CASCADE AND CONVOLUTION:  ONE AND THE SAME

Professor of Civil and Environmental Engineering

San Diego State University

San Diego, California

May 25, 2009

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.

 Table 1.  Effective storm hyetograph. Time(hr) Effective rainfall(cm) 1 1 2 2 3 4 4 3 5 2 6 1

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 C = 1 and number of linear reservoirs N = 2.

For C = 1 and N = 2, the program online_general_uh_cascade gives the general dimensionless unit hydrograph GDUH shown in Table 2.

 Table 2.  General dimensionless unit hydrographfor C = 1 and N = 2. Dimensionless timet* Dimensionless discharge Q* 0 0.0000 1 0.2222 2 0.3704 3 0.2222 4 0.1070 5 0.0466 6 0.0192 7 0.0076 8 0.0029 9 0.0011 10 0.0004 11 0.0002 12 0.0001 13 0.0000

The applicable reservoir storage constant K is:

 K = Δt / C = tr / C = 1 (1)

and total time, in hours:

 t = t* tr (2)

and discharge, in m3/s:

 Q = 2.777778 Q* A / tr (3)

The program online_dimensionless_uh_cascade gives the unit hydrograph shown in Table 3.

 Table 3.  Unit hydrograph for basinof A = 432 km2 and tr = 1 hr. Time t(hr) Discharge Q(m3/s) 0 0.0000 1 266.667 2 444.444 3 266.667 4 128.935 5 55.967 6 23.045 7 9.145 8 3.536 9 1.341 10 0.501 11 0.185 12 0.068 13 0.025 14 0.009 15 0.003 16 0.001 17 0.000

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.

 Table 4.   Composite flood hydrograph by the cascade of linear reservoirs. Time t(hr) Discharge Q(m3/s) 0 0.0000 1 266.667 2 977.778 3 2222.222 4 3239.506 5 3246.091 6 2604.115 7 1642.067 8 805.365 9 354.458 10 146.820 11 58.496 12 22.684 13 8.623 14 3.228 15 1.194 16 0.437 17 0.159 18 0.057 19 0.021 20 0.007 21 0.003 22 0.001 23 0.000

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.

 Table 5.   Composite flood hydrographby convolution. Time t(hr) Discharge Q(m3/s) 0 0.0000 1 266.667 2 977.778 3 2222.223 4 3239.506 5 3246.091 6 2604.115 7 1642.066 8 805.364 9 354.457 10 146.819 11 58.494 12 22.682 13 8.622 14 3.229 15 1.196 16 0.439 17 0.159 18 0.056 19 0.018 20 0.005 21 0.001 22 0.000 23 0.000

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.

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 landsurface slope Cascade parameters GDUHpeak values andtime-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

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: