WATER RESOURCES RESEARCH, VOL. 30, NO. 2, PAGES 237-251, FEBRUARY 1994

Spatial variability of residual nitrate-nitrogen under two tillagesystems in central Iowa: A composite three-dimensionalresistant and exploratory approach

B. P. Mohantyl and R. S. KanwarDepartment of Agricultural and Biosystems Engineering, Iowa State University, Ames

Abstract. Soil nitrate-nitrogen (N03-N) data arranged on a three-dimensional gridwere analyzed to compare tillage effect on the spatial distribution of residual N03-N inthe soil profile of agricultural plots drained by subsurface tiles. A three-dimensionalmedian-based resistant (to outlier(s)) approach was developed to polish the spatiallylocated data on soil NO,-N affected by directional trends (nonstationarity in the mean)in three major directions (row, column, and depth) and along the horizontal diagonaldirections of the grid. Effect of preferential or nonpreferential path of transport ofN03-N in the vertical direction defined as sample hole effect was’also removed tomake the data trend-free across holes. Composite three-dimensional semivariogrammodels (along horizontal and vertical directions) were used to describe the spatialstructure of residual soil nitrate distribution. Two plots in the same field, one undereach tillage system (conventional tillage and no tillage), were studied. In each plot, soilsamples were collected at five depths (30, 60, 90, 120, and 150 cm) from 35 sites (holes)arranged on a 7 x 5 regular grid of 7.6 X 7.6 m. In the conventional tillage plot,residual N03-N concentrations decreased gradually to a depth of 90 cm and increasedbeyond this depth. The coefficient of variation, however, became gradually smaller,showing more uniform distribution for greater depths. In the no-tillage plot, trendswere similar to those in the conventional tillage system, but were spatially more stableacross the profile. Structural analyses indicated that under conventional tillage, thesemivariogram of residual soil nitrate distribution was linear in the horizontal andvertical directions. In contrast, the semivariograms for no-tillage showed nugget-typebehavior, indicating a lack of spatial structure in the residual soil nitrate.

Introduction

In recent years, increased public concern about waterquality has heightened interest in the efficient use of nitrogenfor agricultural production systems. Studies have indicatedthat nitrogen application rates in agricultural fields are oftenmuch higher than the crops can use. Therefore the unusednitrogen either leaves the system through leaching, washoff,and volatilization or remains in the soil profile for possibleleaching to groundwater. Knowledge of spatial structuresand other inducing factors for residual nitrate distribution inthe soil profile under different tillage practices is useful tounderstand its transport, transformation, and retention pat-tern. This will help researchers developing new tillage-basednitrogen management practices to enhance groundwaterquality.

Transport, transformation, and distribution of nitrate-nitrogen (NO,-N) under different field conditions dependupon physical, biological, and chemical processes occurringin the soil profile. Other factors responsible for the spatialdistribution of N03-N are water and solute displacementunder nonsteady flow conditions along preferential and

‘Now at U.S. Salinity Laboratory, Agricultural Research Ser-vice, U.S. Department of Agriculture, Riverside, California.

Copyright 1994 by the American Geophysical Union.

Paper number 93WR02922.0043- 1397/94/93 WR-02922$05.00

nonpreferential flow paths, variable water table depth caus-ing nitrification and denitrification, nitrogen transformationsby enzymatic and bacterial pathways, plant uptake of waterand nitrogen, and soil texture [Wagenet and Rao, 19831.Studies conducted by Burrough [1983], Wagenet and Rao[1983], and Tabor et al. [1985] indicate that each of thesefactors may operate either independently or in conjunctionwith others to cause abrupt or gradual changes in nitrogentransport and retention. These factors must be consideredwhen studying the spatial behavior of residual NOj-N in thesoil profile. Several studies [Walker and Brown, 1983; Rao etal., 1983, 1985; Rao and Wagenet, 19851 have examined thespatial variability and its intrinsic factors influencing herbi-cide degradation rates in the soil profile. Villeneuve et al.[1988] pointed out the intrinsic and extrinsic variability ofsoil chemical properties. For instance, the spatial variabilityof soil organic matter is closely linked to the variability ofsoil composition and structure, which derives from soilpedogenesis. Spatial variability of a pesticide concentrationor flux in soil, which is due to cultivation and pesticideapplication practices, leads to a variation in the adsorptionand degradation parameters.

Macropores are responsible for solute movement [Bevenand Germann, 1982]. Trojan and Linden [1992] report thatmacropore distribution (spatially and along depth), length(vertically), continuity (vertically), and tortuosity (vertical-ly) are functions of depth and affect the spatial variability ofresidual/sorbed solutes (e.g., pesticides). Singh et al. [1991]

237

.238 MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN

showed the tillage-induced differences in spatial and verticaldistribution, length, and continuity of macropores underconventional and no-tillage practices. Many researchers[Gust et al., 1978; Baker et al., 1975; Baker et al., 1989;Kanwar et al., 1985, 1988; Kanwar and Baker, 1991; Kan-war, 1991] have studied the accumulation of NO,-N in soilprofiles and NO,-N loss with tile drainage water undercontinuous corn as a function of different tillage practices,but spatial structure and variability of residual N03-N inthese fields have not been studied.

The objective of this study was to compare the spatialdistribution of residual NO,-N in the (tile) drained soilprofile under conventional tillage and no-tillage systems, byusing a three-dimensional median-based resistant (little af-fected by data outlier(s)) approach coupled with geostatisticsfor the data collected on a three-dimensional spatial grid.The paper also demonstrates how three-dimensional griddeddata give a clearer picture of spatial structure (compositethree-dimensional semivariograms) of residual N03-N dis-tribution in the soil profile under different tillage practicesthan do two-dimensional gridded data producing two-dimensional semivariograms. Moreover, it shows how three-dimensional median polish of gridded spatial data provides arelatively simple, cost-effective, resistant, and almost bias-free procedure in the presence of trend and zonal anisotropywhen compared with more general polynomial modeling andgeneralized least squares fitting of trend.

