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Discussion paper
Compositional analysis for an unbiased measure of soil aggregation
Léon E. Parent a,⁎, Cinara X. de Almeida b, Amanda Hernandes b, Juan J. Egozcue c, Coskun Gülser d,Martin A. Bolinder a,e, Thomas Kätterer e, Olof Andrén e, Serge E. Parent a, François Anctil f,José F. Centurion b, William Natale b
a Department of Soils and Agrifood Engineering, Université Laval, Québec, Canada G1K 7P4b Department of Solos e Adubos, FCAV/Unesp, Jaboticabal, São Paulo, Brazilc Department of Applied Mathematics III, Universitat Politècnica de Catalunya, E-08034 Barcelona, Spaind Soil Science Department, Faculty of Agriculture, Ondokuz Mayıs University, 55139 Samsun, Turkeye Department of Soil Sciences and Environment, SLU, P.O. Box 7014, SE-750 07 Uppsala, Swedenf Department of Civil Engineering, Université Laval, Québec, Canada G1K 7P4
Soil aggregation is an index of soil structure measured by mean weight diameter (MWD) or scaling factorsoften interpreted as fragmentation fractal dimensions (Df). However, the MWD provides a biased estimateof soil aggregation due to spurious correlations among aggregate-size fractions and scale-dependency. Thescale-invariant Df is based on weak assumptions to allow particle counts and sensitive to the selection ofthe fractal domain, and may frequently exceed a value of 3, implying that Df is a biased estimate ofaggregation. Aggregation indices based on mass may be computed without bias using compositional analysistechniques. Our objective was to elaborate compositional indices of soil aggregation and to compare them toMWD and Df using a published dataset describing the effect of 7 cropping systems on aggregation. Sixaggregate-size fractions were arranged into a sequence of D-1 balances of building blocks that portray theprocess of soil aggregation. Isometric log-ratios (ilrs) are scale-invariant and orthogonal log contrasts orbalances that possess the Euclidean geometry necessary to compute a distance between any two aggregationstates, known as the Aitchison distance (A(x,y)). Close correlations (r>0.98) were observed between MWD,Df, and the ilrwhen contrasting large and small aggregate sizes. Several unbiased embedded ilrs can characterizethe heterogeneous nature of soil aggregates and be related to soil properties or functions. Soil bulk density andpenetrater resistance were closely related to A(x,y) with reference to bare fallow. The A(x,y) is easy to imple-ment as unbiased index of soil aggregation using standard sieving methods and may allow comparisonsbetween studies.
A soil aggregate is made of closely packed sand, silt, clay, and or-ganic particles (Cambardella, 2006). Soil aggregation is an index ofsoil structure quality related to crop productivity, water and solutetransport (Mueller et al., 2010), biological habitats (Crawford et al.,1993a), and resistance to erosion (Eash et al., 1994).
Soil aggregation is typically quantified by the distribution of aggre-gate size fractions within a given mass of soil using wet (Angers et al.,2008) or dry (Larney, 2008) sieving techniques. Sieve-size fractionsare computed as the proportion of total mass that is retained on agiven sieve. The mean weight diameter (MWD) (Van Bavel, 1949) andfragmentation fractal dimension (Df) (Rieu and Sposito, 1991a, 1991b)have been suggested as aggregation indices (Anderson et al., 1998;Arshad et al., 1996; Young and Crawford, 2004). Differences in soil
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aggregation measurements before and after application of mechanicalstress have been related to cropping systems, tillage, freezing/thawing,slaking, or dispersion (Diaz-Zorita et al., 2002) and have been used tocompute the probability of aggregate failure (Rasiah et al., 1993).However, measures of distances and dissimilarities across raw propor-tions are non-Euclidian (Mateu-Figueras et al., 2011). Difficulties tointerpret scaling factors as fractal dimensions occur due to weak orinvalid assumptions that produce large uncertainties in particle counts(Young and Crawford, 2004). Thus, MWD and Df are inherently biased.
