Table Of ContentGreat BasinNaturalist54(4),© 1994,pp.313-328
GEOSTATISTICALANALYSIS OF RESOURCE ISLANDS UNDER
ARTEMISIA TRIDENTATA IN THE SHRUB-STEPPE
JonathanJ. Halvorson^ Hai^vey Bolton,Jr.-,Jeffrey L. Smith'^, and Richard E. Rossi-
—
Abstract. Desert plants can influence the pattern ofresources in soil resultinj^ in small-scale enriched zones.
Althoughconceptually simple, theshape, size, andorientationofthese "resourceislands'aredifficulttostudyindetail
usingconventional samplingregimes. Todemonstratean alternativeapproach,wesampled soil underandaroundindi-
vidualArtemisia tridentata (sagebrush), a dominant shrub ofcool desert environments, andanalyzed the samples with
univariate statistics and geostatistics. Univariate statistics revealed that soil variables like total inorganic-N, soluble-C,
andmicrobialbiomass-Cweredistributedwithhighestmeanvalueswithinabout25cmoftheplantaxisandsignificant-
lylowermeanvaluesatdistancesbeyond60cm. However, suchsimpleanalyses restrictedourviewoflesourceislands
toidenticallysized,symmetricalacciunulationsofsoilresourcesundereachplant.
Geostatisticsprovidedadditionalinformationaboutspatialcharacteristicsofsoilvariables.Variographyrevealedthat
samplesseparatedbyadistanceoflessthanabout70cmwerecorrelatedspatially.Over75%ofthesamplevariancewas
attributable to spatial variability. We modeled these spatial relationships and used krigingtopredictvalues forunsam-
pled locations. Resulting maps indicated that magnitude, size, and spatial distribution ofsoil resource islands vary
behveen individual plants and fordifferent soil properties. Maps, togetherwith cross-variography, furtherindicatethat
resourceislandsunderA. tridentata arenotalwaysdistinguishablefrom thesunoundingsoilbyshaiptransitionbound-
ariesandmaybeasymmetricallydistributedaroundtheplanta.\is.
Keywords:resourceislands,geostatistics, Artemisiatridentata, nutrientavailability,kriging,spatialcorrelation.
Recognition that individual plants can sig- Whitford 1986), "resource islands" (Reynolds
nificantly affect the local soil environment et al. 1990), or "ecotessara" (Jenny 1980), are
dates back to at least the middle of the nine- hypothesized to result from several mecha-
teenth centuiy (see Zinke 1962) and has been nisms (Garner and Steinberger 1989), includ-
documented for many plant forms including ing litter-fall or stemflow (Zinke 1962),
broadleaf and coniferous trees (Zinke 1962, decreased erosion or increased deposition
Everett et al. 1986, Doescher et al. 1987, (Coppinger et al. 1991), microclimatological
Belsky et al. 1989), bunch grasses (Hook et al. amelioration of the soil (Smith et al. 1987,
1991), herbaceous legumes (Halvorson et al. Pierson andWight 1991), orinputs ofresources
1991), and, in particular, desert shrubs (e.g., via insects, birds, or animals (Davidson and
Fireman and Hav^ward 1952, Garcia-Moya and Morton 1984).
McKell 1970, Nishitaand Haug 1973, Earthand Detailed knowledge ofthe size and internal
Klemmedson 1978, Burke 1989, Burke et al. dynamics of resource islands is important for
1989, Virginia and Jarell 1983, Bolton et al. understanding energy flu.x, mass transport,
1990, 1993). Soil associated with plants typi- and nutrient cycling processes at the scale of
cally contains greater concentrations of limit- the individual plant. Resource islands mayalso
ing resources (e.g., N, P), contains larger pop- connote a tier in a progressive, hierarchical
ulations of soil microorganisms, and exhibits mosaic ofplant and animal habitats, resource
higher rates ofnutrient cycling processes like distributions, and biogeochemical processes
mineralization (Charley and West 1977, Bolton (i.e., patches sensu Kotliar and Wiens 1990).
et al. 1990) and denitrification (Virginia et al. Estimates of the distribution and numbers of
1982). These small-scale enriched zones, vari- resource islands in the landscape may aid in
ouslytemied "fertile islands" (Gamerand Stein- understanding population level processes and
berger 1989), "isles of fertility" (West 1981, can be used to refine regional estimates of
'Pacific Northwest Laboiaton, Richland,Washington99352. Directcorrespondenceto215Johnson Hall,WashingtonState University, Pullman,
Washington99164-6421.
^PacificNorthwestLaborator\',Richland,Washington99352.
"^LandManagementandWaterConservationResearchUnit,USDA-ARS,WashingtonStateUniversity-,Pullman,Washington99164.
313
314 Great Basin Naturalist [Volume 54
energy flow and mass transfer. Furthermore, correlation ofthe samples, evaluated resource
interrelationships oflarge numbers ofindivid- islands ofindividual plants, nor quantified the
ual resource islands may influence ecosystem scaleofsoil heterogeneit>'beneathA. tridentata.
stiTicture, flinction, and stal)ilit> (Reynolds et al. Geostatistics has previously been used to
1990, Schlesinger et al. 1990, Malvorson et al. describe environmental and soil parameters
1991). associated with A. tridentata. For example,
Although conceptually simple, the size, Pierson and Wight (1991) used one-dimen-
shape, and orientation of resource islands are sional geostatistics to analyze spatial and tem-
not easy to evaluate. Previous studies have poral variability of soil temperature under A.
typically been based on relatively small num- tridentata. Halvorson et al. (1992) demonstrat-
bers ofsamples collected using abinaiy regime ed that geostatistics was an appropriate means
(i.e., samples collectedbeneath theplantversus of measuring resource islands at the scale of
samples collected "away from the plant) or an individual A. tridentata plant Jackson and
"
along a transect passingfrom plant tobare soil. Caldwell (1993a) attempted to quantify the
Such an approach cannot be used to provide a scale of nutrient heterogeneity around indi-
detailed spatial analysis ofresource concentra- vidual A. tridentata and Fseudoroegneria spi-
tions or processes in the soil that are likely to cata in a native sagebrush steppe using semi-
exhibit complex responses to landscape and variograms. They demonstrated increasing
microsite variations (Burke et al. 1989). autocorrelation of soil nutrients at spatial
Additionally, data collected from different scales <1 m but did not determine whether
locations (or depths) have often been analyzed small-scale effects were attributable to individ-
using inferential statistics such as ANOVA or ual plants or an artifact ofthe nested sampling
t tests that assume samples are spatially inde- design used. More recently, they constructed
pendent and identically distributed. However, kriged maps that showed relatively high con-
theseassumptions maybe dubious, ifuntested, centrations of soil variables like soil organic
since ecological phenomena are often spatially matter, extractable phosphate, and potassium
or temporally correlated and their frequency nearPseiidoruegneria tussocksbut notAiieumia
distributions are rarely normal (Rossi et al. shmbs (Jackson and Caldwell 1993b). However,
1992). Recently, a branch ofapplied statistics, these kriged maps did not directly quantify
known as geostatistics, has been demonstrated spatial covariation between locations of indi-
tobe usehil for detennining spatial correlations vidual plants and resource islands. Further,
amongecological dataand forestimatingvalues Jackson and Caldwell did not obsewe autocor-
at unsampled locations (e.g., Robertson 1987, relation for microbial processes at any scale
Robertson et al. 1988, Rossi et al. 1992). thatwas measured.
statOibsjteiccstitvoesdoefsctrhiisbestaudnydwmeordee(l1)tthoeussepagteioa-l tistTiocsmienetthroeuer sotbejpesc.tivFierss,t,weweapcphlaireadctgeeroisztead-
and modeled the similarity between samples
continuity of soil variables around individual
plants, (2) to use this information to produce adsireactfiuonnc.tiSoencoonfdt,hewier usespeadratthiisonredliasttiaonncsehipantdo
graphical representations or maps of specific
interpolate values at unsampled locations
resource islands, and, finally, (3) to quantify directly under and near individual plants.
tvahreiasbplateisa.lWcoerreelxaatmioinnebdetAwreteenmipsliaanttsriadnedntsaotial Finally, we quantified spatial covariation be-
tween soil properties and plants.
(sagebrush), a prominent shrub ofcool desert
environments (West 1983) previously known Study Site
to affect the distribution of resources in the
soil. Several workers have measured higher The study was conducted at the Arid Land
concentrations of resources such as total-C, Ecology (ALE) Reserve, located on the
total-N, inorganic-N, and higher rates of N Hanford Site in south central Washington (see
cycling in soil beneath A. tridentata than in Bolton et al. 1990 for details). There, remnants
nearby open soil using a binary sampling of the native Artemisia tridentata-Elytrigia
regime (e.g., Burke 1989, Burke et al. 1989, spicata association occur on silt loams of the
Bolton etal. 1990, 1993). However, these stud- Warden or Ritzville series. This perennial
ieshave notaccountedforpossible spatial auto- shrub-steppe is the largest grassland-type in
1994] GeostatisticalAnalysis of Resource Islands 315
North America and covers more than 640,000 high. All plots were located within approxi-
km- of the Intermountain Pacific Northwest mately 20 m of each other within a flat area
too chy to support forests (Daubenmire 1970, with randomly spaced plants. Multiple plots
Rogers and Rickard 1988). In an undisturbed were sampled for two reasons: first, to assess
state the A. tridentata-E. spicata association spatial characteristics of resource islands by
would be composed typically ofthree layers of basing our calculations on several examples
vegetation: an overstory shrub {Artemisia tri- rather than a single instance; and, second, to
dentata tridentata), a large caespitose perenni- provide replicates in the event that no spatial
al grass {Eh/trigio spicata [formerly Agropyron dependence of soil properties was observed.
spicatiim]), and a small caespitose perennial Data fiom all plots were combined to consider
grass {Poasecunda) growing on soil with a thin spatial dependence of soil properties around
cryptogamic crust (Daubenmire 1970). several A. tridentata plants simultaneously.
However, following disturbance such as This approach was chosen because it provided
tillage, grazing, or fire, the alien annual grass a more generalized evaluation of resource
Bromus tectorum becomes established. islands under individual A. tridentata and
greatly increased the number of data pairs at
Methods
any separation distance.
Estimates of plant location were required
Soil Collection andAnalysis for cross variography (see below). Therefore,
Cores ofsurface soil (10.5 cm dia. X 5 cm vegetation maps were produced from vertical
deep) were collected at 41 specific locations photographs. Each plot was divided into 5 X
within 2 X 2-m plots centered on mature A. 5-cm squares. Each squ—are was classified into
tridentata individuals (Fig. 1). Samples were one of three gr—oupings bare, grass species,
located so as to minimize the number ofdata or A. tridentata based on predominant cov-
points needed for analysis ofspatial character- erage. For this work no attempt was made to
istics and to avoid preferential clustering. We distinguish amonggrass species.
sampled five identically oriented plots (205 Each soil samplewas sieved (5 mm), mixed,
points) in March 1991, when levels of soil and analyzed for avariety ofsoil variables. For
moisture and microbial biomass activity were this work we present data only for water-solu-
ble forms of C, total inorganic-N (i.e., nitrate
+ ammonium), and soil microbial biomass-C.
Soluble soil C (H2O-C) was extracted with
N
room temperature deionized water and ana-
— —
lyzed using an infrared gas analyzer (Ionics
2.0 ^-e he h -e^ e-® Inc., Watertown, Massachusetts). Total inor-
Q o o d) ganicnitrogen (TI-N) was extractedwithin 48h
o of collection from 10-g subsamples of soil
1.5 -- o o o using 25 ml 2M KCl and analyzed colorimetri-
cally (Alpkem Coip., Clackamas, Oregon). Soil
o o o microbial biomass-C (SIR-C) was estimated
1.0 o o o o o from the respiratoiy response ofsoil to glucose,
o o o a source ofC readily utilized by heterotrophic
O O soil microorganisms (Anderson and Donisch
0.5 (-)- o () i1n97480)-.mTlegnl-agssravmialss,ammpoliessteonfedsoiwlitwherdeeipolnaiczeedd
O— —O H2O, covered with Parafilm, and incubated in
CD ()
0.0 ^-o he \ e^ e-e ptlheewdaarsktahten23a.5me±nd0e.d5°wCitfhora1glwukc.osEeacsohlustaimo-n
0.0 0.5 1.0 1.5 2.0 at the rate of600 mg glucose (240 mg C) kg-1
East (m) soil, bringing the final H2O content ofthe soil
to 20-25% (w/w; equivldent to 30-50 kPa).
(fiFvieg.totLal)ScwhaesmacteincteorfedaotnypiacnalArstaemmpilsiinagtrpliodte.ntEataachplpalnto.t Glass vials were sealed with silicone septa and
Dashedcircles showthelocationof41 soilcores (10..5cm incubated for 3 h. Soil respiration was mea-
dia. X 5cmdeep).Allfiveplotswereorientedasshown. sured by gas chromatography and related to
316 Great Basin Natur.\list [Volume 54
estimates of soil microbial biomass-C with deviations of tail and head values of the lag,
equations developed by Anderson and Domsch respectively. We chose the correlogram
(1978). because it removes the effects of lag means
and standardizes by the lag variances (Rossi et
Univariate Statistics al. 1992). For this work we express correlo-
Univariate statistics were calculated for grams in the form ofa standardized variogram
each soil parameter. For classical inferential by subtracting each p*(h) from 1 (Isaaks and
statistics, data were also assigned to one offive Srivastava 1989, Rossi et al. 1992).
distance classes depending upon sample loca- Correlograms were first calculated solely as
tion within a plot. These classes can be envi- a function oflag distance (i.e., the omnidirec-
sioned as concentric rings located at increas- tional case) without considering any differ-
ing distances from the center ofthe plot. The ences in spatial continuity with direction (i.e.,
first distance class comprised samples collected anisotropy). However, since resource islands
directly underA. tridentata (distance = cm, need not be symmetric (e.g., Zinke 1962), we
n = 1 per plot), followed by the second also calculated directional correlograms. For
(approximate distance = 25 cm, n = 8 per each soil property a separate correlogram was
plot), third (approximate distance = 60 cm, n calculated for samples oriented 0°, 45°, 90°,
= 8 per plot), fourth (approximate distance = and 135° (±15° tolerance) from each other.
110cm, n = 12 perplot), and fifth (appro.ximate Since correlograms are symmetric about the
distance = 130 cm, n = 12 per plot). Average origin (i.e., 0° = 180°, 45° = 225°, etc.), 0°,
values ofsoil properties in each distance class 45°, 90°, and 135° directions correspond to
were plotted as a function of radial distance samples aligned along east-west, northeast-
from the plant axis. Followingvariography (see southwest, north-south, and northwest-south-
below), logjo transformed samples deemed east axes.
spatially independent were compared with Because the dataofeach plotwere concate-
ANOVA usingplot as ablockingfactor. nated during this analysis, local anisotropics
(i.e., anisotropics specific to each plot) were in
Geostatistics
— effect combined. Thus, any directional effects
Variography. We evaluated spatial char- we obsei-vedwere acomposite ofthe five plots
acteristics ofeach soil parameterwith the non- and presumably indicative ofoverall direction-
ergodic autocorrelation function (Srivastava al trends. To identify directions of maximum
and Parker 1989) and summarized results and minimum continuity; we estimated the lag
graphically as correlograms. Like variograms, distance corresponding to a common value for
correlograms represent the average degree of each directional correlogram (Isaaks and
similarity between samples as a function of Srivastava 1989). The directional correlogram
their separation distance (lag) and direction. with the greatest lag associated with a correlo-
Unlike the variogram, the correlogram filters gram value of1 was identified as the direction
out the effects of changes in both lag means ofgreatest continuity. The conelogramwith the
and lag variances. Each point in a correlogram smallest lag corresponding to 1 was deemed
was calculated from this equation: the direction ofminimum continuity.
