An analysis of MLR and NLP for use in river flood routing and comparison with the Muskingum method Mohammad Zare Manfred Koch Dept. of Geotechnology and Geohydraulics, University of Kassel, Kassel, Germany

Contents Introduction Literature review Study methods Study area and flood events used Results and discussion Conclusions 1

Introduction occurrence of floods has resulted in tremendous economic damages and life losses. the correct prediction of the rise and fall of a flood (flood routing) is important. The fundamental differential equation to describe one-dimensional unsteady river flow is the Saint- Venant equation. 2

Introduction Because of the nonlinearity of the convective acceleration term in the Saint-Venant equation, in its most complete form it can only be solved numerically. Nowadays, dynamic wave, diffusion wave and kinematic wave method are widely used in practice. 3

Introduction The numerical implementation of the kinematic wave approximation is usually the Muskingum or the Muskingum-Cunge method. Although Muskingum method is not a very accurate Method, this routing method is alive and well and by no means exhausted. 4

Introduction the determination of the routing coefficients in the Muskingum model is solved by trial and error method and graphical method. In this study, two new parameter estimation techniques, namely, nonlinear programming (NLP) and multiple linear regression (MLR), will be applied to the routing of three flood events in a river section. 5

Literature Review 6 YearResearcherDescription 1951HayamiFirstly, presented Diffusive wave theory 1955 to 1989 Various researcher There have been a lot of investigations since then to what extent the various simplifications in the Saint-Venant Eqs. are valid for routing in a particular channel 1978Gill used linear least squares to determine the two unknown parameters in the prism/wedge storage term which is the basis of the Muskingum method

Literature Review 7 YearResearcherDescription 1997Mohan applied a genetic algorithm to estimate the parameters in a nonlinear Muskingum model 2004Das estimated parameters for Muskingum models using a Lagrange multiplier formulation to transform the constrained parameter optimization problem into an unconstrained one 2009 Oladghafari & Fakheri determined the routing parameters of the Muskingum model for three flood events (which are also at the focus of the present study) in a reach of the Mehranrood river in northwestern Iran by the classical (graphical) procedure

Study methods Kinematic/diffusion wave / Muskingum wave routing method One of the most widely used methods for river flood routing is the Muskingum method. Eq.1 Using the concept of a wedge-/prism storage for a stream reach, whereby the total actual storage is written as a weighted average of the prism-storage S prism =KQ and the wedge storage S wedge =KX(I-Q) Eq.2 8

Study methods Kinematic/diffusion wave / Muskingum wave routing method where K is a reservoir constant, (about equal to the travel-time of a flood wave through the stream reach), and X a weighting factor, both of which a usually determined in an iterative manner from observed input- and output hydrographs, the discrete Muskingum- equations are directly obtained from the time- discretization of Eq. 1 Eq.3 where j = (1,…,m) indicates the time step and C 1, C 2 and C 3 are the routing coefficients, which include the two constants K and X, as well as the time step Δt 9

Study methods General formulation of a constrained nonlinear programming problem (NLP) The main purpose of nonlinear programming (NLP) is to find the optimum value of a functional variation, while respecting certain constraints. The NLP-problem is generally formulated as Eq.4 10

Study methods General formulation of a constrained nonlinear programming problem (NLP) There are lot of methods for solving the NLP problems. One of these methods that is used by WinQSB- software is penalty function method Penalty function method is transformed constrained problem into an unconstrained one. 11

Study methods General formulation of a constrained nonlinear programming problem (NLP) In penalty function method the constrained problem changes to Eq.6 for each stage k, where x(k) is the solution at that stage; μ(k) is the penalty parameter; and E(x(k)) is the sum of all constraint-violations to the power of ρ Iterations k=1,2..k max continue until μ(k)*E(x(k)) ρ <δ, then x(k+1) is the optimal solution, otherwise μ(k+1) = β*μ(k), with β a constant 12

Study methods NLP-formulation of Muskingum flood routing The NLP -problem for flood routing is formulated here as follows under the constraints for the input and output discharges measured at the discrete times j=1,2,…,m. 13

Study methods Multiple linear regression (MLR) method In the multiple linear regression (MLR) model, the Muskingum equation is read like a linear regression equation for the dependent output variable Q i+1 as a function of the three independent variables I i+1, I i (measured input hydrograph) and Q i. (measured output hydrograph). With this the MLR-model can be stated as: 14

Study area and flood events used The study area is located along the reach of the Mehranrood River in the Azarbayejan-e-Sharghi province in northwestern Iran between the two hydrometer stations Hervi (upstream) and Lighvan (downstream). The stream distance between these two stations is meters Three flood events that occurred on April 6, 2003, June 9, 2005 and May 4, 2007, respectively, were selected for the flood routing experiments. Input hydrographs for the simulations are the observed dis-charges at Hervi gage station and the output hydrographs those at station Lighvan 15

Results and discussion General set-up of the flood routing computations The NLP- and the MLR- flood routing method have been applied to the three flood hydrographs the optimal calibration of the three routing coefficients C i=1,2,3 (the decision variables in NLP, or the regression coefficients in MLR) have been done with the observed input and the routed output hydrographs of the April 6, 2003 flood event After calibration, these routing coefficients have been used in the subsequent verification of the other two flood events. 16

Results and discussion Optimal calibration of routing coefficients using the April 6, 2003 flood event Parameters used in the NLP-penalty function method: Optimal NLP-, MLR-, and classical Muskingum(Oladghaffari et al. (2009), who used a classical graphical procedure) routing coefficients for the April 6, 2003 flood event 17

Results and discussion Optimal calibration of routing coefficients using the April 6, 2003 flood event 18

Results and discussion Verification of the flood routing methods with the June 9, 2005 flood event 19

Results and discussion Verification of the flood routing methods with the May 4, 2007 flood event 20

Results and discussion Observed and calculated peak discharges and errors for NLP, MLR and Muskingum flood routing 21

Conclusions Based on the results it can be concluded that the NLP and MLR methods proposed here for the automatic calibration of the routing coefficients in the widely used Muskingum flood routing method, are powerful and reliable procedures for flood routing in rivers. These two methods may be more conveniently used than Muskingum, where suitable routing coefficients (usually the storage parameter K and the weighting factor X) are often obtained only after some lengthy trial and error process 22