ANALYZING SPATIAL AND TEMPORAL VARIABILITY OF

SOIL WATER CONTENT (1)

SIDNEY ROSA VIEIRA e); CÉUA REGINA GREGO e); GEORGE CLARI<E TOPP (4)

ABSTRACT

During the last two decades geoestatistical methods have been intensively used for in-depth descriptionsof spatial variability. The objective of this study was to assess the spatial and temporal variability of soilwater content. The measurements were taken with a TDR equipment to a 20 em depth, in a nearly flat 1.2 hafield at the Central Experimental Farm of the Agriculture Canada, Ottawa. The soil classified as a Rideau soilseries, is a clay loam soil. A square grid with 10 m spacing was laid out, resulting in 164 sampling points atwhich two TDR rods were installed to measure the water content down to 20 em depth. Measurements weretaken on 33 dates duríng the frost free months in 1987, 1988 and 1989. The spatial variability was ana1yzedexamining the scaled semivariograms, the statistical parameters and the parameters of the models fit toindividual semivariograms as a function of time. It was concluded that spatial dependence decreases as thesoil gets drier and that results from one year connect almost continuously to other years. The topographyand structure of topsoil horizon was the primary cause for the repeating spatial pattern of soil water contentin successive samplings. The places where the mean value occurred in the field were more stable in timewhen there was spatial dependence. As the soil gets dryer the temporal stability of the spatial distributiontends to disappear due to the hydraulic conductivity controlling the water evaporation over the field.

Key words: TDR, temporal stability, semívaríogram, geostatistics.

RESUMO

ANÁLISE DA VARIABILIDADE ESPACIAL E TEMPORAL DO TEOR DE ÁGUA DO SOLO

Durante as ultimas duas décadas métodos geoestatísticos têm sido intensamente adotados para descrevera variabilidade espacial em profundidade. O objetivo deste estudo foi avaliar a variabilidade espacial e temporaldo teor de água do solo. As medições foram feitas com TDR a 20 em de profundidade, em uma área plana de1,2 ha no Centro Experimental do Ministério da Agricultura do Canadá, Ottawa, no solo de textura franco-argilosa. Fez-se um quadriculado com pontos distanciados em 10 m, resultando em 164 pontos de amostragem,nos quais duas hastes do TDR foram instaladas para medir a umidade. As medições foram realizadas em 33datas durante os meses livres de gelo na superfície do solo em 1987, 1988 e 1989. A variabilidade espacial foianalisada através de semivariogramas escalonados, de parâmetros estatísticos e de parâmetros de ajuste demodelos para os semivariogramas individuais em função do tempo. Concluiu-se que a dependência espacialdiminuiu conforme o solo tornou-se seco e os resultados de um ano conectam-se quase que continuamentecom os dos outros anos. A topografia e a estrutura do horizonte superficial tiveram influência na repetiçãotemporal do padrão de distribuição espacial do teor de água do solo. Os locais no campo onde o valor médioocorreu tiveram maior estabilidade no tempo quando não existiu dependência espacial. Na medida em queo solo seca e a condutividade hidráulica passa a controlar a perda de água para a atmosfera, a estabilidadetemporal da ocorrência de valores médios em determinados locais tende a desaparecer.

Palavras-chave: TDR, estabilidade temporal, semivariograma, geoestatístíca.

e) Received for publication ín March 24,2006 and accepted in October 18, 2007.e) Centro de Pesquisa e Desenvolvimento de Solos e Recursos Ambientais, lAC, Caixa Postal 28, 130U-970 Campinas (SP).

E-mail: sidney@iac.sp.gov.br (>t) Correspondent author.e) Centro Nacional de Pesquisa de Monitorameoto por Satélite, Embrapa, Av. Soldado Passarinho, 303,Fazeoda Chapadão,

13070-115 Campinas (SP). E-mai1: crgrego@cnpm.embrapa.br(4) Eastern Cereal & Oilseed Research Centre, Agriculture & Agri-Food Canada, %0 Carling Avenue, Ottawa, Canada, K1A OC6.

