Directory UMM :Data Elmu:jurnal:S:Soil & Tillage Research:Vol55.Issue1-2.May2000:
Soil & Tillage Research 55 (2000) 43±61
Mass fractal dimensions and some selected physical properties
of contrasting soils and sediments of Mexico
K. Oleschkoa,*, B. Figueroa, S.b, M.E. Mirandac, M.A. Vuelvasd, E. Solleiro, R.a
a
Instituto de GeologõÂa, Universidad Nacional AutoÂnoma de MeÂxico (UNAM), Aportado Postal 70-296, Ciudad Universitaria, C.P. 04510,
Coyacan, MeÂxico, D.F., Mexico
b
Colegio de Postgraduados, km.35.5 Carretera MeÂxico-Texcoco, Montecillos, Estado de MeÂxico, C.P. 56230, Mexico
c
Posgrado en Ciencias de la Tierra, Instituto de GeologõÂa, Universidad Nacional AutoÂnoma de MeÂxico (UNAM), Aportado Postal 70-296,
Ciudad Universitaria, C.P. 04510, CoyacaÂn, MeÂxico, D.F., Mexico
d
Instituto Nacional de las Investigaciones Forestales y Agropecuarias (INIFAP), km.6.5, Carretera Celaya ± San Miguel Allende, Celaya,
Guanajuato 38001, Mexico
Received 2 February 1999; received in revised form 17 December 1999; accepted 28 January 2000
Abstract
The fractal approach to the study of soil structure, its dynamics, and physical processes appears to be a useful tool in
reaching a better understanding of system performance. Nevertheless, the precise concurrent analysis of soil physical
properties and fractal parameters seems to be important in order to evaluate the applicability of fractal concepts to soil science.
The objective of this study was to relate the solid and pore mass fractal dimensions of some soils of Mexico to in situ
measured bulk density, electrical permittivity and mechanical resistance. For this purpose, the fractal structure of some soils
and sediments with contrasting physical properties is documented, using microscopic images, in the scale range from 0.009 to
0.2 cm. A single and statistically representative mass fractal dimension, independent of scale, was found for analysed solid
and pore networks. Non-invasive techniques were used for in situ measurement of selected soil physical properties. The
physical measurements were made simultaneously with sampling for the micromorphological analysis. Simple but not unique
relations were established between the solid and pore mass fractal dimensions, soil bulk density, apparent dielectric constant
(except for Eutric Vertisol), and mechanical resistance (except for Tepetates). A high correlation was found between mass
fractal dimension and horizon depth. We conclude that the fractal analysis is a useful tool to distinguish between soils and
sediments of different genesis, and that the solid and pore mass fractal dimensions may be useful indicators of the horizon
compaction status. # 2000 Elsevier Science B.V. All rights reserved.
Keywords: Mass fractal dimension; Compaction status; Porosity; Mechanical resistance
*
Corresponding author. Tel.: 52-5622-4267;
fax: 52-5622-4317.
E-mail addresses: [email protected] (K. Oleschko),
[email protected] (K. Oleschko),
[email protected] (B. Figueroa, S.),
[email protected] (M.A. Vuelvas)
1. Introduction
Fractal geometry may be a powerful tool in describing heterogeneity and understanding the relationships
between the soil structure and physical, chemical and
0167-1987/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 1 9 8 7 ( 0 0 ) 0 0 0 9 7 - 0
44
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
biological processes (Anderson et al., 1998). However,
Baveye and Boast (1998) have emphasised that, in
spite of the impressive volume of literature on the
subject, the lack of a consensus on what it means for a
soil to be or to behave ``like a fractal'', is apparent. In
this paper the original de®nition of fractals, regarding
them as geometrical constructions that are similar to
their parts (Baveye and Boast, 1998) is accepted.
Therefore, the fractal dimension of all analysed sets
is strictly constrained to be between the topological
dimension of the fractal and the Euclidean dimension
of the space in which this is embedded. Notwithstanding, the solid and pore networks are physical objects,
and therefore the Pareto distribution (or power law
relationship) can be established for them with known
precision only between lower and upper cut-offs
(length scale limits, Anderson et al., 1998). For the
mentioned scale range, the solid and pore geometrical
properties are statistically similar to those of mathematical (or deterministic) fractals (Baveye and Boast,
1998), and the respective networks can be described as
statistical fractals (Oleschko et al., 1998).
Rieu and Sposito (1991) have stated that precise
concurrent analysis of soil physical properties and
fractal parameters are required in order to evaluate
the applicability of fractal concepts to soil science.
There are several fractal dimensions needed for the
complete characterisation of soil structure (Anderson
et al., 1996). Experimental evidence shows fractal
scaling of mass, pore space, pore surface and the size
distribution of fragments, between upper and lower
limits of scale, but typically there is no coincidence in
the values of the fractal dimensions characterising
different properties (GimeÂnez et al., 1997). The static
structural properties, including heterogeneity and
space-®lling ability of an object, are described by
the mass fractal dimension Dm, and showed to be
directly related to the material compactness (Bartoli
et al., 1991; Anderson et al., 1998; Lipiec et al., 1998).
Anderson et al. (1996) have speci®ed that objects
regarded as mass fractals are those that are porous
and do not have a uniform internal mass distribution.
The more general ``pore±solid fractal'' model (PSF)
was proposed for soil by Perrier et al. (1999). Neither
the pore phase nor the solid phase of a general PSF
exhibits mass fractal scaling. Two groups of empirical
methods are useful to functionally characterise a
porous medium using fractal approach (Crawford
et al., 1999). The ®rst one is based on the direct
imaging and analysis of soil structural units. In the
second group all indirect evaluations of structural
parameters are undertaken, including water retention,
water vapour adsorption data, mercury porosimetry
and X-ray scattering (Gomendy et al., 1999; Rice et al.,
1999; Sokolowska and Sokolowski, 1999). At present,
the analysis of soil images using computer programs is
potentially the most reliable method for measuring the
geometry of fractals (Crawford et al., 1993; Oleschko
et al., 1998). The objective of this study is to relate the
solid and pore mass fractal dimensions of some soils
and sediments of Mexico, obtained from microscopic
images, to in situ measured bulk density, electrical
permittivity and mechanical resistance. These physical properties are more common measures of soil
compaction status and can be measured in situ by
modern, indirect and non-invasive techniques (Jury
et al., 1991). The relation between the mass fractal
dimensions and the horizon depth was also investigated.
2. Materials
Four contrasting geosystems, in steady- or stationary-state moisture conditions (steady state ¯uxes),
were selected for the present study. Their location
is shown in Fig. 1. The complete pro®les of three
representative soils and one set of lacustrine sediments, containing the materials with a broad variety
of physical and morphological properties, were studied.
2.1. Geosystems description
2.1.1. Melanic Andosol, Veracruz state
This site is located on the eastern watershed of the
Cofre de Perote volcano, Veracruz state, at an altitude
of 2500 m above sea level (Fig. 1). The Melanic
Andosol develops from volcanic ashes of different
age. The last eruption is dated at 10 000 years. A 1.5 m
pro®le was opened and used for the morphological
description, micromorphological sampling, and physical measurements. Two horizons were identi®ed by
colour and morphology differences. The melanic layer
is black and deep, extending down to 0.95 m. The next
layer, located from 0.95 to 1.5 m, is more friable and is
yellow in colour. However, four different horizons
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
45
Fig. 1. Map of the experimental areas: (A) Eutric Vertisol; (B) Texcoco Lake and Tepetates; (C) Melanic Andosol.
were distinguished by in situ bulk density measurement. The ®rst of these, at 0±18 cm, was more compacted and corresponded to the Andosol arable
horizon. The last one corresponded to the fresh ash
deposit.
2.1.2. Eutric Vertisol, Guanajuato state
This study area is located in Celaya, Guanajuato
state at the National Institute of Forest and Agricultural Research (INIFAP) experimental ®eld (BajõÂo).
Eutric Vertisols are the typical soils for this area, with
a high shrink±swell capacity which is dependent
on the water content (Oleschko et al., 1993, 1996;
Coulombe et al., 1996). The soils of this zone originated from different types of alluvium derived from
basalt and other igneous extrusive rocks.
The study area is representative of the Bajio agricultural zone, under continuous drip irrigation systems. A 1.5 m deep pro®le was opened, and four
different horizons were identi®ed. The arable layer
was limited by the plough-pan at approximately 30 cm
depth. This horizon is typical for Vertisols under
intensive agricultural use (two crops per year during
the last 50 years). The boundary with the next horizon
is at 80 cm depth, and is characterised by a change in
colour from black to deep brown. A change in soil
46
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
texture from clay to sandy loam marks the transition to
the third horizon at 120 cm. The detailed physical
characterisation of the Eutric Vertisol was accomplished only in the upper 45 cm of the pro®le.
2.1.3. Texcoco lake sediments, Mexico state
This sampling area is located in Montecillo, Mexico
state, and covers the eastern zone of the dry Texcoco
lake bank. The lake is a ¯at geomorphological zone
with total surface area of 1000 ha (Luna, 1980). The
Texcoco lake stratigraphy is composed of a basaltic
bed, and limestone and marine deposits of the Upper
Cretaceous period (at 2000 m depth below the surface). The upper strata are dominated by lacustrine
sediments (from 40 to 20 m) and by speci®c lacustrine
clay accumulations (from 6 to 0 m). The amorphous
clay, locally referred to as ``jaboncillo'', is characterised by speci®c physical and chemical properties:
an extremely low bulk density (280±430 kg mÿ3), a
high speci®c surface of 225 m2 gÿ1 and a very high
water retention capacity (e.g. more than 3 g of water is
retained by 1 g of clay).
This area represents the most interesting case for the
micromorphological and physical studies. Six horizons with clear differences in morphology as well as
physical and chemical properties and regular horizontal boundaries were identi®ed in the reference pro®le.
The natural water table occurs at 1.8 m depth and is
the lower boundary for the morphological description
which follows.
