SELECTED RESEARCH OPPORTUNITIES IN SOIL PHYSICS

D.R. NIELSEN1; M. KÜTILEK2; O. WENDROTH3; J.W. HOPMANS1

1Department of Land, Air and Water Resources, University of California, Davis, California 95616, EUA.2Faculty of Civil Engineering, Technical University Prague, Thakurova 7,16629 Prague 6-Dejvice, Czech Republic3Physicist, Center of Agrolandscape and Land Use Research, Institute f or Soil Research, Ebenwalder Strasse 84,15374Mfincheberg, Germany.

ABSTRACT: Selected research opportunities are discussed in order to guide soil science research, with emphasis onsoil physics, with the aim of improving agricultural productivity and environmental quality.Key Words: research, soil science, soil physics

OPORTUNIDADES SELECIONADAS DE PESQUISA EM FÍSICA DO SOLO

RESUMO: Oportunidades selecionadas de pesquisa são discutidas para orientar a pesquisa em ciência do solo,comênfase na física do solo, com o objetivo de melhorar a produtividade agrícola e a qualidade do ambiente.Descritores: pesquisa, ciência do solo, física do solo

INTRODUCTION

Today, an appreciation of environmentalcitizenship exists throughout the world. Soilscientists are in a pivotal position to help theglobal community improve its environment whileat the same time accelerate the quality andquantity of agricultural production. Optimizinginputs at the farm level, reducing agrochemicalapplications and maximizing agriculturalproduction without soil and water degradation arecommon goals. As we enter the next millennium,we are mindful of the delicate balance betweenfood production and the quality of land and waterresources. Our achievements in production duringthis century were aided by laboratory and smallscale landscape research efforts. Small, replicatedplots established on sites believed to be "typical"or "representative" of a farmer's field providedresponses to fertilizers, pesticides and irrigations.With replicated treatments established in threeto five statistical blocks, the goal was to establishthe relation between crop yield and treatment.For example, having completed the regressionof the data or the analysis of variance, arecommendation was usually made regarding thelevel of fertilizer that should be used for a desiredcrop yield. With an increased yield being obtainedwhen the recommendation was followed, thefarmer was happy and the researcher publishedthe results in a peer-reviewed research journal!

What was generally the next step? It varied fromrepeating the same experiment or modifying itstreatment levels, going to another field or soilcondition, or doing nothing more at that location.Doing nothing more does not benefit the farmernor does it benefit agriculture or the environment.Another sampling across the entire field coulddetect specific locations within the field where,for example, crop production could be increased,excessive levels of the fertilizer nutrient anddeficiencies of other plant nutrients prevail orchanges in soil quality indicative of achieving ordenying sustainable agriculture. Yes, anothersampling across the entire field would ascertain ifthe farmer could make still further improvementsin crop and soil management without necessarilyimposing different treatments in still anotherreplicated small plot experiment.

Today's agricultural research can also beimproved in other ways. Too often it is assumedthat steady-state conditions prevail. Lackingmethods to analytically deal with short-termperturbations of local conditions, we are forced tothink in terms of long-time averages. We applymass balance equations for relatively short timeperiods - minutes, days, weeks or times no longerthan a growing season. We also believe thatdeterministic concepts are adequate to guide ourthinking and explain our experimental results.Attempts to assess the impacts of agriculturalmethods and weather events between and during

several growing seasons have usually only beenmade through long-term experimental plotsmanaged for decades, and in a few cases for morethan a century. Although these kinds ofexperiments persist today, they do liffle forimproving our understanding of how agriculturalpractices impact on the quality of water leaving acultivated field. Moreover, they provide no directinformation regarding the all-too-often subtlechanges in soil quality occurring on or within afarm or an agricultural region. Opportunities toanalyze our efforts outside of those small plotsand stretch our consideration quantitatively acrossregions cultivated by one or many farmers arereadily available. New theoretical opportunities toanalyze the processes occurring even in thosesmall plots are also available.

Researchers have a tendency to rely oninformation and methods learned at the beginningof their careers without spending sufficient timeto learn enough about new ideas and evolvingcontemporary opportunities just outside the nicheof their own usually narrow scientific focus. Thepurpose of this presentation is to provide ourperspective of a few challenges and opportunitieswe believe are potentially fruitful to thoseinterested in soil physics.

LEACHING THEORIES AND THEFATE OF AGROCHEMICALS

Brief History

The mixing and interactive processes ofa solute described for simple, well-definedgeometries and materials provide a basis forunderstanding transport in soils. Unfortunately,the rigor of such solutions exemplified by those ofTaylor (1953) gives way to that of empirical orstatistical formulations owing to our inability tomathematically define the geometry of the soilpore system or to measure parameters descriptiveof the displacement processes that can betranslated from the pore scale throughintermediate scales to that of a pedon or field.These pathways and pore water velocities,severely altered with slight changes of watercontent, have yet to be quantitatively evaluated. Inthe near future, computer-aided micro tomographyand nuclear magnetic resonance techniques willprovide an opportunity to ascertain the exactnature of the velocities at the pore scale. Without

such observations, our understanding of certainfacets of miscible displacement in soils has beenenhanced by considering surrogate porous mediahaving simplified or empirical pore geometries.

Descriptions of idealized soil poreshaving capillary shapes include some mechanismfor transport between parallel capillaries or allowone or more capillary tubes of differing radii to bejoined at their ends at common junctions (e. g.Marie and Defrenne, 1960). The concept ofrandom networks of capillary tubes providesinsights to the meandering paths of displacingfluids. The capillary tube network (de Josselin deJong, 1958) stemmed from the tetrahedral porebetween four closely packed spheres beingrepresented by a junction of four capillaries. Therandomness of the capillaries originated from theassumption that their positions were dictated by arandom arrangement of soil particles. De Josselinde Jong neglected molecular diffusion andassumed that the velocity of a fluid particle wasthat of the mean velocity across the capillarydiameter. With this simple capillary network, deJosselin de Jong was the first to show that thetransverse apparent diffusion coefficient issmaller than the longitudinal apparent diffusioncoefficient. He also showed that the magnitude ofthe longitudinal apparent diffusion coefficientdepends upon the distance traveled.

Statistical concepts have been applied tosolute and water transport through porous mediaat the pore scale primarily because of thedifficulty of integrating differential equations ofmotion with poorly or undefined complexboundary conditions. Danckwerts (1953),Scheidegger (1954) and others assumed that asimple random walk stochastic process can beused to describe transport in a fluid-saturatedhomogeneous, isotropic porous medium generallyconsidered chemically inert. The exact nature ofthe path followed by fluid particles theoreticallyobeying Navier-Stokes equation in the poroussystem is not known. The velocity or position of awater or solute particle is the random variableand as the particle passes through the poroussystem, it eventually encounters all situations thatare possible at any one given time. Thatprobability distribution function was given byScheidegger (1954). Day (1956) described indetail the connection between that probabilitydistribution function and the macroscopic concen-tration of a solute being displaced in a satura-

ted sand. Recognizing that the probability distri-bution function was proportional to the soluteconcentration and knowing that it satisfiesclassical diffusion equations, Danckwerts (1953)used solutions of

subject to appropriate initial and boundaryconditions to describe the displacement ofsolutes through fluid-saturated porous mediawhere X = (x - vxt). He noted that the valueof D must be determined empirically andwould presumably depend upon the viscosity,density and velocity of the fluid, and on thesize and shape of the solid particles. Hecalled D the "diffusivity" while Scheideggernamed it the "factor of dispersion".

