J. Hydrol. Hydromech., 67, 2019, 2, 179–190
DOI: 10.2478/johh-2018-0009

Comparison of alternative soil particle-size distribution models and theircorrelation with soil physical attributes
Fatemeh Afrasiabi1, Habib Khodaverdiloo1*, Farrokh Asadzadeh1, Martinus Th. van Genuchten2, 3
1

Department of Soil Science, Urmia University, Urmia 57135-165, Iran.
Department of Earth Sciences, Utrecht University, Utrecht, Netherlands.
3
Center for Environmental Studies, CEA, São Paulo State University, UNESP, Rio Claro, SP, Brazil.
*
Corresponding author. E-mail: h.khodaverdiloo@urmia.ac.ir
2

Abstract: Complete descriptions of the particle-size distribution (PSD) curve should provide more information about
various soil properties as opposed to only the textural composition (sand, silt and clay (SSC) fractions). We evaluated theperformance of 19 models describing PSD data of soils using a range of efficiency criteria. While different criteria produced different rankings of the models, six of the 19 models consistently performed the best. Mean errors of the six
models were found to depend on the particle diameter, with larger error percentages occurring in the smaller size range.
Neither SSC nor the geometric mean diameter and its standard deviation correlated significantly with the saturated
hydraulic conductivity (Kfs); however, the parameters of several PSD models showed significant correlation with Kfs. Porosity, mean weight diameter of the aggregates, and bulk density also showed significant correlations with PSD model
parameters. Results of this study are promising for developing more accurate pedotransfer functions.
Keywords: Lake Urmia (Iran); Particle-size distribution models; Pedotransfer functions; Soil physical properties.
INTRODUCTION
The particle-size distribution (PSD) is a fundamental physical property of soils. Knowing the soil PSD is critical for arange of hydrological, geological, agricultural, and geotechnicalapplications, including soil classification, as summarized wellby Bayat et al. (2015). For example, the soil PSD is widelyused to estimate such key soil attributes as infiltration (e.g.,Ghorbani-Dashtaki et al., 2016; Parchami-Araghi et al., 2013),
the saturated hydraulic conductivity (Jabro, 1992; Saxton et al.,1986), soil water retention (Arya and Paris, 1981; Ghorbani
Dashtaki et al., 2010; Khodaverdiloo et al., 2011) and movement (Nemes et al., 2003), soil erosion (Beke et al., 1989; Xu etal., 2013), the cation exchange capacity of soils (Ersahin et al.,2006; Liao et al., 2015; Manrique et al., 1991) and various
chemical sorption parameters of soils (Khodaverdiloo andSamadi, 2011).
Predictions of many soil properties and functions often needmore detailed soil textural data than only the sand, silt and clay
(SSC) fractions. Therefore, many studies were undertaken toprovide more detail regarding the soil PSD. For example, Shirazi and Boersma (1984) and Shirazi et al. (1988) transformed
the standard textural triangle into a new diagram based on mean
particle size and standard deviations of particle size to provide
greater resolution in detecting classified soils within a texturalclass. Accordingly, parameters other than SSC are often used toreflect the soil PSD in various predictive functions (e.g. Ghorbani Dashtaki et al., 2010). Examples are the geometric mean
diameter (dg) and its standard deviation (σg), the coefficient of
uniformity (Cu), the coefficient of curvature (Cc), and the diameter of soil particles having a specific percentage of smaller
particles (e.g., D10, D30, D50, D60, etc.), which then may serve as
measures of soil gradation or other properties. Mathematically
representing PSD data hence provides several benefits in soil
mechanics, physics, hydrodynamic and geotechnical applications (Fredlund et al., 2000), as well as helps to convert PSD
data of various particle-size classification systems to the desired
formulation (Nemes et al., 1999; Skaggs et al., 2001). These

applications require the definition of complete PSD curves
using some mathematical model that best fits experimental
data.
Consequently, various equations have been introduced over
the years to describe the soil PSD, each of which often has its
own specific advantages and limitations (Bayat et al., 2015;
Fredlund et al., 2000). A careful selection of the most appropriate PSD model may have a significant impact on the estimatedsoil particle distribution density at a given particle size (Nemes
et al., 1999) and hence is important to more precisely estimate
soil properties (e.g., Bittelli et al., 1999; Hwang and Powers,
2003). Several studies have attempted to compare different
PSD mathematical expressions for soils (e.g., Bayat et al.,2015; Buchan et al., 1993; Hwang et al., 2002; Hwang, 2004;
Bah et al., 2009). However, these studies generally compared
only a limited number of PSD models (Bagarello et al., 2009;
Buchan et al., 1993; Hwang, 2004; Vipulanandan and Ozgurel,2009), invoked only few statistical criteria for the comparisons(Bagarello et al., 2009; Bayat et al., 2015; Buchan et al., 1993;
Hwang, 2004; Vipulanandan and Ozgurel, 2009) and/or applied
the analyses to a relatively narrow range of soil textural classes(Buchan et al., 1993; Vipulanandan and Ozgurel, 2009; Zhao etal., 2011). Since the performance of a PSD model depends on
the type and range of soil textures involved (e.g., Bagarello etal., 2009; Buchan, 1989; Fredlund et al., 2000; Hwang, 2004;
Hwang et al., 2002; Zhao et al., 2011), conclusions obtainedwith such studies may not necessarily be extended or generalized to soil textures other than those used in the comparisons.
Additionally, PSD models have often been evaluated in termsof their general performance for describing the entire PSD
curve, but frequently not for predicting specific (and practically-important) PSD points or parameters. A given model may
well be accurate globally in terms of generating the entire PSD
curve, while failing locally to predict specific points or selectedparameters of the curve.
Soil PSD data are commonly used to derive pedotransfer
functions (PTFs) for estimating soil hydraulic and other properties that are more difficult to measure directly. Traditionally,179

