Models for Estimating Soil Particle-Size Distributions
Sang Il Hwang, Kwang Pyo Lee, Dong Soo Lee, and Susan E. Powers*
ABSTRACT
models for experimental PSD data. Each of the five
models accounted for ⬎90% of the variance in the PSD
of most of the soils examined, with the bimodal lognormal model proposed by Shiozawa and Campbell (SC,
1991) providing the best fit. In addition to lognormal
models, diverse model forms have been proposed, including models based on the water retention curve (Haverkamp and Parlange, 1986; Smettem and Gregory, 1996;
Fredlund et al., 2000), the Gompertz model (Johnson
and Kotz, 1970; Nemes et al., 1999), the fragmentation
model (Bittelli et al., 1999), and the model estimating
PSD from limited soil texture data (Skaggs et al., 2001).
The assumption of lognormality for the soil PSD is
the basis for defining the geometric mean diameter and
geometric standard deviation as mean size and spread
parameters (Buchan, 1989). These parameters are often
used for estimating soil hydraulic properties (Mishra et
al., 1989; Rajkai et al., 1996; GonÇalves et al., 1997; Scheinost, 1997; Minasny et al., 1999). However, the use of
geometric statistical parameters would not provide accurate estimates of soil hydraulic properties if the soil
PSD is not lognormally distributed. For example, Buchan (1989) found that only about half of the USDA
texture triangle can be adequately described with a lognormal PSD. Nemes et al. (1999) evaluated four different procedures to interpolate PSDs to achieve compatibility within soil databases. A loglinear interpolation
procedure was the least accurate for estimating missing
particle size classes for the soil databases they studied.
The suitability of different PSD models appears to
be influenced by soil texture classes and/or by clay content of a soil. Buchan (1989) pointed out that all of silty
clay, silty clay loam, and silt loam among the USDA
texture triangle could be properly modeled with lognormal PSDs; but more complex distributions were required for sandy clay loam, sandy clay, and much of the
clay region. Rousseva (1997) defined two closed-form
models based on exponential and power law distributions and investigated the suitability of these models to
fit the PSDs with different shapes and varying numbers
of measured points. Results showed that the suitability
of the models seemed to be affected by texture type
(coarse or fine textured soils) rather than by measured
size ranges. Fredlund et al. (2000) showed that their
own model provided a better fit as the clay content of
soils increased.
Only a few comparative studies of PSD models have
been conducted in soil science (Buchan et al., 1993;
Rousseva, 1997). Buchan et al. (1993) analyzed 71 PSD
data sets collected from two regions of New Zealand.
Their comparison was limited to the lognormal models.
An accurate mathematical representation of particle-size distributions (PSDs) is required to estimate soil hydraulic properties or to
compare texture measurements from different classification systems.
The objective of this study was to evaluate the ability of seven models
(i.e., five lognormal models, the Gompertz model, and the Fredlund
model) to fit PSD data sets from a wide range of soil textures. Special
attention was given to the effect of texture on model performance.
Several criteria were used to determine the optimum model with the
least number of fitting parameters when other conditions are equal.
The Fredlund model with four parameters showed the best performance with the majority of soils studied, even when three criteria
that impose a penalty for additional fitting parameters were used. Especially, the relative performance of the Fredlund model in regard
to other models increased with increase of clay content. Among all soil
classes, the lognormal models with two or three parameters showed
better fits for silty clay, silty clay loam, and silt loam soils, and worse
fit for sandy clay loam soil.
T
he particle-size distribution of a soil is a fundamental soil physical property. It is widely used as
a basis for estimating soil hydraulic properties such as
the water retention curve and saturated as well as unsaturated hydraulic conductivities (Gupta and Larson,
1979a,b; Arya and Paris, 1981; Campbell, 1985; Schuh
and Bauder, 1986; Vereecken et al., 1989). Mathematical
relationships between the particle-size and hydraulic
properties tend to be fairly good for sandy soils, but not
as accurate for soils with larger fractions of clay (Cornelis et al., 2001). The PSD prediction is also needed
when comparing texture measurements from different
classification systems (Shirazi et al., 1988). A conventional particle-size analysis involves the measurement
of the mass fraction of clay, silt, and sand and use of
these fractions to find the textural class using a textural
diagram. A more complete description of texture is obtained by using a PSD model that best fits experimental
data. Commonly, PSDs are reported as cumulative distributions with different models proposed to fit experimental data. Selecting the most appropriate model to
represent the soil PSD may be important to more precisely estimate the soil hydraulic properties (Bittelli et
al., 1999).
