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DIVISION S-1 - SOIL PHYSICS

Multifractal Characterization of Soil Particle-Size Distributions

^a Dep. of Environmental Sciences, Rutgers Univ., 14 College Farm Road, New Brunswick, NJ 08901
^b Embrapa Agricultural Instrumentation Center, P.O. Box 741, 13560-970, São Paulo, S.P., Brazil

* Corresponding author (gimenez{at}envsci.rutgers.edu)

	ABSTRACT

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

 
A particle-size distribution (PSD) constitutes a fundamental soil property correlated to many other soil properties. Accurate representations of PSDs are, therefore, needed for soil characterization and prediction purposes. A power-law dependence of particle mass on particle diameter has been used to model soil PSDs, and such power-law dependence has been interpreted as being the result of a fractal distribution of particle sizes characterized with a single fractal dimension. However, recent studies have shown that a single fractal dimension is not sufficient to characterize a distribution for the entire range of particle sizes. The objective of this study was to apply multifractal techniques to characterize contrasting PSDs and to identify multifractal parameters potentially useful for classification and prediction. The multifractal spectra of 30 PSDs covering a wide range of soil textural classes were analyzed. Parameters calculated from each multifractal spectrum were: (i) the Hausdorff dimension, f(); (ii) the singularities of strength, ; (iii) the generalized fractal dimension, D_q; and (iv) their conjugate parameter the mass exponent, (q), calculated in the range of moment orders (q) of between -10 and +10 taken at 0.5 lag increments. Multifractal scaling was evident by an increase in the difference between the capacity D₀ and the entropy D₁ dimensions for soils with more than 10% clay content. Soils with <10% clay content exhibited single scaling. Our results indicate that multifractal parameters are promising descriptors of PSDs. Differences in scaling properties of PSDs should be considered in future studies.

Abbreviations: , singularity of strength • f(), Hausdorff dimension • D₀, capacity dimension • D₁, entropy dimension • D₂, correlation dimension • D_q, the generalized fractal dimension • PSD, particle-size distribution • (q), correlation exponent of the q^th order • q, moment order of a distribution

	INTRODUCTION

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

 
A SOIL PSD is one of the fundamental properties characterizing a soil. Information on PSD defined as a combination of clay, silt, and sand contents, is often reduced to a textural class (Soil Survey Division Staff, 1993). Soils grouped in a textural class, however, exhibit a wide range of PSD, making the information contained in a textural triangle suitable only for general classification purposes.

Different functions are used to fit detailed experimental data on PSD (Buchan et al., 1993). Many of these functions are based on a power-law dependence of particle mass on particle diameter (Turcotte, 1986). Such power-law dependence has been interpreted as being the result of a fractal distribution characterized with a single fractal dimension (Matsushita, 1985; Turcotte, 1986; Tyler and Wheatcraft, 1992). Other results, however, have shown that a single fractal dimension is not sufficient to describe PSD in soil (Wu et al., 1993; Grout et al., 1998; Bittelli et al., 1999). Wu et al. (1993) combined several methods to obtain particle sizes ranging from 0.02 µm to several millimeters. They identify three domains with different power exponents. Bittelli et al. (1999), using the model of Turcotte (1986) and Tyler and Wheatcraft (1992), also found that three domains characterized the cumulative PSD of 19 soils. They associated the power exponent in each domain with fractal dimensions defining scaling in the clay, silt, or sand domains.

A distribution of particle sizes reflects the relative balance of weathering and pedogenetic processes. Primary minerals, generally present in the sand- and silt-size fractions, originate from weathering of a parent material, while clay minerals are the result of weathering and synthesis of new minerals. The different origin of the various size fractions may explain the various scaling domains observed in soil PSD (Wu et al., 1993; Bittelli et al., 1999). Grout et al. (1998) found that a single fractal dimension obtained from the model of Tyler and Wheatcraft (1992) did not describe adequately the distribution of particle sizes of three soils, and proposed multifractal techniques as a promising alternative to characterize PSD. Multifractal distributions may be best suited to represent the multiplicative action of the various pedogenetic processes acting on a parent material and resulting in a given distribution of particle sizes.

