Nonlinear and scaling spatial properties of soil ... - Wiley Online Library

WATER RESOURCES RESEARCH, VOL. 37, NO. 4, PAGES 1031-1042,APRIL 2001

Nonlinear and scalingspatial propertiesof soil geochemical element

contents

JonasOlsson, • RonnyBerndtsson, 2 AkissaBahri,3 MagnusPersson, 2 and Kenji Jinno•

Abstract.Thepresent studyaimedat investigating whethera nonlinear andscaling approach issuitable for statistically characterizing thespatial variability of soil geochemical element contents at fieldconditions. Spatialdistributions of 20 soil geochemical elements' contents in anagricultural fieldsoilwereinvestigated. Two indicators of nonlinear scaling wereemployed, empiricalprobability distribution functions (pdfs)andstructure functions. Forthepdfs,a trendwasdiscovered withmajorelements naturally occurring in thesoilbeinghyperbolic andminortraceelements beingcloseto Gaussian. Further,fertilizermanagement wasfoundto modifythebehaviorof related elements. The structure functions weregenerallynonscaling for majorelementsand

scaling forminorelements, however, withoutanynoticeable effectof fertilizerapplication. Thescaling wasof bothmonoaffine andmultiaffine type.Thissuggests thattherandom cascade modelsrecently usedwithingeophysics maybe usefulalsofor geochemical elementvariabilityin soil. exhibitssymmetry acrossscales,typicallymanifested in some statistical fieldpropertybeingrelatedto observation scalein a Soil chemicalvariabilityis an integratedresultof several power-law fashion. If thepowerlawexponent isnoninteger, it interacting genetic(soilforming)andintrinsic (soilmodifying) is termedfractaldimension[Mandelbrot,1982].Scalinglaws processes [e.g.,Thompson etal., 1987;Blocketal., 1991].Par- have been discovered,and fractal dimensionshave been estiticularlytheinherentheterogeneity of thesoilstructure affects matedfor a varietyof soilproperties,for example,porestrucandlimitsthe predictability of physical, chemical, andbiolog- ture,particle-size andaggregate distribution, permeability, and icalprocesses [e.g.,Crawford et al., 1999].Studieson spatial flow pathways[e.g.,Katz and Thompson, 1985;Tylerand soil chemical content have in most cases indicated a tremenWheatcraft, 1990;YoungandCrawford, 1991;Kemblowski and dousvariation,especially for heavymetalssuchasCd andCr Wen,1993;Baveye etal., 1998].Also,waterandsolutetransport [Beckett and Webster, 1971;Wopereis et al., 1988;Bahriet al., properties havebeencharacterized byfractalscaling, andfrac1993].Because of theextreme variation andnoobvious spatial tal transport modelshavebeendeveloped [e.g.,Wheatcraft and structure,thismaylead to unrealistic samplingneedsfor de- Tyler,1988;FluryandFhihler,1995;Mukhopadhyay andCushtermining, forexample, representative meanvaluesbya purely man, 1998;Pachepsky and Timlin,1998]. stochastic approach [Wopereis et al., 1988;Bahriet al., 1993]. Althoughservingas a usefulmeasureof irregularity,a Spatialvariability of soilchemical elements at fieldconditions uniquefractaldimension isoftennotsufficient to characterize limitsthe possibility to describe andmodelprocesses in agri- highlyvariablefields.Thereasonistheadditive natureof this cultural lands such as solute characteristics of the soil and oftentermedsimplescaling(or monofractal)approach.The potentialcropuptakeof hazardous components. corresponding multiplicative approach is termedmultiscaling Considering the closeconnections betweenthe chemical (or multifractal)[Frischand Parisi,1985].For multiscaling 1.

