Geoscience Reference
In-Depth Information
G
n
G
a
=
m
(
x
)
−
=
λ
m
(
x
)
o
o
i
i
i
1
(
m
G
= m and a
o
= 0) as:
and can be easily rewritten (since
)
n
=
λ
=
1
i
1
The minimization of the variance of estimation leads to what is commonly
known as a kriging system:
n
G
G
G
G
∑
=
λ
C
(
x
,
x
)
−
μ
=
C
(
x
,
x
)
j
j
i
o
i
j
1
This equation can also be written as a matrix:
G
G
G
G
G
G
G
G
G
G
C
(
x
,
x
)
C
(
x
,
x
)
...
C
(
x
,
x
)
...
C
(
x
,
x
)
1
C
(
x
,
x
)
1
1
1
2
1
j
1
n
1
o
G
G
G
G
G
G
G
G
G
G
λ
1
C
(
x
,
x
)
C
(
x
,
x
)
...
C
(
x
,
x
)
...
C
(
x
,
x
)
1
C
(
x
,
x
)
2
1
2
2
2
j
2
n
2
o
2
...
...
...
...
...
...
...
...
(
...
G
G
G
G
G
G
G
G
G
G
C
(
x
,
x
)
C
(
x
,
x
)
...
C
(
x
,
x
)
...
C
(
x
,
x
)
1
λ
=
C
x
,
x
)
i
1
i
2
i
j
i
n
i
i
o
...
...
...
...
...
...
...
...
...
G
G
G
G
G
G
G
G
G
G
λ
C
(
x
,
x
)
C
(
x
,
x
)
...
C
(
x
,
x
)
...
C
(
x
,
x
)
n
C
(
x
,
x
)
n
1
n
2
n
j
n
n
n
o
1
1
...
1
...
1
0
1
It is, therefore, necessary to develop a second hypothesis: the values of m,
σ
and
(
C
G
, which are normally defined from different events referred to as
ω
(climatological estimations), can now be defined from one spatial distribution of the
estimations that are given for the entire domain of D. This hypothesis is known as
the Ergodic hypothesis.
)
It is, therefore, assumed that:
G
G
G
{
}
{
}
m
=
E
R
(
x
,
ω
)
=
∫
Ω
R
(
x
,
ω
)
P
(
d
ω
)
=
E
R
(
x
,
ω
)
i
i
i
k
Ω
D
G
G
G
G
G
G
{
}
{
}
E
R
(
x
,
ω
)
R
(
x
,
ω
)
−
m
2
=
E
R
(
x
,
ω
)
R
(
x
,
ω
)
−
m
2
=
σ
2
(
−
ρ
(
h
))
=
C
(
h
)
D
i
j
i
k
j
k
Ω
The field average is evaluated by the following equation:
1
n
G
∑
=
m
≈
R
(
x
,
ω
)
n
i
k
l
1
G
is evaluated in terms of distance
Δ
(in the case
of an anisotropic function phases of azimuth are used instead). The couplets of
)
The covariance function
C
(
h
)
i
x
G
G
(
,
are represented as follows:
j
G
−
G
Δ
h
x
x
=
h
±
i
j
2
where the equation is a number of t and from which it is possible to obtain one of
the points of the covariance function:
