fractal dimension of soil aggregates as an index of soil erodibility

Journal of Hydrology 400 (2011) 305–311

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Journal of Hydrology

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Fractal dimension of soil aggregates as an index of soil erodibility

Abbas Ahmadi a,⇑, Mohammad-Reza Neyshabouri a, Hassan Rouhipour b, Hossein Asadi c

a Department of Soil Science, Faculty of Agriculture, Tabriz University, Tabriz, Iranb Research Institute of Forests and Rangelands, Tehran, Iranc Department of Soil Science, Faculty of Agriculture, University of Guilan, Rasht, Iran

a r t i c l e i n f o s u m m a r y

Article history:Received 8 October 2009Received in revised form 10 January 2011Accepted 24 January 2011Available online 1 March 2011

This manuscript was handled by P. Baveye,Editor-in-Chief

Keywords:Aggregate size distributionFractal dimensionsInterrill erosionSplash erosionWater aggregate stability

0022-1694/$ - see front matter Crown Copyright � 2doi:10.1016/j.jhydrol.2011.01.045

⇑ Corresponding author. Tel.: +98 914 402 3260; faE-mail address: [email protected] (A. Ahmad

Aggregate stability is an influential factor governing soil erodibility. The fractal dimension of soil aggre-gates has been related to their size distributions and stabilities. Several fractal models have been pro-posed for estimating fractal dimension of soil aggregates. This study was conducted to investigate howclosely the soil interrill erodibility factor in WEPP model can be correlated to and predicted from soilaggregate size distribution or from their fractal dimensions. Samples from 36 soil series with contrastingproperties were collected from northwest of Iran. The fractal dimensions of soil aggregates were calcu-lated from Rieu and Sposito (Dn), Tyler and Wheatcraft (DmT), and Young and Crawford (DmY) modelsusing aggregate size distribution (ASD) data. A rainfall simulator with drainable tilting flume(1 � 0.5 m) at slope of 9% was employed and total interrill erosion (TIE), total splashed soil (TS) and inter-rill erodibility factor (Ki) were calculated for 20, 37, and 47 mm h�1 rainfall intensities. Results showedthat both Dn and DmT estimated from aggregate wet-sieving data characterized ASD of the examined soilsand significantly (p < 0.01) correlated to TS, TIE and Ki. Values of Dn and DmT estimated from dry-sievingdata only correlated to TS but not to TIE and Ki. Using air-dried aggregates of 4.75–8 mm size range,instead of aggregates <4.75 mm, in wet-sieving was better for estimating Dn as an index for the predica-tion of TIE, TS and Ki. Correction of ASD for the particle fraction greater than lower sieve mesh size in eachsize class decreased the correlation coefficient between TIE, TS or Ki and Dn or DmT. The values of DmY werenot correlated to TS, TIE and Ki. The correlation coefficient TIE and Ki with Dn and DmT derived from wet-sieving data, were higher than those with wet-aggregate stability (WAS), mean weight diameter (MWD)and geometric mean diameter (GMD), implying that Dn and DmT may be better alternative variables forempirically predicting soil erodibility factor and hence interrill erosion.

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1. Introduction

The description and quantification of soil structure is importantbecause the size, shape and stability of the secondary soil particles(aggregates, peds or clods) affect many agronomic and environ-mental processes (Díaz-Zorita et al., 2002). Aggregate stability, asan influential factor governing soil erodibility (Bryan, 1968), notonly decreases particles detachment and transport by raindrop im-pact and overland flow, but also reduces formation of surface seals(Martinez-Mena et al., 1998).

Several procedures have been proposed for characterizing soilaggregate size distribution (van Bavel, 1949; Kemper and Rosenau,1986; Perfect et al., 1998). Fractal parameters have increasingly re-ceived attention for quantifying and describing aggregate size dis-tribution of soils (Sepaskhah et al., 2000). Fractal parametersdemonstrate dependencies of soil properties on the scale of mea-surement, and may be represented by a power-law relationship,

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x: +98 411 334 5332.i).

either between mass and diameter and/or between the numberand diameter (Turcotte, 1992) of the aggregates. Lower values ofaggregates fractal dimension representing aggregates size distribu-tion dominated by larger fragments while higher values reflectsdistribution dominated by smaller particles (Tyler and Wheatcraft,1989).

