TECHNICAL ARTICLE

Reduction of Feasible Parameter Space of the Inverted SoilHydraulic Parameter Sets for Kosugi Model

Joseph Alexander Paul Pollacco,1 Paolo Nasta,2 José M. Soria-Ugalde,3 Rafael Angulo-Jaramillo,4

Laurent Lassabatere,4 Binayak P. Mohanty,1 and Nunzio Romano,2

Abstract: Effective soil hydraulic parameters of soil vegetation atmo-sphere transfer (SVAT) models can be derived in a cost-efficient way byinverse modeling. Nevertheless, a serious drawback of SVAT modelsbased on Richards' equation is that they require as many as five unex-ploited correlated hydraulic parameters. To reduce the feasible parame-ter space, we propose a method to prevent nonphysical combinationsof soil hydraulic parameter sets obtained by optimization. We adoptthe soil hydraulic analytical model by Kosugi because it enables thefeasible parameter space to be reduced by predicting parameter σ fromRm, which are the variance and mean of the log-transformed soil pore ra-dius, respectively. To further decrease the parameter space, we derivetwo models to predict saturated hydraulic conductivity, Ks, from threeor four Kosugi soil water retention parameters, respectively. These twomodels are based on the combination of the Hagen-Poiseuille and Darcyequations that use three semiempirical parameters (τ1, τ2, and τ3) cali-brated on large UNSODA and HYPRES databases. Our derived modelsare compared with a version of the Mishra and Parker (1990. GroundWater. 28:775–777) Ks model being modified to account for the pa-rameters of Kosugi’s relationships. The results show that the uncertain-ties of the developed Ks model are comparable to the uncertainties ofKs measurements. Moreover, the developed Ks model outperforms theMishra and Parker model. Therefore, the developed method will enableone to substantially reduce the feasible range of the inverted Kosugi’shydraulic parameters.

Key Words: Saturated hydraulic conductivity, Kosugi model,tortuosity, hydraulic parameters, uncertainties, inverse modeling

(Soil Sci 2013;178: 267–280)

An attractive cost-efficient alternative to derive the effectivesoil hydraulic parameters of hydrological models using

Richards’ equation for layered soil profiles (e.g., Sonnleitneret al., 2003; Mohanty and Zhu, 2007; Wollschläger et al., 2009)is by numerical inversion of space-time series data, for example,volumetric soil water content and/or actual evapotranspiration(e.g., Romano, 1993; Romano and Santini, 1999; Pollacco,

1Department of Biological and Agricultural Engineering, Texas A&M Uni-versity, College Station, Texas, USA.2Department of Agriculture, Division of Agricultural, Forest and BiosystemsEngineering-University of Napoli Federico II, Naples, Italy.3Departamento de Ingeniería en Geomática e Hidráulica-Universidad deGuanajuato, Guanajuato, Mexico.4Université de Lyon, CNRS, UMR 5023 LEHNA, ENTPE, F‐69518, Vaulx-en-Velin, France.Address for correspondence: Dr. Joseph Alexander Paul Pollacco, PhD, De-partment of Biological and Agricultural Engineering, Texas A&M Univer-sity, College Station, TX 77843, USA; E-mail: pollacco.water@gmail.comThe authors report no conflicts of interest.Received December 7, 2012.Accepted for publication June 26, 2013.Copyright © 2013 by Lippincott Williams & WilkinsISSN: 0038-075XDOI: 10.1097/SS.0b013e3182a2da21

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2005; Das and Mohanty, 2006; Das et al., 2008; Ines andMohanty, 2008a, 2008b, 2008c; Pollacco et al., 2008; Pollaccoand Mohanty, 2012; Shin et al., 2012) or from water infiltra-tion experiments (e.g., Lassabatere et al., 2006, 2009, 2010;Cannavo et al., 2010; Yilmaz et al., 2010, 2013; Nasta et al.,2012). Nevertheless, one of the drawbacks of using Richards’equation is that analytical, generally unimodal, expressions arerequired to describe the soil water retention, θ(h), and hydrau-lic conductivity, K(θ), functions (e.g., Brooks and Corey,1964; van Genuchten, 1980; Kosugi, 1994). These functionsare identified by as many as five fitting hydraulic parametersfor each layer of the considered soil profile. Moreover, the fea-sible range of the effective parameters has to be set within itsfull feasible range because, in most cases, there is no avail-ability of additional basic soil information that helps to betterconstrain the parameter space (Scharnagl et al., 2011). There-fore, the inverted hydraulic parameters produce inevitably non-unique solutions (Pollacco et al., 2008; Pollacco and Mohanty,2012). In addition, the boundary and initial conditions are oftennot well defined, and the observed data are also affected by rel-atively large measurement errors (Vrugt et al., 2008). To obtaina unique set of solutions when the hydraulic parameters ofsoil vegetation atmosphere transfer (SVAT)–type models areinverted from observed root-zone soil moisture data, Pollaccoet al. (2008) showed that soil water content, θ , data are required,ranging from very dry to very wet soil conditions. If this condi-tion is not met, then the inversion is prone to yield nonuniquehydraulic parameter sets (e.g., Kabat et al., 1997; Abbaspouret al., 1999; Jhorar et al., 2002; Binley and Beven, 2003; Ritteret al., 2003; Minasny and Field, 2005; Beydoun and Lehmann,2006; Pollacco et al., 2008). Consequently, high uncertainties inthe forwarded water fluxes occur (Pollacco and Mohanty, 2012).Thus, it is important to propose new approaches to reduce thenonuniqueness of the inverted hydraulic parameter sets.

A crucial part of the success of an inverse modeling pro-cess is to tighten the feasible parametric range such that anadequate minimum and maximum range for each individual pa-rameter is provided. This can be performed by taking into con-sideration that the value of the saturated hydraulic conductivity,Ks (L T−1), is highly dependent on the values of the other pa-rameters describing the soil water retention θ(h) function. Theproposed method would thus enable to reject all those nonbehav-ioral parameter sets generated by the optimization algorithm,which do not satisfy soil physically based relations. It is expectedthat running only those sequences that satisfy this conditionwould certainly concentrate the optimization search and thereforedecrease the uncertainty of the inverted effective parameters.

Several efforts can be found in the literature for estimatingKs from parameters describing θ(h). For instance, Guarracino(2007) and Mishra and Parker (1990) derived models of Ks forvan Genuchten’s θ(h) based on some of the assumptions ofChilds and Collis-George (1950) who considered that the soilpore network is made up of a bundle of capillary tubes of dif-ferent sizes and that the water flow rate can be computed by

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Pollacco et al. Soil Science • Volume 178, Number 6, June 2013

Hagen-Poiseuille’s equation combined with the Darcy law.Guarracino (2007) derived a relation between Ks and vanGenuchten’s shape parameter, α, assuming that the soil porosityis represented by an equivalent bundle of parallel capillary tubeswith a distribution of pore sizes following a fractal law (Tylerand Wheatcraft, 1990; Yu et al., 2003). Mishra and Parker(1990) developed a simple closed-form expression of Ks by us-ing the model by Mualem (1976) coupled to the capillary waterretention function of van Genuchten (1980). An additional ex-ample of a Ks model is that of Han et al. (2008) based on the in-flection point of θ(h) curve. It is interesting to note that bothmodels proposed by Guarracino (2007) and Mishra and Parker(1990) are similar, although they were developed from differentprinciples.

In this study, we develop a novel model to predict Ks (here-inafter referred as the developed Ks_dev model) from knowledgeof the four parameters featured in Kosugi’s water retention func-tion, namely, θs, θr, hm, and σ (Kosugi, 1994, 1996). We se-lected Kosugi’s θ(h) and K(θ) lognormal functions partlybecause Hayashi et al. (2006) found a correlation between thevariance (σ) of the log-transformed soil pore radius and themean log-transformed soil pore radius (Rm) for forest soils, thusenabling to further reduce the feasible parameter space. A pecu-liarity of Kosugi’s model is that its parameters possess physicalmeanings and have a close link to the soil pore-size distribution.The Ks_dev model is based on the approach of Childs and Collis-George (1950), except that the soil water flux goes through apore network having the hypothetical pore-size distributionof Kosugi (1996). We combined the Hagen-Poiseuille equationwith Darcy law and incorporated three tortuosity parameters.The first two parameters are those of Fatt and Dykstra (1951),whereas the third one is derived according to Vervoort and Cattle(2003) to account for higher porosity media that tend to be moreconnected than lower porosity media. We compare Ks_dev withthe Mishra and Parker (1990) (Ks_mish) model but adapted it soas it can account for the Kosugi model.

Moreover, we later present a simplified Ks_dev (Ks_dev_σ) byexploiting the correlation between σ and Rm (Hayashi et al.,2006). Thus, Ks_dev_σ will be computed from three parametersonly, which are θs, θr, and hm. The predicted values are com-pared with experimental data from the UNSODA (e.g., Leijet al., 1999; Schaap and van Genuchten, 2006) and theHYPRES (e.g., Wösten et al., 1998, 1999; Lilly et al., 2008)large databases. We selected soil samples that contain data onθ(h), K(θ), Ks, and θs (or porosity).

In summary, the article is organized as follows:1) To develop a new saturated hydraulic conductivity model

based on the Kosugi (1996) model (Ks_dev);2) To use the Kosugi (1996) analytical relationship in the

Mishra and Parker (1990) Ks model (Ks_mish);3) To propose a Ks model that makes use of only three

parameters of the Kosugi analytical relations (Ks_dev_σ);4) To select data sets from UNSODA and HYPRES according

to set criteria;5) To optimize the Kosugi hydraulic parameters simultaneously

from available θ(h) and K(θ) data sets;6) To determine the uncertainty bands of the model predictions.

THEORY

Kosugi ModelKosugi (1996) developed a physically based lognormal

function p(R) = dθ/dR (L-1) for describing the distribution of

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pore sizes (R). Specifically, the lognormal probability densityfunction of pore radius can be written as follows:

dθdR

¼ θs − θr

Rσffiffiffiffiffiffiffiffiffiffiffi2 π

p exp −ln R – ln Rmð Þ

2σ 2

� �(1)

where θr and θs (L3 L−3) are the residual and saturated water

contents, respectively, such that θr ≤ θ ≤ θs; ln(Rm) and σ2 arethe mean and variance of the log-transformed soil pore radius,ln(R), respectively.

Let Se denote the effective saturation, such that 0 ≤ Se ≤ 1.Therefore p(R)⋅dR represents the contribution of the filled poresof radius R → (R + dR) to the effective saturation. IntegratingEq.(1) from 0 to R yields the water retention curve, which is afunction of R:

Se Rð Þ ¼ 12erfc

ln Rm − ln R

σffiffiffi2

p� �

(2a)

or

R ¼ Rm

exp erf c−1 2 Seð Þ σffiffiffi2

p� � (2b)

where

Se ¼ θ−θr

θs−θr(3)

In Eq.(2), erfc denotes the complementary error function.The well-known Young-Laplace capillary equation enables

the soil pores radius, R, to be uniquely related to the equivalentmatric suction head, h (cm), at which the pore is filled ordrained (i.e., R = Y/h, with Y = 0.149 cm2). The Kosugi waterretention function has the following analytical expression for Se:

Se ¼ 12erfc

ln h − ln hm

σffiffiffi2

p� �

(4a)

orh ¼ hm exp erf c−1 2 Seð Þ σ

ffiffiffi2

ph i(4b)

where ln(hm) and σ denote the mean and S.D. of ln(h),respectively.

The relative hydraulic conductivity, Kr (0, 1), can be writ-ten in terms of pore-size distribution of soil, as follows(Mualem (1976):

Kr hð Þ ¼ S 0:5e

∫Se0Y

hdSe

∫10Y

hdSe

264

375

2

(5)

where Y is the Young-Laplace constant described above.The unsaturated hydraulic conductivity function can be

obtained (Kosugi et al., 2002) by introducing h of Eq.(4) intoEq.(5):

K Seð Þ ¼ Ks Kr Seð Þ¼ Ks S 0:5

e

12

erfc erf c−1 2Seð Þþ σffiffiffi2

p� �� �2

(6)

Development of a Saturated HydraulicConductivity Model: Ks_dev

Under full saturation condition, Ks can be obtained fromthe Darcy (1856) law as follows:

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Soil Science • Volume 178, Number 6, June 2013 Reducing Parameter Space of Kosugi Hydraulic Parameter

Ks ¼ qL

ΔH(7)

where q (LT�1) is the average water flux, ΔH is the total hydrau-lic head (L), L (L) is the path length in the direction of the flow.

An alternative model to quantify the flow rate into the soilis based on the representation of the porous medium as a bundleof parallel nonintersecting capillary tubes. The volumetric flowrate, Qi (L

3 T�1), in each capillary tube of radius Ri is calculatedthrough Hagen-Poiseuille equation (Bear, 1972):

Qi ¼ C πΔHLτ i

Ri4 (8)

with

C ¼ 18

ρw g

η

where, for water at 20°C, ρw = 0.998 g cm�3 density of wa-ter, g = 980.66 cm s�2 is the acceleration caused by gravity;η = 0.0102 g cm�1 sec�1 is the dynamic viscosity of water,where C is a constant equal to 1.03663 × 109 cm d�1; ΔH/Lτi(L L�1) represents the gradient of the total hydraulic head be-tween the two ends of the capillary tube; and Lτi (cm) is theeffective twisted path length of the ith capillary tube over whichthe fluid travels so Lτi>L.

The total flux density through the column, when all thetubes are filled, is the sum of the flux densities passing througheach pore size class. Let Ni be the number of capillary tubeshaving a radius Ri. Therefore, the specific flux of water, q(L T�1) is defined as the volumetric discharge per unit cross-sectional area (A) and can be expressed as:

q ¼ Q

A¼ C ∑

I

i¼1

ΔHLτ i

Ni

Aπ R4

i

¼ C ∑I

i¼1

ΔHLτ i

ni π R4i (9)

where I is the total number of class Ri in the bundle and whereni = Ni/A is the number of capillary tubes per unit area in eachclass.

Ks is computed by introducing q given by Eq.(9) into Eq.(7):

Ks ¼ C ∑I

i¼1

L

Lτ i

ni π R4i (10)

The change of soil moisture Δθi (L3 L−3) when ni capillary

tubes of cross-sectional area π R2i drain is computed for each

pore size class i by:

Δθ i ¼ ΔSe;i θs−θrð Þ (11)

orΔθ i ¼ ni π R2

i

We introduced a tortuosity model into the Ks model. Fattand Dykstra (1951) proposed the following equation writtenwith our terminology:

Lτ i ¼ L τ01 Riτ 02 (12)

or

L

Lτ i

¼ Ri−τ 0

2

τ 01

where τ′1 , τ′2 [-] are optimized tortuosity parameters that weretaken from Eq.(11) and Eq.(12) of Fatt and Dykstra (1951) for

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which the constant a is substituted for τ′1 and the exponent bis substituted for τ′2.

Isolating ni of Eq.(11), and substituting it into Eq.(10) andintroducing L= Lτ i (Eq.(12)) into Eq.(10) gives:

Ks ¼ 1τ 01

C θs−θrð Þ ∑I

iR2−τ 0

2i ΔSe;i (13)

or

Ks ¼ τ1 C θs−θrð Þ ∑I

iRτ2i ΔSe;i

where τ2 is a tortuosity parameter that adjusts the shape ofthe capillary tube such that τ2 = 2 - τ′2 with 0 < τ2 < 2. Whenτ′2 = 0, the capillary tube is perfectly cylindrical, whereas when2 > τ′2 > 0, the tube changes into the “actual” shape. The tortu-osity of the tubes with the radius Ri defined by L=Lτ i shouldbe smaller than 1. Therefore, because R < 1, then τ′1 >1 and τ1 ≤ 1 since τ1 = (τ1′)

�1.Vervoort and Cattle (2003) state that in high-porosity me-

dia, the large effective pores tend to be more connected thanthe smaller effectives pores. Thus, smaller effective poresshould have a smaller connectivity than larger effective pores.Therefore, we introduce another tortuosity parameter τ3 to thepower of (θs - θr) as in many empirical Ks models (e.g.,Messing, 1989; Mishra and Parker, 1990; Han et al., 2008).The Ks model is then written as:

Ks ¼ τ1 C θs−θrð Þτ3 ∑I

iRτ2 ΔSe;i (14)

where τ3 is a fitting parameter, such that (θs-θr) > (θs–θr)τ3

hence τ3 ≥ 1, to take into account a reduction of Ks caused byreduced connectivity of smaller pores.

Isolating R of Eq.(2b) and substituting it into Eq.(14) andwriting the equation into a continuous derivative form give thedeveloped Ks_dev model:

Ks ̲dev ¼ τ1 C θs−θrð Þτ3 Rmτ2 ∫10

1

exp erf c−1 2 Seð Þ σffiffiffi2

ph i8<:

9=;

τ2

dSe (15a)

Or likewise isolating h of Eq.(2b) and substituting it into Eq.(14) by using the Young-Laplace capillary equation to relate Rwith h and writing the equation into a continuous derivativeform gives the developed Ks_dev model:

Ks ̲dev ¼ τ1 C θs−θrð Þτ3Y

hm

� τ2

∫10

1

exp erf c−1 2 Seð Þ σffiffiffi2

ph i8<:

9=;

τ2

dSe (15b)

In summary, the three tortuosity parameters of the Ks model inEq.(15): τ1 ∈ (0, 1) , τ2 ∈ (0, 2), and τ3 ∈ (1, 10), will be op-timized for different soil types found in two large databases de-scribed below.

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TABLE 2. Feasible Range of the Parameters of the Ks Modelsand the Predictive σp Model

Model Equation Parameters Minimum Maximum

Ks Eq.(15) and Eq.(17) τ1 0.1 1.0Eq.(15) τ2 0.1 1.9

τ3 1.0 10.0σp Eq.(18) Pσ1 0.1 1.0

Pσ2 0.1 1.0

Pollacco et al. Soil Science • Volume 178, Number 6, June 2013

Modified Mishra and Parker Saturated HydraulicConductivity Model: Ks_mish

Mishra and Parker (1990) proposed a Ks model that iscomputed from knowledge of the van Genuchten (1980) soil hy-draulic parameters. We have adjusted the Mishra and Parker(1990) Ks model so as it can account for the Kosugi (1996) soilhydraulic parameters. An integral formulation of the unsatu-rated hydraulic conductivity of Mualem (1976) and Mualemand Dagan (1978) can be described following Mishra andParker (1990) as:

K Seð Þ ¼ τ1 C θs−θrð Þ5=2ffiffiffiffiffiSe

p∫Se0

1h

dSe

� � 2(16)

where τ1 is a tortuosity parameter for which Corey (1979) givesa value of 2/5. Nevertheless, in this study, τ1 is taken as a fittingparameter.

Isolating h of the characteristic curve (Eq.(4b)), introduc-ing it into Eq.(16), and considering the specific case of saturatedconditions (Se = 1) give the modified Mishra and Parker Ks_mish

model:

Ks ̲mish ¼ τ1C θs−θrð Þ5=2

Rm2 ∫10

1

exp erf c−1 2 Seð Þ σffiffiffi2

ph i dSe

0@

1A

2

(17a)

or

Ks ̲mish¼ τ1C θs−θrð Þ5=2

Y

hm

� 2

∫101

exp erf c−1 2 Seð Þ σffiffiffi2

ph i dSe

0@

1A

2

(17b)

We highlight (as shown in Table 1) that Ks_mish is similar toKs_dev. The major difference is because the exponent τ2 of

TABLE 1. Summary of Models Tested in This Work

Model

Ks ̲dev ¼ τ1 C θs−θrð Þ τ3 Rmτ2 ∫10

1

exp erf c−1 2 Seð Þ σ�

(

Ks ̲mish ¼ τ1 C θs−θrð Þ5=2 Rm2 ∫10

1

exp erf c−1 2 Seð Þ σp�

σp ¼ Pσ 1 Ln YRm

�−1

h iPσ2

Ks ̲devσ ¼ τ1 C θs−θrð Þ τ3 Rmτ2 ∫10

exp erf c−1 2Seð Þ Ph

8<:

The feasible ranges of the related parameters are presented in Table 2.

Ks_dev: the developed Ks model; Ks_mish: the modified Mishra and Parker Kσp into Ks_dev.

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Ks_dev is inside the integral, whereas the “τ2” = 2 of Ks_mish isoutside the integral. The Ks_dev has three fitting parameters,whereas Ks_mish has only one. The feasible range of the pa-rameters of the two models is summarized in Table 2.

Reducing the Feasible Parameter Space of σTo reduce the feasible parameter space of σ, we exploit the

finding of Hayashi et al. (2006) who found a correlation be-tween Rm and σ for forest soils (high porosity). Nevertheless,Hayashi et al. (2006) did not verify their relationship for othersoils. To verify this relationship for contrasting soils, we plottedin Fig. 1 the relationship between log10 Rm and σ. The Rm and σKosugi parameters values are obtained from 73 soil data sets,which will be described later. Figure 1 clearly shows that thereis a negative linear correlation between log10 Rm and σ withR2 = 0.63. Thus, the tendency is that the larger median pore size(Rm), which is representative of coarse structure soils, is relatedto the smaller standard distribution (dispersion) of the pore sizeσ. This behavior can be explained by the fact that when Rm islarge, the soil tends to be composed of a single-grained struc-ture (monodisperse) and thus σ tends to be small. Conversely,when Rm is reduced, which is representative of finer material(characterized by a more tortuous structure (Fatt and Dykstra,1951; also refer to Eq.(12)), then the soil structure is aggregatedand the soil is mostly composed of an array of grain sizes (poly-disperse) and, therefore, σ has a larger dispersion. To reduce the

Equation Parameter

ffiffiffi2

p �) τ2

dSe Eq.(15) 3

ffiffiffi2� dSe

!2

Eq.(17) 1

Eq.(18) 2

1

σ 1 Ln YRm

�−1

h iPσ2

ffiffiffi2

p i9=;

τ2

dSe Eq.(19) 5

s model; σp: the predictive σ model; Ks_dev_σ: the implementation of the

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Soil Science • Volume 178, Number 6, June 2013 Reducing Parameter Space of Kosugi Hydraulic Parameter

feasible parameter space of Kosugi parameters, we hereby pro-pose the following equation based on Fig. 1, which linkspredicted σ (σp) with Rm:

σ p ¼ Pσ 1 lnY

Rm

� −1

� �Pσ 2

(18a)

or

σ p ¼ Pσ 1 ln hm− 1ð Þ Pσ2 (18b)

where Pσ1 and Pσ2 are two fitting parameters. Because 0 < σ <5 and ln hm > 2.5, then Pσ1 and Pσ2 should be between 0 and 1.The summary of the feasible range of the parameters is de-scribed in Table 2.

Reducing the Feasible Parameter Space of Ks_dev:Ks_dev_σ

The Ks_dev_σ exploits the relationship of Eq.(18) to reducethe number of input Kosugi parameters from four (θr, θs, σ, Rm

or hm) to three (θr, θs, Rm or hm). The Ks_dev_σ is computed byinputting σp(hm) (Eq.(18)) into Eq.(15):

Ks ̲dev ̲σ ¼ τ1 C θs−θrð Þ τ3 Rmτ2 ∫10

1

exp erf c−1 2Seð Þ Pσ 1 ln YRm

�−1

h iPσ2

ffiffiffi2

ph i8<:

9=;τ2

dSe (19a)

or

FIG. 1. Linear relationship between the S.D. of log-transformedmatric potential head (σ) and the parameter of pore-radiusdistribution Rm (log10 scale).

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Ks ̲dev ̲σ ¼ τ1 C θs−θrð Þ τ3Y

hm

� τ2

∫101

exp erf c−1 2 Seð ÞPσ 1 ln hmð Þ−1½ � Pσ2ffiffiffi2

p� �( )τ2

dSe (19b)

The optimal parameters τ1, τ2, τ3 are retrieved from Ks_dev

model (Eq.(15)), and the parameters Pσ1, Pσ2 are taken fromσp (Eq.(18)). A summary of the different models is shown inTable 1.

MATERIALS AND METHODS

Step 1: Criteria for Selecting Soils From theUNSODA and HYPRES Databases

We selected 318 soil data from the UNSODA database(Leij et al., 1999; Schaap and van Genuchten, 2006) and theHYPRES (Wösten et al., 1998, 1999; Lilly et al., 2008), providingθ(h) and K(θ) measured data points, including measured valuesfor Ks and θs. In the case where θs was not available, this param-eter was calculated from knowledge of porosity, ϕ, as follows:

θs ¼ 0:95 φ (20)

where 0.95 was obtained through a linear regression analysis(results not shown).

From the 318 soils, we selected 73 data sets that complywith the selection criteria described in Table 3. The number ofsoils selected after we performed the different filtering clearlyshows that criterion A is the most important to ensure the re-quired standards of the data sets. The 73 selected data sets werecorrected so that if θi(h) > θs, then θs = θi(h), and likewise if Ki

(θ) > Ks, then Ks = Ki(θ).

Step 2: Inverse ModelingA robust global optimization algorithm AMALGAM written

in MATLAB (http://faculty.sites.uci.edu/jasper/sample/) (e.g.,ter Braak and Vrugt, 2008; Vrugt and ter Braak, 2011) was usedto: (i) inversely estimate the Kosugi parameters from the ob-served θ(h) and K(θ) data points; (ii) optimize the parametersof the saturated hydraulic conductivity models (Ks_dev, Ks_dev_σ,Ks_Mish; Table 1); (iii) optimize the parameters of the regressionsigma model (σp). For each of these models, we required tomaximize the objective functions described below, which cor-responds to minimize the distance between predicted and ob-served values.

Inverting the Soil Hydraulic Parameters From theCharacteristic and Unsaturated HydraulicConductivity Curves

The θr, hm, and σ parameters are optimized by maximizinga weighted objective function (WOF), which is described below.The feasible parameter space of these parameter sets is de-scribed in Table 4. The WOF is composed of two objectivefunctions (OF): the first term, OFθ, is computed with observedand predicted θ(h) values, and the second term, OFk, is calcu-lated with observed and predicted K(θ) values. To account forthe differences in magnitude between the two criteria, they arenormalized according to the Nash-Sutcliffe (1970) efficiency(NSE) formulation such that the optimal minimum is reachedwhen the WOF is equal to 1. The Nash and Sutcliffe (1970)WOF is described as:

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Pollacco et al. Soil Science • Volume 178, Number 6, June 2013

WOF ¼ 12

OFθ þ 12

OFk (21a)

with:

OFθ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−

∑i¼Nθ

i¼1θp hi;Pθð Þ−θobs hið Þ� �2

∑i¼Nθ

i¼1θobs hið Þ−θobs hið Þ�h i2

vuuuuut

OFk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−

∑j¼Nk

j¼1ln Kp Sei;Pkð Þ−ln Kobs Seið Þ� �2

∑j¼Nk

j¼1ln Kobs Seið Þ− ln Kobs Seið Þ

�h i2vuuuuuut

(21b)

where the subscripts p and obs are predicted and observed, re-spectively. The Pθ and Pk are the set of predicted parameters(hm , σ, Rm, θr) of θ(h) and K(θ), respectively. One is remindedthat θs and Ks are directly obtained from the data; values for Nθ

and NK correspond to the total number of data pairs for (hi, θi)and (Sei, Ki), respectively. The θp and Kp are described throughEq.(4) and Eq.(6), respectively. The log transformation of OFkputs relatively more weight on the lower K(θ) and thereforeminimizes the bias toward high conductivity (e.g., VanGenuchten et al., 1991). Also the log transformation takes intoaccount that the uncertainties in measuring the unsaturated hy-draulic conductivity increases as K(θ) increases. It is to be notedthat the global optimizer obtains better results when the naturallogarithm transformation (ln) is used instead of the decimal log-arithm transformation (log10) of K(θ). This is because, for thecomputation of OFk, the natural log is more sensitive than thelog10.

The usage of Nash and Sutcliffe (1970) objective functionenables us to reject nonbehavioral soils, that is, soils that do notcomply with the θ(h) and K(θ) Kosugi model. Thus, soils werealso rejected on the premise that OFθ < 0.1 or OFK < 0.1 so thatonly 73 suitable soil data sets were retained to parameterize theKs models (Table 3).

Inverting the Parameters of the Saturated HydraulicConductivity Models

The “physical” feasible range of the fitting parameters ofthe Ks_dev (Eq.(15)) and Ks_mish (Eq.(17)) models are providedin Table 2. The parameters are optimized by maximizing the ob-jective function OFks of Nash and Sutcliffe (1970):

TABLE 3. Criteria for the Selection of Soils From the UNSODA an

Descript

A In soils, the effective water-filled pore sizes decrease with desaturadecrease as the capillary suction increases (Peters et al., 2011). E

B Soils with at least 6 θ(h) and 6 K(θ) data points.C Soils that have at least one data point near saturation such that Se ≥D Soils in which 0.3 < θs < 0.8 m3 m−3

E A ∩ B ∩ C ∩ DF Soils matching E and with OFθ > 0.1 and OFK > 0.1 after inverse

Where OFθ and OFK are the Nash and Sutcliffe (1970) objective functionumber of soils.

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OFks ̲dev ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−

∑j¼Nks

j¼1ln Ks ̲dev Pksð Þ−ln Ks ̲obs½ �2

∑j¼Nks

j¼1ln Ks ̲obs− ln Ks ̲obs

�h i2

vuuuuuut (22a)

OFks ̲mish ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−

∑j¼Nks

j¼1ln Ks ̲mish Pksð Þ−ln Ks ̲obs½ � 2

∑j¼Nks

j¼1ln Ks ̲obs− ln Ks ̲obs

�h i2

vuuuuuut (22b)

where Pks is the vector of the unknown parameters of the Ks_dev

(i.e., (τ1, τ2 , τ3), and τ1 for Ks_mish; Ks_obs is observed satu-rated hydraulic; Nks is the number of Ks measured data).

Inverting the Parameters of the (Rm and σ)Regression Model

The feasible range of the fitting parameters of the σ modeldescribed in Eq.(18) is provided in Table 2. The optimization ofthe parameters for the σ model is performed by maximizing theNash and Sutcliffe (1970) objective function, which is com-puted as:

OFσ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−

∑i¼Nσ

i¼1σ p pσð Þ−σ� �2

∑i¼Nσ

i¼1σ− �σ� �2

vuuuuuut (23)

where Pσ is the vector of unknown parameters (Pσ1 and Pσ2) ofthe regression sigma model; σp and σ (cm) are predicted and fit-ted σ (by maximizing WOF) parameters respectively; Nσ is thenumber of samples.

Uncertainties in Saturated Hydraulic ConductivityMeasurements

Estimating the uncertainties embedded in the measurementof Ks is challenging because Ks is highly variable (e.g.,Mohanty et al., 1994a; Upchurch et al., 1988 ; Suwardji andEberbach, 1998; Bormann and Klaassen, 2008) and is also scaledependent (e.g., Mallants et al., 1997; Sobieraj et al., 2004; DasGupta et al., 2006). In addition, the Ks value is specially depen-dent on the measurement method used, as shown by Mohantyet al. (1994b) and Fodor et al. (2011) who evaluated the

d HYPRES Databases

ion n

tion. Therefore, the measured θ(h) and K(θ) can only strictlyxception is given for the first two points near saturation.

138

2710.7 and Kr(h) ≥ 0.7 301

30488

modeling 73

ns for θ(h), K(θ), respectively, and described in Eq.(21), where n is the

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TABLE 4. Typical Feasible Range of the Optimized Parametersof the Kosugi Model Described in Eq.(4) and Eq.(6) That WasUsed for Optimization

θr, m3 m−3 log10 hm, cm log10 Rm, cm σ, cm

Minimum 0 1.1 −6.5 0.7Maximum 0.25 6 −2 5

Where hm and Rm are linked through the Young-Laplace capillaryequation.

Soil Science • Volume 178, Number 6, June 2013 Reducing Parameter Space of Kosugi Hydraulic Parameter

measurement errors by applying different methods (double ring,Guelph, velocity, tension disc, and mini disc infiltrometers andother laboratory methods). Fodor et al. (2011) found that theerrors in measuring Ks are estimated to be ±1 × 101.3 cm d−1.On the other hand, Minasny and Field (2005) assessed the errorsof deriving Ks from laboratory experiments (like the evapora-tion method) combined with the generalized likelihood un-certainty estimation (GLUE) inverse procedure for estimatingK(θ) and θ(h), and they found that the uncertainties are approx-imately ±1 × 101.0 cm d−1. Moreover, to take the uncertaintiesassociated to the scale effect into account, as well as the spatialand temporal variability of Ks, we assumed that the errors inobtaining Ks could be of the order of ±1 × 101.8 cm d−1, a valueto which we will refer to evaluate the goodness of our devel-oped Ks model.

It is interesting to note that, although there are largeuncertainties in measuring Ks because it is retrieved indirectlythrough the Darcy law, that Ks is also the least sensitive soil pa-rameter when inverted (e.g., Ines and Droogers, 2002; Beydounand Lehmann, 2006; Pollacco et al., 2008; Pollacco andMohanty, 2012). Furthermore, Pollacco et al. (2008) showedthat if Ks contains uncertainty, then the other parameters ofthe θ(h) would be adjusted during the optimization to correctfor the modeled Ks without greatly influencing the computationof the water fluxes (Pollacco and Mohanty, 2012).

RESULTS

Optimized Hydraulic ParametersThe UNSODA and HYPRES soil databases were filtered

with the rules described in Table 3 and from the resulting 73data sets; the Kosugi hydraulic parameters (θr, hm, and σ) wereoptimized by maximizing WOF (Eq.(21)). The descriptive sta-tistics of the optimized hydraulic parameters as well as of themeasured θs and Ks parameters (obtained directly from thedatabases) are summarized in Table 5. Although only 73 soilsremained after the previously mentioned filtering phase, we ob-served that the ranges of the hydraulic parameters of the 73 soils

TABLE 5. Descriptive Statistics of the Selected (Table 3) Optimize

θs, m3 m−3 θr, m

3 m−3 log10 hm, c

Minimum 0.32 0.00 1.24Maximum 0.71 0.25 5.41Mean 0.47 0.11 2.66S.D. 0.09 0.08 0.84

The θs and Ks values were directly obtained from the data set. The parameThe maximum value of θr was obtained from Table 4. This table shows that

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are representative of all soil textures and therefore very effectivefor our evaluations.

The statistical information of the optimization process de-scribed in Table 6 shows an overall good agreement betweenobserved and fitted θ(h) and K(θ) points. As expected, the fittingof θ(h) represented by OFθ (Eq.(21)) is better than the fitting ofthe log-transformed K(θ) described by OFk (Eq.(21)). These fitsallow the optimization of σ. The values of Ks_obs were directlyderived from the measured values available in the database.The values of σ and Ks_obs are then considered as observed ex-perimental data to be modeled with the proposed models de-scribed in Table 1. Moreover, statistical analyses show that theprobability distribution of Ks_obs values follows a lognormaldistribution (data not shown).

Results of the Sigma ModelIn this section, we model the relationship σp(Rm) (Eq.(18))

between the average pore radius, Rm, and the pore radius S.D. σby maximizing OFσ (Eq.(23)) with respect to the twoparameters Pσ1 and Pσ2. The fit between observed σ andpredicted σp is portrayed in Fig. 2A, and the descriptive statis-tics of the performance of σp are described in Table 7. Figure 2Aand Table 7 show that σp has an acceptable performance, withroot mean square errors (RMSE) = 0.45 and NSE = 0.44. In ad-dition, the model can be considered valid because there is no ev-ident correlation between the model errors, Er, and the predictedvalues (Fig. 2B).

To compute the 95% confidence interval of σp (CL95), weneed to determine the type of distributions the residuals of themodel, Er, obey. This is performed by plotting the probabilityplots of Er described by the histograms in Fig. 2C, which sug-gest a normal distribution. This is further confirmed by theKolmogorov-Smirnov tests ((KST) in Table 7), which attest thatthe hypothesis of normality cannot be rejected (critical probabil-ity KST > 5%). In addition, the residual Er exhibits mean valuesclose to zero (mean in Table 7). Thus, the residuals of σp arenormally distributed, and CL95 interval can be safely computedas approximately twice the S.D. of Er (Table 7), which is usedto define the uncertainty bands of the proposed sigma model(Fig. 2A). When performing inverse modeling, the feasiblerange of σp, summarized in Table 10, depends on the value ofhm and on the CL95. Thus, σ = σp(hm) ± 0.89.

Results of the Saturated Hydraulic ConductivityModels

The observed Ks_obs is plotted against Ks_mish, Ks_dev,Ks_dev_σ in Fig. 3A, B, C, respectively. The optimal tortuosityparameters τ1, τ2, τ3 of Ks_dev and τ1 of the Ks_mish modelsare optimized by maximizing OFks (Eq.(22)). The optimal τ1,τ2, τ3 values, descriptive statistics of the models, and their

d θr, hm, and σ Kosugi Parameters

m log10 Rm, cm σ, cm log10 Ks, cm d−1

−6.24 0.80 0.19−2.07 3.99 4.62−3.49 2.06 3.220.84 0.73 0.78

ters hm and Rm are linked through the Young-Laplace capillary equation.the selected soils cover practically the full soil spectrum.

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TABLE 6. Statistical Information on the Selected (Table 3)Observed and Fitted θ(h) and K(θ) by Using the ObjectiveFunction Described in Eq.(21)

I Δθ(h)I I ΔLog10 K(θ)I

NSE 0.92 0.70RMSE 0.023 0.61Bias 0.018 0.47SD_BIAS 0.008 0.60

Where NSE is the Nash-Sutcliffe (1970) efficiency, RMSE is the rootmean square error, and STD_BIAS is the S.D. of the bias. The resultssuggest that there is an excellent fit between observed and simulatedθ(h) and K(θ).

TABLE 7. Optimal Parameters of σp (Eq.(18))

σmod

Pσ1 0.5920Pσ2 0.7679NSE 0.44RMSE 0.446R2 0.624Bias 0.36Mean −3 10�3

σe 0.44995% CI ±0.89KST 0.89Outliers 4

Where NSE is the Nash-Sutcliffe (1970) efficiency coefficient;RMSE is the root mean square error; R2 is the coefficient of determina-tion; Bias is the bias; Outliers are the number of data points that differsignificantly from model predictions. The following statistics are basedon the modeled residuals (Er) computed as the differences betweenpredicted data and observations: KST is the critical probability forKolmogorov-Smirnov tests test of normality; σe is the S.D.; 95% CI isthe 95% confidence interval by assuming the errors are lognormaldistributed.

Pollacco et al. Soil Science • Volume 178, Number 6, June 2013

performance are provided in Table 8, which clearly showsthe ranking of the models based on the NSE coefficient, thecorrelation coefficient (R2), RMSE, and bias. All these perfor-mance measures show the same conclusions and enable us torank these models as follows: Ks_dev (NSE = 0.71) > Ks_dev_σ(NSE = 0.53) > Ks_mish (NSE = 0.39), and therefore, Ks_dev

outperforms Ks_mish. This is mainly because Ks_dev has two ex-tra degrees of freedom compared with Ks_mish. It would havebeen expected that the implementation of σp model (Eq.(18))

FIG. 2. A, Predicted variance of the log-transformed soil pore radius σp (Eq.(18)) with the optimized σ (Eq.(23)), where the dotted linesrepresent the 95% CI. B, Predicted values, σp, as a function of residuals (Erσ). C, Statistic distribution of residuals (bars) and normaldistribution (line).

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FIG. 3. Uncertainty bands related to the 95% CI for (A) theKs_mish, (B) the proposed Ks_dev, and (C) the Ks_dev_σ models.

TABLE 8. Optimal Parameters of the Saturated HydraulicConductivity Models Described in Table 1

Ks_dev Ks_mish Ks_dev _σ

τ1 0.761 1.083 0.761τ2 1.022 2.000 1.022τ3 5.072 2.500 5.072Pσ1 — — 0.592Pσ2 — — 0.768NSE 0.71 0.39 0.53RMSE 0.42 0.61 0.54R2 0.71 0.39 0.53Bias 0.35 0.50 0.43Mean 5.86 10−12 −2.26 10−12 1.82 10−12

σe 0.42 0.62 0.5495% CI ±0.85 ±1.23 ±1.08KST 0.68 0.77 0.82Outliers 2 2 2

The statistical information of the different models is provided forlog10 Ks. Where NSE is the Nash-Sutcliffe (1970) efficiency coefficient;RMSE is the root mean square error; R2 is the coefficient of determina-tion; Outliers are the number of data points that differ significantly frommodel predictions. The following statistics are based on the modeledresiduals (Er) computed as the differences between predicted data andobservations: KST is the critical probability for Kolmogorov-Smirnovtests of normality; σe is the S.D.; 95% CI is the 95% confidence intervalby assuming that the errors are lognormal distributed. The optimalparameters are well within the feasible range described in Table 2.

Soil Science • Volume 178, Number 6, June 2013 Reducing Parameter Space of Kosugi Hydraulic Parameter

into Ks_dev (Ks_dev_σ; Eq.(19)), which reduces one degree offreedom, would have drastically deteriorated the predictions.Nonetheless, Fig. 3B, C shows that Ks_dev and Ks_dev _σ are wellaligned to the line 1:1, defining a “cloud of points” within theconfidence intervals (described later), with less than two out-liers. The proposed implementation of σp into the Ks modelcan be considered as efficient because Ks_dev_σ leads to a reduc-tion of the degree of freedom without reducing significantly theaccuracy of the model.

Analysis of residuals with regard to the predictions assistsin detecting model structural errors. Good models must ensurethat the residuals have no correlation with either observed orpredicted values. The residuals of the Ks models are computedas follows:

Er ¼ log10 Ks ̲p−log Ks ̲obs (24)

where Ks_p and Ks_obs are predicted and observed values, re-spectively. For the Ks_mish, Ks_dev, Ks_dev_σ models, Er is plottedversus predicted values in Fig. 4B, D, F, respectively. All the de-veloped models, except Ks_mish model, show clearly that Er andthe predicted values are uncorrelated, which suggests that thedeveloped models contain no evident structural errors. In con-trast, Ks_mish exhibits a clear linear dependency of Er withpredicted values with an underestimation for low hydraulic

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conductivities and an overestimation of high hydraulic conduc-tivities (Fig. 4B). This clearly demonstrates the inadequacy ofKs_mish in comparison with Ks_dev, Ks_dev_σ. This may be be-cause Ks_mish lacks degree of freedom compared with the othermodels.

The probability plots of Er show normal distribution(Fig. 4A, C, E). This is confirmed by the KST (in Table 8) thatshow that the critical probabilities are higher than the thresholdof 5%. Furthermore, the residual Er are centered, with meansclose to zero (Mean in Table 8). Clearly, the three models obeya lognormal distribution. Therefore, the 95% confidence in-tervals for the Ks models (95% CI), which define the uncertaintybands of the models, are computed as approximately twicethe S.D. and depicted in Fig. 3A to C. Values for S.D. and95% CI are detailed in Table 8 and show Ks_dev (95% CI 10±0.85) < Ks_dev_σ (95% CI 10±1.08) < Ks_mish (95% CI 10±1.23).Thus, the range of confidence intervals of Ks_dev and Ks_dev_σlies well within the ranges of the uncertainties related to Ks

measurement methods from the field/laboratory that are inthe order of log10 Ks ± 101.8 cm d−1. In contrast, for Ks_mish,95% CI is wider, leading to a less accurate prediction of satu-rated hydraulic conductivity (Fig. 4A).

Thus, when performing inverse modeling, the minimumand maximum values of Ks_dev, summarized in Table 10, dependon the value of θr, θs, hm, and σ and on 95% CI of the Ks_dev,which is 10±0.85. To visualize the reduction of the feasible pa-rameter space (Table 5), we plot in Fig. 5 the feasible parameterspace of Ks, which is in between the upper and lower limits of10±1.08 Ks_dev_σ. Figure 5 clearly shows that the usage of Ks_dev_σreduces dramatically the feasible parameter space of Ks.

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FIG. 4. A, C, E, Residual distributions (bars), along with normal distributions (lines). B, D, F, Residuals versus predicted values. Cases ofMish Model (A, B) and the proposed Ks_dev (C, D) and Ks_dev_σ (E, F). models.

FIG. 5. Three-dimensional plot showing that the feasible rangeof Ks is in-between the lower and upper limits of Ks_dev_σ, whichdepends on the values of θs and σp.

Pollacco et al. Soil Science • Volume 178, Number 6, June 2013

Sensitivity Analysis and Physical Interpretation of theTortuosity Parameters of the Developed Ks Model

To establish if Ks_dev is not overparameterized and to deter-mine the sensitivity of τ1, τ2, τ3, we carried out a sensitivityanalysis described in Table 9. In the sensitivity analysis, one pa-rameter is “removed” in turns and the remaining two parametersare optimized. The sensitivity of the parameters is proportionalto how much the results deteriorate (detailed in difference in ac-curacy Δ) when the parameter in question is not used comparedwith optimizing all the parameters (Step A of Table 9).

The results shown in Table 9 suggest that τ1, τ2, τ3 arehighly sensitive (ΔNSElog10 ≤ −0.2) especially for the tortuosityfactor τ2 of Fatt and Dykstra (1951), with ΔNSElog10 ≤ −3.8and not τ1, as it may be expected, which corrects for the slopeof the curve. Table 9 suggests that the most sensitive parametersin increasing order are τ2 followed by τ1 and τ3. Thus, furthersimplification of the model would considerably deteriorate thepredictability of the Ks_dev.

These results shed light to why Ks_mish gives poorer resultscompared with Ks_dev. This is because Ks_mish has only one de-gree of freedom, which is τ1. Nevertheless, this study showsthat τ2 is a more sensitive parameter. Furthermore, this explainswhy there is a linear correlation between Er and Ks_mish, whichshows that Ks_mish structure is not wrong but that Ks_mish isunderparameterized.

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TABLE 9. Sensitivity Analysis of the Parameters τ1, τ2, and τ3 of Ks_dev by Determining the Decrease of Accuracy (Δ) When theParameter in Question Is Not Used Compared With A Where All the Parameters Are Optimized

τ1 τ2 τ3 NSElog10 RMSElog10 Bias log10 Δ NSElog10 Δ RMSElog10 Δ Biaslog10

A 0.761 1.022 5.072 0.71 0.42 0.35 — — —B 1.00 0.983 7.177 0.47 0.57 0.47 −0.24 −0.15 −0.11C 0.464 2.00 1.01 −4.53 1.85 1.35 −3.82 −1.43 −1.05D 0.549 1.201 1.00 0.50 0.56 0.45 −0.21 −0.14 −0.11

Where NSE is the Nash-Sutcliffe (1970) efficiency and RMSE is the root mean square error.

The results show that the most sensitive parameters in the increasing order is τ2, followed by τ1 and τ3.

Soil Science • Volume 178, Number 6, June 2013 Reducing Parameter Space of Kosugi Hydraulic Parameter

Table 8 shows that for Ks_dev, Ks_dev_σ the tortuosityparameters τ1, τ2, τ3 are well within the physical limits recordedin Table 2. Therefore, this study complies with that of Fatt andDykstra (1951), which states that liquid flowing in the crevicesand small pores will travel a more tortuous path than liquidflowing through the large pores. Our results are also in agree-ment with those of Vervoort and Cattle (2003), which state that,in high-porosity media, the large effective pores tend to be moreconnected than the smaller effective pores. Thus, we can attri-bute a physical interpretation to the τ1, τ2, τ3 parameters.

CONCLUSIONSAn increasingly attractive cost-efficient alternative is to de-

rive the effective hydraulic parameters by inverting the Richardsequation that governs water flow in unsaturated soil. Neverthe-less, relatively large uncertainties can occur when predictingwater fluxes because of nonuniqueness of the inverted hydraulicparameters. We proposed an algorithm to reduce the nonunique-ness of the inverted hydraulic parameters.

Traditionally, the minimum and maximum ranges for eachindividual parameter are provided before an inverse modelingexercise (e.g., Table 4). Nevertheless, this traditional method

TABLE 10. Summary of the Feasible Parameter Space of the KosugKs, Where C = 1.03663 × 109 cm d−1

Minimum θs m3 m�3 0.32θr m3 m�3 0.00hm cm 17σp cm 0.592 (Ln hm − 1) 0.7679 − 0.

Ks_dev cm d�1 0:761 C θs−θrð Þ5:072 0:14hm

Maximum θs m3 m�3 0.71θr m3 m�3 0.25hm cm 2.57 105

σp cm 0.592 (Ln hm − 1) 0.7679 + 0.

Ks_dev cm d�1 0:761 C θs−θrð Þ 5:072 0:h

The feasible range of θs , θr, and hm is obtained from Table 5.

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has the disadvantage that it includes nonphysical parameter com-bination sets. We therefore propose methods to narrow down thefeasible range by developing algorithms to preclude nonphysicalcombinations of hydraulic parameters.

We selected the Kosugi model, which describes the waterretention and the hydraulic conductivity relationships, becausethere is a negative linear correlation between two of its pa-rameters (hm and σ), and therefore, enabling it to reduce the fea-sible parameter space. To further reduce the feasible parameterspace, we developed two physically based models to predict sat-urated hydraulic conductivity from four or three parameters de-scribing the Kosugi characteristic curve.

The proposed saturated hydraulic conductivity model isdeveloped by modifying the approach of Childs and Collis-George (1950) by estimating the soil water flux through a con-tinuous function of the pore-size distribution of Kosugi(1996). We combined the Hagen-Poiseuille equation with theDarcy law and introduced constant empirical universal tortu-osity parameters of Fatt and Dykstra (1951) and incorporatedanother tortuosity parameter to take into account that high-porosity media tend to be more connected and should have alower tortuosity (Vervoort and Cattle, 2003). The Kosugi soilhydraulic parameters were calibrated by selecting 73 water

i Hydraulic Parameters That Is Reduced Dynamically for σ and

89

9�1:022

∫101

exp erf c−1 2 Seð Þ σffiffiffi2

ph i8<:

9=;

1:022

dSe

100:85

89

149m

�1:022∫10

1

exp erf c−1 2 Seð Þ σffiffiffi2

ph i8<:

9=;

1:022

dSe 100:85

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Pollacco et al. Soil Science • Volume 178, Number 6, June 2013

retention and unsaturated curves data sets from the UNSODAand HYPRES database, which also contain measured saturatedhydraulic conductivity and saturated soil moisture.

The developed saturated hydraulic conductivity model(Ks_dev) has shown good performance with an NSE of 0.71. Be-cause no correlation was obtained between residuals and ob-served saturated hydraulic conductivity, we suggest that thedeveloped model has no obvious structural errors. A sensi-tivity analysis was performed with the three fitting tortuosityparameters, and it was found that all the parameters are neces-sary to give good performance and their optimal values werefound to comply with the soil physics theory. The developedsaturated hydraulic conductivity model errors were found tobe log-normally distributed, with the 95% CI of Ks being ±1 × 100.85 cm d−1, which is in the same order of magnitudeof field and/or laboratory Ks measurements. The proposed satu-rated hydraulic conductivity model was compared with the mo-dified saturated hydraulic conductivity of Mishra and Parker(1990) model (Ks_mish), such that it uses the Kosugi (1996)parameters instead of the van Genuchten (1980) hydraulicparameters. We found that the developed saturated hydraulicconductivity model outperforms Ks_mish.

To further reduce the nonuniqueness, we built a second mo-del (Ks_dev_σ), which exploits the correlation between log10 hmand σ and found a reasonable relationship. We introduced theσ model into the saturated hydraulic conductivity models,Ks_dev, and found that it leads to estimate quasi as good as theoriginal model. The proposed Ks_dev_σ can be considered parsi-monious because it leads to a reduction of the degree of free-dom without reducing significantly the accuracy of the model.A summary of the developed model with the dynamic feasiblerange of its parameters is provided in Table 10.

The reduction in uncertainties of the forwarded water fluxby using the proposed methodology of restraining the feasibleparameter space dynamically, summarized in Table 1, shouldbe assessed. Without considering the scale issues, this can beperformed with precise weighing lysimeters for which the storageand water fluxes are measured (e.g., drainage and evapotranspira-tion) (e.g., Abbaspour et al., 1999; Kosugi and Katsuyama, 2001,2004; Scanlon et al., 2005; Durner et al., 2008). Differentexperiments should be performed under contrasting hydroclimateconditions because Pollacco and Mohanty (2012) showed thatthe sensitivity of hydrological parameters is highly dependenton the hydroclimate.

ACKNOWLEDGMENTSWe acknowledge the partial support of National Science

Foundation (CMG/DMS Grant No. 0934837) and NASA THPs(NNX09AK73G) grants. We also thank Lilly A. (Macaulay LandUse Research Institute) for entrusting us with the HYPRES dataset. We thank J. Vrugt for having provided the global optimiza-tion code AMALGAM and assisted us in its implementation.

ABBREVIATIONSθ: soil moisture; θ(h): soil water retention functions; K(θ):

hydraulic conductivity functions; Ks_dev: developed saturatedhydraulic conductivity model that predicts Ks from four Kosugiparameters (θr, θr, hm, and σ); Ks_dev_σ: proposed model thatpredicts Ks from 3 Kosugi parameters (θs, θr and hm); Ks_mish:modified model of Mishra and Parker (1990) that predicts Ks

from 4 Kosugi parameters (θs, θr, hm and σ); OF: objectivefunction; OFks: objective function of the fitted parameters ofthe Ks models; OFσ: objective function of the fitted parametersof the σ models; WOF: weighted objective functionσp: predictsσ from hm Kosugi parameter.

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ABBREVIATIONSVariables and symbols used in the study:σ: S.D. of log-transformed matric potential head (-);σp: predicted value for σ (-);θ: soil water content (L L3);θ r: residual water content (L L3);θs: saturated water content (L L3);A: unit cross-sectional area (L2);H: total hydraulic head (L);h: soil matric potential head (L);hm : median soil matric potential head (L);K: soil hydraulic conductivity (L T−1);Kr: relative hydraulic conductivity (-);Ks: saturated hydraulic conductivity (L T−1);Ks_dev: developed model that predicts Ks from θr, θr, hm,

and σ;Ks_dev_σ: proposedmodel that predictsKs from θs, θr, and hm;Ks_mish: modified model of Mishra and Parker (1990) that

predicts Ks from θs, θr, hm, and σ;Ks_obs: observed value for Ks (-);L: path length in the direction of the flow (L);τ1: macroscopic tortuosity-connectivity parameter (L L�1);τ2: microscopic tortuosity-connectivity parameter to adjusts

the shape of the capillary tube (-);τ3: pore-connectivity parameter (-);R: pore-radius (L);Rm: pore radius distribution parameter (L);Se: degree of saturation (-);Y: Young-Laplace constant of the capillary equation (L2);q: water flux (L T−1);Q: water volumetric flow rate (L3 T�1);C: constant in the Poiseuille equation 1.03663 × 109 cm d�1;Pσ1: first fitting parameter of Eq.(18) (-);Pσ2: second fitting parameter of Eq.(18) (-);ϕ: soil porosity (L3 L�3)

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