Geoscience Reference
In-Depth Information
G
G
⎩
⎨
⎧
∗
⎭
⎬
⎫
[7.3]
2
2
δ
=
E
(
R
(
x
,
ω
)
−
R
(
x
,
ω
))
=
min
o
o
7.2.2.2.
Autocorrelation
No hypothesis will be made about stationarity, and it is assumed that the
unbiased equation [7.2] has already been examined and verified. The interpolation
error can be represented by several different equations:
G
∗
G
G
n
G
G
n
G
but
R
(
x
,
ω
)
−
R
(
x
,
ω
)
=
R
(
x
,
ω
)
−
ao
−
=
λ
R
(
x
,
ω
)
m
(
x
)
=
ao
+
=
λ
m
(
x
)
o
o
o
i
i
o
i
i
i
1
i
1
G
∗
G
G
G
n
G
G
=
R
(
x
,
ω
)
−
R
(
x
,
ω
)
=
R
(
x
,
ω
)
−
m
(
x
)
−
λ
(
R
(
x
,
ω
)
−
m
(
x
))
o
o
o
o
i
i
i
i
1
(
r
G
will be used to refer to the difference that exists between total rainfall and
average rainfall (
)
G
G
G
):
r
(
x
)
=
R
(
x
,
ω
)
−
m
(
x
)
G
∗
G
G
n
G
[7.4]
R
(
x
,
ω
)
−
R
(
x
,
ω
)
=
r
(
x
,
ω
)
−
=
λ
r
(
x
,
ω
)
o
o
o
i
i
i
1
Equation [7.3] can, therefore, be represented as:
⎩
⎨
⎧
G
n
G
⎭
⎬
⎫
2
=
δ
=
E
(
r
(
x
,
ω
)
−
λ
r
(
x
,
ω
))
=
min
o
i
i
2
i
1
It is then possible to determine the n parameters of
λ by canceling the n
δ in relation to
parameters that have been partially derived from
λ :
G
n
G
G
n
n
G
G
{
}
{
}
{
}
2
δ
=
E
(
r
(
x
,
ω
)
−
2
∑
λ
E
r
(
x
,
ω
)
r
(
x
,
ω
)
+
∑
∑
λ
E
r
(
x
,
ω
)
r
(
x
,
ω
)
o
2
i
o
i
i
j
j
i
i
=
1
j
=
1
i
=
1
[
][
]
{
}
{
}
G
G
G
G
G
G
E
r
(
x
o
,
ω
)
r
(
x
,
ω
)
=
E
R
(
x
o
,
ω
)
−
m
(
x
)
R
(
x
,
ω
)
−
m
(
x
)
i
o
i
i
G
G
{
} {
}
{
}
G
G
G
G
E
r
(
x
,
ω
)
r
(
x
,
ω
)
=
E
R
(
x
o
,
ω
)
R
(
x
,
ω
)
−
m
(
x
)
E
R
(
x
,
ω
)
...
o
i
i
o
i
G
G
G
G
{
} {
}
...
−
m
(
x
)
E
R
(
x
,
ω
)
+
E
m
(
x
)
m
(
x
)
i
0
i
o
G
G
{
} {
}
G
G
G
G
E
r
(
x
,
ω
)
r
(
x
,
ω
)
=
E
R
(
x
,
ω
)
R
(
x
,
ω
)
−
m
(
x
)
m
(
x
)
o
i
0
i
i
o
G
G
G
{
}
G
E
r
(
x
,
ω
)
r
(
x
,
ω
)
=
C
(
x
,
x
)
o
i
o
i
where the equation:
G
n
G
G
n
n
G
G
2
δ
=
σ
(
x
o
)
−
2
∑
λ
C
(
x
,
x
)
+
∑
∑
λ
C
(
x
,
x
)
2
i
o
i
i
j
j
i
i
=
1
j
=
1
i
=
1
and the derivatives can be written as follows:
2
∂
(
δ
)
G
G
n
G
G
=
−
2
C
(
x
,
x
)
+
2
∑
=
λ
C
(
x
,
x
)
o
i
j
j
i
∂
(
λ
)
i
j
1