The PSD in soil is frequently assumed to be approximately lognormal (Shirazi and Boersma, 1984; Campbell, 1985; Buchan, 1989), although soils with bimodal
PSDs also occur (Walker and Chittleborough, 1986).
Buchan et al. (1993) compared five different lognormal
S.I. Hwang and S.E. Powers, Dep. of Civil and Environmental Engineering, Clarkson Univ., 8 Clarkson Ave., Potsdam, NY 13699-5710;
K.P. Lee and D.S. Lee, Graduate School of Environ. Stud., Seoul
National Univ., Seoul, 151-742, South Korea. Received 9 July 2001.
*Corresponding author ([email protected]).
Abbreviations: PSD, particle-size distribution; AIC, Akaike’s information criterion; SSE, sum of squared errors; SL, simple lognormal;
ORL, offset-renormalized lognormal; ONL, offset-nonrenormalized
lognormal; SC, Shiozawa and Campbell.
Published in Soil Sci. Soc. Am. J. 66:1143–1150 (2002).
1143
1144
SOIL SCI. SOC. AM. J., VOL. 66, JULY–AUG 2002
To our knowledge, there has been no attempt to compare a variety of PSD models with different underlying
assumptions. Also, the effect of texture on the performance of PSD models has not been investigated fully.
Thus, the objectives of this study were to test a variety
of models with different underlying assumptions to determine which model best represents soil PSDs, and to
investigate if soil texture significantly affects the performance of models. To achieve these objectives, we compared five lognormal models proposed by Buchan et al.
(1993), plus two models recently proposed by Nemes
et al. (1999) and Fredlund et al. (2000). The performance
of these later two models has not previously been investigated intensively. The PSD data sets of 1387 Korean
soils were used in this study. Four comparison techniques were considered to define the best models: the
coefficient of determination (r 2 ), the F statistic, the Cp
statistic of Mallows (1973), and Akaike’s information
criterion (AIC).
MATERIALS AND METHODS
The Korean Soil Database
The Korean soil physical database contains data from 1387
soil layers representing ≈378 soil profiles. The data have been
published by the Rural Development Administration (1971,
1975, 1977a, 1977b, 1980) and are available in Microsoft Excel
worksheet file format. The PSDs in the database were determined using conventional methods following H2O2 pretreatment to eliminate organic matter. Fine size fractions were
determined by pipette method, whereas the coarse fractions
were obtained by sieving. Particle-size fraction data were classified by USDA standard. The particle-size limits were 2, 1,
0.5, 0.25, 0.10, 0.05, and 0.002 mm.
The Korean soils represent a wide range of soil textures
and organic matter content. Silt loam (24.2%), loam (20.2%),
sandy loam (16.7%), and silty clay loam (15.8%) were predominant among USDA texture classes in the 1387 soil layers
(Fig. 1). Textural composition of the Korean soils is similar to the Unsaturated Soil Database (UNSODA; Leij et al.,
Fig. 2. Cumulative particle-size distribution for 1347 Korean soils.
1999) and that of the Netherlands (Nemes et al., 1999). The
fraction of silt (0.002–0.05 mm) showed the highest variation
among the soils and the range of 0.05 to 0.25 mm among the
sand fraction also showed large variation (Fig. 2). Clay content
varies between 0.0 and 82.6%. The highest organic carbon
content was 14.9%, and dry bulk density ranged between 0.9
and 1.9 g cm⫺3.
Particle-Size Distribution Models
There are two basic approaches for representing the PSD:
parametric models of the full distribution, or more simple
statistical transformation of limited three-fraction texture data
(e.g., Shirazi and Boersma, 1984). We considered the first
category for PSD models because the parametric models can
provide complete information on the soil PSD. Seven parametric models were tested, including six unimodal models and one
bimodal model. We adopted five lognormal models previously
studied by Buchan et al. (1993): the Jaky one-parameter model
(Jaky, 1944) with a sigmoid half of a Gaussian lognormal
distribution; a simple lognormal (SL) model with two parameters (Buchan, 1989); and three modified lognormal models
with three parameters, that is, an offset-renormalized lognormal (ORL) model (Buchan et al., 1993), an offset-nonrenormalized lognormal (ONL) model (Buchan et al., 1993), and
a bimodal lognormal model (Shiozawa and Campbell, 1991).
