multifractal matlab

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Multifractal Characterization of Soil Pore Systems

Adolfo N. D. Posadas, Daniel Gimenez,* Roberto Quiroz, and Richard Protz

ABSTRACT and biological activity (Bouma et al., 1977; Ringrose-Voase, 1987; Pagliai and De Nobili, 1993). The selectionSpatial arrangement of soil pores determines soil structure and isand interpretation of the morphological parameters ofimportant to model soil processes. Geometric properties of individualsoil pores that best characterize soil structure, however,pores can be estimated from thin sections, but there is no satisfactory

method to quantify the complexity of their spatial arrangement. The is still a subject of research (Droogers et al., 1998;objective of this work was to apply a multifractal technique to quantify Holden, 2001). Statistical methods of characterizationproperties of ten contrasting soil pore systems. Binary images (500 of pore-solid arrangement from images emphasize theby 750 pixels, 74.2 �m pixel�1) were obtained from thin sections and spatial structure of pores with the advantage of beinganalyzed to obtain f(�) spectra. Pore area and pore perimeter were more amenable to modeling soil structure and processesmeasured from each image and used to estimate a shape factor for (Dexter 1976, 1977; Moran and McBratney, 1997; Gar-pores with area larger than 0.27 � 106 �m2. Mean area of the lower

boczi et al., 1999; Horgan, 1999; Vogel and Roth, 2001).(MAL) and upper (MAU) one-half of cumulative pore area distribu-Among the statistical methods used to characterizetions were calculated. Pore structures with large (MAU � 10 �

soil structure, fractal techniques are relatively common106 �m2) and elongated pores exhibited “flat” f(�)-spectra typical ofin soil science (Anderson et al., 1996; Pachepsky et al.,homogenous systems (three soils). Massive type structure with small

(MAU � 1 � 106 �m2) rounded and irregular pores resulted in asym- 1996; Gimenez et al., 1997). Fractal geometry assumesmetric f(�)-spectra (two soils). Well defined and symmetric f(�)- that the dependence of the properties of a system withspectra were obtained with soil structures having elongated pores of scale (scaling) can be represented with a power law,intermediate size (1 � 106 � MAU � 10 � 106 �m2) clustered around with the exponent being typically a function of a fractalrelatively small structural units (five soils). Multifractal parameters dimension. However, it has become increasingly evidentdefining the maximum of the f(�)-spectra were correlated to total

that knowledge on the fractal dimension of a set isporosity (P � 0.001), and silt content (P � 0.05). This study demon-insufficient to characterize its geometry (Loehle andstrates that the spatial arrangement of contrasting soil structures canLi, 1996). The fractal dimension, D, characterizes thebe quantified and separated by the properties of their f(�)-spectra.average properties of a set and cannot provide informa-Multifractal parameters quantifying spatial arrangement of soil pores

could be used to improve classifications of soil structure. tion on deviations from the average behavior of a powerlaw. For example, the box-counting technique is usedto estimate D from images of pore systems by coveringthem with a grid of “boxes” of various sizes. The tech-Early attempts to classify the arrangement of poresnique ignores the variations in pore density within aand solids from thin sections led to the develop-box other than categorizing boxes as empty or occupiedment of semi-quantitative classification systems based(Vicsek, 1992). As a result, sets with different appear-mainly in the relationship between fine (i.e., inorganic

