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459RESPONSE TO COMMENT

the other hand [and bearing in mind point (2)8) above],Figs 3 and 4 suggest that the "ts were obtained with thesmall potential points ignored.

2.10. ¹he values of the ,tted parameters

Diiwu et al. (1998) present the values for the "ttedparameters in Table 2 of their paper. However, whenthese values are used in the equations, they appear not togive curves that "t the data at all well. This is demon-strated in Fig. 1 of this comment, which shows the con-ventional tillage, A horizon data from Table 1 of Diiwuet al. (1998), together with three curves for each of thethree equations.

Figure 1(a) shows the curves resulting from the correctform of the Hutson and Cass equation [see point (1)above]. The solid curve is a best "t, ignoring the datapoint corresponding to saturation (which, for conveni-ence when using logarithms, is plotted at 0)01 m on thematric potential axis). The "t was performed by choosing(via a minimization algorithm) the values of h

s, a and

b that resulted the smallest sum of squares of deviationsof the measured points from the "tted curve. The twobroken curves result from using the a and b values givenin Table 2 of Diiwu et al. (1998), in the one case taking h

sto equal the water content at zero matric potential and inthe other case treating h

sas a "tting parameter. Diiwu

et al. (1998) did not state what they used for hs.

Figure 1(b) shows the analogous curves resulting fromthe Verma and Brutsaert equation, in the form given byDiiwu et al. (1998) [see point (2) above]. The best-"tcurve (solid line) was obtained again by ignoring the datapoint corresponding to saturation and treating h

sas

a "tting parameter (as well as a and b). Fixing hsat the

water content at saturation, which appears more reason-able, results in a poor "t for the rest of the curve. The twobroken curves were obtained using the a and b valuesgiven in Table 2 of Diiwu et al. (1998), with h

seither "xed

doi:10.1006/jaer.2000.0615, available online at http://www.iSW*Soil and Water

at the water content at zero matric potential or treated asa "tting parameter.

Figure 1(c) shows the analogous curves resulting fromthe van Genuchten equation. The best-"t curve (solidline) was obtained again by ignoring the data pointcorresponding to saturation and treating h

sand h

ras

"tting parameters (as well as a and g). The two brokencurves were obtained using the a and b values given inTable 2 of Diiwu et al. (1998) (and assuming that g"b),either with h

s"xed at the water content at zero matric

potential and hr"xed at the water content at a matric

potential of 150 m [as described by Diiwu et al. (1998)],or treating both as "tting parameters.

The conventional tillage B horizon and the two no-tilldata sets result in similar poor "ts with the a and b para-meters of Diiwu et al. (1998). On the face of it, then, theparameter values given by Diiwu et al. (1998) appear tobe in error. However, the point is not clear because of theinadequate explanations of procedure and the apparentdi!erences among the &measured data' in their Table 1,Figs 1 and 2 and Figs 3 and 4.

References

Diiwu J Y; Rudra R P; Dickinson W T; Wall G J (1998). Tillageand heterogeneity e!ects on the performance of soil watercharacteristic models. Journal of Agricultural EngineeringResearch, 71(3), 307}313, doi: 10.1006/jaer. 1998.0358

Hutson J L; Cass A (1987). A retentivity function for use insoil}water simulation models. Journal of Soil Science, 38,105}113

Ross P J; Smettem K R J (1993). Describing soil hydraulicproperties with sums of simple functions. Soil Science ofAmerica Journal, 57, 1519}1524

van Genuchten M T (1980). A closed-form equation for predic-ting the hydraulic conductivity of unsaturated soils. SoilScience Society of America Journal, 44, 892}898

Verma R D; Brutsaert W (1970). Uncon"ned aquifer seepage bycapillary #ow theory. Journal of the Hydraulics Division,Proceedings of the American Society of Civil Engineers, 96,1331}1344

dealibrary.com on

Response to Comment

J. Y. Diiwu1; R. P. Rudra1; W. T. Dickinson1; G. J. Wall2

1School of Engineering, University of Guelph, Guelph, Ontario, Canada, N1G 2W1; e-mail of corresponding author: jdiiwu@uoguelph.ca2Land Resource Research Centre, Agriculture and Agri-Food Canada, Guelph, ON, Canada N1H 6N1; e-mail: gwald@uoguelph.ca

(Received 25 February 2000; accepted in revised form 22 July 2000; published online 25 October 2000)

Fig. 1. Comparison of observed and predicted soil water charac-teristics in A horizon for no tillage treatment: , observed;

, two-equation; , van Genuchten (1980); , Verma and Brut-saert (1970)

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RESPONSE TO COMMENT460

1. Introduction

We thank Kirby for the comment on the researcharticle by Diiwu et al. (1998). As noted in the comment,soil water characteristic models are useful for the solutionof irrigation and drainage problems, but the e!ect ofheterogeneity on the performance of these models re-mains a thorny issue. This is why we are pleased to notethat our manuscript has attracted attention in the litera-ture. We therefore wish to respond to the issues raised inthe comment as follows.

2. Performance of the models

Part of the objective of the manuscript was to demon-strate the poor performance of soil water characteristicmodels in heterogeneous soil, including soil having mac-ropores. This objective could not have been achieved ifthe macropore region had been ignored in "tting thethree models discussed or in the illustrations presentedand discussed. The poor performance is attributable tothe fact that the models, in their present forms, do notincorporate heterogeneity in the soil system. To improvethe performance of the models in the entire range ofmatric potentials (0}150 m of water) would probablyrequire a two-component approach or some other modi-"ed versions of the models such as those proposed byRoss and Smettem (1993).

In "tting the models, the physical signi"cance of thesoil water content at saturation and at a matric potentialof 150 m of water h

sand h

rcannot be overlooked. While

it is possible to consider hsand h

ras "tting parameters,

we rather preferred to represent them with observedvalues from the soil system response in order not to losethe physical relevance of the soil water characteristicmodels. For all three models discussed in the manuscript,hs

was taken as soil water content measured at zeromatric potential. The soil water content measured at150 m of water was taken as the value for h

rfor the van

Genuchten model. The three models were then "ttedthrough the intermediate points between h

sand h

rby

a least-squares technique, using the optimum values oftheir corresponding parameters.

The optimum values of the parameters for each modelwere obtained by a non-linear least-squares regression,using the SYSTAT statistical program (Wilkinson et al.,1992). The optimum values obtained were those reportedin Table 2 of Diiwu et al. (1998). The standard error ofestimate (SEE) values and the coe$cient of determina-tion (R2) values were also presented to give a statisticalmeasure of the reliability of the "tted parameters. As hasbeen discussed in Section 4.3 of the manuscript, none ofthe three models gives a good "t in the macropore range.

This also happens to be the range that Kirby (this issue)has chosen to demonstrate his point about the "ttedparameters of Diiwu et al. (1998). Not withstandingwhether h

sand h

rare treated as "tted parameters or

assigned measured values, the three models would mostlikely still perform poorly in the macropore range. That iswhy the bias and e$ciency of the models as presentedand discussed using results reported in Tables 3 and 4 ofDiiwu et al. (1998) are relevant. In case the models haveto be used in their present forms in heterogeneous soil,a discussion of the bias and e$ciency would help to givean indication of how reliable the results are.

In the manuscript (Diiwu et al., 1998) selected graphsfor the A horizon were presented to support the dis-cussions due to limitation of space. The remaining graphsfor the B horizon are presented in this response alongwith those reported in Diiwu et al. (1998), as shown inFigs 1}4 of this response. In the macropore range thecurves for no tillage treatment are steeper than those forconventional tillage treatment in both A and B horizons.This probably indicates that the models perform better inthe conventional tillage site than the no tillage site. Thisis probably because soil under no tillage treatment tendsto contain more macropores and more stable aggregatesthan soil under conventional tillage treatment. Theseresults are supported by the bias and e$ciency valuesreported in Tables 3 and 4 of Diiwu et al. (1998).

Fig. 2. Comparison of observed and predicted soil water character-istics in B horizon for no tillage treatment: , observed; , two-equation; , van Genuchten (1980); , Verma and Brutsaert (1970)

Fig. 3. Comparison of observed and predicted soil water charac-teristics in A horizon for conventional tillage treatment: , ob-served; , two-equation; , van Genuchten (1980); , Verma and

Brutsaert (1970)

Fig. 4. Comparison of observed and predicted soil water charac-teristics in B horizon for conventional tillage treatment: , ob-served; , two-equation; , van Genuchten (1980); , Verma and

Brutsaert (1970)

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461RESPONSE TO COMMENT

3. Shapes of 5gures

Figures 1 and 2 of Diiwu et al. (1998) were drawn onlinear scale, including the entire range of matric poten-tials (0}150 m of water) to illustrate the e!ect of hetero-geneity (including macroporosity) on the performance ofthe models. Figures 3 and 4 were also drawn for the entirerange of matric potentials, but on semi-logarithmic scale.The apparent contradiction in shapes of Figs 1 and 2compared to the shapes of Figs 3 and 4 stems from thefact that the two sets of graphs were based on di!erentscales.

4. Use of symbols and notation

The models presented and discussed by Diiwu et al.(1998) are exactly as reported in the literature, except forvariation in use of symbols and notation from one authorto the other. The parameters a and b in Eqn (1) of Diiwuet al. (1998) are related to the parameters a and b in Campbellequation as given by Felton and Nieber (1991) thus,

a"a (1)

b"b (2)

Equation (1) is not used in subsequent discussions inthe manuscript, only the Hutson and Cass (1987) (two-equation), Verma and Brutsaert (1970) and van Genuch-ten (1980) models are discussed.

Equations (2) to (4) of Diiwu et al. (1998) correspond toEqns (8) to (10) of Hutson and Cass (1987) with

a"a (3)

b"b (4)

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RESPONSE TO COMMENT462

As Kirby (this issue) has rightly noted Eqn (3a)of Verma and Brutsaert (1970) reduces to Eqn (5) ofDiiwu et al. (1998) with

A"a (5)

B"b (6)

and assuming residual soil water content hrto be zero. It

would therefore be fair to say that Eqn (5) in the manu-script is a special case of Eqn (3a) of Verma and Brutsaert(1970), and not that the two equations are di!erent. Also,Eqn (5) in the manuscript is the same as Eqn (2) of Feltonand Nieber (1991) with

a"A (7)

b"B (8)

5. Corrigendum

The parameters of the van Genuchten model as pre-sented by Diiwu et al. (1998) should be a and g, not a andb. The typing error is regretted.

In Figs 1}4 of Diiwu et al. (1998) the x-axis valuesshould be divided by 100 to convert to metres. Theomission is regretted. However, the results on thosegraphs are not altered by the magni"cation.

6. Conclusion

We again thank Kirby (this issue) for the interestshown in our work. We hope that all issues raised in the

comment have been addressed to clarify the content ofthe manuscript. However, the corrections made in thetypographical errors and in the units of matric potentialused in the "gures do not change the overall results andconclusions on the performance of soil water character-istic models in heterogeneous soil and under tillage treat-ment. We look forward to further discussion of the sub-ject matter in the literature.

References

Diiwu J Y; Rudra R P; Dickinson W T; Wall G J (1998). Tillageand heterogeneity e!ects on the performance of soil watercharacteristic models. Journal of Agricultural EngineeringResearch, 71(3), 307}313, doi:10.1006/jaer.1998.0358

Felton G K; Nieber J L (1991). Four soil moisture characteristiccurve functions for numerical modelling of sand. Transac-tions of American Society of Agricultural Engineering, 34(2),417}422

Hutson J L; Cass A (1987). A retentivity function for use insoil water simulation models. Journal of Soil Science, 38,105}113

Ross P J; Smettem K R J (1993). Describing soil hydraulicproperties with sums of simple functions. Soil Science Societyof America Journal, 57, 26}29

van Genuchten M T (1980). A closed-form equation for predic-ting the hydraulic conductivity of unsaturated soils. SoilScience Society of America Journal, 44, 892}898

Verma R D; Brutsaert W (1970). Uncon"ned aquifer seepageby capillary #ow theory. Journal of Hydraulics Division,American Society of Civil Engineers, 96(6), 1331}1344

Wilkinson L; Hill M A; Welna J P; Birkenbeuel G K (1992).SYSTAT for Windows: Statistics, Version 5.2 Edition, 724pp.SYSTAT, Inc, Evanston, IL