spatial prediction of soil particle-size fractions as compositional dataabtmartins:... ·...

0038-075X/03/16807-501-515Soil ScienceCopyright C 2003 by Lippincott Williams &-Wilkins, Inc.

SPATIAL PREDICTION OF SOIL PARTICLE-SIZE FRACTIONS ASCOMPOSITIONAL DATA

Inakwu O.A. Odehl, Alison J. Todd', and John Triantafilisl

Particle-size fractions (psf) of mineral soils and, hence, soil texture, arethe most important attributes affecting physical and chemical processesin the soil. More often, psf data are available only at a few locations fora given area and, therefore, require some form of interpolation or spatialprediction. However, psf data are compositional and, therefore, requirespecial treatment before spatial prediction. This includes ensuring posi-tive definiteness and a constant sum of interpolated values at a given lo-cation, error minimization, and lack of bias. In order to meet these re-quirements, this study applied two methods of data transformation priorto kriging of the psf of soils in two regions of eastern Australia. The twomethods are additive log-ratio transformation of the psf (ALROK) andmodified log-ratio transformation (mALROK). The performance of thetransformed values by ordinary kriging was compared with the spatialprediction of the untransformed psf data using ordinary kriging, com-positional kriging (CK) (UTOK), and cokriging, based on the criteria-prediction bias or mean error (ME) and precision (root mean square er-ror (RMSE)), and validity of textural classification. ALROK and mALROKoutperformed UTOK and CK in terms of prediction ME and RMSE. Be-cause of the closure effect on the psf data, UTOK, and, to a lesser extent,CK, did not meet all of the requirements for spatially predicting compo-sitional data and, therefore, performed poorly. mALROK outperformedall of the interpolation methods in terms of misclassification of soils intotextural classes. The results show that without considering the special re-quirements of compositional data, spatial interpolation of psf data willnecessarily produce uncertain and unreliable interpolated psf values. (SoilScience 2003;168:501-515)

Key words: Compositional soil data, particle-size fractions, log-ratiotransformation, kriging, spatial prediction.

A regionalized composition is characterized bycomponents that (i) can be modeled by a spa-

tial random ifiunctioni, (ii) are positive definite, and (iii)suimI to a conistanit (Pawlowsky, 1984). The study ofcompositional data therefore should be con-cerned with the relative values or ratios of thecomponents. It is meaningless to evaluate each

'Australian Cotton Cooperative Research Centre, Faculty of Agriculture, Food andNatural Resources, The University of Sydney, Ross St. Building A03, NSW, 2006,Australia. Dr. Odeh is conesponding author. E-mail: [email protected]

Received Sept. 12, 2002; accepted March 24, 2003.

DOI: 10.1097/01.ss.0000080335.10341.23

component in isolation. Pearson (1897) was thefirst to identify this problem of statistical analysisof compositional data. The problem has beenreferred to as a "fallacy of interpretation"(Woronow and Love, 1990) or "spurious spatialcorrelation" as a result of the "closure effect"(Pawlowsky, 1984). That a composition must haveat least one negative correlation between a pair ofits components is caused by the closure effect. Achange in one component, therefore, results in ashift of all other components. However, an infi-nite number of different combinations or changesof the composition of the components could pro-duce the same shifts without showing evidence ofwhat changes have actually occurred. This invari-ably leads to difficulties in interpreting the corre-

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July 2003Vol. 168, No.7

Printed in U.S.A.

ODEH, TODD, AND TRIANTAFILIS

lations. In geostatistical parlnegative, and in some instarcovariance matrices cause tlto be singular as well. In adarate components of a reg:may not produce estimatesa strict requirement of coithe unbiased constraint fumay not be fulfilled.

The spurious spatial Inents of compositional datlems stated above can be avquirements are met (de Gr

Z8ij(x) 2 0

jZ*(X) 0j=l

and for:

Z*> () - 7, Ai Z0 (x)i=l

E[Z. (x)-Z.(x)] 2 = 0

where Z*.. (x) is the estim:regionalized variable, of thof k components in the cclocation.

The first requirement,;

ance, the covariance is (Sinowski et al., 1997). In some situations it mayices, singular variance- be necessary to predict spatially all of the compo-ie cokriging equations nents of the particle-size data at unknown loca-.dition, kriging of sep- tions before they are used for pedotransfer func-ionalized composition tions or modeling of other soil processes.that sum to a constant. Either ordinary kriging or cokriging has gen-npositions. Therefore, erally been used by researchers in the spatial analy-ndamental to kriging sis ofpsf (e.g., Lookman et al., 1995; Mapa and Ku-

maragamage, 1996; Oberthur et al., 1999: Odeh)rediction of compo- and McBratney, 2000). These kriging techniquesa caused by the prob- do not take into consideration requirements 1-4voided if four basic re- (Eqs. (1-4)), especially requirement (2) above.uijter et al., 1997): Choice of an appropriate geostatistical method for

spatial analysis is, therefore, critical for producing) valid estimates with minimal prediction error vari-ance. A number of statistical methods have been

- constant (2) demonstrated to meet some of the requirements inEqs. (1-4). The first is data transformation beforekriging or cokriging, as suggested by several au-thors (McBratney et al., 1992; Pawlowsky et al.,

Ai-1j -. k (3) 1993). Another method is compositional kriging,developed by de Gruijter et al. (1997). In soil sci-ence, these methods were developed for spatial in-terpolation of fuzzy (continuous) soil class mem-

(4) bership values, a form of composition. However,they have rarely been used for interpolation of

ite of a compositional particle-size data. The aim of this study is to assesse jth component (out the performance of these methods on particle-size)mposition) at the ith data obtained for a region in New South Wales

Australia. First, however, the statistical theory andas indicated in Eq. (1), the methods will be described briefly.

means that each of the components of a region-alized composition must be nonnegative. In thecase of the second requirement (Eq. (2)), a re-gionalized composition must sum to a constant atevery location. A third requirement (Eq. (3)) en-sures that the estimate, Z* (x) are unbiased, andthe fourth (Eq. (4)) is indicative of variance min-imization in the kriging system of equations.Most of the spatial interpolation methods usedfor regionalized compositions in soil studies donot meet all four requirements.

Particle-size data are the most familiar com-position in soil science. The relative proportionsof the individual particle-size fraction (psf) arewhat constitute the soil texture. The importanceof soil texture cannot be overemphasized. Thesoil texture, and indeed the particle-size distribu-tion, determine, in part, water, heat, and nutrientfluxes, water and nutrient holding capacity, andsoil structural form and stability. The clay frac-tion, in particular, as the active constituent of thecomposition, could be incorporated in pedo-transfer functions to predict material fluxes (e.g..Arya, et al., 1999) and other soil properties

PREDICTION METHODS USED ONCOMPOSITIONAL SOIL DATA

Log-Ratio Transfonnation before Krigingor Cokriging

Modeling any data requires identification ofthe appropriate sample space.A restricted part ofreal space (R), termed the positive simplex (Rh, isidentified by Aitchison (1986, 1990a) as the ap-propriate sample space for compositions. A sim-plex is a geometric representation of attributespace, where a composition Z of D parts is rep-resented by a minimum number of vertices for aspace of a given number of dimensions (McBrat-ney et al., 1992). To gain the advantage of sym-metry, the d-dimensional simplex (in terms ofthe subvector) is embedded in D-dimensionalreal space (Aitchison, 1986):

Sd= (Z -, Z.D) Zl >.ZD > O; zl+PPP+zD- 11 (5)

The symmetric positive simplex dovetails wellwith the requirements defined in Eq. (1). How-

502 SOIL SCIENCE

n

SPATIAL PREDICTION OF COMPOSITIONAL SOIL DATA

ever, the simplex does not, cater adequately forindependence and measures of dependence ofthe components in the absence of a satisfactoryclass of distributions (St (Aitchison, 1982). And,although the constant-sum constraint (R" con-fines compositional vectors to a simplex, there isno guarantee that the patterns perceived in sucha constrained space will necessarily have the sameinterpretation as in the more familiar spaces suchas (Rd) (Aitchison, 1986). A solution to the prob-lem is the transformation of the natural space (S'to the real space (R') via additive log-ratio trans-formation (ALR), defined as:

zi

yi = ln Zd+1

where y, is the log-ratio transformation of zi. Ia regionalized composition this is expressed as

y, (x) ln fU x k = d + 1; i1.

j = l,...k

with inverse transformation:

(X) exp y((x)

exp yk (x)j=I

The effect of ALR transformation is twofoldthe closure effect is removed (Aitchison,191990b), and (ii) through perturbation, traformed values may be closer to a normal distrution than the untransformed data. Transforrtion, therefore, makes the data better suitedclassical statistical procedures (Aitchison, 19EALR has been applied to compositional geolccal data (e.g., Zhou et al., 1991; Pawlowsky et1993; Cardenas et al., 1996) and has been uonly rarely in the analysis of soil particle-size c(e.g.,Lark, 1999).

Membership grades resulting from fuclassification of soil types into continuous claare the new compositional data in soil scierContinuous classes have membership valuesthat describe the degree of membership to a c(j, j = 1, . . ., c) (McBratney et al., 1992; Ode]al., 1992). The result of fuzzy classificationmatrix of membership values, such as a comp,tional matrix, Z = c x n. The matrix Z = Ziisfies all the requirements of a composition.resolve the specific case of transforming mebership values of k continuous classes belkriging, McBratney et al. (1992) modifiedAitchison (1986) log-ratio transformation eq

(6)

tion. Because of the presence of zero membershipdata, it is necessary to replace the zeros beforetransformation (Martin-Fernandez et al., 2000;Fry et al., 2000). The constant rj was, therefore,introduced to cater for zero values in the data, rbeing one-half of the smallest membership otherthan zero, and Eq. (7) was modified accordingly.The modified log-ratio transformation (mALR)is expressed as:

yY(x) = ln k i k

T1et(Zik (X)+i)n

The inverse trarnsformation is defined as:

(9)

For z e(X)( y -(-) )(j ) (10)

E~ exp YJ.(X) +D,7 j=1j=i j=-

For strictly positive compositional values, inclu-(7) sion of the constant X is unnecessary, and Eqs. (7)

and (8) are more appropriate.

Comipositional Kriging

(8) Compositional kriging (CK) was recently de-veloped by de Gruijter et al. (1997) and used onmembership classes resulting from fuzzy k meansof classification of soil. They needed to produce

(i) continuous soil maps that relate soil distribution82, patterns to the general landscape structure. Com-nbs- positional kriging is an extension of OK. It is dis-ib similar to cokriging, however, in that it does not

na- assume linear correlations among the composi-for tional components. Moreover, cross-variogram?6) models are not required.

)gi- As with OK, CK also minimnizes the erroral.. variance with respect to the unbiased constraint. Insed this case, the error variance can be minimized bydata setting its partial first derivatives, with respect to its

associated Lagrange multip]ier (i.e. w, a , or f3zY equal to zero with the following linear equations:

ice. n:laes j-I Aj Qj + ,a + a z. + 6 zic = Cio, Vi,

lass. i=l

2 Ai, = 1i=1

AjCi, = 0 and a,- 0i=l

Vc(11)

Vc

k )I,

C I Aiz-i = Ic=l i=l

503VOL. 168 -No. 7


where X. = weight assigned to observation pointj for the membership (composition) class c; C; =covariance between observation points i andj forthe membership class c; C,0. = covariance be-tween observation point i and the predictionpoint for the membership class c; and it, = num-ber of observation points used to predict themembership class c.

Therefore, the memberships at a specific pre-diction point are estimated by:

n

z, = I Ao1i=l

Vc

and the error variances obtained by substitutingweights via algebraic manipulation (de Gruijteret al., 1997) into a computationally more efficientexpression as:

02 ,= 2-, AC, - (- (a+/3)z Vc (13)i=l

where o2 = variance of the prediction error inmembership class c and ou2 = variance of thememberships to class c.

MATERIALS AND METHODS

The methods described above were appliedto the spatial analysis of particle-size data ob-tained for the two regions, the lower Macintyreand Namoi valleys, both situated in east centralAustralia. First, however, we describe the two re-gions and also the methods of data analysis andvalidation.

The Study Regions

The first study region, the lower Macintyrevalley, is approximnately 5100 km2 in extent and islocated on the border between New SouthWalesand Queensland, two of the eastern states ofAus-tra]ia (Fig. 1). The second region is in the lowerNamoi valey, which is about 200 km south ofthe Macintyre.Both regions are part of the Mur-ray Darling Basin, just west of Great DividingRange of Eastern Australia. They are of very lowrelief: a gentle east-west slope of approximately1:3000. The dominant soil on the plains is deep,self-mulching cracking clays; gray clays are foundon the open plains and in depressions, and brownclays are found on the slightly elevated areas(Odeh et al., 1998). Soil information for the twoareas was sparse and, until recently, was sourcedmainly from the "Atlas of Australian Soils"(Northcote, 1966) at a scale of 1:2 mifDion. Thisstudy was part of a bigger project initiated to pro-vide soil attribute information important for sus-

tainable cotton production, a major agriculturalactivity in both regions.

