Water Resources Research

Rainfall extremes, excesses, and intensity-duration-frequency curves: A unified asymptotic framework and new nonasymptotic results based on multifractal measures


  • Daniele Veneziano,

    Corresponding author
    1. Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
    • Corresponding author: D. Veneziano, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Rm. 1-348, Cambridge, MA 02139, USA. (venezian@mit.edu)

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  • Seonkyoo Yoon

    1. Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
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[1] We present a unified asymptotic theory of rainfall extremes including annual maxima, excesses above high thresholds, and intensity-duration-frequency (IDF) curves that builds on previous findings and derive new nonasymptotic results. The analysis is based on stationary multifractal representations of rainfall and produces extensions of the familiar results from extreme value (EV) and extreme excess (EE) theories. The latter results apply to the T-yr maximum as T→∞ and the excess above z as z→∞. By exploiting the scaling relationship among the distributions of rainfall intensity for different averaging durations d, the multifractal asymptotics include, in addition, results in the small-scale limits d→0 and inline image with α > 0. In all cases, the maximum distributions are of the generalized extreme value (GEV) type, but the index k depends on the limit considered. Multifractal models also produce asymptotic scaling results for the IDF curves. For the nonasymptotic case (d and T finite), we obtain accurate approximations of the IDF curves and derive a semitheoretical formula for the index k of the GEV model that best approximates the distribution of the annual maximum over a finite range of return-period intensities. The nonasymptotic analysis explains several observed deviations of rainfall extremes from the asymptotic predictions, such as the tendency of k to decrease as the averaging duration d increases and the tendency of the IDF curves to converge as d or the return period T increase.

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 Id 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 Id,1 (or more in general the T-yr maximum Id,T), the T-year intensity id,T defined as the value exceeded by Id with annual rate 1/T, and the distribution of the excess inline image above high thresholds z. The so-called intensity-duration-frequency (IDF) curves are logarithmic plots of id,T against 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 Id,z [see for example 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 Id, the distribution of the maximum of independent observations inline image is attracted as n → ∞ to a GEV distribution with a certain shape parameter (index) k and the distribution of Id,z is attracted as 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 Id,T. Here we take 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 Id,T and Id,z can be derived. Since multifractality establishes a link among the marginal distributions of Id for different d, it is possible to study the distribution of Id,T not only under 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 inline image, 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 Id,T is GEV, but the index 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 id,T scales in approximation as inline image [e.g., Burlando and Rosso, 1996; Willem, 2000]. Explanations for this power law relationship and theoretical predictions of ξ and δbased 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 inline image, 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 Id (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.

Figure 1.

Empirical IDF curves for Heathrow Airport.

[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.

2. Scaling Models and Marginal Properties of Rainfall

2.1. Scaling Properties and Multifractal Models

[14] Our approach to rainfall extremes is based on two observations [Langousis and Veneziano, 2007; Veneziano and Lepore, 2012]:

[15] 1. Inside rainstorms, the average rainfall intensity over an interval of duration d, Id, has multifractal dependence on d, of the type

display math(1)

where the averaging duration d does not exceed some upper scaling limit dmax, s ≥ 1 is a scale contraction factor, and As is a non-negative random variable independent of Id. The variable Ae for a contraction factor s = e is referred to as the generator of the multifractal process.

[16] The analysis that follows applies to any admissible distribution of Ae, but for illustration, we consider the case when Ae has “beta-lognormal distribution” with parameters Cβ and CLN. This means that Ae has a probability mass inline image at 0 and inline image has lognormal distribution with inline image and inline image. The parameters Cβ and CLN are non-negative and satisfy Cβ + CLN < 1. Representative values inside rainstorms are Cβ = 0 − 0.10 and CLN = 0 10− 0.20 [Veneziano and Lepore, 2012]. Beta-lognormal generators have been found to accurately describe rainfall scaling especially during intense precipitation events, but other distributions have also been proposed [e.g., Tessier et al., 1993; Verrier et al., 2011]. Relationships that are specific to beta-lognormal generators are given in Appendix A.

[17] 2. The distribution of the positive intensity inline image does not significantly depend on the duration D ≥ d of the host storm.

[18] While they are satisfied in approximation, over a range of duration d of practical interest (from below 1 h to about 1 day), these properties are robust enough to be used as the foundation of a theory of rainfall extremes. The properties are consistent with the view that rainstorms are wet D-segments of independent and identically distributed (iid) multifractal cascades with some mean intensity m, upper limit of multifractal scaling dmax, and generator Ae (see Figure 2a).

Figure 2.

Schematic illustration of (a) the storm model and (b) the canonical model of rainfall.

[19] In sections 'Asymptotic Results for the Canonical Model' and 'Nonasymptotic IDF Curves and Maxima', we consider the extreme problem for an even simpler rainfall model, in which there is just one storm (one cascade) per year and the storms are iid with duration D = dmax and mean intensity m = 1. We refer to this as the canonical multifractal model of rainfall (see Figure 2b). In the canonical model, T measures time in number of storms. Adaptation of the results to the case of ns storms per year, variable storm duration D, and m ≠ 1 and hence to the more realistic representation of rainfall in Figure 2a is done in section 'A More Realistic Model of Rainfall'.

2.2. Moments and Upper Tail Behavior

[20] An important property of Id in equation (1) is that its nondiverging moments scale as

display math(2)

with moment scaling function inline image.

[21] For q greater than or equal to some q* > 1, the moments inline image diverge. The critical order q* is found from the condition inline image [Kahane and Peyriere, 1976]. If such q* exists, then Id has a power law upper tail of the type

display math(3)

[22] Approximately, where this power law tail begins is an issue of much interest in extreme rainfall analysis, as it is linked to the value T* beyond which the T-year maximum intensity Id,T has the distribution predicted by EV theory and the threshold z* beyond which the excess has the GP distribution predicted by EE theory. Expressions for T* and z* will be obtained later.

[23] In the analysis of the canonical model, the durations d and dmax appear through the resolution r = dmax/d. That is why in the remainder of this section and in sections 'Asymptotic Results for the Canonical Model' and 'Nonasymptotic IDF Curves and Maxima' we replace d with r and use the notation Ir, Ir,T, Ir,z, ir,T… in place of Id, Id,T, Id,z, id,T

2.3. Marginal Distributions

[24] In the canonical model, the average intensity Ir inside storms is given by

display math(4)

where Ir,b is the “bare” intensity obtained by ignoring the rainfall fluctuations at scales smaller than dmax/r and Z is the so-called dressing factor, which accounts for those fluctuations and is responsible for the power law upper tail of Ir [Schertzer and Lovejoy, 1987; Veneziano and Furcolo, 2003]. To distinguish Ir from Ir,b, we call Ir the “dressed” intensity.

[25] The distribution of the bare intensity Ir,b is the same as the distribution of Ar in equation (1) and is easily obtained. For the dressed intensity, one can numerically calculate the distribution of Z and find the distribution of Ir in equation (4) through convolution [Veneziano and Furcolo, 2003], but these calculations are impractical if one needs high accuracy in the extreme upper tail region. An accurate approximation to Ir, first proposed by Veneziano and Langousis [2005], is obtained in two steps:

[26] First Step: Replace Z with a variable inline image where the resolution rZ is chosen to match the moment of Z of some order q < q*. For example, matching the moment of order 2 gives inline image. With rd = rrZ, the first-step approximation is

display math(5)

[27] Second Step: Approximation 1 is accurate in the body of the distribution, but not in the tail, which due to moment divergence has power law form inline image. One can improve Approximation 1 by grafting a q* power law tail at the intensity inline image that satisfies inline image. Hence, the final approximation is

display math(6)

[28] In what follows we refer to results obtained under Approximation 2 as “exact” (quotes are used to indicate that this is actually an approximation). For example, Figure 3 shows the “exact” IDF curves ir,T for the canonical model with beta-lognormal generator and parameters Cβ = 0.05 and CLN = 0.10. In the canonical model, the return period T equals the number of storms. Since in many climates the number of storms per year is on the order of 100, T = 100 in Figure 3 corresponds to about one physical year. The reason why Figure 3 presents results up to extremely high values of T is that we wanted to include the “ inline image-line,” which is the line above which the classical EV results are accurate; see section 'Asymptotic Results for the Canonical Model'. Below the inline image-line, the IDF curves from Approximations 1 and 2 in equations (5) and (6) are identical.

Figure 3.

“Exact” IDF curves for the canonical model with parameters Cβ = 0.05 and CLN = 0.10.

Figure 4.

Illustration of the Legendre transform relationship between K(q) and Cb(γ).

3. Asymptotic Results for the Canonical Model

[29] The asymptotics of rainfall under the canonical model follow from large-deviation (LD) theory and more specifically from the behavior of inline image for different γ > 0 as r → ∞ [see, e.g., Veneziano and Furcolo, 2002; Veneziano et al., 2006, 2009; Langousis et al., 2009; Veneziano and Langousis, 2010]. We review these LD properties in section 'Large Deviations of Ir' and their consequences on the IDF curves and the distribution of maxima and excesses in section 'Asymptotic Properties of IDF Curves, Maxima, and Excesses'. This section follows closely Veneziano et al. [2009], while casting the results in the context of the canonical model.

