Evaluation of models for description of wet aggregate size distribution from soils of different land uses

Abstract

A proper description of aggregate size distribution (ASD) with an optimum mathematical model would be useful in modeling and monitoring land use effect. The objective of this study was to evaluate the suitability of six cumulative distribution models, namely, Jaky, normal, log-normal, Rosin-Rammler, Fredlund and a mass-based fractal model with wet aggregate size distribution (WASD) data sets from a given range of soil structural properties. The models were tested on wet sieving data of samples that had been collected from a number of different land use types (dry farmland, rangeland and forestland). Three statistical criteria, namely, coefficient of determination (R²), Mallows statistics (C_p), and Akaike’s information criterion (AIC), were used for evaluating model performance, based on the least sum of square error and number of fitting parameters. Analysis of R² showed that the Fredlund three-parameter model showed the best performance in all of the soils apart from the number of parameters. The log-normal model gave a good fit on WASD from rangeland and forestland; it was the best especially in dry farmland. The normal model provided a good description of WASD from the rangeland and forest. However, it failed in dry farmland. According to C_p and AIC as the evaluation criteria, the fractal model was the optimum to describe WASD for all of the land uses. The Fredlund, log-normal, Jaky and Rosin-Rammler models ranked next in the given order.

Key words:

• Different land uses
• evaluation of model performance
• optimum model
• statistical criteria
• wet aggregate size distribution models

INTRODUCTION

Aggregate size distribution (ASD) provides an effective implement to describe soil structure by which the effects of land management on soil physical and hydrological properties can be quantified. Soil structure characterization is needed to succeed in implementation of soil management and to meet the current challenges of sustainable development. Many agricultural and environmental processes are related to the arrangement and stability of soil structural units. Soil structural properties are also influenced by many physical and biogeochemical processes in soils (Amezketa 1999). Many soil functions such as seed placement and germination (Kay and Angers 1999; Reuss et al. 2001), root growth (Lipiec et al. 2007), water and oxygen availability, resistance to root penetration in seedbed (Schneider and Gupta 1985; Nasr and Selles 1995), soil hydraulic properties and solute transport processes (Diaz-Zorita et al. 2002) are affected by ASD. Soil aggregation is strongly influenced by land use (John et al. 2005; Ashagrie et al. 2007). Land use changes, especially cultivation of deforested land, may rapidly diminish soil quality (Islam and Weil 2000). Individual management practices including fertilization, tillage, residue management, amendments and crop rotation influence the ASD (Bronick and Lal 2005).

Evaluating the influence of management practices on soil needs to quantify the variation of the soil structure (Danielson and Sutherland 1986). One of the well-known approaches to represent soil structure is characterization of ASD by appropriate mathematical models (Diaz-Zorita et al. 2002). The parameters of these models are often used as an index of soil structural stability. Accordingly, several parametric models have been proposed based on the statistical distribution of aggregates (e.g., normal and log-normal distribution), empirical power functions (e.g., Baldock and Kay 1987) and fractal geometry (e.g., Tyler and Wheatcraft 1992). Generally, the parameters of models are estimated by using the algorithms which minimize the total differences between observed and predicted data (Buchan et al. 1993). The certainty of estimated parameters highly depends on the nature of experimental data, model characteristics and fitting procedures. However, in many cases, increasing the number of fitting parameters will improve the fitting accuracy, but this can result in the loss of simplicity and subsequently reduce applicability of the model (Buchan et al. 1993). Thus, it would be convenient to select an optimum model which comprises the proper number of fitting parameters with suitable certainty. The objective of this study is to test a variety of parametric models to represent wet aggregate size distribution (WASD) in three land use types: dry farmland, rangeland and forestland. We also intend to find an optimum model that properly describes the effect of land use on WASD. This model provides both the good description and simple applicability of detailed representation of WASD in the land use types.

