Bivariate Statistical Approach to Check Adequacy of Dam Spillway C. De Michele 1 ; G. Salvadori 2 ; M. Canossi 3 ; A. Petaccia 4 ; and R. Rosso 5 Abstract: The problem of selecting the appropriate design flood is a constant concern to dam engineering and, in general, in the hydrological practice. Overtopping represents more than 40% of dam failures in the world. The determination of the design flood is based in some cases on the T-year quantile of flood peak, and in other cases considering also the T-year quantile of flood volume. However, flood peak and flood volume have a positive (strong or weak) dependence. To model properly this aspect a bivariate probability distribution is considered using the concept of 2-Copulas, and a bivariate extreme value distribution with generalized extreme value marginals is proposed. The peak–volume pair can then be transformed into the correspondent flood hydrograph, representing the river basin response, through a simple linear model. The hydrological safety of dams is considered checking adequacy of dam spillway. The reservoir behavior is tested using a long synthetic series of flood hydrographs. An application to an existing dam is given. DOI: 10.1061/(ASCE)1084-0699(2005)10:1(50) CE Database subject headings: Dam safety; Hydrographs; Probability distribution; Spillways; Statistics. Introduction A large number of existing dams in the world were built during the 20th century under engineering, social, economic, and climate conditions different from those to be faced in this century. Data availability, process knowledge, and modeling techniques were at that time less sophisticated than today. During the last century, a large amount of historical and proxy information, i.e., hourly res- ervoir levels and the rules to the operating outflows, were col- lected by the dam regulators. This information can improve the knowledge of the dam inflows and consequently the dam safety issues. Dam failures have been significantly reduced in the last few decades (Berga 1998); the percentage of failures before 1950 was 2.3%, while for dams constructed from 1951 to 1982 it reduced to 0.2%, and since 1982 is only 0.09%. This reduction indicates that progress has been achieved in dam safety. Recent regulations, codes, and guidelines emphasize the importance of the spillway design flood as a key factor to dam safety (De Almeida and Viseu 1997). It is interesting to note that overtopping represents more than 40% of dam failures in the world and has been the cause of many other accidents (Committee on Failures and Accidents to Large Dams of the United States Committee on Large Dams 1975). In the United States, over 2,000 dams (3% of the 75,000 United States dams) have been identified as potential hazards to lives in upstream or downstream areas, due to problems of inad- equate spillway capacity (ASCE 2000). The main purpose of the paper is to outline a very general model describing the possible bivariate behavior of the random variables flood peak and flood volume, which are of primary in- terest in hydrological practice. In particular, here the attention is focused on testing the adequacy of dam spillway. A methodology for evaluating flood hydrographs is provided: it is based on a bivariate analysis of the maximum annual values of flood peak and flood volume. A bivariate extreme value distribution is con- sidered using the mathematical concept of 2-Copulas (see details below). A hydrograph is obtained using the flood-peak and flood- volume pair with the river basin response represented through a lumped model. Successively, a synthetic series of flood-peak and flood-volume pairs is generated using Monte Carlo simulation. From this, a series of flood hydrographs is then obtained. Oper- ating the reservoir routing, it is possible to test adequacy of the dam spillway. An application to the Ceppo Morelli dam located in the Anza river basin in northern Italy is presented. Calculating Flood Hydrograph The flood peak has a fundamental role in both assessing the hy- drologic safety of dams and checking adequacy of the dam spill- way. However, the flood volume can play an important role in the definition of the spillway design flood, and, consequently, may significantly influence the hydrologic safety of the dam. Gener- ally, flood peak and flood volume are two statistically dependent random variables. A joint analysis of flood peak and flood volume can be used to determine the design flood hydrograph. Below, a statistical procedure for the evaluation of flood hydrograph is pro- 1 Assistant Professor, DIIAR-Politecnico di Milano, 32 Piazza Leonardo da Vinci, Milano I-20133, Italy. E-mail: carlo.demichele@polimi.it 2 Assistant Professor, Dip. di Matematica, Univ. di Lecce, Provinciale Lecce-Arnesano, P.O. Box 193, I-73100 Lecce, Italy. E-mail: gianfausto.salvadori@unile.it 3 Graduate Student, DIIAR-Politecnico di Milano, 32 Piazza Leonardo da Vinci, Milano I-20133, Italy. E-mail: michele.canossi@polimi.it 4 Director of Hydraulic Section-RID, Via Curtatone 3, I-00185 Roma, Italy. E-mail: alberto.petaccia@registroitalianodighe.it 5 Professor, DIIAR-Politecnico di Milano, 32 Piazza Leonardo da Vinci, Milano I-20133, Italy. E-mail: renzo.rosso@polimi.it Note. Discussion open until June 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on March 17, 2003; approved on May 6, 2004. This paper is part of the Journal of Hydrologic Engineering, Vol. 10, No. 1, January 1, 2005. ©ASCE, ISSN 1084-0699/2005/1-50–57/$25.00. 50 / JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / JANUARY/FEBRUARY 2005