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Comparison of rainfall-runoff models for urban areas
Journal of Hydrology, 18 (1973) 329-347 © North-Holland Publishing Company, Amsterdam - Printed in The Netherlands
COMPARISON OF R A I N F A L L - R U N O F F MODELS FOR U R B A N AREAS P.B.S. SARMA 1, J.W. DELLEUR 2 and A.R. RAO 2
l lndian Institute of Technology, Kanpur (India) 2School of Civil Engineering, Purdue University, Lafayette, Ind, (U.S.A.J (Accepted for publication May 18, 1972)
ABSTRACT Sarma, P.B.S., Delleur, J.W. and Rao, A.R., 1973. Comparison of rainfall--runoff models for urban areas. J. Hydrol., 18: 329-347. The relative regeneration performances of five linear rainfall excess-direct runoff models are compared for several urban watersheds with varying degrees of development. The five models considered are the single linear reservoir, the Nash model, the double routing method, the linear channel-linear reservoir model and the instantaneous unit hydrograph (IUH) obtained by the Fourier transform method. The IUH always gives the best regeneration performance among the four conceptual models tested. The optimized single linear reservoir constant differs from the theoretical time lag value, but is related to the latter, and for each watershed varies from storm to storm. For larger watersheds the Nash model gives the best regeneration performance among the four conceptual models tested. The model parameters for each watershed are found to vary from storm to storm. The quality of regeneration for larger basins is less than that found for the smaller basins.
INTRODUC~ON
The assumption o f the linearity of the rainfall excess-direct runoff process (hereafter referred to as the rainfall-runoff process) forms the basis of several c o m m o n l y used methods in urban hydrology. Linear conceptual and/or mathematical models of the rainfall-runoff process have been used successfully in recent investigations of small experimental impervious watersheds (less than 1 acre in area) by Willeke (1962), Viessman (1966, 1968), Eagleson and March (1965). Delleur and Vician ( 1 9 6 6 ) have used two conceptua 1 models to simulate the rainfall-runoff process in two actual urban watersheds which are partially impervious. As several linear conceptual models can be formulated by appropriate combinations of linear elements such as the linear reservoir and the linear channel, the question of comparative suitability o f any of these models to simulate the urban rainfall-runoff process arises. The work reported herein deals with an investigation o f the relative regeneration performance of some of the commonly used linear hydrologic system models in an attempt to select an appropriate model to simulate the rainfall-runoff process in a given urban watershed.
330
P.B.S. SARMA, J.W. D E L L E U R AND A.R. RAO
U R B A N WATERSHEDS INVESTIGATED
Long-term records of hydrologic data are not usually available for urban watersheds. In order to select a suitable simulation model of the rainfall-runoff process in watersheds with different stages of urban development, it is necesary to analyze data from several watersheds with varied degrees of imperviousness. In the present study, hydrologic data from the following 13 watersheds, listed in Table I, have been collected and analyzed. The data collection from the Purdue Swine Farm Watersheds in West Lafayette was initiated as part of the present study and is being continued. The studies of the upper and lower Ross Ade watersheds were commissioned in 1964. The hydrologic data from the other watersheds in Indiana and Texas were acquired from the records and reports of the U.S. Geological Survey (1962-1966). Details of the instrumentation of the West Lafayette watersheds can be found elsewhere (Sarma et al., 1969). For each of the watersheds under study, the values of the area of the drainage basin A, the percentage of the impervious area U, and the number of storms analyzed are presented in Table I.
TABLE I Watersheds investigated Watershed number
Name of the watershed*
Area of the watershed A (sq.miles)
Percentage impervious area U(%)
No. of storms used for analysis
1
Ross Ade (upper) 1
0.0455
38.0
21
2
Ross Ade (lower) 1
0.6125
37.4
10
3
Purdue Swine Farm (upper) 1
0.2776
21.3
4
4
Purdue Swine F a r m (lower) 1
0.4562
13.3
2
5
Pleasant R u n (Arlington) 2
7.580
10.5
13
6
Pleasant R u n (Brookville) 2
10.10
15.5
12
7
Little Eagle Creek 2
19.31
2.1
8
8
Lawrence Creek 2
2.86
0.0
12
9
Bear Creek 3
5
7.00
0.0
14.60
0.0
7
2.31
27.0
17
Waller Creek (23rd Street) 4
4.13
37.0
12
Wilbarger Creek 4
4.61
0.0
8
10
Bean Blossom Creek 3
11
Waller Creek (38th Street) 4
12 13
* 1 = West Lafayette, Indiana; 2 = Indianapolis, Indiana; 3
=
Indiana; 4 = Austin, Texas.
COMPARISON OF RAINFALL-RUNOFF MODELS FOR URBAN AREAS
331
Rainfall and runoff data For each watershed about ten hydrographs which, in general, satisfied the following criteria were selected for analysis: (1) the storms which were relatively isolated; (2) storms which exhibited approximate uniform areal distribution over the entire watershed; and (3) stage hydrographs which had a well defined rising limb culminating in a single peak followed by sustained recession. Rainfall data in the form of continuous traces of cumulative rainfall were available for most of the watersheds used in the present study. F o r the rural watersheds in Indiana, where no rain gages were located within the watershed, hourly rainfall data from stations near the watershed boundaries were obtained from the U.S. Weather Bureau. The arithmetic averages of the hourly rainfall from these stations were used to obtain the rainfall mass curves.
Baseflow separation and excess rainfall The procedure adopted in this study to separate the baseflow is based on the premise that the urban watersheds have a very nearly constant rate of baseflow. In a hydrograph, let A and B respectively represent the points at which the direct runoff begins and ends (Fig. 1). The point A is easily located in isolated I~- TIME
I (IN/HR) ]
TOTALRAINFALL
]
EXCESSRAINFALL
DT '
/
/
o
/
A
(0,0)
TIME
Fig. 1. Baseflow separation and determination of rainfall excess.
332
P.B.S. SARMA, J.W. DELLEUR AND A.R. RAO
storms and the value of the maximum discharge Qm is determined by drawing a horizontal line through the point A. The point B is then located on the recession limb of the hydrograph such that the discharge at B in excess of that at A (i.e., QB) is one hundredth of the peak discharge Qm. The points A and B are joined by a straight line which represents the baseflow separation line. The ordinates of the direct runoff hydrograph are then obtained by subtracting the ordinates below the straight line AB from the correponding ordinates of the total runoff hydrograph. The peak of the direct runoff hydrograph is designated by Qp. The excess rainfall was estimated assuming that the sum of the interception, evaporation and depression storage is linearly related to the rainfall intensity. The rainfall which occurred before the time of commencement of direct runoff was assumed to be the initial abstraction. The beginning of the time of rise of runoff was taken as the beginning time of excess rainfall. The ordinates of the excess rainfall hyetograph were obtained by multiplying the corresponding ordinates of the total rainfall hyetograph by the ratio of the volume of direct runoff to the volume of the total rainfall. Thus the time distribution of excess rainfall and the total rainfall are similar after the time of rise (Fig. 1). MODEI~LING THE RAINFALL RUNOFF PROCESS
For small experimental impervious areas, lumped linear system models have been found to be satisfactory in modelling the excess rainfall-direct runoff process (Willke, 1962; Eagleson and March, 1965, Viessman, 1966, 1968). The lumped linear system model may be closer to reality for urban watersheds than for rural watersheds, because the former are often smaller in area and more uniform in surface characteristics. Linear systems are uniquely characterized by their response h(t) to a unit impulse function 6(t), or Dirac-delta function. If X(t) represents the input, Y(t) represents the output, and h(t) represents the unit impulse response, the i n p u t output relationship for a lumped linear system can be expressed by the convolution (Duhamel) integral: t
Y(t) = f x(r) h(t-r)dr
( 1)
0
Thus the response of a lumped linear system to any input can be evaluated if the unit impulse response function is known. Conceptual models and the Fourier transform method were used in the present study to identify the impulse response function h(t). The relative performances of the models were tested by "regenerating" the outflow data which were used to obtain the response function of the model.
COMPARISON OF R A I N F A L L - R U N O F F MODELS FOR URBAN AREAS
333
"Regeneration", as used in this study, involves the following computations: (1) the parameters of the conceptual models are determined from the excess rainfall and the direct runoff data; (2) the IUH or the unit impulse response function of the conceptual models is calculated in terms of the parameters estimated above or directly from the excess rainfall and direct runoff data for the Fourier transform method; and (3) the direct runoff hydrograph is computed or "regenerated" by using the convolution integral (eq. 1) in which X(t) is the observed rainfall and h(t) is the kernel function computed in step 2 above. The following five models were analyzed to compare their relative regeneration performance. The first four are conceptual models in which the system structure is prescribed a priori: (1) the single linear reservoir model (Viessman, 1966, 1968); (2) the Nash (1959) model; (3) the double routing model proposed by Holtan and Overton (1964); and (4) the Clark (1945) model. The fifth model, which does not prescribe the system structure a priori, is that of the "system identification" by evaluating the kernel function in the convolution integral (eq. 1). This was done by the use of the Fourier transform (Blank et al., 1971 ). The parameters of the conceptual models were computed by using the observed data and the regeneration performances of these models were then compared. There is no unique method of comparing the observed and the regenerated direct runoff hydrographs. Besides the qualitative comparisons based on visual observation, peak reproduction, etc., certain statistical measures such as (1) the correlation coefficient; (2) the integral square error; and (3) the special correlation coefficient (Eagleson and March, 1965), were employed in the present study for a quantitative comparison of the observed and the regenerated direct runoff hydrographs. The correlation coefficient is given by:
R =
(2)
The integral square error is defined as: 1/2
N
,,E
°c i"21 N
Qo (i) i=1
x lOO
(3)
334
P.B.S. SARMA,J.W. DELLEURAND A.R. RAO
The special correlation coefficient is estimated from:
2 ~/..~Qo (i)Qc (i) - ~ (Qc (i))2
N R = i i=1 s
N
N i=1
(4)
~(Qo(i)) 2 i=1 In eq. 2, 3 and 4, Qo (i) and Qc (i) are respectively the i-th value of the observed and regenerated outflow, respectively, and N is the number of values in the outflow series.