The random capillary models describedabove were made somewhat more physicallyrealistic (Bear & Bachmat, 1967) by deriving theidea of a representative elementary volume at themacroscopic scale from microscopic quantities atthe pore scale averaged over many pores. Molecu-lar diffusion and convection of solutes and waterflowing within individual pores were described atthe pore scale while the spreading or dispersionof solutes with water as it curves around andbetween soil particles through sequences of poresoccurred at the macroscopic scale. Their equationof mass conservation in one direction was

where u and C are average values within the REVand Dc and Dm are the coefficient of convective(or mechanical) dispersion and coefficient ofmolecular diffusion, respectively. Combining thelatter coefficients into a single term Da

(commonly called the hydrodynamic dispersioncoefficient or the apparent diffusion coefficient),we have

for which many investigators have soughttheoretical or experimental relationships between

the value of Da (which embraces solute mixing atboth the pore scale and the pore-sequence scalewithin an REV) and the value of u (the averagepore water velocity usually estimated by the ratioof the Darcian flux density q and water content

q . Recognizing that the effects of both moleculardiffusion and convection on solute mixing intypical one-dimensional experiments weredifficult separate, Fried & Combarnous (1971)suggested five ranges of Peclet numbers or zonesto delineate the relative magnitudes of eachprocess. Today, with the molecular diffusion termoften neglected the value of Da is frequentlyassumed to be defined as

with the value of n taken as unity. In suchinstances, b is called the dispersivity.

CONTEMPORARY THEORIES

Convective-diffusion equation: Recognizing itsform is tentative in several aspects besides beingfraught with uncertainties of applicable temporal(Skopp, 1986) and spatial scales (Dagan, 1986)that are not easily resolved, the cornerstone ofmost theoretical descriptions of solute transport inporous media is

where rT is the soil bulk density, Cs the soluteadsorbed or exchanged on the soil solids, 6 thevolumetric soil water content and C the solute insolution. Although the source-sink term (fi hasmost often been considered in the absence of therest of the equation in many disciplines, it is oftenapproximated by zero- or first-order rate terms.Microbiologists, considering organic andinorganic transformations of soil solutes inrelation to growth, maintenance and wastemetabolism of soil microbes as a Michaelis-Menten process, often simplify theirconsiderations to that of fi. McLaren (1970)provided incentives to study such reactions asfunctions of both space and time in soil systems -

a task not yet achieved by soil microbiologists,especially when the individual characteristics ofeach microbial species is quantified and notlumped together as a parameter of the entiremicrobial community. Agronomic or plantscientists consider (fi as an irreversible sink andsource of solutes taking place in the vicinity of therhizosphere of cultivated or uncultivated plants asa function of soil depth and time as well as someempirical function defining the root distribution.

The primary fault of (5) is that the sizesof the REVs of the dependent variables andparameters are generally not the same for allterms as well as being unknown for a particularlaboratory column or field soil. Note according to(4) that Da is composed of two terms - one basedupon mixing at the molecular scale and the otherrepresenting mixing at much greater order ofscales. Hence, it is not surprising that fieldstudies yield values of Da two orders ofmagnitude greater than those found in thelaboratory. Because the sizes of the REVs of thedependent variables are not the same, it is notobvious to most investigators that the volume-averaged concentration C measured within acolumn or profile at a particular location is notthe same as the flux density-averagedconcentration moving along at the same location.The same problem arises when sampling the soilsolution using suction lysimeters or extracting thesoil solution from a soil sample removed from thecolumn or profile.

Solutes in Continual Equilibrium with theSolid Phase. Instantaneous adsorption orexchange reactions included in the first term of(5) are described by equilibrium isotherms ofseveral different forms - mass action, linear,Freundlich, Langmuir or other functionalrelations. The most common approach has been toassume instantaneous adsorption or exchange aswell as simple linearity between Cs and C [Cs =kC where k is the slope of the isotherm Cs(C)often referred to as the distribution coefficientKd]. Using this linear isotherm leads to thedefinition of the retardation factor R = 1 +rTkq-1. While the simplicity of a linear iso-thermis a convenient feature for mathematicallymodeling, its limitations are learly apparentowing to adsorption and exchange processesusually being nonlinear and depending upon thecompeting species in the soil solution.

Solutes Not in Equilibrium with the SolidPhase. Diffusion-controlled or chemicallycontrolled kinetic rate reactions included in thefirst term of (5) have been described in a varietyof equations. The most simplest formulation isthat of a first order linear kinetic reaction where(5) is replaced by two coupled equations.Here, for steady state flow through ahomogeneous soil without sources or sinks, wehave diffusion.

Were a is a first order rate constant.The success of this and similar rate models hasbeen best when miscible displacementexperiments have been carried our at relativelyslow velocities when mixed in dominated bymolecular diffusion.

Nevertheless, under such conditions thevalues of a and k may indeed be biased owing tothe use of an average value of a hich does notembrace the spatial distribution of the soluteinfluencing the rate reaction within soil pores atthe microscopic scale.

A second formulation gives moreconsideration to the microscopic pore watervelocity by defining a bimodal distribution whichpartitions the soil water into mobile and immobilephases. In the mobile phase where soil waterflows, solute behavior is described by aconvective-diffusion equation. Inasmuch as wateris stagnant in the immobile phase, solutes movein and out of this phase only by moleculardiffusion. Zones of stagnant water derive fromthin liquid films around soil particles, dead-endpores, non moving intra-aggregate water orisolated regions associated with unsaturatedconditions. Miscible displacement equationsbased on first order exchange of solute betweenmobile and immobile phases initially discussedby Coats & Smith (1964) were extended by vanGenuchten & Wierenga (1976) to includeFreundlich type equilibrium adsorption-desorption processes.

Their equations are

where the subscripts m and im refer to the mobileand immobile phases, respectively. Theretardation factors account for equilibrium typeadsorption while the mass transfer coefficient aembraces a diffusion coefficient and an averagedivisional path length. Although (7) was usedsuccessfully by van Genuchten and Wierenga andby Gaudet et al. (1977) as well as many othersmore recently to describe laboratory columnstudies, its use in structured field soils has beenlimited owing to the difficulty of obtainingreliable values of a which depend upon thegeometry of the soil pore structure (vanGenuchten, 1985). For laboratory experiments,the value of a may well be confounded withnonlinear isotherm and chemically kineticexchange effects. Moreover, the fraction of 6considered to be immobile is sensitive tohysteresis, the concentration of the soil solution,the soil water content and the soil water flux.