Compositional data analysis is an evolving field of mathematics(Aitchison, 1986; Buccianti et al., 2006; Pawlowsky-Glahn andBuccianti, 2011; Pawlowsky-Glahn and Egozcue, 2006) that providestools to project the aggregate data into the Euclidean space and avoidassumptions on particle counts. Particle-size fractions based on massare compositional, i.e. data in which the components represent somepart of total mass. They convey relative information and are con-strained to a constant sum, generally 100%, a characteristic that im-parts important numerical properties. Raw compositional data
124 L.E. Parent et al. / Geoderma 179-180 (2012) 123–131
have an inherently non-normal distribution (Aitchison, 1986; Butleret al., 2005). Confidence intervals about means of raw concentrationdata may take values that are negative or exceed 100% (Diaz-Zoritaet al., 2002). The closed space represented by compositional datagenerates redundancy among data since one fraction can be comput-ed by difference. There is at least one negative correlation sincechange in one proportion must affect other proportions. The prob-lem of spurious correlations due to ratio selection and the closedspace was first raised by Karl Pearson (1897) in evolutionary biologystudies and has been also found to generate bias in medical sciences(Tanner, 1949) and geosciences (Chayes, 1960).
Aitchison (1986) proposed using additive log-ratio (alr) and cen-tered log-ratio (clr) transformations to avoid bias in the linear statisticalanalysis of compositional data. A log-ratio is a log-contrast or balancebetween two parts (ratio) or groups of parts (partitions). Log ratioscanmove freely in the real space (±∞) since both negative and positivevalues are obtained about a ratio of one. Transformations of the lowerand upper confidence limits about means of log ratios back to originalunits maintain positive definiteness. The alr transformations havebeen applied to the spatial analysis of soil texture (Odeh et al., 2003;Walvoort and de Guijter, 2001). The clr transformations have beenused in soil classification studies (McBratney et al., 1992), soil biogeo-chemistry (Daunis-I-Estadella et al., 2006; Duguet et al., 2006; Lopezet al., 2002; Parent et al., 1992), and plant nutrition (Parent et al., 2009).
Log-ratio transformations have specific geometric characteristicsthat may deform the distances between coordinates (Egozcue andPawlowsky-Glanh, 2006; Mateu-Figueras et al., 2011; Pawlowsky-Glahn and Egozcue, 2001). The simple log-ratio transformation is non-orthogonal. The alr transformation generates D-1 variables with obli-que geometry. The clr transformation generates D non-orthogonalvariables that lead to singular matrices in multivariate analysis.Moreover, there is no ad hoc theory behind alr or clr. Egozcue et al.(2003) and Egozcue and Pawlowsky-Glahn (2005) developed thetransformation of isometric log-ratio (ilr) coordinates using D-1 se-quential binary partitions with orthogonal basis. The difference be-tween a given state and a reference one may be computed as theAitchison distance across the selected ilr coordinates using Euclideangeometry (Egozcue et al., 2003).
In recent reviews on soil aggregation and related processes, it hasbeen emphasized that in order to facilitate comparisons betweenstudies, there is a need to better account for the different scales and di-mensions involved, as well as to improve the calculations and expres-sions of aggregate stability (Amézketa, 1999). While neither MWDnor Df have solved such problems up till now, the scale-invariantilrs that explore geometric dimensions of soil aggregation may pro-vide a solution to those concerns.
The objective of this paper is to present the theoretical frameworkbehind the ilr concept that provides definite advantages over MWDand Df for hypothesis testing, using a published dataset. We hypothe-sized that spurious correlations between aggregate-size fractionsoccur due to difference in measurement scale, that cropping systemsinfluence synthetic indices of soil aggregation and that due to self-similarity of aggregate-size fractions, the distance between aggregationstates upon data amalgamation remains the same in the Euclideanspace however many sieve fractions are used to compute it.
2. Theory: standard approach
2.1. Mean weight diameter (MWD)
Van Bavel (1949) suggested MWD as a measure of soil aggrega-tion. The MWD is computed across D sieve size fractions using therelation (Larney, 2008):
MWD ¼ ∑Di¼1�xiwi ð1Þ
where �xi is the assumed diameter for the ith fraction and wi is theweight fraction retained on the ith sieve-size. In soil aggregate stud-ies, the sand fraction (>53 μm) is subtracted from the total soil mass(but not always: Perfect and Kay, 1991) to provide an estimate ofaggregate mass (Kemper and Rosenau, 1986). The MWD approachassumes that the mean diameter between two consecutive sieves isa fair estimate of the average aggregate particle diameter.
Larger-size categories are given more weight, resulting in size-specific scales of measurement. Macro-aggregation is emphasizedupon scaling andMWD overestimates the degree of soil aggregation.When the MWD is computed from a few broad size fractions thereis an even greater overestimation because curvilinear relationshipsbetween mass fractions and sieve size are assumed to be linear be-tween two consecutive sieve sizes (Diaz-Zorita et al., 2002). Gardner(1956) argued that the distribution curves for soil aggregates generallyfollow a logarithmic probability function similar to that of crushed lime-stone (Hatch, 1933).
TheMWD is biased since (1) proportions have an inherently non-normal distribution, (2) one proportion is redundant but not omit-ted, (3) the size of the index largely depends on sieve number andsize since greater sieve size provides more weight to large-sizeaggregate and (4) differences in measurement scales generates spu-rious correlations.
2.2. Soil aggregates as fractal objects
A fractal is a rough object that may be broken into a number ofreduced-size copies of the original object. The fractal theory applied togeology (Turcotte, 1986, 1989) and soil sciences (Rieu and Sposito,1991a, 1991b) is based on a log-distribution model that describesthe clustering, fragmentation, and fragility of materials. If the bound-ary for contact between a fractal object and its immediate environ-ment is incompletely utilized, the 'effective' boundary of the fractalobject has a fractal dimension less than 3 (Mandelbrot, 1983). There-fore, as soil structural units build up from primary particles, the frac-tal dimension must decrease below 3 with aggregation. Fractals areassumed to be scale-invariant (Anderson et al., 1998), allowing thecomputation of scaling factors (Crawford et al., 1993b) over a givensieve-size range or fractal domain.
Mass fractals are porous objects such as soil aggregates that havea non-uniformbut self-similar or self-affine distribution of internalmass(i.e. bulk density decreases as aggregate size increases) (Anderson et al.,1998). Bulk density of soil aggregates decreases as aggregate size in-creases (Rieu and Sposito, 1991b) as follows:
ρi=ρo ¼ di=doð ÞDm−3 ð2Þ
Dm is the mass fractal dimension, ρi is the bulk density of aggre-gates in the ith size class, ρo is the bulk density of aggregates in thelargest size class, di is the diameter of particles in the ith size class,and do is the diameter of particles in the largest size class. The massfractal dimension is computed from the slope of the log-log relation-ship between ρi and di.
Fragmentation fractals are objects fragmented according to thedistribution of joints and preexisting planes of weakness. The frag-mentation fractal dimension Df was defined by Rieu and Sposito(1991b) as:
S dkð Þ ¼ ∑ki¼0N dið Þ ¼ αdk
−Df ð3Þ
In which S(dk) is the number of particles with a diameter≤dk,N(di) is the number of particles in the ith size fraction, α is a propor-tionality parameter, and Df is the scaling factor computed from thelog–log relationship between S(dk) and dk. The number of particles
125L.E. Parent et al. / Geoderma 179-180 (2012) 123–131
N(di) in the ith size fraction is related to the total mass of aggregatesM(di) of bulk density ρi and shape coefficient ci in the ith size class by(Hatch, 1933; Kozak et al., 1996; Perfect et al., 1992) :
N dið Þ ¼ M ið Þ= d3i ciρi
� �ð4Þ
The sand fraction (>53 μm) is subtracted from total soil mass toestimate the aggregate mass retained on a given sieve (Kemper andRosenau, 1986). The average particle size required to count particleswithin a size class has been defined by the diameter of a hypotheticalparticle which in some way represents a set of particles of irregularshape and non-uniform size (Green, 1923). The arithmetic mean be-tween two sieve sizes is an arbitrary measure of di. Assumption of adiscrete distribution of aggregates rather than a continuum impliesthat the value of Df is strongly dependent on the choice of di becausedi is raised to the third power (Tyler and Wheatcraft, 1992).
To avoid assumptions on particle diameter and shape, Eq. (4) canbe reformulated as a difference in aggregate counts between soilstates τ and ν:
N diiτð Þ−N divð Þ ¼ln M diτð Þ.
d3i ciτρiτð Þ� �M diτð Þ� d3i civρivð Þ
0BB@
1CCA ¼ ln
M diτð ÞM divð Þ− ln
ρiτ
ρiv
� �ð5Þ
In which N(diτ) and N(diν) are aggregate numbers for treatments τand ν, respectively. The average aggregate diameter for a given sizeclass cancels out. Anderson et al. (1998) reported that aggregatesare assumed to be cubic, canceling the shape coefficients as well.
Eq. (5) may be reduced to mass fractions only for surface fractalssuch as primary minerals where the mass distribution is compact(Anderson et al., 1998). For mass fractals such as soil aggregates, themethods to measure aggregate ρi are not precise (Logsdon, 1995)and introduce error when calculating bulk density–size relationships(Anderson et al., 1998). Total C and polysaccharide contents ofaggregates that were reported to vary with cropping systems(Martins et al., 2009) may affect the bulk densities of similar-sizedaggregates (Gülser, 2006). On the other hand, tillage disrupts soil ag-gregates and usually leads to increased decomposition of soil organicmatter compounds because of reduced physical protection (Bronikand Lal, 2005; Six et al., 2004). The bias due to the bulk densityratio in Eq. (5) represents a difference in mass fractal dimension(Eq. (2)).
The particle counts required when computing scaling factorsare also biased by the effect of mechanical stresses imposed bysome sieving methods (Diaz-Zorita et al., 2002). Aggregate stabilitymediated by roots, hyphae (Tisdall and Oades, 1982; Tisdall et al.,1978) and the activity of soil fauna (Shipitalo and Le Bayon, 2004)is highly sensitive to mechanical stress. Since larger aggregates aremore likely to break than smaller ones at the same level of appliedstress, scaling factors usually vary between 0 and 3+r, where r deter-mines the extent of scale dependency of the probability of initiator fail-ure (Perfect et al., 1993a, 1993b). However, Kozak et al. (1996)criticized the interpretation of scaling factors exceeding 3 as fractal di-mensions, which is a theoretical impossibility for geometric fractals(Rieu and Perrier, 1997), and considered the estimated scaling fac-tors in aggregation studies as curve fitting parameters for the loga-rithmic probability function.
Finally, the boundaries of fractal domains are visually selected afterexamining the data (Rieu and Sposito, 1991b; Tyler and Wheatcraft,1992), and the correlations between scaling factors and soil physicalproperties will vary with the subjective selection of the fractaldomain. This subjectivity may be controlled by establishing an adhoc theory of soil aggregation with a priori defined disjoint sub-
domains constrained by the principle of orthogonality (Parent et al.,2011).
Fragmentation fractal analysis is thus biased since (1) particlecounts are based on weak or invalid assumptions on particle diame-ter, shape and bulk density and (2) the fractal domains are chosenarbitrarily.
3. Theory: the compositional approach
3.1. Compositional space of soil aggregates
Since the sieve-size fractions are closed to the initial total mass ofsoil, the D-part composition x=[x1,x2,…,xD] of the soil mass may bedescribed by its parts according to the following simplex:
x ¼ C x1; x2;…; xDð Þ ¼ x1κ∑D
i¼1xi;
x2κ∑D
i¼1xi;…;
xDκ∑D
i¼1xi
!ð6Þ
where C is a closure operator to unit κ (Aitchison, 1986). Since thesimplex is closed to total mass rather than counts, the assessment ofsoil aggregation using compositional data analysis techniques is notbiased by assumptions on particle diameter, bulk density, shape, orprobability of failure.
3.2. Early log-ratio transformations of compositional data
Aitchison (1986) proposed additive log-ratios (alr) and centeredlog-ratios (clr) to free compositional data from their constrainedspace. The Aitchison (1986) log-ratios are computed as: alri(x)=ln(xi/xp), and clrj(x)=ln(xj/g(x)), where xi and xj are fractions of thesimplex with i [1, D] \ p and j [1, D]. xp is the p-fraction in thecomposition used as a reference and g(x) is the geometric meanacross fractions 1 through D. The alr may compute the log-ratio ofbulk densities using the largest fraction as denominator and is thusanalog to Eq. (2). In clr, each component is divided by the geometricmean rather than a specific fraction.
3.3. Isometric log ratio transformation
To provide the geometry required to manipulate data in Euclideanspace, Egozcue et al. (2003) and Egozcue and Pawlowsky-Glahn(2005) arranged the D parts of a composition into a sequential binarypartition (SBP) with an orthonormal basis. In soil aggregation studies,the SBP must reflect the main biological, chemical, and physical pro-cesses mediating aggregate formation (Parent et al., 2011). The D-1isometric log-ratio (ilr) coordinates are data transformations mappedaccording to the SBP and computed as contrasts between two groupsof aggregate-size fractions (tagged with + and − signs) as follows(Egozcue and Pawlowsky-Glahn, 2006):
The parameter ilrk is the kth ilr coordinate with k [1, D-1], r is thenumber of parts (sieves) in the + group, s is the number of parts inthe − group, g(x+) is the geometric mean of aggregate-size fractionsin group x+ and g(x−) is the geometric mean of aggregate-size frac-tions in group x−.
The ilr transformation is a special case of log transformation thatpreserves the original information as a result of orthogonality. Parentet al. (2011) derived eight ilr coordinates from nine aggregate-sizefractions in Rieu and Sposito (1991b) compared to two scaling fac-tors, hence capturing many more facets of soil architecture. Hypoth-eses can be tested on the effect of soil and crop management systemson each geometric dimension of soil aggregation computed as ilrcoordinate.
126 L.E. Parent et al. / Geoderma 179-180 (2012) 123–131
The ilr coordinates provide the Euclidean geometry required tocompute the Aitchison distance A(x,y) between two equal-lengthcompositions x and y (Egozcue and Pawlowsky-Glahn, 2006):
d2a x; yð Þ ¼ ∑D−1i¼1 ilrk−ilr�k
� 2and A x; yð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2a x; yð Þ
qð8Þ
in which ilrk* is the reference composition in the kth sieve-size class.A(x,y) can be used as a measure of structure aggradation or degrada-tion for a composition x with coordinates ilrk, relative to a referencecomposition y with coordinates ilrk*. The sign of the (ilri–ilri*) differ-ence indicates in what direction soil aggregation changes in responseto a treatment or over time. A computational example is provided inAppendix 1.
4. Case study
4.1. Dataset
Gülser (2006)measured the effect of six forage cropping treatmentsand a bare fallow control on the distribution of dry aggregate-sizefractions, bulk density of aggregates, and MWD in vertic hapludolls(0–0.15 m) after 3 years of cropping. The soil contained 555 g claykg−1, 249 g silt kg−1, and 192 g sand kg−1.
The treatments were as follows: perennial ryegrass (Lolium perenneL.) (RG), alfalfa (Medicago sativa L.) (AL), bromegrass (Bromus inermisLeyss) (BG), small burnet (Sanquisorba minor Scop.) (SB), subterraneanclover (Trifolium subterraneum) (SC), purple crownvetch (Coronillavaria L.) (CV), and a fallow control (FC). Perennial ryegrass has ashallow fibrous root system while that of bromegrass is deep andabundant; alfalfa has a large dominant taproot digging deeply intothe soil while that of small burnet is shallow to deep; subterraneanclover has a taproot with many supporting roots; purple crownvetchhas a shallow multi-branch root system. It was hypothesized thatcropping systems influence soil aggregation as a result of differentrooting habits. Mass fraction and bulk density data used to computeaggregation indices are presented in Tables 1 and 2. The FC
Table 1Raw data for mass fractions in Gülser (2006) study.
exhibiting the lowest structural indices was selected as the referencetreatment used in Eq. (8).
4.2. Statistical analyses
Statistical analyses were conducted using the programminglanguage R. The R ‘compositions’ package was used to compute com-positional data transformations (Van Den Boogart and Tolosana-Delgado, 2008).
5. Results
5.1. Spurious correlations caused by scale dependency
Spurious correlations were obtained by changing the measure-ment scale of the original data. For example, one might considerinvestigating correlations between each aggregate size fraction andbulk density of the arable layer for the 7 crop treatments presentedin Table 1. Soil bulk densities in the top layer (0–0.15 m) varied asfollows: 1.37 g cm−3 for RG, 1.31 g cm−3 for AL, 1.28 g cm−3 forBG, 1.29 g cm−3 for SB, 1.28 g cm−3 for SC, 1.29 g cm−3 for CV, and1.45 g cm−3 for FC (Gülser, 2006).
We obtained the following correlation vector between soil bulkdensity and the 6 aggregate-size fractions: [0.850*, 0.697, 0.674, 0.129,−0.870*, −0.720], where * indicates significance at the 0.05 level.One might also consider computing the correlations across the macro-aggregates after discarding the micro-aggregate fraction (0–0.25 mm).The five fractions were rescaled to 100%. For example, the first compo-sitional vector in Table 1 is [9.06, 4.79, 7.43, 26.01, 23.18, 29.53]. Aftereliminating 9.06 (the micro-aggregate fraction) and rescaling, weobtained: [omitted, 4.79, 7.43, 26.01, 23.18, 29.53] / (100−9.06)=[omitted, 5.27, 8.17, 28.60, 25.49, 32.47]. The corresponding correlationvector was [omitted, 0.750, 0.780*, 0.489, −0.864*,−0.643].
The differences between correlation coefficients reflect the bias ofomitting the micro-aggregate fraction or macro-aggregate composi-tion after closure. Correlations become scale-dependent (total massvs. mass of macro-aggregates as the scale of measurement) and spu-rious (coefficients differed for the same sieve-size fraction). This
Table 3Correlations between isometric log-ratios (ilrk), Aitchison distance from bare fallow (A(x,y), fragmentation fractal dimension (Df), mean weight diameter (MWD), bulk density(BD), and penetrometer resistance (PNTM) using data from Gülser (2006).
