Abstract
 Top of page
 Abstract
 1. Introduction
 2. An Overview of 2Copulas
 3. IntensityDuration Rainfall Model
 4. Data Analysis
 5. Conclusions
 Appendix A
 Acknowledgments
 References
 Supporting Information
[1] Stochastic models of rainfall, usually based on Poisson arrivals of rectangular pulses, generally assume exponential marginal distributions for both storm duration and average rainfall intensity, and the statistical independence between these variables. However, the advent of stochastic multifractals made it clear that rainfall statistical properties are better characterized by heavy tailed Paretolike distributions, and also the independence between duration and intensity turned out to be a nonrealistic assumption. In this paper an improved intensityduration model is considered, which describes the dependence between these variables by means of a suitable 2Copula, and introduces Generalized Pareto marginals for both the storm duration and the average storm intensity. Several theoretical results are derived: in particular, we show how the use of 2Copulas allows reproducing not only the marginal variability of both storm average intensity and storm duration, but also their joint variability by describing their statistical dependence; in addition, we point out how the use of heavy tailed Generalized Pareto laws gives the possibility of modeling both the presence of extreme values and the scaling features of the rainfall process, and has interesting connections with the statistical structure of the process of rainfall maxima, which is naturally endowed with a Generalized Extreme Value law. Finally, a case study considering rainfall data is shown, which illustrates how the theoretical results derived in the paper are supported by the practical analysis.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. An Overview of 2Copulas
 3. IntensityDuration Rainfall Model
 4. Data Analysis
 5. Conclusions
 Appendix A
 Acknowledgments
 References
 Supporting Information
[2] The Poisson Rectangular Pulses (PRP) model describes the temporal evolution of rainfall as a sequence of rectangular pulses, whose occurrences are driven by a Poisson process with a given arrival rate. This model is characterized by two important features: (1) the choice of the marginal distributions of the random variables involved and (2) the structure of the statistical dependence between these variables.
[3] In his pioneering work, Eagleson [1972] assumed both the average intensity and the duration of the storm as independent and exponentially distributed. Such an assumption was later preserved by other researchers who tried to improve both the theoretical and the operational features of the PRP model [see, e.g., RodriguezIturbe et al., 1984; RodriguezIturbe, 1986], and by researchers who tried to calculate the flood frequency distribution from a simplified schematization of storm rainfall and basin characterization [see, e.g., Wood, 1976; Klemeš, 1978; DiazGranados et al., 1984; Cordova and RodriguezIturbe, 1985; Wood and Hebson, 1986; Raines and Valdes, 1993].
[4] However, Bacchi et al. [1987] showed that the PRP model with independent exponential marginals may provide a poor representation of the rainfall process. Such a failure might be a consequence of the assumptions adopted; although such hypotheses may not always be statistically rejected from data analysis [Grace and Eagleson, 1966], their (suspicious) reliability still represents a problem [see, e.g., RestrepoPosada and Eagleson, 1982; RodriguezIturbe, 1986; Bonta and Rao, 1988]. Indeed, such hypotheses are not realistic in quite a few cases. Cordova and RodriguezIturbe [1985] studied the effects of positive correlation between rainfall intensity and duration on storm surface runoff, and concluded that such a correlation may have an important impact on the runoff itself. Bacchi et al. [1994] considered a bivariate exponential distribution introduced by Gumbel [1960] to model storm duration and average intensity. Singh and Singh [1991] derived several bivariate distributions, with exponential marginals, to represent the positive correlation between the variables. More recently, Kurothe et al. [1997] used a bivariate distribution with exponential marginals to model also the negative correlation. In addition, Robinson and Sivapalan [1997] modeled the dependence between rainfall intensity and storm duration assuming a shiftedexponential law for the storm duration, and a Gamma law for the conditional distribution of the average rainfall intensity given the storm duration; then, they calculated the unconditional moments of the rainfall intensity, and the correlation between intensity and duration.