The empirically determined scatter ofdata
p*(h) = points in each correlogram was fitwith models
N(h) known to produce a positi\'e definite kriging
V^Nfh) system (i.e., matrices diatprovideboth aunique
2^i=l{[z(xi)-m_iJ[z(xi+h)-m+i,]} solution and a positive estimation variance;
S-liS+ii (1) Isaaks and Srivastava 1989). Such models typi-
cally contain several salient features known as
where z(xi) and z(xj -I- h) are two data points nugget, range, and sill. The nugget is the
separated by the distance (lag) h. Datum z(xj) amount ofvariance not explained or modeled
is the tail and z(xj + h) is the head ofthe vec- as spatial correlation. It is the apparent ordi-
tor, N(h) is the total numberofdatapairs sepa- nate intercept and is due to (1) unsampled cor-
rated by lag h, m_j^ and m+j^ are means of the relation below the smallest lag and (2) experi-
points that correspond to tail and head ofthe mental error (Rossi et al. 1992). A small nugget
lag, respectively, and S_i, and S+|^ are standard relative to the sill indicates that a large pro-
1 "
1994] GeostatisticalAnalysis of Resource Islands 317
—
portion ofthe sample variability^ is modeled as Cross-variography. In addition to spatial
spatial dependence. Conversely, alarge nugget characteristics of single soil properties, we
indicates less sample variability can be mod- also determined how soil properties covaried
eled as spatial dependence. The sill is charac- with plants. We modeled spatial covariation
terized by a leveling off of the correlogram with p*^g(h), the estimated nonergodic cross-
model. Ifpresent, it indicates that spatial cor- correlogram. Like the correlogram, it accounts
relation is, on average, constant. However, if for both variables' fluctuating lag means and
spatial correlation continues to change at lags variances (Isaaks and Srivastava 1989, Rossi et
greater than those considered in the correlo- al. 1992). Because comparisons between contin-
gram, then a sill will not be apparent. The lag uous variables (i.e., TI-N, SIR-C, and H2O-C
value when the correlogram model reaches data ) and discrete variables (i.e., plant data)
might be complicated by a "contact effect
tthhee smilalxisimknuomwnseapsartahteioranngdei.stIatnrceeprweistehnitns (Luster 1985), we converted TI-N, SIR-C, and
H2O-C data to binary variables using an indi-
which samples are spatially correlated. At lags
cator transformation (Journel 1983). For this
> the range, the sill of the variogram may work, continuous data values ofTI-N, SIR-C,
approach the sample variance (Barnes 1991). and H2O-C were coded 1 ifthey were greater
Kriging to estimate data at unsampled
— than thelocal (within-plot) median, or follow-
locations. Kriging has been likened to ing Halvorson et al. (in review). Cross-correlo-
"multiple linear regression with a few twists" grams were then calculated for grass species
(Rossi 1989). In classical multiple linearregres- orA. tridentata and indicatortransformedTI-N,
sion, an estimate oftiie dependent variable, Y, SIR-C, and H2O-C datausingthe equation,
is calculated from a weighted linear combina-
tion of independent variables where each is =
P*AB(h)
measured at about the same location in time N(h)
or space. Usually, only a single value of Y is
N(h) N(h)
estimated. Similarly, in kriging, z'''(Xq), the Z X [Ia (Xi,ZA)-mA_J[lB(xk^ZB)-niB^i
estimated value of the variable for an unsam- i=l k= (3)
pled location (x^), is calculated as a weighted
linear combination of the surrounding sam-
pled neighbors: where N(h) is the total number of data pairs
separated by vector h, I^(xj,z^) is the coded
N plant data (equal to 1 if the specified plant
z^ixj = I=gi • z{xi) (2) ttiyopne w(xaj)s,pmr^esientanord S^i_f,absaernet)thaet smoemaenloacnad-
/ 1
standard deviation, respectively, for the plant
where the z(x;)'s are the sampled values at variable at those data locations that are -h
away from a soil property data location.
their respective locations, and the gj's are the
weights associated with each sample value. In Similarly, Ig(xi^,ZB) is the coded soil variable
data (equal to 1 if the data value is greater
ordinary kriging, weights used to estimate
than the local plot median orelse 0) atlocation
izs*(uXqn)biaraesecdhoasnendshoasthaatmthienriemsuulmtinegsteismtaitmiaotne d(xej^v),iamtgion, oafntdhSegso.navraertihaeblmeeianndiacnadtosrtacnadlacrud-
variance and sum to unity. Kriging incorpo- lated for those locations that are +h away
rates amodelofspatialcontinuity' (here the cor- from a plant variable data location. Note,
relogram model) and accounts for the degree when h is 0, equation 3 is equivalent to the
ofclustering ofnearby samples and their dis- Pearson correlation coefficient (Isaaks and
tance to the point being estimated (Isaaks and Srivastava 1989).
