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A Scaling Model of a Storm Hyetograph
Abstract
Empirical evidence suggests that statistical properties of storm rainfall at a location and within a homogeneous season have a well-structured dependence on storm duration. To explain this dependence, a simple scaling model for rainfall intensity within a storm was hypothesized. It was shown both analytically and empirically that such a model can explain reasonably well the observed statistical structure in the interior of storms, thus providing an efficient parametrization of storms of varying durations and total depths. This simple scaling model is also consistent with, and provides a theoretical basis for, the concept of mass curves (normalized cumulative storm depth versus normalized cumulative time since the beginning of a storm) which are extensively used in hydrologic design. In contrast, popular stationary models of rainfall intensity are shown unable to capture the duration dependent statistical structure of storm depths and also inconsistent with the concept of mass curves.
... The theoretical bases for the use of scale invariance in the field of extreme precipitation were comprehensively discussed by Koutsoyiannis and Foufoula-Georgiou (1993), Burlando and Rosso (1996) and Menabde et al. (1999). Koutsoyiannis and Foufoula-Georgiou (1993) derived a stochastic model, underpinned by scaling relationships among events with different observation scales, for obtaining design hyetographs. ...
... The theoretical bases for the use of scale invariance in the field of extreme precipitation were comprehensively discussed by Koutsoyiannis and Foufoula-Georgiou (1993), Burlando and Rosso (1996) and Menabde et al. (1999). Koutsoyiannis and Foufoula-Georgiou (1993) derived a stochastic model, underpinned by scaling relationships among events with different observation scales, for obtaining design hyetographs. Burlando and Rosso (1996), in turn, evaluated the construction of depth-duration-frequency curves based on the application of scaling theory, which proved to be an efficient and parsimonious for this purpose, while Menabde et al. (1999) applied a simple scaling model, based on normalized quantile functions, to develop intensity-duration-frequency (IDF) curves in regions with different climates. ...
The modeling of sub-daily extreme rainfall has long constituted a challenge for hydrologists in view of the limited availability of data, both in time and space. In this context, the scale invariance principle, which formally links precipitation intensities across a range of time scales under a rigorous theoretical underpin, comprises a useful and parsimonious tool for deriving probabilistic models for the referred stochastic variables. This paper aims at reviewing the history of application of the scale invariance principle to the marginal distributions of intense precipitation, as well as presenting recent developments in this field. We first address the basic concepts and the mathematical formalism of scale invariant models, discussing distinct scaling regimes across durations and estimation procedures. Next, we present a comprehensive review on at-site and regional stationary scaling models applied in several regions of the world, with focus on their underlying assumptions and main limitations. Finally, we address extensions of the rationale for nonstationary models as a means of accommodating potential effects of climate change in extreme short-duration rainfall. While discussing each of these frameworks, we indicate potential research gaps and modeling developments which could further advance the understanding of scaling behavior of extreme precipitation and improve statistical inference. Hence, this review may constitute a useful guide for practitioners and motivate future research in the modeling of short-duration extreme rainfall.
... Besides the reconstruction theorem and the multifractal analysis, another monofractal analyses has been widely used in precipitation and hydrological research along the last few decades (Gupta and Waymire, 1990;Koutsoyiannis and Foufoula-Georgiou, 1993;Burlando and Rosso, 1996;Menabde et al., 1999;Nhat et al., 2007;Bara et al., 2010;Rodríguez-Solà et al., 2017;Casas-Castillo et al., 2018a;2022), contributing to quantify the evolution of the time series irregularity, and a good interpretation of fractal scaling also would be found in the research of Zhang et al. (2017). As other natural phenomena looking statistically the same regardless of the temporal scale of observation, rainfall often exhibits self-similarity, with some properties accomplishing power laws of a scale parameter λ which is the ratio between any two temporal scales. ...
A long and complete series of monthly rainfall amounts corresponding to Barcelona city (Catalonia, NE Spain), exceeding two centuries (years 1786‐2019), is analysed in detail. The obtained results of periodicity (annual scale), time trends (monthly, seasonal and annual scales), statistical distribution (seasonal and annual scale) and fractal/multifractal structures and self‐similarity at monthly scale, depict the complex structure of this pluviometric regime, which is characterised by moderate increasing and decreasing trends on rain amounts, varying from +0.08 mm/year (February) to ‐0,07 mm/year (September) and quite evident changes on the pluviometric trends at annual and seasonal scales when the rainfall data is analysed for segments of 50 years from 1800 up to 2019. A good example could be the relevant change on the annual scale time trend, from +0.77 mm/year (years 1800‐1850) to ‐0.17 mm/year (years 1950‐2019). Clear evidences of decreasing pluviometry for spring, summer and autumn for the last segment (1950‐2019) in comparison with the other three segments, including years 1800‐1950, are also detected. Additionally, increasing rainfall patterns complexity, expected difficulties on monthly rainfall forecasting and the increasing irregularity of monthly amounts is also detected by interpreting fractal and multifractal results. Irregularity increases on the monthly rainfall series and on the rainfall regime complexity derived from multifractal parameters, could be associated with the very notable increase of CO₂ emissions into the atmosphere, globally varying from 51.1 x 10⁶ metric tonnes (year 1820) to 36.6 x 10⁹ metric tonnes (year 2019) and the tropospheric concentration increasing from 280.8 ppm (year 1850) to 397.5 ppm (year 2014), being the probable relationship between tropospheric concentrations and changes on rainfall patterns the objective of future researches. This article is protected by copyright. All rights reserved.
... Foufoula-Georgiou (1989) defined the storm transposition area (STA) as the area within which all the occurred storms can be transposed anywhere with the same occurrence probability but with an adjustment to their depth. Indeed, in homogeneous regions, storms are expected to have similarities in their internal structure despite their different durations and total rainfall depths (Koutsoyiannis & Foufoula-Georgiou, 1993). ...
Flash flood risk management is among the key topics of the European Flood Directive. Design hydrographs of small river basins could underestimate the hydraulic risk in case of intense and short‐duration events because of the intensification of hourly rainfall and the design rainfall depths uncertainty in often ungauged river basins. The use of synthetic hyetographs obtained from past observed events can be useful for a better evaluation of the flash flood hazard at the river basin scale. For this reason, a simple approach based on rainfall scaling and storms transposition of occurred events is presented to identify areas where design discharges underestimate the flash flood risk. The methodology considers the spatial distribution of the observed storms over a specific zone and the areal reduction factors to scale the observed hyetographs for different river basins extensions. Hyetographs of past observed events are used as input of a hydrological model in a set of river basins located in Northern Tuscany (Italy). The results show how peak discharges of short‐duration events are usually greater than the design floods of the small river basins with an area generally less than 30 km2.
... Applying simple scaling to the relationship between the IDF properties of short-term rainfall is possible. The determination of the rainfall scaling properties is based on the general shape of the following IDF formula in the form [32]: ...
The paper aims to assess the impact of deforestation due to windstorms on runoff in small mountain river basins. In the Boca and Ipoltica River basins, changes in forested areas were assessed from available historical and current digital map data. Significant forest losses occurred between 2004 and 2012. During the whole period of 1990–2018, forested areas in the Boca river decreased from 83% to 47% and in the Ipoltica River basin from 80% to 70%. Changes in runoff conditions were assessed based on an assessment of changes in the measured time series of the hydrometeorological data for the years 1981–2016. An empirical hydrological model was used to determine the design peak discharges before and after significant windstorms were estimated for different rain intensities and return periods. The regional climate scenario for the period 2070–2100 was used to assess the current impact of climate change and river basin deforestation on predicted changes in design floods in the coming decades. The effect of deforestation became evident in the extreme discharges, especially in future decades. In the Boca River basin, the estimated design floods increased by 59%, and in the Ipoltica River basin by 172% in the case of the 100-year return period.
... The exponential distribution is connected to the Poisson arrival process, and commonly used to model the storm duration (e.g. Eagleson 1972;Koutsoyiannis & Georgiou 1993;Lambert & Kuczera 1998). A two-parameter exponential distribution is adopted in this study to model the windstorm duration (Dur) with the CDF given by ...
Non-cyclonic windstorm is a major natural peril that causes substantial economic losses to housing in New South Wales, Victoria and southeastern Queensland where the majority of Australia’s population live. Housing in non-cyclonic regions of Australia comprises a large portion of metal-clad contemporary houses with complex hip-roof geometries. According to post-damage surveys, wind-induced losses to Australian contemporary houses mainly result from direct wind damage to roof and windows as well as associated rainwater damage to building interior and contents. Construction defects have also been observed as a major contributor to housing damage during windstorms. There is a lack of systematic approaches to assess wind and rainfall losses for metal-clad contemporary houses in non-cyclonic regions of Australia with an explicit modelling of construction defects. Risk mitigation and climate adaptation aim to improve building resilience to wind hazards and reduce economic losses associated with wind damage under a changing climate. Although several mitigation/adaptation measures for Australian housing have been proposed in the literature, quantitative evaluations of their cost-effectiveness are still limited. There is a need for a quantitative decision support model to assist relevant decision-makers and stakeholders in choosing appropriate mitigation/adaptation measures for the protection of houses against wind hazards.
This PhD research develops a probabilistic risk assessment (PRA) and decision support framework for metal-clad contemporary houses subjected to non-cyclonic windstorms. The PRA framework integrates hazard modelling for extreme wind speed and associated rainfall, reliability-based wind damage assessment for roof system and windows, rainwater intrusion evaluation and economic loss estimation. A probabilistic construction defect model including five types of defects in roof connections is also developed, which can be readily integrated into the PRA framework to account for the reduced roof reliability and performance due to defective roof components. A scenario-based approach is adopted to include climate change impact on extreme wind speed and associated rainfall. The developed PRA framework is illustrated on representative metal-clad contemporary houses in two Australian cities − Brisbane and Melbourne (i.e. capital cities in Queensland and Victoria). Risk-based decision models are employed to provide decision support to identify cost-effective measures for risk mitigation and climate adaptation. The magnitude of uncertainty and decision-makers’ risk preferences (i.e. risk aversion, risk-neutrality and risk proneness) are taken into account in the decision-making. The implications for mitigation/adaptation decisions with the consideration of insurance and economic incentives are also discussed.
The PRA and decision analysis results suggest that rainwater damage to building interior and contents is a major contributor to economic risks for Australian contemporary houses subjected to non-cyclonic windstorms. Installing window shutters is a promising mitigation/adaptation measure for homeowners in Brisbane to implement. Climate change has a marginal influence on the cost-effectiveness of mitigation/adaptation measures. The outputs of this PhD research can assist insurance and re-insurance industries in catastrophe risk management, government agencies in disaster planning and management, and homeowners in choosing cost-effective mitigation/adaptation measures to protect their home against windstorms. This research paves the way towards a more resilient residential community under wind hazards.
... As many other complex phenomena in nature, showing remarkable regularities in terms of scaling relationships with size (Mandelbrot 1982;Feder 1988;West 2017), rainfall processes likewise follow a similar law with duration. Regularities in the temporal pattern exhibited by storm records, known as scale-invariant properties of rainfall (Lovejoy and Schertzer 1985;Gupta and Waymire 1990;Koutsoyiannis and Foufoula-Georgiu 1993;Willems 2000), could help in characterizing extreme storms at partially gauged sites better than the application of traditional statistical techniques (Burlando and Rosso 1996). The most important practical implication of probabilistic models based on scaling properties of hydrologic processes is that statistical features of the process at finer resolutions can be inferred by the probabilistic models of the process at higher temporal aggregation (Burlando and Rosso 1996;Menabde et al. 1999;Nguyen et al. 1998Nguyen et al. , 2002Borga et al. 2005). ...
