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Keywords:

  • rainfall modeling;
  • 2-Copulas;
  • storm intensity-duration

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. 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 Pareto-like distributions, and also the independence between duration and intensity turned out to be a nonrealistic assumption. In this paper an improved intensity-duration model is considered, which describes the dependence between these variables by means of a suitable 2-Copula, 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 2-Copulas 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. 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., Rodriguez-Iturbe et al., 1984; Rodriguez-Iturbe, 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; Diaz-Granados et al., 1984; Cordova and Rodriguez-Iturbe, 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., Restrepo-Posada and Eagleson, 1982; Rodriguez-Iturbe, 1986; Bonta and Rao, 1988]. Indeed, such hypotheses are not realistic in quite a few cases. Cordova and Rodriguez-Iturbe [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 shifted-exponential 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 Exponential-like statistical framework. Indeed, a completely different class of probability distributions, i.e. the Pareto-like laws, turns out to be a suitable and promising candidate to statistically model phenomena characterized by heavy upper-tails 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 Pareto-like 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 Rodriguez-Iturbe et al. [1987]; however, an exponential pulse intensity was used in that work. More recently, Menabde and Sivapalan [2000] modeled the intensity-duration relation via Lévy-stable 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 2-Copulas. (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 Pareto-like marginals, yields a rainfall pulse depth with a heavy upper-tail (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 intensity-duration model of storm rainfall using 2-Copulas 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 meteo-hydrology community, especially for the generation of random rainfall simulators.

2. An Overview of 2-Copulas

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. Supporting Information

[9] The problem of specifying a probability model for dependent bivariate observations (X1,Y1), …, (Xn, Yn) from a population with nonnormal distribution function FXY can be simplified by expressing FXY in terms of its marginals, FX and FY, and an associated dependence function C, implicitly defined through the functional identity FXY = C (FX, FY) - see below. A natural way of studying bivariate data thus consists of estimating the dependence function and the marginals separately. This two-step 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 two-dimensional copula (or 2-Copula) is a bivariate function C : I × Iequation imageI such that (1) for all u, vI it holds C (u, 0) = 0, C (u, 1) = u, C (0, v) = 0, and C (1, v) = v; (2) for all u1, u2, v1, v2I such that u1u2 and v1v2 it holds C (u2, v2) − C (u2, v1) − C (u1, v2) + C (u1, v1) ≥ 0. Extensions to the multidimensional case are possible, but these are not of interest here. The link between 2-Copulas and bivariate distributions is provided by the following simplified version of Sklar's theorem: Let X, Y be continuous r.v.'s, and let FXY be their joint distribution function with marginals FX and FY. Then there exists a unique copula C such that

  • equation image

for all reals x, y. Conversely, if C is a copula and FX and FY are distribution functions, then FXY is a joint distribution function with marginals FX and FY.

[12] The interesting point is that the properties of FXY 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

  • equation image

where u, vI 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 C0 (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 Cδ is the solution of the functional equation γ(Cδ (u, v)) = γ(u) + γ(v), where the generator γ : Iequation image [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 tI. The interesting point is that Kendall's τ rank correlation coefficient can be expressed as a one-to-one function of δ via

  • equation image

Note that τ is odd, i.e. τ(−δ) = −τ(δ), and that τ [RIGHTWARDS ARROW] ±1 as δ [RIGHTWARDS ARROW] ±∞, which corresponds to a complete (positive or negative) dependence. Moreover, a Maclaurin series expansion of equation (3) up to the seventh order yields

  • equation image

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 ρS grade correlation coefficient.

image

Figure 1. Plot of the function τ(δ) for Frank's copulas (thick line) and its approximation provided by equation (4) (thin line). Also shown is a first order linear approximation (dashed line).

