The unit hydrograph model for hydrograph separation

~

Pergamon

Environment International, Vol. 21, No. 5, pp. 509-515, 1995 Copyright 01995 Elsevier Science Ltd Printed in the USA. All rights reserved 0160-4120/95 $9.50+.00

0160-4120(95)00050-X THE UNIT HYDROGRAPH MODEL FOR HYDROGRAPH SEPARATION

Ninghu Su Landcare Research New Zealand Ltd., Hamilton, New Zealand

E19502-132 M (Received 14 February 1995; accepted 29 May 1995)

Procedures are presented for baseflow separation based on the unit hydrograph theory. When a mathematical model is used to represent the baseflow unit hydrograph, the parameters derived from a recession limb of the baseflow hydrograph can be further used to determine the rising limb of the baseflow hydrograph. The simple procedures involve estimation of the two parameters of the model using discharge data with simple equations. Once parameters have been estimated, the mathematical model for an entire baseflowhydrograph can be used to determine both the temporal processes and total quantity ofa baseflowhydrograph. Using this method the separated componentof baseflowduring study periods is about 63% of the total storm runoff from two small rural catchments in Australia.

studying baseflow processes can be justified by several earlier studies. Eagleson (1970) showed the general applicability of the kinematic-wave method to the idealised subsurface as well as surface contribution to streamflow. This implies that the temporal variation in baseflow can be examined by a method similar to that used for examining surface flow. Jakeman at al. (I 990) have combined a unit hydrograph and a filtering algorithm for baseflow separation. In subsequent discussions, the unit hydrograph concept is first extended and then a mathematical model, the Nash's model, is used to represent the unit hydrograph. The Nash's model, which was originally developed and traditionally used for surface flow, is selected in this paper for this purpose. It is used to examine both the overall temporal processes and total quantity of a baseflow hydrograph.

INTRODUCTION

The unit hydrograph, originally named unit-graph, was first proposed by Sherman (1932), who classified runoff into surface and groundwater runoff and defined the unit hydrograph for use only with surface runoff. From this beginning, it has been extensively studied and used in hydrologic computation over the past six decades. As a hydrologic model, it falls into the category of the deterministic lumped unsteady flow model as classified by Chow et al. (1988) who also detailed the evolution of the concept, definition, and application of the model. A baseflow hydrograph is usually produced by separating the different components of a storm hydrograph using various methods, such as those used by Chow et al. (1988). The unit hydrograph model, traditionally used for studying overland flow, can be extended to study baseflow. The extension of the unit hydrograph theory to 509

510

N. Su

flow ceased. The difference between the lag of observed total runoffand the starting time ofbaseflow is the lag of baseflow with respect to total streamflow. THE lUll MODEL FOR BASEFLOW HYDROGRAPHS

Since the unit hydrograph is a model for a linear hydrologic system, its solutions follow two basic principles for linear system operations, namely the principles of proportionality and superposition. As one of its transfer functions for the linear system, the instantaneous unit hydrograph (IUH) is extended to examine baseflow and used in the following discussions. The reader is referred to Chow et al. (1988) for more details. The principle involved in this approach is that an impulse input from the infiltrated water to subsurface flow produces an impulse response output. This is analogous to the principle of the IUH which reasons that an impulse input from rainfall produces an impulse response in streamflow. I I I

t=O Tit

I I t

Tlb

I I

I I

I

,

Ix

tpb

Nash's IUH model for baseflow unit hydrographs t

Fig. 1. Sketch for defining storm hydrograph parameters.

DEFINITIONS USED IN THE SUBSEQUENT DISCUSSIONS

In order to clarify the various concepts, Fig. 1 is used to define the parameters used in this paper. L~ = initial loss; K¢ = average continuing loss rate; t, = duration of effective rainfall; tr = time when the effective rainfall ceases; tc = time of concentration of the overland flow generated from the effective rainfall during the final period of a rainfall event; Point A = discharge at the time when rainfall starts, t = 0; Point B = time at which the stream hydrograph starts to rise; Point C = time at which the baseflow hydrograph starts to rise; Point D = stream flow discharge at time t~ when overland flow ceases; Ttt = lag time of the total storm hydrograph with respect to the starting time of rainfall; and Tlb = lag time of the baseflow hydrograph with respect to the starting time of rainfall. Based on the above convention, the procedures for baseflow separation is dieussed: The starting point of baseflow is defined by subtracting the duration of recharge from the time when overland

The IUH is a theoretical hydrograph resulting from an excess rainfall of unit amount applied over a catchment with an infinitesimal small duration. Nash's (Nash 1957) linear reservoir model is among the best of the linear IUH models of this kind. In Nash's model, a catchment is represented by a series of e identical linear reservoirs each having the same storage constant kl. The output from the first reservoir in this model constitutes the inflow to the second reservoir which is further used as the inflow to the third reservoir. Continuation of this procedure with appropriate mathematical operations yields the outflow from the e-th reservoir as U(i) -