The single linear resevoir model The linear reservoir is characterized by the linear relationship between the storage S, and the outflow Q described by the equation:
S = KQ where K is known as the storage coefficient. The instantaneous unit hydrograph (IUH) for the linear reservoir model can be shown to be:
h(t) = ~1 e_t/K
(5)
The outflow Q(t) due to any given excess rainfall input I(t), can be obtained by using the convolution integral (eq. 1), which now takes the form:
t Q(t) = f I(r) le-(t-r)/K dr 0
(6)
The parameter K and its determination The storage coefficient K of the single linear reservoir model has units of time, and can be shown to be equal to the time lag T L which is defined as the time interval between the center of mass of the excess rainfall hyetograph and the center of mass o f the direct r u n o f f hydrograph. Hence: K = TQ - T , = T L
(7)
where TQ is the time to center o f mass of direct r u n o f f hydrograph, and T l is the time to center o f mass of excess rainfall hyetograph. If the excess rainfall-direct r u n o f f process is truly linear, then the parameter
COMPARISON OF R A I N F A L L - R U N O F F MODELS FOR URBAN AREAS
335
K, computed by eq. 7, should be a constant for all storms on a given watershed. However, when the storage coefficient K was computed from the observed data, values of K were found to vary from storm to storm. This variation of K has been previously observed by various investigators (Eagleson and March, 1965; Delleur and Vician, 1966; Viesmann, 1966, 1968). The average value of K for each watershed was determined and used as the representative value for that particular watershed for regeneration of direct runoff hydrographs. This regeneration proved to be unsatisfactory. Consequently, for regeneration, the time lag value for each storm was used as the parameter K in eq. 5, instead of the average value. A study of regenerated hydrographs so obtained indicated that the regeneration could be improved by a slight change in the value of K. The ~hange in the value of K, however, has to be made according to some well defined criteria. Two criteria were established to optimize the value of K, one of which was to minimize the sum of the squares of the deviations between the observed and the regenerated hydrograph ordinates, i.e., to minimize the quantity: N
. ~ [Qo ('i) - Qc(i)] 2 . i=1
The second criterion selected was to minimize the sum of the squares of the relative deviations between the observed and regenerated values of the peak discharge and of the observed and regenerated peak values of the time to the peak discharge, i.e., to minimize the quantity: N
1/2 ([(QPo
i=1
-
QPc)/QPo ]2 + [ ( T p o
-
Tpc)/Tpo ]2}
where QPo and QPc are respectively the magnitudes of the peak discharge of the observed and of the regenerated hydrographs whereas Tpo and Tpc are the times corresponding to QPo and QPc" The first of the above mentioned two criteria yields an optimum value of K designated as KI, which ensures the regeneration of the time distribution of the outflow with best agreement between the observed and the regenerated hydrographs. The value of K designated as K: obtained by using the second criterion yields the best regeneration of the peak discharge both in magnitude and time. A typical example of observed and regenerated hydrographs obtained by using the following different values of the storage coefficient, namely: (1) assuming K to be equal to T L (method 1); (2) using the value o f K 1 (method 2); and (3) using the value o f K 2 (method 3), are presented in Fig. 2.
336
P.B.S. SARMA, J.W. D E L L E U R AND A.R. RAO
1.071
WATERSHED
NO I
DATE OF STORM , 4 / 2 0 / 1 9 6 6 0.918
~
0.765-
0
OBSERVED
/"
K :
4-
K = Kt ( M E T H O D 2)
x
K = Kz (METHOD
T4(METHOD
I)
3)
03 0.612LIA C.D CE -r 0,q59(23
0.306
0.153
O.OOO
O,OO
10~.86
21 .?l
31~1-$7
q3W,43
TIME(MINI
b-'41.89
.... v.'L2T .'. ~ r 65,1q
"/6-00
Fig. 2. Example of regeneration obtained by the single linear reservoir model.
The Nash m o d e l The conceptual model consisting of a cascade of n equal linear reservoirs, each witfi a storage coefficient K N was proposed by Nash (1959). The expression for the IUH of the Nash model is:
1 e-t/KN h(t) - KN F(n) (t/KN)n-I
(8)
where I'(') is the gamma I~anction. The IUH of the Nash model can be described in terms of the two parameters K N and n. Nash suggested that the two parameters can be evaluated from the first and the second moments o f the excess rainfall and direct runoff distributions. t The values of n and K N so determined were found to vary from storm to
COMPARISON OF RAINFALL-RUNOFF MODELS FOR URBAN AREAS
337
storm, and this variation has also been reported by several previous investigators (Diskin, 1964; Wu et al., 1964; Blank and Delleur, 1968). Since it was obvious that a unique set of n and K N values were not applicable to a watershed no attempt was made to estimate such a unique set of values. Instead, by using the data from a watershed, the values of n and K N were estimated for each storm and were used for regeneration of the storm hydrograph by applying eqs. 1 and 8. A typical result o f regeneration is presented in Fig. 3.
The double routing method Holtan and Overton (1964) proposed a "double routing" method to compute the direct runoff from a storm. In this method, the excess rainfall is routed through two linear reservoirs in series having the same storage constant K D . Although Holtan and Overton used the Muskingum routing procedure, the "double routing" method is actually a special case of the Nash model in which n = 2. Judging from the results of all the data analyzed the results of regeneration obtained by using this double routing method were much less satisfactory
453.2 -
WATERSHED NO. 5 DATE OF STORM : 5 /
I /1962
:3eB.5-
0 Z~
OBSERVED HYDROGRAPH
REGENERATEDHYDROGRAPH
9L~1.7 -
co
259.0l,zJ n,rr
"I-
cO Q
129.5
64.7
o.0 $
0.00
21.64
51.29
71.93
101-57
131.21
TIME(HRSI
Fig. 3. Example of regeneration obtained by using the Nash model.
•
151.86
1B.50
338
P.B.S. SARMA, J.W. DELLEUR AND A.R. RAO
(Sarma et al., 1969) than for the previous methods. This is perhaps due to the fact that Holtan and Overton have suggested the evaluation of K D from the recession limb of the hydrograph, whereas the recession segment or the hydrograph, in general, is a complex combination of the surface flow, interflow and the groundwater flow, each having different lag characteristics.
Linear-channel, linear-reservoir model In Clark's (1945) conceptual model, which consists of a series combination of a linear-channel and a linear-reservoir, the translation effects of the linearchannel and the storage effects of the linear-reservoir are combined. Since the order of the linear operations is immaterial, the flow may be considered as passing through a linear-channel first and then through a linear-reservoir. Considering the time-area-concentration diagram w(t) as inflow to the linear reservoir with a storage coefficient K, the IUH of the conceptual model can be written as: t<~T
/~(t) = f
c ( ~1) ~ -a (t - T)/K ~ ( , ) d r
(9)
0
which can be shown to be a special case of the general expression for the IUH derived by Dooge (1959).
Dimensionless IUH In order to analyze the data from actual watersheds by this method the derivation of the time-area-concentration diagram requires the determination of the isochrones. However, no reliable method is available to accurately determine the isochrones for any watershed. O'Kelly (1955) suggested that the smoothing involved in the routing is sufficient to permit replacement of the time-areaconcentration diagram by a well defined geometrical shape such as a rectangle or an isosceles triangle, with a base equal to the time of concentration, Tc . Mathematical expressions of the IUH for the case of rectangular and isosceles triangular shapes of the time-area-concentration diagram were also presented by O'Kelly. Dooge (1959) and O'Kelly (1955) have routed the time-area-concentration diagram through a linear-reservoir to obtain the IUH. However, the IUH so obtained was not used for regeneration either by Dooge or O'Kelly. Four different shapes of the time-area-concentration diagrams were considered: rectangular, isosceles triangular, and triangular shapes with the peaks occurring at the beginning and at the end of the time Tc . In order to apply the dimensionless instantaneous unit hydrographs men-
COMPARISON OF RAINFALL-RUNOFF MODELS FOR URBAN AREAS
339
tioned above for data from actual watersheds, the values of K and T C are to be estimated. The value of K can be estimated from eq. 7. The time of concentration T c is defined as the time required for the surface runoff from the remotest part of the watershed to reach the outlet. In natural watersheds, accurate evaluation of the time of concentration is not possible. Based on the premise that the time of concentration T c is the time taken for the last drop of the excess rainfall to reach the outlet of the watershed, Snyder (1958) suggested that T c can be estimated as the time elapsed between the end of the rainfall and the point of inflection on the recession limb of the hydrograph. To test the regeneration performance of this conceptual model, the parameters K and T c were estimated for each storm by using eq. 7 and Snyder's definition, respectively, and the instantaneous unit hydrographs were derived for all four cases mentioned above. The instantaneous unit hydrographs were then used successively with eq. 1, to c o m p u t e the corresponding outflow hydrographs due to the excess rainfall of the storm. The outflow hydrographs thus obtained for the four cases, were compared with the corresponding observed hydrographs. The results showed that: (1) the peak discharge of the regenerated hydrographs in all the four cases were very nearly the same; and (2) the computed peak discharges of the regenerated hydrographs were less by 5 0 - 1 0 0 % than the peak discharges of the corresponding observed hydrographs.. Thus, the IUH derived by using the value of T c obtained by adopting Snyder's definition did not satisfactorily regenerate the hydrograph of t h e storm from which it was derived. The above results are subjected to the assumption that the storage coefficient K of the linear reservoir is equal to the time lag T L , although as mentioned earlier, the optimum value of K was slightly different from T L . However, the assumption is valid as the storage coefficient K is theoretically equal to the time lag T L . Thus these results indicate that an alternate approach for a better estimation of T c was necessary. As the conceptual model has two parameters, K and T C , and assuming the value of K can be correctly obtained from eq. 7, only the parameter T C remains to be determined in order to obtain an " o p t i m u m " regeneration. Hence for each storm, and " o p t i m u m " value of T c was computed according to the criterion that the sum of the squares of the deviation of the observed and the corresponding computed hydrograph ordinates is minimized. It was observed that the optimum value of T c varied from storm to storm on any watershed. The optimum values of T C were consistently greater than those obtained by Snyder's definition. Typical results of regeneration, obtained by using Snyder's T C and the optimum values of Tc are presented in Fig. 4.
340
P,B.S. SARMA, J.W. DELLEUR AND A.R. RAO
18,q . 7 -
WATERSHED NO. 5 DATE OF STORM , 4 / 2 2 / 1 9 6 5
156.6 -
o •'~ 4130.5
OBSERVED Tc = SNYDER'S Tc Tc = OPTIMUM T¢
-
70 lO~.q-
f-9
n,,CE "7-
L)
O'3
78.3-
C3
,59..2-
26.1
0.0 T"~.
o.oo
2'.1,
4'.m
s'.~
e'.,~
TIME[ HRS)
lo'.s~
-
12'.s~
t~'.Ts.
Fig. 4. Example of regeneration obtained by using the linear-channel-linear-reservoir model.
Fourier
transform method
In all the previous cases the excess rainfall-direct runoff transformation was modelled by lumped linear conceptual systems in which the model structure is specified. An alternative approach in which a system structure is not specified is for example, evaluation of the kernel function in the convolution integral (eq. 1) by the method of integral transforms. The only assumption made in this method is that the excess rainfall--direct runoff process is linear. The fact that the convolution operation in the time domain is equivalent to a multiplication operation in the transformed domain lends convenience to this method. The Fourier transform method of analysis developed by Blank et al. (1968, 1971) was used in the present study. The Fourier transform (FT) of a function f(t) is defined as:
FT[f(t)l = F(co) = y f(t)
e -/~°t d t
COMPARISON OF RAINFALL-RUNOFF MODELS FOR URBAN AREAS
341
and the inverse Fourier transform (FT -1) is: Do
FT-'[F(w)] = f(t) =
(10)
F ( ~ ) ~ ~ ' dco
Applying the Fourier transform to both sides of eq. 1 results in: Q(co) = X(o~) H(o~) or:
H(w ) = O(co )/X(co ) where X(w), Q(co) and H(co) are the transforms of the rainfall excess, of the direct runoff and of the IUH, respectively. The impulse response function (or the IUH) can be obtained by taking the inverse transform of H(c~) using eq. 10. The details of computational procedure can be found elsewhere (Blank and Delleur, 1968; Rao and Delleur, 1971). Typical results of regeneration are presented in Fig. 6 for a large urban watershed. 6.369
~
WATERSHED NO. I DATE
6.q69
OF STORM : 8 / 2 0 / 1 9 6 7
-
,~,
SINGLE LINEAR RESERVOIR MODEL
+
FOURIER TRANSFORM METHOD
A
z
. ~ q.Eq9
-r 3.6q00
e'z r
,-' 2.730(::3 0::: C)
4.÷
! .fRO -
o.g10 -
÷44"4-
0.000
-1-
0.00
. . . . . .
-1
.
.
.
.
.
.
.
I
TIME(MIN)
Fig. 5. Comparison between instantaneous unit hydrograph obtained by the Fourier transform method and by the single linear reservoir model.
342
P.B.S. SARMA, J.W. DELLEUR AND A.R. RAO
866.8 -
WATERSHED NO. 5 DATE OF STORM :10/13/1962
571.6
-
O
OBSERVED REGENERATED
q?S.9 -
381.1t.u rr "1o'J
285.8-
190.5
-
95.9
-
o.o ~-~
0.®
e'.~0
~'.~
~'.,
0'.?, T IME( HRS )
10'.~
---~
......
1s.,8
~;~.~,
Fig. 6. Comparison between the observed hydrograph and the regenerated hydrograph obtained by the Fourier transform method.
EVALUATION OF METHODS OF ANALYSIS
The regeneration performances of the five methods, four of which employ conceptual models and the fifth, the Fourier transform method o f evaluating the kernel function, have been tested. F o r some of the conceptual models, the optimized parameter values were computed according to the several criteria, and were used in the regeneration. The statistical parameters which are defined by eq. 2, 3, and 4 were c o m p u t e d for each set of data tested in order to obtain a quantitative measure of the regeneration performance of the various methods. Based on the values of the statistical parameters the regeneration performance of the various methods were classified according to the ratings shown in Table II. Typical results for some of the watersheds are shown in Table III. The rows in Table III refer to the ratings mentioned in Table II. The columns refer to the different models and to the different methods used for regeneration and also to the statistical measures used to compare the results of regeneration. The num-
COMPARISON OF R A I N F A L L - R U N O F F MODELS F O R U R B A N A R E A S
343
T A B L E II Ratings of the statistical measures Correlation coefficient (R)
Rating
0.99 ~< R < 1.0
excellent
0.95 ~< R < 0.99
very good
0.90 ~< R < 0.95
good
0.85 ~< R < 0.90
fair
0.00 <~ R < 0.85
poor
Special correlation coefficient (Rs)
Rating
0.99 ~< R < 1.0 s 0.95 ~< R < 0.99
very good
0.90 ~< R < 0.95
good
0.85 ~< R < 0.90 s 0.00 ~< R < 0.85
fair
Integral square error ( I S E )
Rating
excellent
S S
poor
S
0% < I S E <<. 3.0%
excellent
3.0% < I S E <~ 6.0%
very good
6.0% < I S E ~ 10.0%
good
10.0% < I S E <<. 25.0%
fair
25.0% < I S E
poor
-
bers in Table III refer to the numbers of storms which satisfy the criterion to qualify for the ratings such as excellent, E, very good, V.G., etc. For example, for watershed no. 1 when the single linear reservoir model (with K = T L ) was used for regeneration, and when the criterion used for comparison between the observed and the regenerated hydrographs was the coefficient of linear correlation, R, seven storms deserved the rating E, whereas thirteen storms deserved the same rating E, when the criterion used for comparison was the integral square error USE). The coefficient o f linear correlation, R, between the observed and the regenerated hydrographs was greater than 0.99 for seven storms whereas the ISE was less than 3% for thirteen storms. The total number of storms satisfying a given rating or a rating superior to it can be obtained by adding the number of storms above the particular rating. For example, for the Nash model used in watershed no. 2 the total number of storms satisfying the criterion F or above was 8 when the statistical measure used for comparison was R, whereas it was 10 when the statistical measures R s and ISE were used, which means
1
3
2
1
4
E
V.G.
G
F
P
1
0
F
P
3
1
2
3
2
0
0
1
0
0
2
3
6
0
0
1
4
1
2
3
1
0
1
1
3
1
2
4
1
0
1
0
6
9
1
8
6
G
2
13
9
7
6
7
E
S
V.G.
R
R
ISE
R
R *4
S
K = Tl
Single linear reservoir model
K = TL
Rating .2
0
1
3
2
5
0
0
1
5
10
ISE
2
5
2
1
1
1
3
3
6
3
R
4
4
1
2
0
1
3
4
6
2
R S
K D = TL/2
1
4
3
2
1
0
0
4
7
5
ISE
Double routing model
3
4
1
2
1
4
0
1
4
7
R
1
4
4
2
0
6
0
0
4
6
R S
Nash model
* 1 Similar results for watersheds 5, 6, 7, 11, 12 and 13 may be found in Sarma et aL (1969). * 2 For explanation of ratings E, V.G., G, F and P, see Table II. *3 Equilateral triangular time-area-concentration diagram. *4 R = linear correlation coefficient (eq. 2). R s = special correlation coefficient (eq. 3). 1SE = integral square error (eq. 4).
