Once we get the \(s_f\) (remind that here t stands for \(T\)) that can be considered continuous when interpolated appropriately, the previous theory applies verbatim.
Numerics
We have now all the tools for implementing the numerics of the IUH, at least in the case of a single rain record. The main choice to do is to select a discretization of the time axis. A reasonable choice is to select a \(\delta t\ <\ \Delta t\) and such not to loose the information derived from the \(f\) function, especially if it is given as a discrete set of points. In this case the continuous variables are set to:
- \(t\ =\ j\ \delta t\ \ \ \ j=0,\ ...,\ n\) with \(n\ \delta t\ =\ \Delta t\)
- \(t_{in}=i\ \delta t\)
- \(T\ =t-t_{in}=\left(j-i\right)\delta t\)
- \(t_c=m\ \delta t\)
With these choices, equations \ref{eq5} and \ref{eq6} becomes: