Review and Analysis Using Pedotransfer FuncƟ ons to EsƟ mate the van Genuchten– H. Vereecken* M. Weynants Mualem Soil Hydraulic ProperƟ es: M. Javaux A Review Y. Pachepsky We reviewed the use of the van Genuchten–Mualem (VGM) model to parameterize soil M. G. Schaap hydraulic proper es and for developing pedotransfer func ons (PTFs). Analysis of litera- M.Th. van Genuchten ture data showed that the moisture reten on characteris c (MRC) parameteriza on by se ng shape parameters m = 1 − 1/n produced the largest devia ons between fi ed and measured water contents for pressure head values between 330 (log10 pressure head [pF] 2.5) and 2500 cm (pF 3.4). The Schaap–van Genuchten model performed best in describing the unsaturated hydraulic conduc vity, K. The classical VGM model using fi xed parame- ters produced increasingly higher root mean squared residual, RMSR, values when the soil became drier. The most accurate PTFs for es ma ng the MRC were obtained when using textural proper es, bulk density, soil organic ma er, and soil moisture content. The RMSR values for these PTFs approached those of the direct fi t, thus sugges ng a need to improve both PTFs and the MRC parameteriza on. Inclusion of the soil water content in the PTFs for K only marginally improved their predic on compared with the PTFs that used only textural proper es and bulk density. Including soil organic ma er to predict K had more eff ect on the predic on than including soil moisture. To advance the development of PTFs, we advocate the establishment of databases of soil hydraulic proper es that (i) are derived from standardized and harmonized measurement procedures, (ii) contain new predictors such as soil structural proper es, and (iii) allow the development of me-dependent PTFs. Successful use of structural proper es in PTFs will require parameteriza ons that account for the eff ect of structural proper es on the soil hydraulic func ons.

Abbrevia ons: Bd, bulk density; HCC, hydraulic conduc vity curve; ME, mean error; MRC, moisture reten- on characteris c; OM, organic ma er; PTF, pedotransfer func on; RMSD, root mean squared devia on; Pedotransfer functions (PTFs) based RMSR, root mean squared residual; RMSRA, adjusted root mean squared residual; SSC, sand, silt, and clay; VGM, van Genuchten–Mualem; WC, water content. on the van Genuchten–Mualem (VGM) model have shown great value in the widespread applica on of water fl ow Knowledge of the unsaturated soil hydraulic properƟ es is mandatory and solute transport models at the for describing and predicting water fl ow and solute transport in the vadose zone. In spite fi eld and larger scales. We reviewed the of major advances in the development of measurement techniques, direct determination state of the art with respect to using the is still expensive, time consuming, and, especially for larger scale applications, impracti- VGM model to describe hydraulic prop- cal. Th erefore, PTFs have been developed that relate the soil hydraulic functions to more er es and to develop PTFs. easily measurable soil properties. While various types of PTFs exist in terms of input and output (e.g., Pachepsky and Rawls, 2004), most PTFs relate parameters in the VGM model H. Vereecken and M. Javaux, Agrosphere Ins tute, Ins tute of Chemistry and Dynamics to specifi c soil properties. Th e development of PTFs has greatly facilitated the widespread of the Geosphere, Forschungszentrum Jülich, application of water fl ow and solute transport models at the fi eld and larger scales. Recent Jülich, Germany; M. Weynants and M. Javaux, Earth and Life Ins tute— Environmental Sci- developments in digital soil mapping and large-scale modeling again emphasize a need ences, Université Catholique de Louvain, for reliable PTFs. Louvain-la-Neuve, Belgium; Y. Pachepsky, USDA-ARS, Environmental Microbial and Food Safety Lab., Beltsville, MD; M.G. Schaap, Dep. of Soil, Water and Environmental Sci- ence, Univ. of Arizona, Tucson, AZ; and M.Th. 6 van Genuchten, Dep. of Mechanical Engi- Pedotransfer FuncƟ ons neering, Federal Univ. of Rio de Janeiro, RJ, Types of Pedotransfer FuncƟ ons Brazil. *Corresponding author (h.vereecken@ fz-juelich.de). Th e past several decades have seen the development of diff erent types of PTFs (e.g., van Genuchten et al., 1992, 1999). Recent reviews of the state of the art in this fi eld of Vadose Zone J. 9:795–820 doi:10.2136/vzj2010.0045 research have been given by Wösten et al. (2001), Pachepsky and Rawls (2004), and Shein Received 7 Mar. 2010. and Arkhangel’skaya (2006). Basically, two main types of PTFs can be distinguished: Published online 27 Oct. 2010. point and parametric PTFs. Point PTFs estimate values of soil moisture at fi xed pres- © Soil Science Society of America 5585 Guilford Rd. Madison, WI 53711 USA. sure head values (e.g., de Jong and Loebel, 1982; Rawls and Brakensiek, 1982; Puckett All rights reserved. No part of this periodical may be reproduced or transmi ed in any form or by any et al., 1985), whereas parametric PTFs estimate parameters of functions that describe means, electronic or mechanical, including photo- copying, recording, or any informa on storage and the observed data across a range of pressure heads (e.g., McCuen et al., 1981; Cosby et retrieval system, without permission in wri ng from the publisher. al., 1984; Wösten and van Genuchten, 1988; Vereecken et al., 1989). A few researchers

2010, Vol. 9 www.VadoseZoneJournal.org | 795 have combined the latter approach with a direct fi t of the para- metric function, such as the VGM model, to the estimated soil Methods for Pedotransfer moisture values (e.g., Baumer, 1992; Tomasella and Hodnett, FuncƟ on Development 1998; van den Berg et al., 1997; Tomasella et al., 2000). Most We only briefl y summarize here the various methods that may be eff orts during the past several years have focused on the develop- used to derive PTFs. An extensive review was given by Pachepsky ment of parametric PTFs because they provide a mathematical and Rawls (2004). Two main categories can be distinguished: sta- function for the MRC or hydraulic conductivity curve (HCC) tistical regression techniques (linear and nonlinear models) and that can be used directly in mathematical models. Also, multi- data mining and exploration techniques (artifi cial neural networks modeling approaches have been developed recently that combine and group methods of data handling). In general, methods based predictions with the diff erent PTFs to either derive a single set on artifi cial neural networks have led to PTFs that performed best of hydraulic parameters or aggregate the output of model runs in terms of basic indicators such as the RMSR. Th is is largely due that were obtained for each of the individual PTFs (Guber et al., to the fact that this approach does not require an a priori func- 2006, 2009). Guber et al. (2009) used 19 published PTFs and the tional form with which to relate PTF input to PTF output. HYDRUS-1D code to model a fi eld soil moisture regime. They evaluated diff erent methods for combining the simulation results Recently, new methods such as nearest neighbor methods (Nemes of the 19 diff erent individual models to obtain the best accuracy et al., 2006a,b; Jagtap et al., 2004), support vector machines in terms of soil moisture content at diff erent depths. (Lamorski et al., 2008; Twarakavi et al., 2009), and multiscale Bayesian neural networks (Jana et al., 2007) have been intro- Pedotransfer FuncƟ on Predictors duced. Nemes et al. (2006a,b) applied a nearest neighbor method Extensive research in the past has focused on improving estimates to estimate the water retention at 330 and 15,000 cm using a of the hydraulic properties using PTFs. One key area of research hierarchical set of predictor variables. Th is method is based on has dealt with the identifi cation of additional soil information pattern recognition and similarity rather than fi tting equations to that could improve the accuracy of the PTFs, besides such clas- data. Lamorski et al. (2008) introduced support vector machines sical PTF predictors as sand, silt, and clay contents, bulk density, (SVMs), an alternative data-driven method developed in the fi eld and C content. Additional information may involve more detailed of machine learning, to improve the effi ciency of artifi cial neural textural information (e.g., Vereecken et al., 1989, 1990; Børgesen networks. Th is approach was successfully used to predict soil water et al., 2008), terrain attributes and vegetation (e.g., Romano and contents at various pressure head levels using simple soil properties. Palladino, 2002; Sharma et al., 2006; Leij et al., 2004), soil struc- Twarakavi et al. (2009) used a SVM-based approach to develop tural information (e.g., Rawls and Pachepsky, 2002; Pachepsky and PTFs for the VGM parameterization model using the database Rawls, 2004; Pachepsky et al., 2006), the water content at selected from Schaap and Leij (1998a) and Schaap et al. (2001). Results of pressure heads (e.g., Al-Majou et al., 2008a; Børgesen and Schaap, their analysis have been included here. 2005; Børgesen et al., 2008), morphological properties (e.g., Lin et al., 1999), or taxonomic information (e.g., Lilly et al., 2008; Unexplained Variability Al-Majou et al., 2008b). Despite much progress in identifying appropriate PTF predictors that explain the observed variability in the estimated soil hydrau- Scaling Pedotransfer FuncƟ ons lic properties, substantial unresolved or unexplained variability Recent attempts to improve PTF predictions has involved the still exists at the level of the soil sample and the database used to scaling of PTFs using coarse- or local-scale data such as the water derive the PTFs (Schaap and Leij, 1998a). Th is unexplained vari- content. Gribb et al. (2009) evaluated the quality of PTFs and ability still plays an important role when functional aspects of soils laboratory-scale measurement techniques (e.g., the multistep out- are being studied and analyzed using mathematical models (e.g., fl ow method) to predict soil moisture dynamics at the fi eld scale. Christiaens and Feyen, 2001). Th ese functional aspects include, Th ey found that inversely estimated hydraulic properties derived e.g., vulnerability to leaching of contaminants (Wösten et al., from in situ measurements of the water content and the pressure 2001), workability (Wösten et al., 2001), moisture supply capac- head provided better descriptions of the soil water dynamics than ity (Vereecken et al., 1992), groundwater recharge (Vereecken et PTFs and the laboratory-scale measurements. Model predic- al., 1992), and aeration (Wösten et al., 2001). Wösten et al. (1999) tions substantially improved, however, by scaling the PTF and showed that in the case of workability and aeration, about 60% of laboratory-estimated soil moisture retention characteristics using the total variability is caused by errors related to the estimation minimum and maximum values of the fi eld soil water content. of soil hydraulic properties using PTFs. In terms of the predic- Pringle and Lark (2007) and Pringle et al. (2007) stressed the tion of the breakthrough of an inert tracer, this value was about need for site-specifi c PTF development, while Jana et al. (2007) 25%. Vereecken et al. (1992) showed that 90% of the variation in used PTFs derived from very coarse-scale data (watershed or sub- the predicted moisture supply was attributed to estimation errors watershed data) to predict the soil water retention characteristics in hydraulic properties using the PTFs developed by Vereecken at the plot and fi eld scales. et al. (1989, 1990). Recently, Chirico et al. (2010) evaluated the

www.VadoseZoneJournal.org | 796 eff ect of PTF prediction uncertainty on the components of the the dry and wet moisture range of soils. In the following, we defi ne soil water balance at the hillslope scale. Th ey found the simulated the dry range as the range of pressure head values larger than pF 3.0 evaporation to be much more aff ected by the PTF model error and the wet range as the range of values smaller than the inverse than by errors due to uncertainty in the input data. Less informa- of the α parameter in Eq. [1] (i.e., water content values close to tion is available about the role of PTF uncertainty in the prediction saturation). Our choice to focus on the traditional VGM model of the leaching susceptibility of soils to chemicals and hazardous is justifi ed by the fact that this parameterization has evolved to a substances. It is generally accepted that, in these cases, preferential de facto standard, including its frequent use as a benchmark for fl ow phenomena and soil structural properties may play an impor- comparison with other parameterizations. tant role; however, frequently used PTFs do not account for these eff ects in the estimation of hydraulic properties. The van Genuchten–Mualem Model ParameterizaƟ on Several reasons have been advanced to explain the presence of Th e VGM model (van Genuchten, 1980) was developed by com- this unresolved variance (Wösten et al., 2001; Minasny and bining an empirically based power law equation describing the McBratney, 2002; Pachepsky and Rawls, 2004). First, predic- relationship between pressure head, h, and moisture content, θ, tors used in the PTFs are not always determined directly on the with the Mualem (1976) predictive pore-size distribution model soil sample used for measuring the hydraulic properties. Th is for the unsaturated hydraulic conductivity. Th e van Genuchten becomes more important as the spatial and temporal variability model for θ(h) is written as of additional soils information increases and the information θ−θ n −m content is not directly related anymore to the sample on which Sh==+r ⎡⎤1 ()α [1] ⎣⎦⎢⎥ the hydraulic properties were determined. Second, both Wösten θ−θsr et al. (2001) and Pachepsky and Rawls (2004) pointed out that the database size should be large enough and contain all the nec- where S is eff ective saturation (dimensionless), h is the pressure essary parameters that impact predictions of the soil hydraulic head (cm), considered here to be positive under unsaturated condi- properties. Most available databases contain only such classical tions, θr is the residual moisture content (dimensionless), θs is the information as textural properties, organic C, and bulk density saturated moisture content (dimensionless), α is a parameter corre- while lacking relevant information related to structural proper- sponding approximately to the inverse of the air-entry value (cm−1), ties, clay mineralogy, support scale, or other data. A third issue and n and m are shape parameters. In its most general formulation, is the appropriateness and accuracy of the functional relation- the Mualem model can be written as (Mualem and Dagan, 1978) ships used to describe the measured soil moisture retention 2 and hydraulic conductivity data. Despite considerable research ⎛⎞θ h1+b ⎜ 1/ dθ ⎟ KKSl ⎜ ∫0 ⎟ on this issue, we feel that too much attention has focused on ()θ = s ⎜ ⎟ [2] ⎜ θsat 1+b ⎟ ⎜ 1/h dθ ⎟ PTFs that use empirical unimodal equations to describe the soil ⎝⎠∫0 moisture retention characteristic. Because of the relatively suc- K l cessful performance of the traditional VGM parameterization, where s is the saturated hydraulic conductivity, is the pore size most studies of the hydraulic conductivity have focused on this interaction term, and b is the tortuosity. Based on a data set of 45 model, while alternative options have not been explored enough, disturbed and undisturbed samples, Mualem (1976) derived an l K especially for structured media. A fi nal issue is the accuracy, optimal value of 0.5 for when using measured s values as the consistency, and quality of the measurement methods. Accurate matching point and setting the value of b = 1. To obtain a relatively measurement of the MRC and HCC remain very diffi cult at high simple closed-form expression for the hydraulic conductivity, van and low pressure heads, thus causing a lack of reliable data in Genuchten assumed that m = 1 − 1/n. He obtained, in that case, these ranges. Wösten et al. (2001) pointed out that accurate mea- the closed form equation surement of the hydraulic properties is probably the single most ⎪⎪⎧⎫11/− n 2 ⎪⎪l ⎡⎤nn/1()− important factor that could improve the accuracy and reliability KS()= K⎨⎬ S⎢⎥11−− S [3] s ⎪⎪⎢⎥() of PTFs. Noticeably lacking thus far, however, are standardized ⎩⎭⎪⎪⎣⎦ methods that could be used to ascertain the quality of measured data and which could be considered as benchmark techniques. EvaluaƟ on of Pedotransfer FuncƟ ons To materially advance the development of PTFs, we believe that and the Direct Fits progress needs to be made in the four areas of research identifi ed Donatelli et al. (2004) reviewed various indicators that have above. In this review, we have addressed issues associated with the been used to evaluate the performance of PTFs for predicting status and further development of PTFs with regard to the VGM soil water contents and unsaturated hydraulic conductivities. Th e parameterization of the hydraulic properties. In terms of the VGM most frequently used indicator for PTF performance using the parameterization, we have focused especially on issues related to VGM model is the RMSR (see Tables 1 and 2). Th is indicator has

www.VadoseZoneJournal.org | 797 Table 1. Overview of van Genuchten–Mualem parameterizations used to describe the moisture retention characteristic and to develop pedotransfer functions (PTFs), including the mean error (ME), the root mean squared residual for the direct fi t (RMSR), and the root mean squared residual adjusted for the direct fi t (RMSRA). Soil properties Method, No. of Pressure Method, sample used for Study parameters† Database samples n head levels state, sample size‡ ME RMSR RMSRA PTFs§ cm M1, Al-Majou et RA; m = 1 − 1/n, Al-Majou et 320 7 10, 33, 100, PPM; UD; 30–70 cm3 −0.008 0.04 0.053 T, Bd, al. (2008a) θr = fi xed al. (2008a) 330, 1000, OM 3300, 15,000 M2, Børgesen and NN; m = 1 − 1/n Børgesen and 1618 5 10, 30, 100, HW (10–100, UD); –¶ 0.012 0.026# T, Bd, Schaap (2005) Schaap (2005) 1000, 15,000 PPM (400–1000,UD); OM, PPM(15,000, HOR, D); 100 cm3 WC M3, Cornelis et RA; 5 parameters Cornelis et al. 48 9 Five between Sand box (10–100, UD); –0.0070.011 al. (2005) (2005) 10 and 100, PPM (200/300, UD); and 200, 300, PPM (1000/15000, 1000, 15,000 D); 100/20 cm3 M4, Cornelis et RA; m = 1 − 1/n Cornelis et al. 48 9 Five between Sand box (10–100, UD); – 0.0101 0.014 al. (2005) (2005) 10 and 100, PPM (200/300, UD); and 200, 300, PPM (1000/15000, 1000, 15,000 D); 100/20 cm3 M5, Nemes et NN; m = 1 − 1/n Nemes et al. 235 10 0, 2.5, 10, 30, – – 0.013 0.018 T, Bd, al. (2003) (2001) 100, 200, 500, OM, 2500, 15,000, WC 1.5 × 106 M6, Schaap et NN; m = 1 − 1/n Rawls (Schaap 1209 6–13 – – – – 0.02 T, Bd, al. (1998) and Leij, 1998a) WC

M7, Schaap and NN; m = 1 − 1/n UNSODA 554 6–52 multiple ranges 12 diff erent methods – – 0.0404# T, Bd Leij (1998a) (Schaap et al., 0–1.5 × 104 (pressure based, HW, 1998) tensiometer-based, and sand box) M8, Schaap and NN; m = 1 − 1/n Rawls (Schaap 1209 6–12 multiple ranges mixed – – 0.0158# T, Bd Leij (1998a) and Leij, 1998a) 0–1.5 × 104 M9, Schaap and NN; m = 1 − 1/n Ahuja (Schaap 371 6–15 multiple ranges mixed – – 0.0113# T, Bd Leij (1998a) and Leij, 1998a) 0–1.5 × 104

M10, Schaap and NN; m = 1 − 1/n Schaap and 204 9 3.5, 7.5, 13, hanging water column – – 0.011# T, Bd Bouten (1996) Bouten (1996) 20, 30, 40, 60, 90, 150 M11, Schaap and NN; m = 1 − Schaap and 204 9 3.5, 7.5, 13, hanging water column – – 0.016# T, Bd n Bouten (1996) 1/ , θr = 0 Bouten (1996) 20, 30, 40, 60, 90, 150 M12, Schaap and NN; m = 1 − 1/n UNSODA 235 6–52 multiple ranges 12 diff erent methods ––0.012T, Bd Leij (2000) 0–1.5 × 104 (pressure based, HW, tensiometer based, and sand box) M13, Schaap et NN; m = 1 − 1/n ROSETTA 2134 6–52 multiple ranges – ?0 0.012 0.0214 T, Bd WC al. (2001) 0–1.5 × 104 M14, Twarakavi SVM; m = ROSETTA 2134 6–52 multiple ranges – ?0, 0.012 0.0214 T, Bd, et al. (2009) 1 − 1/n 0–1.5 × 104 phdiff .†† WC M15, Weynants RA; m = 1 Vereecken et 166 9 0, 3, 10, 31, Sand box (0–98); PPM –0.00970.013 T, Bd, et al. (2009) al. (1989) 98, 196, 619, (196–619); PPM (0.091) OM 2461, 15,542 (2461.5–15,542); 100 cm3 M16, Weynants RA; m = 1 − 1/n Vereecken et 166 9 0, 3, 10, 31, Sand box (0–98); PPM –0.01340.018 T, Bd, et al. (2009) al. (1989) 98, 196, 619, (196–619); PPM (0.099) OM 2461, 15,542 (2461.5–15,542); 100 cm3 M17, Scheinost RA; m = 1 Scheinost et 397 8 4, 30, 60, 300, PPM (4–600, UD), 365 ––RMSD: T, Bd, et al. (1997) al. (1997) 600, 3000, cm3; PPM (600– 0.019 OM 6000, 15,000 15,000, UD), 10 cm3

www.VadoseZoneJournal.org | 798 Table 1. Continued.

Soil properties Method, No. of Pressure Method, sample used for Study parameters† Database samples n head levels state, sample size‡ ME RMSR RMSRA PTFs§ cm M18, Stumpp NN; m = 1 − 1/n Stumpp et al. 125 7 0, 3, 10, 31.6, suction plate (Hartge – – – T, Bd, et al. (2009) (2009) 100, 316, and Horn, 1971) WC 15,800 M19, Ye et al. NN; m = 1 − 1/n Hanford (Ye et 70 15–25 multiple ranges multistep outfl ow (each ––0.008#T, Bd (2007) al., 2007) 0–1.5 × 104 step to equilibrium), pressure plate M20, Schaap and NN; m = 1 − 1/n UNSODA (Leij 554 6–52 multiple ranges 12 diff erent methods – 0.013 T, Bd, Leij (1998b) et al., 1996) 0–1.5 × 104 (pressure based, HW, WC tensiometer based, and sand box) M21, Wösten RA; m = 1 − 1/n Hypress 5521 14 0–1.6 × 104 multiple methods – – – T, Bd, et al. (1999) OM, HOR n m † RA = regression analysis, NN = neural network analysis, SVM = support vector machines; and are shape parameters, θr = residual moisture content. ‡ UD = undisturbed, D = disturbed, HW = hanging water, PPM = pressure plate extraction, numbers in parentheses refer to pressure head levels (cm). § T = texture, OM = organic matter, Bd = bulk density, HOR = horizon, WC = water content. ¶ No information provided for the study. # Made available by the researchers. †† phdiff = ME available as a function of the pressure head.

Table 2. Overview of diff erent van Genuchten–Mualem parameterizations used to describe the hydraulic conductivity curve (HCC) and to the predict HCC using pedotransfer functions (PTFs), including the mean error (ME) and the root mean squared residual for the direct fi t of the VGM model (RMSR). Soil properties Study Parameterization† Database No. of samples‡ ME RMSR Methods used for PTFs§ K K1, Schaap et al. (2001) NN, VGM, s ROSETTA 235 (4117 data 1.4 mixed –¶ and l fi x e d points) K K2, Schaap et al. (2001) NN, VGM, s ROSETTA 235 (4117 data ?0 0.41 mixed T, Bd, WC and l fi tted points) K K3, Schaap and Leij (1998b) NN, VGM, s UNSODA 245 – 1.08 mixed T, Bd, WC fi tted, l = 0.5 K K4, Schaap and Leij (1998b) NN, VGM, s UNSODA 245 – 0.77 mixed texture, Bd, WC fi tted, l = −1 K K5, Schaap and Leij (2000) NN, VGM, s and UNSODA 235 (at least 5 –1.31#mixed – l = 0.5 fi xed K points) K K6, Schaap and Leij (2000) NN, VGM, s UNSODA 235 (at least 5 –0.75#mixed T, Bd fl exible, l = −1 K points) K K7, Schaap and Leij (2000) NN, VGM, s UNSODA 235 (at least 5 –0.41#mixed T, Bd and l fl exible K points) K K8, Børgesen et al. (2008) NN, VGM, s 152 (4 geometric – 0.81# constant-head and drip method T, Bd, OM, WC and l fl exible mean values) (up to 80 cm) + PTF of HYPRES for 100, 400, 600 cm, 20- by 20-cm cylindrical cores K K9, Weynants et al. (2009) RA, VGM, s Vereecken et 166 – 0.73# crust method and hot air method T, Bd, OM and l fl exible al. (1989) K10, Schaap and van modifi ed VGM UNSODA 235 (at least 5 phdiff .†† 0.26# mixed – Genuchten (2006) K points) K l † NN = neural network analysis, VGM = van Genuchten–Mualem model, RA = regression analysis, s = saturated hydraulic conductivity, and = tortuosity parameter. ‡ K = conductivity. § T = texture, Bd = bulk density, OM = organic matter, WC = water content. ¶ No information provided. # RMSR adjusted using Eq. [5]. †† phdiff . = ME available for specifi c pressure head ranges.

www.VadoseZoneJournal.org | 799 also been used frequently to evaluate the performance of the VGM and coeffi cient of determination (Cornelis et al., 2005) have also model in terms of describing the measured soil hydraulic proper- been used (as shown in Table 1) to evaluate the performance of ties. Th e RMSR is defi ned as direct fi ts of Eq. [1] to measured data. Th is has not been done in a systematic enough manner, however, to make it possible to com- N EM2 ∑i ()ii− RMSR = =1 [4] pare results across databases. N E i M i where i is the th estimated value, i is the th measured value, 6 van Genuchten–Mualem and N is the number of data pairs consisting of Ei and Mi. When the number of parameters is not small compared with the number ParameterizaƟ on of the of available data points, the RMSR can be adjusted according to Hydraulic ProperƟ es ⎛⎞N Here we review available information on the performance of Eq. [1] ⎜ ⎟ RMSRA= RMSR⎜ ⎟ [5] and [3] in describing measured hydraulic properties in databases of ⎜⎝⎠Np− ⎟ soil hydraulic properties and basic soil properties that were estab- where p is the number of parameters. lished to derive PTFs Th e performance was evaluated mainly using the RMSRA and RMSR values, for reasons explained above. We Th e adjusted root mean squared residual (RMSRA) is especially also included ME values when stratifi ed data were available (e.g., useful to compare direct fi ts of the VGM model to MRC data diff erent pressure heads). because the number of measurements per curve for some data sets is oft en quite low. Th e adjustment transforms the RSMR into an Performance of the Classical van Genuchten– unbiased estimator of the goodness-of-fi t that can be compared Mualem ParameterizaƟ on directly with the pure error or variability in the data (Whitmore, Th e VGM model defi ned by Eq. [1] and [3] has been widely used 1991). Diff erences between the RSMRA and RSMR become sub- in the past to describe measured hydraulic functions and is now stantial only when p is comparable to N. the most common parameterization of the hydraulic properties in mathematical models for fl ow and transport in porous media. A few researchers have used the root mean squared deviation Equation [1] has been successfully used to parameterize measured (RMSD) as defi ned by Tietje and Tapkenhinrichs (1993): soil moisture retention data for the development of PTFs for the soil hydraulic properties. As discussed above, its attractiveness resides 1 b 2 RMSD= ()EMii− d h [6] in part in the fact that for specifi c assumptions with respect to its ba− ∫a parameters, a closed-form solution as given by Eq. [3] is obtained. We have include RMSD values in the tables whenever avail- In most comparative studies, the van Genuchten model with fi ve able; however, there is no simple conversion to RMSRA to parameters has been found to provide the largest flexibility in perform comparisons. describing observed moisture retention data (Wösten and van Genuchten, 1988; Vereecken et al., 1989; Cornelis et al., 2005). Bias is oft en evaluated in the literature by calculating the mean Although Eq. [1] with fi ve independent parameters (including m) error (ME), which is defi ned as provides the most fl exibility in terms of describing measured data (van Genuchten and Nielsen, 1985), it is not the most effi cient model N EMii− ME = [7] for deriving PTFs (e.g., Cornelis et al., 2005). Reducing the number ∑ N i=1 of parameters from fi ve to four, for example by assuming m = 1 − 1/n or m = 1, generally leads to improved PTF predictions at the expense Negative ME values indicate an average underestimation of the of less fl exibility in describing the observed data (Vereecken et al., observed moisture content or hydraulic conductivity values, while 1990; Cornelis et al., 2005; Lennartz et al., 2008). Th e diff erence positive values indicate overestimation of the target variables. As between the two most oft en used four-parameter parameterizations errors may cancel out, this indicator is of limited use to evaluate is illustrated in Fig. 1. Figure 1a shows that setting m = 1 creates a bias in the MRC and HCC. Th e ME has also been used in the past symmetrical MRC shape, while S is always equal to 0.5 when h = to show the bias of fi ts or PTF predictions for stratifi ed data such as α−1 (i.e., αh = 1). On the other hand, assuming m = 1− 1/n leads to textural class or pressure head class (Schaap et al., 2001; Stumpp et asymmetrical MRC plots, as shown in Fig. 1b. al., 2009; Twarakavi et al., 2009). With the ME it is also possible to bias-correct the RMSR (i.e., RMSR with the bias removed) (e.g., Al Majou et al., 2008a). In this way, the PTF error can be split into Comparison of van Genuchten–Mualem Model a bias and a level of “uncertainty.” Performance across Databases Tables 1 and 2 provide an overview of the diff erent databases that Other indicators such as the Akaike information criterion were used to either evaluate the quality of Eq. [1] to describe MRC (Cornelis et al., 2005), mean absolute error (Weynants et al., 2009), data or to derive PTFs. Each study or database is referred to below

www.VadoseZoneJournal.org | 800 1, 3, and 4, were converted to centimeters by multiplying by a factor of 10. We used this factor (rather than 9.788) for reasons of convenience.