Aggregate size distribution models

The results of aggregate size analysis can be presented as the arithmetic or geometric mean weight diameter (MWD or GMD), and range of aggregation (Ghildyal and Tripathi 1987). Many authors have characterized WASD by fitting distribution functions to the available data sets (e.g., Marshall and Quirk 1950; Gardner 1956). We used the following models for representing soil structure status through the fitting on cumulative aggregate mass-size distribution data.

Jaky model (J), one-parameter

Jaky (1944) proposed a simple model with a sigmoid half of a Gaussian log-normal size distribution:

(1)

where M (< x) is the cumulative mass of objects (aggregates in the present investigation) under size x, p is a size distribution index characterizing the stretching of the curve shape, and x_max is the diameter of the largest aggregate.

Normal distribution model (N), two-parameter

Normal distribution is an absolutely continuous probability distribution of random variable that is normally distributed. The mean diameter and standard deviation parameters can be determined by fitting of this distribution function to ASD data (Allen 1997):

(2)

where µ and σ are the mean diameter and standard deviation, respectively, and can be determined with fitting of Eq. 2 to the WASD data, and erf [ ] is the error function defined as:

(3)

Log-normal distribution model (LN), two-parameter

A log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. The geometric mean diameter (µ_g) and geometric standard deviation (σ_g) have been suggested as two indices of ASD because the aggregate mass-size distribution is often log-normal rather than normal (Gardner 1956). In Eq. 2, × was replaced with ln(x) to obtain Eq. (4):

(4)

Rosin-Rammler model (R), two-parameter

Rosin-Rammler distribution function is used to describe the particle size distribution (PSD) data (Irani and Callis 1963):

(5)

where the parameter α denotes the aggregate size corresponding to 36.78% of the cumulative distribution, similar to the mean diameter in normal distribution. The β parameter is analogous to the standard deviation of a normal distribution. A small value of β indicates a wide spread of ASD (Diaz-Zorita et al. 2002).

Fredlund model (FL), three- and four-parameter

Fredlund et al. (2000) developed a four-parameter model to represent grain size distribution based on the soil moisture characteristic model (Fredlund and Xing 1994):

(6)

where α is a parameter denoting the inflection point of distribution curve, n is a parameter related to the steepest slope on the curve, m is a parameter related to the shape of the curve as it approaches the fine fraction, x_f is a parameter related to the fine fraction in a soil, and x_min is the diameter of the smallest particle.

The expression of has often a negligible amount and, so, the value of the second part of Eq. 6 can be assumed to be 1. Then, Eq. 6 can be simplified as a three- parameter model:

(7)

Fractal model (FR), one-parameter

Fractal geometry has been widely applied to characterize ASD. It represents a power law relationship between number-diameter, mass-diameter and bulk density-diameter of aggregates (Young and Crawford 1991; Baveye and Boast 1998). Fractal parameters have been frequently used to investigate the effects of tillage and other cropping practices on soil structure (Perfect and Blevins 1997). Tyler and Wheatcraft (1992) developed a mass-based fractal model to characterize the PSD:

(8)

where M_T is the total mass of aggregates; x_max is the maximum size of aggregates and D is the fractal dimension. It is instructive to recall here that Eq. 8 can be fitted as a one-parameter model under the assumption of variable standardization (e.g. and .

