Soil Science Society of America Journal 65:956-957 (2001)
© 2001 Soil Science Society of America
COMMENTS & LETTERS TO THE EDITOR
On the Characteristic Aggregate Size for Estimating Fractal Dimensions of Soils
H. Millan
Departamento de Ciencias Básicas Universidad de Granma, Apdo. 21, Bayamo 85100 Granma, Cuba
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INTRODUCTION
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Fractal concepts have been widely applied in Soil Science. I have revised approximately 250 published papers having to do with the parameterization of aggregate-size distributions using fractal dimensions. More than 95% of the revised works used the arithmetic average between successive sieves as a characteristic aggregate size. Many competent soil scientists have questioned this approach. Perfect and Kay (1991) have shown that the lower sieve size is a better estimator of the characteristic aggregate size; more recently, Kozak et al. (1996) derived a sound theoretical equation to estimate the characteristic aggregate size. However, the geometric mean diameter (GMD) has never been considered.
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Where xi-1 and xi are the upper and the lower sieve apertures, respectively. If one introduces the well-known q ratio as:
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then, by combining Eq. [1] and Eq. [2] with the Eq. [14] derived by Kozak et al. (1996), it is possible to derive another explicit expression for the ratio of the characteristic aggregate size calculated from Eq. [14] (xi [14] hereafter) to the GMD as a function of q and the fractal dimension, D.
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I want to make some personal comments on the relation between Eq. [2] and Eq. [3] as represented by Fig. 1
. We soil physicists must keep in mind that each sieve aperture in a nest of sieves is a controllable variable. It means that, for aggregate-size distributions, the soil scientist can select his or her own nest of sieves. Of course, this approach does not apply to particle-size distributions. If the aforementioned selection procedure can keep the q ratio between successive size classes within a range from 1.5 to 2, then the error introduced is <5% as can be deduced from Fig. 1. Furthermore, I selected a nest formed by those more commonly used sieves: 0.125 to 0.25, 0.25 to 0.50, 0.50 to 1.0, 1.0 to 2.0, 2.0 to 3.0, 3.0 to 5.0, 5.0 to 7.0, 7.0 to 10.0, 10.0 to 16.0, and 16.0 to 31.5 mm. Note that q is always 
2. When I correlated the GMD values with the values calculated from Eq. [14] of Kozak et al. (1996) for three different values of fractal dimensions (2.1, 2.5, and 2.9), I found a one-to-one relationship
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In summary, if we do not have conditions to implement the algorithm of Kozak et al. (1996) (remember that a previous knowledge of the aggregate bulk density is required), the GMD can be used as a consistent, unbiased estimator of the characteristic aggregate size (if and only if q 2).
Received for publication July 12, 2000.
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REFERENCES
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- Kozak, E., Ya.A. Pachepsky, S. Sokolowski, Z. Sokolowska, and W. Stepniewski. 1996. A modified number-based method for estimating fragmentation fractal dimensions of soils. Soil Sci. Soc. Am. J. 60:1291-1297.[Abstract/Free Full Text]
- Perfect, E., and B.D. Kay. 1991. Fractal theory applied to soil aggregation. Soil Sci. Soc. Am. J. 55:1552–1558.[Abstract/Free Full Text]
Response: Comments on "On the Characteristic Aggregate Size for Estimating Fractal Dimensions of Soils"
Yakov Pachepsky
b USDA/ARS/BA/RSM, B-007 10300 Baltimore Avenue, BARC-Wes Beltsville, MD 20705
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INTRODUCTION
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The letter proposes defining the characteristic size of grains within a range of sizes as the geometric mean of the upper and the lower boundary sizes for the range. The equation is correct, and the numerical experiment is sound. However, the presented development seems to be an aimless one.
There is no justified theoretical or conceptual need to introduce the geometric mean radius. The letter opens with applications of fractal models to particle- and aggregate-size distributions. Therefore, a reader should expect that the geometric mean radius is suggested to be used in estimations the fragmentation fractal dimensions. I will proceed assuming that this is the main pathos of the letter. Kozak et al. (1996) defined the characteristic size assuming that the fractal scaling law is applicable not only among the size ranges but also within the size ranges. This is why Eq. [14] from Kozak's work gives values of the characteristic size dependent on the fractal dimension. This constitutes the difference between Kozak's work and works of other authors who did not assume the scaling valid within the size ranges and were free to choose a convenient formula for the characteristics size. From the point of view of those other authors, there is no advantage in selecting the geometric mean radius, the minimum radius, or the mean radius; it is just a matter of convenience (Rieu and Sposito, 1991). So, using the geometric mean radius is not correct from the standpoint of Kozak et al. (1996) and is not advantageous from the standpoints of other authors.
There is no practical need in introducing the geometric mean radius. The author indicates that the geometric mean radius can be used if there are no conditions to implement the algorithm of Kozak et al. (1996). I argue that there is no difference between conditions to use geometric mean radius and the Kozak's characteristic size. The author's remark about the "previous knowledge of the aggregate bulk density" is irrelevant. One has to know aggregate bulk density to convert masses to counts regardless of using geometric mean radius or Kozak's characteristic size afterwards to reveal the scaling law.
The author of the letter has probably overlooked the fact that the Kozak's Eq. [14] gives the geometric mean radius in a special case of the fractal dimension D = 1.5.
To summarize, the letter to the editor does not contribute to the methodology of studying scaling in soils.
Received for publication July 12, 2000.
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REFERENCES
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- Kozak, E., Ya.A. Pachepsky, S. Sokolowski, Z. Sokolowska, and W. Stepniewski. 1996. A modified number-based method for estimating fragmentation fractal dimensions of soils. Soil Sci. Soc. Am. J. 60:1291–1297.
- Rieu, M., and G. Sposito. 1991. Fractal fragmentation, and soil porosity, and soil water properties: II. Applications. Soil Sci. Soc. Am. J. 55:1231–1244.[Abstract/Free Full Text]
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INTRODUCTION
REFERENCES