A third formulation considers the firstterm of (5) having two components - one forexchange sites (type 1) on a fraction of the soilparticle surfaces that involve instantaneous,equilibrium reactions and another for type 2exchange sites involving first order kinetics orthose assumed to be time-dependent (Selim et al.,1976; Cameron & Klute, 1977). Following Nkedi-Kizza et al. (1984) we have

where F is the mass fraction of all sites beingoccupied with type 1 sites, and where subscript 2refers to type 2 sites. With values of a and Fusually being dependent upon the average

displacement velocity u, values of F appropriatefor (8) cannot be obtained from equilibrium batchstudies. The use of an average value of u masksany effect of the microscopic pore water velocitydistribution on the exchange process. Because thedimensionless forms of (7) and (8) are identical,effluent curves from laboratory soil columns alonecannot be used to differentiate between thespecific physical and chemical phenomena thatcause an apparent non equilibrium situationduring leaching. The similarity of the two sets ofequations allows an oftentimes satisfactoryempirical description of the mixing of solutes atthe macroscopic scale by either equation withoutascertaining the exact nature of the particularchemical or physical process at the microscopicscale. The exact nature of the processes awaitsfurther research using micro tomography or othertechniques of observation at the microscopicscale.

Dual-Porosity Considerations for StructuredSoils. Dual porosity or bi-modal porosity modelsassume that a soil can be separated into twodistinct pore systems superpositioned over thesame soil volume with each system being ahomogeneous medium having its own water andsolute transport properties (Dykhuizen, 1987).With the two systems exchanging water andsolutes in response to hydraulic and concentrationgradients, the soil is characterized by two watervelocities, two hydraulic heads, two watercontents and two solute concentrations. Here weuse subscript M to denote the inter-aggregatesystem and subscript m to denote the intra-aggregate system. Gerke & van Genuchten (1993)provide a comprehensive review of varioustheoretical and experimental attempts to deal withwater and solute movement in saturated andunsaturated structured soils during steady andtransient water flow conditions. See Gerke & vanGenuchten (1993) for a complete description ofthe hydraulic properties of the structured soil andthe details of the solute transport equations. Forexample, solute transport with linear adsorptionand first order decay is described by

where m is a first order decay coefficient, Gs asolute mass transfer term to which both moleculardiffusion and convectivo transport contributes andWM is the ratio of the volume of the macroscopicpores to that of both macroscopic and microscopicpores.

With the exception of Gs which isdefined as the mass of solutes per unit volume ofbulk soil per unit time, all variables in (9) aredefined relative to the partial volume of each poresystem. Although this and other similar modelscan simulate transport related to specific chemicaland physical properties of the soil, its largenumber of parameters not easily measuredexperimentally remain a topic of future laboratoryand field research.

Consecutive Reactions. Equation (5) isoftentimes sequentially repeated when organic orinorganic products are of interest, especially whenthe products form a consecutive chain ofreactions, e. g. for nitrogen (urea ® NH4

+ ® NO3-

® N2). In such cases a set of simultaneousequations stemming from (5) becomes

where j = 2, 3, n when n is the number of speciesconsidered in the reaction chain. The equationsare linked to each other by their mutual (fj terms.These equations have been applied to consecutivedecay reactions of soil nitrogen species (e. g. Cho,1971; Misra et at., 1974; Starr et al., 1974),organic phosphates (Castro & Rolston, 1977) andpesticides (Bromilow and Leistra, 1980).

With consecutive equations such as(10), field studies in the presence of higher plantsprovide opportunities to better understand agro-ecosystems. For example, Mishra & Misra (1993)learned how liming a cultivated field of cornmodified the values of k2 and k3 as a function of

soil depth and time in the presence and absenceof crop roots. A better understanding ofmicrobial-induced transforma-tions of otherchemical species for transient flow andnonisothermal conditions await investigation inboth the laboratory and the field.

CHROMATOGRAPHlC FORMULATIONS

Descriptions of the transport of fluidswith their dissolved constituents through beds ofreactive porous solids based uponchromatographic plate formulations stemprimarily from those derived by chemicalengineers nearly one-half century ago (e.g.Wilson, 1940; DeVault, 1943; Thomas, 1944;Glueckauf, 1949; Lapidus & Amundson, 1952;Heister & Vermeulen, 1952). The chromato-graphic formulation introduced by Dutt & Tanji(1962) is that of a vertical, homogeneous soilcolumn of length L of unit cross-sectional area ismade up of n segments (plates) each of lengthAzi. The concentration Cj of a number of solutespecies j entering the column in each leachingaliquot DQm (where m is the number of thealiquot) changes as the solutes mix, react andpass through each segment. The first aliquotinfiltrates into the first segment, and fills it tosome prescribed soil water content. The secondaliquot of infiltrating water displaces the soilwater from the first segment into the secondsegment, and so forth. If the amount of solutionAQ in each segment Dzi is identical duringinfiltration, the final concentration Cj; of solutespecies y in the first aliquot DQ1 leached from thecolumn will be

where DCij is the change in concentration ofsolute j when the aliquot DQ1 is passed throughsegment AZj. As n ® ¥. the last term in (11) isthe integral of the change in solute concentrationfrom z = 0 to z = L. Assuming that the solution isin chemical equilibrium with the soil in eachsegment, the difference in concentration betweenthe equilibrium solution and that entering eachsegment is calculated. If n in (11) is consideredfinite, the average concentration of the aliquotsDQm is calculated by progressively equilibrating

the solution of DQm with each of the n segmentsassuming that piston flow takes place within eachsegment. The dispersion of the solutes associatedwith pore water velocity distributions andmolecular diffusion are implicitly and empiricallyincluded by choosing the number of segments orplates n. Tanji et al (1972) utilized the concept ofholdback to allow only a fraction of the soil waterin each segment to be displaced into the next afterchemical equilibrium. They also designatedvariable segment thicknesses Dzi corresponding tosoil sampling depth intervals or soil horizons aswell as choosing the value of n based upon thedispersion of measured chloride breakthroughcurves.

Stochastic Considerations: Because of thenaturally occurring heterogeneity of field soils,deterministic formulations of solute transportprocesses presented above generally must bemodified to describe field scale solute transport.Contemporary research efforts are based upon theconsideration that transport phenomena areintrinsically erratic processes susceptible toquantitative characterization by stochastic models.Common to all stochastic models of field scaletransport is the assumption that parametersobserved in the field are functions with valuesdistributed in space represented as randomvariables with discrete values assigned accordingto a probability distribution. The probabilitydistribution functions at each point in space areusually unknown and cannot be evaluated fromonly one or a few observations within closeproximity of the location. Reviews by Jury (1983)and Dagan (1986) provide details. Many otherstatistical approaches are also described in theliterature [e.g. continuous Markov processes(Knighton & Wagenet, 1987), random walkformulations (Kinzelbach, 1988), momentanalyses (Cvetkovic, 1991) and hierarchicalmethods (Wheatcraft & Cushman, 1991). Here,we consider three approaches to deal with spatialand temporal variability.

Monte Carlo Simulations: Monte Carlosimulations of a solution of a deterministicequation such as (3) allow coefficients to berandom variables of the nature expected within aheterogeneous field soil. The variable may beindependent, spatially or temporally correlatedand perhaps manifest a variance structure. Based

upon an initial sampling, parameters selected forthe assumed probability density function (pdf)permit repeated solutions (i = 1, 2, ...) of thedeterministic equation [e.g. Ci(z, t)] to becalculated. These solutions Ci(z, t) are then usedto calculate sample moments (mean, variance)which are assumed to represent the statisticalproperties of the underlying stochastic process.