Th e results from diff erent databases used to fi t Eq. [1] to measured soil moisture retention data are given in Table 1. In each case, the calculated or reported RMSR, ME, and RMSRA values were averaged across N samples. Th e RMSRA values ranged between 0.008 and 0.053, with an average value of 0.019 and a standard deviation of 0.011. Th e lowest RMSRA was obtained for Eq. [1] with fi ve parameters (i.e., independent m and n values), which refl ects the considerable fl exibility of the fi ve-parameter version in describing observed retention data. Despite this Fig. 1. Eff ect of diff erent van Genuchten–Mualem (VGM) parameterizations on the fl exibility, this parameterization has not been used in the shape of the moisture retention curve, where S is eff ective saturation, h is the pressure head, and α, n, and m are VGM shape parameters: (a) eff ect of the value of n assuming literature to develop PTFs. Several reasons are probable. m = 1: n = 1.1 (C), n = 1.3 (K), n = 1.5 (G), n = 1.7 (I), and n = 1.9 (8); (b) diff er- First, no clearly defi ned relationship between m and easily ences between two four-parameter versions: n = 1.3, m = 1 (C); n = 1.3, m = 1 − 1/n measured properties could be derived (e.g., Cornelis et al., n m n m n (@); = 1.7, = 1 (G); and = 1.7, = 1 − 1/ (X). 2001). Second, substantial correlations oft en exist among the VGM parameters, oft en leading to sets of nearly equi- by using M or K followed by a number, depending on whether the fi nal combinations of ttedfi parameters, which in turn study referred to the moisture retention characteristic (M) or the increases ambiguity in the relationships between the VGM param- hydraulic conductivity (K). A distinction was made between data eters and soil properties. Th ird, in most cases only a marginal loss sets used for developing the PTFs and those used for PTF valida- in the RMSR occurred when describing the MRC with four rather tion. Th roughout this review we have used pressure head having a than fi ve parameters (see also van Genuchten and Nielsen, 1985). unit length (cm) to be consistent with Eq. [1]. Matric potentials in Finally, no closed-form equation for K(h) exists when fi ve indepen- kilopascals, as used in some of the studies that are listed in Tables dent parameters are used in the VGM model. Th is hampers the

Table 3. Accuracy of pedotransfer functions (PTFs) for the moisture retention characteristic expressed as the root mean squared residual (RMSR) using diff erent combinations of predictor variables, including textural class (TC), sand, silt, and clay content (SSC), bulk density (Bd), water content (WC), and organic matter content (OM); best refers to the PTF using WC values providing the lowest RMSR. Numbers in parentheses aft er the RMSR indicate pressure head values of the water contents that were used in the PTF. RMSR Study TC SSC SSCBd SSCBdOM SSCBdWC (best) SSCBdOMWC (best) M1, Al-Majou et al. (2008a) 0.045 – – 0.04 – – M2, Børgesen and Schaap (2005) – 0.048† 0.042†‡ 0.038 (100, 1000, 15000)†‡ M3, Cornelis et al. (2005) – – – – – – M4, Cornelis et al. (2005) – – – – – – M5, Nemes et al. (2003) 0.058§ 0.042§ 0.041§ 0.026§ (330, 15000) 0.02–0.026 (100, 330, 15,000) ?0.02 (100, 330, 15,000) M6, Schaap et al. (1998) 0.107 0.104 0.087 – 0.058 (100, 330) – M7, Schaap and Leij (1998a) – – 0.096 (0.093)¶ – – – M8, Schaap and Leij (1998a) – – 0.086 (0.093)¶ – – – M9, Schaap et al. (1998a) 0.061 (0.093)¶ – – – M12, Schaap and Leij (2000) – – – – – – M13, Schaap et al. (2001) 0.078 0.076 0.068 – 0.044 (330, 15800) – M14, Twarakavi et al. (2009) – 0.062 0.053 – 0.038 (330, 15800) – M15, Weynants et al. (2009) – – – 0.036 – – M20, Schaap and Leij (1998b) 0.109 0.096 – 0.068 (100, 330) – † Seven textural classes were used. ‡ Information on soil horizon was included. § Provided by the researchers. Values are calculated as averages across pressure head levels. ¶ Overall average value obtained for the three databases (M7, M8, and M9).

www.VadoseZoneJournal.org | 801 straightforward calculation of unsatu- Table 4. Accuracy of pedotransfer functions (PTFs) expressed as the root mean squared residual (RMSR) rated hydraulic conductivity values. for the hydraulic conductivity curve for diff erent sets of predictors: textural class (TC), sand, silt, and clay contents (SSC), bulk density (Bd), water content (WC), and organic matter content (OM); best refers to the lowest RMSR found for this combination. Numbers in parentheses aft er the RMSR indicate pressure The databases with fewer than 1000 head values of the water contents that were used in the PTF. samples typically provided RMSRA RMSR values <0.02. Two reasons can be invoked to explain this fi nding. First, Study TC SSC SSCBd SSCBdOM SSCBdWC (best) SSCBdOMWC (best) additional samples may create addi- K1, Schaap et al. (2001) – – – – – – tional variability because the space of K2, Schaap et al. (2001) 1.06 1.02 1.05 – 0.9 (330, 15000) – the soil properties is more extensively K3, Schaap and Leij (1998b) – 1.76 1.7 – 1.62 (100, 330) – sampled. Second, smaller databases K4, Schaap and Leij (1998b) – 1.36 1.29 – 1.18 (100, 330) – have typically used the same measure- K5, Schaap and Leij (2000) – – – – – – ment methodology for all MRCs, thus K6, Schaap and Leij (2000) – – 1.16† – – – eliminating the eff ect of measurement K7, Schaap and Leij (2000) – – 1.18† – – – method and possibly also minimizing K8, Børgesen et al. (2008) – – 0.71 0.598 – 0.614 (100) operator errors. Th e eff ect of the mea- K9, Weynants et al. (2009) – – – 1.07† – – surement method needs to be further † RMSR adjusted using Eq. [5]. substantiated. For example, Børgesen and Schaap (2005) and Al Majou et al. (2008a; M1 and M2 in Tables 1 and 3) used a standard set of methods but obtained larger RMSRA values Samples subsequently are moved into low-pressure chambers for than the RMSRAs found by Schaap and Leij (1998a), who analyzed measurements up to approximately 500 cm (pF 2.7). Finally, high- a database consisting of hydraulic properties obtained with 12 dif- pressure chambers are commonly used to determine moisture ferent measurement techniques. content values up to the wilting point and above using disturbed soil samples. Sometimes small undisturbed samples taken from In addition to diff erences in the measurement method, the data- larger samples are used at the lower pressure heads. In employ- bases in Table 1 show considerable variations in the number of ing the diff erent pressure head steps, it is not clear how long the data points used to determine the MRC, the values of pressure equilibration procedure has to be, and generally no or very little head used to determine the soil moisture contents, and the size of information about this is available in the literature or the databases. the sample used at the diff erent pressure head values. Treatment of In case the experiments are terminated too early at high pressure the samples and operation at medium and high pressure head levels head levels equilibrium conditions do not prevail and moisture was also found to deviate considerably among researchers. Some contents are prone to be systematically biased. A lack of equilib- researchers used undisturbed soil samples up to 15,800 cm (pF 4.2), rium may also be an issue, however, at lower pressures in pressure whereas others used small disturbed soil samples at this pressure chambers when relatively large soil cores are used. Unfortunately, head (e.g., Vereecken et al., 1989). It is not clear how much these information about the duration that was used to obtain equilib- factors aff ect a quality assessment of the VGM model in describ- rium conditions at each imposed pressure head was usually not ing moisture retention data and how much variability could be given in the databases. explained by these various measurement and sampling factors. Values of the ME for direct fi ts have, in a few cases, been reported In addition, the measurement method itself oft en diff ered among also, either for the entire database (Schaap et al., 2001) or as a func- researchers. Typically, a combination of three or more techniques tion of the pressure head (e.g., Twarakavi et al., 2009). Mean error was used to obtain the complete soil moisture retention charac- results obtained at diff erent pressure heads are discussed below. teristic in the laboratory (Dane and Topp, 2002). Table 1 shows We included in this analysis data from Weynants et al. (2009) to the diff erent methods that have been used to measure the mois- provide a broader base for comparison. ture retention data. For composite databases such as UNSODA, a mixture of different methods was used for the various soil Th e various databases shown in Table 1 diff er substantially in samples. Other databases (e.g., Vereecken et al., 1989) used the terms of their textural composition, as shown in Fig. 2. Th e data- same set of measurement techniques for all samples. Combining bases of Nemes et al. (2003) and Stumpp et al. (2009) contained diff erent measurement techniques may well introduce substantial many more fi ne-textured soils and fewer coarse-textured soils inconsistencies in the data. For pressure head values up to 100 cm, than the other databases. Extreme examples are the databases soil cores are typically equilibrated on a sandbox apparatus or a of Schaap and Bouten (1996) and Ye et al. (2007), which con- pressure cell or plate at selected pressure heads. Alternatively, or tained only sandy materials. Th e Rawls database (Schaap and additionally, some have used the hanging water column technique. Leij, 1998a) also contained a large number of coarse-textured

www.VadoseZoneJournal.org | 802 soils and practically no silty soils. Th e diff er- ences in textural composition of the various databases partly explain the observed variation in RMSRA values given in Table 1. Hydraulic properties of certain textural classes are some- times more diffi cult to estimate by direct fi tting of Eq. [1] to the available retention data or from derived PTFs, which suggests that the percent- age share of these textural classes in the overall database may aff ect the overall RMSRA. For example, silts oft en show the largest RMSRA values when fi tting Eq. [1] to retention data. Schaap and Leij (2000) found an RMSRA of Fig. 2. Variation in textural composition of samples in the different databases used to fit the van Genuchten–Mualem (VGM) parameterization and to develop pedotransfer 0.0141 for silts compared with 0.0096 for clays. functions (PTFs). M7 refers to UNSODA, and M9 refers to the Ahuja database (Schaap Soils with a sandy and loamy texture produced and Leij, 1998a). M8 (Schaap and Leij, 1998a) and Schaap et al. (1998) both refer to the values of 0.0122 and 0.0119, respectively. Rawls database.

Pedotransfer function estimates of the hydraulic Table 2 shows that the best results for the hydraulic conduc- K l parameters also tend to diff er among textural classes. Schaap et al. tivity data were obtained when both s and were fitted to (1998) showed that water content values could be estimated well the observed hydraulic conductivity data. In many cases, this for sandy loam and clay loam soils, with an RMSRA of 0.05. Th e improved fl exibility has led to physically unrealistic values ofl , RMSRA values for other textural classes ranged between 0.052 as pointed out by Kosugi (1999), Schaap and Leij (2000), and and 0.08, with the largest values obtained for sands and clays. Schaap and van Genuchten (2006). In an analysis of a subset Clays and sands made up only 12.6% of the samples in their data- of the UNSODA database using a similar approach to that base. As such, the proportion of sandy loam and clay loam soils of Mualem (1976), Schaap and Leij (2000) found an optimal was much larger (34.2%), thus infl uencing the overall RMSRA value of −1 for l. Comparisons of the performance of the VGM more than other textural classes. Results obtained with diff erent model using UNSODA have been partly hampered by the fact databases may therefore be strongly aff ected by the proportion of that diff erent subsets with various sample sizes and proportions the various textural classes. While similar eff ects can be expected in textural composition have been used for the evaluations. with respect to the distribution of organic C content and the bulk Information on how these subsets diff er in the composition of density of soils in the various databases, information about this soils and soil properties is not available in the literature. Figure is mostly missing. Literature data mainly provide information on 3 shows that mostly sandy soils have been used in the past to the range and mean values of C content and bulk density but not evaluate the VGM parameterizations. Schaap and Leij (2000) on the frequency distributions of these properties in the databases provided data about variations in the RMSRA across textural and variations in the RMSRA for specifi c ranges. classes. They found that clays and silts produced the largest RMSRA values (0.481 and 0.403, respectively) when the VGM Table 2 summarizes RMSR values obtained by directly fi tting the was fi tted directly to the data, compared with 0.395 and 0.398 VGM model to hydraulic conductivity data. For the HCC, there is for sands and loams, respectively. Th e data set of Børgesen et al. typically only a small diff erence between the RMSR and RMSRA values due to the large number of data points compared with the number of parameters. In the case of the HCC, a direct comparison between the RMSRA and RMSR is therefore allowable.

In general, less information is available on the perfor- mance of the VGM for describing hydraulic conductivity compared with the moisture retention data. Basically, four databases have been used to evaluate direct fi ts of the VGM model to measured data: ROSETTA (Schaap et al., 2001), UNSODA as a subset of ROSETTA (Schaap and Leij, 1998a), and the databases of Børgesen et al. (2008) and Vereecken et al. (1989). Fig. 3. Distribution of the main textural fractions within the diff erent databases for the hydraulic conductivity curve.

www.VadoseZoneJournal.org | 803 (2008) also contained mostly sandy material but no quantifi ca- Earlier work by Vereecken et al. (1989) using 182 measured tion of variations in the RMSRA across textural classes. soil moisture retention data sets of soils with diff erent textures showed that the restriction m = 1 − 1/n led to a somewhat worse Similarly as for the retention data, most databases comprised mixed description than the fi ve-parameter model or the model with m methods for measuring the hydraulic conductivity. Th e databases of = 1 as originally proposed by Brutsaert (1966). Vereecken et al. Børgesen et al. (2008) and Vereecken et al. (1989) are the only ones (1989) therefore suggested not to use the restriction proposed by that have hydraulic properties that were measured on every sample van Genuchten but to fi xm at 1.0 and leave n as a free parameter. with the same measurement techniques within the entire database. As a consequence, n values <1 were frequently obtained for fi ne- An evaluation of the possible eff ect of measurement technique on textured soils, which implies that the diff erential soil moisture the RMSRAs is hence not possible due to a lack of data. capacity goes to infi nity, as discussed by Nielsen and Luckner (1992). Th e main reason for these low n values were caused by the sharp drop in the soil moisture content in the very wet end of the Analysis of the van Genuchten–Mualem moisture retention characteristic, probably caused by structural Parameteriza on at Diff erent Pressure Heads properties producing a heterogeneous porous system. Th e pres- The Soil Moisture Reten on Characteris c. Several studies ence of a heterogeneous porous system may not have been detected, have pointed out weaknesses in the VGM parameterization for however, due to a lack of suffi cient retention data pairs close to describing the wet and dry parts of the soil moisture retention saturation and the eff ect of sample height. Indeed, as shown by characteristic (e.g., Khlosi et al., 2008). To verify these fi ndings, Peters and Durner (2006), when measuring retention curves, the we analyzed literature data that provided information about the water content should be explicitly considered as an integral of the RMSRA obtained by fi tting the VGM model to observed data and equilibrium water content distribution with depth. plotting the RMSRA as a function of the pressure head. Only a few studies provided this information for the MRC, while only Figure 5 shows the ME for diff erent parameterizations of Eq. [1] two published studies could be found where this information as obtained for two models derived from one database. Weynants was available for the hydraulic conductivity. For the purpose of et al. (2009) fi tted Eq. [1] with m = 1 and m = 1 − 1/n to the data- this review, we derived the variation in the RMSRA with pres- base of Vereecken et al. (1989). Both parameterizations tend to sure head using the data of Weynants et al. (2009). Figure 4 shows underestimate the observed soil moisture content close to satura- RMSRA values for the MRC at diff erent pressure heads obtained tion. Th is may be explained by the inability of Eq. [1] to account for for four data sets: UNSODA (Nemes et al., 2003), Tomasella et macroporous fl ow domains. In the very dry range, Eq. [1] tended al. (2003), the Vereecken database (Weynants et al., 2009); and to overestimate the observed moisture content. In the medium Børgesen and Schaap (2005). For Eq. [1] with m = 1 − 1/n and pressure head range, the parameterization with m = 1 provided m = 1, the RMSRA tended to increase with increasing pressure the lowest ME values. head. Th e largest RMSRA values were found for pressure heads between 500 (pF 2.7) and 2500 cm (pF 3.4), with values ranging An accurate description of the MRC in the wet range remains a between 0.015 and 0.037. Th e largest RMSRAs were obtained for challenge and has not yet been resolved. Th is is due not only to the data from Børgesen and Schaap (2005). Relatively low RMSRA problems with the parameterization itself but also to diffi culties in values for Eq. [1] were found at pressure heads in the very wet range, precisely determining the soil water content at low pressure heads with the lowest values obtained by Schaap et al. (2001) for m = near saturation. Incomplete saturation may occur because of wet- 1 − 1/n, and for m = 1 obtained by Weynants et al. (2009) using tability problems and the presence of entrapped or dissolved air, the data of Vereecken et al. (1989). In addition, the lowest variation in RMSRA among the diff erent databases was found at pressure head values between 2.5 (pF 0.4) and 30 cm (pF 1.5). Th e VGM parameterization (Eq. [1]) applied to the UNSODA data of Nemes et al. (2003) gave the largest deviation for the moisture content at saturation. Th e researchers attributed this deviation to the presence of macroporosity. Durner (1994) previously pointed out that for heterogeneous pore size domains, the classical VGM parameterization may fail. Th e limitation of the restriction m = 1 − 1/n in fi tting the wet part was also shown in the data of Vereecken et Fig. 4. Root mean squared residual adjusted (RMSRA) values for the direct fi t of Eq. [1] to al. (1989) and Cornelis et al. (2005). moisture retention characteristic data as a function of the pressure head.

www.VadoseZoneJournal.org | 804 both of which will aff ect the measured satu- rated water content, θs. In addition, exact measurement of changes in the water content at low pressure heads is very diffi cult, par- ticularly when the pressure head is less than the sample height. Moreover, inverse meth- ods such as the multi-step outfl ow method are not model free and typically invoke a relationship between the pressure head and moisture content that ignores the presence of macroporous regions.

We note that air entrapment is generally Fig. 5. Mean error (ME) of the water content at diff erent pressure head values for two diff erent far more of an issue at the plot or fi eld scale, parameterizations of the van Genuchten–Mualem (VGM) model. thus causing laboratory measurements of the saturated water content, or even of the labo- ratory-measured satiated water content that accounts for air entrapment (e.g., Luckner et al., 1989; Hillel, 2004), not to be representa- tive of fi eld conditions. Th is also suggests that extrapolation of laboratory-measured θs and K s values, as well as PTFs derived from such measurements, to transient fi eld moisture con- ditions may be problematic.

Weaknesses in describing the dry end of the MRC has been mainly attributed to the fact that water retention in that part is no Fig. 6. Root mean squared residual (RMSR) values of the unsaturated hydraulic conductivity (K) longer determined by capillary forces but by for the direct fi t of the van Genuchten–Mualem (VGM) model to hydraulic conductivity curve adsorptive forces (Li and Wardlaw, 1986; data as a function of the pressure head (pF). “Gardner, Vereecken” refers to the Gardner equation Lenormand, 1990; Nimmo, 1991). In addi- (Eq. [8]) as it was used by Vereecken et al. (1990) and Weynants et al. (2009). tion, there are only a few data sets containing measurements for pressure head data beyond 15,800 cm (pF 4.2). Khlosi et al. (2008) pointed out diffi culties in of Schaap and van Genuchten (2006) to account for the eff ects measuring moisture contents at these high values. Several studies of macroporosity near saturation. Th e RMSR values of Schaap et have shown that the residual moisture content in Eq. [1] is diffi - al. (2001) were taken from their Fig. 4. We included in Fig. 6 also cult to determine using inverse methods. As such, Nimmo (1991), the results for the Gardner equation of the hydraulic conductiv- as well as many others (e.g., van Genuchten and Nielsen, 1985) ity because this equation was found to perform equally well or suggested considering θr, which strongly infl uences the MRC better than the VGM parameterization when evaluated across the description in the dry range, as a VGM fi tting parameter. entire database (Vereecken et al., 1990; Schaap and Leij 1998b). Th e Gardner HCC is given by Th e above considerations suggest that the invoked experimental K Kh s techniques, procedures, and protocols for measuring moisture ()= n [8] retention data probably contribute to the RMSRA when fi tting 1+()Bh g Eq. [1] to data and that they may introduce inaccuracies when B n estimating parameters. where and g are fi tting parameters.

K l The Hydraulic Conduc vity Curve. Figure 6 shows variations Except for the classical VGM model with fi xed s and parameters, in RMSR values for the hydraulic conductivity as a function of all parameterizations in Fig. 6, including the Gardner HCC, gen- the pressure head for fi ve cases. eTh particular cases involve the erated lower RMSR values with increasing pressure heads up to a K l classical VGM parameterization with fi xed s and parameters, value of 2500 cm (pF 3.4). Th e best results across the entire range K l the VGM with adjustable s and values (Schaap et al., 2001; of pressure head values were obtained with the model proposed by Weynants et al., 2009), and the modifi ed VGM parameterization Schaap and van Genuchten (2006) that included provisions for

www.VadoseZoneJournal.org | 805 fl ow through macropores (see also below). All models, except for the classical VGM model, produced the highest RMSR in the wet and very dry ranges. Th e lowest RMSR values were obtained in the pressure head range between 100 (pF 2) and 2500 cm (pF 3.4). Th is diff ers from analyses of the MRC, which showed the largest residuals in this pressure head range (Fig. 4). Notice that the Gardner equation performed quite well in the very wet range, provid- ing RMSR values that were very similar to those of the improved model of Schaap Fig. 7. Mean error (ME) values of the unsaturated hydraulic conductivity (K) at diff erent pressure and van Genuchten (2006). Th is may be heads (pF) for the direct fi t of the van Genuchten–Mualem (VGM) model using a fl exible saturated explained by the presence of the B param- K l hydraulic conductivity s and tortuosity parameter to observed hydraulic conductivity data (Wey- eter in Eq. [8], which may be interpreted in nants et al., 2009). a way similar to the α parameter in Eq. [1].

Figure 7 shows the variation in ME across diff erent pressure head a multimodal retention function by superposition of subcurves levels for the database of Vereecken et al. (1989) as analyzed by described by Eq. [1]. Using example soils, he showed that conduc- K Weynants et al. (2009) using the VGM model with variable s tivity estimates based on the classical VGM model can diff er by and l parameters. At relatively low pressure head values, the VGM orders of magnitude compared with the van Genuchten model model clearly underestimated the observed hydraulic conductivity with a multimodal pore size distribution. Recently, Priesack and as indicated by the negative ME values. Th e ME for pressure head Durner (2006) extended this approach of heterogeneous pore sys- values between 30 (pF 1.5) and 200 cm (pF 2.3) changed from a tems to the mathematical description of the VGM HCC. negative to a positive value. For larger pressure head values (except at pF 4.2), the ME remained positive. Th is was confi rmed by the In addition to a unimodal pore size distribution function, the analysis of Schaap and van Genuchten (2006) of the UNSODA VGM parameterization assumes that the pore system can be database, which revealed a substantial underestimation between 0 described as a bundle of capillaries with fl ow processes governed and 30 cm (pF 1.5), a slight overestimation between 30 and 1000 cm, by the principles of capillary fl ow. Moreover, capillaries are either and again an underestimation beyond 1000 cm (pF 3) (Fig. 7). Th ey fi lled with water or with air, without accounting for fi lm fl ow also showed that the traditional VGM model substantially overes- that may dominate in relatively dry soils. Tuller et al. (1999) pro- timated the hydraulic conductivity between 0 and 1000 cm and posed a new concept of pore space geometry that was composed severely underestimated the hydraulic conductivity beyond 1000 cm. of angular pore cross-sections connected to slit-shaped spaces with an internal surface area to led to a better description of liquid dis- Extensions of the Classical van Genuchten– tribution, water fl ow, and solute transport in partially saturated Mualem Model porous media. Th eir approach provides a better means of combin- Th e following is an overview of several approaches that were developed ing adsorption and capillary processes and is more consistent with to overcome the limitations of the VGM model in describing and pre- the observed pore shapes obtained from soil core slices. dicting the hydraulic properties across the entire range of moisture contents, with specifi c attention to the wet and dry ends of the curves. Tuller and Or (2001) compared their approach with the VGM model to describe experimental data of the hydraulic conductiv- ity. Th ey considered the MRC to consist of diff erent ranges based Heterogeneous Pore Size Systems on liquid-filling stages, each described using a simple algebraic and Pore Geometry expression relating the relative liquid saturation to the pressure Th e use of the VGM model assumes that the pore size distribution head. Th ey fi tted the parameters of these algebraic expressions to of the soil can be described by a single unimodal pore size dis- the measured soil moisture retention data, which were then used as tribution function. Durner (1994) emphasized that undisturbed input parameters in the derived sample-scale hydraulic conductivity soils frequently have pore systems that cannot be described with equations. Very good agreement was found between the predictions a unimodal pore size distribution function. Aggregation and bio- and measured data. For each of the analyzed soil samples, additional logical soil-forming processes may lead to secondary pore systems measured or predicted soil properties were available, which makes involving larger pores. To describe the presence of a heterogeneous it possible to derive PTFs between the MRC parameters and clay pore size system in a satisfactory manner, Durner (1994) proposed content, specifi c surface area, and porosity or other data. Four model

www.VadoseZoneJournal.org | 806 parameters were adjusted to the data: a dimensionless slit length fi ndings that the VGM model does not perform well for macropo- parameter β, a γ distribution parameter ω, the air entry value μb, and rous soils or soils with heterogeneous pore size distributions near a distribution overlap parameter λ that relates a dimensionless slit- saturation (e.g., Durner, 1994). Mohanty et al. (1997) used visual spacing parameter αs to the largest pore and the mean pore length. inspection of the hydraulic functions to identify a breakpoint in Analysis of their data (Tuller and Or, 2001, p. 1268, Tables 2 and 3) the pressure head that separates the fl ow domain into capillary- showed high positive correlation between the β and λ parameters driven and a non-capillary-driven regimes. Th e VGM hydraulic and the clay content, while a negative correlation was found between parameters were used to estimate the hydraulic properties of the μb and the surface area. Th ese results are encouraging and suggest capillary fl ow domain, whereas an exponential equation with three that further work is needed to relate these parameters to MRC data parameters was used to represent the non-capillary-fl ow domain. obtained on natural soil samples. Th ese parameters included the breakpoint soil water pressure head, its corresponding hydraulic conductivity, and a fi tting parameter representing the eff ective macroporosity or other structural fea- Improving the Descrip on in the Wet Range tures contributing to the non-capillary-dominated fl ow domain. Two problems may arise close to saturation and question the use of the VGM model: (i) structural pores in the soil are activated and Finally, Schaap and van Genuchten (2006) introduced a small may lead to noncapillary fl ow, for which the VGM model is not but constant pressure head in the VGM model, as suggested by appropriate; and (ii) noncontinuity of the air phase may lead to Vogel and Cislerova (1988), to account for the eff ects of macro- erroneous estimates of the hydraulic conductivity. porosity on K(h) near saturation, thereby removing several of the shortcomings in the original VGM model. Using a detailed neural Macroporous Flow and the Air-Entry Value. Diff erent solu- network analysis of part of the UNSODA database, they found the tions have been proposed for the fi rst problem. Dual-permeability macropore contribution to the hydraulic conductivity to be most models have been developed over the years in attempts to predict signifi cant between 0 and 4 cm (pF 0.6), although the eff ects of fl ow and transport in macroporous soils and fractured rock (Tsang macroporosity were visible until pressure heads of about 40 cm and Pruess, 1987; Gerke and van Genuchten, 1993; Larsbo et al., (pF 1.6). Th e modifi ed conductivity model signifi cantly reduced 2005; Jarvis, 2008). Th ese models assume that structured media the RMSRA for all textures (range 0.242–0.275), while also pro- consist of two overlying but intertwined continua at the macro- ducing only very small systematic errors between 0 and 15,800 cm scopic scale: a macropore or facture pore system and a permeable (pF 4.2) (Fig. 6). Schaap and van Genuchten (2006) used a value h matrix pore system. Th e Richards equation with VGM parame- of 4 cm (pF 0.6) for the minimum capillary height, a, in their terization is typically used for the matrix domain, while diff erent modifi ed model. Børgesen et al. (2006) found a similar value for h options exist for simulating water fl ow in the macropore domain. a in their analysis of a diff erent database. Th ey similarly focused Gerke and van Genuchten (1993) also used VGM parameteriza- on the HCC near saturation by implementing a scaling approach tions and the Richards equation for the high-permeability domain, to account for soil structural eff ects on K. Th e value of 4 cm (pF along with a coupling term to allow water exchange between the 0.6) is similar to the value of 6 cm (pF 0.8) used by Weynants et al. macropore and matrix domains. Jarvis (2008) based his model on (2009) to exclude the eff ects of macroporosity in their analysis of the generalized Kozeny–Carman equation, along with a pore-scale the Vereecken database, and the values of 6 (pF 0.8) to 10 cm (pF 1) tortuosity factor initially suggested by Fatt and Dykstra (1951) and used by Jarvis (2007) as limits of the macropore region (equivalent the soil water retention function model of Ross et al. (1991). Using to macropores having equivalent cylindrical diameters of 0.3–0.5 a data set of 12 Swedish soils, Jarvis (2008) was able to relate the mm, as suggested by the Laplace–Young equation). pore size distribution index in the Ross et al. (1991) model to clay content and macroporosity. Unfortunately, broad applicability of In addition, introduction of an air-entry value provides better this approach using PTFs is questionable because of the current numerical stability to numerical solutions of the Richards equa- lack of reliable hydraulic data in the wet range (see also below). tion, in particular when applied to the hydraulic properties of very fi ne-textured soils, when n is less than about 1.2. Th e discon- Using the saturated hydraulic conductivity as a matching factor for tinuity in the diff erential soil water capacity at saturation leads predicting the unsaturated hydraulic conductivity function with to strong nonlinearities in the hydraulic conductivity function, Eq. [3] or some other equations for K(h) is problematic and of little K(h), near saturation (Vogel and Cislerova, 1988; Luckner et al., value in macroporous soils (e.g., Ehlers, 1977; van Genuchten and 1989; Fuentes et al., 1992). Th erefore, Vogel and Cislerova (1988) Nielsen, 1985; Jarvis et al., 2002). Based on an analysis of fi eld and Vogel et al. (2000) introduced a very small non-zero mini- h data, Mohanty et al. (1997) suggested that the VGM parameteriza- mum capillary height, a, in Eq. [1]. Very similar to the approach tion should only be used for capillary-dominated fl ows at pressure by Vogel et al. (2000), Ippisch et al. (2006) introduced an air- n h heads >3 cm (pF 0.5). When gravity fl ow dominates (pressure entry value in Eq. [1] for cases when < 2 or α a > 1. Th ese heads <3 cm), other more appropriate models for retention and modifi cations do not cause a major change in the retention func- the hydraulic conductivity should be used. Th is again supports tion as such but lead to improved predictions of the hydraulic

www.VadoseZoneJournal.org | 807 conductivity function while also improving the performance of values obtained with Eq. [1] were generally larger than the inverse methods for measuring the hydraulic properties of fi ne- RMSRs obtained using the PTF point estimation method. Th ey textured soils. Weynants et al. (2009) compared the model of attributed this to imperfect fi ts of the VGM parameterization to Ippisch et al. (2006) with the MVG model, the Mualem–Kosugi retention data at 15,000 cm (?pF 4.2). model (Kosugi, 1996), and the Gardner (1958) model using the database of Vereecken et al. (1989). For the MRC, they found Several studies have addressed the dry region issue by proposing that Eq. [1] with m = 1 and the MVG parameterization as used by improved MRC descriptions across the whole range between satu- Weynants et al. (2009) provided the lowest RMSRA values. For ration and oven dryness. Improvements in the dry end have been the hydraulic conductivity data, only values for pressure heads obtained by incorporating an additional expression that describes >6 cm (pF 0.8) were used. In this case, the lowest RMSRA was the adsorption of water (Campbell and Shiozawa, 1992; Fayer and found for the model of Ippisch et al. (2006). Th ey found, however, Simmons, 1995; Cornelis et al., 2005). that introduction of the air-entry value improved the prediction of only the HCC and not the MRC. In a more recent study, Khlosi et al. (2008) compared the perfor- mance of eight diff erent MRC models (including those that account Desatura on and Pore Size Distribu on. Discontinuities in the for adsorption of water) in describing retention data between sat- non-wetting (air) phase in the near-saturated part of the MRC may uration and oven dryness. Th ey found that a combination of the lead to erroneous estimates of the hydraulic conductivity, irrespec- adsorption equation of Campbell and Shiozawa (1992) with the tive of whether one or many pore size distribution models are used model of Kosugi (1999) provided the best description for 137 MRCs (Corey, 1992; Jana et al., 2007). Th is is due to the fact that, at high taken from the UNSODA database. Rossi and Nimmo (1994) also water contents (above approximately 85% saturation), air is not free proposed two models for the dry end, both based on the observation to enter all parts of the medium. As explained by Corey (1992), that the retention curve in the dry range is approximately propor- regions containing large pores may be entirely isolated from the air tional to the logarithm of the suction. Th ey indicated that their phase by surrounding water-fi lled pores. One may therefore question models can be easily added to the VGM parameterization. whether data close to saturation should be used to predict hydrau- lic conductivities based on capillary phenomena only, and whether or not two-phase fl ow models should be used. Th e uncertainty in Flexible Interpreta ons of the Classical van determining the very wet part of the MRC motivated Corey (1992) Genuchten–Mualem Parameteriza on to extrapolate retention values down to the saturated value. Earlier Since its publication in 1980, the VGM model has been the subject work by Henderson (1949) and White et al. (1972) demonstrated of many studies analyzing its capability to describe the unsaturated that retention of soil water in the very wet range is controlled by the hydraulic conductivity of natural soils. Mualem (1976) initially tested boundary area to volume ratio of the sample but not by the pore size his predictive hydraulic conductivity model using a limited database distribution. At water contents small enough to provide a fi nite air involving 45 MRC and HCC data sets mostly obtained from coarse- permeability, the curves are no longer aff ected by the boundary area textured, uniformly packed soil materials. By fi tting the model with to volume ratio. Th e presence of boundary eff ects seems to have been l as a variable parameter, he found an average value of 0.5. Th e model confi rmed by lattice Boltzmann simulations of pore-scale air and of Mualem (1976) was tested on diff erent soils and found to perform water redistribution in a glass bead system on wetting and drying better than other theoretical models such as the models proposed by (Schaap et al., 2007). Better correlations between water retention Burdine (1953) and Childs and Collis-George (1950) (Mualem, 1976; and hydraulic conductivity curves were obtained by excluding water van Genuchten and Nielsen, 1985; Vereecken, 1995). content data >85% of the saturated value. Using the VGM model with l = 0.5 to predict the measured unsat- urated hydraulic conductivity, however, frequently leads to over- or Improving the Descrip on in the Dry Range underpredictions (Schaap and Leij, 2000). In addition, consider- Several researchers have reported diffi culty in describing the dry ing l as a fi tting parameter brings much more fl exibility, as shown end of the MRC using Eq. [1]. Th e residual soil moisture con- in Fig. 8. An increase in l causes a steeper decrease in the relative tent, being the moisture content when the pressure head goes hydraulic conductivity as a function of relative saturation, S. Th e to infi nity, has been found to be an ill-defi ned parameter, with decrease in the hydraulic conductivity near full saturation is con- values oft en becoming negative during optimization unless spe- trolled by the value of n to a much larger extent than by the value cial precautions are taken (e.g., Khlosi et al., 2008). To prevent of l (e.g., Wösten and van Genuchten, 1988). negative values, the lower boundary of the residual moisture con- tent is usually fi xed to zero, which frequently leads to inaccurate Th erefore, many studies have revisited the value of l in Eq. [2] descriptions of the observed data and unrealistic paths of the and [3]. For example, Vereecken (1995) found values of l strongly curve (Pachepsky et al., 1996; Khlosi et al., 2006; Haverkamp deviating from 0.5, including many negative values, using a set et al., 2005). Børgesen and Schaap (2005) showed that RMSR of 185 measured hydraulic conductivity data sets determined on

www.VadoseZoneJournal.org | 808 conductivity can be described. Th e Gardner equation was also found to better describe the hydraulic conductivity data of 245 K(h) data sets from the UNSODA database (Schaap and Leij, 1998a). Th e RMSR of the direct fi t was equal to 0.22, compared with 1.08 for the VGM model with l = 0.5, and 0.77 for the VGM model with l = −1. Th e PTFs using textural properties and soil moisture data were found to better predict the unsaturated hydraulic conductivity than the traditional VGM parameterization. Decoupling of the MRC and HCC functions was also suggested by Fuentes et al. (1992).

Pachepsky et al. (1999) noted that while much eff ort has focused on l Fig. 8. Eff ect of the tortuosity parameter l on the relative hydraulic the parameter in Mualem’s predictive model for the unsaturated K K conductivity ( / s) computed with the van Genuchten–Mualem hydraulic conductivity, almost no research has been performed on (VGM) model at diff erent eff ective saturation (S) values. Results the tortuosity parameter b in Eq. [2]. Pachepsky et al. (2000) related n are plotted for shape parameter values of 1.3 (solid lines) and 1.7 the unsaturated hydraulic conductivity to soil water retention using (dashed lines). a capillary bundle model with a microscopic connectivity correction, γ, expressed as a power law function of the pore radii: undisturbed soil columns. Negative values for l in the Mualem model were also found by Schuh and Cline (1990), Kaveh and van KKθ h−−γ2 = s ∫ ()ττd [9] Genuchten (1992), Yates et al. (1992), Kosugi (1999), and Schaap 0 and Leij, (2000), among others. A negative value of l implies that the connectivity of water-fi lled pores increases with decreasing soil where τ is the integration variable. Th e term Sl was excluded to moisture content, which intuitively does not seem realistic. avoid using two diff erent terms in an equation to simulate the same phenomenon of tortuosity. Th e model was applied to data from Improved descriptions of measured hydraulic conductivity data are 147 soil horizons, providing an accurate approximation so that the l K K possible by treating both and s as adjustable parameters in the RMSR in the estimated log10( ) was 0.21. Two parameters of the VGM parameterization. Schaap and Leij (2000) showed that opti- model appeared to be closely correlated so that only one connec- mization of both parameters improved the RMSR by a factor of 3.2. tivity parameter was needed. Reducing the number of parameters l K Th e parameters and s hence should be treated as empirical shape from two to one and refi tting the model to the data decreased the factors without much physical meaning. Th is ultimately implies that accuracy of the estimates, as the RMSR increased from 0.21 to the model of Mualem (1976), as well as related statistical pore-size 0.31; however, the RMSR remained about 1.5 to 2 times lower than distribution models published previously or later, provides a sim- the RMSR of the best saturated hydraulic conductivity estimates plistic conceptualization of the hydraulic conductivity of a porous obtained with the UNSODA database. medium (Schaap and Leij, 2000; Tuller and Or, 2001).

K Treating s, similarly to the saturated or satiated water content, 6 Accuracy and ValidaƟ on of θ s, as an empirical parameter with only limited physical meaning Pedotransfer FuncƟ ons for the had already been advocated earlier by van Genuchten and Nielsen (1985) and Luckner et al. (1989). In a more recent study, Wagner van Genuchten–Mualem Model et al. (2001) compared diff erent PTFs to predict the unsaturated Accuracy hydraulic conductivity. Th ey found that the best estimates of K(h) Several reviews about the accuracy and reliability of PTFs for the K were obtained when s was predicted from soil properties rather VGM model have been provided (e.g., Wösten et al., 2001; Schaap, than using the measured saturated values. 2004; Donatelli et al., 2004). In the literature, the term accuracy is related to the comparison between predicted and measured values of Another option to improve descriptions of the soil water retention moisture content or hydraulic conductivity that were used to develop and unsaturated hydraulic conductivity functions is by decoupling the PTF. Reliability is related to the evaluation of PTFs on measured the retention and conductivity predictions. Th is decoupling would values that are diff erent from those that were used to develop the PTFs provide more fl exibility in describing both properties. For example, (Wösten et al., 2001). Reliability studies are typically validation studies and as mentioned above, Vereecken et al. (1990) showed that the such as those performed by Tietje and Tapkenhinrichs (1993), Wösten Gardner equation for K(h) provides better prediction of the hydrau- et al. (2001), and Wagner et al. (2001). lic conductivity than the VGM model. Th is is partly due to the Gardner (1958) equation providing more fl exibility in fi tting the As for the evaluation of the direct fi t of the VGM model to hydraulic wet range. By letting B in Eq. [8] become >1, sharp drops in hydraulic data, evaluations of the performance of PTFs have been performed

www.VadoseZoneJournal.org | 809 mainly in terms of the RMSR. Other indicators are the ME (Nemes et al. (2003). Th ey obtained the lowest RMSR value of all the studies et al., 2003), the coeffi cient of determination (e.g., Stumpp et al., presented in Table 3 using three moisture content values in addition 2009; Cornelis et al., 2001), the RMSD (Tietje and Tapkenhinrichs, to SSC and Bd. Th e combined use of OM and soil water content, in 1993), the mean absolute error (Weynants et al., 2009), the mean addition to texture and bulk density, showed a somewhat higher relative error (Weynants et al., 2009), and the model effi ciency RMSR than the case with only SSCBdWC (see Nemes et al., 2003, (Weynants et al., 2009). In our analysis, we have focused on the p. 1097, Fig. 2a). Th e data show that OM as a predictor variable RMSR because this indicator has been reported widely in the litera- relative to the introduction of WC data seems to be less important; ture. In addition, we have included data on the ME as a function of however, not all moisture content values are equally eff ective in the pressure head for both the MRC and the HCC. reducing the RMSR. Børgesen et al. (2008) also showed that, in general, the lowest RMSR values (highest accuracy) were obtained Pedotransfer functions providing point estimates of the water for the data set of Nemes et al. (2001) compared with the values content or hydraulic conductivity tend to be more accurate than obtained for a European data set derived from HYPRES, an inter- estimates based on parametric PTFs (e.g., Pachepsky et al., 1996). continental data set, or a combination of these data sets using PTFs For example, Tomasella et al. (2003) compared two different with diff erent input levels. approaches for obtaining analytical expressions of the whole MRC. In the fi rst approach, moisture content values at specifi c pressure Study M6 (Schaap et al., 1998) provided information on the distri- heads were estimated by PTFs, aft er which the VGM model was bution of the RMSRA across the USDA textural classes. For the fi tted to these data. In a second approach, the VGM model was PTFs using the highest level of input data (SSC, Bd, and WC at 330 directly fi tted to the data, aft er which the fi tted parameters were cm), the lowest RMSRA was obtained for the sandy loam and clay related to soil properties. Th ey found that the RMSR of the direct loam classes with a value of 0.050. Th e overall proportion of the two estimation technique (the second approach) was 0.098, whereas textural classes in the data set was 34%, thus infl uencing strongly the RMSR of the fi rst approach was 0.036. Th e largest diff erence the overall RMSRA value. Th e largest RMSRA was found for sandy in RMSR between the two approaches was found for pressure head soils (0.080). Th is can be explained by the large variation in particle values of 0, 60 (pF 1.8), and 15,800 cm (pF 4.2). Th e RMSR values sizes (fi ne to coarse) within the sand fraction. Th is will lead to a of the development data set for the two approaches were 0.04 and larger air-entry value for the sandy soils within the coarser fraction. 0.182, 0.035 and 0.111, and 0.025 and 0.182, respectively. Of the validation data set, the RMSRs for the parametric approach were Table 4 shows that even the best PTF models for the HCC still again larger, but the pressure heads at which this occurred were in showed RMSR or RMSRA values that were twice as large as the the midrange, i.e., 100 cm (pF 2), 330 cm (pF 2.5), and 1000 cm RMSR or RMSRA values obtained for the best direct fi t (Table (pF 3). Tomasella et al. (2003) suggested that moisture contents at 2). Using moisture content values and a more detailed particle size diff erent pressure heads may be controlled by diff erent soil proper- distribution (e.g., Børgesen et al. [2008], not shown in the table) ties and thus that using PTFs for a specifi c pressure head allows a did not necessarily improve the estimates. Th is is diff erent from the better combination of these properties. Th e large errors obtained case of the PTFs used to estimate soil moisture contents, as pre- with the parametric methods for specifi c pressure heads may well sented in Table 3. Here the introduction of moisture content and be due to the inappropriate parameterization in these regions. organic matter substantially reduced the RMSR values. Th e RMSR or RMSRA values of PTFs using textural information, bulk den- Table 3 shows that the largest improvements in predictions of the sity, and organic matter (K8 and K9 in Table 4) were of the same MRC have been obtained when water content information was order as the PTF using only textural information and Bd. Although included in the PTF. Th e RMSR values for that case are still con- Weynants et al. (2009) only considered hydraulic conductivity data siderably larger than the values obtained for the direct fi t as given in the matrix fl ow domain, their obtained RMSR value was con- in Table 1. Th e potential of combining organic matter (OM) and siderably larger than the value obtained in Study K8. On the other the soil water content to improve a PTF has not yet been fully hand, K8 used only a limited number of measured data points for explored. Only a few researchers (e.g., Nemes et al., 2003; Børgesen the hydraulic conductivity, eventually leading to lower RMSR values. et al., 2008) have evaluated the combined eff ect of using OM and Studies K1 and K5 could not be evaluated because no PTF was K water content data in addition to textural data and bulk density. In required due to the fact that s was fi xed to the measured value and the study of Børgesen et al. (2008), the RMSR obtained with this l was fi xed at 0.5. Only Study K6 provided information on the dis- information was not better, or was even slightly worse, than PTFs tribution of the RMSRA across textural classes. No distinct pattern based on either sand, silt, and clay (SSC), bulk density (Bd) and OM could be identifi ed. For the highest level of input that included the (SSCBdOM) or SSC, Bd, and water content (WC) (SSCBdWC). It VGM parameters, the lowest RMSRA (0.67) was obtained for the should be noted that also information on the type of soil horizon was class of silty soils. In the case of Study K8, the use of seven textural used as a predictor but its impact on reducing the RMSRs appeared classes and water contents measured at 100 (pF 2), 500 (pF 2.7), and to be relatively small based on the data presented in Børgesen et al. 15,000 cm (?pF 4.2) did not improve the results compared with the (2008). A similar result with respect to OM was obtained by Nemes PTF using the water content at 100 cm.

www.VadoseZoneJournal.org | 810 In the following, we analyzed the accuracy of PTFs to estimate spe- intervals coincided with the number of data available. Textural cifi c ranges of the MRC and HCC with due attention to the wet and classes with more data (sandy loam and loam) resulted in narrower dry ranges using RMSR or RMSRA values as the basic indicator. confi dence intervals than textural classes with fewer data (e.g., clay, Wösten et al. (2001) provided RMSR values for moisture contents at silty loam). In addition, they suggested that not only the number 100 (pF 2) to 330 cm (pF 2.5) and 15,000 cm (?pF 4.2) of 19 diff er- of samples is important but also the variability in soil hydraulic ent PTFs based on point estimation. Th e mean value of the RMSR properties within a textural class. It should be noted, however, that across all PTFs, MRMSR, was approximately 0.05. Evaluating only the largest uncertainty occurred for the fi ner textured soils. As the RMSR for the directly estimated soil water contents at 330 and mentioned above, the standard VGM parameterization does not 15,000 cm resulted in a MRMSR of 0.049 (SD = 0.02, n = 7) and perform well for those soils because of pore heterogeneity and pos- 0.45 (SD = 0.03, n = 11). Calculating the MRSR for the complete sible limitations in the VGM model for small n values. It would moisture retention characteristic gave a value of 0.059 (SD = 0.03, n therefore be worthwhile to examine diff erent parameterizations = 10). A comparison of the RMSR for the point estimations with the that may perform better in describing the wet range of the MRC. MRMSR for the complete MRC obtained from the Ahuja (0.061), In addition to a statistical analysis of the residuals between the Rawls (0.086), and UNSODA (0.096) databases, all cited in Schaap measured and the fi tted values, the VGM parameterization may and Leij (1998a), shows that the latter are still larger. Th is may be due provide insight into the soil properties infl uencing the behavior of to inappropriate parameterization or having a combination of diff er- the MRC in the wet range. Th is requires, of course, that relevant ent MRC measurement methodologies. Inspection of the fi gures in structural information is available. Schaap and Leij (1998b) suggests that the largest uncertainties still occur in the wet and dry range extremes. Analyzing Schaap and Leij (1998a, Fig. 4) showed that, in gen- eral, either the dry or wet range of the MRC showed the largest Rajkai and Várallyay (1992) used regression equations to estimate uncertainty. Th e range between 30 (pF 1.5) and 1000 cm (pF 3) soil water contents at diff erent pressure heads for 270 soil samples. showed, in most cases, the lowest uncertainty, indicating that Th ey found that the largest uncertainty, expressed as the coeffi cient relevant information was available and that the VGM parameter- of determination (R2), occurred at intermediate pressure heads (30 ization performed well. Th is contradicts the fi ndings of Rajkai and [pF 1.5] to 1000 cm [pF 3]), with the lowest value of R2 around Várallyay (1992) and Nemes et al. (2003), among others. 100 cm (pF 2). Th ey did not make a distinction between textural properties of the soils. Donatelli et al. (2004) developed a set of Figure 9 shows the variation in RMSR values as a function of pres- basic indices to evaluate 14 PTFs to estimate moisture content sure head for PTFs using texture and bulk density as predictor values at 330 and 15,000 cm. Th e indices were mainly used to rank variables. As expected, the highest RMSR values were found for the diff erent PTFs in terms of their ability to estimate moisture PTFs that used only soil texture (Fig. 9). For most of the PTFs, we content values. No evaluation of the VGM approach related to the observed an initial increase in the RMSR with pressure head, fol- complete MRC and HCC was performed. lowed by a decrease in the RMSR for pressure head values >2500 cm (pF 3.4). Th is behavior was also observed for the RMSRA in Schaap and Leij (1998a) analyzed the accuracy of PTFs at estimat- the case of the direct VGM fi t to measured moisture retention ing the MRC based on the VGM parameterization using three data. Th e PTFs of Stumpp et al. (2009) and Nemes et al. (2003) diff erent data sets. As predictors for the PTFs they required the showed a slight decrease in the RMSR with increasing pressure availability of sand, silt, clay, and bulk density, and they required head in the very wet range. The higher values near saturation at least six water retention points for the MRC. Th ey found that calibrated PTFs predicted the soil moisture con- tent with RMSRs between 0.06 and 0.10 cm3 cm−3. Cross-validating PTFs among the data sets showed that the RMSR increased to values between 0.08 and 0.13 cm3 cm−3. The extent of the confidence intervals for water retention data was found to vary among the textural classes. Th e intervals were found to be relatively narrow for the loamy sand and loam samples (0.04 3 −3 3 cm cm ) but increased to 0.10 cm Fig. 9. Variation in the root mean squared residual (RMSR) value as a function of pressure head for cm−3 for the silty loam and clay sam- pedotransfer functions using texture and bulk density as predictors: SSC = sand, silt and clay; Bd = ples. Th ey also showed that narrower bulk density; TC = textural class; TC(7) = textural distribution using seven classes.

www.VadoseZoneJournal.org | 811 may be due to the inability of Eq. [1] to capture the eff ects of macroporosity. Th e values in the dry range were on the same order as in the wet range. Th e only exception was for the data measured by Stumpp et al. (2009). Th is may partly be explained by the diff erent nature of their soils compared with most data- bases. The database of Stumpp et al. (2009) contained a large contribution of fi ne-textured soils, especially clay soils, as well as a large number of soils with very low bulk densities of about 0.2 g cm−3 and very high organic matter con- Fig. 10. Variation in the root mean squared residual (RMSR) value as a function of pressure head for pedo- transfer functions using texture, bulk density, and organic matter in various combinations: SSC = sand, tents (up to 50%) (see also Fig. 2). silt and clay; Bd = bulk density; OM = organic matter; TC(7) = textural distribution using seven classes.

Using bulk density improved predic- tions at the lower pressure head values (Sharma et al., 2006; Schaap et al., 2001). Th e RMSR values at 15,800 cm (pF 4.2) were very similar for all PTFs except Børgesen and Schaap (2005). In the case of Stumpp et al. (2009), we did fi nd very large RMSR values compared with the other databases. Evaluation of the ME showed that water contents were underestimated at all pressure head levels. Th e high RMSR values for the PTFs using sand, silt, and clay were found mainly for soils rich in organic Fig. 11. Variation in the root mean squared residual (RMSR) value as a function of pressure head for matter. Cornelis et al. (2001) analyzed pedotransfer functions using texture, bulk density, organic matter, and soil water content in diff erent various PTFs for the VGM model (see combinations: SSC, sand, silt and clay; Bd, bulk density; OM, organic matter; TC(7), textural distribu- Table 2) and found that most of them tion using seven classes; WC, water content at pressure head values of 30 cm (3), 320 cm (32), 330 cm (33), 15,000 cm (1580), and 15,800 cm (1580). also underestimated the MRC. The mean diff erence varied between −0.058 and 0.028, with the largest mean diff er- in this range. Th is pressure head value falls within the range of ence value being for the PTFs derived by Scheinost et al. (1997) pressure head values where the largest RMSRs were observed. and the smallest one for Rawls and Brakensiek (1985). Th erefore, the moisture content corresponding to this pressure head value was very eff ective in reducing the RMSRA. Th e best Figure 10 shows the eff ect of including organic matter on the PTFs produced RMSR values very close to those of the direct fi t, RMSR across pressure heads. Including organic matter led to showing the same order of magnitude. improved predictions, with the lowest RMSR values occurring in the wet range and the very dry range for the models of Vereecken Sharma et al. (2006) provided information on the variation et al. (1989) and Weynants et al. (2009) using the same database. in the RMSRA across textural classes for PTFs using texture All models showed an increase in the RMSR between 10 (pF 1) and bulk density with different sets of remotely sensed param- and 500 cm (pF 2.7). Th e RMSRs tended to decrease or remain eters. The PTFs were derived from the database established stable at the larger pressure heads, depending on the model. by Mohanty et al. (1997). The remotely sensed data used in the PTFs were slope, aspect, normalized difference vegeta- Figure 11 shows the eff ect on the RMSR of including the soil water tion index, aspect, flow accumulation, and digital elevation. content at certain pressure heads. In general, including the water Figure 12 shows the RMSR for the PTF using the largest set content produced the lowest RMSRs in the range between 100 (pF of remotely sensed predictors in addition to texture and bulk 2) and 3000 cm (pF 3.5). Apparently the water content at 330 cm density. A substantial improvement between 23 and 27% was (pF 2.5) pressure head was very eff ective in reducing the RMSR obtained for the RMSR in the medium pressure head range

www.VadoseZoneJournal.org | 812 (330–8000 cm) compared with when only textural information and bulk density were used. The improvement decreased with increasing pressure head, indicat- ing that remotely sensed information is less capable of improving the predictions for dry soils. For the dry and wet ranges, practically no improvement was obtained (compare with Fig. 9).

Figure 13 shows variations in the RMSR/ RMSRA as a function of the pressure head for PTFs predicting VGM hydraulic con- ductivity using textural information, bulk Fig. 12. Variation of the root mean squared residual (RMSR) value as a function of pressure head for pedotransfer functions using textural data (sand, silt, and clay content [SSC]), bulk density density, and organic matter. The model (Bd), and a maximum set of remotely sensed data (RS) consisting of slope, aspect, normalized dif- of Weynants et al. (2009) excluded values ference vegetation index, fl ow accumulation, and digital elevation. below a pressure head of 6 cm. Th e PTFs using textural information, bulk density, and organic matter or a combination of these predictors showed the lowest RMSR values for pressure heads between 10 (pF 1) and 500 cm (pF 2.7) (Fig. 13). A comparison of the results of Weynants et al. (2009) and Schaap et al. (2001) shows that the eff ect of including organic matter is the largest in the dry range.

Figure 14 shows the eff ect of including water content at diff erent pressure head values as an additional variable to texture and bulk Fig. 13. Variation in the root mean squared residual or root mean squared residual adjusted (RMSR density. We included the data of Schaap et or RMSRA) of the unsaturated hydraulic conductivity (K) as a function of pressure head (pF) al. (2001) for the PTF using texture and for pedotransfer functions using textural information (textural class [TC] or sand, silt, and clay bulk density as a reference. Including the [SSC]), bulk density (Bd), and organic matter (OM) as predictors. Weynants et al. (2009) used water content at 330 cm (pF 2.5) did not the RMSRA and Schaap et al. (2001) the RMSR. improve the hydraulic conductivity predic- tion. Both lines are on top of each other (Fig. 14). Introducing the water content at 15,800 cm (pF 4.2) mainly improved the RMSR values in the dry range. For this reason, it seems that organic matter is a better predic- tor of the hydraulic conductivity in the dry range than the water content at the wilting point (see also Fig. 12). Still, care should be taken in extrapolating these results because we compared the outcomes of two differ- ent databases. Further research is therefore needed to confi rm these fi ndings.

Fig. 14. Variation in the root mean squared residual (RMSR) of the unsaturated hydraulic conduc- In addition to the RMSR, we also evaluated tivity (K) as a function of pressure head (pF) for pedotransfer functions using textural information, the dependency of the ME on the pressure bulk density and water content: SSC = sand, silt and clay; Bd = bulk density; WC2.5 and WC2.5, head for the PTFs of Weynants et al. (2009) 4.2 = water content at pressure head levels of only pF 2.5 (330 cm) and pF 2.5 and 4.2 (15,800 cm), respectively. for predicting the MRC and HCC. Figure 15 shows the ME across diff erent pressure

www.VadoseZoneJournal.org | 813 head values for selected PTFs. Th e results indicate that the ME values were lowest for PTFs using SSCBdOM or SSCBd and water content data. Including OM or water content information led to ME values that evened out across the zero axis. Pedotransfer functions based on texture alone or texture and bulk den- sity tended to underestimate the water contents. Figure 16 shows the ME as a function of pressure head for PTFs pre- dicting the HCC. Th e largest deviations were found in the wet and the very dry Fig. 15. Mean error (ME) values for water content as a function of pressure head for pedotransfer func- range, refl ecting the diffi culty of PTFs to tions using various combinations of predictors: SSC = sand, silt, and clay; Bd = bulk density; Om = correctly estimate hydraulic conductiv- organic matter; WC33 = water content at a pressure head of 330 cm; WC33, 1500 = water content at ity near saturation and at pressure head pressure heads of 330 and 15,000 cm. values around 15,800 cm (pF 4.2).

ValidaƟ on of Pedotransfer FuncƟ ons Using the van Genuchten–Mualem ParameterizaƟ on We reviewed the data available for evalu- ating the accuracy and performance of a PTF to predict the VGM moisture retention and hydraulic conductivity functions using independent data sets.

Valida on of Pedotransfer Fig. 16. Plot of the mean error (ME) of the unsaturated hydraulic conductivity (K) as a function of Func ons for the Moisture pressure head (pF) for two pedotransfer functions using either sand, silt, and clay content, bulk den- Reten on Characteris c sity, and organic matter (SSCBdOm) or the van Genuchten–Mualem (VGM) model with unsaturated K l Table 5 provides an overview of the hydraulic conductivity ( s) and a tortuosity parameter ( ). various PTF validation studies that were conducted for the VGM soil water reten- tion function using independent data sets. Overall, the RMSR for 0.1402, whereas the RMSR values of Børgesen and Schaap (2005) PTF validation ranged between 0.037 and 0.143. Again, the lowest and Børgesen et al. (2008) ranged between 0.037 and 0.064. Th is values were obtained for those PTFs that used soil moisture con- suggests that validation of PTFs is best performed on completely tent, such as in the case for the Børgesen and Schaap (2005) and independent data sets rather than on data sets that are split into a Børgesen et al. (2008) data sets. Th ese values were at the same level subset for development and a subset for validation. as the RMSR obtained for the development data set using the PTFs that included soil moisture. Th e other PTFs using either (i) texture In attempts to validate diff erent PTFs, Cornelis et al. (2001) found and bulk density or (ii) texture, bulk density, and organic matter that clay soils showed the largest RMSD values for all PTFs based produced similar RMSR values as those obtained for the data set on the VGM parameterization, except for the PTF of Scheinost et used for PTF development. Th is indicates that PTFs show a high al. (1997). Th e latter had the largest deviations for the loamy sand reliability when applied to independent data sets. It appears, however, and sand textural classes. Th e better performance of Scheinost et that there is a diff erence between PTF validation based on splitting al. (1997) for clay soils may be due to the larger number of clay soils the data set and validation based on completely independent data. compared with the sandy soils. Scheinost et al. (1997) developed Th is can be seen by comparing the RMSR values of Børgesen and their database and PTFs for specifi c regional conditions involving Schaap (2005) and Børgesen et al. (2008) with the data obtained soils having high organic matter contents (up to 8%), substantial by Weynants et al. (2009) using the database of Cornelis et al. gravel content, a large range in bulk density (0.8–1.85 g cm−3), and (2001). Th e values for the PTF validation derived from Vereecken a textural composition with clay, silt, and sand contents of up to 60, et al. (1989), HYPRES, and ROSETTA ranged between 0.091 and 76, and 97%, respectively. Th ey did not include the saturated water

www.VadoseZoneJournal.org | 814 Table 5. Literature fi ndings dealing with pedotransfer function (PTF) validation using the van Genuchten–Mualem (VGM) parameterization for the moisture retention characteristic, including the root means squared residual (RMSR) or root mean squared deviation (RMSD). No. of Study PTF validated, parameterization, type† Database samples RMSR Weynants et al. (2009) Weynants et al. (2009), m = 1 − 1/n, RA Cornelis et al. (2001) 48 0.099 Weynants et al. (2009) Vereecken et al. (1989), m = 1, RA Cornelis et al. (2001) 48 0.091 Weynants et al. (2009) HYPRES, m = 1 − 1/n, RA Cornelis et al. (2001) 48 0.117 Weynants et al. (2009) ROSETTA, m = 1 − 1/n, NN Cornelis et al. (2001) 48 0.140 Weynants et al. (2009) Wösten et al. (1999), m = 1 − 1/n, RA Vereecken et al. (1989) 166 0.046 Weynants et al. (2009) Schaap et al. (1999), m = 1 − 1/n, NN Vereecken et al. (1989) 166 0.059 Tietje and Tapkenhinrichs (1993) Vereecken et al. (1989), m = 1, RA Tietje and Tapkenhinrichs (1993) – 0.05 (RMSD) Cornelis et al. (2001) Vereecken et al. (1989), RA Cornelis et al. (2001) 298 0.0312(RMSD) Cornelis et al. (2001) Wösten et al. (1994), m = 1 − 1/n, RA Cornelis et al. (2001) 298 0.033 (RMSD) Cornelis et al. (2001) Scheinost et al. (1997), m = 1, RA Cornelis et al. (2001) 298 0.0373 (RMSD) Cornelis et al. (2001) Schaap et al. (1999), m = 1 − 1/n, NN Cornelis et al. (2001) 298 0.0553 (RMSD) Cornelis et al. (2001) Wösten et al. (1999), m = 1 − 1/n, RA-cont Cornelis et al. (2001) 298 0.0388 (RMSD) Cornelis et al. (2001) Wösten et al. (1999), m = 1 − 1/n, RA-class Cornelis et al. (2001) 298 0.0389 (RMSD) Schaap et al. (1998) Schaap et al. (1998), m = 1, RA Vereecken et al. (1989) 1209 0.098‡ Schaap and Leij (1998a) Schaap and Leij (1998a), m = 1 − 1/n, NN Rawls (Schaap and Leij (1998a) 1209 0.087‡ (0.127/0.119)§ Schaap and Leij (1998a) Schaap and Leij (1998a), m = 1 − 1/n, NN Ahuja (Schaap and Leij (1998a) 371 0.062‡ (0.092/0.079)§ Schaap and Leij (1998a) Schaap and Leij (1998a), m = 1 − 1/n, NN UNSODA (Schaap and Leij, 1998a) 554 0.087‡ (0.105/106)§ Børgesen and Schaap (2005) Børgesen and Schaap (2005), m = 1 − 1/n, NN Børgesen and Schaap (2005) 1608 0.038–0.055¶ Børgesen et al. (2008) Børgesen et al. (2008), m = 1 − 1/n, NN Børgesen and Schaap (2005) 1608 0.037–0.064¶ Scheinost et al. (1997) m = 1, RA Scheinost et al. (1997) 396 0.017/0.035 (RMSD)# (3 replicates) † RA = regression analysis, RA-cont = regression analysis–continuous PTF, RA-class, regression analysis–class PTF, NN = neural network analysis, m and n = shape parameters. ‡ RMSR adjusted using Eq. [5]. § Cross-validation between data sets. ¶ Data range refers to diff erent predictor variables (three [Børgesen and Schaap (2005)] or seven textural classes [Børgesen et al., 2008]). # Data range refers to diff erent validation databases.

content when fi tting the VGM equation. Th e samples that were used. Vereecken et al. (1989) used the methods given in Table 1 used were quite large (365 cm3), which raises concerns about the to determine the MRC. For Schaap et al. (1998) and Wösten et long time periods needed to reach equilibrium, especially for the al. (1999), a great variety of methods and sample sizes was used clay soils. Th ey did not provide information about the equilibra- in these large multi-institute databases. Th e possible eff ects of tion time used at the diff erent pressure heads. diff erent measurement techniques on the determination of the MRC were already discussed above. Diff erences in sample size, the For the Vereecken PTFs using the VGM model, Tietje and duration of equilibration, and water removal procedures (pressure Tapkenhinrichs (1993) found that, on average, the moisture increase vs. imposed evaporative fl ux) may be key factors. retention data were underestimated. Deviations from the mea- sured moisture content data were largest for soils having a low Clearly, the lack of information on the type of methods used and the bulk density. Samples with high organic matter content were still diff erence in accuracy of the methods themselves makes it diffi cult to well predicted, although no information on the range of organic evaluate the performance of the PTFs and the VGM parameteriza- C was given (the database had a range of 0–22%). Cornelis et al. tion. More information is needed on the impact of the measurement (2001) found that the PTFs listed in Table 2 generally underes- method on the MRC and HCC. Only a few studies have addressed timated the measured soil moisture content, except the PTFs of the issue of measurement technique comparison (Stolte et al., 1994; Scheinost et al. (1997). Th is may have been due to diff erent mea- Mallants et al., 1997, 1998; Gribb et al., 2004). In addition, diff erent surement techniques. Wösten and van Genuchten (1988) used the methods have been used to determine the predictor variables, which evaporation method of Boels et al. (1978) and Bouma (1983) to may also have an eff ect. Th is is especially the case for the textural determine the MRC for pressure head values <800 cm (pF 2.9). composition of soils (e.g., Weynants et al., 2009) but also for the For the higher pressure heads, standard pressure chambers were determination of the bulk density. Benchmark studies have shown

www.VadoseZoneJournal.org | 815 large diff erences in determining the size Table 6. Validation of pedotransfer functions (PTFs) expressed as the root mean squared residual (RMSR) of the clay fraction. A thorough analysis for the van Genuchten–Mualem model to predict hydraulic conductivity curve using independent data. of the impact of these diff erent methods No. of data on the textural predictors used in PTFs Study PTF validated, type† Database points RMSR has, to our knowledge, not yet been per- Schaap and Leij (1998b) Schaap and Leij (1998b), l = −1 UNSODA 1.65–1.77‡ formed. Studies that compare diff erent Schaap and Leij (1998b) Schaap and Leij (1998b), l = 0.5 UNSODA 1.24–1.37‡ measurement methods for the hydraulic Weynants et al. (2009) HYPRES, Wösten et al. (1999) Vereecken et al. (1989) 136 2.54§ properties in terms of their suitability to Weynants et al. (2009) ROSETTA, Schaap et al. (2001) Vereecken et al. (1989) 136 3.01§ develop PTFs are lacking. Studies of this † l is the tortuosity parameter. type have focused mainly on comparisons ‡ Ranges evaluated the eff ect of diff erent predictor sets. of field measurement techniques with § RMSR adjusted using Eq. [5]. laboratory-scale methods.

