COPULAS: A NEW WAY FOR INFLOW DESIGN FLOOD DETERMINATION Ross D. Zhou, P. Eng., Hatch Energy C. Richard Donnelly, P. Eng.

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1 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 COPULAS: A NEW WAY FOR INFLOW DESIGN FLOOD DETERMINATION Ross D. Zhou, P. Eng., Hatch Energy C. Richard Donnelly, P. Eng., Hatch Energy ABSTRACT In any dam safety assessment a key issue is the determination of the inflow design flood (IDF). Although there are many ways for the computation of IDF, flood frequency analyses based on long term flood records, remains one of the most frequently used methods. To simplify the analyses, flood frequency analyses mainly focus on one variable, the peak discharge. In fact, however, the actual magnitude of the IDF is dependent on a number of variables including the peak flow, the time to peak, the total volume and the duration of the flood. There have been several models developed for multiple variable frequency analysis. However, the usefulness of these approaches has been limited by the fact that they require that the variables all follow the same frequency distribution curve (e.g. multiple normal or multiple Gumbel). Therefore, if one of these variables is normally distributed and the other(s) belong(s) to a different frequency distribution, the models will not work properly. In this paper, a new method called copulas is described that provides a general way to construct a multivariate frequency distribution which overcomes the difficulties with the other methods providing an effective tool to hydrologists for the analysis of multivariate flood frequency distribution that can be very useful in dam safety assessments. RÉSUMÉ Dans n'importe quelle évaluation de sécurité de barrage, une question clé est la détermination de la crue maximale de dimensionnement. Bien qu'il y ait beaucoup de façons pour évaluer la crue de dimensionnement, la méthode d analyse de fréquence d'inondation basée sur des données d'inondation de long terme, reste une des méthodes la plus fréquemment utilisée. Pour simplifier les analyses, cette méthode se concentre principalement sur une variable, laquelle est la décharge maximale. En fait, cependant, l'importance réelle de la crue maximale de dimensionnement dépend d'un certain nombre de variables comprenant le débit de pointe, l'heure de pointe, le volume total et la durée de l'inondation. Il y a eu plusieurs modèles qui on été développés pour l'analyse de fréquence avec plusieurs variables. Cependant, l'utilité de ces modèles a été limitée par le fait qu'ils exigent que toutes les variables suivent la même courbe de distribution de fréquence (par exemple multiple normal ou multiple Gumbel). Par conséquent, si une de ces variables est normalement distribuée et les autres appartiennent à une distribution de fréquence différente, les modèles ne fonctionneront pas correctement. Dans cet article, on décrit une nouvelle méthode appelée la copule qui fournit une manière générale de construire une distribution multi-variable de fréquence et surmonte ainsi les difficultés des autres méthodes en fournissant un outil efficace aux hydrologistes pour l'analyse de la distribution multi-variable de fréquence d'inondation qui peut être très utile dans les évaluations de sécurité de barrage.. INTRODUCTION Determination of inflow design flood (IDF) is an important task in the assessment of the safety of dams. The inflow design flood is defined as most severe inflow flood (volume, peak shape, duration, timing) for which a dam and associated facilities are designed (CDA, 999). According to the definition, an IDF includes the total volume, the peak flow, the duration and the timing as the basic parameters to completely describe the entire hydrograph. When long term streamflow data is available, flood frequency analysis remains one of the most frequently applied methods to establish the IDF and there have been ongoing improvements in the methods available for these analyses. However, most of these developments focused on one variable, normally the peak flow or volume. The flood frequency analysis technique includes parameter estimation, estimation of confidence intervals, selection of suitable theoretical frequency distributions and an assessment of how well the data fits the distribution. In fact, however, the maximum water level that a reservoir may rise to during a flood event may also be affected by the peak flow, the flood volume, the rate of rise and the duration of flood, depending on the characteristics of the particular dam site. The IDF is, therefore, controlled by a number of variables, as recognized by the Canadian Dam Association (CDA). Therefore, analytical methods that account for only the peak flow or volume may not be sufficient to undertake dam design and dam safety assessments. For example, in the case of small

2 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 reservoir without storage, the peak flow may be the determining factor for establishing the maximum water level at a given damsite. On the other hand, for medium to large storage reservoirs, the volume of flood, how quickly the flows enter into the reservoir, the peak flow, the shape of the reservoir itself, how the reservoir is operated and the available spillway capacity will contribute to the maximum water level. Therefore, it is important to know the peak flow, volume of flood and the timing factors of the IDF. The theoretical development of the copulas method for performing multivariate frequency distribution analysis makes it possible for hydrologists to use a generalized multivariate flood frequency model for IDF determination. In this paper, the general method is discussed and applications described. For details of the derivations of the method, the reader is referred to papers by Favre et. al (2004) and Salvadori and Michele (2004). 2. STATE OF THE ART There are many methods available for flood frequency analysis and to allow flood hydrographs to be constructed. For example, the U.S. Department of the Interior, Bureau of Reclamation (989) described a balanced flood hydrograph method. Under the assumption that the maximum peak and maximum volume occur within the same flood event, the flood volume for all unit durations can be analyzed by frequency analysis and then to fit a single hydrograph including all of the volume data from the various unit times. A symmetrical or other reasonable shape is usually selected as a guideline for the fitting of the volumes. One of the problems associated with this method is associated with the inconsistencies that often exist between the volume frequency values for the different unit times. For example, the 00-year volume, obtained from a 5-day frequency analysis, might be higher than the 00-year 30-day volume because of the extrapolating calculations in the derivation. In addition, this method avoided multivariate frequency analysis. In 2004, the U.S. Department of the Interior Bureau of Reclamation released a research report that described four procedures used in their dam safety program to derive a hydrologic hazard curve. Some of the approaches produced a flood hydrograph by applying concepts very similar to the balanced hydrograph approach. Maione et. al. (2000 (A)(B), 2003) developed a procedure to estimate the synthetic design hydrograph (SDH) in gauged river where a long series of recorded data are available using a regional estimation method. The method is based on the analysis of recorded floods through the construction of the flow duration frequency reduction curve. Balocki and Burges (994) evaluated relationships between n-day flood volumes for extreme floods. They concluded that constructing design hydrographs for high return periods, using measured flood flow-volume-duration-frequency data, may provide an adequate estimation of extreme floods in some, but not in all cases, because of the limitations and assumptions needed to apply this approach. In reviewing these available methods, all were found to use the same concept in that only one variable evaluated in the frequency analysis. All of the other parameters are assumed to be associated with the main variable in various ways based on empirical relationships or assumptions. The drawback is that, in reality, these assumptions may not be valid such that the true probability characteristics of multivariate of flood events may not be fully accounted for. A more appropriate approach is the construction of a multivariate frequency distribution for analysis of flood events. However, due to the difficulties in modeling non-normally distributed multivariate, most of early works focused on the use of a bivariate normal distribution as the parent distribution function for the flood peaks and volumes. Therefore, to make the normal distribution applicable, some kind of data transformation is needed. Goel, Seth and Chandra (998) studied the method by applying the normalization procedure. They used a two step power transformation (TSPT) for normalization of the variables and applied the bivariate normal distribution to the transformed variables. One of the problems in data transformation is that the transformed data may not belong to a normal distribution. In addition, there are infinite numbers of potential transformation functions that can be used and there is no theoretical basis on the identification of the best transformation function. Correia (987) assumed that both flood peaks and durations are exponentially distributed and the conditional distribution of flood peaks for given flood duration is normal so that a bivariate distribution could be fitted to flood time series. Yue, et. Al. (999) developed a Gumbel mixed model for multivariate flood frequency analysis under the assumption that flood peaks, volumes and distributions could be represented by bivariate extreme value distribution with Gumbel marginal. With these assumptions, when the product moment correlation is greater than 2/3 or less than zero, the model fails to work. This can represent a serious problem since flood peaks, volumes and durations may be highly linearly correlated or negatively correlated in some time series. The problems related to the multivariate flood frequency distribution are due to the lack of suitable multivariate frequency distribution which has a strong theoretical basis for non-normally distributed random variables. Only the multivariate normal distribution has been systematically studied. Therefore, hydrologists did not have many choices in selecting an appropriate distribution until a method called copulas functions appeared. Sklar (959) showed that all finite dimensional probability laws have a copula function associated with them and, therefore, a multivariate probability distribution can easily be constructed

