Linear reservoir and Clark UH from time-area

Linear reservoir

A linear reservoir is a reservoir (or storage) in which the storage is directly proportional to the outflow:

$$S = K\,O$$

where $K$ is the storage constant (units of time). With continuity $I - O = dS/dt$, the response is $K\,dO/dt + O = I$, and its IUH is $u(t) = \dfrac{1}{K}e^{-t/K}$.

Procedure for Clark UH from the time-area method

1. Isochrones and areas. Divide the catchment by isochrones at interval $\Delta t$ up to $t_c$ and find the strip areas $A_1, A_2, \dots, A_N$.
2. Time-area (translation) inflow. For 1 cm excess rain:

$$I_i = \frac{2.78\,A_i}{\Delta t}\ \text{m}^3/\text{s}$$

This is the inflow hydrograph to an imaginary linear reservoir at the outlet. 3. Storage constant. Get $K$ from the recession of an observed hydrograph, or by a regional formula. 4. Route through the linear reservoir:

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

starting with $O_1 = 0$. The outflow ordinates are the IUH. 5. $\Delta t$-hour unit hydrograph. The ordinates are the average of successive IUH ordinates, $U_n = (O_n + O_{n-1})/2$. 6. Longer-duration UH. For a unit hydrograph of duration $D$ (a multiple of $\Delta t$), use the S-curve or average successive ordinates.

 Time-area        Linear reservoir      IUH / UH
 inflow I(t) ---> S = K O  ----------> O(t), U(t)
 (translation)    (storage effect)