Unit hydrograph

Remarks

The total volume under a unit hydrograph equals the volume of one unit of excess rainfall over the basin: $\int_0^\infty U_D(t)\,dt = A \cdot (\text{unit depth})$. For example, take $D = 1$ hour and sampled values $U_D = 0, 2, 5, 3, 1, 0$ at hours $0, \dots, 5$. For depths $P_1 = 2$ and $P_2 = 1$, the runoff at hour $2$ is $2 \cdot 5 + 1 \cdot 2 = 12$, a discrete convolution. A non-example is the hydrograph of a storm of duration $D$ with two units of excess rainfall. It is not a unit hydrograph, since the rainfall depth is not one unit; under linearity it equals $2 U_D$. The limit $D \to 0$ gives the instantaneous unit hydrograph $u(t)$, the impulse response of the basin, and then $Q(t) = \int_0^t i(\tau)\, u(t - \tau)\, d\tau$ for an excess rainfall intensity $i$.