ETL 1110-1-175
30 Jun 97
(3) Zonal anisotropy is indicated by direc-
(6) The computed sample variograms for the
tional variograms that have the same range but
general north-south and east-west directions for the
different sills. Pure zonal anisotropy is usually not
Saratoga data are shown in Figure 4-6. The north-
seen in practice; typically it is found in combina-
south variogram is specified by a direction angle of
tion with geometric anisotropy. Such mixed
90 deg and a window angle of 45 deg. The north-
anisotropy may be encountered if evaluating the
south variogram reveals the preferential north-
variograms of three-dimensional HTRW sampling
south data alignment by mimicking the omni-
results. Variability of such data (as indicated by
directional (direction angle = 0 deg and window
the sill of the variogram) may be significantly
angle = 90 deg) sample variogram of Figure 4-3.
higher and the range significantly shorter in the
The east-west variogram is specified by a direction
vertical direction than in the horizontal direction.
angle of 0 deg and a window angle of 45 deg. The
In order to model this mixture of anisotropic vari-
lack of pairs of locations for the east-west vario-
ograms, the overall variogram is set to a weighted
gram precludes a good analysis for this direction,
sum of individual models of the directional vario-
but the overlap of the few sufficiently defined
grams scaled by their ranges. In this process,
variogram points with the north-south variogram
called nesting, the choice of weights requires a trial
indicates a consistency of drift in the two direc-
and error approach with a constraint that the sum
tions. Because of this consistency, an isotropic
of the weights equals the sill of the overall vario-
variogram is assumed for the Saratoga residuals.
gram. The reader is referred to Isaaks and
An example of anisotropic variograms is described
Srivastava (1989, pp. 377-390) for further infor-
in Chapter 5.
mation on both types of anisotropy.
4-6. Fitting a Theoretical Variogram to the
(4) For a given number of data locations,
Sample Variogram Points
directional sample variograms will necessarily
have fewer points for any lag when compared to
a. General.
the points for the same lag in the omnidirectional
variogram. Hence, there will be less reliability in
the directional-variogram point values, which
(1) The importance of adequately defining the
would be a critical constraining factor for small
bin values of a sample variogram is substantiated
data sets or for a data pattern that does not con-
by the need to accurately generalize the data-based
form to a direction of anisotropy. For a general
behavior of the sample variogram by a theoretical
variogram (. The parameters controlling the spe-
idea of the sufficiency of the data to adequately
determine any anisotropy, the computations of
cific behavior of theoretical variograms are the
anisotropic sample variograms can be initially
nugget value, the range, the sill, or in the case of a
limited to two orthogonal directions with window
linear variogram, a slope parameter. Of these
angles of 45 deg.
parameters, the nugget and the sill can be related to
properties and statistics of the data.
(5) Directional sample variograms also can be
used to further delineate nonstationarity of the
(2) The nugget is essentially the extrapolation
spatial mean. If the omnidirectional sample vario-
of the sample variogram to a lag of zero. It
gram indicates a drift in the data, the directional
reflects the uncertainty of the variogram at lags
variograms may determine the dimensionality of
that are much smaller than the minimum separation
the drift. That is, although they may not establish
between any two data locations. The nugget value
the degree of the polynomial in the drift equation,
can include measurement error variance, and an
the directional sample variograms can indicate the
estimate of this variance will approximate a mini-
relative strengths of the drift in the u and v
mum value of the extrapolation.
directions.
4-10
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