Multifractal methods have been applied in many areas of applied sciences. Muller and McCauley (1992) used multifractal for characterization of pore systems in rocks. Kropp et al. (1994) quantified spatial variability in bioactive marine sediments. Appleby (1996) characterized distribution patterns of the human population, and Cheng and Agterberg (1995) described the underlying spatial structure of Au mineral occurrence using multifractal models. On the other hand, there are few studies using multifractal methods to characterize soil properties. Most of the applications are related to the spatial variability of soil properties (e.g., surface strength) (Folorunso et al., 1994), hydraulic conductivity (Liu and Molz, 1997), and various soil properties (Kravchenko et al., 1999). Recently, Martin and Taguas (1998) suggested the entropy dimension, D₁, as a useful parameter for classifying soil texture within the classical textural triangle.

Grout et al. (1998) applied multifractal techniques to the study of PSDs, but their analyses were limited to three size distributions of mostly clayey soils. Without a more comprehensive study it is difficult to assess the applicability of the multifractal model to the description of PSD in soils.

The objective of this work was to apply multifractal methods to the characterization of contrasting soil PSDs, and to identify trends in the multifractal parameters related to textural separates.

	Theory

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

 
Multifractal models provide more information about a distribution of a physical system than fractal models (Voss, 1988). A brief overview of multifractal theory is presented here, but more detailed discussions can be found in Feder (1988), Voss (1988), Evertsz and Mandelbrot (1992), and Gouyet (1996).

In a homogeneous system, the probability (P) of a measure varies with scale L as (Chhabra et al., 1989; Evertsz and Mandelbrot, 1992; and Vicsek, 1992):

(1)

where D is a fractal dimension. For heterogeneous or nonuniform systems, the probability within the i^th region P_i scales as:

(2)

where _i is the Lipschitz-Holder exponent or singularity strength, characterizing scaling in the i^th region (Feder, 1988). A technique to determine multifractal parameters is to cover a measure with boxes of size L. Similar _i values can be found at different positions within a distribution. The number of boxes N() where the P_i has singularity strengths between and + d is found to scale as (Chhabra, et al., 1989; Hasley et al., 1986):

(3)

where f() can be defined as the fractal dimension of the set of boxes with singularities . The exponent can take on values from the interval [_-, ₊], and f() is usually a single humped function with a maximum at df((q))/d(q) = 0, where q is the moment order of a distribution. Thus, when q = 0, f_max is equal to the box-counting or D₀ (Gouyet, 1996; Viksec, 1992).

Multifractal sets can also be characterized on the basis of the generalized dimensions of the q^th moment orders of a distribution, D_q, defined as (Hentchel and Procaccia, 1983):

(4)

where µ(q, L) is the partition function defined as (Chhabra et al., 1989):

(5)

The generalized dimension D_q is a monotone decreasing function for all real q values within the interval [-, +]. When q < 0, µ emphasizes regions in the distribution with less concentration of a measure, whereas the opposite is true for q > 0 (Chhabra and Jensen, 1989).

The partition function scales as:

(6)

where (q) is the mass or correlation exponent of the q^th order defined as (Hasley et al., 1986; Viksec, 1992):

(7)

The connection between the power exponents f() (Eq. [3]) and (q) (Eq. [6]) is made via the Legendre transformation (Callen, 1985; Chhabra and Jensen, 1989; Hasley et al., 1986):

(8)

and,

(9)

The f() spectrum and the generalized dimensions contain the same information, both characterizing interwoven ensemble of fractals of dimension f(_i). In each of the i^th fractals, the observable P_i scales with the Lipschitz-Hölder-exponent _i.