Srivastava 1989). Weusedordinaiypointkiiging Unlike the correlogram, values calculated for
to estimate values ofsoil properties at unsam- the cross-correlogram may not be symmetric
pled locations. For each plot we estimated val- about the origin because both the order and
ues for the nodes of a 5 X 5-cm- grid. Each direction are switched when variables are re-
predicted value was based on a minimum of6 versed (Isaaks and Srivastava 1989). Conse-
and a maximum of 12 neighbors located with- quently, we calculated individual cross-correl-
in a0.8-m circular search radius. ograms for the 0°, 45°, 90M35M80°, 225°,
318 Cheat Basin Naturalist [Volume54
270°, and 315° directions (±30° tolerance).
These correspond to soil samples aligned to uu -
the east, northeast, north, northwest, west,
southwest, south, or southeast ofa plant.
Results
Summarx' statistics indicated that samples
ofTI-N and H2O-C were positively skewed,
while samples of SIR-C were more normally
distributed (Figs. 2A-C). Total inorganic-N
ranged from 0.6 mg / kg soil to a maximum of
23.6 mg /kg soil (Fig. 2A). The mean value for
TI-N of3.8 mg / kg soil compared reasonably
to the values reportedby Bolton etal. (1990) of
4.1 and 4.9 mg/kg soil for open soil crust and
A. tridentata soil, respectively. Values obsei^ved
for H2O-C ranged widely from 9.8 mg/kg soil
to 633.9 mg / kg soil (Fig. 2B). Estimates of
SIR-C ranged from less than 200 to over 1800
mg / kg soil (Fig. 2C). The average value for
SIR-C, 750 mg/ kg, was within the range
reported by Burke et al. (1989) and equivalent
to about 980 kg C / ha soil assuming a bulk
densit\'of1.3 (Bolton etal. 1990). Comparative-
ly, Smith and Paul (1990) reported average
microbial biomass pool size for grassland sys-
tems of1090kg C /ha.
Univariate statistics also indicated how soil
properties varied with distance from the A.
tridentata axis (i.e., center of the plot; Figs.
3A-C). Concentrations of HgO-C and SIR-C
were greatest within 25 cm of the plant axis
and lowest at distances beyond 60 cm (Figs.
3B, C). A similarpattern was obsei^ved forTI-N
except that mean concentration was low in soil
collected from directly beneath the A. triden-
tata plant (Fig. 3A) and from distances beyond
60 cm. This somewhat unexpected finding ofa
resource "hole" in the center of the resource
island may be indicative ofdifferences in the
cyclingofN under sagebrush and grass plants.
Variography indicated that samples ofTI-N,
H2O-C, and SIR-C were spatially correlated
(Fig. 4). Correlograms for SIR-C and TI-N
exhibited similar ranges ofabout 0.7 or 0.8 m.
The correlogram for H2O-C was similar to the
others at small lag distances and equaled the
sample variance at a range near 0.7 m. How-
ever, at greater lags, correlogram values for
H2O-C increased above the sample variance
and did not appear to reach a sill until lags
were greater than 1 m. A correlogram sill
greater than 1 for H.9O-C can occur if the
1994] GeostatisticalAnalysis of Resource Islands 319
«n
320 Great Basin Naturalist [Volume 54
Tablic 1. Summatystatisticsand randomizedcompleteblockANOV'As forlog],,transformedTI-N, HgO-C, and SIR-
C. Data arc summarized into two sample location classes: Near (all measurements collected from within 29 cm ofthe
plantaxis;n — 45)andAway(samplescollectedatdistancesgreaterthan 107cm from the plantaxis;n — 120).Average
separationdistancebetweenthetwolocationclasseswas99.4cm.
1994] GeostatisticalAnalysis of Resource Islands 321
1.5 n
322 Great Basin Naturalist [Volume54
TaBLF2. Model'"parametersused forordinan kri^ini obsened out to a lag ofabout 0.5 m. Beyond
ALEsoilproperties. this, correlations of grass species with SIR-C
Soilparameter remained approximately constant.