*, **: significant at the 0.05 and 0.01 levels, respectively (n=7 treatments).
Table 4ANOVA results of crop treatment influence on aggregation indices using data fromGülser (2006).
Cropping system A(x,y) ‡ Df MWD (mm)
Ryegrass 0.470 b† 2.509 b† 0.991 b†Alfalfa 0.513 b 2.517 b 1.014 bBromegrass 1.294 a 2.124 c 1.162 aSmall burnet 0.604 b 2.443 b 1.014 bSubterranean clover 1.167 a 2.218 c 1.099 aPurple crownvetch 0.767 b 2.397 b 1.018 bFallow 0 c 2.734 a 0.902 cLsd (0.01) 0.316 0.145 0.066
‡ A(x,y)fallow: Aitchison distance from bare fallow; Df=fragmentation fractaldimension; MWD=mean weight diameter.† Means with the same letter on the same column are not significantly different at the0.01 level.
Table 2Raw data for aggregate bulk density in Gülser (2006) study.
127L.E. Parent et al. / Geoderma 179-180 (2012) 123–131
emphasizes the fact that compositional data must be analyzed usingratios rather than raw proportions to maintain scale invariance andsub-compositional coherence in the compositional vector.
5.2. Indices of soil aggregation
The synthetic indices of soil aggregation MWD, Df, and A(x,y)were significantly correlated with soil bulk density and penetrome-ter resistance (Table 3) and significantly influenced by cropping sys-tem selection (Table 4).
The log–log plot presented in Fig. 1 indicates average scaling fac-tors of 2.12 for BG and 2.73 for FC, the most contrasting treatments(Gülser, 2006), but there was evidence of several fractal patterns ata finer scale. There were two main fractal domains with specific proba-bilities of failure above and below the aggregate size of 0.50 mm. Rieuand Sposito (1991b) also noted an ‘apparent break’ in the number-size log–log distribution curve of nine aggregate fractions of Sharpsburgsoil and computed fragmentation fractal dimensions intersecting at0.111 mm aggregate diameter, a subjective selection driven by thedata. Since two fractal domains sharing a common sieve size werenot orthogonal, their scaling factors were non-Euclidean. Scaling fac-tors, computed from the slope of a log–log relationship, are very sen-sitive to the selection of the fractal domain (Parent et al., 2011).
Based on an ad hoc theory, two disjoint domains can be embeddedinto the main one using SBP (Parent et al., 2011).
The SBP presented in Fig. 2 provides the orthonormal basis for twofractal domains in Fig. 1 as well as their embedded sub-domains.Positive signs were assigned to larger aggregates to provide a positivedirection for the process of soil structure building. Should the ilr coor-dinates embedded into the SBP reflect the Df of embedded fractal
Log [ Sieve (mm) ]
Log
[ Nb
Agg
rega
tes
/ g ]
2.0
2.5
3.0
3.5
4.0
y = −2.12x + 2.37
R² = 0.981
0.5
mm
−0.8 −0.6 −0.4 −0.2 0.0 0.2
Log [ Sieve (mm) ]
Log
[ Nb
Agg
rega
tes
/g ]
2.0
2.5
3.0
3.5
4.0
4.5y = −2.73x − 2.22R² = 0.986
0.5
mm
−0.8 −0.6 −0.4 −0.2 0.0 0.2
a
b
Fig. 1. Fractal domains covered by embedded ilr balance coordinates for two contrast-ing treatments in Gülser (2006): (a) Bromegrass and (b) Fallow.
128 L.E. Parent et al. / Geoderma 179-180 (2012) 123–131
dimensions, then the ilr concept would capture the fine-scale multi-fractal patterns generated by each cropping system.
With this SBP model, ilr1 represents a general contrast or balancebetween aggregate particlesb0.50 and≥0.50 mm in size. The valueof ilr1 was highly correlated to Df since they were both computedacross the same range of aggregate sizes (Table 3). The parameter
Fig. 2. Sequential binary partition of bala
ilr2 covered the sub-domain of lower-size fractions and was signifi-cantly correlated to ilr1 (Table 3). The ilr3 was embedded into ilr2and the ilr5 embedded into ilr4 was well correlated to it (Table 3).Only ilr1 was significantly correlated to soil bulk density and pene-trometer resistance (Table 3). The greatest contribution to the in-creasing Aitchison distance from bare fallow was provided by ilr1and ilr2 (Table 3). The large gains in ilr1 indicated a general increasein soil aggregation.