3.1. Large Deviations of Ir

[30] As a first step, consider the probabilities inline image for the bare intensities Ir,b = Ar. A result in large deviation theory known as rough Cramer's theorem [Cramer, 1938; Den Hollander, 2000] gives that, for any γ > 0 and r → ∞,

display math(7)

where Cb(γ) is the Legendre transform of the moment scaling function K(q) and ∼ stands for equality up to a factor g(γ,r) that varies slowly with r at infinity. The Legendre transform is given in Appendix A and is illustrated in Figure 4.

Figure 5.

Parameters associated with q* and α. ξ(α) and δ(α) are the scaling exponents of the IDF values in equations (10) and (11).

[31] For the dressed intensities Ir, an extension of Cramer's theorem to account for the dressing factor Z [Lovejoy and Schertzer, 1995; Veneziano, 2002, 2005] gives

display math(8)


display math

q* = order of moment divergence, inline image where ′ denotes differentiation inline image

[32] For a geometrical interpretation of q*, c*, and γ*, see Figure 5a. For r large, the inline image-line in Figure 3 is given by inline image.

3.2. Asymptotic Properties of IDF Curves, Maxima, and Excesses

[33] A key step in assessing the asymptotic properties of rainfall is to determine the limiting behavior of the return-period values ir,T as one moves along different directions in the [log(r), log(T)] plane (see Figure 6). One can span the different directions by considering the “α limit” (r → ∞, inline image) for different α ≥ 0. α = 0 corresponds to (r → ∞, T finite), whereas α → ∞ corresponds to (Τ → ∞, r finite), which is the case considered by classical EV theory.

Figure 6.

Asymptotic conditions considered in section 'Asymptotic Results for the Canonical Model'.

[34] Recalling that inline image is the value exceeded by Ir with probability inline image, it follows from equation (8) that at high resolutions inline image is a power law function of r,

display math(9)

where γ1+α is the value of γ such that C(γ) in equation (8) equals 1+α. Equation (9) has important implications on the IDF curves, the T-yr maximum distribution, and the distribution of the excess above high thresholds, which we discuss later.

3.2.1. IDF Curves

[35] One can use equation (9) to determine how the IDF values scale with r and T. Let M >1 be any finite number and denote by Ωα,M the region of the (r, T)-plane with resolution between r/M and Mr and return period between rα/M and Mrα. By giving perturbations to log(r) and log(T) in equation (9) one obtains that, as r → ∞, the IDF values inside Ωα,M scale as

display math(10)

with exponents

display math(11)

where c = α + 1, γc is the value of γ such that C(γ) = c, inline image is the corresponding value of q, α* = c* − 1, and q*, c*, and γ* are the same as in equation (8). A geometrical interpretation of these various quantities is shown in Figure 5 and explicit expressions for the beta-lognormal case are given in Appendix A.

[36] In essence, equation (10) says that in the α-limit, the IDF curves are straight lines in log(r) with even spacing in log(T). The slope ξ and spacing δ depend on α = logr(T) and the multifractal properties of rain according to equation (11).

[37] Two special cases of equation (10) are worth mentioning. One is the scaling inline image for α = 0. This is important because, for any given T, α = logr(T) → 0 as r → ∞. Therefore, for T in any finite range, in the small-scale limit the IDF curves scale as inline image. This scaling can be seen approached in Figure 3 for small T and large r. Appendix A gives explicit expressions of γ1 and q1 for the beta-lognormal case.

[38] The second special case of equation (10) is when α exceeds α* = q*(γ* − 1) (this is the region above the inline image-line in Figure 3). In this case, the dressed IDF values scale as inline image. The scaling with T indicates that for these values of α the IDF intensities ir,T are in the power law tail of Ir. Proportionality to r further says that the intensities ir,T for different resolutions r come with probability 1 from the same rainfall event and that in those events, the total rainfall inside intervals of different duration d = dmax/r is the same. In essence, the extraordinary events that produce intensities in the power law tail of Ir act like delta functions in time.

[39] Figure 7 shows plots of ξ(α) and δ(α) in equation (11) as a function of α/α* for the four combinations of Cβ = 0, 0.10 and CLN = 0.10, 0.15. The value of α* and other important parameters are listed in the figure. The slope of the IDF curves, ξ(α), varies almost linearly with α = logr(T) between the values γ1 for α = 0 and 1 for α = α*. This variation explains the tendency of the IDF curves to converge as the averaging duration increases. The spacing of the IDF curves, δ(α), is a nonlinear function of α, with a high gradient for small α. This explains the common observation that, when plotted for equal increments of log(T), the IDF curves become closer as T increases. Both features are clearly visible in Figure 3.

Figure 7.

Slope ξ(α) and spacing δ(α) of the IDF curves in the α-limit. In the small-scale limit r → ∞, the index k of the GEV distribution of the rα-year maximum equals δ(α).