[5] Furthermore, it is well known that the rainfall process often features an extreme behavior, which cannot be modeled within an Exponentiallike statistical framework. Indeed, a completely different class of probability distributions, i.e. the Paretolike laws, turns out to be a suitable and promising candidate to statistically model phenomena characterized by heavy uppertails and scaling properties, such as the ones often encountered while investigating the rainfall process (for a recent analysis see, e.g., Salvadori and De Michele [2001]). The Paretolike laws we shall consider in the present paper have already been used in Extreme Values analysis and in hydrology [see, e.g., Davison, 1984; van Montfort and Witter, 1985, 1986; Hosking and Wallis, 1987; Castillo, 1994]. For instance, the possibility of modeling storm duration by means of heavy tailed distributions was explored by RodriguezIturbe et al. [1987]; however, an exponential pulse intensity was used in that work. More recently, Menabde and Sivapalan [2000] modeled the intensityduration relation via Lévystable independent variables. Therefore, the possibility of using heavy tailed bivariate laws for both the average intensity and the duration of storm pulses still needs exploration.
[6] The scope of the present paper is twofold. (1) Model the statistical dependence (positive and/or negative) between average rainfall intensity and storm duration using 2Copulas. (2) Model the marginal distributions of these two variables using a heavy tailed law, namely, a Generalized Pareto (GP) distribution.
[7] We anticipate that the model we shall illustrate later, featuring a dependence structure belonging to the Frank's family and Paretolike marginals, yields a rainfall pulse depth with a heavy uppertail (at least asymptotically): thus, as shown by Salvadori and De Michele [2001], not only scaling properties and easy generation of extreme values might be preserved, but also the corresponding process of rainfall maxima (under the standard assumption of a Poissonian chronology) is naturally endowed with a Generalized Extreme Value law, whose parameters can be directly calculated from those of the parent rainfall process considered.
[8] Overall, the present paper shows how an intensityduration model of storm rainfall using 2Copulas allows reproducing not only the marginal variability of both storm average intensity and storm duration, but also their joint variability by describing their statistical dependence; in addition, the use of heavy tailed GP laws allows modeling both the presence of extreme values and scaling features of the rainfall process. Clearly, these issues are of great importance and relevance in the hydrology and meteohydrology community, especially for the generation of random rainfall simulators.
2. An Overview of 2Copulas
 Top of page
 Abstract
 1. Introduction
 2. An Overview of 2Copulas
 3. IntensityDuration Rainfall Model
 4. Data Analysis
 5. Conclusions
 Appendix A
 Acknowledgments
 References
 Supporting Information
[9] The problem of specifying a probability model for dependent bivariate observations (X_{1},Y_{1}), …, (X_{n}, Y_{n}) from a population with nonnormal distribution function F_{XY} can be simplified by expressing F_{XY} in terms of its marginals, F_{X} and F_{Y}, and an associated dependence function C, implicitly defined through the functional identity F_{XY} = C (F_{X}, F_{Y})  see below. A natural way of studying bivariate data thus consists of estimating the dependence function and the marginals separately. This twostep approach to stochastic modeling is often convenient, since many tractable models are readily available for the marginal distributions: it is clearly appropriate when the marginals are known, and it is invaluable as a general strategy for data analysis in that it allows to investigate the dependence structure independently of marginal effects.
[10] Before introducing our model, it is necessary to illustrate the concept of copula. As we shall show below, the statistical dependence between different r.v.'s can indeed be modeled through a suitable copula; for all the mathematical details omitted we refer to Joe [1997] and Nelsen [1999].
[11] Let I = [0, 1]; a twodimensional copula (or 2Copula) is a bivariate function C : I × II such that (1) for all u, v ∈ I it holds C (u, 0) = 0, C (u, 1) = u, C (0, v) = 0, and C (1, v) = v; (2) for all u_{1}, u_{2}, v_{1}, v_{2} ∈ I such that u_{1} ≤ u_{2} and v_{1} ≤ v_{2} it holds C (u_{2}, v_{2}) − C (u_{2}, v_{1}) − C (u_{1}, v_{2}) + C (u_{1}, v_{1}) ≥ 0. Extensions to the multidimensional case are possible, but these are not of interest here. The link between 2Copulas and bivariate distributions is provided by the following simplified version of Sklar's theorem: Let X, Y be continuous r.v.'s, and let F_{XY} be their joint distribution function with marginals F_{X} and F_{Y}. Then there exists a unique copula C such that
for all reals x, y. Conversely, if C is a copula and F_{X} and F_{Y} are distribution functions, then F_{XY} is a joint distribution function with marginals F_{X} and F_{Y}.
