an entropy-based parametrization of soil texture via fractal modelling of particle-size distribution

An Entropy-Based Parametrization of Soil Texture via Fractal Modelling of Particle-SizeDistributionAuthor(s): Miguel Angel Martíin, José-Manuel Rey and Fco. Javier TaguasSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 457, No. 2008 (Apr.8, 2001), pp. 937-947Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/3067485 .

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[rrn THE ROYAL 10.1098/rspa.2000.0699 B.I SOCIETY

An entropy-based parametrization of soil texture via fractal modelling of particle-size distribution

BY MIGUEL ANGEL MARTIN1, JOSE-MANUEL REY2 AND FCO. JAVIER TAGUAS1

1Departamento de Matemdtica Aplicada, Escuela Tecnica Superior de Ingenieros Agronomos,

Universidad Politecnica de Madrid, 28040 Madrid, Spain 2Departamento de Andlisis Economico, Universidad Complutense,

Campus de Somosaguas, 28223 Madrid, Spain

Received 6 April 2000; accepted 2 October 2000

Classification of soil texture is usually carried out by considering different textural classes which group together soils with mass percentages of clay, silt and sand within certain ranges. Soil samples of diverse composition, however, become indistinguishable from each other under the grouping that these classes establish. A parametrization of soil texture based on the entropy dimension is proposed. Following Martin & Taguas, a fractal modelling allows us to compute the entropy dimension of the modelled particle-size distribution (PSD) from readily available textural data. First, the role and meaning of the entropy dimension computed in that manner for soil PSD is discussed. Then the fractal model is applied to compute the value of the entropy dimension for a large number of soils from the Soil Conservation Service (SCS). Soil textural classes can be characterized by the average value of the entropy dimension of soils belonging to each class. Soils with entropy dimensions between prescribed limits are located within a well-defined region of the textural triangle. Thus, entropy dimension, via the fractal model of Martin & Taguas, provides a continuous parameter which is suitable for a fine quantitative characterization of soil textures using conventional textural data.

Keywords: particle-size distribution; fractal distribution; entropy dimension

1. Introduction

Classification of soil textures is an important goal in the soil sciences. Soil particle-size distribution (PSD) is usually reported in terms of the mass percentages of clay, silt and sand. Fixing percentage limits of these fractions defines different textural classes in the US Department of Agriculture (USDA) textural triangle (Soil Conservation Service (SCS) 1975). Other classifications of soil texture (Folk 1954; Shepard 1954; Baver et al. 1972; Vanoni 1980) follow the above scheme in the definition of the textural classes, although these systems differ in the limits chosen to define clay, silt and sand and in the percentage limits chosen to define each textural class.

Each textural class groups together many different combinations of clay, silt and sand. As a result, soil samples of diverse composition become indistinguishable from each other under the grouping that these classes establish. Shirazi & Boersma (1984)

Proc. R. Soc. Lond. A (2001) 457, 937-947 ? 2001 The Royal Society 937



M. A. Martin, J.-M. Rey and F. J. Taguas

extended the textural classification by integrating the information obtained from mechanical analysis of soil samples in the conventional texture diagram used by the USDA. They converted the textural triangle into a new diagram by considering the geometric mean particle size and the geometric standard deviation of soil samples, obtained by mechanical analysis.

During the last decade, new ideas coming from fractal geometry have been useful in characterizing highly irregular media, in which the apparent disorder present in the system appears to be structured and ordered by laws which reflect the way the disorder itself is reproduced at different scales. In particular, in soil, the number N(R) of particles of size greater than R and, consequently, the mass M(R) of particles of size less than R, have been shown to obey power-scaling laws of the type N(R) oc R-D and M(R) oc R3-D (Turcotte 1986; Tyler & Wheatcraft 1989, 1992; Wu et al. 1993). Such scaling reveals a scale invariance or self-similarity which is characteristic for many heterogeneous media and processes in nature (Mandelbrot 1982). The scaling exponent D (or 3 - D), usually called the fractal dimension, becomes a characteristic parameter of the PSD. However, soils with quite different percentages of clay, silt and sand may have similar fractal dimensions (Tyler & Wheatcraft 1992), and thus this parameter, which was useful in predicting other soil properties, fails in characterizing soil texture.