I

kiF(e)

r t I e'i

k~J

I- t l expt~J

(1)

where U{i~is the IUH ordinate; e and k~ are the model parameters; l'(e) is the Gamma function of e; r'(e) = (e - 1)! for integers; and for decimals, r(e) can be determined by tables. Following the similar procedures for deriving Eq. 1, the IUH can also derived for baseflow as 1

] - t l p'I

tz j where

Unit hydrograph model for hydrograph separation

511

U(o is the baseflow unit hydrograph ordinate; p and k2 are the model parameters; r'(p) is the Gamma function of p. In Eq. 2, the two parameters, k2 and p, need to be determined.

and

(11)

Qb3= B3h°exp ('B2h)

Solving Eqs. 9, 10, and 11 for B2, B3, and 0 gives,

(

,.rgm

0 = ln[Qb3Qb2"]" (t2"tl) ""L Qb2J Parameter estimation

ln[~]

(t3"t2), rt_j.1

In order to evaluate the parameters, according to the principle of proportionality of a linear system (Linsley et al. 1982; Chow et al. 1988), Eq. 2 can be written Qb = R~t B, t o exp (-B2t)

B2 =

O)

where Qb represents actual baseflow discharge; Rg is the total depth of baseflow; 0 = p- 1

k2F(p)

t2 - t 1

Qb2- Qbl t2°exp[-B2 t2] - h°exp[-B2 h]

B2 =

1

In [Qb212

(5) (6)

0, B i, and B 2 are also parameters for the actual baseflow hydrograph. Equation 3 may be further written Qb =B3 to exp(-B2 t)

(7)

(14)

If the sampling time intervals are equal, that is At = h t, and At = h - h , then Eq. 12 can be simplified to

O=

and

(13)

,,Zb.2

and /33 =

(4)

(12)

- (t2.h) mtt2-1

QblQb3

(15)

ha[tz]2 tlt3

To make sure that the three readings of baseflow from these three points on a baseflow recession curve represent storm baseflow only, the readings for these three points should be taken from the recession limb of the hydrograph after time tpb, which are detailed in the following section. Time of concentration

where =

R, BI

(8)

Since Eq. 7 is an expression for actual baseflow, its parameters can be directly evaluated by fitting the equation to the readings of the recession limb of a baseflow hydrograph. The simplest way to evaluate the parameters is to make use of the following procedures. Because there are three unknowns in Eq. 7, three equations are needed to derive these three parameters. The three parameters can be derived in the following way: after time tpb, which isthe time required for 'the last drop of effective rainfall' to reach the catchment outlet (the time of concentration is used as the criterion), given three readings on the recession limb of a hydrograph, Qbl, Qb2, and Qb3, at times h, h, and h, respectively, one can write three equations for the three points Qbl = B3 tlOexp('B2h)

(9)

Qb2 = B3 t2Oexp(-Bzh)

(10)

There is a number of formulae for time of concentration. Here, a formula developed by Kadoya and Fukushima (1977) is used, which is based on the kinematical wave theory and physiographic characteristics of the slope-channel network in a catchment. It is physically based and is also simple in form for practical use. The equation is: t, = CAt°22 ro° " (16) where A t is the catchment area in km2; ro is the effective rainfall intensity in mmh~; and C is a constant equal to 290 for a mountainous catchment and 190~210 for a grazing area. This model was tested in natural catchments ranging in size from 0.5 km 2 to 143.0 kmL An average of 200 for C is used in the calculations for this analysis as the study catchment has been grazed for many decades. The effective rainfall is determined using the following procedures (Fig. 1):

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Other procedures for estimating the parameters Li = Pt

t < Tit

(17)

Lc = Pt, - Rt

t > Ta

(18)

Kr = L c te

t > T a,

(19)

where Pt is the gross rainfall before the time to rise of the streamflow hydrograph, mm; Pte is the total rainfall after the time to rise of the hydrograph, mm; Lc is the continuing loss, mm; Kr is the average continuing loss rate, mmht; R~ is the total storm runoff with antecedent baseflow removed, ram; and te is the duration of effective rainfall, h, and to= t,- Tit, h. Then, the effective rainfall intensities can be evaluated: r e = r - K,

