## 1. Introduction

[2] Extreme rainfall events have been a frequent cause of economic, social, and environmental disasters and are critical to hydrologic design and risk assessment. This is why the quantitative modeling of intense rainfall has been a major focus of modern hydrologic science and engineering [e.g., *Chow et al*., 1988; *Singh*, 1992; *Stedinger et al*., 1993; *Smith*, 2001].

[3] If *I _{d}* is the average rainfall intensity in a generic interval of duration

*d*, the quantities of interest in extreme rainfall analysis are typically the distribution of the annual maximum

*I*

_{d}_{,1}(or more in general the

*T*-yr maximum

*I*

_{d}_{,}

*), the*

_{T}*T*-year intensity

*i*

_{d}_{,}

*defined as the value exceeded by*

_{T}*I*with annual rate 1/

_{d}*T*, and the distribution of the excess above high thresholds

*z*. The so-called intensity-duration-frequency (IDF) curves are logarithmic plots of

*i*

_{d}_{,}

*against*

_{T}*d*for different fixed

*T*.

[4] In standard practice and much previous theoretical work, the distributions used are of the generalized extreme value (GEV) type for the annual or high local maxima [see, e.g., *Koutsoyiannis et al*., 1998; *Martins and Stedinger*, 2000; *Gellens*, 2002; *Baillon et al*., 2004; *Koutsoyiannis*, 2004, 2007; *Overeem et al*., 2008] and the generalized Pareto (GP) type for the excesses *I _{d}*

_{,}

*[see for example*

_{z}*Davison and Smith*, 1990;

*Rosbjerg et al*., 1992;

*Stedinger et al*., 1993;

*Rosbjerg and Madsen*, 1995;

*Madsen et al*., 1997a, 1997b; and

*Madsen et al*., 2002]. These distributional choices are based on extreme value (EV) theory and extreme excess (EE) theory, which show that under certain conditions on the upper tail of

*I*, the distribution of the maximum of independent observations is attracted as

_{d}*n*→ ∞ to a GEV distribution with a certain shape parameter (index)

*k*and the distribution of

*I*

_{d}_{,}

*is attracted as*

_{z}*z*→ ∞ to a GP distribution with the same index [

*Gumbel*, 1958;

*Pickands*, 1975;

*Stedinger et al*., 1993;

*Leadbetter*, 1991]. Notice that

*n*→ ∞ is analogous to

*T*→ ∞ for the

*T*-yr maximum

*I*

_{d}_{,}

*. Here we take*

_{T}*k*to be positive for GEV distributions in the Frechet range.

[5] In the context of multifractal representations of rainfall [e.g., *Gupta and Waymire*, 1993; *Tessier et al*., 1993; *Lovejoy and Schertzer*, 1995; *Deidda*, 2000; *Veneziano and Langousis*, 2010], additional asymptotic results on *I _{d}*

_{,}

*and*

_{T}*I*

_{d}_{,}

*can be derived. Since multifractality establishes a link among the marginal distributions of*

_{z}*I*for different

_{d}*d*, it is possible to study the distribution of

*I*

_{d}_{,}

*not only under*

_{T}*T*→ ∞ for given

*d*(this reproduces the EV predictions), but also in the small-scale limit

*d*→ 0 for fixed

*T*and, importantly, under asymptotic conditions involving both

*d*and

*T*, such as (

*d*→ 0 and , for some

*α*> 0 [see

*Veneziano and Furcolo*, 2002;

*Langousis and Veneziano*, 2007;

*Veneziano et al*., 2006, 2009;

*Langousis et al*., 2009]). In all these cases, the asymptotic distribution of

*I*

_{d}_{,}

*is GEV, but the index*

_{T}*k*depends on the specific limit considered.

[6] Under multifractality, one can also derive asymptotic results for the IDF curves. It has long been known that the *T*-yr intensity *i _{d}*

_{,}

*scales in approximation as [e.g.,*

_{T}*Burlando and Rosso*, 1996;

*Willem*, 2000]. Explanations for this power law relationship and theoretical predictions of

*ξ*and

*δ*based on multifractal models were first proposed by

*Hubert et al*. [1998] and

*Veneziano and Furcolo*[2002], with focus on the small-scale limit

*d*→ 0. Additional asymptotic results, under , were derived by

*Langousis and Veneziano*[2007] and

*Langousis et al*. [2007]. In all cases, the IDF curves approach power laws, but the exponents

*ξ*and

*δ*depend on the limit considered.

[7] The first part of the paper presents these asymptotic multifractal results in a unified way. First, we consider an idealized “canonical model” in which rainstorms are well separated and have the same duration *D* and the same unit mean intensity. This choice of model simplifies the notation as well as the illustration of results. Then, we extend the canonical results to the more realistic case of random storm durations and intensities. The canonical results are based on prior research, whereas the extension to more realistic models is new.

[8] The existence of multiple asymptotic results motivates our study of intense rainfall under nonasymptotic conditions (finite *T* and *z* and noninfinitesimal *d*). The nonasymptotic analysis sheds light on a variety of issues, including the following:

[9] 1. For practical application, it is important to determine the properties of intense rainfall (*T*-yr maxima, excesses, IDF curves) for finite ranges of (*d*, *T*, *z*) and possibly find which asymptotic result, if any, produces accurate approximations.

[10] 2. One should explain why the empirical estimates of the index *k* are generally positive (in the Frechet range) whereas, based on frequently assumed distributions of *I _{d}* (exponential, gamma, lognormal, etc.), EV/EE theory gives

*k*= 0.

[11] 3. Estimates of *k* from annual maxima and excesses tend to decrease as the averaging duration *d* increases [e.g., *Baillon et al*., 2004]. However, any dependence of *k* on *d* leads to mathematical inconsistencies, such as making it more likely to exceed a high total precipitation in a short time interval than in a longer interval. One should find the source of this dependence and either impose *k* to be the same for all *d* or limit the range of use of models with variable *k*.

[12] 4. For the IDF curves, observed deviations from exact scaling include a tendency for the curves to converge as the averaging duration *d* or the return period *T* increase. As an example, Figure 1 shows empirical IDF curves for a 52 year hourly rainfall record from Heathrow airport, which display these characteristics. No asymptotic result produces such deviations from scaling.

[13] Section 'Scaling Models and Marginal Properties of Rainfall' recalls some basic scaling properties of rainfall within storms, introduces the canonical model, and presents basic results for moments and marginal distributions that are needed in the analysis that follows. Section 'Asymptotic Results for the Canonical Model' presents known asymptotic results on the IDF curves, maxima and excesses in the context of the canonical model. Section 'Nonasymptotic IDF Curves and Maxima' derives new nonasymptotic results on the IDF curves and annual maxima of the canonical model, including a nonasymptotic formula for the GEV index *k* as a function of *d*, the range of return periods *T* of interest, and the multifractal properties of rain. Explanations are given for the empirical observations on *k* and the IDF curves mentioned above. Section 'A More Realistic Model of Rainfall' adapts the canonical model results to more realistic representations of rainfall, and section 'Conclusions' summarizes the main findings and points at future research needs.