Multifractal Characterization of Soil Pore Systems
Adolfo N.D.Posadas,Daniel Gime´nez,*Roberto Quiroz,and Richard Protz
ABSTRACT and biological activity(Bouma et al.,1977;Ringro-
Voa,1987;Pagliai and De Nobili,1993).The lection Spatial arrangement of soil pores determines soil structure and is
and interpretation of the morphological parameters of important to model soil process.Geometric properties of individual
soil pores that best characterize soil structure,however, pores can be estimated from thin ctions,but there is no satisfactory
method to quantify the complexity of their spatial arrangement.The is still a subject of rearch(Droogers et al.,1998; objective of this work was to apply a multifractal technique to quantify Holden,2001).Statistical methods of characterization properties of ten contrasting soil pore systems.Binary images(500of pore-solid arrangement from images emphasize the by750pixels,74.2m pixelϪ1)were obtained from thin ctions and spatial structure of pores with the
advantage of being analyzed to obtain f(␣)spectra.Pore area and pore perimeter were more amenable to modeling soil structure and process measured from each image and ud to estimate a shape factor for(Dexter1976,1977;Moran and McBratney,1997;Gar-pores with area larger than0.27ϫ106m2.Mean area of the lower boczi et al.,1999;Horgan,1999;Vogel and Roth,2001). (MA L)and upper(MA U)one-half of cumulative pore area distribu-
Among the statistical methods ud to characterize tions were calculated.Pore structures with large(MA UϾ10ϫ
soil structure,fractal techniques are relatively common 106m2)and elongated pores exhibited“flat”f(␣)-spectra typical of
in soil science(Anderson et al.,1996;Pachepsky et al., homogenous systems(three soils).Massive type structure with small
(MA UϽ1ϫ106m2)rounded and irregular pores resulted in asym-1996;Gime´nez et al.,1997).Fractal geometry assumes metric f(␣)-spectra(two soils).Well defined and symmetric f(␣)-that the dependence of the properties of a system with spectra were obtained with soil structures having elongated pores of scale(scaling)can be reprented with a power law, intermediate size(1ϫ106ϽMA
UϽ10ϫ106m2)clustered around with the exponent being typically a function of a fractal relatively small structural units(five soils).Multifractal parameters dimension.However,it has become increasingly evident
defining the maximum of the f(␣)-spectra were correlated to total
that knowledge on the fractal dimension of a t is porosity(PϽ0.001),and silt content(PϽ0.05).This study demon-
insufficient to characterize its geometry(Loehle and strates that the spatial arrangement of contrasting soil structures can
Li,1996).The fractal dimension,D,characterizes the be quantified and parated by the properties of their f(␣)-spectra.
average properties of a t and cannot provide informa-Multifractal parameters quantifying spatial arrangement of soil pores
could be ud to improve classifications of soil structure.tion on deviations from the average behavior of a power
law.For example,the box-counting technique is ud
to estimate D from images of pore systems by covering E arly attempts to classify the arrangement of pores nique ignores the variations in pore density within a
them with a grid of“boxes”of various sizes.The tech-and solids from thin ctions led to the develop-
box other than categorizing boxes as empty or occupied ment of mi-quantitative classification systems bad
(Vick,1992).As a result,ts with different appear-mainly in the relationship between ,inorganic
and organic soil colloid)and coar material(Jim,1988).ance or textures may have similar fractal dimensions Application of the systems to the characterization of(Mandelbrot,1982;Voss,1988).On the other hand,a soil thin ctions have shown that arrangement of soil multifractal analysis captures the inner variations in a material is correlated to particle-size distribution,or-system by resolving local densities and express them ganic matter content,and pedogenic process(Es-in the shape of a multifractal spectrum(Hentschel and waran and Banos,1976;Brewer,1979;Goenadi and Procaccia,19