Previous ApplicationsTransport and retention capacities for water and chemi-

cals in an agricultural field are spatially and temporallyvariable. Spatial variability, as used here, includes area1variations at a given depth and also variations with depth ata given location in the field. Both types of variability need tobe considered when assessing the fate of NO,-N underdifferent field conditions. Exploratory data analysis, relyingon resistant measures and graphical tools, and robustnessconcepts in geostatistics, offer a variety of ways to modelsuch processes as realizations of space-time random func-tions. Dagan [1986], Ginn and Cushman [1990], Rouhaniand Wackernagel [1990], Shafer and Varljen [1990], Rubinand Journel [1991], and others have successfully appliedcovariance-related stochastic methods in groundwater mod-eling. Further efforts to develop better spatial characteriza-tion methods of water and chemical transport and retentionproperties seem warranted, not only in the saturated zonebut also for extending these methods to transport modelingin the vadose zone. However, few studies [Russo, 1984;Unlu et al., 1990] of the spatial variability of unsaturatedflow parameters have concentrated on quantifying heteroge-neity in the horizontal plane of fields. A recent study byRusso and Bouton [ 1992] focused on the spatial variability ofthese flow parameters in the depth direction. Cressie andHorton [ 1987] and Onofiok [ 1988] studied the tillage effect onsoil water infiltration and other soil properties and observedsome interesting differences.

Sweeping the localized effects by exploratory approaches[Tukey, 1977: Velleman and Hoaglin, 1981] such as row andcolumn median polish [ffamlett et al., 1986; Cressie andHorton. 1987]. winsorization [Gotway and Cressie, 1990].and split-window median polish [Mohanty et al., 1991]coupled with classical geostatistics [Matheron, 1963] or

robust geostatistics [Cressie and Hawkins, 1980] is becomingpopular in soil science. Among others, one important advan-tage of these approaches is that they allow visual examina-tion of data for normality, constancy in the mean and thevariance, and trend freeness. Trends are often unverified bygeologists and soil scientists, who confuse inherent spatialstructure with the effects of extrinsic factors and drift[Horowitz and Hillel, 1983; Hamlett et al., 1986]. In thepresent study, data arranged on a three-dimensional grid andresistant (to outlier(s)) schemes were investigated with the“three-way main effects only” median polish approach[Cook, 1985], an extension of “two-way” median polish[Tukey, 1977; Velleman and Hoaglin, 1981; Emerson andHoaglin, 1983]. This approach removes trend due to croprow effects and/or other effects (parallel to grid rows),drainage tile induced water table effects at deeper depthsand/or other effects at shallow depths (parallel to gridcolumns), and surface/subsurface soil characteristics effectsand/or soil water content effects (in terms of advection anddispersion) (parallel to grid depth) from the data beforespatial analyses of residual NOJ-N in the soil profile weremade to compare distribution patterns under conventionaltillage with those under no tillage.

Experimental DesignTwo 0.4-ha (1 acre) plots, each under a different tillage

system (conventional tillage and no tillage), were selected atthe Agronomy and Agricultural Engineering’ Research Cen-ter near Ames, Iowa. The plots had been continuouslyplanted to corn under the same tillage system for 5 years andhad received an application of 175 kg/ha liquid nitrogenfertilizer every year. For both plots, a single application(soon after planting) of nitrogen was adopted. Soils of theseplots were Nicollet loam (fine loamy, mixed, mesic aquicHapludolls) and Clarion loam (fine loamy, mixed, mesictypic Hapludolls) in the Nicollet-Clarion-Webster Associa-tion and had a maximum slope of 2%. Each plot was drainedby a single subsurface tile drain at 120 cm depth to eliminateperiodic buildup of excessive wetness.

Samples were collected from both plots by using a 5 X 7grid network of 35 sites, with a regular grid spacing of 7.6 m.Sampling scheme was limited to plot boundary in horizontaldirections and close to subsurface tile in the vertical direc-tion. Soil samples were taken to a depth of 150 cm a t 30-cmintervals with a 3.2-cm probe. There were 175 data points intotal. Figure 1 shows the location of sampling points withinthe plot arranged on a three-dimensional grid. Under thepractical geostatistical rule for the minimum number of pairsfor semivariogram estimation [Journel and Huijbregts, 1978;N. A. C. Cressie, personal communication, 1992], our num-ber of samples is considered adequate. The labor and budgetrequired in conducting a study on NO,-N transport andretention in soil profile were also considered in determiningthe number of samples. Each soil sample represents averageN03-N over 10 cm depth at a particular spatial location(x, y, z). A sample of 10-cm length, however, was consid-ered a point sample for the two- and one-dimensional typesemivariogram estimation in horizontal directions. In verti-cal direction, however, regularized semivariogram models[Journel and Huijbregts, 1978, p. 801 were calculated follow-ing the theoretical model fitting, to compensate for thevertical sample length (i.e., 10 cm). Because we focused on

MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN 239

,’Y +

N-E Direction (Mb

5.6

Figure 1. Arrangement of nitrate sampling locations in the experimental plots; 35 sites on a 7 X 5 gridand 5 depths. .

the spatial behavior of residual nitrate (i.e., NO,-N retainedin the soil mostly from previous crop years), samples werecollected during the period when no appreciable rain fell tominimize the contribution of surface-applied nitrogen fertil-izer (a few days before soil sampling). This was necessarybecause of the limited advection and dispersion processesoccurring during this rain-free period. Thus the amount ofNOs-N in the soil samples indicates mostly the residualamount retained from previous years after leaching togroundwater body and tile discharge, and uptake by plants.All samples were collected within a 13-day period in June tominimize temporal variability among samples. The sampleswere analyzed for soil water content and for soil waterNOj-N concentration. Colorimetric automated Cadmiumreduction method (EPA 353.2) was used to determine theNO,-N concentration in the soil samples.