[12] The interesting point is that the properties of F_{XY} can be discussed in terms of the structure of C: in fact, it is precisely the copula which captures many of the features of a joint distribution, and measures of association and dependence properties between r.v.'s can be investigated in terms of copulas (see Appendix A). Actually, a copula exactly describes and models the dependence structure between random variables, independently of the marginal laws of the variables involved: clearly, this gives a large freedom in choosing the univariate marginal distributions once the desired dependence framework has been selected, and it usually makes it easier to formulate bivariate (and/or multivariate) models. Incidentally, we observe that all the bivariate models cited in the Introduction can easily be described in terms of proper copulas.
[13] In the sequel we shall mainly be concerned with a particular class of copulas, i.e. the Frank's family [Frank, 1979; see also Nelsen, 1986; Hutchinson and Lai, 1990; Joe, 1997; Nelsen, 1999]. The analytical expression of a generic member is
where u, v ∈ I and δ ∈ R. Here δ represents the dependence parameter (see Appendix A): the case δ < 0 corresponds to a negative dependence, the case δ > 0 corresponds to a positive dependence, and the (limit) case δ = 0 corresponds to independent variables, with C_{0} (u, v) = uv; thus, members of the Frank's family may model both negatively and positively dependent variables.
[14] A further property of Frank's copulas is that these are Archimedean [Nelsen, 1999]; this means that a copula C_{δ} is the solution of the functional equation γ(C_{δ} (u, v)) = γ(u) + γ(v), where the generator γ : I [0, ∞] is a continuous, convex, strictly decreasing function such that γ(1) = 0. In the present case one has γ(t) = ln (e^{−δ}−1) −ln (e^{−δt}−1), where t ∈ I. The interesting point is that Kendall's τ rank correlation coefficient can be expressed as a onetoone function of δ via
Note that τ is odd, i.e. τ(−δ) = −τ(δ), and that τ ±1 as δ ±∞, which corresponds to a complete (positive or negative) dependence. Moreover, a Maclaurin series expansion of equation (3) up to the seventh order yields
which provides a good approximation to τ for ∣δ∣ < 5; practically, the linear approximation τ(δ) ≈ δ/9 may suffice for ∣δ∣ < 3. As an illustration, in Figure 1 we compare the function τ with both a linear approximation and the approximation provided by equation (4). Clearly the above relation between δ and τ confers a natural interpretation to δ as an association parameter; a similar relation also holds for the Spearman's ρ_{S} grade correlation coefficient.
[15] A key result, useful for fitting a Frank's copula to experimental data, is provided by the following formulas
where δ' = e^{−δ}, and D_{1} and D_{2} are, respectively, the Debye functions [Luke, 1969] of order 1 and 2. Indeed, since τ (and ρ_{S}) are measures of association based on the ranks, this suggests how δ might be estimated in situations where the marginals are unknown [see, e.g., Genest, 1987; Genest and Rivest, 1993; Carriere, 1994; Nelsen, 1999]. Indeed, using the empirical copula frequency c_{n} introduced in Appendix A, a sample estimation of τ is given by
Since the mapping δ τ(δ) is onetoone (and the same also holds for ρ_{S}), once an estimation of τ (ρ_{S}) is obtained, it is then possible to calculate an estimation of δ (usually via numerical algorithms), and thus select a well defined copula from Frank's family. Most importantly, we observe that such a procedure does not depend upon the marginal laws involved, which then need not be known or estimated in advance; indeed, in practical applications, empirical copulas represent a fundamental tool for fitting a given dependence structure to the available data [see, e.g., Deheuvels, 1979; Genest and Rivest, 1993; Nelsen, 1999].