Many researchers have used the fractal parameters such as frag-mentation fractal dimension (Dn) and mass fractal dimension (Dm)as indices to evaluate the influences of cropping practices, wettingtreatments and soil amendments on the size distribution of waterstable aggregates (Perfect and Kay, 1991; Rasiah et al., 1995;Gülser, 2006). Toledo et al. (1990) developed models for the soilwater characteristic curve and unsaturated hydraulic conductivityusing fractal geometry and thin-film physics. There are, however, afew studies about using fractal dimensions of the aggregates inerosion study.

Martinez-Mena et al. (1999) used Dn as an indicator to deter-mine soil erodibility. They estimated Dn from the fractal modelproposed by Rieu and Sposito (1991) using dry-sieving of soilaggregates data. Their study confirmed that higher Dn values were

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306 A. Ahmadi et al. / Journal of Hydrology 400 (2011) 305–311

associated with the lower aggregate stabilities. They also pointedout that fractal dimension of soil aggregates (Dn) can be a useful in-dex for characterizing and describing soil erodibility. At the presenttime, the question which aggregate stability parameter may be amore suitable erodibility index remains unanswered. Thus, thisstudy was conducted to investigate how closely the soil interrillerodibility factor in WEPP model can be correlated to and predictedfrom the soil aggregate size distribution or from their fractaldimensions.

2. Materials and methods

2.1. Soil sampling and analysis

In order to provide wide ranges of soil aggregate size distribu-tion and stability, 36 soil series with diverse properties were se-lected from the Northwest of Iran and 36 samples were takenfrom the Ap or A horizon of each soil profile. The samples wereair-dried in room temperature. A sub-sample of about 2 kg fromeach soil was sieved using a 2-mm sieve aperture. Soil texture, or-ganic matter (OM), cation exchange capacity (CEC), sodium adsorp-tion ratio (SAR), calcium carbonate equivalent (CCE) CaSO4, EC andpH were determined using the standard laboratory methods(Klute, 1986). Wet-aggregate stability (WAS) was measured byKemper and Rosenau (1986) method and aggregates density byChepil (1950) method.

Aggregates size distributions (ASD) were determined by bothdry and wet-sieving methods (Nimmo and Perkins, 2002). Indry-sieving, the air-dried soils were gently passed through an 8-mm sieve. A sub-sample (<8 mm) of 500 g was placed on a nestof sieves with opening sizes of 4.75, 2.0, 1.0, 0.5, and 0.25 mmand was shaken for two minutes on a Fritsch laboratory sieve sha-ker at a frequency of approximately 50 Hz and an oscillationamplitude of 2 mm (Perfect et al., 2002). The mass remained oneach sieve was separately collected, oven-dried and weighed(Wi), then were used to compute ASD without adjusting Wi forthe accompanied sand fraction. Each dry mass was then dispersedby adding Na-hexametaphosphate (Na-HMP) solution and handstirring. Then the material washed through the same sieve fromwhich it was collected. The remained sand in each sieve was gath-ered, oven-dried and weighed (Wsi). The difference Wi �Wsi wasregarded as water stable aggregates in the ith size class and theywere used to obtain sand corrected ASD. The two ASDs were spec-ified by using SC (sand correction) and NSC (no sand correction) inthe tables.

In wet-sieving, 30 g of air dry soil sample (<4.75 mm) wasplaced on a nest of sieves with opening sizes of 2.00, 1.00, 0.50,and 0.25 mm arranged from top to bottom respectively, and slowlysubmerged in tab water (EC = 0.5 ds m�1 and SAR = 1.2) while con-nected to a wet-sieving apparatus. The apparatus had a verticalstroke (the vertical distance that the sieve set moved up and downin water) of 38.1 mm and was operated for 10 min at a speed of30 Hz. Wet-sieving was also accomplished for the 4.75–8.0 mmsize range aggregates on the same sieve set. ASD were determinedwith and without correction for the sand fractions as was carriedout for the dry sieving. Aggregates <4.75 mm were also employedand ASD data derived from wet-sieving of the aggregates(<4.75 mm) with correction for the sand fraction was used to com-pute MWD and GMD (Kemper and Rosenau, 1986). Both ASD dataderived with and without correction for the sand fraction wereused to compute fractal dimensions of the aggregates.

The number of aggregates, N(xi), within each size fraction wascomputed using Eq. (1) (Rieu and Sposito, 1991):

NðxiÞ ¼MðxiÞx3

i qai

i ¼ 1;2;3; . . . ð1Þ

M(xi) and qai are the mass and density of aggregates in the ith sizeclass, respectively. xi is mean diameter of aggregates in the ith sizeclass, and was assumed equal to the arithmetic mean of the upperand lower sieve apertures used to separate the ith fraction. Class1 contained the largest aggregate class.

2.2. Fractal parameters estimations

Although a diagnostic of fractal behavior is power-law scaling,such behavior is not necessarily indicative of a structure exhibitingfractal properties. However on the base mathematical concept offractals, fractal dimension of aggregates could be estimated frompower-law relationship between either number-diameter ormass–diameter or bulk density–diameter of the aggregates (Baveyeand Boast, 1998; Young and Crawford, 1991). Number-basedfragmentation fractal dimension (Dn), was calculated by Rieu andSposito (1991) model:

Nð> xiÞ ¼ knðxiÞ�Dn ð2Þ

where N(>xi) is the cumulative number of aggregates greater than xi.Dn and kn are the slope and intercept of the regression line of N ver-sus xi in the log–log scale.

Mass fractal dimension according to Tyler and Wheatcraft(1992), designated as DmT, was calculated from:

Mðx < XÞMt

¼ xXL

� �3�DmT

ð3Þ

where M(x < X) is the cumulative mass of aggregates remained oneach sieve and summed up from the bottom to the top of the nestof sieves; Mt is the total mass of aggregates; XL is the diameter ofthe largest aggregates and DmT is the mass fractal dimension.

Mass fractal dimension according to Young and Crawford(1991), designated as DmY was computed from:

MðxiÞ ¼ kmxDmYi ð4Þ

where M(xi) is the mean individual aggregate mass for ith fractionwith diameter of xi, DmY is the fractal dimension of mass, and km

is the mass of an aggregate of unit diameter which were regarded,respectively, as the slope and intercept of the regression line be-tween M(xi) and xi drown in the log–log scale. When DmY < 3, thereis a decrease in bulk density with aggregate size (Anderson andMcBratney, 1995).

2.3. Soil erosion test

A rainfall simulator with a single scanning nozzle located 4 me-ters above the soil surface, and a drainable tilting flume (1-m-long,0.5-m-wide and 0.1-m-depth), which manufactured by Delta LabCompany, was employed. The flume provided sufficient runoffand soil erosion for interrill erodibility analysis, but could not con-centrate flow and produce bed shear stress sufficiently to inducerill erosion. To prepare erosion test sample, flume was laid in a hor-izontal position and a water-permeable mat with 1 cm thicknesswas placed on the flume bed. Air-dried, soil that passed througha 4.75-mm sieve, was loosely packed in the flume with 0.09-m-thick layer and then was saturated from the base by a constant-head water supply for a 24 h. Excess water was allowed to drainfrom the soil by gravity before the commencement of each exper-iment, and the drainage outlet remained open during the experi-ment. After this period, the slope of the flume was adjusted to 9%and was subjected to the rainfall for at least 90 min. Rainfall inten-sity treatments were 20, 37 and 47 mm h�1, which will be desig-nated hereafter as IA, IB and IC, respectively. Outflow runoffsamples were collected manually at different time intervals, beingless than 60 s at the beginning, up to 15 min near the end of the

A. Ahmadi et al. / Journal of Hydrology 400 (2011) 305–311 307

test. At the end of experiment, the volumes of runoff samples (V)were measured and they allowed to evaporate. The remainingmass was oven dried at 105 �C for 24 h and weighed (Md), to deter-mine the sediment load at each time interval during the erosiontest. Sediment concentration in each runoff sample was computedas Md/V. These data were used to calculate runoff and erosion rates.The total oven-dried sediment mass for each erosion test was re-garded as TIE.

The soil particles splashing out of the flume during the experi-ment were collected at a splash board attached to the front and thetwo lateral sides of the flume. The collected soil materials weightedafter drying at 105 �C for 24 h; it was regarded as TS.

The observed interrill erodibility values were calculated usingEq. (5) (Kinnell, 1993):

Ki ¼Di

IerirSfð5Þ

where Ki is interrill erodibility (kg s m�4), Di is interrill erosion rate(kg m�2 s�1), Ie is rainfall intensity (m s�1), rir is interrill runoff rate(m s�1) and Sf is slope factor (dimensionless) calculated as (Liebe-now et al., 1990):

Sf ¼ 1:05� 0:85 expð�4 sinðhÞÞ ð6Þ

where h is the slope angle (degrees).

