DIVISION S-6—SOIL & WATER MANAGEMENT& CONSERVATION

Fractal Description of Soil Fragmentation for Various Tillage Methodsand Crop Sequences

Bahman Eghball,* Lloyd N. Mielke, Guillermo A. Calvo, and W. W. Wilhelm

ABSTRACTSoil structure has been difficult to quantify and, at best, has been

studied semiquantitatively. Fractal representation of soil fragmenta-tion can provide an indication of soil structure. The purpose of ourstudy was to use fractal analysis to quantify soil fragmentation undervarious tillage and crop sequence treatments at different times duringthe growing season. We collected soil samples from four tillage treat-ments (established 10 yr earlier) of chisel, disk, no-till, and moldboardplow in factorial arrangement with two crop sequences of corn (Zeamay L.)-soybean [Glycine max (L.) Merr.]-corn (C-S-C), and soy-bean-corn-soybean, (S-C-S) on a Sharpsburg (fine, montmorilloni-tic, mesic Typic Argiudoll) soil. Aggregate-size distribution was usedto calculate fractal dimension (D) for each treatment. Higher D valuesindicate greater soil fragmentation and a soil dominated by smalleraggregates. The opposite is true for lower D values. Differences in soilfragmentation observed for tillage treatments after autumn tillage be-came even greater over winter. Soil fragmentation increased over au-tumn and winter, with D increasing in the order of plow > chisel >disk > no-till. Formation of larger soil aggregates increased duringthe growing season for all tillage systems. The D values for C-S-Cwere smaller than S-C-S in the no-till, indicating that the previousyear's corn in C-S-C provided more large aggregates. Soybean ap-pears to have negative effects on large-aggregate formation in no-till.Aggregate densities, averaged across tillage and crop sequence, in-creased from 1.25 to 1.77 Mg m~3 as the aggregate diameter decreasedfrom 6.38 to 0.162 mm. Fractal analysis was found to be useful indetermining soil fragmentation differences due to different tillagemethods and crop sequences.

'T'HE DISTRIBUTION OF AGGREGATES of different sizes,A abbreviated to aggregate-size distribution, is a

consequence of soil structure and is a potentially use-ful way of expressing structure quantitatively. Fractalanalysis, which is based on self-similarity (the mannerin which variations on one scale are repeated at an-other), has been applied to soil since soil is both afragmented material and a porous medium (Mandel-brot, 1983; Tyler and Wheatcraft, 1989; Rieu andSposito, 1991a). Fractal representation of soil is basedon pore-space and particle-size distribution, which areuseful ways of quantifying soil structure. In this rep-resentation, the fractal dimension is an indicator ofsoil fragmentation and subsequently of soil structure.

Rieu and Sposito (1991a) mathematically developedtwo fractal dimensions for the soil system based onthe degree of soil fragmentation. These are Dt, the

USDA-ARS, Dep. of Agronomy, Univ. of Nebraska, Lincoln,NE 68583. Contribution of the USDA-ARS in cooperation withthe Nebraska Exp. Stn., Lincoln, NE, as Paper no. 9909. Re-ceived 3 Dec. 1992. 'Corresponding author.

Published in Soil Sci. Soc. Am. J. 57:1337-1341 (1993).

bulk fractal dimension of an incompletely fragmentedsoil system, and D, the fractal dimension of a com-pletely fragmented soil or a soil whose aggregate-sizedistribution is being measured. A soil with clusters ofsize-scaled, similar partial volumes separated from oneanother by a network of size-scaled similar fracturesis considered to have complete fragmentation (Rieuand Sposito, 1991a). Incomplete fragmentation is thecase where there are interaggregate bridges in the frac-tures holding the aggregates together. According toRieu and Sposito (1991aJ, the fractal dimension in soilis expected to be <3 since a solid volume with noporosity would have a dimension of 3. Perfect andKay (1991), however, indicated that the D value de-termined from aggregate-size distribution is a measureof soil fragmentation and showed that it can be as highas 3.5. Generally, improving the structure of a soilwould result in the formation of larger aggregates anda decrease in the soil fractal dimension.