In the lower Macintyre valley, a total of 119sites (Fig. la) were visited and sampled, and-moredetailed sampling was conducted in the lowerNamoi (Fig. lb)).Auger samples were obtained atsix prespecified depths down to 2 meters. Thesampling strategy adopted for the lower Macin-tyre valley and part of the lower Namoi valleyis described elsewhere (Odeh and McBratney,1994). The sampling design used for the easternpart of the lower Namoi valley is described inMcGarry et al., 1989. The particle-size fractionswere determined by an in-house developed mi-cropipette method. We utilized only the topsoil(0-10 cm) particle-size data for this study. Eachsoil sample was assigned to one of 12 textureclasses as described in Soil Survey Staff (1962).Clay was differentiated into light-medium clay(40-50% clay) and heavy clay (>50% clay), mak-ing 13 overall potential classes for the purposes ofthe study.

Data AnalysisThe following methods were examined and

compared:

* kriging of each untransformed (UTOK) psf di-rectly as has been the practice;

* additive log-ratio (ALR) and modified log-ratio transformation (mALR) prior to ordinarykriging (ALROK, rnALROK) and cokrigingof the transformed value, followed by back-transformation of the kriged results;

* compositional kriging (CK) (de Gruijter et al.,1997) using all of the fractions.

Ordinary kriging (OK) and cokriging arewell known within the soil science community(e.g., Wackernagel, 1995; Goovaerts, 1997) andthus will not be described here. It should benoted, however, that cokriging compositionaldata make sense only after log-ratio transforma-tion as ALR removes the effect of closure (Zhouet al., 1991). Isotropic spherical variogram mod-els were fitted to the experimental variogram ofthe untransformed and transformed data for all ofthe kriging methods.

The FORTRAN program, COKRIG (Carr etal., 1985) was used for generalized cokriging.Compositional kriging was performed using theprogram developed by de Gruijter et al. (1997).The semivariogram and cross-variogram neededfor cokriging and compositional kriging werecomputed and modeled using the geostatistical

504 SOIL SCIENCE

SPATIAL PRPEDICTION OF COMPOSITIONAL SOIL DATA

700M0 72 740000 7W(XO 7WCX a8o

6= F/= 6= -= = 7,- MM Z 7 75 7M 7A= 7= 79=

Fig. 1. Location of study area and sampling layout.

N | Key Features- - Road

10 0 10 20 Kilometers E - River+ . ~~~~Sample point

a

505VOL. 168 - No. 7


package VESPER (Minasny et al., 1999). Anotherprogram, ISATIS (Geovariances, 1997) was usedfor OK analysis of both the untransformed andtransformed particle-size data.

Validationi of the Prediction jlWetlhods

The primary validations used to test the qual-ity of spatial prediction of soil attributes are themean error (ME) and root mean square error(RMSE) (Voltz and Webster, 1990). Mean errorcan be estimated using Eq. (13):

,,

z'i-~

ME i=

RMSE is expressed as:

(13)

RMSE (14)

Mean error is a measure of bias (or unbias), andRMSE is a measure of precision and bias. AsRMSE is sensitive to both systematic and randomerrors, it could be used to estimate the accuracyof prediction (Atkinson and Foody, 2002), and itcould be based on a validation sample set selectedindependent of the training set.

When the data set used is large, simple di-chotomous sets, one for validation and the otherfor training, would not pose a problem. However,when the sample size is small (as is the case here:119 for the Macintyre valley and 237 for theNamoi valley), it becomes difficult to have suffi-cient (and separate) sample data sets for modelingand validation. Moreover it has been suggested inthe literature that a sample size of 100 is the barestminimum required for variogram estimation(Webster and Oliver, 1992). For this reason a

modified jackknifing technique (Good, 1999)was used to resample the base sample data 20times for the purpose of validation. The resam-pling size for validation was maintained at ap-proximately one-sixth of the available data, i.e.,19 of 119 for the lower Macintyre valley and 58of 328 for the lower Namoi valley. Mean error(Eq. 13), as a measure of bias, and RMSE (Eq. 14),as a measure of precision and bias, were esti-mated, for each of the resampled validation sets.The prediction quality of each predictionmethod was determined by averaging the MEsand RMSEs of the 20 jackknifed samples.

RESULTS AND DISCUSSION

The summaly statistics of the two sets of dataare shown in Table 1. The distributions of thesample data are typical of a composition. None ofthe fractions (clay, silt or sand) approximates anormal distribution. All are skewed, either posi-tively (silt and sand) or negatively (clay). Percentclay content is by far the most variable of thefractions (SD = 13.6%), followed by sand (SD =10.6%) and, lasdy, silt (SD = 9.6%). The standarddeviation (SD) is higher than one may expect forparticle-size data, probably because of the largegeographical extent of the study area (Fig. 1).This occurs because, although the area is rela-tively flat and geologically homogenous, smalldepressions (gilgai) are pedologically diverse fromtheir surroundings (Hubble et al., 1983).

Illustrated here as examplesi the histograms ofthe untransformed and transformed data for thelower Mcintyre valley are shown in Fig. 2. mALRimproved normality slightly. Both transformationmethods reduce skewness for sand and siltmarkedly, but ALR actually increased skewnessfor clay. Similar results were obtained for thelower Namoi valley. The correlation coefficient

TABLE ISummary statistcs of the topsoil (0-10cm) particle size fl-actions data

Lower Macintyre Lowor Namoi

Clay (/o) Silt Sand Clay Silt SandMinimum (/O) 10.3 18.1 1.7 2.5 0.1 0.11st quartile (Q1) (/O) 44.7 26.9 11.1 39.1 16.1 17.9Median (Q2) '/o) 51.4 31.5 16.3 51.3 19.6 25.5Mean (%o) 48.0 32.6 17.7 47.4 20.9 29.63rd quartile (Q3) (%O) 56.3 38.5 22.8 58.4 24.4 38.1Maximum (%) 77.4 60.0 62.2 71.8 73.3 95.5Range (%) 67.1 41.9 60.5 69.3 73.2 95.4((Q3-Q1)/2) (%) 23.0 10.2 10.6 28.0 12.2 19.0Std (/o) 13.9 9.6 10.6 14.6 8.0 17.1Skewness -1.03 0.95 131 -0.76 1.47 1.02

506 SOIL SCIENCE


Clay Silt Sand

,----. I

60 80 0 20 40 60 6 ~' ' o 4o0' ~

-1.8 -1.4 -1.0 -0.6

-3I

-0.4I -. 0I0. 0. 1.2 -2-0.4 -0.0 0.O4 0.18 1.2 -2.0

I I II I I I .-4.0 -3.0 -2.0 -1.0

-. n----..

-1.0 0.0

Fig. 2. Histograms and boxplots of the Macintyre particle-size data (n= 119), including clay (%),sand (%), and silt(%), comparing (a) untransformed (UT), (b) additive log-ratio (ALR), and (c) modified additive log-ratio (mALR)transformations.

of the untransformed (UT) with transformeddata are shown in Table 2. ALR decreased thecorrelation between clay and silt and clay andsand but improved slightly the correlation be-tween silt and sand. iALR improved consider-ably the correlation beween sand and clay, and

between sand and silt, but the correlation be-tween silt and clay deteriorated.

A map of the spatial distribution of the sumof psf predicted by ordinary kriging of untrans-formed psf (UTOK) for the lower Macintyre val-ley is shown in Fig. 3. It is evident that (UT0 1,)

TABLE2

Correlation matrix of untransformed (UTOK) and transformed sample data (ALROK and JiiALROK)

Variable UtOK ALROR siiALROK

Clay Silt Sand Clay Silt Sand Clay Silt Sand

a) Lower MacintyreClay 1.00 1.00 1.00

Silt -0.57 1.00 -0.49 1.00 -0.01 1.00Sand -0.73 -Q.15 1.00 -0.60 -0.18 1.00 -0.81 -0.57 1.00

b) Lower NamoiClay 1.00 1.00 1.00

Silt 0.12 1.00 0.23 1.00 -0.01 1.00Sand 0.81 -0.54 1.00 -0.33 -0.58 1.00 -0.81 -0.57 1.00

(a)

i%0%; 20 40 60%

0

1.0

� ___M

5()7VOL. 1 68 -NO. 7

--E��


6O22 7702m9 722022 7491OX0 o 60 780C 8s2.2 82002 840000 86=

6850220

6820203

6s2C0 7000 720022 74002 7600 78W= &=x7 8200 4ro 860C0

20 0 20 40 60 Kloniters

Fig. 3. Sum of the particle-size fraction produced by ordinary kriging of the untransformed particle-size data fromthe Macintyre valley (UTOK); expected sum = 100%.

both overestimated and underestimated values ofsoil psf. only 35% of the predicted sites sum to100%. It is not surprising that the constant sumrequirement is not wholly met by (UTOK) as thedependencies between fractions are discardedand the separate fractions are estimated indepen-dently at each point on the spatial grid. The sameresults were obtained for the lower Namoi valley.The results of (UTOK) concur with those ofother researchers who have investigated the effi-cacy of the approach in compositional analysis(see Pawlowsky et al., 1995; de Gruijter et al..1997).

An operation to overcome the problem of in-terpolated values of untransformed psf not sum-ming to a constant involves kriging all the frac-tions but one and then calculating the remainingfraction by difference. This operation. however, isnot order invariant. By contrast ALROK (andn2ALROK) are order invariant (McBratney et al.,1992; de Gruijter et al., 1997) in the sense that themultivariate normality is preserved under permu-tation of the components of the composition(Barcel6 et al., 1996). These methods (and CK)produced kriged particle-size values that sum toa constant (100%). Therefore, the methods basedon log-ratio transformation and CK meet the re-quirements in Eqs. (1)-(3). This outcome high-lights the need to reassess applying OK to psfwithout transformation, especially psf commonlyimbedded in pedotransfer functions (e.g., Sin-owski et al., 1997).

Cokriging of the log-transformed two datasets produced very poor results. There are severalpossible reasons for this. As Table 1 and Fig. 2show, the sample data are highly skewed, and allfractions have outliers. Although the (cross-) var-iogram estimator is unbiased, it is susceptible tooutliers and shows nonrobust behavior towarddistributional deviations (Armstrong, 1984;Dowd, 1984; Omre, 1984). In addition to out-liers, the data also shows a weak trend and lack ofspatial co-dependence. Surprisingly, the more in-tensively sampled data in the lower Namoi valleydid not produce better cokriging results.

Figure 4 shows the relative proportions of thetexture classes identified for the sample dataset and the results of UTOK CK, and the back-transformed particle-size values resulting fromALROK, nlALROK for the lower Macintyre val-ley. For the sample data set (n = 119) the pro-portions are sandy loam (SaLm) 2% of sites, loam(Lm) 7%, silty loam (SiLm) 2.5%, clay loam(CLm) 2.5%, silty clay loam (SiCLm) 4%. siltyclay (SiC) 8%, light-medium clay (LMC) 21%.and heavy clay (HC) 53%. UTOK had only half(4) the number of classes of the original samplesites. There are only five texture classes resultingfrom each ofALROK, miiALROK, and CK. Figure4 also shows the similarity of texture class distri-bution between ALROK and inALROK and thesample data set. However, whether kriging trans-formed or untransformed data, all of the krigingmethods have the effect of smoothing the final

508 SOIL SCIENCF


80

60

% 40

20

0

* Sanpled

Elu(mOALI(oK)

ElmALIR(ok)

o CK

Saim Lm SiLm CILm SiClhLm SiCI LMC

Texture Class

Fig. 4. Histogram of texture classes of sample particle-size data from the Macintyre valley (n = 119) and of the tex-ture classes produced by compositional kriging (CK - n = 18722), and ordinary kriging of untransformed data(UTOK -n = 18722) and kriging of transformed data (additive log-ratio transformation - ALRoK; modified additivelog-ratio transformation - mALR n = 18722 for both)

results. This is illustrated in maps of the soil tex-ture classes presented in Fig. 5.

The patterns of texture class distribution pro-duced by UTOK (Fig. 5a) and CK (Fig. 5d) aresimilar. ALROK1 and mALROK have similar pat-terns also (Fig. 5 a and c), which is not surprisingconsidering the similarity of the two methods.However, nALROK gives slightly better spatialcontinuity in the prediction. To a lesser extent,the spatial pattern of texture classes produced byCK is similar to those of ALROK and mALROK(Figs. 5b and c) except for the northeastern cor-ner of the map of the CK results (Fig. 5d). Somepatches in Fig. 5d also show some discrepancycompared with the other two methods. The his-togram in Fig. 4 illustrates this, with CK clearlyoverestimating LMC and underestimating HCcompared with ALROK and iiALROK. This issupported by the results of assessing the accuracyof textural classification for the two study areas aspresented in Table 3.

Validationi Results

Recall that we used repeated resampling of theavailable data to validate the prediction method. Asan example, Fig. 6 shows the histograms of theRMSE of prediction for the resampled validationsets for the Macintyre valley using different pre-diction methods. Only the RMSE of the clay frac-tion exhibits distribution close to normal. This isnot surprising as the clay fraction is generally char-acterized by a large range ofvalues compared withsilt and sand fractions (Table 1). The study regionsare characterized mainly by Vertisols (which havea preponderance of clay) and a few Alfisols and In-

ceptisols (Soil Survey Staff, 1998). However, thehistograms demonstrate that RMSE obtained byrepeated resampling (multiple jackknifing) is morerepresentative of the population than that obtainedby a single jackknifmg. The average values ofRMSE are used to compare the quality of the pre-diction methods.

