A comparison of algorithms for stream flow recession and baseflow separation

A comparison of algorithms for stream ¯ow recessionand base¯ow separation

Tom Chapman*School of Civil and Environmental Engineering, The University of New South Wales, Sydney 2052, Australia

Abstract:Simple hydraulic models for out¯ow of groundwater into a stream suggest that the form of the storage±discharge relationship for groundwater changes from linear, for a con®ned aquifer, to quadratic, for an

uncon®ned ¯ow. Tests of the form of stream ¯ow recessions in 11 streams, during periods of no recharge, showthat for most catchments the storage±discharge relationship is more strongly non-linear than the quadraticform. However, for the commonly occurring case of recessions of duration up to about 10 days, the linearmodel remains a very good approximation, using a biased value of the groundwater turnover time. In contrast,

estimates, from the stream hydrograph, of recharge during a storm event are very sensitive to the form of thestorage±discharge relationship. The results of this study also show great variability in the parameters of therecession algorithm from one recession to another, attributable to spatial variability in groundwater recharge.

An extension of the linear model to `leaky' catchments, where the recession reaches zero ¯ow, has been tested ontwo data sets.The second part of the paper deals with algorithms for base¯ow during surface runo� events Ð the problem

of hydrograph separation. Algorithms with one, two and three parameters have been compared, using data forthe same 11 streams, and the results show signi®cant di�erences in the base¯ow index (BFI) predicted forsome catchments. The two-parameter algorithm, which is ®tted subjectively, is more consistent in providingplausible results than either the one- or three-parameter algorithms, both of which can be ®tted objectively.

Copyright # 1999 John Wiley & Sons, Ltd.

KEY WORDS hydrograph; base¯ow; recession; separation; groundwater; algorithms

INTRODUCTION

The characteristics of ¯ow in perennial streams during extended dry spells have long been recognized asdi�erent from those experienced during and following storm rainfall. Since the development of modernhydrology, these di�erences have been studied in contrasting ways by engineering hydrologists, systemsanalysts and scienti®c hydrologists.

The engineering hydrology approach has resulted from the need to develop models for the ¯ood runo�from storm events. Here the underlying dry weather runo� (or `base¯ow') is subtracted from the total stream¯ow, and the di�erence (called `direct runo�' or `storm runo�') is related to the causative rainfall. Thebase¯ow is generally regarded as being a result of groundwater discharging into the stream, while the directruno� is considered to result from overland or near-surface ¯ow.

In the systems analysis approach, an attempt is made to develop a rainfall±runo� model from the time-series of rainfall and stream ¯ow. Jakeman and Hornberger (1993) have shown that, after applying a non-linear loss function to the rainfall data, the response of a wide range of catchments is well represented by a

CCC 0885±6087/99/050701±14$17�50 Received 5 January 1998Copyright # 1999 John Wiley & Sons, Ltd. Revised 1 July 1998

Accepted 11 August 1998

HYDROLOGICAL PROCESSESHydrol. Process. 13, 701±714 (1999)

*Correspondence to: Dr T. Chapman, School of Civil and Environmental Engineering, The University of New South Wales,Sydney 2052, Australia.

linear model with two components, interpreted as de®ning a `quick ¯ow' and `slow ¯ow' response to the®ltered rainfall.

Scienti®c hydrologists, interested in understanding the processes by which rainfall becomes stream ¯ow,have used tracers to determine the time-varying proportion of òld ¯ow' and `new ¯ow' in the streamhydrograph. Here the old ¯ow is identi®ed as being water that was already in the catchment before the startof rainfall, while the new ¯ow has the same quality characteristics as the incoming rainfall. Thus it would beexpected that the old ¯ow would result from subsurface pathways, while the new ¯ow would be identi®edwith surface runo�. However, the result of this work has been the recognition that the old ¯ow has many ofthe characteristics of quick ¯ow, and although it can be modelled by algorithms used for base¯ow separa-tion, selection of parameter values requires experimental data from tracer experiments (Chapman andMaxwell, 1996).

Linear models underlie both the usual engineering equations for base¯ow recession during non-rechargeperiods and the systems analysis of slow ¯ow. The ®rst part of this paper explores the extent to which theassumption of linearity is valid. An extension of the linear model is then applied to the case of a leakycatchment, where stream base¯ow may be zero in extended dry periods.

The second part of the paper deals with the behaviour of base¯ow during a storm runo� event, and makescomparisons between the separation algorithms used by engineering hydrologists and the hydrographs ofslow ¯ow obtained by the systems analysis approach.

MODEL DEVELOPMENT

Recession curves

The equation most used for base¯ow during a non-recharge period is

Qt � Q0 eÿt=t � Q0k

t �1�

where Q0 ,Qt are the ¯ows at times 0 and t, t is the turnover time of the groundwater storage and k is therecession constant for the selected time units. The ®rst, and physically more meaningful, form of Equation (1)was based on the analysis by Boussinesq (1877) of ¯ow in aquifers. Its ®rst application to stream ¯ow data isusually credited to Maillet (1905), although Horton (1933) claims to have applied it to several streams in1904. The second form, more familiar to engineering hydrologists, was popularized by Barnes (1939).

Equation (1) is readily shown to be the result of a linear storage, in which the groundwater storage S isrelated to the stream ¯ow Q by

Q � S=t � aS �2a�

where a � 1=t. Linear behaviour of groundwater in a con®ned aquifer of constant thickness would beexpected a priori from the Darcy equation, and Werner and Sundquist (1951) showed that Equation (1) canbe derived from the equation for one-dimensional ¯ow in such an aquifer. This may also be regarded as areasonable approximation for uncon®ned ¯ow when the underlying impermeable layer is well below thestream bed, resulting in little spatial variation of ¯ow depth (Figure 1a).

For shallower bedrock, the spatial variation in the groundwater ¯ow depth must be taken into account.The vertical plane analysis by Werner and Sundquist (1951), for the case where the stream bed intersectsimpermeable bedrock (Figure 1b), was shown by Chapman (1963) to infer that the ¯ow would beproportional to the square of the volume of groundwater storage

Q � aS2 �2b�

Copyright # 1999 John Wiley & Sons, Ltd. HYDROLOGICAL PROCESSES, VOL. 13, 701±714 (1999)

702 T. CHAPMAN

These results can be generalized into the non-linear relationship (Coutagne, 1948)

Q � aSn �2c�

Combining Equation (2c) with the water balance equation

Q � ÿ dS

dt�3�

results in the recesssion equation

Qt � Q0�1 � �n ÿ 1�t=t0�ÿn=�nÿ1� �4�

where t0 � S0=Q0 is now the turnover time at time 0. Equation (1) is the special case of Equation (4) whenn � 1. Boughton (1986) obtained a value of n � 1.78 for a small catchment in Queensland, while Padillaet al. (1994) documented three karst springs with n equal to 1.5 or 2.

Extension to leaky catchments

The above equations are based on the assumption [Equation (3)] that all the groundwater storage in thecatchment is released as stream ¯ow recorded at the gauging station. This is not necessarily the case, and it isproposed that the appropriate model for a leaky catchment could be as shown in Figure 2, where a lineargroundwater storage feeding the stream overlies a storage that has its out¯ow outside the catchmentboundaries.

Figure 1. Schematic cross-sections from drainage divide to stream channel: (a) little variation in ¯ow depth H; (b) signi®cant variationin ¯ow depth

Figure 2. Linear storages model for a leaky catchment


5: STREAM FLOW RECESSION 703

While there is base¯ow, the upper storage Ss 4 0, and the lower storage Sd is constant. Under theseconditions, it can be shown (see Appendix) that the recession equation for Qt � Qs takes the form

Qt � q � �Q0 � q�eÿt=t* �5�where q � Sd=�ts � td� and t* � tstd=�ts � td�. The ¯ow becomes zero at time

t � t* ln�1 � Q0=q� �6�Boughton (1995) obtained an equation of the same form as Equation (5) by regarding the ¯ow q as a

transmission loss in the stream bed, resulting from evapotranspiration.Figure 3 shows that there is little obvious departure from linearity of the graph of log Qt against t for short

recession curves of the form of Equations (4) and (5).

Base¯ow separation algorithms

Separation of base¯ow from a stream hydrograph starts with identifying the points at which the directruno� starts and ends. The start point is readily identi®ed as the time when the ¯ow starts to increase, whilethe end-point is usually taken as the time when a plot of log Q against time becomes a straight line.

Having established the end-points for the separation, a wide range of graphical techniques is availablefor de®ning the base¯ow between these points (Dickinson et al., 1967). These techniques are inconvenientwhen separations are to be undertaken on a long continuous record of stream ¯ow, rather than just afew storm period hydrographs, and this has led to the development of numerical algorithms, as describedbelow.

The one-parameter algorithm. Lyne and Hollick (1979) appear to have been the ®rst to suggest the use of adigital ®lter, which has been used in Australia (Chapman, 1987; Nathan and McMahon, 1990; O'Loughlinet al., 1982) and overseas (Arnold et al., 1995). Chapman (1990, 1991) showed that the form of the®lter implied that the base¯ow would be constant when there was no direct runo�, and proposed areformulation, which was subsequently simpli®ed (Chapman and Maxwell, 1996), to a form that is based on

Figure 3. Comparison of Equations (1) (linear model), (4) (non-linear model with n � 2) and (5) (leaky catchment model), fort � t0 � ts � 30 days, td � 100 days, Sd � 10 mm


704 T. CHAPMAN

the base¯ow being a simple weighted average of the direct runo� and the base¯ow at the previous timeinterval, i.e.

Qb�i� � kQb�i ÿ 1� � �1 ÿ k�Qd�i� �7�

where Qb�i� and Qd�i� are the base¯ow and direct runo�, respectively, at time interval i, and the parameter kis the recession constant during periods of no direct runo�. As the total stream ¯ow Q is the sum of thebase¯ow, Qb, and direct runo�, Qd, the latter can be eliminated to give the recession algorithm

Qb�i� �k

2 ÿ kQb�i ÿ 1� � 1 ÿ k

2 ÿ kQ�i� �8�

subject to

Qb�i�4Q�i� �8a�

The Boughton two-parameter algorithm.More ¯exibility is provided by introducing a second parameter, C,in place of (1ÿ k) in Equation (7). The separation algorithm then becomes

Qb�i� �k

1 � CQb�i ÿ 1� � C

1 � CQ�i� �9�

[subject also to Equation (8a)], which is the algorithm used in the AWBM model developed by Boughton(1993).

The IHACRES three-parameter algorithm. In the usually chosen form of the linear module of theIHACRES model (Jakeman and Hornberger, 1993), the rainfall excess u is partitioned into slow and quickcomponents, here called Qb and Qd, by the equations

Qb�i� � bsu�i� ÿ asQb�i ÿ 1� �10a�Qd�i� � bqu�i� ÿ aqQd�i ÿ 1� �10b�

where the a's and b's are parameters, and the su�xes q and s refer to quick and slow ¯ow, respectively. Itshould be noted that the a's are negative.

Eliminating u from these equations, and expressing the direct runo� as the di�erence between the base¯owand the total stream ¯ow, results in the following equation for base¯ow

Qb�i� � ÿasbq � aqbsbq � bs

Qb�i ÿ 1� � bsbq � bs

�Q�i� � aqQ�i ÿ 1��

It should be noted that the Q in this equation is strictly the modelled ¯ow rather than the observed ¯ow usedin separation algorithms such as Equations (8) and (9).

By putting C � bs=bq and k � ÿas ÿ aqbs=bq in the above equation, we obtain

Qb�i� �k

1 � CQb�i ÿ 1� � C

1 � C�Q�i� � aqQ�i ÿ 1�� 11�

which can be seen as an extension of Equation (9), with an additional parameter. Given that aq 5 0,Equation (11) implies that the rate of change of base¯ow is positively linked to the rate of change of stream¯ow.



The base¯ow index. The base¯ow index (BFI) is de®ned as the long-term ratio of base¯ow to total stream¯ow. For the IHACRES model, it can be obtained by summation of the terms in Equation (11) over asu�ciently long period that the di�erence between the initial and ®nal ¯ows is negligible relative to the otherterms in the ®nal expression (see Appendix). This gives

BFI � C�1 � aq�1 � C ÿ k

�12�

Putting aq � 0 gives the corresponding result for the two-parameter algorithm, Equation (9), and thenputting C � 1 ÿ k gives the BFI for the one-parameter algorithm, Equation (8), as 0.5. This clearlyshows the importance of the constraint in Equation (8a) that the modelled base¯ow must not exceed theobserved stream ¯ow. It is the extent of operation of this constraint that causes di�erences, from catch-ment to catchment, in the BFI obtained by this algorithm. Nevertheless, the value of 0.5 is an upper boundto the BFI for the one-parameter algorithm, and this is unrealistic in catchments fed largely by ground-water.

DATA

The data used in this study were stream ¯ow records for the Queensland and New South Wales (NSW)catchments in the data set of Australian catchments prepared by Chiew andMcMahon (1993). The locationsof the gauging stations are shown in Figure 4, and details of the catchments are given in Table 1.

The record length varied from 8 to 16 years, with very few missing data. Flows for the 24 h period up to 9a.m. were used for the NSW catchments, and up to midnight for the Queensland sites. Daily ¯ows, in Ml,were converted to an equivalent depth in mm over the catchment. No base¯ows were identi®ed for theMitchell Grass or Naradhan sites, and these were not examined further.

Figure 4. Location of study catchments


706 T. CHAPMAN

METHODOLOGY AND RESULTS

Computer programs were written to display plots of observed ¯ow and modelled base¯ow (usually on a logscale) against time for any selected period.

Base¯ow recessions

Periods of base¯ow were identi®ed as sections of the hydrograph, of at least 4 days' duration, that wereapparently linear on the semi-log plot. While this subjective technique may be considered as possibly biasingthe selection in favour of log-linear recessions, it is the procedure that has been traditionally used byengineering hydrologists to identify a base¯ow sequence. Each base¯ow period was then ®tted to Equa-tion (4), using as an objective function the mean sum of squares of di�erences between the logs of theobserved and modelled ¯ows. This objective function gives equal weight to a given proportional error in themodelled ¯ows, which corresponds to a roughly constant proportional error in the measurement of stream¯ows. The value ofQ0 in Equation (4) was taken as a parameter to be optimized, in addition to t0 and n. Theoptimization technique used was a modi®cation of the simplex method (Nelder and Mead, 1965).

The response surface is very ¯at, resulting in a range of values of n, which have little e�ect on the objectivefunction. From the results of these tests, summarized in Table II, it can be shown that the 95% lowercon®dence limit for the mean value of n, calculated from the sample size and standard error, is greater than 1for all sites, and greater than 2 for all but one site (Jardine R.). There appeared to be no relationship betweenthe value of n and the ¯ow Q0 at the start of the recession.

The exponential recession, Equation (1), was also ®tted to the data, using the same objective function(OF). Student's t-test showed that the mean of this OF was signi®cantly greater than that for ®ttingEquation (4) at better than the 90% level for four catchments, 95% for two and 99% for ®ve.

Figures 5 and 6 demonstrate the variability from event to event of the turnover time parameter t0 inEquation (4) and t in Equation (1), with some tendency for the turnover time to decrease with an increase inthe ¯ow at the start of the recession. There was no signi®cant correlation between any of the parameters n, t0or Q0 or the derived quantity S0 .

The mean value of the storage, S0 , at the beginning of each recession (Table II) is shown in Figure 7 to bestrongly related to the mean catchment rainfall for the period of record. With one exception (Styx), values ofS0 below 150 mm were associated with catchments that occasionally experienced zero ¯ow.

Tests of the leaky catchment model

Of the six catchments that had zero ¯ow at some time, only two (Alligator and Corang) had recessions ofsu�cient duration for analysis by the leaky catchment model. Four recessions were identi®ed for each

Table I. Some details of catchments used in the study

Map ref. National no. Catchment Area (km2) Mean rain (mm)

1 927001 Jardine R. at Telegraph Line 2500 17002 111105 Babinda Ck at the Boulders 39 54003 113004 Cochable Ck at Powerline 93 24004 118106 Alligator Ck at Allendale 69 11005 915001 Mitchell Grass at Richmond 3 4506 120204 Broken R. at Crediton 41 21007 145103 Cainable Ck at Good Dam Site 41 9008 206001 Styx R. at Jeogla 163 13009 420003 Belar Ck at Warkton 133 110010 210022 Allyn R. at Halton 215 120011 412093 Naradhan Ck at Naradhan 44 45012 215004 Corang R. at Hockeys 166 80013 401554 Tooma R. above Tooma Res. 114 1700



Table II. Summary of base¯ow recession characteristics, showing values of the three parameters n, Q0 and t0 ®tted toEquation (4). Also given is the mean of the derived quantity S0 � Q0t0

Mapref.