Buchan et al. (1993) provides details of the five lognormal
models. The Gompertz (Nemes et al., 1999) and Fredlund et
al. (2000) models were tested as four-parameter models. The
seven models considered in this study are listed in Table 1.
Model Comparison and Fitting Techniques
Fig. 1. Textural composition of the soil data set.
Several approaches have been reported for selection of a
suitable model. The simplest approach is to find the best model
that minimizes the discrepancy between observed and predicted data. For example, a model with larger r 2 may be preferred more than one with smaller r 2. However, it must be recognized that as the number of fitted parameters increases, the
fitting performance generally improves. This occurs at the
expense of a corresponding increase in the possibility of overparameterization. The PSD models considered here required
between one and four fitting parameters. Therefore, a better
approach is to define the optimum model as the model that
fits data well with the least number of fitting parameters when
1145
HWANG ET AL.: PARTICLE-SIZE DISTRIBUTION MODELS
Table 1. Particle-size distribution models tested for texture data of 1387 soils.
Name
Model†
Jaky
(Jaky, 1944)
Parameters
p
(do ⫽ 2 mm)
冦 p1 冤ln冢dd 冣冥 冧
2
F(d) ⫽ exp ⫺
2
o
Simple Lognormal (SL)
(Buchan, 1989)
F(X) ⫽ (1 ⫹ erf [(X ⫺ ␮)/␴√2])/2 (X ⱖ ␮)‡
F(X) ⫽ (1 ⫺ erf [(X ⫺ ␮)/␴√2])/2 (X ⬍ ␮)
X ⫽ ln(d)
␮, ␴
Offset-Renormalized Lognormal (ORL)
(Buchan et al., 1993)
G(X) ⫽ (1 ⫺ ε)F(X) ⫹ ε
[F(X) defined by SL model]
␮, ␴, ε
Offset-Nonrenormalized Lognormal (ONL)
(Buchan et al., 1993)
G(X) ⫽ F(X) ⫹ c
[F(X) defined by SL model]
␮, ␴, c
Shiozawa and Campbell (SC) (Shiozawa and Campbell, 1991)
G(X) ⫽ εF1 (X) ⫹ (1 ⫺ ε)F2(X)§
␮ 2, ␴ 2, ε
Gompertz (Nemes et al., 1999)
F (d) ⫽ ␣ ⫹ ␥ exp{⫺ exp[⫺␤(d ⫺ ␮)]}
␣, ␤(⬍0), ␥, ␮
Fredlund
(Fredlund et al., 2000)
F(d) ⫽
1
冦 冤
␣, n, m, df
(dm ⫽ 0.001 mm)
冢 冣
冤
冦
冢 冣 冥冧
冢
冣冥 冧
␣
ln exp(1) ⫹
d
df
d
1⫺
df
ln 1 ⫹
dm
7
ln 1 ⫹
n
m
† d, particle diameter in mm.
‡ erf [ ], error function.
§ F1(X ), cumulative function as defined by the SL model for clay fraction (␮1 ⫽ ⫺1.96, ␴1 ⫽ 1.00 as suggested by Shiozawa and Campbell (1991)), F2(X ),
cumulative function as defined by the SL model for sand and silt fractions.
other conditions are equal. In addition to r 2, we need to adopt
additional model comparison criteria that have a penalty for
additional fitting parameters. Several researchers have used
this kind of criteria for selecting the best model. Vereecken
et al. (1989) used the F statistic (Green and Caroll, 1978) to
compare models fit to soil moisture characteristic data. Buchan
et al. (1993) used both the F statistic and the Cp statistic of
Mallows (1973) to select the best PSD model; they advocated
the use of Cp as the best model selection criterion because
the F cannot be used to compare equiparameter models. The
AIC (Carrera and Neuman, 1986) has been used by others to
select the best predictive function of soil moisture characteristic indirectly determined from other easily measured soil
properties (Minasny et al., 1999), and to select the best soil
hydraulic function (Chen et al., 1999). The four comparison
criteria we adopted (r 2, F statistic, Cp statistic, AIC) are listed
in Table 2.
In this study, the Fredlund model with four fitting parameters was used as a referenced model to compare with results
from the other six models. Three criteria were used to compare
among the models. In case of the F statistic, the sums of
squared errors (SSEs) of the referenced model and the comparison model were calculated for a soil, with df ⫽ (N ⫺ p ),
where N is the number of observed PSD data points (i.e.,
seven) and p is the number of model parameters (see Table 2).
