Interpolation of water surface profile in unsteady open channel flow using the adjoint method based on 2D shallow water equations

Real-time interpolation of high-water stage data obtained at separate water gauges was investigated to provide visual information for evacuation decision-making by residents near a river. The adjoint method was applied to a set of shallow water equations in a generalized coordinate system in which the hydraulic effect of tree communities is included as semi-permeable blocks. The cost function to be minimized was assumed as the summation of four misfits for water level and discharge from observation and for Manning's roughness (n) and tree community permeability (K) from standard values. The Lagrange multipliers method was adopted to determine the uncertain quantities, upstream discharge and spatial distributions of n and K, for minimizing the total cost under the constraint of the shallow water equations. The method was applied to a flood event observed in 2011 over a 20-km reach of the Tone River, where four water gauges are installed. The computation results successfully explained the time series of river discharge observed at stations located in the river reach, as well as the longitudinal profile of the trace water levels measured at 500-m intervals just after the flood event.