279
Precipitation Estimate Using NEXRAD Ground-Based Radar Images
2008) showed that the performance of kriging methods can be improved by using
external information to estimate m(x).
In OK, a common type of kriging practice, the trend is considered unknown and
constant. OK estimates the unknown precipitation depth at the unsampled location
u as a linear combination of neighboring observations: Z u
Z
ui
i
n
i
( )
[ ( )]
=
=
∑ λ
1
x . The
optimal weights are obtained through solving a series of linear functions known as
the OK system (Goovaerts 2000):
λ γ
µ
γ
λ
uj
j
n
ij
ui
uj
j
n
h
u
h
i
n
=
=
∑
−
=
=
1
1
1
( ) ( )
( )
,...,
∑ ∑ =









1,
(12.4)
where μ(u) is the Lagrange parameter accounting for the constraint on the weights,
and h ij denotes the separation distance between sampled location x i and x j . The semivariance γ(h) is computed using the following:
γ ( )
( )
( ( ) (
)) ,
( )
h
N h
z
z
h
i
i
i
N h
=
−
+
∑
1
2
2
x
x
(12.5)
where h is the difference between two point locations, N(h) is the number of pairs of
points separated by h, and z(x i ) – z(x i + h) is the value difference between point x i and
another point separated by distance h.
12.3.2.3  Simple Kriging 
The SK estimator is Z u m u
Z
m i
ui
i
n
i
( )
( )
[ ( )
( )]
−
=
−
=
∑ λ
1
x
. SK assumes that the trend
of the random variable is known and constant. The equation system used to estimate
the weights in Equation 12.1 is
λ uj
j
n
ij
ui
C h
C h
=
∑
=
1
( )
( ) i = 1,…,n,
(12.6)
where C(h) is the spatial covariance between two points separated by distance h.
12.3.2.4  Regression Kriging
RK is a technique that combines the theory of generalized linear models with kriging
(Hengl et al. 2004). In RK, the trend m(x) is commonly fitted using linear regression