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SPECIAL SECTION: MULTISCALE MAPPING

Fractal Mapping of Pore and Solid Attributes

^a Dep. de Estudios de Posgrado de la Facultad de Ingeniería, Univ. Nacional Autónoma de Querétaro, Santiago de Querétaro, Mexico
^b Centro de Geociencias, Univ. Nacional Autónoma de México (UNAM), Juriquilla, Querétaro, Mexico
^c Dep. de Edafología, Instituto de Geología, Univ. Nacional Autónoma de México, CU, 04510, Mexico, D.F., Mexico
^d Instituto de Geología, Univ. Nacional Autónoma de México, CU, 04510, Mexico, D.F., Mexico
^* Corresponding author (olechko{at}servidor.unam.mx).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Received 24 January 2007.

	ABSTRACT

TOP ABSTRACT INTRODUCTION Materials and Methods Results and Discussion Conclusions REFERENCES

 
Soil structure depends on its genesis and consists of highly variable pore and solid networks. Several internal and external factors affect the attributes of these networks, with water being the most aggressive agent. In this study, we used selected fractal parameters (called descriptors) to quantify the basic topological attributes—compactness and connectedness—as well as lacunarity and roughness of porous materials, with special attention to sampling error and population variance dynamics. Four microhorizons were sampled from a 1.0- by 1.0- by 1.6-m monolith during a long-term drying period under controlled conditions. A comparative fractal analysis of selected multitemporal (11 mo) and multiscale (50x, 500x, 1000x, and 5000x) scanning electron microscopy (SEM) images was accomplished for the following layers with contrasting texture or structure: loam topsoil, pure basaltic compacted sand, and two amorphous lacustrine clays. The scale invariance (self-similarity) of pore and solid networks was documented. A complex relationship was found between the descriptor mean values and corresponding variances. Simple power-law relations were established between the standard deviation (and variance) and layer depth, as well as scale and sampling time. Pearson's r correlation criterion was applied to measure the degree of association between multiobjective fractal descriptors. Systematic and standardized fractal analysis of multiscale images is recommended for the statistical quantitative description of complex and dynamic pore and solid networks.

Abbreviations: BI, binarized image • OI, original image • SEM, scanning electron microscope

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Materials and Methods Results and Discussion Conclusions REFERENCES

 
SOIL STRUCTURE comprises a complex and highly heterogeneous set of attributes whose topology results from the spatial arrangement of solid particles and aggregates of different nature, morphology, and size. A soil structural pattern can be visualized as a multiscale, hierarchic, and multifaceted combination of solid and pore networks. The basic soil properties (physical, chemical, and biological) depend on the exact geometry, topology (compactness and connectedness), and stability of these networks (Whiting and Wilson, 2006). The dynamic proportion among the solids, liquids, and gases derived from these structural pattern attributes (Whitlow, 2000) result in a typical physical and mechanical behavior for each soil. Recent applications of fractal techniques to the analysis of soil structure have proved the efficiency of nonlinear models to quantify and map the basic attributes of pore and solid networks as well as the topology (Xu, 2005), giving more realistic and dynamic statistical results (Cheng, 1999) in comparison with traditional methods.

Soil pore space is always perceived as the most variable part of the system (Sovik and Aagaard, 2003), with attributes oscillating within a wide range of length scales in response to continuous modifications of the physical background. Changes in the pore volume occupied by air and water become more obvious in high-clay soils (Güntner and Bronstert, 2004). Differential porosity is the most complex property inherited from soil genesis, while topology, pore size distribution, and morphology result from interactions between the soil structure and mass and energy fluxes through the profile (Flores, 2003). A relatively high proportion of macropores favors preferential water flow. In contrast, a predominance of capillary pores enhances water storage capacity, optimizing the functioning of the soil–water–plant system as well as root development and growth (Gil, 2001) and favoring gas and heat diffusion and exchange between the soil and physical media (Armatas, 2006). It should be emphasized that any change in porosity is accompanied by modifications in the mechanical properties of the porous medium. Sometimes a lower porosity produces a smaller shear stress and a lower elastic modulus (Braja, 2000), accelerating system degradation. The dynamics of physical and mechanical properties in soil, including clay compressibility, have been described and modeled intensively by fractal techniques (Xu and Xia, 2006).

But what is a fractal? Mandelbrot (1983) believed that one would do better without a unique definition of this concept. To him, fractals are irregularly shaped and fragmented sets, either mathematical or real, with parts similar to the whole in appearance or in a statistical sense. Recently, Mandelbrot (2002) informally defined fractal geometry as "the study of scale-invariant roughness," which is closer to the main objectives of the present research and will be used for the following discussion.

Fractal analysis has become a common practice for structural pattern description in medicine (Gamba et al., 2003; Manousaki et al., 2006), physics (Martínez-Lopez et al., 2002), the geosciences (Kalda, 2003) including especially geology (Plotnick et al., 1993), art (Voss and Wyatt, 1993), ecology (Kirkpatrick and Weishampel, 2005; Rauch, 2005), and archeology (Oleschko et al., 2000; Brown et al., 2005), proving to be suitable for withdrawing realistic attributes of pore and solid networks from images (Raoush and Willson, 2005), time series, and fields of numbers (Korvin and Oleschko, 2004).

In soil science, the fractal models of pore (or solid) size and frequency distributions are used to predict highly variable (Richter, 1987) and characteristic parameters such as permeability and conductivity (Sobieraj et al., 2004). These parameters are extracted from the log–log plots of pore (or solid) size and scale (or size and frequency) distributions (Perrier et al., 1999). Fractal models are applied to the description of the scale invariance of soil retention and conduction curves (Tuli et al., 2001; Xu, 2004a; Millan and Gonzalez-Posada, 2005; Rojas and Rojas, 2005), while the fractal analysis of multiscale and multitemporal images allows attainment of statistically precise quantitative information about retracted object structure in terms of gray grade statistics, from which the main features of structural patterns are obtained (Oleschko et al., 2002, 2003; Korvin and Oleschko, 2004). The time series analysis is associated with the space–time complexity and multifractal predictability (Schertzer and Lovejoy, 2004) of the studied processes, converting the fractal techniques in a highly precise toolbox to measure the basic attributes of porous media. This efficiency is derived from the scale invariance (scaling) of pore and solid networks in space and time, documented for a wide range of scales (Perfect and Kay, 1995; Pachepsky et al., 1995). The main structure attributes show some degree of association with physical and mechanical properties of natural systems; however, the influence of image resolution, time series length, and the algorithm used for each measurement determines the quality of the results (Pendleton et al., 2005).

The main objective of the present study was to select a toolbox of reference techniques (Mandelbrot, 2002) for a comparative, multiscale, and multitemporal analysis of the dynamics of solid and pore network attributes in materials subjected to long-term drying under controlled conditions. The sampling error of selected fractal descriptors was estimated by standard deviation (SD), while the population variation (extracted from 50 data for each descriptor) was measured by variance. The degree of fractal descriptor association was quantified by Pearson's r correlation analysis. The scaling of SD and variance as well as the power-law relation between them and the mean value of each studied fractal variable is shown.

	Materials and Methods

TOP ABSTRACT INTRODUCTION Materials and Methods Results and Discussion Conclusions REFERENCES

 
Study Site and Materials Description
The experimental setup below was described in Flores (2003). The sampling area is located on a flat zone near Montecillo, State of Mexico, at the eastern dried bank of Texcoco Lake (19°27' N, 98°54' W). A monolith with dimensions of 1.0- by 1.0- by 1.6-m depth was extracted from these lacustrine deposits under field-moist conditions, with the water profile in a quasi-steady state. Water content of the topsoil was quantified as field capacity, while the bottom of the profile was saturated. Under these moisture conditions, the monolith was transported to a greenhouse of the Institute of Geology, National Autonomous University of Mexico (UNAM), where it was turned in its side and dried for 11 mo (Flores, 2003; Fig. 1 ). The monolith is composed of several microhorizons (layers) of contrasting genesis, with lake sediments dominant in the lower part (Oleschko et al., 2002). Each layer was dried independently by capillary flux through the surface exposed to physical evaporation, all other sides being protected by a wooden box (Flores, 2003).

View larger version (154K): [in this window] [in a new window]   FIG. 1. The 1.0- by 1.0- by 1.6-m depth monolith is composed of several microhorizons with contrasting structure. It was dried in a greenhouse for 11 mo. Each layer was dried independently by capillary flux through the surface exposed to physical evaporation (Flores, 2003).

The dynamics of microstructure, in time and space, was compared for four contrasting microhorizons of the monolith. The first layer (I) corresponds to the topsoil (0–10-cm depth), is classified as a Udoll (Phaeozem), and has the maximum organic matter content in comparison with the other layers (Gutiérrez, 1997). The second layer (II), located between 30 and 40 cm, is composed of basaltic sand. The third (III) and fourth (IV) layers, located at 130- to 140- and 150- to 160-cm depth, respectively, consist of lacustrine clay sediments in direct contact with highly saline underground water. The former horizon consists of a mixture of pure clay and a high quantity of silicified diatom bodies ranging in size from silt to sand (Flores, 2003). The diatom content is used in prehistoric research as an indicator of climatic changes that occurred in an area during sediment deposition. During drying of the monolith, each body was the principal modifier of the topology of developing mechanical tension fields. Diatom bodies act like pseudo-sand and -silt particles, defining the observed differences in topology (and so in fractal descriptors) among structural patterns of compared clays (Layers III and IV).

Undisturbed samples were taken daily from each studied layer and were imaged at four different resolutions using a low-vacuum scanning electron microscope (SEM). Sampling was initiated in April 1999 and ended in February 2000. Most of the SEM images were captured in gray-scale thermal impressions using magnifications of 50x, 500x, 1000x, and 5000x. A multiscale fractal analysis of pore and solid patterns was performed during the drying of the monolith. The selection of magnification range was arbitrary, based on two considerations: (i) the volume of data to be analyzed (50 images for each one of 13 descriptors used)—each additional magnification meant adding 16 measurements for each fractal parameter; and (ii) only the data coming from the same physical source (SEM) were compared. All studied images form part of our multiscale and multitemporal data bank gathered during previous research in which the fractal nature of Texcoco Lake materials was documented for optic and electron microscopy images within a wide range of scales (0.008–3 mm), using magnifications from 5x to 20,000x and 12 images per layer (Oleschko et al., 1998, 2000, 2002, 2003; Flores 2003).

Fractal Analysis
It is well known that several fractal dimensions are required to characterize the fractal set (Korvin, 1992). In the present research, we chose those techniques recommended as reference (Seffens, 1999) and designed for the analysis of self-similar as well as self-affine fractals. These techniques constitute the BENOIT (Version 1.3) commercial software and were calibrated in previous research (Oleschko et al., 1998, 2000, 2002, 2003; Flores, 2003), showing their high suitability to extract the basic attributes of solid and pore networks of diverse natural systems with known sampling error (quantified in terms of the SD). The fractal analysis was accomplished on digital bidimensional images. In addition to BENOIT, other algorithms specially designed for different projects at the Laboratory for Fractal Analysis of Natural Systems were used. These algorithms were written in the Borland environment and encoded in C++ during several years by Dr. Jean-Francois Parrot and collaborators (1998–2003). The basic features of the algorithms applied to the present research are described below.

Binar3
The fractal dimension is the basic parameter used to describe the grade of irregularity, fragmentation, or roughness of the fractal set. Voss and Wyatt (1993) emphasized that this concept is only applicable to sets, collections of points or regions that are specified according to some membership rule. Therefore, the use of a fractal dimension assumes a binary view of the world, where each specific point is either a member of the set of interest (solid or pore in our case) or not (Voss and Wyatt, 1993), which implies the need for image binarization previous to fractal analysis. From the beginning, the image segmentation is assumed to be biased, since the output results depend on the user's ability to define a threshold between the gray levels, dividing the image into two parts (solids and pores in the present research). The threshold between sets was defined in Paint Shop Pro (Version 7.0), passing the dropper tool manually through the interface between two previously specified parts of the image. The Binar3 algorithm (Parrot, 1997) transforme the image into a black and white map with similar appearance to the original one. A representative interval of gray levels, corresponding to the interfaces, was selected and introduced into Binar3 to create a binary map, referred to as the measure image, from which the fractal parameters of pore and solid networks were extracted. All these parameters were compared with those extracted by BENOIT without image binarization.

Box-Counting Dimension
The box counting dimension, D_box, is defined as the exponent of the power law

[1]

where N(d) is the number of boxes of linear size d necessary to cover a set of points of interest distributed across the bidimensional plane (the image, in our case). The box-counting technique of BENOIT 1.3 measures the dimension of the white part of a bitmap image. Therefore, to extract the dimension of solid as well as pore networks, two measurements should be made for each image, using first the original SEM image (where the pores appear in black) and then its negative (with solids in black, measuring the D_box of the pores). The analysis is performed automatically, selecting all input parameters by default (including the threshold value of the boundary between solids and pores). The output file contains the D_box and SD values, displaying point by point in a log–log plot the number of boxes occupied by the set of interest vs. the box size (Fig. 2 ). Data fit to a straight line can be improved by eliminating the outliers. The elimination of points results in changes in the D_box and SD values. The algorithm covers the image with grids of varying box size (Fig. 2) placed at any random position and orientation, looking for the minimum number of boxes needed to cover the set of interest and ensuring <5% overestimation of N(d). BENOIT accomplished this procedure by rotating the grid for the selected box size (90° by default) and plotting the minimum value of N(d); however, the user can change the angular increment of rotation as well as the interval of box sizes.

View larger version (67K): [in this window] [in a new window]   FIG. 2. Four output files of the box counting technique (BENOIT, Version 1.3) applied on scanning electron micrograph images of the same 1000x magnification showing the details of this procedure and making visible some attributes of the pore and solid sets of the (a) Udoll topsoil, (b) basaltic sand, and (c and d) clays.

The designed path is recognized as valid only if the whole vector is kept within the set of interest, i.e., inside the range of grays selected during the image segmentation. If this condition is met, the point reached by a walk becomes the starting point of the new path (segment). If the path lies outside the analyzed set, the design would return to the starting point and the procedure would be repeated in a random way until all initial conditions are satisfied and a new walk is initiated. The user defines the number of iterations (n) to be performed.