Design of urban drainage systems or flood risk assessment in small catchments often requires knowledge of very short-duration rainfall events (less than 1 h). Unfortunately, data for these events are often unavailable or too scarce for a reliable statistical inference. However, regularities in the temporal pattern exhibited by storm records, known as scaling properties of rainfall, could help in characterizing extreme storms at partially gauged sites better than the application of traditional statistical techniques. In this work, a scaling approach for estimating the distribution of sub-hourly extreme rainfall in Sicily (Italy) is presented based on data from high-resolution rain gauges with a short functioning period and from low-resolution rain gauges with longer samples. First, simple scaling assumption was tested for annual maxima rainfall (AMR) data from 10 min to 24 h duration, revealing that the simple scaling regime holds from 20 to 60 min for most of the stations. Then, scaling homogeneous regions were classified based on the values of the scaling exponent. In each region, this parameter was regionalized through power-law relationships with the median of 1 h AMR data. After that, regional Depth Duration Frequency (DDF) curves were developed by combining the scale-invariant framework with the generalized extreme value (GEV) probability distribution and used to estimate T-year sub-hourly extreme rainfalls at sites where only rainfall data for longer durations (≥ 1 h) were available. The regional GEV simple scaling model was validated against sub-hourly historical observations at ten rain gauges, generally yielding, in relation to the scaling exponent value, to similar or better sub-hourly estimates than empirical approach.
... If the basic probability distribution of some physical measurements at one scale is similar to the distribution at another scale, a regular procedure fulfills the simple scaling property (Nhat et al. 2007). Many researchers have studied the main theoretical development of scaling (e.g., Gupta and Waymire (1990); Koutsoyiannis and Foufoula-Georgiou 1993;Menabde et al. 1999 andKuzuha et al. 2005). ...
Increasing the concentration of CO2 and other greenhouse gases in the atmosphere makes a large-scale change in atmospheric processes. Among the most dramatic effects of climate change, its impact on the type, amount, and intensity of rainfall is of particular importance. The goal of the present study was to provide a new method to investigate climate change impacts on rainfall intensity—duration—frequency (IDF) in the southwest of Iran. For this purpose, IDF curves were determined using recorded short-term rainfall from 15 stations. Then, the efficiency of the scale invariance theory was evaluated in determining IDFs by comparing IDFs resulting from observed data and IDFs estimated from the scaling model. Results demonstrated that two sets of IDFs had a good coincidence. In the next step, the data from three RCM models of the CORDEX project under two scenarios (RCP4.5 and RCP8.5) were used, and daily rainfall was downscaled by the musica package. Finally, IDFs were determined for two future periods including 2025–2049 as near future and 2075–2099 as far future. The result showed that rainfall intensity decreased in the north and central parts of the region.
A simple scaling analysis was performed in Andalusia (Spain) using daily records from 377 selected stations covering the temporal period between 1870 and 2018. Since Andalusia is a region of considerable climatic variety, with notably wet areas as well as extremely dry zones, this study is useful to investigate the relationship between the simple scaling parameter value and the characteristic rainfall regime of a place. Despite the great correspondence with the average annual precipitation (PRCPTOT), a clear dependence on rainfall irregularity was observed, revealed by the ratio of the maximum daily precipitation and PRCPTOT, as well the wet spells frequency index CWD. The spatial distribution of the simple scaling parameter captured the increasing influence of the Mediterranean Sea towards the East. The easternmost dry areas are clearly influenced by Mediterranean disturbances, with a high proportion of convective rainfall and an irregular rainfall pattern. Using a simple scaling parameter, the generalized equations of the intensity-duration- frequency (IDF) curves, of great hydrological interest were calculated for the eight Andalusian provincial capitals. Moreover, the temporal trends of this parameter in the four past decades were studied in the different areas with the aim of determining if changes in their rainfall patterns due to global warming could be detected.
