estimating water infiltration and retention characteristics using a computer program in r

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Contents lists available at ScienceDirect

Computers & Geosciences

Computers & Geosciences 35 (2009) 579–585

0098-30

doi:10.1

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HydroM� Cor

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journal homepage: www.elsevier.com/locate/cageo

Estimating water infiltration and retention characteristics using acomputer program in R$

C.T. Omuto �, L.O. Gumbe

Department of Environmental and Biosystems Engineering, University of Nairobi, P.O. Box 30197-0100, Nairobi, Kenya

a r t i c l e i n f o

Article history:

Received 8 September 2007

Received in revised form

20 August 2008

Accepted 21 August 2008

Keywords:

HydroMe

Infiltration

Water retention

Pedometrics

Hydraulic parameters

04/$ - see front matter & 2008 Elsevier Ltd. A

016/j.cageo.2008.08.011

e available on server at http://cran.r-project

e/index.html.

responding author. Tel.: +254 721723492.

ail address: thineomuto@yahoo.com (C.T. Om

a b s t r a c t

Infiltration and water retention functions are widely used soil hydraulic properties in

the geosciences. They contain coefficients known as hydraulic parameters that are

traditionally obtained through curve-fitting. Computer programs for the curve-fitting

process are available for certain infiltration or water retention models. However, these

programs are either not freely accessible or do not estimate certain hydraulic

parameters. They are also inefficient and prone to errors for applications involving

large datasets. This paper discusses the use of a freely accessible HydroMe package for

fast, efficient, and accurate estimation of soil hydraulic parameters in some commonly

used infiltration and water retention models. The package is executable in the freely

downloadable R programming software. It contains a program for estimating the

parameters in infiltration models previously proposed. The program is capable of

estimating parameters from arrays of grouped data in one single pass without having to

enter the data each time for the parameter estimation. It incorporates mixed-effects and

covariate modelling techniques for improved estimation accuracy. These techniques are

not common in any other computer programs in the geosciences. Through covariate

modelling, the package provides opportunity to include environmental correlates in

the estimation of soil hydraulic parameters. Therefore, HydroMe not only improves the

estimation accuracy and efficiency, but also provides insight into environmental risk

factors that influence the management of soil and water resources.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Infiltration and water retention characteristics are soilhydraulic properties that govern the entry of water intothe soil surface and its subsequent movement or storagein the soil. Infiltration functions are models for predictingthe infiltration characteristics, while water retentionfunctions are models of the water retention characteristics(Kutilek and Nielsen, 1994; Marshall et al., 1996; Raats,2001). Many infiltration and water retention models have

ll rights reserved.

.org/web/packages/

uto).

been proposed in the literature for varied applications insoil hydrology, environmental modelling, and crop-yieldestimation (Kutilek and Nielsen, 1994; Nielsen andWendroth, 2003; Dexter, 2004; Nearing et al., 2005;Dexter et al., 2008). These models contain shape para-meters known as hydraulic parameters (van Genuchtenet al., 1991). Applications of infiltration and water retentionmodels require knowing these hydraulic parameters.

Soil hydraulic parameters are traditionally obtained bycurve-fitting the infiltration and water retention functionsusing experimental data (van Genuchten et al., 1991).Various computer programs and statistical packages havebeen presented in the literature for the curve-fittingprocess. For example, RETC and HYDRUS-1D are freelyaccessible computer programs for estimating the hydrau-lic parameters (van Genuchten et al., 1991; Bohne et al.,

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Table 1Overview of different hydraulic functions, related R functions, and estimated parameters included in the HydroMe Package.