For each designed walk, the total number of visited pixels (S_t), as well as the number of "null" steps (S₀) and the number of steps located within the set of interest (S_n) are counted. The null steps are defined as the paths that intercept some other previously accomplished trajectory. If the step coincides with a previously visited pixel (a null step), one is added to the number of total visited steps (S_t), while S_n is kept unchanged. More details about this method can be found in Anderson et al. (1996). The random walk is finished when the cumulative length reaches a maximum or when a maximum number of null steps (100 for our case) is reached. Both values are established by the user at the beginning of the program. The maximum number of steps >1 (crossings) was 5 and the minimum number of pixels in a path was 200. When either the maximum length or the number of null steps is reached, a new random walk is initiated. The user defines the maximum number of pathways that will be run across the image, taking into account that the greater the number of steps, the bigger will be the coverage of the selected feature by the Brownian movement and the closer to the original the measure image will be (Fig. 3 ).

View larger version (80K): [in this window] [in a new window]   FIG. 3. The (a) original scanning electron micrograph image of the Udoll topsoil (50x) compared with (b) the measured image composed by continuous time random walks (in red) drawn by the Bromov_8a algorithm (Parrot and Taud, 1998).

Lac_Grid
Lac_Grid is an algorithm written by Parrot and Taud (1998), following the method of Allain and Cloitre (1991), to calculate the lacunarity () of an image using gliding boxes of regular shape. Currently, the so-called self-referenced technique is designed to estimate by approximating the shape of the gliding boxes as a function of the pattern's original morphology (Rodríguez et al., 2005). The algorithm applied in our study analyzes the variability of gray levels across the image, within moving square windows of variable size. The image exploration starts from the minimum window size of three pixels and finishes with the maximum size selected by the user at the beginning of the program (11 in the present research). In the method proposed by Allain and Cloitre (1991), the lacunarity is calculated by comparing the first- and second-order central moments (the mean and variance) of the gray distribution across the image. Equation [2] describes as a function of window size (r), by which these statistical parameters are calculated:

[2]

where (r) tends to 1 when the mass distribution of grays across a fractal set is homogeneous. The range of pore, gap, or lacune sizes in the image increases with (r). All following comparisons of lacunarity values extracted from SEM images correspond to a (r) discrete value calculated for a three-pixel window. This size was chosen as representative of the whole lacunarity curve after comparing all window sizes.

Rescaled Range Fractal Dimension
Rescaled range (R/S) analysis is used to calculate the Hurst exponent (H) of a time series, H being the best measure of its roughness. The technique is applied to the output file of the algorithm Histo_Gene (Parrot, 2003) obtained with extension .ts.

The latter algorithm translates a raw image into one column of gray values, preserving their original space arrangement. Once the data are processed by the R/S program, a new output file displays the H value, the corresponding fractal dimensions (D_R/S) and the SDs. The data fit to a straight line is given by default, but the best fit can be found by eliminating the outliers, and looking for a more precise value for H and D_R/S and therefore diminishing SD.

In our study, the raw data of R/S analysis were used without any outlier elimination or changes in data fitting. The R/S algorithm divides the studied time series into intervals or windows of length w, measuring two quantities:

• R(w): the range of y values in the interval w, which is analyzed as related to the general tendency of variation within the window. This tendency is estimated simply as the line connecting the first and last point of the corresponding window. This procedure subtracts the average trend for each window (Benoit, 1999).
  
• S(w): the standard deviation of the first differences in y (dy) within the window, defined as the differences between the values of y at some location x and at the previous location (x – dx) along the x axis:
  

  where dx is the sampling interval or the interval between two consecutive x values (BENOIT, Version 1.3).
  

The equation used to calculate R/S(w) is

[3]

where the brackets denote the averaged value.

The Hurst exponent is extracted from the following power law, which expresses the self-affine behavior of R/S(w), when the range yielded by values of y in a window of length w becomes proportional to the window length with a power equal to H:

[4]

The rescaled range fractal dimension (D_R/S) is calculated from the Hardy relation (Mandelbrot, 1983) established between the Hurst exponent and the rescaled range fractal dimension:

[5]

Sampling Errors in Fractal Descriptors Measurement
The BENOIT software calculates the current SD of the least-squares fitting of the points (at least 18 for each image) by a line in log–log space, measuring the sample deviation from fractal behavior. The SD values are displayed for each measured parameter (Table 2 ). After this, the mean and variance of the SD are calculated from 50 values for all used descriptors, independently on the material genesis, water content, and scale of observation.

[6]

where m = is the sample mean and N is the sample size (50 in our case).

To estimate the population variance (µ₂ = ²) from a sample of N elements with a priori unknown mean (i.e., the mean is estimated from the sample itself) is of common use to calculate an unbiased estimator for µ₂ (Weisstein, 2007). This estimator is given by k statistics k₂, defined by the following equation (Kenney and Keeping, 1951, cited by Weisstein, 2007):

[7]

In the present research, we used the sample variance as an unbiased estimator for the population (N = 50) variance of each fractal descriptor, accepting the following definition (Zwillinger, 1995, cited by Weisstein, 2007):

[8]

The unbiased sample variance s_N–1² was implemented as the variance in agreement with Mathematica, Version 5 (Weisstein, 2007).

Scaling of Standard Deviation and Population Variance
A comparative analysis of SD and variance dynamics with scale was also accomplished. For this purpose, plots of mean variance vs. the scale of observation were constructed in log–log space. All mean values were calculated from 50 values of fractal descriptors at each of the four compared SEM resolutions. The data are plotted on double logarithmic graphs, looking for the fit to the straight line that is measured in terms of the coefficient of determination, R². The error dynamics with scale are discussed from the point of view of the multiplicative erodic theory of Schertzer and Lovejoy (2004).

Pearson's r Correlation
A Pearson's r correlation matrix was constructed for all studied variables, comparing their means pair by pair (X_i vs. Y_i) using the software Minitab (Minitab, 1998). The correlation coefficient r is computed as

[9]

	Results and Discussion

TOP ABSTRACT INTRODUCTION Materials and Methods Results and Discussion Conclusions REFERENCES

 
To reveal the tendencies in structure and space–time dynamics of compared porous materials, we applied numerical tests, called fractal descriptors, separately to solid and pore networks. Both networks were extracted from SEM multiscale images analyzing the mass and gap properties (Mandelbrot, 2002) in space and time, with special attention to their dynamics with respect to depth, sampling time, and scale of observation. Results are presented in Table 2, where the selected fractal descriptor values correspond to different dates along the drying cycle (only data of three, in some cases four, months are presented). The general trends of pore and solid network behavior are discussed below in relation to material genesis (and depth inside the profile) and scale invariance.