ABSTRACT
Short-duration rainfall data are basic inputs to many rainfall-runoff models for generating the flood
hydrographs. But these data are scarce in comparison to daily data, which is abundantly available, but
cannot be directly used unless disaggregated to a shorter duration. This paper adopts a simple scaling
approach for disaggregation of daily design precipitation into the desired duration and the time
distribution of precipitation based on 3p Beta distribution herein referred to as the scaled 3p-Beta
approach. The rainfall quantiles and their temporal distribution, when used as an input to the
rainfall-runoff model, generate the design flood hydrograph at the ungauged location. The Upper
Baitarani Basin under Mahanadi Sub-zone (3d) of India has been selected as the test catchment.
The efficacy of the proposed scaled 3p-Beta approach, and also the rational approach coupled
with CWC-2p Gamma UH was verified in the test catchment for design flood computation. The
findings reveal that the methodology ensures encouraging results directly derivable from the
abundantly available daily rainfall data and can be applied in any ungauged catchment in a region.
Intensity–duration–frequency (IDF) analyses of rainfall extremes provide critical information to mitigate, manage, and adapt to urban flooding. The accuracy and uncertainty of IDF analyses depend on the availability of historical rainfall records, which are more accessible at daily resolution and, quite often, are very sparse in developing countries. In this work, we quantify performances of different IDF models as a function of the number of available high-resolution (Nτ) and daily (N24h) rain gauges. For this aim, we apply a cross-validation framework that is based on Monte Carlo bootstrapping experiments on records of 223 high-resolution gauges in central Arizona. We test five IDF models based on (two) local, (one) regional, and (two) scaling frequency analyses of annual rainfall maxima from 30-min to 24-h durations with the generalized extreme value (GEV) distribution. All models exhibit similar performances in simulating observed quantiles associated with return periods up to 30 years. When Nτ > 10, local and regional models have the best accuracy; bias correcting the GEV shape parameter for record length is recommended to estimate quantiles for large return periods. The uncertainty of all models, evaluated via Monte Carlo experiments, is very large when Nτ ≤ 5; however, if N24h ≥ 10 additional daily gauges are available, the uncertainty is greatly reduced and accuracy is increased by applying simple scaling models, which infer estimates on subdaily rainfall statistics from information at daily scale. For all models, performances depend on the ability to capture the elevation control on their parameters. Although our work is site specific, its results provide insights to conduct future IDF analyses, especially in regions with sparse data.
The influence of time-varying rainfall on overland flow is investigated. The kinematic wave equations for turbulent flow (across a plane, impermeable surface) are solved using a time-varying rainfall appropriate for thunderstorms. The peak discharge is shown to be a function of surface length, total precipitation, storm duration, and time to equilibrium for rainfall of constant intensity. -from ASCE Publications Abstracts
ABSTRACT Hershenhorn, J. and Woolhiser, D.A., 1987. Disaggregation ofdaily rainfall. J. Hydrol., 95:299-322. A parameter-efficient model,for disaggregating,daily,rainfall,into individual,storms,is presen ted. This model allows simulation of the number of rainfall events (storms) in a day, and the amount, duration, and starting time ofeach event, given only the total rainfall on that day and on the preceding and following days. Twenty-three,years ofdata for July and August, from a gage on the Walnut Gulch Experimental,Watershed, were used to find the appropriate model structure and to estimate,parameters.,Statistical,tests indicate,that simulated,sequences,of storms compare favorably with observed sequences, and that the disaggregation model structure and parameters identified,for one,gage,provide,a satisfactory,fit for,three,stations,within,a 121 km,radius,where elevation differs by as much as 244m, and mean annual rainfall differs by up to 76mm.
