CE322 Basic Hydrology
Jorge A. Ramírez
Unit Hydrographs - Example

A. Obtain a Unit Hydrograph for a basin of 315 km² of area using the rainfall and streamflow data tabulated below.

1. Separate the baseflow from the observed streamflow hydrograph in order to obtain the Direct Runoff Hydrograph (DRH).
   For this example, use the horizontal line method to separate the baseflow. From observation of the hydrograph data, the streamflow at the start of the rising limb of the hydrograph is 100 m³/s.
2. Compute the volume of Direct Runoff. This volume must be equal to the volume of the Effective Rainfall Hyetograph (ERH).
    

   

   Thus, for this example:

   V_DRH = (200+600+900+700+500+300+200+100) m³/s (3600) s = 12'600,000 m³

3. Express V_DRH in equivalent units of depth:
   V_DRH in equivalent units of depth = V_DRH/A_basin = 12'600,000 m³/(315000000 m²) = 0.04 m = 4 cm.
4. Obtain a Unit Hydrograph by normalizing the DRH. Normalizing implies dividing the ordinates of the DRH by the V_DRH in equivalent units of depth.

   Time (h)	Observed Hydrograph (m3/s)	Direct Runoff Hydrograph (DRH) (m3/s)	Unit Hydrograph (m3/s/cm)
   0	100	0	0
   1	100	0	0
   2	300	200	50
   3	700	600	150
   4	1000	900	225
   5	800	700	175
   6	600	500	125
   7	400	300	75
   8	300	200	50
   9	200	100	25
   10	100	0	0
   11	100	0	0

   
    

   

   
   
   

5. Determine the duration D of the ERH associated with the UH obtained in 4. In order to do this:

1. Determine the volume of losses, V_Losses which is equal to the difference between the volume of gross rainfall, V_GRH, and the volume of the direct runoff hydrograph, V_DRH .

   
   V_Losses = V_GRH - V_DRH = (0.5 + 2.5 + 2.5 +0.5) cm/h 1 h - 4 cm = 2 cm

2. Compute the f-index equal to the ratio of the volume of losses to the rainfall duration, t_r. Thus,

   
   f-index = V_Losses/t_r = 2 cm / 4 h = 0.5 cm/h

3. Determine the ERH by subtracting the infiltration (e.g., f-index) from the GRH:

As observed in the table, the duration of the effective rainfall hyetograph is 2 hours. Thus, D = 2 hours, and the Unit Hydrograph obtained above is a 2-hour Unit Hydrograph. Therefore, it can be used to predict runoff from precipitation events whose effective rainfall hyetographs can be represented as a sequence of uniform intensity (rectangular) pulses each of duration D. This is accomplished by using the principles of superposition and proportionality, encoded in the discrete convolution equation:



where Q_n is the n^th ordinate of the DRH, P_m is the volume of the m^th rainfall pulse expressed in units of equivalent depth (e.g., cm or in), and U_n-m+1 is the (n-m+1)^th ordinate of the UH, expressed in units of m³/s/cm.

B. Using the UH obtained in A., predict the total streamflow that would be observed as a result of the following ERH:

1. Determine the volume of each ERH pulse, P_m, expressed in units of equivalent depth:

   Time (h)	Pm (cm)
   0 - 2	1.0
   2 - 4	3.0
   4 - 6	4.0
   6 - 8	2.0

2. Use superposition and proportionality principles:

1. Columns 2 - 5: Apply the proportionality principle to scale the UH by the actual volume of the corresponding rectangular pulse, P_m. Observe that the resulting hydrographs are lagged so that their origins coincide with the time of occurrence of the corresponding rainfall pulse.
2. Column 6: Apply the superposition principle to obtain the DRH by summing up Columns 2 - 5.
3. Column 7: Add back the baseflow in order to obtain the Total Streamflow Hydrograph.