Geoscience Reference
In-Depth Information
smoothing, with a resulting loss of detail. h e separation distance and the
lag width must therefore be chosen with care. Another option is to use
a more robust variogram estimator (Cressie 1993, Deutsch and Journel
1998).
•
Calculation of separation distance
- If the observations cover a large area,
for example more than 1,000 km
2
, spherical distances should be calculated
instead of Pythagorean distances from a planar Cartesian coordinate
system.
Kriging
We will now interpolate the observations onto a regular grid by
ordinary
point kriging
, which is the most popular kriging method. Ordinary point
kriging uses a weighted average of the neighboring points to estimate the
value of an unobserved point:
where ʻ
i
are the weights that have to be estimated. h e sum of the weights
should be equal to one in order to guarantee that the estimates are unbiased:
h e expected (average) error for the estimation must be zero. h at is:
where
z
x
0
is the true, but unknown value. We can use the above equations
to compute algebraically the mean-squared error in terms of the variogram:
where
E
is the estimation or
kriging variance
(which must be minimized),
ʳ(
x
i
,
x
0
) is the variogram (semivariance) between the data points and the
unobserved points, ʳ(
x
i
,
x
j
) is the variogram between the data points
x
i
and
x
j
, and ʻ
i
and ʻ
j
are the weights of the
i
th and
j
th data points.
