405
12
Geostatistics
Kriging
12.1 KRIGING
In this chapter, we will continue the study of geostatistics that we commenced in Chapter 8. We
will look at Kriging methods, which are techniques to predict values of regionalized variables in
nonsampled points of a spatial domain using a collection of sampled points. This set of sampled
points form a marked point pattern, which can be regular or irregular (Figure 12.1). As explained
in Chapter 8, we calculate similarity (using the covariance) or dissimilarity (using the semivariance,
Figure 12.2) between points separated at given distances and generate a model of the covariance and
semivariance as a function of lag (Figure 12.3).
We then conduct kriging, which consists of using the covariance model to predict the values of
the variables at the nonsampled points. A regular grid like the one shown in Figure 12.4 determines
the target points for prediction. This way, we have values distributed in the entire domain; some of
these are measured values while others are kriging estimates. We visualize the results as a grid or
raster image with overlaid contour lines (Figure 12.5). By decreasing step size between the predic-
tion grid points, we produce a higher resolution image (Figure 12.6).
There is a variety of Kriging procedures, including kriging in three-dimensional spatial domains.
In this book, we cover only two basic methods and limit ourselves to two-dimensional domains.
12.2 ORDINARY KRIGING
Ordinary kriging helps to interpolate values of the regionalized variable Z(x, y) assuming that there
is not a trend, that is, the regionalized variable is random with constant mean. This assumes that
Z is stationary, that is, its mean and variance do not change with location x, y. Since we now know
matrices and vectors, we will use vector notation x for a generic point at coordinates x, y. Thus, x is
a 2 × 1 vector with entries x and y.
More specically, denote x
0
as a point x where we want to estimate the value of Z. The estimate
of Z(x
0
) is obtained by linear combination of n known values Z(x
i
) around the target location x
0
(Figure 12.7). This procedure requires a set of weights or coefcients λ
i
of this linear combination
to form the equation
ZZ
i
i
k
()
()xx
0i
=
=
∑
λ
1
(12.1)
Note that the kriging error is the difference between the estimate and the real value
eZ Z
=−
()
()xx
00
We want to preserve the so-called intrinsic hypothesis; that is, that the mean and variance
are constant. Thus, the expected value of the estimate should be equal to the mean or expected
406 Data Analysis and Statistics for Geography, Environmental Science, and Engineering
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
(70, 92)
(92, 103)
(103, 110.5)
(110.5, 123)
Dataset
x
y
0.00.1 0.20.3
0
50
100
150
200
250
Lag
Classical semivariogram estimator
Variogram estimator: dataset.v
0.0 0.1 0.2 0.3 0.4
Lag
Square root dierence
Point.pair
12345678910
0
2
4
6
0
2
4
6
Lag
Square root dierence
Boxplot by bin
FIGURE 12.2 Semivariogram diagnostics of point pattern of previous gure.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
(70, 92)
(92, 103)
(103, 110.5)
(110.5, 123)
FIGURE 12.1 Sampled regionalized variable as marked point pattern.
407Geostatistics
0.0 0.1 0.2 0.3
0
50
100
150
200
Lag distance (h)
Semivariance gamma (h)
0.0 0.1 0.2 0.3
0
50
100
150
Lag distance (h)
Covariance c(h)
FIGURE 12.3 Semivariance and covariance model to be used for kriging.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
(70, 92)
(92, 103)
(103, 110.5)
(110.5, 123)
FIGURE 12.4 Prediction grid and sampled points.
408 Data Analysis and Statistics for Geography, Environmental Science, and Engineering
0.0 0.2 0.4 0.6 0.8
0.0
0.2
0.4
0.6
0.8
x
y
90
95
95
95
100
100
100
105
105
105
105
105
110
110
110
115
FIGURE 12.5 Kriged regionalized variable using 10 × 10 grid.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
y
80
85
90
90
90
95
95
95
95
95
95
95
100
100
100
100
100
100
100
100
100
100
100
105
105
105
105
105
105
105
105
105
105
105
110
110
110
110
110
110
110
110
115
115
FIGURE 12.6 Kriged regionalized variable using 100 × 100 grid.

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