Linear diffusion-wave channel routing using a discrete Hayami convolution method

The convolution of an input with a response function has been widely used in hydrology as a means to solve various problems analytically. Due to the high computation demand in solving the functions using numerical integration, it is often advantageous to use the discrete convolution instead of the integration of the continuous functions. This approach greatly reduces the amount of the computational work; however, it increases the possibility for mass balance errors. In this study, we analyzed the characteristics of the kernel function for the Hayami convolution solution to the linear diffusion-wave channel routing with distributed lateral inflow. We propose two ways of selection of the discrete kernel function values: using the exact point values or using the center-averaged values. Through a hypothetical example and the applications to Asotin Creek, WA and the Clearwater River, ID, we showed that when the point kernel function values were used in the discrete Hayami convolution (DHC) solution, the mass balance error of channel routing is dependent on the number of time steps on the rising limb of the Hayami kernel function. The mass balance error is negligible when there are more than 1.8 time steps on the rising limb of the kernel function. The fewer time steps on the rising limb, the greater risk of high mass balance errors. When the average kernel function values are used for the DHC solution, however, the mass balance is always maintained, since the integration of the discrete kernel function is always unity.