(20)

where r is the rainfall intensity. Using the above procedures, the two parameters derived from observed hydrographs in Sherwood catchment, ACT, Australia are listed in Table 1. Tablel. P~am~rsof~ebaseflow unit hydrograph model. Da~

k2

p

11 April 13 April 18 April 20 April 22 May 24 May 30 May 26 June 1 July 4 July 5 Aug. 12 Sept.

0.956 1.812 2.506 0.957 6.258 0.259 3.198 1.823 2.257 0.971 4.615 0.316

13.980 4.080 4.601 6.963 1.085 1.011 2.947 3.267 9.642 41.320 4.358 29.900

7 June 9 June 11 June 12 June 8 July 9 July 10 July 11 July 5 Aug. 7 Aug.

2.148 0.963 4.070 5.513 7.452 2.044 2.336 0.841 2.397 9.206

2.133 19.442 1.769 1.823 1.163 3.484 8.883 2.410 1.386 2.091

1990

1991

There are alternative methods for estimating the parameters. One of the options is to estimate the parameters by rearranging Eq. 7 to fit the data. Eq. 7 can be arranged as Qbexp(Bzt) = ~ t o

(21)

which can be rearranged to give lnQb+B2t = lnB3+01nt

(22)

For the two equations to be equal, both sides of Eq. 22 must be equal to the same constant, which is taken as A, i.e., A = lnQb+B2t (23) or lnQb = A-B2t (24) Equation 24 is a simple linear equation and its constants A and B2 can be estimated from streamflow data using least square procedures. Once B~ is determined, Eq. 21 can be used to estimate B3 and 0, i.e., ln[Qb exp(B2 t)] = lnB3 + 0t

(25)

The stability of baseflow parameters could be improved by using this procedure since the fitting procedure involves all the data taken from the recession hydrograph after time tpb.

Three components of a storm hydrograph In this paper, only the first procedure (the three-point method in the section containing Eqs. 3-15) is used to derive parameters. As far as the sampling interval and the time are concerned, there should be no difference in the values of the parameters resulting from the sampling. There are, however, slight differences in the parameters because of the effects of the irregular rainfall bursts and evapotranspiration during the recession. Theoretically speaking, the parameters of the equations for the unit hydrograph should be the same as those for actual baseflow hydrographs from which the unit baseflow hydrograph is derived. As an example, the baseflow hydrographs constructed using these approaches are depicted in Fig. 2. It can be seen from this figure that the different sampling time intervals used in taking readings from the same recession limb cause minor deviations between the hydrographs but generally, these curves fit the data

Unit hydrograph model for hydrograph separation

"7

513

Qt •............... Qb2 ............ QblO ........ Qb20

°' t 0.6

E

0.4

1 ~,

0.2 0.0 ~ 0

,r.,

I

5

I

10

15

I

I

|

I

|

I

I

20

25

30

35

40

45

50

Time, h Fig. 2. Baseflowseparationusing the unit hydrographmodelfor runoffcommencingat 22:30 on 13 April 1990, Sherwoodcatchment,ACT, Australia. The subscriptsin numbersreferto samplingtime intervalsin hours. 0.8 ,-a

0.7 0.6

-

0.5

.....

0.4

......... Qb

Qt

-

Qs

0.3 0.2 0.1 0

'

0

I

5

'

I

10

'

I

'

15

I

20

'

t

25

'

I

30

'

I

35

'

I

40

'

I

45

'

I

50

Fig. 3. Three components of flow separated using the unit hydrograph model for runoff commencing 22:30 on 13 April 1990 (sampling interval is 2 h).

well. The three components of a storm hydrograph separated using the unit hydrograph method are depicted in Fig. 3. It is obvious that the three components of a hydrograph flow accord with both the rising and receding limbs and are characterised with inflections points.

Relationships between baseflowparameters There are several interrelationships between the observed and computed parameters of hydrographs. Once physically-based models have been developed, these interrelationships between model parameters can be evaluated. Equation 2 is used to examine the time to peak of the baseflow after the parameters have been evaluated. To obtain the time to peak and peak discharge of the unit hydrograph as represented by Eq. 2, differentiating Eq. 2 with respect to t and equating its result to zero yields tpu = (p - 1)k 2

(26)

Inserting Eq. 26 into Eq. 2, yields the peak discharge of the unit hydrograph of baseflow 1 ) [p-l] p-1 exp[- (p-l)] U(t)max - k21_,(p

(27)

Equation 26 indicates that the parameters k2 and p are interrelated. Further, Eq. 27 gives the peak discharge of the baseflow unit hydrograph once the model parameters have been derived. The two parameters derived using data have been listed in Table 1. Another important relationship exists between the time to peak of the baseflow hydrograph, tpu, calculated using Eq. 26 and tpb. the time when the recession limb of a hydrograph comprises only baseflow, as described earlier. Analysis shows that the calculated time to peak of the unit hydrograph baseflow, tpu, is shorter than tpb as shown in Fig. 4 indicating that if the unit hydrograph model is regarded as the representative of the true baseflow processes, baseflow hydrograph would peak before the end of the overland flow.

