Selecting the best IDF model by using the multifractal approach

In this work, the multifractal properties of hourly rainfall data recorded at a location in Southern Spain have been related to the scale properties of the corresponding intensity–duration–frequency (IDF) curves. Four parametric models for the IDF curves have been fitted to the quantiles of rainfall obtained using the generalized Pareto frequency distribution function with the extreme data series obtained for the same place. The scaling of the rainfall intensity moments has been analysed, and the empirical moments scaling exponent function has been obtained. The corresponding values of q₁ and γ₁ have been empirical and theoretically calculated and compared with some characteristics of the different IDF models. Thus, the scaling behaviour of IDF curves has been analysed, and the best model has been selected. Copyright © 2012 John Wiley & Sons, Ltd.     

Simple generalization approach for intensity–duration–frequency relationships

Rainfall intensity–duration–frequency (IDF) relationships describe rainfall intensity as a function of duration and return period, and they are significant for water resources planning, as well as for the design of hydraulic constructions. In this study, the two‐parameter lognormal (LN2) and Gumbel distributions are used as parent distribution functions. Derivation of the IDF relationship by this approach is quite simple, because it only requires an appropriate function of the mean of annual maximum rainfall intensity as a function of rainfall duration. It is shown that the monotonic temporal trend in the mean rainfall intensity can successfully be described by this parametric function which comprises a combination of the parameters of the quantile function a(T) and completely the duration function b(d) of the separable IDF relationship. In the case study of Aegean Region (Turkey), the IDF relationships derived through this simple generalization procedure (SGP) may produce IDF relationships as successfully as does the well‐known robust estimation procedure (REP), which is based on minimization of the nonparametric Kruskal–Wallis test statistic with respect to the parameters θ and η of the duration function. Because the approach proposed herein is based on lower‐order sample statistics, risks and uncertainties arising from sampling errors in higher‐order sample statistics were significantly reduced. The authors recommend to establish the separable IDF relationships by the SGP for a statistically favorable two‐parameter parent distribution, because it uses the same assumptions as the REP does, it maintains the observed temporal trend in the mean additionally, it is easy to handle analytically and requires considerably less computational effort. Copyright © 2012 John Wiley & Sons, Ltd.     

A spatial temporal downscaling approach to development of IDF relations for Perth airport region in the context of climate change

ABSTRACT

Downscaling of climate projections is the most adapted method to assess the impacts of climate change at regional and local scales. This study utilized both spatial and temporal downscaling approaches to develop intensity–duration–frequency (IDF) relations for sub-daily rainfall extremes in the Perth airport area. A multiple regression-based statistical downscaling model tool was used for spatial downscaling of daily rainfall using general circulation models (GCMs) (Hadley Centre’s GCM and Canadian Global Climate Model) climate variables. A simple scaling regime was identified for 30 minutes to 24 hours duration of observed annual maximum (AM) rainfall. Then, statistical properties of sub-daily AM rainfall were estimated by scaling an invariant model based on the generalized extreme value distribution. RMSE, Nash-Sutcliffe efficiency coefficient and percentage bias values were estimated to check the accuracy of downscaled sub-daily rainfall. This proved the capability of the proposed approach in developing a linkage between large-scale GCM daily variables and extreme sub-daily rainfall events at a given location. Finally IDF curves were developed for future periods, which show similar extreme rainfall decreasing trends for the 2020s, 2050s and 2080s for both GCMs.

Editor M.C. Acreman; Associate editor S. Kanae     

Pooled frequency analysis for intensity–duration–frequency curve estimation

Rainfall intensity–duration–frequency (IDF) curves are used in the design of urban infrastructure. Their estimation is based on rainfall frequency analysis, usually performed on rainfall records from a single gauged station. However, available at‐site record length is often too short to provide accurate estimates for long return periods. In the present study, a general framework for pooled rainfall frequency analysis based on the index‐event model is proposed for IDF estimation at gauged stations. Pooling group formation is defined by the region of influence approach on the basis of the geographical distance similarity measure. Several pooled approaches are defined and evaluated by a procedure through which quantile estimation and uncertainty are assessed. Alternate approaches for the definition of a pooling group are based on different criteria regarding initial pooling group size (and the relationship between size and return period), approaches for assessing pooling group homogeneity, and the use of macroregions in pooling group formation. The proposed framework is applied to identify the preferred approach for pooled rainfall intensity frequency analysis in Canada. Pooled approaches are found to provide more precise estimates than the at‐site approach, especially for long return periods. Pooled parent distribution selection supported the use of the generalized extreme value distribution across the country. Recommendations for pooling group formation include increasing the pooling group size with increases in return period and identifying an appropriate trade‐off between pooling group homogeneity and size for long return periods.     