In Martin & Taguas (1998), a measure-theoretic self-similar modelling for PSD was proposed based in a new interpretation of the invariance of PSD with respect to the scale. This model, defined by means of two coupled sets of functions and probabilities, determines how the PSD reproduces its structure at different length- scales and allows one to simulate intermediate values from common textural data. The ability of the model to characterize soil textures in that way, and to reconstruct the entire PSD, were successfully tested in Taguas et al. (1999).

Soil PSDs are complex distributions, revealing new details when explored at de- creasing length-scales. Entropy, or rather its rate of growth with respect to the scale (entropy dimension), is a natural parameter to evaluate the degree of heterogeneity of any complex distribution. A natural attempt to characterize PSDs thus consists of estimating their entropy dimension. Computing entropy dimension, however, requires scanning the structure of the PSD within a range of scales (in particular, at small

scales) so that a large amount of textural data is required. Since textures are usually reported in terms of the content of a small number of soil fractions (e.g. clay, silt and sand content), an estimation of entropy dimension seems impracticable for classification purposes. The fractal model of Martin & Taguas (1998) permits us to avoid this problem by supplying a self-similar distribution for PSD whose entropy dimension is given by a closed simple formula (see (3.1) below). This is a well-established result from theoretical fractal geometry (Young 1982; Deliu et al. 1991). In this paper the entropy dimension of the self-similar modelled PSD is proposed as a parameter to characterize soil textures. The aim of the paper is, first, to analyse and to interpret some aspects of this approach as well as different meanings of the entropy dimension formula in the context of the mentioned modelling of PSD and, second, to test the ability of this entropy-based method to make a quantitative characterization of soil textures using the commonly available textural data.

The paper is organized as follows. The self-similar model of Martin & Taguas (1998) is described in ? 2. In ? 3 the entropy dimension formula for PSD is analysed in detail. An interpretation of the formula and some of its properties with regard to

Proc. R. Soc. Lond. A (2001)

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An entropy-based parametrization of soil texture

the characterization of the PSD's heterogeneity are considered. In particular, results of Young (1982), together with the main result of Deliu et al. (1991), provide an

interesting interpretation of the entropy dimension number as a meaningful parameter that measures an essential feature of PSD: its richness in terms of the diversity of particle sizes that contribute to the whole soil mass. In ?? 4 and 5 the model is applied to compute the entropy dimensions for a large number of soils and to supply statistical evidence of the validity of this parameter to characterize soil textures for

practical purposes.

2. Measure-theoretic self-similar modelling for PSD

Commonly in practice, equivalent masses of soil corresponding to the clay, silt and sand fractions are associated with intervals of sizes of quite different lengths. This remark suggests a crucial feature of PSD, namely the disagreement or absence of pro- portionality between the mass of soil formed by particles with characteristic diame- ters in an interval [11,12] and the length 12 - 11 of such an interval. This single fact, together with scale-invariant arguments suggested by fractal scalings obtained from real data, permits one to suppose that the mentioned disagreement remains over a certain range of scales. Moreover, it allows one to assume that the disagreement holds at every scale (this is called the self-similarity hypothesis), which is the basic ingredient of the fractal model for PSD introduced by Martin & Taguas (1998). This model permits one to define, from usual textural data, a set of functions and probabilities which is called an iterated function system. The iterated function system determines how a self-similar distribution reproduces its fractal structure at different length-scales (see, for example, Falconer 1990).

Assume that a set of N proportions of mass of soil, corresponding to N consecutive size classes, have been selected from soil textural data. To keep things simple, assume that N = 3. The fractal model for PSD derived from the self-similarity hypothesis is constructed as follows (see Martin & Taguas (1998) for the details). Let I1 = [0, a1], I2 = [ail, a2] and I3 = [a2, c] be the sub-intervals of sizes corresponding to the three size classes and let P1, P2 and P3 be the proportions of mass for the intervals I1, I2 and I3, respectively. Notice that pi > 0 and P1 + P2 + p3 = 1. Associated with these definitions, we consider the following functions

p(1(x) = rlx, 92(x) = r2x + a1, 3(x) = r3x + a2,

where rl = al/c, r2 = (a2 - a1)/c, r3 = (c - a2)/c and x is any point (or particle size value) in the interval I = [0, c]. That is, P1, T2 and T3 are the linear mappings (similarities) that transform the points of the interval [0, c] in the points of the sub- intervals Ix, I2 and I3, respectively. The set {pi, pi, i = 1, 2, 3} is called an iterated function system (IFS) and it defines a self-similar distribution p supported on [0, c], which satisfies p(Ii) = pi for every i (Barnsley & Demko 1985).

