JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION VOL. 37, NO. 5 AMERICAN WATER RESOURCES ASSOCIATION OCTOBER 2001 PARAMETER ESTIMATION OF THE NONLINEAR MUSKINGUM MODEL USING HARMONY SEARCH 1 Joong Hoon Kim, Zong Woo Geem, and Eung Seok Kim2 ABSTRACT: A newly developed heuristic algorithm, Harmony Search, is applied to the parameter estimation problem of the non- linear Muskingum model. Harmony Search found better values of parameters in the nonlinear Muskingum model than five other methods including another heuristic method, genetic algorithm, in terms of SSQ (the sum of the square of the deviations between the observed and routed outflows), SAD (the sum of the absolute value of the deviations between the observed and routed outflows), DPO (deviations of peak of routed and actual flows), and DPOT (devia- tions of peak time of routed and actual outflow). Harmony Search also has the advantage that it does not require the process of assuming the initial values of design parameters. The sensitivity analysis of Harmony Memory Considering Rate showed that rela- tively large values of Harmony Memory Considering Rate makes the Harmony Search converge to a better solution. (KEY TERMS: Harmony Search; nonlinear Muskingum method; parameter calibration; genetic algorithm.) INTRODUCTION There are two types of basic approaches to route floods: the hydrologic routing approach and the hydraulic routing approach. The former routes using the storage-continuity equation, and the latter routes using the Saint-Venant equation (Fread, 1976). One of the hydrologic routing techniques, the Muskingum method (McCarthy, 1938) has been fre- quently used to route floods in natural channels and rivers. In the Muskingum model, the following conti- nuity and storage equations are used -o dt S, = K[xI + (1 - (2) in which S is the channel storage at time t; I and Ot are the rates of inflow and outflow at time t, respec- tively; K is the storage-time constant for the river reach, which has a value close to the flow travel time through the river reach; and x is a weighting factor varying between 0 and 0.3 for stream channels. Commonly, the parameters K and x in the Musk- ingum model are graphically estimated by a trial and error procedure. After x is assumed, the values of EXIt + (1 - X)Ot] are computed using recorded data and plotted against St. The value of x that minimizes the width of the plotted loop can be chosen as the correct value of x, and the line slope for the correct value of x can be chosen as K. However, the above procedure is subjective and inefficient, and the relationship between EXIt + (1 - x)O} and S is not always linear. Gill (1978) proposed a nonlinear Muskingum model as S =K[xI +(1—x)O]m (3) This nonlinear Muskingum model has an addition- al parameter m used as an exponent, which presum- ably makes the model fit closer to the nonlinear relation between accumulated storage and weighted flow. However, the calibration procedure for finding the correct values of the three parameters K, x, m becomes more complicated. These values cannot be determined graphically from historical inflow and outflow hydrographs. Alternative parameter estima- (1) tion methods are required. Several endeavors have 1Paper No. 00005 of the Journal of the American Water Resources Association. Discussions are open until June 1, 2002. 2Respectively, Associate Professor and Graduate Research Assistant, Department of Civil and Environmental Engineering, Korea Univer- sity, Seoul, South Korea 136-70 1; and Faculty Researcher, Department of Civil and Environmental Engineering, University of Maryland, Col- lege Park, Maryland (former Graduate Research Assistant at Korea University) (E-MaillJoong Hoon Kim: jaykim@korea.ac.kr). JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION 1131 JAWRA