Geography Reference
In-Depth Information
n
Â
(9.13)
l
=
1
i
i
=
1
h e objective of the kriging system is to i nd appropriate weights by which the avail-
able observations will be multiplied before summing them to obtain the predicted
value. h ese weights are determined using the coei cients of a model i tted to the
variogram (or another function such as the covariance function). h is is in contrast to
IDW (Section 9.5), where the weights are selected arbitrarily (i.e. not using information
about the spatial variation in the data).
h e weights are obtained by solving (i.e. i nding the values of unknown coei cients
in) the OK system:
Ï
n
Â
lg
(
xx
-+= -
)
y
g
(
xx
)
i
=
1,...,
n
Ì
Ô
j
i
j
i
0
j
=
1
(9.14)
n
Â
l
=
1
Ó
j
j
=
1
where y is the Lagrange multiplier. h is equation may seem at i rst sight complicated.
In words, it says that the sum of the weights multiplied by the modelled semivariance
for the lag separating locations x i and x j plus the Lagrange multiplier equals the semi-
variance between locations x i and the prediction location x 0 with the constraint that
the weights must sum to 1. h e way we i nd the weights and the Lagrange multiplier is
outlined below.
Computing the weights and a value of the Lagrange multiplier, y , allows us to obtain
the prediction variance of OK, a by-product of OK, which can be given as:
n
Â
ˆ
2
OK
(9.15)
s
=
l g
(
xx
-
)
+
y
i
i
0
i
=
1
h e kriging variance is a measure of coni dence in predictions and is a function of
the form of the variogram, the sample coni guration, and the sample support (the area
over which an observation is made, which may be approximated as a point or may be
an area) (Journel and Huijbregts, 1978). If the variogram model range is short then the
kriging variance will increase markedly with distance from the nearest samples. h ere
are two varieties of OK: punctual OK and block OK. With punctual OK the predic-
tions cover the same area (the support, V ) as the observations. In block OK, the pre-
dictions are made to a larger support than the observations (e.g. prediction from
points to areas of 2 m by 2 m). h e system presented here is for the more commonly
used form, punctual OK.
Returning to Equation 9.14, using matrix notation, the OK system can be written as:
Kk
l
(9.16)
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