R function Hydraulic function Parameters Reference

SSvan y(h) ¼ yr+(ys�yr)[1+(ah)n]�(1�1/n) yr, ys, a, n Van Genuchten (1980)

SSbrook y(h) ¼ yr+(ys�yr)(ah)�n yr, ys, a, n Brooks and Corey (1964)

SScamp y(h) ¼ ys(ah)n ys, a, n Campbell (1974)

SSgard y(h) ¼ yr+(ys�yr)[1+(ah)n]�1 yr, ys, a, n Gardner (1958)

SSphilip i(t) ¼ fc+0.5St�0.5 fc, S Philip (1957)

SShort i(t) ¼ fc+(fc�fc)exp(�kt) fc, fo, k Horton (1940)

SSGampt i(t) ¼ Ks+A/I(t) Ks, A Green and Ampt (1911)

C.T. Omuto, L.O. Gumbe / Computers & Geosciences 35 (2009) 579–585580

1992; Simunek et al., 1998). In addition, statisticalpackages, such as GENSTAT, S-Plus, and Microsoft Excel,have also been used by many researchers to estimate thehydraulic parameters (Rawls et al., 2003; Gitriana andFredlund, 2004; Kosuke, 2004; Whalley et al., 2004), andother researchers have written their own programs in C,C++, or FORTRAN for the same purpose (Deuchers et al.,1999). The existence of these examples shows that there isstill unsatisfied demand for computer programs forestimating soil hydraulic parameters.

There are a few important limitations with the existingcomputer programs or statistical packages. First, moststatistical packages are not freely available and weredeveloped for general computations. Thus, programs muststill be written within them for hydraulic parameterestimation. Second, freely accessible computer programsin the literature do not allow users enough flexibilityfor choosing different models that they may find moreappropriate for their applications. Furthermore, the pro-grams cannot estimate parameters for groups of measureddata simultaneously. They require data entry and para-meter estimation processes to be repeated for each group.This can be tedious and is prone to errors, especially withlarge datasets such as the ISRIC database for waterretention (http://www.isric.org/NR/exeres/545B0669-6743-402B-B79A-DBF57E9FA67F.htm).1 Finally, the availableprograms lack the ability to incorporate environmentalcorrelates (e.g., soil type and depth, land use, andtopography) into the parameter estimation process. Manyusers agree that the environmental correlates are im-portant management risk factors in soil and waterresources and that they can significantly influence theestimation of soil hydraulic parameters (Rawls andPachepsky, 2002; Omuto et al., 2006; Dexter et al., 2008).

Recent advances in the language interfaces of somestatistical freeware have facilitated tremendous improve-ments in modelling soil hydrology (Grunsky, 2002;Schlather and Huwe, 2004). R is an example of freestatistical software with a programming environment thatallows linkage with external C, C++, and FORTRANsubroutines and functions (R Development Core Team,2007). It can therefore provide an environment for C, C++,

1 ISRIC-WISE International Soil Profile Dataset.

and FORTRAN subroutines and functions that have beenseparately developed for accurately estimating soil hy-draulic parameters (Schlather and Huwe, 2004). Recently,a HydroMe package was included in R for estimating soilhydraulic parameters in some commonly used infiltrationand water retention models (Omuto, 2007). This packagecontains a program for estimating parameters in theinfiltration models proposed by Philip (1957), Horton(1940), and Green and Ampt (1911) and in the waterretention models proposed by van Genuchten (1980),Campbell (1974), Brooks and Corey (1964), and Gardner(1958). It is capable of estimating the parameters forgrouped datasets in a single pass without having to repeatdata entry during parameter estimation. The package alsointegrates mixed-effects and covariate modelling techni-ques to improve the accuracy of the estimation process(Omuto et al., 2006). Mixed-effects and covariate model-ling approaches are not common in any other computerprograms in soil hydrology. This paper discusses thestructure, computational advantages, and hydraulic para-meter estimation process using HydroMe.

2. HydroMe

HydroMe is an add-on package in R. It is freelydownloadable from the website http://cran.r-project.org/web/packages/HydroMe/index.html (Omuto, 2007) andcontains a program for estimating hydraulic parametersin some common infiltration and water retention models(Table 1).