Watershed No. * 1
Sample of evaluation of various methods of analysis
TABLE IlI
1
3
4
1
2
0
3
3
2
8
ISE
2
4
2
2
1
2
0
4
5
5
R
1
4
3
3
0
4
0
2
5
5
R S
0
4
3
3
1
1
3
2
3
7
ISE
Linear channelreservoir model *3
0
0
0
2
9
1
0
1
0
14
R
S
0
0
0
2
9
1
0
1
0
14
R
0
0
0
1
10
1
0
0
1
14
ISE
Fourier transform method
> ©
>.
> Z
t"
>
COMPARISON OF R A I N F A L L - R U N O F F MODELS FOR URBAN AREAS
345
that there were 8 storms for which the value of R was greater than 0.85 whereas there were 10 storms for which ISE was less than 25%. DISCUSSION OF THE RESULTS
The Fourier transform method is perhaps the most general method of obtaining the response function (IUH) of linear systems, whereas the use of conceptual models to represent linear systems is neither unique nor general. For some of the data tested, the impulsive response functions obtained by the Fourier transform m e t h o d exhibited high frequency oscillations. As the hydrologic systems are stable, their lUll or impulsive response are expected to be stable also. Similar oscillations have been reported by Blank et al. (1971). Methods of controlling these computational instabilities by means of the proper choice of the discretization interval and by digital filtering have been discussed by Delleur and Rao (1971a, b) and Rao and Delleur (1971). The kernel functions obtained by the Fourier transform method for small watersheds were of the exponential decay type as shown in Fig. 5. The instantaneous unit hydrographs which result from using a single linear reservoir model, are, by definition (eq. 5) of the same type. This concurrence of the response functions obtained by the general method and by the single linear reservoir model substantiates the use of the single linear reservoir model in the analysis of small urban watersheds of area less than about 5 sq.miles. For larger watersheds (greater than, say, 5 sq.miles in area) the choice of a particular model which can be used to simulate the rainfall-runoff process was not so easy. By an examination of the regeneration performance, it is evident that if at all possible, use of response functions obtained by the Fourier transform methods would be the best choice. In the double routing model, there was considerable ambiguity in the selection of an appropriate value of the storage coefficient K D . There is no unique value of K D that can be obtained by considering any portion of the recession curve, as assumed by this model. Apart from this difficulty, this method is but a special case of the Nash model, with n = 2. Whereas in the Nash model, the values of both n and K N can be determined to give a better regeneration, in the double routing method this freedom is lost without any advantage gained. In the analysis using the linear-channel linear-reservoir model the parameter T c was evaluated by the m e t h o d suggested by Snyder. Although the results were slightly improved when the parameter T c was optimized such that the sum of the squares of the differences between the corresponding ordinates of the observed and the regenerated hydrographs was minimized, the regeneration was not as good as that obtained by the Nash model. Thus, among the four conceptual models considered for analysis of data from
346
P.B.S. SARMA, J.W. DELLEUR AND A.R. RAO
larger watersheds, the Nash model yielded a relatively better regeneration performance. CONCLUSIONS
(1) The linear system methods can be usefully employed in the analysis of direct runoff from urban watersheds. The methods of linear system analysis are relatively more accurate for smaller watersheds (less than about 5 sq.miles). (2) The estimation of the response function by the Fourier transform method and its use in prediction is very promising. On the other hand, the nonlinear behavior of larger watersheds may be so pronounced that no linear model, however sophisticated it might be, could give accurate results. (3) For small watersheds, (less than 5 sq.miles approximately) the response functions computed by the Fourier transform method and the instantaneous unit hydrographs obtained by using the single linear reservoir model were similar. This similarity supports the use of the single linear reservoir model in small urban watersheds. (4) The regeneration performance of the single linear reservoir model for smaller watersheds can be improved by suitably optimizing the value of the parameter K (5) For watersheds of area larger than about 5 sq.miles, of all the conceptual models tested, the Nash model gave relatively the best regeneration and hence can be used for simulating the rainfall-runoff process in such watersheds. However, the regeneration performance of the Nash model, by itself was not very good. This is perhaps due to the fact that for larger urban watersheds the hydraulics of the sewer system which is strictly not represented in the Nash model predominates over the overland flow phase of the runoff cycle. This suggests that further investigation is necessary to select a suitable model to simulate the rainfall-runoff process in larger urban watersheds. ACKNOWLEDGMENTS
This paper is based on the doctoral dissertation of Mr. P.B.S. Sarma, "Effects of Urbanization on Runoff from Small Watersheds" submitted to Purdue University in January 1970. The support of Dr. J.F. McLaughlin, Head, School of Civil Engineering, Purdue University, in completing the research reported herein is gratefully acknowledged. The District Offices of the U.S. Geol. Surv. at Indianapolis, Indiana, and Austin, Texas, are thanked for their ready help in data collection. The research was supported as a cooperative project by the OWRR under project OWRR-B-002-IND, the Indiana Department of Natural Resourches, and Purdue
COMPARISONOF RAINFALL-RUNOFFMODELSFOR URBAN AREAS
347
University from July 1966 to June 1969 and it was further supported by Purdue University until September 1969. We would like to thank our sponsors for their support.
REFERENCES Blank, D. and Delleur, J.W., 1968. A program for estimating runoff from Indiana watersheds, Part I: Linear system analysis in surface water hydrology and its applications to indiana watersheds. Purdue Univ. Water Resour. Res. Cent., Tech. Rep. 4. Blank, D., Delleur, J.W. and Giorgini, A., 1971. Oscillatory kernel functions in linear hydrologic models. WaterResour. Res., 7 (5): 1102-1117. Clark, C.O., 1945. Storage and the Unit Hydrograph. Trans. Am. Soc. Cir. Eng., 110: 1419-1446. Delleur, J.W. and Vician, E.B., 1966. Discussion on the paper "Time in urban hydrology" by G.E. Willeke. Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 92 (HY5): 243-251. Delleur, J.W. and Rao, R.A,, 1971a. Linear systems analysis in hydrology - the transform approach, the kernel oscillations and the effect of noise. In:Systems Approach to Hydrology. Proc. 1st Bilat. U.S. Japan Sem. HydroL, Univ. HawaiL Honolulu, January 11 17, 1971. Water Resour. PubL, Fort Collins, Colo., pp. 116-142. Delleur, J.W. and Rao, R.A., 1971b. Characteristics and filtering of noise in linear hydrologic systems. Proc. Int. Syrup. Math. ModelsHydroL, Warsaw, Poland, July 2 6 - 3 1 , 1971. Vol. 2(1), Pap. 4/13, 14 pp. Diskin, M.H., 1964. A basic Study o f the Linearity o f the R a i n f a l l - R u n o f f Process in Watersheds. Thesis, University of lllinois, Urbana, Ill. Dooge, J.C.I., 1959. A general theory of unit hydrograph. J. Geophys. Res., 64(1): 241-256. Eagleson, P.S. and March, F., 1965. Approaches to the linear synthesis of urban runoff systems. Hydrodvn. Lab. Rep. 85. M.I.T., Cambridge, Mass. Holtan, H.N. and Overton, D.E., 1964. Storage flow hysteresis in hydrograph synthesis. J. tlydroL, 2: 309-323. Nash, J.W., 1959. Systematic determination of unit hydrograph parameters. J. Geophys. Res., 64( 1): 111 115. O'Kelly, J.J., 1955. The employment of unit hydrographs to determine the flows of Irish arterial drainage channels. Proc. Inst. Civ. Eng., 4(3): 365-412. Rao, R.A. and Delleur, J.W., 1971. The instantaneous unit hydrograph: its calculation by the transfornr method and noise control by digital filtering. Purdue University, Water Resour. Res. Cent., Tech. Rep. 20. Sarma, P.B.S., Delleur, J.W. and Rao, A.R., 1969. A program in urban hydrology, part II: An evaluation of rainfall-runoff models for small urbanized watersheds and the effect of urbanization on runoff. Purdue Water Resour. Res. Cent., Tech. Rep. 9 : 2 4 0 pp. Snyder, F.F., 1958. Synthetic flood frequency. Proc. Am. Soc. Cir. Eng., J. Hydraul. Div., 84 (HY5): 1808-1-1802-22. U.S. Geological Survey, 1962-1966. Compilation o f Hydrologic Data, Waller and Wilbarger Creeks, Colorado River Basin, Texas. Rep. Water Resour. Div., Texas Dist., Austin, Texas. Viessman Jr., W., 1966. The hydrology of small impervious areas. Water Resour. Res., 2 (3): 4 0 5 - 4 1 2 . Viessman Jr., W., 1968. Runoff estimation for very small drainage areas. Water Resour. Res., 4 (1): 87-93. Willeke, G.E., 1962. The prediction of runoff hydrographs for urban watersheds from precipitation data and watershed characteristics. J. Geophys. Res., 67 (9): 3610. Wu, I.P., Delleur, J.W. and Diskin, M.H., 1964. Determination o f Peak Discharge and Design Hydrographs Cbr Small Watersheds in Indiana. Hydromech. Lab., School Cir. Eng., Purdue Univ., Lafayette, Ind., 106 pp.