and organic soil colloid) and coarse material (Jim, 1988). ance or textures may have similar fractal dimensionsApplication of these systems to the characterization of (Mandelbrot, 1982; Voss, 1988). On the other hand, asoil thin sections have shown that arrangement of soil multifractal analysis captures the inner variations in amaterial is correlated to particle-size distribution, or- system by resolving local densities and expresses themganic matter content, and pedogenic processes (Es- in the shape of a multifractal spectrum (Hentschel andwaran and Banos, 1976; Brewer, 1979; Goenadi and Procaccia, 1983; Frisch and Parisi, 1985; Halsey et al.,Tan, 1989). The development of the general field of 1986; Chhabra et al., 1989; Chhabra and Jensen, 1989).mathematical morphology allowed quantifying the The multifractal concept has been useful in studies ofshape, size, and connectivity of soil pores (Ringrose- spatial arrangement of physical and chemical quantitiesVoase, 1987; Horgan, 1998; Holden, 2001). Size, shape, (Stanley and Meakin, 1988; Feder, 1988; Evertsz andand spatial arrangement of pores have been used to Mandelbrot, 1992; Cheng and Agterberg, 1996), turbu-classify soil structure (Ringrose-Voase and Bullock, lence (Meneveau and Sreenivasan, 1991), and geology1984; Pagliai and De Nobili, 1993). Pore shape and size (Muller and McCauley, 1992; Cheng, 1999). In soil sci-have been related to water flow, pedogenetic processes, ence, multifractal techniques have been applied to the

characterization of particle- and pore-size distributions(Grout et al., 1998; Caniego et al., 2001; Posadas etA.N.D. Posadas, International Potato Center (CIP), P.O. Box 1558,al., 2001; Martın and Montero, 2002), surface strengthLima 12– Peru and Universidad Mayor de San Marcos, FCF-DAFI,

P.O. Box 10584, Lima 1- Peru; D. Gimenez, Dep. of Environmental (Folorunso et al., 1994), and spatial variability of soilSciences, Rutgers Univ., 14 College Farm Road, New Brunswick, NJ properties (Kravchenko et al., 1999). The only published08901; R. Quiroz, International Potato Center (CIP), P.O. Box 1558, report on the application of a multifractal method toLima 12– Peru,; R. Protz (deceased), Dep. of Land Resource Science,

soil pores is an analysis of the histogram of pore areaUniv. of Guelph, Guelph, ON, Canada. Received 27 Aug. 2003. *Cor-responding author ([email protected]).

Abbreviations: MAL, mean area of the lower one-half of cumulativePublished in Soil Sci. Soc. Am. J. 67:1361–1369 (2003). Soil Science Society of America pore-area distributions; MAU, mean area of the upper one-half of

cumulative pore-area distributions.677 S. Segoe Rd., Madison, WI 53711 USA

1361

1362 SOIL SCI. SOC. AM. J., VOL. 67, SEPTEMBER–OCTOBER 2003

Table 1. Soil classification and selected properties of the studied soil horizons.

HorizonOrganic

Group Soil Soil classification† Type Depth matter Sand Silt Clay

cm %1 1 Spodosol C 65–73 0.1 54.0 15.0 31.0

2 Argiaquic Argialboll B 34–40 0.4 3.1 66.2 25.13 Typic Haplorthox B 30–38 1.2 20.0 22.0 58.04 Typic Haplorthox B 70–78 0.7 22.0 6.0 72.05 Inceptisol C 20–25 0.3 94.6 4.9 0.4

2 6 Orthoxic Tropudult B 33–40 0.3 48.0 21.0 31.07 Mollic Albaqualf B 47–52 0.4 2.7 67.1 30.28 Petric Plinthudult A 1–8 1.7 12.0 51.0 37.0

3 9 Entisol BC 20–32 1.3 2.7 63.5 33.810 Entisol B 6–14 1.1 1.4 65.2 33.4

† According to the soil taxonomy.

that did not consider the spatial arrangement of pores Pi(L) � L�i [4](Caniego et al., 2001).

where �i is the Lipschitz-Holder exponent characterizing scal-A multifractal analysis done on binary images hasing in the ith region or spatial location (Halsey et al., 1986).shown that spatial patterns of pores in rocks are multi- In our case, these exponents reflect the local behavior of the

fractal (Muller and McCauley, 1992). This conclusion measure Pi (L) around the center of a box with diametercannot be extrapolated to soil pore systems because of L, and can be estimated from Eq. [4] as �i � log Pi (L)/the differences in genesis between rocks and soils. Thus, log(L)(Fig. 1d and 1e). Note that similar �i values are foundthe objective of this paper was to apply a multifractal at different positions in an image. The number of boxes N(�)

where the probability Pi has exponent values between � andmethod to binary images of soil pores representing con-� � d� is found to scale as (Chhabra et al., 1989; Halsey ettrasting soil structures. This approach has the advantageal., 1986):of integrating the spatial properties of pore systems.

