A bivariate frequency analysis of extreme rainfall with implications for design

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2007jd008522, 2007 A bivariate frequency analysis of extreme rainfall with implications for design Shih-Chieh Kao 1 and Rao S. Govindaraju 1 Received 8 February 2007; revised 18 April 2007; accepted 30 April 2007; published 13 July [1] Analysis of extreme rainfall events has conventionally been performed by prespecifying rainfall duration as a filter to abstract annual maximum rainfall depths as the only variable for analysis. However, this univariate approach does not account for dependence between rainfall properties. To characterize extreme rainfall events, a bivariate analysis is conducted in this study using hourly precipitation data from Indiana, USA. Samples of extreme rainfall events are chosen on the basis of three different criteria: annual maximum volume (AMV), annual maximum peak intensity (AMI), and annual maximum cumulative probability (AMP) based on empirical copulas. Rainfall characteristics, such as total depth, duration, and peak intensity are analyzed using copulas to describe the dependence structures between rainfall variables and to construct their joint distribution for extreme rainfall events. Results from the derived bivariate models are compared to those from conventional univariate analysis by computing the corresponding conditional distributions. Traditional univariate analysis seems to provide reasonable estimates of rainfall depths for durations greater than 10 hours. For shorter durations, a bivariate analysis with extreme events defined on the basis of AMP is recommended. The univariate analysis combined with Huff curves grossly underestimates peak intensities, and again AMP estimates are recommended. Results of this study have implications for current hydrologic design in that they provide better estimates of design rainfall. Citation: Kao, S.-C., and R. S. Govindaraju (2007), A bivariate frequency analysis of extreme rainfall with implications for design, J. Geophys. Res., 112,, doi: /2007jd Introduction 1 School of Civil Engineering, Purdue University, West Lafayette, Indiana, USA. Copyright 2007 by the American Geophysical Union /07/2007JD [2] In order to prevent loss of property and human life, designs of hydraulic and hydrologic structures are based on extreme rainfall estimates. These estimates cannot be obtained in a deterministic manner, and statistical methods are generally adopted to quantify rainfall by probability of exceedance (for example, the use of return period and hydrologic risk). The reliability of rainfall estimates will depend on the appropriateness of the derived probability distribution, the sufficiency of observations, the data processing algorithm, and the selected method for analysis. Since extreme rainfall estimates are fundamental to many hydrologic projects, their estimation has always been an important issue for hydrologic engineers. [3] Rainfall frequency analysis is currently performed through univariate approaches, i.e., by treating the total rainfall depth as the only variable and finding the underlying probability distribution (see Chow et al. [1988] for an overview of this subject). To relate rainfall depth to duration, a prespecified duration is declared as a filter to find the annual maxima as samples for analysis. In this sense, stochastic rainfall models are constructed for various durations [e.g., Hershfield, 1961; Frederick et al., 1977; Huff and Angel, 1992; Bonnin et al., 2004]. However, it should be noted that this duration is artificially prescribed and does not reflect the actual duration of rainfall events. When using a shorter prescribed duration (say 1 hour), the selected maximum event may be from a longer-duration extreme rainfall event, and possibly represents the peak intensity part. On the other hand, when using a longer prescribed duration (say 48 hours), the selected maximum may cover several short-term events with periods of rainfall hiatus. While the current practice provides estimates for various artificial durations, it is not able to truly characterize the behavior of extreme rainfall events. Rainfall records reveal that rainfall events exhibit high variability in their properties such as total depth (volume), duration, and peak intensity. Therefore there is a need to perform a multivariate analysis to construct a more realistic stochastic model for extreme rainfall events. [4] The main challenge of applying multivariate stochastic models stems from the mathematical complexity of the joint probability distribution that encompasses knowledge of both marginal distributions and dependence structure. In earlier studies [Eagleson, 1972; Carlson and Fox, 1976; Wood, 1976; Chan and Bras, 1979; Díaz-Granados et al., 1984], the use of joint distribution was usually accompanied with the assumption of independence between different variables (i.e., by neglecting the possible dependence structure). For instance, rainfall was described as a rectangular 1of15