The sediment pro®le is divided into two contrasting
parts by an intermediate horizon of compacted basaltic
volcanic ash deposit with a bulk density of
1420 kg mÿ3 and a particle size varying from silt to
sand, at a depth of 39 and 69 cm (Fig. 2). This horizon
represents the textural screen between Mollic surface
layer (from 0 to 39 cm), and the clay accumulation
(from 69 to 180 cm). The detailed seasonal studies
have shown that these two parts of the pro®le are not
hydrologically connected. Therefore, the moisture
content in the upper part of the pro®le depends only
on the precipitation regime, whereas the water content
in the lower part is determined by the ground water
table ¯uctuations. The ®eld survey was carried out
in the dry season (January), after more than four
months had elapsed since the last rain. Thus, the
studied area was in quasi-equilibrium with respect
to water content.
Fig. 2. Texcoco Lake pro®le with abrupt limits between horizons.
On one side of the reference pro®le, three ¯at
surfaces 5 m5 m were opened for detailed physical
measurements. The ®rst surface was at 15 cm depth,
while the second coincided with the centre of the
volcanic ash deposits (50 cm), and the third corresponded to the pure lacustrine clay deposit at 80 cm
depth.
The dielectric constants of these materials ranged
from 13.01 in the upper zone to 65.9 in the water
saturated lower part of the pro®le. The bulk density
varied from 980 to 280 kg mÿ3 and the gravimetric
water content in steady-state conditions from 21.48 to
363.35% (estimated in grams of water retained by
100 g of solid material). These data emphasise the
high porosity of clays in the lower part of the studied
pro®le. However, microscopic observations of this
zone disclosed a predominance of pores of sub-microscopic size as well as presence of a few ®ssures.
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
2.1.4. Tepetates, Mexico state
In Latin America, hardened soils of volcanic origin
are often referred to by their vernacular names (Zebrowski, 1992). In Mexico, these formations are called
``Tepetates'', which means ``hard'' in the NaÂhuatl
Indian language. Commonly, Tepetates are interstrati®ed with volcanic and sedimentary deposits. All
pro®les include different paleosols characterised by
clear and abrupt horizontal limits between the layers.
In general, Tepetates in the Mexican high plateau are
natural, massive, compact and hard formations,
cemented by different chemical agents, including
clays and silica. Hardness is the distinctive characteristic of these materials.
All Tepetates have a speci®c geomorphological
position. Hardened horizons are inserted in piedmont
and glacis soils, and associated with brown clay soils,
de®ned as Cambisols (Zebrowski et al., 1991). This
location coincided with sub-humid and sub-arid
regimes (ustic), and suggests the climate and pedogenic effects on Tepetates formation (Quantin, 1992).
The opened pro®le presents several advantages for
the fractal study: (1) All cemented horizons are separated by paleosols with vertic properties. (2) The limits
between horizons are abrupt and horizontal, since
Tepetates are related to ancient volcanic events. (3)
The period of moisture quasi-equilibrium can be
selected easily. (4) Tepetates are characterised by
speci®c pore space morphology: main pores are isolated and occupied by cementing agents (Fig. 3d). The
apparent dielectric constant was measured only in the
upper 15 cm (Ka4.4), since the hardness and the
presence of numerous ®ssures make the measurements
in other layers statistically incoherent.
Nine horizons of different origin with contrasting
properties were described in this pro®le, and three of
them were identi®ed as cemented layers (Tepetates),
separated by clayey layers.
3. Methods
3.1. Micromorphological analysis
Three undisturbed samples (8 cm4 cm) were collected with metal samplers from each horizon of the
four opened pro®les. All samples were taken at ®eld
moisture and carried in plastic bags without drying. In
47
the laboratory, samples were dried by the acetone
replacement (in liquid phase) method and then impregnated with a 1:1 polyester resin (HU-543) and acetone
mixture (Murphy, 1986). Soil cores were re-impregnated with the same resin under vacuum conditions.
When the resin was suf®ciently hardened, samples were
horizontally sectioned parallel to the soil surface.
Thin sections (2 cm4 cm, with average thickness
of 30 mm) were prepared by standard petrographic
procedures (Brewer, 1964) and analysed under the
petrographic microscope (Olympus, BH-2). Four black
and white photographs were taken at the same scale
from each thin section. In total 12 microphotographs
were used for the fractal analysis of each soil horizon.
More contrasting examples are shown in Fig. 3.
3.2. Fractal analysis
Each photograph (15 cm10 cm) of the thin section
was scanned, using a 600 dpi resolution commercial
scanner. The obtained image was rendered on a grid of
10001000 pixels, and used for the automatic fractal
analysis. Adobe(TM) software was used to change all
grey scale ``tiff'' images into ``raw'' format. The
distribution of grey tones, varied from 0 (totally black)
to 255 (completely white) and was estimated using the
image histogram. On the microphotograph the boundary between the solid and pore networks is fuzzy;
therefore the limit between both sets was quanti®ed by
a range of grey tones. The Adobe PhotoShop software
was used to measure with pipette the grey values
corresponding to the solid and pore networks. The
obtained grey tone distributions were compared with
histogram data. The fractal programme, designed by
Parrot and Rico (1997) in Borland environment and
coded using C, was used for the mass fractal
analysis. This programme is based on the traditional
box-counting technique, and was calibrated with
deterministic fractals in previous research (Oleschko,
1999). Two ``box'', capacity or mass fractal dimensions were estimated for each image, and used to
describe the space-®lling ability of solid (Dms) and
pore (Dmp) networks, respectively. The values of Dms
and Dmp for each two-dimensional thin section image
is always less than two, since two would be expected
for a completely ®lled rectangle (Anderson et al.,
1996). The solid and pore mass fractal dimensions
were estimated by ®lling the corresponding image part
48
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 3. Microscopic images of the more contrasting horizons of: (a) Melanic Andosol; (b) Eutric Vertisol; (c) Texcoco Lake pro®le; (d)
Tepetates pro®le.
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 3. (Continued ).
49
50
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 3. (Continued ).
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 3. (Continued ).
51
52
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
progressively with larger boxes. The box size was
equal to m-pixels. The size of m begins at 1 and was
increased to a maximum value of 1000. Sixteen
different box sizes are used in the Parrot and Rico
(1997) programme. The number of m-pixels or boxes
that ®t in the solid (or pore) part of the image, when m
is increased, was counted. This method was described
and shown diagrammatically by Anderson et al.
(1996), who have mentioned that, the value of Dms
(and Dmp) can be estimated from a plot of ln (number
of m-pixels corresponding to the solid or pore network) versus ln(m). The theoretical line was adjusted
by least-squares regression and Dms and Dmp were
estimated from the negative of the slope.
3.3. Physical measurements
3.3.1. Time domain re¯ectometry
The time domain re¯ectometry technique (TDR),
was used to measure the apparent dielectric constant
(Ka) and the volumetric water content (yi) of the
studied soils and sediments. The Trase System Model
6050x1 is designed to measure dielectric constants
over a frequency bands between 100 and 1000 MHz.
A step pulse of electromagnetic radiation was sent
along the two parallel waveguides placed in the soil.
The permittivity of the material between the waveguides causes the pulse velocity deviation from the
known light velocity in vacuum (Jury et al., 1991). The
permittivity is estimated from the pulse travel time,
and the water content is calculated by a known
empirical third-order polynomial regression, by relating these two parameters. This model was proposed by
Topp et al. (1980). The obtained calibration curve was
used for the in situ volumetric content measurements.
In total 16 measurements were accomplished by TDR
on each horizon of interest.
Simultaneously to the TDR measurements, three
samples were taken for gravimetric water content (Wi)
and soil bulk density (rb). In the arable horizon of
Eutric Vertisol more detailed analysis was carried out,
taking more than 50 samples for Ka, Wi and rb (Fig. 4d).
Fig. 4. Relation between the material bulk density (rb) and apparent dielectric constant (Ka): (a) all materials compared together; (b) Melanic
Andosol; (c) Texcoco Lake pro®le; (d) Eutric Vertisol.
53
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
cone index (CI) data down to a depth of 600 mm to
give CI values up to 5000 kPa. The most important
feature of this penetrometer is that it uses an ultrasonic
method for measuring depth. CI data at all of the
studied sites were recorded on each of 16 sampling
points, every 15 mm down to 600 mm depth, and are
presented in Fig. 7 in values divided by 100.
The undisturbed samples (10 cm10 cm10 cm) were
obtained by using a coring tube of known volume
suitable to measure rb. The gravimetric water and bulk
density data were used for the volumetric water calculation. The volumetric water content, measured by
TDR and estimated from rb and Wi data, were compared, and used: (1) to de®ne the precision of the TDR
calibration curve in contrasting compactness conditions; (2) to establish the relation between yi and rb.
Subsequently, the last relation was used for the indirect bulk density estimation from the volumetric water
content data.