N(�) � L�f(�) [5]THEORY

where f(�) can be defined as the fractal dimension of the setFractal dimensions offer a systematic approach to quantify- of boxes with exponent �. Equation [5] generalizes Eq. [1] by

ing irregular patterns that contain an internal structure re- including several indices to quantify the scaling of a system.peated over a range of scales (Meakin, 1991). For a fractal Multifractal measures can also be characterized throughobject the number of features of a certain size ε, N(ε), varies as: the scaling of the qth moments of Pi distributions in the form

(Chhabra et al., 1989; Korvin, 1992):N(ε) � ε�D [1]

where D is the fractal dimension. Equation [1] is a scaling (or �N(L)

i�1

P qi (L) � L(q�1)Dq [6]

power) law that has been shown to describe the size distribu-tion of many objects in nature. The box-counting technique

where Dq are the generalized fractal dimensions defined fromis used to obtain the scaling properties of two-dimensionalEq. [6] as:fractal objects by covering a measure with boxes of size L and

counting the number of boxes containing at least one pixelrepresenting the object under study, N(L):

Dq � limL→0

1q � 1

log �N(L)

i�1

P qi (L)

logL[7]

D0 � limL→0

logN(L)log(1/L)

[2]The exponent in Eq. [6] is known as the mass exponent of

the qth order moment, �(q) (Halsey et al., 1986; Vicsek, 1992):Using Eq. [2], the box-counting dimension D0 can be deter-mined as the negative slope of log N(L) versus log(L) mea-

�(q) � (q � 1) Dq [8]sured over a range of box sizes. The disadvantage of the box-counting technique is that the process does not consider the From Eq. [7] we see than when q � 0 all the boxes haveamount of mass inside a box Ni (L) and is, therefore, not able a weight of unity, the numerator becomes N(L), and Dq be-to resolve regions with high or low density of mass. Multifractal comes the capacity dimension, D0 (Eq. [2]). Similarly, whenmethods are suited for characterizing complex spatial arrange- all the boxes have the same probability, that is, Pi �1/N, Dq �ment of mass because they can resolve local densities (Vicsek, D0 for all values of q and �(q) becomes a linear function of1992). In practice, a way to quantify local densities is by esti- q (homogeneous fractal). Two other special cases are for q � 1mating the mass probability in the ith box as: and q � 2. The values D1 and D2 are known as the entropy

dimension and the correlation dimension, respectively. ThePi(L) � Ni(L)/NT [3] entropy dimension is related to the information entropy ofShannon and Weaver (1949), which quantifies the decrease inwhere Ni(L) is the number of pixels containing mass in the

ith box and NT is the total mass of the system. Examples of information as the size of the boxes increases. The correlationdimension D2 is mathematically associated with the correlationspatial patterns of Pi (L � 10) and Pi (L � 50) for Soil 7

(Fig. 4, Table 1) are shown in Fig. 1b and 1c, respectively (L function and computes the correlation of measures containedin a box of size L.is expressed in pixels). Also important is to quantify the scaling

(or dependence) of Pi with box size L. For heterogeneous The connection between the power exponents f(�) (Eq. [5])and �(q) (Eq. [8]) is made via the Legendre transformationor non-uniform systems the probability in the ith box Pi(L)

varies as: (Callen, 1985; Halsey et al., 1986; Chhabra and Jensen, 1989):

POSADAS ET AL.: MULTIFRACTAL CHARACTERIZATION OF SOIL PORE SYSTEMS 1363

Fig. 1. Illustration of multifractal theory applied to a binary image. (a) binary image of Soil 7 (500 � 750 pixels), spatial pattern of probabilitiesPi(L ) calculated with Eq. [3] using (b) L � 10 and (c) L � 50 pixels, and spatial pattern of the exponent �i estimated with Eq. [4] using (d)L � 10 and (e) L � 50 pixels.