A comparison model was accepted for the soil if its F value
did not exceed the F value at the 95% significance level (ob-
tainable from standard tables, e.g., Snedecor and Cochran,
1989), otherwise the reference model was defined as optimal.
Two important points have to be considered here concerning
the F statistics (Vereecken et al., 1989; Buchan et al., 1993).
First, for nonlinear models, the F statistic equation (Table 2)
is known to be approximate because the numerator and denominator no longer contain independent chi-square distributions (Beck and Arnold, 1977). The F statistic is still a useful
criterion to judge the necessity of model parameters through
the increase of the SSE. Secondly, the F statistic can be used
only when the error term is normally and independently distributed with zero mean and constant variance. Violation of
these conditions are most frequently checked through visual
inspection of the residuals plotted against the independent
variable or through an examination of the unit normal deviate
of the residuals in an overall plot. Using this latter test, we
found no indication that the assumptions regarding the error
term were violated. Note that the F statistic does not permit
intercomparison of equiparameter models, whereas Cp and
AIC do allow this comparison.
In the case of the Cp statistic, the Cp value equals four when
the Fredlund model is artificially compared with itself. The
comparison model was accepted for a particular soil if the
calculated Cp value was significantly lower than four. In this
case, the condition of acceptance with statistical significance
for the comparison model was defined as Cp ⬍ 3.8, thereby
defining those with Cp values within 5% of four to be too
Table 2. Four criteria for particle-size distribution model comparison.
Criteria
Equations
冦√
N(兺YoYp) ⫺ (兺Yo)(兺Yp)
Coefficient of determination (r 2)
r2 ⫽
F statistic (Green and Carol, 1978)
F ⫽ [(SSEc ⫺ SSEr)/SSEr][dr /(dc ⫺ dr)]
[N兺Y 2o ⫺ (兺Yo)2]
SSEj ⫽
√
[N兺Y 2p ⫺ (兺Yp)2]
Explanation
冧
2
N
兺 (Yp ⫺ Yo)2
i⫽1
SSEc
⫺ (N ⫺ 2p)
SSEr /(N ⫺ pr)
Cp statistic (Mallows, 1973)
Cp ⫽
Akaike’s information criterion (AIC)
(Carrera and Neuman, 1986)
AIC ⫽ N{ln(2␲) ⫹ ln[SSE/(N ⫺ p)] ⫹ 1} ⫹ p
N, number of observed data points.
Yo, observed data values.
Yp, predicted data values.
j ⫽ c, comparison model.
j ⫽ r, referenced model.
dr, degree of freedom of the referenced model.
dc, degree of freedom of the comparison model.
p, the number of model parameters.
pr, the number of parameters of the referenced model.
1146
SOIL SCI. SOC. AM. J., VOL. 66, JULY–AUG 2002
Fig. 3. Box plot for r 2 percentiles as the goodness-of-fit of seven
models for all soils. Jaky ⫽ Jaky model, SL ⫽ simple lognormal
model, ORL ⫽ offset-renormalized lognormal model, ONL ⫽ offset-nonrenormalized lognormal model, SC ⫽ Shiozawa and Campbell model, GOM ⫽ Gompertz model, and Fred ⫽ Fredlund model.
similar to differentiate. In case of the AIC criterion, the model
having the smallest AIC value was selected as best for a soil
considering that the AIC value of the Fredlund model has a
5% decrease from its original value.
Five models used in our study have same underlying assumption that the PSD in soil is lognormal, implying the possibility
of identical performance among these models. Therefore, to
determine whether a statistically identical performance exists
in each of all possible model pairs, including the Gompertz
Fig. 4. Statistical analyses of the particle-size distribution models for
all soils using the F statistic, the Cp statistic (Mallows, 1973), and
the Akaike’s information criterion (AIC). The number above bars
indicates the number of soils for which a model can be accepted
instead of the Fredlund model. Jaky ⫽ Jaky model, SL ⫽ simple
lognormal model, ORL ⫽ offset-renormalized lognormal model,
ONL ⫽ offset-nonrenormalized lognormal model, SC ⫽ Shiozawa
and Campbell model, and Gom ⫽ Gompertz model.
and Fredlund models, we conducted paired t-tests on each of
the above three criteria.
All models were fit to experimental PSD data of 1387 Korean soils using an iterative nonlinear regression procedure,
Fig. 5. Comparative fit of seven models: (a) sand (Jangcheon soil, 20–65 cm), (b) loamy sand (Kocheon soil, 55–100 cm), and (c) sandy loam
(Kwangpo soil, 60–120 cm). Obs. ⫽ observed, SL ⫽ simple lognormal model, SC ⫽ Shiozawa and Campbell model, Gomp. ⫽ Gompertz
model, ORL ⫽ offset-renormalized lognormal model, ONL ⫽ offset-nonrenormalized lognormal model, and Fred. ⫽ Fredlund model.
