THE MODEL-BASED APPROACH TO GEOSTATISTICAL ANALYSIS:

A Case Study of Micro-Nutrient Content in the Soils of the Witwatersrand Area

Mzabalazo Z. Ngwenya a, Christien Thiart b & Linda M. Haines b

a Biometry Unit, Agricultural Research Council (ARC), b Department of Statistical Sciences, University of Cape Town (UCT)

OUTLINE1. Introduction

o Geostatisticso Model-based geostatistics

2. Spatial prediction & Krigingo Model-based approacho Kriging predictor & varianceo Estimation of parameters

3. Case Studyo Descriptiono Motivation for studyo Analysis

4. Conclusions

INTRODUCTION

1.1 Geostatistics: The branch of spatial statistics concerned with continuous spatial variation

Traditional geostatistics developed largely independently outside mainstream spatial statistics hence the approach lacks statistical rigor

1.2 Model-based Geostatistics: The application of formal statistical methods of modeling and inference to geostatistical problems;

Analyses are carried out under explicitly assumed stochastic models

SPATIAL PREDICTION & KRIGING

2.1 Model-based approach

2.2 Kriging predictor & variance

2.3 Estimation of parameters

0.0 0.5 1.0 1.5 2.0 2.5

0.4

00

.45

0.5

00

.55

0.6

00

.65

0.7

0

distance

sem

iva

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nce

CASE STUDY

3.1 Description

• 214 soil samples collected at 1000 locations between 2005 & 2008

• Various micronutrients measured; Fe, Zn, Pb, Cd, Co, Cr, Ni, Mn

3.2 Motivation for study

• Iron (Fe) has important role in plant health. Plays major role in Energy transfer within plant Nitrogen fixation Plant respiration & metabolism Chlorophyll development & function

• However its accumulation within plant cells can be toxic

STUDY AREA

East (km)

No

rth

(km

)

-29

-28

-27

-26

-25

27 28 29 30

1.8277.08612.520.186236.85

3.3 Analysis

27 28 29 30 31

-29

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X Coord

Y C

oord

0 50 100 150 200

-29

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data

Y C

oord

27 28 29 30 31

050

100

150

200

X Coord

data

data

Den

sity

0 50 100 150 200

0.00

0.01

0.02

0.03

27 28 29 30 31

-29

-28

-27

-26

-25

X Coord

Y C

oord

1 2 3 4 5

-29

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data

Y C

oord

27 28 29 30 31

12

34

5

X Coord

data

data

Density

1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5

0.4

50

.50

0.5

50

.60

0.6

50

.70

distance

sem

iva

ria

nce

exponentialmatern (kappa=1.5)matern (kappa=2.5)sphericalgaussian

ESTIMATES OF PARAMETERS FOR FITTED ORDINARY KRIGING

MODELS

PROFILE LOG-LIKELIHOODS OF ESTIMATED PARAMETERS OF BEST MODEL

LOG SCALE

KRIGING PREDICTIONS

KRIGING VARIANCES

2.0

2.5

3.0

3.5

27 28 29 30

-29

-28

-27

-26

-25

X Coord

Y C

oo

rd

0.52

0.54

0.56

0.58

0.60

0.62

0.64

27 28 29 30

-29

-28

-27

-26

-25

X Coord

Y C

oo

rd

ORIGINAL SCALE

KRIGING PREDICTIONS

KRIGING VARIANCES

10

20

30

40

50

27 28 29 30

-29

-28

-27

-26

-25

X Coord

Y C

oo

rd

640

650

660

670

680

690

700

27 28 29 30

-29

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-27

-26

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X Coord

Y C

oo

rd

CONCLUSIONS

• Clear methodology

• None of the subjectivity of empirical semivariogram construction

• Models for kriging selected on the basis of established statistical criterion

• Can obtain confidence intervals for parameters in models

• Methods extendable to multivariate and non-Gaussian cases

REFERENCES Cressie, N.A.C (1993).

Statistics for Spatial Data (revised edn). John Wiley & Sons, New York.

DeVillers, S. , Thiart, C. & Basson, N.C. (2010). Identification of sources of environmental lead in South Africa from surface soil geochemical maps. Environmental Geochemistry and Health, 32, 451-459.

Diggle, P.J. , Riberio, P.J. & Christensen, O.F. (2003). An introduction to model-based geostatistics. In Spatial Statistics and Computational Methods, Møller, J. (ed), Springer, 43-86.

Diggle, P.J. & Riberio, P.J. (2007). Model-Based Geostatistics. Springer, New York.

Riberio, P.J. Jr. & Diggle, P. (2001). geoR: a package for geostatistical analysis. R News, 2, 14-18.

Schabenberger, O. & Gotway, C.A. (2005). Statistical Methods for Spatial Data Analysis. Chapman & Hall, New York.