MATERIALS AND METHODS

Sampling area and land properties

This study was conducted on different land use types from the Mazandaran province in northern Iran. A gographical area of 5000 hectares of the Chalousrood and the Sardabrood watersheds comprising dry farmland, rangeland and forestland was selected. Accordingly, in the present research, 14 land uses including four dry farmland, three rangeland and seven forestland (four coniferous and three deciduous forests) sites were evaluated in WASD. The farmland had been conventionally cultivated in the crop rotation pattern dry wheat or barley – fallow, for more than 100 years. The rangeland included natural pastures covered with some species of annual, biennial and perennial Gramineae and legumes. The coniferous stands had been planted for 50 years and comprised four sites that were different in terms of dominant species and densities. In this case, two different sites had been afforested with the dominant species Cupressus sempervirens L. and the others had been planted with Picea abies (L.) Karst. stands at different densities. All of the deciduous forest sites (three sites) included the natural mixed oriental beech (Fagus orientalis L.) stand with associated the following species: Carpinus betulus L., Parrotia persica (DC.) C. A. May., Acer velutinum Boiss. and Acer cappadocicum Gled. in different densities. All of the soils in the sampling area were formed on calcareous bedrock. The average slope of the study area was 18–25% for the rangeland and forestland, and 5–10% for the dry farmland. Soil moisture and thermal regimes were Xeric and Mesic in both the rangeland and forestland identically, so that the soils were classified as Typic Haploxeralfs (Soil Survey Staff 1999). The dry farmland soil with Ustic and Mesic regimes was classified as Typic Haplustepts. Sampling was carried out in sunny days of summer when the matric potential of topsoil was approximately −300 to −400 kPa. Three hundred sampling segments in dry farmland as well as rangeland sites, and 400 sampling segments in forest sites, were selected. Undisturbed soil samples in each segment were collected in five replicates from the A pedogenic horizon of forest and rangeland sites, and from the plough layer (A_p horizon) of dry farmland sites. The undisturbed samples were gently removed with a shovel and carried in specific plastic boxes to the lab and then air dried at room temperature. A subsample under 2 mm in size was prepared from each sample. PSD was determined using the standard laboratory method (Gee and Bouder 1986). Soil organic carbon (SOC) and calcium carbonate equivalent (CCE) were determined using the wet combustion (Nelson and Sommers 1982) and volumetric methods (Allison and Moodie 1965), respectively. Electrical conductivity (EC) and pH were measured with a Jeneway instrument (model, 4330). Wet sieving was performed on soil aggregates smaller than 8 mm (Yoder 1936). A 50-g air dried soil sample (< 8 mm) was placed on the nest of sieves consisting of seven aperture sizes, 4.75, 2, 1, 0.5, 0.25, 0.15 and 0.053 mm, and then gently submerged in water (EC = 0.6 dS.m⁻¹, SAR = 1.1). The samples were shaken for 10 min with a vertical stroke (30 mm distance and speed of 35 cycles min^–1). Afterwards, the oven-dried masses of material passed through every individual aperture size on the sieve set were weighted. These values were normalized with respect to the total mass. Aggregate size (each size fraction), x_i and normalized cumulative mass of aggregates under the individual size fraction, M (< x_i) were considered as the independent and dependent variables, respectively. Nonlinear regression analysis was also used to fit Eqs. 1, 2, 4, 5, 7 and 8 to the 1000 WASD data from the study land uses, using Matlab 7.1 software (Matlab 7.1, the Mathworks Inc., Natick, MA) and the Maraquardt-Levenberg (Marquardt 1963) algorithm. For each model, the R² values were tested by paired t-student.

Model comparison methods

There are several approaches to evaluate the model performance. One of the simplest methods is to find the best model using the sum of square error (SSE) and/or coefficient of determination (R²) statistics. In general, increasing the number of fitting parameters improves the agreement between the modeled and measured values, but it can reduce consistent physical interpretations. Furthermore, fitting models with a large number of parameters can complicate the inferences. Therefore, a valuable method can be to conduct a balance between goodness of fit and number of fitting parameters (Buchan et al. 1993). Buchan et al. (1993) tested five models on 71 PSD data sets and applied three statistical criteria: R², F statistic (Green and Caroll 1978) and the C_p statistic of Mallows (1973) to determine the optimum model. They defined the optimum model based on balancing the minimization of measuring errors against minimization of the number of fitting parameters. They explained that the R² is not a powerful tool for relative discrimination of nonlinear models. Since both SSE and number of parameters, P, have been contributed for calculating F and Cp criteria, these can provide a better indicator of model performance. The criteria are capable of discriminating statistically-significant differences between models. The Akaike’s information criterion (AIC) (Carrera and Neuman 1986) has been applied by researchers in order to evaluate and compare the parametric models’ performance. Hwang et al. (2002) used the AIC to select the optimum model for PSD data on 1387 Korean soils. Others applied the AIC to determine the optimum models for the soil moisture characteristic curve (Minasny et al. 1999) and soil permeability (Chen et al. 1999). In the present study, the model performance was evaluated using the three statistical criteria. The models were also ranked in the given order and determined the optimum model for representing WASD from the study land uses. The three criteria are as follows:

Coefficient of determination (R²)

The absolute amount of variability accounted for by a model can be calculated as:

(9)

where n is the number of ASD data points and Y_o and Y_p are the observed and predicted data values, respectively.

Mallows C_p statistic

Mallows (1973) proposed a statistical criterion, C_p, which has been used to compare the goodness of fit of different models:

(10)

where SSE_c and SSE_r are the sums of square errors of the comparison and reference models, respectively, n is the number of data points and p_r and p_c are the numbers of parameters of reference and comparison model, respectively. The SSE is given as follows:

(11)

Snedecor and Cochran (1989) advocated the use of C_p to overcome the inherent imperfections of using R².

Akaike’s information criterion (AIC)

The AIC was proposed by Carrera and Neuman (1986). It has been applied by many researchers (Chen et al. 1999; Hwang et al. 2002) for selecting the optimum model:

(12)

where n is the number of data points and p is the number of the model parameters. In our study, AIC was used to verify the C_p results in the evaluation of the model performance.

In order to compare the models using C_p and AIC statistics, one model was considered as a reference, and other models were compared to it. In this study, the N model was assumed as the reference model, and then the optimum model was determined for all studied land uses. In the case of the C_p, first, the reference model must be compared with itself and consequently the value of C_p will be the same number of parameters (see Eq. 10). Hence, for the N model in the present study, the C_p value was equal to 2. Then the C_p was calculated for the other comparison models (five models). The comparison model would be better than the N model if its C_p value was significantly (5%) lower than 2. The AIC was used as an additional statistic to confirm the results of the C_p statistic in evaluation of the model performance. The model with the smallest AIC value was selected as the best considering that the AIC value of this model must be smaller than the 0.95 AIC value of the N model. Finally, the optimum model is the one that yielded the smallest C_p and/or AIC values.

RESULTS AND DISCUSSION

Soil properties and land uses

The main soil properties are summarized for the three types of land uses in Table 1. The soil textural classes ranged from clay to clay loam. No significant difference was observed in mean values of the clay content between the dry farmland, rangeland and forestland, but in the rangeland, sand content was significantly higher, and silt was lower than in the dry farmland and forestland (P < 0.05). In the forestland, the high annual precipitation with high soil permeability has resulted in intensive soil solute leaching and subsequently lower CCE and pH values (Table 1). Moreover, decomposition of litter may produce organic acids and lower pH. Soil organic carbon (SOC) is one of the most fundamental factors in the formation and stability of aggregates (Amezketa 1999; Bronick and Lal 2005). Diversity of plant species in the studied land uses may lead to differences in the quantity and quality of the SOC. Accordingly, the soil structure was highly affected by this variation. There was a significant difference (P < 0.01) in SOC content between dry farmland and the other studied areas. Results showed that the SOC value for the dry farmland (1.36%) is less than the rangeland (2.79%) and forestland (2.83%). In the farmland, removing crop residue and conventional tillage operations caused exposure of soil organic matter to atmosphere, and oxidation and depletion of the SOC (Kay 1990). This can also lead to destruction of soil structure and degradation of the land (Amezketa 1999).