Two kinds of solute distributions are ofinterest. The first kind is that of C(z, t)/Corealized at any location within a field, and thesecond is that of C(z,t)c0 obtained by averaginga large number of Ci(z, t)C0 across a field. Theformer is important because it is associated with aparticular soil pedon location, a single crop plantor a small community of plants. With site-specificcrop and soil management practices, each locationcan indeed be treated and managed individuallyacross the entire field in order to account for localvariations of soil properties. The latter isimportant because it is the expectation of soluteretention and emission of solutes from an entirefield considered as a single domain. Althoughthere is general appreciation of the latter, afanner is also appreciative of the former owing tothe desire to provide optimum growing conditionsfor each and every crop plant within the cultivatedfield. The opportunities afforded by Monte Carlosimulations hinge upon the development ofmethods to measure and ascertain pdfs of thetransport coefficients within prescribed limits ofvadose zone depths and times.

Stochastic Continuum Equations: Recognizingthe paucity of solute concentration data usuallyavailable in soils and other subsurfaceenvironments coupled with their naturalgeometric complexities and heterogeneity, Gelharet al. (1979) initiated stochastic continuumformulations to describe transport processes inwater saturated aquifers. Unlike the above MonteCarlo methods which assume that the randomparameters or variables have no spatialcorrelation, stochastic continuum formulationsassume that a random variable can be representedby the sum of its mean and a spatially correlatedrandom fluctuation.

When a random parameter such as Da in(3) is represented by the sum of its mean valueand a random fluctuation, a mean transport modelwith additional terms is obtained. By solving thestochastic equation of the local-scale water and

solute transport, the functional form of Da formacroregions is related to the statistics describingthe variability. A macro-scale value of Da isreached asymptotically as distance and/or timeincrease. From such an analysis the longterm,large-scale solute transport can be described usingthe stochastically derived value of Da in thedeterministic equation (3). The application ofstochastic continuum research for unsaturatedsoils appears promising (e.g. Russo, 1993, andYeh et al., 1985a, b and c) but not yet sufficientlydeveloped to be a proven field technology.

Stochastic Convectiva Equations: Thedisplacement and attenuation of a solutedistribution within a vertical soil profile duringinfiltration can be considered the result of astochastic convective flow process with itsformulation based upon a solute travel timeprobability density function. The advantage ofsuch formulations is that they do not require anexplicit accounting of all of the various physical,chemical and biological processes occurring in thecomplex, heterogeneous soil environment.Although many stochastic convective models havebeen used in different scientific disciplines, thoseinitiated by Simmons (1982) and Jury (1982)stimulated research in soil hydrology during thepast decade. Here we describe the stochasticconvective concept as a transfer function Qury(1982) which can easily be obtained for anonreactive solute by a single, simple fieldcalibration to measure the travel timedistribution. It is assumed that no dispersion ofthe solute takes place other than that which isrepresented by the travel time variations withinthe soil.

Assuming that the depth reached by asolute applied in water at the soil surface dependsupon the net amount of water applied, theprobability that the solute will reach depth L aftera net amount of water I has been applied to thesoil surface is

where fL(I) is the probability density function, fL

(l) is the average concentration at soil depth z = Lin response to a narrow pulse (Dirac d-function)of solute CIN = C0d(I) applied at I = 0 at the soil

surface. A set of observations of fi(I) can beobtained by measuring the soil solutionconcentration at depth L at various locationswithin a field to determine the amount ofuniformly applied water I required to move thesolute pulse from the soil surface to depth L. Theaverage concentration CL (I) at z = L for arbitraryvariations of solute CIN applied at the soil surfaceis

The integrand is the probability fL, (I') ofreaching z = L between I' and (I' + dI'), andmultiplied by the concentration CIN (I - I') ofsolution displacing at I. For spatially variablewater application rates Jury used a jointprobability function in (13).

We assume that the distribution ofphysical processes contributing to the probabilitydensity fL, (l) between z = 0 and z = L is the samefor all soil depths. Hence the probability that anapplied solute will reach any depth z after anamount of water / = /; has infiltrated the soilsurface is equal to the probability of reaching z =L after I = I/Lz'1 has infiltrated. For example, theprobability of reaching a depth of 50 cm with 10cm of infiltrated water is equal to the probabilityof reaching 100 cm with 20 cm of infiltratedwater. Hence, from (12)

To predeict the average soluteconcentration as a function of any depth z ¹ L, werelate the probality density function fz(I) to thereference density function fL by

If the transport properties arestatistically similar for al depths, only onecalibration fL is needed. On the other hand, ifthey are dissimilar owing to strongly developed

and obtain

horizons or textural and structural differences,additional calibrations f are required or a morerobust calibration fL for all depths can be made atdepth L below the strongly stratified soil.

Relatively few distributions of solutetransport parameters have been measured in thefield. Presently, stochastic-convective formula-tions are being extended to include the transportof adsorbing and decaying solutes, two componentchemical nonequilibrium models, physicalnonequilibrium models and other nonlinearprocesses. See a review by Sardin et al. (1991) aswell as a more recent contribution by Roth & Jury(1993).

SPATIAL VARIABILITY ANDGEOSTATISTICS

Semivariagrams and Crossvariograms: Theconcept of variance known from classicalstatistics is extended in geostatistics to considerthe location of the observations [A(xi), A(xi + h)]separated by a distance h. The equation for theconstruction of the semivariogram is

As the distance between pairs ofobservations or lag h increases, g(h) rises andasymptotically approaches the value of thevariance called the sill. The sill is approached ath = l denoted as the range or scale of thevariogram as well as the zone of correlation. For h< l, the variance is deformed by the position ofthe sampling points, or in other words, by thespatial dependence otherwise called the spatialstructure. Differently shaped semivariograms areobtained for spatially independent and dependentdata as well as for a spatially changing domain. Ifthe domain is spatially changing and notstatistically homogeneous, l(h) increases and doesnot approach a sill. The intercept at h = 0 iscalled the nugget and usually appears as aconsequence of fine scale estimates not beingavailable. The nugget also includes themeasurement error. In a non-isotropic domainvariograms differ for different directions. Forsampling on a rectangular grid, constructingvariograms along the main two directions of thegrid and on the two diagonals is a logical first

choice to identify the presence of non-isotropicbehavior. For the construction of the appropriatesemivariograms, computed data l(h) are fitted to asimple curve, usually the segment of a circle, orthat of an exponential or hyperbolic curve.Interpreting semivariograms is made somewhatmore reliable if a couple of "rules of thumbs" arefollowed. First, the minimum number of samplesalong a transect should be in the range of 50 to100 (Gutjahr, 1985; Webster, 1985). And second,the estimation of l(h) is considered reliable forlags not exceeding 20% of the total transectlength. The term support refers to the size, shapeand orientation of samples. An increase ofsupport, called regularization, generally leads to adecrease of variation.