Derivation of rainfall IDF relations by third-order polynomial normal transform

Establishing the rainfall intensity–duration–frequency (IDF) relations by the conventional method, the use of parametric distribution models has the advantage of automatic compliance of monotonicity condition of rainfall intensity and frequency. However, fitting rainfall data to a distribution separately by individual duration may possibly produce undulation and crossover of IDF curves which does not comply physical reality. This frequently occurs when rainfall record length is relatively short which often is the case. To tackle this problem this study presents a methodological framework that integrates the third-order polynomial normal transform (TPNT) with the least squares (LS) method to establish rainfall IDF relations by simultaneously considering multi-duration rainfall data. The constraints to preserve the monotonicity and non-crossover in the IDF relations can be incorporated easily in the LS-based TPNT framework. Hourly rainfall data at Zhongli rain gauge station in Taiwan with 27-year record are used to establish rainfall IDF relations and to illustrate the proposed methodology. Numerical investigation indicates that the undulation and crossover behavior of IDF curves can be effectively circumvented by the proposed approach to establish reasonable IDF relations.     

POTENTIAL OF EXTENDING THE RAINFALL INTENSITY–DURATION–FREQUENCY RELATIONSHIP TO NON-RECORDING RAIN GAUGES

Maximum rainfall intensity–duration–frequency (IDF) curves are commonly applied to determine the design rainfall in water resource projects. Normally, the IDF relationship is derived from recording rain gauges. As the network of non-recording rain gauges (daily rainfall) in Taiwan has a higher density than recording rain gauges, attempts were made in this study to extend the IDF relationship to non-recording rain gauges. Eighteen recording rain gauges and 99 non-recording rain gauges over the Chi-Nan area in Southern Taiwan provide the data sets. The regional IDF formulae were generated for ungauged areas to estimate rainfall intensity for various return periods and rainfall durations larger than or equal to one hour. For rainfall durations less than one hour, a set of adjustment formulae were applied to modify the regional IDF formulae. The method proposed in this study had reasonable application to non-recording rain gauges, which was concluded from the verification of four additional recording rain gauges. © 1997 John Wiley & Sons, Ltd.     

Estimation of rainfall intensity–duration–frequency curves at ungauged locations using quantile regression methods

Intensity–duration–frequency (IDF) curves of extreme rainfall are used extensively in infrastructure design and water resources management. In this study, a novel regional framework based on quantile regression (QR) is used to estimate rainfall IDF curves at ungauged locations. Unlike standard regional approaches, such as index-storm and at-site ordinary least-squares regression, which are dependent on parametric distributional assumptions, the non-parametric QR approach directly estimates rainfall quantiles as a function of physiographic characteristics. Linear and nonlinear methods are evaluated for both the regional delineation and IDF curve estimation steps. Specifically, delineation by canonical correlation analysis (CCA) and nonlinear CCA (NLCCA) is combined, in turn, with linear QR and nonlinear QR estimation in a regional modelling framework. An exhaustive comparative study is conducted between standard regional methods and the proposed QR framework at sites across Canada. Overall, the fully nonlinear QR framework, which uses NLCCA for delineation and nonlinear QR for estimation of IDF curves at ungauged sites, leads to the best results.     