2 pulse with duration as the temporal width and average rainfall intensity as the magnitude of this pulse. By identifying the marginals and assuming independence between duration and average intensity, the joint rainfall distribution simplified to a simple product of two marginals. Though this assumption was fairly convenient, it is frequently not supported by data. Using exponential probability density functions for marginals, Córdova and Rodríguez-Iturbe [1985] showed that the correlation between rainfall duration and average intensity had a nonnegligible effect on the storm surface runoff, and they suggested that the assumption of independence may not be appropriate. The importance of dependence structure in flood frequency analysis was further emphasized by Singh and Singh [1991], Bacchi et al. [1994], Kurothe et al. [1997], and Goel et al. [2000]. [5] Subsequently, numerous multivariate rainfall models have been proposed to include the dependence between average intensity and duration, but limitations have persisted. Usually, the proposed procedure was only suitable for particular forms of marginals, such as exponential distribution used by Córdova and Rodríguez-Iturbe [1985], Singh and Singh [1991], Bacchi et al. [1994], Kurothe et al. [1997], and Goel et al. [2000], and normal distribution with Box-Cox transformation used by Yue [2000]. Also, the dependence structure could not be expressed clearly in these models. For a complicated problem like extreme rainfall behavior, more choices of marginal distributions and dependence structures are desired for wider applicability. Over the last decade, copulas have emerged as a method for addressing multivariate problems in several disciplines. Using Sklar s [1959] theorem, the analysis of joint distributions can be performed separately for the marginal distributions and for the dependence structure. Nelsen [2006] provided a theoretical background and description on the use of copulas. [6] De Michele and Salvadori [2003] were perhaps the first to apply copulas in hydrology to analyze the joint behavior between rainfall duration and average intensity. Hourly precipitation data from two rain gauges at La Presa (Italy) for 7 years (from 1990 to 1996) were utilized to construct a bivariate model for regular storms. For studying extreme rainfall behavior, Grimaldi and Serinaldi [2006b] discussed the relationship between design rainfall depth (critical depth, obtained from intensity-duration-frequency (IDF) curves by specifying design duration and return period) and the actual features of extreme rainfall events. Half-hourly rainfall data from 10 rain gauges at Umbria (Italy) from 1995 to 2001 were combined with the assumption of regional homogeneity to form a 70-year annual maximum series for analysis. The trivariate model containing critical depth, actual total depth, and peak intensity was constructed via copulas. By providing critical depth, it was expected that important features of extreme rainfall could be obtained. Zhang and Singh [2007] performed multivariate analysis for extreme rainfall events via copulas. Hourly precipitation data from three rain gauges at Amite River basin in Louisiana (US) for 42 years were analyzed. Bivariate rainfall models between total depth (volume), duration, and average intensity were constructed. Several types of conditional and joint return periods were illustrated in their study. For other applications in hydrology, Favre et al. [2004], De Michele et al. [2005], Grimaldi and Serinaldi [2006a], and Zhang and Singh [2006] applied copulas in the multivariate flood frequency analysis. Salvadori and De Michele [2004] discussed the use of copulas to assess the return period of hydrological events using bivariate models. Salvadori and De Michele [2006] applied trivariate copulas to examine the role of antecedent moisture conditions for regular storms. Kao and Govindaraju [2007] quantified the effect of dependence between rainfall duration and average intensity on surface runoff, and demonstrated that use of copulas results in simpler mathematical treatment of zero runoff probabilities. [7] Though these rainfall studies [De Michele and Salvadori, 2003; Grimaldi and Serinaldi, 2006b; Salvadori and De Michele, 2006; Zhang and Singh, 2007] advocated the use of copula techniques for studying extreme rainfall, several questions remain. A minimum record length of 50 years has been suggested for constructing reliable at-site rainfall estimates using univariate analysis [Bonnin et al., 2004]. This criterion was rarely met in these earlier studies. Furthermore, it should be noted that the definition of extreme rainfall in a multivariate setting is ambiguous and deserves clarification. For instance, an annual maximum peak intensity (AMI) event may produce more stress to a watershed than an annual maximum volume (AMV) event with longer duration (and lower average intensity), and vice versa. Thus what constitutes an extreme event needs more careful attention. Besides, it is useful to know whether the copula procedure can provide reasonable results for a large region, or if its applicability is restricted to a few select stations. We prepare to address these lingering issues. [8] In the spirit of Zhang and Singh [2007], three defining characteristics of rainfall: total depth (volume) P, duration D, and peak intensity I are utilized to perform multivariate frequency analysis in this study. Sufficiently long (over 50 years) hourly precipitation data sets within Indiana are adopted to provide a more statistically reliable description of extreme rainfall behavior. Bivariate frequency analysis is conducted by using copulas to represent the dependencies between P, D, and I. For reasons to be discussed later, we have computed bivariate distributions, as opposed to a trivariate model. To be exhaustive, extreme events are defined according to (1) annual maximum peak intensity, (2) annual maximum volume, and (3) annual maximum cumulative probability. Extreme rainfall estimates are then computed from their conditional distribution in each case and compared to their univariate counterparts. Arguments are presented for selecting the proper choice of definition for extreme rainfall in multivariate analysis. Results from this study are expected to provide a method for evaluating extreme rainfall estimates via copulas in practice. The method may be applied to other parts of the world. 2. A Brief Introduction to Copulas [9] A copula C is a function composed of marginals. Sklar [1959] showed that for continuous random variable X and Y with marginals (cumulative density functions, CDFs) F X (x) =u and F Y (y) =v, there exists one unique C UV such that: C UV ðu; vþ ¼ C UV ðf X ðþ; x F Y ðyþþ ¼ H XY ðx; yþ ð1þ 2of15

