1. 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)

2. OUTLINE
1. Introduction
o Geostatistics
o Model-based geostatistics
1. Spatial prediction & Kriging
o Model-based approach
o Kriging predictor & variance
o Estimation of parameters
1. Case Study
o Description
o Motivation for study
o Analysis
1. Conclusions

3. 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

4. SPATIAL PREDICTION & KRIGING

5.  2.1 Model-based approach

6.  2.2 Kriging predictor & variance

8.  2.3 Estimation of parameters
0.70
0.65
0.60
semivariance
0.55
0.50
0.45
0.40
0.0 0.5 1.0 1.5 2.0 2.5
distance

12. 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

13. STUDY AREA

14.  3.3 Analysis
-25
-26
1.827
North (km)
7.086
12.5
20.186
-27 236.85
-28
-29
27 28 29 30
East (km)

15. Y Coord
data -29 -28 -27 -26 -25
0 50 100 150 200
27
27
28
28
X Coord
X Coord
29
29
30
30
31
31
Density
0.00 0.01 0.02 0.03 Y Coord
-29 -28 -27 -26 -25
0
0
50
50
100
100
data
data
150
150
200
200

16. Y Coord
data -29 -28 -27 -26 -25
1 2 3 4 5
27
27
28
28
X Coord
X Coord
29
29
30
30
31
31
Density
0.0 0.1 0.2 0.3 0.4 0.5 Y Coord
-29 -28 -27 -26 -25
1
1
2
2
3
3
data
data
4
4
5
5

17. 0.70
0.65
0.60
semivariance
0.55
0.50
exponential
matern (kappa=1.5)
0.45
matern (kappa=2.5)
spherical
gaussian
0.0 0.5 1.0 1.5 2.0 2.5
distance

18. ESTIMATES OF PARAMETERS FOR FITTED ORDINARY KRIGING MODELS

19. PROFILE LOG-LIKELIHOODS OF ESTIMATED PARAMETERS OF BEST MODEL

20. LOG SCALE
KRIGING PREDICTIONS
3.5
-25
3.0
-26
KRIGING VARIANCES
Y Coord
-27 0.64
2.5
-25
0.62
-28
-26 0.60
2.0
Y Coord
-29 0.58
27 28 29 30 -27
X Coord 0.56
-28 0.54
0.52
-29
27 28 29 30
X Coord

21. ORIGINAL SCALE
KRIGING PREDICTIONS
50
-25
40
-26
KRIGING VARIANCES
Y Coord
-27 30
700
-25
-28 20
690
-26 680
-29 10
Y Coord
27 28 29 30 670
X Coord -27
660
-28 650
640
-29
27 28 29 30
X Coord

22. 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

23. 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.