## 1. Introduction

[2] In hydrological risk analysis and design, one is often interested in the rainfall intensity averaged over a region of area *a* and duration *d*, with return period *T*. Plotting such extreme rainfall intensity *I*(*a*, *d*, *T*) against *d* for given *a* and *T* produces so-called intensity duration area frequency (IDAF) curves. For *a* ? 0 (precipitation at a point), the IDAF curves reduce to the familiar intensity duration frequency (IDF) curves [see, e.g., *Singh*, 1992, pp. 905-908]. While various definitions of *T* are in use, the one that makes best sense for many hydrologic applications and is easiest to handle analytically is the reciprocal of the exceedance rate [e.g., *Willems*, 2000; *Veneziano and Furcolo*, 2002a]. Accordingly, *I*(*a*, *d*, *T*) is the intensity *i* that satisfies

where *I*(*a*, *d*) is the average intensity in (*t*, *t* + *d*).

[3] Direct estimation of the IDAF curves from rainfall data requires very long records from spatially dense rain gauge networks or radar, which are seldom available. A common strategy to avoid direct estimation is to express *I*(*a*, *d*, *T*) as the product of the IDF value *I*(*d*, *T*) and the areal reduction factor (ARF) ?(*a*, *d*, *T*) = . Advantages of this factored approach are that the IDF values can be found using long records from single pluviometric stations (which are available at many locations) and, if the ARF does not vary much in space, the function ?(*a*, *d*, *T*) needs be estimated just once. In the literature, two different types of ARFs are found: the storm centered ARF and the fixed area ARF [*Hershfield*, 1962; *Omolayo*, 1993]. The storm centered ARF is associated with rainfall intensity within the isohyets of specific storm events, and does not have a precise return period interpretation. By contrast the fixed area ARF is obtained as the ratio of return period rainfall intensities over a fixed area and at a point. Hence the fixed-area definition of the ARF is better suited for hydrologic risk analysis, and is the one used throughout this paper.

[4] General properties of empirical ?(*a*, *d*, *T*) functions are that (1) ? increases as *a* decreases or *d* increases approaching unity as *a* ? 0 or *d* ? 8 and (2) for large *a* and small *d*, ? depends on *a* and *d* through /*d* [see *Natural Environmental Research Council* (*NERC*), 1975, vol. II, p. 40]. Whether and how ? depends on the return period *T* is less clear. *NERC* [1975] reports a weak dependence, whereas *Bell* [1976], *Asquith and Famiglietti* [2000], and *De Michele et al.* [2001] found that ? decreases significantly as *T* increases. ARF charts for routine hydrologic design [e.g., *Leclerc and Schaake*, 1972; *NERC*, 1975; *Koutsoyiannis*, 1997] typically give ? as a function of only *a* and *d*.

[5] An alternative to empirical IDF, IDAF and ARF estimation is to use theoretical analysis based on a random process representation of rainfall. Some studies have derived properties of the IDAF curves and the ARFs using nonscaling representations of rainfall. An early attempt in this direction was made by *Roche* [1966], who developed a theoretical approach to point and areal rainfall based on the correlation structure of intense storms. *Rodriguez-Iturbe and Mejia* [1974] extended *Roche*'s [1966] approach by assuming that the rainfall field is a zero mean stationary Gaussian process. A different approach to ARF estimation, based on crossing properties of random fields, was proposed by *Bacchi and Ranzi* [1996]. Properties of extremes of random functions were used also by *Sivapalan and Blöschl* [1998]. Finally, *Asquith and Famiglietti* [2000] derived the ARF as the catchment average of the ratio between the *T*-year rainfall depths at distance *r* from the centroid of the storm and at the centroid itself.

[6] Several other studies have assumed that rainfall intensity has scale invariance and used multifractal analysis to derive scaling properties of the IDF curves and ARFs with *a*, *d* and *T*. *De Michele et al.* [2001] have argued directly that the annual maximum value of *I*(*a*, *d*) could scale in a self similar or multifractal way with *a* and *d*. They then focus on the self similar case and specify the form of *I*(*a*, *d*, *T*) by reasoning on the limiting behavior when *a* ? 0 and *a* or *d* ? 8. By contrast, *Hubert et al.* [1998] and *Veneziano and Furcolo* [2002a] derive scaling properties of the IDF curves with *d* and *T* from the condition that rainfall has multifractal scale invariance. In a recent study, *Castro et al.* [2004] developed a multiftractal approach to explain how the IDAF values scale with *a*, *d* and *T*. Although not explicitly stated, the analysis of *Castro et al.* [2004] is valid only for large values of *T* [see *Langousis*, 2004].

[7] Multifractal models are attractive for studying rainfall scaling since they provide parsimonious representations of space-time rainfall fields [*Lovejoy and Schertzer*, 1995; *Gupta and Waymire*, 1993; *Deidda*, 2000], and possess scaling properties that likely determine the power law behaviors of empirical IDF, IDAF curves and ARFs. However, several studies [e.g., *Fraedrich and Larnder*, 1993; *Olsson et al.*, 1993; *Olsson*, 1995; *Menabde et al.*, 1997] have shown that temporal rainfall ceases to be multifractal for aggregation periods larger than about 2 weeks or smaller than several minutes. Also the analysis of rainfall fields in space and space-time reveals systematic deviations from exact multifractality [*Veneziano et al.*, 2005].

[8] In this paper, we study the behavior of the IDAF curves and the areal reduction factor ? under exact and approximate multifractality. We also analyze how these quantities depend on basin shape and rainfall advection and quantify the distortions in ARF scaling caused by the common practice of estimating area rainfall from sparse pluviometric networks. Finally, we show how the theoretical results explain many features of empirical ARFs.

[9] Since in certain limiting cases to be considered later (such as sampling along a line segment) the basin has finite extent but zero area, it is convenient to parameterize the basin through its shape *S* and largest linear dimension *l* rather than area *a*. For example, a basin could have in approximation the shape *S* of a disc, a square or a rectangle with a given aspect ratio. Together, *S* and *l* define the planar geometry of the basin (except for rigid translation, rotation, or reflection). Accordingly, we use the notation *I*(*S*, *l*, *d*, *T*) and ?(*S*, *l*, *d*, *T*) in place of the less descriptive *I*(*a*, *d*, *T*) and ?(*a*, *d*, *T*).

[10] In hydrologic applications, the averaging duration *d* is often related to the response time of the basin, for example the concentration time, but in general the region of interest needs not be a river basin and *d* needs not refer to the travel time of water particles. While hydrologic extremes remain the main practical focus of our analysis and, consequently, we refer to the geographical region as a “basin,” results hold beyond this application context.