3 Table 1. One-Parameter Archimedean Copulas Used in This Study Family q (t) a Range of q a K C (t) b t(t) b,c Frank ln e qt 1 e q 1 ( 1, 0)[ (0, 1) t + eqt 1 q ln e q 1 e qt q [D 1(q) 1] d Clayton 1 q (t q 1) [ 1, 0) [ (0, 1) t(1 + 1 tqþ1 q ) q Genest-Ghoudi (1 t 1/q ) q [1, 1) t 1 1/q 2q 3 2q 1 1 q 1 t Ali-Mikhail-Haq ln ð Þ t [ 1, 1) t + tð1 qþqtþ 1 q ln 1 qþqt t 1 2 3q q ln (1 q) c a Column q (t), range of q adapted from Nelsen [2006]. b Column K C (t) derived in this study, t(t) of the Genest-Ghoudi family derived in this study. c t(t) of the Frank, and Ali-Makhail-Haq families adopted from Zhang and Singh [2007], t(t) of the Clayton family adopted from Grimaldi and Serinaldi [2006b]. R d q D 1 is the Debye function of order 1, D 1 (q) = 0 (t/q(et 1))dt. q qþ2 where H XY is the joint cumulative distribution. Since probability measurements are absolutely increasing (for absolutely increasing continuous random variables) from 0 to 1, copulas C UV can be regarded as a transformation of H XY from [ 1, 1] 2 to [0, 1] 2. In other words, it simplifies the joint distribution to a bounded domain, and attention can be focused on the dependence structure described by copulas. [10] Among various types of copulas, one-parameter Archimedean copulas have attracted the most attention owing to their possessing several convenient properties. For an Archimedean copula, there exists a generator 8 such that the following relationship holds: 8ðCu; ð vþþ ¼ 8ðÞþ8 u ðþ v In (2), the generator 8 is a convex and decreasing function defined in [0, 1], and 8(1) = 0. A special case is the independent copula P(u, v)=uv with generator 8(t) = ln t. For Archimedean copulas, several statistical properties can be simply expressed in terms of 8, such as the distribution function K C of copulas (i.e., K C (t) =P[C(u, v) t]) and the concordance measure Kendall s t: ð2þ K C ðþ¼t t 8 qðþ t 8 0 qðþ; t t 2 ½ 0; 1 Š ð3þ Z 1 t ¼ 1 þ q ðþ t 8 0 q ðþdt t where q is a parameter of the generator 8. The distribution function K C offers a cumulative probability measure for the set {(u, v) 2 [0, 1] 2 jc(u, v) t} and therefore can be applied for examining the goodness of fit of copulas onto one single dimension (along t). By using (3), the theoretical Kendall s t can be derived as (4). The statistic Kendall s t is a concordance measurement defined in [ 1, 1], where 1 represents total concordance, 1 represents total discordance, and 0 represents zero concordance (conceptually similar to the traditional correlation coefficient r). Apart from being a better measure of dependence than r, Kendall s t has also been extensively used for obtaining a nonparametric estimator for the dependence parameter q by equating sample ^t to theoretical t. Letting (x 1, y 1 ), (x 2, y 2 ) ð4þ be two observations from a size n sample space, ^t can be estimated by: ^t ¼ ðc dþ n 2 where c denotes concordant pairs ((x 2 x 1 )(y 2 y 1 )>0), and d denotes discordant pairs ((x 2 x 1 )(y 2 y 1 )<0).It should be noted that Kendall s t is defined for continuous random variables. Therefore all pairs should be either concordant or discordant (c + d equals totals number of pairs). However, though rainfall properties are naturally continuous random variables, they are recorded in pulse format [see Chow et al., 1988]. This will cause some pairs to be neither concordant nor discordant, i.e., pairs with (x 2 x 1 )(y 2 y 1 ) = 0. With the assumption for this special case that the possibility of being concordant or discordant is the same, the quantity (c d) can be regarded as the expectation of difference between the concordant and discordant pairs. This nonparametric estimator for the dependence parameter does not rely on prior information of marginal distributions, and hence provides a more objective measure of dependence structure. [11] The choice of a copula function depends on the range of dependence level it can describe. Four commonly used families of one-parameter Archimedean copulas are adopted and examined in this study, namely: Frank, Clayton, Genest- Ghoudi, and Ali-Mikhail-Haq, and are shown in Table 1. All of these are valid both for positive and negative dependence (note that Ali-Mikhail-Haq is valid only for < t < ). The parameters are estimated by nonparametric procedure using Kendall s t. For sample size n, empirical copulas C n described by Nelsen [2006] are computed for examining goodness of fit: C n i n ; j n ¼ a n where a is the number of pairs (x, y) in the sample with x x (i) and y y (j), and x (i), y (j),1 i, j n, isthe order statistics from the sample. Similarly, empirical distribution function K Cn can be written as: K Cn k n ¼ b n ð5þ ð6þ ð7þ 3of15

4 Figure 1. AMV, AMP, and AMI events of station Alpine 2 NE (COOPID: ). where b is the number of pairs (x, y) in the sample with C n (i/n, j/n) k/n. More details on copulas are available from standard references such as Nelsen [2006]. 3. Selection of Extreme Events [12] A total of 53 hourly rain gauges from Hourly Precipitation Database (TD 3240) of National Climate Data Center (NCDC, in Indiana were selected in this study. Each selected station possesses 50 to 55 years of data, which should be sufficient for performing univariate at-site frequency analysis for each station (criterion taken from Bonnin et al. [2004]). A minimum rainfall hiatus of six hours between nonzero records was selected to abstract rainfall events [Huff, 1967]. An average of about 4800 observed events were available for each station. [13] As mentioned in section 1, unlike the definition of annual maximum precipitation series used in conventional analysis, the definition of annual maximal events for mutlivariate problems is somewhat ambiguous. Depending on the problem at hand, total rainfall depth (volume) P or peak intensity I may govern hydrologic design. Further, extreme events could be defined as those storms that have both high volume and peak intensity. However, there exists a bivariate stochastic relationship H PI between P and I. Extreme events selected on the basis of AMI or AMV may not be extremal in the other characteristic as well. Therefore we consider a third definition of extreme rainfall based on events with annual maximum cumulative probability (AMP), where the joint cumulative probabilities of samples can be estimated directly via the empirical copulas C n introduced in (6). In this study, the empirical copulas C PI,n between P and I are computed to determine the AMP events using all rainfall events within a year for each station. [14] An example plot of the selected extreme events for station Alpine 2 NE (COOPID: ) is shown in Figure 1. It can be observed that these three criteria result in different distributions. A large fraction of AMV events tend to be associated with larger durations (half of them are over 20 hours), while AMI events typically correspond to shorter durations (half of them are less than 6 hours), and AMP events tend to be more evenly distributed. As expected, the total depth of AMV events is higher than the other two series of events, while peak intensity distribution of AMI events is the highest. It should be noted that the distributional properties of AMP events always fall between AMV and AMI events. Clearly, AMV and AMI represent two contrasting types of extreme rainfall: the former involving longer-duration events with high accumulated precipitation, and the latter involving typically shortduration events with high peak intensity. From a statistical sampling point of view, an AMV-based model is expected to be better for describing long-duration rainfall behavior, while an AMI-based model would be better for shortduration rainfall. The AMP-based model should work well for a wide range of durations and span the AMV and AMI results. The selected AMP, AMV and AMI events are analyzed further. 4. Analysis of Marginal Distributions [15] The process of constructing joint distribution through copulas can be decomposed into two parts: marginal distributions and dependence structure. Marginal distributions are analyzed through the conventional univariate approach for each station. In this study, six candidate probability density functions (PDFs) are tested for their applicability to individual rainfall attributes. They are extreme value type I (EV1), generalized extreme value (GEV), Pearson type III (P3), log-pearson type III (LP3), generalized Pareto (GP), and the lognormal (LN) distribution. The theoretical background for univariate analysis can be obtained from Rao and Hamed [2000]. Model parameters are estimated primarily by maximum likelihood (ML) method or by method of moments (MOM). Gringorton formula was chosen to estimate the empirical cumulative probability F N (x). F N i 0:44 x ðþ i ¼ N þ 0:12 where i is the rank in ascending order, N is the number of observations, and x (i) are the order statistics from the sample. The Chi-square and Kolmogorov-Smirnov (KS) tests were applied for goodness of fit at the 10% significance level. The summary of test results for the 53 selected stations in Indiana is shown in Table 2. [16] On the basis of the rejection rate in Table 2, it can be observed that EV1, GEV, LP3, and LN generally provide better fits than P3 and GP for extreme rainfall characteristics. The LN distribution was found to yield good fits for extreme rainfall marginals in some other studies as well [Grimaldi and Serinaldi, 2006b; Zhang and Singh, 2007]. ð8þ 4of15

5 Table 2. Summary of Chi-Square and Kolmogorov-Smirnov (KS) Test Results for Marginal Distributions Rejection Rate (%) of Chi-Square Test Rejection Rate (%) of KS Test EV1 GEV P3 LP3 GP LN EV1 GEV P3 LP3 GP LN AMV Events Depth, P Duration, D Intensity, I AMI Events Depth, P Duration, D Intensity, I AMP Events Depth, P Duration, D Intensity, I EV1 and LN would be recommended for use in practice because of good performance measures (Table 2) despite having smaller number of parameters. The marginals of depth P, duration D, and peak intensity I are expressed as u = F P (p), v = F D (d), and w = F I (i) in the following discussion. As an example, Figure 2 shows the fits provided for these marginals using EV1 distribution. [17] It should be noticed that though GP was adopted by De Michele and Salvadori [2003] for their model describing regular rainfall events, it was found to be the weakest distribution in this study. This highlights the different nature of extreme rainfall behavior when compared to regular rainfall. It was also observed that fitting for duration of AMI events did not yield very good results. This may be attributed to the fact that most AMI events are associated with short durations, and therefore the recording unit (hour) used in this study is not able to resolve it at lower durations, thus affecting the statistics adversely. This problem is also observed for AMP events, but to a smaller degree. 