subsurface flow and stormwater pipes, (ii) what is the threshold magnitude of change in extreme precipitation intensities that will necessarily result in change in flood events in catchments? and (iii) what are the effects of change in temperature and hence evapotranspiration? Many studies on non-stationarity and trend detection from systematic records (e.g., see a review by Madsen et al., 2014) were performed separately either for extreme precipitation events or for floods. Such studies would not help to identify whether concurrent trends are observed in extreme precipitation and runoff events in the catchments. The results from the present study suggests that for catchments sufficiently gauged for both precipitation and runoff, non-stationarity and trend detection studies should be done simultaneously for both extreme precipitation and runoff in order to understand the relationships between the changes in the two variables and probably to identify the roles of catchment runoff response, storage mechanisms and evapotranspiration in transforming the change in precipitation to the change in runoff.

distributions were found to be ‘‘best-fit” for the extreme precipitation events of different durations and there is no systematic patterns from the shorter durations (higher intensities) to the longer durations (low intensities) (Table 1). The GLO distribution is the ‘‘best-fit” for P:1min, P:2min, P:3min, P:5min, P:45min, P:60min, P:90min and P:1440min. The GEV distribution is the ‘‘best-fit” for P:30min, P:120min, P:180min, P:360min and P:720min. The Pearson type 3 (P3) distribution is the ‘‘best-fit” for P:10min, P:15min, P:20min and P:1080min. The P3 distribution is also the ‘‘best-fit” for the extreme runoff event (Q:1min). The results comply with previous studies (e.g., Hailegeorgis et al., 2013), which substantiate the need for selection of frequency distributions for each durations of extreme precipitation intensities than fitting a single commonly used distribution for all durations. However, for some durations, there are marginal differences between the goodness-of-fit of the ‘‘best-fit” distributions compared to the next fit distributions (e.g., GEV distribution for P:1min, P:45min and P:90min) (Fig. 3). This can be explained as the uncertainty due to the choice of distributions, which its effect on quantile estimation may be negligible compared to the uncertainty due to sampling variability quantified in the present study. There is the same ‘‘best-fit” distribution (P3) for extreme runoff events and precipitation intensities of 10-min, 15-min and 20-min durations that are expected to be within the ranges of the time of concentration of the Risvollan catchment.

4.3. Estimation of quantiles and uncertainties 4.2. Selection of frequency distribution Fig. 3 provides the L-moment ratio diagrams, which was estimated from the original samples of extreme precipitation and runoff events and used for selection of the ‘‘best-fit” distributions. Table 1 provides the lists of the ‘‘best-fit” distributions and values of index storm (mm/h) or index flood (l/s) respectively for precipitation intensities of different durations and runoff. Different

We illustrated the uncertainty due to short length of the samples of extreme precipitation intensities for P:1min, P:30min, P:60min and P:1440min in Fig. 4 using the Monte Carlo simulation. The simulation provided marked percentage of times the original samples are exceeded (Perexc), for instance, for the smallest and largest samples respectively of 99.17% and 0.98% for the P:1min, 95.96% and 1.83% for the P:30min, 94.21% and 1.38% for P:60min,

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Fig. 3. The L-moment ratio diagrams (LMRD) for distribution selection from the original samples.

Table 1 Index storms or flood (l1) and ‘‘best-fit” distributions for extreme precipitation intensities of different durations (P) and extreme runoff (Q) for the Risvollan catchment. Variables

l1

Unit

‘‘Best-fit” distribution

P:1min P:2min P:3min P:5min P:10min P:15min P:20min P:30min P:45min P:60min P:90min P:120min P:180min P:360min P:720min P:1080min P:1440min Q:1min

61.56 52.67 47.85 40.89 28.56 22.46 18.68 14.24 10.88 9.06 7.20 6.85 5.54 4.00 2.93 2.25 1.93 316.93

mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h mm/h l/s

GLO GLO GLO GLO P3 P3 P3 GEV GLO GLO GLO GEV GEV GEV GEV P3 GLO P3

and 98.91% and 2.52% for P:1440min (Fig. 4). The uncertainty due to short record length is a potential source of uncertainty on the quantile estimation. Therefore, the uncertainty ranges of quantile estimation due to sampling variability, which was quantified in the present study using the BBRS, would allow sampling of several combinations of extreme events from the original samples and probably covers the uncertainty due to short sample data. Summary of the estimated quantiles of extreme precipitation intensities (IDF curves) and their 95% confidence intervals in units of liters per second per hectar or l/s ha (1 l/s ha = 2.778 mm/h) are presented in Fig. 5. Table 2 provides estimated flood (Q:1min) quantiles for the Risvollan outlet in l/s from the FFA of the original sample, and their 95% confidence intervals, minimum and

maximum values. We presented relative percentage differences of the quantiles estimated from the original sample from the maximum, minimum, 95% UCL and 95% LCL quantiles obtained from the BBRS for P:15min, P:30min, P:1440min and Q:1min (floods) in Table 3. The 95% confidence bounds of the IDF curves (Fig. 5) and runoff quantiles (Table 2) indicated wide uncertainty ranges. The maximum and minimum quantiles indicated further wider uncertainty bounds. For instance, quantile estimates of the original samples for P:15min 2y, P:60min 25y, P:1440min 25y and Q:1min 25y respectively are relatively less than their corresponding maximum quantile estimates by
43%,
44%,
38.6% and
35.4% (Table 3). Based on the equation of flow in circular pipes Q ¼ V pD2 =4 and assuming the same average velocity (V) of flow in the pipe, underestimation of a design flood by 35.4% would underestimate the diameter (D) of stormwater pipe by 16.4% that may trigger risks of surcharging and flooding. Similarly, overestimation of a design flood would result in overestimation of sizes of the pipes that may result in uneconomical solutions. Similarly, the considerable uncertainty observed in the IDF curves showed the need for quantification of uncertainties in the design storm hyetographs and design flood hydrographs that are estimated based on the IDF curves. Several countries suggested national guidelines on climate change adjustment factors on design rainfall and design floods, for instance among others, 0–40% increase on design floods in Norway (Lawrence and Hisdal, 2011), 5–30% increase on design rainfall in Sweden (SWWA, 2011), 30% increase on design rainfall in Belgium (Willems, 2011) and 10–30% on design rainfall in UK (Defra, 2006). Gado and Nguyen (2016b) reported a difference (a decrease) in a predicted 0.01-annual probability (T = 100 years) flood magnitude of the non-stationary models in some years from the flood magnitude predicted using the stationary models by 24%, 35% and 38.5% for three stations in Quebec. However, in the present study, more than 35–40% relative differences were obtained for quantiles of extreme runoff and precipitation intensities due

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Fig. 4. Uncertainty due to short record lengths for P:1min, P:30min, P:60min and P:1440min estimated from the Monte Carlo simulation in terms of the percentage of times the original samples are exceeded (Perexc).

to only the sampling variability of systematic observed records without including the uncertainty due to the climate change. These

results pose important questions if the uncertainty quantified from the observed records are sufficient to account for the uncertainty

Fig. 5. The IDF curves of precipitation intensities (l/s ha) including their 95% lower and upper confidence intervals for return periods of 2–25 years.

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Table 2 Estimated runoff quantiles from frequency analysis of the original sample, and their 95% confidence intervals, minimum and maximum values from the BBRS for Risvollan catchment. Return periods in years (y)

Minimum (l/s) LCL (l/s) Estimates (l/s) UCL (l/s) Maximum (l/s)

2

5

10

15

20

25

225.24 255.47 299.37 342.23 387.16

300.07 347.30 404.62 466.07 524.73

331.09 395.44 470.64 548.93 623.16

347.37 418.27 506.57 596.38 678.28

358.37 433.96 531.33 630.12 716.17

366.61 443.90 550.09 654.99 744.87

Table 3 Relative percentage differences of the quantiles estimates of the original samples from the maximum, minimum, 95% UCL and 95% LCL quantiles estimates from the BBRS. Return periods (years) 2

