variogram. At smaller lags, the numbers of data

consider in interpreting the kriging results of the

pairs per bin can be nearer the minimum *N(h*k) to

transformed data or in back-transforming kriging

define more bins. At larger lags, a smaller number

results into the untransformed (original) units, as

of wider bins would be adequate. Knowing that

discussed in Chapter 1. If a satisfactory variogram

the variogram should be a smooth function, ulti-

of the original data cannot be achieved and a trans-

mately the analyst visually decides when the sam-

formation is indicated, the sample variogram com-

ple variogram is sufficiently defined at all lags to

putation process must begin again with Equa-

adequately approximate a theoretical variogram.

tion 4-2. Even though no transformation was

needed for the Saratoga data, an example using a

logarithmic transformation and an example using

the indicator transformation are presented in

Chapter 5.

(1) A transformation is applied to a data set

(1) Anisotropy in the data can be investigated

generally for one of two interrelated purposes.

by computing sample variograms for specific

First, a transformation can reduce the scale of

directions. Locations included in a given direction

variability of highly fluctuating data. This varia-

from any other location are contained in a sector of

a circle of radius *h*max centered on the location.

bility would especially occur with contaminant

concentrations in which order of magnitude

The sector is specified by two angular inputs. The

changes in data at proximate sites are not uncom-

first is a bearing defining the specific direction of

mon. The effects of such data would be erratic

interest [measured counterclockwise from east

(=0o)] and the second is a (window) angle defining

sample variogram points as exhibited by a large-

amplitude, ill-defined sawtooth pattern of the lines

an arc of rotation swept in both directions from the

connecting the points.

bearing. Thus, in the terminology used here, the

total angle defining a direction is equal to twice the

(2) Second, a proper transformation of data

window angle. Differences in sample variograms

whose probability distribution is highly skewed

computed using these angle windows specified for

often produces a set of values that is approxi-

different directions can be an indication of

mately normally distributed by mitigating the

anisotropy.

influence of problematic extreme data values. A

data set with a normal distribution is important in

(2) Anisotropy is generally either geometric or

kriging when confidence levels of the estimates are

zonal. Geometric anisotropy is indicated by direc-

desired. This usage of confidence levels in a

tional theoretical variograms that have a common

kriging analysis will be illustrated in Chapter 5.

sill value, but different ranges. The treatment of

geometric anisotropy is dependent on the software

(3) Among the more common transformations

used. The lags of the directional variograms can

is the natural log transform. As an example, for

be scaled by the ratio of their ranges to the range

8

this transformation, the ( will be the sample vari-

of a standard or common variogram. In some

ogram values of logarithms, and subsequent kriged

cases, the lags of all directional variograms are

estimates will be logarithms. Another transfor-

scaled by their respective ranges, and a common

mation that is often used, especially in spatial

variogram with a range of 1 is used. Groundwater

analyses of contaminant levels, is the indicator

contaminant plumes often have geometric aniso-

transformation described in Chapter 2. Although a

tropy in which the prevailing plume direction

transformation might achieve better-behaved sam-

would have a greater range than that of the transect

ple variogram points, there are subtleties to

of the plume.

4-9

Integrated Publishing, Inc. |