rényi dimensions of soil pore size distribution

Renyi dimensions of soil pore size distribution

F.J. Caniego*, M.A. Martın, F. San Jose

Dpto. Matematica Aplicada a la Ing. Agronomica, E.T.S.I. Agronomos, Universidad Politecnica de Madrid,

28040 Madrid, Spain

Received 31 October 2001; accepted 14 October 2002

Abstract

The existence of fractal scaling for pore volume-size distribution has been empirically shown, and

several fractal models have been constructed to replicate it, deriving also other important physical

properties.

One step forward may be given by using multifractal analysis, which achieves a better charac-

terization of a distribution by means of a singularity spectrum whenever suitable scaling properties

are present. Renyi dimensions are a classical tool to characterize complex distributions that are

closely related to the multifractal singularity spectrum but having an easier handling and

interpretation.

In this article, Renyi dimensions of pore size distributions are computed from data obtained by 2-D

image analysis of soil samples. Analyzed samples show suitable scaling properties. The Renyi

dimensions Dq appear defined with R2 greater than 0.95 within a significative range of q’s. The

variation ofDqwith respect to q and the shape of the Renyi spectrum reveal that pore size distributions

have properties close to multifractal self-similar measures.

These results show that Renyi dimensional analysis is an appropriate tool for characterizing soil

pore size distribution and thus may be used as indicator of soil structure as well as for deriving

important soil physical properties. Moreover, this characterization opens the possibility of simulating

the real distributions by adequate self-similar models of fractal geometry, as iterated function systems

and related ones, with the corresponding benefits for prediction purposes.

D 2002 Elsevier Science B.V. All rights reserved.

Keywords: Pore size distribution; Multifractal analysis; Renyi dimensions

0016-7061/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.

PII: S0016 -7061 (02 )00307 -5

* Corresponding author. Fax: +34-91-3365817.

E-mail addresses: [email protected] (F.J. Caniego), [email protected] (M.A. Martın),

[email protected] (F. San Jose).

www.elsevier.com/locate/geoderma

Geoderma 112 (2003) 205–216

1. Introduction

Soil physical configuration may be conceptualized as a heterogeneous arrangement of

solid and void components displaying an intricate geometry, which reveals new details

when it is observed at decreasing scales of resolution in the tridimensional space. A great

number of studies (see Anderson et al., 1998 for a review) have shown the existence of

power laws

MðRÞ~RDM PðRÞ~RDP SðRÞ~RDS

for the mass of solid component, volume of porosity or surface interface within a portion

of matrix of characteristic size R. These scaling laws reflect the existence of scale

invariance in the spatial heterogeneity of those solid and void components, appearing

thus a new symmetry called self-similarity. The power scaling laws thus become the by-

products of an irregular, fractal-shaped tridimensional geometry, and the scaling exponent,

called fractal dimension, quantifies the regularity trend in the reproduction of geometric

irregularities along a certain range of scales.

Also, different power scaling laws of the type

Nðx > X Þ~X�Df V ðx < X Þ~X d ð1Þ

have been reported for the number of aggregates or particles with diameter greater than a

characteristic size X and the volume (or mass) of the same elements of size less than X,

respectively (Turcotte, 1986; Perfect and Kay, 1991; Perfect et al., 1992; Tyler and

Wheatcraft, 1992; Anderson et al., 1998). The exponent Df is referred to as the

fragmentation dimension. Although particle or aggregate size distribution has an obvious

influence on the physical configuration of soil as heterogeneous porous medium, there is

not a direct conceptual relation between size distributions and the geometric configuration

of soil that are in fact totally (or partially) destroyed when analyzing them. Thus, those

distributions are, strictus sensus, independent of the way the soil is structured. Above

exponents characterize the scale dependence that takes place for the number of grains

accumulated with respect to the size scale. The ‘‘scaling space’’ is now the interval of size

particles instead of the tridimensional space, as occurs above for M(R), P(R) and S(R).

One step forward in the scaling analysis of soil properties may be given by using

multifractal analysis which characterizes a measure or mass distribution by means of the

singularity spectrum (Mandelbrot, 1974, 1989; Everstz and Mandelbrot, 1992). Now the

considered ‘‘measure’’ is supported in the corresponding scaling space. Multifractal

analysis has been successfully used to characterize spatial variation of pore space in chalk

(Saucier and Muller, 1993; Muller, 1996) by considering the measure v(X) rendering the

pore space volume enclosed in the tridimensional domain X. The singularity spectrum of

this measure adds valuable information of spatial porosity distribution to that obtained by

means of the single scaling P(R)~RDP. Multifractal analysis has been also used to

characterize mineral deposits (Cheng et al., 1994; Agterberg et al., 1996), analysis of

vegetation patterns (Scheuring and Riedi, 1994), spatial variation in soil (Forolunso et al.,

1994; Kravchenko et al., 1999) and grain volume-size soil distributions (Martın and

Montero, in press).

F.J. Caniego et al. / Geoderma 112 (2003) 205–216206

Pore size distributions have been shown to obey scaling rules similar to the above

mentioned for aggregates and soil particles (Diamond, 1970; Katz and Thompson, 1985;

Ahl and Niemeyer, 1989; Bouabid et al., 1992; Pachepsky, 1995; Perrier et al., 1996).

Fractal dimensions computed by Eq. (1) for soil pore size distributions may be seen as a

rough characterization of the respective scaling in the interval of pore sizes. In mathematical

terms, pore size distribution may be considered as a measure or distribution on the interval

of sizes, being the measure of a subinterval the total volume of pores with diameter within

such subinterval. A closer look to different subintervals in the size domain reveals, in

general, existence of denser and rarer regions with respect to the above measure, displaying

a behavior close to multifractal singular measures (Caniego et al., 2001). Standard

statistical parameters and distributions fail in describing such kind of measures (Everstz

and Mandelbrot, 1992), and the global fractal scaling is a rough description of the scale

dependence. Renyi dimensions (Renyi, 1970) are a classical tool to characterize complex

distributions closely related to the multifractal singularity spectrum (Everstz and Mandel-

brot, 1992; Scheuring and Riedi, 1994).

The goal of this work is to evaluate the applicability of Renyi dimensions to pore size

distributions in soil with data obtained by 2-D image analysis of soil samples. Important

benefits may be derived from this characterization. In the same spirit of fractal models

proposed to predict power law expressions for aggregate and soil pore size distributions

(Rieu and Sposito, 1991; Perrier et al., 1999), demonstration of multifractal behavior and its

characterization by means of the Renyi spectrum of dimensions open the possibility of

using powerful techniques of fractal geometry to simulate soil properties and generate

pedotransfer functions. One of these techniques uses iterated function systems (IFS) to

simulate a real distribution for which certain parameters has been previously obtained

(Forte and Vrscay, 1994a,b). Iterated function systems have been already used to simulate

particle size distributions (PSD) in order to infer the entire distribution from a limited

description, consisting of the knowledge of common textural data, i.e. percentages of clay,

silt and sand (Martın and Taguas, 1998; Taguas et al., 1999). Since Renyi dimensions

characterize the type of distributions that IFSs produce (Peitgen et al., 1992), when a proper

characterization through Renyi dimensions has been previously provided for pore size

distributions, an adequate IFS fitting those dimensions may lead to a profitable simulation

of related soil properties.

2. Theory

To implement the scaling analysis of a general mass distribution or measure l supported

on the interval I=[a,b], a set of different meshes with cells or subintervals of I of equal

length is required. A common choice is to consider dyadic scaling down, that is, successive

partitions of I of size e= 2� kL, being L the length of I and k = 0, 1, 2. . .. At each size scale e,a number N(e) = 2k of cells are considered and their respective measure li(e) are found fromthe data.

The number log li/log e is the singularity strength or Holder exponent of the ith cell.

This exponent may be interpreted as a crowding index or a degree of concentration of l: thegreater this value is, the smaller is the concentration of the measure and vice versa.

F.J. Caniego et al. / Geoderma 112 (2003) 205–216 207

Multifractal analysis aims to find scaling parameters or dimensions for characterizing

measures displaying high irregularity or variation of singularity strength values.

The Shannon entropy of the measure defined by

HðeÞ ¼ �XNðeÞ

i¼1

liðeÞlogliðeÞ

is a measure of heterogeneity or unevenness of the measure. When the limit

DE ¼ lime!0

XNðeÞ

i¼1

liðeÞlogliðeÞ

loge

exists, its value is called the entropy dimension or information dimension of the

distribution (Renyi, 1970; Billingsly, 1965), quantifying the growth rate of the entropy

with respect to e. This expression can be seen as l-weighted average of the singularity

strength values. It is also related to the size level (dimension) of the minimal set where the

whole measure is concentrated (Everstz and Mandelbrot, 1992).

Entropy dimension DE is a special case of the Renyi dimensions defined by

Dq ¼1

q� 1lime!0

logXNðeÞ

i¼1

liðeÞq

logeð2Þ

whenever this limit is properly defined.

The dimension D0 is called the capacity dimension, D1 is just the entropy dimension and

D2 is the correlation dimension (see Peitgen et al., 1992 for details). In general, Dq is a

nonincreasing function that is constant in the case of standard smooth distributions. The

greater the variation of Dq with respect to q, the higher the degree of heterogeneity of the

measure. The parameter q provides a way to scan the denser and rarer regions of the support

interval. For qH1, the dimension quantifies the scaling behavior of denser cells, while for

qb� 1, the dimension measures the scaling properties of the rarer ones. When an adequate

scaling behavior takes place for an experimental measure, the spectrum of Renyi dimensions

Dq provides a valuable characterization of the singular behavior of the measure and the

respective interpretation within each context (see Harte, 2001 for further details).

3. Materials and methods

3.1. Soil samples

Soil samples analyzed in the present study were obtained in different experimental fields

at Sistema Central (Spain). Ten samples were taken from forested soils developed over

slates, graywackes and granite in four profiles, one for each of the horizons that were found.


Two of the profiles present three different horizons while the others have only two. Textures

vary from sandy loam to silty loam in the USDA system. Description of the soil samples is

summarized in Table 1.

3.2. Image analysis of thin sections

Soil structure was maintained undisturbed taking samples into standard metal containers.

They were immediately dried during 48 h at 40 jC and impregnated with a resin inside a

void camera for 7 h. The resin was previously mixed with a fluorescent substance and

introduced into the pores by dripping technique which allows filling pores of sizes ranging

from the greatest ones to microscopic scale (Protz et al., 1987). When hard, blocks were cut

into thin sections of 5� 4 cm size and 0.2 mm thickness.

These sections were photographed using a color film and ultraviolet light. Photographs

were scanned in Prodislete to 210 dpi and transferred to Adobe Photoshop program.

Digitalized images were processed by IMAGE 1.55 (developed at the U.S. National

Institutes of Health and available at http://rsb.info.nih.gov/nih-image/). This image-pro-

cessing program evaluates the area of each pore of a given sample whenever that area is

greater than 0.037 mm2 (this corresponds to a circular pore radius of approximately 108

Am). This lower limit of pore size comes imposed by the image technique here followed.

The output of the program is the set of sizes (areas) of the pores of the sample.

3.3. Estimation of Renyi spectrum

For each sample, an experimental measure l is considered whose support is the interval

determined by the extreme values of the list of pore sizes. Binary partitions of the interval I

of pore sizes have been considered with e = 2� kL for k = 1 to 12. For any subinterval

Ii=[ai,bi]so determined, l(Ii) = li is the area fraction of pores with sizes greater than ai and

less than bi. It may be thought about as a measure on the interval of pore sizes. This measure

built over two-dimensional data provides an indirect method of evaluation of the real soil

pore size distribution (Protz et al., 1987; Anderson et al., 1996).

