Principles of IUH development by Clark's method

Clark's method develops the IUH of a catchment by treating the transformation of excess rainfall into runoff as two separate processes.

1. Translation (pure delay, no storage effect)

• The catchment is divided by isochrones, lines of equal travel time to the outlet, up to the time of concentration $t_c$.
• The area between consecutive isochrones ($A_i$) gives the time-area diagram. Excess rain falling on a strip reaches the outlet after a time equal to its travel time.
• For 1 cm excess rain, the inflow to the outlet from each strip is $I_i = 2.78\,A_i/\Delta t$ (m³/s), which is the translation hydrograph.

2. Attenuation (storage effect)

• All storage of the basin is lumped into one linear reservoir at the outlet, with $S = KO$ ($K$ = storage constant, in hours).
• The translation hydrograph is routed through it using continuity $I - O = dS/dt$:

$$O_2 = C_0 (I_1 + I_2) + C_2 O_1,\quad C_0 = \frac{\Delta t}{2K + \Delta t},\quad C_2 = \frac{2K - \Delta t}{2K + \Delta t}$$

Result

The outflow ordinates form the IUH; the average of two successive ordinates, $U_n = (O_n + O_{n-1})/2$, gives the $\Delta t$-hour unit hydrograph.

Basic assumptions: uniform excess rain over the basin, constant travel velocity (fixed isochrones), a single linear reservoir, and $t_c$ and $K$ constant. The parameters $t_c$ and $K$ are obtained from observed hydrographs or regional relations.