Table 1Values, minimum, maximum, mean, standard deviation (SD) and coefficient of variation (

Soil sample pH SARa CECb EC (dS m�1) SP (%) C

1 7.88 3.2 14.8 2.3 24.01 02 7.95 4.75 17.5 1.35 36.37 03 7.72 1.99 17.8 1.84 24.48 04 7.48 1.33 16.5 4.67 29.85 05 7.83 4.82 15.3 2.18 32.07 06 7.85 3.01 14.1 1.07 26.06 07 8.3 2.43 11 0.79 25.62 08 7.99 34.72 25.6 8.18 69.08 09 7.77 5.3 43.1 0.87 67.49 010 7.95 6.68 24.9 0.95 43.97 011 7.85 16.53 24.6 4.17 57.38 012 7.9 7.77 22.4 1.71 52.41 014 7.77 7.92 21.3 2.06 47.73 015 7.79 5.49 20.7 1.72 25.85 016 7.76 22.49 21.5 8.56 56.98 017 7.82 13.87 23 4.85 45.45 0B1 7.57 2.73 49.4 0.94 54.69 0B2 7.76 4.42 22.3 1.45 40.06 0B3 7.61 3.35 18.7 3.91 40.41 0B4 7.54 3.36 24.4 4.05 35.24 0H 7.55 3.27 19.3 2.72 35.53 0J1 7.95 0.86 18 0.53 25.82 0J2 8.17 5.76 21.1 0.79 36.52 0J3 8.11 1.1 6.8 0.77 49.76 0J4 8.19 6.73 19.8 1.64 31.81 0J5 8.14 0.31 19.4 0.49 31.76 0K1 7.49 0.62 27.1 0.93 64.73 0K2 8.07 8.03 17.5 0.8 27.98 0K3 7.6 0.56 26.1 0.78 53.5 0K4 7.75 0.37 42.2 0.41 34.99 0K5 6.81 0.42 59.9 0.83 50.73 0K6 7.17 0.53 46.4 0.93 49.88 0K7 7.59 0.93 24.4 1.12 46.15 0K8 7.96 0.52 15.9 0.46 30.35 0K9 7.87 0.32 26.8 0.96 46.28 0P1 7.77 23.86 23.1 7.15 54.23 0Min 6.81 0.31 6.8 0.41 24.01 0Max 8.3 34.72 59.9 8.56 69.08 0Mean 7.79 5.84 23.96 2.19 41.81 0SD 0.29 7.61 11.07 2.15 12.94 0CV (%) 3.7 130.3 46.2 98.7 31.0 5

a Sodium adsorption ratio (mmol l�1)0.5.b Cation exchange capacity (cmolc kg�1).c Calcium carbonate equivalent (%).

Di and rir were considered, respectively, as the ratios of meansediment mass (Md) per unit area and the mean runoff volume(V) per unit area to the mean time intervals (�t) at which steady-state conditions of the runoff generation rate were realized andthe sediments and runoffs were measured.

2.4. Statistical analysis

Statistical analysis of the experimental data was accomplishedusing the SPSS software package (SPSS Inc., 2007). It included nor-mality test for all collected data using Kolmogorov–Smirnov test.Pearson correlation analysis was also carried out between TS, TIEand Ki as dependent variables and MWD, GMD, WAS and fractaldimensions of aggregates as independent variables.

3. Result and discussion

3.1. Soil properties

Table 1 summarizes range, mean and standard deviation of se-lected physical and chemical properties of the examined soils.There were considerable differences in SAR, EC, organic matter,CCE, clay, silt and sand contents among the soils used in this exper-iment. The two highest coefficients of variation related to SAR

CV) of some physical and chemical properties of the soils.

aSO4 (%) CCEc (%) OM (%) Clay (%) Silt (%) Sand (%)