E-mail: toppc@agr.gc.ca

Bragantia, Campinas, v.67, 0.2, p.463-469, 2008

464 S.R Vieira et al,

1. INTRODUCTION

During the last two decades geoestatisticalmethods have been intensively used for in-depthdescriptions of spatial variability (BURGESSand WEBSTER,1980; VIElRAet al., 1981; NIElSENet al., 1983; VIElRAetal., 1983; VIEIRA,2000; VIElRAet aI., 2002). Someresearchers used autocorrelograms (WEBSTERandCUANAW,1975), others semivariograms and krigingestimation (VlEIRAet al., 1981; VIElRAETAL.,1983), andanother ones cokriging estimation (VAUCLINEet al.,1983). Kriged contour or three-dimensional maps arepopular representations of spatial variability results(HAJRASULlliAet al., 1980; BURGESSand WEBSfER,1980;MAcBRATNEYet aI., 1982; VAUCLINet al., 1983) andprovide quantitative assessments of variability.

Variability in time, and in particular, withrepeating patterns is being a challenge to soil research.VACHAUDet al. (1985) showed that time stability of thespatial variability may exist for water contentmeasured on the same locations at different times.Scaling semivariograms of several variables measuredover the same field provides a simple but powerfulintegration method (VIEIRAet al., 1988, VIEIRAet al.,1991; VIEIRAet al., 1997) fi the sense that the more thesemivariograms scale the more similar the variabilityof the corresponding variables are. Therefore, ifsemivariograms scale it indicates that not only themean values and dispersion coefficients occur at thesame locations but all variability repeats in time,although the absolute values may be different. Besides,the analysis of the parameters of the models fitted tothe semivariograms as a function of the time ofsuccessive samplings may be of help in assessing thetemporal stability of the spatial variability.

The objective of this study was to assess thespatial variability as well as the temporal stability ofspatial distribution for soil water contento

2. MATERIAL AND METHODS

Theory

The experimental semivariogram, y(h), of nspatial observations z(xJ, i=L, ... , n, can be estimatedusing:

1 N(h)

Y (h)= 2N(h) !;fz( x.) - z( x, +h)r (1)

where N(h) is the number of pairs of observationsseparated by a distance h. Experimentalsemivariograms need to be fitted to some mathematicalmodel which must meet thecriteria of conditional

Bragantia, Campinas, v.67, n.2, p.463-469, 2008

positive definiteness (MACBRATNEYET AL., 1982).Amongst all the variety of models which satisfy thatcondition, the fitting parameters that describe themare: the nugget effect Co, the sill (CO+C1) (C1 is thestructured variance coefficient to be defined later), andthe range of spatial dependence a.

With the objective of comparing the variabilityof different samplings, VIElRAETAL. (1997) proposeda scaling technique for the semivariogram expressedby:

y se (h)=y Jh)/ eu 1= l,2, ..,m (2)

where m indicates the number of measured variables.The scale factor a, is a constant that can take the valueof the calculated variance, of the sill when it exists,of the square of the mean values or of the highestvalue of the semivariogram y(h). The scaling conceptproposed by VIElRAETAL.(1997) may be helpful in theanalysis of the temporal stability of the spatialvariability of soil water content corresponding todifferent sampling dates for the same location. Withineach data set (year of sampling) the scaledsemivariograms for all sampling dates can be plottedtogether in order to make comparisons and verify ifand when the spatial variability looses temporalstability. When data of several dates coalesce into aunified semivariogram structure, it is possible to takeadvantage of variables having the same spatialstructure and, hence, reduce the number ofsemivariograms needed to analyze and drawinterpretations regarding their spatial values.

When the semivariogram of any particularvariable does not stabilize at constant value for thesill, it is an indication that the sill does not exist, andtherefore, the stationarity of the mean cannot beguaranteed because the variableincreases unlimitedlyin some direction. In this situation, it is necessary toremove the trend before any geostatistical applicationbased on the intrinsic hypothesis is made (VIEIRA,2000). One possible way of removing the trend froma data set is using a trend surface fit to the datathrough minimized squared deviations of thedifference between the surface and the original data,producing a new residual variable (VIElRA,2000).Trend surfaces are polynomials of some degree,usually first (linear), second (parabolic) or third(cubíc), depending on which fits better the data. In thisstudy, all soil water content data showed some trendthat was best fit to a parabolic surface, (Zest), usingthe equation:

Analyzing spatial and temporal variability of soil water content 465

where Ao, A J, Az, A3, A4 and A5 are the parameters fitoThe residual variable, (Zres)' can be obtained bysubtracting the estimated trend surface from theoriginal values for each point:

Zres(x,y) Z(x,y) Zest(X'y) (4)

where the variable Zres is subsequently tested for theexistenceofa definedsíllín the semivariogram.Thekrigingestimation is done on the residual values to which theestimated surfaceis added after the estimationis done.