Table 1 Average soil physical and chemical properties by the land use types

Land use types
	Dry farmland (n = 300)			Rangeland (n = 300)			Forestland (n = 400)
Soil properties^†	Mean	Min–Max	SD	Mean	Min–Max	SD	Mean	Min–Max	SD
pH	8.32	8.2–8.4	0.0957	8.20	8–8.3	0.1732	7.08	6.8–7.5	0.2714
EC (dS m^−1)	1.08	0.89–1.56	0.3244	1.57	0.87–2.56	1.20	1.26	0.87–2.4	0.5145
CCE (%)	16.04	10.5–23.2	5.414	5.25	0–15.15	8.5816	3.51	0–16.36	6.4384
SOC (%)	1.36	1.2–1.62	0.2435	2.79	2.65–2.93	0.1401	2.83	2.62–3.23	0.2067
Sand (%)	18.25	10–41	15.196	21	12–36	13.077	16.86	6–40	12.144
Silt (%)	36.75	31–46	6.898	35	34–36	1.041	38.86	31–50	6.067
Clay (%)	45	27–59	13.880	44	29–52	13	44.28	29–53	8.40
MWD (mm)	1.73	1.26–2.16	0.5171	3.85	3.80–3.92	0.0625	4.14	3.69–4.90	0.4861

^†EC: electrical conductivity, CCE: calcium carbonate equivalent, SOC: soil organic carbon, Min–Max: minimum and maximum values of soil parameters by the samples, SD: standard deviation of data for the samples, n: number of soil samples; MWD: mean weight diameter.

Statistical modeling of WASD

Inspection of the results (MWD values and shape of the WASD curves) confirmed the impact of studied land use types on soil structure. The samples provided a varying range of WASD and wet-aggregate stability. Table 1 shows that the average values of MWD are 3.85 and 4.14 mm for the rangeland and forestland, respectively. In addition, these land use types indicated a higher SOC than the dry farmland. The result of aggregate size analysis also showed that the dry farmland had the lowest average MWD (1.73 mm) and so the least aggregate stability among all of the land uses (Table 1). The relationship between SOC and developing water stable aggregates is a well-known process (Amezketa 1999) and so it is anticipated that current status of the dry farmland may have provided an effective ground for susceptibility to land degradation. Six et al. (2000) reported that land uses can strongly influence WASD and structural stability.

Coefficient of determination (R²) as a comparative criterion

Table 2 shows the comparisons of the models’ performance in terms of the statistical parameters R² and SSE. Among all of the land uses and models, R² values ranged from 0.8735 to 0.9996. The lowest R² value was obtained for the N model for a dry farmland site, and the largest R² value was observed for the FR model for a site in deciduous forest. Figure 1 presents an overview of the performances of six models fit on the examples of WASD from the study land use types. Statistical analysis (paired t-test) of R² showed that the FL model was significantly better than the J, R, N, and LN models (P < 0.01 and n = 1000) (Table 2). This superiority can be attributed to its larger number of fitting parameters. The results indicated that, in terms of R² values, the FR model was significantly better than the FL model to describe the WASD in forest and rangeland. Statistical analysis of R² also showed that the FR model was significantly better than the J, R and N models for all of the studied land use types (P < 0.01). The J model had the smallest average value of R² in the rangeland and forestland and it was the worst model for describing the WASD in these land uses (Table 2). The uncertainty of the J model in expression of PSD data was also shown by Hwang et al. (2002). According to R² and SSE values, the R two-parameter model was better than the J model. The R model was also significantly (P < 0.01) weaker than the LN model for all land uses. The lowest average value of R² (0.8909) was observed for N model performance in dry farmland (Table 2). However, this model could describe more than 98% of variations on WASD curves in all of the rangeland and forest sites (Table 2). It was the best in these land uses after the FR and FL model. As expected, between two log-normal models, J and LN, the LN two-parameter model was significantly better than the J one-parameter model (p < 0.01). Both of these models were significantly weaker than the N model for the rangeland and forestland. Among the models, the LN model had the largest average R² values (0.9847) in dry farmland (Table 2). In this land use, the FL three-parameter model could accurately represent the WASD as well as the LN model. Although the FR model fitted adequately to the data of dry farmland, it was still weaker than the LN and FL models in this land use.

Table 2 Average values of goodness of fit criteria from the six models for each type of land use

Land use types
		Dry farmland (n = 300)		Rangeland (n = 300)		Forestland (n = 400)
Models		R^2	SSE	R^2	SSE	R^2	SSE
J	Mean	0.9612	0.040	0.9478	0.057	0.9493	0.055
	Min–Max	0.9455–0.9894	0.012–0.052	0.9282–0.9666	0.035–0.085	0.9344–0.9601	0.045–0.063
N	Mean	0.8909	0.098	0.9805	0.018	0.9814	0.017
	Min–Max	0.8735–0.9092	0.086–0.108	0.9790–0.9821	0.016–0.022	0.9560–0.9924	0.008–0.040
LN	Mean	0.9847	0.014	0.9726	0.025	0.9695	0.029
	Min–Max	0.9777–0.9942	0.006–0.019	0.9659–0.9785	0.020–0.029	0.9588–0.9775	0.020–0.035
R	Mean	0.9723	0.026	0.9517	0.045	0.9503	0.047
	Min–Max	0.9603–0.9902	0.010–0.033	0.9415–0.9591	0.037–0.051	0.9313–0.9608	0.038–0.058
FL	Mean	0.9808	0.014	0.9849	0.012	0.9851	0.012
	Min–Max	0.9718–0.9921	0.007–0.020	0.9830–0.9861	0.011–0.015	0.9811–0.9896	0.009–0.014
FR	Mean	0.9641	0.038	0.9969	0.003	0.9981	0.002
	Min–Max	0.9189–0.9827	0.018–0.091	0.9947–0.9983	0.002–0.006	0.9918–0.9996	0.0005–0.009

Models: Jaky (J), normal (N), log-normal (LN), Rosin-Rammler (R), Fredlund (FL); fractal (FR); R²: coefficient of determination; SSE: sum of square error; n: number of soil samples.

Figure 1 Comparative fit of six models on examples of wet sieving data for (a) dry farmland, (b) rangeland, (c) forestland. The models: FR (fractal model), FL (Fredlund model), R (Rosin-Rammler model), N (normal model), LN (log-normal model), J (Jaky model).

In summary, comparing the models’ performances for the three types of land uses (dry farmland, rangeland and forestland) indicated that both FL and LN were the best models for quantifying WASD variations in farmland, whereas the FR and N models were the best for the rangelands and forests. Among the models, FL was the best one, which gave a good fit on WASDs with the normal or log-normal statistical pattern, whereas the FR model yielded the largest R² values only when the aggregates sizes were normally distributed. There are several reports that the WASD follows a power function (Baldock and Kay 1987; Caruso et al. 2011). In this case, the exponent of the function can be a descriptor of soil structure status. Assuming that the fragmentation processes in soil aggregates is scale invariant, the exponent of the power function may be an indication of the fractal dimension, D (Turcotte 1986).

Normal or log-normal distribution of WASD

MWD and GMD are the statistical indices which are derived from the normal and log-normal functions, respectively (µ and µ_g in Eqs. 2 and 4). Those have been widely used as suitable indices for assessing and comparing soil aggregates and structural stability in different soils. The proper application of these parameters requires recognizing the normality or log-normality in distribution of aggregate size. A detailed overview on normality and log-normality of the studied WASD data is given in the following.