The generalized dimensions for q = 0, q = 1, and q = 2 are known as D₀, D₁, and the correlation dimension, D₂. The capacity dimension is independent of q and provides global (or average) information of a system (Voss, 1988). The D₁ is related to the information or Shannon entropy (Shannon and Weaver, 1949), and quantifies the degree of disorder present in a distribution, by measuring the way the Shannon entropy {i.e., µ(L) log[µ(L)]} scales as L tends to 0. According to Gouyet (1996) and Andraud et al. (1994)( 1997) for a measure µ [0, 1], the value of D₁ is in the range of 0 < D₁ < 1. A D₁ value close to 1.0 characterizes a system uniformly distributed throughout all scales, whereas a D₁ close to 0 reflects a subset of the scale in which the irregularities are concentrated. The D₂ is mathematically associated to the correlation function (Grassberger and Procaccia, 1983) and computes the correlation of measures contained in intervals of size L. The relationship between D₀, D₁, and D₂ is,

where the equality D₀ = D₁ = D₂ occurs only if the fractal is statistically or exact self similar and homogeneous (Korvin, 1992).

	MATERIALS AND METHODS

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

 
Particle-Size Distributions
A total of 30 PSDs were analyzed. The database encompassed eight soils from the USA, 14 soils from Switzerland, and eight soils sampled from different regions of the state of São Paulo, Brazil. Five of the USA soils and the Swiss soils were taken from Bittelli et al. (1999). The remaining three PSDs from the USA soils were obtained for this study using a Leeds and Northrup MICROTRAC Standard Range particle-size Analyzer (SRA) (Microtrac, Inc., Montgomeryville, PA). The analyzer reads wet and slurry mixtures from 0.20 to 704 µm. Soil samples were crushed and placed in a muffle furnace at 600°C for 12 to 16 h to burn the organic matter, and then dispersed using sodium hexametaphosphate (NaHMP). Particle-size distributions in Bittelli et al. (1999), covering a size range from 0.12 to 1000 µm, were obtained using a small-angle light scattering apparatus. For the Brazilian soils, a gamma ray attenuation method was used to obtain distribution of particle sizes in the range from 2 to 250 µm (Vaz et al., 1992; Oliveira et al., 1997; Naime et al., 2001). Percentages of clay, silt, and sand of the soils in this study are summarized in Table 1, and their distribution in the textural triangle is shown in Fig. 1 . The textural triangle shows three groups following an increase of either clay, silt, or sand while maintaining one of the fractions more or less constant (Fig. 1). Between 50 and 80 particle-size measurements were available in each PSD, covering the range from 0.5 to 250 µm (except for the Brazilian soils that were between 2.2 and 250 µm), with the largest concentration of measurements for particles smaller than 50 µm.

View larger version (33K): [in this window] [in a new window]   Fig. 1. Soil textural triangle according to the Soil Survey Division Staff (1993), showing the range of soils studied. Three groups of selected soils are distinguished based on variations in the content of two soil separates while keeping the third one approximately constant. Soils in Group 1 () vary along the silt and sand axis. Soils in Group 2 () vary along the clay and sand axis. Soils in Group 3 (•) vary along the silt and clay axis. Open squares represent all other soils in the database.

(10)

where P_i(L) is the probability of finding a class-i measure in the interval L. In our case, a PSD was partitioned in intervals of size L, and µ_i was constituted by the percentage of mass contained in each i^th interval. The multifractal spectrum, f(q) vs. (q), was calculated as in (Chhabra et al., 1989; Chhabra and Jensen, 1989):

(11)

(12)

