Analytical calculation of storm volume statistics involving Pareto-like intensity-duration marginals (original) (raw)
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A Generalized Pareto intensity-duration model of storm rainfall exploiting 2Copulas
Journal of Geophysical Research, 2003
1] Stochastic models of rainfall, usually based on Poisson arrivals of rectangular pulses, generally assume exponential marginal distributions for both storm duration and average rainfall intensity, and the statistical independence between these variables. However, the advent of stochastic multifractals made it clear that rainfall statistical properties are better characterized by heavy tailed Pareto-like distributions, and also the independence between duration and intensity turned out to be a nonrealistic assumption. In this paper an improved intensity-duration model is considered, which describes the dependence between these variables by means of a suitable 2-Copula, and introduces Generalized Pareto marginals for both the storm duration and the average storm intensity. Several theoretical results are derived: in particular, we show how the use of 2-Copulas allows reproducing not only the marginal variability of both storm average intensity and storm duration, but also their joint variability by describing their statistical dependence; in addition, we point out how the use of heavy tailed Generalized Pareto laws gives the possibility of modeling both the presence of extreme values and the scaling features of the rainfall process, and has interesting connections with the statistical structure of the process of rainfall maxima, which is naturally endowed with a Generalized Extreme Value law. Finally, a case study considering rainfall data is shown, which illustrates how the theoretical results derived in the paper are supported by the practical analysis.
Revista Brasileira De Meteorologia, 2022
The design and management of various hydraulic structures (such as stormwater drains, bridges and dams) require the estimation of rainfall with duration of a few minutes up to 24 h or more. Intensity-duration-frequency (IDF) curves links probability of occurrence to a given rainfall intensity. The procedure for obtaining IDF curves basically involves two steps: (i) frequency analysis for different durations and (ii) modeling of IDF curves. In the first step, this study aimed to adequately select the upper tail weight of the following distributions: generalized extreme value (GEV), generalized logistic (GLO) and generalized Pareto (GPA). In the second step, this study aimed to evaluate the performance of three models of IDF curves. The traditional model (M1) was compared with empirical model (M2) and a second-order polynomial model (M3). To perform this study, rainfall data from the city of Caraguatatuba (São Paulo state, Brazil) for the period between 1971 and 2001 were used, for time intervals between 10 and 1440 min. The main conclusions were: (i) GLO and GEV had heavy upper tail while GPA had light upper tail, impacting quantiles with T > 100 years; (ii) M3 presents errors lower than M1 for return periods greater than 100 years.
Journal of Hydrology, 2002
In this paper we present an analytical formulation of the derived distribution of peak¯ood and maximum annual peak¯ood, starting from a simpli®ed description of rainfall and surface runoff processes, and we show how such a distribution is useful in practical applications. The assumptions on rainfall dynamics include the hypotheses that the maximum storm depth has a Generalized Pareto distribution, and that the temporal variability of rainfall depth in a storm can be described via power±law relationships. The SCS-CN model is used to describe the soil response, and a lumped model is adopted to transform the rainfall excess into peak¯ood; in particular, we analyse the in¯uence of antecedent soil moisture condition on the¯ood frequency distribution. We then calculate the analytical expressions of the derived distributions of peak¯ood and maximum annual peak ood. Finally, practical case studies are presented and discussed. q
2016
Hourly archived rainfall records are separated into individual rainfall events with an Inter-Event Time Definition. Individual storms are characterized by their depth, duration, and peak intensity. Severe events are selected from among the events for a given station. A lower limit, or threshold depth is used to make this selection, and an upper duration limit is established. A small number of events per year are left, which have relatively high depth and average intensity appropriate to small to medium catchment responses. The Generalized Pareto Distributions are fitted to the storm depth data, and a bounded probability distribution is fitted to storm duration. Peak storm intensity is bounded by continuity imposed by storm depth and duration. These physical limits are used to develop an index measure of peak storm intensity, called intensity peak factor, bounded on (0, 1), and fitted to the Beta distribution. The joint probability relationship among storm variables is established, combining increasing storm depth, increasing intensity peak factor, with decreasing storm duration as being the best description of increasing rainstorm severity. The joint probability of all three variables can be modelled with a bivariate copula of the marginal distributions of duration and intensity peak factor, combined simply with the marginal distribution of storm depth. The parameters of the marginal distributions of storm variables, and the frequency of occurrence of threshold-excess events are used to assess possible shifts in their values as a function of time and temperature, in order to evaluate potential climate change effects for several stations. Example applications of the joint probability of storm variables are provided that illustrate the need to apply the methods developed. I would like to thank Prof. Yiping Guo for his ongoing support over the years. My family has been an inspiration, and without their acceptance of my late return to school, I could never have continued. v Table of Contents Abstract iv Acknowledgements v Chapter 1. Introduction 1. Problem statement 2. Threshold analysis of rainstorm depth and duration statistics at Toronto, Canada 3. A probabilistic description of rain storms incorporating their peak intensity 4. Changes in heavy rain storm characteristics with time and temperature at four locations. 