The concept of covariance known fromclassical statistics is also extended in geostatisticsto consider the location of two sets ofobservations [A(xi), A(xi + h)] an [B(Xi), B(Xi +h)] separated by a distance h. The equation for theconstruction of the crossvariogram is

The crossvariogram shows over whatdistances the two different kinds of observationsare related to each other. The spatial covariancebetween soil and crop attributes as well as itstransformation as a function of time remainslargely unexplored in the agricultural sciences.Fundamental research is needed to even identifydifferent scales of space and time to bestinvestigate such covariance relationships. Wechoose soil erosion, a major impediment tosustainable agriculture, to illustrate the need formeasuring spatial covariance structures. Soilerosion has long been studied using analysis ofvariance of observations from replicated plotscarefully placed at different positions on thelandscape e.g., the knoll, shoulder, back, foot ortoe (Rune, 1960). On a rolling or hilly landscape,eroded material moves from higher landscapepositions downhill and, as a result, regions withinthe field are "enriched" or "depleted".

Miller et al. (1988) used geostatisticalanalyses to evaluate the spatial variability ofwheat yield and soil properties on complex hillsunder dry land agriculture conditions. Theyassumed that the variability of crop and soil

properties was the result of erosion. Five 400-mlong transects, each 50 m apart, were made overhills ranging in slope from 1 to 30%. Individualsite descriptions and surface soil samples weretaken every 20 m along each transect. Wheat(Triticum aestivum L. cv. Anza) was aerially sownat a rate of 135 kg.ha"!. A 1-m2 sample of wheatwas harvested at each of the 100 sites and theabove ground biomass measured. Although a highcorrelation between percent slope and surface soilproperties is generally expected, this was not thecase. Neither were any simple relationships foundbetween wheat growth and soil properties.Standard regression analysis for above groundwheat biomass versus percentage clay in the soilsurface showed no relationship. On the otherhand, when the spatial locations of these sameobservations were included, a strong spatialdependency exists. Cross-variograms, calculatedusing geostatistical methods, showed well-defineddependencies with the average range of influenceof both crop and soil properties roughlycorresponding to the diameter of the hills. Cross-variograms are proving useful in determiningspatial correlations of other field measuredproperties. For example, Nielsen & Alemi (1989)used cross-variograms to relate cotton yield tonematode infestation in a 40 ha field. They alsoexamined the spatial covariance structure ofnematode infestation at different times during thegrowing season and the competition betweendifferent kinds of nematodes within the field.Such relations were not evident using classicalregression techniques.

Spatio-Temporal Development of Disease andPest Damage of Field Crops. Fanners must notonly grow a crop, but they must protect it fromdisease and pest damage. Small plot designsprovide little help for farmers wishing to monitorthe origin and spread of a disease or pestpopulation within their fields. Research hasshown the importance of analyzing the spatial andtemporal distributions of pathogenic organisms orpests to understand their ecology andepidemiology. Similarly, the behavior,distribution and foraging of beneficial insects andpredators should be better understood forimproving agricultural production. Samplingpatterns in a fanner's field may lack order andregularity both in space and time. Where andwhen should observations be taken to gain themost knowledge about potential crop damage andto select optimum management alternatives for its

minimization? Answers to these questions andrelated issues are far from resolution in theacademic community. Statistical modeling ofobservations varying in both space and time as afunction of scale was the subject of a symposiumin the 1996 annual meetings of the AmericanSociety of Agronomy. Concepts and proceduresfor analyzing such data from farmers' fields areonly beginning to emerge. Here we give only oneexample of such research.

Spatial and temporal data on theincidence of downy mildew of red cabbagedeveloped from a point source in a field plot wasanalyzed with a geostatistical model (Stein et al.,1994). The objectives of the research includedmodeling and predicting the spatial pattern of thedisease at any time, developing sampling schemesfor future assessment and determining the initiallocation and spreading rate of the disease. Thefield plot consisted of plants in a 0.5 m2 gridplanted on 15 May, 1991 with rows oriented in anortheasterly direction. One plant inoculated andinfected with the downy mildew pathogen(Peronosora parasítica) was placed in the centerof the plot on 29 May and removed on 3 June.Observations of diseased plants were madeperiodically for more than two months. On thefirst observation date (5 June), the red cabbagecrop had a disease incidence of 7%. The patternsof infection are shown in Fig. 1 for differenttimes. By 29 July, the disease incidence hadincreased to 95%. By 12 August, the disease haddeclined to 50%. Disease incidence was high nearthe source and the spatial pattern of the diseasedplants was not stationary over time.

Spectral and cospectral analyses arepotentially powerful tools for managing andincreasing our knowledge of land resources. Withthem, we can spatially link observations ofdifferent physical, chemical, and biologicalphenomena. We can identify the existence andpersistence of cyclic patterns across thelandscape. In some cases, the cyclic behavior ofsoil attributes may be of more or equalimportance than the average behavior. From aspectral analysis, some insights may be gainedrelative to the distances over which a meaningfulaverage should be calculated. And, with spectralanalyses, it is possible to filter out trends across afield to examine local variations more closely, orvice versa. Examples will illustrate the utility ofspectral and cospectral analyses for improvingfield technology without dwelling on fundamen-tal theoretical assumptions

Spatial semivariograms for each of thepatterns of Fig. 1 are given in Fig. 2. The smallrelative nugget for the period from 5 to 25 Juneindicates strong spatial dependence with limitedrandom variation. The nugget was higher on 18and 25 June and associated with high sill valueson 10, 18 and 25 June. This behavior reflects thefavorable weather conditions (cool rain, highhumidity and prolonged periods of leaf wetness)which stimulated a rapid disease spread resultingin increased spatial dependence. With thesemivariograms being nonisotropic during June,the data showed that the disease spread mainlyalong the southwest-northeast diagonal associatedwith the prevailing winds. Spatial variabilitydecreased by 1 July and remained low for 5 weeksbecause the plot was nearly completely diseased.During this period the semivariograms were flatshowing a pure nugget and absence of spatialdependence. Defoliation by loss of older leavesled to a decline in disease incidence duringAugust. The distribution of healthy plants wasrandom on 5 August and spatial dependenceoccurring on 12 August showed a clustering ofrecovering plants during warm and dry days. Aperusal of their original paper will reveal howStein et al. used space-time kriging to predict theincidence of disease within the plot and toestimate optimal sampling schemes.

Many agricultural and environmentalphenomena exhibit spatial patterns ofdevelopment during the growing season or overthe course of time like those of the downymildew. They transcend regions within andamong farmers' fields. As technology improves,we expect that spatio-temporal analyses willbecome an integral part of agriculture.

Kriging & Cokriging: Derived information onspatial variability in the form of g(h) can beadvantageously used for estimating a soil propertyat locations where it is not measured. Kriging is aweighted interpolation named after the geologicalprospector D.G. Krige. If A(x1) and A(X2) aremeasured values of A at locations xi and x2,respectively, we seek an unbiased estimate of Ain between x1 and x2. We interpolate with weightsfx and v for each of the positions xi and x2. Valuesof m, and v depend upon the covariance function orthe semivariogram as well as upon the location ofthe interpolated value. Note that the weights donot depend upon the actual values of A. The

kriging variance or the minimum square error s2is a measure of the precision of the interpolatedvalue. With kriging, additional optimal locationsof sampling can be gained inasmuch as the krigedisolines depict more objectively the district ofsoils than an interpolation done by eye, or bylinear interpolation between the measured data.The results of kriging depend upon the fittedsemivariogram and can be easily validated using a"jack-knifing" procedure. The computationalprocedure is readily available in the literature(e.g. Journel & Huijbregts, 1978; and Webster,1985). An oral discussion of future researchutilizing kriging and cokriging will be presentedat the conference.

