Single and Joint Multifractal Analysis of Soil Particle Size Distributions

Pedosphere 21(1): 75–83, 2011 ISSN 1002-0160/CN 32-1315/P c 2011 Soil Science Society of China  Published by Elsevier B.V. and Science Press

Single and Joint Multifractal Analysis of Soil Particle Size Distributions∗1 LI Yi1,∗2 , LI Min1,2 and R. HORTON3 1 Northwest

Agriculture and Forestry Sci-Tech University, Yangling 712100 (China) of Soil Science, University of Saskatchewan, Saskatoon, SK S7N 5A8 (Canada) 3 Department of Agronomy, Iowa State University, Ames IA 50011 (USA) 2 Department

(Received April 14, 2010; revised September 21, 2010)

ABSTRACT It is noted that there has been little research to compare volume-based and number-based soil particle size distributions (PSDs). Our objectives were to characterize the scaling properties and the possible connections between volume-based and number-based PSDs by applying single and joint multifractal analysis. Twelve soil samples were taken from selected sites in Northwest China and their PSDs were analyzed using laser diffractometry. The results indicated that the volume-based PSDs of all 12 samples and the number-based PSDs of 4 samples had multifractal scalings for moment order −6 < q < 6. Some empirical relationships were identified between the extreme probability values, maximum probability (Pmax ), minimum probability (Pmin ), and Pmax /Pmin , and the multifractal indices, the difference and the ratio of generalized dimensions at q = 0 and 1 (D0 − D1 and D1 /D0 ), maximum and minimum singularity strength (αmax and αmin ) and their difference (αmax − αmin , spectrum width), and asymmetric index (RD ). An increase in Pmax generally resulted in corresponding increases of D0 − D1 , αmax , αmax − αmin , and RD , which indicated that a large Pmax increased the multifractality of a distribution. Joint multifractal analysis showed that there was significant correlation between the scaling indices of volume-based and number-based PSDs. The multifractality indices indicated that for a given soil, the volume-based PSD was more homogeneous than the number-based PSD, and more likely to display monofractal rather than multifractal scaling. Key Words:

distribution probability, generalized dimensions, laser diffractometry, scaling, singularity strength

Citation: Li, Y., Li, M. and Horton, R. 2011. Single and joint multifractal analysis of soil particle size distributions. Pedosphere. 21(1): 75–83.

INTRODUCTION The soil particle size distribution (PSD) is useful for quantifying soil properties, because it is related to soil pore space, water movement, productivity, and erosion. The laser diffraction (LD) method has been widely adopted in recent years for analyzing soil PSDs because it has several advantages over the classical methods of sieving and sedimentation with either pipettes or hydrometers (Gee and Or, 2002): short time for analysis (5–10 min per sample), higher repeatability, smaller quantities of samples (< 1 g) needed (Beuselinck et al., 1998), and smaller detected particles of sizes around 0.1 μm which can be divided into a larger number of size fractions (Eshel et al., 2004). Typically, the software in combination with an LD instrument permits the user to have up to 100 sizerange classes between the lower radius limit (Rlower ) and the upper radius limit (Rupper ). ∗1 Supported

The soil PSD is often observed to be self-similar, following a power-law scaling of mass or number with respect to length scale, at least within a subset of the particle size ranges (Wu et al., 1993; Bittelli et al., 1999). But closer examination of soil PSD may indicate that PSD is not exactly fractal, or that it does not conform to a single power-law scaling (Grout et al., 1998). It has also been argued that the presence of power-law scaling does not imply applicability of a single-fractal (monofractal) model (Pachepsky et al., 1997). In general, the mass (or number) of soil particles shows both denser and sparser regions across different size intervals (Montero, 2005). This indicates its heterogeneity and complexity, which are present in most measures in nature, displaying a behavior called multifractal (Mandelbrot, 1989). Multifractal analysis has been successfully applied in several scientific fields such as turbulence (Chhabra et al., 1989) and sea-ice (Falco et al., 1996). Recent

by the National Natural Science Foundation of China (No. 50709028) and the Basic Foundation for Scientific Research of Northwest Agriculture and Forestry Sci-Tech University, China (No. QN2009087). ∗2 Corresponding author. E-mail: [email protected].