Fatemeh Afrasiabi, Habib Khodaverdiloo, Farrokh Asadzadeh, Martinus Th. van Genuchten

soil SSC percentages have been used as input data for the PTFs
(e.g., Carsel and Parrish, 1988; Chapuis, 2012; Jabro, 1992;
Patil and Singh, 2016; Saxton et al., 1986). While attempts havebeen made to test parameters other than SSC to represent soil
particle size information in PTFs, it is not obvious which PSD
models are most suitable for this purpose. Also, there is often
no clear physical basis of the parameters in some of PSD models. Many PSD model parameters frequently are purely empirical without any physical significance. For these reasons, studies
are still needed to obtain optimal correlations of PSD parameters with soil properties in terms of underlying physical concepts.
The objectives of this study were to (1) evaluate the performance of 19 models for describing PSD data of selected soils innorthwestern Iran using a broad database in terms of soil textures, (2) provide a functional evaluation of the models to predict selected practically-important PSD points or parameters
using different efficiency criteria, (3) compare results obtainedwith the general and functional evaluations, and (4) correlate
physically based PSD model parameters with selected soil
physical properties to examine their suitability to serve as input
for pedotransfer functions.
MATERIALS AND METHODS
Soil datasets
Twenty four locations neighboring the western edge of
threatened hypersaline Lake Urmia in northwestern Iran were
sampled for this study. The locations were selected such thatsoils with a wide range of soil textures were included in the
database. Figure 1 shows the textural distributions of the samples. The study area was located between geographical coordinates of 45° 05′ and 45° 08′ E, and 37° 32′ and 37° 38′ N, at
1290 to 1350 m above mean sea level, and with mean annual
precipitation and temperature of approximately 300 mm and
11.8°C, respectively, with xeric soil moisture and mesic soil
temperature regimes (Banaei, 1998).
Three replicate surface (0–30 cm depth) soil samples werecollected at each site. After clay deflocculation using sodiumhexametaphosphate and mechanical agitation, soil PSD curves
were determined using a combination of hydrometer (Gee andOr, 2002) and mechanical sieving methods. Fine particle fractions (<0.15 mm) were determined by sedimentation, whereas
the coarser fractions (≥ 0.15 mm) were obtained by sieving.
The sieve analysis involved nested columns of sieves (2, 1, 0.5,
0.25 and 0.15 mm), which were shaken with a mechanicalshaker. Twelve data points were obtained from the hydrometer
analysis by recording the soil suspension density at times of 30,
40 and 60 s, and at 3, 10, 15, 30, 60, 90, 120, 240 and 1440
min. A total of 16 hydrometer particle size limits hence wereobtained for each soil sample.
Once the cumulative PSD curves (i.e., F(d) as a function of
the particle diameter, d) were obtained, the geometric mean dg
and geometric standard deviation σg of the soil particle diameters were determined using methods proposed by Shirazi andBoersma (1984). We also determined the mean weight diameter(MWD) of soil aggregates after dry sieving.
Soil particle data larger than 53 μm as measured with thesieve method were used to obtain direct estimates of the “measured” sand content. These measured values were then used totest the accuracy of the PSD models to predict the sand content
from the complete grain size distribution using complete F (d ≥53 μm). We further compared the SSC contents obtained withthe hydrometer data at sedimentation times of 40 s and 7 hr,further referred to as the “nominal” SSC contents, with PSD

180

model-predicted F(d) values corresponding to diameters (d) of
2–0.05, 0.05–0.002, and < 0.002 mm for sand, silt, and clay,
respectively.
We also estimated “measured” particle-size diameters corresponding to 10, 30, and 60% of the cumulative PSD curve (i.e.
D10, D30, and D60, respectively) by linear interpolation from thenearest lower and higher measured data points (a common
procedure to determine D10, D30, and D60 values). These measured values were compared with D10, D30, and D60 values “predicted” with the complete F(d) models.
Several other soil physical parameters were obtained at the
study site. The field-saturated soil hydraulic conductivity (Kfs)
was measured using constant head single-ring methods (Razzaghi et al., 2016; Reynolds, 2008), while the soil bulk density(ρb) was determined using core sampling (Blake and Hartge,1986). To determine total porosity (Φ), the soil cores were
slowly saturated by soaking them from below to a water level
just below the top of the core. This procedure was carried out
over a period of 2 days to obtain as complete saturation as
possible by minimizing entrapped and dissolved air in the
cores. Soil water contents as determined subsequently by ovendrying the saturated samples at 105°C for 24 hr were regarded
to be equivalent to porosity, Φ (Khodaverdiloo et al., 2011).
Particle-size distribution models
In this study we compared the performance of 19 different
mathematical models for F(d) in describing observed cumulative PSD data of the sampled Urmia soils. The models are listed
in Table 1. The references cited in the second column of Table
1 provide details of the corresponding model shown in the thirdcolumn.
Fitting procedure
An iterative nonlinear optimization procedure was used to fit
the parametric functions F(d) in Table 1 to the observed cumulative PSD data. The optimization procedure produced valuesof the unknown parameters that provided the best fit with thedata (Bagarello et al., 2009; Bah et al., 2009; Hwang et al.,2002). Optimized values of the model parameters were determined by minimizing the following objective function:
n

SSE =

 (O − P )
i

i

2

i =1

(1)

where SSE represents the sum of squared errors between themeasured (Oi) and predicted (Pi) cumulative PSD fractions, andn the number of particle size data points.
The optimization procedure was implemented using theleast-square curve fitting toolbox in the MATLAB R2011aenvironment (The MathWorks Inc., Natick, MA). The final
fitted parameter values in each case were tested by using at least
three different initial parameter estimates as done by Hwang etal. (2002) and Bah et al. (2009), among others. The optimization
process in most or all cases converged to the same final values,
thus avoiding the problem of selecting local minima.
Criteria for model comparison
After optimization of the model parameters, values of the
cumulative PSD fractions as calculated with the F(d) models
were compared with their corresponding observed values to test
how well the calibrated models fitted the observed PSD data.

Comparison of alternative soil particle-size distribution models and their correlation with soil physical attributes
Table 1. Particle-size distribution models, F(d), tested in this study.