APPLIED TIME SERIES AND STATE-SPACE ANALYSES

Spectral and Cospectral Analyses: Anopportunity to discern repetitious irregularities orcyclic patterns in soil or plant communities acrossa field exists with spectral analysis that utilizesthe autocorrelation function r (h), or withcospectral analysis that utilized the crosscorrelation function rc(h). A spectral analysisidentifies periodicities and can be calculated by

where f is the frequency equal to p-1 and p is thespatial period. In a similar manner, rc(h) is usedto partition the total covariance for two sets ofobservations across a field. A cospectral analysisis made by

where:

Spectral and cospectral analyses arepotentially powerful tools for managing andincreasing our knowledge of land resources. Withthem, we can spatially link observations ofdifferent physical, chemical, and biologicalphenomena. We can identify the existence andpersistence of cyclic patterns across thelandscape. In some cases, the cyclic behaviorof soil attributes may be of more or equal

importance than the average behavior. From aspectral analysis, some insights may be gainedrelative to the distances over which a meaningfulaverage should be calculated. And, with spectralanalyses, it is possible to filter out trends across afield to examine local variations more closely, orvice versa. Examples will illustrate the utility ofspectral and cospectral analyses for improvingfield technology without dwelling on fundamentaltheoretical assumptions.

State-Space Analysis: Thirty-seven years agoKalman (1960) introduced the use of a linearestimation theory with stochastic regressioncoefficients to filter noisy electrical data in orderto extract a clear signal or to predict ahead of thelast observation. Kalman filtering iscommonplace in electrical and mechanicalengineering. Shumway & Stoffer (1982) extendedthe concept of Kalman filtering to smooth andforecast relatively short and non stationaryeconomic series observed in time. Using thisstate-space model, a series of observations can besmoothed, missing observations can be estimated,and values outside the domain of observation canbe predicted. For the purpose of prediction onecan calculate confidence intervals to demonstratethe reliability of the forecasting. Reliability of theprediction will depend on the adequacy of theavailable model and number of predictionsoutside the domain of observations. Morkoc et al.(1985) utilized the state-space model to analyzespatial variations of surface soil water content andsoil surface temperature as a dynamic bivariatesystem. Following the suggestion of Shumway &Stoffer (1982), Morkoc et al. (1985) described thevariables of the observation model as

where H is a known q x p observation matrixwhich expresses the pattern and converts theunobserved stochastic vector Z, into qx observedseries gi and vi is the observation noise or measu-rement error. Equation (21) written in matrixform for the bivariate system of Morkoc et al. is

where Wi0 and Ti

0 are the ith soil water content

and the ith soil surface temperature observations,respectively. Equation (14) indicates that anobservation of either soil variable consists of twoparts. The observational noise or measurementerrors may be generated either by errors in

measuring Wi0 or Ti

0 or by ignoring other

variables which affect either soil water content orsoil surface temperature. For example, soiltemperature is not only affected by incoming solarradiation but also by mineralógica! composition,microtopography, texture, structure and color ofthe soil surface. Hence, any omitted variablessuch as texture or color enter into the state spacesystem as observational noise or measurementerror vi.

Values of soil water content and surfacetemperature were modeled as a firstordermulti variate process of the form

where f is a p x p matrix of state-spacecoefficients. Many other functions of multivariateprocesses equally applicable to site-specific cropmanagement await future investigation. Thematrix form of the above equation is

Values of fij and the error terms of theabove equation can be calculated using recursiveprocedures (Shumway & Stoffer, 1982).

Details and the results of the state-spaceanalysis of water content and temperaturerelations of a soil surface (without vegetation andhaving been nonuniformly sprinkler irrigated) areavailable (Morkoc et al, 1985). More directlyrelevant to the topic of this conference is theapplication of equations similar to (21) and (23)to improve our understanding and management ofbiological processes as they occur in the field andto prioritize those soil attributes that contributethe most to crop productionand water quality.

Physically Based State-Space Analyses: Webelieve the continued success of soil physicists tolift and accelerate research into the nextmillennium depends upon the development of

new technologies for simultaneously examining(1) alternative formulations of theoreticalequations to describe soil and plant processes, (2)alternative functions for parameters contained inthose equations, (3) alternative frequencies ofspatial and temporal measurements to match thetheoretical considerations in (1) and (2), and (4)the adequacy of accepting different levels ofuncertainty always inherent in instruments andtheir calibration. Theoretical descriptions toooften match the narrow academic specialty of thescientist rather than an on-site assessment of aspectrum of local processes having various levelsof importance that impact a particular fanner'sfield. Justification for the adoption of functionsdescriptive of soils and crops generally remainsinadequate. Optimal frequencies in time andspace for making observations of soil and cropattributes remain unknown for most agronomists.This puzzle persists owing to our inadequatecharacterization within fields of spatial andtemporal variances which embrace theuncertainties of our instrumentation and those ofsoil taxonomy.

With the availability of new instrumentsand alternative spatial and temporal statisticalmethods to be used in farmers' fields underpotentially known local conditions, soil scientistsare poised for accelerated progress. State-spacemodels can be used to simultaneously examine atheoretical equation, its empirical parameters, andfield measurements which embrace theuncertainties of soil heterogeneity and instrumentcalibration. From a presentation (Nielsen et al.1994) of the details showing how to formulatestate space equations for a vertical soil profile, weillustrate how to compute evaporation betweenirrigation events as well as to determine adiffusivity function from field-measured soilwater profiles. The details of the experimentalsetup have been fully described by Parlange et al.(1993). Five neutron access tubes positionedevery 18 m along a transect allowed soil watercontent to be monitored with a neutron probe at15 cm depth intervals within the soil profile ateach location. The level site was free ofvegetation and equipped with a sprinklerirrigation system that was used to apply 15 smallirrigations (each <20 mm) during a 3-monthperiod. In addition to the neutron probemeasurements of soil water to estimate evapora-

tion, 20 min weighings of a 50 t capacitylysimeter in the same field were integrated toobtain daily values of evaporation.

The spatially averaged amounts of waterstored within the five depth intervals of the soilprofile measured with the neutron probe as afunction of time together with the amounts ofwater applied with the sprinklers are shown inFig. 3. Evaporation and infiltration from theapplied water are the primary physical processesthat create changes in stored water and thosechanges occur primarily in the 0-22.5 cm depth.Variations in soil water stored in the top 22.5 cmof the profile as a function of time as well aslocation are shown in Fig. 4.