MethodologyThe objective of this study was to present simple, physi-

cally interpretable. resistant schemes for spatial analyses.The basic purpose of resistant schemes is to clean or topolish field data to satisfy the basic assumptions for theestimation of semivariograms, that is, for second-orderstationarity or the ueaker intrinsic hypothesis. Second-orderstationarity implied that the mathematical expectationE[Z(x)] = p exists and does not depend upon the positionx and that for each pair of regionalized variables [Z(X), Z(x+ h)], the covariance exists and depends only upon the

separation vector h. On the other hand, the weaker intrinsichypothesis implied that the mathematical expectationE[Z( x)] = p exists, and for all vectors h the increment [Z( x+ h) - Z(X)] has a finite variance that does not depend onx [Journel and Huijbregts, 1978, p. 32]. These basic assump-tions of regional variable studies, however, are often over-looked or rarely verified by researchers, who may introduceartifacts and confound the interpretation of results, as de-scribed in the work by Cressie and Horton [ 1987]. Thereforeit is important to run a thorough data analysis before andduring geostatistical analysis to filter out any site-specific potential problems (producing trends) related to the knownregionatized variable (i.e., soil property under investigation).

This study was conducted to identify the physically recog-nizable extrinsic factors existing on the site and to isolate theireffects in the distribution of residual N03-N in the soil profile.These effects usually produce nonstationarity in the mean andthe variance along the major axes of experimental design.Besides these main effects, the intrinsic/cross-product trendsalong the diagonal directions in a grid-type sampling patterncould also be identified and filtered out. Once spatial station-arity of the regionalized variable is attained, the measurementsover all depths are combined into a composite data set, andsemivariogram analyses are conducted. This study shows howeasily exploratory data analysis techniques could assure crucialstationarity assumptions in regionalized variable and make thedata free of extrinsic factors and intrinsic trends before furthergeostatistical analyses.

240 MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN

Exploratory Analysis

Variability in geo-related properties such as transport,transformation, and retention of N03-N is likely to bethree-dimensional in the three-dimensional space of a soilprofile. In our study, we identified the effects of crop rowparallel to the grid row, of tile-induced water table elevationsparallel to the grid column at deeper depths, and of surface/subsurface soil and soil water content parallel to the griddepth in residual NOJ-N distribution in the soil profile. Alsothe effects of (tillage-induced) pore size distribution, porelength, and their continuity were found in residual NO,-Ndistribution. Although spatial field experiments are usuallydesigned with their grids in meaningful directions, residualsshould still be checked for cross-product trends or interac-tions, e.g., between row and column factors, along thediagonals. Unlike a mining problem extending over greatdepths in which drift might occur in any direction in three-dimensional space, this study has been kept simple, anddiagonal trends have been assumed only in horizontal direc-tions because of the shallow depth (1.5 m), the limited scale(plot size, 46 m x 31 m), and the type (vertical transport ofNOs-N in the soil profile) of the problem. Scheffe [1959, p.130] and Cressie [1991, p. 190] modeled this diagonal trendthrough an extra parameter g in the additive models, whichaccounts for a quadratic term in the fit. Consequently, thisstudy, a modified three-dimensional additive model based onCook’s [I985] “three-way main effects only” model andC r e s s i e ' s [1991] diagonal drift term was developed:

Zijn=CL+ai+Pj+5n+g(Xi-~)(yj-~)+&Eijn, (1)

i= 1, .-. ) I j= 1, *-*, J; k= 1, --*, K,

where _ -

Zijk

9f

Y

&ijk

regionalized variable (residual NOj-N) of the cell (i,j, k) corresponding to row i, column j, and layer(depth) k;common value;row effect (i.e., crop row effect and/or other effect);column effect (i.e., elliptical water table effect and/or other effect);layer effect (i.e., surface/subsurface soil effect and/or soil water content effect);diagonal drift parameter;ave {Xi, i = 1 * * * I};ave {Yj,j = 1 a-*J};a random component that may or may not inherit the

under the tillage practice) in study in cell (i, j, k).spatial structure of the property (residual NO,-N

Exploratory data analyses that rely on graphing are espe-cially suitable for spatial data. Identifying the contaminatingoutlier(s), directional trends (drift), normality, and depen-dency of the mean and the variance in the data throughsuitable plots and graphs yield valuable insights into theproblem and aids analysis and interpretation. Frequencyplots and median versus interquartile range-squared plotsare used to assess data distributions and median and vari-ance (interquartile range squared) constancy or dependency.Plots of median across rows and columns can identifydirectional trends along both horizontal grid directions. Plotsof medians of horizontal layers against depth are used toidentify trend in the vertical direction. Diagonal intrinsictrends or cross-product trends of row and column effectsalong the diagonals of the grid can be examined by plottingthe residuals of row and column median polish versus thequadratic term [(Xi - ~)( Yj - y)].

Median Polish

Representing the raw (for more normally distributed) orlog,-transformed (for more lognormally distributed) data bya modified additive model will lead to an interpretation of acombined output of some stochastic random fluctuation anddeterministic physical realities existing at a particular site.The next logical step would be to remove the effects of thesephysical factors (i, j, k, and ij) one by one to leave behindthe residual (Eijk). The residual would be analyzed for theinherent spatial structure of the variable at that location. Theless formal approach to resistant methods by Cressie [ 1984,19861 was adopted. It is, however, always a concern that theremoval of the mean or the median might introduce bias into

Spatial Variability

After log, transformation and resistant schemes of medianpolish were used to remove nonstationarity (of the mean andof the variance) and any nonnormality of the data due toextrinsic factors and intrinsic trends, a geostatistical analysiswas made to study the spatial variability of residual soilnitrate concentration based on this three-dimensional ho-mogenized residual set. Figure 2b shows the steps involvedin computing the composite three-dimensional semivario-gram. Initially proposed by Myers and Journel [1990] tomodel zonal anisotropy, the composite three-dimensionalsemivariogram is primarily a combination of the averageone-dimensional horizontal semivariograms (?(/I,) in the xdirection and y(/zY) in the y direction) and the averageone-dimensional vertical semivariogram (gh,) in the z di-

the data. However, Cressie and Glonek [1984] have shownthat known bias problems in estimation of the semivario-grams are greatly ameliorated if medians instead of meansare used to define the residuals. Moreover, early studiesshowed that median polish is better than the mean because itis little affected by outlier(s), common in field data [Hoaglinet al., 1983; Cressie and Glonek, 1984; Mohanty et al., 1991].Therefore the median-polishing techniques were used topolish the data for different additive directional components.Detailed median polish algorithm for a two-way table isdefined by Emerson and Hoaglin [1983, p. 166] and Cressie[1991, p. 186]. AS in the work by Cook [1985], a medianpolish algorithm for a three-dimensional grid was establishedin this study. This polishing could be achieved either (1) withthe algorithm that successively removes the row and columnmedians until there is no further change in residuals for eachhorizontal layer and then removes the layer median or (2)with the algorithm that successively removes the row, col-umn, and layer medians until no further change in residuals.Whatever route we took, however, the residuals remain thesame at end of the median polishing scheme. A schematicflowchart (Figure 2) illustrates the approach adopted forexploratory data analyses and semivariogram estimation. Insummary, a ladder of median-polishing algorithms has beendeveloped to iterate the procedure until the mean and thevariance along rows, columns, diagonals, and layers attainconstancy (i.e., independent of each other) as well as thehistogram of residuals looks normal.