Soil structure is difficult to quantify and has oftenbeen presented and discussed semiquantitatively, i.e.,geometric mean diameter (Fahad et al., 1982). Thepurpose of this study was to quantify soil fragmenta-tion, as an indicator of soil structure, for various til-lage methods and crop sequences at different timesduring the growing season using fractal analysis.

MATERIALS AND METHODSAggregate-Size Distribution

A tillage-crop sequence experiment was started in 1978 atthe University of Nebraska, Rogers Memorial Farm, near Lin-coln, NE, on a Sharpsburg soil. The tillage treatments werechisel, disk, no-till, and plow in a randomized complete blockwith six replicates. Continuous corn was planted on the ex-perimental area through 1984. In 1985, each tillage plot wasdivided into four subplots (23 m long by 4.6 m wide each) towhich four crop sequences were assigned. The crop sequenceswere C-C-C, C-S-C, S-C-S, and S-S-S. This made the ex-periment a split plot in a randomized complete-block designwith tillage as the main plot and crop sequence as the subplot.Plow and chisel treatments were applied in the autumn anddisk treatment was applied in the spring. All tilled treatmentswere disked in the spring prior to planting. Corn was plantedon 2 May 1988 and 3 May 1989 and soybean was planted on14 May in both years. Plots were under rainfed conditions.Weeds were controlled by herbicides in the no-till and by her-bicides and cultivation in the tilled treatments. In 1988 and1989, soil samples were collected from the four tillage andtwo crop sequence (C-S-C and S-C-S) treatments (six repli-cates) for determination of aggregate-size distribution. Soy-Abbreviations: C-S-C, corn-soybean-corn sequence; S-C-S,soybean-corn-soybean; C-C-C, corn-corn-corn; S-S-S, soy-bean-soybean-soybean .

1337

1338 SOIL SCI. SOC. AM. J., VOL. 57, SEPTEMBER-OCTOBER 1993

bean was growing in C-S-C and corn in S-C-S in 1988 whilethese were reversed in 1989. Three soil subsamples were takenfrom each plot with a bulk hand sampler, 108-mm i.d., to adepth of 80 mm and processed for aggregate-size distributionindividually. Soil samples were collected at precultivation (Day158), flowering (both crops, Day 205), physiological maturity(both crops, Day 256) and post autumn-tillage (Day 303) in1988 and at prespring-tillage (Day 98), emergence (Day 144),and physiological maturity (Day 265) in 1989. Soil sampleswere stored in double plastic bags in a cold room at 4 °Cconstant temperature before determining aggregate-size distri-bution. The soil samples were spread on a shallow aluminumplate (230-mm diam.) and air dried in a glasshouse.

After air drying, =750 g of soil aggregates were separatedby shaking the samples on nine sieves with different sizedopenings (16, 8, 4.76, 4, 2, 1, 0.5, 0.25, and 0.074 mm) for30 s. This was found to be the best shaking time to ensurepassage of aggregates through the sieve openings and to min-imize aggregate disruption (Calvo, 1991). Aggregates on eachsieve were collected and weighed.

Soil samples were taken from three replicates of the tillageand crop sequence treatments in the spring of 1990 and wereused to determine the density (a; Mg m~3) of aggregates indifferent size classes. The samples were sieved to the samesize classes as above and the density of each aggregate-sizeclass was determined based on the bulk density method de-scribed by Chepil (1950). Aggregate density was determinedfor all size classes except aggregates of >8 and < 0.074 mmin diameter. Aggregate density was assumed to be similar forall sampling times.