5

10

15

20

25

P:15min Maximum Minimum UCL LCL


43.0 34.6
20.5 25.7


31.8 35.5
20.5 19.0


29.3 40.3
21.7 15.4


28.1 43.2
22.4 13.9


27.4 45.0
23.1 13.2


26.9 46.2
23.4 12.7

P:60min Maximum Minimum UCL LCL


34.5 25.9
14.0 15.8


35.1 27.0
15.6 16.9


39.6 29.1
19.1 18.7


41.7 30.8
22.1 20.7


43.0 32.2
24.3 22.2


44.0 33.5
26.1 23.6

P:1440min Maximum Minimum UCL LCL


34.6 24.2
10.3 12.8


35.8 29.2
12.9 13.4


34.0 33.7
15.5 13.9


33.6 39.8
17.4 14.4


36.4 44.8
19.0 14.9


38.6 48.8
20.3 15.4

Q:1min Maximum Minimum UCL LCL


29.3 24.8
14.3 14.7


29.7 25.8
15.2 14.2


32.4 29.7
16.6 16.0


33.9 31.4
17.7 17.4


34.8 32.6
18.6 18.3


35.4 33.4
19.1 19.3

due to the climate change or not, if not how the uncertainty in the IDF curves that are obtained from the observed systematic records could be incorporated to the uncertainty due to the climate change. The L-moment ratio diagrams for P:30min and P:60min, which are obtained from the BBRS, are presented in Fig. 6. The results indicate that ‘‘best-fit” frequency distributions for the resamples appear to be different from the ‘‘best-fit” distribution for the original samples. For the P:30min, the theoretical LMRD for the ‘‘bestfit” GEV distribution passes nearly through the center of the scatter plots of the LMRD obtained from the BBRS. However, for the P:60min, the theoretical LMRD of the ‘‘best-fit” GLO distribution passes far from the center of the scatter plots of the LMRD obtained from the BBRS compared to the theoretical LMRD of the GEV distribution. These particular results show robustness of the GEV distribution to the sampling variability than the GLO distribution. The results also indicated that assuming the same ‘‘best-fit” frequency distribution in the BBRS (e.g., Burn, 2003; Hailegeorgis et al., 2013) could affect the quantile estimation. The effect of this assumption on the quantile estimation may be small compared to the estimated uncertainty ranges due to the sampling variability. Nevertheless, identification of a ‘‘best-fit” distribution would be important for each bootstrap resamples for reliable estimation of quantiles and uncertainty bounds. 4.4. Uncertainty in the design storm and implications to the design flood The synthetic design storm hyetographs for 10y and 20y return periods and their 95% lower and upper confidence intervals, which are derived from the IDF curves using the alternating block

method, are given in Fig. 7 for P:20min and P:30min in terms of precipitation depth (mm). The results showed considerable uncertainty ranges in the design storm hyetographs. For instance, there are +23% and +20.4% relative differences between the UCL and quantile estimates from the original samples respectively for P:30min 20y and P30min 10y, which can result in considerable uncertainty ranges in simulation of design flood hydrographs using precipitation-runoff models. The design storm hyetographs and their 95% confidence intervals are useful in order to simulate the design flood hydrographs including their uncertainty bounds. However, in addition to the uncertainties in estimation of IDF curves and design storm, there are many other sources of uncertainties in simulation of design floods or urban runoff time series using the precipitation-runoff models (e.g. see Deletic et al., 2012). The major sources of uncertainties that need to be addressed are: (i) lack of good quality input climate and runoff data used for parameter calibration; (ii) poor performance of temporally transferable calibrated parameters; (iii) non identifiability of the calibrated parameters and (iv) poor transferability of the parameters calibrated based on relationships between observed precipitation and runoff time series to transformation of the synthetic design storm hyetographs to design flood hydrographs. For the simplest model known as the rational method (Kuichling, 1889) Q ¼ CIA, the results from the quantile estimates from the original sample for the Risvollan catchment (A = 21.255 ha) of the present study indicated that the value of the runoff coefficient parameter (C) does not remain constant for different durations of precipitation intensities (I) and return periods (T). For instance, the value C decreases from 0.25 to 0.20, from 0.29 to 0.25 and from 0.34 to

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Fig. 6. The L-moment ratio diagrams (LMRD) from the results of the BBRS for P:30min and P:60min.

Fig. 7. The synthetic design storm hyetographs including their 95% lower and upper confidence intervals derived from the IDF curves for P:20min and P:30min for return periods of 10 and 20 years.