Data Check

MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN 241

ro(i=

YES

REPEAT THE STEPS FOREVERY INDIVI UAL LAYER

tCHECK TRENDALONG DEPTH I

I

YES

OF RESIDUALS OF EARLIER STEPS

Figure 2a. Schematic flow chart for the three-dimensional resistant median-polishing scheme.

rection) in their respective lag scales, according to thethree-dimensional homoscedastic data (residuals). The semi-variogram models constructed in this manner will result inconditionally nonnegative definite models and as a conse-quence to noninvertible kriging matrices. However, nonne-gative definite models are sufficient conditions in this appli-cation as we did not use semivariogram models for krigingpurposes. This three-dimensional spatial variability phenom-enon can also be interpreted as geometric anisotropy in threedirections (i.e., x, y, and 2).

Composite Three-Dimensional Semivariogram

Define

y(h,, $9 hJ = Y(h,) + Y(h,) + YW,). (2)

In addition, two-dimensional horizontal isotropic samplesemivariograms (y*(h,, h,)) were computed for comparisonpurposes.

The classical semivariogram estimator as developed by

Mutheron [1963] was used to estimate two- and one-dimensional semivariograms:

NW;)y*(hi) = 1/2N(hi) C [Z(X ) - Z(X + h;)J*

where

Y*(h)Z(x)

Z(X + h i )N(hi)

I i=l

estimate of r(h) for lag distance class hi;measured sample value at point x;measured sample value at point x + hi;total number of sample pairs for the lag classhi.

Average two-dimensional horizontal isotropic samplesemivariogram ?*(/I~) was calculated as the weighted aver-age of the individual two-dimensional horizontal isotropicsample semivariograms y:(hi) at different soil horizons,based on the number of pairs Nk(hi) at each lag class (hi)[Journel and Huijbregts, 1978, p. 213; Cressie, 1985]. Define

242 MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGENl .

ro(i= layer k(k = 1..5)

SHED SPATIAL RESIDUALSROW, COLUMN,

AND LAYER EFFECTS)

ESTIMATE 1-D (IN x- AND y- DIRECTIONS)SEMIVARIOGRAMS FOR EACH DEPTH (k= 1..5)BASED ON THE POLISHED RESIDUAL SETS R:;L

CALCULATE WEIGHTED AVERAGE (k= 1.5)-INo+,

TO THE EXPERIMENTAL VARIOGRAMS

ISEMIVARIOGRAM MODELIN THE HORIZONTAL DIRECTIONS

III

POLISH THE RESIDUALS FOR THE HOLE EFFECTBY CORRESPONDING MEDIAN OF EACH HOLE

ESTIMATE 1-D (IN z-DIRECTION)

VERTICAL CORE (i, j: i = 1..5 j = 1..7)BASED ON THE NEW POLISHED RESIDUAL SETS

1-D SEMIVARIOGRAM IN Z-DIRECTION

3-D COMPOSITE SEMIVARIOGRAM= GAMMA (x-dir) + GAMMA (y-dir) + GAMMA (z-dir)

GAMMA (hx, hy, hz) = GAMMA (hx) + GAMMA (hy) + GAMMA (hz)

Figure 2b. Schematic flowchart for the semivariogram analyses adopted for the data when arranged ona three-dimensional grid.

number of pairs lag class Similarly, one-dimensional semivariograms (y*(h,),Layer 1:

y*k=l(hi) Nk=l(hi)i= 1 -a-n; y*(h,,), and y*(h,)) can be estimated for each soil layer and

Layer 2:Y*k=2thi) Nk=2(hi)

i= 1 -e-n; vertical cross-sectional face parallel to rows (or columns). . . .. . .. . . .

Layer K: Y:=K(hi) Nk=K(hi) i= 1 we-n;

Average two-dimensional horizontal isotropic sample semi-variogram:

K

2 y;(hj)Nk(hi)k=I

T*fhi) = K i= 1 *e-n. (4)

C Nk(hi)k=l

using the two-dimensional semivariogram estimators butlimiting the estimation to one direction (x, y, or z). Beforeestimating the one-dimensional vertical semivariogram,however, the preferential or nonpreferential movement ofNOs-N, i.e., sample hole effect, should be polished bymedians, if present, by examining the median of each samplehole across the vertical cross-sectional face. Like the aver-age two-dimensional horizontal isotropic sample semivario-gram, the average one-dimensional sample semivariograms(y*(h,), Y*(hy), and 7*(/r,)) were calculated as a weightedaverage of one-dimensional sample semivariograms over thenumber of horizontal layers or vertical faces. This averagesemivariogram estimation (of the residuals of three-dimensional median polish approach) is, however, based onthe assumption that the individual horizontal layers or ver-

MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN 243

Table 1. Statistical Parameters of N03-N Concentration Under ConventionalTillage and No-Tillage Systems

Depth,cm

306090

120150

306090

120150

Coefficientof

Standard Variation,Mean Median Deviation % Minimum Maximum

Conventional Tillage, NOj-N Concentration, mg/L19.34 17.30 13.69 70.79 5.50 83.3014.39 13.40 6.75 46.93 7.20 38.808.16 7.70 2.81 34.45 3.50 14.20