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applications of multifractal analysis in soil and agronomic sciences have concentrated on soil pore size distribution (Caniego et al., 2003), topographic indices and crop yield (Zeleke and Si, 2004), spatial scaling of soil physical properties (Zeleke and Si, 2005, 2006), soil surface structure by image analysis (Guan et al., 2007), mass-based soil PSDs (Posadas et al., 2001), volume-based soil PSDs (Montero and Martin, 2003; Montero, 2005; Wang et al., 2007, 2008), and numberand mass-based soil PSDs (Grout et al., 1998). Combining multifractal techniques with laser diffractometry has a potential to detect scaling properties of soil PSDs in rarely explored regions of particle size (Montero, 2005). Grout et al. (1998) concluded that soil PSD data were more appropriately described as multifractal than simple power law for three clayey soils. Posadas et al. (2001) found soil PSDs with both single-fractal and multifractal scalings, and concluded that multifractality was generally related to clay content. Wang et al. (2008) characterized soil PSDs of five land-use types and found a significant correlation between the entropy dimension or the ratio of entropy dimension to capacity dimension and the contents of both fine particles and organic matter in soil. For determining whether a given natural distribution is monofractal or multifractal, a universal multifractal scaling model was developed by Liu and Molz (1997). A further extension of the multifractal technique is joint multifractal analysis, which characterizes the degree of correlation between two or more different quantities, each of which is singly represented by a multifractal (Meneveau et al., 1990). For example, a joint multifractal analysis may be performed on the volumeand number-based PSDs of a single soil to describe their joint intermittent distributions. To date, little research has been conducted on simultaneous detection of multi-scale properties of volume- and number-based PSDs using multifractal and joint multifractal techniques. Multifractal analysis which concludes that a given distribution has only monofractal scaling can be useful because a monofractal soil PSD may allow closed form solutions of properties such as the water retention curve. Our objectives were to characterize multifractality of volume- and number-based PSDs through the use of single and joint multifractal methods, and to find possible connections between their scaling properties. THEORY Let NT be the total number of particles with radius larger than that of the ith particle (Ri ), NT = lim N (R > Ri ). Assuming that the particles are

Ri →Rlower

Y. LI et al.

PV (Ri ) , where N (Ri ) is 4πRi3 /3 the number of particles per unit volume at Ri , and PV (Ri ) denotes the volume probability of particles at Ri . PN (Ri ) is the number probability of particles at Ri , calculated as:

spherical, we have N (Ri ) =

Rupper PV (Ri )   PV (Ri ) PN (Ri ) = N (Ri )/NT = = 4πRi3 /3 4πRi3 /3 Rlower

Rupper PV (Ri )   PV (Ri ) Ri3 Ri3

(1)

Rlower

Thus, a number-based PSD could be obtained from a volume-based PSD. In this study we used both and comparisons were made between them. Single multifractal analysis The multifractal formalism describes the statistical properties of the singular spectrum of a physical system whose singularity spectrum exhibits chaotic behaviors (Chhabra et al., 1989). A relatively simple and accurate method developed by Chhabra and Jensen (1989) was followed for directly computing the multifractal spectrum. Within an interval I, a dyadic partition in k stages (k = 1, 2, 3, . . .) generates a number of cells N (ε) (N (ε) = 2k ) with diameter (scale of the interval) ε (ε. = L × 2k , where L is the total length of the interval I). In a homogeneous system, the volumetric probability Pi , the integral of particle density over the radius range from Ri to Ri+1 , of a class i measure varies with ε as Pi (ε) ∼ εD

(2)

where D is the fractal dimension. When the system is heterogeneous or nonuniform, Pi scales as Pi (ε) ∼ εαi

(3)

where αi is the Lipschitz-H´older exponent, also known as the singularity strength (Posadas et al., 2001), which characterizes scaling in the ith region. If we count the number of boxes N (α) within which Pi has singularity strength between α and α+dα (with size ε), then f (α) is defined as the fractal dimension of the set of boxes with singularity strength α by N (α) ∼ ε−f (α)

