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Anatomy of an anisotropic correlation
model.With increasing distance, ?(h) (or
C(h)) tends to reach a constant value,
known as the sill (dashed horizontal
line in Figure 2). For a variogram, the
sill is the variance (2) of measured
data. The distance at which the sill is
reached by the variogram is called the
range or correlation length. The covariance
reaches its range when C(h) = 0.
The range for a variogram and the
covariance should be the same for a
given set of search parameters. The sill
and range are useful properties when
one compares directional trends in the
data. Often the variogram and covariance
show a discontinuity at the origin,
termed the nugget effect. The
nugget effect is considered random
noise and may represent short-scale
variability, measurement error, sample
rate, etc.
Spatial continuity analysis is one
of the most important steps in a geostatistical
study, because it strongly
influences the kriging and conditional
simulation results and associated
uncertainties.
Kriging and conditional simulation
applications require knowledge
of the correlation function for all possible
distances and azimuths. This
requires a model of the experimental
variogram (covariance) in order to
know the variance (covariance) at any
location, not just along specific interdistance
vectors corresponding to
angular/distance classes. Spatial modeling
is not curve fitting, in the leastsquares
sense, because the selected
model must ensure that the kriging
variance is ? 0-a condition not necessarily
satisfied by least-squares or
other fitting methodologies.
Search ellipse design. Because computers
are used in mapping, we must
instruct the program how to gather
and use control points during interpolation.
Most familiar with computer
mapping know that this involves
designing a search ellipse or neighborhood.
We must specify the length
of the search radius, the number of
sectors (typically four or eight), and
the number of data points per sector.
Most common mapping programs
allow the user to specify only one
radius; thus, the search ellipse is circular
(isotropic). However, during
geostatistical analysis, we often find
that the spatial model is anisotropic.
Thus, we should design the search
ellipse based on the spatial model
correlation scales, aligning the search
ellipse azimuth with the major axis
of anisotropy (Figure 3).
Model crossvalidation. Crossvalidation
tests the "goodness" of the spatial
model and the search ellipse
design. The procedure compares estimated
values with measured values,
just as one computes residuals
between predicted and observed values
in regression or analysis of variance.
The procedure is:
For each sample in the data set,
compute a kriged estimate at the
same location, using the spatial
model and search ellipse parameters
but ignoring that sample value
during reestimation. Thus, each
sample value of a data set has a
reestimated value and a measure of
the kriging variance. From this
information, various displays are
created which are often humbling.
One common display is a scatterplot
of measured versus reestimated
values. If the model were
perfect, the scatterplot would be a
straight line, but this never occurs.
However, if the kriged estimates
are unbiased, averages of the estimated
and measured values should
be equal.
Another usual display is the histogram
of the standardized estimation
error, which is the reestimated
minus the observed values, divided
by the kriging variance. If the histogram
is symmetrical about a mean
of 0, the estimates are unbiased. This
ensures that anywhere in the mapped
area, interpolated values have an
equal chance of being over- or underestimates
of the true value.
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