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0.32 respectively for P:15min, P:20min and P:30min for return periods of 2–25 years. 4.5. Relationships among extreme precipitation intensities and runoff events Fig. 8 presents seasonality of the AMS of precipitation intensities of 30 min (left) and 1440 min (middle) durations, and runoff (right). The AMS events occur between May and August for extreme precipitation intensities of durations less than 180 min while there are some AMS events during winter season for precipitations of durations greater or equal to 180 min. For instance, for P:1440min some AMS events occur between October and February (Fig. 8, middle). Most of the AMS events of runoff occur between April and August, but there are some AMS events of runoff in winter between October and December (Fig. 8, right). This also show that the snowmelt or a combination of rainfall and snowmelt events may cause winter AMS runoff events. There are high seasonality or less variability in occurrence time of extreme precipitation events of 1-min to 120-min durations. The r ranges from 0.86 to 0.93 (e.g., r = 0.90 for P:30min in Fig. 8, left). There are less seasonality or high variability throughout the year in occurrence time of extreme precipitation events of 180-min to 1440-min durations. The r ranges from 0.10 to 0.24 (e.g., r = 0.10 for P:1440min in Fig. 8, middle). The regularity or seasonality for extreme runoff events is moderate (r = 0.71). Only 46% of the AMS events of the P:1min, P:15min and P:30min are found to occur concurrently with the AMS of the Q:1min. Only 3.70% of the AMS events of P:1440min are found to be concurrent with the AMS of Q:1min. In the present study, we defined concurrence of records based on a time span of one day (1440min), which is the maximum aggregation time of the extreme precipitation records used in the present study. In many cases of the conventional design flood estimation approach, the extreme rainfall events are assumed to be the only causes of extreme runoff events. However, for the 15-min and 30-min durations, which are in the range of time of concentration for the Risvollan catchment, the AMS of extreme precipitation intensities caused the extreme runoff events only for 46% of the time. Table 4 shows the magnitudes and return periods of the concurrent extreme precipitation intensities (P:1min) and runoff records (Q:1min). The results in Table 4 further show in most of the cases that the assumption of T-year extreme precipitation leading to a Tyear runoff is not valid and challenge the concepts of design rainfall, time of concentration, historical storm and lag-time in design

301

flood estimation. For instance, the maximum of the AMS runoff observed at the Risvollan outlet is 645.3 l/s on 12/08/2013 at 16:16, which corresponds to a 90y flood as estimated from the original AMS runoff sample. However, the recorded concurrent AMS of extreme rainfall intensities are low, for instance, for P:1min, P:15min and P:30min respectively are 42 mm/h (T < 2y), 22.40 mm/h (T < 5y) and 14.8 mm/h (T < 2y). Similarly, higher rainfall intensities of 102 mm/h (T = 10y), 41.2 mm/h (T < 15y) and 31.2 mm/h (T < 25y) respectively for P:1min, P:15min and P:30min that occurred on 10/06/2002 resulted in concurrent runoff record of only 333.8 l/s (T < 5y). These results challenge the approach of determining the design storm from the historical storm (e.g., Reilly and Piechota, 2005). Table 4 also shows that the lag-time between concurrent AMS events of rainfall and runoff is variable and in some cases, peak runoff occurs earlier to peak rainfall. This also challenges the lag-time based computations of peak runoff. Therefore, the results probably show that there are marked effects of catchment moisture state and interactions between the subsurface flow and stormwater pipes by infiltration and exfiltration processes through cracked pipes on the catchment runoff response. Furthermore, the AMS runoff events between April and May and between October and December (Fig. 8, right) challenge the concept of time of concentration of design storm since the runoff events occurred from snowmelt or combined snowmelt and rainfall events of longer durations. The results support the need for two design events, a storm rainfall event and a snowmelt plus rain event suggested by Watt and Marsalek (2013). Therefore, the results from the present study substantiate the importance of further understanding of the processes of extreme runoff generation in urban catchments and estimation of design floods from continuous simulation using precipitation-runoff models. However, as discussed earlier this approach would also involve various sources of uncertainties. More importantly, the approach would require long and continuous observations of high quality runoff, precipitation and other climate variables to simulate long enough series for flood estimations. There is also another source of uncertainty relevant to the frequency analysis of extreme precipitation events of longer durations (P180-min) for the snow-influenced catchment since some of the AMS records of extreme precipitation intensities occurred during winter periods. These winter events are dominated by snowfall rather than rainfall. Therefore, there may be violation of the key assumption of identically distributed data in frequency analysis and hence another source of uncertainty because the