514

N. Su

80" 1:1LineJ

O

o

60'

"~

40

°,j/ o

o

20

0

0

I

I

I

I

I

I

i

I

10

20

30

40

50

60

70

80

Fig. 4. The relationship between assumed time to peak of baseflow tpb and ~ .

The equation that best fits the data in Fig. 4 is in the form of tp, = 0.538tpb - 0.902 (28) In Eq. 28, tpb has been calculated from the following formula according to definitions at the beginning of this paper tpb = t~+ tc

(29)

where to is the duration of effective rainfall and to is the time of concentration. Analysis is also made of the relationship between tpu and to. The analysis indicates that the baseflow unit hydrograph peaks before cessation of effective rainfall, that is tpu is shorter than to. For the actual baseflow hydrographs, tpb is longer than the duration of effective rainfall, to. This relationship is represented by an equation of the form tpb = 7.83 + 0.68t~

(30)

Based on the two results, it is clear that if the unit hydrograph model is regarded as a representative of the baseflow hydrograph, the baseflow peaks before the end of effective rainfall and prior to cessation of overland flow. Total depth of baseflow

After readings are taken and0, B=, and B 3are determined from Eqs. 12 to 15,the parameters p, BI, and k~ can be derived easily from Eqs. 4, 5, and 6, respectively. Then, the total depth ofbaseflow is computed using Eq. 8. Further,

Further, one can derive the various relationships between different parameters. The equation fitted to the data shown for the relationships between the total runoff and baseflow calculated using this procedure is Rg= 0.659Rt - 1.53

R 2= 0.748

(31)

which indicates that within the data set, about 63% of streamflow is baseflow. CONCLUSIONS AND DISCUSSIONS

In this paper, simple procedures for baseflow separation by extension of the instantaneous unit hydrograph theory to baseflow processes were introduced. The procedure can be essentially described as a baseflow unit hydrograph model. Nash's cascade reservoir model is used to represent mathematically the baseflow unit hydrograph with its parameters derived from a recession limb of the baseflow hydrograph. The parameters can then be used to determine the rising limb of the baseflow hydrograph. The two parameters of the model can be estimated from discharge data using simple equations. Once the parameters have been estimated, the mathematical model for an entire baseflow hydrograph can be used to determine both the temporal processes and total quantity of baseflow. The time of concentration of overland flow after cessation of rainfall is used as the criterion for timing the end of overland flow. It is accepted in formulating the model that at time tpb, i.e. after the travel time of the overland flow of the 'last drop of effective rain', the overland flow ceases. This procedure ensures that the overland flow from a given storm event is completely

Unit hydrograph model for hydrograph separation

drained and that the start time of baseflow recession for analysis would be more reliable. Following this convention, data on the recession limb of a hydrograph are used to derive parameters. The baseflow hydrographs constructed with the unit hydrograph theory are reasonably characterised with inflection points on both the rising and receding limbs of the baseflow hydrographs. The procedures developed for separating simple baseflow hydrographs can also be used to separate complex hydrographs by defining tpb and estimating model parameters for each sub-hydrograph. As an example, the separated component of baseflow using this method is about 63% of the total storm runoff during study periods for 22 hydrographs from two small rural catchments in ACT, Australia.

515

REFERENCES Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied hydrology. Singapore: McGraw-Hill; 1988. Eagleson, P. S. Dynamic hydrology. New York, NY: McGraw-Hill; 1970. Jakeman, A. J.; Littlewood, I. G.; Whitehead, P. G, Computation of the instantaneous unit hydrograph and identifiable component flows with application to two upland catchments. J. Hydrol. 117: 275-300; 1990. Kadoya, M.; Fukushima, A. Concentration time of flood runoff in smaller river basins. In: Morel-Seytoux, H. J.; Salas, J. D.; Sanders, T. G.; Smith, R.E., eds. Surface and subsurface hydrology. In: Proc. Fort Collins 3rd int. hydrology symposium on theoretical and applied hydrology, held at the Colorado State Univ., Fort Collins, Co., USA, July 27-29, 1977. Littleton, CO: Water Resources Publication; 1979: 75-88. Linsley, R.K.; Kohler, M. A.; Paulhus, J.L.H. Hydrology for engineers. New York, NY: McGraw-Hill; 1982. Nash, J.E. The form of the instantaneous unit hydrograph. In: Proc. General Assembly, Toronto. IASH 3:114-121; 1957. Sherman, L.K. Streamflow from rainfall by the unit.graph method. Eng. News Rec. 108: 501-505; 1932.