#

Model

Reference

Equation#AD

Andersson (1990)( )=+ arctan(

G (Gompertz)

Nemes et al. (1999)

F(d) =+ exp{− exp[− ( − )]}Fred-3p (Fredlund with 3
parameters)

Fredlund et al. (2000)( )=Fred-4p (Fredlund with 4parameters)

Fredlund et al. (2000)( )=ORL (Offset-Renormalized
Lognormal)

Buchan et al. (1993)( ) = (1 − ) ( ) +F(X) defined by SLN modelμ, σ,

ORN (OffsetNonrenormalized Lognormal)

Buchan et al. (1993)

G(X) = F(X) +F(X) defined by SLN model

μ, σ,

SH-C

Shiozawa and Campbell(1991)( ) + (1 − ) ( )( )=F(X) defined by SLN modelμ, σ,

MLG (Modified Logistic
Growth function)

Liu et al. (2004)( )=Wei (Weibull)

Assouline et al. (1998)( ) = + (1 − ){1 − exp(−
)
( −
=
−
)
(

SLN (Simple Lognormal)

Buchan (1989)

Parameters)1{ln[exp(1) + ( ) ]}1{ln[exp(1) + ( ) ]}

[1 +

1
exp(−

{1 − [{1 − [

ln 1 +
ln 1 +
ln 1 +
ln 1 +

−

0.001
0.001

] }

] }

a, n, mdm = 0.0001

n, m, df, α
dm = 0.0001)}
c, a, b≤√2α, β, , γa, b, c)]−1( ) = (1 − erf2= ln( )1( ) = (1 + erf
2= ln( )

fo, b, c, do

>

μ, σ

μ, σ

Norm (Normal)

Buchan et al. (1993)

VG

Haverkamp and Parlange (1986)( )= 1+ ()

dg, n(m = 1–1/n)BEST (Beerkan Estimationof Soil Transfer)

Lassabatère et al. (2006)( ) = [1 + () ]dg, N, M
M = 1–2/N

Fr(B) (Fractal)

Bird et al. (2000)

( )=

)

c, Dm

Fr (T-W) (Fractal)

Tyler and Wheatcraft(1992)( ) = ( )(1.5

L-P (Log-power)

Kolev et al. (1996)( ) = exp( log )A, BExp (Exponential)

Gimenez et al. (2001)

( )=

c, β

Log (Logarithmic)

Zhuang et al. (2001)

( )=

J

Jaky (1944)1d( ) = exp{
ln
p
d

(

√2

)

Dm

ln +

a, b
}

p >1
(d0 = 2 mm)

d: particle diameter in mm; F(d): cumulative fraction with a diameter smaller than d; erf: error function

The accuracy of the models was evaluated using the NashSutcliffe Efficiency (NSE), the root mean square error (RMSE),
the coefficient of determination (R2), the index of agreement(δ), the relative error (Er), Akaike’s information criterion(AIC), the standard deviation of error (STDEV), the maximum
error (MAXe), the mean absolute error percentage (MAEP), themaximum absolute error percentage (MAXaep), and the geomet-

ric mean of error ratio (GMER). Mathematical representationsof these criteria are given in Table 2.
We used the various statistical measures in Table 2 to evaluate different aspects of the optimizations. This was done sincesome of the criteria emphasize the optimizations more from aperspective of variances, while other statistics capture the bias
better. For example, RMSE is an index of the absolute error.

181

Fatemeh Afrasiabi, Habib Khodaverdiloo, Farrokh Asadzadeh, Martinus Th. van Genuchten
Table 2. Efficiency criteria used to evaluate the accuracy of the
PSD models.
Criteria
R2

Model Equation#
={
[∑

RMSE

=

Er

=

NSE
δ

(

(

− )]

1

=

AIC

∑

(

(

−

∑
=1−

=1−{
∑

−

[∑

(

}
− )]

)
2( + 1)( + 2)
− −2

)

( )
∑
(

)
(
− )

∑ ( − )}
(ǀ − ǀ + ǀ − ǀ)1GMER

= exp(

STDEV

=

(

−

MAXe

= MAX|

−

|

MAEP

=(

MAX aep− )) + 2( + 1) +

ln(
∑

− )(

= Max

ln( ))−))/ × 100−/ × 100#n refers to the number of calculated (Pi) and measured (Oi) valuesof the cumulative PSD curve; over-lined symbols represent meanvalues; SSE is the sum of squared errors; P is the number of model
parameters.

However, models with more fitting parameters will almost
always result in a better overall fit. The AIC is for this reason
often used to compare models with different numbers of fitting
parameters. In this study we used the corrected AIC as suggested by Bolster and Hornberger (2007).
NSE values were selected to indicate whether or not the observed mean is as good a predictor as the model. On the other
hand, R2 is an indicator of the extent to which the model explains the total variance of the observed data. To overcome
some of the limitations of R2, Willmott (1981) proposed theindex of agreement, δ, to better reflect the degree at whichobservations are estimated accurately by the predictions. However, relatively high values (more than 0.65) of δ may be obtained even for poor model fits, and δ hence is not overly sensitive to systematic over- or under-predictions. We used GMER
to test the possible under- or over-estimations in the model
predictions, while MAXe is used to show the outliers. Since
errors within the lower range of the model output may bemasked by relatively high errors in the higher range of theoutput, MAEP and MAXaep have been used to compare theerrors and outliers, respectively, of the models as percentagesof the observed values. For a perfect prediction RMSE, SD,
MAXe, MAEP, MAXaep and Er equal zero and NSE, δ, R2, and
GMER equal 1. Models with lower AIC values are considered
to be the more likely to be accurate.

182

Since most or all statistical criteria evaluate only some aspect of the correspondence between measured and calculated
values, we suggest that a complete assessment of model performance should include a combination of criteria. We evaluated for this reason initially all of the criteria in Table 2 for theF(d) models. The criteria subsequently were classified into
groups using a hierarchical cluster analysis (HCA) based on thelevels of similarity among their standardized values. Finally, a
subset of the criteria was selected among the distinct groups toevaluate and compare the performance of the models. The
selected criteria were then used to classify the models into
groups using HCA. The cluster analysis was conducted using
the Minitab 16 software.
Once the different F(d) models and their optimized parameters were determined, measured Kfs, log Kfs, MWD, and Φ
values were correlated with either the conventional PSD parameters (i.e., sand/silt/clay percentages), geometric representations of the PSDs (dg and σg), selected PSD mechanical parameters (notably D50 and D60), and the optimized parameters of themost accurate F(d) models. This is to find the best representation(s) of the soil PSDs to predict relevant soil physical quality
indices, and to interpret correlations of soil PSD data with field
properties. For the functional evaluation of the models against
the field data we used selected practically-important PSD parameters such as sand, silt and clay percentages, dg and σg, andD30, D50 and D60 as calculated with the F(d) models.
The accuracy of the models in terms of predicting Kfs,
log Kfs, MWD, and Φ values was assessed using a weighted
version of R2 (i.e., wR2) following Krause et al. (2005):

 b .R 2 for b ≤ 1
wR 2 =  −1
 b .R2 for b > 1(2)where the weighting factor, b, and R2 are, respectively, thegradient and the coefficient of determination of the regressionof predicted versus observed values of a given parameter. The
advantage of Eq. (2), as opposed to R2, is that only a single but
more effective criterion is used for the comparisons. Models
which systematically over- or under-predict observed data all
the time will still result in R2 values close to 1.0, even if all
predictions are incorrect. This problem can be avoided by using
additional information about the regression. Such information isprovided by the gradient b and the intercept a of the regression
on which R2 is based. For good agreement the intercept ashould be close to zero (which means that an observed value ator near zero would also result in a prediction at or near zero),
while the gradient b should be close to 1.0. By using wR2, systematic under- or over-predictions are quantified, together withthe regression as such, thus leading to a more comprehensive
reflection of model results (Krause et al., 2005).
RESULTS AND DISCUSSION
Statistical descriptions of selected physical and chemical
properties of the soils used in this study are presented in Table
3. The data show that the soils differed widely in their PSD. For
example, CV values of clay and silt contents were 48 and 55%,
respectively, while the soil textural classes ranged from sandy
loam to clay as shown in Fig. 1. On the other hand, pH values
were within a relatively narrow range among the soils studied.
The calcium carbonate equivalent (CCE) of all soils was above
7.3% and the pH was equal to 7.5 or more, which indicates thatthe soils were calcareous and alkaline. The organic matter(OM) content varied widely among the soils (CV = 53%). Most

Comparison of alternative soil particle-size distribution models and their correlation with soil physical attributes
Table 3. Maximum (Max), Minimum (Min), Mean and Standard Deviation (STDEV) of selected physico-chemical properties of the soil
samples.
Soil property
Clay (%)
Silt (%)
Sand (%)
pH
CCE (%)OM (%)ρb (g cm–3)
EC (dS m–1)
ESP (cmol(+) kg–1)
Kfs (cm day–1)

Maximum
52
60
59
8.5
33.5
5.98
1.46
85.4
71.0
970.6

Minimum
8
30
8
7.5
17.3
0.89
0.94
0.8
4.0
0.34

Mean
27
44
29
8.2
23.3
2.94
1.22
14.4
25.5
201.2

STDEV
13
8
16
0.5
7.2
1.55
0.11
22.9
18.0
260.4

Range
45
30
51
1
16
5
0.52
85
67
970

CV
48
18
55
5.5
31
53
9.31
159
71
129.4

EC, Electrical conductivity; ESP, Exchangeable sodium percentage; CCE, Calcium carbonate equivalent; OM, Organic matter content; ρb, Bulk density, Kfs,
Saturated hydraulic conductivity.

of the soils were either saline/sodic (n = 14) or non-saline/nonsodic (n = 8).
Comparison of the fitting ability of the PSD models
Box plots of the statistical criteria of the applied PSD models for the 24 soils are given in Fig. 2. Results indicate that thevalues of R2, δ, NSE, and GMER differed in a narrow range
among the models (CV < 6.2%). This shows that these criteria
are not very effective in representing differences in the accuracy of the models. By comparison, AIC varied moderately (CV =18.7%), whereas RMSE, MAXe, MAEP, STDEV, and MAXaep
varied more widely (CV > 50%) among the models.
Mean values of R2 ranged from 0.820 to 0.994 (Fig. 2). The
Wie and Fred-4p models for F(d) showed the highest R2 values,
ONL, Fred-3p and MLG models showed similar intermediate
performance, while the Fr(T-W), Fr(B), Exp, and L-E models
had the lowest R2 values. The favorable fitting capability of
Wie and Fred-4p reflects their flexibility in terms of describing
a wide range of PSD shapes (Bayat et al., 2015). The Wei model also has a physical basis and no overlap in the effects of its
parameters on the shape and position of the PSD curve. The
Fred-4p model performed slightly better than the Fred-3p model. This indicates that Fred-3p is probably accurate enough formost soils, with the advantage of having one parameter less(Fredlund et al., 2000). These results are consistent with thoseby Bayat et al. (2015) who found no difference between theFred-4p and Fred-3p models in terms of R2 and RMSE whenapplied to the UNSODA database with a broad range of soil
textural classes. However, they suggested using the Fred-3p
model according to the AIC criterion. The favorable performance of Fred-4p in describing PSD data was reported also by
Hwang et al. (2002) for soils covering a wide range of textures.
Of the three models assuming lognormal curves containing
three parameters (i.e., ORL, ONL and SH-C), the R2 values
were higher for ONL as compared to the ORL and SH-C models. These results agree with those by Hwang et al., (2002), butdiffer from findings by Buchan et al. (1993) who, for a limited
number of soils, found higher average R2 values for the SH-C
model than the two other models. However, we note that the
CV of the R2 values in our study was only about 6.2% among
the studied models. This shows that the models are difficult torank conclusively in terms of their R2 values. Similarly, the low
values of the coefficient of variation of δ, NSE, and GMER(<4.3%) makes it difficult to properly rank those models.
RMSE values of the models ranged from a low of 0.019 for
AD to a high of 0.106 for Fr(T-W). Fr (T-W) is a single powerlow exponent model which could not properly characterize the

PSD across the whole range of measured data points. For thisreason Bitteli et al. (1999) proposed the use of three separate
power laws for the clay, silt and sand particle sizes.
The G, Norm, J, Exp, Log, L-E, Fr(B), and Fr(T-W) models
performed the poorest as reflected by their high mean AIC
values, whereas the other models performed well with very
similar low mean AIC values (AIC < –109), as can be seen in
Figure 2. In terms of their AIC, the Fred-3p, MLG, Wei andFred-4p models ranked first to fourth. This result differs fromthat of Shangguan et al. (2014), who found that AD and Wei
had the lowest mean AIC values and hence were considered thebest PSD models among eleven models that they compared.
Considering the average AIC values of all of the soils theystudied, Botula et al. (2013) suggested that the number of fitting
parameters does not always explain the differences in the quality of the fittings. They found lower AIC values for Fred-4p
relative to the models having three, two and one parameters.
Bayat et al. (2015) similarly suggested that an increase in thenumber of fitting parameters in the PSD models does not necessarily lead to better performance. The Gompertz model withfour parameters showed the poorest performance in their study.
Even though the AIC has a penalty on the number of parameters, the Wei, Fred-3p, MLG models with three parameters
were still among the best PSD models in our study.

Fig. 1. Textural distribution of the 24 soil samples.

183

Fatemeh Afrasiabi, Habib Khodaverdiloo, Farrokh Asadzadeh, Martinus Th. van Genuchten

Fig. 2. Box plots of the statistical criteria used to describe the accuracy of 19 models in fitting the cumulative particle-size distribution dataof 24 soil samples. The box plots show medians, interquartile ranges, and outliers. The full names and other information about the models(horizontal axes) are given in Table 1. The statistical criteria (vertical axes) are listed in Table 2.

184

Comparison of alternative soil particle-size distribution models and their correlation with soil physical attributes
Table 4. Rankings of the six most accurate PSD models in terms of various statistical criteria.
Ranking

R2

AIC

RMSE

Er

d

MAXaep

NSE

STDEV

GMER

MAXe

MAEP

1
2
3
4
5
6

AD
Wei
Fred-4p
Fred-3p
MLG
ONL

Wei
Fred-3p
AD
MLG
Fred-4p
ORL

AD
Fred-4p
Wei
Fred-3p
MLG
ONL

Fred-4p
MLG
Fred-3p
AD
ONL
ORL

Fred-4p
Wei
Fred-3p
MLG
ONL
ORL

Fred-4p
Wei
MLG
Fred-3p
AD
ONL

VG
Fred-4p
Wei
ADFred-3p
MLG

Fred-4p
AD
Wei
Fred-3p
MLG
ONL

ONL
ORL
AD
Wei
VG
Log

Fred-4p
Fred-3p
MLG
AD
Wei
ORL

AD
Wei
Fred-4p
ONL
Fred-3P
MLG

Regarding the relative error (Er), Bagarello et al. (2009)
considered that Er values of less than 5% indicate a satisfactory
fitting performance of the PSD function. In our study Fred-4p,
MLG, Fred-3p, ONL, AD, ORL, vG, BEST, Wei, SH-C, and
SLN all had Er values less than 5%, which would suggest thatthese models all performed well. Other models had Er values
larger than 5%. The Er values of Exp, L-E, Fr(B) and Fr(T-W)
in particular were very high (≥ 13%).
The mean value of MAXe among the models differed widelyfrom 0.042 to 0.207, indicating outlier error values of 4.2 to20.7 percent in the cumulative fractions. The Fred-4p, Fred-3p,
MLG, and Wei models showed the lowest (and very similar)
outlier errors (MAXe = 0.042 to 0.046). On the other hand, theFr(T-W), Fr(B), L-E, and Exp models had the highest outliers(MAXe≥ 0.136).
The highest MAEP values were obtained with the Exp, Log,
L-E, Fr(B), Fr(T-W), and Norm models (MAEP values between
12.5 and 21.5%). MAEP values of the G and J models weremoderate (about 10%) and those of the other models about 6%
or less. The lowest MAEP values were obtained for Fred-4p,
Fred-3p, Wei, MLG, and AD; MAEP values of these models
were very similar, varying between 3.91 and 4.41%.
The tendency of the models to predict cumulative fractions
with outlying errors is most effectively described using MAXaep.
This efficiency criterion varied widely among the models from
6.7 to 47.6% (Fig. 2). The lowest values were obtained for theFred-4p, Fred-3p, and MLG models, which produced fairly
similar values between 6.7 and 8.2%. A relatively large groupof models (i.e. G, Wei, BEST, AD, and ONL) had moderate
MAXaep values ranging from 12.9 to 16.5%. MAXaep values ofthe other models were higher than 23.6%.
The STDEV of the errors of the models ranged from 0.02 to
0.11 (Fig. 2). The models having the highest standard deviations were VG, L-E, Fr(B), and Fr(T-W), whereas Fred-4p,
Fred-3p, MLG, and AD models had the lowest variations in theerror (i.e. the lowest STDEV values). These and several otherefficiency criteria are partly in agreement with published results. For example, in a study evaluating the accuracy of 11
PSD models for the conversion of Chinese data to the
FAO/USDA System, Shangguan et al. (2014) reported muchbetter performance of the AD, Fred-4p, MLG, and Wei models
compared to the other models. Botula et al. (2013), Zhao et al.,(2011) and Hwang (2004) found the fractal models as well as
the Exp and L-E models to be less accurate in their study. Evaluating the performance of 14 PSD models for characterizing thesize distribution of sediments of dams in China, Zhao et al.
(2011) identified Fred-4p as the best performing model, whileof the 3-parameter formulations the Wei model performed thebest. The J model was found to perform the best among theone-parameter models. Zhao et al. (2011) furthermore found thefractal models, Exp, Log and L-E to be the least accurate.
Buchan et al. (1993) reported similar results. However, ourstudy indicated ORL, ONL and SLN to be superior to the J

model. Hwang et al. (2002) and Liu et al. (2003) identified
Fred-4p as the best PSD model for a wide range of soil types.
Table 4 shows the ranking of the six most accurate models
of our study in terms of the various efficiency criteria we used.
The data show that using different criteria will lead to different
rankings of the models. For example, if one uses NSE to rank
the models, VG would be the most accurate model followed byFred-4p, Wei, AD, Fred-3p, and MLG. On the other hand,according to the MAXaep criterion, the Fred-4p model was themost accurate followed by Wei, MLG, Fred-3p, and AD, while
RMSE values ranked the models as Wei, Fred-4p, Fred-3p,
MLG, and ONL (Table 4). However, we note that the differences in the values of most of the criteria among the most accurate models are generally not very substantial.
Mean absolute error percentages over all soils as obtainedwith the six most accurate models are plotted against the particle diameter in Fig. 3. The plots indicate that the prediction
error of the models depends upon the diameter at which the
cumulative PSD fraction is calculated. Much higher error percentages occurred when predicting the cumulative fraction forsmaller particles. When evaluating the accuracy of a certain
PSD model, an important issue here is that one should consider
the range of soil particle sizes for which the model will be used.
A model may be accurate enough for predicting the particle
diameter at a certain point or within a certain range, or maybe
even on average over the entire PSD curve, but not within
another range of particular sizes. Applying the model to inappropriate ranges of particle sizes may lead to large errors.

