Kriging Approach for Estimating Deficient Micronutrients in the Soil: A Case Study

Methodology

Theȱpresentȱstudyȱisȱbasedȱuponȱtheȱsecondaryȱdataȱ

Whereȱ σ _ok 2 varianceȱofȱordinaryȱKriging,ȱw’ȱisȱvectorȱ

ofȱ weightsȱ andȱ Lagrangeȱ multiplierȱ andȱ Dȱ isȱ theȱ

ofȱ 42ȱ soilȱ testedȱ forȱ deficiencyȱ ofȱ micronutrientsȱ

vectorȱofȱcovariancesȱatȱtheȱpredictionȱlocation.ȱThus,ȱ

namelyȱZincȱ(Zn),ȱBoronȱ(B)ȱandȱIronȱ(Fe)ȱ(mg/kg)ȱ

experimentalȱsemivarianceȱvaluesȱwereȱdeterminedȱ

testedȱbyȱSinghȱ(2012).ȱForȱtheȱanalysisȱofȱtheȱspatialȱ

andȱ theȱ semivariogramȱ modelȱ wasȱ fittedȱ withȱ theȱ

variabilityȱ ofȱ micronutrients,ȱ theȱ experimentalȱ

validationȱ methodȱ “Jack-knifing”.ȱ Forȱ theȱ modelȱ

semivariogramsȱ wereȱ calculatedȱ withȱ theȱ helpȱ ofȱ

fittingȱ toȱ theȱ experimentalȱ semivariograms,ȱ theȱ

GS+ȱusingȱtheȱequation:

followingȱmodelsȱwereȱconsidered:ȱlinear,ȱspherical,ȱ

exponentialȱ andȱ Gaussian.ȱ Theȱ semivariogramȱ isȱ

1

y ( h ) =

theȱ plotsȱ ofȱ theȱ semivarianceȱ againstȱ theȱ distanceȱ

2 N ( h )

∑ i N = 1( h ) [ Z ( x i ) − Z ( x i + h ) ] 2 ȱ _..(1)

(lag),ȱandȱitsȱshapeȱindicatesȱwhetherȱtheȱvariablesȱ

Where:ȱ yȱ (h)ȱ isȱ experimentalȱ semivariance,ȱ Nȱ (h)ȱ areȱspatiallyȱconditionalȱorȱnot,ȱe.g.ȱthereȱisȱspatialȱ

isȱ theȱ numberȱ ofȱ pairsȱ ofȱ measuredȱ valuesȱ Z(x),ȱ autocorrelation.ȱ Theȱ descriptiveȱ parametersȱ ofȱ

Z(x _i ȱ +ȱ h)ȱ separatedȱ byȱ aȱ vectorȱ (h).ȱ Theȱ predictionȱ semivariogramsȱareȱnuggetȱ(C 0 ),ȱsillȱ(C 0 +C)ȱandȱrange,ȱ

weightsȱ inȱ Krigingȱ interpolationȱ (Krigeȱ 1951)ȱ areȱ ofȱspatialȱdependenceȱAȱ(Camberdellaȱ et al .ȱ1994).ȱTheȱ

basedȱonȱspatialȱdependenceȱbetweenȱobservationsȱ nuggetȱvarianceȱ(C0)ȱexpressesȱtheȱvariabilityȱdueȱtoȱ

modelledȱbyȱtheȱvariogram.ȱGivenȱspatialȱdataȱZȱ(s _i )ȱ unseenȱpatternsȱ(samplingȱerrorsȱandȱscalesȱshorterȱ

thatȱ followsȱ anȱ intrinsicallyȱ stationaryȱ process,ȱ i.e.ȱ thanȱ minimumȱ inter-sampleȱ distance).ȱ Theȱ sillȱ

havingȱ constantȱ unknownȱ meanȱ µ,ȱ knownȱ spatialȱ varianceȱminusȱtheȱnuggetȱvarianceȱisȱtheȱstructuralȱ

covarianceȱfunctionȱCȱ(h)ȱforȱspatialȱlagsȱh=s -s ,ȱandȱ varianceȱ(C).ȱThisȱtermȱaccountsȱforȱtheȱpartȱofȱtheȱ

i j

canȱbeȱwrittenȱasȱZȱ(s _i )ȱ=ȱµ+ε(s _i ),ȱweȱtypicallyȱwantȱtoȱ totalȱ varianceȱ thatȱ canȱ beȱ modelledȱ byȱ theȱ spatialȱ

predictȱvaluesȱofȱtheȱprocessȱatȱunobservedȱlocations,ȱ structure.ȱSelectionȱofȱmodelsȱwasȱmadeȱprincipallyȱ

s _0ȱ ∈ ȱD.ȱKrigingȱisȱaȱmethodȱthatȱenablesȱpredictionȱofȱ onȱvisualȱfit,ȱregressionȱcoefficientȱ(R 2 )ȱandȱresidualȱ

aȱspatialȱprocessȱbasedȱonȱaȱweightedȱaverageȱofȱtheȱ sumȱofȱsquareȱ(SSR),ȱwhichȱprovidedȱanȱindicationȱ

observations.ȱInȱtheȱcaseȱofȱanȱintrinsicallyȱstationaryȱ ofȱ howȱ wellȱ theȱ modelȱ fitsȱ theȱ semivariogramsȱ

processȱwithȱconstantȱunknownȱmean,ȱtheȱordinaryȱ data.ȱ Theȱ degreeȱ ofȱ spatialȱ dependenceȱ (GD)ȱ wasȱ

Krigingȱ(OK)ȱmethodȱisȱused.

calculatedȱusingȱequation:

N

GD=ȱ(C _0ȱ /C+C ₀ )*100ȱ

…ȱ(5)

Z ˆ ( s 0 ) = ∑ w i Z ( s i )

i = 1

Itȱ isȱ alsoȱ knownȱ asȱ goodnessȱ ofȱ prediction.ȱ Anȱ

ȱ

…ȱ(2)

ordinaryȱ Krigingȱ wasȱ usedȱ forȱ constructingȱ ofȱ

Nowȱ forȱ findingȱ bestȱ linearȱ unbiasedȱ predictorȱ surfaceȱmapsȱtoȱprovideȱenoughȱestimatedȱdata.

(BLUP)ȱ varianceȱ ofȱ interpolationȱ errorȱ willȱ beȱ

minimized.ȱThusȱMeanȱsquareȱerrorȱofȱtheȱvarianceȱofȱ Results and Discussion

anȱordinaryȱKrigingȱwasȱcalculatedȱusingȱequation:

Characterizationȱ andȱ rationalȱ managementȱ ofȱ

σ 2 = σ 2 − ∑ i N = 1 w i (cov[ Z ( s i ), Z ( s 0 )] − λ

micronutrientȱ behaviourȱ requireȱ anȱ understandingȱ

ok

ȱ ...(3)

ofȱhowȱtotalȱandȱavailableȱsoilȱmicronutrientsȱvaryȱ

acrossȱtheȱland.ȱIntegratedȱnutrientȱmanagementȱisȱ

Inȱmatrixȱform

importantȱ forȱ sustainableȱ agriculturalȱ productionȱ

andȱ protectingȱ environmentȱ qualityȱ andȱ hasȱ beenȱ

σ _ok 2 = σ 2 − w ' D

…ȱ(4)

widelyȱinvestigatedȱaroundȱtheȱworld.

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