Time area method for runoff hydrograph

The time-area method estimates the direct runoff hydrograph at the outlet of a catchment from effective rainfall, taking into account only the translation of runoff to the outlet and ignoring storage effects.

Basic idea

• Lines of equal travel time to the outlet (isochrones) are drawn at interval $\Delta t$: $t_1, t_2, \dots, t_c$.
• The area between two consecutive isochrones is $A_1, A_2, \dots, A_N$. Rain falling in $A_1$ reaches the outlet in the first interval, in $A_2$ in the second interval, and so on. The last strip takes time $t_c$ (time of concentration).
• The plot of cumulative area against time (or the histogram of strip areas) is the time-area diagram.

Time-area histogram

 Area
  A3 |      ____
  A2 |  ___|    |___
  A1 | |    |    |   |
     | |    |    |   |___
     +-+----+----+----+---> t
       0   Δt  2Δt 3Δt  tc

Procedure

1. Prepare the isochrone map and find the strip areas $A_1, \dots, A_N$.
2. Convert the effective rainfall hyetograph into blocks $r_1, r_2, \dots, r_M$ (cm or mm) of duration $\Delta t$.
3. The discharge at the end of interval $n$ is the sum of contributions of every rainfall block over its corresponding strip:

$$Q_n = \frac{10^4}{3600\,\Delta t}\sum_{i=1}^{n} r_i\,A_{n-i+1}\quad (A \text{ in km}^2, r \text{ in cm})$$

i.e. $Q_n = 2.78 \sum r_i A_{n-i+1}/\Delta t$ for $r$ in cm and $\Delta t$ in h. For example, $Q_1 = 2.78\,r_1A_1/\Delta t$, $Q_2 = 2.78\,(r_1A_2 + r_2A_1)/\Delta t$, $Q_3 = 2.78\,(r_1A_3 + r_2A_2 + r_3A_1)/\Delta t$. 4. Plot $Q_n$ against time to get the direct runoff hydrograph.

The calculation is a convolution of the rainfall blocks with the area strips (reversed order), and the base flow is added afterwards.

Limitations

• Storage (channel and catchment) is ignored, so peaks are overestimated and the hydrograph is too steep.
• Isochrones are hard to draw accurately and the velocity is assumed constant, independent of flow depth and rainfall intensity.
• Uniform rainfall over each strip is assumed.

Clark improved the method by routing the result through a linear reservoir to include storage.