Statistical modeling of flood discharges and volumes in Continental Portugal: convencional and bivariate analyses

Transcription

1 Statistical modeling of flood discharges and volumes in Continental Portugal: convencional and bivariate analyses Filipa Leite Rosa Extended Abstract Dissertation for obtaining the degree of master in Civil Engineering Jury President: Professor António Jorge Silva Guerreiro Monteiro Supervisor: Professor Maria Manuela Portela Correia dos Santos Ramos da Silva Co-supervisor: Eng. João Filipe Fragoso dos Santos Vowel: Professor Maria Madalena Vitório Moreira Vasconcelos October 011

2 Abstract The frequency distribution analysis applied to annual maximum instantaneous discharges is a common procedure in the estimation of peak flood discharges for different return periods. However, the length of the available samples of such discharges is often too small to validate the statistical inference. The present study provides an alternative approach that allows overcoming such circumstance by also applying such analysis to samples of discharges above given thresholds, which normally have a longer length as they incorporate more than one event per year. Thus, besides the development of models of frequency distribution based on samples of annual maximum instantaneous discharge and on samples of discharge above given thresholds, the present study has as a second objective the comparison between discharges and volumes provided by each one of those types of samples. However, the majority of the hydrological events cannot be fully characterized based on only one of the variables involved as, in general, they depend on a diverse set of variables. This is the case of flood events in which, in addition to the peak discharges, the volumes associated with such discharges or even the duration of the floods may also be important. Thus, the study of the joint probability distribution of the variables involved in these events may provide a better understanding of their probabilistic features. In that context, the third main objective involved the development of statistical models to characterize the joint probability distribution of flood discharge-flood volume, namely applying a joint distribution and the copulas theory. Keywords: samples of annual maximum values; samples of values above given thresholds; frequency analysis of flood discharges and of flood volumes; copulas. 1. Introduction In hydrologic design, the frequency distribution analysis is regularly applied and many authors have studied this subject and analyzed its applications. Specifically in Portugal, NUNES CORREIA, 1983, developed statistical models to estimate maximum annual daily average discharges based on samples collected at hydrometric stations purposely scattered throughout the country, in order to enable a characterization of the suitability of the proposed techniques in Portugal. The author tested the Gumbel, log-gumbel, log-normal, Pearson III, log- Pearson III and Extreme Generalized distributions and concluded that, in general, the Pearson III function was the best model of frequency distribution for such applications, followed by, in order of preference, the Gumbel and log-pearson III distributions. Equivalently, HENRIQUES, 1981 and 1990, conducted studies comparing different statistical models, with the aim of selecting the most suitable for frequency distribution analysis of annual maximum instantaneous discharges, also in Portugal. In the set of both papers, the following distributions were tested: Pearson III, log-pearson III, Fisher-Tippett generalized, Fisher-Tippett type I, three parameters log-normal, two parameters log-normal, extreme type I and Extreme Generalized. The distributions that showed a better behavior were the Extreme Generalized, the Pearson III, the three parameters log-normal and the Generalized Fisher-Tippett. However, the multivariate analysis of flood events, that, in fact, cannot be fully characterize by considering only one of the variables involved, is also subject of study. Until recently, the most common procedure was to adopt a known joint probability distribution, such as the Normal distribution, to model the dependence between two or more random variables (GRIMALDI et al., 005). This method implied the assumption that every variable involved had the same marginal distribution, not being possible to distinguish the behavior of the marginal from the joint distribution. To overcome such limitation, an alternative approach has been introduced and it involves the copulas theory, where the construction of the multivariate model is completely independent from the marginal distributions. The application of such functions in Hydrology has been fairly reduced and, until the date of the present study, there are no known works that apply the copulas theory in a Portuguese context. The following synthesis resumes some of the works that were in the scientific base of the present investigation: SALVADORI and DE MICHELE, 004 presented theoretical aspects of frequency analysis based on copulas. They also presented case studies to illustrate the power of the copula- based analysis. FAVRE et al., 004, developed a methodology for modeling extreme values using copulas. DE MICHELE et al., 005 derived a bivariate extreme 1