A Bayesian hierarchical spatio-temporal model for extreme rainfall in Extremadura (Spain)

A statistical study was made of the temporal trend in extreme rainfall in the region of Extremadura (Spain) during the period 1961–2009. A hierarchical spatio-temporal Bayesian model with a GEV parameterization of the extreme data was employed. The Bayesian model was implemented in a Markov chain Monte Carlo framework that allows the posterior distribution of the parameters that intervene in the model to be estimated. The results show a decrease of extreme rainfall in winter and spring and a slight increase in autumn. The uncertainty in the trend parameters obtained with the hierarchical approach is much smaller than the uncertainties obtained from the GEV model applied locally. Also found was a negative relationship between the NAO index and the extreme rainfall in Extremadura during winter. An increase was observed in the intensity of the NAO index in winter and spring, and a slight decrease in autumn.     

A simplified storm index method to extrapolate intensity–duration–frequency (IDF) curves for ungauged stations in central Chile

Most of meteorological stations in Chile register rainfall amounts once every 24 h. The creation of intensity–duration–frequency (IDF) curves requires continuous recorded data, and this insufficiency of proper instrumentation has resulted in a lack of IDF curves nationwide. The objective of this study is to further develop and evaluate the feasibility of a new method to estimate IDF curves in ungauged stations under Mediterranean climates of central Chile. A technique used to address this problem is the use of a storm index (S_I), also known as the ‘K’ method, which allows the construction of IDF curves from stations with discontinuous data, by extrapolating data from stations with continuous records, as long as daily rainfall intensities for both stations differ by less than 2 mm h^?1. To test the applicability of this method, S_I values were calculated for 40 meteorological stations located throughout Central Chile (latitudes 30°S to 40°S). The extrapolated IDF curves were then compared with observed data, and the goodness of fit was determined. The results indicate that the storm index method can adequately estimate hourly IDF curve values for stations lacking of continuous rainfall data. Copyright © 2014 John Wiley & Sons, Ltd.     

Regionalization of heavy rainfall to improve climatic design values for infrastructure: case study in Southern Ontario,Canada

Abstract

Southern Ontario, Canada, has been impacted in recent years by many heavy rainfall and flooding events that have exceeded existing historical estimates of infrastructure design rainfall intensity–duration–frequency (IDF) values. These recent events and the limited number of short-duration recording raingauges have prompted the need to research the climatology of heavy rainfall events within the study area, review the existing design IDF methodologies, and evaluate alternative approaches to traditional point-based heavy rainfall IDF curves, such as regional IDF design values. The use of additional data and the regional frequency analysis methodology were explored for the study area, with the objective of validating identified clusters or homogeneous regions of extreme rainfall amounts through Ward's method. As the results illustrate, nine homogeneous regions were identified in Southern Ontario using the annual maximum series (AMS) for daily and 24-h rainfall data from climate and rate-of-rainfall or tipping bucket raingauge (TBRG) stations, respectively. In most cases, the generalized extreme value and logistic distributions were identified as the statistical distributions that provide the best fit for the 24-h and sub-daily rainfall data in the study area. A connection was observed between extreme rainfall variability, temporal scale of heavy rainfall events and location of each homogeneous region. Moreover, the analysis indicated that scaling factors cannot be used reliably to estimate sub-daily and sub-hourly values from 24- and 1-h data in Southern Ontario.

Citation Paixao, E., Auld, H., Mirza, M.M.Q., Klaassen, J. & Shephard, M.W. (2011) Regionalization of heavy rainfall to improve climatic design values for infrastructure: case study in Southern Ontario, Canada. Hydrol. Sci. J. 56(7), 1067–1089.     

Assessment of design rainfall uncertainty for hydrologic engineering applications in Hong Kong

Hydrologic engineering designs and analyses often require the specification of design storm which involves rainfall amount, duration and hyetograph. In practice, the determination of design rainfall in hydrologic engineering applications involves the frequency analysis of extreme rainfalls of different durations and the establishment of rainfall hyetograph for the design event under consideration. Sampling errors exist in the estimation of rainfall depth (or intensity) quantiles from frequency analysis, which will be transmitted in the process of determining the design rainfall hyetograph. This paper presents a practical methodological framework based on the bootstrap resampling scheme to assess the uncertainty features associated with the magnitude of estimated rainfall depth/intensity quantiles and the corresponding design hyetographs. The procedure is implemented to quantify uncertainty of design rainfall hyetograph following the Stormwater Drainage Manual of Hong Kong involving the use of rainfall intensity–duration–frequency (IDF) model. Of particular interesting is that the bootstrap resampling scheme implemented herein is modified to handle unequal record period of annual maximum rainfall data series of different durations and to account for their intrinsic correlations. According to the adopted rainfall IDF model, the design rainfall hyetograph is a function of the IDF model coefficients. Due to the correlation among rainfall quantiles of different durations, the IDF coefficients are found to be strongly related in a nonlinear fashion which should not be ignored in the establishment of the design hyetographs.     