As the semivariograms for the original datashowed a very strong trend violating the intrinsichypothesis of geostatistics OOURNEL and HUIJBREGTS,1978), parabolic trend surface equations were fittedto the data and subtracted from them. The residualsgenerated by the difference between origmals andtrend surface produced semivariograms that showeda very well defined sill, and for this reason theresiduaIs were used in the remaining analysis.

Once the semivariogram has a defined sill, amodel must be fit to its experimental values in orderto provi de a continuous function over the wholedistance range. The spherical model was fit to allsemivariograms in this study, using trial and errortechnique and jack knifing procedure for validation(VIEIRA, 2000).The equation for the spherical model is:

[3 h 1 (h)3]y(h)=C. +C, 2-;';-2 -;.; for h 5. a

y(h)= c. +C, for h >- a

Where Co, C1and a are, respectively,the nugget effect,thestructuredvariancecontrlbutionand therange.ThequantityCo+C1 is the sill of the semivariogram.Comparisonof thesemivariogramparameters for differentdates of sarnplingis a way of supporting or not the changes in spatialvariabilitywithtime.lhe dependenceratio(DR)definedbyCAMIlARDEuAEf AL (1994)quantitiestherelationshipbetweenCOand CI. Thedependenceratiomeasuresthedependencedegreeofa variableand iscalculatedby:

DR = _10_0_x_C_oCO+C1

Two factsare very important about the above theoreticalconcepts: i) The semivariogram y(h) is assumed to beisotropic, i.e., either there is no significant anisotropy orthere is a transfonnation to remove the anisotropy beforescaling is applied; ii) y(h) can take the value of thecalculatedvarianceVar(z)no matter whether it representsthe true variance or not, since the sca1ingfactor is simplyanumber chosentomakethe semivariogramscoalesceintoa singlecurve.Thereasonwhy scaledsemivariogramsmayprovide an adequate way to analyze temporal evolution

of the spatial variability is that it will be noticeablewhenthe spatial variability pattem changes, and thus thepossiblecausecanbe examined.Therefore,sca1ingis usedin this paper only for the comparison betweensemivariogramsfordifferentsarnplingson the sarnefield.

Data sets

One hundred and sixty four TDR (timedomain reflectometry) rods were installed in a Rideauday loam soil in the Central Experimental Farm ofAgriculture Canada, Ottawa, for measuring volumetricwater content of the surface 0.2 m of soil. Therefore,soil water content measurements could be obtained onthe same points as many times as wanted since theTDRrods remained in-place at the same points in thefield. The triangular field, kept vegetated with naturalpasture grass, measured 110m in the x-direction (base)and 220m in the y-direction (height), was divided intoa 10 x 10 m grid, as shown in figure Ia. All watercontent measurements over the field were collectedwithin a two-hour time period. The topographic mapof the study area is shown in figure 1b.

20

(5)

+++ a++++++++++++++++~ +++++

.! 1 ++++++~ ++++++·12 +++++++

>- +++++++~ 1 ++++++++lij ++++++++o; +++++++++õ +++++++++++++++++++++++++++++++++++++++++++++++++++++

2 ++++++++++++++++++++++++++++++++++++o 20 40 60 80 100 120

Dístanoe X, meterso 20 40 60 80 100

b

120

100

80

60

40

oHeight, em

Figure 1. 5tudied field settings: a) 5ampling scheme; b)Topography.

(6)50 ___ 1987

•E 40 ~1988E.~30

---.-1989

iií Â'6.20ÕQ)

ô:: 10o

100 150 200Julian day

Figure 2. Rainfall as a function of time during the years.

250 300

Bragantia, Campinas, v.67, n.2, p.463-469, 2008

466 S.R. Vieira et al.

The study period extended from earlySeptember of 1987 until early May 1989 during theground surface frost-free months. The dates ofsamplings in 1987 were September 11, 14, 17, 21, 24,and October 29 with two samplings on the same daywith different TDR instruments.

