An Integrated Extreme Rainfall Modeling Tool (SDExtreme) for Climate Change Impacts and Adaptation

The estimation of the Intensity–Duration–Frequency (IDF) relation is often necessary for the planning and design of various hydraulic structures and design storms. It has been an increasingly greater challenge due to climate change conditions. This paper therefore proposes an integrated extreme rainfall modeling software package (SDExtreme) for constructing the IDF relations at a local site in the context of climate change. The proposed tool is based on a temporal downscaling method to describe the relationships between daily and sub-daily extreme precipitation using the scale-invariance General Extreme Value (GEV) distribution. In addition, SDExtreme provides a modified bootstrap technique to determine confidence intervals (CIs) of the estimated IDF curves for current and the future climate conditions. The feasibility and accuracy of SDExtreme were assessed using rainfall data available from the selected rain gauge stations in Quebec and Ontario provinces (Canada) and climate simulations under three different climate change scenarios provided by the Canadian Earth System Model (CanESM2) and the Canadian Regional Climate Model (CanRCM4).


Introduction
The rainfall intensity-duration frequency (IDF) relationship is a commonly used standard in urban hydrology and drainage applications, such as the design and management of stormwater works. For locations where observed annual maximum precipitations (AMPs) are available, frequency analysis is implemented to estimate design rainfall intensities for a specific duration and a given return period (Buishand 1989;Zalina et al. 2002). The computational procedure of constructing the IDF relation can be summarized as: i) select the appropriate distribution; ii) estimate the parameters of the selected distribution for each duration; and iii) calculate quantiles (or intensities) with respect to the desired return periods.
The conventional procedure, however, has inherent limitations -time scaling and spatial scale limitations, and no consideration of external forcing effects such as climate change conditions. The inferences from the procedure are applicable only to the particular time scale associated with the data used and at a local site where data are available. To transfer at-site information to a region or other stations, via satellite, utilizing the Weather Research and Forecasting (WRF) model and some statistical techniques are proposed to overcome the spatial scale limitation (Liew et al. 2014;Ombadi et al. 2018;Ouali and Cannon 2018;Pizarro et al. 2018). While efforts to describe the association of extreme rainfalls at different time scales have been proposed and tested (Burlando and Rosso 1996;Casas-Castillo et al. 2018;Gupta and Waymire 1990;Menabde et al. 1999), it still is a challenge to incorporate climate change signals in the IDF relationship for the future. Over the last decade, researchers have suggested various linking methods between the IDF relationship and climate change scenarios (Vu et al. 2018;Yeo et al. 2021b).
Urbanization and land-use changes further increase vulnerability by creating more potential for catastrophic impacts from climate extremes with a large loss of human life, excessive economic losses, and uncertain long-term consequences to ecosystems. The increase in magnitude and frequency of extreme rainfalls are expected under climate change. Hence, it is important to understand not only the current patterns of extreme rainfalls, but also how they are likely to change in the future.
Nonstationary conditions of extreme precipitation raise the need to update the current IDF relationships. Because the definition of climate change infers the changes in mean and/ or standard deviation of climate variables, the theoretical distribution for the future IDF curves under the presence of climate change will be different from the current distribution. Therefore, Global Climate Models (GCMs) have been widely used to assess the various cliamte change impact studies. Outputs from these models, however, are not usually applicable for frequency analysis at a local site due to their coarse temporal and spatial resolutions, respectively.
Spatial downscaling approaches such as regional climate models (RCMs) and various statistical downscaling methods have been proposed and implemented for linking the large-scale atmospheric conditions simulated by GCMs to local weather conditons. In spite of the remarkable question of how efficiently current storm drain systems work under climate change conditions, relatively fewer studies for temporal downscaling methods have been made than the spatial downscaling efforts due to mathematical complexity and limitations (e.g., a specific extreme value distribution). Although studies have suggested theoretical approaches to incorporate the climate change conditions into frequency analyses (Alam and Elshorbagy 2015;Sarhadi and Soulis 2017;Shrestha et al. 2017;Simonovic et al. 2016;Yeo et al. 2021b), IDF_CC tool (Simonovic et al. 2016) is the only one to account for the temporal scale limitations and climate change issues. Therefore, this paper introduces another software package that enables the construction of IDF curves with various climate change scenarios.
In this study, the Statistical Downscaling for Extreme Rainfall (SDExtreme) tool is designed to easily perform climate change adaptation and mitigation practices. It allows users to automatically construct IDF curves and the confidence intervals (CIs) for the current and future periods. SDExtreme offers two versions: graphic user interface (SDExtreme-GUI) and MATLAB source codes (SDExtreme-MATLAB). The two versions are available at the Mendeley Dataset (https:// data. mende ley. com/ datas ets/ kc9fr pgfvs/2; Yeo (2022)). Two example files from the three selected rain gauge stations from Ontario and Quebec are included for practical purposes. Because the main source codes are compiled by MATLAB 2014a, the requirement for running this software is to install MATLAB Runtime version 8.3 (See the website https:// www. mathw orks. com/ produ cts/ compi ler/ matlab-runti me. html).
The proposed assessment tool is tested using an historical annual maximum precipitation series available for the period of 1961-1990 from sixteen rain gauge networks located in southern Ontario and Quebec Provinces of Canada. As for the effects of climate change on extreme rainfall systems, future weather conditions are projected using the same set of variables taken from three climate change scenarios (RCP 2.6,RCP 4.5,and RCP 8.5) given by the second-generation Canadian Earth System Model (CanESM2) and two climate change scenarios (RCP 4.5 and RCP 8.5) by the Canadian Regional Climate Model (CanRCM4). The proposed tool is used to generate future IDF curves and CI associated with both CanESM2 and CanRCM4 climate change scenarios.
The remainder of this paper is organized as follows. Section 2 provides a brief overview of the theoretical approach, Sect. 3 describes the design and application, and the last section presents our conclusion.