COMPARISON OF R A I N F A L L - R U N O F F MODELS FOR U R B A N AREAS P.B.S. SARMA 1, J.W. DELLEUR 2 and A.R. RAO 2
l lndian Institute of Technology, Kanpur (India) 2School of Civil Engineering, Purdue University, Lafayette, Ind, (U.S.A.J (Accepted for publication May 18, 1972)
ABSTRACT Sarma, P.B.S., Delleur, J.W. and Rao, A.R., 1973. Comparison of rainfall--runoff models for urban areas. J. Hydrol., 18: 329-347. The relative regeneration performances of five linear rainfall excess-direct runoff models are compared for several urban watersheds with varying degrees of development. The five models considered are the single linear reservoir, the Nash model, the double routing method, the linear channel-linear reservoir model and the instantaneous unit hydrograph (IUH) obtained by the Fourier transform method. The IUH always gives the best regeneration performance among the four conceptual models tested. The optimized single linear reservoir constant differs from the theoretical time lag value, but is related to the latter, and for each watershed varies from storm to storm. For larger watersheds the Nash model gives the best regeneration performance among the four conceptual models tested. The model parameters for each watershed are found to vary from storm to storm. The quality of regeneration for larger basins is less than that found for the smaller basins.
INTRODUC~ON
The assumption o f the linearity of the rainfall excess-direct runoff process (hereafter referred to as the rainfall-runoff process) forms the basis of several c o m m o n l y used methods in urban hydrology. Linear conceptual and/or mathematical models of the rainfall-runoff process have been used successfully in recent investigations of small experimental impervious watersheds (less than 1 acre in area) by Willeke (1962), Viessman (1966, 1968), Eagleson and March (1965). Delleur and Vician ( 1 9 6 6 ) have used two conceptua 1 models to simulate the rainfall-runoff process in two actual urban watersheds which are partially impervious. As several linear conceptual models can be formulated by appropriate combinations of linear elements such as the linear reservoir and the linear channel, the question of comparative suitability o f any of these models to simulate the urban rainfall-runoff process arises. The work reported herein deals with an investigation o f the relative regeneration performance of some of the commonly used linear hydrologic system models in an attempt to select an appropriate model to simulate the rainfall-runoff process in a given urban watershed.
330
P.B.S. SARMA, J.W. D E L L E U R AND A.R. RAO
U R B A N WATERSHEDS INVESTIGATED
Long-term records of hydrologic data are not usually available for urban watersheds. In order to select a suitable simulation model of the rainfall-runoff process in watersheds with different stages of urban development, it is necesary to analyze data from several watersheds with varied degrees of imperviousness. In the present study, hydrologic data from the following 13 watersheds, listed in Table I, have been collected and analyzed. The data collection from the Purdue Swine Farm Watersheds in West Lafayette was initiated as part of the present study and is being continued. The studies of the upper and lower Ross Ade watersheds were commissioned in 1964. The hydrologic data from the other watersheds in Indiana and Texas were acquired from the records and reports of the U.S. Geological Survey (1962-1966). Details of the instrumentation of the West Lafayette watersheds can be found elsewhere (Sarma et al., 1969). For each of the watersheds under study, the values of the area of the drainage basin A, the percentage of the impervious area U, and the number of storms analyzed are presented in Table I.
TABLE I Watersheds investigated Watershed number
Name of the watershed*
Area of the watershed A (sq.miles)
Percentage impervious area U(%)
No. of storms used for analysis
1
Ross Ade (upper) 1
0.0455
38.0
21
2
Ross Ade (lower) 1
0.6125
37.4
10
3
Purdue Swine Farm (upper) 1
0.2776
21.3
4
4
Purdue Swine F a r m (lower) 1
0.4562
13.3
2
5
Pleasant R u n (Arlington) 2
7.580
10.5
13
6
Pleasant R u n (Brookville) 2
10.10
15.5
12
7
Little Eagle Creek 2
19.31
2.1
8
8
Lawrence Creek 2
2.86
0.0
12
9
Bear Creek 3
5
7.00
0.0
14.60
0.0
7
2.31
27.0
17
Waller Creek (23rd Street) 4
4.13
37.0
12
Wilbarger Creek 4
4.61
0.0
8
10
Bean Blossom Creek 3
11
Waller Creek (38th Street) 4
12 13
* 1 = West Lafayette, Indiana; 2 = Indianapolis, Indiana; 3
=
Indiana; 4 = Austin, Texas.
COMPARISON OF RAINFALL-RUNOFF MODELS FOR URBAN AREAS
331
Rainfall and runoff data For each watershed about ten hydrographs which, in general, satisfied the following criteria were selected for analysis: (1) the storms which were relatively isolated; (2) storms which exhibited approximate uniform areal distribution over the entire watershed; and (3) stage hydrographs which had a well defined rising limb culminating in a single peak followed by sustained recession. Rainfall data in the form of continuous traces of cumulative rainfall were available for most of the watersheds used in the present study. F o r the rural watersheds in Indiana, where no rain gages were located within the watershed, hourly rainfall data from stations near the watershed boundaries were obtained from the U.S. Weather Bureau. The arithmetic averages of the hourly rainfall from these stations were used to obtain the rainfall mass curves.
Baseflow separation and excess rainfall The procedure adopted in this study to separate the baseflow is based on the premise that the urban watersheds have a very nearly constant rate of baseflow. In a hydrograph, let A and B respectively represent the points at which the direct runoff begins and ends (Fig. 1). The point A is easily located in isolated I~- TIME
I (IN/HR) ]
TOTALRAINFALL
]
EXCESSRAINFALL
DT '
/
/
o
/
A
(0,0)
TIME
Fig. 1. Baseflow separation and determination of rainfall excess.
332
P.B.S. SARMA, J.W. DELLEUR AND A.R. RAO
storms and the value of the maximum discharge Qm is determined by drawing a horizontal line through the point A. The point B is then located on the recession limb of the hydrograph such that the discharge at B in excess of that at A (i.e., QB) is one hundredth of the peak discharge Qm. The points A and B are joined by a straight line which represents the baseflow separation line. The ordinates of the direct runoff hydrograph are then obtained by subtracting the ordinates below the straight line AB from the correponding ordinates of the total runoff hydrograph. The peak of the direct runoff hydrograph is designated by Qp. The excess rainfall was estimated assuming that the sum of the interception, evaporation and depression storage is linearly related to the rainfall intensity. The rainfall which occurred before the time of commencement of direct runoff was assumed to be the initial abstraction. The beginning of the time of rise of runoff was taken as the beginning time of excess rainfall. The ordinates of the excess rainfall hyetograph were obtained by multiplying the corresponding ordinates of the total rainfall hyetograph by the ratio of the volume of direct runoff to the volume of the total rainfall. Thus the time distribution of excess rainfall and the total rainfall are similar after the time of rise (Fig. 1). MODEI~LING THE RAINFALL RUNOFF PROCESS
For small experimental impervious areas, lumped linear system models have been found to be satisfactory in modelling the excess rainfall-direct runoff process (Willke, 1962; Eagleson and March, 1965, Viessman, 1966, 1968). The lumped linear system model may be closer to reality for urban watersheds than for rural watersheds, because the former are often smaller in area and more uniform in surface characteristics. Linear systems are uniquely characterized by their response h(t) to a unit impulse function 6(t), or Dirac-delta function. If X(t) represents the input, Y(t) represents the output, and h(t) represents the unit impulse response, the i n p u t output relationship for a lumped linear system can be expressed by the convolution (Duhamel) integral: t
Y(t) = f x(r) h(t-r)dr
( 1)
0
Thus the response of a lumped linear system to any input can be evaluated if the unit impulse response function is known. Conceptual models and the Fourier transform method were used in the present study to identify the impulse response function h(t). The relative performances of the models were tested by "regenerating" the outflow data which were used to obtain the response function of the model.
COMPARISON OF R A I N F A L L - R U N O F F MODELS FOR URBAN AREAS
333
"Regeneration", as used in this study, involves the following computations: (1) the parameters of the conceptual models are determined from the excess rainfall and the direct runoff data; (2) the IUH or the unit impulse response function of the conceptual models is calculated in terms of the parameters estimated above or directly from the excess rainfall and direct runoff data for the Fourier transform method; and (3) the direct runoff hydrograph is computed or "regenerated" by using the convolution integral (eq. 1) in which X(t) is the observed rainfall and h(t) is the kernel function computed in step 2 above. The following five models were analyzed to compare their relative regeneration performance. The first four are conceptual models in which the system structure is prescribed a priori: (1) the single linear reservoir model (Viessman, 1966, 1968); (2) the Nash (1959) model; (3) the double routing model proposed by Holtan and Overton (1964); and (4) the Clark (1945) model. The fifth model, which does not prescribe the system structure a priori, is that of the "system identification" by evaluating the kernel function in the convolution integral (eq. 1). This was done by the use of the Fourier transform (Blank et al., 1971 ). The parameters of the conceptual models were computed by using the observed data and the regeneration performances of these models were then compared. There is no unique method of comparing the observed and the regenerated direct runoff hydrographs. Besides the qualitative comparisons based on visual observation, peak reproduction, etc., certain statistical measures such as (1) the correlation coefficient; (2) the integral square error; and (3) the special correlation coefficient (Eagleson and March, 1965), were employed in the present study for a quantitative comparison of the observed and the regenerated direct runoff hydrographs. The correlation coefficient is given by:
R =
(2)
The integral square error is defined as: 1/2
N
,,E
°c i"21 N
Qo (i) i=1
x lOO
(3)
334
P.B.S. SARMA,J.W. DELLEURAND A.R. RAO
The special correlation coefficient is estimated from:
2 ~/..~Qo (i)Qc (i) - ~ (Qc (i))2
N R = i i=1 s
N
N i=1
(4)
~(Qo(i)) 2 i=1 In eq. 2, 3 and 4, Qo (i) and Qc (i) are respectively the i-th value of the observed and regenerated outflow, respectively, and N is the number of values in the outflow series.
The single linear resevoir model The linear reservoir is characterized by the linear relationship between the storage S, and the outflow Q described by the equation:
S = KQ where K is known as the storage coefficient. The instantaneous unit hydrograph (IUH) for the linear reservoir model can be shown to be:
h(t) = ~1 e_t/K
(5)
The outflow Q(t) due to any given excess rainfall input I(t), can be obtained by using the convolution integral (eq. 1), which now takes the form:
t Q(t) = f I(r) le-(t-r)/K dr 0
(6)
The parameter K and its determination The storage coefficient K of the single linear reservoir model has units of time, and can be shown to be equal to the time lag T L which is defined as the time interval between the center of mass of the excess rainfall hyetograph and the center of mass o f the direct r u n o f f hydrograph. Hence: K = TQ - T , = T L
(7)
where TQ is the time to center o f mass of direct r u n o f f hydrograph, and T l is the time to center o f mass of excess rainfall hyetograph. If the excess rainfall-direct r u n o f f process is truly linear, then the parameter
COMPARISON OF R A I N F A L L - R U N O F F MODELS FOR URBAN AREAS
335
K, computed by eq. 7, should be a constant for all storms on a given watershed. However, when the storage coefficient K was computed from the observed data, values of K were found to vary from storm to storm. This variation of K has been previously observed by various investigators (Eagleson and March, 1965; Delleur and Vician, 1966; Viesmann, 1966, 1968). The average value of K for each watershed was determined and used as the representative value for that particular watershed for regeneration of direct runoff hydrographs. This regeneration proved to be unsatisfactory. Consequently, for regeneration, the time lag value for each storm was used as the parameter K in eq. 5, instead of the average value. A study of regenerated hydrographs so obtained indicated that the regeneration could be improved by a slight change in the value of K. The ~hange in the value of K, however, has to be made according to some well defined criteria. Two criteria were established to optimize the value of K, one of which was to minimize the sum of the squares of the deviations between the observed and the regenerated hydrograph ordinates, i.e., to minimize the quantity: N
. ~ [Qo ('i) - Qc(i)] 2 . i=1
The second criterion selected was to minimize the sum of the squares of the relative deviations between the observed and regenerated values of the peak discharge and of the observed and regenerated peak values of the time to the peak discharge, i.e., to minimize the quantity: N
1/2 ([(QPo
i=1
-
QPc)/QPo ]2 + [ ( T p o
-
Tpc)/Tpo ]2}
where QPo and QPc are respectively the magnitudes of the peak discharge of the observed and of the regenerated hydrographs whereas Tpo and Tpc are the times corresponding to QPo and QPc" The first of the above mentioned two criteria yields an optimum value of K designated as KI, which ensures the regeneration of the time distribution of the outflow with best agreement between the observed and the regenerated hydrographs. The value of K designated as K: obtained by using the second criterion yields the best regeneration of the peak discharge both in magnitude and time. A typical example of observed and regenerated hydrographs obtained by using the following different values of the storage coefficient, namely: (1) assuming K to be equal to T L (method 1); (2) using the value o f K 1 (method 2); and (3) using the value o f K 2 (method 3), are presented in Fig. 2.
336
P.B.S. SARMA, J.W. D E L L E U R AND A.R. RAO
1.071
WATERSHED
NO I
DATE OF STORM , 4 / 2 0 / 1 9 6 6 0.918
~
0.765-
0
OBSERVED
/"
K :
4-
K = Kt ( M E T H O D 2)
x
K = Kz (METHOD
T4(METHOD
I)
3)
03 0.612LIA C.D CE -r 0,q59(23
0.306
0.153
O.OOO
O,OO
10~.86
21 .?l
31~1-$7
q3W,43
TIME(MINI
b-'41.89
.... v.'L2T .'. ~ r 65,1q
"/6-00
Fig. 2. Example of regeneration obtained by the single linear reservoir model.
The Nash m o d e l The conceptual model consisting of a cascade of n equal linear reservoirs, each witfi a storage coefficient K N was proposed by Nash (1959). The expression for the IUH of the Nash model is:
1 e-t/KN h(t) - KN F(n) (t/KN)n-I
(8)
where I'(') is the gamma I~anction. The IUH of the Nash model can be described in terms of the two parameters K N and n. Nash suggested that the two parameters can be evaluated from the first and the second moments o f the excess rainfall and direct runoff distributions. t The values of n and K N so determined were found to vary from storm to
COMPARISON OF RAINFALL-RUNOFF MODELS FOR URBAN AREAS
337
storm, and this variation has also been reported by several previous investigators (Diskin, 1964; Wu et al., 1964; Blank and Delleur, 1968). Since it was obvious that a unique set of n and K N values were not applicable to a watershed no attempt was made to estimate such a unique set of values. Instead, by using the data from a watershed, the values of n and K N were estimated for each storm and were used for regeneration of the storm hydrograph by applying eqs. 1 and 8. A typical result o f regeneration is presented in Fig. 3.
The double routing method Holtan and Overton (1964) proposed a "double routing" method to compute the direct runoff from a storm. In this method, the excess rainfall is routed through two linear reservoirs in series having the same storage constant K D . Although Holtan and Overton used the Muskingum routing procedure, the "double routing" method is actually a special case of the Nash model in which n = 2. Judging from the results of all the data analyzed the results of regeneration obtained by using this double routing method were much less satisfactory
453.2 -
WATERSHED NO. 5 DATE OF STORM : 5 /
I /1962
:3eB.5-
0 Z~
OBSERVED HYDROGRAPH
REGENERATEDHYDROGRAPH
9L~1.7 -
co
259.0l,zJ n,rr
"I-
cO Q
129.5
64.7
o.0 $
0.00
21.64
51.29
71.93
101-57
131.21
TIME(HRSI
Fig. 3. Example of regeneration obtained by using the Nash model.
•
151.86
1B.50
338
P.B.S. SARMA, J.W. DELLEUR AND A.R. RAO
(Sarma et al., 1969) than for the previous methods. This is perhaps due to the fact that Holtan and Overton have suggested the evaluation of K D from the recession limb of the hydrograph, whereas the recession segment or the hydrograph, in general, is a complex combination of the surface flow, interflow and the groundwater flow, each having different lag characteristics.
Linear-channel, linear-reservoir model In Clark's (1945) conceptual model, which consists of a series combination of a linear-channel and a linear-reservoir, the translation effects of the linearchannel and the storage effects of the linear-reservoir are combined. Since the order of the linear operations is immaterial, the flow may be considered as passing through a linear-channel first and then through a linear-reservoir. Considering the time-area-concentration diagram w(t) as inflow to the linear reservoir with a storage coefficient K, the IUH of the conceptual model can be written as: t<~T
/~(t) = f
c ( ~1) ~ -a (t - T)/K ~ ( , ) d r
(9)
0
which can be shown to be a special case of the general expression for the IUH derived by Dooge (1959).