5. Analysis of Dependence Structure Using Copulas [18] Dependence between variables is typically quantified by the correlation coefficient (Pearson s product moment) r, which can be regarded as the ratio of cross product moment to product of individual moments. The correlation coefficient is known to be influenced by outliers. Moreover, except for Gaussian (and some elliptic) distributions, the correlation coefficient is not an appropriate measure of dependence between the marginals [see Nelsen, 2006]. In contrast to r, copulas provide a complete and robust description of dependence between correlated quantities irrespective of their marginal distributions, and therefore are appropriate for multivariate stochastic analysis. This section will focus on the selection of Archimedean copulas. [19] As mentioned in section 2, since there exists a oneto-one correspondence between the generator of Archimedean copulas and the concordance measure Kendall s t, sample ^t PD, ^t DI, and ^t PI, for each pair of variables P, D, and I were computed for all selected stations. Mean and standard deviations of these sample statistics were computed and tabulated in Table 3. The areal distributions of ^t PD, ^t DI, and ^t PI, constructed from point values from each station for AMP events are shown in Figure 3 as an example. It can be observed that depth and duration are positively correlated, duration and peak intensity are negatively correlated, and depth and peak intensity are positively correlated. It is important to notice that the dependence levels are generally not high (close to ±1) between any two variables, nor are they very low (close to 0) except for duration and peak intensity for AMI events. When dependence level is close to ±1, the number of variables can be reasonably reduced and replaced by a reliable regression formula. On the other hand, low dependence validates the assumption of independence and hence the joint distribution reduces to a simple Figure 2. EV1 fitting for marginal distributions of station Alpine 2 NE (COOPID: ). The light lines are the empirical CDFs, and the darker lines are the theoretical fits. 5of15

6 Figure 3. Kendall s ^t between extreme rainfall features (AMP events) of Indiana: (a) ^t PD between depth and duration, (b) ^t DI between duration and peak intensity, and (c) ^t PI between depth and peak intensity. 6of15

7 Table 3. Statistics of Kendall s ^t Between Extreme Rainfall Properties in Indiana ^t PD ^t DI ^t PI Mean stdev Mean stdev Mean stdev AMV events AMI events AMP events product of marginals. These two limiting approximations are common in engineering applications, but are not appropriate for the variables P, D, and I, being considered in this study. When analyzing important problems like extreme rainfall behavior, the construction of dependent joint distributions of rainfall characteristics is inevitable. [20] From Table 3, it can be observed that the dependence levels for AMV, AMI, and AMP events are not similar. Consequently, stochastic models based on events selected by different definitions of extreme events will lead to different models, and the choice of which one to adopt should be based on the nature of the problem at hand. The values of dependence level for AMP events always fall in between AMV and AMI events. [21] The spatial variation of ^t PD, ^t DI, and ^t PI in Figure 3 show that while there are some similarities between ^t DI, and ^t PI, in general, these are not alike. For a statistically homogeneous region, the rainfall parameters need not be the same, but they should show similar trends. In this respect, because of the discrepancies in the behavior of ^t PD, ^t DI, and ^t PI, we find that rainfall properties are not homogeneous over Indiana. This conclusion is contradictory to what has been suggested in recent univariate rainfall studies [Bonnin et al., 2004; Rao and Kao, 2006] that found rainfall to be homogeneous over Indiana on the basis of homogeneity tests [Hosking and Wallis, 1997]. However, these homogeneity tests were performed on annual maximum depth series under prespecified durations, and their conclusions cannot be extended to multivariate cases, and do not appear to be valid for extreme events as defined in this study. This further illustrates the need to conduct multivariate analyses to understand the behavior of extreme rainfall events. [22] There are numerous Archimedean copulas, such as Gumbel-Hougaard, Gumbel-Barnett, etc. (refer to Nelsen [2006]). However, only a small subset of Archimedean copulas can accommodate negative or unlimited range of dependence levels, as is the case between D and I for Indiana rainfall data. Therefore Frank, Clayton, Genest- Ghoudi, and Ali-Mikhail-Haq families (Table 1) are selected as possible candidate copula models (we note that Ali- Mikhail-Haq is not valid for some stations). The dependence parameters are estimated by nonparametric procedure for Kendall s ^t, and, as an example, the estimators ^q PD, ^q DI and ^q PI for Frank family (see Table 1) are shown in Figure 4. As expected, the spatial trends of these parameters are similar to those of Kendall s t in Figure 3. [23] Figure 5 demonstrates an example of visual examination of goodness of fit for AMP events of station Alpine 2 NE. Three types of plots are provided: empirical and theoretical distribution function of copulas (K Cn (t) and K C (t) introduced in (7) and (2)), diagonal section of copulas d(t) =C(t, t), and section with one marginal as median (when one marginal equals 0.5). It is found that the plot of K Cn (t) and K C (t) is the most discriminating for selection of the appropriate copula function. We note that K C (t) = P[C(u, v) t] is a projection from the entire two-dimensional field of u and v into a one-dimensional axis that abstracts all features of copulas. On the other hand, diagonal section and the section with one specified marginal reveal partial but relevant information about dependence between the variables. Furthermore, K C (t) can also be utilized for the estimation of return period for bivariate stochastic model (defined as secondary return period by Salvadori and De Michele [2004]). [24] Generally, it is observed that Clayton and Ali- Mikhail-Haq families perform well for positive dependence cases (C UV and C UW ), and Frank family performs well for both positive and negative dependence. In fact, Frank family is the only Archimedean copula that satisfies radial symmetry [see Nelsen, 2006], and is suitable for the entire range of dependence. Not surprisingly, the Frank family has been a popular choice for constructing dependence structure [De Michele