3 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 using copula function without assumptions and limitations. He proved that for any real valued random variables, x, x 2,, x p, with joint distribution F 2 p (X,X 2,,X p ), there is a copula C such that [ ] F L p ( x, x2, L, x p ) = C( F ( x ), F2 ( x2 ), L, Fp ( x 2 p )) Where; F (x ), F 2 (x 2 ),, F p (x p ) are the marginal distribution functions of THE variable x, x 2,, x p. Note that there is no limitation on the type of the marginal distributions so that each of the variables can have a different marginal distribution if so desired. Sklar provided a general way to construct a multivariate distribution that separates the joint distribution into two contributions: the marginal distribution of the individual variable, and the interdependency of the probabilities. Since the appearance of this general theory, many copulas functions have been studied and applied in statistics and other fields. However, although the work undertaken by Sklar has a long history, application of the theory in other scientific fields has only recently begun. In 2004, Favre, A. et. al. introduced copulas into hydrological analyses. They used two case studies to demonstrate the applicability of the methodology in hydrological studies and showed that the key problem in the use of copula is the selection of the best copula function to fit the data. Salvadori, G. and Michele, C.D. (2004) realized that hydrologists are familiar with the concept of return period for defining extreme flood events. Therefore, they derived the expression of return period for multivariate flood frequency distributions based on copula function for bivariate conditions. They showed that several definitions of return period are possible due to the complicated relationships between the variables. The joint probability approach makes risk assessments an easier task since the probability of events can be investigated using the joint distribution function and the definition of return period under various conditions. 3. MULTIVARIATE FLOOD FREQUENCY DISTRIBUTION COPULA Sklar s work (959) proved that for any given individual marginal probability distribution function composed of a number of variables, the joint probability function of these variables can be constructed using a copula function that captures the features of the joint distribution. There are many types of copula functions which satisfy equation [] and they have been extensively analysed. A few important bivariate copulas are briefly listed as follows: Frank s Copula (979) For two unit uniform random variables u=f (x ) and v=f 2 (x 2 ), which are the marginal probability distribution of X and X 2, a function of the form: [ 2] C( u, v) = ln + α α e αu αν ( e )( e ) Where, α 0 is the parameter of the function independence When two variables are independent, u and v have a copula function of: [ 3] C ( u, v) = uv The Clayton Family (978) has the following form: [4] α α ( u + ) α C ( u, v) = v

4 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 The Gumbel (960) family takes the form: α α [( ln u) + ( ln ) ] [ 5] C ( u, v) exp v = α Heavy Right Tail (HRT) Copula Flood events may be highly correlated in the right tail that is at the extreme peak and extreme volume. In such cases, the HRT copula would be more suitable for modeling this type of flood event. The HRT has the following form: [6] C ( u, v) = u + v + ( ) α ( ) α u + v α Given two variables of extreme floods, for example peak flows and total volumes, the probability distribution of the two variables can be established using one of the above copula functions. As pointed out by Salvadori and Michele (2004), hydrologists often use return period to define extreme flood events. Therefore, for this method to be widely applicable, it is necessary to have expressions for flood return period. Salvadori and Michele discussed eight types of return period definitions. For IDF derivation by bivariate distribution, accounting for peak flow and total volume, two of these return period functions are of interest that can be used to define two types of events: ) the maximum peak flow and maximum flood volume; 2) the maximum peak flow or maximum volume. Using the OR operator V and the AND operator Λ, the probability distributions for maximum peak OR maximum volume event can be defined as (Salyaadori and Michele (2004): p u v ( > u V > v) = C( u, ) [ 7], = P U v And, therefore, the return period of the OR event is T v [ 8] u, = C( u, v) The probability distributions for maximum peak AND maximum volume event can be defined as (Salvadori and Michele (2004): p u v ( > u V > v) = C ( u, ) [ 9], = P U v And, therefore, the return period of the AND event is T v [ 0] u, = C ( u, v)