.17 12.4 2.89 11.0 25.1 64.0

.21 18.4 0.69 31.9 41.5 26.7

.24 10.7 3.6 15.7 24.6 59.7

.13 12.5 1.55 21.0 32.5 46.5

.21 20.1 0.96 19.6 42.7 37.7

.41 17.1 0.69 18.8 29.5 51.7

.29 7.5 0.06 8.6 1.4 90.1

.03 21.3 0.63 50.2 40.6 9.2

.24 18.5 1.44 46.0 47.5 6.5

.14 14.9 1.5 35.2 41.6 23.2

.17 17.4 4.32 36.6 51.5 11.9

.19 25.7 2.49 25.6 44.8 29.6

.22 18.8 1.33 38.5 48.5 13.0

.13 12.3 1.04 22.4 29.8 47.820.2 1.36 38.4 48.1 13.5

.08 16.8 2.11 40.1 32.9 27.1

.08 11.1 0.96 38.0 19.7 42.3

.24 6 1.21 19.5 33.1 47.4

.33 9.9 0.61 14.0 39.1 46.9

.17 13.3 0.68 19.9 30.2 49.9

.22 7.6 3.09 11.9 14.1 74.1

.27 25.1 0.85 16.5 33.6 49.9

.2 22 0.12 22.2 39.7 38.1

.23 25.3 1.38 38.5 53.0 8.5

.61 25.5 0.81 18.9 38.5 42.6

.18 26.1 0.85 23.2 41.2 35.6

.23 23 4.17 38.1 49.3 12.7

.18 10.4 1.93 9.8 15.9 74.3

.39 19 3.98 30.0 38.8 31.2

.23 7.7 3.1 15.2 24.8 60.0

.4 6.3 4.38 14.3 31.4 54.2

.16 3.7 3.59 30.4 26.2 43.4

.18 26.3 2.33 29.8 43.2 27.0

.19 17.8 0.52 24.8 23.4 51.8

.2 14.8 4.38 26.1 33.7 40.1

.01 16.9 3.3 36.1 33.5 30.43.7 0.06 8.55 1.37 6.53

.61 26.3 4.38 50.17 52.99 90.08

.21 16.18 1.91 26.01 34.58 39.41

.12 6.48 1.35 10.95 11.40 20.177.1 40.1 70.7 42.1 33.0 51.2

Table 2Range; mean; standard deviation (SD) and coefficient of variation (CV) and mean coefficient of determination (R2) of fractal dimensions estimated from three fractal models.

Aggregate size range (mm) Sieving method Sand correction Fractal parameters Minimum Maximum Mean SD CV (%) Mean-R2

Dn 2.63 5.78 3.89 0.75 19.28 0.985<4.75 Wet SC DmT 2.07 3.00 2.68 0.23 8.58 0.906

DmY 2.83 3.06 2.97 0.05 1.68 0.998

Dn 2.79 5.28 3.73 0.59 15.82 0.987<4.75 Wet NSC DmT 2.39 2.95 2.79 0.15 5.38 0.869

DmY 3.05 3.62 3.19 0.12 3.76 0.989

Dn 1.14 4.91 2.90 0.89 30.69 0.9734.75–8 Wet SC DmT 1.20 2.93 2.38 0.46 19.33 0.966

DmY 2.78 3.05 2.95 0.06 2.03 0.999

Dn 1.17 5.06 3.08 0.81 26.30 0.9744.75–8 Wet NSC DmT 2.27 2.96 2.77 0.20 7.22 0.912

DmY 2.84 3.11 2.97 0.06 2.02 0.999

Dn 2.84 4.37 3.50 0.35 10.00 0.991<8 Dry SC DmT 2.21 2.83 2.56 0.17 6.64 0.953

DmY 2.78 3.30 2.96 0.08 2.70 0.999

Dn 2.79 4.40 3.49 0.40 11.46 0.989<8 Dry NSC DmT 1.94 2.78 2.45 0.22 8.98 0.929

DmY 2.86 3.11 2.97 0.05 1.68 0.999

SC and NSC mean correction and no correction for the sand fraction in ASD computation (see the text) n = 36.


(130.3%) and EC (98.7%). The wide range of soils properties impartsa greater generality to the findings and makes them to be appliedwith more reliability to the other soils.

3.2. Fractal parameters

The values for Dn, DmT and DmY, ranged from 1.14 to 5.78, 1.20 to3.0 and 2.78 to 3.62, respectively (Table 2). The ranges of these val-ues are comparable to those reported by Bayat (2009) for the 148soil samples from west and northwest of Iran. The coefficients ofdetermination, R2, for three models fitted to the fragment-size dis-tributions were highly significant (Table 2). Some of the fractal

Table 3Range, mean, standard deviation (SD) and coefficient of variation (CV) of theaggregates density in each size classes.

Size class Aggregates density (g cm�3) SD (g cm�3) CV (%)

Minimum Maximum Mean

4.75–8.0 mm 1.31 2.39 1.69b,A 0.26 15.482–4.75 mm 1.31 2.1 1.70b 0.21 12.141–2 mm 1.38 2.05 1.71b 0.16 9.340.5–1 mm 1.54 2.42 1.81a 0.19 10.550.25–0.5 mm 1.55 2.43 1.82a 0.20 10.99

A Based on the least significant difference (LSD) test means followed by the sameletters (a or b) are not significantly different at 5% level of probability.

