Accounting for Backwater Effects in Flow Routing by the Discrete Linear Cascade Model Jozsef Szilagyi 1 and Pal Laurinyecz 2 Abstract: Flow-routing at a tributary (Koros River) of the Tisza River in Hungary was achieved by relating the storage coefficient (k) of the state-space formulated discrete linear cascade model (DLCM) to the concurrent discharge rate of the Tisza. As a result, the root mean square error of the 1-day forecasts decreased from 25 m 3 ·s -1 (k ¼ 1.7 days -1 and the number of storage elements is 2) with the corresponding Nash-Sutcliffe-type performance value of 0.95 to 11 m 3 ·s -1 in the calibration period and to 15 m 3 ·s -1 in the val- idation period (the corresponding Nash-Sutcliff-type performance values are 0.99 and 0.98, respectively). During floods of the Tisza, the k value decreased to as little as 0.35 days -1 , indicating a significant slowdown of the tributary flood-wave because of the resulting backwater effect. Subsequent stage-forecasts were aided by a coupled autoregressive moving-average (1,1) model of the DLCM error sequence and the application of the Jones formula in addition to a conveyance curve, the latter yielding the most accurate 1-day forecasts with a root mean square error of 28 cm and Nash-Sutcliff-type performance value of 0.99 for the combined (validation and calibration) time periods. The method requires no significant change in the mathematical structure of the original DLCM and thus is well-posed for inclusion of existing operational streamflow-forecasting schemes. DOI: 10.1061/(ASCE)HE.1943-5584.0000771. © 2014 American Society of Civil Engineers. Author keywords: Flood routing; Streamflow; Rivers; Unsteady flow. Introduction Almost every textbook of hydrology (e.g., Beven et al. 2009) mentions that traditional flow-routing methods, because of their inherent assumptions, are not recommended for river reaches significantly affected by backwater effects. Yet, traditional flow- routing methods are still widely used in hydrological forecasting centers such as the National Hydrological Forecasting Service (NHFS) in Hungary because of the minimal data-requirement of the flow-routing methods (typically only inflow) and their very fast numerical (or analytical) solutions. One would think that such con- siderations would not matter in the 21st century of fast computers, but they become a factor when one deals with several hundreds of gauging stations and performs operational forecasts 2 ×=day (7 days=week) from 12 h up to 6 days in advance, and all that with minimal human and financial resources such as the practice at NHFS. The discrete linear cascade model (DLCM; e.g., Szollosi-Nagy 1982) in use at NHFS is a spatially (using a backward difference scheme) and temporally discretized form of the linear kinematic wave equation (Lighthill and Witham 1955) written in a state- space form; see Szilagyi and Szollosi-Nagy (2010), Theorem 3, p. 34. Because of the finite spatial differences involved, DLCM also approximates the diffusion wave equation (Szilagyi and Szollosi-Nagy 2010, p. 59) in its flow-routing either in a pulsed [i.e., the last measured upstream discharge rate (q in ) held constant in time, as piece-wise continuous input to the river reach, until the next measurement arrives (Szollosi-Nagy 1982)] or linearly interpolated (between consecutive inflow measurements) data- framework (Szilagyi 2003). In this context, the latter approach is summarized next. For a rigorous mathematical treatment on the theory, see Szilagyi and Szollosi-Nagy (2010). Over the past decade DLCM has also been applied to account for and infer stream-aquifer interactions (Szilagyi 2004; Szilagyi et al. 2006) and to detect historical changes in channel properties (Szilagyi et al. 2008). The state and output equations of the DLCM for a river reach comprised of n number of subreaches can be written as _ SðtÞ¼ FSðtÞþ Gq in ðtÞ ð1aÞ q out ðtÞ¼ HSðtÞ ð1bÞ where q out = outflow of the stream reach; the dot denotes the time- rate of change; t = time-reference, F and S are the n × n state matrix and n × 1 state variable, respectively; and G and H are the n × 1 input and 1 × n output vectors, defined as F ¼ 2 6 6 4 -k k -k . . . . . . k -k 3 7 7 5 ð2aÞ SðtÞ¼ 2 6 6 6 4 S 1 ðtÞ S 2 ðtÞ . . . S n ðtÞ 3 7 7 7 5 ð2bÞ 1 Professor, Dept. of Hydraulic and Water Resources Engineering, Budapest Univ. of Technology and Economics, H-1111 Muegyetem Rakpart 1-3, Budapest, Hungary; and School of Natural Resources, Univ. of Nebraska-Lincoln, 3310 Holdrege St., Lincoln, NE 68583 (correspond- ing author). E-mail: jszilagyi1@unl.edu 2 Junior Researcher, Dept. of Flood-Protection, Koros-Valley Water Authority, H-5700 Varoshaz utca 27, Gyula, Hungary. E-mail: laurinyecz .pal@hotmail.com Note. This manuscript was submitted on January 18, 2012; approved on December 16, 2012; published online on December 18, 2012. Discussion period open until June 1, 2014; separate discussions must be submitted for individual papers. This paper is part of the Journal of Hydrologic Engi- neering, Vol. 19, No. 1, January 1, 2014. © ASCE, ISSN 1084-0699/2014/ 1-69-77/$25.00. JOURNAL OF HYDROLOGIC ENGINEERING © ASCE / JANUARY 2014 / 69 J. Hydrol. Eng. 2014.19:69-77. Downloaded from ascelibrary.org by University of Nebraska-Lincoln on 01/03/14. Copyright ASCE. For personal use only; all rights reserved.