Fig. 8. The rose diagrams showing seasonality of the AMS of P:30min (left), P:1440min (middle) and Q:1min or runoff (right). Plotted on GeoRose (http://www. yongtechnology.com). The 0° corresponds to January 1st, 270° corresponds to April 1st, 180° corresponds to July 1st, 90° corresponds to October 1st and so on (the 30° divisions on the circumference represents one month). The lengths in the radial direction show the r that is a measure of the degree of variability or regularity of occurrences (r = 0 or low regularity corresponds to the center of the circles while r = 1 or high regularity corresponds to the outer circle).

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Table 4 The AMS precipitation intensity and runoff, and their corresponding return periods and lag-time for concurrent records. Year

1988 1992 1994 1999 2002 2005 2008 2010 2011 2012 2013 2014 a

P:1min

Lag-timea (min)

Q:1min

mm/h

Incident time

T (y)

(l/s)

Incident time

T (y)

54.00 84.00 42.00 24.00 102.00 66.00 90.00 60.00 60.00 36.00 42.00 102.00

20.07.1988 13.07.1992 06.08.1994 20.06.1999 10.06.2002 07.08.2005 28.08.2008 01.08.2010 08.08.2011 01.07.2012 12.08.2013 10.07.2014

<5 <10 <2 <2 10 <5 <10 <5 <5 <2 <2 10

325.85 469.94 241.01 307.94 333.81 242.38 299.34 279.28 240.12 315.52 645.29 385.47

19.07.1988 13.07.1992 06.08.1994 20.06.1999 10.06.2002 07.08.2005 28.08.2008 01.08.2010 08.08.2011 01.07.2012 12.08.2013 10.07.2014

<5 10 <2 <5 <5 <2 2 <2 <2 <5 90 <5

01:34 14:07 03:55 17:03 15:51 16:05 01:04 12:57 18:41 12:53 15:59 17:32

20:52 13:16 02:26 17:12 16:23 16:09 01:11 13:03 18:49 12:56 16:16 17:36

9 32 4 7 6 8 3 17 4

Lag-time of peak runoff after peak precipitation intensity. The lag-time is not provided for cases when peak runoff occurred before the peak precipitation intensity.

distributions of snowfall and rainfall may not be identical. In addition, the magnitude and timing of the snowmelt is more important for runoff generation than the magnitude and timing of snowfall. 4.6. Implications of uncertainty in flood quantiles to reliability-based design and management of stormwater pipes Reliability values for hypothetical capacities of the stormwater pipe at the outlet of Risvollan catchment for 2y to 25y floods that are computed based on the uncertainty in quantiles obtained from the BBRS, is presented in Fig. 9. The estimates from the original sample correspond to reliability of only 0.5–0.6 for return periods of 2y to 25y. Discharge capacities of pipes of 387 l/s, 524 l/s, 623 l/s, 716 l/s and 745 l/s respectively would provide a reliability of 1.0 or zero probability of failure for 2y, 5y, 10y, 15y, 20y and 25y floods (Fig. 9). Therefore, decision making under uncertainty due to sampling variability indicate that a capacity of 745 l/s would provide a 100% reliability for up to 25y floods. Based on the BBRS, minimum, estimate from the original sample and maximum flood quantiles respectively of 366 l/s, 550 l/s and 745 l/s were obtained for the 25y return period (Table 2). The 95% LCL and UCL for the 25y flood

correspond to 443.9 l/s and 654.99 l/s (Table 2). Based on flood frequency analysis from the original sample, flood magnitude of 745 l/s corresponds to a return period of greater than 200 years. The maximum of the AMS runoff observed at the Risvollan outlet in 2013 of 645.3 l/s corresponds to a return period of about 90 years (Table 4). The diameter of outlet pipe is 0.50 m and hence based on the equation of flow in circular pipes Q ¼ V pD2 =4, the discharge capacity of 645.3 l/s corresponds to an average velocity of flow in the outlet pipe of 3.29 m/s. Sizing of the stormwater pipes based on the guidelines of 2y–30y design floods (Nie et al., 2011) may result in underestimation and hence risks of flooding. Decision making on sizing of pipes under uncertainty with implications to reliability would allow more reliable results. Fig. 10 presents a time-dependent reliability and corresponding probability of failure of different hypothetical capacities of the stormwater pipe at the outlet of Risvollan catchment. The discharge capacity of 745 l/s, which corresponds to a 100% reliability for up to 25y flood quantiles, provides time dependent reliabilities of about 0.99, 0.98, 0.96, 0.92 and 0.84 respectively for lifetimes of 2, 5, 10, 25 and 50 years while a discharge capacity of 1000 l/s would provide a reliability nearly equal to 1.0 or a probability of

Fig. 9. Reliability of hypothetical capacities of the stormwater pipe at the outlet of Risvollan catchment for floods of 2–25 years return periods based on the BBRS results for runoff.

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Fig. 10. Time-dependent reliability and corresponding probability of failure of hypothetical capacities of the stormwater pipe at the outlet of Risvollan catchment for 2–50 years time based on original sample of runoff.

failure nearly equal to 0.0 for lifetimes of up to 50 years (Fig. 10). Therefore, the time-dependent reliability based approach would indicate the lifetime probability of failure and guides upgrading of the stormwater systems that the return period based method would not allow. However, the 100% reliability or lifetime reliability solution may be uneconomical and hence use of optimization techniques with the reliability approaches would be advantageous for optimal solutions to the design and management problems. 5. Conclusions Analysis of patterns of non-stationarity in the extreme events are found to be dependent on the period of the data used. Patterns of non-stationarity were observed in extreme precipitation events for some periods of data, but these patterns do not result in patterns of non-stationarity in the runoff series. Therefore, nonstationarity or trend detection studies should be done simultaneously for both extreme precipitation and runoff records in catchments to understand the relationships between the changes in the two variables. Analyses of the full length of both extreme precipitation intensities and runoff events used in the present study do not indicate significant patterns of non-stationarity and hence the stationarity assumption is valid for the frequency analysis. Different distributions are found to be the ‘‘best-fit” for extreme precipitation intensities of different durations. Moreover, there are no systematic variations of ‘‘best-fit” distributions from shorter to longer durations. Therefore, thorough selection of frequency distributions is important for different durations. The considerable uncertainty in quantiles and synthetic design storm hyetographs due to the sampling variability of the systematic observations that was obtained from the balanced bootstrap

resampling (BBRS) showed the importance of: (i) quantifying the uncertainties in the design flood hydrographs that is obtained from the IDF curves-derived synthetic design storm hyetographs, (ii) relating or incorporating the uncertainty ranges observed from the systematic records to other sources of uncertainty like the effects of future climate change. The ‘‘best-fit” distributions vary during the BBRS and hence identification of ‘‘best-fit” distribution is required for each bootstrap resamples even if it may result in small effect on quantile estimation compared to the wide ranges of uncertainty due to the sampling variability. The differences in seasonality of extreme precipitation and runoff events indicate that the extreme runoff generation in urban catchments is also affected by snowmelt, and probably by the catchment moisture state and infiltration through cracked pipes. Therefore, the results from the present study substantiate the importance of understanding the effects of these processes on extreme runoff generation in urban catchments and estimation of design floods from continuous simulation using precipitationrunoff models. Decision-making under uncertainty and the reliability-based sizing of stormwater pipes would probably provide more reliable solutions to urban flood risk management than the return period based method. The time-dependent reliability based procedure can be more applicable for assessing the capacities of stormwater pipes in management cases (e.g., capacity upgrading of the existing stormwater network) when there are continuous runoff records to estimate and update the flood quantiles. However, the method can also be applied for design purposes and for catchments with no runoff records if the design floods including their uncertainty bounds can be estimated, for instance, by transforming the design