3 value distribution for modeling peak and volume, using the copula concept. The marginals were the generalized extreme value distribution. They applied the bivariate distribution to assess the adequacy of spillway. GRIMALDI et al., 005, employed a 3-dimensional copula for modeling peak, volume and duration, simultaneously.. General concepts.1. Series of annual maximum values and series of values above given thresholds In the present study two types of data series were considered: annual maximum series and series of values above given thresholds. This type of series can constitute a better source of information and a stronger basis for the statistical model of frequency distribution than annual maximum series, since in the later only the highest event of each hydrological year is considered, not taking into account the following major events that, in fact, can exceed the maximum events of other years. In addition, according to HENRIQUES (1990, p.17), the uncertainty associated to the estimation models based on series of values above given thresholds is lower for small return periods, especially in the case of small size samples. However, the determination of such thresholds constitutes an arbitrary factor (HENRIQUES, 1990, p. 38), which represents a disadvantage of these types of series when compared with annual maximum series. In the series of values above given thresholds, a frequency distribution is fitted to the series of all floods which exceed a certain base or threshold value. If there are M such events in N years of record, then the average rate of occurrence is M N. Therefore, as is considered more than one event per year, a correction in the return period is necessary: to an event in the series of values above given thresholds corresponds a return period measured in years of approximately T 1 1 F' associated with the considered event., where F is the non-exceedance probability.. Frequency distribution analysis The application of a frequency distribution model to a random variable sample is a phased process that can be developed in accordance with the following steps (HENRIQUES, 1990; and TAKARA STEDING, 1994, in DIAS, 003, p. 16): (i) list the theoretical distributions applicable to quantiles determination; (ii) obtain the parameters for each distribution; (iii) verify the goodness of fit of each postulated distribution to the sample. In the present study, the theoretical distributions considered are suggested, in the specified literature, as the most suitable distributions for adjustment to hydrologic series, which were: Normal, log-normal, Pearson III, log-pearson III, Gumbel and Goodrich. The mathematical expressions of the cumulative distribution functions of each theoretical function considered are listed in table 1. However, for quantile estimation the probability factor method was applied. This method, which stems directly from the method of moments (HENRIQUES, 1990, p. 319), allows the estimation of the value of a random variable with a given return period from the mean, μ, and standard deviation, σ, of the corresponding sample, which are equaled to the mean and to the standard deviation of the distribution, respectively. The equation that defines the application of the probability factor is given by CHOW, 1964: x T K T (1) where x T represents the estimated value of the variable X for the return period T and K T is the corresponding probability factor that depends on the probability distribution and the considered return period. In table 1, the mathematical expressions for the probability factor of each distribution are also listed. The method of moments was applied for the parameters estimation of each distribution, which are: α position parameter, β scale parameter and θ shape parameter.

4 Table 1 Cumulative distribution functions and probability factor of the theoretical distributions considered. Distribution Cumulative distribution functions Probability factor Normal* 1 Pearson III* 1 * 1 The log-normal and log-pearson III distributions are defined by the same expressions as the correspondent Normal and Pearson III functions, the difference relies in the sample, which must be transformed in its logarithmic form: y ln( x) * 77 (Euler-Mascheroni constant) * 3 Γ is the Gamma function * 4 For the Goodrich distribution, the probability factor method was not applied. For simplicity reasons only, the following formulation was used instead: 1 ln 1 x T x 1 1 x F x exp dx X 1 x x 1 F ( x) exp X dx ( ) x Gumbel F ( x) exp exp X Goodrich* 4 F ( x) 1 exp ( x ) 1 X In the present study, the evaluation of the adjustment of each distribution to the random variable consisted in three steps: visual adjustment, chi-square test and Kolmogorov-Smirnov test. For the visual adjustment the sample values and estimates obtained by the postulated probability distributions were displayed in the same graphic, which permits to visually identify the distribution with the most similar development to the hydrologic series. In order to automate the procedure of graphical representation on Normal probability paper inherent in the visual examination, the successive values of cumulative probability functions, whether provided by the distributions of theoretical or empirical probabilities, were expressed in terms of the reduced values of the Normal variable, z, to which they relate. Thus, the ordinates axis of the graphics represent the sample values or estimates obtained by the postulated probability distributions and the abscises axis represent the reduced Normal values, z. The hypotheses tests of chi-square and Kolmogorov Smirnov consist in the evaluation of the initial premise that the variable analyzed follows the frequency distribution put to test. The results are the rejection or not rejection of the premise, taking into account that the not rejection result does not indicate the suitability of the theoretical distribution. The methodology of the chi-square test requires the division of the cumulative distribution function domain in M intervals to, subsequently, compare the number of sample elements contained in each j interval, O j, with the mathematical expectation, as indicated by the model, of the number of elements corresponding to that interval, E j. The test statistic is defined as (NAGHETTINI and PINTO, 007):, , 80853z0, 01038z K z z 8969 z 0, z K z z 1 z 6z z 1 z K 6 T ln ln T 1 * 1 K 1 Γ( 1) Γ 1 Γ ( 1) 1 1 T Γ 1 Γ ( 1) ln 1 * 3 M ( Oj E j ) E j j1 () 3

5 The statistical test can then be formulated as follows: the hypothesis is rejected with a confidence level of (1-α) if χ > χ 1-α, in which χ 1-α is the quantile (1-α) of the χ distribution (unilateral test). In the present work it was adopted a significance level of α = The Kolmogorov-Smirnov test consists in the determination of the statistic D N given by the maximum difference between the functions of accumulated empirical and theoretical probabilities. The test statistic is defined by (NAGHETTINI and PINTO, 007): D máx F ( x) F ( x) N N x (3) where F N (x) is the empirical non-exceedance probability assigned to each sample element and F X (x) is the corresponding theoretical estimate, calculated according to the probability distribution model put to test. The formulation of the Kolmogorov-Smirnov test is analogous to the chi-square test: the hypothesis is rejected with a confidence level of (1-α) if D N >D N,1-α (unilateral test), in which D N,1-α is the critical value, the maximum acceptable for that level of confidence. The critical value of D N depends on the sample size, N, and can be obtained by consulting the respective tables of values. The empirical non-exceedance probabilities were estimated for each rainfall variable using CUNNANNE S, 1978, formulation, as the author suggests that it is the best choice when a single simple compromise formula for use with all distributions is required: / 5 1/ 5 F x i N i (4) x i is the element i of the random variable X, i is the order number of the element, for which the sample should be sorted from its lowest to its highest value, and N is the sample size..3. Multivariate analysis The multivariate analysis consists in obtaining a statistical model that can represent the joint probability distribution of two or more dependent random variables. The conventional process uses the very limited number of multivariate models available that, generally, are not well suited to represent extreme values. The Normal, the Gumbel and the Exponentional models have fairly dominated the statistical study of multivariate distributions (GRIMALDI et al., 005). In addition, those models have usually been derived using one of three fundamental assumptions (ZHANG and SINGH, 006): (i) either rainfall variables have each the same type of the marginal probability distribution; (ii) the variables have been assumed to have a known joint probability distribution, or (iii) they have been assumed independent. However, in reality, rainfall variables are dependent, do not have, in general, the same type of marginal distribution and their joint probability distribution do not correspond to any of the mentioned models. To overcome such limitations, the copulas have been recently introduced in the hydrologic multivariate analysis, with the exceptional advantage that no assumption is needed and the selection of the copula and of the marginal distributions are totally independent from each other. Concept of copula: Let there be two random variables, X and Y, with joint distribution function given by F X,Y (x,y) and the respective marginal distribution functions given by F X (x) and F Y (y). According to Sklar's theorem (ZHANG and SINGH, 006), for any pair of random variables there is a dependence function C(u,v) such that: where U = F X (x) and V = F Y (y), so that: FX, Y ( x, y) C FX x, FY y (5) 4

6 FUV, ( u, v) C( u, v) (6) Using such concept, it is possible to define the joint distribution of random variables with the marginal distributions of each variable - which completely characterize the behavior of the variables when considered isolated - and with a dependence C function - which contains all the information of how the variables depend on each other. Therefore, the problem of the joint distribution definition is reduced to the determination of those two components: the marginal distribution functions and the dependency function C, the copula. For the present study, three Archimedean copulas were considered (Clayton, Frank and Gumbel-Hougaard), since the Archimedean family has been suggested as appropriated for hydrologic applications (ZHANG e SINGH, 006). The Archimedean copula is a one-parameter function, parameter θ, which expresses the dependence between the variables analyzed, in the copula function. This parameter depends on the copula function considered and on the Kendall s coefficient, τ. The Kendall s coefficient is a measure of agreement between two variables: there is agreement between observations when the values of a given observation (x, y) are both higher or lower than the values of another observation, i.e., if the trends between each two observations are equal for x and y. Kendall s coefficient can be determined with the following formulation (ZHANG e SINGH, 006): N 1 sign xi x j yi y j (7) i j where N is the sample size and sign = 1 if x x e y y and sign = - 1, otherwise, for i, j = 1,,, N. x and y i j i j are the values of the random variables X and Y, respectively. In table 3, the mathematical expressions of the three Archimedean copulas applied are listed, as well as the expression corresponding to the bivariate Gumbel function (MELGHIORI, 003; ZHANG and SINGH, 006 and YUE and RASMUSSEN, 00). This model was also applied in order to compare the results provided by the two methods Gumbel and copulas and, thus, identify the model that produces better results. In the same table, the equations that relate the copula parameter, θ, and the Kendall s coefficient, τ, are also listed, as well as the equivalent equation and parameters of the bivariate Gumbel function. For the selection of the best joint distribution model, the evaluation of the adjustment of the theoretical functions to the observed data implied the visual adjustment and the application of the Kolmogorv-Smirnov test. Using the same technique as for a single variable, the empirical joint distribution for a pair of dependent variables based on ordered values was calculated as (YUE, 1999): F( u, v) NP 4 N (8) where N represents the sample size and NP represents the sequential order number of each pair of sample elements. Since the sample now consists of the pairs of values of the random variables analyzed, in the present study volume and flood discharge, it is not possible to sort increasingly the sample values, simultaneously. Therefore, the NP number associated to the element (x i,y i ) is given by the number of pairs (x j, y j ) that verify the following conditions: x x and y y for i, j 1,..., N. j i j i 5

7 Table Mathematical expressions of the Archimedean copulas, the bivariate Gumbel function and the dependence parameter equations Function * 1 D 1 is the first order Debye function D k. The resolution of this function is not analytically possible, therefore the parameter θ F was obtained resorting to the commercial software ModelRisk * ρ is the Pearson s correlation coefficient 3. Application Function mathematical expression Clayton copula 1 Frank copula Gumbel-Hougaard 1 C ( u, v) ln 1 F F C( u, v) u v Rainfall data From the hydrometric stations listed in the Sistema Nacional de Informação de Recursos Hídricos, SNIRH (responsibility of the Instituto da Água, INAG) four in natural regime were selected. No other special selection criteria were considered other than the need to ensure series as long as possible and with recognized quality. For these stations INAG provided tables containing the instantaneous records of the hydrometric heights (collected every hour) and the discharge curve equations. The records of annual maximum instantaneous discharge listed in SNIRH were also consulted. The hydrometric stations selected were: Albernoa (6J/01H), Couto de Andreiros (18L/01H), Monforte (19M/01H) and Torrão do Alentejo (4H/03H). The hydrometric records between the hydrological year of 1961 /196 and 1999/000 were obtained. However, not all hydrological years included in the previous period of data acquisition were considered. Indeed, the years that presented register failures that could not ensure the record of the annual maximum instantaneous discharge were eliminated. In general, the data collection and processing had as major steps: Dependence parameter equation C C C C C u v F F exp 1 exp 1 exp 1 1 C ( u, v) exp ln u ln v copula G Bivariate Gumbel (i) the digitization of hydrometric records, (ii) the completion of the series that presented gaps in the records, by interpolation of successive registered hydrometric heights; (iii) the selection of the flood events through the thresholds definition. For that limit It was adopted a flood discharge of 5 times de modular discharge, as QUINTELA, 1996 (p. 10.1), suggests that threshold as an indicative of the occurrence of a flood event; (iv) the time "delimitation" of the independent flood events: it was assumed that a given flood hydrograph, with one or more peak discharges, separated from the next hydrograph by a recession period sufficiently long to ensure the annulment of the direct runoff, would constitute a independent flood event. For the peak discharge of each event it was adopted the maximum instantaneous discharge registered during the considered event; (v) the estimation of the direct runoff volume associated with the flood events as defined, through the numerical integration of the corresponding hydrograph. These calculations implied the identification of the base runoff so it would be deducted from the total flood volume. For the base runoff it was adopted a very simple model, given by the union, through a line segment, of the lower water discharge between two independent events. F G G G 4 1 D 1 * 1 1 F F ( x, y) exp ln F ( x) m ln F ( y) m m function X, Y X Y m F 1 1 G * 6

8 From the information collected as indicated, for each hydrometric station, the following four types of samples were constructed: i) samples of flood peak discharges above the threshold adopted for each hydrometric station ATQ, ii) samples of the flood volumes corresponding to the previous discharge samples ATV, iii) samples of annual maximum instantaneous discharge AMQ and v) samples of the flood volumes corresponding to the previous discharge samples AMV. 3.. Identification of marginal distributions of rainfall variables Using the methodology presented in section., the six theoretical distributions considered were fitted to each sample of flood discharge and volume. The evaluation of the goodness of fit of those theoretical distributions conduced to the selection of the best fitting model to each sample of three of the hydrometric stations considered: Albernoa, Couto de Andreiros and Torrão do Alentejo. For the samples constructed based on the gathered information from Monforte s hydrometric station it was not possible to select a theoretical model based on the visual adjustment. Once the hypotheses tests only indicate the rejection or not rejection of the hypothesis, they cannot be used to identify the statistical distribution with better adjustment as well as the error measurements applied which only provide complementary information. In result, the samples of flood discharge and volume of Monforte s hydrometric station were excluded from the subsequent study. The selected statistical distributions and the corresponding parameters are systematized in table 4. In order to simplify the subsequent applications of the statistical models systematized in the previous table, were also included the means and standard deviations of the different samples, so that the general equation of the probability factor method (1) can be applied. The probability factor equations can be consulted in table 1. Table 3 Identification of the statistical distributions selected to characterize each random variable. Statistical parameters values of each distribution. Albernoa Couto de Andreiros Torrão do Alentejo ATQ ATV AMQ AMV Distribution Goodrich log-pearson III log-pearson III log-normal α ,8145,3039 8,3587 β θ ,5139, λ 3,4000 3, Mean 53,004 (m 3 /s) 359,781 (1000 m 3 ) 954 (m 3 /s) 5901,911 (1000 m 3 ) Standard-deviation 53,648 (m 3 /s) (1000 m 3 ) 69,58 (m 3 /s) 6119,76 (1000 m 3 ) Distribution log-pearson III log-pearson III Gumbel log-pearson III α ,7335 5,054 β , θ 9, , ,568 λ,3333, Mean 8,157 (m 3 /s) 10061,857 (1000 m 3 ) 116,651 (m 3 /s) 1434,861 (1000 m 3 ) Standard-deviation 734 (m 3 /s) 10694,666 (1000 m 3 ) 84,51 (m 3 /s) 11968,866 (1000 m 3 ) Distribution Goodrich log-pearson III Pearson III log-pearson III α -3,1377 6, ,940 β , θ 831 3,7070,333 4,8973 λ,941, Mean 103,11 (m 3 /s) 11853,590 (1000 m 3 ) 156,414 (m 3 /s) 19339,619 (1000 m 3 ) Standard-deviation 75,854 (m 3 /s) 10947,13 (1000 m 3 ) 73,597 (m 3 /s) (1000 m 3 ) 7

9 3.3. Comparison between models based on series of annual maximum values and based on series of values above given thresholds As two types of series of values were considered, from the frequency analysis applied resulted two types of statistical models: one based on annual maximum series (method I) and the other based on series of values above given thresholds (method II). Therefore, for each hydrometric station, it is now possible to estimate values of flood discharge and volume for different return periods using two different approaches. Thus, an appropriate exercise would be to compare the estimates of flood discharge or volume provided by the two methods. In fact, if both methods are valid, it is expected that for the same return period and for the same variable flood discharge or volume they provide similar estimates and, therefore, the relationship between these estimates is approximately linear. The exercise consisted in the comparison of the estimates in question for a set of values of return period with great range - in particular for, 4, 10, 0, 5, 50, 100 and 1000 years - in order to evaluate the adequacy of the linear relationship across the range of values expected for T in hydraulic design. In order to simplify the exercise, the estimates of the three hydrometric stations were expressed in terms of their specific values which are obtained dividing each estimate by the watershed area to which they relate. Figure 1 contains two graphics, one for each variable flood discharge or volume where the abscises axis refers to the estimates provided by method I and the ordinates axis to the estimates resulting from method II. Each graphic also contains the line segment that represents the linear regression between the estimates and the corresponding determination coefficient, R, which is the square of the Pearson correlation coefficient, ρ. R allows to quantitatively evaluate the quality of the linear relationship: the coefficient varies between 0 and 1, where the maximum value can be interpreted as 100% of the variance of the variable referred to the y-axis is explained by the variance of the variable referred to the x-axis. Specific flood flow - method I (m 3 /s/km ),5 Specific flood volume - method I (1000 m 3 /km ) 800,0 1,5 R² = R² = ,5,0,5 Specific flood flow - method I (m 3 /s/km ) Specific flood volume - method II (1000 m 3 /km ) Figure 1 Hydrometric stations of Albernoa, Couto de Andreiros and Torrão do Alentejo. Relationship between estimates of the specific values of flood discharge and volume, with the same return period, provided by method I (left) and method II (right). Line segments representing the linear regressions and corresponding determination coefficients, R. The graphics in figure 1 clearly indicate that the two approaches are very similar, as it is possible to establish determination coefficients higher than 6 for both discharge and volume. Therefore, the obtained results conduce to the conclusion that, for the three hydrometric stations analyzed, the two processes of sampling provide very similar estimates of the same variable, discharge or volume. Finally, it is possible to observe that the point dispersion around the regression line is more pronounced for higher return periods. This may be a reflection of the imprecision of statistical inference which arises from the small size of the samples Identification of the best bivariate model The multivariate analysis is applied to series of values of more than one variable. In the present case, the analysis was applied to the pair of samples of flood discharge and volume annual maximum instantaneous discharges/corresponding volumes (AMQ/AMV) and flood peak discharges above the threshold /corresponding 8