In terms of PSD, the set of textural data, together with the self-similarity hypothesis mentioned above, thus determine a self-similar mass distribution ,u for particle soil size defined in I, which fits exactly the input data, that is, '(Ii) = pi. From the self-similarity, the contribution to soil mass of particles with sizes in the interval

Iij = (pj(I) is given by i'(Iij) = PiPj. In general, for any size interval J C I, the IFS model allows one to determine the mass of soil 1p(J) corresponding to the soil particles with sizes in the interval J. Moreover, theoretical results from fractal geometry allows one to simplify the computer simulation of IL(J) (see Martin & Taguas 1998).


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The self-similarity hypothesis for PSD was tested in Taguas et al. (1999), where soil data from SCS (1975) were used to check the ability of different IFSs to reconstruct the entire PSD. The results obtained in that paper strongly support the modelling, in the sense that IFSs are able to effectively reconstruct soil PSDs using commonly available data.

3. Entropy dimension for PSD

Given a self-similar mass distribution u for PSD defined on the interval of sizes I, the next step is to evaluate its complexity. Shannon's entropy is a natural measure of the degree of complexity or heterogeneity of any distribution (Shannon 1948a, b). If the probability space I is decomposed into a finite collection of disjointed subsets A = {Ai, i = 1,..., n} and it(Ai) denotes the mass (or probability) of the subset Ai, the entropy of the partition A is defined by

n

H(A) = - E (Ai) log u(Ai) i=l

(with the convention that OlogO = 0). A uniform distribution (with respect to A, i.e. /t(Ai) = 1/n for all i) is of maximum entropy log n, whereas an uneven or highly non-uniform distribution (i.e. fi(Aj) = 1 for some j) is of null entropy.

The entropy of the self-similar distribution /L for PSD defined in ? 2 can be interpreted in terms of mass contribution as follows. All sizes contribute equally to the total soil mass when p is uniformly distributed on I and thus on any uniform partition of I. This is the most heterogeneous case in the sense that the whole variety of sizes contribute to the mass (highest entropy case). On the other hand, the full soil mass is contributed only by one particle size when p, is concentrated on one point in I (i.e. fu is a Dirac delta), which is the most homogeneous case (lowest entropy case).

A partition-free definition of the entropy of /p is obtained by setting (see, for example, Young 1982)

H(E) = inf{H(A): diamAi < e, A = {Ai}i,

where 'inf' means 'infimum' and 'diam' stands for diameter in the Euclidean space. The value H(E) estimates the minimum amount of heterogeneity of AL with respect to a partition of size E (whose elements have diameter at most E).

Usually, the complexity of f increases when the size of the partitions decreases so that H(E) - +00 as e approaches 0. This motivates the computation of

H(e) DI = lim H

E-o0 - log E

which is called the information or entropy dimension of the distribution ,t (Renyi 1957). The value DI measures the rate of growth of the entropy of f with respect to e. It is known that 0 DI < 1 (e.g. Young 1982).

The value DI provides a measure of the heterogeneity of a PSD: the most heterogeneous case (as explained above) gives DI = 1, whereas the most homogeneous distribution (Dirac PSD as above) satisfies DI = 0.

It turns out that there is a simple closed formula for the entropy dimension of the self-similar measure associated with the IFS {oi, pi, i = 1,2,... , N in terms


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of its structural parameters (ri,pi). Specifically, it follows from a theorem by Young (1982) and from the dimension computation in Deliu et al. (1991) that the entropy dimension of p is given by the number d defined as

d= E i Pi ogpi (3.1) Ni=1 Pi log ri

The formula is valid if pj = 0 for some j, with the agreement that 0 log 0 = 0. This dimension formula has a rich interpretation from both a geometric and a dynamical point of view (see, for example, Moran & Rey 1998).