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storm to design flood by including the uncertainty bounds. In the present study, the reliability approach was illustrated only for the outlet pipe using the runoff data at the outlet but the reliability of several bottleneck pipes and hence probability of surcharge or flooding at the corresponding nodes can be assessed if the uncertainty in design floods can be estimated for the bottleneck pipes in the network. The procedures proposed in the present study were applied for a small snow-influenced urban catchment in mid-Norway. However, they are also applicable to urban catchments in different climate regimes for further understanding of the relationships between extreme precipitation and runoff events, and for further evaluation of the existing approaches or assumptions in the design and management of urban drainage infrastructure. Acknowledgements We are very thankful to Knut Iden and Jostein Mamen of the Norwegian Meteorological Institute for providing us the annual maximum series of the extreme precipitation data for the Risvollan catchment. We are also very thankful to Christian Sveen of ViaNova Trondheim AS for providing us the GIS maps of the Risvollan catchment and stormwater sewer network. We found runoff data of the catchment from the Norwegian water and Energy Directorate (NVE). We are very thankful to the anonymous reviewers for their constructive comments, which helped to improve the paper. The Norwegian University of Science and Technology (NTNU) – Norway provided funding for the first author through a strategic scholarship. References Arnbjerg-Nielsen, K., 2006. Significant climate change of extreme rainfall in Denmark. Water Sci. Technol. 54 (6–7), 1–8. Arnbjerg-Nielsen, K., Willems, P., Olsson, J., Beecham, S., Pathirana, A., Bülow Gregersen, I., Madsen, H., Nguyen, V.-T.-V., 2013. Impacts of climate change on rainfall extremes and urban drainage systems: a review. Water Sci. Technol. 68 (1), 16–28. Ashley, R.M., Balmforth, D.J., Saul, A.J., Blanksby, D.J., 2005. Flooding in the futurepredicting climate change, risks and responses in urban areas. Water Sci. Technol. 55 (5), 265–273. Asquith, W.H., 2007. L-moments and TL-moments of the generalized lambda distribution. Comput. Stat. Data Anal. 51, 4484–4496. Bayliss, A.C., Jones, R.C., 1993. Peaks-over-threshold flood database: summary statistics and seasonality. Institute of Hydrology, Wallingford, UK (Report 121). Bell, F.C., 1969. Generalized rainfall–duration–frequency. J. Hydr. Div. ASCE 95, 311–327. Brunetti, M., Maugeri, M., Nanni, T., 2001. Changes in total precipitation, rainy days and extreme events in northeastern Italy. Int. J. Climatol. 21, 861–871. Burn, D.H., 1997. Catchment similarity for regional flood frequency analysis using seasonality measures. J. Hydrol. 202, 212–230. Burn, D.H., 2003. The use of resampling for estimating confidence intervals for single site and pooled frequency analysis. Hydrol. Sci. J. 48 (1), 25–38. Cannon, A.J., 2010. A flexible nonlinear modelling framework for nonstationary generalized extreme value analysis in hydroclimatology. Hydrol. Process. 24 (6), 673–685. Caroletti, G.N., Barstad, I., 2010. An assessment of future extreme precipitation in western Norway using a linear model. Hydrol. Earth Syst. Sci. 14, 2329–2341. Carpenter, J., 1999. Test inversion bootstrap confidence intervals. J. R. Stat. Soc. (Ser. B): Stat. Methodol. 61 (1), 159–172. Cheng, L., AghaKouchak, A., 2014. Nonstationary precipitation intensity-durationfrequency curves for infrastructure design in a changing climate. Sci. Rep. 4, 7093. http://dx.doi.org/10.1038/srep07093. Chow, V.T., Maidment, D.R., Mays, L.W., 1988. Applied Hydrology. McGraw-Hill, New York, p. 572. Cox, D.R., Isham, V.S., Northrop, P.J., 2002. Floods: some probabilistic and statistical approaches. Philos. Trans. R. Soc. Lond., Ser. A 360, 1389–1408. Coron, L., Andréassian, V., Perrin, C., Lerat, J., Vaxe, J., Bourqui, M., Hendrickx, F., 2012. Crash testing hydrological models in contrasted climate conditions: an experiment on 216 Australian catchments. Water Resour. Res. 48 (5), W05552. Cunderlik, J.M., Burn, D.H., 2003. Non-stationary pooled flood frequency analysis. J. Hydrol. 276, 210–223. Cunderlik, J.M., Burn, D.H., 2006. Site-focused nonparametric test of regional homogeneity based on flood regime. J. Hydrol. 318, 301–315. Cunnane, C., 1973. A particular comparison of annual maxima and partial duration series methods of flood frequency predictions. J. Hydrol. 18, 257–271.

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