(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) For a given number of data locations,

directional sample variograms will necessarily

have fewer points for any lag when compared to

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