Table 1

Description of soil samples

Soil

sample

Depth

(cm)

Organic

matter (%)

Texture

(USDA)

Coarse

elements (%)

Horizon Taxonomic name

(FAO, 1998)

1 10–20 7.8 Sandy loam 58 A Haplic Umbrisol

2 50–60 3.1 Sandy loam 62 Bw Haplic Umbrisol

3 80–90 1.2 Sandy loam 59 C Haplic Umbrisol

4 10–20 3.2 Sandy loam 42 A Gleyic Umbrisol

5 40–50 1.3 Sandy loam 33 Bw Gleyic Umbrisol

6 80–90 0.3 Sandy loam 25 Cg Gleyic Umbrisol

7 10–20 7.4 Silt loam 41 A Haplic Umbrisol

8 30–40 1.3 Silt loam 45 Bw Haplic Umbrisol

9 10–20 9.8 Loam 38 A Haplic Umbrisol

10 40–50 0.8 Loam 47 Bw Haplic Umbrisol


http:\\rsb.info.nih.gov\nih-image\

The limit in Eq. (2) is seen in applications as an asymptotic scaling behavior, that is

XNðeÞ

i¼1

lqi~eðq�1ÞDq and

XNðeÞ

i¼1

lilogli~D1loge

for q p 1 and for q = 1, respectively. The evaluation of Dq, as other fractal dimensions, is

currently implemented by means of the linear fitting in a log-log plotting for decreasing

values of e (Peitgen et al., 1992). Thus, for q p 1, the dimensions Dq are estimated by least

square fitting of logPNðeÞ

i¼1 lqi against log e modified by the factor 1/( q� 1). Plots of

PNðeÞi¼1

lilogli against log e are used when q = 1. Coefficients of determination R2 of those fits

were obtained. This process is repeated for q ranging from � 4 to 4 in increments of 0.5.

Plots for sample 6 are shown in Fig. 1 for q =� 1, 0, 1, 2.

4. Results and discussion

For the studied samples, the values of Renyi dimensions as well as the corresponding

coefficients of determination R2 are summarized in Tables 2 and 3, respectively.

The values of R2 vary between 0.8 and 0.995. ForD0 andD1, they are greater than 0.928,

showing always better fits for D0. Samples 2, 5, 6 and 7 show coefficients of determination

Fig. 1. Example of plots of fittings for evaluating of Dq for q=� 1, 0, 1, 2.


greater than 0.9 in the whole range of q’s. In the other samples, the range of q’s with R2

greater than such value appears slightly shortened, especially for positive q’s.

The values of D0 range from 0.475 (sample 8) to 0.565 (sample 6). For D1, they vary

between 0.339 (sample 8) and 0.542 (sample 6). In fact, as theory predicts, they are always

Table 3

Coefficients of determination (R2)