dure outlined in VandenBygaart and Protz (1999). Imagesf [�(q)] � q�(q) � �(q) [9a]had a size of 500 � 750 pixels, with a pixel size of 74.2 m.

and Distributions of pore area, A, and perimeter, P, were calcu-lated from each image using NIH-Image software (Rasband,

�(q) �d�(q)

dq[9b] 1993). Pore area was used to calculate cumulative distribu-

tions. Mean pore areas of the lower one-half of a distribution,MAL, and upper one-half of a distribution, MAU, were calcu-The function f(�) is concave downward with a maximum atlated. Pore shapes were estimated by calculating a shape index,q � 0. In natural systems, � and f(�) are not evaluated at theF, for pores with pore area larger than 0.27 � 106 m2 orlimit L → 0, but rather in the scaling region in which � and50 pixels2:f(�) can be described as powers of L, which also restricts the

range of q values that can be used.

F �4A

P2[10]

MATERIALS AND METHODSThree basic pore shapes were defined based on the valuesTen thin sections of soils with contrasting soil structure and

F: planar when F � 0.2, irregular when 0.2 � F � 0.5, andsoil texture were selected for this study (Table 1). Binaryimages of soil thin sections were produced following the proce- rounded when F � 0.5 (Bouma et al., 1977). For each image,


Fig. 3. Example of f(q ) and �(q ) functions estimated in the range ofq values in which the numerators of Eq. [12] and [13] were linearwith log L (see Fig. 2).

�(q) � limL→0

�N(L)

i�1

i(q,L) log[Pi(L)]

logL[13]

For each q, values of f(q) and �(q) were obtained fromthe slope of plots of the numerators of Eq. [12] and [13] vs.log L over the entire range of L values considered (2–250pixels). The range of q values over which both functions werelinear, �q, was selected considering the coefficients of determi-nation (R2) of both fits (Fig. 2). The f(q) and �(q) functionsobtained over a given �q (Fig. 3), were used to construct thef(�)-spectra as an implicit function of q and L. In addition, we

Fig. 2. Examples of application of (a) Eq. [12], and (b) Eq. [13] to tested the validity of the results by verifying that the tangent ofthe binary image of Soil 7 for selected q values. Values of f(q )the graph f(�) vs. � at �(q � 1) is the bisector defined byand �(q ) were obtained from the slope of plots similar to thosedf(�)/d� � q. The point of intersection corresponds toin (a) and in (b), respectively. The plot of �(q � �1.0) illustratesf[�(1)] � �[1] � D1 (Evertsz and Mandelbrot, 1992). Thedata that resulted in R2 � 0.88, one of the lowest found in thissymmetry of multifractal spectra was evaluated by comparingstudy (R2 for the rest of the plots can be found in Table 3).the width of the spectra from their center [�(0)] to �(|qi|).Values of |qi| were the same in both the positive and negativedomains and equal to the smaller of the two defining a �q in-the percentage of total porosity falling in each one of the poreterval.shape categories was determined.

The method developed by Chhabra and Jensen (1989) wasimplemented in MatLab 6.0 (The Math Works Inc., Natick, RESULTS AND DISCUSSIONMA) and used to calculate the f(�)-spectra. Images were parti-tioned in boxes of size L, for L � 2, 3, 5, 10, 25, 50, 100, 125, Thin sections for this study were primarily selectedand 250 pixels. A family of normalized measures i(q,L) was to include contrasting soil structures (Fig. 4). Soil textureconstructed for positive and negative values of q covering among selected soils showed wide ranges in the percent-variable ranges in steps of 0.1: ages of sand (1.4–94.6%), silt (4.9–67.1%), and clay (0.4–

72%). Organic matter content was relatively low (0.1–i(q,L) �

P qi (L)