HWANG ET AL.: PARTICLE-SIZE DISTRIBUTION MODELS
which finds the values of the fitting parameters that give the
best fit between the model and the data. This procedure was
done using the SOLVER routine of Microsoft Excel software
(Microsoft Company, Redmond, WA). To evaluate whether
the final optimized parameter values depend on the initial
estimates of model parameters, we carried out nonlinear optimization runs using the SOLVER routine with at least three
different initial parameter estimates for all soils. The final
solutions for each soil converged to similar parameter values.
RESULTS AND DISCUSSION
Defining the Optimum Model
Among all of the soils and all of the models, values
of r 2 ranged from 0.561 to 1.000 (Fig. 3). As expected,
the lowest r 2 values were obtained with the Jaky model
because it only has one fitting parameter. The Fredlund
model had the highest r 2 values and the four-parameter
Gompertz model yielded lower r 2 values than two- or
three-parameter lognormal models. Among the three
models with three parameters, the r 2 values were higher
for the ONL model than for ORL and SC model. This
result differs from that of Buchan et al. (1993). On the
basis of their limited number of soil data sets, they
showed that the SC model had higher average r 2 values
than the other two models. Our result may be more
comprehensive because of the use of a much larger soil
data set.
1147
The relative model performance using the F statistic,
calculated with the four-parameter Fredlund model as
the reference model, showed differences among the
models (Fig. 4). A comparison model can be accepted
when the F value of the model is lower than the F value
calculated at the 95% significance level. The Gompertz
model was not included in Fig. 4 because the F statistic
cannot be used to compare equiparameter models. The
ONL model was better than the Fredlund model for 673
soils (≈48.5%) among 1387 soils. As with the previous r 2
analysis, the ONL model was again identified as the best
model among one- through three-parameter models.
Results of Cp analysis also showed differences among
the models (Fig. 4). The ORL and ONL models had Cp
⬍3.8 (i.e., significantly lower than the Cp value of the
Fredlund model itself) for ≈268 (≈19.3%) of the 1387
soils, indicating that the ORL and ONL models were
better than the Fredlund model for these soils. Other
models were superior compared with the Fredlund
model for smaller number of soils. On the basis of the
Cp statistic, the Fredlund model described the PSDs best
for most (≈81%) of the soils studied. The Gompertz
model with four fitting parameters showed a lower percentage (≈4.7%) than two- and three-parameter lognormal models, indicating that increasing the number of
parameters cannot always guarantee improved performance. It was noted that all soils for which a model was
found to be accepted by the F statistic were also found
Fig. 6. Comparative fit of seven models: (a) loam (Manseoung soil, 0–12 cm), (b) silt loam (Yihyun soil, 11–20 cm), and (c) sandy clay loam
(Kangseo soil, 28–78 cm). Obs. ⫽ observed, SL ⫽ simple lognormal model, SC ⫽ Shiozawa and Campbell model, Gomp. ⫽ Gompertz model,
ORL ⫽ offset-renormalized lognormal model, ONL ⫽ offset-nonrenormalized lognormal model, and Fred. ⫽ Fredlund model.
1148
SOIL SCI. SOC. AM. J., VOL. 66, JULY–AUG 2002
Fig. 7. Comparative fit of seven models: (a) clay loam (Misan soil, 23–47 cm), (b) silty clay loam (Wunkyo soil, 25–65 cm), (c) silty clay (Seotan
soil, 84–100 cm), and (d) clay (Bankok soil, 0–12 cm). Obs. ⫽ observed, SL ⫽ simple lognormal model, SC ⫽ Shiozawa and Campbell model,
Gomp. ⫽ Gompertz model, ORL ⫽ offset-renormalized lognormal model, ONL ⫽ offset-nonrenormalized lognormal model, and Fred. ⫽
Fredlund model.
to be accepted by the Cp statistic, but not vise versa
(data are not shown). This indicated that the F statistic
has a greater penalty for additional fitting parameters.
To confirm the performance test from F and Cp, the
AIC was used as an additional statistic. The AIC value
for the ONL model was significantly smaller than the
Fredlund model in 167 (≈12.0%) soils, indicating its
superiority over the Fredlund model for these soils (Fig.