[16] Below we show how Frank's copulas are suitable candidates to model the dependence between average rainfall intensity and storm duration, and also to reproduce the asymptotic statistical distribution of the storm depth. As a matter of fact, several other copulas were tested (see Hutchinson and Lai [1990] and Nelsen [1999] for a full list of possibilities); as a result, Frank's copulas turned out to provide the most valuable model of the phenomena considered here. Indeed, Frank's copulas may provide stronger forms of associations between the variables X and Y than other families of copulas [Genest, 1987].
3. IntensityDuration Rainfall Model
 Top of page
 Abstract
 1. Introduction
 2. An Overview of 2Copulas
 3. IntensityDuration Rainfall Model
 4. Data Analysis
 5. Conclusions
 Appendix A
 Acknowledgments
 References
 Supporting Information
[17] We assume here that the sequence of storms can be described as a series of independent pulses, whose arrivals are ruled by a Poissonian chronology. Each pulse is characterized by an average intensity I and a duration D, so that the overall contribution of the pulse to the rainfall process is given by the pulse depth (or volume) P = ID.
[18] Given the merits of the Generalized Pareto distribution (see, in particular, Salvadori and De Michele [2001]), we choose for both the r.v.'s I and D a GP law as follows:
where b_{I}, b_{D} ≥ 0 are position parameters, c_{I}, c_{D} > 0 are scale parameters, and k_{I}, k_{D} < 0 are shape parameters. Note that b_{I}, b_{D} are nonnegative, since so are both I and D; moreover, the shape parameters are negative, since we assume both I and D unbounded above. Incidentally, this latter constraint on k_{I}, k_{D} also yields (asymptotic) excess probability functions for I, D with an algebraic falloff, i.e.,
and hence only the moments of order less than −1/k_{I, D} exist. It is just the presence of such heavy tails which makes GP laws useful for describing extreme phenomena: in fact, such a tail behavior is typical of Lévystable random variables [Feller, 1971; Samorodnitsky and Taqqu, 1994], playing a fundamental role in modeling extreme events and stochastic multifractal processes [Schertzer and Lovejoy, 1987]. It is important to emphasize that, under the standard assumption of a Poissonian chronology for the arrivals of the storms, if the pulse depth P has an asymptotic heavy tail (as shown below), then the corresponding process of maxima pulse depths has a limit Paretolike behavior with the same decay rate [Salvadori and De Michele, 2001]: more precisely, it has a Generalized Extreme Value distribution with the same shape parameter as the pulse intensity I.
[19] In order to model the dependence between I and D we use a copula from Frank's family and Sklar's theorem. Thus, the joint distribution F_{ID} of I and D is given by
where the constraints on i, d and the six GP parameters involved are the same as the ones previously considered. Then, obtaining the joint density f_{ID} is simply a matter of differentiating F_{ID} with respect to i and d; in turn, once f_{ID} is made available, the law F_{P} of the pulse depth P = ID can be derived via standard procedures, integrating f_{ID} over proper regions of the idplane. As an illustration, in Figure 2 we show two such regions, corresponding to two different arguments p_{1} < p_{2} of the distribution function F_{P}: the first one is the curvilinear triangle Δ(1, 2, 3) (yielding F_{P}(p_{1})), and the second one is the curvilinear triangle Δ(1, 4, 5) (yielding F_{P}(p_{2})).
[20] However, a different approach is possible exploiting copulas. Let (U, V) be a random vector, where the r.v.'s U and V have uniform marginals f_{U}, f_{V} on I, and joint distribution F_{UV} = C_{δ}. Making a proper use of the Probability Integral Transform, the statistical behavior of the couple (I, D), and hence of P, can be modeled as a function of the couple (U, V). In fact, for fixed p > b_{I}b_{D}, we may write
where the function g_{p}(v) = F_{I}({p}/{F_{D}^{−1}(v)}) is given by, for v ∈ (0, 1),
Using equation (A7) given in Appendix A, we see that
Note that the last integrand is a function of v only, since p acts as a parameter. Therefore, the calculation of the distribution function F_{P}, for any p > b_{I}b_{D}, reduces to a onedimensional path integration in the unit square I × I along the curve u = g_{p}(v). As an illustration, in Figure 3 we plot the function g_{p} for two different values p_{1}< p_{2} of the parameter p, and we show the “slices” of the integrand function in equation (14) cut along g_{p1} and g_{p2}: the areas of the vertical surfaces give, respectively, F_{P}(p_{1}) and F_{P}(p_{2}).