Non-crossing nonlinear regression quantiles by monotone composite quantile regression neural network,with application to rainfall extremes

The goal of quantile regression is to estimate conditional quantiles for specified values of quantile probability using linear or nonlinear regression equations. These estimates are prone to “quantile crossing”, where regression predictions for different quantile probabilities do not increase as probability increases. In the context of the environmental sciences, this could, for example, lead to estimates of the magnitude of a 10-year return period rainstorm that exceed the 20-year storm, or similar nonphysical results. This problem, as well as the potential for overfitting, is exacerbated for small to moderate sample sizes and for nonlinear quantile regression models. As a remedy, this study introduces a novel nonlinear quantile regression model, the monotone composite quantile regression neural network (MCQRNN), that (1) simultaneously estimates multiple non-crossing, nonlinear conditional quantile functions; (2) allows for optional monotonicity, positivity/non-negativity, and generalized additive model constraints; and (3) can be adapted to estimate standard least-squares regression and non-crossing expectile regression functions. First, the MCQRNN model is evaluated on synthetic data from multiple functions and error distributions using Monte Carlo simulations. MCQRNN outperforms the benchmark models, especially for non-normal error distributions. Next, the MCQRNN model is applied to real-world climate data by estimating rainfall Intensity–Duration–Frequency (IDF) curves at locations in Canada. IDF curves summarize the relationship between the intensity and occurrence frequency of extreme rainfall over storm durations ranging from minutes to a day. Because annual maximum rainfall intensity is a non-negative quantity that should increase monotonically as the occurrence frequency and storm duration decrease, monotonicity and non-negativity constraints are key constraints in IDF curve estimation. In comparison to standard QRNN models, the ability of the MCQRNN model to incorporate these constraints, in addition to non-crossing, leads to more robust and realistic estimates of extreme rainfall.     

Approaches for defining thresholds and return periods for rainfall‐triggered shallow landslides

Probabilistic thresholds for triggering shallow landslides by rainfall are developed using two approaches: a logistic regression model and Iverson's physically based model. Both approaches are applied to a 180 km² area in northern Italy. For the physically based model a Monte Carlo approach is used to obtain probabilities of slope failure associated with differing combinations of rainfall intensity and duration as well as differing topographic settings. For the logistic regression model hourly and daily rainfall data and split‐sample testing are used to explore the effect of antecedent rainfall on triggering thresholds. It is demonstrated that both the statistical and physically based models provide stochastic thresholds that express the probability of landslide triggering. The resulting thresholds are comparable, even though the two approaches are conceptually different. The physically based model also provides an estimate of the percentage of potentially unstable areas in which failure can be triggered with a certain probability. The return period of rainfall responsible for landslide triggering is studied by using a Gumbel scaling model of rainfall intensity–duration–frequency curves. It is demonstrated that antecedent rainfall must be taken into account in landslide forecasting, and a method is proposed to correct the rainfall return period by filtering the rainfall maxima with a fixed threshold of antecedent rainfall. This correction produces an increase of the return periods, especially for rainstorms of short duration. Copyright © 2009 John Wiley & Sons, Ltd.     

Incorporating uncertainty analysis into a regional IDF formula

A rainfall intensity–duration–frequency (IDF) relationship was generated by pooling annual maximum rainfall series from 14 recording rain gauges in southern Taiwan. Dimensionless frequency curves, plotted by the growth curve method, can be well fitted by regression equations for a duration ranging from 10 mins to 24 hours. As the parameters in regression equations have a good statistical relationship with average annual rainfall, a generalized regional IDF formula was then formulated. The formula, based on average annual rainfall as an index, can be easily applied to non-recording rain gauges. This paper further applies the mean value first-order second moment (MFOSM) method to estimate the uncertainty of the proposed regional IDF formula. From a stochastic viewpoint, the generalized regional IDF formula can accurately simulate the IDF relationship developed using frequency analysis (EV1) at individual stations. The method can provide both rainfall intensity and variance isohyetal maps for various rainfall durations and return periods over the study area. © 1998 John Wiley & Sons, Ltd.     