First we give an interpretation of the structure of the formula (3.1). For the sake of simplicity, assume momentarily that the interval of sizes [0, c] has been partitioned in N = 3 sub-intervals I1, 12, I3 corresponding to clay, silt and sand. Notice that the numerator in (3.1) is the opposite of the entropy H of this partition. Recall that the ri are the lengths of the intervals Ii normalized to add up 1. The denominator in (3.1) can be interpreted as the weighted averaged value of the logarithms of the lengths of the sub-intervals in the partition. If this mean value is denoted by (log ri), formula (3.1) can be read as (logri)-d. The result in Deliu et al. (1991) implies that this law also holds asymptotically for arbitrarily small partitions.

There is an interesting interpretation of the entropy dimension formula (3.1) that comes from the fact that the value d agrees with the dimension of the fractal distribution p. Specifically, from Deliu et al. (1991) and another result of Young (1982), it follows that the entropy dimension DI coincides with the dimension of the measure ,u (dim,u), which is the Hausdorff dimension of a 'minimal' set S that concentrates the full mass. In general, S and the interval of sizes [0, c] are different sets (see Martin & Taguas (1998) for details). So the set S may be seen as the 'set of sizes' and the number DI gives the geometric size of this set. Again, the larger the number d, the bigger the set S, and thus the wider the continuum of sizes that contribute to the soil mass, and the richer the texture of the soil. Notice that, since all the PSDs are supported in the same interval [0, c] (Barnsley & Demko 1985), the set S and, more specifically, its dimension, play an important role by gauging the geometric size of the set of sizes S that actually contribute to the dimension.

Next we analyse the extremal values of (3.1) and the properties of the related distributions. First notice that the parameter choice pi = ri corresponds to the most heterogeneous PSD, which carries full dimension DI = dimp/ = 1. Actually, it follows from Moran & Rey (1998) that this is the only possible parameter choice that generates the most heterogeneous PSD, that is, if dim / = d = 1, then pi = ri (and ,u is the Lebesgue measure on I). In the case N = 3 with the ri corresponding to the sub-intervals I1 = [0, 0.002] (clay), I2 = [0.002, 0.05] (silt) and I3 = [0.05, 2] (sand), the most heterogeneous PSD is given by P1 = 0.001, P2 = 0.024 and P3 = 0.975. Now observe that d = 0 if and only if Ei pi log pi = 0, which holds if and only if pj = 1 for some j and so pi = 0 for i $& j. Thus totally homogeneous distributions (with a unique contributing size interval) have null dimension. This is related to the fact that the value d properly reflects the heterogeneity in the PSD: whereas percentages of sand close to 97.5 yield highly heterogeneous PSDs of nearly maximal dimension, the dimension must drop from one to zero in the narrow interval [97.5,100] of sand percentages, which accounts for the high homogeneity of these soils.

Commonly, in real soils, the pi and ri disagree widely, so that a large proportion of soil mass corresponds, say, to a large amount of sizes located in the interval [0, 0.002]


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M. A. Martzn, J.-M. Rey and F. J. Taguas

80

60

40

20 9' * \X

KXX

20- ? ........

0 20 40 60 80 100

sand

Figure 1. Soils with entropy dimension greater than 0.6 (0), between 0.6 and 0.4 (*), between 0.4 and 0.2 (e) and less than 0.2 (+).

(clay). In these cases, certain areas of the interval [0, c] accumulate much more mass than others, and this is related to the fact that d is a value well below 1. In general, the more important the disagreement between the values of the Pi and ri, the more rarified the set of sizes contributing to the soil mass, and so the smaller the value of the dimension of p/. It follows from results in Moran & Rey (1998) that any soil texture model different from the most heterogeneous case pi = ri is described by a self-similar PSD which is singular, i.e. the mass within small size intervals and the length of the intervals have different power-scaling behaviour. The number d quantifies the singular behaviour of the measure. According to this interpretation, the dimension of p, summarizes important features of PSD and, in turn, of soil texture.

