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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
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.65
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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
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X Coord
Y C
oord
0 50 100 150 200
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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
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-29
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X Coord
Y C
oord
1 2 3 4 5
-29
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data
Y C
oord
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5
X Coord
data
data
Density
1 2 3 4 5
0.0
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0.0 0.5 1.0 1.5 2.0 2.5
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.60
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.70
distance
sem
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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
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-26
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X Coord
Y C
oo
rd
0.52
0.54
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0.58
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0.62
0.64
27 28 29 30
-29
-28
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-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
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rd
640
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-27
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X Coord
Y C
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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.
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