Convolution, deconvolution, the unit hydrograph and flood routing

Convolution equations are used to relate the input and the output of a system such as rainfall and runoff, or inflow and outflow of a river reach. There have been numerous reports of unsatisfactory results from the deconvolution necessary to calculate the connecting transfer function. The cause is that the equations are ill-conditioned, and it is shown here that the fundamental theoretical solution is that of wild oscillations such as has often been found computationally. A spectral method is proposed for numerical solution, where, instead of individual point values, the transfer function is expressed as series of given continuous functions, where the problem is to determine the coefficients of those functions. The resulting equations have been found to be well-conditioned, and solutions obtained were smooth, bounded, and enabled a certain amount of physical interpretation of the transfer function. The method has been applied to several problems, including typical rainfall–runoff ones and flood routing and wave propagation problems, with quite satisfactory results. Another problem for deconvolution is found to be the traditional use of truncated equations. A remedy is only to use later output data points where convolution with input data does not reach back beyond the initial one. For the routing of larger flood events, the linear methods employed were found to be not so accurate. However as they are a first approximation that requires no knowledge of stream geometry or resistance, and as either discharge or water level hydrographs can be used, they may be useful. •System identification, deconvolution, has always been problematical.•A new computational method is suggested that overcomes problems.•Results are good, giving smooth bounded system functions.•A useful technique for simulation where river properties are not known.•For large floods, problem nonlinearity means results are not as accurate.