Fractal AnalysisAggregate-size distribution was determined based on the

weight of soil in each size class (> 16, 8-16, 4.76-8, 4—4.76,2-4, 1-2, 0.5-1, 0.25-0.5, 0.074-0.25, and <0.074 mm)with respect to the total soil sample weight. In the fractalanalysis, all the soil samples were adjusted to a total soil sam-ple mass of 750 g. Average weight of three subsamples foreach plot was used for fractal analysis. A quantity proportionalto the number of aggregates of each size class, N(d^, wascalculated based on the following relationship given by Rieuand Sposito (1991b):

where M(df), d,, and a; are the mass, mean diameter, anddensity of aggregates in the ith size class, respectively. SizeClass 0 contains the largest aggregates. Mean aggregate di-ameters and aggregate densities of size classes for the tillagemethods and crop sequences that were used in the fractal analy-sis are given in Table 1.

The quantity (N(dk), whereN(dk) = (i = 0, .... *) [2]

N(d.) =M(di)/dJ<rl (/ = 0,1.....) [1]

and dk is the mean diameter of Size Class / = k, was used toestimate the fractal dimension, D, which is the negative slopeof regression line of log N(dt) vs. log (dk/dg) (Perfect and Kay,1991) for the tillage and crop sequence treatments. In thisanalysis, ds is the diameter of the largest aggregates, while theregression line intercept indicates the quantity of the largestaggregates.

A test of homogeneity was performed on the data to deter-mine if the D values for the repeated observations of eachtreatment were homogeneous. After homogeneity was estab-lished, analysis of covariance (Winer, 1971), with log (djd^as covariant, was performed on the data to estimate D andintercept for each treatment and also to determine differencesbetween values of D for different tillage and crop sequencesusing SAS (Miles-McDermott et al., 1988; SAS Institute, 1985).Interactions of log (dk/dg) with tillage, crop sequence, and til-lage x crop sequence were used to determine if values of Dwithin each treatment were different. Orthogonal contrasts wereused to compare D between levels of tillage and tillage x cropsequence. In the analysis of covariance, the tillage and cropsequence main effects and the tillage x crop sequence inter-action test the equality of aggregate numbers at midpoints ofthe regression lines. A probability level of <0.10 was consid-ered significant.

RESULTS AND DISCUSSIONAggregate densities for the two crop sequence systems

were not significantly different at all size classes (Table1). The density of aggregates from no-till (lowest den-sity) was smaller by 6.5, 3,9, and 3.4% than the tillagesystem with the highest density at 1- to 2-, 0.5- to 1-,and 0.25- to 0.5-mm aggregate-size classes, respec-tively. This may have occurred because organic C in theno-till tended to be greater than for other tillage treat-

Table 1. Aggregate density of size classes at different tillage and crop sequence treatments in 1990.Aggregate density

4.76-8 ramVariable (6.38)t

Tillage (Till)ChiselDiskNo-tillPlowLSD(0.10)

Crop sequence (CS)C-S-C*S-C-S

Analysis of variance, P > FReplicate (R), 2 dfTillage, 3 dfR x Tillage, 6 dfCS, 1 dfTillage x CS, 3 dfCV, %

1.261.231.231.250.08

1.241.25

0.09H0.89

0.710.324.3

4-4.76 mm(4.38)

1.291.281.261.340.10

1.301.29

0.360.50

0.720.405.6

2-4 mm(3)

1.451.441.421.500.10

1.441.47

0.940.49

0.300.273.9

1-2 mm(1.5)

——— Mg m-3

1.651.601.571.680.04

1.621.63

0.350.01

0.640.673.1

0.5-1 mm(0.75)

1.781.751.711.740.04

1.751.74

0.160.08

0.410.552.5

0.25-0.5 mm(0.375)

1.771.761.711.730.03

1.751.73

0.090.02

0.230.231.9

0.074-0.25 mm(0.162)

1.791.771.761.770.06

1.781.76

0.750.72

0.320.562.9

t Numbers in parentheses are mean aggregate diameter.t C-S-C is corn-soybean-corn, and S-C-S is soybean-corn-soybean in 1987, 1988, and 1989, respectively.II A probability level sO.10 was considered significant.