[21] In the following we shall mainly be interested in calculating the law of large pulse depths P under “extremal” conditions, i.e. when I is large and D is small [Monda, 2002]: this corresponds to considering extreme precipitations in short time periods, as it might be of interest, e.g., in hydrologic design and natural hazards survey and modeling. In particular, we may condition with respect to the event {I ≫ 1, D ≈ b_{D}}, and focus the attention on the behavior of F_{ID}(i, d) within the region R_{ID}* defined by i ≫ 1 and d ≈ b_{D}. Thanks to Sklar's theorem and equation (11), this is equivalent to analyzing the behavior of C_{δ}(u,v) within the region defined by u ≈ 1 and v ≈ 0; there C_{δ} reduces to
as can be derived via first order approximations. Thus, again by recourse to Sklar's theorem, it is possible to calculate the asymptotic behavior of F_{P} simply considering the local structure of C_{δ}, given by equation (15), and the marginal distributions of I and D. In turn, in R_{ID}* the joint density f_{ID} behaves approximately as
where the function h represents a multiplicative factor. Note how the “extremal” behavior of f_{ID} does not depend upon k_{D}. Indeed, from a physical point of view, under “extremal” conditions the severity of the rainfall storm events is mainly influenced by the strong storm intensity I, and not by the duration D (which is considered as small). From a statistical point of view, this implies that the joint distribution of the couple (I, D) is influenced by the shape parameter k_{I}, and is practically independent of the shape parameter k_{D}. As an illustration, in Figure 2 we show the region R_{ID}* (corresponding to the stripe (i > i*, b_{D} < d < d*), with i* ≫ 1 and d* ≈ b_{D}), and the integration region for a value p* ≫ 1, corresponding to the curvilinear triangle Δ(6, 7, 8) (yielding F_{P}(p*)). As a consequence, for large values of p, also the function F_{P} shows a Paretolike behavior, with an algebraic heavy uppertail falling off at the rate 1/k_{I}, the same as the one of the rainfall average intensity I. Note that, with some further mathematical effort, the same result could have been derived via a proper analysis of equation (14).
4. Data Analysis
 Top of page
 Abstract
 1. Introduction
 2. An Overview of 2Copulas
 3. IntensityDuration Rainfall Model
 4. Data Analysis
 5. Conclusions
 Appendix A
 Acknowledgments
 References
 Supporting Information
[22] In this Section an application to the rainfall data of the Bisagno drainage basin at La Presa (Thyrrhenian Liguria, Northwestern Italy) is given. Hourly recorded rainfall depth data, collected by two raingauges, are made available for a period of 7 years, from 1990 to 1996. Assuming homogenous climatic conditions within the drainage basin, we can consider the available data as a sample extracted from the same statistical population; thus, this is essentially equivalent to analyzing a data set of 14 years of measurements. Note that, given the present conditions, we may legitimately assume statistical independence between the data considered. According to Salvadori and De Michele [2001], we use a dry period equal to 7 hours to separate different storms, the choice being motivated by the known behavior of the meteorology in the region considered; such a value is consistent with those reported in the literature: Huff [1967] estimates the dry period as 6 hours; Koutsoyiannis and FoufoulaGeorgiou [1993] and Menabde and Sivapalan [2000] as 7 hours; Cordova and RodriguezIturbe [1985] as 12 hours. In addition, the selection of the storm event is made conditioning upon the average storm intensity I: here we consider values of I larger than 1, 3, and 5 mm/h. From a physical point of view, this practically corresponds to investigating different climate scenarios, the present situation being fixed to a 1 mm/h threshold, and two other possible (climate change) scenarios corresponding to 3 and 5 mm/h thresholds. In Figure 4 we show a pictorial representation of a representative portion of the time series considered: the horizontal bases of the rectangles represent the storm duration D, and the vertical heights the average storm intensity I. Evidently, an extreme event is present, corresponding to a storm lasting only two hours, with an intensity of 48 mm/h.