Model and parameter uncertainty in IDF relationships under climate change

Quantifying distributional behavior of extreme events is crucial in hydrologic designs. Intensity Duration Frequency (IDF) relationships are used extensively in engineering especially in urban hydrology, to obtain return level of extreme rainfall event for a specified return period and duration. Major sources of uncertainty in the IDF relationships are due to insufficient quantity and quality of data leading to parameter uncertainty due to the distribution fitted to the data and uncertainty as a result of using multiple GCMs. It is important to study these uncertainties and propagate them to future for accurate assessment of return levels for future. The objective of this study is to quantify the uncertainties arising from parameters of the distribution fitted to data and the multiple GCM models using Bayesian approach. Posterior distribution of parameters is obtained from Bayes rule and the parameters are transformed to obtain return levels for a specified return period. Markov Chain Monte Carlo (MCMC) method using Metropolis Hastings algorithm is used to obtain the posterior distribution of parameters. Twenty six CMIP5 GCMs along with four RCP scenarios are considered for studying the effects of climate change and to obtain projected IDF relationships for the case study of Bangalore city in India. GCM uncertainty due to the use of multiple GCMs is treated using Reliability Ensemble Averaging (REA) technique along with the parameter uncertainty. Scale invariance theory is employed for obtaining short duration return levels from daily data. It is observed that the uncertainty in short duration rainfall return levels is high when compared to the longer durations. Further it is observed that parameter uncertainty is large compared to the model uncertainty.     

Variable power‐law scaling of hillslope Hortonian rainfall–runoff processes

Hydrological studies focused on Hortonian rainfall–run‐off scaling have found that the run‐off depth generally declines with the plot length in power‐law scaling. Both the power‐law proportional coefficient and the scaling exponent show great variability for specific conditions, but why and how they vary remain unclear. In the present study, the scaling of hillslope Hortonian rainfall–run‐off processes is investigated for different rainfall, soil infiltration, and hillslope surface characteristics using the physically based cell‐based rainfall‐infiltration‐run‐off model. The results show that both temporally intermittent and steady rainfalls can result in prominent power‐law scaling at the initial stage of run‐off generation. Then, the magnitude of the power‐law scaling decreases gradually due to the decreasing run‐on effect. The power‐law scaling is most sensitive to the rainfall and soil infiltration parameters. When the ratio of rainfall to infiltration exceeds a critical value, the magnitude of the power‐law scaling tends to decrease notably. For different intermittent rainfall patterns, the power‐law exponent varies in the range of ?1.0 to ?0.113, which shows an approximately logarithmic increasing trend for the proportional coefficient as a function of the run‐off coefficient. The scaling is also sensitive to the surface roughness, soil sealing, slope angle, and hillslope geometry because these factors control the run‐off routing and run‐on infiltration processes. These results provide insights into the variable scaling of the Hortonian rainfall–run‐off process, which are expected to benefit modelling of large‐scale hydrological and ecological processes.     

Construction of Intensity–Duration–Frequency (IDF) curves for precipitation at Lubumbashi,Congo, under the hypothesis of inadequate data

Abstract

Intensity–Duration–Frequency (IDF) curves for precipitation constitute a probabilistic tool and have proven useful in water resources management. In particular, IDF curves for precipitation enable questions on the extreme character of precipitation to be answered. The construction of IDF curves for precipitation is difficult or impossible in tropical areas due to the lack of long-term extreme precipitation data. A technique is proposed to overcome this shortcoming by combining limited high-frequency information on rainfall extremes with long-term daily rainfall information. It may be regarded as an extension of Koutsoyiannis' approach. Using this technique, IDF curves for precipitation are produced for Lubumbashi in Congo.

