Geoscience Reference
In-Depth Information
for min(
z
)+
m
>0. h is is known as the called Box-Cox transform, with the
special case
k
=0 when a logarithm transformation is used. In the logarithm
transformation,
m
should be added if
z
is zero or negative. Interpolation
results of power-transformed values can be back-transformed directly at er
kriging. h e back-transformation of log-transformed values is slightly
more complicated and will be explained later. h e procedure is known as
lognormal kriging
. It can be important because lognormal distributions are
not uncommon in geology.
Variography with the Classical Variogram
A variogram describes the spatial dependency of referenced observations in
a unidimensional or multidimensional space. Since the true variogram of the
spatial process is usually unkown, it has to be estimated from observations.
h is procedure is called variography. Variography starts by calculating the
experimental variogram
from the raw data. In the next step, the experimental
variogram is summarized by the
variogram estimator
. h e variography then
concludes by i tting a variogram model to the variogram estimator. h e
experimental variogram is calculated as the dif erences between pairs of
observed values and is dependent on the
separation vector h
(Fig. 7.17). h e
classical experimental variogram is dei ned by the
semivariance
,
where
z
x
is the observed value at location
x
and
z
x+h
is the observed value at
another point at a distance
h
from
x
. h e length of the separation vector
h
is called the
lag distance
, or simply the
lag
. h e correct term for ʳ(
h
) is the
semivariogram
(or
semivariance
), where
semi
refers to the fact that it is half
of the variance in the dif erences between
z
x
and
z
x+h
. It is, nevertheless, the
variance per point when points are considered in pairs (Webster and Oliver
2001). Conventionally, ʳ(
h
) is termed a
variogram
instead of a semivariogram,
a convention that we shall follow for the rest of this section. To calculate the
experimental variogram we i rst need to group pairs of observations. h is is
achieved by typing
[X1,X2] = meshgrid(x);
[Y1,Y2] = meshgrid(y);
[Z1,Z2] = meshgrid(z);
h e matrix of separation distances
D
between the observation points is
D = sqrt((X1 - X2).^2 + (Y1 - Y2).^2);
