Geography Reference
In-Depth Information
a nugget ef ect (with a value of 195.227) and a spherical component (with a structured
component of 320.56 and a range of 42428.3 m).
h e model i tted to the variogram can be used to determine the weights assigned to
observations using a geostatistical prediction procedure (or family of procedures)
called kriging. In kriging, the variogram model is used to obtain values of the semi-
variance for the lags by which observations are separated and for the lags that separate
the prediction location from the observation.
For the variogram model in Figure 9.12 there is a nugget ef ect and a spherical
component. Combining Equations 9.10 and 9.11 this gives:
Ï
3
cc
+◊
[1 . 5
h
-
0 . 5
h
] i f
ha
£
=
Ì
0
a
a
g
()
h
cc
+
if
ha
>
Ó
0
For a lag of 6481.996 m (which is less than the range value of 42428.3 m) the
modelled semivariance is obtained from:
È
3
˘
g
(6481.996)
=
195.227
+
320.560
◊
1.5
6481.996
-
0.5
6481.996
=
268.116 m
Î
˚
42428.3
42428.3
h is will be coni rmed by examining Figure 9.12 and reading upwards from a lag of
6482 m to the variogram model and then let to read of the semivariance value. Mulla
(1988) used variograms, along with other measures, to characterize landforms. As
noted previously, in terms of landforms, a mountainous area would have a short range,
since there are large changes in elevation over small distances. In contrast, a river l ood
plain would have a long range as elevations tend to be similar over quite large dis-
tances. h e variogram is, therefore, a useful tool for measuring the scale of spatial
variation in a property. Prediction using kriging, which makes use of the variogram
model, is the subject of the following section.
9.7.2
Kriging
h ere are many varieties of kriging. Its simplest form is called simple kriging (SK). To
use SK it is necessary to know the mean of the property of interest and this must be
constant across the region of interest. In practice this is rarely the case. h e most widely
used variant of kriging, ordinary kriging (OK), allows the mean to vary and the mean
is estimated for each prediction neighbourhood. OK predictions are weighted aver-
ages of the
n
available data (i.e. the predictions are based on the
n
nearest neighbours
of the prediction location). h e OK prediction,
ˆ
()
z
x
, is dei ned as:
0
n
=
Â
ˆ
()
(9.12)
z
x
l
z
()
x
0
i
i
i
=
1
with the constraint that the weights,
l
i
, sum to 1 (this is to ensure an unbiased
prediction):
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