Fig. 3. Mean absolute error percentages of the six most accurate
models as a function of soil particle diameter.

Classifying the evaluation criteria and PSD models
Figure 4 shows results when classifying the evaluation criteria in groups based on the level of similarity among their values. If, in this figure, the similarity of two criteria is high, theircorrelation will also be high. In general, the efficiency criteria
were classified in two major groups, each having six subgroups.

185

Fatemeh Afrasiabi, Habib Khodaverdiloo, Farrokh Asadzadeh, Martinus Th. van Genuchten

3.29

Similarity

35.52

67.76

100.00
R2

d

E
NS

SE
M
R

Er

P
AE
M

EV
D
ST

AX
M

e

C
AI

ER
GM

p
ae
AX
M

Fig. 4. Hierarchical cluster analysis of the criteria used to evaluate
the accuracy of the particle-size distribution models. Information
regarding the statistical criteria is given in Table 2.

Similarity

23.23

48.82

74.41

100.00G LN -C 4P AD 3 p LG NLS SH eded M O
Fr
Fr

vG

ei T LW BES OR

J
N

)
) -E
m g xpor Lo E Fr(B T-W L(
r
F

Fig. 5. Hierarchical cluster analysis of the 16 particle-size distribution models in groups based on the level of similarity among their
R2, NSE, STDEV, AIC, and GMER values. The full names andother information about the models are given in Table 1; information regarding the statistical criteria is given in Table 2.

According to these results, six criteria (i.e., R2, NSE, STDEV,
AIC, GMER and MAXaep) were selected. The selected criteria
were then used to classify the models into groups using a hierarchical cluster analysis based on the level of similarity in theirperformance (Fig. 5). According to Fig. 5, the PSD models
could be classified into eight distinct group/subgroups. The
similarity among the Fred-4p, Fred-3p, AD, MLG, ONL, vG,
and Wei models was more than 80%. This reflects the similar
performance of these seven models in terms of characterizing
the PSD of our soils.
Evaluating the soil texture-dependency of the PSD models
In order to evaluate model performance per textural class,soil samples were divided into two groups based on their textural class (i.e., fine-textured and coarse-textured soils). The
fine-textured (FT) soils group (n = 9) included 3 silty clay
loams, 2 clay loams, one silty clay and one clay, whereas thecoarse-textured (CT) soils group (n = 14) included 3 silty
loams, 8 loams and 3 sandy loams. The minimum, maximum
and mean contents of sand were, respectively, 8, 29, and 16%
for the fine-textured and 30, 59, and 38% for the coarsetextured soils. The values for clay content of the FT and CT soil
groups were 28, 52, and 41% and 8, 27, and 18%, respectively.
Several comparative studies have shown that performance of
the PSD models may depend upon soil texture (e.g. Bagarello et186al., 2009; Buchan, 1989; Fredlund et al., 2000; Hwang, 2004;
Hwang et al., 2002; Zhao et al., 2011). These studies revealed
that many or most PSD models performed better for finetextured soils, which would imply that their performance
should improve with an increase in the clay content in the soils(e.g. Fredlund et al., 2000; Hwang, 2004; Hwang et al., 2002;
Zhao et al., 2011). We compared the efficacy of the best models(Fred-4p, Fred-3p, AD, MLG, ONL, and Wei) in the two soil
textural groups using the t-test (P≤0.05) in terms of their mean
RMSE, MAEP and STDEV values over all of the soils involved(Fig. 6). No matter which criterion was used for the comparison, all of the models performed significantly (P ≤ 0.05) betterin the fine textured (FT) soils group as compared to the coarse
textured (CT) soils (Fig. 6).
The above results partially confirm literature findings. For
example, Buchan (1989) pointed out that a log-normal (L-N)
model could properly explain all regions of silty clay, silty clay
loam, and silt loam soils, whereas sandy clay loams, sandy
clays, and much of the clay soils should not be modeled with
the L-N model. Liu et al., (2003) similarly found that the Fred3p and MLG model produced comparable results for silty clay
loam and silt loam soils, but yielded worse results for sandy
loam soils. In another study, Botula et al., (2013) found that the
AD model performed better for relatively fine-textured soils.
The generally better performance of the PSD models for finetextured soils may be attributed to the fact that the values of the
cumulative fractions for finer particles are lower than those ofcoarser particles, which may lead to higher prediction errors forcoarser particles. This then would produce higher values of theprediction errors for soils containing coarser particles.
We note that the rankings of the models in terms of their
RMSE values were not the same for the fine-textured (FT) andcoarse-textured (CT) groups. The ranking was Wei> Fred-4p >AD > MLG> Fred-3p > ONL for the CT group, and AD > ONL> Fred-4p > Fred-3p > MLG > Wei for the FT group. However,
the ranking for all the soils combined (regardless of their texture) was AD > Fred-4p > Wei >Fred-3p > MLG > ONL.
Functional evaluation of the PSD models
The data in Table 5 indicate the different performance of the
PSD models when estimating different soil PSD indicators or
parameters. For example, while the AD and SH-C models predicted the nominal sand fraction (as calculated using hydrometer records at 40 s) more accurately, the ONL, Wei and VG
models showed better predictions of the silt and clay fractions.
The measured sand fraction (particles larger than 53 μm as
estimated using sieving) was predicted more accurately with theFred-3p and VG models. Generally, the models were moreaccurate in predicting the sand and clay fractions than the silt
content (Table 5).
While several models were relatively accurate in estimatingdg, σg, D50, and D60 values of the soils, no model was able topredict D30 values with acceptable accuracy (Table 5). We notethat a common procedure to measure SSC fractions is to usehydrometer readings at specific times of particle settlement insoil suspensions. Resulting size fractions may not necessarilybe the same as their actual values (i.e. particles of specific sizegroups). Also, a common procedure to measure D30, D50, andD60 is to linearly interpolate between measured cumulative PSD
data that are immediately higher and lower than the required
point on the PSD curve. Since the PSD curve is generally nonlinear rather than linear, linear interpolation is not always appropriate and hence values derived from the fitted PSD models
may be preferred.