4. Materials and methods

Textural data of 143 soils were used in this research. These data correspond to the two first horizons of soils described in SCS (1975, pp. 486-742).

The data refer to the percentages mi of mass of soil corresponding to particles whose sizes are hein the following size classes (mm): clay (less than 0.002); silt (0.002- 0.02) and (0.02-0.05); very fine sand (0.05-0.1); fine sand (0.1-0.25); medium sand

(0.25-0.5); coarse sand ad (0.5-1) and very coarse sand (1-2). To construct an associated IFS from these data according to the model of ? 2, notice that N = 8 and c = 2 in this case. Let [a, b] denote tos the sizes greater than or equal to a and less than or equal to b. These size classes determine a set of seven intermediate boundary points ci, namely


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80

60

40

0.2-0.4

20 - 0.4-0.6

> 0.6

0 20 40 60 80 100 sand

Figure 2. Regions defined by soils represented in figure 1.

{0.002, 0.02,0.05,0.1,0.25, 0.5, 1}, and eight consecutive intervals corresponding to the eight size classes I, = [0,0.002], I2 = [0.002,0.02],..., I8 = [1,2].

Denote ao = 0 and as = 2 for convenience. For each i = 1, 2,..., 8, let

Wi(x) = rix + ai-l, (4.1)

where ri = (oai - ai-), and let pi = mi/100. For any given soil, the IFS {(i, pi, i = 1,..., 8} determines the self-similar PSD.

The entropy dimension computed via formula (3.1) gives, in this case,

d =i1 pi logpi (4.2) Ei=lPilogri

5. Results and discussion

The PSD data allow us to represent each soil by means of a point in the textural triangle. Soils with entropy dimension greater than 0.6, between 0.6 and 0.4, between 0.4 and 0.2 and, finally, less than 0.2, are represented in figure 1. In each case, soils are located in a well-defined region of the textural triangle. Figure 2 is a superposition of those regions. Notice that the regions do not overlap much and, when combined, give a good coverage of the whole region where the soils analysed are represented.


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100.

80

60

40

20

0 20 40 60 80 100 sand

Figure 3. Subregions where soils with entropy dimension between 0.2 and 0.3, and between 0.3 and 0.4, are represented.

Moreover, further subregions can be defined inside each primary region by setting new bounds for the entropy dimension, also obtaining similar nice results. For instance, figure 3 shows the subregions where soils with entropy dimension between 0.2 and 0.3, and between 0.3 and 0.4, are represented.

These results amount to a successful attempt in the aim of classifying soil texture by means of fractal parameters. Previous works (Tyler & Wheatcraft 1992) have shown that using usual scaling fractal dimensions is inappropriate for that goal.

Table 1 shows the average entropy dimension

En di d= i=l n

obtained from the dimensions di of n soils within each textural class, the average fractal dimension

.n= D D = i=l

n obtained by the method proposed by Tyler & Wheatcraft (1992), and their respective standard deviations

e= ,l(di- d)2 n- 1

First, a clear trend can be observed in the variation of the entropy dimension. An increase in the proportion of sand leads to an increase in the value of entropy dimen-


944

a* =Ej=l(Di - D)2

n-1




Table 1. Average entropy dimension d, average fractal dimension (cf. Tyler & Wheatcraft 1992) D and their standard deviations a and r*, respectively, organized by soil textural classes

entropy fractal dimension dimension

textural class l~-~

(number of soils) d a D a*

sand (6) 0.59 0.026 2.25 0.128 loamy sand (5) 0.56 0.035 2.70 0.232 sandy loam (38) 0.54 0.079 2.82 0.238 loam (17) 0.44 0.070 2.79 0.204 silt loam (26) 0.30 0.071 2.80 0.150 silt (2) 0.27 0 2.9 0.028 sandy clay loam (5) 0.45 0.016 2.89 0.075 clay loam (10) 0.35 0.031 2.81 0.186 silty clay loam (15) 0.28 0.040 2.77 0.272 silty clay (7) 0.24 0.053 2.84 0.155 clay (12) 0.20 0.061 2.86 0.086

