cascade and convolution: one and the same



CASCADE AND CONVOLUTION:  ONE AND THE SAME


Victor M. Ponce

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.


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 widely used 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 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 online example.


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) 1 2 3 4 5 6
Effective rainfall (cm) 1 2 4 3 2 1


UNIT HYDROGRAPHS

We assume a large basin of basin drainage area A = 432 km2. The applicable unit hydrograph duration tr is the same as the [effective] storm hyetograph time interval (Table 1), i.e., tr = Δt = 1 hr. The basin is assumed to have relatively steep relief, with cascade parameters C = 1 and N = 2.

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

Table 2.  General dimensionless unit hydrograph for C = 1 and N = 2.
Dimensionless time t* 0 1 2 3 4 5 6
Dimensionless discharge Q* 0.0000 0.2222 0.3704 0.2222 0.1070 0.0466 0.0192

Table 2.  Continued.
Dimensionless time t* 7 8 9 10 11 12 13
Dimensionless discharge Q* 0.0076 0.0029 0.0011 0.0004 0.0002 0.0001 0.0000

With reservoir storage constant

K = Δt / C = tr / C = 1(1)

and time

t = t* tr(2)

and discharge

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 basin of A = 432 km 2 and tr = 1 hr.
Time t (hr) 0 1 2 3 4 5 6 7
Discharge Q (m3/s) 0.0000 266.667 444.444 266.667 128.935 55.967 23.045 9.145

Table 3.  Continued.
Time t (hr) 8 9 10 11 12 13 14 15 16 17
Discharge Q (m3/s) 3.536 1.341 0.501 0.185 0.068 0.025 0.009 0.003 0.001 0.000


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.

Table 4.  Composite flood hydrograph by the cascade of linear reservoirs.
t (hr) 0 1 2 3 4 5 6 7 8 9
Q (m3/s) 0.0000 266.667 977.778 2222.222 3239.506 3246.091 2604.115 1642.067 805.365 354.458

Table 4.  Continued.
t (hr) 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Q (m3/s) 146.82 58.496 22.684 8.623 3.228 1.194 0.437 0.159 0.057 0.021 0.007 0.003 0.001 0.000


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 hydrograph by convolution.
t (hr) 0 1 2 3 4 5 6 7 8 9
Q (m3/s) 0.0000 266.667 977.778 2222.223 3239.506 3246.091 2604.115 1642.066 805.364 354.457

Table 5.  Continued.
t (hr) 10 11 12 13 14 15 16 17 18 19 20 21 22
Q (m3/s) 146.819 58.494 22.682 8.622 3.229 1.196 0.439 0.159 0.056 0.018 0.005 0.001 0.000


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 and convolution methods give the same results.


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 great diffusion can 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, as evaluated by the square-grid overlay method (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
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


CONCLUSIONS

The methods of cascade of linear reservoirs and unit hydrograph convolution are shown to be one and the same, and to give 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_general_uh_cascade

online_dimensionless_uh_cascade

online_all_series_uh_cascade

online_routing08

online_convolution


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