The Aitchison distance from bare fallow differed significantly be-tween cropping systems (Table 4). Most ilr values also differed signif-icantly among cropping systems. This indicated that crops modifieddifferently the geometric dimensions of soil architecture. Comparedto the fallow, there were significant gains (Pb0.01) in ilr1 and ilr2for all cropping systems except AL, in ilr4 and ilr5 for AL and BG, inilr3 and ilr5 for SC, and in ilr3 for SB. The magnitude of these perturba-tions may impact on soil bulk density and penetrometer resistance(ilr1) and other soil properties as well.
5.3. Aitchison distance computed from original or amalgamated sieve-sizefractions: sensitivity analysis
To facilitate comparisons between studies (Amézketa, 1999), itwould be advantageous to derive a scale-invariant synthetic indexthat is independent of the number of sieve sizes used to computeit. Where standardmethods are used to determine aggregate-size dis-tribution, effect sizes can be compared directly for each specified ilr(Eq. (7)) and the Aitchison distance (Eq. (8)). However, the numberand range of sieve sizes vary widely among studies. When aggregatesizes are measured across different ranges, the synthetic index shouldbe computed across the same fractal domain. Using a smaller numberof sieves within the same range of sieve sizes, some fractions areamalgamated.
If the parts of a D-part composition are separated into C mutuallyexclusive and exhaustive subsets of reduced dimensionality (C≤D)and the components within each subset are added together, theresulting C-part composition is termed an amalgamation (Aitchison,1986). A smaller number of sieve-size fractions reduce the numberof ilr coordinates used to compute the Aitchison distance. Since thefractal distribution of particles is cumulative (Eq. (3)), the scaling fac-tor must depend on the C-part composition. While some informationon parts is lost upon amalgamation, the question is to what extentthis information loss distorts synthetic aggregation indices. On the
nces among aggregate-size fractions.
129L.E. Parent et al. / Geoderma 179-180 (2012) 123–131
other hand, if only total mass of sieve-size fractions is reported in stud-ies, aggregates and solid particles are amalgamated. To what extent theAitchison distance is distorted upon this type of amalgamation isbeyond the scope of this paper.
In the main fractal domain (Gülser, 2006), the set of 6 sieve-sizefractions can be reduced by amalgamation to obtain sets of 3 to 5sieve-size fractions. The Aitchison distances computed from the 3 to5 sieve-size sets were then compared to the A(x,y) computed fromthe 6-sieve set using regression analysis. Although the R2 valueswere close to one (>0.98) and highly significant (P≤0.01), the slopeswere 0.71–0.86 for the 3-sieve sets, 0.91–0.92 for 4-sieve sets, and0.95–0.98 for 5-sieve sets, hence systematically below the A(x,y)value computed from the 6-sieve set. At least 4 sieves were necessaryto obtain A(x,y) values within the experimental error of 0.316 forthe 6-sieve set (Table 4).
In the present case (Gülser, 2006) there were one main fractaldomain across 6 aggregate sizes and two disjoint fractal domainseach containing 3 aggregate sizes. To compare two fractal domainsembedded into the main one, at least 8 sieves would thus be neces-sary. In aggregation studies, 3 to 7 sieves, most frequently non uni-form in size, are currently used. Studies could thus be comparedusing A(x,y) computed over one fractal domain as identified by pre-liminary fragmentation fractal analysis.
On the other hand, when relating scaling factors across the 5-sievesets and that of the 6-sieve set, the R2 values always exceeded 0.99and were highly significant (P≤0.01). However, the slope varied be-tween 0.91 for the amalgamation of the two largest fractions and1.08 for the amalgamation of the two smallest fractions. Comparedto A(x,y), the scaling factors thus appeared to be more sensitive tothe amalgamation of sieve-size fractions.
6. Discussion
The MWD used to synthesize aggregate-size fractions is an empir-ical index of soil aggregation that is biased by spurious correlations,redundancy, and subjective weighing. We recommend discontinuingits use as an index of soil aggregation.
Despite large differences in failure probability among aggregatesizes (Perfect et al., 1993a), fractal analysis offers a basic understand-ing of aggregate fragmentation. The scaling factor was related to soilprocesses and functions (Anderson et al., 1998; Crawford et al.,1993b; Guber et al., 2005; Young and Crawford, 2004). While a fractalstructure undergoing complete fragmentation produces a power-lawnumber-size distribution function, such a function does not neces-sarily imply fractal structure (Crawford et al., 1993b; Kozak et al.,1996). Crawford et al. (1993b) considered that embedded fractaland non-fractal rather than multi-fractal dimensions can cohabit inthe soil matrix. Anderson and McBratney (1995) explained that acollection of various fragments may bear no resemblance to the orig-inal arrangement of the soil matrix.
