UH derivation from complex storms

A unit hydrograph (UH) derived from a complex storm uses a direct runoff hydrograph (DRH) that results from several successive periods of rainfall excess of different intensity. Because the UH ordinates cannot be read off directly as in a single burst, the DRH is deconvoluted using the convolution (discrete) equation.

Basic equation

If the net rain in successive periods of duration $D$ is $P_1, P_2, \dots, P_M$ (cm), the UH of duration $D$ has ordinates $U_1, U_2, \dots, U_N$, and the DRH has ordinates $Q_1, Q_2, \dots, Q_n$, then

$$Q_n = \sum_{m=1}^{n\le M} P_m\, U_{n-m+1}, \qquad n = M+N-1$$

Writing it out for each ordinate:

$$\begin{aligned} Q_1 &= P_1U_1 \\ Q_2 &= P_1U_2 + P_2U_1 \\ Q_3 &= P_1U_3 + P_2U_2 + P_3U_1 \\ &\ \ \vdots \end{aligned}$$

Procedure

1. Select a storm with several periods of rain of uniform duration $D$ over the whole basin, and its stream-flow record.
2. Separate base flow from the storm hydrograph to get the DRH ordinates $Q_n$ at intervals $D$.
3. Compute the rainfall excess $P_m$ of each period by deducting losses (φ-index or a loss model). A period with no excess stays in the sequence as $P_m=0$, because it only shifts the later periods in time.
4. Number of UH ordinates $N = n - M + 1$ (n = number of DRH ordinates, M = number of excess-rain periods).
5. Solve the equations successively (forward substitution):

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

6. Check: the area of the derived UH must equal 1 cm of runoff over the basin: $\sum U_i \cdot D\cdot 3600 = A\times 0.01$ m. Adjust (smooth) the ordinates if needed.
7. Plot the UH and, if necessary, re-compute the DRH from it to confirm that it reproduces the observed DRH.

Remarks

• Forward substitution accumulates errors, so ordinates may oscillate or become negative. In that case the least-squares (matrix) form $\mathbf{Q}=[\mathbf{P}]\mathbf{U}$, with $\mathbf{U}=(\mathbf{P}^T\mathbf{P})^{-1}\mathbf{P}^T\mathbf{Q}$, or a trial-and-error / optimisation approach with a non-negative constraint is used (Collins' method: assume a UH, compute the DRH, correct the UH, and repeat).
• The UH obtained from the average of several storms is more reliable.