Dimensionless IUH In order to analyze the data from actual watersheds by this method the derivation of the time-area-concentration diagram requires the determination of the isochrones. However, no reliable method is available to accurately determine the isochrones for any watershed. O'Kelly (1955) suggested that the smoothing involved in the routing is sufficient to permit replacement of the time-areaconcentration diagram by a well defined geometrical shape such as a rectangle or an isosceles triangle, with a base equal to the time of concentration, Tc . Mathematical expressions of the IUH for the case of rectangular and isosceles triangular shapes of the time-area-concentration diagram were also presented by O'Kelly. Dooge (1959) and O'Kelly (1955) have routed the time-area-concentration diagram through a linear-reservoir to obtain the IUH. However, the IUH so obtained was not used for regeneration either by Dooge or O'Kelly. Four different shapes of the time-area-concentration diagrams were considered: rectangular, isosceles triangular, and triangular shapes with the peaks occurring at the beginning and at the end of the time Tc . In order to apply the dimensionless instantaneous unit hydrographs men-
COMPARISON OF RAINFALL-RUNOFF MODELS FOR URBAN AREAS
339
tioned above for data from actual watersheds, the values of K and T C are to be estimated. The value of K can be estimated from eq. 7. The time of concentration T c is defined as the time required for the surface runoff from the remotest part of the watershed to reach the outlet. In natural watersheds, accurate evaluation of the time of concentration is not possible. Based on the premise that the time of concentration T c is the time taken for the last drop of the excess rainfall to reach the outlet of the watershed, Snyder (1958) suggested that T c can be estimated as the time elapsed between the end of the rainfall and the point of inflection on the recession limb of the hydrograph. To test the regeneration performance of this conceptual model, the parameters K and T c were estimated for each storm by using eq. 7 and Snyder's definition, respectively, and the instantaneous unit hydrographs were derived for all four cases mentioned above. The instantaneous unit hydrographs were then used successively with eq. 1, to c o m p u t e the corresponding outflow hydrographs due to the excess rainfall of the storm. The outflow hydrographs thus obtained for the four cases, were compared with the corresponding observed hydrographs. The results showed that: (1) the peak discharge of the regenerated hydrographs in all the four cases were very nearly the same; and (2) the computed peak discharges of the regenerated hydrographs were less by 5 0 - 1 0 0 % than the peak discharges of the corresponding observed hydrographs.. Thus, the IUH derived by using the value of T c obtained by adopting Snyder's definition did not satisfactorily regenerate the hydrograph of t h e storm from which it was derived. The above results are subjected to the assumption that the storage coefficient K of the linear reservoir is equal to the time lag T L , although as mentioned earlier, the optimum value of K was slightly different from T L . However, the assumption is valid as the storage coefficient K is theoretically equal to the time lag T L . Thus these results indicate that an alternate approach for a better estimation of T c was necessary. As the conceptual model has two parameters, K and T C , and assuming the value of K can be correctly obtained from eq. 7, only the parameter T C remains to be determined in order to obtain an " o p t i m u m " regeneration. Hence for each storm, and " o p t i m u m " value of T c was computed according to the criterion that the sum of the squares of the deviation of the observed and the corresponding computed hydrograph ordinates is minimized. It was observed that the optimum value of T c varied from storm to storm on any watershed. The optimum values of T C were consistently greater than those obtained by Snyder's definition. Typical results of regeneration, obtained by using Snyder's T C and the optimum values of Tc are presented in Fig. 4.
340
P,B.S. SARMA, J.W. DELLEUR AND A.R. RAO
18,q . 7 -
WATERSHED NO. 5 DATE OF STORM , 4 / 2 2 / 1 9 6 5
156.6 -
o •'~ 4130.5
OBSERVED Tc = SNYDER'S Tc Tc = OPTIMUM T¢
-
70 lO~.q-
f-9
n,,CE "7-
L)
O'3
78.3-
C3
,59..2-
26.1
0.0 T"~.
o.oo
2'.1,
4'.m
s'.~
e'.,~
TIME[ HRS)
lo'.s~
-
12'.s~
t~'.Ts.
Fig. 4. Example of regeneration obtained by using the linear-channel-linear-reservoir model.
Fourier
transform method
In all the previous cases the excess rainfall-direct runoff transformation was modelled by lumped linear conceptual systems in which the model structure is specified. An alternative approach in which a system structure is not specified is for example, evaluation of the kernel function in the convolution integral (eq. 1) by the method of integral transforms. The only assumption made in this method is that the excess rainfall--direct runoff process is linear. The fact that the convolution operation in the time domain is equivalent to a multiplication operation in the transformed domain lends convenience to this method. The Fourier transform method of analysis developed by Blank et al. (1968, 1971) was used in the present study. The Fourier transform (FT) of a function f(t) is defined as:
FT[f(t)l = F(co) = y f(t)
e -/~°t d t
COMPARISON OF RAINFALL-RUNOFF MODELS FOR URBAN AREAS
341
and the inverse Fourier transform (FT -1) is: Do
FT-'[F(w)] = f(t) =
(10)
F ( ~ ) ~ ~ ' dco
Applying the Fourier transform to both sides of eq. 1 results in: Q(co) = X(o~) H(o~) or:
H(w ) = O(co )/X(co ) where X(w), Q(co) and H(co) are the transforms of the rainfall excess, of the direct runoff and of the IUH, respectively. The impulse response function (or the IUH) can be obtained by taking the inverse transform of H(c~) using eq. 10. The details of computational procedure can be found elsewhere (Blank and Delleur, 1968; Rao and Delleur, 1971). Typical results of regeneration are presented in Fig. 6 for a large urban watershed. 6.369
~
WATERSHED NO. I DATE
6.q69
OF STORM : 8 / 2 0 / 1 9 6 7
-
,~,
SINGLE LINEAR RESERVOIR MODEL
+
FOURIER TRANSFORM METHOD
A
z
. ~ q.Eq9
-r 3.6q00
e'z r
,-' 2.730(::3 0::: C)
4.÷
! .fRO -
o.g10 -
÷44"4-
0.000
-1-
0.00
. . . . . .
-1
.
.
.
.
.
.
.
I
TIME(MIN)
Fig. 5. Comparison between instantaneous unit hydrograph obtained by the Fourier transform method and by the single linear reservoir model.
342
P.B.S. SARMA, J.W. DELLEUR AND A.R. RAO
866.8 -
WATERSHED NO. 5 DATE OF STORM :10/13/1962
571.6
-
O
OBSERVED REGENERATED
q?S.9 -
381.1t.u rr "1o'J
285.8-
190.5
-
95.9
-
o.o ~-~
0.®
e'.~0
~'.~
~'.,
0'.?, T IME( HRS )
10'.~
---~
......
1s.,8
~;~.~,
Fig. 6. Comparison between the observed hydrograph and the regenerated hydrograph obtained by the Fourier transform method.
EVALUATION OF METHODS OF ANALYSIS
The regeneration performances of the five methods, four of which employ conceptual models and the fifth, the Fourier transform method o f evaluating the kernel function, have been tested. F o r some of the conceptual models, the optimized parameter values were computed according to the several criteria, and were used in the regeneration. The statistical parameters which are defined by eq. 2, 3, and 4 were c o m p u t e d for each set of data tested in order to obtain a quantitative measure of the regeneration performance of the various methods. Based on the values of the statistical parameters the regeneration performance of the various methods were classified according to the ratings shown in Table II. Typical results for some of the watersheds are shown in Table III. The rows in Table III refer to the ratings mentioned in Table II. The columns refer to the different models and to the different methods used for regeneration and also to the statistical measures used to compare the results of regeneration. The num-
COMPARISON OF R A I N F A L L - R U N O F F MODELS F O R U R B A N A R E A S
343
T A B L E II Ratings of the statistical measures Correlation coefficient (R)
Rating
0.99 ~< R < 1.0
excellent
0.95 ~< R < 0.99
very good
0.90 ~< R < 0.95
good
0.85 ~< R < 0.90
fair
0.00 <~ R < 0.85
poor
Special correlation coefficient (Rs)
Rating
0.99 ~< R < 1.0 s 0.95 ~< R < 0.99
very good
0.90 ~< R < 0.95
good
0.85 ~< R < 0.90 s 0.00 ~< R < 0.85
fair
Integral square error ( I S E )
Rating
excellent
S S
poor
S
0% < I S E <<. 3.0%
excellent
3.0% < I S E <~ 6.0%
very good
6.0% < I S E ~ 10.0%
good
10.0% < I S E <<. 25.0%
fair
25.0% < I S E
poor
-
bers in Table III refer to the numbers of storms which satisfy the criterion to qualify for the ratings such as excellent, E, very good, V.G., etc. For example, for watershed no. 1 when the single linear reservoir model (with K = T L ) was used for regeneration, and when the criterion used for comparison between the observed and the regenerated hydrographs was the coefficient of linear correlation, R, seven storms deserved the rating E, whereas thirteen storms deserved the same rating E, when the criterion used for comparison was the integral square error USE). The coefficient o f linear correlation, R, between the observed and the regenerated hydrographs was greater than 0.99 for seven storms whereas the ISE was less than 3% for thirteen storms. The total number of storms satisfying a given rating or a rating superior to it can be obtained by adding the number of storms above the particular rating. For example, for the Nash model used in watershed no. 2 the total number of storms satisfying the criterion F or above was 8 when the statistical measure used for comparison was R, whereas it was 10 when the statistical measures R s and ISE were used, which means
1
3
2
1
4
E
V.G.
G
F
P
1
0
F
P
3
1
2
3
2
0
0
1
0
0
2
3
6
0
0
1
4
1
2
3
1
0
1
1
3
1
2
4
1
0
1
0
6
9
1
8
6
G
2
13
9
7
6
7
E
S
V.G.
R
R
ISE
R
R *4
S
K = Tl
Single linear reservoir model
K = TL
Rating .2
0
1
3
2
5
0
0
1
5
10
ISE
2
5
2
1
1
1
3
3
6
3
R
4
4
1
2
0
1
3
4
6
2
R S
K D = TL/2
1
4
3
2
1
0
0
4
7
5
ISE
Double routing model
3
4
1
2
1
4
0
1
4
7
R
1
4
4
2
0
6
0
0
4
6
R S
Nash model
* 1 Similar results for watersheds 5, 6, 7, 11, 12 and 13 may be found in Sarma et aL (1969). * 2 For explanation of ratings E, V.G., G, F and P, see Table II. *3 Equilateral triangular time-area-concentration diagram. *4 R = linear correlation coefficient (eq. 2). R s = special correlation coefficient (eq. 3). 1SE = integral square error (eq. 4).