5 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 Where; C is the survival copula of C which is defined by: [ ] C ( u, v) = u + v + C( u, v) In addition, for flood assessment, it is often necessary to define conditional flood events. For example, given the condition that a certain peak flood already occurred one needs to find the probability of the total volume of certain magnitude. In this case, conditional return period would be useful. Details of this type of return period can also be found in Salvadori and Michele s (2004) paper. The expressions of return period show that the return period of bivariate frequency distribution is determined by two variables. In many cases, one variable alone can not fully describe the return period of a flood event. As traditional methods can only account for one variable, either the peak flow or peak volume, to describe the recurrence interval of a flood event, this newer approach can offer some significant advantages in establishing the IDF for a particular dam site. 4. DEGREE OF ASSOCIATION AND PARAMETER ESTIMATION As described by Frees and Valdez (997), the parameter α of the copulas ise related to the association of the variables which can be defined by Kendall s τ or by Spearman s rank correlation. Kendall s τ estimator is defined as follows: For a random sample of n observations {(X,Y ),(X 2,Y 2 ),,(X n,y n )} obtained from a vector (X,Y) of continuous random variables, c denotes the number of concordant pairs, and d the number of discordant pairs. For this condition, an estimate of Kendall s τ is as follows: c d [ 2] τ = c + d ( c d) = n 2 = n 2 i< j sign [( X X ( Y Y )] i j i j The sample estimator of Spearman s rank function ρ s (X,Y) is defined as: [ 3] ρ s 2 n( n ) ( X, Y ) = 2 n i= n + n + rank( X i ) rank( Yi ) 2 2 Now the relationship between the degree of association and the parameters can be defined for each copula family as follows: Independency Copula: as indicated by the name of the copula, the variables are independent each other and hence the Kendall s τ = 0 and Spearman s ρ s = 0. Frank s Copula: the parameter α has the following relationship with Kendall s τ as 4 [ 4] τ = D α α { ( ) }

6 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 and with the Spearman s rank correlation ρ s as 2 [ 5] ρ s 2 D α ( X, Y ) { D ( α ) ( α )} = Where; for k= and 2, D k (x) is called the Debye function [ 6] D ( x) = k k k x k x t dt 0 t e Gumbel copula family: there is no closed form for relationship between Spearman s rank correlation and the parameter but the relationship between Kendall s τ is: [ 7] τ = α Clayton copula family: the parameter α is related to Kendall s τ as [ 8] α τ = 2 + α The Spearman s ρ s has very complicated form. HRT Copula: the HRT copula has the following relationship between α and Kendall s τ: [9] τ = 2 α + Parameter estimation can be achieved by maximum likelihood method. Details of the estimation can be found in many papers, for example, Favre (2004). This is also called one step estimation method because the parameters of the marginal distribution functions and the copula function are estimated simultaneously. However, this one step method requires extensive computational efforts. Another method of parameter estimation is the so-called two step method. In this method, the parameters of the marginal distributions are estimated first. The parameter of the copula function α is then estimated in the second step. The two step method is relatively simple and easy to understand but has the disadvantage that possible deviations in the first step will be magnified in the second step. In one step method, the log likelihood for the observation is denoted by l i (θ), where θ is the vector of the parameters to be estimated. Given n independent observations, the log likelihood function is:

7 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 i= n i= n j= p i i i ln c( F ( x ),, Fp ( x p )) + ln f j ( x j ) i= i= j= [ 20] l( θ ) = L Maximization of equation [20] by finding the parameters θ yields the estimation for the copula model. The reader should note that different copula are suitable for different applications. The next question then is: which copula function should be used for a particular data set and problem? An obvious answer is that one should use a copula function that has the capability to model extreme events since most of the floods that would be selected for use as the IDF represent extremely rare events. Venter (2002) discussed the properties of a few copula functions and revealed that copulas differ not so much in the degree of association but rather in which part of the distributions the association is strongest. Venter shown that the HRT copula is heavier in the right tail than Gumbel, but Gumbel is right tail correlated. The Frank copula do not show tail dependence. Therefore, the suitability of this copula function for flood frequency analysis needs to be assessed by the hydrologist for each particular situation. The Clayton copula has a heavy left tail dependence and hence is very suitable for low flow frequency analysis. Favre et al (2004) discussed the applications of elliptical copulas, Archimedean copulas and copulas with quadratic section in flood frequency analysis. They used four bivariate Archimedean copulas namely: independence, Clayton, Gumbel and Frank families in their application. The selection of the best copula function for IDF determination needs further study. 5. PROCEDURE FOR IDF DERIVATION As discussed, Copula functions and the return period expressions can be used to form the basis of IDF determination. A general procedure for such a derivation is described as follows:. Obtain the flood time series for the hydrometric station and determine the annual maximum flood event for each year. The flood characteristics include peak flow (Q p ), total volume (V) and time base (T b ) which is the duration of the flood event in days or hours. These parameters are illustrated in Figure. If a triangular hydrograph is assumed, one of these parameters can be estimated given two other parameters using the relationship: V=Q p T b /2 as shown in Figure. In this case, a bivariate frequency analysis will be sufficient to define the IDF. The peak flow and volume are used in this paper to illustrate the concept. Other combinations could be used depending on the problem to be assessed. 2. Select a copula function for the data sets 3. In step 3, there are two options available to estimate bivariate flood frequency distribution for the time series as outlined in section 3. Either the one step or two step estimating method could be used depending on the nature of the problem. 4. The bivariate flood frequency distribution defined in step 3 is used to define the return period for the flood event to determine the peak flow and volume. It should be noted that there are indefinite number of combinations corresponding to a return period. 5. Construct a synthetic flood hydrograph for a given peak flow and volume and a given recurrence interval meeting the design criteria. 6. Undertake a flood routing simulation to determine the peak water level. Step 5 and 6 are repeated as needed to find the pair of peak flow and volume pairs that lead to the highest water level in the reservoir. This event would then be the IDF for the damsite. 6. AN ILLUSTRATING EXAMPLE For the purpose of illustrating the concept outlined in the previous sections, a hydrometric station in Ontario (station 02CE00) was analyzed. This station started operation in 96 and has flow records from 96 to 994. The flood in the watershed generally occurs in spring as snowmelt or rainfall plus snowmelt event each year. The peak flow and maximum volume event normally occurs at the same time. The daily and peak flow was obtained from HYDAT (Environment Canada, 200). The annual peak flow and its associated flood volume were extracted from the daily flow records. Marginal flood frequency analysis for both the peak flow and the flood volume were undertaken assuming that both the flood peak and volume belong to Gumbel distribution without goodness of fit testing since this is only meant to demonstrate the use of the method. When doing real data analysis, standard procedures in performing frequency analysis must be followed. The fitted frequency curves for the two variables are plotted in Figure 2 for peak flow and in Figure 3 for flood volume.

8 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 Flow Q p V Time T b Figure Parameters Needed to Describe Flood Event Peak Flow Frequency Distribution (Gumbel) Peak Flow (m 3 /s) Exceedance Probability (%). Figure 2 Peak Flow Marginal Frequency Distribution The fitted frequency curves have the following forms: For peak flow (m 3 /s) { ln( ln )} [ 2] = p Q p

9 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 Flood Volume Marginal Distribution (Gumbel) Flood Volume (Million m 3 ) Exceedance Probability (%). Figure 3 Flood Volume Marginal Frequency Distribution Where; p is the probability of occurrence, and for flood volume (million m 3 ) { ln( ln )} [ 22] V = p The revised functions of equation [2] and [22] are: [23] u = exp exp Q p [24] V v = exp exp It is assumed that the Gumbel copula function (equation 5) is suitable for modeling the joint probability distribution. The Kendall s τ was estimated to be 0.4 and the Gumbel copula function parameter α was estimated equals to The copula function is then: [25] Q C( u, v) = exp exp p.6667 V + exp

10 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 Equation 25 represents the joint probability of peak flow OR maximum volume to occur. The risk of this type of event can be then be evaluated. For example, a peak flow of 462 m 3 /s OR total flood volume of 6492 million m 3 event has a return period of once in a hundred years. A flow of 6 m 3 /s OR volume of 4866 million m 3 would also be a once in a hundred year event. However, there are many other combinations of peak flow and total volume that has a recurrence interval of :00 years. For this reason, one has to undertake flood routing analysis to find the pair of variables that cause the maximum water level at a particular dam site. In the actual analysis, an isoline of recurrence interval as proposed by Salvadori and Michele (2004) would help to speed up the process in selecting the most severe combination as the IDF for the dam. It is also interesting to compare the AND operator with the OR operator. For a peak flow of 462 m 3 /s and total volume of 6492 million m 3 to occur in the same flood event, the probability would be once in twelve hundred and fifty (:,250) years. This is understandable since there is less chance for the two extreme variables to occur at the same time than the event either peak of 462 m 3 /s or flood volume of 6492 million m 3 to occur. 7. CONCLUSIONS It is concluded that the use of copula functions for multivariate frequency analysis provides a general method to construct flood frequency relationships and allows for an improved capability for hydrologists to determine inflow design floods for dams. Traditional flood frequency analyses using either peak flow or volume alone could, in some circumstances, lead to underestimates for IDF since the marginal frequency of one variable does not necessarily describe the characteristics of flood events due to the fact that floods are, in reality, multivariate events to a greater or lesser degree with peak flow, volume and duration all important factors to be considered in real world flood assessments. Because the method is new, there are areas where additional work is needed. For example, currently available computer programs for flood frequency analysis are all designed for one variable frequency analysis. Extending the capability of these computer programs to include copula functions for multivariate distributions would be very helpful and useful. It is hoped that other will use this introduction to more fully develop this concept. In this paper, only the frequency analysis method was discussed. In most actual studies, rainfall runoff simulation methods are an important method needed for IDF assessment, especially in the case where the Probable Maximum Flood (PMF) represents the IDF. This is the subject of a future paper. 8. REFERENCES Canadian Dam Association, 999, Dam Safety Guidelines Clayton, D.G., 978, a Model for Association in Bivariate Life Tables and its Application in Epidemiological Studies of Familial Tendency in Chronic Disease Incidence, Biometrika 65: 4-5 Correia, F. N., 987, Multivariate Partial Duration Series in Flood Risk Analysis, In: Singh, V.P. (Ed.). Hydrologic Frequency Modeling, Reidel, Dordrecht, pp Environment Canada, 200, HYDAT CD-ROM User s Manual, Water Survey of Canada Favre, A., Adlouni, S.E., Perreault, L., Thiémonge, N. and Bobée, B., 2004, Multivariate Hydrological Frequency Analysis Using Copulas, Water Resources Research, Vol. 40, W00, doi:0.029/2003wr002456, 2004 Frank, M.J., 979, On the Simultaneous Associativity of F(x,y) and x+y-f(x,y), Equations Mathematicae 9: Frees, E.W. and Valdez, E.A., 997, Understanding Relationships Using Copulas, presented at the 32 nd Actuarial Research Conference, August 6-8, 997, at the University of Calgary, Calgary, Alberta, Canada Gumbel, E.J., 960, Bivariate Exponential Distributions, Journal of the American Statistical Association 55: Maione, U., Mignosa, P. and Tomirotti, M., 2000 (A), Synthetic Design Hydrographs for Flood-Control Reservoirs and Flood- Plain Management, in Maione, U., Majone Lehto, B. and Monti, R. (eds), New Trends in Water and Environmental Engineering for Safety and Life : Eco-compatible Solutions for Aquatic Environments, Proceedings of the International Conference, Capri, 3-7 July, Rotterdam, Balkema Goel, N. K., Seth, S. M. and Chandra S., 998, Multivariate Modeling of Flood Flows, Journal of Hydraulic Engineering, Vol. 24, No. 2 Maione, U., Mignosa, P. and Tomirotti, M., 2000(B), Synthetic Design Hydrographs for River Flood Management, in Toensmann, F. and Koch, M. (eds), River Flood Defence, Proceedings of the International Symposium, Kassel, September, Vol., Kassel, Herkules Verlag

11 Congrès annuel 2006 de l ACB CDA 2006 Annual Conference Québec, Québec City, Canada CANADIAN DAM ASSOCIATION 30 Septembre 5 Octobre, 2006 ASSOCIATION CANADIENNE DES BARRAGES September 30 October 6, 2006 Maione, U., Mignosa, P. and Tomirotti, M., Regional Estimation of Synthetic Design Hydrogrphs, International River Basin Management, Vol., No. 2, pp5-63 Salvadori, G. and Michele, C. De., 2004, Frequency Analysis Via Copulas: Theoretical Aspects and Applications to Hydrological Events, Water Resource Research, Vol. 40, W25, doi:0.029/2004wr00333, 2004 Skar, A., 959, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, U.S. Department of the Interior, Bureau of Reclamation, 989, Flood Hydrology Manual, A Water Resources Technical Publication, Denver Office Venter, G.G., 2002, Tails of Copulas, Proceedings of the Casualty Actuarial Society, Vol. LXXXIX Yue, S., Ouarda, T.B.M.J., Bobée, B., Legendre, P. and Bruneau, P., 999, The Gumbel Mixed Model for Flood Frequency Analysis, Journal of Hydrology 226, 88-00