10.19 9.30 3.87 37.97 4.90 22.3011.85 11.30 2.43 20.48 6.60 17.70

No-Tillage, NOj-N Concentration, mg/L18.40 18.20 5.58 30.32 8.40 33.3011.74 10.70 4.89 41.67 3.70 24.4010.11 10.30 3.30 32.58 1.90 16.509.21 9.00 3.04 32.98 3.30 16.309.14 8.80 2.86 31.31 4.00 17.70

tical cross sections are intrinsic random fields. By thiscomposite three-dimensional semivariogram approach notonly did we increase the dimensions of prediction but alsowe achieved more accurate semivariogram estimates foreach lag class as they are based on a greater numbers of pairs(by averaging over horizontal layers or vertical faces).Different theoretical models with or without sill and definiterange, such as nugget, linear, and spherical models wereused to describe spatial structure fitting the one-dimensionalaverage sample semivariograms. Vertical semivariogrammodels, however, were regularized to compensate for thevertical sample length of 10 cm.

Results and DiscussionVarious statistical parameters for NO,-N levels at various

depths and tillage systems were estimated and compared(Table 1). Some important observations were made based onthe values of these statistical parameters. Consistent de-creases in nitrate levels with depth to 90 cm and reaccumu-lation and more stable distribution of N03-N between 90-and 150-cm depths for conventional tillage system are twonoticeable features. Under the no-tillage system, a similarphenomenon was observed, but it was more stable withdepth.

As was mentioned earlier, our experimental grid designwas made in purposeful directions with grid rows parallel tocrop rows in the plot. Hamlett et al. [1986] and Cressie andHorton [1987] have reported that crop rows contribute todifferences in soil hydrologic properties. Similar effects arealso possible on the distribution of NOj-N in the soil profilebecause of plant extraction and biological processes. There-fore medians of parallel rows were used to “polish” the datato remove the possible crop row effect. Figure 1 shows thelocation of existing subsurface tile lines in the tillage plots.Because tile lines are present in the center of plots andparallel to row crops, an elliptical shape water table developsduring high rainfall (and so high water table). This causespreferential movement of NO,-N toward the tile line in thesoil profile in a parallel study by the authors (B. P. Mohantyand R. S. Kanwar. Spatial variability of nitrate-nitrogen andmoisture content in the soil profile of a tile-drained agricul-

tural field: Coregionalization, submitted to Water ResourcesResearch, 1993). This movement may cause an unevendistribution of residual NOs-N in the soil profile. Thereforepolishing the columns perpendicular to the tile line wouldremove any possible tile-induced effect or any other trend inthat direction.

Initially, soil NOs-N concentration data were examinedfor additivity of small effects (normality) at each horizontallayer at different depths under both conventional tillage andno tillage. In most instances, these conditions could not besatisfied. Therefore log, transformations were made toachieve normality if the data were lognormally distributed.Because most soil NOs-N concentration data were foundlognormally distributed, all further analyses of resistanttechniques were made using the log,-transformed data.Besides additivity in small effects (normality), additivity ofsmall effects to large effects (homogenous variances) isusually achieved by log, transformation of the data [Cressie,1985; Hamlett et al., 1986]. Cressie [1985] clearly showedthe advantage of (combined) weighted average semivario-grams based on the transformed grades over scaled (relative)semivariograms across regions. Our study also revealed thatlog, transformation can be used as a variance-controllingtransformation.

Detrending- - .

Log,-transformed soil nitrate concentration data wereexamined for stationarity or possible trend in the medianalong the two major directions of the grid (i.e., along the rowand the column). Medians of five rows and seven columnswere plotted across the respective rows and columns toinvestigate any nonstationarity (or trend) along directionsparallel to grid columns and rows, respectively. In bothtillage practices (no tillage and conventional tillage) and allfive horizontal layers (at 30-, 60-, 90-, 120-, and 150-cmdepths), nonstationarity was more the rule than the excep-tion. We picked up one example, arbitrarily, to show thenonstationarity in the median along rows and columns.Figure 3 shows that medians along rows and columns for thehorizontal layer at 90-cm depth under conventional tillageare nonstationary and have definite trends in both row and

244 MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN

CONVENTIONAL TILLAGE ( 9 0 CM DEPTH)

MEDIANSAMPLING GRID - - - rQI.J

26 21 I6 11 6 1 bbbrI.Jb&31

MEDIANP 0

MEDIAN

After 1 full row and columnmedian polish

1 3 5 7COLUMN NO --->

OF/ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _iji_5~,a”dco,“m”1 3 5 7

COLUMN NO --->

Figure 3. (a) Nonstationarity in the median of log,-transformed data, indicating trend in row and columndirections; (b) medians along row and column directions after one full step sweep; (c) medians along rowand column directions after two full step sweeps, indicating complete stationarity in the median.

.

column directions with occasional bounce. The directions ofincreasing or decreasing trend, however, were inconsistentfor different horizontal layers, under both tillage practices.Medians of soil nitrate concentration of horizontal layers atdifferent depths were plotted against the depth (e.g., notillage; Figure 4) to illustrate any trend in the verticaldirection. As expected, nonstationarity in the median (trendin the vertical direction) prevailed in the data.

To check the homogeneity of variance, median versusinterquartile range squared was plotted; these plots areshown in Figure 5 for no tillage and conventional tillage.Each point results from every row and every column ofevery horizontal layer for a total of 60 [= (5 + 7) X 5 ] points.Nonstationarity in variance for the measured NOs-N data isevident from Figure 5. The degree of nonstationarity, how-ever, is more prominent under conventional tillage than notillage. The plots also show that the relation between medianand interquartile range squared is proportional for conven-tional tillage, whereas the relation is relatively more stablefor no tillage. Our objective is to use a resistant approach toremove these mean and variance nonstationarities andtrends along the three major axes (row, column, and depth).Once trends along three major axes are removed from data,any diagonal trends present (trends along the diagonal direc-tions of the horizontal grid) need to be investigated andremoved. This whole procedure will generate a trend-free(composite) data set for geostatistical analysis.