Cumulative mass size distribution curves were interpolated using a spline technique, and the amount of mass determined for each interval of size L. The total number of points used to compute a multifractal spectrum varied between 120 and 200. The maximum value of L that can be used in Eq. [11] and [12] and the range of q values was assessed by the linear behavior of the function ^N_i=1 µ_i (q, L) log[µ_i (q, L)] vs. log (L) for all the values of q used (Chhabra et al., 1989; Evertsz and Mandelbrot, 1992). The values of q considered were between –10 and +10 taken at 0.5 lag increments. In addition, we tested the validity of the results by verifying that the tangent of the graph f() vs. at = 1 is the bisector defined by df()/d = q. The point of intersection corresponds to f[(1)] = (1) = D₁ (Evertsz and Mandelbrot, 1992). The D_q and (q) were obtained with Eq. [7] and [8], respectively. The errors in the D_q's were estimated using the theory of propagation of error considering the square root of the error mean square (standard deviation) in Eq. [7] and [8] (Beers, 1953).

	RESULTS AND DISCUSSION

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Values of the regression coefficients, R², of the linear fit of the summation term of the right-hand-side of Eq. [11]{^N_i=1) µ_i log vs. log } were highest for q = 0 and q = 1 (Table 2) and decreased for higher q's. For most soils, the maximum value of L used was 50 µm covered with 25 steps. The R² varied between 0.988 and 0.999 for q = 0, and between 0.968 to 0.999 for q = 1 (Table 2). For samples with <10% clay, the R² dropped to about 0.6 for both q = –10 and q = +10. The drop of R² with higher q's was more pronounced for samples with >10% clay. For a homogeneous system, the summation term of the right-hand side of Eq. [11] is linear and with similar slopes for a wide range of q's (Les Berges, Fig. 2a) . For a multifractal system, on the other hand, the linear behavior covers a limited range of scales and it changes for different q's (Dusk Red Latosol 1, Fig. 2b) (Voss, 1988).

View larger version (15K): [in this window] [in a new window]   Fig. 2. Plots of Ni=1 µi (q, L) log[µi (q, L)] vs. log (L) for q = 0 and q =1 for the particle-size distributions of (a) the Les Barges, and (b) the Dusk Red Latosol 1 soils. For explanation see text.

View larger version (12K): [in this window] [in a new window]   Fig. 3. Example of f()-spectra for the Les Barges and Dusk Red Latosol 1 soils.

View larger version (13K): [in this window] [in a new window]   Fig. 4. Plot of differences (D0 - D1) and {f[(-1)] - D0} as a function of clay content.

View larger version (18K): [in this window] [in a new window]   Fig. 5. Plots of (q) vs. q for soil groups shown in Fig. 1, (a) Group 1, (b) Group 2, and (c) Group 3. All soils with clay content <10% exhibited spectra similar to those in Group 1 (data not shown).

View larger version (15K): [in this window] [in a new window]   Fig. 6. Plot of the capacity dimension, D0, the entropy dimension, D1, and the correlation dimension, D2, as a function of clay content for all soils.

The results presented in this paper do not invalidate the fractal analyses done in the past, but it suggests that multifractal techniques provide additional information to characterize a distribution. Models based on single fractal are limited to PSDs with clay contents of <10%. Further studies including samples representing all regions of the textural triangle may refine the limits of applicability of single fractal models in soils.

	CONCLUSIONS

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Multifractal analyses revealed PSDs with single and multifractal scaling. In general, multifractal parameters were related to clay content. A clay content of 10% separated single from multifractal scaling. A further refinement in such study will be to redefine size boundaries of soil separates (e.g., in the size range between clay and silt) based on their effect on the scaling of a distribution.

The entropy dimension, D₁, is a useful parameter can be used to distinguish single from multifractal scaling, and the difference (D₀ - D₁) constitutes a measure of the degree of heterogeneity of a distribution and may be useful as a predictive parameter.

Models of PSD should recognize differences in scaling properties of particle distributions, and research should be devoted to link these differences with pedogenetic processes to fully interpret the significance and potential of multifractal parameters as a representation of a PSD.

Received for publication July 25, 2000.

	REFERENCES

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

 

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