IDF Curves with GEV Distribution
IDF curves are a mathematical tool to represent the likelihood of rainfall intensity corresponding to duration and frequency of extreme rainfall occurrence. The generalized extreme value (GEV) distribution is widely applied to construct the IDF curves. The cumulative distribution function, F(x) , for the GEV distribution is given as (Eq. (1)): where , , and are the location, scale and shape parameter, respectively. The GEV distribution can be subdivided into three different distributions, which are Gumbel, Fréchet, and Weibull, with respect to the value of the shape parameter. In the proposed SDExtreme, the shape parameter is required always as a positive condition. (1)

Parameter Estimation
In frequency analyses, the parameters of the selected distribution can be estimated by three methods of fitting: method of moments (MOM), maximum likelihood estimator (MLE), and probability weighted moments (PWM). Assumption of MOM is that parameters of the sample distribution are equal to those of population distribution. Given asymmetric conditions for extreme rainfalls and flood, this method is not efficient. MLE generally is recognized as the most effective parameter estimation method, but additional complicated techniques such as the Newton-Raphson method are required due to non-linear equations. While MOM and MLE give same weight to all values, PWM does various weights with respect to ordering. L-moment method as one PWM method is widely used around the globe. To describe the linkage of distribution parameters for different timeframes, we suggest the use of tools available with the method of moments (MOM) using the first three non-central moments (NCMs).
A detailed description of the use of NCMs can be found in Yeo et al. (2021b). After converting return periods (e.g., 2, 5, 10, 25, 50, and 100 years) to the exceedance probability ( p ), we can estimate the quantiles ( X ) by the inverse distribution function as follows (Eq. (2)):

Consideration of Climate Change
The primary assumption of conventional frequency analysis is the stationary condition. Non-stationary conditions caused by climate change raises the question of how reliable the current drainage system mitigates damages by stormwater. Although GCMs explicitly account for climate change conditions with respect to different greenhouse gas emissions scenarios, three steps are required to conduct the adaptation and mitigation studies: i) spatial downscaling modeling to describe the linkage of large-scale atmospheric variables to at-site weather conditions, ii) temporal downscaling modeling to estimate sub-daily/hourly AMPs from the downscaled daily AMPs, and iii) construction of design storms for the future and application to stormwater management models for calculating future peak discharge magnitudes. For estimating sub-daily and hourly AMPs, researchers have been proposing various statistical approaches: the quantile mapping approach (Simonovic et al. 2016); the K-nearest neighbor technique (Alam and Elshorbagy 2015); rainfall disaggregation based on Bartlett-Lewis Rectangular Pulse approach (Shrestha et al. 2017); changing location and scale parameters of GEV distribution (Sarhadi and Soulis 2017); and scaling-invariant properties of AMPs (Yeo et al. 2021b). Although these methods are mathematically complicated for the practical applications, only IDF_CC tool (Simonovic et al. 2016) based on Gumbel distribution is accessible to get the updated IDF curves in accordance with the climate change scenarios. The tool is only available for Canadian stations, unfortunately. Therefore, SDExtreme, which is based on scaling-invariant properties with GEV distribution, is propsoed to help the users to estimate sub-daily/hourly AMPs and to update IDF curves for the future.