83;Frisch and Parisi,1985;Haly et al., Tan,1989).The development of the general field of1986;Chhabra et al.,1989;Chhabra and Jenn,1989). mathematical morphology allowed quantifying the The multifractal concept has been uful in studies of shape,size,and connectivity of soil pores(Ringro-spatial arrangement of physical and chemical quantities Voa,1987;Horgan,1998;Holden,2001).Size,shape,(Stanley and Meakin,1988;Feder,1988;Evertsz and and spatial arrangement of pores have been ud to Mandelbrot,1992;Cheng and Agterberg,1996),turbu-classify soil structure(Ringro-Voa and Bullock,lence(Meneveau and Sreenivasan,1991),and geology 1984;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 process,ence,multifractal techniques have been applied to the
characterization of particle-and pore-size distributions
(Grout et al.,1998;Caniego et al.,2001;Posadas et A.N.D.Posadas,International Potato Center(CIP),P.O.Box1558,
al.,2001;Martı´n and Montero,2002),surface strength Lima12–Peru´and Universidad Mayor de San Marcos,FCF-DAFI,
P.O.Box10584,Lima1-Peru´;D.Gime´nez,Dep.of Environmental(Folorunso et al.,1994),and spatial variability of soil Sciences,Rutgers Univ.,14College Farm Road,New Brunswick,NJ properties(Kravchenko et al.,1999).The only published 08901;R.Quiroz,International Potato Center(CIP),P.O.Box1558,report on the application of a multifractal method to Lima12–Peru´,;R.Protz(decead),Dep.of Land Resource Science,
soil pores is an analysis of the histogram of pore area Univ.of Guelph,Guelph,ON,Canada.Received27Aug.2003.*Cor-
responding author(gimenez@envsci.rutgers.edu).
Published in Soil Sci.Soc.Am.J.67:1361–1369(2003).
Abbreviations:MA L,mean area of the lower one-half of cumulative Soil Science Society of America pore-area distributions;MA U,mean area of the upper one-half of 677S.Segoe Rd.,Madison,WI53711USA
cumulative pore-area distributions.
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SOIL SCI.SOC.AM.J.,VOL.67,SEPTEMBER–OCTOBER 2003
Table 1.Soil classification and lected properties of the studied soil horizons.
Horizon
Organic Group Soil Soil classification†Type Depth matter Sand
Silt
Clay
cm %1
1Spodosol
C 65–730.154.015.031.02Argiaquic Argialboll B 34–400.4 3.166.225.13Typic Haplorthox B 30–38 1.220.022.058.04Typic Haplorthox B 70–780.722.0 6.072.05Inceptisol
C 20–250.394.6 4.90.426Orthoxic Tropudult B 33–400.348.021.031.07Mollic Albaqualf B 47–520.4 2.767.130.28Petric Plinthudult A 1–8 1.712.051.037.03
9Entisol BC 20–32 1.3 2.763.533.810
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 P i (L )ෂL ␣i
[4]
(Caniego et al.,2001).
where ␣i is the Lipschitz-Holder exponent characterizing scal-A multifractal analysis done on binary images has ing in the i th region or spatial location (Haly et al.,1986).shown that spatial patterns of pores in rocks are multi-In our ca,the exponents reflect the local behavior of the fractal (Muller and McCauley,1992).This conclusion measure P i (L )around the center of a box with diameter cannot be extrapolated to soil pore systems becau of L ,and can be estimated from Eq.[4]as ␣i ϭlog P i (L )/the differences in genesis between rocks and soils.Thus,log(L )(Fig.1d and 1e).Note that similar ␣i values are found the objective of this paper was to apply a multifractal at different positions in an image.The number of boxes N (␣)where the probability P i has exponent values between ␣and method to binary images of soil pores reprenting con-␣ϩd ␣is found to scale as (Chhabra et al.,1989;Haly et trasting soil structures.This approach has the advantage al.,1986):
数学家的故事of integrating the spatial properties of pore systems.