(4)

This formalism leads to the description of a multifractal measure in terms of interwoven sets of Hausdorff dimension f (α) possessing singularity strength (Chhabra

ANALYSIS OF SOIL PARTICLE SIZE DISTRIBUTIONS

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et al., 1989). The generalized dimensions (Chhabra and Jensen, 1989) at moment order q, Dq , are exponents that characterize nonuniformity of the measure, defined as the limit of ε going to zero. When q = 1, N (ε)

ln 1 lim Dq = q − 1 ε→0



[Pi (ε)]q

i=1

(5)

lnε

When q = 1, N (ε)



D1 = lim

Pi (ε)lnPi (ε)

i=1

ε→0

(6)

lnε

where q is a integer that acts as a scanning tool, identifying denser and sparser regions of a distribution. For q  1, dense regions are amplified, whereas the opposite is true for q  1. D0 , the D value at q = 0, is called the capacity dimension, which provides average information of a distribution. D1 is called the entropy dimension, which indicates the degree of heterogeneity in a measure. Their ratio, D1 /D0 , is an indicator for the heterogeneity of a PSD (Wang et al., 2008). Values of D1 /D0 close to 1 indicate sets with similar dimension, while values close to 0 will be found in distributions with most of the measures concentrated in a small region of sizes. D2 , the D value at q = 2, is called the correlation dimension. When a particle size distribution has a monofractal scaling, D0 = D1 = D2 ; when the distribution has a multifractal scaling, D0 > D1 > D2 . The partition function μ(q, ε), a family of normalized measures, is constructed as follows: [Pi (ε)]q

μi (q, ε) =

∝ (ε)τ (q)

N (ε)

ln



(7)

q

[Pi (ε)]

where τ (q) is called qth mass exponent (Grout et al., 1998) and is related to Dq as (8)

When a plot of τ (q) vs. q has curvature that is both significant and negative, the PSD has a multifractal type of distribution; otherwise, the distribution is monofractal (Zeleke and Si, 2005). A universal model (UM) derived by Schertzer and Lovejoy (1987) simulates τ (q) function in Eq. 8 of a cascade process with just a few parameters (Falco et al., 1996). According to this model, the monofractal extreme gives

(9)

Because Eq. 9 indicates the monofractal case, it may be plotted with calculated values of τ (q). Eq. 9 is a simplification of Eq. 8 at Dq = 1. A stricter test of whether a measure is monofractal or multifractal can be made using computed τ (q) in a χ2 (chi-square) test, comparing the best-fit segmented linear model (different linear equations for τ (q) at q > 0 and q < 0) to the best-fit simple linear model for τ (q) at all values of q. If the χ2 test indicates a significant difference, the measure is multifractal. The average value of the singularity strength α within an interval I, αI (αI = lnPi /lnε), with respect to μ(q, ε) is computed by evaluating a H´older spectrum (Montero and Martin, 2003): N (ε)



α(q) = − lim

μi (q, ε)lnPi (ε)

i=1

ε→0

lnε

(10)

Let αmax and αmin denote the maximum and minimum values of α, respectively. The wider the spectrum (αmax − αmin ), the greater the heterogeneity in the local scaling indices of the variable (Zeleke and Si, 2005). The multifractal spectrum, f (q), is calculated by Eq. 11 (Chhabra et al., 1989): N (ε)



f (q) = − lim

μi (q, ε)lnμi (q, ε)

i=1

ε→0

lnε

(11)

The plots of f (q) vs. α(q) are always asymmetrical, reflecting more pervasively distributions of high values. The asymmetry index, RD , is defined to quantify deviations in the spectrum (Xie and Bao, 2004): RD =

i=1

τ (q) = (q − 1)Dq

τ (q) = q − 1

ΔαL − ΔαR ΔαL + ΔαR

(12)

where ΔαL = α0 − αmin , ΔαR = αmax − α0 , and α0 is the value of α at q = 0. ΔαL and ΔαR are the ranges of the left and right portions of the multifractal spectrum curve, respectively. The parameter RD can take values between −1 and 1. RD > 0 implies a left deviation of f (α), and RD = 0 represents a completely symmetrical case. Joint multifractal analysis For two distinct multifractal measures coexisting in the same radius domain (here, the volume-base PSD and number-based PSD of a single soil), the multifractal formula is extended by implementing a new norma-

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lized multi-variable μ measure (the partition function introduced in Eq. 7) by using two different moment orders q and m. This new partition function, μ(q, m, ε), is calculated by Eq. 13 (Meneveau et al., 1990): μj (q, m, ε) =

[PV,j (ε)]q [PN,j (ε)]m

(13)

N (ε)



q

m

[PV,j (ε)] [PN,j (ε)]

j=1

The H´older exponents α and β, with respect to the partition function, are computed by Eqs. 14 and 15, respectively: N (ε)



α(q, m) = lim

μj (q, m, ε)ln[PV,j (ε)]

j=1

ε→0

(14)

lnε N (ε)



β(q, m) = lim

μj (q, m, ε)ln[PN,j (ε)]

j=1

ε→0

(15)

lnε

The dimension of the measure-theoretic support of μ(q, m, ε) is calculated as follows: N (ε)



f (q, m) = lim

μj (q, m, ε)lnμj (q, m, ε)

j=1

ε→0

MATERIALS AND METHODS Twelve soil samples were taken from different selected geographical locations in Northwest China for this study. All samples except Samples 3, 7, and 9 were collected from the 0 to 10 cm depth in a field in Manasi County, Xinjiang Uygur Autonomous Region (44◦ 19 N, 86◦ 12 E, 472 m above sea level). Sample 3 was from the 90 to 100 cm depth in a field near Urumqi, Xinjiang (43◦ 48 N, 87◦ 36 E, 654 m above sea level). Samples 7 and 9 were from the 0 to 10 cm depth in a field in Shenmu County, Shaanxi Province (38◦ 49 N, 110◦ 26 E, 941 m above sea level). All the samples were air-dried, crushed, and sieved to pass a 2-mm screen to remove roots, stones, and debris. Aggregates were dispersed in distilled water for 2 d, sonicated for 30 s, and then analyzed using laser diffractometry with a Longbench Mastersizer 2000 (Malvern Instruments, England). The measured soil PSDs represent Pv vs. Ri along 64 subintervals (Wang et al., 2008). The length of the ith subinterval Ii (Ii = [Ri , Ri+1 ], i = 1, 2, . . . , 64) followed a logarithmic scale with log(Ri+1 /Ri ) being constant. The maximum particle radius Rmax is 1 mm and the minimum particle radius Rmin varies from 0.1 to 0.5 nm depending on the soil.

(16)

lnε

Therefore, f (q, m) is the dimension of a set on which α(q, m) and β(q, m) are the mean local exponents of two measures. Joint multifractal measures are characterized by their spectra of dimensions, f (q, m) vs. f (α, β). Notice that Eq. 10 is equivalent to Eqs. 14 and 15, and Eq. 11 to Eq. 16, for the special case that m = 0.

RESULTS AND DISCUSSION Volume- and number-based PSDs Basic textural properties and extreme probability values for the samples are presented in Table I. For a given soil, the maximum probability for a volumebased PSD was obviously smaller than that of a nu-

TABLE I Textures of the soil samples and the extreme probabilitya) values for soil particle size distributions (PSDs) Sample

1 2 3 4 5 6 7 8 9 10 11 12 a) P max

Sand

202 165 46 186 535 243 181 82 110 222 18 48

Silt

g kg−1 506 564 676 529 425 492 626 603 715 583 709 868

Clay

292 271 278 285 40 265 193 315 175 195 274 85

Texture

Clay loam Silty clay loam Silty clay loam Silty clay loam Sandy loam Loam Silt loam Silty clay loam Silt loam Sandy loam Silty clay loam Silt