Modified Bootstrapping for CIs
Confidence intervals (CIs) have been used to indicate the uncertainty of quantiles in hydrological frequency analysis. The uncertainty in estimating extreme rainfall events is the result of insufficient data size, the procedure for selecting appropriate probability distribution, and the estimation of parameters of the selected distribution. Non-stationary conditions for extreme rainfalls become a critical challenge in the construction of future IDF curves. SDExtreme uses the modified bootstrapping method for constructing the CIs for current and future IDF curves. and (iv) Confidence Intervals. Figure 2 is the main menu of SDExtreme-GUI. When a user executes SDExtreme-MATLAB, it allows the user to choose four options, as follows.

IDF Current Period
The 'IDF Current Period' operation generates current IDF curves given historical AMPs records. Because both the L-moment and Three-NCM parameter estimation methods are used here, the procedure enables the verification of the quantiles estimated by Three-NCM methods. The user must specify the duration for observed data and the return period to 'SDExtreme.ini' file. After reading the historical records and specified information file, the tool automatically shows the information on data length, the number of intervals, duration and return period. Finally, this tool provides both graphical and numerical IDF curves so that the user can easily compare the quantiles (see Fig. 3). After creating a sub-folder named "Current," both SDExtreme-GUI and SDExtreme-MATLAB automatically save IDF curves, parameter sets of GEV distributions and quantiles estimated by two parameter estimators, which are the L-moments method and the NCMs method, for the current period.

Scaling GEV Model
Given the historical AMP series, the 'Scaling GEV Model' operation investigates their simple scaling properties. Two different scaling regimes raised by the transition of extreme rainfall systems are observed all over the world (Bougadis and Adamowski 2006;Chang and Hiong 2013;Nhat et al. 2007;Rodríguez et al. 2014). The presence of the breakpoint would imply the transition of extreme rainfall systems. Because greater understanding of extreme rainfall behavior results in more accurate estimations, SDExtreme allows the user to investigate the transition of storm systems by the first three NCMs. If a user hits 'Find Best Break-Point' of SDExtreme-GUI or inputs '3' value on the Command Window of MATLAB, SDExtreme creates six plots to determine the storm system transition and the break-point. Once the breakpoint is identified, the operation generates a directory, named 'Scaling Exponent,' and saves the estimated scaling exponents ( ) for both short and long durations. The user can examine the linearity of the scaling exponents for both durations for the simple properties (Fig. 4). The parameters of the downscaled GEV distributions for the duration of sub-hourly and sub-daily rainfall can be estimated by the function of 'Estimating Parameters and Quantiles' and saved in the directory of 'Scaling-GEV' with the name of 'Scaled-GEV_Param_Obs.txt.' Finally, the user can compare the estimated quantiles by the scaling-GEV models to the observed values, graphically (Fig. 5).

IDF Climate Change
SDExtreme version 2.0 uses the second-degree polynomial equation to improve the accuracy of the downscaled AMPs at a given site. Before the error-adjustment, the 'IDF Climate Change' divides GCM's output series (1961-2100) into four time periods: 1961-1990, 2020s, 2050s and 2080s. The length of each period is identical, at 30 years. After making the bias-correction adjustment, SDExtreme provides the relative root mean square of errors (RMSE) to account for accuracy improvement between the unadjusted and the adjusted downscaled NCMs based on the historical values.
After selecting GCM's outputs and its type, SDExtreme calculates the RMSE values with and without bias correction and provides probability plots and effort adjustment function (Fig. 6). The constructed error adjustment function is applied to the extracted daily AMP series for each period (current, 2020s, 2050s, and 2080s). The adjusted AMP series are saved in the 'Adjusted-NCM' directory. Finally, the tools allow the user to compare the IDF curves with respect to time periods (e.g., calibration period, 2020s, 2050s and 2080s) after estimating parameters of the downscaled GEV distributions for sub-daily and subhourly durations (Fig. 7). All parameters, quantiles and IDF curves for each period are saved in the 'Scaling-GEV' directory.