Watershed No. * 1
Sample of evaluation of various methods of analysis
TABLE IlI
1
3
4
1
2
0
3
3
2
8
ISE
2
4
2
2
1
2
0
4
5
5
R
1
4
3
3
0
4
0
2
5
5
R S
0
4
3
3
1
1
3
2
3
7
ISE
Linear channelreservoir model *3
0
0
0
2
9
1
0
1
0
14
R
S
0
0
0
2
9
1
0
1
0
14
R
0
0
0
1
10
1
0
0
1
14
ISE
Fourier transform method
> ©
>.
> Z
t"
>
COMPARISON OF R A I N F A L L - R U N O F F MODELS FOR URBAN AREAS
345
that there were 8 storms for which the value of R was greater than 0.85 whereas there were 10 storms for which ISE was less than 25%. DISCUSSION OF THE RESULTS
The Fourier transform method is perhaps the most general method of obtaining the response function (IUH) of linear systems, whereas the use of conceptual models to represent linear systems is neither unique nor general. For some of the data tested, the impulsive response functions obtained by the Fourier transform m e t h o d exhibited high frequency oscillations. As the hydrologic systems are stable, their lUll or impulsive response are expected to be stable also. Similar oscillations have been reported by Blank et al. (1971). Methods of controlling these computational instabilities by means of the proper choice of the discretization interval and by digital filtering have been discussed by Delleur and Rao (1971a, b) and Rao and Delleur (1971). The kernel functions obtained by the Fourier transform method for small watersheds were of the exponential decay type as shown in Fig. 5. The instantaneous unit hydrographs which result from using a single linear reservoir model, are, by definition (eq. 5) of the same type. This concurrence of the response functions obtained by the general method and by the single linear reservoir model substantiates the use of the single linear reservoir model in the analysis of small urban watersheds of area less than about 5 sq.miles. For larger watersheds (greater than, say, 5 sq.miles in area) the choice of a particular model which can be used to simulate the rainfall-runoff process was not so easy. By an examination of the regeneration performance, it is evident that if at all possible, use of response functions obtained by the Fourier transform methods would be the best choice. In the double routing model, there was considerable ambiguity in the selection of an appropriate value of the storage coefficient K D . There is no unique value of K D that can be obtained by considering any portion of the recession curve, as assumed by this model. Apart from this difficulty, this method is but a special case of the Nash model, with n = 2. Whereas in the Nash model, the values of both n and K N can be determined to give a better regeneration, in the double routing method this freedom is lost without any advantage gained. In the analysis using the linear-channel linear-reservoir model the parameter T c was evaluated by the m e t h o d suggested by Snyder. Although the results were slightly improved when the parameter T c was optimized such that the sum of the squares of the differences between the corresponding ordinates of the observed and the regenerated hydrographs was minimized, the regeneration was not as good as that obtained by the Nash model. Thus, among the four conceptual models considered for analysis of data from
346
P.B.S. SARMA, J.W. DELLEUR AND A.R. RAO
larger watersheds, the Nash model yielded a relatively better regeneration performance. CONCLUSIONS
(1) The linear system methods can be usefully employed in the analysis of direct runoff from urban watersheds. The methods of linear system analysis are relatively more accurate for smaller watersheds (less than about 5 sq.miles). (2) The estimation of the response function by the Fourier transform method and its use in prediction is very promising. On the other hand, the nonlinear behavior of larger watersheds may be so pronounced that no linear model, however sophisticated it might be, could give accurate results. (3) For small watersheds, (less than 5 sq.miles approximately) the response functions computed by the Fourier transform method and the instantaneous unit hydrographs obtained by using the single linear reservoir model were similar. This similarity supports the use of the single linear reservoir model in small urban watersheds. (4) The regeneration performance of the single linear reservoir model for smaller watersheds can be improved by suitably optimizing the value of the parameter K (5) For watersheds of area larger than about 5 sq.miles, of all the conceptual models tested, the Nash model gave relatively the best regeneration and hence can be used for simulating the rainfall-runoff process in such watersheds. However, the regeneration performance of the Nash model, by itself was not very good. This is perhaps due to the fact that for larger urban watersheds the hydraulics of the sewer system which is strictly not represented in the Nash model predominates over the overland flow phase of the runoff cycle. This suggests that further investigation is necessary to select a suitable model to simulate the rainfall-runoff process in larger urban watersheds. ACKNOWLEDGMENTS
This paper is based on the doctoral dissertation of Mr. P.B.S. Sarma, "Effects of Urbanization on Runoff from Small Watersheds" submitted to Purdue University in January 1970. The support of Dr. J.F. McLaughlin, Head, School of Civil Engineering, Purdue University, in completing the research reported herein is gratefully acknowledged. The District Offices of the U.S. Geol. Surv. at Indianapolis, Indiana, and Austin, Texas, are thanked for their ready help in data collection. The research was supported as a cooperative project by the OWRR under project OWRR-B-002-IND, the Indiana Department of Natural Resourches, and Purdue
COMPARISONOF RAINFALL-RUNOFFMODELSFOR URBAN AREAS
347
University from July 1966 to June 1969 and it was further supported by Purdue University until September 1969. We would like to thank our sponsors for their support.
REFERENCES Blank, D. and Delleur, J.W., 1968. A program for estimating runoff from Indiana watersheds, Part I: Linear system analysis in surface water hydrology and its applications to indiana watersheds. Purdue Univ. Water Resour. Res. Cent., Tech. Rep. 4. Blank, D., Delleur, J.W. and Giorgini, A., 1971. Oscillatory kernel functions in linear hydrologic models. WaterResour. Res., 7 (5): 1102-1117. Clark, C.O., 1945. Storage and the Unit Hydrograph. Trans. Am. Soc. Cir. Eng., 110: 1419-1446. Delleur, J.W. and Vician, E.B., 1966. Discussion on the paper "Time in urban hydrology" by G.E. Willeke. Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 92 (HY5): 243-251. Delleur, J.W. and Rao, R.A,, 1971a. Linear systems analysis in hydrology - the transform approach, the kernel oscillations and the effect of noise. In:Systems Approach to Hydrology. Proc. 1st Bilat. U.S. Japan Sem. HydroL, Univ. HawaiL Honolulu, January 11 17, 1971. Water Resour. PubL, Fort Collins, Colo., pp. 116-142. Delleur, J.W. and Rao, R.A., 1971b. Characteristics and filtering of noise in linear hydrologic systems. Proc. Int. Syrup. Math. ModelsHydroL, Warsaw, Poland, July 2 6 - 3 1 , 1971. Vol. 2(1), Pap. 4/13, 14 pp. Diskin, M.H., 1964. A basic Study o f the Linearity o f the R a i n f a l l - R u n o f f Process in Watersheds. Thesis, University of lllinois, Urbana, Ill. Dooge, J.C.I., 1959. A general theory of unit hydrograph. J. Geophys. Res., 64(1): 241-256. Eagleson, P.S. and March, F., 1965. Approaches to the linear synthesis of urban runoff systems. Hydrodvn. Lab. Rep. 85. M.I.T., Cambridge, Mass. Holtan, H.N. and Overton, D.E., 1964. Storage flow hysteresis in hydrograph synthesis. J. tlydroL, 2: 309-323. Nash, J.W., 1959. Systematic determination of unit hydrograph parameters. J. Geophys. Res., 64( 1): 111 115. O'Kelly, J.J., 1955. The employment of unit hydrographs to determine the flows of Irish arterial drainage channels. Proc. Inst. Civ. Eng., 4(3): 365-412. Rao, R.A. and Delleur, J.W., 1971. The instantaneous unit hydrograph: its calculation by the transfornr method and noise control by digital filtering. Purdue University, Water Resour. Res. Cent., Tech. Rep. 20. Sarma, P.B.S., Delleur, J.W. and Rao, A.R., 1969. A program in urban hydrology, part II: An evaluation of rainfall-runoff models for small urbanized watersheds and the effect of urbanization on runoff. Purdue Water Resour. Res. Cent., Tech. Rep. 9 : 2 4 0 pp. Snyder, F.F., 1958. Synthetic flood frequency. Proc. Am. Soc. Cir. Eng., J. Hydraul. Div., 84 (HY5): 1808-1-1802-22. U.S. Geological Survey, 1962-1966. Compilation o f Hydrologic Data, Waller and Wilbarger Creeks, Colorado River Basin, Texas. Rep. Water Resour. Div., Texas Dist., Austin, Texas. Viessman Jr., W., 1966. The hydrology of small impervious areas. Water Resour. Res., 2 (3): 4 0 5 - 4 1 2 . Viessman Jr., W., 1968. Runoff estimation for very small drainage areas. Water Resour. Res., 4 (1): 87-93. Willeke, G.E., 1962. The prediction of runoff hydrographs for urban watersheds from precipitation data and watershed characteristics. J. Geophys. Res., 67 (9): 3610. Wu, I.P., Delleur, J.W. and Diskin, M.H., 1964. Determination o f Peak Discharge and Design Hydrographs Cbr Small Watersheds in Indiana. Hydromech. Lab., School Cir. Eng., Purdue Univ., Lafayette, Ind., 106 pp.