N (␣)ෂL Ϫf (␣)
[5]
THEORY
where f (␣)can be defined as the fractal dimension of the t Fractal 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 veral 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 through object the number of features of a certain size ε,N (ε),varies as:
the scaling of the q th moments of P i distributions in the form (Chhabra et al.,1989;Korvin,1992):
N (ε)ෂεϪD
[1]
where D is the fractal dimension.Equation [1]is a scaling (or ͚
N (L )i ϭ1
P q i (L )ϭL (q Ϫ1)D q [6]
power)law that has been shown to describe the size distribu-tion of many objects in nature.The box-counting technique where D q are the generalized fractal dimensions defined from is ud to obtain the scaling properties of two-dimensional Eq.[6]as:
fractal objects by covering a measure with boxes of size L and counting the number of boxes containing at least one pixel reprenting the object under study,N (L ):
D q ϭlim
L →0
1q Ϫ1
log ͚N (L )
i ϭ1
P q i (L )
log L
[7]
D 0ϭlim
L →0log N (L )
log(1/L )
[2]
The exponent in Eq.[6]is known as the mass exponent of the q th order moment,(q )(Haly et al.,1986;Vick,1992):
Using Eq.[2],the box-counting dimension D 0can be deter-mined as the negative slope of log N (L )versus log(L )mea-(q )ϭ(q Ϫ1)D q
[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 e than when q ϭ0all the boxes have amount of mass inside a
box N i (L )and is,therefore,not able a weight of unity,the numerator becomes N (L ),and D q be-to resolve regions with high or low density of mass.Multifractal comes the capacity dimension,D 0(Eq.[2]).Similarly,when methods are suited for characterizing complex spatial arrange-all the boxes have the same probability,that is,P i ϭ1/N,D q ϭment of mass becau they can resolve local densities (Vick,D 0for all values of q and (q )becomes a linear function of 1992).In practice,a way to quantify local densities is by esti-q (homogeneous fractal).Two other special cas are for q ϭ1mating the mass probability in the i th box as:
and q ϭ2.The values D 1and D 2are known as the entropy dimension and the correlation dimension,respectively.The P i (L )ϭN i (L )/N T [3]
entropy dimension is related to the information entropy of Shannon and Weaver (1949),which quantifies the decrea in where N i (L )is the number of pixels containing mass in the i th box and N T is the total mass of the system.Examples of information as the size of the boxes increas.The correlation dimension D 2is mathematically associated with the correlation spatial patterns of P i (L ϭ10)and P i (L ϭ50)for Soil 7(Fig.4,Table 1)are shown in Fig.1b and 1c,respectively (L function and computes the correlation of measures contained in a box of size L .
is expresd in pixels).Also important is to quantify the scaling (or dependence)of P i with box size L .For heterogeneous The connection between the power exponents f (␣)(Eq.[5])and (q )(Eq.[8])is made via the Legendre transformation or non-uniform systems the probability in the i th box P i (L )varies as:
(Callen,1985;Haly et al.,1986;Chhabra and Jenn,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(500ϫ750pixels),spatial pattern of probabilities P i (L )calculated with Eq.[3]using (b)L ϭ10and (c)L ϭ50pixels,and spatial pattern of the exponent ␣i estimated with Eq.[4]using (d)L ϭ10and (e)L ϭ50pixels.
年会节目速成
dure outlined in VandenBygaart and Protz (1999).Images f [␣(q )]ϭq ␣(q )Ϫ(q )
[9a]
had a size of 500ϫ750pixels,with a pixel size of 74.2m.and
Distributions of pore area,A ,and perimeter,P ,were calcu-lated from each image using NIH-Image software (Rasband,␣(q )ϭ
d (q )d q
[9b]
1993).Pore area was ud to calculate cumulative distribu-tions.Mean pore areas of the lower one-half of a distribution,MA L ,and upper one-half of a distribution,MA U ,were calcu-The function f (␣)is c
oncave downward with a maximum at lated.Pore shapes were estimated by calculating a shape index,q ϭ0.In natural systems,␣and f (␣)are not evaluated at the F ,for pores with pore area larger than 0.27ϫ106m 2or limit L →0,but rather in the scaling region in which ␣and 50pixels 2:
f (␣)can be described as powers of L ,which also restricts the range of q values that can be ud.
F ϭ
4A P 2
[10]
MATERIALS AND METHODS
Three basic pore shapes were defined bad on the values Ten thin ctions of soils with contrasting soil structure and F :planar when F Ͻ0.2,irregular when 0.2ϽF Ͻ0.5,and soil texture were lected for this study (Table 1).Binary images of soil thin ctions were produced following the proce-
rounded when F Ͼ0.5(Bouma et al.,1977).For each image,
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SOIL SCI.SOC.AM.J.,VOL.67,SEPTEMBER–OCTOBER
2003
Fig.3.Example of f (q )and ␣(q )functions estimated in the range of q values in which the numerators of Eq.[12]and [13]were linear with log L (e Fig.2).