and Salvadori, 2003; Favre et al., 2004; Grimaldi and Serinaldi, 2006b; Salvadori and De Michele, 2006; Zhang and Singh, 2006, 2007]. We also note that the Frank family of copulas passed the multidimensional KS test proposed by Saunders and Laud [1980] for the entire state of Indiana at the 10% significant level. Therefore Frank family of Archimedean copulas was adopted in this study, and is recommended for use in practice. A summary of parameter estimates for Frank family is presented in Table Joint Distribution and Applications [25] Results from Figure 3 and 4 show that the nature of dependence is quite complex, i.e., ^q PD 6¼ ^q DI 6¼ ^q PI. Apart from mathematical simplicity offered by a bivariate analysis (compared to a trivariate case), the results we want to develop are more easily expressed through the examination of bivariate behaviors of depth P, duration D, and peak intensity I. The three bivariate joint cumulative distributions between P, D, and I can be constructed by merging marginal distributions and dependence structure obtained in sections 4 and 5: H PD ðp; dþ ¼ C UV ðf P ðpþ; F D ðdþþ ¼ C UV ðu; vþ ð9þ H DI ðd; iþ ¼ C VW ðf D ðdþ; F I ðþ i Þ ¼ C VW ðv; wþ ð10þ H PI ðp; iþ ¼ C UW ðf P ðpþ; F I ðþ i Þ ¼ C UW ðu; wþ ð11þ These bivariate models can be helpful for many purposes, such as risk assessment, flood frequency derivation, and computation of expectation for rainfall-related properties. Three applications in evaluating extreme rainfall estimates are presented in this study Estimate of Depth for Known Duration [26] For a known (or measured) d-hour rainfall event, the conditional cumulative distribution for depth P can be 7of15

8 Figure 4. Dependence parameter q of AMP events for Frank family of Archimedean copulas: (a) ^q PD between depth and duration, (b) ^q DI between duration and peak intensity, and (c) ^q PI between depth and peak intensity. 8of15

9 Figure 5. Visual examination of goodness of fit of copulas (AMP events) for station Alpine 2 NE (COOPID: ). Emp, empirical copulas; FRK, Frank family; CLT, Clayton family; GeG, Genest- Ghoudi family; AMH, Ali-Mikhail-Haq family. (top) Empirical K Cn (t) and K(t). (middle) Diagonal section d(t) = C(t, t). (bottom) Section with one marginal being median (one marginal equals 0.5). written as: F P ðpd j 1 < D dþ ¼ H PDðp; dþ H PD ðp; d 1Þ F D ðdþ F D ðd 1Þ ¼ C UV ðf P ðpþ; F D ðdþþ C UV ðf P ðpþ; F D ðd 1ÞÞ F D ðdþ F D ðd 1Þ ð12þ For a given return period T, the T-year, d-hour rainfall estimate p T will satisfy F P (p T jd 1<D d) =1 1/T. This forms a general equation for evaluating rainfall estimates, and can be solved after specification of marginal distributions and copulas. Applying EV1 and Frank family of Archimedean copulas, 10-year and 100-year estimates for three sample stations are shown in Figure 6 as an example, along with the corresponding conventional univariate rainfall estimates using GEV distribution fitted separately for various durations (according to Rao and Kao [2006], GEV was the recommend distribution for at-site analysis in Indiana). [27] It is interesting to note that though the samples from AMV, AMI and AMP events are quite different, most of them possess similar estimates for durations greater than 10 hour, and are also close to their univariate counterparts. Table 4. Statistics of Dependence Parameter ^q for Frank Family of Archimedean Copulas in Indiana ^quv ^qvw ^quw Mean stdev Mean stdev Mean stdev AMV events AMI events AMP events of15

10 Figure 6. Rainfall depth estimates for various durations for station Alpine 2 NE (COOPID: ), Anderson sewage plant (COOPID: ), and Angola (COOPID: ). This consistency underscores the applicability of multivariate rainfall analysis. Larger discrepancies are found for durations less than 10 hours. AMV estimates generally provide larger rainfall depths than either AMP or AMI in this range. It may be recalled that the abstracted AMV samples generally correspond to longer durations, and hence should be less reliable than AMP and AMI estimates for shorter durations. On the basis of statistical sampling, AMI estimates should be the most appropriate for shorter durations. However, it may also be recalled that the marginal fitting of duration for AMI events may not yield good results because of the insufficient recording precision of duration (i.e., 1-hour data), and would likely cause some error in evaluating estimates at low durations. This could be rectified by using observations with finer precision (like 5-min data) if they were available over long periods (>50 years), but that is currently a data limitation. On the other hand, the marginal distribution of duration for AMP events showed better fits, and hence should be a good estimate for durations less than 10 hours. [28] The ratios of conditional estimates in (12) and their univariate counterparts were computed for each selected station. Table 5 shows three average ratios (AMV/GEV, AMI/GEV, and AMP/GEV) for entire Indiana under various durations. Though bivariate estimates are derived in a different way than univariate estimates, most of the average regional ratios approach 1 (except for shorter duration of Table 5. Mean of Ratio of Estimates Constructed Through (11) to Its Univariate Counterpart Over Entire Indiana Duration, hour AMV/GEV AMP/GEV AMI/GEV of 15

