A unit hydrograph represents the runoff hydrograph from a drainage basin subjected to one inch of runoff (effective precipitation) applied over a duration of Δt. However, a runoff (effective rainfall) hyetograph typically contains many Δt time periods, each of which has its own associated runoff depth. In discrete convolution, the direct runoff hydrograph resulting from a complete rainfall hyetograph is computed by applying a unit hydrograph to each discrete time step within the hyetograph.
To conceptualize discrete convolution, refer to Figures 2-21 and 2-22. The first figure illustrates a runoff (effective rainfall) hyetograph consisting of np = 3 rainfall pulses, each of duration Δt = 10 minutes, with the depth of each pulse denoted by Pi (i = 1, 2 … np).
The second figure shows a unit hydrograph with nu = 11 non-zero ordinates shown at Δt time intervals, with the nonzero ordinates denoted by Uj (j = 1, 2 … nu). (Note that the ordinates of the unit hydrograph are in units of cfs per inch of runoff, and thus the notation of this subsection is different from the previous subsection in which unit hydrograph ordinates were denoted by discharge Q.) The time increment Δt used for development of the runoff (effective rainfall) hyetograph must be the same as the Δt duration of excess rainfall used to create the unit hydrograph.
Figure 2-21: Effective Rainfall (Runoff) Hyetograph
Figure 2-22: Unit Hydrograph Ordinates
Because unit hydrograph theory assumes that drainage basins behave linearly, the ordinates Qj of the runoff hydrograph produced by P1 inches of effective rainfall must be equal to Qj = P1Uj. These ordinates are shown in the third column of Table 2-14 (the first and second columns of the table reproduce the information in Figure 2-22). These ordinates were calculated by multiplying each unit hydrograph ordinate by the runoff depth for pulse P1 = 0.2 in. The values in column three are shifted down such that the time at which the runoff hydrograph in column three begins is the same as the time at which the first pulse of effective rainfall begins.
The fourth and fifth columns in Table 2-14 are computed by multiplying the unit hydrograph ordinates in the second column by the effective rainfall depths P2 = 1.5 in.
and P3 = 0.7 in., respectively. Again, the time at which each runoff pulse’s hydrograph begins corresponds to the time at which each runoff (effective rainfall) pulse begins. The sum of the runoff hydrograph ordinates in each row of the table (shown in the sixth column) is an ordinate of the direct runoff hydrograph caused by the complete rainfall event.
Table 2-14: Discrete Convolution
(1) (2) (3) (4) (5) (6) t (min) U (cfs/in) P1U (cfs) P2U (cfs) P3U (cfs) Q (cfs)
0 0 0
10 20 0
20 70 0 0
30 160 4 0 4
40 240 14 30 0 44
50 180 32 105 14 151
60 110 48 240 49 337
70 75 36 360 112 508
80 50 22 270 168 460
90 33 15 165 126 306
100 21 10 113 77 200
110 10 7 75 53 134
120 0 4 50 35 89
130 2 32 23 57
140 0 15 15 30
150 0 7 7
160 0 0
Figure 2-23 shows the runoff hydrograph resulting from each rainfall pulse and the total runoff hydrograph for the entire rainfall event.
Figure 2-23: Summing of Hydrographs from Individual Rainfall Pulses
Example 2-13: Computing a Runoff Hydrograph
Compute the direct runoff hydrograph for the Memphis, Tennessee drainage basin described in Example 2-12. Use the unit hydrograph developed for the basin in that example, and use the runoff (effective rainfall) hyetograph developed in Example 2-11.
Solution
There are a total of np = 4 runoff (effective rainfall) pulses, and a total of nu = 26 non-zero unit hydrograph ordinates. Thus, there will be a total of nq = 29 non-zero direct runoff hydrograph ordinates.
Table 2-15 illustrates the tabular calculation (discrete convolution). The first column is the time since the beginning of rainfall in Δt = 10 minute increments (the duration of the unit hydrograph and the duration of the effective rainfall pulses). The second column contains the ordinates of the unit hydrograph from Example 2-12. The third through the sixth columns are the unit hydrograph ordinates multiplied by the effective rainfall depths. Note that the first entry (the first zero) in each column corresponds to the time at which the corresponding effective rainfall pulse begins (t = 20 min for the first pulse, t = 30 min for the second pulse, and t = 40 min for the third pulse; see Figure 2-16). The seventh column is the sum of the previous four, and is the direct runoff hydrograph.
Table 2-15: Discrete Convolution to Obtain Direct Runoff Hydrograph
(1) (2) (3) (4) (5) (6) (7)
t (min) U (cfs/in) 0.99U 0.45U 0.25U 0.09U Q (cfs)
0 0 0
10 59 0
20 178 0 0
30 363 58 0 58
40 585 176 27 0 203
50 710 359 80 15 0 454
60 740 579 163 45 5 792
70 688 703 263 91 16 1073
80 599 733 320 146 33 1231
90 474 681 333 178 53 1244
100 348 593 310 185 64 1152
110 259 469 270 172 67 977
120 192 345 213 150 62 769
130 155 256 157 119 54 585
140 118 190 117 87 43 436
150 89 153 86 65 31 336
160 67 117 70 48 23 258
170 52 88 53 39 17 197
180 37 66 40 30 14 150
190 30 51 30 22 11 115
200 22 37 23 17 8 85
210 15 30 17 13 6 65
220 15 22 14 9 5 49
230 7 15 10 8 3 36
240 7 15 7 6 3 30
250 7 7 7 4 2 19
260 7 7 3 4 1 15
270 0 7 3 2 1 13
280 7 3 2 1 12
290 0 3 2 1 6
300 0 2 1 2
310 0 1 1
320 0 0
2.5 Problems
1. The following data is from Huff and Angle (1992). The numbers indicate the total rainfall (inches) expected in Chicago, Illinois for storm recurrence intervals of 2, 5, 10, 25, 50 and 100 years and durations from 5 minutes to 24 hours. Plot a series of IDF curves showing the rainfall intensities for each recurrence interval for storm durations between 5 minutes and 2 hours.
Recurrence Interval (years) Duration
2 5 10 25 50 100 24 hrs 3.11 3.95 4.63 5.60 5.63 7.36 2 hr 1.83 2.33 2.74 3.31 3.86 4.47 1 hr 1.46 1.86 2.18 2.63 3.07 3.51 30 min 1.15 1.46 1.71 2.07 2.42 2.77 15 min 0.84 1.07 1.25 1.51 1.76 1.99 10 min 0.68 0.87 1.02 1.23 1.44 1.62 5 min 0.37 0.47 0.56 0.67 0.78 0.89
2. Use the data in Problem 1 and the SCS Type II distribution to develop a 25-year, 24-hour storm for Chicago.
3. During a 40-mm rainfall event, the interception capacity of the watershed is estimated to be 11 mm, the depression storage is 9 mm, and the infiltration is 7 mm. What is the total volume of runoff from the 2.56 ha watershed?
4. A proposed development consists of the following land use areas:
Land Use Area (ac) Condominiums 2.75
Greenbelt 0.89 Commercial 1.37
The time of concentration is estimated to be 25 min. Using the Rational Method IDF data from Problem 1, calculate the peak runoff resulting from a 10-yr storm.
Use the midpoint in the range of C coefficients in Table 2-5.
5. A suburban watershed has a time of concentration of 2.5 hours and the following land use characteristics:
Land Cover Area (mi2) CNII
Commercial & business 4 92
¼ acre housing 14 75
Parking lots, roofs & driveways 0.78 98
a) Use the NRCS (SCS) CN method to determine volume of runoff resulting from a 6-in storm. Assume that AMC-I applies.
b) Use the SCS Method to determine the peak discharge from the watershed.
6. Develop a unit hydrograph for a drainage basin with an area of 3.5 mi2. Use a time of concentration of 120 min. Assume that a shape factor of 484 applies and that the time of concentration is 2 hours.