[23] We estimate the statistical dependence between the average storm intensity I and the storm duration D considering different measures of association: the canonical (Pearson's) coefficient of linear correlation ρ_{P}, the Kendall's τ, and the Spearman's ρ_{S}. Note that, from a statistical viewpoint, considering values of I larger than increasing thresholds corresponds to investigating different “strengths” of the dependence between I and D. In Table 1 we show the estimates of ρ_{P}, τ, and ρ_{S}, as a function of the thresholds considered. As a general comment, in our case the dependence between I and D is always negative, and it increases (in an absolute sense, i.e. in “strength”) by increasing the threshold. For the latter two measures, shown are the corresponding estimates of the dependence parameter δ of Frank's copula, calculated using equations (4)–(6). For the sake of comparison, also shown are the estimates of δ calculated via the Maximum Likelihood method. Finally, we compute the average estimate of δ, and the corresponding standard error. Evidently, all the estimates reported are consistent with one another, and precisely fix the degree of dependence between I and D independently of the marginal laws of such variables. As a further analysis, in Figure 5 we plot several Frank's copulas (calculated via equation (2) using the three different estimates of given in Table 1), and compare them with the corresponding empirical copula functions C_{n} calculated via equation (A3) given in Appendix A. Clearly, the agreement between the data and the model is good. A proper twodimensional KolmogorovSmirnov test is performed to check the goodnessoffit [see Peacock, 1983; Fasano and Franceschini, 1987; Press et al., 1992]. As a result, the test is passed for all the three data sets considered at a 5% level of significance.
Table 1. Estimated Values of Different Measures of Association: The Pearson's ρ_{P}, the Kendall's τ, and the Spearman's ρ_{S}, as a Function of the Threshold T on the Intensity I^{a}T  ρ_{P}  τ  δ_{τ}  ρ_{S}  δ_{ρS}  δ_{ML}  


1  −0.13  −0.09  −0.83  −0.13  −0.79  −0.86  −0.82 ± 0.04 
3  −0.14  −0.11  −0.99  −0.17  −1.04  −1.06  −1.05 ± 0.04 
5  −0.23  −0.27  −2.38  −0.39  −2.47  −2.66  −2.50 ± 0.14 
[24] In Table 2 we show the estimates of the parameters of the Generalized Pareto marginal laws of I and D; here the Lmoments technique is used [Hosking and Wallis, 1987]. Without loss of generality, the estimates of b_{I} are naturally fixed to the values of the three thresholds on I considered (i.e., b_{I} = 1, 3, 5 mm/h), and those of b_{D} are fixed to the lower bound of the distribution (i.e., b_{D} = 0 hours). The shape parameter k turns out to be always negative for both variables, which then exhibit an asymptotic heavy tail behavior. In particular, k decreases by increasing the threshold on I or, in other words, by increasing the severity of the storm event. It is important to note that, as a consequence of equation (10), the skewness of I does not exist at all, as well as its variance considering the largest threshold; thus, apparently, the traditional Method of Moments technique could not be used to estimate the parameters of interest (as opposed to the Lmoments technique adopted here). In Figures 6–8 we compare the marginal distributions of I and D, and the corresponding GP fits (using the parameters of Table 2), for all the three thresholds on I considered. Evidently, the agreement is good in all cases, a conclusion also supported by the fact that both the KolmogorovSmirnov and AndersonDarling goodnessoffit tests [see, e.g., Kottegoda and Rosso, 1997] were passed at a 5% level of significance.