Table 4Range, mean, standard deviation (SD) and coefficient of variation (CV) of total interrill ero

Parameters Rainfall intensity Minimum M

IA 2.5 2TIE (g event�1) IB 10.19 4

IC 45.91 9

IA 1.82 5TS (g event�1) IB 10.19 6

IC 11.47 8

IA 0.21 � 106 7Ki (kg s m�4) IB 0.10 � 106 3

IC 0.32 � 106 4

IA, IB and IC present rainfall intensities of 20, 37 and 47 mm h�1, respectively.A Based on the least significant difference (LSD) test means followed by the same le

probability.

dimension values particularly Dn exceeded 3, which theoreticallyis unsounded (Perfect et al., 1993). The values greater than 3 havebeen explained as an artifact due to the measurement error, modelestimation and their underlying assumptions (Anderson et al.,1998; Martinez-Mena et al., 1999). On the other hand, Perfectet al. (1992) stated that if the fragmentation probability of aggre-gates assumed scale-invariant, then the fractal dimension rangewould be 0 6 D 6 3. However the basic idea is that fragmentationprobability of aggregates is size dependent and bigger aggregatesimply greater stability (Nimmo and Perkins, 2002). In this case,the fractal dimension range would be 0 6 D 6 3 + r. Thereforedue to positive assign of ‘r’, D > 3 can be expected.

The coefficient of variations (CVs) of the fractal dimension val-ues estimated from three different models could be ranked as fol-lows: Dn > DmT > DmY (Table 2). Probably the difference in themodels structure and the underlying assumptions, contributed totheir CVs. For example in the model of Tyler and Wheatcraft, thedensity of aggregates is assumed to be scale-invariant, while inthe model of Rieu and Sposito model it was measured for each sizeclass separately.

The values of DmT generally were less than DmY and the latterwere less than Dn values. Lower values of DmT and Dn indicate thatthe aggregate size distribution is dominated by the larger frag-ments. DmY values (representing distribution of aggregate densityin different size classes) smaller than 3 implies that aggregate den-sity is not uniform within aggregates (Giménez et al., 2002). This

sion (TIE), total splash (TS) and interrill soil erodibility (Ki).

aximum Mean SD CV (%)

54.62 49.48c,A 45.80 92.5654.76 149.88b 94.62 63.1340.27 341.24a 223.14 65.39

9.30 19.67c 11.58 58.874.37 30.89b 12.25 39.661.08 36.55a 15.78 43.17

.97 � 106 1.53 � 106a 1.45 � 106 94.40

.97 � 106 1.33 � 106a 0.87 � 106 65.95

.44 � 106 1.51 � 106a 0.89 � 106 58.54

tters (a or b or c) for each parameter are not significantly different at 5% level of

Table 5Correlation between soil erosion data (TIE, TS and Ki) and fractal dimension parameters estimated by different models using air-dried aggregates in size range <4.75 mm in wet-sieving.

Sand correction Fractal parameters TIE TS Ki

IA IB IC IA IB IC IA IB IC

Dn 0.027 0.252 0.357* 0.130 0.083 0.204 0.285 0.408* 0.500**

SC DmT 0.325 0.402* 0.541** 0.511** 0.373* 0.449** 0.172 0.495** 0.686**

DmY �0.432** �0.101 �0.095 0.011 0.265 0.199 �0.336 �0.066 �0.084

Dn 0.124 0.344* 0.486** 0.260 0.219 0.292 0.223 0.488** 0.592**

NSC DmT 0.639** 0.772** 0.809** 0.508** 0.312 0.259 0.434* 0.660** 0.657**

DmY �0.220 �0.063 �0.066 �0.179 �0.061 0.048 0.285 �0.084 �0.140

IA, IB and IC present rainfall intensities of 20, 37 and 47 mm h�1, respectively.SC and NSC: see Table 2 n = 36.* Mean significant at 0.01 level.** Mean significant at 0.05 level.

Table 6Correlation between soil erosion data (TIE, TS and Ki) and fractal dimension parameters estimated by different models using air-dried aggregates in size range 4.75–8.0 mm inwet-sieving.