Despite this large variety of fractal and non-fractal patterns, asingle scaling factor is generally computed to interpret changingfractal patterns (Bartolli et al., 2005; Perfect and Kay, 1995). Proce-dures to compute the scaling factor is sensitive to bias on aggregatecounts, to the choice of the fractal domain and to the number ofsieve-size fractions used for curve fitting.We recommend using frag-mentation fractal analysis as preliminary test to determine the frac-tal domains of disaggregation under mechanical stress. Thereafter,the scale-invariant isometric log-ratio coordinates can explore themain fractal domain and the embedded subdomains.
The ilr concept is convenient to handle aggregate-size fractionssince such data carry relative information (i.e. they are proportionsof a whole). They are constrained to a closed space and compositional.Balances between aggregate sizes are portrayed by ilr coordinateselaborated according to an ad hoc theory. Since the closed space is de-fined as total aggregate mass, the bias on aggregate counts is avoided.
Orthogonally arranged fractal domains also avoid the bias due to arbi-trary choices of the fractal domain and data-sharing domains.
Anderson et al. (1998) reported that although the number of clas-ses in aggregation studies is generally less than 10, a set of 5 size frac-tions is thought to be the absolute minimum of size classes thatshould be used for drawing a cumulative particle distribution func-tion (Jelinek, 1970). The Aitchison distance appeared to vary withinexperimental error when computed from sets of 4 or 5 size fractions.Otherwise, ilr coordinates can be compared one-to-one between stud-ies that use the same methodology. This suggests that the Aitchisondistance could facilitate comparisons among studies using standardmethods however many sieve fractions are used to compute it.
Meta-analysis (e.g. Gurevitch et al., 2001) could synthesize theeffect size of soil and crop management systems on the ilr coordinatesand A(x,y) in studies from the peer-review and gray literature pursu-ing the same objective and using the same methodology (wet or drysieving, mechanical stress applied, …). Since each ilr is a balance be-tween building blocks of soil architecture (Fig. 2), they could be relat-ed to soil attributes and functions such as (1) characteristics of rootsystems driving biological, physical, and chemical processes duringsoil aggradation, (2) specific biological niches in the soil, (3) soilhydrological properties, (4) SOC sequestration, and (5) selected soilproperties for use in pedotransfer functions. Critical A(x,y) and ilrindices could be developed to manage sustainable soil conservationsystems. The computations shown in Appendix 1 can be easily per-formed using spreadsheets or statistical packages.
7. Conclusion
Soil aggregates have been described by their mean weight diame-ter and by fractal dimension, requiring assumptions for aggregate di-ameter, bulk density, shape coefficient, and probability of failure,that generate bias. In contrast, the ilr approach does not require suchassumptions and can be computed from mass only. The A(x,y) metricorders the effect of management systems on soil aggregation and canbe segmented into differential ilr coordinates that indicate the direc-tion of change. The ilrs are interpreted in terms of balances betweengroups of aggregate-size fractions that may impact soil physical,chemical and biological processes and functions. The unbiased A(x,y)and ilr indices are easy to obtain using standard sieving methods.They could be related to soil attributes and functions and facilitate in-tegrating studies to develop critical values for the management of soiland crop systems.
Acknowledgements
We thank the Natural Sciences and Engineering Research Councilof Canada, the Government of Canada International ScholarshipProgram, and contributing partners (Cultures Dolbec Inc., GroupeGosselin FG Inc., Agriparmentier Inc., and Ferme Daniel Bolduc et FilsInc.) for financial support. This research was also conducted under pro-jects MTM2009-13272 and Ingenio Mathematica (i-MATH) (CSD2006-00032) (Consolider - Ingenio 2010) supported by the Spanish Ministryof Education and Science, and project 2009SGR424 supported by theAgència de Gestió d'Ajuts Universitaris i de Recerca of the Generalitatde Catalunya.
Appendix 1
Using the first replicate of the ryegrass treatment (Table 1) asan example, the ilr1 to ilr5 coordinates are computed according toEq. (7):
The corresponding ilr values for the first replicate of the fallowcrop are: ilr1=1.018, ilr2=−0.573, ilr3=0.175, ilr4=−0.033, andilr5=−0.184. The Aitchison distance (Eq. (8)) using fallow as refer-ence composition is computed using:
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