Volume-based PSD

Number-based PSD

Pmax

Pmin

Pmax /Pmin

Pmax

Pmin

Pmax /Pmin

0.032 0.047 0.045 0.050 0.061 0.045 0.032 0.052 0.057 0.025 0.051 0.034

1.2 × 10−6 1.9 × 10−5 4.2 × 10−6 1.1 × 10−5 1.2 × 10−3 4.4 × 10−4 1.3 × 10−4 2.0 × 10−4 5.3 × 10−5 1.1 × 10−3 4.4 × 10−5 1.6 × 10−3

2.7 × 104 2.5 × 103 1.1 × 104 4.5 × 103 4.9 × 10 1.0 × 102 2.5 × 102 2.6 × 102 1.1 × 103 2.2 × 10 1.2 × 103 2.1 × 10

0.336 0.324 0.313 0.325 0.275 0.268 0.295 0.299 0.405 0.248 0.338 0.172

6.1 × 10−13 9.9 × 10−16 8.9 × 10−13 6.1 × 10−16 1.3 × 10−9 1.7 × 10−14 1.6 × 10−10 5.4 × 10−12 3.2 × 10−15 1.2 × 10−9 6.9 × 10−12 2.4 × 10−8

5.5 × 1011 3.3 × 1014 3.5 × 1011 5.4 × 1014 2.1 × 108 1.6 × 1013 1.8 × 109 5.6 × 1010 1.3 × 1014 2.0 × 108 4.9 × 1010 7.3 × 106

= maximum probability; Pmin = minimum probability.

ANALYSIS OF SOIL PARTICLE SIZE DISTRIBUTIONS

mber-based PSD, indicating that they were different. The probability values for the number-based PSDs of Samples 1 and 8 ranged from 10−1 to 10−16 , approximately three times the range of the volume-based PSDs which ranged from 10−1 to 10−5 (Fig. 1). For a given increase in particle radius, probability values for the number-based PSDs decreased more rapidly than those for the volume-based PSDs of Samples 1 and 8. These patterns were generally representative of those for the other samples as well.

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dex for describing effect of increasing q values on Dq . For comparing the trend of Dq with that of Dq−1 , plots of Dq /Dq−1 vs. q were made (Fig. 3). Values of

Fig. 1 Volume- and number-based particle size distributions for 2 soil samples.

Power-law scaling of soil PSDs A soil PSD is considered to be fractal (whether monofractal or multifractal) when its moments obeyed power laws—the log-log plots of normalized measures against their measurement scales are approximately straight lines (Zeleke and Si, 2006). By this criterion, both the volume-based and number-based PSDs for Sample 1 were fractal, but they clearly had different scaling characteristics (Fig. 2). There were generally high coefficients of determination (R2 ) for linear fits of ln[Σ P (ε)] vs. lnε for the volume-based PSDs of all 12 samples and the numberbased PSDs of 4 samples. For these 16 PSDs, R2 decreased with increasing absolute values of q, being greatest at q = 0. R2 was generally larger for a numberbased PSD than for a volume-based PSD. The eight remaining number-based PSDs did not follow powerlaw scaling, so they were not well described by a fractal model.

Fig. 2 Power law plots of natural logarithms of the probability values with diameter ε, P (ε), at different moment order (q) values for the volume-based and number-based particle size distributions (PSDs) of Sample 1.

Generalized dimensions of soil PSDs Generalized dimensions Dq (Eq. 4, −6 < q < 6) were calculated for all 16 fractal PSDs. Plots of Dq vs. q all had a non-increasing sigmoid shape (Montero, 2005), but they supplied limited information for variations of Dq with Dq−1 , a promising in-

Fig. 3 Plots of Dq /Dq−1 vs. q for volume- and numberbased particle size distributions (PSDs). Dq and Dq−1 are the fractal dimensions at the moment order q and q − 1, respectively.

