Geology Reference
In-Depth Information
n
)
= 1
(13)
j
j
1
and the variance of the estimation error becomes:
n
)
k
2
=
(
x
i
,
x
o
) +
(14)
i
i
1
Expression in a matrix form:
................
1
11
12
13
1n
1
10
..............
1
21
22
23
2n
2
3
20
...............
1
!"
31
32
33
3n
30
=
(15)
.................
1
n
n1
n 2
n3
nn
!"
n0
1
1.....1......1.......................1...0
!
"
where
nm
means
(
x
n
,
x
m
). In the short form it is written as:
] = [
B
]
or [] = [
A
]
-1
[
B
] (16)
We have seen previously that after a certain distance, called the range,
which also shows the zone of influence, the variogram generally takes an
asymptotic value. Thus for estimating any variable at a particular point, it is
not very useful to consider the data points beyond the zone of influence,
called the neighbourhood of the estimated point
x
0
. Therefore, if we consider
the points, which lie inside the zone of influence for estimating the variable
at any given point, a smaller matrix will be used. This procedure is called
the 'moving neighbourhood'. However, when we used the moving
neighbourhood for each point of estimation, a different matrix has to be
inverted separately for calculating the situation and this becomes very time
consuming. If we work with the so-called 'unique neighbourhood', where all
the available data points are considered simultaneously, it is sufficient to
calculate the inverse of matrix [
A
] only once. A simple multiplication is
needed to get the kriged estimate to any location.
It is also clear that the value of
z
*(
x
0
) is equal to []
T
[
Z
] where [
Z
] is
the column matrix of the measured values at the measurement points; therefore
we can easily write:
z
*(
x
o
) = [
[
A
][
]
T
[
Z
] = {[
A
]
-1
[
B
]}
T
[
Z
]