Catchment Recessions Parameter n Other parameters

No. Meanlength(d)

Mean SD MeanQ0

(mm/d)

Meant0(d)

MeanS0

(mm)

1 Jardine 25 72.2 1.62 0.34 1.66 148.7 225.92 Babinda 119 9.1 2.61 0.48 10.48 45.6 383.23 Cochable 98 8.1 2.57 0.43 4.55 62.3 210.54 Alligator 52 8.0 2.46 0.63 1.69 30.7 37.06 Broken 41 9.3 2.91 0.58 3.70 51.6 134.27 Cainable 20 9.7 2.68 0.86 0.29 33.7 7.18 Styx 95 9.2 3.16 0.68 0.96 66.5 48.010 Allyn 65 7.5 3.07 0.84 0.37 46.6 15.69 Belar 57 8.5 2.87 0.82 0.28 37.9 6.112 Corang 86 9.1 3.22 0.75 0.32 41.6 10.713 Tooma 95 7.1 3.24 0.74 3.43 74.8 167.9

Figure 5. Typical scatter diagram of parameter t0 in the non-linear model [Equation (4)] against ¯ow Q0 at the start of recession

Figure 6. Typical scatter diagram of parameter t in the linear model [Equation (1)] against ¯ow Q0 at the start of recession


708 T. CHAPMAN

catchment, and these were ®tted to Equation (5) by determining the value of q that gave the best ®t to astraight line when Qt � q was plotted against t. The value of ts was obtained from the previously ®ttedexponential recession. The results (Table III) suggest that the Alligator has a relatively small deep storagewitha turnover time about the same as that of the storage feeding the base¯ow recession, while the Corang has avery small deep storage with a relatively low turnover time. G. Pocock (1997, personal communication) statesthat the bed of the Alligator Creek at Allendale is of a sandy to light gravelly material, which would be inagreement with the calculated turnover time; the depth to bedrock is unknown, so that the plausibility of theestimated storage depth cannot be checked. I. Milroy (1997, personal communication) states that the gaugingstation on the Corang at Hockeys is on a rock bar, with a downstream pool; the results may therefore indicateleakage through ®ssures in the rock bar, but with negligible storage due to the underlying rock.

Base¯ow separation

The one-parameter algorithm [Equation (8)] was ®tted to the data by minimizing the di�erences betweenthe modelled and recorded ¯ows during the periods of recession previously identi®ed, using the sameobjective function as before. The ®tting was generally done on three years of data, and the average values of kthen applied to the whole data set. This had a minimal e�ect on the calculated value of the BFI.

This optimization technique cannot be used for the two-parameter algorithm [Equation (9)], as theoptimization simply increases the value of the parameter C until the modelled data coincide with theobserved data over the whole of the hydrograph, thereby interpreting all the stream ¯ow as base¯ow. The®tting is therefore subjective, and Boughton (1994, personal communication) recommends basing the choice`mainly on the endings of surface runo�, particularly in the big runo� events'. The values of the parametersand the BFI shown in Table IV have been kindly provided by Dr Boughton (1997, personal communication)from calculations performed previously by him on the same data set.

Figure 7. Relationship between mean catchment storage at the beginning of recession and mean annual rainfall

Table III. Average parameter values for the leaky catchment model

Map ref. Catchment td (d) ts (d) Sd (mm)

Mean SD Mean SD Mean SD

4 Alligator 16.5 14.9 17.2 4.6 2.26 0.9612 Corang 4.6 2.5 58.3 8.0 0.10 0.07



For the same reasons, the three-parameter separation cannot be optimized in the form of Equation (11),but the IHACRES parameters and the BFI have been determined by Ye (1996), and the parametersconverted to the form of Equation (11) for display in Table IV.