10 volumes (ATQ/ATV), as constructed in section 3.1. However, previously, in order to verify the applicability of the multivariate analysis to such series of values the correlation between samples was evaluated by determining the Pearson coefficients of each pair of samples. The results are presented in table 5 below. Table 4 Hydrometric stations of Albernoa, Couto de Andreiros and Torrão do Alentejo. Pearson correlation coefficients between samples of AMQ and AMV and samples of ATQ and ATV. AMQ/AMV ATQ/ATV Albernoa Couto de Andreiros 4 0 Torrão do Alentejo 18 In all three hydrometric stations, only the samples of flood discharge above thresholds and the samples of corresponding volumes have correlation coefficients with statistical significance for present objective, only those samples were considered for the application of the multivariate analysis. The application of the copula functions presented in section involves the determination of the parameter θ of each copula, which can be defined as an estimator of the dependence between the two variables involved and it requires the determination of Kendall coefficient, τ. Similarly, the parameter m is the estimator of dependence used in bivariate Gumbel distribution and its determination depends on the Pearson correlation coefficient, ρ. Given the importance of the above parameters, table 6 shows the values obtained for each sample discharge/volume, where are also included the previously presented values of the Pearson correlation coefficients. Table 5 Hydrometric stations: Albernoa, Couto de Andreiros and Torrão do Alentejo. Values of: Pearson correlation coefficient, ρ, parameter m of the bivariate Gumbel function; Kendall coefficient, τ, and θ parameters of the copula functions Clayton, θ Clayton, Frank, θ Frank, and Gumbel, θ Gumbel. Albernoa Couto de Andreiros Torrão do Alentejo ρ 94 0 m 1,809 1,889 1,896 τ θ Clayton 5,7 3,591 5,800 θ Frank 13,574 9,18 13,73 θ Gumbel 3,861,795 3,900 The observation of the previous table allows the following observation: it has been claimed that the construction of bivariate models depends mainly on the Pearson correlation coefficient, ρ, and Kendall's coefficient, τ, for the bivariate Gumbel and copula functions, respectively. The fact that these coefficients do not show the same trend for example, the lowest values of ρ and τ do not correspond to the same sample means that the coefficients evaluate differently the dependencies between variables and, therefore, the bivariate model constructions rely on bases of different nature. For the selection of the best bivariate model the Kolmogorov-Smirnov test was performed as well as the visual adjustment between observed (empirical) and theoretical joint non-exceedance probability of each element of samples ATQ/ATV, obtained from equation (8) and from the functions presented on table 3, respectively. However the four curves representing the theoretical functions presented adjustments to the sampling points virtually indistinguishable, therefore, in order to identify the best fitting function, the selection criterion consisted in the identification of the minimum Kolmogorov-Smirnov test statistic, since no bivariate model was rejected, a result consistent with the visual adjustment. The Kolmogorov-Smirnov test results are presented in table 7 and the minimum test statistic of each sample is indicated by gray fill cells. In table 8, is presented the 9

11 bivariate distribution selected for each discharge/volume sample as well as the corresponding parameters necessary to the distributions definition. Table 6 Kolmogorov-Smirnov test results Albernoa Couto de Andreiros Torrão do Alentejo Clayton copula Gumbel copula Frank copula Bivariate Gumbel function Test statistic (D N ) D N,1-α, α= Test result Test statistic (D N ) D N,1-α, α= Test result Test statistic (D N ) D N,1-α, α= Test result Note: 1 hypothesis is not rejected; 0 hypothesis is rejected Table 7 Bivariate function selected for each sample Selected function θ parameter Albernoa Frank copula 13,574 Couto de Andreiros Clayton copula 3,591 Torrão do Alentejo Clayton copula 5,800 It is important to note that the multivariate analysis never resulted in the selection of the bivariate Gumbel function to describe the probability distribution of the variables under evaluation. However, the fact that only three case studies were analyzed, it is not true to say that the copulas theory may provide a more appropriate model than the bivariate Gumbel function. In this context, it is still necessary the following observation: the application of the Gumbel distribution in the bivariate form implies the assumption that the marginal probability distributions of the variables in presence also follow that distribution, which may be incorrect, and indeed, in the procedures implemented, the univariate analysis applied to the samples never resulted in the selection of the Gumbel function. In conclusion, for the three hydrometric stations considered, and through copulas theory, it was possible to establish a descriptive model of the joint probability of the variables of flood discharge above a given threshold, Q, and the volumes associated with those discharges, V. Figure, contains the surfaces for each of the copula functions obtained. The horizontal axes correspond to the marginal probabilities of non-exceedance of discharge and volume and the vertical axis, to the joint probabilities of these two variables. For the previous curves were generated randomly marginal probabilities of non-exceedance and calculated the corresponding theoretical joint probabilities. 10