Confidence Intervals
The main purpose of the 'ConfIdence Intervals' operation is to quantify the uncertainties of IDF curves. Given the number of resampling and their significant levels, users can construct current and future uncertainties. The procedure for calculating CIs for the present is conducted using the historical record, while those for the future are done with climate change scenario outputs. After choosing the 'ConfIdence Intervals' option, SDExtreme asks the user to select one of CIs for the current or the future IDF curves. The users input additional information. Here is the example CIs for the future IDF curves.
The estimated upper and lower limits of the IDF curves are saved in the 'Confidence-Interval' directory.

Study Area and Data
The feasibility of the proposed SDExtreme is demonstrated with at-site AMP data available at 16 rain gauge stations in Ottawa, Ontario, as shown in Fig. 8. Table 1 provides the geographical information on the rain gauge stations used in this study. The historical at-site AMP series are obtained from Canadian Weather Energy and Engineering Datasets (CWEEDS). They are made up of nine durations (5-, 10-, 15-and 30-min; 1-, 2-, 6-and 12-h; and 1 day).

Verification of the Three-NCM Parameter Estimation Method
The 'IDF Current Period' procedure provides numerical and graphical results (e.g., quantile plots and IDF curves) to evaluate the performance of the Three-NCM parameter estimation method using the historical and estimated quantiles by L-moment method. Figure 9 shows the quantile plots for durations of 5 and 30 min for the selected representative stations (Toronto International Airport for Ontario, and St. Hubert for Quebec). Black, blue and red dots represent observed, estimated quantiles by L-moments and the Three-NCMs methods, respectively. Congruency between the estimated amounts using two estimation   Figure 10 shows the scaling relationships with respect to all durations. SDExtreme provides the R-square plot for determining the breakpoint to identify scaling properties at two different regimes of durations. From Fig. 10B, D, case 2 (30-min.) are determined for both Toronto and St. Hubert stations. Table 4 shows scaling exponents and the duration when breakpoints are observed. The breakpoints at thirteen stations are located at 30-min. While the points at Ottawa and Hamilton are of 60-min, the point at Kingston is located at 3 h. The spatial distributions of exponents are shown in Fig. 11A is for the scaling exponents for the duration before breakpoints; and Fig. 11B is for the duration after breakpoints. Because the scaling exponent is a ratio of extreme precipitation intensities to duration, a station with a bigger value of the scaling exponent would have more intensive extreme rainfall than a station with a lower value of the scaling exponent. In general, rain gauge stations in the southern portion have more intensive rainfall during short periods.

Investigation of Scale-invariant Properties
To examine the simple scaling properties of the AMP series, SDExtreme carries out the simple scale test graphically and numerically. As shown in Fig. 12, the linearity of the scaling exponents with the order of moments supports the assumption that the extreme rainfall series can be described by a simple scaling model for both stations.
With the simple scaling relationship, Fig. 13 shows the comparison between the observed and estimated AMPs by L-moments and scaling GEV distributions for 5-and 30-min durations for Toronto and St. Hubert stations. It can be seen that the quantiles Hubert. Blue diamonds denote the first order NCMs, green triangles note the second order NCMs, while red dots represent the third order NCMs. R-square plots are provided for detecting the best case to demonstrate the two scaling regimes derived from the daily AMPs using the established scaling relationships align with those values given by the conventional fitted GEV distribution, as well as with the observed values. Similar results were found for other durations and stations.