The results show some large discrepancies between the estimated values of the BFI for the three algo-rithms. There is no correlation between the BFI values for the three-parameter algorithm and the values forthe other two algorithms. However, as pointed out by R. J. Nathan (1998, personal communication), there isa strong correlation (Figure 8) between the BFI values for the one- and two-parameter algorithms, with amuch greater range of BFI's for the two-parameter model.

Figure 9 illustrates the case where the one-parameter model reaches the total ¯ow hydrograph well downthe recession curve, resulting in an unrealistically low BFI for the Jardine catchment. In contrast, the slow

Table IV. Base¯ow separation parameters and base¯ow index (BFI) for the one-parameter [Equation (8)],two-parameter [Equation (9)] and three-parameter [Equation (11)] algorithms

Mapref.

Catchment Equation (8) Equation (9) (Boughton) Equation (11) (IHACRES)

k BFI k C BFI k C aq BFI

1 Jardine 0.993 0.48 0.990 0.019 0.63 1.145 0.197 ÿ0.81 0.722 Babinda 0.971 0.46 0.975 0.069 0.62 0.975 0.033 ÿ0.26 0.433 Cochable 0.987 0.45 0.975 0.047 0.60 *4 Alligator 0.953 0.38 0.965 0.021 0.30 *6 Broken 0.977 0.39 0.985 0.042 0.44 0.974 0.030 ÿ0.01 0.537 Cainable 0.956 0.29 0.970 0.018 0.24 0.984 0.012 ÿ0.02 0.438 Styx 0.981 0.43 0.980 0.066 0.60 0.976 0.026 ÿ0.52 0.2410 Allyn 0.976 0.30 0.950 0.033 0.28 0.989 0.011 ÿ0.43 0.289 Belar 0.949 0.39 0.960 0.050 0.40 0.907 0.105 ÿ0.17 0.4412 Corang 0.972 0.32 **13 Tooma 0.971 0.49 0.970 0.085 0.70

* IHACRES model could not be ®tted satisfactorily.**No slow ¯ow identi®ed by the IHACRES model.

Figure 8. Relationship between BFI values estimated by the one- and two-parameter algorithms


710 T. CHAPMAN

¯ow from the IHACRES model is seen to have what an engineering hydrologist would regard as implausiblysharp peaks and rates of rise, and meets the total ¯ow hydrograph before what would be usually regarded asthe start of recession; the BFI is consequently high.

For the period on the Styx shown in Figure 10, the peaky nature of the IHACRES slow ¯ow is again seen,and the predicted recession is well below the observed hydrograph, resulting in a low BFI for this catchment.Again, the ¯exibility of the two-parameter algorithm appears to provide plausible results, particularly duringthe ®rst peak event.

Figure 9. Comparison of separation algorithms for Jardine River at Telegraph Line

Figure 10. Comparison of separation algorithms for Styx River at Jeogla



DISCUSSION AND CONCLUSIONS

Base¯ow recessions

With hindsight, the high values identi®ed for the index n in the non-linear recession model [Equation (4)]can be explained in terms of ¯ow convergence in the groundwater system. When using the vertical planemodel of Werner and Sundquist (1951), the value of n changes from 1, for con®ned ¯ow with uniform ¯owdepth, to 2, for uncon®ned ¯ow in which the ¯ow depth decreases downstream. It is reasonable then toexpect n to increase further in the three-dimensional situation, owing to the horizontal convergence of ¯owlines.

Even when the value of n is high, the linear storage model can still be used as a very good approximation inmost cases. Noting that, with one exception, the mean duration of recessions in the study catchments isabout 7±10 days, reference to Figure 3 shows that the recession can be ®tted very well by Equation (1), with alower value of the turnover time t. Only in the very long recessions of the Jardine catchment does the non-linearity have signi®cance in modelling, as can be seen in Figure 9 from the divergence between the modelledand observed stream ¯ows after about day 200.

However, estimates of the recharge during a runo� event are very sensitive to the non-linearity of thestorage. Such estimates are based on the water balance, so that the recharge R is given by

R � S2 ÿ S1 �Z t2

t1

Qb dt �13�

where S1 and S2 are the groundwater storages at times t1 and t2 , respectively, before and after the period ofsurface runo�. For the event shown in Figure 11, the recharge calculated using the linear storage model was9.5 mm, while the non-linear storage gave a value of 19.8 mm. Such di�erences far outweigh any e�ects dueto selecting a di�erent separation algorithm (for the separation used in Figure 11, the base ¯ow outputduring the period shown was 7.5 mm).

This study has shown that a major di�culty in modelling base¯ow recessions, irrespective of the linearityor otherwise of the model, is the variation in values of turnover time from one event to the next, as evidencedby an average coe�cient of variation for all sites of 0.61 for this parameter. It can be hypothesized that this is

Figure 11. An event on the Allyn River at Halton, used for recharge estimation. Separation is by the one-parameter algorithm[Equation (8)]


712 T. CHAPMAN

due to spatial variations in groundwater storage, occurring as a result of spatial variations in rainfall andsubsequent recharge. As noted earlier, there is a tendency for the turnover time to decrease with an increasein the base¯ow Q0 at the start of a recession, but this explains only a small part of the variance in t0 or t.The leaky catchment model requires testing and comparison with site conditions on more catchments

before its merits can be evaluated. It should be noted that the data used in this study have a discrimination of0.01 Ml dÿ1, which is usually too large to determine the hydrograph in the last few days before `cease to¯ow' occurs. However, the results obtained for the two catchments studied are in general accordance withobserved site data.

Base¯ow separation

Of the algorithms tested, the two-parameter Boughton algorithm appears to be the most satisfactoryoverall, although it su�ers from the disadvantage that the parameter selection is subjective. However, thevariations from one operator to another seem likely to have less e�ect on the BFI than use of the objectively®tted one- and three-parameter algorithms, both of which can result in implausible separations on somecatchments, with consequently high or low BFI values.

While one would expect a three-parameter algorithm to have considerable ¯exibility, the form of thealgorithm derived from the IHACRES model results in sharp peaks in the ¯ow ¯ow hydrograph. As shownby P. C. Young and A. J. Jakeman (1996, personal communication), the unit hydrograph for the two-parameter model has a smooth peak, while that for the IHACRES model is structured to have a sharp peakat one time unit. There may therefore be merit in recasting the linear component of the IHACRES model insuch a way that the derived separation algorithm has the same form as the Boughton two-parameteralgorithm.

REFERENCES

Arnold, J. G., Allen, P. M., Muttiah, R., and Bernhardt, G. 1995. Àutomated base ¯ow separation and recession analysis techniques',Ground Water, 33, 1010±1018.

Barnes, B. S. 1939. `The structure of discharge recession curves', Trans. Am. Geophys. Union, 20, 721±725.Boughton, W. C. 1986. `Linear and curvilinear base¯ow recessions', J. Hydrol. (N.Z.), 25, 41±48.Boughton, W. C. 1993. À hydrograph-based model for estimating the water yield of ungauged catchments', in Hydrol. and Water

Resour. Symp., Institution of Engineers Australia, Newcastle, NSW. pp. 317±324.Boughton, W. C. 1995. `Base¯ow recessions', Aust. Civ. Eng. Trans., CE37, 9±13.Boussinesq, J. 1877. Èssai sur la theÂ orie des eaux courantes', MeÂm. preÂs. par divers savants aÁ l'Acad. des Sci. de l'Inst. Nat. de France,

23, 1±680.Chapman, T. G. 1963. È�ects of ground-water storage and ¯ow on the water balance', in Water Resources, Use and Management.