12 F(Q,V) F(Q,V) F(Q,V) a) b) F(V) F(V) c) F(V) Figure Surfaces of joint probability distribution of discharge and volume in the hydrometric stations: a) Albernoa b) Couto de Andreiros c) Torrão Alentejo. Although these graphics represent faithfully the curves to which they refer, their usefulness is somewhat reduced. In addition three graphics of equal joint probability curves (probability isolines) were also obtained. In such graphics, the axes correspond to the non-exceedance probability of flood discharge above threshold variables,, and of the associated volume variables, F(V). Each curve respects to a given joint probability, F(Q,V).The graphics mentioned are presented in figure graphics a1), b1) and c1). In addition, it was also considered appropriate to seek other ways to illustrate the curves obtained in order to facilitate their understanding and provide perspectives for future applications of the copulas theory. Thus, three other graphics, one for each hydrometric station, were constructed with the assumption that the nonexceedance probability associated with the variable volume is pre-set, for what it was considered the probabilities of 5, 0 and 5. By varying the non-exceedance probability associated with the discharge variable three curves were obtained, for each hydrometric station, that provide the joint probability for those non-exceedance probabilities of the volume variable. The graphics containing the curves are presented in figure graphics a), b) and c). In each graphic, the x-axis corresponds to the non-exceedance probability of the flood discharge above given threshold and the y-axis to the joint probability of the two variables, with pre-set non-exceedance probability of volume. 11

13 F(V) F(Q,V) F(V) F(V) F(Q,V) F(V) F(Q,V) a1) a) b1) b) c1) c) F(Q,V)= F(Q,V)= F(Q,V)= F(Q,V)= F(Q,V)= F(Q,V)= F(Q,V)= F(Q,V)= F(Q,V)= F(Q,V<5) F(Q,V<0) F(Q,V<5) Figure 3 Curves of equal joint probability: a1) Hydrometric station of Albernoa; b1) Hydrometric station of Couto de Andreiros; c1) Hydrometric Station of Torrão do Alentejo. Joint probability curves of flood discharge above threshold and volume with non-exceedance probability of 5, 0 and 5: a) Hydrometric Station of Albernoa; b) Hydrometric station of Couto of Andreiros; c) of hydrometric station of Torrão do Alentejo. 1

14 The progress of each of the curves in figure 4 reflects the expected behavior for the two variables at hand: as the non-exceedance probability of the discharge variable tends to unity, the curves tend to the value of the non-exceedance probability inherent to the variable volume (vertical cuts on the surfaces of figure for the non-exceedance probabilities stipulated, 5, 0 and 5). In fact, non-exceedance probabilities values of the discharge variable in close proximity to one correspond to very high discharge values and therefore may be associated with almost any value of the volume variable. In this extreme case, and for each curve, the joint probability is "independent" of the non-exceedance probability of the discharge variable and tend, as much as possible, to the set value for the non-exceedance probability of the variable volume. 4. Conclusions The frequency analysis performed resulted in the selection of the log-pearson III distribution for the majority of the analyzed samples. This result is consisting to that reported in previous studies. In particular, HENRIQUES, 1990, specifically refers that this distribution has a good adaptability to the generality of the annual maximum instantaneous discharge series. In addition, with the aim of selecting the most appropriate frequency model of annual maximum instantaneous discharge, HENRIQUES, 1981, performs a comparative analysis of different models, concluding that there is a clear superiority of the models based on distributions of three parameters (such as log-pearson III), in relation to models based on distributions of two parameters. The comparison between models based on series of annual maximum values and based on series of values above given thresholds led to the following conclusions: a) the estimates derived from both samples are very similar, corresponding to determination coefficients higher than or equal to 3, and b) with one exception, the flood volume in Couto de Andreiros, the use of samples of values above threshold provided estimates of discharge and flood volume higher than those obtained from samples of maximum annual values. However, such differences are not significant, so it can be said that the two approaches provide equivalent results, which validates the use of sample values above thresholds, particularly in cases involving short periods of records. Regardless of this conclusion, the following comments on the advantages and/or disadvantages of two the types of samples analyzed are considered relevant: a) compared to maximum annual samples, the samples of values above the thresholds contain broader information since those first samples only consider the major hydrological event of each year, not taking into account the following major events that may be superior than the maximum events from other years; b) the need to set a limit to form a sample of values above threshold is, generally, an arbitrary factor (Henriques, 1990, p. 38), requiring further investigation; c) the constitution of samples of values above thresholds is clearly more difficult and laborious since the information required for this purpose is not usually, accessible, whereas the collection of annual maximum instantaneous discharge is immediate through the Sistema Nacional de Informação de Recursos Hídricos, SNIRH (available by Instituto da água; d) by definition, the samples of values above thresholds are always larger than annual maximum ones, which induces a greater confidence in the selection of models to be applied in statistical inference. Concerning to the bivariate analysis, the methodology applied in the present study allowed the selection of a copula function to characterize the joint probability distribution of each flood discharge-flood volume sample. Once more, it is extremely important to note that the bivariate Gumbel function was never selected, based on the selection criterion adopted. However, taking into account that only three case studies were analyzed, it is not possible to declare that the copulas theory provide a better modeling process than the one involving a known joint distribution. Nevertheless, to choose the bivariate Gumbel function for the characterization of the joint distribution of any of the samples would constitute an error, as the frequency analysis applied to the isolated samples never resulted in the selection of the Gumbel function for the marginal distributions. The practical applications of this type of analysis may take the form presented in chapter 4.4. where, for example, were obtained graphics that provide non-exceedance probabilities of flood discharge for a given maximum value of the non-exceedance probability of flood volume by selecting a non-exceedance probability of the joint event. On the other hand, obtaining the value of non-exceedance probability of the joint event is 13