Citation Van de Vyver, H. & Demarée, G. R. (2010) Construction of Intensity–Duration–Frequency (IDF) curves for precipitation at Lubumbashi, Congo, under the hypothesis of inadequate data. Hydrol. Sci. J. 55(4), 555–564.     

How to construct future IDF curves,under changing climate,for sites with scarce rainfall records?

Optimal designs of stormwater systems rely very much on the rainfall Intensity–Duration–Frequency (IDF) curves. As climate has shown significant changes in rainfall characteristics in many regions, the adequacy of the existing IDF curves is called for particularly when the rainfall are much more intense. For data sparse sites/regions, developing IDF curves for the future climate is even challenging. The current practice for such regions is, for example, to ‘borrow’ or ‘interpolate’ data from regions of climatologically similar characteristics. A novel (3‐step) Downscaling‐Comparison‐Derivation (DCD) approach was presented in the earlier study to derive IDF curves for present climate using the extracted Dynamically Downscaled data an ungauged site, Darmaga Station in Java Island, Indonesia and the approach works extremely well. In this study, a well validated (3‐step) DCD approach was applied to develop present‐day IDF curves at stations with short or no rainfall record. This paper presents a new approach in which data are extracted from a high spatial resolution Regional Climate Model (RCM; 30 × 30 km over the study domain) driven by Reanalysis data. A site in Java, Indonesia, is selected to demonstrate the application of this approach. Extremes from projected rainfall (6‐hourly results; ERA40 Reanalysis) are first used to derive IDF curves for three sites (meteorological stations) where IDF curves exist; biases observed resulting from these sites are captured and serve as very useful information in the derivation of present‐day IDF curves for sites with short or no rainfall record. The final product of the present‐day climate‐derived IDF curves fall within a specific range, +38% to +45%. This range allows designers to decide on a value within the lower and upper bounds, normally subjected to engineering, economic, social and environmental concerns. Deriving future IDF curves for Stations with existing IDF curves and ungauged sites with simulation data from RCM driven by global climate model (GCM ECHAM5) (6‐hourly results; A2 emission scenario) have also been presented. The proposed approach can be extended to other emission scenarios so that a bandwidth of uncertainties can be assessed to create appropriate and effective adaptation strategies/measures to address climate change and its impacts. Copyright © 2013 John Wiley & Sons, Ltd.     

Risk assessment of extreme air pollution based on partial duration series: IDF approach

The occurrences of extreme pollution events have serious effects on human health, environmental ecosystems, and the national economy. To gain a better understanding of this issue, risk assessments on the behavior of these events must be effectively designed to anticipate the likelihood of their occurrence. In this study, we propose using the intensity–duration–frequency (IDF) technique to describe the relationship of pollution intensity (i) to its duration (d) and return period (T). As a case study, we used data from the city of Klang, Malaysia. The construction of IDF curves involves a process of determining a partial duration series of an extreme pollution event. Based on PDS data, a generalized Pareto distribution (GPD) is used to represent its probabilistic behaviors. The estimated return period and IDF curves for pollution intensities corresponding to various return periods are determined based on the fitted GPD model. The results reveal that pollution intensities in Klang tend to increase with increases in the length of time between return periods. Although the IDF curves show different magnitudes for different return periods, all the curves show similar increasing trends. In fact, longer return periods are associated with higher estimates of pollution intensity. Based on the study results, we can conclude that the IDF approach provides a good basis for decision-makers to evaluate the expected risk of future extreme pollution events.     

Constrained scaling approach for design rainfall estimation

Rainfall depth (or intensity) of the same frequency should follow a non-decreasing relationship with rainfall duration. However, due to the use of finite samples and sampling error, rainfall frequency analysis could yield rainfall intensity (depth)–frequency (IDF, DDF) curves of different durations that might intersect among them. Results of this kind violate physical reality and it is more likely to occur when rainfall record length gets shorter. To ensure the compliance of the physical reality, this paper applied the scale-invariant approach, in conjunction with constrained regression analysis, to circumvent intersections in rainfall IDF or DDF curves. Rainfall data of various durations at rain gauge in Hong Kong are used to demonstrate the procedure. Numerical investigation indicates that the proposed procedure yields more reasonable results than those based on the conventional frequency analysis, especially when only a small sample of data are available.