␣(q )ϭlim
什么斟句酌
L →0
͚
N (L )
i ϭ1
i (q ,L )log[P i (L )]
log L
[13]
For each q ,values of f (q )and ␣(q )were obtained from the 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 value
s over which both functions were linear,⌬q ,was lected considering the coefficients of determi-nation (R 2)of both fits (Fig.2).The f (q )and ␣(q )functions obtained over a given ⌬q (Fig.3),were ud to construct
the f (␣)-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 of the binary image of Soil 7for lected q values.Values of f (q )the graph f (␣)vs.␣at ␣(q ϭ1)is the bictor defined by and ␣(q )were obtained from the slope of plots similar to tho d f (␣)/d ␣ϭq .The point of interction corresponds to in (a)and in (b),respectively.The plot of ␣(q ϭϪ1.0)illustrates f [␣(1)]ϭ␣[1]ϭD 1(Evertsz and Mandelbrot,1992).The data that resulted in R 2ϭ0.88,one of the lowest found in this symmetry of multifractal spectra was evaluated by comparing study (R 2for the rest of the plots can be found in Table 3).
the width of the spectra from their center [␣(0)]to ␣(|q i |).Values of |q i |were the same in both the positive and negative domains and equal to the smaller of the two defining a ⌬q in-the percentage of total porosity falling in each one of the pore terval.
shape categories was determined.
The method developed by Chhabra and Jenn (1989)was implemented in MatLab 6.0(The Math Works Inc.,Natick,RESULTS AND DISCUSSION
MA)and ud to calculate the f (␣)-spectra.Images were parti-tioned in boxes of size L ,for L ϭ2,3,5,10,25,50,100,125,Thin ctions for this study were primarily lected and 250pixels.A family of normalized measures i (q,L )was to include contrasting soil structures (Fig.4).Soil texture constructed for positive and negative values of q covering among lected 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 )ϭ
P q i (L )
͚
N (L )i ϭ1
P q i (L )
[11]
1.7%)becau mainly subsurface horizons were sampled (Table 1).In this study,distributions of cumulative pore area ranged from relatively balanced distributions to where P i (L )is the fraction (or probability)of pores contained bimodal ones with few pores concentrating relatively in each i th box of size L (Eq.[3]).Note that for any value of large percentage of the porosity (Fig.5).Formation of q ,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 Jenn,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 wide f (q )ϭlim
L →0
͚
N (L )
i ϭ1
i (q ,L )log[i (q ,L )]
log L
[12]
range of variation in the values of the MA U (Table 2).Typically,macropores resulting from the formation of
In addition,values of ␣(q )were computed by evaluating:
POSADAS ET AL.:MULTIFRACTAL CHARACTERIZATION OF SOIL PORE SYSTEMS1365
Fig.5.Selected cumulative distributions of pore area measured on
binary images.
to Soil5)have relatively higher porosities and smaller
MA U values than soils in Group2(Soil6to Soil8),but
in both groups elongated pores concentrate the largest
proportion of soil porosity(Table2).On the other hand,
soils in Group3(Soil9and Soil10)have the lowest
porosity and smallest values of MA U.Pore shapes in
this group were predominately irregular and rounded
(Table2).Rounded pores are typically the result of
random packing of particles or aggregates,whereas ir-
regular voids may originate from the compaction of Fig.4.Binary images of the studied soil thin ctions parated in rounded pores or from biological activity(Ringro-groups of similar pore properties.Voa,1987;Pagliai and De Nobili,1993).A visual
asssment of soil structure shows that structural units
in Group1were smaller than in Group2,but in both
cas more developed than the two samples in Group structural units are elongated in shape and intercon-
3that exhibited massive structure characterized by a nected(Bouma et al.,1977;Ringro-Voa and Bul-
coherent mass and the abnce of structural units lock,1984).Except for Soil9and Soil10,between50
(Fig.4).
and74%of the total porosity contributed by pores with六年级体育教案
area larger than0.27ϫ106m2was formed by elongated Multifractal Analysis
pores suggesting that pedogenetic process were active
in the soils(Table2).A crucial step in multifractal analysis is to determine Three groups of soils were distinguished using the
the range of both L and(negative and positive)mo-values of porosity and MA U.Soils in Group1(Soil1ments of order q over which a multifractal method is
Table2.Total porosity,mean pore area of the lower(MA L)and upper(MA U)one-half cumulative distribution of this property,and pore shape class(expresd as percentage of total porosity)measured from binary images.