There were 21 samplings dates in 1988: May06, 13 and 30, June 06, 13, 21 and 27, July 04, 15, 22and 28, August 11, 18 and 25, September 02, 08, 15,22 and 29, and October 06. The dates of samplings in1989 were April 14, 18, 21, 25 and 28 and May OI.Figure 2 shows the precipitation in mm, during thestudy period.

The sampling dates were transformed in Juliandays in order to make the graphs of the parametersas a function of time during the year.

The examination of the spatial variability forsoil water content as the time during the yearprogresses may reveal information about some watercontent threshold value at which the soil hydraulicconductivity begins to cause changes in the spatialvariability patterns.

3. RESULTS AND DISCUSSION

Figure 3 shows the general behaviour of thewater content changes and the main descriptivestatistical moments as a function of the dates ofsampling. Data corresponding to September andOctober (end of the frost-free period of 1987, and fromApril and May (beginning of the frost-free period of1989 season as the result show continuity of thestatistcal parameters from one year to the other. Thedata from 1989 had a similar pattern to those at thebeginning of 1988 and the data from 1987 had similarpattern to those at the end of 1988. These qualitativesimilarities indica te continuity of water contentpatterns from one year to the next. The mean soil watercontents shown in figure 3a vary up and down about35%. Being highest at the first ground frost-free day,the soil water content then starts decreasing until abig rain occurs. Proportionally the CV values increaseat the same times (Figure 3b), as it is noticeable froma sudden decrease in CV values around julian day180, following a large rain (Figure 2). The skewnesscoefficients (Figure 3c), except for one date in 1987,are all slightly negative. The coefficients of kurtosis,on the other hand, are all dose to 3 (Figure 3d).Therefore, it is easy to see that most of the dataanalyzed approach the normal distribution. Fromfigures 3e and 3f it seems that the minimum values(Figure 3e) are more sensitive to the rainfall variationin time than the maximum values (figure 3f).

Bragantia, Campinas, v.67, n.2, p.463-469, 2008

50

-4-1987 -+--1988 --.-1989

õ; 40

i" 3028 20.!i~ 10

O+--r--.-~-'--~~--.--r--r-~100 120 140 160 180 200 220 240 260 280 300

Jufian day

403530

b) CV values

ti! 25;; 20<..i 15

105 _4_1987 -+--1988 --.-1989O+--r--r--r--r--r--.--.--r--.--r-

100 120 140 160 180 200 220 240 260 280 300

Julian day

0,4 c) Skewness coefficients0,20,0 +--r--.--.~r--,--,--~---tr,----~-r-

i -0,21~ -0,4.,~ -0,6

-0,8

-1,0 _4_1987-1,2

5,0

4,0

Julian day

d) Kurtosis coeflicients

..~ 3,0

~ 2,0

1,0_4_1987 -+--1988 --.-1989

0,0 +-~--r--r--.---.--r-~--'r--.--'-100 120 140 160 180 200 220 240 260 280 300

Julian day40

g 35~ 30

i" 25 \~ 208 15., 10

~ 5O+--r--.--~--'r-~~--.--r--.-~

100 120 140 160 180 200 220 240 260 280 300

e) Minimum values

JuBan day

;:~-i:': ~~~30 '~'-Jê 202~ 10 _4_1987 -+--1988 --.-1989

O+--r--r--.--'r-.-~--~-r--.-~100 120 140 160 180 200 220 240 260 280 300

JuBan day

Figure 3. Statistical parameters as a function of timeduring the years: a) Mean values; b) CV values; c)Skewness coefficients; d) Kurtosis coefficients; e)Minimum values; f) Maximum values.

Analyzing spatial and temporal varíability of soil water content 467

It is possible too that for low water contentvalues the TDRmethod fails to work properly as thecontact between the wave-guides and the soil maybecome limiting in soil of a day loam texture.