As shown in Table 3, ALRoK is the most ac-curate predictor of textural classification for thelower Macintyre valley, with 72% of the valida-tion sites classified correctly compared with 65%with mALROK and 44% for UTOK. Only 55% ofthe validation sites were correctly classified byCK. This trend is also repeated for the lowerNamoi valley. Evidently, the ME values, shown inTable 3, also indicate that all methods overesti-mated the silt fraction, except for UTOK of thepercent silt for the lower Namoi. Conversely, inmost cases the methods underestimated the clayfraction. RMSE indicates that ALROK and mAL-ROK are equally accurate for predicting percentclay, with UT as the most precise for predictingsand. Statistical testing, using the mean RMSE forordinary kriging of the raw psf, indicates that theperformance by ALROK and mALROK is signi-ficantly superior (P = 0.05) to that ofUTOK(Table 3). Although the difference be-tween RMSE of ALROK rnALROK, and CK isminimal, what is surprising is the greater misclas-sification of texture classes by CK compared withthe other methods. The poor performance intextural classification by CK is probably causedby the algebraic manipulation in the CK pro-gram, which was probably not as order invariantas we would like it to be.

509VOL. 168 - No. 7


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6860000

684Q000

6820000

680X0DO

Easting (m)

Fig. 5. Comparison of the Macintyre texture classes as predicted by a) ordinary kriging of untransformed data (UToK);b) kriging after additive log-ratio transformation (ALROK); c) kriging after modified additive log-ratio transformation(mALROK); and d) compositional kriging (CK).

, ., -+

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510 SOIL SCIENCE

1- I- -1. .___

| . .e 'P.: .3e,.1itE: ;e :as.la.Vf: u E X tj * s,- T-aR@J9- %


a) Ordinary kriging of untransformed psf data6 Clay 8 ~~~~~~Silt 8Sand

6> Cla 8- 77 8S4- ~~~~~6 *6

02

2.5 5.0 7.5 10.0 12.5 15.017.5 2.5 5.0 7.5 10.0 12.5 15.0 2 4 6 8 10 12

b) Compositional kriging of psf data

Clay ~~~~8 * Silt 8- Sand6 - ] 8 j 80s6

0 4- 42..:2. 2- 7F7 J

5 10 15 20 2 4 6 8 10 12 0.0 2.5 5.0 7.5 10.0 12.5

c) Modified log-ratio kriging of psf data

00

2.5 5.0 7.5 1o.0 12.5 15.0 2 4 6 8 10 12RMSE (%) RMSE (%)

0.0 2.5 5.0 7.5 10.0 12.5RMSE (%)

Fig. 6. Histograms of root mean square errors (RMSE) for a) Ordinary kriging of particle-size fractions (psf); b) Com-positional kriging of psf; and c) Ordinary kriging of modified log-ratio transformed psf.

Figure 7 shows some of the results of mAL-ROK on the lower Namoi psf data. Figure 7a alsoshows the distribution pattern of the textureclasses produced by niALROK. It is apparent thata broad band of heavier clay soil runs diagonallyfrom the southwest corner near Pilliga to thenortheast corner north of Edgeroi. Most of theirrigated cotton-growing farms are located in thisarea, particularly to the northeast and northwestof Wee Waa. To the south, the soil is muchcoarser in texture, ranging from sand to loamysand and sandy clay loam. This is consistent withthe soils derived from Pilliga Sandstone (Tri-antafilis et al., 2001). The area' near Edgeroi issimilarly characterized by silt loams and sandyclay loams, which were probably derived fromwashed sediments from the nearby NandewarRange (Triantafilis et al., 2001).

The spatial distribution pattern of each of thetopsoil clay fractions, as produced by inALROKfor the lower Namoi valley, is shown in Fig. 7b.In the areas south of Edgeroi and southeast ofWee Waa, the clay fraction (Fig., 7d) is low(<40%). In the area to the north of Wee Waa,however, the clay fraction is relatively large(>55%). This is because this area coincides withthe area where the Namoi River flows into a very

flat alluvial plains landscape, and, as a result, thearea has seen various depositional and erosionalevents, and the soil tends to be predominantlyfine-textured (Triantafilis et al., 2001).

In general, critics of CK may not be support-ive of the embedded model in the CK program,which involves algebraic manipulation in orderto solve the problem of the constraint caused byerror minimization. Even though CK producednumerically valid output, inasmuch as our resultsare concerned, the statistical validity of themethod is more or less heuristic and may not jus-tify its being more computationally efficient. Thetime involved in data transformation bebforekriging or the overparameterization and the la-boriousness of constructing so many variogramsor cross-variograms for kriging (de Gruijter etal., 1997) may not be reason enough to discardthese techniques. However, the data transforma-tion methods are not without problems.