3.5. Statistical analysis
The data variation was estimated by classical statistical procedures. For all mass fractal dimensions the
mean value of 12 samples was used (Table 1). The
number of physical data has varied, depending on the
used technique, and not always has coincided with the
number of analysed fractal parameters, in this case
only mean values of variables were used to establish
the relationships of interest, where it was possible all
3.4. Mechanical resistance
A cone penetrometer (Rimik CP20) was used to
measure the mechanical resistance of soils and sediments. The cone penetrometer measures and records
Table 1
Solid (Dms) and pore (Dmp) fractal dimensions of studied materialsa
Da
(kg mÿ3)b
Kac
Morphological description
(0.009)
(0.012)
(0.009)
(0.010)
0.39
0.37
0.32
0.3
18.7
25.6
36.0
44.4
Melanic horizon
Sandy loam
Sandy loam
Sandy
1.9310 (0.009)
1.9363 (0.011)
1.9350 (0.008)
1.8845 (0.090)
1.8352 (0.003)
1.7987 (0.007)
1.07
1.23
1.35
29.55
Arable layer
Arable layer
Plough-pan
0±10
27±38
56±67
69±80
90±100
110±120
1.9385
1.9479
1.9351
1.9570
1.9697
1.9580
(0.005)
(0.010)
(0.009)
(0.012)
(0.009)
(0.008)
1.8233
1.8620
1.8741
1.7651
1.7174
1.7808
(0.01)
(0.020)
(0.019)
(0.002)
(0.006)
(0.020)
0.98
1.42
1.21
0.43
0.37
0.28
13.01
16.43
21.84
44.85
56.09
65.9
Mollic horizon
Loamy horizon
Basaltic volcanic ashes
Loamy with small clay subhorizons
Clay
Clay
0±25
69±87
87±124
124±150
200±220
220±235
235±260
260±420
420±460
1.9520
1.9537
1.9600
1.9408
1.9593
1.9773
1.9150
1.9645
1.9888
(0.001)
(0.004)
(0.004)
(0.005)
(0.010)
(0.002)
(0.004)
(0.010)
(0.007)
1.8360
1.8033
1.8875
1.7768
1.7815
1.8013
1.9079
1.7606
1.6219
(0.009)
(0.032)
(0.006)
(0.003)
(0.050)
(0.008)
(0.004)
(0.016)
(0.165)
1.57
1.63
1.15
1.25
1.68
1.68
1.19
1.30
1.66
4.4
Soil
Depth
(cm)
Dms
Melanic Andosol
0±18
18±55
55±95
95±130
1.9374
1.9304
1.9246
1.9278
Eutric Vertisol
0±15
15±30
30±45
Texcoco Lake deposits
Tepetate profile
a
Dmp
(0.007)d
(0.008)
(0.008)
(0.009)
1.8864
1.8768
1.8969
1.9082
Some data of this table were published before by Oleschko (1999).
Bulk density.
c
Apparent dielectric constant.
d
The standard deviation is in parentheses.
b
Colluvium
Vertic properties
Vertic properties
Tepetate, t2a
Tepetate, t2b, hard
Tepetate, t2b, hard
Paleosoil
Paleosoil
Tepetate, t3, very hard
54
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
measured values were graphed (Fig. 4d, Figs. 7a±d,
8a±d and 9a±c).
4. Results and discussion
4.1. Fractal analysis of the microscopic images
Pentland (1984) has shown that a fractal surface and
its bi-dimensional photographic image are mathematically equivalent. In the present study, the fractal
structure of the solid and pore networks of soils and
sediments in four contrasting geosystems, was documented in the scale range from 0.009 to 0.2 cm, using
their microscopic images. All analysed networks were
characterised by a single, near ``ideal'' fractal dimension, independent of scale. The solid (Dms) and pore
(Dmp) mass fractal dimensions have shown a clear
dependence on the soil and sediment genesis (Table 1).
A maximum fractal dimension of 1.9888 was obtained
for the solid set of fragipan (t3), in the Tepetate pro®le,
identi®ed as the hardest horizon. This means that the
solids in this layer occupy considerably more space
than the pores. The minimum value of the pore mass
fractal dimension (Dmp1.6219) was found for the
same horizon. The Melanic Andosol and the paleosol
with vertic properties in the Tepetates pro®le have the
lowest values of the solid set fractal dimension
(1.9246 and 1.9150, respectively). These results coincide with our previous data (Oleschko et al., 1997,
1998), and showed that the fractal dimension, within
the studied scale range, is a useful parameter for
distinguishing between materials of different genesis.
It means that this parameter may possibly be used to
monitor the tillage in¯uence on soil physical properties, including the soil compaction status. Nevertheless, from time to time the differences among the
fractal dimension of extremely contrasting materials,
were small. For instance, the solid set fractal dimensions in the arable horizon of the Melanic Andosol and
Eutric Vertisol soils with clear bulk density differences
(390 compared to 1070 kg mÿ3), were 1.9374 and
1.9310, respectively. The highest contrasting example
is related to Dmp of the amorphous clay layer (Texcoco
Lake, Dmp1.7651), compared with the cemented
horizons (Tepetates, t2a formation, Dmp1.7768),
which are estimated in the same range of scales and
are not statistically different. These results agree with
Anderson et al. (1998), who have concluded that the
common box-counting technique takes into account
only pores that are visible at the resolution of the
analysed photograph. The visible porosity is similar in
the compacted clay and cemented Tepetate horizon,
when the sub-microscopic porosity is out of the
method resolution. Therefore, it should be emphasised, that it is essential to refer all fractal analysis
conclusions to the exact scale range at which the data
were acquired, and that more than one fractal parameter is needed for soil structure description. The
standard deviation varied between 0.001 and 0.011 for
the solid mass fractal dimension, and between 0.003
and 0.165 for the pore mass fractal dimension
(Table 1).
4.2. Relationship between the soil and sediment
physical properties and mass fractal dimensions
For a better understanding of the origin of fractal
dimension dependence on the material genesis, the
relationships between some physical properties of the
studied soils and sediments and their solid and pore
fractal dimensions were analysed. A number of dif®culties, including some incoherent data and results
opposite to the theoretically expected tendencies, were
encountered. This was due to either the speci®c properties of all studied materials, which vary widely, or
independence of fractal parameters from soil genesis.
Nevertheless, it was considered that even if relations
between solid and pore mass fractal dimensions and
physical properties were not clearly obvious, it would
be very helpful to know some important trends in the
dynamics of these fractal parameters in soils and
sediments with different compaction status. A representative analysis of both fractal dimensions and
physical properties would ensure, whether such correlation of fractal dimensions and physical properties
exists, or it is underlined by large measurement noise
and random sample spatial variation.
Two types of analysis were carried out. First, each
of the relations of interest was established for soils and
sediments, analysed all together, and then, the same
analysis was applied for each one of the studied
pro®les. Special attention was given to Texcoco Lake
deposits, where the layers with the strongest differences in physical and morphological properties were
found.
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
4.2.1. Relation between bulk density, apparent
dielectric constant and mass fractal dimensions
Dirksen and Dasberg (1993) tried to establish a
single relation between the soil permittivity and volumetric water content related directly to the material
bulk density. They concluded that the bulk densities
differed not only between soils, but also among arti®cially packed columns of the same soil: bulk densities
®rst decreased and then increased with water content
increase. Therefore, the relationship between rb and
Ka has two components with completely opposite
tendencies. Dirksen and Dasberg (1993) strongly
recommended comparing this pair of variables individually for each studied material.
In the present study, a nearly linear relation
(R2=0.55) was obtained among rb and Ka when all
horizons of the four pro®les were compared (Fig. 4a).
The established trend has shown that the higher the
value of rb (the more compact a soil), the lower the
dielectric constant. Therefore, the more porous material has a greater permittivity. All compared materials,
except the Tepetates pro®le, had water contents near
®eld capacity at the time of sampling. Therefore, it
seems reasonable that, lower values of bulk density
correlated with higher values of the volumetric water
content, and also, with larger apparent dielectric constants.
In the Melanic Andosol, this relation has shown the
best ®t (R20.98). The layer with maximum bulk
density (390 kg mÿ3) has the minimum dielectric
constant (18.7, Fig. 4b). It can be noted that in this
volcanic and highly porous soil, a change in bulk
density from 390 to 300 kg mÿ3 coincides with a
drastic change in the volumetric water content (from
25.3 to 58.3%) and a considerable increase in apparent
dielectric constant (from 18.7 to 44.4).
In order to validate the established relation in more
contrasting materials, detailed studies were conducted
in the Texcoco Lake and in the Eutric Vertisol. In the
latter area, the only arable layer was analysed in detail.
A strong direct linear relation (R20.96) was obtained
for the Texcoco lacustrine deposits (Fig. 4c). The near
saturation clay horizons with bulk densities of 280 and
370 kg mÿ3 were characterised by the maximum
values of apparent dielectric constant (65.9 and
56.09, respectively).
Nevertheless in contrast, an inverse tendency was
observed in the arable layer of the Eutric Vertisol,
55
where more than 100 TDR measurements were made,
with simultaneous sampling for bulk density (the
coring tube method) and gravimetric water content.
High data variability was noted, but the dielectric
constant increase with bulk density was statistically
con®rmed (R20.57, Fig. 4d). This atypical behaviour
of Vertisol with high shrink±swell capacity is well
documented in the literature (Oleschko et al., 1993,
1996; Coulombe et al., 1996), and is considered one of
the possible reasons of the reported differences. The
unsuitability of the commonly used tube method to
obtain the precise Vertisol bulk density measurements,
and the absence of some alternative techniques may be
the other reason for the observed trend.
The mass fractal dimensions of soil solid (Dms) and
pore (Dmp) networks were related to the bulk density
(rb) and apparent dielectric constant (Ka) of the
studied materials. A good ®t (R2=0.75) was obtained
for the Melanic Andosol, where a direct linear relation
between the Dms and Ka was observed (Fig. 5a). The
horizon with the maximum value of solid fractal
dimension (1.9374) had the lowest apparent dielectric
constant (18.7). From a theoretical point of view, the
larger the solid fractal dimension the higher is the
solid occupied space, and as a consequence the larger
the bulk density and the lower the volumetric water
content. A similar, but only slightly correlated statistical relationship (R2=0.46) was observed between
the Andosol bulk density and the solid set mass
fractal dimension: the more compacted horizon
(Da390 kg mÿ3) was characterised by the maximum
Dms (1.9374, Fig. 5b). This conclusion agrees with
general assessments about the solid set mass fractal
dimension dynamics. Previously, comparing the same
soils under different tillage conditions, it was established that the solid mass fractal dimensions were
higher and the pore mass fractal dimensions were
lower for the more compacted soil (Bartoli et al.,
1991; Oleschko et al., 1997, 1998). Therefore, it
was shown that Dms was inversely related to average
soil porosity.
In the Melanic Andosol, a reasonably good direct
linear ®t (R2=0.75 and 0.79) was obtained between the
pore fractal dimensions, material bulk density and
apparent dielectric constant (Fig. 5c and d, respectively). The layer with the highest mean pore fractal
dimension (1.9082) and the largest porosity has the
highest apparent dielectric constant (44.4).