3.2.2. T-yr Maxima and Excesses

[40] It follows from equation (10) that, for any given M > 1 and as r → ∞,

display math(12)

[41] Simply put, since in the α-limit, the IDF curves become parallel and evenly spaced over any finite range inline image of T, the exceedance probability inline image over the corresponding range of intensities inline image must approach a power function. The exponent of the power function is the reciprocal of the spacing δ(α) of the IDF curves.

[42] The power law behavior in equation (12) implies that, in the small-scale limit:

[43] 1. The distribution of the excess Id,z above thresholds z on the order of the rα-year return period value inline image approaches a Pareto distribution with index k(α) = δ(α), and

[44] 2. The rα-year maximum inline image approaches an EV2 distribution with the same index. In particular, by setting α = 0 one obtains that, in the small-scale limit, the distribution of the annual maximum (and more in general the distribution of the T-yr maximum for any given T) is EV2 with index inline image.

[45] The previous asymptotic results are summarized in Figure 8. The results encompass maxima, excesses, and the scaling of the IDF curves, thus giving a complete picture of the extreme rainfall problem under asymptotic conditions (high-resolution and/or high-return period). Section 'Nonasymptotic IDF Curves and Maxima' shows how these asymptotic results can be used to produce nonasymptotic approximations to the IDF curves and the annual maximum distribution. Importantly, we find that accurate approximations for finite r and T follow not from the EV/EE results, but from the LD results for α = logr(T). This is why the present extension of classical asymptotic analysis is of much practical relevance.

Figure 8.

Schematic illustration of the asymptotic EV, EE, and LD results on the scaling of the IDF values ir,T, the distribution of the T-year maximum intensity Ir,T, and the distribution of the excess Ir,z.

4. Nonasymptotic IDF Curves and Maxima

[46] We turn now to the study of nonasymptotic properties, specifically the IDF values ir,T over finite ranges of r = dmax/d and T and the distribution of the annual maximum Ir,1 for finite r. A challenge is that equation (8) no longer suffices because the unspecified prefactor g(γ,r) in that equation, which is unimportant for the asymptotic properties, has significant effects on the nonasymptotic results. Hence, one needs to develop refinements of equation (8) in which the prefactor is included, at least in approximation.

4.1. A Refined Large-Deviation Formula

[47] We start by noting that under Approximations 1 (equation (5)) one can use the so-called refined version of Cramer's theorem [Cramer, 1938], which includes an asymptotic first-order expansion of g(γ,r). This gives [Veneziano and Furcolo, 2002]

display math(13)

where rd = rrZ and ′ and ″ denote first and second derivative. An explicit expression for the beta-lognormal model is given in Appendix A.

[48] One can also extend Cramer's refined result to account for the dressing factor Z [Veneziano, 2002, 2005], but this result does not produce useful approximations. Rather, to account for the algebraic tail of Ir due to dressing, we simply change the exponents on the right hand side of equation (13) to C(γ) in equation (8). The modified refined limit is

display math(14)

[49] Although somewhat heuristic and asymptotic, equation (14) produces very accurate approximations for finite r (see below). It is also possible to develop higher-order approximations to g(γ,r), but the first-order formula in equation (14) is sufficiently accurate not to warrant additional refinements.

4.2. IDF Curves

[50] By setting the exceedance probability in equation (14) to the value inline image that characterizes the IDF intensity ir,T, one obtains the refined IDF approximation

display math(15)

where γ is a function of r, rd, and T defined implicitly by the condition

display math(16)

[51] For any given (r, rd, α = logr(T)), one can solve equation (16) for γ using an iterative procedure. Denote by h(r, rd, α, γ) the expression on the right hand side of equation (16). One may start from γ(0) = γ1 (this is the value of γ for r → ∞) and update γ as inline image for i = 1, 2, … until convergence.

[52] Figure 9 compares the “exact” dressed IDF curves for Cβ = 0.05 and CLN = 0.10 with the refined approximations from equation (15). For these and other values of Cβ and CLN, the refined approximation is very accurate. Therefore, it is appropriate to use equation (15) with α = logr(T) to approximate the IDF curves ir,T under nonasymptotic conditions.

Figure 9.

Comparison of “exact” IDF curves with the refined approximation in equations (15) and (16). Canonical beta-lognormal model with parameters Cβ = 0.05 and CLN = 0.10. Return periods inline image

[53] If desired, the slope ξ and spacing δ in the local power law approximation ir,TrξTδ can be found numerically by giving small perturbations to log(r) and log(T) in equation (15). One finds that the spacing δof the IDF curves decreases as T increases or r decreases, while the opposite is true for the slope of the curves ξ. These trends are consistent with those of typical empirical IDF curves.

[54] We made an extensive comparison of the local refined exponents obtained in this way with the “exact” exponents found numerically from equation (6). Except for T less than 100 (storms) and α very close to zero, the exponents are practically identical. We also compared the “exact” local exponents with rough asymptotic exponents calculated using equation (11) with inline image (this correction of c, which is based on equations (5) and (6), improves the accuracy of the asymptotic values in equation (11) when r is finite). While less accurate, the rough approximations from equation (11) capture well the dependence of ξ and δ on r and T. This is important because the rough approximations to ξ and δ are analytic.