Soil water contents at this site in Ottawashowed similar sets of semivariograms for all datesfor the three years (Figure 4). It can be seen that thescaled semivariograms coalesced without muchscatter. All semivariograms are for residuals ofparabolic trend. Although the soil map for this fieldis not available, there was a regíon located about midway dose to the diagonal side which showed probablya sandier surface texture. This regíon is also wherethe highest elevation occurs, as shown in figure lb.These two factors are undoubtedly the cause for thesimilarity in spatial variability in different samplingdates. In general, semivariograms show that a11datessampled have strong spatial dependence to about 40m. This distance for spatial dependence may alsoresult from the location of the higher elevation inrelation to the shape and length of the field.

1,6 a) 19871,4

!i - ,,: o o o1,2 .•'fi!'B ;*IG9! • Ia: •.• o

c 1,0 •• .o"ca •• " ~ ~-c ·ca 0,8 e •.2:

"E 0,6 ~ • Sep11 o Sep14 -Sep17 o Sep21.,rn

0,4 o Sep24 o0c29F -Oc29S0,20,0

o 20 40 60 80 100

Distance, m

1,4 b)1968 .1,2 - ·· .ii.liliI I~..o., 1.0 i 'a,o 11 :~. .i ~,i, : :. Ic

ca 0,8 I !\,

1ii rj.2: 0,6 6 ~E • May06 o May13 -May30 o Jun06 o Jun13.,rn 0,4 o Jun21 - Jun27 o JulO4 x Jul15 x Jul22

. Ju128 -Aug11 • Aug18 Aug25 oSep020,2 A SepOB D Sep15 Sep22 Sep290,0

o 20 40 60 80 100

Distance, m

1,4 c) 19891,2 .

li! I '. ";: • I.

• t., 1,0 .1 l' · ,J'~ 'ta~o

c •ca 0,8 ·'t: I!l 8~ o 'lE 0,6 !l.,rn 0,4 • Apr14 o Apr18 "Apr21 o Apr25 o Apr28 o May01

0,2

0,0o 20 40 60 80 100

Distance, m

Figura 4. Scaled semivariograms for water content values:a) 1987;b) 1988;c) 1989.

The qualitative seasonal similarity of the meanwater content noted above is confirmed quantitativelyby the repeated and continuous similarity of thesemivariogram parameters (Figure 5). Overall, thedays that showed larger ranges of spatial dependencealsohad higher nugget effectvalues as the dependenceratio indicates.

8070 a60

~ 50~ 40c:

~ 302010 -11-1987 -+0-1988 -.ir-1989

04---,- ....•.--r---.---,--,----,---,----;.------.--100 120 140 160 180 200 220 240 260 280 300

Julian day

70.2 60f! 50Q)

g 40Q)-g 30~ 20Q)

o 10 -11-1987 -+0-1988 -.ir-198904---,- ....•.--r---.---,--,----,---,---.------.--

100 120 140 160 180 200 220 240 260 280 300Julian day

Figura 5. Semivariograms parameters as function ofsampling time: a) Ranges; b) Dependence ratios.

Figure 6 shows the linear relationshipbetween mean values and dependence ratio andbetween skewness coefficientsand dependence ratio.These two graphs were constructed because theresults indica te that they may contribute to showthat the spatial variability of soil water content isstable in time within some limits of threshold value.Although this threshold value may vary from one soilto another, it seems quite evident that it does existoThe Dependence Ratio (DR)decreases linearly as themean soil water content increases (Figure 6a). Thereason for it is that as the soil water contentdecreases, the spatial variability of topsoilcharacteristics such as texture implies a different soilwater content spatial distribution pattern, probablyarising from the water content of coarser texturedregion(s) decreasing more rapidly than those of themore clayey region(s). Because the soil has not beencultivated for a long time, the variability expressedhere is natural to the soil characteristics andrelatively stable in time.

Bragantia, Campinas, v.67, n.2, p.463-469,2008

468 S.R. Vieira et a1.

70 y = -O,1509x+ 38,19860 a R2= 0,076

~~ 50 , •••• ••~"Ê 40 • •:5.!! 30 • :~.. • ••coC::::I~ 8 20 • •

10o

o 10 20 30 40 50 60Dependence ratio

Dependence ratioo 10 20 30 40 50 60

0,4 •0,2b

U) 0,0U) •• • •~ -0,2 •:;: -0,4 • •~ -0,6 •••CJ) -0,8 • • y = 0,0086x- 0,7364

-1,0R2= 0,1872-1,2

Figura 6. Relationship between statistical parameters anddependence ratio: a) Mean and b) Skewness coefficientof soil water contento

Because this soil unit has a very welldeveloped blocky structured top soil (about 30 em),and a massive layer with no structure at deeperhorizons, its hydraulic conductivity is much higherat the surface horizon than at deeper depths,according to the results reported by VIEIRA ET AL. (1988).

