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Soil Science Society of America Journal 66:1143-1150 (2002)
© 2002 Soil Science Society of America

DIVISION S-1—SOIL PHYSICS

Models for Estimating Soil Particle-Size Distributions

Sang Il Hwanga, Kwang Pyo Leeb, Dong Soo Leeb and Susan E. Powers*,a

a Dep. of Civil and Environmental Engineering, Clarkson Univ., 8 Clarkson Ave., Potsdam, NY 13699-5710
b D.S. Lee, Graduate School of Environ. Stud., Seoul National Univ., Seoul, 151-742, South Korea

* Corresponding author (sep{at}clarkson.edu)


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
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.

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


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
THE 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 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. 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 (r2), the F statistic, the Cp statistic of Mallows (1973), and Akaike's information criterion (AIC).


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The Korean Soil Database
The Korean soil physical database contains data from 1387 soil layers representing {approx}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., 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.



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  		Fig. 1. Textural composition of the soil data set.

 


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  		Fig. 2. Cumulative particle-size distribution for 1347 Korean soils.

 
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.


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  		Table 1. Particle-size distribution models tested for texture data of 1387 soils.

 
Model Comparison and Fitting Techniques
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 r2 may be preferred more than one with smaller r2. 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 other conditions are equal. In addition to r2, 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 (r2, F statistic, Cp statistic, AIC) are listed in Table 2.


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  		Table 2. Four criteria for particle-size distribution model comparison.

 
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 (obtainable 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 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 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, 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
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Defining the Optimum Model
Among all of the soils and all of the models, values of r2 ranged from 0.561 to 1.000 (Fig. 3) . As expected, the lowest r2 values were obtained with the Jaky model because it only has one fitting parameter. The Fredlund model had the highest r2 values and the four-parameter Gompertz model yielded lower r2 values than two- or three-parameter lognormal models. Among the three models with three parameters, the r2 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 r2 values than the other two models. Our result may be more comprehensive because of the use of a much larger soil data set.



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  		Fig. 3. Box plot for r2 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.

 
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 ({approx}48.5%) among 1387 soils. As with the previous r2 analysis, the ONL model was again identified as the best model among one- through three-parameter models.



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  		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.

 
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 {approx}268 ({approx}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 ({approx}81%) of the soils studied. The Gompertz model with four fitting parameters showed a lower percentage ({approx}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 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 ({approx}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 ({approx}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.



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  		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.

 


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  		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.

 


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  		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.

 
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 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. 5–7). 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 r2 percentiles shown in Fig. 9 . The r2 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.



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  		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.

 


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  		Fig. 9. Box plot for r2 percentiles as the goodness-of-fit of seven models for textural classes. ORL = offset-renormalized lognormal model, and ONL = offset-nonrenormalized lognormal model.

 
The r2 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 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 r2 range among lognormal models across most texture classes except for the sand soil for which the SC model had the highest r2 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
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
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 r2 values indicated that the four-parameter Fredlund model provided the best fit across all samples, and the ONL model showed higher r2 values than other lognormal models. The r2 values of the Gompertz four-parameter 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, {approx}81% based on the Cp test, and {approx}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 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 r2 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).

Received for publication July 9, 2001.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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