[1] The multifractal representation of rainfall and its use to predict rainfall extremes have advanced significantly in recent years. This paper summarizes this body of work and points at some open questions. The need for a coherent overview comes in part from the use of different terminology, notation, and analysis methods in the literature and in part from the fact that results are dispersed and not always readily available. Two important trends have marked the use of multifractals for rainfall and its extremes. One is the recent shift of focus from asymptotic scaling properties (mainly for the intensity-durationfrequency curves and the areal reduction factor) to the exact extreme distribution under nonasymptotic conditions. This shift has made the results more relevant to hydrologic applications. The second trend is a more sparing use of multifractality in modeling, reflecting the limits of scale invariance in space-time rainfall. This trend has produced models that are more consistent with observed rainfall characteristics, again making the results more suitable for application. Finally, we show that rainfall extremes can be analyzed using rather rough models, provided the parameters are fitted to an appropriate range of large-deviation statistics.Citation: Veneziano, D., A. Langousis, and P. Furcolo (2006), Multifractality and rainfall extremes: A review, Water Resour. Res., 42, W06D15,
[1] We derive scaling properties of the intensity-duration-frequency (IDF) curves under the condition that temporal rainfall is a stationary multifractal process. We find that under limiting conditions (very short durations D or very long return periods T), the IDF values are simple scaling with a power law dependence on D and T. The exponents of D and T differ in the two limiting cases of very short D and very long T and can be calculated from the moment scaling function K(q) of the rainfall time series. These theoretical findings differ from those of previous studies. They are validated through direct calculation and analysis of an actual rainfall record.
INDEX T ERMS:1854 Hydrology: Precipitation (3354); 1869 Hydrology: Stochastic processes; 1833 Hydrology: Hydroclimatology; 3250 Mathematical Geophysics: Fractals and multifractals; KEYWORDS: rainfall process, extreme precipitations, scaling processes Citation: Veneziano, D., and P. Furcolo, Multifractality of rainfall and scaling of intensity-duration-frequency curves, Water Resour.
[1] Practical methods for the estimation of the intensity-duration-frequency (IDF) curves are usually based on the observed annual maxima of the rainfall intensity I(d) in intervals of different duration d. Using these historical annual maxima, one estimates the IDF curves under the condition that the rainfall intensity in an interval of duration d with return period T is the product of a function a(T) of T and a function b(d) of d (separability condition). Various parametric or semiparametric assumptions on a(T) and b(d) produce different specific methods. As alternatives we develop IDF estimation procedures based on the marginal distribution of I(d). If the marginal distribution scales in a multifractal way with d, this condition can be incorporated. We also consider hybrid methods that estimate the IDF curves using both marginal and annual maximum rainfall information. We find that the separability condition does not hold and that the marginal and hybrid methods perform better than the annual maximum estimators in terms of accuracy and robustness relative to outlier rainfall events. This is especially true for long return periods and when the length of the available record is short. Marginal and hybrid methods produce accurate IDF estimates also when only a few years of continuous rainfall data are available.Citation: Veneziano, D., C. Lepore, A. Langousis, and P. Furcolo (2007), Marginal methods of intensity-duration-frequency estimation in scaling and nonscaling rainfall, Water Resour.
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