Reverse flood routing with the inverted Muskingum storage routing scheme

Abstract. This work treats reverse flood routing aiming at signal identification: inflows are inferred from observed outflows by orienting the Muskingum scheme against the wave propagation direction. Routing against the wave propagation is an ill-posed, inverse problem (small errors amplify, leading to large spurious responses); therefore, the reverse solution must be smoothness-constrained towards stability and uniqueness (regularised). Theoretical constrains on the coefficients of the reverse routing scheme assist in error control, but optimal grids are derived by numerical experimentation. Exact solutions of the convection-diffusion equation, for a single and a composite wave, are reverse-routed and in both instances the wave is backtracked well for a range of grid parameters. In the arduous test of a square pulse, the result is comparable to those of more complex methods. Seeding outflow data with random errors enhances instability; to cope with the spurious oscillations, the reversed solution is conditioned by smoothing via low-pass filtering or optimisation. Good-quality inflow hydrographs are recovered with either smoothing treatment, yet the computationally demanding optimisation is superior. Finally, the reverse Muskingum routing method is compared to a reverse-solution method of the St. Venant equations of flood wave motion and is found to perform equally well, at a fraction of the computing effort. This study leads us to conclude that the efficiently attained good inflow identification rests on the simplicity of the Muskingum reverse routing scheme that endows it with numerical robustness.