12 Presentation to CDA 2006 Conference COPULAS: A NEW WAY FOR IDF DETERMINATION Ross Zhou and Rick Donnelly October 2, 2006

13 COPULAS: A NEW WAY FOR IDF DETERMINATION Enter Subject Here Inflow design flood is required in any dam safety assessment Flood frequency analysis has been one of the major method for determining the inflow design flood when long term stream flow data are available Currently, flood frequency analyses were undertaken mainly for one variable, such as peak flow or flood volume because of lacking suitable generalized method for the derivation of multivariate frequency distributions

14 COPULAS: A NEW WAY FOR IDF DETERMINATION As we all know, floods are multivariate events which have a few important characteristics, including peak flow, duration, volume and time to rise etc. Knowing only one variable is not sufficient for determining the maximum water levels for most storage dams For this reason, many people tried several ways to overcome the problem

15 COPULAS: A NEW WAY FOR IDF DETERMINATION USBR (989) developed a balanced flood hydrograph method under the assumption that maximum peak and maximum volume occur within the same flood event Maione (2000, 2003) developed a procedure for synthetic design hydrograph (SDH) using regional flood information Balocki and Burges (994) evaluated the relationships between n-day flood-volume-duration-frequency All of above methods have the same concept in that only one variable frequency analysis was undertaken and the other variables were related to the main variable in some way based on some assumptions

16 COPULAS: A NEW WAY FOR IDF DETERMINATION There also some bivariate flood frequency models. However, due to the difficulties in modeling nonnormally distributed multivariate, most of early works focused on the use of a bivariate normal distribution as the parent distribution function. To make the normal distribution applicable, some kind of data transformation is needed. One of the problem is that the transformed data may not belong to a normal distribution and there are infinite number of potential ways for the data transformation. But there are no theoretical basis on the identification of the best transformation function Recently, there are new models to use other distributions such as Gumbel mixed model. But for the models to be applicable, the data must be satisfy certain conditions. In other words, these model can be used in limited conditions

17 COPULAS: A NEW WAY FOR IDF DETERMINATION Copulas is a generalized method for constructing multivariate frequency distribution without limitations Any two or more variables, belonging to any parent distributions, can be used in a copulas function as multivariate frequency distribution Sklar (959) proven that for any given real random variables, x, x 2,..,x p, each variable has a marginal distribution function F i (x i ), there is a copulas function C such that ( x x, L, x ) C( F ( x ), F ( x ), F ( x ) F, 2L p, 2 p = 2 2 L p p The best property of a copula function is that there is no any limitations on the type of the marginal functions. Therefore the variables can belong to different distribution

18 COPULAS: A NEW WAY FOR IDF DETERMINATION Sklar provided a general way to construct a multivariate distribution that separates the joint distribution into two parts: the marginal distribution of the individual variable, and the interdependency of the variables This is a significant and very useful development in statistics and probability theory Now, the problem for constructing a multivariate frequency distribution becomes very easy: to find a suitable copula function and the joint probability function is fully determined! The Sklar method has been widely used in many fields

19 COPULAS: A NEW WAY FOR IDF DETERMINATION A few useful bivariate, for uniform random variable s u and v, copula functions Frank s Copula (979) C ( u, v ) = ln + α α e Heavy Right Tail (HRT) Copula C ( u, v ) = u + v + ( u ) + ( ) α v α Independent Copula C ( u, v ) = Clayton Copula C Gumbel Copula C uv α u α v ( e )( e ) α α ( u + v ) α ( u, v ) = α α [( ln u ) + ( ln ) ] ( u, v ) exp v = α