Table 2 shows that the WASD data could be properly represented by the LN model. For the soils with low structural stability (e.g., dry farmland) which were highly fragmented through the wet sieving, the LN model was the best in terms of R² (0.99) and SSE (0.014) (Table 2). The fragmentation yielded an increase in the number of fine aggregates such that the final distribution resembled a log-normal curve. As the soils with low SOC are rapidly slaked upon submergence in water (Kemper and Rosenau, 1986), it seems that in the dry farmland, the slaking might be an important event to disrupt the coarser aggregates to the finer fragments. Through the fragmentation process due to an unlimited stress, the secondary soil units are involved in a progressive descent to the lowest hierarchical level or primary particles (Díaz-Zorita et al., 2002). Also, there are several reports that many of PSDs are approximately log-normal (Shirazi and Boresma 1984; Campbell 1985). Accordingly, in the farmland, the upper tail of the cumulative WASD curve, representing the coarsest fragments, tended to form a straight line (Df in Fig. 2) similar to a PSD curve. Since the WASD in dry farmland indicated a log-normal pattern (Fig. 2), it is expected that the statistical parameter GMD will describe the mean aggregate size more properly than the MWD. Figure 2 presents the examples of fitting of the LN and N models on WASDs from three sites in dry farmland (Df) and three sites in forestland (F). The results of all the samples from dry farmland showed that the N model could not adequately converge to the WASD data (R² = 0.8909) (Table 2). However, the N model more properly fitted on WASD data from the forest (R² = 0.9814) and rangeland (R² = 0.9805) (Table 2) than the dry farmland. Although both the N and LN models gave a reasonable fit on WASDs for some of the samples (Table 2), our results showed that the performance of the N model will fail with increasing aggregates fragmentation, from the soils with more to less stable aggregates (e.g., the rangeland and forest to the dry farmland) (Table 2). Given that the current status of soil structure arises from interaction between internal (i.e. stabilizing material) and external (i.e. environmental stresses) factors (Amezketa 1999), a good performance of the N model to a WASD data set can suggest a favorable balance between internal and external factors in the soil. In other words, the normality in WASD might address the numerous stable aggregates or high aggregation in the soil. In such cases, MWD can be also used as a good index for expressing mean aggregate size.

Figure 2 An example of appropriate fitting of the normal (N) or log-normal (LN) models on wet sieving data from dry farmland (Df) and forest (F).

C_p and AIC statistics as comparative criteria

Our results indicated that the FR model could accurately represent soil with higher structural stability. In this case, any change in WASD will be reflected by a single value, D. Therefore the parameter D provides a suitable index to quantify the environmental stress and land use effects on soil structure. In the case of the FL model which had the best fitting to all of the data sets, the characteristics of the WASD curve were represented by several parameters (a, n, m in Eq.7) which can be intercorrelated. Thus, it is difficult to find an appropriate individual index for describing WASD. Moreover, initial estimation of model parameters, which is necessary for fitting of this model, adds a burden to the modeling process. These can lead to difficulties in application and complexity of interpretation of results. Balancing between the number of fitting parameters and goodness of fit was performed by analyzing statistical criteria, C_p and AIC values. The C_p and AIC values for each comparison model were calculated for 300, 300 and 400 WASDs data from dry farmland, rangeland and forest, respectively. The values are summarized in Table 3.

Table 3 Average values of Mallows statistics (C_p) and Akaike’s information criterion (AIC) calculated for all of the soils from the five comparison models (normal model was the reference model)

Models
	J		LN		R		FL		FR
Statistics	Cp	AIC	Cp	AIC	Cp	AIC	Cp	AIC	Cp	AIC
Mean	13.87	−20.15	5.52	−24.38	12.00	−19.55	1.48	−27.17	−5.59	−42.35
Min	−6.17	−32.06	−4.61	−36.51	−4.34	−31.83	−2.54	−32.76	−6.69	−60.64
Max	46.78	−14.40	21.68	−20.22	37.17	−16.84	5.83	−22.68	−0.72	−13.80
SD	17.14	4.38	8.97	4.29	13.49	4.04	2.99	2.52	1.52	14.20

Models: Jaky (J), log-normal (LN), Rosin-Rammler (R), Fredlund (FL); fractal (FR). Mean, Min, Max and SD are respectively average, minimum, maximum and standard deviation of C_p and AIC values for all WASD data sets. Underlined mean values denote the optimum model in all of the studied soils (FR was the optimum model with the lowest C_p and/or AIC values).