Update IDF Curve for the Future
To illustrate SDExtreme for future IDF curves, two different downscaling methods were used. For statistical downscaling, SDRain (Yeo et al. 2021a) was calibrated and used to generate a daily precipitation time series for an Ottawa station. A set of significant global atmospheric reanalysis variables of the NCEP/NCAR (Kalnay et al. 1996) given by the second-generation Canadian Earth System Model (CanESM2) was used to establish statistical downscaling models. Once the spatial downscaling model with SDRain was calibrated, future weather conditions were projected with the three climate change scenarios (RCP2.6, RCP4.5, and RCP8.5), where RCP denotes a representative concentration pathway. As a dynamical downscaling method, the Canadian Regional Climate Model (CanRCM4) with 0.22° grid resolution was used. The downscaled daily precipitation series of climate variables for the grid point nearest to each rain gauge station were used. A secondary quantile mapping was implemented to account for uncertainty in the CanRCM4 coming from climate systems.  SDExtreme estimates the IDF curves for the 2080s using five different greenhouse gas emission scenarios (RCP2.6, RCP4.5, and RCP8.5) provided by CanESM2 and (RCP4.5 and RCP8.5) by CanRCM4 at Ottawa (Fig. 14). In the plots, the black asterisks denote quantiles corresponding to a 100-year return period . Quantiles for the 1-h duration and 100-year return period are 54.01 mm (calibration period), 59.99 mm (CanESM2 RCP2.6), 61.47 mm (CanESM2 RCP4.5), 65.39 mm (CanESM2 RCP8.5), 58.47 mm (CanRCM4 RCP4.5), and 60.60 mm (CanRCM4 RCP8.5), respectively. While the IDF values calculated by the CanESM2 scenarios are increased overall, those by CanRCM4 are not significantly different from the current quantiles. For assessing how different IDF curves result within shorter return periods, the comparison studies are conducted (Figs. 15 and 16). Contrary to the longer return period conditions, CanRCM4's climate change scenarios provide higher values of IDF quantiles than do CanESM2's scenarios. For illustration purposes, Fig. 17 shows the plot of daily AMPs corresponding to a 100-year return period simulated by SDExtreme and the three greenhouse gas emission scenarios given by CanESM2 and CanRCM4. It is found that the estimated 100-year daily AMPs exhibit similar, continuously increasing trends from the current period through the 2020s, the 2050s and 2080s. With CanESM2, the intensity increases from about 3.6 mm/h (current) to 3.70 mm/h (RCP 2.6), to 3.74 mm/h (RCP 4.5), and to 3.79 mm/h (RCP 8.5), respectively. However, CanRCM4 shows different patterns from the estimated intensities by CanESM2.

Construction of Confidence Intervals
SDExtreme was used to construct CIs of AMPs for the present and future periods. The function of 'Confidence Interval' was used to generate 1,000 sets of AMPs for a day's duration using observed and synthesized daily AMPs by SDRain and quantile mapping with the three climate change scenarios by CanESM2 and CanRCM4. The significance level was set up as 0.05 for constructing the 95% CIs in this study. The Grey shade represents the range of CIs illustrated at the selected significance level, and the black lines denote the estimated IDF curves. CIs were constructed by not using historical records to evaluate the performance of SDExtreme CI methods. As shown in Fig. 18, all observed values fall into the CI ranges for both durations of 5-and 30-min. This implies that SDExtreme could provide robust CIs of extreme rainfall values without the observed sub-daily and sub-hourly AMPs. With the verification of the proposed CIs method, SDExtreme constructed CIs for IDF curves for the period of 2080s, estimated under four climate change scenarios (i.e., CanESM2 RCP 4.5 & 8.5 and CanRCM4 RCP 4.5 & 8.5), corresponding to the 50-year return period. As shown in Fig. 19, it has been found that the range of CIs for RCP 8.5 is quite narrow, compared to those for other climate change scenarios. In addition, the ranges given by CanRCM4 are thinner than those by CanESM2 scenarios. Because the thin CIs imply low variability of the estimated AMPs, the result could imply that a significant increase in extreme rainfall is highly likely under the RCP 8.5 climate change scenario.