4.3. Annual Maxima and a Semitheoretical Formula for the GEV Index k

[55] The distribution of the annual maximum Ir,1 is linked to the IDF values ir,T as follows. In the canonical model, there are r intervals of duration d = dmax/r inside each storm and there is one storm per year. Assuming independence of the intensities Ir in those r intervals (Veneziano and Langousis [2005] have shown that this assumption produces accurate annual maximum approximations), the distributions of Ir,1 and Ir are related as inline image. Using the definition of the IDF value ir,T as the intensity for which inline image, the probability that ir,T is not exceeded in 1 year is inline image, which for T large is close to 1−1/T. Using these relationships, one can infer the shape of the annual maximum distribution from how ir,T varies with T. In particular, in what follows we relate the spacing of the IDF curves to the index k of the GEV distribution that best approximates the distribution of the annual maximum over a finite range inline image of intensities.

[56] To evaluate the optimal k for given (r, Cβ, CLN, Tmin, Tmax), we consider a large number n of evenly spaced values inline image, j = 1, …, n between inline image and inline image. For each αj, we calculate the corresponding return period inline image, the “exact” IDF value inline image from the distribution in equation (6), and the associated annual nonexceedance probability inline image. Finally, we plot the points inline image on EV2(k) paper [this corresponds to plotting inline image against inline image and estimate k as the index for which the coefficient of determination R2 of the linear least-squares regression through the points is maximum. This is essentially the procedure used in Veneziano et al. [2009].

[57] An example of optimal fit (with k = 0.1401) is shown in Figure 10. An important consideration is that, for any given set of parameters (r, Cβ, CLN, Tmin, Tmax), there is a wide range of k values for which the regression fit is good. For example, for the case in Figure 10, coefficients of determination R2 in excess of 0.995 are obtained for k in the range 0.0243–0.2580. Figure 10 show EV2-paper plots for the extremes of this range. We conclude that even small changes in the annual maximum distribution can produce best GEV fits with very different indices. This well-known sensitivity makes the development of theoretical approaches to k particularly important.

Figure 10.

Canonical model with Cβ = 0.05 and CLN = 0.10. Index k of the best fitting GEV distribution to the annual maximum for r = 10, Tmin = 102 and Tmax = 104. k = 0.1401 produces the best linear regression fit. For k = 0.0243 and 0.2580, the coefficient of determination of the regression R2 is 0.995.

[58] To develop an analytical expression for k, we start from the first-order estimate

display math(17)

where inline image is given by equation (11) for inline image, inline image, and inline image. This estimator is based on asymptotic properties (see section 'Asymptotic Properties of IDF Curves, Maxima, and Excesses') and ignores the fact that for finite r and T the spacing of the IDF curves is not constant. Figure 11a compares optimal values of k calculated as explained above under different combinations of (r, Cβ, CLN, Tmin, Tmax) with the corresponding values of inline image. The parameter combinations used in this analysis have been chosen as follows. Since k is insensitive to Cβ, we have fixed this parameter to the frequently observed intrastorm value 0.05. For CLN, we have considered the values 0.10 and 0.20, which bracket what is typically observed inside rainstorms [Veneziano and Lepore, 2012]. The resolution r has been made to vary from 2 to 100, with equal log increments. Since the outer limit of scaling dmax is about 24 h, this range of r encompasses most durations d = dmax/r of practical interest. For the return periods Tmin and Tmax, we have considered combinations of typical hydrologic interest (Cases 1–8 in Figure 11) as well as combinations with very large values (Cases 9–12). Figure 11a shows that inline image performs well for extremely long return periods, whereas for return periods of hydrologic interest inline image overestimates k. The discrepancy is especially large when r and inline image are small.

Figure 11.

Performance of the predictors of k in equations (17), (19), and (20). See text for an explanation.

[59] To improve on the estimation, we include information on how the spacing of the IDF curves varies with T by using the slope inline image. One can show that for large inline image,

display math(18)

[60] These conditions are satisfied by estimators of the type inline image where the function g is such that g(0) = 0 and g(1) = 1. For example, power functions g(x) = xc with c > 0 satisfy these conditions. Using least squares and the optimal k values calculated numerically, one obtains c = 0.86 and the estimator

display math(19)

[61] As Figure 11b shows, this estimator is far more accurate than inline image over all parameter combinations. The performance remains near optimal if one sets c = 1 and uses

display math(20)

[62] (see Figure 11c). In the present beta-lognormal case, the quantities inline image, inline image, and inline image that appear in equations (17)-(20) are given by

display math(21)

where rd = rrZ and rZ is given just before equation (5).

[63] Figure 12 shows how the optimal estimator in equation (19) varies with resolution r for Cβ = 0, 0.10, CLN = 0.1, 0.2, and different values of inline image (divide inline image by about 100 to express the return period in physical years). As anticipated, sensitivity to Cβ is low. The index k is nearly proportional to inline image (because of the leading factor in the expression of inline image in equation (21)). The dependence of k on r is consistent with what is found when fitting GEV distributions to annual maximum data [e.g., Baillon et al., 2004]. As r → ∞, k approaches the limiting value inline image, which is shown in Figure 12 as a horizontal dashed line. The index k decreases as inline image increases (as Tmin and Tmax increase). For inline image with inline image, k equals CLN/(1 − Cβ). Also, this value is shown in Figure 12 as a horizontal line. The curves in Figure 12 do not approach this limit, because the values of inline image used in the figure are all far below inline image.

Figure 12.

Plots of the estimator of k in equation (19) against resolution r for Cβ = 0, 0.10, CLN = 0.1, 0.2, and different values of inline image.

[64] Given the difficulty of estimating k from annual maximum or marginal excess data and the fact that scaling parameters can be obtained using also lower rainfall intensities, the availability of semitheoretical formulas like equation (19) should prove valuable. If as expected the scaling parameters Cβ and CLN are geographically stable within a given rainfall climate, equation (19) could be used as either a default value for k or as the basis of a prior distribution in extreme rainfall analysis. The fact that equation (19) is theoretically based and can account for the effects of r and inline image makes this an attractive alternative to other proposed methods to constrain k [Coles and Dixon, 1999; Martins and Stedinger, 2000; Gellens, 2002; Yoon et al., 2010].

5. A More Realistic Model of Rainfall

[65] All previous results were derived under the canonical model, a simple idealization of rainfall in which there is one storm per year and the storms are iid multifractal cascades with mean intensity 1 and duration D equal to the outer scale of multifractality dmax (see Figure 2b). Now, we consider the problem of extremes for the more realistic model in Figure 2a. While the asymptotic results of section 'Asymptotic Results for the Canonical Model' still hold (a proof is given below), one should make adjustments to the nonasymptotic results in section 'Nonasymptotic IDF Curves and Maxima'. The adjustments concern the mean rainfall intensity within storms m, the number of storms per year ns, and the variability of storm duration D.

[66] The effect of m is simply to scale the intensities of the canonical model. The effect of ns is also simple to account for: the IDF value ir,T equals the value inline image in the canonical model and the annual maximum ir,1 equals the ns-year maximum inline image in the canonical model. Section 'The Effect of Variable Storm Duration on the Marginal Distribution of Id' considers the effect of variable storm duration, section 'Asymptotic Small-Scale Properties' relates the asymptotic properties of the two models, and section 'Nonasymptotic IDF Curves and Annual Maximum Distribution' discusses the IDF curves and annual maximum distribution for the new model. As this section deals with less idealized representations of rainfall, we revert to using the average duration d rather than the resolution r = dmax/d.

5.1. The Effect of Variable Storm Duration on the Marginal Distribution of Id

[67] Accounting for the variability of D is somewhat complicated. In the frequent case when interest is in averaging durations d that do not exceed the upper limit of multifractal scaling dmax (in practice, d less than about 1 day) and only individual storms contribute significantly to the extremes of Id (hence large rainfall intensities do not come from events where d overlaps two or more storms), the marginal distribution of inline image inside storms can be found as a mixture of the conditional distributions inline image for different storm durations D. Note that

display math(22)

where inline image is the average intensity in a wet within-storm d-interval in the canonical model and inline image is the same as inline image for d = D. For storms of duration D < d, equation (22) considers one d interval that includes the entire storm. The weights of the distribution mixture are proportional to the product of the probability density of D times the number of wet d intervals for a storm of duration D. Since a storm of duration D ≥ d contains (D/d) subintervals of duration d and each such d-interval is wet with probability inline image, the number of wet d intervals for a storm of duration D, inline image, is inline image for D ≥ d and 1 for D < d.

[68] These considerations produce the following expressions for the marginal distribution of inline image and the probability inline image that a generic within-storm d-interval is wet:

display math(23)

where inline image is the partial expectation of g(D) between a and b, inline image is the expected number of d-intervals per storm, and inline image is the expected number of wet d-intervals per storm.

5.2. Asymptotic Small-Scale Properties

[69] To relate the small-scale properties of the present model to those of the canonical model in section 'Asymptotic Results for the Canonical Model' (see also Figure 8), note that in the small-scale limit d→0 the marginal distribution of inline image from equation (23) approaches the distribution of inline image and

display math(24)

[70] In the canonical model, inline image has the same behavior ∼ inline image. Since the asymptotic small-scale properties depend on the distribution of inline image and the probability inline image as d → 0 and these small-scale properties are shared by the two models, we conclude that the small-scale results in Figure 8 continue to hold.

5.3. Nonasymptotic IDF Curves and Annual Maximum Distribution

[71] In the present model, the IDF intensity id,T is the value exceeded by inline image in equation (23) with probability inline image. For d smaller than the average storm duration E[D], this value is close to m times the IDF intensity inline image in the canonical model, with inline image, inline image in equation (23), and inline image.

[72] For the annual maximum Id,1, if one ignores dependencies within storms, one obtains

display math(25)

[73] For d < E[D], this distribution is close to the distribution of inline image (m times the bd-year maximum) in the canonical model, with bd given above.

[74] As an illustration, Figure 13a compares the IDF curves calculated from the marginal distribution and probability of wet interval in equation (23) with the empirical IDF curves from the Heathrow airport record. The return periods are T = 1, 2, 7, 17, and 52 years. The distribution of storm duration D and the multifractal parameters that describe the intensity fluctuations inside the storms have been estimated from the Heathrow record as described in Veneziano and Lepore [2012]. This gives Cβ = 0.08, CLN = 0.11, and dmax = 38 h. The agreement is quite good, except at short durations for T = 17 and 52 years. The reason is that the Heathrow record includes two “outlier” events that have produced very intense short-duration rainfalls with return period likely above 100 years.

Figure 13.

Comparison of empirical IDF curves and annual maximum distributions for d = 2, 8 h with model-generated results. Rainfall record from Heathrow airport.

[75] Figures 13b and 13c compare the empirical annual maximum distributions for d = 2, 8 h with the distributions calculated through equation (25) (solid lines) and the approximations based on the canonical model with m and bd corrections, as described following equation (25) (dashed lines). The plots are on Gumbel paper. The largest recorded hourly intensities are associated with the “outlier events” mentioned above. Except for those events, there is a good correspondence between the empirical and model-based annual-maximum distributions. For Heathrow, E[D] = 4 h. For averaging durations, d < E[D], for example d = 2 h, the distribution of the annual maximum obtained through equation (25) is very close to the canonical approximation (in Figure 13b, the two curves overlap), whereas for d > E[D], for example d = 8 h, the canonical approximation is conservative. The reason is that in the canonical approximation all the wet d-intervals are considered to be entirely within storms of duration D > d. For d large, this is a conservative assumption. One could improve on the proposed approximation by reducing the value of bd, but here we do not pursue this issue further. Rather we notice that the shape of the annual distribution from equation (25) is similar to that of the canonical approximation. Therefore, the estimator of k in equation (19), which was based on the canonical model, is accurate also for the more realistic rainfall model in Figure 2a.

6. Conclusions

[76] We have presented a unified framework of extreme rainfall based on multifractal models that elucidates fundamental properties such as the form of the annual maximum distribution and the variation of the IDF values with averaging duration d and return period T. Previously obtained asymptotic results (see in particular Veneziano et al. [2009]) have been presented first in the context of a simple representation of rainfall called the canonical model and then extended to more realistic representations. The main new results concern the distribution of the annual maxima for finite d, the shape of the IDF curves when d and T are finite, and the derivation of a semitheoretical formula for the GEV index k of the annual maximum as a function of d, the range of return periods of interest, and the multifractal properties of rain.

[77] The canonical model is quite simple and indeed simplistic: it assumes that all rainstorms are iid realizations of a single multifractal cascade and are widely separated (there is just one storm per year; see Figure 2b). In spite of this drastic simplification, the model suffices to study fundamental issues in rainfall extremes that have previously eluded clear understanding. Some of the questions we have posed and answers we have provided are:

[78] 1. Does the distribution of the T-yr maximum of the average intensity in d, Id, come from the extreme upper tail of the marginal distribution, so that its shape can be predicted using EV and EE theories?

[79] In general, it does not. For values of d and T of practical interest, the distribution of the T-yr maximum reflects the upper body not the extreme upper tail of the marginal distribution. This is true also—in fact especially—for small averaging durations d, although in this case the number or rainy d-intervals in T years is large and one might feel justified in using EV theory. As d → 0, the T-yr maximum distribution is GEV, but its index k must be found from LD theory not (EV/EE) theory. The LD value is positive (in the Frechet range) and larger than that from EV/EE theory.

[80] 2. What explains the approximate scaling of the IDF curves?

[81] In the small-scale limit d → 0, the IDF curves scale exactly with d and T and LD analysis gives the power law exponents in terms of the scaling properties of rainfall inside the storms. For d finite, LD analysis predicts deviations from exact scaling, like the fact that the spacing of the IDF curves decreases as d increases or T increases. These deviations from scaling are frequently observed in practice.