sion, and an increase in the proportion of clay implies a lower value for the entropy dimension. As a general rule, the more average agreement between percentage mass in each interval of sizes and the length of the interval, the greater the entropy dimension. This seems quite natural since the entropy dimension measures the degree of heterogeneity of the PSD. It does not seem possible to make an equivalent interpretation of the variation of the usual fractal dimension, as Tyler & Wheatcraft (1992) have already pointed out. Secondly, the standard deviation takes much smaller values in the case of entropy dimension parameter. Notice that the greatest values of standard deviation for the entropy dimension correspond to textural classes which occupy broader regions in the textural triangle. This has a quite natural explanation, since soils in the same textural class may have rather different mass proportions in the different size classes. Further, soils belonging to different textural classes may be close in the textural triangle, that is, they may have close values of the mentioned proportions. However, these results reflect the nice ability of the entropy dimension to characterize soil textural classes quantitatively.

We end this section with some final remarks on the role of the fractal model and the entropy dimension formula in the parametrization proposed in the paper. First notice that the entropy dimension formula can be applied without any reference to the IFS model. However, a naive use of the formula would lack the rich interpretation of entropy dimension as well as a plausible explanation of its genesis. Secondly, in strict a sense, the dimension formula under IFS modelling has full meaning only for soils whose texture has been successfully simulated by the corresponding IFS. Obviously, whereas a universal self-similar behaviour cannot be guaranteed for all soils, the results in Taguas et al. (1999) are compatible with that behaviour and thus make self-similarity a sensible working hypothesis. Finally, although the definition of the entropy dimension DI given above makes sense for PSD without assuming the fractal model, estimating DI for real soils becomes a difficult task. However, assuming the self-similarity hypothesis, the model of Martin & Taguas (1998) provides the simple


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exact formula (3.1) to compute DI and, furthermore, the interpretations in terms of PSD heterogeneity explained above.

6. Conclusions

In this paper, the entropy dimension is proposed as a parameter to characterize textural heterogeneity of soil PSDs.

Entropy and entropy dimension are natural measures of the complexity of general distributions and thus are candidates to provide a fine quantitative characterization of individual soil textures. A fair direct computation of entropy dimension from real soils, however, would require textural data for a wide range of scales, which are not usually available in practice.

This difficulty may be overcome via the fractal model for PSD introduced by Martin & Taguas (1998). This model, plus well-known results from theoretical fractal geometry, further provides a closed formula to compute the entropy dimension from textural data.

The computation of the entropy dimension for different soils using commonly available data demonstrates that entropy dimension actually gives a fine quantitative parametrization of soil texture within the textural triangle. It is shown how pre- scribing bounds for entropy dimension groups together soils in well-defined regions of the textural triangle.

Entropy dimensions computed in the proposed way for different soils within a textural class give different average values for each class. The variations of these mean values can be interpreted in terms of the mass proportions in the different size classes. So, by gauging different PSD heterogeneities within the same class, entropy dimension appears to be useful in the quantitative characterization within standard soil textural classes.

A main conclusion of this research is that entropy dimension, computed via the PSD fractal model of Martin & Taguas, permits a fine characterization of soil textures from commonly available textural data.

This research is partly supported by Comunidad de Madrid, under 07M/0048/2000, and Plan Nacional de Investigaci6n Cientifica, Desarrollo e Innovaci6n Tecnol6gica (I+D+I), Spain, under REN2000-1542.

Nomenclature

N number of textural classes in the fractal model for PSD I = [0, c] whole interval of sizes for PSD

Ii = [ci-i, i] ith sub-interval of sizes for PSD

mi percentages of mass of soil corresponding to sizes in Ii

pi proportion of mass contributed by sizes inside the interval Ii

ri ratio between the length of the sub-interval Ii and the length of the interval I

cpi(x) = rix + qci- linear mapping that maps I onto Ii /u self-similar distribution for soil PSD


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DI entropy dimension of a distribution (defined for an

arbitrary distribution) dim p dimension of the distribution p d the number E pi log pi/ pi log ri (which coincides

with the entropy dimension DI of the self-similar distribution associated with {(i, pi})

References

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an entropy-based parametrization of soil texture via fractal modelling of particle-size distribution

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