射雕英雄传人物Pore-shape class†
FϽ0.20.2ϽFϽ0.5FϾ0.5 Total MA L MA U
关于垃圾分类的内容Group Soil porosity106m2106m2%Total porosity
110.170.051 5.04950.218.6 1.7
20.150.046 3.83255.211.4 2.3
30.200.068 3.20449.823.8 1.9
40.160.0370.89934.916.9 2.4
50.210.066 1.77051.717.1 1.2 260.090.0538.60861.613.3 3.0
70.130.05477.64673.5 3.35 2.8
80.130.08312.69967.49.5 4.2 390.060.0250.34511.718.3 2.4 100.060.0340.69213.228.18.8†Only pores with areaϾ0.27ϫ106m2are included.
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SOIL SCI.SOC.AM.J.,VOL.67,SEPTEMBER–OCTOBER 2003
Table 3.Selected multifractal parameters†Ϯthe residual error of the estimates from the analysis of binary images.Also shown are the values of the coefficients of determination of the fits (R 2).
刑天的故事
⌬q
Group Soil q Ϫq ϩD 0R 2D 1R 2f [␣(Ϫ1)]R 2␣(0)R 2␣max R 2␣min R 21
1Ϫ1.4 2.3 1.75Ϯ0.030.99 1.70Ϯ0.090.99 1.70Ϯ0.180.99 1.81Ϯ0.040.98 1.94Ϯ1.300.88 1.67Ϯ0.050.992Ϫ2.5 2.8 1.73Ϯ0.040.99 1.67Ϯ0.020.99 1.66Ϯ0.100.99 1.80Ϯ0.050.98 2.01Ϯ1.400.87 1.60Ϯ0.090.993Ϫ1.4 3.5 1.79Ϯ0.030.99 1.73Ϯ0.020.99 1.73Ϯ0.100.99 1.85Ϯ0.040.98 2.00Ϯ1.300.90 1.67Ϯ0.080.994Ϫ2.0 3.0 1.78Ϯ0.030.99 1.72Ϯ0.020.99 1.71Ϯ0.100.99 1.84Ϯ0.040.98 2.03Ϯ1.000.92 1.65Ϯ0.060.995Ϫ1.3 5.0 1.82Ϯ0.030.99 1.77Ϯ0.020.99 1.74Ϯ0.100.99 1.88Ϯ0.040.99 2.06Ϯ1.000.93 1.66Ϯ0.100.9926Ϫ1.0 1.3 1.58Ϯ0.050.98 1.56Ϯ0.080.99 1.57Ϯ0.100.99 1.61Ϯ0.060.95 1.65Ϯ1.300.85 1.56Ϯ0.050.997Ϫ1.3 1.0 1.64Ϯ0.040.98 1.63Ϯ0.070.99 1.62Ϯ0.120.99 1.67Ϯ0.060.96 1.74Ϯ1.200.87 1.63Ϯ0.070.998Ϫ1.0 1.4 1.65Ϯ0.040.99 1.63Ϯ0.090.99 1.64Ϯ0.160.99 1.68Ϯ0.050.97 1.72Ϯ1.300.85 1.63Ϯ0.020.993
9Ϫ1.0 2.4 1.53Ϯ0.050.98 1.49Ϯ0.030.99 1.51Ϯ0.170.99 1.59Ϯ0.060.97 1.65Ϯ1.100.85 1.44Ϯ0.050.9910
Ϫ2.0
4.0
1.56
Ϯ0.04
0.98
1.50
Ϯ0.04
0.99
1.52
Ϯ0.12
0.99
1.63
Ϯ0.06
0.96
1.77
Ϯ1.40
0.87
1.36
Ϯ0.16
0.99
†D 0,D 1,and f [␣(Ϫ1)]are fractal dimensions at q ϭ0,q ϭ1,and q ϭϪ1,respectively;␣(0),␣max ,and ␣min are the values of the Lipschitz-holder exponent at q ϭ0,and at the most negative (q Ϫ)and most positive (q ϩ)values that defined the range ⌬q over which the multifractal method was applied.
applicable (Saucier and Muller,1999).In our ca,this and f [␣(Ϫ1)].In the ␣axis,the difference (␣max Ϫ␣min )is ud as an indication of the heterogeneity of ant 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 D 0values were highly correlated (Table 4).The results are similar to the log L .A constant L range (
2–250pixels)was ud in this study to avoid reprenting multifractal properties ones reported by Lipiec et al.(1998),who correlated box-counting dimensions of pore volume with porosity over a small interval of scales.The range of q values was lected considering the coefficients of determina-for a soil compacted at various levels (R ϭ0.97).Corre-lation between D 0and porosity provides no information tion (R 2)of the fits (e Fig.2),which in general were equal or larger than 0.95(Table 3).In a few cas,lower on the scaling properties of a pore structure becau calculation of D 0assumes a homogeneous soil structure values of R 2were 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 among largest variation in ⌬q was obrved in the range of positive q values (Table 3).Moments q Ͼ0magnify soil structures by quantifying the average scaling of mass density (or probability)with L .Porosity and ␣(0)were the 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 D 0and ␣(0)values reveal differences in pore site is true for q Ͻ0.Soils in Group 1and Group 3exhibited multifractal properties over a wider range of structure among soils.Soils in Group 2exhibited the most homogeneous pore system as demonstrated by the moments q than soils in Group 2(Table 3).Since soils in Group 2have larger pores than soils in Group 1and proximity of their values to the 1:1line (Fig.7a).This tendency to homo
geneity (or simple scaling)in soils of Group 3,it is possible that a complete characterization of pore scaling in soils of Group 2would require sam-Group 2is further confirmed by the notably smaller values of the differences D 0ϪD 1,D 0Ϫf [␣(Ϫ1)],and pling an area larger than the provided by a thin ction.Analys of local porosity and of the mean entropy of (␣max Ϫ␣min )for soils in this group (Table 3).The implica-tion of this finding is that pore systems in Group 2soils a soil structure have shown that,in general,the size of thin ctions is enough for an appropriate reprenta-can be characterized almost entirely by the capacity dimension D 0.This interpretation needs to be weighted tion of the lower moments of porosity distributions (Dexter,1977;VandenBygaart and Protz,1999),but by considering that soils in this group scaled over a narrow range of q values (Table 3),which might indicate there are no studies on the relationship between sam-pled area and the higher moments of a measure of po-that in the prence of macropores sampling area should be larger than the one ud here.This hypothesis is rosity.
The f (␣)-spectra among groups showed distinctively supported by the negative correlation between log of MA U and the largest value of positive q over which the different shape and symmetry (Fig.6).The curvature and the symmetry of the f (␣)-spectra provide informa-multifractal method applies (Table 4).张继枫桥夜泊
In a symmetric spectrum,the widths from ␣(0)to tion on the heterogeneity of a system defined by the diversity of scaling exponents needed to characterize it.␣(|q i |)are equal (or very similar)implying that regions with 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 1and Group 2had f (␣)-spectra that were more symmetrical than spectra from soils in Group 3the maximum value of f (␣)found at q ϭ0,that is,f [␣(0)]ϭD 0.Heterogeneity can be assd at q ϭ0as indicated by the proximity of the ␣-intervals to the 1:1line (Fig.7b).The left-hand side of the f (␣)-spectra by the magnitude of the differences in the values of D 0and ␣(0),or more generally,by the magnitude of of soils in Group 3exhibited the lowest f (␣)Ϫ␣values in this region of the spectrum.Fractal dimensions with changes around D 0in both the f (␣)and ␣axes.Values to the right and left of D 0reprent negative and positive values similar to the obrved f (␣)values have been obtained in analys of dye-stained patterns,which are q values,respectively (e Fig.3).In the f (␣)axis,com-parison is between D 0and the values of f [␣(1)]or (D 1)typically disconnected in the plane perpendicular to flow
