Geology Reference
In-Depth Information
=[
B
]
T
[
A
]
-1
[
Z
] (17)
Since [
A
] is a square symmetric matrix, it is unchanged when transposed
also
z
*
(
x
0
) = [
B
]
T
[
C
] where [
C
] = [
A
]
-1
[
Z
]
So in the case of the unique neighbourhood, matrix [
C
] is calculated once
and used for the estimation of all the points.
Similarly the matrix equation for the variance of the estimation error can
be written as follows:
k
2
=[
]
T
[
B
] = [
B
]
T
[
A
]
-1
[
B
]
(18)
A few remarks about the Kriging
1. If we estimate the value of the variable
Z
at a point
X
m
, which also is one
of the data locations, i.e.
x
m
{
x
1
…..
x
n
}, the kriging system provides an
estimated value
z
*
(
x
m
) of the data and nil variance of estimation error.
Hence the kriging estimator is called an “exact” estimator. It is also called
a Best Linear Unbiased Estimator (BLUE) since it is indeed a linear
estimator defined by the conditions of universality and of optimality.
2. Instead of working with the natural values of the variable, it is sometimes
preferable to work with its logarithm; this is often the case with
transmissivity, because when the natural values have a lognormal
probability distribution the spatial structure, e.g. variogram or covariance,
is better defined if the logarithms of the values are used. Moreover, the
arithmetic mean of the logarithm gives, in fact, the geometric mean of the
natural value, which is a better estimator of the true spatial average in the
case of transmissivity (Matheron, 1967).
3. Very often the observed field data are approximate and unreliable but the
kriging theory permits the use of data with uncertainties. For this we
assume that the observed value of the variable
z
consists of two components
T
and
e
(true part and uncertainty).
4. The variogram calculated on the raw data uses the average variability of
the variable for any particular separation
h
, and the theoretical model
fitted through the experimental variograms makes it more approximate.
Thus it is necessary to check the validity of the adjusted theoretical model.
In other words, the variogram model and the kriging procedure should
reproduce the observed data. To do so we remove one data point and
estimate the variable at that point from the rest of the measured values
and calculate the difference between the estimated and observed values
(i.e. error).
5. The variance of the kriging estimation error depends on the geometry of
the data network and on the structural model, i.e., the covariance or
variogram but not on the measured value
z
(
x
i
) at the measurement points.
We have seen that the variance decreases as the number of points in the
near-neighbourhood is increased.
E