The program uses the Gauss–Newton algorithm forparameter estimation (Subramanian and Xiu, 1997).Gauss–Newton is an iterative algorithm for least-squarescurve-fitting of continuous and continuously differenti-able mathematical models (Gourdin and Boumahrat,2002). The inputs for this algorithm include a mathema-tical model with parameters to be estimated, startingvalues for the parameters, first-order partial derivativesof the model with respect to the parameters, andexperimental data to facilitate the estimation process(Subramanian and Xiu, 1997). Apart from the experimen-tal data, the other three Gauss–Newton algorithm require-ments have been written as R functions in the package.For example, the function SSvan in Table 1 for estimatingsoil hydraulic parameters in the van Genuchten (1980)

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model contains the following expressions for modeldefinition and first-order partial derivatives:

fun4-function (x, Thr, Ths, alp, scal) {expr1 4-Ths - Thr

expr2 4-x * alp

expr3 4-.expr2^scal

expr4 4-1+.expr3

expr7 4-�(1-1/scal)

expr8 4-.expr4^.expr7

expr13 4-.expr4^(.expr7 - 1)

value 4-Thr + .expr1 * .expr8

grad 4-array(0, c(length(.value), 4L), list(NULL,

c(‘‘Thr’’, ‘‘Ths’’, ‘‘alp’’, ‘‘scal’’)))

grad[, ‘‘Thr’’] 4- 1-.expr8

grad[,‘‘Ths’’] 4-.expr8

grad[,‘‘alp’’] 4-.expr1 * (.expr13 * (.expr7 * (.ex-

pr2^(scal -1) * (scal * x))))

grad[,‘‘scal’’] 4- .expr1 * (.expr13 * (.expr7 * (.expr3 *

log(.expr2))) - .expr8 * (log(.expr4) * (1/scal^2)))

attr(.value, ‘‘gradient’’) 4- .grad

.value}

The .value expression in the function returns thehydraulic model, while .grad expressions return the first-order partial derivatives of the model with respect to itsparameters. Since the Gauss–Newton algorithm requiresstarting values to be as close to the exact solutionsas possible, the choice for starting values is criticalto guaranteeing success (Subramanian and Xiu, 1997;Gourdin and Boumahrat, 2002). In the HydroMe package,the guidelines for choosing the starting values are basedon the recommendations from the original references forthe hydraulic models (Table 1). These guidelines have alsobeen written as R functions in the package. For example,Philip (1957) recommends that sorptivity (S) be estimatedfrom the slope of the infiltration rate versus the squareroot of time and that the steady infiltration rate (fc) beestimated from the asymptote of the infiltration curvewith time. The following is the R function incorporatingguidelines from Philip (1957) for obtaining the startingvalues of S and fc:

Initial 4-function (mCall, data, LHS) {xy4-data.frame(sortedXyData(mCall[[‘‘input’’]], LHS,

data))

if(nrow(xy)43) {stop(‘‘Too few distinct input values to fit a Philip

model’’)}ndistinct 4-nrow(xy)

fc 4-mean(xy[(ndistinct-2):ndistinct,][[‘‘y’’]])

lfirst 4-xy[1:(ndistinct/4), ]

pars24-coef(lm(exp(y) �sqrt(x), data ¼ lfirst))

S4-(�1/pars2[2])

value4-c(fc ¼ fc, S ¼ S)

names (value)4-mCall[c(‘‘fc’’, ‘‘S’’)]

value}

The functions for first-order partial derivatives andstarting values are combined to produce self-starting R

functions. Table 1 shows the names for the self-starting Rfunctions corresponding to the hydraulic models in thepackage. These self-starting functions are integrated intoone program for estimating the soil hydraulic parameters,and the program is made available within the R environ-ment by the statement library (HydroMe). Its execution isinternalized in R and any of its preferred soil hydraulicmodels can be called by the corresponding R name (Table 1),just like any other R function.

3. Estimating hydraulic parameters using HydroMe

3.1. Estimating the parameters for grouped and

ungrouped data

There are two scenarios in the literature that dominateuser needs for soil hydraulic parameters (Omuto et al.,2006): the need for average soil hydraulic parameters andthat for hydraulic parameters for individual samples fromspecific locations or land-use types or hydrologic units,etc., in a study area. HydroMe provides users with theflexibility to choose the scenario for their applicationsand the hydraulic models containing the parameters ofinterest.

