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Published online 1 January 2007
Published in Soil Sci Soc Am J 71:15-25 (2007)
DOI: 10.2136/sssaj2006.0132
© 2007 Soil Science Society of America
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SOIL PHYSICS

A Multifractal Approach for Assessing the Structural State of Tilled Soils

Christian J. C. Roisin*

Walloon Agricultural Research Centre, Dep. of Crop Production, 4, rue du Bordia, B-5030 Gembloux, Belgium

* Corresponding author (roisin{at}cra.wallonie.be).


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Although there are several methods for assessing the structural state of highly structured tilled soils, this task remains challenging because of the great heterogeneity of the solid phase caused by tillage tools and biological activity. This paper describes a method for quantifying the inner heterogeneity of elementary soil volumes large enough to be representative of the modifications caused by tillage implements. It is based on soil strength measurements collected at the intersection points of a 5 x 5 cm2 grid on the surface of an 80 x 80 cm2 area. This study shows that multifractal formalism is appropriate for analyzing the variability of such close measurements and could be a practical way of assessing the structural heterogeneity of agricultural soils. With this aim, two dimensionless parameters were defined to provide a characterization of the soil layers that captures field observations. The first one, Ir, called the regularity index, indicates the possible presence of strong irregularities caused by tillage implements while the second, Ih, called the homogeneity index, is related to the local variation of the structure. It expresses the structural homogeneity of soil layers not showing strong irregularity. The results produced from a series of 672 datasets highlight the good discriminating power of this last parameter. Due its high sensitivity to the spatial organization hidden in a dataset, it appears to be a better suited parameter to compare differently managed plots than the classical coefficient of variation (CV).


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
A number of studies have shown that root growth and therefore plant development depend largely on the quality of the structural state of the arable layer (Lows, 1973; Ellis and Barnes, 1980; Tardieu, 1988a; Unger and Kaspar, 1994; Ley et al., 1995). Although this concept dates back to the start of agricultural activity and is widely used in soil tillage studies, neither a clear definition nor a useful quantitative index has been developed for its characterization. The difficulties lie in the complexity of the organization of soil particles (aggregates, clods) and the associated pore space, and therefore in the high temporal and spatial variability of soil physical parameters (Nielsen et al., 1973; Cassel and Nelson, 1979; Cassel, 1983; Ley and Larya, 1994; Clark, 1999). As stated by Addiscott and Dexter (1994), the spatial variability of any soil property measurements, such as soil strength data for example, reflects the heterogeneity of the structural state of the arable layer. This physical heterogeneity seems to be a key soil characteristic affecting its functioning (Dexter, 1997). It affects root development and water or nutrient uptake (Tardieu and Manichon, 1987; Tardieu, 1988b; Hadas, 1997), as well as biological activity (Young and Ritz, 2000).

Manichon (1982a, 1982b) developed a method to analyze soil structure in the field and characterize its diversity. Known as the Cropping Profile Method, it uses trenches to evaluate changes induced by agricultural machines in the Ap horizon. It is an interesting diagnostic tool, but is fundamentally qualitative (Neves et al., 2003). Among the quantitative approaches for assessing the structural state of a soil, the penetrometer is the most widely used instrument (Ehlers et al., 1983; Martino and Shaykewitch, 1994) but because of its high spatial variability, the mechanical impedance alone is difficult to use for assessing the effect of different management and/or tillage practices (Cassel and Nelson, 1979). According to Clark (1999) who showed that the variability of such measurements was high even over very short distances (<7.5 cm), this technique could nonetheless be useful for elucidating the spatial organization of soil components and for quantitatively characterizing a tilled soil in terms of its structural heterogeneity as long as many close measurements are made and the results are analyzed using a suitable mathematical procedure.

The statistical distribution of such measurements may or may not be normal. Several researchers have studied the spatial variability of soil strength measurements at different scales (McIntyre and Tanner, 1959; Cassel and Nelson, 1979; Cassel, 1983; Hewit and Dexter, 1984; Perfect et al., 1990) and all concluded that the statistical distribution of such measurements is log-normal, rather than normal. These findings, associated with the fact that close spacing of penetration points likely leads to autocorrelated data, mean that such measurements cannot be treated stochastically by using classical statistical approaches that assume a normal distribution and independent measurements. An alternative approach is the use of geostatistics (Matheron, 1965). This is based on the idea that samples taken over small distances are more likely to be similar than those taken over large distances. Nevertheless, statistical procedures such as autocorrelation analysis or geostatistics are more difficult to implement with two-dimensional signals, especially when periodicity or anisotropy is present (Webster, 1977; Journel and Huijbregts, 1978; Cressie and Hawkins, 1980; Voltz and Webster, 1990), and yet periodicity and anisotropy are frequent in tilled soils. Several authors (Webster, 1977; Burrough et al., 1985; Grant et al., 1985; Hadas and Shmulewitch, 1990) used spectral analysis to process penetration resistance data gathered along a transect. Although spectral analysis is well suited for signals in which periodicity appears and can be easily transposed for a two-dimensional analysis, it was not appropriate for processing our datasets because the number of points (16) in each direction was too small.

Burrough (1983a, 1983b, 1983c) first suggested that fractal theory (Mandelbrot, 1982) may be appropriate for describing the variation of soil properties because looking at a finer scale reveals more details. Fractal theory has also been used to study soil properties (Armstrong, 1986; Culling, 1986; Eghball et al., 1993). These fractal theory applications used a monofractal approach, which assumes that soil spatial distribution can be uniquely characterized by a single fractal dimension. Folorunso et al. (1994) found multifractal formalism (Evertsz and Mandelbrot, 1992) to be superior to a single fractal dimension for qualifying the intrinsic variability of soil surface strength measurements. Similarly, Kravchenko et al. (1999) showed that a multifractal formalism reflected many of the major aspects of soil chemical data variability. These papers indicated that the multifractal spectra allow discrimination among soils or soil property distribution maps.

Our objective is to test the possibility and practicality of using multifractal formalism to quantify the structural heterogeneity of soil layers at a scale related to modifications caused by tillage implements and using in situ penetration resistance measurements.


   THEORY
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Background information on the multifractal concept and on procedures to characterize multifractals can be found in several books (Feder, 1988; Falconer, 1990; Evertsz and Mandelbrot, 1992). The essential difference between a fractal and a multifractal model is that the former refers to a set and can be characterized by a single parameter D, such as the fractal dimension, while the latter refers to a measure and can be characterized by a continuous spectrum of fractal dimension usually referred to as f({alpha}). If µ is a measure supported by a bounded region, the quantity {alpha}, called the coarse Hölder exponent, is defined as

Formula 1[1]
where µ(B{delta}) is the measure concentration contained in ‘boxes’ of size {delta}.

For multifractal measures, the number N{delta}({alpha}) of boxes of size {delta} having a coarse Hölder exponent equal to {alpha} increases for decreasing {delta} and obeys a power law,

Formula 2[2]
where the exponent f({alpha}) is a continuous function of {alpha}. The graph of f({alpha}) versus {alpha}, called the multifractal spectrum, has a concave downward curvature, with the range of {alpha}-values increasing with the increase in the heterogeneity of the distribution of the measure concentration (Agterberg, 1995).

An empirical estimate of f({alpha}) can be obtained using the method of moments (Evertsz and Mandelbrot, 1992). This method is based on a quantity called the partition function and is defined as

Formula 3[3]
where N{delta} is the total number of the cells of size {delta} and the moment order q is any real number in the range –{infty} to +{infty} (Feder, 1988). In our study, where the studied region is a square area of initial size {delta} = 80 cm and the dataset is made up of 4k = 4 data, µ{delta}, i = µi/µ* where µ* is the sum of all the data contained in the set and µi is the sum of all the data contained in the ith cell of size {delta} = 80 x 2–h with h being an integer ranging from 1 to k.

The multifractal model applies for any value of q,

Formula 4[4]
if the plots of log Mq ({delta})versus log {delta} are straight lines. If this is the case, {tau}(q) is the slope of the line corresponding to the exponent q and the method of moments is justified (Evertsz and Mandelbrot, 1992). {tau}(q) is called the qth mass exponent of the measure (Feder, 1988) and is related to the generalized fractal dimensions (Hentschel and Procaccia, 1983) defined as

Formula 5[5]
For a fractal measure, the D(q) curve is called the spectrum of fractal dimensions. When q = 0, D(0) is the fractal dimension of the geometric support of the measure which, in our study, is the Euclidian dimension of a plane (i.e., 2). D(1) = {alpha}(1) = f[{alpha}(1)] is called the entropy (or information) dimension and D(2) = {tau}(2) is known as the correlation dimension. For a multifractal set, the function D(q) decreases monotonically with increasing q. In the limit q -> {infty}, D({infty}) = {alpha}min, whereas D(–{infty}) = {alpha}max. The multifractal spectrum f({alpha})and the function D(q) curve are in fact two equivalent ways of characterizing multifractal measures (Feder, 1988). So, as well as the range of {alpha}-values increases with the increase in the heterogeneity of the distribution of the measure, the range of D(q)-values can be regarded as an appropriate heterogeneity parameter.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Soil Strength Measurements
Understanding the effect of tillage on structural state requires a three-dimensional approach and taking into account a large volume (at least a few decimeters) of soil. Thus, a fully automated penetrometer (30° angle cone with a base area of 10 mm2) mounted on a small vehicle was used (Fig. 1 ). With this equipment, we covered a 80 x 80 cm2 area, which we estimate to be sufficiently wide to evaluate the effect of the most commonly used tillage implements. This square was divided along a 16 x 16 lattice (with 5-cm spacing between neighboring points) yielding a total of 162 = 256 nodes. At each node, a penetration was performed, and data were collected every 2.5 cm from 5 cm below surface down to a depth of 55 cm. This procedure resulted in a 16 x 16 matrix of resistance values at each of 21 depth levels.


Figure 1
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  		Fig. 1. View of the apparatus used for the soil strength measurements.

 
The study was conducted in two successive stages in 2000 and 2001. All field measurements were performed on the same field at Gembloux, in the loess belt of central Belgium. The field soil was a loam soil classified as a Luvisol Orthique (FAO). The soil characteristics were approximately 0.14 clay, 0.80 silt, and 0.06 Mg Mg–1 sand at depths to 0.33 m, and 0.23 clay, 0.74 silt, and 0.03 Mg Mg–1 sand at depths between 0.33 and 0.90 m.

In 2000, measurements were made in September in two adjacent plots retained from an experiment involving different cultivation systems. The first plot was characterized by a continuous ploughing system (conventional plow, depth of ploughing about 30 cm); the second one had been managed using a tine cultivator (heavy wide-blade sweep share cultivator made up of seven tines spaced of 45 cm, depth of tillage about 20 cm) for 5 yr. Resistance measurements were made on the same day in both plots, several months after the last soil preparation. At that time, soil gravimetric water content was 0.21 Mg Mg–1 for depths 0 to 40 cm in the two plots. Resistance measurements were made in the central zone of small bare subplots (3 x 3 m). Just after the measurements, the sampling areas were dug up and a photograph of the soil profile was taken. Among the two series of 21 collected datasets, five particular ones (three, 125, 200, and 325 mm down, named P125, P200, and P325 respectively, for the plowed plot, and two, 125 and 200 mm down, named T125 and T200 for the tine-cultivated plot) were retained and submitted to multifractal analysis. They were selected because they allowed to investigate several features observed in soil profiles, such as anisotropy, periodicity or the presence of a severely compacted zone related to wheel tracks through a tilled horizon (Fig. 2 ). The objective of this first stage of the study was to apply the method of moments and to deduce one or two heterogeneity parameters suitable for characterizing those particular penetration-resistance datasets.


Figure 2
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  		Fig. 2. Cropping profiles of the two selected plots and the origin of the five datasets used during the first stage of the study.

 
During the measurements in 2001, eight plots from a 23 factorial experiment involving different soil management practices (three factors at two variants: compacted versus noncompacted initial soil, ploughing versus tine cultivation, two soil water contents at the time of soil tillage) were selected for measurements (Table 1). The whole field was well loosened in September 2000. Eight plots (4 m x 12 m) were delimited and four of them were immediately compacted by making, perpendicularly to their length, single wheel-beside-wheel passes with a 6910 John Deere tractor (John Deere Co., Moline, IL) weighted down with a heavy cultivator (9780 kg total mass). The plots were then left without any vegetation until spring 2001. Four plots (two non-compacted and two compacted ones) were tilled on May 7 and the other four were tilled on May 30 when the soil was drier (Table 1). Tillage was performed either with a conventional plow (depth of ploughing about 40 cm) or with the tine cultivator used in the first stage of the study (maximum depth of tillage 27.5 cm). To level the soil surface, a shallow harrowing (depth of harrowing about 10 cm) was performed on all eight plots. Each final plot was therefore 3 m wide by 12 m long. The plots were again left without any vegetation, and resistance measurements were not performed until September or October (Table 2). The measurements were made in the center (the unwheeled zone) of each plot and were repeated four times so that 672 (8 plots x 4 replications x 21 layers) datasets were collected. To prevent rain disturbance during the measurements and to reach similar soil water content values for all the sampling spots (Table 2), the plots were protected by plastic-covered greenhouses, all set up on the same day, about 4 wk before the series of measurements began. At the end of the penetration experiment, each sampling spot was dug up perpendicular to the wheel track and a photograph of the soil profile was taken. The objective of this second stage was only to assess the variability of the heterogeneity parameters retained at the end of the first stage and to test their ability to discriminate, in a same field, plots with possibly different soil structural states.


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  		Table 1. Description of the eight plots used during the second stage of the study.

 

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  		Table 2. Planning of measurements and mean gravimetric water content (%) for plots used during the second stage of the study (between brackets: standard deviation, n = 10 replications per date).

 
Data Analysis
The first step in the data analysis consisted of drawing two-dimensional contour plots for each horizon to display the structure of penetration data. These plots were prepared with SURFER software using the nearest neighbors interpolation modality and a high smoothing procedure. These were used only to provide a basic idea of the structural state of the soil volume. Classical statistical analysis (mean penetration resistance, standard deviation, CV, skewness, and kurtosis) and multifractal analysis (method of moments) were performed at each layer. With regard to the latter, the mathematical procedure used to build the multifractal spectra had to be simplified to achieve one or two usable parameters that would be more useful than a spectrum for making statistical comparisons between experimental plots.

Muller (1996) showed that the width of the multifractal spectrum referred to as the deviation of the D(q > 0) values from the D(0) value could be a practical parameter for characterizing and comparing the pore space in chalk samples. Since the structural state of a soil layer depends not only on the distribution of the highest resistance values but also on the distribution of the lowest ones, it seemed appropriate to consider both the cases q < 0 and q > 0. We defined then the following heterogeneity parameter {Omega} as

Formula 6[6]
where qmax can take any real positive value. In our study, to avoid numerical complications that arise with large values of qmax, positive or negative, qmax = 10 was retained.

The proposed parameter {Omega} has the advantage of taking into account negative as well positive moments, which is not the case for the parameter proposed by Muller (1996) or for the entropy dimension D(1) and the correlation dimension D(2), which are often used in the characterization of multifractals (Folorunso et al., 1994; Kravchenko et al., 1999).

As mentioned above, the numerical procedure used to estimate the multifractal spectrum is based on covering the studied area by a succession of square lattices with the width halved at each step. Since in our study, each dataset could be split into four cells of 8 x 8 data, 16 cells of 4 x 4 data, 64 cells of 2 x 2 data or 256 cells of 1 datum, there were only four partitions that could be used to estimate the slopes {tau}(q) that might seem rather poor for checking the linearity of the plots of log Mq ({delta}) versus log {delta} as recommended by Evertsz and Mandelbrot (1992). Thus, a patch made of several high or low values within a dataset might so affect the regularity of the scaling trends that multifractality would not truly take place even if the plots of log Mq({delta})versus log {delta} looked like straight lines. Furthermore, since there was only one possibility for superimposing the four lattices on a given dataset, we could expect that a patch would affect the multifractal spectrum differently and lead to different results, depending on whether it fit in a single cell of a lattice or straddled two adjacent cells of the same lattice, all else being equal.

It is worth noting that an alternative way for implementing multifractals is the gliding box algorithm in which square boxes of linear size {delta} "glide" over the grid map in all possible ways provided that the boxes are bounded completely by the grid map (Allain and Cloitre, 1991; Cheng, 1997). This method generates many more possible number of boxes of size {delta}. However, it does not allow the detection of contentious cases such as mentioned above and edge effects are not negligible because more boxes result from gliding over the central cells than over cells close to the edges. Consequently, we devised another procedure that gives the same importance to all grid cells and that we call the wraparound technique. It involves treating the dataset as if the first datum of any row or column is the neighbor of the last datum in the same row or column. When the wraparound technique is performed only in one direction, the manipulation is similar to a paper cylinder cut along its length and unfolded; the edge is immaterial because what lies behind the edge is found on the other side. For a two-dimensional-set made up of 16 x 16 data, the wraparound technique can be performed in both directions and provides 256 variants from the same dataset. Whatever the variant, each point keeps the same neighbors as in the original set, neighbors of edge points being simply taken at the opposite side of the array.

For every variant having the same number of data than the original dataset, the same splitting procedure as previously is possible but the multifractal spectrum differ of course from one variant to each other. Thus each variant leads to one value for {Omega}. In our study, the wraparound technique should theoretically provide 256 different values for {Omega} but, given that the largest subset of data used to estimate the multifractal spectrum is an 8 x 8 grid, only 64 different ones are obtained from which it is possible to calculate the following parameter

Formula 7[7]
where {Omega}max = max {{Omega}i:i = 1 to 64} and {Omega}min = min{{Omega}i:i = 1 to 64}.

When {Delta}{Omega} {approx} 0, there is no reason to doubt the multifractal behavior of the dataset and thus {Omega}min can be considered as a relevant structural heterogeneity parameter of a soil layer. The case of {Delta}{Omega} >> 0 implies the presence within the investigated soil layer of a structural peculiarity. While, in this case, it is more difficult to give a physical meaning to {Omega}min, {Delta}{Omega} remains interesting because its value is a good indication of the extent of the structural abnormality revealed by the penetrometric measurements. As this structural abnormality is related to soil management practices, it is clearly a point to include in an agronomic plan.

Statistical analysis have been performed with MINITAB statistical software on the results from the 672 datasets (8 plots x 4 replications x 21 layers) gathered during the second stage of the study. The parameters {Omega}min and {Delta}{Omega} were first checked for normality to determine whether a transformation should be conducted before using them for statistical comparisons between plots. They were then submitted to a two-way ANOVA (the model being: Plot, Depth and Plot x Depth interaction), and a Kolmogorov–Smirnov normality test was performed on residuals.

Subsequently, the discriminating capacity of the possibly transformed parameters was assessed and compared with that one of the CV. In a replicated experiment, the treatment effect could be tested using an F-test. When the null hypothesis (equality of means) cannot be rejected, this is either because there was essentially no difference between plots (they are all identical from the considered point of view), or because the method used to characterize plots was not appropriate or not sensitive enough, or, ultimately, because there were not enough replications. In these last two cases, the parameter supplied by the method must not be regarded as a discriminant depending on the aim of the experiment. Given the lack of treatment replications, it was not possible here, and it would even have been inappropriate, to assess the sensitivity of our parameters to the tillage-induced differences in structural heterogeneity. Consequently, their discriminating capacity was only assessed through their ability to detect some significant between-plots differences without making any assumption about the causes of these possible differences. Therefore no conclusions were made regarding the effect of management practices and each of the 21 successive soil layers was analyzed in turn by a simple one-way ANOVA.


   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
First Stage of the Study
The characteristics of the five datasets in terms of penetration resistance are displayed in Table 3. These results are quite comparable with those obtained for a similar loess soil by Ehlers et al. (1983) with a penetrometer having the same characteristics. The contour maps (Fig. 3 ) show clear differences between the five layers considered. P125, from the central part of the plowed layer, was characterized by a fine and regular structure and a fairly homogenous dispersion of resistance values. P200, situated at the bottom of this layer, was slightly different because of a particular periodicity, probably due to plow bodies appearing in a perpendicular direction to the tracks (x-direction). P325, from the plow pan, exhibited a massive structure and high values of resistance which were quite evenly spread. The structure of T125 was fine and regular in the left part of the profile but a highly compacted block appeared in the right part. The compacted zone was formed during the harvest of the previous crop and was not completely broken by the tines of the cultivator. This zone was clearly visible in the contour map. T200 was exceptional because it consisted of two alternating kinds of structure (massive or crumb), due to the passage of the tines in the soil. The data were therefore clearly affected by a certain periodicity.


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  		Table 3. Characteristics of the five datasets used during the first stage of the study (P refers to plowed plot, T refers to tine-cultivated, 125, 200, and 325 refer to depth in millimeters).

 

Figure 3
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  		Fig. 3. Contour maps of the five datasets used during the first stage of the study.

 
According to Evertsz and Mandelbrot (1992), the original datasets as well as the wraparound-generated variants, can be regarded as multifractal measures within the range of sampling scales. Indeed, the graphs of loge Mq({delta})versus loge {delta} appear practically linear (Fig. 4 ), the CV relating to these graphs are always very close to 1 in absolute value (Fig. 5 ) and D(q + 1) < D(q) (Fig. 6 ). The D(q) curve relative to P200 displays the possible asymmetry of the multifractal spectrum justifying the choice of a parameter taking into account both the cases q < 0 and q > 0. Furthermore, considering only q > 0 would have brought us to conclude that P200 and P125 were nearly identical whereas it arises from the left part of the graph that impact of low penetration resistance values was probably more important in P200, which, besides, is in accordance with the low values periodicity brought out by the contour map (Fig. 3). Finally, the arbitrary value qmax = 10 can in retrospect be regarded as an optimal choice in these data sets because the higher |q|, the worse the linearity of plots of log Mq({delta}) versus log {delta} (Fig. 5), but the better the differentiation between layers (Fig. 6).


Figure 4
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  		Fig. 4. Examples of graphs expressing loge Mq({delta}) versus loge ({delta}): graphs relating to the original datasets supplying the worst and the best coefficient of variation when the moment order |q| = qmax = 10.

 

Figure 5
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  		Fig. 5. Coefficients of correlation associated with the graphs of loge Mq({delta}) versus loge {delta}) for the five datasets used during the first stage of the study and relating to: (a) the original datasets, (b) the wraparound-generated variants supplying {Omega} max, (c) the wraparound-generated variants supplying {Omega} min.

 

Figure 6
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  		Fig. 6. Spectra of fractal dimensions [D(q) curves] for the five datasets used during the first stage of the study and relating to: (a) the original datasets, (b) the wraparound-generated variants supplying {Omega} max, (c) the wraparound-generated variants supplying {Omega} min.

 
The statistical characteristics of the five datasets and results of the multifractal analysis performed on the original datasets and on wraparound-generated variants are displayed in Table 3. It is worth noting that, regarding the five soil layers, {Omega}0does not supply the same ranking as the CV. This is due to the fact that multifractal parameters reflect many of the major aspects of soil spatial variability (Kravchenko et al., 1999) unlike CV, which does not account for the spatial arrangement of the points in a dataset.

The importance of the parameter {Delta}{Omega} is evident in the study even if high values raise some questions about the real interpretation to give to the parameter {Omega}min. Layers T125 and T200 are characterized by {Delta}{Omega} values about three to eight times larger than for the other layers, suggesting the presence of a peculiarity. There is an obvious connection between these high {Delta}{Omega} values and the peculiarities visible on the contour maps (Fig. 3). Indeed, as it can be observed in Tables 4 and 5, the {Omega} values differ much more from one variant of the dataset to the next when the wraparound technique is used in the direction parallel to the wheel track of the tractor. Moreover, it is worth noting that, when the data are wrapped perpendicularly to the wheel track, the differences between variants in terms of {Omega} are similar to the {Delta}{Omega} value observed for the layer P325, which was not affected by tillage. Thus, not only can the parameter {Delta}{Omega} be used as an index indicating that, according to the Cropping Profile Method (Gautronneau and Manichon, 1987), the profile should be vertically divided into different homogenous morphological units, but its magnitude is a measure of the contrast between these units from a structural point of view. It appears therefore that the wheel track had a stronger impact on the structural irregularity of the arable layer than the tines passages were the cause of a certain periodicity. {Delta}{Omega}could therefore be integrated into the compartmental model developed by Roger-Estrade et al. (2000) to simulate changes in tilled topsoil over time.


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  		Table 4. Values of the parameter {Omega} for the 64 variants of T125 provided by the wraparound technique (horizontally: values when the data are wrapped in the direction parallel to the wheel track; vertically: values when the data are wrapped in the direction perpendicular to the wheel track).

 

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  		Table 5. Values of the parameter {Omega} for the 64 variants of T200 provided by the wraparound technique (horizontally: values when the data are wrapped in the direction parallel to the wheel track; vertically: values when the data are wrapped in the direction perpendicular to the wheel track).

 
For the three other layers, P125, P200, and P325 for which {Delta}{Omega} is lower, {Omega}min could be regarded as a measure of their structural heterogeneity. The smallest value was obtained for P325, which was collected from the plow pan and appeared morphologically to be the most homogenous soil layer as also confirmed by the very low CV. Within the plowed layer, P200 seems slightly more heterogeneous than P125, in which the crumbling is more intense because of the shallow harrowing.

Second Stage of the Study
The Kolmogorov-Smirnov tests computed on the 672 residuals of the two-way ANOVA indicated that the two parameters {Delta}{Omega} and {Omega}min were not normally distributed, but that they became so after logarithmic transformation. Thus, with regard to using them in the future to compare plots or tillage treatments by means of classical F-tests, we thought it preferable to replace {Delta}{Omega} and {Omega}min with two normally distributed parameters. The first one, relating to the regularity of a soil horizon, was defined as

Formula 8[8]
This parameter was called the regularity index because it increases with the regularity of the horizon. For a sufficiently regular horizon, the second parameter, defined as

Formula 9[9]
expresses the structural homogeneity of a horizontal soil layer. It is called the homogeneity index because the more homogeneous the soil structure the higher its value.

With regard to the regularity index Ir, results displayed in Table 6 show that the undisturbed layers are generally characterized by values greater than 3, while, in the tilled layers, Ir values vary from 1.36 to 3.21. The lowest values, smaller than 2.3, occur in some particular layers and are easily explained by the results of the in situ observations made just after the measurements and at the same places. Thus, in Plots 3, 4, 7, and 8, the lower values in layers 20.0 or 22.5 cm stem from the undulating bottom of the tilled stratum created by the tine-cultivator used in these plots (Fig. 7 ). In Plots 5 and 6, the very low values at depths of between 37.5 and 42.5 are due to the fact that the penetration measurements straddled two successive passes of the plow while a faulty adjustment of the plow led to a strong difference in ploughing depth depending on the forward movement of the tractor (Fig. 8 ). It is worth noting that differences between plots are significant only for layers mentioned above, that is, depths 22.5 cm and depths of between 30.0 and 47.5 cm (Table 7). Ir appears then, not only a good discriminant, but also a better suited parameter than CV to highlight strong irregularities due to tillage tools. These findings strengthen the originality of parameter Ir, which could be very useful for designing new tillage tools by testing them for their ability to affect the whole working width. It could also be used to compare different adjustments of the same tool and to assess the impact of various working conditions on the regularity of the tilled stratum.


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  		Table 6. Mean values obtained for the regularity index Ir. {dagger}

 

Figure 7
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  		Fig. 7. Soil profile in Plot 8 illustrating the undulating bottom of the tilled stratum in a tine-cultivated plot.

 

Figure 8
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  		Fig. 8. Soil profile in Plot 5 showing the two different ploughing depths.

 

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  		Table 7. ANOVA for testing plot effect in terms of the coefficient of variation (CV), the regularity index (Ir) and the homogeneity index (Ih) at each studied depth: Fobserved values.

 
With regard to the homogeneity index Ih, the observed values are greater than 1.5 in undisturbed layers, while they are generally much lower in tilled layers (Table 8). This index appears then as a parameter capable of distinguishing among tilled and undisturbed layers. Moreover, in the upper tilled horizon where the impact of the managements practices is the most pronounced (depths of between 5 and 25 cm), Ih is more helpful than CV to detect significant differences between plots. Not only, it generally supplies higher Fobs values (Table 7), but, using Tukey's WSD procedure (Tukey, 1953, Maxwell and Delaney, 1989) to perform all possible pairwise comparisons, it also detects about three times more significant differences between plots (Table 9). It is worth remembering that CV is a simple measure of data variability while Ih is a more complex parameter sensitive not only to the variability (there is no heterogeneity without variability) but also to the spatial organization (structure) hidden in the dataset and revealed by the multifractal formalism. To illustrate this comment from a structural point of view, comparison between Plots 5 and 6 is interesting because they look fairly comparable at first glance: their mean penetration resistances, CV and regularity index Ir are very close and far from being statistically different (Table 10). However, at depths of between 17.5 to 32.5 cm, their homogeneity index Ih are significantly different (Table 10) suggesting some structural differences. These results agree with morphological observations showing that compacted clods buried in the profile are more numerous in Plot 6 than in Plot 5 (Fig. 9 ).


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  		Table 8. Mean values obtained for the homogeneity index Ih. {dagger}

 

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  		Table 9. Pairwise comparisons between plots (error rate P provided by the Tukey's procedure). {dagger}

 

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  		Table 10. Comparison between plots 5 and 6 in terms of the mean penetration resistance (RP), the coefficient of variation (CV), the regularity index (Ir) and the homogeneity index (Ih) at each studied depth: error rate P provided by the Tukey's WSD procedure.

 

Figure 9
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  		Fig. 9. Detail of the structure for depths between 15 and 35 cm in: (a) Plot 5 and (b) Plot 6

 

   CONCLUSIONS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
This study shows that penetration measurements within a relatively small area can be a very practical way of assessing the structural state of agricultural soils, which has been difficult to achieve quantitatively. It also shows that multifractal formalism is appropriate for analyzing the variability of such measurements when the wraparound technique is used to provide a characterization of the soil structural state that is less dependent on sampling chance, more consistent with field observations and better suited to comparing experimental plots in management studies.

Comparatively with the more classical CV, the two proposed parameters, Ir and Ih are found to be better discriminants for revealing differences between plots and, jointly used, to give a more accurate idea about the spatial organization of resistance penetration data. The first one, Ir, is well suited for expressing the structural irregularities caused by tillage implements. It could therefore be useful in designing new tillage tools by testing them for their ability to affect the whole working width. It could also be used to compare different adjustments of the same tool and to assess the impact of various working conditions on the regularity of the tilled stratum. As for the second parameter, Ih, there is every indication that it is a very sensitive parameter well suited to determining soil structural heterogeneities. However, from the soil management point of view, more sophisticated field experiments are still needed to better assess its sensitivity to tillage-induced differences. Moreover, further investigations should be performed to standardize the two proposed indices, Ir and Ih, before they can be applied as general structural state indicators.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Abbreviations: CV, coefficient of variation.

Received for publication March 28, 2006.


   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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R. G. Moreno, M. C. D. Alvarez, A. S. Requejo, and A. M. Tarquis
Multifractal Analysis of Soil Surface Roughness
Vadose Zone J., May 27, 2008; 7(2): 512 - 520.
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