EGHBALL ET AL.: FRACTAL DESCRIPTION OF SOIL FRAGMENTATION 1339

ments. Organic C content for the top 80 mm of soil was16.4 ± 0.5, 15.9 ± 1.1, 17.2 ± 1.2, and 15.7 ± 0.9g kg"1 for the chisel, disk, no-till, and plow treatments,respectively. Wittmuss and Mazurak (1958) found theorganic matter content of intermediate-sized aggregateswas greater than either smaller or larger aggregates in aSharpsburg soil. Aggregate densities, averaged acrosstillage and crop sequence, significantly (P < 0.01) in-creased from 1.25 to 1.77 Mg m~3 as the aggregatediameter decreased from 6.38 to 0.162 mm, respec-tively.

Covariance analysis is usually performed on a data setto adjust the treatment means for the covariant effect. Inour analyses, however, the procedure was performed tocompare values of D for different levels of each treat-ment (Table 2). In these analyses the main effects oftillage and crop sequence and the interaction of tillagex crop sequence test the equality of aggregate numbersat midpoints of the regression lines. Our discussion,however, will focus on the interactions of the covariantwith tillage, crop sequence, and tillage x crop se-quence, which indicate the differences between levels ofeach treatment for D.

Fractal dimension, which is calculated from aggre-gate-size distribution and is an indicator of soil frag-mentation and subsequently of soil structure, was usedto determine differences between tillage treatments andcrop sequences. Lower D values indicate a soil domi-nated by larger aggregates. For example, at prespringtillage (Day 98) in 1989, no-till (D = 2.767) had 2.3times more 6.38-mm aggregates than plow (D = 3.306),while plow had three times more 0.162-mm aggregatesthan no-till. These values were calculated from theregression values in Table 3. Higher D values indicatea soil dominated by smaller aggregates. Soil fragmen-tation was found to be fractal in the range of the aggre-gate sizes used.

Fractal dimension was significantly different amongthe tillage treatments at the post-autumn-tillage sampling

time as indicated by the significant log d x tillage in-teraction (Table 2). The fractal dimensions for plow anddisk were smaller than for chisel at post-autumn tillage,indicating that the inversion of soil in these tillage sys-tems created a medium dominated by large aggregates(Table 3). By early spring, however, the plowed soilwas dominated by smaller aggregates, indicating a de-terioration of aggregation over winter since the D forplow at prespring-tillage in 1989 was 25% greater thanat post-autumn tillage in 1988. The D values for plowat all sampling times, except post-autumn tillage, weregreater than no-till, indicating more fragmentation in theplow treatment. The low values of D for plow at post-autumn tillage in 1988 was probably caused by movingthe lower layer of soil, with larger granules, to the sur-face during the plowing operation.

Analysis of variance performed on the D values overtime indicated a significant sampling time x tillage in-teraction (P < 0.01). The D values determined in springof 1989 (prespring tillage and emergence) were greaterthan post-autumn tillage for all tillage systems (Table 3),indicating soil fragmentation over winter in the order ofplow > chisel > disk > no-till. By physiological ma-turity, however, the D values were significantly smallerthan early spring in both years, indicating formation oflarger aggregates during the growing season for all til-lage systems. It seems that increased biological activityand subsequent production of metabolic byproducts, andalso living plant roots, encouraged formation of largeraggregates during the growing season. The D values forthe tillage treatments at physiological maturity in 1989followed the same trend as in 1988, but the values weresmaller. The soil water content was greater in 1989 com-pared with 1988 (data not shown), which would increasethe activity of microorganisms (Doran, 1987) and resultin greater degradation of plant residues with a concom-itant release of substrates, which may enhance the soilaggregation processes.

There were significant plow vs. no-till x crop se-

Table 2. Analysis of covariance for determination of fractal dimensions of different tillage and crop sequence treatments atdifferent sampling times in 1988 and 1989.