56
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 5. Relation between Melanic Andosol: (a) solid set fractal dimension (Dms) and apparent dielectric constant (Ka); (b) Dms and bulk
density (rb); (c) Dmp and Ka; (d) pore set fractal dimension (Dmp) and bulk density (rb).
Relationships between the solid fractal dimension
and the bulk density observed in the Texcoco Lake
pro®le were unclear (Fig. 6a). The high data dispersion
was typical for this experimental area. However,
R20.59 suggests that one of the horizons with the
highest clay content (90±100 cm) had the largest Dms
(1.9697), high apparent dielectric constant (56.09,
Fig. 6b), and low bulk density (370 kg mÿ3). This is
in line with the ®ndings of Perfect and Kay (1995) and
others who also found that Dms increased with increasing clay content.
The relationship between the pore mass fractal
dimension and bulk density in this pro®le, was opposite to the theoretical trend. The horizons with the
largest Dmp (and therefore with the smallest Dms and
highest porosity) were characterised by higher bulk
density (R20.86, Fig. 6c). The speci®c nature of this
lacustrine amorphous clay with bulk density decreasing down to 280 kg mÿ3, is that of a high invisible submicroscopic porosity. This was considered responsible
for the observed tendency, where the increment of
pore mass fractal dimension coincided with the
increase in horizon compactness. This is consistent
with the relation established in the same pro®le
between the pore fractal dimension and the apparent
dielectric constant: the larger the Dmp, the smaller is
the Ka (R20.60, Fig. 6d).
Rieu and Sposito (1991), using the Chepil data,
came to a similar conclusion, establishing that
increases in Dm (only solid dimension was estimated
in their study) were correlated to increases in clay
content. Crawford et al. (1993) and Anderson et al.
(1996) have concluded that, generally, a soil structure
with large continuous pores had larger values of Dmp
than a structure with small discrete pores and low
lacunarity. However, Anderson et al. (1996) have
underlined that Dmp does not depend wholly on porosity, and different values of Dmp are obtained for
structures of equal porosity but with different spatial
distribution of pore sets (distinct lacunarity). It was not
possible to check the above trend for the Tepetates
pro®le, since the data of the apparent dielectric con-
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
57
Fig. 6. Texcoco Lake pro®le, relation between: (a) solid set fractal dimension (Dms) and bulk density (rb); (b) Dms and apparent dielectric
constant (Ka); (c) pore set fractal dimension (Dmp) and bulk density (rb); (d) Dmp and Ka.
stant were obtained only for the upper, unconsolidated
horizon. In the Vertisol, the high data dispersion,
explained by the presence of ®ssures, was responsible
for the statistically incoherent ®ts between the variables of interest. However, if only mean values of
variables were considered, the largest pore fractal
dimension (1.8845) would coincide with the minimal
bulk density (1070 kg mÿ3 and R2=0.83).
In the present study it was impossible to establish
some general relation between Dms and Ka, for the
strongly contrasting soils and sediments studied. This
relation should be established individually for each
pro®le.
It was concluded that, in general, for the studied
soils and sediments with water contents near ®eld
capacity, the apparent dielectric constant tends to
increase with decrease in bulk density (except the
Eutric Vertisol). However, the relation between the
material bulk density, apparent dielectric constant and
mass fractal dimensions strongly depends on the soil
and sediment origin.
4.2.2. Relation between the mechanical resistance
and mass fractal dimensions
An inverse intuitively valid linear relation was
detected in the Texcoco Lake pro®le between the soil
apparent dielectric constant and mechanical resistance
(R20.717, Fig. 7a). The clay horizon has signi®cantly lower CI than the compacted basaltic ash
deposits. Nevertheless, in this pro®le, all obtained
data were derived from materials with extremely
contrasting properties, therefore the experimental
points form two clouds concentrated at both extreme
ends of Ka versus CI graph, and never form a real
continuous function. The mass fractal dimension of
the pore set also shows the inverse, contrary to the
theoretically expected, relation: the layer with bigger
pore dimension (1.8620) has the largest mechanical
resistance (8.1 kPa cmÿ2, Fig. 7c). The same unexpected trend was found for the relation between solid
mass fractal dimension and mechanical resistance of
the Texcoco Lake sediments (Fig. 7b). The decrease in
the Dms values (in the upper part of the pro®le with less
58
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 7. Relation between material mechanical resistance (CI/100) and: (a) apparent dielectric constant (Ka); (b) solid set fractal dimension
(Dms); (c) pore set fractal dimension (Dmp), in Texcoco Lake pro®le; (d) CI versus Dmp in Eutric Vertisol.
clay content) coincided with the increase in the material's mechanical resistance (R20.50). Similar relations were observed for the solid and pore mass fractal
dimensions versus mechanical resistance in the Melanic Andosol pro®le. The deeper horizons (from 55 to
130 cm) have larger CI (600 kPa cmÿ2 in comparison
with 380 kPa cmÿ2 in the melanic horizon) and smaller solid set mass fractal dimensions (1.9278 versus
1.9374, R20.92). Therefore, the mass fractal dimension is not the direct indicator of soil compaction
status, depending more on the material genesis.
The results obtained for Eutric Vertisol con®rm this
hypothesis. The solid mass fractal dimension in the
®rst 45 cm of Eutric Vertisol, sampled at 15 cm intervals, was almost constant at 1.93, however the pore
dimension has diminished from 1.8845 (0±15 cm) to
1.7987 (30±45 cm), and the mechanical resistance
increased from 797 to 1317 kPa cmÿ2. Changes in
the pore mass fractal dimension and mechanical
resistance re¯ect the presence of a plough-pan, while
Dms is independent of this. An inverse linear relation
was established between CI and Dmp: higher mechanical resistance coincided with smaller pore mass fractal dimensions (R20.96, Fig. 7d). These results are in
agreement with bulk density data and the soil morphological description.
In the Tepetates pro®le, it was not possible to
analyse the relation between the mechanical resistance
and mass fractal dimensions of different layers,
because the cone penetrometer was unsuitable for
use in these consolidated materials.
4.2.3. Relation between horizon depth and mass
fractal dimensions
Hatano and Booltink (1992) have estimated the
mass fractal dimension (Dmp) from the stained ¯ow
pattern, using two-dimensional images. They concluded that, in general, values of Dmp decrease with
depth. Similar results were obtained in the present
study for the pore fractal sets of the Texcoco Lake,
Tepetates and Vertisol soils, where the pore set fractal
dimension decreased with depth (R20.94, 0.63 and
0.82, respectively, Fig. 8b±d). These data indicate a
lower porosity in the deeper layers, linked with a solid
set fractal dimension increase in the same direction.
However, the inverse trend was obtained for the
Andosol pro®le (Fig. 8a) with the deeper and more
Fig. 8. The dynamic of the pore set fractal dimension with depth, in: (a) Melanic Andosol; (b) Texcoco Lake pro®le and (c) Tepetates pro®le;
(d) Eutric Vertisol.
Fig. 9. The dynamic of the solid set fractal dimension with depth, in: (a) Melanic Andosol; (b) Texcoco Lake pro®le; (c) Tepetates pro®le.
60
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
porous horizons having higher pore fractal dimensions
(R20.61).
A trend opposite to the pore networks, was obtained
for the solid set fractal dimensions in relation to pro®le
depth (except Vertisol, Fig. 9a±c). Different and sometimes opposite tendencies, depending on the origin of
soil and sediment, were detected. In the Melanic
Andosol (R20.71) Dms decreased with depth
(Fig. 9a). Then the more recent, more porous and
unweathered ashes constitute the deeper layers. The
opposite trend, and therefore the increase of the solid
mass fractal dimension with depth, was detected in the
Texcoco Lake and Tepetates pro®les (R20.81 and
0.70, respectively, Fig. 9b and c). These relationships
are strongly linked with the deposition history of the
materials. For instance, in the Texcoco Lake the clay
horizon comprises a more recent undisturbed material,
that was never in direct interaction with the surface. In
the Tepetates pro®le, the deep Tepetate (t3) is the
hardest layer with clear geological origin, which
shows only a few features of perturbation. In the
Eutric Vertisol, the pedoturbation is responsible for
the continuous mixing of materials, and the homogenisation of the pro®le up to one or more meters of
depth. In this case, the Dms appears to be independent
of depth (Table 1).
compaction status. Nevertheless, the solid mass fractal
dimension depends more on material genesis. On the
same image, the pore mass fractal dimension is always
smaller than the solid one. A complete description of
this pattern requires the multiscale fractal analysis,
and the use of some additional fractal parameters (i.e.
lacunarity). It seems clear that for many soil properties
affecting water movement and root growth, the poresize distribution pattern has much greater importance
than total porosity, bulk density and mass fractal
dimensions. It is possible to establish simple and
statistically coherent relations between fractal parameters and some selected physical properties of materials. These relations should be established
individually for each soil and sediment of interest.
Nevertheless, for the same soil or sediment, the pore
mass fractal dimension is a useful indicator of soil
compaction status and visible porosity.
Acknowledgements
This research was supported by DGAPA (PAPIT
programme), UNAM (project IN-106697) and CONACYT (project 3617P-A), Mexico. We thank A.M.
Rocha T. for technical support and M. Alcayde O. and
Dr. G. Tolson for improving the English version of the
manuscript.
5. Conclusions
The fractal dimensions of solid and pore networks
are useful parameters capable of distinguishing
between materials with contrasting genesis. The
cemented horizons (Tepetates) are characterised by
the maximum solid and minimum pore fractal dimensions. The solid mass fractal dimension near 1.99, and
the pore mass fractal dimension near 1.62, determined
for these hardest and extremely compacted layers,
may be proposed as the bounded values for the soil
solid and pore networks. The former value is close to
the upper topological limit of a two-dimensional
image. However, the traditional box-counting technique applied to image of one scale is able to estimate
only visible porosity, and therefore, on occasions, the
soils and sediments with extremely contrasting physical properties may be described by similar mass
fractal dimensions, inside the studied scale range.