�N(L)

i�1

P qi (L)

[11] 1.7%) because mainly subsurface horizons were sampled(Table 1). In this study, distributions of cumulative porearea ranged from relatively balanced distributions to

where Pi(L) is the fraction (or probability) of pores contained bimodal ones with few pores concentrating relativelyin each ith box of size L (Eq. [3]). Note that for any value oflarge percentage of the porosity (Fig. 5). Formation ofq, the normalized measures take values in the interval [0,1].soil structure implies development of a bimodal pore-The direct computation of f(q) values is (Chhabra et al., 1989;size distribution with the largest pores (macropores)Chhabra and Jensen, 1989):being found between structural units. The largest vari-ability in the distribution of pore area across soils oc-curred in the macropore size, as indicated by the widef(q) � lim

L→0

�N(L)

i�1

i(q,L)log[i(q,L)]

logL[12]

range of variation in the values of the MAU (Table 2).Typically, macropores resulting from the formation ofIn addition, values of �(q) were computed by evaluating:


Fig. 5. Selected cumulative distributions of pore area measured onbinary images.

to Soil 5) have relatively higher porosities and smallerMAU values than soils in Group 2 (Soil 6 to Soil 8), butin both groups elongated pores concentrate the largestproportion of soil porosity (Table 2). On the other hand,soils in Group 3 (Soil 9 and Soil 10) have the lowestporosity and smallest values of MAU. Pore shapes inthis group were predominately irregular and rounded(Table 2). Rounded pores are typically the result ofrandom packing of particles or aggregates, whereas ir-regular voids may originate from the compaction of

Fig. 4. Binary images of the studied soil thin sections separated in rounded pores or from biological activity (Ringrose-groups of similar pore properties. Voase, 1987; Pagliai and De Nobili, 1993). A visual

assessment of soil structure shows that structural unitsin Group 1 were smaller than in Group 2, but in bothcases more developed than the two samples in Groupstructural units are elongated in shape and intercon-3 that exhibited massive structure characterized by anected (Bouma et al., 1977; Ringrose-Voase and Bul-coherent mass and the absence of structural unitslock, 1984). Except for Soil 9 and Soil 10, between 50(Fig. 4).and 74% of the total porosity contributed by pores with

area larger than 0.27 � 106 m2 was formed by elongated Multifractal Analysispores suggesting that pedogenetic processes were activein these soils (Table 2). A crucial step in multifractal analysis is to determine

the range of both L and (negative and positive) mo-Three groups of soils were distinguished using thevalues of porosity and MAU. Soils in Group 1 (Soil 1 ments of order q over which a multifractal method is

Table 2. Total porosity, mean pore area of the lower (MAL) and upper (MAU) one-half cumulative distribution of this property, andpore shape classes (expressed as percentage of total porosity) measured from binary images.

Pore-shape classes†

F � 0.2 0.2 � F � 0.5 F � 0.5Total MAL MAU

Group Soil porosity 106�m2 106�m2 % Total porosity

1 1 0.17 0.051 5.049 50.2 18.6 1.72 0.15 0.046 3.832 55.2 11.4 2.33 0.20 0.068 3.204 49.8 23.8 1.94 0.16 0.037 0.899 34.9 16.9 2.45 0.21 0.066 1.770 51.7 17.1 1.2

2 6 0.09 0.053 8.608 61.6 13.3 3.07 0.13 0.054 77.646 73.5 3.35 2.88 0.13 0.083 12.699 67.4 9.5 4.2

3 9 0.06 0.025 0.345 11.7 18.3 2.410 0.06 0.034 0.692 13.2 28.1 8.8

† Only pores with area � 0.27 � 106 �m2 are included.


Table 3. Selected multifractal parameters† � the residual error of the estimates from the analysis of binary images. Also shown are thevalues of the coefficients of determination of the fits (R2).