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based PSDs, D2 values clearly differed from D1 . The volume-based PSDs of Samples 10, 11, and 12 had very similar values of D0 , D1 and D2 , showing approximately monofractal scaling. The other PSDs showed varying degrees of multifractal scaling. In general, αmax had a relatively large range from 1.36 to 3.13, but the other multifractal parameters had relatively small ranges. So αmax was selected as a parameter for grouping the soil samples. If αmax > 3, soil PSDs showed complex multifractal structures, simplified as CMS; if αmax was between 2 and 3, soil PSDs showed middle multifractal structures, simplified as MMS; and when αmax < 2, soil PSDs showed simple multifractal structures or tended to be homogeneous, simplified as SMS. For the volume-based PSDs, Sample 1 showed CMS, Samples 2, 3, 4, 5, and 6 showed MMS, and Samples 7, 8, 9, and 10 showed SMS. For the number-based PSDs, Samples 1 and 8 showed CMS, Samples 10 and 12 showed CMS. Larger αmax also tended to have a smaller D1 /D0 and showed a more complex multifractal structure. Some simple relationships were found between D0 − D1 and the extreme probability values Pmax , Pmin , and Pmax /Pmin ; e.g., D0 − D1 = 0.0462lnPmax + 0.1768, with R2 = 0.566. This empirical equation suggested that values of D0 and D1 were affected by the extreme probability values of distributions.

Dq /Dq−1 ranged between 0.5 and 1, and generally decreased with increases in q when q < −1. Dq /Dq−1 generally had a minimum at q = −1. A relatively flat plot of Dq /Dq−1 vs. q indicates little change in Dq with Dq−1 , implying a monofractal distribution. Only the volume-based PSDs for Sample 12 had a flat Dq /Dq−1 vs. q plot. Capacity dimension (D0 ), entropy dimension (D1 ), and correlation dimension (D2 ) can be used to describe the homogeneity of a distribution. The difference, D0 −D1 , and the ratio, D1 /D0 , also indicate homogeneity of a distribution. Smaller values of D0 − D1 and values of D1 /D0 near 1 indicate a more homogeneous distribution. Values of D0 , D1 , D2 , D0 − D1 , and D1 /D0 for the selected soil samples are given in Table II. The capacity dimension (D0 ) ranged from 0.94 to 1.0 for the volume-based PSDs and from 0.99 to 1.0 for the number-based PSDs. Correlation dimension (D1 ) values for the volume-based PSDs ranged from 0.90 to 0.99, and from 0.81 to 0.97 for the numberbased PSDs. For the difference D0 − D1 , the volumebased PSD values ranged from 0.01 to 0.08, while the number-based PSD values ranged from 0.03 to 0.19. The volume-based PSDs thus tended to be more homogeneous and monofractal than the number-based PSDs. The ratio D1 /D0 had values ranging from 0.92 to 0.99 for the volume-based PSDs, and from 0.81 to 0.97 for the number-based PSDs, again showing greater heterogeneity in the number-based PSDs than in the volume-based PSDs. For the volume-based PSDs, D2 values were generally close to D1 , but for the volume-

Mass exponents of soil PSDs A comparison between the mass exponents τ (q) and the simulated monofractal distribution (the simple

TABLE II Generalized dimensionsa) and spectra parametersb) for the volume-based particle size distributions (PSDs) of all soil samples and the number-based PSDs of selected soil samplesc) PSDs

Sample

D0

D1

D0 − D1

D1 /D0

D2

αmax

αmin

αmax − αmin

RD

Volume-based

1 2 3 4 5 6 7 8 9 10 11 12 1 8 10 12

1.00 0.94 0.98 0.95 1.00 0.96 1.00 0.98 0.94 1.00 0.95 0.99 1.00 1.00 1.00 0.99

0.97 0.91 0.96 0.91 0.92 0.93 0.98 0.95 0.90 0.99 0.94 0.97 0.97 0.81 0.86 0.88

0.03 0.03 0.02 0.04 0.08 0.03 0.02 0.03 0.04 0.01 0.01 0.02 0.03 0.19 0.14 0.11

0.97 0.97 0.98 0.96 0.92 0.97 0.98 0.97 0.96 0.99 0.99 0.98 0.97 0.81 0.86 0.89

0.97 0.91 0.95 0.91 0.87 0.91 0.98 0.93 0.90 0.99 0.94 0.97 0.97 0.68 0.77 0.83

3.13 2.98 2.95 2.75 2.73 2.19 1.90 1.91 1.86 1.74 1.59 1.36 3.25 3.00 2.72 2.10

0.94 0.84 0.94 0.80 0.76 0.76 0.83 0.94 0.81 0.98 0.80 0.92 0.44 0.46 0.52 0.66

2.19 2.14 2.01 1.94 1.96 1.43 1.07 0.97 1.05 0.76 0.79 0.45 2.82 2.54 2.21 1.44

0.03 0.05 0.02 0.08 0.09 0.16 0.16 0.06 0.08 −0.01 0.22 0.22 0.25 0.26 0.28 0.31

Number-based

a) D = capacity dimension; D = entropy dimension; D = correlation dimension. 0 1 2 b) α max = maximum value of singularity strength α; αmin = minimum value of α; RD = asymmetric index. c) The coefficients of determination for obtaining D , D , D , α max , and αmin are all larger than 0.95. 0 1 2

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linear fit computed by Eq. 9) for the volume- and number-based PSDs are shown in Fig. 4. All τ (q) curves have negative curvature. In the range q < 0, curves of τ (q) have different slopes from the monofractal type for q < 0, implying a multiple scaling nature where low and high density regions of the variable scale differently. The τ (q) curves of both volume- and number-based PSDs are not straight lines or alien from monofractal type of distributions. The τ (q) curves of number-based PSDs show more pronounced deviations from a straight line (monofractal scaling type) than the volume-based PSDs. The χ2 test was used to detect if the τ (q) curves were significantly different from the monofractal type of distributions. The results showed that all 16 τ (q) curves were significantly different from the monofractal scaling type and showed multifractal scaling. For each soil sample, the volume-based PSDs had a lower degree of multifractality than the numberbased PSDs. Fig. 5 Single multifractal spectra f (q) vs. singularity strength α(q) at moment order q for volume- and number-based particle size distributions (PSDs) of 4 soil samples.

Fig. 4 Mass exponents for soil volume- and number-based particle size distributions (PSDs).

Single multifractal spectra of soil PSDs The multifractal spectra f (q) vs. α(q) of the volume-based PSDs and number-base PSDs for 4 soil samples are given in Fig. 5. The smallest coefficient of determination for calculating f (q) and α(q) was 0.95. Almost all the multifractal spectra are skewed to the left. For each soil sample, the multifractal spectrum of the number-based PSDs had a larger range of

α than that of the volume-based PSDs, indicating greater multifractality of the number-based PSDs than the volume-based PSDs. For further description of the multifractal spectra, the extreme α values and the calculated asymmetric index RD values are presented in Tables II. For the volume-based PSDs, values of αmax ranged widely from 1.36 to 3.13 and the range of αmin values was small, from 0.76 to 0.98. For the number-based PSDs, values of αmax ranged from 2.1 to 3.25 and values of αmin ranged small from 0.44 to 0.66. There were generally larger αmin values for the volume-based and smaller αmax values fro the number-based PSDs for any given sample. Consequently, the difference αmax − αmin had smaller values for the volume-based PSDs (0.45–2.19) than for the number-based PSDs (1.44–2.82). Because a wider spectrum (αmax − αmin ) indicates greater heterogeneity in the local scaling indices (Zeleke and Si, 2006), the number-based PSDs had a more heterogeneous distribution than the volume-based PSDs in each soil. Several correlations were noted between the spectrum parameters and the extreme values of particle size probabilities: αmin = −0.193lnPmax + 0.253 (R2 = 0.895), αmin = 0.0218lnPmin + 1.06 (R2 = 0.735), αmin = −0.02ln(Pmax /Pmin ) + 0.977 (R2 = 0.765), αmax − αmin = −0.078lnPmin + 0.609 (R2 = 0.555), and αmax − αmin = 0.069ln(Pmax /Pmin ) + 0.907 (R2 = 0.545). These equations showed that αmin generally decreased with increasing Pmax and Pmax /Pmin , but increased with increasing Pmin .

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The asymmetric index RD ranged from −0.01 to 0.22 for the volume-based PSDs, and from 0.25 to 0.31 for the number-based PSDs. There was a correlation between RD and extreme probability as follows: RD = 0.0903lnPmax + 0.385 (R2 = 0.537). The equation showed that a higher Pmax value in particle size distribution would generally resulted in a larger values of RD which corresponded to a more skewed f (α) curve. In general, as Pmax increased, values of D0 −D1 , αmax , αmax − αmin , and RD increased, while values of D1 /D0 and αmin decreased. When Pmin was smaller than 10−5 , values of D0 − D1 , αmax , αmax − αmin , and RD decreased rapidly with increasing Pmin ; in contrast, values of D1 /D0 and αmin increased gradually with the increase in Pmin . When Pmin was greater than 10−5 , the increase or decrease of the multifractal parameters slowed down with the increase in Pmin . When Pmax /Pmin was smaller than 108 , values of D0 − D1 , αmax , αmax − αmin , and RD increased rapidly, while values of D1 /D0 and αmin decreased rapidly with the increase in Pmax /Pmin . The empirical relationships between the extreme values of distribution probabilities (Pmax , Pmin , and Pmax /Pmin ) and the multifractal indices (D0 − D1 , D1 /D0 , αmax , αmin , and αmax − αmin ) indicated that multifractality of soil PSDs was generally affected by the extreme values of probabilities. Joint multifractal spectra of soil PSDs The single multifractal theory can be extended to characterize the joint distribution of two interacting variables along a common geometric support. f (α, β) was obtained as an explicit function of α and β by interpolating the f values on an equally spaced mesh in α and β. The resulting spectra f (α, β) for the joint multifractal analysis between the volume- and numberbased PSDs are shown in Fig. 6. The contour lines are the joint dimensions of the volume- and number-based PSDs, f (α, β). The bottom left section of the contours shows the joint dimension of the high values of PV and PN , while the top right section shows that of the low values. For Samples 1 and 12, there were some relationships between the scaling dimensions of PV and PN for both high and low values, as is evident from the slightly diagonal feature of the plots and strongly negative correlation coefficients between α and β for the volumeand number-based PSDs (R = −0.61 for Sample 1, and R = −0.76 for Sample 12). For Samples 8 and 10, the scaling indices of PV were not related to those of PN , with R being −0.37 and −0.28, respectively. All the plots were slightly distorted diagonally, indicating different degrees of multifractality in the scaling indices of the volume- and number-based PSDs. It appeared

Y. LI et al.

Fig. 6 Joint multifractal spectra of volume- and number-based probability of soil particle size distributions (PSDs). Contour lines show the joint multifractal dimensions f (α, β), where α and β are the H´ older exponents.

that there were some associations between scaling indices of the volume- and number-based PSDs. CONCLUSIONS The continuous volume-based soil particle size distributions observed by the laser diffractometry method could be used to obtain continuous number-based distributions. The volume-based PSDs of 12 samples and the number-based PSDs of 4 samples followed power laws, with logarithmic plots of moments (−6 < q < 6) against measurement scales strongly linear. Single multifractal analysis showed that all of the distributions were multifractal, but the volume-based PSDs tended to be less multifractal than the numberbased PSDs. Some empirical relationships were found between the distribution probability extreme values (Pmax , Pmin , and Pmax /Pmin ) and the multifractal indices (D0 − D1 , D1 /D0 , αmax , αmin and αmax − αmax ). This indicated that multifractality of soil PSDs was affected by the extreme values of particle size distribution probability. Joint multifractal analysis connecting the volume probability (PV ) with the number probability (PN ) confirmed different degrees of associations between the scaling indices of the volume and number-based PSDs. REFERENCES Beuselinck, L., Govers, G., Psesen, J., Degraer, G. and Froyen, L. 1998. Grain-size analysis by laser diffractometry: comparison with the sieve-pipette method. Catena. 32: 193–208. Bittelli, M., Campbell, G. S. and Flury, M. 1999. Characteriza-

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