The pore mass fractal dimension re¯ects the soil
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Mass fractal dimensions and some selected physical properties
of contrasting soils and sediments of Mexico
K. Oleschkoa,*, B. Figueroa, S.b, M.E. Mirandac, M.A. Vuelvasd, E. Solleiro, R.a
a
Instituto de GeologõÂa, Universidad Nacional AutoÂnoma de MeÂxico (UNAM), Aportado Postal 70-296, Ciudad Universitaria, C.P. 04510,
Coyacan, MeÂxico, D.F., Mexico
b
Colegio de Postgraduados, km.35.5 Carretera MeÂxico-Texcoco, Montecillos, Estado de MeÂxico, C.P. 56230, Mexico
c
Posgrado en Ciencias de la Tierra, Instituto de GeologõÂa, Universidad Nacional AutoÂnoma de MeÂxico (UNAM), Aportado Postal 70-296,
Ciudad Universitaria, C.P. 04510, CoyacaÂn, MeÂxico, D.F., Mexico
d
Instituto Nacional de las Investigaciones Forestales y Agropecuarias (INIFAP), km.6.5, Carretera Celaya ± San Miguel Allende, Celaya,
Guanajuato 38001, Mexico
Received 2 February 1999; received in revised form 17 December 1999; accepted 28 January 2000
Abstract
The fractal approach to the study of soil structure, its dynamics, and physical processes appears to be a useful tool in
reaching a better understanding of system performance. Nevertheless, the precise concurrent analysis of soil physical
properties and fractal parameters seems to be important in order to evaluate the applicability of fractal concepts to soil science.
The objective of this study was to relate the solid and pore mass fractal dimensions of some soils of Mexico to in situ
measured bulk density, electrical permittivity and mechanical resistance. For this purpose, the fractal structure of some soils
and sediments with contrasting physical properties is documented, using microscopic images, in the scale range from 0.009 to
0.2 cm. A single and statistically representative mass fractal dimension, independent of scale, was found for analysed solid
and pore networks. Non-invasive techniques were used for in situ measurement of selected soil physical properties. The
physical measurements were made simultaneously with sampling for the micromorphological analysis. Simple but not unique
relations were established between the solid and pore mass fractal dimensions, soil bulk density, apparent dielectric constant
(except for Eutric Vertisol), and mechanical resistance (except for Tepetates). A high correlation was found between mass
fractal dimension and horizon depth. We conclude that the fractal analysis is a useful tool to distinguish between soils and
sediments of different genesis, and that the solid and pore mass fractal dimensions may be useful indicators of the horizon
compaction status. # 2000 Elsevier Science B.V. All rights reserved.
Keywords: Mass fractal dimension; Compaction status; Porosity; Mechanical resistance
*
Corresponding author. Tel.: 52-5622-4267;
fax: 52-5622-4317.
E-mail addresses: [email protected] (K. Oleschko),
[email protected] (K. Oleschko),
[email protected] (B. Figueroa, S.),
[email protected] (M.A. Vuelvas)
1. Introduction
Fractal geometry may be a powerful tool in describing heterogeneity and understanding the relationships
between the soil structure and physical, chemical and
0167-1987/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 1 9 8 7 ( 0 0 ) 0 0 0 9 7 - 0
44
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
biological processes (Anderson et al., 1998). However,
Baveye and Boast (1998) have emphasised that, in
spite of the impressive volume of literature on the
subject, the lack of a consensus on what it means for a
soil to be or to behave ``like a fractal'', is apparent. In
this paper the original de®nition of fractals, regarding
them as geometrical constructions that are similar to
their parts (Baveye and Boast, 1998) is accepted.
Therefore, the fractal dimension of all analysed sets
is strictly constrained to be between the topological
dimension of the fractal and the Euclidean dimension
of the space in which this is embedded. Notwithstanding, the solid and pore networks are physical objects,
and therefore the Pareto distribution (or power law
relationship) can be established for them with known
precision only between lower and upper cut-offs
(length scale limits, Anderson et al., 1998). For the
mentioned scale range, the solid and pore geometrical
properties are statistically similar to those of mathematical (or deterministic) fractals (Baveye and Boast,
1998), and the respective networks can be described as
statistical fractals (Oleschko et al., 1998).
Rieu and Sposito (1991) have stated that precise
concurrent analysis of soil physical properties and
fractal parameters are required in order to evaluate
the applicability of fractal concepts to soil science.
There are several fractal dimensions needed for the
complete characterisation of soil structure (Anderson
et al., 1996). Experimental evidence shows fractal
scaling of mass, pore space, pore surface and the size
distribution of fragments, between upper and lower
limits of scale, but typically there is no coincidence in
the values of the fractal dimensions characterising
different properties (GimeÂnez et al., 1997). The static
structural properties, including heterogeneity and
space-®lling ability of an object, are described by
the mass fractal dimension Dm, and showed to be
directly related to the material compactness (Bartoli
et al., 1991; Anderson et al., 1998; Lipiec et al., 1998).
Anderson et al. (1996) have speci®ed that objects
regarded as mass fractals are those that are porous
and do not have a uniform internal mass distribution.
The more general ``pore±solid fractal'' model (PSF)
was proposed for soil by Perrier et al. (1999). Neither
the pore phase nor the solid phase of a general PSF
exhibits mass fractal scaling. Two groups of empirical
methods are useful to functionally characterise a
porous medium using fractal approach (Crawford
et al., 1999). The ®rst one is based on the direct
imaging and analysis of soil structural units. In the
second group all indirect evaluations of structural
parameters are undertaken, including water retention,
water vapour adsorption data, mercury porosimetry
and X-ray scattering (Gomendy et al., 1999; Rice et al.,
1999; Sokolowska and Sokolowski, 1999). At present,
the analysis of soil images using computer programs is
potentially the most reliable method for measuring the
geometry of fractals (Crawford et al., 1993; Oleschko
et al., 1998). The objective of this study is to relate the
solid and pore mass fractal dimensions of some soils
and sediments of Mexico, obtained from microscopic
images, to in situ measured bulk density, electrical
permittivity and mechanical resistance. These physical properties are more common measures of soil
compaction status and can be measured in situ by
modern, indirect and non-invasive techniques (Jury
et al., 1991). The relation between the mass fractal
dimensions and the horizon depth was also investigated.
2. Materials
Four contrasting geosystems, in steady- or stationary-state moisture conditions (steady state ¯uxes),
were selected for the present study. Their location
is shown in Fig. 1. The complete pro®les of three
representative soils and one set of lacustrine sediments, containing the materials with a broad variety
of physical and morphological properties, were studied.
2.1. Geosystems description
2.1.1. Melanic Andosol, Veracruz state
This site is located on the eastern watershed of the
Cofre de Perote volcano, Veracruz state, at an altitude
of 2500 m above sea level (Fig. 1). The Melanic
Andosol develops from volcanic ashes of different
age. The last eruption is dated at 10 000 years. A 1.5 m
pro®le was opened and used for the morphological
description, micromorphological sampling, and physical measurements. Two horizons were identi®ed by
colour and morphology differences. The melanic layer
is black and deep, extending down to 0.95 m. The next
layer, located from 0.95 to 1.5 m, is more friable and is
yellow in colour. However, four different horizons
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
45
Fig. 1. Map of the experimental areas: (A) Eutric Vertisol; (B) Texcoco Lake and Tepetates; (C) Melanic Andosol.
were distinguished by in situ bulk density measurement. The ®rst of these, at 0±18 cm, was more compacted and corresponded to the Andosol arable
horizon. The last one corresponded to the fresh ash
deposit.
2.1.2. Eutric Vertisol, Guanajuato state
This study area is located in Celaya, Guanajuato
state at the National Institute of Forest and Agricultural Research (INIFAP) experimental ®eld (BajõÂo).
Eutric Vertisols are the typical soils for this area, with
a high shrink±swell capacity which is dependent
on the water content (Oleschko et al., 1993, 1996;
Coulombe et al., 1996). The soils of this zone originated from different types of alluvium derived from
basalt and other igneous extrusive rocks.
The study area is representative of the Bajio agricultural zone, under continuous drip irrigation systems. A 1.5 m deep pro®le was opened, and four
different horizons were identi®ed. The arable layer
was limited by the plough-pan at approximately 30 cm
depth. This horizon is typical for Vertisols under
intensive agricultural use (two crops per year during
the last 50 years). The boundary with the next horizon
is at 80 cm depth, and is characterised by a change in
colour from black to deep brown. A change in soil
46
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
texture from clay to sandy loam marks the transition to
the third horizon at 120 cm. The detailed physical
characterisation of the Eutric Vertisol was accomplished only in the upper 45 cm of the pro®le.
2.1.3. Texcoco lake sediments, Mexico state
This sampling area is located in Montecillo, Mexico
state, and covers the eastern zone of the dry Texcoco
lake bank. The lake is a ¯at geomorphological zone
with total surface area of 1000 ha (Luna, 1980). The
Texcoco lake stratigraphy is composed of a basaltic
bed, and limestone and marine deposits of the Upper
Cretaceous period (at 2000 m depth below the surface). The upper strata are dominated by lacustrine
sediments (from 40 to 20 m) and by speci®c lacustrine
clay accumulations (from 6 to 0 m). The amorphous
clay, locally referred to as ``jaboncillo'', is characterised by speci®c physical and chemical properties:
an extremely low bulk density (280±430 kg mÿ3), a
high speci®c surface of 225 m2 gÿ1 and a very high
water retention capacity (e.g. more than 3 g of water is
retained by 1 g of clay).
This area represents the most interesting case for the
micromorphological and physical studies. Six horizons with clear differences in morphology as well as
physical and chemical properties and regular horizontal boundaries were identi®ed in the reference pro®le.
The natural water table occurs at 1.8 m depth and is
the lower boundary for the morphological description
which follows.