CONCLUSIONSThe two study sites were large geographical

areas, one of which was sparsely sampled. Weidentified five main texture classes in the surface-layer of the soil, the dominant class of which washeavy clay. The performance of each of the pre-

VOL. 168 -No. 7 511


TABLE 3

Validity of textural classification, and bias and accuracy of the prediction methods

Texture correcdy Bias AccuracyMethod classified ME (%) -- -RMSE (%)-

(%6) Clay Silt Sand Clay Silt Sand

a) Lower MacintyreUTOK 44 -0.83 1.40 0.86 12.13 9.26 738ALROK 72 0.75 1.85 -1.01 9.56' 5.26*' 5.89"SmALROK 65 -0.58 1.38 0.52 9.40*' 5.17*" 6.07n'CK 55 -1.43 2.54 0.85 10.58n, 7.17' 6.36"'

b) Lower NamoiUTOK 49 -1.12 -0.02 0.11 11.10 8.58 8.18ALROK 69 -0.25 1.84 -0.89 8.66"6 6.01" 8.12"niALROK 66 -1.00 0.56 -0.73 8.85- 5.81' 8.26a"CK 52 -4.58 2.15 -0.94 10.47"1 7.18'" 6.36*

"Statistically different from mean for UTOK (bold underlined) at 0.01 alpha levelStatistically different from mean for UTOK (bold underlined) at 0.05 alpha level

ns - Not statistically different from mean for UToK (bold underlined) at 0.05 alpha level

sat 6M M 6 7a= 77t= 72a7 73 7 47 7WKr5 7a10 7TAM 7WC72 7X

Fig. 7. Some results of mALROK on particle-size fractions of soils in the lower Namoi valley (a) pattems of topsoil tex-tural classes, and (b) spatial patterns of topsoil % clay fraction. (continued)

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10 0 10 20 Kilometers W*s

512 SOIL SCIENCE


7ax= 74= = 71r= 7/ 7a= ;9

Fig. 7. (b) spatial patterns of topsoil % clay fraction.

diction methods was assessed based on four con-straints: (i) predictions should be greater than orequal to zero, (ii) each prediction point shouldsum to a constant, (iii) the predictions should beunbiased, and (iv) the variance of the predictionerrors should be minimized.

We, like other authors (Pawlowsky et al.,1995; de Gruijter et al., 1997), were unable to as-sess requirement (iv) using ALRoK and inALROK,although the problem has recently been solved byPawlowsky-Glahn and Egozcue (2002). However,all methods predicted the particle-size valueswhich were greater than zero. Ordinary krigingon the untransformed psf led to output values notsumming to a constant at many locations. Allother methods (CK,ALROK,and inALROK) pro-duced results that sum to a constant at every lo-cation. Nevertheless, although the accuracy ofCK was similar to that of the other methods, itperformed poorly in predicting texture classes.ALR and tnALR transformation before krigingusually outperformed UTOK and CK. In this casewe believe the success was the result of the trans-formation before kriging. There was evidence ofa slight trend in the data for both study areas. Per-

haps further improvement can be achieved usinguniversal cokriging after ALR (Stein andCorsten, 1991). Nevertheless, data cannot alwaysbe modeled adequately using log-ratio transfor-mation, and alternatives such as Box-Cox faniily(Barcel6 et al., 1996) or an adaptation of theRenner (1996) transformation may need to beassessed. The use of standardized residual sum ofsquare (STRESS) has also recently been proposed(Martin-Fernandez et al., 2001) as an alternativecriterion for testing prediction quality. We willfocus on these issues in a future work.

This study also highlights the problem of re-lying on one method to validate a particularmethod of data analysis. An obvious example isthe use of RMSE. RMSE of prediction forUTOK and CK seems not to reflect the inaccu-racy of predicting the texture classes across thestudy area.

The analysis and regionalization of composi-tional data present specific problems for soil sci-entists. There are new compositional soil data(e.g., fuzzy membership dlasses). There is also aneed to incorporate components of psf in pedo-transfer functions. Therefore, to ensure that we

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513VOL. 168 -NO. 7


do not make use of spurious interpolated results,it is important that we use the appropriatemethod (such as ALR or, perhaps, CK) ratherthan rely on interpolation of untransformedcomponents of compositions, which has previ-ously been the practice.

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