A concise stochastic model for the nondimensional thunderstorm rainfall
process at a point is proposed. The accumulated precipitation process
for individual thunderstorms is nondimensionalized by dividing the
precipitation at any time by the total precipitation and the elapsed
time by the total duration. The dimensionless process is divided into
100 equal time increments, and the depth increments are rescaled to
range between 0 and 1. The sequence of rescaled increments
Z1, Z2,…, Z9 are assumed to
represent a nonhomogeneous Markov process in discrete time with
continuous state space. The expected value of the kth rescaled
increment, given the k-1st increment, is assumed to be a linear function
of that increment, and the marginal distribution of the first increment
and the conditional distributions are assumed to be described by the
beta distribution. An analyses of data for 275 thunderstorms observed at
the Walnut Gulch Experimental Watershed in southeastern Arizona showed
that the proposed model structure is a good approximation for this
region. The number of model parameters can be reduced from 26 to a
minimum of 10 by approximating the 2 parameters in the conditional
expectation function and the conditional beta parameter as polynomial
functions of the dimensionless time. Likelihood ratio tests and the
Akaike information criterion suggest that the dependence parameters are
independent of storm amount and duration, but the conditional beta
parameter αk is larger for short-duration storms than
for long-duration storms. A 13-parameter model is recommended for
disaggregating thunderstorm rainfall in southeastern Arizona.
Point precipitation is represented by Poisson arrivals of rectangular
intensity pulses that have random depth and duration. By assuming the
storm depths to be independent and identically gamma distributed, the
cumulative distribution function for normalized annual precipitation is
derived in terms of two parameters of the storm sequence, the mean
number of storms per year and the order of the gamma distribution. In
comparison with long-term observations in a subhumid and an arid climate
it is demonstrated that when working with only 5 years of storm
observations this method tends to improve the estimate of the variance
of the distribution of the normalized annual values over that obtained
by conventional hydrologic methods which utilize only the observed
annual totals.
A two-level point stochastic model for the rainfall occurrences at a given rainfall station is constructed in the time dimension. The model is a cluster process of the Neyman-Scott type. The model has the rainfall-generating mechanisms as its primary level and the rainfalls that are generated by these mechanisms as the secondary level. It uses infinite superposition of rainfalls and has a very flexible dependence structure. The model is fitted to daily rainfall sequences in Indiana after these are stationarized by a transformation. The fit of the model is then tested in terms of its correlation and marginal probability characteristics. The present form of the Neyman-Scott cluster model is time homogeneous. Therefore the Neyman-Scott process, as presented in this paper, may be of practical use only for modeling the stationary rainfall occurrences.
The rainfalls used in the practical estimation of design floods are generally based on frequency-duration relationships derived from recorded intense burst of rainfall of various durations rather than from complete storms. These recorded intense bursts are therefore used in the derivation of the temporal patterns. The method produces patterns that incorporate average variability of intense rainfall and also the most likely sequence of intensities. Use of these patterns should minimize the introduction of joint probabilities into the design flood model and aid in estimation of a flood with the same frequency as the design rainfall. The method provides patterns with average or typical variations in intensity, in contrast to simple averaging which is shown to be generally unlikely yield satisfactory patterns. Frequency distributions of rainfall during various periods immediately antecedent to the recorded intense bursts are also derived, and these antecedent rainfalls in flood estimation are considered.
Observations on temporal rainfall are invariably taken in cumulative amounts over disjoint intervals representing different time scales such as hours, days, etc. At these scales a mathematical description of the rainfall process is shown to depend on the scale of measurements. This illustration is carried out employing three stochastic models having different dependence structures which range from complete independence and Markovian dependence to non-Markovian clustering dependence. The problem of estimating the model parameters is also shown to be tied to the scale of measurements. Hourly and daily rainfall data from Denver, Colorado, and daily rainfall data from Agua Fria, Venezuela, are employed for this illustration.