1988

Variable

Replicate (Rep)Tillage (Till)Rep x Till

Precultivationdf (158)t

5 0.65$3 0.08(0.59)

Flowering(205)

0.600.59(0.53)

Physiologicalmaturity

(256)

0.390.01(0.62)

Post autumntillage (303)

———— P ———0.76

0.02(0.63)

Prespringtillage (98)

0.960.34(0.01)

1989

Emergence(144)

0.200.81(0.18)

Physiologicalmaturity

(256)

0.010.01(0.01)

15Crop sequence (CS) 1Till x CS

0.01(0.17)3 0.99

Log diameter (Log </)HLog d x Till

Disk vs. no-tillPlow vs. no-tillChisel vs. plow

Log d x CS

0.013 0.88

Log d x Till x CSDisk vs. no-tillPlow vs. no-tillChisel vs. plow

R2

by CSbyCSby CS

0.460.730.760.160.320.690.070.330.981

0.03(0.44)0.700.010.230.810.100.400.440.280.360.050.420.979

0.02(0.26)0.560.010.190.530.050.720.670.370.180.090.560.979

0.12(0.76)0.220.010.060.540.940.020.110.840.500.490.900.979

0.91(0.87)0.930.010.010.310.010.620.770.910.590.840.660.972

0.86(0.54)0.750.010.360.560.240.430.370.480.480.230.170.975

0.14(0.09)0.780.010.050.440.080.580.340.170.280.240.260.978

t Numbers in parentheses are days of the year.t A probability level <0.10 is considered significant; for Till and CS, probabilities were determined from Type I sum of squares testing equality of

the midpoints of regression lines, and probabilities in parentheses are determined from Type III sum of squares testing the equality of theintercepts.

t Covariant, log d = log (<V<y where dk is the aggregate diameter of Class k and ds is diameter of the largest aggregates.

1340 SOIL SCI. SOC. AM. J., VOL. 57, SEPTEMBER-OCTOBER 1993

Table 3. Fractal dimension (D = -slope) and intercept (Int.) of regression line of log of aggregate number vs. log of aggregatediameter/diameter of the largest aggregates for the tillage and crop sequence treatments at different sampling times in 1988and 1989.

1988Precultivation

(158)tVariable DTillage

Chisel 2.793Disk 2.799No-till 2.752Plow 2.774SEE| 0.045

Crop sequencellC-S-C 2.748S-C-S 2.811SEE 0.032

Int.

2.3102.2962.2462.3420.040

2.2712.3260.028

Flowering(205)

D

2.7652.6982.7132.2810.046

2.7312.7670.033

Int

2.3272.3982.3332.3020.041

2.3242.3560.029

Physiologicalmaturity (256)

D

2.8082.7432.7012.8320.047

2.7612.7810.034

Int.

2.2832.3002.2302.2940.042

2.2532.3000.030

Post autumntillageD

2.7982.6902.6512.6460.045

2.6602.7330.032

(303)Int.

2.3482.3972.3812.3140.040

2.3662.3540.028

Prespringtillage (98)D

3.2642.8542.7673.3060.060

3.0563.0390.042

Int.

1.9192.2422.2641.8930.053

2.0752.0840.037

1989Emergence

(144)D

2.8892.8272.8712.9580.053

2.9122.8640.038

Int.

2.2632.3002.2582.2060.047

2.2422.2710.033

Physiologicalmaturity (265)

D

2.7562.6262.6772.7920.047

2.6902.7350.033

Int.