Figure 5 was used to compare nonstationarity in variancesbefore and after the log, transformation. It is evident from

these plots that log, transformation has reduced nonstation-arity in the variance. Median polish was conducted for rowsand columns at each depth for both the no tillage andconventional tillage plots. Medians are plotted across rowsand columns after one full sweep (polished for rows followedby columns) as we11 as for two full sweeps (polished twice,for rows followed by columns) when constancy in medianwas attained in this particular instance (90 cm depth, con-ventional tillage). In most layers, no change in mediansoccurred after two full sweeps. In a few layers, constancy inmedian was attained after one full or one and o n e halfsweeps. Usually, however, the directional trends are re-moved in the first sweep, and (any) residual effects areremoved in the following sweeps. By comparing Figures 3a,3b, and 3c, the measured data were gradually cleaned up forthe trends along crop row and its perpendicular direction.Moreover, Figure 6 was plotted to provide a three-dimensional view of the measured data and the residualsachieved along the route of resistant analysis; it shows thatthe log, transformation [log,(Zjik)] tends to squeeze the highvalues and spread out the small values of raw (measured)data, which makes better stationarity in the variance. Fur-thermore, median polish by row and column sweeps re-moves directional trends (nonstationarity in the median) andmakes the data [log,( sijk)] smoother and more homoscedas-tic, except for a few spatial outliers. However, Gotway andCressie [1990] pointed out that these spatial outliers needcareful examination before removing them from furtheranalyses, which might provide some important information.

MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN 245

NO TILLAGE

-AFTER 2 FULL

RAW DATA ROW AND COLUMN SWEEPS

Figure 4. Layer medians in the depth direction before and after log, transformation and two fulI stepsweeps. Shows nonstationarity in the median is completely removed by the median polish.

Holes and bumps in residual soil NOs-N data might beshowing macropore or channel (causing preferential flow ofsoluble nitrate) effect and an impervious layer or incompletemixing of water and nitrate in a given layer due to the effectsof macropore flow. Corresponding frequency plots are alsopresented to show the gradual improvement of normality,and nonstationarity (i.e., the secondary peaks in the histo-

Iya 150

100I

..

. .= l..50

l m. .. ’&&*t: l ,+ -O . . F4 6 8 10 12 14 16 19 20 22

-

I .‘.

1.1. .

l-0.9 -0.8 -0.7 - .0.6r .

$1 ,.. .., --j’;\/,~ ym]: :.1.4 1.6 1.9 2 2.; 2.4 2.6 2.0 3 3.2

MEDIAN

NO TILLAGE IRAW DATAI

MEDIAN

NO TILLAGE (LOG DATA). .”10090 t

.

IyP

00 - .70. .60- * . ?50- - a40 - .

MEDIAN MEDIANFigure 5. Check for nonstationarity in the variance. Interquartile range squared versus median (spreadversus level) demonstrates log, transformation as a variance stabilizing transformation.

CONVENTIONAL TILLAGE (RAW DATA) .

gram) is removed, thereby partly fulfilling our stationarityassumption.

After removing the trends along the rows and columns, thepossible presence of diagonal trend was examined by re-gressing these residuals versus [(xi - x)( yj - ~31. In Figure7, diagnostic plots show individual depths under both tillagesystems. No trend was evident along any diagonal direction

CONVENTIONAL TILLAGE (LOG DATA)

246 MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN

N03-N (mg/l) NOJ-N (log(mg/l)) N03-N (log(mg/t))

1.67

40

0.0

6, I 6,

8 10 12 14 17 1.8 2.1 2.4 2.7 O -1.1 -0.4 -0.2 0 0.2 0.5nitrate concentration (mg/l) nitrate concentration Ilog(mg/l)) nitrate concentration (log~mg/l)~

IAl RAW DATA (81 LOG TRANSFORMED DATA ICI RESIDUALS AFTER 2 FVLLROW AND COLUMN SWEEPS

Figure 6. Three-dimensional surface plots and frequency histograms show the gradual improvement ofdata (normality and stationarity in the mean and the variance) by log, transformation and two full step rowand column sweeps. This particular case is for no-tillage plot at 120-cm depth. Any hole or bump on thesurface (Figure 6c) indicates the location of preferential or nonpreferential NOs-N transport path.

by visual inspection. A maximum fit of t-’ = 0.15 with a g(equation (1)) = 0.0068 was evaluated by linear regressionfor 90-cm depth under the no-tillage system. This reconfirmsour visual judgement of no trend along the diagonal (i.e., g =0). After close stationarity was achieved in the mean and thevariance in individual layers, in all horizontal directions, thenext step was to remove the layer effect (i.e., trend invertical direction) from the five horizontal data sets, thusmaking a three-dimensional composite homogenized dataset. Individual horizontal layers at different depths werepolished by their respective layer medians. Interestingly, wedid not find any layer effect remaining in the residuals afterone or two rounds of row and column median polish in eachindividual soil layer. Residuals must become so small andclustery around the median(O) that an evident layer effect nolonger exists.

The spatial distribution of residual NOs-N is representedby the additive model (equation (l)), a combination ofextrinsic factor(s) contribution(s) along different direc-tion(s), and the spatial structure of the residual term (&ijL).Separating the common term (p) and directional components((u, p, and 0 from the data leaves the residual term (E~~) tobe analyzed for inherent tillage effect in the spatial variabilityof residual NOs-N. Before the semivariograms for residuals(Eijk) were estimated, composite residual sets for both tillagepractices in three dimensions were reexamined to investigatewhether the pooled residuals were near normal with meanzero and variance CT’. Residuals had homogenous variance

due to log, transformation (Figure 5) and a mean near zeroafter row, column, and layer medians were removed (Figure8). These residuals were used to diagnose the spatial struc-ture of residual soil nitrate distribution in horizontal direc-tions under both no tillage and conventional tillage. For thesemivariogram in the vertical direction, however, we needone further refinement in the data residuals. Although theresiduals have no more directional effects (in x, y, and tdirections), they may have some sample hole effects (non-stationarity in the median across sample holes). Hole medi-ans were plotted to investigate the sample hole effect across

the vertical faces (i = 1, - * * , 5), and nonstationarity wasevident in the hole medians shown in Figure 9. This effectwas removed by sweeping the holes once by their respectivemedians. Now these new residuals were used to infer thespatial structure of soil nitrate distribution in the verticaldirection (t direction) under both tillage practices.