4). Other models outperformed the Fredlund model for
a smaller number of soils.
On the basis of the Cp and AIC tests, the Fredlund
model performed best for the majority (≈81%) of the
soils studied. Buchan et al. (1993) showed through the
F and Cp statistic that the SC model performed slightly
better than other lognormal models for the majority of
the soils studied. However, in this study, the ORL and
ONL models were the best among the lognormal models, based on all three criteria (Fig. 4).
As a result of paired t-tests on each of the three
criteria, we found that several pairs of models performed
identically based on both the F and Cp statistics. However, only the ORL-ONL and SL-SC pairs performed
identically based on all three criteria. The statistical
identity of each of these can be confirmed by the comparative fits in Fig. 5 through 7, which show that the
ORL-ONL pair made no distinct differences in predicted values at seven measurement points in all texture
classes, and the SL-SC pair showed no great differences
in half of 10 texture classes (e.g., sand, silt loam, silty
clay loam, silty clay, and clay soils). This result indicated
that models within each pair (ORL-ONL; SL-SC) may
have statistically identical performance even though the
equations are a little different.
Effect of Texture on Performance of Models
An illustration of the comparative fit of different
models is shown in Fig. 5 through 7 for 10 soils with
HWANG ET AL.: PARTICLE-SIZE DISTRIBUTION MODELS
1149
Fig. 8. Statistical analyses of the particle-size distribution models using the Cp statistic (Mallows, 1973) according to clay content of
soils. SL ⫽ simple lognormal model, ORL ⫽ offset-renormalized
lognormal model, ONL ⫽ offset-nonrenormalized lognormal
model, SC ⫽ Shiozawa and Campbell model, and Gom ⫽ Gompertz model.
different textures. Note that the Fredlund model gave
a very good fit for all 10 soils. In sand, there was not a
significant difference among models except for the poor
fit of the Jaky model (Fig. 5a). The Fredlund model
performed best compared with other models, especially
in loamy sand, sandy loam, sandy clay loam, and clay
loam soils (Fig. 5 through 7). For silty clay and clay soils
with relatively high clay content (⬎40%), it was surprising
to find that models except for the Jaky model showed
no distinct differences in their predicted values at each
measurement points (see Fig. 7c and d). It seems to be
because the PSD in high clay texture has simpler shape
that can be represented easily by various models. Note
that the greatest differences among the models generally
occurred for the silt fraction (0.002 ⬍ d ⬍ 0.05 mm) (Fig.
5 through 7). This variability could be due to the lack of
experimental data for this size range (Fig. 2).
The number of soils that the comparison models could
be accepted, instead of the Fredlund model, decreased
with increase of clay content, according to analyses on
all three criteria (see example for Cp values in Fig. 8).
This result indicated that the relative performance of the
Fredlund model over the comparison models increased
with increase of clay content. It may be because the
performance of the Fredlund model was much higher
than those of the comparison model for soil with high
clay content, although there were no great fitting differences between the Fredlund and comparison models for
these soils (Fig. 7c and d). This trend was confirmed by
the r 2 percentiles shown in Fig. 9. The r 2 percentiles of
the Fredlund model were very close to 1.000 than those
of other models for silty clay and clay soils with high
clay content.
The r 2 percentiles indicated that the Fredlund model
again showed the best fitting ability for all soil textural
classes (Fig. 9). Buchan (1989) pointed out that a lognormal model could properly explain all regions of silty
clay, silty clay loam, and silt loam soils, whereas sandy
Fig. 9. Box plot for r 2 percentiles as the goodness-of-fit of seven
models for textural classes. ORL ⫽ offset-renormalized lognormal
model, and ONL ⫽ offset-nonrenormalized lognormal model.
clay loams, sandy clays, and much of clay soils should
not be modeled with a lognormal model. Our results
also showed that the two or three parameter lognormal
models (SL, ORL, ONL, SC) were excellent for silty
clay, silty clay loam, and silt loam soils and not good
for sandy clay loam soils. We found, however, that these
models were good for clay soils in the Korean soil database. The ONL model had the highest r 2 range among
lognormal models across most texture classes except for
the sand soil for which the SC model had the highest
r 2 range. Note that the Gompertz four-parameter model
showed a lower fitting ability than the Jaky one-parameter model for loam, clay loam, and clay soils. For sand and
loamy sand soils, the Gompertz model’s fitting ability,
however, was better compared with other one- through
three-parameter models.
CONCLUSIONS
We compared seven models for soil PSD, including
five lognormal models (Jaky, SL, ORL, ONL, SC), the
Gompertz model, and the Fredlund model. The results
from r 2 values indicated that the four-parameter Fredlund model provided the best fit across all samples, and
the ONL model showed higher r 2 values than other
lognormal models. The r 2 values of the Gompertz fourparameter model were not much higher than other models with fewer parameters, indicating that increasing
the number of parameters does not always guarantee a
better fit. On the basis of the criteria with a penalty for
additional fitting parameters, the Fredlund model again
performed best for the majority of the soils studied (i.e.,
best for 52% of soils based on the F statistic, ≈81%
based on the Cp test, and ≈88% based on the AIC test).
Using paired t-tests for each of the three criteria, we
found that both ORL-ONL and SL-SC model pairs can
1150
SOIL SCI. SOC. AM. J., VOL. 66, JULY–AUG 2002
be considered to perform identically at the 95% significance level.
Our results showed that texture could affect the performance of PSD models. The Fredlund model again
showed the best fit for all soil textural classes. Especially,
the relative performance of the Fredlund model over
the comparison models increased with increase of clay
content. Among all soil classes, the lognormal models
with two or three parameters showed better fits for silty
clay, silty clay loam, and silt loam soils, and worse fit
for sandy clay loam soil. This result conforms to that of
Buchan (1989). The ONL model had the highest r 2 range
among lognormal models across most soil classes except
for the sandy soil.
Our results provide a starting point for the selection
of soil PSD models for the estimation of soil hydraulic
properties and for the comparison of texture measurements from different classification systems. To do these
completely, other more complex models such as the
fragmentation model (Bittelli et al., 1999) and other
models based on the water retention curve (Haverkamp
and Parlange, 1986) deserve further testing for soils.
ACKNOWLEDGMENTS
This research was partly supported by the Korea Science
and Engineering Foundation (no. KOSEF-971-1106-043-2)
and the U.S. Department of Energy, Environmental Management and Science Program (no. DE-FG07-99ER 15006).
REFERENCES
Arya, L.M., and J.F. Paris. 1981. A physicoempirical model to predict
the soil moisture characteristic from particle-size distribution and
bulk density data. Soil Sci. Soc. Am. J. 45:1023–1030.
Beck, V.J., and K.J. Arnold. 1977. Parameter estimation in engineering
and science. John Wiley and Sons, New York.
Bittelli, M., G.S. Campbell, and M. Flury. 1999. Characterization of
particle-size distribution in soils with a fragmentation model. Soil
Sci. Soc. Am. J. 63:782–788.
Buchan, G.D. 1989. Applicability of the simple lognormal model to
particle-size distribution in soils. Soil Sci. 147:155–161.
Buchan, G.D., K.S. Grewal, and A.B. Robson. 1993. Improved models
of particle-size distribution: An illustration of model comparison
techniques. Soil Sci. Soc. Am. J. 57:901–908.
Campbell, G.S. 1985. Soil physics with BASIC: Transport models for
soil-plant systems. Elsevier, Amsterdam.
Carrera, J., and S.P. Neuman. 1986. Estimation of aquifer parameters
under transient and steady state conditions: 1. Maximum likelihood
incorporating prior information. Water Resour. Res. 22:199–210.
Chen, J., J.W. Hopmans, and M.E. Grismer. 1999. Parameter estimation of two-fluid capillary pressure-saturation and permeability functions. Adv. Water Resour. 22:479–493.
Cornelis, W.M., J. Ronsyn, M. Van Meirvenne, and R. Hartmann.
2001. Evaluation of pedotransfer functions for predicting the soil
moisture retention curve. Soil Sci. Soc. Am. J. 65:638–648.
Fredlund, M.D., D.G. Fredlund, and G.W. Wilson. 2000. An equation
to represent grain-size distribution. Can. Geotech. J. 37:817–827.
GonÇalves, M.C., L.S. Pereira, and F.J. Leij. 1997. Pedo-transfer functions for estimating unsaturated hydraulic properties of Portuguese
soils. Eur. J. Soil Sci. 48:387–400.
Green, P.E., and J.D. Caroll. 1978. Analyzing multivariate data. John
Wiley and Sons, New York.
Gupta, S.C., and W.E. Larson. 1979a. Estimating soil-water retention
characteristics from particle size distribution, organic matter percent, and bulk density. Water Resour. Res. 15:1633–1635.
Gupta, S.C., and W.E. Larson. 1979b. A model for predicting packing
density of soils using particle-size distribution. Soil Sci. Soc. Am.
J. 43:758–764.
Havercamp, R., and J.Y. Parlange. 1986. Predicting the water-retention curve from a particle-size distribution: 1. Sandy soils without
organic matter. Soil. Sci. 142:325–339.
Jaky, J. 1944. Soil mechanics. (In Hungarian.) Egyetemi Nyomda,
Budapest.
Johnson, N.L., and S. Kotz. 1970. Distributions in statistics. John
Wiley, New York.
Leij, F.J., W.J. Alves, M.Th. Van Genuchten, and J.R. Williams. 1999.
The UNSODA unsaturated soil hydraulic database. p. 1269–1281.
In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media.
Univ. of California, Riverside, CA.
Mallows, C.L. 1973. Some comments on Cp. Technometrics 15:661–675.
Minasny, B., A.B. McBratney, and K.L. Bristow. 1999. Comparison
of different approaches to the development of pedotransfer functions for water-retention curves. Geoderma. 93:225–253.
Mishra, S., J.C. Parker, and N. Singhal. 1989. Estimation of soil hydraulic properties and their uncertainty from particle size distribution
data. J. Hydrol. 108:1–18.
Nemes, A., J.H.M. Wösten, A. Lilly, and J.H.O. Voshaar. 1999. Evaluation of different procedures to interpolate particle-size distributions to achieve compatibility within soil databases. Geoderma
90:187–202.
Rajkai, K., S. Kabos, M.Th. Van Genuchten, and P.-E. Jansson. 1996.
Estimation of water-retention characteristics from the bulk density
and particle-size distribution of Swedish soils. Soil Sci. 161:832–845.
Rousseva, S.S. 1997. Data transformations between soil texture
schemes. Eur. J. Soil Sci. 48:749–758.
Rural Development Administration. 1971. Official Soil Series Description: Vol. 1. South Korea. (In Korean.) RDA, Seoul, South Korea.
Rural Development Administration. 1975. Official Soil Series Description: Vol. 2. South Korea. (In Korean.) RDA, Seoul, South Korea.
Rural Development Administration. 1977a. Official Soil Series Description: Vol. 3. South Korea. (In Korean.) RDA, Seoul, South Korea.
Rural Development Administration. 1977b. Official Soil Series Description: Vol. 4. South Korea. (In Korean.) RDA, Seoul, South Korea.
Rural Development Administration. 1980. Official Soil Series Description: Vol. 5. South Korea. (In Korean.) RDA, Seoul, South Korea.
Scheinost, A.C., W. Sinowski, and K. Auerswald. 1997. Regionalization of soil water retention curves in a highly variable soilscape—I.
Developing a new pedotransfer function. Geoderma 78:129–143.
Schuh, W.M., and J.W. Bauder. 1986. Effect of soil properties on
hydraulic conductivity-moisture relationship. Soil Sci. Soc. Am. J.
50:848–855.
Shiozawa, S., and G.S. Campbell. 1991. On the calculation of mean
particle diameter and standard deviation from sand, silt, and clay
fractions. Soil Sci. 152:427–431.
Shirazi, M.A., and L. Boersma. 1984. A unifying quantitative analysis
of soil texture. Soil Sci. Soc. Am. J. 48:142–147.
Shirazi, M.A., L. Boersma, and J.W. Hart. 1988. A unifying quantitative analysis of soil texture: Improvement of precision and extension of scale. Soil Sci. Soc. Am. J. 52:181–190.
Skaggs, T.H., L.M. Arya, P.J. Shouse, and B.P. Mohanty. 2001. Estimating particle-size distribution from limited soil texture data. Soil
Sci. Soc. Am. J. 65:1038–1044.
Smettem, K.R.J., and P.J. Gregory. 1996. The relation between soil
water retention and particle size distribution parameters for some
predominantly sandy Western Australian soils. Aust. J. Soil Res.
34:695–708.
Snedecor, G.W., and W.G. Cochran. 1989. Statistical methods. Iowa
State Univ. Press, Ames, IA.
Vereecken, H., J. Maes, J. Feyen, and P. Darius. 1989. Estimating
the soil moisture retention characteristic from texture, bulk density,
and carbon content. Soil Sci. 148:389–403.
Walker, P.H., and D.J. Chittleborough. 1986. Development of particlesize distributions in some alfisols of Southeastern Australia. Soil
Sci. Soc. Am. J. 50:394–400.