q Soil samples

1 2 3 4 5 6 7 8 9 10

� 4 0.951 0.918 0.980 0.987 0.981 0.987 0.944 0.992 0.982 0.988

� 3.5 0.956 0.923 0.980 0.980 0.981 0.988 0.947 0.993 0.983 0.988

� 3 0.963 0.930 0.981 0.982 0.982 0.990 0.952 0.994 0.985 0.988

� 2.5 0.971 0.938 0.982 0.984 0.984 0.991 0.956 0.995 0.986 0.989

� 2 0.981 0.949 0.983 0.987 0.985 0.993 0.962 0.995 0.988 0.989

� 1.5 0.989 0.963 0.984 0.980 0.986 0.993 0.967 0.995 0.989 0.990

� 1 0.993 0.977 0.985 0.990 0.988 0.994 0.972 0.993 0.991 0.991

� 0.5 0.991 0.984 0.985 0.989 0.990 0.992 0.976 0.988 0.992 0.992

0 0.983 0.973 0.985 0.987 0.992 0.990 0.978 0.977 0.992 0.992

0.5 0.968 0.951 0.981 0.980 0.994 0.984 0.977 0.955 0.988 0.988

1 0.947 0.938 0.968 0.965 0.991 0.975 0.971 0.928 0.975 0.973

1.5 0.923 0.936 0.939 0.944 0.980 0.962 0.961 0.909 0.948 0.946

2 0.898 0.935 0.899 0.923 0.965 0.946 0.947 0.897 0.919 0.914

2.5 0.872 0.930 0.862 0.907 0.951 0.932 0.934 0.889 0.897 0.886

3 0.848 0.924 0.834 0.897 0.940 0.920 0.923 0.881 0.883 0.866

3.5 0.827 0.919 0.814 0.890 0.933 0.912 0.915 0.876 0.874 0.852

4 0.810 0.915 0.801 0.885 0.928 0.906 0.909 0.871 0.868 0.843

Table 2

Renyi dimensions (Dq)

q Soil samples

1 2 3 4 5 6 7 8 9 10

� 4 0.732 0.586 0.646 0.577 0.552 0.616 0.560 0.700 0.575 0.591

� 3.5 0.715 0.580 0.635 0.572 0.548 0.607 0.556 0.688 0.569 0.584

� 3 0.695 0.572 0.623 0.566 0.544 0.598 0.551 0.673 0.562 0.576

� 2.5 0.671 0.565 0.610 0.560 0.541 0.589 0.547 0.657 0.555 0.567

� 2 0.643 0.556 0.598 0.553 0.538 0.582 0.542 0.636 0.547 0.557

� 1.5 0.613 0.547 0.585 0.544 0.535 0.575 0.536 0.610 0.538 0.545

� 1 0.588 0.537 0.572 0.533 0.533 0.571 0.529 0.577 0.526 0.532

� 0.5 0.569 0.524 0.556 0.518 0.529 0.569 0.520 0.533 0.510 0.515

0 0.556 0.503 0.534 0.497 0.521 0.565 0.508 0.475 0.486 0.494

0.5 0.544 0.466 0.500 0.469 0.505 0.557 0.492 0.406 0.448 0.465

1 0.529 0.412 0.449 0.435 0.478 0.542 0.472 0.339 0.398 0.425

1.5 0.508 0.350 0.387 0.402 0.444 0.522 0.449 0.286 0.347 0.381

2 0.483 0.300 0.330 0.376 0.412 0.500 0.428 0.251 0.309 0.344

2.5 0.457 0.266 0.289 0.356 0.387 0.482 0.411 0.229 0.284 0.317

3 0.433 0.245 0.261 0.341 0.369 0.467 0.397 0.215 0.267 0.298

3.5 0.413 0.230 0.243 0.330 0.356 0.455 0.387 0.205 0.256 0.285

4 0.397 0.221 0.230 0.321 0.346 0.446 0.380 0.198 0.249 0.276


smaller than the corresponding values of D0. The capacity dimension D0 alludes to the

scaling of the number of cells containing some pore sizes under successive finer partitions,

and then, it provides information about the geometry of the set supporting the pore sizes (non

necessarily the whole interval of sizes). The entropy dimensionD1 gauges the concentration

degree of the distribution of the porosity on the above set. For distributions with a given

value of D0, the maximum possible value of D1 is just D0, which corresponds to the least

heterogeneous distributionwith suchD0. This suggests that the ratioD1/D0may be seen as a

measure of the dispersion of the porosity relative to the dispersion of the pore sizes. The

closer to 1 this ratio is, the more evenly distributed is the porosity over the set of pore

sizes. In this work, the extreme values correspond to sample 8 with D1/D0 = 0.712 and to