Semivariogram Analysis

Two-dimensional horizontal isotropic sample semivario-gram. After examining and removing nonnormality and non-stationarity in the mean and the variance of data sets, semi-variograms were estimated by the classical semivariogramestimator (equation (3)). Two-dimensional horizontal isotropic(assuming isotropy) as well as directional sample semivario-grams (in x and y direction) were computed for individualhorizontal layers. Average two-dimensional horizontal isotro-pic and average directional sample semivariograms (in x and Y

MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN 247

CONVENTIONAL TILLAGE

(60 CM DEPTH)

Rvk

(DO CM DEPTH)

‘r

(30 CM DEPTH)

“ ‘ I . ’

I20 CM DEPTH) -1160 CM DEPTH)

R..Ilk o

NO TILLAGE

ID0 CM DEPTH)0.f

R iik

(120 CM DEPTH)

o’*r

(150 CM DEPTH10.8 *.

(60 CM DEPTH)0.7

I30 CM DEPTH)0.8 ,

’.R.. ’

Ilk;.: a

.0 .a: :.,.: I

l

-0.6.

.-50 -30 -10 10 30 5<

Figure 7. Diagnostic plots to examine the presence of any diagonal trends in the horizontal layers.

direction) were calculated as the weighted average (equation(4)) of these five sets of sample semivariograms. Two-dimensional horizontal isotropic sample semivatiograms atdifferent depths and “average two-dimensional horizontal iso-tropic” sample semivariograms are presented in Figure 10 forboth no tillage and conventional tillage plots. A maximum of760 to a minimum of 360 pairs were used to calculate the

CONVENTIONAL TILLAGESO,

CON”ENrlONAL TILLAOE (FACE I,

RESIDUAL

ND TILLAGE

8 0

70

60

t,?I 60

2E 40

30

20

10

0-1.6 -1.2 -0.8 -0.4 0 0.4 0.8

0.2.

0.1 -.

0-m .

-0.1 -

-0.2 -.

.

-0.3 -

.

-0.4 ’1 3 6 7

HOLE I ON FACE 1 IS-W DIRECTIONI ->RESIDUAL

Figure 8. Frequency histogram of the 175 pooled residualsover the three-dimensional grid after executing three-dimensional median-nolishine scheme.

Figure 9. An example of sample hole effect on the verticalface after two full steps of row and column median polish.

248 MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN

CONVENTIONAL TILLAGE0.6

n 3 0 C M

:/, j

6 12 16 20 24 28 32 36 40

0.5

0.4

0.3

0.2

0.1

0

IAD DISTANCE I,“,

NO TILLAGE

n 3 o C Ml 60CM

r - l

0 SOCM0 1 2 0 C Mx 15OCM

6 12 16 20 24 28 32 36 40,.A0 DISTANCE I”,,

Figure 10. Two-dimensional horizontal isotropic samplesemivariograms at different depths and the average two-dimensional horizontal isotropic sample semivariograms un-der conventional tillage and no-tillage systems. Semivario-grams are estimated based on three-dimensional median-polished residuals of the log,-transformed NO,-N (milligramsper liter) data.

average horizontal isotropic sample semivariograms at each lagdistance. The number of pairs is about 5 times greater than thenumber of pairs used for individual horizontal isotropic semi-variograms, making the average horizontal isotropic semivari-ograms more accurate.

Examination of these individual and average isotropichorizontal semivariograms revealed some interesting facts.In conventional tillage, the sill of the semivariogram wasrelatively much higher at the 30-cm depth than at deeperdepths, although the range and the trend (shape) are similarfor all these semivariograms. Greater structural variation atthe 30-cm depth indicates the tillage-induced variability,which reduced gradually (with the depth with some differ-ences at 90 and 120 cm depths). We suspect that a trace ofthe effect of tile line (at 120-cm depth) still persists becauseof the “elliptical” shape of water table (governs the prefer-ential path of NOs-N transport during high water tablecondition), which could not be removed by three-way maineffects only median polish approach. For no tillage, semi-variograms are more uniform for all depths except at 90 cm.We suggest the same reasoning of tile-induced “elliptical”water table variability as in case of conventional tillage.

Furthermore, comparing the average two-dimensional hori-zontal isotropic semivariograms under no tillage with thoseunder conventional tillage (Figure IO), we can infer that soilnitrate distribution under conventional tillage has a smalltransitive spatial structure and that the distribution under notillage tends to be more randomly distributed. Under con-ventional tillage, however, nugget variance is a significantcomponent of the sample semivariogram.

Composite three-dimensional semivariogram. Directionalhorizontal semivariograms (in x and y directions) wereestimated, and “average directional horizontal semivario-grams” were calculated as their weighted average (Figure11). We limited our semivariogram calculation to smallerlags (approximately two thirds of the total lag distance in aparticular direction) because they are more accurate andcritical for geostatistical interpretation. Visual examinationof Figure 11, however, shows evidence of some anisotropyin the semivariograms for conventional tillage practice.Linear structure in the y direction and nugget behavior in xdirection were evident from these sample semivariograms.In contrast, the semivariograms showed nugget behaviorwithin very close limits for the no-tillage practice. Similarly,one-dimensional vertical semivariograms (in z direction)were estimated for each vertical cross section. An averageone-dimensional vertical semivariogram was calculated asthe weighted average over five vertical faces (Figure 11).Figure 11 shows that the spatial structure of soil nitratedistribution is linear on the vertical lag scale for conventionaltillage and nugget for no tillage. This structural pattern mightresult because plowing with conventional tillage distributessoil N03-N in a more structural fashion than no tillage.Because of plowing, macropores and cracks are destroyed atsurface soil layers and become discontinuous at the deeperlayers, whose presence or absence is reason for preferentialor nonpreferential transport of water and chemicals causingrandom distribution in space. On the other hand, under notillage, these factors contribute to nugget behavior in thesemivariograms. Similar phenomena were observed in thetillage-induced infiltration studies by Cressie and Horton[1987]. Moreover, under conventional tillage, better spatialstructure with relatively low nugget effect was found in thevertical direction than in the horizontal directions.