The relative performances of models were compared in terms of the C_p and AIC. Figure 3 represents the superiority of the comparison models with the N model assumed as the reference model. The C_p and AIC criteria provide similar results for all cases (Fig. 3). The results showed that the one-parameter FR model was better than the N model for all of the 1000 WASDs data, because the C_p values of all data sets were less than 1.9 (5% less than 2). In this case, AIC values were significantly less than the calculated AIC from the N model for all data (P < 0.05) (Table 3). Although analyzing of the R² and SSE showed that the FR model isn’t the best model in dry farmland, it is the optimum model to characterize the WASD for all of the study land uses in terms of the C_p and AIC (Fig. 3). Furthermore, the fractal dimension D can be used as a parameter to describe the effects of cropping and tillage on soil structure (Pirmoradian et al. 2005). This parameter can be also used to estimate some hydraulic properties such as pore size distribution, hydraulic conductivity, and the soil moisture characteristic curve (Perfect et al. 2002). The FL three- parameter model exhibited a good flexibility to fit on WASD data sets (Table 2), but in terms of the C_p and AIC criteria, it properly represented WASD only for 65% of the studied soils (650 soil samples). This result was a penalty due to additional fitting parameters in this model. Hwang et al. (2002) showed that the FL four-parameter model had the best performance for majority of PSDs soil samples. Both the J one-parameter and LN two-parameter models were better than the N model for 35% of soils (350 soil samples) (Fig. 3). The paired t-test on AIC values was conducted in order to compare the J and LN models, and the results showed that the LN is better than the J model for all WASD. The R model was better than the N model only for 30% of the soils (300 soil samples) so it was the most inappropriate model to characterize the WASDs in studied soils.

Figure 3 Relative superiority of the comparison models in all of the study soils using Mallows statistics (Cp) and Akaike’s information criterion (AIC) statistics. The number above bars indicates the number of soils for which a model can be accepted instead of the normal model (N). The comparison model: FR (fractal model), FL (Fredlund model), LN (log-normal model), J (Jaky model), R (Rosin-Rammler model).

CONCLUSION

A variety of parametric models have already been fitted to PSD data for different soil textures, but there still has been no attempt to evaluate the ability of these models in optimum performance on WASD data from different land use types. We compared six parametric models with different underlying assumptions for WASD, including four statistical models (J, LN, N and R), a power function (FR) and the FL model. As expected, both the FL three-parameter and LN two-parameter models showed a good flexibility for fitting experimental WASD data. Regardless of the penalty of additional fitting parameters, these models were the best for the majority of the studied soils (in terms of C_p and AIC). Although both the N and LN models showed a good fit on the data sets for the most samples, our results indicated that the performance of the N model will fail with increasing aggregate fragmentation. We also suggest this issue needs to be investigated for WASD curves induced at different levels of incoming energy impact for a given soil. Log-normality in WASD in soil with low structural stability (e.g., dry farmland in this study) confirmed that GMD was better than MWD for expressing mean aggregate size. However, normality in distribution of aggregates in soil with high structural stability (e.g. forest and rangeland in this study) indicated that MWD can be used as an accurate structural index in these land uses. The FR one-parameter model was also a proper choice for characterizing WASD curves. In this case, a good performance on the data with the single parameter model provided a suitable quantification. According to C_p and AIC, the FR model was an optimum descriptor for WASD. It could result in the best quantification of land use effects on soil structure. Apparently the fractal nature of soils produces a fractal behavior of aggregates which does support the FR model to better characterize the WASD (Millan et al. 2003). The FL and R models were better than the N model for 65 and 30% of soils, respectively. Both the LN and J models, for 35% of the soils, were identically better than the N model. Although the wide range of structural properties of the studied soils can provide a generality to apply the results for similar land uses, these results may change with more extensive sampling and so need further investigation.

ACKNOWLEDGMENTS

This project was supported by the Department of Soil Science Engineering, Faculty of Agricultural Engineering and Technology, University of Tehran, Iran. We would like to thank Dr. Hossein Mirhosseini for help and reviewing this paper.