The skewness of the soil water content alsochanges proportionally to the continuity of the spatialvariability (Figure6b).Therefore,the unique conditionimplied by both the topography and the soil strueturefavors faster loss of water from the soil at higherelevation region resulting in the picture patternexpressed by the parameters shown in figure 6.As thesoil gets dryer the temporal stability of the spatialdistribution tends to disappear due to the hydraulicconductivity controlling the water evaporation overthe field.

As the soil water content for alI datespresented parabolic trend, the parabolic trend surfaceparameters of equation (3)were plotted as a functionof the date of sampling. The results are shown infigure 7. The Aovalues shown in figure 7a are veryclosely related to the mean soil water contentoOn theother hand, the AI and A2 values, respectively shownin figures 7b and 7c, indicate that the variability inthe Xand Ydirections is stronger at beginning of theseason than in the summer time.

Bragantia, Campinas, v.67, n.2, p.463-469, 2008

60

50

40

a

< 30

20

10 1987 -+-1988 •..••. 1989

O+-~-'--~~-'--T-~-'--~~-'100 120 140 160 180 200 220 240 260 280 300 320

Julian day

0.2

0.1

0.0

.:( ~.1

~.2

~.3

~.4 +-""T"""-.---.----,.--T- ...•........•..-,--r--r--.100 120 140 160 180 200 220 240 260 280 300 320

b ...•.• 1987 •..••.1988 •..••.1989

Julian day

0.10 C ...•.• 1987 •..••. 1988 •..••. 19890.05

0.00

~.05s ~.10

~.15

~.20

~.25+-...•........•..-,--..--r-~--.----,.--~...•...-,100 120 140 160 180 200 220 240 260 280 300 320

Julian day

35 ...•.• 1987 •..••.1988 •..••. 19890.00 d0.00300.00250.00200.0015

:( 0.0010OJlOO50.0000-0.0005-0.0010 +--r-...•...~--.---.---.-r-~-r-...•...-.

100 120 140 160 180 200 220 240 260 280 300 320

JuIian day

0.0010 eOJJ005

0.0000

.:[ ~.OOO5

-0.0010

___ 1987 •..••. 1988 •..••. 1989

~.0015

~.0020 +-""T"""-.---.---.-.--......-""T"""-.---.--'---'100 120 140 160 180 200 220 240 260 280 300 320

Julian day

0.0010

0.0008

0.0006

< 0.0004

0.0002

0.0000

~.0002 +-~-.--.---.---..--.....-..-...-""T"""-.-....,100 120 140 160 180 200 220 240 260 280 300 320

f

Juianday

Figura 7. Parameter A for parabolic trend surface usedto remove trend: a) Ao; b) A1; c) A2; d) A3; e) A4; f) As.

4. CONCLUSIONS

1. Spatial dependence decreases as the soilgets drier, and results from one year connect almostcontinuously to other years. As the soil gets dryer thetemporal stability of the spatial distribution tends todisappear due to the hydraulic conductivitycontrolling the water evaporation over the field.

2. The field shows strong parabolic trend andspatial pattem of soil water content repeating in time.

3. The topographical heights are identified asbeing responsible for the parabolic trend and for thetime stability of the spatial pattem for soil watercontento

ACKNOWLEDGEMENTS

The first author wants to acknowledgeConselho Nacional de Desenvolvimento Científico eTecnológico (CNPq) for the bilateral agreement grantprovided for this work.

REFERENCES

BURGESS, T.M.; WEBSTER, R Optimal interpolation andisarithmic mapping of soil properties. I. The semivariogramand punctual kriging. Soil Science of America Joumal,MaClison,v. 31,p. 315-331, 1980.

CAMBARDEU.A.CA;MOOMAN, T.B.;NOVAK.J.M;PARKIN,T.B.;KARLEM, D.L.; TURVO, RF.; KONOPA, A.E. Field scalevariability of soil properties in central Iowa soil. Soil Science ofAmericaJoumal, Madison, v.47, p.1501-1.511,1994.