5. Example Applications Chapter 2. Threshold analysis of rainstorm depth and duration statistics at List of Tables 2.1 Rainstorm depth parameters and test statistics. 2.2 Rainstorm duration parameters and test statistics. 2.3 Statistical moments for conventional DDF annual maximum series. 3.1 Correlation of intensity peak factor with storm depth and duration. 3.2 Beta distribution parameters for intensity peak factor. 3.3 Correlation of intensity peak factor with storm depth (I pf − V) for three separate ranges of storm duration.. 3.4 Kendall's τ for intensity peak factor and storm duration (I pf-T), Ali-Mikhail-Haq and Frank copula parameters α. 3.5 Relative Goodness-of-fit, Theoretical models compared to Empirical Plotting Point Distribution. 3.6 Table A1. Summary of statistical analysis results for two stations in Toronto, Canada. 4.1 Station Descriptions, Threshold Analysis of Rainstorm Events March-November, ≤ 24hours 4.2 Threshold statistics 4.3 Difference in values, pre-and post-1980, of statistical parameters. 4.4 J u , average probability of exceedence of u v , pre-and post-1980. 4.5 χ 2 analysis of difference between pre-and post-1980 empirical distributions of storm variables. 4.6 Correlation analysis between V , T , and I pf , pre-1980 and post-1980. 4.7 Correlation analysis between the mean monthly temperature (M M T) when storm events occur and V , T , I pf , pre-and post-1980. 4.8 Average of mean monthly temperature when threshold-excess storm events occurred, pre-and post-1980. 4.9 Summary of major time-span and temperature changes and trends 5.1 Intensity peak factor and return periods of different design storm hyetographs. viii List of Figures 2.1 The plotting point distribution of rainstorm data is compared with the one-parameter exponential distribution. Crosses are used to represent actual measures of storm depth. 2.2 Comparison of the plotting point distribution with Type I and III distributions, storm threshold depth uv = 25 mm. Solid line shows GPD Type I modeled storm depths, dashed line plots Type III. 2.3 Figure 3a (left). Storm duration histogram based upon hourlyrainfall records, using IETD = 6 hours to define individual events. Figure 3b (right). Storm duration histogram, adjusted for start and finish time uncertainty. 2.4 100, 25 and 5-year return period storm-event depth-durations are compared with conventional DDF rainfall accumulation. Box symbol indicates conventional DDF GEV I-modeled rainfall accumulation. Cross symbol is used to show SEA GPD Type I-modelled depth. 2.5 Comparison of event-based and DDF annual maximum rainfall depths, for durations/time intervals of one hour. Actual conventional DDF depths are shown with a box symbol. SEA measures of storm depth are shown with a cross. 2.6 Comparison of event-based annual maximum storm depths for durations of 6 hours or less and DDF annual maximum depths for the 6-hour time interval. 2.7 Comparison of event-based annual maximum storm depths for durations of 12 hours or less, and DDF annual maxima rainfall accumulations for the 12-hour time interval. 3.1 Plot showing actual peak discharge for Mimico Creek (Ontario, Canada) catchment on y-axis, and predicted by regression equation on x-axis. Independent variables for regression equation are runoff depth and peak hour intensity. 3.2 Plotting point or empirical distribution of intensity peak factor (I pf , i pf), based upon Eq. (5), together with Beta distribution in solid line, fitted by method of moments. ix 3.3 Comparisons of plotting position and theoretical joint probability distributions. Plotting position empirical distribution established with Eq. (23) plotted on y-axis. Theoretical distributions based upon Eq. (19) for AMH copula, Eq. (20) for Frank copula, and Eq. (24) for simple joint probability plotted along x-axis. Figure 3(a) for Toronto station, Figure 3(b) for TPIA. Points falling closest to 45-degree line indicate best fit between empirical and theoretical models. 4.1 Storm duration frequency by 3-hour duration increments, pre-and post-1980 4.2 Mean storm depth versus mean monthly temperature 4.3 Mean storm duration and mean intensity peak factor, versus mean monthly temperature 4.4 Storm frequency versus temperature ranges. p-value of difference in proportions shown on reverse scale, significance assessed as p < 0.10 4.5 Mann-Kendall test statistic for Peoria station, on a 20-year moving window basis. 5.1 Maximization of peak discharge for 100-year joint return period. x Ph.D. Thesis-Barry Palynchuk McMaster-Civil Engineering 2. Threshold analysis of rainstorm depth and duration statistics at Toronto, Canada Chapter 2 in this thesis introduces the initial application of threshold analysis to the definition and characterization of extreme rainstorm events. One of the key contributions is the discovery that, for storms with depths exceeding a high threshold, storm depth and duration are not correlated, a rupture with the conventional paradigm of an assumed coupling between the two variables. This chapter has been reformatted from a paper presented by the author and Dr.
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