Introduction

elementdistributionand the soilstructure,key featuresof soil characteristics suchasporosityandhydraulicconductivity are

likelyto bereflected alsoin theelement contents [e.g.,Blocket al., 1991].It couldthereforebe fruitfulto consider novelapproaches to characterize soil structuralpropertiesas potentially usefulalsofor the containedchemicalelements.One suchapproach thatreceives a constantly increasing attention is basedon the conceptof nonlinearscaling[e.g.,Mandelbrot, 1982;Schertzer and Lovejoy,1991].Generally,a scalingfield

fields, the fractal dimensionis not a unique constantbut a functionof the field intensitylevel [e.g.,Feder,1988].While

simplescalingis limitedto presence or absence of a certain object,multiscaling isbasedona probability measure andthus includesintensityinformation.Concerning soil,one interpretationof thisdistinctionis thatwhilea singlefractaldimension maysufficeto characterize simplegeometrical featuresof the

support (i.e.,thesoilmatrix),a multifractal scaling framework is requiredfor describing physical, biological, and chemical processes takingplacein it [Blocket al., 1991].Scalingproqnstituteof Environmental Systems, KyushuUniversity,Fukuoka, cesses based onmultiplicative contributions generally displaya Japan. variability thandotheiradditive counterpart, andthey 2Department of Water Resources Engineering, Lund University, greater Lund, Sweden. haveprovedto morerealistically characterize nonlinearvari3NationalInstitutefor Researchon Rural Engineering,Water, and abilityin manygeophysical fields[e.g.,Schertzer andLovejoy, Forestry,Tunis,Tunisia.

1991].

Copyright 2001by theAmericanGeophysical Union.

In recentyears,multiscaling approaches to analyzeand modelsoilproperties havebeengrowing. Perfectet al. [1993]

Papernumber2000WR900354. 0043-1397/01/2000WR900354509.00

foundevidenceof multifractalfragmentation of soilaggregates 1031

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As an alternativeto Gaussianmodelingof soil properties, and developeda multifractalaggregationmodel. Multifractal scalinghavebeen found to accuratelydescribethe distribution the useof so-calledhyperbolic(or Pareto) distributions charof fracture apertures [Belfield, 1994] and surface strength acterizedby slowlydecaying"fat" tails has recentlybeen ad[Folorunso et al., 1994].Liu andMolz [1997b]andGim•nezet vocated[e.g.,Painter,1996].For a randomvariableR (or, more al. [1999]founda multifractalscalingbehaviorof soilhydraulic generally, fieldR (x)), thetermhyperbolic refersto a pdfwhose conductivity.Groutet al. [1998]demonstratedthe insufficiency tail exhibitsa powerlaw form of a singlefractal dimensionto describethe particle size disPr (R > r)ocr-qcr (1) tribution in clayeyand silty soil,which exhibiteddistinctmultifractality.From theseand other findings,it maybe concluded for high thresholdvaluesr [e.g.,Mandelbrot,1974]. From a that the scalingof subsurface propertiesmay be of both simple theoreticalpoint of view, a hyperbolicbehaviorimpliesdiverand multiscalingtype and that it is important to employ a gence of high-order statisticalmoments, which means that methodologyallowingfor both typeswhen analyzingnew data. samplemomentsincreasewith the samplesize and may beOne ultimate aim of statistical characterization of soil struccome arbitrarily large. For a particular sample, the critical tural properties,suchas the efforts basedon scalingsummamoment order abovewhich divergenceoccursis specifiedby rized above, is to achieve a better understandingof solute the power law exponentqcr, in practiceimplyingthat largertransportand associated contaminantredistribution.In light of order momentsare dominatedby the very extremevaluesonly. this, it is clear that direct investigationsof chemicalelement 1•o.-ontly,i....... tlcmwith the development"f distributionsare important to complementthis research.If stochasticmodels,hyperbolicityhasbeen observedfor rainfall similaritiescan be found betweenthe variability of soil strucdata sets [e.g., Lovejoyand Mandelbrot,1985;Fraedrichand tural propertiesand of chemicalelement contents,respecLarnder, 1993; Olsson,1995]. Although not necessary,hypertively, this would verify the relevanceand usefulnessof the bolic behavioris characteristic of multiscalingprocesses (e.g., approachin question.However,whileconcepts of nonlinearity multiplicativecascades) widelyusedin modelingof geophysiand scalingare increasingly beingappliedto soilstructure,only cal fields [e.g., Schertzerand Lovejoy,1987, 1991; Gupta and very few similar studiesof soil chemicalelementsappear to Waymire,1990]and recentlyalsoappliedto soil-relatedprophave been carried out. One natural reason for this is that erties [e.g.,Liu and Molz, 1997b]. scalinganalysesrequirelargeamountsof data and this is cumThe family of so-calledL6vy-stabledistributionsconstitutes bersomeand expensiveto collect for soil chemicalcontent, a well-knownexampleof hyperbolictail behavior.A L6vyparticularly for scalescorrespondingto entire agricultural stable processis parameterizedby a, 0 < a -< 2, the L•vy fieldsor infiltrationareas.On a microscale,Blocket al. [1991] index,and the width parameterC (C > 0). L•vy-stablepdfs found the spatial distributionof Al, Fe, and Si in German do not in generalhave a closedanalyticform (exceptfor a = marine sedimentsto exhibit multifractal scaling.Kropp et al. 2, whichproducesthe Gaussiandistributionas a specialcase) [1997] performeda similar analysisof Ca from the samesite, but usuallydisplayhyperbolictailswith qcr = a. Painter[1996] which alsodisplayedan unambiguousmultifractalbehavior. found spatial incrementsof permeability and porosityin a In this paper, we investigatenonlinearand scalingfeatures sedimentaryformationto be substantially better approximated in the spatialvariationof 20 major and trace elementcontents by a Lfivy-stablethan a Gaussiandistribution,and Liu and in the top layer of a representative agriculturalsoil.Two exMolz [1997a]reporteda goodfit of Lfivy-stabledistributionsto perimental fields with different fertilizer managementwere incrementsof (naturallogarithmsof) hydraulicconductivity, studied.The samplingpattern in each field was specifically ln(K). designedto allow for analysesover a range of length scales, As mentionedabove,hyperbolictails may be viewedas one between-0.2 and 60 m. The analysiswasperformedwith the facet of a wider conceptreferredto as a multiscalingor mulgoalof findingan appropriatemodelfor describingthe spatial tifractal behavior. This is often defined in terms of statistical variationof soilchemicalcontent.This included(1) establishmomentsand impliesthat for a spatialrandomfieldR(x), the ingwhethernonlinearscalingis presentor not and,if scalingis averageqth order moment of the coarse-grainedfield R x represent,(2) determiningthe characteristics and estimatingthe lates to the coarse-graining scaleh as parametersof scaling-based models.The paperconcludeswith a discussionon the variation of model parameters. (2)

whereanglebracketsdenote(ensemble)averagingandK(q) is a nonlinearfunction(see,e.g.,Daviset al. [1994]for detailsof We use two different but related indicatorsto explorethe the coarse-graining procedure).The functionK(q) essentially nonlinear and scalingpropertiesof the spatial variation in representsan infinite hierarchyof fractal dimensions,each and elementcontent:(1) the tail behaviorof empiricalprobability characterizedby a different field intensitylevel. Schertzer distributionfunctions(pdfs)and (2) the scalingcharacteristics Lovejoy[1987] made a parameterizationof K(q) under the of structurefunctions.The spatialvariability of geochemical assumptionof an underlyingmultiplicativecascadeprocess elementsin soil, aswell as manyother soil properties,is gen- with a L6vy-distributedgenerator.Multiplicativecascadeproerally either normallyor lognormallydistributed;this hasbeen cessesoriginatedasconceptualmodelsof statisticalturbulence supported by empirical investigations [e.g., Shaw, 1961; [e.g., Yaglom, 1966; Mandelbrot, 1974]. The approach of McBratneyet al., 1982;Wopereis et al., 1988;Bahri et al., 1993]. Schertzerand Lovejoy [1987] was termed a "universalmultiGaussian-type pdfs (includinglognormal)often approximate fractal model" with parametersa, the L•vy index introduced the main bodyof the distributionreasonablywell and is there- above,C•, related to fractal propertiesof the mean process, fore sometimesusedfor reasonsof simplicityevenif the tails andH relatedto the scalingof structurefunctions(seebelow) fail to properlyreproducethe most extremevalues[e.g.,Ein- [seealso,e.g., Schertzerand Lovejoy,1991; Tessieret al., 1993, steinand Baecher,1983]. 1996]. It shouldbe noted that q = qcr definesan upper limit

2.