TIN
l-p*45 = .L54 + 0.428Sph(0.27) + 0.524Sph(L80) Discussion
l.p*j3^ = .L54 + 0.428Sph(0.71) + 0.524Sph(L20)
Geostatistics can be applied to resource
SIR-C island datato provide several useful diagnostic
ll..pp**^^535 == ..007722 ++ 00..338811 SSpplhi((00..7243)) ++ 00..771177SSpphh((01..7857)) features prior to actual mapping of the land-
scape itself. For example, variography can
H.,0-C define the presence and extent of spatial cor-
"l-p* = 1.40Sph(1.3) relation and alert the researcher to apply with
'ModelsshownforTI-NandSlR-Cart-aconiliinationolanuggi'tconstant caution classical statistical comparisons that
andtwosphericalmodels.Thesphericalmodel,denoted"Sph, isanautho- assume samples are independent and from
rizedmodelcommonlyusedtomodelvariograms.Thenumberthatprecedes
"Sph"canbethoughtofasthelocalsillforthatmodelwhilethenumberin identically distributed populations. For these
pSrairveanstthaevsaes198is9)t.heForranagceorartewlohgircahm,thtehelosctaalndsialrldiiszerdeafcohmeidi(ssele-p*I(sha)aks=a1n5d methods to be more properly applied to spa-
(lag/range)-0.5(lag/range)'^iflag^=range,else1ifotherwise. tial data, comparisons should probably be lim-
^NOTE:Geostatisticiansoftendistinguishbetweenthenuggetusedfordiag-
nosticpurposes,which istheapparentordinateintercept,andthenugget ited to those samples separated l^y distances
\'alue.whichisusedinmodeling. > range ofthe correlogram (Table 1; Webster
1985, Robertson 1987). This is true for studies
Indicator transformed HoO-C data were that compare samples collected along a con-
positiveK' correlated withA. tridentata but not tinuum such as distance, depth, or concentra-
with grass species at a lag of (Figs. 7C,D). As tion (i.e., resource gradient) or as a function of
with TI-N, highest correlations withA. triden- time.
tata were observed in the 45° and 90° direc- Another promising use ofvariography is to
tions (Fig. 7C). Similar patterns of spatial relate spatial continuityoftwoormorevariables
dependence were obsei'ved for all directions. at the same site or the same variable at two or
The distance to which H^O-C remained posi- more sites by comparing variograms, covario-
tively correlated with A. tridentata ranged grams, or correlograms with one another This
from a minimum ofabout 0.5 m to a maximum approach may be useful for comparing the
ofnear 0.75 m in the 45° and 90° directions. scale of ecological processes or ecosystem
Unlike A. tridentata, H2O-C was not positive- boundaries, but should be approached with
ly correlated with grass species (Fig. 7D). caution for several reasons. First, each pointin
Instead, H2O-C was moderately negatively a traditional variogram represents the average
correlated in the 0°, 45°, 90°, and 135° direc- value ofthe squared difference between many
tions, meaning grass species were more asso- pairs of data points. While an average value
ciated with below-median concentrations of maybe appropriate for modeling spatial conti-
H2O-C. At lags greater than about 0.2 m, little nuity (as a summary statistic), it does not indi-
change in cross-correlograms was observed, cate the range of individual squared differ-
indicatingonlyaweak spatial dependence. ences or provide an estimate ofthe "goodness
In contrast to H2O-C, cross-correlograms of fit" about each point in a variogram. The
indicated that SIR-C was slightly more corre- range ofthe deviation about the average value
lated with grass species than with A. tridenta- maybe large or small (see Webster and Oliver
ta at a lag of (Figs. 7E,F). Like other soil 1992), complicating comparisons ofvariograms.
properties, strongest positive correlation Thus, for comparative purposes other more
between A. tridentata and SIR-C was "robust" representations ofspatial dependence
observed in the 45° direction, which also such as Journel s niAD estimator, Cressie-
remained positively correlated to lags in Hawkins' robust estimator, or the rodogram
excess of 1 m (Fig. 7E). Lowest correlations (see Rossi etal. 1992) maybe moreappropriate
with A. tridentata were to the 270° and 225° choices. Second, even if variograms for two
directions. Indicator transformed SIR-C data properties are similar, resultant estimates may
were most correlated to grass species in the yieldveiy different maps (e.g., TI-N and SIR-C
225°, 270°, and 315° directions (Fig. 7F). in this study). This is because estimation of un-
Spatial dependence of correlations was known datavalues bykrigingdepends not only