Conclusions
Climate change is defined as statistical changes in the mean or standard deviation of climatic variables. Consequently, probability distributions for heavy rainfalls shall be altered under the climate change conditions. The conventional methods are unable to demonstrate the external forcing effects such as climate change because the current frequency analysis is carried out using only historical data. An integrated extreme rainfall modeling tools (SDExtreme-GUI and SDExtreme-MATLAB) are therefore proposed to assess the impacts induced by climate change on extreme rainfalls.
SDExtreme is used to update the IDF curves in accordance with various climate change scenarios. The tool is composed of four steps: IDF curve current, Scaling GEV, IDF climate change, and CIs. The first step is to construct IDF curves using the L-moment and Three-NCM parameter estimators for GEV distributions. If the good agreement of the Fig. 19 The 95% confidence intervals (CIs) for IDF curves (for the period of 2080s) corresponding to 50-year return periods estimated under two greenhouse gases emission scenarios (RCP 4.5,and RCP 8.5) given by CanESM2 and two (RCP 4.5 and RCP 8.5) by CanRCM4, respectively. (A) under RCP 4.5 by CanESM2, (B) under RCP 5.5 by CanESM2, (C) under RCP 4.5 by CanRCM4, and (D) under RCP 8.5 by CanRCM4. Grey shade regions are the 95% CIs and black lines are the estimated IDF curves by SDExtreme quantiles estimated by two estimators, SDExtreme allows the user to investigate scaling properties of historical AMP series for building up a temporal downscaling model based on a simple scaling property of AMPs. With statistically or dynamically downscaled daily precipitation series, the user can obtain four sets of IDF curves with respect to different time frames (current, 2020s, 2050s, and 2080s). Finally, the upper and lower intensities of the CIs for current and future periods can be estimated for the uncertainty study.
In this study, the proposed tool is implemented to climate simulation outputs from CanESM2 and CanRCM4 under three different greenhouse gas emission scenarios, and available AMP series for durations ranging from 5 min to 24 h at 16 stations located in southern Quebec and Ontario, Canada, over three decades, from 1961 to 1990. It has been found that the AMP series at all stations display a simple scaling behavior within two different time intervals. Based on this scaling property, the scaling GEV distribution has been shown to be able to provide accurate estimates of sub-daily AMPs from observed and GCM-downscaled daily AMP amounts. Therefore, it can be concluded that it is feasible to use SDExtreme to describe the relationship between large-scale climate predictors for daily scale given by GCM and RCM simulation outputs, and the daily and sub-daily AMPs at a local site.
Furthermore, the proposed assessment tool was implemented to construct the IDF relations for a given site for the 1961-1990 period and the future (2020s, 2050s and 2080s), using climate predictors given by CanESM2 and CanRCM4 simulations. Results show the significant increasing trend in daily AMPs through, and beyond, the 2080s. The highest increase in extreme rainfall was observed in the estimate by RCP 8.5 scenario given by CanESM2.
This SDExtreme software package provides the CIs using only daily AMP series and simple scaling properties. The CI estimation method was implemented for the synthesized future daily AMPs, as well as the observed value for the uncertainty study. It was found that the CIs vary with respect to climate change scenarios.
The proposed SDExtreme tools are tested using historical AMP series available in Canada and South Korea. Due to the limited application of the tools, it is likely to expect the debugging procedure with data sets from different regions. Since SDExtreme-GUI doesn't allow the user to debug it, SDExtreme-MATLAB is more accessible for MATLAB users to modify the source codes.

Availability of Data and Material
The Mendeley Dataset (https:// data. mende ley. com/ datas ets/ kc9fr pgfvs/2) contains several data sets, including the historical annual maximum precipitation and downscaled daily precipitation series. More specifically, here is the detailed information; Historical annual maximum precipitation series -the three rain gauge stations (Ottawa, St. Hubert and Toronto) were downloaded from Canadian Weather Energy and Engineering Datasets (CWEEDS). Downscaled precipitation series -In this study, two downscaling methods were implemented for comparison purposes. The statistical downscaling model, SDRain (Yeo et al. 2021a), was used for spatially downscaling the CanESM2 model with three greenhouse gas emission scenarios (RCP2.6, RCP4.5 and RCP8.5). The file names are CanESM2_R**_Ottawa.OUT. For the dynamical downscaling model, CanRCM4 is used. Because of low value events, an additional statistical procedure (quantile mapping) was used. The file names are CanRCM4_His_R**_Ottawa.OUT.

Declarations
Ethics Approval Approved. Consent for Publication All authors have consented.

Conflicts of Interest/Competing Interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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