Melbourne University Press, Canberra. pp. 290±301.Chapman, T. G. 1987. Ùnit hydrograph identi®cation using only stream¯ow data', Civ. Eng. Trans. Inst. Eng. Aust., CE29, 187±191.Chapman, T. G. 1990. `Natural processes of groundwater recharge and discharge', in Groundwater and the Environment. Universiti

Kebangsaan Malaysia, Kota Bahru, Malaysia. pp.C1±C21.Chapman, T. G. 1991. `Comment on `Èvaluation of automated techniques for base¯ow and recession analyses'' by R. J. Nathan and

T. A. McMahon', Wat. Resour. Res., 27, 1783±1784.Chapman, T. G. and Maxwell, A. I. 1996. `Base¯ow separation Ð comparison of numerical methods with tracer experiments', in

Hydrol. and Water Resour. Symp. Institution of Engineers Australia, Hobart. pp. 539±545.Chiew, F. H. S. and McMahon, T. A. 1993. `Complete set of daily rainfall, potential evapotranspiration and stream¯ow data for twenty

eight unregulated Australian catchments', Centre for Environmental Applied Hydrology, University of Melbourne, Melbourne.Coutagne, A. 1948. `Les variations de deÂ bit en peÂ riode non in¯uenceÂ e par les preÂ cipitations', La Houille Blanche, 3, 416±436.Dickinson, W. T., Holland, M. E., and Smith, G. L. 1967. Àn experimental rainfall±runo� facility', Hydrology Paper 25. Colorado

State University, Fort Collins.Horton, R. E. 1933. `The roÃ le of in®ltration in the hydrologic cycle', Trans. Am. Geophys. Union, 14, 446±460.Jakeman, A. J. and Hornberger, G. M. 1993. `How much complexity is warranted in a rainfall±runo� model?' Wat. Resour. Res., 29,

2637±2649.Lyne, V. D. and Hollick, M. 1979. `Stochastic time-variable rainfall±runo� modelling', in Hydrol. and Water Resour. Symp. Institution

of Engineers Australia, Perth. pp. 89±92.Maillet, E. 1905. Essais d'Hydraulique Souterraine et Fluviale. Hermann, Paris. 218 pp.Nathan, R. J. and McMahon, T. A. 1990. Èvaluation of automated techniques for base ¯ow and recession analyses', Wat. Resour.

Res., 26, 1465±1473.



Nelder, J. A. and Mead, R. 1965. À simplex method for function minimisation', Comp. J., 7, 308±313.O'Loughlin, E. M., Cheney, M. P., and Burns, J. 1982. `The Bushrangers experiment: hydrological response of a eucalypt catchment to

®re', in Hydrol. and Water Resour. Symp. Institution of Engineers Australia, Melbourne. pp. 132±138.Padilla, A., Pulido-Bosch, A., and Mangin, A. 1994. `Relative importance of base¯ow and quick¯ow from hydrographs of karst spring',

Ground Water, 32, 267±277.Werner, P. W. and Sundquist, K. J. 1951. Òn the groundwater recession curve for large water-sheds', in IASH General Assembly,

Brussels, IAHS Publ., 33, 202±212.Ye, W. 1996. `Climate impacts on stream¯ow in Australian catchments', PhD Thesis, Australian National University, Canberra.

APPENDIX

Derivation of Equation (5)

From Figure 2, and noting that Sd is constant as long as Ss 4 0, the continuity equation is

ÿ dSs

dt� Qs � Qd � Qs �

Ss � Sd

td

and since Ss � tsQs, this becomes

ÿtsdQs

dt� Qs 1 � ts

td

� �� Sd

td

By putting t* � tstd=�ts � td� and q � Sd=�ts � td�, this takes the formdQs

dt� ÿ 1

t*�Qs � q�

which has Equation (5) as its solution.

Derivation of Equation (12)

Equation (11) can be expressed in the form

Qb�i� ÿ Qb�i ÿ 1� � k ÿ C ÿ 1

1 � CQb�i ÿ 1� � C

1 � C�Q�i� ÿ Q�i ÿ 1� � �1 � aq�Q�i ÿ 1��

Taking the sum from i � 1 to i � n,

Qb�n� ÿ Qb�0� �k ÿ C ÿ 1

1 � C

Xnÿ1i�0

Qb�i� �C

1 � CQ�n� ÿ Q�0� � �1 � aq�

Xnÿ1i�0

Q�i�" #

For a su�ciently long series, only the summation terms remain signi®cant, and then the base¯ow index is

BFI � SQb�i�SQ�i� �

C�1 � aq�1 � C ÿ k


714 T. CHAPMAN

Documents

A comparison of algorithms for stream flow recession and baseflow separation