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[15] A key result, useful for fitting a Frank's copula to experimental data, is provided by the following formulas

  • equation image
  • equation image

where δ' = e−δ, and D1 and D2 are, respectively, the Debye functions [Luke, 1969] of order 1 and 2. Indeed, since τ (and ρS) 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 cn introduced in Appendix A, a sample estimation equation image of τ is given by

  • equation image

Since the mapping δ equation image τ(δ) is one-to-one (and the same also holds for ρS), once an estimation equation image equation image of τ (ρS) is obtained, it is then possible to calculate an estimation equation image 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. Intensity-Duration Rainfall Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. 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:

  • equation image
  • equation image

where bI, bD ≥ 0 are position parameters, cI, cD > 0 are scale parameters, and kI, kD < 0 are shape parameters. Note that bI, bD 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 kI, kD also yields (asymptotic) excess probability functions for I, D with an algebraic falloff, i.e.,

  • equation image

and hence only the moments of order less than −1/kI, 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 Lévy-stable 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 Pareto-like 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 FID of I and D is given by

  • equation image

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 fID is simply a matter of differentiating FID with respect to i and d; in turn, once fID is made available, the law FP of the pulse depth P = ID can be derived via standard procedures, integrating fID over proper regions of the id-plane. As an illustration, in Figure 2 we show two such regions, corresponding to two different arguments p1 < p2 of the distribution function FP: the first one is the curvilinear triangle Δ(1, 2, 3) (yielding FP(p1)), and the second one is the curvilinear triangle Δ(1, 4, 5) (yielding FP(p2)).

image

Figure 2. Standard integration regions of the joint density fID corresponding to two different arguments p1 < p2 of the distribution function FP: the curvilinear triangle Δ(1, 2, 3) (yielding FP(p1), and the curvilinear triangle Δ(1, 4, 5) (yielding FP(p2). Also shown is the limit integration region of fID (considering the stripe (i > i*, bD< d < d*), with i* ≫ 1 and d* ≈ bD) for a value p* ≫ 1: this corresponds to the curvilinear triangle Δ(6, 7, 8) yielding FP(p*). Note that bI, bD are the (lower) bounds of the position parameters of the GP laws involved, while i*, d* play the role of constraints on the variability of the random variables I, D.

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[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 fU, fV on I, and joint distribution FUV = Cδ. 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 > bIbD, we may write

  • equation image

where the function gp(v) = FI({p}/{FD−1(v)}) is given by, for v ∈ (0, 1),

  • equation image

Using equation (A7) given in Appendix A, we see that

  • equation image

Note that the last integrand is a function of v only, since p acts as a parameter. Therefore, the calculation of the distribution function FP, for any p > bIbD, reduces to a one-dimensional path integration in the unit square I × I along the curve u = gp(v). As an illustration, in Figure 3 we plot the function gp for two different values p1< p2 of the parameter p, and we show the “slices” of the integrand function in equation (14) cut along gp1 and gp2: the areas of the vertical surfaces give, respectively, FP(p1) and FP(p2).

image

Figure 3. (a) Plot of the function gp for two different values of the parameter p, with p1 < p2: gp1 (continuous line) and gp2 (dashed line). (b) Illustration of the path integration using copulas: the dark surface in the foreground represents the area below the integrand in equation (14) when p = p1 (yielding FP(p1)), and the light surface in the background represents the same area when p = p2 (yielding FP(p2)); the projections on the uv-plane correspond to, respectively, the functions gp1 and gp2 plotted in (a).

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[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, DbD}, and focus the attention on the behavior of FID(i, d) within the region RID* defined by i ≫ 1 and dbD. Thanks to Sklar's theorem and equation (11), this is equivalent to analyzing the behavior of Cδ(u,v) within the region defined by u ≈ 1 and v ≈ 0; there Cδ reduces to

  • equation image

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 FP simply considering the local structure of Cδ, given by equation (15), and the marginal distributions of I and D. In turn, in RID* the joint density fID behaves approximately as

  • equation image

where the function h represents a multiplicative factor. Note how the “extremal” behavior of fID does not depend upon kD. 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 kI, and is practically independent of the shape parameter kD. As an illustration, in Figure 2 we show the region RID* (corresponding to the stripe (i > i*, bD < d < d*), with i* ≫ 1 and d* ≈ bD), and the integration region for a value p* ≫ 1, corresponding to the curvilinear triangle Δ(6, 7, 8) (yielding FP(p*)). As a consequence, for large values of p, also the function FP shows a Pareto-like behavior, with an algebraic heavy upper-tail falling off at the rate 1/kI, 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. 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 Foufoula-Georgiou [1993] and Menabde and Sivapalan [2000] as 7 hours; Cordova and Rodriguez-Iturbe [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.

image

Figure 4. Illustration of one year of storms extracted from the available data: the horizontal bases of the rectangles represent the storm duration D, and the vertical heights the average storm intensity I.