20 COPULAS: A NEW WAY FOR IDF DETERMINATION Each of these copula functions has some useful properties The user can select the most suitable copula function for their conditions For inflow design flood determination, the HRT and Gumbel copula functions are very useful since they are suitable for combining extreme distributions

21 COPULAS: A NEW WAY FOR IDF DETERMINATION Hydrologists like to use the concept of return period in flood frequency analysis Salvadori and Michele (2004) defined the return period for bivariate conditions There are a few ways to define return: Return period of peak flow 2 Return period of flood volume 3 Return period of peak flow OR flood volume 4 Return period of peak flow AND flood volume From dam safety point of view, it is more useful to know the return period of peak flow and flood volume occur at the same time

22 COPULAS: A NEW WAY FOR IDF DETERMINATION For the u AND v occurring at the same time, the return period is defined as: The probability of the two variables occur at the same time is p u v, = P( U > u, V > v) = C ( u, v) And the return period is T u, v = C ( u, v ) Where C ( u, v) = u + v + C( u, v)

23 COPULAS: A NEW WAY FOR IDF DETERMINATION Parameter estimation can be achieved by maximum likelihood method which is also called as one step estimation. This means that all parameters (the marginal frequency curve of each variable and the copula function parameters) are estimated at the same time Another method is called two step method. The frequency curve parameters of the marginal distributions are estimated first and the copula function parameter is estimated after the first step. The advantage of this method is that it is easy to do. The disadvantage is that possible deviations in the first step might be magnified in the second step

24 COPULAS: A NEW WAY FOR IDF DETERMINATION Procedure for Inflow Design Flood derivation obtain the flood time series for the hydrometric station and determine the annual maximum flood event for each year. For example peak flow and total flood volume 2 select a copula function for the data sets 3 Estimate the parameters for the marginal distributions of peak flow and flood volume and the copula function parameter 4 Select an appropriate definition of return period for the bivariate flood event 5 Construct a synthetic flood hydrograph for the given pair of peak flow and volume 6 Routing the design flood hydrograph into the reservoir to obtain the maximum water level. This will be the IDF water level

25 COPULAS: A NEW WAY FOR IDF DETERMINATION Example Flow Q p Time T b Assuming triangular hydrograph: Flood Volume V = Q p T b /2

26 COPULAS: A NEW WAY FOR IDF DETERMINATION A hydrometric station (02CE00) was analyzed to illustrate the concept The station has flow records from 96 to 994 Flood generally occur in spring Peak flow and maximum flood volume event normally occur at the same flood event Marginal flood volume and peak flow assumed to belong to Gumbel distribution and parameters were estimated for the two variables

27 Peak Flow Frequency Distribution (Gumbel) Peak Flow (m 3 /s) Exceedance Probability (%).

28 Flood Volume Marginal Distribution (Gumbel) Flood Volum e (M illion m 3 ) Exceedance Probability (%).

29 COPULAS: A NEW WAY FOR IDF DETERMINATION The fitted frequency curves have the following forms ( Q p in m 3 /s and V in million m 3 ) = ln( ln p ) Q p V { } { ln( ln )} = p The uniform random variables are then u = exp exp Q p v = exp exp V

30 COPULAS: A NEW WAY FOR IDF DETERMINATION It is assumed that the Gumbel copula function can be used and the parameter of the Gumbel copula function was estimated to be.6667 Hence the copula function is: Q C( u, v) = exp exp p.6667 V + exp It is interesting to note that peak flow of 462 m 3 /s OR total volume of 6492 million m 3 event has a return period of :00 year. But for the event of 462 m 3 /s AND total volume of 6492 million m 3 to occur at the same time the return period is 250 year This is understandable since there is less chance for the two extreme variables to occur at the same time than the event either peak of 462 m 3 /s OR flood volume of 6492 million m 3 to occur

31 COPULAS: A NEW WAY FOR IDF DETERMINATION Conclusions The use of copula functions for multivariate frequency analysis provides a general method to construct flood frequency relationships 2 This allow hydrologists to determine inflow design flood for dams more confidently 3 The method would be very useful for risk analysis 4 Because the method is new, there are areas where additional works are needed (parameter estimation, the selection of best copula function for particular application, computer program etc.)

32 COPULAS: A NEW WAY FOR IDF DETERMINATION QUESTIONS? Thank You