When estimating average parameters, replicate mea-surements of infiltration or water retention characteristicsare consolidated and treated as though they belong to onegroup. Hydraulic parameter estimation for this scenario isachieved using the nls function in HydroMe. The followingexample with Philip’s infiltration model illustrates howthe nls function is used to obtain the average parameterestimates:

> data(Infilt) # Infilt is sample data stored in HydroMe> philip.nls <-nls(log(Rate)~SSphilip(Time, fc, S), data=Infilt)> summary(philip.nls)

Formula: log(Rate) ~ SSphilip(Time, fc, S)Parameters: Estimate Std. Error t value Pr(>|t|) fc −1.63666 0.04447 −36.80 <2e−16 ***S 4.48017 0.27679 16.19 <2e−16 ***---Residual standard error: 0.8012 on 1103 degrees of freedom

Number of iterations to convergence: 1 Achieved convergence tolerance: 1.562e−16

Infilt in this example is a data frame in the packagecontaining 30 groups of infiltration measurements. Thedata include records for Rate (cm min�1) and Time (min),which are used in this example for fitting the SSphilip

function. The estimated parameters are retrieved usingthe coef function in R (e.g., coef(philip.nls) for thisexample).

HydroMe has two steps for estimating parameters forgrouped data. The first step makes a first approximation ofthe parameter for each group using the nlsList function.This function treats each group as an individual entity,even though in reality, it belongs to a population withsimilar characteristics. The function can sometimes

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produce estimates with large standard errors for groupswith high scatter or outlying observations. Estimates withlarge standard errors are generally the results of lowestimation accuracy (Gourdin and Boumahrat, 2002). Ifconvergence errors are not reported for such cases, thelow estimation accuracy may not be noticed. This explainsthe occurrence of low accuracy in existing computerprograms where the estimation process treats eachsample as an independent individual. The Gauss–Newtonalgorithm in HydroMe always reports convergence errorsif the estimation accuracy exceeds a preset tolerance.These convergence errors should not raise too much alarmother than a warning to go back and check the reasonsbehind the scatter. Otherwise, the package takes careof such convergence errors in the next step. The focus inthe second step is to improve the estimation accuracy.This step considers the individual groups as belonging to a

> data(isric)> van.lis <-nlsList(y~SSvan(x,Thr, Ths, alp, scal)|Sample,data=isric)#The 1st stepError in nlsModel(formula, mf, start, wts) : singular gradient matrix at initial parameter estimates # 4 errors are returned…………………………> van.nlme <-nlme(van.lis)# The 2nd step> summary(van.nlme)Nonlinear mixed-effects model fit by maximum likelihood Model: y ~ SSvan(x, Thr, Ths, alp, scal) Data: isric

AIC BIC logLik −1211.601 −1155.076 620.8003Random effects:Formula: list(Thr ~ 1, Ths ~ 1, alp ~ 1, scal ~ 1)Level: SampleStructure: General positive-definite, Log-Cholesky parameterization

StdDev Corr Thr 0.10302531 Thr Ths alp Ths 0.11085289 0.834 alp 0.91786081 −0.449 −0.324 scal 0.31850225 0.131 0.060 −0.338Residual 0.01508187

Fixed effects: list(Thr ~ 1, Ths ~ 1, alp ~ 1, scal ~ 1) Value Std.Error DF t-value p-valueThr 0.1930440 0.01752351 277 11.01628 0Ths 0.5386298 0.01776416 277 30.32115 0alp −3.0192858 0.16161970 277 −18.68142 0scal 1.4988464 0.05531627 277 27.09594 0……………………….