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[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 ρP, the Kendall's τ, and the Spearman's ρS. 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 ρP, τ, and ρS, 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 equation image 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 equation image given in Table 1), and compare them with the corresponding empirical copula functions Cn calculated via equation (A3) given in Appendix A. Clearly, the agreement between the data and the model is good. A proper two-dimensional Kolmogorov-Smirnov test is performed to check the goodness-of-fit [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.

image

Figure 5. Comparison between theoretical Frank's copulas (solid lines) and the corresponding empirical copulas (circles) generated using the available (I, D) data with different theresholds on I: (a) δ = −0.82 and I > 1 mm/h; (b) δ = −1.05 and I > 3 mm/h; (c) δ = −2.50 and I > 5 mm/h.

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Table 1. Estimated Values of Different Measures of Association: The Pearson's ρP, the Kendall's τ, and the Spearman's ρS, as a Function of the Threshold T on the Intensity Ia
TρPτδτρSδρSδMLequation image
  • a

    Units are in mm/h. For the latter two measures, shown are the corresponding estimates of the dependence parameter δ of Frank's copula. For the sake of comparison, also shown is the estimate of δ calculated via the Maximum Likelihood (ML) method. The last column shows the average estimate equation image of δ, together with the corresponding standard error.

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 L-moments technique is used [Hosking and Wallis, 1987]. Without loss of generality, the estimates of bI are naturally fixed to the values of the three thresholds on I considered (i.e., bI = 1, 3, 5 mm/h), and those of bD are fixed to the lower bound of the distribution (i.e., bD = 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 L-moments technique adopted here). In Figures 68 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 Kolmogorov-Smirnov and Anderson-Darling goodness-of-fit tests [see, e.g., Kottegoda and Rosso, 1997] were passed at a 5% level of significance.

image

Figure 6. Plot of the survival function of the storm average intensity I (circles) and the storm duration D (squares); here the threshold on I equals 1 mm/h. Also shown are the marginal GP distributions (solid lines) fitted on the intensity and duration data (see Table 2).

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image

Figure 7. Plot of the survival function of the storm average intensity I (circles) and the storm duration D (squares); here the threshold on I equals 3 mm/h. Also shown are the marginal GP distributions (solid lines) fitted on the intensity and duration data (see Table 2).

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image

Figure 8. Plot of the survival function of the storm average intensity I (circles) and the storm duration D (squares); here the threshold on I equals 5 mm/h. Also shown are the marginal GP distributions (solid lines) fitted on the intensity and duration data (see Table 2).

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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 Ia
TIntensity IDuration D
kIcIbIkDcDbD
  • a

    Units are in mm/h. The parameters kI, kD are adimensional, whereas cI, bI are in mm/h, and cD, bD are in hours.

1−0.3831.2851−0.17210.5680
3−0.4721.4753−0.2287.7610
5−0.5711.8215−0.2926.3820

[25] In passing, we observe a further important feature of the present approach: the possibility of selecting a suitable 2-Copula via the estimate of the Kendall's τ or the Spearman's ρS, as explained in Section 2, may not be affected at all by the possible nonexistence of lower-order moments. On the contrary, the use of other measures of association such as the canonical Pearson's coefficient of linear correlation ρP (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 ρP 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, kI ≈ −0.57< −1/2; nevertheless, the “suspicious” value of ρP 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 QQ-plot technique [see Falk et al., 1994; Beirlant et al., 1996]). Clearly, for values large enough, both I and P show a Pareto-like behavior, with an algebraic upper-tail falling off at the same rate: in fact 1/kI ≈ −0.63 and 1/kP ≈ −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.

image

Figure 9. Plot of the extremal behavior of the survival functions of both the storm average intensity I (circles) and the pulse depth P (triangles) – see text.

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5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. 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 2-Copula 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 intensity-duration models present in the literature could be reformulated using suitable 2-Copulas. 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 intensity-duration 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 upper-tail 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 2-Copula is well suited to describe the dependence structure between the available intensity-duration 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 intensity-duration 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 2-Copula. 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.