PTFs. Here we give a few examples of possible strategies for further Valida on of Pedotransfer Func ons for the improving PTFs in the future. Unsaturated Hydraulic Conduc vity Func on Only a few studies are available that have attempted to validate a Obtaining More Reliable PTF for the VGM unsaturated hydraulic conductivity function Hydraulic Measurements using independent data sets (Table 6). Independent refers to data To be predicted accurately, measurements of the hydraulic param- that have not been used in the development process. As such, inde- eters must be repeatable, scale consistent, and associated with the pendent data sets may include the eff ects of measurement method, lowest possible experimental error. If these three criteria are not diff erences in soil type, soil composition, and sampling size, and met, measurements result in incompatible data, which compli- diff erences in soil physical and chemical properties (texture, struc- cates PTF development and decreases PTF predictive power. For K ture, C content, bulk density, and clay mineralogy). Th is is, of course, example, s has been known to be intrinsically scale dependent also valid and applicable to data sets for the MRC. Weynants et (Timlin and Pachepsky, 1998; Pachepsky and Rawls, 2004). Th is al. (2009) found that the PTFs from HYPRES performed better was also reported for water retention (Pachepsky et al., 2001). than the PTFs derived from ROSETTA. Th ey suggested that this Th e type of experiment and experimental setup may also aff ect was due to the fact that HYPRES represents mainly European soils the assessment of hydraulic properties, particularly with inverse and as such is more affi liated with the database of Vereecken et al. modeling (Javaux and Vanclooster, 2006). Some experimental (1989) than ROSETTA, which represents a mixture of predomi- procedures may also result in non-unique identifi able soil proper- nantly European and American soils. Probably the distribution of ties. Standardization of soil hydraulic characterization methods soil types and soil properties and the underlying correlations in the is sorely needed. Developing databases with harmonized support data are substantially diff erent among the three databases. Figure 1 scales and experimental protocols could signifi cantly improve the already indicated that substantial diff erences among databases exist quality of PTF predictions. with respect to their textural composition. Developing More Comprehensive Databases Procedures for the development and validation of PTFs may well Current soil PTF databases do not homogeneously cover all soil diff er also depending on the invoked regression and optimization types. Larger and more complete databases, with a uniform distri- approach. For example, Schaap and Leij (1998a) used the bootstrap bution of samples among soil classes, should increase the predictive method to select data sets for development and validation from the capability of a PTF obtained with these databases, in particular same database. Because many of the samples in the database were for fi ne-textured soils. not completely independent (e.g., most databases consist of clus- ters of samples taken from the same soil or research area), however, New Data ExploraƟ on Techniques the data sets for development and validation were not completely In addition to establishing comprehensive databases, there is still independent and some bias may exist. a need to further evaluate and analyze the existing databases by developing and applying new data exploration methods with 6 a specifi c emphasis on sparse or unevenly distributed data sets. Prospects in Pedotransfer Techniques such as nearest neighbor methods (e.g., Nemes et al., FuncƟ on Development 2006a,b) and support vector machine approaches (e.g., Twarakavi In this review, we have highlighted some of the approaches that et al., 2009) based on machine learning methods seem to be very have been used to develop PTFs for the VGM hydraulic property promising in this respect. parameterization and indicated possible areas that need improved

www.VadoseZoneJournal.org | 816 Defi ning Comprehensive Metadata 6 Conclusions The term metadata refers to information about the data. To Analysis of the RMSRA values obtained by directly fi tting the improve database quality, it is important that users know precisely VGM parameterization to measured moisture retention data the experimental conditions under which the data have been col- produced an average value of 0.017 for the analyzed databases. lected. Databases should include information on sample scale and Th e accuracies of PTFs for the moisture retention characteristic experimental protocols, but also on the experimental error associ- obtained from databases in the literature are now approaching ated with the measurements, the number of replicates, and the those obtained with direct VGM model fi ts to moisture retention uncertainty associated with the data. For example, the hydraulic data but the RMSRA values are still two to fi ve times larger than properties of the upper soil horizon may depend on soil manage- the RMSRA obtained with direct fi ts. Th e lowest values were ment. Information on the period of the year, the state of the soil, obtained for PTFs using textural properties, bulk density, organic the history of tillage practices, and the type of crop or vegetation matter, and water content. Th e accuracy of the best PTF for the should therefore be indicated to be able to further develop time- or HCC was approximately twice as large as the accuracy obtained structure-dependent PTFs. with the direct VGM fi t to hydraulic conductivity data. Th e most accurate PTFs are based on fl exible Ks and l parameter while allow- Adding Relevant Predictors ing negative values for l. Like the measured hydraulic properties, predictors need also to be unique for a given spatial location and optimally should be obtained A pressure-head-dependent analysis of the RMSRA showed that directly on the sample used for the determination of the hydraulic the largest deviations occurred at intermediate pressure heads properties. In addition, they need to be easily identifi ed (in terms for the MRC in the case of a direct fi t, as well as for PTFs using of time and experimental eff orts) and bring relevant information. combinations of soil texture, bulk density, and organic matter as Predictors that are expensive to obtain and are not unique or are predictors. Including soil moisture measurements as a predictor user dependent should be avoided. Various studies have shown the in PTFs was shown to improve the prediction of soil moisture, potential of predictors such as soil structure, clay mineralogy, and although the improvements were not equally eff ective for all pres- taxonomic information, for the prediction of soil hydraulic properties sure heads. Including soil measurements at intermediate pressure (e.g., Pachepsky and Rawls, 2004). No unifi ed approach exists thus far head values (between approximately pF 1.5–3.0) was successful in on how to best include structural properties in PTFs. Soil structure reducing uncertainty in that specifi c range. In the case of the HCC, predictors in particular can suff er from a lack of uniform protocol the largest deviations were found in the wet and dry ranges for or defi nition, or may depend on the experience of the observer. Th is both the direct fi ts and for all PTFs. Th is suggests that particular is typically the case for soil aggregation. Recent developments in the attention should be paid to improving both the parameterization fi eld of noninvasive methods may be valuable and provide proxies for of the HCC as well as the accuracy of PTFs in these ranges. improved PTFs such as x-ray tomography, spectral induced polariza- tion, and nuclear magnetic resonance. Further research is needed Our analysis also indicated that soil organic matter is an impor- on how to extract relevant information from these new techniques tant predictor for hydraulic parameters and soil moisture contents, and to fully explore their potential in improving PTF predictions. especially in the wet and dry ranges. Organic matter may help Electromagnetic signals may contain relevant information on soil more in reducing RMSRs in the dry soil moisture range (pres- pore distribution and hydraulic properties. Photographs can also give sure head values around the wilting point) than remotely sensed relevant information aft er image analysis or as metadata. Pictures of attributes, compared with cases where only soil texture and bulk simple dye tracer experiments could be used similarly. Current com- density were used. Soil organic matter reduced substantially the puter data storage capacities allow the addition of such information RMSRA of PTFs predicting the hydraulic conductivity compared in databases with almost no limitations. with cases where only soil texture and bulk density were used. Th is is especially the case for the dry range (around the wilting point). Toward Time-Dependent Including soil moisture content in PTFs predicting the HCC was Pedotransfer FuncƟ ons found to be less eff ective in reducing the RMSR values than the Pedotransfer functions have traditionally been developed for the eff ect of soil moisture in PTFs for the MRC. upper soil layers, which are prone to rapid changes due to human activities. With the inclusion of more information on the history of Th e RMSRs of PTFs for the MRC in validation studies were found the soil management practices at a site, the development of temporal to be similar to the RMSRs obtained from the data sets for the devel- pedotransfer functions (TPTFs) could explicitly take into account opment of PTFs using only soil texture and bulk density. For the soil tillage practices to predict soil hydraulic properties at given times. HCC, the RMSRs obtained in validation analyses were larger than To include a temporal component in PTFs would require a priori the values obtained from the data sets used for PTF development. an understanding of the temporal variation of the MRC and HCC. Th ese diff erences may be as large as a factor of seven. Only a few PTF validation studies for the HCC exist in the literature. Databases that

www.VadoseZoneJournal.org | 817 have been used for the development of PTFs for the HCC at present Al Majou, H., A. Bruand, O. Duval, C. Le Bas, and A. Vau er. 2008b. Predic on of soil water reten on proper es a er stra fi ca on by combining tex- contain mostly coarse-textured soils. Th ere is a need to include more ture, bulk density and the type of horizon. Soil Use Manage. 24:383–391. soils with fi ner textures and diff erent mineralogies. Baumer, O.W. 1992. Predic ng unsaturated hydraulic parameters. p. 341–355. In M.Th. van Genuchten et al. (ed.) Indirect methods for es ma ng the hydraulic proper es of unsaturated soils. U.S. Salinity Lab., Riverside, CA. Th e contribution of measurement errors to the RMSR obtained by Boels, D., J. Van Gils, G.J. Veerman, and K.E. Wit. 1978. Theory and system of automa c determina on of soil moisture characteris cs and unsatu- directly fi tting the VGM parameterization to measured data has rated hydraulic conduc vi es. Soil Sci. 126:191–199. not yet been quantifi ed. More attention should be paid to this issue Børgesen, C.D., B.V. Iversen, O.H. Jacobsen, and M.G. Schaap. 2008. Pedo- transfer func ons es ma ng soil hydraulic proper es using diff erent soil because this may well lead to much improved PTF predictions. parameters. Hydrol. Processes 22:1630–1639. Børgesen, C.D., O.H. Jacobsen, S. Hansen, and M.G. Schaap. 2006. Soil hy- draulic proper es near satura on, an improved conduc vity model. J. Databases used to identify the quality of the VGM model in Hydrol. 324:40–50. describing the soil hydraulic properties and to develop PTFs diff er Børgesen, C.D., and M.G. Schaap. 2005. Point and parameter pedotransfer func ons for water reten on predic ons for Danish soils. Geoderma considerably in terms of the proportion of soil textural classes, the 127:154–167. availability soil properties, and information on measurement meth- Bouma, J. 1983. 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Hydraulic conduc vity es ma on for soils with heteroge- parameters with respect to bias are only rarely included in the eval- neous pore structure. Water Resour. Res. 30:211–223. uation. A pressure-head-dependent analysis of bias indicators or Ehlers, W. 1977. Measurement and calcula on of hydraulic conduc vity in horizons of lled and loess-derived soil, Germany. Geoderma 19:293–306. any other relevant information may especially provide new insights Fa , I., and H. Dykstra. 1951. Rela ve permeability studies. Trans. Am. Inst. for the improvement of PTFs. Min. Metall. Eng. 192:249–256. Fayer, M.J., and C.S. Simmons. 1995. Modifi ed soil water reten on func ons for all matric suc ons. Water Resour. Res. 31:1233–1238. Acknowledgments Fuentes, C., R. Haverkamp, and J.-Y. Parlange. 1992. Parameter constraints Attila Nemes (Agricultural Research Station, Beltsville, MD), Wim Cornelis (Uni- on closed form soil water rela onships. J. Hydrol. 134:117–142. versity of Ghent, Belgium), Christine Stumpp (University of California, Riverside), Navin Kumar Twarakavi (University of Alabama at Auburn) and Christen Børge- Gardner, W.R. 1958. Some steady state solu ons of the unsaturated mois- sen (University of Aarhus, Denmark) are gratefully thanked for providing the ture fl ow equa on with applica on to evapora on from a water table. original data. We thank the Transregional Collaborative Research Centre, TR-32, Soil Sci. 85:228–232. “Patterns in Soil–Plant–Atmosphere Systems: Monitoring, Modelling and Data As- Gerke, H.H., and M.Th. van Genuchten. 1993. A dual-porosity model for similation” for supporting this research. simula ng the preferen al movement of water and solutes in structured porous media. Water Resour. Res. 29:305–319. Gribb, M.M., I. Forkutsa, A. Hansen, D.G. Chandler, and J.P. McNamara. 2009. The eff ect of various soil hydraulic property es mates on soil moisture simula ons. Vadose Zone J. 8:321–331. 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