The generalization to cases for which \(t_{in}\ne0\) is left for exercise.

The width function case

Another interesting and classical case is the case in which the IUH is given by a width function \cite{Rigon_2015}. This is defined as the area of the catchment at a certain distance from the outlet measured along the drainage directions. A width function is usually derived from the analysis of a digital elevation model (DEM) and given at discrete time distance steps, according to the resolution of the DEM. To obtain from it a IUH, the geomorphic width function has to be properly normalised by the catchment area and space (distance to outlet) has to be mapped to time. Procedures to do it are illustrated, for instance in \cite{analysis} and in \cite{Rigon_2015}. The resulting IUH looks like the one in Figure \ref{973251} below: