Geology Reference
In-Depth Information
hydrology has been described by a number of authors, viz. Delhomme (1976,
1978, 1979), Delfiner and Delhomme (1973), Marsily et al. (1984), Marsily
(1986), Aboufirassi and Marino (1983, 1984), Gambolti and Volpi (1979 and
1979a) to name a few.
If
Z
(
x
) represents any random function (for instance, the transmissivity
field in an aquifer) with values measured at
n
locations in space
z
(
x
i
),
i
= 1….
n
and if the value of the function
Z
has to be estimated at the point
x
0
, which has not been measured, the kriging estimate is defined as:
n
)
Zx
()
z
*(
x
0
) =
(1)
i
i
i
1
where
z
*(
x
0
) is the estimation of function
Z
(
x
) at the points
x
0
and
i
are the
weighting factors. Now we impose two conditions to equation (1), i.e. the
unbiased condition and the condition of optimality.
The unbiased condition (also called universality condition) means that the
expected value of estimation error or the difference between the estimated
z
*(
x
0
) and the true (unknown)
z
(
x
0
) value should in the average be zero:
E
[
z
*(
x
o
) -
z
(
x
o
)] = 0 (2)
The condition of optimality means the variance of the estimation error
should be minimum.
2
= var [
z
*(
x
o
) -
z
(
x
o
)] minimum
(3)
2
of the random variable
z
. It is instead a measure of the uncertainty in the
estimation of
Z
at an unmeasured location. By definition
2
k,
, is not a priori variance
Note that the variance of estimation error
2
k
at a measured
location is zero.
Using the expected value
E
[
Z
(
x
)] =
m
and rewriting equation (2) with the
help of equation (1) we get:
n
)
= 1
(4)
i
i
1
Developing equation (3) we get:
n
n
n
)
)
)
k
2
=
E
[
Z
(
x
i
)
Z
(
x
j
)] +
E
[
Z
(
x
o
)
2
] - 2
E
[
Z
(
x
i
)
Z
(
x
o
)]
(5)
i
j
i
i
1
j
1
i
1
Introducing a coefficient called Lagrange multiplier and adding the term
n
)
-2
1
to the above equation we get:
i
i
1
n
)
Q
=
k
2
- 2
1
i
i
1