The hydrologic balance in the absenceof lateral flow and negligible drainage is

where S is the depth of stored water between thesoil surface (z = 0) and some depth b (b is thedepth assumed to be uniformly wet), E, theevaporation rate and F the rate at which water isapplies to the soil surface (precipitation orirrigation). The evaporation rate can be calculatedusing the diffusion equation (assuming isothermaland homogeneous soil conditions)

where D(q) is the soil water diffusivity which isa highly nonlinear function of q. An approximatesolution of (26) is

where the diffusivity is a function of stored waterS. Combining (25) and (27) the hydrologicbalance becomes the ordinary differentialequation

For an exponential diffusivity functionD(S / b) = A exp(BS / b), the state-space equationfor (28) is

where Xt is the state variable representing thestored water S and vsdt is a stochastic noise owingto the uncertainties in the proposed equation. Thenoise vsdt arises owing to the various simplifyingassumptions of the physical model such asneglecting thermal and salinity effects, swellingand shrinkage, hysteresis, etc. The correspondingobservation equation at time tk (k = 1,2,3,) is

where Zm is the observed amount of stored waterat time tk and vm is the observation noise.

The diffusivity parameters A and Bwere calculated assuming that the observationvariance R' can be estimated from spatiallyaveraging the neutron probe readings from thefive access tubes located 18 m apart. It can beseen in Fig. 5 that the observation variance R' ismuch larger than the neutron probe calibrationvariance. Values of A and B were estimated to be0.0292 mm2 d-1 and 32.59, respectively, with thestate variance per unit time Q being 18.87 mm2.d-1 . The value of Q is of the same order ofmagnitude as the neutron probe calibration

variance, while the model uncertainty for storedwater prediction on a daily basis is within theneutron probe noise. The agreement foundbetween the state space estimation of cumulativeevaporation and that measured with the lysimeter(r2=0.98) indicates that the estimated diffusivityfunction provides an adequate description ofevaporation and the soil water transport process.The usefulness of these models and similarapproaches lies in their capability to incorporatethe uncertainty in both the model and statemeasurements. Here the advantage over mostother field methods is that it is acknowledged thatthe model is only approximate and hence,contains an explicitly identified model error. Bysolving and examining the model error orvariance, realistic improvements in the model canbe achieved. Another feature of the state-spaceapproach is the inclusion of an explicitobservation error which can be treated as a knownand measured quantity, or alternatively, treated asan unknown for which a solution is found. Themagnitude of a known observation error allows areconsideration of the state variable in theequation or an improvement in instrumentation orcalibration. On the other hand, treating theobservation error as an unknown, its behavior inspace and time can be related to spatial andtemporal correlation lengths that may manifestthemselves within the domain of the field beingstudied. We also expect that progress will bemade using time-averaged equations to examinecritical periods during which soil processes occur,these soil processes may be related to particularsoil locations, mapping units or regions. A largearray of combinations of state and observationvariables together with different functions forparameter estimation used in a state-spaceapproach provides attractive opportunities toenhance soil physics research.

SCALING CONCEPTS IN SOIL SCIENCE

In 1955, Miller & Miller created a newavenue for research in soil physics when theypresented their pioneering concepts for scalingcapillary flow phenomena. Their description ofself-similar microscopic soil particle structure andits implications for the retention and transport ofsoil water stimulated many studies to test howwell laboratory-measured soil water retentioncurves could be coalesced into a single scale

mean function. Because the results of ensuingtests were not particularly encouraging except forsoils composed of graded sands, their scalingconcepts lay idle for several years. During the1960s, the development and accepted use of theportable neutron meter to measure soil watercontent spurned research on field-measured soilwater properties. With its availability combinedwith the well-known technology of tensiometry,field studies of soil water behavior wereaccelerated in the 1970s. However, soil physicistswere soon faced with a dilemma - how to dealwith the naturally occurring variability of fieldsoils (Nielsen et al, 1973) and concomitantlymeasure within reliably prescribed fiducial limits,the much needed soil water functions associatedwith the Darcy-Buckingham equation and that ofRichards. Extending the concepts of Millerscaling was thought to be a promising answer.With many different kinds of invasive and non-invasive techniques available today to measuresoil water and related soil properties (Hopmans,Hendricks, & Selker 1997), scaling opportunitiescontinue to appear both promising andprovoking.

Miller & Miller Scaling: Scale-invariantrelationships for water properties of homogeneoussoils based upon the microscopic arrangement oftheir soil particles and the viscous flow of waterwithin their pores was proposed by Miller &Miller (1955a, b). Each soil was assumed to becharacterized by a soil water retention curve0 (h) where 0 is the volumetric soil watercontent and h the soil water pressure head.Through the law of capillarity, the value of h for aparticular 0 is related to a function of r-1 where ris the effective radius of the largest soil poresremaining filled with water. According to Miller& Miller, two soils or porous media are similarwhen scale factors exist which will transform thebehavior of one of the porous media to that of theother. Fig. 6 illustrates their concept of self-similar microscopic soil particle structure for twosoils. The relative size of each of thegeometrically identical particles is defined by theparticular value of the microscopic scale length litThis kind of similarity leads to the constantrelation r1/l1 = r2/l2 = r3/l3 = ... = ri/li and to theformulation of a scaled, invariant soil waterpressure head h* such that

where h* is the scale mean pressure head and l*the mean scale length. Dividing each scale lengthby the mean scale length reduces (31) to

where ai are the scale factors having a meanvalue of unity. The hydraulic conductivityfunction K(q) which relates the soil-water fluxdensity to the force acting on the soil water isanalogously scaled

where K* is the scale mean hydraulicconductivity function. Written in terms of scalefactors ai (33) becomes

Note that the scale length li has aphysical interpretation and that the porosity ofeach soil is assumed identical. A constant porosityacross "similar" soils is an important assumptionmade in this approach.

Early Attempts to Scale Field-Measured SoilWater Properties: Scaling field-measuredfunctions K(q) and q (h) was based upon theassumption that a field soil is ensemble ofmutually similar homogeneous domains. Owing tothe fact that the total porosity of a field soil ishighly variable even within a given soil mappingunit, Warrick, Mullen, and Nielsen (1977) foundit necessary to modify the restrictive, constantporosity microscopic scaling concept of Miller &Miller. By introducing the degree of watersaturation s (=q.q S

- 1 ) with qs becoming asecond scaling factor, they provided a morerealistic description of field soils by relaxing theconstraint of constant porosity. Moreover, theyavoided a search for a microscopic physical lengthby merely deriving values of a that minimized thesum of squares

for N macroscopic locations within a field soiland M observations of h. For example, with thisminimization, 840 measurements of (q, h)[samples taken at 6 soil depths and 20 sites (N =120) within an agricultural field and analyzed inthe laboratory with 7 values of h(M=7] shownin Fig. 7a as h(s) were coalesced into the singlecurve

in Fig. 7b Warrick et al (1977). The 2640 valuesof (K, q) stemming from field measurementsanalyzed by the instantaneous profile method for6 soil depths and 20 locations shown in Fig. 8awere coalesced and described by the regressionexpression

as shown in Fig. 8b. Although Warrick et al.(1977) abandoned the microscopic geometricalsimilarity concept of Miller & Miller (1955a, b)and based their scaling method on the similaritybetween soil hydraulic functions, they noted thatvalues of ar required for scaling h in (36) werenot equal to those for scaling K in (37).