[82] 3. How is it possible for the extreme value index k to depend on d, when such dependence produces paradoxical results, such as making it more likely to exceed a high total precipitation volume in a smaller interval than in a longer interval?

[83] The fitted GEV distributions are only local approximations to the exact annual maximum distributions, which are not GEV. The index k of the best GEV approximation decreases as the range of intensities over which the GEV distribution is fitted becomes more extreme. As the intensities diverge, the best-fitting index k approaches the EV/EE value, which is the same for all d. The paradox does not arise when one uses the exact non-GEV distributions.

[84] 4. Why does the empirical extreme value index k tend to decrease for longer averaging durations d?

[85] LD analysis predicts (and observations confirm) that, as d increases, the spacing of the IDF curves becomes smaller and more variable with T. These features are consistent with GEV distributions with smaller index k.

[86] For quantitative predictions, one must of course use more realistic representations of rainfall, like the one illustrated in Figure 2a. We have shown that the asymptotic small-scale properties of the canonical model still apply and that, after simple adjustments, also the nonasymptotic approximations from that model are accurate. These findings give generality to the results based on the canonical model.

[87] Another important result of this work is the dependence of the extreme value index k of the annual maximum distribution and the scaling exponents ξ and δ of the IDF curves (expressed locally as inline image) on the averaging duration d, return period T, and scaling properties of the rainfall process. In particular, we have derived an explicit semitheoretical formula for k, which expresses this dependence. The formula may be used to fix or formulate prior probability distributions of k when estimating extremes from historical annual maxima or excesses.

[88] The theory of extremes proposed in this paper is built on the assumption that rainfall inside storms is multifractal. While this assumption is generally accurate from about 1 h to about 1 day, there is evidence of scaling violations outside this range. Lack of scaling at small scales would of course invalidate our asymptotic results, which apply under d → 0. However, if one is not interested in such small temporal scales, this violation is of little concern because the nonasymptotic results are largely unaffected.

[89] Much remains to be done to extend our theoretical findings and translate them into practical instruments. On the theoretical side, our rainfall model suffers from a few limitations. Since rainfall generally scales only within storms, we had to limit consideration to average rainfall intensities Id over short durations d. The extremes of such average intensities come from d-intervals inside individual storms and to them we could apply scaling arguments. As d increases, the extremes may be contributed by multiple storms; hence, for them one must model also the dry interstorm intervals and the dependence of the rainfall intensities in consecutive storms. This extension would add significant complications.

[90] Another limitation of the present analysis is that it considers rainfall in time and ignores the spatial extent of storms. A simple way to produce space-time results is to apply empirical area reduction factors to the temporal extremes, while a more satisfactory approach is to explicitly model rainfall in space and time. In analogy with rainfall at a single site, one would have to distinguish between average intensities Id,A over space-time regions (d, A) small enough that their extremes are controlled by individual storms and average intensities in larger regions for which this assumption does not hold. For the former case, the theory of temporal extremes can be extended in a relatively straightforward way, whereas the extension to large space-time regions is far more involved.

[91] From a practical point of view, a still unsettled problem is the estimation of the scaling properties of rainfall within storms. These properties likely vary with the type of storm (frontal storms, thunderstorms of different types, tropical cyclones, etc.) and possibly with climate and season, and are affected by the amount of convection inside a storm. No comprehensive study has been made yet to assess these variations. Once the scaling properties have been assessed, results like those derived here in section 'Nonasymptotic IDF Curves and Maxima' could be used to theoretically estimate and map the shape parameter k of the annual maximum distribution and the scaling exponents of the IDF curves.

[92] Finally, we should mention that our extreme analysis applies under current climatic conditions. If one is interested in extremes during a period of time when significant climatic changes are expected, one should first consider how parameters like the mean positive rainfall intensity, the duration of storms, and the scaling parameters of rainfall might vary under those scenarios.

Appendix A: Legendre Transform and Beta-Lognormal Formulas

[93] Legendre transform relationship between K(q) and Cb(γ) are shown below

display math(A1)

[94] Some formulas for the beta-lognormal case are the following:

[95] Equation (A2) shows K(q) and related functions (for simplicity, the subscript b is omitted).

display math(A2)

[96] Equations (A3) shows critical values at moment divergence

display math(A3)

[97] Note: the values of α* and inline image use the approximation in equation (6).

[98] Equation (A4) shows scaling exponents in equation (11)

display math(A4)

where c = 1 + α. In particular, for α = 0,

display math(A5)


[99] This work was sponsored by the National Science Foundation under Grant EAR-0910721. The second author was also partially supported by the National Institute for International Education Development of South Korea, under scholarship 2010-36. The authors are grateful to Chiara Lepore for assistance in preparing some of the figures. The UK Meteorological Office provided the Heathrow data set, which was used as part of collaborative work with Christian Onof.