ETL 1110-1-175
30 Jun 97
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(hk) 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
4-5. Transformations and Anisotropy
the indicator transformation are presented in
Considerations
Chapter 5.
a. Transformations.
b. Directional variograms and anisotropy.
(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 hmax 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
Previous Page