Box-Counting Dimension
Compactness was the first studied topological feature of the images, being measured in terms of D_box. The value of D_box is an indicator of the network's ability to fill the common space, which measures the heterogeneity of the Euclidean space occupied by the set of interest (solids or pores) and its tendency to form clusters; D_box describes the static properties of the fractal set, especially its topography. The microscopic studies were applied to areas (microscope view field) ranging from a few square angstroms (Xu, 2004b) to hundreds of square millimeters (Oleschko et al., 2000), depending on the image resolution. The box dimensions of the studied materials cover the broad range of values: from 1.61 to 1.89 for solid sets and from 1.87 to 1.39 for the corresponding pore networks. First of all, these data confirm the veracity of the well-known inverse relationship supposed between solid and pore box-counting dimensions: the greater the former, the smaller the latter. When data are analyzed all together, independently on layer genesis, the scale of the image, and the sampling data (Fig. 4a ), the inverse relation between solid and pore networks is established with R² = –0.84 (linear model) and –0.93 (polynomial model). For individual layers, this relation is proved by a linear model with R² = –0.79, –0.89, –0.88, and –0.85 for Layers I, II, III, and IV, respectively. The best fit was obtained by polynomial regression with R² = –0.92, –0.90, –0.97, and –0.96, respectively. In Fig. 4b, the best example of an inverse relation between solid and pore box-counting dimensions is shown for the clay Layer IV. Note that, in general, the SD was bigger for pore sets (p) than solids (s) for the topsoil and basaltic sand (with mean SDs of 0.03–0.0063 and 0.01–0.005 for OI and BI, respectively) and for clayey layers (SD means of 0.02–0.0089 and 0.01–0.0057, respectively, Table 2). The inverse relation between D_boxs and D_boxp is not so strong for the binarized images, decreasing R² to –0.58 and –0.72 for linear and polynomial models, respectively. Therefore, in spite of the same documented tendencies for the s–p relation, the box fractal dimensions extracted from the BI show more bias than those from the OI. The differences between the box dimensions extracted from the OI and BI were compared by Pearson's r analysis, showing statistically low associations between values of interest (Table 1). The Student's t-test analysis documented the statistically significant differences between the box-counting dimensions of solids, this difference not being significant for the pore sets extracted from the original and binarized images (Table 3 ).

View larger version (26K): [in this window] [in a new window]   FIG. 4. The inverse relationship between the solid (Dboxs) and pore (Dboxp) box-counting dimensions: the larger the solid dimension, the smaller the pore dimension. Two regression models are compared for (a) all data sets and (b) for the clay Layer IV. In both cases, the polynomial model is the best fit, with a high level of significance.

Box-Counting Dimension and Material Genesis
The box-counting fractal dimension of the solids varied drastically during the drying period, showing opposite tendencies for silty and sandy materials compared with the clayey ones. In the topsoil (50x), the value of D_box extracted from the OI decreased from 1.85 (June) to 1.67 (September) during drying, and a similar tendency was observed in compacted basaltic sand (for the same period and scale of observation). Notwithstanding, in both clayish microhorizons (50x), the solid box-counting fractal dimension increased from 1.61 (May) to 1.72 (September) and from 1.62 to 1.82 (50x) for Layers III and IV, respectively. The D_box mean values, extracted from the OI and estimated taking into account all sampled data and the entire scale range, show a slightly greater magnitude for the box-counting dimension of the topsoil solid networks (1.82); its values were smaller but similar for solids from the basaltic layer (1.78) and clayey microhorizons (1.79 in Layer III and 1.77 in Layer IV). This tendency inverts, however, when the box-counting fractal dimensions of binary images are compared. In this case, the topsoil layer shows the minimum mean box-counting dimension of solids (1.81), while the basaltic horizon and the bottom clay horizon have similar and larger D_box values (1.84), this dimension being slightly smaller (1.82) in Layer III.

These results confirm once more that the largest pores (fractures) that have been generated during drying were left out of the sample. This may be the main reason for the opposite tendencies observed in box-counting dynamics during drying for loam and sandy materials (where these new gaps were not big enough to be excluded from the analysis) in comparison with the clayey ones (where the fractures were not sampled). If some macrofractures are not sampled, the layers (III and IV) should show an increase in the degree of Euclidean space occupied by solids, which implies a greater compaction of the solid matrix. It should be mentioned that the studied clayey layers have simple packing voids, whose attributes differ significantly in comparison with those of the topsoil enriched with organic matter, where complex packing microstructure is derived from the well-developed aggregation processes (Fig. 2). The greater values of the solid box-counting dimensions are in agreement with the lower dimensionality of the pore space (i.e., the smaller porosity). The generic tendency of standard deviation to decrease with drying, and therefore decrease in sampling error, was detected for the solids D_box values measured on the OI (R² = 0.55). The regression coefficient improved significantly for the BI (R² = 0.87). The opposite tendency was documented for the pore sets, where only a slight tendency for SD to increase with drying was detected (R² = 0.46). This observation coincides with the higher variability of clay pore dimensionality observed at more advances stages of drying (November).

The box-counting dimension of pores in the topsoil varied during the drying period from1.38 to1.84 for the OI (entire scale range) and from 1.45 to 1.86 for the BI. In the basaltic sand the changes were smaller, varying from 1.52 to 1.87 (OI) and from 1.49 to 1.76 (BI). The D_box of pores for the clayey horizons varied from 1.47 to 1.86 (OI) and from 1.53 to 1.79 (BI), for Layer III. For Layer IV, this fractal descriptor fluctuated from 1.48 to 1.87 (OI) and from 1.48 to 1.87 (BI), where the sampling error for the box-counting dimensions extracted from the binarized images was always smaller. The more drastic differences were documented for clays. For instance, the mean SD of all images from Layer III changed from 0.023 to 0.007 when manually binarized images were used for estimating the D_box, while in Layer IV this change was from 0.014 to 0.007.

The mean values of D_boxp for the clayey Layer III were similar for the original and binarized images (1.66 vs. 1.67, respectively), while for Layer IV this difference became significant, with D_boxp values fluctuating between 1.72 and 1.63. All mentioned D_boxp values were slightly higher than the corresponding fractal dimensions of the topsoil (1.59 vs. 1.62, respectively) and close to the basaltic sand (1.70 vs. 1.63).

The dynamics of the box-counting dimension (mean values) of solids and pores with respect to depth is shown in Fig. 5 , where the subtle significant tendency to decrease with depth can be observed for the solid set (R² = 0.53, Fig. 5a) and the tendency to increase with depth can be observed for pores (R² = 0.47, Fig. 5c). The best fit of data was obtained in log–log space and for a polynomial regression model (R² = 0.81, Fig. 5b, and R² = 0.75, Fig. 5d, respectively, for compared sets).

View larger version (23K): [in this window] [in a new window]   FIG. 5. The dynamics of mean values of box-counting dimension of (a and b) solids (Dboxs) and (c and d) pores (Dboxp) established with respect to layer depth. The best fit of the data in log–log space was by the polynomial model.

View this table: [in this window] [in a new window]   TABLE 4. Pearson's r correlation for all fractal parameters. Data in italics are the best-fit regressions.

View larger version (28K): [in this window] [in a new window]   FIG. 6. Dynamics of (a) the bulk density of the studied porous materials with depth; (b) the best fit to a polynomial model and good fit to the straight line in log–log space are comparable with fractal descriptor changes with depth, sampling time, and scale, as shown in Fig. 4 to 12.

View larger version (35K): [in this window] [in a new window]   FIG. 7. (a) Scale invariance of the solid set box-counting dimension (Dboxs) is visible in spite of the high data dispersion; (b) mean values vs. scale (); (c) the good fit of data to the straight line is documented in log–log space for the solid set; (d) this behavior is more pronounced for the pore set box-counting dimension (Dboxp), (e) with a good fit of data to the straight line in log–log space for the pore set; and (f) standard deviations (SD) for Dboxp vs. .

View larger version (19K): [in this window] [in a new window]   FIG. 8. The change in (a and b) solid and (c and d) pore spectral dimensions () with depth.

View larger version (20K): [in this window] [in a new window]   FIG. 9. Relationship between the spectral dimension () and scale () of (a and b) solids and (c and d) pores with best fit to the polynomial model being in (b and d) log–log space (R2 = 0.99).

View larger version (24K): [in this window] [in a new window]   FIG. 10. Testing of (a and b) solid and (c and d) pore set continuity for scale invariance. despite the high original data dispersion, (a) the mean values of the spectral dimension/box-counting dimension ratio (/Dbox) fitted well to the polynomial model (R2 = 0.97 and 0.99, respectively), the best fit being in (b and d) log–log space (R2 = 0.99) for linear model.

View larger version (26K): [in this window] [in a new window]   FIG. 11. Testing scale () invarience of pore set continuity as the spectral dimension/box-counting dimension ratio (/Dbox) by (a) mean standard deviation (SD) analysis and (b) data fitting in log–log space.

View larger version (34K): [in this window] [in a new window]   FIG. 12. Despite high data variation, (a) the scale () invariance of the Hurst exponent (H) is documented by (b) fitting the H mean values to power-law, linear, and polynomial models. Similar tendencies are observed in the plots of (c) the SD (calculated for the rescaled range fractal dimension, DR/S) dynamics with scale as well as (d) mean DR/S values. These plots are compared with the variance dynamics (e) with time, also presented in log–log space (f) for time as well as (g) for scales ().

An opposite tendency was documented for pore patterns (Fig. 7d and 7e), with a smaller fractal dimension at the higher magnifications (1.78, 1.71, 1.66, and 1.55 for 50x, 500x, 1000x and 5000x magnification, respectively). This fact is in agreement with our theoretical assumptions because the drying process consolidates solid particles, implying the loss of the largest pores, especially fractures or cracks, leaving them out of the SEM view field. With a high level of statistical significance (R² = 0.998 for the polynomial model), it was concluded that the mean box-counting dimension of pores decreased with scale growth (Fig. 7d), despite the high data dispersion inside each compared scale (similar to that shown for the solid set, Fig. 7a). The minimum SDs, and therefore the smallest sampling errors, correspond to D_boxp at the magnifications of 50x and 5000x, where SD became twice as small (0.01) as at 500x and 1000x. Therefore, the dynamics of the SD of the pores with respect to scale was similar to the tendency of the log–log plot of mean D_boxp values (R² = 0.998 for the polynomial model, Fig. 7f).

Mean D_boxp values for all compared materials analyzed at magnifications of 50x, 500x, 1000x, and 5000x were 1.78, 1.71, 1.66, and 1.55, respectively.

These results support the scale invariance of the studied networks (solids and pores), showing slightly a better fit to the fractal behavior for the solid networks than the pore sets, with a similar tendency for the SD.

Student's t-test (Table 3) was applied to estimate the paired sample correlation between the box-counting dimension extracted from the OI and the BI, as calculated by the BENOIT 1.3 software, with statistically significant differences between the D_box values of the solid networks at the 95% confidence level (with a two-tailed significance of 0.003). Differences between the box-counting dimensions of pores were not significant statistically (with a two-tailed significance of 0.065). In spite of the higher sampling error of data derived from the manually binarized images, however, both sources of information provided similar, statistically coherent tendencies for the solid and pore network dynamics with respect to layer depth (and therefore its genesis), sampling time, and scale of observation.

Spectral Dimension or Fracton
Connectivity is the other distinctive topological attribute of solid and pore patterns, quantifiable by the spectral dimension or fracton (). In contrast to the box-counting dimension, which measures static properties of a fractal set, the spectral dimension measures the dynamic attributes of the network (Alexander and Orbach, 1982). The connectivity of pore networks determines the velocity of water and mass fluxes through the porous medium, its magnitude being the precise indicator of how easily particles move through a network. The spectral dimension of a two-dimensional image completely filled by a specific set is close to 2 (the upper topologic limit of ). This limit is characteristic of classic Brownian movement in Euclidean space, where its trajectories completely fill the space. The white noise model or Gaussian normal distribution is characterized by a fracton value of 2. The greater the value of the spectral dimension, the fewer obstacles will be encountered by fluxes passing through the pores. The opposite is true for the porous medium with a low fracton value, which means that the medium imposes severe limitations on flow. In this case, the flow becomes less continuous and more tortuous, with tending to reach the lower topologic limit of zero (Oleschko et al., 1998). The mean spectral dimensions of solids, _s, and pores, _p, varied from 1.44 to 1.82 for the studied monolith, with fracton values always smaller than the mass fractal dimension of the same materials (Table 2). The lower value of 1.44 was characteristic for the solid set of clayey Layer III (May, 50x), while the higher value of 1.82 was found in the pore set of the topsoil (June, 50x). The generic tendency of an increase in pore continuity accompanied by a decrease in solid continuity was documented by spectral dimension analysis during the drying process, this tendency being in agreement with the visual testing of fragmentation observed in different layers of the monolith (Fig. 2).

Fracton and Genesis of Materials
Orbach (1986) proposed the spectral dimension as a generic fractal parameter correlated only with the degree of percolation of a fractal aggregate or cluster and independent of the pore network genesis. For soils, however, Anderson et al. (1996) showed clear visual differences and a statistical correlation between the appearance and genesis of the pore network. To test this relationship, we measured fracton values in contrasting materials during the drying period. The range of values varied from 1.47 to 1.71 for the Udoll topsoil solid set, with the lowest fracton value corresponding to the beginning of the drying process and the highest value documented for the final step of the experiment. The basaltic sand layer had the widest range of values, from 1.48 to 1.82, while the clayey horizons had a more constricted range of , fluctuating between 1.44 and 1.66 for the amorphous clay enriched with diatoms and between 1.47 to 1.68 for the pure clay (Layer IV). The mean fracton values were similar for the topsoil and basaltic sand (1.56 and 1.57, respectively) and lower means correspond to the clays, where the mean spectral dimensions were 1.53 and 1.54 for Layers III and IV, respectively.

Lower fracton values for solids were generally found in the clayey layers; however, the coefficient of determination for the relation between the spectral dimension of solids and profile depth was lower for solids (R² = 0.67) and increased (R² = 0.94) for pores (Fig. 8a and 8b ), being statistically significant in log–log space (Fig. 8c and 8d).

The spectral dimensions of the pore network varied between 1.47 and 1.82 for the Udoll topsoil, 1.49 and 1.76 for the basaltic sand, 1.45 and 1.67 for Layer III, and between 1.44 and 1.62 for Layer IV. The dynamic tendency of the pore fracton, _p, with depth was similar to the solid networks (Fig. 9a, 9b, 9c, and 9d ), its values always being inverse to the solid fraction, _s, with higher pore continuity in the topsoil and basaltic sand, where the mean fracton values were 1.56 and 1.53, respectively. These results confirm the well-known fact that the pore pattern of pure clay (without aggregation) is significantly less continuous and more tortuous than that of sandy soils. On the other hand, the mean spectral dimension in clay enriched with diatomaceous material, forming a set of obstacles for the flow, decreased to 1.50, increasing again in Layer IV to 1.53 (Fig. 8). In general, the greatest continuity was documented for the pore network, with a mean value of 1.55, in comparison to the solid network, where the corresponding fracton value was 1.53. None of these differences, however, were statistically significant.

Scale Invariance of the Spectral Dimension
Fracton scaling was studied in this research using the complete range of SEM image magnifications (50x–5000x); the spectral dimension and sampling error (SD) were higher in clay layers. The spectral dimension of the solids varied from 1.44 to 1.58 for 50x magnification, with a mean value of 1.49 and a SD of 0.0016. The corresponding values for 500x, 1000x and 5000x magnifications (SD in parentheses) were 1.54 (0.0026), 1.54 (0.0036), and 1.61 (0.0069). Note the increase of variance with magnification, which will be discussed separately. The lowest mean fracton value (1.49) was found for the 50x magnification. A linear relationship was established between the spectral dimension and SEM magnification, confirming the fact that the higher fracton of solids corresponds to a higher magnification (R² = 0.88; Fig. 9a), and a smaller pore spectral dimension (R² = 0.78). This finding agrees with our original hypothesis that the main cracks formed during the drying process were excluded from the clods analyzed by SEM. For both relationships, the best fit was obtained in log–log space (R² = 0.99 by polynomial model, Fig. 9b and 9d).

The range of pore spectral dimensions varied during the drying time from 1.45 to 1.82, 1.47 to 1.64, 1.44 to 1.62, and 1.47 to 1.61 for magnifications of 50x, 500x, 1000x, and 5000x, respectively. The corresponding mean and SD values were 1.58 (0.013), 1.51 (0.002), 1.51 (0.002), and 1.53 (0.003), respectively.

Pearson's r correlation analysis (Table 4) showed statistical significance for the inverse relation between the solid spectral dimension and the pore box-counting dimension in the basaltic sand (r = –0.78) and pure clay (Layer IV, r = –0.85), as well as a direct relation between the solid fracton and image global roughness measured in terms of the Hurst exponent (r = 0.80 for both layers). The other two studied layers showed similar tendencies, which were not statistically significant, with r values between 0.20 and 0.40 for established relations.

Solid and Pore Network Continuity
The relation between and D_box (or D_m, the mass fractal dimension) is used as a precise measure of the diffusion velocity through a pore network, quantifying the competitive effect of heterogeneity and tortuosity of space. It measures the mutual dynamic tendencies of the box-counting (D_box) and spectral () dimensions as a function of soil genesis (related to the depth, in our special case) and the physical conditions at the time of sampling. In general, the smaller the value of /D_box the shorter the distance a particle follows when diffusing during a given time period. The topological limits of the /D_box relation are: 1 for Euclidean space completely occupied by the set of interest; and 0 for a medium that absolutely inhibits flow (Anderson et al., 1996). This relation is <1 for all fractal networks, tending asymptotically to the upper or lower topological limits of /D_box.

The /D_box values obtained in our study varied from 0.79 to 1.0 for sets of solids and pores, showing the generic tendency to high continuity of all compared materials. The upper topological limit was reached by the solid set of clayey Layer III before drying and by the pore sets of the Udoll topsoil (three samples out of 11, all of them before drying), as well as by a pore set of the basaltic layer (one sample with 5000x magnification, after drying), and five samples of clayey layers, four of them also after drying (Table 2). A value of 1 means no restriction to flow and theoretically should be related to the simple packing type of voids. Figure 3 shows an example of the original (Fig. 3a) and measure images (Fig. 3b). The latter is constituted by the trajectories of Brownian motion generated by the algorithm Bromov_8a (Parrot and Taud, 1998), completed through the pore space limited by the grays derived from the image threshold (Fig. 3b).

The Pearson's r correlation analysis (Table 4) showed a statistically significant correlation between /D_box of solid and pore sets and the box-counting dimensions of the solids (r = –0.67 and 0.89, respectively) and pores (r = 0.62 and –0.84), this correlation being always greater for pores and inverse for solids (the increase of the former resulted in the decrease of the latter). An important association was found between /D_box (continuity) and set tortuosity, with r increasing to –0.998 for solid and pore tortuosity.

Analysis by Layers
The trends documented for the variations in continuity data as a function of material genesis confirmed the theoretical expectations. A higher heterogeneity and consequently a smaller continuity of pore networks is detected in the clayey microhorizons (Layers III and IV) where the mean values of /D_boxp were smaller and equal to 0.90 and 0.93, respectively. For the Udoll topsoil and basaltic sand layers, the corresponding mean values were slightly greater and equal to 0.94, indicating a greater pore continuity. In general, a greater heterogeneity and a lower continuity are derived from an increase in clay content. This statement was confirmed by the highly significant adjustment of mean /D_boxp for pores vs. layer depth to linear regression (R² = 0.92), the power distribution being the best fit (R² = 0.998). Both models indicate a decrease in pore network continuity with depth. Notwithstanding, the change in solids continuity with depth is not so convincing, the polynomial regression model being again the best fit (R² = 0.55).

Test for Scale Invariance of Pore and Solid Continuity
Pore continuity was measured in terms of the fractal descriptor (_p/D_boxp) and varied from 0.82 to 1 for 50x, and from 0.83 to 1, 0.82 to 0.99, and 0.87 to 1 for 500x, 1000x, and 5000x magnifications, respectively. In spite of a generally large spread of the original data for each studied magnification (Fig. 10a ), it is possible to conclude that the uniformity of images tended to maximize at 5000x magnification, in which the upper topological limit of _p/D_boxp was almost reached and a minimum mean SD was obtained, decreasing from 0.0032 (50x) to 0.0019 (5000x). The dynamics of the mean _p/D_boxp values with respect to scale show a clear tendency to increase from 0.92 (50x) to 0.96 (5000x) independently of layer genesis and sampling date. The mean values of set continuity fitted well to a polynomial model with R² = 0.99 for the pore set (Fig. 10d), this model being also the best fit for solid continuity with R² = 0.97 (Fig. 10b). When these data are plotted in log–log space, a straight line with a high coefficient of determination for pores (R² = 0.99) is obtained, in contrast to a low coefficient of determination (R² = 0.37) for the solid set. When, regardless of mean D_box values, the mean SDs are plotted vs. scale, the data fit well to a power-law model (R² = 0.98, Fig. 11a ) for pores, the coefficient of determination being the same for the log–log plot (Fig. 11b). We interpreted this behavior as statistical evidence for a power-law behavior of the relation between pore continuity and scale, where an exponent of –0.13 was found for the pore set.

Tortuosity
The tortuosity () of pore and solid networks is related to the diffusion velocity through a fractal network, integrating the multiscale heterogeneities of flow paths (Armatas, 2006). The lower topological limit of is 0, which corresponds to flat pores (or solids) embedded in the Euclidean space. In the studied layers, this limit was reached sometimes by pores (Table 2), with pore tortuosity, _p, fluctuating within a wide range of values between 0 and 0.57; the minimum magnitudes were found in the basaltic sand (0 and 0.02) and the Udoll topsoil (0 and 0.04). These layers have perfectly developed smooth interfaces between solids and pores, distinguishable on the SEM images with 5000x magnification. The maximum tortuosity of the solid set occurred in Layer III (0.57) and the basaltic horizon (0.50) at 50x magnification. The following discussion shows how the material genesis (depth) and image magnification affect tortuosity.

Tortuosity and Material Genesis
The largest mean value of _p (0.26) corresponded to the pure clay Layer IV, this fractal descriptor being significantly smaller in the Udoll topsoil (0.12) and equal to 0.21 and 0.23, respectively, in the other two compared layers. The diatom-bearing clay (Layer III) showed the highest tortuosity of the solid set (0.36, Table 2). The generic tendency of solid sets to increase tortuosity with respect to depth was documented (R² = 0.96) with a best fit of the data to a power-law model. A significant difference in pore tortuosity between the pure clay layer and the diatom-bearing clay was observed. The smallest population variance was found for the clay (Layer IV) solid set (0.004) and the pore set from the basaltic sand (0.023), while the largest variance was detected in the pores of the clay Layer III (0.036).

Scale Invariance of Tortuosity
The values of tortuosity for solids and pores (50x magnification) varied from 0.00 to 0.57 and 0.00 to 0.43, respectively. In the case of pores, _p ranged from 0.00 to 0.40 for 500x magnification, from 0.02 to 0.43 for 1000x magnification, and from 0.00 to 0.30 for 5000x magnification. In general, it was found that at greater scales the tortuosity became smaller for the same material genesis and sampling time. Mean values of this descriptor for the pore set were 0.28, 0.30, 0.22, and 0.05 for 50x, 500x, 1000x, and 5000x magnification, respectively. The linear regression constructed in log–log space from 50 experimental data between _p and the scale of observation showed the best fit with R² = 0.71, which confirms the pore pattern tortuosity invariance to scaling.

Lacunarity
The concept of lacune came from pure mathematics and has been in use since the last century in different areas; for example, in the study of compact groups (Edwards et al., 1971), lacune is the measure of the gap (void) distribution across a trigonometric series, prime distribution, etc. Therefore, the term lacunarity, proposed by Mandelbrot (1983), has deep roots and is applicable not only to fractals. For the fractal, lacunarity is a counterpart to the fractal dimension useful in describing fractal set texture (Rauch, 2005). In the present study, it was used to quantify the variation of pore network attributes from the point of view of translational invariance and its dynamics in space and time. Lacunarity is defined as the prefactor of generic power laws established by Mandelbrot (1983) for fractals, which relate their attributes to the scale of observation, where the fractal dimension (D) plays the role of power (Henebry and Kux, 1995). Lacunarity is a measure of dual nature, which quantifies the degree of space occupied by a set of interest as well as its internal structure (Chmiela et al., 2006). This parameter is variable in space, dynamic in time (Hai-Tao et al., 2005), and scale dependent (Mandelbrot, 1983). Currently, it is widely accepted that lacunarity is a precise indicator of the efficiency and velocity of physical processes occurring in natural systems constituted by pores and solids. It has also been documented that structural patterns with similar fractal dimensions can appear different because of the differences in the pore distribution (or gaps or lagoons) across the image (Saleh et al., 2001). To distinguish these structures quantitatively, the lacunarity is used (Li and Nekka, 2003). In this research, the range of lacunarity values varied from 1.05 to 1.56 for compared materials. The former value is closer to the lower topological limit of this fractal descriptor and coincides with a maximum homogeneity of the object. Note that both extreme values were extracted from the images of the topsoil, a soil with high organic matter content and consequently with a well-developed and stable structural pattern. Therefore in this case, detected the original large space variability of the pore pattern in this soil.

Lacunarity and Material Genesis
The lacunarity of layers contrasting by genesis showed large variation during the drying period, the values fluctuating between 1.05 and 1.16, 1.05 and 1.28, and 1.09 and 1.25 for basaltic sand and amorphous clays (Layers III and IV), respectively; the mean lacunarity values were 1.17, 1.13, and 1.15 for the topsoil, basaltic sand, and clays, respectively. In both clayey layers, was similar. Therefore, both the largest and the smallest gaps were characteristic of the highly variable surface soil, in which the elemental mineral and organic particles are combined in aggregates. Drying significantly increased the number of gaps quantifiable in terms of lacunarity. Basaltic sand had the lowest mean lacunarity in comparison with the other horizons, in agreement with its simple packing structure and homogeneous pore shape and size.

Lacunarity decreased with depth. In spite of high data dispersion (compare its variance with the other descriptors described above, Fig. 12 ), the relation between mean values and depth fit well (R² = 0.91) to a polynomial regression model. Its coefficient of determination improved still more in log–log space (R² = 0.98). This tendency is related to the layers' physical properties (Fig. 6) as well as to the gravity effect on porosity and lacunarity.

Pearson's r correlation analysis showed the high inverse association between lacunarity and the solid box-counting dimension of the topsoil (r = –0.93) and clayey Layer III (r = –0.85), this relation not being significant for the basaltic sand (r = –0.21) and slightly significant for Layer IV (r = –0.40) (Table 4). For the pore sets, this negative correlation became positive. In the same horizons, a high correlation was detected between and pore continuity (negative) and tortuosity (positive), these being highly statistically significant in the topsoil (r = –0.87 and 0.88, respectively) and clayey Layer III, characterized by the mixture of clays and pseudo-sands (diatoms), where r was –0.73 and 0.74, respectively. All other possible correlations between the lacunarity and other fractal descriptors were not statistically significant.

Scale-Dependent Lacunarity
According to previous studies and contrary to the well-known scaling behavior of the fractal dimension, lacunarity is a scale-dependent parameter (Mandelbrot, 1983; Lee and Main, 1989). Our data did not show any statistical relationship between and image magnification (Table 2). Only a subtle tendency toward smaller mean lacunarity at 5000x magnification was observed, decreasing from 1.15 at 50x, 1.16 at 500x, and 1.17 at 1000x, to 1.11 at 5000x. This can be explained by the fact that microscopic features of the images are seen with more detail at greater scales, with the macropores being left out of the view field of SEM. Lacunarity was the only fractal descriptor studied that did not show a statistical relation with the scale of observation.

Roughness
Roughness is the last, but not least, fractal descriptor used in the present research for joint description of solids and pores embedded in the Euclidian space (two-dimensional SEM image). This analysis does not require a threshold and therefore was considered an unbiased fractal descriptor. It can be applied to self-affine sets, characterized by invariance under some linear reductions and dilatations, implying uniform global statistical dependence (Mandelbrot, 1983, 2002). In this context, the following discussion deals with the Hurst exponent in spite of the fractal dimension, the former being a co-dimension of the latter. The H exponent measures the statistical properties of a random process at t >> 1 (Denisov, 1998), and joins their local and global features inside a unique multiscale variable called roughness, which refers to the increased level of complexity of this fractal descriptor.

In the present research, the Hurst exponent was used to measure the type of memory present inside the images. An H value of 0.50 corresponds to the random or memoryless sequence (white noise or normal Gaussian distribution), while the major or minor Hurst exponents characterize a sequence with short-range (H > 0.50) or long-range (H < 0.50) memories (Mandelbrot, 2002), respectively. In statistical terms, the latter two sequences are known as persistence and antipersistence, representing the convergent and divergent time series. The memory behavior of time or space sequences is strongly dependent on the nature and magnitude of system perturbation (long-term drying, in our case), showing the fluctuation–periodicity correlation (Akin et al., 2006). Note that the relation between the Hurst exponent and roughness is inverse; therefore, the bigger H is, the smoother the image and the lower its roughness.

Roughness and Material Genesis
Antipersistence behavior was found for SEM images of all studied layers, in which values of H fluctuated between 0.07 and 0.20, always being significantly <0.50. Due to high data dispersion, the mean roughness was similar for the four compared horizons, being slightly higher in both clayey layers (0.12) followed by the topsoil and the basaltic sand (0.13). Notwithstanding, the regression analysis confirmed the statistically strong decrease of H (increase in roughness) with depth (R² = 0.96), which agrees with theoretical expectations: the clays are characterized by maximum roughness. The log–log plot of H vs. depth showed a similar tendency, with R² = 0.86 for linear and 0.99 for polynomial models. The standard deviation of H values calculated by the rescaled range technique of BENOIT shows the same dynamics with depth (R² = 0.94 for a polynomial model), decreasing from 0.53 (topsoil) to 0.45 (clayey Layer IV). The minimum SD and therefore the minimum sampling error was documented for Layer III (0.42), where the best fit of a polynomial model (R² = 0.94) was obtained in comparison with the simple linear regression (R² = 0.85).

Scale Invariance of Roughness
The Hurst exponent shows clear statistical scale invariance. The H dynamics had the same tendency as all other fractal descriptors used (except scale-dependent lacunarity), high dispersion of values extracted from the individual images (R² = 0.51, Fig. 12a), strong fitting of means to power-law (R² = 0.86) and linear (R² = 0.93) models, as well as the best fit of the data to a polynomial (R² = 0.94) regression (Fig. 12b); the coefficient of determination increased still more in log–log space. These groups of data (discrete values and their means) showed an increase of H with image magnification (scale) that testifies to the smoothing of information with scale.

The degree of Hurst exponent association with other fractal descriptors was measured by Pearson's r, varying from 0.991 for the rescaled range dimension (D_R/S) to 0.049 for lacunarity. The former fact confirmed once more the veracity of the strong relation between fractal dimension and the Hurst exponent, while the latter documented the lack of correlation between H and . All other pairs of variables showed different degrees of association, with the highest r for the association between H and continuity of the solid set (Table 4).

Heterogeneity vs. Homogeneity of Fractal Descriptors: Sampling Error, Scale Invariance, and Correlations
In the present research, the concepts of heterogeneity and homogeneity of fractal parameters are outlined in terms of SD (sampling error), population variance (unbiased sample variance), SD and variance scaling (the relation between the SD and variance vs. the image magnification, Fig. 12c, 12d, and 12g), and finally, the degree of association between the pairs of descriptors (Pearson's r correlation matrix). The behavior of the sampling error as well as some established correlations between selected fractal descriptors, visualized by Pearson's r, has been widely exemplified in this study.

The unbiased variance of all studied fractal descriptors was calculated for 50 values, independently for porous material genesis, sampling time, and image magnification. The highest values of variance for the box-counting dimension of pore sets were extracted from the OI (0.020), followed by the tortuosity of pores (0.016), and solids (0.013). The Hurst exponent had the minimum variance for the entire data population, followed by the continuity of solid sets (0.0020), the box-counting dimension of binarized images (0.0025), and continuity of pores (0.0028). The graphic comparison between the variances of all studied fractal parameters shows that these are always higher for pores than solids, independently of the descriptor nature (Fig. 13a ). It is interesting to compare the mean dynamic of the descriptor with unbiased oscillation of variance (Fig. 13b).

View larger version (18K): [in this window] [in a new window]   FIG. 13. Comparison among the (a) mean values and (b) variances of the fractal descriptors: the original images of the box-counting dimensions for solids (Dboxs) and pores (Dboxp) as well as the binarized images (Dboxsb and Dboxpb, respectively), the spectral dimension of solids (ds) and pores (dp), their ratios with the box-counting dimensions for solids (Dboxs) and pores (Dboxp), tortuosity of solids (s) and pores (p), lacunarity (), rescaled range fractal dimension (DR/S), and the Hurst exponent (H); H shows the minimum variance and sampling error.

The relation found in the present research between the variance of fractal parameters and the scale of observation (Fig. 11 and 12) is in agreement with the Schertzer and Lovejoy (2004) multiplicative erodic theory, which describes exponential error growth as

[10]

for the amplitude of the infinitesimal separation X(0), a pair of points X_–1(t), X_–2(t) = X(0), with a finite Lyapunov exponent µ (directly related to the fractal dimension).

And the last point derived from our research is the nature of the best fit of a polynomial model to the mean values of all compared fractal descriptors and their SDs, as well as corresponding variances. To make clear the genetic nature of these relations, we have analyzed the relationship between the bulk density of the studied layers and depth. A highly significant statistical correlation was documented for these variables, with the same best-fit polynomial regression (R² = 0.99), conserving its statistical significance in log–log space (R² = 0.98, Fig. 6) and proving once more the fractal nature of the studied porous materials.

	Conclusions

TOP ABSTRACT INTRODUCTION Materials and Methods Results and Discussion Conclusions REFERENCES

 
Selected fractal descriptors, as well as sampling error (estimated by SD) and population variance, imprint genetic features of the studied porous materials, showing scale invariance and correlation with depth, image magnification, and sampling time. For all selected fractal descriptors extracted from the original SEM images, SD and variance were always higher for pore sets than for solids. The image roughness measured in terms of the Hurst exponent showed the highest precision in comparison with the other descriptors, its variance being 2.44 to 24.67 times smaller than the variance of, for instance, the solid set continuity or the pore box-counting dimension.

The selected "toolbox" of fractal techniques is useful to describe, in quantitative terms, the diverse attributes of soil and sediment structural patterns, and is able to fit to their profile of physical properties. The dynamics of the studied structural patterns was documented statistically in time and space. The power-law relations between the means of the fractal descriptors, SDs, and variances vs. layer depth, scale, and sampling time have been established. The scaling properties of all mentioned parameters were exemplified. The complex, power-type relation was found between the mean values and variances of the studied fractal descriptors.

	ACKNOWLEDGMENTS

 
This work is supported by the Consejo Nacional de la Ciencia y Tecnología, México, under Grant no. 42571. K. Oleschko thanks Dr. Mariano Cerca from the Geosciences Center, UNAM, México, for fruitful discussion.

	REFERENCES

TOP ABSTRACT INTRODUCTION Materials and Methods Results and Discussion Conclusions REFERENCES

 

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