sample 6 with D1/D0 = 0.960 (see Table 4). Fig. 2 shows how D1/D0 discriminates

heterogeneity of pore size distributions of studied samples. Similar ratios between

entropy and maximum possible entropy of distributions have been widely used in

Table 4

Dimensions of capacity and entropy, ratio and difference

Soil D0 D1 D1/D0 D0�D1

1 0.556 0.529 0.951 0.027

2 0.503 0.412 0.818 0.092

3 0.534 0.449 0.841 0.085

4 0.497 0.435 0.875 0.062

5 0.521 0.478 0.918 0.043

6 0.565 0.542 0.960 0.023

7 0.508 0.472 0.928 0.037

8 0.475 0.339 0.712 0.137

9 0.486 0.398 0.819 0.088

10 0.494 0.425 0.860 0.069

Fig. 2. Plot of the graphs of the Renyi dimensions for q= 0, 1, the ratio D1/D0 and the difference D0�D1. All the

samples are represented in the x axis.


Fig. 3. Plots of the spectra of Renyi dimensions for all samples.


Ecology as an index of evenness, i.e. an aspect of diversity in the species abundance setting

(Pielou, 1975; Magurran, 1988). Other indexes as D1�D0 have been used in literature; this

difference is shown in Fig. 2 for all samples. However, here the ratio D1/D0 is preferred

since it provides information about the proportional variation instead of the absolute

variation, i.e. same differences D0�D1 have different meaning for dimensions close to 1 or

close to 0, being this fact reflected by the ratio D1/D0. Moreover, D1/D0 has the additional

meaning above mentioned.

Spectra curves (Fig. 3) follow similar trend to those obtained for multinomial measures

(Beck and Schlogl, 1995), showing a significant variation of Dq with respect to q. The

spectrum of sample 1 follows a quasi-linear variation, while, in the opposite, the spectrum

of sample 8 has a nearly sigma shape. In the other cases, spectra follow the same pattern:

on the left of the vertical axis, they are quasi-linear and concave up on the right (although,

it is very attenuated for sample 6). Quasi-linear shape for the negative q’s may be

interpreted in our context as a rather homogeneous distribution of the small concentrations

of porosity along the interval of pore sizes, whereas a concave up shape for positive q’s

would indicate a more heterogeneous distribution of the high concentrations.

Though sample 6 has not a clear quasi-linear shape on the right of the vertical axis, it

shows the smallest overall variation. In fact, the narrowest range of Dq corresponds to

sample 6, and the widest range to sample 8. Moreover, as it has been pointed out before,

the ratio D1/D0 is maximum for sample 6 and minimum for sample 8. It could suggest that

this parameter might be a good candidate in order to evaluate the overall homogeneity.

5. Conclusions

Computing Renyi dimensions of soil pore size distributions requires using new tech-

niques, as image analysis, to provide the amount of data needed to obtain a significative

spectrum of dimensions.

The Renyi dimensions Dq appear defined with R2 greater than 0.95 within a

significative range of q’s. In particular, the capacity dimension and the entropy dimension

showed the best fits. The ratio D1/D0 shows a suitable discriminatory ability for the studied

samples. The variation of Dq with respect to q and the shape of the Renyi spectrum reveal

that pore size distributions have properties close to multifractal self-similar measures.

These results show that Renyi dimensional analysis is an appropriate tool for character-

izing soil pore size distribution and thus may be used as indicator of soil structure as well as

for deriving important soil physical properties. Moreover, this characterization opens the

possibility of simulating the real distributions by adequate self-similar models of fractal

geometry, as iterated function systems and related ones, with the corresponding benefits for

the prediction of soil water and other physical properties.

Acknowledgements

This work has been partially supported by Comunidad de Madrid (Project Ref. 07M/

0048/2000) and Plan Nacional de Investigacion Cientifica, Desarrollo e Investigacion

Tecnologica (Spain) (REN2000-1542).


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