A simple scaling model for extreme rainfall

Abstract

Abstract. The simple scaling hypothesis is applied to the intensity-duration-frequency (IDF) description of rainfall. It is shown that the cumulative distribution function for the annual maximum series of mean rainfall intensity has a simple scaling property over the range 30 min to 24 hours and in some instances to 48 hours. This behavior is demonstrated through an examination of the scaling properties of the moments and the scaling of the parameters of an extreme value distribution fitted to the data. A simple analytical formula for the IDF relationship is proposed, which embodies the scaling behavior. Once the scaling parameter has been obtained for a gauge or set of gauges in a region, this formula enables the calculation of rainfall amounts, of a chosen return period and duration shorter than a day, directly from the information obtained from the analysis of daily data. IntroductionThe relationship between rainfall intensity, duration, and frequency has been of considerable interest to practicing engineers and hydrologists for over a century. Sherman [1905] where 0 and ,/ are phenomenological parameters to be estimated.Daily rainfall is by far the most accessible form of rainfall data, and long sequences of rainfall data at higher time resolution are still relatively rare despite the fact that the technology to record such data has been available since the 1880s. There is much to be gained therefore in developing a methodology that is able to use daily rainfall statistics to infer the IDF characteristics for short duration rainfall. One such attempt was made by Adamson [1981], who compiled the statistics for 2500 rain gauges in South Africa. For each of the stations, he listed (inter alia) the estimates of the 1-, 2-, 3-, and 7-day total rainfall occurring with return periods of 2-200 years. To make these useful for intervals of less than a day, he compiled a table of disaggregation coefficients for the coastal and inland regions of South Africa, based on digitized data, fitting a model of the type [Bell, 1969] referred to above and purporting to be independent of return periods. In contrast to the above treatments, which depend on curvefitting techniques, a natural source for theories regarding the rescaling of rainfall statistics is to be found in the scaling hypotheses popularized by Mandelbrot [1982] and Lovejoy and Schertzer [1985]. Burlando and Rosso [1996] in a pioneering paper sought to apply the scaling hypotheses to annual maximum series of rainfall depth. In their work the scaling and multiscaling properties of the statistical moments of rainfall depth of different duration were analyzed and a lognormal probability distribution was used to model its statistical properties.In the present paper it will be shown that based on the empirically observed scaling properties of rainfall and some general assumptions about the cumulative distribution function (CDF) for the annual maximum of the mean rainfall 335

Estimation of IDF Curves of Extreme Rainfall by Simple Scaling in Northern Oueme Valley, Benin Republic (West Africa)

“…The basic theoretical development of scaling has been investigated by many authors and considerable amount of studies were devoted to extreme rainfall and its scaling properties, including Waymire and Gupta (1981) ;Waymire, et al(1984); ; Marien and Vandewiele (1986); Sivapalan and Wood (1987); Gupta and Waymire(1990); Rosso and Burlando (1990); Smith (1992); Koutsoyiannis and Foulfoula-Georgiu (1993); Burlando and Rosso (1996); Veneziano et al (1996), Bendjoudi et al (1997) Ghanmi (2014). In this work the simple scaling hypothesis is adopted to test the scaling behavior of rainfall in northern of Benin, following the methods described in Menabde et al (1999); Yu et al (2004); Kuzuha et al (2002); Nhat et al (2007); Bara et al (2009) and Ghanmi (2014) and are briefly outlined in the ensuing sections. The equality refers to identical propability distributions in both sides of the equations; denotes a scale factor and k is a scaling exponent.…”

“…For example the fractal properties of rainfall which implies scaling invariance were developed (e.g., Menabde, et al, 1999) where the scaling properties of the statistical moments of rainfall in South Africa over a range of durations from 30 min to 1440 min were examined. Yu, et al, (2004) used regional IDF formulas for estimating scaling exponents for 46 non-recording rain gauge stations in Taiwan.…”

“…Gupta and Waymire (1990) studied the concepts of simple and multiple scaling to characterize the probabilistic structure of the precipitation process, Koutsoyiannis and Foulfoula-Georgiu (1993) used a scaling model to predict storm hyetographs. Menabde et al (1999) showed that based on the empirically observed scaling properties of rainfall and some general assumptions about the cumulative distribution function for the annual maxima of mean rainfall intensity, it is possible to derive simple IDF relationships. Bendjoudi, et al (1997) used a multifractal point of view on rainfall IDF curves.…”

Development of IDF Curve for Dhaka City Based on Scaling Theory under Future Precipitation Variability Due to Climate Change

“…According to the scaling theory by Menabde [18], the scaling property in a strict sense can be written explicitly using the CDF:…”

The scaling law relating world point‐precipitation records to duration

“…The most richly developed approach to precipitation extremes attempts to estimate return period and related statistics for flood and water resource prediction. In this kind of analysis, represented by the work of Menabde et al (1999), the statistics of extreme values of precipitation are fitted to extreme value distributions, the objectives being to generate tools for prediction of the statistical properties of extremes, including maxima for given durations. This class of approaches is unsuitable for elucidating mechanisms underlying the scaling law for records.…”