Sand correction Fractal parameter TIE TS Ki


Dn 0.399* 0.585** 0.698** 0.504** 0.349* 0.455** 0.35 0.708** 0.791**

SC DmT 0.401* 0.561** 0.685** 0.547** 0.377* 0.486** 0.327 0.688** 0.776**

DmY �0.205 0.055 0.066 �0.113 0.159 0.092 �0.003 0.158 0.077

Dn 0.403* 0.581** 0.716** 0.587** 0.431** 0.549** 0.263 0.684** 0.830**

NSC DmT 0.507** 0.650** 0.747** 0.534** 0.344* 0.346* 0.104 0.507** 0.636**

DmY �0.254 �0.263 �0.390* �0.384* �0.190 �0.206 0.013 �0.219 �0.523**



finding confirms the non-uniform distribution of aggregates den-sity within different size classes. Aggregates density (Table 3) sig-nificantly (p < 0.05) increased as size classes decreased. No cleartrend was found for the variation of the Dn and DmT values, withand without sand correction, but the DmY values decreased withsand correction.

3.3. Erosion test

Total interrill erosion (TIE), total splash (TS) and interrill soilerodibility (Ki) values (Table 4) were calculated from soil erosiontest data. Their values ranged from 2.5 to 940.27 g event �1, 1.82to 81.08 g event�1 and 0.10 � 106 to 7.97 � 106 kg s m�4, respec-tively, depending on soil and rainfall intensity.

According to Table 4 even though the effect of rainfall intensityon TIE and TS is highly significant (p < 0.01), but its effect on Ki

turned to be insignificant. This implies that Ki was independent

y = 63568x -51523R² = 0.689

0

1

2

3

4

1.0 2.0 3.0 4.0 5.0

Ki (

kg s

m-4

)×

106

Dn

Fig. 1. Relation between interrill erodibility (Ki) and fragmentation fractal dimen-sion parameters (Dn) estimated by Rieu and Sposito model using air-driedaggregates in size range 4.75–8.0 mm in wet-sieving.

of rainfall intensities. According to Eq. (5) it appears that Di andrir were varied almost proportional to the variation in the rainfallrate (Ie) in a way that Ki remained unaltered. Kinnell (1993) andFoster et al. (1995) have also considered Ki as an independent fac-tor from rainfall intensity at the steady-state condition of soil ero-sion. Our finding, however, is different than that was reported byAsadi et al. (2008). In their study Ki increased with rainfall rate.They interpreted this as a result of structural uncertainty in Eq. (5).

3.4. Relationship between soil erosion and fractal parameters

Results showed that TIE, TS and Ki values were significantly cor-related to Dn and DmT computed from the wet-sieving data (Tables5 and 6; Figs. 1 and 2). The Pearson correlations were greater forTIE and Ki at the highest rainfall intensity (47 mm h�1). This maybe due to the rapid approach of steady-state conditions at that

y = 2×106 x -5 ×106

R² = 0.404

0

1

2

3

4

2.0 2.5 3.0

Ki

(kg

s m

-4)

×106

DmT

Fig. 2. Relation between interrill erodibility (Ki) and mass fractal dimensionparameters (DmT) estimated by Tyler and Wheatcraft model using air-driedaggregates in size range 4.75–8.0 mm in wet-sieving.

Table 7Correlation between soil erosion data (TIE, TS and Ki) and fractal dimension parameters estimated by different models using dry-sieving data for the soil samples <8-mm.

Sand correction Fractal parameters TIE TS Ki


Dn �0.044 0.072 0.172 0.388* 0.392* 0.393* �0.333 0.080 0.228SC DmT 0.035 0.091 0.218 0.502** 0.488** 0.483** �.426* 0.128 0.285

DmY �0.196 0.071 0.091 �0.118 0.150 0.095 0.005 0.171 0.106

Dn 0.012 0.084 0.245 0.455** 0.382* 0.447** �0.301 0.120 0.352*

NSC DmT 0.092 0.170 0.246 0.605** 0.636** 0.477** �0.619** 0.119 0.232DmY �0.260 �0.263 �0.388* �0.319 �0.196 �0.210 0.005 �0.219 �0.320


Table 8Correlation between soil erosion data and aggregate stability indices of wet-aggregate stability (WAS), mean weight diameter (MWD) and geometric mean diameter (GMD).

Parameters TIE TS Ki


WAS �0.349* �0.492** �0.525** �0.401* �0.19 �0.183 �0.189 �0.267 �0.421*

MWD �0.279 �0.362* �0.425** �0.348* �0.08 �0.11 �0.203 �0.32 �0.388**

GMD �0.414** �0.597** �0.597** �0.538** �0.174 �0.19 �.394* �0.547** �0.519**

IA, IB and IC present rainfall intensities of 20, 37 and 47 mm h�1, respectively.* Mean significant at 0.01 level.** Mean significant at 0.05 level.


rainfall intensity. The formation of a protective water layer on topof the soil surface may explain the lower correlation between TSand fractal dimensions (Dn and DmT) at higher rainfall intensity.Sutherland and Ziegler (1998) found for a rainfall intensity of120 mm h�1 that splash erosion decreased with decreasing aggre-gates stability due to rapid formation of a water layer on the soilsurface.