HAJRASULlHA, S.; BANIABBASSI, N.; METTHEY, J.;NIELSEN, D.R Spatial variability of soil sampling for salinitystudies in southwest lran. Irrigation Science, Heidelberg, v.l,p.I97-208,198O

JOURNEL, AG.; HUIJBREGIS, CH. J.Mining Geoestatistics.London: Academic Press, 1978. 600p.

MacBRATNEY, AB.; WEBSfER R.; McLAREN, RG.; SPIERS,RE.B. Regional variation of extractable copper and cobalt inthe topsoil of southwest Scotland. Agronomie, Paris, v. 2, n.10, p. 969-982, 1982.

NIELSEN, D.R; TILWTSON, P.M; VIEIRA, S.R Analyzingfield-measured soil-water properties. Agricultura! WaterManagement, Amsterdam, v. 6, p. 93-109, 1983.

TOPP, G.c.; DA VIS, J.L. Measurement of soil water contentusing time domain reflectometry (TDR): a field evaluation.Soil Science of America Joumal, Madison, v.49, p. 574-582,1985.

VACHAUD, G.;PASSERAT, S.A; BALABANIS,P.;VAUCLlN,M. Temporal stability of spatially measured soil waterprobability density function. Soil Science of America Journal,Madison, v. 49, p. 822-827, 1985.

VAUCLIN, M; VIEIRA, S.R; VACHAUD, G.; NIELSEN,D.R. The use of cokriging with limited field soilobservation. Soil Science of America Joumal, Madison, v.47, p. 175-184, 1983.

VIEIRA, S.R Geoestatística em estudos de variabilidadeespacial do solo. In: NOVAIS, RF.; ALVAREZ, V.H.;SCHAEFER, G.R (Ed.). Tópicos em Ciência do solo. Viçosa:Sociedade Brasileira de Ciência do solo, v.I, 2000. p. 1-54.

VIElRA,S.R.;NIELSEN, D.R.;BIGGAR,J.W.;TILLOTSON, P.M.The Scaling of semivariograms and the kriging estimation.Revista Brasileira de Ciencia do Solo, Campinas, v.21, n. 2,p.521-533,1997.

VIEIRA, S.R; HA TFIELD, J.L.; NIELSEN, D.R; BIGGAR, j.w.Geoestatistical theory and application to variability of someagronomical properties. Hilgardia, Oakland, v. 51,n. 3, p. 1-75,1983.

VIEIRA, S.R.; LOMBARDI NETO, F.; BURROWS; I.T.Mapeamento das chuvas máximas prováveis para o Estado deSão Paulo. Revista Brasileira de Ciencia do Solo, Campinas,v.15,n. 2, p.219-224, 1991.

VIEIRA, S.R; MILLETE, J.; TOPP, c.c, REYNOLDS, W.D.Handbook for geostatistical analysis of variability in soil andclimate data. In: ALVAREZ, V. VH.; SCHAEFER, C.E.G.R;BARROS,N.F.; MELW,J.W.V.; COSTA, LM. (Ed). Tópicos emCiência do Solo. Viçosa: Sociedade Brasileira de Ciência dosolo, v.2, 2002. p. 1-45.

VIEIRA, S.R; NIELSEN, D.R; BIGGAR J.W. Spatial variabilityof field-measured infiltration rate. Soil Science of AmericaJoumal, Madison, v. 45, p. 1040-1048, 1981.

VIEIRA,S.R; REYNOLDS, W.D.; TOPP, G.c. Spatial variabilityof hydraulic properties in a highly structured day soil. In:WIERENGA, P.J.; BACHELET, D. (Ed.). SYMPOSIUMVALIDATION OF FLOW ANO TRANSPORT MODELS FORTHE UNSATURATED 2ONE, 1988.Proceedings ... Las Cruces,NM: Department of Agronomy and Horticulture; Ruidoso:New Mexico State University, New Mexico, 1988. p.471-483.

WEBSTER, R;. CUANALO, H.E. de Ia C. Soil transects ofcorrelograms of north Oxfordshire and their interpretation ..Soil Science of Am.erica Joumal, Madison, v.26, n. 2. p. 176-194,1975.

Bragantia, Campinas, v.67, n.2, p.463-469, 2008