During the next decade, several othersattempted to scale field-measured hydraulicproperties (e.g. Ahuja et al., 1984a; Ahuja et al,1989b; Hills et al,. 1989). Rao et al. (1983) aswell as others found that scale factors thatcoalesced field-measured functions of K(q)differed from those that coalesced fieldmeasuredfunctions of q (h).

Future Research: We expect avenues ofintellectual curiosity supported or derived fromobservations in the field and the laboratory tocontinue to kindle investigations of scaling soilwater regimes. Because potential avenues for thedevelopment of a comprehensive set of differentkinds of scaling theories remain largelyunexplored, opportunities to quantitativelyascertain the efficacy of scaling field soil waterregimes must await additional inquiry andcreativity. Without a unified comprehensivetheory, fragmented, theoretical considerationsprovide inadequate criteria for success.

We do not anticipate abundant progressuntil a complete set of field-measured soil waterproperties for several locations within at least onefield is simultaneously and directly observed,analyzed and published. To date, in everyreference cited or omitted in this chapter, criticalfield measurements have been lacking. Forexample, Nielsen et al (1973) and Shouse et al.(1992a, b) estimated field soil water contentsfrom laboratory measurements of q (h) on soilcores. Russo & Bresler (1980b) estimatedfunctions q(h) and K(q) from fieldobservations of sorptivity and other parameters.Ahuja & Williams (1991) and Rockhold et al.(1996) also estimated field values of q (h) frommeasurements on soil cores analyzed in thelaboratory. Although Eching et al. (1994)measured B(z, t) in the field with a neutronmeter, they made no observations of h(z, t).Moreover, they made no independentconfirmation that the functional relations assumedfor the hydraulic properties in the inversemethod were descriptive of the field soil studied.On the other hand, both Eching et al (1994) and

Rockhold et al. (1996) explicitly showed thatvalues of Ks measured in the laboratory on soilcores were different than those measured in situ.

Progress toward improved scalingconcepts should be accelerated as investigatorstake the opportunity to simultaneously studydetails of both an experiment and a theory (e.g.Flühler et al 1976). Present-day scaling attemptsare confounded by not recognizing that mostexperimental observations are subject to space-and time-dependent instrumental responses(Baveye & Sposito 1985). And, more attentionshould be given to the consequences of selectingsimplified theoretical models to analyze and scalefield-measured data (Tseng & Jury 1993).Presently, no criteria are established to ascertainappropriate soil depth or time intervals at whichobservations should be taken. The choice ofhorizontal spacings between observations remainsad hoc. Functional forms of soil hydraulicproperties remain without theoretical foundation.Indeed, a dilemma persists regarding how toinclude "preferential" flow near water saturation.If Ks is dominated by "preferential" flow, shouldthe relative hydraulic conductivity functionK^Ks'1 be scaled (Jury et al 1987; Ahuja &Williams 1991) or should that "preferential" flowbe described by equations other than that inRichards' equation (e.g. Germann & Beven 1985)and scaled independently? When and how shouldlaboratory studies complement fieldinvestigations? Equipment and methods forascertaining the essential observations are readilyavailable to those wishing to make a contributionto the development of scaling technology.Paradigms for scaling steady-state, one directionalBuckingham-Darcy flow are anticipated to be lessrestrictive than those for Richards' equationdescribing transient flow in one or moredirections.

We believe information derived fromlaboratory investigations at the soil pore scaleobtained with computed microtomography,magnetic resonance imaging and othernoninvasive techniques will improve the use offractal concepts by Tyler & Wheatcraft (1990) todescribe O(h) and by Shepard (1993) to calculateK(~). The logical next step based on fractalswould extend the descriptions and calculations toa field scale as other fractal properties andprocesses within field soils become better knownand understood (Burrough 1983a, b).

Eventually, appropriate scale factors offield-measured soil water properties andprocesses will be measured in sufficient quantityand detail to analyze and document their spatialand temporal statistical variance structures acrossand within the landscape. With their values beinglinked to other soil properties through statespaceand other regionalized variable analyses (e.g.Wendroth et al, 1993), we anticipate that newparadigms for local and regional scales ofhomogeneity in pedology and soil classificationwill emerge. With soil mapping units embracingmagnitudes and distributions of spatial andtemporal soil-water scale factors, unlimitedopportunities will unfold. We expect thenumerous uniquely scaled solutions ofBuckingham-Darcy and Richards' equations nowonly theoretically available (e.g. Kutilek et al1991; Warrick & Hussen 1993; Nachabe 1996) tobe extended to specific landscape and fieldregions categorized by mapping units describedby information containing scale factors for theirsoil water properties.

SOIL CLASSIFICATION ALTERNATIVES

Presently, there is a debate emergingamongst soil scientists regarding conventional andcontinuous methods of soil classification.Conventional classification methods establish aseries of subdivisions which place individual soilprofile descriptions into a hierarchical schemesuch as that of Soil Taxonomy of the USDA (SoilSurvey Staff, 1975). Such schemes prevalent inmany countries, methodically constructed anddeveloped during the past half century, rely on amodal (most frequently expected) soil profile torepresent the land unit. The precision of anyprediction using this classification is dependenton the homogeneity of the mapping units andhence on the within unit spatial variance that istypically not ascertained (Trangmar et al, 1985).And without a measure of the spatial variancestructure within each mapping unit, little isknown regarding the reliability of the modal soilprofile to represent the mapping unit (McBratney& Webster, 1981). Data from measuring field soilproperties generally exhibit both short and longrange variations, are highly irregular and aremultivariate. Various methods for obtainingoptimal sampling strategies for mapping soiltypes according to hierarchical schemes based

upon spatial distribution functions are available(e.g. Webster, 1973). On the other hand, thehierarchical classification of Soil Taxonomy doesnot account for the gradational nature of the soilcontinuum when crossing a soil mapping unitboundary. Hence, proponents of continuousmethods of soil classification argue for a soilclassification system which accounts for thecontinuous nature of soils in both vertical andhorizontal directions. The method of fuzzy k-means analysis is a technique capable ofaccounting for the continuous nature of the soil(McBratney & DeGruijter, 1992) and allows anindividual soil profile to belong totally, partiallyor not at all to a particular class (Zadeh, 1965). Infuzzy or continuous classification, soils beingclassified may well be a member, to a greater orlesser degree, of every class (Bezdek, 1981), andaccording to this definition, fuzzy classes are ageneralization of discrete classifications (Odeh etal., 1990).

For either classification method -conventional or continuous - there remains noconsensus as regards the choice of the statevariables used to describe the soil profiles or amethod to discern them. Here we suggest thatthose variables initially perceived as beingdescriptive of a soil (and its chemical, physicaland biological properties) be observed andanalyzed in state-space models similar to (24). Byexamining the relative values of fi, such aprocedure would help identify the choice of thedistance-dependent metrics or state variables aswell as the development of functions of the statevariables that could be used to describe soilswithin mapping units of either classificationmethod. Covariance and cokriging of such statevariables within the domain of a mapping unit ofeither classification method would beimmediately helpful to soil physics as well as toother earth sciences.

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Enviado para publicação em 30.04.97Aceito para publicação em 05.05.97