In wet-sieving using air-dried aggregates in the size range4.75–8 mm instead of aggregates <4.75 mm was more reliable forestimating Dn as an index for predication of TIE, TS and Ki. This isbecause aggregates <4.75 mm may contain a greater fraction ofsmall particles or aggregates resulting from disintegration by soilsampling or preparation (Le Bissonnais and Le Souder, 1995). Someauthors also have proposed using calibrated aggregates (aggre-gates with a specified size range) instead of entire soil sample inthe wet-sieving analysis (Barthès and Roose, 2002).

Table 7 shows correlation of TIE, TS and Ki with the fractalparameters estimated using dry-sieving data. Unlike the resultsof Martinez-Mena et al. (1999), no significant correlation wasfound between TIE and Dn obtained from the dry-sieving data. Cor-relations of the Ki values with Dn and DmT obtained from the dry-sieving data were also insignificant (Table 7). Cantón et al. (2009)also reported that ASD based on dry-sieving was not correlatedto the runoff and water erosion in Rambla Honda soils. Unger(1997) has pointed out that ASD based on dry-sieving is mainly re-lated to wind erosion.

The TS values were correlated to Dn and DmT obtained from dry-sieving data (Table 7). Barthès and Roose (2002) reported that thebreakdown of aggregates by raindrop impact mainly occurs duringthe early stage of the rainfall event when the surface of the soil isapproximately dry. In this condition disintegration of the aggre-gates occurred as mechanical breakdown by raindrop impact. Dryaggregate stability has been suggested as an indicator of soil resis-tance to mechanical force (Daraghmeh et al., 2009) e.g., splash andwind erosion. This is corroborated by our finding that the degree ofcorrelation of TS to Dn and Dm predicted from dry-sieving was high-er than those predicted from wet-sieving data (Tables 5–7).

Tables 4, 5 and 7 compare the correlation coefficient betweenthe soil erosion data and fractal dimensions of the aggregatesobtained with and without correction for sand. Apparently theexclusion of the sand fraction for computing ASD and the relatedfractal dimensions did not improve the correlation with soil ero-sion data for the most cases, particularly for Ki. Therefore, it seemsthat this correction is not required for predicting soil erosion.

The DmY values represent variation of aggregates density in dif-ferent size classes (Giménez et al., 2002). In contrast to Dn and DmT,DmY was not correlated to Ki, TIE and TS. This indicates that theaggregate density alone may not be an appropriate criterion forinterrill erodibility prediction. Aggregates with high density andsmall size, may have identical mass comparing to those with lowdensity and greater size and, consequently, behave similarly con-sidering their susceptibility to erosion.

Table 8 depicts correlation of WAS, MWD and GMD with theerosion parameters. Comparing R values in Table 8 with those inTables 5–7 indicates that TIE, TS and Ki were better correlated tothe fractal dimension of the aggregates derived from wet-sievingdata, than to the conventional stability indices such as WAS,MWD and GMD. Martinez-Mena et al. (1998) stated that Dn wasbetter than MWD or GMD for soil erodibility study. Sepaskhahet al. (2000) also reported that for some of their treatments Dn re-spond better than MWD.

4. Conclusion

Both number- and mass-based fragmentation fractal dimension(Dn and DmT) may characterize the aggregate size distribution, andthey appear to be appropriate indices in estimating splash andinterrill soil erosion. Results showed that TIE and Ki values weresignificantly correlated to Dn and DmT for wet-sieving data. The cor-relations were greater for TIE and Ki at the highest rainfall intensity(47 mm h�1).

There were no significant correlations for TIE and Ki with Dn andDmT when obtained from dry-sieving data. However TS values were


significantly correlated to Dn and DmT obtained from dry-sievingdata.

The correlation coefficient of TIE and Ki with Dn and DmT derivedfrom wet-sieving data, were higher than those with WAS, MWDand GMD, implying that Dn and DmT may be better alternative vari-ables for empirically predicting soil erodibility factor and henceinterrill erosion.

Using Young and Crawford model for estimation of fractal dimen-sion of aggregates as an index of soil erodibility may not berecommended.

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