The sediment pro®le is divided into two contrasting
parts by an intermediate horizon of compacted basaltic
volcanic ash deposit with a bulk density of
1420 kg mÿ3 and a particle size varying from silt to
sand, at a depth of 39 and 69 cm (Fig. 2). This horizon
represents the textural screen between Mollic surface
layer (from 0 to 39 cm), and the clay accumulation
(from 69 to 180 cm). The detailed seasonal studies
have shown that these two parts of the pro®le are not
hydrologically connected. Therefore, the moisture
content in the upper part of the pro®le depends only
on the precipitation regime, whereas the water content
in the lower part is determined by the ground water
table ¯uctuations. The ®eld survey was carried out
in the dry season (January), after more than four
months had elapsed since the last rain. Thus, the
studied area was in quasi-equilibrium with respect
to water content.
Fig. 2. Texcoco Lake pro®le with abrupt limits between horizons.
On one side of the reference pro®le, three ¯at
surfaces 5 m5 m were opened for detailed physical
measurements. The ®rst surface was at 15 cm depth,
while the second coincided with the centre of the
volcanic ash deposits (50 cm), and the third corresponded to the pure lacustrine clay deposit at 80 cm
depth.
The dielectric constants of these materials ranged
from 13.01 in the upper zone to 65.9 in the water
saturated lower part of the pro®le. The bulk density
varied from 980 to 280 kg mÿ3 and the gravimetric
water content in steady-state conditions from 21.48 to
363.35% (estimated in grams of water retained by
100 g of solid material). These data emphasise the
high porosity of clays in the lower part of the studied
pro®le. However, microscopic observations of this
zone disclosed a predominance of pores of sub-microscopic size as well as presence of a few ®ssures.
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
2.1.4. Tepetates, Mexico state
In Latin America, hardened soils of volcanic origin
are often referred to by their vernacular names (Zebrowski, 1992). In Mexico, these formations are called
``Tepetates'', which means ``hard'' in the NaÂhuatl
Indian language. Commonly, Tepetates are interstrati®ed with volcanic and sedimentary deposits. All
pro®les include different paleosols characterised by
clear and abrupt horizontal limits between the layers.
In general, Tepetates in the Mexican high plateau are
natural, massive, compact and hard formations,
cemented by different chemical agents, including
clays and silica. Hardness is the distinctive characteristic of these materials.
All Tepetates have a speci®c geomorphological
position. Hardened horizons are inserted in piedmont
and glacis soils, and associated with brown clay soils,
de®ned as Cambisols (Zebrowski et al., 1991). This
location coincided with sub-humid and sub-arid
regimes (ustic), and suggests the climate and pedogenic effects on Tepetates formation (Quantin, 1992).
The opened pro®le presents several advantages for
the fractal study: (1) All cemented horizons are separated by paleosols with vertic properties. (2) The limits
between horizons are abrupt and horizontal, since
Tepetates are related to ancient volcanic events. (3)
The period of moisture quasi-equilibrium can be
selected easily. (4) Tepetates are characterised by
speci®c pore space morphology: main pores are isolated and occupied by cementing agents (Fig. 3d). The
apparent dielectric constant was measured only in the
upper 15 cm (Ka4.4), since the hardness and the
presence of numerous ®ssures make the measurements
in other layers statistically incoherent.
Nine horizons of different origin with contrasting
properties were described in this pro®le, and three of
them were identi®ed as cemented layers (Tepetates),
separated by clayey layers.
3. Methods
3.1. Micromorphological analysis
Three undisturbed samples (8 cm4 cm) were collected with metal samplers from each horizon of the
four opened pro®les. All samples were taken at ®eld
moisture and carried in plastic bags without drying. In
47
the laboratory, samples were dried by the acetone
replacement (in liquid phase) method and then impregnated with a 1:1 polyester resin (HU-543) and acetone
mixture (Murphy, 1986). Soil cores were re-impregnated with the same resin under vacuum conditions.
When the resin was suf®ciently hardened, samples were
horizontally sectioned parallel to the soil surface.
Thin sections (2 cm4 cm, with average thickness
of 30 mm) were prepared by standard petrographic
procedures (Brewer, 1964) and analysed under the
petrographic microscope (Olympus, BH-2). Four black
and white photographs were taken at the same scale
from each thin section. In total 12 microphotographs
were used for the fractal analysis of each soil horizon.
More contrasting examples are shown in Fig. 3.
3.2. Fractal analysis
Each photograph (15 cm10 cm) of the thin section
was scanned, using a 600 dpi resolution commercial
scanner. The obtained image was rendered on a grid of
10001000 pixels, and used for the automatic fractal
analysis. Adobe(TM) software was used to change all
grey scale ``tiff'' images into ``raw'' format. The
distribution of grey tones, varied from 0 (totally black)
to 255 (completely white) and was estimated using the
image histogram. On the microphotograph the boundary between the solid and pore networks is fuzzy;
therefore the limit between both sets was quanti®ed by
a range of grey tones. The Adobe PhotoShop software
was used to measure with pipette the grey values
corresponding to the solid and pore networks. The
obtained grey tone distributions were compared with
histogram data. The fractal programme, designed by
Parrot and Rico (1997) in Borland environment and
coded using C, was used for the mass fractal
analysis. This programme is based on the traditional
box-counting technique, and was calibrated with
deterministic fractals in previous research (Oleschko,
1999). Two ``box'', capacity or mass fractal dimensions were estimated for each image, and used to
describe the space-®lling ability of solid (Dms) and
pore (Dmp) networks, respectively. The values of Dms
and Dmp for each two-dimensional thin section image
is always less than two, since two would be expected
for a completely ®lled rectangle (Anderson et al.,
1996). The solid and pore mass fractal dimensions
were estimated by ®lling the corresponding image part
48
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 3. Microscopic images of the more contrasting horizons of: (a) Melanic Andosol; (b) Eutric Vertisol; (c) Texcoco Lake pro®le; (d)
Tepetates pro®le.
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 3. (Continued ).
49
50
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 3. (Continued ).
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 3. (Continued ).
51
52
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
progressively with larger boxes. The box size was
equal to m-pixels. The size of m begins at 1 and was
increased to a maximum value of 1000. Sixteen
different box sizes are used in the Parrot and Rico
(1997) programme. The number of m-pixels or boxes
that ®t in the solid (or pore) part of the image, when m
is increased, was counted. This method was described
and shown diagrammatically by Anderson et al.
(1996), who have mentioned that, the value of Dms
(and Dmp) can be estimated from a plot of ln (number
of m-pixels corresponding to the solid or pore network) versus ln(m). The theoretical line was adjusted
by least-squares regression and Dms and Dmp were
estimated from the negative of the slope.
3.3. Physical measurements
3.3.1. Time domain re¯ectometry
The time domain re¯ectometry technique (TDR),
was used to measure the apparent dielectric constant
(Ka) and the volumetric water content (yi) of the
studied soils and sediments. The Trase System Model
6050x1 is designed to measure dielectric constants
over a frequency bands between 100 and 1000 MHz.
A step pulse of electromagnetic radiation was sent
along the two parallel waveguides placed in the soil.
The permittivity of the material between the waveguides causes the pulse velocity deviation from the
known light velocity in vacuum (Jury et al., 1991). The
permittivity is estimated from the pulse travel time,
and the water content is calculated by a known
empirical third-order polynomial regression, by relating these two parameters. This model was proposed by
Topp et al. (1980). The obtained calibration curve was
used for the in situ volumetric content measurements.
In total 16 measurements were accomplished by TDR
on each horizon of interest.
Simultaneously to the TDR measurements, three
samples were taken for gravimetric water content (Wi)
and soil bulk density (rb). In the arable horizon of
Eutric Vertisol more detailed analysis was carried out,
taking more than 50 samples for Ka, Wi and rb (Fig. 4d).
Fig. 4. Relation between the material bulk density (rb) and apparent dielectric constant (Ka): (a) all materials compared together; (b) Melanic
Andosol; (c) Texcoco Lake pro®le; (d) Eutric Vertisol.
53
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
cone index (CI) data down to a depth of 600 mm to
give CI values up to 5000 kPa. The most important
feature of this penetrometer is that it uses an ultrasonic
method for measuring depth. CI data at all of the
studied sites were recorded on each of 16 sampling
points, every 15 mm down to 600 mm depth, and are
presented in Fig. 7 in values divided by 100.
The undisturbed samples (10 cm10 cm10 cm) were
obtained by using a coring tube of known volume
suitable to measure rb. The gravimetric water and bulk
density data were used for the volumetric water calculation. The volumetric water content, measured by
TDR and estimated from rb and Wi data, were compared, and used: (1) to de®ne the precision of the TDR
calibration curve in contrasting compactness conditions; (2) to establish the relation between yi and rb.
Subsequently, the last relation was used for the indirect bulk density estimation from the volumetric water
content data.
3.5. Statistical analysis
The data variation was estimated by classical statistical procedures. For all mass fractal dimensions the
mean value of 12 samples was used (Table 1). The
number of physical data has varied, depending on the
used technique, and not always has coincided with the
number of analysed fractal parameters, in this case
only mean values of variables were used to establish
the relationships of interest, where it was possible all
3.4. Mechanical resistance
A cone penetrometer (Rimik CP20) was used to
measure the mechanical resistance of soils and sediments. The cone penetrometer measures and records
Table 1
Solid (Dms) and pore (Dmp) fractal dimensions of studied materialsa
Da
(kg mÿ3)b
Kac
Morphological description
(0.009)
(0.012)
(0.009)
(0.010)
0.39
0.37
0.32
0.3
18.7
25.6
36.0
44.4
Melanic horizon
Sandy loam
Sandy loam
Sandy
1.9310 (0.009)
1.9363 (0.011)
1.9350 (0.008)
1.8845 (0.090)
1.8352 (0.003)
1.7987 (0.007)
1.07
1.23
1.35
29.55
Arable layer
Arable layer
Plough-pan
0±10
27±38
56±67
69±80
90±100
110±120
1.9385
1.9479
1.9351
1.9570
1.9697
1.9580
(0.005)
(0.010)
(0.009)
(0.012)
(0.009)
(0.008)
1.8233
1.8620
1.8741
1.7651
1.7174
1.7808
(0.01)
(0.020)
(0.019)
(0.002)
(0.006)
(0.020)
0.98
1.42
1.21
0.43
0.37
0.28
13.01
16.43
21.84
44.85
56.09
65.9
Mollic horizon
Loamy horizon
Basaltic volcanic ashes
Loamy with small clay subhorizons
Clay
Clay
0±25
69±87
87±124
124±150
200±220
220±235
235±260
260±420
420±460
1.9520
1.9537
1.9600
1.9408
1.9593
1.9773
1.9150
1.9645
1.9888
(0.001)
(0.004)
(0.004)
(0.005)
(0.010)
(0.002)
(0.004)
(0.010)
(0.007)
1.8360
1.8033
1.8875
1.7768
1.7815
1.8013
1.9079
1.7606
1.6219
(0.009)
(0.032)
(0.006)
(0.003)
(0.050)
(0.008)
(0.004)
(0.016)
(0.165)
1.57
1.63
1.15
1.25
1.68
1.68
1.19
1.30
1.66
4.4
Soil
Depth
(cm)
Dms
Melanic Andosol
0±18
18±55
55±95
95±130
1.9374
1.9304
1.9246
1.9278
Eutric Vertisol
0±15
15±30
30±45
Texcoco Lake deposits
Tepetate profile
a
Dmp
(0.007)d
(0.008)
(0.008)
(0.009)
1.8864
1.8768
1.8969
1.9082
Some data of this table were published before by Oleschko (1999).
Bulk density.
c
Apparent dielectric constant.
d
The standard deviation is in parentheses.
b
Colluvium
Vertic properties
Vertic properties
Tepetate, t2a
Tepetate, t2b, hard
Tepetate, t2b, hard
Paleosoil
Paleosoil
Tepetate, t3, very hard
54
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
measured values were graphed (Fig. 4d, Figs. 7a±d,
8a±d and 9a±c).
4. Results and discussion
4.1. Fractal analysis of the microscopic images
Pentland (1984) has shown that a fractal surface and
its bi-dimensional photographic image are mathematically equivalent. In the present study, the fractal
structure of the solid and pore networks of soils and
sediments in four contrasting geosystems, was documented in the scale range from 0.009 to 0.2 cm, using
their microscopic images. All analysed networks were
characterised by a single, near ``ideal'' fractal dimension, independent of scale. The solid (Dms) and pore
(Dmp) mass fractal dimensions have shown a clear
dependence on the soil and sediment genesis (Table 1).
A maximum fractal dimension of 1.9888 was obtained
for the solid set of fragipan (t3), in the Tepetate pro®le,
identi®ed as the hardest horizon. This means that the
solids in this layer occupy considerably more space
than the pores. The minimum value of the pore mass
fractal dimension (Dmp1.6219) was found for the
same horizon. The Melanic Andosol and the paleosol
with vertic properties in the Tepetates pro®le have the
lowest values of the solid set fractal dimension
(1.9246 and 1.9150, respectively). These results coincide with our previous data (Oleschko et al., 1997,
1998), and showed that the fractal dimension, within
the studied scale range, is a useful parameter for
distinguishing between materials of different genesis.
It means that this parameter may possibly be used to
monitor the tillage in¯uence on soil physical properties, including the soil compaction status. Nevertheless, from time to time the differences among the
fractal dimension of extremely contrasting materials,
were small. For instance, the solid set fractal dimensions in the arable horizon of the Melanic Andosol and
Eutric Vertisol soils with clear bulk density differences
(390 compared to 1070 kg mÿ3), were 1.9374 and
1.9310, respectively. The highest contrasting example
is related to Dmp of the amorphous clay layer (Texcoco
Lake, Dmp1.7651), compared with the cemented
horizons (Tepetates, t2a formation, Dmp1.7768),
which are estimated in the same range of scales and
are not statistically different. These results agree with
Anderson et al. (1998), who have concluded that the
common box-counting technique takes into account
only pores that are visible at the resolution of the
analysed photograph. The visible porosity is similar in
the compacted clay and cemented Tepetate horizon,
when the sub-microscopic porosity is out of the
method resolution. Therefore, it should be emphasised, that it is essential to refer all fractal analysis
conclusions to the exact scale range at which the data
were acquired, and that more than one fractal parameter is needed for soil structure description. The
standard deviation varied between 0.001 and 0.011 for
the solid mass fractal dimension, and between 0.003
and 0.165 for the pore mass fractal dimension
(Table 1).
4.2. Relationship between the soil and sediment
physical properties and mass fractal dimensions
For a better understanding of the origin of fractal
dimension dependence on the material genesis, the
relationships between some physical properties of the
studied soils and sediments and their solid and pore
fractal dimensions were analysed. A number of dif®culties, including some incoherent data and results
opposite to the theoretically expected tendencies, were
encountered. This was due to either the speci®c properties of all studied materials, which vary widely, or
independence of fractal parameters from soil genesis.
Nevertheless, it was considered that even if relations
between solid and pore mass fractal dimensions and
physical properties were not clearly obvious, it would
be very helpful to know some important trends in the
dynamics of these fractal parameters in soils and
sediments with different compaction status. A representative analysis of both fractal dimensions and
physical properties would ensure, whether such correlation of fractal dimensions and physical properties
exists, or it is underlined by large measurement noise
and random sample spatial variation.
Two types of analysis were carried out. First, each
of the relations of interest was established for soils and
sediments, analysed all together, and then, the same
analysis was applied for each one of the studied
pro®les. Special attention was given to Texcoco Lake
deposits, where the layers with the strongest differences in physical and morphological properties were
found.
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
4.2.1. Relation between bulk density, apparent
dielectric constant and mass fractal dimensions
Dirksen and Dasberg (1993) tried to establish a
single relation between the soil permittivity and volumetric water content related directly to the material
bulk density. They concluded that the bulk densities
differed not only between soils, but also among arti®cially packed columns of the same soil: bulk densities
®rst decreased and then increased with water content
increase. Therefore, the relationship between rb and
Ka has two components with completely opposite
tendencies. Dirksen and Dasberg (1993) strongly
recommended comparing this pair of variables individually for each studied material.
In the present study, a nearly linear relation
(R2=0.55) was obtained among rb and Ka when all
horizons of the four pro®les were compared (Fig. 4a).
The established trend has shown that the higher the
value of rb (the more compact a soil), the lower the
dielectric constant. Therefore, the more porous material has a greater permittivity. All compared materials,
except the Tepetates pro®le, had water contents near
®eld capacity at the time of sampling. Therefore, it
seems reasonable that, lower values of bulk density
correlated with higher values of the volumetric water
content, and also, with larger apparent dielectric constants.
In the Melanic Andosol, this relation has shown the
best ®t (R20.98). The layer with maximum bulk
density (390 kg mÿ3) has the minimum dielectric
constant (18.7, Fig. 4b). It can be noted that in this
volcanic and highly porous soil, a change in bulk
density from 390 to 300 kg mÿ3 coincides with a
drastic change in the volumetric water content (from
25.3 to 58.3%) and a considerable increase in apparent
dielectric constant (from 18.7 to 44.4).
In order to validate the established relation in more
contrasting materials, detailed studies were conducted
in the Texcoco Lake and in the Eutric Vertisol. In the
latter area, the only arable layer was analysed in detail.
A strong direct linear relation (R20.96) was obtained
for the Texcoco lacustrine deposits (Fig. 4c). The near
saturation clay horizons with bulk densities of 280 and
370 kg mÿ3 were characterised by the maximum
values of apparent dielectric constant (65.9 and
56.09, respectively).
Nevertheless in contrast, an inverse tendency was
observed in the arable layer of the Eutric Vertisol,
55
where more than 100 TDR measurements were made,
with simultaneous sampling for bulk density (the
coring tube method) and gravimetric water content.
High data variability was noted, but the dielectric
constant increase with bulk density was statistically
con®rmed (R20.57, Fig. 4d). This atypical behaviour
of Vertisol with high shrink±swell capacity is well
documented in the literature (Oleschko et al., 1993,
1996; Coulombe et al., 1996), and is considered one of
the possible reasons of the reported differences. The
unsuitability of the commonly used tube method to
obtain the precise Vertisol bulk density measurements,
and the absence of some alternative techniques may be
the other reason for the observed trend.
The mass fractal dimensions of soil solid (Dms) and
pore (Dmp) networks were related to the bulk density
(rb) and apparent dielectric constant (Ka) of the
studied materials. A good ®t (R2=0.75) was obtained
for the Melanic Andosol, where a direct linear relation
between the Dms and Ka was observed (Fig. 5a). The
horizon with the maximum value of solid fractal
dimension (1.9374) had the lowest apparent dielectric
constant (18.7). From a theoretical point of view, the
larger the solid fractal dimension the higher is the
solid occupied space, and as a consequence the larger
the bulk density and the lower the volumetric water
content. A similar, but only slightly correlated statistical relationship (R2=0.46) was observed between
the Andosol bulk density and the solid set mass
fractal dimension: the more compacted horizon
(Da390 kg mÿ3) was characterised by the maximum
Dms (1.9374, Fig. 5b). This conclusion agrees with
general assessments about the solid set mass fractal
dimension dynamics. Previously, comparing the same
soils under different tillage conditions, it was established that the solid mass fractal dimensions were
higher and the pore mass fractal dimensions were
lower for the more compacted soil (Bartoli et al.,
1991; Oleschko et al., 1997, 1998). Therefore, it
was shown that Dms was inversely related to average
soil porosity.
In the Melanic Andosol, a reasonably good direct
linear ®t (R2=0.75 and 0.79) was obtained between the
pore fractal dimensions, material bulk density and
apparent dielectric constant (Fig. 5c and d, respectively). The layer with the highest mean pore fractal
dimension (1.9082) and the largest porosity has the
highest apparent dielectric constant (44.4).
56
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 5. Relation between Melanic Andosol: (a) solid set fractal dimension (Dms) and apparent dielectric constant (Ka); (b) Dms and bulk
density (rb); (c) Dmp and Ka; (d) pore set fractal dimension (Dmp) and bulk density (rb).
Relationships between the solid fractal dimension
and the bulk density observed in the Texcoco Lake
pro®le were unclear (Fig. 6a). The high data dispersion
was typical for this experimental area. However,
R20.59 suggests that one of the horizons with the
highest clay content (90±100 cm) had the largest Dms
(1.9697), high apparent dielectric constant (56.09,
Fig. 6b), and low bulk density (370 kg mÿ3). This is
in line with the ®ndings of Perfect and Kay (1995) and
others who also found that Dms increased with increasing clay content.
The relationship between the pore mass fractal
dimension and bulk density in this pro®le, was opposite to the theoretical trend. The horizons with the
largest Dmp (and therefore with the smallest Dms and
highest porosity) were characterised by higher bulk
density (R20.86, Fig. 6c). The speci®c nature of this
lacustrine amorphous clay with bulk density decreasing down to 280 kg mÿ3, is that of a high invisible submicroscopic porosity. This was considered responsible
for the observed tendency, where the increment of
pore mass fractal dimension coincided with the
increase in horizon compactness. This is consistent
with the relation established in the same pro®le
between the pore fractal dimension and the apparent
dielectric constant: the larger the Dmp, the smaller is
the Ka (R20.60, Fig. 6d).
Rieu and Sposito (1991), using the Chepil data,
came to a similar conclusion, establishing that
increases in Dm (only solid dimension was estimated
in their study) were correlated to increases in clay
content. Crawford et al. (1993) and Anderson et al.
(1996) have concluded that, generally, a soil structure
with large continuous pores had larger values of Dmp
than a structure with small discrete pores and low
lacunarity. However, Anderson et al. (1996) have
underlined that Dmp does not depend wholly on porosity, and different values of Dmp are obtained for
structures of equal porosity but with different spatial
distribution of pore sets (distinct lacunarity). It was not
possible to check the above trend for the Tepetates
pro®le, since the data of the apparent dielectric con-
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
57
Fig. 6. Texcoco Lake pro®le, relation between: (a) solid set fractal dimension (Dms) and bulk density (rb); (b) Dms and apparent dielectric
constant (Ka); (c) pore set fractal dimension (Dmp) and bulk density (rb); (d) Dmp and Ka.
stant were obtained only for the upper, unconsolidated
horizon. In the Vertisol, the high data dispersion,
explained by the presence of ®ssures, was responsible
for the statistically incoherent ®ts between the variables of interest. However, if only mean values of
variables were considered, the largest pore fractal
dimension (1.8845) would coincide with the minimal
bulk density (1070 kg mÿ3 and R2=0.83).
In the present study it was impossible to establish
some general relation between Dms and Ka, for the
strongly contrasting soils and sediments studied. This
relation should be established individually for each
pro®le.
It was concluded that, in general, for the studied
soils and sediments with water contents near ®eld
capacity, the apparent dielectric constant tends to
increase with decrease in bulk density (except the
Eutric Vertisol). However, the relation between the
material bulk density, apparent dielectric constant and
mass fractal dimensions strongly depends on the soil
and sediment origin.
4.2.2. Relation between the mechanical resistance
and mass fractal dimensions
An inverse intuitively valid linear relation was
detected in the Texcoco Lake pro®le between the soil
apparent dielectric constant and mechanical resistance
(R20.717, Fig. 7a). The clay horizon has signi®cantly lower CI than the compacted basaltic ash
deposits. Nevertheless, in this pro®le, all obtained
data were derived from materials with extremely
contrasting properties, therefore the experimental
points form two clouds concentrated at both extreme
ends of Ka versus CI graph, and never form a real
continuous function. The mass fractal dimension of
the pore set also shows the inverse, contrary to the
theoretically expected, relation: the layer with bigger
pore dimension (1.8620) has the largest mechanical
resistance (8.1 kPa cmÿ2, Fig. 7c). The same unexpected trend was found for the relation between solid
mass fractal dimension and mechanical resistance of
the Texcoco Lake sediments (Fig. 7b). The decrease in
the Dms values (in the upper part of the pro®le with less
58
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
Fig. 7. Relation between material mechanical resistance (CI/100) and: (a) apparent dielectric constant (Ka); (b) solid set fractal dimension
(Dms); (c) pore set fractal dimension (Dmp), in Texcoco Lake pro®le; (d) CI versus Dmp in Eutric Vertisol.
clay content) coincided with the increase in the material's mechanical resistance (R20.50). Similar relations were observed for the solid and pore mass fractal
dimensions versus mechanical resistance in the Melanic Andosol pro®le. The deeper horizons (from 55 to
130 cm) have larger CI (600 kPa cmÿ2 in comparison
with 380 kPa cmÿ2 in the melanic horizon) and smaller solid set mass fractal dimensions (1.9278 versus
1.9374, R20.92). Therefore, the mass fractal dimension is not the direct indicator of soil compaction
status, depending more on the material genesis.
The results obtained for Eutric Vertisol con®rm this
hypothesis. The solid mass fractal dimension in the
®rst 45 cm of Eutric Vertisol, sampled at 15 cm intervals, was almost constant at 1.93, however the pore
dimension has diminished from 1.8845 (0±15 cm) to
1.7987 (30±45 cm), and the mechanical resistance
increased from 797 to 1317 kPa cmÿ2. Changes in
the pore mass fractal dimension and mechanical
resistance re¯ect the presence of a plough-pan, while
Dms is independent of this. An inverse linear relation
was established between CI and Dmp: higher mechanical resistance coincided with smaller pore mass fractal dimensions (R20.96, Fig. 7d). These results are in
agreement with bulk density data and the soil morphological description.
In the Tepetates pro®le, it was not possible to
analyse the relation between the mechanical resistance
and mass fractal dimensions of different layers,
because the cone penetrometer was unsuitable for
use in these consolidated materials.
4.2.3. Relation between horizon depth and mass
fractal dimensions
Hatano and Booltink (1992) have estimated the
mass fractal dimension (Dmp) from the stained ¯ow
pattern, using two-dimensional images. They concluded that, in general, values of Dmp decrease with
depth. Similar results were obtained in the present
study for the pore fractal sets of the Texcoco Lake,
Tepetates and Vertisol soils, where the pore set fractal
dimension decreased with depth (R20.94, 0.63 and
0.82, respectively, Fig. 8b±d). These data indicate a
lower porosity in the deeper layers, linked with a solid
set fractal dimension increase in the same direction.
However, the inverse trend was obtained for the
Andosol pro®le (Fig. 8a) with the deeper and more
Fig. 8. The dynamic of the pore set fractal dimension with depth, in: (a) Melanic Andosol; (b) Texcoco Lake pro®le and (c) Tepetates pro®le;
(d) Eutric Vertisol.
Fig. 9. The dynamic of the solid set fractal dimension with depth, in: (a) Melanic Andosol; (b) Texcoco Lake pro®le; (c) Tepetates pro®le.
60
K. Oleschko et al. / Soil & Tillage Research 55 (2000) 43±61
porous horizons having higher pore fractal dimensions
(R20.61).
A trend opposite to the pore networks, was obtained
for the solid set fractal dimensions in relation to pro®le
depth (except Vertisol, Fig. 9a±c). Different and sometimes opposite tendencies, depending on the origin of
soil and sediment, were detected. In the Melanic
Andosol (R20.71) Dms decreased with depth
(Fig. 9a). Then the more recent, more porous and
unweathered ashes constitute the deeper layers. The
opposite trend, and therefore the increase of the solid
mass fractal dimension with depth, was detected in the
Texcoco Lake and Tepetates pro®les (R20.81 and
0.70, respectively, Fig. 9b and c). These relationships
are strongly linked with the deposition history of the
materials. For instance, in the Texcoco Lake the clay
horizon comprises a more recent undisturbed material,
that was never in direct interaction with the surface. In
the Tepetates pro®le, the deep Tepetate (t3) is the
hardest layer with clear geological origin, which
shows only a few features of perturbation. In the
Eutric Vertisol, the pedoturbation is responsible for
the continuous mixing of materials, and the homogenisation of the pro®le up to one or more meters of
depth. In this case, the Dms appears to be independent
of depth (Table 1).
compaction status. Nevertheless, the solid mass fractal
dimension depends more on material genesis. On the
same image, the pore mass fractal dimension is always
smaller than the solid one. A complete description of
this pattern requires the multiscale fractal analysis,
and the use of some additional fractal parameters (i.e.
lacunarity). It seems clear that for many soil properties
affecting water movement and root growth, the poresize distribution pattern has much greater importance
than total porosity, bulk density and mass fractal
dimensions. It is possible to establish simple and
statistically coherent relations between fractal parameters and some selected physical properties of materials. These relations should be established
individually for each soil and sediment of interest.
Nevertheless, for the same soil or sediment, the pore
mass fractal dimension is a useful indicator of soil
compaction status and visible porosity.
Acknowledgements
This research was supported by DGAPA (PAPIT
programme), UNAM (project IN-106697) and CONACYT (project 3617P-A), Mexico. We thank A.M.
Rocha T. for technical support and M. Alcayde O. and
Dr. G. Tolson for improving the English version of the
manuscript.
5. Conclusions
The fractal dimensions of solid and pore networks
are useful parameters capable of distinguishing
between materials with contrasting genesis. The
cemented horizons (Tepetates) are characterised by
the maximum solid and minimum pore fractal dimensions. The solid mass fractal dimension near 1.99, and
the pore mass fractal dimension near 1.62, determined
for these hardest and extremely compacted layers,
may be proposed as the bounded values for the soil
solid and pore networks. The former value is close to
the upper topological limit of a two-dimensional
image. However, the traditional box-counting technique applied to image of one scale is able to estimate
only visible porosity, and therefore, on occasions, the
soils and sediments with extremely contrasting physical properties may be described by similar mass
fractal dimensions, inside the studied scale range.
The pore mass fractal dimension re¯ects the soil
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