Two composite three-dimensional semivariograms(-y*(h,, h,, h,)) are plotted in Figure 12 for both tillagemethods by superimposing three one-dimensional averagesemivariograms (y*(h,), r*(h,), and r*(h,)) on the sameplot. Horizontal scale indicates both lag depth and lagdistance. Lag depth and distances, however, must not bemixed when horizontal or vertical semivariograms are used.Standard theoretical models were used to fit the horizontaland vertical semivariograms to describe their structuralpatterns. Linear models were adequate for horizontal andvertical semivatiograms in conventional tillage, whereasnugget-type models had good fit for no tillage. Interestingly,semivariograms for no tillage in all (three) directions (x. y,and Z) were close to one another. The fitted horizontal andvertical semivariogram models for conventional and notillage are given below:

Conventional tillage, semivariogram model in x direction:

Nugget model

Y(h,) = 0, h, = 0, (5a)

MOHANTY AND KANWAR:

. - .

Egg, ,II 10 20 30 40LAG OISTANCE ,M)

. . - . -

_.LAG DISTANCE 04

SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN

CONVENTIONAL TILLAGE=?I$ 0.26 0.24L 0.22E 0.2(1 0.18

s ;;z

r 0:124 0.1

% t:::6f2 0.04 0.02

z O 0 4 6 12 16 2 0 2 4B

LAG DISTANCE (Ml

NO TILLAGE

InLAG DISTANCE IMI

=P$ 0 . 2 6 0 . 2 4 g a m m a (hz); 0.223 0.2

:: 0.16 0.16

z “,::‘:d0222

P

0.08 ~ .0.1 . .

0.060.040.02

O 0 0.2 0.4 0.6 0.6 1

LAG DISTANCE IM,

249

Figure 11. Average directional (x, y, and Z) sample semivariograms under conventional tillage andno-tillage systems.

y(h,) = 0.11, h, > 0. (5b)

Conventional tillage, semivariogram model in y direction:

Linear model (r* = 0.92)

Y(h,J = 0, h, = 0, (6a)

y(h,) = 0.076 + 0.0027 hy, h, > 0. (6b)

Conventional tillage, semivariogram model in z direction:

Linear model (r* = 0.99)

r(h,) = 0, h, = 0, (7a)

y(h,) = 0.041 + 0.088 h,, h, > 0. (7b)

Regularizing y(h,) to compensate for the vertical samplelength (0.1 m), the regularized semivariogram (y, (h,))

y,(h,) = 0, h, = 0, (8a)

y,(h,)=(0.41 +0.088 h,)2(0.259-0.088 hJ0.03, (8b)

h, 0.1 m. (8c)

No-tillage, semivariogram model in x direction:

Nugget model

Y(h,) = 0, h, = 0, (9a)

y(.fl,) = 0.09, h, > 0. (9b)

No-tillage, semivariogram model in y direction:

Nugget model

Ah,) = 0, h, = 0, (10a)

y(h,) = 0.091, h, > 0. (10b)

No-tillage, semivariogram model in z direction:

Nugget model

y(h,) = 0, h, = 0, (11a)

y(h,) = 0.089, h, > 0. (11b)

Regularizing y(h,) to compensate for the vertical samplelength (0.1 m), the regularized semivariogram ( yI (h,))

y,(h,) = 0, h, = 0, (12a)

yl(h,) = 0.055, h, > 0. (12b)

Composite three-dimensional semivariogram models(-y(h,, h,, h,) = y(h,) + y(h,) + y(h,)) in conjunctionwith the additive model (equation (1)) could be used torepresent the variability of residual nitrate content in the soilprofile. Besides the net advantage of three over two dimen-sionality, this approach helps to unmask the directionaltrends of external influences.

Conclusions

Experimental data on NO,-N distribution, arranged on athree-dimensional grid, were used to study the spatial struc-ture of residual NO,-N distribution in the soil profile underconventional tillage and no-tillage pIots drained by subsur-face tiles. A three-dimensional additive model with quadraticdrift (equation (1)) was used to describe the spatial data.Log, transformation with a three-dimensional median pol-

250

0.26

0.24

0.22

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

MOHANTY AND KANWAR: SPATIAL VARIABILITY OF RESIDUAL NITRATE-NITROGEN

CONVENTIONAL TILLAGE

&I0 g*mm* (hz, - - - - LINEAR MODEI. w-2 - 0.99,

__I--_:.

gammelhx,hy,hzl - gammdhx) + gammathvl + gammafhzl

0 10 20 30 4 0

LAG DISTANCE (M)

NO TILLAGE

0.12

o.lf c*.

+ +E-- - - - - + - - - - - - - t - - - _

0.08 Q . .

0.0s -

0.04 -o.02 _ gamm~lhx.hy.hzl = gammalhxl + gunmalhvl + glmmalhd

0 a0 10 2 0 3 0 4 0

LAG DISTANCE (M)

Figure 12. Three-dimensional composite semivariograms(composed of X-, y-, z_- directional semivariograms) andfitted models under conventional tillage and no-tillage sys-tems.

ish-resistant scheme was adopted to clean up the data; freethe data from directional trends, nonstationarity in the mean,and nonstationarity in the variance; and make the data setmore homoscedastic and suitable for geostatistical analyses.The trend-free residuals were used to compute the experi-mental semivariograms for horizontal layers and verticalfaces, and average semivariograms in horizontal and verticaldirections were calculated as weighted averages of theindividual semivariograms. Two composite three-dimen-

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Acknowledgments. We sincerely acknowledge the valuable sugges-tions of Noel Cressie, Department of Statistics, Iowa State Univer-sity, for data handling and analyses. Also we express our appreci-ation to all our anonymous reviewers whose excellent suggestionshelped to improve the quality of this paper. This study was partlyfunded by the Iowa Department of Natural Resources through grantcontract 90-6257H-02. Journal paper J-15039 of the Iowa Agricultureand Home Economics Experiment Station, Ames, Iowa.

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(Received June 8, 1993; revised September 15, 1993;accepted October 15, 1993.)