11 Figure 7. The 6-hour, 10-year AMP rainfall depth estimates over Indiana. AMV and 1 hour of AMP) which indicates both algorithms (univariate and bivariate analyses) yield similar estimates. Thus the traditional univariate analysis is perhaps adequate for estimating rainfall depths over longer durations, but is not appropriate for durations less than 10 hours. We note that each bivariate result presented in Figure 6 is based on 5 parameters (4 for EV1 marginals, and 1 for Frank family copula expressing dependence). However, the univariate procedure requires 3 different parameters (GEV) for each individual duration. We expect the bivariate approach to replicate rainfall behavior better, and is a promising algorithm for extreme rainfall analysis. [29] In order to recommend a suitable model for design, we studied results along the lines of Figure 6 for numerous Indiana stations, and for different return periods as well. On the basis of our observations, we recommend the use of the AMP definition for extreme rainfall events. As argued, the sampling statistics for AMP are better over a wide range of durations. Furthermore, as evidenced in Figure 6, the AMP definition provides more conservative estimates of rainfall depths for shorter durations than does AMI. Adopting the AMP definition, Figure 7 shows the 6-hour, 10-year rainfall depth estimates for Indiana. These were constructed from point estimates at each of the 53 rainfall stations. It can be observed that a general areal trend exists, where southwestern Indiana has higher rainfall estimates than northeastern Indiana. This areal trend corresponds to what has been reported in the previous studies [Hershfield, 1961; Huff and Angel, 1992]. [30] It is of interest to investigate the consequences if the given condition in (12) changes from d 1<D d to D d (the former characterizes the measured d-hour event and the latter describes all possible events less than d-hour). Mathematically, this is expressed as F P ðpd j dþ ¼ H PDðp; dþ F D ðdþ ¼ C UV ðf P ðpþ; F D ðdþþ F D ðdþ ¼ C UV ðu; vþ v ð13þ which results in a simpler expression than (12), and it is the conditional distribution discussed by Zhang and Singh [2007]. An example using AMP events for station Anderson Sewage Plant is shown in Figure 8. It can be seen that (13) yields smaller estimates than (12) as expected because depth typically increases with duration. For design, once the time of concentration for a certain watershed is known, the most potentially damaging events should be considered given the probability of exceedance. In this sense, condition D d provides a less conservative estimate, and equation (12) is recommended Estimate of Peak Intensity for Known Duration [31] Similar to (12), for a given (or measured) d-hour rainfall event, the conditional cumulative distribution for Figure 8. Comparison of rainfall estimates by equations (12) and (13). 11 of 15

12 Figure 9. Peak intensity estimates for various durations for station Alpine 2 NE (COOPID: ), Anderson sewage plant (COOPID: ), and Angola (COOPID: ). peak intensity I can be written as: F I ðid j 1 < D dþ ¼ H DIðd; iþ H DI ðd 1; iþ F D ðdþ F D ðd 1Þ ¼ C VW ðf D ðdþ; F I ðþ i Þ C VW ðf D ðd 1Þ; F I ðþ i Þ F D ðdþ F D ðd 1Þ ð14þ For a given return period T, the peak intensity estimate i T for a T-year, d-hour rainfall event can be solved from F I (i T jd 1<D d) =1 1/T. Peak intensity I is not mentioned independently in the literature. Thus there is no direct approach for obtaining estimates through univariate analysis. In practice, after the rainfall depth is estimated, the average intensity is defined as the ratio of depth to duration. For a longer-duration event, the average intensity will become smaller (following the IDF relationship). In this construct, interception losses will become unreasonably high because ponding time increases with decreasing rainfall intensity [see Chow et al., 1988]. Even if some rainfall temporal distribution is used (like Huff [1967]), peak intensity is a crucial aspect. [32] Using this bivariate procedure (14), the peak intensity can be directly constructed from observations without adopting any empirical relationship. For comparison purposes, the conventional peak intensity estimates based on univariate approach were evaluated using Huff s [1967] temporal distribution derived at each station of Indiana [Rao and Kao, 2006]. Using 50% Huff curve (common choice in practice), the maximum percentage rainfall increment (within 10% storm duration increments) for all 4 different quartiles was found, multiplied with the GEV rainfall depth, and divided by the peak duration (10% storm duration) to get the peak intensity estimates. It is noted that the estimated peak intensity in this manner is 4 5 times larger than the average intensity. [33] An example of peak intensity estimates for three sample stations is shown in Figure 9. It can be observed that 12 of 15

13 However, what usually concerns us is not the condition P = p but the condition P > p (extreme events greater than a threshold p). Hence the conditional cumulative probability of I becomes: F I ðip> j pþ ¼ F IðÞ i H PI ðp; iþ ¼ w C UW ðu; wþ 1 F P ðpþ 1 u ð15þ Since the depth threshold p is usually related to a return period, it would not be recommended to use the concept of return period again in (15) to estimate i (though it can be done, it would be conceptually quite confusing). What would be more useful is the conditional expectation E[IjP > p], EIP> ½ j pš ¼ Z 1 0 if I ðip> j pþdi ¼ Z 1 F IðiP> j pþdi ð16þ Figure 10. The 6-hour, 10-year AMP peak intensity estimates over Indiana. these three sets of definitions (AMI, AMV, or AMP) yield similar trends. AMI events generally possess the highest estimates as expected on the basis of their definition, followed by AMP and then AMV events (expect for the rare cases when positive dependence exists between D and I, such as the AMI estimates for station Alpine). We again recommend the adoption of AMP estimates for design because AMP criterion selects events with joint extreme behaviors and would not rely solely on any single variable. It can be also observed that the conventional univariate estimators fail to capture the peak intensity. The univariate procedure generally underestimates peak intensities for durations greater than 4 hours (around 4 times smaller), and overestimates for shorter duration (especially