2.0482.2502.1742.0710.041

2.1702.1010.029

t Numbers in parentheses are days of the year.| SEE is standard error of estimates.II C-S-C is corn-soybean-corn, and S-C-S is soybean-corn-soybean in 1987, 1988, and 1989, respectively.

quence contrasts for D at precultivation, flowering, andphysiological maturity in 1988 (Tables 2 and 4). Thefractal dimensions in no-till were lower for C-S-C thanS-C-S, indicating a better soil structure for the C-S-Ccrop sequence than S-C-S at these sampling times (Ta-ble 4). These differences may reflect the effect of theprevious year's crop on soil aggregate formation in thecrop sequences. For example, the previous crop was cornin C-S-C and, subsequently, the lower D value for thistreatment in the no-till may indicate the positive effectof corn on soil structure compared with soybean. Thegreater D value for S-C-S, however, may indicate anegative effect of the previous year's soybean crop onsoil structure. Fahad et al. (1982) concluded that the lowgeometric mean diameter of soil aggregates in continu-ous soybean was an indication of the negative effects ofsoybean roots in building a stable soil structure. In theplow system, however, the D values were not differentfor the two crop sequence systems, indicating that theprevious year's crop had little effect on soil structure inthis tillage treatment. It appears that the greater mixingof residue with soil and the more rapid residue decom-position with plow reduced the effects of the previouscrop on soil structure.

The intercepts of the regression lines represent theabundance of the largest aggregates [inverse log at log(dk/dg) = 0] in each treatment and are given in Table 3.Eghball et al. (1993) showed that the intercept of theregression line, which was used to determine fractal di-mension, indicated the abundance of the material beingmeasured and could be used to determine quantitativedifferences between treatments. In this study, the inter-cepts were significantly different among tillage treat-

Table 4. Fractal dimension (D) for no-till and plow under twocrop sequences at various times in 1988.

PhysiologicalPrecultivation Flowering maturity

Crop sequencef no-till plow no-till plow no-till plowC-S-CS-C^S

2.672 2.808 2.630 2.865 2.622 2.8652.831 2.739 2.797 2.777 2.780 2.799

t C-S-C is corn-soybean-corn, and S-C-S is soybean-corn-soybeanin 1987, 1988, and 1989, respectively.

ments at prespring tillage and at physiological maturityin 1989 (Table 2). Tillage treatments with lower fractaldimensions had higher intercepts, indicating a greaterabundance of the largest aggregates. The intercepts werenot different among the crop sequences at all samplingtimes except at physiological maturity in 1989, whereC-S-C had a larger intercept than S-C-S.

Tyler and Wheatcraft (1992) suggested that the use ofa mass-based equation to estimate fractal dimension forparticle-size distribution (assuming a constant particledensity) would constrain D between 0 and 3, unlike thenumber-based equations. They argued that D values > 3observed in other studies (Perfect and Kay, 1991; Rasiahet al., 1992; Tyler and Wheatcraft, 1989) may be arti-facts of grain density and size assumptions. In these stud-ies, the aggregate densities were assumed to be the samefor all size classes and, as indicated in Table 1, that maynot be a valid assumption. In our study, however, wherewe actually determined the density of aggregate sizeclasses, we still observed D values >3 for two tillagetreatments and the crop sequences at one sampling time(prespring tillage). Comparison of the intercepts indi-cated that these treatments indeed had a lower numberof larger aggregates and the results seem to be a validindicator of the soil status. The D values for the treat-ments at all other sampling times were <3.

CONCLUSIONSFractal analysis was a useful method of quantifying

soil fragmentation and subsequently of expressing thequality of soil structure, which has previously been dif-ficult to achieve. Fractal dimension was used as an in-dicator of soil fragmentation and of structure for detectingdifferences between tillage methods and crop sequences.Soil fragmentation differences observed between tillagetreatments after autumn tillage became even greater overwinter. In all tillage systems, the D value increased overwinter, indicating that the soil fragmentation increasedand soil became dominated by smaller aggregates. Theincrease in D over winter was in the order of plow >chisel > disk > no-till. Formation of larger aggregatesincreased as the growing season proceeded for all tillagesystems. This was presumably because of breakdown of

SIDLE ET AL.: EROSION PROCESSES ON ARID MINESPOIL SLOPES 1341

plant residue by microorganisms and living plant roots,which provided aggregate forming byproducts. It ap-pears that corn increased formation of larger aggregateswhile soybean plants decreased it.