Number of Observations: 320Number of Groups: 40

population and gives them weights so that the influenceof any outlying observation is minimized (Omuto et al.,2006). It uses the approximations from the first step as thestarting values to estimate the average hydraulic para-meters (also known as fixed-effects). Then, randomvariations of hydraulic parameters for each group aroundthe fixed-effects are iteratively determined (Omuto et al.,2006). These variations are known as random-effects. The

sum of random- and fixed-effects gives the estimatedhydraulic parameters. The second step of the hydraulicparameter estimation is achieved using the nlme function.

The following example with the van Genuchten waterretention model illustrates the implementation of thetwo-step parameter estimation process. This exampleuses a subset of the ISRIC database. The entire ISRICdatabase can be downloaded from http://www.isric.org/NR/exeres/545B0669-6743-402B-B79A-DBF57E9FA67F.htm.The data frame (known as isric) in the package has recordsfor soil moisture contents y (cm3 cm�3) and suctionpotential x (cm), which are used in this example tofit the van Genuchten model. The parameters in thevan Genuchten model (Ths ¼ ys, Thr ¼ yr, alp ¼ loge(a),scal ¼ n) are called by the SSvan function. The finalparameter estimates for each group are retrieved usingthe coef function.

3.2. The use of covariate modelling

Covariates, such as land use/cover, topography, andsoil texture, are related in some ways to soil hydraulicparameters (Rawls and Pachepsky, 2002; Rawls et al.,2003). Therefore, including them in the estimationprocess can help account for more variability andeventually improve the estimation accuracy. HydroMe

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Fig. 1. Plot of random-effects for Campbell’s saturated moisture content

with bulk density and soil depth.

C.T. Omuto, L.O. Gumbe / Computers & Geosciences 35 (2009) 579–585 583

provides allows such covariates to be included in theestimation process.

Covariates can be included directly if their influence onthe hydraulic parameters is already known. Otherwise, anassessment step may be necessary to test their influence.There are two ways to assess these: either through a plotof the random-effects with potential covariates or throughan analysis of variation (ANOVA). A characteristic patternin the plot of the random-effects with the potentialcovariates is taken to imply that there is a significantrelationship between the covariates and soil hydraulicparameters (Omuto et al., 2006). Similarly, a p-valueo5%(at the 95% confidence interval) in the ANOVA indicatesa significant relationship between the covariates andsoil hydraulic parameters. Once the covariates havebeen selected, they should be included together withnew starting values for the hydraulic parameters. Newstarting values enable the Gauss–Newton algorithmto estimate the effects due to the added covariates.These starting values are extracted from the previousfixed-effects.

The following example with the Campbell (1974) waterretention model illustrates how covariates are testedand included in the parameter estimation process. Theexample uses Wret data in the package. It contains29 groups of water retention measurements and soil bulkdensity (g cm�3) for two horizons of the soil profile(topsoil: 0–20 cm and subsoil: 20–50 cm). The recordsfor soil moisture contents y (in cm3 cm�3) and suctionpotential x (in cm) are used in this example to fit theCampbell (1974) model. The parameters in the Campbellmodel (ys ¼ Ths, a ¼ alp, n ¼ scal) are called using theSScamp function.

>data(Wret)>camp.lis <-nlsList(y~SScamp(x, Ths, alp, scal)|Points, data=Wret)>camp.nlme <-nlme(camp.lis)…..>plot(ranef(camp.nlme,augFrame=T), form=Ths~BULK+Depth,col=1, lwd=2, +

par.settings=list(box.rectangle=list(col=1,lwd=2),box.umbrella=list(col=1,lwd=1), + plot.symbol=list(pch=1,col=1),plot.line=list(col=1,lwd=2))) # Figure 1

#If the relationship is known a priori, the covariate is included as follows> camp.nlme$coef$fixed # Extract new estimates for the covariate using fixed effects Ths alp scal 0.4937684 0.1841466 0.1648822

> camp2.nlme<-update(camp.nlme,fixed=list(Ths~BULK,alp~1,scal~1),+start=c(camp.nlme$coef$fixed[1],0,camp.nlme$coef$fixed[2:3]))…….> anova(camp.nlme,camp2.nlme) Model df AIC BIC logLik Test L.Ratio p-valuecamp.nlme 1 10 −795.5871 −761.1197 407.7935 camp2.nlme 2 11 −838.4622 −800.5480 430.2311 1 vs 2 44.8751 <.0001

Fig. 1 shows the relationship between ys random-effects and bulk density and soil depth. The characteristicnegative correlation with bulk density indicates its choiceas a covariate to model the ys parameter. Similarly, if thisrelationship was known a priori, then bulk density couldhave been included in the model directly and the results

tested using ANOVA. A low p-value (at 5% level ofsignificance) for the ANOVA test implies that the covariateis indeed significant. After including the covariate, thestandard deviation for the ys parameter decreased from0.0912 to 0.032 cm3 cm�3 and the predictive accuracyimproved by about 60%.

4. Comparison with RETC program

HydroMe has many features and advantages that makeits parameter estimation more accurate, efficient, and

flexible than other computer programs. The followingcomparison with the RETC program illustrates how itoutperforms other computer packages.

We used both HydroMe and RETC to estimate vanGenuchten parameters on a dataset of 30 samples for waterretention characteristics. Each sample had eight records of

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Fig. 2. Relative performance of HydroMe and RETC program in fitting van Genuchten (1980) model to soil measured moisture content. HydroMe had the

highest r2 and the lowest residual standard error (RSE).

C.T. Omuto, L.O. Gumbe / Computers & Geosciences 35 (2009) 579–585584

soil moisture contents and suction potential. It took 43 minon average to estimate the van Genuchten parameters usingthe RETC program, while HydroMe took less than 3 min. TheRETC program does not provide for one-pass parameterestimation for grouped data (van Genuchten et al., 1991).It requires data entry and parameter estimation processesto be repeated for each sample. Hence, it took longer tocomplete the 30 samples with RETC than with HydroMe.RETC is therefore likely to be much more inefficient forlarge number of samples. Fig. 2 shows the outputs forHydroMe and RETC on the same dataset.

HydroMe further improved its accuracy by usingcovariate modelling (r2

¼ 0.996, residual standarderror ¼ 1.8�10�4), while RETC could not improve itsperformance any further since it does not provide forcovariate modelling. Furthermore, apart from the modelsby van Genuchten (1980) and Brooks and Corey (1964)in RETC, the program also lacks the flexibility for differentwater retention models that can adequately fit thedata. However, it has models for unsaturated hydraulicconductivity functions that are not yet incorporatedin HydroMe. In addition to improving the estimationaccuracy through covariate modelling, HydroMe alsoprovides insight into the factors that influence soilhydraulic parameters. Consequently, the package can givea first approximation of the factors to target in managingwatershed hydrology. This opportunity is not availablewith the RETC program.

5. Conclusions

The HydroMe package presents an efficient andimproved approach for estimating soil hydraulic para-meters using experimental data. The package and itssource program are freely downloadable from www.cran.r-project.org. It provides a new method for estimat-ing soil hydraulic parameters for grouped data in a singlepass without having to repeat data entry and parameterestimation processes for each group. Not only does thepackage eliminate drudgery and data transfer errors, but italso expedites the estimation process. The package also

contains mixed-effects and covariate modelling techni-ques for improved estimation accuracy.

HydroMe does not include models for estimat-ing unsaturated hydraulic conductivity. Future versionsof the package should consider such functions as wellas other models for water retention and infiltrationfunctions.

Acknowledgements

Some of the water retention dataset used in thispackage was obtained from the ISRIC database (http://www.isric.org/NR/exeres/545B0669-6743-402B-B79A-DBF57E9FA67F.htm). The International Foundation ofSciences (IFS) supported the development of the HydroMepackage through project number C-3953/1. Supportfrom the ICRAF through Dr. Markus Walsh, Mr. RichardCoe, and Dr. Keith Shepherd is also acknowledged.Dr. Budiman Minasny of ACPA, University of Sydney andDr. W. Venables of the R Development Core Team providedvaluable comments on the development of the HydroMepackage.

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