Appendix A

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. Supporting Information

[32] We illustrate here some further properties of copulas; for all the mathematical details omitted we refer to Joe [1997] and Nelsen [1999]. When considering bivariate distributions, an important concept is that of concordance. Intuitively, a pair of r.v.'s are concordant if “large” values of one tend to be associated with “large” values of the other, and “small” values of one with “small” values of the other; discordant r.v.'s are defined in an obvious way. Now, let (X1,Y1) and (X2,Y2) be independent vectors of continuous r.v.'s with common marginals FX and FY and respective copulas C1 and C2. Let Q denote the concordance function given by

  • equation image

In terms of copulas, it can be shown that

  • equation image

Clearly, Q is symmetric in its arguments, and since it is the difference of two probabilities it turns out that Q ∈ [−1, 1]. Well known measures of association, such as Kendall's τ, Spearman's ρS, and Blomqvist's β [see, e.g., Krustal, 1958; Lehmann, 1966], which generalize Pearson's product moment correlation coefficient ρP, are easily expressed in terms of Q; i.e., they are also measures of concordance. In fact, let C denote an arbitrary copula for the random vector (X,Y); then, τXY = Q(C, C), ρXY = 3Q(C,Π), and βXY = 4 C(1/2, 1/2) − 1, where Π(u,v) = uv represents the copula associated with independent r.v.'s, since in this case Sklar's theorem yields FXY(x, y) = Π(FX(x), FY(y))= FX(x) FY(y).

[33] Closely related to the notion of concordance is that of dependence, as positive dependence properties express the fact that “large” (or “small”) values of the variables tend to occur together, whereas negative dependence properties are featured when “large” values of one variable tend to occur with “small” values of the other. More precisely, the continuous r.v.'s X and Y are said to be positively quadrant dependent (PQD) if P{X ≤ x, Y ≤ y} ≥ P{X ≤ x}P{Y ≤ y} or, equivalently, FXY(x, y) ≥ FX(x) FY(y) for all (x, y) ∈ R2 or, in terms of the copula C of X and Y, C (u,v) ≥ u v for all (u,v) ∈ I2. Reversing the sense of the inequalities in the above relations gives the definition of negatively quadrant dependent (NQD) r.v.'s. We observe that some copulas only model either PQD or NQD cases, while others are able to model both cases (like, e.g., the copulas belonging to the Frank's family used in this paper).

[34] In the same way as the population versions of the various measures of association outlined above can be expressed in terms of copulas, the corresponding sample versions can be calculated via the empirical copulas defined as follows. Let {(xm, ym)}, m = 1,…, n, denote a sample of size n from a continuous bivariate distribution. The empirical copula Cn is the function given by, for i, j = 1,…, n,

  • equation image

where nij is the number of sample pairs (x, y) such that xx(i) and yy(j), where the x(i)'s and the y(j)'s are the order statistics from the sample. A further important function is the empirical copula frequency cn given by, for i, j = 1,…, n,

  • equation image

if (x(i),y(j)) is in the sample, and zero otherwise. Note that Cn and cn are related via

  • equation image
  • equation image

[35] Finally, numerical simulations are often a useful tool in applications. To this purpose, an algorithm to generate r.v.'s (U, V) with copula C is as follows. Let

  • equation image

which exists and is nondecreasing almost everywhere in I; then it suffices to (1) generate two independent r.v.'s, r1 and r2, both with a Uniform distribution on I; (2) set u = r1 and v = su−1(r2).

[36] The desired pair is then (u,v). Finally, to generate a pair (x,y) of observations from the joint law FXY = C(FX,FY), where both FX and FY are invertible, it simply suffices to set x = FX−1(u) and y = FY−1(v).

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. Supporting Information

[37] The research was partially supported by MURST via the project “Hydrological Safety of Impounded Rivers.” The support of “Progetto Giovani Ricercatori” is also acknowledged.

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  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Overview of 2-Copulas
  5. 3. Intensity-Duration Rainfall Model
  6. 4. Data Analysis
  7. 5. Conclusions
  8. Appendix A
  9. Acknowledgments
  10. References
  11. Supporting Information

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