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Professor of Civil and Environmental Engineering
San Diego State University
San Diego, California
Victor M. Ponce
ABSTRACT: The methods of cascade of linear reservoirs and unit hydrograph convolution are shown to be one and the same when the cascade parameters are used to calculate the unit hydrograph of the convolution. In the absence of gaged data, the cascade parameters may be estimated based on geomorphology. Once the parameters are established, the composite flood hydrograph is uniquely determined.
1. INTRODUCTION
The convolution of a unit hydrograph is an established method to calculate a composite flood hydrograph (Sherman 1932; Ponce 1989). Likewise, the cascade of linear reservoirs is at the core of many hydrologic models, among them, notably, the SSARR model (U.S. Army Corps of Engineers, 1954; Ponce, 1989). Lesser known is the fact that these two apparently different methods lead to exactly the same composite flood hydrograph, provided the dimensionless unit hydrograph of the cascade is used to develop the unit hydrograph for the convolution. These propositions are now substantiated with an example.
2. EFFECTIVE STORM HYETOGRAPH
An effective storm hyetograph is derived from a total storm hyetograph by using a hydrologic abstraction method such as the NRCS runoff curve number (Ponce, 1989). For our example, we assume the 6-hr effective storm hyetograph shown in Table 1.
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3. UNIT HYDROGRAPHS
Assume a relatively large basin of drainage area A = 432 km2.
The applicable unit hydrograph duration tr is assumed to be
the same as the
storm hyetograph time interval (Table 1),
i.e., tr = Δt = 1 hr.
The basin is assumed to have relatively steep relief, with
cascade parameters Courant number
For C = 1 and N = 2, the program online_general_uh_cascade gives the general dimensionless unit hydrograph GDUH shown in Table 2.
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The applicable reservoir storage constant K is:
K = Δt / C = tr / C = 1 | (1) |
and total time, in hours:
t = t* tr | (2) |
Q = 2.777778 Q* A / tr | (3) |
The program online_dimensionless_uh_cascade gives the unit hydrograph shown in Table 3.
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4. CASCADE OF LINEAR RESERVOIRS
With C = 1 (i.e., K = 1), N = 2, and the given effective storm hyetograph (Table 1), we use the program online_routing_08 to calculate the composite flood hydrograph by the cascade of linear reservoirs (Ponce 1989). The composite flood hydrograph is shown in Table 4.
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5. CONVOLUTION
The convolution of the unit hydrograph (Table 3) with the effective storm hyetograph (Table 1) is accomplished using program online_convolution. For this example, use CN = 100. The composite flood hydrograph is shown in Table 5. It is seen that the results of Tables 4 and 5 are essentially the same.
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6. DISCUSSION
The method of cascade of linear reservoirs calculates a composite flood hydrograph for the given effective storm hyetograph. The convolution of the unit hydrograph with the effective storm hyetograph gives the same composite flood hydrograph, provided the cascade parameters are used to derive the unit hydrograph for the convolution. Thus, once the applicable cascade parameters are established, the cascade of linear reservoirs and convolution methods give exactly the same results.
7. CASCADE PARAMETERS FROM GEOMORPHOLOGY
The cascade parameters are estimated based on the runoff diffusion properties of the basin under consideration. The runoff diffusion properties depend on the terrain's topography and geomorphology. Steep basins have little or no diffusion; conversely, mild basins have substantial amounts of diffusion. The case of zero diffusion is modeled with C = 2 and N = 1. Conversely, the case of large diffusion may be modeled with C = 0.1 and N = 10 (Ponce 1980).
In nature, basins are classified for runoff diffusion on the basis of mean land surface slope (Ponce, 1989). A preliminary classification is proposed in Table 6. The range in cascade parameters and corresponding GDUH peak discharge Q *p and associated time t *p were obtained from the program online_all_series_uh_cascade. In the absence of gaged data, Table 6 may be used as a reference for the preliminary appraisal of C and N for a given basin.
Table 6. Basin classification for runoff diffusion based on mean
land surface slope.
| Class |
Mean land |
surface slope Cascade parameters |
GDUH | peak values and time-to-peak C |
N |
Q*p |
t*p |
Very steep |
> 0.1 |
2 |
1 |
1 |
1 |
Steep |
0.01 - 0.1 |
1.5 |
2 |
0.472 |
2 |
Average |
0.001 - 0.01 |
1 |
4 |
0.224 |
4 |
Mild |
0.0001 - 0.001 |
0.5 |
6 |
0.088 |
11 |
Very mild |
0.00001 - 0.0001 |
0.2 |
8 |
0.03 |
36 |
Extremely mild |
< 0.00001 |
0.1 |
9 |
0.014 |
81 |
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8. CONCLUSIONS
The methods of cascade of linear reservoirs and unit hydrograph convolution are shown to be one and the same, and to give exactly the same results, provided the cascade parameters are used to calculate the unit hydrograph of the convolution. In the absence of gaged data, the cascade parameters may be estimated based on geomorphology. Once the parameters are established, the composite flood hydrograph is uniquely determined by either method.
REFERENCES
Ponce, V. M., 1980. Linear reservoirs and numerical diffusion. Journal of the Hydraulics Dvision, ASCE, Vol. 106, HY5, May, 691-699.
Ponce, V. M., 1989. Engineering Hydrology: Principles and Practices. Prentice Hall, Upper Saddle River, New Jersey.
Sherman, L. K., 1932. Streamflow from rainfall by unit-graph method. Engineering News-Record, Vol. 108, April 7, 501-505.
U.S. Army Corps of Engineers, North Pacific Division, 1975. Program description and user's manual for SSARR Model: Streamflow Synthesis and Reservoir Regulation. Portland, Oregon, September 1972, revised June 1975.
NOTATION
The following symbols are used in this publication:
A = basin drainage area (km2);
C = Courant number, dimensionless;
CN = (NRCS runoff) curve number;
K = (linear) reservoir storage constant (hr), Eq. 1;
N = number of linear reservoirs in series;
Q = unit hydrograph discharge (m3/s), Eq. 3;
Q*p = maximum dimensionless discharge, Table 6;
Q* = dimensionless discharge;
t = time (hr), Eq. 2;
t*p = dimensionless time associated with maximum discharge, Table 6;
tr = unit hydrograph duration (hr);
t* = dimensionless time; and
Δt = hyetograph time interval (hr).
ONLINE PROGRAMS
The following programs are used in this publication:
online_dimensionless_uh_cascade