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of nonlinearityof K(q); for q > qcr the functionis linear [e.g., Mandelbrot,1974]. Related to soil,moment-basedmultifractal analyseshavebeen carriedout for variationsof, for example, fracture aperturesand soil surfacestrength[Belfield,1994; Folorunsoet al., 1994]. Since a coarse-graining procedure requires reasonably

evenlydistributedmeasurement points,whichis not the case for our measurements (seesection3.1), rather than statistical momentswe chooseto studythe related scalingbehaviorof structurefunctions.For the fieldR (x), the qth order structure function may be defined as

([AR(h.)]q) = (IR(x + X) - R(x) whereA -

(3)

denotesseparation distance. Thisis a generali-

zation of the semivariogramusedin geostatistics, whichcorrespondsto the caseq = 2 [e.g.,Journeland Huijbregts,1978].

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1033

eral fertilizerson field A and sludgeon field B were manually applied. The soil was subsequentlymixed with the fertilizers and the sludge,attempting to reduce effects of the uneven original application. During the 1988-1989 and 1989-1990 croppingseasons, durumwheatwascultivated.The fieldswere sprinklerirrigatedduringthe former season,whereasno irrigation wasperformedduringthe latter. Soil samplingtook place immediatelyafter harvestin June 1990,by a steelaugerdownto a depthof 0.20m (diameter7 cm) from the soilsurface. The samplingprocedurewas a combinationof nested and rectangularsamplinggrid specificallydesignedfor spatialscaling analyses[e.g., Oliver and Webster,1986; Wopereiset al., 1988]. Sampleswere taken with the same spatial pattern at each of three different scalesrepresentedby squaresof side length40 m (61 samples),10 m (48 samples),and 2.5 m (48

samples),respectively (Figure 1). In total, 314 soil samples weretakenin the twofields(157 samplesin eachfield). In each of the sixscaledsquares(three scaledsquaresin eachfield), 61 to the separationdistanceA as sampleswith the samesamplingpatterncanbe compared(61 ([ AR(/•)]q)oc/• •(q), (4) unique samplesat the largestscale,48 at the intermediate scale,and 48 at the smallestscale).It shouldbe emphasized where •(q) is a nonlinear(concave)functioncorresponding to that this multiscalesamplingstrategy,which providesan apK(q) in the caseof statisticalmoments[e.g.,Vicsekand Barapropriate resolutionacrossa wide range of scales,makesthe btisi, 1991]. Similar to K(q), the entire •(q) functionis redata particularlysuitedfor the presenttype of analysis. quired to completelyspecifythe multiaffinity,i.e., the multiA mixed sifted (0.2 mm), 1 g subsamplewas completely scalingbehaviorof the structurefunctions.However, a useful digestedby mixingthe samplewith 10 mL of 20 M HF and 5 characterizationof the field is providedby the parameterHx = mL of 7 M HC10 4 in closedTeflon beakersset to boil over•(1), 0 -< Hx -< 1. This may be viewed as a measureof night at 100øC.The sampleswere then heatedat 15øCuntil no "smoothness" (increasingHx, increasingsmoothness) and is liquidremained.The dry samplewascooledto roomtemperrelated to the power spectrum exponent /• widely used to ature, after which 70 mL of 1.3M HC10 4 was addedto dissolve characterizescalingfields[e.g.,Daviset al., 1994;Olsson,1995]. the dry material.Sampleswere finallydilutedby distilledwater It further specifiespropertiesof a cascadeprocessthat maybe to 100 mL and analyzedby inductivelycoupledplasmamass usedto reproducethe multiaffinity[e.g., Tessletet al., 1993; spectrometry.The following 20 elements were analyzed for Daviset al., 1994].Also •(q) canbe expressed in termsof the total content;Al, Ba, Be, Ca, Cd, Co, Cr, Cu, Fe, K, Mg, Mn, parametersa, C•, and H; for detailson the estimationproceNa, Ni, P, Pb, S, Sr, V, and Zn. dure, seeSchmittet al. [1995] and Liu and Molz [1997b].Liu The different chemicalelementsanalyzedcan roughlybe and Molz [1997b]analyzedthe horizontalvariationin In (K) divided into two major groupsbased on element origin and usingstructurefunctionsbut found no accuratescalingbehavThe field is called multiaffine

ior.

if the structure function

relates

spatial variation pattern: majorchemical elements andminor

trace elements.The major elementsare naturallyoccurringat highcontentsin the soilbecauseof the mineralcompositionof parentrock.Minor traceelements,however,shouldbe present 3. Experimental Setup only in very smallquantities(even someof them are not nat3.1. Material ural constituentsin the soil). Major elementsare Al, Ca, Fe, Soil sampleswere taken at the Cherfech agriculturalfield Mg, and K. Consequently,minor elementsare Ba, Be, Cd, Co, research station located at Cherfech, 25 km north of Tunis in Cr, Cu, Mn, Na, Ni, P, Pb, S, Sr, V, and Zn. Concerningoverall the Lower Medjerda Valley, Tunisia.The soil conditionshere concentrationlevels,the mineralfertilizer applicationon field in terms of texture and chemicalcompositionare representa- A affected mainly the Cd content,whereasthe sludgeapplitive for the region [UnitedNationsDevelopmentProgramme cationon field B affected,in decreasingorder, Cu, S, Zn, P, Pb, (UNDP), 1970].The U.S. Departmentof Agriculture(USDA) and Ba [seeBahri et al., 1993]. In the presentstudy,all 20 classification of the soilis VerticXerofiuvent.The topographyis elementsare analyzedin orderto identifytrendsand depena plane, and the microtopographyin the experimentalplots denciesof the nonlinearand scalingproperties.For a complete wasat the samplingtime almostlevel.The alternatelyfine and descriptionof agriculturaloperations,soil sampling,chemical coarse-texturedsoils are of fluvatile origin and have been analyses, analytical procedures, and basic statistical and formed on alluvial sedimentsof the Medjerda river. Owing to geostatistical analysesof the data, seeBahri et al. [1993] and agriculturalpractices,the upper 0.20 m of the soil profile is Berndtsson et al. [1993]. spatiallyhomogeneouswith a clay content (q=O -1 q=ll

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3

,

2

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• -4

q=2© e._e.6 -½ v

a

q=3&• q=4©• Log[X]

smoothedout. It is reasonableto assumethat sucha processis

lesslikelyto producea highvariabilitywith prominentextreme valuesthan, for example,the concentratedapplicationof fertilizersat a limited time periodprior to sampling. For the hyperbolicelements,the overallqcr varieswidely, reachingvaluesabove20 (Table 1). As mentionedpreviously, this is generallyhigherthan what has been found for some geophysical processes (e.g.,rain and river flow). No relevant comparablestudyof the horizontalvariationof elementcontentsor othersoilpropertiesexiststo our knowledge.Regardlessof the valueof qcr, for manyelementsa qualitativehyperbolicbehavioris stronglysuggested with extremevaluesbeing far more commonthan predictedby a Gaussiandistribution. This indicatesthat a (multi)scaling modelmaybe relevantfor the elementcontents'spatialdistribution,and we investigate thisfurtherby determiningthe structurefunctions. Similarlyto the pdfs,the structurefunctionsof variouselementsdisplaya varietyof shapes,and sometypicalexamples are givenin Figures5-7. In the figures,the separationdistance X rangesbetween0.25 m (log X = -1.38) and 48 m (log X = 3.87). Figure 5 showsthe result for K, representinga group of elementswithclearlyscalingstructurefunctions. For q = 0 the pointsare naturallyhorizontallyaligned,and for 1