15 essential for the study of joint return periods, a concept that has been recently analyzed by some researchers. Understandably, to a joint return period corresponds more than one pair of discharge/volume. Thus, it becomes interesting to obtain analytical expressions and graphics of isolines of joint non-exceedance probability, as presented in chapter 4.4. As a suggestion for future work, it is considered appropriate the development of the concept of joint return period based on a multivariate analysis equivalent to the subject of this research. The isolines joint return period can be extremely useful to simulate different scenarios to assess the risk associated with flood events. The determination of the conditional probability distribution of hydrologic variables have also been developed by several authors, particularly by those whose work formed the basis of the scientific research presented. Thus, the study of conditional analysis, in the Portuguese case, will also constitute an interesting approach. Acknowledgments To the Instituto Nacional da Água (INAG) for supporting this research by providing the required data. References CHOW, V.T., 1964, Section 8-I. Statistical and Probability Analysis of Hydrologic Data. Part I Frequency Analysis. In: Handbook of Applied Hydrology. McGraw-Hill, USA. CUNNANE, C., Unbiased Plotting Positions a Review. Journal of Hydrology, 37, pp DE MICHELE, C., SALVADORI, G., CANOSSI, M., PETACCIA, A., ROSSO, R., 005. Bivariate statistical approach to check adequacy of dam spillway. Journal of Hydrologic Engineering ASCE 10 (1), DIAS, A. T. G. S., 003. Caudais Instantâneos Máximos Anuais em Portugal Continental, Proposta de Regionalização. Masters Degree Dissertation in Civil Engineering. Instituto Superior Técnico, Lisboa. FAVRE A-C, EL ADLOUNI S, PERREAULT L, THIE MONGE N, BOBE E B., 004. Multivariate hydrological frequency analysis using copulas. Water Resour Res ;40 GRIMALDI, S. and SERINALDI, F., 005. Asymmetric copula in multivariate flood frequency analysis. Advances in Water Resources, 9, pp HENRIQUES, A. G., Modelos de Distribuição de Frequências de Caudais de Cheia. Dissertação de Doutoramento em Engenharia Civil no Instituto Superior Técnico, Lisboa. MELGHIORI, M. R., 003. Which Archimedean Copula is the Right One? YieldCurve.com e-journal. NAGHETTINI, M. and PINTO, E. J. A., 007. Hidrologia Estatística. Serviço Geológico do Brasil. QUINTELA, A.C., Hidrologia e Recursos Hídricos. Secção de Folhas. Instituto Superior Técnico. SALVADORI G, DE MICHELE C., 004. Frequency analysis via copulas: theoretical aspects and applications to hydrological events. Water Resources Research; 40:W1511. YUE S, OUARDA TBMJ, BOBE E B, LEGENDRE P, BRUNEAU P., The Gumbel mixed model for flood frequency analysis. J Hydrol; 6(1 ): ZHANG, L. and SINGH, V.P., 006. Bivariate Rainfall Frequency Distributions using Archimedean Copulas. Journal of Hydrology, 33, pp