Table 2. Estimated Values of the GP Parameters {k, c, b} for Both the Average Storm Intensity I and the Storm Duration D, as a Function of the Threshold T on the Intensity I^{a}T  Intensity I  Duration D 

k_{I}  c_{I}  b_{I}  k_{D}  c_{D}  b_{D} 


1  −0.383  1.285  1  −0.172  10.568  0 
3  −0.472  1.475  3  −0.228  7.761  0 
5  −0.571  1.821  5  −0.292  6.382  0 
[25] In passing, we observe a further important feature of the present approach: the possibility of selecting a suitable 2Copula via the estimate of the Kendall's τ or the Spearman's ρ_{S}, as explained in Section 2, may not be affected at all by the possible nonexistence of lowerorder moments. On the contrary, the use of other measures of association such as the canonical Pearson's coefficient of linear correlation ρ_{P} (which, instead, is often used in common practice), necessarily requires the existence of (at least) the second order moment. However, it is clear that meaningless estimates of ρ_{P} may be obtained if, as in the third case reported above considering a 5 mm/h threshold on I, the existence of the variance is questionable: in fact, k_{I} ≈ −0.57< −1/2; nevertheless, the “suspicious” value of ρ_{P} is also reported in Table 1 for the sake of completeness.
[26] Finally, we investigate the behavior of both the average storm intensity I and the pulse depth P under “extremal conditions”, i.e. when I is large and the storm duration D is short. As already pointed out, this is of great importance in hydrologic modeling. In particular, given the experimental nature of the available data, we select as representative events those storms with {I > 5 mm/h, 1h ≤ D ≤ 3h}, which represents the conditioning event. In Figure 9 we show the limiting behavior of I and P, and the corresponding asymptotic fits (using a QQplot technique [see Falk et al., 1994; Beirlant et al., 1996]). Clearly, for values large enough, both I and P show a Paretolike behavior, with an algebraic uppertail falling off at the same rate: in fact 1/k_{I} ≈ −0.63 and 1/k_{P} ≈ −0.61. This result supports the assumptions made throughout the paper, and the conclusions drawn from equation (16) about the extremal asymptotic behavior of P.
5. Conclusions
 Top of page
 Abstract
 1. Introduction
 2. An Overview of 2Copulas
 3. IntensityDuration Rainfall Model
 4. Data Analysis
 5. Conclusions
 Appendix A
 Acknowledgments
 References
 Supporting Information
[27] In this paper we propose an original intensity−duration model of storm rainfall, which describes the dependence between these variables by means of a 2Copula belonging to Frank's family, and introduces Generalized Pareto (GP) marginals for both the storm duration and the average storm intensity.
[28] First, we briefly introduce the mathematics of copulas, and we show how (via Sklar's theorem) all the bivariate intensityduration models present in the literature could be reformulated using suitable 2Copulas. In particular, we illustrate the theoretical properties of Frank's family of copulas, and discuss the relationships with standard measures of association and dependence currently used in practical applications.
[29] Then, we introduce a bivariate intensityduration model of storm rainfall, whose dependence structure is described by a Frank's copula and featuring GP marginals, and discuss its properties in details. In particular, we show how the use of copulas may simplify the calculations involved, and especially those concerning the asymptotic behavior of the pulse depth. As a result we show how, under “extremal” conditions, the pulse volume may feature the same heavy uppertail as the storm intensity. In addition, we emphasize how scaling properties and easy generation of extreme values might be preserved, and also how (under the standard assumption of a Poissonian chronology) the process of rainfall maxima is naturally endowed with a Generalized Extreme Value law, whose parameters can be directly calculated from those of the parent rainfall process.
[30] Finally, a practical case study is illustrated considering rainfall data. As a result, we show how a Frank's 2Copula is well suited to describe the dependence structure between the available intensityduration data, and how the assumptions of GP marginals are well respected. In addition, also the theoretical results concerning the asymptotic behavior of the pulse depth are well supported by the analysis.
[31] In conclusion, the present approach may provide a very general model of the intensityduration relationship of storm rainfall: on the one hand, the marginal variability of both storm average intensity and storm duration can easily be reproduced; on the other hand, also the joint variability of such variables can be precisely modeled by describing their statistical dependence via a suitable 2Copula. In addition, the use of heavy tailed GP laws gives the possibility of modeling both the presence of extreme values, and the scaling features of the rainfall process. Clearly, not only may it be useful for a proper statistical description of the rainfall phenomenon, but it may also be of interest for the simulation of rainfall fields and the calculation of the derived distribution of flood data.