for 1 hour). This observation further emphasizes the need for adopting multivariate analysis in order to better evaluate peak rainfall intensity that is sometimes needed for hydrologic design. Similar to Figure 7, the 6-hour, 10-year AMP peak intensity estimates for Indiana are presented in Figure 10. A similar areal trend can be observed Estimate of Peak Intensity for Known Depth [34] Since the stochastic relationship (11) between P and I through copulas has been obtained, peak intensity can be estimated on the basis of information about given depth. This is useful for instances when rainfall depth controls the design (for example, Soil Conservation Service [1972] curve number method) and (11) can provide a stochastic approach for estimating the corresponding peak intensity. This conditional expectation represents the most likely peak intensity for events greater than a depth threshold p. The difficulty for (16) is that an analytical solution is usually unavailable and the partial derivative term could result in a fairly complicated formula even for numerical integration. An alternative numerical procedure is introduced here. After deciding a small probability increment Dw and integer n with ndw =1,forw j = jdw, j =1,, n 1, solve F I (i j jp > p)=w j for i j. Then E[IjP > p] will approximately be the mean of series i j. The underlying idea is to sample for each equally incremental probability interval in (15). For a small Dw, the result should be close to (16). An example for station Anderson Sewage Plant is shown in Figure 11. With increasing threshold, the expected peak intensity would increase as in Figure 11. This provides another way of studying peak intensity in relation to storm depth. It should be noticed that the peak intensity estimated by (16) has a different statistical meaning than the result provided by (14). 7. Conclusions [35] Traditional risk-based hydrologic design has relied on an artificial prescription of durations to analyze rainfall data using a univariate statistical analysis with depth as the primary variable. With more data now becoming available, Figure 11. Expectation of peak intensity for events greater than a given depth threshold for Anderson sewage plant (COOPID: ). 13 of 15

14 we use copulas to reconstruct extreme rainfall events in a bivariate fashion in this paper. The following conclusions are presented on the basis of this study. [36] 1. Samples of annual extreme rainfall were selected on the basis of volume (AMV), peak intensity (AMI), and joint probability (AMP) criteria based on empirical copulas. It was found that the duration of AMV events is generally longer than AMP and then AMI events, and that AMV estimates are suspected to be less reliable for short durations. As for the AMI definition, the hourly recording precision used in this study was found to be limiting. This study concluded that AMP criterion is an appropriate indicator for defining extreme events. [37] 2. The total volume (depth), duration, and peak intensity were selected as variables of interest for describing rainfall in this study. EV1, GEV, LP3, and LN distributions were found to be appropriate marginal models for extreme rainfall, with EV1 and LN being recommended for use in practice. While GP was found to perform well for regular rainfall models in previous studies, it had the poorest performance statistics in this study for extreme rainfall events. [38] 3. Though Indiana was identified to be a statistically homogenous region for rainfall [Bonnin et al., 2004; Rao and Kao, 2006] because of the univariate homogeneity test [Hosking and Wallis, 1997], it was observed that the spatial variabilities of various rainfall parameters are not similar to each other, and hence homogeneity may not exist in this multivariate sense. [39] 4. The dependence between volume and duration was found to be positively correlated, between duration and peak intensity to be negatively correlated, and between volume and peak intensity to be positively correlated. The Frank family of Archimedean copulas was shown to be an appropriate model for characterizing these dependence behaviors. [40] 5. The bivariate joint distribution can be constructed by merging marginal distribution and dependence structure. The application of conditional distribution of depth given a known measured duration yielded rainfall estimates that were qualitatively similar to those obtained through the conventional univariate approach for durations larger than 10 hours. For smaller durations, the AMP definition for extreme events is recommended for estimating design rainfall depths. The regional average depth ratio indicates that these two algorithms provide same level of estimates for longer durations. This proposed bivariate stochastic rainfall model is expected to provide a better characterization for extreme rainfall behavior. [41] 6. For peak intensity, a comparison between bivariate estimates given a known measured duration and univariate estimates combined with Huff curves was performed. The conventional approach failed to capture peak intensities for a majority of rainfall durations. The AMP definition is again recommended for estimating peak intensity. [42] The copula method with the AMP definition for extreme events was examined over Indiana in this study. We emphasize that the method can be translated to other geographical areas, where perhaps different marginal distributions and copulas may apply. We believe that hydrologic design, which is largely driven by design rainfall, will be affected by this new methodology. References Bacchi, B., G. Becciu, and N. T. Kottegoda (1994), Bivariate exponential model applied to intensities and durations of extreme rainfall, J. Hydrol., 155(1 2), Bonnin, G. M., D. Martin, B. Lin, T. Parzybok, M. Yekta, and D. 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