�q

Group Soil q� q� D0 R2 D1 R2 f [�(�1)] R2 �(0) R2 �max R2 �min R2

1 1 �1.4 2.3 1.75 � 0.03 0.99 1.70 � 0.09 0.99 1.70 � 0.18 0.99 1.81 � 0.04 0.98 1.94 � 1.30 0.88 1.67 � 0.05 0.992 �2.5 2.8 1.73 � 0.04 0.99 1.67 � 0.02 0.99 1.66 � 0.10 0.99 1.80 � 0.05 0.98 2.01 � 1.40 0.87 1.60 � 0.09 0.993 �1.4 3.5 1.79 � 0.03 0.99 1.73 � 0.02 0.99 1.73 � 0.10 0.99 1.85 � 0.04 0.98 2.00 � 1.30 0.90 1.67 � 0.08 0.994 �2.0 3.0 1.78 � 0.03 0.99 1.72 � 0.02 0.99 1.71 � 0.10 0.99 1.84 � 0.04 0.98 2.03 � 1.00 0.92 1.65 � 0.06 0.995 �1.3 5.0 1.82 � 0.03 0.99 1.77 � 0.02 0.99 1.74 � 0.10 0.99 1.88 � 0.04 0.99 2.06 � 1.00 0.93 1.66 � 0.10 0.99

2 6 �1.0 1.3 1.58 � 0.05 0.98 1.56 � 0.08 0.99 1.57 � 0.10 0.99 1.61 � 0.06 0.95 1.65 � 1.30 0.85 1.56 � 0.05 0.997 �1.3 1.0 1.64 � 0.04 0.98 1.63 � 0.07 0.99 1.62 � 0.12 0.99 1.67 � 0.06 0.96 1.74 � 1.20 0.87 1.63 � 0.07 0.998 �1.0 1.4 1.65 � 0.04 0.99 1.63 � 0.09 0.99 1.64 � 0.16 0.99 1.68 � 0.05 0.97 1.72 � 1.30 0.85 1.63 � 0.02 0.99

3 9 �1.0 2.4 1.53 � 0.05 0.98 1.49 � 0.03 0.99 1.51 � 0.17 0.99 1.59 � 0.06 0.97 1.65 � 1.10 0.85 1.44 � 0.05 0.9910 �2.0 4.0 1.56 � 0.04 0.98 1.50 � 0.04 0.99 1.52 � 0.12 0.99 1.63 � 0.06 0.96 1.77 � 1.40 0.87 1.36 � 0.16 0.99

† D0, D1, and f [�(�1)] are fractal dimensions at q � 0, q � 1, and q � �1, respectively; �(0), �max, and �min are the values of the Lipschitz-holder exponentat q � 0, and at the most negative (q�) and most positive (q�) values that defined the range �q over which the multifractal method was applied.

applicable (Saucier and Muller, 1999). In our case, this and f[�(�1)]. In the � axis, the difference (�max � �min)is used as an indication of the heterogeneity of a system.meant determining the range of L and q in which the

numerators of Eq. [12] and [13] are linear functions of In our analysis, porosity and D0 values were highlycorrelated (Table 4). These results are similar to thelog L. A constant L range (2–250 pixels) was used in

this study to avoid representing multifractal properties ones reported by Lipiec et al. (1998), who correlatedbox-counting dimensions of pore volume with porosityover a small interval of scales. The range of q values

was selected considering the coefficients of determina- for a soil compacted at various levels (R � 0.97). Corre-lation between D0 and porosity provides no informationtion (R2) of the fits (see Fig. 2), which in general were

equal or larger than 0.95 (Table 3). In a few cases, lower on the scaling properties of a pore structure becausecalculation of D0 assumes a homogeneous soil structurevalues of R2 were accepted when the relatively poorer

fit involved only one of the two functions (Fig. 2). The (at every partition level, all boxes have the same proba-bility). On the other hand, �(q) can distinguish amonglargest variation in �q was observed in the range of

positive q values (Table 3). Moments q � 0 magnify soil structures by quantifying the average scaling of massdensity (or probability) with L. Porosity and �(0) werethe contribution of boxes with high concentration of

pores to the estimates of either f(q) or �(q). The oppo- also highly correlated (Table 4), but a comparison be-tween D0 and �(0) values reveal differences in poresite is true for q � 0. Soils in Group 1 and Group 3

exhibited multifractal properties over a wider range of structure among soils. Soils in Group 2 exhibited themost homogeneous pore system as demonstrated by themoments q than soils in Group 2 (Table 3). Since soils

in Group 2 have larger pores than soils in Group 1 and proximity of their values to the 1:1 line (Fig. 7a). Thistendency to homogeneity (or simple scaling) in soils ofGroup 3, it is possible that a complete characterization

of pore scaling in soils of Group 2 would require sam- Group 2 is further confirmed by the notably smallervalues of the differences D0 � D1, D0 � f [�(�1)], andpling an area larger than the provided by a thin section.

Analyses of local porosity and of the mean entropy of (�max� �min) for soils in this group (Table 3). The implica-tion of this finding is that pore systems in Group 2 soilsa soil structure have shown that, in general, the size of

thin sections is enough for an appropriate representa- can be characterized almost entirely by the capacitydimension D0. This interpretation needs to be weightedtion of the lower moments of porosity distributions

(Dexter, 1977; VandenBygaart and Protz, 1999), but by considering that soils in this group scaled over anarrow range of q values (Table 3), which might indicatethere are no studies on the relationship between sam-

pled area and the higher moments of a measure of po- that in the presence of macropores sampling area shouldbe larger than the one used here. This hypothesis isrosity.

The f(�)-spectra among groups showed distinctively supported by the negative correlation between log ofMAU and the largest value of positive q over which thedifferent shape and symmetry (Fig. 6). The curvature

and the symmetry of the f(�)-spectra provide informa- multifractal method applies (Table 4).In a symmetric spectrum, the widths from �(0) totion on the heterogeneity of a system defined by the

diversity of scaling exponents needed to characterize it. �(|qi|) are equal (or very similar) implying that regionswith high and low concentrations of mass scale similarly.The f(�)-spectra of homogeneous systems with fractal

support is reduced to a single point corresponding to Soils in Group 1 and Group 2 had f(�)-spectra that weremore symmetrical than spectra from soils in Group 3the maximum value of f(�) found at q � 0, that is,

f[�(0)] � D0. Heterogeneity can be assessed at q � 0 as indicated by the proximity of the �-intervals to the1:1 line (Fig. 7b). The left-hand side of the f(�)-spectraby the magnitude of the differences in the values of D0

and �(0), or more generally, by the magnitude of of soils in Group 3 exhibited the lowest f(�) � � valuesin this region of the spectrum. Fractal dimensions withchanges around D0 in both the f(�) and � axes. Values

to the right and left of D0 represent negative and positive values similar to the observed f(�) values have beenobtained in analyses of dye-stained patterns, which areq values, respectively (see Fig. 3). In the f(�) axis, com-

parison is between D0 and the values of f[�(1)] or (D1) typically disconnected in the plane perpendicular to flow


(Table 3 and Fig. 6). On the other hand, several studieshave demonstrated relationships between textural com-position and soil microstructure (Eswaran and Banos,1976; Brewer, 1979). If general relationships betweenmultifractal parameters of soil structure and soil proper-ties are established, it could serve as a first approxima-tion to the characterization and modeling of soil poresystems. In our study silt content was significantly corre-lated to D0, �(0), and D1 (Table 4). These parametersrepresent the behavior at or around the maximum pointof the f(�)-spectra, and the correlations could be theresult of silt (and to a lesser extent sand) content beingcorrelated to total porosity. Even though more researchis needed to understand the mechanisms underlyingthese correlations, they illustrate the potential for devel-oping characterization schemes of soil structure thatincludes soil texture as a general framework (Eswaranand Banos, 1976; Brewer, 1979). More accurate charac-terization of relationships between texture and soilstructure may need to involve the parameterization ofthe particle-size distribution (Posadas et al., 2001).