Comparison of alternative soil particle-size distribution models and their correlation with soil physical attributes

Value

0.06
0.04

a

0.08

a

AD

CT

FT

b
a

b

b

0.06

Value

0.08

0.04

0

Fred-4p

MAXe

STDEV
0.08

a
b

a b

a b

0.02

MAXe

0.04

STDEV

a
ONL
a

b
b

a

b

RMSE

STDEV

MAXe

STDEV

0.08

MLG
a

a
b
a b

b

0.02

Value

0.08

Value

b

0
RMSE

0.04

MAXe

0.06

0.02

0

0.06

a

a b

RMSE

Value

Value

0.04

b

0
RMSE

0.06

a

0.02

0.02

0.08

Wei

0.06
0.04

Fred-3p

a
b
a

a b

0.02

b

0

0
RMSE

MAXe

RMSE

STDEV

MAXe

STDEV

Fig. 6. Comparison of the efficacy of the six most promising particle-size distribution models for the fine-textured (FT) and coarse-textured(CT) soils. The full names and other information about the models are given in Table 1. Information regarding the statistical criteria isgiven in Table 2. For each criterion the non-similar letters show statistically significant differences (P ≤ 0.05).
Table 5. Weighted R2 (equation 2, wR2) values of the studied particle-size distribution models to predict selected parameters of soil PSD.
AD
BEST
Exp
Fred-4p
Fred-3p
Fr(B)
Fr(T-W)
G
J
SLN
Log
L-E
MLG
Norm
ONL
ORL
SH-C
VG
Wei

Sand
0.991
0.209
0.807
0.883
0.893
0.807
0.761
0.786
0.826
0.962
0.694
0.807
0.903
0.692
0.937
0.928
0.963
0.889
0.913

Silt
0.786
0.003
0.004
0.767
0.731
0.004
0.00
0.554
0.005
0.590
0.000
0.004
0.825
0.406
0.892
0.779
0.592
0.741
0.883

Clay
0.881
0.151
0.652
0.952
0.936
0.652
0.615
0.818
0.822
0.876
0.790
0.652
0.961
0.591
0.933
0.882
0.867
0.962
0.967

Sand measured
0.928
0.231
0.694
0.938
0.946
0.695
0.647
0.831
0.665
0.833
0.603
0.694
0.938
0.731
0.901
0.917
0.834
0.940
0.933

Correlation of PSD parameters with soil physical properties
We next correlated various representations of the soil PSD(e.g. SSC, dg and σg, D50 and D60, as well as the model parameters of the most accurate PSD models) with selected soil physical properties. While neither SSC nor dg, σg, D50 and D60
showed significant statistical correlations with either Kfs or
logKfs, the fitted parameters of several PSD models correlated
significantly with Kfs and/or logKfs, as shown in Table 6. Mostof the correlations were stronger with logKfs as compared to Kfs.
Since Kfs is known to be one of the most variable soil properties,
finding effective inputs to derive PTFs for Kfs remains a challenge (e.g. Mbonimpa et al., 2002). Our results suggest that the

dg
0.981
0.984
0.852
0.982
0.974
0.852
0.819
0.901
0.497
0.827
0.806
0.852
0.988
0.746
0.948
0.995
0.827
0.927
0.999

σg
0.669
0.823
0.303
0.900
0.873
0.302
0.113
0.929
0
0.877
0.432
0.303
0.891
0.727
0.808
0.567
0.877
0.855
0.950

D60
0.946
0.953
0.701
0.962
0.974
0.016
0.641
0.930
0.529
0.770
0.727
0.696
0.966
0.778
0.908
0.918
0.770
0.911
0.961

D50
0.853
0.90
0.579
0.880
0.850
0.013
0.496
0.835
0.532
0.768
0.850
0.560
0.860
0.811
0.867
0.824
0.769
0.857
0.873

D30
0.005
0.010
0.016
0.008
0.010
0.016
0.018
0.001
0.010
0.006
0.004
0.017
0.009
0.006
0.007
0.008
0.006
0.009
0.008

use of fitting parameters of the PSD models provides an attractive alternative to account for the effects of soil texture on Kfs.
Correlations between the mean weight diameter (MWD) of
soil aggregates with selected PSD model parameters were generally higher for either SSC, dg or D50 and D60 (Table 6). While,Φ and ρb correlated significantly with selected fitting parameters of the PSD models, ρb showed also significant correlation
with many or most geometric (dg), mechanical (D50 and D60),
and agronomic (clay and sand) representations of the PSDs
(Table 6), with the exception of the parameters m (Fred-3p) andc (Wei) which showed essentially no correlations. The parameters df (Fred-4p), δ (ONL), b0 (AD), c0 (AD) correlated onlywith σg, while m (Fred-4p) had a significant positive correlation

187

Fatemeh Afrasiabi, Habib Khodaverdiloo, Farrokh Asadzadeh, Martinus Th. van Genuchten
Table 6. Correlations between various PSD model parameters and selected soil physical properties.
Kfs

logKfs

MWD

Φ

ρb

dg

σg

D50

D60

Clay

Silt

Sand

α (Fred-4p)

–0.36

–0.26

–0.67**

–0.27

0.52*

0.87**

0.61**

0.78***

0.87***–0.67***–0.66***

0.87***

df (Fred-4p)

0.18

0.23

–0.14

–0.03

–0.06

0.21

0.50*

0.08

0.17

–0.02

–0.24

0.13

m (Fred-4p)

0.39

0.30

0.17

–0.19

-0.11

–0.17

–0.08

–0.08

–0.13

–0.19

0.55**–0.07

n (Fred-4p)–0.38–0.66*

–0.27

–0.14

0.48*

0.54**–0.36

0.58**

–0.28

0.61

**

0.88

***

0.68

**

0.95

***

c(ONL)μ (ONL)

–0.48

*

–0.32

–0.44

*

–0.38

–0.63

**

–0.62

**

–0.43

*

0.05

0.64**

0.64**
0.85

***

0.91

***

0.50

*

0.78

***

0.46

*

0.87

***

**

–0.48*
–0.66

***

–0.92

***

–0.68

***

0.87***–0.45*

0.97***

δ (ONL)

–0.01

–0.28

–0.16

–0.07

–0.19

–0.11

0.67

–0.37

–0.22

0.19

–0.09

b0 (AD)

0.11

0.24

–0.22

-0.20

0.01

0.20

0.57**

–0.04

0.05

–0.25

–0.03

–0.11
0.22

c0(AD)

–0.25

–0.29

0.22

0.32

–0.15

–0.23

–0.60**

0.08

0.01

0.21

–0.03

–0.16

d0(AD)

–0.37

–0.30

–0.62**–0.27

0.57**

0.96***

0.41

0.94***

0.96***–0.73***–0.68**

0.91***

f0 (AD)

–0.23

–0.25

–0.36

0.19

0.09

0.41

0.02

0.40

0.44*

–0.05

–0.58**

0.31

α (Fred-3p)

–0.40

–0.38

–0.65**–0.28

0.60**

0.95***

0.48*

0.91***

0.96***–0.73***–0.68***

0.93***m (Fred-3p)

0.38

0.39

0.43

0.05

–0.40

–0.36

–0.02

–0.35

–0.41

0.24

0.31

–0.34

n (Fred-3p)–0.42*–0.65**

–0.38

–0.17

0.49*

0.60**

0.04

0.72***

0.70***–0.61**–0.27

0.65**

**

*

–0.25

–0.53

**

–0.60

**

0.57

**

0.31

–0.58

**

–0.71

***

0.63

**

0.46

***

–0.39

a (MLG)

0.59*

b (MLG)

0.46

c (MLG)*a (Wei)b (Wei)

–0.43

–0.02
–0.33

0.53
0.39
–0.60

**

–0.20
–0.53

**

0.47

*

0.59

**

–0.52

*

0.32
–0.43

0.40
0.37
–0.20
0.29

–0.72

***

–0.62

**

0.55

**

–0.31

–0.19

0.48

–0.59

**

–0.68

***

0.75
–0.50

*

***

*

0.60

**

–0.64

**

0.33
–0.79

***

0.77

***

–0.36

–0.00

0.72

–0.15

–0.01

0.79
–0.43

***

***

*

0.69

***

–0.00

–0.71

0.32
–0.75

***

0.37

0.57

–0.62***–0.75***
0.80*****–0.54**–0.07

0.67***

–0.34

–0.12

c (Wei)

–0.37

–0.25

–0.19

0.27

–0.08

–0.05

dg

–0.27

–0.35

–0.53*–0.42*

0.65**

1.00

σg

–0.11

0.04

–0.40

–0.27

0.26

–0.40

D50

–0.33

–0.37

–0.53*–0.29

0.63**

0.97***

0.25

1.00

D60

–0.35

–0.34

–0.61**–0.28

0.63**

0.97***

0.40

0.95***

1.00

Clay

0.18

0.33

0.51**

0.44*–0.61**–0.82***–0.32–0.76***–0.77***

Silt

0.39

0.22

0.35

0.11

–0.37

–0.54**–0.36–0.63**–0.63**

0.11

1.00

Sand

–0.34

–0.38

–0.59**–0.41*

0.68***

0.97***

0.43*

0.93***

0.94***–0.88***–0.58**

1.00

1.00
1.00

, , and ***: indicate significant correlation at P ≤ 0.05, P ≤ 0.01, and P ≤ 0.001, respectively. Soil bulk density (ρb), soil total porosity (Φ), Field saturated
soil hydraulic conductivity (Kfs), Mean Weight Diameter of soil aggregates (MWD).

* **only with silt content (Table 6). The values of α (Fred-4p), df(Fred-4p), n (Fred-4p), c (ONL), μ (ONL), d0 (AD), α (Fred3p), n (Fred-3p), c (MLG), and b (Wei) correlated positively
with soil particle size. These parameters increased with the sizeof soil particles, while the parameters a (MLG), b (MLG), anda (Wei) decreased with increasing soil particle size. Furthermore, as shown in Table 6, values of α (Fred-4p), df (Fred-4p),c(ONL), μ (ONL), δ (ONL), b0 (AD), α (Fred-3p) increased witha widening of the PSD range (as reflected by the standard deviation of the soil particle diameter, σg). However, c0 (AD), b(MLG), and a (Wei) decreased with increasing σg values.
CONCLUSIONSIn this preliminary study we investigated the feasibility of 19
models to describe measured soil particle-size distributions(PSDs) of 24 different soils in terms of different efficiency
criteria. Results indicate that rankings of the model performance differed depending upon the particular efficiency criteria
considered. For example, RMSE, MAXe, and STDEV values
varied widely among the PSD models, which implies that thesecriteria all represented different aspects of the accuracy of themodels. Six models were found to provide the most accurate
results: a modified logistic growth function (MLG), Fredlund
type models with 3 and 4 parameters (Fred-3p and Fred-4p), an
Anderson type model (AD), an offset nonrenormalized lognormal function (ONL), and a Weibull type function (Wei). Prediction errors of these models depended on the particle diameter at which the cumulative PSD fraction was estimated. Accu-

188

racies were found to decrease with increasing soil particle diameter. This implies that one should consider the range of sizesof the soil particles for which the model will be used. A model
may be acceptable for some range of the particle diameter, orthe entire PSD, but not necessarily for other particle sizes.
Based on the mean values of the RMSE, MAXe, and STDEV, allof the selected models performed significantly better for finetextured soils as compared to coarse-textured soils.
The fitting parameters of selected PSD models showed statistically significant correlations with several soil quality indices such as Kfs, MWD, soil porosity, and soil bulk density.
These results may be used to advantage in future studies toderive more accurate pedotransfer functions (PTFs) for estimating the unsaturated soil hydraulic properties and/or related soil
physical parameters from PSD data. Still, we acknowledge thatthe number of soil samples (n = 24) used in this study is not
sufficient to draw strong conclusions. A similar analysis asfollowed herein, but with a much larger soils database, may beneeded to obtain more definite conclusions.
Acknowledgements. This work was supported by the Iran National Science Foundation [grant No. 91002216] and Urmia
University [grant No. 93/K/023].
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Received 28 June 2017
Accepted 19 October 2017