Unit hydrograph and UH for multiple storms

Unit hydrograph

A unit hydrograph is the direct runoff hydrograph produced by 1 cm of effective rainfall, falling uniformly over the basin at a constant rate during a specified duration $D$. It assumes linearity and time invariance.

Preparing a UH from a multiple (complex) storm

When a storm has several periods of effective rainfall, the observed direct runoff hydrograph (DRH) is the sum of the responses to each period. The UH is obtained by deconvolution.

Steps

1. Separate the baseflow from the storm hydrograph to get the DRH ordinates $Q_n$.
2. Compute the effective rainfall for each period of duration $D$ by deducting losses ($\phi$-index): $P_1, P_2, \dots, P_M$.
3. The DRH ordinate at time $n$ is

$$Q_n=\sum_{m=1}^{\min(n,M)} P_m\,U_{n-m+1}$$

4. Solve for the UH ordinates $U_n$ successively:

$$U_1=\frac{Q_1}{P_1},\quad U_2=\frac{Q_2-P_2U_1}{P_1},\quad U_3=\frac{Q_3-P_2U_2-P_3U_1}{P_1},\ \dots$$

5. Check that the area under the UH equals 1 cm over the basin: $\sum U\,\Delta t = 1\ \text{cm}\times A$.
6. For more accuracy, use many storms and the least-squares method, and average the UH ordinates.

Example

Two periods of 1 h with effective rainfall $P_1=2$ cm and $P_2=3$ cm. Observed DRH: $Q_1=20$, $Q_2=90$, $Q_3=130$, $Q_4=80$, $Q_5=30$ m$^3$/s.

The 1-h UH ordinates are 10, 30, 20, 10 m$^3$/s at 1, 2, 3, 4 h.