CONCLUSIONSThe multifractal f(�)-spectra successfully separated

contrasting soil structural types. Three soil groups, dis-tinguished based on size and shape of pores, resultedin f(�)-spectra that were distinctively different. Proper-ties of the f(�)-spectra important to distinguish amonggroups were: (i) the range of negative and positive qvalues over which scaling was verified, and (ii) the shapeand symmetry of the spectra. Soil pore structure con-taining clusters of relatively large pores exhibited scalingover a reduced range of moment q and relatively “flat”f(�)-spectra, which might have resulted from a limitedsampling area in relation to the size of the largest pores.Asymmetrical f(�)-spectra were obtained in samples ofmassive structure characterized by low porosity domi-nated by small rounded and irregular pores and a matrixwithout noticeable structural units. The lowest valuesof � and f(�) obtained in this group suggest a poresystem with disconnected pores and low spatial density.Symmetric f(�)-spectra exhibiting the typical concavedownward shape were obtained with soil structures thatcombined small structural units and relatively smallpores.

The advantage of multifractal techniques is that af(�)-spectrum integrates and quantifies spatial proper-ties of a pore system. Conceivably, multifractal para-

Fig. 6. f(�)-spectra for soils in (a) Group 1, (b) Group 2, and (c) meters could be used to improve classifications of soilGroup 3, as shown in Fig. 4. structure. Thus, an approximation to the shape and

properties of a spectrum could be useful as an approxi-mation to pore structure. The relationship between siltcontent and multifractal parameters found in this study(Lipiec et al., 1998). The absence of structural unitsindicate the potential for such approach, but a system-(massive structure) and pore clustering is also evidentatic and more extensive analysis of images of soil sec-in the lowest � values obtained for these two soils.tions representing various levels of the soil taxonomyIt is generally recognized that particle- and pore-sizeis needed to identify and interpret parameters from thedistributions are not well correlated in structured soils.f(�)-spectrum and their relation to soil properties andFor instance, Soil 7 and Soil 9 have very similar contents

of sand, silt, and clay (Table 1), but different f(�)-spectra processes.


Table 4. Correlation coefficients between selected multifractal parameters and pore properties measured in thin sections (total porosity,log MAL, and log MAU) or on different soil samples (organic matter, sand, silt, and clay contents).

Multifractalparameters‡ Total porosity log MAL log MAU Organic matter Sand Silt Clay

D0 0.940*** 0.493 0.039 �0.335 0.520 �0.669* 0.090D1 0.987*** 0.613† 0.196 �0.364 0.547 �0.669* 0.050�(0) 0.940*** 0.387 �0.079 �0.321 0.492 �0.641* 0.086�max 0.844** 0.189 �0.225 �0.323 0.403 �0.544 0.084�min 0.505 0.062 �0.016 �0.666 0.351 �0.438 0.045q� �0.118 0.446 0.350 0.139 0.281 �0.147 �0.181q� 0.368 �0.237 �0.715* 0.022 0.407 �0.342 �0.181

* Significant at the 0.05 probability level.** Significant at the 0.01 probability level.*** Significant at the 0.001 probability level.† Significant at the 0.1 probability level.‡ D0 and D1 are fractal dimensions at q � 0 and q � 1, respectively; �(0), �max, and �min are the values of the Lipschitz-holder exponent at q � 0, and at

the most negative (q�) and most positive (q�) values that defined the range �q over which the multifractal method was applied.

Fig. 7. Plots of (a) �(0) vs. D0 and (b) widths of �-intervals: �(0) � �(qi,�) vs. �(qi,�) � �(0) for soils in the three groups considered. Thenumerical values of qi,� and qi,� were the same and equal to the smallest of the two values defining a �q interval (see Table 3). Proximityto the 1:1 line implies (a) more homogeneous distributions, or (b) more symmetric f(�)-spectra. In plot (b) two soils of Group 2 had identicalvalues and show as one point in the graph.


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